Sequentially equidistant points in metric spaces

A sequence (z ₀,z ₁,z ₂,, ...,z _n, z _n+1) of points fromp=z ₀ toq=z _n+1 in a metric spaceX is said to besequentially equidistant ifd(z _i−1,z _i)=d(z _i,z _i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR ^m ,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the positive results. Supported in part by NSF Grant DMS-8701666.     

Realizability of Graphs in Three Dimensions

This paper concludes the characterization of 3-realizable graphs begun by Belk and Connelly. A graph is 3-realizable if, for every configuration of its vertices in E^N with N ≥ 3, there exists a corresponding configuration in E³ with the same edge lengths. In this paper the two graphs V₈ and C₅ × C₂ are shown to be 3-realizable. As shown by Belk and Connelly, this means that the forbidden minors for 3-realizability are K₅ and K_2,2,2.A graph is d-realizable if, for every configuration of its vertices in E^N, there exists a another corresponding configuration in E^d with the same edge lengths.     

Realizability of Graphs

A graph is d-realizable if, for every configuration of its vertices in E^N, there exists a another corresponding configuration in E^d with the same edge lengths. A graph is 2-realizable if and only if it is a partial 2-tree, i.e., a subgraph of the 2-sum of triangles in the sense of graph theory. We show that a graph is 3-realizable if and only if it does not have K₅ or the 1-skeleton of the octahedron as a minor.     

The Zero-Divisor Graphs Which Are Uniquely Determined By Neighborhoods

A nonempty simple connected graph G is called a uniquely determined graph, if distinct vertices of G have distinct neighborhoods. We prove that if R is a commutative ring, then Γ(R) is uniquely determined if and only if either R is a Boolean ring or T(R) is a local ring with x² = 0 for any x ∈ Z(R), where T(R) is the total quotient ring of R. We determine all the corresponding rings with characteristic p for any finite complete graph, and in particular, give all the corresponding rings of K_n if n + 1 = p^q for some primes p, q. Finally, we show that a graph G with more than two vertices has a unique corresponding zero-divisor semigroup if G is a zero-divisor graph of some Boolean ring.     

Eine Spektralsequenz in der stetigen Kohomologietheorie topologischer Gruppen

Let H^*(G; M) be the continuous cohomology of a locally compact group G with coefficients in a topological RG-module M. If G operates without fixed points on a R-paracompact space X such that there is a slice through each point and X/G is R-paracompact, then there exists a spectral sequence converging to the equivariant cohomology H^*(X,G; M) of X with second term E₂ ^p.q?H^p(G; H^qX; M)) where the sheaf theoretical cohomology of X is suitable topologized. Several applications and a generalization to actions of G with non-empty fixed point sets are given.     

Linear Optimization Queries

Let Γ₀ be a set of n halfspaces in E^d (where the dimension d is fixed) and let m be a parameter, n ≤ m ≤ n^d/2. We show that Γ₀ can be preprocessed in time and space O(m^1+δ) (for any fixed δ > 0) so that given a vector c E^d and another set Γ_q of additional halfspaces, the function c · x can be optimized over the intersection of the halfspaces of Γ₀ Γ_q in time O((n/m^1/d/2 + |Γ_q|)log^4d+3n). The algorithm uses a multidimensional version of Megiddo′s parametric search technique and recent results on halfspace range reporting. Applications include an improved algorithm for computing the extreme points of an n-point set P in E^d, improved output-sensitive computation of convex hulls and Voronoi diagrams, and a Monte-Carlo algorithm for estimating the volume of a convex polyhedron given by the set of its vertices (in a fixed dimension).     

On tricyclic graphs of a given diameter with minimal energy

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G(n,d) be the class of tricyclic graphs G on n vertices with diameter d and containing no vertex disjoint odd cycles C_p,C_q of lengths p and q with p+q≡2(mod4). In this paper, we characterize the graphs with minimal energy in G(n,d).     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

Self-similar fractals: An algorithmic point of view

This paper studies the self-similar fractals with overlaps from an algorithmic point of view.A decidable problem is a question such that there is an algorithm to answer"yes"or"no"to the question for every possible input.For a classical class of self-similar sets{E b.d}b,d where E b.d=Sn i=1(E b,d/d+b i)with b=(b1,...,b n)∈Qn and d∈N∩[n,∞),we prove that the following problems on the class are decidable:To test if the Hausdorff dimension of a given self-similar set is equal to its similarity dimension,and to test if a given self-similar set satisfies the open set condition(or the strong separation condition).In fact,based on graph algorithm,there are polynomial time algorithms for the above decidable problem.     

球上Zygmund 型空间上的加权Cesàro 算子

该文在Cⁿ中单位球上讨论了Zygmund 型空间(小Zygmund 型空间)之间的加权Cesàro 算子T_g 的有界性和紧性特征, 得到了以下的结果: (1) T_g 是Z^p 到Z^q的有界算子或紧算子的充要条件; (2) T_g 是 Z^p₀ 到Z^q₀ 的有界算子或紧算子的充要条件.     

Spherical Maximal Operators on Radial Functions

Let A_tf(x) denote the mean of f over a sphere of radius t and center x. We prove sharp estimates for the maximal function M_E f(X) = sup_t∈_E |Atf(x)| where E is a fixed set in IR⁺ and f is a radial function ∈ L^p(IR^d). Let P_d = d/(d?1) (the critical exponent for Stein's maximal function). For the cases (i) p < p_d, d ? 2, and (ii) p = p_d, d ? 3, and for p ? q ? ∞ we prove necessary and sufficient conditions on E for M_E to map radial functions in L^p to the Lorentz space L^P,q.     

Note on induced subgraphs of the unit distance graphE n

The unit distance graphE ⁿ is the graph whose vertices are the points in Euclideann-space, and two vertices are adjacent if and only if the distance between them is 1. We prove that for anyn there is a finite bipartite graph which cannot be embedded inE ⁿ as an induced subgraph and that every finite graph with maximum degreed can be embedded inE ^N ,N=(d ³ –d)/2, as an induced subgraph.     

On super edge-antimagic total labelings of

A graph G of order p and size q is called (a,d)-edge-antimagic total if there exists a bijective function f:V(G)E(G)→{1,2,…,p+q} such that the edge-weights w(uv)=f(u)+f(v)+f(uv), uvE(G), form an arithmetic sequence with first term a and common difference d. The graph G is said to be super (a,d)-edge-antimagic total if the vertex labels are 1,2,…,p. In this paper we study super (a,d)-edge-antimagic properties of mK_n, that is, of the graph formed by the disjoint union of m copies of K_n.     

Slight improvements of the Singleton bound

The problem of determining A_q(n,d), the maximum cardinality of a q-ary code of length n with minimum distance at least d, is considered in some cases where corresponding MDS codes do not exist. Slight improvements of the Singleton bound are given, including A_q(q+2,q)?q³-3 if q is odd, A₅(7,5)?5³-4 and A₁₆(18,15)?18⁴-4.     

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.     

单位球上小Bloch型空间之间的加权复合算子

对所有的0p、q∞,该文得到了Cn中单位球上小Bloch型空间β_0~p到β_0~q之间的加权复合算子T_ψ,φ为有界算子或紧算子的充要条件.     

On exact order estimates of N-widths of classes of functions analytic in a simply connected domain

In the spaces E _q(Ω), 1 < q < ∞, introduced by Smirnov, we obtain exact order estimates of projective and spectral n-widths of the classes W ^r E _p(Ω) and W ^r E _p(Ω)Ф in the case where p and q are not equal. We also indicate extremal subspaces and operators for the approximative values under consideration.     

VECTOR BUNDLES ON PROJECTIVE SPACES OVER A DISCRETE VALUATION RING

Let E be a vector bundle on P ⁿ _D, n ≥ 2, with D discrete valuation ring. Here we give a necessary and sufficient condition for the splitting of E as a direct sum of line bundles.     

Labelling of some planar graphs with a condition at distance two

The problem of vertex labeling with a condition at distance two in a graph, is a variation of Hale’s channel assignment problem, which was first explored by Griggs and Yeh. For positive integerp ≥q, the λ _p,q -number of graph G, denoted λ(G;p, q), is the smallest span among all integer labellings ofV(G) such that vertices at distance two receive labels which differ by at leastq and adjacent vertices receive labels which differ by at leastp. Van den Heuvel and McGuinness have proved that λ(G;p, q) ≤ (4q-2) Δ+10p+38q-24 for any planar graphG with maximum degree Δ. In this paper, we studied the upper bound of λ _p ,q-number of some planar graphs. It is proved that λ(G;p, q) ≤ (2q?1)Δ + 2(2p?1) ifG is an outerplanar graph and λ(G;p,q) ≤ (2q?1) Δ + 6p - 4q - 1 if G is a Halin graph.