Finite Order Research Articles

On Distance Average Degree Independent Resolving Sets of Some Algebraic Graphs

Objectives: This study introduces the concept of distance average degree independent resolving sets and its dimensions for identity graphs of finite groups and order prime graphs of finite groups as a theorem in detail. Methods: The methodology involves first constructing the subset from the given graph, then finding the distance between each and every vertice of the graph and the subset, and also finding the average of the distance such that each and every vertice should receive a distinct code. The minimum number of vertices that satisfy this condition, their cardinality, is called the dimension. And then check various conditions like adjacency, equitability, independence, etc. in order to find various types of dimensions for the considered graphs. Findings: In this article, the concept of distance average degree resolving sets has been found, and further distance average degree independent resolving sets have been found, with their corresponding dimensions also found for algebraic graphs like identity graphs of finite groups and order prime graphs of finite groups. Novelty: The novelty in this article is the concept of distance-average-degree independent resolving sets. Particularly, this concept has been applied to the algebraic graphs, namely the identity graph and the order prime graph of a finite group. In addition to that, the concept of distance has been added to this resolving sets area, which is totally new to this area of graph theory. AMS Subject Classification: 05C12, 05C50. Keywords: Resolving set, Independent set, Distance average degree independent resolving set, Identity graphs, Order prime graphs

Zeros of higher derivatives of Riemann zeta function

In this article, we extend the result of Conrey [4, Theorem 2] to shorter intervals for higher-order derivatives of the zeta function. That is we study the mean value of the product of two finite order derivatives of the zeta function multiplied by a mollifier in short intervals. In this process, we obtain better mollifier length in some short intervals compared to the length of mollifier implied by Conrey's result. These finer studies allow us to refine the error term of some classical results of Levinson and Montgomery [13], Ki and Lee [11] on zero density estimates of ζ(k). Further, we showed that almost all non-trivial zeros of Matsumoto-Tanigawa's ηk-function cluster near the critical line.

Zero Distribution of Finite Order Bank–Laine Functions

Zero Distribution of Finite Order Bank–Laine Functions

Symplectic rigidity of O’Grady’s tenfolds

We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O’Grady’s 10 10 -dimensional deformation type is trivial.

Transcendental meromorphic solutions of certain nonlinear difference equations

In this paper, we mainly explore the following finite order meromorphic solutions of the nonlinear difference equationfn(z)+Pd(z,f)=p1eα1zk+p2eα2zk, where n≥2, k≥1 are two integers, Pd(z,f) is a difference polynomial in f of degree d≤n−1 with polynomials as its coefficients, and pj, αj(j=1,2) are nonzero constants satisfying certain conditions. Furthermore, we obtain the exact forms of meromorphic solutions of the above difference equation. The result of this paper extends and improves previous results, and some examples are given to show the accuracy of the result.

A simple construction of finitely generated infinite torsion groups

The goal of this note is to provide yet another proof of the following theorem of Golod: there exists an infinite finitely generated group G such that every element of G has finite order. Our proof is based on the Nielsen–Schreier index formula and is simple enough to be included in a standard group theory course.

Nonlocal Massive Gravity from Einstein Gravity.

We present a top-down construction of a three-dimensional nonlocal theory of massive gravity. This "nonlocal massive gravity" (NLMG) is obtained as the gravitational theory induced by Einstein gravity on a brane inserted in anti-de Sitter space modified by an overall minus sign. The theory involves an infinite series of increasingly complicated higher-derivative corrections to the Einstein-Hilbert action, with the quadratic term coinciding with new massive gravity. We obtain an analytic formula for the quadratic action of NLMG and show that its linearized spectrum consists of an infinite tower of positive-energy massive spin-2 modes. We compute the Newtonian potential and show that the introduction of the infinite series of terms makes it behave as ∼1/r at short distances, as opposed to the logarithmic behavior encountered when the series is truncated at any finite order. We use this and input from brane-world holography to argue that the theory may contain asymptotically flat black hole solutions.

Geometrically parametrised reduced order models for studying the hysteresis of the Coanda effect in finite element-based incompressible fluid dynamics

This article presents a general reduced order model (ROM) framework for addressing fluid dynamics problems involving time-dependent geometric parametrisations. The framework integrates Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) hyper-reduction techniques to effectively approximate incompressible computational fluid dynamics simulations. To demonstrate the applicability of this framework, we investigate the behaviour of a planar contraction-expansion channel geometry exhibiting bifurcating solutions known as the Coanda effect. By introducing time-dependent deformations to the channel geometry, we observe hysteresis phenomena in the solution.The paper provides a detailed formulation of the framework, including the stabilised finite elements full order model (FOM) and ROM, with a particular focus on the considerations related to geometric parametrisation. Subsequently, we present the results obtained from the simulations, analysing the solution behaviour in a phase space for the fluid velocity at a probe point, considered as the Quantity of Interest (QoI). Through qualitative and quantitative evaluations of the ROMs and hyper-reduced order models (HROMs), we demonstrate their ability to accurately reproduce the complete solution field and the QoI.While HROMs offer significant computational speedup, enabling efficient simulations, they do exhibit some errors, particularly for testing trajectories. However, their value lies in applications where the detection of the Coanda effect holds paramount importance, even if the selected bifurcation branch is incorrect. Alternatively, for more precise results, HROMs with lower speedups can be employed.

Simple model for the gap in the surface states of the antiferromagnetic topological insulator MnBi[formula omitted]Te[formula omitted]

We study the influence of the antiferromagnetic order on the surface states of topological insulators. We derive an effective Hamiltonian for these states, taking into account the spatial structure of the antiferromagnetic order. We obtain a typical (gapless) Dirac Hamiltonian for the surface states when the surface of the sample is not perturbed. Gapless spectrum is protected by the combination of time-reversal and half-translation symmetries. However, a shift in the chemical potential of the surface layer opens a gap in the spectrum away from the Fermi energy. Such a gap occurs only in systems with finite antiferromagnetic order. We observe that the system topology remains unchanged even for large values of the disorder. We calculate the spectrum using the tight-binding model with different boundary condition. In this case we get a gap in the spectrum of the surface states. This discrepancy arises due to the violation of the combined time-reversal symmetry. We compare our results with experiments and density functional theory calculations.

Limit Cycles and Local Bifurcation of Critical Periods in a Class of Switching Z2 Equivariant Quartic System

In this paper, the limit cycles and local bifurcation of critical periods for a class of switching Z2 equivariant quartic system with two symmetric singularities are investigated. First, through the computation of Lyapunov constants, the conditions of the two singularities to become the centers are determined. Then, we prove that there are at most 18 limit cycles with a distribution pattern of 9-9 around the two symmetric singular points of the system. Numerical simulation is conducted to validate the obtained results. Furthermore, by calculating the period constants, we determine the conditions for the critical point to be a weak center of finite order. Finally, the number of local critical periods that bifurcate from the equilibrium point under the center conditions is discussed. This study presents the first example of a quartic switching smooth system with 18 limit cycles and 4 local critical periods bifurcating from two symmetric singular points.

Sign involutions on para-abelian varieties

We study the so-called sign involutions on twisted forms of abelian varieties, and show that such a sign involution exists if and only if the class in the Weil–Châtelet group is annihilated by two. If these equivalent conditions hold, we prove that the Picard scheme of the quotient is étale and contains no points of finite order. In dimension one, such quotients are Brauer–Severi curves, and we analyze the ensuing embeddings of the genus-one curve into twisted forms of Hirzebruch surfaces and weighted projective spaces.

Characterizations of Finite Order Solutions of Circular Type Partial Differential-Difference Equations in $$ \mathbb {C}^n $$

Characterizations of Finite Order Solutions of Circular Type Partial Differential-Difference Equations in $$ \mathbb {C}^n $$

First-Order Temporal Logic on Finite Traces: Semantic Properties, Decidable Fragments, and Applications

Formalisms based on temporal logics interpreted over finite strict linear orders, known in the literature as finite traces , have been used for temporal specification in automated planning, process modelling, (runtime) verification and synthesis of programs, as well as in knowledge representation and reasoning. In this article, we focus on first-order temporal logic on finite traces . We first investigate preservation of equivalences and satisfiability of formulas between finite and infinite traces, by providing a set of semantic and syntactic conditions to guarantee when the distinction between reasoning in the two cases can be blurred. Moreover, we show that the satisfiability problem on finite traces for several decidable fragments of first-order temporal logic is ExpSpace -complete, as in the infinite trace case, while it decreases to NExpTime when finite traces bounded in the number of instants are considered. This leads also to new complexity results for temporal description logics over finite traces. Finally, we investigate applications to planning and verification, in particular by establishing connections with the notions of insensitivity to infiniteness and safety from the literature.

On Fixed Point of Difference Polynomials with Meromorphic Function of Finite Order

On Fixed Point of Difference Polynomials with Meromorphic Function of Finite Order

Entire solutions to Fermat-type difference and partial differential-difference equations in C^n

In this article, we study the existence and the form of finite order transcendental entire solutions of systems of Fermat-type difference and partial differential-difference equations in several complex variables. Our results extend previous theorems given by Xu-Cao [49], Xu et al [52], and Zheng-Xu [55]. We give some examples to illustrate the content of this article. For more information see https://ejde.math.txstate.edu/Volumes/2024/26/abstr.html

Geometric Quantization Results for Semi-positive Line Bundles on a Riemann Surface

In earlier work (Marinescu and Savale in Math Ann. https://doi.org/10.1007/s00208-023-02750-3, 2023) the authors proved the Bergman kernel expansion for semi-positive line bundles over a Riemann surface whose curvature vanishes to at most finite order at each point. Here we explore the related results and consequences of the expansion in the semi-positive case including: Tian’s approximation theorem for induced Fubini-Study metrics, leading-order asymptotics and composition for Toeplitz operators, asymptotics of zeroes for random sections, and the asymptotics of holomorphic torsion.

Big Ramsey degrees in ultraproducts of finite structures

We develop a transfer principle of structural Ramsey theory from finite structures to ultraproducts. We show that under certain mild conditions, when a class of finite structures has finite small Ramsey degrees, under the (Generalized) Continuum Hypothesis the ultraproduct has finite big Ramsey degrees for internal colorings. The necessity of restricting to internal colorings is demonstrated by the example of the ultraproduct of finite linear orders. Under CH, this ultraproduct L⁎ has, as a spine, η1, an uncountable analogue of the order type of rationals η. Finite big Ramsey degrees for η were exactly calculated by Devlin in [5]. It is immediate from [39] that η1 fails to have finite big Ramsey degrees. Moreover, we extend Devlin's coloring to η1 to show that it witnesses big Ramsey degrees of finite tuples in η on every copy of η in η1, and consequently in L⁎. This work gives additional confirmation that ultraproducts are a suitable environment for studying Ramsey properties of finite and infinite structures.

Adjacency preserving maps on symmetric tensors

Let r and s be positive integers such that r⩾2. Let U and V be vector spaces over fields F and K, respectively, such that dim⁡U⩾3 and F has at least r+1 elements. In this paper, we characterize surjective maps ψ:SrU→SsV preserving adjacency in both directions on symmetric tensors of finite order, which generalizes Hua's fundamental theorem of geometry of symmetric matrices. We give examples showing the indispensability of the assumptions dim⁡U⩾3 and the cardinality |F|⩾r+1 in our result.

Phase retrieval of entire functions and its implications for Gabor phase retrieval

We characterise all pairs of finite order entire functions whose magnitudes agree on two arbitrary lines in the complex plane by means of the Hadamard factorisation theorem. Building on this, we also characterise all pairs of second order entire functions whose magnitudes agree on infinitely many equidistant parallel lines. Furthermore, we show that the magnitude of an entire function on three parallel lines, whose distances are rationally independent, uniquely determines the function up to global phase, and that there exists a first order entire function whose magnitude on the lines R+inZ does not uniquely determine it up to global phase, for all positive integers n. Our results have direct implications for Gabor phase retrieval.

Coupled thermal-mechanical analysis of power electronic modules with finite element method and parametric model order reduction

This work presents a new approach for performing a parametric study and examining nonlinear material behaviours of a coupled thermal-mechanical model of a Power Electronics Module (PEM) by integrating the Finite Element Method (ANSYS-FEM) with Parametric Model Order Reduction (pMOR). The considered coupling method solves the thermal and structural models concurrently compared to the widely practised sequential coupling method. Instead of constant parameter values, which are generally regarded for pMOR studies, the temperature-dependent material properties of the wire material have been parametrised in the work using the pMOR method. A generalised 2D model has been regarded here for thermal-mechanical analysis with the pMOR approach, parametrising temperature-dependent coefficient of thermal expansion (CTE) and Young’s modulus (E) of the wire material to explore their impact on wire bonds. The matrix interpolation method has been applied here for the pMOR study, and PRIMA, a Krylov subspace-based model order reduction (MOR) technique, has been exercised for local model order reductions. A new efficient process of matrix interpolation, based on the Lagrange interpolation technique, has been developed to implement matrix interpolation in the parametric reduced order model (pROM). The local reduced order models (ROMs) have a degree of freedom (DOF) of just 8, compared to the full-order models’ (FOMs) of 50,602. The pROM provides an excellent solution and reduces computational time by 84% for the presented case.