Proper holomorphic self-maps of smooth bounded Reinhardt domains in ?2

It is proved that every proper holomorphic self-map of a smooth bounded Reinhardt domain in ℂ² is an automorphism. The first author’s work was supported in part by the National Natural Science Foundation of China (Grant No. 10571135) and the Doctoral Program Foundation of the Ministry of Education of China (Grant No. 20050240711)     

Bi-Lipschitz maps in <Emphasis Type="Italic">Q</Emphasis>-regular Loewner spaces

By applying the theory of quasiconformal maps in measure metric spaces that was introduced by Heinonen-Koskela, we characterize bi-Lipschitz maps by modulus inequalities of rings and maximal, minimal derivatives in Q-regular Loewner spaces. Meanwhile the sufficient and necessary conditions for quasiconformal maps to become bi-Lipschitz maps are also obtained. These results generalize Rohde’s theorem in ℝ ⁿ and improve Balogh’s corresponding results in Carnot groups. This research is supported by China NSF (Grant No. 10271077)     

Forward equations for reflected diffusions with jumps

In this paper we obtain the forward equations associated with the evolution of the density, if it exists, of reflected diffusions on the positive orthant with jumps which form a marked point process whose random jump measure possesses a stochastic intensity. These results generalize the so-called generalized Dynkin equations for piecewise deterministic jump processes due to Davis. We then consider the stationary case where the existence of a stochastic intensity is not needed. The techniques are based on local times and the use of random jump measures. We discuss the application of these results to problems arising in queuing and storage processes as well as stationary distributions of diffusions with delayed and jump reflections at the origin.This research was supported in part by the Quebec-France Cooperative Research Program and by the Natural Sciences and Engineering Research Council of Canada under Grant OGP 0042024.     

On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

On the connected components of the space of line transversals to a family of convex sets

Let ℒ be the space of line transversals to a finite family of pairwise disjoint compact convex sets in ℝ³. We prove that each connected component of ℒ can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets. The research of J. E. Goodman was supported in part by NSF Grant DMS91-22065 and by NSA Grant MDA904-92-H-3069. R. Pollack's research was supported in part by NSF Grant CCR91-22103 and by NSA Grant MDA904-92-H-3075. The research of R. Wenger was supported in part by NSA Grant MDA904-93-H-3026 and by the NSF Regional Geometry Institute (Smith College, July 1993) Grant DMS90-13220.     

A Note on Asymptotic Normality of Kernel Estimation for Linear Random Fields on <Emphasis Type="Italic">Z</Emphasis><Superscript>2</Superscript>

This note considers the kernel estimation of a linear random field on Z ². Instead of imposing certain mixing conditions on the random fields, it is assumed that the weights of the innovations satisfy a summability property. By building a martingale decomposition based on a suitable filtration, asymptotic normality is proven for the kernel estimator of the marginal density of the random field. T.-L. Cheng’s research is supported in part by NSC 94-2118-M-018-001, Taiwan. Also, he is indebted to Department of Mathematics and Statistics, University of Calgary, for their hospitality during his visit. X. Lu’s research is supported in part by NSERC Discovery Grant of Canada.     

The Odd-Distance Plane Graph

The vertices of the odd-distance graph are the points of the plane ℝ². Two points are connected by an edge if their Euclidean distance is an odd integer. We prove that the chromatic number of this graph is at least five. We also prove that the odd-distance graph in ℝ² is countably choosable, while such a graph in ℝ³ is not. The research of J. Maňuch was supported in part by MITACS (Mathematics of Information Technology and Complex Systems). The research of M. Rosenfeld was supported in part by the Chancellor Research Grant and the Institute of Technology, UWT. The research of S. Shelah was supported by the United States-Israel Binational Science Foundation (Grant no. 2002323), and by NSF grant No. NSF-DMS 0600940. No. 923 on Shelah’s publication list. The research of L. Stacho was supported in part by NSERC (Natural Science and Engineering Research Council of Canada) grant.     

The LIL for <Emphasis Type="Italic">U</Emphasis>-Statistics in Hilbert Spaces

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U-statistics of arbitrary order, which are of independent interest. R. Adamczak’s research partially supported by MEiN Grant 2 PO3A 019 30. R. Latała’s research partially supported by MEiN Grant 1 PO3A 012 29.     

An integral estimate of Bessel function and its application

In this paper the authors give a new integral estimate of the Bessel function,which is an extension of Calder(?)n-Zygmund's result.As an application of this result,we prove that the parameterized Marcinkiewicz integralμ_Ω~p with variable kernels is of type (2,2),where the kernel functionΩdoes not have any smoothness on the unit sphere in R~n.     

A quantum model of a real scalar field

A quantum model of a real scalar field with local operator gauge symmetry is discussed. In the localized theory, in order to keep the local operator gauge symmetry, an operator gauge potential BB _μ, is needed. By combining the constraint of operator gauge potentialB _μ, and the microscopic causality theorem, the usual canonical quantization condition of a real scalar field is obtained. Therefore, a quantum model of a real scalar field without the usual procedure of quantizing a related classical model can be directly constructed. Project supported in part by T.D. Lee’s NNSF Grant, National Natural Science Foundation of China, Foundation of Ph. D. Directing Programme of Chinese Universities and the Chinese Academy of Sciences.     

Harmonic maps into loop spaces of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. We conjecture that harmonic maps of the Riemann sphere ℂℙ¹ into ΩG are related to Yang-Mills G-fields on ℝ⁴. This work was partly supported by the RFBR (Grant Nos. 04-01-00236, 06-02-04012), by the program of Support of Scientific Schools (Grant No. 1542.2003.1), and by the Scientific Program of RAS “Nonlinear Dynamics”     

Vacuum states on compressible Navier-Stokes equations with general density-dependent viscosity and general pressure law

In this paper,we study the one-dimensional motion of viscous gas with a general pres- sure law and a general density-dependent viscosity coefficient when the initial density connects to the vacuum state with a jump.We prove the global existence and the uniqueness of weak solutions to the compressible Navier-Stokes equations by using the line method.For this,some new a priori estimates are obtained to take care of the general viscosity coefficientμ(ρ)instead ofρ~θ.     

Boundary Behavior of Harmonic Functions for Truncated Stable Processes

For any α∈(0,2), a truncated symmetric α-stable process in ℝ ^d is a symmetric Lévy process in ℝ ^d with no diffusion part and with a Lévy density given by c|x|^−d−α 1_{|x|<1} for some constant c. In (Kim and Song in Math. Z. 256(1): 139–173, [2007]) we have studied the potential theory of truncated symmetric stable processes. Among other things, we proved that the boundary Harnack principle is valid for the positive harmonic functions of this process in any bounded convex domain and showed that the Martin boundary of any bounded convex domain with respect to this process is the same as the Euclidean boundary. However, for truncated symmetric stable processes, the boundary Harnack principle is not valid in non-convex domains. In this paper, we show that, for a large class of not necessarily convex bounded open sets in ℝ ^d called bounded roughly connected κ-fat open sets (including bounded non-convex κ-fat domains), the Martin boundary with respect to any truncated symmetric stable process is still the same as the Euclidean boundary. We also show that, for truncated symmetric stable processes a relative Fatou type theorem is true in bounded roughly connected κ-fat open sets. The research of P. Kim is supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-331-C00037). The research of R. Song is supported in part by a joint US-Croatia grant INT 0302167.     

Cauchy-riemann operator and decomposition ofL 2 space on ℝ + n+1

We decompose L_α²(ℝ₊ ⁿ⁺¹) into direct sum of closed subspaces which are isomorphic to the monogenic Bergman space. Application on commutators is also discussed. Research supported in part by the NSF Grant DMS-9622890     

Asymptotic distributions of non-central studentized statistics

Let X1,...,Xn be independent and identically distributed random variables and Wn = Wn(X1,...,Xn) be an estimator of parameter θ.Denote Tn =(Wn - θ0)/sn,where sn2 is a variance estimator of Wn.In this paper a general result on the limiting distributions of the non-central studen-tized statistic Tn is given.Especially,when s2n is the jacknife estimate of variance,it is shown that the limit could be normal,a weighted χ2 distribution,a stable distribution,or a mixture of normal and stable distribution.Applicati...     

Sufficient and Necessary Conditions for Stochastic Comparability of Jump Processes

Abstract This note is devoted to the study of the stochastic comparability of jump processes. On the basis of [2] and [3], it is proved that two jump processes are stochatically comparable if and only if their q-pairs are comparable. Meanwhile, the result concerning the uniqueness given in [6] is also improved upon. Research supported in part by DPFIHE(Grant No.96002704), NNSFC(Grant No.19771008), MCSEC, Ying-Tung Fok Educational Foundation and Youth Science Foundation of BNU     

The zero-velocity limit solutions of the Navier-Stokes equations of compressible fluid revisited

For an isentropic fluid flow driven by a potential force ∇F, the set of the socalled rest states (zero-velocity stationary solutions) is investigated. A sufficient condition is found, in terms of the potentialF, for the problem to possess at most two distinct solutions with given mass and total energy.

---

Sunto Com'è noto, per un fluido isentropico viscoso, soggetto ad una forza esterna di tipo potenzialeF, solo la quiete è possibile. La struttura topologica dell’insieme di tutte le soluzioni di quiete è studiata come l’ω-limite delle traiettorie. è provato che se gli insiemi di livello superiore diF sono connessi, edF è localmente Lipschitziana, allora la soluzione (densità) è univocamente determinata dalla sua massa. Inoltre, se l’energia potenzialee è assegnata, la condizione geometrica sulle superficie di livello può essere rilassata.

---

---

The work was supported by Grant 201/98/1450 GA ČR and Grant A1019703 GA AV ČR.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

<Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Bold">1,</Emphasis><Emphasis Type="Italic">α</Emphasis></Superscript> regularity for infinity harmonic functions in two dimensions

We propose a new method for showing C ^{1, α} regularity for solutions of the infinity Laplacian equation and provide full details of the proof in two dimensions. The proof for dimensions n ≥ 3 depends upon some conjectured local gradient estimates for solutions of certain transformed PDE. LCE is supported in part by NSF Grant DMS-0500452. OS was supported in part by the Miller Institute for Basic Research in Science, Berkeley.     

On distance sets of large sets of integer points

Distance sets of large sets of integer points are studied in dimensions at least 5. To any ε > 0 a positive integer Q _ɛ is constructed with the following property; If A is any set of integer points of upper density at least ε, then all large multiples of Q _ɛ ² occur as squares of distances between the points of the set A. Research supported in part by NSF Grant DMS-0456490.