The singular linear preservers of non-singular matrices

Given an arbitrary field K, we reduce the determination of the singular endomorphisms f of M_n(K) such that f(GL_n(K))⊂GL_n(K) to the classification of n-dimensional division algebras over K. Our method, which is based upon Dieudonné’s theorem on singular subspaces of M_n(K), also yields a proof for the classical non-singular case.     

Sampling Theorem for Entire Functions of Exponential Growth

Applying the theory of generalized functions we obtain the Shannon sampling theorem for entire functions F(z) of exponential growth and give its error estimate which shows how much the error depends on the sampling size and bandwidth for given domain of the signal F(z). As an application we obtain a uniqueness theorem for entire functions and temperature functions.     

A Liouville theorem for conformal Gaussian curvature type equations in {{\mathbb{R}}^2}

In this paper, we obtain a Liouville type theorem for a class of elliptic equations including the conformal Gaussian curvature equation $$-\Delta u=K(x)e^{2u}\quad {\rm in}\,\, {\mathbb{R}}^2,$$ where K(x) is a H?lder continuous function in ${{\mathbb{R}}^2}$ that does not have a fixed sign near infinity. The main tool in our approach is an asymptotic formula for the solution at infinity and the method of moving planes. We also show how our Liouville theorem can be used to obtain a priori bound for solutions of the prescribing Gaussian curvature equation in S ², namely $$\Delta\, u+K(x)e^{2u}=1\, {\rm in}\, S^2,$$ where K(x) is H?lder continuous and nonnegative in S ² but vanishes on a set with nonempty interior, a case left open in previous research.     

Un raffinement du caractère hilbertien du corps K(X)

Given an infinite field K. We will show that for the elements of a hilbertian set of K(X) associated to a family of polynomials with coefficients in K, the specialized polynomials remain also primitive knowing that the starting polynomials are it. This result refines the hilbertian character of the field K(X). Moreover, by an application of this result, one obtains an analog of the famous arithmetic progression theorem.     

On the controlled separable projection property for some C(K) spaces

We give necessary and sufficient conditions for a compact K in order to guarantee that C(K) enjoys the controlled separable projection property (CSPP). As a consequence, we obtain an example of a scattered compact K such that C(K) is not injected into a Hilbert space, although C(K)* is, and nevertheless C(K) does not have the CSPP.     

Weak topology and Browder-Kirk's theorem on hyperspace

Let K be a weakly compact, convex subset of a Banach space X with normal structure. Browder-Kirk's theorem states that every non-expansive mapping T which maps K into K has a fixed point in K. Suppose now that WCC(X) is the collection of all non-empty weakly compact convex subsets of X. We shall define a certain weak topology Tw on WCC(X) and have the above-mentioned result extended to the hyperspace (WCC(X);Tw).     

A Newton-like method and its application

In this paper we prove an existence and uniqueness theorem for solving the operator equation F(x)+G(x)=0, where F is a Gateaux differentiable continuous operator while the operator G satisfies a Lipschitz-condition on an open convex subset of a Banach space. As corollaries, a theorem of Tapia on a weak Newton's method and the classical convergence theorem for modified Newton-iterates are deduced. An existence theorem for a generalized Euler-Lagrange equation in the setting of Sobolev space is obtained as a consequence of the main theorem. We also obtain a class of Gateaux differentiable operators which are nowhere Frechet differentiable. Illustrative examples are also provided.     

The Bishop-Phelps-Bollobás theorem for operators

We prove the Bishop-Phelps-Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop-Phelps-Bollobás theorem holds for operators from ?₁ into Y. Several examples of classes of such spaces are provided. For instance, the Bishop-Phelps-Bollobás theorem holds when the range space is finite-dimensional, an L₁(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.     

The problem of determining the one-dimensional kernel of the electroviscoelasticity equation

We consider the problem of finding the kernel K(t), for t ∈ [0, T], in the integrodifferential system of electroviscoelasticity. We assume that the coefficients depend only on one spatial variable. Replacing the inverse problem with an equivalent system of integral equations, we apply the contraction mapping principle in the space of continuous functions with weighted norms. We prove a global unique solvability theorem and obtain a stability estimate for the solution to the inverse problem.     

C0 coarse geometry and scalar curvature

In this paper we introduce an alternative form of coarse geometry on proper metric spaces, which is more delicate at infinity than the standard metric coarse structure. There is an assembly map from the K-homology of a space to the K-theory of the C∗-algebra associated to the new coarse structure, which factors through the coarse K-homology of the space (with the new coarse structure). A Dirac-type operator on a complete Riemannian manifold M gives rise to a class in K-homology, and its image under assembly gives a higher index in the K-theory group. The main result of this paper is a vanishing theorem for the index of the Dirac operator on an open spin manifold for which the scalar curvature κ(x) tends to infinity as x tends to infinity. This is derived from a spectral vanishing theorem for any Dirac-type operator with discrete spectrum and finite dimensional eigenspaces.     

Stability of multi-compacton solutions and Backlund transformation in K(m,n,1)

We introduce a fifth-order K(m,n,1) equation with nonlinear dispersion to obtain multi-compacton solutions by Adomian decomposition method. Using the homogeneous balance (HB) method, we derive a Backlund transformation of a special equation K(2,2,1) to determine some solitary solutions of the equation. To study the stability of multi-compacton solutions in K(m,n,1) and to obtain some conservation laws, we present a similar fifth-order equation derived from Lagrangian. We finally show the linear stability of all obtained multi-compacton solutions.     

On implicit function theorems at abnormal points

We consider the equation F(x, σ) = 0, x ∈ K, in which σ is a parameter and x is an unknown variable taking values in a specified convex cone K lying in a Banach space X. This equation is investigated in a neighborhood of a given solution (x*, σ*), where Robinson’s constraint qualification may be violated. We introduce the 2-regularity condition, which is considerably weaker than Robinson’s constraint qualification; assuming that it is satisfied, we obtain an implicit function theorem for this equation. The theorem is a generalization of the known implicit function theorems even in the case when the cone K coincides with the whole space X.     

On support points and continuous extensions

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by ${\mathcal{BCC}(X)}A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by BCC(X){\mathcal{BCC}(X)} the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S : BCC(X) ? X{S : \mathcal{BCC}(X) \rightarrow X} such that S(K) is a support point of K for each K ? BCC(X){K \in \mathcal{BCC}(X)}. Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X){\mathcal{BCC}(X)}.     

Little Grothendieck's theorem for sublinear operators

Let SB(X,Y) be the set of the bounded sublinear operators from a Banach space X into a Banach lattice Y. Consider π₂(X,Y) the set of 2-summing sublinear operators. We study in this paper a variation of Grothendieck's theorem in the sublinear operators case. We prove under some conditions that every operator in SB(C(K),H) is in π₂(C(K),H) for any compact K and any Hilbert H. In the noncommutative case the problem is still open.     

On the enumeration of finite Abelian and solvable groups

Let a(n) be the number of nonisomorphic abelian groups of order n. We obtain a short interval result for the local density of a(n). More generally, we get short interval version of results of Ivi? on the local density of prime independent multiplicative functions. Also we prove a short interval version of the theorem of Erdös and Szekeres on the summatory function of a(n) and the theorem of Greenberg and Newman on the enumeration of a certain type of finite solvable groups.     

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K (K(z, z′) = ?K(z′, z)) and i.i.d. random variates Z_n, n?1, such that EK²(Z₁, Z₂)<∞, set A_n = ∑₁?i?j?nK(Z_i,Z_j), n?1. If the Z_n's are two-dimensional and K is the determinant function, A_n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A_n's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.     

Properties of random walks on discrete groups: Time regularity and off-diagonal estimates

In this paper we study some properties of the convolution powers K(n)=K∗K∗?∗K of a probability density K on a discrete group G, where K is not assumed to be symmetric. If K is centered, we show that the Markov operator T associated with K is analytic in Lp(G) for 1<p<∞, and prove Davies-Gaffney estimates in L² for the iterated operators Tn. This enables us to obtain Gaussian upper bounds for the convolution powers K(n). In case the group G is amenable, we discover that the analyticity and Davies-Gaffney estimates hold if and only if K is centered. We also estimate time and space differences, and use these to obtain a new proof of the Gaussian estimates with precise time decay in case G has polynomial volume growth.     

Cyclic isogenies and nonstandard arithmetic

For a prime N we denote by X₀(N)(K) the set of K-rational points on the modul curve of elliptic curves with isogenies of degree N. We formulate arithmetical axioms for number fields K that imply finiteness properties of X₀(N)(K). To prove the results we use the nonstandard version of the Siegel-Mahler theorem (A. Robinson and P. Roquette, J. Number Theory7 (1975), 121–176) and the nonstandard interpretation of a sum formula derived from the local heights on elliptic curves.     

A logarithm type mean value theorem of the Riemann zeta function

For any integer K?2 and positive integer h, we investigate the mean value of |ζ(σ+it)|^2k×logh|ζ(σ+it)| for all real number 0<k<K and all σ>1−1/K. In case K=2, h=1, this has been studied by Wang in [F.T. Wang, A mean value theorem of the Riemann zeta function, Quart. J. Math. Oxford Ser. 18 (1947) 1-3]. In this note, we give a new brief proof of Wang's theorem, and, with this method, generalize it to the general case naturally.     

Vanishing of some cohomology groups and bounds for the Shafarevich-Tate groups of elliptic curves

Let E be an elliptic curve over Q and ? be an odd prime. Also, let K be a number field and assume that E has a semi-stable reduction at ?. Under certain assumptions, we prove the vanishing of the Galois cohomology group H¹(Gal(K(E[?ⁱ])/K),E[?ⁱ]) for all i?1. When K is an imaginary quadratic field with the usual Heegner assumption, this vanishing theorem enables us to extend a result of Kolyvagin, which finds a bound for the order of the ?-primary part of Shafarevich-Tate groups of E over K. This bound is consistent with the prediction of Birch and Swinnerton-Dyer conjecture.