On the Hardness of Computing Intersection,Union and Minkowski Sum of Polytopes

For polytopes P ₁,P ₂⊂ℝ ^d , we consider the intersection P ₁∩P ₂, the convex hull of the union CH(P ₁∪P ₂), and the Minkowski sum P ₁+P ₂. For the Minkowski sum, we prove that enumerating the facets of P ₁+P ₂ is NP-hard if P ₁ and P ₂ are specified by facets, or if P ₁ is specified by vertices and P ₂ is a polyhedral cone specified by facets. For the intersection, we prove that computing the facets or the vertices of the intersection of two polytopes is NP-hard if one of them is given by vertices and the other by facets. Also, computing the vertices of the intersection of two polytopes given by vertices is shown to be NP-hard. Analogous results for computing the convex hull of the union of two polytopes follow from polar duality. All of the hardness results are established by showing that the appropriate decision version, for each of these problems, is NP-complete.     

Embeddings of bipartite graphs

If G is a bipartite graph with bipartition A, B then let G_m,n(A, B) be obtained from G by replacing each vertex a of A by an independent set a₁, …, a_m, each vertex b of B by an independent set b₁,…, b_n, and each edge ab of G by the complete bipartite graph with edges a_ib_j (1 ≤ i ≤ m and 1 ≤ j ≤ n). Whenever G has certain types of spanning forests, then cellular embeddings of G in surfaces S may be lifted to embeddings of G_m,n(A, B) having faces of the same sizes as those of G in S. These results are proved by the technique of “excess-current graphs.” They include new genus embeddings for a large class of bipartite graphs.     

Localisation in rings equipped with a solvable Lie algebra action

Let k be an algebraically closed uncountable field of characteristic 0,g a finite dimensional solvable k-Lie algebraR a noetherian k-algebra on which g acts by k-derivationsU(g) the enveloping algebra of g,A=R*g the crossed product of R by U(g)P a prime ideal of A and Ω(P) the clique of P. Suppose that the prime ideals of the polynomial ring R[x] are completely prime. If R is g-hypernormal, then Ω(P) is classical. Denote by A_T the localised ring and let M be a primitive ideal of A_T Set Q=P∩R In this note, we show that if R is a strongly (R,g)-admissible integral domain and if QR_Q is generated by a regular g-centralising set of elements, then

(1)M is generated by a regular g-semi-invariant normalising set of elements of cardinald = dim (R_Q 0 + ∣X_A (P)∣

(2)d gldim(A_T ) = Kdim(A_T ) = ht(M) = ht(P).     

Partially Ordered Monoids Generated by Operators H , S , P , and by H , S , P f Are Isomorphic

Let I , H , S , P , P _f be the usual operators on classes of algebras of the same type (P _f for filtered products). The partially ordered monoid generated by the operators H , S , P with respect to composition of operators, I as an identity element, and a natural ordering between operators is described by Pigozzi (Algebra Universalis 2 (1972), 346—353). Let us denote by \cal M =\langle H, S, P\rangle and by \cal M _f =\langle H, S, P _f \rangle the partially ordered monoids generated by {H, S, P} and by {H, S, P _f } respectively. The aim of this paper is to prove that \cal M is isomorphic to \cal M _f . October 29, 1999     

Lattices Generated by Orbits of Flats Under Finite Affine Groups

Let AG(n, _q ) be the n-dimensional affine space over  _q . For a given integer m with 0 ≤ m ≤ n, all m-flats form an orbit, denoted by ?(m,n), under the action of the affine group AGL _n ( _q ) of AG(n, _q ). Denote the set of all intersections of m-flats in ?(m,n) by ?(m,n). By ordering ?(m,n) by ordinary or reverse inclusion, two classes of lattices are obtained. This article discusses their geometricity, and computes their character polynomials.     

Almost sure exponential stabilization by stochastic feedback control based on discrete-time observations

Since Mao initiated the study of stabilization of ordinary differential equations (ODEs) by stochastic feedback controls based on discrete-time state observations in 2016, no more work on this intriguing topic has been reported. This article investigates how to stabilize a given unstable linear non-autonomous ODE by controller σ(t)x(δ_t)dB(t), and how to stabilize an unstable nonlinear hybrid SDE by controller G(r(δ_t))x(δ_t)dB(t), where δ_t represents time points of observation with sufficiently small observation interval, B(t) is a Brownian motion and r(t) is the Markov Chain, in the sense of pth moment (0 < p < 1) and almost sure exponential stability.     

On Genuine Bernstein–Durrmeyer Operators

We continue the studies on the so–called genuine Bernstein–Durrmeyer operators U _n by establishing a recurrence formula for the moments and by investigating the semigroup T(t) approximated by U _n . Moreover, for sufficiently smooth functions the degree of this convergence is estimated. We also determine the eigenstructure of U _n , compute the moments of T(t) and establish asymptotic formulas. Received: January 26, 2007.     

Subdirect decompositions and the radical of a generalized Boolean algebra extension of a lattice ordered group

The extension of a lattice ordered group A by a generalized Boolean algebra B will be denoted by A _B . In this paper we apply subdirect decompositions of A _B for dealing with a question proposed by Conrad and Darnel. Further, in the case when A is linearly ordered we investigate (i) the completely subdirect decompositions of A _B and those of B, and (ii) the values of elements of A _B and the radical R(A _B ).     

Extensions and biextensions of locally constant group schemes,tori and abelian schemes

Let S be a scheme. We compute explicitly the group of homomorphisms, the S-sheaf of homomorphisms, the group of extensions, and the S-sheaf of extensions involving locally constant S-group schemes, abelian S-schemes, and S-tori. Using the obtained results, we study the categories of biextensions involving these geometrical objects. In particular, we prove that if G _i (for i = 1, 2, 3) is an extension of an abelian S-scheme A _i by an S-torus T _i , the category of biextensions of (G ₁, G ₂) by G ₃ is equivalent to the category of biextensions of the underlying abelian S-schemes (A ₁, A ₂) by the underlying S-torus T ₃.      

The intrinsic dimensionality of graphs

We resolve the following conjecture raised by Levin together with Linial, London, and Rabinovich [Combinatorica, 1995]. For a graph G, let dim(G) be the smallest d such that G occurs as a (not necessarily induced) subgraph of ℤ_∞ ^d , the infinite graph with vertex set ℤ ^d and an edge (u, v) whenever ∥u − v∥_∞ = 1. The growth rate of G, denoted ρ _G , is the minimum ρ such that every ball of radius r > 1 in G contains at most r ^ρ vertices. By simple volume arguments, dim(G) = Ω(ρ _G ). Levin conjectured that this lower bound is tight, i.e., that dim(G) = O(ρ _G ) for every graph G. Previously, it was unknown whether dim(G) could be bounded above by any function of ρ _G . We show that a weaker form of Levin’s conjecture holds by proving that dim(G) = O(ρ _G log ρ _G ) for any graph G. We disprove, however, the specific bound of the conjecture and show that our upper bound is tight by exhibiting graphs for which dim(G) = Ω(ρ _G log ρ _G ). For several special families of graphs (e.g., planar graphs), we salvage the strong form, showing that dim(G) = O(ρ _G ). Our results extend to a variant of the conjecture for finite-dimensional Euclidean spaces posed by Linial and independently by Benjamini and Schramm. Supported by NSF grant CCR-0121555 and by an NSF Graduate Research Fellowship.     

Extremal cover times for random walks on trees

Let C_ν(T) denote the “cover time” of the tree T from the vertex v, that is, the expected number of steps before a random walk starting at v hits every vertex of T. Asymptotic lower bounds for C_ν(T) (for T a tree on n vertices) have been obtained recently by Kahn, Linial, Nisan and Saks, and by Devroye and Sbihi; here, we obtain the exact lower bound (approximately 2n In n) by showing that C_ν(T) is minimized when T is a star and v is one of its leaves. In addition, we show that the time to cover all vertices and then return to the starting point is minimized by a star (beginning at the center) and maximized by a path (beginning at one of the ends).     

Topological properties of the multifunction space <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">X</Emphasis>) of cusco maps

A set-valued mapping F from a topological space X to a topological space Y is called a cusco map if F is upper semicontinuous and F(x) is a nonempty, compact and connected subset of Y for each x ∈ X. We denote by L(X), the space of all subsets F of X × ℝ such that F is the graph of a cusco map from the space X to the real line ℝ. In this paper, we study topological properties of L(X) endowed with the Vietoris topology. The second author is supported by the SPM fellowship awarded by the Council of Scientific and Industrial Research, India.     

On 2-Detour Subgraphs of the Hypercube

A spanning subgraph H of a graph G is a 2-detour subgraph of G if for each x, y ∈ V(G), d _H (x, y) ≤ d _G (x, y) + 2. We prove a conjecture of Erdős, Hamburger, Pippert, and Weakley by showing that for some positive constant c and every n, each 2-detour subgraph of the n-dimensional hypercube Q _n has at least clog₂ n · 2 ⁿ edges. József Balogh: Research supported in part by NSF grants DMS-0302804, DMS-0603769 and DMS-0600303, UIUC Campus Reseach Board #06139 and #07048, and OTKA 049398. Alexandr Kostochka: Research supported in part by NSF grants DMS-0400498 and DMS-0650784, and grant 06-01-00694 of the Russian Foundation for Basic Research.     

A generalized Turán problem in random graphs

We study a generalization of the Turán problem in random graphs. Given graphs T and H, let ex(G(n,p),T,H) be the largest number of copies of T in an H‐free subgraph of G(n,p). We study the threshold phenomena arising in the evolution of the typical value of this random variable, for every H and every 2‐balanced T. Our results in the case when m₂(H) > m₂(T) are a natural generalization of the Erd?s‐Stone theorem for G(n,p), proved several years ago by Conlon‐Gowers and Schacht; the case T = K_m was previously resolved by Alon, Kostochka, and Shikhelman. The case when m₂(H) ≤ m₂(T) exhibits a more complex behavior. Here, the location(s) of the (possibly multiple) threshold(s) are determined by densities of various coverings of H with copies of T and the typical value(s) of ex(G(n,p),T,H) are given by solutions to deterministic hypergraph Turán‐type problems that we are unable to solve in full generality.     

Procreating tiles of double commutative-step digraphs

Double commutative-step digraph generalizes the double-loop digraph. A double commutative-step digraph can be represented by an L-shaped tile, which periodically tessellates the plane. Given an initial tile L（l, h, x, y）, Aguil5 et al. define a discrete iteration L（p） = L（l ＋ 2p, h ＋ 2p, x ＋ p, y ＋ p）, p = 0, 1, 2,..., over L-shapes （equivalently over double commutative-step digraphs）, and obtain an orbit generated by L（l, h, x,y）, which is said to be a procreating k-tight tile if L（p）（p = 0, 1, 2, ~ ~ ~ ） are all k-tight tiles. They classify the set of L-shaped tiles by its behavior under the above-mentioned discrete dynamics and obtain some procreating tiles of double commutative-step digraphs. In this work, with an approach proposed by Li and Xu et al., we define some new discrete iteration over L-shapes and classify the set of tiles by the procreating condition. We also propose some approaches to find infinite families of realizable k-tight tiles starting from any realizable k-tight L-shaped tile L（l, h, x, y）, 0 ≤ y - x ≤ 2k ＋ 2. As an example, we present an infinite family of 3-tight optimal double-loop networks to illustrate our approaches.     

Constructive characterizations of （γp, γ）- and （γp, γpr）-trees

Let G = （V, E） be a graph without isolated vertices. A set S lohtain in V is a domination set of G if every vertex in V - S is adjacent to a vertex in S, that is N[S] = V. The domination number of G, denoted by γ（G）, is the minimum cardinality of a domination set of G. A set S lohtain in V is a paired-domination set of G if S is a domination set of G and the induced subgraph G[S] has a perfect matching. The paired-domination number, denoted by γpr（G）, is defined to be the minimum cardinality of a paired-domination set S in G. A subset S lohtain in V is a power domination set of G if all vertices of V can be observed recursively by the following rules： （i） all vertices in N[S] are observed initially, and （ii） if an observed vertex u has all neighbors observed except one neighbor v, then v is observed （by u）. The power domination number, denoted by γp（G）, is the minimum cardinality of a power domination set of G. In this paper, the constructive characterizations for trees with γp=γ and γpr = γp are provided respectively.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

The number of labeled graphs placeable by a given permutation

Let S be a finite set and σ a permutation on S. The permutation σ* on the set of 2-subsets of S is naturally induced by σ. Suppose G is a graph and V(G), E(G) are the vertex set, the edge set, respectively. Let V(G) = S. If E(G) and σ*(E(G)), the image of E(G) by σ*, have no common element, then G is said to be placeable by σ. This notion is generalized as follows. If any two sets of {E(G), (σ¹)*(E(G)),…,(σ^l−1)* (E(G))} have no common element, then G is said to be I-placeable by σ. In this paper, we count the number of labeled graphs which are I-placeable by a given permutation. At first, we introduce the interspaced Ith Fibonacci and Lucas numbers. When I = 2 these numbers are the ordinary Fibonacci and Lucas numbers. It is known that the Fibonacci and Lucas numbers are rounded powers. We show that the interspaced Ith Fibonacci and Lucas numbers are also rounded powers when I = 3. Next, we show the number of labeled graphs which are I-placeable by a given permutation is a product of the interspaced Ith Lucas numbers. Finally, using a property of the generalized binomial series, we count the number of labeled graphs of size k which are I-placeable by σ. © 1996 John Wiley & Sons, Inc.     

ON THE PROJECTION OF THE EFFICIENT SET AND POTENTIAL APPLICATIONS

Abstract

We study the efficient set X _E for a multiple objective linear program by using its projection into the linear space L spanned by the independent criteria. We show that in the orthogonally complementary space of L, the efficient points form a polyhedron, while in L an efficiency-equivalent polyhedron for the projection P(X _E ) of X _E can be constructed by algorithms of outer and inner approximation types. These algorithms can be also used for generating all extreme points of P(X _E ). Application to optimization over the efficient set for a multiple objective linear program is considered.