the classification of complete stable area stationary surfaces in the heisenberg group h1 abstract we prove that any c 2 complete orientable connected stable area stationary surface in the sub riemannian heisenberg group h 1 is either a euclidean plane or congruent to the hyperbolic paraboloid t x y

on the topology and index of minimal surfaces we show that for an immersed two sided minimal surface in r 3 there is a lower bound on the index depending on the genus and number of ends using this we show the nonexistence of an embedded minimal surface in r 3 of index 2 as conjectured by choe cho90 moreover we show that the index of an immersed two sided minimal surface with embedded ends is bounded from above and below by a linear function of the total curvature of the surface

singular perturbations of degenerate diffusions ventsel and freidlin studied small random perturbations dx varepsilon t b x varepsilon t dt sqrt varepsilon sigma x varepsilon t d beta t of the dynamical system dx t b x t dt they proved a large deviations theorem for the family x varepsilon t and used the large deviations estimates to prove among other things a result about exit from a bounded domain containing an equilibrium of the unperturbed system fleming rederived this exit theorem using techniques from stochastic control theory in this paper we study analogous questions for singular perturbations of degenerate diffusions we consider the family of processes dx varepsilon t b x varepsilon t dt tau x varepsilon t circ dz t sqrt varepsilon sigma x varepsilon t d beta t for varepsilon geq 0 where the process x 0 t is degenerate we show that a large deviations principle need not hold but we derive bounds sufficient for obtaining some estimates on probabilities we also adapt fleming s approach to prove an exit theorem

quantum automorphism groups of homogeneous graphs associated to a finite graph x is its quantum automorphism group g the main problem is to compute the poincare series of g meaning the series f z 1 c1z c2z2 whose coefficients are multiplicities of 1 into tensor powers of the fundamental representation in this paper we find a duality between certain quantum groups and planar algebras which leads to a planar algebra formulation of the problem together with some other results this gives f for all homogeneous graphs having 8 vertices or less

existence for 2 point boundary value problems in two ion electrodiffusion abstract in this paper we consider the two point boundary value problem y y y 0 2 y 2 2 l y 0 2 y 1 2 2 x l y 0 2 y 1 2 2 d x 0 1 y 0 0 y 1 which arises when two ions with the same valency diffuse and migrate across a liquid junction under the influence of an electric field e here y is proportional to the electric field e and after scaling the junction occupies the region 0 x 1 the parameters l and d are functions of the physical parameters of the problem and the range of physical interest is l 0 1 0 using the maximum principle we show that positive solutions are strictly decreasing and satisfy y 0 y 1 1 l and that there are no negative solutions using schauder degree theory and upper and lower solutions we show positive solutions exist if formula these results extend and unify those of the author acta math sci 8 1988 373 387 we briefly discuss the corresponding model for two ions of different valencies

one sided complete stable minimal surfaces we prove that there are no complete one sided stable minimal surfaces in the euclidean 3 space we classify least area surfaces in the quotient of mathbb r 3 by one or two linearly independent translations and we give sharp upper bounds of the genus of compact two sided index one minimal surfaces in non negatively curved ambient spaces finally we estimate from below the index of complete minimal surfaces in flat spaces in terms of the topology of the surface

quantitative index bounds for translators via topology we obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form the estimate involves the dimension of the space of weighted square integrable f harmonic 1 forms by the adaptation to the weighted setting of li tam theory developed in previous works this yields estimates in terms of the number of ends of the hypersurface when this is contained in a upper halfspace with respect to the translating direction when there exists a point where all principal curvatures are distinct we estimate the nullity of the stability operator this permits to obtain quantitative estimates on the stability index via the topology of translators with bounded norm of the second fundamental form which are either two dimensional or in higher dimension have finite topological type and are contained in a upper halfspace

quantum isometry groups examples and computations in this follow up of 4 where the quantum isometry group of a noncommutative manifold has been defined we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori it is also proved that the quantum isometry group of an isospectral deformation of a classical or noncommutative manifold is a suitable deformation of the quantum isometry group of the original undeformed manifold

differential equations of electrodiffusion the equations governing one dimensional steady state electrodiffusion are considered when there are arbitrarily many mobile ionic species present in any number of valence classes possibly also with a uniform distribution of fixed charges exact constant field solutions and new formulas of goldman hodgkin katz type are found all of these formulas are exact unlike the usual approximate ones corresponding boundary conditions on the ionic concentrations are identified the question of uniqueness of constant field solutions with such boundary conditions is considered and is reposed in terms of an autonomous ordinary differential equation of order n 1 for the electric field where n is the number of valence classes when there are no fixed charges the equation can be integrated once to give the nonautonomous equation of order n considered previously in the literature including in the case n 2 the form of painleve s second equation considered first in the context of electrodiffusion by one of us when n 1 the new equation is a form of lie nard s equation uniqueness of the constant field solution is established in this case

immersions minimales premiere valeur propre du laplacien et volume conforme generalisation de quelques resultats de li yau en toute dimension caracterisation de la metrique admettant des immersions minimales par ses premieres fonctions propres

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

index one minimal surfaces in spherical space forms abstract we prove that orientable index one minimal surfaces in spherical space forms with large fundamental group have genus at most two this confirms a conjecture of r schoen for an infinite class of 3 manifolds

local removable singularity theorems for minimal laminations in this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a riemannian three manifold this removable singularity theorem is the key result used in our proof that a complete embedded minimal surface in r 3 with quadratic decay of curvature has nite total curvature

min max minimal hypersurface in manifolds with convex boundary and hbox ric ge 0 let m n 1 partial m g be a compact manifold with non negative ricci curvature convex boundary and 2 le n le 6 we show that the min max minimal hypersurface with respect to one parameter families of hypersurfaces in m partial m is orientable of index one and multiplicity one

a family of differential equations arising from multi ion electrodiffusion the system of nernst planck equations and gauss s law describing an arbitrary mixture of ions in steady state drift and diffusion is shown to be equivalent to a single ordinary differential equation of order equal to the number of distinct ionic charges in the system

a poisson nernst planck model for biological ion channels an asymptotic analysis in a three dimensional narrow funnel we wish to predict ionic currents that flow through narrow protein channels of biological membranes in response to applied potential and concentration differences across the channel when some features of channel structure are known we propose to apply singular perturbation analysis to the coupled poisson nernst planck equations which are the basic continuum model of ionic permeation and semiconductor physics in semiconductor physics the problem is a singular perturbation because the ratio of the debye length to the width of the channel is a very small parameter that multiplies the laplacian term in the poisson equation in contrast to semiconductors the atomic scale geometry of narrow ion channels sometimes makes this ratio a large parameter which surprisingly renders the problem a singular perturbation in a different sense we construct boundary layers and match them asymptotically across the different regions of the channel to derive good approximations for fick s and ohm s laws our aim is to e

quantum automorphism group of the lexicographic product of finite regular graphs we study the quantum automorphism group of the lexicographic product of two finite regular graphs providing a quantum generalization of sabidussi s structure theorem on the automorphism group of such a graph

the classification of cmc foliations of bbb r 3 and bbb s 3 with countably many singularities in this paper we generalize our previous local removable singularity theorem for minimal laminations to the case of weak h laminations with h in bbb r constant in a punctured ball of a riemannian three manifold we also obtain a curvature estimate for any weak cmc foliation with possibly varying constant mean curvature from leaf to leaf of a compact riemannian three manifold n with boundary solely in terms of a bound of the absolute sectional curvature of n and of the distance to the boundary of n we then apply these results to classify weak cmc foliations of bbb r 3 and bbb s 3 with a closed countable set of singularities

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

on a neumann boundary value problem for the painleve ii equation in two ion electro diffusion abstract a two point neumann boundary value problem for a two ion electro diffusion model reducible to the painleve ii equation is investigated the problem is unconventional in that the model equation involves yet to be determined boundary values of the solution in prior work by thompson the existence of a solution was established subject to an inequality on the physical parameters here a two dimensional shooting method is used to show that this restriction may be removed a practical algorithm for the solution of the boundary value problem is presented in an appendix

simple compact quantum groups i the notion of simple compact quantum group is introduced as non trivial noncommutative and noncocommutative examples the following families of compact quantum groups are shown to be simple a the universal quantum groups bu q for q gl n c satisfying qq in n 2 b the quantum automorphism groups aaut b of finite dimensional c algebras b endowed with the canonical trace when dim b 4 including the quantum permutation groups aaut xn on n points n 4 c the standard deformations kq of simple compact lie groups k and their twists kqu as well as rieffel s deformation kj

representations of quantum permutation algebras we develop a combinatorial approach to the quantum permutation algebras as hopf images of representations of type pi a s n to b h we discuss several general problems including the commutativity and cocommutativity ones the existence of tensor product or free wreath product decompositions and the tannakian aspects of the construction the main motivation comes from the quantum invariants of the complex hadamard matrices we show here that under suitable regularity assumptions the computations can be performed up to n 6

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

two parameter families of quantum symmetry groups we introduce and study natural two parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups specifically for each pair p q of non negative integers we define and investigate quantum groups o p q b p q s p q and h p q corresponding to respectively orthogonal groups bistochastic groups symmetric groups and hyperoctahedral groups in the first three cases the new quantum groups turn out to be related to the dual free products of free quantum groups studied earlier for h p q the situation is different and we show that h p 0 qiso fp where the latter can be viewed as a liberation of the classical isometry group of the p dimensional torus

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

quantum group of orientation preserving riemannian isometries we formulate a quantum group analogue of the group of orientation preserving riemannian isometries of a compact riemannian spin manifold more generally of a possibly r twisted and of compact type spectral triple the main advantage of this formulation which is directly in terms of the dirac operator is that it does not need the existence of any good laplacian as in our previous works on quantum isometry groups several interesting examples including those coming from rieffel type deformation as well as the equivariant spectral triples on su 2 and s c2 are discussed

flow decomposition and large deviations we study large deviations properties related to the behavior asegoes to 0 of diffusion processes generated bye2l1 l2 wherel1andl2are two second order differential operators extending recent results of doss and stroock and rabeherimanana the main tool is the decomposition theorem for flows of stochastic differential equations proved by bismut and kunita we give another application of flow decomposition in a nonlinear filtering problem

on stable compact minimal submanifolds of riemannian product manifolds abstract in this paper we prove a classification theorem for stable compact minimal submanifolds of a riemannian product of an m 1 dimensional m 1 3 hypersurface m 1 in euclidean space and any riemannian manifold m 2 when the sectional curvature k m 1 of m 1 satisfies 1 m 1 1 k m 1 1 in particular when the ambient space is an m dimensional m 3 compact hypersurface m in euclidean space if the sectional curvature k m of m satisfies 1 m 1 k m 1 then we conclude that there exist no stable compact minimal submanifolds in m

integration over quantum permutation groups abstract we find a combinatorial formula for the haar measure of quantum permutation groups this leads to a dynamic formula for laws of diagonal coefficients explaining the poisson free poisson convergence result for characters

a singularly perturbed boundary value problem modelling a semiconductor device this paper is concerned with the static one dimensional modelling of a semiconductor device namely the pn junction when a bias is applied the governing equations are the well known equations describing carrier transport in a semiconductor which consist of a system of ordinary differential equations subject to boundary conditions imposed at the contacts because of the different orders of magnitude of the solution components at the boundaries we scale the components individually and obtain a singular perturbation problem we analyse the equilibrium case zero bias applied and set up approximate models posed as singularly perturbed second order equations by neglecting the hole and electron current densities this makes sense for small forward bias and for moderate reverse bias for the full problem we prove an a priori estimate on the number of electron hole carrier pairs and derive asymptotic expansions as the perturbation parameter tends to zero by setting up the reduced system and the boundary

a compositional approach to quantum functions we introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2 category of quantum sets and quantum functions we use this framework to formulate a 2 categorical theory of quantum graphs which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups we show that the 2 categories of quantum sets and quantum graphs are semisimple we analyze dualisable and invertible 1 morphisms in these 2 categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs

the classical theory of minimal surfaces we present here a survey of recent spectacular successes in classical minimal surface theory we highlight this article with the theorem that the plane the helicoid the catenoid and the one parameter family frtgt2 0 1 of riemann minimal examples are the only complete properly embedded minimal planar domains in r 3 the proof of this result depends primarily on work of colding

quantum isometries of the finite noncommutative geometry of the standard model we compute the quantum isometry group of the finite noncommutative geometry f describing the internal degrees of freedom in the standard model of particle physics we show that this provides genuine quantum symmetries of the spectral triple corresponding to m f where m is a compact spin manifold we also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries

minimal hypersurfaces with bounded index we prove a structural theorem that provides a precise local picture of how a sequence of closed embedded minimal hypersurfaces with uniformly bounded index and volume if the ambient dimension is greater than three in a riemannian manifold m n g 3 le n le 7 can degenerate loosely speaking our results show that embedded minimal hypersurfaces with bounded index behave qualitatively like embedded stable minimal hypersurfaces up to controlled errors several compactness finiteness theorems follow from our local picture

nouveaux resultats concernant les petites perturbations de systemes dynamiques resume nous etudions les proprietes de grandes deviations associees au comportement quand e 0 du processus de diffusion x e de generateur l e e 2 l l ou l 1 2 i 1 r i 2 l 1 2 j 1 l j 2 b 1 r 1 l et b etant des champs de vecteurs reguliers sur r n les estimations obtenues permettent de resoudre le probleme des goulots de sortie hors d un ouvert d borne lorsque les champs de vecteurs gs 1 gs l sont a support compact k regulier contenu dans d nous formulons enfin et demontrons sous certaines conditions un principe de grandes deviations qui rend compte de la vitesse de convergence de x e vers x 0 e 0 sur un espace approprie valable meme quand x 0 est non degenere

long time estimates for the heat kernel associated with a uniformly subelliptic symmetric second order operator abstract second order subelliptic operators have been the subject of a considerable amount of research in recent years starting with the paper rothschild and stein in which the sharp form of hormander s famous subellipticity theorem is proved and continuing through the work of fefferman and phong and sanchez calle it has become increasingly clear that precise regularity estimates for these operators depend intimately on the geometry associated with the operator under consideration the main purpose of this article is to obtain bounds from above and below on p t x y t epsilon 1 infinity in terms of standard heat kernels

general curvature estimates for stable h surfaces in 3 manifolds applications we obtain an estimate for the norm of the second fundamental form of stable h surfaces in riemannian 3 manifolds with bounded sectional curvature our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself we give some applications in particular we obtain an interior gradient estimate for h sections in killing submersions

quantum groups acting on 4 points we classify the compact quantum groups acting on 4 points these are the quantum subgroups of the quantum permutation group q4 our main tool is a new presentation for the algebra c q4 corresponding to an isomorphism of type q4 so 1 3 the quantum subgroups of q4 are subject to a mckay type correspondence that we describe at the level of algebraic invariants

electrical structures of interfaces a painleve ii model a painleve ii model derived out of the classical nernst planck system is applied in the context of boundary value problems that describe the electric field distribution in a region x 0 occupied by an electrolyte for privileged flux ratios of the ion concentrations the auto backlund transformation admitted by the painleve ii equation may be applied iteratively to construct exact solutions to classes of physically relevant boundary value problems these representations involve in turn either yablonskii vorob ev polynomials or classical airy functions the requirement that the electric field distribution and ion concentrations in these representations be non singular imposes constraints on the physical parameters these are investigated in detail along with asymptotic properties

geometric singular perturbation approach to steady state poisson nernst planck systems boundary value problems of a one dimensional steady state poisson nernst planck pnp system for ion flow through a narrow membrane channel are studied by assuming the ratio of the debye length to a characteristic length to be small the pnp system can be viewed as a singularly perturbed problem with multiple time scales and is analyzed using the newly developed geometric singular perturbation theory within the framework of dynamical systems the global behavior is first studied in terms of limiting fast and slow systems it is rather surprising that a complete set of integrals is discovered for the nonlinear limiting fast system this allows a detailed description of the boundary layers for the problem the slow system itself turns out to be a singularly perturbed one too which indicates that the singularly perturbed pnp system has three different time scales a singular orbit zeroth order approximation of the boundary value problem is identified based on the dynamics of limiting fast and slow

an operator space characterization of dual operator algebras this paper is devoted to dual algebras that is w closed algebras of bounded operators on hilbert space we give abstract characterizations of unital dual algebras resp dual algebras up to w homeomorphic unital complete isometry resp complete isomorphism the results we obtain are natural w versions of analogous characterizations due to blecher ruan sinclair and blecher respectively in the category of operator algebras

hopf coactions on odd spheres abstract we prove that the q deformed unitary group i e u q n is the universal compact quantum group in the category of compact quantum groups which coact on the q deformed odd sphere s q 2 n 1 leaving the space spanned by the natural set of generators invariant and preserving the unique s u q n invariant functional on s q 2 n 1 using this we identify u q n as the quantum group of orientation preserving isometries in the sense of bhowmick and goswami 5 for a natural spectral triple associated with s q 2 n 1 constructed by chakraborty and pal 9

index estimates for free boundary minimal hypersurfaces we show that the morse index of a properly embedded free boundary minimal hypersurface in a strictly mean convex domain of the euclidean space grows linearly with the dimension of its first relative homology group which is at least as big as the number of its boundary components minus one in ambient dimension three this implies a lower bound for the index of a free boundary minimal surface which is linear both with respect to the genus and the number of boundary components thereby the compactness theorem by fraser and li implies a strong compactness theorem for the space of free boundary minimal surfaces with uniformly bounded morse index inside a convex domain our estimates also imply that the examples constructed in the unit ball by fraser schoen and folha pacard zolotareva have arbitrarily large index extensions of our results to more general settings including various classes of positively curved riemannian manifolds and other convexity assumptions are discussed

on integrable ermakov painleve iv systems novel hybrid ermakov painleve iv systems are introduced and an associated ermakov invariant is used in establishing their integrability backlund transformations are then employed to generate classes of exact solutions via the linked canonical painleve iv equation

ergodic coactions with large multiplicity and monoidal equivalence of quantum groups we construct new examples of ergodic coactions of compact quantum groups in which the multiplicity of an irreducible corepresentation can be strictly larger than the dimension of the latter these examples are obtained using a bijective correspondence between certain ergodic coactions on c algebras and unitary fiber functors on the representation category of a compact quantum group we classify these unitary fiber functors on the universal orthogonal and unitary quantum groups the associated c algebras and von neumann algebras can be defined by generators and relations but are not yet well understood

on a painleve ii model in steady electrolysis application of a backlund transformation a model equation of painleve ii type was introduced by bass in 1964 in connection with a boundary value problem which describes the electric field distribution in a region x 0 occupied by an electrolyte this is possibly the earliest explicit physical application of a painleve equation to be found in the literature here we return to this problem informed by the subsequent discovery of a backlund transformation for painleve ii this enables us to construct exact representations for the electric field and ion distributions for boundary value problems wherein the ratio of fluxes of the positive and negative ions at the boundary adopts one of an infinite sequence of values

stokes phenomena in discrete painleve i in this study we consider the asymptotic behaviour of the first discrete painleve equation in the limit as the independent variable becomes large using an asymptotic series expansion we identify two types of solutions which are pole free within some sector of the complex plane containing the positive real axis using exponential asymptotic techniques we determine stokes phenomena effects present within these solutions and hence the regions in which the asymptotic series expression is valid from a careful analysis of the switching behaviour across stokes lines we find that the first type of solution is uniquely defined while the second type contains two free parameters and that the region of validity may be extended for appropriate choice of these parameters

duality and normal parts of operator modules for an operator bimodule x over von neumann algebras a b h and b b k the space of all completely bounded a b bimodule maps from x into b k h is the bimodule dual of x basic duality theory is developed with a particular attention to the haagerup tensor product over von neumann algebras to x a normal operator bimodule xn is associated so that completely bounded a b bimodule maps from x into normal operator bimodules factorize uniquely through xn a construction of xn in terms of biduals of x a and b is presented various operator bimodule structures are considered on a banach bimodule admitting a normal such structure

nonlocal games and quantum permutation groups we present a strong connection between quantum information and quantum permutation groups specifically we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game moreover we show that two connected graphs x and y are quantum isomorphic if and only if there exists x v x and y v y that are in the same orbit of the quantum automorphism group of the disjoint union of x and y this connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours we exploit this link by using ideas and results from quantum information in order to prove new results about quantum automorphism groups and about quantum permutation groups more generally in particular we show that asymptotically almost surely all graphs have trivial quantum automorphism group furthermore we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive answering a question from the quantum group literature our main tool for proving these results is the introduction of orbits and orbitals orbits on ordered pairs of quantum permutation groups we show that the orbitals of a quantum permutation group form a coherent configuration algebra a notion from the field of algebraic graph theory we then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group we furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other

quantum group of isometries in classical and noncommutative geometry we formulate a quantum generalization of the notion of the group of riemannian isometries for a compact riemannian manifold by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples and then proving the existence of a universal object called the quantum isometry group in the category of compact quantum groups acting smoothly and isometrically on a given possibly noncommutative manifold satisfying certain regularity assumptions the idea of quantum families due to woronowicz and soltan are relevant to our construction a number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed

on the quantum symmetry of distance transitive graphs abstract in this article we study quantum automorphism groups of distance transitive graphs we show that the odd graphs the hamming graphs h n 3 the johnson graphs j n 2 and the kneser graphs k n 2 do not have quantum symmetry we also give a table with the quantum automorphism groups of all cubic distance transitive graphs furthermore with one graph missing we can now decide whether or not a distance regular graph of order 20 has quantum symmetry moreover we prove that the hoffman singleton graph has no quantum symmetry on a final note we present an example of a pair of graphs with the same intersection array the shrikhande graph and the 4 4 rook s graph where one of them has quantum symmetry and the other one does not

a singular perturbation analysis of the fundamental semiconductor device equations in this paper we present a singular perturbation analysis of the fundamental semiconductor device equations which form a system of three second order elliptic differential equations subject to mixed neumann dirichlet boundary conditions the system consists of poisson s equation and the continuity equations and describes potential and carrier distributions in an arbitrary emiconductor device the singular perturbation parameter is the minimal normed debye length of the device under consideration using matched asymptotic expansions we demonstrate the occurrence of internal layers at surfaces across which the impurity distribution appearing as an inhomogeneity of poisson s equation has a jump discontinuity these surfaces are called junctions and the occurrence of boundary layers at semiconductor oxide interfaces we derive the layer equations and the reduced problem charge neutral approximation and give existence proofs for these problems the layer solutions which characterize the solutions of the s

the morita theory of quantum graph isomorphisms we classify instances of quantum pseudo telepathy in the graph isomorphism game exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups specifically we show that graphs quantum isomorphic to a given graph are in bijective correspondence with morita equivalence classes of certain frobenius algebras in the category of finite dimensional representations of the quantum automorphism algebra of that graph we show that such a frobenius algebra may be constructed from a central type subgroup of the classical automorphism group whose action on the graph has coisotropic vertex stabilisers in particular if the original graph has no quantum symmetries quantum isomorphic graphs are classified by such subgroups we show that all quantum isomorphic graph pairs corresponding to a well known family of binary constraint systems arise from this group theoretical construction we use our classification to show that of the small order vertex transitive graphs with no quantum symmetry none is quantum isomorphic to a non isomorphic graph we show that this is in fact asymptotically almost surely true of all graphs

series solution of painleve ii in electrodiffusion conjectured convergence a perturbation series solution is constructed with the use of airy functions for a nonlinear two point boundary value problem arising in an established model of steady electrodiffusion in one dimension with two ionic species carrying equal and opposite charges the solution includes a formal determination of the associated electric field which is known to satisfy a form of the painleve ii differential equation comparisons with the numerical solution of the boundary value problem show excellent agreement following termination of the series after a sufficient number of terms for a much wider range of values of the parameters in the model than suggested by previously presented analysis or admitted by previously presented approximation schemes these surprising results suggest that for a wide variety of cases a convergent series expansion for the painleve transcendent describing the electric field has been obtained a suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases it is conjectured that the series does converge for this subset

quantum isometry groups of the podles spheres abstract for 0 1 c 0 we identify the quantum group so 3 as the universal object in the category of compact quantum groups acting by orientation and volume preserving isometries in the sense of bhowmick and goswami 2009 4 on the natural spectral triple on the podles sphere s c 2 constructed by dabrowski d andrea landi and wagner 2007 in 9