Programmable Restoration Granularity in Constraint Programming

In most constraint programming systems, a limited number of search engines is offered while the programming of user-customized search algorithms requires low-level efforts, which complicates the deployment of such algorithms. To alleviate this limitation, concepts such as computation spaces have been developed. Computation spaces provide a coarse-grained restoration mechanism, because they store all information contained in a search tree node. Other granularities are possible, and in this paper we make the case for dynamically adapting the restoration granularity during search. In order to elucidate programmable restoration granularity, we present restoration as an aspect of a constraint programming system, using the model of aspect-oriented programming. A proof-of-concept implementation using Gecode shows promising results

Stochastic completeness for graphs with curvature dimension conditions

We prove pointwise gradient bounds for heat semigroups associated to general (possibly unbounded) Laplacians on infinite graphs satisfying the curvature dimension condition CD(K,\infty). Using gradient bounds, we show stochastic completeness for graphs satisfying the curvature dimension condition.Comment: 20 pages. A new lemma, Lemma 3.6, is introduce

Equivalent Properties of CD Inequality on Graph

We study some equivalent properties of the curvature-dimension conditions $CD(n,K)$ inequality on infinite, but locally finite graph. These equivalences are gradient estimate, Poincar\'e type inequalities and reverse Poincar\'e inequalities. And we also obtain one equivalent property of gradient estimate for a new notion of curvature-dimension conditions $CDE'(\infty, K)$ at the same assumption of graphs.Comment: 13 page

Recollection: an Alternative Restoration Technique for Constraint Programming Systems

Search is a key service within constraint programming systems, and it demands the restoration of previously accessed states during the exploration of a search tree. Restoration proceeds either bottom-up within the tree to roll back previously performed operations using a trail, or top-down to redo them, starting from a previously stored state and using suitable information stored along the way. In this paper, we elucidate existing restoration techniques using a pair of abstract methods and employ them to present a new technique that we call recollection. The proposed technique stores the variables that were affected by constraint propagation during fix points reasoning steps, and it conducts neither operation roll-back nor recomputation, while consuming much less memory than storing previous visited states. We implemented this idea as a prototype within the Gecode solver. An empirical evaluation reveals that constraint problems with expensive propagation and frequent failures can benefit from recollection with respect to runtime at the expense of a marginal increase in memory consumption, comparing with the most competitive variant of recomputation

Properties for CD Inequalities with Unbounded Laplacians

The CD equalities were introduced to imply the gradient estimate of laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,$\infty$)and CD(K,n).Comment: 12 page

Controlled Singular Volterra Integral Equations and Pontryagin Maximum Principle

This paper is concerned with a class of controlled singular Volterra integral equations, which could be used to describe problems involving memories. The well-known fractional order ordinary differential equations of the Riemann--Liouville or Caputo types are strictly special cases of the equations studied in this paper. Well-posedness and some regularity results in proper spaces are established for such kind of questions. For the associated optimal control problem, by using a Liapounoff's type theorem and the spike variation technique, we establish a Pontryagin's type maximum principle for optimal controls. Different from the existing literature, our method enables us to deal with the problem without assuming regularity conditions on the controls, the convexity condition on the control domain, and some additional unnecessary conditions on the nonlinear terms of the integral equation and the cost functional.Comment: 30 page

Blow-up problems for nonlinear parabolic equations on locally finite graphs

Let $G=(V,E)$ be a locally finite connected weighted graph, $\Delta$ be the usual graph Laplacian. In this paper, we study the blow-up problems for the nonlinear parabolic equation $u_t=\Delta u + f(u)$ on $G$. The blow-up phenomenons of the equation are discussed in terms of two cases: (i) an initial condition is given; (ii) a Dirichlet boundary condition is given. We prove that if $f$ satisfies appropriate conditions, then the solution of the equation blows up in a finite time.Comment: 14 page

Trust Region Subproblem with a Fixed Number of Additional Linear Inequality Constraints has Polynomial Complexity

The trust region subproblem with a fixed number m additional linear inequality constraints, denoted by (Tm), have drawn much attention recently. The question as to whether Problem (Tm) is in Class P or Class NP remains open. So far, the only affirmative general result is that (T1) has an exact SOCP/SDP reformulation and thus is polynomially solvable. By adopting an early result of Martinez on local non-global minimum of the trust region subproblem, we can inductively reduce any instance in (Tm) to a sequence of trust region subproblems (T0). Although the total number of (T0) to be solved takes an exponential order of m, the reduction scheme still provides an argument that the class (Tm) has polynomial complexity for each fixed m. In contrast, we show by a simple example that, solving the class of extended trust region subproblems which contains more linear inequality constraints than the problem dimension; or the class of instances consisting of an arbitrarily number of linear constraints is NP-hard. When m is small such as m = 1,2, our inductive algorithm should be more efficient than the SOCP/SDP reformulation since at most 2 or 5 subproblems of (T0), respectively, are to be handled. In the end of the paper, we improve a very recent dimension condition by Jeyakumar and Li under which (Tm) admits an exact SDP relaxation. Examples show that such an improvement can be strict indeed.Comment: 18 pages, 0 figure

GHZ States, Almost-Complex Structure and Yang--Baxter Equation (I)

Recent study suggests that there are natural connections between quantum information theory and the Yang--Baxter equation. In this paper, in terms of the generalized almost-complex structure and with the help of its algebra, we define the generalized Bell matrix to yield all the GHZ states from the product base, prove it to form a unitary braid representation and present a new type of solution of the quantum Yang--Baxter equation. We also study Yang-Baxterization, Hamiltonian, projectors, diagonalization, noncommutative geometry, quantum algebra and FRT dual algebra associated with this generalized Bell matrix.Comment: 17 pages, late

On RIC bounds of Compressed Sensing Matrices for Approximating Sparse Solutions Using ℓq\ell_q Quasi Norms

This paper follows the recent discussion on the sparse solution recovery with quasi-norms $\ell_q,~q\in(0,1)$ when the sensing matrix possesses a Restricted Isometry Constant $\delta_{2k}$ (RIC). Our key tool is an improvement on a version of "the converse of a generalized Cauchy-Schwarz inequality" extended to the setting of quasi-norm. We show that, if $\delta_{2k}\le 1/2$, any minimizer of the $l_q$ minimization, at least for those $q\in(0,0.9181]$, is the sparse solution of the corresponding underdetermined linear system. Moreover, if $\delta_{2k}\le0.4931$, the sparse solution can be recovered by any $l_q, q\in(0,1)$ minimization. The values $0.9181$ and $0.4931$ improves those reported previously in the literature.Comment: 16page