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17 pages, 374 KiB

Open AccessArticle

Construction of Inequalities for Network Quantum Steering Detection

by Jia Ji and Kan He

Axioms 2025, 14(8), 615; https://doi.org/10.3390/axioms14080615 - 7 Aug 2025

Quantum network correlations are crucial for long-distance quantum communication, quantum cryptography, and distributed quantum computing. Detecting network steering is particularly challenging in complex network structures. We have studied the steering inequality criteria for a 2-forked 3-layer tree-shaped network. Assuming the first and third [...] Read more.

Quantum network correlations are crucial for long-distance quantum communication, quantum cryptography, and distributed quantum computing. Detecting network steering is particularly challenging in complex network structures. We have studied the steering inequality criteria for a 2-forked 3-layer tree-shaped network. Assuming the first and third layers are trusted and the second layer is untrusted, we derived a steering inequality criterion using the correlation matrix between trusted and untrusted observables. In particular, we apply the steering criterion to three classes of measurements which are of special significance: local orthogonal observables, mutually unbiased measurements, and general symmetric informationally complete measurements. We further illustrate the effectiveness of our method through an example. Full article

(This article belongs to the Special Issue Mathematical Foundations of Quantum Computing)

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9 pages, 236 KiB

Open AccessArticle

Full Automorphism Group of (m,2)-Graph in Finite Classical Polar Spaces

by Yang Zhang, Shuxia Liu and Liwei Zeng

Axioms 2025, 14(8), 614; https://doi.org/10.3390/axioms14080614 - 6 Aug 2025

Abstract

Let \( \mathcal{Q} \) be the finite classical polar space of rank \( \nu\geq 1 \) over \( \mathbb{F}_q \), and \( \mathcal{Q}_m \) be the set of all m-dimensional subspaces of \( \mathcal{Q} \). In this paper, we introduce the \( [...] Read more.

Let \( \mathcal{Q} \) be the finite classical polar space of rank \( \nu\geq 1 \) over \( \mathbb{F}_q \), and \( \mathcal{Q}_m \) be the set of all m-dimensional subspaces of \( \mathcal{Q} \). In this paper, we introduce the \( (m,2) \)-graph with \( \mathcal{Q}_m \) as its vertex set, and two vertices \(P,Q\) are adjacent if and only if \( P+Q \) is an \( (m+2) \)-dimensional subspace of \( \mathcal{Q} \). The full automorphism group of \( (m,2)\)-graph is determined. Full article

22 pages, 481 KiB

Open AccessArticle

Fuzzy Signature from Computational Diffie–Hellman Assumption in the Standard Model

by Yunhua Wen, Tianlong Jin and Wei Li

Axioms 2025, 14(8), 613; https://doi.org/10.3390/axioms14080613 - 6 Aug 2025

Abstract

Fuzzy signature (

{SIG}_{F}

) is a type of digital signature that preserves the core functionalities of traditional signatures, while accommodating variations and non-uniformity in the signing key. This property enables the direct use of high-entropy fuzzy data, such as biometric information, [...] Read more.

Fuzzy signature (

{SIG}_{F}

) is a type of digital signature that preserves the core functionalities of traditional signatures, while accommodating variations and non-uniformity in the signing key. This property enables the direct use of high-entropy fuzzy data, such as biometric information, as the signing key. In this paper, we define the m-existentially unforgeable under chosen message attack (

m {- EUF - CMA}^{*}

) security of fuzzy signature. Furthermore, we propose a generic construction of fuzzy signature, which is composed of a homomorphic secure sketch (

SS

) with an error-recoverable property, a homomorphic average-case strong extractor (

Ext

), and a homomorphic and key-shift* secure signature scheme (

SIG

). By instantiating the foundational components, we present a

m {- EUF - CMA}^{*}

secure fuzzy signature instantiation based on the Computational Diffie–Hellman (CDH) assumption over bilinear groups in the standard model. Full article

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20 pages, 356 KiB

Open AccessArticle

Parabolic and Linear Rotational Motions on Cones and Hyperboloids

by Harun Barış Çolakoğlu, Mehmet Duru and Ayşe Yılmaz Ceylan

Axioms 2025, 14(8), 612; https://doi.org/10.3390/axioms14080612 - 6 Aug 2025

Abstract

In this study, we consider the Lorentzian rotation about a lightlike axis. First, we introduce a geometric characterization for the rotation angle between two vectors that can overlap each other under a Lorentzian rotation about a lightlike axis. Then, we give a definition [...] Read more.

In this study, we consider the Lorentzian rotation about a lightlike axis. First, we introduce a geometric characterization for the rotation angle between two vectors that can overlap each other under a Lorentzian rotation about a lightlike axis. Then, we give a definition for the angle measurement between two spacelike vectors whose vector product is lightlike. Later, we generalize the Lorentzian rotation about a lightlike axis, and determine matrices of these transformations using the Cartan frame and the well-known Rodrigues formula, then using the Cayley map, and finally using the generalized split quaternions. We see that such transformations give parabolic rotational motions on general cones or general hyperboloids of one or two sheets, while they also give linear rotational motions on general hyperboloids of one sheet. Full article

(This article belongs to the Section Geometry and Topology)

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23 pages, 311 KiB

Open AccessArticle

Some Results on Maxima and Minima of Real Functions of Vector Variables: A New Perspective

by Bibiano Martin Cerna Maguiña, Dik Dani Lujerio Garcia, Victor Pocoy Yauri, Vladimir Giovanni Rodriguez Sabino and Ruben Mario Leiva Bernuy

Axioms 2025, 14(8), 611; https://doi.org/10.3390/axioms14080611 - 6 Aug 2025

Abstract

This article presents a method for proving several theorems that enable the determination of the maxima and minima of certain classes of real-valued functions with vector variables, without relying on the classical theory based on partial derivatives. However, the main objective of this [...] Read more.

This article presents a method for proving several theorems that enable the determination of the maxima and minima of certain classes of real-valued functions with vector variables, without relying on the classical theory based on partial derivatives. However, the main objective of this work is not to compare this approach with existing methods but, rather, to extend the study of extrema in real functions of vector variables that are not differentiable, as illustrated in Example 1. Each theorem is accompanied by various examples that demonstrate its applicability. The results are based on Theorems 1 and 2, as well as the selection of an appropriate connection between Theorem 2 and the functions to be optimized. Additionally, definitions related to the hierarchy between variables within a given domain are introduced, providing the theoretical framework necessary for the development of the proposed results. Full article

25 pages, 4865 KiB

Open AccessArticle

Mathematical Modeling, Bifurcation Theory, and Chaos in a Dusty Plasma System with Generalized (r^⋆, q^⋆) Distributions

by Beenish, Maria Samreen and Fehaid Salem Alshammari

Axioms 2025, 14(8), 610; https://doi.org/10.3390/axioms14080610 - 5 Aug 2025

Abstract

This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized

(r^{⋆}, q^{⋆})

distributions. First, boundary conditions are applied to reduce the model to a second-order nonlinear ordinary differential equation. [...] Read more.

This study investigates the dynamics of dust acoustic periodic waves in a three-component, unmagnetized dusty plasma system using generalized

(r^{⋆}, q^{⋆})

distributions. First, boundary conditions are applied to reduce the model to a second-order nonlinear ordinary differential equation. The Galilean transformation is subsequently applied to reformulate the second-order ordinary differential equation into an unperturbed dynamical system. Next, phase portraits of the system are examined under all possible conditions of the discriminant of the associated cubic polynomial, identifying regions of stability and instability. The Runge–Kutta method is employed to construct the phase portraits of the system. The Hamiltonian function of the unperturbed system is subsequently derived and used to analyze energy levels and verify the phase portraits. Under the influence of an external periodic perturbation, the quasi-periodic and chaotic dynamics of dust ion acoustic waves are explored. Chaos detection tools confirm the presence of quasi-periodic and chaotic patterns using Basin of attraction, Lyapunov exponents, Fractal Dimension, Bifurcation diagram, Poincaré map, Time analysis, Multi-stability analysis, Chaotic attractor, Return map, Power spectrum, and 3D and 2D phase portraits. In addition, the model’s response to different initial conditions was examined through sensitivity analysis. Full article

(This article belongs to the Special Issue Trends in Dynamical Systems and Applied Mathematics)

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16 pages, 343 KiB

Open AccessArticle

Structured Distance to Normality of Dirichlet–Neumann Tridiagonal Toeplitz Matrices

by Zhaolin Jiang, Hongxiao Chu, Qiaoyun Miao and Ziwu Jiang

Axioms 2025, 14(8), 609; https://doi.org/10.3390/axioms14080609 - 5 Aug 2025

Abstract

This paper conducts a rigorous study on the spectral properties and operator-space distances of perturbed Dirichlet–Neumann tridiagonal (PDNT) Toeplitz matrices, with emphasis on their asymptotic behaviors. We establish explicit closed-form solutions for the eigenvalues and associated eigenvectors, highlighting their fundamental importance for characterizing [...] Read more.

This paper conducts a rigorous study on the spectral properties and operator-space distances of perturbed Dirichlet–Neumann tridiagonal (PDNT) Toeplitz matrices, with emphasis on their asymptotic behaviors. We establish explicit closed-form solutions for the eigenvalues and associated eigenvectors, highlighting their fundamental importance for characterizing matrix stability in the presence of perturbations. By exploiting the structural characteristics of PDNT Toeplitz matrices, we obtain closed-form expressions quantifying the distance to normality, the deviation from normality. Full article

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19 pages, 317 KiB

Open AccessFeature PaperArticle

New Class of Specific Functions with Fractional Derivatives

by Hatun Özlem Güney and Shigeyoshi Owa

Axioms 2025, 14(8), 608; https://doi.org/10.3390/axioms14080608 - 5 Aug 2025

Viewed by 53

Abstract

Let

A_{n}

be the class of specific analytic functions [...] Read more.

Let

A_{n}

be the class of specific analytic functions

f (z) = z + \sum_{k = 1}^{\infty} a_{1 + \frac{k}{n}} z^{1 + \frac{k}{n}} (n \in N = {1, 2, 3, \dots})

in the open unit disk

U .

For

f (z) \in A_{n},

fractional derivatives

D_{z}^{λ} f (z)

and

D_{z}^{j + λ} f (z)

(0 \leq λ < 1, j \in N)

are defined by using Gamma functions. Applying such fractional derivatives, we introduce a new subclass

A_{n} (j, λ, α, β)

of

A_{n} .

In this paper, we establish sufficient conditions for

f (z)

for

A_{n} (j, λ, α, β),

coefficient inequalities for

| a_{1 + \frac{1}{n}} |

and

| a_{1 + \frac{k}{n}} |

(k = 2, 3, 4, \dots)

of

f (z) \in A_{n} (j, λ, α, β),

and some interesting argument properties of fractional derivatives for

f (z) \in A_{n}

through an example. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Related Topics)

24 pages, 362 KiB

Open AccessArticle

Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5

by Anna Louise Skelt, Stephanie Perkins and Paul Alun Roach

Axioms 2025, 14(8), 607; https://doi.org/10.3390/axioms14080607 - 4 Aug 2025

Viewed by 122

Abstract

There has been much interest in the mathematical investigation of critical sets and unavoidable sets in Latin Squares, Sudoku, and their applications to practical problems in areas such as agriculture and cryptology. This paper considers the associated structures of Strictly Concentric Magic Squares [...] Read more.

There has been much interest in the mathematical investigation of critical sets and unavoidable sets in Latin Squares, Sudoku, and their applications to practical problems in areas such as agriculture and cryptology. This paper considers the associated structures of Strictly Concentric Magic Squares (SCMSs) and Prime Strictly Concentric Magic Squares (PSCMSs). A framework of formal definitions is given that leads to the definitions of critical sets and unavoidable sets. Minimal critical sets are of interest in Latin Squares, and in this article, the cardinality of minimal critical sets of SCMS is given for all n, n odd. Two families of unavoidable sets are established for SCMS, leading to a complete classification of unavoidable sets of minimum PSCMS of order 5. Full article

(This article belongs to the Section Algebra and Number Theory)

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20 pages, 390 KiB

Open AccessArticle

Injective Hulls of Infinite Totally Split-Decomposable Metric Spaces

by Maël Pavón

Axioms 2025, 14(8), 606; https://doi.org/10.3390/axioms14080606 - 4 Aug 2025

Viewed by 168

Abstract

We extend the theory of splits in finite metric spaces to infinite ones. Within this more general framework, we investigate the class of spaces having metrics that are integer-valued and totally split-decomposable, as well as the polyhedral complex structure of their injective hulls. [...] Read more.

We extend the theory of splits in finite metric spaces to infinite ones. Within this more general framework, we investigate the class of spaces having metrics that are integer-valued and totally split-decomposable, as well as the polyhedral complex structure of their injective hulls. For this class, we provide a characterization for the injective hull to be combinatorially equivalent to a CAT(0) cube complex. Intermediate results include the generalization of the decomposition theory introduced by Bandelt and Dress in 1992 as well as results on the tight span of totally split-decomposable metric spaces proved by Huber, Koolen, and Moulton in 2006. Next, using results of Lang from 2013, we obtain proper actions on CAT(0) cube complexes for finitely generated groups endowed with a totally split-decomposable word metric and for which the associated splits satisfy a simple combinatorial property. In the case of Gromov hyperbolic groups, the obtained action is both proper aand co-compact. Finally, we obtain as an application that injective hulls of odd cycles are cell complexes isomorphic to CAT(0) cube complexes. Full article

(This article belongs to the Section Geometry and Topology)

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Graphical abstract

17 pages, 2693 KiB

Open AccessArticle

Mitigating the Drawbacks of the L₀ Norm and the Total Variation Norm

by Gengsheng L. Zeng

Axioms 2025, 14(8), 605; https://doi.org/10.3390/axioms14080605 - 4 Aug 2025

Viewed by 144

Abstract

In compressed sensing, it is believed that the L₀ norm minimization is the best way to enforce a sparse solution. However, the L₀ norm is difficult to implement in a gradient-based iterative image reconstruction algorithm. The total variation (TV) norm minimization [...] Read more.

In compressed sensing, it is believed that the L₀ norm minimization is the best way to enforce a sparse solution. However, the L₀ norm is difficult to implement in a gradient-based iterative image reconstruction algorithm. The total variation (TV) norm minimization is considered a proper substitute for the L₀ norm minimization. This paper points out that the TV norm is not powerful enough to enforce a piecewise-constant image. This paper uses the limited-angle tomography to illustrate the possibility of using the L₀ norm to encourage a piecewise-constant image. However, one of the drawbacks of the L₀ norm is that its derivative is zero almost everywhere, making a gradient-based algorithm useless. Our novel idea is to replace the zero value of the L₀ norm derivative with a zero-mean random variable. Computer simulations show that the proposed L₀ norm minimization outperforms the TV minimization. The novelty of this paper is the introduction of some randomness in the gradient of the objective function when the gradient is zero. The quantitative evaluations indicate the improvements of the proposed method in terms of the structural similarity (SSIM) and the peak signal-to-noise ratio (PSNR). Full article

(This article belongs to the Special Issue Advances in Mathematical Methods in Signal Processing and Its Applications)

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14 pages, 403 KiB

Open AccessArticle

An Inexact Nonsmooth Quadratic Regularization Algorithm

by Anliang Wang, Xiangmei Wang and Chunfang Liao

Axioms 2025, 14(8), 604; https://doi.org/10.3390/axioms14080604 - 4 Aug 2025

Viewed by 120

Abstract

The quadratic regularization technique is widely used in the literature for constructing efficient algorithms, particularly for solving nonsmooth optimization problems. We propose an inexact nonsmooth quadratic regularization algorithm for solving large-scale optimization, which involves a large-scale smooth separable item and a nonsmooth one. [...] Read more.

The quadratic regularization technique is widely used in the literature for constructing efficient algorithms, particularly for solving nonsmooth optimization problems. We propose an inexact nonsmooth quadratic regularization algorithm for solving large-scale optimization, which involves a large-scale smooth separable item and a nonsmooth one. The main difference between our algorithm and the (exact) quadratic regularization algorithm is that it employs inexact gradients instead of the full gradients of the smooth item. Also, a slightly different update rule for the regularization parameters is adopted for easier implementation. Under certain assumptions, it is proved that the algorithm achieves a first-order approximate critical point of the problem, and the iteration complexity of the algorithm is

O (ε^{- 2})

. In the end, we apply the algorithm to solve LASSO problems. The numerical results show that the inexact algorithm is more efficient than the corresponding exact one in large-scale cases. Full article

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16 pages, 281 KiB

Open AccessArticle

Existence and Uniqueness of Solutions for Impulsive Stochastic Differential Variational Inequalities with Applications

by Wei Liu and Kui Liu

Axioms 2025, 14(8), 603; https://doi.org/10.3390/axioms14080603 - 3 Aug 2025

Viewed by 222

Abstract

This paper focuses on exploring an impulsive stochastic differential variational inequality (ISDVI), which combines an impulsive stochastic differential equation and a stochastic variational inequality. Innovatively, our work incorporates two key aspects: first, our stochastic differential equation contains an impulsive term, enabling better handling [...] Read more.

This paper focuses on exploring an impulsive stochastic differential variational inequality (ISDVI), which combines an impulsive stochastic differential equation and a stochastic variational inequality. Innovatively, our work incorporates two key aspects: first, our stochastic differential equation contains an impulsive term, enabling better handling of sudden event impacts; second, we utilize a non-local condition

z (0) = χ_{0} + ϑ (z)

that integrates measurements from multiple locations to construct superior models. Methodologically, we commence our analysis by using the projection method, which ensures the existence and uniqueness of the solution to ISDVI. Subsequently, we showcase the practical applicability of our theoretical findings by implementing them in the investigation of a stochastic consumption process and electrical circuit model. Full article

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22 pages, 463 KiB

Open AccessArticle

Improved Bounds for Integral Jensen’s Inequality Through Fifth-Order Differentiable Convex Functions and Applications

by Sidra Nisar, Fiza Zafar and Hind Alamri

Axioms 2025, 14(8), 602; https://doi.org/10.3390/axioms14080602 - 2 Aug 2025

Viewed by 239

Abstract

The main objective of this research is to obtain interesting estimates for Jensen’s gap in the integral sense, along with their applications. The convexity of a fifth-order absolute function is used to established proposed estimates of Jensen’s gap. We performed numerical computations to [...] Read more.

The main objective of this research is to obtain interesting estimates for Jensen’s gap in the integral sense, along with their applications. The convexity of a fifth-order absolute function is used to established proposed estimates of Jensen’s gap. We performed numerical computations to compare our estimates with previous findings. With the use of the primary findings, we are able to obtain improvements of the Hölder inequality and Hermite–Hadamard inequality. Furthermore, the primary results lead to some inequalities for power means and quasi-arithmetic means. We conclude by outlining the information theory applications of our primary inequalities. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities, 2nd Edition)

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23 pages, 1480 KiB

Open AccessArticle

Operator Newton Method for Large-Scale Coupled Riccati Equations Arising from Jump Systems

by Bo Yu, Yiwen Liu and Ning Dong

Axioms 2025, 14(8), 601; https://doi.org/10.3390/axioms14080601 - 1 Aug 2025

Viewed by 187

Abstract

Consider a class of coupled discrete-time Riccati equations arising from jump systems. To compute their solutions when systems reach a steady state, we propose an operator Newton method and correspondingly establish its quadratic convergence under suitable assumptions. The advantage of the proposed method [...] Read more.

Consider a class of coupled discrete-time Riccati equations arising from jump systems. To compute their solutions when systems reach a steady state, we propose an operator Newton method and correspondingly establish its quadratic convergence under suitable assumptions. The advantage of the proposed method lies in the fact that its subproblems are solved using the operator Smith method, which allows it to maintain quadratic convergence in both the inner and outer iterations. Moreover, it does not require the constant term matrix of the equation to be invertible, making it more broadly applicable than existing inverse-free iterative methods. For large-scale problems, we develop a low-rank variant by incorporating truncation and compression techniques into the operator Newton framework. A complexity analysis is also provided to assess its scalability. Numerical experiments demonstrate that the presented low-rank operator Newton method is highly effective in approximating solutions to large-scale structured coupled Riccati equations. Full article

(This article belongs to the Special Issue Advances in Linear Algebra with Applications, 2nd Edition)

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