Fischer Theorem Research Articles

On the Properties of Quasi-Banach Function Spaces

In this paper we explore some basic properties of quasi-Banach function spaces which are important in applications. Namely, we show that they possess a generalised version of Riesz–Fischer property, that embeddings between them are always continuous, and that the dilation operator is bounded on them. We also provide a characterisation of separability for quasi-Banach function spaces over the Euclidean space. Furthermore, we extend the classical Riesz–Fischer theorem to the context of quasinormed spaces and, as a consequence, obtain an alternative proof of completeness of quasi-Banach function spaces.

Spectral analysis of Markov switching GARCH models with statistical inference

AbstractWe derive matrix expressions in closed form for the autocovariance function and the spectral density of Markov switching GARCH models and their powers. For this, we apply the Riesz–Fischer theorem which defines the spectral representation as the Fourier transform of the autocovariance function. Under suitable assumptions, we prove that the sample estimator of the spectral density is consistent and asymptotically normally distributed. Further statistical implications in terms of order identification and parameter estimation are discussed. A simulation study confirms the validity of the asymptotic properties. These methods are also well suited for financial market applications, and in particular for the analysis of time series in the frequency domain, as shown in some proposed real‐world examples.

Hermitian normalized Laplacian matrix for directed networks

In this paper, we extend and generalize the spectral theory of undirected networks towards directed networks by introducing the Hermitian normalized Laplacian matrix for directed networks. In order to start, we discuss the Courant–Fischer theorem for the eigenvalues of Hermitian normalized Laplacian matrix. Based on the Courant–Fischer theorem, we obtain a similar result towards the normalized Laplacian matrix of undirected networks: for each i ∈ {1, 2,…, n}, any eigenvalue of Hermitian normalized Laplacian matrix λi ∈ [0, 2]. Moreover, we prove some special conditions if 0, or 2 is an eigenvalue of the Hermitian normalized Laplacian matrix L(X). On top of that, we investigate the symmetry of the eigenvalues of L(X) and the edge-version for the eigenvalue interlacing result. Finally we present two expressions for the coefficients of the characteristic polynomial of the Hermitian normalized Laplacian matrix. As an outlook, we sketch some novel and intriguing problems to which our apparatus could generally be applied.

Birational classification of fields of invariants for groups of order 128

Let G be a finite group acting on the rational function field C(xg:g∈G) by C-automorphisms h(xg)=xhg for any g,h∈G. Noether's problem asks whether the invariant field C(G)=k(xg:g∈G)G is rational (i.e. purely transcendental) over C. By Fischer's theorem, C(G) is rational over C when G is a finite abelian group. Saltman and Bogomolov, respectively, showed that for any prime p there exist groups G of order p9 and of order p6 such that C(G) is not rational over C by showing the non-vanishing of the unramified Brauer group: Brnr(C(G))≠0, which is an avatar of the birational invariant H3(X,Z)tors given by Artin and Mumford where X is a smooth projective complex variety whose function field is C(G). For p=2, Chu, Hu, Kang and Prokhorov proved that if G is a 2-group of order ≤32, then C(G) is rational over C. Chu, Hu, Kang and Kunyavskii showed that if G is of order 64, then C(G) is rational over C except for the groups G belonging to the two isoclinism families Φ13 with Brnr(C(G))=0 and Φ16 with Brnr(C(G))≃C2. Bogomolov and Böhning's theorem claims that if G1 and G2 belong to the same isoclinism family, then C(G1) and C(G2) are stably C-isomorphic. We investigate the birational classification of C(G) for groups G of order 128 with Brnr(C(G))≠0. Moravec showed that there exist exactly 220 groups G of order 128 with Brnr(C(G))≠0 forming 11 isoclinism families Φj. We show that if G1 and G2 belong to Φ16, Φ31, Φ37, Φ39, Φ43, Φ58, Φ60 or Φ80 (resp. Φ106 or Φ114), then C(G1) and C(G2) are stably C-isomorphic with Brnr(C(Gi))≃C2. Explicit structures of non-rational fields C(G) are given for each cases including also the case Φ30 with Brnr(C(G))≃C2×C2.

A generalization of the Riesz–Fischer theorem and linear summability methods

We extend the classical Riesz–Fischer theorem to biorthogonal systems of functions in Orlicz spaces: from a given double series (not necessarily convergent but satisfying a growth condition) we construct a function (in a given Orlicz space) by a linear summation method, and recover the original double series via the coefficients of the expansion of this function with respect to the biorthogonal system. We give sufficient conditions for the regularity of some linear summation methods for double series. We are inspired by a result of Fomin who extended the Riesz–Fischer theorem to Lp spaces.

The spectral representation of Markov switching ARMA models

In this paper we propose a method to derive the spectral density function of Markov switching ARMA models. We apply the Riesz–Fischer theorem which defines the spectral representation as the Fourier transform of the autocovariance functions.

On variational eigenvalues of the p -Laplacian which are not of Ljusternik-Schnirelmann type

In this paper we demonstrate the fact that the famous Ljusternik–Schnirelmann characterization of some eigenvalues of nonlinear elliptic problems (by a minimax formula) has a global variational character. Indeed, we show that, for some homogeneous quasilinear elliptic eigenvalue problems, there are variational eigenvalues other than those of the Ljusternik–Schnirelmann type. In contrast, these eigenvalues have a local variational character. Such a phenomenon does not occur in typical linear elliptic eigenvalue problems, owing to the Courant–Fischer theorem which is the linear analogue and predecessor of the Ljusternik–Schnirelmann theory.

The Monotonicity Theorem, Cauchy's Interlace Theorem, and the Courant- Fischer Theorem

(1987). The Monotonicity Theorem, Cauchy's Interlace Theorem, and the Courant-Fischer Theorem. The American Mathematical Monthly: Vol. 94, No. 4, pp. 352-354.

Relationship between the Karhunen- Loéve transform and the Courant - Fischer theorem (Corresp.)

It is shown that the Karhunen-Loeve transform problem can be formulated as a matrix approximation problem with Hilbert-Schmidt error norm. On the other hand, the Courant-Fischer minimax theorem provides a characterization for the best matrix approximation when the spectral norm is used. It appears that the optimality conditions of the Karhunen-Loeve problem lead to the selection of a particular solution among the set of solutions to the Courant-Fischer problem.

Generalization of fischer's theorem for distributive quasigroups

We generalize Fischer's well-known theorem which asserts that the right associated group of a finite distributive quasigroup is solvable. We prove that the associated group of a finitely generated distributive quasigroup is solvable and that the associated group of any distributive quasigroup is locally solvable.

Finite distributive quasigroups

AbstractThe Bruck-Slaby theorem on nilpotence of commutative Moufang loops may be combined with a theorem of Belousov to give a quick elementary proof of a stronger version of Fischer's Theorem on the solubility of right multiplication groups of finite distributive quasigroups.