Antiperiodic Boundary Conditions Research Articles

Lyapunov-type inequality for higher order nonlinear time scale systems with anti-periodic boundary conditions

Lyapunov-type inequality for higher order nonlinear time scale systems with anti-periodic boundary conditions

Twisted circle compactification of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM and its holographic dual

We consider a compactification of 4D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM, with SU(N) gauge group, on a circle with anti-periodic boundary conditions for the fermions. We couple the theory to a constant background gauge field along the circle for an abelian subgroup of the R-symmetry which allows to preserve four supersymmetries. The 3D effective theory exhibits gapped and ungapped phases, which we argue are holographically dual, respectively, to a supersymmetric soliton in AdS5 × S5, and a particular quotient of AdS5 × S5. The gapped phase corresponds to an IR 3D N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric Yang-Mills-Chern-Simons theory at level N, while the ungapped phase is naturally identified with the root of a Higgs branch in the 3D theory. We discuss the extension of the twisting procedure to maximally SUSY Yang-Mills theories in different dimensions, obtaining the relevant duals for 2D and 6D, and comment on the odd dimensional cases.

Multimaterial filtering applied to the topology optimization of a permanent magnet synchronous machine

PurposeThis study aims to propose a general methodology to handle multimaterial filtering for density-based topology optimization containing periodic or antiperiodic boundary conditions, which are expected to reduce the simulation time of electrical machines. The optimization of the material distribution in a permanent magnet synchronous machine rotor illustrates the relevance of this approach.Design/methodology/approachThe optimization algorithm relies on an augmented Lagrangian with a projected gradient descent. The 2D finite element method computes the physical and adjoint states to evaluate the objective function and its sensitivities. Concerning regularization, a mathematical development leads to a multimaterial convolution filtering methodology that is consistent with the boundary conditions and helps eliminate artifacts.FindingsThe method behaves as expected and shows the superiority of multimaterial topology optimization over bimaterial topology optimization for the chosen test case. Unlike the standard approach that uses a cropped convolution kernel, the proposed methodology does not artificially reflect the limits of the simulation domain in the optimized material distribution.Originality/valueAlthough filtering is a standard tool in topology optimization, no attention has previously been paid to the influence of periodic or antiperiodic boundary conditions when dealing with different natures of materials. The comparison between the bimaterial and multimaterial topology optimization of a permanent magnet machine rotor without symmetry constraints constitutes another originality of this work.

Existence and stability results for a Langevin system with Caputo–Hadamard fractional operators

This paper is devoted to analyzing a new model of a coupled Langevin system with fractional operators under nonlocal antiperiodic integral boundary conditions. This model involves nonlinear Langevin fractional equations with Caputo–Hadamard and Caputo fractional operators. Also, the existence and uniqueness of solutions to the suggested model have been investigated by the fixed-point technique. Moreover, the Hyers–Ulam stability of the solutions has been discussed. Finally, we provide an illustrative example to support the theoretical results.

Study of Hybrid Problems under Exponential Type Fractional-Order Derivatives

In this investigation, we develop a theory for the hybrid boundary value problem for fractional differential equations subject to three-point boundary conditions, including the antiperiodic hybrid boundary condition. On suggested problems, the third-order Caputo–Fabrizio derivative is the fractional operator applied. In this regard, the corresponding hybrid fractional integral equation is obtained by the Caputo–Fabrizio operator’s properties with the Green function’s aid. Then, we apply Dhage’s nonlinear alternative to the Schaefer type to prove the existence results. Finally, two examples are provided to confirm the validity of our main results.

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.

Topological defects in Floquet circuits

We introduce a Floquet circuit describing the driven Ising chain with topological defects. The corresponding gates include a defect that flips spins as well as the duality defect that explicitly implements the Kramers-Wannier duality transformation. The Floquet unitary evolution operator commutes with such defects, but the duality defect is not unitary, as it projects out half the states. We give two applications of these defects. One is to analyze the return amplitudes in the presence of “space-like” defects stretching around the system. We verify explicitly that the return amplitudes are in agreement with the fusion rules of the defects. The second application is to study unitary evolution in the presence of “time-like” defects that implement anti-periodic and duality-twisted boundary conditions. We show that a single unpaired localized Majorana zero mode appears in the latter case. We explicitly construct this operator, which acts as a symmetry of this Floquet circuit. We also present analytic expressions for the entanglement entropy after a single time step for a system of a few sites, for all of the above defect configurations.

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.

Solving nonlinear matrix and Riesz-Caputo fractional differential equations via fixed point theory in partial metric spaces

A modified implicit relation and ?-implicit contractive condition are introduced in the setting of relational partial metric spaces and some related fixed point results are derived. Two suitable examples are provided. As an application, sufficient conditions are derived for the existence of a unique positive definite solution of the non-linear matrix equation X = B + Pk i=1A* i T(X)Ai. An example is given, using matrices that are randomly generated, as well as convergence and error analysis and average CPU time analysis. Solving fractional differential equations of Riesz-Caputo type with anti-periodic boundary conditions is also discussed, followed by two illustrations.

A study of a self-adjoint coupled system of three nonlinear ordinary differential inclusions with cyclic anti-periodic boundary conditions

A study of a self-adjoint coupled system of three nonlinear ordinary differential inclusions with cyclic anti-periodic boundary conditions

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.