gauge theories labelled by three manifolds we propose a dictionary between geometry of triangulated 3 manifolds and physics of three dimensional n 2 gauge theories under this duality standard operations on triangulated 3 manifolds and various invariants thereof classical as well as quantum find a natural interpretation in field theory for example independence of the sl 2 chern simons partition function on the choice of triangulation translates to a statement that s 3 b partition functions of two mirror 3d n 2 gauge theories are equal three dimensional n 2 field theories associated to 3 manifolds can be thought of as theories that describe boundary conditions and duality walls in four dimensional n 2 scfts thus making the whole construction functorial with respect to cobordisms and gluing

a weak formulation of roe s approximate riemann solver abstract this paper presents a weak formulation of roe s approximate riemann solver based on a notion of weak solution to nonlinear hyperbolic systems in nonconservative form this weak formulation is applied to the euler equations for real gases and the homogeneous equilibrium two phase flow model numerical results for a two phase flow heat addition problem are given

analyticity of resonances and eigenvalues and spectral properties of the massless spin boson model abstract we extend the method of pizzo multiscale analysis for resonances introduced in 5 in order to infer analytic properties of resonances and eigenvalues and their eigenprojections as well as estimates for the localization of the spectrum of dilated hamiltonians and norm bounds for the corresponding resolvent operators in neighborhoods of resonances and eigenvalues we apply our method to the massless spin boson model assuming a slight infrared regularization we prove that the resonance and the ground state eigenvalue and their eigenprojections are analytic with respect to the dilation parameter and the coupling constant moreover we prove that the spectrum of the dilated spin boson hamiltonian in the neighborhood of the resonance and the ground state eigenvalue is localized in two cones in the complex plane with vertices at the location of the resonance and the ground state eigenvalue respectively additionally we provide norm estimates for the resolvent of the dilated spin boson hamiltonian near the resonance and the ground state eigenvalue the topic of analyticity of eigenvalues and resonances has let to several studies and advances in the past however to the best of our knowledge this is the first time that it is addressed from the perspective of pizzo multiscale analysis once the multiscale analysis is set up our method gives easy access to analyticity essentially it amounts to proving it for isolated eigenvalues only and use that uniform limits of analytic functions are analytic the type of spectral and resolvent estimates that we prove are needed to control the time evolution including the scattering regime the latter will be demonstrated in a forthcoming publication the introduced multiscale method to study spectral and resolvent estimates follows its own inductive scheme and is independent and different from the method we apply to construct resonances

a p adic regulator problem in algebraic k theory and group cohomology is injective where p is a fixed rational prime and n 1 is odd observe that each p is clearly injective for n 1 because ki 0 0 and k op op a much harder problem is whether zp is injective for n 1 and f totally real abelian injectively of zp on the subgroup of 0 consisting of those elements congruent to lmodp for each p p is equivalent to nonvanishing of the p adic regulator br c as an example of 1 let f be quadratic imaginary then is

elliptic hypergeometry of supersymmetric dualities ii orthogonal groups knots and vortices we consider seiberg electric magnetic dualities for 4d mathcal n 1 sym theories with so n gauge group for all such known theories we construct superconformal indices scis in terms of elliptic hypergeometric integrals equalities of these indices for dual theories lead both to proven earlier special function identities and new conjectural relations for integrals in particular we describe a number of new elliptic beta integrals associated with the s confining theories with the spinor matter fields reductions of some dualities from sp 2n to so 2n or so 2n 1 gauge groups are described interrelation of scis and the witten anomaly is briefly discussed possible applications of the elliptic hypergeometric integrals to a two parameter deformation of 2d conformal field theory and related matrix models are indicated connections of the reduced scis with the state integrals of knot theory generalized agt duality for 3 3 d theories and a 2d vortex partition function are described

estimates for the norms of products of sines and cosines abstract in this paper we prove asymptotic formulas for the l p norms of p n k 1 n 1 e i k and q n k 1 n 1 e i k these products can be expressed using k 1 n sin k 2 and k 1 n cos k 2 respectively we prove an estimate for p n at a point near where its maximum occurs finally we give an asymptotic formula for the maximum of the fourier coefficients of q n

trigonometry of spacetimes a new self dual approach to a curvature signature in dependent trigonometry a new method to obtain trigonometry for the real spaces of constant curvature and metric of any even degenerate signature is presented the method could be described as curvature signature in dependent trigonometry and encapsulates trigonometry for all these spaces into a single basic trigonometric group equation this brings to its logical end the idea of an absolute trigonometry and provides equations which hold true for the nine two dimensional spaces of constant curvature and any signature this family of spaces includes both relativistic and non relativistic spacetimes therefore a complete discussion of trigonometry in the six de sitter minkowskian newton hooke and galilean spacetimes follow as particular instances of the general approach distinctive traits of the method are universality and self duality every equation is meaningful for the nine spaces at once and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves these basic structural properties allow a complete study of trigonometry and in fact any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes in most cases these equations are new

a construction of killing spinors on s n we derive simple general expressions for the explicit killing spinors on the n sphere for arbitrary n using these results we also construct the killing spinors on various ads sphere supergravity backgrounds including ads5 s5 ads4 s7 and ads7 s4 in addition we extend previous results to obtain the killing spinors on the hyperbolic spaces hn

a geometrical angle on feynman integrals a direct link between a one loop n point feynman diagram and a geometrical representation based on the n dimensional simplex is established by relating the feynman parametric representations to the integrals over contents of n 1 dimensional simplices in non euclidean geometry of constant curvature in particular the four point function in four dimensions is proportional to the volume of a three dimensional spherical or hyperbolic tetrahedron which can be calculated by splitting into birectangular ones it is also shown that the known formula of reduction of the n point function in n 1 dimensions corresponds to splitting the related n dimensional simplex into n rectangular ones

a comparison of the computational efficiency of three averaging procedures in a numerical scheme for the shallow water equations abstract three numerical schemes all based on numerical characteristic decomposition are compared in terms of their computational efficiency when applied to the shallow water equations the schemes differ through the use of alternative averaging procedures

a variational principle for weighted delaunay triangulations and hyperideal polyhedra we use a variational principle to prove an existence and uniqueness theorem for planar weighted delaunay triangulations with non intersecting site circles with prescribed combinatorial type and circle intersection angles such weighted delaunay triangulations may be interpreted as images of hyperbolic polyhedra with one vertex on and the remaining vertices beyond the infinite boundary of hyperbolic space thus the main theorem states necessary and sufficient conditions for the existence and uniqueness of such polyhedra with prescribed combinatorial type and dihedral angles more generally we consider weighted delaunay triangulations in piecewise flat surfaces allowing cone singularities with prescribed cone angles in the vertices the material presented here extends work by rivin on delaunay triangulations and ideal polyhedra

an efficient riemann solver for unsteady flows with non ideal gases a riemann solver is presented for the euler equations of gas dynamics with real gases this represents a more efficient version of an algorithm originally presented by the author

the poincare metric and isoperimetric inequalities for hyperbolic polygons we prove several isoperimetric inequalities for the conformal radius or equivalently for the poincare density of polygons on the hyperbolic plane our results include as limit cases the isoperimetric inequality for the conformal radius of euclidean n gons conjectured by g polya and g szego in 1951 and a similar inequality for the hyperbolic n gons of the maximal hyperbolic area conjectured by j hersch both conjectures have been proved in previous papers by the third author our approach uses the method based on a special triangulation of polygons and weighted inequalities for the reduced modules of trilaterals developed by a yu solynin we also employ the dissymmetrization transformation of v n dubinin as an important part of our proofs we obtain monotonicity and convexity results for special combinations of the euler gamma functions which appear to have a significant interest in their own right

positive commutators at the bottom of the spectrum abstract bony and hafner have recently obtained positive commutator estimates on the laplacian in the low energy limit on asymptotically euclidean spaces these estimates can be used to prove local energy decay estimates if the metric is non trapping we simplify the proof of the estimates of bony hafner and generalize them to the setting of scattering manifolds i e manifolds with large conic ends by applying a sharp poincare inequality our main result is the positive commutator estimate i h 2 g i 2 h 2 g a i h 2 g c i h 2 g 2 where h is a large parameter i is a compact interval in 0 and i its indicator function and where a is a differential operator supported outside a compact set and equal to 1 2 r d r r d r near infinity the laplacian can also be modified by the addition of a positive potential of sufficiently rapid decay the same estimate then holds for the resulting schrodinger operator

on a three dimensional symmetric ising tetrahedron and contributions to the theory of the dilogarithm and clausen functions perturbative quantum field theory for the ising model at the three loop level yields a tetrahedral feynman diagram c a b with masses a and b and four other lines with unit mass the completely symmetric tetrahedron ctet c 1 1 has been of interest from many points of view with several representations and conjectures having been given in the literature we prove a conjectured exponentially fast convergent sum for c 1 1 as well as a previously empirical relation for c 1 1 as a remarkable difference of clausen function values our presentation includes propositions extending the theory of the dilogarithm li2 and clausen cl2 functions as well as their relation to other special functions of mathematical physics the results strengthen connections between feynman diagram integrals volumes in hyperbolic space number theory and special functions and numbers specifically including dilogarithms clausen function values and harmonic numbers

an analysis of arithmetic averaging in approximate riemann solvers with an application to steady supercritical flows an analysis of various arithmetic averaging procedures for approximate riemann solvers is made with a specific emphasis on efficiency and a jump capturing property the various alternatives discussed are intended for future work as well as the more immediate problem of steady supercritical free surface flows numerical results are shown for two test problems

the relationship between the direct and weak formulations of a linearised riemann solver for systems of conservation laws abstract the direct and weak formulations of a linearised riemann solver attributable to roe and toumi are shown to be equivalent for certain systems of conservation laws and particular parameter vectors it is demonstrated however that in other cases this is not necessarily true

resolvent smoothness and local decay at low energies for the standard model of non relativistic qed we consider an atom interacting with the quantized electromagnetic field in the standard model of non relativistic qed the nucleus is supposed to be fixed we prove smoothness of the resolvent and local decay of the photon dynamics for quantum states in a spectral interval i just above the ground state energy our results are uniform with respect to i their proofs are based on abstract mourre s theory a mourre inequality established by frohlich griesemer and sigal see frohlich et al 2008 14 hardy type estimates in fock space and a low energy dyadic decomposition

a geometric approach to scalar field theories on the supersphere following a strictly geometric approach we construct globally supersymmetric scalar field theories on the supersphere defined as the quotient space s2 2 uosp 1 2 u 1 we analyze the superspace geometry of the supersphere in particular deriving the invariant vielbein and spin connection from a generalization of the left invariant maurer cartan form for lie groups using this information we proceed to construct a superscalar field action on s2 2 which can be decomposed in terms of the component fields yielding a supersymmetric action on the ordinary two sphere we are able to derive lagrange equations and noether s theorem for the superscalar field itself

partition functions of n 2 2 gauge theories on s 2 and vortices we apply localization techniques to compute the partition function of a two dimensional mathcal n 2 2 r symmetric theory of vector and chiral multiplets on s2 the path integral reduces to a sum over topological sectors of a matrix integral over the cartan subalgebra of the gauge group for gauge theories which would be completely higgsed in the presence of a fayet iliopoulos term in flat space the path integral alternatively reduces to the product of a vortex times an antivortex partition functions weighted by semiclassical factors and summed over isolated points on the higgs branch for applications we evaluate the partition function for some u n gauge theories showing equality of the path integrals for theories conjectured to be dual by hori and tong and deriving new expressions for vortex partition functions

shock capturing for supercritical free surface flows in curvilinear channel geometries abstract a shock capturing scheme is developed for the shallow water equations governing steady supercritical flow in two dimensions in channels with a curvilinear coordinate geometry the scheme incorporates a special averaging procedure of the flow variables as well as details of the transformation that maps a nonrectangular physical domain for a general channel transition

the invariant of hyperbolic 3 manifolds in this paper we use the formula in the above theorem to study t m for hyperbolic 3 manifolds and calculate the right hand side of the formula explicitly in a special case our main task is to represent the integral of 1 by some more tractible ones in sect 1 we deal with general compact oriented riemannian manifolds m of dimension 3 let f m be the so 3 oriented frame bundle of m let l be a

absence of singular continuous spectrum for certain self adjoint operators we give a sufficient condition for a self adjoint operator to have the following properties in a neighborhood of a pointe of its spectrum a its point spectrum is finite b its singular continuous spectrum is empty c its resolvent satisfies a class of a priori estimates

stratified fiber bundles quinn homology and brane stability of hyperbolic orbifolds we revisit the problem of stability of string vacua involving hyperbolic orbifolds using methods from homotopy theory and k homology we propose a definition of type ii string theory on such backgrounds that further carry stratified systems of fiber bundles which generalize the more conventional orbifold and symmetric string backgrounds together with a classification of wrapped branes by a suitable generalized homology theory for spaces stratified fibered over hyperbolic orbifolds we use the algebraic k theory of their fundamental groups and quinn homology to derive criteria for brane stability in terms of an atiyah hirzebruch type spectral sequence with its lift to k homology stable d branes in this setting carry stratified charges which induce new additive structures on the corresponding k homology groups we extend these considerations to backgrounds which support h flux where we use k groups of twisted group algebras of the fundamental groups to analyze stability of locally symmetric spaces with k amenable isometry groups and derive stability conditions for branes wrapping the fibers of an eilenberg maclane spectrum functor

an efficient numerical scheme for the two dimensional shallow water equations using arithmetic averaging a finite difference scheme based on flux difference splitting is presented for the solution of the two dimensional shallow water equations of ideal fluid flow a linearised problem analogous to that of riemann for gas dynamics is defined and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem and incorporates the technique of operator splitting an average of the flow variables across the interface between cells is required and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging this is in contrast to usual square root averages found in this type of riemann solver where the computational expense can be prohibitive the method of upwind differencing is used for the resulting scalar problems together with a flux limiter for obtaining a second order scheme which avoids nonphysical spurious oscillations an extension to the two dimensional equations with source terms is included the scheme is applied to the one dimensional problems of a breaking dam and reflection of a bore and in each case the approximate solution is compared to the exact solution of ideal fluid flow the scheme is also applied to a problem of stationary bore generation in a channel of variable cross section finally the scheme is applied to two other dam break problems this time in two dimensions with one having cylindrical symmetry each approximate solution compares well with those given by other authors

a comparison of the different extensions of a weak formulation of an approximate riemann solver for supercritical flows and their relationship to existing schemes abstract a weak formulation of the roe linearised riemann solver proposed recently is examined in the context of the equations governing steady supercritical flows two possible alternatives present themselves and it is shown that these are equivalent and moreover are equivalent to existing schemes

killing spinors on spheres and hyperbolic manifolds properties of killing spinors on spheres and hyperbolic manifolds are investigated with an emphasis on the relations to killing vectors conformal killing vectors and solutions of maxwell s equations

a finite difference scheme for steady supersonic two dimensional compressible flow of real gases a finite difference scheme is presented for the solution of the two dimensional equations of steady supersonic compressible flow of real gases the scheme incorparates numerical characteristic decomposition is shock capturing by design and incorporates space marching as a result of the assumption that the flow is wholly supersonic in at least one space dimension results are shown for problems involving oblique hydraulic jumps and reflection from a wall

quantum electrodynamics of confined nonrelativistic particles abstract we consider a system of finitely many nonrelativistic quantum mechanical electrons bound to static nuclei the electrons are minimally coupled to the quantized electromagnetic field but we impose an ultraviolet cutoff on the electromagnetic vector potential appearing in covariant derivatives and the interactions between the radiation field and electrons localized very far from the nuclei are turned off for a class of hamiltonians we prove exponential localization of bound states establish the existence of a ground state and derive sufficient conditions for its uniqueness furthermore we show that excited bound states of the unperturbed system become unstable and turn into resonances when the electrons are coupled to the radiation field to this end we develop a novel renormalization transformation which acts directly on the space of hamiltonians

prediction of supercritical flow in open channels a finite difference scheme based on flux difference splitting is presented for the solution of the one dimensional shallow water equations in open channels a linearised problem analogous to that of riemann for gas dynamics is defined and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem the method of upwind differencing is used for the resulting scalar problems together with a flux limiter for obtaining a second order scheme which avoids non physical spurious oscillations the scheme is applied to a problem of flow in a river whose geometry induces a region of supercritical flow

integrating over higgs branches we develop some useful techniques for integrating over higgs branches in supersymmetric theories with 4 and 8 supercharges in particular we define a regularized volume for hyperkahler quotients we evaluate this volume for certain ale and alf spaces in terms of the hyperkahler periods we also reduce these volumes for a large class of hyperkahler quotients to simpler integrals these quotients include complex coadjoint orbits instanton moduli spaces on 4 and ale manifolds hitchin spaces and moduli spaces of parabolic higgs bundles on riemann surfaces in the case of hitchin spaces the evaluation of the volume reduces to a summation over solutions of bethe ansatz equations for the non linear schrodinger system we discuss some applications of our results

ground states in non relativistic quantum electrodynamics the excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay but such a particle should have a true ground state one that minimizes the energy and satisfies the schrodinger equation we prove quite generally that this state exists for all values of the fine structure constant and the ultraviolet cutoff we also show the same thing for a many particle system under physically natural conditions

an efficient shock capturing scheme for two dimensional open channel unsteady flows in a generalised coordinate system abstract an efficient algorithm based on flux difference splitting is presented for the solution of two dimensional open channel flows a transformation maps a nonrectangular physical domain into a rectangular one the governing equations are then the shallow water equations including terms of slope and friction in a generalised coordinate system a regular mesh on a rectangular computational domain can then be employed the resulting scheme is efficient has good jump capturing properties and the advantage of using boundary body fitted meshes

an efficient numerical scheme for the euler equations a finite difference scheme based on flux difference splitting is presented for the solution of the euler equations for the compressible flow of an ideal gas a linearised riemann problem is defined and a scheme based on numerical characteristic decomposition is presented for obtaining approximate solutions to the linearised problem an average of the flow variables across the interface between cells is required and this average is chosen to be the arithmetic mean for computational efficiency leading to arithmetic averaging this is in contrast to the usual square root averages found in this type of riemann solver where the computational expense can be prohibitive the method of upwind differencing is used for the resulting scalar problems together with a flux limiter for obtaining a second order scheme which avoids nonphysical spurious oscillations the scheme is applied to a shock tube problem and a blast wave problem each approximate solution compares well with those given by other schemes and for the shock tube problem is in agreement with the exact solution

maximal velocity of photons in non relativistic qed abstract we consider the problem of propagation of photons in the quantum theory of non relativistic matter coupled to electromagnetic radiation which is presently the only consistent quantum theory of matter and radiation assuming that the matter system is in a localized state i e for energies below the ionization threshold we show that the probability to find photons at time t at the distance greater than c t where c is the speed of light vanishes as t as an inverse power of t

similarity solutions for multi component flows a one dimensional shock reflection test problem in the case of slab cylindrical or spherical symmetry is discussed for multi component flows the differential equations for a similarity solution are derived and then solved numerically in conjunction with the rankine hugoniot shock relations

spectral theory of massless pauli fierz models we study the spectral theory of massless pauli fierz models using an extension of the mourre method we prove the local finiteness of point spectrum and a limiting absorption principle away from the eigenvalues for an arbitrary coupling constant in addition we show that the expectation value of the number operator is finite on all eigenvectors

exponential decay and ionization thresholds in non relativistic quantum electrodynamics abstract spatial localization of the electrons of an atom or molecule is studied in models of non relativistic matter coupled to quantized radiation we give two definitions of the ionization threshold one in terms of spectral data of cluster hamiltonians and one in terms of minimal energies of non localized states we show that these two definitions agree and that the electrons described by a state with energy below the ionization threshold are localized in a small neighborhood of the nuclei with a probability that approaches 1 exponentially fast with increasing radius of the neighborhood the latter result is derived from a new general result on exponential decay tailor made for our problem but applicable to many non relativistic quantum systems outside quantum electrodynamics as well

an analysis of arithmetic averaging in a riemann solver for the one dimensional euler equations abstract an analysis of various arithmetic averaging procedures is undertaken for an approximate riemann solver for the one dimensional euler equations governing the compressible flow of an ideal gas this study draws on earlier work for both the reduced set of equations governing shallow water flows and the full euler equations and is ultimately intended for use with the equations governing two dimensional steady and unsteady compressible flows

a weak formulation of roe s scheme for two dimensional unsteady compressible flows and steady supersonic flows abstract a weak formulation of roe s approximate riemann solver developed recently for one dimensional unsteady compressible flows is extended to two dimensions using operator splitting and to steady supersonic flows and it is shown that the resulting schemes are analogous to other methods for such flows the technique for steady flows draws on recent work in this area

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

simplified godunov schemes for 2 x 2 systems of conservation laws some godunov type schemes for solving isentropic gas dynamic equations are presented after analyzing a one intermediary state scheme and its properties we prove consistency with entropy condition for time continuous limit solutions of a simplified godunov scheme by techniques due to osher we prove also similar results for high order versions of these numerical schemes

l2 torsion invariants we discuss a generalisation of reidemeister franz torsion which applies to infinite dimensional representations of the fundamental group by utilising the theory of finite von neumann algebras our results apply to a class of closed oriented manifolds which are l2 acyclic in the sense of having trivial l2 cohomology evidence is presented for equality of the essentially combinatorial torsion introduced here with an analytic l2 ray singer torsion defined by the second named author