Axioms doi: 10.3390/axioms15110832

Authors: Özlem Orhan Oya Mert Coşkun Yasemin Demirel

Scalar centrality measures assign one score to each sector of a production network and cannot compare its exposure to two shock sources. We develop a matrix-algebraic framework for this comparison. The Asymmetry Multiplier γj(α,β)=Ljα⊤/Ljβ⊤, the ratio of the price responses of a target sector to equal increments in unit value added at two sources, factors into a direct ratio and a network amplification ratio. Using sub-eigenvectors of the technical coefficient matrix, we derive lower and upper bounds that hold for every productive network and reduce to Perron–Frobenius bounds under irreducibility. We also prove that linear price responses bound those of competitive constant-returns economies from above and that no labeling-independent node-level measure determines the multiplier. A symmetric structural decomposition separates technological and final demand effects. As a proof of concept, the framework is applied to electricity and mining cost shocks in Türkiye. In 2022, the multiplier ranges from 0.20 to 1.86 across eleven sectors, indirect paths attenuate the direct ratio by a factor of 8 to 22 in six sectors, and Basic Metals is more exposed to mining than to electricity cost shocks, through imported mining inputs. For most sectors, the dominant channel is robust to disaggregation, coefficient perturbation and the annual tables of 2012–2022.

Axioms doi: 10.3390/axioms15110831

Authors: Branislav D. Vlahović Nenad O. Vesić Dušan J. Simjanović Branislav M. Randjelović

This study extends our previous investigations of (m¯,m)-conformal mappings. While the earlier studies were primarily theoretical and included only limited applications, the present study focuses on the applicability of these mappings to problems in mechanics. First, we derive the fundamental invariants associated with (m¯,m)-conformal mappings of two-dimensional manifolds. Based on these invariants, we formulate the corresponding Lagrangean and investigate its variational properties. By combining the resulting Euler–Lagrange equations with the transformation laws for the Christoffel symbols induced by transformations between manifolds, we derive the corresponding Einstein field equations within the proposed geometric framework. Subsequently, we analyze the transformation properties of the Einstein equations and investigate the influence of the relevant geometric areas on these transformations. Particular attention is given to the extent to which the resulting equations preserve the fundamental physical structure. Finally, the theoretical framework is illustrated through an application to the motion of a leaf. The corresponding velocity, acceleration, jerk, force, momentum, and mass are derived within the proposed model. This example serves to demonstrate that the fundamental laws of classical mechanics remain invariant under the considered transformations, thereby providing an illustrative validation of the physical consistency of the proposed geometric approach.

Axioms doi: 10.3390/axioms15110830

Authors: Al-Mehdi Mohamed Ibrahem Lotfi Tadj

The generalized-cμ (Gcμ) rule is asymptotically optimal for a single multiclass queue with convex delay costs, no setups and traffic intensity near one. We ask what it is worth when used only at the bottleneck of a job shop with sequence-dependent family setups. Without setups, its heavy-traffic cost ratio to FIFO has a closed form (0.66 for our shop), which simulation confirms. For a saturated two-class chain we derive the switching frequency exactly, with setups, and show that its ordering against FIFO can fail when setups are present. We then simulate a five-machine job shop from the literature, with three part families and one bottleneck machine, against a heuristic that is aware of both setups and weights. The advantage of the plain rule disappears at setup-to-processing ratios of about 0.1. A hysteresis variant, with its margin tuned on separate seeds, extends this to about 0.25–0.4 and gains 5–16% at 90% bottleneck occupancy. At the ratio of the source shop (0.5) it is slightly worse than the baseline, and weighted shop flow time is 0.5–2.4% worse at ratios of 0.125 and above. The benchmark and real-plant instances we found have ratios of about 0.7 to 9, where the exhaustive and hysteresis variants lose 4–14%. The rule suits bottlenecks with small setups.

Axioms doi: 10.3390/axioms15110829

Authors: Florin Felix Nichita

The Yang–Baxter equation is one of the most celebrated equations in science. There exist some correspondences between Yang–Baxter operators and Hopf algebras. For example, the quasitriangular Hopf algebras are used to define ribbon Hopf algebras and knot invariants. Finding new solutions for the Yang–Baxter equation and relating it to other algebraic structures is of great interest for scientists, as the classification of its solutions is out of reach for the moment. In the current paper, the new solutions arise from Boolean algebras and from racks. Our new solutions of the Yang–Baxter equation are generalizations of well-known solutions and they are related to the Euler formula. There exist self-dual Yang–Baxter operators as well as self-dual geometry theorems. There exists a partial dualization in the definition of entwining structures and there exists a partial dualization in our new theorem. Finding geometric representations for the phenomena from the Yang–Baxter theory is very useful at conferences and congresses. In geometry, the Morley’s trisector theorem was proved by the Fields medalist Alain Connes. We propose a similar theorem, which is a many circles version of a famous theorem. Connecting celebrated topics, the current paper also deals with an Euler formula proof for the original Johnson–Tzitzeica theorem.

Axioms doi: 10.3390/axioms15110828

Authors: Chuanjun Yan Lin Chen

Motivated by sparse principal component analysis (PCA), this paper proposes ATE-ManPG, a safeguarded transport-extrapolated proximal gradient method for minimizing the sum of a smooth function and a convex, possibly nonsmooth function over the Stiefel manifold. The method augments the manifold proximal gradient method (ManPG) with two ingredients: momentum obtained by transporting the previous proximal search direction to the current tangent space, protected by a monotone restart safeguard, and a bounded adaptive proximal parameter. For the exact idealized iteration, global convergence to stationary points and O(1/ε2) iteration complexity are proved; under a proximal Kurdyka–Łojasiewicz condition with exponent 1/2 and a relative-error condition, both stated as explicit hypotheses, the convergence is R-linear. Experiments on a sparse-PCA model, validated by an external residual certificate at the returned point, show a 3.6× mean iteration-count reduction relative to fixed-parameter ManPG; a factorial ablation attributes most of this reduction to the adaptive parameter, while transported momentum adds a further 6.4% on average, concentrated in one paired trial.

Axioms doi: 10.3390/axioms15110827

Authors: Zijian Luo Hongyu Yang Qianqian Zhang

This work addresses the issue of robust iterative learning tracking for heterogeneous multi-agent systems in the presence of random impulsive disturbances. Unlike existing literature focusing solely on bounded deterministic impulsive perturbations, this work explicitly models the stochastic nature of practically prevalent random impulses, which emerge randomly in the iteration domain and impair tracking performance. A key mathematical observation is that these persistent non-repetitive random impulses preclude asymptotic zero convergence of the tracking error. To tackle this fundamental limitation, a distributed iterative learning control strategy is proposed to realize robust ultimate bounded tracking. We establish sufficient conditions that guarantee the ultimate boundedness of the tracking error in expectation by applying the Gronwall inequality and mathematical analysis tools. The derived theoretical results generalize the classical convergence criteria developed under deterministic impulsive disturbance frameworks. Finally, numerical simulations are conducted to verify the effectiveness of the established theoretical results.

Axioms doi: 10.3390/axioms15110826

Authors: Ravindra K. Bisht Manuel De La Sen

In this paper, we introduce a generalized (ε-δ)-type contractive condition and develop a unified framework that extends a wide spectrum of classical fixed-point results for self-mappings in complete metric spaces, with or without continuity assumptions. The framework accommodates both contractive and nonexpansive mappings and unifies a broad class of classical fixed-point theorems, including those of Banach, Kannan, Chatterjea, Bianchini, Matkowski, Boyd–Wong, Meir–Keeler, Reich, Ćirić, Ćirić–Jachymski–Matkowski, rational contractions, and Pant, within a multiple-fixed-point setting. One of the established theorems yields a characterization of metric completeness that is different from that of Subrahmanyam in his research in 1975, which covers mappings that are discontinuous at their fixed points as well as mappings admitting multiple fixed points. The theoretical developments are illustrated through several nontrivial examples, including a geometric example in the complex plane whose fixed points are the Nth roots of unity. Following the initial 2024 resolution by Pant and collaborators, this paper provides an alternative constructive solution to the strong version of Rhoades’ 1988 open problem. An example on the integer solutions of a nonlinear Diophantine equation illustrates the results in a discrete arithmetic setting.

Axioms doi: 10.3390/axioms15110825

Authors: Angelo Plastino

We formulate an Information-Geometric Grammar Principle (IGGP) for adaptive systems in which grammatical admissibility, probabilistic realization, and statistical geometry are treated as distinct but connected levels of description. A grammar is represented by a state space together with a set of admissible productions and a probabilistic rule assigning weights to those productions. The resulting family of probability distributions defines a statistical manifold endowed with the Fisher information metric. We distinguish probabilistic adaptation, which changes probabilities while preserving the admissible grammar, from structural adaptation, which changes the set of admissible productions itself. For a fixed statistical manifold, an information-geometric action leads to geodesic equations and provides a coordinate-invariant criterion for locally economical statistical evolution. Interacting components generate a collective statistical manifold, where off-diagonal Fisher components quantify statistical coupling. We further identify the additional convexity and integrability conditions required for a Legendre representation and emphasize that a thermodynamic interpretation is conditional rather than automatic. The resulting framework provides a hierarchy from grammar to statistics, Fisher geometry, adaptive dynamics, collective organization, and, when additional conditions hold, information-geometric thermodynamics.

Axioms doi: 10.3390/axioms15110823

Authors: T. M. G. Ahsanullah Jawaher Al-Mufarrij

Let Q be an integral, commutative quantale with a completely distributive underlying lattice. It is known that there is a one-to-one correspondence between the category of quantale-valued tolerance spaces and tolerance-preserving mappings, denoted by Q-TOL, and the category of quantale-valued tolerance convergence spaces and continuous mappings, denoted by Q-TOLCONV. Our motivations in this article include introducing Q-tolerance structures on quantale-valued tolerance function spaces as well as on quantale-valued tolerance convergence function spaces. We prove that the category Q-TOLCONV and the category Q-TOLCONVGRP, consisting of quantale-valued tolerance convergence groups, are topological categories, and we provide examples of natural transformations in the category Q-TOL. Moreover, we show that both categories are cartesian closed; in particular, for the category Q-TOLCONV, we present a categorical proof of cartesian closedness via an adjoint functor. Furthermore, we study quantale-valued tolerance convergence function space structures on groups of homeomorphisms, where the homeomorphism group serves as a canonical framework for analyzing the compatibility of quantale-valued convergence with topological symmetries. This analysis clarifies how quantale-valued convergence interacts with algebraic symmetry and provides the structural foundation for our final result that quantale-valued tolerance convergence groups are group objects in Q-TOLCONV.

Axioms doi: 10.3390/axioms15110824

Authors: Cheng Zeng Dongxu Wang

For a connected graph, the Fermat difference of three vertices is defined as the minimum sum of their distances to a common vertex minus half the sum of their pairwise distances. In this paper, we investigate the average Fermat difference of unicyclic graphs with prescribed order and cycle length. By establishing a projection identity, we reduce the problem to a weighted cubic form on the unique cycle. We obtain an exact formula for cycles and establish a sharp lower bound together with a characterization of all extremal graphs attaining equality. We further derive a general upper bound for unicyclic graphs with prescribed order and cycle length. Finally, we determine the maximum average Fermat difference among all unicyclic graphs of fixed order and characterize the corresponding extremal graphs.

Axioms doi: 10.3390/axioms15110822

Authors: Rongxia Shao Xuanle Guo Shuo Li Jinfeng Zhang Guodong Li

We study a parameterized SOR-like (PSOR-like) representation of the parameterized inexact Uzawa (PIU) iteration for generalized saddle point systems. An explicit parameter change establishes equivalence to PIU with an exact first-block solve, and to GSOR when stabilization is zero. A quadratic matrix polynomial characterizes the complete iteration spectrum, including unit relaxation. We give a self-contained necessary and sufficient matrix-inequality convergence criterion for this specialization, considered alongside existing generalized PIU analyses without claiming a new iteration or priority for Uzawa convergence theory. The derivation avoids Rayleigh-quotient denominator assumptions and yields computable bounds for definite preconditioners and an exact relation to the preconditioned spectrum. Sparse experiments on two synthetic families compare the stationary iteration, an explicitly defined CGPIU* predictor–corrector comparator and GMRES. Archived tests use a common true residual tolerance and report setup and solve times separately. New manufactured-solution tests use a common forward-error threshold. At the common forward-error threshold, PSOR/PIU and CGPIU* require identical stationary counts; P-GMRES uses fewer inner steps on both synthetic families.

Axioms doi: 10.3390/axioms15110821

Authors: Muhammad Umar Farooq Muhammad Arif Hari M. Srivastava Mohamed Biomy Muhammad Abbas

The present article is mainly concerned with radius problems for a newly defined subclass of meromorphic starlike functions having the punctured open unit disk as its domain. This subclass is introduced by using differential subordination with the generalized tangent function q0(z)=1+tan(μz) as the superordinate function. Sharp radius constants are established for starlikeness and convexity of order ρ, and the MStanμ*-radii are determined for several notable subclasses, including Janowski-type classes and classes represented through quotients of meromorphic functions. We further introduce a tangent-order subclass MStanμ,σ*, obtain its representation formula, and derive corresponding starlikeness and convexity radius estimates. Finally, the superordinate tangent function is studied from the viewpoint of conformal mapping theory: its conformality, inverse mapping, boundary parametrization, disk inclusions, and conformal range criterion are established and illustrated graphically.

Axioms doi: 10.3390/axioms15110820

Authors: Oksana Kharchenko Volodymyr Chumakov Zlatinka Kovacheva Liudmyla Zabrodina Velin Andonov

The article examines alpha-rhythm spindles. The importance of studying alpha rhythm spindles for improving the reliability of nervous system diagnostics is substantiated. The analysis employs a mathematical framework based on the Hilbert transform as well as spectral analysis based on the classical Fourier transform. Using the Hilbert transform, the instantaneous frequency of the spindles is determined. It is shown that a spindle is not an amplitude-modulated oscillation and a spindle can be described using the beating method. Two frequencies are obtained that determine the frequency composition of the spindles. It is shown that the beating method is more accurate compared to the Fourier transform.

Axioms doi: 10.3390/axioms15110819

Authors: Hector Eduardo Roman

μ–τ flavor reflection symmetry (RS) of the PMNS mixing matrix implies |Uμi| = |Uτi|, leading to the atmospheric mixing angle θ23=45° and maximal Dirac CP violation phase, δCP=±90°. Here, we establish an analytical connection between leptonic CP violation and the breaking of μ–τ RS through the effective flavor squared masses mα2=∑i|Uαi|2mi2. We first examine the symmetric condition Δmτμ2≡mτ2−mμ2=0, which constrains θ23 to a highly localized weakly broken-RS geometric band, ±0.123°, centered at θ23=45° and δCP=−90° over the continuous phase interval −180°≤δCP≤0°. The lower edge of the joint T2K–NOvA benchmark interval, θ23(exp)=48.45−2.9°+1.7°, lies above this symmetric boundary (45.55°>45.123°), motivating an extension to finite RS breaking. We therefore introduce the dimensionless RS-breaking indicator (RSBI), β0=Δmτμ2/Δm212≠0, and characterize the departure from RS through the μ–τ asymmetries Δi=|Uμi|2−|Uτi|2 and their flavor-asymmetry quadratic deviation σ2(δCP)=∑iΔi2. This leads us to conjecture that the physical value of the Dirac CP-violating phase corresponds to a minimum of σ2(δCP). At leading-order, the resulting stationary condition provides a closed-form relation between cosδCP and the three mixing angles. This leading order relation lies midway between the partial-symmetry results of Xing and Zhou. Evaluated at the empirical joint T2K–NOvA benchmark, the framework gives |β0| ≃3.8 and δCP=−100−5°+9°, placing the phase in the third quadrant near-maximal CP violation. Including sub-leading corrections associated with the neutrino mass ordering gives the conditional estimates δCP=−101° for Normal Ordering (NO) and δCP=−99° for Inverted Ordering (IO), consistent with the corresponding direct numerical minimizations. These results suggest that δCP may be constrained by collective relations among the neutrino mixing parameters rather than behaving as an entirely independent free parameter.

Axioms doi: 10.3390/axioms15110818

Authors: Sidney A. Morris

A dense Gδ subset of a Polish group is large in the sense of Baire category. We study a configuration consequence of this largeness. If G is a Polish abelian group, R⊆G is dense Gδ, and A⊆G is countable, then the set of translations {t∈G:A+t⊆R} is itself dense Gδ. Thus, every prescribed countable configuration occurs inside R in comeagrely many translated positions. We record labelled, affine, and difference consequences of this classical Baire category observation and relate them to the realization of Cantor configurations through Mycielski’s theorem. Combining the classical finite-arity theorem with an analytic preimage lemma gives a simultaneous analytic realization formulation. For any prescribed countable family of suitable real analytic maps and any dense Gδ target in the real line, there is a Cantor set on which all corresponding admissible analytic expressions simultaneously take values in the target. Repeated arguments are handled by restricting variable-identification maps to connected components on which they remain nonconstant. Applications are given to Diophantine approximation. In particular, Liouville and ultra-Liouville numbers exhibit the same countable translation universality despite their extreme metric smallness. More generally, approximation classes determined by arbitrary countable families of nonnegative rational approximation functions, each positive at infinitely many integers, are dense Gδ and inherit both the translation and analytic Cantor realization principles. The results emphasize the distinction between metric smallness and Baire category configuration largeness.

Axioms doi: 10.3390/axioms15110817

Authors: Silvestru Sever Dragomir Ghada N. Alnemer Ahad Hamoud Alotaibi

We establish several upper bounds for the Berezin number and Berezin norms of products of three bounded operators. Let (HΓ,⟨·,·⟩) be a reproducing kernel Hilbert space (RKHS) on a set Γ with inner product ⟨·,·⟩ and associated norm ∥·∥. Berezin number and the Berezin norms of the operator V∈B(HΓ) are, respectively, defined by ber(V):=supλ∈Γ|⟨Vh^λ,h^λ⟩|, ∥V∥ber:=sup|⟨Vh^λ,h^μ⟩|;λ,μ∈Γ, and ∥V∥ber˜:=supλ∈Γ∥Vh^λ∥. In this paper, we show among others that, if the functions φ and ψ are continuous nonnegative on [0,∞) and satisfying the relation φ(t)ψ(t)=t for all t∈[0,∞), then for any X, Y, V∈BHΓ, we have that berYVX2r≤121pφVX2pr+1qψV*Y*2qrber+berψV*Y*2φVX2r for r≥1, p, q>1 with 1p+1q=1 and pr, qr≥2. Inequalities involving the Moore–Penrose inverse of closed range operators V are provided as well. The sharpness of some obtained inequalities for three, two, and one operator are considered with details in the last section of the paper.

Axioms doi: 10.3390/axioms15110816

Authors: Pingping Liu Shuanghua Luo Junxiang Lu

We study bias-corrected empirical-likelihood inference for single-index expectile regression when responses are missing at random. Local-linear expectile estimation is combined with conditional centering to construct three procedures based on complete cases, inverse-probability weighting, and cross-fitted imputation. The expectile score is continuous and globally Lipschitz, which permits the bias-correction argument without further smoothing of the score. Under the stated conditions, the index estimators are asymptotically normal and each empirical-likelihood ratio has a standard chi-squared limit. Joint confidence regions can therefore be formed without estimating the limiting covariance matrix. For the imputation construction considered here, I-BCEL and CC-BCEL have the same first-order limiting covariance. In the simulations, the estimation error decreases with sample size; differences among the incomplete-data methods are more apparent when the missing-response rate is 40%, where small-sample undercoverage also remains. An analysis of the New York air-quality data illustrates the three procedures with naturally missing responses.

Axioms doi: 10.3390/axioms15110814

Authors: Yulong Qiao Qian Zhang Bin Dai Qin Xu Kaimeng Ding

Gamma models are natural for positive, right-skewed quality characteristics and grouped time-between-event observations. Fixed Gamma cumulative sum (CUSUM) charts can detect persistent scale changes, but their deployed performance depends on the in-control (IC) scale estimated from a finite Phase-I sample. This study considers one-sided fixed Gamma CUSUM charts with known shape and a scale estimated once in Phase I. Conditional on this estimate, Phase-II observations follow a Gamma distribution with an effective scale ratio, allowing a one-dimensional Markov approximation to the complete run length (RL) distribution. Integration over the exact estimator law gives the marginal RL distribution while retaining the same estimate throughout each Phase-II path. Although every conditional average run length (ARL) is finite, the marginal ARL can be infinite; the marginal median run length (MRL) remains finite. Explicit integrability conditions characterize this distinction. The marginal MRL, the mean conditional MRL, and the distribution of conditional MRL are treated separately. Pathwise ordering yields the conditional-MRL distribution through scalar estimator crossings and reduces its probability constraint to one evaluation at a known Gamma quantile. Design selection profiles the reference value over a prespecified finite candidate grid, calibrates the corresponding control limits, and enforces finite marginal in-control ARL. The Python implementation reproduces published exponential and fixed-Gamma benchmarks. Across the eligible paired scenarios, conditional-MRL protection increased the probability of meeting the in-control conditional MRL target. The protected profiles had larger target-shift marginal out-of-control (OC) MRLs and larger marginal IC MRLs, so these comparisons do not hold in-control performance fixed. Independent continuous-state simulation assessed the finite-state probabilities at prespecified validation points; no exact-binomial comparison was rejected after Holm adjustment. A protected design whose integer MRL crossing remained sensitive to state resolution was excluded from the paired comparison. Supplementary sensitivity analysis shows reduced protection under downward shape misspecification, while matched-IC comparisons show shift-dependent detection tradeoffs.

Axioms doi: 10.3390/axioms15110815

Authors: Vladimir Orlov Liubov Markovich

We further develop Kernel Quantum State Estimation (KQSE), a fully data-driven nonparametric method for continuous variable quantum state reconstruction and characterization, introduced in our recent work and rooted in nonparametric kernel density estimation (KDE). Unlike approaches relying on finite-dimensional basis truncations, parametric ansätze, or prior models, KQSE combines a new kernel estimator of the characteristic function of the tomographic quadrature distribution with suitable kernel integral transformations. The characteristic function is estimated directly from experimental homodyne or heterodyne data and subsequently transformed to estimate quantum state representations and characteristics, including the Wigner function and the density matrix kernel. Observing that several other physically relevant representations and characteristics admit transformations of a closely related form, we derive convergence rates for the corresponding broad class of KQSE-based estimators. The resulting framework covers all phase-space representations, the photon-number tomogram, trace products of quantum states, and purity. We derive mean squared error convergence rates for these kernel transformed estimators and show that the corresponding KQSE applications inherit the near-optimal rate O˜(T−1), where T is the total number of measurements. The proposed theory applies equally to Gaussian and non-Gaussian continuous variable quantum states without imposing a Fock space cutoff. Numerical experiments on simulated Gaussian and non-Gaussian states and real homodyne data demonstrate that KQSE is competitive with state-of-the-art methods, establishing it as a statistically consistent and computationally modest framework for continuous variable quantum state estimation and characterization, particularly in the non-Gaussian regime.

Axioms doi: 10.3390/axioms15110813

Authors: Marian Milev Radka Koleva Yuri Dimitrov Mehran Paziresh

In this paper, a second-order implicit difference scheme for the time-fractional subdiffusion equation is constructed using a corrected L1 approximation of the fractional derivative. Estimates for the generating functions and resolvent coefficients are derived and used to establish the stability and convergence of the scheme. Numerical results are consistent with the theoretical error estimates and confirm the second-order accuracy of the scheme.

Axioms doi: 10.3390/axioms15110812

Authors: Julio Marny Hoff da Silva Eduardo Notte-Cuello

Starting from a general analysis of obstruction classes, we develop the investigation of obstructions associated with the bundle structure of the hyperbolic Clifford algebra. By taking into account particularities arising from the Whitney sum, it is shown that, unlike classical tangent bundle cases, the hyperbolic frame bundle of an oriented manifold admits lifting without any topological obstruction; that is, it admits a structure of the connected double covering of GL+(2m,R), and the hyperbolic tangent bundle carries a Spinm,me structure (with m=dimM), whatever the value of w2(M). These conclusions are known from the work of Karoubi and Gualtieri, and the point of this note is that they follow from the Whitney product formula alone, or from the additivity formula of Whitney sums with an explicit covering group, without recourse to Karoubi’s criterion for indefinite signature. We also describe the spinor bundles involved, which are bundles of differential forms and not Dirac spinor bundles, and we show that orientability is needed for the covering of the full linear group.

Axioms doi: 10.3390/axioms15110811

Authors: Savita Rathee Priyanka Lathwal Muhammad Waseem Asghar

In this research paper, we develop a relation-theoretic Φ-contraction framework in multiplicative metric spaces. We introduce the concept of a relation-theoretic Φ-contraction, in which the contractive condition is imposed only on pairs satisfying a prescribed binary relation. The existence and uniqueness of fixed point results are established under suitable relation-dependent completeness, continuity, and closedness assumptions. Two finite examples are given. The first shows that the logarithmic quantity associated with the chosen Φ-function need not satisfy the triangle inequality; in the second, the map is neither a global Φ-contraction nor a contraction of the logarithmic metric, so that the relation is essential. The theorem is then applied to a nonlinear Volterra-type integral equation whose kernel is only locally contractive, the region of contractivity being encoded by the binary relation.

Axioms doi: 10.3390/axioms15110810

Authors: Cheng-Cheng Zhu Jiang Zhu

Secondary infections and reactivation of latent pathogens pose prominent public-health challenges in the post-pandemic era. To account for spatiotemporal heterogeneity, host relapse transitions, and the coupling of multi-stage infection processes, this paper develops a six-compartment reaction-diffusion dynamical model consisting of susceptible, primary-infected, primary-recovered, secondary-infected, secondary-recovered, and fully recovered compartments. The model captures two distinct transmission pathways: exogenous secondary infection and endogenous reactivation of latent pathogens within hosts. Within the framework of infinite-dimensional dynamical systems, we rigorously prove the existence of a global exponential attractor for this non-autonomous spatiotemporal model, which constrains the long-term behavior of all solution trajectories. By means of upper–lower control systems and the Krein–Rutman theorem, we derive threshold criteria based on principal eigenvalues to judge disease extinction and endemic persistence. Numerical simulations are carried out using epidemiological data of herpes zoster following COVID-19 infection in China. We compare epidemic dynamics under homogeneous and spatiotemporally heterogeneous conditions, investigate the regulatory effects of the transmission coefficient, relapse rate, and state-transition parameters, and implement sensitivity analysis for key model parameters. Numerical experiments demonstrate that spatiotemporal heterogeneity triggers periodic spatiotemporal oscillations of infection burden; relapse elevates the magnitude of secondary infections; secondary cases exhibit remarkable time-lag responses even when the primary epidemic is well controlled. This work provides theoretical foundations for understanding spatiotemporal transmission of multi-stage infectious diseases, establishing early-warning frameworks and designing targeted intervention strategies for secondary infectious diseases.

Axioms doi: 10.3390/axioms15110809

Authors: Dong-il Lee

We compute the generalized noncommutative Castelnuovo-Mumford regularity of irreducible highest-weight modules over the Virasoro algebra Vir. Working over the graded negative universal enveloping algebra A=U(n−)⊂U(Vir) equipped with the level grading (degL−n=n), we analyze three structural classes of representations: generic Verma modules, irreducible quotients with a single singular generator, and rational minimal models. In particular, for the irreducible quotients L(cp,q,hr,s) in minimal models (1≤r≤p−1, 1≤s≤q−1), we show, using the Feigin–Fuchs structure theorem for Verma modules, that the BGG-type resolution of Feigin–Fuchs and Rocha-Caridi–Wallach restricts to a minimal graded free resolution over A with exactly two free summands in each positive homological degree, and we determine all generator degrees separately for odd and even homological degrees. The degree bounds are Ti=(ipq+ηi)2−(rq−sp)2/(4pq), where ηi=|rq−sp| for even i, and ηi=|pq−rq−sp| for odd i. Consequently, the asymptotic regularity exponent is e=2, and the regularity rate is r=pq/4, giving regAL(cp,q,hr,s)=(2,pq/4). The asymptotic regularity rate depends exclusively on the minimal model parameters (p,q) (and thus on the central charge cp,q), remaining invariant under the choice of highest weight hr,s, whereas the linear correction term does depend on hr,s. For (p,q)=(2,3) the result recovers Goncharova’s theorem on the homology of the Lie algebra of vector fields on the line.

Axioms doi: 10.3390/axioms15110808

Authors: Ali A. Al-Shomrani

This paper develops a unified framework for continuous distributional characterizations based on conditional expectations of functions of a random variable. The proposed framework expresses several characterization identities through a single differentiable characterizing function ζ(v), thereby providing necessary and sufficient conditions under which a distribution is uniquely determined by identities of the form Eg(V)|V≤v. Unlike earlier treatments in which related conditional expectation characterizations were handled separately, the present approach places power-, nonlinear power-, exponential-, and logarithmic-type identities under one common formulation. Propositions 1–3 extend earlier conditional expectation characterization results, whereas Proposition 4 provides a related necessary and sufficient characterization with an explicit Lambert W inversion. Propositions 5 and 6 introduce additional necessary and sufficient identities whose explicit inversions involve the Lambert W function. Propositions 2, 5, and 6 are obtained as special cases of a common transformation theorem. A distinctive feature of the proposed results is the systematic use of the Lambert W function to obtain explicit forms of the characterizing function when nonlinear exponential or logarithmic equations arise. The results also clarify cases in which closed-form inversion is difficult and numerical solutions may be required. These characterizations provide useful theoretical tools for identifying continuous probability models and for adapting conditional expectation characterizations to modern generated distribution families. The study fills a gap in the literature by connecting conditional expectation characterizations with modern generated distribution families through a single adaptable function rather than through separate distribution-specific derivations. Analytical applications are provided for several generated families, including TL-G, AGT-G, and OK-G families, illustrating how the proposed propositions can be used to verify and construct distributional characterizations.

Axioms doi: 10.3390/axioms15110807

Authors: Hessah Alqahtani

The averaged Gauss–Radau quadrature rule assigns equal weights to the left and right Gauss–Radau formulas regardless of the behavior of the integrand. This choice is generally not optimal for integrands possessing algebraic endpoint singularities. In this paper, we introduce an oracle-weighted averaging strategy in which the averaging parameter is chosen to minimize the quadrature error. For a class of integrands consisting of an algebraic endpoint singularity multiplied by a smooth regular factor, we establish an asymptotic invariance property of the oracle averaging parameter. Its leading asymptotic behavior is determined by the endpoint singularity exponent, while the regular factor contributes only to higher-order terms. Using the established Gauss–Radau remainder estimates, together with the boundedness of the oracle parameter associated with the principal singular component, we show that the difference between the oracle parameters associated with the general integrand and its principal singular component is O(n−2) as the quadrature order increases. Motivated by these results, we construct a simple rational approximation of the oracle averaging parameter depending solely on the endpoint exponent. Numerical experiments support the theoretical analysis and show that the resulting model-weighted averaged Gauss–Radau rule improves upon equal weighting for the test problems considered, without requiring knowledge of the exact integral.

Axioms doi: 10.3390/axioms15110806

Authors: Azhar Rashad Jan

This paper investigates a mixed Volterra–Fredholm integral equation of the first kind involving a hypergeometric spatial kernel and a continuous Volterra kernel in time. The analysis is formulated in an admissible even spatial space: the unweighted even subspace ofwhen appropriate, or the corresponding weighted even space for weakly singular cases, together with continuous time dependence and the associated supremum-in-time norm. The temporal and spatial contributions are separated at the operator level. Under appropriate temporal regularity and nondegeneracy assumptions, together with the regularity, endpoint, and integrability conditions required for the successive Abel-type inversions, the temporal first-kind equation is transformed into an equivalent Volterra equation of the second kind, while the reduced spatial equation is inverted analytically through successive Abel-type transformations. The logarithmic, Carleman, and elliptic kernels are recovered as important special cases of the hypergeometric structure. For the numerical treatment, structured Toeplitz–Hankel discretizations are considered for the logarithmic and Carleman kernels. Weighted solutions are introduced to account for their different endpoint singularities, and the computations are assessed through maximum and relative errors, discrete relative L2-errors, independent residuals, experimental convergence rates, and spectral condition numbers. The results show that the Carleman approximation has more uniform maximum-norm convergence, whereas the logarithmic error is strongly localized near the singular endpoint. Over the tested discretization range, the least-squares fitted maximum-norm rates are approximately 0.086 for the logarithmic kernel and 0.469 for the Carleman kernel. A manufactured space–time benchmark is also considered for an exponentially decaying memory kernel. The response curves illustrate the effect of the memory-decay parameter, and the trapezoidal temporal discretization exhibits approximately second-order convergence. The combined analytical and numerical results demonstrate how the spatial singularity structure and temporal memory influence the accuracy, conditioning, and evolution of the mixed integral problem.

Axioms doi: 10.3390/axioms15110805

Authors: Pei-Pei Zhang Bao-Ting Zhang Na Li

This paper investigates optimal resource allocation in multi-agent networks subject to simultaneous measurement and communication delays. We propose a distributed stochastic extremum seeking (SES) algorithm for connected undirected graphs with heterogeneous constant communication delays. The algorithm uses only delayed pointwise measurements of the local objective values; neither gradient nor Hessian information is fed to the controller. It combines a proportional–integral (PI) consensus protocol, which enforces the supply–demand balance, with a one-stage sequential predictor that compensates for the measurement delays. The convergence analysis is developed for local objectives of quadratic type; this restriction, which is required by the stochastic-averaging argument, is stated explicitly and discussed. Using stochastic averaging theory and Lyapunov–Krasovskii functionals for delay systems, we prove that, for every finite initial history, the estimates converge almost surely to an arbitrarily small neighborhood of the optimum for every fixed tolerance, and we derive an explicit admissible range for the constant communication delays together with a sequential procedure for selecting the design parameters. Two case studies, namely a three-generator microgrid and a six-generator microgrid, are used to illustrate the theoretical results.

Axioms doi: 10.3390/axioms15110804

Authors: Xun Yang Xaychalern Kongkeo Guaiqi Tian

The present work is concerned with a class of biharmonic elliptic equations involving critical and singular nonlinearities. Under the assumption that p(x) attains its global maximum at some point, we derive that the problem admits a mountain pass type solution via the Brezis-Lieb lemma. Meanwhile, if p(x) has k global maximum points, we establish the existence of k nontrivial solutions by virtue of variational methods and the Nehari manifold.

Axioms doi: 10.3390/axioms15110803

Authors: Abdelhamid Zaidi Nouf Almozaini

Clausen’s series 3F2(a,b,c;d,e;1), a generalized hypergeometric series with three numerator and two denominator parameters taken at unit argument, has classical closed forms when its denominator parameters exceed two of its numerator parameters by integers. These closed forms contain gamma or beta factors that become infinite when the remaining numerator parameter a equals 1, so that the sum at a=1 cannot be read off from them, although the series converges there. We give an explicit finite formula for 3F2(1,b,c;b+1+m,c+1+r;1), where m and r are non-negative integers, as a combination of digamma values with coefficients that are rational functions of b and c. For the integer values of b−c at which this formula breaks down we prove a companion formula containing trigamma values. Together the formulas cover the non-terminating domain b,c∈C∖Z≤0. We also show that the same expression is the limit, as a→1, of the formula of Shpot and Srivastava. As an application, we study a two-parameter family of such series indexed by integers n≥2 and p≥1. Each member of the family equals a rational number plus a non-zero rational multiple of πcot(π/n); hence, its sum is rational for n=2 and transcendental for every n≥3. Explicit formulas are given for p=1,2,3, the limit n→∞ is determined, and all results are confirmed by high-precision computation.

Axioms doi: 10.3390/axioms15110802

Authors: Gengran Hu

The Shortest Vector Problem (SVP) is the computational foundation of lattice-based cryptography, and its hardness on structured lattices directly impacts the security of NIST-standardized schemes such as Kyber and Dilithium. This paper investigates the SVP on anti-circulant lattices, studies the algebraic structure underlying the polynomial ring Z[x]/(xn+1), and reveals a fundamental distinction from their circulant counterparts. Using the convex body restriction strategy, we prove that for n≤3, every SVP solution has coefficients in {0,±1} unconditionally, and for n=4,5,6, explicit coefficient bounds hold under mild conditions on the lattice parameters. For higher dimensions up to n=50, experiments with an exact SVP show that with probability ≥95%, there exists an SVP solution with coefficients in {0,±1}, and with probability ≥97%, and there exists one with coefficients in {0,±1,±2}. All experiments are computed with the exact SVP, without using any approximation algorithm as the primary oracle.

Axioms doi: 10.3390/axioms15110801

Authors: Zhiyan Quan Yunfei Guo

In this paper, we investigate robust model averaging for high-dimensional data with outlier contamination. Most existing model averaging methods rely on ordinary least squares (OLS) estimation and are therefore highly sensitive to outliers. Moreover, these methods are generally not designed to accommodate high-dimensional settings. To overcome these limitations, we propose a robust fast greedy model averaging (Robust FGMA) method. The proposed approach incorporates the Huber robust loss function and fold-specific median absolute deviation (MAD) scale estimation into the cross-validation weight-selection criterion within the fast greedy model averaging (FGMA) framework. Simulation studies demonstrate that Robust FGMA achieves superior predictive accuracy and stability under various contamination and high-dimensional scenarios. Furthermore, empirical analysis based on the real biological dataset GSE10245 supports the effectiveness of the proposed method in the presence of outliers. The proposed approach retains the accelerated weight optimization structure of FGMA and is applicable to high-dimensional regression settings.

Axioms doi: 10.3390/axioms15110800

Authors: Mostafa Behtouei Carlos Salgado Lopez Giancarlo Gatti

In this paper, we develop an operator-based framework for laser–plasma interactions, providing a compact and systematic description of the coupled dynamics of laser fields and plasma response, with particular relevance to laser wakefield acceleration (LWFA). The formalism employs key operators: the transverse modal operator K^, the nonlinear plasma operator N^[Ψ], the plasma oscillation operator Ω^p2, and the ponderomotive source operator α^, which together describe mode coupling, plasma oscillations, and nonlinear feedback induced by the ponderomotive force. In the linear regime, the system is characterized by invariant subspaces associated with stable modal structures, while nonlinear interactions break these invariances, leading to mode mixing. The operators are defined on explicit Hilbert spaces. We establish the unitarity of linear propagation, the conservation of modal power under the nonlinear flow, the equivalence between the operator and PDE descriptions, and an explicit error bound for finite-mode truncations. We also characterize invariant subspaces through selection rules. Local well-posedness of the nonlinear problem is established under stated regularity assumptions. A reduced two-mode model is validated against direct numerical integration of the paraxial equation, and its accuracy and computational cost are quantified, the validation does not cover wakefield generation (transient plasma wave, pump depletion, injection or acceleration). The formalism provides a unified operator-based framework for the cold-fluid Maxwell–plasma model.

Axioms doi: 10.3390/axioms15110799

Authors: Osama Moaaz Asma Al-Jaser

This paper investigates the oscillatory behavior of second-order noncanonical neutral differential equations involving a neutral term with continuous delay and distributed deviating arguments. The main contribution is the establishment of a new and sharper relation between the solution and the corresponding neutral function, a problem that has not been addressed previously for noncanonical neutral equations with continuous delay. By combining iterative techniques with refined inequalities, we derive several new oscillation criteria, including Kneser-type conditions. The obtained results significantly extend the existing theory and improve a number of known oscillation tests in the literature. Illustrative examples and detailed comparisons with recent criteria demonstrate the effectiveness and superiority of the proposed approach in detecting oscillatory solutions.

Axioms doi: 10.3390/axioms15110798

Authors: Lulah Alnaji

We study estimation of the integrated squared latent density from two measurements per subject with independent, unknown measurement errors. The error distributions may differ and need not be symmetric or centered, but their characteristic functions must satisfy prespecified polynomial lower bounds. Gaussian errors and characteristic-function zeros are excluded. A ratio of observed characteristic functions identifies the target without division by the latent characteristic function. For a regularized estimator using the same subjects throughout, we derive an exact linearization and a weighted integrated remainder bound. Under sufficient Sobolev smoothness, these yield a root-n normal limit and mean squared error of order n−1. Conditional projection identities prove positivity of the limiting variance. A consistent variance estimator permits logarithmic and sufficiently slow polynomial cutoffs, giving confidence intervals with pointwise asymptotic coverage under prespecified regularization. Full-pair likelihood bounds identify a low-smoothness obstruction, but the available lower and upper rates do not match. Simulations examine bias, coverage, interval length, and numerical sensitivity. In the simulated regime, logarithmic intervals overcover, whereas some smaller-cutoff settings undercover, emphasizing the distinction between asymptotic validity and finite-sample calibration. The results do not establish adaptation, a sharp smoothness threshold, or semiparametric efficiency.

Axioms doi: 10.3390/axioms15110797

Authors: Meiqi Yan Hailou Yao

In this paper, we generalize the notions of pure derived and pure singularity categories by introducing X-pure derived and X-pure singularity categories for a module class X satisfying suitable conditions. We prove that the main classical results for pure derived and pure singularity categories extend naturally to the X-pure framework.

Axioms doi: 10.3390/axioms15110796

Authors: Abdelhamid Taieb Zaidi

We study the two-point problem −u′′(t)=w(t)f(t,u(t)), a<t<b, u(a)=M1, u(b)=M2, where the weight w is positive and integrable on (a,b) but may be unbounded near one or both endpoints. We prove that the problem has exactly one solution whenever f satisfies a one-sided Lipschitz condition with constant K<λ1(w) and the first eigenvalue of −φ′′=λwφ with Dirichlet conditions, and no condition depending only on K and w can do better: for K=λ1(w), there are nonlinearities for which the problem has no solution and others for which it has infinitely many. For globally Lipschitz f, the successive approximations converge geometrically in a weighted L2 metric, in which the Green operator is self-adjoint, with a ratio of at most K/λ1(w). For power weights w(t)=(t−a)γ−2, γ>1, the threshold is λ1=γ2j1/γ,12/(4(b−a)γ), where j1/γ,1 is the first positive zero of the Bessel function J1/γ, that is, jν,1 with ν=1/γ. Spectral sums of the Green operator give elementary two-sided bounds, and λ1 strictly exceeds the supremum-norm, Hilbert–Schmidt and Lyapunov-type thresholds. As an application, we obtain sharp uniqueness thresholds, within criteria depending only on the Lipschitz constant, the order and the interval length, for conformable boundary value problems of order γ∈(1,2] and for sequential problems of every order δ∈(0,1], enlarging the admissible Lipschitz constant for the problem of order γ by up to 35% relative to the supremum-norm criterion and by up to 22% relative to the Hilbert–Schmidt criterion, with the largest gains for orders close to 1. Numerical experiments confirm these results.

Axioms doi: 10.3390/axioms15110795

Authors: Khaled Boukerrioua Mohsen Dlala Taoufik Ghrissi Mohamed Ali Hammami

We establish new nonlinear retarded integral inequalities of Gronwall–Bihari type involving time-dependent kernels, variable delays, nonconstant free terms, and iterated Volterra-type contributions. The first class of results provides explicit estimates for inequalities with kernels depending on both the observation time and the integration variable. A second class treats composite nonlinearities through successive comparison transformations and extends known retarded inequalities to the presence of an additional iterated kernel. The obtained estimates are then applied to nonlinear Volterra integral equations and neutral functional differential equations. For the Volterra problem, sufficient conditions for uniqueness, global existence, and boundedness are derived under Osgood-type assumptions. For neutral functional differential equations, the retarded estimates are combined with Driver’s continuation framework to obtain a global existence criterion. A delayed Liénard system is included as an illustrative application. These results extend several classical Gronwall–Bihari and retarded integral inequalities and provide a flexible tool for the qualitative analysis of nonlinear integral and functional differential equations.

Axioms doi: 10.3390/axioms15110794

Authors: Abu Zaid Ansari Mukhtar Ahmad Majed Albaity Faiza Shujat

Let n≥2 be fixed, and let Λ be a Lie superalgebra over a commutative ring R. We investigate the structure of the algebra Dern(Λ) of n-derivations. We show that Dern(Λ) is a Z2-graded Lie superalgebra and that its elements preserve n-perfect graded ideals, thereby inducing n-derivations on the corresponding quotients. We obtain a direct-sum decomposition with explicit cross terms and describe n-derivations of split central extensions, including the contributions from maps into the n-center. For perfect centerless Lie superalgebras that are (n−1)-torsion-free, we prove that every n-derivation is an ordinary derivation; consequently, all n-derivations are inner when all ordinary derivations are inner. We also exhibit solvable examples for which Der(Λ) is properly contained in Dern(Λ). These results clarify the roles of the grading, ideals, centers, and split central extensions in the theory of n-derivations.

Axioms doi: 10.3390/axioms15110793

Authors: Özlem Arık İbrahim Arık Yeliz M. Kantar

In this study, we introduce and study the record-based transmuted Pareto distribution (RBTPD), a three-parameter model obtained by applying the record-based transmutation map to the classical two-parameter Pareto distribution. Its relationship to the record-based transmuted power Lomax distribution is set out explicitly, and the contribution of this paper is located in the Pareto parameterization itself: the closed-form properties it permits, the fact that its lower endpoint is a threshold parameter with a substantive interpretation and requires non-regular inference, and a systematic comparison of five estimation methods. The additional parameter is the weight attached to the second upper record value of the baseline; it is confined to the unit interval and returns the Pareto distribution at zero. The proposed model is more flexible than the baseline and, unlike several existing extensions, has a density that remains bounded on the whole support. We derive the distribution; density; survival and hazard rate functions; the quantile function, in terms of the Lambert W function; moments and incomplete moments; mean deviations; Bonferroni and Lorenz curves; order statistics; record values; and Renyi and Shannon entropies. Five estimation methods are compared in a Monte Carlo study: maximum likelihood, least squares, weighted least squares, Anderson–Darling, and maximum product of spacings. For α and for p away from the boundaries of its range, the mean squared errors of all five estimators decrease with increasing sample size, a pattern consistent with consistency, though a finite range of sample sizes cannot establish it; the threshold is a support-boundary parameter and is treated separately, as the model is non-regular in it. Maximum likelihood performs best overall and is markedly the most stable numerically. The model is fitted to two real data sets, one medical and one hydrological, and gives a better fit than both the baseline Pareto and the transmuted Pareto distributions on every criterion considered. Compared with competing models outside the Pareto family, the proposed model gives generally comparable results and performs relatively well in some cases. Bootstrap-calibrated goodness-of-fit tests indicate adequate absolute fit for the medical data but not for the hydrological series, and the conditions under which the model is appropriate are stated explicitly.

Axioms doi: 10.3390/axioms15110792

Authors: Muhammad Imran Liaqat Abdulaziz Khalid Alsharidi

We investigate nonlinear Caputo–nabla fractional stochastic difference systems driven by independent Gaussian increments. The fractional summation identity yields an exact pathwise Volterra representation whose strictly causal structure establishes finite-horizon existence and pathwise uniqueness under measurability and linear-growth assumptions, without requiring a contraction condition. Under global Lipschitz assumptions, separate weighted ℓ1 and ℓ2 kernel estimates provide quantitative mean-square bounds for perturbations of initial data, drift and diffusion coefficients, and the fractional order. The dependence of these estimates on the exponential weight and the observation horizon is made explicit. We also establish residual-based Ulam-type stability estimates, finite-horizon moment bounds, and finite-step exactness of Picard iteration. For long-time analysis, we prove discrete Caputo–nabla comparison principles, establish decay of the associated scalar resolvent, and derive an exact quadratic identity for the corresponding dissipativity condition. The Mittag–Leffler stability result remains conditional on a history-dependent dissipativity inequality. Independently, a directly verifiable coefficient-level criterion establishes asymptotic mean-square stability and is applied to a three-dimensional illustrative macroeconomic model at fractional order ϑ=0.75. Numerical experiments, including a nonlinear Monte Carlo simulation with sampling uncertainty estimates, illustrate the theoretical results.

Axioms doi: 10.3390/axioms15110791

Authors: Xingyue He Jingjing Wang

In this paper, we aim to investigate the boundary Hölder regularity for a class of Monge–Ampère-type equations in a bounded convex domain. We study the local regularity of convex solutions at special (a,η)-type boundary points. By constructing appropriate auxiliary functions and sub-solutions, we investigate the local regularity of convex solutions at some special points, which are called (a,η)-type points. It is emphasized that the regularity of the solution at the boundary depends essentially on the convexity of the domain, and is independent of the smoothness of the domain.

Axioms doi: 10.3390/axioms15110790

Authors: Lina Ji Huili Liu

We study the martingale problem for two mutually competing superprocesses in independent Brownian random environments. The competition rates depend on the spatial position and both population measures. Under suitable assumptions, we establish well-posedness for every pair of finite initial measures and derive moment estimates for the total masses. Existence is proved by constructing branching particle systems and establishing the tightness of their empirical measure processes. Uniqueness in law follows from pathwise uniqueness for the system of stochastic partial differential equations satisfied by the associated distribution-function-valued processes.

Axioms doi: 10.3390/axioms15110789

Authors: Khaled Aldwoah Inzamam ul Huque Hafedh Alnoghashi Muntasir Suhail Amel Touati Radwan M. Al-omary

In the setting of nonzero semigroup ideals, Jordan ideals, and Lie ideals of prime near-rings (under 2-torsion-free and 3-torsion-free conditions), this study investigates the structure of a prime near-ring that admits a right generalized derivation K associated with a derivation σ. Some of the obtained results generalize earlier works under comparable assumptions on the whole near-ring. In addition, examples are provided to illustrate why the hypotheses of the results are necessary.

Axioms doi: 10.3390/axioms15100788

Authors: Christopher S. Withers Saralees Nadarajah Paul Teal

Consider a multivariate function f that maps an open region of Rp into Rp, together with its inverse function. This paper presents a general method for obtaining the partial derivatives of the inverse function, of any order, expressed in terms of the partial derivatives of the inverse of the Jacobian matrix of f. The results are local. They require only that f have continuous partial derivatives up to the order of interest in a neighbourhood of a point, and a nonsingular Jacobian at that point, so global one-to-one-ness is not needed. A recursion for the derivatives of the inverse is proved by induction. Explicit formulas are derived for all partial derivatives up to the fourth order, both for the inverse function and for the inverse Jacobian. The symmetry of the mixed partial derivatives of the inverse is established. The computational cost is discussed: the explicit formulas grow factorially with the order, whereas a Leibniz-type recursion gives polynomial cost for numerical evaluation. A recursive scheme implemented in Python using symbolic computation is provided for derivatives of arbitrary order, together with a program that verifies the explicit formulas independently against direct differentiation. The univariate case, published previously by Withers, Nadarajah and Teal in the journal Axioms, is reviewed first, recovering known results expressed in terms of Bell polynomials. The method is illustrated by worked examples in two dimensions. Its use in mathematical statistics is shown by casting maximum likelihood estimation in canonical exponential families, including a canonical regression setting, in the framework of the paper, and by deriving explicitly the bias-corrected maximum likelihood estimator with bias On−2, which is checked against an exact calculation for the normal distribution.

Axioms doi: 10.3390/axioms15100787

Authors: Hong Yang Yurong Wu Yin Wu

Elliptic fuzzy PDEs bridge deterministic models and real-world uncertainty, enabling rigorous modelling of steady-state systems with imprecise parameters. The Green’s function method has been widely recognized as an effective technique for investigating boundary value problems associated with the Poisson and Laplace equations. This method relies on the principle of superposition: a complex field source is decomposed into a superposition of several simple point or line sources. Once the field produced by a point source is known, the field generated by an arbitrary source can be computed via superposition. In the present study, the Green’s function method is primarily applied to solve the first boundary value problems for both the fuzzy Laplace equation and the fuzzy Poisson equation. Furthermore, we derive the fundamental integral formulation for the fuzzy Poisson and Laplace equations. At the same time, the fundamental integral formulation is used to obtain the solutions of the fuzzy Poisson and fuzzy Laplace equations. Several illustrative examples are also provided to demonstrate the solution procedure for these BVPs using the Green’s function method.

Axioms doi: 10.3390/axioms15100786

Authors: Lingling Ma Xiangyan Qiu

In this paper, an efficient iterative method based on the generalized multiscale finite element (Iter-GMs) method is applied to solve the distributed optimal control problems governed by stochastic parabolic equations. Within this framework, we firstly reformulate the time-dependent optimality system into an equivalent form, which contains the parameter-independent diffusion coefficients and the parameter-dependent right-hand side. Then, the fixed-point iteration method is used to solve the reformulated optimality system. Quantifying the statistics of stochastic optimal control problems requires solving a coupled optimality system across numerous samples in the stochastic space, which incurs a prohibitively high computational cost and may even trigger the so-called “curse of dimensionality.” To mitigate this issue, we exploit the multiscale and high-contrast nature of the physical parameters by invoking the generalized multiscale finite element method (GMsFEM) to construct a reduced-order space. We divide the proposed iterative method into two stages: an offline stage and an online stage. In the offline stage, we construct local multiscale basis functions for each coarse-grid region based on the deterministic multiscale optimality system, and then build the corresponding offline spaces. In the online stage, we project the reformulated model onto the reduced multiscale space and apply the fixed-point iteration method to obtain the optimal solutions of the reduced system. Finally, two numerical experiments are presented to validate the rationality and effectiveness of the proposed method.

Axioms doi: 10.3390/axioms15100785

Authors: Zhenghao Hu Suixin He

We determine the exact operator norms of the weighted Hardy operator Uψ and the weighted Cesàro operator Vψ on Total Morrey spaces Lp,λ,μ(Rn). We also study their commutators with symbols b∈BMO(Rn) satisfying the additional dilation condition sup0<t<1∥b−b(t·)∥L∞(Rn)log(2/t)<∞. Assuming in addition that the corresponding underlying weighted Hardy or Cesàro operator is bounded, we obtain necessary and sufficient integral conditions on the weight function ensuring the boundedness of the first-order and higher-order commutators for every symbol satisfying this dilation condition. The results reflect the two-scale structure of Total Morrey spaces: the weighted Hardy operator is governed by the large-scale parameter μ, whereas the weighted Cesàro operator is governed by the small-scale parameter λ. The exact operator norm formulas extend their classical Morrey-space counterparts, while the commutator criteria are obtained for BMO symbols satisfying the stated additional dilation condition.

Axioms doi: 10.3390/axioms15100784

Authors: Fatimah M. Alghamdi Abdulhameed Ado Osi Muftahu Zubairu Ringim Aadil Ahmad Mir Souha K. Badr Rana Bakoban Gamal A. Abd-Elmougod

This paper introduces the Transmuted Unit Burr XII (TrUBXII) distribution, a flexible probability model for analyzing bounded data that exhibit skewness and heavy-tailed characteristics. The proposed distribution is derived by incorporating a transmutation parameter into the unit-support Burr XII framework, extending existing bounded distributions and enhancing modeling flexibility for data on the interval (0, 1). Several fundamental statistical properties of the distribution are derived, including its moments, quantile function, and hazard rate function, which provide insight into its structural features. Parameter estimation is conducted using the maximum likelihood approach, and the finite-sample behavior of the estimators is examined through an extensive Monte Carlo simulation study with improved design and performance evaluation metrics. The practical relevance of the TrUBXII distribution is illustrated through an application to a real dataset of residence times under three watering regimes (clean air, high-watered, and low-watered plants), which exhibits the asymmetry and boundary accumulation that motivate the proposed model. Comparative analyses with established bounded distributions, including the Beta and Kumaraswamy models, indicate that the TrUBXII distribution provides a superior fit, particularly for datasets characterized by asymmetry and extreme observations. Therefore, the proposed model offers a robust and versatile framework for bounded data analysis, supported by reproducible implementation in the R programming environment.

Axioms doi: 10.3390/axioms15100783

Authors: Hong Wang Xiaoyuan Wang

In the present paper, we derive sharp bounds for Schippers–Schwarzian derivatives of functions belonging to the two core Ma–Minda subclasses associated with the canonical half-plane majorant ψ(z)=1+z1−z: the convex class C(ψ) and starlike class S*(ψ), and the generalized Robertson class SPα(β). All derivations avoid Carathéodory coefficients cn; we only express the Taylor coefficients of Schwarz functions via Schur polynomials Fk(γ1,…,γk), then reduce each functional to a single-variable optimization, and construct explicit extremal Schwarz kernels as well as integral-form extremal univalent mappings.

Axioms doi: 10.3390/axioms15100782

Authors: Anwar Almualim Fethi Bouzeffour Mhamed Eddahbi

We developed a random-field theory for the Euler–Poisson–Darboux equation on Euclidean space under additive Gaussian forcing that was white in the evolution variable and spatially homogeneous. An explicit two-parameter Bessel Green kernel provided the basic deterministic propagator. Sharp high-frequency bounds for its integrated energy yielded a necessary and sufficient Dalang criterion for the existence of the stochastic convolution and showed that the singular Darboux coefficient did not change the wave-type spectral threshold. A dissipative energy identity for the Green kernel provided temporal increment estimates which, together with the spatial Fourier bounds, led to joint space–time Hölder regularity under a natural strengthened spectral moment. We also distinguished between the two initial traces: the Dalang condition guaranteed the mean-square displacement trace, whereas a mean-square zero-velocity trace required additional spectral regularity; finite total spectral mass provided a simple sufficient condition. Space–time white noise and Riesz-colored noise were treated explicitly. Finally, the mean solution was identified with the Darboux spherical mean, Brownian first-exit sampling, and the Pizzetti expansion.

Axioms doi: 10.3390/axioms15100781

Authors: Milena Dimova Natalia Kolkovska Nikolai Kutev

In this paper, we investigate wave equations with a general class of Lipschitz nonlinearities combining power and logarithmic terms. Local existence and uniqueness of weak solutions are established via the Galerkin method. For nonlinearities of defocusing type, we prove global existence of the weak solutions for arbitrary initial energy. In particular, we provide examples of globally defined solutions with negative initial energy.

Axioms doi: 10.3390/axioms15100780

Authors: Alexander Khludnev Evgeny Rudoy

This paper addresses the investigation of an equilibrium problem describing an elastic body with an incorporated thin elastic Timoshenko inclusion, provided that the Neumann boundary condition is imposed on the external boundary of the body. The inclusion is delaminated from the elastic body, thus forming an interfacial crack. A restriction of the inequality type considered at the crack faces does not allow the opposite faces to penetrate each other. Moreover, the crack faces’ boundary conditions depend on a positive parameter characterizing the adhesion. We find the condition imposed on the external forces, proving the existence of a solution to the boundary value problem analyzed, and prove the solution’s existence. The asymptotic analysis of the solutions with respect to the adhesion parameter and the stiffness parameter of the inclusion is investigated. Limit models are analyzed. Both variational and differential statements of the considered problems are obtained, and a connection between them is established.

Axioms doi: 10.3390/axioms15100779

Authors: Ziyu Zhang Yan Wang Jingqi Yan Zhongguo Xu Rubing Xue Wei Dong Zhenzhe Zhang

The functional linear model Y=∫TX(t)β0(t)dt+ε is widely used in various fields, including medical image analysis, economics, and climate science. Reliable statistical inference for β0(t) across the domain t∈T is essential in these applications. While substantial progress has been made in addressing prediction problems in offline settings, hypothesis testing for β0(t), especially in the context of streaming and functional data, continues to present considerable challenges. In the paper, we propose a novel statistical inference framework designed specifically for streaming data based on functional stochastic gradient descent algorithms. Our approach constructs an online sequence of randomly perturbed estimators and incorporates a two-armed bandit process to build an adaptive test statistic for hypothesis testing. Moreover, we establish the limiting distribution of this statistic under both the null and alternative hypotheses, thus ensuring its statistical validity. Simulation studies examine the power and null calibration of the framework, and a real-world application illustrates its use for streaming functional data.

Axioms doi: 10.3390/axioms15100778

Authors: Gyurhan Nedzhibov

Accurate data-driven identification and long-term forecasting of surface ship motions in realistic ocean environments are essential for advanced autonomous navigation and predictive control systems. However, robust system identification in such conditions remains challenging due to high-frequency wave-induced disturbances, which contaminate measured signals and may contribute to error accumulation in recursive forecasting. This paper proposes a control- and wave-aware data-driven framework for modeling and prediction of ship maneuvering dynamics in stochastic wave environments. The approach integrates Hankel dynamic mode decomposition with control (Hankel-DMDc) with a spectral Mode Attribution Layer that characterizes modal associations with commanded control inputs and measured wave excitation. A dual-criterion selection rule, based on the Wave Energy Projection (WEP) and Control Correlation Index (CCI), is introduced to characterize and separate maneuvering-related and wave-associated modal content without relying on a predefined physical model. The framework is evaluated on a multidimensional ship-motion dataset from the DaRUS repository, using routine maneuvering trajectories for model identification and independent out-of-distribution test trajectories for prediction evaluation. One-step prediction analysis indicates high local accuracy, particularly for rotational states, while recursive multi-step forecasting shows gradual error growth over a 200-step prediction horizon. A quantitative comparison with unfiltered Hankel-DMDc, conventional Butterworth filtering, and direct wave-input modeling further shows that the proposed framework retains predictive performance comparable to direct Hankel-DMDc for the present dataset, while providing explicit modal attribution and wave/maneuvering characterization. The identified discrete-time operator also exhibits a spectral radius below unity, while no rapid divergence is observed over the evaluated recursive prediction horizon. The proposed methodology provides an interpretable data-driven framework for modeling marine systems, with potential applicability to control-oriented system identification, predictive modeling, autonomous navigation, and Model Predictive Control (MPC).

Axioms doi: 10.3390/axioms15100777

Authors: Junkee Jeon Geonwoo Kim

We study the valuation of European exchange options subject to the issuer’s terminal default risk under correlated skew Brownian motions. The two underlying assets and the issuer’s asset value are exposed to correlated Gaussian risks and a common reflected Brownian factor, with different skewness parameters. The contract allows full payment when the issuer is solvent and proportional recovery otherwise. Conditioning on the reflected factor gives a joint Gaussian distribution for the log asset values. Using Gaussian exponential tilting, we derive a pricing formula consisting of four truncated moments, each expressed as a one-dimensional integral involving the bivariate normal distribution function. The formula treats the exercise and default conditions exactly. Its mathematical contribution is the joint evaluation of the exchange and default regions, including the asset–issuer products arising in recovery, without a payoff approximation. This retains a one-dimensional integration structure despite the three-asset dependence. Numerical experiments compare the resulting prices with direct Monte Carlo estimates and illustrate the effects of the issuer’s asset value, bankruptcy costs, skewness parameters, and credit-related correlations.

Axioms doi: 10.3390/axioms15100776

Authors: Marlina Slamet Viraht Sahni

We prove the Wave Function Identity (WFI), a newly discovered property of solutions to the two-dimensional Schrödinger–Pauli equation for the N=3 electron Coulombically interacting harmonically bound ‘artificial atom’ or quantum dot in a uniform magnetic field. The symmetry is comprised of an interchange of the spatial coordinates of two electrons whilst keeping their spin moments unchanged, followed by an inversion of all three electron coordinates. The proof is achieved by first deriving the general form of the exact correlated wave function in the high-electron-correlation Wigner regime and then proving the WFI employing this form for a ground and quartet state. The satisfaction of the Pauli principle (PP), and the odd parity of the quartet state which follows from the WFI, are also proved. An exact analytical expression for a quartet state in the Wigner regime is derived, and the properties of the PP, WFI, parity and other properties exhibited diagrammatically. Together with our prior proof that the exact wave functions of the N=2 electron systems satisfy the WFI, we conjecture the ad hoc validity of the WFI for all bound-state N≥2 electron systems for arbitrary symmetric binding potential, arbitrary electron-interaction of the form ω(|r−r′|) and arbitrary dimensionality.

Axioms doi: 10.3390/axioms15100775

Authors: Luca Grilli Domenico Santoro Giacinto Angelo Sgarro

A network risk associated with persistent exposure depends on both the history of the exposure and the points at which decisions can be made. These two sources of dependence must remain distinguishable when an arc is divided into shorter segments. Substochastic operators provide a finite-state description in which homogeneous subdivision preserves propagation, while changes in exposure and in the available continuations remain visible. Building on continuous semigroups and linear realization theory, this paper studies the interaction between refinement and the information required for network decisions. At each physical vertex, the pairing of reachable states with admissible continuation observations determines the dimension of an exact linear state. Its transitions respect arc subdivision when the physical decision and stopping locations are preserved. The same observations determine which partial paths can be discarded without worsening any permitted continuation. Worked finite-state networks demonstrate why a smaller prefix risk may lead to a larger final risk and how recovery, exposure ordering, and generator uncertainty affect that comparison. They also show that allowing uncertainty to change at an artificial breakpoint alters the decision problem even when local propagation remains consistent. These relationships connect physical exposure, state reduction, and path optimization within a formulation whose invariance and computational requirements can be stated explicitly.

Axioms doi: 10.3390/axioms15100774

Authors: Zuhur Alqahtani Belgees Qaraad Areej Almuneef Fahd Masood

This paper investigates the oscillatory behavior of a new class of second-order nonlinear neutral advanced differential equations with mixed superlinear and sublinear neutral terms in the noncanonical case. The coexistence of these nonlinearities substantially increases the analytical complexity and limits the applicability of many existing oscillation criteria. To address this challenge, a new analytical framework is developed by establishing qualitative properties of eventually positive solutions. Based on this framework, new oscillation criteria are derived by combining the comparison method with the generalized Riccati transformation. The proposed results provide a unified treatment of the cases m>n, m=n, and m<n. Several explicit criteria are further obtained as direct consequences through suitable choices of auxiliary functions. Illustrative examples demonstrate the applicability and effectiveness of the theoretical results. Overall, the proposed framework broadens the scope of existing oscillation theory and provides a unified approach for this class of nonlinear advanced neutral differential equations in the noncanonical case.

Axioms doi: 10.3390/axioms15100773

Authors: Jiju Gillariose Joshin Joseph Hanan Alahmadi Mahmoud E. Bakr Noha A. Tashkandi Mustafa M. Hasaballah

In this study, we investigate a Slash Marshall–Olkin (SMO) family obtained by dividing a positive Marshall–Olkin random variable by an independent beta variable. Its four-parameter exponential member (SMOE) has a regularly varying power-law tail and belongs to the subexponential class. Integral representations are provided for all positive parameters. The original hypergeometric expansions are retained within their convergence domain, and an alternative convergent representation is derived for the remaining Marshall–Olkin parameter values. The density and hazard have finite, strictly positive limits at the origin. The density is decreasing for η≤2 and has at least one positive interior mode for η>2. We establish regular variation of the survival function with exponent −α and show that a positive-order moment exists exactly below order α. In addition, an exact generator, an observed-information representation, and precautions for likelihood optimization are provided. The simulations and two economic applications are also examined.

Axioms doi: 10.3390/axioms15100772

Authors: Areej Almuhaimeed Abu Zaid Ansari Faiza Shujat

This paper investigates cyclic codes over Fqn×Pm×Qk, where Fq is a finite field of odd characteristic, P=Fq[v]/⟨v2−τ2v⟩, and Q=Fq[u,v]/⟨u3−τ12u,v2−τ2v⟩, with τ1,τ2∈Fq*. Using complete systems of orthogonal idempotents, we decompose (P) and (Q) into two and six constituent codes over Fq, respectively, and define matrix-based Gray maps. The induced Gray map from Fqn×Pm×Qk to Fqn+2m+6k is shown to be Fq-linear and Lee–Hamming distance preserving. We prove that a mixed-alphabet code is cyclic if and only if its deinterleaved Gray image is a generalized quasi-cyclic code with block-length profile (n, m, m, k, k, k, k, k, k). A polynomial-module characterization, constituent generator polynomials, dimension bounds, shift-generated spanning sets, and structural results for the Euclidean dual and hull are also established. Furthermore, the generator structure and cardinality of cyclic codes over P, Q, and the mixed alphabet are determined through their field constituents. As an application, scalar-orthogonal Gray matrices and dual-containing constituent cyclic codes are combined with the Calderbank–Shor–Steane construction. An explicit construction over F5 yields a dual-containing generalized quasi-cyclic code with parameters [90, 75, 3]5, and a quantum error-correcting code with parameters [[90, 60, 3]]5.

Axioms doi: 10.3390/axioms15100771

Authors: Javier Rodrigo Sagrario Lantarón María Dolores López

In this paper, we deal with a challenging problem in Discrete Geometry. The problem is to maximize the number of halving lines for finite sets in the plane. With respect to this, we improve the lower bounds for the maximum number of halving lines for sets in the plane of n points (hn), for some values of n with n ≤ 100, expanding the range of the values of n for which the authors achieved an improvement in preliminary works. To do this, we strengthen previous results of the authors about recursive bounds concerning hn. These improvements are important due to the relation of the number of halving lines for finite sets with another classical problem in Discrete Geometry, the rectilinear crossing number problem.

Axioms doi: 10.3390/axioms15100770

Authors: Hualei Zhang

In this paper, we study the energy decay rate for a porous-elastic system with fractional infinite memory, which acts on the material displacement equation. Using the frequency domain method, we show that the energy decay rate of the system depends on the order θ of the fractional operator and the wave speeds μρ and δJ, where θ∈[0,1]. Specifically, the energy of the system decays exponentially when θ=1 and μρ=δJ. The energy of the system decays polynomially when θ∈[0,1) and μρ=δJ or θ∈[0,1] and μρ≠δJ. In addition, we also show that the polynomial decay rates obtained are optimal in some sense.

Axioms doi: 10.3390/axioms15100769

Authors: Mohammad Akram Inna Kalchuk Muhammad Tariq Waqar Afzal

In this paper, we introduce and investigate a refined generalization of convexity, termed the multiplicatively strongly s-convex function, and establish several of its fundamental algebraic properties. Within the framework of multiplicative calculus, this generalized convexity structure is employed to derive new midpoint-, trapezoid-, and Simpson-type inequalities, thereby providing unified refinements of several classical integral inequalities. Particular choices of the underlying parameters yield a variety of meaningful special cases and corresponding novel inequalities. Furthermore, we demonstrate the applicability of the established results through inequalities involving several important special means, including the arithmetic, geometric, logarithmic, and m-logarithmic means. To complement the theoretical developments, selected consequences are illustrated numerically and graphically, providing comchosenational evidence for the validity and effectiveness of the proposed inequalities. The results contribute to the development of multiplicative convexity theory and offer a flexible framework for deriving further inequalities within multiplicative calculus.

Axioms doi: 10.3390/axioms15100768

Authors: Rocío Fernández-Iglesias Sonia Pérez-Fernández Miguel García-Villarino Pablo Martínez-Camblor

Quantile regression (QR) provides a flexible framework for characterizing conditional distributions and can offer useful insights for quantile estimation. In this paper, we provide intuition for the interpretation of QR by highlighting the approach that it is a reparameterization of more popular conditional cumulative distribution function regression. From this approach, we naturally explore the use of the proportional hazards Cox regression model in the QR context. Therefore, we propose a framework based on the Cox regression model for estimating conditional quantiles, obtained through a reparameterization of the conditional cumulative distribution function under a proportional hazards structure. We establish the large-sample properties of the resulting estimator and investigate its behavior in finite samples through a Monte Carlo simulation study. An empirical application illustrates its practical implementation in a real-world context. Furthermore, we introduce the average quantile effect, a summary measure for quantifying the effects of covariates across conditional quantiles. The proposed estimator behaves adequately under the required assumptions, although inferences on the associated parameters strongly depend on these premises.

Axioms doi: 10.3390/axioms15100767

Authors: Ayman Elsharkawy Murat Turan Eman Ghareeb Rezk Hülya Gün Bozok Sıddıka Özkaldı Karakuş

This paper develops a comprehensive quasi-frame theory for inextensible curve flows in the dual space D3. While the Frenet frame has been the traditional tool for studying curve evolution, it encounters singularities when curvature vanishes. We introduce a quasi-frame (q-frame) formulation that overcomes these limitations and provides a robust framework for analyzing inextensible flows, flows that preserve arc length throughout evolution. We derive the fundamental evolution equations for quasi-curvatures under inextensible flow conditions and establish necessary and sufficient conditions for inextensibility in dual space. In contrast to the Euclidean, Minkowski, and Galilean settings, the quasi-curvatures in dual space are dual-valued, so they simultaneously encode the direction of the oriented line and the evolution of its moment vector. Our approach extends inextensible flows using the Frenet frame by providing a more general framework that remains well-defined even at singular points. We demonstrate the advantages of our method through detailed worked examples, including complete computations of time-evolving curves with explicit verification of arc-length preservation. These results contribute to the theoretical foundations of curve dynamics in dual spaces.

Axioms doi: 10.3390/axioms15100766

Authors: Hamid Reza Asadi Dehaghi Mohammad Mahdi Zahedi Hashem Bordbar Ali Iranmanesh

Nodal filters provide an order-theoretic approach to the study of the filter structure of algebraic systems. In this paper, we investigate nodal filters in equality algebras, with particular emphasis on nodal filters of types 1, 2, and 3. We first establish several structural properties of these filters and examine their relationships with positive implicative and implicative filters. In particular, we show that, under suitable conditions, such as purity and density, important classes of filters are nodal. We also provide examples and counterexamples to illustrate the scope and limitations of these results. We then introduce operations on the set of all nodal filters of type 1 and study the resulting algebraic structure. We prove that this structure forms an L-algebra and a bounded distributive lattice. Furthermore, by defining a suitable operation, we show that it is a residuated lattice, an MTL-algebra and a BL-algebra. We also establish that the resulting structure is a semi-De Morgan algebra, a bounded equality algebra, and a Stonean equality algebra. Finally, several additional algebraic identities and order-theoretic properties are obtained, providing further insight into the interaction between nodal filters and the algebraic structure induced by their operations.

Axioms doi: 10.3390/axioms15100764

Authors: Mahmoud Ali Bakhit Munirah Aljuaid

In this paper, we obtain new criteria for the boundedness, the compactness and the essential norm of composition operators acting from the space H∞ of bounded harmonic mappings into weighted harmonic Bloch spaces BHα, 0<α<∞, on the unit disk. These criteria are expressed in terms of the asymptotic behavior of the sequence {∥Cψqk∥BHα}k≥0, where qk(z)=zk+z¯k, and of a family of test functions Fa,n normalized by Fa,n(a)=2. We further relate these quantities to an explicit function of the symbol ψ, which yields symbol-based characterizations of boundedness and compactness, and two-sided estimates with absolute constants for the essential norm of Cψ:H∞→BHα.

Axioms doi: 10.3390/axioms15100765

Authors: Paulius Drungilas Artūras Dubickas

We study short multiplicative relations among additive translates of algebraic numbers. Every quadratic algebraic number admits infinitely many relations with three factors on each side and arbitrarily large integer shifts. For quadratic algebraic integers, we classify the minimum total length over integer tails and show that it is eventually five. We also construct infinitely many reduced relations of length eight for every cubic algebraic number using rational shifts, and prove that the integer-shift length of 23 is exactly eight.

Axioms doi: 10.3390/axioms15100763

Authors: Christopher G. Provatidis

This paper addresses the treatment of large T-elements, based on Lagrange interpolation, through the application of constraints along the horizontal and vertical mesh lines of a quadrilateral patch. Instead of resorting to manual or symbolic factorization to determine explicit closed-form expressions of the shape functions, we propose constructing a transformation matrix and applying it directly to the background tensor-product grid, thereby establishing a linear relationship between the active (primary or master) and the auxiliary (secondary or slave) degrees of freedom. Software implementation is greatly simplified when all data (boundary conditions, slave-node definitions, etc.) refer directly to the background tensor-product mesh, while the subsequent computation of the corresponding nodal values streamlines post-processing operations. Representative numerical examples illustrate the correctness of the proposed formulation and demonstrate its practical effectiveness in the analysis of field, elasticity and eigenvalue problems.

Axioms doi: 10.3390/axioms15100762

Authors: Svetlana Topalova Stela Zhelezova

A point-cyclic resolution of a 2-(v,k,1) design is invariant under a cyclic automorphism group of order v. Point-cyclic resolutions are of interest for their relations to many other combinatorial structures, and for their various applications. Resolutions of 2-(v,4,1) designs can exist if v≡4(mod 12). There are six point-cyclic resolutions of 2-(v,4,1) designs with v<64. Before the present study, v=64 was the smallest v for which neither theoretical nor constructive results on the existence of point-cyclic resolutions of 2-(v,4,1) designs were known. We establish that there are 258 point-cyclic resolutions of 2-(64,4,1) designs. The investigation is computer-aided.

Axioms doi: 10.3390/axioms15100761

Authors: Yaxuan Shen Renxia Wan

Dempster–Shafer evidence theory and rough set theory are two important mathematical tools for dealing with uncertain information, and their integration provides a powerful theoretical framework for quantifying cognitive uncertainty. The classical D-S rough set model usually adopts the basic probability assignment BPA construction method based on the cardinality of equivalence classes, which fails to fully explore the correlation information between the internal structure of knowledge granularity and the target concept, thus limiting its representation and discrimination ability. To break through this limitation, this paper proposes a novel BPA construction theory based on the inclusion-degree gradient, aiming to significantly enhance the flexibility and semantic richness of the model by finely modeling the “partial inclusion” relationship. Following the progressive logic of “from classic to extended”, this study systematically constructs a hierarchical extended model system of D-S rough sets. First, within the classical framework, the equivalent relationships between the lower approximation and belief function, and the upper approximation and plausibility function are clarified. The core innovation lies in reconstructing the traditional fixed BPA assignment into a dynamic weighting process with inclusion degree as the core variable. Then, this innovative framework is systematically extended to variable precision rough sets and probabilistic rough sets: in variable precision rough sets, a continuous weight function with parameter β as the threshold is designed to realize adaptive reward for knowledge granularity with high inclusion degree, thus obtaining the β-weighted BPA; in probabilistic rough sets, a three-region BPA assignment model of positive region, boundary region, and negative region based on (α,β) threshold division is constructed, and the differentiated mass functions of evidence for each region are given. Through rigorous mathematical derivation, the proposed models are shown to satisfy the basic axioms of Dempster–Shafer evidence theory and to possess theoretical properties including monotonicity, continuity, and compatibility with classical constructions. The numerical example further illustrates how the introduction of inclusion-degree information redistributes evidence mass and changes the resulting belief–plausibility intervals. In particular, under the considered illustrative setting, the β-weighted BPA model produces a more concentrated belief assignment and a narrower belief–plausibility interval than the classical cardinality-based construction, while the three-region BPA model explicitly represents boundary-induced nonspecific evidence. These numerical results characterize changes in the representation and concentration of evidential uncertainty and should not be interpreted as evidence of improved predictive or diagnostic accuracy.

Axioms doi: 10.3390/axioms15100760

Authors: John R. Fanchi

Parametrized Relativistic Quantum Theory (PRQT) extends conventional relativistic quantum mechanics by introducing an invariant evolution parameter (s) and a broader symmetry group. This paper presents the axiomatic and probabilistic foundations of PRQT. The resulting Stueckelberg equation is applied to the study of barrier penetration, with explicit calculations for scanning tunneling microscopy (STM). The STM application demonstrates high sensitivity to barrier width and quantitative agreement between exact and thick-barrier approximations.

Axioms doi: 10.3390/axioms15100759

Authors: Manuel Úbeda-Flores

We study bivariate copulas that have a singular component on the main diagonal and are generated by a single univariate function, the correlation function. The resulting class is that of lower semilinear copulas; in this parametrisation, known facts about it become statements about a single function, which is the conditional probability that the two coordinates coincide. Our main result is that the canonical correlation operator of each copula in the class is unitarily equivalent to multiplication by its correlation function, the unitary operator being the identity minus the Hardy averaging operator. Consequently, the spectrum of the canonical correlation operator is the essential range of the correlation function and the maximal correlation is its essential supremum; canonical functions exist if and only if the correlation function is constant on a set of positive measure, and the maximal correlation is attained if and only if the correlation function equals its essential supremum on such a set. We also treat copulas with singular components supported on a curve, a family that contains all Marshall–Olkin copulas and whose admissibility is governed by the elasticity of the curve, and derive Bayes factors for testing the equality of two binomial proportions, in closed form for the Fréchet family and as one-dimensional integrals in general.

Axioms doi: 10.3390/axioms15100758

Authors: Arifuzzaman Syed Shakaib Irfan Ahmad Almutlg Iqbal Ahmad

In this paper, we study a variational inclusion problem in a real ordered Hilbert space via an averaged-operator-based approach. The proposed framework combines Yosida and Cayley approximation operators with the logical XOR and XNOR operations, resulting in a unified iterative procedure for obtaining solutions. Using the relevant properties of averaged operators and the associated Yosida and Cayley operators, we derive conditions that guarantee the existence of solutions and the convergence of the resulting iterative process. The performance of the proposed schemes is further assessed against several existing iterative methods in terms of convergence characteristics and computational cost. Finally, we present numerical results obtained by MATLAB to corroborate the theoretical analysis and to demonstrate the effectiveness of the proposed methodology.

Axioms doi: 10.3390/axioms15100757

Authors: Hongbo Hua

The Sombor index of a graph G is SO(G)=∑uv∈E(G)(dG(u)2+dG(v)2)1/2, where dG(v) denotes the degree of v. The repetition number of G is the maximum multiplicity of an entry in its degree sequence. In this paper, we determine the maximum Sombor index among connected graphs with given repetition number r and cyclomatic number c, where c≥1 and r≥max{3,2c}. We characterize all graphs attaining the maximum value of the Sombor index. These extremal graphs consist of c cycles sharing one common vertex, together with r pendant vertices adjacent to the common vertex. For graphs outside the extremal family, we establish an explicit upper bound smaller than the maximum value. When r≥2c+1, we characterize all graphs attaining this bound, which also yields the second largest Sombor index. If r=2c, the bound is strict. We further prove a general lower bound in terms of the repetition number and a stronger bound involving both r and c for graphs with positive cyclomatic number. For c=1, the lower and upper bounds are both sharp for unicyclic graphs with given repetition number.

Axioms doi: 10.3390/axioms15100756

Authors: Sanaa Ahmed Bajri Mohammad Shane Alam

Let ℑ and ℜ be unital complex ∗-algebras, and suppose that ℑ contains a nontrivial projection ℘1. Writing ℘2=ℷℑ−℘1, assume the annihilator conditions ϑℑ℘1=0⇒ϑ=0 and ϑℑ℘2=0⇒ϑ=0. If a bijection Φ:ℑ→ℜ preserves the mixed ternary product [ϑ,ς*∘ϱ]* for all ϑ,ς,ϱ∈ℑ, then Φ is additive. Moreover, if Φ(ℷℑ)=ℷℜ, then Φ is a ∗-preserving ring isomorphism. The result yields corresponding statements for prime ∗-algebras, factor von Neumann algebras, and standard operator algebras. We also show that the identity normalization cannot in general be omitted and indicate several directions for extending the argument to broader preserver problems.

Axioms doi: 10.3390/axioms15100755

Authors: Sami H. Saif Yousef Alkhamees

Let p be a prime, let n≥2, and let R be a finite commutative local ring that is free of rank three over R0=Zpn and has residue field Fp. We show that R/pR is isomorphic either to Fp[t]/(t3) or to Fp[s,t]/(s,t)2, and that this dichotomy governs the ideal theory of R. In the first case, R≅R0[X]/(f) for a monic cubic polynomial f≡X3(modp); the ring R is a chain ring if p2∤f(0) and has embedding dimension two if p2∣f(0). In embedding dimension two, the nilpotency index t of the Jacobson radical J satisfies the sharp bounds n+2≤t≤2n+1, and we characterize the rings with J3=pJ2, for which all powers of J are given explicitly. In the second case, the embedding dimension is three, Ji=pi−1J for all i≥1, and t=n+1. In particular, no ring of embedding dimension two satisfies p∈J2. Using the left regular representation, we derive and solve the associativity equations for the structure constants and obtain an explicit six-parameter matrix model, together with matrix criteria for locality and for isomorphism. Finally, we show that R is a Frobenius ring if and only if its embedding dimension is at most two.

Axioms doi: 10.3390/axioms15100754

Authors: Swapnil Swagat Phookan Sajadul Hussain Partha Jyoti Hazarika Diego I. Gallardo Osvaldo Venegas

Circular data arise naturally in a wide range of scientific fields, including biology, geology, meteorology, and environmental sciences, where conventional linear statistical methods are inappropriate unless the circular structure of the observations is taken into account. Motivated by the need for parsimonious, analytically tractable models for unimodal directional data, we introduce a new one-parameter circular model, the Wrapped Q-Lindley (WQL) distribution. The proposed model is obtained by wrapping the Q-Lindley distribution and has a specific unimodal asymmetric shape. Closed-form or analytically tractable expressions are derived for several important distributional properties, including the characteristic function, trigonometric moments, circular descriptive measures, and Rényi entropy. We investigate parameter estimation using ten classical estimation methods, along with Bayesian inference, under four symmetric and asymmetric loss functions. A Monte Carlo simulation study evaluates and compares the finite-sample performance of the competing estimators. The practical usefulness of the proposed model is illustrated through two real directional datasets involving sun compass orientation measurements of starhead topminnows and wind direction observations. Comparisons with several established circular models indicate that the WQL distribution provides competitive fitting performance for the considered datasets. We discuss its parsimonious structure and analytical tractability, along with the implications of its structural discontinuity at the wrap point, in the context of directional data analysis.

Axioms doi: 10.3390/axioms15100753

Authors: Marta Na Chen Wenchang Chu

Infinite series about reciprocal products of Fibonacci and Lucas numbers are systematically examined. This is done by incorporating the telescoping method, Cassini-like identities and partial fractions, as well as the induction principle. Numerous closed formulae are presented with most of them not having appeared previously.

Axioms doi: 10.3390/axioms15100752

Authors: Richard F. Patterson Ekrem Savaş Sami M. Hamid

Let F be a complete, non-trivially valued, non-Archimedean field of characteristic zero. This paper studies the space V^λ of λ-almost convergent sequences over F and its paranormed generalization V^λ(p), establishing it as a complete non-Archimedean topological vector space for every admissible λ. The classical inclusion c⊂V^λ holds if and only if F has residue characteristic zero and m−λm is unbounded. The matrix class (c,V^λ) is characterized by means of an ultrametric gliding hump argument. The strong almost space [V^,λ] and the almost λ-statistical space s^λ coincide on l∞, yet over Qp this common space is not contained in V^λ, a separation that reflects the arithmetic of the base field.

Axioms doi: 10.3390/axioms15100751

Authors: Maoqin Yuan Juan Zhang Mingyang Li Lixiu Dong

We develop an energy-stable variable-step BDF2-based scalar auxiliary variable (SAV) finite element scheme for a droplet thin-film coarsening model with a Lennard–Jones singular potential. The method uses continuous piecewise linear finite elements in space and retains the original linear SAV update. We prove discrete mass conservation, modified-energy decay under an admissible step-ratio condition, and solvability of the coupled algebraic system. A Sherman–Morrison realization exploits the rank-one coupling and reduces each accepted time step to two solves with the same sparse base matrix. Numerical experiments assess temporal and spatial convergence, SAV-shift sensitivity, two-droplet mesh dependence, long-time coarsening, and solver efficiency. The film-height variable exhibits second-order temporal convergence and the expected spatial rates, whereas the auxiliary variable is first-order accurate; the consistency audit shows no hidden cancellation. The computations also verify mass conservation, modified-energy decay, positive film heights in the reported runs, and the efficiency of the sparse solver.

Axioms doi: 10.3390/axioms15100750

Authors: Ziyang Pan Zhiping Xiong Yingying Qin

By using the forward order law for the Drazin inverse of matrix products, we investigate the Drazin inverse of 2×2 block triangular matrices. A necessary and sufficient condition for ABOCD=ADBDOCD is presented. Notably, this condition is expressed exclusively in terms of rank equalities.

Axioms doi: 10.3390/axioms15100749

Authors: Mona Alsulami Mariam Al-Mazmumy Bashair Banweer

This paper proposes two modified Adomian decomposition method-based algorithms for solving fractional delay boundary value problems. The algorithms effectively handle linear and nonlinear problems by constructing rapidly convergent series solutions using Adomian polynomials. Several examples are presented to showcase the exactness and effectiveness of the devised algorithms, where the results portray that the new methods achieve exact solutions with fewer components compared to existing numerical techniques.

Axioms doi: 10.3390/axioms15100748

Authors: Jing Tang Xing

An energy-flow approach (EFA) is developed to reveal the critical flutter speed (CFS) of an air–structure electromagnetic interactive flutter system. To illustrate the theory and method, a two-DOF integrated linear/nonlinear coupling system is investigated, in which the electromagnetic isolator adds damping, increasing CFS and collecting vibration energy. Following a summary of EFA applicable to the system, numerical simulations on a simplified aeroelastic system of natural frequencies 3 and 0.6 are completed. The results reveal that (a) EFA can effectively identify CFSs of linear/nonlinear systems, at which the systems show non-damping periodical motions with periodical phase space diagrams and time histories of generalised potential energy (GPE)/kinetic energy (GKE) and zero time-averaged energy flows; (b) under and over CFSs, the systems respectively show damped and convergence motions; and (c) the electromagnetic isolator clearly increases CFSs and enables the collection of vibration energy. To check possible chaotic motions for a non-damping flutter system of two natural frequencies without their common rational period, a system with two frequencies 1 and 3 is examined. At zero wind speed, its free vibrations under the initial conditions are chaotic. For cases of air excitation, given the inertia, stiffness, and damping coupling mechanism contributed by wind flows, the original two frequencies with no common period gradually close in on each other until they reach the corresponding flutter speeds, producing periodical motions of single frequency. The results under/over CFS for linear systems show similar damped/convergence oscillations. However, for nonlinear systems, the under-flutter speed gives damped oscillation, while the over-flutter one still shows periodical motion consisting of two mixed frequencies with their common period. The results confirm that the proposed EFA is an effective general means to tackle any complex linear/nonlinear integrated flutter-control system for aircraft designs as well as green energy collection.

Axioms doi: 10.3390/axioms15100747

Authors: Ibrahim Alliu Lili Yu Hani Samawi

This study investigates the comparative performance of the Weibull accelerated failure time (AFT) model and the Cox proportional hazards (PH) model for interval-censored Weibull data. The study emphasizes baseline survival function precision and confidence interval coverage at key quantile-based time points across the survival curve. Through Monte Carlo simulation experiments and applications to the Breast Cosmesis and Diabetic Retinopathy datasets, we compare model performance using bias, empirical variance, estimated variance, and 95% coverage probability. We also evaluate two bootstrap confidence interval methods, the bias-corrected and accelerated (BCa) bootstrap and the Wald bootstrap, across the survival, log-survival, and complementary log–log scales. Results show that while both models provide reasonable regression coefficient inference, the Weibull AFT model consistently yields lower empirical variance for baseline survival estimation when the Weibull assumption holds. The complementary log–log scale provides the most stable coverage overall. These findings support the Weibull AFT model when its distributional assumption is plausible.

Axioms doi: 10.3390/axioms15100746

Authors: Ayşe Metin Karakaş Fatma Bulut Murat Karakaş

This study introduces the Cauchy–Modified Garima (CMG) distribution, a flexible two-parameter lifetime model obtained by applying a Cauchy-type generator to the Modified Garima distribution. Several statistical properties of the model are derived, and parameter estimation is investigated using maximum likelihood and Bayesian methods. Monte Carlo simulations show that both estimation procedures become more accurate as the sample size increases. Applications to real lifetime data demonstrate that the CMG distribution provides a competitive and flexible alternative to several classical lifetime models.

Axioms doi: 10.3390/axioms15100745

Authors: Edoardo Ballico

Chiantini and others in Arxiv:2026.25060v1 introduced the Weddle locus W(Z,d) of a finite set in the framework of Veronese embeddings of Pn. It is the Zariski closure in Pn of an even more interesting algebraic object, W0(Z,d). In this paper, we give several generalizations of this definition using zero-dimensional schemes: an abstract set-up and the extensions of the classical case to multiprojective spaces and projections from more than one point. One of our results includes “the sets are as small as possible for the given numerical data”. We raise a few open questions. Some of these are for general finite sets with prescribed cardinality. For the classical Weddle locus, we show that W(Z,d)=∅ if Z is a general subset of Pn with at least n+(n+d−1n−1) points. For arbitrary zero-dimensional schemes Z, we describe it if deg(Z)≤2d+1. Over a smaller range, we describe it for multiprojective spaces.

Axioms doi: 10.3390/axioms15100744

Authors: Wolf-Dieter Richter

We start from a pure vector update of the concepts of complex numbers and the characteristic function of a probability distribution, free from the imaginary unit, and update some basic properties of the latter. In particular, we derive vector inverse formulas for probability distributions and density functions, provide a new geometric proof of the convergence of suitably centered and normalized vector-powers of characteristic functions, and thus provide an update to the proof of the central limit theorem. Moreover, we demonstrate how to eliminate the imaginary unit from various equations—such as Schrödinger’s equation—by transitioning to vector equations.

Axioms doi: 10.3390/axioms15100743

Authors: Shatha Qandos Osama Moaaz

This study investigates the dynamics of solutions to nonlinear difference equations. The equation under study is distinguished by its dynamical richness and contains many interesting special cases, including May’s Host–Parasitoid Model. We establish sufficient criteria to ensure the stability, periodicity, boundedness, and convergence of their solutions. Furthermore, to support the theoretical results, we apply the theoretical results to some special cases and perform numerical simulations of some solutions.

Axioms doi: 10.3390/axioms15100742

Authors: Sibel Başkal Marilyn E. Noz

In this review article, we first introduce the density matrix, along with its properties and time evolution. We note that one of the principal uses of the density matrix is to calculate measurement probabilities in quantum mechanics. When only some of the variables can be measured, it is necessary to integrate out the variables that are inaccessible to measurement. The resulting subsystem is then analogous to an open quantum system, where the unmeasured variables belong to an unspecified environment. Coupled and Lorentz-covariant harmonic oscillators are described and their differences are explained. We introduce the density matrix for two harmonic oscillators and demonstrate how they become entangled as well as entangled excited states of the Lorentz-covariant harmonic oscillator. We elaborate on the concept of von Neumann entropy and show how entropy increases when not all the variables of the system can be measured. We use the timeseparation between quarks as an example of an unmeasurable quantity and then present a discussion of von Neumann entropy and entanglement. We include “boiling quarksgluons”, whose most interesting feature is the transition from the confinement phase to the plasma phase, to illustrate the application of the density function and entropy. In this context, we discuss the quark-parton puzzle and a possible explanation.

Axioms doi: 10.3390/axioms15100741

Authors: Shanshan Li Zaizai Yan

This paper investigates degradation modeling and failure time prediction for constant-stress accelerated degradation tests (CSADT) using a generalized Wiener process. Owing to differences in manufacturing processes and operating environments, the degradation trajectories may vary across individual units of the same product. To more accurately characterize such complex degradation behavior, this paper proposes a generalized Wiener process model with mixed random effects that account for heterogeneity in both the drift coefficient and the diffusion coefficient, and develops corresponding parameter estimation methods. Using the total probability formula, we derive an approximate expression for the probability density function (PDF) of the failure time distribution, thereby providing a theoretical basis for product reliability assessment. The necessity of the proposed model is verified through simulation examples, and a case study on circuit voltage performance degradation demonstrates the effectiveness and practicality of the proposed model. The results show that the proposed model can improve the accuracy of product failure time and reliability predictions, demonstrating its practical value in engineering applications.

Axioms doi: 10.3390/axioms15100739

Authors: Refah Alotaibi Mazen Nassar Ahmed Elshahhat

The analysis of lifetime data in the presence of multiple competing causes of failure remains a fundamental problem in reliability and survival studies, particularly when observations are obtained under complex and resource-constrained censoring mechanisms. In this context, the adaptive progressive first-failure censoring scheme has emerged as a flexible design that allows the removal mechanism to adapt dynamically to the observed evolution of failures, making it well suited for modern life-testing experiments involving grouped units and limited resources. This paper develops statistical inference for the Xgamma competing risks model under the adaptive progressive first-failure censoring scheme. Both classical and Bayesian approaches are employed for parameter estimation and reliability analysis. Maximum likelihood estimation is carried out despite the inherent analytical complexity of the resulting nonlinear likelihood function, and two types of approximate confidence intervals are constructed based on direct and transformation-based normal approximations. From a Bayesian perspective, independent gamma prior distributions are assumed, and posterior inference is performed using Markov chain Monte Carlo methods to address the intractability of the posterior distribution. Bayesian point estimators, along with symmetric credible intervals and highest posterior density intervals, are obtained from the generated posterior samples. The proposed methods are evaluated through an extensive simulation study conducted under a variety of experimental settings. In addition, the practical applicability of the developed methodology is illustrated through the analysis of real data involving relapse times of multiple myeloma patients. The results demonstrate that the proposed approach yields accurate estimation and reliable uncertainty quantification, confirming its effectiveness for analyzing competing risks data under advanced censoring schemes.

Axioms doi: 10.3390/axioms15100740

Authors: Atanas Ilchev Vanya Ivanova Rumen Marinov Diana Nedelcheva Angel Todorov Boyan Zlatanov

This paper introduces and studies alternating points of two set-valued mappings acting between two 0-complete strong partial b-metric spaces. An alternating point is a mutually compatible pair of states, each of which belongs to the image of the other under the corresponding mapping. Such a pair simultaneously generates fixed points of the two set-valued compositions. We establish a local existence theorem under nonlinear contractive conditions expressed by one-sided excess functionals on prescribed closed balls. The two successive transitions are controlled by different comparison functions, while the Bianchini–Grandolfi summability condition is imposed only on their composition, which represents one complete alternating cycle. The proof constructs two interdependent sequences, shows that they remain within their respective localization regions, and establishes their zero-Cauchy convergence by means of chain estimates adapted to strong partial b-metrics. A finite discrete model is developed to verify the assumptions of the theorem explicitly and to demonstrate that several alternating points may coexist. The same abstract model is then interpreted in two different settings: as a set-valued price–demand adjustment process and as an alternating process of lexical enrichment and grammatical stabilization. These applications illustrate how the proposed framework can describe nonunique adjustment and refinement processes with multiple mutually compatible terminal configurations.

Axioms doi: 10.3390/axioms15100738

Authors: David J. Olive Sanjuka Johana Lemonge

We consider hypothesis tests for the multiple linear regression model with ordinary least squares if the predictor variables have been scaled to have unit sample variance. Some tests are unchanged, but confidence intervals, confidence regions, and some tests are no longer valid. Similar results are obtained for a robust regression estimator.

Axioms doi: 10.3390/axioms15100737

Authors: Muhammad Imran Liaqat Abdulaziz Khalid Alsharidi

We consider a coupled system of Caputo–Katugampola integro–fractional stochastic differential equations with Volterra memory, nonlocal integral boundary conditions, Wiener perturbations, and compensated Poisson jumps. The main purpose is to give a mathematically explicit fixed-point treatment in which the two components are solved simultaneously on a product space. A Green representation is derived directly from the two boundary conditions; this derivation fixes the boundary denominator and the complete Green kernel used in all subsequent estimates. Integral contractors are formulated with a correction operator and a residual modulus. In particular, a cubic nonlinearity is used to show that a coefficient need not be globally Lipschitz on the whole state space in order to satisfy the contractor hypothesis on an invariant ball. The moment order p≥2 is fixed throughout each theorem, so all contractor and contraction constants are explicitly allowed to depend on p. The coupled fixed-point problem is represented by a nonnegative 2×2 matrix Qp and the condition ρ(Qp)<1 is used as the actual contraction condition. Ulam–Hyers stability is obtained from the same matrix by an explicit Neumann-series estimate. For trajectory control we distinguish ordinary predictable finite-energy controls from generalized semimartingale controls; this distinction is necessary because a two-point stochastic boundary-value problem is generally noncausal and an ordinary drift control cannot cancel independent martingale terms pathwise. A numerical experiment based on the same Green representation verifies the contraction criterion, the non-Lipschitz contractor construction, the Ulam–Hyers estimate, and the generalized trajectory-control mechanism. The presentation also records the precise scope of the adaptedness assumption for the stochastic integrals.

Axioms doi: 10.3390/axioms15100736

Authors: Dong-Fang Xie Wei-Xue Shi Wei-Feng Xuan

In this paper, we investigate how the properties of the nowhere dense subspaces reflect those of the entire space. We call a space X an SAC-P space if the set of accumulation points of any infinite family of mutually disjoint open subsets of X is a non-empty set having property P. We prove that for a regular space X, X is SAC-compact precisely when it is compact. In general, for a regular space X, X is SAC-P (here, P is a property which closed subsets inherit) exactly when X is feebly compact and each closed nowhere dense subspace of X has property P. Similar to discrete reflexivity, we introduce the notion of nowhere dense reflexivity. A property P is nowhere densely reflexive inA if each space X∈A has property P precisely when each closed nowhere dense subspace of X has property P. We prove that compactness, linear Lindelöfness and initial κ-compactness are all nowhere densely reflexive both in the class of feebly compact spaces and in that of dense-in-itself spaces. For topological groups, we prove that if G is a non-compact connected feathered topological group, then G is metrizable precisely when each closed nowhere dense subgroup of G is metrizable.

Axioms doi: 10.3390/axioms15100735

Authors: José Luis Díaz Palencia

Quantum fields on a time-dependent geometry can undergo particle production, but this does not establish a self-amplifying response of the geometry. We study a conditional two-amplitude closure for a signed curvature perturbation and a signed renormalized stress response. The physical assumptions are specified through constrained projection, a low-frequency response expansion, and a stable single-pole approximation; none is asserted to hold for every semiclassical state. A quantitative memory-remainder estimate and a conditional scalar-field example make the scope of this approximation explicit. For the resulting cooperative reaction–diffusion system, a weighted spectral argument identifies the first loss of stability with the homogeneous mode. For the homogeneous cubic completion, a coercive gradient potential proves global existence and convergence to an equilibrium, including asymptotic stability at the critical parameter. We also establish the local spatial bifurcation, distinguish quintic corrections from a degenerate bifurcation, and classify a constant symmetry-breaking perturbation. Renormalization of the cosmological constant, conservation, higher-curvature terms, and the distinction between physical and effective-theory runaway solutions are treated explicitly. The contribution is a response-level diagnostic and its mathematical limitations, rather than a new microscopic theory or an observational cosmological prediction.

Axioms doi: 10.3390/axioms15100734

Authors: Mario Lefebvre Antoine Habib

Let τ(x) be the first time that a jump-diffusion process with positive and negative jumps leaves the interval (a,b). The moment-generating function M(x) of τ(x) satisfies an integro-differential equation (IDE). In the case of exponential jumps, we transform this IDE into an ordinary differential equation (ODE). This is similar to the IDE satisfied by m(x):=E[τ(x)]. Explicit solutions to particular problems are obtained for both M(x) and m(x) by solving the appropriate ODEs.

Axioms doi: 10.3390/axioms15100733

Authors: Ivan Ginchev Stoil I. Ivanov Boyan Zlatanov

In the present paper, we investigate the local and the semilocal convergence of a recently constructed family of accelerated simultaneous methods with two corrections applied to polynomials with simple and multiple zeros. A new local convergence theorem that provides explicit domains of initial conditions and error estimates ensuring the Q-convergence of the methods from the first iteration is established. Based on this result, we derive a semilocal convergence theorem, for simple zeros, with computable initial conditions and an error estimate, making it directly applicable in practice. Three numerical tests illustrate the applicability of the theoretical results.