A simple proof for the Chernoff-Savage theorem

A proof for the Chernoff-Savage theorem is given based on differentiable statistical functionals. It employs no second order differentiability condition.     

基于分解测度的模糊积分及收敛定理Mamadou Traore

为了解决一些收敛定理,我们给出基于半环([0,1],, )的伪可加分解测度的积分这种模糊积分被深入研究.在给出这种积分的性质的基础上,我们得到一些收敛定理,它们是经典收敛定理的扩张,同时我们得到关于这种模糊测度的Egorof定理.     

A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations

The theorem in question is that the group of automorphisms of a partially ordered set (X,π), π denoting the order relation on the set X, is isomorphic to the maximal subgroup of ℬ_x containing π, where ℬ_x is the semigroup of all binary relations on X. This theorem is due to Montague and Plemmons [1] for the case X finite or countably infinite, and was extended by Schein to the general case, using a theorem due to Zaretsky [4]. A proof of the general case, based on [1] and results due to Plemmons and West [3], is also given in the preceding note by Plemmons and Schein [2]. The purpose of this note is to give an entirely self-contained proof of this intersesting theorem.     

Construction and properties of quasi-linear functionals

Quasi-linear functionals are shown to be uniformly continuous and decomposable into a difference of two quasi-integrals. A predual space for the quasi-linear functionals inducing the weak*-topology is given. General constructions of quasi-linear functionals by solid set-functions and q-functions are given.

     

An eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities

A multiplicity result for an eigenvalue Dirichlet problem involving the p-Laplacian with discontinuous nonlinearities is obtained. The proof is based on a three critical points theorem for nondifferentiable functionals.     

The duals of almost completely decomposable groups

In this paper, we classify the direct products of one-dimensional compact connected abelian groups by cardinal invariants dualizing Baer’s classification theorem of completely decomposable groups. Almost completely decomposable groups are finite rank torsion-free abelian groups which contain a completely decomposable group of finite index. An isomorphism theorem for their Pontrjagin dual groups is given by using the dual concept of a regulating subgroup.     

Some remarks on conditional entropy

Summary Using a definition of conditional entropy given by Hanen and Neveu [5, 10, 11] we discuss in this paper some properties of conditional entropy and mean entropy, in particular an integral representation of conditional entropy (§ 2), and the decomposition theorem of the KolmogorovSina¯i invariant (§ 3) (see also [6–8] and [12]).There is an essential difference between Jacob's proof of the last called theorem [6] and the proof given below. The definition of a Lebesgue space, given by Rokhlin [7–9] and [12], is not used in this paper.     

Linear operators preserving the decomposable numerical radius

The characterization of all linear operators on matrices which preserve the decomposable numerical radius is obtained. This result refines those of Tam. Marcus and Filippenko on the topic. The proof of the main theorem depends on a characterization of scalar multiples of unitary matrices in terms of decomposable numerical radius that is of independent interest.     

Minimum of a decomposable cubic form of three variables

There is given a proof of the theorem, asserting that if for all X∈?³(X≠0) we have ∥F(X)∥≥m>0, where F(X) is a decomposable cubic form of three variables, then F(X) is proportional to an integral form. Making use of this result, the author gives a proof to Littlewood's problem: can one find α, β ∈ ? such that q‖qα‖·‖qβ‖>x>0 for all natural numbers g? From the result of the paper there follows that such (α, β) do not exist.     

Über den Satz von Weil-Cartier

It is shown that the theorem ofWeil-Cartier ([10, Th. 5], [4, Th. 3]) is connected with a homomorphism of groups of unitary operators. The existence proof for this homomorphism is based on simple results in harmonic analysis and on an extension property of the Schwartz-Bruhat functions. Some applications are given, including a result ofIgusa's [6, Th. 3] and the reciprocity formula ofKrazer-Siegel [9, Th. 2]. An outline of the proof has been given in [8].     

On a Class of Functionals Whose Local Minima are Global

In this note we introduce a suitable class of functionals, including the class of integral functionals, and prove that any (strict) local minimum of a functional of this class, defined on a decomposable space, is a (strict) global minimum. So, the recent result obtained by Giner in [1] is specified and extended.     

Time dependent critical fluctuations of a one dimensional local mean field model

Summary One-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.     

Partial symmetry and asymptotic behavior for some elliptic variational problems

A short elementary proof based on polarizations yields a useful (new) rearrangement inequality for symmetrically weighted Dirichlet type functionals. It is then used to answer some symmetry related open questions in the literature. The non symmetry of the Hénon equation ground states (previously proved in [19]) as well as their asymptotic behavior are analyzed more in depth. A special attention is also paid to the minimizers of the Caffarelli-Kohn-Nirenberg [8] inequalities.Received: 5 May 2002, Accepted: 3 September 2002, Published online: 17 December 2002Mathematics Subject Classification (2000): 35B40 - 35J20     

A proof of strongly uniform termination for Gödel's T by methods from local predicativity

We estimate the derivation lengths of functionals in G?del's system of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for . By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a collapsing f unction so that for (closed) terms of G?del's we have: If reduces to then By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in , hence new proof of strongly uniform termination of . A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in , hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for . A new proof of Troelstra's result on strong normalization for . Additionally, a slow growing analysis of G?del's is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed -calculus. Received August 4, 1995     

Non-analytic functional calculi in several variables

Generalizing a result of [1] and [2], we show that every-scalar system (see Def. 3) is decomposable (in the sense of [6], [7]). By means of this fact and by some results on decomposable n-tuples (contained in [6] and [7]) we prove the theorem of support and (in the case of an inverse closed admissible algebra) the spectral mapping theorem for-functional calculi in several variables (see Def. 3). In the case of a single operator we obtain a simplification of the definition of an-spectral function (in the sense of [5], Def. 3.1.3).     

A stability criterion

We come up with an independent proof for a corollary to the main theorem in [1]. This corollary is the degenerate case of the main theorem (with empty sets B ₀ and B ₁) and establishes a stability criterion for a Henselian valued field. Such a proof is given here based on an analysis of tame and purely wild extensions made in [2].     

An existence theorem for generalized variational inequalities with discontinuous and pseudomonotone operators

This paper gives a solution existence theorem for a generalized variational inequality problem with an operator which is defined on an infinite dimensional space, which is C-pseudomonotone in the sense of Inoan and Kolumbán [D. Inoan, J. Kolumbán, On pseudomonotone set-valued mappings, Nonlinear Analysis 68 (2008) 47-53], but which may not be upper semicontinuous on finite dimensional subspaces. The proof of the theorem provides a new technique which reduces infinite variational inequality problems to finite ones. Two examples are given and analyzed to illustrate the theorem. Moreover, an example is presented to show that the C-pseudomonotonicity of the operator cannot be omitted in the theorem.     

Brownian analogues of Burke's theorem

We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke's theorem for the Brownian queue was stated and proved by Harrison (Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the two-dimensional case. In all of the above, reversibility plays a key role.     

Ein globales Ergebnis für Bifurkationsaufgaben mit schwach differenzierbaren Operatoren

In this paper we are dealing with positive linear functionals on W-algebras. We introduce the notion of a positive linear functional with ∑-property (see Definition 1.1). It is shown that each positive linear functional on a W-algebra possesses the ∑-property. This fact allows to give a short proof of UHLMANN's conjecture on unitary mixing in the state space of a W-algebra. In proving our main theorem (see Theorem 1.2.) we obtain some results on positive linear functionals and orthoprojections which are useful in other context, too.     

A Tree Version of Kőnig's Theorem

Kőnig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell [2]. As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed. Received January 27, 2000/Revised November 2, 2000 RID=" " ID=" " The research of the first author was supported by grants from the Israel Science Foundation, the M. & M.L Bank Mathematics Research Fund and the fund for the promotion of research at the Technion.