Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

Let , where is a random symmetric matrix, a random symmetric matrix, and with being independent real random variables. Suppose that , and are independent. It is proved that the empirical spectral distribution of the eigenvalues of random symmetric matrices converges almost surely to a non-random distribution.     

Remarks on Hardy spaces defined by non-smooth approximate identity

We study in this paper some relations between Hardy spaces which are defined by non-smooth approximate identity ?(x), and the end-point Triebel-Lizorkin spaces (1?q?∞). First, we prove that for compact ? which satisfies a slightly weaker condition than Fefferman and Stein's condition. Then we prove that non-trivial Hardy space defined by approximate identity ? must contain Besov space . Thirdly, we construct certain functions and a function such that Daubechies wavelet function but .     

A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property .In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber ℵ₁ and for forcing notions with the property fails. This negatively answers a part of one of the classical problems about implications between fragments of .     

Classical predicative logic-enriched type theories

A logic-enriched type theory (LTT) is a type theory extended with a primitive mechanism for forming and proving propositions. We construct two LTTs, named and , which we claim correspond closely to the classical predicative systems of second order arithmetic and . We justify this claim by translating each second order system into the corresponding LTT, and proving that these translations are conservative. This is part of an ongoing research project to investigate how LTTs may be used to formalise different approaches to the foundations of mathematics.The two LTTs we construct are subsystems of the logic-enriched type theory , which is intended to formalise the classical predicative foundation presented by Herman Weyl in his monograph Das Kontinuum. The system has also been claimed to correspond to Weyl’s foundation. By casting and as LTTs, we are able to compare them with . It is a consequence of the work in this paper that is strictly stronger than .The conservativity proof makes use of a novel technique for proving one LTT conservative over another, involving defining an interpretation of the stronger system out of the expressions of the weaker. This technique should be applicable in a wide variety of different cases outside the present work.     

Higher Kurtz randomness

A real x is -Kurtz random (-Kurtz random) if it is in no closed null set ( set). We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.     

Normal cones to infinite intersections

For sets given as finite intersections the basic normal cone is given as , but such a result is not, in general, available for infinite intersections. A comparable characterization of is obtained here for a class of such infinite intersections.     

Markov interlacing property for exponential polynomials

Let Un be an extended Tchebycheff system on the real line. Given a point , where x₁<?<xn, we denote by the polynomial from Un, which has zeros x₁,…,xn. (It is uniquely determined up to multiplication by a constant.) The system Un has the Markov interlacing property (M) if the assumption that and interlace implies that the zeros of and interlace strictly, unless . We formulate a general condition which ensures the validity of the property (M) for polynomials from Un. We also prove that the condition is satisfied for some known systems, including exponential polynomials and . As a corollary we obtain that property (M) holds true for Müntz polynomials , too.     

Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

For a pseudovariety of ordered semigroups, let be the class of sofic subshifts whose syntactic semigroup lies in . It is proved that if contains then is closed under taking shift equivalent subshifts, and conversely, if is closed under taking conjugate subshifts then contains and . Almost finite type subshifts are characterized as the irreducible elements of , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.     

Efficient algorithms for robust generalized cross-validation spline smoothing

Generalized cross-validation (GCV) is a widely used parameter selection criterion for spline smoothing, but it can give poor results if the sample size n is not sufficiently large. An effective way to overcome this is to use the more stable criterion called robust GCV (RGCV). The main computational effort for the evaluation of the GCV score is the trace of the smoothing matrix, , while the RGCV score requires both and . Since 1985, there has been an efficient O(n) algorithm to compute . This paper develops two pairs of new O(n) algorithms to compute and , which allow the RGCV score to be calculated efficiently. The algorithms involve the differentiation of certain matrix functionals using banded Cholesky decomposition.     

On the numerical radius of operators in Lebesgue spaces

We show that the absolute numerical index of the space Lp(μ) is (where ). In other words, we prove that

     

On Γ-convergence of pairs of dual functionals

The paper considers a slightly modified notion of the Γ-convergence of convex functionals in uniformly convex Banach spaces and establishes that under standard coercitivity and growth conditions the Γ-convergence of a sequence of functionals {Fj} to implies that the corresponding sequence of dual functionals converges in an analogous sense to the dual to functional .     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .     

On the Wiener integral with respect to a sub-fractional Brownian motion on an interval

The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , k≠0, is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions.     

Kirillov-Reshetikhin crystals for nonexceptional types

We provide combinatorial models for all Kirillov-Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types , , we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type we use a Dynkin diagram folding and for types , a similarity construction. We also show that for types and the analog of the Dynkin diagram automorphism exists on the level of crystals.     

Nonstandard arithmetic and recursive comprehension

First order reasoning about hyperintegers can prove things about sets of integers. In the author’s paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 (2006) 100-125, it was shown that each of the “big five” theories in reverse mathematics, including the base theory , has a natural nonstandard counterpart. But the counterpart of has a defect: it does not imply the Standard Part Principle that a set exists if and only if it is coded by a hyperinteger. In this paper we find another nonstandard counterpart, , that does imply the Standard Part Principle.     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .