Quantum secret sharing with continuous variable graph state

In this paper, we study several physically feasible quantum secret sharing (QSS) schemes using continuous variable graph state (CVGS). Their implementation protocols are given, and the estimation error formulae are derived. Then, we present a variety of results on the theory of QSS with CVGS. Any $(k,n)$ threshold protocol of the three specific schemes satisfying $\frac{n}{2}<k\le n$ , where $n$ denotes the total number of players and $k$ denotes the minimum number of players who can collaboratively access the secret, can be implemented by certain weighted CVGS. The quantum secret is absolutely confidential to any player group with number less than threshold. Besides, the effect of finite squeezing to these results is properly considered. In the end, the duality between two specific schemes is investigated.     

Some representations for the Drazin inverse of a modified matrix

In this paper, we give some results for the Drazin inverse of a modified matrix \(M=A-CD^dB\) with the generalized Schur complement \(Z=D-BA^dC\) under some conditions. Further, we present some new results for the Drazin inverse of the modified matrix \(M=A-CD^dB\) , when the generalized Schur complement \(Z=0\) under some conditions. As a result, some conclusions are obtained directly from our results.     

Growth rates for persistently excited linear systems

We consider a family of linear control systems \(\dot{x}=Ax+\alpha Bu\) on \(\mathbb {R}^d\) , where \(\alpha \) belongs to a given class of persistently exciting signals. We seek maximal \(\alpha \) -uniform stabilization and destabilization by means of linear feedbacks \(u=Kx\) . We extend previous results obtained for bidimensional single-input linear control systems to the general case as follows: if there exists at least one \(K\) such that the Lie algebra generated by \(A\) and \(BK\) is equal to the set of all \(d\times d\) matrices, then the maximal rate of convergence of \((A,B)\) is equal to the maximal rate of divergence of \((-A,-B)\) . We also provide more precise results in the general single-input case, where the above result is obtained under the simpler assumption of controllability of the pair \((A,B)\) .     

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.     

Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE’s by a single control

We study the exact controllability, by a reduced number of controls, of coupled cascade systems of PDE’s and the existence of exact insensitizing controls for the scalar wave equation. We give a necessary and sufficient condition for the observability of abstract-coupled cascade hyperbolic systems by a single observation, the observation operator being either bounded or unbounded. Our proof extends the two-level energy method introduced in Alabau-Boussouira (Siam J Control Opt 42:871–906, 2003) and Alabau-Boussouira and Léautaud (J Math Pures Appl 99:544–576, 2013) for symmetric coupled systems, to cascade systems which are examples of non-symmetric coupled systems. In particular, we prove the observability of two coupled wave equations in cascade if the observation and coupling regions both satisfy the Geometric Control Condition (GCC) of Bardos et al. (SIAM J Control Opt 30:1024–1065, 1992). By duality, this solves the exact controllability, by a single control, of $2$ -coupled abstract cascade hyperbolic systems. Using transmutation, we give null-controllability results for the multidimensional heat and Schrödinger $2$ -coupled cascade systems under GCC and for any positive time. By our method, we can treat cases where the control and coupling coefficients have disjoint supports, partially solving an open question raised by de Teresa (CPDE 25:39–72, 2000). Moreover we answer the question of the existence of exact insensitizing locally distributed as well as boundary controls of scalar multidimensional wave equations, raised by Lions (Actas del Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Universidad de Málaga, pp 43–54, 1989) and later on by Dáger (Siam J Control Opt 45:1758–1768, 2006) and Tebou (C R Acad Sci Paris 346(Sér I):407–412, 2008).     

A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation

A $C^0$ -weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for $C^0$ functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete $H^2$ norm and the standard $H^1$ and $L^2$ norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order $(k+1-d)$ and the surface mass of order $(k+2-d)$ for the $P_{k+2}$ finite element functions in $d$ -dimensional space.     

An algebraic axiomatization of the Ewald’s intuitionistic tense logic

Ewald (J Symbolic Logic 51(1):166–179, 1986) considered tense operators \(G\) , \(H\) , \(F\) and \(P\) on intuitionistic propositional calculus and constructed an intuitionistic tense logic system called IKt. The aim of this paper is to give an algebraic axiomatization of the IKt system. We will also show that the algebraic axiomatization given by Chajda (Cent Eur J Math 9(5):1185–1191, 2011) of the tense operators \(P\) and \(F\) in intuitionistic logic is not in accordance with the Halmos definition of existential quantifiers. In this paper, we will study the IKt variety of IKt-algebras. First, we will introduce some examples and we will prove some properties. Next, we will prove that the IKt system has IKt-algebras as algebraic counterpart. We will also describe a discrete duality for IKt-algebras bearing in mind the results indicated by Or?owska and Rewitzky (Fundam Inform 81(1–3):275–295, 2007) for Heyting algebras. We will also get a general construction of tense operators on a complete Heyting algebra, which is a power lattice via the so-called Heyting frame. Finally, we will introduce the notion of tense deductive system which allowed us both to determine the congruence lattice in an IKt-algebra and to characterize simple and subdirectly irreducible algebras of the IKt variety.     

Optimal Error Estimates of Two Mixed Finite Element Methods for Parabolic Integro-Differential Equations with Nonsmooth Initial Data

In the first part of this article, a new mixed method is proposed and analyzed for parabolic integro-differential equations (PIDE) with nonsmooth initial data. Compared to the standard mixed method for PIDE, the present method does not bank on a reformulation using a resolvent operator. Based on energy arguments combined with a repeated use of an integral operator and without using parabolic type duality technique, optimal $L^2$ L 2 -error estimates are derived for semidiscrete approximations, when the initial condition is in $L^2$ L 2 . Due to the presence of the integral term, it is, further, observed that a negative norm estimate plays a crucial role in our error analysis. Moreover, the proposed analysis follows the spirit of the proof techniques used in deriving optimal error estimates for finite element approximations to PIDE with smooth data and therefore, it unifies both the theories, i.e., one for smooth data and other for nonsmooth data. Finally, we extend the proposed analysis to the standard mixed method for PIDE with rough initial data and provide an optimal error estimate in $L^2,$ L 2 , which improves upon the results available in the literature.     

Gaussian,Lobatto and Radau positive quadrature rules with a prescribed abscissa

For a given $\theta \in (a,b)$ , we investigate the question whether there exists a positive quadrature formula with maximal degree of precision which has the prescribed abscissa $\theta $ plus possibly $a$ and/or $b$ , the endpoints of the interval of integration. This study relies on recent results on the location of roots of quasi-orthogonal polynomials. The above positive quadrature formulae are useful in studying problems in one-sided polynomial $L_1$ approximation.     

Combinatorial algorithms for distributed graph coloring

Numerous problems in Theoretical Computer Science can be solved very efficiently using powerful algebraic constructions. Computing shortest paths, constructing expanders, and proving the PCP Theorem, are just few examples of this phenomenon. The quest for combinatorial algorithms that do not use heavy algebraic machinery, but are roughly as efficient, has become a central field of study in this area. Combinatorial algorithms are often simpler than their algebraic counterparts. Moreover, in many cases, combinatorial algorithms and proofs provide additional understanding of studied problems. In this paper we initiate the study of combinatorial algorithms for Distributed Graph Coloring problems. In a distributed setting a communication network is modeled by a graph $G=(V,E)$ of maximum degree $\varDelta $ . The vertices of $G$ host the processors, and communication is performed over the edges of $G$ . The goal of distributed vertex coloring is to color $V$ with $(\varDelta + 1)$ colors such that any two neighbors are colored with distinct colors. Currently, efficient algorithms for vertex coloring that require $O(\varDelta + \log ^* n)$ time are based on the algebraic algorithm of Linial (SIAM J Comput 21(1):193–201, 1992) that employs set-systems. The best currently-known combinatorial set-system free algorithm, due to Goldberg et al. (SIAM J Discret Math 1(4):434–446, 1988), requires $O(\varDelta ^2+\log ^*n)$ time. We significantly improve over this by devising a combinatorial $(\varDelta + 1)$ -coloring algorithm that runs in $O(\varDelta + \log ^* n)$ time. This exactly matches the running time of the best-known algebraic algorithm. In addition, we devise a tradeoff for computing $O(\varDelta \cdot t)$ -coloring in $O(\varDelta /t + \log ^* n)$ time, for almost the entire range $1 < t < \varDelta $ . We also compute a Maximal Independent Set in $O(\varDelta + \log ^* n)$ time on general graphs, and in $O(\log n/ \log \log n)$ time on graphs of bounded arboricity. Prior to our work, these results could be only achieved using algebraic techniques. We believe that our algorithms are more suitable for real-life networks with limited resources, such as sensor networks.     

Budget-bounded model-checking pushdown systems

We address the verification problem for concurrent programs modeled as multi-pushdown systems (MPDS). In general, MPDS are Turing powerful and hence come along with undecidability of all basic decision problems. Because of this, several subclasses of MPDS have been proposed and studied in the literature (Atig et al. in LNCS, Springer, Berlin, 2005; La Torre et al. in LICS, IEEE, 2007; Lange and Lei in Inf Didact 8, 2009; Qadeer and Rehof in TACAS, LNCS, Springer, Berlin, 2005). In this paper, we propose the class of bounded-budget MPDS, which are restricted in the sense that each stack can perform an unbounded number of context switches only if its depth is below a given bound, and a bounded number of context switches otherwise. We show that the reachability problem for this subclass is Pspace-complete and that LTL-model-checking is Exptime-complete. Furthermore, we propose a code-to-code translation that inputs a concurrent program \(P\) and produces a sequential program \(P'\) such that running \(P\) under the budget-bounded restriction yields the same set of reachable states as running \(P'\) . Moreover, detecting (fair) non-terminating executions in \(P\) can be reduced to LTL-Model-Checking of \(P'\) . By leveraging standard sequential analysis tools, we have implemented a prototype tool and applied it on a set of benchmarks, showing the feasibility of our translation.     

Isomorphisms and functors of fuzzy sets and cut systems

Any fuzzy set \(X\) in a classical set \(A\) with values in a complete (residuated) lattice \( Q\) can be identified with a system of \(\alpha \) -cuts \(X_{\alpha }\) , \(\alpha \in Q\) . Analogical results were proved for sets with similarity relations with values in \( Q\) (e.g. \( Q\) -sets), which are objects of two special categories \({\mathbf K}={Set}( Q)\) or \({SetR}( Q)\) of \( Q\) -sets, and for fuzzy sets defined as morphisms from an \( Q\) -set into a special \(Q\) -set \(( Q,\leftrightarrow )\) . These fuzzy sets can be defined equivalently as special cut systems \((C_{\alpha })_{\alpha }\) , called f-cuts. This equivalence then represents a natural isomorphism between covariant functor of fuzzy sets \(\mathcal{F}_{\mathbf K}\) and covariant functor of f-cuts \(\mathcal{C}_{\mathbf K}\) . In this paper, we prove that analogical natural isomorphism exists also between contravariant versions of these functors. We are also interested in relationships between sets of fuzzy sets and sets of f-cuts in an \(Q\) -set \((A,\delta )\) in the corresponding categories \({Set}( Q)\) and \({SetR}( Q)\) , which are endowed with binary operations extended either from binary operations in the lattice \(Q\) , or from binary operations defined in a set \(A\) by the generalized Zadeh’s extension principle. We prove that the resulting binary structures are (under some conditions) isomorphic.     

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.     

Universal regular control for generic semilinear systems

We consider discrete-time projective semilinear control systems \(\xi _{t+1} = A(u_t) \cdot \xi _t\) , where the states \(\xi _t\) are in projective space \(\mathbb {R}\hbox {P}^{d-1}\) , inputs \(u_t\) are in a manifold \(\mathcal {U}\) of arbitrary finite dimension, and \(A :\mathcal {U}\rightarrow \hbox {GL}(d,\mathbb {R})\) is a differentiable mapping. An input sequence \((u_0,\ldots ,u_{N-1})\) is called universally regular if for any initial state \(\xi _0 \in \mathbb {R}\hbox {P}^{d-1}\) , the derivative of the time- \(N\) state with respect to the inputs is onto. In this paper, we deal with the universal regularity of constant input sequences \((u_0, \ldots , u_0)\) . Our main result states that generically in the space of such systems, for sufficiently large \(N\) , all constant inputs of length \(N\) are universally regular, with the exception of a discrete set. More precisely, the conclusion holds for a \(C^2\) -open and \(C^\infty \) -dense set of maps \(A\) , and \(N\) only depends on \(d\) and on the dimension of \(\mathcal {U}\) . We also show that the inputs on that discrete set are nearly universally regular; indeed, there is a unique non-regular initial state, and its corank is 1. In order to establish the result, we study the spaces of bilinear control systems. We show that the codimension of the set of systems for which the zero input is not universally regular coincides with the dimension of the control space. The proof is based on careful matrix analysis and some elementary algebraic geometry. Then the main result follows by applying standard transversality theorems.     

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.     

Single-processor scheduling with time restrictions

We consider the following list scheduling problem. We are given a set \(S\) of jobs which are to be scheduled sequentially on a single processor. Each job has an associated processing time which is required for its processing. Given a particular permutation of the jobs in \(S\) , the jobs are processed in that order with each job started as soon as possible, subject only to the following constraint: For a fixed integer \(B \ge 2\) , no unit time interval \([x, x+1)\) is allowed to intersect more than \(B\) jobs for any real \(x\) . It is not surprising that this problem is NP-hard when the value \(B\) is variable (which is typical of many scheduling problems). There are several real world situations for which this restriction is natural. For example, suppose in addition to our jobs being executed sequentially on a single main processor, each job also requires the use of one of \(B\) identical subprocessors during its execution. Each time a job is completed, the subprocessor it was using requires one unit of time in order to reset itself. In this way, it is never possible for more than \(B\) jobs to be worked on during any unit interval. In this paper we carry out a classical worst-case analysis for this situation. In particular, we show that any permutation of the jobs can be processed within a factor of \(2-1/(B-1)\) of the optimum (plus an additional small constant) when \(B \ge 3\) and this factor is best possible. For the case \(B=2\) , the situation is rather different, and in this case the corresponding factor we establish is \(4/3\) (plus an additional small constant), which is also best possible. It is fairly rare that best possible bounds can be obtained for the competitive ratios of list scheduling problems of this general type.     

Finding large k-clubs in undirected graphs

Finding cohesive subgroups is an important issue in studying social networks. Many models exist for defining cohesive subgraphs in social networks, such as clique, $k$ -clique, and $k$ -clan. The concept of $k$ -club is one of them. A $k$ -club of a graph is a maximal subset of the vertex set which induces a subgraph of diameter $k$ . It is a relaxation of a clique, which induces a subgraph of diameter $1$ . We conducted algorithmic studies on finding a $k$ -club of size as large as possible. In this paper, we show that one can find a $k$ -club of maximum size in $O^{*}(1.62^n)$ time where $n$ is the number of vertices of the input graph. We implemented a combinatorial branch-and-bound algorithm that finds a $k$ -club of maximum size and a new heuristic algorithm called IDROP given in this paper. To speed up the programs, we introduce a dynamic data structure called $k$ -DN which, under deletion of vertices from a graph, maintains for a given vertex $v$ the set of vertices at distances at most $k$ . From the experimental results that we obtained, we concluded that a $k$ -club of maximum size can be easily found in sparse graphs and dense graphs. Our heuristic algorithm finds, within reasonable time, $k$ -clubs of maximum size in most of experimental instances. The gap between the size of a $k$ -club of maximum size and a $k$ -club found by IDROP is a constant for the number of vertices that we are able to test.     

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.     

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.