Frame Self-orthogonal Mendelsohn Triple Systems

A Mendelsohn triple system of order ν, MTS(ν) for short, is a pair (X, B) where X is a ν-set (of points) and B is a collection of cyclic triples on X such that every ordered pair of distinct points from X appears in exactly one cyclic triple of B. The cyclic triple (a, b, c) contains the ordered pairs (a, b), (b, c) and (c, a). An MTS(ν) corresponds to an idempotent semisymmetric Latin square (quasigroup) of order ν. An MTS(ν) is called frame self-orthogonal, FSOMTS for short, if its associated semisymmetric Latin square is frame self-orthogonal. It is known that an FSOMTS(1 ⁿ ) exists for all n≡1 (mod 3) except n=10 and for all n≥15, n≡0 (mod 3) with possible exception that n=18. In this paper, it is shown that (i) an FSOMTS(2 ⁿ ) exists if and only if n≡0,1 (mod 3) and n>5 with possible exceptions n∈{9, 27, 33, 39}; (ii) an FSOMTS(3 ⁿ ) exists if and only if n≥4, with possible exceptions that n∈{6, 14, 18, 19}. *Research supported by NSFC 10371002 *Partially supported by National Science Foundation under Grant CCR-0098093     

On additive functionals of Markov chains

Let{S _n } _n=0 ^∞ be a Harris-recurrent Markov chain on a measurable state space. We prove strong approximation results for the additive functionals . Research supported by the Hungarian National Foundation for Scientific Research, Grant No. 1905. Mathematical Institute of the Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary. Research supported by an NSERC Canada Grant, Carleton University. Department of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6.     

Shadowing and Inverse Shadowing for C^1 Endomorphisms

In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also ＂uniform＂ with respect to C^1 perturbation.     

Semyanistyi’s Integrals and Radon Transforms on Matrix Spaces

We introduce a new analytic family of intertwining operators which include the Radon transform over matrix planes and its inverse. These operators generalize integral transformations introduced by Semyanistyi (Dokl. Akad. Nauk SSSR 134:536–539, [1960]) in his research related to the hyperplane Radon transform in ℝ ⁿ . We obtain an extended version of Fuglede’s formula, connecting generalized Semyanistyi’s integrals, Radon transforms and Riesz potentials on the space of real rectangular matrices. This result combined with the matrix analog of the Hilbert transform leads to variety of new explicit inversion formulas for the Radon transform of functions of matrix argument. The authors were supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). The first author was also supported by Abraham and Sarah Gelbart Research Institute for Mathematical Sciences. The second author was also supported by the NSF grants EPS-0346411 (Louisiana Board of Regents) and DMS-0556157).     

A remark on Weierstrass points and linear series on multiple covering curves

Here we study multiple coverings of rational and irational curves. We give a theorem about the non-gap sequence on m-gonal curves. We then study general irrational covering f ： X→ C, and say when h^0（X, f^＊（L）） = h^0（C,L） for L line bundle on C.     

On the equivalence of extremal Teichmüller mapping

Let T(Δ) and B(Δ) be the Teichmüller space and the infinitesimal Teichmüller space of the unit disk Δ respectively. In this paper, we show that [ν] _B(Δ) being an infinitesimal Strebel point does not imply that [ν] _T(Δ) is a Strebel point, vice versa. As an application of our results, problems proposed by Yao are solved. This work was supported by the National Natural Science Foundation of China (Grant No. 10571028)     

Self-similar sets of zero Hausdorff measure and positive packing measure

We prove that there exist self-similar sets of zero Hausdorff measure, but positive and finite packing measure, in their dimension; for instance, for almost everyu ∈ [3, 6], the set of all sums ∑ ₀ ⁸ a _n 4⁻ⁿ a _n 4⁻ⁿ with digits witha _n ∈ {0, 1,u} has this property. Perhaps surprisingly, this behavior is typical in various families of self-similar sets, e.g., for projections of certain planar self-similar sets to lines. We establish the Hausdorff measure result using special properties of self-similar sets, but the result on packing measure is obtained from a general complement to Marstrand’s projection theorem, that relates the Hausdorff measure of an arbitrary Borel set to the packing measure of its projections. Research of Y. Peres was partially supported by NSF grant #DMS-9803597. Research of K. Simon was supported in part by the OTKA foundation grant F019099. Research of B. Solomyak was supported in part by NSF grant #DMS 9800786, the Fulbright Foundation, and the Institute of Mathematics at The Hebrew University of Jerusalem.     

On closures of orbits and arithmetic of quaternions

ForG=PGL₂(ℚ _p )×PGL₂ ℚ we study the closures of orbits under the maximal split Cartan subgroup ofG in homogeneous spacesΓ\G. We show that if a closure of an orbit contains a closed orbit then the orbit is either dense or closed. We show the relation of this to divisibility properties of integral quaternions and other lattices. Sponsored in part by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Germany). Research at MSRI supported by NSF grant DMS8505550.     

A cardinal preserving extension making the set of points of countable V cofinality nonstationary

Assuming large cardinals we produce a forcing extension of V which preserves cardinals, does not add reals, and makes the set of points of countable V cofinality in κ⁺ nonstationary. Continuing to force further, we obtain an extension in which the set of points of countable V cofinality in ν is nonstationary for every regular ν ≥ κ⁺. Finally we show that our large cardinal assumption is optimal. This material is based upon work supported by the National Science Foundation under Grant no. DMS-0094174.     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

On orderable set functions and continuity I

A set functionv (which is not necessarily additive) on a measurable spaceI is called orderable if for each measurable order ℛ onI there is a measureϱ ^ℛ ν(J) =ν(J) onI such that for all subsetsJ ofI that are initial segements,ϱ ^ℛ ν. Properties such as nonatomicity, nullness of sets, and weak continuity are shown to be inherited from orderable set functionsv toϱ ^ℛ ν and vice versa. A characterization of set functions which are absolutely continuous (with respect to some positive measure) in the set of orderable set functions is also given. Reporduction of this report was partially supported by the Office of Naval Research under contract N-000 14-67-0112-0011; The U.S. Atomic Energy Commission contract AT (04-3)-326-PA #18; and The National Science Foundation, Grant GP 31393X. Reproduction in whole or in part is permitted for any purposes of the United States Government. This document has been approved for public release and sale; its distribution is unlimited. Research of this report was carried out at Tel Aviv University.     

Locally Closed Semirings

 We call a semiring S locally closed if for all a ∈ S there is some integer k such that 1 + a + ⋯ + a ^k  =1 + a + ⋯ + a ^k + 1 . In any locally closed semiring we may define a star operation a ↦ a ^*, where a ^* is the above finite sum. We prove that when S is locally closed and commutative, then S is an iteration semiring. Partially supported by grant no. T30511 from the National Foundation of Hungary for Scientific Research and the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Partially supported by the Austrian–Hungarian Bilateral Research and Development Fund, no. A-4/1999, and by the Austrian–Hungarian Action Foundation. Received March 16, 2001     

Branched two-sheeted covers

In this paper we explore the connection between Weierstrass points of subspaces of the holomorphic differentials and the geometry of the canonical curve inPC ^g−1. In particular, we consider non-hyperelliptic Riemann surfaces with involution and the Weierstrass points of the −1 eigenspace of the holomorphic differentials. The case of coverings of a torus is considered in detail. Research of the first author supported in part by the Paul and Gabriella Rosenbaum Foundation, the Landau Center for Research in Mathematical Analysis (supported by Minerva Foundation-Germany) and a US-Israel BSF grant. Research by the second author supported in part by NSF Grant DMS 9003361 and a Lady Davis Visiting Professorship at the Hebrew University.     

Orthogonal graphs of characteristic 2 and their automorphisms

The (singular) orthogonal graph O(2ν + δ,q) over a field with q elements and of characteristic 2 (where ν 1, and δ = 0,1 or 2) is introduced. When ν = 1, O(2 · 1,q), O(2 · 1 + 1,q) and O(2 · 1 + 2,q) are complete graphs with 2, q + 1 and q2 + 1 vertices, respectively. When ν 2, O(2ν + δ,q) is strongly regular and its parameters are computed. O(2ν + 1,q) is isomorphic to the symplectic graph Sp(2ν,q). The chromatic number of O(2ν + δ,q) except when δ = 0 and ν is odd is computed and the group of graph automo...     

Every null-additive set is meager-additive

It is proved that every null-additive subset of^ω2 is meager-additive. Several characterizations of the null-additive subsets of^ω2 are given, as well as a characterization of the meager additive subsets of^ω2. Under CH, an uncountable null-additive subset of^ω2 is constructed. Publ. No. 445. First version written in April 1991. Partially supported by the Basic Research Fund of the Israel Academy of Sciences and the Edmund Landau Center for Research in Mathematical Analysis, supported by the Minerva Foundation (Germany).     

The speed of convergence of a martingale

LetX _n, n≧0, be a martingale with respect to the σ-fieldsF _n and letB _n ² =Σ_1≧n E{(X ₁−X ₁₋₁)²|F ₁₋₁} It is known that ifB ₁ ² <∞ on some set Ω₀ thenX _∞=limX _n exists and is finite a.e. on Ω₀ We show that under suitable conditions there exists a constant ν<∞ for which lim supB _n ⁻¹ {log logB _n ² }^−1/2|X _∞−X _n−1 | ≦ √2(η+1). If “the fluctuations ofB _n are small” (in the sense of the Corollary) then ν=0 and the usual upper bound of a law of the iterated logrithm results. This upper bound is not necessarily achieved, though. Research supported in part by the NSF under Grant No. MCS 72-04534A04.     

Hausdorff dimension of the sample path of the Westwater process

In this paper, we prove that the Hausdorff dimension of the sample path of the three dimensional coordinate process is exactly two a.e. −ν(g), where ν(g) is the polymer measure constructed by Westwater. Furthermore, we prove that the sample path of the Westwater process has also zero Hausdorff 2-measure. This answers a problem suggested by E. Nelson. Research supported in part by the National Natural Science Foundation of China     

Asymptotics for Christoffel functions for general measures on the real line

We consider asymptotics of Christoffel functions for measures ν with compact support on the real line. It is shown that under some natural conditionsn times thenth Christoffel function has a limit asn→∞ almost everywhere on the support, and the limit is the Radon-Nikodym derivative of ν with respect to the equilibrium measure of the support of ν. The case in which the support is an interval was settled previously by A. Máté, P. Nevai and the author. The present paper solves the general problem. Work was supported by the National Science Foundation, DMS 9801435 and by the Hungarian National Science Foundation for Research, T/022983.     

A characterization of vector measure games inpNA

We give a complete characterization of games inpNA of the formf ^o μ (where μ is a vector of finite number of non-atomic probability measures, andf is a real valued function on the range of μ withf(0)=0). Specifically, we show thatf ^o μ is inpNA iff “f is continuous at μ” (the definition of the latter is given in the paper). This work was supported by National Science Foundation Grant SOC 75-21820-A01 at the Institute for Mathematical Studies in the Social Sciences, Stanford University.     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999