Journal of Pure and Applied Mathematics and Information Science Open Access Journal
Dipartimento di Matematica e Informatica
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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
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
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
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
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
From Feynman diagrams to the amplituhedron: a gentle review
In this article we review, for a mathematical audience, the computation of (tree-level) scattering amplitudes in Yang-Mills theory in detail. In particular we demonstrate explicitly how the same formulas for six-particle NMHV helicity amplitudes are obtained from summing Feynman diagrams and from computing the canonical form of the n=6, k=1, m=4 amplituhedron.
Copyright (c) 2025 S. De, D. Pavlov, M. Spradlin, A. Volovich http://creativecommons.org/licenses/by/4.0
We study the set of linear subspaces of a fixed dimension intersecting a given polytope. To describe this set as a semialgebraic subset of a Grassmannian, we introduce a Schubert arrangement of the polytope, defined by the Chow forms of the polytope’s faces of complementary dimension. We show that the set of subspaces intersecting a specified family of faces is defined by fixing the sign of the Chow forms of their boundaries. We give inequalities defining the set of stabbing subspaces in terms of sign conditions on the Chow form.
Copyright (c) 2025 S. Seemann, F. Zaffalon http://creativecommons.org/licenses/by/4.0
Kinematic varieties for massless particles
We study algebraic varieties that encode the kinematic data for n massless particles in d-dimensional spacetime subject to momentum conservation. Their coordinates are spinor brackets, which we derive from the Clifford algebra associated to the Lorentz group. This was proposed for d=5 in the recent physics literature. Our kinematic varieties are given by polynomial constraints on tensors with both symmetric and skew symmetric slices.
Copyright (c) 2025 S. Rajan, B. Sturmfels, S. Sverrisdottir http://creativecommons.org/licenses/by/4.0
Hyperplane arrangements in the grassmannian
The Euler characteristic of a smooth very affine variety encodes the algebraic complexity of solving likelihood (or scattering) equations on this variety. We study this quantity for the Grassmannian with d hyperplane sections removed. We provide a combinatorial formula, and explain how to compute this Euler characteristic in practice, both symbolically and numerically. Our particular focus is on generic hyperplane sections and on Schubert divisors. We also consider special Schubert arrangements relevant for physics. We study both the complex and the real case.
Copyright (c) 2025 E. Mazzucchelli, D. Pavlov, K. Wang http://creativecommons.org/licenses/by/4.0
We study the conditions under which a subvariety of the Grassmannian may be recovered from certain of its linear projections. In the special case that our Grassmannian is projective space, this is equivalent to asking when a variety can be recovered from its Chow form; the answer is "always" by work of Chow in 1937. In the general Grassmannian setting, the analogous question is when a variety can be recovered from its Chow-Lam form. We give both necessary conditions for recovery and families of examples where, in contrast with the projective case, recovery is not possible.
Copyright (c) 2025 E. Pratt, K. Ranestad http://creativecommons.org/licenses/by/4.0
An explicit lower bound for large gaps between some consecutive primes
The main purpose of this paper is to clarify the numerical value of the constant cLG such that the above in- equality holds. We see that cLG is determined by several factors related to analytic number theory, for example, the ratio of integrals of functions in the multidimensional sieve of Maynard [14], the distribution of primes in arithmetic progressions to large moduli, and the coefficient of upper bound sieve of Selberg. We prove that the above inequality is valid at least for some cLG ≥ 2.0 × 10−17 .
Copyright (c) 2025 K. Sono http://creativecommons.org/licenses/by/4.0
Bivariate exponential integrals and edge-bicolored graphs
We show that specific exponential bivariate integrals serve as generating functions of labeled edge-bicolored graphs. Based on this, we prove an asymptotic formula for the number of regular edge-bicolored graphs with arbitrary weights assigned to different vertex incidence structures. The asymptotic behavior is governed by the critical points of a polynomial. As an application, we discuss the Ising model on a random 4-regular graph and show how its phase transitions arise from our formula.
Copyright (c) 2025 M. Borinsky, C. Meroni, M. Wiesmann http://creativecommons.org/licenses/by/4.0
Sign-changing solutions for a nonlinear degenerate elliptic system
In this article, we study the multiplicity of weak solutions to the non-linear degenerate elliptic system. The existence of sign-changing solutions is proved by the truncation method and the invariant sets of descending flow method.
method of invariant sets of descending flow
Copyright (c) 2025 L. Thi Hong Hanh, D. Trong http://creativecommons.org/licenses/by/4.0
Global solvability of the Laplace equation in weighted Sobolev spaces
We consider a non-local boundary value problem for the Laplace equation in an unbounded strip, studying the weak and strong solvability of the problem within the framework of the weighted Sobolev space W1,pν with a Muckenhoupt weight. Utilising tools from non-harmonic analysis, we prove that any weak solution belonging to W2,pν is also a strong solution and satisfies the corresponding boundary conditions. It is worth noting that such problems do not fall within the scope of the general theory of elliptic equations and therefore require a specialized approach.
Copyright (c) 2025 B. Bilalov, N. Nasibova, L. Softova, S. Tramontano http://creativecommons.org/licenses/by/4.0
Construction of 3-helix systems of any index
Let H(3) be a uniform of rank 3 hypergraph. A 3-helix S(3)(1,3) of \emph{centre} {c} is a 3-uniform hypergraph, with 3 hyperedges, all having in common exactly the centre {c}, with c of degree 3 and the remaining vertices of degree 1. In this paper we determine the spectrum of all S(3)(1,3)-designs, for every index λ.
Copyright (c) 2024 A. Causa, M. Gionfriddo, E.M. Guardo http://creativecommons.org/licenses/by/4.0