Dimensional Case Research Articles

Excitations in the higher-lattice gauge theory model for topological phases. III. The (3+1)-dimensional case

In this, the third paper in our series describing the excitations of the higher-lattice gauge theory model for topological phases, we will examine the (3+1)-dimensional case in detail. We will explicitly construct the ribbon and membrane operators which create the topological excitations, and use these creation operators to find the pattern of condensation and confinement. We also use these operators to find the braiding relations of the excitations, and to construct charge measurement operators which project to states of definite topological charge. Published by the American Physical Society 2024

Lifespan estimates for 1d damped wave equation with zero moment initial data

In this manuscript, a sharp lifespan estimate of solutions to semilinear classical damped wave equation is investigated in one dimensional case when the Fourier 0th moment of sum of initial position and speed is 0. Especially, it is shown that the behavior of lifespan changes with p=3/2 with respect to the size of the initial data.

Anisotropic regularity for elliptic problems with Dirac measures as data

We study the Possion problem with singular data given by a source supported on a one dimensional curve strictly contained in a three dimensional domain. We prove regularity results for the solution on isotropic and on anisotropic weighted spaces of Kondratiev type. Our technique is based on the study of a regularized problem. This allows us to exploit the local nature of the singularity. Our results hold with very few smoothness hypotheses on the domain and on the support of the data. We also discuss some extensions of our main results, including the two dimensional case, sources supported on closed curves and on polygonals.

On the depth of simplicial affine semigroup rings

AbstractWe recall and delve into the different characterizations of the depth of an affine semigroup ring, providing an original characterization of depth two in three and four dimensional cases which are closely related to the existence of a maximal element in certain Apéry sets.

Unconditional energy conservation and conditional regularity for the incompressible Navier-Stokes Maxwell system

In this paper, we study a hydrodynamic system modeling the evolution of a plasma subject to a self-induced electromagnetic Lorentz force in incompressible viscous fluids. The system consists of the Navier–Stokes equations coupled with a Maxwell equation. In the three dimensional case, we show that every weak solution verifies the energy equality for the incompressible Navier–Stokes–Maxwell equations with damping. We also establish some non-explosion criteria in terms of the velocity and magnetic of local strong solution for standard Navier–Stokes–Maxwell system.

How is a black hole created from nothing?

Using the synchronous coordinates, the creation of a Schwarzschild black hole immersed in a de Sitter spacetime can be viewed as a coherent creation of a collection of timelike geodesics. The previously supposed conical singularities do not exist at the horizons of the constrained instanton. Instead, the unavoidable irregularity is presented as a non-vanishing second fundamental form elsewhere at the quantum transition 3-surface. The same arguments can be applied to charged, topological or higher dimensional black hole cases.

Solvability of a third-order system of nonlinear difference equations via a generalized Fibonacci sequence

In this paper, we solve in closed-form the following third-order system of nonlinear difference equations xn+1 = ynyn−1xn−1p xn(anyn−2q + bnynyn−1),yn+1 = xnxn−1yn−1q yn(cnxn−2p + dnxnxn−1),p,q ∈ ℕ,n ∈ ℕ0 where the initial values x−i,y−i,i = 0,1,2 and the parameters (an)n∈ℕ0,(bn)n∈ℕ0,(cn)n∈ℕ0, (dn)n∈ℕ0 are non-zero real numbers. The form of the solutions of the one dimensional case of our system and a more general system defined by one to one functions are also presented.

Sharp Global Well-Posedness and Scattering for the Radial Conformal Nonlinear Wave Equation

Abstract In this paper we prove global well-posedness and scattering for the conformal, defocusing, nonlinear wave equation with radial initial data in the critical Sobolev space, for dimensions $d \geq 4$. This result extends a previous result proving sharp scattering in the three dimensional case.

Randomized complexity of parametric integration and the role of adaption I. Finite dimensional case

We study the randomized n-th minimal errors (and hence the complexity) of vector valued mean computation, which is the discrete version of parametric integration. The results of the present paper form the basis for the complexity analysis of parametric integration in Sobolev spaces, which will be presented in Part 2. Altogether this extends previous results of Heinrich and Sindambiwe (1999) [12] and Wiegand (2006) [27]. Moreover, a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting is solved.

Logarithmic Gross-Pitaevskii equation

We consider the Schrödinger equation with a logarithmic nonlinearty and non-trivial boundary conditions at infinity. We prove that the Cauchy problem is globally well posed in the energy space, which turns out to correspond to the energy space for the standard Gross-Pitaevskii equation with a cubic nonlinearity, in small dimensions. We then characterize the solitary and traveling waves in the one dimensional case.

Mathematical discoveries from program search with large language models

Large language models (LLMs) have demonstrated tremendous capabilities in solving complex tasks, from quantitative reasoning to understanding natural language. However, LLMs sometimes suffer from confabulations (or hallucinations), which can result in them making plausible but incorrect statements1,2. This hinders the use of current large models in scientific discovery. Here we introduce FunSearch (short for searching in the function space), an evolutionary procedure based on pairing a pretrained LLM with a systematic evaluator. We demonstrate the effectiveness of this approach to surpass the best-known results in important problems, pushing the boundary of existing LLM-based approaches3. Applying FunSearch to a central problem in extremal combinatorics—the cap set problem—we discover new constructions of large cap sets going beyond the best-known ones, both in finite dimensional and asymptotic cases. This shows that it is possible to make discoveries for established open problems using LLMs. We showcase the generality of FunSearch by applying it to an algorithmic problem, online bin packing, finding new heuristics that improve on widely used baselines. In contrast to most computer search approaches, FunSearch searches for programs that describe how to solve a problem, rather than what the solution is. Beyond being an effective and scalable strategy, discovered programs tend to be more interpretable than raw solutions, enabling feedback loops between domain experts and FunSearch, and the deployment of such programs in real-world applications.

Double-parameter regularization for solving the backward diffusion problem with parallel-in-time algorithm

We propose a double-parameter regularization scheme for dealing with the backward diffusion process. Considering the smoothing effect of Yosida approximation for PDE, we propose to regularize this ill-posed problem by modifying original governed system in terms of a pseudoparabolic equation together with a quasi-boundary condition simultaneously, which consequently contains two regularizing parameters. Theoretically, we establish the optimal error estimates between the regularizing solution and the exact one in terms of suitable choice strategy for the regularizing parameters, under a-priori regularity assumptions on the exact solution. The a-posteriori choice strategy for the regularizing parameters based on the discrepancy principle is also studied. To weaken the heavy computational cost for solving the discrete nonsymmetric linear regularizing system by finite difference scheme, especially in higher spatial dimensional cases, the block divide-and-conquer method together with the properties of the Schur complement is applied to decompose the linear system into two half-size linear systems, one of which can be solved by the diagonalization technique, and consequently an efficient parallel-in-time algorithm originally developed for direct problem is applicable. Our proposed method is of much lower complexity than the standard solver for the corresponding linear system. Finally, some numerical examples are presented to verify the efficiency of our proposed method.

Hypocoercivity in Algebraically Constrained Partial Differential Equations with Application to Oseen Equations

AbstractThe long-time behavior of solutions to different versions of Oseen equations of fluid flow on the 2D torus is analyzed using the concept of hypocoercivity. The considered models are isotropic Oseen equations where the viscosity acts uniformly in all directions and anisotropic Oseen-type equations with different viscosity directions. The hypocoercivity index is determined (if it exists) and it is shown that similar to the finite dimensional case of ordinary differential equations and differential-algebraic equations it characterizes its decay behavior.

SIR model on one dimensional small world networks

We study the nonequilibrium phase transition for the model of epidemic spreading, Susceptible–Infected–Refractory (SIR), on one dimensional small world networks. This model belongs to the universality class of dynamical percolation class (DyP) and the upper critical dimension corresponding to this class is dc=6. One dimensional case is special case of this class in which the percolation threshold goes to one (boundary value) in thermodynamic limit. This behavior resembles slightly the behavior of equilibrium phase transition in a one dimensional Ising and XY models where the critical thresholds for both models go to zero temperature (boundary value) in thermodynamic limit. By analytical arguments and numerical simulations we demonstrate that, increasing the connectivity (2k) of this model on regular one dimensional lattice does not alter the criticality of the model. However the phase transition study shows that, this model crosses from a one dimensional structure to mean field like for any finite value of the rewiring probability (p). This behavior is similar to what happened in the equilibrium phase transition for the Ising and XY models on small world networks. Thus, this model is a one of nonequilibrium models which behaves similarly to the equilibrium systems on small world networks. Unlike of many nonequilibrium systems on small world networks which have been found to display a mean field like behavior only at finite values of p or even show critical exponents depend on p. Furthermore, we calculate the critical exponents and the full critical phase space of this model on small world network. We also introduce the crossover scaling function of this model from one dimensional behavior to mean field behavior and reveal the similarity between this model and the equilibrium models on the small world networks.

DFT studies of 2D materials inspired by Lie algebras

Inspired by the root systems of Lie algebras of rank 2, we propose a mathematical method to engineer new 2D materials with double periodic structures tessellating the plane. Concretely, we investigate two geometries relaying on squares and hexagons exhibiting the D 4 × D 4 and D 6 × D 6 dihedral group invariances, respectively. Due to lack of empirical verifications of such double configurations, we provide a numerical investigation by help of the open source quantum espresso. Motivated by hybrid structures of the graphene, the silicene, and the germanene, we investigate two models involving the D 4 × D 4 and D 6 × D 6 dihedral symmetries which we refer to as Si4Ge4 and Si6C6 compounds, respectively. For simplicities, we study only the opto-electronic physical properties by applying an electromagnetic source propagating in linear and isotropic mediums. Among others, we find that such 2D materials exhibit metallic behaviors with certain optical features. Precisely, we compute and discuss the relavant optical quantities including the dielectric function, the absorption spectra, the refractive index, and the reflectivity. We believe that the Lie algebra inspiration of such 2D material studies, via density functional theory techniques, could open new roads to think about higher dimensional cases.

Operators that are Weakly Coercive and a Compact Perturbation

ABSTRACT Let X, Y be two Banach spaces over K = ℝ \[\mathbb{K}=\mathbb{R}\] or ℂ \[\mathbb{C}\] , and let f := F+C be a weakly coercive operator from X onto Y, where F is a Fredholm proper operator, and C is a C 1-compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C 1-diffeomorphism. When one of these conditions does not hold and instead y is a regular value, the equation f(x) = y has at most finite number of solutions. As a consequence of the main result two corollaries are given. A second theorem studies the finite dimensional case. As an application, one example is given. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping theorems.

Rational tensegrities through the lens of toric geometry

A classical tensegrity model consists of an embedded graph in a vector space with rigid bars representing edges, and an assignment of a stress to every edge such that at every vertex of the graph the stresses sum up to zero. The tensegrity frameworks have been recently extended from the two dimensional graph case to the multidimensional setting. We study the multidimensional tensegrities using tools from toric geometry. We introduce a link between self-stresses and Chow rings on toric varieties. More precisely, for a given rational tensegrity framework F, we construct a glued toric surface XF. We show that the abelian group of tensegrities on F is isomorphic to a subgroup of the Chow group A1(XF;Q). In the case of planar frameworks, we show how to explicitly carry out the computation of tensegrities via classical tools in toric geometry.

A note on factorization of vector valued functions involving the extraction of elementary factors

This note contains reflections on a type of factorization that came up in the work of analytic operator valued functions by I.C. Gohberg and E.I. Sigal (cf. [14], [10], [11]). The determining characteristic of the factorizations in question is that they are obtained by the extraction of elementary factors of the form I−P+λP where P is an idempotent in the underlying Banach algebra. It will be demonstrated here that there is a fundamental difference between the finite dimensional (matrix) case and the situation where infinite dimensional Banach algebras are taken into consideration.

A multidimensional objective prior distribution from a scoring rule

The construction of objective priors is, at best, challenging for multidimensional parameter spaces. A common practice is to assume independence and set up the joint prior as the product of marginal distributions obtained via “standard” objective methods, such as Jeffreys or reference priors. However, the assumption of independence a priori is not always reasonable, and whether it can be viewed as strictly objective is still open to discussion. In this paper, by extending a previously proposed objective approach based on scoring rules for the one dimensional case, we propose a novel objective prior for multidimensional parameter spaces which yields a dependence structure. The proposed prior has the appealing property of being proper and does not depend on the chosen model; only on the parameter space considered.

Internal symmetry in Poincarè gauge gravity

We find a large internal symmetry within 4-dimensional Poincarè gauge theory.In the Riemann-Cartan geometry of Poincaré gauge theory the field equation and geodesics are invariant under projective transformation, just as in affine geometry. However, in the Riemann-Cartan case the torsion and nonmetricity tensors change. By generalizing the Riemann-Cartan geometry to allow both torsion and nonmetricity while maintaining local Lorentz symmetry the difference of the antisymmetric part of the nonmetricity Q and the torsion T is a projectively invariant linear combination S = T - Q with the same symmetry as torsion. The structure equations may be written entirely in terms of S and the corresponding Riemann-Cartan curvature. The new description of the geometry has manifest projective and Lorentz symmetries, and vanishing nonmetricity.Torsion, S and Q lie in the vector space of vector-valued 2-forms. Within the extended geometry we define rotations with axis in the direction of S. These rotate both torsion and nonmetricity while leaving S invariant. In n dimensions and (p, q) signature this gives a large internal symmetry. The four dimensional case acquires SO(11,9) or Spin(11,9) internal symmetry, sufficient for the Standard Model.The most general action up to linearity in second derivatives of the solder form includes combinations quadratic in torsion and nonmetricity, torsion-nonmetricity couplings, and the Einstein-Hilbert action. Imposing projective invariance reduces this to dependence on S and curvature alone. The new internal symmetry decouples from gravity in agreement with the Coleman-Mandula theorem.