Dimensional Case Research Articles

Unital positive Schur multipliers on $S_n^p$ with a completely isometric dilation

Let $1<p\neq 2<\infty $ and let $S^p_n$ be the associated Schatten von Neumann class over $n\times n$ matrices. We prove new characterizations of unital positive Schur multipliers $S^p_n\to S^p_n$ which can be dilated into an invertible complete isometry acting on a non-commutative $L^p$-space. Then we investigate the infinite dimensional case.

Efficient Convex PCA with Applications to Wasserstein GPCA and Ranked Data

Convex PCA, which was introduced by Bigot et al. modifies Euclidean PCA by restricting the data and the principal components to lie in a given convex subset of a Hilbert space. This setting arises naturally in many applications, including distributional data in the Wasserstein space of an interval, and ranked compositional data under the Aitchison geometry. Our contribution in this article is 3-fold. First, we present several new theoretical results including consistency as well as continuity and differentiability of the objective function in the finite dimensional case. Second, we develop a numerical implementation of finite dimensional convex PCA when the convex set is polyhedral, and show that this provides a natural approximation of Wasserstein GPCA. Third, we illustrate our results with two financial applications, namely distributions of stock returns ranked by size and the capital distribution curve, both of which are of independent interest in stochastic portfolio theory. Supplementary materials for this article are available online.

$2$-nodal domain theorems for higher-dimensional circle bundles

We prove that the real parts of equivariant (but non-invariant) eigenfunctions of generic bundle metrics on non-trivial principal S^{1} bundles over manifolds of any dimension have connected nodal sets and exactly 2 nodal domains. This generalizes earlier results of the authors in the 3 -dimensional case. The failure of the results on for non-free S^{1} actions is illustrated on even-dimensional spheres by one-parameter subgroups of rotations whose fixed point set consists of two antipodal points.

Two Numerical Schemes Via an Auxiliary Function for Nonlocal Problems With Integrable Kernels

ABSTRACTIn this study, we explore volume constrained nonlocal problem with integrable kernels. A central challenge addressed is the discontinuity of true solutions across the boundary, posing difficulties for designing efficient numerical methods. To address this issue, we introduce an auxiliary function which serves as the solution of an auxiliary problem. This approach effectively transforms the original problem to a new nonlocal problem with a true solution which is continuous across the boundary. We propose two numerical schemes to solve the new nonlocal problem: one is exclusively applicable to one‐dimensional case, while the other is suitable for general dimensional cases. Experimental results show that both schemes exhibit optimal convergence order under mild condition.

The signaling dimension of two-dimensional and polytopic systems

The signaling dimension of any given physical system represents its classical simulation cost, that is, the minimum dimension of a classical system capable of reproducing all the input/output correlations of the given system. The signaling dimension landscape is vastly unexplored; the only non-trivial systems whose signaling dimension is known -- other than quantum systems -- are the octahedron and the composition of two squares.}{Building on previous results by Matsumoto, Kimura, and Frenkel, our first result consists of deriving bounds on the signaling dimension of any system as a function of its Minkowski measure of asymmetry. We use such bounds to prove that the signaling dimension of any two-dimensional system (i.e. with two-dimensional set of admissible states, such as polygons and the real qubit) is two if and only if such a set is centrally symmetric, and three otherwise, thus conclusively settling the problem of the signaling dimension for such systems.} {Guided by the relevance of symmetries in the two dimensional case, we propose a branch and bound division-free algorithm for the exact computation of the symmetries of any given polytope, in polynomial time in the number of vertices and in factorial time in the dimension of the space. Our second result then consist of providing an algorithm for the exact computation of the signaling dimension of any given system, that outperforms previous proposals by exploiting the aforementioned bounds to improve its pruning techniques and incorporating as a subroutine the aforementioned symmetries-finding algorithm. We apply our algorithm to compute the exact value of the signaling dimension for all rational Platonic, Archimedean, and Catalan solids, and for the class of hyper-octahedral systems up to dimension five.

Toward Exact Critical Exponents from the low-order loop expansion of the Effective Potential in Quantum Field Theory

The asymptotic strong-coupling behavior as well as the exact critical exponents from scalar field theory even for the simplest case of 1+1 dimensions have not been obtained yet. Hagen Kleinert has linked both critical exponents and strong coupling parameters to each other. He used a variational technique ( back to kleinert and Feynman) to extract accurate values for the strong coupling parameters from which he was able to extract precise critical exponents. In this work, we suggest a simple method of using the effective potential ( low order) to obtain exact values for the strong-coupling parameters for the ϕ4 scalar field theory in 0+1 and 1+1 space–time dimensions. For the 0+1 case, our results coincide with the well-known exact values already known from literature while for the 1+1 case we test the results by obtaining the corresponding exact critical exponent. As the effective potential is a well-established tool in quantum field theory, we expect that the results can be easily extended to the most important three dimensional case and then the dream of getting exact critical exponents is made possible.

Global large solutions for the nonlinear dissipative system modeling electro-hydrodynamics

In this paper, we are concerned with global existence of large solutions for a dissipative model arising from electro-hydrodynamics, which is the nonlinear nonlocal system coupled by the Poisson–Nernst–Planck equations and the incompressible Navier–Stokes equations through charge transport and external forcing terms. By introducing some proper weighted functions and fully using the algebraic structure of the system, we prove that, under some conditions imposed on the indices p, p1, q, r, α, there exist two positive constants c0, C0 such that if the initial data u0=(u0h,u03) and (v0, w0) satisfy ‖u0h‖Ḃp1,∞−1+3p1+‖u0h‖Ḃp1,∞−1+3p1α‖u03‖Ḃp1,∞−1+3p11−α+K0≤c0 with K0≔‖v0‖Ḃq,1−2+3q⁡expC0‖u0‖Ḃp,1−1+3p+C0‖w0‖Ḃr,1−2+3r+1expC0‖u0‖Ḃp,1−1+3p, then the system admits a unique global solution. Moreover, the global existence of large solution was also established in two dimensional case.

Optimal error bounds of the time-splitting sine-pseudospectral method for the biharmonic nonlinear Schrödinger equation

We propose a time-splitting sine-pseudospectral (TSSP) method for the biharmonic nonlinear Schrödinger equation (BNLS) and establish its optimal error bounds. In the proposed TSSP method, we adopt the sine-pseudospectral method for spatial discretization and the second-order Strang splitting for temporal discretization. The proposed TSSP method is explicit and structure-preserving, such as time symmetric, mass conservation and maintaining the dispersion relation of the original BNLS in the discretized level. Under the assumption that the solution of the one dimensional BNLS belongs to Hm with m≥9, we prove error bounds at O(τ2+hm) and O(τ2+hm−1) in L2 norm and H1 norm respectively, for the proposed TSSP method, with τ time step and h mesh size. For general dimensional cases with d=1,2,3, the error bounds are at O(τ2+hm) and O(τ2+hm−2) in L2 and H2 norm under the assumption that the exact solution is in Hm with m≥10. The proof is based on the bound of the Lie-commutator for the local truncation error, discrete Gronwall inequality, energy method and the H1- or H2-bound of the numerical solution which implies the L∞-bound of the numerical solution. Finally, extensive numerical results are reported to confirm our optimal error bounds and to demonstrate rich phenomena of the solutions including rapidly dispersion in space of high frequency waves and soliton collisions.

The role of dimensions in gravitating relativistic shear-free fluids

We study the dynamics of relativistic shear-free gravitating fluids in higher dimensions for both neutral and charged matter. We reduce the Einstein–Maxwell equations to a single second order nonlinear partial differential equation which contains two arbitrary functions. This generalizes the condition of pressure isotropy to higher dimensions; the new condition is functionally different from four dimensions. Our result in higher dimensions reduces to known results in four dimensions. The presence of higher dimensions affects the dynamics of relativistic fluids in general relativity. The dynamical behaviour of the gravitating fluid in higher dimensions is qualitatively different to the four dimensional case. Higher dimensions affect astrophysical and cosmological processes in gravitating shear-free fluids.

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.

Classical solvability to the free boundary problem for a foam drainage equation. II. From the interface to the bottom

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. in 1988 in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region below an interface to the bottom in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle. Moreover, the existence of its steady solution and its stability are shown.

Classical solvability to the free boundary problem for a foam drainage equation. I. From the top to the interface

We study a nonlinear partial differential equation describing the evolution of a foam drainage in one dimensional case which was proposed by Goldfarb et al. [Izv. Akad. Nauk SSSR Mekh. Ghidk. Gaza 2, 103–108 (1988)] in order to investigate the flow of liquid through channels (Plateau borders) and nodes (intersections of four channels) between the bubbles, driven by gravity and capillarity. Its mathematical studies so far are mainly restricted within numerical and particular solutions; as for mathematical analysis of it there are only a few. We prove that the free boundary problem for the foam drainage equation in the region from the top to the interface in a foam column admits a unique global-in-time classical solution by the standard classical mathematical method, the maximum principle and the comparison theorem. Moreover, the existence of its steady solution and its stability are discussed.

Model to identify comfortable cloth fabric for human body during thermal stress due to physical exercise

Physical exercise enhances heat generation in human skin and subcutaneous tissues causing thermal stress. This thermal stress causes discomfort in the human body and deteriorates the physical performance of the human subjects. Appropriate choice of cloth for outer covering of human body is necessary to prevent this to provide comfort during the thermal stress. A mathematical model is proposed in this study to analyze the thermal stresses in human skin and subcutaneous tissues covered with various different cloth fabrics. The model considers four compartments namely cloth, epidermis, dermis, subcutaneous tissues. The bioheat equation along with appropriate boundary condition is employed in this model for one dimensional unsteady state case. The numerical simulation is performed using forward and central finite difference formulas. The findings have been utilized to examine how different types of fabric impact thermal stress experienced by the layers of human skin during physical activity. The polyester is found to be superior in comparison to cotton fabric as it shows less thermal stress and thus can be concluded to be more comfortable as a dress material.

Shape optimization for a nonlinear elliptic problem related to thermal insulation

In this paper, we consider a minimization problem of a nonlinear functional I_{\beta,p}(D,\Omega) related to a thermal insulation problem with a convection term, where \Omega is a bounded connected open set in \R^{n} and D\subset\overline{\Omega} is a compact set. The Euler–Lagrange equation relative to I_{\beta,p} is a p -Laplace equation, 1<p<\infty , with a Robin boundary condition with parameter \beta>0 . The main aim is to study extremum problems for I_{\beta,p}(D,\Omega) , among domains D with given geometrical constraints and \Omega\setminus D of fixed thickness. In the planar case, we show that under perimeter constraint the disk maximizes I_{\beta,p} . In the n -dimensional case we restrict our analysis to convex sets showing that the same is true for the ball but under different geometrical constraints.

Boundedness and asymptotic behavior in the higher dimensional fully parabolic attraction-repulsion chemotaxis system with nonlinear diffusion

In this paper, we consider the initial-boundary value problem for the fully parabolic attraction-repulsion chemotaxis model with nonlinear diffusion{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w),x∈Ω,t>0,vt=D1Δv+αu−βv,x∈Ω,t>0,wt=D2Δw+γu−δw,x∈Ω,t>0, in a bounded domain Ω⊂Rn(n≥3) with smooth boundary subject to homogeneous Neumann boundary conditions, where χ,ξ,D1,D2,α,β,γ,δ are positive parameters. The function D satisfies D(u)≥CDuθ for all u>0 with constant CD>0. We study the higher dimensional case with D1≠D2 and obtain that the problem possesses a global bounded solution under ξγχα≥max⁡{D1D2,D2D1,βδ,δβ} and θ>1−4n+2. If the parameter further satisfies θ≤1, then the bounded solution converges to the constant steady state (u‾0,αβu‾0,γδu‾0) as t→∞, where u¯0=1|Ω|∫Ωu0dx and u0 is the initial data of u. The result extends some existing conclusions on two fronts: we obtain the boundedness with D1≠D2 in the higher dimensional case and expand the range of θ due to the repulsion effect.

On the geometry of zero sets of central quaternionic polynomials

Let R be the ring H[x1,…,xn] of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if p∈R vanishes at all the common zeros of I in Hn with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in Hn. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.

Infinite dimensional analogues of Choi matrices

For a class of linear maps on a von Neumann factor, we associate two objects, bounded operators and trace class operators, both of which play the roles of Choi matrices. Each of them is positive if and only if the original map on the factor is completely positive. They are also useful to characterize positivity of maps as well as complete positivity. It turns out that such correspondences are possible for every normal completely bounded map if and only if the factor is of type I. As an application, we provide criteria for Schmidt numbers of normal positive functionals in terms of Choi matrices of k-positive maps, in infinite dimensional cases. We also define the notion of k-superpositive maps, which turns out to be equivalent to the property of k-partially entanglement breaking.

Thermodynamics of the 3-dimensional Einstein-Maxwell system

Recently, I studied the thermodynamical properties of the Einstein-Maxwell system with a box boundary in 4-dimensions [1]. In this paper, I investigate those in 3-dimensions using the zero-loop saddle-point approximation and focusing only on a simple topology sector as usual. Similar to the 4-dimensional case, the system is thermodynamically well-behaved when Λ < 0 (due to the contribution of the “bag of gold” saddles). However, when Λ = 0, a crucial difference to the 4-dimensional case appears, i.e. the 3-dimensional system turns out to be thermodynamically unstable, while the 4-dimensional one is thermodynamically stable. This may offer two options for how we think about the thermodynamics of 3-dimensional gravity with Λ = 0. One is that the zero-loop approximation or restricting the simple topology sector is not sufficient for 3-dimensions with Λ = 0. The other is that 3-dimensional gravity is really thermodynamically unstable when Λ = 0.

Pinning-depinning transitions in two classes of discrete elastic-string models in (2+1)-dimensions

The pinning-depinning phase transitions of interfaces for two classes of discrete elastic-string models are investigated numerically. In the (1+1)-dimensions, we revisit these two elastic-string models with slight modification to the growth rule, and compare the estimated values with the previous numerical and experimental results. For the (2+1)-dimensional case, we perform extensive simulations on pinning-depinning transitions in these discrete models with quenched disorder. For full comparisons in the physically relevant spatial dimensions, we also perform numerically two distinct universality classes, including the quenched Edwards–Wilkinson, and the quenched Kardar–Parisi–Zhang equations with and without external driving forces. The critical exponents of these systems in the presence of quenched disorder are numerically estimated. Our results show that the critical exponents satisfy scaling relations well, and these two discrete elastic-string models do not fall into the existing universality classes. In order to visually comparisons of these discrete systems with quenched disorder in the (2+1)-dimensional cases, we present surface morphologies with various external driving forces during the saturated time regimes. The relationships between surface morphologies, scaling exponents and correlation length are also revealed.

A shortest‐path‐aided fast‐sweeping method to improve the accuracy of traveltime calculation in vertically transverse isotropic media

AbstractThe high accuracy and efficiency of traveltime calculation are critical in seismic tomography, migration, static corrections, source locations and anisotropic parameter estimation. The fast‐sweeping method is an efficient upwind finite‐difference approach for solving the eikonal equation. However, the fast‐sweeping method is accurate only along the axis directions. In two‐dimensional or higher dimensional cases, the accuracy is severely decreased in the diagonal directions due to the numerical errors in these directions. These similar numerical errors also arose in higher order fast‐sweeping method and anisotropic fast‐sweeping method. To improve the accuracy of traveltime calculation in two‐dimensional or higher dimensional space, a shortest‐path‐aided fast‐sweeping method is proposed. The shortest‐path‐aided solution is embedded into the sweeping process of the standard fast‐sweeping method to improve the traveltime accuracy in the diagonal directions. Shortest‐path‐aided fast‐sweeping method is very easy to implement nearly without additional computational cost and memory consumption. Furthermore, this method is easy to extend from two‐dimensional to higher dimensional, from low‐order to higher‐order and from isotropic to anisotropic cases.