quantum algebras associated with bell states the bell matrix has become an interesting interdisciplinary topic involving quantum information theory and the yang baxter equation it is an antisymmetric unitary solution of the braided yang baxter equation and yields all the bell states by acting on the product basis in this paper using the faddeev reshetikhin takhtadjian frt construction we obtain a quantum algebra associated with the bell matrix we explore two characteristic algebraic structures in its four dimensional representation one is a representation with a composition series namely it has irreducible subrepresentations but is not completely reducible the other is a direct sum of two dimensional cyclic representations and can be spanned by four maximally entangled states as local unitary transformations of the bell states both of them are expected to be realized in physical systems and exploited in quantum information theory besides we present the other quantum algebra associated with the unitary evolution of the bell states or the yang baxterization of the bell matrix

isoperimetry for wreath products of markov chains and multiplicity of selfintersections of random walks we prove an isoperimetric inequality for wreath products of markov chains with variable fibers we use isoperimetric inequalities for wreath products to estimate the return probability of random walks on infinite groups and graphs drift of random loops the expected value e exp trn where rn is the number of distinct sites visited up to the moment n and more generally open image in new window where lz n is the number of visits of z up to the moment n and f x y is some non negative function

unusual solutions to the yang baxter equation the complete list of non symmetric eight vertex constant solutions to the yang baxter equation is obtained and new solutions dependent on more than one variable are presented the corresponding spin hamiltonians are derived by sutherland s method 1970

yang baxter operators need quantum entanglement to distinguish knots any solution to the yang baxter equation yields a family of representations of braid groups under certain conditions identified by turaev 1988 inventiones math 92 527 53 the appropriately normalized trace of these representations yields a link invariant any yang baxter solution can be interpreted as a two qudit quantum gate here we show that if this gate is non entangling then the resulting invariant of knots is trivial we thus obtain a connection between topological entanglement and quantum entanglement

teleportation braid group and temperley lieb algebra we explore algebraic and topological structures underlying the quantum teleportation phenomena by applying the braid group and temperley lieb algebra we realize the braid teleportation configuration teleportation swapping and virtual braid representation in the standard description of the teleportation we devise diagrammatic rules for quantum circuits involving maximally entangled states and apply them to three sorts of descriptions of the teleportation the transfer operator quantum measurements and characteristic equations and further propose the temperley lieb algebra under local unitary transformations to be a mathematical structure underlying the teleportation we compare our diagrammatical approach with two known recipes to the quantum information flow the teleportation topology and strongly compact closed category in order to explain our diagrammatic rules to be a natural diagrammatic language for the teleportation

solving the two dimensional constant quantum yang baxter equation a detailed analysis of the constant quantum yang baxter equation rk1k2j1j2 rl1k3k1j3rl2l3k2k3 rk2k3j2j3 rk1l3j1k3rl1l2k1k2 in two dimensions is presented leading to an exhaustive list of its solutions the set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the grobner basis methods each solution was then transformed into a canonical form based on the various trace matrices of r for final elimination of duplicates and subcases if we use homogeneous parametrization the solutions can be combined into 23 distinct cases modulo the well known c p and t reflections and rotations and scalings r q q r q q 1

chiral potts model as a descendant of the six vertex model we observe that then state integrable chiral potts model can be considered as a part of some new algebraic structure related to the six vertex model as a result we obtain a functional equation which is supposed to determine all the eigenvalues of the chiral potts model transfer matrix

duality and representations for new exotic bialgebras we find the exotic matrix bialgebras which correspond to the two nontriangular nonsingular 4 4 r matrices in the classification of hietarinta namely rs0 3 and rs1 4 we find two new exotic bialgebras s03 and s14 which are not deformations of the classical algebras of functions on gl 2 or gl 1 1 with this we finalize the classification of the matrix bialgebras which unital associative algebras generated by four elements we also find the corresponding dual bialgebras of these new exotic bialgebras and study their representation theory in detail we also discuss in detail a special case of rs1 4 in which the corresponding algebra turns out to be a special case of the two parameter quantum group deformation glp q 2

multiplicity results for some nonlinear schrodinger equations with potentials multiple semiclassical standing waves for a class of nonlinear schrodinger equations with potentials are found by means of a perturbative variational method

random walks among time increasing conductances heat kernel estimates for any graph having a suitable uniform poincare inequality and volume growth regularity we establish two sided gaussian transition density estimates and parabolic harnack inequality for constant speed continuous time random walks evolving via time varying uniformly elliptic conductances provided the vertex conductances i e reversing measures increase in time such transition density upper bounds apply for discrete time uniformly lazy walks with the matching lower bounds holding once the parabolic harnack inequality is proved

multiparameter solutions of the yang baxter equation the multiparameter solutions 4 4 r matrices of the quantum yang baxter equation have been systematically investigated and more than thirty selected solutions are presented by using wu elimination the free parameters and non zero eigenvalues of these r matrices vary from one to five and zero to four respectively they are classified according to the corresponding algebraic structures

piecewise automatic groups we introduce a notion of a piecewise automatic group among these groups we describe a new class of groups of intermediate growth we show that for any function f n n there exists a finitely generated torsion group of intermediate growth g for which the folner function satisfies folg s n f n for some generating set s and all sufficiently large n as a corollary we see that the asymptotic entropy of simple random walks on these groups could be arbitrarily close to being linear while the poisson boundary is trivial

exponential decay of the solutions of quasilinear second order equations and pohozaev identities keywords pohozaev identities decay nonlinear elliptic equation reference ana article 2000 002doi 10 1006 jdeq 1999 3749view record in web of science record created on 2008 12 10 modified on 2016 08 08

duality for exotic bialgebras in the classification of hietarinta three triangular 4 4 r matrices lead via the frt formalism to matrix bialgebras which are not deformations of the trivial one in this paper we find the bialgebras which are in duality with these three exotic matrix bialgebras we note that the l t duality of frt is not sufficient for the construction of the bialgebras in duality we find also the quantum planes corresponding to these bialgebras both by the wess zumino r matrix method and by manin s method

on multiplicity and concentration behavior of solutions for a critical system with equations in divergence form abstract we consider the system e 2 div a x u u q u u v 1 2 k u u v in r n e 2 div b x v v q v u v 1 2 k v u v in r n u v h 1 r n u x v x 0 for each x r n where 2 2 n n 2 n 3 e 0 a and b are positive continuous potentials and q and k are homogeneous function with k having critical growth we obtain existence of a ground state solution and relate the number of solutions with the topology of the set where the potentials a and b attain their minima we also show that at the maximum points of each solution the potentials a and b converge to their points of minima points when e converges to zero

new constant and trigonometric 4 4 solutions to the yang baxter equations a classification of triangular constant solutions of the yang baxter equations is performed the formulae for the baxterization of braid group representation are used for finding trigonometric solutions of the yang baxter equations from the constant ones some of the obtained solutions correspond to known solutions but several new ones are obtained as well the calculations confirm that the baxterization formulae for more than two eigenvalues of the r matrix are not universal

all teleportation and dense coding schemes we establish a one to one correspondence between 1 quantum teleportation schemes 2 dense coding schemes 3 orthonormal bases of maximally entangled vectors 4 orthonormal bases of unitary operators with respect to the hilbert schmidt scalar product and 5 depolarizing operations whose kraus operators can be chosen to be unitary the teleportation and dense coding schemes are assumed to be tight in the sense that all hilbert spaces involved have the same finite dimension d and the classical channel involved distinguishes d 2 signals a general construction procedure for orthonormal bases of unitaries involving latin squares and complex hadamard matrices is also presented

on a class of singularly perturbed elliptic equations in divergence form existence and multiplicity results the main purpose of this paper is to study the existence of single peaked positive solutions of the singularly perturbed elliptic equation e2div j x u v x u f u inrn where j is a symmetric uniformly elliptic matrix and v is a positive potential possibly unbounded from above if f u up then solutions concentrate at non degenerate critical points of x v x p 1p 1 n2 detj x 1 2

modified braid equations for soq 3 and noncommutative spaces general solutions of the rtt equation with a maximal number of free parameters in the specrtal decomposition of vector soq 3 r matrices are implemented to construct modified braid equations mbe these matrices conserve the given standard group relations of the nine elements of t but are not constrained to satisfy the standard braid equation be apart from q and a normalization factor our r contains two free parameters instead of only one such parameter for deformed unitary algebras studied in a previous paper math qa 0009178 where the nonzero right hand side of the mbe had a linear term proportional to r 12 r 23 in the present case the rhs is in general nonlinear several particular solutions are given sec ii and the general structure is analyzed appendix a our formulation of the problem in terms of projectors yields also two new solutions of standard nonmodified braid equation sec ii which are further discussed appendix b the noncommutative three spaces obtained by i

on random walks on wreath products wreath products are a type of semidirect product they play an important role in group theory this paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting somewhat exotic behaviors the crucial fact is that the probability of return to the starting point of certain walks on wreath products is closely related to some functionals of the local times of a walk taking place on a simpler factor group

random walks driven by low moment measures we study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition for any locally compact unimodular group g and any positive function g 0 we introduce a function g which describes the fastest possible decay of n 2n e when is a symmetric continuous probability density such that is finite we estimate g for a variety of groups g and functions when is of the form with 0 0 a fixed increasing function and g 0 a natural word length measuring the distance to the identity element in g g can be thought of as a group invariant

metrics on the real quantum plane using the frame formalism we determine some possible metrics and metric compatible connections on the noncommutative differential geometry of the real quantum plane by definition a metric maps the tensor product of two 1 forms into a function on the quantum plane it is symmetric in a modified sense namely in the definition of symmetry one has to replace the permutator map with a deformed map fulfilling some suitable conditions correspondingly also the definition of the hermitian conjugate of the tensor product of two 1 forms is modified but reduces to the standard one if coincides with the permutator the metric is real with respect to such modified structure

on the number of solutions of nls equations with magnetics fields in expanding domains abstract in this paper we look for multiple weak solutions u c for the complex equation i a x 2 u u f u 2 u in the set r n is a smooth bounded domain 0 is a parameter a is a regular magnetic field and f is a superlinear function with subcritical growth our main result relates for large values of the number of solutions with the topology of the set in the proof we apply minimax methods and ljusternik schnirelmann theory

multiplicity of positive solutions for a class of elliptic equations in divergence form we prove results concerning the existence and multiplicity of positive solutions for the quasilinear equation div a ex u p 2 u u p 2 u f u in r n u w 1 p r n where 2 p n a is a positive potential and f is a superlinear function we relate the number of solutions with the topology of the set where a attains its minimum the results are proved by using minimax theorems and ljusternik schnirelmann theory

difference equations isoperimetric inequality and transience of certain random walks the difference laplacian on a square lattice in rn has been stud ied by many authors in this paper an analogous difference operator is studied for an arbitrary graph it is shown that many properties of the laplacian in the continuous setting e g the maximum principle the harnack inequality and cheeger s bound for the lowest eigenvalue hold for this difference oper ator the difference laplacian governs the random walk on a graph just as the laplace operator governs the brownian motion as an application of the theory of the difference laplacian it is shown that the random walk on a class of graphs is transient the random walks we consider are defined as follows let k be a connected graph i e a one dimensional simplicial complex for a vertex x e k let m x denote the number of edges emanating from x the probability that a particle moves from x to another vertex y e k is l m x if x and y are connected by an edge and it is zero otherwise as observed by courant friedrichs and lewy cfl for the case of a square lattice in the plane this random walk is intimately related to the difference analog of the laplacian

higher dimensional unitary braid matrices construction associated structures and entanglements we construct 2n 2 2n 2 unitary braid matrices r for n 2 generalizing the class known for n 1 a set of 2n 2n matrices i j k l is defined r is expressed in terms of their tensor products such as k j leading to a canonical formulation for all n complex projectors p provide a basis for our real unitary r baxterization is obtained diagonalizations and block diagonalizations are presented the loss of braid property when r n 1 is block diagonalized in terms of r n 1 is pointed out and explained for odd dimension 2n 1 2 2n 1 2 a previously constructed braid matrix is complexified to obtain unitarity rll and rtt algebras chain hamiltonians potentials for factorizable s matrices and complex noncommutative spaces are all studied briefly in the context of our unitary braid matrices turaev construction of link invariants is formulated for our case we conclude with comments concerning entanglements

ultracontractivity and embedding into l given a self adjoint semigroup e ta satisfying an ultracontractivity bound of the type e ta 2 rightarrow infty leq e m t we find conditions on the sequence a n f 2 1 n that imply that f is a bounded function sobolev s classical embedding theorem says that when a is the laplace operator on mathbb r d a kf 2 d 4 suffices to imply that f is bounded in the cases we are interested in the desired condition involves the whole sequence a n f 2 1 n and depends on the behavior of the ultracontractivity function m

spike solutions for a class of singularly perturbed quasilinear elliptic equations abstract by means of a penalization scheme due to del pino and felmer we prove the existence of single peaked solutions for a class of singularly perturbed quasilinear elliptic equations associated with functionals which lack of smoothness we do not require neither uniqueness assumptions on the limiting autonomous equation nor monotonicity conditions on the nonlinearity compared with the semilinear case some difficulties arise and the study of concentration of the solutions needs a somewhat involved analysis in which the pucci serrin variational identity plays an important role

a new approach to pointwise heat kernel upper bounds on doubling metric measure spaces abstract on doubling metric measure spaces endowed with a strongly local regular dirichlet form we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale invariant inequalities that correspond respectively to the nash inequality and to a gagliardo nirenberg type inequality when the volume growth is polynomial this yields a new proof and a generalisation of the well known equivalence between classical heat kernel upper bounds and relative faber krahn inequalities or localised sobolev or nash inequalities we are able to treat more general pointwise estimates where the heat kernel rate of decay is not necessarily governed by the volume growth a crucial role is played by the finite propagation speed property for the associated wave equation and our main result holds for an abstract semigroup of operators satisfying the davies gaffney estimates

variational perturbative methods and bifurcation of bound states from the essential spectrum this paper consists of two main parts the first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results the second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum

random walks on free solvable groups for any finitely generated group g let n mapsto phi g n be the function that describes the rough asymptotic behavior of the probability of return to the identity element at time 2n of a symmetric simple random walk on g this is an invariant of quasi isometry we determine this function when g is the free solvable group mathbf s d r of derived length d on r generators and some related groups

multiple solutions for schrodinger poisson type equation with magnetic field in this paper we consider the multiplicity and concentration behavior for a nonlinear schrodinger poisson type equation with magnetic field by penalization method and ljusternik schnirelmann theory we present the relationship between the number of solutions and the topology of the set where the potential attains its minimum value

a new braid like algebra for baxterisation we introduce a new baxterisation for r matrices that depend separately on two spectral parameters the baxterisation is based on a new algebra close to but different from the braid group we study representations of this new algebra on the vector space mathbb c m otimes n when the generators act locally the ones for m 2 are completely classified we also introduce some representations for generic m they allow us to recover the r matrix of the multi species generalization of the totally asymmetric simple exclusion process with different hopping rates

solutions to the quantum yang baxter equation arising from pointed bialgebras let r m m m m be a solution to the quantum yang baxter equation where m is a finite dimensional vector space over a field k we introduce a quotient ana r of the bialgebra a r constructed by fadeev reshetihkin and takhtajan whose characteristics seem to more faith fully reflect properties r possesses as a linear operator we characterize all r such that ared r is a pointed bialgebra and we determine all solutions r to the quantum yang baxter equation when ared r is pointed and dim m 2 with a few technical exceptions when k has characteristic 2 extensions of such solutions to the quantum plane are studied in this paper we study the quantum yang baxter equation from the point of view of the faddeev reshetikhin and takhtajan construction 2 to refine techniques for constructing solutions using bialgebra methods applying our methods in the two dimensional case we generate with relative ease six extensive classes of solutions in particular we find all solutions r m m m m where m is a two dimensional vector space over a field k of characteristic not 2 which are represented by an invertible upper triangular matrix with respect to a basis of the form m m n m m n n n included among these is the familiar quantum yang baxter operator

spectral distribution and l2 isoperimetric profile of laplace operators on groups we give a formula relating the l2 isoperimetric profile to the spectral distribution of a laplace operator on a finitely generated group we prove the asymptotic stability of the spectral distribution under changes of measures with finite second moment as a consequence we can apply techniques from geometric group theory to estimate the spectral distribution of the laplace operator in terms of the growth and the folner s function of the group this leads to upper bounds on spectral distributions of some non solvable amenable groups and to sharp estimates of the spectral distributions of some solvable groups with exponential growth