an algorithm for a constrained weber problem we present an algorithm for solving the weber problem when the set of feasible locations is the union of a finite number of convex polygons

a projected weiszfeld algorithm for the box constrained weber location problem abstract the weber problem consists of finding a point in r n that minimizes the weighted sum of distances from m points in r n that are not collinear an application that motivated this problem is the optimal location of facilities in the 2 dimensional case a classical method to solve the weber problem proposed by weiszfeld in 1937 is based on a fixed point iteration in this work we generalize the weber location problem considering box constraints we propose a fixed point iteration with projections on the constraints and demonstrate descending properties it is also proved that the limit of the sequence generated by the method is a feasible point and satisfies the kkt optimality conditions numerical experiments are presented to validate the theoretical results

distribution of values of bounded generalized polynomials a generalized polynomial is a real valued function which is obtained from conventional polynomials by the use of the operations of addition multiplication and taking the integer part a generalized polynomial mapping is a vector valued mapping whose coordinates are generalized polynomials we show that any bounded generalized polynomial mapping u zd rl has a representation u n f n x n zd where f is a piecewise polynomial function on a compact nilmanifold x x x and is an ergodic zd action by translations on x this fact is used to show that the sequence u n n zd is well distributed on a piecewise polynomial surface mathcal s subset mathbf r l with respect to the borel measure on mathcal s that is the image of the lebesgue measure under the piecewise polynomial function defining mathcal s as corollaries we also obtain a von neumann type ergodic theorem along generalized polynomials and a result on diophantine approximations extending the work of van der corput and of furstenberg weiss

a bi objective median location problem with a line barrier the multiple objective median problem momp involves locating a new facility with respect to a given set of existing facilities so that a vector of performance criteria is optimized a variation of this problem is obtained if the existing facilities are situated on two sides of a linear barrier such barriers like rivers highways borders or mountain ranges are frequently encountered in practice in this paper theory of an momp with line barriers is developed as this problem is nonconvex but specially structured a reduction to a series of convex optimization problems is proposed the general results lead to a polynomial algorithm for finding the set of efficient solutions the algorithm is proposed for bicriteria problems with different measures of distance

on the use of the varignon frame for single facility weber problems in the presence of convex barriers abstract this paper presents a new experimental approach to the weber problem in the presence of convex barriers by using the varignon frame the varignon frame is a mechanical system of strings weights and a board with holes that has been used to identify an optimal location for the classical weber problem we show through analytical results that the same analog can also be used for some of the weber problems in the presence of barriers some examples from the literature are revisited through experiments findings are compared to those found in the literature practical use of the analog is discussed as it provides rapid solutions allows for flexibility and enables one to visualize the problem

finding shortest paths in the plane in the presence of barriers to travel for any lp norm abstract this paper develops an efficient algorithm for the computation of the shortest paths between given sets of points origins and destinations in the plane when these paths are constrained not to cross any of a finite set of polygonal open or closed barriers it is proved that when distances are measured by an 1p norm with 1 it is also shown that optimal solutions when distances are measured according to the rectilinear or max norm i e lp norm with p 1 or p can be deduced from the results of the algorithm

an efficient solution method for weber problems with barriers based on genetic algorithms in this paper we consider the problem of locating one new facility with respect to a given set of existing facilities in the plane and in the presence of convex polyhedral barriers it is assumed that a barrier is a region where neither facility location nor travelling are permitted the resulting non convex optimization problem can be reduced to a finite series of convex subproblems which can be solved by the weiszfeld algorithm in case of the weber objective function and euclidean distances a solution method is presented that by iteratively executing a genetic algorithm for the selection of subproblems quickly finds a solution of the global problem visibility arguments are used to reduce the number of subproblems that need to be considered and numerical examples are presented

mixed stabilized finite element method for the stationary stokes dual permeability fluid flow model abstract this article proposes a stabilized mixed finite element method to investigate the stationary stokes dual permeability fluid flow model which couples the free flow region modeled by the stokes equations and dual porous medium modeled by the dual permeability equations the robust mixed finite element method developed here does not require any lagrange multiplier but a mesh dependent stabilization term with a relaxation parameter is introduced essentially to ensure the numerical stability of the algorithm the well posedness and the optimal error estimate of the finite element scheme are derived furthermore an iterative scheme is developed based on the traditional coupled algorithm we prove that the iterative scheme converges to the coupled scheme and the rate of convergence is estimated the feature of the model is illustrated by considering five numerical examples which reveal the applicability of the fluid flow model as well as the accuracy of the numerical method

a reduction result for location problems with polyhedral barriers abstract in this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facilities where a set of polyhedral barriers restricts traveling this non convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities like e g the median and center objective functions an exact algorithm and a heuristic solution procedure based on this reduction result are developed

zimmer s conjecture for actions of mathrm sl m mathbb z we prove zimmer s conjecture for c 2 actions by finite index subgroups of mathrm sl m mathbb z provided m 3 the method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in mathrm sl m mathbb r brown et al in zimmer s conjecture subexponential growth measure rigidity and strong property t 2016 arxiv 1608 04995 but new ideas are needed to overcome the lack of compactness of the space g times m gamma admitting the induced g action non compactness allows both measures and lyapunov exponents to escape to infinity under averaging and a number of algebraic geometric and dynamical tools are used control this escape new ideas are provided by the work of lubotzky mozes and raghunathan on the structure of nonuniform lattices and in particular of mathrm sl m mathbb z providing a geometric decomposition of the cusp into rank one directions whose geometry is more easily controlled the proof also makes use of a precise quantitative form of non divergence of unipotent orbits by kleinbock and margulis and an extension by de la salle of strong property t to representations of nonuniform lattices

algebraic properties of location problems with one circular barrier abstract the consideration of barriers to travel plays an increasingly important role in the transportation and location literature in one of the classical papers on location problems with barriers katz and cooper eur j oper res 6 1981 166 considered the weber problem often also referred to as median problem with one circular barrier region considering the same problem we develop new structural results showing that the set of feasible solutions can be subdivided into a polynomial number of cells of algebraic invariance on every convex subset of which the generally non convex objective function is convex these results imply improved exact and heuristic solution procedures based on convex optimization methods

placing a finite size facility with a center objective on a rectangular plane with barriers abstract this paper addresses the finite size 1 center placement problem on a rectangular plane in the presence of barriers barriers are regions in which both facility location and travel through are prohibited the feasible region for facility placement is subdivided into cells along the lines of larson and sadiq r c larson g sadiq facility locations with the manhattan metric in the presence of barriers to travel operations research 31 4 1983 652 669 to overcome complications induced by the center minimax objective we analyze the resultant cells based on the cell corners we study the problem when the facility orientation is known a priori we obtain domination results when the facility is fully contained inside 1 2 and 3 cornered cells for full containment in a 4 cornered cell we formulate the problem as a linear program however when the facility intersects gridlines analytical representation of the distance functions becomes challenging we study the difficulties of this case and formulate our problem as a linear or nonlinear program depending on whether the feasible region is convex or nonconvex an analysis of the solution complexity is presented along with an illustrative numerical example

an efficient algorithm for facility location in the presence of forbidden regions abstract this paper investigates a constrained form of the classical weber problem specifically we consider the problem of locating a new facility in the presence of convex polygonal forbidden regions such that the sum of the weighted distances from the new facility to n existing facilities is minimized it is assumed that a forbidden region is an area in the plane where travel and facility location are not permitted and that distance is measured using the euclidean distance metric a solution procedure for this nonconvex programming problem is presented it is shown that by iteratively solving a series of unconstrained problems this procedure terminates at a local optimum to the original constrained problem numerical examples are presented

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

expanding translates of curves and dirichlet minkowski theorem on linear forms we show that a multiplicative form of dirichlet s theorem on simulta neous diophantine approximation as formulated by minkowski cannot be improved for almost all points on any analytic curve on r k which is not contained in a proper affine subspace such an investigation was initiated by davenport and schmidt in the late sixties the diophantine problem is then settled via showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in r k 1 gets equidistributed in the limit we use ratner s theorem on unipotent flows linearization techniques and a new observation about intertwined linear dynamics of various sl m r s contained in sl k 1 r

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

facility location in the presence of forbidden regions i formulation and the case of euclidean distance with one forbidden circle abstract a constrained form of the weber problem is formulated in which no path is permitted to enter a prespecified forbidden region r of the plane using the calculus of variations the shortest path between two points x y r which does not intersect r is determined if d x y is unconstrained distance we denote the shortes distance along a feasible path by d x y the constrained weber problem is then given points x j r and positive weights wj j 1 2 n find a point x r such that f x n j 1 d x x j is a minimum an algorithm is formulated for the solution of this problem when d x y is euclidean distance and r is a single circular region numerical results are presented