Large deviations for independent random walks

Summary We consider a system of independent random walks on . Let _n (x) be the number of particles atx at timen, and letL _n (x)=₀(x)+ ... + _n (x) be the total occupation time ofx by timen. In this paper we study the large deviations ofL _n (0)–L _n (1). The behavior we find is much different from that ofL _n (0). We investigate the limiting behavior when the initial configurations has asymptotic density 1 and when ₀(x) are i.i.d Poisson mean 1, finding that the asymptotics are different in these two cases.This work was done while the first author was on sabbatical at Cornell University. Both authors were partially supported by the National Science Foundation and the Army Research Office through the Mathematical Sciences Institute at Cornell     

On countably compact,locally countable spaces

We study the cardinalities of countably compact, locally countableT ₃ spaces. For alln(<), there exists one of cardinality _n . IfV=L, then there exists one of cardinalityx iffx= orx =x. MA implies that there exists one of cardinality>2.     

A Bernstein Property of Affine Maximal Hypersurfaces

Let x: M A ^{n + 1} be a locally strongly convex hypersurface, given as a graph of a locally strongly convexfunction x _{n + 1} =f(x ₁, ..., x _n )defined in a domain A ⁿ . We introduce a Riemannian metricG ^# = (² f/x _i x _j )dx _i dx _j on M. In this paper, we investigate the affine maximalhypersurfaces which are complete with respect to the metricG ^# and prove a Bernstein property for the affine maximalhypersurfaces.     

A note onn-groupoids linearly defined on fields

On the real fieldR and the Galois fields GF(p), define operations by [x₁ x₂ ···x_n]=₁x₁+₂x₂+ ··· +_nx_n, where ₁,₂, ...,_n are elements of the relevant fields. LetB be the class of alln-groupoids defined on Galois fields in this way. In this paper, we will study the variety generated byB and the variety generated by the algebra (R, [ ]), where ₁,...,_n are algebraically independent inR. We will study also varieties defined in a similar way with the operation [x₁, x₂,..., x_n]=(x₁+x₂+ ···+x_n).Presented by Jan Mycielski.The author thanks Professor T. Evans for his suggestions in developing this article.     

The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

Über die Lösungsanzahl der Gleichungf(n)=ϰ·n für gewisse Klassen multiplikativer Funktionen

Denote byV _æ (f,x) the number of solutionsnx of the equationf(n)=æ·n. Then there are multiplicative functions, such that the lower estimate holds for infinitely many pairs (æ,x),x.     

Binomial-coefficient multiples of irrationals

Denote byx a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and byd _n (x) the binomial coefficient at then'th level along the pathx. Then for a denseG set of in the unit interval, {d _n (x)} is almost surely dense but not uniformly distributed modulo 1.     

A Criterion of Sign Definiteness for Quadratic and Quasi-Quadratic Forms in a Cone

A new criterion of sign definiteness is given for quadratic and quasi-quadratic forms in a cone in space ⁿ . The basic results are proved for the nonnegative cone K ₊ ⁿ : x 0, x ⁿ . A change of variables reduces the case of an arbitrary octant of space ⁿ to K ₊ ⁿ .     

A Family of Asymptotically e(n − 1) ! Polynomial Orders of Nn

Here, N is the set of nonnegative integers, while an order in N ⁿ is a bijective function : N ⁿ N. Two orders are equivalent if they differ only by a permutation of their arguments. Let s(x)=x₁+ ··· +x _n for 0 < n N and x =(x ₁, ···, x _n ) N ⁿ ; such an is a diagonal order if (x) < (y) whenever x,y N ⁿ , and s(x) < s(y). Lew composed Skolem"s diagonal polynomial orders to construct c _n inequivalent nondiagonal polynomial orders in N ⁿ . Afterwards, Morales and Lew did the same with respect to the Morales–Lew"s diagonal orders, obtaining additional d _n inequivalent nondiagonal polynomial orders. Moreover, they proved that d _n / c _n as n . Recently, Sanchez obtained a family of (n – 1) ! inequivalent diagonal orders in N ⁿ . In this paper, we compose the Sanchez diagonal polynomial orders to construct e _n inequivalent nondiagonal polynomial orders with e _n e(n – 1) !, where e is the base of natural logarithms. Furthermore, we prove that e _n / d _n as n .     

Invariant equations defining coincident root loci

Let F(x) = xⁿ+₁ x^n-1+₂ x^n-2+ ··· +_n be a polynomial with complex coefficients, and suppose we are given a partition (₁,...,_r) of n. It is a classical problem to determine explicit algebraic conditions on the _i so that F may have roots with multiplicities ₁,...,_r. We give an invariant theoretic solution to this problem, to wit, we exhibit a set of covariants of F whose vanishing is a necessary and sufficient condition. The construction of such covariants is combinatorial, and involves associating a set of graphs on n vertices (called decisive graphs) to each .Received: 28 September 2003     

Onp-adic functions preserving Haar measure

Let {a _n } _n ^=0/ be a uniformly distributed sequence ofp-adic integers. In the present paper we study continuous functions close to differentiable ones (with respect to thep-adic metric); for these functions, either the sequence {f(a _n )} _n ^=0/ is uniformly distributed over the ring ofp-adic integers or, for all sufficiently largek, the sequences {f _k (_k(a_n))} _n ^=0/ are uniformly distributed over the residue class ring modp ^k , where _k is the canonical epimorphism of the ring ofp-adic integers to the residue class ring modp ^k andf _k is the function induced byf on the residue class ring modp ^k (i.e.,f k _(x) =f( _k (x)) (modp ^k )). For instance, these functions can be used to construct generators of pseudorandom numbers.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 935–950, June, 1998.In conclusion, the author wishes to express his deep gratitude to V. S. Anashin for permanent attention to this research and for support.     

Sets associated with the farthest point problem

LetS be a convex compact set in a normed linear spaceX. For each cardinal numbern, defineS _n = {x X:x has exactlyn farthest points inS} andT _n = _kn S _k. It is shown that ifX =E thenT ₃ is countable andT ₂ is contractible to a point. Properties of associated level curves are given.     

The Spectrum of a Random Geometric Graph is Concentrated

Consider n points, x ₁,... , x _n , distributed uniformly in [0, 1] ^d . Form a graph by connecting two points x _i and x _j if . This gives a random geometric graph, , which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on is concentrated, and in fact converges to that of the graph on the deterministic grid.      

Distribution of values of σ2 (n) in residue classes

IntegersN for which ₂ (n has weak uniform distribution (modN) are determined.     

Finite cutsets and finite antichains

An ordered set (P,) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} Fx meets every maximal chain of (P,). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We specify n for m < w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n₁ if m 4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4m₀, n=₀. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.This work was supported by the NSERC of Canada.     

On the finite sum representations of the Lauricella functions <Emphasis Type="Italic">F</Emphasis><Subscript>D</Subscript>

By using divided differences, we derive two different ways of representing the Lauricella function of n variables F_D⁽ⁿ⁾(a,b₁,b₂,...,b_n;c;x₁,x₂,...,x_n) as a finite sum, for b₁,b₂,...,b_n positive integers, and a,c both positive integers or both positive rational numbers with c–a a positive integer. AMS subject classification 33D45, 40B05, 40C99Jieqing Tan: Research supported by the National Natural Science Foundation of China under Grant No. 10171026 and Anhui Provincial Natural Science Foundation under Grant No. 03046102.Ping Zhou: Corresponding author. Research supported by NSERC of Canada.     

Punkt- und Geradenabbildungen,die das volumen 1 vonn- dimensionalen simplices im ℝn erhalten

A mappingf of ⁿ ,n3, into itself such thatf(x ₁),f(x ₂, ...,f(x _n+1 ) are the vertices of a simplex of volume 1 ifx ₁,...,x _n+1 are the vertices of a simplex of volume 1, must be equi-affine. (This theorem is also true in casen= 2 as it was proved by Gil Martin, see W. Benz [4].)LetM ⁿ be the set of lines of ⁿ . A mapping: M ⁿ M ⁿ ,n3 such that(a ₁ ),...,(a _n(n+1)/2 ) are the edges of a simplex of volume 1 ifa ₁,...,a _n(n+1)/2 are the edges of a simplex of volume 1, must be induced by an equi-affine mapping of ⁿ .     

Kernel-type density and failure rate estimation for associated sequences

Let {X _n ,n1} be a strictly stationary sequence of associated random variables defined on a probability space (,B, P) with probability density functionf(x) and failure rate functionr(x) forX ₁. Letf _n (x) be a kerneltype estimator off(x) based onX ₁,...,X _n . Properties off _n (x) are studied. Pointwise strong consistency and strong uniform consistency are established under a certain set of conditions. An estimatorr _n (x) ofr(x) based onf _n (x) andF _n (x), the empirical survival function, is proposed. The estimatorr _n (x) is shown to be pointwise strongly consistent as well as uniformly strongly consistent over some sets.     

Asymptotic Recurrence and Waiting Times for Stationary Processes

Let be a discrete-valued stationary ergodic process distributed according to P and let x=(..., x _–1, x ₀, x ₁,...) denote a realization from X. We investigate the asymptotic behavior of the recurrence time R _n defined as the first time that the initial n-block reappears in the past of x. We identify an associated random walk, on the same probability space as X, and we prove a strong approximation theorem between log R _n and . From this we deduce an almost sure invariance principle for log R _n. As a byproduct of our analysis we get unified proofs for several recent results that were previously established using methods from ergodic theory, the theory of Poisson approximation and the analysis of random trees. Similar results are proved for the waiting time W _n defined as the first time until the initial n-block from one realization first appears in an independent realization generated by the same (or by a different) process.     

Delay optimization of linear depth boolean circuits with prescribed input arrival times

We consider boolean circuits C over the basis Ω={,} with inputs x₁, x₂,…,x_n for which arrival times are given. For 1in we define the delay of x_i in C as the sum of t_i and the number of gates on a longest directed path in C starting at x_i. The delay of C is defined as the maximum delay of an input.Given a function of the form

f(x₁,x₂,…,x_n)=g_n−1(g_n−2(…g₃(g₂(g₁(x₁,x₂),x₃),x₄)…,x_n−1),x_n)