K-theory of twisted differential operators on flag varieties

Let ? be a semisimple Lie algebra over k, an algebraically closed field of characteristic zero, and let ?⊂? be a Cartan subalgebra inside a Borel subalgebra of ?. Let ? be the enveloping algebra of ?. For μ∈? ^* let M(μ) denote the corresponding Verma module and let ? _μ = ?/ Ann  M(μ). Let W be the Weyl group and let W ⁰ _μ be the stabiliser of μ in W. We prove the following theorem, which affirms a conjecture of T.J. Hodges. Oblatum 16-XII-1994     

Structures and Chromaticity of Extremal 3-Colourable Sparse Graphs

 Assume that G is a 3-colourable connected graph with e(G) = 2v(G) −k, where k≥ 4. It has been shown that s ₃(G) ≥ 2 ^{k −3}, where s _r (G) = P(G,r)/r! for any positive integer r and P(G, λ) is the chromatic polynomial of G. In this paper, we prove that if G is 2-connected and s ₃(G) < 2 ^{k −2}, then G contains at most v(G) −k triangles; and the upper bound is attained only if G is a graph obtained by replacing each edge in the k-cycle C _k by a 2-tree. By using this result, we settle the problem of determining if W(n, s) is χ-unique, where W(n, s) is the graph obtained from the wheel W _n by deleting all but s consecutive spokes. Received: January 29, 1999 Final version received: April 8, 2000     

Hypergraphs with independent neighborhoods

For each k ≥ 2, let ρ _k ∈ (0, 1) be the largest number such that there exist k-uniform hypergraphs on n vertices with independent neighborhoods and (ρ _k + o(1))( _k ⁿ ) edges as n → ∞. We prove that ρ _k = 1 − 2logk/k + Θ(log log k/k) as k → ∞. This disproves a conjecture of Füredi and the last two authors.     

Bivariate spline interpolation with optimal approximation order

Let Δ be a triangulation of some polygonal domain Ω ⊂ R² and let S_q^r(Δ) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to Δ. We develop the first Hermite-type interpolation scheme for S _q ^r (Δ), q ≥ 3r + 2, whose approximation error is bounded above by Kh ^q +1, where h is the maximal diameter of the triangles in Δ, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and near-singular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of S _q ^r (Δ). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [7] and [18].     

Tiling Transitive Tournaments and Their Blow-ups

Let TT _k denote the transitive tournament on k vertices. Let TT(h,k) denote the graph obtained from TT _k by replacing each vertex with an independent set of size h≥1. The following result is proved: Let c ₂=1/2, c ₃=5/6 and c _k =1−2^{−k−log k} for k≥4. For every ∈>0 there exists N=N(∈,h,k) such that for every undirected graph G with n>N vertices and with δ(G)≥c _k n, every orientation of G contains vertex disjoint copies of TT(h,k) that cover all but at most ∈n vertices. In the cases k=2 and k=3 the result is asymptotically tight. For k≥4, c _k cannot be improved to less than 1−2^{−0.5k(1+o(1))}. This revised version was published online in June 2006 with corrections to the Cover Date.     

Universal Deformation Rings for the Symmetric Group <Emphasis Type="Italic">S</Emphasis><Subscript>4</Subscript>

Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Let S ₄ denote the symmetric group on 4 letters. We determine the universal deformation ring R(S ₄,V) for every kS ₄-module V which has stable endomorphism ring k and show that R(S ₄,V) is isomorphic to either k, or W[t]/(t ²,2t), or the group ring W[ℤ/2]. This gives a positive answer in this case to a question raised by the first author and Chinburg whether the universal deformation ring of a representation of a finite group with stable endomorphism ring k is always isomorphic to a subquotient ring of the group ring over W of a defect group of the modular block associated to the representation.     

An asymptotic approximation of Wallis’ sequence

An asymptotic approximation of Wallis’ sequence W(n) = Π _k=1 ⁿ 4k ²/(4k ² − 1) obtained on the base of Stirling’s factorial formula is presented. As a consequence, several accurate new estimates of Wallis’ ratios w(n) = Π _k=1 ⁿ (2k−1)/(2k) are given. Also, an asymptotic approximation of π in terms of Wallis’ sequence W(n) is obtained, together with several double inequalities such as, for example,

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     

Torseurs sous une variété abélienne sur R((t))

Let V be an henselian discrete valuation ring with real closed residue field and let k be its quotient ring; we denote by k ⁺ and k ⁻ the two real closures of k. Consider a k-abelian variety A. We compute the Galois-cohomology group H ¹(k,A) in terms of the reduction of the dual variety of A and of the semi-algebraic topology of A(k ⁺) and A(k ⁻). The tools we need are Ogg's results concerning valuation rings with algebraically closed residue field, Hochschild–Serre spectral sequence and Scheiderer's local-global principles. At the end we study more precisely the case of an elliptic curve. Received: 23 October 2000     

A nodal basis of Cμ-rational spline functions on triangulations

Let ρ be a triangulation of a polygonal domain D⊂R² with vertices V={v_i:l≤i≤N_v} and RS^k(D, ρ)={u∈C^k(D): ≠ T∈ρ, u/_T is a rational function}. The purpose of this paper is to study the existence and construction of C^μ-rational spline functions on any triangulation ρ for CAGD. The Hermite problem H^μ(V,U)={find u∈U: D^αu(v_i)=D^αf(v_i),|α|≤μ} is solved by the generalized wedge function method in rational spline function family, i.e. U=RS^μ. this solution needs only the knowledge of partial derivatives of order≤μ at v_i. The explicit repesentations of all C^μ-GWF(generalized wedge functions)and the interpolating operator with degree of precision at least 2μ+1 for any triangulation are given.     

Embeddings ofl p m intol ∞ n , 1≦p≦2

Isomorphic embeddings ofl _l ^m intol _∞ ⁿ are studied, and ford(n, k)=inf{‖T ‖ ‖T ⁻¹ ‖;T varies over all isomorphic embeddings ofl ₁ ^[klog2n] intol _∞ ⁿ we have that lim _n→∞ d(n, k)=γ(k)⁻¹,k>1, whereγ(k) is the solution of (1+γ)ln(1+γ)+(1 −γ)ln(1 −γ)=k ⁻¹ln4. Here [x] denotes the integer part of the real numberx.     

On the distribution of reducible polynomials

Let ℛ _n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ _n (t)| of the set ℛ _n (t) by showing that, as t → ∞, t ² log t ≪ |ℛ₂(t)| ≪ t ² log t and t ⁿ ≪ |ℛ _n (t)| ≪ t _n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ _k,n (t) ⊂ ℛ _n (t) such that p(X) ∈ ℛ _k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t ^k+1 ≪ |ℛ _k,n (t)| ≪ t ^k+1 and hence |ℛ _n−1,n (t)| ≫ |ℛ _n (t)| so that ℛ _n−1,n (t) is the dominating subclass of ℛ _n (t) since we can show that |ℛ _n (t)∖ℛ _n−1,n (t)| ≪ t ⁿ⁻¹(log t)².On the contrary, if R _n ^s (t) is the total number of all polynomials in ℛ _n (t) which split completely into linear factors over ℤ, then t ²(log t) ⁿ⁻¹ ≪ R _n ^s (t) ≪ t ² (log t) ⁿ⁻¹ (t → ∞) for every fixed n ≥ 2.      

A Continuous Analogue of the Upper Bound Theorem

For an absolutely continuous probability measure μ on R ^d and a nonnegative integer k , let \tilde s _k (μ ,\origin ) denote the probability that the convex hull of k+d+1 random points which are i.i.d. according to μ contains the origin \bf 0 . For d and k given, we determine a tight upper bound on \tilde s _k (μ ,\origin ) , and we characterize the measures in R ^d which attain this bound. As we will see, this result can be considered a continuous analogue of the Upper Bound Theorem for the maximal number of faces of convex polytopes with a given number of vertices. For our proof we introduce so-called h -functions, continuous counterparts of h -vectors of simplicial convex polytopes. Received April 14, 2000, and in revised form October 6, 2000. Online publication June 20, 2001.     

On thek dimensional Piatetski-Shapiro prime number theorem

Thek-dimensional Piatetski-Shapiro prime number problem fork⩾3 is studied. Let π(x ₁ c ₁,⋯,c _k ) denote the number of primesp withp⩽x, , where 1<c ₁<⋯<c _k are fixed constants. It is proved that π(x;c ₁,⋯,c _k ) has an asymptotic formula ifc ₁ ⁻¹ +⋯+c _k ⁻¹ >k−k/(4k ²+2). Project supported by the National Natural Science Foundation of China (Grant No. 19801021) and the Natural Science Foundation of Shandong Province (Grant No.Q98A02110).     

Median spheres: theory, algorithms, applications

A generalized hypersphere is either a hyperplane or a hypersphere, which consists of all points equidistant from a center. Geometrically, a weighted median hypersphere minimizes a weighted average of the distances from it to finitely many data points. As proved here, for each finite data set there exists at least one weighted median generalized hypersphere. Moreover, denote the sums of the weights of the data points inside by W ⁻, outside by W ⁺, and on the hypersphere by W ⁰. The present results show that each weighted median hypersphere is a weighted pseudo-halving hypersphere, in the sense that |W ⁻ − W ⁺| < W ⁰, and passes through at least two distinct data points. Combinatorically, a hypersphere is blocked if and only if it passes through data points in general position, in the sense that no other hypersphere passes through the same data points. A hypersphere is a halving hypersphere if and only if it is blocked, contains exactly k data points inside, confines exactly ℓ data points outside, and |k − ℓ| ≤ 1. In the plane, the present results also show that if a median circle is not a halving circle, then moving its center along a median between two data points on it until it passes through the next data point yields a halving circle. Relative to the center, if the direction cosines of the external and internal data points have the same mean and variance, then the median circle must be blocked, and stays so under sufficiently small perturbations of the data. Moreover, for every set of four points, at least one unweighted median circle is blocked. These results lend credence to a variant of a method used by archaeologists, and explain some findings from operations research.     

On the difference between the number of prime divisors at consecutive integers

Suppose thatg(n) is equal to the number of divisors ofn, counting multiplicity, or the number of divisors ofn, a≠0 is an integer, andN(x,b)=|{n∶n≤x, g(n+a)−g(n)=b orb+1}|. In the paper we prove that sup _b N(x,b)≤C(a)x)(log log 10 _x )^−1/2 and that there exists a constantC(a,μ)>0 such that, given an integerb |b|≤μ(log logx)^1/2,x≥x _o, the inequalityN(x,b)≥C(a,μ)x(log logx(^−1/2) is valid. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 579–595, October, 1999.     

Categoricity,amalgamation, and tameness

Theorem: For each 2 ≤ k < ω there is an -sentence ϕ_k such that (1) ϕ_k is categorical in μ if μ≤ℵ_k−2; (2) ϕ_k is not ℵ_k−2-Galois stable (3) ϕ_k is not categorical in any μ with μ>ℵ_k−2; (4) ϕ_k has the disjoint amalgamation property (5) For k > 2 (a) ϕ_k is (ℵ₀, ℵ_k−3)-tame; indeed, syntactic first-order types determine Galois types over models of cardinality at most ℵ_k−3; (b) ϕ_k is ℵ_m-Galois stable for m ≤ k − 3 (c) ϕ_k is not (ℵ_k−3, ℵ_k−2). The first author is partially supported by NSF grant DMS-0500841.     

Bounding the number of connected components of a real algebraic set

For every polynomial mapf=(f ₁,…,f _k): ℝ ⁿ →ℝ ^k , we consider the number of connected components of its zero set,B(Z _f) and two natural “measures of the complexity off,” that is the triple(n, k, d), d being equal to max(degree off _i), and thek-tuple (Δ₁,...,Δ₄), Δ _k being the Newton polyhedron off _i respectively. Our aim is to boundB(Z _f) by recursive functions of these measures of complexity. In particular, with respect to (n, k, d) we shall improve the well-known Milnor-Thom’s bound μ _d (n)=d(2d−1) ⁿ⁻¹. Considered as a polynomial ind, μ _d (n) has leading coefficient equal to 2 ⁿ⁻¹. We obtain a bound depending onn, d, andk such that ifn is sufficiently larger thank, then it improves μ _d (n) for everyd. In particular, it is asymptotically equal to 1/2(k+1)n ^k−1 dⁿ, ifk is fixed andn tends to infinity. The two bounds are obtained by a similar technique involving a slight modification of Milnor-Thom's argument, Smith's theory, and information about the sum of Betti numbers of complex complete intersections.     

Subgraphs of colour-critical graphs

Some problems and results on the distribution of subgraphs in colour-critical graphs are discussed. In section 3 arbitrarily largek-critical graphs withn vertices are constructed such that, in order to reduce the chromatic number tok−2, at leastc _k n ² edges must be removed. In section 4 it is proved that a 4-critical graph withn vertices contains at mostn triangles. Further it is proved that ak-critical graph which is not a complete graph contains a (k−1)-critical graph which is not a complete graph.     

The total reducibility order of a polynomial in two variables

LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ _i=1 ^n(λ) A _iλ ^k _μ whereA _iλ are distinct primes. Let ϱ_λ(A) =n(λ) − 1 and let ρ(A) = Σ_λɛσ(A)ϱ_λ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.