Journal Description

Axioms

Axioms is an international, peer-reviewed, open access journal of mathematics, mathematical logic and mathematical physics, published monthly online by MDPI. The International Fuzzy Systems Association (IFSA), Union of Slovak Mathematicians and Physicists (JSMF) and Eurekas Community are affiliated with Axioms and their members receive discounts on the article processing charges.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High visibility: indexed within SCIE (Web of Science), dblp, and other databases.
Journal Rank: JCR - Q2 (Mathematics, Applied)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 21.6 days after submission; acceptance to publication is undertaken in 2.9 days (median values for papers published in this journal in the first half of 2026).
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.
Companion journal: Logics.
Journal Cluster of Mathematics and Its Applications: AppliedMath, Axioms, Computation, Fractal and Fractional, Geometry, International Journal of Topology, Logics, Mathematics and Symmetry.

Impact Factor: 1.5 (2025); 5-Year Impact Factor: 1.5 (2025)

Imprint Information Journal Flyer Open Access ISSN: 2075-1680

Latest Articles

22 pages, 345 KB

Open AccessArticle

New Cryptanalysis of Two Families of RSA-like Schemes

by Brahim Chnioune, Mohammed Rahmani, Abderrahmane Nitaj and Mhammed Ziane

Axioms 2026, 15(7), 543; https://doi.org/10.3390/axioms15070543 (registering DOI) - 19 Jul 2026

Abstract

In certain variants of the RSA cryptosystem with a modulus

N = p q

, the public exponent e and the private exponent d are related by the equation

e d \equiv 1 (mod φ_{n} (N))

or [...] Read more.

In certain variants of the RSA cryptosystem with a modulus

N = p q

, the public exponent e and the private exponent d are related by the equation

e d \equiv 1 (mod φ_{n} (N))

e d \equiv 1 (mod ψ_{n} (N))

, where

φ_{n} (N) = (p^{n} - 1) (q^{n} - 1)

and

ψ_{n} (N) = \frac{(p^{n} - 1) (q^{n} - 1)}{(p - 1) (q - 1)}

are defined for a positive integer

n \geq 1

. In this paper, we introduce a new attack against the RSA variants when two public exponents

e_{1}

and

e_{2}

are given, satisfying

e_{i} d_{i} \equiv 1 (mod φ_{n} (N))

e_{i} d_{i} \equiv 1 (mod ψ_{n} (N))

for

i = 1, 2

. Specifically, we show that when the prime factors of N share an amount of their least significant bits, and

d_{1}

and

d_{2}

share an amount of their most significant bits, the factorization of N can be computed in polynomial time. Our method is based on an extension of Coppersmith’s technique and lattice basis reduction and achieves improved bounds as the number of shared bits increases. The proposed attacks are heuristic in nature, relying on the standard assumption that the polynomials obtained after lattice reduction are algebraically independent. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 3rd Edition)

30 pages, 691 KB

Open AccessArticle

Local-Time Sensitivity and Burst Instability for Threshold Functionals of One-Dimensional Diffusions

by Tristan Guillaume

Axioms 2026, 15(7), 542; https://doi.org/10.3390/axioms15070542 (registering DOI) - 19 Jul 2026

Abstract

Let

X = {(X_{t})}_{0 \leq t \leq T}

be a real-valued continuous process. For a threshold a, the sub-threshold time set [...] Read more.

Let

X = {(X_{t})}_{0 \leq t \leq T}

be a real-valued continuous process. For a threshold a, the sub-threshold time set

E_{T} (a) = {t \in [0, T] : X_{t} \leq a}

encodes several different threshold observables. The most elementary one is the cumulative occupation time

A_{T} (a) = \int_{0}^{T} 1_{{X_{t} \leq a}} d t .

For a regular one-dimensional diffusion, the classical occupation density formula gives

A_{T} (a) = \int_{- \infty}^{a} \frac{L_{T}^{y} (X)}{σ^{2} (y)} d y,

and hence

\frac{\partial A_{T}}{\partial a} (a) = \frac{L_{T}^{a} (X)}{σ^{2} (a)} .

Thus additive threshold occupation admits a local-time sensitivity calculus. In the terminology of barrier contracts, this additive clock is the cumulative, non-resetting Parisian clock, also called the Parasian clock. The purpose of this paper is to contrast this additive/Parasian regime with the behavior of resetting Parisian burst functionals. The connected components of

E_{T} (a)

represent sub-threshold episodes. We study in particular the longest burst

M_{T} (a) = sup {| I | : I is a connected component of E_{T} (a)} .

While

A_{T}

is locally controlled by local time,

M_{T}

is governed by the connectivity of the sub-threshold time set. We prove that

M_{T}

is monotone, that its supremum is attained, and that the weak-sublevel version is right-continuous with left limits, while the strict-sublevel version is its left-continuous regularization. The jump at a level is the increase in the maximal connected-component length produced by adjoining the level set. This gives a deterministic càdlàg/càglàd calculus for longest-burst profiles. For regular one-dimensional diffusions, this yields a sharp structural contrast. At deterministic levels which are almost surely not local-extreme values, the weak and strict longest bursts agree almost surely. Whenever the path has a unique interior maximum, the level-indexed longest-burst profile has a positive jump at the maximum level and is therefore not absolutely continuous. Brownian motion satisfies this criterion almost surely. We further identify the deterministic mechanism behind this instability: small threshold increases may fill short temporal bridges and merge large sub-threshold components. Finally, we show that the longest burst is exactly a one-sided continuous Parisian functional. This yields an exact Laplace-transform representation of its Brownian law through the Chesney–Jeanblanc-Picqué–Yor Parisian transform, and an excursion-measure formulation in which local time enters only as the Itô excursion intensity. We also discuss smoothed burst statistics, moving thresholds, and diffusion examples. The paper is intended as a threshold-sensitivity comparison: local time controls cumulative Parasian occupation, whereas resetting Parisian burst observables are controlled by component mergers and excursion structure. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

24 pages, 350 KB

Open AccessArticle

Weighted Higher-Order Beesack–Opial Inequalities on Time Scales

by Ramy R. Mahmoud, Samir H. Saker, Douglas R. Anderson and Khadega R. Abdo

Axioms 2026, 15(7), 541; https://doi.org/10.3390/axioms15070541 (registering DOI) - 19 Jul 2026

Abstract

This work constructs a family of weighted higher-order Opial-type estimates on an arbitrary time scale

T

, with the weight structure generated by a non-decreasing auxiliary function

ω

and its delta derivative

ω^{Δ}

. For n-times delta differentiable functions whose delta [...] Read more.

This work constructs a family of weighted higher-order Opial-type estimates on an arbitrary time scale

T

, with the weight structure generated by a non-decreasing auxiliary function

ω

and its delta derivative

ω^{Δ}

. For n-times delta differentiable functions whose delta derivatives of orders

0, 1, \dots, n - 1

vanish at the left endpoint, the mixed functional

\int_{a}^{b} {v (t) | u (t) |}^{p} {| u^{Δ^{n}} (t) |}^{q} Δ t

is bounded by expressions involving only the highest-order delta derivative

u^{Δ^{n}}

. Three forms are obtained: a Hölder-type product estimate, a Young-type two-term estimate with a free balancing parameter

θ > 0

, and an optimized single-integral estimate with the explicit constant

K = {(r - 1)}^{1 - 1 / r} / r

, where

r = (p + q) / q

. The proofs rely on the Taylor representation on time scales, Hölder’s inequality, Jensen’s inequality, and Fubini’s theorem. The resulting bounds are governed by computable kernels that encode the interaction between the Taylor monomials, the auxiliary weight, and the external weight. Specializations to

T = R

T = Z

, and the quantum lattice

q_{0}^{N_{0}}

recover classical, discrete, weighted, and quantum Opial inequalities with their standard constants and also yield several higher-order weighted estimates. As an application, a uniqueness criterion for higher-order dynamic initial value problems is established. Full article

(This article belongs to the Section Mathematical Analysis)

17 pages, 286 KB

Open AccessArticle

Some Properties of Noncompact Ricci Solitons

by Hana Al-Sodais and Sharief Deshmukh

Axioms 2026, 15(7), 540; https://doi.org/10.3390/axioms15070540 (registering DOI) - 18 Jul 2026

Abstract

In Hamilton’s quest to resolve the Poincare conjecture, he introduced the Ricci flow on an n-dimensional Riemannian manifold

(M, h)

as the heat equation (cf [...] Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 3rd Edition)

22 pages, 443 KB

Open AccessArticle

Exponential Decay Rate of a One-Dimensional Tree-Structured Timoshenko Beam System with Boundary Feedbacks

by Yaru Xie, Mengjiao Gao and Yanfang Li

Axioms 2026, 15(7), 539; https://doi.org/10.3390/axioms15070539 (registering DOI) - 18 Jul 2026

Abstract

In this paper, we investigate a one-dimensional tree-structured Timoshenko beam system composed of three beams with boundary feedback controls and derive its explicit exponential decay rate. We first formulate a dynamic model incorporating end loads, boundary feedback, and multi-beam coupling relationships. Then, by applying Lyapunov’s method, we construct an energy functional suitable for this coupled system. This study not only provides rigorous proof of the exponential stability of the coupled Timoshenko beam system, but also yields an explicit decay rate under prescribed conditions. Our results offer theoretical support and a quantitative foundation for vibration suppression of multi-beam-coupled flexible structures. Full article

38 pages, 1581 KB

Open AccessArticle

Dynamics and Density Function of a Stochastic Tuberculosis Model of Home Treatment with the Ornstein–Uhlenbeck Process

by Zhenwen Liu, Daqing Jiang and Zengchao Wu

Axioms 2026, 15(7), 538; https://doi.org/10.3390/axioms15070538 (registering DOI) - 17 Jul 2026

Abstract

In this paper, we consider a randomized TB model with two treatment modalities, home treatment and hospital treatment, in which

β_{1}

and

β_{2}

satisfy the log-normal Ornstein–Uhlenbeck process. First, we prove the local stability of the positive equilibrium point, and in [...] Read more.

In this paper, we consider a randomized TB model with two treatment modalities, home treatment and hospital treatment, in which

β_{1}

and

β_{2}

satisfy the log-normal Ornstein–Uhlenbeck process. First, we prove the local stability of the positive equilibrium point, and in order to analyze the dynamic properties of the stochastic model, we prove the existence and uniqueness of the global positive solution for any initial value. In the persistent case, by constructing a suitable random Lyapunov function, we establish sufficient conditions for the existence of the stationary distribution of the positive solution of the model. Then, after proving the existence of stationary distribution, we investigate the solution of the corresponding local normal density function of the stochastic system near the quasi-equilibrium point and get the conditions for disease extinction. Finally, numerical simulations are used to verify the results we obtain. Full article

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22 pages, 324 KB

Open AccessArticle

Nonlinear Left η-∗-Jordan n-Derivations on ∗-Algebras

by Shengsheng Wu, Quanyuan Chen and Rundong Zheng

Axioms 2026, 15(7), 537; https://doi.org/10.3390/axioms15070537 (registering DOI) - 17 Jul 2026

Abstract

Let

A

be a ∗-algebra with the unit I and a nontrivial projection P. Assume that

A

satisfies the condition that

X A P = 0

implies

X = 0

and

X A (I - P) = 0

implies [...] Read more.

Let

A

be a ∗-algebra with the unit I and a nontrivial projection P. Assume that

A

satisfies the condition that

X A P = 0

implies

X = 0

and

X A (I - P) = 0

implies

X = 0

. Let

η

be a nonzero complex number. For any

A, B \in A

, the left

η

-Jordan ∗-product is defined by

A ◊_{η} B = A^{*} B + η B^{*} A

. In this study, it is shown that if a map

φ : A \to A

satisfies

φ (A_{1} ◊_{η} A_{2} ◊_{η} \dots ◊_{η} A_{n}) = \sum_{k = 1}^{n} A_{1} ◊_{η} \dots ◊_{η} A_{k - 1} ◊_{η} φ (A_{k}) ◊_{η} A_{k + 1} ◊_{η} \dots ◊_{η} A_{n} (n \geq 3)

for all

A_{1}, A_{2}, \dots, A_{n} \in A

, then

φ

is an additive ∗-derivation and

φ (η A) = η φ (A)

for all

A \in A .

Full article

(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)

28 pages, 6945 KB

Open AccessArticle

Emergent Geometry in Complex Systems from Relational Dynamics: The Classical Setting

by Ernesto Estrada

Axioms 2026, 15(7), 536; https://doi.org/10.3390/axioms15070536 - 17 Jul 2026

Abstract

Geometry is typically treated as a primitive background structure within which physical processes unfold. In this work, we investigate an alternative possibility: that geometry can arise as a structural consequence of relational dynamics. We study the framework of Graph Dynamical Geometrization (GDG), in which a system is specified by a relational structure encoded by a graph, together with a dynamical law acting on that structure. The dynamics define a symmetric operator whose spectral evolution generates a positive–definite kernel. From this kernel, one obtains a squared Euclidean distance matrix that induces a canonical embedding of the relational system into a Euclidean space. Therefore geometry appears not as an assumed background but as a structure generated by the dynamics of relations. This construction motivates a philosophical interpretation that we call relational–dynamical structural realism, according to which the fundamental ontology consists of entities connected by relations and governed by a dynamical law, while geometric structure emerges from the spectral organization of those dynamics. We analyze the status of the induced geometry within several forms of structural realism. The GDG framework shows that invariant metric relations generated by relational dynamics can possess explanatory and predictive significance. More generally, the results suggest a structural principle: geometry may emerge whenever relational dynamics admit a spectral articulation. Full article

(This article belongs to the Section Mathematical Physics)

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3 pages, 149 KB

Open AccessEditorial

Recent Advances in Complex Analysis and Related Topics

by Andriy Bandura

Axioms 2026, 15(7), 535; https://doi.org/10.3390/axioms15070535 - 17 Jul 2026

Abstract

In this Editorial, we introduce the Special Issue of Axioms titled “Recent Advances in Complex Analysis and Related Topics [...] Full article

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

29 pages, 735 KB

Open AccessArticle

A Semiparametric Empirical Likelihood Estimation Method for Mixed Integer-Valued Autoregressive Model

by Qi Li, Xiaolei Zhang, Jianhua Cheng and Dehui Wang

Axioms 2026, 15(7), 534; https://doi.org/10.3390/axioms15070534 - 16 Jul 2026

Abstract

This paper investigates a first-order mixed integer-valued autoregressive model based on the binomial thinning operator. The proposed MINAR(1) model allows the count process to evolve under heterogeneous autoregressive mechanisms and therefore provides greater flexibility for modeling count time series with structural heterogeneity. To estimate the unknown parameters, a maximum empirical likelihood estimation (MEL) procedure is developed using conditional moment restrictions. Compared with conventional likelihood-based methods, the proposed approach does not require a full specification of the innovation distribution and is therefore more robust to potential distributional misspecification. The consistency and asymptotic normality of the proposed MEL estimator are established under suitable regularity conditions. Simulation studies demonstrate that the MEL estimator performs satisfactorily in finite samples and exhibits competitive estimation accuracy compared with the maximum likelihood estimation (MLE). Finally, an application to monthly insurance claim counts illustrates the practical usefulness of the proposed method. Full article

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34 pages, 11420 KB

Open AccessArticle

Algebraically Stabilizing Blocks for Quantized and Finite-Precision Neural Networks

by Kostadin Yotov, Emil Hadzhikolev and Stanka Hadzhikoleva

Axioms 2026, 15(7), 533; https://doi.org/10.3390/axioms15070533 - 16 Jul 2026

Abstract

This paper proposes a construction of algebraically stabilizing blocks for quantized and finite-precision neural networks. The approach is based on linear transformations defined by integer-valued matrices satisfying a condition of the form

W^{k} = I + μ D

, which specifies algebraically [...] Read more.

W^{k} = I + μ D

, which specifies algebraically controlled

k

-step behavior and modular periodicity in the integer-valued setting. The proposed property is invariant under conjugation, allowing the stabilizing construction to be transferred across equivalent linear representations. The resulting module can be integrated locally into existing neural architectures without requiring the algebraic structure to be imposed globally. The theoretical results concern algebraic and modular stabilization of the linear block and do not constitute a general guarantee of classical spectral or asymptotic stability in real-valued space. The approach is evaluated experimentally under multi-component harmonic, impulsive, and noisy inputs in both floating-point and INT8-quantized settings. Across several experimental configurations, the proposed block reduces output energy, component-wise variation, selected amplitude-related measures, and finite-precision deviation relative to floating-point reference trajectories. Norm-matched control experiments further suggest that the observed effects are not attributable solely to a reduction in operator magnitude, but may also reflect structural properties of the algebraically constructed operator. The proposed construction is particularly relevant to neural systems operating under limited numerical precision, including FPGA-, ASIC-, and edge-oriented implementations. It provides a structural approach for incorporating formally specified algebraic properties into the design of neural-network architectures. Full article

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

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21 pages, 1641 KB

Open AccessArticle

The Unit Extended Exponential Distribution: Properties, Inference and Applications

by Katherine A. Gaete, Yolanda M. Gómez, Inmaculada Barranco-Chamorro, Francisco A. Segovia and Héctor W. Gómez

Axioms 2026, 15(7), 532; https://doi.org/10.3390/axioms15070532 - 15 Jul 2026

Abstract

In this paper, we introduce a new probability distribution defined on the unit interval

(0, 1)

, called the unit extended exponential (UEE) distribution. The proposed model is obtained through a transformation of a random variable following the extended exponential [...] Read more.

In this paper, we introduce a new probability distribution defined on the unit interval

(0, 1)

, called the unit extended exponential (UEE) distribution. The proposed model is obtained through a transformation of a random variable following the extended exponential distribution, leading to a flexible family able to describe skewed and boundary-concentrated data. The structural properties of this distribution are derived, including its density, cumulative distribution, survival, hazard rate and quantile function, along with the mode and moments. A characterization based on its hazard function is also presented. Parameter estimation is performed via maximum likelihood. Under standard regularity conditions, the asymptotic properties of the estimators are established. A Monte Carlo simulation study is carried out to assess the finite-sample performance of the estimators. Finally, the usefulness of the model is illustrated through two applications to real proportion data, where our proposal is compared with competing models such as the unit-Lindley and Beta distributions using information criteria. The results suggest that the proposed distribution provides a competitive and often superior fit. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimations, 3rd Edition)

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24 pages, 310 KB

Open AccessArticle

The Dawson Transform and Its Operational Properties

by Osman Yürekli

Axioms 2026, 15(7), 531; https://doi.org/10.3390/axioms15070531 - 15 Jul 2026

Abstract

We introduce and study an integral transform whose kernel is Dawson’s integral. The proposed transform is connected to the Fourier sine transform through a classical integral representation of Dawson’s function and to the Glasser transform through an iteration formula. Basic operational properties are developed, including derivative identities, delta-derivative identities, Mellin-transform representations, and an inversion theorem. Several transform pairs are obtained for elementary functions, Gaussian functions, powers multiplied by trigonometric functions, and Bessel functions. The results place the Dawson transform within the broader theory of special-function integral transforms and Parseval–Goldstein-type identities. Full article

(This article belongs to the Section Mathematical Analysis)

18 pages, 487 KB

Open AccessArticle

A Practical Stabilization Controller Design for Positive Linear Time-Varying Systems with Multiple Time Delays

by Hongli Yang, Chunlan Gao and Ivan Ganchev Ivanov

Axioms 2026, 15(7), 530; https://doi.org/10.3390/axioms15070530 - 15 Jul 2026

Abstract

In this brief paper, we investigate the practical stabilization problem of multiple time-delayed positive linear time-varying systems. Firstly, a sufficient condition for the practical stability of the system is derived using the maximal separable Lyapunov–Krasovskii functional. Then, a state feedback controller is designed for the considered positive linear time-varying system with multiple time delays. Building on this method, the study further addresses the practical stabilization problem for positive linear time-varying systems with multiple state time delays and input time delays. The positivity of the linear time-varying system with input time delays is verified by designing a state feedback controller, and the practical stability is proven using the maximal separable Lyapunov–Krasovskii functional method. Finally, numerical examples are presented to validate the feasibility of the proposed method. Full article

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25 pages, 3188 KB

Open AccessArticle

Chaotic Motions in Linear/Nonlinear and Quasi-Periodical Dynamic Systems Revealed by Energy-Flow Investigations

by Jing Tang Xing and Wei Dai

Axioms 2026, 15(7), 529; https://doi.org/10.3390/axioms15070529 - 14 Jul 2026

Abstract

By using 1~2 degrees of freedom (DOF) examples, linear/nonlinear and quasi-periodical systems (QPS) are investigated based on the energy-flow theory (EFT), which reveals the following new findings on chaotic motions. (a) A 1-DOF linear non-damped-forced motion is chaotic if there is no least common multiple (LCM) for the periods of natural and force frequencies. (b) For a 1-DOF nonlinear non-damping-forced system with nonlinear stiffness of

γ x^{2}, (γ > 0),

both the free vibration by initial conditions and the composed motion with forced one excited by a force of given frequency are chaotic; while, when damping is added, the free vibration is damped to zero, but the forced vibration consists of different frequency components showing chaotic characteristics. (c) For a 1-DOF system of natural frequency 1 with nonlinear damping

2 ε (\dot{x}) = - 2 + \dot{x} + {\dot{x}}^{2}, ε (\dot{x}) < 0, {ε (\dot{x})}^{2} < 1,

the free vibration by the initial conditions

(x_{0} = 1, p_{0} = 0)

is along a non-repeating orbit towards a zero energy-flow limit circle of

\dot{E} = 0, p = - 1.

The tackled 1-DOF systems of no LCMs belong to a type of QPS, which implies that the orbit of a QPS in the phase space is also a non-closed curve, and based on the EFT, its motion is chaotic. The chaotic orbits of above 1-DOF cases in the phase space, as well as a 2-DOF free vibration case, are numerically examined to confirm the theoretical prediction. The findings may suggest that the LCMs could be an important factor in identifying chaotic motions. It is demonstrated that the generalised potential energy (GPE) and the energy-flow characteristic factors (EFCF), respectively, play a Lyapunov function and components to tackle the stabilities and chaotic motions of dynamic systems. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Numerical Modeling)

19 pages, 491 KB

Open AccessArticle

Geometric Properties of New Subclasses of the (λ, q)-Fractional Differential Operator of Analytic Functions Associated with Euler Generating Function

by Suha B. Al-Shaikh

Axioms 2026, 15(7), 528; https://doi.org/10.3390/axioms15070528 - 14 Jul 2026

Abstract

In this paper, we introduce and investigate a new subclass of analytic functions in the open unit disk by combining the

(λ, q)

-fractional differintegral operator with the Euler generating function through the technique of subordination. This class, denoted by [...] Read more.

In this paper, we introduce and investigate a new subclass of analytic functions in the open unit disk by combining the

(λ, q)

-fractional differintegral operator with the Euler generating function through the technique of subordination. This class, denoted by

K_{q, λ} (τ)

, unifies and extends several families of q-starlike and q-convex functions as special cases corresponding to particular choices of the parameter

λ

. For this class, coefficient estimates for the initial Taylor coefficients and a Fekete–Szegő inequality are established. Furthermore it is shown that the transformed function

D_{q}^{λ} f

is q-starlike in the unit disk under suitable assumptions on the parameters. In addition, first-order distortion estimates are obtained for the associated q-starlike and q-convex subclasses. Several consequences are derived for the special cases

λ = 0

and

λ = 1

, corresponding, respectively, to q-starlike and q-convex function classes. The classical limit

q \to 1^{-}

is also investigated and establish a new connection between Euler generating functions, fractional q-operators, and subclasses of analytic and univalent functions. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 4th Edition)

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42 pages, 1072 KB

Open AccessArticle

A Block Encryption Construction Based on Bi-Periodic Pell Matrices and Elliptic Curve Group Operations

by Ersen Akinci and Sukran Uygun

Axioms 2026, 15(7), 527; https://doi.org/10.3390/axioms15070527 - 14 Jul 2026

Abstract

This paper presents a theoretical algebraic block transformation based on bi-periodic Pell matrices and elliptic-curve point-valued blocks. The construction uses block-dependent scalar key matrices [...] Read more.

This paper presents a theoretical algebraic block transformation based on bi-periodic Pell matrices and elliptic-curve point-valued blocks. The construction uses block-dependent scalar key matrices

K_{i} = {\tilde{P}}_{m + i - 1} (a, b) \in G L_{2} (F_{q})

to transform plaintext blocks whose entries lie in a selected elliptic-curve subgroup

G = ⟨ G_{0} ⟩ \subseteq E (F_{p})

of prime order

q

. The encryption operation is defined by the elliptic-curve scalar matrix action

M_{i} = B_{i} ⋆ K_{i},

where the entries of

K_{i}

act as scalar coefficients modulo

q

, and point additions are performed in

G

. The main purpose of the construction is to formalize the interaction between recursive Pell-type matrix sequences over

F_{q}

and elliptic-curve point-valued block transformations. The algebraic correctness follows from the invertibility of

K_{i}

over

F_{q}

and the compatibility of the

⋆

-action with the matrix multiplication. The deterministic version is not claimed to provide semantic security or IND-CPA security without additional randomization, nonce-dependent masking, and formal security proofs. Full article

(This article belongs to the Special Issue Elliptic Curves, Modular Forms, L-Functions and Applications)

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18 pages, 322 KB

Open AccessArticle

Geometric Decomposition of Force and Yank for Variable-Mass Systems in Minkowski 3-Space

by Fatimah Alghamdi and Ayman Elsharkawy

Axioms 2026, 15(7), 526; https://doi.org/10.3390/axioms15070526 - 14 Jul 2026

Abstract

We develop a differential-geometric framework for variable-mass particles moving along non-lightlike curves with non-vanishing curvature in Minkowski 3-space

E_{1}^{3}

, employing the Frenet–Serret apparatus adapted to a Lorentzian signature. The force is defined as the time derivative of momentum, [...] Read more.

We develop a differential-geometric framework for variable-mass particles moving along non-lightlike curves with non-vanishing curvature in Minkowski 3-space

E_{1}^{3}

, employing the Frenet–Serret apparatus adapted to a Lorentzian signature. The force is defined as the time derivative of momentum,

F = d (m v) / d t

, incorporating mass variation through a Meshchersky-type reactive term; no covariant four-momentum formulation is assumed. Explicit closed-form expressions are derived for the momentum vector

P (t)

, force

F (t)

, and yank

Y (t) = d F / d t

for three distinct causal types of regular Frenet curves: spacelike curves with a spacelike principal normal, spacelike curves with a timelike principal normal, and timelike curves. The tangential yank component carries the causal sign factor

δ_{B}

, reflecting the type of curve. A theorem on the evolution of kinetic energy separates the inertial contribution

m v \dot{v}

from the reactive contribution

\frac{1}{2} \dot{m} v^{2}

due to mass variation. A radial decomposition of the force in the osculating plane generalizes Siacci’s classical theorem to Lorentzian geometry and variable-mass systems. When the rectifying coordinate b is non-zero, a corresponding decomposition of the yank is also obtained. Three illustrative physical scenarios are discussed: rocket motion with variable mass (with potential future relevance to trajectory prediction, stability analysis, and motion-anomaly assessment in unmanned systems), a geometric analogy for orbital parameter changes, and particle motion in a magnetic monopole field. Two fully worked examples (a Lorentzian helix and a logarithmic spiral) provide explicit closed-form expressions for all geometric and dynamical quantities, accompanied by numerical plots. The results recover the Euclidean case in the appropriate signature limit. Full article

(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology, 2nd Edition)

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24 pages, 379 KB

Open AccessArticle

Jacobi–Sobolev Orthogonal Polynomials, Differential Properties and Structural Formulas

by Héctor Pijeira-Cabrera, Javier Quintero-Roba and Juan Toribio-Milane

Axioms 2026, 15(7), 525; https://doi.org/10.3390/axioms15070525 - 13 Jul 2026

Abstract

In this paper, we extend some differential and structural results for monic Jacobi–Sobolev orthogonal polynomials, associated with a general discrete Sobolev inner product, with a Jacobi continuous part. We consider finitely many exterior mass points and a positive semidefinite Sobolev product matrix. Using Christoffel–Darboux kernels, we derive several structure and connection formulas involving two consecutive Jacobi and Jacobi–Sobolev polynomials. This representation leads to lowering and raising operators with rational coefficients; a second-order ordinary differential equation and a three-term recurrence relation, with polynomial coefficients. These coefficients depend on n. These results extend several classical structural properties of Jacobi polynomials to a general discrete Sobolev setting. Full article

(This article belongs to the Section Mathematical Analysis)

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Open AccessArticle

Model-Free Prediction of Multivariate Time Series

by Hanieh Saeidi and Adel Mohammadpour

Axioms 2026, 15(7), 524; https://doi.org/10.3390/axioms15070524 - 13 Jul 2026

Abstract

This paper extends a model-free prediction framework from univariate to multivariate time series. We show that, under a mild uniformly bounded first-moment condition, a multivariate time series admits a VARMA-type representation and an associated infinite-past linear solution. These results are obtained without assuming that the observed data are generated by a parametric VAR, VARMA, or other specifically postulated stochastic model. Motivated by this representation, we develop a practical forecasting method based on truncating the infinite-past solution and estimating the resulting coefficient matrices by multivariate least squares. The proposed procedure is simple to implement and is applicable to a broad class of stationary or nonstationary, linear or nonlinear multivariate time series when the required moment, identification, and empirical design conditions are plausible. We also describe recursive multi-step forecasting and give multivariate analogues of the main theoretical accuracy statements. The representation results require only a uniformly bounded first moment, whereas the forecast-accuracy results are stated under an explicit selected-solution identification assumption and additional empirical design conditions needed for multivariate least-squares estimation. Monte Carlo results, a real-data application, and additional sensitivity checks show that, in the lag-based implementation studied here, the method is empirically comparable to standard linear and regularized forecasting approaches, while not uniformly dominating them, and retains the conceptual advantage of a model-free formulation. Full article