non deterministic polynomial optimization problems and their approximations abstract a unified and general framework for the study of nondeterministic polynomial optimization problems npop is presented and some properties of npop s are investigated a characterization of npop s with regard to their approximability properties is given by proving necessary and sufficient conditions for two approximability schemes known approximability results are shown to fit within the general frame developed in the paper finally npop s are classified and studied with regard to the possibility or impossibility of reducing certain types of npop s to other types in a sense specified in the text

hybrid fourier pseudospectral discontinuous galerkin time domain method for wave propagation the fourier pseudospectral time domain fourier pstd method was shown to be an efficient way of modelling acoustic propagation problems as described by the linearized euler equations lee but is limited to real valued frequency independent boundary conditions and predominantly staircase like boundary shapes this paper presents a hybrid approach to solve the lee coupling fourier pstd with a nodal discontinuous galerkin dg method dg exhibits almost no restrictions with respect to geometrical complexity or boundary conditions the aim of this novel method is to allow the computation of complex geometries and to be a step towards the implementation of frequency dependent boundary conditions by using the benefits of dg at the boundaries while keeping the efficient fourier pstd in the bulk of the domain the hybridization approach is based on conformal meshes to avoid spatial interpolation of the dg solutions when transferring values from dg to fourier pstd while the data transfer from fourier pstd to dg is done utilizing spectral interpolation of the fourier pstd solutions the accuracy of the hybrid approach is presented for one and two dimensional acoustic problems and the main sources of error are investigated it is concluded that the hybrid methodology does not introduce significant errors compared to the fourier pstd stand alone solver an example of a cylinder scattering problem is presented and accurate results have been obtained when using the proposed approach finally no instabilities were found during long time calculation using the current hybrid methodology on a two dimensional domain a method for wave propagation in fluid domains with complex geometries is presented the method combines fourier pstd in the bulk of the domain and dg at boundaries the main application of interest is atmospheric sound propagation in cities the method shows no significant additional error compare with a fourier pstd solver no indication of instability in long time calculations has been detected

approximability of the vertex cover problem in power law graphs in this paper we construct an approximation algorithm for the minimum vertex cover min vc problem with an expected approximation ratio of email protected b 1 12 b2 email protected b 1 z b for random power law graphs in the p a b model due to aiello et al here z b is the riemann zeta function of b we obtain this result by combining the nemhauser and trotter approach for min vc with a new deterministic rounding procedure which achieves an approximation ratio of 32 on a subset of low degree vertices for which the expected contribution to the cost of the associated linear program is sufficiently large

superconvergence and a posteriori error estimates of the dg method for scalar hyperbolic problems on cartesian grids in this paper we analyze the discontinuous galerkin dg finite element method for the steady two dimensional transport reaction equation on cartesian grids we prove the l2 stability and optimal l2 error estimates for the dg scheme we identify a special numerical flux for which the l2 norm of the solution is of order p 1 when tensor product polynomials of degree at most p are used we further prove superconvergence towards a particular projection of the directional derivative the order of superconvergence is proved to be p 1 2 we also provide a very simple derivative recovery formula which is o h p 1 superconvergent approximation to the directional derivative moreover we establish an o h 2 p 1 global superconvergence for the solution flux at the outflow boundary of the domain these results are used to construct asymptotically exact a posteriori error estimates for the directional derivative approximation by solving a local problem on each element finally we prove that the proposed a posteriori dg error estimates converge to the true errors in the l2 norm at o h p 1 rate and that the global effectivity index converges to unity at o h rate our results are valid without the flow condition restrictions we perform numerical experiments to demonstrate that theoretical rates proved in this paper are optimal

minimum vertex cover in generalized random graphs with power law degree distribution in this paper we study the approximability of the minimum vertex cover problem in power law graphs in particular we investigate the behavior of a standard 2 approximation algorithm together with a simple pre processing step when the input is a random sample from a generalized random graph model with power law degree distribution more precisely if the probability of a vertex of degree i to be present in the graph is c i where 2 and c is a normalizing constant the expected approximation ratio is 1 1 l i e where is the riemann zeta function of li is the polylogarithmic special function of and li 2 1 e 1

an order based algorithm for minimum dominating set with application in graph mining abstract dominating set is a set of vertices of a graph such that all other vertices have a neighbour in the dominating set we propose a new order based randomised local search rlso algorithm to solve minimum dominating set problem in large graphs experimental evaluation is presented for multiple types of problem instances these instances include unit disk graphs which represent a model of wireless networks random scale free networks as well as samples from two social networks and real world graphs studied in network science our experiments indicate that rlso performs better than both a classical greedy approximation algorithm and two metaheuristic algorithms based on ant colony optimisation and local search the order based algorithm is able to find small dominating sets for graphs with tens of thousands of vertices in addition we propose a multi start variant of rlso that is suitable for solving the minimum weight dominating set problem the application of rlso in graph mining is also briefly demonstrated

a threshold of ln n for approximating set cover given a collection f of subsets of s 1 n setcover is the problem of selecting as few as possiblesubsets from f such that their union covers s and maxk cover is the problem of selecting k subsets from f such that their union has maximum cardinality both these problems arenp hard we prove that 1 o 1 ln n is a threshold below which setcover cannot be approximated efficiently unless np has slightlysuperpolynomial time algorithms this closes the gap up to low orderterms between the ratio of approximation achievable by the greedyalogorithm which is 1 o 1 lnn and provious results of lund and yanakakis that showed hardness ofapproximation within a ratio of log 2 n 2s0 72 ln n for max k cover we show an approximationthreshold of 1 1 e up tolow order terms under assumption that p np

a note on the convergence of the discontinuous galerkin method for a scalar hyperbolic equation a numerical example is presented which shows that the known l 2 error estimate left u u h right leqq ch k 1 2 left u right k 1 for the discontinuous galerkin method cannot be improved within the class of quasi uniform meshes not even for smooth exact solutions u

an optimal order error estimate for the discontinuous galerkin method in this paper a new approach is developed for analyzing the discontinuous galerkin method for hyperbolic equations for a model problem in r2 the method is shown to converge at a rate 0 hn l when applied with nth degree polynomial approxi mations over a semiuniform triangulation assuming sufficient regularity in the solution

existence and uniqueness of indexed multiparametric eigenvalues we study indexed eigenvalues 0 k rk 1 for the multiparameter problem wm um 0 um 1 m 1 k where wm 0tm n 1k nvmn tm vm0 im and the vmn are compact symmetric operators on complex hilbert spaces hm let vmn be the quadratic form induced by vmn and let v u be the k k matrix with m n th element vmn um 1 m n k the index of at least in the case 0 0 depends on the sign of det v u for um in and on the dimensions of maximal subspaces on which the wm are positive definite conditions are imposed on the vmn to ensure existence and uniqueness of eigenvalues of specified norm and index in particular the known results for the left and right definiteness conditions in the literature whether strong or not are extended in a unified fashion

domination number and minimum dominating sets in pseudofractal scale free web and sierpinski graph abstract the minimum dominating set mds problem is a fundamental subject of theoretical computer science and has found vast applications in different areas including sensor networks protein interaction networks and structural controllability however the determination of the size of a mds and the number of all mdss in a general network is np hard and it thus makes sense to seek particular graphs for which the mds problem can be solved analytically in this paper we study the mds problem in the pseudofractal scale free web and the sierpinski graph which have the same number of vertices and edges for both networks we determine explicitly the domination number as well as the number of distinct mdss we show that the pseudofractal scale free web has a unique mds and its domination number is only half of that for the sierpinski graph which has many mdss we argue that the scale free topology is responsible for the difference of the size and number of mdss between the two studied graphs which in turn indicates that power law degree distribution plays an important role in the mds problem and its applications in scale free networks

second root vectors for multiparameter eigenvalue problems of fredholm type a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied a basis for the second root subspace at eigenvalues of fredholm type is computed in terms of the underlying multiparameter system a self adjoint version of this result is given under a weak definiteness condition and sturm liouville and finitedimensional examples are considered

maximum matchings and minimum dominating sets in apollonian networks and extended tower of hanoi graphs abstract the apollonian networks display the remarkable power law and small world properties as observed in most realistic networked systems their dual graphs are extended tower of hanoi graphs which are obtained from the tower of hanoi graphs by adding a special vertex linked to all its three extreme vertices in this paper we study analytically maximum matchings and minimum dominating sets in apollonian networks and their dual graphs both of which have found vast applications in various fields e g structural controllability of complex networks for both networks we determine their matching number domination number the number of maximum matchings as well as the number of minimum dominating sets

left definite multiparameter eigenvalue problems we study the problem k tmxm anvmnxm o xmehm m i k n 1 where tm and vmn are selfadjoint linear operators on separable hilbert spaces hm with tm positive r i compact and vmn bounded we assume left definiteness which involves positivity of certain linear combinations of cofactors in the determi nant with m n th entry xm vmnxm we establish a spectral theory for that is in some way simpler and more complete than those hitherto available for this case in particular we make use of operators bn a iao where the an are determinantal operators on i hm this complements an established approach to the alternative right definite problem where ao is positive via the operators fn aoian 1 introduction in this section we shall discuss how the left and right definiteness conditions arose and what methods and results are currently available for the left definite case then in 1 3 we shall summarise our approach here throughout we shall be considering the system of the abstract 1 1 definiteness conditions early work on was motivated by separation of variables in certain classical boundary value problems the resulting ordinary differential equations in most cases do satisfy the conditions we impose on the operators tm and vmn the hm being l2 spaces the bulk of the literature which also covers finite dimensional equations including difference approximations involves sign definiteness of the determinant 80 x whose m n th entry is xm vmnx m atkinson 1 has treated such problems systematically and his work together with various recent extensions now provides an analogue of much of the basic spectral theory for a single equation tx avx x 1 0 with x vx sign definite such problems are nowadays called right definite actually various right definiteness conditions have been used and two which are relevant to our discussion are the strong 80 condition

root vectors for geometrically simple two parameter eigenvalues a class of multiparameter eigenvalue problems involving generally non self adjoint and unbounded operators is studied bases for lower order root subspaces at geometrically simple eigenvalues of fredholm type of arbitrary finite index are computed in terms of the underlying multiparameter system

on the hardness of optimization in power law graphs our motivation for this work is the remarkable discovery that many large scale real world graphs ranging from internet and world wide web to social and biological networks appear to exhibit a power law distribution the number of nodes y i of a given degree i is proportional to i b where b 0 is a constant that depends on the application domain there is practical evidence that combinatorial optimization in power law graphs is easier than in general graphs prompting the basic theoretical question is combinatorial optimization in power law graphs easy does the answer depend on the power law exponent b our main result is the proof that many classical np hard graph theoretic optimization problems remain np hard on power law graphs for certain values of b in particular we show that some classical problems such as clique and coloring remain np hard for all b 1 moreover we show that all the problems that satisfy the so called optimal substructure property remain np hard for all b 0 this includes classical problems such as minimum vertex cover maximum independent set and minimum dominating set our proofs involve designing efficient algorithms for constructing graphs with prescribed degree sequences that are tractable with respect to various optimization problems

multiparameter definiteness conditions a previous attempt to systematize various conditions from multiparameter spectral theory is extended to weaker forms of definiteness the latter not only occur in the established literature but are also under active investigation at present several algebraic and geometrical formulations exist and questions concerning their equivalence are approached in a unified fashion where possible

nonuniform right definiteness abstract the multiparameter eigenvalue problem wm x m xm w m t m n 1 k n v mn m 1 k where gl ge c k xm is a nonzero element of the separable hilbert space hm and tm and vmn are compact symmetric is studied various properties including existence and uniqueness of i c k for which the imth greatest eigenvalue of wm i equals one are proved right definiteness is assumed which means positivity of the determinant with m n th entry ym vmnym for all nonzero ym hm m 1 k this gives a klein oscillation theorem for systems of an o d e satisfying a definiteness condition that is usefully weaker than in previous such results an expansion theorem in terms of the corresponding eigenvectors xmi is also given thereby connecting the abstract oscillation theory with a result of atkinson

an analysis of the discontinuous galerkin method for a scalar hyperbolic equation we prove lp stability and error estimates for the discontinuous galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain using finite element analysis techniques we obtain l2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials lp estimates for p 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree the latter estimates are proved by combining finite difference and finite element analysis techniques

inapproximability of dominating set on power law graphs we prove the first logarithmic lower bounds for the approximability of the minimum dominating set problem for the case of connected power law graphs for being a size parameter and the power law exponent we give also a best up to now upper approximation bound for this problem in the case of the parameters 2 we develop also a new functional method for proving lower approximation bounds and display a sharp approximation phase transition area between approximability and inapproximability of the underlying problems our results depend on a method which could be also of independent interest

asymptotic expansion of the variational eigencurve for two parameter simple pendulum type equations abstract we consider the nonlinear two parameter simple pendulum type equation u t u p sin u t u t 0 t i t t u t 0 where p 1 is a constant and 0 are parameters for a given 0 there exists a solution triple u r 2 c 2 i which is obtained by a variational method such that u 0 as we establish the precise asymptotic formula for the variational eigencurve as whose growth order is sublinear

on multiparameter theory abstract in this paper we point out a mistake in the paper abstract multiparameter theory i of p j browne we give new proofs for some of the results of p j browne

perturbation theory of multiparameter eigenvalue problems abstract we investigate the multiparameter eigenvalue problem in the case where some of the operators depend holomorphically on a perturbation parameter the behaviour of eigenvalues of finite multiplicity together with the associated eigenvectors is considered as the perturbation parameter is varied in addition we discuss the effect of the perturbation on a complete orthonormal system of eigenvectors the theory allows fairly general holomorphic perturbations and only requires a weak multiparameter definiteness condition

optimal convergence of the original dg method for the transport reaction equation on special meshes we show that the approximation given by the original discontinuous galerkin method for the transport reaction equation in d space dimensions is optimal provided the meshes are suitably chosen the l 2 norm of the error is of order k 1 when the method uses polynomials of degree k these meshes are not necessarily conforming and do not satisfy any uniformity condition they are required only to be made of simplexes each of which has a unique outflow face we also find a new element by element postprocessing of the derivative in the direction of the flow which superconverges with order k 1

on root vectors of self adjoint pencils let a and c be compact symmetric operators on a hilbert space h and let b i c be 1 1 then the eigenvectors for the generalised eigenvalue problem a b x 0 x h and the associated root vectors coincide with those of the operator q b 1 a which is self adjoint in the pontryagin space hb under the inner product given by x y b x by the main result is equivalence of i completeness of these root vectors in h ii nondegeneracy of a particular root subspace l of q in hb iii decomposability of hb into a q invariant orthogonal direct sum f b g b l where dim f is finite and gb is a hilbert space this includes iohvidov s theorem a complete proof of which we could not find in print in addition to precise bounds on dim f we also give various special cases and related results e g for the case where a is self adjoint and bounded below with compact resolvent and b is bounded and symmetric

uniform convergence of multiparameter eigenfunction expansions abstract we study the uniform convergence of the eigenfunction expansion associated with a multiparameter system of differential equations a coercive inequality together with known results on the l 2 convergence of the expansions is used to obtain uniform convergence we deal specifically with the case of systems of second order ordinary differential equations satisfying right definiteness or left definiteness however the methods used can be applied to more general situations

multiparameter sturm liouville problems with eigenparameter dependent boundary conditions abstract a system of ordinary differential equations y j q j y j k 1 n k r jk y j j 1 n 0 1 with real valued and continuous coefficient functions q j r jk is studied on 0 1 subject to boundary conditions y j 0 y j 0 cot j and b j y j 1 d j y j 1 e t j c j y j 1 a j y j 1 0 2 for j 1 n here e t e 1 e 2 e n is an arbitrary n n matrix of real numbers and j a j d j b j c j 0 a point 1 n t c n satisfying 0 1 and 0 2 is called an eigenvalue of the system results are given on the existence and location of the eigenvalues and completeness and oscillation of the eigenfunctions

new techniques for approximating optimal substructure problems in power law graphs the remarkable discovery of many large scale real networks is the power law distribution in degree sequence the number of vertices with degree i is proportional to i b for some constant b 1 a lot of researchers believe that it may be easier to solve some optimization problems in power law graphs unfortunately many problems have been proved np hard even in power law graphs intuitively a theoretical question is raised are these problems on power law graphs still as hard as on general graphs in this paper we show that many optimal substructure problems such as minimum dominating set minimum vertex cover and maximum independent set are easier to solve in power law graphs by illustrating better inapproximability factors an optimization problem has the property of optimal substructure if its optimal solution on some given graph is essentially the union of the optimal sub solutions on all maximal connected components in particular we prove the above problems and a more general problem r minimum dominating set remain apx hard and their constant inapproximability factors on general power law graphs by using the cycle based embedding technique to embed any d bounded graphs into a power law graph in addition in simple power law graphs we further prove the corresponding inapproximability factors of these problems based on the graphic embedding technique as well as that of maximum clique and minimum coloring using the embedding technique in 1 as a result of these inapproximability factors the belief that there exists some 1 o 1 approximation algorithm for these problems on power law graphs is proven to be not always true at last we do in depth investigations in the relationship between the exponential factor b and constant greedy approximation algorithms

on a problem of b d sleeman abstract the problem is to derive a spectral theory including eigenvector completeness for the system 1 i t m x m m 1 k n v mn x m 0 x m h m m 1 k where tm and vmn are self adjoint operators on separable hilbert spaces with the vmn bounded sleeman imposes the definiteness conditions 2 ii t m 0 i e t m i for some 0 m 1 k 3 iii n 0 k n n 0 on d h m 1 k h m where the n are operators formally defined by tensor determinants expression iii includes both left and right definiteness conditions under which i is frequently studied we point out and with a slight strengthening of iii correct errors in sleeman s analysis we also make extensions obviating the need for ii and giving expansion theorems for cases with discrete and continuous spectra

dual variational approaches to multiparameter eigenvalue problems abstract we study the abstract multiparameter problem t m x m k n 1 n v mn x m 0 x m h m m 1 k for selfadjoint operators t m and v mn on separable hilbert spaces h m we develop dual variational approaches related to the polar duality between certain convex sets the primal approach has been the basis of several investigations particularly in the so called right definite case here we use the dual approach to derive existence and comparison results for the left definite case

layer structures for the solutions to the perturbed simple pendulum problems we consider the perturbed simple pendulum equation u t f u t sinu t t i t t u t 0 t i u t 0 where 0 and r are parameters the typical example of f is f u u p 1u p 1 the purpose of this paper is to study the shape of the solutions when 1 more precisely by using a variational approach we show that there exist two types of solutions one is almost flat inside i and another is like a step function with two steps

classification of eigentuples for uniformly elliptic multiparameter problems abstract let tm vmn be self adjoint operators in hilbert spaces hm the t3 having compact resolvents and lower bounds m 0 and the vmn satisfying uniform ellipticity 1 m n k when m 0 for each m these conditions are called uniform left definiteness and guarantee that all eigenvalues for the simultaneous equations wm xm 0 xm 0 wm tm n 1k nvmn belong to r k and are semi simple in an appropriate sense when some m 0 these conclusions may fail and we use an embedding technique to discuss for example algebraic multiplicities of eigenvalues transitions between real and non real eigenvalues and bounds for the numbers of eigenvalues of various types

optimal error estimate and superconvergence of the dg method for first order hyperbolic problems we consider the original discontinuous galerkin method for the first order hyperbolic problems in d dimensional space we show that when the method uses polynomials of degree k the l 2 error estimate is of order k 1 provided the triangulation is made of rectangular elements satisfying certain conditions further we show the o h 2 k 1 order superconvergence for the error on average on some suitably chosen subdomains including the whole domain and their outflow faces moreover we also establish a derivative recovery formula for the approximation of the convection directional derivative which is superconvergent with order k 1