relativistic chern simons higgs vortex equations an existence theorem is established for the solutions to the non abelian relativistic chern simons higgs vortex equations over a doubly periodic domain when the gauge group g assumes the most general and important prototype form g su n mathematics subject classification 2010 35c08 35j50 35q70 81e13

central sets and a non commutative roth theorem in 1952 k roth showed that any subset of n having positive upper density contains arithmetic progressions of length 3 a result e szemer edi extended to progressions of arbitrary length in 1975 an approach involving multiple recurrence for measure preserving systems developed by h furstenberg led to extensions to linear configurations in more general groups while demonstrating families of such configurations to be large in various senses using ultrafilter techniques for doing ergodic theory without averaging we prove a version of roth s theorem in its ergodic theoretic formulation for general countable groups g a combinatorial consequence valid in amenable groups is the presence of many configurations of the form a b ag b ag bg in positive density subsets of g g here many implies in particular central in the variable g

symmetric vortices for the ginzberg landau equations of superconductivity and the nonlinear desingularization phenomenon abstract the existence of countably many distinct symmetric vortices for the nonlinear ginzberg landau equations of superconductivity with an arbitrary positive parameter is proven we then prove the existence of a linearization phenomenon as the parameter in this case the linearized equations contain an appropriate dirac delta function these equations are known as the london equations

existence of dyons in minimally gauged skyrme model via constrained minimization abstract we prove the existence of electrically and magnetically charged particle like static solutions known as dyons in the minimally gauged skyrme model developed by brihaye hartmann and tchrakian the solutions are spherically symmetric depend on two continuous parameters and carry unit monopole and magnetic charges but continuous skyrme charge and non quantized electric charge induced from the t hooft electromagnetism the problem amounts to obtaining a finite energy critical point of an indefinite action functional arising from the presence of electricity and the minkowski spacetime signature the difficulty with the absence of the higgs field is overcome by achieving suitable strong convergence and obtaining uniform decay estimates at singular boundary points so that the negative sector of the action functional becomes tractable

the relativistic non abelian chern simons equations we study ther xr system of nonlinear elliptic equations delta u a lambda sum limits b 1 r k ab e u b lambda sum limits b 1 r sum limits c 1 r e u b k ab e u c k bc 4 pi sum limits j 1 n a delta p aj a 1 2 r x r 2 where 0 is a constant parameter k kab is the cartan matrix of a semi simple lie algebra and p is the dirac measure concentrated atp r 2 this system of equations arises in the relativistic non abelian chern simons theory and may be viewed as a nonintegrable deformation of the integrable toda system we establish the existence of a class of solutions known as topological multivortices the crucial step in our method is the use of the decomposition theorem of cholesky for positive definite matrices so that a variational principle can be formulated

a system of elliptic equations arising in chern simons field theory we prove the existence of topological vortices in a relativistic self dual abelian chem simons theory with two higgs particles and two gauge fields through a study of a coupled system of two nonlinear elliptic equations over r 2 we present two approaches to prove existence of solutions on bounded domains via minimization of an indefinite functional and via a fixed point argument we then show that we may pass to the full r 2 limit from the bounded domain solutions to obtain a topological solution in r 2

papr problem for walsh systems and related problems high peak values of transmission signals in wireless communication systems lead to wasteful energy consumption and degradation of several transmission performances we continue the theoretical contributions made by boche and farell toward the understanding of peak value reduction using the strategy known as tone reservation for orthogonal transmission schemes there it was shown that for orthogonal frequency division multiplexing ofdm systems the combinatorial object called arithmetic progression plays an important role in setting limitations for the applicability of the tone reservation method in this paper we show that the combinatorial object introduced as perfect walsh sum pws plays a similar role for code division multiple access cdma systems as arithmetic progression for ofdm systems by specific construction we show that for a chosen numbers m and n all subsets mathcal i of the set n of the first n 2 n natural numbers which has the density in n larger than a given delta in 0 1 i e left mathcal i right n geq delta and which is sufficiently large enough in the sense that left i right geq 2 2 delta 2 m 1 contains a pws of size 2 m by means of this result and motivated by the previously mentioned connection between arithmetic progression and pws we show results for the pws which are analogous to the famous szemeredi theorem on arithmetic progressions conlon gower s theorem on probabilistic construction of sparse sets containing an arithmetic progression and even a solution of an analogon to the erd s conjecture on arithmetic progressions those results give in particular an insight into the asymptotic limitations of tone reservation method for the cdma systems besides we show that a subset i of n is a pws if and only if the embedding inequality of the subspace of l 1 0 1 containing linear combinations of walsh functions indexed by elements of mathcal i holds with the minimum possible embedding constant sqrt left mathcal i right the corresponding approach based in particular by the fact that the pwss are the only walsh sums having unit l 1 norm proven in this paper by means of that results we show that the minimum possible threshold constant for which the tone reservation method is applicable yields sqrt left mathcal i right if and only if the information set mathcal i is a pws

non local boundary integral formulation for softening damage a strongly non local boundary element method bem for structures with strain softening damage treated by an integral type operator is developed a plasticity model with yield limit degradation is implemented in a boundary element program using the initial stress boundary element method with iterations in each load increment regularized integral representations and boundary integral equations are used to avoid the difficulties associated with numerical computation of singular integrals a numerical example is solved to verify the physical correctness and efficiency of the proposed formulation the example consists of a softening strip perforated by a circular hole subjected to tension the strain softening damage is described by a plasticity model with a negative hardening parameter the local formulation is shown to exhibit spurious sensitivity to cell mesh refinements localization of softening damage into a band of single cell width and excessive dependence of energy dissipation on the cell size by contrast the results for the non local theory are shown to be free of these physically incorrect features compared to the classical non local finite element approach an additional advantage is that the internal cells need to be introduced only within the small zone or band in which the strain softening damage tends to localize within the structure copyright 2003 john wiley sons ltd

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

resolution of chern simons higgs vortex equations it is well known that the presence of multiple constraints of non abelian relativisitic chern simons higgs vortex equations makes it difficult to develop an existence theory when the underlying cartan matrix k of the equations is that of a general simple lie algebra and the strongest result in the literature so far is when the cartan subalgebra is of dimension 2 in this paper we overcome this difficulty by implicitly resolving the multiple constraints using a degree theorem argument utilizing a key positivity property of the inverse of the cartan matrix deduced in an earlier work of lusztig and tits which enables a process that converts the equality constraints to inequality constraints in the variational formalism thus this work establishes a general existence theorem that settles a long standing open problem in the field regarding the general solvability of the equations

a multidimensional szemeredi theorem for hardy sequences of different growth abstract we prove a variant of the multidimensional polynomial szemeredi theorem of bergelson and leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some hardy field and satisfy certain growth conditions we do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula a consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences another consequence is that every syndetic set of integers contains certain nonshift invariant patterns and every finite coloring of n with each color class a syndetic set contains certain polychromatic patterns results very particular to our non polynomial setup

independent sets in hypergraphs many important theorems and conjectures in combinatorics such as the the orem of szemeredi on arithmetic progressions and the erd os stone theorem in extremal graph theory can be phrased as statements about families of independent sets in certain uniform hypergraphs in recent years an important trend in the area has been to extend such classical results to the so called sparse random setting this line of research has recently culminated in the breakthroughs of conlon and gowers and of schacht who de veloped general tools for solving problems of this type although these two papers solved very similar sets of longstanding open problems the methods used are very different from one another and have different strengths and weaknesses in this paper we provide a third completely different approach to proving extremal and structural results in sparse random sets that also yields their natural counting coun terparts we give a structural characterization of the independent sets in a large class of uniform hypergraphs by showing that every independent set is almost contained in one of a small number of relatively sparse sets we then derive many interesting results as fairly straightforward consequences of this abstract theorem in particular we prove the well known conjecture of kohayakawa luczak and rodl a probabilistic embedding lemma for sparse graphs for all 2 balanced graphs we also give alternative proofs of many of the results of conlon and gowers and schacht such as sparse random versions of szemeredi s theorem the erd os stone theorem and the erd os simonovits stability theorem and ob tain their natural counting versions which in some cases are considerably stronger we also obtain new results such as a sparse version of the erd os frankl rodl theorem on the number of h free graphs and as a consequence of the k lr conjecture we extend a re sult of rodl and rucinski on ramsey properties in sparse random graphs to the general non symmetric setting similar results have been discovered independently by saxton and thomason

small coupling limit and multiple solutions to the dirichlet problem for yang mills connections in four dimensions ii in this paper we complete the proof of the existence of multiple solutions and in particular non minimal ones to the e dirichlet problem obtained as a variational problem for the su 2 e yang mills functional this is equivalent to proving the existence of multiple solutions to the dirichlet problem for the su 2 yang mills functional with small boundary data in the first paper of this series this non compact variational problem is transformed into the finite dimensional problem of finding the critical points of the function je q which is essentially the yang mills functional evaluated on the approximate solutions constructed via a gluing technique in the present paper we establish a morse theory for je q by means of ljusternik schnirelmann theory thus complete the proofs of theorems 1 3 given by isobe and marini small coupling limit and multiple solutions to the dirichlet problem for yang mills connections in 4 dimensions part i j math phys 53 063706 2012 10 1063 1 4728211

boundary element method for thermoelastoplastic problems in this paper we present the boundary integral formulation for inelastic stationary thermoelasticity including thermally induced non homogeneity generally the unique bem formulation for crack problems is achieved by using the traction boundary integral equations which are now considered in a non singular form both the two and three dimensional boundary value problems are analysed simultaneously numerical examples for 2 d problems are presented in order to illustrate the suitability of this boundary element approach in the inelastic thermoelasticity

rudin shapiro sequences along squares we estimate exponential sums of the form sigma n x f n 2 e nu n for a large class of digital functions f and nu is an element of r we deduce from these estimates the distribution along squares of this class of digital functions which includes the rudin shapiro sequence and some of its generalizations

extremal results in sparse pseudorandom graphs abstract szemeredi s regularity lemma is a fundamental tool in extremal combinatorics however the original version is only helpful in studying dense graphs in the 1990s kohayakawa and rodl proved an analogue of szemeredi s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs many of the key applications of szemeredi s regularity lemma use an associated counting lemma in order to prove extensions of these results which also apply to sparse graphs it remained a well known open problem to prove a counting lemma in sparse graphs the main advance of this paper lies in a new counting lemma proved following the functional approach of gowers which complements the sparse regularity lemma of kohayakawa and rodl allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph we use this to prove sparse extensions of several well known combinatorial theorems including the removal lemmas for graphs and groups the erd s stone simonovits theorem and ramsey s theorem these results extend and improve upon a substantial body of previous work

revisiting the nilpotent polynomial hales jewett theorem abstract answering a question posed by bergelson and leibman in 6 we establish a nilpotent version of the polynomial hales jewett theorem that contains the main theorem in 6 as a special case important to the formulation and the proof of our main theorem is the notion of a relative syndetic set relative with respect to a closed non empty subsets of g 25 as a corollary of our main theorem we prove an extension of the restricted van der waerden theorem to nilpotent groups which involves nilprogressions

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

non singular boundary integral representation of stresses this paper presents the derivation of the non singular integral representation of stresses in two and three dimensional elastostatics in contrast to the strongly singular and weakly singular integral representations the numerical computation of the nearly singular integrals is eliminated because all the integrands are made finite in this new formulation even if the internal point approaches the boundary thus the method gives accurate numerical results even in that portion of a solid which is very close to a discretized boundary three test problems are analysed in which we present a comparison of the accuracies achieved by the numerical computations based on the use of strongly singular weakly singular and non singular integral representations of stresses

dirichlet and neumann boundary value problems for yang mills connections the yang mills equations have been extensively studied on compact riemannian manifolds without boundary a substantial amount of work has also been done on non compact examples with a boundary component at infinity see 5 10 11 14 15 here we define and describe a solution of dirichlet and neumann boundary value problems for yang mills connections on a compact four dimensional manifold with smooth boundary the equations in the interior are the full yang mills equations we do not restrict ourselves to self dual or anti self dual solutions in the case where the structure group in the gauge theory is abelian our results are contained in the hodge de rham theory for manifolds with boundary see 6 9 4 3 in the general case our results describe a non linear version of this theory the base manifold m of the theory is a compact orientable four dimensional manifold with smooth boundary m we assume that m is endowed with a riemannian metric we consider a principal fiber bundle p over m with structure group g a compact lie group which will have a metric invariant under the adjoint action in the usual example this comes from trace the unknown in the problem is the connection a i e a differential form in p ad g where g is the lie algebra of g there are two natural boundary value problems one of them requires fixing the tangential part of the connection a on the bundle restricted to m if m is locally described as x 0 this amounts to prescribing a1 x dx a2 x dx 2 a3 x dx 3 on m the connection is fixed in the sense that a particular choice is fixed in the gauge equivalence class of a on the boundary the yang mills equations are satisfied in the interior this corresponds to the dirichlet problem in the other boundary value problem we consider a is free at the boundary but the normal part of the curvature is set equal to zero we call this the neumann problem for the dirichlet problem we consider a sequence of l1 connections that minimizes the yang mills functional with prescribed admissible tangential boundary data we show that there exists a weak limiting connection a by finding local good gauges the limiting connection is only locally defined we show that a solves the yang mills equations and satisfies the boundary conditions up to

yang mills fields which are not self dual the purpose of this paper is to prove the existence of a new family of non self dual finite energy solutions to the yang mills equations on euclidean four space withsu 2 as a gauge group the approach is that of equivariant geometry attention is restricted to a special class of fields those that satisfy a certain kind of rotational symmetry for which it is proved that 1 a solution to the yang mills equations exists among them and 2 no solution to the self duality equations exists among them the first assertion is proved by an application of the direct method of the calculus of variations existence and regularity of minimizers and the second assertion by studying the symmetry properties of the linearized self duality equations the same technique yields a new family of non self dual solutions on the complex projective plane

intersective polynomials and the polynomial szemeredi theorem abstract let p p 1 p r q n 1 n m be a family of polynomials such that p i z m z i 1 r we say that the family p has the psz property if for any set e z with d e lim sup n m e m n 1 n m 0 there exist infinitely many n z m such that e contains a polynomial progression of the form a a p 1 n a p r n we prove that a polynomial family p p 1 p r has the psz property if and only if the polynomials p 1 p r are jointly intersective meaning that for any k n there exists n z m such that the integers p 1 n p r n are all divisible by k to obtain this result we give a new ergodic proof of the polynomial szemeredi theorem based on the fact that the key to the phenomenon of polynomial multiple recurrence lies with the dynamical systems defined by translations on nilmanifolds we also obtain as a corollary the following generalization of the polynomial van der waerden theorem if p 1 p r q n are jointly intersective integral polynomials then for any finite partition z i 1 k e i of z there exist i 1 k and a n e i such that a a p 1 n a p r n e i

on the efficient computation of the stress components near a closed boundary in plane elasticity by using classical complex boundary integral equations complex boundary integral equations fredholm type regular or cauchy type singular or even hadamard mangler type hypersingular have been used for the numerical solution of general plane isotropic elasticity problems the related muskhelishvili and particularly lauricella sherman equations are famous in the literature but several more extensions of the lauricella sherman equations have also been proposed in this paper it is just mentioned that the stress and displacement components can be very accurately computed near either external or internal simple closed boundaries for anyone of the above equations regular or singular or hypersingular but with a prerequisite their actual numerical solution through the appropriate use of the even more classical elementary cauchy theorem in complex analysis this approach has been already used for the accurate numerical computation of analytic functions and their derivatives by ioakimidis papadakis and perdios bit 1991 31 276 285 without applications to elasticity problems but here the much more complicated case of the elastic complex potentials is studied even when just an appropriate non analytic complex density function such as an edge dislocation loading distribution density is numerically available on the boundary the present results are also directly applicable to inclusion problems anisotropic elasticity antiplane elasticity and classical two dimensional fluid dynamics but unfortunately not to crack problems in fracture mechanics brief numerical results for the complex potentials showing the dramatic increase of the computational accuracy are also displayed and few generalizations proposed copyright 2000 john wiley sons ltd

proof of the julia zee theorem it is a well accepted principle that finite energy static solutions in the classical relativistic gauge field theory over the 2 1 dimensional minkowski spacetime must be electrically neutral we call such a statement the julia zee theorem in this paper we present a mathematical proof of this fundamental structural property

a symmetric family of yang mills fields we examine a family of finite energyso 3 yang mills connections overs4 indexed by two real parameters this family includes both smooth connections when both parameters are odd integers and connections with a holonomy singularity around 1 or 2 copies ofrp2 these singular ym connections interpolate between the smooth solutions depending on the parameters the curvature may be self dual anti self dual or neither for the anti self dual connections we compute the formal dimension of the moduli space for the non self dual connections we examine the second variation of the yang mills functional and count the negative and zero eigenvalues each component of the non self dual moduli space appears to consist only of conformal copies of a single solution

the ergodic theoretical proof of szemeredi s theorem partial results were obtained previously by k f roth 1952 who established the existence of arithmetic progressions of length three in subsets of z of positive upper density and by e szemeredi 1969 who proved the existence of progressions of length four in 1976 furstenberg noticed that the statement of theorem i is equivalent to a statement about multiple recurrence of measure preserving transformations namely

existence theorems for non abelian chern simons higgs vortices with flavor abstract in this paper we establish the existence of vortex solutions for a chern simons higgs model with gauge group su n u 1 and flavor su n these symmetries ensure the existence of genuine non abelian vortices through a color flavor locking under a suitable ansatz we reduce the problem to a 2 2 system of nonlinear elliptic equations with exponential terms we study this system over the full plane and over a doubly periodic domain respectively for the planar case we use a variational argument to establish the existence result and derive the decay estimates of the solutions over the doubly periodic domain we show that the system admits at least two gauge distinct solutions carrying the same physical energy by using a constrained minimization approach and the mountain pass theorem in both cases we get the quantized vortex magnetic fluxes and electric charges

doubly periodic vortices for a chern simons model abstract in this paper we establish the existence of vortex solutions for a relativistic self dual u 1 u 1 chern simons system over a doubly periodic domain we use a constrained minimization approach to obtain the existence result the system admits a solution when the chern simons coupling parameter 0 is sufficiently small while no solution exists when 0 is not sufficiently small and we also get the quantized integral

a new proof of the density hales jewett theorem the hales jewett theorem asserts that for every r and every k there exists n such that every r colouring of the n dimensional grid 1 k contains a monochromatic combinatorial line this result is a generalization of van der waerden s theorem and it is one of the fundamental results of ramsey theory the theorem of van der waerden has a famous density version conjectured by erd s and turan in 1936 proved by szemeredi in 1975 and given a different proof by furstenberg in 1977 the hales jewett theorem has a density version as well proved by furstenberg and katznelson in 1991 by means of a significant extension of the ergodic techniques that had been pioneered by furstenberg in his proof of szemeredi s theorem in this paper we give the first elementary proof of the theorem of furstenberg and katznelson and the first to provide a quantitative bound on how large n needs to be in particular we show that a subset of 1 2 3 of density contains a combinatorial line if n is at least as big as a tower of 2s of height o 1 our proof is surprisingly simple indeed it gives arguably the simplest known proof of szemeredi s theorem

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

the primes contain arbitrarily long polynomial progressions we establish the existence of infinitely many polynomial progressions in the primes more precisely given any integer valued polynomials p 1 p k z m in one unknown m with p 1 0 p k 0 0 and given any e 0 we show that there are infinitely many integers x and m with 1 leqslant m leqslant x varepsilon such that x p 1 m x p k m are simultaneously prime the arguments are based on those in 18 which treated the linear case p j j 1 m and e 1 the main new features are a localization of the shift parameters and the attendant gowers norm objects to both coarse and fine scales the use of pet induction to linearize the polynomial averaging and some elementary estimates for the number of points over finite fields in certain algebraic varieties

a framework for morse theory for the yang mills functional on explique comment la theorie de morse pour la fonctionnelle de yang mills peut se retrouver en examinant la restriction de la fonctionnelle a un ensemble denombrable de varietes non compactes de dimension finie

a self adaptive co ordinate transformation for efficient numerical evaluation of general boundary element integrals almost all general purpose boundary element computer packages include a curved geometry modelling capability thus numerical quadrature schemes play an important role in the efficiency of programming the technique the present work discusses this problem in detail and introduces efficient means of computing singular or nearly singular integrals currently found in two dimensional axisymmetric and three dimensional applications emphasis is given to a new third degree polynomial transformation which was found greatly to improve the accuracy of gaussian quadrature scheme s within the near singularity range the procedure can easily be implemented into existing be codes and presents the important feature of being self adaptive i e it produces a variable lumping of the gauss stations toward the singularity depending on the minimum distance from the source point to the element the self adaptiveness of the scheme also makes it inactive when not useful large source distances which makes it very safe for general usage

topological multivortex solutions of the self dual maxwell chern simons higgs system abstract we study the existence and various behaviors of topological multivortex solutions of the relativistic self dual maxwell chern simons higgs system we first prove the existence of general topological solutions by applying variational methods to the newly discovered minimizing functional then by an iteration method we prove the existence of topological solutions satisfying some extra conditions which we call admissible solutions we establish asymptotic exponential decay estimates for these topological solutions we also investigate the limiting behavior of the admissible solutions as parameters in our system go to some limits for the abelian higgs limit we obtain strong convergence result while for the chern simons limit we only obtained that our admissible solutions weakly approximate one of the chern simons solutions

existence of charged vortices in a maxwell chern simons model abstract in this paper we prove the existence of charged vortex solitons in a maxwell chern simons model we establish the main existence theorem by a constrained minimization method applied on an indefinite action functional which is induced from the original field theoretical lagrangian we also show that the solutions obtained are smooth

topological multiple recurrence for polynomial configurations in nilpotent groups abstract we establish a general multiple recurrence theorem for an action of a nilpotent group by homeomorphisms of a compact space this theorem can be viewed as a nilpotent version of our recent polynomial hales jewett theorem ann math 150 1999 33 and contains nilpotent extensions of many known abelian results as special cases

the geometry of the yang mills moduli space for definite manifolds on etudie la geometrie riemannienne de l espace des modules des connexions auto duales su 2 k 1 sur une classe tres generale de 4 varietes des 4 varietes compactes orientees simplement connexe m a forme d intersection definie positive

existence theorems for a general 2 2 non abelian chern simons higgs system over a torus in this paper we study a general 2 2 non abelian chern simons higgs system of the form ui 1e2 j 12kjieuj j 12 k 12kkjkjieujeuk 4 j 1ni pij x i 1 2 over a flat 2 torus t2 where e 0 p is the dirac measure at p ni n i 1 2 k is a non degenerate 2 2 matrix of the form k 1 a a b1 b which may cover the physically interesting case when k is a cartan matrix of a rank 2 semisimple lie algebra concerning the existence results of this type system over t2 usually in the literature there is a requirement that a b 0 however it is an open problem so far for the solvability about such system with a b 0 such that this system admits a solution over the torus if 0 e e0 provided a b are suitably small furthermore if ab 0 in addition with suitable condition on a b n1 n2 this system admits a mountain pass solution our argument is based on a perturbation approach and the mountain pass lemma

extremal results for random discrete structures we study thresholds for extremal properties of random discrete structures we determine the threshold for szemer edi s theorem on arithmetic progressions in random subsets of the integers and its multidimensional extensions and we determine the threshold for tur an type problems for random graphs and hypergraphs in particular we verify a conjecture of

non singular boundary integral representation of potential field gradients in this paper we derive the non singular boundary integral representation of the field gradients for two dimensional problems of classical potential field theory numerical implementation of this representation is developed too the proposed method eliminates the most inaccurate influence coefficients which arise when singular integral representations are used and the internal point approaches the boundary since the integrands in this new method are finite at any internal point accurate numerical results are achieved even in that portion of a solid which is very close to a discretized boundary two test problems are analysed in which the numerical results computed by strongly singular weakly singular and non singular integral representations are compared mutually and with exact solutions

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

the existence of nonminimal solutions to the yang mills equation with group rm su 2 on s 2 times s 2 and s 1 times s 3

the number of c2l free graphs abstract one of the most basic questions one can ask about a graph h is how many h free graphs on n vertices are there for non bipartite h the answer to this question has been well understood since 1986 when erd s frankl and rodl proved that there are 2 1 o 1 ex n h such graphs for bipartite graphs however much less is known even the weaker bound 2 o ex n h has been proven in only a few special cases for cycles of length four and six and for some complete bipartite graphs for even cycles bondy and simonovits proved in the 1970s that ex n c 2 l o n 1 1 l and this bound is conjectured to be sharp up to the implicit constant in this paper we prove that the number of c 2 l free graphs on n vertices is at most 2 o n 1 1 l confirming a conjecture of erd s our proof uses the hypergraph container method which was developed recently and independently by balogh morris and samotij and by saxton and thomason together with a new balanced supersaturation theorem for even cycles we moreover show that there are at least 2 1 c ex n c 6 c 6 free graphs with n vertices for some c 0 and infinitely many values of n n disproving a well known and natural conjecture as a further application of our method we essentially resolve the so called turan problem on the erd s renyi random graph g n p for both even cycles and complete bipartite graphs

non self dual yang mills connections with quadrupole symmetry we prove the existence of non self dual yang mills connections onsu 2 bundles over the four sphere specifically on all bundles with second chern number not equal 1 we study connections equivariant under ansu 2 symmetry group to reduce the effective dimensionality from four to one and then use variational techniques the existence of non self dualsu 2 ym connections on the trivial bundle second chern number equals zero has already been established by sibner sibner and uhlenbeck via different methods

multiplicative richness of additively large sets in zd abstract in their proof of the ip szemeredi theorem a far reaching extension of the classic theorem of szemeredi on arithmetic progressions furstenberg and katznelson 14 introduced an important class of additively large sets called ip r sets which underlies recurrence aspects in dynamics and is instrumental to enhanced formulations of combinatorial results the authors recently showed that additive ip r subsets of z d are multiplicatively rich with respect to every multiplication on z d without zero divisors e g multiplications induced by degree d number fields in this paper we explain the relationships between classes of multiplicative largeness with respect to different multiplications on z d we show for example that in contrast to the case for z there are infinitely many different notions of multiplicative piecewise syndeticity for subsets of z d when d 2 this is accomplished by using the associated algebra representations to prove the existence of sets which are large with respect to some multiplications while small with respect to others in the process we give necessary and sufficient conditions for a linear transformation to preserve a class of multiplicatively large sets one consequence of our results is that additive ip r sets are multiplicatively rich in infinitely many genuinely different ways we conclude by cataloging a number of sources of additive ip r sets from combinatorics and dynamics

solutions to yang mills equations that are not self dual abstract the yang mills functional for connections on principle su 2 bundles over s4 is studied critical points of the functional satisfy a system of second order partial differential equations the yang mills equations if in particular the critical point is a minimum it satisfies a first order system the self dual or anti self dual equations here we exhibit an infinite number of finite action nonminimal unstable critical points they are obtained by constructing a topologically nontrivial loop of connections to which min max theory is applied the construction exploits the fundamental relationship between certain invariant instantons on s4 and magnetic monopoles on h3 this result settles a question in gauge field theory that has been open for many years

an improved exponential transformation for nearly singular boundary element integrals in elasticity problems abstract this paper presents an improved exponential transformation for nearly singular boundary element integrals in elasticity problems the new transformation is less sensitive to the position of the projection point compared with the original transformation in our work the conventional distance function is modified into a new form in the polar coordinate system based on the refined distance function an improved exponential transformation is proposed in the polar coordinate system moreover to perform integrations on irregular elements an adaptive integration scheme considering both the element shape and the projection point associated with the improved transformation is proposed furthermore when the projection point is located outside the integration element another nearest point is introduced to subdivide the integration elements into triangular or quadrilateral patches of fine shapes numerical examples are presented to verify the proposed method results demonstrate the accuracy and efficiency of our method

on a sharp sobolev type inequality on two dimensional compact manifolds motivated by the asymptotic analysis of double vortex condensates in the chern simons higgs theory we construct a suitable minimizing sequence for a sharp sobolev inequality a la moser for two dimensional compact manifolds as a consequence we first obtain a direct proof of the sharp character of such an inequality secondly and more interestingly we use such minimizing sequence to show that for the flat torus the corresponding extremal problem attains its infimum

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

blow up at infinity in the su 3 chern simons model part i abstract we consider non topological solutions of a nonlinear elliptic system problem see 1 4 below derived from the s u 3 chern simons models in r 2 the existence of non topological solutions even for radial symmetric case has been a long standing open problem recently choe kim and lin in 7 8 showed the existence of radial symmetric non topological solution when the vortex points collapse however the arguments in 7 8 cannot work for an arbitrary configuration of vortex points in this paper we develop a new approach by using different scalings for different components of the system to construct a family of non topological solutions which blows up at infinity

a general algorithm for evaluating nearly singular integrals in anisotropic three dimensional boundary element analysis abstract this paper presents a new general method for the evaluation of nearly singular boundary element integrals arising in anisotropic three dimensional 3d boundary element analysis it is shown that the original nearly singular integrals can be transformed into using a sinh function an element by element sum of regular integrals each one expressed in terms of intrinsic local coordinates as a consequence the actual computation can be performed by using standard n n gaussian quadrature and the procedure can be easily included in any existing computer code this new method has full generality and therefore can be applied to a wide class of integrals the numerical results demonstrate the accuracy and efficiency of the method along with its insensitivity to the location of the nearly singular points it is shown that several orders of magnitude improvement in relative errors can be obtained using this transformation when compared to a straightforward implementation of gaussian quadrature

hypergraph regularity and the multidimensional szemeredi theorem we prove analogues for hypergraphs of szemeredi s regularity lemma and the associated counting lemma for graphs as an application we give the first combinatorial proof of the multidimensional szemeredi theorem of furstenberg and katznelson and the first proof that provides an explicit bound similar results with the same consequences have been obtained independently by nagle rodl schacht and skokan

combinatorial theorems in sparse random sets we develop a new technique that allows us to show in a unified way that many well known combinatorial theorems including turan s theorem szemeredi s theorem and ramsey s theorem hold almost surely inside sparse random sets for instance we extend turan s theorem to the random setting by showing that for every 0 and every positive integer t 3 there exists a constant c such that if g is a random graph on n vertices where each edge is chosen independently with probability at least cn 2 t 1 then with probability tending to 1 as n tends to infinity every subgraph of g with at least 1 1 t 1 e g edges contains a copy of kt this is sharp up to the constant c we also show how to prove sparse analogues of structural results giving two main applications a stability version of the random turan theorem stated above and a sparse hypergraph removal lemma many similar results have recently been obtained independently in a different way by schacht and by friedgut rodl and schacht

an immersed boundary element method for level set based topology optimization summary in this paper we propose a new bem for level set based topology optimization in the proposed bem the nodal coordinates of the boundary element are replaced with the nodal level set function and the nodal coordinates of the eulerian mesh that maintains the level set function because this replacement causes the nodal coordinates of the boundary element to disappear the boundary element mesh appears to be immersed in the eulerian mesh therefore we call the proposed bem an immersed bem the relationship between the nodal coordinates of the boundary element and the nodal level set function of the eulerian mesh is clearly represented and therefore the sensitivities with respect to the nodal level set function are strictly derived in the immersed bem furthermore the immersed bem completely eliminates grayscale elements that are known to cause numerical difficulties in topology optimization by using the immersed bem we construct a concrete topology optimization method for solving the minimum compliance problem we provide some numerical examples and discuss the usefulness of the constructed optimization method on the basis of the obtained results copyright 2012 john wiley sons ltd

a novel boundary element approach for solving the 2d elasticity problems abstract the presentation is mainly devoted to the research on the regularized bem formulations with indirect unknowns for two dimensional 2d elasticity problems a novel regularization technique in which regularized forms of the gradient equations without involving the direct calculation of cpv and hfp integrals are derived and shown to be independent of displacement equations is pursued in this paper after that a numerically systematic scheme with generality is established by adopting quadratic lagrange s elements moreover considering the special geometric domain such as circular or elliptic arcs a new boundary geometric approximate technique named as exact elements is presented and thus by the utilization of these elements the error of the results will arise mainly from the approximation of boundary quantities numerical examples show that a better precision and high computational efficiency can be achieved

a refinement of the cameron erdős conjecture in this paper we study sum free subsets of the set 1 n that is subsets of the first n positive integers which contain no solution to the equation x y z and erdos conjectured in 1990 that the lumber of such sets is o 2 n 2 this conjecture was confirmed by green and independently by sapozhenko here we prove a refined version of their theorem by showing that the number of sum free subsets of n of size in is 2 o n m n 2 for every 1 to 17 21 for m root t this result is sharp up to the constant implicit in the oh our proof uses a general bound on the number of independent sets of size to in 3 uniform hypergraphs proved recently by the authors and new bounds on the number of integer partitions with small sumset