On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

Concerning the order of approximation of periodic continuous functions by trigonometric interpolation polynomials

Letx _kn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetR _n(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:R _n(x_kn)=f(x_kn),R _n ^(j)(x_kn)=0,j=1,2,4,k=0,1…,n−1. We setw ₂(t,f) as the second modulus of continuity off(x). Then we prove that |R _n(x)-f(x)|=0(nw₂(1/nf)). We also examine the question of lower estimate of ‖R _n-f‖. This generalizes an earlier work of the author.     

Multidimensional version of M. A. Krasnoseľskii’s generalized contraction principle

Let M be a complete K-metric space with n-dimensional metric ρ(x, y): M × M → R ⁿ , where K is the cone of nonnegative vectors in R ⁿ . A mapping F: M → M is called a Q-contraction if ρ (Fx,Fy) ⩽ Qρ (x,y), where Q: K → K is a semi-additive absolutely stable mapping. A Q-contraction always has a unique fixed point x* in M, and ρ(x*,a) ⩽ (I - Q)^-1 ρ(Fa, a) for every point a in M. The point x* can be obtained by the successive approximation method x _k = Fx _k-1, k = 1, 2,..., starting from an arbitrary point x ₀ in M, and the following error estimates hold: ρ (x*, x _k ) ⩽ Q ^k (I - Q)^-1ρ(x ₁, _x ⁰) ⩽ (I - Q)^-1 Q ^k ρ(x ₁, x ₀), k = 1, 2,.... Generally the mappings (I - Q)^-1 and Q ^k do not commute. For n = 1, the result is close to M. A. Krasnosel’skii’s generalized contraction principle.     

Multiplicative arithmetic of finite quadratic forms over Dedekind rings

Letq(X) be a quadratic form in an even numberm of variables with coefficients in a Dedekind ringK. Let us assume that the setsR(q,a) = {N∈K ^m ;q(N) = a} of representations of elementsa ofK by the formq are finite. Then certain multiplicative relations are obtained by elementary means between the setsR(q,a) andR(q,ab), whereb is a product of prime elementsρ ofK with finite coefficientsK/ρK. The relations imply similar multiplicative relations between the numbers of elements of the setsR(q,a), which formerly could be obtained only in some special cases like the case whenK = ℤ is the ring of rational integers and only by means of the theory of Hecke operators on the spaces of theta-series. As an application, an almost elementary proof of the Siegel theorem on the mean number of representations of integers by integral positive quadratic forms of determinant 1 is given. Dedicated to the memory of Professor K G Ramanathan     

On convolution powers on semidirect products

LetK be a compact group of linear operators of thed-dimensional spaceR ^d andG _K,d denote the semidirect productK byR ^d . It is shown that if an adapted probability measureμ onG _K,d is not scattered (i.e. for some compactF we havex ₀ ∈ R ^d (gF)>0), then there exists a nonzero vectorx ₀ ∈R ^d such thatk ₁(x ₀)=k ₂(x ₀) holds for all (k ₁,x ₁) and (k ₂,x ₂) belonging to the topological supportS(μ) of the measureμ. As a result we obtain that every adapted and strictly aperiodic probability measure on the group of all rigid motions of thed-dimensional Euclidian space is scattered. I thank the Foundation for Research Development for financial support.     

On conditional edge-connectivity of graphs

1. IntroductionIn this paper, a graph G ~ (V,E) always means a simple graph (without loops andmultiple edges) with the vertex-set V and the edge-set E. We follow [1] for graph-theoreticalterllilnology and notation not defined here.It is well known that when the underlying topology of a computer interconnectionnetwork is modeled by a graph G, the edge-connectivity A(G) of G is an important measurefor fault-tolerance of the network. However, it has many deficiencies (see [2]). MotiVatedby t…     

On subgroups of the general linear group that contain a maximal nonsplit torus

The paper deals with the structure of intermediate subgroups of the general linear group GL(n, k) of degree n over a field k of odd characteristic that contain a nonsplit maximal torus related to a radical extension of degree n of the ground field k. The structure of ideal nets over a ring that determine the structure of intermediate subgroups containinga transvection is given. Let K = k( ⁿ?{d} ) K = k\left( {\sqrt[n]{d}} \right) be a radical degree-n extension of a field k of odd characteristic, and let T =(d) be a nonsplit maximal torus, which is the image of the multiplicative group of the field K under the regular embedding in G =GL(n, k). In the paper, the structure of intermediate subgroups H, T ≤ H ≤ G, that contain a transvection is studied. The elements of the matrices in the torus T = T (d) generate a subring R(d) in the field k.Let R be an intermediate subring, R(d) ⊆ R ⊆ k, d ∈ R. Let σ_R denote the net in which the ideal dR stands on the principal diagonal and above it and all entries of which beneath the principal diagonal are equal to R. Let σ^R denote the net in which all positions on the principal diagonal and beneath it are occupied by R and all entries above the principal diagonal are equal to dR. Let E(σ_R) be the subgroup generated by all transvections from the net group G(σ_R). In the paper it is proved that the product TE(σ_R) is a group (and thus an intermediate subgroup). If the net σ associated with an intermediate subgroup H coincides with σ_R,then TE(σ_R) ≤ H ≤ N(σ_R),where N(σ_R) is the normalizer of the elementary net group E(σ_R) in G. For the normalizer N(σ_R),the formula N(σ_R)= TG(σ^R) holds. In particular, this result enables one to describe the maximal intermediate subgroups. Bibliography: 13 titles.     

Higher order Riesz transforms associated with Bessel operators

In this paper we investigate Riesz transforms R _μ ^(k) of order k≥1 related to the Bessel operator Δ_μ f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We obtain that for every k≥1, R _μ ^(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x ^2μ+1  dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R _μ ^(k) maps L ^p (ω) into itself and L ¹(ω) into L ^1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman.     

On Cramer Approximations under Violation of Cramer's Condition

Let X ₁, X ₂,... be independent identically distributed random variables with distribution function F, S ₀ = 0, S _n = X ₁ + ⋯ + X _n , and Sˉ _n = max_1⩽k⩽n S _k . We obtain large-deviation theorems for S _n and Sˉ _n under the condition 1 − F(x) = P{X ₁ ⩾ x} = e^−l(x), l(x) = x ^α L(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 447–456, October–December, 2005.     

On tense modules for extended algebras over rational fields

Let k(x) be the field of fractions of the polynomial algebra k[x] over the field k. We prove that, for an arbitrary finite dimensional k-algebra Λ, any finitely generated Λ ⊗_k k(x)-module M such that its minimal projective presentation admits no non-trivial selfextension is of the form M ≅ N^k(x), for some finitely generated Λ-module N. Some consequences are derived for tilting modules over the rational algebra Λ ⊗_k k(x) and for some generic modules for Λ. Received: 24 November 2003; revised: 11 February 2005     

On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

Derivations with power central values on multilinear polynomials in prime rings

Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C. Suppose that x ^s d(x)x ^t ∈ Z(R) for all x ∈ {d(f(x ₁, ..., x _n ))|x ₁, ..., x _n ∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ C = eRC for some idempotent e in the socle of RC and f(x ₁, ..., x _n ) ^N is central-valued in eRCe, where N = s + t + 1.      

Constrained Minkowski Sums: A Geometric Framework for Solving Interval Problems in Computational Biology Efficiently

In this paper, we introduce the notion of a constrained Minkowski sum: for two (finite) point-sets P,Q⊆ℝ² and a set of k inequalities Ax≥b, it is defined as the point-set (P ⊕ Q) _Ax≥b ={x=p+q∣p∈P,q∈Q,Ax≥b}. We show that typical interval problems from computational biology can be solved by computing a set containing the vertices of the convex hull of an appropriately constrained Minkowski sum. We provide an algorithm for computing such a set with running time O(Nlog N), where N=|P|+|Q| if k is fixed. For the special case where P and Q consist of points with integer x ₁-coordinates whose absolute values are bounded by O(N), we even achieve a linear running time O(N). We thereby obtain a linear running time for many interval problems from the literature and improve upon the best known running times for some of them. The main advantage of the presented approach is that it provides a general framework within which a broad variety of interval problems can be modeled and solved. T. Bernholt gratefully acknowledges the Deutsche Forschungsgemeinschaft for the financial support (SFB 475, “Reduction of complexity in multivariate data structures”).     

A bounded compactness theorem forL 1-embeddability of metric spaces in the plane

LetM=(W, d) be a metric space. LetL ¹ denote theL ¹ metric. AnL ¹-embedding ofM into Cartesiank-space ℝ ^k is a distance-preserving map from (W, d) into (ℝ ^k ,L ¹). Letc(k) be the smallest integer such that for every metric spaceM, M isL ¹-embeddable inR ^k iff everyc(k)-sized subspace ofM isL ¹-embeddable inR ^k. A special case of a theorem of Menger (see p. 94 of [5]) says thatc(1) exists and equals 4. We show thatc(2) exists and satisfies 6≦c(2)≦11. Whether or notc(k) exists for anyk≧3 is an open question. The research of S. M. Malitz was partially supported by NSF Grant CCR-8909953.     

Van der waerden and ramsey type games

Let us consider the following 2-player game, calledvan der Waerden game. The players alternately pick previously unpicked integers of the interval {1, 2, ...,N}. The first player wins if he has selected all members of ann-term arithmetic progression. LetW*(n) be the least integerN so that the first player has a winning strategy. By theRamsey game on k-tuples we shall mean a 2-player game where the players alternately pick previously unpicked elements of the completek-uniform hypergraph ofN verticesK _N ^k , and the first player wins if he has selected allk-tuples of ann-set. LetR _k*(n) be the least integerN so that the first player has a winning strategy. We prove (W* (n))¹/n → 2,R ₂*(n)<(2+ε) ⁿ andR _k ^* n<2 ^nk / ^k! fork ≧3.     

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

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.     

The asymptotic formula in Waring’s problem for one square and seventeen fifth powers

Let R _k,s (n) denote the number of solutions of the equation n = x² + y₁^k + y₂^k + ?+ y_s^k{n= x^2 + y_1^k + y_2^k + \cdots + y_s^k} in natural numbers x, y ₁, . . . , y _s . By a straightforward application of the circle method, an asymptotic formula for R _k,s (n) is obtained when k ≥ 3 and s ≥ 2 ^k–1 + 2. When k ≥ 6, work of Heath-Brown and Boklan is applied to establish the asymptotic formula under the milder constraint s ≥ 7 · 2 ^k–4 + 3. Although the principal conclusions provided by Heath-Brown and Boklan are not available for smaller values of k, some of the underlying ideas are still applicable for k = 5, and the main objective of this article is to establish an asymptotic formula for R _5,17(n) by this strategy.     

A characterization of convexity and central symmetry for planar polygonal sets

LetS be a compact set inR ² with nonempty interior,L(u,k) be a line 〈u, x〉 =k, and ζ _u (k) be the linear Lebesgue measure ofS∩L(u,k). It is well known that for a convexS, ζ _u (k) is unimodal, that is, as a function ofk, it is first non-decreasing and then nonincreasing for everyu∈R ². Further, ifS is centrally symmetric with respect toM, ζ _u (k) achieves maximum whenL(u, k) passes throughM. Converse propositions are considered in this paper for polygonalS with Jordan boundary. It is shown that unimodality alone does not suffice for convexity. However, if ζ _u (k) achieves maximum wheneverL(u, k) passes through some fixed pointM then unimodality yields convexity as well as central symmetry. It is also shown that continuity of ζ _u (k) in the interior of its support implies convexity ofS. This last result, however, is false for non-polygonal sets. Research supported by National Science Foundation Grant GP-28154.     

Nonstandard arithmetic of iterated polynomials

LetK be an algebraic number field of finite degree andf(X,T) a polynomial overK. For eachφ(X)∈Z[X], we denote byE(φ) the set of all integersa with φ ^m (a) =φ ⁿ (a) for somem≠n. In this paper, we give a condition for a polynomialφ(X)∈Z[X] to satisfy the following; If forn∈N, there existr∈K anda∈Z−E(φ) such thatf r, φ ^m (a)=0, then there exists a rational functiong(X) overK andk∈N such thatf(g(T)), φ ^k (T))=0 .