The chromatic numbers of random hypergraphs

For a pair of integers 1≤γ<r, the γ-chromatic number of an r-uniform hypergraph H=(V, E) is the minimal k, for which there exists a partition of V into subsets T₁,…,T_k such that |e∩T_i|≤γ for every e∈E. In this paper we determine the asymptotic behavior of the γ-chromatic number of the random r-uniform hypergraph H_r(n, p) for all possible values of γ and for all values of p down to p=Θ(n^−r+1). © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 12: 381–403, 1998     

A numerical method for computing radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation

In this article we develop a finite‐difference scheme to approximate radially symmetric solutions of a dissipative nonlinear modified Klein‐Gordon equation subject to smooth initial conditions ? and ψ in an open sphere D around the origin, with constant internal and external damping coefficients—β and γ, respectively—, and nonlinear term of the form G′(w) = w^p, with p > 1 an odd number. The functions ? and ψ are radially symmetric in D, and ?, ψ, r?, and rψ are assumed to be small at infinity. We prove that our scheme is consistent order ??(Δt²) + ??(Δr²) for G′ identically equal to zero and provide a necessary condition for it to be stable order n. Part of our study will be devoted to compare the physical effects of β and γ. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Negative Latin square type partial difference sets and amorphic association schemes with Galois rings

A partial difference set (PDS) having parameters (n², r(n?1), n+r²?3r, r²?r) is called a Latin square type PDS, while a PDS having parameters (n², r(n+1), ?n+r²+3r, r² +r) is called a negative Latin square type PDS. There are relatively few known constructions of negative Latin square type PDSs, and nearly all of these are in elementary abelian groups. We show that there are three different groups of order 256 that have all possible negative Latin square type parameters. We then give generalized constructions of negative Latin square type PDSs in 2‐groups. We conclude by discussing how these results fit into the context of amorphic association schemes and by stating some open problems. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 266‐282, 2009     

Hypersurfaces with null higher order mean curvature

A hypersurface M ⁿ immersed in a space form is r-minimal if its (r + 1) ^th -curvature (the (r + 1) ^th elementary symmetric function of its principal curvatures) vanishes identically. Let W be the set of points which are omitted by the totally geodesic hypersurfaces tangent to M. We will prove that if an orientable hypersurface M ⁿ is r-minimal and its r ^th -curvature is nonzero everywhere, and the set W is nonempty and open, then M ⁿ has relative nullity n − r. Also we will prove that if an orientable hypersurface M ⁿ is r-minimal and its r ^th -curvature is nonzero everywhere, and the ambient space is euclidean or hyperbolic and W is nonempty, then M ⁿ is r-stable.     

Balloons,cut‐edges,matchings, and total domination in regular graphs of odd degree

A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}A balloon in a graph G is a maximal 2‐edge‐connected subgraph incident to exactly one cut‐edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut‐edges, and let α′(G) be the maximum size of a matching. Let ${\mathcal{F}}_{{{n}},{{r}}}$ be the family of connected (2r+1)‐regular graphs with n vertices, and let ${{b}}={{max}}\{{{b}}({{G}}): {{G}}\in {\mathcal{F}}_{{{n}},{{r}}}\}$. For ${{G}}\in{\mathcal{F}}_{{{n}},{{r}}}$, we prove the sharp inequalities c(G)?[r(n?2)?2]/(2r²+2r?1)?1 and α′(G)?n/2?rb/(2r+1). Using b?[(2r?1)n+2]/(4r²+4r?2), we obtain a simple proof of the bound proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number γ_t(G) of a cubic graph, we prove γ_t(G)?n/2?b(G)/2 (except that γ_t(G) may be n/2?1 when b(G)=3 and the balloons cover all but one vertex). With α′(G)?n/2?b(G)/3 for cubic graphs, this improves the known inequality γ_t(G)?α′(G). © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 116–131, 2010     

Coloring uniform hypergraphs with few colors

Let m(r, k) denote the minimum number of edges in an r‐uniform hypergraph that is not k‐colorable. We give a new lower bound on m(r, k) for fixed k and large r. Namely, we prove that if k ≥ 2ⁿ, then m(r, k) ≥ ?(k)k^r(r/ln r)^n/(n+1). © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

A shortness exponent for r-regular r-connected graphs

Let r≧ 3 be an integer. It is shown that there exists ε= ε(r), 0 < ε < 1, and an integer N = N(r) > 0 such that for all n ≧ N (if r is even) or for all even n ≧ N(if r is odd), there is an r-connected regular graph of valency r on exactly n vertices whose longest cycles have fewer than n^ε vertices.     

On explicit factors of cyclotomic polynomials over finite fields

We study the explicit factorization of 2 ⁿ r-th cyclotomic polynomials over finite field \mathbbF_q{\mathbb{F}_q} where q, r are odd with (r, q) = 1. We show that all irreducible factors of 2 ⁿ r-th cyclotomic polynomials can be obtained easily from irreducible factors of cyclotomic polynomials of small orders. In particular, we obtain the explicit factorization of 2 ⁿ 5-th cyclotomic polynomials over finite fields and construct several classes of irreducible polynomials of degree 2 ^n–2 with fewer than 5 terms.     

Symplectic embeddings of ellipsoids

We study the rigidity and flexibility of symplectic embeddings in the model case in which the domain is a symplectic ellipsoid. It is first proved that under the conditionr _n ² ≤2r ₁ ² the symplectic ellipsoidE(r ₁,…,r _n)with radiir ₁≤…≤r _ndoes not symplectically embed into a ball of radius strictly smaller thanr _n.We then use symplectic folding to see that this condition is sharp. We finally sketch a proof of the fact that any connected symplectic 4-manifold of finite volume can be asymptotically filled with skinny ellipoids.     

Embeddings on a boolean cube

In this paper, we characterize a class of graphs which can be embedded on a boolean cube. Some of the graphs in this class are identified with the well known graphs such asmulti-dimensional mesh of trees, tree of meshes, etc. We suggest (i) an embedding of anr-dimensional mesh of trees ofn ^r (r+1)–rn ^r–1 nodes on a boolean cube of (2n) ^r nodes, and (ii) an embedding of a tree of meshes with 2n ² logn+n ² nodes on a boolean cube withn ² exp₂ (log (2 logn+1)]) nodes.     

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ<Superscript>3</Superscript>

Symplectic instanton vector bundles on the projective space ℙ³ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I _n;r of rank-2r symplectic instanton vector bundles on ℙ³ with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I _n;r ^* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).     

Equiangular lines in C (part II)

A subset S of a complex projective space is F-regular provided each two points of S have the same non-zero distance and each subset of three points of S has the same shape invariant. The aim of this paper is the determination for any odd integer r, of the largest integer n(r) such tht CP^r−1 contains an F-regular subset of n(r) points.It is established that n(r) ≤ 2r − 2 for any odd integer r and n(1 + 2^s) = 2^s+1 for any integer s.     

Some Estimates for Radial Fourier Multiplier Operators with Slowly Decaying Kernels

We consider the restriction to radial functions of a class of radial Fourier multiplier operators containing the Bochner-Riesz multiplier operator. The convolution kernel K(x) of an operator in this class decays too slowly at infinity to be integrable, but has enough oscillation to achieve L^p -boundedness for p inside a suitable interval (a, b). We prove boundedness results for the maximal operator Kf(x) = sup_{r>0 r}ⁿ∣K(r) * f(x)∣ associated with such a kernel. The maximal operator is shown to be weak type bounded at the lower critical index a, restricted weak type bounded at the upper critical index b, and strong type bounded between. This together with our assumptions on K(x) leads to the pointwise convergence result lim_γ→ γⁿ K(γ·) * f(x) = cf(x) a. e. for radial f ? L^P(?ⁿ), a ≥ p > b.     

On Order Types of Systems of Segments in the Plane

Let r(n) denote the largest integer such that every family C\mathcal{C} of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Pach and Tóth gave a construction that shows r(n) < n ^log8/log9 (Pach and Tóth 2009). They also stated that one can apply the Erdős–Szekeres theorem for convex sets in Pach and Tóth (Discrete Comput Geom 19:437–445, 1998) to obtain r(n) > log₁₆ n. In this note, we will show that r(n) > cn ^1/4 for some absolute constant c.     

Covering Complete Hypergraphs with Cuts of Minimum Total Size

A cut [X, V − X] in a hypergraph with vertex-set V is the set of all edges that meet both X and V − X. Let s _r (n) denote the minimum total size of any cover of the edges of the complete r-uniform hypergraph on n vertices K_n^r{K_n^r} by cuts. We show that there is a number n _r such that for every n > n _r , s _r (n) is uniquely achieved by a cover with ?\fracn-1r-1?{\lfloor \frac{n-1}{r-1}\rfloor} cuts [X _i , V − X _i ] such that the X _i are pairwise disjoint sets of size at most r − 1. We show that c₁r2^r < n_r < c₂r⁵2^r{c_1r2^r < n_r < c_2r^52^r} for some positive absolute constants c ₁ and c ₂. Using known results for s ₂(n) we also determine s ₃(n) exactly for all n.     

Pavages des simplexes, schémas de graphes recollés et compactification des PGLrn+1/PGLr

Some equivariant compactifications of the quotients PGL _r ^{n +1}/PGL _r are constructed. Each one is decomposed into locally closed strata which are smooth, are indexed by the entire convex pavings of the simplex of dimension n and admit a modular interpretation deduced from that of the Grassmann varieties. Together, they form a simplicial scheme which “compactifies” the classifying simplicial scheme of PGL _r consisting of all the quotients PGL _r ^{n +1}/PGL _r , n≥0.

Oblatum 8-IV-1998 & 8-X-1998 / Published online: 28 January 1999     

Regular factors in K1,n free graphs

A graph is said to be K_1,n-free, if it contains no K_1,n as an induced subgraph. We prove that for n ? 3 and r ? n ?1, if G is a K_1,n-free graph with minimum degree at least (n²/4(n ?1))r + (3n ?6)/2 + (n ?1)/4r, then G has an r-factor (in the case where r is even, the condition r ? n ?1 can be dropped).     

Cuttings for Disks and Axis-Aligned Rectangles in Three-Space

We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈?, a $\frac{1}{r}We present new asymptotically tight bounds on cuttings, a fundamental data structure in computational geometry. For n objects in space and a parameter r∈ℕ, a \frac1r\frac{1}{r} -cutting is a covering of the space with simplices such that the interior of each simplex intersects at most n/r objects. For n pairwise disjoint disks in ℝ³ and a parameter r∈ℕ, we construct a \frac1r\frac{1}{r} -cutting of size O(r ²). For n axis-aligned rectangles in ℝ³, we construct a \frac1r\frac{1}{r} -cutting of size O(r ^3/2). As an application related to multi-point location in three-space, we present tight bounds on the cost of spanning trees across barriers. Given n points and a finite set of disjoint disk barriers in ℝ³, the points can be connected with a straight line spanning tree such that every disk is stabbed by at most O(?n)O(\sqrt{n}) edges of the tree. If the barriers are axis-aligned rectangles, then there is a straight line spanning tree such that every rectangle is stabbed by O(n ^1/3) edges. Both bounds are best possible.     

The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions

In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S ^r−1 ⊂ R^r. The hyperinterpolation approximation L _n ƒ, where ƒ ∈ C(S ^r −1), is derived from the exact L ² orthogonal projection Π ƒ onto the space P _n ^r (S ^r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n ^r /2−1), which is the same as that of the orthogonal projection Π_n, and best possible among all linear projections onto P _n ^r (S ^r −1).