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Journal Description

Mathematics

Mathematics is a peer-reviewed, open access journal which provides an advanced forum for studies related to mathematics, and is published semimonthly online by MDPI. The European Society for Fuzzy Logic and Technology (EUSFLAT) and International Society for the Study of Information (IS4SI) are affiliated with Mathematics and their members receive a discount on article processing charges.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), RePEc, and other databases.
Journal Rank: JCR - Q1 (Mathematics) / CiteScore - Q1 (General Mathematics)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 18.3 days after submission; acceptance to publication is undertaken in 1.9 days (median values for papers published in this journal in the second half of 2024).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Sections: published in 17 topical sections.
Companion journals for Mathematics include: Foundations, Analytics, International Journal of Topology, Geometry and Logics.

Impact Factor: 2.3 (2023); 5-Year Impact Factor: 2.2 (2023)

Imprint Information Journal Flyer Open Access ISSN: 2227-7390

Latest Articles

19 pages, 339 KiB

Open AccessArticle

On the Monogenity of Quartic Number Fields Defined by x⁴ax²b⁺⁺

by Lhoussain El Fadil and István Gaál

Mathematics 2025, 13(6), 905; https://doi.org/10.3390/math13060905 - 7 Mar 2025

For any quartic number field K generated by a root

α

of an irreducible trinomial of type

x^{4} + a x^{2} + b \in Z [x]

, we characterize when

Z [α]

is integrally closed. Also for [...] Read more.

For any quartic number field K generated by a root

α

of an irreducible trinomial of type

x^{4} + a x^{2} + b \in Z [x]

, we characterize when

Z [α]

is integrally closed. Also for

p = 2, 3

, we explicitly give the highest power of p dividing

i (K)

, the common index divisor of K. For a wide class of monogenic trinomials of this type, we prove that up to equivalence, there is only one generator of power integral bases in

K = Q (α)

. We illustrate our statements with a series of examples. Full article

(This article belongs to the Section A: Algebra and Logic)

31 pages, 848 KiB

Open AccessFeature PaperArticle

Research on the Mechanism of Social Emotion Formation in Public Emergencies Based on the DeGroot Model

by Xiaohan Yan, Yi Liu, Tiezhong Liu and Yan Chen

Mathematics 2025, 13(6), 904; https://doi.org/10.3390/math13060904 - 7 Mar 2025

In recent years, the frequent occurrence of public emergencies has often triggered the rapid spread and amplification of social emotions. The accumulation and intensification of negative emotions can lead to collective behaviors and even pose a threat to social stability. To better understand [...] Read more.

In recent years, the frequent occurrence of public emergencies has often triggered the rapid spread and amplification of social emotions. The accumulation and intensification of negative emotions can lead to collective behaviors and even pose a threat to social stability. To better understand the formation and evolution of social emotions in such contexts, this study constructs a theoretical framework and simulation approach that combines opinion dynamics with emotional and trust interactions. First, we propose a clustering method that incorporates emotional similarity and trust relationships among users to delineate group structures involved in social emotion formation. Second, a dynamic trust adjustment mechanism is also proposed to capture how trust evolves as individuals interact emotionally. Third, a large-scale group emotional consensus decision-making approach, based on the DeGroot model, is developed to simulate how emotional exchanges and resonance drive groups toward consensus in public emergencies. Additionally, we present a strategy for guiding emotional interactions to reach a desired consensus that ensures minimal modifications to collective preference values while achieving an acceptable consensus level, helping to manage emotional escalation. To validate the proposed model, we conduct simulations using the “Fat Cat” incident as a case study. The results reveal key mechanisms underlying social emotion formation during public emergencies and highlight critical influencing factors, including user participation, opinion leader influence, and trust relationships. This study provides a clear understanding of how social emotions are generated and offers practical insights for managing emotional dynamics and improving group decision-making during crises. Full article

14 pages, 267 KiB

Open AccessArticle

Second-Order Neutral Differential Equations with Sublinear Neutral Terms: New Criteria for the Oscillation

by Meraa Arab, Hajer Zaway, Ali Muhib and Sayed K. Elagan

Mathematics 2025, 13(6), 903; https://doi.org/10.3390/math13060903 - 7 Mar 2025

This paper aims to study the oscillatory behavior of second-order neutral differential equations. Using the Riccati substitution technique, we introduce new oscillation criteria that essentially improve some related criteria from the literature. We provide some examples and compare the results in this paper [...] Read more.

This paper aims to study the oscillatory behavior of second-order neutral differential equations. Using the Riccati substitution technique, we introduce new oscillation criteria that essentially improve some related criteria from the literature. We provide some examples and compare the results in this paper with earlier results to illustrate the importance of our results. Full article

16 pages, 264 KiB

Open AccessArticle

Geometry of LP-Sasakian Manifolds Admitting a General Connection

by Rajesh Kumar, Laltluangkima Chawngthu, Oğuzhan Bahadır and Meraj Ali Khan

Mathematics 2025, 13(6), 902; https://doi.org/10.3390/math13060902 - 7 Mar 2025

This paper concerns certain properties of projective curvature tensor, conharmonic curvature tensor, quasi-conharmonic curvature tensor, and Ricci semi-symmetric conditions with respect to the general connection in an LP-Sasakian manifold. We also provide the applications of LP-Sasakian manifolds admitting general connections in the context [...] Read more.

This paper concerns certain properties of projective curvature tensor, conharmonic curvature tensor, quasi-conharmonic curvature tensor, and Ricci semi-symmetric conditions with respect to the general connection in an LP-Sasakian manifold. We also provide the applications of LP-Sasakian manifolds admitting general connections in the context of the general theory of relativity. Full article

(This article belongs to the Special Issue Differential Geometry, Geometric Analysis and Their Related Applications)

18 pages, 373 KiB

Open AccessFeature PaperArticle

Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations

by Petar Popivanov and Angela Slavova

Mathematics 2025, 13(6), 901; https://doi.org/10.3390/math13060901 - 7 Mar 2025

This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of [...] Read more.

This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude Q multiplied by

e^{i Φ}

,

Φ

being linear. Then, Q is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included. Full article

(This article belongs to the Section C1: Difference and Differential Equations)

10 pages, 256 KiB

Open AccessArticle

Properties for Close-to-Convex and Quasi-Convex Functions Using q-Linear Operator

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Wael W. Mohammed

Mathematics 2025, 13(6), 900; https://doi.org/10.3390/math13060900 - 7 Mar 2025

In this work, we describe the

q

-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators

I_{q, ρ}^{s} (ν, τ)

, and we obtain findings related to geometric function theory (GFT) by utilizing approaches [...] Read more.

In this work, we describe the

q

-analogue of a multiplier–Ruscheweyh operator of a specific family of linear operators

I_{q, ρ}^{s} (ν, τ)

, and we obtain findings related to geometric function theory (GFT) by utilizing approaches established through subordination and knowledge of

q

-calculus operators. By using this operator, we develop generalized classes of quasi-convex and close-to-convex functions in this paper. Additionally, the classes

K_{q, ρ}^{s} (ν, τ) (φ)

,

Q_{q, ρ}^{s} (ν, τ) (φ)

are introduced. The invariance of these recently formed classes under the

q

-Bernardi integral operator is investigated, along with a number of intriguing inclusion relationships between them. Additionally, several unique situations and the beneficial outcomes of these studies are taken into account. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

28 pages, 8592 KiB

Open AccessArticle

Sensorless Control of Permanent Magnet Synchronous Motor Drives with Rotor Position Offset Estimation via Extended State Observer

by Ramón Ramírez-Villalobos, Luis N. Coria, Paul A. Valle and Christian Aldrete-Maldonado

Mathematics 2025, 13(6), 899; https://doi.org/10.3390/math13060899 - 7 Mar 2025

The aim of this study is to develop sensorless high-speed tracking control for surface-mounted permanent magnet synchronous motors by taking the rotor position offset error and time-varying load torque into consideration. This proposal combines an extended state observer with an adaptation position algorithm, [...] Read more.

The aim of this study is to develop sensorless high-speed tracking control for surface-mounted permanent magnet synchronous motors by taking the rotor position offset error and time-varying load torque into consideration. This proposal combines an extended state observer with an adaptation position algorithm, employing only the measurement of electrical variables for feedback. First, a rotatory coordinate model of the motor is proposed, wherein the rotor position offset error is considered as a perturbation function within the model. Second, based on the aforementioned model, a rotary coordinate model of the motor is extended in one state to estimate the load torque, as well as the rotor’s position and speed, despite the presence of the rotor position offset error. Through Lyapunov stability analysis, sufficient conditions were established to guarantee that the error estimations were bounded. Finally, to validate the feasibility of the proposed sensorless scheme, experiments were conducted on the Technosoft^® development platform, where the alignment routine was disabled and an intentional misalignment between the magnetic north pole and the stator’s south pole was established. Full article

(This article belongs to the Special Issue Nonlinear Dynamical Systems: Modeling, Control and Applications)

24 pages, 1737 KiB

Open AccessFeature PaperArticle

Interpretable Evaluation of Sparse Time–Frequency Distributions: 2D Metric Based on Instantaneous Frequency and Group Delay Analysis

by Vedran Jurdana

Mathematics 2025, 13(6), 898; https://doi.org/10.3390/math13060898 - 7 Mar 2025

Compressive sensing in the ambiguity domain offers an efficient method for reconstructing high-quality time–frequency distributions (TFDs) across diverse signals. However, evaluating the quality of these reconstructions presents a significant challenge due to the potential loss of auto-terms when a regularization parameter is inappropriate. [...] Read more.

Compressive sensing in the ambiguity domain offers an efficient method for reconstructing high-quality time–frequency distributions (TFDs) across diverse signals. However, evaluating the quality of these reconstructions presents a significant challenge due to the potential loss of auto-terms when a regularization parameter is inappropriate. Traditional global metrics have inherent limitations, while the state-of-the-art local Rényi entropy (LRE) metric provides a single-value assessment but lacks interpretability and positional information of auto-terms. This paper introduces a novel performance criterion that leverages instantaneous frequency and group delay estimations directly in the 2D time–frequency plane, offering a more nuanced evaluation by individually assessing the preservation of auto-terms, resolution quality, and interference suppression in TFDs. Experimental results on noisy synthetic and real-world gravitational signals demonstrate the effectiveness of this measure in assessing reconstructed TFDs, with a focus on auto-term preservation. The proposed metric offers advantages in interpretability and memory efficiency, while its application to meta-heuristic optimization yields high-performing reconstructed TFDs significantly quicker than the existing LRE-based metric. These benefits highlight its usability in advanced methods and machine-related applications. Full article

19 pages, 1581 KiB

Open AccessArticle

A Structural Credit Risk Model with Jumps Based on Uncertainty Theory

by Hong Huang, Meihua Jiang, Yufu Ning and Shuai Wang

Mathematics 2025, 13(6), 897; https://doi.org/10.3390/math13060897 - 7 Mar 2025

This study, within the framework of uncertainty theory, employs an uncertain differential equation with jumps to model the asset value process of a company, establishing a structured model of uncertain credit risk that incorporates jumps. This model is applied to the pricing of [...] Read more.

This study, within the framework of uncertainty theory, employs an uncertain differential equation with jumps to model the asset value process of a company, establishing a structured model of uncertain credit risk that incorporates jumps. This model is applied to the pricing of two types of credit derivatives, yielding pricing formulas for corporate zero-coupon bonds and Credit Default Swap (CDS). Through numerical analysis, we examine the impact of asset value volatility and jump magnitude on corporate default uncertainty, as well as the influence of jump magnitude on the pricing of zero-coupon bonds and CDS. The results indicate that an increase in volatility levels significantly enhances default uncertainty, and an expansion in the magnitude of negative jumps not only directly elevates default risk but also leads to a significant increase in the value of zero-coupon bonds and the price of CDS through a risk premium adjustment mechanism. Therefore, when assessing corporate default risk and pricing credit derivatives, the disturbance of asset value jumps must be considered a crucial factor. Full article

(This article belongs to the Special Issue Uncertainty Theory and Applications)

► Show Figures

Figure 1

Figure 1
<p>Pictorial representation of the proposed work.</p> Full article ">Figure 2
<p>The variation in <math display="inline"><semantics> <msub> <mi mathvariant="script">M</mi> <mi>T</mi> </msub> </semantics></math> with respect to <math display="inline"><semantics> <mi>η</mi> </semantics></math>.</p> Full article ">Figure 3
<p>The variation in <math display="inline"><semantics> <msub> <mi mathvariant="script">M</mi> <mi>T</mi> </msub> </semantics></math> with respect to <math display="inline"><semantics> <mi>σ</mi> </semantics></math>.</p> Full article ">Figure 4
<p>The variation in <math display="inline"><semantics> <msub> <mi mathvariant="script">M</mi> <mi>T</mi> </msub> </semantics></math> with respect to <math display="inline"><semantics> <mi>μ</mi> </semantics></math>.</p> Full article ">Figure 5
<p>The research approach of this section.</p> Full article ">Figure 6
<p>The variation in <math display="inline"><semantics> <mrow> <mi>C</mi> <mi>S</mi> <mfenced separators="" open="(" close=")"> <mrow> <mn>0</mn> <mo>,</mo> <mi>T</mi> </mrow> </mfenced> </mrow> </semantics></math> with respect to <math display="inline"><semantics> <mi>η</mi> </semantics></math>.</p> Full article ">Figure 7
<p>The variation in <math display="inline"><semantics> <mi>ω</mi> </semantics></math> with respect to <math display="inline"><semantics> <mi>η</mi> </semantics></math>.</p> Full article ">

9 pages, 245 KiB

Open AccessArticle

A Study of Geodesic (E, F)-Preinvex Functions on Riemannian Manifolds

by Ehtesham Akhter, Mohd Bilal and Musavvir Ali

Mathematics 2025, 13(6), 896; https://doi.org/10.3390/math13060896 - 7 Mar 2025

In this manuscript, we define the

(E, F)

-invex set,

(E, F)

-invex functions, and

(E, F)

-preinvex functions on Euclidean space, i.e., simply vector space. We extend these concepts on the Riemannian manifold. [...] Read more.

In this manuscript, we define the

(E, F)

-invex set,

(E, F)

-invex functions, and

(E, F)

-preinvex functions on Euclidean space, i.e., simply vector space. We extend these concepts on the Riemannian manifold. We also detail the fundamental properties of

(E, F)

-preinvex functions and provide some examples that illustrate the concepts well. We have established a relation between

(E, F)

-invex and

(E, F)

-preinvex functions on Riemannian manifolds. We introduce the conditions

A

and define the

(E, F)

-proximal sub-gradient.

(E, F)

-preinvex functions are also used to demonstrate their applicability in optimization problems. In the last, we establish the points of extrema of a non-smooth

(E, F)

-preinvex functions on

(E, F)

-invex subset of the Riemannian manifolds by using the

(E, F)

-proximal sub-gradient. Full article

(This article belongs to the Section C: Mathematical Analysis)

26 pages, 1606 KiB

Open AccessArticle

Formalization of Side-Aware DNA Origami Words and Their Rewriting System, and Equivalent Classes

by Da-Jung Cho

Mathematics 2025, 13(6), 895; https://doi.org/10.3390/math13060895 - 7 Mar 2025

DNA origami is a powerful technique for constructing nanoscale structures by folding a single-stranded DNA scaffold with short staple strands. While traditional models assume staples bind to a fixed side of the scaffold, we introduce a side-aware DNA origami framework that incorporates the [...] Read more.

DNA origami is a powerful technique for constructing nanoscale structures by folding a single-stranded DNA scaffold with short staple strands. While traditional models assume staples bind to a fixed side of the scaffold, we introduce a side-aware DNA origami framework that incorporates the directional binding of staples to either the left or right side. The graphical representation of DNA origami is described using rectangular basic modules of scaffolds and staples, which we refer to as symbols in side-aware DNA origami words. We further define the concatenation of these symbols to represent side-aware DNA origami words. A set of rewriting rules is introduced to define equivalent words that correspond to the same graphical structure. Finally, we compute the number of possible structures by determining the equivalence classes of these words. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

► Show Figures

Figure 1

Figure 1
<p>Graphical representation of the Jones monoid <math display="inline"><semantics> <msub> <mi mathvariant="script">J</mi> <mn>5</mn> </msub> </semantics></math>. (<b>a</b>) The generators <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>h</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, and <math display="inline"><semantics> <msub> <mi>h</mi> <mn>3</mn> </msub> </semantics></math> in <math display="inline"><semantics> <msub> <mi mathvariant="script">J</mi> <mn>5</mn> </msub> </semantics></math>. (<b>b</b>) The relation <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>2</mn> </msub> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> </mrow> </semantics></math>. (<b>c</b>) The relation <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>h</mi> <mn>1</mn> </msub> </mrow> </semantics></math>. (<b>d</b>) The relation <math display="inline"><semantics> <mrow> <msub> <mi>h</mi> <mn>1</mn> </msub> <msub> <mi>h</mi> <mn>3</mn> </msub> <mo>=</mo> <msub> <mi>h</mi> <mn>3</mn> </msub> <msub> <mi>h</mi> <mn>1</mn> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>An example of four different types of staples segments over <math display="inline"><semantics> <msub> <mi>E</mi> <mn>5</mn> </msub> </semantics></math>. Scaffolds are represented by black plain lines, virtual staples are represented by grey dotted lines, left and right staples are represented by red dotted lines, and block staples are represented by pink dotted lines. The virtual staple is not present and can be extended to left, right, or block according to their neighbor endpoints. In the figure, we arbitrarily draw the virtual staple on the right side of the scaffolds. For convenience, representation of virtual staples can be omitted.</p> Full article ">Figure 3
<p>Graphical structure of <math display="inline"><semantics> <msub> <mi>α</mi> <mi>i</mi> </msub> </semantics></math>s and <math display="inline"><semantics> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </semantics></math>, where <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Real scaffolds are represented by black lines and real staples are represented by red dashed lines. Virtual staples are grey dashed lines. Virtual staples can be assigned to either the left or right side of a scaffold, but in this figure, we assign the virtual staples to the right side of a scaffold. In the graphical structure of <math display="inline"><semantics> <msub> <mi>α</mi> <mn>1</mn> </msub> </semantics></math>, the context <math display="inline"><semantics> <msub> <mi mathvariant="script">T</mi> <mn>1</mn> </msub> </semantics></math> contains <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mn>3</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>3</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="script">R</mi> <mi>c</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msubsup> <mn>3</mn> <mi>b</mi> <mi>ϵ</mi> </msubsup> <mo>,</mo> <msubsup> <mn>3</mn> <mi>t</mi> <mi>ϵ</mi> </msubsup> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="script">V</mi> <mi>p</mi> </msub> </mrow> </semantics></math>. The pair <math display="inline"><semantics> <mrow> <mo>(</mo> <msubsup> <mn>3</mn> <mi>b</mi> <mi>ϵ</mi> </msubsup> <mo>,</mo> <msubsup> <mn>3</mn> <mi>t</mi> <mi>ϵ</mi> </msubsup> <mo>)</mo> </mrow> </semantics></math> shows that the virtual staple has possible strands to either the left or right of the scaffold <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mn>3</mn> <mi>t</mi> </msub> <mo>,</mo> <msub> <mn>3</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The set <math display="inline"><semantics> <mi mathvariant="script">C</mi> </semantics></math> of scaffolds of a graphical structure <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>w</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mi mathvariant="script">C</mi> <mo>,</mo> <mi mathvariant="script">P</mi> <mo>)</mo> </mrow> </semantics></math> of a word <math display="inline"><semantics> <mrow> <mi>w</mi> <mo>=</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <msub> <mi>β</mi> <mn>1</mn> </msub> </mrow> </semantics></math>. We omit the representation of staples. For all scaffolds in <math display="inline"><semantics> <msub> <mi mathvariant="script">C</mi> <mn>1</mn> </msub> </semantics></math>, replace the subscript <span class="html-italic">b</span> by <span class="html-italic">m</span>, and for all staples in <math display="inline"><semantics> <msub> <mi mathvariant="script">C</mi> <mn>2</mn> </msub> </semantics></math>, replace the subscript <span class="html-italic">t</span> by <span class="html-italic">m</span>. Given set <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">C</mi> <mn>1</mn> </msub> <mo>∪</mo> <msub> <mi mathvariant="script">C</mi> <mn>2</mn> </msub> </mrow> </semantics></math>, for a sequence <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mn>2</mn> <mi>t</mi> </msub> <mo>,</mo> <msub> <mn>2</mn> <mi>m</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mn>2</mn> <mi>m</mi> </msub> <mo>,</mo> <msub> <mn>1</mn> <mi>m</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mn>1</mn> <mi>m</mi> </msub> <mo>,</mo> <msub> <mn>1</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, add <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mn>2</mn> <mi>t</mi> </msub> <mo>,</mo> <msub> <mn>1</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mi mathvariant="script">C</mi> </semantics></math>. For a sequence <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mn>3</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>3</mn> <mi>m</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mn>3</mn> <mi>m</mi> </msub> <mo>,</mo> <msub> <mn>3</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math>, add <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mn>3</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>3</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mi mathvariant="script">C</mi> </semantics></math>. Furthermore, add <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mn>1</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>2</mn> <mi>b</mi> </msub> <mo>)</mo> </mrow> </semantics></math> to <math display="inline"><semantics> <mi mathvariant="script">C</mi> </semantics></math>.</p> Full article ">Figure 5
<p>An example of concatenation of two graphical structures. Each case (<b>a</b>–<b>d</b>) describes the concatenation process of staples.</p> Full article ">Figure 6
<p>An example of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> for the cap case and the graphical structures after concatenating six different generators. The top two images illustrate the graphical structures for <math display="inline"><semantics> <msub> <mi>w</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>w</mi> <mn>2</mn> </msub> </semantics></math>. (<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>vi</b></span>) illustrate the concatenation of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with each of the six different generators <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math>, and the concatenation of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with each of the six different generators <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>An example of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> for the cup case, and the graphical structures after concatenating six different generators. (<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>vi</b></span>) illustrate the concatenation of each of the six different generators <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math>, and the concatenation of each of the six different generators <math display="inline"><semantics> <mrow> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>α</mi> <mi>i</mi> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>An example of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is right and <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is block, and the graphical structures before and after concatenating four different generators. (<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>viii</b></span>) illustrate the case when the right staple from <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> affects the concatenation of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with four different generators and the case when the block staple from <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> affects the concatenation of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with four different generators.</p> Full article ">Figure 9
<p>An example of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>s</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is left and <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is block, and the graphical structures before and after concatenating four different generators. (<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>viii</b></span>) illustrate the case when the left staple from <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> affects the concatenation of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with four different generators and the case when the block staple from <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> affects the concatenation of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> with four different generators.</p> Full article ">Figure 10
<p>(<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>viii</b></span>) illustrate the graphical structures after concatenating four different generators on top, see (<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>iv</b></span>), and the bottom, see (<span class="html-italic"><b>v</b></span>–<span class="html-italic"><b>viii</b></span>), with <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is virtual and <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is right.</p> Full article ">Figure 11
<p>(<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>viii</b></span>) illustrate the graphical structures after concatenating four different generators on top, see (<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>iv</b></span>), and the bottom, see (<span class="html-italic"><b>v</b></span>–<span class="html-italic"><b>viii</b></span>), with <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is virtual and <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is left.</p> Full article ">Figure 12
<p>(<span class="html-italic"><b>i</b></span>–<span class="html-italic"><b>viii</b></span>) illustrate the graphical structures after concatenating four different generators on top, see (<b>i</b>–<span class="html-italic"><b>iv</b></span>), and the bottom, see (<span class="html-italic"><b>v</b></span>–<span class="html-italic"><b>viii</b></span>), with <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> when <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is virtual and <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>(</mo> <mi>i</mi> <mo>)</mo> </mrow> </semantics></math> is block.</p> Full article ">Figure 13
<p>The transition graph of the changes in the different staple types for <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 14
<p>The transition graph of the changes in the different staple types for <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>w</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>β</mi> <mi>i</mi> </msub> <msub> <mi>β</mi> <mrow> <mi>i</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <msub> <mi>β</mi> <mi>i</mi> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 15
<p>An example of <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <mi>w</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>G</mi> <mo>(</mo> <msup> <mi>w</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </semantics></math>. Each number <span class="html-italic">i</span> illustrates the <span class="html-italic">i</span>th property in the lemma.</p> Full article ">Figure 16
<p>(<b>a</b>) Shows an example of staple segments <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>e</mi> <mi>n</mi> <mi>d</mi> <mi>e</mi> <mi>d</mi> <mspace width="3.33333pt"/> <mi>c</mi> <mi>u</mi> <mi>p</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>e</mi> <mi>n</mi> <mi>d</mi> <mi>e</mi> <mi>d</mi> <mspace width="3.33333pt"/> <mi>c</mi> <mi>a</mi> <mi>p</mi> </mrow> </semantics></math>. Given a graphical structure with 6 columns, staple segments <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mn>1</mn> <mi>t</mi> </msub> <mo>,</mo> <msub> <mn>6</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mn>3</mn> <mi>t</mi> </msub> <mo>,</mo> <msub> <mn>2</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mn>5</mn> <mi>t</mi> </msub> <mo>,</mo> <msub> <mn>4</mn> <mi>t</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> are <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>e</mi> <mi>n</mi> <mi>d</mi> <mi>e</mi> <mi>d</mi> <mspace width="3.33333pt"/> <mi>c</mi> <mi>u</mi> <mi>p</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <msub> <mn>2</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>1</mn> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mn>4</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>3</mn> <mi>b</mi> </msub> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msub> <mn>6</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>5</mn> <mi>b</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> are <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>e</mi> <mi>n</mi> <mi>d</mi> <mi>e</mi> <mi>d</mi> <mspace width="3.33333pt"/> <mi>c</mi> <mi>a</mi> <mi>p</mi> </mrow> </semantics></math>. (<b>b</b>) Shows an example of staple segments <math display="inline"><semantics> <mrow> <mi>j</mi> <mi>u</mi> <mi>n</mi> <mi>c</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>e</mi> <mi>n</mi> <mi>d</mi> <mi>e</mi> <mi>d</mi> <mspace width="3.33333pt"/> <mi>c</mi> <mi>a</mi> <mi>p</mi> </mrow> </semantics></math>. With 5 columns, <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mn>2</mn> <mi>b</mi> </msub> <mo>,</mo> <msub> <mn>5</mn> <mi>b</mi> </msub> <mo>)</mo> </mrow> </semantics></math> is <math display="inline"><semantics> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>e</mi> <mi>n</mi> <mi>d</mi> <mi>e</mi> <mi>d</mi> <mspace width="3.33333pt"/> <mi>c</mi> <mi>a</mi> <mi>p</mi> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mo>(</mo> <msub> <mn>5</mn> <mi>t</mi> </msub> <mo>,</mo> <msub> <mn>1</mn> <mi>b</mi> </msub> <mo>)</mo> </mrow> </semantics></math> is <math display="inline"><semantics> <mrow> <mi>j</mi> <mi>u</mi> <mi>n</mi> <mi>c</mi> <mi>t</mi> <mi>i</mi> <mi>o</mi> <mi>n</mi> </mrow> </semantics></math>.</p> Full article ">Figure 17
<p>An example of a bridge of width <span class="html-italic">n</span>. (<b>a</b>) A set <math display="inline"><semantics> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mn>1</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>3</mn> <mi>b</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>5</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>2</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>4</mn> <mi>b</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>6</mn> <mi>t</mi> <mi>R</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> of staple segments is a bridge of width 6. (<b>b</b>) There exists a set <math display="inline"><semantics> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mn>1</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>5</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>6</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>4</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>4</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>1</mn> <mi>b</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> that violates the property (v) since a proper subset <math display="inline"><semantics> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mn>6</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>4</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>4</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>1</mn> <mi>b</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> is a bridge of width 6. Therefore, the set <math display="inline"><semantics> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mn>1</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>5</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>6</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>4</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>4</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>1</mn> <mi>b</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> is not bride of width 6 whereas the set <math display="inline"><semantics> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mn>1</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>5</mn> <mi>b</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>6</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>4</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> is a bridge. (<b>c</b>) A set <math display="inline"><semantics> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mn>1</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>3</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>)</mo> </mrow> <mo>,</mo> <mrow> <mo>(</mo> <msubsup> <mn>6</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>,</mo> <msubsup> <mn>2</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> is a bridge of width 6. (<b>d</b>) A staple segment <math display="inline"><semantics> <mrow> <mo>(</mo> <msubsup> <mn>5</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>2</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>)</mo> </mrow> </semantics></math> violates the property (ii) for a bridge of width 7. A set <math display="inline"><semantics> <mrow> <mo>{</mo> <mrow> <mo>(</mo> <msubsup> <mn>1</mn> <mi>t</mi> <mi>L</mi> </msubsup> <mo>,</mo> <msubsup> <mn>7</mn> <mi>b</mi> <mi>R</mi> </msubsup> <mo>)</mo> </mrow> <mo>}</mo> </mrow> </semantics></math> is a bridge of width 7.</p> Full article ">

16 pages, 2979 KiB

Open AccessArticle

Learning High-Dimensional Chaos Based on an Echo State Network with Homotopy Transformation

by Shikun Wang, Fengjie Geng, Yuting Li and Hongjie Liu

Mathematics 2025, 13(6), 894; https://doi.org/10.3390/math13060894 - 7 Mar 2025

Learning high-dimensional chaos is a complex and challenging problem because of its initial value-sensitive dependence. Based on an echo state network (ESN), we introduce homotopy transformation in topological theory to learn high-dimensional chaos. On the premise of maintaining the basic topological properties, our [...] Read more.

Learning high-dimensional chaos is a complex and challenging problem because of its initial value-sensitive dependence. Based on an echo state network (ESN), we introduce homotopy transformation in topological theory to learn high-dimensional chaos. On the premise of maintaining the basic topological properties, our model can obtain the key features of chaos for learning through the continuous transformation between different activation functions, achieving an optimal balance between nonlinearity and linearity to enhance the generalization capability of the model. In the experimental part, we choose the Lorenz system, Mackey–Glass (MG) system, and Kuramoto–Sivashinsky (KS) system as examples, and we verify the superiority of our model by comparing it with other models. For some systems, the prediction error can be reduced by two orders of magnitude. The results show that the addition of homotopy transformation can improve the modeling ability of complex spatiotemporal chaotic systems, and this demonstrates the potential application of the model in dynamic time series analysis. Full article

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Figure 1

Figure 1
<p>Echo state network architecture: (<b>a</b>) training phase, and (<b>b</b>) testing phase. <math display="inline"><semantics> <mrow> <mi mathvariant="bold">I</mi> <mo>/</mo> <mi mathvariant="bold">R</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi mathvariant="bold">R</mi> <mo>/</mo> <mi mathvariant="bold">O</mi> </mrow> </semantics></math> denote the input-to-reservoir and reservoir-to-output couplers, respectively. <math display="inline"><semantics> <mi mathvariant="bold">R</mi> </semantics></math> denotes the reservoir.</p> Full article ">Figure 2
<p>Transition of <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics></math> from <math display="inline"><semantics> <mrow> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>h</mi> </mrow> </semantics></math> to <span class="html-italic">x</span> under different values of <math display="inline"><semantics> <mi>θ</mi> </semantics></math>.</p> Full article ">Figure 3
<p>Prediction results of the ESN, H-ESN, and DeepESN for each dimension of the Lorenz system. (<b>a</b>) Lorenz-x, (<b>b</b>) Lorenz-y, and (<b>c</b>) Lorenz-z.</p> Full article ">Figure 4
<p>EPT variation curves of the three dimensions of the Lorenz system with respect to <math display="inline"><semantics> <mi>θ</mi> </semantics></math> are shown, with blue for Lorenz-x, red for Lorenz-y, and green for Lorenz-z.</p> Full article ">Figure 5
<p>Comparison of the prediction results for the MG time series between the ESN and H-ESN; the upper panel shows the ESN predictions, and the lower panel shows the H-ESN predictions.</p> Full article ">Figure 6
<p>Prediction error curves of the H-ESN with <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>1.2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>1.25</mn> </mrow> </semantics></math> as functions of varying reservoir sizes <math display="inline"><semantics> <msub> <mi>D</mi> <mi>r</mi> </msub> </semantics></math>.</p> Full article ">Figure 7
<p>Variation curves of the prediction errors of the ESN, H-ESN, and DeepESN at different spectral radius <math display="inline"><semantics> <mi>ρ</mi> </semantics></math> values.</p> Full article ">Figure 8
<p>Comparison of the prediction results for the KS system between the ESN and H-ESN: the left panel shows the ESN predictions, while the right panel shows the H-ESN predictions, where <math display="inline"><semantics> <mrow> <msub> <mo>Λ</mo> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mi>t</mi> </mrow> </semantics></math> represents the Lyapunov time.</p> Full article ">Figure 9
<p>MSE plot of the predicted values and true values for different dimensions of the KS system using the ESN and H-ESN.</p> Full article ">Figure 10
<p>Comparison of prediction errors of the H-ESN under different Gaussian noise intensities.</p> Full article ">

52 pages, 29859 KiB

Open AccessReview

2D Object Detection: A Survey

by Emanuele Malagoli and Luca Di Persio

Mathematics 2025, 13(6), 893; https://doi.org/10.3390/math13060893 - 7 Mar 2025

Object detection is a fundamental task in computer vision, aiming to identify and localize objects of interest within an image. Over the past two decades, the domain has changed profoundly, evolving into an active and fast-moving field while simultaneously becoming the foundation for [...] Read more.

Object detection is a fundamental task in computer vision, aiming to identify and localize objects of interest within an image. Over the past two decades, the domain has changed profoundly, evolving into an active and fast-moving field while simultaneously becoming the foundation for a wide range of modern applications. This survey provides a comprehensive review of the evolution of 2D generic object detection, tracing its development from traditional methods relying on handcrafted features to modern approaches driven by deep learning. The review systematically categorizes contemporary object detection methods into three key paradigms: one-stage, two-stage, and transformer-based, highlighting their development milestones and core contributions. The paper provides an in-depth analysis of each paradigm, detailing landmark methods and their impact on the progression of the field. Additionally, the survey examines some fundamental components of 2D object detection such as loss functions, datasets, evaluation metrics, and future trends. Full article

(This article belongs to the Special Issue Advanced Research in Image Processing and Optimization Methods)

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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>Object detection pipeline. The model processes an input image and detects instances of predefined object classes (here, <span class="html-italic">dog</span> and <span class="html-italic">cat</span>), predicting bounding boxes, class labels, and confidence scores.</p> Full article ">Figure 2
<p>Milestones of 2D generic object detection. AlexNet [<a href="#B6-mathematics-13-00893" class="html-bibr">6</a>] marks the transition from traditional methods, based on handcrafted features, to deep learning-based approaches, based on learned features. Among the latter, three distinct colors identify the paradigms examined in this survey: blue for two-stage detectors, red for one-stage detectors, and light blue for transformer-based detectors, based on the transformer architecture [<a href="#B11-mathematics-13-00893" class="html-bibr">11</a>], represented on the timeline. Milestone detectors in this figure: Viola–Jones Detector [<a href="#B4-mathematics-13-00893" class="html-bibr">4</a>,<a href="#B5-mathematics-13-00893" class="html-bibr">5</a>], HOG Detector [<a href="#B1-mathematics-13-00893" class="html-bibr">1</a>], DPM [<a href="#B12-mathematics-13-00893" class="html-bibr">12</a>], R-CNN [<a href="#B7-mathematics-13-00893" class="html-bibr">7</a>], SPPNet [<a href="#B13-mathematics-13-00893" class="html-bibr">13</a>], Fast R-CNN [<a href="#B14-mathematics-13-00893" class="html-bibr">14</a>], Faster R-CNN [<a href="#B15-mathematics-13-00893" class="html-bibr">15</a>,<a href="#B16-mathematics-13-00893" class="html-bibr">16</a>], R-FCN [<a href="#B17-mathematics-13-00893" class="html-bibr">17</a>], FPN [<a href="#B18-mathematics-13-00893" class="html-bibr">18</a>], Mask R-CNN [<a href="#B19-mathematics-13-00893" class="html-bibr">19</a>], Cascade R-CNN [<a href="#B20-mathematics-13-00893" class="html-bibr">20</a>,<a href="#B21-mathematics-13-00893" class="html-bibr">21</a>], OverFeat [<a href="#B22-mathematics-13-00893" class="html-bibr">22</a>], SSD [<a href="#B9-mathematics-13-00893" class="html-bibr">9</a>], DSSD [<a href="#B23-mathematics-13-00893" class="html-bibr">23</a>], R-SSD [<a href="#B24-mathematics-13-00893" class="html-bibr">24</a>], FSSD [<a href="#B25-mathematics-13-00893" class="html-bibr">25</a>], RefineDet [<a href="#B26-mathematics-13-00893" class="html-bibr">26</a>], EFGRNet [<a href="#B27-mathematics-13-00893" class="html-bibr">27</a>], ASSD [<a href="#B28-mathematics-13-00893" class="html-bibr">28</a>], RetinaNet [<a href="#B29-mathematics-13-00893" class="html-bibr">29</a>], CornerNet [<a href="#B30-mathematics-13-00893" class="html-bibr">30</a>], CenterNet [<a href="#B31-mathematics-13-00893" class="html-bibr">31</a>], ExtremeNet [<a href="#B32-mathematics-13-00893" class="html-bibr">32</a>], FCOS [<a href="#B33-mathematics-13-00893" class="html-bibr">33</a>], FoveaBox [<a href="#B34-mathematics-13-00893" class="html-bibr">34</a>], FSAF [<a href="#B35-mathematics-13-00893" class="html-bibr">35</a>], YOLOv1 [<a href="#B8-mathematics-13-00893" class="html-bibr">8</a>], YOLOv2 [<a href="#B36-mathematics-13-00893" class="html-bibr">36</a>], YOLOv3 [<a href="#B37-mathematics-13-00893" class="html-bibr">37</a>], YOLOv4 [<a href="#B38-mathematics-13-00893" class="html-bibr">38</a>], YOLOv6 [<a href="#B39-mathematics-13-00893" class="html-bibr">39</a>], YOLOv7 [<a href="#B40-mathematics-13-00893" class="html-bibr">40</a>], YOLOv9 [<a href="#B41-mathematics-13-00893" class="html-bibr">41</a>], DETR [<a href="#B10-mathematics-13-00893" class="html-bibr">10</a>], Deformable DETR [<a href="#B42-mathematics-13-00893" class="html-bibr">42</a>], DAB-DETR [<a href="#B43-mathematics-13-00893" class="html-bibr">43</a>], DN-DETR [<a href="#B44-mathematics-13-00893" class="html-bibr">44</a>], DINO [<a href="#B45-mathematics-13-00893" class="html-bibr">45</a>], ViT-FRCNN [<a href="#B46-mathematics-13-00893" class="html-bibr">46</a>], ViTDet [<a href="#B47-mathematics-13-00893" class="html-bibr">47</a>].</p> Full article ">Figure 3
<p>On the <b>left</b>: the first and second rectangular features selected by AdaBoost [<a href="#B50-mathematics-13-00893" class="html-bibr">50</a>] in Viola–Jones detector [<a href="#B4-mathematics-13-00893" class="html-bibr">4</a>,<a href="#B5-mathematics-13-00893" class="html-bibr">5</a>]. The two features are shown in the top row and then overlaid on a typical training face in the bottom row. On the <b>right</b>: an example detection obtained using the deformable part-based model (DPM) [<a href="#B12-mathematics-13-00893" class="html-bibr">12</a>]. The model consists of a coarse template (blue rectangle), several higher-resolution part templates (yellow rectangles), and a spatial model that defines the relative location of each part.</p> Full article ">Figure 4
<p>The architecture of R-CNN [<a href="#B7-mathematics-13-00893" class="html-bibr">7</a>], which, in order, takes an input image, extracts around 2000 bottom-up region proposals, computes features for each proposal using a backbone, and then classifies each region using class-specific linear SVMs.</p> Full article ">Figure 5
<p>Mask R-CNN results on the MS-COCO [<a href="#B121-mathematics-13-00893" class="html-bibr">121</a>] dataset. For each image, masks are shown in color, along with the corresponding bounding boxes, category labels, and confidence scores.</p> Full article ">Figure 6
<p>SSD architecture: SSD adds several feature layers to the end of a backbone, which predict the offsets to default boxes of different scales and aspect ratios, along with their associated confidence scores.</p> Full article ">Figure 7
<p>The three images show anchor-free keypoint-based methods which use different combinations of keypoints (red circles) and then group them for bounding box prediction. A pair of corners, a triplet of keypoints, and extreme points on the object are respectively used in CornerNet [<a href="#B30-mathematics-13-00893" class="html-bibr">30</a>], CenterNet [<a href="#B31-mathematics-13-00893" class="html-bibr">31</a>], and ExtremeNet [<a href="#B32-mathematics-13-00893" class="html-bibr">32</a>].</p> Full article ">Figure 8
<p>On the <b>left</b>: FCOS [<a href="#B33-mathematics-13-00893" class="html-bibr">33</a>] works by predicting a 4D vector <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>l</mi> <mo>,</mo> <mi>t</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>b</mi> <mo>)</mo> </mrow> </semantics></math> encoding the location of a bounding box at each foreground pixel. The second plot on the left illustrates the ambiguity that arises when a location resides in multiple bounding boxes. On the <b>right</b>: the center-ness of FCOS [<a href="#B33-mathematics-13-00893" class="html-bibr">33</a>] is shown, where red, blue, and other colors denote <math display="inline"><semantics> <mrow> <mn>1</mn> <mo>,</mo> <mn>0</mn> </mrow> </semantics></math>, and the values between them, respectively. Center-ness decays from 1 to 0 as the location deviates from the center of the object.</p> Full article ">Figure 9
<p>YOLOv1 [<a href="#B8-mathematics-13-00893" class="html-bibr">8</a>] detection pipeline: it divides the image into an <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>×</mo> <mi>S</mi> </mrow> </semantics></math> grid and for each grid cell predicts <span class="html-italic">B</span> bounding boxes, their associated confidence scores, and <span class="html-italic">C</span> class probabilities. These outputs are encoded as an <math display="inline"><semantics> <mrow> <mi>S</mi> <mo>×</mo> <mi>S</mi> <mo>×</mo> <mo>(</mo> <mi>B</mi> <mo>×</mo> <mn>5</mn> <mo>+</mo> <mi>C</mi> <mo>)</mo> </mrow> </semantics></math> tensor.</p> Full article ">Figure 10
<p>DETR [<a href="#B10-mathematics-13-00893" class="html-bibr">10</a>] detection pipeline. DETR [<a href="#B10-mathematics-13-00893" class="html-bibr">10</a>] combines a common CNN with a transformer architecture and directly predicts (in parallel) the final set of detections. During training, bipartite matching is used to uniquely pair predictions with ground truth boxes; unmatched predictions should yield a “no object” (Ø) class prediction.</p> Full article ">Figure 11
<p>ViT-FRCNN [<a href="#B46-mathematics-13-00893" class="html-bibr">46</a>] detection pipeline. ViT [<a href="#B359-mathematics-13-00893" class="html-bibr">359</a>] backbone is extended to perform object detection by making use of the per-patch outputs in the final transformer layer. These outputs are reinterpreted as spatial feature map and passed to a detection network.</p> Full article ">Figure 12
<p>Examples of images and their corresponding annotations from some of the most widely used object detection datasets. From left to right: Pascal VOC [<a href="#B81-mathematics-13-00893" class="html-bibr">81</a>,<a href="#B82-mathematics-13-00893" class="html-bibr">82</a>,<a href="#B83-mathematics-13-00893" class="html-bibr">83</a>,<a href="#B384-mathematics-13-00893" class="html-bibr">384</a>,<a href="#B385-mathematics-13-00893" class="html-bibr">385</a>,<a href="#B386-mathematics-13-00893" class="html-bibr">386</a>,<a href="#B387-mathematics-13-00893" class="html-bibr">387</a>,<a href="#B388-mathematics-13-00893" class="html-bibr">388</a>,<a href="#B389-mathematics-13-00893" class="html-bibr">389</a>,<a href="#B390-mathematics-13-00893" class="html-bibr">390</a>], MS-COCO [<a href="#B121-mathematics-13-00893" class="html-bibr">121</a>], and Open Images [<a href="#B391-mathematics-13-00893" class="html-bibr">391</a>,<a href="#B392-mathematics-13-00893" class="html-bibr">392</a>,<a href="#B393-mathematics-13-00893" class="html-bibr">393</a>].</p> Full article ">

10 pages, 252 KiB

Open AccessArticle

Generalized Local Charge Conservation in Many-Body Quantum Mechanics

by F. Minotti and G. Modanese

Mathematics 2025, 13(5), 892; https://doi.org/10.3390/math13050892 - 6 Mar 2025

In the framework of the quantum theory of many-particle systems, we study the compatibility of approximated non-equilibrium Green’s functions (NEGFs) and of approximated solutions of the Dyson equation with a modified continuity equation of the form [...] Read more.

In the framework of the quantum theory of many-particle systems, we study the compatibility of approximated non-equilibrium Green’s functions (NEGFs) and of approximated solutions of the Dyson equation with a modified continuity equation of the form

\partial_{t} ⟨ ρ ⟩ + (1 - γ) \nabla \cdot ⟨ J ⟩ = 0

. A continuity equation of this kind allows the e.m. coupling of the system in the extended Aharonov–Bohm electrodynamics, but not in Maxwell electrodynamics. Focusing on the case of molecular junctions simulated numerically with the Density Functional Theory (DFT), we further discuss the re-definition of local current density proposed by Wang et al., which also turns out to be compatible with the extended Aharonov–Bohm electrodynamics. Full article

(This article belongs to the Special Issue Mathematics and Applications)

34 pages, 592 KiB

Open AccessArticle

A Fractional Adams Method for Caputo Fractional Differential Equations with Modified Graded Meshes

by Yuhui Yang and Yubin Yan

Mathematics 2025, 13(5), 891; https://doi.org/10.3390/math13050891 - 6 Mar 2025

In this paper, we introduce an Adams-type predictor–corrector method based on a modified graded mesh for solving Caputo fractional differential equations. This method not only effectively handles the weak singularity near the initial point but also reduces errors associated with large intervals in [...] Read more.

In this paper, we introduce an Adams-type predictor–corrector method based on a modified graded mesh for solving Caputo fractional differential equations. This method not only effectively handles the weak singularity near the initial point but also reduces errors associated with large intervals in traditional graded meshes. We prove the error estimates in detail for both

0 < α < 1

and

1 < α < 2

cases, where

α

is the order of the Caputo fractional derivative. Numerical experiments confirm the convergence of the proposed method and compare its performance with the traditional graded mesh approach. Full article

(This article belongs to the Section E: Applied Mathematics)

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42 pages, 2300 KiB

Open AccessArticle

Pricing and Return Strategies in Omni-Channel Apparel Retail Considering the Impact of Fashion Level

by Yanchun Wan, Zhiping Yan and Shudi Wang

Mathematics 2025, 13(5), 890; https://doi.org/10.3390/math13050890 - 6 Mar 2025

In the context of new retail, the development of omni-channels is flourishing. The entry threshold for the clothing industry is low, and the popularity of online shopping has, to some extent, reduced consumers’ perception of the authenticity of clothing. As a result, returns [...] Read more.

In the context of new retail, the development of omni-channels is flourishing. The entry threshold for the clothing industry is low, and the popularity of online shopping has, to some extent, reduced consumers’ perception of the authenticity of clothing. As a result, returns are a serious issue in the clothing industry. This article focuses on a clothing retailer while addressing retail and return issues in the clothing industry. It develops and analyzes models for an online single-channel strategy and two omni-channel showroom strategies: “Experience in Store and Buy Online (ESBO)” with an experience store and “Buy Online and Return in Store (BORS)” with a physical store. These models are used to examine the pricing and return decisions of the retailer in the three strategic scenarios. Additionally, this study considers the impact of fashion trends on demand. It explores pricing and return strategies in two showroom models under the influence of the fashion trend decay factor. Moreover, sensitivity analyses and numerical analyses of the important parameters are performed. This research demonstrates the following: (1) In the case of high return transportation costs and online return hassle costs, clothing retailers can attract consumers to increase profits through establishing offline channels; (2) extending the sales time of fashionable clothing has a positive effect on profits, but blindly prolonging the continuation of the sales time will lead to a decrease in profits; (3) the larger the initial fashion level or the smaller the fashion level decay factor, the greater the optimal retailer profits. The impacts of the initial fashion level and fashion level decay factor on profits are more significant in omni-channel operations. This article aims to identify optimal strategies for retailers utilizing omni-channel operations and offer managerial insights for the sale of fashionable apparel. Full article

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Figure 1
<p>Buying procedure of the consumer with the “Experience in Store and Buy Online (ESBO)” omni-channel strategy.</p> Full article ">Figure 2
<p>Buying procedure of consumers with the “Buy Online and Return in Store (BORS)” omni-channel strategy.</p> Full article ">Figure 3
<p>Consumer’s channel options in the ESBO model.</p> Full article ">Figure 4
<p>Consumer segmentation when opening an experience store: (<b>a</b>) Case <math display="inline"><semantics> <msubsup> <mi>S</mi> <mi>i</mi> <mi>e</mi> </msubsup> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>≤</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>θ</mi> <mi>v</mi> <mo>−</mo> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mo>−</mo> <msub> <mi>h</mi> <mi>o</mi> </msub> </mrow> <mi>θ</mi> </mfrac> </mstyle> </mrow> </semantics></math>; (<b>b</b>) Case <math display="inline"><semantics> <msubsup> <mi>S</mi> <mrow> <mi>i</mi> <mi>i</mi> </mrow> <mi>e</mi> </msubsup> </semantics></math>: <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>></mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>θ</mi> <mi>v</mi> <mo>−</mo> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mo>−</mo> <msub> <mi>h</mi> <mi>o</mi> </msub> </mrow> <mi>θ</mi> </mfrac> </mstyle> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Consumer’s channel options in the BORS model.</p> Full article ">Figure 6
<p>Consumer segmentation when opening a physical store: (<b>a</b>) <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>≤</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>θ</mi> <mi>v</mi> <mo>−</mo> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mo>−</mo> <msub> <mi>h</mi> <mi>o</mi> </msub> </mrow> <mi>θ</mi> </mfrac> </mstyle> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>≤</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>h</mi> <mi>o</mi> </msub> <mi>θ</mi> </mfrac> </mstyle> <mo>−</mo> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> </mrow> </semantics></math>; (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>≤</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>θ</mi> <mi>v</mi> <mo>−</mo> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mo>−</mo> <msub> <mi>h</mi> <mi>o</mi> </msub> </mrow> <mi>θ</mi> </mfrac> </mstyle> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>></mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>h</mi> <mi>o</mi> </msub> <mi>θ</mi> </mfrac> </mstyle> <mo>−</mo> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> </mrow> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi>p</mi> <mo>></mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mi>θ</mi> <mi>v</mi> <mo>−</mo> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <mn>1</mn> <mo>−</mo> <mi>θ</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">(</mo> </mstyle> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> <mo>+</mo> <mi>f</mi> <mstyle displaystyle="true"> <mo stretchy="false">)</mo> </mstyle> </mrow> <mo>−</mo> <msub> <mi>h</mi> <mi>o</mi> </msub> </mrow> <mi>θ</mi> </mfrac> </mstyle> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>f</mi> <mo>></mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <msub> <mi>h</mi> <mi>o</mi> </msub> <mi>θ</mi> </mfrac> </mstyle> <mo>−</mo> <msub> <mi>h</mi> <mrow> <mi>r</mi> <mi>o</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Comparisons of retailer profits as return transportation costs change.</p> Full article ">Figure 8
<p>Comparisons of retailer profits as online return hassle costs change.</p> Full article ">Figure 9
<p>Comparisons of retailer profits as retail prices change.</p> Full article ">Figure 10
<p>Changes in retailer profits under the influence of the fashion level existence time and retail price: (<b>a</b>) in the BORO strategy; (<b>b</b>) in the ESBO strategy; (<b>c</b>) in the BORS strategy.</p> Full article ">Figure 11
<p>Changes in retailer profits under the influence of the continuing sales time and retail price: (<b>a</b>) in the BORO strategy; (<b>b</b>) in the ESBO strategy; (<b>c</b>) in the BORS strategy.</p> Full article ">Figure 12
<p>Changes in the retailer’s profits with the retail price at different initial fashion levels: (<b>a</b>) in the BORO strategy; (<b>b</b>) in the ESBO strategy; (<b>c</b>) in the BORS strategy.</p> Full article ">Figure 13
<p>Changes in retailer’ profits with the retail price with different fashion level decay factors: (<b>a</b>) in the BORO strategy; (<b>b</b>) in the ESBO strategy; (<b>c</b>) in the BORS strategy.</p> Full article ">Figure A1
<p>Image of <math display="inline"><semantics> <msubsup> <mo>Π</mo> <mi>i</mi> <mi>e</mi> </msubsup> </semantics></math> changing with <span class="html-italic">p</span> and <span class="html-italic">f</span>.</p> Full article ">

12 pages, 251 KiB

Open AccessArticle

On the Complexity of Computing a Maximum Acyclic Matching in Undirected Graphs

by Samer Nofal

Mathematics 2025, 13(5), 889; https://doi.org/10.3390/math13050889 - 6 Mar 2025

The problem of finding a maximum acyclic matching in a simple undirected graph is known to be NP-complete. In this paper, we present new results; we show that a maximum acyclic matching in a given undirected graph (with n vertices and m edges) [...] Read more.

The problem of finding a maximum acyclic matching in a simple undirected graph is known to be NP-complete. In this paper, we present new results; we show that a maximum acyclic matching in a given undirected graph (with n vertices and m edges) can be computed recursively with a recursion depth

O (ln m)

in expectation. Consequently, employing a recursive computation of a maximum acyclic matching in a given graph, if the recursion depth meets the expectation

O (ln m)

, then a maximum acyclic matching can be computed in time

O (n^{3.4})

and space

O (m ln m)

. However, for the general case, the complexity of the recursive computation of a maximum acyclic matching is in

O (n^{2} 2^{m})

time and in

O (m^{2})

space. Full article

17 pages, 591 KiB

Open AccessFeature PaperArticle

Enhancing Uplink Communication in Wireless Powered Communication Networks Through Rate-Splitting Multiple Access and Joint Resource Optimization

by Iqra Hameed, Mario R. Camana, Mohammad Abrar Shakil Sejan and Hyoung Kyu Song

Mathematics 2025, 13(5), 888; https://doi.org/10.3390/math13050888 - 6 Mar 2025

Wireless powered communication networks (WPCNs) provide a sustainable solution for energy-constrained IoT devices by enabling wireless energy transfer (WET) in the downlink and wireless information transmission (WIT) in the uplink. However, their performance is often limited by interference in uplink communication and inefficient [...] Read more.

Wireless powered communication networks (WPCNs) provide a sustainable solution for energy-constrained IoT devices by enabling wireless energy transfer (WET) in the downlink and wireless information transmission (WIT) in the uplink. However, their performance is often limited by interference in uplink communication and inefficient resource allocation. To address these challenges, we propose an RSMA-aided WPCN framework, which optimizes rate-splitting factors, power allocation, and time division to enhance spectral efficiency and user fairness. To solve this non-convex joint optimization problem, we employ the simultaneous perturbation stochastic approximation (SPSA) algorithm, a gradient-free method that efficiently estimates optimal parameters with minimal function evaluations. Compared to conventional optimization techniques, SPSA provides a scalable and computationally efficient approach for real-time resource allocation in RSMA-aided WPCNs. Our simulation results demonstrate that the proposed RSMA-aided framework improves sum throughput by 12.5% and enhances fairness by 15–20% compared to conventional multiple-access schemes. These findings establish RSMA as a key enabler for next-generation WPCNs, offering a scalable, interference-resilient, and energy-efficient solution for future wireless networks. Full article

(This article belongs to the Special Issue Advanced Algorithms in Wireless Communication and Internet of Things (IoT))

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Figure 1
<p>The RSMA-aided wireless powered communication network (WPCN) framework. The hybrid access point (H-AP) first transmits energy to users <math display="inline"><semantics> <msub> <mi>U</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>U</mi> <mn>2</mn> </msub> </semantics></math> via wireless energy transfer (WET), followed by wireless information transmission (WIT) in the uplink. Each user employs RSMA, where messages are split into two parts, optimally allocated for power and transmitted to the H-AP.</p> Full article ">Figure 2
<p>Convergence of proposed SPSA-based algorithm for different transmit power levels.</p> Full article ">Figure 3
<p>Sum throughput versus transmit power at H-AP for different schemes.</p> Full article ">Figure 4
<p>Throughput of <math display="inline"><semantics> <msub> <mi>U</mi> <mn>2</mn> </msub> </semantics></math> versus transmit power at H-AP for different schemes.</p> Full article ">Figure 5
<p>Sum throughput versus distance between <math display="inline"><semantics> <msub> <mi>U</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>U</mi> <mn>2</mn> </msub> </semantics></math> for different schemes.</p> Full article ">Figure 6
<p>Throughput of <math display="inline"><semantics> <msub> <mi>U</mi> <mn>2</mn> </msub> </semantics></math> versus distance between <math display="inline"><semantics> <msub> <mi>U</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>U</mi> <mn>2</mn> </msub> </semantics></math> for different schemes.</p> Full article ">Figure 7
<p>Sum throughput versus transmit power at H-AP.</p> Full article ">

30 pages, 1422 KiB

Open AccessFeature PaperArticle

A Comparative Analysis of Compression and Transfer Learning Techniques in DeepFake Detection Models

by Andreas Karathanasis, John Violos and Ioannis Kompatsiaris

Mathematics 2025, 13(5), 887; https://doi.org/10.3390/math13050887 - 6 Mar 2025

DeepFake detection models play a crucial role in ambient intelligence and smart environments, where systems rely on authentic information for accurate decisions. These environments, integrating interconnected IoT devices and AI-driven systems, face significant threats from DeepFakes, potentially leading to compromised trust, erroneous decisions, [...] Read more.

DeepFake detection models play a crucial role in ambient intelligence and smart environments, where systems rely on authentic information for accurate decisions. These environments, integrating interconnected IoT devices and AI-driven systems, face significant threats from DeepFakes, potentially leading to compromised trust, erroneous decisions, and security breaches. To mitigate these risks, neural-network-based DeepFake detection models have been developed. However, their substantial computational requirements and long training times hinder deployment on resource-constrained edge devices. This paper investigates compression and transfer learning techniques to reduce the computational demands of training and deploying DeepFake detection models, while preserving performance. Pruning, knowledge distillation, quantization, and adapter modules are explored to enable efficient real-time DeepFake detection. An evaluation was conducted on four benchmark datasets: “SynthBuster”, “140k Real and Fake Faces”, “DeepFake and Real Images”, and “ForenSynths”. It compared compressed models with uncompressed baselines using widely recognized metrics such as accuracy, precision, recall, F1-score, model size, and training time. The results showed that a compressed model at 10% of the original size retained only 56% of the baseline accuracy, but fine-tuning in similar scenarios increased this to nearly 98%. In some cases, the accuracy even surpassed the original’s performance by up to 12%. These findings highlight the feasibility of deploying DeepFake detection models in edge computing scenarios. Full article

(This article belongs to the Special Issue Ambient Intelligence Methods and Applications)

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39 pages, 806 KiB

Open AccessArticle

Limit Theorems for Kernel Regression Estimator for Quasi-Associated Functional Censored Time Series Within Single Index Structure

by Said Attaoui, Oum Elkheir Benouda, Salim Bouzebda and Ali Laksaci

Mathematics 2025, 13(5), 886; https://doi.org/10.3390/math13050886 - 6 Mar 2025

In this paper, we develop kernel-based estimators for regression functions under a functional single-index model, applied to censored time series data. By capitalizing on the single-index structure, we reduce the dimensionality of the covariate-response relationship, thereby preserving the ability to capture intricate dependencies [...] Read more.

In this paper, we develop kernel-based estimators for regression functions under a functional single-index model, applied to censored time series data. By capitalizing on the single-index structure, we reduce the dimensionality of the covariate-response relationship, thereby preserving the ability to capture intricate dependencies while maintaining a relatively parsimonious form. Specifically, our framework utilizes nonparametric kernel estimation within a quasi-association setting to characterize the underlying relationships. Under mild regularity conditions, we demonstrate that these estimators attain both strong uniform consistency and asymptotic normality. Through extensive simulation experiments, we confirm their robust finite-sample performance. Moreover, an empirical examination using intraday Nikkei stock index returns illustrates that the proposed method significantly outperforms traditional nonparametric regression approaches. Full article

(This article belongs to the Special Issue Probability, Stochastic Processes and Machine Learning)

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