The weighted Fermat-Torricelli problem on a surface and an “inverse” problem

We study the weighted Fermat-Torricelli (w.F-T) problem for geodesic triangles on a C² complete surface and on an Aleksandrov space of curvature bounded above by a real number K and solve an “inverse” problem on a C² complete surface. The solution of the w.F-T problem and the inverse w.F-T problem on a C² complete surface is based on the differentiation of the length of geodesics with respect to the arc length.     

A Plasticity Principle of Convex Quadrilaterals on a Convex Surface of Bounded Specific Curvature

We derive the plasticity equations for convex quadrilaterals on a complete convex surface with bounded specific curvature and prove a plasticity principle which states that: Given four shortest arcs which meet at the weighted Fermat-Torricelli point their endpoints form a convex quadrilateral and the weighted Fermat-Torricelli point belongs to the interior of this convex quadrilateral, an increase of the weight corresponding to a shortest arc causes a decrease of the two weights that correspond to the two neighboring shortest arcs and an increase of the weight corresponding to the opposite shortest arc by solving the inverse weighted Fermat-Torricelli problem for quadrilaterals on a convex surface of bounded specific curvature. The invariance of the weighted Fermat-Torricelli point(geometric plasticity principle) and the plasticity principle of quadrilaterals characterize the evolution of quadrilaterals on a complete convex surface. Furthermore, we show a connection between the plasticity of convex quadrilaterals on a complete convex surface with bounded specific curvature with the plasticity of some generalized convex quadrilaterals on a manifold which is certainly composed by triangles. We also study some cases of symmetrization of weighted convex quadrilaterals by introducing a new symmetrization technique which transforms some classes of weighted geodesic convex quadrilaterals on a convex surface to parallelograms in the tangent plane at the weighted Fermat-Torricelli point of the corresponding quadrilateral. This geometric method provides some pattern for the variable weights with respect to the 4-inverse weighted Fermat-Torricelli problem such that the weighted Fermat-Torricelli point remains invariant. By introducing the notion of superplasticity, we derive as an application of plasticity the connection between the Fermat-Torricelli point for some weighted kites with the fundamental equation of P. de Fermat for real exponents in the two dimensional Euclidean space. By using as an initial condition to the 3 body problem the solution of the 3-inverse weighted Fermat-Torricelli problem we give some future perspectives in plasticity, in order to derive new periodic solutions (chronotrees). We conclude with some philosophical ideas regarding Leibniz geometric monad in the sense of Euclid which use as an internal principle the plasticity of quadrilaterals.     

A Plasticity Principle of Closed Hexahedra in the Three-Dimensional Euclidean Space

We prove a plasticity principle of closed hexahedra in the three dimensional Euclidean space which states that: Suppose that the closed hexahedron A ₁ A ₂?A ₅ has an interior weighted Fermat-Torricelli point A ₀ with respects to the weights B _i and let α _i0j =∠A _i A ₀ A _j . Then these 10 angles are determined completely by 7 of them and considering these five prescribed rays which meet at the weighted Fermat-Torricelli point A ₀, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. By applying the plasticity principle of closed hexahedra to the n-inverse weighted Fermat-Torricelli problem, we derive some new evolutionary structures of closed polyhedra for n>5. Finally, we derive some evolutionary structures of pentagons in the two dimensional Euclidean space from the plasticity of weighted hexahedra as a limiting case.     

On the three-dimensional Wigner-Poisson problem with inflow boundary conditions

We study the Wigner-Poisson problem in a bounded spatial domain, with non-homogeneous and time-dependent “inflow” boundary conditions. This system is a quantum model of charge transport in a semiconductor device coupled with reservoirs, in presence of a self-consistent potential and of an external one. We state a local-in-time well-posedness result for the problem. The main difficulty is proving in the three-dimensional case that the non-linear potential term is a Lipschitz perturbation of the “affine” streaming operator, in an appropriately weighted L²-space.     

Shifted lattice rules based on a general weighted discrepancy for integrals over Euclidean space

We approximate weighted integrals over Euclidean space by using shifted rank-1 lattice rules with good bounds on the “generalised weighted star discrepancy”. This version of the discrepancy corresponds to the classic L_∞ weighted star discrepancy via a mapping to the unit cube. The weights here are general weights rather than the product weights considered in earlier works on integrals over R^d. Known methods based on an averaging argument are used to show the existence of these lattice rules, while the component-by-component technique is used to construct the generating vector of these shifted lattice rules. We prove that the bound on the weighted star discrepancy considered here is of order O(n^−1+δ) for any δ>0 and with the constant involved independent of the dimension. This convergence rate is better than the O(n^−1/2) achieved so far for both Monte Carlo and quasi-Monte Carlo methods.     

Inverse 1-median problem on trees under weighted Hamming distance

The inverse 1-median problem consists in modifying the weights of the customers at minimum cost such that a prespecified supplier becomes the 1-median of modified location problem. A linear time algorithm is first proposed for the inverse problem under weighted l _?? norm. Then two polynomial time algorithms with time complexities O(n log n) and O(n) are given for the problem under weighted bottleneck-Hamming distance, where n is the number of vertices. Finally, the problem under weighted sum-Hamming distance is shown to be equivalent to a 0-1 knapsack problem, and hence is ${\mathcal{NP}}$ -hard.     

Control parameter estimation in a semi-linear parabolic inverse problem using a high accurate method

Parabolic inverse problems have an important role in many branches of science and technology. The aim of this research work is to solve these classes of equations using a high order compact finite difference scheme. We consider the following inverse problem for finding u(x, t) and p(t) governed by u_t = u_xx + p(t)u + φ(x, t) with an over specified condition inside the domain. Spatial derivatives are approximated using central difference scheme. The time advancement of the simulation is performed using a “third order compact Runge-Kutta method”. The convergence orders for the approximation of both u and p are of o(k³ + h²) which improves the results obtained in the literature. An exact test case is used to evaluate the validity of our numerical analysis. We found that the accuracy of the results is better than that of previous works in the literature.     

The inequality-satisfiability problem

We define a generalization of the satisfiability problem (SAT) where each “clause” is an or-list of inequalities in n variables. This problem is NP-complete even when containing only two inequalities per “clause”, but solvable in polynomial time when either the number of variables or the number of “clauses” is fixed.     

B-splines and discretization in an inverse problem for Poisson processes

Recent results on quasi-maximum likelihood histogram sieve estimators in inverse problems for Poisson processes are generalized to B-spline sieves. The impact of discretization effects on strong L² consistency and convergence rates are studied in detail. In particular, a “rates saturation effect”, caused by discretization, is demonstrated. Finite-sample implementation is proposed and tested in a Monte Carlo experiment with the Wicksell problem, which shows a superior performance of the new approach, when compared to other methods commonly used in that context. The proposed algorithm can also be used in cases with only approximately known folding kernel.     

The problem of harmonic analysis on the infinite-dimensional unitary group

The goal of harmonic analysis on a (noncommutative) group is to decompose the most “natural” unitary representations of this group (like the regular representation) on irreducible ones. The infinite-dimensional unitary group U(∞) is one of the basic examples of “big” groups whose irreducible representations depend on infinitely many parameters. Our aim is to explain what the harmonic analysis on U(∞) consists of.We deal with unitary representations of a reasonable class, which are in 1-1 correspondence with characters (central, positive definite, normalized functions on U(∞)). The decomposition of any representation of this class is described by a probability measure (called spectral measure) on the space of indecomposable characters. The indecomposable characters were found by Dan Voiculescu in 1976.The main result of the present paper consists in explicitly constructing a 4-parameter family of “natural” representations and computing their characters. We view these representations as a substitute of the nonexisting regular representation of U(∞). We state the problem of harmonic analysis on U(∞) as the problem of computing the spectral measures for these “natural” representations. A solution to this problem is given in the next paper (Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes, math/0109194, to appear in Ann. Math.), joint with Alexei Borodin.We also prove a few auxiliary general results. In particular, it is proved that the spectral measure of any character of U(∞) can be approximated by a sequence of (discrete) spectral measures for the restrictions of the character to the compact unitary groups U(N). This fact is a starting point for computing spectral measures.     

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-λ inequality with two parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all Lp spaces for 1<p<∞. Pointwise estimates are then replaced by appropriate localized Lp-Lq estimates. We obtain weighted Lp estimates for a range of p that is different from (1,∞) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.     

Minmax regret location–allocation problem on a network under uncertainty

We consider a robust location–allocation problem with uncertainty in demand coefficients. Specifically, for each demand point, only an interval estimate of its demand is known and we consider the problem of determining where to locate a new service when a given fraction of these demand points must be served by the utility. The optimal solution of this problem is determined by the “minimax regret” location, i.e., the point that minimizes the worst-case loss in the objective function that may occur because a decision is made without knowing which state of nature will take place. For the case where the demand points are vertices of a network we show that the robust location–allocation problem can be solved in O(min{p, n − p}n³m) time, where n is the number of demand points, p (p < n) is the fixed number of demand points that must be served by the new service and m is the number of edges of the network.     

A naturally ordered enumeration of compositions

A composition of the positive integer M, with N-parts, is a vector of N non-negative integer components the sum of which is M. The paper presents a transformation that “enumerates” (assigns serial numbers) the set of all possible compositions most densely so that their “natural” ordering is preserved.     

Polynomial cases of graph decomposition: A complete solution of Holyer’s problem

Let H be a fixed graph. A graph G has an H-decomposition if the edge set of G can be partitioned into subsets inducing graphs isomorphic to H. Let P_H denote the following decision problem: “Does an instance graph G admit H-decomposition?” In this paper we prove that the problem P_H is polynomial time solvable if H is a graph whose every component has at most 2 edges. This way we complete a solution of Holyer’s problem which is the problem of classifying the problems P_H according to their computational complexities.     

Tractability of Multivariate Integration for Periodic Functions

We study multivariate integration in the worst case setting for weighted Korobov spaces of smooth periodic functions of d variables. We wish to reduce the initial error by a factor ε for functions from the unit ball of the weighted Korobov space. Tractability means that the minimal number of function samples needed to solve the problem is polynomial in ε⁻¹ and d. Strong tractability means that we have only a polynomial dependence in ε⁻¹. This problem has been recently studied for quasi-Monte Carlo quadrature rules and for quadrature rules with non-negative coefficients. In this paper we study arbitrary quadrature rules. We show that tractability and strong tractability in the worst case setting hold under the same assumptions on the weights of the Korobov space as for the restricted classes of quadrature rules. More precisely, let γ_j moderate the behavior of functions with respect to the jth variable in the weighted Korobov space. Then strong tractability holds iff ∑^∞_j=1 γ_j<∞, whereas tractability holds iff lim sup_d→∞ ∑^d_j=1 γ_j/ln d<∞. We obtain necessary conditions on tractability and strong tractability by showing that multivariate integration for the weighted Korobov space is no easier than multivariate integration for the corresponding weighted Sobolev space of smooth functions with boundary conditions. For the weighted Sobolev space we apply general results from E. Novak and H. Woźniakowski (J. Complexity17 (2001), 388–441) concerning decomposable kernels.     

An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem

We construct a generalized solution of the Riemann problem for strictly hyperbolic systems of conservation laws with source terms, and we use this to show that Glimm's method can be used directly to establish the existence of solutions of the Cauchy problem. The source terms are taken to be of the form a^′G, and this enables us to extend the method introduced by Lax to construct general solutions of the Riemann problem. Our generalized solution of the Riemann problem is “weaker than weak” in the sense that it is weaker than a distributional solution. Thus, we prove that a weak solution of the Cauchy problem is the limit of a sequence of Glimm scheme approximate solutions that are based on “weaker than weak” solutions of the Riemann problem. By establishing the convergence of Glimm's method, it follows that all of the results on time asymptotics and uniqueness for Glimm's method (in the presence of a linearly degenerate field) now apply, unchanged, to inhomogeneous systems.     

Wiener’s criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries

This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of R^N${\mathbb{R}}^N$ (N≥3$N\ge 3$) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.     

Leading coefficient problem for polynomial-like iterative equations

Because of difficulties in applying fixed point theorems, most of known results on the polynomial-like iterative equation were given by assuming λ₁>0 and the existence of solutions under the most natural assumption λn>0 is an interesting problem, called “Leading Coefficient Problem”. For this problem locally expansive invertible C¹ solutions are obtained in the expansive case and the non-hyperbolic case in [W. Zhang, On existence for polynomial-like iterative equations, Results Math. 45 (2004) 185-194] and C⁰ increasing solutions are constructed in [B. Xu, W. Zhang, Construction of continuous solutions and stability for the polynomial-like iterative equation, J. Math. Anal. Appl. 325 (2007) 1160-1170]. In this paper we discuss C¹ solutions for more combinations between expansive mappings and contractive ones and combinations between increasing mappings and decreasing ones.     

Spectral Asymptotics for Schrödinger Operators with Periodic Point Interactions

Spectrum of the second-order differential operator with periodic point interactions in L₂(R) is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spectrum of the operators with anomalous spectral asymptotics (when the ratio between the lengths of the bands and gaps tends to zero at infinity) can be approximated by standard periodic “weighted” operators with step-wise density functions. It is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal (generalized) resolvent of the periodic “Schrödinger operator” with certain energy-dependent boundary conditions. The operator acting in an extended Hilbert space such that its resolvent restricted to L₂(R) coincides with the formal resolvent is constructed explicitly.     

A new look at the role of players’ weights in the weighted Shapley value

everal new families of semivalues for weighted n-person transferable utility games are axiomatically constructed and discussed under increasing collections of axioms, where the weighted Shapley value arises as the resulting one member family. A more general approach to such weighted games defined in the form of two components, a weight vector λ and a classical TU-game v, is provided. The proposed axiomatizations are done both in terms of λ and v. Several new axioms related to the weight vector λ are discussed, including the so-called “amalgamating payoffs” axiom, which characterizes the value of a weighted game in terms of another game with a smaller number of players. They allow for a new look at the role of players’ weights in the context of the weighted Shapley value for the model of weighted games, giving new properties of it. Besides, another simple formula for the weighted Shapley value is found and examples illustrating some surprising behavior of it in the context of players’ weights are given. The paper contains a wide discussion of the results obtained.