Showing papers on "Disjoint sets published in 2003"

A Preconditioner for Substructuring Based on Constrained Energy Minimization

Abstract: A preconditioner for substructuring based on constrained energy minimization concepts is presented. The preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second- and fourth-order structural mechanics problems. The approach involves constraints associated with disjoint sets of nodes on substructure boundaries. These constraints provide the means for preconditioning at both the substructure and global levels. Numerical examples are presented that demonstrate the good performance of the method in terms of iterations, compute time, and condition numbers of the preconditioned equations.

Wiener's lemma for twisted convolution and Gabor frames

Abstract: As a consequence, Ca is invertible and bounded on all ?p(Zd) for 1 < p < oo simultaneously. In this article we study several non-commutative generalizations of Wiener's Lemma and their application to Gabor theory. The paper is divided into two parts: the first part (Sections 2 and 3) is devoted to abstract harmonic analysis and extends Wiener's Lemma to twisted convolution. The second part (Section 4) is devoted to the theory of Gabor frames, specifically to the design of dual windows with good time-frequency localization. In particular, we solve a conjecture of Janssen, Feichtinger and one of us [17], [18], [9]. These two topics appear to be completely disjoint, but they are not. The solution of the conjectures about Gabor frames is an unexpected application of methods from non-commutative harmonic analysis to application-oriented mathematics. It turns out that the connection between twisted convolution and the Heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis. To be more concrete, we formulate some of our main results first and will deal with the details and the technical background later.

Approximating the Domatic Number

Abstract: A set of vertices in a graph is a dominating set if every vertex outside the set has a neighbor in the set. The domatic number problem is that of partitioning the vertices of a graph into the maximum number of disjoint dominating sets. Let n denote the number of vertices, $\delta$ the minimum degree, and $\Delta$ the maximum degree. We show that every graph has a domatic partition with $(1 - o(1))(\delta + 1)/\ln n$ dominating sets and, moreover, that such a domatic partition can be found in polynomial-time. This implies a $(1 + o(1))\ln n$-approximation algorithm for domatic number, since the domatic number is always at most $\delta + 1$. We also show this to be essentially best possible. Namely, extending the approximation hardness of set cover by combining multiprover protocols with zero-knowledge techniques, we show that for every $\epsilon > 0$, a $(1 - \epsilon)\ln n$-approximation implies that $NP \subseteq DTIME(n^{O(\log\log n)})$. This makes domatic number the first natural maximization problem (known to the authors) that is provably approximable to within polylogarithmic factors but no better. We also show that every graph has a domatic partition with $(1 - o(1))(\delta + 1)/\ln \Delta$ dominating sets, where the "o(1)" term goes to zero as $\Delta$ increases. This can be turned into an efficient algorithm that produces a domatic partition of $\Omega(\delta/\ln \Delta)$ sets.

Equilibrium statistical mechanics of fermion lattice systems

Abstract: We study equilibrium statistical mechanics of Fermion lattice systems which require a different treatment compared with spin lattice systems due to the non-commutativity of local algebras for disjoint regions. Our major result is the equivalence of the KMS condition and the variational principle with a minimal assumption for the dynamics and without any explicit assumption on the potential. Its proof applies to spin lattice systems as well, yielding a vast improvement over known results. All formulations are in terms of a C*-dynamical systems for the Fermion (CAR) algebra with all or a part of the following assumptions: (I) The interaction is even, namely, the dynamics αt commutes with the even-oddness automorphism Θ. (Automatically satisfied when (IV) is assumed.) (II) The domain of the generator δα of αt contains the set of all strictly local elements of . (III) The set is the core of δα. (IV) The dynamics αt commutes with lattice translation automorphism group τ of . A major technical tool is the conditional expectation from onto its C*-subalgebras for any subset I of the lattice, which induces a system of commuting squares. This technique overcomes the lack of tensor product structures for Fermion systems and even simplifies many known arguments for spin lattice systems. In particular, this tool is used for obtaining the isomorphism between the real vector space of all *-derivations with their domain , commuting with Θ, and that of all Θ-even standard potentials which satisfy a specific norm convergence condition for the one point interaction energy. This makes it possible to associate a unique standard potential to every dynamics satisfying (I) and (II). The convergence condition for the potential is a consequence of its definition in terms of the *-derivation and not an additional assumption. If translation invariance is imposed on *-derivations and potentials, then the isomorphism is kept and the space of translation covariant standard potentials becomes a separable Banach space with respect to the norm of the one point interaction energy. This is a crucial basis for an application of convex analysis to the equivalence proof in the major result. Everything goes in parallel for spin lattice systems without the evenness assumption (I).

Lyapunov exponents for products of matrices and multifractal analysis. Part II: General matrices

Abstract: We continue the study in [15, 18] on the upper Lyapunov exponents for products of matrices. Here we consider general matrices. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. In any case, we focus our interest on a special case where the matrix function M(x) takes finite values M 1, ..., M m . In this case, we prove the variational formula under an additional irreducibility condition. This extends a previous result of the author and Lau [18]. As an application, we prove a new multifractal formalism for a certain class of self-similar measures on ℝ with overlaps. More precisely, let μ be the self-similar measure on ℝ generated by a family of contractive similitudes {S j = ρx + b j } =1 l which satisfies the finite type condition. Then we can construct a family (finite or countably infinite) of closed intervals {I j } j∈Λ with disjoint interiors, such that μ is supported on ⋃ j∈Λ I j and the restricted measure $$ \mu |_{I_j } $$ of μ on each interval I j satisfies the complete multifractal formalism. Moreover, the dimension spectrum dim H $$ E_{\mu |_{I_j } } $$ (α) is independent of j.

Normal and Non-Normal Points of Self-Similar Sets and Divergence Points of Self-Similar Measures

Abstract: Let and be the self-similar set and the self-similar measure associated with an IFS (iterated function system) with probabilities satisfying the open set condition. Let denote the full shift space and let denote the natural projection. The (symbolic) local dimension of at is defined by , where for . A point for which the limit does not exist is called a divergence point. In almost all of the literature the limit is assumed to exist, and almost nothing is known about the set of divergence points. In the paper a detailed analysis is performed of the set of divergence points and it is shown that it has a surprisingly rich structure. For a sequence , let denote the set of accumulation points of . For an arbitrary subset of , the Hausdorff and packing dimension of the set \[ \left\{\omega\in\Sigma\left\vert {\sf A}\left(\frac{\log\mu K_{\omega\mid n}}{\log\hbox{ diam }K_{\omega\mid n}}\right)\right.=I\right\} \]and related sets is computed. An interesting and surprising corollary to this result is that the set of divergence points is extremely ‘visible’; it can be partitioned into an uncountable family of pairwise disjoint sets each with full dimension.In order to prove the above statements the theory of normal and non-normal points of a self-similar set is formulated and developed in detail. This theory extends the notion of normal and non-normal numbers to the setting of self-similar sets and has numerous applications to the study of the local properties of self-similar measures including a detailed study of the set of divergence points.

Harmonic analysis on the infinite symmetric group

Abstract: Let S be the group of finite permutations of the naturals 1,2,... The subject of the paper is harmonic analysis for the Gelfand pair (G,K), where G stands for the product of two copies of S while K is the diagonal subgroup in G. The spherical dual to (G,K) (that is, the set of irreducible spherical unitary representations) is an infinite-dimensional space. For such Gelfand pairs, the conventional scheme of harmonic analysis is not applicable and it has to be suitably modified. We construct a compactification of S called the space of virtual permutations. It is no longer a group but it is still a G-space. On this space, there exists a unique G-invariant probability measure which should be viewed as a true substitute of Haar measure. More generally, we define a 1-parameter family of probability measures on virtual permutations, which are quasi-invariant under the action of G. Using these measures we construct a family {T_z} of unitary representations of G depending on a complex parameter z. We prove that any T_z admits a unique decomposition into a multiplicity free integral of irreducible spherical representations of (G,K). Moreover, the spectral types of different representations (which are defined by measures on the spherical dual) are pairwise disjoint. Our main result concerns the case of integral values of parameter z: then we obtain an explicit decomposition of T_z into irreducibles. The case of nonintegral z is quite different. It was studied by Borodin and Olshanski, see e.g. the survey math.RT/0311369.

Hypothesis generation guided by co-word clustering

Abstract: Co-word analysis was applied to keywords assigned to MEDLINE documents contained in sets of complementary but disjoint literatures. In strategical diagrams of disjoint literatures, based on internal density and external centrality of keyword-containing clusters, intermediate terms (linking the disjoint partners) were found in regions of below-median centrality and density. Terms representing the disjoint literature themes were found in close vicinity in strategical diagrams of intermediate literatures. Based on centrality-density ratios, characteristic values were found which allow a rapid identification of clusters containing possible intermediate and disjoint partner terms. Applied to the already investigated disjoint pairs Raynaud"s Disease - Fish Oil, Migraine - Magnesium, the method readily detected known and unknown (but relevant) intermediate and disjoint partner terms. Application of the method to the literature on Prions led to Manganese as possible disjoint partner term. It is concluded that co-word clustering is a powerful method for literature-based hypothesis generation and knowledge discovery.

Null systems and sequence entropy pairs

Abstract: A measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained.Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering a question in Blanchard et al (F. Blanchard, B. Host and A. Maass, Topological complexity. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662). For a group action we give a direct proof of the fact that the factor induced by the smallest invariant equivalence relation containing the regionally proximal relation is equicontinuous. Furthermore, we show that a non-equicontinuous minimal distal system is not null.

Edge disjoint Hamiltonian cycles in k-ary n-cubes and hypercubes

Abstract: Solutions for decomposing a higher dimensional torus to edge disjoint lower dimensional tori, in particular, edge disjoint Hamiltonian cycles are obtained based on the coding theory approach. First, Lee distance Gray codes in Z/sub k//sup n/ are presented and then it is shown how these codes can directly be used to generate edge disjoint Hamiltonian cycles in k-ary n-cubes. Further, some new classes of binary Gray codes are designed from these Lee distance Gray codes and, using these new classes of binary Gray codes, edge disjoint Hamiltonian cycles in hypercubes are generated.

Approximating Node Connectivity Problems via Set Covers

Abstract: Given a graph (directed or undirected) with costs on the edges, and an integer $k$, we consider the problem of finding a $k$-node connected spanning subgraph of minimum cost. For the general instance of the problem (directed or undirected), there is a simple $2k$-approximation algorithm. Better algorithms are known for various ranges of $n,k$. For undirected graphs with metric costs Khuller and Raghavachari gave a $( 2+{2(k-1)}/{n})$-approximation algorithm. We obtain the following results: (i) For arbitrary costs, a $k$-approximation algorithm for undirected graphs and a $(k+1)$-approximation algorithm for directed graphs. (ii) For metric costs, a $(2+({k-1})/{n})$-approximation algorithm for undirected graphs and a $(2+{k}/{n})$-approximation algorithm for directed graphs. For undirected graphs and $k=6,7$, we further improve the approximation ratio from $k$ to $\lceil (k+1)/2 \rceil=4$; previously, $\lceil (k+1)/2 \rceil$-approximation algorithms were known only for $k \leq 5$. We also give a fast $3$-approximation algorithm for $k=4$. The multiroot problem generalizes the min-cost $k$-connected subgraph problem. In the multiroot problem, requirements $k_u$ for every node $u$ are given, and the aim is to find a minimum-cost subgraph that contains $\max\{k_u,k_v\}$ internally disjoint paths between every pair of nodes $u,v$. For the general instance of the problem, the best known algorithm has approximation ratio $2k$, where $k=\max k_u$. For metric costs there is a 3-approximation algorithm. We consider the case of metric costs, and, using our techniques, improve for $k \leq 7$ the approximation guarantee from $3$ to $2+{\lfloor (k-1)/2 \rfloor}/{k} < 2.5$.

Communications network system and method for routing based on disjoint pairs of paths

Abstract: Methods for determining at least two pre-computed paths to a destination in a communications network are provided. The two paths are maximally disjoint. Maximally disjoint paths are paths where the number of links or nodes common to the two paths is minimized. This minimization is given a priority over other path considerations, such as bandwidth or cost metrics. By pre-computing a maximally disjoint pair of paths, the probability that an inoperable link or node is in both paths is minimized. The probability that the inoperable link or node blocks a transfer of data is minimized. Additionally, a pair of maximally disjoint paths is determined even if absolutely disjoint paths are not possible. The communications network may include at least four nodes, and maximally disjoint pairs of paths are pre-computed from each node to each other node. A third path from each node to each other node may also be computed as a function of bandwidth or a cost metric. Therefore, the advantages of the maximally disjoint pair of paths are provided as discussed above and a path associated with a higher bandwidth or lower cost is provided to more likely satisfy the user requirements of a data transfer.

On reducibility and symmetry of disjoint NP pairs

Abstract: We consider some problems about pairs of disjoint NP sets. The theory of these sets with a natural concept of reducibility is, on the one hand, closely related to the theory of proof systems for propositional calculus, and, on the other, it resembles the theory of NP completeness. Furthermore, such pairs are important in cryptography. Among others, we prove that the Broken Mosquito Screen pair of disjoint NP sets can be polynomially reduced to Clique-Coloring pair and thus is polynomially separable and we show that the pair of disjoint NP sets canonically associated with the Resolution proof system is symmetric.

Parameterized Tractability of Edge-Disjoint Paths on Directed Acyclic Graphs

Abstract: Given a graph and pairs s i t i of terminals, the edge-disjoint paths problem is to determine whether there exist s i t i paths that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time n O(k) where k is the number of paths. It has been a long-standing open question whether it is fixed-parameter tractable in k. We resolve this question in the negative: we show that the problem is W[1]-hard. In fact it remains W[1]-hard even if the demand graph consists of two sets of parallel edges.

Uniform Decay Rates of Solutions to a Nonlinear Wave Equation with Boundary Condition of Memory Type

Abstract: In this article we study the hyperbolic problem (1) where Ω is a bounded region in R n whose boundary is partitioned into disjoint sets Γ0, Γ1. We prove that the dissipation given by the memory term is strong enough to assure exponential (or polynomial) decay provided the relaxation function also decays exponentially (or polynomially). In both cases the solution decays with the same rate of the relaxation function.

Manifold structure of spaces of spherical tight frames

Abstract: We consider the space F^E_{k,n} of all spherical tight frames of k vectors in real or complex n--dimensional Hilbert space E^n, i.e. E=R or E=C, and its orbit space G^E_{k,n}=F^E_{k,n}/O^E_n under the obvious action of the group O^E_n of structure preserving transformations of E^n. We show that the quotient map F^E_{k,n} -> G^E_{k,n} is a locally trivial fiber bundle (also in the more general case of ellipsoidal tight frames) and that there is a homeomorphism G^E_{k,n} -> G^E_{k,k-n}. We show that G^E_{k,n} and F^E_{k,n} are real manifolds whenever k and n are relatively prime, and we describe them as disjoint unions of finitely many manifolds (of various dimensions) when when k and n have a common divisor. We also prove that F^R_{k,2} is connected (k >= 4) and F^R_{n+2,n} is connected, (n >= 2). The spaces G^R_{4,2} and G^R_{5,2} are investigated in detail. The former is found to be a graph and the latter is the orientable surface of genus 25.

Method for providing QoS (quality of service) - guaranteeing multi-path and method for providing disjoint path using the same

Abstract: A method for providing a QoS-guaranteeing multi-path and a method for providing disjoint paths using the same are provided. The method configures the shortest path tree by adapting the start node “s” and the destination node “d”. When a new node is selected as a tree node in a tree configuration process according to the “s” or the “d”, the closest node to either the “s” or the “d” is selected. If a specified node “v” is contained in both a tree oriented from the “s” and another tree oriented from the “d”, one path of “s”-“v”-“d” is created. If all nodes are contained in either the tree of “s” or the other tree of “d”, then a program of path creation process is terminated. Further, the method further includes a step for determining two disjoint paths from the “s” to the “d” among the found multiple paths above.

On Maximal Frequent and Minimal Infrequent Sets in Binary Matrices

Abstract: Given an m×n binary matrix A, a subset C of the columns is called t-frequent if there are at least t rows in A in which all entries belonging to C are non-zero. Let us denote by α the number of maximal t-frequent sets of A, and let β denote the number of those minimal column subsets of A which are not t-frequent (so called t-infrequent sets). We prove that the inequality αl(m−t+1)β holds for any binary matrix A in which not all column subsets are t-frequent. This inequality is sharp, and allows for an incremental quasi-polynomial algorithm for generating all minimal t-infrequent sets. We also prove that the analogous generation problem for maximal t-frequent sets is NP-hard. Finally, we discuss the complexity of generating closed frequent sets and some other related problems.

Equilibrium relations and bipolar cognitive mapping for online analytical processing with applications in international relations and strategic decision support

Abstract: Bipolar logic, bipolar sets, and equilibrium relations are proposed for bipolar cognitive mapping and visualization in online analytical processing (OLAP) and online analytical mining (OLAM). As cognitive models, cognitive maps (CMs) hold great potential for clustering and visualization. Due to the lack of a formal mathematical basis, however, CM-based OLAP and OLAM have not gained popularity. Compared with existing approaches, bipolar cognitive mapping has a number of advantages. First, bipolar CMs are formal logical models as well as cognitive models. Second, equilibrium relations (with polarized reflexivity, symmetry, and transitivity), as bipolar generalizations and fusions of equivalence relations, provide a theoretical basis for bipolar visualization and coordination. Third, an equilibrium relation or CM induces bipolar partitions that distinguish disjoint coalition subsets not involved in any conflict, disjoint coalition subsets involved in a conflict, disjoint conflict subsets, and disjoint harmony subsets. Finally, equilibrium energy analysis leads to harmony and stability measures for strategic decision and multiagent coordination. Thus, this work bridges a gap for CM-based clustering and visualization in OLAP and OLAM. Basic ideas are illustrated with example CMs in international relations.

Temporal logic with cyclic counting and the degree of aperiodicity of finite automata

Abstract: We define the degree of aperiodicity of finite automata and show that for every set M of positive integers, the class QAM of finite automata whose degree of aperiodicity belongs to the division ideal generated by M is closed with respect to direct products, disjoint unions, subautomata, homomorphic images and renamings These closure conditions define q-varieties of finite automata We show that q-varieties are in a one-to-one correspondence with literal varieties of regular languages We also characterize QAM as the cascade product of a variety of counters with the variety of aperiodic (or counter-free) automata We then use the notion of degree of aperiodicity to characterize the expressive power of first-order logic and temporal logic with cyclic counting with respect to any given set M of moduli It follows that when M is finite, then it is decidable whether a regular language is definable in first-order or temporal logic with cyclic counting with respect to moduli in M

Location-based localized alternate, disjoint and multi-path routing algorithms for wireless networks

Abstract: Recently, several fully distributed (localized) location-based routing protocols for a mobile ad hoc network were reported in literature. They are variations of directional (DIR), geographic distance (GEDIR) or progress-based (MFR) routing methods. In DIR methods, each node A (the source or intermediate node) transmits a message m to several neighbors whose direction is closest to the direction of D. In MFR (most forward progress within radius), and GEDIR (GEographic Distance Routing) methods, when node A wants to send m to node D, it forwards m to its neighbor C whose projection or distance (respectively) is closest to D among all neighbors of A. The same procedure is repeated until D, if possible, is eventually reached. In this paper, we introduce three variants of multiple path c-GEDIR, c-DIR and c-MFR methods, in which m is initially sent to c best neighbors according to corresponding criterion, and afterwards, on intermediate nodes, it is forwarded to only the best neighbor. In the original c-path method, only the first received copy at intermediate nodes is forwarded to the best neighbor. In the alternate c-path method, the i th received copy is forwarded to i the best neighbor, according to the selected criterion. In the disjoint c-path method, each intermediate node, upon receiving the message, will forward it to its best neighbor among those who never received the message (thus, in effect, the methods attempts to create c disjoint paths). The simulation experiments with random graphs show that disjoint multiple path methods provide high success rates, and small hop counts for small values of c. They also have reduced flooding rates compared to the best existing multiple-path methods and/or methods that require memorizing past traffic, such as recently proposed LAR2, f-GEDIR, and DFS based routing, and can serve as a basis for scalable QoS routing in wireless networks.