Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contr...
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Given a collection of test functions, one defines the associated Schur-Agler class as the intersection of the contractive multipliers over the collection of all positive kernels for which each test function is a contractive multiplier. We indicate extensions of this framework to the case where the test functions, kernel functions, and Schur-Agler-class functions are allowed to be matrix- or operator-valued. We illustrate the general theory with two examples: (1) the matrix-valued Schur class over a finitely-connected planar domain and (2) the matrix-valued version of the constrained Hardy algebra (bounded analytic functions on the unit disk with derivative at the origin constrained to have zero value). Emphasis is on examples where the matrix-valued version is not obtained as a simple tensoring with of the scalar-valued version.
We give a set of test functions for H-1(infinity), the algebra of bounded holomorphic functions on the disk with first derivative equal to 0;whose interpolation problem was studied by Davidson, Paulsen;Raghupathi and ...
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We give a set of test functions for H-1(infinity), the algebra of bounded holomorphic functions on the disk with first derivative equal to 0;whose interpolation problem was studied by Davidson, Paulsen;Raghupathi and Singh (2009). We show that this set of test functions is minimal by relating these ideas to realization and interpolation problems.
Let {A(n)}(n) be a sequence of square matrices such that size(A(n)) = d(n) -> infinity as n -> infinity. We say that {A(n)}(n) has an asymptotic spectral distribution described by a Lebesgue measurable function ...
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Let {A(n)}(n) be a sequence of square matrices such that size(A(n)) = d(n) -> infinity as n -> infinity. We say that {A(n)}(n) has an asymptotic spectral distribution described by a Lebesgue measurable function f: D subset of R-k -> C if, for every continuous function F: C -> C with bounded support, lim(n ->infinity) 1/d(n) Sigma(dn)(i=1) F(lambda(i)(A(n))) = 1/mu(k)(D) integral F-D(f(x))dx, where mu(k) is the Lebesgue measure in R-k and lambda(1)(A(n)),..., lambda(dn)(A(n)) are the eigenvalues of A(n). In the last decades, the asymptotic spectral distribution of increasing size matrices has become a subject of investigation by several authors. Special attention has been devoted to matrices A(n) arising from the discretization of differential equations, whose size d(n) diverges to infinity along with the mesh-fineness parameter n. However, despite the popularity that the topic has reached nowadays, the role of the so-called test functions F appearing in the above limit relation has been inexplicably neglected so far. In particular, a natural question such as "Which is the largest set of test functions F for which the above limit reis satisfied?" is still unanswered. In the present paper, we provide a definitive answer to this question by identifying the largest set of test functions F for which the above limit relation is satisfied. We also present some applications of this result to the analysis of spectral clustering, including new asymptotic estimates on the number of outliers. A special attention is devoted to the so-called "inner outliers" of Hermitian block Toeplitz matrices. We conclude the paper with an interpretation of the main result in the context of the vague convergence of probability measures. (c) 2024 The Author(s). Published by Elsevier Inc.
Surge control is investigated in conjunction with rotating stall control for axial flow compressors using a bifurcation approach. test functions are developed to determine the existence and stability of the Hopf bifur...
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Surge control is investigated in conjunction with rotating stall control for axial flow compressors using a bifurcation approach. test functions are developed to determine the existence and stability of the Hopf bifurcation associated with surge for closed-loop systems under linear state feedback. A control design method is proposed for the synthesis of linear feedback laws that eliminate surge, coupled with rotating stall for any given compact parameter set. Comparisons are made with existing results. Stabilization results are demonstrated with numerical simulations. (C) 1999 Elsevier Science Ltd. All rights reserved.
In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free ...
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In this paper, the governing equations for free vibration of a non-homogeneous rotating Timoshenko beam, having uniform cross-section, is studied using an inverse problem approach, for both cantilever and pinned-free boundary conditions. The bending displacement and the rotation due to bending are assumed to be simple polynomials which satisfy all four boundary conditions. It is found that for certain polynomial variations of the material mass density, elastic modulus and shear modulus, along the length of the beam, the assumed polynomials serve as simple closed form solutions to the coupled second order governing differential equations with variable coefficients. It is found that there are an infinite number of analytical polynomial functions possible for material mass density, shear modulus and elastic modulus distributions, which share the same frequency and mode shape for a particular mode. The derived results are intended to serve as benchmark solutions for testing approximate or numerical methods used for the vibration analysis of rotating non-homogeneous Timoshenko beams.
We prove the exponential law A(E x F, G) congruent to A(E, A(F, G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W-infinity,W-P (globally p-integrable der...
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We prove the exponential law A(E x F, G) congruent to A(E, A(F, G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W-infinity,W-P (globally p-integrable derivatives), S (Schwartz space), 1) (compact support), B-[M] (globally Denjoy-Carleman), W-[M],W-P (Sobolev-Denjoy-Carleman), S-[L]([M]) (Gelfand-Shilov), and D-[M] (Denjoy-Carleman with compact support). Here E, F, G are convenient vector spaces which are finite dimensional in the cases of D, W-infinity,W-P, D-[M], and W-[M],W-P. Moreover, M = (M-k) is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms Diff B, Diff W-infinity,W-P, Diff S, and Diff D are C-infinity Lie groups, and that Diff B-{M}, Diff W-{M},W-P, Diff S-{L}({M}), and Diff D-{M}, for non-quasianalytic M, are C-{M} Lie groups, where Diff A = {Id +f : f is an element of A(R-n , R-n), inf(x is an element of Rn) det (IIn + df (x)) > 0}. We also discuss stability under composition. (C) 2015 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
We provide necessary and sufficient conditions for operator-valued functions on arbitrary sets associated with a collection of test functions to have factorizations in several situations.
We provide necessary and sufficient conditions for operator-valued functions on arbitrary sets associated with a collection of test functions to have factorizations in several situations.
We uniformly find suitable functions, which are called test functions, in the space of joint harmonics of Schrodinger model such that the Godement-Jacquet zeta integrals are equal to the principal L-functions for GL(n...
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We uniformly find suitable functions, which are called test functions, in the space of joint harmonics of Schrodinger model such that the Godement-Jacquet zeta integrals are equal to the principal L-functions for GL(n)(R). We give the explicit expressions for test functions by using Langlands parameters and harmonic functions. (C) 2018 Elsevier Inc. All rights reserved.
We suggest weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization. The functions are easy to code, cheap to calculate, and have important appli...
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We suggest weighted least squares scaling, a basic method in multidimensional scaling, as a class of test functions for global optimization. The functions are easy to code, cheap to calculate, and have important applications in data analysis. For certain data these functions have many local minima. Some characteristic features of the test functions are investigated.
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