In this paper, we study the (newly introduced) class of functions called semi-c-periodic functions of type I with values in a Banach space. We first investigate their basic properties, including a convolution result and invertibility of abstract convolution operators on suvh function space. We prove that an important subclass of such functions is a Banach space under the sup-norm. We then study the existence and uniqueness of semi-cperiodic mild solutions for both autonomous and non-autonomous linear evolution equations. We achieve the existence results using the Banach fixed point theorem and the method of reduction.
The rings considered in this paper are commutative with identity and are nonzero. An ideal of a ring is said to be nonnil if it contains an element that is not nilpotent. Let R be a ring. We say that R is nonnil-Noetherian (resp., nonnil-Laskerian) if each proper nonnil ideal of R is finitely generated (resp., each proper nonnil ideal of R admits a primary decomposition). Whenever R is a subring of a ring T, then it is assumed that R contains the identity element of T. Let R subset of T be rings. We say that (R, T) is a nonnil-Noetherian pair (resp., nonnil-Laskerian pair) if each intermediate ring A between R and T is nonnil-Noetherian (resp., nonnil-Laskerian). 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 pairs of the form (R, R[X]) such that each intermediate ring A between R and R[X] has a property that is related to being nonnil-Noetherian (resp., nonnil-Laskerian).
This paper is concerned with the positive solutions of a Matukumatype nonlinear equation with double singular terms, triangle(p)u+ |x|(m1) u(delta 1) + |x|(m2) u(delta 2) = 0, x is an element of R-N, where p > 2, N >= 1, delta(2) > delta(1)>= 1, -p
Let S = K[x(1), ... ,x(n)] be the ring of polynomials over a field K and let I be a monomial ideal of S. We prove that the following are equivalent: (i) I is principal, (ii) hdepth(I) = n, (iii) hdepth(S/I) = n-1. If I is squarefree, we prove that if hdepth(S/I) < 3 or n <= 5, then hdepth(I) >= hdepth(S/I) + 1. Also, we prove that if hdepth(S/I) <5 or n <= 7, then hdepth(I) >= hdepth(S/I).
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.
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.
In this article, we study the multiplicity of weak solutions to the non-linear degenerate elliptic system -partial derivative(2)u(j)/partial derivative x(2 )-( )|x|(2) partial derivative(2)u(j)/partial derivative y(2 )+ lambda(j)u(j) = Sigma(k )(i=1)beta(i,j)u(i)(2)u(j),( ) in Omega ( )uj(x, y) = 0, on partial derivative Omega, j = 1, ... ,k, where Omega subset of R-2 is a bounded smooth domain, lambda(j) > 0, k >= 2, j = 1... ,k, beta(ij) are constants satisfying beta(jj) > 0, beta(ij )= beta(ji) <= 0 for 1 <= i < j <= k. The existence of sign-changing solutions is proved by the truncation method and the invariant sets of descending flow method.
This paper deals with the attraction-repulsion chemotaxis system { u(t) = a triangle u-chi del (u del v) +xi del (u del w), x is an element of ohm, t > 0, tau v(t )= b triangle v+alpha u-beta v, x is an element of ohm, t > 0, 0 = c triangle w+gamma u-delta w, x is an element of ohm, t>0 under homogeneous Neumann initial-boundary conditions, where ohm subset of R-n (n <3) is a smoothly bounded domain and a, b, c,chi,xi,alpha, beta, gamma, delta >0 and tau is an element of {0,1} are constants. The purpose of the present paper is to construct a local solution of this system for any L-2-initial data without additional conditions on chi and xi 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.
This research delves into the exact controllability of semilinear measure driven integrodifferential systems in nonlocal settings. We give enough controllability requirements using the measure of noncompactness and the Monch fixed point theorem without making any assumptions about how compact the evolution system is in relation to the linear part of the measure system. We find results here that both generalize and improve upon many prior findings.
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.
We show that a unital complex normed Q-algebra (A, .) in which the spectral radius satisfies: rho(Lambda)(x)= inf{p(x):p is an element of E-un(A) , p < || .|| }, where Eun (A) denotes the set of all algebra-norms p on A equivalent to the given algebra-norm . such that p(e) = 1, is commutative modulo its Jacobson radical. The same conclusion is obtained if (A, ||.||) satisfies: rho(Lambda)b(xy) < rho(Lambda)b(x) ||y || for every x is an element of Ab, y is an element of A, where A is the completion of (A, || .||).
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 to this, 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.
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 M-0,M-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 MC(& ell; )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 MC & ell; as the defined space and its signed tropicalizations as the aforementioned subfans.
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 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.
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.
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.
The loop-Amplituhedron A(n )((L)) is a semialgebraic set in the product of Grassmannians Gr(R()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(4)((2))
The volume of a cyclic polytope can be obtained by forming an iterated integral 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 linear combinations of iterated integrals that are invariant under the subgroup action. This yields interesting polynomial attributes of the cyclic polytope. We prove that there are infinitely many of these invariants which are algebraically independent in the shuffle algebra.
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.
The Plucker 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).
