on the liouville type theorems for self similar solutions to the navier stokes equations we prove liouville type theorems for the self similar solutions to the navier stokes equations one of our results generalizes the previous ones by necas ru icka sverak and tsai using a liouville type theorem we also remove a scenario of asymptotically self similar blow up for the navier stokes equations with the profile belonging to l p infty mathbb r 3 with p frac 3 2

on the singular set in the navier stokes equations we consider suitably weak solutions u p to the incompressible navier stokes equations and under various assumptions on u obtain estimates for the size of its singular set one of our results improves a well known theorem of caffarelli kohn and nirenberg

arctic octahedron in three dimensional rhombus tilings and related integer solid partitions three dimensional integer partitions provide a convenient representation of codimension one three dimensional random rhombus tilings calculating the entropy for such a model is a notoriously difficult problem we apply transition matrix monte carlo simulations to evaluate their entropy with high precision we consider both free and fixed boundary tilings our results suggest that the ratio of free and fixed boundary entropies is free fixed 3 2 and can be interpreted as the ratio of the volumes of two simple nested polyhedra this finding supports a conjecture by linde moore and nordahl concerning the arctic octahedron phenomenon in three dimensional random tilings

estimating the asymptotics of solid partitions we study the asymptotic behavior of solid partitions using transition matrix monte carlo simulations if p 3 n denotes the number of solid partitions of an integer n we show that lim n rightarrow infty n 3 4 log p 3 n sim 1 822 pm 0 001 this shows clear deviation from the value 1 7898 attained by macmahon numbers m 3 n that was conjectured to hold for solid partitions as well in addition we find estimates for other sub leading terms in log p 3 n in a pattern deviating from the asymptotics of line and plane partitions we need to add an oscillatory term in addition to the obvious sub leading terms the period of the oscillatory term is proportional to n 1 4 the natural scale in the problem this new oscillatory term might shed some insight into why partitions in dimensions greater than two do not admit a simple generating function

algebraic approximants a new method of series analysis the authors propose a new method of series analysis in which the available series coefficients are fitted to an algebraic equation such a proposal restricts solutions to have algebraic singularities with rational critical exponents the advantage of the method over the method of differential approximants is that in those cases where the two methods are exact the algebraic approximant method usually though not invariably requires far fewer series coefficients for its exact specification and hence discovery the method is therefore potentially valuable in the study of two dimensional critical systems in particular

efficient multiple range random walk algorithm to calculate the density of states we present a new monte carlo algorithm that produces results of high accuracy with reduced simulational effort independent random walks are performed concurrently or serially in different restricted ranges of energy and the resultant density of states is modified continuously to produce locally flat histograms this method permits us to directly access the free energy and entropy is independent of temperature and is efficient for the study of both 1st order and 2nd order phase transitions it should also be useful for the study of complex systems with a rough energy landscape

solution of the dimer problem on a hexagonal lattice with boundary the dimer problem is solved exactly for a hexagonal lattice with general boundary using a known generating function from the theory of partitions it is shown that the leading term in the entropy depends on the shape of the boundary

on ultrametric 1 median selection abstract consider the problem of finding a point in a finite ultrametric space with the minimum average distance to all points we give this problem a monte carlo o log 2 1 3 time 1 approximation algorithm with success probability 1 o for all 0

asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals there is an exact one to one correspondence between the number of d 1 dimensional partitions of an integer and the number of directed compact lattice animals in d dimensions using enumeration techniques we obtain upper and lower bounds for the number of multidimensional partitions both restricted and unrestricted we show that asymptotically the number of unrestricted d 1 dimensional partitions of an integer n goes as exp we also show that for restricted partitions in d 1 dimensions with j dimensions finite 0 j d 1 this number goes as where is the extent of the lattice along the kth axis

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

data stability in clustering a closer look we consider the model introduced by bilu and linial 2010 13 who study problems for which the optimal clustering does not change when distances are perturbed they show that even when a problem is np hard it is sometimes possible to obtain efficient algorithms for instances resilient to certain multiplicative perturbations e g on the order of o n for max cut clustering awasthi et al 2012 6 consider center based objectives and balcan and liang 2012 9 analyze the k median and min sum objectives giving efficient algorithms for instances resilient to certain constant multiplicative perturbations here we are motivated by the question of to what extent these assumptions can be relaxed while allowing for efficient algorithms we show there is little room to improve these results by giving np hardness lower bounds for both the k median and min sum objectives on the other hand we show that constant multiplicative resilience parameters can be so strong as to make the clustering problem trivial leaving only a narrow range of resilience parameters for which clustering is interesting we also consider a model of additive perturbations and give a correspondence between additive and multiplicative notions of stability our results provide a close examination of the consequences of assuming stability in data

a nearly linear time approximation scheme for the euclidean k median problem this paper provides a randomized approximation scheme for the k median problem when the input points lie in the d dimensional euclidean space the worst case running time is o 2 o log 1 epsilon varepsilon d 1 n log d 6 n which is nearly linear for any fixed varepsilon and d moreover our method provides the first polynomial time approximation scheme for and uncapacitated facility location instances in d dimensional euclidean space for any fixed d 2 our work extends techniques introduced originally by arora for the euclidean traveling salesman problem tsp to obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions the theorem is based on an adaptive decomposition scheme which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition we believe that our methodology is of independent interest and may find applications to further geometric problems

on the number of singular points of weak solutions to the navier stokes equations we consider a suitable weak solution to the three dimensional navier stokes equations in the space time cylinder 0 t let be the set of singular points for this solution and t x t for a given open subset and for a given moment of time t 0 t we obtain an upper bound for the number of points of the set t 2001 john wiley sons inc

on the asymptotics of higher dimensional partitions we conjecture that the asymptotic behavior of the numbers of solid three dimensional partitions is identical to the asymptotics of the three dimensional macmahon numbers evidence is provided by an exact enumeration of solid partitions of all integers 68 whose numbers are reproduced with surprising accuracy using the asymptotic formula with one free parameter and better accuracy on increasing the number of free parameters we also conjecture that similar behavior holds for higher dimensional partitions and provides some preliminary evidence for four and five dimensional partitions

flip dynamics in three dimensional random tilings we study single flip dynamics in sets of three dimensional rhombus tilings with fixed polyhedral boundaries this dynamics is likely to be slowed down by so called cycles such structures arise when tilings are encoded via the partition on tiling method and are susceptible to break connectivity by flips or at least ergodicity because they locally suppress a significant amount of flip degrees of freedom we first address the so far open question of the connectivity of tiling sets by elementary flips we prove exactly that sets of tilings of codimension 1 and 2 are connected for any dimension and tiling size for higher codimension tilings of dimension 3 the answer depends on the precise choice of the edge orientations which is a non trivial issue in most cases we can prove connectivity despite the existence of cycles in the few remaining cases among which one is the icosahedral symmetry the question remains open we also study numerically flip assisted diffusion to explore the possible effects of the previously mentioned cycles cycles do not seem to slow down significantly the dynamics at least as far as self diffusion is concerned

interior regularity criteria for suitable weak solutions of the navier stokes equations we present new interior regularity criteria for suitable weak solutions of the 3 d navier stokes equations a suitable weak solution is regular near an interior point z if either the scaled l p q x t norm of the velocity with 3 p 2 q 2 1 q or the l p q x t norm of the vorticity with 3 p 2 q 3 1 q or the l p q x t norm of the gradient of the vorticity with 3 p 2 q 4 1 q 1 p is sufficiently small near z

dimer problem on the cylinder and torus institute of mathematics academia sinica nsfc 10771086 10671162 fujian province university nsc nsc96 2115 m 001 005

some results on approximate 1 median selection in metric spaces a 1 median in a finite metric space is a point with the minimum average distance to all other points given a positive integer n and oracle access to a distance metric on 1 2 n we study the problem of finding a 1 median in particular we show the nonexistence of 1 deterministic o 1 approximation o n query algorithms 2 deterministic email protected 1 approximation o n 2 query algorithms for graph metrics 3 deterministic email protected 1 approximation o n 2 query algorithms and 4 monte carlo email protected 1 approximation o n query algorithms with an w 1 probability of success we also show a monte carlo email protected approximation o log 2 1 e e 3 query algorithm with a 1 o e probability of success where email protected 0 1

an algorithm to generate exactly once every tiling with lozenges of a domain we first show that the tilings of a domain d form a lattice using the same kind of arguments as in research report no 1999 25 which we then undertake to decompose and generate without any redundance to this end we study extensively the relatively simple case of hexagons and their deformations we show that general domains can be broken up into hexagon like parts finally we give an algorithm to generate exactly once every element in the lattice of the tilings of a general domain d

ramsey theory integer partitions and a new proof of the erdős szekeres theorem abstract let h be a k uniform hypergraph whose vertices are the integers 1 n we say that h contains a monotone path of length n if there are x 1 x 2 x n k 1 so that h contains all n edges of the form x i x i 1 x i k 1 let n k q n be the smallest integer n so that every q coloring of the edges of the complete k uniform hypergraph on n vertices contains a monochromatic monotone path of length n while the study of n k q n for specific values of k and q goes back implicitly to the seminal 1935 paper of erd s and szekeres the problem of bounding n k q n for arbitrary k and q was studied by fox pach sudakov and suk our main contribution here is a novel approach for bounding the ramsey type numbers n k q n based on establishing a surprisingly tight connection between them and the enumerative problem of counting high dimensional integer partitions some of the concrete results we obtain using this approach are the following we show that for every fixed q we have n 3 q n 2 n q 1 thus resolving an open problem raised by fox et al we show that for every k 3 n k 2 n 2 2 2 o 1 n where the height of the tower is k 2 thus resolving an open problem raised by elias and matousek we give a new pigeonhole proof of the erd s szekeres theorem on cups vs caps similar to seidenberg s proof of the erd s szekeres lemma on increasing decreasing subsequences

critical endpoint behavior in an asymmetric ising model application of wang landau sampling to calculate the density of states using the wang landau sampling method with a two dimensional random walk we determine the density of states for an asymmetric ising model with two and three body interactions on a triangular lattice in the presence of an external field with an accurate density of states we were able to map out the phase diagram accurately and perform quantitative finite size analyses at and away from the critical endpoint we observe a clear divergence of the curvature of the spectator phase boundary and of the magnetization coexistence diameter derivative at the critical endpoint and the exponents for both divergences agree well with previous theoretical predictions

backward uniqueness for parabolic equations it is shown that a function u satisfying t u m u u u x t mem x 2 in n br 0 t and u x 0 0 for x n br must vanish identically in n br 0 t

on regularity and singularity for l 0 t l3 w r3 solutions to the navier stokes equations we study local regularity properties of a weak solution u to the cauchy problem of the incompressible navier stokes equations we present a new regularity criterion for the weak solution u satisfying the condition l infty 0 t l 3 w mathbb r 3 without any smallness assumption on that scale where l 3 w mathbb r 3 denotes the standard weak lebesgue space as an application we conclude that there are at most a finite number of blowup points at any singular time t

numerical estimation of the asymptotic behaviour of solid partitions of an integer the number of solid partitions of a positive integer is an unsolved problem in combinatorial number theory in this paper solid partitions are studied numerically by the method of exact enumeration for integers up to 50 and by monte carlo simulations using wang landau sampling method for integers up to 8000 it is shown that limn ln p3 n n3 4 1 79 0 01 where p3 n is the number of solid partitions of the integer n this result strongly suggests that the macmahon conjecture for solid partitions though not exact could still give the correct leading asymptotic behaviour

an alternative view on quasicrystalline random tilings we apply a framework for the description of random tilings without height representation which was proposed recently to the special case of quasicrystalline random tilings two important examples are discussed thereby demonstrating the consistency of this alternative description with the conventional one we also clarify the latter by deriving a group theoretic criterion for the validity of the first random tiling hypothesis

random tilings of high symmetry i mean field theory we study random tiling models in the limit of high rotational symmetry in this limit a mean field theory yields reasonable predictions for the configurational entropy of free boundary rhombus tilings in two dimensions we base our mean field theory on an iterative tiling construction inspired by the work of de bruijn in addition to the entropy we consider correlation functions phason elasticity and the thermodynamic limit tilings of dimension other than two are considered briefly

clustering through continuous facility location problems we consider the continuous facility location problem conflp given a finite set of clients c r d and a number f r conflp consists in opening a set f r d of facilities each at cost f and connecting each client to an open facility the objective is to minimize the costs of opening facilities and connecting clients we reduce conflp to the standard facility location problem flp by using the so called approximate center sets this reduction preserves the approximation except for an error e and implies 1 488 e and 2 04 e approximations when the connection cost is given by the euclidean distance and the squared euclidean distance respectively moreover we obtain approximate center sets for the case that the connection cost is the th power of the euclidean distance achieving approximations for the corresponding problems for any 1 as a byproduct we also obtain a polynomial time approximation scheme for the k median problem with this cost function for any fixed k

motivic degree zero donaldson thomas invariants given a smooth complex threefold x we define the virtual motive operatorname hilb n x operatorname vir of the hilbert scheme of n points on x in the case when x is calabi yau operatorname hilb n x operatorname vir gives a motivic refinement of the n point degree zero donaldson thomas invariant of x the key example is x 3 where the hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety and its virtual motive is defined in terms of the denef loeser motivic nearby fiber a crucial technical result asserts that if a function is equivariant with respect to a suitable torus action its motivic nearby fiber is simply given by the motivic class of a general fiber this allows us to compute the generating function of the virtual motives operatorname hilb n mathbb c 3 operatorname vir via a direct computation involving the motivic class of the commuting variety we then give a formula for the generating function for arbitrary x as a motivic exponential generalizing known results in lower dimensions the weight polynomial specialization leads to a product formula in terms of deformed macmahon functions analogous to gottsche s formula for the poincare polynomials of the hilbert schemes of points on surfaces

polynomial time approximation schemes for euclidean traveling salesman and other geometric problems we present a polynomial time approximation scheme for euclidean tsp in fixed dimensions for every fixed c 1 and given any n nodes in r 2 a randomized version of the scheme finds a 1 1 c approximation to the optimum traveling salesman tour in o n log n o c time when the nodes are in r d the running time increases to o n log n o d c d 1 for every fixed c d the running time is n poly log n that is nearly linear in n the algorithmm can be derandomized but this increases the running time by a factor o n d the previous best approximation algorithm for the problem due to christofides achieves a 3 2 aproximation in polynomial time we also give similar approximation schemes for some other np hard euclidean problems minimum steiner tree k tsp and k mst the running times of the algorithm for k tsp and k mst involve an additional multiplicative factor k the previous best approximation algorithms for all these problems achieved a constant factor approximation we also give efficient approximation schemes for euclidean min cost matching a problem that can be solved exactly in polynomial time all our algorithms also work with almost no modification when distance is measured using any geometric norm such as e p for p 1 or other minkowski norms they also have simple parallel i e nc implementations

enumeration of octagonal tilings random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms of particular interest is the tiling entropy density which measures the relation of the number of distinct tilings to the number of constituent tiles tilings by squares and 45 rhombi receive special attention as presumably the simplest model that has not yet been solved exactly in the thermodynamic limit however an exact enumeration formula can be evaluated for tilings in finite regions with fixed boundaries we implement this algorithm in an efficient manner enabling the investigation of larger regions of parameter space than previously were possible our new results appear to yield monotone increasing and decreasing lower and upper bounds on the fixed boundary entropy density that converge toward s 0 36021 3

linear time approximation schemes for clustering problems in any dimensions we present a general approach for designing approximation algorithms for a fundamental class of geometric clustering problems in arbitrary dimensions more specifically our approach leads to simple randomized algorithms for the k means k median and discrete k means problems that yield 1 e approximations with probability 1 2 and running times of o 2 k e o 1 dn these are the first algorithms for these problems whose running times are linear in the size of the input nd for n points in d dimensions assuming k and e are fixed our method is general enough to be applicable to clustering problems satisfying certain simple properties and is likely to have further applications