Search

Stoilow's theorem from 1928 states that a continuous, open, and light map between surfaces is a discrete map with a discrete branch set. This result implies that such maps between orientable surfaces are locally modeled by power maps z -> z(k) and admit a holomorphic factorization. The purpose of this expository article is to give a proof of this classical theorem having readers in mind that are interested in continuous, open and discrete maps. (C) 2019 Elsevier GmbH. All rights reserved.

In this chapter, the first approach is made to establishing the convergence of scaled random sums, considering independent sequences. The classic Lindeberg–Lévy, Khinchine, Lindeberg–Feller, and Liapunov theorems are proved. The main focus is on the treatment of heterogeneous summands, applying the Lindeberg condition, and extensions are given to allow trending (growing or shrinking) variances. The final sections review cases of the central limit theorem under non-standard conditions and α-stable convergence.

Elkik established a remarkable theorem that can be applied for any noetherian henselian ring. For algebraic equations with a formal solution (restricted by some smoothness assumption), this theorem provides a solution adically close to the formal one in the base ring. In this paper, we show that the theorem would fail for some non-noetherian henselian rings. These rings do not satisfy several conditions weaker than noetherianness, such as weak proregularity (due to Grothendieck et al.) of the defining ideal. We describe the resulting pathologies., 14 pages, to appear in Journal of Pure and Applied Algebra

In this article we prove a general result on a nef vector bundle $E$ on a projective manifold $X$ of dimension $n$ depending on the vector space $H^{n,n} (X, E). $ It is also shown that $H^{n,n} (X, E)=0$ for an indecomposable nef rank 2 vector bundles $E$ on some specific type of $n$ dimensional projective manifold $X.$ The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.

In 1995, Palfy and Szabo stated the theorem that a projective space satisfies the six-cross theorem if and only if it is Desarguesian. A mistake in their proof is corrected. Moreover, for projective planes, the three-cross theorem of Palfy and Szabo is identified as the Reidemeister condition.

We study the following fractional Yamabe-type equation: { A s ⁢ u = u n + 2 ⁢ s n - 2 ⁢ s , u > 0 in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right. Here Ω is a regular bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 2 {n\geq 2} , and A s {A_{s}} , s ∈ ( 0 , 1 ) {s\in(0,1)} , represents the fractional Laplacian operator ( - Δ ) s {(-\Delta)^{s}} in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

We use suitably Page’s theorem to get effective results for interesting problems, by avoiding the ineffective Siegel’s theorem.

The Valiron–Mohon’ko theorem plays an important role in the theory of complex differential equations. In this paper, a difference analogue of the Valiron–Mohon’ko theorem is established, which can be used to get the characteristic functions on irreducible rational functions of f(z) and its shifts. Using our results and some properties of periodic functions, the proof of [10, Theorem 1.1] can be organized in a short way.

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.

Let ℋ be a complex separable Hilbert space, e the collection of orthogonal projections in ℋ, and (X, Ɓ) a Borel space. A function E : Ɓ → e is called a spectral measure if E(X) = I and E(∪ i=1 ∞ A i ) = Σ i=1 ∞ (A i ), for any pairwise disjoint collection A1, A2, A3, …, of sets in Ɓ.

A non-Euclidean analog of the generalized Darboux equation is considered. For the case where its solutions are radial functions of second variable we obtain a uniqueness result (Theorem 1) which deals with zero sets of these solutions. The example of the function in Theorem 2 of the paper shows that Theorem 1 cannot be essentially reinforced.

The paper examines different kinds of p-convexity of a function g which are sufficient for the existence of a linear functional such that in Theorem 1.13 of Simons in his monograph ‘From Hahn–Banach to monotonicity’, published in Springer lecture notes (2008). We replace sublinearity of p with convexity, the field with Dedekind vector lattice and present -convexity which is also necessary. In Theorem 4.7 we also generalize a result of MM. Neumann from 1991 published in Czech. Mathem. Journal Vol 41 on the Mazur–Orlicz theorem.

Summary This article formalizes the proof of Niven’s theorem [12] which states that if x/π and sin(x) are both rational, then the sine takes values 0, ±1/2, and ±1. The main part of the formalization follows the informal proof presented at Pr∞fWiki (https://proofwiki.org/wiki/Niven’s_Theorem#Source_of_Name). For this proof, we have also formalized the rational and integral root theorems setting constraints on solutions of polynomial equations with integer coefficients [8, 9].

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincare 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.

A theorem by K. Meyer and D. Schmidt says that The reduced three-body problem in two or three dimensions with one small mass is approximately the product of the restricted problem and a harmonic oscillator [7]. This theorem was used to prove dynamical continuation results from the classical restricted circular three-body problem to the three-body problem with one small mass. We state and prove a similar theorem applicable to a larger class of mechanical systems. We present applications to spatial $(N+1)$-body systems with one small mass and gravitationally coupled systems formed by a rigid body and a small point mass.

The purpose of this paper is to show that the Rudin–Carleson interpolation theorem is a direct corollary of Fatou’s much older interpolation theorem (of 1906).

Starting from one extension of the Hahn---Banach theorem, the Mazur---Orlicz theorem, and a not very restrictive concept of convexity, that arises naturally in minimax theory, infsup-convexity, we derive an equivalent version of that fundamental result for finite dimensional spaces, which is a sharp generalization of Konig's Maximum theorem. It implies several optimal statements of the Lagrange multipliers, Karush/Kuhn---Tucker, and Fritz John type for nonlinear programs with an objective function subject to both equality and inequality constraints.

In this paper, we consider the Bessel–Struve transform FBSα and we establish an analogous of Beurling-Hormander's theorem for each α>−12. More precisely, we determine the form of nonzero functions satisfying weaken condition of Beurling–Hormander's theorem which differ from α half integer or non-half integer. As applications, we obtain a Cowling–Price-type theorem and analogous of Hardy's theorem.

We give a new proof of a classical uniqueness theorem of Alexandrov [4] using the weak uniqueness continuation theorem of Bers–Nirenberg [8] . We prove a version of this theorem with the minimal regularity assumption: the spherical Hessians of the corresponding convex bodies as Radon measures are nonsingular.

It was proved earlier that, for multiple Fourier series whose coefficients are monotone in each index, the classicalHardy–Littlewood theorem is not valid for p ≤ 2m/(m+1), where m is the dimension of the space. We establish how the theorem must be modified in this case.

We study the relation between Stein’s theorem and Stokes’ theorem (or the divergence theorem) and show, using completeness of certain exponential families, that they are equivalent, in a certain sense, by using each to prove a version of the other.

The central sets theorem originally proven by H. Furstenburg is a powerful result which is applicable to derive many combinatorial conclusions. Furstenburg's original theorem applied to N and finitely many sequences in Z . Some strengthenings of this theorem have been derived first by V. Bergelson and N. Hindman in 1990. Later in 2008, D. De, N. Hindman, and D. Strauss proved a stronger version of the central sets theorem for arbitrary semigroups S which applied to all sequences in S. We provide here a generalization of the stronger version and some applications of this new generalization.

We prove a generalized inverse function theorem in a neighborhood of a singular point of a mapping. As corollaries to this theorem, we obtain an inverse function theorem, an error bound theorem, and a tangent cone theorem that extend and strengthen the corresponding classical results in the irregular case. Using these corollaries, we establish necessary extremum conditions that are meaningful for abnormal problems.

This paper provides a novel proof of a generalization of Pappus’ centroid theorem on [math] -dimensional tubes using Stokes’ theorem on manifolds.

In this note we present a counterexample to Theorem 24 in G.S.F. Frederico and D.F.M. Torres (2010) [3] .

We present a new uniqueness result for first order systems of ordinary differential equations which contains a generalization of Montel–Tonelli's Uniqueness Theorem as a particular case. An example is given to illustrate its applicability.

A uniqueness theorem is proved for the space of functions analytic in the right half plane and satisfying the condition $$ \underset{\left|\upvarphi \right

In this paper, we prove a central limit theorem for m-dependent random variables under sublinear expectations. This theorem can be regarded as a generalization of Peng’s central limit theorem.

The Funk’s section theorem in the n-complex space Cn is investigated. It turns out that this theorem does not admit an extension for the class of general origin-symmetric star bodies in Cn but for a class of star bodies called generalized complex intersection bodies. A quasi-version of Funk’s section theorem in Cn is established then.

The Matrix-Tree Theorem is a formula for the number of spanning trees of a graph in terms of the determinant of a certain matrix. We begin with the necessary graph-theoretical background. Let G be a finite graph, allowing multiple edges but not loops. (Loops could be allowed, but they turn out to be completely irrelevant.)

We prove an inequality for concave functions and partitioned matrices whose numerical ranges lie in a sector. This complements a theorem by E.Y. Lee concerning the positive semi-definite case.

Berge’s maximum theorem is an important statement in set-valued analysis that has significant applications in mathematical economics, operations research, and control. For a minimization problem for a continuous function of two variables and continuous compact-valued set-valued mapping defining feasible sets, this theorem states continuity of the value function and upper semi-continuity of the solution multifunction. One of the main limitations of this theorem is that the set-valued mapping is compact-valued. Recently the authors of this paper and their coauthors generalized Berge’s maximum theorem to set-valued mappings that may not be compact-valued. Here we formulate and prove the local Berge’s maximum theorem for possibly noncompact feasible sets and show that it is more general than the recently established Berge’s maximum theorem for possibly noncompact feasible sets and than the known formulations of the local Berge’s maximum theorem.

Ramsey’s theorem states that each coloring has an infinite homogeneous set, but these sets can be arbitrarily spread out. Paul Erdős and Fred Galvin proved that for each coloring f f , there is an infinite set that is “packed together” which is given “a small number” of colors by f f . We analyze the strength of this theorem from the perspective of computability theory and reverse mathematics. We show that this theorem is close in computational strength to the standard Ramsey’s theorem by giving arithmetical upper and lower bounds for solutions to computable instances. In reverse mathematics, we show that that this packed Ramsey’s theorem is equivalent to Ramsey’s theorem for exponents n ≠ 2 n\neq 2 . When n = 2 n=2 , we show that it implies Ramsey’s theorem and that it does not imply A C A 0 \mathsf {ACA}_0 .

In this paper, with the motivation from Diophantine approximation, a truncated second main theorem is established for meromorphic maps from M into P ( V ) with moving targets g j : M → P ( V ⁎ ) , 1 ≤ j ≤ q , where M is a parabolic manifold and V is a Hermitian vector space. As an application of this second main theorem, a uniqueness theorem without counting multiplicities is given.