Characterizing compact unions of two starshaped sets inR d

SetS inR ^d has propertyK ₂ if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c₁,c₂ (depending onF) such that each point ofF is clearly visible viaS from at least one c_i,i = 1,2. The following characterization theorem is established: Let , d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S _j } converging toS (relative to the Hausdorff metric) such that each setS _j satisfies propertyK ₂. For , the sufficiency of the condition above still holds, although the necessity fails.     

Selmer groups of abelian varieties in extensions of function fields

Let k be a field of characteristic q, a smooth geometrically connected curve defined over k with function field . Let A/K be a non-constant abelian variety defined over K of dimension d. We assume that q = 0 or >  2d + 1. Let p ≠ q be a prime number and a finite geometrically Galois and étale cover defined over k with function field . Let (τ′, B′) be the K′/k-trace of A/K. We give an upper bound for the -corank of the Selmer group Sel _p (A × _K K′), defined in terms of the p-descent map. As a consequence, we get an upper bound for the -rank of the Lang–Néron group A(K′)/τ′B′(k). In the case of a geometric tower of curves whose Galois group is isomorphic to , we give sufficient conditions for the Lang–Néron group of A to be uniformly bounded along the tower. This work was partially supported by CNPq research grant 305731/2006-8.     

Error estimates in S P for cubature formulas exact for haar polynomials in the two-dimensional case

On the spaces S _p , an upper estimate is found for the norm of the error functional δ _N (f) of cubature formulas possessing the Haar d-property in the two-dimensional case. An asymptotic relation is proved for $ \left\| {\delta _N (f)} \right\|_{S_p^* } On the spaces S _p , an upper estimate is found for the norm of the error functional δ _N (f) of cubature formulas possessing the Haar d-property in the two-dimensional case. An asymptotic relation is proved for with the number of nodes N ∼ 2 ^d , where d → ∞. For N ∼ 2 ^d with d → ∞, it is shown that the norm of δ _N for the formulas under study has the best convergence rate, which is equal to N ^−1/p . Original Russian Text ? K.A. Kirillov, M.V. Noskov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 1, pp. 3–13.     

Kernels of Toeplitz operators,smooth functions,and Bernstein type inequalities

Let ϕ be a unimodular function on the unit circle and let K_p(ϕ) denote the kernel of the Toeplitz operator T_ϕ in the Hardy space H^p, p≥1; . Suppose K_p(ϕ)≠{0}. The problem is to find out how the smoothness of the symbol ϕ influences the boundary smoothness of functions in K_p(ϕ). One of the main results is as follows. Theorem 1 Let 1<p, q<+∞, 1<r≤+∞, q⁻¹=p⁻¹+r⁻¹. Suppose |ϕ|≡1 on and ϕ∈W _r ¹ (i.e., ). Then K_p(ϕ)⊂W _q ¹ . Moreover, for any f∈K_p(ϕ) we have ‖f′‖_q≤c(p, r)‖ϕ′‖_r ‖f‖. Bibliography: 19 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 5–21. Translated by K. M. D'yakonov.     

Classification of terminal simplicial reflexive d-polytopes with 3d − 1 vertices

We classify terminal simplicial reflexive d-polytopes with 3d − 1 vertices. They turn out to be smooth Fano d-polytopes. When d is even there is one such polytope up to isomorphism, while there are two when d is uneven.     

TWO-DISTANCE PRESERVING FUNCTIONS FROM EUCLIDEAN SPACE

Let 0 < c < s be fixed real numbers such that , and let f : E² → E ^d for d ≥ 2 be a function such that for every p, q ∈ E ² if |p − q| = c, then |f(p) − f(q)| ≤ c, and if |p − q| = s, then |f(p) − f(q)| ≥ s. Then f is a congruence. This result depends on and expands a result of Rádo et. al. [9], where a similar result holds, but for replacing . We also present a further extensions where E² is replaced by E ⁿ for n > 2 and where the range of c/s is enlarged. This revised version was published online in August 2006 with corrections to the Cover Date.     

Upper bounds on a two-term exponential sum*

We obtain upper bounds for mixed exponential sums of the type where p^m is a prime power with m⩾ 2 and X is a multiplicative character (mod p^m). If X is primitive or p⫮(a, b) then we obtain |S(χ,f,p ^m)| ⩽2np ^{2/3 m}. If X is of conductor p and p⫮( a, b) then we get the stronger bound |S(χ,f,p ^m)|⩽np ^m/2. This paper is dedicated to Prof. Wang Yuan on the occasion of his 70th birthday.     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

Dilatation estimates for quasiconformal extensions

LetD andD′ be ring domains inB ⁿ , withS ⁿ⁻¹ as one boundary component, and let be a homeomorphism which isK-quasiconformal inD and withf(S ⁿ⁻¹)=S ⁿ⁻¹. According to a result of Gehringf÷S ⁿ⁻¹ admits an extension which is quasiconformal inB ⁿ . We find here an upper bound for the dilatation ofg in terms ofn, K, and modD. This work was started during a visit to Université de Paris, financed by a cultural exchange program between France and Finland.     

Highly saturated packings and reduced coverings

We introduce and study certain notions which might serve as substitutes for maximum density packings and minimum density coverings. A body is a compact connected set which is the closure of its interior. A packingP with congruent replicas of a bodyK isn-saturated if non–1 members of it can be replaced withn replicas ofK, and it is completely saturated if it isn-saturated for eachn1. Similarly, a coveringC with congruent replicas of a bodyK isn-reduced if non members of it can be replaced byn–1 replicas ofK without uncovering a portion of the space, and its is completely reduced if it isn-reduced for eachn1. We prove that every bodyK ind-dimensional Euclidean or hyperbolic space admits both ann-saturated packing and ann-reduced covering with replicas ofK. Under some assumptions onKE ^d (somewhat weaker than convexity), we prove the existence of completely saturated packings and completely reduced coverings, but in general, the problem of existence of completely saturated packings, and completely reduced coverings remains unsolved. Also, we investigate some problems related to the the densities ofn-saturated packings andn-reduced coverings. Among other things, we prove that there exists an upper bound for the density of ad+2-reduced covering ofE ^d with congruent balls, and we produce some density bounds for then-saturated packings andn-reduced coverings of the plane with congruent circles.     

Mordell-Weil groups and Selmer groups of twin- prime elliptic curves

Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained.     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

Gauss Sum of Index 4： （1） Cyclic Case

Let p be a prime, m ≥ 2, and （m,p（p - 1）） = 1. In this paper, we will calculate explicitly the Gauss sum G（X） = ∑x∈F＊qX（x）ζ^Tp^（x） in the case of [（Z/mZ）＊ ： （p）] = 4, and -1 （不属于） （p）, where q P^f, f =φ（m）/4, X is a multiplicative character of Fq with order m, and T is the trace map for Fq/Fp. Under the assumptions [（Z/mZ）＊ ： （p）] = 4 and 1（不属于） （p）, the decomposition field of p in the cyclotomic field Q（ζm） is an imaginary quartic （abelian） field. And G（X） is an integer in K. We deal with the case where K is cyclic in this oaDer and leave the non-cvclic case to the next paper.     

Construction of somep-extensions of the rational number field

In this paper, we constructp-extensionsK _a ,a(modp ^r ), of degreep ^3r,p≠2, r>0, of the field ℚ of rational numbers with ramification pointsp andq. The Galois groupG(K _a )/ℚ of the extensionK _a /ℚ,a(modp ^r ), is defined by the generators and relations

一类粗糙核参数型Marcinkiewicz积分

In this paper,the L2-boundedness of a class of parametric Marcinkiewicz integral μρΩ,h with kernel function Ω in Bq0.0 (Sn-1) for some q＞ 1,and the radial function h (x)∈ l∞ (Ls) (R+) for 1＜s≤∞ are given. The Lp(Rn) (2≤p＜∞) boundedness of μ*Ω,ph,λ and μρΩ,h,s with Ω in Bq0,0(Sn-1) and h(|x|)∈l∞(Ls)(R+) in application are obtained. Here μ*Ω,p h,λ and μpΩ,h,s are parametric Marcinkiewicz integrals corresponding to the Littlewood-Paley gλ* function and the Lusin area function S,respectively.     

Coincidences of Centers of Edge-Incentric, or Balloon, Simplices

An edge-incentric d-simplex is defined to be a d-simplex S which admits a (d − 1)-sphere that touches all the edges of S internally. The center of such a sphere is called the edge-incenter of S and is denoted by . Equivalently, S is edge-incentric if and only if its vertices are the centers of d + 1 (d − 1)-spheres in mutual external touch, and for this reason one may call such an S a balloon d-simplex. An orthocentric d-simplex is a d-simplex in which the altitudes are concurrent. The point of concurrence is called the orthocenter and is denoted by . The spaces of edge-incentric and of orthocentric d-simplices have the same dimension d in the sense that a d-simplex in either space can be parametrized, up to shape, by d numbers. Edge-incentric and orthocentric tetrahedra are the first two of the four special classes of tetrahedra studied in [1, Chapter IX.B, pp. 294–333]. The degree of regularity implied by the coincidence of two or more centers of a general d-simplex is investigated in [8], where it is shown that the coincidence of the centroid , the circumcenter , and the incenter does not imply much regularity. For an orthocentric d-simplex S, however, it is proved in [9] that if any two of the centers , and coincide, then S is regular. In this paper, the same question is addressed for edge-incentric d-simplices. Among other things, it is proved that if any three of the centers , and of an edge-incentric d-simplex S coincide, then S is regular, and it is also shown that none of the coincidences , and implies regularity (except when d ≤ 3, d ≤ 4, and d ≤ 6, respectively). In contrast with the afore-mentioned results for orthocentric d-simplices, this emphasizes once more the feeling that, regarding many important properties, orthocentric d-simplices are the true generalizations of triangles. Several open questions are posed. Received: June 19, 2006.     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

Sausages are good packings

LetB ^d be thed-dimensional unit ball and, for an integern, letC _n ={x ¹,...,x ⁿ } be a packing set forB ^d , i.e.,|x ⁱ −x ^j |≥2, 1≤i<j≤n. We show that for every a dimensiond(ρ) exists such that, ford≥d(ρ),V(conv(C _n )+ρB ^d )≥V(conv(S _n )+ρB ^d ), whereS _n is a “sausage” arrangement ofn balls, holds. This gives considerable improvement to Fejes Tóth's “sausage” conjecture in high dimensions. Further, we prove that, for every convex bodyK and ρ<1/32d ⁻²,V(conv(C _n )+ρK)≥V(conv(S _n )+ρK), whereC _n is a packing set with respect toK andS _n is a minimal “sausage” arrangement ofK, holds.     

A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra

A study of the set of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p > 0 was initiated by Shalev and continued by the present author. The main goal of this paper is to produce more elements of . Our main result shows that any divisor n of q − 1, where q is a power of p, such that n ≥ (p − 1)^1/p (q − 1)^1−1/(2p), necessarily belongs to . This extends its special case for p = 2 which was proved in a previous paper by a different method.     

Detection of Some Elements in the Stable Homotopy Groups of Spheres

Let A be the mod p Steenrod algebra and S be the sphere spectrum localized at an odd prime p. To determine the stable homotopy groups of spheres π＊S is one of the central problems in homotopy theory. This paper constructs a new nontrivial family of homotopy elements in the stable homotopy groups of spheres πp^nq＋2pq＋q-3S which isof order p and is represented by kohn ∈ ExtA^3,P^nq＋2pq＋q（Zp,Zp） in the Adams spectral sequence, wherep 〉 5 is an odd prime, n ≥3 and q = 2（p-1）. In the course of the proof, a new family of homotopy elements in πp^nq＋（p＋1）q-1V（1） which is represented by β＊i＇＊i＊（hn） ∈ ExtA^2,pnq＋（p＋1）q＋1 （H^＊V（1）, Zp） in the Adams sequence is detected.