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

regularity of quasi linear equations in the heisenberg group we prove the c regularity of the gradient of weak solutions of a class of quasi linear equations in nilpotent stratified lie groups of step two as applications we prove higher regularity theorems and a liouville type theorem for 1 quasi conformal mappings between domains of the heisenberg group 1997 john wiley sons inc

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

w2 1 e estimates for the monge ampere equation abstract we study strictly convex alexandrov solutions u of the real monge ampere equation det 2 u f where f is measurable positive and bounded away from 0 and under only these assumptions we prove interior w 2 1 e regularity of u

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

partial regularity for degenerate subelliptic systems associated with hormander s vector fields abstract the paper is devoted to partial holder estimates for weak solutions of a class of degenerate subelliptic systems constructed by hormander s vector fields assumptions on the systems are that coefficient matrix satisfies vmo condition in the sense of carnot caratheodory distance in variables x for fixed u and lower order terms satisfy the natural growth condition and the controllable growth condition respectively

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

optimal partial regularity for sub elliptic systems with dini continuous coefficients under the superquadratic natural growth the paper is concerned with nonlinear sub elliptic systems with dini continuous coefficients under super quadratic m 2 natural growth conditions in the heisenberg group a general criterion for weak solutions to be regular in the neighborhood of a given point is established by a generalization of method of a harmonic approximation in particular our result is optimal in the sense that in the case of holder continuous coefficients we obtain directly the optimal holder exponent for horizontal gradient of weak solutions on its regular set

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

isoperimetric and sobolev inequalities for carnot caratheodory spaces and the existence of minimal surfaces after hormander s fundamental paper on hypoellipticity 54 the study of partial differential equations arising from families of noncommuting vector fields has developed significantly in this paper we study some basic functional and geometric properties of general families of vector fields that include the hormander type as a special case similar to their classical counterparts such properties play an important role in the analysis of the relevant differential operators both linear and nonlinear to motivate our results we recall some classical inequalities let e c r be a caccioppoli set a measurable set having a locally finite perimeter then one has the isoperimetric inequality

on the regularity of subelliptic p harmonic functions in carnot groups abstract in this paper we prove second order horizontal differentiability and c 1 regularity results for subelliptic p harmonic functions in carnot groups for p close to 2

regularity results for quasilinear elliptic equations in the heisenberg group we prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the heisenberg group mathbb h n the model case is the non degenerate p laplacean operator sum i 1 2n x i left left mu 2 left mathfrak x u right 2 right frac p 2 2 x i u right 0 where mu 0 and p is not too far from 2

a constructive version of the boyle handelman theorem on the spectra of nonnegative matrices abstract a constructive version of the celebrated boyle handelman theorem on the non zero spectra of nonnegative matrices is presented

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

strong shift equivalence and positive doubly stochastic matrices abstract we give sufficient conditions for a positive stochastic matrix to be similar and strong shift equivalent over r to a positive doubly stochastic matrix through matrices of the same size we also prove that every positive stochastic matrix is strong shift equivalent over r to a positive doubly stochastic matrix consequently the set of nonzero spectra of primitive stochastic matrices over r with positive trace and the set of nonzero spectra of positive doubly stochastic matrices over r are identical we exhibit a class of 2 2 matrices pairwise strong shift equivalent over r through 2 2 matrices for which there is no uniform upper bound on the minimum lag of a strong shift equivalence through matrices of bounded size in contrast we show for any n n primitive matrix of positive trace that the set of positive n n matrices similar to it contains only finitely many sse r classes

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

fundamental solutions for non divergence form operators on stratified groups we construct the fundamental solutions and for the non divergence form operators i j a i j x t x i x j t and i j a i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we also provide gaussian estimates of r and its derivatives and some results for the relevant cauchy problem suitable long time estimates of allow us to construct using both t saturation and approximation arguments

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

holder regularity for weak solutions to divergence form degenerate quasilinear parabolic systems abstract in this paper we establish holder regularity of weak solutions to the diagonal divergence form degenerate quasilinear parabolic system related to hormander type vector fields by deriving a parabolic poincare inequality and the higher integrability of gradients of weak solutions

partial regularity for subquadratic homogeneity elliptic system with vmo coefficients abstract in this paper we are concerned with subquadratic homogeneity elliptic problems with vmo coefficients in divergence form we obtain that the weak solution u is locally holder continuous besides a singular set by using a harmonic approximation where the holder exponent 0 1 the hausdorff dimension of the singular set is less than n p

partial regularity for discontinuous sub elliptic systems with subquadratic growth in the heisenberg group abstract we consider nonlinear sub elliptic systems with vmo coefficients in the heisenberg group and prove partial holder continuity for discontinuous sub elliptic problems with subquadratic growth by a generalization of the technique of a harmonic approximation the model case is the sub elliptic p laplace system with vmo coefficients in the form i 1 2 n x i a i 1 x u 2 p 2 2 x i u 0 where a i v m o and 1 p 2

regularity for weak solutions to nondiagonal quasilinear degenerate elliptic systems abstract the aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to hormander s vector fields where the coefficients are bounded with vanishing mean oscillation we first prove l p p 2 estimates for gradients of weak solutions by using a priori estimates and a known reverse holder inequality and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems then we get higher morrey and campanato estimates for gradients of weak solutions to original systems and holder estimates for weak solutions

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

williams s conjecture is false for reducible subshifts we show that for two subshifts of finite type having exactly two irreducible components strong shift equivalence is not the same as shift equiv alence this refutes the williams conjecture w in the reducible case the irreducible case remains an open problem mathematics research group alabama state university montgomery alabama 36101 this content downloaded from 207 46 13 180 on thu 08 sep 2016 04 22 09 utc all use subject to http about jstor org terms

equivariant flow equivalence for shifts of finite type by matrix equivalence over group rings let g denote a nite group in this paper by a g shift of nite type g sft we will mean a shift of nite type sft together with a continuous g action which commutes with the shift where in addition the action is free and the sft is irreducible and is non trivial contains more than one orbit we will classify these systems up to g 5ow equivalence this equivalence relation can be described in terms of g sfts skew products or suspension 5ows x 2 for example two g sfts are g 5ow equivalent if and only if there exists an orientation preserving homeomorphism between their mapping tori which commutes with the induced g actions a g sft can be presented by a nite square matrix a over z g the positive cone of the integral group ring zg w parry personal communication 2001 let ia 1 denote the nn matrix whose upper left corner is ia and which otherwise equals the in nite identity matrix let e zg be the group of nn matrices generated by basic elementary matrices those which di er from i in at most one entry which must be o diagonal over zg let w a denote the weight class of a de nition 4 1 the conjugacy class in g of the group of weights of loops based at a xed vertex we show that the weight class is an invariant of g 5ow equivalence when w a w b g we will show that g sfts presented by matrices a and b are g 5ow equivalent if and only if there are matrices u and v in e zg such that u ia 1v ib 1 theorem 6 1 the complete classi cation up to g 5ow equivalence which allows the possibility w a g has a more complicated statement theorem 6 4 in the case that g is trivial our classi cation reduces to the familiar classi cation of franks 14 by cokernel group and determinant when g is non trivial the classi cation up to e zg is much more dibcult and interesting and remains an open problem we consider these algebraic issues inxx 8 and 9 in x 8 we give the modest requisite k theory terminology and background and for the case g z 2 we give a constructive partial result theorem 8 1 and some very concrete illustrative examples examples 8 6 and 8 7 which indicate how the zg equivalence problem becomes more dibcult when g is non trivial that is zg6 z in x 9 we consider e zg equivalence of injective matrices in this case gl zg equivalence amounts to isomorphism of cokernel modules and the re nement toe zg equivalence is classi ed by k 1 zg h for an associated subgroup h ofs k 1 zg as one consequence if g is abelian and det ia is not a zerodiviso r in zg then det ia determines the g 5ow equivalence class up

analysis on lie groups theorie locale diffusion de la chaleur sur des groupes de lie connexes la theorie globale les inegalites de sobolev bases graduees sur des corps lisses construction des bases graduees estimation locale du noyau diffusion de la chaleur demonstration de l inegalite de sobolev locale formes de dirichlet sur des groupes le semi groupe de convolution les conditions de sobolev modifiees

harnack inequality for non divergence form operators on stratified groups we prove lower bounds for the fundamental solutions of the non divergence form operators j i j x t x i x j t and i j i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we then prove an invariant harnack inequality for such operators as a byproduct we also study some relevant properties of the green functions on bounded domains

c1 partial regularity for sub elliptic systems with dini continuous coefficients in carnot groups abstract in this paper we consider nonlinear sub elliptic systems with dini continuous coefficients for the case 1 m 2 in carnot groups based on a generalization of the technique of a harmonic approximation introduced by duzaar grotowski kronz and a sobolev poincare type inequality established in carnot groups a c 1 partial regularity result for weak solutions of sub elliptic systems with natural growth terms is established in particular our result is optimal in the sense that in the case of holder continuous coefficients it yields directly the optimal holder exponent for horizontal gradients of weak solutions on its regular set

gradient regularity for elliptic equations in the heisenberg group abstract we give dimension free regularity conditions for a class of possibly degenerate sub elliptic equations in the heisenberg group exhibiting super quadratic growth in the horizontal gradient this solves an issue raised in j j manfredi g mingione regularity results for quasilinear elliptic equations in the heisenberg group math ann 339 2007 485 544 where only dimension dependent bounds for the growth exponent are given we also obtain explicit a priori local regularity estimates and cover the case of the horizontal p laplacean operator extending some regularity proven in a domokos j j manfredi c 1 regularity for p harmonic functions in the heisenberg group for p near 2 in contemp math vol 370 2005 pp 17 23 in turn using some recent techniques of caffarelli and peral l caffarelli i peral on w 1 p estimates for elliptic equations in divergence form comm pure appl math 51 1998 1 21 the a priori estimates found are shown to imply the suitable local calderon zygmund theory for the related class of non homogeneous possibly degenerate equations involving discontinuous coefficients these last results extend to the sub elliptic setting a few classical non linear euclidean results t iwaniec projections onto gradient fields and l p estimates for degenerated elliptic operators studia math 75 1983 293 312 e dibenedetto j j manfredi on the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems amer j math 115 1993 1107 1134 and to the non linear case estimates of the same nature that were available in the sub elliptic setting only for solutions to linear equations

schauder estimates for parabolic nondivergence operators of hormander type let x1 x2 xq be a system of real smooth vector fields satisfying hormander s rank condition in a bounded domain of rn let a aij t x i j 1q be a symmetric uniformly positive definite matrix of real functions defined in a domain u r for operators of kind h t i j 1qaij t x xixj i 1qbi t x xi c t x we prove local a priori estimates of schauder type in the natural parabolic ck u spaces defined by the vector fields xi and the distance induced by them namely for aij bi c ck u and u u we prove u ck 2 u c hu ck u u l u

partial regularity for sub elliptic systems with dini continuous coefficients involving natural growth terms in the heisenberg group abstract we consider nonlinear sub elliptic systems with dini continuous coefficients in divergence form in the heisenberg group based on a generalization of the method of a harmonic approximation introduced by duzaar and steffen partial regularity of weak solutions for sub elliptic systems under natural growth conditions is established in particular our result is optimal in the sense that in the case of holder continuous coefficients we obtain directly the optimal holder exponent for the horizontal gradients of weak solutions on the regular set

vertical versus horizontal sobolev spaces abstract let 0 1 p and let h n be the heisenberg group folland in 1975 showed that if f h n r is a function in the horizontal sobolev space s 2 p h n then f belongs to the euclidean sobolev space s p r 2 n 1 for any test function in short s 2 p h n s loc p r 2 n 1 we show that the localisation can be omitted if one only cares for sobolev regularity in the vertical direction the horizontal sobolev space s 2 p h n is continuously contained in the vertical sobolev space v p h n our search for the sharper result was motivated by the following two applications first combined with a short additional argument it implies that bounded lipschitz functions on h n have a 1 2 order vertical derivative in bmo h n second it yields a fractional order generalisation of the non endpoint vertical versus horizontal poincare inequalities of v lafforgue and a naor

partial regularity for discontinuous sub elliptic systems with vmo coefficients involving controllable growth terms in heisenberg groups abstract in this paper we consider discontinuous sub elliptic systems with vmo coefficients for the controllable growth case of p 2 in the heisenberg group based on a generalization of the technique of a harmonic approximation with the superquadratic growth case a partial holder continuity result for weak solutions of discontinuous sub elliptic systems with vmo coefficients is established in particular the primary model covered by our analysis is the non degenerate p sub laplace system involving super quadratic controllable growth terms

optimal partial regularity for weak solutions of nonlinear sub elliptic systems in carnot groups abstract this paper is concerned with partial regularity for weak solutions to nonlinear sub elliptic systems in divergence form in carnot groups the technique of a harmonic approximation introduced by simon and developed by duzaar and grotowski is adapted to our context we establish caccioppoli type inequalities and partial regularity with optimal local holder exponents for horizontal gradients of weak solutions to systems under super quadratic natural structure conditions and super quadratic controllable structure conditions respectively

quasiregular mappings of the heisenberg group the condition 1 1 means that for all d g the coordinate functions o f f belong to the sobolev space w d of functions in l d whose distributional first order partial derivatives are also l integrable in d in the above definition f x is the formal derivative of f a t x defined by means of the partial derivatives i f x l is the operator norm of f x and ji x is the jacobian determinant of f a t x these are uniquely defined almost everywhere by 1 1 it turns out that quasiregular mappings are differentiable almost everywhere and a nonconstant quasiregular map is discrete open and sense preserving for these and other basic properties of qr maps we refer to books re by reshetnyak vu by vuorinen and also to a forthcoming book r6 by the second author the definition extends immediately to the case f m n where m and n are connected oriented riemannian n manifolds we say that a continuous map f m n is quasiregular if it has a local representation at each point of m which is quasiregular in the above sense and if there exists k 1 k oe such that

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

path components of matrices and strong shift equivalence over q abstract we show that two positive matrices over q or r which lie in the same component of a set m m n r m is conjugate over r to a are strong shift equivalent it is a simple corollary that 2 2 matrices over q are strong shift equivalent if they are shift equivalent conversely we prove that if two matrices a b over r are strong shift equivalent then a 0 b 0 can be joined by a path of nonnegative real matrices within the strong shift equivalence class of a up to conjugation of b 0 by a permutation matrix

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

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

conformality and q harmonicity in carnot groups we show that if f is a 1 quasiconformal map defined on an open subset of a carnot group g then composition with f preserves q harmonic functions we combine this with a regularity theorem for q harmonic functions and an algebraic regularity theorem for maps between carnot groups to show that f is smooth we give some applications to the study of rigidity

interior w 1 p x t regularity for parabolic divergence equations of hormander s type abstract let x 1 x 2 x q be a system of real smooth vector fields satisfying hormander s condition in a bounded domain r n we consider the following parabolic equations u t x i a i j x t x j u x i f i in t where x i is the formal adjoint of x i the coefficients a i j x t are real valued measurable functions defined in t 0 t satisfying the uniform parabolic condition 0 1 2 i j 1 q a i j x t i j 1 2 for almost every x t r n r every r q and some constant we prove the interior w 1 p x t regularity for weak solutions to the parabolic equations under the assumptions that p x t satisfies the strong log holder continuity condition and the coefficients a i j x t belong to the space v m o l o c l our method relies on a gagliardo nirenberg inequality constituted on hormander s vector fields and a certain vitali covering lemma

optimal partial regularity for sub elliptic systems with sub quadratic growth in carnot groups abstract this paper is devoted to partial regularity for weak solutions to nonlinear sub elliptic systems for the case 1 m 2 under natural growth conditions in carnot groups the method of a harmonic approximation introduced by simon and developed by duzaar grotowski and kronz is adapted to our context and then partial regularity with the optimal local holder exponent for horizontal gradients of weak solutions to the systems is established