An elementary approach to gap theorems

Using elementary comparison geometry, we prove: Let (M, g) be a simply-connected complete Riemannian manifold of dimension ≥ 3. Suppose that the sectional curvature K satisfies −1 − s(r) ≤ K ≤ −1, where r denotes distance to a fixed point in M. If lim _{r → ∞} e^2r s(r) = 0, then (M, g) has to be isometric to ℍ ⁿ . The same proof also yields that if K satisfies −s(r) ≤ K ≤ 0 where lim _{r → ∞} r ² s(r) = 0, then (M, g) is isometric to ℝ ⁿ , a result due to Greene and Wu. Our second result is a local one: Let (M, g) be any Riemannian manifold. For a ∈ ℝ, if K ≤ a on a geodesic ball B _p (R) in M and K = a on ∂B _p (R), then K = a on B _p (R).     

A Polynomial-Time Algorithm to Approximate the Mixed Volume within a Simply Exponential Factor

Let K=(K ₁,…,K _n ) be an n-tuple of convex compact subsets in the Euclidean space R ⁿ , and let V(⋅) be the Euclidean volume in R ⁿ . The Minkowski polynomial V _K is defined as V _K (λ ₁,…,λ _n )=V(λ ₁ K ₁+⋅⋅⋅+λ _n K _n ) and the mixed volume V(K ₁,…,K _n ) as

Contact Elements on Fibered Manifolds

For every product preserving bundle functor T ^μ on fibered manifolds, we describe the underlying functor of any order (r, s, q), s ≥ r ≤ q. We define the bundle K_k,l^r,s,q YK_{k,l}^{r,s,q} Y of (k, l)-dimensional contact elements of the order (r, s, q) on a fibered manifold Y and we characterize its elements geometrically. Then we study the bundle of general contact elements of type μ. We also determine all natural transformations of K_k,l^r,s,q YK_{k,l}^{r,s,q} Y into itself and of T( K_k,l^r,s,q Y )T\left( {K_{k,l}^{r,s,q} Y} \right) into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms from Y to K_k,l^r,s,q YK_{k,l}^{r,s,q} Y .     

MAXIMAL CALDERON-ZYGMUND SINGULAR INTEGRAL ON RBMO

Given a positive Radon measure μ on R^d satisfying the linear growth condition μ（B（x,r））≤C0r^n,x∈R^d,r〉0,（1） where n is a fixed number and O〈n≤d. When d-1〈n,it is proved that if Tt,N1=0,then the corresponding maximal Calderon-Zygmund singular integral is bounded from RBMO to itself only except that it is infinite μ-a. e. on R^d.     

Regelflächenscharen

Let K be any commutative field and V:=K ⁴. A collection of ruled quadrics in V is called a flock of ruled quadrics if the following holds true. (1) ⋃_{ℱ∈ G} ℱ = V; (2) There is a line S⊂V such that ℱ₁⋂ℱ₂= S for all distinct ℱ₁, ℱ₂∈ . The group ΓL(V) decomposes the set of all those flocks into equivalence classes. Besides that, we consider any cone R in V, say R:= {x∈V|x ₁ x ₃ - x ₂ ² = 0}. Let R denote the set of all regular points of R. Plane sections of R which do not contain the singular point of ℜ are called regular sections. We consider decompositions of R ^* by regular sections and their equivalence classes with respect to the symmetry group ΓL(V)R of the cone ℜ. The main result is as follows. There is a (natural) bijection between the classes of equivalent flocks of rules quadrics and the classes of equivalent decompositions of R ^* by regular sections. A brief discussion of those flocks of ruled quadrics on which the construction of the so-called Betten-Walker planes is based ends the paper. Provided that char K≠3, these planes exist if and only if x∈K→x ³∈K is bijective.      

ON THE NORMALITY OF MONOMIAL IDEALS OF MIXED PRODUCTS

Let R = K[x, y] be a polynomial ring in two disjoint sets of variables x, y over a field K. We study ideals of mixed products L = I_kJ_r + I_sJ_t such that k + r = s + t, where I_k (resp. J_r ) denotes the ideal of R generated by the square-free monomials of degree k (resp. r) in the x (resp. y ) variables. Our main result is a characterization of when a given ideal L of mixed products is normal.

     

The natural functions on the cotangent bundle of higher order vector tangent bundles over fibered manifolds

For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T ^*(J ^(r,s,q)(Y, R ^1,1)₀)^*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T ^*(J ^(r,s) (Y, R)₀)^*→R for any Y as above.     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.     

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.     

Rationally isotropic quadratic spaces are locally isotropic

Let R be a regular local ring, K its field of fractions and (V,ϕ) a quadratic space over R. Assume that R contains a field of characteristic zero we show that if (V,ϕ)⊗ _R K is isotropic over K, then (V,ϕ) is isotropic over R. This solves the characteristic zero case of a question raised by J.-L. Colliot-Thélène in [3]. The proof is based on a variant of a moving lemma from [7]. A purity theorem for quadratic spaces is proved as well. It generalizes in the charactersitic zero case the main purity result from [9] and it is used to prove the main result in [2].     

Some estimation theory on the sphere

Summary We consider distributions with densities of the formf(μ′x) andf(‖x _v ‖) where μ andx are unit vectors inR ^q and ‖x _v ‖ is the norm of the part ofx in somes dimensional subspaceV ofR ^q . For several loss functions, optimal Bayesian and Pitman estimators of μ andV are given. When uniform priors are used, these estimators are identical. Then the infinitesimal robustness characteristics of several special cases of these estimators are calculated.     

On the number of certain subgraphs contained in graphs with a given number of edges

All graphs considered are finite, undirected, with no loops, no multiple edges and no isolated vertices. For two graphsG, H, letN(G, H) denote the number of subgraphs ofG isomorphic toH. Define also, forl≧0,N(l, H)=maxN(G, H), where the maximum is taken over all graphsG withl edges. We determineN(l, H) precisely for alll≧0 whenH is a disjoint union of two stars, and also whenH is a disjoint union ofr≧3 stars, each of sizes ors+1, wheres≧r. We also determineN(l, H) for sufficiently largel whenH is a disjoint union ofr stars, of sizess ₁≧s ₂≧…≧s _r>r, provided (s ₁−s _r)²<s ₁+s _r−2r. We further show that ifH is a graph withk edges, then the ratioN(l, H)/l ^k tends to a finite limit asl→∞. This limit is non-zero iffH is a disjoint union of stars.     

Identity orientation of complete bipartite graphs

An identity orientation of a graph G=(V,E) is an orientation of some of the edges of E such that the resulting partially oriented graph has no automorphism other than the identity. We show that the complete bipartite graph K_s,t, with st, does not have an identity orientation if t3^s-log₃(s-1). We also show that if (r+1)(r+2)2s then K_s,3^s-r does have an identity orientation. These results improve the previous bounds obtained by Harary and Jacobson (Discuss. Math. - Graph Theory 21 (2001) 158). We use these results to determine exactly the values of t for which an identity orientation of K_s,t exists for 2s17.     

Inequalities for convex bodies and polar reciprocal lattices inR n

LetL be a lattice and letU be ano-symmetric convex body inR ⁿ . The Minkowski functional ∥ ∥ _U ofU, the polar bodyU ⁰, the dual latticeL ^*, the covering radius μ(L, U), and the successive minima λ _i (L,U)i=1,...,n, are defined in the usual way. Let ℒ _n be the family of all lattices inR ⁿ . Given a pairU,V of convex bodies, we define and kh(U, V) is defined as the smallest positive numbers for which, given arbitraryL∈ℒ _n andu∈R ⁿ /(L+U), somev∈L ^* with ∥v∥ _V ≤sd(uv, ℤ) can be found. Upper bounds for jh(U, U ⁰), j=k, l, m, belong to the so-called transference theorems in the geometry of numbers. The technique of Gaussian-like measures on lattices, developed in an earlier paper [4] for euclidean balls, is applied to obtain upper bounds for jh(U, V) in the case whenU, V aren-dimensional ellipsoids, rectangular parallelepipeds, or unit balls inl _p ⁿ , 1≤p≤∞. The gaps between the upper bounds obtained and the known lower bounds are, roughly speaking, of order at most logn asn→∞. It is also proved that ifU is symmetric through each of the coordinate hyperplanes, then jh(U, U ⁰) are less thanCn logn for some numerical constantC.     

Subvarieties of the matrix variety of second order

LetR _s be the subalgebra ofM ₂(K[t]/(t ^s )) generated bye ₁₁,e ₂₂,te ₁₂ andte ₂₁, whereK is a field of characteristic 0,K[t] is the polynomial algebra in one variablet and (t ^s ) is the principal ideal inK[t], generated byt ^s . The main result of this paper is that we have described theT-idealT(R _s ). Besides the two matrix polynomial identities — the standart identityS ₄ and the identity of Hall, thisT-ideal is generated by one more explicitly given identity. The algebrasR _s are interesting due to the fact that the proper identities of any subvarietyu of the variety ℳ=varM ₂(K), generated by the matrix algebraM ₂(K) of second order overK, asymptoticaly coincide with the proper identities of someR _s . Partially supported by Grant MM605/96 of the Bulgarian Foundation for Scientific Research.     

Equitable versus nearly equitable coloring and the Chen-Lih-Wu conjecture

Chen, Lih, and Wu conjectured that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are K _r,r (for odd r) and K _r+1. If true, this would be a strengthening of the Hajnal-Szemerédi Theorem and Brooks’ Theorem. We extend their conjecture to disconnected graphs. For r ≥ 6 the conjecture says the following: If an r-colorable graph G with maximum degree r is not equitably r-colorable then r is odd, G contains K _r,r and V(G) partitions into subsets V ₀, …, V _t such that G[V ₀] = K _r,r and for each 1 ≤ i ≤ t, G[V _i ] = K _r . We characterize graphs satisfying the conclusion of our conjecture for all r and use the characterization to prove that the two conjectures are equivalent. This new conjecture may help to prove the Chen-Lih-Wu Conjecture by induction.     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

Kakeya-type sets in finite vector spaces

For a finite vector space V and a nonnegative integer r≤dim V, we estimate the smallest possible size of a subset of V, containing a translate of every r-dimensional subspace. In particular, we show that if K⊆V is the smallest subset with this property, n denotes the dimension of V, and q is the size of the underlying field, then for r bounded and r<n≤rq ^r−1, we have |V∖K|=Θ(nq ^n−r+1); this improves the previously known bounds |V∖K|=Ω(q ^n−r+1) and |V∖K|=O(n ² q ^n−r+1).     

Strict Betweennesses Induced by Posets as well as by Graphs

For a finite poset P = (V, ≤ ), let _s(P){\cal B}_s(P) consist of all triples (x,y,z) ∈ V ³ such that either x < y < z or z < y < x. Similarly, for every finite, simple, and undirected graph G = (V,E), let B_s(G){\cal B}_s(G) consist of all triples (x,y,z) ∈ V ³ such that y is an internal vertex on an induced path in G between x and z. The ternary relations B_s(P){\cal B}_s(P) and B_s(G){\cal B}_s(G) are well-known examples of so-called strict betweennesses. We characterize the pairs (P,G) of posets P and graphs G on the same ground set V which induce the same strict betweenness relation B_s(P)=B_s(G){\cal B}_s(P)={\cal B}_s(G).