Critical sinh-Gordon flow with non-negative weight functions

It is well known that the zeros of successive orthogonal polynomials, associated with a nonnegative weight function, are interlaced. Conversely, any two polynomials with interlaced zeros may be regarded as orthogonal polynomials associated with a nonnegative weight function. A simple construction of such a weight function is given.

On Games Arising from Multi-Depot Chinese Postman Problems

Given a space (Ω, d, μ) of homogeneous type, necessary and sufficient conditions are obtained which ensure that given a non-negative weight function U(x) there is a non-negative weight function V(x) which is finite a.e. and the fractional maximal function operator is bounded from Lp(Vdμ) to Lq(Udμ). The dual problem and the analogous problems for non-isotropic fractional integrals are also solved.

Let T be a positive linear operator defined for nonnegative functions on a rj-finite measure space {X,m,fi).Given 1 < p < oo and a nonnegative weight function w on X , it is shown that there exists a nonnegative weight function v , finite /-almost everywhere on X , such that (1) I \Tf)*wdfi< j fvd/i, for all/>0, J x J x tere exists <j> posi ( h if and only if there exists <j> positive /-almost everywhere on X with(2) / {T(t>)pwd<oo J xIn case (2) holds, we may take v = 4>l~pT'[(T(t>)P~]w] in (1).This partially answers a question of B. Muckenhoupt in [5].Applications to some specific operators are also given.

Let G be a locally compact totally disconnected Abelian group with dual group T. Let U and V be nonnegative measurable functions on T and G, respectively.In this paper we give, in terms of U and V, a necessary condition and some sufficient conditions for the inequality ||/i/||4 < C7||/V||p to hold for all / in Li(G), where / denotes the Fourier transform of / and 1 < p < q < oo.If U and V are both radial, we give a necessary and sufficient condition for the above norm inequality to hold for all / in LX(G).1. Introduction.In 1978 B. Muckenhoupt [6] posed the problem of characterizing, for given p and q with 1 < p, q < oo, those nonnegative weight functions U and V on the real line R so that the inequality (1.1) \\fU\\q<C\\fV\\p holds for all / in Li(R).This problem has been studied by, among others, Muckenhoupt himself [7, 8], W.

Introduction. This paper is concerned with an extension of certain studies of the degree of convergence of methods of approximation to a given function. The approximating functions are polynomials determined so as to minimize the integral over an infinite interval of the product of a non-negative weight function and the mth power of the magnitude of the error. The essential point of this discussion is that the interval of integration is infinite. The case of a finite interval has been considered by Jacksont and others for polynomials as well as for other types of approximating functions. The problem may be formulated more precisely as follows. Let f(x) be a given function of x defined on an infinite interval S. Under suitable restrictions onf(x) and the non-negative weight function r(x), there exists a determination of the coefficients in the polynomial

Gauss-type quadrature rules for variable-sign weight functions

We consider the following generalization of the binary search problem. A search strategy is required to locate an unknown target node $t$ in a given tree $T$. Upon querying a node $v$ of the tree, the strategy receives as a reply an indication of the connected component of $T\setminus\{v\}$ containing the target $t$. The cost of querying each node is given by a known non-negative weight function, and the considered objective is to minimize the total query cost for a worst-case choice of the target. Designing an optimal strategy for a weighted tree search instance is known to be strongly NP-hard, in contrast to the unweighted variant of the problem which can be solved optimally in linear time. Here, we show that weighted tree search admits a quasi-polynomial time approximation scheme: for any $0 \textless{} \varepsilon \textless{} 1$, there exists a $(1+\varepsilon)$-approximation strategy with a computation time of $n^{O(\log n / \varepsilon^2)}$. Thus, the problem is not APX-hard, unless $NP \subseteq DTIME(n^{O(\log n)})$. By applying a generic reduction, we obtain as a corollary that the studied problem admits a polynomial-time $O(\sqrt{\log n})$-approximation. This improves previous $\hat O(\log n)$-approximation approaches, where the $\hat O$-notation disregards $O(\mathrm{poly}\log\log n)$-factors.

Closed form expressions and Monte Carlo estimates for the Bayes factor are obtained for selection among weighted and unweighted models. Weighted distributions occur naturally in contexts where the probability that a particular observation enters the sample gets multiplied by some non-negative weight function. Suppose a realizationy of Y under the generalized densityf(y|ϑ) enters the investigator’s record with probability proportional tow(y,τ). Clearly, the recorded y is not an observation onY, but on the random variableY w , say, having pdf: $$f^w (y|\theta ,\tau ) = w(y|\tau )f(y|\theta )/E_{y|\theta } [w(y,\tau )],$$ which is called a weighted distribution. Closed form expressions for the Bayes factor are obtained for models arising from the exponential family for commonly used weight functions, and the behavior of these expressions is analyzed. Unknown weight functions are also considered. A convenient form for Monte Carlo estimation of the Bayes factor is presented, and a computational example is discussed, which uses this method to compare weighted mixture models of some aircraft data from Proschan (1963).

The prize-collecting dominating set (PCDS) problem is the generalization of the minimum dominating set (MDS) problem. The MDS problem requires a dominating set in a given graph, namely a subset of vertexes. Any vertex in the graph belongs to the closed neighborhood of the subset of vertexes, where the subset of vertexes with the smallest cardinality is the MDS. In the PCDS problem: given an undirected graph, its vertex setV has nonnegative weighting function&omega; and nonnegative penalty function&pi; . Finding a vertex subset D⊂V, for vertexes do not belong to the closed neighborhood of D , we need to pay penalties for them. The objective of this issue is to minimize the sum of weights and penalties. As we all know that the MDS problem is NP-hard in general graph, obviously, PCDS problem is NP-hard, too., In certain cases, the PCDS problem can achieve better results than the MDS problem, which has a certain value in practical applications such as facility location and logistics management. Whether the problem is limited to a special graph will get useful structural characterization and corresponding algorithm results is considered, designed polynomial time exact algorithm on the star and according to the r -LMP approximation algorithm, designed the 2-LMP approximation algorithm on the path and cycle.

This paper presented the optimal minimax and weighted least squares (WLS) methods for designing digital finite impulse response (FIR) filters to reduce the aliasing errors generated by the non-ideality of analog filters and mixers in bandwidth interleaving digital-to-analog converter (BI-DAC). To satisfy the given expected spurious free dynamic range (SFDR), we formulated these optimal designs of digital FIR filters in BI-DAC as a convex optimization problem-second-order cone programming (SOCP) that allowed the linear equality and convex quadratic inequality constraints including the magnitude flatness and the peak aliasing errors constraints to be merged. Furthermore, we derived the computational complexity of our presented optimal design. Several design examples were given to evaluate the performance of our presented unconstrained and constrained minimax and WLS designs using SOCP including their effectiveness and computational complexity. The simulation results showed that, in our presented unconstrained minimax and WLS designs using SOCP, the maximum distortion errors were all around 0.02 dB. The maximum aliasing errors were-73.9 and -80.5 dB, which satisfied the expected SFDR of a 12-bit BI-DAC system. In addition, we analyzed the influence of different values of the nonnegative weighting function on our presented unconstrained minimax and WLS designs using SOCP, and we found that there was a tradeoff among the nonnegative weighting function's value, and the distortion and aliasing errors. Moreover, when the constraints were imposed in our presented constrained minimax and WLS designs using SOCP in the selected frequency bands, the distortion errors were equal to zero and the aliasing errors were reduced below -110 dB, but the expense was that the larger distortion and aliasing errors achieved out of these selected frequency bands. Finally, we gave the computational complexity comparisons among our presented unconstrained and constrained minimax and WLS design using SOCP, we also compared the influence of the digital FIR filters' length on our presented designs' worst-case passband ripple and stopband roll-off, and we found that there was a tradeoff among the digital FIR filters' length, the passband ripple, the stopband roll-off, the computational complexity, and the actual hardware cost.

The COST-DISTANCE network design problem is the following. We are given an undirected graph G = (V,E), a designated root vertex r ∈ V, and a set of terminals S ⊂ of V. We are also given two non-negative real valued functions defined on E, namely, a cost function c and a length function l, and a non-negative weight function w on the set S. The goal is to find a tree T that connects the terminals in S to the root r and minimizes σ e ∈ Tc(e) + σ t ∈ Sw(t)lT(r,t), where lT(r,t) is the length of the path in T from t to r. We give a deterministic O(log k) approximation algorithm for the COST-DISTANCE network design problem, in a sense derandomizing the algorithm given in [4]. Our algorithm is based on a natural linear programming relaxation of the problem and in the process we show that its integrality gap is O(log k). Comments Copyright SIAM, 2001. Published in Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pages 232-233. This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/cis_papers/77 A Deterministic Algorithm for the COST-DISTANCE Problem Chandra Chekuri Sanjeev Khanna Joseph (Seffi) Naor

In this paper we are engaged with classical orthogonal polynomials. We deduce relations for the reproducing kernels for these polynomials without derivations. By the using this form of reproducing kernels of classical orthogonal polynomials we deduce some inequalities for polynomials with special weight function. This enabled us to derive some inequalities for orthonormal polynomial with special weight function. It allows us to deduce general statement for every orthonormal polynomial with a nonnegative weight function on a finite interval of orthogonality. Mathematics Subject Classification: 33C45, 33C47

For centered stationary Gaussian fields $X(s), s\in {\bf R}^d, d>1,$ we show that properly centered and normalized $\sup_{s\in {\bf R}^d}w(s/T)|X(s)|$ with a nonnegative weight function $w$ converges in distribution to a double exponential law as $T\to\infty$, provided that $w$ and the covariance function of $X$ satisfy certain smoothness and regularity conditions. This limit theorem extends the results for compact sets without weight functions obtained earlier in [V. Piterbarg, Asymptotics Methods in the Theory of Gaussian Processes and Fields, Amer. Math. Soc., Providence, RI, 1996]. This new result for Gaussian fields is then applied to obtain necessary and sufficient conditions for the properly centered and normalized sequence $\sup_{x\in {\bf R}^d}|\Psi(x)(\widehat{f}_n(x)-\e\,\widehat{f}_n(x))|$ to converge in distribution to a double exponential law under natural smoothness conditions, where $\widehat{f}_n$ denotes the kernel density estimator of the bounded continuous density $f$ on ${\bf R}^d$ based on the sample of size $n$, and $\Psi$ is a positive continuous function such that $\sup_{x\in {\bf R}^d}|\Psi(x) f(x)^\beta|<\infty$ for some $\beta\in (0, 1/2)$. This paper extends results of the paper [E. Giné, V. Koltchinskii, and L. Sakhanenko, Probab. Theory Related Fields, 130 (2004), pp. 167--198] to the case of $d>1$. It also extends the results of [E. Rio, Probab. Theory Related Fields, 98 (1994), pp. 21--45] for $\Psi(\cdot)=f^{-1/2}(\cdot)I_S(\cdot)$ with some compact set $S$ in ${\bf R}^d$ to a general class of functions $\Psi$. An example of an application completes this work.

A.O. Ivanov and A.A. Tuzhilin proposed a particular case of Gromov’s minimal fillings problem generalized to the case of stratified manifolds using weighted graphs with a nonnegative weight function as minimal fillings of finite metric spaces. In this paper we introduce generalized minimal fillings, i.e., the minimal fillings where the weight function is not necessarily nonnegative. We prove that for any finite metric space its minimal filling has the minimum weight in the class of its generalized fillings.