A comparison result for elliptic equations in the infinite dimensional Gauss space

We obtain a comparison result for a class of Dirichlet problems for the operator in an infinite dimensional separable Hilbert space X with the Gauss measure γ and a suitable differentiable structure.     

Euler–Lehmer constants and a conjecture of Erdös

The Euler–Lehmer constants γ(a,q) are defined as the limits We show that at most one number in the infinite list is an algebraic number. The methods used to prove this theorem can also be applied to study the following question of Erdös. If f:Z/qZ→Q is such that f(a)=±1 and f(q)=0, then Erdös conjectured that If , we show that the Erdös conjecture is true.     

Modulation spaces, Wiener amalgam spaces, and Brownian motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces and Wiener amalgam spaces . We show that the periodic Brownian motion belongs locally in time to and for (s−1)q<−1, and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T, we show that and form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space . Specifically, we prove that the Brownian motion belongs to for (s−1)p=−1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces , and indicate the endpoint large deviation estimates.     

CERES in higher-order logic

We define a generalization of the first-order cut-elimination method CERES to higher-order logic. At the core of lies the computation of an (unsatisfiable) set of sequents (the characteristic sequent set) from a proof π of a sequent S. A refutation of in a higher-order resolution calculus can be used to transform cut-free parts of π (the proof projections) into a cut-free proof of S. An example illustrates the method and shows that can produce meaningful cut-free proofs in mathematics that traditional cut-elimination methods cannot reach.     

On the layered nearest neighbour estimate, the bagged nearest neighbour estimate and the random forest method in regression and classification

Let be identically distributed random vectors in R^d, independently drawn according to some probability density. An observation is said to be a layered nearest neighbour (LNN) of a point if the hyperrectangle defined by and contains no other data points. We first establish consistency results on , the number of LNN of . Then, given a sample of independent identically distributed random vectors from R^d×R, one may estimate the regression function by the LNN estimate , defined as an average over the Y_i’s corresponding to those which are LNN of . Under mild conditions on r, we establish the consistency of towards 0 as n→∞, for almost all and all p≥1, and discuss the links between r_n and the random forest estimates of Breiman (2001) [8]. We finally show the universal consistency of the bagged (bootstrap-aggregated) nearest neighbour method for regression and classification.     

Bogomolov restriction theorem for Higgs bundles

Let (E,θ) be a stable Higgs bundle of rank r on a smooth complex projective surface X equipped with a polarization H. Let C⊂X be a smooth complete curve with [C]=n⋅H. If where , then we prove that the restriction of (E,θ) to C is a stable Higgs bundle. This is a Higgs bundle analog of Bogomolov's restriction theorem for stable vector bundles.     

Independence results for variants of sharply bounded induction

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic (with ⌊x/2⌋ but not ⌊x/2^y⌋), has recently been unconditionally separated from full bounded arithmetic S₂. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer many results and methods from open to sharply bounded induction with relative ease.We provide two applications: (i) the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; (ii) extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes.     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.     

On the irrationality of factorial series III

In this paper we derive some irrationality and linear independence results for series of the form where is either a non-negative integer sequence with υ_n = o(log n/log log n) or a non-decreasing integer sequence with .     

Smoothness of the moduli space of complexes of coherent sheaves on an abelian or a projective K3 surface

For an abelian or a projective K3 surface X over an algebraically closed field k, consider the moduli space of the objects E in Db(Coh(X)) satisfying and Hom(E,E)≅k. Then we can prove that is smooth and has a symplectic structure.     

A Morse complex for infinite dimensional manifolds—part I

In this paper and in the forthcoming Part II, we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M, possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle—or more generally an essential subbundle—of the tangent bundle of M, suitably related with the gradient flow of f. This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y: finite dimensionality of W, possibility that different components of W have different dimension, orientability of W and coherence in the choice of an orientation, compactness of the closure of W, classification, up to topological conjugacy, of the gradient flow on the closure of W, in the case .     

An infinite dimensional Schur-Horn Theorem and majorization theory

The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences ξ and η that converge to 0, there exists a positive compact operator A with eigenvalue list η and diagonal sequence ξ if and only if for every n if and only if ξ=Qη for some orthostochastic matrix Q. When ξ and η are summable, requiring additionally equality of their infinite series obtains the same conclusion, extending a theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices.     

Malmheden's theorem revisited

In 1934 Malmheden [16] discovered an elegant geometric algorithm for solving the Dirichlet problem in a ball. Although his result was rediscovered independently by Duffin (1957) [8] 23 years later, it still does not seem to be widely known. In this paper we return to Malmheden's theorem, give an alternative proof of the result that allows generalization to polyharmonic functions and, also, discuss applications of his theorem to geometric properties of harmonic measures in balls in .     

Approximation of the finite dimensional distributions of multiple fractional integrals

We construct a family Inεt(f) of continuous stochastic processes that converges in the sense of finite dimensional distributions to a multiple Wiener-Itô integral with respect to the fractional Brownian motion. We assume that and we prove our approximation result for the integrands f in a rather general class.     

C-Weierstrass for compact sets in Hilbert space

The C¹-Weierstrass approximation theorem is proved for any compact subset X of a Hilbert space . The same theorem is also proved for Whitney 1-jets on X when X satisfies the following further condition: There exist finite dimensional linear subspaces such that ?_n?1H_n is dense in and π_n(X)=X∩H_n for each n?1. Here, is the orthogonal projection. It is also shown that when X is compact convex with and satisfies the above condition, then C¹(X) is complete if and only if the C¹-Whitney extension theorem holds for X. Finally, for compact subsets of , an extension of the C¹-Weierstrass approximation theorem is proved for C¹ maps with compact derivatives.     

A higher rank version of Kitai's theorem

We prove that each linear action of on an infinite-dimensional Banach space generated by compact operators cannot be hypercyclic. This result generalizes a theorem of Kitai for the case of Z₊ actions. Contrary to the case of infinite dimension, a hypercyclic action of on C is given.     

High dimensional random sections of isotropic convex bodies

We study two properties of random high dimensional sections of convex bodies. In the first part of the paper we estimate the central section function for random F∈Gn,k and K⊂Rn a centrally symmetric isotropic convex body. This partially answers a question raised by V.D. Milman and A. Pajor (see [V.D. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Lecture Notes in Math., vol. 1376, Springer, 1989, p. 88]). In the second part we show that every symmetric convex body has random high dimensional sections F∈Gn,k with outer volume ratio bounded by

     

Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Let be a sequence of d-dimensional stationary Gaussian vectors, and let denote the partial maxima of . Suppose that there are missing data in each component of and let denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector where the correlation and cross-correlation satisfy some dependence conditions.     

Extracting information is hard: A Turing degree of non-integral effective Hausdorff dimension

We construct a infinite binary sequence with effective Hausdorff dimension 1/2 that does not compute a sequence of higher dimension. Introduced by Lutz, effective Hausdorff dimension can be viewed as a measure of the information density of a sequence. In particular, the dimension of A∈ω² is the lim inf of the ratio between the information content and length of initial segments of A. Thus the main result demonstrates that it is not always possible to extract information from a partially random source to produce a sequence that has higher information density.