Sense Of Weyl Research Articles

The β maps: Strong clustering and distribution results on the complex unit circle

In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter β>1, of the basic Toeplitz matrix-sequence {Tn(eiθ)}n∈N, i2=−1. The latter of which has obviously all eigenvalues equal to zero for any matrix order n, while for the matrix-sequence under consideration we will show a strong clustering on the complex unit circle. A detailed discussion on the outliers is also provided. The problem appears mathematically innocent, but it is indeed quite challenging since all the classical machinery for deducing the eigenvalue clustering does not cover the considered case. In the derivations, we resort to a trick used for the spectral analysis of the Google matrix plus several tools from complex analysis. We only mention that the problem is not an academic curiosity and in fact stems from problems in dynamical systems and number theory. Additionally, we also provide numerical experiments in high precision, a distribution analysis in the Weyl sense concerning both eigenvalues and singular values is given, and more results are sketched for the limit case of β=1.

GLT sequences and automatic computation of the symbol

Spectral and singular value symbols are valuable tools to analyse the eigenvalue or singular value distributions of matrix-sequences in the Weyl sense. More recently, Generalized Locally Toeplitz (GLT) sequences of matrices have been introduced for the spectral/singular value study of the numerical approximations of differential operators in several contexts. As an example, such matrix-sequences stem from the large linear systems approximating Partial Differential Equations (PDEs), Fractional Differential Equations (FDEs), Integro Differential Equations (IDEs), using any discretization on reasonable grids via local methods, such as Finite Differences, Finite Elements, Finite Volumes, Isogeometric Analysis, Discontinuous Galerkin etc. Studying the asymptotic spectral behaviour of GLT sequences is useful in analysing classical techniques for the solution of the corresponding PDEs/FDEs/IDEs and in designing novel fast and efficient methods for the corresponding large linear systems or related large eigenvalue problems. The theory of GLT sequences, in combination with the results concerning the asymptotic spectral distribution of perturbed sequences of matrices, is one of the most powerful and successful tools for computing the spectral symbol f. In this regard, it would be beneficial to design an automatic procedure to compute the spectral symbols of such matrix-sequences and Ahmed Ratnani partially pursued it. Here, in the case of one-dimensional and two-dimensional differential problems, we continue in this direction by proposing an automatic procedure for computing the symbol of the underlying sequences of matrices, assuming that it is a GLT sequence satisfying mild conditions.

Publisher's Note

In the current work, we study the eigenvalue distribution results of a class of non-normal matrix-sequences which may be viewed as a low rank perturbation, depending on a parameter β>1, of the basic Toeplitz matrix-sequence {Tn(eiθ)}n∈N, i2=−1. The latter of which has obviously all eigenvalues equal to zero for any matrix order n, while for the matrix-sequence under consideration we will show a strong clustering on the complex unit circle. A detailed discussion on the outliers is also provided. The problem appears mathematically innocent, but it is indeed quite challenging since all the classical machinery for deducing the eigenvalue clustering does not cover the considered case. In the derivations, we resort to a trick used for the spectral analysis of the Google matrix plus several tools from complex analysis. We only mention that the problem is not an academic curiosity and in fact stems from problems in dynamical systems and number theory. Additionally, we also provide numerical experiments in high precision, a distribution analysis in the Weyl sense concerning both eigenvalues and singular values is given, and more results are sketched for the limit case of β=1.

Asymptotic spectra of large (grid) graphs with a uniform local structure, Part II: Numerical applications

In the current work we are concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain Ω⊂Rd, d≥1. When Ω=[0,1], such graphs include the standard Toeplitz graphs and, for Ω=[0,1]d, the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and it has been shown in the theoretical part of this work that we can associate to it a symbol f. The knowledge of the symbol and of its basic analytical features provides key information on the eigenvalue structure in terms of localization, spectral gap, clustering, and global distribution. In the present paper, many different applications are discussed and various numerical examples are presented in order to underline the practical use of the developed theory. Tests and applications are mainly obtained from the approximation of differential operators via numerical schemes such as Finite Differences, Finite Elements, and Isogeometric Analysis. Moreover, we show that more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks, whenever the involved matrices enjoy a uniform local structure.

Multisymplectic constraint analysis of scalar field theories, Chern-Simons gravity, and bosonic string theory

The (pre)multisymplectic geometry of the De Donder–Weyl formalism for field theories is further developed for a variety of field theories including a scalar field theory from the canonical Klein-Gordon action, the electric and magnetic Carrollian scalar field theories, bosonic string theory from the Nambu-Goto action, and 2+1 gravity as a Chern-Simons theory. The Lagrangians for the scalar field theories and for 2+1 Chern-Simons gravity are found to be singular in the De Donder–Weyl sense while the Nambu-Goto Lagrangian is found to be regular. Furthermore, the constraint structure of the premultisymplectic phase spaces of singular field theories is explained and applied to these theories. Finally, it is studied how symmetries are developed on the (pre)multisymplectic phase spaces in the presence of constraints.

A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses

The fractional oscillator equation with the sinusoidal excitation mx″(t)+bDtαx(t)+kx(t)=Fsin(ωt), m,b,k,ω>0 and 0<α<2 is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation xW(t). In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case α∈(0,1)∪(1,2), the response to excitation, S(t), is a superposition of three parts: the steady periodic oscillation xW(t), an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values, S0(t) and S1(t), either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense.

Asymptotic Spectra of Large (Grid) Graphs with a Uniform Local Structure (Part I): Theory

We are mainly concerned with sequences of graphs having a grid geometry,with a uniform local structure in a bounded domain Ω⊂Rd,d≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\Omega} {\\subset} \\mathbb{R}^{d}, d \\geq 1$$\\end{document}. When Ω=[0,1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega = [0, 1]$$\\end{document}, such graphs include the standard Toeplitz graphs and, for Ω=[0,1]d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega = [0, 1]^{d}$$\\end{document},the considered class includes d-level Toeplitz graphs. In the general case, the underlyingsequence of adjacency matrices has a canonical eigenvalue distribution, inthe Weyl sense, and we show that we can associate to it a symbol f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak{f}$$\\end{document}. The knowledgeof the symbol and of its basic analytical features provides many information onthe eigenvalue structure, of localization, spectral gap, clustering, and distributiontype.Few generalizations are also considered in connection with the notion of generalizedlocally Toeplitz sequences and applications are discussed, stemming e.g.from the approximation of differential operators via numerical schemes. Nevertheless,more applications can be taken into account, since the results presentedhere can be applied as well to study the spectral properties of adjacency matricesand Laplacian operators of general large graphs and networks.

Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis

In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.

Spectral analysis of Pk Finite Element matrices in the case of Friedrichs–Keller triangulations via Generalized Locally Toeplitz technology

SummaryIn the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(−a(x)∇·), with a continuous and positive over , Ω being an open and bounded subset of , d≥1. For the numerical approximation, we consider the classical Finite Elements, in the case of Friedrichs–Keller triangulations, leading, as usual, to sequences of matrices of increasing size. The new results concern the spectral analysis of the resulting matrix‐sequences in the direction of the global distribution in the Weyl sense, with a concise overview on localization, clustering, extremal eigenvalues, and asymptotic conditioning. We study in detail the case of constant coefficients on Ω=(0,1)2 and we give a brief account in the more involved case of variable coefficients and more general domains. Tools are drawn from the Toeplitz technology and from the rather new theory of Generalized Locally Toeplitz sequences. Numerical results are shown for a practical evidence of the theoretical findings.

Non‐Hermitian perturbations of Hermitian matrix‐sequences and applications to the spectral analysis of the numerical approximation of partial differential equations

SummaryThis article concerns the spectral analysis of matrix‐sequences which can be written as a non‐Hermitian perturbation of a given Hermitian matrix‐sequence. The main result reads as follows. Suppose that for every n there is a Hermitian matrix Xn of size n and that {Xn}n∼λf, that is, the matrix‐sequence {Xn}n enjoys an asymptotic spectral distribution, in the Weyl sense, described by a Lebesgue measurable function f; if with ‖·‖2 being the Schatten 2 norm, then {Xn+Yn}n∼λf. In a previous article by Leonid Golinskii and the second author, a similar result was proved, but under the technical restrictive assumption that the involved matrix‐sequences {Xn}n and {Yn}n are uniformly bounded in spectral norm. Nevertheless, the result had a remarkable impact in the analysis of both spectral distribution and clustering of matrix‐sequences arising from various applications, including the numerical approximation of partial differential equations (PDEs) and the preconditioning of PDE discretization matrices. The new result considerably extends the spectral analysis tools provided by the former one, and in fact we are now allowed to analyze linear PDEs with (unbounded) variable coefficients, preconditioned matrix‐sequences, and so forth. A few selected applications are considered, extensive numerical experiments are discussed, and a further conjecture is illustrated at the end of the article.

Approximation of Even Functions with Nonnegative Fourier Coefficients by Fractional Riesz Sums

We establish two-sided estimates for deviations of Riesz sums on the class of even functions with nonnegative Fourier coefficients. Estimates are obtained by using the fractional modulus of continuity of the function itself and the fractional Hilbert transform of its derivative in the sense of Weyl. The constants are expressed in an explicit form.

On the discrepancy of random subsequences of $\{n\alpha\}$

For irrational $\alpha$, $\{n\alpha\}$ is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences $\{n_k \alpha\}$, with the exception of metric results for exponentially growing $(n_k)$. It is therefore natural to consider random $(n_k)$, and in this paper we give nearly optimal bounds for the discrepancy of $\{n_k \alpha\}$ in the case when the gaps $n_{k+1}-n_k$ are independent, identically distributed, integer-valued random variables. As we will see, the discrepancy behavior is determined by a delicate interplay between the distribution of the gaps $n_{k+1}-n_k$ and the rational approximation properties of $\alpha$. We also point out an interesting critical phenomenon, a sudden change of the order of magnitude of the discrepancy of $\{n_k \alpha\}$ as the Diophantine type of $\alpha$ passes through a certain critical value.

Bounded error uniformity of the linear flow on the torus

A linear flow on the torus $$\mathbb {R}^d / \mathbb {Z}^d$$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $$\mathbb {Q}$$ . In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne–Ehrenfest type theorem for the mod 1 discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.

Eigenvalue asymptotics for the Sturm-Liouville operator with potential having a strong local negative singularity

We find asymptotic formulas for the eigenvalues of the Sturm-Liouville operator on the finite interval, with potential having a strong negative singularity at one endpoint. This is the case of limit circle in H. Weyl sense. We establish that, unlike the case of an infinite interval, the asymptotics for positive eigenvalues does not depend on the potential and it is the same as in the regular case. The asymptotics of the negative eigenvalues may depend on the potential quite strongly, however there are always asymptotically fewer negative eigenvalues than positive ones. By unknown reasons this type of problems had not been studied previously.

About Classical to Quantum Weyl Correspondence

After developing the mathematical means for the correspondence of classical phase-space function to quantum-mechanical operators with symmetrical ordering of the basic canonical operators in the sense of Weyl the approach is applied to an infinite series of classical monomial functions of the canonical variables. These include as well as pure powers of the amplitude as also basic periodic functions of the phase φ with their quantum-mechanical correspondence. In the representation by number states, all the considered operators involve the Jacobi polynomials as the essential formative element. Whereas the quantity in normal ordering due to its indeterminacy leads to the introduction of the notions of sub- and super-Poissonian statistics the analogous quantity in (Weyl) symmetrical orderingis positive definite and satisfies an inequality. The notions of sub- and super-Poissonian statistics are problematic when they are used for the definition of nonclassicality of states since the mentioned measure in normal ordering does not determine the Poisson statistics in their middle in unique way but determines only a large set of statistics which may be very far in the sense of the Hilbert-Schmidt distance from a Poisson statistics that is discussed.

Formula omitted]-widths of certain function classes defined by the modulus of continuity

Several exact inequalities are found between the best approximation by trigonometric polynomials in L2 of differentiable periodic functions, in the sense of Weyl, averaged with the weight of the modulus of continuity of arbitrary fractional order. For some classes of functions, defined by specific moduli of continuity, the exact values of the various n-widths in L2 are calculated. In particular the problem of minimizing the constants in inequalities of Jackson–Stechkin type over all subspaces of dimension N is solved. It is proved that this value is equal to the exact value of the different widths of the class L2(α)(β,p,h,φ), where φ is a non-negative integrable weight function on (0,h](0<h≤π/n).

Well-posedness of degenerate differential equations with fractional derivative in vector-valued functional spaces

In this paper, we study the well-posedness of the degenerate differential equations with fractional derivative Dα(Mu)(t)=Au(t)+f(t),(0≤t≤2π) in Lebesgue–Bochner spaces Lp(T;X), periodic Besov spaces Bp,qs(T;X) and periodic Triebel–Lizorkin spaces Fp,qs(T;X), where A and M are closed linear operators in a complex Banach space X satisfying D(A)⊂D(M), α>0 and Dα is the fractional derivative in the sense of Weyl. Using known operator-valued Fourier multiplier results, we completely characterize the well-posedness of this problem in the above three function spaces by the R-bounedness (or the norm boundedness) of the M-resolvent of A.

Weighted pseudo antiperiodic solutions for fractional integro-differential equations in Banach spaces

In this paper we prove the existence of weighted pseudo antiperiodic mild solutions for fractional integro-differential equations in the formDαu(t)=Au(t)+∫-∞ta(t-s)Au(s)ds+f(t,u(t)),t∈R,where f(·,u(·)) is Stepanov-like weighted pseudo antiperiodic and A generates a resolvent family of bounded and linear operators on a Banach space X,a∈Lloc1(R+) and α>0. Here the fractional derivative is considered in the sense of Weyl. Also, we give a short proof to show that the vector-valued space of Stepanov-like weighted pseudo antiperiodic functions is a Banach space.

Spectral Analysis and Spectral Symbol of $d$-variate $\mathbb Q_{\boldsymbol p}$ Lagrangian FEM Stiffness Matrices

We study the spectral properties of the stiffness matrices coming from the approximation of a $d$-dimensional second order elliptic differential problem by the $\mathbb Q_{\boldsymbol p}$ Lagrangian finite element method (FEM); here, ${\boldsymbol p}=(p_1,\ldots,p_d)$ and $p_j$ represents the polynomial approximation degree in the $j$th direction. After presenting a construction of these matrices, we investigate the conditioning and the spectral distribution in the Weyl sense, and we determine the so-called (spectral) symbol describing the asymptotic spectrum. We also study the properties of the symbol, which turns out to be a $d$-variate function taking values in the space of $N({\boldsymbol p})\times N({\boldsymbol p})$ Hermitian matrices, where $N({\boldsymbol p})=\prod_{j=1}^d p_j$. Unlike the stiffness matrices coming from the ${\boldsymbol p}$-degree B-spline isogeometric analysis approximation of the same differential problem, where a unique $d$-variate real-valued function describes all the spectrum, here the spectrum is described by $N({\boldsymbol p})$ different functions, i.e., the $N({\boldsymbol p})$ eigenvalues of the symbol, which are well-separated, far away, and exponentially diverging with respect to ${\boldsymbol p}$ and $d$. This very involved picture provides an explanation of: (a) the difficulties encountered in designing robust solvers, with convergence speed independent of the matrix size, of the approximation parameters ${\boldsymbol p}$, and of the dimensionality $d$; (b) the convergence deterioration of known iterative methods, already observed in practice for moderate ${\boldsymbol p}$ and $d$. (The PDF has been changed.)

On the spectrum of stiffness matrices arising from isogeometric analysis

We study the spectral properties of stiffness matrices that arise in the context of isogeometric analysis for the numerical solution of classical second order elliptic problems. Motivated by the applicative interest in the fast solution of the related linear systems, we are looking for a spectral characterization of the involved matrices. In particular, we investigate non-singularity, conditioning (extremal behavior), spectral distribution in the Weyl sense, as well as clustering of the eigenvalues to a certain (compact) subset of $$\mathbb C$$ . All the analysis is related to the notion of symbol in the Toeplitz setting and is carried out both for the cases of 1D and 2D problems.