Comparison of the Bergman and Szegö kernels

The quotient of the Szegö and Bergman kernels for a smooth bounded pseudoconvex domains in Cn is bounded from above by a constant multiple of for any p>n, where δ is the distance to the boundary. For a class of domains that includes those of D?Angelo finite type and those with plurisubharmonic defining functions, the quotient is also bounded from below by a constant multiple of for any p<−1. Moreover, for convex domains, the quotient is bounded from above and below by constant multiples of δ.     

Existence for nonlinear evolution inclusions with nonlocal retarded initial conditions

In this paper, we prove a sufficient condition for the global existence of bounded C⁰-solutions for a class of nonlinear functional differential evolution equation of the form where X is a real Banach space, A is the infinitesimal generator of a nonlinear compact semigroup, is a nonempty, convex, weakly compact valued, and almost strongly–weakly u.s.c. multi-function, and is nonexpansive.     

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.     

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces and Triebel–Lizorkin spaces for all s∈(0,1) and p,q∈(n/(n+s),∞], both in Rn and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve on Rn for all s∈(0,1) and q∈(n/(n+s),∞]. A metric measure space version of the above morphism property is also established.     

Radially increasing minimizing surfaces or deformations under pointwise constraints on positions and gradients

In this paper, we prove existence of radially symmetric minimizersu_A(x)=U_A(|x|), having U_A(⋅)AC monotone and increasing, for the convex scalar multiple integral(∗ ) among those u(⋅) in the Sobolev space. Here, |∇u(x)| is the Euclidean norm of the gradient vector and B_R is the ball ; while A is the boundary data.Besides being e.g. superlinear (but no growth needed if (∗) is known to have minimum), our Lagrangian?^∗∗:R×R→[0,∞] is just convex lsc and and ?^∗∗(s,⋅) is even; while ρ₁(⋅) and ρ₂(⋅) are Borel bounded away from .Remarkably, (∗) may also be seen as the calculus of variations reformulation of a distributed-parameter scalar optimal control problem. Indeed, state and gradient pointwise constraints are, in a sense, built-in, since ?^∗∗(s,v)=∞ is freely allowed.     

Unconditional constants and polynomial inequalities

If P is a polynomial on R^m of degree at most n, given by , and P_n(R^m) is the space of such polynomials, then we define the polynomial |P| by . Now if is a convex set, we define the norm on P_n(R^m), and then we investigate the inequality providing sharp estimates on for some specific spaces of polynomials. These ’s happen to be the unconditional constants of the canonical bases of the considered spaces.     

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.     

Reduced class groups grafting relative invariants

Let (X,T) be a regular stable conical action of an algebraic torus on an affine normal conical variety X defined over an algebraically closed field of characteristic zero. We define a certain subgroup of Cl(X//T) and characterize its finiteness in terms of a finite T-equivariant Galois descent of X. Consequently we show that the action (X,T) is equidimensional if and only if there exists a T-equivariant finite Galois covering such that is cofree. Moreover the order of is controlled by a certain subgroup of Cl(X). The present result extends thoroughly the equivalence of equidimensionality and cofreeness of (X,T) for a factorial X. The purpose of this paper is to evaluate orders of divisor classes associated to modules of relative invariants for a Krull domain with a group action. This is useful in studying on equidimensional torus actions as above. The generalization of R.P. Stanley?s criterion for freeness of modules of relative invariants plays an important role in showing key assertions.     

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.     

Isochronicity conditions for some planar polynomial systems

We study the isochronicity of centers at O∈R² for systems , , where A,B∈R[x,y], which can be reduced to the Liénard type equation. Using the so-called C-algorithm we have found 27 new multiparameter isochronous centers.     

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.     

Sub-criticality of non-local Schrödinger systems with antisymmetric potentials and applications to half-harmonic maps

We consider non-local linear Schrödinger-type critical systems of the type(1) where Ω is antisymmetric potential in L²(R,so(m)), v is an Rm valued map and Ωv denotes the matrix multiplication. We show that every solution v∈L²(R,Rm) of (1) is in fact in , for every 2?p<+∞, in other words, we prove that the system (1) which is a-priori only critical in L² happens to have a subcritical behavior for antisymmetric potentials. As an application we obtain the regularity of weak 1/2-harmonic maps into C² compact sub-manifolds without boundary.     

Lower classes of the Riemann–Liouville process

Let be a Riemann–Liouville process with index H>0. We characterize the lower classes of its sup-norm statistic by a unique integral test and thus measure the influence of the non-stationarity of increments.     

Linear choosability of sparse graphs

A linear coloring is a proper coloring such that each pair of color classes induces a union of disjoint paths. We study the linear list chromatic number, denoted , of sparse graphs. The maximum average degree of a graph G, denoted mad(G), is the maximum of the average degrees of all subgraphs of G. It is clear that any graph G with maximum degree Δ(G) satisfies . In this paper, we prove the following results: (1) if and Δ(G)≥3, then , and we give an infinite family of examples to show that this result is best possible; (2) if and Δ(G)≥9, then , and we give an infinite family of examples to show that the bound on cannot be increased in general; (3) if G is planar and has girth at least 5, then .     

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.     

Continuous extension operators and convexity

Given a nonempty closed subset A of a Hilbert space X, we denote by L(A) the space of all bounded Lipschitz mappings from A into X, equipped with the supremum norm. We show that there is a continuous mapping F_c:L(A)?L(X) such that for each g∈L(A), F_c(g)|_A=g, , and . We also prove that the corresponding set-valued extension operator is lower semicontinuous.     

Holomorphic cohomology groups on compact Kähler complex spaces

Let (X,OX) be a compact (reduced) complex space, bimeromorphic to a Kähler manifold. The singular cohomology groups Hq(X,C) carry a mixed Hodge structure. In particular they carry a weight filtration W−lHq(X,C) (l=0,…,q), and the graded quotients have a direct sum decomposition into holomorphic invariants as . Here we investigate the relationships between the above invariants for r=0 and the cohomology groups , where is the sheaf of weakly holomorphic functions on X. Moreover, according to the smooth case, we characterize the topological line bundles L on X such that the class of c₁(L) in has pure type (1,1).     

The higher-order Picone identity and comparison of half-linear differential equations of even order

An identity of the Picone type for higher-order half-linear ordinary differential operators of the form and where p_j and P_j, j=0,…,n, are continuous functions defined on [a,b] and , is derived and then the Sturmian comparison theory for the corresponding 2nth-order equations l_α[x]=0 and L_α[y]=0 based on this identity is developed.     

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 .     

Isometric reflection vectors in Banach spaces

The aim of this paper is to study the set of isometric reflection vectors of a real Banach space X. We deal with geometry of isometric reflection vectors and parallelogram identity vectors, and we prove that a real Banach space is a Hilbert space if the set of parallelogram identity vectors has nonempty interior. It is also shown that every real Banach space can be decomposed as an Ir-sum of a Hilbert space and a Banach space with some points which are not isometric reflection vectors. Finally, we give a new proof of the Becerra-Rodríguez result: a real Banach space X is a Hilbert space if and only if is not rare.