Antiperiodic Boundary Conditions Research Articles

Mode recombination formula and nonanalytic term in an effective potential at finite temperature on a compactified space

We develop a new formula called a mode recombination formula, and we can recast the effective potential at finite temperature in one-loop approximation for fermion and scalar fields on the D-dimensional spacetime, Sτ1×RD−(p+1)×∏i=1pSi1 into a convenient form for discussing nonanalytic terms, which cannot be written in the form of any positive integer power of the field-dependent mass squared, in the effective potential. The formula holds irrespective of whether the field is a fermion or a scalar and of boundary conditions for spatial Si1 directions and clarifies the importance of zero modes in the Matsubara and Kaluza-Klein modes for the existence of the nonanalytic terms. The effective potential is drastically simplified further to obtain the nonanalytic terms in easier and more transparent way. In addition to reproducing previous results, we find that there exists no nonanalytic term for the fermion field with arbitrary boundary condition for the spatial Si1 direction, which is also the case for the scalar field with the antiperiodic boundary condition for the spatial direction. Published by the American Physical Society 2024

Lyapunov-type inequalities for a nonlinear system including operators

UDC 517.9 We obtain new Lyapunov-type inequalities for a nonlinear system including p -relativistic operator and q -prescribed curvature operator under the Dirichlet or antiperiodic boundary condition.

Casimir and Helmholtz forces in one-dimensional Ising model with Dirichlet (free) boundary conditions

Attention in the literature has increasingly turned to the issue of the dependence on ensemble and boundary conditions of fluctuation-induced forces. We have recently investigated this problem in the one-dimensional Ising model with periodic and antiperiodic boundary conditions (Annals of Physics 459, 169533 (2023)). Significant variations of the behavior of Casimir and Helmholtz forces was observed, depending on both ensemble and boundary conditions. Here we extend our study by considering the problem in the important case of Dirichlet (also termed free, or missing neighbors) boundary conditions. The advantage of the mathematical formulation of the problem in terms of Chebyshev polynomials is demonstrated and, in this approach, expressions for the partition functions in the canonical and the grand canonical ensembles are presented. We prove analytically that the Casimir force is attractive for all values of temperature and external ordering field, while the Helmholtz force can be both attractive and repulsive.

Existence and stability of a q-Caputo fractional jerk differential equation having anti-periodic boundary conditions

In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder’s alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study.

Lyapunov-type inequalities for Hilfer fractional boundary value problems

This paper deals with fractional boundary value problems involving the Hilfer–Hadamard fractional differential operator of order [Formula: see text] and type [Formula: see text]. We derive the corresponding Lyapunov-type inequalities for two prominent classes of Hilfer–Hadamard fractional boundary value problems (HFBVPs) involving separated and anti-periodic boundary conditions. For this purpose, we construct the associated Green’s functions and deduce their important properties.

Casimir versus Helmholtz forces: Exact results

Recently, attention has turned to the issue of the ensemble dependence of fluctuation induced forces. As a noteworthy example, in O(n) systems the statistical mechanics underlying such forces can be shown to differ in the constant M→ magnetic canonical ensemble (CE) from those in the widely-studied constant h→ grand canonical ensemble (GCE). Here, the counterpart of the Casimir force in the GCE is the Helmholtz force in the CE. Given the difference between the two ensembles for finite systems, it is reasonable to anticipate that these forces will have, in general, different behavior for the same geometry and boundary conditions. Here we present some exact results for both the Casimir and the Helmholtz force in the case of the one-dimensional Ising model subject to periodic and antiperiodic boundary conditions and compare their behavior. We note that the Ising model has recently being solved in Phys.Rev. E 106 L042103 (2022), using a combinatorial approach, for the case of fixed value M of its order parameter. Here we derive exact result for the partition function of the one-dimensional Ising model of N spins and fixed value M using the transfer matrix method (TMM); earlier results obtained via the TMM were limited to M=0 and N even. As a byproduct, we derive several specific integral representations of the hypergeometric function of Gauss. Using those results, we rigorously derive that the free energies of the CE and grand GCE are related to each other via Legendre transformation in the thermodynamic limit, and establish the leading finite-size corrections for the canonical case, which turn out to be much more pronounced than the corresponding ones in the case of the GCE.

Existence of Solutions for a Coupled System of Nonlinear Implicit Differential Equations Involving $$\varrho $$-Fractional Derivative with Anti Periodic Boundary Conditions

Existence of Solutions for a Coupled System of Nonlinear Implicit Differential Equations Involving $$\varrho $$-Fractional Derivative with Anti Periodic Boundary Conditions

Qualitative analysis of tripled system of fractional Langevin equations with cyclic anti-periodic boundary conditions

Qualitative analysis of tripled system of fractional Langevin equations with cyclic anti-periodic boundary conditions

Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings

We introduce a notion of nonlinear cyclic orbital (xi -mathscr{F})-contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for:1. The nonlinear multi-order fractional differential equation L(D)θ(ς)=σ(ς,θ(ς)),ς∈J=[0,A],A>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{L}(\\mathcal{D})\ heta (\\varsigma )=\\sigma \\bigl(\\varsigma , \ heta ( \\varsigma ) \\bigr), \\quad \\varsigma \\in \\mathscr{J}=[0,\\mathscr{A}], \\mathscr{A}>0, $$\\end{document} where L(D)=γwcDδw+γw−1cDδw−1+⋯+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3,…,w),γw≠0,0≤δ0<δ1<δ2<⋯<δw−1<δw<1;\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})=\\gamma _{w} \\,{}^{c} \\mathcal{D}^{\\delta _{w}}+ \\gamma _{w-1} \\,{}^{c} \\mathcal{D}^{\\delta _{w-1}}+\\cdots+\\gamma _{1} \\,{}^{c} \\mathcal{D}^{\\delta _{1}}+\\gamma _{0} \\,{}^{c} \\mathcal{D}^{\\delta _{0}},\\\\ &\\gamma _{\\flat}\\in \\mathbb{R}\\quad (\\flat =0,1,2,3,\\ldots,w), \\qquad \\gamma _{w} \ eq 0, \\\\ &0\\leq \\delta _{0}< \\delta _{1}< \\delta _{2}< \\cdots< \\delta _{w-1}< \\delta _{w}< 1; \\end{aligned}$$ \\end{document}2. The nonlinear multi-term fractional delay differential equation L(D)θ(ς)=σ(ς,θ(ς),θ(ς−τ)),ς∈J=[0,A],A>0;θ(ς)=σ¯(ς),ς∈[−τ,0],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})\ heta (\\varsigma ) =\\sigma \\bigl(\\varsigma , \ heta ( \\varsigma ),\ heta (\\varsigma -\ au ) \\bigr), \\quad \\varsigma \\in \\mathscr{J}=[0, \\mathscr{A}], \\mathscr{A}>0; \\\\ &\ heta (\\varsigma ) =\\bar{\\sigma}(\\varsigma ),\\quad \\varsigma \\in [-\ au ,0], \\end{aligned}$$ \\end{document} where L(D)=γwcDδw+γw−1cDδw−1+⋯+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3,…,w),γw≠0,0≤δ0<δ1<δ2<⋯<δw−1<δw<1;\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})=\\gamma _{w} \\,{}^{c} \\mathcal{D}^{\\delta _{w}}+ \\gamma _{w-1} \\,{}^{c} \\mathcal{D}^{\\delta _{w-1}}+\\cdots+\\gamma _{1} \\,{}^{c} \\mathcal{D}^{\\delta _{1}}+\\gamma _{0} \\,{}^{c} \\mathcal{D}^{\\delta _{0}},\\\\ &\\gamma _{\\flat}\\in \\mathbb{R}\\quad (\\flat =0,1,2,3,\\ldots,w), \\qquad \\gamma _{w} \ eq 0, \\\\ &0\\leq \\delta _{0}< \\delta _{1}< \\delta _{2}< \\cdots< \\delta _{w-1}< \\delta _{w}< 1; \\end{aligned}$$ \\end{document} moreover, here {}^{c}mathcal{D}^{delta} is predominantly called Caputo fractional derivative of order δ.

Three types of boundary conditions in a four-fermion interacting model for quarks

In this note, we study the phase transition of an effective four-fermion model describing interactions between quarks, taking into account finite size and chemical potential effects on the system, with periodic, quasi-periodic, and anti-periodic boundary conditions in D - Euclidean dimensions through Jacobi theta functions. Strictly speaking, this is performed by the generalized Matsubara formalism and the Schwinger proper-time representation. We consider the system at temperature T and the spatial coordinates compacted along the direction j with length Lj. The gap equation of the system is solved numerically and we investigate its behavior when the thermodynamic parameters of the model are changed.

A coupled system of fractional difference equations with anti-periodic boundary conditions

"In this article, we give su cient conditions for the existence, uniqueness and Ulam{Hyers stability of solutions for a coupled system of two-point nabla fractional di erence boundary value problems subject to anti-periodic boundary conditions, using the vector approach of Precup [4, 14, 19, 21]. Some examples are included to illustrate the theory."

The finite volume effects on the critical endpoint of chiral phase transition in Nambu–Jona-Lasinio model with different regularization schemes

In this paper, we study the influence of different regularization schemes on the critical endpoint (CEP) of chiral phase transition within a cubic box with volume [Formula: see text]. A two-flavor Nambu–Jona-Lasinio model at finite temperature [Formula: see text] and chemical potential [Formula: see text] is adopted as the effective model of the strong interacting matter. Due to the finite volume of the box, the momentum integral in gap equation is replaced by discrete summation, and an anti-periodic boundary condition for quark field is applied. We employ the Schwinger’s proper time and the Pauli–Villars regularization (PVR) schemes, respectively. It is found that the first-order phase transition line displays an intriguing “staircase” behavior, and eventually disappears as [Formula: see text] increases. In particular, there is no existence of the CEP for both regularization schemes in infinite volume limit [Formula: see text]. However, for the finite volume, the locations of the CEPs with proper time and PVR are determined, respectively.

Finite-volume effects in baryon number fluctuations around the QCD critical endpoint

We present results for the volume dependence of baryon number fluctuations in the vicinity of the (conjectured) critical endpoint of QCD. They are extracted from the nonperturbative quark propagator that is obtained as a solution to a set of truncated Dyson–Schwinger equations of (2+1)-flavor QCD in Landau gauge, which takes the backcoupling of quarks onto the Yang–Mills sector explicitly into account. This well-studied system predicts a critical endpoint at moderate temperatures and rather large chemical potential. We investigate this system at small and intermediate finite, three-dimensional, cubic volumes and study the resulting impact on baryon number fluctuations and ratios thereof up to fourth order in the region of the critical endpoint. Due to the limitations of our truncation, the results are quantitatively meaningful only outside the critical scaling region of the endpoint. We find that the fluctuations are visibly affected by the finite volume, particularly for antiperiodic boundary conditions, whereas their ratios are practically invariant.

Characterizing the delocalized–localized Anderson phase transition based on the system’s response to boundary conditions

A new characterization of the Anderson phase transition, based on the response of the system to the boundary conditions is introduced. We change the boundary conditions from periodic to antiperiodic and look for its effects on the eigenstate of the system. To characterize these effects, we use the overlap of the states. In particular, we numerically calculate the overlap between the ground-state of the system with periodic and antiperiodic boundary conditions in one-dimensional models with delocalized-localized phase transitions. We observe that the overlap is close to one in the localized phase, and it gets appreciably smaller in the delocalized phase. In addition, in models with mobility edges, we calculate the overlaps between single-particle eigenstate with periodic and antiperiodic boundary conditions to characterize the entire spectrum. By this single-particle overlap, we can locate the mobility edges between delocalized and localized states.

Existence of anti-periodic solutions for Ψ-Caputo-type fractional p-Laplacian problems via Leray--Schauder degree theory

Abstract The main crux of this work is to study the existence of solutions for a certain type of nonlinear Ψ-Caputo fractional differential equations with anti-periodic boundary conditions and p-Laplacian operator. The proofs are based on the Leray–Schauder degree theory and some basic concepts of Ψ-Caputo fractional calculus. As an application, our theoretical result has been illustrated by providing a suitable example.

Periodic and antiperiodic boundary value problems for the Sturm-Liouville equation with the discontinuous coefficient

The present paper studies the boundary-value problem in a finite segment generating with the discontinuous Sturm-Liouville equation and periodic(antiperiodic) boundary conditions. We prove completeness of the system of eigenfunctions and generalized eigenfunctions in the space L2(0, π; ρ), obtain the asymptotic formulas for the solution and prove the asymptotic formulas for the eigenvalues of the periodic and antiperiodic boundary value problems

Problem on piecewise Caputo-Fabrizio fractional delay differential equation under anti-periodic boundary conditions

This manuscript considers a class of piecewise differential equations (DEs) modeled with the Caputo-Fabrizio differential operator. The proposed problem involves a proportional delay term and is equipped with anti-periodic boundary conditions. The piecewise derivative can be applied to model many complex nature real-world problems that show a multi-step behavior. The existence theory and Hyer-Ulam (HU) stability results are studied for the proposed problem via fixed point techniques such as Banach contraction theorem, Schauder’s fixed point theorem and Arzelá Ascoli theorem. A numerical problem is presented as an example to see the validity and effectiveness of the applied concept.

Existence and Uniqueness Results for Different Orders Coupled System of Fractional Integro-Differential Equations with Anti-Periodic Nonlocal Integral Boundary Conditions

This paper presents a new class of boundary value problems of integrodifferential fractional equations of different order equipped with coupled anti-periodic and nonlocal integral boundary conditions. We prove the existence and uniqueness criteria of the solutions by using the Leray-Schauder alternative and Banach contraction mapping principle. Examples are constructed for the illustration of our results.

Existence and Stability of Solutions for Nabla Fractional Difference Systems with Anti-periodic Boundary Conditions

In this paper, we propose sufficient conditions on existence, uniqueness and Ulam-Hyers stability of solutions for coupled systems of fractional nabla difference equations with anti-periodic boundary conditions, by using fixed point theorems. We also support these results through a couple of examples.

Study of a coupled system with anti-periodic boundary conditions under piecewise Caputo-Fabrizio derivative

A coupled system under Caputo-Fabrizio fractional order derivative (CFFOD) with antiperiodic boundary condition is considered. We use piecewise version of CFFOD. Sufficient conditions for the existence and uniqueness of solution by ap?plying the Banach, Krasnoselskii?s fixed point theorems. Also some appropriate results for Hyers-Ulam (H-U) stability analysis is established. Proper example is given to verify the results.