Finite-dimensional Space Research Articles

Lineynaya zadacha gruppovogo presledovaniya s drobnymi proizvodnymi, prostymi matritsami i raznymi vozmozhnostyami igrokov

In a finite-dimensional Euclidean space, we consider the problem of pursuit by a group of pursuers of one evader, which is described by a system of equations with a Caputo derivative of order a , where the sets of feasible controls are convex compact sets. We obtain sufficient conditions for the solvability of pursuit and evasion problems, in the study of which the method of resolving functions is used.

Estimates of Integrally Bounded Solutions of Linear Differential Inequalities

We study integrally bounded solutions of the differential equation A(x)=z, where A is a linear differential operator of order l defined on functions x:R→H (R=(−∞,∞), H () and H is a finite-dimensional Euclidean space). The right-hand side z is an integrally bounded function on R ranging in H and satisfying the inequality (ψ(t),z(t))≥δ|z(t)|, t∈R, δ0. Conditions are given on the operator A and the function ψ:R→H that guarantee an inverse inequality of the following form for the solutions x under consideration: ∫τ+1τ|x(l)(t)|dt≤c∫τ+2τ−1|x(t)|dt, where the constant is independent of the choice of a real number t and function x.

The quadratic sum problem for symplectic pairs

Let (b,u) be a pair consisting of a symplectic form b on a finite-dimensional vector space V over a field F, and of a b-alternating endomorphism u of V (i.e. b(x,u(x))=0 for all x in V).Let p and q be arbitrary polynomials of degree 2 with coefficients in F. We characterize, in terms of the invariant factors of u, the condition that u splits into u1+u2 for some pair (u1,u2) of b-alternating endomorphisms such that p(u1)=q(u2)=0.

Contractively decomposable projections on noncommutative Lp-spaces

We describe and characterize the contractively decomposable projections on noncommutative Lp-spaces. Our result relies on a new lifting result for decomposable maps of independent interest and on some tools from ergodic theory. Our theorem is new even for finite-dimensional Schatten spaces. Our description allows us to connect this topic with W⁎-ternary rings of operators and a slight generalization of our result for more general projections makes JBW⁎-triples appear in this context. We also prove that all rectangular Lp-spaces associated with W⁎-ternary rings of operators arise as contractively decomposable complemented subspaces of noncommutative Lp-spaces. Finally, we introduce a notion of Lp-space associated to each σ-finite JBW⁎-triple and we explain the link with the context of this paper.

Approximate Solution of Optimal Pulse Control Problem Associated with the Heat Conduction Process

Abstract We consider the approximate solution of the control problem with minimum energy for an object described by the heat equation, with the process described by the linear equation of parabolic type and the system controlled by impulsive external influences. Our optimal control problem deals with finding a control parameter belonging to the class of admissible controls that provides the desired temperature distribution in a finite time with minimal energy consumption (energy consumption is described by the quadratic functional). Previous works dedicated to optimal impulse control problems have mostly used the Pontryagin’s maximum principle. However, from a practical point of view, this approach does not lead to satisfactory results. This is due to the fact that the corresponding boundary value problems in this case have no solution in a traditional class of absolutely continuous trajectories. In this work, we propose a method based on the moment relations. We seek for the approximate solution of the corresponding boundary value problem in the form of finite Fourier sum and state our optimal control problem in a finite-dimensional phase space. As a result, we obtain an optimal impulse control problem in a finite-dimensional function space. Taking into account the given condition for a finite time, we reduce the obtained problem to the L-problem of moments. Thus, the problem of finding a control parameter is reduced to the solution of the system of Fredholm integral equations of the first kind, with the norm of the sought solution not exceeding a given number. By Levi’s theorem, every element of Hilbert space can be represented by the sum of the elements of two orthogonal subspaces. This assertion makes it possible to find control parameters in analytical form. We also establish the convergence of the chosen approximation.

Uniform estimates for solutions of a class of nonlinear equations in a finite-dimensional space

Uniform estimates for solutions of a class of nonlinear equations in a finite-dimensional space

Soft particles and infinite-dimensional geometry

In the sigma model, soft insertions of moduli scalars enact parallel transport of S-matrix elements about the finite-dimensional moduli space of vacua, and the antisymmetric double-soft theorem calculates the curvature of the vacuum manifold. We explore the analogs of these statements in gauge theory and gravity in asymptotically flat spacetimes, where the relevant moduli spaces are infinite-dimensional. These models have spaces of vacua parameterized by (trivial) flat connections on the celestial sphere, and soft insertions of photons, gluons, and gravitons parallel transport S-matrix elements about these infinite-dimensional manifolds. We argue that the antisymmetric double-soft gluon theorem in d + 2 bulk dimensions computes the curvature of a connection on the infinite-dimensional space Map, where G is the global part of the gauge group. The analogous metrics in abelian gauge theory and gravity are flat, as indicated by the vanishing of the antisymmetric double-soft theorems in those models. In other words, Feynman diagram calculations not only know about the vacuum manifold of Yang–Mills theory, they can also be used to compute its curvature. The results have interesting implications for flat space holography.

Solving Fractional Gas Dynamics Equation Using Müntz–Legendre Polynomials

To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method’s effectiveness and accuracy.

Generalized black hole clustering algorithm

The Black Hole Clustering (BHC) algorithm is a density-based partitional clustering method inspired by the Density-based Spatial Clustering of Applications with Noise (DBSCAN). It does not require the number of clusters nor the computation of the pair-wise distance matrix between the data points, making it faster than DBSCAN. Also, it only needs one parameter that is intuitively easier to set than the epsilon parameter of DBSCAN. However, BHC needs the allocation of the so-called black holes that have to be linearly independent, making the algorithm in its current version suitable only for two or three-dimensional data sets. In this paper, we propose a generalized version of the black hole clustering algorithm (GBHC) by introducing a novel black hole allocation procedure for higher-dimensional data spaces. Furthermore, the proposed method is data-independent, so we have to run it once to obtain the black hole positions for all finite-dimensional metric spaces. We performed extensive computational experiments to compare GBHC with DBSCAN. The results show that both algorithms obtain comparable clustering solutions. GBHC, however, outperforms DBSCAN in computational complexity and explainability.

Mapping cones of monomial ideals over exterior algebras

Let K be a field, V a finite dimensional K-vector space and E the exterior algebra of V. We analyze iterated mapping cone over E. If I is a monomial ideal of E with linear quotients, we show that the mapping cone construction yields a minimal graded free resolution F of I via the Cartan complex. Moreover, we provide an explicit description of the differentials in F when the ideal I has a regular decomposition function. Finally, we get a formula for the graded Betti numbers of a new class of monomial ideals including the class of strongly stable ideals.

Enhancing Equation Solving: Extending the Applicability of Steffensen-Type Methods

Local convergence analysis is mostly carried out using the Taylor series expansion approach, which requires the utilization of high-order derivatives, not iterative methods. There are other limitations to this approach, such as the following: the analysis is limited to finite-dimensional Euclidean spaces; no a priori computable error bounds on the distance or uniqueness of the solution results are provided. The local convergence analysis in this paper positively addresses these concerns in the more general setting of a Banach space. The convergence conditions involve only the operators in the methods. The more important semi-local convergence analysis not studied before is developed by using majorizing sequences. Both types of convergence analyses are based on the concept of generalized continuity. Although we study a certain class of methods, the same approach applies to extend the applicability of other schemes along the same lines.

Quantum detection problem for fusion frames

A quantum injective frame is a frame whose measurements for density operators can be used as a distinguishing feature in a quantum system, and the frame quantum detection problem demands a characterization of all such frames. Very recently, the quantum detection problem for continuous as well as discrete frames in both finite and infinite dimensional Hilbert spaces received significant attention. The quantum detection problem pertaining to the characterization of informationally complete positive operator-valued measures (POVM) can be split into two cases: The quantum injectivity or state separability problem and the rang analysis or quantum state estimation problem. Building upon this notion, this note is aimed at the quantum detection problem for fusion frames. The injectivity of a family of vectors and a family of closed subspaces is characterized in terms of some operator equations in Hilbert–Schmidt and trace classes.

Some problems of target approach for nonlinear control system at a fixed time moment

The game problem of target approach for nonlinear control system with a target set in a finite-dimensional phase space at a fixed time is studied. The problem is formulated and studied within the framework of concepts and constructions of the theory of positional differential games, created by N.N. Krasovskii and A.I. Subbotin in the second half of the 20th century. One of the central problems of the theory of positional differential games is the problem of calculating positional sets of uptake in convergence game problems. The paper investigates a key stability property in the theory of positional differential games, which is a characteristic of some closed sets in the space of positions of a controlled system that are convenient for the first player to conduct the game. It is important that this property is also characteristic for solvability sets in convergence problems: the inclusion of the concept of stability in research makes it possible to obtain analytical descriptions of solvability sets in some specific convergence problems and to develop algorithms for approximate calculation of the solution in a number of specific problems. Some modifications of the definition of a $u$-stable bridge in the considered convergence problem and a system of sets approximating the attainability set are given. Three specific problems on the convergence of mechanical systems are also presented, computer solutions are simulated, and graphical simulation results are presented.

The differential game “Cossacks–robbers” on time scales

In finite-dimensional Euclidean space, we address the problem of simple pursuit of a group of evaders by a group of pursuers in a given time scale with equal opportunities for all participants. The set of controls of each participant is a sphere of unit radius with its center at the origin. The goal of the group of pursuers is to catch all evaders. The goal sets are the origin. The goal of the evaders is the opposite one, namely, for at least one evader to avoid capture. Conditions for solvability of the local and global problems of evasion and the upper and lower estimates of the minimal number of evaders avoiding a given number of pursuers from any initial positions are obtained.

A linear group pursuit problem with fractional derivatives and different player capabilities

In a finite-dimensional Euclidean space, the problem of pursuit of one evader by a group of pursuers is considered, described by a system of the form $$ D^{(\alpha)}x_i = a_i x_i + u_i, \ u_i \in U_i,\quad D^{(\alpha)}y = ay + v, \ v \in V, $$ where $D^{(\alpha)}f$ is the Caputo derivative of order $\alpha \in (1, 2)$ of the function $f$. Sets of admissible controls~$U_i$, $V$ are convex compacts, $a_i$, $a$ are real numbers. Terminal sets are convex compacts. Sufficient conditions for the solvability of the problems of pursuit and evasion are obtained. In the study, the method of resolving functions is used as the basic one.

Three-Level Schemes with Double Change in the Time Step

Nonstationary problems are solved numerically by applying multilevel (with more than two levels) time approximations. They are easy to construct and relatively easy to study in the case of uniform grids. However, the numerical study of application-oriented problems often involves approximations with a variable time step. The construction of multilevel schemes on nonuniform grids is associated with maintaining the prescribed accuracy and ensuring the stability of the approximate solution. In this paper, three-level schemes for the approximate solution of the Cauchy problem for a second-order evolution equation are constructed in the special case of a doubled or halved step size. The focus is on the approximation features in the transition between different step sizes. The study is based on general results of the stability (well-posedness) theory of operator-difference schemes in a finite-dimensional Hilbert space. Estimates for stability with respect to initial data and the right-hand side are obtained in the case of a doubled or halved time step.

Spectra of quotient modules

We determine the Taylor spectra of quotient tuples of the d-shift on Drury-Arveson spaces with finite-dimensional coefficient spaces. We show that the Taylor spectrum can be described in terms of the approximate zero set of the annihilator ideal, and in terms of the pointwise behavior of the inner multiplier associated with the quotient tuple.

Calculus and Fine Properties of Functions of Bounded Variation on RCD Spaces

We generalize the classical calculus rules satisfied by functions of bounded variation to the framework of {textrm{RCD}} spaces. In the infinite dimensional setting, we are able to define an analogue of the distributional differential and, on finite dimensional spaces, we prove fine properties and suitable calculus rules, such as the Vol’pert chain rule for vector valued functions.

Subproduct systems with quantum group symmetry

We introduce a class of subproduct systems of finite dimensional Hilbert spaces whose fibers are defined by the Jones–Wenzl projections in Temperley–Lieb algebras. The quantum symmetries of a subclass of these systems are the free orthogonal quantum groups. For this subclass, we show that the corresponding Toeplitz algebras are nuclear C ^* -algebras that are KK -equivalent to \mathbb C and obtain a complete list of generators and relations for them. We also show that their gauge-invariant subalgebras coincide with the algebras of functions on the end compactifications of the duals of the free orthogonal quantum groups. Along the way we prove a few general results on equivariant subproduct systems, in particular, on the behavior of the Toeplitz and Cuntz–Pimsner algebras under monoidal equivalence of quantum symmetry groups.

Representations of map extended Witt algebras

In this paper, we classify irreducible modules for map extended Witt algebras with finite dimensional weight spaces. They turn out to be either modules with uniformly bounded weight spaces or highest weight modules. We further prove that all these modules are single point evaluation modules (n≥2). So they are actually irreducible modules for extended Witt algebras.