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40 pages, 1733 KB

Open AccessArticle

Fine Stability Properties of the Hankel and Wiener–Khinchin Transforms

by François Vigneron

Axioms 2026, 15(3), 194; https://doi.org/10.3390/axioms15030194 - 6 Mar 2026

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The Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles [...] Read more.

The Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles

(| x |^{- α}, c_{d} | ξ |^{d - α})

on

R^{d}

that encompasses the case where the homogeneous profiles exist only on a large but finite range. In this case, largely overlooked in the literature, we provide precise error estimates in terms of the size of the tails outside the homogeneous range. We also prove a series of refined properties of the Fourier transform on related questions including criteria that ensure an approximate homogeneous behavior asymptotically near the origin or at infinity. The sharpness of our results is checked with numerical simulations. We also investigate briefly how these results consolidate the mathematical foundations of turbulence theory. Full article

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

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13 pages, 257 KB

Open AccessArticle

On sM-Prime Ideals in Commutative Rings

by Gülşen Ulucak, Violeta Leoreanu-Fotea, Seçil Çeken Güneş and Ünsal Tekir

Axioms 2026, 15(2), 142; https://doi.org/10.3390/axioms15020142 - 15 Feb 2026

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Abstract

All rings considered are commutative with identity, and all modules are assumed to be unital. In this paper, we study R-modules in which every quasi-primary submodule is also primary; we refer to such modules as satisfying condition (*). We present several structural [...] Read more.

All rings considered are commutative with identity, and all modules are assumed to be unital. In this paper, we study R-modules in which every quasi-primary submodule is also primary; we refer to such modules as satisfying condition (*). We present several structural properties of these modules and investigate when the direct sum of two modules

M_{1}

and

M_{2}

inherits condition (*). In addition, we focus on prime ideals P of a ring R with the property that any P-quasi-primary submodule of an R-module M is automatically P-primary. Prime ideals exhibiting this behaviour are introduced as weak

s M

-prime ideals relative to M. Our results provide a framework for understanding the interaction between the quasi-primary structure of modules and the prime spectrum of the underlying ring. Full article

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

18 pages, 305 KB

Open AccessArticle

Colour Algebras over Rings

by Susanne Pumplün

Axioms 2026, 15(2), 139; https://doi.org/10.3390/axioms15020139 - 14 Feb 2026

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Abstract

Colour algebras are noncommutative Jordan algebras and closely related to octonion algebras. They initially emerged in physics since the multiplication of a colour algebra over the real or complex numbers describes the colour symmetry of the Gell–Mann quark model. Over fields of characteristic [...] Read more.

Colour algebras are noncommutative Jordan algebras and closely related to octonion algebras. They initially emerged in physics since the multiplication of a colour algebra over the real or complex numbers describes the colour symmetry of the Gell–Mann quark model. Over fields of characteristic not equal to two, their structure is now well-known. We initiate the study of colour algebras over a unital commutative base ring R where two is an invertible element, and show when colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, their automorphism group and their derivations. We show that there is again a close connection between the colour algebras obtained from hermitian forms and certain types of octonion algebras. Full article

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

19 pages, 335 KB

Open AccessArticle

A Note on Truncated Exponential-Based Appell Polynomials via Fractional Operators

by Waseem Ahmad Khan, Francesco Aldo Costabile, Khidir Shaib Mohamed, Alawia Adam and Shahid Ahmad Wani

Axioms 2026, 15(2), 111; https://doi.org/10.3390/axioms15020111 - 2 Feb 2026

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Abstract

In this work, we construct a new class of Appell-type polynomials generated through extended truncated and truncated exponential kernels, and we analyze their core algebraic and operational features. In particular, we establish a suitable recurrence scheme and obtain the associated multiplicative and differential [...] Read more.

In this work, we construct a new class of Appell-type polynomials generated through extended truncated and truncated exponential kernels, and we analyze their core algebraic and operational features. In particular, we establish a suitable recurrence scheme and obtain the associated multiplicative and differential operators. By confirming the quasi-monomial structure, we further deduce the governing differential equation for the proposed family. In addition, we present both a series expansion and a determinant formulation, providing complementary representations that are useful for symbolic manipulation and computation. As special cases, we introduce and study subfamilies arising from this setting, namely, extended truncated exponential versions of the Bernoulli, Euler, and Genocchi polynomials, and discuss their structural identities and operational behavior. Overall, these developments broaden the theory of special polynomials and furnish tools relevant to problems in mathematical physics and differential equations. Full article

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

12 pages, 272 KB

Open AccessArticle

Upper Semicontinuous Representations of Semiorders as Interval Orders

by Gianni Bosi, Gabriele Sbaiz and Magalì Zuanon

Axioms 2026, 15(1), 53; https://doi.org/10.3390/axioms15010053 - 10 Jan 2026

Viewed by 531

Abstract

We characterize the upper semicontinuous representability of a semiorder ≺ as an interval order (namely, by a pair

(u, v)

of upper semicontinuous real-valued functions) on a topological space with a countable basis of open sets, where one of the [...] Read more.

We characterize the upper semicontinuous representability of a semiorder ≺ as an interval order (namely, by a pair

(u, v)

of upper semicontinuous real-valued functions) on a topological space with a countable basis of open sets, where one of the representing functions is a one-way utility for the characteristic weak order

≺^{0}

associated with the semiorder. Such a description generalizes the upper semicontinuous threshold representation. To this end, we introduce a suitable upper semicontinuity condition concerning a semiorder, namely strict upper semicontinuity. We further characterize the mere existence of an upper semicontinuous one-way utility for this characteristic weak order, with a view to the identification of maximal elements on compact metric spaces. Full article

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

9 pages, 321 KB

Open AccessEditor’s ChoiceArticle

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

Viewed by 2320

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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16 pages, 296 KB

Open AccessArticle

Averaged Iterative Algorithms for Convex Optimization Problems over a Common Fixed-Points Set of Demicontractive Mappings

by Vasile Berinde and Khairul Saleh

Axioms 2026, 15(1), 8; https://doi.org/10.3390/axioms15010008 - 25 Dec 2025

Viewed by 557

Abstract

In this article, we introduce a novel averaged-type iterative scheme designed for solving convex minimization problems over the set of common fixed points of a pair of demicontractive mappings. Under suitable assumptions, we prove that the proposed algorithm converges strongly to the solution [...] Read more.

In this article, we introduce a novel averaged-type iterative scheme designed for solving convex minimization problems over the set of common fixed points of a pair of demicontractive mappings. Under suitable assumptions, we prove that the proposed algorithm converges strongly to the solution of the considered problem in a Hilbert space setting. We further demonstrate the applicability of our method to quadratic optimization problems with a bounded linear operator. In addition, we also report the numerical experiments that were performed in order to demonstrate the convergence behavior of the algorithm and to highlight its superiority over related existing methods. Full article

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

19 pages, 836 KB

Open AccessArticle

A Hybrid Walrus Optimization-Based Fourth-Order Method for Solving Non-Linear Problems

by Aanchal Chandel, Eulalia Martínez, Sonia Bhalla, Sattam Alharbi and Ramandeep Behl

Axioms 2026, 15(1), 6; https://doi.org/10.3390/axioms15010006 - 23 Dec 2025

Viewed by 664

Abstract

Non-linear systems of equations play a fundamental role in various engineering and data science models, where accurate solutions are essential for both theoretical research and practical applications. However, solving such systems is highly challenging due to their inherent non-linearity and computational complexity. This [...] Read more.

Non-linear systems of equations play a fundamental role in various engineering and data science models, where accurate solutions are essential for both theoretical research and practical applications. However, solving such systems is highly challenging due to their inherent non-linearity and computational complexity. This study proposes a novel hybrid iterative method with fourth-order convergence. The foundation of the proposed scheme combines the Walrus Optimization Algorithm and a fourth-order iterative technique. The objective of this hybrid approach is to enhance global search capability, reduce the likelihood of convergence to local optima, accelerate convergence, and improve solution accuracy in solving non-linear problems. The effectiveness of the proposed method is checked on standard benchmark problems and two real-world case studies, hydrocarbon combustion and electronic circuit design, and one non-linear boundary value problem. In addition, a comparative analysis is conducted with several well-established optimization algorithms, based on the optimal solution, average fitness value, and convergence rate. Furthermore, the proposed scheme effectively addresses key limitations of traditional iterative techniques, such as sensitivity to initial point selection, divergence issues, and premature convergence. These findings demonstrate that the proposed hybrid method is a robust and efficient approach for solving non-linear problems. Full article

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

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38 pages, 532 KB

Open AccessArticle

Stone and Flat Topologies on the Minimal Prime Spectrum of a Commutator Lattice

by George Georgescu, Leonard Kwuida and Claudia Mureşan

Axioms 2025, 14(11), 803; https://doi.org/10.3390/axioms14110803 - 30 Oct 2025

Viewed by 674

Abstract

In previous work we have studied minimal prime spectra, as well as extensions of universal algebras whose term condition commutator behaves like the modular commutator in the sense that it is commutative and distributive with respect to arbitrary joins, while modularity does not [...] Read more.

In previous work we have studied minimal prime spectra, as well as extensions of universal algebras whose term condition commutator behaves like the modular commutator in the sense that it is commutative and distributive with respect to arbitrary joins, while modularity does not even need to be enforced on their congruence lattices, let alone on those of the members of the variety they generate. Commutator lattices, defined by Czelakowski in 2008, are commutative multiplicative lattices having as prototype the algebraic structure of the congruence lattice of a such an algebra. Considering the prime elements with respect to the commutator operation, we obtain algebraic characterizations for minimal primes, then study the Stone and flat topologies on the set of minimal primes in a commutator lattice. We also prove abstract versions of congruence extension properties, actually of the general case of arbitrary morphisms instead of algebra embeddings, by means of complete join–semilattice morphisms between commutator lattices. We thus obtain abstractions for our results on congruence lattices and generalizations for results on frames and quantales, but also further cases in which these results hold. Furthermore, we investigate the lattice structures of these topologies as sublattices of the power sets of the sets of (minimal) primes. Full article

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

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