A Cl closing lemma for equivariant endomorphisms under actions of finite groups

A C^l closing lemma is proved for equivariant endomorphisms under actions of finite groups. Our result shows that, for such an endomorphism, a nonwandering orbit together with its symmetric conjugates can be closed up under a Cl-small equivariant perturbation Project partially supported by the National Natural Science Foundation of China (Grant No. 19531070).     

The Schwarz-Pick lemma for planar harmonic mappings

The classical Schwarz-Pick lemma for holomorphic mappings is generalized to planar harmonic mappings of the unit disk D completely. (I) For any 0 < r < 1 and 0 ρ < 1, the author constructs a closed convex domain Er,ρ such that F((z,r))  eiαEr,ρ = {eiαz : z ∈ Er,ρ} holds for every z ∈ D, w = ρeiα and harmonic mapping F with F(D)D and F(z) = w, where △(z,r) is the pseudo-disk of center z and pseudo-radius r; conversely, for every z ∈ D, w = ρeiα and w ∈ eiαEr,ρ, there exists a harmonic mapping F such that ...     

The averaging lemma

Averaging lemmas deduce smoothness of velocity averages, such as

from properties of . A canonical example is that is in the Sobolev space whenever and are in . The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove versions of the averaging lemma. For example, it is shown that implies that is in the Besov space , . Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint .

     

Johnson‐Lindenstrauss lemma for circulant matrices**

We prove a variant of a Johnson‐Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimize the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(ε^?2 log³ n) instead of the classical bound k = O(ε^?2 log n). © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

Wiener's lemma for infinite matrices

The classical Wiener lemma and its various generalizations are important and have numerous applications in numerical analysis, wavelet theory, frame theory, and sampling theory. There are many different equivalent formulations for the classical Wiener lemma, with an equivalent formulation suitable for our generalization involving commutative algebra of infinite matrices . In the study of spline approximation, (diffusion) wavelets and affine frames, Gabor frames on non-uniform grid, and non-uniform sampling and reconstruction, the associated algebras of infinite matrices are extremely non-commutative, but we expect those non-commutative algebras to have a similar property to Wiener's lemma for the commutative algebra . In this paper, we consider two non-commutative algebras of infinite matrices, the Schur class and the Sjöstrand class, and establish Wiener's lemmas for those matrix algebras.

     

An algorithmic hypergraph regularity lemma

Szemerédi 's Regularity Lemma is a powerful tool in graph theory. It asserts that all large graphs admit bounded partitions of their edge sets, most classes of which consist of uniformly distributed edges. The original proof of this result was nonconstructive, and a constructive proof was later given by Alon, Duke, Lefmann, Rödl, and Yuster. Szemerédi's Regularity Lemma was extended to hypergraphs by various authors. Frankl and Rödl gave one such extension in the case of 3‐uniform hypergraphs, which was later extended to k‐uniform hypergraphs by Rödl and Skokan. W.T. Gowers gave another such extension, using a different concept of regularity than that of Frankl, Rödl, and Skokan. Here, we give a constructive proof of a regularity lemma for hypergraphs.     

New light on Hensel''s lemma

The historical development of Hensel's lemma is briefly discussed (Section 1). Using Newton polygons, a simple proof of a general Hensel's lemma for separable polynomials over Henselian fields is given (Section 3). For polynomials over algebraically closed, valued fields, best possible results on continuity of roots (Section 4) and continuity of factors (Section 6) are demonstrated. Using this and a general Krasner's lemma (Section 7), we give a short proof of a general Hensel's lemma and show that it is, in a certain sense, best possible (Section 8). All valuations here are non-Archimedean and of arbitrary rank. The article is practically self-contained.     

The closing lemma for nonsingular endomorphisms equivariant under free actions of finite groups

In this paper a closing lemma for nonsingular endomorphisms equivariant under free actions of finite-groups is proved. Hence a recurrent trajectory, as well as all of its symmetric conjugates, of a nonsingular endomorphism equivariant under a free action of a finite group can be closed up simultaneously by an arbitrarily small equivariant perturbation.

     

Perturbations of Hemivariational Inequalities with Constraints and Applications

The aim of the present paper is to discuss the influence which certain perturbations have on the solution of the eigenvalue problem for hemivariational inequalities on a sphere of given radius. The perturbation results in adding a term of the type >g ⁰(x, u(x); v(x)) to the hemivariational inequality, where g is a locally Lipschitz nonsmooth and nonconvex energy functional. Applications illustrate the theory.     

A variant of the hypergraph removal lemma

Recent work of Gowers [T. Gowers, A new proof of Szemerédi's theorem, Geom. Funct. Anal. 11 (2001) 465-588] and Nagle, Rödl, Schacht, and Skokan [B. Nagle, V. Rödl, M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, in press; V. Rödl, J. Skokan, Applications of the regularity lemma for uniform hypergraphs, preprint] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975) 299-345], and Furstenberg and Katznelson [H. Furstenberg, Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Anal. Math. 34 (1978) 275-291] concerning one-dimensional and multidimensional arithmetic progressions, respectively. In this paper we shall give a self-contained proof of this hypergraph removal lemma. In fact we prove a slight strengthening of the result, which we will use in a subsequent paper [T. Tao, The Gaussian primes contain arbitrarily shaped constellations, preprint] to establish (among other things) infinitely many constellations of a prescribed shape in the Gaussian primes.     

Morse-Palais lemma for nonsmooth functionals on normed spaces

Using elementary differential calculus we get a version of the Morse-Palais lemma. Since we do not use powerful tools in functional analysis such as the implicit theorem or flows and deformations in Banach spaces, our result does not require the -smoothness of functions nor the completeness of spaces. Therefore it is stronger than the classical one but its proof is very simple.

     

The Dedekind-Mertens lemma and the contents of polynomials

Let be a commutative ring, let be an indeterminate, and let . There has been much recent work concerned with determining the Dedekind-Mertens number =min , especially on determining when = . In this note we introduce a universal Dedekind-Mertens number , which takes into account the fact that deg() + for any ring containing as a subring, and show that behaves more predictably than .

     

Minimal surfaces and Schwarz lemma

We prove a sharp Schwarz lemma type inequality for the Weierstrass–Enneper parameterization of minimal disks. It states the following. If $F:\mathbf{D}\to \Sigma$ is a conformal harmonic parameterization of a minimal disk $\Sigma \subset {\mathbf{R}}^3$, where $\mathbf{D}$ is the unit disk and $\left|\Sigma \right|=\pi R^2$, then $\left|F_x\left(z\right)\right|(1-{\left|z\right|}^2)\le R$. If for some $z$ the previous inequality is equality, then the surface is an affine image of a disk, and $F$ is linear up to a Möbius transformation of the unit disk.     

A short proof of the wonderful lemma

The Wonderful Lemma, that was first proved by Roussel and Rubio, is one of the most important tools in the proof of the Strong Perfect Graph Theorem. Here we give a short proof of this lemma.     

The twisted conjugacy problem for endomorphisms of metabelian groups

Let M be a finitely generated metabelian group explicitly presented in a variety of all metabelian groups. An algorithm is constructed which, for every endomorphism φ ∈ End(M) identical modulo an Abelian normal subgroup N containing the derived subgroup M′ and for any pair of elements u, v ∈ M, decides if an equation of the form (xφ)u = vx has a solution in M. Thus, it is shown that the title problem under the assumptions made is algorithmically decidable. Moreover, the twisted conjugacy problem in any polycyclic metabelian group M is decidable for an arbitrary endomorphism φ ∈ End(M). Supported by RFBR (project No. 07-01-00392). (V. A. Roman’kov) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 157–173, March–April, 2009.     

A general Farkas lemma

An abstract version of the classical Farkas lemma for locally convex spaces is given. The abstract Farkas lemma is shown to imply Farkas-type results which have been obtained by Shimizu-Aiyoshi-Katayama, Schecter, Eisenberg, Zalinescu, and Smiley.     

A Gronwall-type lemma with parameter and dissipative estimates for PDEs

We discuss the problem of establishing dissipative estimates for certain differential equations for which the usual methods apparently do not work. The main tool is a new Gronwall-type lemma with parameter. As an application, we consider a semilinear equation of viscoelasticity with low dissipation.