Two vertex-disjoint cycles in a graph

LetG be a graph of ordern 6 with minimum degree at least (n + 1)/2. Then, for any two integerss andt withs 3,t 3 ands + t n, G contains two vertex-disjoint cycles of lengthss andt, respectively, unless thatn, s andt are odd andG is isomorphic toK _{(n–1)/2,(n–1)/2} + K₁. We also show that ifG is a graph of ordern 8 withn even and minimum degree at leastn/2, thenG contains two vertex-disjoint cycles with any given even lengths provided that the sum of the two lengths is at mostn.     

A critical case for Brownian slow points

Summary LetX _t be a Brownian motion and letS(c) be the set of realsr0 such that üX _r+t –X _r üct, 0th, for someh=h(r)>0. It is known thatS(c) is empty ifc<1 and nonempty ifc>1, a.s. In this paper we prove thatS(1) is empty a.s.This research was partially supported by NSF Grant 9322689.     

Small ball probabilities for a Wiener process under weighted sup-norms,with an application to the supremum of bessel local times

LetR be the radial part of ad-dimensional Wiener process, starting from 0. In this paper, small ball probabilities are evaluated for sup_0<11(t ^–p R(t)) and sup _t 0(e ^–1 R(t)), withp[0, 1/2]. Chung's law of the iterated logarithm is established for the supremum of the local times of a two-dimensional Bessel process.     

On long cycles in a 2-connected bipartite graph

Letk be an integer withk 2. LetG = (A, B; E) be a 2-connected bipartite graph. Supposed(x) + d(y) k + 1 for every pair of non-adjacent verticesx andy. ThenG contains a cycle of length at leastmin(2a, 2k) wherea = min(|A|,|B|), unlessG is one of some known exceptions. We conjecture that if|A| = |B| andd(x) + d(y) k + 1 for every pair of non-adjacent verticesx andy withx A andy B, thenG contains a cycle of length at leastmin(2a, 2k).     

Connected components of arithmetic graphs

Let . LetG _m (R) be the graph whose vertices are the numbers 1, 2, ...,m and whose edges are all pairs {a, b} such thata+br (modm) for somerR. LetC _m (R) be the number of connected components ofG _m (R). Letd be the greatest common divisor ofm and the differencesr _j –r _j or allr _i ,r _j R. ThenC _m (R) is equal to (i) (d+1)/2 ifd is odd, (ii)d/2 ifd is even andr is odd for allrR, or (iii) (d/2)+1 ifd is even andr is even for allrR.This research was supported in part by the National Science Foundation under grant No. MCS78-07908.     

Independent sets,cliques and hamiltonian graphs

LetG be ak-connected (k 2) graph onn vertices. LetS be an independent set ofG. S is called essential if there exist two distinct vertices inS which have a common neighbor inG. LetV _r, be a clique which is a complete subgraph ofG. In this paper it is proven that if every essential independent setS ofk + 1 vertices satisfiesS V _r , thenG is hamiltonian, orG{u} is hamiltonian for someu V _r, orG is one of three classes of exceptional graphs. Our theorem generalizes several well-known theorems.     

A note on induced cycles in Kneser graphs

Letg(n,r) be the maximal order of an induced cycle in the Knesser graph Kn([n] r), whose vertices are ther-sets of [n]={1, ...,n} and whose adjacency relation is disjointness. Thusg(n, r) is the largestm for which there is a sequenceA ₁,A ₂,...,A _m [n] ofr-sets withA _i A _j= if and only if |i-j|=1 orm–1. We prove that there is an absolute constantc>0 for which

On independent statistical decision problems and products of diffusions

Summary LetZ _t be a null recurrent diffusion on ^p with generatorG=(1/r)·(r ) for smooth positiver. This note constructs an independent recurrent diffusionZ _ton ¹ such that (Z _t, Z_t)is transient in ^p+1. This resolves negatively an old question in simultaneous estimation: Is there an admissible but not Bayes estimator(X) of the mean of a multivariate normal distribution for quadratic loss with the property: for every admissible (X), whereX is normal and independent ofX, (, ) remains admissible in the combined problem obtained by summing the component losses?Work supported by NSF at Mathematical Sciences Research Institute, Berkeley     

Graphs of small dimensions

LetG=(V, E) be a graph withn vertices. The direct product dimension pdim (G) (c.f. [10], [12]) is the minimum numbert such thatG can be embedded into a product oft copies of complete graphsK _n.In [10], Lovász, Neetil and Pultr determined the direct product dimension of matchings and paths and gave sharp bounds for the product dimension of cycles, all logarithmic in the number of vertices.     

The order of the differentials in the Atiyah-Hirzebruch spectral sequence

LetF _* be the homology theory corresponding to a spectrumF and consider the Atiyah-Hirzebruch spectral sequenceE _s,t ² H _s (X; _t F) F _s+t (X) for a bounded below spectrum (or CW-complex)X. This paper shows that the images of the differentials d _s,t ^r :E _s,t ^r E _s ^{–r,t+r–1r} in this spectral sequence are always torsion groupsof finite exponent and that this exponent isbounded in a very universal way: we prove the existence of integersR _r forr2 such thatR _r d _s,t ^r =0 for any spectrumF, for any bounded below spectrumX and for all integersr2,s andt. The interesting point is that these upper boundsR _r for the additive order of the differentials d _s,t ^r dependonly onr, and that the result holds without any hypothesis on the spectrumF. In certain special cases, this implies that the spectral sequence collapses and even that the extension problems given by itsE -term are trivial.     

A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations

LetDR ^N be a region with smooth boundaryD. Letp·q>1,p, q1. We consider the system:u _t=u+v ^p,v _t=u+u ^q inD×[0, ) withu=v=0 inD×[0, ) andu ₀,v ₀ nonnegative. Let=max(p, q). We show that ifD isR ^N, a cone or the exterior of a bounded domain, then there is a numberpc(D) such that (a) if (+1)/(pq–1)>pc(D) no nontrivial global positive solutions of the system exist while (b) if (+1)/(pq–1)<pc(D) both nontrivial global and nonglobal solutions exist. In caseD is a cone orD=R ^N, (a) holds with equality. An explicit formula forpc(D) is given.This research was supported in part by NSF Grant DMS-8822788 and in part by the Air Force Office of Scientific Research.     

A law of the iterated logarithm for processes with independent increments

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Transformation of Wiener measure under anticipative flows

Summary LetT()=+F() be a transformation from the Wiener space to itself with the range ofF() assumed to be in the Cameron-Martin space. The absolute continuity and the density function associated withT is considered;T is assumed to be embedded in or defined through a parameterizationT _t =+F _t () andF _t is assumed to be differentiable int. The paper deals first with the case where the range of thet-derivative ofF _t () is also in the Cameron-Martin space and new representations for the Radon-Nikodym derivative and the Carleman-Fredholm determinant are derived. The case where thet-derivative ofF _t is not in the Cameron-Martin space is considered next and results on the absolute continuity and the density function, under conditions which are considerably weaker than previously known conditions, are presented.The work of the second author was supported by the fund for promotion of research at the Technion     

The asymptotic distribution of magnitude trimmed sums for distributions in the Feller class

LetX, X ₁,X ₂,... be i.i.d. with common distribution functionF. Rather than study limit behavior of the sum,S _n =X ₁++X _n , under constant normalizations, we consider the sum with ther _n summands largest in magnitude removed from the sumS _n , wherer _n andr _n n ^–10. This is known as an intermediate magnitude trimmed sum. LetF be such that lim inf_t lim inf _t EX ² I(|X|t/)t ² P((|X|>t)>0. The collection of suchF is known as the Feller class, a large class of distributions which includes all domains of attraction (in particular the stable laws). Pruitt(¹³) showed that asymptotic normality always holds for the trimmed sums ifF is in the Feller class and ifF is symmetric. Here, for anyF in the Feller class, we obtain complete results including the form of the possible limit laws and their convergence criteria, thus generalizing Pruitt's result to the asymmetric setting.This paper forms a portion of the author's Ph.D. dissertation under the supervision of Daniel C. Weiner.     

On the maximal number of certain subgraphs inK r -free graphs

Given two graphsH andG, letH(G) denote the number of subgraphs ofG isomorphic toH. We prove that ifH is a bipartite graph with a one-factor, then for every triangle-free graphG withn verticesH(G) H(T ₂(n)), whereT ₂(n) denotes the complete bipartite graph ofn vertices whose colour classes are as equal as possible. We also prove that ifK is a completet-partite graph ofm vertices,r > t, n max(m, r – 1), then there exists a complete (r – 1)-partite graphG* withn vertices such thatK(G) K(G*) holds for everyK _r -free graphG withn vertices. In particular, in the class of allK _r -free graphs withn vertices the complete balanced (r – 1)-partite graphT _r–1(n) has the largest number of subgraphs isomorphic toK _t (t < r),C ₄,K _2,3. These generalize some theorems of Turán, Erdös and Sauer.Dedicated to Paul Turán on his 80th Birthday     

A convexity-preservingC 2 parametric rational cubic interpolation

Summary AC ² parametric rational cubic interpolantr(t)=x(t) i+y(t) j,t[t ₁,t _n] to data S={(x_j, y_j)|j=1,...,n} is defined in terms of non-negative tension parameters _j ,j=1,...,n–1. LetP be the polygonal line defined by the directed line segments joining the points (x _j ,y _j ),t=1,...,n. Sufficient conditions are derived which ensure thatr(t) is a strictly convex function on strictly left/right winding polygonal line segmentsP. It is then proved that there always exist _j ,j=1,...,n–1 for whichr(t) preserves the local left/righ winding properties of any polygonal lineP. An example application is discussed.This research was supported in part by the natural Sciences and Engineering Research Council of Canada.     

Density estimation for linear processes

Summary LetX _t , ...,X _n be random variables forming a realization from a linear process where {Z _t } is a sequence of independent and identically distributed random variables with E|Z _t |<∞ for some ε>0, andg _r →0 asr→∞ at some specified rate. LetX ₁ have a probability density functionf. It is then established that for every realx, the standard kernel type estimator based onX _t (1≦t≦n) is, under some general regularity conditions, asymptotically normal and converges a.s. tof(x) asn→∞. Research was supported in part by the Air Force Office of Scientific Research Grant No. AFOSR-81-0058.     

Global null controllability of nonlinear delay equations with controls in a compact set

The question of controlling some nonlinear retarded functional differential equations from an initial function to the zero function is considered. The control setsM() are square integrable functions with values in the unit closed sphereL ₂([t ₀, ),E ^m ) with center the origin. Assuming that the linear approximation of the nonlinear equation is null controllable with some integrable controls on some interval [t ₀,t ₁–2r], wheret ₀ is sufficiently large and wherer>0 is the delay, and assuming that the nonlinear system, withu=0, is uniformly globally asymptotically stable, we show that the nonlinear control process is globally null controllable with controlsuM(). The paper gives conditions which guarantee the stability assumptions, and also indicates conditions which yield the null controllability assumptions of the linear approximation. Our research extends known results on ordinary differential processes.     

An extreme value theory for long head runs

Summary For an infinite sequence of independent coin tosses withP(Heads)=p(0,1), the longest run of consecutive heads in the firstn tosses is a natural object of study. We show that the probabilistic behavior of the length of the longest pure head run is closely approximated by that of the greatest integer function of the maximum ofn(1-p) i.i.d. exponential random variables. These results are extended to the case of the longest head run interrupted byk tails. The mean length of this run is shown to be log(n)+klog(n)+(k+1)log(1–p)–log(k!)+k+/–1/2+ r₁(n)+ o(1) where log=log_1/p , =0.577 ... is the Euler-Mascheroni constant, =ln(1/p), andr ₁(n) is small. The variance is ₂/6²+1/12 +r ₂(n)+ o(1), wherer ₂(n) is again small. Upper and lower class results for these run lengths are also obtained and extensions discussed.This work was supported by a grant from the System Development Foundation     

Upper classes for the increments of the fractional Wiener process

Summary LetX(t) be a fractional Wiener process, i.e., a centered Gaussian process on [0, ) with stationary increments and varianceEX ² (t)=t ², anda(t) a positive nondecreasing function witha(t)t. We investigate the a.s. asymptotic behaviour of the incrementsI(t, a (t))=max{X{u+a(t))–X(u): 0ut–a(t)} (and some others that are similarly defined) ast.