Search results for "Monotone polygon"

Optimal Guard Placement Problem Under L-Visibility

2006

Two points a and b in the presence of polygonal obstacles are L-visible if the length of the shortest path avoiding obstacles is no more than L. For a given convex polygon Q, Gewali et al [4]. addressed the guard placement problem on the exterior boundary that will cover the maximum area exterior to the polygon under L-visibility. They proposed a linear time algorithm for some given value of L. When the length L is greater than half of the perimeter, they declared that problem as open. Here we address that open problem and present an algorithm whose time complexity is linear in number of vertices of the polygon.

The Monotone Area-preserving Flux-Form Advection Algorithm: Reducing the Time-splitting Error in Two-Dimensional Flow Fields

1993

Operational Quantification of Continuous-Variable Correlations

2007

We quantify correlations (quantum and/or classical) between two continuous variable modes in terms of how many correlated bits can be extracted by measuring the sign of two local quadratures. On Gaussian states, such `bit quadrature correlations' majorize entanglement, reducing to an entanglement monotone for pure states. For non-Gaussian states, such as photonic Bell states, ideal and real de-Gaussified photon-subtracted states, and mixtures of pure Gaussian states, the bit correlations are shown to be a {\em monotonic} function of the negativity. This yields a feasible, operational way to quantitatively measure non-Gaussian entanglement in current experiments by means of direct homodyne d…

Monogamy Inequality for Distributed Gaussian Entanglement

2007

We show that for all n-mode Gaussian states of continuous variable systems, the entanglement shared among n parties exhibits the fundamental monogamy property. The monogamy inequality is proven by introducing the Gaussian tangle, an entanglement monotone under Gaussian local operations and classical communication, which is defined in terms of the squared negativity in complete analogy with the case of n-qubit systems. Our results elucidate the structure of quantum correlations in many-body harmonic lattice systems.

Positive Versions of Polynomial Time

1998

Abstract We show that restricting a number of characterizations of the complexity class P to be positive (in natural ways) results in the same class of (monotone) problems, which we denote by posP . By a well-known result of Razborov, posP is a proper subclass of the class of monotone problems in P . We exhibit complete problems for posP via weak logical reductions, as we do for other logically defined classes of problems. Our work is a continuation of research undertaken by Grigni and Sipser, and subsequently Stewart; indeed, we introduce the notion of a positive deterministic Turing machine and consequently solve a problem posed by Grigni and Sipser.

Relaxation for a Class of Control Systems with Unilateral Constraints

2019

We consider a nonlinear control system involving a maximal monotone map and with a priori feedback. We assume that the control constraint multifunction $U(t,x)$ is nonconvex valued and only lsc in the $x \in \mathbb{R}^{N}$ variable. Using the Q-regularization (in the sense of Cesari) of $U(t,\cdot )$, we introduce a relaxed system. We show that this relaxation process is admissible.

A General Framework for the One Center Location Problem

1992

This paper deals with an optimization problem where the objective function F is defined on a real vector space X by F(x) = γ(w 1║x - a 1║1, ⋯, w n ║x - a n║ n ), a formula in which a 1, ⋯, a n are n given points in X, ║∙║1, ⋯, ║∙║ n n norms on X, w 1, ⋯, w n positive numbers and γ a monotone norm on ℝ n . A geometric description of the set of optimal solutions to the problem min F(x) is given, illustrated by some examples. When all norms ║∙║i are equal, and γ being successively the l 1 , l ∞ and l 2-norm, a particular study is made, which shows the peculiar role played by the l 1-norm.

From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture

2020

Abstract In this paper we study the joint convexity/concavity of the trace functions Ψ p , q , s ( A , B ) = Tr ( B q 2 K ⁎ A p K B q 2 ) s , p , q , s ∈ R , where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of ( p , q , s ) ∈ R 3 for Ψ p , q , s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of ( α , z ) for α-z Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψ p…

Variable exponent p(x)-Kirchhoff type problem with convection

2022

Abstract We study a nonlinear p ( x ) -Kirchhoff type problem with Dirichlet boundary condition, in the case of a reaction term depending also on the gradient (convection). Using a topological approach based on the Galerkin method, we discuss the existence of two notions of solutions: strong generalized solution and weak solution. Strengthening the bound on the Kirchhoff type term (positivity condition), we establish existence of weak solution, this time using the theory of operators of monotone type.

Multiple solutions with sign information for semilinear Neumann problems with convection

2019

We consider a semilinear Neumann problem with convection. We assume that the drift coefficient is indefinite. Using the theory of nonlinear operators of monotone type, together with truncation and comparison techniques and flow invariance arguments, we prove a multiplicity theorem producing three nontrivial smooth solutions (positive, negative and nodal).