Journal of Pure and Applied Mathematics and Information Science Open Access Journal
Dipartimento di Matematica e Informatica
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Square-mean S-asymptotically (ω,Q)-periodic processes and mild solutions to some neutral fractional stochastic evolution equations
Recent results concerning (ω,Q)-periodic functions are extended tore current stochastic processes in the square mean sense, termedS-asymptotic (ω,Q)-periodic processes, where Q denotes a linear isomorphism from a Hilbert space to itself. These quasi-periodic stochastic processes covers S-asymptotic ω-(anti-)periodic processes, Bloch and(ω,c)-periodic stochastic processes in Hilbert spaces. We prove the completeness, convolution and superposition theorems for theS-asymptotic (ω,Q)-periodic process in abstract spaces. We also consider some existence results forS-asymptotically(ω,Q)-periodic mild solutions to stochastically forced fractional evolution equations under some different conditions.Somme examples are given to illustrate the existence results
Copyright (c) 2026 Ousmane Touré, Mamadou Moustapha Mbaye, Amadou Diop http://creativecommons.org/licenses/by/4.0
Stability of constant equilibria in a diffusive logistic SIR model with saturated treatment and death rates
This paper is concerned with the Neumann initial-boundary valueproblem for a diffusive logistic SIR model with saturated treatment anddeath rates. The purpose of this paper is to study stability of constant equi-libria under some conditions on parameters. Moreover, several numerical simulations are given
Copyright (c) 2026 Yutaro Chiyo, Koyuki Yamamoto, Tomomi Yokota http://creativecommons.org/licenses/by/4.0
Weak solutions for nonlocal problems in fractional Orlicz-Sobolev spaces
We study a class of nonlocal boundary value problems involving the fractional a(·)-Laplacian operator in fractional Orlicz–Sobolev spaces.The main equation features nonlinear source terms with a nonnegative parameterμand is posed on a bounded domain with homogeneous Dirichlet conditions in the nonlocal sense. Using variational approaches, we establish the existence of at least one nontrivial, non negative weak solution when the parameter μ is sufficiently small. Furthermore, we identify conditions under which the solution is strictly positive. Two examples are included to demonstrate the applicability of the main result
Fractional order Sobole spaces, nonlocal problems, weak solutions, critical points.
Copyright (c) 2026 Lingju Kong http://creativecommons.org/licenses/by/4.0
On minimum embeddings of Pk(u, λ) into KS(u+w,μ)
Let G be a simple finite graph and H be a subgraph of G. We say that an H-design (U,C) of order u and index λ is embedded into a G-design (V,B) of order u+w and index μ, λ ≤ μ, if there is an injective function f : C → B such that C is a subgraph of f (C) for every C ∈ C. The mapping f is called the embedding of (U,C) into (V,B). If w attains the minimum possible value, then f is a minimum embedding. In this paper we study the minimum embedding of a Pk(u,λ) into a KS(u+w,μ), for k = 3,4.
graph decomposition, embedding, minimum embedding
Copyright (c) 2026 Giorgio Ragusa, Antoinette Tripodi http://creativecommons.org/licenses/by/4.0
On the lattice formulation of the union-closed sets conjecture
The union-closed sets conjecture, also known as Frankl's conjecture, is a well-studied problem with various formulations. In terms of lattices, the conjecture states that every finite lattice L with more than one element contains a join-irreducible element that is less than or equal to at most half of the elements in L. In this work, we obtain several necessary conditions for any counterexample L of minimum size.
Copyright (c) 2026 Christopher Bouchard http://creativecommons.org/licenses/by/4.0
On semi-c-periodic functions of type I and applications
In this paper, we study the (newly introduced) class of functions called semi-c-periodic functions of type I in a Banach space. We first investigate their basic properties, including a convolution result. Using a suitable norm we prove that the set of such functions is a Banach space. We then study the existence and uniqueness of semi-c-periodic mild solutions for both autonomous and non-autonomous linear evolution equations. We achieve the existence results using the Banach fixed point theorem and the principle of reduction
C-periodic; Semi-c-periodic; Uniformly close c-periodic; Volterra integral equation; Banach fixed point principle
Copyright (c) 2025 R. Aldrsoni, G. N'Guerekata http://creativecommons.org/licenses/by/4.0
Elastic beam qquations with variable coefficients: multiple solutions under mixed nonlinearities
This paper investigates the existence of multiple solutions for a fourth-order differential equation modelling an elastic beam, where the coefficients are variable, and the nonlinearities exhibit both concave and convex characteristics. Our approach is based on variational methods and critical point theorems, particularly those formulated by Ricceri, which provide a powerful framework for proving the existence of solutions in reflexive Banach spaces. By leveraging these mathematical tools, we establish that the considered problem admits at least three distinct weak solutions under specific conditions. To validate our theoretical findings, we present an illustrative example demonstrating how our results can be applied in practice.
Copyright (c) 2025 S. Heidarkhani, S. Moradi, A. L. A. De Araujo, D. Barilla http://creativecommons.org/licenses/by/4.0
Disjoint weak star p-convergent operators on Banach lattices, 1 ≤ p ≤ ∞
We introduce and study the disjoint weak star p-convergent operators on Banach lattices, and we explore some characterizations of them in terms of disjoint sequences in the positive cones. As an application, we examine the domination and the duality properties of the class of positive disjoint weak star p-convergent operators. Next, we investigate the connections between our operators and disjoint p-convergent operators. Finally, we deduce some important results about the positive Schur property of order p.
Banach lattice, order continuous norm, p-convergent operator, disjoint p-convergent operator, positive Schur property of order p
Copyright (c) 2026 Abderrahman Retbi http://creativecommons.org/licenses/by/4.0
This article serves as an introduction to the special volume on Positive Geometry in the journal Le Matematiche. We attempt to answer the question in the title by describing the origins and objects of positive geometry at this early stage of its development. We discuss the problems addressed in the volume and report on the progress. We also list some open challenges.
Copyright (c) 2025 K. Ranestad, B. Sturmfels, S. Telen http://creativecommons.org/licenses/by/4.0
On the complements of union of open balls and related set classes
Let a R-body be a closed set, complement of union of open balls of radius R in the Euclidean space. Properties generalizing similar ones for convex sets are proved for the family of R-bodies; properties for the family of sets supported by sphere of radius R (extension of the supporting hyperplane to convex bodies) are investigated. Comparison of that family with the sets of reach R and with the R-rolling sets ( G. Walther, On a generalization of Blachke's Rolling Theorem and the Smoothing Surfaces 1999) are studied. New properties for the previous familes are proved, by using the generalization of the convex cones.
generalized convexity, convex hull, rolling bodies, reach positive sets, support cones
Copyright (c) 2026 Marco Longinetti, Paolo Manselli, Adriana Venturi http://creativecommons.org/licenses/by/4.0
Nonnil-Noetherian pairs of the form (R, R[X]) and some related results
The rings considered in this paper are commutative with identity and are nonzero. Let R be a ring. An ideal I of R is said to be nonnil if it is not contained in the nilradical of R. We say that R is nonnil-Noetherian (resp., nonnil-Laskerian) if each proper nonnil ideal of R is finitely generated (resp., admits a primary decomposition). Whenever T is an extension ring of R, we assume that R contains the identity element of T. Let T be an extension ring of R. We say that (R, T) is a Nonnil-Noetherian pair (resp., Nonnil-Laskerian pair) if f A is nonnil-Noetherian (resp., nonnil-Laskerian) for any intermediate ring A between R and T. This paper aims to characterize R such that (R, R[X]) is a nonnil-Noetherian pair (resp., nonnil-Laskerian pair), where R[X] is the polynomial ring in one variable X over R. Also, this paper aims to characterize R such that each intermediate ring A between R and R[X] posses a property which is related to being nonnil-Noetherian (resp., nonnil-Laskerian).
Copyright (c) 2025 S. Visweswaran http://creativecommons.org/licenses/by/4.0
Moments and (p; q)-Integral Representations of (p; q)-Classical Forms
In [6], the authors introduce a new technical method for studying the Dp,q-classical orthogonal polynomials, where Dp,q is the(p,q)-difference operator, using an algebraic approach. In this paper, we investigate Dp,q-classical orthogonal polynomials associated with the(p,q)-difference op-eratorDp,q. Using Pearson-type functional equations and a suitable de-composition of Dp,q, we simplify the description of the canonical Dp,q-classical forms and clarify their relation with theq-classical framework.We also study the moments and modified moments of the correspondingforms, derive recurrence relations, and establish several explicit (p,q)-integral representations for canonical cases
(p; q) calculus; (p; q)-di erence operator; (p; q)-classical polynomial
Copyright (c) 2026 Mabrouk Sghaier, Mohamed Zaatra http://creativecommons.org/licenses/by/4.0
Local and global solvability in attraction-repulsion chemotaxis systems with L2-initial data
This paper deals with the attraction-repulsion chemotaxis system under homogeneous Neumann initial-boundary conditions, where Ω ⊂ Rn (n ≤ 3) is a smoothly bounded domain and a,b,c,χ,ξ,α,β,γ,δ > 0 and τ ∈ {0, 1} are constants. The purpose of the present paper is to construct a local solution of this system for any L2-initial data without additional conditions on χ and ξ by using the theory for abstract evolution equations and to extend the local solution globally in the repulsion-dominant case by relying on a priori estimates.
Copyright (c) 2025 Y. Chiyo, T. Saga, T. Yokota http://creativecommons.org/licenses/by/4.0
On all-path convex, gated and Chebyshev sets in graphs
We present new characterizations for trees, block graphs, and geodetic graphs using all-path convex, gated and Chebyshev sets. Specifically, we prove that trees are exactly the graphs in which all-path convexity is a convex geometry. Block graphs are characterized as graphs in which all balls are all-path convex (equivalently, gated), and geodetic graphs are exactly those graphs where all balls (equivalently, closed neighborhoods) are Chebyshev. Additionally, we prove that almost all graphs have geodesically convex Chebyshev sets, provide a characterization of bipartite graphs with connected Chebyshev sets, and establish a criterion for graphs with trivial Chebyshev sets in the class of graph joins. Finally, we show that graphs of odd order with maximal number of edges under the Seidel switching operation always have trivial Chebyshev sets.
Copyright (c) 2025 V. Haponenko, S. Kozerenko http://creativecommons.org/licenses/by/4.0
On locally factorial Fano fourfolds of Picard number two
We classify the locally factorial Fano fourfolds of Picard number two with a hypersurface Cox ring that admit an effective action of a three- dimensional torus.
Copyright (c) 2026 Andreas Baeuerle, Christian Mauz http://creativecommons.org/licenses/by/4.0
Voronovskaya-type asymptotic expansions and convergence analysis for neural network operators in complex domains
This paper extends the classical theory of Voronovskaya-type asymptotic expansions to generalized neural network operators defined on non-Euclidean and fractal domains. We introduce and analyze smooth operators activated by modified and generalized hyperbolic tangent functions, extending their applicability to manifold and fractal geometries. Key theoretical results include the preservation of density properties, detailed convergence rates, and asymptotic expansions. Additionally, we explore the role of fractional derivatives in defining neural network operators, which capture non-local behavior and are particularly useful for modeling systems with long-range dependencies or fractal-like structures. Our findings contribute to a deeper understanding of neural network operators in complex, structured spaces, offering robust mathematical tools for applications in signal processing on manifolds and solving partial differential equations on fractals.
Copyright (c) 2026 Romulo D. Chaves dos Santos, Jorge H. O. Sales http://creativecommons.org/licenses/by/4.0
These are lecture notes for five lectures given at MPI Leipzig in May 2024. We study the moduli space M0,n of n distinct points on P1 as a positive geometry and a binary geometry. We develop mathematical formalism to study Cachazo-He-Yuan's scattering equations and the associated scalar and Yang-Mills amplitudes. We discuss open superstring amplitudes and relations to tropical geometry.
Copyright (c) 2025 T. Lam http://creativecommons.org/licenses/by/4.0
The loop-Amplituhedron A(L)n is a semialgebraic set in the product of Grassmannians GrR(2,4)L. Recently, many aspects of this geometry for the case of L=1 have been elucidated, such as its algebraic and face stratification, its residual arrangement and the existence and uniqueness of the adjoint. This paper extends this analysis to the simplest higher loop case given by the two-loop four-point Amplituhedron A(2)4$.
Copyright (c) 2025 G. Dian, E. Mazzucchelli, F. Tellander http://creativecommons.org/licenses/by/4.0
The Plücker positive region OGr+(k,2k) of the orthogonal Grassmannian emerged as the positive geometry behind the ABJM scattering amplitudes. In this paper we initiate the study of the positive orthogonal Grassmannian OGr+(k,n) for general values of k, n. We determine the boundary structure of the quadric OGr+(1,n) in Pn-1+ and show that it is a positive geometry. We show that OGr+(k,2k+1) is isomorphic to OGr+(k+1, 2k+2) and connect its combinatorial structure to matchings on [2k+2]. Finally, we show that in the case n > 2k+1, the positroid cells of Gr+(k,n) do not induce a CW cell decomposition of OGr+(k,n).
Copyright (c) 2025 Y. El Maazouz, Y. Mandelshtam http://creativecommons.org/licenses/by/4.0
Points on rational normal curves and the ABCT variety
The ABCT variety is defined as the closure of the image of G(2, n) under the Veronese map. We realize the ABCT variety V(3,n) as the determinantal variety of a vector bundle morphism. We use this to give a recursive formula for the fundamental class of V(3,n). As an application, we show that special Schubert coefficients of this class are given by Eulerian numbers, matching a formula by Cachazo-He-Yuan. On the way tothis, we prove that the variety of configuration of points on a common divisor on a smooth variety is reduced and irreducible, generalizing a result of Caminata-Moon-Schaffler.
rational normal curve, configurations, Grassmannian, positive geometry, spinor helicity, scattering equations
Copyright (c) 2025 D. Agostini, L. Ramesh, D. Shen http://creativecommons.org/licenses/by/4.0
Cyclic polylopes through the lense of iterated integrals
The volume of a cyclic polytope can be obtained by forming an it- erated integral, known as the path signature, along a suitable piecewise linear path running through its edges. Different choices of such a path are related by the action of a subgroup of the combinatorial automorphisms of the polytope. Motivated by this observation, we look for other polyno- mials in the vertices of a cyclic polytope that arise as path signatures and are invariant under the subgroup action. We prove that there are infinitely many such invariants which are algebraically independent in the shuffle algebra.
Copyright (c) 2025 F. Lotter, R. Preiß http://creativecommons.org/licenses/by/4.0
The type A cluster configuration space, commonly known as M0,n, is the very affine part of the binary geometry associated with the associahedron. The tropicalization of M0,n can be realized as the space of phylogenetic trees and its signed tropicalizations as the dual-associahedron subfans. We give a concise overview of this construction and propose an extension to type C. The type C cluster configuration space MCl arises from the binary geometry associated with the cyclohedron. We define a space of axially symmetric phylogenetic trees containing many dual-associahedron and dual-cyclohedron subfans. We conjecturally realize the tropicalization of MCl as the defined space and its signed tropicalizations as the aforementioned subfans.
Copyright (c) 2025 S. Cox, I. Makhlin http://creativecommons.org/licenses/by/4.0
We investigate MHV tree-level gravity amplitudes as defined on the spinor-helicity variety. Unlike their gluon counterparts, the gravity amplitudes do not have logarithmic singularities and do not admit Amplituhedron-like construction. Importantly, they are not determined just by their singularities, but rather their numerators have interesting zeroes. We make a conjecture about the uniqueness of the numerator and explore this feature from a more mathematical perspective. This leads us to a new approach for examining adjoints. We outline steps of our proposed proof and provide computational evidence for its validity in specific cases.
scattering amplitudes, adjoints, commutative algebra
Copyright (c) 2025 J. Koefler, U. Oktem, S. Paranjape, J. Trnka, B. Zacovic http://creativecommons.org/licenses/by/4.0
Recent developments in particle physics have revealed deep connections between scattering amplitudes and tropical geometry. From the heart of this relationship emerged the chirotropical Grassmannian TropχG(k,n) and the chirotropical Dressian Drχ(k,n), polyhedral fans built from uniform realizable chirotopes that encode the combinatorial structure of Generalized Feynman Diagrams. We prove that TropχG(3,n) = Drχ(3,n) for $n = 6,7,8$, and develop algorithms to compute these objects from their rays modulo lineality. Using these algorithms, we compute all chirotropical Grassmannians TropχG(3,n) for n = 6,7,8 across all isomorphism classes of chirotopes. We prove that each chirotopal configuration space Xχ(3,6) is diffeomorphic to a polytope and propose an associated canonical logarithmic differential form. Finally, we show that the equality between chirotropical Grassmannian and Dressian fails for (k,n) = (4,8).
Copyright (c) 2025 D. Antolini, N. Early http://creativecommons.org/licenses/by/4.0
Proudfoot-Speyer degenerations of scattering equations
We study scattering equations of hyperplane arrangements from the perspective of combinatorial commutative algebra and numerical algebraic geometry. We formulate the problem as linear equations on a reciprocal linear space and develop a degeneration-based homotopy algorithm for solving them. We investigate the Hilbert regularity of the corresponding homogeneous ideal and apply our methods to CHY scattering equations.
Copyright (c) 2025 B. Betti, V. Borovik, S. Telen http://creativecommons.org/licenses/by/4.0
Logarithmic discriminants of hyperplane arrangements
A recurring task in particle physics and statistics is to compute the complex critical points of a product of powers of affine-linear functions. The logarithmic discriminant characterizes exponents for which such a function has a degenerate critical point in the corresponding hyperplane arrangement complement. We study properties of this discriminant, exploiting its connection with the Hurwitz form of a reciprocal linear space.
Copyright (c) 2025 L. Kayser, A. Kretschmer, S. Telen http://creativecommons.org/licenses/by/4.0
Binary geometries have recently been introduced in particle physics in connection with stringy integrals. In this work, we study a class of simple polytopes, called \emph{pellytopes}, whose number of vertices are given by Pell's numbers. We provide a new family of binary geometries determined by pellytopes as conjectured by He--Li--Raman--Zhang. We relate this family to the moduli space of curves by comparing the pellytope to the ABHY associahedron.
Copyright (c) 2025 L. Bossinger, M.L. Telek, H. Tillmann-Morris http://creativecommons.org/licenses/by/4.0
Differential equations for moving hyperplane arrangements
We investigate Mellin integrals of products of hyperplanes, raised to an individual power each. We refer to the resulting functions as combinatorial correlators. We investigate their behavior when moving the hyperplanes individually. To encode these functions as holonomic functions in the constant terms of the hyperplanes, we aim to construct a holonomic annihilating D-ideal purely in terms of the hyperplane arrangement.
Copyright (c) 2025 A. Pfister, A.-L. Sattelberger http://creativecommons.org/licenses/by/4.0
Algebraic approaches to cosmological integrals
Cosmological correlators encode statistical properties of the initial conditions of our universe. Mathematically, they can often be written as Mellin integrals of a certain rational function associated to graphs, namely the flat space wavefunction. The singularities of these cosmological integrals are parameterized by binary hyperplane arrangements. Using different algebraic tools, we shed light on the differential and difference equations satisfied by these integrals. Moreover, we study a multivariate version of partial fractioning of the flat space wavefunction, and propose a graph-based algorithm to compute this decomposition.
Copyright (c) 2025 C. Fevola, G. L. Pimentel, A.-L. Sattelberger , T. Westerdijk http://creativecommons.org/licenses/by/4.0
Multiple solutions for a dynamic Sturm-Liouville boundary value problem on time scales with impulsive effects
In this study, we explore the existence of multiple solutions for a dynamic Sturm-Liouville boundary value problem on time scales that incorporate impulsive effects. Utilizing variational methods and applying certain critical point theorems from Ricceri for smooth functionals, we demonstrate the existence of at least three solutions to the problem. To illustrate the practical relevance of our findings, we provide an example at the end.
Copyright (c) 2025 M. Ferrara, S. Heidarkhani, S. Moradi, A.L.A. de Araujo http://creativecommons.org/licenses/by/4.0