The Rank-Width of Edge-Coloured Graphs

A C-coloured graph is a graph, that is possibly directed, where the edges are coloured with colours from the set C. Clique-width is a complexity measure for C-coloured graphs, for finite sets C. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss some possible extensions of the notion of rank-width to C-coloured graphs. There is not a unique natural notion of rank-width for C-coloured graphs. We define two notions of rank-width for them, both based on a coding of C-coloured graphs by ${\mathbb{F}}^{*}$ -graphs— $\mathbb {F}$ -coloured graphs where each edge has exactly one colour from $\mathbb{F}\setminus \{0\},\ \mathbb{F}$ a field—and named respectively $\mathbb{F}$ -rank-width and $\mathbb {F}$ -bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for $\mathbb{F}^{*}$ -graphs and prove that $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width are characterised by a list of $\mathbb{F}^{*}$ -graphs to exclude as vertex-minors (this list is finite if $\mathbb{F}$ is finite). An algorithm that decides in time O(n ³) whether an $\mathbb{F}^{*}$ -graph with n vertices has $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) at most k, for fixed k and fixed finite field $\mathbb{F}$ , is also given. Graph operations to check MSOL-definable properties on $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) are presented. A specialisation of all these notions to graphs without edge colours is presented, which shows that our results generalise the ones in undirected graphs.     

Quantum discord in spin systems with dipole–dipole interaction

The behavior of total quantum correlations (discord) in dimers consisting of dipolar-coupled spins 1/2 are studied. We found that the discord $Q=0$ at absolute zero temperature. As the temperature $T$ increases, the quantum correlations in the system increase at first from zero to its maximum and then decrease to zero according to the asymptotic law $T^{-2}$ . It is also shown that in absence of external magnetic field $B$ , the classical correlations $C$ at $T\rightarrow 0$ are, vice versa, maximal. Our calculations predict that in crystalline gypsum $\hbox {CaSO}_{4}\cdot \hbox {2H}_{2}{\hbox {O}}$ the value of natural $(B=0)$ quantum discord between nuclear spins of hydrogen atoms is maximal at the temperature of 0.644  $\upmu $ K, and for 1,2-dichloroethane $\hbox {H}_{2}$ ClC– $\hbox {CH}_{2}{\hbox {Cl}}$ the discord achieves the largest value at $T=0.517~\upmu $ K. In both cases, the discord equals $Q\approx 0.083$  bit/dimer what is $8.3\,\%$ of its upper limit in two-qubit systems. We estimate also that for gypsum at room temperature $Q\sim 10^{-18}$  bit/dimer, and for 1,2-dichloroethane at $T=90$  K the discord is $Q\sim 10^{-17}$  bit per a dimer.     

Simple characterization of strict residuated lattices with an involutive negation

We consider property of strict residuated lattices (SRL-algebras) with a new involutive negation $\lnot, $ called here by SRL $_{\lnot }$ -algebras, and give a simple characterization of SRL $_{\lnot }$ -algebras. We also prove a prime filter theorem of SRL $_{\lnot }$ -algebras, from which we conclude that every linearly ordered SRL $_{\lnot }$ -algebra is simple. As a corollary to this fact, we have a well-known result that every SML $_{\lnot }$ -algebra (SBL $_{\lnot }$ -algebra) is a subdirect product of linearly ordered SML $_{\lnot }$ -algebras (SBL $_{\lnot }$ -algebras).     

Hyper-Hamiltonian laceability of balanced hypercubes

The balanced hypercube, proposed by Wu and Huang, is a new variation of hypercube. The particular property of the balanced hypercube is that each processor has a backup processor that shares the same neighborhood. A Hamiltonian bipartite graph with bipartition $V_{0}\cup V_{1}$ is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices $x\in V_{0}$ and $y\in V_{1}$ . A graph $G$ is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for any vertex $v\in V_{i}$ , $i\in \{0,1\}$ , there is a Hamiltonian path in G–v between any pair of vertices in $V_{1-i}$ . In this paper, we mainly prove that the balanced hypercube is hyper-Hamiltonian laceable.     

On reducing factorization to the discrete logarithm problem modulo a composite

The discrete logarithm problem modulo a composite??abbreviate it as DLPC??is the following: given a (possibly) composite integer n??? 1 and elements ${a, b \in \mathbb{Z}_n^*}$ , determine an ${x \in \mathbb{N}}$ satisfying a ^x ?=?b if one exists. The question whether integer factoring can be reduced in deterministic polynomial time to the DLPC remains open. In this paper we consider the problem ${{\rm DLPC}_\varepsilon}$ obtained by adding in the DLPC the constraint ${x\le (1-\varepsilon)n}$ , where ${\varepsilon}$ is an arbitrary fixed number, ${0 < \varepsilon\le\frac{1}{2}}$ . We prove that factoring n reduces in deterministic subexponential time to the ${{\rm DLPC}_\varepsilon}$ with ${O_\varepsilon((\ln n)^2)}$ queries for moduli less or equal to n.     

The Nonconvergence of h-Refinement in Prolate Elements

Prolate elements are a “plug-compatible” modification of spectral elements in which Legendre polynomials are replaced by prolate spheroidal wave functions of order zero. Prolate functions contain a“bandwidth parameter” $c \ge 0 $ c ≥ 0 whose value is crucial to numerical performance; the prolate functions reduce to Legendre polynomials for $c\,=\,0$ c = 0 . We show that the optimal bandwidth parameter $c$ c not only depends on the number of prolate modes per element $N$ N , but also on the element widths $h$ h . We prove that prolate elements lack $h$ h -convergence for fixed $c$ c in the sense that the error does not go to zero as the element size $h$ h is made smaller and smaller. Furthermore, the theoretical predictions that Chebyshev and Legendre polynomials require $\pi $ π degrees of freedom per wavelength to resolve sinusoidal functions while prolate series need only 2 degrees of freedom per wavelength are asymptotic limits as $N \rightarrow \infty $ N → ∞ ; we investigate the rather different behavior when $N \sim O(4-10)$ N ～ O ( 4 ? 10 ) as appropriate for spectral elements and prolate elements. On the other hand, our investigations show that there are certain combinations of $N,\,h$ N , h and $c>0$ c > 0 where a prolate basis clearly outperforms the Legendre polynomial approximation.     

Optimal quadrature formulas in the sense of Sard in W_2^{(m,m-1)} space

This paper studies the problem of construction of optimal quadrature formulas in the sense of Sard in the $W_2^{(m,m-1)}(0,1)$ space. Using the Sobolev’s method we obtain new optimal quadrature formulas of such type for $N+1\ge m$ , where $N+1$ is the number of the nodes. Moreover, explicit formulas of the optimal coefficients are obtained. We investigate the order of convergence of the optimal formula for $m=1$ and prove an asymptotic optimality of such a formula in the Sobolev space $L_2^{(1)}(0,1)$ . It turns out that the error of the optimal quadrature formula in $W_2^{(1,0)}(0,1)$ is less than the error of the optimal quadrature formula given in the $L_2^{(1)}(0,1)$ space. The obtained optimal quadrature formula in the $W_2^{(m,m-1)}(0,1)$ space is exact for $\exp (-x)$ and $P_{m-2}(x)$ , where $P_{m-2}(x)$ is a polynomial of degree $m-2$ . Furthermore, some numerical results, which confirm the obtained theoretical results of this work, are given.     

Efficient broadcasting in radio networks with long-range interference

We study broadcasting, also known as one-to-all communication, in synchronous radio networks with known topology modeled by undirected (symmetric) graphs, where the interference range of a node is likely exceeding its transmission range. In this model, if two nodes are connected by a transmission edge they can communicate directly. On the other hand, if two nodes are connected by an interference edge they cannot communicate directly and transmission of one node disables recipience of any message at the other node. For a network $G,$ we term the smallest integer $d$ , s.t., for any interference edge $e$ there exists a simple path formed of at most $d$ transmission edges connecting the endpoints of $e$ as its interference distance $d_I$ . In this model the schedule of transmissions is precomputed in advance. It is based on the full knowledge of the size and the topology (including location of transmission and interference edges) of the network. We are interested in the design of fast broadcasting schedules that are energy efficient, i.e., based on a bounded number of transmissions executed at each node. We adopt $n$ as the number of nodes, $D_T$ is the diameter of the subnetwork induced by the transmission edges, and $\varDelta $ refers to the maximum combined degree (formed of transmission and interference edges) of the network. We contribute the following new results: (1) We prove that for networks with the interference distance $d_I\ge 2$ any broadcasting schedule requires at least $D_T+\varOmega (\varDelta \cdot \frac{\log {n}}{\log {\varDelta }})$ rounds. (2) We provide for networks modeled by bipartite graphs an algorithm that computes $1$ -shot (each node transmits at most once) broadcasting schedules of length $O(\varDelta \cdot \log {n})$ . (3) The main result of the paper is an algorithm that computes a $1$ -shot broadcasting schedule of length at most $4 \cdot D_T + O(\varDelta \cdot d_I \cdot \log ^4{n})$ for networks with arbitrary topology. Note that in view of the lower bound from (1) if $d_I$ is poly-logarithmic in $n$ this broadcast schedule is a poly-logarithmic factor away from the optimal solution.     

Completions of cut systems in Q-sets

Let $ Q$ be a complete residuated lattice. Let $\text {SetR}(Q)$ be the category of sets with similarity relations with values in $ Q$ (called $ Q$ -sets), which is an analogy of the category of classical sets with relations as morphisms. A cut in an $ Q$ -set $(A,\delta )$ is a system $(C_{\alpha })_{\alpha \in Q}$ , where $C_{\alpha }$ are subsets of $A\times Q$ . It is well known that in the category $\text {SetR}(Q)$ , there is a close relation between special cuts (called f-cuts) in an $ Q$ -set on one hand and fuzzy sets in the same $ Q$ -set, on the other hand. Moreover, there exists a completion procedure according to which any cut $(C_{\alpha })_{\alpha }$ can be extended onto an f-cut $(\overline{C_{\alpha }})_{\alpha }$ . In the paper, we prove that the completion procedure is, in some sense, the best possible. This will be expressed by the theorem which states that the category of f-cuts is a full reflective subcategory in the category of cuts.     

Arithmetizing Classes Around \textsf{NC} 1 and \textsf{L}

The parallel complexity class $\textsf{NC}$ ¹ has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. (J. Comput. Syst. Sci. 57:200–212, 1992) considered arithmetizations of two of these classes, $\textsf{\#NC}$ ¹ and $\textsf{\#BWBP}$ . We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in $\textsf{FLogDCFL}$ , while counting proof-trees in logarithmic width formulae has the same power as $\textsf{\#NC}$ ¹. We also consider polynomial-degree restrictions of $\textsf{SC}$ ⁱ , denoted $\textsf{sSC}$ ⁱ , and show that the Boolean class $\textsf{sSC}$ ¹ is sandwiched between $\textsf{NC}$ ¹ and $\textsf{L}$ , whereas $\textsf{sSC}$ ⁰ equals $\textsf{NC}$ ¹. On the other hand, the arithmetic class $\textsf{\#sSC}$ ⁰ contains $\textsf{\#BWBP}$ and is contained in $\textsf{FL}$ , and $\textsf{\#sSC}$ ¹ contains $\textsf{\#NC}$ ¹ and is in $\textsf{SC}$ ². We also investigate some closure properties of the newly defined arithmetic classes.     

Pseudorandom generators for CC0[p] and the Fourier spectrum of low-degree polynomials over finite fields

In this paper, we give the first construction of a pseudorandom generator, with seed length O(log n), for CC⁰[p], the class of constant-depth circuits with unbounded fan-in MOD _p gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ${\epsilon > 0}$ . In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed (Luby et al. 1993) or could only fool the distribution generated by linear functions over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route of constructing our PRG, we prove two structural results for low-degree polynomials over finite fields that can be of independent interest.

1. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . Then, for every ${\epsilon > 0}$ , there exists a subset ${S \subset [n]}$ , whose size depends only on d and ${\epsilon}$ , such that ${\sum_{\alpha \in \mathbb{F}_p^n: \alpha \ne 0, \alpha_S=0}|\hat{f}(\alpha)|^2 \leq \epsilon}$ . Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small.
2. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . If the distribution of f when applied to uniform zero–one bits is ${\epsilon}$ -far (in statistical distance) from its distribution when applied to biased bits, then for every ${\delta > 0}$ , f can be approximated over zero–one bits, up to error δ, by a function of a small number (depending only on ${\epsilon,\delta}$ and d) of lower degree polynomials.

     

Null controllability of Kolmogorov-type equations

We study the null controllability of Kolmogorov-type equations $\partial _t f + v^\gamma \partial _x f - \partial _v^2 f = u(t,x,v) 1_{\omega }(x,v)$ in a rectangle $\Omega $ , under an additive control supported in an open subset $\omega $ of $\Omega $ . For $\gamma =1$ , with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support $\omega $ . This improves the previous result by Beauchard and Zuazua (Ann Ins H Poincaré Anal Non Linéaire 26:1793–1815, 2009), in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if $\gamma =1$ or if $\gamma =2$ and $\omega $ contains the segment $\{v=0\}$ , and only in large time if $\gamma =2$ and $\omega $ does not contain the segment $\{v=0\}$ . Our approach, inspired from Benabdallah et al. (C R Math Acad Sci Paris 344(6):357–362, 2007), Lebeau and Robbiano (Commun Partial Differ Equ 20:335–356, 1995), is based on two key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components.     

A note on the factorization conjecture

We give partial results on the factorization conjecture on codes proposed by Schützenberger. We consider a family of finite maximal codes $C$ over the alphabet $A = \{a, b\}$ and we prove that the factorization conjecture holds for these codes. This family contains $(p,4)$ -codes, where a $(p,4)$ -code $C$ is a finite maximal code over $A$ such that each word in $C$ has at most four occurrences of $b$ and $a^p \in C$ , for a prime number $p$ . We also discuss the structure of these codes. The obtained results once again show relations between factorizations of finite maximal codes and factorizations of finite cyclic groups.     

On matrices, automata, and double counting in constraint programming

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables $\mathcal{M}$ , with the same constraint C defined by a finite-state automaton $\mathcal{A}$ on each row of $\mathcal{M}$ and a global cardinality constraint $\mathit{gcc}$ on each column of $\mathcal{M}$ . We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the $\mathit{gcc}$ constraints from the automaton $\mathcal{A}$ . The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We also provide a domain consistency filtering algorithm for the conjunction of lexicographic ordering constraints between adjacent rows of $\mathcal{M}$ and (possibly different) automaton constraints on the rows. We evaluate the impact of our methods in terms of runtime and search effort on a large set of nurse rostering problem instances.     

Self-Referential Justifications in Epistemic Logic

This paper is devoted to the study of self-referential proofs and/or justifications, i.e., valid proofs that prove statements about these same proofs. The goal is to investigate whether such self-referential justifications are present in the reasoning described by standard modal epistemic logics such as  $\mathsf{S4}$ . We argue that the modal language by itself is too coarse to capture this concept of self-referentiality and that the language of justification logic can serve as an adequate refinement. We consider well-known modal logics of knowledge/belief and show, using explicit justifications, that $\mathsf{S4}$ , $\mathsf{D4}$ , $\mathsf{K4}$ , and  $\mathsf{T}$ with their respective justification counterparts  $\mathsf{LP}$ , $\mathsf{JD4}$ , $\mathsf{J4}$ , and  $\mathsf{JT}$ describe knowledge that is self-referential in some strong sense. We also demonstrate that self-referentiality can be avoided for  $\mathsf{K}$ and  $\mathsf{D}$ . In order to prove the former result, we develop a machinery of minimal evidence functions used to effectively build models for justification logics. We observe that the calculus used to construct the minimal functions axiomatizes the reflected fragments of justification logics. We also discuss difficulties that result from an introduction of negative introspection.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler–Bernoulli beam equation in one space dimension. We prove the $L^2$ stability of the scheme and several optimal $L^2$ error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are $\mathcal O (h^{k+3/2})$ super close to particular projections of the exact solutions for $k$ th-degree polynomial spaces while computational results show higher $\mathcal O (h^{k+2})$ convergence rate. Our proofs are valid for arbitrary regular meshes and for $P^k$ polynomials with $k\ge 1$ , and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the $L^2$ -norm under mesh refinement.     

Distinguishing and relating higher-order and first-order processes by expressiveness

This is a paper on distinguishing and relating two important kinds of calculi through expressiveness, settling some critical but long unanswered questions. The delimitation of higher-order and first-order process calculi is a basic and pivotal topic in the study of process theory. Particularly, expressiveness studies mutual encodability, which helps decide whether process-passing or name-passing is more fundamental, and the way they ought to be used in both theory and practice. In this paper, we contribute to such demarcation with three major results. Firstly $\pi $ (first-order pi-calculus) can faithfully express $\varPi $ (basic higher-order pi-calculus). The calculus $\varPi $ has the elementary operators (input, output, composition and restriction). This actually is a corollary of a more general result, that $\pi $ can encode $\varPi ^r$ ( $\varPi $ enriched with the relabelling operator). Secondly $\varPi $ cannot interpret $\pi $ reasonably. This is of more significance since it separates $\varPi $ and $\pi $ by drawing a well-defined boundary. Thirdly an encoding from $\pi $ to $\varPi ^r$ is revisited and discussed, which not only implies how to make $\varPi $ more useful but also stresses the importance of name-passing in $\pi $ .     

Computational complexity and approximability of social welfare optimization in multiagent resource allocation

A central task in multiagent resource allocation, which provides mechanisms to allocate (bundles of) resources to agents, is to maximize social welfare. We assume resources to be indivisible and nonshareable and agents to express their utilities over bundles of resources, where utilities can be represented in the bundle form, the $k$ -additive form, and as straight-line programs. We study the computational complexity of social welfare optimization in multiagent resource allocation, where we consider utilitarian and egalitarian social welfare and social welfare by the Nash product. Solving some of the open problems raised by Chevaleyre et al. (2006) and confirming their conjectures, we prove that egalitarian social welfare optimization is $\mathrm{NP}$ -complete for the bundle form, and both exact utilitarian and exact egalitarian social welfare optimization are $\mathrm{DP}$ -complete, each for both the bundle and the $2$ -additive form, where $\mathrm{DP}$ is the second level of the boolean hierarchy over  $\mathrm{NP}$ . In addition, we prove that social welfare optimization by the Nash product is $\mathrm{NP}$ -complete for both the bundle and the $1$ -additive form, and that the exact variants are $\mathrm{DP}$ -complete for the bundle and the $3$ -additive form. For utility functions represented as straight-line programs, we show $\mathrm{NP}$ -completeness for egalitarian social welfare optimization and social welfare optimization by the Nash product. Finally, we show that social welfare optimization by the Nash product in the $1$ -additive form is hard to approximate, yet we also give fully polynomial-time approximation schemes for egalitarian and Nash product social welfare optimization in the $1$ -additive form with a fixed number of agents.     

Two-Grid Methods for Hermitian positive definite linear systems connected with an order relation

Given a multigrid procedure for linear systems with coefficient matrices $A_n,$ we discuss the optimality of a related multigrid procedure with the same smoother and the same projector, when applied to properly related algebraic problems with coefficient matrices $B_n$ : we assume that both $A_n$ and $B_n$ are Hermitian positive definite with $A_n\le \vartheta B_n,$ for some positive $\vartheta $ independent of $n.$ In this context we prove the Two-Grid Method optimality. We apply this elementary strategy for designing a multigrid solution for modifications of multilevel structured linear systems, in which the Hermitian positive definite coefficient matrix is banded in a multilevel sense. As structured matrices, Toeplitz, circulants, Hartley, sine ( $\tau $ class) and cosine algebras are considered. In such a way, several linear systems arising from the approximation of integro–differential equations with various boundary conditions can be efficiently solved in linear time (with respect to the size of the algebraic problem). Some numerical experiments are presented and discussed, both with respect to Two-Grid and multigrid procedures.     

A new PAC bound for intersection-closed concept classes

For hyper-rectangles in $\mathbb{R}^{d}$ Auer (1997) proved a PAC bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ , where $\varepsilon$ and $\delta$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $d$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$ and $O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ . For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$ examples to learn some particular maximum intersection-closed concept classes.