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Symmetry, Volume 14, Issue 1 (January 2022) – 181 articles

Cover Story (view full-size image): Liquid crystal director distributions have been numerically analyzed between asymmetric anchoring surfaces, that is, infinite strong and very weak anchoring surfaces in a hybrid aligned nematic (HAN) cell and a twisted nematic (TN) cell. When the weak anchoring strength is lower than a critical one, HAN and TN orientations turn to a homogeneous orientation. Such quasi-homogeneous orientations return to original HAN and TN orientations under the application of voltage. A unique voltage transmission curve of 0–100–0% appears in the quasi-homogeneous TN cell between crossed polarizers. View this paper
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11 pages, 374 KiB  
Article
Description for N = 126 Isotones 210Po and 212Rn with Particle-Hole Excited Nucleon-Pair Approximation and Realistic Effective Interaction
by Yi-Xing Wang, Yi-Yuan Cheng and Thomas T. S. Kuo
Symmetry 2022, 14(1), 181; https://doi.org/10.3390/sym14010181 - 17 Jan 2022
Viewed by 2237
Abstract
In this paper, we study yrast states of two N=126 isotones 210Po and 212Rn using the nucleon-pair approximation with particle–hole excitations and using a low-momentum interaction Vlowk renormalized from the free CD-Bonn NN potential. An [...] Read more.
In this paper, we study yrast states of two N=126 isotones 210Po and 212Rn using the nucleon-pair approximation with particle–hole excitations and using a low-momentum interaction Vlowk renormalized from the free CD-Bonn NN potential. An overall good agreement with experimental level structures, B(E2)s, and B(E3)s, is achieved. We also calculate the probabilities of neutron particle–hole excitations in these yrast states, with a focus on negative-parity states, which reflect the roles played by the neutron negative-parity configurations of one-particle-one-hole excitations across the N=126 shell gap and the negative-parity configurations of valence proton particles involving the 0i13/2 orbit. The N=126 shell gap is discussed in terms of energies of neutron one-particle-one-hole excitations. Full article
(This article belongs to the Special Issue Experiments and Theories of Radioactive Nuclear Beam Physics)
Show Figures

Figure 1

Figure 1
<p>Calculated energy levels of <math display="inline"><semantics> <msup> <mrow/> <mn>210</mn> </msup> </semantics></math>Po are in the upper panel and those of <math display="inline"><semantics> <msup> <mrow/> <mn>212</mn> </msup> </semantics></math>Rn are in the lower panel, in comparison with experimental data [<a href="#B4-symmetry-14-00181" class="html-bibr">4</a>,<a href="#B51-symmetry-14-00181" class="html-bibr">51</a>]. In the calculation, we use the low-momentum interaction <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>low</mi> <mo>–</mo> <mi>k</mi> </mrow> </msub> </semantics></math> [<a href="#B24-symmetry-14-00181" class="html-bibr">24</a>,<a href="#B25-symmetry-14-00181" class="html-bibr">25</a>,<a href="#B26-symmetry-14-00181" class="html-bibr">26</a>] renormalized from the CD-Bonn potential [<a href="#B23-symmetry-14-00181" class="html-bibr">23</a>], as our two-body effective interaction. The adopted single-particle energies with respect to the <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mn>82</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>82</mn> </mrow> </semantics></math> core, together with monopoles of <math display="inline"><semantics> <mrow> <mo>〈</mo> <mo>|</mo> <msub> <mi>V</mi> <mrow> <mi>low</mi> <mo>–</mo> <mi>k</mi> </mrow> </msub> <mo>|</mo> <mo>〉</mo> </mrow> </semantics></math>s, give corresponding single-particle energies with respect to the <math display="inline"><semantics> <mrow> <mi>Z</mi> <mo>=</mo> <mn>82</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>126</mn> </mrow> </semantics></math> core, as well as the <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>126</mn> </mrow> </semantics></math> shell gap, consistent with Figure 1 of Reference [<a href="#B10-symmetry-14-00181" class="html-bibr">10</a>]. We classify the discussed yrast states into three cases which are drawn in black, red, and blue, respectively. See the texts for details.</p>
Full article ">Figure 2
<p>The probability of neutron one-particle-one-hole (1p1h) excitations across the <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>126</mn> </mrow> </semantics></math> shell gap, as well as the probability of the closed-shell (i.e., 0p0h) state, in the yrast states of <math display="inline"><semantics> <msup> <mrow/> <mn>210</mn> </msup> </semantics></math>Po and <math display="inline"><semantics> <msup> <mrow/> <mn>212</mn> </msup> </semantics></math>Rn.</p>
Full article ">Figure 3
<p>The energy differences (in units of MeV), <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>E</mi> <mn>1</mn> </msub> <mo>=</mo> <mi>E</mi> <mrow> <mo>(</mo> <msubsup> <mn>12</mn> <mn>1</mn> <mo>−</mo> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mi>E</mi> <mrow> <mo>(</mo> <msubsup> <mn>8</mn> <mn>1</mn> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>E</mi> <mn>2</mn> </msub> <mo>=</mo> <mi>E</mi> <mrow> <mo>(</mo> <msubsup> <mn>13</mn> <mn>1</mn> <mo>−</mo> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mi>E</mi> <mrow> <mo>(</mo> <msubsup> <mn>8</mn> <mn>1</mn> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics></math> of <math display="inline"><semantics> <msup> <mrow/> <mn>210</mn> </msup> </semantics></math>Po, as well as <math display="inline"><semantics> <mrow> <mo>Δ</mo> <msub> <mi>E</mi> <mn>3</mn> </msub> <mo>=</mo> <mi>E</mi> <mrow> <mo>(</mo> <msubsup> <mn>19</mn> <mn>1</mn> <mo>−</mo> </msubsup> <mo>)</mo> </mrow> <mo>−</mo> <mi>E</mi> <mrow> <mo>(</mo> <msubsup> <mn>14</mn> <mn>1</mn> <mo>+</mo> </msubsup> <mo>)</mo> </mrow> </mrow> </semantics></math> of <math display="inline"><semantics> <msup> <mrow/> <mn>212</mn> </msup> </semantics></math>Rn, which we use to represent the energies of corresponding neutron 1p1h excitations, versus the change of the <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>126</mn> </mrow> </semantics></math> shell gap (denoted as <math display="inline"><semantics> <mi>δ</mi> </semantics></math> and in units of MeV) adopted in the calculation. In all the calculations, the configuration spaces are constructed in the same way. For comparison, we also present the corresponding experimental values. See the texts for details.</p>
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109 pages, 4632 KiB  
Article
Notes on Confinement on R3 × S1: From Yang–Mills, Super-Yang–Mills, and QCD (adj) to QCD(F)
by Erich Poppitz
Symmetry 2022, 14(1), 180; https://doi.org/10.3390/sym14010180 - 17 Jan 2022
Cited by 17 | Viewed by 4084
Abstract
This is a pedagogical introduction to the physics of confinement on R3×S1, using SU(2) Yang–Mills with massive or massless adjoint fermions as the prime example; we also add fundamental flavours to conclude. The small- [...] Read more.
This is a pedagogical introduction to the physics of confinement on R3×S1, using SU(2) Yang–Mills with massive or massless adjoint fermions as the prime example; we also add fundamental flavours to conclude. The small-S1 limit is remarkable, allowing for controlled semiclassical determination of the nonperturbative physics in these, mostly non-supersymmetric, theories. We begin by reviewing the Polyakov confinement mechanism on R3. Moving on to R3×S1, we show how introducing adjoint fermions stabilizes center symmetry, leading to abelianization and semiclassical calculability. We explain how monopole–instantons and twisted monopole–instantons arise. We describe the role of various novel topological excitations in extending Polyakov’s confinement to the locally four-dimensional case, discuss the nature of the confining string, and the θ-angle dependence. We study the global symmetry realization and, when available, present evidence for the absence of phase transitions as a function of the S1 size. As our aim is not to cover all work on the subject, but to prepare the interested reader for its study, we also include brief descriptions of topics not covered in detail: the necessity for analytic continuation of path integrals, the study of more general theories, and the ’t Hooft anomalies involving higher-form symmetries. Full article
(This article belongs to the Special Issue New Applications of Symmetry in Lattice Field Theory)
Show Figures

Figure 1

Figure 1
<p>The one-loop heavy <italic>W</italic>-boson contribution to the photon kinetic term. From (<xref ref-type="disp-formula" rid="FD1-symmetry-14-00180">1</xref>), all vertices come with a factor <inline-formula><mml:math id="mm3025"><mml:semantics><mml:mfrac><mml:mn>1</mml:mn><mml:msubsup><mml:mi>g</mml:mi><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:msubsup></mml:mfrac></mml:semantics></mml:math></inline-formula> and the <italic>W</italic>-boson propagator is <inline-formula><mml:math id="mm3026"><mml:semantics><mml:mrow><mml:mo>∼</mml:mo><mml:mfrac><mml:msubsup><mml:mi>g</mml:mi><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:mfrac></mml:mrow></mml:semantics></mml:math></inline-formula>. Thus, the loop integral scales as <inline-formula><mml:math id="mm3027"><mml:semantics><mml:mrow><mml:mo>∫</mml:mo><mml:mfrac><mml:mrow><mml:msup><mml:mi>d</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mi>k</mml:mi></mml:mrow><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>k</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:mfrac><mml:mo>∼</mml:mo><mml:mfrac><mml:mn>1</mml:mn><mml:mi>v</mml:mi></mml:mfrac></mml:mrow></mml:semantics></mml:math></inline-formula>, producing the second term in <inline-formula><mml:math id="mm3028"><mml:semantics><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>f</mml:mi><mml:mi>f</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> of (<xref ref-type="disp-formula" rid="FD3-symmetry-14-00180">3</xref>), with a calculable coefficient <italic>C</italic>. (For this estimate, the momenta at the interaction vertices can be taken to act on the external <inline-formula><mml:math id="mm3029"><mml:semantics><mml:msubsup><mml:mi>A</mml:mi><mml:mi>μ</mml:mi><mml:mn>3</mml:mn></mml:msubsup></mml:semantics></mml:math></inline-formula> to produce the field strength in (<xref ref-type="disp-formula" rid="FD3-symmetry-14-00180">3</xref>)).</p>
Full article ">Figure 2
<p>A representation of a monopole–instanton solution in <inline-formula><mml:math id="mm3030"><mml:semantics><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula>, showing the scales characterizing the <inline-formula><mml:math id="mm3031"><mml:semantics><mml:mrow><mml:mi>λ</mml:mi><mml:mo>&lt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> spherically-symmetric monopole–instanton: the nonabelian core of size <inline-formula><mml:math id="mm3032"><mml:semantics><mml:msup><mml:mi>v</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:semantics></mml:math></inline-formula>, the outside core region of size <inline-formula><mml:math id="mm3033"><mml:semantics><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msqrt><mml:mi>λ</mml:mi></mml:msqrt><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:semantics></mml:math></inline-formula> with abelian <inline-formula><mml:math id="mm3034"><mml:semantics><mml:msub><mml:mi>E</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> field, and the “long-range” region <inline-formula><mml:math id="mm3035"><mml:semantics><mml:mrow><mml:mi>r</mml:mi><mml:mo>&gt;</mml:mo><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:msqrt><mml:mi>λ</mml:mi></mml:msqrt><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> where only an abelian <inline-formula><mml:math id="mm3036"><mml:semantics><mml:mover accent="true"><mml:mi>B</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:semantics></mml:math></inline-formula> field in the unbroken-<inline-formula><mml:math id="mm3037"><mml:semantics><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> is present. For our applications, only the long-range region will be relevant, as the monopole–instantons are well separated in the dilute-gas approximation. The action of the <inline-formula><mml:math id="mm3038"><mml:semantics><mml:mrow><mml:mi>λ</mml:mi><mml:mo>≪</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> solution is approximately the BPS action (<xref ref-type="disp-formula" rid="FD9-symmetry-14-00180">9</xref>), <inline-formula><mml:math id="mm3039"><mml:semantics><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:mi>v</mml:mi></mml:mrow><mml:msubsup><mml:mi>g</mml:mi><mml:mn>3</mml:mn><mml:mn>2</mml:mn></mml:msubsup></mml:mfrac></mml:mrow></mml:semantics></mml:math></inline-formula>, and the magnetic charge (<xref ref-type="disp-formula" rid="FD14-symmetry-14-00180">14</xref>) is unity.</p>
Full article ">Figure 3
<p>The photon–dual photon duality (<xref ref-type="disp-formula" rid="FD21-symmetry-14-00180">21</xref>) maps an electric charge <inline-formula><mml:math id="mm3040"><mml:semantics><mml:msub><mml:mi>Q</mml:mi><mml:mi>e</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> into a vortex of the dual photon field, with monodromy determined by the charge. As shown in Exercise 3, the monodromy equals <inline-formula><mml:math id="mm3041"><mml:semantics><mml:mrow><mml:mo>±</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> for “quarks” in the fundamental of <inline-formula><mml:math id="mm3042"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>. A fundamental Wilson loop is thus mapped to a disorder operator for the <inline-formula><mml:math id="mm3043"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> field, defined by imposing <inline-formula><mml:math id="mm3044"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> monodromy of the dual photon around the loop.</p>
Full article ">Figure 4
<p>An illustration of the scale hierarchy, controlled by the exponentially large <inline-formula><mml:math id="mm3045"><mml:semantics><mml:msup><mml:mi>e</mml:mi><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:msup></mml:semantics></mml:math></inline-formula>, between the various length scales characterizing the the dilute <italic>M</italic>–<inline-formula><mml:math id="mm3046"><mml:semantics><mml:msup><mml:mi>M</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula> monopole–instanton gas. The dual photon Compton wavelength (the Debye screening length) is the largest length scale. A Debye volume contains a large number of <italic>M</italic> and <inline-formula><mml:math id="mm3047"><mml:semantics><mml:msup><mml:mi>M</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula> fluctuations.</p>
Full article ">Figure 5
<p>The rectangular Wilson loop used to study the potential between probe fundamental sources.</p>
Full article ">Figure 6
<p>The charge-<italic>q</italic> Wilson loop <inline-formula><mml:math id="mm3048"><mml:semantics><mml:mrow><mml:msub><mml:mi>W</mml:mi><mml:mi>q</mml:mi></mml:msub><mml:mrow><mml:mo>[</mml:mo><mml:mi>C</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> requires a <inline-formula><mml:math id="mm3049"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mi>q</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> monodromy of <inline-formula><mml:math id="mm3050"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> around any loop <italic>L</italic> linked with <italic>C</italic>. To obtain an analytic expression, we imagine that <italic>C</italic> is in the <inline-formula><mml:math id="mm3051"><mml:semantics><mml:mrow><mml:mi>x</mml:mi><mml:mi>y</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>-plane and take it to be infinitely large, running around the perimeter of the plane.</p>
Full article ">Figure 7
<p>Continuing the Euclidean picture to Minkowski space. Taking <italic>x</italic> to be the Euclidean time, we intersect the Wilson loop with a fixed-time plane (shown by the dotted line) and plot the resulting field configuration in <xref ref-type="fig" rid="symmetry-14-00180-f008">Figure 8</xref>.</p>
Full article ">Figure 8
<p>A plot of the spatial field configuration of static quark/antiquark sources taken at a finite separation, obtained as explained in <xref ref-type="fig" rid="symmetry-14-00180-f007">Figure 7</xref>, after taking <italic>T</italic> to infinity. The potential energy density of the <inline-formula><mml:math id="mm3052"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> field is shown. Far from the sources, one can check that the <inline-formula><mml:math id="mm3053"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> field profile approaches that of the analytic solution from <xref ref-type="fig" rid="symmetry-14-00180-f009">Figure 9</xref>. The gradient of the <inline-formula><mml:math id="mm3054"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> field is nonzero only in a flux tube region of width <inline-formula><mml:math id="mm3055"><mml:semantics><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>σ</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> connecting the static sources. In terms of the electric variables, recall the duality relation (<xref ref-type="disp-formula" rid="FD21-symmetry-14-00180">21</xref>), the <inline-formula><mml:math id="mm3056"><mml:semantics><mml:mover accent="true"><mml:mi mathvariant="script">E</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:semantics></mml:math></inline-formula>-field flux of the sources is collimated in a flux tube (line in <inline-formula><mml:math id="mm3057"><mml:semantics><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula>), the semiclassical confining string of the Polyakov model. (This Figure is taken from [<xref ref-type="bibr" rid="B35-symmetry-14-00180">35</xref>,<xref ref-type="bibr" rid="B62-symmetry-14-00180">62</xref>] with the sole purpose to qualitatively illustrate the physics. The numerical methods used to obtain them are explained in these references).</p>
Full article ">Figure 9
<p>The domain wall solution (<xref ref-type="disp-formula" rid="FD51-symmetry-14-00180">51</xref>) plotted for <inline-formula><mml:math id="mm3058"><mml:semantics><mml:mrow><mml:msub><mml:mi>m</mml:mi><mml:mi>σ</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. It is clear that the thickness of the wall is a few times the Compton wavelength of the dual photon, as the naive estimate shows.</p>
Full article ">Figure 10
<p>The segment <inline-formula><mml:math id="mm3059"><mml:semantics><mml:mrow><mml:mrow><mml:mo>〈</mml:mo><mml:mi>L</mml:mi><mml:msubsup><mml:mi>A</mml:mi><mml:mn>4</mml:mn><mml:mn>3</mml:mn></mml:msubsup><mml:mo>〉</mml:mo></mml:mrow><mml:mo>∈</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> known as the “moduli space” or “Weyl chamber” of the <inline-formula><mml:math id="mm3060"><mml:semantics><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula>-holonomy of the <inline-formula><mml:math id="mm3061"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> theory. All points inside the Weyl chamber are physically distinct. That there are no further gauge identifications of points inside the Weyl chamber follows the fact that they are distinguished by the different values of the gauge invariant operator <inline-formula><mml:math id="mm3062"><mml:semantics><mml:mrow><mml:mo>〈</mml:mo><mml:mi>tr</mml:mi><mml:mo>Ω</mml:mo><mml:mo>/</mml:mo><mml:mn>2</mml:mn><mml:mo>〉</mml:mo><mml:mo>∈</mml:mo><mml:mo>[</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>. As explained in the text, on the two edges of the Weyl chamber, the <inline-formula><mml:math id="mm3063"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula> center symmetry is maximally broken (see also Exercise 10). Center symmetry acts on the Weyl chamber as a reflection w.r.t. the middle point. This point corresponds to the center-symmetric value of the holonomy, <inline-formula><mml:math id="mm3064"><mml:semantics><mml:mrow><mml:mo>〈</mml:mo><mml:mi>tr</mml:mi><mml:mo>Ω</mml:mo><mml:mo>〉</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, a major player in our study of the dynamics.</p>
Full article ">Figure 11
<p>The asymptotically-free running coupling <inline-formula><mml:math id="mm3065"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>g</mml:mi><mml:mn>4</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>μ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of the 4D theory approaches strong coupling at IR scales of order <inline-formula><mml:math id="mm3066"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>, as shown by the dashed line. At (or near) the center-symmetric point on <inline-formula><mml:math id="mm3067"><mml:semantics><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm3068"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo><mml:mo stretchy="false">→</mml:mo><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> Higgsing takes place at a scale <inline-formula><mml:math id="mm3069"><mml:semantics><mml:mrow><mml:mi>v</mml:mi><mml:mo>∼</mml:mo><mml:mfrac><mml:mi>π</mml:mi><mml:mi>L</mml:mi></mml:mfrac></mml:mrow></mml:semantics></mml:math></inline-formula>. The long-distance 3D <inline-formula><mml:math id="mm3070"><mml:semantics><mml:mrow><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> theory is free, with 3D gauge coupling determined by matching to the 4D theory and given by <inline-formula><mml:math id="mm3071"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>g</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:msubsup><mml:mi>g</mml:mi><mml:mn>4</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>μ</mml:mi><mml:mo>∼</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>/</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. The weak-coupling analysis is justified if the scale of the breaking is larger than <inline-formula><mml:math id="mm3072"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>, i.e., when <inline-formula><mml:math id="mm3073"><mml:semantics><mml:mrow><mml:mo>Λ</mml:mo><mml:mi>L</mml:mi><mml:mo>≪</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure 12
<p>A cartoon of the spacetime structure of the <inline-formula><mml:math id="mm3074"><mml:semantics><mml:msup><mml:mi>x</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula>-independent <italic>M</italic> monopole–instanton solution on <inline-formula><mml:math id="mm3075"><mml:semantics><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. This is essentially the solution localized in <inline-formula><mml:math id="mm3076"><mml:semantics><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula> shown in <xref ref-type="fig" rid="symmetry-14-00180-f002">Figure 2</xref>, now trivially embedded in <inline-formula><mml:math id="mm3077"><mml:semantics><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, by allowing it to propagate in <inline-formula><mml:math id="mm3078"><mml:semantics><mml:msup><mml:mi>x</mml:mi><mml:mn>4</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula> without change. The <inline-formula><mml:math id="mm3079"><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:mi>K</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> monopole–instanton, on the other hand, is twisted in the <inline-formula><mml:math id="mm3080"><mml:semantics><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula> direction by the improper gauge transform <inline-formula><mml:math id="mm3081"><mml:semantics><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mi>A</mml:mi></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> of (<xref ref-type="disp-formula" rid="FD59-symmetry-14-00180">59</xref>).</p>
Full article ">Figure 13
<p>Illustrating the construction of the <inline-formula><mml:math id="mm3082"><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:mi>K</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> monopole–instanton. The arrow of length <italic>v</italic> denoted by <inline-formula><mml:math id="mm3083"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> shows the variation of the <inline-formula><mml:math id="mm3084"><mml:semantics><mml:msub><mml:mi>A</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> field of the <inline-formula><mml:math id="mm3085"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> solution: from 0 at its center to its asymptotic value <italic>v</italic> at <inline-formula><mml:math id="mm3086"><mml:semantics><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula> infinity. To construct the <inline-formula><mml:math id="mm3087"><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:mi>K</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> monopole, start with an <inline-formula><mml:math id="mm3088"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> solution (denoted by <inline-formula><mml:math id="mm3089"><mml:semantics><mml:msup><mml:mi>M</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula> in the text), where <inline-formula><mml:math id="mm3090"><mml:semantics><mml:msub><mml:mi>A</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> varies from 0 at the center to <inline-formula><mml:math id="mm3091"><mml:semantics><mml:msup><mml:mi>v</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula> asymptotically, as shown by the lower arrow. Then apply a global <inline-formula><mml:math id="mm3092"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula> (represented by the improper gauge transformation <inline-formula><mml:math id="mm3093"><mml:semantics><mml:msub><mml:mi>g</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> of (<xref ref-type="disp-formula" rid="FD59-symmetry-14-00180">59</xref>)) and a Weyl reflection on the <inline-formula><mml:math id="mm3094"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>v</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> solution, a procedure schematically denoted by <inline-formula><mml:math id="mm3095"><mml:semantics><mml:mrow><mml:mi>i</mml:mi><mml:msup><mml:mi>σ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>∘</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> on the Figure. As symmetries map solutions into solutions, the result is the monopole–instanton <inline-formula><mml:math id="mm3096"><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:mi>K</mml:mi><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, in the same vacuum as <inline-formula><mml:math id="mm3097"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:mo>(</mml:mo><mml:mi>v</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>.</p>
Full article ">Figure 14
<p>A plot of the one-loop effective potential on the Weyl chamber in pure YM theory, computed using supersymmetry (Equation (<xref ref-type="disp-formula" rid="FD81-symmetry-14-00180">81</xref>) is plotted with <inline-formula><mml:math id="mm3098"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>). In the pure-YM theory, the one-loop potential is given by the top curve (<xref ref-type="disp-formula" rid="FD81-symmetry-14-00180">81</xref>) and the energy is minimized at <inline-formula><mml:math id="mm3099"><mml:semantics><mml:mrow><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm3100"><mml:semantics><mml:mrow><mml:mi>v</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow><mml:mi>L</mml:mi></mml:mfrac></mml:mrow></mml:semantics></mml:math></inline-formula>, i.e., at <inline-formula><mml:math id="mm3101"><mml:semantics><mml:mrow><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac><mml:mrow><mml:mo>〈</mml:mo><mml:mi>tr</mml:mi><mml:mo>Ω</mml:mo><mml:mo>〉</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The <inline-formula><mml:math id="mm3102"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula> symmetry is thus maximally broken. The physical interpretation of this phase is a deconfined phase of YM theory, see <xref ref-type="sec" rid="sec3dot4dot3-symmetry-14-00180">Section 3.4.3</xref>.</p>
Full article ">Figure 15
<p>A fundamental quark–antiquark pair inserted in the thermal <inline-formula><mml:math id="mm3103"><mml:semantics><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> theory (where <inline-formula><mml:math id="mm3104"><mml:semantics><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula> size is <inline-formula><mml:math id="mm3105"><mml:semantics><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>T</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>) is represented by two Polyakov loops a distance <inline-formula><mml:math id="mm3106"><mml:semantics><mml:mrow><mml:mi>R</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> apart. It measures the <italic>T</italic>-dependent interaction potential <inline-formula><mml:math id="mm3107"><mml:semantics><mml:mrow><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>R</mml:mi><mml:mo>,</mml:mo><mml:mi>T</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> between the two sources. In the low-<italic>T</italic> confined phase, <inline-formula><mml:math id="mm3108"><mml:semantics><mml:mrow><mml:mi>V</mml:mi><mml:mo>∼</mml:mo><mml:mo>Σ</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, giving an area-like behaviour of the correlation function and implying, by (<xref ref-type="disp-formula" rid="FD84-symmetry-14-00180">84</xref>), <inline-formula><mml:math id="mm3109"><mml:semantics><mml:mrow><mml:mo>〈</mml:mo><mml:mi>tr</mml:mi><mml:mo>Ω</mml:mo><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mo>〉</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, an unbroken <inline-formula><mml:math id="mm3110"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula>. In a deconfined phase, instead, the correlator is nonzero at large <italic>R</italic> implying that <inline-formula><mml:math id="mm3111"><mml:semantics><mml:mrow><mml:mo>〈</mml:mo><mml:mi>tr</mml:mi><mml:mo>Ω</mml:mo><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mo>〉</mml:mo><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, a broken center symmetry.</p>
Full article ">Figure 16
<p>The one-loop effective potential on the Weyl chamber for <inline-formula><mml:math id="mm3112"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> massless adjoint fermions, shown by the thick curve (the same <inline-formula><mml:math id="mm3113"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> scale as in <xref ref-type="fig" rid="symmetry-14-00180-f014">Figure 14</xref> is used). For <inline-formula><mml:math id="mm3114"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>&gt;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, the massless fermions dominate over the gauge contribution in the total potential. The potential on the Weyl chamber is minimized at the center-symmetric point <inline-formula><mml:math id="mm3115"><mml:semantics><mml:mrow><mml:mo>〈</mml:mo><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mo>Ω</mml:mo><mml:mo>〉</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Thus the IR theory is abelian and weakly-coupled at <inline-formula><mml:math id="mm3116"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>Λ</mml:mo><mml:mo>≪</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, as discussed in <xref ref-type="sec" rid="sec3dot2-symmetry-14-00180">Section 3.2</xref> and <xref ref-type="sec" rid="sec3dot4dot1-symmetry-14-00180">Section 3.4.1</xref>.</p>
Full article ">Figure 17
<p>The one-loop effective potential (<xref ref-type="disp-formula" rid="FD86-symmetry-14-00180">86</xref>) on the Weyl chamber for the <inline-formula><mml:math id="mm3117"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> gauge theory with <inline-formula><mml:math id="mm3118"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> massive adjoint fermions, periodic on the <inline-formula><mml:math id="mm3119"><mml:semantics><mml:msup><mml:mi>S</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm3120"><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. The potential on the Weyl chamber is minimized at the center-symmetric point <inline-formula><mml:math id="mm3121"><mml:semantics><mml:mrow><mml:mrow><mml:mi mathvariant="normal">t</mml:mi><mml:mi>r</mml:mi></mml:mrow><mml:mo>Ω</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, and the abelianized IR theory is valid. The same scale, with <inline-formula><mml:math id="mm3122"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, as in <xref ref-type="fig" rid="symmetry-14-00180-f014">Figure 14</xref> and <xref ref-type="fig" rid="symmetry-14-00180-f016">Figure 16</xref> is used.</p>
Full article ">Figure 18
<p>The energies of the two extrema of the leading order semiclassical potential (<xref ref-type="disp-formula" rid="FD92-symmetry-14-00180">92</xref>), <inline-formula><mml:math id="mm3123"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm3124"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, plotted as a function of <inline-formula><mml:math id="mm3125"><mml:semantics><mml:mi>θ</mml:mi></mml:semantics></mml:math></inline-formula>. At <inline-formula><mml:math id="mm3126"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, the <inline-formula><mml:math id="mm3127"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> extremum is a minimum, while the one at <inline-formula><mml:math id="mm3128"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> is a maximum. Conversely, at <inline-formula><mml:math id="mm3129"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, the minimum is at <inline-formula><mml:math id="mm3130"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, while <inline-formula><mml:math id="mm3131"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> is now a maximum of the potential. The level crossing occurs near <inline-formula><mml:math id="mm3132"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. To study the physics in a finite region <inline-formula><mml:math id="mm3133"><mml:semantics><mml:mrow><mml:mo>|</mml:mo><mml:mi>θ</mml:mi><mml:mo>−</mml:mo><mml:mi>π</mml:mi><mml:mo>|</mml:mo><mml:mo>=</mml:mo><mml:mi mathvariant="script">O</mml:mi><mml:mo>(</mml:mo><mml:msup><mml:mi>e</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, one has to account of the next order in the semiclassical expansion, due to the fact that the leading-order potential in (<xref ref-type="disp-formula" rid="FD92-symmetry-14-00180">92</xref>) vanishes at <inline-formula><mml:math id="mm3134"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. As discussed in <xref ref-type="sec" rid="sec4dot2dot1-symmetry-14-00180">Section 4.2.1</xref>, the next-order contribution implies that there are two ground states at <inline-formula><mml:math id="mm3135"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with broken parity. This reflects the generalized ’t Hooft anomaly involving parity and the center symmetry.</p>
Full article ">Figure 19
<p>The double-string configuration confining fundamental quarks in the <inline-formula><mml:math id="mm3136"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> dYM theory, embedded in the <inline-formula><mml:math id="mm3137"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> vacuum. The chromoelectric flux of the fundamental quarks is equally split between the two domain walls. Inside the double-string, the <inline-formula><mml:math id="mm3138"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> field is in the other vacuum. The transverse size of the double string grows similar to <inline-formula><mml:math id="mm3139"><mml:semantics><mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mi>R</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> with the source separation <italic>R</italic>. (A similar configuration, but with <inline-formula><mml:math id="mm3140"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> inside the double-string and <inline-formula><mml:math id="mm3141"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> at the top, is responsible for the confinement of <inline-formula><mml:math id="mm3142"><mml:semantics><mml:msup><mml:mi>W</mml:mi><mml:mo>±</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula> bosons in the Polyakov model and in dYM at <inline-formula><mml:math id="mm3143"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>≠</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. As discussed in <xref ref-type="sec" rid="sec2dot6-symmetry-14-00180">Section 2.6</xref>, pair production of <italic>W</italic>-boson will lead to a breaking of the adjoint string).</p>
Full article ">Figure 20
<p>Deconfinement of quarks on the domain wall between the two vacua of <inline-formula><mml:math id="mm3144"><mml:semantics><mml:mrow><mml:mi>θ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> dYM. The location of the quark sources is indicated by the two dots; the dotted lines are unphysical <inline-formula><mml:math id="mm3145"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> discontinuities of <inline-formula><mml:math id="mm3146"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>. The gradient of the <inline-formula><mml:math id="mm3147"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> field and the <inline-formula><mml:math id="mm3148"><mml:semantics><mml:mover accent="true"><mml:mi mathvariant="script">E</mml:mi><mml:mo stretchy="false">→</mml:mo></mml:mover></mml:semantics></mml:math></inline-formula> field are shown by arrows. The quarks exhibit no force while on the wall (provided their separation is larger than the screening length) as the tensions of the domain walls to the left and right of each quark are equal.</p>
Full article ">Figure 21
<p>The action of a <inline-formula><mml:math id="mm3149"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula> center-symmetry transformation and a subsequent <inline-formula><mml:math id="mm3150"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> shift of <inline-formula><mml:math id="mm3151"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> on the domain wall solution connecting <inline-formula><mml:math id="mm3152"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> to <inline-formula><mml:math id="mm3153"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (the arrow shows the values <inline-formula><mml:math id="mm3154"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> takes as the wall is traversed). This solution is thus mapped to the domain wall interpolating from <inline-formula><mml:math id="mm3155"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> to <inline-formula><mml:math id="mm3156"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. As the walls are related by a symmetry, they have the same tension, ensuring deconfinement of quarks due to tension balance, as shown in <xref ref-type="fig" rid="symmetry-14-00180-f020">Figure 20</xref>. The existence of two inequivalent domain walls between the same vacua can be formalized by saying that “a topological QFT lives on the domain wall worldvolume” (for the case at hand, this is a QFT with only two states in its Hilbert space).</p>
Full article ">Figure 22
<p>A picture of the <inline-formula><mml:math id="mm3157"><mml:semantics><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msubsup><mml:mi>S</mml:mi><mml:mi>β</mml:mi><mml:mn>1</mml:mn></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula> vacuum. The <italic>M</italic> and <inline-formula><mml:math id="mm3158"><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:mi>K</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> monopole–instantons, represented by dots, are much smaller than the size of the thermal circle at the temperatures of interest. In addition, static electrically-charged <italic>W</italic>-bosons, whose worldlines are shown by the blue lines, are excited with nonzero probability. In the temperature range (<xref ref-type="disp-formula" rid="FD103-symmetry-14-00180">103</xref>), the typical distance between either kind of objects is much larger than the size of the thermal circle (this is not depicted to scale on the drawing and has to be imagined), allowing the gas of to be considered as approximately 2D. The resulting 2D gas of electrically (squares) and magnetically (circles) charged particles is shown on the right. These interact by their respective (magnetic and electric) Coulomb interactions, and also by a mutual Aharonov–Bohm interaction. (Briefly, see, e.g., [<xref ref-type="bibr" rid="B98-symmetry-14-00180">98</xref>,<xref ref-type="bibr" rid="B99-symmetry-14-00180">99</xref>,<xref ref-type="bibr" rid="B100-symmetry-14-00180">100</xref>], the latter is a term in the Boltzmann partition function that depends on the 2D angular position of the electric and magnetic charges, but not on the distance). The physics can be studied by various means that we do not have time to discuss, resulting in a deconfinement phase transition at the critical temperature (<xref ref-type="disp-formula" rid="FD102-symmetry-14-00180">102</xref>).</p>
Full article ">Figure 23
<p>A plot of the separation quasizero mode integrand in (<xref ref-type="disp-formula" rid="FD125-symmetry-14-00180">125</xref>), plotted for <inline-formula><mml:math id="mm3159"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>g</mml:mi><mml:mn>4</mml:mn><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0.1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm3160"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> QCD (adj) in units of <inline-formula><mml:math id="mm3161"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The bion size, <inline-formula><mml:math id="mm3162"><mml:semantics><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, is <inline-formula><mml:math id="mm3163"><mml:semantics><mml:mrow><mml:mo>∼</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> times larger than the size of the constituent <italic>M</italic> and <inline-formula><mml:math id="mm3164"><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:msup><mml:mi>K</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, indicated by <inline-formula><mml:math id="mm3165"><mml:semantics><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mi>g</mml:mi><mml:mn>4</mml:mn></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> on the figure.</p>
Full article ">Figure 24
<p>The correlated tunnelling events known as “magnetic bions” <italic>B</italic>: the <italic>M</italic> and <inline-formula><mml:math id="mm3166"><mml:semantics><mml:mrow><mml:mi>K</mml:mi><mml:msup><mml:mi>K</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> constituents have equal magnetic charge and thus repel at long distances. However, the hopping of the <inline-formula><mml:math id="mm3167"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:msub><mml:mi>n</mml:mi><mml:mi>f</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> fermions associated to each of the constituents induce attraction, leading to a stable “molecule” of size <inline-formula><mml:math id="mm3168"><mml:semantics><mml:mrow><mml:msub><mml:mi>r</mml:mi><mml:mrow><mml:mi>b</mml:mi><mml:mi>i</mml:mi><mml:mi>o</mml:mi><mml:mi>n</mml:mi></mml:mrow></mml:msub><mml:mo>∼</mml:mo><mml:mi>L</mml:mi><mml:mo>/</mml:mo><mml:msubsup><mml:mi>g</mml:mi><mml:mn>4</mml:mn><mml:mn>2</mml:mn></mml:msubsup></mml:mrow></mml:semantics></mml:math></inline-formula>. The <italic>B</italic> (and <inline-formula><mml:math id="mm3169"><mml:semantics><mml:msup><mml:mi>B</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula>) have magnetic charge 2 but have no topological charge and hence no fermion zero modes. Their proliferation in the vacuum is responsible for the generation of mass gap as shown on <xref ref-type="fig" rid="symmetry-14-00180-f025">Figure 25</xref>.</p>
Full article ">Figure 25
<p>A typical configuration contributing to the vacuum to vacuum amplitude in QCD (adj): a dilute grand-canonical gas of bions <italic>B</italic> and anti-bions <inline-formula><mml:math id="mm3170"><mml:semantics><mml:msup><mml:mi>B</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:semantics></mml:math></inline-formula> and the hierarchy of characteristic scales involved. Compare with <xref ref-type="fig" rid="symmetry-14-00180-f004">Figure 4</xref> for the Polyakov model.</p>
Full article ">Figure 26
<p>The conjectured phase diagram of softly-broken SYM in the <italic>m</italic>-<italic>L</italic> plane. The calculable center-symmetry breaking quantum phase transition, occurring at small-<italic>m</italic>,<italic>L</italic> and shown in Figure 28 is in the left-hand corner of the diagram, shown by a thick red line. The center-breaking transition occurs, in the approximation of (<xref ref-type="disp-formula" rid="FD159-symmetry-14-00180">158</xref>), at the critical value <inline-formula><mml:math id="mm3171"><mml:semantics><mml:mrow><mml:mover accent="true"><mml:mi>m</mml:mi><mml:mo>^</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:msup><mml:mover accent="true"><mml:mo>Λ</mml:mo><mml:mo>^</mml:mo></mml:mover><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. More appropriately, this is written as <inline-formula><mml:math id="mm3172"><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mo>Λ</mml:mo><mml:mn>3</mml:mn></mml:msup><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, which is the equation of the thick red curve shown on the plot. If one keeps <italic>m</italic> fixed, it is clear that the center-broken phase is entered by decreasing <italic>L</italic> (which, in the thermal theory means going to higher temperature). This weak-coupling quantum phase transition is conjectured to be continuously connected, upon decoupling the gaugino, to the thermal deconfinement transition in pure YM theory, shown by the thick black line on the right.</p>
Full article ">Figure 27
<p>The lifting of monopole–instanton fermion zero modes by a small fermion mass term. The fermion zero modes are localized over a distance of order <italic>L</italic>, the <italic>M</italic> size. As <inline-formula><mml:math id="mm3173"><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo>≪</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mi>L</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, the propagators shown by the blue arrows (the same ones responsible for the bion binding of <xref ref-type="fig" rid="symmetry-14-00180-f024">Figure 24</xref>) are massless propagators over a distance <italic>L</italic>, with the divergence cut off by the monopole–instanton core of size <italic>L</italic>. The calculation is sketched in Equation (<xref ref-type="disp-formula" rid="FD156-symmetry-14-00180">155</xref>).</p>
Full article ">Figure 28
<p>A plot of the potential (<xref ref-type="disp-formula" rid="FD159-symmetry-14-00180">158</xref>) for <inline-formula><mml:math id="mm3174"><mml:semantics><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mi>π</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> as a function of <inline-formula><mml:math id="mm3175"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula> for two values of <inline-formula><mml:math id="mm3176"><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mi>L</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>: <inline-formula><mml:math id="mm3177"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mi>L</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&gt;</mml:mo><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mi>L</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mover accent="true"><mml:mo>Λ</mml:mo><mml:mo>^</mml:mo></mml:mover><mml:mn>3</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> (lower curve) with broken <inline-formula><mml:math id="mm3178"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm3179"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mi>L</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>&lt;</mml:mo><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mi>L</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> (top curve) with center-symmetric holonomy vev. The top and bottom curve illustrate the 2nd order of the center-symmetry breaking phase transition, corresponding to the unbroken- and broken-<inline-formula><mml:math id="mm3180"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula> phases, respectively. A study of the <inline-formula><mml:math id="mm3181"><mml:semantics><mml:mi>θ</mml:mi></mml:semantics></mml:math></inline-formula>-dependence reveals that <inline-formula><mml:math id="mm3182"><mml:semantics><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mi>L</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>c</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> is a decreasing function of <inline-formula><mml:math id="mm3183"><mml:semantics><mml:mi>θ</mml:mi></mml:semantics></mml:math></inline-formula>.</p>
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<p>A plot of the one-loop holonomy potential (<xref ref-type="disp-formula" rid="FD164-symmetry-14-00180">163</xref>), the thick line, in an <inline-formula><mml:math id="mm3184"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> gauge theory with a periodic and an antiperiodic massless fundamental flavours. The same scale <inline-formula><mml:math id="mm3185"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> is used, to facilitate comparison with <xref ref-type="fig" rid="symmetry-14-00180-f016">Figure 16</xref> and <xref ref-type="fig" rid="symmetry-14-00180-f017">Figure 17</xref>. The P and A contributions, favouring the broken <inline-formula><mml:math id="mm3186"><mml:semantics><mml:msubsup><mml:mi>Z</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:msubsup></mml:semantics></mml:math></inline-formula>, are also shown separately.</p>
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<p>The one-loop graph generating mixed Chern–Simons (CS) coupling of the dynamical Cartan gauge field <inline-formula><mml:math id="mm3187"><mml:semantics><mml:msub><mml:mi>A</mml:mi><mml:mi>ν</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm3188"><mml:semantics><mml:msub><mml:mi>X</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula>, the <inline-formula><mml:math id="mm3189"><mml:semantics><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> chiral-symmetry background field. To make the result (<xref ref-type="disp-formula" rid="FD166-symmetry-14-00180">165</xref>) plausible, using 4D <inline-formula><mml:math id="mm3190"><mml:semantics><mml:mi>γ</mml:mi></mml:semantics></mml:math></inline-formula>-matrix notation, we have schematically indicated the relevant factors leading to the 3D CS coupling. The <inline-formula><mml:math id="mm3191"><mml:semantics><mml:msub><mml:mi>X</mml:mi><mml:mi>μ</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> coupling to the A and P Dirac fermions is chiral, while an insertion of the holonomy vev in the numerator of one of the fermion propagators gives a <inline-formula><mml:math id="mm3192"><mml:semantics><mml:msub><mml:mi>γ</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> factor. Altogether, this leads to <inline-formula><mml:math id="mm3193"><mml:semantics><mml:mrow><mml:mi>tr</mml:mi><mml:msub><mml:mi>γ</mml:mi><mml:mn>5</mml:mn></mml:msub><mml:msub><mml:mi>γ</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msub><mml:mi>γ</mml:mi><mml:mi>μ</mml:mi></mml:msub><mml:msub><mml:mi>γ</mml:mi><mml:mi>ν</mml:mi></mml:msub><mml:msub><mml:mi>γ</mml:mi><mml:mi>λ</mml:mi></mml:msub><mml:mo>∼</mml:mo><mml:msub><mml:mi>ϵ</mml:mi><mml:mrow><mml:mi>μ</mml:mi><mml:mi>ν</mml:mi><mml:mi>λ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>. To obtain Equation (<xref ref-type="disp-formula" rid="FD166-symmetry-14-00180">165</xref>), one has to carefully sum the loop contributions of the heavy A and P Dirac fermions, including their KK mode tower. The result for each Dirac fermion is given in Equation (B.4) in [<xref ref-type="bibr" rid="B37-symmetry-14-00180">37</xref>] (one has to substitute <inline-formula><mml:math id="mm3194"><mml:semantics><mml:mrow><mml:mi>q</mml:mi><mml:mo>=</mml:mo><mml:mo>±</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for the <inline-formula><mml:math id="mm3195"><mml:semantics><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>X</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> charges, use the <inline-formula><mml:math id="mm3196"><mml:semantics><mml:mrow><mml:mi>S</mml:mi><mml:mi>U</mml:mi><mml:mo>(</mml:mo><mml:mn>2</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> fundamental-representation weights, and account for the different A-fermion boundary condition by inserting the <inline-formula><mml:math id="mm3197"><mml:semantics><mml:mrow><mml:mi>μ</mml:mi><mml:mo>=</mml:mo><mml:mfrac><mml:mi>π</mml:mi><mml:mi>L</mml:mi></mml:mfrac></mml:mrow></mml:semantics></mml:math></inline-formula><inline-formula><mml:math id="mm3198"><mml:semantics><mml:mrow><mml:mi>U</mml:mi><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mi>V</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>-Wilson line, similar to (<xref ref-type="disp-formula" rid="FD162-symmetry-14-00180">161</xref>)).</p>
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19 pages, 2255 KiB  
Article
Adaptive Autonomous Robot Navigation by Neutrosophic WASPAS Extensions
by Rokas Semenas and Romualdas Bausys
Symmetry 2022, 14(1), 179; https://doi.org/10.3390/sym14010179 - 17 Jan 2022
Cited by 1 | Viewed by 1871
Abstract
In this research, a novel adaptive frontier-assessment-based environment exploration strategy for search and rescue (SAR) robots is presented. Two neutrosophic WASPAS multi-criteria decision-making (MCDM) method extensions that provide the tools for addressing the inaccurate input data characteristics are applied to measure the utilities [...] Read more.
In this research, a novel adaptive frontier-assessment-based environment exploration strategy for search and rescue (SAR) robots is presented. Two neutrosophic WASPAS multi-criteria decision-making (MCDM) method extensions that provide the tools for addressing the inaccurate input data characteristics are applied to measure the utilities of the candidate frontiers. Namely, the WASPAS method built under the interval-valued neutrosophic set environment (WASPAS-IVNS) and the WASPAS method built under the m-generalised q-neutrosophic set environment (WASPAS-mGqNS). The indeterminacy component of the neutrosophic set can be considered as the axis of symmetry, and neutrosophic truth and falsity membership functions are asymmetric. As these three components of the neutrosophic set are independent, one can model the input data characteristics applied in the candidate frontier assessment process, while also taking into consideration uncertain or inaccurate input data obtained by the autonomous robot sensors. The performed experiments indicate that the proposed adaptive environment exploration strategy provides better results when compared to the baseline greedy environment exploration strategies. Full article
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<p>The proposed adaptive environment exploration strategy.</p>
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<p>(<b>a</b>) The 1st simulated SAR environment representing 26 by 17 m exploration space with an open topology. (<b>b</b>) The 2nd simulated SAR environment representing 32 by 26 m exploration space with the separated area topology. (<b>c</b>) The 3rd simulated SAR environment representing 43 by 28 m exploration space with a mirrored loop-type topology.</p>
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<p>The considered candidate frontier assessment problem. The red markers represent dangerous areas. The yellow marker represents visible survivor and white markers represent priority locations that should be visited by the robot. Frontier regions are defined by blue lines and the candidate frontiers <span class="html-italic">f</span> are marked by the green markers.</p>
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<p>The information discovered in the 2nd environment.</p>
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<p>The penalty received in the 2nd environment.</p>
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<p>The information discovered in the 3rd environment.</p>
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<p>The penalty received in the 3rd environment.</p>
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11 pages, 320 KiB  
Article
On the Optimality of the LR Test for Mediation
by Kees Jan Van Garderen and Noud Van Giersbergen
Symmetry 2022, 14(1), 178; https://doi.org/10.3390/sym14010178 - 17 Jan 2022
Viewed by 1743
Abstract
Testing for mediation, or indirect effects, is empirically very important in many disciplines. It has two obvious symmetries that the testing procedure should be invariant to. The ordered absolute t-statistics from two ordinary regressions are maximal invariant under the associated groups of transformations. [...] Read more.
Testing for mediation, or indirect effects, is empirically very important in many disciplines. It has two obvious symmetries that the testing procedure should be invariant to. The ordered absolute t-statistics from two ordinary regressions are maximal invariant under the associated groups of transformations. Sobel’s (1982) Wald-type and the LR test statistic are both functions of this maximal invariant and satisfy two logical coherence requirements: (1) size coherence: rejection at level α implies rejection at all higher significance levels; and (2) information coherence: more (less) evidence against the null implies continued (non) rejection of the null. The LR test statistic is simply the smallest of the two absolute t-statistics, and we show that the LR test is the Uniformly Most Powerful (information and size) Coherent Invariant (UMPCI) test. In short: the LR test for mediation is simple and best. Full article
(This article belongs to the Section Mathematics)
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Figure 1
<p>Boundaries <math display="inline"><semantics> <mrow> <mo>∂</mo> <mi>C</mi> <mi>R</mi> </mrow> </semantics></math> for Sobel-(Wald) and LR tests in <math display="inline"><semantics> <msubsup> <mi mathvariant="double-struck">R</mi> <mrow> <mo>+</mo> </mrow> <mn>2</mn> </msubsup> </semantics></math>, the sample space for <math display="inline"><semantics> <mrow> <mo>|</mo> <mi>t</mi> <mo>|</mo> </mrow> </semantics></math>.</p>
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17 pages, 375 KiB  
Article
Symmetry, Confinement, and the Higgs Phase
by Jeff Greensite and Kazue Matsuyama
Symmetry 2022, 14(1), 177; https://doi.org/10.3390/sym14010177 - 17 Jan 2022
Cited by 13 | Viewed by 2055
Abstract
We show that the Higgs and confinement phases of a gauge Higgs theory, with the Higgs field in the fundamental representation of the gauge group, are distinguished both by a broken or unbroken realization of the global center subgroup of the gauge group, [...] Read more.
We show that the Higgs and confinement phases of a gauge Higgs theory, with the Higgs field in the fundamental representation of the gauge group, are distinguished both by a broken or unbroken realization of the global center subgroup of the gauge group, and by the type of confinement in each phase. This is color confinement in the Higgs phase, and a stronger property, which we call “separation-of-charge” confinement, in the confining phase. Full article
(This article belongs to the Special Issue New Applications of Symmetry in Lattice Field Theory)
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<p>The location of remnant global gauge symmetry breaking in Landau and Coulomb gauges, in the <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>−</mo> <mi>γ</mi> </mrow> </semantics></math> coupling plane, for the SU(2) gauge Higgs theory in (<a href="#FD4-symmetry-14-00177" class="html-disp-formula">4</a>). Figure from ref. [<a href="#B9-symmetry-14-00177" class="html-bibr">9</a>].</p>
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<p>(<b>a</b>) Extrapolation of <math display="inline"><semantics> <mrow> <mo>〈</mo> <mo>Φ</mo> <mo>〉</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mrow> <mi>s</mi> <mi>y</mi> <mi>m</mi> </mrow> </msub> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math> above (<math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1.5</mn> </mrow> </semantics></math>) and below (<math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1.1</mn> <mo>,</mo> <mn>1.25</mn> </mrow> </semantics></math>) the custodial symmetry breaking transition at <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>1.2</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>1.4</mn> </mrow> </semantics></math>, in SU(2) gauge Higgs theory. The lattice volume is <math display="inline"><semantics> <msup> <mn>16</mn> <mn>4</mn> </msup> </semantics></math>; error bars are smaller than the symbol sizes. (<b>b</b>) The custodial symmetry breaking/spin glass transition line joins the filled squares; the Coulomb gauge transition line, joining the open triangles, lies entirely within the broken custodial symmetry phase, as it must from Theorem 1. Figures from ref. [<a href="#B12-symmetry-14-00177" class="html-bibr">12</a>].</p>
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<p>Unlike Hydrogen, where the ionization energy can be measured experimentally, there is no experimental procedure for creating an “ionized” hadron. There are, however, physical states (see next figure) in the Hilbert space which do correspond to widely separated but interacting quarks.</p>
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<p>Decay of a state with widely separated quark-antiquark color charges and fractional electric charge into a set of color neutral hadrons of integer electric charge. The property of S<math display="inline"><semantics> <msub> <mrow/> <mi mathvariant="normal">c</mi> </msub> </semantics></math> confinement is related to the energy of the color charge separated state <math display="inline"><semantics> <mrow> <msub> <mo>Ψ</mo> <mi>V</mi> </msub> <mrow> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, in the limit of color charge separation <math display="inline"><semantics> <mrow> <mi>R</mi> <mo>→</mo> <mo>∞</mo> </mrow> </semantics></math>. Figure from [<a href="#B15-symmetry-14-00177" class="html-bibr">15</a>].</p>
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<p>Contrasting properties of charged (<math display="inline"><semantics> <msub> <mo>Φ</mo> <mn>1</mn> </msub> </semantics></math>) and neutral (<math display="inline"><semantics> <msub> <mo>Φ</mo> <mn>4</mn> </msub> </semantics></math>) fermion–antifermion states in the confinement and Higgs phases of an SU(3) gauge Higgs theory. (<b>a</b>) Energy expectation value <math display="inline"><semantics> <mrow> <msup> <mi>E</mi> <mo>Φ</mo> </msup> <mrow> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> vs. separation <span class="html-italic">R</span> of the <math display="inline"><semantics> <msub> <mo>Φ</mo> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mo>Φ</mo> <mn>4</mn> </msub> </semantics></math> states in the confined phase, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>5.5</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. (<b>b</b>) Same as subfigure (<b>a</b>), but in the Higgs phase at <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>5.5</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>. (<b>c</b>) Overlap vs. <span class="html-italic">R</span> of normalized charge (<math display="inline"><semantics> <msub> <mo>Φ</mo> <mn>1</mn> </msub> </semantics></math>) and neutral (<math display="inline"><semantics> <msub> <mo>Φ</mo> <mn>4</mn> </msub> </semantics></math>) states in the confined phase, at <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>5.5</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. (<b>d</b>) Same as subfigure (<b>c</b>), but in the Higgs phase at <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>5.5</mn> <mo>,</mo> <mi>γ</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math>. Figures from ref. [<a href="#B15-symmetry-14-00177" class="html-bibr">15</a>].</p>
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8 pages, 448 KiB  
Article
On Mikheyev–Smirnov–Wolfenstein Resonance Widths
by Mihail Chizhov
Symmetry 2022, 14(1), 176; https://doi.org/10.3390/sym14010176 - 17 Jan 2022
Viewed by 1280
Abstract
The aim of the present paper is the evaluation of the resonance half-widths of the first maximum for the probability of the total neutrino conversion in a medium. We consider the simplest case of two-neutrino mixing in matter with a constant refraction length. [...] Read more.
The aim of the present paper is the evaluation of the resonance half-widths of the first maximum for the probability of the total neutrino conversion in a medium. We consider the simplest case of two-neutrino mixing in matter with a constant refraction length. The results can be applied, for example, to studies of neutrino oscillations in the Earth’s mantle and elsewhere. Full article
(This article belongs to the Special Issue Recent Advances in Neutrino Physics)
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<p>A comparison between the half-widths: The sign-dependent ratio of the exact numerical solution of Equation (<a href="#FD8-symmetry-14-00176" class="html-disp-formula">8</a>) to the analytical guess of Equation (<a href="#FD5-symmetry-14-00176" class="html-disp-formula">5</a>).</p>
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<p>The difference between the upper and lower curves presents the absolute half-width dependence, with respect to <math display="inline"><semantics> <mrow> <mo form="prefix">tan</mo> <mn>2</mn> <mi>θ</mi> </mrow> </semantics></math>.</p>
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<p>The total width dependence, with respect to the parameter <math display="inline"><semantics> <mi>θ</mi> </semantics></math> for the variable <span class="html-italic">y</span> (solid line) and maximal width for the variable <span class="html-italic">x</span> (dashed line).</p>
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<p>Resonance shapes (solid lines for <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) for the mixing parameters <math display="inline"><semantics> <mrow> <mi>π</mi> <mo>/</mo> <mn>100</mn> </mrow> </semantics></math> (<b>a</b>), <math display="inline"><semantics> <mrow> <mn>2</mn> <mi>π</mi> <mo>/</mo> <mn>25</mn> </mrow> </semantics></math> (<b>b</b>), and <math display="inline"><semantics> <mrow> <mi>π</mi> <mo>/</mo> <mn>4</mn> </mrow> </semantics></math> (<b>c</b>). Dashed lines correspond to Equation (<a href="#FD16-symmetry-14-00176" class="html-disp-formula">16</a>), and the dots show the absolute maxima.</p>
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<p>Comparison of the minimal width of the medium in units of <math display="inline"><semantics> <msub> <mi>d</mi> <mn>0</mn> </msub> </semantics></math> from Equation (<a href="#FD21-symmetry-14-00176" class="html-disp-formula">21</a>) (dashed line) and direct calculations (solid line).</p>
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<p>Resonance shape of the probability distribution (solid lines for <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics></math>) for the atmospheric neutrinos in Earth. Dashed line corresponds to Equation (<a href="#FD16-symmetry-14-00176" class="html-disp-formula">16</a>) and dot shows absolute maximum.</p>
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17 pages, 2064 KiB  
Article
Analysis of Urban Visual Memes Based on Dictionary Learning: An Example with Urban Image Data
by Ming Zhang, Xin Gu, Jun Xiao, Pu Zou, Zuoqin Shi, Silu He, Haifeng Li and Sumin Li
Symmetry 2022, 14(1), 175; https://doi.org/10.3390/sym14010175 - 17 Jan 2022
Cited by 2 | Viewed by 2725
Abstract
The coexistence of different cultures is a distinctive feature of human society, and globalization makes the construction of cities gradually tend to be the same, so how to find the unique memes of urban culture in a multicultural environment is very important for [...] Read more.
The coexistence of different cultures is a distinctive feature of human society, and globalization makes the construction of cities gradually tend to be the same, so how to find the unique memes of urban culture in a multicultural environment is very important for the development of a city. Most of the previous analyses of urban style have been based on simple classification tasks to obtain the visual elements of cities, lacking in considering the most essential visual elements of cities as a whole. Therefore, based on the image data of ten representative cities around the world, we extract the visual memes via the dictionary learning method, quantify the symmetric similarities and differences between cities by using the memetic similarity, and interpret the reasons for the similarities and differences between cities by using the memetic similarity and sparse representation. The experimental results show that the visual memes have certain limitations among different cities, i.e., the elements composing the urban style are very similar, and the linear combinations of visual memes vary widely as the reason for the differences in the urban style among cities. Full article
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<p>A part of urban image data example.</p>
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<p>Research framework.</p>
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<p>Urban dictionary classification result.</p>
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<p>Visualization of comparison on similarity of urban dictionary classification results.</p>
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<p>Urban visual memetic similarity.</p>
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<p>Urban style similarity.</p>
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<p>Visual memetic type.</p>
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<p>Overall sparse representation of cities.</p>
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15 pages, 6261 KiB  
Article
Image Encryption Based on Arnod Transform and Fractional Chaotic
by Chao Chen, Hongying Zhang and Bin Wu
Symmetry 2022, 14(1), 174; https://doi.org/10.3390/sym14010174 - 17 Jan 2022
Cited by 6 | Viewed by 1887
Abstract
An image encryption and decryption algorithm based on Arnod transform and fractional chaos is proposed in this work for solving the problem that the encrypted image is easily cracked and the content of the decrypted image is distorted. To begin with, the Arnold [...] Read more.
An image encryption and decryption algorithm based on Arnod transform and fractional chaos is proposed in this work for solving the problem that the encrypted image is easily cracked and the content of the decrypted image is distorted. To begin with, the Arnold transform is used to encrypt, so that the spatial confidence of the original image has been comprehensively disturbed. Secondly, the XOR involving the fractional order chaotic sequence is used to encrypt. The key sequence is dynamically generated to ensure the randomness and difference of key generation. When decryption is required, the first decryption is performed using the key and XOR. Then, the second decryption is carried out by using the inverse Arnold transform, and finally the decrypted image is obtained. Experimental results show that the improved algorithm has achieved a better performance in encryption and decryption. Full article
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<p>Comparison map before and after encryption of the Lena.</p>
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<p>Histogram of original Lena image.</p>
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<p>Histogram of encrypted Lena image.</p>
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<p>Correlation charts of three directions before and after encrypted Lena image.</p>
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<p>Comparison map before and after encryption of the girl.</p>
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<p>Histogram of original girl image.</p>
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<p>Histogram of encrypted girl image.</p>
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<p>Correlation charts of three directions before and after encrypted girl image.</p>
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<p>Comparison map before and after encryption of the peppers. The Chinese character “Neijiang” is the name of a prefecture-level city in China.</p>
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<p>Histogram of original peppers image.</p>
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<p>Histogram of encrypted peppers image.</p>
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<p>Correlation charts of three directions before and after encrypted peppers image.</p>
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<p>Comparison map before and after encryption of the baboon.</p>
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<p>Histogram of original baboon image.</p>
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<p>Histogram of encrypted baboon image.</p>
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<p>Correlation charts of three directions before and after encrypted baboon image.</p>
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23 pages, 32906 KiB  
Article
A Modification of the Imperialist Competitive Algorithm with Hybrid Methods for Multi-Objective Optimization Problems
by Jianfu Luo, Jinsheng Zhou, Xi Jiang and Haodong Lv
Symmetry 2022, 14(1), 173; https://doi.org/10.3390/sym14010173 - 16 Jan 2022
Cited by 2 | Viewed by 2028
Abstract
This paper proposes a modification of the imperialist competitive algorithm to solve multi-objective optimization problems with hybrid methods (MOHMICA) based on a modification of the imperialist competitive algorithm with hybrid methods (HMICA). The rationale for this is that there is an obvious disadvantage [...] Read more.
This paper proposes a modification of the imperialist competitive algorithm to solve multi-objective optimization problems with hybrid methods (MOHMICA) based on a modification of the imperialist competitive algorithm with hybrid methods (HMICA). The rationale for this is that there is an obvious disadvantage of HMICA in that it can only solve single-objective optimization problems but cannot solve multi-objective optimization problems. In order to adapt to the characteristics of multi-objective optimization problems, this paper improves the establishment of the initial empires and colony allocation mechanism and empire competition in HMICA, and introduces an external archiving strategy. A total of 12 benchmark functions are calculated, including 10 bi-objective and 2 tri-objective benchmarks. Four metrics are used to verify the quality of MOHMICA. Then, a new comprehensive evaluation method is proposed, called “radar map method”, which could comprehensively evaluate the convergence and distribution performance of multi-objective optimization algorithm. It can be seen from the four coordinate axes of the radar maps that this is a symmetrical evaluation method. For this evaluation method, the larger the radar map area is, the better the calculation result of the algorithm. Using this new evaluation method, the algorithm proposed in this paper is compared with seven other high-quality algorithms. The radar map area of MOHMICA is at least 14.06% larger than that of other algorithms. Therefore, it is proven that MOHMICA has advantages as a whole. Full article
(This article belongs to the Special Issue Meta-Heuristics for Manufacturing Systems Optimization)
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<p>Pareto frontiers of SCH benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of FON benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of ZDT1 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of ZDT2 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of ZDT3 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of ZDT4 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of UF1 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of UF2 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of UF3 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of UF7 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of UF8 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Pareto frontiers of UF10 benchmark function obtained by MOHMICA, PESA-II and MOEA\D.</p>
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<p>Comprehensive evaluation radar maps of SCH function (<b>left</b>), FON function (<b>center</b>) and ZDT1 function (<b>right</b>) calculated by eight different algorithms.</p>
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<p>Comprehensive evaluation radar maps of ZDT2 function (<b>left</b>), ZDT3 function (<b>center</b>) and ZDT4 function (<b>right</b>) calculated by eight different algorithms.</p>
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<p>Comprehensive evaluation radar maps of UF1 function (<b>left</b>), UF2 function (<b>center</b>) and UF3 function (<b>right</b>) calculated by eight different algorithms.</p>
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<p>Comprehensive evaluation radar maps of UF7 function (<b>left</b>), UF8 function (<b>center</b>) and UF10 function (<b>right</b>) calculated by eight different algorithms.</p>
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23 pages, 1552 KiB  
Article
Hybrid Analysis of the Decision-Making Factors for Software Upgrade Based on the Integration of AHP and DEMATEL
by Dosung Kim and Mi Kim
Symmetry 2022, 14(1), 172; https://doi.org/10.3390/sym14010172 - 16 Jan 2022
Cited by 4 | Viewed by 2229
Abstract
Software is a very important part to implement advanced information systems, such as AI and IoT based on the latest hardware equipment of the fourth Industrial Revolution. In particular, decision making for software upgrade is one of the essential processes that can solve [...] Read more.
Software is a very important part to implement advanced information systems, such as AI and IoT based on the latest hardware equipment of the fourth Industrial Revolution. In particular, decision making for software upgrade is one of the essential processes that can solve problems for upgrading the information systems. However, most of the decision-making studies for this purpose have been conducted only from the perspective of the IT professional and management position. Moreover, software upgrade can be influenced by various layers of decision makers, so further research is needed. Therefore, it is necessary to conduct research on what factors are required and affect the decision making of software upgrade at various layers of organization. For this purpose, decision factors of software upgrade are identified by literature review in this study. Additionally, the priority, degree of influence and relationship between the factors are analyzed by using the AHP and DEMATEL techniques at the organizational level of users, managers and IT professionals. The results show that the priority, weight value, causal relationship of decision factors of users, managers and IT professionals who constitute the organizational level were very different. The managers first considered the benefits, such as ROI, for organization as a leader. The users tended to consider their work efficiency and changes due to the software upgrade first. Finally, the IT professionals considered ROI, budget and compatibility for the aspect of the managers and users. Therefore, the related information of each organizational level can be presented more clearly for the systematic and symmetrical decision making of software upgrade based on the results of this study. Full article
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<p>The proposed research framework.</p>
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<p>IDM for managers.</p>
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<p>IDM for users.</p>
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<p>IDM for IT professionals.</p>
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17 pages, 3127 KiB  
Article
Symmetry Control of Comfortable Vehicle Suspension Based on H
by Jiguang Hou, Xianteng Cao and Changshu Zhan
Symmetry 2022, 14(1), 171; https://doi.org/10.3390/sym14010171 - 16 Jan 2022
Cited by 4 | Viewed by 2628
Abstract
Suspension is an important part of intelligent and safe transportation; it is the balance point between the comfort and handling stability of a vehicle under intelligent traffic conditions. In this study, a control method of left-right symmetry of air suspension based on H [...] Read more.
Suspension is an important part of intelligent and safe transportation; it is the balance point between the comfort and handling stability of a vehicle under intelligent traffic conditions. In this study, a control method of left-right symmetry of air suspension based on H theory was proposed, which was verified under intelligent traffic conditions. First, the control stability caused by the active suspension control system running on uneven roads needs to be ensured. To address this issue, a 1/4 vehicle active suspension model was established, and the vertical acceleration of the vehicle body was applied as the main index of ride comfort. H performance constraint output indicators of the controller contained the tire dynamic load, suspension dynamic stroke, and actuator control force limit. Based on the Lyapunov stability theory, an output feedback control law with H-guaranteed performance was proposed to constrain multiple targets. This way, the control problem was transformed into a solution to the Riccati equation. The simulation results showed that when dealing with general road disturbances, the proposed control strategy can reduce the vehicle body acceleration by about 20% and meet the requirements of an ultimate suspension dynamic deflection of 0.08 m and a dynamic tire load of 1500 N. Using this symmetrical control method can significantly improve the ride comfort and driving stability of a vehicle under intelligent traffic conditions. Full article
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<p>1/4 Vehicle active suspension model.</p>
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<p>Typical closed-loop control system.</p>
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<p>Output feedback <span class="html-italic">H</span><span class="html-italic">∞</span> control system.</p>
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<p>Speed bump cross-sectional form.</p>
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<p>Vehicle–road coupling excitation curve under simulation condition I.</p>
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<p>Vehicle–road coupling excitation curve under simulation condition II.</p>
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<p>Vehicle–road coupling excitation curve under simulation condition III.</p>
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<p>Comparison of vertical acceleration of vehicle body.</p>
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<p>Comparison of dynamic deflection of suspension.</p>
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<p>Comparison of dynamic deflection of tire.</p>
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<p>Comparison of vertical acceleration of vehicle body.</p>
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<p>Comparison of dynamic deflection of suspension.</p>
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<p>Comparison of dynamic deflection of tire.</p>
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<p>Acceleration power spectral density of two models of vehicle body on class C road.</p>
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<p>Acceleration power spectral density of two models of vehicle body on class B road.</p>
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<p>Body diagram of dynamic deflection/body acceleration of suspension under road pulse input.</p>
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19 pages, 6391 KiB  
Article
On the Analytical Solution of the Kuwabara-Type Particle-in-Cell Model for the Non-Axisymmetric Spheroidal Stokes Flow via the Papkovich–Neuber Representation
by Panayiotis Vafeas, Eleftherios Protopapas and Maria Hadjinicolaou
Symmetry 2022, 14(1), 170; https://doi.org/10.3390/sym14010170 - 15 Jan 2022
Cited by 2 | Viewed by 1954
Abstract
Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. [...] Read more.
Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. Here, the steady Stokes flow of an incompressible, viscous fluid through an assemblage of particles, at low Reynolds numbers, is studied by employing a particle-in-cell model. The mathematical formulation adopts the Kuwabara-type assumption, according to which each spheroidal particle is stationary and it is surrounded by a confocal spheroid that creates a fluid envelope, in which the Newtonian fluid moves with a constant velocity of arbitrary orientation. The boundary value problem in the fluid envelope is solved by imposing non-slip conditions on the surface of the spheroid, which is also considered as non-penetrable, while zero vorticity is assumed on the fictitious spheroidal boundary along with a uniform approaching velocity. The three-dimensional flow fields are calculated analytically for the first time, in the spheroidal geometry, by virtue of the Papkovich–Neuber representation. Through this, the velocity and the total pressure fields are provided in terms of a vector and the scalar spheroidal harmonic potentials, which enables the thorough study of the relevant physical characteristics of the flow fields. The newly obtained analytical expressions generalize to any direction with the existing results holding for the asymmetrical case, which were obtained with the aid of a stream function. These can be employed for the calculation of quantities of physical or engineering interest. Numerical implementation reveals the flow behavior within the fluid envelope for different geometrical cell characteristics and for the arbitrarily-assumed velocity field, thus reflecting the different flow/porous media situations. Sample calculations show the excellent agreement of the obtained results with those available for special geometrical cases. All of these findings demonstrate the usefulness of the proposed method and the powerfulness of the obtained analytical expansions. Full article
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<p>The 3D velocity field <math display="inline"><semantics> <mrow> <mi mathvariant="bold">v</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> in the three-dimensional space <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </semantics></math> in the elliptical ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </semantics></math> in the elliptical ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>2</mn> </msub> </mrow> </semantics></math> in the circular ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </semantics></math> in the elliptical ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </semantics></math> in the elliptical ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </semantics></math> in the circular ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </semantics></math> in the elliptical ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </semantics></math> in the elliptical ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Vector plot (<b>a</b>) and streamlines (<b>b</b>) of velocity components <math display="inline"><semantics> <mrow> <msub> <mi>v</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>v</mi> <mn>3</mn> </msub> </mrow> </semantics></math> in the circular ring on the coordinate plane <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>.</p>
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<p>Total pressure field <math display="inline"><semantics> <mi>Ρ</mi> </semantics></math> on <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> </mrow> </semantics></math>-axis (<b>a</b>), on <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>2</mn> </msub> </mrow> </semantics></math>-axis (<b>b</b>) and on <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>3</mn> </msub> </mrow> </semantics></math>-axis (<b>c</b>).</p>
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22 pages, 5815 KiB  
Review
The CMS Magnetic Field Measuring and Monitoring Systems
by Vyacheslav Klyukhin, Austin Ball, Felix Bergsma, Henk Boterenbrood, Benoit Curé, Domenico Dattola, Andrea Gaddi, Hubert Gerwig, Alain Hervé, Richard Loveless, Gary Teafoe, Daniel Wenman, Wolfram Zeuner and Jerry Zimmerman
Symmetry 2022, 14(1), 169; https://doi.org/10.3390/sym14010169 - 15 Jan 2022
Cited by 2 | Viewed by 2843
Abstract
This review article describes the performance of the magnetic field measuring and monitoring systems for the Compact Muon Solenoid (CMS) detector. To cross-check the magnetic flux distribution obtained with the CMS magnet model, four systems for measuring the magnetic flux density in the [...] Read more.
This review article describes the performance of the magnetic field measuring and monitoring systems for the Compact Muon Solenoid (CMS) detector. To cross-check the magnetic flux distribution obtained with the CMS magnet model, four systems for measuring the magnetic flux density in the detector volume were used. The magnetic induction inside the 6 m diameter superconducting solenoid was measured and is currently monitored by four nuclear magnetic resonance (NMR) probes installed using special tubes at a radius of 2.9148 m outside the barrel hadron calorimeter at ±0.006 m from the coil median XY-plane. Two more NRM probes were installed at the faces of the tracking system at Z-coordinates of −2.835 and +2.831 m and a radius of 0.651 m from the solenoid axis. The field inside the superconducting solenoid was precisely measured in 2006 in a cylindrical volume of 3.448 m in diameter and 7 m in length using ten three-dimensional (3D) B-sensors based on the Hall effect (Hall probes). These B-sensors were installed on each of the two propeller arms of an automated field-mapping machine. In addition to these measurement systems, a system for monitoring the magnetic field during the CMS detector operation has been developed. Inside the solenoid in the horizontal plane, four 3D B-sensors were installed at the faces of the tracking detector at distances X = ±0.959 m and Z-coordinates of −2.899 and +2.895 m. Twelve 3D B-sensors were installed on the surfaces of the flux-return yoke nose disks. Seventy 3D B-sensors were installed in the air gaps of the CMS magnet yoke in 11 XY-planes of the azimuthal sector at 270°. A specially developed flux loop technique was used for the most complex measurements of the magnetic flux density inside the steel blocks of the CMS magnet yoke. The flux loops are installed in 22 sections of the flux-return yoke blocks in grooves of 30 mm wide and 12–13 mm deep and consist of 7–10 turns of 45 wire flat ribbon cable. The areas enclosed by these coils varied from 0.3 to 1.59 m2 in the blocks of the barrel wheels and from 0.5 to 1.12 m2 in the blocks of the yoke endcap disks. The development of these systems and the results of the magnetic flux density measurements across the CMS magnet are presented and discussed in this review article. Full article
(This article belongs to the Section Physics)
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<p>(<b>a</b>) Schematic view of a nuclear magnetic resonance probe. The probe external dimensions in mm (230 × 16 × 12), the position of an active volume (NMR sample) with a radio frequency (RF) coil, as well as the slope of a modulation coil, equal to 45° with respect to the probe axis, are shown. The NMR sample has a diameter of 4 mm and a length of 4.5 mm and is made of either a solid material containing a large amount of hydrogen or a sealed glass tube containing D<sub>2</sub>O. The measured magnetic field direction can be transverse or axial; (<b>b</b>) Automated field-mapping machine [<a href="#B8-symmetry-14-00169" class="html-bibr">8</a>] for measuring the CMS magnetic field, installed inside the barrel hadron calorimeter. A carriage made of aluminum alloy moving by steps of 0.05 m along the rails aligned with the <span class="html-italic">Z</span>-axis, a tower made of durable non-magnetic material, two propeller arms rotating by steps of 7.5° along the azimuth angle in the forward and backward directions, and five 3D B-sensors on the propeller arm viewed from the positive <span class="html-italic">Z</span>-coordinates are visible.</p>
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<p>(<b>a</b>) Hall probes on the B-sensor PCB. Each PCB contains three single-axis Siemens KSY44 Hall chips [<a href="#B23-symmetry-14-00169" class="html-bibr">23</a>] which are glued to a glass cube of 4 × 4 × 2.4 mm<sup>3</sup>. The distance between the <span class="html-italic">b</span><sub>1</sub> (at the cube top) and <span class="html-italic">b</span><sub>3</sub> (at the H1 side) chip centers is 1.8 mm. The distance between the <span class="html-italic">b</span><sub>1</sub> and <span class="html-italic">b</span><sub>2</sub> (at the R18 side) chip centers is 2.6 mm. The B-sensors have an orientation error of about 1 mrad, and the relative orientation error of local <b><span class="html-italic">b</span></b><sub>1</sub>, <b><span class="html-italic">b</span></b><sub>2</sub>, <b><span class="html-italic">b</span></b><sub>3</sub> measured fields is estimated to be approximately 0.2 mrad [<a href="#B24-symmetry-14-00169" class="html-bibr">24</a>]. The analogue voltages from the Hall probes are simultaneously read out by a 24-bit ΔΣ-modulator; (<b>b</b>) The Hall probe calibrator scheme [<a href="#B17-symmetry-14-00169" class="html-bibr">17</a>,<a href="#B18-symmetry-14-00169" class="html-bibr">18</a>,<a href="#B19-symmetry-14-00169" class="html-bibr">19</a>]. The local coordinate system <span class="html-italic">XYZ</span> is rotated with respect to the constant magnetic flux density vector <b><span class="html-italic">B</span></b> in two angular directions: a polar angle <span class="html-italic">θ</span> is counted between <b><span class="html-italic">B</span></b> and the <span class="html-italic">Z</span>-axis, and an azimuthal angle <span class="html-italic">φ</span> is counted between the projection <b><span class="html-italic">B</span></b><span class="html-italic">·sinθ</span> and the <span class="html-italic">X</span>-axis. The rotations are performed with the calibrator outer and inner axis providing the rotations of the calibrator head in <span class="html-italic">θ</span> and <span class="html-italic">φ</span> directions, accordingly. To cover the full 4<span class="html-italic">π</span> space in the local reference frame, 6 turns of the outer axis and 5 turns of the inner axis in the opposite directions are needed. Four B-sensor PCB with the same orientation are mounted by two on each side of the coil support plate. Three coils measure the components <b><span class="html-italic">b</span></b><sub>1</sub>, <b><span class="html-italic">b</span></b><sub>2</sub>, and <b><span class="html-italic">b</span></b><sub>3</sub> of <b><span class="html-italic">B</span></b> in the local coordinate system by the magnetic flux integration. The Hall probe voltages and the coil signals are sampled each 1/15 s and approximated then by the orthogonal spherical harmonics with a set of calibration coefficients at three values of <b><span class="html-italic">B</span></b> and two values of temperature.</p>
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<p>Trajectories of the magnetic flux density unit vector in the calibrator head local coordinate system: (<b>a</b>) in the <span class="html-italic">XY</span>-plane; (<b>b</b>) in the <span class="html-italic">YZ</span>-plane. Different colors correspond to six complete turns of the calibrator head with the outer axes. Markers denote the increments of 9.375° in azimuth <span class="html-italic">φ</span> and 11.25° in polar <span class="html-italic">θ</span> angles used to prepare the plot.</p>
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<p>Modelled (<b>a</b>) CMS coil current fast discharge; (<b>b</b>) minimum (dashed) and maximum (solid) EMF voltages per one-turn flux loop on the blocks of the CMS barrel wheels ; (<b>c</b>) minimum (dashed) and maximum (solid) voltages per one-turn flux loop on the 18° segments of the CMS endcap disks [<a href="#B12-symmetry-14-00169" class="html-bibr">12</a>].</p>
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<p>(<b>a</b>) 3D model for the test magnet with the steel sample disk inserted between the pole tips [<a href="#B12-symmetry-14-00169" class="html-bibr">12</a>]; (<b>b</b>) Induced voltage (curve 1) and the integrated magnetic flux density (curve 2) when the test magnet current ramped down from 320 A to zero during 32 s. Curve 3 shows the requested current from the control software. Curve 4 corresponds to the measured current read-back [<a href="#B12-symmetry-14-00169" class="html-bibr">12</a>].</p>
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<p>ELEKTRA model used for the yoke eddy current calculation [<a href="#B29-symmetry-14-00169" class="html-bibr">29</a>]. CMS coil (1), the yoke sectors of the barrel wheels (2), nose disk (3), and two endcap disks (4) are presented in the model.</p>
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<p>(<b>a</b>) Voltages calculated in the first flux loop on the L2 layer of the external barrel wheel when the eddy currents with realistic electrical resistances (dotted blue line with open diamonds) and infinite resistances (smoothed solid magenta line with filled diamonds) are modelled during the current fast discharge (black solid line with black circles). The dashed light blue line with open circles represents the result of voltage integration when the eddy currents exist. The solid red line with filled triangles displays the result of voltage integration in the model with eddy currents suppressed. The difference between two integrated magnetic flux densities is within 0.3%; (<b>b</b>) Voltages calculated in the middle flux loop on 18° segment of the D−2 endcap disk when eddy currents from realistic electrical resistances (dotted blue line with open diamonds) and eddy currents suppressed by infinite resistances (smoothed solid magenta line with filled diamonds) are modelled during the current fast discharge (black solid line with black circles). The dashed light blue line with open circles represents the result of voltage integration when eddy currents exist. The solid red line with filled triangles displays the result of voltage integration when eddy currents are suppressed. The difference between two integrated magnetic flux densities is within 2.8%.</p>
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<p>(<b>a</b>) Magnetic flux density measured [<a href="#B8-symmetry-14-00169" class="html-bibr">8</a>] at a radius of 0.092 m along the coil axis in the range of ±3.5 m with respect to the coil middle plane for full azimuth coverage; (<b>b</b>) Comparison [<a href="#B10-symmetry-14-00169" class="html-bibr">10</a>] of the measured (smooth curves) and modelled (dashed curve) values of the magnetic flux density (left scale) averaged over the full azimuth angle range. The measurements have been performed with two B-sensors located at a radius of 0.092 m with respect to the coil axis on the negative (thick smooth curve) and positive (thin smooth curve) fieldmapper arms, respectively. The differences between the measured and calculated values (right scale) are shown by square and round dots, respectively.</p>
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<p>(<b>a</b>) Magnetic flux density (left scale) measured [<a href="#B8-symmetry-14-00169" class="html-bibr">8</a>] with the NMR probe (rhombs) along the coil axis in the range of <span class="html-italic">Z</span>-coordinate from −1.675 to 3.025 m and calculated (smooth line) with the magnet model version 16_130503 [<a href="#B1-symmetry-14-00169" class="html-bibr">1</a>]. The difference between the measurements and calculations (right scale) is shown by triangles. Different colors correspond to 4 sets of measurements; (<b>b</b>) Magnetic flux density (left scale) measured [<a href="#B8-symmetry-14-00169" class="html-bibr">8</a>] with the NMR probe (rhombs), B-sensor of negative arm of the fieldmapper (open squares), and B-sensor of positive arm (open circles) all located at the radius of 1.724 m. The measurements are compared with the modelled values (smooth lines). The differences between the measured and calculated values (right scale) are shown by filled triangles, slanted and right crosses, respectively.</p>
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<p>The magnetic flux density distribution in the longitudinal sections of the CMS detector. The colour scale is from zero to 4 T with a unit of 0.5 T. The black lines display twenty-two flux loop cross sections. The black squares denote the projections of the 3D Hall probe positions to the vertical <span class="html-italic">YZ</span>-plane. The values along the vertical <span class="html-italic">Y</span>-axis and horizontal <span class="html-italic">Z</span>-axis are presented in meters and are counted from the centre of the superconducting solenoid.</p>
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<p>Double layer PCB to form the flux loop by connecting forty-five AWG 30 wires on both ends of the 0.635 pitch flat ribbon cable using 3M<sup>TM</sup> 1.27 mm pitch wiremount sockets, boardmount right angle plugs, and a special scheme to offset the individual conductors in the ribbon by one conductor at the ends of the cable. The flux loop relates to the readout AD-USB module by twisted pair screened cable connected to the PCB in two termination points: TP1 (red wire) and TP2 (blue wire). The PCB jack J1 is on the right side and jack J2 is on the left side.</p>
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<p>Voltages (smooth lines) induced in the flux loop of the W−2 barrel wheel second layer L2 (<b>a</b>), and in the middle flux loop of the D−2 endcap disk (<b>b</b>) in the 2006 magnetic field measurements. The integrated flux densities (dashed lines) and the fast discharge of the coil current from 19.14 kA (dotted lines) are also shown. In (<b>a</b>) the dashed-dotted line cuts the contribution of the eddy currents into the barrel wheel flux loop voltage. The dashed-double dotted line shows the integrated magnetic flux density without the eddy current contribution.</p>
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<p>(<b>a</b>) CMS magnet current discharges from 18.164 to 0 kA made on 17 and 18 July 2015 (blue smooth line), 21 and 22 September 2015 (green dashed line), 10 September 2016 (red short-dashed line), and 30 November 2017 (magenta dotted line) [<a href="#B30-symmetry-14-00169" class="html-bibr">30</a>]; (<b>b</b>) Induced voltage (left scale, noisy curve) and the integrated average magnetic flux density (right scale, smooth curve) in the cross section at <span class="html-italic">Z</span> = −2.691 m of the first layer block of the W−1 barrel wheel [<a href="#B30-symmetry-14-00169" class="html-bibr">30</a>].</p>
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<p>Axial magnetic flux density measured at <span class="html-italic">B</span><sub>0</sub> of 3.81 T (filled markers) and modelled (open markers) versus the <span class="html-italic">Z</span>-coordinate (<b>a</b>) in the TC (squares), and the L1 (diamonds), L2 (triangles), and L3 (circles) barrel layers at the yoke near side and the <span class="html-italic">Y</span>-coordinates of −3.958 m (dotted line), −4.805 m (solid line), −5.66 m (dashed line), and −6.685 m (short-dashed line); (<b>b</b>) in the L1 (diamonds), L2 (triangles), and L3 (circles) barrel layers at the yoke far side of and the <span class="html-italic">Y</span>-coordinates of −4.805 m (solid line), −5.66 m (dashed line), and −6.685 m (small dashed line).</p>
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<p>Radial magnetic flux density measured at <span class="html-italic">B</span><sub>0</sub> of 3.81 T (filled markers) and modelled (open markers) versus the <span class="html-italic">Y</span>-coordinate in the D−1 (diamonds) and D−2 (triangles) endcap disks. The lines represent the calculated values along the lines across the centres of the flux loops.</p>
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21 pages, 20276 KiB  
Article
Microgrid Operations Planning Based on Improving the Flying Sparrow Search Algorithm
by Trong-The Nguyen, Truong-Giang Ngo, Thi-Kien Dao and Thi-Thanh-Tan Nguyen
Symmetry 2022, 14(1), 168; https://doi.org/10.3390/sym14010168 - 15 Jan 2022
Cited by 48 | Viewed by 2874
Abstract
Microgrid operations planning is crucial for emerging energy microgrids to enhance the share of clean energy power generation and ensure a safe symmetry power grid among distributed natural power sources and stable functioning of the entire power system. This paper suggests a new [...] Read more.
Microgrid operations planning is crucial for emerging energy microgrids to enhance the share of clean energy power generation and ensure a safe symmetry power grid among distributed natural power sources and stable functioning of the entire power system. This paper suggests a new improved version (namely, ESSA) of the sparrow search algorithm (SSA) based on an elite reverse learning strategy and firefly algorithm (FA) mutation strategy for the power microgrid optimal operations planning. Scheduling cycles of the microgrid with a distributed power source’s optimal output and total operation cost is modeled based on variables, e.g., environmental costs, electricity interaction, investment depreciation, and maintenance system, to establish grid multi-objective economic optimization. Compared with other literature methods, such as Genetic algorithm (GA), Particle swarm optimization (PSO), Firefly algorithm (FA), Bat algorithm (BA), Grey wolf optimization (GWO), and SSA show that the proposed plan offers higher performance and feasibility in solving microgrid operations planning issues. Full article
(This article belongs to the Topic Applied Metaheuristic Computing)
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<p>A typical microgrid structure schematic of distributed power sources.</p>
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<p>The effective applied equation strategies of each of the two mechanisms of the elite reverse-learning (strategy 1) and FA-mutation (strategy 2) with the original SSA in different problem dimension spaces.</p>
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<p>Comparison of convergence curves of seven algorithms obtained on the selected benchmark functions, e.g., F1–F4 and F11–F12.</p>
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<p>Comparison of convergence curves of seven algorithms obtained on the selected benchmark functions, e.g., F1–F4 and F11–F12.</p>
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<p>Comparison of convergence curves of seven algorithms obtained on the selected benchmark functions, e.g., F1–F4 and F11–F12.</p>
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<p>Flowchart of the ESSA for planning microgrid operations.</p>
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<p>Daily demand local load curves of the power system in the island.</p>
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<p>Typical daily recourse load outputs of a microgrid system.</p>
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<p>Typical monthly recourse load outputs of a microgrid system.</p>
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<p>Comparison of the optimization results obtained by the proposed ESSA with the FA, PSO, and SSA schemes in daily schedule cycles.</p>
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<p>Comparison of the optimization results obtained by the proposed ESSA with the FA, PSO, and SSA schemes in monthly schedule cycles.</p>
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<p>The daily load and the microgrid component-distributed power sources’ output curves of grid-connected and off-grid optimizations.</p>
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<p>The daily load and the microgrid component-distributed power sources’ output curves of off-grid optimizations.</p>
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17 pages, 3210 KiB  
Article
Research on Influence Factors of Bearing Capacity of Concrete-Filled Steel Tubular Arch for Traffic Tunnel
by Lei Li and Ke Lei
Symmetry 2022, 14(1), 167; https://doi.org/10.3390/sym14010167 - 14 Jan 2022
Cited by 4 | Viewed by 1909
Abstract
When a traffic tunnel passes through special strata such as soft rock with high geo-stress, expansive rock, and fault fracture zones, the traditional supporting structure is often destroyed due to complicated loads, which threatens the construction and operation safety of tunnel engineering. Concrete-filled [...] Read more.
When a traffic tunnel passes through special strata such as soft rock with high geo-stress, expansive rock, and fault fracture zones, the traditional supporting structure is often destroyed due to complicated loads, which threatens the construction and operation safety of tunnel engineering. Concrete-filled steel tubular (CFST) structure gives full play to the respective advantages of steel and concrete and has better bearing capacity and economic benefits than traditional support structure, which has achieved good results in some underground engineering applications. In order to promote the application of CFST in the construction of traffic tunnels with complex geological conditions and improve the bearing capacity of the initial supporting structure of tunnels, the influencing factors of the bearing capacity of CFST arch were studied by numerical simulation. The main achievements are as follows: (1) The load-displacement curves of CFST members under different material parameters are basically consistent. CFST members have significant restrictions on displacement in the elastic stage and have high ultimate bearing capacity. Although the bearing capacity decreases obviously after reaching the peak, it shows good extension performance. (2) The height of the steel tube section, the thickness of the steel tube wall and the grade of the core concrete have an approximately linear positive correlation with the bearing capacity of CFST arch, but the influence of these three factors on the bearing capacity of CFST arch decreases in turn, and when the grade of core concrete increases above C50, it has no significant effect on the bearing capacity of members. Full article
(This article belongs to the Section Computer)
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<p>Schematic diagram of research object.</p>
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<p>Calculation model and grid division diagram. (<b>a</b>) Circular section CFST arch; (<b>b</b>) Triangular section CFST arch.</p>
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<p>Uniaxial stress-strain curve of concrete. (<b>a</b>) C30; (<b>b</b>) C40; (<b>c</b>) C50; (<b>d</b>) C60; (<b>e</b>) C70; (<b>f</b>) C80.</p>
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<p>Grid size meaning.</p>
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<p>Load-Vault settlement history curves with different grid density.</p>
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<p>Variation curve of ultimate bearing capacity under different grid density.</p>
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<p>Variation curve of vault settlement when reaching the ultimate bearing capacity under different grid density.</p>
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<p>Load-vault settlement curve of CFST members with different section heights of steel tube. (<b>a</b>) Circular section CFST; (<b>b</b>) Triangular section CFST.</p>
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<p>Performance curve of CFST members with different section heights of steel tube. (<b>a</b>) Change curve of earing capacity; (<b>b</b>) Change curve of stiffness; (<b>c</b>) Change curve of DRBC; (<b>d</b>) Change curve of PBPED.</p>
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<p>Load-vault settlement curve of CFST members with different wall thickness of steel tube. (<b>a</b>) Circular section CFST; (<b>b</b>) Triangular section CFST.</p>
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<p>Performance curve of CFST members with different wall thickness of steel tube. (<b>a</b>) Change curve of earing capacity; (<b>b</b>) Change curve of stiffness; (<b>c</b>) Change curve of DRBC; (<b>d</b>) Change curve of PBPED.</p>
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<p>Load-vault settlement curve of CFST members with different core concrete grades. (<b>a</b>) Circular section CFST; (<b>b</b>) Triangular section CFST.</p>
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<p>Performance curve of CFST members with different core concrete grades. (<b>a</b>) Change curve of earing capacity; (<b>b</b>) Change curve of stiffness; (<b>c</b>) Change curve of DRBC; (<b>d</b>) Change curve of PBPED.</p>
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<p>General distribution of bearing capacity index of components. (<b>a</b>) Ultimate bearing capacity; (<b>b</b>) Working stiffness; (<b>c</b>) Residual bearing capacity; (<b>d</b>) Energy consumption capacity.</p>
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19 pages, 12486 KiB  
Article
Friction Characteristics Analysis of Symmetric Aluminum Alloy Parts in Warm Forming Process
by Jiansheng Xia, Jun Zhao and Shasha Dou
Symmetry 2022, 14(1), 166; https://doi.org/10.3390/sym14010166 - 14 Jan 2022
Cited by 11 | Viewed by 2294
Abstract
There are many typical symmetric large plastic deformation problems in aluminum alloy stamping. Warm stamping technology can improve the formability of materials and obtain parts with high-dimensional accuracy. Friction behavior in the stamping process is significant for the forming quality. An accurate friction [...] Read more.
There are many typical symmetric large plastic deformation problems in aluminum alloy stamping. Warm stamping technology can improve the formability of materials and obtain parts with high-dimensional accuracy. Friction behavior in the stamping process is significant for the forming quality. An accurate friction coefficient is helpful in improving the prediction accuracy of forming defects. It is hard to obtain a unified and precise friction model through simple experiments due to the complicated contact conditions. To explore the effect of friction behavior on the forming quality, warm friction experiments of the AA6061 aluminum alloy and P20 steel with different process parameters were carried out using a high-temperature friction tester CFT-I (Equipment Type), including temperatures, the interface load, and sliding speeds. The variation curves of the friction coefficient with various parameters were obtained and analyzed. The results show that the friction coefficient increases with temperature and decreases with the sliding speed and load. Then, the influences of process parameters on the surface morphology of the samples after friction were observed by an optical microscope; adhesive wear occurred when the temperature increased, and the surface scratch increased and deepened with the increase in the load. Finally, the friction coefficient models of the speed and load were established by analyzing the data with Original software. Compared with the experimental and the finite element analysis results of the symmetrical part, the errors of the velocity friction model in thickness and springback angle are less than 4% and 5%, respectively. The mistakes of the load friction model are less than 6% and 7%, respectively. The accuracy of the two friction models is higher than that of the constant friction coefficient. Therefore, those coefficient models can effectively improve the simulation accuracy of finite element software. Full article
(This article belongs to the Section Engineering and Materials)
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<p>A fiction test sample of AA6061 aluminum alloy and P20 steel. (<b>a</b>) Friction specimens; (<b>b</b>) Surface topography of P20.</p>
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<p>CFT-I multifunctional material surface comprehensive performance tester. (<b>a</b>) Friction test; (<b>b</b>) friction test principle; (<b>c</b>) Detailed parts of friction device; (<b>d</b>) temperature control principle.</p>
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<p>CFT-I multifunctional material surface comprehensive performance tester. (<b>a</b>) Friction test; (<b>b</b>) friction test principle; (<b>c</b>) Detailed parts of friction device; (<b>d</b>) temperature control principle.</p>
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<p>Variation curves of the friction coefficient with time at different temperatures.</p>
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<p>Friction coefficient curve with temperature.</p>
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<p>Surface morphology at different temperatures. (<b>a</b>) <span class="html-italic">T</span> = 25 °C; (<b>b</b>) <span class="html-italic">T</span> = 100 °C; (<b>c</b>) <span class="html-italic">T</span> = 150 °C; (<b>d</b>) <span class="html-italic">T</span> = 200 °C; (<b>e</b>) <span class="html-italic">T</span> = 250 °C.</p>
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<p>Relation curves between speed and friction.</p>
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<p>Surface morphology at different sliding speeds. (<b>a</b>) <span class="html-italic">v</span><sub>1</sub> = 20 mm/s; (<b>b</b>) <span class="html-italic">v</span><sub>2</sub> = 30 mm/s; (<b>c</b>) <span class="html-italic">v</span><sub>3</sub> = 50 mm/s; (<b>d</b>) <span class="html-italic">v</span><sub>4</sub> = 60 mm/s.</p>
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<p>Surface morphology at different sliding speeds. (<b>a</b>) <span class="html-italic">v</span><sub>1</sub> = 20 mm/s; (<b>b</b>) <span class="html-italic">v</span><sub>2</sub> = 30 mm/s; (<b>c</b>) <span class="html-italic">v</span><sub>3</sub> = 50 mm/s; (<b>d</b>) <span class="html-italic">v</span><sub>4</sub> = 60 mm/s.</p>
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<p>Relation curve between load and friction.</p>
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<p>Surface morphology at different normal loads. (<b>a</b>) 10 N; (<b>b</b>) 20 N; (<b>c</b>) 40 N; (<b>d</b>) 50 N.</p>
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<p>Surface morphology at different normal loads. (<b>a</b>) 10 N; (<b>b</b>) 20 N; (<b>c</b>) 40 N; (<b>d</b>) 50 N.</p>
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<p>The relationship between friction coefficient and sliding speed.</p>
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<p>The fitting curve of friction coefficient under different loads. (<b>a</b>) 10 N; (<b>b</b>) 20 N; (<b>c</b>) 30 N; (<b>d</b>) 40 N; (<b>e</b>) 50 N.</p>
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<p>The fitting curve of friction coefficient under different loads. (<b>a</b>) 10 N; (<b>b</b>) 20 N; (<b>c</b>) 30 N; (<b>d</b>) 40 N; (<b>e</b>) 50 N.</p>
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<p>Relationship between friction coefficient and sliding speed.</p>
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<p>Friction coefficient curve and function fitting curve under different sliding speeds. (<b>a</b>) 20 mm/s; (<b>b</b>) 30 mm/s; (<b>c</b>) 40 mm/s; (<b>d</b>) 50 mm/s; (<b>e</b>) 60 mm/s.</p>
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<p>Friction coefficient curve and function fitting curve under different sliding speeds. (<b>a</b>) 20 mm/s; (<b>b</b>) 30 mm/s; (<b>c</b>) 40 mm/s; (<b>d</b>) 50 mm/s; (<b>e</b>) 60 mm/s.</p>
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<p>Standard P–P diagram of regression standardized residuals of the model.</p>
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<p>Numerical model of punch. (<b>a</b>) Two-dimensional drawing; (<b>b</b>) three-dimensional picture.</p>
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<p>Diagram of thickness distribution. (<b>a</b>) μ = 0.12; (<b>b</b>) velocity friction model; (<b>c</b>) load friction model; (<b>d</b>) mix friction model.</p>
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<p>Warm stamping experimental device: (<b>a</b>) warm stamping mold; (<b>b</b>) hot stamping part.</p>
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<p>The thickness distribution curve of half of the symmetrical part.</p>
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<p>Springback: (<b>a</b>) springback angle; (<b>b</b>) springback angle measurement under different friction coefficients.</p>
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12 pages, 805 KiB  
Article
Global Well-Posedness and Analyticity of the Primitive Equations of Geophysics in Variable Exponent Fourier–Besov Spaces
by Muhammad Zainul Abidin, Naeem Ullah and Omer Abdalrhman Omer
Symmetry 2022, 14(1), 165; https://doi.org/10.3390/sym14010165 - 14 Jan 2022
Cited by 2 | Viewed by 1954
Abstract
We consider the Cauchy problem of the three-dimensional primitive equations of geophysics. By using the Littlewood–Paley decomposition theory and Fourier localization technique, we prove the global well-posedness for the Cauchy problem with the Prandtl number P=1 in variable exponent Fourier–Besov spaces [...] Read more.
We consider the Cauchy problem of the three-dimensional primitive equations of geophysics. By using the Littlewood–Paley decomposition theory and Fourier localization technique, we prove the global well-posedness for the Cauchy problem with the Prandtl number P=1 in variable exponent Fourier–Besov spaces for small initial data in these spaces. In addition, we prove the Gevrey class regularity of the solution. For the primitive equations of geophysics, our results can be considered as a symmetry in variable exponent Fourier–Besov spaces. Full article
(This article belongs to the Special Issue Symmetry in Nonlinear Functional Analysis and Optimization Theory II)
12 pages, 2087 KiB  
Article
Efficiency Optimization Strategy of Permanent Magnet Synchronous Motor for Electric Vehicles Based on Energy Balance
by Wenhui Pei, Qi Zhang and Yongjing Li
Symmetry 2022, 14(1), 164; https://doi.org/10.3390/sym14010164 - 14 Jan 2022
Cited by 12 | Viewed by 3585
Abstract
This paper presents an efficiency optimization controller for a permanent magnet synchronous motor (PMSM) of an electric vehicle. A new loss model is obtained based on the permanent magnet synchronous motor’s energy balance equation utilizing the theory of the port-controlled Hamiltonian system. Since [...] Read more.
This paper presents an efficiency optimization controller for a permanent magnet synchronous motor (PMSM) of an electric vehicle. A new loss model is obtained based on the permanent magnet synchronous motor’s energy balance equation utilizing the theory of the port-controlled Hamiltonian system. Since the energy balance equation is just the power loss of the PMSM, which provides great convenience for us to use the energy method for efficiency optimization. Then, a new loss minimization algorithm (LMA) is designed based on the new loss model by adjusting the ratio of the excitation current in the d–q axis. Moreover, the proposed algorithm is achieved by the principle of the energy shape method of the Hamiltonian system. Simulations are finally presented to verify effectiveness. The main results of these simulations indicate that the dynamic performance of the drive is maintained and the efficiency increase is up to about 7% compared with the id=0 control algorithm, and about 4.5% compared with the conventional LMA at a steady operation of a PMSM. Full article
(This article belongs to the Special Issue Symmetry in Power Battery Management Systems)
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<p>Dynamic d–q axes equivalent circuits of a PMSM. (<b>a</b>) d axes, (<b>b</b>) q axes.</p>
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<p>The simulation results: (<b>a</b>) the direct-axis and quadrature-axis currents (<math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1500</mn> </mrow> </semantics></math> rad/s, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>L</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> N·m); (<b>b</b>) the excitation currents (<math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1500</mn> </mrow> </semantics></math> rad/s, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>L</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> N·m); (<b>c</b>) the excitation currents (<math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1500</mn> </mrow> </semantics></math> rad/s, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>L</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> N·m); (<b>d</b>) the rotor speeds with (<math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>r</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>r</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>).</p>
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<p>The simulation results: (<b>a</b>) Comparison of the ratio <span class="html-italic">K</span> of rated torque (<math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>L</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> N·m) versus the rotor speed with the three algorithms; (<b>b</b>) Comparison of the motor efficiencies at rated load (<math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>L</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> N·m) versus the angular speed.</p>
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<p>The simulation results: (<b>a</b>) Comparison of the ratio <span class="html-italic">K</span> at rated rotor speed (<math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1500</mn> </mrow> </semantics></math> rad/s) versus the load torque; (<b>b</b>) Comparison of the efficiency at rated rotor speed (<math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1500</mn> </mrow> </semantics></math> rad/s) versus the load torque.</p>
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<p>The simulation results: (<b>a</b>) Efficiency improvement of the proposed LMA at rated load torque (<math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mi>L</mi> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> N·m) versus the rotor speed; (<b>b</b>) Efficiency improvement of the proposed LMA at rated rotor speed (<math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <mn>1500</mn> </mrow> </semantics></math> rad/s) versus the load torque.</p>
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28 pages, 593 KiB  
Review
A Critical Review of Works Pertinent to the Einstein-Bohr Debate and Bell’s Theorem
by Karl Hess
Symmetry 2022, 14(1), 163; https://doi.org/10.3390/sym14010163 - 14 Jan 2022
Cited by 13 | Viewed by 2204
Abstract
This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems [...] Read more.
This review is related to the Einstein-Bohr debate and to Einstein–Podolsky–Rosen’s (EPR) and Bohm’s (EPRB) Gedanken-experiments as well as their realization in actual experiments. I examine a significant number of papers, from my minority point of view and conclude that the well-known theorems of Bell and Clauser, Horne, Shimony and Holt (CHSH) deal with mathematical abstractions that have only a tenuous relation to quantum theory and the actual EPRB experiments. It is also shown that, therefore, Bell-CHSH cannot be used to assess the nature of quantum entanglement, nor can physical features of entanglement be used to prove Bell-CHSH. Their proofs are, among other factors, based on a statistical sampling argument that is invalid for general physical entities and processes and only applicable for finite “populations”; not for elements of physical reality that are linked, for example, to a time-like continuum. Bell-CHSH have, furthermore, neglected the subtleties of the theorem of Vorob’ev that includes their theorems as special cases. Vorob’ev found that certain combinatorial-topological cyclicities of classical random variables form a necessary and sufficient condition for the constraints that are now known as Bell-CHSH inequalities. These constraints, however, must not be linked to the observables of quantum theory nor to the actual EPRB experiments for a variety of reasons, including the existence of continuum-related variables and appropriate considerations of symmetry. Full article
(This article belongs to the Special Issue Mathematical Modelling of Physical Systems 2021)
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<p>Photon impinging on Wollaston prisma (WP) deflected into photon with horizontal polarization H (black) or vertical polarization V (dashed). The Wollaston’s cubic shape is symbolized by the square WP arranged perpendicular to the <span class="html-italic">z</span>-direction; its sides pointing into x- (vertical), y- (horizontal) directions. The incoming photons are separated into two sets <math display="inline"><semantics> <msub> <mi>S</mi> <msub> <mi>H</mi> <mi>x</mi> </msub> </msub> </semantics></math> (black channel) and <math display="inline"><semantics> <msub> <mi>S</mi> <msub> <mi>V</mi> <mi>y</mi> </msub> </msub> </semantics></math> (dashed channel). A subsequent Wollaston leaves this sorting unchanged.</p>
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<p>Correlated photons impinging on two distant Wollaston Prisms are deflected by each into either horizontal polarization H (black) or vertical polarization V (gray). Special preparation of the photon pair guarantees that, in the case where one is evaluated as H, the other is evaluated as V and vice versa.</p>
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23 pages, 49558 KiB  
Article
Experimental and Numerical Peeling Investigation on Aged Multi-Layer Anti-Shatter Safety Films (ASFs) for Structural Glass Retrofit
by Silvana Mattei, Luca Cozzarini and Chiara Bedon
Symmetry 2022, 14(1), 162; https://doi.org/10.3390/sym14010162 - 14 Jan 2022
Cited by 9 | Viewed by 2116
Abstract
Anti-shatter safety films (ASFs) are often used for structural glass applications. The goal is to improve the response of monolithic elements and prevent fragments from shattering. Thus, the main reason behind their use is the possibility to upgrade safety levels against the brittle [...] Read more.
Anti-shatter safety films (ASFs) are often used for structural glass applications. The goal is to improve the response of monolithic elements and prevent fragments from shattering. Thus, the main reason behind their use is the possibility to upgrade safety levels against the brittle failure of glass and minimize the number of possible injuries. However, the impact response of glass elements bonded with Polyethylene terephthalate (PET)-films and pressure sensitive adhesives (PSAs) still represents a research topic of open discussion. Major challenges derive from material characterization and asymmetrical variability under design loads and ageing. In particular, the measurement of interface mechanical characteristics for the adhesive layer in contact with glass is a primary parameter for the ASF choice optimization. For this reason, the present paper presents an experimental campaign aimed at calibrating some basic mechanical parameters that provide the characterization of constitutive models, such as tensile properties (yielding stress and Young modulus) for PET-film and adhesive properties for PSA (energy fracture and peel force). In doing so, both tensile tests for PET-films and peeling specimens are taken into account for a commercially available ASF, given that the peeling test protocol is one of most common methods for the definition of adhesion properties. Moreover, an extensive calibration of the Finite Element (FE) model is performed in order to conduct a parametric numerical analysis of ASF bonded glass solutions. Furthermore, a Kinloch approach typically used to determine the fracture energy of a given tape by considering a variable peel angle, is also adopted to compare the outcomes of calibration analyses and FE investigations on the tested specimens. Finally, a study of the effect of multiple aspects is also presented. The results of the experimental program and the following considerations confirm the rate dependence and ageing dependence in peel tests. Full article
(This article belongs to the Section Computer)
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<p>Percentage difference of fracture energy by IC-Peel software or current approach, as a function of (<b>a</b>) <span class="html-italic">v</span> or (<b>b</b>) ageing time.</p>
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<p>Stratification detail for the examined multi-layer ASF (values in mm).</p>
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<p>Example of (<b>a</b>) PET-film specimens for tensile tests and (<b>b</b>) selection of ASF-bonded glass samples for peel tests.</p>
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<p>FE numerical model (Abaqus/Explicit): (<b>a</b>) assembly, (<b>b</b>) lateral view and (<b>c</b>) traction-separation law in case of linear softening for damage evolution.</p>
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<p>DSC heating thermograms of Layer-1 (blue) and Layer-2 (red).</p>
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<p>FT-IR spectra of Layer-1 (both sides, in red) and Layer-2 internal side (blue).</p>
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<p>FT-IR spectra of Layer-2, external side.</p>
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<p>FT-IR spectra of adhesive layer.</p>
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<p>Tensile test setup (values in mm). (<b>a</b>) 3D view, (<b>b</b>) frontal view, (<b>c</b>) lateral view.</p>
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<p>Tensile test at (<b>a</b>) initial step, (<b>b</b>) yielding step and (<b>c</b>) failure step.</p>
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<p>Tensile test output: (<b>a</b>) experimental stress–strain curves and (<b>b</b>) mechanical behaviour model.</p>
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<p>Configuration of peeling process at (<b>a</b>) t = 0 and (<b>b</b>) t = i.</p>
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<p>Experimental setup of peel test (values in mm). (<b>a</b>) 3D view, (<b>b</b>) frontal view, (<b>c</b>) lateral view.</p>
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<p>Peel curves of unaged samples (specimens 1–4) in terms of (<b>a</b>) <span class="html-italic">P<sub>vert</sub></span> and (<b>b</b>) its component along peel arm direction.</p>
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<p>Experimental results with respect to (<b>a</b>) 2<span class="html-italic">v</span>, (<b>b</b>) 3<span class="html-italic">v</span>; (<b>c</b>) a comparison of mean curves with respect to peel rate and (<b>d</b>) peel force-peel rate correlation.</p>
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<p>Peel force-ageing time for each imposed ageing temperature.</p>
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<p>Comparison between FE numerical and experimental peel curves, as observed at peel rates equal to (<b>a</b>) 2<span class="html-italic">v</span> or (<b>b</b>) 3<span class="html-italic">v</span>.</p>
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<p>Fracture energy calculated (<b>a</b>) by modified Kinloch expressions or (<b>b</b>) by corresponding FE numerical simulations, as a function of peel rate.</p>
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<p>General shape of peel force master curve for peel rate and temperature dependence (reproduced from [<a href="#B48-symmetry-14-00162" class="html-bibr">48</a>] with permission from Elsevier®, license number 5226440524626, January 2022).</p>
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<p>Comparison between selected FE numerical and experimental peel curves at ageing temperatures equal to (<b>a</b>) 50 °C (<b>b</b>) 70 °C.</p>
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<p>Fracture energy as a function of ageing time, as obtained by (<b>a</b>) experimental and theoretical approach or (<b>b</b>) FE numerical method.</p>
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<p>Fracture energy as a function of ageing time at (<b>a</b>) 50 °C or (<b>b</b>) 70 °C, as obtained by experimental and theoretical approach or FE numerical method. In evidence, the fitting curves of average data.</p>
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15 pages, 2494 KiB  
Article
An Efficient Hyperparameter Control Method for a Network Intrusion Detection System Based on Proximal Policy Optimization
by Hyojoon Han, Hyukho Kim and Yangwoo Kim
Symmetry 2022, 14(1), 161; https://doi.org/10.3390/sym14010161 - 14 Jan 2022
Cited by 32 | Viewed by 3303
Abstract
The complexity of network intrusion detection systems (IDSs) is increasing due to the continuous increases in network traffic, various attacks and the ever-changing network environment. In addition, network traffic is asymmetric with few attack data, but the attack data are so complex that [...] Read more.
The complexity of network intrusion detection systems (IDSs) is increasing due to the continuous increases in network traffic, various attacks and the ever-changing network environment. In addition, network traffic is asymmetric with few attack data, but the attack data are so complex that it is difficult to detect one. Many studies on improving intrusion detection performance using feature engineering have been conducted. These studies work well in the dataset environment; however, it is challenging to cope with a changing network environment. This paper proposes an intrusion detection hyperparameter control system (IDHCS) that controls and trains a deep neural network (DNN) feature extractor and k-means clustering module as a reinforcement learning model based on proximal policy optimization (PPO). An IDHCS controls the DNN feature extractor to extract the most valuable features in the network environment, and identifies intrusion through k-means clustering. Through iterative learning using the PPO-based reinforcement learning model, the system is optimized to improve performance automatically according to the network environment, where the IDHCS is used. Experiments were conducted to evaluate the system performance using the CICIDS2017 and UNSW-NB15 datasets. In CICIDS2017, an F1-score of 0.96552 was achieved and UNSW-NB15 achieved an F1-score of 0.94268. An experiment was conducted by merging the two datasets to build a more extensive and complex test environment. By merging datasets, the attack types in the experiment became more diverse and their patterns became more complex. An F1-score of 0.93567 was achieved in the merged dataset, indicating 97% to 99% performance compared with CICIDS2017 and UNSW-NB15. The results reveal that the proposed IDHCS improved the performance of the IDS by automating learning new types of attacks by managing intrusion detection features regardless of the network environment changes through continuous learning. Full article
(This article belongs to the Section Computer)
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<p>Process for intrusion detection hyperparameter control system.</p>
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<p>Deep neural network (DNN)-based feature extractor.</p>
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<p>Experimental operation process.</p>
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<p>IDHCS—CICIDS2017 data detection.</p>
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<p>IDHCS—UNSW-NB15 data detection.</p>
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<p>IDHCS—CICIDS2017 and UNSW-NB15 merged data detection.</p>
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13 pages, 1863 KiB  
Article
Short-Term Energy Forecasting Using Machine-Learning-Based Ensemble Voting Regression
by Pyae-Pyae Phyo, Yung-Cheol Byun and Namje Park
Symmetry 2022, 14(1), 160; https://doi.org/10.3390/sym14010160 - 14 Jan 2022
Cited by 46 | Viewed by 4855
Abstract
Meeting the required amount of energy between supply and demand is indispensable for energy manufacturers. Accordingly, electric industries have paid attention to short-term energy forecasting to assist their management system. This paper firstly compares multiple machine learning (ML) regressors during the training process. [...] Read more.
Meeting the required amount of energy between supply and demand is indispensable for energy manufacturers. Accordingly, electric industries have paid attention to short-term energy forecasting to assist their management system. This paper firstly compares multiple machine learning (ML) regressors during the training process. Five best ML algorithms, such as extra trees regressor (ETR), random forest regressor (RFR), light gradient boosting machine (LGBM), gradient boosting regressor (GBR), and K neighbors regressor (KNN) are trained to build our proposed voting regressor (VR) model. Final predictions are performed using the proposed ensemble VR and compared with five selected ML benchmark models. Statistical autoregressive moving average (ARIMA) is also compared with the proposed model to reveal results. For the experiments, usage energy and weather data are gathered from four regions of Jeju Island. Error measurements, including mean absolute percentage error (MAPE), mean absolute error (MAE), and mean squared error (MSE) are computed to evaluate the forecasting performance. Our proposed model outperforms six baseline models in terms of the result comparison, giving a minimum MAPE of 0.845% on the whole test set. This improved performance shows that our approach is promising for symmetrical forecasting using time series energy data in the power system sector. Full article
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<p>The source information of collected load and weather data.</p>
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<p>Correlations for all variables.</p>
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<p>The workflow of the proposed ML-based ensemble voting regression mechanism.</p>
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<p>The comparison between actual and prediction on best predicted week of all models.</p>
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<p>The best predicted day of the proposed model.</p>
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15 pages, 432 KiB  
Article
Modified One-Sided EWMA Charts without- and with Variable Sampling Intervals for Monitoring a Normal Process
by Xuelong Hu, Suying Zhang, Guan Sun, Jianlan Zhong and Shu Wu
Symmetry 2022, 14(1), 159; https://doi.org/10.3390/sym14010159 - 13 Jan 2022
Cited by 3 | Viewed by 1824
Abstract
Much research has been conducted on two-sided Exponentially Weighted Moving Average (EWMA) control charts, while less work has been devoted to the one-sided EWMA charts. Traditional one-sided EWMA charts involve resetting the EWMA statistic to the target whenever it falls below or above [...] Read more.
Much research has been conducted on two-sided Exponentially Weighted Moving Average (EWMA) control charts, while less work has been devoted to the one-sided EWMA charts. Traditional one-sided EWMA charts involve resetting the EWMA statistic to the target whenever it falls below or above the target, or truncating the observations above or below the target and further applying the EWMA statistic to the truncated samples. In order to further improve the performance of traditional one-sided EWMA mean (X¯) charts, this paper studies the performance of the Modified One-sided EWMA (MOEWMA) X¯ charts to monitor a normally distributed process. The Monte-Carlo simulation method is used to obtain the zero- and steady-state Run Length (RL) properties of the proposed control charts. Through extensive simulations and comparisons with other charts, it is shown that the proposed MOEWMA X¯ charts compare favorably with some existing competing charts. Moreover, by attaching the variable sampling intervals (VSI) feature to the MOEWMA X¯ charts, it is shown that the VSI MOEWMA charts outperform the corresponding charts without the VSI feature. Finally, a real data example from manufacturing process shows the implementation of the proposed one-sided charts. Full article
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<p>One-sided EWMA type charts applied to the dataset in <a href="#symmetry-14-00159-t006" class="html-table">Table 6</a>. (<b>a</b>) SEWMA Chart (<b>b</b>) IEWMA Chart (<b>c</b>) REWMA Chart (<b>d</b>) MOEWMA Chart (<b>e</b>) VSI-MOEWMA Chart.</p>
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19 pages, 1261 KiB  
Article
High-Dimensional Conditional Covariance Matrices Estimation Using a Factor-GARCH Model
by Xiaoling Li, Xingfa Zhang and Yuan Li
Symmetry 2022, 14(1), 158; https://doi.org/10.3390/sym14010158 - 13 Jan 2022
Cited by 3 | Viewed by 2324
Abstract
Estimation of a conditional covariance matrix is an interesting and important research topic in statistics and econometrics. However, modelling ultra-high dimensional dynamic (conditional) covariance structures is known to suffer from the curse of dimensionality or the problem of singularity. To partially solve this [...] Read more.
Estimation of a conditional covariance matrix is an interesting and important research topic in statistics and econometrics. However, modelling ultra-high dimensional dynamic (conditional) covariance structures is known to suffer from the curse of dimensionality or the problem of singularity. To partially solve this problem, this paper establishes a model by combining the ideas of a factor model and a symmetric GARCH model to describe the dynamics of a high-dimensional conditional covariance matrix. Quasi maximum likelihood estimation (QMLE) and least square estimation (LSE) methods are used to estimate the parameters in the model, and the plug-in method is introduced to obtain the estimation of conditional covariance matrix. Asymptotic properties are established for the proposed method, and simulation studies are given to demonstrate its performance. A financial application is presented to support the methodology. Full article
(This article belongs to the Section Mathematics)
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<p>Time series plots of the three factors (Rm − Rf, SMB and HML).</p>
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<p>Time series plots of <math display="inline"><semantics> <msub> <mi>y</mi> <mrow> <mn>1</mn> <mi>t</mi> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>y</mi> <mrow> <mn>2</mn> <mi>t</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>y</mi> <mrow> <mn>3</mn> <mi>t</mi> </mrow> </msub> </semantics></math>.</p>
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<p>Annualized Sharpe ratios.</p>
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<p>Annualized Sharpe ratios for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>500</mn> </mrow> </semantics></math>.</p>
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18 pages, 5763 KiB  
Article
Velocity-Free State Feedback Fault-Tolerant Control for Satellite with Actuator and Sensor Faults
by Mingjun Liu, Aihua Zhang and Bing Xiao
Symmetry 2022, 14(1), 157; https://doi.org/10.3390/sym14010157 - 13 Jan 2022
Cited by 5 | Viewed by 2071
Abstract
A velocity-free state feedback fault-tolerant control approach is proposed for the rigid satellite attitude stabilization problem subject to velocity-free measurements and actuator and sensor faults. First, multiplicative faults and additive faults are considered in the actuator and the sensor. The faults and system [...] Read more.
A velocity-free state feedback fault-tolerant control approach is proposed for the rigid satellite attitude stabilization problem subject to velocity-free measurements and actuator and sensor faults. First, multiplicative faults and additive faults are considered in the actuator and the sensor. The faults and system states are extended into a new augmented vector. Then, an improved sliding mode observer based on the augmented vector is presented to estimate unknown system states and actuator and sensor faults simultaneously. Next, a velocity-free state feedback attitude controller is designed based on the information from the observer. The controller compensates for the effects of actuator and sensor faults and asymptotically stabilizes the attitude. Finally, simulation results demonstrate the effectiveness of the proposed scheme. Full article
(This article belongs to the Special Issue Recent Progress in Robot Control Systems: Theory and Applications)
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<p>The structure of the proposed fault-tolerant attitude control system without velocity measurements.</p>
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<p>The initial response of attitude angle estimation errors. (<b>a</b>) Observer in Equation (27); (<b>b</b>) observer in [<a href="#B26-symmetry-14-00157" class="html-bibr">26</a>].</p>
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<p>The initial response of angular velocity estimation errors. (<b>a</b>) Observer in Equation (27); (<b>b</b>) observer in [<a href="#B26-symmetry-14-00157" class="html-bibr">26</a>].</p>
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<p>The initial response of actuator fault reconstruction errors. (<b>a</b>) Observer in Equation (27); (<b>b</b>) observer in [<a href="#B26-symmetry-14-00157" class="html-bibr">26</a>].</p>
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<p>The initial response of sensor fault reconstruction errors. (<b>a</b>) Observer in Equation (27); (<b>b</b>) observer in [<a href="#B26-symmetry-14-00157" class="html-bibr">26</a>].</p>
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<p>The function <math display="inline"><semantics> <mrow> <mi>Χ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> by the observer in Equation (27) and the observer in [<a href="#B26-symmetry-14-00157" class="html-bibr">26</a>].</p>
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<p>The steady-state behaviors of system state estimation errors by the observer in Equation (27) and the observer in [<a href="#B26-symmetry-14-00157" class="html-bibr">26</a>]. (<b>a</b>) Attitude estimation errors; (<b>b</b>) angular velocity estimation errors.</p>
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<p>The steady-state behaviors of fault reconstruction errors by the observer in Equation (27) and the observer in [<a href="#B26-symmetry-14-00157" class="html-bibr">26</a>]. (<b>a</b>) Actuator faults reconstruction errors; (<b>b</b>) sensor faults reconstruction errors.</p>
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<p>The actuator commanded control torque <math display="inline"><semantics> <mi mathvariant="bold-italic">u</mi> </semantics></math>.</p>
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<p>The total torque <math display="inline"><semantics> <mi mathvariant="bold-italic">τ</mi> </semantics></math>.</p>
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<p>The attitude angle estimation of the satellite. (<b>a</b>) The initial response; (<b>b</b>) the steady-state behavior.</p>
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<p>The angular velocity estimation of the satellite. (<b>a</b>) The initial response; (<b>b</b>) the steady-state behavior.</p>
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<p>The actual attitude angle of the satellite. (<b>a</b>) The initial response; (<b>b</b>) the steady-state behavior.</p>
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<p>The actual angular velocity of the satellite. (<b>a</b>) The initial response; (<b>b</b>) the steady-state behavior.</p>
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16 pages, 430 KiB  
Article
Capturing a Change in the Covariance Structure of a Multivariate Process
by Andriette Bekker, Johannes T. Ferreira, Schalk W. Human and Karien Adamski
Symmetry 2022, 14(1), 156; https://doi.org/10.3390/sym14010156 - 13 Jan 2022
Cited by 1 | Viewed by 1767
Abstract
This research is inspired from monitoring the process covariance structure of q attributes where samples are independent, having been collected from a multivariate normal distribution with known mean vector and unknown covariance matrix. The focus is on two matrix random variables, constructed from [...] Read more.
This research is inspired from monitoring the process covariance structure of q attributes where samples are independent, having been collected from a multivariate normal distribution with known mean vector and unknown covariance matrix. The focus is on two matrix random variables, constructed from different Wishart ratios, that describe the process for the two consecutive time periods before and immediately after the change in the covariance structure took place. The product moments of these constructed random variables are highlighted and set the scene for a proposed measure to enable the practitioner to calculate the run-length probability to detect a shift immediately after a change in the covariance matrix occurs. Our results open a new approach and provides insight for detecting the change in the parameter structure as soon as possible once the underlying process, described by a multivariate normal process, encounters a permanent/sustained upward or downward shift. Full article
(This article belongs to the Special Issue Symmetry in Multivariate Analysis)
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<p>Schematic description of the multivariate process.</p>
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13 pages, 771 KiB  
Article
Practical Criteria for H-Tensors and Their Application
by Min Li, Haifeng Sang, Panpan Liu and Guorui Huang
Symmetry 2022, 14(1), 155; https://doi.org/10.3390/sym14010155 - 13 Jan 2022
Cited by 2 | Viewed by 1515
Abstract
Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the [...] Read more.
Identifying the positive definiteness of even-order real symmetric tensors is an important component in tensor analysis. H-tensors have been utilized in identifying the positive definiteness of this kind of tensor. Some new practical criteria for identifying H-tensors are given in the literature. As an application, several sufficient conditions of the positive definiteness for an even-order real symmetric tensor were obtained. Numerical examples are given to illustrate the effectiveness of the proposed method. Full article
(This article belongs to the Topic Applied Metaheuristic Computing)
20 pages, 1358 KiB  
Article
Smooth Group L1/2 Regularization for Pruning Convolutional Neural Networks
by Yuan Bao, Zhaobin Liu, Zhongxuan Luo and Sibo Yang
Symmetry 2022, 14(1), 154; https://doi.org/10.3390/sym14010154 - 13 Jan 2022
Cited by 4 | Viewed by 2002
Abstract
In this paper, a novel smooth group L1/2 (SGL1/2) regularization method is proposed for pruning hidden nodes of the fully connected layer in convolution neural networks. Usually, the selection of nodes and weights [...] Read more.
In this paper, a novel smooth group L1/2 (SGL1/2) regularization method is proposed for pruning hidden nodes of the fully connected layer in convolution neural networks. Usually, the selection of nodes and weights is based on experience, and the convolution filter is symmetric in the convolution neural network. The main contribution of SGL1/2 is to try to approximate the weights to 0 at the group level. Therefore, we will be able to prune the hidden node if the corresponding weights are all close to 0. Furthermore, the feasibility analysis of this new method is carried out under some reasonable assumptions due to the smooth function. The numerical results demonstrate the superiority of the SGL1/2 method with respect to sparsity, without damaging the classification performance. Full article
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<p>The convolution operation.</p>
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<p>Reshape the input array with a matrix <span class="html-italic">X</span>.</p>
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<p>Reshape the filter with a matrix <span class="html-italic">V</span>.</p>
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<p>The convolution operation is described with the matrix multiplication of the reshaped input array and filter.</p>
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<p>The mean pooling is described with the matrix multiplication of the reshaped feature map and the mean matrix.</p>
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<p>Sparsity of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Mnist dataset. (<b>a</b>) Node sparsity, (<b>b</b>) Weight sparsity.</p>
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<p>Loss and gradient of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Mnist dataset. (<b>a</b>) Loss function, (<b>b</b>) Norm of gradient.</p>
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<p>Sparsity of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Letter Recognition dataset. (<b>a</b>) Node sparsity, (<b>b</b>) Weight sparsity.</p>
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<p>Loss and gradient of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Letter Recognition dataset. (<b>a</b>) Loss function, (<b>b</b>) Norm of gradient.</p>
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<p>Sparsity of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Cifar 10 dataset. (<b>a</b>) Node sparsity, (<b>b</b>) Weight sparsity.</p>
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<p>Loss and gradient of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Cifar 10 dataset. (<b>a</b>) Loss function, (<b>b</b>) Norm of gradient.</p>
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<p>Sparsity of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Crowded Mapping dataset. (<b>a</b>) Node sparsity, (<b>b</b>) Weight sparsity.</p>
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<p>Loss and gradient of <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>S</mi> <mi>G</mi> <msub> <mi>L</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </mrow> </semantics></math> algorithms in Crowded Mapping dataset. (<b>a</b>) Loss function, (<b>b</b>) Norm of gradient.</p>
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17 pages, 6345 KiB  
Article
Operation Zone Analysis of the Voltage Source Converter Based on the Influence of Different Grid Strengths
by Wenning Wang, Kejun Li, Kaiqi Sun and Jianjian Wang
Symmetry 2022, 14(1), 153; https://doi.org/10.3390/sym14010153 - 13 Jan 2022
Viewed by 1699
Abstract
With the increasing penetration of renewable energy into the power system, the voltage source converter (VSC) for integrating renewable energy has become the most common device in the electric network. However, the operating stability of the VSC is strongly dependent on its operating [...] Read more.
With the increasing penetration of renewable energy into the power system, the voltage source converter (VSC) for integrating renewable energy has become the most common device in the electric network. However, the operating stability of the VSC is strongly dependent on its operating control strategy, which is also highly related to the strength of the AC system. Choosing the control strategy of VSC for different strengths of AC systems becomes an essential issue for maintaining the symmetry between high proportion of renewable energy integration and stable operation of AC system. In order to obtain the operation zones of the control strategies of the VSC under different strengths of AC system, in this paper, the two common VSC control strategies, vector current control (VCC) and power synchronization control (PSC), are compared. Firstly, the principle of VCC and PSC are introduced. Then, based on the short circuit ratio (SCR) and the power limit calculation under steady-state conditions of the VSC, the operation zones of the vector current control and power synchronization control are proposed. Finally, a medium voltage modular multilevel converter (MMC) system was built in PSCAD/EMTDC and the proposed operation zones of the VCC and PSC were tested by changing the SCR of the modified IEEE 33 bus system and analyzed via the critical short circuit ratio (CSCR) analysis, the small-signal stability analysis, and transient stability analysis. The results indicate that, as the SCR decreases, the VSC based on VCC is gradually worked into unstable conditions, while the stability of VSC based on PSC gradually increases. The analysis results provide a criterion for the converter operation strategy change that could significantly improve the operating stability of the VSC in the power system and realize the symmetry of the stability of the converter and the change of the strength of the AC system. Full article
(This article belongs to the Section Engineering and Materials)
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<p>Vector current control overview.</p>
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<p>Control loop of vector current control.</p>
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<p>Phase-locked loop.</p>
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<p>Power synchronization control overview.</p>
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<p>Power-synchronization loop.</p>
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<p>Current-limitation controller.</p>
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<p>Current control loop block diagram of power synchronization control.</p>
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<p>Operation zones of two kinds of control: (<b>a</b>) vector current control; (<b>b</b>) power synchronization control.</p>
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<p>Operation zones of two kinds of control: (<b>a</b>) vector current control; (<b>b</b>) power synchronization control.</p>
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<p>Simulation structure diagram of the medium voltage converter.</p>
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<p>CSCR of VSC based on vector current control: (<b>a</b>) active power; (<b>b</b>) reactive power.</p>
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<p>CSCR of VSC based on power synchronization control: (<b>a</b>) active power under low SCR; (<b>b</b>) frequency under low SCR; (<b>c</b>); active power under high SCR; (<b>d</b>) frequency under high SCR.</p>
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<p>Small-signal stability simulation results of VSC based on vector current control: (<b>a</b>) active power; (<b>b</b>) reactive power.</p>
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<p>Small-signal stability simulation results of VSC based on power synchronization control: (<b>a</b>) active power; (<b>b</b>) frequency.</p>
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<p>Small-signal stability simulation results of VSC based on power synchronization control: (<b>a</b>) active power; (<b>b</b>) frequency.</p>
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<p>Transient stability simulation results of VSC based on vector current control: (<b>a</b>) active power; (<b>b</b>) reactive power.</p>
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<p>Transient stability simulation results of VSC based on vector current control: (<b>a</b>) active power; (<b>b</b>) reactive power.</p>
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<p>Transient stability simulation results of VSC based on power synchronization control: (<b>a</b>) active power; (<b>b</b>) frequency.</p>
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8 pages, 729 KiB  
Article
Deep Vision Servo Hand-Eye Coordination Planning Study for Sorting Robots
by Tao Ning, Changcheng Wang, Yumeng Han, Yuchen Jin, Yan Gao, Jizhen Liu, Chunhua Hu, Yangyang Zhou and PinPin Li
Symmetry 2022, 14(1), 152; https://doi.org/10.3390/sym14010152 - 13 Jan 2022
Cited by 5 | Viewed by 2478 | Correction
Abstract
Within the context of large-scale symmetry, a study on deep vision servo multi-vision tracking coordination planning for sorting robots was conducted according to the problems of low recognition sorting accuracy and efficiency in existing sorting robots. In this paper, a kinematic model of [...] Read more.
Within the context of large-scale symmetry, a study on deep vision servo multi-vision tracking coordination planning for sorting robots was conducted according to the problems of low recognition sorting accuracy and efficiency in existing sorting robots. In this paper, a kinematic model of a mobile picking manipulator was built. Then, the kinematics design of the orwX, Y, Z three-dimensional space manipulator was carried out, and a method of deriving and calculating the base position coordinates through the target point coordinates, the current moving chassis center coordinates and the determined manipulator grasping attitude conditions was proposed, which realizes the adjustment of the position and attitude of the moving chassis as small as possible. The multi-vision tracking coordinated sorting accounts 79.8% of the whole cycle. The design of a picking robot proposed in this paper can greatly improve the coordination symmetry of logistic package target recognition, detection and picking. Full article
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<p>Structure diagram of small lifting picking robot. 1. Three-axis rotating parts; 2. Grab manipulator; 3. Autonomous mobile chassis; 4. Moving track; 5. Visual identification equipment.</p>
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<p>Flow chart of picking process.</p>
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<p>Schematic of measurement system.</p>
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