Infinite-dimensional Matrices Research Articles

Statistics of Local Level Spacings in Single- and Many-Body Quantum Chaos.

We introduce a notion of local level spacings and study their statistics within a random-matrix-theory approach. In the limit of infinite-dimensional random matrices, we determine universal sequences of mean local spacings and of their ratios which uniquely identify the global symmetries of a quantum system and its internal-chaotic or regular-dynamics. These findings, which offer a new framework to monitor single- and many-body quantum systems, are corroborated by numerical experiments performed for zeros of the Riemann zeta function, spectra of irrational rectangular billiards, and many-body spectra of the Sachdev-Ye-Kitaev Hamiltonians.

Limits of Bessel functions for root systems as the rank tends to infinity

We study the asymptotic behaviour of Bessel functions associated to root systems of type An−1 and type Bn with positive multiplicities as the rank n tends to infinity. In both cases, we characterize the possible limit functions and the Vershik–Kerov type sequences of spectral parameters for which such limits exist. In the type A case, this gives a new and very natural approach to recent results by Assiotis and Najnudel in the context of β-ensembles in random matrix theory. These results generalize known facts about the approximation of the positive-definite Olshanski spherical functions of the space of infinite-dimensional Hermitian matrices over F=R,ℂ,H (with the action of the associated infinite unitary group) by spherical functions of finite-dimensional spaces of Hermitian matrices. In the type B case, our results include asymptotic results for the spherical functions associated with the Cartan motion groups of non-compact Grassmannians as the rank goes to infinity, and a classification of the Olshanski spherical functions of the associated inductive limits.

On Recursion Operators for Full-Fledged Nonlocal Symmetries of the Reduced Quasi-classical Self-dual Yang–Mills Equation

AbstractWe introduce the idea of constructing recursion operators for full-fledged nonlocal symmetries and apply it to the reduced quasi-classical self-dual Yang–Mills equation. It turns out that the discovered recursion operators can be interpreted as infinite-dimensional matrices of differential functions which act on the generating vector functions of the nonlocal symmetries simply by matrix multiplication. To the best of our knowledge, there are no other examples of such recursion operators in the literature so far, so our approach is completely innovative. Further, we investigate the algebraic properties of the discovered operators and discuss the $${\mathbb {R}}$$ R -algebra structure on the set of all recursion operators for full-fledged nonlocal symmetries of the equation in question. Finally, we illustrate the action of the obtained recursion operators on particularly chosen full-fledged symmetries and emphasize their advantages compared to the action of traditionally used recursion operators for shadows.

A note on Oishi’s lower bound for the smallest singular value of linearized Galerkin equations

Recently Oishi published a paper allowing lower bounds for the minimum singular value of coefficient matrices of linearized Galerkin equations, which in turn arise in the computation of periodic solutions of nonlinear delay differential equations with some smooth nonlinearity. The coefficient matrix of linearized Galerkin equations may be large, so the computation of a valid lower bound of the smallest singular value may be costly. Oishi’s method is based on the inverse of a small upper left principal submatrix, and subsequent computations use a Schur complement with small computational cost. In this note some assumptions are removed and the bounds improved. Furthermore a technique is derived to reduce the total computationally cost significantly allowing to treat infinite dimensional matrices.

Solving Infinite-Dimensional Harmonic Lyapunov and Riccati Equations

In this paper, we address the problem of solving infinite-dimensional harmonic algebraic Lyapunov and Riccati equations up to an arbitrary small error. This question is of major practical importance for analysis and stabilization of periodic systems including tracking of periodic trajectories. We first give a closed form of a Floquet factorization in the general setting of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$L^{2}$</tex-math></inline-formula> matrix functions and study the spectral properties of infinite-dimensional harmonic matrices and their truncated version. This spectral study allows us to propose a generic and numerically efficient algorithm to solve infinite-dimensional harmonic algebraic Lyapunov equations up to an arbitrary small error. We combine this algorithm with the Kleinman algorithm to solve infinite-dimensional harmonic Riccati equations and we apply the proposed results to the design of a harmonic LQ control with periodic trajectory tracking.

Benefits of using matrix methods in teach-ing mechanics in higher education institu-tions

Problem. Basic concepts, concepts, principles of mechanics are mainly taught from simple to complex in institutions of higher technical education, which to some extent resembles a course on the history of mechanics. In today's conditions, the education system is required to take into account current trends in scientific and engineering fields. The use of inherently geometric methods of vector algebra and vector analysis as a basis for studying mechanics is losing its relevance. To a certain extent, this approach interferes with the use of multidimensional spaces (configurational and phase), tensor algebra, and computer technologies in modeling complex mechanical systems. Goal. The goal is to expand the field of application of the matrix formalism to infinite-dimensional spaces, justify its advantages compared to other approaches used in teaching the basics of mechanical disciplines, and demonstrate its capabilities in solving complex practical problems, both in terms of modeling and in terms of using computer technologies. Methodology. The methodological basis for choosing matrix methods is the use of infinite-dimensional and abstract, in physical terms, spaces when studying the movement of mechanical systems. The fundamental impossibility of accurately describing the state of a mechanical system requires the use of statistical modeling methods and the extension of the so-called law of conservation of matter in the form of a balance equation to these models. Results. It is shown that the matrix method, which is widely used in quantum mechanics and very limited in classical mechanics, allows you to use the advantages of the transition to phase space when obtaining canonical equations and when solving problems of kinematics and dynamics of a solid body. Originality. The paper proposes certain improvements of the matrix apparatus, related to the use of geometric parameters of the mechanical system as indexes. That is, infinite-dimensional matrices with real coefficients are used. Practical value. The proposed method allows you to effectively solve the problems of the mechanics of an absolutely rigid body without using any theorems of the general course, remaining within the limits of standard mathematical training.

Power spectra and autocovariances of level spacings beyond the Dyson conjecture.

Introduced in the early days of random matrix theory, the autocovariances δI_{k}^{j}=cov(s_{j},s_{j+k}) of level spacings {s_{j}} accommodate detailed information on the correlations between individual eigenlevels. It was first conjectured by Dyson that the autocovariances of distant eigenlevels in the unfolded spectra of infinite-dimensional random matrices should exhibit a power-law decay δI_{k}^{j}≈-1/βπ^{2}k^{2}, where β is the symmetry index. In this Letter, we establish an exact link between the autocovariances of level spacings and their power spectrum, and show that, for β=2, the latter admits a representation in terms of a fifth Painlevé transcendent. This result is further exploited to determine an asymptotic expansion for autocovariances that reproduces the Dyson formula as well as provides the subleading corrections to it. High-precision numerical simulations lend independent support to our results.

Truncation Number Selection of Harmonic State-Space Model Based on the Floquet Characteristic Exponent

Harmonic state-space (HSS) theory has been an effective method for studying system stability with time-periodic characteristics, such as modular multilevel converters (MMCs) and renewable power generation under unbalanced grid conditions. However, the original state-space formulation of the HSS model comprises infinite-dimensional matrices. Truncation is necessary before practical computing application. Therefore, this letter proposes a truncation number selection method for the HSS model. The theoretical basis of the proposed method is based on the Floquet characteristic exponent of the linear time-periodic model, which has a straightforward physical meaning. Finally, a case study of an MMC grid-tied system with tests in RT-LAB illustrates the validity of the proposed method.

Transfer Matrices of Rational Spin Chains via Novel BGG-Type Resolutions

We obtain BGG-type formulas for transfer matrices of irreducible finite-dimensional representations of the classical Lie algebras $\mathfrak{g}$, whose highest weight is a multiple of a fundamental one and which can be lifted to the representations over the Yangian $Y(\mathfrak{g})$. These transfer matrices are expressed in terms of transfer matrices of certain infinite-dimensional highest weight representations (such as parabolic Verma modules and their generalizations) in the auxiliary space. We further factorise the corresponding infinite-dimensional transfer matrices into the products of two Baxter $Q$-operators, arising from our previous study (arXiv:2001.04929, arXiv:2104.14518) of the degenerate Lax matrices. Our approach is crucially based on the new BGG-type resolutions of the finite-dimensional $\mathfrak{g}$-modules, which naturally arise geometrically as the restricted duals of the Cousin complexes of relative local cohomology groups of ample line bundles on the partial flag variety $G/P$ stratified by $B_{-}$-orbits.

Power spectrum of the circular unitary ensemble

We study the power spectrum of eigen-angles of random matrices drawn from the circular unitary ensemble CUE(N) and show that it can be evaluated in terms of either a Fredholm determinant, or a Toeplitz determinant, or a sixth Painlevé function. In the limit of infinite-dimensional matrices, N→∞, we derive a concise parameter-free formula for the power spectrum which involves a fifth Painlevé transcendent and interpret it in terms of the Sine2 determinantal random point field. Further, we discuss a universality of the predicted power spectrum law and tabulate it for easy use by random-matrix-theory and quantum chaos practitioners.

Nonlinear Random Matrix-Based Intelligent Management Model for Swimming Place Waters

In this paper, a nonlinear random matrix approach is used to analyze the management of swimming place waters, and in this way, an intelligent management model is designed and applied to the actual swimming place waters management. Firstly, some basic small deviation results of the random matrix are presented. Then, several types of small deviation inequalities are obtained for the maximum eigenvalues of independent stochastic semi-positive definite (PSD) matrices. These small deviation inequalities are independent of the matrix dimension, and the results apply to high-dimensional and even infinite-dimensional matrices. For the inverse eigenvalue problem of higher-order doubly random matrices, the conclusion that two symmetric doubly random matrices are combined into a higher-order symmetric doubly random matrix is proved. In other words, the method of constructing new symmetric random matrices using smaller matrices with known spectra is applied to the inverse eigenvalue problem of higher-order symmetric random matrices, and sufficient conditions for the existence of the solution of the inverse eigenvalue problem of higher-order random matrices are derived, avoiding the discussion of the parity of the number of eigenvalues. This paper focuses on realistic management, to explore the standardization of safety management of swimming venues, and is based on modern management theory, risk management theory and methods, using research integration method, fieldwork method, questionnaire survey method, expert interview method, and other research methods, detailing the methods and conclusions of risk identification in safety management of swimming venues, and further exploring the safety management standards of swimming venues on this basis. And, combined with risk management theory, depending on the probability of occurrence and the degree of harm of risk, four basic disposal methods such as risk avoidance, risk control, risk transfer, and risk self-retaining are proposed.

Manin Matrices for Quadratic Algebras

We give a general definition of Manin matrices for arbitrary quadratic algebras in terms of idempotents. We establish their main properties and give their interpretation in terms of the category theory. The notion of minors is generalised for a general Manin matrix. We give some examples of Manin matrices, their relations with Lax operators and obtain the formulae for some minors. In particular, we consider Manin matrices of the types B, C and D introduced by A. Molev and their relation with Brauer algebras. Infinite-dimensional Manin matrices and their connection with Lax operators are also considered.

Kaluza-Klein fermion mass matrices from exceptional field theory and mathcal{N} = 1 spectra

Using Exceptional Field Theory, we determine the infinite-dimensional mass matrices for the gravitino and spin-1/2 Kaluza-Klein perturbations above a class of anti-de Sitter solutions of M-theory and massive type IIA string theory with topologically-spherical internal spaces. We then use these mass matrices to compute the spectrum of Kaluza-Klein fermions about some solutions in this class with internal symmetry groups containing SU(3). Combining these results with previously known bosonic sectors of the spectra, we give the complete spectrum about some mathcal{N} = 1 and some non-supersymmetric solutions in this class. The complete spectra are shown to enjoy certain generic features.

Nonperturbative theory of power spectrum in complex systems

The power spectrum analysis of spectral fluctuations in complex wave and quantum systems has emerged as a useful tool for studying their internal dynamics. In this paper, we formulate a nonperturbative theory of the power spectrum for complex systems whose eigenspectra – not necessarily of the random-matrix-theory (RMT) type – possess stationary level spacings. Motivated by potential applications in quantum chaology, we apply our formalism to calculate the power spectrum in a tuned circular ensemble of random N×N unitary matrices. In the limit of infinite-dimensional matrices, the exact solution produces a universal, parameter-free formula for the power spectrum, expressed in terms of a fifth Painlevé transcendent. The prediction is expected to hold universally, at not too low frequencies, for a variety of quantum systems with completely chaotic classical dynamics and broken time-reversal symmetry. On the mathematical side, our study brings forward a conjecture for a double integral identity involving a fifth Painlevé transcendent.

Light Scattering by Small Multilayer Nonconfocal Spheroids Using Suitable Spheroidal Basis Sets

We construct a Rayleigh approximation for multilayer particles the layer boundaries of which are nonconfocal spheroids. The geometry of the problem is taken into account to the maximum extent by representing the field potentials inside nonconfocal shells as expansions in terms of spheroidal harmonics in different coordinate systems in which the surfaces of the layers are coordinate. To sew two expansions inside each layer, we use relations between spheroidal harmonics of the Laplace equation in systems with different focal lengths that we obtained. The extended boundary conditions method (ЕВСМ) and the separation of variables method (SVM) prove to be equivalent, because they yield the same results. The polarizability of the particle and, therefore, the characteristics of the scattered radiation are written in terms of infinite-dimensional matrices, the elements of which are determined either explicitly or in the form of finite sums. In particular cases of confocal spheroids, this solution is completely consistent with the known results.

2-Local derivations on algebras of matrix-valued functions on a compactum

The present paper is devoted to 2-local derivations. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-lo\-cal derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and \mbox{2-local} derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every \mbox{2-local} derivation on a ∗-algebra C(Q,Mn(F)) or C(Q,Nn(F)), where Q is a compactum, Mn(F) is the ∗-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, Nn(F) is the ∗-subalgebra of Mn(F) defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.

Batch arrival single-server queue with variable service speed and setup time

In this paper, we consider an M $${}^X$$ /M/1/SET-VARI queue which has batch arrivals, variable service speed and setup time. Our model is motivated by power-aware servers in data centers where dynamic scaling techniques are used. The service speed of the server is proportional to the number of jobs in the system. The contribution of our paper is threefold. First, we obtain the necessary and sufficient condition for the stability of the system. Second, we derive an expression for the probability generating function of the number of jobs in the system. Third, our main contribution is the derivation of the Laplace–Stieltjes transform (LST) of the sojourn time distribution, which is obtained in series form involving infinite-dimensional matrices. In this model, since the service speed varies upon arrivals and departures of jobs, the sojourn time of a tagged job is affected by the batches that arrive after it. This makes the derivation of the LST of the sojourn time complex and challenging. In addition, we present some numerical examples to show the trade-off between the mean sojourn time (response time) and the energy consumption. Using the numerical inverse Laplace–Stieltjes transform, we also obtain the sojourn time distribution, which can be used for setting the service-level agreement in data centers.

Stability of milling with non-uniform pitch and variable helix Tools

We study mechanical vibrations in milling with non-uniform pitch and variable helix tools. The process is modeled by a periodic delay differential equation with distributed delay, which takes into account, for example, the nonlinear cutting force behavior, the effect of runout, and the exact delay distribution due to the unequally spaced flutes. We present a new method for the identification of the chatter stability lobes from the linearized system that is based on the multifrequency solution. We give detailed remarks on the truncation of the resulting infinite dimensional matrices and the efficient numerical implementation of the method. Cutting tests for steel milling with a customary end mill with non-uniform pitch and variable helix angle and a conventional end mill with uniform pitch and constant helix angle are performed. The numerical and experimental results coincide well. They reveal a significant increase of the limiting depth of cut for the variable helix tool compared to the conventional tool. Moreover, we show that in contrast to conventional tools, for non-uniform pitch and variable helix tools, an exact model with time-varying coefficients, nonlinear cutting force behavior, and runout is necessary for an accurate prediction of the stability lobes.

Coupled mode theory of scattering by a cylindrically symmetric seamount

The coupled-mode equations of Shevchenko are extended to three-dimensional irregular acoustic waveguides. These equations are solved for a model of a cylindrically symmetric seamount and a time-harmonic point source in a horizontally stratified ocean. An algebraic formula for the scattered field beyond the seamount is obtained from this solution. The formula is characterized by a set of infinite-dimensional matrices that are independent of the source. A stable numerical procedure is developed to compute finite-dimensional approximations to these matrices for use in truncated versions of the formula.

On lattices of closed subgroups in the group of infinite triangular matrices over a field

We investigate a special type of closed subgroups of the topological group UT(∞,K) of infinite-dimensional unitriangular matrices over a field K (|K|>2), considered with the natural inverse limit topology. Namely, we generalize the concept of partition subgroups introduced in [23] and define partition subgroups in UT(∞,K). We show that they are all closed and discuss the problem of their invariance to various group homomorphisms. We prove that a characteristic subgroup of UT(∞,K) is necessarily a partition subgroup and characterize the lattices of characteristic and fully characteristic subgroups in UT(∞,K). We conclude with some implications of the given characterization on verbal structure of UT(∞,K) and T(∞,K) and use some topological properties to discuss the problem of the width of verbal subgroups in groups defined over a finite field K.