mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

a dna based solution to the graph isomorphism problem using adleman lipton model with stickers since the experimental demonstration of its feasibility dna based computing has been applied to a number of decision or combinatorial optimization problems the graph isomorphism problem belongs to the class of np problems and has been conjectured intractable although probably not np complete in this paper we demonstrate the power of dna based computing by showing the graph isomorphism problem can be efficiently solved under this computation model by generating the solution space using stickers we present dna based algorithms to solve the problem using polynomial number of basic biological operations

a dna based graph encoding scheme with its applications to graph isomorphism problems abstract feynman first proposed dna based computation in 1961 but his idea was not implemented by experiment for a few decades by properly manipulating dna strands as the input instance of the hamiltonian path problem adleman succeeded in solving the problem in a test tube since the experimental demonstration of its feasibility dna based computing has been applied to a number of decision or combinatorial optimization problems in this paper we propose a dna based graph encoding scheme which can be used to solve some intractable graph problems such as the subgraph isomorphism problem and its generalized problem the maximum common subgraph problem which are known to be np complete problems in the adleman lipton model using polynomial number of basic biological operations

a computational dna solution approach for the quadratic diophantine equation abstract we present a fast parallel algorithm based on stickers for the dna computing model to solve the classical mathematical quadratic diophantine equation ax 2 by c with a b c x y z this problem also in a decision form is np complete and has no known polynomial time solving within the deterministic turing machine model

mappings of finite distortion discreteness and openness we establish a sharp integrability condition on the partial derivatives of a mapping with l p integrable distortion for some p n 1 to guarantee discreteness and openness we also show that a mapping with exponentially integrable distortion and integrable jacobian determinant is either constant or both discrete and open we give an example demonstrating the preciseness of our criterion

analysis of ray tube area in geometrical shock dynamics abstract we provide a rigorous analysis of structure of a ray tube in geometrical shock dynamics our aim is to shed more lights on the cross sectional area function of a ray tube we have shown that for a given ray tube and a given initial value of cross sectional area then its cross sectional area can be determined uniquely everywhere we give a definition of cross sectional area of a ray tube in precise mathematical terms and from that we derive a set of relations each describing the cross sectional area for an arbitrary ray tube in geometrical shock dynamics we have shown that from our results one can deduce whitham s area function relation as a partial differential equation from our general formulations some applications are discussed

information processing in complex networks graph entropy and information functionals this paper introduces a general framework for defining the entropy of a graph our definition is based on a local information graph and on information functionals derived from the topological structure of a given graph more precisely an information functional quantifies structural information of a graph based on a derived probability distribution such a probability distribution leads directly to an entropy of a graph then the structural information content of a graph will be is interpreted and defined as the derived graph entropy another major contribution of this paper is the investigation of relationships between graph entropies in addition to this we provide numerical results demonstrating not only the feasibility of our method which has polynomial time complexity but also its usefulness with regard to practical applications aiming to an understanding of information processing in complex networks

a fourier mukai approach to the k theory of compact lie groups abstract let g be a compact connected simply connected lie group we use the fourier mukai transform in twisted k theory to give a new proof of the ring structure of the k theory of g

a biological algorithm to solve the assignment problem based on dna molecules computation abstract the assignment problem ap is to optimally solve the task of assigning n jobs to n individuals such that minimum cost or maximum profit can be obtained it is a vitally important np complete problem in operation management and applied mathematics having a wide range of applications in this paper we present a new parallel dna algorithm to solve the assignment problem using dna molecular operations meanwhile we design flexible length dna strands to represent different jobs and individuals take appropriate steps and get the solutions of the ap in proper length range and with o n 2 time complexity we extend the application of dna molecular operations and simultaneity reduce the complexity of the computation

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

injectivity almost everywhere for weak limits of sobolev homeomorphisms abstract let r n be an open set and let f w 1 p r n be a weak sequential limit of sobolev homeomorphisms then f is injective almost everywhere for p n 1 both in the image and in the domain for p n 1 we construct a strong limit of homeomorphisms such that the preimage of a point is a continuum for every point in a set of positive measure in the image and the topological image of a point is a continuum for every point in a set of positive measure in the domain

t equivariant k theory of generalized flag varieties let g be a kac moody group with borel subgroup b and compact maximal torus t analogous to kostant and kumar kostant b kumar s 1986 proc nati acad sci usa 83 1543 1545 we define a certain ring y purely in terms of the weyl group w associated to g and its action on t by dualizing y we get another ring 1 which we prove is canonically isomorphic with the t equivariant ktheory kt g b of gib now kt g b apart from being an algebra over kt pt a t also has a weyl group action and moreover kt g b admits certain operators dw w similar to the demazure operators defined on a t we prove that these structures on kt g b come naturally from the ring y by evaluating the a t module w at 1 we recover k g b together with the above mentioned structures we believe that many of the results of this paper are new in the finite case i e g is a finite dimensional semisimple group over c

regularity properties of deformations with finite energy the purpose of this paper is to point out some regularity properties of a class of functions which play an important role in nonlinear elasticity

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

optical solitons in a parabolic law media with fourth order dispersion abstract this paper obtains the closed form optical 1 soliton solution of the nonlinear schrodinger s equation with parabolic law nonlinearity in presence of the fourth order dispersion the restrictions on the parameters of the soliton are also calculated in course of the derivation of the soliton solution

mappings of finite distortion monotonicity and continuity we study mappings f f1 fn rn in the sobolev space w loc r n where is a connected open subset of rn with n 2 thus for almost every x we can speak of the linear transformation d f x rn rn called differential of f at x its norm is defined by d f x sup d f x h h sn 1 we shall often identify d f x with its matrix and denote by j x f det d f x the jacobian determinant thus using the language of differential forms we can write

radial symmetry of p harmonic minimizers it is still not known if the radial cavitating minimizers obtained by ball philos trans r soc lond a 306 557 611 1982 and subsequently by many others are global minimizers of any physically reasonable nonlinearly elastic energy this quotation is from sivaloganathan and spector ann inst henri poincare anal non lineaire 25 1 201 213 2008 and seems to be still accurate the model case of the p harmonic energy is considered here we prove that the planar radial minimizers are indeed the global minimizers provided we prescribe the admissible deformations on the boundary in the traction free setting however even the identity map need not be a global minimizer

a surface based dna algorithm for the solving binary knapsack problem molecular biology suggests a new method for solving an np complete problem the idea due to leonard adleman is to use strands of dna to encode the instance of the problem and to manipulate using techniques commonly available in any molecular biology laboratory the knapsack problem is an np complete in this paper we solved the binary knapsack problem by dna computing on surface

invertibility and non invertibility in thin elastic structures the nonlinear elastic energy of a thin film of thickness h is given by a functional eh friesecke james and muller derived the limits as h 0 of the functionals h eh for 3 in this article we study the invertibility properties of almost minimizers of these functionals and more generally of sequences with equiintegrable energy density we show that they are invertible almost everywhere away from a thin boundary layer near the film surface moreover we obtain an upper bound for the width of this layer and a uniform upper bound on the diameter of preimages we construct examples showing that these bounds are sharp in particular for all 3 there exist lipschitz continuous low energy deformations which are not locally invertible

homeomorphisms of bounded variation we show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation in higher dimensions we show that f 1 is of bounded variation provided that f w1 1 rn is a homeomorphism with df in the lorentz space ln 1 1

extremal mappings of finite distortion the theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where one finds concrete applications in non linear elasticity and the calculus of variations in this paper we initiate the study of extremal problems for mappings with finite distortion and extend the theory of extremal quasiconformal mappings by considering integral averages of the distortion function instead of its supremum norm for instance we show the following suppose that f o is a homeomorphism of the circle with f o 1 in cal w 1 2 2 then there is a unique extremal extension to the disk which is a real analytic diffeomorphism with non vanishing jacobian determinant the condition on f o is sharp classically the mapping f o is assumed to be quasisymmetric then there is an extremal quasiconformal mapping with boundary values f o but it is not always unique and it is seldom smooth indeed even when f o is quasisymmetric the cal l 1 minimiser for the distortion function will almost never be quasiconformal we further find that there are many new and unexpected phenomena concerning existence uniqueness and regularity for these extremal problems where the functionals are polyconvex but typically not convex these seem to differ markedly from phenomena observed when studying multi well type functionals

twisted k theory and k theory of bundle gerbes in this note we introduce the notion of bundle gerbe k theory and investigate the relation to twisted k theory we provide some examples possible applications of bundle gerbe k theory to the classification of k brane charges in nontrivial backgrounds are briefly discussed

analytic n solitary wave solution of a variable coefficient gardner equation from fluid dynamics and plasma physics in this paper the investigation is focused on a variable coefficient gardner equation with quadric and cubic nonlinearities from fluid dynamics and plasma physics using the hirota bilinear method the one two and three solitary wave solutions of the variable coefficient gardner equation are derived and the analytic n solitary wave solution is presented for the first time in this paper with the aid of symbolic computation figures are plotted to illustrate the solutions obtained in this paper

hyperelastic deformations and total combined energy of mappings between annuli abstract we consider the so called combined energy of a deformation between two concentric annuli and minimize it provided that it keep order of the boundaries it is an extension of the corresponding result of euclidean energy it is interesting that the minimizers are certain radial mappings and they exist if and only if the annulus on the image domain is not too thin provided that the original annulus is fixed this in turn implies a nitsche type phenomenon next we consider the combined distortion and obtain certain related results which are dual to the results for combined energy which also involve some nitsche type phenomenon the main part of the paper is concerned with the total combined energy a certain integral operator defined as a convex linear combination of the combined energy and combined distortion of diffeomorphisms between two concentric annuli a 1 r and b 1 r first we construct radial minimizers of total combined energy then we prove that those radial minimizers are absolute minimizers on the class of all mappings between the annuli under certain constraint this extends the main result obtained by iwaniec and onninen in 9

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

injectivity and self contact in nonlinear elasticity let be a bounded open connected subset of 3 with a sufficiently smooth boundary the additional condition det dx vol is imposed on the admissible deformations of a hyperelastic body whose reference configuration is we show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate we also extend j ball s theorems on existence to this case by establishing the existence of a minimizer of the energy in the space w1 p 3 p 3 that is injective almost everywhere

neohookean deformations of annuli existence uniqueness and radial symmetry let mathbb x x in mathbb r 2 r x r and mathbb y y in mathbb r 2 r ast y r ast be annuli in the plane we consider the class fancyscript f mathbb x mathbb y of all orientation preserving homeomorphisms h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev space fancyscript w 1 2 mathbb x mathbb y which keep the boundary circles in the same order this means that lim x searrow r h x r ast and lim x nearrow r h x r ast we study the neohookean energy integral mathcal e h int limits mathbb x dh 2 phi rm det dh quad rm for h in fancyscript f mathbb x mathbb y quad quad quad 1 where phi in fancyscript c infty 0 infty is positive and strictly convex we assume in addition that the function psi z 1 ddot phi z and its derivative extend continuously to 0 with 0 0 then we prove

non interpenetration of matter for sbv deformations of hyperelastic brittle materials we prove that the ciarlet necas non interpenetration of matter condition 9 can be extended to the case of sbv deformations of hyperelastic brittle materials and can be taken into account for some variational models in fracture mechanics in order to formulate such a condition we define the deformed configuration under an sbv map by means of the approximately differentiable representative and we prove some connected stability results under weak convergence we provide an application to the case of brittle ogden s materials

total energy of radial mappings abstract we prove that the so called total energy functional defined on the class of radial stretchings between annuli attains its minimum on a total energy diffeomorphism between annuli on r n this involves a subtle analysis of some special ode the result is an extension of the corresponding 2 dimensional case obtained by iwaniec and onninen 2009

regularity of the inverse of a planar sobolev homeomorphism let be a domain suppose that f w1 1loc r2 is a homeomorphism such that df x vanishes almost everywhere in the zero set of j f we show that f 1 w1 1loc f r2 and that df 1 y vanishes almost everywhere in the zero set of sharp conditions to quarantee that f 1 w1 q f r2 for some 1 q 2 are also given

uniqueness of regular and singular equilibria for spherically symmetric problems of nonlinear elasticity introduction 97 constitutive assumptions 99 0 the stored energy function and weak equilibrium solutions 101 1 properties of radial equilibrium solutions 105 2 uniqueness of solutions to boundary value problems for shells 111 3 uniqueness of cavitating solutions 117 4 existence of radial equilibria 121 5 asymptotic behaviour of equilibria for shells 128 6 concluding remarks 132 7 appendix 133 references 135

rank one convexity implies polyconvexity for isotropic objective and isochoric elastic energies in the two dimensional case we show that in the two dimensional case every objective isotropic and isochoric energy function which is rank one convex on gl 2 is already polyconvex on gl 2 thus we negatively answer morrey s conjecture in the subclass of isochoric nonlinear energies since polyconvexity implies quasiconvexity our methods are based on different representation formulae for objective and isotropic functions in general as well as for isochoric functions in particular we also state criteria for these convexity conditions in terms of the deviatoric part of the logarithmic strain tensor

optimal assumptions for discreteness there exists in every dimension n geq 3 a lipschitz mapping of finite distortion such that the outer distortion satisfies k in l n 1 but it maps a line segment to a point

monotone sobolev mappings of planar domains and surfaces an approximation theorem of youngs duke math j 15 87 94 1948 asserts that a continuous map between compact oriented topological 2 manifolds surfaces is monotone if and only if it is a uniform limit of homeomorphisms analogous approximation of sobolev mappings is at the very heart of geometric function theory gft and nonlinear elasticity ne in both theories the mappings in question arise naturally as weak limits of energy minimizing sequences of homeomorphisms as a result of this the energy minimal mappings turn out to be monotone in the present paper we show that conversely monotone mappings in the sobolev space fancyscript w 1 p 1 p infty are none other than fancyscript w 1 p weak also strong limits of homeomorphisms in fact these are limits of diffeomorphisms by way of illustration we establish the existence of traction free energy minimal deformations for p harmonic type energy integrals

global invertibility for orientation preserving sobolev maps via invertibility on or near the boundary by a result of ball proc r soc edinb sect a math 88 315 328 1981 https doi org 10 1017 s030821050002014x a locally orientation preserving sobolev map is almost everywhere globally invertible whenever its boundary values admit a homeomorphic extension as shown here for any dimension the conclusions of ball s theorem and related results can be reached while completely avoiding the problem of homeomorphic extension for suitable domains it is enough to know that the trace is invertible on the boundary or can be uniformly approximated by such maps an application in nonlinear elasticity is the existence of homeomorphic minimizers with finite distortion whose boundary values are not fixed as a tool in the proofs strictly orientation preserving maps and their global invertibility properties are studied from a purely topological point of view

on certain hecke rings abstract we examine rings that embed into the smash product of the group algebra of the weyl group with the field of meromorphic functions on the cartan subalgebra and are generated by elements that satisfy braid relations we prove that every such ring is isomorphic to either the hecke algebra the nil hecke ring or the group algebra of the weyl group

an existence theory for nonlinear elasticity that allows for cavitation in this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body such cavities have been observed in experiments on elastomers and a mathematical theory for radially symmetric cavities has been developed by ball here the full three dimensional problem is considered and an additional physically motivated energy term that is proportional to the area of the boundary of the deformed body is included the minimizers lie in a subclass of those maps in w1 p 2 p 3 that are one to one almost everywhere and preserve orientation roughly speaking this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body such maps are shown to be much more regular than expected in particular some ideas of sverak are used to show that each map in this subclass has a representative which is continuous outside a set of hausdorff dimension 3 p and that this representative also satisfies lusin s condition n i e it maps lebesgue null sets onto such sets it is also shown that the distributional jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to lebesgue measure and at most a countable number of dirac measures

the domination numbers of cylindrical grid graphs abstract let p m c n denote the domination number of the cylindrical grid graph formed by the cartesian product of the graphs p m the path of length m m 2 and the graph c n the cycle of length n n 3 in this paper methods to find the domination numbers of graphs of the form p m c n with n 3 and m 2 3 and 4 are proposed moreover bounds on domination numbers of the graphs p 5 c n n 3 are found the methods that are used to prove that results readily lead to algorithms for finding minimum dominating sets of the above mentioned graphs

a polynomial time dna computing solution for the bin packing problem it is suggested here an algorithm based on stickers for the dna computing model s roweis e winfree r burgoyne n chelyapov m goodman p rothemund l adleman a sticker based model for dna computation journal of computational biology 5 1998 615 629 that solves the well known bin packing problem bpp that belongs to the class np hard in the strong sense in a time bounded by o n 2q where n is the quantity of items and q the space requirements expressed in bits to the best of the authors knowledge this is the first polynomial time algorithmic solution for the bpp in such a model

molecular solution to the 0 1 knapsack problem based on dna computing many combinatorial optimization problems are known to be np complete a common point of view is that finding fast algorithms for such problems using a polynomial number of processors is unlikely however facts of this kind are usually established for worst case situations and in practice many instances of np complete problems are successfully solved in polynomial time by such traditional combinatorial optimization techniques such as dynamic programming branch and bound new opportunities for an effective solution of combinatorial problems emerged with the advent of parallel machines in this paper we describe an algorithm which generates an optimal solution for the 0 1 integer knapsack problem on dna computing

local invertibility in sobolev spaces with applications to nematic elastomers and magnetoelasticity we define a class of deformations in w 1 p omega mathbb r n p n 1 with a positive jacobian that do not exhibit cavitation we characterize that class in terms of the non negativity of the topological degree and the equality det det that the distributional determinant coincides with the pointwise determinant of the gradient maps in this class are shown to satisfy a property of weak monotonicity and as a consequence they enjoy an extra degree of regularity we also prove that these deformations are locally invertible moreover the neighbourhood of invertibility is stable along a weak convergent sequence in w 1 p and the sequence of local inverses converges to the local inverse we use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps we apply those results to prove the existence of minimizers in some models for nematic elastomers and magnetoelasticity

elements of a theory of computer simulation i sequential ca over random graphs this paper is a first step in the development of mathematical foundations for a theory of simulation we employ sequentially updated cellular automata sca over arbitrary graphs as a paradigmatic framework in the development of the theory we focus on the properties of causal dependencies among local mappings in a simulation let y be a graph in which a each vertex i has a state x i0 1 and b there exists a local map i defined on the states of the y neighbors and x i that returns the state of y i0 1 suppose i 1 i n is a permutation of the y vertices then the order of application of the local maps i k induces a sequential or asynchronous cellular automaton sca over y in this paper we introduce a mapping between the base graph y over which the sca is defined and which represents the mutual dependencies of the local maps and the update graph u y whose vertices are permutations of all y vertices two permutations p 1 p 2 are adjacent in u y if they differ in exactly two consecutive coordinates i k i k 1 and i k i k 1 ey u y represents a conceptual framework which allows to determine equivalence classes of sca for a given graph x independent of the family of local maps i 1 i n we consider a u as a random variable over g n p and analyze the induced random graph u g n p the update graph and show b that u exhibits properties of a covariant functor

a similarity measure for graphs with low computational complexity we present and analyze an algorithm to measure the structural similarity of generalized trees a new graph class which includes rooted trees for this we represent structural properties of graphs as strings and define the similarity of two graphs as optimal alignments of the corresponding property stings we prove that the obtained graph similarity measures are so called backward similarity measures from this we find that the time complexity of our algorithm is polynomial and hence significantly better than the time complexity of classical graph similarity methods based on isomorphic relations

soliton soliton interaction with dual power law nonlinearity abstract the intra channel collision of optical solitons with dual power law nonlinearity is studied in this paper by the aid of quasi particle theory the perturbation terms that are considered in this paper are of hamiltonian type the suppression of soliton soliton interaction in presence of these perturbation terms is acheived the numerical simulations support the quasi particle theory

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians