a paradox in the approximation of dirichlet control problems in curved domains in this paper we study the approximation of a dirichlet control problem governed by an elliptic equation defined on a curved domain omega to solve this problem numerically it is usually necessary to approximate omega by a typically polygonal new domain omega h the difference between the solutions of both infinite dimensional control problems one formulated in omega and the second in omega h was studied in e casas and j sokolowski siam j control optim 48 2010 pp 3746 3780 where an error of order o h was proved in k deckelnick a gunther and m hinze siam j control optim 48 2009 pp 2798 2819 the numerical approximation of the problem defined in omega was considered the authors used a finite element method such that omega h was the polygon formed by the union of all triangles of the mesh of parameter h they proved an error of order o h 3 2 for the difference between continuous and discrete optimal controls here we show that the estimate obtained in e casas and j sokolowski siam j control optim 48 2010 pp 3746 3780 cannot be improved which leads to the paradox that the numerical solution is a better approximation of the optimal control than the exact one obtained just by changing the domain from omega to omega h

the mean field traveling salesman and related problems the edges of a complete graph on n vertices are assigned i i d random costs from a distribution for which the interval 0 t has probability asymptotic to t as t 0 through positive values in this so called pseudo dimension 1 mean field model we study several optimization problems of which the traveling salesman is the best known we prove that as n the cost of the minimum traveling salesman tour converges in probability to a certain number approximately 2 0415 which is characterized analytically

a necessary condition for the spectrum of nonnegative symmetric 5 5 matrices abstract let a be a nonnegative symmetric 5 5 matrix with eigenvalues 1 2 3 4 5 we show that if i 1 5 i 1 2 1 then 3 i 1 5 i mcdonald and neumann showed that 1 3 4 0 let 1 2 3 4 5 be a list of decreasing real numbers satisfying 1 i 1 5 i 1 2 1 2 3 i 1 5 i 3 1 3 4 0 4 the perron property that is 1 max we show that is the spectrum of a nonnegative symmetric 5 5 matrix thus we solve the symmetric nonnegative inverse eigenvalue problem for n 5 in a region for which a solution has not been known before

the symmetric nonnegative inverse eigenvalue problem for 5 5 matrices abstract the symmetric nonnegative inverse eigenvalue problem sniep asks when a list 1 2 n of n real numbers is the spectrum of an n n symmetric nonnegative matrix this problem is completely solved only for n 4 our main goal here is to contribute to the solution of sniep for n 5 we also give a sufficient condition for a list to be realized as the spectrum of a symmetric positive matrix

approximation of boundary control problems on curved domains in this paper we consider boundary control problems associated to a semilinear elliptic equation defined in a curved domain omega the dirichlet and neumann cases are analyzed to deal with the numerical analysis of these problems the approximation of omega by an appropriate domain omega h typically polygonal is required here we do not consider the numerical approximation of the control problems instead we formulate the corresponding infinite dimensional control problems in omega h and we study the influence of the replacement of omega by omega h on the solutions of the control problems our goal is to compare the optimal controls defined on gamma partial omega with those defined on gamma h partial omega h and to derive some error estimates the use of a convenient parametrization of the boundary is needed for such estimates

on the regularity of the solutions of dirichlet optimal control problems in polygonal domains a linear quadratic dirichlet control problem governed by an elliptic equation posed on a possibly nonconvex polygonal domain is analyzed detailed regularity results are provided in classical sobolev slobodetski spaces in particular it is proved that in the presence of pointwise control constraints the optimal control is continuous despite the nonconvexity of the domain

exact value for the average optimal cost of the bipartite traveling salesman and two factor problems in two dimensions we show that the average cost for the traveling salesman problem in two dimensions which is the archetypal problem in combinatorial optimization in the bipartite case is simply related to the average cost of the assignment problem with the same euclidean increasing convex weights in this way we extend a result already known in one dimension where exact solutions are avalaible the recently determined average cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction overline e n frac 1 pi log n for large number of points 2n as a byproduct of our analysis also the loop covering problem has the same optimal average cost we also explain why this result cannot be extended at higher dimensions we numerically check the exact predictions

the dyck bound in the concave 1 dimensional random assignment model we consider models of assignment for random n blue points and n red points on an interval of length 2n in which the cost for connecting a blue point in x to a red point in y is the concave function x y p for 0 1 where the optimal matching is trivially determined here the optimization is non trivial the purpose of this paper is to introduce a special configuration that we call the dyck matching and to study its statistical properties we compute exactly the average cost in the asymptotic limit of large n together with the first subleading correction the scaling is remarkable it is of order n for p 1 2 and it is universal for a wide class of models we conjecture that the average cost of the dyck matching has the same scaling in n as the cost of the optimal matching and we produce numerical data in support of this conjecture we hope to produce a proof of this claim in future work

a characterization of trace zero nonnegative 5 5 matrices let 1 2 3 4 5 be a list of complex numbers with 1 2 3 4 5 0 necessary and sufficient conditions for the existence of an entry wise nonnegative 5 5 matrix a with spectrum are presented

solution for a bipartite euclidean traveling salesman problem in one dimension the traveling salesman problem is one of the most studied combinatorial optimization problems because of the simplicity in its statement and the difficulty in its solution we characterize the optimal cycle for every convex and increasing cost function when the points are thrown independently and with an identical probability distribution in a compact interval we compute the average optimal cost for every number of points when the distance function is the square of the euclidean distance we also show that the average optimal cost is not a self averaging quantity by explicitly computing the variance of its distribution in the thermodynamic limit moreover we prove that the cost of the optimal cycle is not smaller than twice the cost of the optimal assignment of the same set of points interestingly this bound is saturated in the thermodynamic limit

a robust finite element method for nonhomogeneous dirichlet problems in domains with curved boundaries in this paper the authors consider a simple finite element method on an approximately polygonal domain using linear elements the dirichlet data are transferred in a natural way and the resulting linear system can be solved using multigrid techniques their analysis takes into account the change in domain and data transfer and optimal error estimates are obtained that are robust in the regularity of the boundary data provided they are at least square integrable it is proved that the natural extension of this finite element approximation to the original domain is optimal order accurate

scaling hypothesis for the euclidean bipartite matching problem ii correlation functions we analyze the random euclidean bipartite matching problem on the hypertorus in d dimensions with quadratic cost and we derive the two point correlation function for the optimal matching using a proper ansatz introduced by caracciolo et al phys rev e 90 012118 2014 to evaluate the average optimal matching cost we consider both the grid poisson matching problem and the poisson poisson matching problem we also show that the correlation function is strictly related to the green s function of the laplace operator on the hypertorus

on universal realizability of spectra abstract a list 1 2 n of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix the list is said to be universally realizable ur if it is the spectrum of a nonnegative matrix for each possible jordan canonical form allowed by it is well known that an n n nonnegative matrix a is co spectral to a nonnegative matrix b with constant row sums in this paper we extend the co spectrality between a and b to a similarity between a and b when the perron eigenvalue is simple we also show that if 0 and 1 2 n is ur then 1 2 n is also ur we give counter examples for the cases 1 2 n is ur implies 1 2 3 n is ur and 1 2 are ur implies 1 2 is ur

error estimates for the numerical approximation of dirichlet boundary control for semilinear elliptic equations we study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control the control is the trace of the state on the boundary of the domain which is assumed to be a convex polygonal open set in mathbb r 2 piecewise linear finite elements are used to approximate the control as well as the state we prove that the error estimates are of order o h 1 1 p for some p 2 which is consistent with the w 1 1 p p gamma regularity of the optimal control

ruling out certain 5 spectra for the symmetric nonnegative inverse eigenvalue problem abstract a method is developed to show that certain spectra cannot be realized for the s niep it is applied in the 5 by 5 case to rule out many spectra that were previously unresolved these are all in the case of 3 positive and 2 negative eigenvalues as all other cases are now resolved for spectra of the sort we discuss a diagram is given of the spectra that are excluded here as well as those trivially realizable those realizable because of the trace 0 case and those that may also be excluded because of the j l l conditions a small region remains unresolved it is a very small fraction of the area of those spectra we consider

anomalous scaling of the optimal cost in the one dimensional random assignment problem we consider the random euclidean assignment problem on the line between two sets of n random points independently generated with the same probability density function varrho the cost of the matching is supposed to be dependent on a power p 1 of the euclidean distance of the matched pairs we discuss an integral expression for the average optimal cost for n gg 1 that generalizes a previous result obtained for p 2 we also study the possible divergence of the given expression due to the vanishing of the probability density function the provided regularization recipe allows us to recover the proper scaling law for the cost in the divergent cases and possibly some of the involved coefficients the possibility that the support of varrho is a disconnected interval is also analysed we exemplify the proposed procedure and we compare our predictions with the results of numerical simulations

a characterization of trace zero symmetric nonnegative 5 5 matrices abstract the problem of determining necessary and sufficient conditions for a set of real numbers to be the eigenvalues of a symmetric nonnegative matrix is called the symmetric nonnegative inverse eigenvalue problem sniep in this paper we solve sniep in the case of trace zero symmetric nonnegative 5 5 matrices

finite element approximation of dirichlet boundary control for elliptic pdes on two and three dimensional curved domains we consider the variational discretization of elliptic dirichlet optimal control problems with constraints on the control the underlying state equation which is considered on smooth two and three dimensional domains is discretized by linear finite elements taking into account domain approximation the control variable is not discretized we obtain optimal error bounds for the optimal control in two and three space dimensions and prove a superconvergence result in two dimensions provided that the underlying mesh satisfies some additional condition we confirm our analytical findings by numerical experiments

a characterization of trace zero bisymmetric nonnegative 5 5 matrices abstract let 1 2 3 4 5 1 be real numbers such that i 1 5 i 0 in 16 o spector proved that a necessary and sufficient condition for 1 2 3 4 5 to be the eigenvalues of a traceless symmetric nonnegative 5 5 matrix is 2 5 0 and i 1 5 i 3 0 in this article we show that this condition is also a necessary and sufficient condition for 1 2 3 4 5 to be the spectrum of a traceless bisymmetric nonnegative 5 5 matrix