Constant-approximation for minimum weight partial sensor cover

Consider a set of homogeneous wireless sensors, [Formula: see text] with nonnegative weight [Formula: see text] for each sensor [Formula: see text]. Let [Formula: see text] be a set of target points. Given a integer [Formula: see text], we study the minimum weight partial sensor cover problem, that is, find the minimum total weight subset of sensors covering at least [Formula: see text] target points in [Formula: see text]. In this paper, we show the existence of polynomial-time constant-approximation for this problem.

Minimizing the Maximum Moving Cost of Interval Coverage

In this paper, we consider an interval coverage problem. We are given [Formula: see text] intervals of the same length on a line [Formula: see text] and a line segment [Formula: see text] on [Formula: see text]. Each interval has a nonnegative weight. The goal is to move the intervals along [Formula: see text] such that every point of [Formula: see text] is covered by at least one interval and the maximum moving cost of all intervals is minimized, where the moving cost of each interval is its moving distance times its weight. Algorithms for the “unweighted” version of this problem have been given before. In this paper, we present a first-known algorithm for this weighted version and our algorithm runs in [Formula: see text] time. The problem has applications in mobile sensor barrier coverage, where [Formula: see text] is the barrier and each interval is the covering interval of a mobile sensor.

Existence and Uniqueness of Positive Solution for a Fractional Dirichlet Problem with Combined Nonlinear Effects in Bounded Domains

We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem(-Δ)α/2u=a1(x)uσ1+a2(x)uσ2, in D limx→z∈∂D(δ(x))1-(α/2)u(x)=0,where0<α<2, σ1, σ2∈(-1,1), Dis a boundedC1,1-domain inℝn,n≥2,andδ(x)denotes the Euclidian distance fromxto the boundary ofD.The nonnegative weight functionsa1, a2are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution.

A RANDOMIZED ALGORITHM FOR WEIGHTED APPROXIMATION OF POINTS BY A STEP FUNCTION

The problem considered in this paper is: Given an integer k > 0 and a set P of n points in the plane each with a corresponding nonnegative weight, find a step function f with k steps that minimize the maximum weighted vertical distance between f and all the points in P. We present a randomized algorithm to solve the problem in O(n log n) expected running time. The bound is obviously optimal for the unsorted input. The previously best known algorithm runs in O(n log 2 n) worst-case time. Another merit of the algorithm is its simplicity. The algorithm is just a randomized implementation of Frederickson and Johnson's matrix searching technique, and it only exploits a simple data structure.

WEIGHTED THIN PLATE SPLINES

A widely known method in multivariate interpolation and approximation theory consists of the use of thin plate splines. In this paper, we investigate some results and properties relative to a wide variety of variational splines in some space of functions arising from a nonnegative weight function. This model includes thin plate splines, splines in tension and discusses smoothing and interpolating splines. Pointwise error estimates are given for both problems.

PARALLEL ALGORITHMS FOR LONGEST INCREASING CHAINS IN THE PLANE AND RELATED PROBLEMS

Given a set [Formula: see text] of n points in the plane such that each point in [Formula: see text] is asscociated with a nonnegative weight, we consider the problem of computing the single-source longest increasing chains among the points in [Formula: see text] This problem is a generalization of the plannar maximal layers problem. In this paper, we present a parallel algorithm that computes the single-source longest incresing chains in the plane in [Formula: see text] time using [Formula: see text] processors in the CREW PRAM computational model. We also solve a related problem of computing the all-pairs longest paths in an n-node weighted planar st-graph, in [Formula: see text] time using [Formula: see text] CREW PRAM processors. Both of our parallel algorithms are improvement over the previously best known results.

Weighted Restriction for Curves

We prove weighted norm inequalities for the Fourier transform of the formwhere v is a nonnegative weight function on ℝd and ψ: [— 1,1 ] —> ℝd is a nondegenerate curve. Our results generalize unweighted (i.e. v = 1) restriction theorems of M. Christ, and two-dimensional weighted restriction theorems of C. Carton-Lebrun and H. Heinig.

Weighted Generalized Hardy Inequalities for Nonincreasing Functions

The nonnegative weight function pairs u, v for which the operator maps the nonnegative nonincreasing functions in LP(v) boundedly into weak Lq(u) are characterized. This result is used, in particular, both to generalize and to provide an alternate proof of certain strong type inequalities recently obtained by Ariño and Muckenhouptfor the Hardy averaging operator restricted to nonnegative nonincreasing functions.

Commutator of two projections in prediction theory

Let w be a nonnegative weight function in L1 = L1 (dθ/2π). Let Q and P denote the orthogonal projections to the closed linear spans in L2(wdθ/2π) of {einθ: n ≤ 0} and {einθ: n > 0}, respectively. The commutator of Q and P is studied. This has applications for prediction problems when such a weight arises as the spectral density of a discrete weakly stationary Gaussian stochastic process.