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.     

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 .     

Games with 1-backtracking

We associate with any game G another game, which is a variant of it, and which we call . Winning strategies for have a lower recursive degree than winning strategies for G: if a player has a winning strategy of recursive degree 1 over G, then it has a recursive winning strategy over , and vice versa. Through we can express in algorithmic form, as a recursive winning strategy, many (but not all) common proofs of non-constructive Mathematics, namely exactly the theorems of the sub-classical logic Limit Computable Mathematics (Hayashi (2006) [6], Hayashi and Nakata (2001) [7]).     

Cut elimination and strong separation for substructural logics: An algebraic approach

We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on (associative) substructural logics over the full Lambek Calculus (see, for example, Ono (2003) [34], Galatos and Ono (2006) [18], Galatos et al. (2007) [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic associated with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics.Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit.     

Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L^′ be R-modules. We investigate the properties of the functors and . For instance, we show the following:

(a)

if L and L^′ are artinian, then is artinian, and is noetherian over the completion ;

(b)

if L is artinian and L^′ is Matlis reflexive, then , , and are Matlis reflexive.

Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.     

The Stein phenomenon for monotone incomplete multivariate normal data

We establish the Stein phenomenon in the context of two-step, monotone incomplete data drawn from , a (p+q)-dimensional multivariate normal population with mean and covariance matrix . On the basis of data consisting of n observations on all p+q characteristics and an additional N−n observations on the last q characteristics, where all observations are mutually independent, denote by the maximum likelihood estimator of . We establish criteria which imply that shrinkage estimators of James-Stein type have lower risk than under Euclidean quadratic loss. Further, we show that the corresponding positive-part estimators have lower risk than their unrestricted counterparts, thereby rendering the latter estimators inadmissible. We derive results for the case in which is block-diagonal, the loss function is quadratic and non-spherical, and the shrinkage estimator is constructed by means of a nondecreasing, differentiable function of a quadratic form in . For the problem of shrinking to a vector whose components have a common value constructed from the data, we derive improved shrinkage estimators and again determine conditions under which the positive-part analogs have lower risk than their unrestricted counterparts.     

Finite-sample inference with monotone incomplete multivariate normal data, II

We continue our recent work on inference with two-step, monotone incomplete data from a multivariate normal population with mean and covariance matrix . Under the assumption that is block-diagonal when partitioned according to the two-step pattern, we derive the distributions of the diagonal blocks of and of the estimated regression matrix, . We represent in terms of independent matrices; derive its exact distribution, thereby generalizing the Wishart distribution to the setting of monotone incomplete data; and obtain saddlepoint approximations for the distributions of and its partial Iwasawa coordinates. We prove the unbiasedness of a modified likelihood ratio criterion for testing , where is a given matrix, and obtain the null and non-null distributions of the test statistic. In testing , where and are given, we prove that the likelihood ratio criterion is unbiased and obtain its null and non-null distributions. For the sphericity test, , we obtain the null distribution of the likelihood ratio criterion. In testing we show that a modified locally most powerful invariant statistic has the same distribution as a Bartlett-Pillai-Nanda trace statistic in multivariate analysis of variance.     

Polar syzygies in characteristic zero: The monomial case

Given a set of forms , where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module , whose elements are called differential syzygies of . There is a distinct submodule P⊂Z coming from the polynomial relations of through its transposed Jacobian matrix, the elements of which are called polar syzygies of . We say that is polarizable if equality P=Z holds. This paper is concerned with the situation where are monomials of degree 2, in which case one can naturally associate to them a graph with loops and translate the problem into a combinatorial one. The main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra and that the converse holds provided the graph is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of has diameter at most 2 then is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.     

Critical points between varieties generated by subspace lattices of vector spaces

We denote by the semilattice of all compact congruences of an algebra A. Given a variety V of algebras, we denote by the class of all semilattices isomorphic to for some A∈V. Given varieties V and W of algebras, the critical point of V under W is defined as . Given a finitely generated variety V of modular lattices, we obtain an integer ?, depending on V, such that for any n≥? and any field F.In a second part, using tools introduced in Gillibert (2009) [5], we prove that:

     

Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator

Let γ be the Gauss measure on and the Ornstein-Uhlenbeck operator. For every p in [1,∞)?{2}, set , and consider the sector . The main results of this paper are the following. If p is in (1,∞)?{2}, and , i.e., if M is an Lp(γ)uniform spectral multiplier of in our terminology, and M is continuous on , then M extends to a bounded holomorphic function on the sector . Furthermore, if p=1 a spectral multiplier M, continuous on , satisfies the condition if and only if M extends to a bounded holomorphic function on the right half-plane, and its boundary value M(i·) on the imaginary axis is the Euclidean Fourier transform of a finite Borel measure on the real line. We prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on belonging to a wide class, which contains . From these results we deduce that operators in this class do not admit an H^∞ functional calculus in sectors smaller than .     

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.     

Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H(Γ) and H₊(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of Γ×Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of RP³. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on for finite graphs Γ.     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Stability to the global large solutions of 3-D Navier-Stokes equations

In this paper, we consider the stability to the global large solutions of 3-D incompressible Navier-Stokes equations in the anisotropic Sobolev spaces. In particular, we proved that for any , given a global large solution v∈C([0,∞);H0,s₀(R³)∩L³(R³)) of (1.1) with and a divergence free vector satisfying for some sufficiently small constant depending on , v, and , (1.1) supplemented with initial data v(0)+w₀ has a unique global solution in u∈C([0,∞);H0,s₀(R³)) with ∇u∈L²(R⁺,H0,s₀(R³)). Furthermore, uh is close enough to vh in C([0,∞);H0,s(R³)).     

New applications of Besov-type and Triebel-Lizorkin-type spaces

In this paper, the authors prove that Besov-Morrey spaces are proper subspaces of Besov-type spaces and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces . The authors also establish an equivalent characterization of when τ∈[0,1/p). These Besov-type spaces and Triebel-Lizorkin-type spaces were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces and , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos-Torres by taking τ=0.