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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.

The fundamental theorem of surface theory asserts that a surface in the three-dimensional Euclidean space E 3 can be reconstructed from the knowledge of its two fundamental forms under the assumptions that their components are smooth enough—classically in the space C 2 ( ω ) for the first one and in the space C 1 ( ω ) for the second one—and satisfy the Gauss and Codazzi–Mainardi equations over a simply-connected open subset ω of R 2 ; the surface is then uniquely determined up to proper isometries of E 3 . Then S. Mardare showed in 2005 that this result still holds under the much weaker assumptions that the components of the first form are only in the space W loc 1 , p ( ω ) and those of the second form only in the space L loc p ( ω ) , the components of the immersion defining the reconstructed surface being then in the space W loc 2 , p ( ω ) , p > 2 . The purpose of this paper is to complement this last result as follows. First, under the additional assumption that ω is bounded and has a Lipschitz-continuous boundary, we show that a similar existence and uniqueness theorem holds with the spaces W m , p ( ω ) instead of W loc m , p ( ω ) . Second, we establish a nonlinear Korn inequality on a surface asserting that the distance in the W 2 , p ( ω ) -norm, p > 2 , between two given surfaces is bounded, at least locally, by the distance in the W 1 , p ( ω ) -norm between their first fundamental forms and the distance in the L p ( ω ) -norm between their second fundamental forms. Third, we show that the mapping that uniquely defines in this fashion a surface up to proper isometries of E 3 in terms of its two fundamental forms is locally Lipschitz-continuous.

We use that the n-sphere for \(n\ge 2\) is simply-connected to prove the Poincare-Birkhoff-Witt Theorem.

The generalized commutator of a list of k real matrices is defined as a multilinear skew-symmetric function and the linear operator on the vector space is defined by . The Amitsur–Levitzki ...

In this paper, a generalized fixed point theorem and its results are established in the concept of multiplicative distance which was introduced by Agamirza et.al [3] to improve the non-Newtonian calculus. Our results include some existing results in the concept of multiplicative metric space.

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.

We propose an alternative proof of Pellet’s theorem for matrix polynomials that, unlike existing proofs, does not rely on Rouche’s theorem. A similar proof is provided for the generalization to matrix polynomials of a result by Cauchy that can be considered as a limit case of Pellet’s theorem.

We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E . The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.

The aim of this paper is to explore Karamata’s mean value theorem. In the second section, we reformulate a mean theorem for the convex functions and prove some consequences. Also, we prove an integral version of Karamta theorem. Third section is reserved to the stability results. We establish some conditions for the stability of the intermediate point arising from Karamata, Godner and the integral mean theorem proved in the previous section.

We proved some inequalities for concave functions. Those inequalities complemented a theorem obtained by Lee. Finally, we partially solved an open problem proposed by Zhang P.

The aim of this paper is to generalize Minkowski’s theorem. This theorem is usually stated for a centrally symmetric convex body and a lattice both included in $$\mathbf {R}^n$$ . In some situations, one may replace the lattice by a more general set for which a notion of density exists. In this paper, we prove a Minkowski theorem for quasicrystals, which bounds from below the frequency of differences appearing in the quasicrystal and belonging to a centrally symmetric convex body. The last part of the paper is devoted to quite natural applications of this theorem to Diophantine approximation and to discretization of linear maps.

We hope to initiate a discussion about various methods for introducing Cauchy’s Theorem. Although Cauchy’s Theorem is the fundamental theorem upon which complex analysis is based, there is no “stan...

The author of the above mentioned paper found that pointwise ergodic theorem for bounded additive processes, cf. Theorem 7.11 needs an additional assumption. Including the assumption of approximability, we show that the ergodic theorem is correct. Since we have not found a proof for Poisson point processes being approximable, we withdraw Corollary 7.12 which stated the pointwise almost-everywhere convergence for those processes. A minor flaw in the proof of the abstract mean ergodic theorem (Theorem 5.7) is fixed as well.

In this paper, we study α ¯ -harmonic mappings with an integer exponent and obtain a representation theorem which determines the relation between α ¯ -harmonic mappings and Euclidean harmonic mappings. As two applications of this representation theorem, we obtain a counterexample of the Rado–Kneser–Choquet theorem for α ¯ -harmonic mappings and show that the Lipschitz continuity of an α ¯ -harmonic mapping with an integer exponent is determined by the Euclidean harmonic mapping with same boundary.

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.

We prove constructively that every uniformly continuous convex function f : X -> R+ has positive infimum, where X is the convex hull of finitely many vectors. Using this result, we prove that a separating hyperplane theorem, the fundamental theorem of asset pricing, and Markov's principle are constructively equivalent. This is the first time that important theorems are classified into Markov's principle within constructive reverse mathematics. (C) 2016 Elsevier B.V. All rights reserved.

Each one of six general existence principles of compactness (the extreme value theorem), completeness (the Newton method or the modified Newton method), topology (Brouwer’s fixed point theorem), homotopy (on contractions of a sphere to its center), variational analysis (Ekeland’s principle), and monotonicity (the Minty–Browder theorem)) is shown to lead to the inverse function theorem, each one giving some novel insight. There are differences in assumptions and algorithmic properties; some of the propositions have been constructed specially for this paper. Simple proofs of the last two principles are included. The proof by compactness is shorter and simpler than the shortest and simplest known proof, that by completion. This gives a very short self-contained proof of the Lagrange multiplier rule, which depends only on optimization methods. The proofs are of independent interest and are intended as well to be useful in the context of the ongoing efforts to obtain new variants of methods that are based on the inverse function theorem, such as comparative statics methods.

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.

This paper is devoted to the proof of the infinite-dimensional Borsuk-Ulam theorem for odd completely continuous multivalued maps with convex images which are defined on level sets of even functions. The results obtained in the paper are new even for single-valued maps. In the final section some applications of the theorem to analysis and differential equations are discussed. Bibliography: 12 titles.

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 present a restricted variable generalization of Warning's Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink's restricted variable generalization of Chevalley's Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warning's Second Theorem implies Chevalley's Theorem, our result implies Schauz-Brink's Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems. Let q = pl be a power of a prime number p, and let Fq be "the" finite field of order q. For a1,...,an, NźZ+, we denote by m(a1,...,an;N)źZ+ a certain combinatorial quantity defined and computed in Section 2.1.

The linear dynamic theory of microstretch thermomagnetoelectroelasticity is studied in this paper. First, a reciprocity relation which involves two processes at different instants is established to form the basis of a uniqueness result and a reciprocal theorem. The proof of the reciprocal theorem avoids both using the Laplace transform and incorporating the initial conditions into the equations of motion. The uniqueness theorem is derived with no definiteness assumption on the elastic constitutive coefficients. Then the continuous dependence theorem is discussed upon two external data systems. Finally, the variational principle of Hamilton type which fully characterizes the solution of the mixed boundary-initial-value problem (mixed problem) is obtained. These theorems lay a solid foundation for further theoretical and numerical studies on microstretch thermomagnetoelectroelastic materials.

For an n -dimensional simplex in R n , a generalization of the Bolzano theorem is given. A proof based on the classical Knaster–Kuratowski–Mazurkiewicz covering lemma is obtained.

In this note we give a proof of the Gauss-Bonnet theorem for Riemannian manifolds (of any dimension) using Morse theory.

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.

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.

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.

The variant of Cauchy’s integral theorem and the proof of Morera’s theorem have been checked. The existing proofs need the condition of analyticity of a given function, and while, the proposed method in this article is employed without the condition of independence of path. Mathematics Subject Classification: 34A25, 28A05

In 2008, Evertse and Ferretti stated a quantitative version of the Subspace Theorem for a projective variety with higher degree polynomials instead of linear forms. Our goal is to generalize their results.

We obtain a new infinite-dimensional linking theorem. This theorem replaces the upper semi-continuous assumption in the classical infinite-dimensional linking theorem of Kryszewski and Szulkin by a new assumption. As an application of this theorem, we obtain a nontrivial solution for a strongly indefinite periodic Schrodinger equation with sign-changing nonlinearity.

International audience; Using Baire's theorem, we give a very simple proof of a special version of the Lusin-Privalov theorem and deduce via Abel's theorem the Riemann-Cantor theorem on the uniqueness of the coefficients of pointwise convergent unilateral trigonometric series.

We prove a local saddle point theorem that can be viewed as a generalization of the saddle point theorem of Rabinowitz. A difficulty to overcome is that there isn’t any linking. We then apply the theorem to show the existence of solutions of a nonlocal partial differential equations.

In this chapter, we complete the proof of Theorem 2.2.4. The computations needed for this are delicate and somehow technical. Many calculations are based on general ideas of Taylor expansions and can be adapted to other types of nonlinearities (though other estimates do take into account the precise growth conditions of the main term of the nonlinearity and of its perturbation). Rather than trying to list abstract conditions on the nonlinearity which would allow similar techniques to work (possibly at the price of more careful Taylor expansions), we remark that the case treated here is somehow classical and motivated from geometry.

The proof of stationary phase method for oscillatory integrals over a space of large dimension is divided into two steps. As the first step we shall prove a theorem of Kumano-go-Taniguchi’s type in this chapter.

We give an expository background to the Bishop–Phelps–Bollobás theorem and explore some progress and questions in recently developed areas of related research.

A unification version of the Perron–Frobenius theorem and the Krein–Rutman theorem for increasing, positively 1-homogeneous, compact mappings is given on ordered Banach spaces without monotonic norm. A Collatz-type minimax characterization of the positive eigenvalue with positive eigenvector is obtained. The power method in computing the largest eigenpair is also extended.

In this paper we establish a complete representation theorem for $G$-martingales. Unlike the existing results in the literature, we provide the existence and uniqueness of the second order term, which corresponds to the second order derivative in Markovian case. The main ingredient of the paper is a new norm for that second order term, which is based on an operator introduced by Song [26]., Comment: 22 pages

In this article, we use the translation theorem to obtain several relationships involving integral transforms and convolution products. In particular, we obtain several useful formulas involving various functionals, which arise naturally in quantum mechanics.

We present a new approach to the celebrated theorem of Rado–Kneser–Choquet (RKC) on univalence of planar harmonic mappings. The novelty lies in establishing a continuous path (isotopy) from the given harmonic map to a conformal one. Along this path the mappings retain positive Jacobian determinant by virtue of so-called Minimum Principle. These ideas extend to nonlinear uncoupled systems of partial differential equations, as in Iwaniec, Koski and Onninen [‘Isotropic p-harmonic systems in 2D, Jacobian estimates and univalent solutions’, Rev. Mat. Iberoam, to appear]. Unfortunately, details of such digression would lead us too far afield. Nonetheless, one gains (in particular) the RKC-Theorem for the isotropic p-harmonic deformations.