Solving visibility and separability problems on a mesh-of-processors

In this paper we study parallel algorithms for the Mesh-of-Processors architecture to solve visibility and related separability problems for sets of simple polygons in the plane. In particular, we present the following algorithms:

- AnO( \(\sqrt N\) time algorithm for computing on a Mesh-of-Processors of size N the visibility polygon from a point located in anN-vertex polygon, possibly with holes.

-O( \(\sqrt N\) ) time algorithms for computing on a Mesh-of-Processors of sizeN the set of all points on the boundary of anN-vertex polygonP which are visible in a given directiond as well as the visibility hull ofP for a given directiond.

- AnO( \(\sqrt N\) ) time algorithm for detecting on a Mesh-of-Processors of size 2N whether twoN-vertex polygons are separable in a given direction and anO( \(\sqrt {MN}\) ) time algorithm for detecting on a Mesh-of-Processors of sizeMN whetherM N-vertex polygons are sequentially separable in a given direction.

All proposed algorithms are asymptotically optimal (for the Mesh-of-Processors) with respect to time and number of processors.     

Verallgemeinerte Ausgleichsformeln und relativ optimale Formeln in Hilberträumen

A linear discretisation formula (1) for the approximation of a given linear functionalF over a Hilbert spaceH is called a ρ-optimal formula for ρ≧0, if it minimizes \(\left\| {F - \tilde F} \right\|_{H*} \) under the sidecondition \(r(\tilde F) \leqq \rho \) among all formulas \(\tilde F\) of type (1). Herein \(r(\tilde F)\) , is a suitably chosen parameter of the numerical instability of \(\tilde F\) (see (3)). \(\tilde F\) is called relative-optimal if \(\tilde F\) is ρ-optimal for \(r(\tilde F) \leqq \rho \) . For very general classes of HilbertspacesH _ε, ε>0, of analytic functions (whose regions of regularity cover, the hole complex plane for ε→0) we investigate asymptotic properties of relative-optimal formulas: as a main result it is shown that they converge (for ε→0) to the well-known least-square approximate formulas of to a generalized type of least square formulas.     

Bounds and domains of existence for hyperbolic initial value problems

Nonlinear hyperbolic initial value problems in plane regions are considered. By a discretization method which makes use of certain structure properties of the solutions \(\bar u\) , a finite dimensional technique is constructed which provides pointwise bounds for \(\bar u\) . At the same time, realistic informations on the domain of existence of \(\bar u\) can be obtained. The method's high degree of accuracy is shown by numerical examples.     

Ableitungsfreie Abschätzungen bei trigonometrischer Interpolation und Konjugierten-Bestimmung

Letf be 2 π-periodic,T _v its trigonometric interpolation polynomial, \(\bar f\) and \(\bar T_T \) the conjugates off andT _T, respectively. In this note the maximum norms ‖f-T _T‖, \(\parallel f - \bar T_T \parallel \) ‖f′?T _′‖ and \(\parallel \bar f' - \bar T'_T \parallel \) are estimated by the Fourier coefficients off. Iff is analytic on Φ results by Kress [2], [3] are obtained.     

Projecting CLPR constraints

The presentation of constraints in a usable form is an essential aspect of Constraint Logic Programming (CLP) systems. It is needed both in the output of constraints, as well as in the production of an internal representation of constraints for meta-level manipulation. Typically, only a small subset \(\tilde x\) of the variables in constraints is of interest, and so an informal statement of the problem at hand is: given a conjunction \(c(\tilde x,\tilde y)\) of constraints, express the projection \(\exists \tilde y c(\tilde x,\tilde y)\) ofc onto \(\tilde x\) in the simplest form. In this paper, we consider the constraints of the CLP(R) system and describe the essential features of its projection module. One main part focuses on the well-known problem of projection inlinear arithmetic constraints. We start with a classical algorithm and augment it with a procedure for eliminating redundant constraints generated by the algorithm. A second part discusses projection of the other object-level constraints: equations over trees and nonlinear equations. The final part deals with producing a manipulable form of the constraints, which complicates the projection problem.     

The structure of nonsingular polynomial matrices

LetK be a field and letL ∈ K ^{n × n [z]} be nonsingular. The matrixL can be decomposed as \(L(z) = \hat Q(z)(Rz + S)\hat P(z)\) so that the finite and (suitably defined) infinite elementary divisors ofL are the same as those ofRz + S, and \(\hat Q(z)\) and \(\hat P(z)^T\) are polynomial matrices which have a constant right inverse. If $$Rz + S = \left( {\begin{array}{*{20}c} {zI - A} & 0 \\ 0 & {I - zN} \\ \end{array} } \right)$$ andK is algebraically closed, then the columns of \(\hat Q\) and \(\hat P^T\) consist of eigenvectors and generalized eigenvectors of shift operators associated withL.     

Properties of interval operators

In this paper we give some properties of interval operatorsF which guarantee the convergence of the interval sequence {X _k} defined byX _k+1:=F(X_k)∩X_k to a unique fixed interval \(\hat X\) . This interval \(\hat X\) encloses the “zero-set”X ^* of a function strip \(G(x): = [g(x),\bar g(x)]\) . for some known interval operators we investigate under which assumptions these properties are valid.     

Converging multistep methods for initial value problems involving multivalued maps

Every consistent and strongly stable multistep method of stepnumberk yields a solution, of the setvalued initial value problem \(\dot y \in F(t,y),y(t_0 ) = y_0 \) . The setF(t, z) is assumed to be nonvoid, convex and closed. Upper semicontinuity of F with respect to both variables is not required everywhere. If the initial value problem is uniquely solvable, the solutions of the multistep method will converge to the solution of the continuous problem. These results carry over to functional differential equations \(\dot y \in F(t,M_t y)\) of Volterra type and to discontinuous problems \(\dot y(t) = f(t,M_t y)\) in the sense of A.F. Filippov. A difference method is applied to the discontinuous delay equation \(\ddot x(t) + 2D\dot x(t) + \omega ^2 x(t) = = - \operatorname{sgn} (x(t - \tau ) + \dot x(t - \tau ))\) . In the limit τ→0 we obtain results for the problem \(\ddot x + 2D\dot x + \omega ^2 x = = - \operatorname{sgn} (x + \dot x)\) which cannot be solved classically everywhere.     

A new method for automatical computation of error bounds for the set of all solutions of nonlinear boundary value problems

We consider nonlinear boundary value problems with arbitrarily many solutionsuεC ² [a, b]. In this paper an Algorithm will be established for a priori bounds \(\bar u,\bar d \in C[a,b]\) with the following properties:

1. For every solutionu of the nonlinear problem we obtain $$\bar u(x) \leqslant u(x) \leqslant \bar u(x), - \bar d(x) \leqslant u'(x) \leqslant \bar d(x)$$ for any,xε[a, b].
2. The bounds \(\bar u\) and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmizayaara% aaaa!36EE!\[\bar d\] are defined by the use of the functions exp, sin and cos.
3. We use neither the knowledge of solutions nor the number of solutions.

     

Global structure of families of multivariable linear systems with an application to identification

The identification problem for linear stochastic systems may be stated roughly as follows: given observations on two stochastic processes which are the input and output of some unknown linear system, determine some estimate of the parameters of the system. A set of candidate linear systems which contains the “true” system is introduced, and probabilistic assumptions on the two stochastic processes turn the identification problem into the deterministic problem of minimizing some objective function over this candidate model set. If this set is a manifold, the existence of globally convergent identification algorithms hinges on the critical point behavior of the objective functions which it carries. By way of Morse Theory, the critical point behavior of objective functions on a manifold has implications with regard to the topology of the manifold. This paper analyzes the topology and critical point behavior of objective functions on a specific manifold of linear systems which appears frequently as the candidate model set in identification problems. This manifold is the set Σ of allm-input,p-output linear systems of fixed McMillan degree with real or complex coefficients. Over this manifold sits the principal bundle \(\tilde \Sigma\) of minimal realizations of systems in Σ It is shown that there exist three natural analytic metrics on the associated vector bundle. It is also shown that, in the real case, the first Stiefel-Whitney class of the bundle \(\tilde \Sigma\) has min(m, p)-1 nonvanishing powers; the same conclusion is drawn about the first Chern class of \(\tilde \Sigma\) in the complex case. These results, which follow from Morse Theory and some elementary homotopy and homology theory, imply that the category of the bundle \(\tilde \Sigma\) is at least min(m, p), and hence that the Lusternik-Schnirelmann category of Σ is at least min(m, p). It follows that canonical forms (i.e. sections of \(\tilde \Sigma\) ) may exist only when min(m, p) = 1 and that any objective function on Σ with compact sublevel sets has at least min(m, p) critical points. In particular, there exist on Σ no globally convergent gradient algorithms when min(m, p) > 1.     

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 \((A,B)\) .     

Quantum entropy-typical subspace and universal data compression

The quantum entropy-typical subspace theory is specified. It is shown that any \(\rho ^{\otimes n}\) with von Neumann \(\hbox {entropy}\le h\) can be preserved approximately by the entropy-typical subspace with \(\hbox {entropy}=h\) . This result implies an universal compression scheme for the case that the von Neumann entropy of the source does not exceed \(h\) .     

Parameterized complexity of three edge contraction problems with degree constraints

For any graph class \(\mathcal{H}\) , the \(\mathcal{H}\) -Contraction problem takes as input a graph \(G\) and an integer \(k\) , and asks whether there exists a graph \(H\in \mathcal{H}\) such that \(G\) can be modified into \(H\) using at most \(k\) edge contractions. We study the parameterized complexity of \(\mathcal{H}\) -Contraction for three different classes \(\mathcal{H}\) : the class \(\mathcal{H}_{\le d}\) of graphs with maximum degree at most  \(d\) , the class \(\mathcal{H}_{=d}\) of \(d\) -regular graphs, and the class of \(d\) -degenerate graphs. We completely classify the parameterized complexity of all three problems with respect to the parameters \(k\) , \(d\) , and \(d+k\) . Moreover, we show that \(\mathcal{H}\) -Contraction admits an \(O(k)\) vertex kernel on connected graphs when \(\mathcal{H}\in \{\mathcal{H}_{\le 2},\mathcal{H}_{=2}\}\) , while the problem is \(\mathsf{W}[2]\) -hard when \(\mathcal{H}\) is the class of \(2\) -degenerate graphs and hence is expected not to admit a kernel at all. In particular, our results imply that \(\mathcal{H}\) -Contraction admits a linear vertex kernel when \(\mathcal{H}\) is the class of cycles.     

Strong Conflict-Free Coloring for Intervals

We consider the \(k\) -strong conflict-free ( \(k\) -SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval \(I\) of the family there are at least \(k\) colors each appearing exactly once in \(I\) . We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when \(k=1\) and \(5-\frac{2}{k}\) when \(k\ge 2\) . In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any \(k \ge 1\) . We also provide, in case \(k=O({{\mathrm{polylog}}}(n))\) , a quasipolynomial time algorithm to decide the existence of a \(k\) -SCF coloring that uses at most \(q\) colors.     

Fast reversible coding and indexing of colorations

Given a non-branched tree withn vertices. Then, by ann-g-coloration we understand a partition of the set of vertices into no more thang classes, such that adjacent vertices belong to different classes. Supposed the set \(\mathfrak{S}\) of alln-g-colorations (for givenn andg) is lexicographically ordered, here are given two algorithms: the first directly determines (without using the set proper) the ordinal number of an arbitrary element of \(\mathfrak{S}\) ; the other directly generates an element of \(\mathfrak{S}\) from its given ordinal number.     

The categories L-\mathbf{Top}_{0} and L-\mathbf{Sob} as epireflective hulls

We show that the category \(L\) - \(\mathbf{Top}_{0}\) of \(T_{0}\) - \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}\) of \(L\) -topological spaces and the category \(L\) - \(\mathbf{Sob}\) of sober \(L\) -topological spaces is the epireflective hull of Sierpinski \(L\) -topological space in the category \(L\) - \(\mathbf{Top}_{0}\) .     

Aspects of insertion in random trees

A method formulated by Yao and used by Brown has yielded bounds on the fraction of nodes with specified properties in trees bult by a sequence of random internal nodes in a random tree built by binary search and insertion, and show that in such a tree about bounds better than those now known. We then apply these methods to weight-balanced trees and to a type of “weakly balanced” trees. We determine the distribution of the weight-balance factors of the internal nodes in a random tree built by binary search and insertion and show that in such a tree about 72% of all internal nodes have weight balance factors lying between \(1 - \sqrt 2 /2\) and \(\sqrt 2 /2\) .     

Some kinds of primitive and non-primitive words

If the length of a primitive word \(p\) is equal to the length of another primitive word \(q\) , then \(p^{n}q^{m}\) is a primitive word for any \(n,m\ge 1\) and \((n,m)\ne (1,1)\) . This was obtained separately by Tetsuo Moriya in 2008 and Shyr and Yu in 1994. In this paper, we prove that if the length of \(p\) is divisible by the length of \(q\) and the length of \(p\) is less than or equal to \(m\) times the length of \(q\) , then \(p^{n}q^{m}\) is a primitive word for any \(n,m\ge 1\) and \((n,m)\ne (1,1)\) . Then we show that if \(uv,u\) are non-primitive words and the length of \(u\) is divisible by the length \(v\) or one of the length of \(u\) and \(uv\) is odd for any two nonempty words \(u\) and \(v\) , then \(u\) is a power of \(v\) .