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9 pages, 359 KiB

Open AccessArticle

On the Transition Density of the Time-Inhomogeneous 3/2 Model: A Unified Approach for Models Related to Squared Bessel Process

by Rattiya Meesa, Ratinan Boonklurb and Phiraphat Sutthimat

Mathematics 2025, 13(12), 1948; https://doi.org/10.3390/math13121948 - 12 Jun 2025

Viewed by 377

Abstract

We derive an infinite-series representation for the transition probability density function (PDF) of the time-inhomogeneous 3/2 model, expressing all coefficients in terms of Bell-polynomial and generalized Laguerre-polynomial formulas. From this series, we obtain explicit expressions for all conditional moments of the variance process, [...] Read more.

We derive an infinite-series representation for the transition probability density function (PDF) of the time-inhomogeneous 3/2 model, expressing all coefficients in terms of Bell-polynomial and generalized Laguerre-polynomial formulas. From this series, we obtain explicit expressions for all conditional moments of the variance process, recovering the familiar time-homogeneous formulas when parameters are constant. Numerical experiments illustrate that both the density and moment series converge rapidly, and the resulting distributions agree with high-precision Monte Carlo simulations. Finally, we demonstrate that the same approach extends to a broad family of non-affine, time-varying diffusions, providing a general framework for obtaining transition PDFs and moments in advanced models. Full article

(This article belongs to the Special Issue Probability Statistics and Quantitative Finance)

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18 pages, 7721 KiB

Open AccessArticle

A Novel Lorenz-like Attractor and Stability and Equilibrium Analysis

by Jun Pan, Haijun Wang, Guiyao Ke and Feiyu Hu

Axioms 2025, 14(4), 264; https://doi.org/10.3390/axioms14040264 - 30 Mar 2025

Cited by 12 | Viewed by 369

Abstract

This paper introduces a novel 3D periodically forced extended Lorenz-like system and illustrates a single thick two-scroll attractor with potential unboundedness whose time series of the second state variable present some certain random characteristics rather than pure periodicity yielded by that system itself. [...] Read more.

This paper introduces a novel 3D periodically forced extended Lorenz-like system and illustrates a single thick two-scroll attractor with potential unboundedness whose time series of the second state variable present some certain random characteristics rather than pure periodicity yielded by that system itself. Combining the Lyapunov function and the definitions of both the

α

-limit set and

ω

-limit set, the following rigorous results are proved: infinitely many heteroclinic orbits to two families of parallel parabolic-type non-hyperbolic equilibria, two families of infinitely many pairs of isolated equilibria, an infinite set of isolated equilibria, and infinitely many pairs of isolated equilibria. Full article

(This article belongs to the Section Mathematical Analysis)

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16 pages, 294 KiB

Open AccessArticle

A New Family of Multipartition Graph Operations and Its Applications in Constructing Several Special Graphs

by Qiuping Li, Liangwen Tang, Qingyun Liu and Mugang Lin

Symmetry 2025, 17(3), 467; https://doi.org/10.3390/sym17030467 - 20 Mar 2025

Viewed by 419

Abstract

A new family of graph operations based on multipartite graph with an arbitrary number of parts is defined and their applications are explored in this paper. The complete spectra of graphs derived from multipartite graphs are determined. Because the adjacency matrix of the [...] Read more.

A new family of graph operations based on multipartite graph with an arbitrary number of parts is defined and their applications are explored in this paper. The complete spectra of graphs derived from multipartite graphs are determined. Because the adjacency matrix of the multipartite graph is symmetric, we can use it to generate an unlimited number of special symmetric graphs. Methods for generating countless new families of integral graphs using these multipartite graph operations have been presented. By applying these multipartite graph operations, we can construct infinitely many orderenergetic graphs from orderenergetic or non-orderenergetic graphs. Additionally, infinite pairs of equienergetic and non-cospectral graphs can be generated through these new operations. Moreover, this kind of graph operation can also be used to construct other special graphs related to eigenvalues and energy. Full article

(This article belongs to the Special Issue Symmetry in Combinatorics and Discrete Mathematics)

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43 pages, 521 KiB

Open AccessArticle

On Finite Exceptional Orthogonal Polynomial Sequences Composed of Rational Darboux Transforms of Romanovski-Jacobi Polynomials

by Gregory Natanson

Axioms 2025, 14(3), 218; https://doi.org/10.3390/axioms14030218 - 16 Mar 2025

Cited by 2 | Viewed by 412

Abstract

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. [...] Read more.

The paper presents the united analysis of the finite exceptional orthogonal polynomial (EOP) sequences composed of rational Darboux transforms of Romanovski-Jacobi polynomials. It is shown that there are four distinguished exceptional differential polynomial systems (X-Jacobi DPSs) of series J1, J2, J3, and W. The first three X-DPSs formed by pseudo-Wronskians of two Jacobi polynomials contain both exceptional orthogonal polynomial systems (X-Jacobi OPSs) on the interval (−1, +1) and the finite EOP sequences on the positive interval (1, ∞). On the contrary, the X-DPS of series W formed by Wronskians of two Jacobi polynomials contains only (infinitely many) finite EOP sequences on the interval (1, ∞). In addition, the paper rigorously examines the three isospectral families of the associated Liouville potentials (rationally extended hyperbolic Pöschl-Teller potentials of types a, b, and a′) exactly quantized by the EOPs in question. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

14 pages, 269 KiB

Open AccessFeature PaperArticle

Random Variables Aren’t Random

by Paul W. Vos

Mathematics 2025, 13(5), 775; https://doi.org/10.3390/math13050775 - 26 Feb 2025

Viewed by 419

Abstract

This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We show how measure-theoretic probability provides a framework for modeling populations through distributions, leading [...] Read more.

This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We show how measure-theoretic probability provides a framework for modeling populations through distributions, leading to three key contributions. First, we establish that random variables, properly understood as measurable functions, can be fully characterized without appealing to infinite hypothetical samples. Second, we demonstrate how this perspective enables statistical inference through logical rather than probabilistic reasoning, extending the reductio ad absurdum argument from deductive to inductive inference. Third, we show how this framework naturally leads to an information-based assessment of statistical procedures, replacing traditional inference metrics that emphasize bias and variance with information-based approaches that describe the families of distributions used in parametric inference better. This reformulation addresses long-standing debates in statistical inference while providing a more coherent theoretical foundation. Our approach offers an alternative to traditional frequentist inference that maintains mathematical rigor while avoiding the philosophical complications inherent in repeated sampling interpretations. Full article

(This article belongs to the Section D1: Probability and Statistics)

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19 pages, 955 KiB

Open AccessArticle

Resolving the Open Problem by Proving a Conjecture on the Inverse Mostar Index for c-Cyclic Graphs

by Liju Alex and Kinkar Chandra Das

Symmetry 2025, 17(2), 291; https://doi.org/10.3390/sym17020291 - 14 Feb 2025

Viewed by 547

Abstract

Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index,

M o (G)

, is defined as [...] Read more.

Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index,

M o (G)

, is defined as

M o (G) = \sum_{u v \in E (G)} | n_{u} (e | G) - n_{v} (e | G) |,

where

n_{u} (e | G)

and

n_{v} (e | G)

represent the number of vertices closer to vertex u than v and closer to v than u, respectively, for an edge

e = u v

. The inverse Mostar index problem has gained significant attention recently. In their work, Alizadeh et al. [Solving the Mostar index inverse problem, J. Math. Chem. 62 (5) (2024) 1079–1093] proposed the following open problem: “Which nonnegative integers can be realized as Mostar indices of c-cyclic graphs, for a given positive integer c?”. Subsequently, one of the present authors [On the inverse Mostar index problem for molecular graphs, Trans. Comb. 14 (1) (2024) 65–77] conjectured that, except for finitely many positive integers, all other positive integers can be realized as the Mostar index of a c-cyclic graph, where

c \geq 3

. In this paper, we address the inverse Mostar index problem for c-cyclic graphs. Specifically, we construct infinitely many families of symmetric c-cyclic structures, thereby demonstrating a solution to the inverse Mostar index problem using an infinite family of such symmetric structures. By providing a comprehensive proof of the conjecture, we fully resolve this longstanding open problem. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)

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14 pages, 302 KiB

Open AccessArticle

Multi-Dimensional Markov Chains of M/G/1 Type

by Valeriy Naumov and Konstantin Samouylov

Mathematics 2025, 13(2), 209; https://doi.org/10.3390/math13020209 - 9 Jan 2025

Cited by 1 | Viewed by 730

Abstract

We consider an irreducible discrete-time Markov process with states represented as (k, i) where k is an M-dimensional vector with non-negative integer entries, and i indicates the state (phase) of the external environment. The number n of phases may [...] Read more.

We consider an irreducible discrete-time Markov process with states represented as (k, i) where k is an M-dimensional vector with non-negative integer entries, and i indicates the state (phase) of the external environment. The number n of phases may be either finite or infinite. One-step transitions of the process from a state (k, i) are limited to states (n, j) such that n ≥ k−1, where 1 represents the vector of all 1s. We assume that for a vector k ≥ 1, the one-step transition probability from a state (k, i) to a state (n, j) may depend on i, j, and n − k, but not on the specific values of k and n. This process can be classified as a Markov chain of M/G/1 type, where the minimum entry of the vector n defines the level of a state (n, j). It is shown that the first passage distribution matrix of such a process, also known as the matrix G, can be expressed through a family of nonnegative square matrices of order n, which is a solution to a system of nonlinear matrix equations. Full article

(This article belongs to the Special Issue Queue and Stochastic Models for Operations Research, 3rd Edition)

22 pages, 764 KiB

Open AccessArticle

An Inertial Subgradient Extragradient Method for Efficiently Solving Fixed-Point and Equilibrium Problems in Infinite Families of Demimetric Mappings

by Habib ur Rehman, Fouzia Amir, Jehad Alzabut and Mohammad Athar Azim

Mathematics 2025, 13(1), 20; https://doi.org/10.3390/math13010020 - 25 Dec 2024

Viewed by 658

Abstract

The primary objective of this article is to enhance the convergence rate of the extragradient method through the careful selection of inertial parameters and the design of a self-adaptive stepsize scheme. We propose an improved version of the extragradient method for approximating a [...] Read more.

The primary objective of this article is to enhance the convergence rate of the extragradient method through the careful selection of inertial parameters and the design of a self-adaptive stepsize scheme. We propose an improved version of the extragradient method for approximating a common solution to pseudomonotone equilibrium and fixed-point problems that involve an infinite family of demimetric mappings in real Hilbert spaces. We establish that the iterative sequences generated by our proposed algorithms converge strongly under suitable conditions. These results substantiate the effectiveness of our approach in achieving convergence, marking a significant advancement in the extragradient method. Furthermore, we present several numerical tests to illustrate the practical efficiency of the proposed method, comparing these results with those from established methods to demonstrate the improved convergence rates and solution accuracy achieved through our approach. Full article

(This article belongs to the Special Issue Applied Functional Analysis and Applications: 2nd Edition)

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14 pages, 311 KiB

Open AccessArticle

Kekulé Structure of Angularly Connected Even Ring Systems

by Simon Brezovnik

Axioms 2024, 13(12), 827; https://doi.org/10.3390/axioms13120827 - 26 Nov 2024

Viewed by 714

Abstract

An even ring system G is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. G is angularly connected if all of the [...] Read more.

An even ring system G is a simple 2-connected plane graph with all interior vertices of degree 3, all exterior vertices of either degree 2 or 3, and all finite faces of an even length. G is angularly connected if all of the peripheral segments of G have odd lengths. In this paper, we show that every angularly connected even ring system G, which does not contain any triple of altogether-adjacent peripheral faces, has a perfect matching. This was achieved by finding an appropriate edge coloring of G, derived from the proof of the existence of a proper face 3-coloring of the graph. Additionally, an infinite family of graphs that are face 3-colorable has been identified. When interpreted in the context of the inner dual of G, this leads to the introduction of 3-colorable graphs containing cycles of lengths 4 and 6, which is a supplementation of some already known results. Finally, we have investigated the concept of the Clar structure and Clar set within the aforementioned family of graphs. We found that a Clar set of an angularly connected even ring system cannot in general be obtained by minimizing the cardinality of the set A. This result is in contrast to the previously known case for the subfamily of benzenoid systems, which admit a face 3-coloring. Our results open up avenues for further research into the properties of Clar and Fries sets of angularly connected even ring systems. Full article

(This article belongs to the Special Issue Recent Developments in Graph Theory)

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25 pages, 361 KiB

Open AccessArticle

Reciprocal Hyperbolic Series of Ramanujan Type

by Ce Xu and Jianqiang Zhao

Mathematics 2024, 12(19), 2974; https://doi.org/10.3390/math12192974 - 25 Sep 2024

Cited by 1 | Viewed by 930

Abstract

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known [...] Read more.

This paper presents an approach to summing a few families of infinite series involving hyperbolic functions, some of which were first studied by Ramanujan. The key idea is based on their contour integral representations and residue computations with the help of some well-known results of Eisenstein series given by Ramanujan, Berndt, et al. As our main results, several series involving hyperbolic functions are evaluated and expressed in terms of

z = {}_{2}F_{1} (1 / 2, 1 / 2; 1; x)

and

z^{'} = d z / d x

. When a certain parameter in these series is equal to

π

, the series are expressed in closed forms in terms of some special values of the Gamma function. Moreover, many new illustrative examples are presented. Full article

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

19 pages, 365 KiB

Open AccessArticle

Linear Codes Constructed from Two Weakly Regular Plateaued Functions with Index (p − 1)/2

by Shudi Yang, Tonghui Zhang and Zheng-an Yao

Entropy 2024, 26(6), 455; https://doi.org/10.3390/e26060455 - 27 May 2024

Cited by 2 | Viewed by 1158

Abstract

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting [...] Read more.

Linear codes are the most important family of codes in cryptography and coding theory. Some codes only have a few weights and are widely used in many areas, such as authentication codes, secret sharing schemes and strongly regular graphs. By setting

p \equiv 1 (mod 4)

, we constructed an infinite family of linear codes using two distinct weakly regular unbalanced (and balanced) plateaued functions with index

(p - 1) / 2

. Their weight distributions were completely determined by applying exponential sums and Walsh transform. As a result, most of our constructed codes have a few nonzero weights and are minimal. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

32 pages, 440 KiB

Open AccessArticle

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces

by Zainab Alsheekhhussain, Ahmed Gamal Ibrahim, M. Mossa Al-Sawalha and Khudhayr A. Rashedi

Fractal Fract. 2024, 8(5), 289; https://doi.org/10.3390/fractalfract8050289 - 13 May 2024

Cited by 2 | Viewed by 1554

Abstract

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted,

Φ

-Hilfer, fractional derivative of order

μ \in (1, 2)

with non-instantaneous impulses in Banach spaces [...] Read more.

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted,

Φ

-Hilfer, fractional derivative of order

μ \in (1, 2)

with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted

Φ

-Laplace transform, w-weighted

ψ

-convolution and the measure of non-compactness. Since the operator, the w-weighted

Φ

-Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results. Full article

12 pages, 387 KiB

Open AccessArticle

On a Family of Hamilton–Poisson Jerk Systems

by Cristian Lăzureanu and Jinyoung Cho

Mathematics 2024, 12(8), 1260; https://doi.org/10.3390/math12081260 - 22 Apr 2024

Cited by 2 | Viewed by 906

Abstract

In this paper, we construct a family of Hamilton–Poisson jerk systems. We show that such a system has infinitely many Hamilton–Poisson realizations. In addition, we discuss the stability and we prove the existence of periodic orbits around nonlinearly stable equilibrium points. Particularly, we [...] Read more.

In this paper, we construct a family of Hamilton–Poisson jerk systems. We show that such a system has infinitely many Hamilton–Poisson realizations. In addition, we discuss the stability and we prove the existence of periodic orbits around nonlinearly stable equilibrium points. Particularly, we deduce conditions for the existence of homoclinic and heteroclinic orbits. We apply the obtained results to a family of anharmonic oscillators. Full article

(This article belongs to the Special Issue Advances in Differential Dynamical Systems with Applications to Economics and Biology, 2nd Edition)

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21 pages, 527 KiB

Open AccessFeature PaperArticle

Bipartite Unique Neighbour Expanders via Ramanujan Graphs

by Ron Asherov and Irit Dinur

Entropy 2024, 26(4), 348; https://doi.org/10.3390/e26040348 - 20 Apr 2024

Cited by 6 | Viewed by 2300

Abstract

We construct an infinite family of bounded-degree bipartite unique neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may be closer to being implementable in practice, due to the [...] Read more.

We construct an infinite family of bounded-degree bipartite unique neighbour expander graphs with arbitrarily unbalanced sides. Although weaker than the lossless expanders constructed by Capalbo et al., our construction is simpler and may be closer to being implementable in practice, due to the smaller constants. We construct these graphs by composing bipartite Ramanujan graphs with a fixed-size gadget in a way that generalises the construction of unique neighbour expanders by Alon and Capalbo. For the analysis of our construction, we prove a strong upper bound on average degrees in small induced subgraphs of bipartite Ramanujan graphs. Our bound generalises Kahale’s average degree bound to bipartite Ramanujan graphs, and may be of independent interest. Surprisingly, our bound strongly relies on the exact Ramanujan-ness of the graph and is not known to hold for nearly-Ramanujan graphs. Full article

(This article belongs to the Special Issue Extremal and Additive Combinatorial Aspects in Information Theory)

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7 pages, 231 KiB

Open AccessArticle

On Fall-Colorable Graphs

by Shaojun Wang, Fei Wen, Guoxing Wang and Zepeng Li

Mathematics 2024, 12(7), 1105; https://doi.org/10.3390/math12071105 - 7 Apr 2024

Viewed by 1108

Abstract

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to [...] Read more.

A fall k-coloring of a graph G is a proper k-coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k-coloring is equivalent to having a partition of the vertex set

V (G)

in k independent dominating sets. In this paper, we first prove that for any fall k-colorable graph G with order n, the number of edges of G is at least

(n (k - 1) + r (k - r)) / 2

, where

r \equiv n (mod k)

and

0 \leq r \leq k - 1

, and the bound is tight. Then, we obtain that if G is k-colorable (

k \geq 2

) and the minimum degree of G is at least

\frac{k - 2}{k - 1} n

, then G is fall k-colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where

k \geq 3

. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for

k \in {5, 6}

. Full article

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