trace expansions for elliptic cone operators with stationary domains we analyze the behavior of the trace of the resolvent of an ellip tic cone differential operator as the spectral parameter tends to infinity the resolvent splits into two components one associated with the minimal exten sion of the operator and another of finite rank depending on the particular choice of domain we give a full asymptotic expansion of the first compo nent and expand the component of finite rank in the case where the domain is stationary the results make use and develop further our previous investi gations on the analytic and geometric structure of the resolvent the analysis of nonstationary domains considerably more intricate is pursued elsewhere

quantum spinors and spin groups from quantum clifford algebras a general construction of multiparametric quantum spinors and corresponding quantum groups associated to pseudo euclidean spaces is presented and their homomorphism to the respective groups is discussed this construction is based on a quantum clifford algebra and is described in detail for involutive pure twists intertwining braids for general braid operators that admit abstract volume elements a procedure is also given for deriving quantum analogues of these groups

on the curvature of piecewise flat spaces we consider analogs of the lipschitz killing curvatures of smooth riemannian manifolds for piecewise flat spaces in the special case of scalar curvature the definition is due to t regge considerations in this spirit date back to j steiner we show that if a piecewise flat space approximates a smooth space in a suitable sense then the corresponding curvatures are close in the sense of measures

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

conservation laws for linear equations on braided linear spaces the properties of linear equations on braided linear spaces are investigated and conservation laws for them are derived the conserved currents are given in the explicit form the procedure is then applied to scalar wave equations on a quantum plane and on q minkowski space

conservation laws for linear equations on quantum minkowski spaces the general linear equations with constant coefficients on quantum minkowski spaces are considered and the explicit formulae for their conserved currents are given the proposed procedure can be simplified for invariant equations the derived method is then applied to klein gordon dirac and wave equations on different classes of minkowski spaces in the examples also symmetry operators for these equations are obtained they include quantum deformations of classical symmetry operators as well as an additional operator connected with deformation of the leibnitz rule in non commutative differential calculus

witt spaces a geometric cycle theory for ko homology at odd primes introduction this paper studies a class of stratified piecewiselinear pseudomanifolds which we call witt spaces characterized by natural local intersection homology conditions 11 we compute the cobordism groups by introducing an invariant taking values in the witt group of symmetric bilinear forms over the rationals w q these pseudomanifolds solve a problem posed by d sullivan 25 to construct a class of p l cycles with signature which represent the connected version of ko homology at odd primes ko 0d z 1 2 the result was one of the first applications of the intersection homology theory of goresky and macpherson 11 specifically the rational intersection homology groups of witt spaces satisfy a poincare duality theorem to each witt space x we associate a p l invariant w x with values in w q this invariant generalizes the signature of manifolds and satisfies cobordism invariance additivity and a product formula adapting classical surgery spherical modification to this setting we prove that w x determines the cobordism class of x and obtain an explicit description of the witt cobordism groups the only nontrivial groups occur in dimensions 4k and for k 0 they are w q sullivan 25 27 has shown that a class of p l pseudomanifolds equipped with such an invariant possesses canonical orientations in ko 0 z 1 2 the orientations induce a natural transformation it of homology theories from witt space bordism to ko 0 z 1 2 the structure of w q is known 14 and we conclude that it is an equivalence at odd primes in this paper we study the cobordism theory of a class of stratified p l pseudomanifolds by means of the recent intersection homology theory of goresky and macpherson 11 goresky and macpherson were interested in finding a class of spaces with cobordism invariant signature for the purpose of extending the hirzebruch l class to the setting of manifolds with

quantum lorentz and braided poincare groups quantum lorentz groups h admitting quantum minkowski space v are selected the natural structure of a quantum space is introduced defining a quantum group structure on g only for triangular h q 1 we show that it defines a braided quantum group structure on g for q 1

a new proof of weyl s formula on the asymptotic distribution of eigenvalues one of the more perplexing problems of finite particle quantum mechanics is to find the appropriate quantum mechanical description of a given classical mechanical system unfortunately from the mathematical point of view this problem is not terribly well posed suppose the classical system in question is represented by a symplectic manifold x and a classical observable p x r suppose also for simplicity that this system is a confined system in the sense that the sets p 1 are all compact let h be a hilbert space and let p be a self adjoint operator on h representing a quantum observable without specifying anything further about p for instance its relation to other observables there is practically no way of ascertaining whether p is an appropriate quantum counterpart of p at first glance the only requirements this appears to impose on p are that the spectrum of p be discrete and that the number of eigenvalues n a less than 1 be finite for all 1 this is the analogue of the condtion that p 1 be compact by examining the statistical properties of a large aggregate of classical systems each a copy of x p one can in fact say a little bit more about the relation p to p by comparing classical and quantum partition functions one can show that for iz large

resolvents of cone pseudodifferential operators asymptotic expansions and applications we study the structure and asymptotic behavior of the resolvent of elliptic cone pseudodifferential operators acting on weighted sobolev spaces over a compact manifold with boundary we obtain an asymptotic expansion of the resolvent as the spectral parameter tends to infinity and use it to derive corresponding heat trace and zeta function expansions as well as an analytic index formula

the heat equation with singular coefficients the small time asymptotics of the kernel ofe th is defined and derived for h frac d 2 dx 2 frac kappa x 2 on 1 lemmas on singular asymptotics in the sense of distributions are formulated and used the results are applied to derive an index formula on 1

on crossed product of algebras the concept of a crossed tensor product of algebras is studied from a few points of view some related constructions are considered crossed enveloping algebras and their representations are discussed applications to the noncommutative geometry and particle systems with generalized statistics are indicated

adjoints of elliptic cone operators we study the adjointness problem for the closed extensions of a general b elliptic operator a x diff m b m e 0 initially defined as an unbounded operator a c c m e x l 2 b m e x l 2 b m e inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the case where a is a symmetric semibounded operator is of particular interest and we give a complete description of the domain of the friedrichs extension of such an operator

canonical q deformations of noncompact lie super algebras the author proposes a procedure for q deformations of the real forms g of complex lie super algebras associated with generalized cartan matrices the procedure gives different q deformations for the nonconjugate cartan subalgebras of g the author gives several illustrations e g q deformed lorentz and conformal super algebras the q deformed conformal algebra contains as a subalgebra a q deformed poincare algebra and as hopf subalgebras two conjugate 11 generator q deformed weyl algebras the q deformed lorentz algebra is a hopf subalgebra of both weyl algebras

lipschitz flow box theorem abstract a generalization of the flow box theorem is proven the assumption of a c 1 vector field f is relaxed to the condition that f be locally lipschitz continuous the theorem holds in any banach space

the resolvent expansion for second order regular singular operators abstract the paper gives a scheme for computing the asymptotics of tr e tl as t 0 where l is an elliptic operator of the form l d 2 x 2 a x and a x is a family of operators satisfying appropriate ellipticity and smoothness conditions a principle example is the laplace operator for a manifold with an asymptotically conic singularity the expansion has the usual terms away from the singularity appropriately regularized at x 0 plus singular contributions determined by the function of a 0 1 4 1 2 applications to index theorems are given in a subsequent paper

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

kappa minkowski spacetime kappa poincare hopf algebra and realizations we unify minkowki spacetime and lorentz algebra in unique lie algebra introducing commutative momenta a family of deformed heisenberg algebras and deformed poincare algebras are defined they are specified by the matrix depending on momenta we construct all such matrices realizations and star product are defined and analyzed in general and specially their relation to coproduct of momenta is pointed out hopf algebra of the poincare algebra related to the covariant realization is presented in unified covariant form left right dual realizations and dual algebra are introduced and considered the generalized involution and the star inner product are analyzed and their properties are discussed partial integration and deformed trace property are obtained in general the translation invariance of the star product is pointed out finally perturbative approach up to the first order in a is presented in the appendix

spectral geometry of singular riemannian spaces on considere la trace du noyau de la chaleur et l application de la methode de l equation de la chaleur au theoreme d indice pour la caracteristique d euler et des complexes signatures

quantum 1 1 extended galilei algebras from lie bialgebras to quantum r matrices and integrable systems the lie bialgebras of the 1 1 extended galilei algebra are obtained and classified into four multiparametric families their quantum deformations are obtained together with the corresponding deformed casimir operators for the coboundary cases quantum universal r matrices are also given applications of the quantum extended galilei algebras to classical integrable systems are explicitly developed

groupoids and an index theorem for conical pseudo manifolds we define an analytical index map and a topological index map for conical pseudomanifolds these constructions generalize the analogous constructions used by atiyah and singer in the proof of their topological index theorem for a smooth compact manifold m a main ingredient is a non commutative algebra that plays in our setting the role of c 0 t m we prove a thom isomorphism between non commutative algebras which gives a new example of wrong way functoriality in k theory we then give a new proof of the atiyah singer index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds we thus prove a topological index theorem for conical pseudomanifolds

resolvents of elliptic cone operators we prove the existence of sectors of minimal growth for general closed extensions of elliptic cone operators under natural ellipticity conditions this is achieved by the construction of a suitable parametrix and reduction to the boundary special attention is devoted to the clarification of the analytic structure of the resolvent

on the structure of inhomogeneous quantum groups we investigate inhomogeneous quantum groups g built from a quantum group h and translations the corresponding commutation relations contain inhomogeneous terms under certain conditions which are satisfied in our study of quantum poincare groups 12 we prove that our construction has correct size find the r matrices and the analogues of minkowski space for g

on the resolvent and spectral functions of a second order differential operator with a regular singularity we consider the resolvent of a second order differential operator with a regular singularity admitting a family of self adjoint extensions we find that the asymptotic expansion for the resolvent in the general case presents unusual powers of which depend on the singularity the consequences for the pole structure of the function and for the small t asymptotic expansion of the heat kernel are also discussed

classification of real three dimensional poisson lie groups all real three dimensional poisson lie pl groups are explicitly constructed and fully classified under group automorphisms by making use of their one to one correspondence with the complete classification of real three dimensional lie bialgebras given in gomez 2000 j math phys 41 4939 many of these 3d pl groups are non coboundary structures whose poisson brackets are given here for the first time casimir functions for all three dimensional pl groups are given and some features of several pl structures are commented

solutions of klein gordon and dirac equations on quantum minkowski spaces covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified in the case of quantum minkowski spaces they have the same dimensions as in the classical case formal solutions of the corresponding klein gordon and dirac equations are found the fock space construction is sketched

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

on the classification of quantum poincare groups using the general theory of 10 quantum poincare groups without dilatations are described and investigated the description contains a set of numerical parameters which satisfy certain polynomial equations for most cases we solve them and give the classification of quantum poincare groups each of them corresponds to exactly one quantum minkowski space the poincare series of these objects are the same as in the classical case we also classify possibler matrices for the fundamental representation of the group

quasitriangularity and enveloping algebras for inhomogeneous quantum groups coquasitriangular universal r matrices on quantum lorentz and quantum poincare groups are classified the results are extended under certain assumptions to inhomogeneous quantum groups enveloping algebras on those objects are described

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

commutators of flow maps of nonsmooth vector fields relying on the notion of set valued lie bracket introduced in an earlier paper we extend some classical results valid for smooth vector fields to the case when the vector fields are just lipschitz in particular we prove that the flows of two lipschitz vector fields commute for small times if and only if their lie bracket vanishes everywhere i e equivalently if their classical lie bracket vanishes almost everywhere we also extend the asymptotic formula that gives an estimate of the lack of commutativity of two vector fields in terms of their lie bracket and prove a simultaneous flow box theorem for commuting families of lipschitz vector fields

unusual poles of the ζ functions for some regular singular differential operators we consider the resolvent of a system of first order differential operators with a regular singularity admitting a family of self adjoint extensions we find that the asymptotic expansion for the resolvent in the general case presents powers of which depend on the singularity and can take even irrational values the consequences for the pole structure of the corresponding and functions are also discussed

strichartz estimates and wave equation in a conic singular space consider the metric cone x c y 0 infty r times y with metric g dr 2 r 2h where the cross section y is a compact n 1 dimensional riemannian manifold y h let delta g be the positive friedrichs extension laplacian on x and let delta h be the positive laplacian on y and consider the operator mathcal l v delta g v 0 r 2 where v 0 in mathcal c infty y such that delta h v 0 n 2 2 4 is a strictly positive operator on l 2 y in this paper we prove global in time strichartz estimates without loss regularity for the wave equation associated with the operator mathcal l v it verifies a conjecture in wang remark 2 4 in ann inst fourier 56 1903 1945 2006 for wave equation the range of the admissible pair is sharp and the range is influenced by the smallest eigenvalue of delta h v 0 n 2 2 4 to prove the result we show a sobolev inequality and a boundedness of a generalized riesz transform in this setting in addition as an application we study the well posed theory and scattering theory for energy critical wave equation with small data on this setting of dimension n ge 3

generalized q exponentials related to orthogonal quantum groups and fourier transformations of noncommutative spaces essential prerequisites for the study of q deformed physics are particle states in position and momentum representation in order to relate x and p space by fourier transformations the appropriate q exponential series related to orthogonal quantum symmetries is constructed it turns out to be a new q special function giving rise to q plane wave solutions that transform with a noncommutative phase under translations

the inhomogeneous quantum groups from differential calculi with classical dimension from the bicovariant first order differential calculus on inhomogeneous hopf algebra b we construct the set of right invariant maurer cartan one forms considered as a right invariant basis of a bicovariant b bimodule over which we develop the woronowicz general theory of differential calculus on quantum groups in this formalism we introduce suitable functionals on b which control the inhomogeneous commutation rules in particular we find that the homogeneous part of commutation rules between the translations and those between the generators of the homogeneous part of b and translations are controlled by different r matrices satisfying characteristic equations

weakly parametric pseudodifferential operators and atiyah patodi singer boundary problems this paper introduces a class of pseudodifferential operators depending on a parameter in a particular way the main application is a complete expansion of the trace of the resolvent of a dirac type operator with nonlocal boundary conditions of the kind introduced by atiyah patodi and singer aps this extends the partial expansion in g2 to a complete one and extends the complete expansion in gs 1 to the case where the dirac operator does not have a product structure near the boundary a secondary application is to obtain a complete expansion of the resolvent of a bdo on a compact manifold essentially reproving a result of agranovich agr the resolvent expansion yields immediately an expansion of the trace of the heat kernel and determines the singularities of the zeta function moreover a pseudodifferential factor can be allowed a major motive for these expansions is to obtain index formulas for elliptic operators there are many such applications in the physics and geometry literature the index formula comes from one particular term in the expansion but each term is a spectral invariant and they have been used for other purposes as well as for the index in particular branson and gilkey have a number of papers e g bg and gi analyzing these invariants and drawing geometric consequences interest in the asymptotic behavior of the resolvent goes back to carleman c more recently agmon agm developed it extensively for analytic applications he introduced the fundamental idea of treating the resolvent parameter essentially as another cotangent variable this idea was developed in s1 to analyze the singularities of the zeta function of an elliptic odo on a compact manifold and in 3 to analyze the resolvent of a differential operator with differential boundary conditions the technique works smoothly for differential operators producing so called local invariants integrals over the underlying

on the periodic orbits shadowing and strong transitivity of continuous flows abstract we prove that chaotic flows i e flows that satisfy the shadowing property and have a dense subset of periodic orbits satisfy a reparametrized gluing orbit property similar to the one introduced in bomfim and varandas 2015 in particular these are strongly transitive in balls of uniform radius we also prove that the shadowing property for a flow and a generic time t map and having a dense subset of periodic orbits hold for a c 0 baire generic subset of lipschitz vector fields that generate continuous flows similar results also hold for c 0 generic homeomorphisms and in particular we deduce that chain recurrent classes of c 0 generic homeomorphisms have the gluing orbit property

the dirac operator on spaces with conical singularities and positive scalar curvatures on obtient un theoreme d annulation de type bochner et on demontre un theoreme d indice dans le cas singulier on etudie la relation avec des varietes a bord

q deformed poincare algebra theq differential calculus for theq minkowski space is developed the algebra of theq derivatives with theq lorentz generators is found giving theq deformation of the poincare algebra the reality structure of theq poincare algebra is given the reality structure of theq differentials is also found the real laplacian is constructed finally the comultiplication counit and antipode for theq poincare algebra are obtained making it a hopf algebra

zeta determinant for laplace operators on riemann caps the goal of this paper is to compute the zeta function determinant for the massive laplacian on riemann caps or spherical suspensions these manifolds are defined as compact and boundaryless d dimensional manifolds deformed by a singular riemannian structure the deformed spheres considered previously in the literature belong to this class after presenting the geometry and discussing the spectrum of the laplacian we illustrate a method to compute its zeta regularized determinant the special case of the deformed sphere is recovered as a limit of our general formulas

the expansion of the resolvent near a singular stratum of conical type in descending power of i for suitably large m the coefficients are given by integrals of the usual densities associated with the laplacian at nonsingular points suitably regularized at the singular stratum plus singular terms given as integrals over the singular stratum a concrete example of the situation treated is a wedge wc m in a riemannian manifold m the boundary of w is smooth except along a singular stratum c parametrized by se up near c 3 w z u z with c and c2 intersecting transversally in c in a plane section perpendicular to z at a point s c and z2 meet at an angle cl s 0 measured inside w introduce coordinates x1 x2 s in a neighborhood of c where

on the index of elliptic operators on manifolds with boundary abstract an extension of the index theorem of atiyah patodi singer for dirac type operators on manifolds with boundary to general elliptic b pseudodifferential operators is given first basic results about the complex powers of these operators are established then the main formula for the index of an elliptic b pseudo differential operator acting on an r weighted sobolev space r r is proved this expresses the index as the sum of an interior contribution given in terms of regularized zeta functions and a boundary contribution generalizing the eta invariant of atiyah patodi singer this second term measures the asymmetry of the boundary spectrum of the operator a discrete set in the complex plane with respect to the line z c im z r

conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces the conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived for discrete models the conserved charges are constructed explicitly the applications of the general method include equations on quantum plane supersymmetric equations for chiral and antichiral supermultiplets auxiliary equations of integrable models which means various cases of nonlinear toda lattice equations and anomalous diffusion equation

quantum instantons with classical moduli spaces we introduce a quantum minkowski space time based on the quantum group su 2 q extended by a degree operator and formulate a quantum version of the anti self dual yang mills equation we construct solutions of the quantum equations using the classical adhm linear data and conjecture that up to gauge transformations our construction yields all the solutions we also find a deformation of penrose s twistor diagram giving a correspondence between the quantum minkowski space time and the classical projective space p 3

minimal time for the continuity equation controlled by a localized perturbation of the velocity vector field in this work we study the minimal time to steer a given crowd to a desired configuration the control is a vector field representing a perturbation of the crowd velocity localized on a fixed control set we will assume that there is no interaction between the agents we give a characterization of the minimal time both for microscopic and macroscopic descriptions of a crowd we show that the minimal time to steer one initial configuration to another is related to the condition of having enough mass in the control region to feed the desired final configuration the construction of the control is explicit providing a numerical algorithm for computing it we finally give some numerical simulations