On the normalizer of a Gaschütz system of a finite soluble group

The notion of a Gaschütz system of a finite soluble group was introduced by S. F. Kamornikov in 2008 (this is a set of complements of crowns of pairwise nonisomorphic non-Frattini factors of a chief series of the group). In the present paper, properties of Gaschütz systems are investigated. In particular, we calculate the number of Gaschütz systems in a finite soluble group and prove their conjugacy, obtain a connection between mathfrakN mathfrak{N} -prefrattini subgroups and normalizers of Gaschütz systems, and investigate factorizations of the normalizer of a Gaschütz system.     

On a theorem of M. Satô

In 1953 Nash [7] introduced the class of functions Φ. In this paper the behaviour of generalized Cesàro (C,α _n )-means (α _n ∈(−1,0)) of trigonometric Fourier series of the classes H ^ω ∩Φ in the space of continuous functions is studied. The sharpness of the results obtained is shown.     

A note on chaotic and predictable representations for Itô–Markov additive processes

In this article, we provide predictable and chaotic representations for Itô–Markov additive processes X. Such a process is governed by a finite-state continuous time Markov chain J which allows one to modify the parameters of the Itô-jump process (in so-called regime switching manner). In addition, the transition of J triggers the jump of X distributed depending on the states of J just prior to the transition. This family of processes includes Markov modulated Itô–Lévy processes and Markov additive processes. The derived chaotic representation of a square-integrable random variable is given as a sum of stochastic integrals with respect to some explicitly constructed orthogonal martingales. We identify the predictable representation of a square-integrable martingale as a sum of stochastic integrals of predictable processes with respect to Brownian motion and power-jumps martingales related to all the jumps appearing in the model. This result generalizes the seminal result of Jacod–Yor and is of importance in financial mathematics. The derived representation then allows one to enlarge the incomplete market by a series of power-jump assets and to price all market-derivatives.     

A note on Hardy’s theorem and rotations

Semigroup Forum - We give a very short proof, using the Hermite semigroup, to a generalized version of Hardy’s theorem due to Hogan and Lakey. We characterize $$fin L^2({mathbb {R}}^n)$$...     

A note on the uniqueness theorem of meromorphic mappings

In this article, some uniqueness theorems of meromorphic mappings are proved.     

A note on a fixed point theorem and the Hyers–Ulam stability

We present a theorem that is a simultaneous generalization of the classical Banach fixed point theorem and a theorem of G.L. Forti on the Hyers–Ulam stability (of difference and functional equations). We also give an example that shows how to use that result in finding the solutions of some difference equations.     

A note on conjugate Π-variation and a weak limit theorem for the number of renewals

A weak limit theorem for the number of renewals N(t) when 1 − F(x) is regularly varying with index −1 is given utilizing an alternatemethod of constructing conjugates for functions in the well-known subclass Π of slowly varying functions. An error in the literature concerning the centering term when F(x) has a finite mean is corrected.     

A note on Robinson–Ursescu and Lyusternik–Graves theorem

The aim of this note is twofold. First, we prove an analogue of the well-known Robinson–Ursescu Theorem on the relative openness with a linear rate (restrictive metric regularity) of a multivalued mapping. Second, we prove a generalization of Graves Open Mapping Theorem for a class of mappings which can be approximated at a reference point by a bunch of linear mappings. The approximated non-linear mapping is restricted to a closed convex subset of a Banach space.     

A note on the Fuglede–Putnam theorem

We prove the following generalization of the Fuglede–Puntam theorem. Let N be an unbounded normal operator in the Hilbert space, and let A be an unbounded self-adjoint operator such that $D(N)\subseteq D(A)$ . Then, $ AN\subseteq N^*A \Rightarrow AN^*\subseteq NA.$      

A note on the Engelking-Karłowicz theorem

We investigate the chromatic number of infinite graphs whose definition is motivated by the theorem of Engelking and Kar?owicz (in [?]). In these graphs, the vertices are subsets of an ordinal, and two subsets X and Y are connected iff for some a ∈ X ∩ Y the order-type of a ∩ X is different from that of a ∩ Y.In addition to the chromatic number x(G) of these graphs we study χ _κ (G), the κ-chromatic number, which is the least cardinal µ with a decomposition of the vertices into µ classes none of which contains a κ-complete subgraph.     

A note on a Mar?enko-Pastur type theorem for time series

In this note we develop an extension of the Mar?enko-Pastur theorem to time series model with temporal correlations. The limiting spectral distribution (LSD) of the sample covariance matrix is characterised by an explicit equation for its Stieltjes transform depending on the spectral density of the time series. A numerical algorithm is then given to compute the density functions of these LSD’s.     

A note on Linnik’s theorem on quadratic non-residues

Archiv der Mathematik - We present a short and purely combinatorial proof of Linnik’s theorem: for any $$varepsilon >0$$ there exists a constant $$C_varepsilon $$ such that for any...     

The number of solutions of some equations in finite groups and a new proof of Itô's theorem

We compute the number of solutions to some equations in finite groups and use these computations to give a new proof of Itô's theorem on the degrees of the irreducible characters.     

Notes on a classic theorem of Erdős and Grünwald

We establish a companion result to a classic theorem of Erd?s and Grünwald on the maximum of the fundamental functions of Lagrange interpolation based on the Chebyshev nodes.