new identities between unitary minimal virasoro characters two sets of identities between unitary minimal virasoro characters at levelsm 3 4 5 are presented and proven the first identity suggests a connection between the ising and the tricritical ising models since them 3 virasoro characters are obtained as bilinears ofm 4 virasoro characters the second identity given the tricritical ising model characters as bilinears in the ising model characters and the six combinations ofm 5 virasoro characters which do not appear in the spectrum of the three state potts model the implication of these identities on the study of the branching rules ofn 4 superconformal characters into s widehat u 2 times s widehat u 2 characters is discussed

some hecke rogers type identities abstract recently a new partition function associated with ramanujan s third order mock theta function q was discovered and subsequently its overpartition analogue was introduced in this paper we prove an intriguing identity arising from the study of that analogue which involves a double series of hecke rogers type in addition two further identities will be given our proofs heavily rely on formulas from the work of liu 12

on the complete classification of unitary n 2 minimal superconformal field theories aiming at a complete classification of unitary n 2 minimal models where the assumption of space time supersymmetry has been dropped it is shown that each modular invariant candidate partition function of such a theory is indeed the partition function of a fully fledged unitary n 2 minimal model subject to the assumptions that orbifolding is a physical process and that the space time supersymmetric mathcal a mathcal d mathcal e models are physical a family of models constructed via orbifoldings of either the diagonal model or of the space time supersymmetric exceptional models then demonstrates that there exists a unitary n 2 minimal model for every one of the allowed partition functions in the list obtained from gannon s work gannon in nucl phys b 491 659 688 1997

remarks on the connection between the additive separation of the hamilton jacobi equation and the multiplicative separation of the schrodinger equation ii first integrals and symmetry operators the commutation relations of the first order and second order operators associated with the first integrals in involution of a hamiltonian separable system are examined it is shown that these operators commute if and only if a pre robertson condition is satisfied this condition involves the ricci tensor of the configuration manifold and it is implied by the robertson condition which is necessary and sufficient for the separability of the schrodinger equation

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

superintegrable three body systems on the line we consider classical three body interactions on a euclidean line depending on the reciprocal distance of the particles and admitting four functionally independent quadratic in the momentum first integrals these systems are multiseparable superintegrable and equivalent up to rescalings to a one particle system in the three dimensional euclidean space common features of the dynamics are discussed we show how to determine quantum symmetry operators associated with the first integrals considered here but do not analyze the corresponding quantum dynamics the conformal multiseparability is discussed and examples of conformal first integrals are given the systems considered here in generality include the calogero wolfes and other three body interactions widely studied in mathematical physics

enriched two dimensional mixed finite element models for linear elasticity with weak stress symmetry abstract the purpose of this article is to derive and analyze new discrete mixed approximations for linear elasticity problems with weak stress symmetry these approximations are based on the application of enriched versions of classic poisson compatible spaces for stress and displacement variables and or on enriched stokes compatible space configurations for the choice of rotation spaces used to weakly enforce stress symmetry accordingly the stress space has to be adapted to ensure stability such enrichment procedures are done via space increments with extra bubble functions which have their support on a single element in the case of h 1 conforming approximations or with vanishing normal components over element edges in the case of h div conforming spaces the advantage of using bubbles as stabilization corrections relies on the fact that all extra degrees of freedom can be condensed in a way that the number of equations to be solved and the matrix structure are not affected enhanced approximations are observed when using the resulting enriched space configurations which may have different orders of accuracy for the different variables a general error analysis is derived in order to identify the contribution of each kind of bubble increment on the accuracy of the variables individually the use of enriched poisson spaces improves the rates of convergence of stress divergence and displacement variables stokes enhancement by bubbles contributes to equilibrate the accuracy of weak stress symmetry enforcement with the stress approximation order reaching the maximum rate given by the normal traces which are not affected

de rham diagram for hp finite element spaces abstract we prove that the hp finite elements for h curl spaces introduced in 1 fit into a general de rham diagram involving hp approximations the corresponding interpolation operators generalize the notion of hp interpolation introduced in 2 and are different from the classical operators of nedelec and raviart thomas

quadrilateral h div finite elements we consider the approximation properties of quadrilateral finite element spaces of vector fields defined by the piola transform extending results previously obtained for scalar approximation the finite element spaces are constructed starting with a given finite dimensional space of vector fields on a square reference element which is then transformed to a space of vector fields on each convex quadrilateral element via the piola transform associated to a bilinear isomorphism of the square onto the element for affine isomorphisms a necessary and sufficient condition for approximation of order r 1 in l2 is that each component of the given space of functions on the reference element contain all polynomial functions of total degree at most r in the case of bilinear isomorphisms the situation is more complicated and we give a precise characterization of what is needed for optimal order l2 approximation of the function and of its divergence as applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for some standard finite element approximations of h div we also derive new estimates for approximation by quadrilateral raviart thomas elements requiring less regularity and propose a new quadrilateral finite element space which provides optimal order approximation in h div finally we demonstrate the theory with numerical computations of mixed and least squares finite element approximations of the solution of poisson s equation

two dimensional mixed finite element approximations for elliptic problems with enhanced accuracy for the potential and flux divergence abstract the purpose of the present paper is to analyse two new different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal potential variables one for triangles and the other one for quadrilateral elements to be used in discrete versions of the mixed finite element method for elliptic problems they can be interpreted as enriched versions of b d f m k 1 spaces based on triangles and of r t k spaces for quadrilateral elements the new flux approximations are incremented with properly chosen internal shape functions with vanishing normal components on the edges of degree k 2 and matching primal functions of degree k 1 higher than the border fluxes which are kept of degree k in all these cases the divergence of the flux space coincide with the primal approximation space on the master element producing stable simulations using static condensation the global condensed system to be solved in the enriched cases has same dimension and structure of the original ones which is proportional to the space dimension of the border fluxes for each element geometry measuring the errors with l 2 norms the enriched space configurations give higher convergence rate of order k 2 for the primal variable while keeping the order k 1 for the flux for affine meshes the divergence errors have the same improved accuracy rate as for the error in the primal variable for quadrilateral non affine meshes for instance trapezoidal elements the divergence error has order k 1 one unit more than the order k occurring for r t k spaces on this kind of deformed meshes this fact also holds for a b f k elements but for them the potential order of accuracy does not improve keeping order k 1

quantum r matrix for g2 and a solvable 175 vertex model the quantum r matrix for cartan s exceptional simple lie algebra g2 in its seven dimensional minimal representation is presented the r matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra uq g2 this defines a new solvable seven state 175 vertex model on the planar square lattice whose boltzmann weights satisfy the yang baxter equation for the equivalent face model the local state probability is obtained in terms of a relative of a level 1 string function related to the affine lie algebra pair b3 1 contains implies g2 1

hierarchical high order finite element bases for h div spaces based on curved meshes for two dimensional regions or manifolds the mixed finite element formulation for elliptic problems is characterized by simultaneous calculations of the potential primal variable and of the flux field dual variable this work focuses on new h div conforming finite element spaces which are suitable for flux approximations based on curved meshes of a planar region or a manifold domain embedded in r 3 the adopted methodology for the construction of h div bases consists in using hierarchical h 1 conforming scalar bases multiplied by vector fields that are properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure they are classified as being of edge or internal type the normal component of an edge function coincides on the corresponding edge with the associated scalar shape function and vanishes over the other edges and the normal components of an internal shape function vanishes on all element edges these properties are fundamental for the global assembly of h div conforming functions locally defined by these vectorial shape functions for applications to the mixed formulation the configuration of the approximation spaces is such that the divergence of the dual space and the primal approximation space coincides results of verification numerical tests are presented for curved triangular and quadrilateral partitions on circular cylindrical and spherical regions demonstrating stable convergence with optimal convergence rates coinciding for primal and dual variables innovative two dimensional hdiv approximation spaces for curved quadrilateral and triangular elements hdiv bases properly constructed on the master element and mapped to the geometrical elements by the piola transformation followed by a normalization procedure stable convergence with optimal convergence rates coinciding for primal and dual variables study of number of condensed equations as a function of the polynomial order of the shape functions

three manifold invariants from chern simons field theory with arbitrary semi simple gauge groups invariants for framed links in s 3 obtained from chern simons gauge field theory based on an arbitrary gauge group semi simple have been used to construct a three manifold invariant this is a generalization of a similar construction developed earlier for su 2 chern simons theory the procedure exploits a theorem of lickorish and wallace and also those of kirby fenn and rourke which relate three manifolds to surgeries on framed unoriented links the invariant is an appropriate linear combination of framed link invariants which does not change under kirby calculus this combination does not see the relative orientation of the component knots the invariant is related to the partition function of chern simons theory this thus provides an efficient method of evaluating the partition function for these field theories as some examples explicit computations of these manifold invariants for a few three manifolds have been done

integrability versus separability for the multi centre metrics the multi centre metrics are a family of euclidean solutions of the empty space einstein equations with self dual curvature for this full class we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta induced by a killing stackel tensor our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems within this subclass we analyze on the one hand the separation of coordinates in the hamilton jacobi equation and on the other hand the construction of some new killing yano tensors

three dimensional hierarchical mixed finite element approximations with enhanced primal variable accuracy abstract there are different possibilities of choosing balanced pairs of approximation spaces for dual flux and primal pressure variables to be used in discrete versions of the mixed finite element method for elliptic problems arising in fluid simulations three cases shall be studied and compared for discretized three dimensional formulations based on tetrahedral hexahedral and prismatic meshes the principle guiding the constructions is the property that the divergence of the dual space and the primal approximation space should coincide while keeping the same order of accuracy for the flux variable and varying the accuracy order of the primal variable there is the classic case of b d m k spaces based on tetrahedral meshes and polynomials of total degree k for the dual variable and k 1 for the primal variable showing stable simulations with optimal convergence rates of orders k 1 and k respectively another case is related to r t k and b d f m k 1 spaces for hexahedral and tetrahedral meshes respectively but holding for prismatic elements as well it gives identical approximation order k 1 for both primal and dual variables an improvement in accuracy obtained by increasing the degree of primal functions to k and by enriching the dual space with some properly chosen internal shape functions of degree k 1 while keeping degree k for the border fluxes a new type of approximation is proposed by further incrementing the order of some internal flux functions to k 2 and matching primal functions to k 1 higher than the border fluxes of degree k thus higher convergence rate of order k 2 is obtained for the primal variable using static condensation the global condensed system to be solved in all the cases has same dimension and structure which is proportional to the space dimension of the border fluxes for each element geometry illustrating results comparing the three different space configurations are presented for simulations based on hierarchical high order shape functions for h div conforming spaces which are specially constructed for affine tetrahedral hexahedral and prismatic meshes expected convergence rates are obtained for the flux pressure and divergence variables

quantumr matrices related to the spin representations ofbn anddn we present the explicit form of the trigonometricr matrices related to the spin representations of the simple lie algebrasx n b n d n we conjecture that one dimensional configuration sums of the corresponding vertex models in the face formulation are the string functions ofx 1 modules

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

the problem of quantum integrability abstract the notion of integrability in quantum mechanics is investigated in order to prepare rigorous grounds for the study of regular and irregular behaviour of quantum systems its common sense definition turns out to have deficiencies which are illustrated by various explicit examples part of the ambiguities are shown to have their origin in the difficulty of transferring the concept of independent constants of motion into quantum mechanics due to a fundamental theorem on sets of commuting operators by von neumann taking into account the classical limit with coherent states does not resolve the problems as a result it is pointed out why the appealing phenomenological distinction between regular and chaotic quantum systems cannot be traced back to the present notion of quantum integrability in a mathematically rigorous way

quantum integrability of quadratic killing tensors quantum integrability of classical integrable systems given by quadratic killing tensors on curved configuration spaces is investigated it is proven that using a minimal quantization scheme quantum integrability is ensured for a large class of classic examples

virasoro character identities and artin l functions some identities between unitary minimal virasoro characters at levels m 3 4 5 are shown to arise as a consequence of relations between artin l functions of different quadratic fields the definitions and concepts of number theory necessary to present the theta function identities which can be derived from these relations are introduced a new infinite family of identities between virasoro characters at level 3 and level m 4a 2 for a odd and 1 4a 2 a 2 p where p is prime is obtained as a by product

meromorphic c 24 conformal field theories modular invariant conformal field theories with just one primary field and central charge c 24 are considered it has been shown previously that if the chiral algebra of such a theory contains spin 1 currents it is either the leech lattice cft or it contains a kac moody sub algebra with total central charge 24 in this paper all meromorphic modular invariant combinations of the allowed kac moody combinations are obtained the result suggests the existence of 71 meromorphic c 24 theories including the 41 that were already known

systematic and generic construction of shape functions for p adaptive meshes of multidimensional finite elements this paper presents a methodology for generating high order shape functions for the complete family of finite elements the geometric entities presented are the point line triangle quadrilateral tetrahedron pyramid prism and hexahedron the shape functions constructed are hierarchical and generate continuous approximation spaces the order of interpolation of the shape function can be determined independently for each edge face and volume the space of interpolation of order p for each element is complete this definition allows for the construction of hybrid meshes with any combination of these elements including elements of distinct dimension and or interpolation order the reader will observe that our systematic way of constructing shape functions can be extended to elements of higher dimension

a new procedure for the construction of hierarchical high order hdiv and hcurl finite element spaces this paper considers a systematic procedure for the construction of a hierarchy of high order finite element approximation for hdiv and hcurl spaces based on triangular and quadrilateral partitions of bidimensional domains the principle is to choose an appropriate set of vectors based on the geometry of each element which are multiplied by an available set of h 1 hierarchical scalar basic functions this strategy produces vector basis functions with continuous normal or tangent components on the elements interfaces properties that characterise functions in hdiv or hcurl respectively we also present a numerical study to evaluate the correct balancedness of the resulting hdiv spaces of degree k and l 2 spaces of degree k 1 on the resolution of the mixed formulation for a steklov eigenvalue problem

remarks on the connection between the additive separation of the hamilton jacobi equation and the multiplicative separation of the schrodinger equation i the completeness and robertson conditions the fundamental elements of the variable separation theory are revisited including the eisenhart and robertson theorems kalnins miller theory and the intrinsic characterization of the separation of the hamilton jacobi equation in a unitary and geometrical perspective the general notion of complete integrability of first order normal systems of pdes leads in a natural way to completeness conditions for separated solutions of the schrodinger equation and to the robertson condition two general types of multiplicative separation for the schrodinger equation are defined and analyzed they are called free and reduced separation respectively in the free separation the coordinates are necessarily orthogonal while the reduced separation may occur in nonorthogonal coordinates but only in the presence of symmetries killing vectors

fermionic representations for characters of calm 3 t calm 4 5 calm 5 6 and calm 6 7 minimal models and related dilogarithm and rogers ramanujan type identities characters and linear combinations of characters that admit a fermionic sum representation as well as a factorized form are considered for some minimal virasoro models as a consequence various rogers ramanujan type identities are obtained dilogarithm identities producing corresponding effective central charges and secondary effective central charges are derived several ways of constructing more general fermionic representations are discussed

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

various quantum mechanical aspects of quadratic forms two of the oldest known classical integrable systems are i n decoupled harmonic oscillators constrained to the surface of the n sphere i e the classical c neumann system and ii geodesic flow on an n axial ellipsoid we integrate both these systems at the quantum level that is in both cases we construct n 1 pairwise commuting partial differential operators which in turn commute with the respective quantum hamiltonians moreover the joint eigenvalues of the commuting partial differential operators appear to be encoded linearly as parameters in an n parameter eigenvalue equation given by a second order complex ordinary differential equation with automorphic boundary conditions we then focus on the particular case of two degrees of freedom i e r3 via the microlocal wave averaging ansatz combined with explicit wkb constructions we give a spectral analysis both classical and semiclassical of the above systems and subsequently exhibit connections between the various spectral asymptotics in the course of this analysis we show that the above two systems are intimately related to yet another quantum integrable system the reduced quantum asymmetric rigid body in a vacuum

implementation of continuous h p adaptive finite element spaces without limitations on hanging sides and distribution of approximation orders adaptive techniques using h p refinement are known to be one of the most efficient methodologies to accelerate the convergence of finite element algorithms however the implementation of computational tools for the development of h p adaptive algorithms is intricate and depends strongly on the data structure there exist few computational environments available to the scientific finite element community capable to implement h p adaptive approximation spaces for the complete family of finite element topologies and which implement hanging sides this article describes a methodology for the development of continuous h p adaptive finite element approximation spaces without constraints on the refinement strategy concerning the difference of levels and approximation orders between neighboring elements the shape functions are hierarchical and the coefficient constraints associated with hanging sides that can occur in non conformal geometric meshes are defined using l 2 projections the topological and functional aspects of the construction are described in one two and three dimensions for a variety of geometric entities line triangle quadrilateral tetrahedron pyramid prism and hexahedron the implementation is demonstrated in the object oriented scientific computational environment neopz http github com labmec neopz neopz is a general finite element approximation software which incorporates a variety of variational formulations validation of the refinement methodology is demonstrated by two and three dimensional numerical experiments

the semi infinite cohomology of affine lie algebras we study the semi infinite or brst cohomology of affine lie algebras in detail this cohomology is relevant in the brst approach to gauged wznw models our main result is to prove necessary and sufficient conditions on ghost numbers and weights for non trivial elements in the cohomology in particular we prove the existence of an infinite sequence of elements in the cohomology for non zero ghost numbers this will imply that the brst approach to the topological wznw model admits many more states than a conventional coset construction this conclusion also applies to some non topological models

a remark concerning divergence accuracy order for h div conforming finite element flux approximations abstract the construction of finite element approximations in h div usually requires the piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region it is known that degradation may occur in convergence order if non affine geometric mappings are used on this point we revisit a general procedure for the improvement of two dimensional flux approximations discussed in a recent paper of this journal comput math appl 74 2017 3283 3295 the starting point is an approximation scheme which is known to provide l 2 errors with accuracy of order k 1 for sufficiently smooth flux functions and of order r 1 for flux divergence an example is r t k spaces on quadrilateral meshes where r k or k 1 if linear or bilinear geometric isomorphisms are applied furthermore the original space is required to be expressed by a factorization in terms of edge and internal shape flux functions the goal is to define a hierarchy of enriched flux approximations to reach arbitrary higher orders of divergence accuracy r n 1 as desired for any n 1 the enriched versions are defined by adding higher degree internal shape functions of the original family of spaces at level k n while keeping the original border fluxes at level k the case n 1 has been discussed in the mentioned publication for two particular examples general stronger enrichment n 1 shall be analyzed and applied to darcy s flow simulations the global condensed systems to be solved having same dimension and structure of the original scheme

two families of h div mixed finite elements on quadrilaterals of minimal dimension we develop two families of mixed finite elements on quadrilateral meshes for approximating mathbf u p solving a second order elliptic equation in mixed form standard raviart thomas rt and brezzi douglas marini bdm elements are defined on rectangles and extended to quadrilaterals using the piola transform which are well known to lose optimal approximation of nabla cdot mathbf u arnold boffi falk spaces rectify the problem by increasing the dimension of rt so that approximation is maintained after piola mapping our two families of finite elements are uniformly inf sup stable achieve optimal rates of convergence and have minimal dimension the elements for mathbf u are constructed from vector polynomials defined directly on the quadrilaterals rather than being transformed from a reference rectangle by the piola mapping and then supplemented by two one for the lowest order basis functions that are piola mapped one family has full h div approximation mathbf u p

hierarchical high order finite element spaces in h div ω h1 ω for a stabilized mixed formulation of darcy problem abstract the classical dual mixed finite element method for flow simulations is based on h div conforming approximation spaces for the flux which guarantees continuous normal components on element interfaces and discontinuous approximations in l 2 for the pressure however stability and convergence can only be obtained for compatible approximation spaces stabilized finite element methods may provide an alternative stable procedure to avoid this kind of delicate balance the main purpose of this paper is to present a high order finite element methodology to solve the darcy problem based on the combination of an unconditionally stable mixed finite element method with a hierarchical methodology for the construction of finite dimensional subspaces of h div and h 1 the chosen stabilized method is free of mesh dependent stabilization parameters and allows for the use of different high order finite element approximations for the flux and the pressure variables without requiring any compatibility constraint as required in mixed methods for these problems convergence studies are presented comparing the numerical solutions obtained for different approximation orders on quadrilateral elements with the ones given by classical mixed formulation with raviart thomas elements

representations of affine kac moody algebras and the affine scalar product simple procedures are given for finding all the dominant weights in a highest weight representation of an affine algebra for finding the weyl orbit of an arbitrary weight and for determining whether or not any given weight is in any given representation a simple definition of congruency is given that applies to all affine algebras the standard indefinite scalar product is generalized the generalization is used in the procedures

orientation embedded high order shape functions for the exact sequence elements of all shapes a unified construction of high order shape functions is given for all four classical energy spaces h1 h curl h div and l2 and for elements of all shapes segment quadrilateral triangle hexahedron tetrahedron triangular prism and pyramid the discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element the shape functions are conforming hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape the polynomial order is allowed to differ for each separate topological entity vertex edge face or interior in the mesh so the shape functions can be used to implement local p adaptive finite element methods each topological entity may have its own orientation and the shape functions can have that orientation embedded by a simple permutation of arguments