Characterization of the Points of ϕ-Strong Summability of Fourier–Laplace Series for Functions of the Class L p(S m), p > 1

We consider the behavior of the -strong means of Fourier–Laplace series for functions that belong to L _p(S ^m), p > 1, on a set of points of full measure on an m-dimensional sphere S ^m.     

A Characteristic of the Points of Strong Summability of Fourier–Laplace Series for Functions of the Class L(S m) for a Critical Index

We announce a result that gives a fairly constructive characteristic of the set of points of full measure on a sphere S ^m at which strong means converge to a given function f(·).     

The distribution of dense Sidon subsets of $ \mathbb{Z}_m $

Let S \subseteqq \mathbbZ_m S \subseteqq \mathbb{Z}_m be a Sidon set of cardinality | S | = m^1/₂ + O(1) \mid S \mid = m^{1 \over 2} + O(1) . It is proved, in particular, that for any interval á = {a, a + 1, ?, a + l- 1} {\cal I} = \{a, a + 1, \ldots, a + \ell - 1\} in \mathbbZ_m \mathbb{Z}_m , 0 \leqq l 0 \leqq \ell < m, we have | | S ?á | - | S | l/m | = O( | S | ^1/₂ln m) \big| {\mid S \cap {\cal I} \mid - \mid S \mid \ell/m} \big| = O(\mid S \mid^{1 \over 2}\textrm{ln}\, m) .     

Extreme convex set functions with finite carrier: General theory

Let Ω={1,…,n} and P={X:SΩ}. A mapping e : P→R⁺ is a convex set function if e()=0 and e(S) + e(T)e(S∩T) + e(S T) for all S. TεP. The set of convex set functions for fixed Ω is a convex cone and the paper is dealing with the extreme points of the base of this cone. To this end a representation theorem is proved: every e ε ¹ can be written as e(·)=max(m¹(·)−α₁…m^t(·)−α_t), where m¹,…,m^t are measures on P and α₁,…,α_t are nonnegative reals. Given additional requirements, the representation is unique and called “canonical”. Fix H {1,…,r},|H| 2. There is a certain subsystem of sets SεP such that m^τ(S)−α_τ=e(S) (τε H}, that is, the subsystem of sets S such that m^τ(S)−α_τ(τεH) is a maximal term in the representation of e by m¹,…,m^τ and α₁,…α_t.e is called nondegenerate is these subsystems determine the measures m¹,…,m^τ uniquely and it turns out that nondegeneracy and extremality are equivalent for e ε ¹. Moreover, it is seen that nondegeneracy is closely related to a generalized version of the problem “represent a given integer λ o by means of integer weights g,…,g_r 0 via σ^r=₁ag=λ such that the integer coefficients a satisfy 0ak (=1,…,r), where k are prescribed integer bounds. Find r such representations with the additional property that the coefficients form a nonsingular matrix.” A solution to the generalized version of this number theoretical problem is given and, finally, a few examples are discussed.     

The Coulomb energy of spherical designs on <Emphasis Type="Italic">S</Emphasis><Superscript>2</Superscript>

In this work we give upper bounds for the Coulomb energy of a sequence of well separated spherical n-designs, where a spherical n-design is a set of m points on the unit sphere S ² ⊂ ℝ³ that gives an equal weight cubature rule (or equal weight numerical integration rule) on S ² which is exact for spherical polynomials of degree ⩽ n. (A sequence Ξ of m-point spherical n-designs X on S ² is said to be well separated if there exists a constant λ > 0 such that for each m-point spherical n-design X ∈ Ξ the minimum spherical distance between points is bounded from below by .) In particular, if the sequence of well separated spherical designs is such that m and n are related by m = O(n ²), then the Coulomb energy of each m-point spherical n-design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy of point sets on S ². Dedicated to Edward B. Saff on the occasion of his 60th birthday.     

Belts and κ-invariants of link maps in spheres

Linkings of higher dimensional disjoint singular spheres in euclidean space ℝ^m are compared with linkings in the sphereS ^m. Somewhat surprisingly, link homotopy inS ^m can be interpreted as a special case of link homotopy in ℝ^m. This leads to a considerable refinement of standard invariants such as the generalized Milnor invariants.     

An extension of the Erdős–Ginzburg–Ziv Theorem to hypergraphs

An n-set partition of a sequence S is a collection of n nonempty subsequences of S, pairwise disjoint as sequences, such that every term of S belongs to exactly one of the subsequences, and the terms in each subsequence are all distinct with the result that they can be considered as sets. For a sequence S, subsequence S^′, and set T, |T∩S| denotes the number of terms x of S with xT, and |S| denotes the length of S, and SS^′ denotes the subsequence of S obtained by deleting all terms in S^′. We first prove the following two additive number theory results.(1) Let S be a finite sequence of elements from an abelian group G. If S has an n-set partition, A=A₁,…,A_n, such that

then there exists a subsequence S^′ of S, with length |S^′|≤max{|S|−n+1,2n}, and with an n-set partition, , such that . Furthermore, if ||A_i|−|A_j||≤1 for all i and j, or if |A_i|≥3 for all i, then .(2) Let S be a sequence of elements from a finite abelian group G of order m, and suppose there exist a,bG such that . If |S|≥2m−1, then there exists an m-term zero-sum subsequence S^′ of S with or .Let be a connected, finite m-uniform hypergraph, and be the least integer n such that for every 2-coloring (coloring with the elements of the cyclic group ) of the vertices of the complete m-uniform hypergraph , there exists a subhypergraph isomorphic to such that every edge in is monochromatic (such that for every edge e in the sum of the colors on e is zero). As a corollary to the above theorems, we show that if every subhypergraph of contains an edge with at least half of its vertices monovalent in , or if consists of two intersecting edges, then . This extends the Erdős–Ginzburg–Ziv Theorem, which is the case when is a single edge.     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.     

On the intersection of maximalm-convex subsets

A subsetS of a real linear spaceE is said to bem-convex providedm≧2, there exist more thanm points inS, and for eachm distinct points ofS at least one of the ( ₂ ^m ) segments between thesem points is included inS. InE, letxy denote the segment between two pointsx andy. For any pointx inSυE, letS _x ={y: xyυS}. The kernel of a setS is then defined as {xεS: S _x=S}. It is shown that the kernel of a setS is always a subset of the intersection of all maximalm-convex subsets ofS. A sufficient condition is given for the intersection of all the maximalm-convex subsets of a setS to be the kernel ofS.     

Explicit representations of minimal immersions of 2-spheres intoS 2m

For all minimal immersions of the 2-sphereS ² into the unit 2m-sphereS ^2m with area 2π[m(m+1)+2], we are able to give an explicit representation by some real parameters. This representation is rather important in the study of minimal 2-spheres inS ^2m because the parameters concerned are independent of each other. Particular examples are given and a classification theorem is also obtained. Supported by NNSF of China and NSF of Henan     

A solution to a problem of Cameron on sum-free complete sets

For any subsets A and B of an additive group G, define A + B = { a + b: a ε A and b ε B} and −A = {−a: a ε A}. A subset S of G is said to be sum-free, complete, and symmetric respectively if S + S S^c, S + S S^c, and S = −S. Cameron asked if for all sufficiently large moduli m there exists a sum-free complete set in Z/mZ that is not symmetric. We answer Cameron's question by showing there exists such a set for all moduli greater than or equal to 890626. We also show that every sum-free complete set in Z/mZ that is not symmetric can be used to construct a counter-example to a conjecture of Conway disproved by Marica. Conway conjectured that for any finite set S of integers, |S + S| |S --- S|.     

de Sitter空间S1m+1中具有平行Blaschke张量的正则类空超曲面

本文引入两个以de Sitter空间为模型的非齐性坐标来覆盖共形空间Q₁^m+1.利用球面S^m+1中超曲面的Möbius几何的方法,本文研究了Q₁^m+1中正则类空超曲面的共形几何.作为其结果,本文对所有具有平行Blaschke张量的正则类空超曲面进行了完全分类.     

On the Kronecker pairing for mixed cusp forms

LetG⊃PSL(2,R) be a Fuchsian group of the first kind with no elements of finite order, and letS ^2m V be the 2m-fold symmetric power of the standard representationV ofSL(2,R) on C². We determine the value of the Kronecker pairing between the canonical image of a mixed cusp formf of type (2,2m) inH ¹(G, S ^2m V) and a cycleg ⊗Q _g ^m inH ₁ (G, (S ^2m V)^*) for eachg inG, whereQ _g ^m is an element of (S ^2m V)^* associated tog, m and a monodromy representation ofG.     

On factorizing meromorphic functions

Summary The paper determines all cases when a meromorphic functionF can be expressed both asf ⊗p andf ⊗q with the same meromorphicf and different polynomialsp andq. In all cases there are constantsk, β, a positive integerm, a root λ of unity of orderS and a polynomialr such thatp=(Lr) ^m+k,q=r ^m+k, whereLz=λz+β. We have eitherm=1,S arbitrary orm=2,S=2, which can occur even ifF andf are entire, or, in the remaining casesS=2, 3, 4 or 6,m dividesS andf(k+t ^m) is a doubly-periodic function.     

A decomposition theorem form-convex sets

LetS be a closedm-convex subset of the plane,m≧2,Q the set of points of local nonconvexity ofS, with convQ ⊆S. If there is some pointp in [(bdryS) ∩ (kerS)] ∼Q, thenS is a union ofm−1 closed convex sets. The result is best possible for everym.     

Zero-Semidistributive Inverse Semigroups

An inverse semigroup S is said to be 0-semidistributive if its lattice ?F (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab) ^m  = a ⁿ or (ab) ^m  = b ⁿ , where σ is the minimum group congruence on S.     

On nonnegatively curved 4-manifolds with discrete symmetry

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z _m -action for a positive integer m ≧ 61⁷. Assume that Z _m acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z _m has at most one two-dimensional component, then M is homeomorphic to S ⁴, # _i ^l =1S ² × S ², l = 1, 2, or # _j ^k = 1 ± CP ², k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic χ(M) under the homological assumption of a Z _m -action as above by using the Lefschetz fixed point formula.     

Quadratic forms between euclidean spheres

We study the relationship between the dimensions of euclidean spheres which admit a nonconstant homogeneous quadratic map between them. Givenm (respectivelyn), we determine the least (respectively greatest) possible value ofn (respectivelym) for which there exists a nonconstant homogeneous quadratic mapS ^m →S ⁿ . Research partially supported by NSF DMS-9201204     

Convexity and a certain propertyP m

The propertyP _m (directly analogous to Valentine’s propertyP ₃) is used to prove several curious results concerning subsets of a topological linear space, among them the following: (a) If a closed setS has propertyP _m and containsk points of local nonconvexity no distinct pair of which can see each other viaS, thenS is the union ofm − k − 1 or fewer starshaped sets. (b) Any closed connected set with propertyP _m is polygonally connected. (c) A closed connected setS with propertyP _m is anL _m−1 set (each pair of points may be joined by a polygonal arc ofm − 1 of fewer sides inS). (d) A finite-dimensional set with propertyP _m is anL _{2m − 3} set. A new proof of Tietze’s theorem on locally convex sets is given, and various examples refute certain plausible conjectures.     

Die algebraische Struktur der Homotopiemenge von Produktabbildungen auf Sphären

Bezeichne fürm mitS ^m diem-dimensionale Sphäre. Wir bestimmen in der nachstehenden Arbeit die durch das reflektierende Produkt auf Sphären induzierte algebraische Struktur auf der Homotopiemenge [S ^p×S ^q;S ⁿ].