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.     

A new method for solving an arbitrary fully fuzzy linear system

In this paper, We propose a simple and practical method (that works only for triangular fuzzy numbers) to solve an arbitrary fully fuzzy linear system (FFLS) in the form $\widetilde{A}\otimes \widetilde{x}=\widetilde{b},$ where $\widetilde{A}_{n \times n}$ is a fuzzy matrix, $\widetilde{x}$ and $\widetilde{b}$ are n × 1 fuzzy vectors. The idea of the presented method is constructed based on the extending 0-cut and 1-cut solution of original fully fuzzy linear systems (FFLS). We also define a fuzzy solution of FFLS and establish the necessary and sufficient conditions for the uniqueness of a fuzzy solution.     

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.     

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.     

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).     

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.     

Integrated damping parameter and control design in structural systems for \bf{\mathcal{H}^2} and {\mathcal{H}^{\infty}} specifications

The paper presents a linear matrix inequality (LMI)-based approach for the simultaneous optimal design of output feedback control gains and damping parameters in structural systems with collocated actuators and sensors. The proposed integrated design is based on simplified $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ norm upper bound calculations for collocated structural systems. Using these upper bound results, the combined design of the damping parameters of the structural system and the output feedback controller to satisfy closed-loop $\mathcal{H}^2$ or $\mathcal{H}^{\infty}$ performance specifications is formulated as an LMI optimization problem with respect to the unknown damping coefficients and feedback gains. Numerical examples motivated from structural and aerospace engineering applications demonstrate the advantages and computational efficiency of the proposed technique for integrated structural and control design. The effectiveness of the proposed integrated design becomes apparent, especially in very large scale structural systems where the use of classical methods for solving Lyapunov and Riccati equations associated with $\mathcal{H}^2$ and $\mathcal{H}^{\infty}$ designs are time-consuming or intractable.     

Richardson Extrapolation on Some Recent Numerical Quadrature Formulas for Singular and Hypersingular Integrals and Its Study of Stability

Recently, we derived some new numerical quadrature formulas of trapezoidal rule type for the integrals \(I^{(1)}[g]=\int ^b_a \frac{g(x)}{x-t}\,dx\) and \(I^{(2)}[g]=\int ^b_a \frac{g(x)}{(x-t)^2}\,dx\) . These integrals are not defined in the regular sense; \(I^{(1)}[g]\) is defined in the sense of Cauchy Principal Value while \(I^{(2)}[g]\) is defined in the sense of Hadamard Finite Part. With \(h=(b-a)/n, \,n=1,2,\ldots \) , and \(t=a+kh\) for some \(k\in \{1,\ldots ,n-1\}, \,t\) being fixed, the numerical quadrature formulas \({Q}^{(1)}_n[g]\) for \(I^{(1)}[g]\) and \(Q^{(2)}_n[g]\) for \(I^{(2)}[g]\) are $$\begin{aligned} {Q}^{(1)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2),\quad f(x)=\frac{g(x)}{x-t}, \end{aligned}$$ and $$\begin{aligned} Q^{(2)}_n[g]=h\sum ^n_{j=1}f(a+jh-h/2)-\pi ^2g(t)h^{-1},\quad f(x)=\frac{g(x)}{(x-t)^2}. \end{aligned}$$ We provided a complete analysis of the errors in these formulas under the assumption that \(g\in C^\infty [a,b]\) . We actually show that $$\begin{aligned} I^{(k)}[g]-{Q}^{(k)}_n[g]\sim \sum ^\infty _{i=1} c^{(k)}_ih^{2i}\quad \text {as}\,n \rightarrow \infty , \end{aligned}$$ the constants \(c^{(k)}_i\) being independent of \(h\) . In this work, we apply the Richardson extrapolation to \({Q}^{(k)}_n[g]\) to obtain approximations of very high accuracy to \(I^{(k)}[g]\) . We also give a thorough analysis of convergence and numerical stability (in finite-precision arithmetic) for them. In our study of stability, we show that errors committed when computing the function \(g(x)\) , which form the main source of errors in the rest of the computation, propagate in a relatively mild fashion into the extrapolation table, and we quantify their rate of propagation. We confirm our conclusions via numerical examples.     

A P_N P_M{-} CPR Framework for Hyperbolic Conservation Laws

The correction procedure via reconstruction (CPR) method is a discontinuous nodal formulation unifying several well-known methods in a simple finite difference like manner. The \(P_NP_M{-} CPR \) formulation is an extension of \(P_NP_M\) or the reconstructed discontinuous Galerkin (RDG) method to the CPR framework. It is a hybrid finite volume and discontinuous Galerkin (DG) method, in which neighboring cells are used to build a higher order polynomial than the solution representation in the cell under consideration. In this paper, we present several \(P_NP_M\) schemes under the CPR framework. Many interesting schemes with various orders of accuracy and efficiency are developed. The dispersion and dissipation properties of those methods are investigated through a Fourier analysis, which shows that the \(P_NP_M{-} CPR \) method is dependent on the position of the solution points. Optimal solution points for 1D \(P_NP_M{-} CPR \) schemes which can produce expected order of accuracy are identified. In addition, the \(P_NP_M{-} CPR \) method is extended to solve 2D inviscid flow governed by the Euler equations and several numerical tests are performed to assess its performance.     

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.     

A note on reverse scheduling with maximum lateness objective

The inverse and reverse counterparts of the single-machine scheduling problem $1||L_{\max }$ are studied in [2], in which the complexity classification is provided for various combinations of adjustable parameters (due dates and processing times) and for five different types of norm: $\ell _{1},\ell _{2},\ell _{\infty },\ell _{H}^{\Sigma } $ , and $\ell _{H}^{\max }$ . It appears that the $O(n^{2})$ -time algorithm for the reverse problem with adjustable due dates contains a flaw. In this note, we present the structural properties of the reverse model, establishing a link with the forward scheduling problem with due dates and deadlines. For the four norms $\ell _{1},\ell _{\infty },\ell _{H}^{\Sigma }$ , and $ \ell _{H}^{\max }$ , the complexity results are derived based on the properties of the corresponding forward problems, while the case of the norm $\ell _{2}$ is treated separately. As a by-product, we resolve an open question on the complexity of problem $1||\sum \alpha _{j}T_{j}^{2}$ .     

On the power of unambiguity in log-space

We report progress on the NL versus UL problem.

We show that counting the number of s-t paths in graphs where the number of s-v paths for any v is bounded by a polynomial can be done in FUL: the unambiguous log-space function class. Several new upper bounds follow from this including ${{{ReachFewL} \subseteq {UL}}}$ and ${{{LFew} \subseteq {UL}^{FewL}}}$

We investigate the complexity of min-uniqueness—a central notion in studying the NL versus UL problem. In this regard we revisit the class OptL[log n] and introduce UOptL[log n], an unambiguous version of OptL[log n]. We investigate the relation between UOptL[log n] and other existing complexity classes.

We consider the unambiguous hierarchies over UL and UOptL[log n]. We show that the hierarchy over UOptL[log n] collapses. This implies that ${{{ULH} \subseteq {L}^{{promiseUL}}}}$ thus collapsing the UL hierarchy.

We show that the reachability problem over graphs embedded on 3 pages is complete for NL. This contrasts with the reachability problem over graphs embedded on 2 pages, which is log-space equivalent to the reachability problem in planar graphs and hence is in UL.

     

Superconvergence of Any Order Finite Volume Schemes for 1D General Elliptic Equations

We present and analyze a finite volume scheme of arbitrary order for elliptic equations in the one-dimensional setting. In this scheme, the control volumes are constructed by using the Gauss points in subintervals of the underlying mesh. We provide a unified proof for the inf-sup condition, and show that our finite volume scheme has optimal convergence rate under the energy and $L^2$ norms of the approximate error. Furthermore, we prove that the derivative error is superconvergent at all Gauss points and in some special cases, the convergence rate can reach $h^{r+2}$ and even $h^{2r}$ , comparing with $h^{r+1}$ rate of the counterpart finite element method. Here $r$ is the polynomial degree of the trial space. All theoretical results are justified by numerical tests.     

On the computation of Tricomi's ψ function

Two methods for calculating Tricomi's confluent hypergeometric function are discussed. Both methods are based on recurrence relations. The first method converges like $$\exp ( - \alpha |\lambda |^{1/3} n^{2/3} )for some \alpha > 0$$ and the second like $$\exp ( - \beta |\lambda |^{1/2} n^{1/2} )for some \beta > 0.$$ Several examples are presented.     

Rounds in Combinatorial Search

A set system $\mathcal{H} \subseteq2^{[m]}$ is said to be separating if for every pair of distinct elements x,y∈[m] there exists a set $H\in\mathcal{H}$ such that H contains exactly one of them. The search complexity of a separating system $\mathcal{H} \subseteq 2^{[m]}$ is the minimum number of questions of type “x∈H?” (where $H \in\mathcal{H}$ ) needed in the worst case to determine a hidden element x∈[m]. If we receive the answer before asking a new question then we speak of the adaptive complexity, denoted by $\mathrm{c} (\mathcal{H})$ ; if the questions are all fixed beforehand then we speak of the non-adaptive complexity, denoted by $\mathrm{c}_{na} (\mathcal{H})$ . If we are allowed to ask the questions in at most k rounds then we speak of the k-round complexity of $\mathcal{H}$ , denoted by $\mathrm{c}_{k} (\mathcal{H})$ . It is clear that $|\mathcal{H}| \geq\mathrm{c}_{na} (\mathcal{H}) = \mathrm{c}_{1} (\mathcal{H}) \geq\mathrm{c}_{2} (\mathcal{H}) \geq\cdots\geq\mathrm{c}_{m} (\mathcal{H}) = \mathrm{c} (\mathcal{H})$ . A group of problems raised by G.O.H. Katona is to characterize those separating systems for which some of these inequalities are tight. In this paper we are discussing set systems $\mathcal{H}$ with the property $|\mathcal{H}| = \mathrm{c}_{k} (\mathcal{H}) $ for any k≥3. We give a necessary condition for this property by proving a theorem about traces of hypergraphs which also has its own interest.     

The PSurface library

We describe PSurface, a C $++$ library that allows to store and access piecewise linear maps between simplicial surfaces in $\mathbb{R }^2$ and $\mathbb{R }^3$ . Piecewise linear maps can be used, e.g., to construct boundary approximations for finite element grids, and grid intersections for domain decomposition methods. In computer graphics the maps allow to build level-of-detail representations as well as texture- and bump maps. The PSurface library can be used as the basis for the implementation of a wide range of algorithms that use piecewise linear maps between triangulated surfaces. A few simple examples are given in this work. We document the data structures and algorithms and show how PSurface is used in the numerical analysis framework Dune and the visualization software Amira.     

The Relationships Between Chebyshev,Legendre and Jacobi Polynomials: The Generic Superiority of Chebyshev Polynomials and Three Important Exceptions

We analyze the asymptotic rates of convergence of Chebyshev, Legendre and Jacobi polynomials. One complication is that there are many reasonable measures of optimality as enumerated here. Another is that there are at least three exceptions to the general principle that Chebyshev polynomials give the fastest rate of convergence from the larger family of Jacobi polynomials. When $f(x)$ is singular at one or both endpoints, all Gegenbauer polynomials (including Legendre and Chebyshev) converge equally fast at the endpoints, but Gegenbauer polynomials converge more rapidly on the interior with increasing order $m$ . For functions on the surface of the sphere, associated Legendre functions, which are proportional to Gegenbauer polynomials, are best for the latitudinal dependence. Similarly, for functions on the unit disk, Zernike polynomials, which are Jacobi polynomials in radius, are superior in rate-of-convergence to a Chebyshev–Fourier series. It is true, as was conjectured by Lanczos 60 years ago, that excluding these exceptions, the Chebyshev coefficients $a_{n}$ usually decrease faster than the Legendre coefficients $b_{n}$ by a factor of $\sqrt{n}$ . We calculate the proportionality constant for a few examples and restrictive classes of functions. The more precise claim that $b_{n} \sim \sqrt{\pi /2} \sqrt{n} a_{n}$ , made by Lanczos and later Fox and Parker, is true only for rather special functions. However, individual terms in the large $n$ asymptotics of Chebyshev and Legendre coefficients usually do display this proportionality.