Arbitrary Space Dimension Research Articles

General Relations between Stress Fluctuations and Viscoelasticity in Amorphous Polymer and Glass-Forming Systems.

Mechanical stress governs the dynamics of viscoelastic polymer systems and supercooled glass-forming fluids. It was recently established that liquids with long terminal relaxation times are characterized by transiently frozen stress fields, which, moreover, exhibit long-range correlations contributing to the dynamically heterogeneous nature of such systems. Recent studies show that stress correlations and relaxation elastic moduli are intimately related in isotropic viscoelastic systems. However, the origin of these relations (involving spatially resolved material relaxation functions) is non-trivial: some relations are based on the fluctuation-dissipation theorem (FDT), while others involve approximations. Generalizing our recent results on 2D systems, we here rigorously derive three exact FDT relations (already established in our recent investigations and, partially, in classical studies) between spatio-temporal stress correlations and generalized relaxation moduli, and a couple of new exact relations. We also derive several new approximate relations valid in the hydrodynamic regime, taking into account the effects of thermal conductivity and composition fluctuations for arbitrary space dimension. One approximate relation was heuristically obtained in our previous studies and verified using our extended simulation data on two-dimensional (2D) glass-forming systems. As a result, we provide the means to obtain, in any spatial dimension, all stress-correlation functions in terms of relaxation moduli and vice versa. The new approximate relations are tested using simulation data on 2D systems of polydisperse Lennard-Jones particles.

An approach to modeling the spatial interaction of point sets on integer regular grids

The paper demonstrates a generalized approach to modeling the interaction of points and point sets on integer regular grids in metric spaces. The modeling process is based on the definition of spatial relations in "Point-Set" and "Set-Set" pairs, which allows to determine the spatial interaction between arbitrary geometric objects, defined as grid-based geometric models (GGM), which represent point sets on regular grids, generalized schemes of spatial interaction are developed in the work. As a result of the work, software was developed for conducting computer experiments on simulation of interaction in metric spaces of arbitrary dimensions and visualization of their results. The developed approach was tested on data sets of different dimensions. Figs.: 10. Refs.: 11 titles.

Multidimensional Hyperbolic Equations with Nonlocal Potentials: Families of Global Solutions

In spaces of arbitrary dimensions, hyperbolic differential–difference equations with potentials undergoing translations along arbitrary spatial directions are investigated. The fundamental novelty of this study is that, unlike previous investigations, no restrictions are imposed on the symbol of the differential–difference operator contained in the equation. For the investigated equation, we succeed to explicitly construct a multiparametric family of classical global solutions.

Quantifying the distribution of feature values over data represented in arbitrary dimensional spaces.

Identifying the structured distribution (or lack thereof) of a given feature over a point cloud is a general research question. In the neuroscience field, this problem arises while investigating representations over neural manifolds (e.g., spatial coding), in the analysis of neurophysiological signals (e.g., sensory coding) or in anatomical image segmentation. We introduce the Structure Index (SI) as a directed graph-based metric to quantify the distribution of feature values projected over data in arbitrary D-dimensional spaces (defined from neurons, time stamps, pixels, genes, etc). The SI is defined from the overlapping distribution of data points sharing similar feature values in a given neighborhood of the cloud. Using arbitrary data clouds, we show how the SI provides quantification of the degree and directionality of the local versus global organization of feature distribution. SI can be applied to both scalar and vectorial features permitting quantification of the relative contribution of related variables. When applied to experimental studies of head-direction cells, it is able to retrieve consistent feature structure from both the high- and low-dimensional representations, and to disclose the local and global structure of the angle and speed represented in different brain regions. Finally, we provide two general-purpose examples (sound and image categorization), to illustrate the potential application to arbitrary dimensional spaces. Our method provides versatile applications in the neuroscience and data science fields.

Quantum Many-Body Scars in Dual-Unitary Circuits.

Dual-unitary circuits are a class of quantum systems for which exact calculations of various quantities are possible, even for circuits that are nonintegrable. The array of known exact results paints a compelling picture of dual-unitary circuits as rapidly thermalizing systems. However, in this Letter, we present a method to construct dual-unitary circuits for which some simple initial states fail to thermalize, despite the circuits being "maximally chaotic," ergodic, and mixing. This is achieved by embedding quantum many-body scars in a circuit of arbitrary size and local Hilbert space dimension. We support our analytic results with numerical simulations showing the stark contrast in the rate of entanglement growth from an initial scar state compared to nonscar initial states. Our results are well suited to an experimental test, due to the compatibility of the circuit layout with the native structure of current digital quantum simulators.

Curves whose arcs with a fixed origin are similar

The author previously put forward the hypothesis that in the n-dimensional Euclidean space En, curves, any two oriented arcs of which are similar, are rectilinear. The same statement was proven for dimensions n = 2 and n = 3. In a space of arbitrary dimension, the hypothesis found its confirmation in the class of rectifiable curves. The work provides a complete solution to the problem, and in a stronger version:a) a curve in En, any two oriented arcs of which with a common origin (not fixed) are similar, is rectilinear;b) if a curve in En has a half-tangent at its boundary point and any two of its oriented arcs with a beginning at this point are similar, then the curve is rectilinear;c) if a curve in En has a tangent at an interior point and all its oriented arcs starting at this point are similar, then the curve is rectilinear. Examples of curves in E2 and E3 are given, in which all arcs with a common origin are similar, but they are not rectilinear, and a complete description of such curves in E2 is also given. Research methods are topological, set-theoretic, using the apparatus of functional equations.

Интегральное уравнение Вольтерра в банаховом пространстве

Despite the significant results achieved in the study of operator equations (including Volterra equations) in normalized Banach spaces, fundamental research in this field of mathematics attracts the attention of a huge number of mathematicians around the world. The solutions of the Volterra equation describe many important processes in various fields of science and technology. Studies of various inverse problems, experimental or experimental data processing problems related to the study of spherical or axisymmetric plasma formations, numerous mathematical models of the existence of various biological systems lead to the consideration and solution of this type of integral equations. A great contribution to the development of this theory was made by N.A. Magnitsky, L.I. Panov, A.N. Tikhonov, M.M. Lavrentiev and others. Fundamental results were obtained in the study of multiple operator equations with singularities in various functional spaces. Solutions depending on many parameters were constructed for the above equations. Currently, such problems are considered in spaces of arbitrary dimension and with coefficients having a derivative of finite order. In this paper, a finite set of solutions in a certain functional space is constructed for an integral equation of the first kind. The kernel of the integral operator has a finite order and is sufficiently differentiable near zero. The integral equation under consideration is reduced to an integro-differential equation representing two terms. For the first term, it is possible to solve the corresponding inhomogeneous equation and obtain a set of solutions in some functional normalized space. For the second term, we obtain an equation with an operator whose norm in some operator space is arbitrarily small near zero. Such splitting of the integral operator makes it possible to construct a partial and general solution of the corresponding integro-differential equation in the form of convergent equations. Applying modern methods of functional analysis, it is possible, by studying two separate equations, to construct a multiparametric family of solutions with values in some Banach space with weight for the original equation under consideration.

Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponent γ > 3

We consider the unsteady compressible Navier-Stokes equations in a perforated three-dimensional domain, and show that the limit system for the diameter of the holes going to zero is the same as in the perforated domain provided the perforations are small enough. The novelty of this result is the lower adiabatic exponent γ>3 instead of the known value γ>6. The proof is based on the use of two different restriction operators leading to two different types of pressure estimates. We also discuss the extension of this result for the unsteady Navier-Stokes-Fourier system as well as the optimality of the known results in arbitrary space dimension for both steady and unsteady problems.

Quantum Computing with Dartboards.

We present a physically appealing and elegant picture for quantum computing, using rules constructed for a game of darts. A dartboard is used to represent the state space in quantum mechanics, and the act of throwing the dart is shown to have close similarities to the concept of measurement or collapse of the wave function in quantum mechanics. The analogy is constructed in arbitrary dimensional spaces, that is, using arbitrary dimensional dartboards, and for such arbitrary spaces this also provides us a "visual" description of uncertainty. Finally, connections between qubits and quantum computing algorithms are also made, opening the possibility to construct analogies between quantum algorithms and coupled dart throws.

GTMesh: A Highly Efficient C++ Template Library for Numerical Schemes on General Topology Meshes

This article introduces GTMesh, an open-source C++ library providing data structures and algorithms that facilitate the development of numerical schemes on general polytopal meshes. After discussing the features and limitations of the existing open-source alternatives, we focus on the theoretical description of geometry and the topology of conforming polytopal meshes in an arbitrary-dimensional space, using elements from graph theory. The data structure for mesh representation is explained. The main part of the article focuses on the implementation of data structures and algorithms (computation of measures, centers, normals, cell coloring) by using State-of-the-Art template metaprogramming techniques for maximum performance. The geometrical algorithms are designed to be valid regardless of the dimension of the underlying space. As an integral part of the library, a template implementation of class reflection in C++ has been created, which is sufficiently versatile and suitable for the development of numerical and data I/O algorithms working with generic data types. Finally, the use of GTMesh is demonstrated on a simple example of solving the heat equation by the finite volume method.

Discrete-time analysis of optimized Schwarz waveform relaxation with Robin parameters depending on the targeted iteration count

We propose a new approach that provides new results in the convergence analysis of optimized Schwarz waveform relaxation (OSWR) iterations for parabolic problems, and allows to define efficient optimized Robin parameters that depend on the targeted iteration count, a property that is shared by the actual observed optimal parameters, while traditional Fourier analysis in the time direction leads to iteration independent parameters. This new approach is based on the exact resolution of the time semi-discrete error equations. It allows to recommend a couple (number of iterations, Robin parameter) to reach a given accuracy. While the general ideas may apply to an arbitrary space dimension, the analysis is first presented in the one dimensional case. Numerical experiments illustrate the performance obtained with such iteration-dependent optimized Robin parameters.

Enhanced dissipation and Taylor dispersion in higher‐dimensional parallel shear flows

AbstractWe consider the evolution of a passive scalar advected by a parallel shear flow in an infinite cylinder with bounded cross section, in arbitrary space dimension. The essential parameters of the problem are the molecular diffusivity , which is assumed to be small, and the wave number in the streamwise direction, which can take arbitrary values. Under generic assumptions on the shear velocity , we obtain optimal decay estimates for large times, both in the enhanced dissipation regime and in the Taylor dispersion regime . Our results can be deduced from resolvent estimates using a quantitative version of the Gearhart–Prüss theorem, or can be established more directly via the hypocoercivity method. Both approaches are explored in the present example, and their relative efficiency is compared.

The enumerative geometry of cubic hypersurfaces: point and line conditions

The set of smooth cubic hypersurfaces in Pn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {P}}}^n$$\\end{document} is an open subset of a projective space. A compactification of the latter which allows to count the number of smooth cubic hypersurfaces tangent to a prescribed number of lines and passing through a given number of points is termed a 1–complete variety of cubic hypersurfaces, in analogy with the space of complete quadrics. Imitating the work of Aluffi for plane cubic curves, we construct such a space in arbitrary dimensions by a sequence of five blow-ups. The counting problem is then reduced to the computation of five total Chern classes. In the end, we derive the desired numbers in the case of cubic surfaces.

Validity Index for Clustered Data in Non-negative Space

We propose a novel nonparametric cluster validity index which can be used to evaluate the unknown number of existing clusters prevailing a data set, to assess the quality of classification for a clustered set of data members, or to compare the clustering output obtained from different algorithms. Our efficient measure depends only on the observation-wise distances of the non-negative clustered data from their origin given in an arbitrary dimensional space. Its fast implementation makes it appealing for big data analysis, whereas the high-dimensional applicability widens its usefulness. Easy interpretation, simple algorithm, speedy computation and great performance, shown in terms of data study, establish our advised validity index as a strong cluster accuracy measure among the acknowledged ones from the literature. AMS subject classification: 62H30

Pointwise norm-based clustering of data in arbitrary dimensional space

ABSTRACT In this paper, a novel nonparametric norm-based clustering algorithm is proposed to classify real-valued continuous data sets given in arbitrary dimensional space. For data univariate, multivariate or high-dimensional, with the number of study variables close to or larger than the data size, our straightforward algorithm is implemented throughout under an univariate set-up, where we make use of the observation-wise (or pointwise) norms which quantify the distances of the observations from the origin zero or the null vector. The method begins with determination of the sample quantile using nonparamteric bootstrapping on the computed norms and always converges independently. By its design, the suggested algorithm is fast enough to detect the number of existing clusters itself and to form well-defined groups. Data study demonstrates its competitiveness in comparison to 2 popular clustering algorithms K-means and K-medoids.

Analytical and numerical radially symmetric solutions to a heat equation with arbitrary nonlinearity

The paper deals with the construction of radially symmetric heat waves, which are solutions to the heat conduction equation with an arbitrary form of nonlinearity under nonzero boundary condition specified on a moving manifold. The boundary value problem under study is a generalization of those solved by us earlier. Firstly, the class of the considered parabolic equations is extended; secondly, the boundary condition generating a heat wave in a space of arbitrary dimensionality has a more general form. A new theorem of the existence and uniqueness of the heat-wave-type analytical solution is proved for this problem. An approximate method of constructing solutions of the required form is proposed, which is based on expansion in radial basis functions combined with the collocation method. At each time step, the solution is constructed in two stages. The first stage is solving a problem in the region bounded by a specified moving manifold and a heat wave front, which is a priori unknown and evaluated during solving. Herewith, a special substitution is used, i.e. the required function and the spatial variable change their roles. In the second stage, the solution is completed in the region bounded by the positions of the specified moving manifold on a current step and at the initial time. The boundary conditions are defined from the first-step solution. In the test example, the solutions constructed by the developed algorithm are compared with the known exact solution. Calculations show a good accuracy of the numerical solutions at various values of the numerical parameters, including space dimensionality. The observed numerical convergence with respect to the time step shows the correctness of the proposed computational procedure.

Blow-Up Solutions for the Space-Time Fractional Evolution Equation

This paper focuses on the blow-up solutions of the space-time fractional equations with Riemann–Liouville type nonlinearity in arbitrary-dimensional space. Using the Banach mapping principle and the test function method, we establish the local well-posedness and overcome the difficulties caused by the fractional operators to obtain the blow-up results. Furthermore, we get the precise lifespan of blow-up solutions under special initial conditions.

Hessian matrix non-decomposition theorem and its application to nonlinear filtering

In 1983, Roger Brockett gave his invited lecture at the International Congress of Mathematics about the classification of finite dimensional estimation algebra (FDEA), which plays a vitally important role in the classification of finite dimensional filters (FDFs). The Hessian matrix non-decomposition (HMND) theorem was one of the key steps for the complete classification of FDEAs with maximal rank of the arbitrary state space dimension (Yau, 2003; Yau and Hu, 2005). In (Yau et al., 1999), the HMND theorem for homogeneous polynomials of degree 4 was proven and at the same time, it is firstly proposed that there exists a HMND conjecture for the general polynomials of degree 2d with d≥2. In this paper, we consider the Hessian matrix of a nontrivial real homogeneous polynomial at degree 6. For the state space dimension n≤4, we prove such matrix cannot be decomposed in the form Q(x)Q(x)T, where Q(x) is a skew-symmetric matrix whose all elements are homogeneous polynomials of degree 2, and Q(x)T is the transpose of Q(x). The fundamental strategy we use in this paper is that we generalized the techniques proposed by Yau et al. (1999) for the case of degree 4 and derived identities of coefficient matrices for the case of degree 6. For the arbitrary state space dimension n, the HMND theorem is demonstrated under several conditions by certain basic linear algebra techniques. These results can be closely related to the classification of FDEAs with non-maximal rank in the filtering theory.

Shape modeling with spline partitions

Shape modelling (with methods that output shapes) is a new and important task in Bayesian nonparametrics and bioinformatics. In this work, we focus on Bayesian nonparametric methods for capturing shapes by partitioning a space using curves. In related work, the classical Mondrian process is used to partition spaces recursively with axis-aligned cuts, and is widely lied in multi-dimensional and relational data. The Mondrian process outputs hyper-rectangles. Recently, the random tessellation process was introduced as a generalization of the Mondrian process, partitioning a domain with non-axis aligned cuts in an arbitrary dimensional space, and outputting polytopes. Motivated by these processes, in this work, we propose a novel parallelized Bayesian nonparametric approach to partition a domain with curves, enabling complex data-shapes to be acquired. We apply our method to HIV-1-infected human macrophage image dataset, and also simulated datasets sets to illustrate our approach. We compare to support vector machines, random forests and state-of-the-art computer vision methods such as simple linear iterative clustering super pixel image segmentation. We develop an R package that is available at https://github.com/ShufeiGe/Shape-Modeling-with-Spline-Partitions.

Operator spreading in quantum hardcore gases

In this article we study a set of integrable quantum cellular automata, the quantum hardcore gases (QHCG), with an arbitrary local Hilbert space dimension, and discuss the matrix product ansatz based approach for solving the dynamics of local operators analytically. Subsequently, we focus on the dynamics of operator spreading, in particular on the out-of-time ordered correlation functions (OTOCs), operator weight spreading and operators space entanglement entropy (OSEE). All of the quantities were conjectured to provide signifying features of integrable systems and quantum chaos. We show that in QHCG OTOCs spread diffusively and that in the limit of the large local Hilbert space dimension they increase linearly with time, despite their integrability. On the other hand, it was recently conjectured that operator weight front, which is associated with the extent of operators, spreads diffusively in both, integrable and generic systems, but its decay seems to differ in these two cases (Lopez-Piqueres et al 2021 Phys. Rev. B 104 104307). We observe that the spreading of the operator weight front in QHCG is markedly different from chaotic, generic integrable and free systems, as the front freezes in the long time limit. Finally, we discuss the OSEE in QHCG and show that it grows at most logarithmically with time in accordance with the conjectured behaviour for interacting integrable systems (Alba et al 2019 Phys. Rev. Lett. 122 250603).