On a problem of spencer

LetX ₁, ...,X _n be events in a probability space. Let ϱ_i be the probabilityX _i occurs. Let ϱ be the probability that none of theX _i occur. LetG be a graph on [n] so that for 1 ≦i≦n X _i is independent of ≈X _j ‖(i, j)∉G≈. Letf(d) be the sup of thosex such that if ϱ₁, ..., ϱ _n ≦x andG has maximum degree ≦d then ϱ>0. We showf(1)=1/2,f(d)=(d−1) ^d−1 d ^−d ford≧2. Hence df(d)=1/e. This answers a question posed by Spencer in [2]. We also find a sharp bound for ϱ in terms of the ϱ _i andG.     

Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA ₁, ...,A _k are distinct subsets of a set of cardinalityn, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, andk>n, then some two ofA ₁, ...,A _k have empty intersection. We prove a strengthening, that at leastk /n ofA ₁, ...,A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440     

Two-point concentration in random geometric graphs

A random geometric graph G _n is constructed by taking vertices X ₁,…,X _n ∈ℝ ^d at random (i.i.d. according to some probability distribution ν with a bounded density function) and including an edge between X _i and X _j if ‖X _i -X _j ‖ < r where r = r(n) > 0. We prove a conjecture of Penrose ([14]) stating that when r=r(n) is chosen such that nr ^d = o(lnn) then the probability distribution of the clique number ω(G _n ) becomes concentrated on two consecutive integers and we show that the same holds for a number of other graph parameters including the chromatic number χ(G _n ). The author was partially supported by EPSRC, the Department of Statistics, Bekkerla-Bastide fonds, Dr. Hendrik Muller’s Vaderlandsch fonds, and Prins Bernhard Cultuurfonds.     

Über allgemeine Hyperflächenschnitte einer algebraischen Varietät

LetV/k be an irreducible algebraic variety over a fieldk in an affinen-space andF _u a generic hypersurface defined byu ₁ f ₁ (X)+...+u r f _r(X)=0, whereu ₁...,u _r are indeterminates overk andf ₁(X), ...,f _r(X) are polynomials ink[X ₁, ...,X _n]. Let (E) be a property which an arbitrary algebraic variety could have, e. g. irreducibility, normality (local or global), ... Then it will be studied under which conditions off ₁(X), ...,f _r(X) (E) may be transfered fromV/k toVF _u /k(u) (and conversely).     

Wavelet expansions of fractal measures

Let μ be a measure on ℝⁿ that satisfies the estimate μ(B _r(x))≤cr ^α for allx ∈ ℝⁿ and allr ≤ 1 (B _r(x) denotes the ball of radius r centered atx. Let ϕ _j,k ^(ɛ) (x)=2 ^nj2ϕ^(ɛ)(2 ^j x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤⁿ, and ∈ ∈E, a finite set, and letP _j (T)=Σ_ɛ,k <T,ϕ _j,k ^(ɛ) >ϕ _j,k ^(ɛ) denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls ^p(dμ) for 1 ≤p ≤ ∞, we show thatP _j(f dμ) ∈ L^p(dx) and ||P _j (fdμ)||L _p(dx)≤c2 ^{j((n-α)/p′))}||f||L _p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P _j (fdμ)||L _p(dx) under more restrictive hypotheses. Communicated by Guido Weiss     

Partition conditions and vertex-connectivity of graphs

It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev ₁, ...,v _n ≠V(G) and for any sequence of positive integersk ₁, ...,k _n such thatk ₁+...+k _n =|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv ₁, ...,v(n), and the class of the partition containingv _i induces a connected subgraph consisting ofk _i vertices, fori=1, 2, ...,n. Now fix the integersk ₁, ...,k _n . In this paper we study what can we say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv ₁, ...,v _n ≠V(G). We find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases we give sharp lower bounds for the vertex-connectivity ofG.     

Diperfect graphs

Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ₁, μ₂, ..., μ_k such tht |μ_i ∩S|=1 for alli. Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic number; here again, we do not know if there exists an optimal coloring (S ₁,S ₂, ...,S _k) and a path μ such that |μ ∩S _i|=1 for alli. In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the perfect graph conjecture. Dedicated to Tibor Gallai on his seventieth birthday     

DNA labelled graphs with DNA computing

Let k≥2, 1≤i≤k andα≥1 be three integers. For any multiset which consists of some k-long oligonucleotides, a DNA labelled graph is defined as follows: each oligonucleotide from the multiset becomes a point; two points are connected by an arc from the first point to the second one if the i rightmost uucleotides of the first point overlap with the i leftmost nucleotides of the second one. We say that a directed graph D can be(k, i;α)-labelled if it is possible to assign a label(l_1(x),..., l_k(x))to each point x of D such that l_j(x)∈{0,...,a-1}for any j∈{1,...,k}and(x,y)∈E(D)if and only if(l_k-i 1(x),..., l_k(x))=(l_1(y),..., l_i(y)). By the biological background, a directed graph is a DNA labelled graph if there exist two integers k, i such that it is(k, i; 4)-labelled. In this paper, a detailed discussion of DNA labelled graphs is given. Firstly, we study the relationship between DNA labelled graphs and some existing directed graph classes. Secondly, it is shown that for any DNA labelled graph, there exists a positive integer i such that it is(2i, i; 4)-labelled. Furthermore, the smallest i is determined, and a polynomial-time algorithm is introduced to give a(2i, i; 4)-labelling for a given DNA labelled graph. Finally, a DNA algorithm is given to find all paths from one given point to another in a(2i, i; 4)-labelled directed graph.     

Generalized derivations with Engel condition on multilinear polynomials

Let R be a prime ring with extended centroid C, δ a nonzero generalized derivation of R, f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C, I a nonzero right ideal of R and k ≥ a fixed integer. If [δ(f(r ₁, ..., r _n )), f(r ₁, ..., r _n )] _k = 0, for all r ₁, ..., r _n ∈ I, then either δ(x) = ax, with (a-γ)I = 0 and a suitable γ ∈ C or there exists an idempotent element e ∈ soc(RC) such that IC = eRC and one of the following holds (1) if char(R) = 0 then f(x ₁, ..., x _n ) is central valued in eRCe (2) if char(R) = p > 0 then is central valued in eRCe, for a suitable s ≥ 0, unless when char(R) = 2 and eRCe satisfies the standard identity s ₄ (3) δ(x) = ax−xb, where (a+b+α)e = 0, for α ∈ C, and f(x ₁, ..., x _n )² is central valued in eRCe.     

Accuracy of several multidimensional refinable distributions

Compactly supported distributions f₁,..., f_r on ℝ^d are fefinable if each f_i is a finite linear combination of the rescaled and translated distributions f_j(Ax−k), where the translates k are taken along a lattice Γ ⊂ ∝^d and A is a dilation matrix that expansively maps Γ into itself. Refinable distributions satisfy a refinement equation f(x)=Σ_k∈Λ c_k f(Ax−k), where Λ is a finite subset of Γ, the c_k are r×r matrices, and f=(f₁,...,f_r)^T. The accuracy of f is the highest degree p such that all multivariate polynomials q with degree(q)<p are exactly reproduced from linear combinations of translates of f₁,...,f_r along the lattice Γ. We determine the accuracy p from the matrices c_k. Moreover, we determine explicitly the coefficients y_α,i(k) such that x^α=Σ _i=1 ^r Σ_k∈Γy_α,i(k) fi(x+k). These coefficients are multivariate polynomials y_α,i(x) of degree |α| evaluated at lattice points k∈Γ.     

Hypergraphs with no special cycles

A special cycle in a hypergraph is a cyclex ₁ E ₁ x ₂ E ₂ x ₃ ...x _n E _n x ₁ ofn distinct verticesx _i andn distinct edgesE _j (n≧3) whereE _i ∩{x ₁,x ₂, ...,x _n }={x _i ,x _i+1} (x _n+1=x ₁). In the equivalent (0, 1)-matrix formulation, a special cycle corresponds to a square submatrix which is the incidence matrix of a cycle of size at least 3. Hypergraphs with no special cycles have been called totally balanced by Lovász. Simple hypergraphs with no special cycles onm vertices can be shown to have at most ( ₂ ^m )+m+1 edges where the empty edge is allowed. Such hypergraphs with the maximum number of edges have a fascinating structure and are called solutions. The main result of this paper is an algorithm that shows that a simple hypergraph on at mostm vertices with no special cycles can be completed (by adding edges) to a solution. Support provided by NSERC.     

Highly parity linked graphs

A graph G is k-linked if G has at least 2k vertices, and for any 2k vertices x ₁,x ₂, …, x _k ,y ₁,y ₂, …, y _k , G contains k pairwise disjoint paths P ₁, …, P _k such that P _i joins x _i and y _i for i = 1,2, …, k. We say that G is parity-k-linked if G is k-linked and, in addition, the paths P ₁, …, P _k can be chosen such that the parities of their length are prescribed. Thomassen [22] was the first to prove the existence of a function f(k) such that every f(k)-connected graph is parity-k-linked if the deletion of any 4k-3 vertices leaves a nonbipartite graph. In this paper, we will show that the above statement is still valid for 50k-connected graphs. This is the first result that connectivity which is a linear function of k guarantees the Erdős-Pósa type result for parity-k-linked graphs. Research partly supported by the Japan Society for the Promotion of Science for Young Scientists, by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research and by Inoue Research Award for Young Scientists.     

Existence of subgraph with orthogonal (g,f)-factorization

Simple graphs are considered. Let G be a graph andg(x) andf(x) integer-valued functions defined on V(G) withg(x)⩽f(x) for everyxɛV(G). For a subgraphH ofG and a factorizationF=|F ₁,F ₂,⃛,F ₁| ofG, if |E(H)∩E(F ₁)|=1,1⩽i⩽j, then we say thatF orthogonal toH. It is proved that for an (mg(x)+k,mf(x) -k)-graphG, there exists a subgraphR ofG such that for any subgraphH ofG with |E(H)|=k,R has a (g,f)-factorization orthogonal toH, where 1⩽k<m andg(x)⩾1 orf(x)⩾5 for everyxɛV(G). Project supported by the Chitia Postdoctoral Science Foundation and Chuang Xin Foundation of the Chinese Academy of Sciences.     

Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

Limiting distribution of sums of nonnegative stationary random variables

Summary Let {X _n,j,−∞<j<∞∼,n≧1, be a sequence of stationary sequences on some probability space, with nonnegative random variables. Under appropriate mixing conditions, it is shown thatS _n=X_n,1+…+X _n,n has a limiting distribution of a general infinitely divisible form. The result is applied to sequences of functions {f _n(x)∼ defined on a stationary sequence {X _j∼, whereX _n.f=f_n(X_j). The results are illustrated by applications to Gaussian processes, Markov processes and some autoregressive processes of a general type. This paper represents results obtained at the Courant Institute of Mathematical Sciences, New York University, under the sponsorship of the National Sciences Foundation, Grant MCS 82-01119.     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

Partitioning Posets

Given a poset P = (X, ≺ ), a partition X ₁, ..., X _k of X is called an ordered partition of P if, whenever x ∈ X _i and y ∈ X _j with x ≺ y, then i ≤ j. In this paper, we show that for every poset P = (X, ≺ ) and every integer k ≥ 2, there exists an ordered partition of P into k parts such that the total number of comparable pairs within the parts is at most (m − 1)/k, where m ≥ 1 is the total number of edges in the comparability graph of P. We show that this bound is best possible for k = 2, but we give an improved bound, , for k ≥ 3, where c(k) is a constant depending only on k. We also show that, given a poset P = (X, ≺ ) and an integer 2 ≤ k ≤ |X|, we can find an ordered partition of P into k parts that minimises the total number of comparable pairs within parts in time polynomial in the size of P. We prove more general, weighted versions of these results. Supported by an EPSRC doctoral training grant.     

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x ₁,…,x _n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that ‖x _i −x _j ‖≤‖L(x _i )−L(x _j )‖≤O(1)⋅‖x _i −x _j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion 2^2O(log*n)2^{2^{O(\log^{*}n)}} . On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-dimensional subspace E _n ⊆Y whose Euclidean distortion is at least 2^Ω(α(n)), where α is the inverse Ackermann function.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Cohen-macaulay bipartite graphs

Let G be a graph on the vertex set V={x ₁, ..., x _n}. Let k be a field and let R be the polynomial ring k[x ₁, ..., x _n]. The graph ideal I(G), associated to G, is the ideal of R generated by the set of square-free monomials x _ix_j so that x _i, is adjacent to x _j. The graph G is Cohen-Macaulay over k if R/I(G) is a Cohen-Macaulay ring. Let G be a Cohen-Macaulay bipartite graph. The main result of this paper shows that G{v} is Cohen-Macaulay for some vertex v in G. Then as a consequence it is shown that the Reisner-Stanley simplicial complex of I(G) is shellable. An example of N. Terai is presented showing these results fail for Cohen-Macaulay non bipartite graphs. Partially supported by COFAA-IPN, CONACyT and SNI, México.