On a generalization of distance sets

A subset X in the d-dimensional Euclidean space is called a k-distance set if there are exactly k distinct distances between two distinct points in X and a subset X is called a locally k-distance set if for any point x in X, there are at most k distinct distances between x and other points in X.Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of k-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally k-distance sets on a sphere. In the first part of this paper, we prove that if X is a locally k-distance set attaining the Fisher type upper bound, then determining a weight function w, (X,w) is a tight weighted spherical 2k-design. This result implies that locally k-distance sets attaining the Fisher type upper bound are k-distance sets. In the second part, we give a new absolute bound for the cardinalities of k-distance sets on a sphere. This upper bound is useful for k-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d−1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in d-space with more than d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.     

Tree spanners in planar graphs

A tree t-spanner of a graph G is a spanning subtree T of G in which the distance between every pair of vertices is at most t times their distance in G. Spanner problems have received some attention, mostly in the context of communication networks. It is known that for general unweighted graphs, the problem of deciding the existence of a tree t-spanner can be solved in polynomial time for t=2, while it is NP-hard for any t⩾4; the case t=3 is open, but has been conjectured to be hard. In this paper, we consider tree spanners in planar graphs. We show that even for planar unweighted graphs, it is NP-hard to determine the minimum t for which a tree t-spanner exists. On the other hand, we give a polynomial algorithm for any fixed t that decides for planar unweighted graphs with bounded face length whether there is a tree t-spanner. Furthermore, we prove that it can be decided in polynomial time whether a planar unweighted graph has a tree t-spanner for t=3.     

Set-valued contractions and fixed points

A common fixed point theorem is proved for a family of set-valued contraction mappings in gauge spaces. This result is related to a recent result of Frigon for ‘generalized contractions’ and it includes a method for approximating the fixed point. The remainder of the paper is devoted to results for families of set-valued contraction mappings in hyperconvex spaces. It is proved, for example, that if M is a hyperconvex metric space and f_α is a family of set-valued contractions indexed over a directed set Λ and taking values in the space of all nonempty admissible subsets of M endowed with the Hausdorff metric, then the condition f_β(x)⊆f_α(x) for all x∈M and β⩾α implies that the set of points x∈M for which x∈⋂_α∈Λf_β(x) is nonempty and hyperconvex.     

The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands

By showing that there is an upper bound for the price of anarchyρ(Γ) for a non-atomic congestion game Γ with only separable cost maps and fixed demands, Roughgarden and Tardos show that the cost of forgoing centralized control is mild. This letter shows that there is an upper bound for ρ(Γ) in Γ for fixed demands with symmetric cost maps. It also shows that there is a weaker bound for ρ(Γ) in Γ with elastic demands.     

Optimal algorithms to compute the closure of a set of iso-rectangles

Three varieties of the closure of a set of iso-(oriented) rectangles, i.e., rectilin-early-oriented rectangles, are introduced. These are uni-directional, diagonal, and rectangular closure. First a strong decomposition theorem for diagonal closure in terms of uni-directional closure is proved. Then time and space optimal algorithms to compute uni-directional and diagonal closure, each running in O(nlogn) time and O(n) space, are described. An O(nlogn) time and space algorithm for rectangular closure is also described. The algorithm for diagonal closure has applications in database concurrency control: an O(nlogn) time and O(n) space algorithm for testing for safety and detecting deadlocks in locked transaction systems is obtained.     

A priori bounds and weak solutions for the nonlinear Schrödinger equation in Sobolev spaces of negative order

Solutions to the Cauchy problem for the one-dimensional cubic nonlinear Schrödinger equation on the real line are studied in Sobolev spaces Hs, for s negative but close to 0. For smooth solutions there is an a priori upper bound for the Hs norm of the solution, in terms of the Hs norm of the datum, for arbitrarily large data, for sufficiently short time. Weak solutions are constructed for arbitrary initial data in Hs.     

The weak theory of monads

We construct a ‘weak’ version EMw(K) of Lack and Street's 2-category of monads in a 2-category K, by replacing their compatibility constraint of 1-cells with the units of monads by an additional condition on the 2-cells. A relation between monads in EMw(K) and composite pre-monads in K is discussed. If K admits Eilenberg-Moore constructions for monads, we define two symmetrical notions of ‘weak liftings’ for monads in K. If moreover idempotent 2-cells in K split, we describe both kinds of weak lifting via an appropriate pseudo-functor EMw(K)→K. Weak entwining structures and partial entwining structures are shown to realize weak liftings of a comonad for a monad in these respective senses. Weak bialgebras are characterized as algebras and coalgebras, such that the corresponding monads weakly lift for the corresponding comonads and also the comonads weakly lift for the monads.     

On the Hyers-Ulam-Rassias stability of functional equations in n-variables

In this paper we investigate a generalization of the Hyers-Ulam-Rassias stability for a functional equation of the form f(φ(X))=?(X)f(X)+ψ(X) and the stability in the sense of Ger for the functional equation of the form f(φ(X))=?(X)f(X), where X lie in n-variables. As a consequence, we obtain a stability result in the sense of Hyers-Ulam-Rassias, Gǎvruta, and Ger for some well-known equations such as the gamma, beta, and G-function type's equations.     

Lyapunov,Bézout,and Hankel

Given a polynomial f of degree n, we denote by C its companion matrix, and by S the truncated shift operator of order n. We consider Lyapunov-type equations of the form X?SXC=>W and X?CXS=W. We derive some properties of these equations which make it possible to characterize Bezoutian matrices as solutions of the first equation with suitable right-hand sides W (similarly for Hankel and the second equation) and to write down explicit expressions for these solutions. This yields explicit factorization formulae for polynomials in C, for the Schur-Cohn matrix, and for matrices satisfying certain intertwining relations, as well as for Bezoutian matrices.     

An upper bound for permanents of nonnegative matrices

A recent conjecture of Caputo, Carlen, Lieb, and Loss, and, independently, of the author, states that the maximum of the permanent of a matrix whose rows are unit vectors in lp is attained either for the identity matrix I or for a constant multiple of the all-1 matrix J.The conjecture is known to be true for p=1 (I) and for p?2 (J).We prove the conjecture for a subinterval of (1,2), and show the conjectured upper bound to be true within a subexponential factor (in the dimension) for all 1<p<2. In fact, for p bounded away from 1, the conjectured upper bound is true within a constant factor.     

Steiner intervals and Steiner geodetic numbers in distance-hereditary graphs

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I(S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S={u,v}, then I(S)=I[u,v] is called the interval between u and v and consists of all vertices that lie on some shortest u-v path in G. The smallest cardinality of a set S of vertices such that ?_u,v∈SI[u,v]=V(G) is called the geodetic number and is denoted by g(G). The smallest cardinality of a set S of vertices of G such that I(S)=V(G) is called the Steiner geodetic number of G and is denoted by sg(G). We show that for distance-hereditary graphs g(G)?sg(G) but that g(G)/sg(G) can be arbitrarily large if G is not distance hereditary. An efficient algorithm for finding the Steiner interval for a set of vertices in a distance-hereditary graph is described and it is shown how contour vertices can be used in developing an efficient algorithm for finding the Steiner geodetic number of a distance-hereditary graph.     

Test sets for morphisms with bounded delay

Let p be a fixed nonnegative integer. We prove the Ehrenfeucht Conjecture for morphisms having deciphering delay bounded by p. In other words, we show that for each language L over a finite alphabet there exists a finite subset F of L such that for arbitrary morphisms h and g having deciphering delay bounded by p, the equation h(x) = g(x) holds for all x in L if and only if it holds for all x in F.     

On non-strong jumping numbers and density structures of hypergraphs

Estimating Turán densities of hypergraphs is believed to be one of the most challenging problems in extremal set theory. The concept of ‘jump’ concerns the distribution of Turán densities. A number α∈[0,1) is a jump for r-uniform graphs if there exists a constant c>0 such that for any family F of r-uniform graphs, if the Turán density of F is greater than α, then the Turán density of F is at least α+c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0,1) is a jump for graphs. Erd?s also showed that every number in [0,r!/r^r) is a jump for r-uniform hypergraphs. Furthermore, Frankl and Rödl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in [r!/r^r,1) for r-uniform hypergraphs. But there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we propose a new but related concept-strong-jump and describe several sequences of non-strong-jumps. It might help us to understand the distribution of Turán densities for hypergraphs better by finding more non-strong-jumps.     

Coderivative calculus and metric regularity for constraint and variational systems

This paper provides new developments in generalized differentiation theory of variational analysis with their applications to metric regularity of parameterized constraint and variational systems in finite-dimensional and infinite-dimensional spaces. Our approach to the study of metric regularity for these two major classes of parametric systems is based on appropriate coderivative constructions for set-valued mappings and on extended calculus rules supporting their computation and estimation. The main attention is paid in this paper to the so-called reversed mixed coderivative, which is of crucial importance for efficient pointwise characterizations of metric regularity in the general framework of set-valued mappings between infinite-dimensional spaces. We develop new calculus results for the latter coderivative that allow us to compute it for large classes of parametric constraint and variational systems. On this basis we derive verifiable sufficient conditions, necessary conditions as well as complete characterizations for metric regularity of such systems with computing the corresponding exact bounds of metric regularity constants/moduli. This approach allows us to reveal general settings in which metric regularity fails for major classes of parametric variational systems. Furthermore, the developed coderivative calculus leads us also to establishing new formulas for computing the radius of metric regularity for constraint and variational systems, which characterize the maximal region of preserving metric regularity under linear (and other types of) perturbations and are closely related to conditioning aspects of optimization.     

Equivariant Nielsen fixed point theory for n-valued maps

We develop an equivariant Nielsen fixed point theory for n-valued G-maps by associating (as in Better (2010) [2]) an abstract simplicial complex to any equivariant n-valued map and defining, in terms of this complex, two n-valued continuous G-homotopy invariants that are lower bounds for the number of fixed points and of orbits in the n-valued continuous G-homotopy class of a given n-valued G-map. We also provide an equivariant Hopf construction for n-valued G-maps as well as a minimality result for the Nielsen numbers introduced in this setting.     

Optimum two level fractional factorial plans for model identification and discrimination

Model identification and discrimination are two major statistical challenges. In this paper we consider a set of models M_k for factorial experiments with the parameters representing the general mean, main effects, and only k out of all two-factor interactions. We consider the class D of all fractional factorial plans with the same number of runs having the ability to identify all the models in M_k, i.e., the full estimation capacity.The fractional factorial plans in D with the full estimation capacity for k?2 are able to discriminate between models in M_u for u?k^*, where k^*=(k/2) when k is even, k^*=((k-1)/2) when k is odd. We obtain fractional factorial plans in D satisfying the six optimality criterion functions AD, AT, AMCR, GD, GT, and GMCR for 2^m factorial experiments when m=4 and 5. Both single stage and multi-stage (hierarchical) designs are given. Some results on estimation capacity of a fractional factorial plan for identifying models in M_k are also given. Our designs D4.1 and D10 stand out in their performances relative to the designs given in Li and Nachtsheim [Model-robust factorial designs, Technometrics 42(4) (2000) 345-352.] for m=4 and 5 with respect to the criterion functions AD, AT, AMCR, GD, GT, and GMCR. Our design D4.2 stands out in its performance relative the Li-Nachtsheim design for m=4 with respect to the four criterion functions AT, AMCR, GT, and GMCR. However, the Li-Nachtsheim design for m=4 stands out in its performance relative to our design D4.2 with respect to the criterion functions AD and GD. Our design D14 does have the full estimation capacity for k=5 but the twelve run Li-Nachtsheim design does not have the full estimation capacity for k=5.     

Quasi-median hulls in Hamming space are Steiner hulls

A Hamming space Λⁿ consists of all sequences of length n over an alphabet Λ and is endowed with the Hamming distance. In particular, any set of aligned DNA sequences of fixed length constitutes a subspace of a Hamming space with respect to mismatch distance. The quasi-median operation returns for any three sequences u,v,w the sequence which in each coordinate attains either the majority coordinate from u,v,w or else (in the case of a tie) the coordinate of the first entry, u; for a subset of Λⁿ the iterative application of this operation stabilizes in its quasi-median hull. We show that for every finite tree interconnecting a given subset X of Λⁿ there exists a shortest realization within Λⁿ for which all interior nodes belong to the quasi-median hull of X. Hence the quasi-median hull serves as a Steiner hull for the Steiner problem in Hamming space.     

An upper bound for the minimum rank of a graph

For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields.     

On 3-Steiner simplicial orderings

Let G be a connected graph and S a nonempty set of vertices of G. A Steiner tree for S is a connected subgraph of G containing S that has a minimum number of edges. The Steiner interval for S is the collection of all vertices in G that belong to some Steiner tree for S. Let k≥2 be an integer. A set X of vertices of G is k-Steiner convex if it contains the Steiner interval of every set of k vertices in X. A vertex x∈X is an extreme vertex of X if X?{x} is also k-Steiner convex. We call such vertices k-Steiner simplicial vertices. We characterize vertices that are 3-Steiner simplicial and give characterizations of two classes of graphs, namely the class of graphs for which every ordering produced by Lexicographic Breadth First Search is a 3-Steiner simplicial ordering and the class for which every ordering of every induced subgraph produced by Maximum Cardinality Search is a 3-Steiner simplicial ordering.     

On degrees in random triangulations of point sets

We study the expected number of interior vertices of degree i in a triangulation of a planar point set S, drawn uniformly at random from the set of all triangulations of S, and derive various bounds and inequalities for these expected values. One of our main results is: For any set S of N points in general position, and for any fixed i, the expected number of vertices of degree i in a random triangulation is at least γiN, for some fixed positive constant γi (assuming that N>i and that at least some fixed fraction of the points are interior).We also present a new application for these expected values, using upper bounds on the expected number of interior vertices of degree 3 to get a new lower bound, Ω(N^2.4317), for the minimal number of triangulations any N-element planar point set in general position must have. This improves the previously best known lower bound of Ω(N^2.33).