flow invariance preserving feedback controllers for the navier stokes equation abstract in this paper we develop a concrete procedure for designing feedback controllers to ensure that the resultant dynamics of turbulence will preserve certain prescribed physical constraints examples of such constraints include in particular the level sets of well known invariants of the inviscid flow such as helicity we also bring to light a certain m accretivity property of suitable quantization of the nonlinearity in the navier stokes equation and utilize the theory of nonlinear semigroups to resolve the controlled navier stokes inclusion with a multi valued feedback term

the traveling salesman problem with distances one and two we present a polynomial time approximation algorithm with worst case ratio 7 6 for the special case of the traveling salesman problem in which all distances are either one or two we also show that this special case of the traveling salesman problem is max snp hard and therefore it is unlikely that it has a polynomial time approximation scheme

derivation and analysis of near wall modelsfor channel and recirculating flows the problem of predicting features of turbulent flows occurs in many applications such as geophysical flows turbulent mixing pollution dispersal and even in the design of artificial hearts one promising approach is large eddy simulation les which seeks to predict local spatial averages u of the fluid s velocity u in some applications the les equations are solved over moderate time intervals and the core difficulty is associated with modeling near wall turbulence in complex geometries thus one important problem in les is to find appropriate boundary conditions for the flow averages which depend on the behavior of the unknown flow near the wall inspired by early works of navier and maxwell we develop such boundary conditions of the form email protected email protected email protected d email protected email protected email protected email protected email protected email protected email protected email protected email protected d email protected email protected 0 on the wall we derive effective friction coefficients b appropriate for both channel flows and recirculating flows and study their asymptotic behavior as the averaging radius s 0 and as the reynolds number re oc in the first limit no slip conditions are recovered in the second free slip conditions are recovered our goal herein is not to develop new theories of turbulent boundary layers but rather to use existing boundary layer theories to improve numerical boundary conditions for flow averages

a note on optimal homotopy asymptotic method for the solutions of fractional order heat and wave like partial differential equations optimal homotopy asymptotic method oham is prolifically implemented to find the optimal solutions of fractional order heat and wave like equations we inspect the competence of the method by examining fractional order time dependent partial differential equations it is observed that oham is a prevailing and convergent method for the solutions of linear and nonlinear fractional order time dependent partial differential problems the numerical results rendering that the applied method is explicit effective and easy to use for handling more general fractional order heat and wave like models

more on the complexity of common superstring and supersequence problems the shortest common superstring scst problem is the decision version of the problem to find for a finite set of strings l over an alphabet and an integer k n a string s of length k over such that each string in l is a substring of s analogously the shortest common supersequence scse problem is the decision version of the problem to find for a finite set of strings l over an alphabet and an integer k n a string s of length k over such that each string in l is a subsequence of s our main results are scst is np complete if the given strings have length 3 and the maximal orbit size is 8 the orbit size of a character is the number of its occurrences in the strings in l this partially solves a problem of timkovskii 1990 scst is np complete over the alphabet 0 1 even if each given string contains exactly three ones scse over the alphabet 0 1 is np complete even if the given strings all have the same length and each string contains exactly two ones moreover we introduce cyclic and permutation variants of scst and scse namely cyclic scst cyclic scse permutation scst and permutation scse the permutation scst resp cyclic scst problem is the decision version of the problem to find for a finite set of strings l over an alphabet and an integer k n a string s of length k over such that there exists a cyclic permutation of each string in l that is a substring of s permutation scse and cyclic scse are defined analogously the main results are cyclic scst and cyclic scse are np complete for strings of length 3 and polynomial time solvable for string of length 2 cyclic scst and cyclic scse are np complete for an alphabet of size 2 permutation scst is np complete if the given strings have length 3 and the maximal orbitsize is 8 permutation scst is np complete for an alphabet of size 3 and polynomial time solvable for an alphabet of size 2

boldmath a 2 frac12 approximation algorithm for shortest superstring given a set of strings s s1 s2 sn over a finite alphabet sigma a superstring of s is a string that contains each si as a contiguous substring the shortest superstring ss problem is to find a superstring of minimum length this problem has important applications in computational biology and in data compression see respectively a lesk ed computational molecular biology sources and methods for sequence analysis oxford university press oxford 1988 j storer data compression methods and theory computer science press rockville md 1988 ss is max snp hard a blum et al proc 23rd annual acm symposium on theory of computing acm new york 1991 pp 328 336 so it is unlikely that the length of a shortest superstring can be approximated to within an arbitrary constant several heuristics have been suggested and it is conjectured that greedy achieves an approximation factor of 2 this unfortunately remains an open question several linear approximation algorithms for ss have been proposed the first by blum et al proc 23rd annual acm symposium on theory of computing acm new york 1991 pp 328 336 guarantees a performance factor of 3 the factor has been successively improved to 2 frac 8 9 2 frac 5 6 2 frac 50 63 2 frac 3 4 2 frac 2 3 and 2 596 see respectively s teng and f yao proc 34th annual ieee symposium on foundations of computer science ieee computer society press piscataway nj 1993 pp 158 165 a czumaj et al proc first scandinavian workshop on algorithm theory lecture notes in comput sci 824 springer verlag berlin 1994 pp 95 106 r kosaraju j park and c stein proc 35th annual ieee symposium on foundations of computer science ieee computer society press piscataway nj 1994 pp 166 177 c armen and c stein proc 5th internat workshop on algorithms and data structures lecture notes in comput sci 955 springer verlag berlin 1995 pp 494 505 c armen and c stein proc combinatorial pattern matching lecture notes in comput sci 1075 springer verlag berlin 1996 pp 87 101 and d breslauer t jiang and z jiang j algorithms 24 1997 pp 340 353 in this paper we give an algorithm that guarantees a 2 frac 1 2 approximation factor

on the existence of global attractors of the approximate deconvolution models of turbulence abstract in this report we consider the problem of the existence of global attractors of the weak solution of the approximate deconvolution models of turbulence the l 2 global attractors are known to exist for large eddy simulation models such as leray alpha leray deconvolution or ns omega models in the case of the approximate deconvolution model as the l 2 norm of the weak solution of the adm is not controlled by the l 2 norm of the initial data and the approximate deconvolution model is an approximation of the filtered nse we are led to the question of existence of h 1 global attractors we also prove better regularity of the weak solution of adm

technical note on traveling salesman games with asymmetric costs we consider cooperative traveling salesman games with nonnegative asymmetric costs satisfying the triangle inequality we construct a stable cost allocation with budget balance guarantee equal to the held karp integrality gap for the asymmetric traveling salesman problem using the parsimonious property and a previously unknown connection to linear production games we also show that our techniques extend to larger classes of network design games we then provide a simple example showing that our cost allocation does not necessarily achieve the best possible budget balance guarantee even among cost allocations stable for the game defined by the held karp relaxation and discuss its implications on future work on traveling salesman games

traveling salesman path problems in the traveling salesman path problem we are given a set of cities traveling costs between city pairs and fixed source and destination cities the objective is to find a minimum cost path from the source to destination visiting all cities exactly once in this paper we study polyhedral and combinatorial properties of a variant we call the traveling salesman walk problem in which the objective is to find a minimum cost walk from the source to destination visiting all cities at least once we first characterize traveling salesman walk perfect graphs graphs for which the convex hull of incidence vectors of traveling salesman walks can be described by linear inequalities we show these graphs have a description by way of forbidden minors and also characterize them constructively we also address the asymmetric traveling salesman path problem atspp and give a factor o sqrt n approximation algorithm for this problem

approximation algorithms for the shortest common superstring problem abstract the object of the shortest common superstring problem scs is to find the shortest possible string that contains every string in a given set as substrings as the problem is np complete approximation algorithms are of interest the value of an aproximate solution to scs is normally taken to be its length and we seek algorithms that make the length as small as possible a different measure is given by the sum of the overlaps between consecutive strings in a candidate solution when considering this measure the object is to find solutions that make it as large as possible these two measures offer different ways of viewing the problem while the two viewpoints are equivalent with respect to optimal solutions they differ with respect to approximate solutions we describe several approximation algorithms that produce solutions that are always within a factor of two of optimum with respect to the overlap measure we also describe an efficient implementation of one of these using mccreight s compact suffix tree construction algorithm the worstcase running time is o m log n for small alphabets where m is the sum of the lengths of all the strings in the set and n is the number of strings for large alphabets the algorithm can be implemented in o m log m time by using sleator and tarjan s lexicographic splay tree data structure

length scale estimates for the lans α equations in terms of the reynolds number abstract foias holm and titi c foias d d holm e s titi the three dimensional viscous camassa holm equations and their relation to the navier stokes equations and turbulence theory j dynam differential equations 14 2002 1 35 have settled the problem of existence and uniqueness for the 3d lans equations on periodic box 0 l 3 there still remains the problem first introduced by doering and foias c r doering c foias energy dissipation in body forced turbulence j fluid mech 467 2002 289 306 for the navier stokes equations of obtaining estimates in terms of the reynolds number r e whose character depends on the fluid response as opposed to the grashof number whose character depends on the forcing r e is defined as r e u l where u is a bounded spatio temporally averaged navier stokes velocity field and l the characteristic scale of the forcing it is found that the inverse kolmogorov length is estimated by l k 1 c l 1 4 r e 5 8 moreover the estimate of foias holm and titi for the fractal dimension of the global attractor in terms of r e comes out to be d f a c v v l 1 2 l 2 1 9 8 r e 9 4 where v l l 1 2 3 and v l l l 3 it is also shown that there exists a series of time averaged inverse squared length scales whose members n 0 2 are estimated as n 1 l 2 n 0 2 c n v n 1 n r e 11 4 7 4 n ln r e 1 n c 1 r e ln r e the upper bound on the first member of the hierarchy 1 0 2 coincides with the inverse squared taylor micro scale to within log corrections

on the relationship between atsp and the cycle cover problem in this paper we study the relationship between the asymmetric traveling salesman problem atsp and the cycle cover problem in terms of the strength of the triangle inequality on the edge costs in the given complete directed graph instance g v e the strength of the triangle inequality is captured by parametrizing the triangle inequality as follows a complete directed graph g v e with a cost function c e r is said to satisfy the c parametrized triangle inequality if c c u w c w v c u v for all distinct u v w v then the graph g is called a c triangular graph for any c triangular graph g for c 1 the ratio atsp g ap g can become unbounded the upper bound is shown constructively and can also be viewed as an approximation algorithm for atsp with parametrized triangle inequality we also consider the following problem in a c triangular graph does there exist a function f c such that c m a xc m i n is bounded above by f c here c m a x and c m i n are the costs of the maximum cost and minimum cost edges respectively we show that when c 13 no such function f c exists

large eddy simulation for turbulent flows with critical regularization abstract in this paper we establish the existence of a unique regular weak solution to the large eddy simulation les models of turbulence with critical regularization we first consider the critical les for the navier stokes equations and we show that its solution converges to a solution of the navier stokes equations as the averaging radii converge to zero then we extend the study to the critical les for magnetohydrodynamic equations

on the complexity of general graph factor problems for arbitrary graphs g and h a g factor of h is a spanning subgraph of h composed of disjoint copies of g g factors are natural generalizations of l factors or perfect matchings in which g replaces the complete graph on two vertices our results show that the perfect matching problem is essentially the only instance of the g factor problem that is likely to admit a polynomial time bounded solution specifically if g has any component with three or more vertices then the existence question for g factors is np complete in all other cases the question can be resolved in polynomial time br the notion of a g factor is further generalized by replacing g by an arbitrary family of graphs this generalization forms the foundation for an extension of the traditional theory of matching this theory whose details will be developed elsewhere includes in addition to further np completeness results new polynomial algorithms and simple duality results some indication of the nature and scope of this theory are presented here

on the leray deconvolution model for the incompressible magnetohydrodynamics equations abstract we extend leray deconvolution modeling to the incompressible magnetohydrodynamics mhd the resulting model is shown to be well posed and have attractive limiting behavior both in its filtering radius and order of deconvolution additionally we present and study a numerical scheme for the model based on an extrapolated crank nicolson finite element method we show the numerical scheme is unconditionally stable preserves energy and cross helicity and optimally converges to the mhd solution numerical experiments are provided that verify convergence rates and test the scheme on benchmark problems of channel flow over a step and the orszag tang vortex problem

bounds on moments of the energy spectrum for weak solutions of the three dimensional navier stokes equations we consider weak solutions of the incompressible navier stokes equations in a finite three dimensional volume without boundaries for flows driven by a time independent body force we focus on some long time averaged ratios of norms of derivatives of the velocity field corresponding to moments of the fourier energy spectrum of the solutions the rigorous bounds derived here without making any additional assumptions on regularity of solutions are consistent with an asymptotic k 8 3 energy spectrum in contrast if spatial fluctuations in velocity gradients are suppressed by assumption in the analysis then the bounds are consistent with the k 5 3 turbulent kolmogorov energy spectrum these results are interpreted and discussed in the context of spatial intermittency and the energy cascade picture of turbulent dynamics

control of error in the homotopy analysis of nonlinear klein gordon initial value problems in the present paper we discuss the application of homotopy analysis to general nonlinear klein gordon type equations we first outline the method for general forms of the nonlinearity as well as for general functional forms of the initial conditions in particular we discuss a method of controlling the residual error in approximate solutions which may be found via homotopy analysis through adequate selection of the convergence control parameter with the general problem outlined we apply the method to various equations including the quasilinear cubic klein gordon equation the modified liouville equation the sinh gordon equation and the tanh gordon equation for each of these equations and related initial data we obtain residual error minimizing solutions which demonstrate the qualitative behavior of the true solutions in each case

on legendre polynomial approximation with the vim or ham for numerical treatment of nonlinear fractional differential equations the variational iteration method and the homotopy analysis method as alternative methods have been widely used to handle linear and nonlinear models the main property of the methods is their flexibility and ability to solve nonlinear equations accurately and conveniently this paper deals with the numerical solutions of nonlinear fractional differential equations where the fractional derivatives are considered in caputo sense the main aim is to introduce efficient algorithms of variational iteration and homotopy analysis methods that can be simply used to deal with nonlinear fractional differential equations in these algorithms legendre polynomials are effectively implemented to achieve better approximation for the nonhomogeneous and nonlinear terms that leads to facilitate the computational work the proposed algorithms are capable of reducing the size of calculations improving the accuracy and easily overcome the difficulty arising in calculating complicated integrals numerical examples are examined to show the efficiency of the algorithms

why greed works for shortest common superstring problem the shortest common superstring problem scs has been extensively studied for its applications in string compression and dna sequence assembly although the problem is known to be max snp hard the simple greedy algorithm performs extremely well in practice to explain the good performance previous researchers proved that the greedy algorithm is asymptotically optimal on random instances unfortunately the practical instances in dna sequence assembly are very different from the random instances in this paper we explain the good performance of the greedy algorithm by using the smoothed analysis we show that for any given instance i of scs the average approximation ratio of the greedy algorithm on a small random perturbation of i is 1 o 1 the perturbation defined in the paper is small and naturally represents the mutations of the dna sequence during evolution due to the existence of the uncertain nucleotides in the output of a dna sequencing machine we also proposed the shortest common superstring with wildcards problem scsw we prove that in the worst case scsw cannot be approximated within the ratio n 1 7 e while the greedy algorithm still has 1 o 1 smoothed approximation ratio

lower and upper competitive bounds for online directed graph exploration abstract we study the problem of exploring all nodes of an unknown directed graph a searcher has to construct a tour that visits all nodes but only has information about the parts of the graph it already visited analogously to the traveling salesman problem the goal is to minimize the cost of such a tour in this article we present upper and lower bounds for the competitive ratio of both the deterministic and the randomized online version of exploring all nodes of directed graphs our bounds are sharp or sharp up to a small constant depending on the specific model as it turns out restricting the diameter the incoming outgoing degree or randomly choosing a starting point does not improve lower bounds beyond a small constant factor even supplying the searcher in a planar euclidean graph with the nodes coordinates does not help essentially exploring a directed graph has a multiplicative overhead linear in the number of nodes furthermore if one wants to search for a specific node in unweighted directed graphs a greedy algorithm with quadratic multiplicative overhead can only be improved by a small constant factor as well

complexity of approximating bounded variants of optimization problems we study low degree graph problems such as maximum independent set and minimum vertex cover the goal is to improve approximation lower bounds for them and for a number of related problems like max b set packing min b set cover and max b dimensional matching b 3 we prove for example that it is np hard to achieve an approximation factor of 95 94 for max 3 dm and a factor of 48 47 for max 4 dm in both cases the hardness result applies even to instances with exactly two occurrences of each element

efficient unconditionally stable and optimally accurate fe algorithms for approximate deconvolution models this paper addresses an open question of how to devise numerical schemes for approximate deconvolution fluid flow models that are efficient unconditionally stable and optimally accurate we propose analyze and test a scheme for these models that has each of these properties for the case of homogeneous dirichlet velocity boundary conditions there are several important components of the derivation both at the continuous and discrete levels which allow for these properties to hold the proofs of stability and convergence are carried out through the use of a special choice of test function and some technical estimates numerical tests are provided that confirm the effectiveness of the scheme

truncation of scales by time relaxation abstract we study a time relaxation regularization of flow problems proposed and tested extensively by stolz and adams the aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term our aim herein is to understand how this term by itself acts to truncate solution scales and to use this understanding to give insight into parameter selection

length scale estimates for the 3d simplified bardina magnetohydrodynamic model we consider the mhd model known as double viscous simplified bardina mhd sbmhd on a 3d periodic box mhd stands for magnetohydrodynamic this system is a large eddy simulation model useful to approximate the turbulent behavior of an incompressible homogeneous magnetofluid because of the actual impossibility to handle the mhd model neither analytically nor via direct numerical simulation in a previous paper joint work with secchi the global existence of strong solutions to the sbmhd has been proved as well as the existence of a global attractor of finite fractal dimension upper bounds for such dimension are provided both in terms of the modified grashof number and the dissipation length associated to the mean rate of energy dissipation in this paper we commute the same bound in an estimate in terms of the modified reynolds number r this result is useful because the classical kolmogorov theory of turbulence is expressed using the reynolds number the global attractor estimate in terms of r is c

existence of solutions for the mhd leray alpha equations and their relations to the mhd equations in this paper we derive some new equations and we call them mhd leray alpha equations which are similar to the mhd equations we put forward the concept of weak and strong solutions for the new equations whether the 3 dimensional mhd equations have a unique weak solution is unknown however there is a unique weak solution for the 3 dimensional mhd leray alpha equations the global existence of strong solution and the gevrey class regularity for the new equations are also obtained furthermore we prove that the solutions of the mhd leray alpha equations converge to the solution of the mhd equations in the weak sense as the parameter e in the new equations converges to zero

an improved optimal homotopy analysis algorithm for nonlinear differential equations abstract this paper presents an improved optimal homotopy analysis algorithm to deal with nonlinear first order differential equations we discuss the optimal selection of the auxiliary linear operator in the application of the homotopy analysis method that will accelerate the convergence of series solutions then a powerful algorithm that employed the optimal auxiliary linear operator of the homotopy analysis method is presented to facilitate the calculations numerical illustrations are examined to show the efficiency and the pertinent features of the proposed algorithm this algorithm could lead to a promising approach for many nonlinear problems in applied sciences

an improved integrality gap for asymmetric tsp paths the asymmetric traveling salesperson path problem atspp is one where given an asymmetric metric space v d with specified vertices s and t the goal is to find an s t path of minimum length that passes through all the vertices in v this problem is closely related to the asymmetric tsp atsp which seeks to find a tour instead of an s t path visiting all the nodes for atsp a approximation guarantee implies an o approximation for atspp however no such connection is known for the integrality gaps of the linear programming lp relaxations for these problems the current best approximation algorithm for atspp is o ln n ln ln n whereas the best bound on the integrality gap of the natural lp relaxation the subtour elimination lp for atspp is o ln n in this paper we close this gap and improve the current best bound on the integrality gap from o ln n to o ln n ln ln n the resulting algorithm uses the structure of narrow s t cuts in the lp solution to construct a random tree spanning

on a leray α model of turbulence in this paper we introduce and study a new model for three dimensional turbulence the leray model this model is inspired by the lagrangian averaged navier stokes model of turbulence also known navier stokes model or the viscous camassa holm equations as in the case of the lagrangian averaged navier stokes model the leray model compares successfully with empirical data from turbulent channel and pipe flows for a wide range of reynolds numbers we establish here an upper bound for the dimension of the global attractor the number of degrees of freedom of the leray model of the order of l l d 12 7 where l is the size of the domain and l d is the dissipation length scale this upper bound is much smaller than what one would expect for three dimensional models i e l l d 3 this remarkable result suggests that the leray model has a great potential to become a good sub grid scale large eddy simulation model of turbulence we support this observation by studying analytically and computationally the energy spectrum and show that in addition to the usual k 5 3 kolmogorov power law the inertial range has a steeper power law spectrum for wavenumbers larger than 1 finally we propose a prandtl like boundary layer model induced by the leray model and show a very good agreement of this model with empirical data for turbulent boundary layers

a two level newton finite element algorithm for approximating electrically conducting incompressible fluid flows we consider the approximation of stationary electrically conducting incompressible fluid flow problems at small magnetic reynolds number the finite element discretization of these systems leads to a very large system of nonlinear equations we consider a solution algorithm which involves solving a much smaller number of nonlinear equations on a coarse mesh then one large linear system on a fine mesh under a uniqueness condition this one step two level newton fem procedure is shown to produce an optimally accurate solution this result extends both the two level method of xu 1 2 from elliptic boundary value problems to mhd problems and the energy norm error analysis of peterson 3 see also 4 of mhd problems at a small magnetic reynolds number to l2 error estimates and multilevel discretization and solution procedures

some apx completeness results for cubic graphs four fundamental graph problems minimum vertex cover maximum independent set minimum dominating set and maximum cut are shown to be apx complete even for cubic graphs therefore unless p np these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three

large eddy simulation for turbulent magnetohydrodynamic flows abstract we investigate the mathematical properties of a model for the simulation of large eddies in turbulent electrically conducting viscous incompressible flows we prove existence and uniqueness of solutions for the simplest zeroth closed mhd model 1 7 we show that its solutions converge to the solution of the mhd equations as the averaging radii converge to zero and derive a bound on the modeling error furthermore we show that the model preserves the properties of the 3d mhd equations the kinetic energy and the magnetic helicity are conserved while the cross helicity is approximately conserved and converges to the cross helicity of the mhd equations and the model is proven to preserve the alfven waves with the velocity converging to that of the mhd as 1 2 tend to zero we perform computational tests that verify the accuracy of the method and compare the conserved quantities of the model to those of the averaged mhd

on approximating restricted cycle covers a cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle an l cycle cover is a cycle cover in which the length of every cycle is in the set l the weight of a cycle cover of an edge weighted graph is the sum of the weights of its edges we come close to settling the complexity and approximability of computing l cycle covers on the one hand we show that for almost all l computing l cycle covers of maximum weight in directed and undirected graphs is mathsf apx hard most of our hardness results hold even if the edge weights are restricted to zero and one on the other hand we show that the problem of computing l cycle covers of maximum weight can be approximated within a factor of 2 for undirected graphs and within a factor of 8 3 in the case of directed graphs this holds for arbitrary sets l

self adjusting binary search trees the splay tree a self adjusting form of binary search tree is developed and analyzed the binary search tree is a data structure for representing tables and lists so that accessing inserting and deleting items is easy on an n node splay tree all the standard search tree operations have an amortized time bound of o log n per operation where by amortized time is meant the time per operation averaged over a worst case sequence of operations thus splay trees are as efficient as balanced trees when total running time is the measure of interest in addition for sufficiently long access sequences splay trees are as efficient to within a constant factor as static optimum search trees the efficiency of splay trees comes not from an explicit structural constraint as with balanced trees but from applying a simple restructuring heuristic called splaying whenever the tree is accessed extensions of splaying give simplified forms of two other data structures lexicographic or multidimensional search trees and link cut trees

approximating shortest superstrings with constraints various versions of the shortest common superstring problem play important roles in data compression and dna sequencing only recently the open problem of how to approximate a shortest superstring given a set of strings was solved in 1 9 1 shows that several greedy algorithms produce a supeistring of length o n where n is the optimal length however a major problem remains open can we still linearly approximate a superstring in polynomial time when the superstring is required to be consistent with some given negative strings i e it must not contain any negative string the best previous algorithm group merge given in 6 9 produces a consistent superstring of length n log n the negative strings make the problem much more difficult and as we will show a greedy style algorithm cannot achieve linear approximation for this problem

activation of effector immune cells promotes tumor stochastic extinction in this article we provide homotopy solutions of a cancer nonlinear model describing the dynamics of tumor cells in interaction with healthy and effector immune cells we apply a semi analytic technique for solving strongly nonlinear systems the step homotopy analysis method sham this algorithm based on a modification of the standard homotopy analysis method ham allows to obtain a one parameter family of explicit series solutions by using the homotopy solutions we first investigate the dynamical effect of the activation of the effector immune cells in the deterministic dynamics showing that an increased activation makes the system to enter into chaotic dynamics via a period doubling bifurcation scenario then by adding demographic stochasticity into the homotopy solutions we show as a difference from the deterministic dynamics that an increased activation of the immune cells facilitates cancer clearance involving tumor cells extinction and healthy cells persistence our results highlight the importance of therapies activating the effector immune cells at early stages of cancer progression

the effects of triangle inequality on the vehicle routing problem abstract routing problems often utilize experimental networks to represent real world scenarios however most ignore the inclusion of triangle inequality violations a phenomenon resulting from delays or rounding errors within a network this work evaluates the effect of both frequency the number of violations and severity the degree of intensity of a violation of triangle inequality and evaluates both solution quality and solution time based on simulated annealing ant colony optimization and savings algorithm methods findings indicate that while both frequency and severity degrade solution quality increased levels of frequency and severity together result in significant adverse affects to solution quality solution time however is not impacted by the presence of triangle inequality violations within the network this information should encourage practitioners to identify delays and maintain the presence of triangle inequality violations in a network to ensure accuracy of solution quality

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

on the existence uniqueness and finite element approximation of solutions of the equations of stationary incompressible magnetohydrodynamics the authors consider the equations of stationary incompressible magneto hydrodynamics posed in a bounded domain in three dimensions and treat the full coupled system of equations with inhomogeneous boundary conditions under certain conditions on the data they show that the existence and uniqueness of the solution of a weak formulation of the equations can be guaranteed they discuss a finite element discretization of the equations and prove an optimal estimate for the error of the approximate solution

numerical approach to differential equations of fractional order in this paper the variational iteration method and the adomian decomposition method are implemented to give approximate solutions for linear and nonlinear systems of differential equations of fractional order the two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of differential equations in these schemes the solution takes the form of a convergent series with easily computable components this paper presents a numerical comparison between the two methods for solving systems of fractional differential equations numerical results show that the two approaches are easy to implement and accurate when applied to differential equations of fractional order

on the well posedness and conservation laws of a family of multiscale deconvolution models for the magnetohydrodynamics equations abstract we present a mathematical study of a large eddy simulation les model for the incompressible magnetohydrodynamics equations the classical closure problem arising for les models is solved with the multiscale deconvolution technique developed by dunca in 11 we prove the model admits unique regular weak solutions and provide a mathematical study of the modeling error

a probabilistic ptas for shortest common superstring we consider approximation algorithms for the shortest common superstring problem scs it is well known that there is a constant f 1 such that there is no efficient approximation algorithm for scs achieving a factor of at most f in the worst case unless p np we study scs on random inputs and present an approximation scheme that achieves for every e 0 a 1 e approximation in expected polynomial time we also show that the greedy algorithm achieves approximation ratio 1 e with probability exponentially close to 1 these results apply not only if the letters are chosen independently at random but also to the more realistic mixing model which allows dependencies among the letters of the random strings our results are based on a sharp tail bound on the optimal compression which improves a previous result by frieze and szpankowski 1998

linear approximation of shortest superstrings we consider the following problem given a collection of strings s 1 s m find the shortest string s such that each s i appears as a substring a consecutive block of s although this problem is known to be np hard a simple greedy procedure appears to do quite well and is routinely used in dna sequencing and data compression practice namely repeatedly merge the pair of distinct strings with maximum overlap until only one string remains let n denote the length of the optimal superstring a common conjecture states that the above greedy procedure produces a superstring of length o n in fact 2 n yet the only previous nontrivial bound known for any polynomial time algorithm is a recent o n log n result we show that the greedy algorithm does in fact achieve a constant factor approximation proving an upper bound of 4 n furthermore we present a simple modified version of the greedy algorithm that we show produces a superstring of length at most 3 n we also show the superstring problem to be maxsnp hard which implies that a polynomial time approximation scheme for this problem is unlikely

the shortest common superstring problem average case analysis for both exact and approximate matching the shortest common superstring problem and its extension to approximate matching are considered in the probability model where each string in a given set has the same length and letters of strings are drawn independently from a finite set in the exact matching case several algorithms proposed in the literature are shown to be asymptotically optimal in the sense that the ratio of the savings resulting from the superstring constructed by each of these algorithms that is the difference between the total length of the strings in the given set and the length of the superstring to the optimal savings from the shortest superstring approaches in probability to 1 as the number of strings in the given set increases in the approximate matching case a modified version of the shortest common approximate matching superstring problem is analyzed it is demonstrated that the optimal savings in this case is given approximately by nlogn i sub l q q 2d where n is the number of strings in the given set q is the probability distribution governing the selection of letters of strings i sub l q q 2d is the lower mutual information between q and q with respect to 2d and d spl ges 0 is the distortion allowed in approximate matching in addition an approximation algorithm is proposed and proved asymptotically optimal

no wait flowshop scheduling is as hard as asymmetric traveling salesman problem in this paper we study the classical no wait flowshop scheduling problem with makespan objective f no wait cmax in the standard three field notation this problem is well known to be a special case of the asymmetric traveling salesman problem atsp and as such has an approximation algorithm with logarithmic performance guarantee in this work we show a reverse connection we show that any polynomial time approximation algorithm for the no wait flowshop scheduling problem with makespan objective implies the existence of a polynomial time 1 approximation algorithm for the atsp for any 0 this in turn implies that all nonapproximability results for the atsp current or future will carry over to its special case in particular it follows that the no wait flowshop problem is apx hard which is the first nonapproximability result for this problem

a constant factor approximation algorithm for the asymmetric traveling salesman problem we give a constant factor approximation algorithm for the asymmetric traveling salesman problem atsp our approximation guarantee is analyzed with respect to the standard lp relaxation and thus

analysis of numerical errors in large eddy simulation we consider the question of numerical errors in large eddy simulation it is often claimed that straightforward discretization and solution using centered methods of models for large eddy motion can simulate the motion of turbulent flows with complexity independent of the reynolds number and dependence only on the resolution delta of the eddies sought this report considers this question analytically is it possible to prove error estimates for discretizations of actually used large eddy models whose error constants depend only on delta but not re we consider the most common simplest and most mathematically tractable model and the most mathematically clear discretization in two cases we prove such an error estimate and try to explain why our technique of proof fails in the most general case our analysis aims to assume as little time regularity on the true solution as possible

well posedness of a reduced order approximate deconvolution turbulence model abstract we prove well posedness for a recently proposed turbulence model and derive an estimate for its modeling error the model is derived from the fourth order representation of the stolz adams approximate deconvolution model by making an additional deconvolution approximation to reduce the model to second order which also allows the model to be computed with efficient and unconditionally stable algorithms

a study on the convergence of homotopy analysis method in this study a reliable approach for convergence of the homotopy analysis method when applied to nonlinear problems is discussed first we present an alternative framework of the method which can be used simply and effectively to handle nonlinear problems then mainly we address the sufficient condition for convergence of the method the convergence analysis is reliable enough to estimate the maximum absolute truncated error of the homotopy series solution the analysis is illustrated by investigating the convergence results for some nonlinear differential equations the study highlights the power of the method

on the homotopy analysis method for nonlinear problems a powerful easy to use analytic tool for nonlinear problems in general namely the homotopy analysis method is further improved and systematically described through a typical nonlinear problem i e the algebraically decaying viscous boundary layer flow due to a moving sheet two rules the rule of solution expression and the rule of coefficient ergodicity are proposed which play important roles in the frame of the homotopy analysis method and simplify its applications in science and engineering an explicit analytic solution is given for the first time with recursive formulas for coefficients this analytic solution agrees well with numerical results and can be regarded as a definition of the solution of the considered nonlinear problem

an o log n log log n approximation algorithm for the asymmetric traveling salesman problem we present a randomized o log n log log n approximation algorithm for the asymmetric traveling salesman problem atsp this provides the first asymptotic improvement over the long standing log n approximation bound stemming from the work of frieze et al 1982 frieze am galbiati g maffioki f 1982 on the worst case performance of some algorithms for the asymmetric traveling salesman problem networks 12 1 23 39 the key ingredient of our approach is a new connection between the approximability of the atsp and the notion of so called thin trees to exploit this connection we employ maximum entropy rounding a novel method of randomized rounding of lp relaxations of optimization problems we believe that this method might be of independent interest

the optimal homotopy asymptotic method for solving blasius equation starting from the reality that many known methods fail in the attempt to obtain analytic solutions of blasius type equations in this work a new procedure namely optimal homotopy asymptotic method oham is proposed to obtain an explicit analytical solution of the blasius problem comparison with howarth s numerical solution as well as the obtained residual reveals that the proposed method is highly accurate this proves the validity and great potential of the proposed procedure oham as a new kind of powerful analytical tool for nonlinear problems

on the integrality ratio for the asymmetric traveling salesman problem we improve the lower bound on the integrality ratio of the held karp bound for asymmetric tsp with triangle inequality from 4 3 to 2

numerical analysis of leray tikhonov deconvolution models of fluid motion this report develops and studies a new family of navier stokes equation regularizations leray tikhonov regularizations with time relaxation models this new family of turbulence models is based on a modification consistent with the large scales of tikhonov lavrentiev regularization with this approach we obtain an approximation of the unfiltered solution by one filtering step we introduce the modified tikhonov deconvolution operator and study its mathematical properties we also perform rigorous numerical analysis of a computationally attractive algorithm for this family of models and present numerical experiments using it

convergence of approximate deconvolution models to the mean magnetohydrodynamics equations analysis of two models abstract we consider two large eddy simulation les models for the approximation of large scales of equations of magnetohydrodynamics mhd in the sequel we study two models which are obtained adapting to the mhd the approach by stolz and adams with van cittert approximate deconvolution operators first we prove the existence and uniqueness of a regular weak solution for a system with filtering and deconvolution in both equations then we study the behavior of solutions as the deconvolution parameter goes to infinity the main result of this paper is the convergence to a solution of the filtered mhd equations next we also study the problem with filtering acting only on the velocity equation

approximation algorithms for asymmetric tsp by decomposing directed regular multigraphs a directed multigraph is said to be d regular if the indegree and outdegree of every vertex is exactly d by hall s theorem one can represent such a multigraph as a combination of at most n2 cycle covers each taken with an appropriate multiplicity we prove that if the d regular multigraph does not contain more than d 2 copies of any 2 cycle then we can find a similar decomposition into n2 pairs of cycle covers where each 2 cycle occurs in at most one component of each pair our proof is constructive and gives a polynomial algorithm to find such a decomposition since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs we also give an alternative simpler algorithm to extract a single such pair this combinatorial theorem then comes handy in rounding a fractional solution of an lp relaxation of the maximum traveling salesman problem tsp problem the first stage of the rounding procedure obtains two cycle covers that do not share a 2 cycle with weight at least twice the weight of the optimal solution then we show how to extract a tour from the 2 cycle covers whose weight is at least 2 3 of the weight of the longest tour this improves upon the previous 5 8 approximation with a simpler algorithm utilizing a reduction from maximum tsp to the shortest superstring problem we obtain a 2 5 approximation algorithm for the latter problem which is again much simpler than the previous one for minimum asymmetric tsp the same technique gives two cycle covers not sharing a 2 cycle with weight at most twice the weight of the optimum assuming triangle inequality we then show how to obtain from this pair of cycle covers a tour whose weight is at most 0 842 log 2 n larger than optimal this improves upon a previous approximation algorithm with approximation guarantee of 0 999 log 2 n other applications of the rounding procedure are approximation algorithms for maximum 3 cycle cover factor 2 3 previously 3 5 and maximum asymmetric tsp with triangle inequality factor 10 13 previously 3 4

a new modification of the homotopy perturbation method for linear and nonlinear operators in this paper we present an efficient modification of the homotopy perturbation method that will facilitate the calculations we then conduct a comparative study between the new modification and the homotopy perturbation method numerical illustrations are investigated to show the features of the technique the modified method accelerates the rapid convergence of the series solution and reduces the size of work the new modification introduces a promising tool for many linear and nonlinear problems

free bits pcps and nonapproximability towards tight results this paper continues the investigation of the connection between probabilistically checkable proofs pcps and the approximability of np optimization problems the emphasis is on proving tight nonapproximability results via consideration of measures such as the free bit complexity and the amortized free bit complexity of proof systems the first part of the paper presents a collection of new proof systems based on a new error correcting code called the long code we provide a proof system that has amortized free bit complexity of 2 epsilon implying that approximating maxclique within n frac13 e and approximating the chromatic number within n frac15 e are hard assuming np neq corp for any e 0 we also derive the first explicit and reasonable constant hardness factors for min vertex cover msat 2 and max cut and we improve the hardness factor for msat 3 we note that our nonapproximability factors for maxsnp problems are appreciably close to the values known to be achievable by polynomial time algorithms finally we note a general approach to the derivation of strong nonapproximability results under which the problem reduces to the construction of certain gadgets the increasing strength of nonapproximability results obtained via the pcp connection motivates us to ask how far this can go and whether pcps are inherent in any way the second part of the paper addresses this the main result is a reversal of the connection due to feige et al fglss connection j acm 43 1996 pp 268 292 where the latter had shown how to translate proof systems for np into np hardness of approximation results for maxclique we show how any np hardness of approximation result for maxclique yields a proof system for np roughly our result says that for any constant f if maxclique is np hard to approximate within n 1 1 f then np subseteq overline fpcp log f the latter being the class of languages possessing proofs of logarithmic randomness and amortized free bit complexity f this suggests that pcps are inherent to obtaining nonapproximability results furthermore the tight relation suggests that reducing the amortized free bit complexity is necessary for improving the nonapproximability results for maxclique the third part of our paper initiates a systematic investigation of the properties of pcp and fpcp free pcp as a function of the following various parameters randomness query complexity free bit complexity amortized free bit complexity proof size etc we are particularly interested in triviality results which indicate which classes are not powerful enough to capture np we also distill the role of randomized reductions in this area and provide a variety of useful transformations between proof checking complexity classes

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

motion planning algorithms for the dubins travelling salesperson problem two motion planning algorithms for the so called dubins travelling salesperson problem are presented and compared via simulations with existing algorithms from the literature the first algorithm dubbed k step look ahead algorithm stems naturally from the formulation of the dubins travelling salesperson problem as a minimum time control problem and is suitable for obtaining short tours when the number of cities is relatively small the second algorithm is an adaptation of the classic 2 opt algorithm for the travelling salesperson problem and can be applied to larger instances of the dubins travelling salesperson problem in this sense the two algorithms complement each other in terms of their range of applicability instead of being decoupled the combinatorial and motion planning aspects of the dubins travelling salesperson problem are treated in an integrated manner by both algorithms and no assumptions are made on the magnitude of the intercity distances

accurate analytical solutions to oscillators with discontinuities and fractional power restoring force by means of the optimal homotopy asymptotic method in this paper a new approach combining the features of the homotopy concept with an efficient computational algorithm which provides a simple and rigorous procedure to control the convergence of the solution is proposed to find accurate analytical explicit solutions for some oscillators with discontinuities and a fractional power restoring force which is proportional to sign x a very fast convergence to the exact solution was proved since the second order approximation lead to very accurate results comparisons with numerical results are presented to show the effectiveness of this method four numerical applications prove the accuracy of the method which works very well for the whole range of initial amplitudes the obtained results prove the validity and efficiency of the method which can be easily extended to other strongly nonlinear problems

numerical approximation of the voigt regularization for incompressible navier stokes and magnetohydrodynamic flows we study voigt regularizations for the navier stokes equations nses and magnetohydrodynamic mhd equations in the presence of physical boundary conditions in particular we develop the first finite element numerical algorithms for these systems prove stability and convergence of the algorithms and test them computationally on problems of practical interest it is found that unconditionally stable implementations of the voigt regularization can be made from a simple change to existing nse and mhd codes and moreover optimal convergence of the developed algorithms solutions to physical solutions can be obtained if lower order mixed finite elements are used finally we show that for several benchmark problems the voigt regularization on a coarse mesh produces good approximations to nse and mhd systems that is the voigt regularization provides accurate reduced order models for nse and mhd flows