Axioms

16 pages, 311 KB

Open AccessArticle

Robust Synchronization of Time-Fractional Memristive Hopfield Neural Networks

by Yuncheng You

Axioms 2026, 15(1), 37; https://doi.org/10.3390/axioms15010037 - 2 Jan 2026

We introduce and study robust synchronization of time-fractional Hopfield neural networks with memristive synapses and Hebbian learning. This novel model of artificial neural networks exhibits strong memory and long-range path dependence. By scaled group estimates and analysis of fractional differencing equations, it is [...] Read more.

We introduce and study robust synchronization of time-fractional Hopfield neural networks with memristive synapses and Hebbian learning. This novel model of artificial neural networks exhibits strong memory and long-range path dependence. By scaled group estimates and analysis of fractional differencing equations, it is proved that under rather general assumptions the solution dynamics are globally dissipative and there exists a threshold condition for achieving robust synchronization of the entire neural networks if this condition is satisfied by the interneuron coupling strength. The synchronizing threshold is explicitly expressed in terms of the original parameters in the model equations and strictly decreasing for the fractional order

α \in (0, 1)

. This result makes a breakthrough in the exploration of fractional global and longtime dynamics for AI mathematical models. Full article

(This article belongs to the Section Mathematical Analysis)

14 pages, 292 KB

Open AccessArticle

Radical Numerical Semigroups

by María Ángeles Moreno-Frías and José C. Rosales

Axioms 2026, 15(1), 36; https://doi.org/10.3390/axioms15010036 - 2 Jan 2026

Abstract

This work contributes to the study of radical numerical semigroups. If

n \in Z

where

n \geq 2

, then the product of all its prime positive divisors is called the radical of n. It is denoted by

r (n)

[...] Read more.

This work contributes to the study of radical numerical semigroups. If

n \in Z

where

n \geq 2

, then the product of all its prime positive divisors is called the radical of n. It is denoted by

r (n)

. A radical numerical semigroup is a numerical semigroup S such that

s + r (s) \in S

for every

s \in S ∖ {0}

. We present three algorithms that will help us understand the structure of radical semigroups. These algorithms allow us to calculate all radical numerical semigroups with a fixed genus, with a fixed Frobenius number, as well as with a fixed multiplicity. Furthermore, given X, a set of positive integers such that

gcd (X) = 1

, we will prove the existence of the smallest radical semigroup containing X. We will also present an algorithm to obtain it. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 2nd Edition)

15 pages, 288 KB

Open AccessArticle

On the Existence of Optimal (v,5,1) and (v,6,1) Binary Cyclically Permutable Constant-Weight Codes

by Tsonka Baicheva and Svetlana Topalova

Axioms 2026, 15(1), 35; https://doi.org/10.3390/axioms15010035 - 1 Jan 2026

Abstract

The problem of the existence of optimal

(v, k, 1)

binary cyclically permutable constant-weight (CPCW) codes has been completely solved for codeword weights

k < 5

. We consider the smallest open cases, namely

k = 5

and [...] Read more.

The problem of the existence of optimal

(v, k, 1)

binary cyclically permutable constant-weight (CPCW) codes has been completely solved for codeword weights

k < 5

. We consider the smallest open cases, namely

k = 5

and

k = 6

. We present such codes for small values of the code length v and derive necessary conditions for the existence of optimal

(k (k - 1) t + 2, k, 1)

CPCW codes. These necessary conditions can be used to construct such codes, as well as to show that optimal codes with some parameters do not exist. In particular, we use them to prove that an optimal

(92, 6, 1)

CPCW code does not exist. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 2nd Edition)

64 pages, 11024 KB

Open AccessArticle

Geometric and Topological Analysis of Financial Market Structure: Evidence from Turkish Markets and the 2022–2023 Structural Break

by Larissa Margareta Batrancea, Mehmet Ali Balcı, Ömer Akgüller and Lucian Gaban

Axioms 2026, 15(1), 34; https://doi.org/10.3390/axioms15010034 - 1 Jan 2026

Abstract

The financial market correlation structure exhibits dynamic evolution with implications for systemic risk and diversification benefits; yet, conventional approaches using correlation magnitude monitoring may miss fundamental organizational changes. This study develops an integrated geometric–topological framework synthesizing Riemannian manifold geometry with discrete network topology [...] Read more.

The financial market correlation structure exhibits dynamic evolution with implications for systemic risk and diversification benefits; yet, conventional approaches using correlation magnitude monitoring may miss fundamental organizational changes. This study develops an integrated geometric–topological framework synthesizing Riemannian manifold geometry with discrete network topology to characterize market structure transformations, applying the methodology to Turkish financial markets spanning May 2015–May 2025. The analysis employs intrinsic dimensionality estimation, curvature tensor computation, and Forman–Ricci curvature calculation alongside correlation network topology measures across seventeen variables, including sectoral equity indices and macroeconomic indicators. The results document an extraordinary mid-2022 structural break featuring dimensional collapse from

2.4

to

0.43

dimensions, network density surge to

0.97

, and Ricci curvature spike to

16.0

, establishing a persistent hypersynchronized regime evident through 2024–2025 with density elevated by

44 %

and the Ricci curvature being amplified

258 %

above the baseline. A comparative crisis analysis reveals heterogeneous structural signatures distinguishing COVID-19 geometric vulnerability from currency crisis modular reorganization. Macro-financial linkages identify credit growth as primary structural driver, achieving

r = 0.504

correlation with geometric resilience. The findings demonstrate that markets undergo permanent structural transformations responding systematically to monetary–credit conditions, with implications for surveillance frameworks, stress testing design, and macroprudential policy. Full article

(This article belongs to the Special Issue Recent Developments in Mathematical and Statistical Finance)

18 pages, 296 KB

Open AccessArticle

Parameterized Anti-Periodic Problems: Existence and Ulam-Hyers Stability for Fractional p(t)-Laplacian Langevin Equations

by Fangfang Hu, Weimin Hu and Xiaoxiao Cui

Axioms 2026, 15(1), 33; https://doi.org/10.3390/axioms15010033 - 1 Jan 2026

Abstract

This paper investigates a novel class of fractional Langevin equations, which introduces a time-varying p(t)-Laplacian operator and parameterized anti-periodic boundary conditions. This approach overcomes the limitations of traditional models characterized by constant diffusion exponents and fixed boundary locations. Under non-compactness conditions, the existence [...] Read more.

This paper investigates a novel class of fractional Langevin equations, which introduces a time-varying p(t)-Laplacian operator and parameterized anti-periodic boundary conditions. This approach overcomes the limitations of traditional models characterized by constant diffusion exponents and fixed boundary locations. Under non-compactness conditions, the existence of solutions is established by applying Schaefer’s fixed-point theorem, which significantly relaxes the conventional constraints on the nonlinear term. Moreover, by imposing a Lipschitz condition on the nonlinear term, a Ulam–Hyers-type stability criterion for the coupled system is derived. This work not only extends the relevant stability theory but also provides a rigorous theoretical foundation for error control in practical applications. The effectiveness of the theoretical results is validated through numerical examples. Full article

25 pages, 2309 KB

Open AccessArticle

Dynamics of a Fishery Management Model with Predation-Induced Fear Effect, Impulsive Nonlinear Harvesting Prey and Predator Seasonally Migrating Between Two Patches

by Bingying Gao, Jianjun Jiao, Zeli Zhou and Shirui Zhang

Axioms 2026, 15(1), 32; https://doi.org/10.3390/axioms15010032 - 31 Dec 2025

Abstract

This paper proposes a novel fishery management model that integrates three critical ecological factors: the prey (fish) fear effect, impulsive nonlinear harvesting, and predator seasonal migration. It is demonstrated that all solutions of the proposed system are uniformly ultimately bounded. We establish the [...] Read more.

This paper proposes a novel fishery management model that integrates three critical ecological factors: the prey (fish) fear effect, impulsive nonlinear harvesting, and predator seasonal migration. It is demonstrated that all solutions of the proposed system are uniformly ultimately bounded. We establish the conditions for the local and global asymptotic stability of the prey-extinction periodic solution, derive the permanence criteria for the system, and determine the threshold condition for prey extinction. Numerical simulations are conducted to validate the theoretical findings. These simulations also help identify the key parameters influencing the threshold condition and reveal the complex dynamics of the system. The results provide significant insights for fishery management, suggesting that regulating harvesting intensity and timing, as well as considering predator migration mortality, are crucial for sustaining fish populations and preventing overexploitation. Full article

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18 pages, 1458 KB

Open AccessArticle

Bifurcation Analysis of a Generalist Predator-Prey Model with Holling Type II Harvesting

by Mengxin He and Yiqin Wang

Axioms 2026, 15(1), 31; https://doi.org/10.3390/axioms15010031 - 30 Dec 2025

Abstract

In this paper, we consider a generalist predator–prey model with nonlinear harvesting, which has at most eight non-negative equilibria. We prove that the double positive equilibrium is a cusp of codimension up to 3; therefore, the system exhibits a cusp-type degenerate Bogdanov–Takens bifurcation [...] Read more.

In this paper, we consider a generalist predator–prey model with nonlinear harvesting, which has at most eight non-negative equilibria. We prove that the double positive equilibrium is a cusp of codimension up to 3; therefore, the system exhibits a cusp-type degenerate Bogdanov–Takens bifurcation of the same codimension. The elementary antisaddle equilibrium can act as a weak focus of the order of no more than two, giving rise to a degenerate Hopf bifurcation of codimension up to two. These high-codimension bifurcations identify organizing centers in parameter space, indicating regions where the ecosystem is highly sensitive and prone to abrupt regime shifts. Our results indicate that the generalist predator can induce a richer bifurcation phenomenon and more complex dynamics and can drive the system to certain desired stable states. Full article

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20 pages, 2273 KB

Open AccessArticle

The Optimal Robust Investment Problem in the Foreign Stock Market of an Ambiguity-Averse Insurer

by Linlin Tian, Yixuan Tian and Xiaoyi Zhang

Axioms 2026, 15(1), 30; https://doi.org/10.3390/axioms15010030 - 29 Dec 2025

Abstract

To address the need for robust investment strategies in an increasingly uncertain global market, this study focuses on an ambiguity-averse insurer facing exchange rate uncertainty while investing in a foreign stock market. The insurer’s surplus is modeled via a classical compound Poisson process, [...] Read more.

To address the need for robust investment strategies in an increasingly uncertain global market, this study focuses on an ambiguity-averse insurer facing exchange rate uncertainty while investing in a foreign stock market. The insurer’s surplus is modeled via a classical compound Poisson process, and exchange rate dynamics are captured using an Ornstein–Uhlenbeck process for the drift component. Within the framework of maximizing expected exponential utility of terminal wealth, we derive and solve the Hamilton–Jacobi–Bellman equation to characterize the optimal investment strategy and the associated value function. Finally, a numerical example illustrates how varying model parameters influences the insurer’s optimal investment behavior. Full article

(This article belongs to the Special Issue Advances in Financial Mathematics and Stochastic Processes)

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28 pages, 1567 KB

Open AccessArticle

Entropy and Heat Transfer in Casson-MHD Flow Through Inclined Permeable Channels via Fibonacci Wavelets

by Azhar Iqbal, Vivek, Khidir Shaib Mohamed, Alawia Adam, Waseem Ahmad Khan and Wei Sin Koh

Axioms 2026, 15(1), 29; https://doi.org/10.3390/axioms15010029 - 29 Dec 2025

Abstract

This article investigates the influence of wall permeability on channel flows and addresses the lack of studies that quantify entropy generation in magnetized Casson fluid models using wavelet-based numerical schemes. We introduce a Fibonacci Wavelet Collocation Method (FWCM) to efficiently solve the transformed [...] Read more.

This article investigates the influence of wall permeability on channel flows and addresses the lack of studies that quantify entropy generation in magnetized Casson fluid models using wavelet-based numerical schemes. We introduce a Fibonacci Wavelet Collocation Method (FWCM) to efficiently solve the transformed nonlinear ordinary differential equations and demonstrate its applicability to the coupled momentum and energy equations. The analysis includes detailed graphical and numerical evaluations of entropy generation, temperature, and velocity fields, along with the Bejan number, Nusselt number, and skin-friction variations. The results reveal that entropy generation increases by approximately 18–22% with a higher Biot number and by nearly 15% with increasing Grashof number, while it decreases by about 12% for higher Eckert numbers. Magnetic field strength exhibits a dual effect, producing both suppressing and enhancing behaviors depending on parameter ranges. The FWCM solutions show strong agreement with previously published data, confirming both accuracy and robustness. Full article

(This article belongs to the Special Issue Recent Progress in Computational Fluid Dynamics)

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51 pages, 5351 KB

Open AccessArticle

Isogeometric Transfinite Elements: A Unified B-Spline Framework for Arbitrary Node Layouts

by Christopher G. Provatidis

Axioms 2026, 15(1), 28; https://doi.org/10.3390/axioms15010028 - 29 Dec 2025

Abstract

This paper presents a unified framework for constructing partially unstructured B-spline transfinite finite elements with arbitrary nodal distributions. Three novel, distinct classes of elements are investigated and compared with older single Coons-patch elements. The first consists of classical transfinite elements reformulated using B-spline [...] Read more.

This paper presents a unified framework for constructing partially unstructured B-spline transfinite finite elements with arbitrary nodal distributions. Three novel, distinct classes of elements are investigated and compared with older single Coons-patch elements. The first consists of classical transfinite elements reformulated using B-spline basis functions. The second includes elements defined by arbitrary control point networks arranged in parallel layers along one direction. The third features arbitrarily placed boundary nodes combined with a tensor-product structure in the interior. For all three classes, novel macro-element formulations are introduced, enabling flexible and customizable nodal configurations while preserving the partition of unity property. The key innovation lies in reinterpreting the generalized coefficients as discrete samples of an underlying continuous univariate function, which is independently approximated at each station in the transfinite element. This perspective generalizes the classical transfinite interpolation by allowing both the blending functions and the univariate trial functions to be defined using non-cardinal bases such as Bernstein polynomials or B-splines, offering enhanced adaptability for complex geometries and nonuniform node layouts. Full article

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24 pages, 359 KB

Open AccessArticle

On m-Isometric and m-Symmetric Operators of Elementary Operators

by Bhagwati Prashad Duggal

Axioms 2026, 15(1), 27; https://doi.org/10.3390/axioms15010027 - 29 Dec 2025

Abstract

Given Hilbert space operators

A, B

, and X, let

▵_{A, B}

and

δ_{A, B}

denote, respectively, the elementary operators

▵_{A, B} (X) = I - A X B

and the generalised derivation [...] Read more.

Given Hilbert space operators

A, B

, and X, let

▵_{A, B}

and

δ_{A, B}

denote, respectively, the elementary operators

▵_{A, B} (X) = I - A X B

and the generalised derivation

δ_{A, B} (X) = A X - X B

. This paper considers the structure of operators

D_{d_{1}, d_{2}}^{m} (I) = 0

and

D_{d_{1}, d_{2}}^{m}

compact, where m is a positive integer,

D = ▵

or

δ

,

d_{1} = ▵_{A^{*}, B^{*}}

or

δ_{A^{*}, B^{*}}

and

d_{2} = ▵_{A, B}

or

δ_{A, B}

. This is a continuation of the work performed by C. Gu for the case where

▵_{δ_{A^{*}, B^{*}}, δ_{A, B}}^{m} (I) = 0

, and the author with I.H. Kim for the cases where

▵_{δ_{A^{*}, B^{*}}, δ_{A, B}}^{m} (I) = 0

or

▵_{δ_{A^{*}, B^{*}}, δ_{A, B}}^{m}

is compact, and

δ_{▵_{A^{*}, B^{*}}, ▵_{A, B}}^{m} (I) = 0

or

δ_{▵_{A^{*}, B^{*}}, δ_{A, B}}^{m}

is compact. Operators

D_{d_{1}, d_{2}}^{m} (I) = 0

are examples of operators with a finite spectrum; indeed, the operators

A, B

have at most a two-point spectrum, and if

D_{d_{1}, d_{2}}^{m}

is compact, then (the non-nilpotent operators)

A, B

are algebraic.

D_{d_{1}, d_{2}}^{m} (I) = 0

implies

D_{d_{1}, d_{2}}^{n} (I) = 0

for integers

n \geq m

: the reverse implication, however, fails. It is proved that

D_{d_{1}, d_{2}}^{m} (I) = 0

implies

D_{d_{1}, d_{2}} (I) = 0

if and only if of A and B (are normal and hence) satisfy a Putnam–Fuglede commutativity property. Full article

(This article belongs to the Section Mathematical Analysis)

9 pages, 321 KB

Open AccessArticle

The Complete Strong Version of Blundon’s Inequality

by Dorin Andrica, Ovidiu Bagdasar, Cătălin Barbu and Laurian-Ioan Pişcoran

Axioms 2026, 15(1), 26; https://doi.org/10.3390/axioms15010026 - 29 Dec 2025

Abstract

In this paper we present some key results related to Blundon’s inequality, its long history, geometric interpretations and implications, as well as highlight some connections to results in other fields of mathematics. We make a case that this is a fundamental inequality in [...] Read more.

In this paper we present some key results related to Blundon’s inequality, its long history, geometric interpretations and implications, as well as highlight some connections to results in other fields of mathematics. We make a case that this is a fundamental inequality in triangle geometry. Also, we provide a new proof for the inequalities (8) and we generalize the strong version of Blundon’s inequalities presented in Theorem 1. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics, 2nd Edition)

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19 pages, 21031 KB

Open AccessArticle

Bifurcation Analysis of a Semilinear Generalized Friction System with Time-Delayed Feedback Control

by Haicheng Liu, Yanfeng Li and Xuejiao Liu

Axioms 2026, 15(1), 25; https://doi.org/10.3390/axioms15010025 - 28 Dec 2025

Abstract

In this paper, we investigate a semilinear parabolic friction system with time-delay feedback control and diffusion. This model more accurately describes the coupled dynamic behavior between vibrations induced by time-delayed control forces and the diffusion-driven evolution of material surface properties in practical friction [...] Read more.

In this paper, we investigate a semilinear parabolic friction system with time-delay feedback control and diffusion. This model more accurately describes the coupled dynamic behavior between vibrations induced by time-delayed control forces and the diffusion-driven evolution of material surface properties in practical friction processes. Through eigenvalue analysis, it is proven that the system’s stability does not vary monotonically with parameters. Instead, as the time delay varies, the system undergoes a finite number of alternating switches between stability and instability, before eventually losing its stability. The established stability criteria and bifurcation formulae can provide a predictive basis for and strategies to avoid the frictional vibration caused by time-delayed feedback in mechanical systems, providing significant guidance for vibration-reducing design and control parameter optimization in equipment such as braking systems and precision machine tools. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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22 pages, 425 KB

Open AccessArticle

Fractional Black–Scholes Under Memory Effects: A Sixth-Order Local RBF–FD Scheme with Integrated Multiquadric Kernels

by Yutong Li, Mingqian Zhang, Ruosong Cao, Tao Liu, Xiaoxi Hu and Yakun Li

Axioms 2026, 15(1), 24; https://doi.org/10.3390/axioms15010024 - 27 Dec 2025

Abstract

In this work, a high-order meshless framework is developed for the numerical resolution of the temporal–fractional Black–Scholes equation arising in option pricing with long-memory effects. The spatial discretization is carried out with a local radial basis function produced finite difference (RBF–FD) method on [...] Read more.

In this work, a high-order meshless framework is developed for the numerical resolution of the temporal–fractional Black–Scholes equation arising in option pricing with long-memory effects. The spatial discretization is carried out with a local radial basis function produced finite difference (RBF–FD) method on seven-node stencils. Analytical differentiation weights are constructed by employing closed-form second integrations of a variant of the inverse multiquadric kernel, which yields sparse differentiation matrices. Explicit formulas are derived for both first- and second-order operators, and a detailed truncation error analysis confirms sixth-order convergence in space. Numerical experiments for European options discuss better accuracy per spatial node than standard finite difference schemes. Full article

(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)

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40 pages, 577 KB

Open AccessArticle

Variational Quantum Eigensolver for Clinical Biomarker Discovery: A Multi-Qubit Model

by Juan Pablo Acuña González, Moisés Sánchez Adame and Oscar Montiel

Axioms 2026, 15(1), 23; https://doi.org/10.3390/axioms15010023 - 27 Dec 2025

Abstract

We formalize an inverse, data-conditioned variant of the Variational Quantum Eigensolver (VQE) for clinical biomarker discovery. Given patient-encoded quantum states, we construct a task-specific Hamiltonian whose coefficients are inferred from clinical associations and interpret its expectation value as a calibrated energy score for [...] Read more.

We formalize an inverse, data-conditioned variant of the Variational Quantum Eigensolver (VQE) for clinical biomarker discovery. Given patient-encoded quantum states, we construct a task-specific Hamiltonian whose coefficients are inferred from clinical associations and interpret its expectation value as a calibrated energy score for prognosis and treatment monitoring. The method integrates coefficient estimation, ansatz specification with basis rotations, commuting-group measurements, and a practical shot budget analysis. Evaluated on public infectious disease datasets under severe class imbalance, the approach yields consistent gains in balanced accuracy and precision–recall over strong classical baselines, with stability across random seeds and feature ablations. This variational energy scoring framework bridges Hamiltonian learning and clinical risk modeling, offering a compact, interpretable, and reproducible route to biomarker prioritization and decision support. Full article

(This article belongs to the Special Issue A Century of Quantum Mechanics: Mathematical Foundations and Computational Applications)

23 pages, 466 KB

Open AccessArticle

Dimension-Free Estimators of Gradients of Functions with(out) Non-Independent Variables

by Matieyendou Lamboni

Axioms 2026, 15(1), 22; https://doi.org/10.3390/axioms15010022 - 27 Dec 2025

Abstract

This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using

ℓ_{p}

-spherical distributions on

R^{d}

with

d, p \geq 1

. The upper-bounds of the bias of the [...] Read more.

This study proposes a unified stochastic framework for approximating and computing the gradient of every smooth function evaluated at non-independent variables, using

ℓ_{p}

-spherical distributions on

R^{d}

with

d, p \geq 1

. The upper-bounds of the bias of the gradient surrogates do not suffer from the curse of dimensionality for any

p \geq 1

. Additionally, the mean squared errors (MSEs) of the gradient estimators are bounded by

K_{0} N^{- 1} d

for any

p \in [1, 2]

, and by

K_{1} N^{- 1} d^{2 / p}

when

2 \leq p ≪ d

with N the sample size and

K_{0}, K_{1}

some constants. Taking

max \{2, log (d)\} < p ≪ d

allows for achieving dimension-free upper-bounds of MSEs. In the case where

d ≪ p < + \infty

, the upper-bound

K_{2} N^{- 1} d^{2 - 2 / p} / {(d + 2)}^{2}

is reached with

K_{2}

a constant. Such results lead to dimension-free MSEs of the proposed estimators, which boil down to estimators of the traditional gradient when the variables are independent. Numerical comparisons show the efficiency of the proposed approach. Full article

(This article belongs to the Special Issue Recent Stochastic and Statistical Approaches for Modeling Complex Systems and Dependent Variables)

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21 pages, 497 KB

Open AccessArticle

Optimization Problems of Hermitian Quadratic Matrix-Valued Functions and Applications

by Sihem Guerarra

Axioms 2026, 15(1), 21; https://doi.org/10.3390/axioms15010021 - 27 Dec 2025

Abstract

In this paper, we investigate optimization problems for a Hermitian quadratic matrix-valued function involving two variable matrices. We derive algebraic formulas for the maximal and minimal ranks and partial inertias of this function based on a linearization method and specific block matrix rank [...] Read more.

In this paper, we investigate optimization problems for a Hermitian quadratic matrix-valued function involving two variable matrices. We derive algebraic formulas for the maximal and minimal ranks and partial inertias of this function based on a linearization method and specific block matrix rank and inertia transformations. Further, we establish necessary and sufficient conditions for the existence of solutions satisfying the associated matrix equation and various Hermitian inequalities. As applications, we obtain extremal ranks and inertias for the Hermitian generalized Schur complement with respect to Hermitian reflexive generalized inverses, and provide conditions for Hermitian reflexive generalized inverses to satisfy specific Hermitian properties. In addition, we apply these results to analyze optimization, stability, and congestion in a traffic flow network modeled by this quadratic function. Full article

(This article belongs to the Special Issue Recent Advances in Matrix Generalized Inverses and Applications)

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31 pages, 443 KB

Open AccessArticle

Asymptotic Formulas for the Haezendonck–Goovaerts Risk Measure of Sums with Consistently Varying Increments

by Jonas Šiaulys, Mantas Dirma, Neda Nakliuda and Luca Zanardelli

Axioms 2026, 15(1), 20; https://doi.org/10.3390/axioms15010020 - 26 Dec 2025

Abstract

The Haezendonck–Goovaerts (HG) risk measure defined on Orlicz spaces via the so-called normalised Young function is a direct generalisation of the Expected Shortfall risk measure. The HG measure is known to be a coherent one, thus making it more robust than some of [...] Read more.

The Haezendonck–Goovaerts (HG) risk measure defined on Orlicz spaces via the so-called normalised Young function is a direct generalisation of the Expected Shortfall risk measure. The HG measure is known to be a coherent one, thus making it more robust than some of the alternatives, such as Value-at-Risk, for aggregating and comparing risks, and at the same time more flexible for capital allocation problems, risk premium estimation, solvency assessment, and stress testing in insurance and finance. As random risk in practical applications is often assessed in a portfolio setting—a collection of insurance policies or financial assets, like stocks or bonds—we examine the situation in which the total portfolio risk is expressed as the sum of individual random risks. For this, we consider the sum

S_{n}^{(ξ)} = ξ_{1} + \dots + ξ_{n}

of possibly dependent and non-identically distributed real-valued random variables

ξ_{1}, \dots, ξ_{n}

with consistently varying distributions. Assuming that the collection

{ξ_{1}, \dots, ξ_{n}}

follows the dependence structure, similar to the asymptotic independence, we obtain the asymptotic estimations of the HG risk measure for the sum

S_{n}^{(ξ)}

when the confidence level tends to 1. The formulas presented in our work show that in the case where a portfolio of random losses contains consistently varying losses and the others are asymptotically negligible, it is sufficient for risk assessment to consider only the tails of those dominant losses. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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15 pages, 3769 KB

Open AccessArticle

A Unified Framework for the Upper and Lower Discretizations of Fuzzy Implications

by Songsong Dai

Axioms 2026, 15(1), 19; https://doi.org/10.3390/axioms15010019 - 26 Dec 2025

Abstract

In this paper, we investigate the discretization of fuzzy implications using rounding functions. Our discretization method is a unified framework of the upper and lower discretization methods of Munar-Covas et al. Furthermore, we examine the extent to which the essential properties of these [...] Read more.

In this paper, we investigate the discretization of fuzzy implications using rounding functions. Our discretization method is a unified framework of the upper and lower discretization methods of Munar-Covas et al. Furthermore, we examine the extent to which the essential properties of these fuzzy implications are preserved in our discretization process. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic and Fuzzy Implications)

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