A generalization of Sen–Brinon’s theory

Let K be a complete discrete valuation field of mixed characteristic and k be its residue field of prime characteristic p > 0. We assume that [k : k ^p ] = p ^h < ∞. Let G _K be the absolute Galois group of K and ${\mathcal{R}}$ be a Banach algebra over ${\mathbb{C}_p:=\widehat{\overline{K}}}$ with a continuous action of G _K . When k is perfect (i.e. h = 0), Sen studied the Galois cohomology ${{\rm H}^1(G_K, \mathcal{R}^\ast)}$ and Sen’s operator associated to each class (Sen Ann Math 127:647–661, 1988). In this paper we generalize Sen’s theory to the case h ≥ 0 by using Brinon’s theory (Brinon Math Ann 327:793–813, 2003). We also give another formulation of Brinon’s theorem (à la Colmez).     

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 $L^p$ spaces.     

A generalization of the Beurling—Lax theorem

We obtain conditions for the completeness of the system {G(z)e ^τz , τ ≤ 0} in the space H _σ ² (?₊), 0 < σ < + ∞, of functions analytic in the right-hand half-plane for which $$\parallel f\parallel : = \mathop {\sup }\limits_{ - \pi /2 < \varphi < \pi /2} \left\{ {\int_0^{ + \infty } {|f(re^{i\varphi } )|^2 } e^{ - 2r\sigma |\sin \varphi |} dr} \right\}^{1/2} < + \infty $$ .     

A torsion—theoretic generalization of fully bounded Noetherian rings

ABSTRACT

We introduce semidirect and wreath products of finite ordered semigroups and extend some standard decomposition results to this case.     

A generalization of Baer’s theorem

In this paper, we studied the supersolvability of the product of two subgroups and got a generalization of Baer’s theorem.     

A generalization of Menonʼs identity

In this note we give a generalization of the well-known Menon?s identity. This is based on applying the Burnside?s lemma to a certain group action.     

A generalization of d’Alembert formula

In this paper we find a closed form of the solution for the factored inhomogeneous linear equation

A generalization of Obata’s theorem

In a complete Riemannian manifold (M, g) if the hessian of a real-valued function satisfies some suitable conditions, then it restricts the geometry of (M, g). In this paper we characterize all compact rank-one symmetric spaces as those Riemannian manifolds (M, g) admitting a real-valued functionu such that the hessian ofu has at most two eigenvalues ?u and $ - \frac{{u + 1}}{2}$ under some mild hypotheses on (M, g). This generalizes a well-known result of Obata which characterizes all round spheres.     

A generalization of Gauss-Kuzmin-Lévy theorem

We prove a generalized Gauss-Kuzmin-Lévy theorem for the generalized Gauss transformation

$T_p\left(x\right)=\left\{\frac{p}{x}\right\}.$

In addition, we give an estimate for the constant that appears in the theorem.     

A generalization of Euler’s constant

The purpose of this paper is to evaluate the limit γ(a) of the sequence , where a ∈ (0, + ∞ ).      

A generalization of rummer’s congruences

Suppose thatm, n are positive even integers andp is a prime number such thatp-1 is not a divisor ofm. For any non-negative integerN, the classical Kummer’s congruences on Bernoulli numbersB _n(n = 1,2,3,...) assert that (1-p ^m-1)B _m/m isp-integral and

A generalization of Clausen’s identity

The paper gives an extension of Clausen’s identity to the square of any Gauss hypergeometric function. Accordingly, solutions of the related third-order linear differential equation are found in terms of certain bivariate series that can reduce to ₃F₂ series similar to those in Clausen’s identity. The general contiguous variation of Clausen’s identity is found as well. The related Chaundy’s identity is generalized without any restriction on the parameters of the Gauss hypergeometric function. The special case of dihedral Gauss hypergeometric functions is underscored.     

A generalization of Menger’s Theorem

This paper generalizes one of the celebrated results in Graph Theory due to Karl. A. Menger (1927), which plays a crucial role in many areas of flow and network theory. This paper also introduces and characterizes strength reducing sets of nodes and arcs in weighted graphs.     

A generalization of Forelli’s theorem

The purpose of this paper is to present a generalization of Forelli’s theorem. In particular, we prove an all dimensional version of the two-dimensional theorem of Chirka (Kompleks. Anal. i Prilozh, 232–240, 2006) of 2005.     

A generalization of Nekhoroshev’s theorem

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincaré theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.