On a problem of Erdős and Lovász: Random lines in a projective plane

Letn(k) be the least size of an intersecting family ofk-sets with cover numberk, and let _k denote any projective plane of orderk–1.Theorem There is a constant A such that ifH is a random set ofm Aklogk lines from _k then P_r(H<)0(k).Corollary If there exists a _k thenn(k)=O(klogk). These statements were conjectured by P. Erds and L. Lovász in 1973.Supported in part by NSF-DMS87-83558 and AFOSR grants 89-0066, 89-0512 and 90-0008     

Convergence to equilibrium of critical branching particle systems and superprocesses,and related nonlinear partial differential equations

We consider a class of multitype particle systems in ^d undergoing spatial diffusion and critical stable multitype branching, and their limits known as critical stable multitype Dawson-Watanabe processes, or superprocesses. We show that for large classes of initial states, the particle process and the superprocess converge in distribution towards known equilibrium states as time tends to infinity. As an application we obtain the asymptotic behavior of a system of nonlinear partial differential equations whose solution is related to the distribution of both the particle process and the superprocess.Research partially supported by CONACyT (Mexico), CNRS (France) and BMfWuF (Austria).     

Designs as maximum codes in polynomial metric spaces

Finite and infinite metric spaces % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] that are polynomial with respect to a monotone substitution of variable t(d) are considered. A finite subset (code) W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is characterized by the minimal distance d(W) between its distinct elements, by the number l(W) of distances between its distinct elements and by the maximal strength (W) of the design generated by the code W. A code W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is called a maximum one if it has the greatest cardinality among subsets of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with minimal distance at least d(W), and diametrical if the diameter of W is equal to the diameter of the whole space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. Delsarte codes are codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with (W)2l(W)–1 or (W)=2l(W)–2>0 and W is a diametrical code. It is shown that all parameters of Delsarte codes W) % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are uniquely determined by their cardinality |W| or minimal distance d(W) and that the minimal polynomials of the Delsarte codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are expansible with positive coefficients in an orthogonal system of polynomials {Q _i(t)} corresponding to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. The main results of the present paper consist in a proof of maximality of all Delsarte codes provided that the system {Q _i)} satisfies some condition and of a new proof confirming in this case the validity of all the results on the upper bounds for the maximum cardinality of codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \]% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with a given minimal distance, announced by the author in 1978. Moreover, it appeared that this condition is satisfied for all infinite polynomial metric spaces as well as for distance-regular graphs, decomposable in a sense defined below. It is also proved that with one exception all classical distance-regular graphs are decomposable. In addition for decomposable distance-regular graphs an improvement of the absolute Delsarte bound for diametrical codes is obtained. For the Hamming and Johnson spaces, Euclidean sphere, real and complex projective spaces, tables containing parameters of known Delsarte codes are presented. Moreover, for each of the above-mentioned infinite spaces infinite sequences (of maximum) Delsarte codes not belonging to tight designs are indicated.     

Primes,irreducibles and extremal lattices

This paper studies certain types of join and meet-irreducibles called coprimes and primes. These elements can be used to characterize certain types of lattices. For example, a lattice is distributive if and only if every join-irreducible is coprime. Similarly, a lattice is meet-pseudocomplemented if and only if each atom is coprime. Furthermore, these elements naturally decompose lattices into sublattices so that often properties of the original lattice can be deduced from properties of the sublattice. Not every lattice has primes and coprimes. This paper shows that lattices which are long enough must have primes and coprimes and that these elements and the resulting decompositions can be used to study such lattices.The length of every finite lattice is bounded above by the minimum of the number of meet-irreducibles (meet-rank) and the number of join-irreducibles (join-rank) that it has. This paper studies lattices for which length=join-rank or length=meet-rank. These are called p-extremal lattices and they have interesting decompositions and properties. For example, ranked, p-extremal lattices are either lower locally distributive (join-rank=length), upper locally distributive (meet-rank=length) or distributive (join-rank=meet-rank=length). In the absence of the Jordan-Dedekind chain condition, p-extremal lattices still have many interesting properties. Of special interest are the lattices that satisfy both equalities. Such lattices are called extremal; this class includes distributive lattices and the associativity lattices of Tamari. Even though they have interesting decompositions, extremal lattices cannot be characterized algebraically since any finite lattice can be embedded as a subinterval into an extremal lattice. This paper shows how prime and coprime elements, and the poset of irreducibles can be used to analyze p-extremal and other types of lattices.The results presented in this paper are used to deduce many key properties of the Tamari lattices. These lattices behave much like distributive lattices even though they violate the Jordan-Dedekind chain condition very strongly having maximal chains that vary in length from N-1 to N(N-1)/2 where N is a parameter used in the construction of these lattices.     

Fourier analysis of the approximation power of principal shift-invariant spaces

The approximation order provided by a directed set {S _h } _h>0 of spaces, each spanned by thehZ ^d -translates of one function, is analyzed. The nearoptimal approximants of [R2] from eachs _h to the exponential functions are used to establish upper bounds on the approximation order. These approximants are also used on the Fourier transform domain to yield approximations for other smooth functions, and thereby provide lower bounds on the approximation order. As a special case, the classical Strang-Fix conditions are extended to bounded summable generating functions.The second part of the paper consists of a detailed account of various applications of these general results to spline and radial function theory. Emphasis is given to the case when the scale {s _h } is obtained froms ₁ by means other than dilation. This includes the derivation of spectral approximation orders associated with smooth positive definite generating functions.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Some properties of the riesz charge associated with a δ-subharmonic function

The first property is a refinement of earlier results of Ch. de la Vallée Poussin, M. Brelot, and A. F. Grishin. Let w=u–v with u, v superharmonic on a suitable harmonic space (for example an open subset of R ⁿ ), and let [w]=[u]–[v] denote the associated Riesz charge. If w0, and if E denotes the set of those points of at which the lim inf of w in thefine topology is 0, then the restriction of [w] to E is 0. Another property states that, if e denotes a polar subset of such that the fine lim inf of |w| at each point of e is finite, then the restriction of [w] to e is 0.     

Problème inverse de la théorie du potentiel logarithmique pour les lemniscates

The paper is related to the question of uniqueness in the inverse logarithmic potential problem. This question is to find the conditions on which two domains D ₁ and D ₂ producing the same external potential must coincide. Assuming the general hypothesis of regularity and an additional condition of connectivity of (D₁D₂)_c, we prove a theorem of uniqueness in the case when one of the domains is a lemniscate. The main tool is one lemma for Cauchy's potential due to M. Sakai. We give a simple proof of its extension to Newtonian potential.

     

Dirichlet norms,capacities and generalized isoperimetric inequalities for Markov operators

We define the notion of p-capacity for a reversible Markov operator on a general measure space and prove that uniform estimates for the ratio of capacity and measure are equivalent to certain imbedding theorems for the Orlicz and Dirichlet norms. As a corollary we get results on connections between embedding theorems and isoperimetric properties for general Markov operators and, particularly, a generalization of the Kesten theorem on the spectral radius of random walks on amenable groups for the case of arbitrary graphs with non-finitely supported transition probabilities.     

Harmonic maps with fixed singular sets

Suppose Ω is a smooth domain in R^m,N is a compact smooth Riemannian manifold, andZ is a fixed compact subset of Ω having finite (m − 3)-dimensional Minkowski content (e.g.,Z ism − 3 rectifiable). We consider various spaces of harmonic mapsu: Ω →N that have a singular setZ and controlled behavior nearZ. We study the structure of such spacesH and questions of existence, uniqueness, stability, and minimality under perturbation. In caseZ = 0,H is a Banach manifold locally diffeomorphic to a submanifold of the product of the boundary data space with a finite-dimensional space of Jacobi fields with controlled singular behavior. In this smooth case, the projection ofu εH tou |ϖΩ is Fredholm of index 0. R. H.’s research partially supported by the National Science Foundation.