An extension of Barta’s Theorem and geometric applications

We prove an extension of a theorem of Barta and we give some geometric applications. We extend Cheng’s lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We show that the spectrum of the Nadirashvili bounded minimal surfaces in have positive lower bounds. We prove a stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove generalization of a result of Kazdan–Kramer about existence of solutions of certain quasi-linear elliptic equations. Bessa and Montenegro were partially supported by CNPq Grant.     

An extension of Jung’s theorem

Theorem. Let a set X?Rⁿ have unit circumradius and let B be the unit ball containing X. Put C =conv \(\bar X\) D =diam C (=diam X), k =dim C,d _i = √(2i + 2)/i. Then: (i) D∈[d_n, 2]; (ii) k≧m where m∈{2,3,...,n} satisfies D∈[d_m, d_m?1) (d_i decreases by i); (iii) In case k=m (by (ii), this is always the case when m=n), C contains a k-simplex Δ such that: (α) its vertices are on δB; (β) the centre of B belongs toint Δ; (γ) the inequalitiesλ _k (D) ≦l ≦D with $$\lambda _k (D) = D\sqrt {\frac{{4k - 2D^2 (k - 1)}}{{2 - (k - 2)(D^2 - 2)}}, D \in (d_k ,d_{k - 1} )} $$ are unimprovable estimates for length l of any edge of Δ.     

An extension of Babbage’s criterion for primality

Let n > 1 and k > 1 be positive integers. We show that if $$\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)$$ for each integer m with 0 ≤ m ≤ n ? 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.     

An extension of Mercer’s theory to L p

Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure ?? and ${\mathcal{K}:L^2(X,\nu)\to L^2(X,\nu)}$ an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer??s theory for K and ${\mathcal{K}}$ under the assumption that the function ${x\in X\to K(x,x)}$ belongs to some L ^p/2(X, ??), p??? 1. In particular, we obtain series representations for K and some powers of ${\mathcal{K}}$ , with convergence in the p-mean, and show that the range of certain powers of ${\mathcal{K}}$ contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of ${\mathcal{K}}$ acting on L ^p (X, ??).     

An extension of Wilf’s conjecture to affine semigroups

Let \(\mathcal {C}\subset \mathbb {Q}^p_+\) be a rational cone. An affine semigroup \(S\subset \mathcal {C}\) is a \(\mathcal {C}\)-semigroup whenever \((\mathcal {C}\setminus S)\cap \mathbb {N}^p\) has only a finite number of elements. In this work, we study the tree of \(\mathcal {C}\)-semigroups, give a method to generate it and study the \(\mathcal {C}\)-semigroups with minimal embedding dimension. We extend Wilf’s conjecture for numerical semigroups to \(\mathcal {C}\)-semigroups and give some families of \(\mathcal {C}\)-semigroups fulfilling the extended conjecture. Other conjectures formulated for numerical semigroups are also studied for \(\mathcal {C}\)-semigroups.     

An extension of Fujita’s non-extendability theorem for Grassmannians

In this paper we study smooth complex projective varieties X containing a Grassmannian of lines ${{\mathbb G}(1, r)}$ which appears as the zero locus of a section of a rank two nef vector bundle E. Among other things we prove that the bundle E cannot be ample.     

An Analogue of Bochner’s Theorem for Damek-Ricci Spaces

We characterize the image of radial positive measures θ’s on a harmonic NA group S which satisfies ∫ _S ? ₀(x)?dθ(x)<∞ under the spherical transform, where ? ₀ is the elementary spherical function.     

An Extension of Mercer’s Theorem via Pontryagin Spaces

We extend Mercer’s theorem to a composition of the form RS, in which R and S are integral operators acting on a space L ²(X) generated by a locally finite measure space (X, ν). The operator R is compact and positive while S is continuous and having spectral decomposition based on well distributed eigenvalues. The proof is based on a Pontryagin space structure for L ²(X) constructed via the operators R and S themselves.     

An extension of Günther’s volume comparison theorem

The aim of this note if to give an extension of a classical volume comparison theorem for Riemannian manifolds with sectional curvature bounded above (see Günther, P. Einige Sätze über das Volumenelement eines Riemannschen Raumes, Publ. Math. Debrecen 7, 78–93 (1960)). For the case of a n-dimensional simply connected complete Riemannian manifold with nonpositive sectional curvature our theorem states that the function tarea(S _t (p))/t ^n–2 is convex for every pM where S _t (p) denotes the sphere of radius t with center p. In view of area(S ₀ (p))=0, it is easy to see that our theorem implies the classical result. A similar result holds true for simply connected manifolds with sectional curvature bounded above by a negative constant.Research partially supported by Fondecyt Grant # 1000713 and by UTFSM Grant # 120023     

An extension of Greenberg’s theorem to general valuation rings

We extend Greenberg’s strong approximation theorem to schemes of finite presentation over valuation rings with arbitrary value group. As an application, we prove a closed image theorem (in the strong topology on rational points) for proper morphisms of varieties over valued fields.     

An extension of one direction in Marty’s normality criterion

We prove the following extension of one direction in Marty’s theorem: If $k$ is a natural number, $\alpha >1$ and $\mathcal{F }$ is a family of functions meromorphic on a domain $D$ all of whose poles have multiplicity at least $\frac{k}{\alpha -1}$ , then the normality of $\mathcal{F }$ implies that the family $$\begin{aligned} \left\{ \frac{|f^{(k)}|}{1+|f|^\alpha }\,:\, f\in \mathcal{F }\right\} \end{aligned}$$ is locally uniformly bounded.     

An example related to Gregory’s Theorem

In this paper, we give an example of a complete computable infinitary theory T with countable models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ and T has no uncountable model. In fact, ${\mathcal{M}}$ and ${\mathcal{N}}$ are (up to isomorphism) the only models of T. Moreover, for all computable ordinals α, the computable ${\Sigma_\alpha}$ part of T is hyperarithmetical. It follows from a theorem of Gregory (JSL 38:460–470, 1972; Not Am Math Soc 17:967–968, 1970) that if T is a Π ₁ ¹ set of computable infinitary sentences and T has a pair of models ${\mathcal{M}}$ and ${\mathcal{N}}$ , where ${\mathcal{N}}$ is a proper computable infinitary extension of ${\mathcal{M}}$ , then T would have an uncountable model.     

An Extension of Polyak’s Theorem in a Hilbert Space

Let H be an infinite-dimensional real Hilbert space equipped with the scalar product (⋅,⋅) _H . Let us consider three linear bounded operators,

Limits on an extension of Carleson’s\bar \partial - Theorem

We study a theorem essentially due to Carleson about solving the $\bar \partial - equation$ on the unit disk. We show that this theorem generalizes to bordered Riemann surfaces with finitely generated fundamental groups. However, our main result is that the constant appearing in the generalized theorem cannot be taken to be independent of the bordered Riemann surface in question. We exhibit a sequence of (topologically equivalent) Riemann surfaces on which the constant tends to ∞. Since Carleson’s $\bar \partial - theorem$ depends on the notion of a Carleson measure, we also discuss Carleson measures at some length in order to define them appropriately on arbitrary Riemann surfaces.     

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.     

An analog of shemetkov’s conjecture for fischer classes of finite groups

We describe some methods for constructing Fischer classes of finite groups by means of the operators defined by given properties of Hall π-subgroups. It is in particular proved that, for a Fischer class $\mathfrak{F}$ and a set of primes π, the class of all finite π-soluble $C_\pi \mathfrak{F}$ -groups, i.e., of all groups whose Hall π-subgroups belong to $\mathfrak{F}$ , is a Fischer class.     

An Integral Version of ?iri?’s Fixed Point Theorem

We establish a new fixed point theorem for mappings satisfying a general contractive condition of integral type. The presented theorem generalizes the well known Ćirić’s fixed point theorem [Lj. B. Ćirić, Generalized contractions and fixed point theorems, Publ. Inst. Math. 12 (26) (1971) 19-26]. Some examples and applications are given.