a note on clique web facets for multicut polytopes in this note we provide a previously undiscovered necessary condition for the facet defining property of clique web inequalities for the multicut polytope this condition imposes a minimum cardinality requirement on the node set of the clique thus implying in general that clique web inequalities associated with relatively small cliques are not facet defining for multicut polytopes

on the stable set polytope of a series parallel graph we give a short proof of chvatal s conjecture that the nontrivial facets of the stable set polytope of a series parallel graph all come from edges and odd holes

the steiner tree problem i formulations compositions and extension of facets in this paper we give some integer programming formulations for the steiner tree problem on undirected and directed graphs and study the associated polyhedra we give some families of facets for the undirected case along with some compositions and extensions we also give a projection that relates the steiner tree polyhedron on an undirected graph to the polyhedron for the corresponding directed graph this is used to show that the lp relaxation of the directed formulation is superior to the lp relaxation of the undirected one

asymptotic behavior of solutions for the damped wave equation with slowly decaying data abstract we consider the cauchy problem for the damped wave equation u t t u u t u 1 u t x r r n and the heat equation t 1 t x r r n if the data is small and slowly decays likely c 1 1 x k n 0 k 1 then the critical exponent is c k 1 2 k n for the semilinear heat equation in this paper it is shown that in the supercritical case there exists a unique time global solution to the cauchy problem for the semilinear heat equation in any dimensional space r n whose asymptotic profile is given by 0 t x r n e x y 2 4 t 4 t n 2 c 1 1 y 2 k n 2 d y provided that the data 0 satisfies lim x x k n 0 x c 1 0 even in the semilinear damped wave equation in the supercritical case a time global solution u with the data u u t 0 x u 0 u 1 x is shown in low dimensional spaces r n n 1 2 3 to have the same asymptotic profile 0 t x provided that lim x x k n u 0 u 1 x c 1 0 those proofs are given by elementary estimates on the explicit formulas of solutions

the steiner tree polytope and related polyhedra we consider the vertex weighted version of the undirected steiner tree problem in this problem a cost is incurred both for the vertices and the edges present in the steiner tree we completely describe the associated polytope by linear inequalities when the underlying graph is series parallel for general graphs this formulation can be interpreted as a partial extended formulation for the steiner tree problem by projecting this formulation we obtain some very large classes of facet defining valid inequalities for the steiner tree polytope

critical exponents for semilinear dissipative wave equations in rn abstract we shall present new critical exponents 1 2m n with m 1 2 to the cauchy problem utt u ut u p 1u with the initial data u 0 u 1 h 1 r n l m r n l 2 r n l m r n that is the small data global existence property can be derived to the cauchy problem above with power 1 2m n

0 1 2 chva tal gomory cuts given the integer polyhedronp t conv x n ax b wherea m n andb m achvatal gomory cg cut is a valid inequality forp 1 of the type ax b for some such that a n in this paper we study 0 1 2 cg cuts arising for 0 1 2 m we show that the associated separation problem 0 1 2 sep is equivalent to finding a minimum weight member of a binary clutter this implies that 0 1 2 sep is np complete in the general case but polynomially solvable whena is related to the edge path incidence matrix of a tree we show that 0 1 2 sep can be solved in polynomial time for a convenient relaxation of the systemax b this leads to an efficient separation algorithm for a subclass of 0 1 2 cg cuts which often contains wide families of strong inequalities forp 1 applications to the clique partitioning asymmetric traveling salesman plant location acyclic subgraph and linear ordering polytopes are briefly discussed

on the cut polytope the cut polytopep c g of a graphg v e is the convex hull of the incidence vectors of all edge sets of cuts ofg we show some classes of facet defining inequalities ofp c g we describe three methods with which new facet defining inequalities ofp c g can be constructed from known ones in particular we show that inequalities associated with chordless cycles define facets of this polytope moreover for these inequalities a polynomial algorithm to solve the separation problem is presented we characterize the facet defining inequalities ofp c g ifg is not contractible tok 5 we give a simple characterization of adjacency inp c g and prove that for complete graphs this polytope has diameter one and thatp c g has the hirsch property a relationship betweenp c g and the convex hull of incidence vectors of balancing edge sets of a signed graph is studied

wave equations with time dependent dissipation ii effective dissipation abstract this article is intended to present a construction of structural representations of solutions to the cauchy problem for wave equations with time dependent dissipation above scaling these representations are used to give estimates of the solution and its derivatives based on l q r n q 2 the article represents the second part within a series in jens wirth wave equations with time dependent dissipation i non effective dissipation j differential equations 222 2 2006 487 514 weak dissipations below scaling were discussed

polyhedral characterization of discrete dynamic programming many interesting combinatorial problems can be optimized efficiently using recursive computations often termed discrete dynamic programming in this paper we develop a paradigm for a general class of such optimizations that yields a polyhedral description for each model in the class the elementary concept of dynamic programs as shortest path problems in acyclic graphs is generalized to one seeking a least cost solution in a directed hypergraph sufficient conditions are then provided for binary integrality of the associated hyperflow problem given a polynomially solvable dynamic program the result is a linear program in polynomially many variables and constraints having the solution vectors of the dynamic program as its extreme point optima that is the linear program provides a succinct characterization of the solutions to the underlying optimization problem expressed through an appropriate change of variables we also discuss projecting this formulation to recover constraints on the original variables and illustrate how some important dynamic programming solvable models fit easily into our paradigm a classic multiechelon lot sizing problem in production and a host of optimization problems on recursively defined classes of graphs are included

dimensional reduction of the master equation for stochastic chemical networks the reduced multiplane method chemical reaction networks which exhibit strong fluctuations are common in microscopic systems in which reactants appear in low copy numbers the analysis of these networks requires stochastic methods which come in two forms direct integration of the master equation and monte carlo simulations the master equation becomes infeasible for large networks because the number of equations increases exponentially with the number of reactive species monte carlo methods which are more efficient in integrating over the exponentially large phase space also become impractical due to the large amounts of noisy data that need to be stored and analyzed the recently introduced multiplane method a lipshtat and o biham phys rev lett 93 170601 2004 is an efficient framework for the stochastic analysis of large reaction networks it is a dimensional reduction method based on the master equation which provides a dramatic reduction in the number of equations without compromising the accuracy of the results the reduction is achieved by breaking the network into a set of maximal fully connected subnetworks maximal cliques a separate master equation is written for the reduced probability distribution associated with each clique with suitable coupling terms between them this method is highly efficient in the case of sparse networks in which the maximal cliques tend to be small however in dense networks some of the cliques may be rather large and the dimensional reduction is not as effective furthermore the derivation of the multiplane equations from the master equation is tedious and difficult here we present the reduced multiplane method in which the maximal cliques are broken down to the fundamental two vertex cliques the number of equations is further reduced making the method highly efficient even for dense networks moreover the equations take a simpler form which can be easily constructed using a diagrammatic procedure for any desired network architecture it is shown that the method provides accurate results for the population sizes of the reactive species and their reaction rates

new bounds for the max k cut and chromatic number of a graph we consider several semidefinite programming relaxations for the max k cut problem with increasing complexity the optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest laplacian eigenvalue of the graph under consideration this is the first known eigenvalue bound for the max k cut when k 2 that is applicable to any graph this bound is exploited to derive a new eigenvalue bound on the chromatic number of a graph for regular graphs the new bound on the chromatic number is the same as the well known hoffman bound however the two bounds are incomparable in general we prove that the eigenvalue bound for the max k cut is tight for several classes of graphs we investigate the presented bounds for specific classes of graphs such as walk regular graphs strongly regular graphs and graphs from the hamming association scheme

weighted l2 estimates for dissipative wave equations with variable coefficients abstract we establish weighted l 2 estimates for the wave equation with variable damping u t t u a u t 0 in r n where a x a 0 1 x with a 0 0 and 0 1 in particular we show that the energy of solutions decays at a polynomial rate t n 2 1 if a x a 0 x for large x we derive these results by strengthening significantly the multiplier method this approach can be adapted to other hyperbolic equations with damping

asymptotic behaviors of solutions for viscoelastic wave equation with space time dependent damping term abstract in this paper we consider a viscoelastic wave equation with an absorbing term and space time dependent damping term based on the weighted energy method and by assuming that the kernel decaying exponentially we obtain the l 2 decay rates of the solutions more precisely we show that the decay rates are the same as those obtained in lin et al 2010 15 for the semilinear wave equation with absorption term

transitive packing a unifying concept in combinatorial optimization this paper attempts to provide a better understanding of the facial structure of polyhedra previously investigated separately it introduces the notion of transitive packing and the transitive packing polytope polytopes that turn out to be special cases of the transitive packing polytope include the node packing acyclic subdigraph bipartite subgraph planar subgraph clique partitioning partition transitive acyclic subdigraph interval order and relatively transitive subgraph polytopes we give cutting plane proofs for several rich classes of valid inequalities of the transitive packing polytope thereby introducing generalized cycle generalized clique generalized antihole generalized antiweb and odd partition inequalities on the one hand these classes subsume several known classes of valid inequalities for several special cases on the other hand they yield many new inequalities for several other special cases for some of the classes we also prove a lower bound on their gomory chvatal rank finally we relate the concept of transitive packing to generalized set packing and covering as well as to balanced and ideal matrices

the traveling salesman problem on a graph and some related integer polyhedra given a graphg n e and a length functionl e the graphical traveling salesman problem is that of finding a minimum length cycle goingat least once through each node ofg this formulation has advantages over the traditional formulation where each node must be visited exactly once we give some facet inducing inequalities of the convex hull of the solutions to that problem in particular the so called comb inequalities of grotschel and padberg are generalized some related integer polyhedra are also investigated finally an efficient algorithm is given wheng is a series parallel graph

facets of the clique partitioning polytope a subseta of the edge set of a graphg v e is called a clique partitioning ofg is there is a partition of the node setv into disjoint setsw 1 w k such that eachw i induces a clique i e a complete but not necessarily maximal subgraph ofg and such thata 1 uv u v w i u v given weightsw e for alle e the clique partitioning problem is to find a clique partitioninga ofg such that e a w e is as small as possible this problem known to be hard see wakabayashi 1986 comes up for instance in data analysis and here the underlying graphg is typically a complete graph in this paper we study the clique partitioning polytope of the complete graphk n i e is the convex hull of the incidence vectors of the clique partitionings ofk n we show that triangles 2 chorded odd cycles 2 chorded even wheels and other subgraphs ofk n induce facets of the theoretical results described here have been used to design an empirically efficient cutting plane algorithm with which large real world instances of the clique partitioning problem could be solved these computational results can be found in grotschel and wakabayashi 1989

efficient semidefinite branch and cut for map mrf inference we propose a branch and cut b c method for solving general map mrf inference problems the core of our method is a very efficient bounding procedure which combines scalable semidefinite programming sdp and a cutting plane method for seeking violated constraints in order to further speed up the computation several strategies have been exploited including model reduction warm start and removal of inactive constraints we analyze the performance of the proposed method under different settings and demonstrate that our method either outperforms or performs on par with state of the art approaches especially when the connectivities are dense or when the relative magnitudes of the unary costs are low we achieve the best reported results experiments show that the proposed algorithm achieves better approximation than the state of the art methods within a variety of time budgets on challenging non submodular map mrf inference problems

the partition problem in this paper we describe several forms of thek partition problem and give integer programming formulations of each case the dimension of the associated polytopes and some basic facets are identified we also give several valid and facet defining inequalities for each of the polytopes

extended formulations for the a cut problem letg v e be an undirected graph anda v we consider the problem of finding a minimum cost set of edges whose deletion separates every pair of nodes ina we consider two extended formulations using both node and edge variables an edge variable formulation has previously been considered for this problem chopra and rao 1991 cunningham 1991 we show that the lp relaxations of the extended formulations are stronger than the lp relaxation of the edge variable formulation even with an extra class of valid inequalities added this is interesting because while the lp relaxations of the extended formulations can be solved in polynomial time the lp relaxation of the edge variable formulation cannot we also give a class of valid inequalities for one of the extended formulations computational results using the extended formulations are performed

the equipartition polytope i formulations dimension and basic facets the following basic clustering problem arises in different domains ranging from physics statistics and boolean function minimization given a graphg v e and edge weightsce partition the setv into two sets of 1 2 v and 1 2 v nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized anequicut is a feasible solution of the above problem and theequicut polytope q g is the convex hull of the incidence vectors of equicuts ing in this paper we give some integer programming formulations of the equicut problem study the dimension of the equicut polytope and describe some basic classes of facet inducing inequalities forq g

damped wave equation in the subcritical case abstract we study large time asymptotics of small solutions to the cauchy problem for the one dimensional nonlinear damped wave equation 1 v tt v t v xx v 1 0 x r t 0 v 0 x v 0 x v t 0 x v 1 x in the sub critical case 2 3 2 we assume that the initial data v 0 1 x 1 v 1 l l 1 a a 0 1 where l 1 a l 1 l 1 a a l 1 x 1 x 2 also we suppose that the mean value of initial data r v 0 x v 1 x dx 0 then there exists a positive value such that the cauchy problem 1 has a unique global solution v t x c 0 l l 1 a satisfying the following time decay estimate v t l c t 1 for large t 0 here 2 3 2

global asymptotics of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u 1 u 0 t x r r n the behavior of u as t is expected to be same as that for the corresponding heat equation t 1 0 t x r r n in the subcritical case 1 c n 1 2 n there exists a similarity solution w b t x with the form t 1 1 f x t depending on b lim x x 2 1 f x 0 our first aim is to show the decay rates u t l 2 u t l 1 u t l 2 o t 1 1 n 4 t 1 1 n 2 1 t 1 1 1 2 n 4 provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order the decay rates are sharp in the sense that they are same as those of the similarity solution the second aim is to show that the gauss kernel is the asymptotic profile in the supercritical case which has been shown in case of one dimensional space by hayashi kaikina and naumkin n hayashi e i kaikina p i naumkin asymptotics for nonlinear damped wave equations with large initial data preprint 2004 we show this assertion in two and three dimensional space to prove our results both the weighted l 2 energy method and the explicit formula of solutions will be employed the weight is an improved one originally developed in y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

the equipartition polytope ii valid inequalities and facets theequipartition problem is defined as follows given a graphg v e and edge weightsce partition the setv into two sets of 1 2 v and 1 2 v nodes in such a way that the sum of the weights of edges not having both endnodes in the same set is maximized or minimized anequicut is a feasible solution of the above problem and theequicut polytope q g is the convex hull of the incidence vectors of equicuts ing in this paper we describe some facet inducing inequalities of this polytope

an extended formulation of the convex recoloring problem on a tree we introduce a strong extended formulation of the convex recoloring problem on a tree which has an application in analyzing phylogenetic trees the extended formulation has only a polynomial number of constraints but dominates the conventional formulation and the exponentially many valid inequalities introduced by campelo et al math progr 156 303 330 2016 we show that all valid inequalities introduced by campelo et al can be derived from the extended formulation we also show that the natural restriction of the extended formulation provides a complete inequality description of the polytope of subtrees of a tree the solution time using the extended formulation is much smaller than that with the conventional formulation moreover the extended formulation solves all the problem instances attempted in campelo et al 2016 and larger sized instances at the root node of the branch and bound tree without branching

hierarchical image segmentation using correlation clustering in this paper we apply efficient implementations of integer linear programming to the problem of image segmentation the image is first grouped into superpixels and then local information is extracted for each pair of spatially adjacent superpixels given local scores on a map of several hundred superpixels we use correlation clustering to find the global segmentation that is most consistent with the local evidence we show that although correlation clustering is known to be np hard finding the exact global solution is still feasible by breaking the segmentation problem down into subproblems each such sub problem can be viewed as an automatically detected image part we can further accelerate the process by using the cutting plane method which provides a hierarchical structure of the segmentations the efficiency and improved performance of the proposed method is compared to several state of the art methods and demonstrated on several standard segmentation data sets

stochastic analysis of dimerization systems the process of dimerization in which two monomers bind to each other and form a dimer is common in nature this process can be modeled using rate equations from which the average copy numbers of the reacting monomers and of the product dimers can then be obtained however the rate equations apply only when these copy numbers are large in the limit of small copy numbers the system becomes dominated by fluctuations which are not accounted for by the rate equations in this limit one must use stochastic methods such as direct integration of the master equation or monte carlo simulations these methods are computationally intensive and rarely succumb to analytical solutions here we use the recently introduced moment equations which provide a highly simplified stochastic treatment of the dimerization process using this approach we obtain an analytical solution for the copy numbers and reaction rates both under steady state conditions and in the time dependent case we analyze three different dimerization processes dimerization without dissociation dimerization with dissociation and heterodimer formation to validate the results we compare them with the results obtained from the master equation in the stochastic limit and with those obtained from the rate equations in the deterministic limit potential applications of the results in different physical contexts are discussed

formulations and valid inequalities for the node capacitated graph partitioning problem we investigate the problem of partitioning the nodes of a graph under capacity restriction on the sum of the node weights in each subset of the partition the objective is to minimize the sum of the costs of the edges between the subsets of the partition this problem has a variety of applications for instance in the design of electronic circuits and devices we present alternative integer programming formulations for this problem and discuss the links between these formulations having chosen to work in the space of edges of the multicut we investigate the convex hull of incidence vectors of feasible multicuts in particular several classes of inequalities are introduced and their strength and robustness are analyzed as various problem parameters change

asymptotic behaviors of solutions for time dependent damped wave equations abstract in this paper we consider the cauchy problem for the wave equation with time dependent damping b t u t and absorbed semilinear term u 1 u here b t b 0 1 t with 1 1 and b 0 0 using the weighted energy method we obtain the l 1 and l 2 decay rates of the solution which coincide to those for self similar solutions to the corresponding parabolic equation when 1 f n 1 2 n

lagrangian relaxation versus genetic algorithm based matheuristic for a large partitioning problem abstract this paper is concerned with a partitioning problem one of the applications and the motivation for this research is the problem of class formation for training and retraining sessions at large electricity distributors two different approaches are developed one is based on the quadratic multiple knapsack formulation and lagrangian relaxation the other is a matheuristic developed as an amalgamation of genetic algorithms and integer programming the approaches are tested by means of computational experiments both heuristics outperformed the direct application of quadratic programming with the lagrangian relaxation based approach performing the best on average and the genetic algorithm based approach performing the best on the larger test cases

facets of the bipartite subgraph polytope the bipartite subgraph polytope pbg of a graph g v e is the convex hull of the incidence vectors of all edge sets of bipartite subgraphs of g we show that all complete subgraphs of g of odd order and all so called odd bicycle wheels contained in g induce facets of pbg moreover we describe several methods with which new facet defining inequalities of pbg can be constructed from known ones examples of these methods are contraction of node sets in odd complete subgraphs odd subdivision of edges certain splittings of nodes and subdivision of all edges of a cut using these methods we can construct facet defining inequalities of pbg having coefficients of order v 2

on the partial order polytope of a digraph we introduce the partial order polytope of a digraphd defined as the convex hull of the incidence vectors of all transitive acyclic arc sets ofd for this polytope we prove some classes of inequalities to be facet defining and show that there is a polynomial separation algorithm for each of these classes the results imply a polynomial separation algorithm for a class of valid inequalities of the clique partitioning polytope that includes the two chorded odd cycle inequalities

min cut clustering we describe a decomposition framework and a column generation scheme for solving a min cut clustering problem the subproblem to generate additional columns is itself an np hard mixed integer programming problem we discuss strong valid inequalities for the subproblem and describe some efficient solution strategies computational results on compiler construction problems are reported

l2 estimates of solutions for damped wave equations with space time dependent damping term abstract in this paper we consider the damped wave equation with space time dependent potential b t x and absorbing semilinear term u 1 u here b t x b 0 1 x 2 2 1 t with b 0 0 0 and 0 1 based on the local existence theorem we obtain the global existence and the l 2 decay rate of the solution by using the weighted energy method the decay rate coincides with the result of nishihara k nishihara decay properties for the damped wave equation with space dependent potential and absorbed semilinear term preprint in the case of 0 and coincides with the result of nishihara and zhai k nishihara j zhai asymptotic behaviors of time dependent damped wave equations preprint in the case of 0

total energy decay for semilinear wave equations with a critical potential type of damping abstract we consider the cauchy problem for semilinear wave equations with a critical potential type of damping coefficient 1 x 1 in r n and a power nonlinearity u p 1 u we shall derive total energy and l p decay estimates in the case when the initial data have a compact support

validity of rate equation results for reaction rates in reaction networks with fluctuations systems of reacting particles which are well mixed or distributed homogeneously in space are commonly modeled by deterministic rate equations these equations which are based on the mean field approximation are valid for macroscopic systems however they neglect fluctuations in the particle populations as a result under conditions of strong fluctuations the reaction rates obtained from the rate equations are highly inaccurate to account for the fluctuations stochastic methods are required however these methods are computationally intensive and may become infeasible for complex reaction networks therefore it is useful to identify the conditions under which the rate equations provide accurate results naively one expects strong fluctuations when the average population sizes of some of the reactants are of order one or lower here we present a systematic approach for testing the validity of the rate equations in which we define characteristic scales in terms of the rate constants of the network we show that the rate equations fail to accurately reproduce the reaction rates when the system size is reduced below these scales surprisingly the rate equations are found to be applicable in a wider range than expected their validity depends not only on the population sizes of the reactive species but also on the kinetic properties of the reaction network

a cutting plane algorithm for a clustering problem in this paper we consider a clustering problem that arises in qualitative data analysis this problem can be transformed to a combinatorial optimization problem the clique partitioning problem we have studied the latter problem from a polyhedral point of view and determined large classes of facets of the associated polytope these theoretical results are utilized in this paper we describe a cutting plane algorithm that is based on the simplex method and uses exact and heuristic separation routines for some of the classes of facets mentioned before we discuss some details of the implementation of our code and present our computational results we mention applications from e g zoology economics and the political sciences

evaluation of the multiplane method for efficient simulations of reaction networks reaction networks in the bulk and on surfaces are widespread in physical chemical and biological systems in macroscopic systems which include large populations of reactive species stochastic fluctuations are negligible and the reaction rates can be evaluated using rate equations however many physical systems are partitioned into microscopic domains where the number of molecules in each domain is small and fluctuations are strong under these conditions the simulation of reaction networks requires stochastic methods such as direct integration of the master equation however direct integration of the master equation is infeasible for complex networks because the number of equations proliferates as the number of reactive species increases recently the multiplane method which provides a dramatic reduction in the number of equations was introduced lipshtat and biham phys rev lett 93 170601 2004 the reduction is achieved by breaking the network into a set of maximal fully connected subnetworks maximal cliques lower dimensional master equations are constructed for the marginal probability distributions associated with the cliques with suitable couplings between them in this paper we test the multiplane method and examine its applicability we show that the method is accurate in the limit of small domains where fluctuations are strong it thus provides an efficient framework for the stochastic simulation of complex reaction networks with strong fluctuations for which rate equations fail and direct integration of the master equation is infeasible the method also applies in the case of large domains where it converges to the rate equation results

global existence of solutions for semilinear damped wave equations in rn with noncompactly supported initial data abstract we consider the n dimensional cauchy problem in r n for a semilinear damped wave equation with a power type nonlinearity u p for a noncompactly supported initial data which has a small energy we shall derive a global in time existence result in the case when the power of the nonlinear term satisfies 1 2 n p n n 2 this generalizes a previous result due to todorova yordanov j differential equations 174 2001 464 489 which dealt with a solution in the framework of compactly supported initial data

clique web facets for multicut polytopes let g v e be a graph an edge set uv e u si v sj i j where s1 sk is a partition of v is called a multicut with k shores we investigate the polytopes mck n and mck n that are defined as the convex hulls of the incidence vectors of all multicuts with at most k shores and at least k shores respectively of the complete graph kn we introduce a large class of inequalities called clique web inequalities valid for these polytopes and provide a quite complete characterization of those clique web inequalities that define facets of mck n and mck n using general facet manipulation techniques like collapsing and node splitting we construct further new classes of facets for these multicut polytopes we also exhibit a class of clique web facets for which the separation problem can be solved in polynomial time

decay properties of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u p 1 u 0 t x r r n the behavior of u as t is expected to be the same as that for the corresponding heat equation t p 1 0 t x r r n which has the similarity solution w a t x with the form t 1 p 1 f x t depending on a lim x x 2 p 1 f x 0 provided that p is less than the fujita exponent p c n 1 2 n in this paper as a first step if 1 p p c n and the data u 0 u 1 x decays exponentially as x without smallness condition the solution is shown to decay with rates as t u t l 2 u t l p 1 u t l 2 o t 1 p 1 n 4 t 1 p 1 n 2 p 1 t 1 p 1 1 2 n 4 those of which seem to be reasonable because the similarity solution w a t x has the same decay rates as for the proof the weighted l 2 energy method will be employed with suitable weight similar to that in todorova and yordanov y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

the steiner tree problem ii properties and classes of facets this is the second part of two papers addressing the study of the facial structure of the steiner tree polyhedron in this paper we identify several classes of facet defining inequalities and relate them to special classes of graphs on which the steiner tree problem is known to be np hard

a dual ascent approach for steiner tree problems on a directed graph the steiner tree problem on a directed graph stdg is to find a directed subtree that connects a root node to every node in a designated node setv we give a dual ascent procedure for obtaining lower bounds to the optimal solution value the ascent information is also used in a heuristic procedure for obtaining feasible solutions to the stdg computational results indicate that the two procedures are very effective in solving a class of stdg s containing up to 60 nodes and 240 directed 120 undirected arcs