quantitative phase field modeling of dendritic growth in two and three dimensions we report the results of quantitative phase field simulations of the dendritic crystallization of a pure melt in two and three dimensions these simulations exploit a recently developed thin interface limit of the phase field model a karma and w j rappel phys rev e 53 r3017 1996 which is given here a detailed exposition this limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient simulations in one and two dimensions are first carried out to test the accuracy of phase field computations performed within this limit dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method both with and without interface kinetics simulations in three dimensions exploit in addition to the asymptotics a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies they are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three dimensional morphology of needle crystals without sidebranches for small crystalline anisotropy simulated values of ensuremath sigma are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy in contrast for large anisotropy simulated ensuremath sigma values are significantly larger than the predicted values this disagreement however does not signal a breakdown of solvability theory it is consistent with the finding that the amplitude of the mathrm cos 4 ensuremath varphi mode which measures the departure of the tip morphology from a shape of revolution increases with anisotropy this departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy finally the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law r ensuremath sim z with a slope dr dz that is nearly equal to the ratio of the two dimensional and three dimensional steady state tip velocities furthermore the evolution of the cross section of a three dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two dimensional growth shape in time in accord with the current theory of the three dimensional needle crystal shape

universal spectral statistics in wigner dyson chiral and andreev star graphs ii semiclassical approach a semiclassical approach to the universal ergodic spectral statistics in quantum star graphs is presented for all known ten symmetry classes of quantum systems the approach is based on periodic orbit theory the exact semiclassical trace formula for star graphs and on diagrammatic techniques the appropriate spectral form factors are calculated up to one order beyond the diagonal and self dual approximations the results are in accordance with the corresponding random matrix theories which supports a properly generalized bohigas giannoni schmit conjecture

on the relationship between monstrous moonshine and the uniqueness of the moonshine module we consider the relationship between the conjectured uniqueness of the moonshine module open image in new window and monstrous moonshine the genus zero property of the modular invariance group for each monster group thompson series we first discuss a family of possiblezn meromorphic orbifold constructions of open image in new window based on automorphisms of the leech lattice compactified bosonic string we reproduce the thompson series for all 51 non fricke classes of the monster groupm together with a new relationship between the centralisers of these classes and 51 corresponding conway group centralisers generalising a well known relationship for 5 such classes assuming that open image in new window is unique we consider meromorphic orbifoldings of open image in new window and show that monstrous moonshine holds if and onlyzr if the only meromorphic orbifoldings of open image in new window are open image in new window itself or the leech theory this constraint on the meromorphic orbifoldings of open image in new window therefore relates monstrous moonshine to the uniqueness of open image in new window in a new way

modularity in orbifold theory for vertex operator superalgebras this paper is about the orbifold theory for vertex operator superalgebras given a vertex operator superalgebra v and a finite automorphism group g of v we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in g under the c2 cofinite condition we also establish that these functions afford a representation of the full modular group if v is c2 cofinite and g rational for any g g

group theoretic characterizations of almost open immersions of curves abstract let p be a prime number and k either a finite field of characteristic p or a generalized sub p adic field let x 1 and x 2 be hyperbolic curves over k in the present paper we introduce a kind of morphism between x 1 and x 2 called an almost open immersion and give some group theoretic characterizations for the set of almost open immersions between x 1 and x 2 via their arithmetic fundamental groups this result generalizes the isom version of grothendieck s anabelian conjecture for curves over k which has been proven by s mochizuki and a tamagawa to the case of almost open immersions

mirror symmetry on kummer type k3 surfaces we investigate both geometric and conformal field theoretic aspects of mirror symmetry on n 4 4 superconformal field theories with central charge c 6 our approach enables us to determine the action of mirror symmetry on non stable singular fibers in elliptic fibrations of n orbifold limits of k3 the resulting map gives an automorphism of order 4 8 or 12 respectively on the smooth universal covering space of the moduli space we explicitly derive the geometric counterparts of the twist fields in our orbifold conformal field theories the classical mckay correspondence allows for a natural interpretation of our results

bosonic and fermionic realizations of the affine algebra g hat l_n we give an explicit description of all inequivalent heisenberg subalgebras of the affine lie algebra g hat l n mathbb c and the associated vertex operator constructions of the level one integrable highest weight representations of this algebra the construction uses multicomponent fermionic fields and yields a correspondence between bosons elements of the heisenberg subalgebra and fermions

streamer branching rationalized by conformal mapping techniques spontaneous branching of discharge channels is frequently observed but not well understood we recently proposed a new branching mechanism based on simulations of a simple continuous discharge model in high fields we here present analytical results for such streamers in the lozansky firsov limit where they can be modeled as moving equipotential ionization fronts this model can be analyzed by conformal mapping techniques which allow the reduction of the dynamical problem to finite sets of nonlinear ordinary differential equations our solutions illustrate that branching is generic for the intricate head dynamics of streamers in the lozansky firsov limit

twisted quantum affinizations and their vertex representations in this paper we generalize drinfeld s twisted quantum affine algebras to construct twisted quantum algebras for all simply laced generalized cartan matrices and present their vertex representation realizations

thin interface asymptotics for an energy entropy approach to phase field models with unequal conductivities karma and rappel phys rev e 57 1998 4342 recently developed a new sharp interface asymptotic analysis of the phase field equations that is especially appropriate for modeling dendritic growth at low undercoolings their approach relieves a stringent restriction on the interface thickness that applies in the conventional asymptotic analysis and has the added advantage that interfacial kinetic effects can also be eliminated however their analysis focussed on the case of equal thermal conductivities in the solid and liquid phases when applied to a standard phase field model with unequal conductivities anomalous terms arise in the limiting forms of the boundary conditions for the interfacial temperature that are not present in conventional sharp interface solidification models as discussed further by almgren siam j appl math 59 1999 2086 in this paper we apply their asymptotic methodology to a generalized phase field model which is derived using a thermodynamically consistent approach that is based on independent entropy and internal energy gradient functionals that include double wells in both the entropy and internal energy densities the additional degrees of freedom associated with the generalized phase field equations can be used to eliminate the anomalous terms that arise for unequal conductivities

morphological instability and dynamics of fronts in bacterial growth models with nonlinear diffusion depending on the growth condition bacterial colonies can exhibit different morphologies as argued by ben jacob et al there is biological and modeling evidence that a nonlinear diffusion coefficient of the type d b d 0 b k is a basic mechanism that underlies almost all of the patterns and generates a long wavelength instability we study a reaction diffusion system with a nonlinear diffusion coefficient and find that a unique planar traveling front solution exists whose velocity is uniquely determined by k and d d 0 d n where d n is the diffusion coefficient of the nutrient due to the fact that the bacterial diffusion coefficient vanishes when stackrel ensuremath rightarrow b 0 in the front solution b vanishes in a singular way as a result the standard linear stability analysis for fronts cannot be used we introduce an extension of the stability analysis that can be applied to singular fronts and use the method to perform a linear stability analysis of the planar bacteriological growth front we show that a nonlinear diffusion coefficient generates a long wavelength instability for kg0 and dl d c k we map out the region of stability in the d ensuremath k plane and determine the onset of stability that is given by d c k both for stackrel ensuremath rightarrow d 0 and stackrel ensuremath rightarrow k ensuremath infty the dynamics of the growth zone essentially reduces to that of a sharp interface problem that is reminiscent of a so called one sided growth problem where the growth velocity is proportional to the gradient of a diffusion field ahead of the interface the moving boundary approximation that we derive in these limits is quite accurate but surprisingly does not become a proper asymptotic theory in the strict mathematical sense in the limit stackrel ensuremath rightarrow d 0 due to lack of full separation of scales on all dynamically relevant length scales our linear stability analysis and sharp interface formulation will also be applicable to other examples of interface formation due to nonlinear diffusion like in porous media or in the problem of vortex motion in superconductors

a construction of lower bounded generalized twisted modules for a grading restricted vertex super algebra we give a general direct and explicit construction of lower bounded generalized twisted modules satisfying a universal property for a grading restricted vertex super algebra v associated to an automorphism g of v in particular when g is the identity we obtain lower bounded generalized v modules satisfying a universal property let w be a lower bounded graded vector space equipped with a set of generating twisted fields and a set of generator twist fields satisfying a weak commutativity for generating twisted fields a generalized weak commutativity for one generating twisted field and one generator twist field and some other properties that are relatively easy to verify we first prove the convergence and commutativity of products of an arbitrary number of generating twisted fields one twist generator field and an arbitrary number of generating fields for v then using the convergence and commutativity we define a twisted vertex operator map for w and prove that w equipped with this twisted vertex operator map is a lower bounded generalized g twisted v module using this result we give an explicit construction of lower bounded generalized g twisted v modules satisfying a universal property starting from vector spaces graded by weights mathbb z 2 fermion numbers and g weights eigenvalues of g and real numbers corresponding to the lower bounds of the weights of the modules to be constructed in particular every lower bounded generalized g twisted v module every lower bounded generalized v module when g is the identity is a quotient of such a universal lower bounded generalized g twisted v module a universal lower bounded generalized v module

cusp development in hele shaw flow with a free surface free surface flow in a hele shaw cell is considered it has been known for some time that when the fluid region is contracting a finite time blow up can occur in which a cusp forms in the free surface the solution does not exist beyond the time of blow up examples are presented of this blow up and also of a new process in which a cusp of a different type forms in the free surface but after which the solution continues to exist after a transformation of the dependent variable the pressure the new cusps are related to permissible singularities in the free boundary of the obstacle problem

vertex operators and symmetric functions an algorithm for the calculation of q dependent spin characters of the symmetric group is given with an explicit example of s4 a method for constructing the q analogue of the vertex operators is developed a 1 1 correspondence between the space v of twisted q vertex operators and the ring of q deformed symmetric functions lambda x zq q t is established and a mapping from v to lambda x zq q t is defined a number of relevant theorems are given

anabelian geometry with etale homotopy types anabelian geometry with etale homotopy types generalizes in a natural way classical anabelian geometry with etale fundamental groups we show that both in the classical and the generalized sense any point of a smooth variety over a eld k that is nitely generated over q has a fundamental system of ane anabelian zariski neighborhoods this was predicted by

on orbifold theory abstract let v be a simple vertex operator algebra and g a finite automorphism group of v such that v g is regular it is proved that every irreducible v g module occurs in an irreducible g twisted v module for some g g moreover the quantum dimensions of irreducible v g modules are determined and a global dimension formula for v in terms of twisted modules is obtained in particular the orbifold theory conjecture is completely solved if g is solvable

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

the local pro p anabelian geometry of curves let x be a connected scheme then one can associate after grothendieck to x its algebraic fundamental group 1 x this group 1 x is a profinite group which is uniquely determined up to inner automorphisms by the property that the category of finite discrete sets equipped with a continuous 1 x action is equivalent to the category of finite etale coverings of x moreover the assignment x 1 x is a functor from the category of connected schemes and morphisms of schemes to the category of profinite topological groups and continuous outer homomorphisms i e continuous homomorphisms of topological groups where we identify any two homomorphisms that can be obtained from one another by composition with an inner automorphism

numerical studies of the steklov eigenvalue problem via conformal mappings abstract in this paper spectral methods based on conformal mappings are proposed to solve the steklov eigenvalue problem and its related shape optimization problems in two dimensions to apply spectral methods we first reformulate the steklov eigenvalue problem in the complex domain via conformal mappings the eigenfunctions are expanded in fourier series so the discretization leads to an eigenvalue problem for coefficients of fourier series for shape optimization problem we use a gradient ascent approach to find the optimal domain which maximizes kth steklov eigenvalue with a fixed area for a given k the coefficients of fourier series of mapping functions from a unit circle to optimal domains are obtained for several different k

on spinor vertex operator algebras and their modules abstract in this paper we first present a generator theorem and introduce an asymmetric tensor of two colored vertex operator superalgebras with the same grading and supermap then we give axioms which characterize a class of vertex operator superalgebras constructed from clifford algebras in particular we are able to give a shorter argument for the jacobi identity than the existing proofs by using elements analogous to coherent states and our generator theorem using the new tensor and the same techniques we classify twisted irreducible modules of these algebras in a constructive way

second order phase field asymptotics for unequal conductivities we extend karma and rappel s improved asymptotic analysis of the phase field model to different diffusivities in solid and liquid we consider both second order classical asymptotics in which the interface thickness is taken much smaller than the capillary length and the new isothermal asymptotics in which the two lengths are considered comparable in the first case if the phase field model is required to be gradient flow for an entropy functional then for unequal diffusivities it is impossible to construct a phase equation with finite kinetics which converges with second order accuracy to a gibbs thomson equilibrium condition with infinitely fast kinetics in the second case some error terms are pushed to higher orders and it is easy to eliminate the remaining errors with finite phase kinetics

global existence singular solutions and ill posedness for the muskat problem the muskat or muskat leibenzon problem describes the evolution of the interface between two immiscible fluids in a porous medium or hele shaw cell under applied pressure gradients or fluid injection extraction in contras t to the hele shaw problem the one phase version of the muskat problem there are few nontrivial exact solutions or analytic results for the muskat problem for the stable forward muskat problem in which the higher viscosity fluid ex pands into the lower viscosity fluid we show global in time existence for initial data that is a small perturbation of a flat interface the initial data in this result ma y contain weak e g curvature singularities for the unstable backwa rd problem in which the higher viscosity fluid contracts we construct singular solution s that start off with smooth initial data but develop a point of infinite curvature at fi nite time c 2004 wiley periodicals inc

twisted modules over vertex algebras on algebraic curves abstract we extend the geometric approach to vertex algebras developed by the first author to twisted modules allowing us to treat orbifold models in conformal field theory let v be a vertex algebra h a finite group of automorphisms of v and c an algebraic curve such that h aut c we show that a suitable collection of twisted v modules gives rise to a section of a certain sheaf on the quotient x c h we introduce the notion of conformal blocks for twisted modules and analyze them in the case of the heisenberg and affine kac moody vertex algebras we also give a chiral algebra interpretation of twisted modules

convective stabilization of a laplacian moving boundary problem with kinetic undercooling we study the shape stability of disks moving in an external laplacian field in two dimensions the problem is motivated by the motion of ionization fronts in streamer type electric breakdown it is mathematically equivalent to the motion of a small bubble in a hele shaw cell with a regularization of kinetic undercooling type namely a mixed dirichlet neumann boundary condition for the laplacian field on the moving boundary using conformal mapping techniques linear stability analysis of the uniformly translating disk is recast into a single pde which is exactly solvable for certain values of the regularization parameter we concentrate on the physically most interesting exactly solvable and nontrivial case we show that the circular solutions are linearly stable against smooth initial perturbations in the transformation of the pde to its normal hyperbolic form a semigroup of automorphisms of the unit disk plays a central role it mediates the convection of perturbations to the back of the circle where

on the section conjecture in anabelian geometry let x be a smooth projective curve of genus 1 over a field k which is finitely generated over the rationals the section conjecture in grothendieck s anabelian geometry says that the sections of the canonical projection from the arithmetic fundamental group of x onto the absolute galois group of k are up to conjugation in one to one correspondence with k rational points of x the birational variant conjectures a similar correspondence where the fundamental group is replaced by the absolute galois group of the function field k x the present paper proves the birational section conjecture for all x when k is replaced e g by the field of p adic numbers it disproves both conjectures for the field of real or p adic algebraic numbers and it gives a purely group theoretic characterization of the sections induced by k rational points of x in the birational setting over almost arbitrary fields as a biproduct we obtain galois theoretic criteria for radical solvability of polynomial equations in more than one variable and for a field to be pac to be large or to be hilbertian

the krichever map vector bundles over algebraic curves and heisenberg algebras we study the grassmanniangr x n consisting of equivalence classes of rankn algebraic vector bundles over a riemann surfacex with an holomorphic trivialization at a fixed pointp commutative subalgebras ofgl n h h being the ring of functions holomorphic on a punctured disc aboutp define flows on the grassmannian giving rise to classes of solutions to multi component kp hierarchies these commutative subalgebras correspond to heisenberg algebras in the kac moody algebra associated togl n h one can obtain by the krichever map points ofgr x n and solutions of mckp from coveringsf y x and other geometric data conversely for every point ofgr x n and for every choice of heisenberg algebra we construct using the cotangent bundle ofgr x n an algebraic curve coveringx and other data thus inverting the krichever map we show the explicit relation between the choice of heisenberg algebra and the geometry of the covering space

mapping class group action versus galois action on profinite fundamental groups let x be an algebraic curve of genus g n punctured defined over a number field k then the profinite or the pro l completion of the topological fundamental group of x admits two actions the action of the profinite completion of the mapping class group of the orientable surface of topological type g n and the action of the absolute galois group of k this paper compares these two in the profinite case it is shown that the intersection of the images of these two actions is trivial if x is affine and its fundamental group is nonabelian on the contrary in the pro l case there are many curves such that the image of the galois action contains the image of the mapping class group action it is proved that the set of points corresponding to such curves is dense in the moduli space of g n curves

twisted γ lie algebras and their vertex operator representations let be a generic subgroup of the multiplicative group c of nonzero complex numbers we define a class of lie algebras associated to called twisted lie algebras which are natural generalizations of the twisted affine lie algebras starting from an arbitrary even sublattice q of zn and an arbitrary finite order isometry of zn preserving q we construct a family of twisted vertex operators acting on generalized fock spaces which afford irreducible representations for certain twisted lie algebras as an application this recovers a number of known vertex operator realizations for infinite dimensional lie algebras such as twisted affine lie algebras extended affine lie algebras of type a trigonometric lie algebras of series a and b unitary lie algebras and bc graded lie algebras

pursuing the double affine grassmannian i transversal slices via instantons on a_k singularities this article is the first in a series that describes a conjectural analog of the geometric satake isomorphism for an affine kac moody group for simplicity we only consider the untwisted and simply connected case here the usual geometric satake isomorphism for a reductive group g identifies the tensor category rep g of finite dimensional representations of the langlands dual group g with the tensor category pervg o grg of g o equivariant perverse sheaves on the affine grassmannian grg g k g o of g here k c t and o c t as a by product one gets a description of the irreducible g o equivariant intersection cohomology ic sheaves of the closures of g o orbits in grg in terms of q analogs of the weight multiplicity for finite dimensional representations of g the purpose of this article is to try to generalize the above results to the case when g is replaced by the corresponding affine kac moody group gaff we refer to the not yet constructed affine grassmannian of gaff as the double affine grassmannian more precisely in this article we construct certain varieties that should be thought of as transversal slices to various gaff o orbits inside the closure of another gaff o orbit in grgaff we present a conjecture that computes the intersection cohomology sheaf of these varieties in terms of the corresponding q analog of the weight multiplicity for the langlands dual affine group gaff and we check this conjecture in a number of cases some further constructions such as convolution of the corresponding perverse sheaves analog of the beilinson drinfeld grassmannian and so forth will be addressed in another publication

2 permutations of lattice vertex operator algebras higher rank abstract this paper is a continuation of our investigation on 2 permutation of lattice vertex operator algebras in particular the quantum dimensions of irreducible modules and the fusion rules are determined if the rank of the lattice is one these results have been obtained previously in 27

the cuspidalisation of sections of arithmetic fundamental groups abstract in this paper which is a sequel to 7 we investigate the theory of cuspidalisation of sections of arithmetic fundamental groups of hyperbolic curves to cuspidally i th and 2 p th step prosolvable arithmetic fundamental groups as a consequence we exhibit two necessary and sufficient conditions for sections of arithmetic fundamental groups of hyperbolic curves over p adic local fields to arise from rational points we also exhibit a class of sections of arithmetic fundamental groups of p adic curves which are orthogonal to pic and which satisfy unconditionally one of the above conditions

local constancy in families of non abelian galois representations if mathcal s is a a scheme of finite type over a local field f and mathcal x longrightarrow mathcal s is a proper smooth family then to each rational point s in mathcal s one can assign an extension of the absolute galois group of f by the geometric fundamental group g of the fibre mathcal x s if f has uniformiser pi and residue characteristic p we show that the corresponding extension of the absolute galois group of mathcal s by the maximal prime to p quotient of g is locally constant in the pi adic topology of mathcal s we give a similar result in the case of non proper families and families over pi adic analytic spaces

bimodules and g rationality of vertex operator algebras this paper studies the twisted representations of vertex operator algebras let v be a vertex operator algebra and g an automorphism of v of finite order t for any m n 1 tz an a g n v a g m v bimodule a g n m v is constructed the collection of these bimodules determines any admissible g twisted v module completely a verma type admissible g twisted v module is constructed naturally from any a g m v module furthermore it is shown with the help of bimodule theory that a simple vertex operator algebra v is g rational if and only if its twisted associative algebra ag v is semisimple and each irreducible admissible g twisted v module is ordinary

sharp interface limits of phase field models the use of continuum phase field models to describe the motion of well defined interfaces is discussed for a class of phenomena that includes order disorder transitions spinodal decomposition and ostwald ripening dendritic growth and the solidification of eutectic alloys the projection operator method is used to extract the sharp interface limit from phase field models which have interfaces that are diffuse on a length scale xi in particular phase field equations are mapped onto sharp interface equations in the limits xi kappa 1 and xi v d 1 where kappa and v are respectively the interface curvature and velocity and d is the diffusion constant in the bulk the calculations provide one general set of sharp interface equations that incorporate the gibbs thomson condition the allen cahn equation and the kardar parisi zhang equation

multiple steady bubbles in a hele shaw cell this paper presents new solutions in analytical form for the shapes of an assembly of steady co travelling bubbles in a hele shaw cell the associated velocity field is also derived the assembly can consist of any finite number of bubbles the solutions are expressed in terms of schottky klein prime functions and are derived by exploiting results on multiply connected conformal slit mappings

void electromigration as a moving free boundary value problem abstract the void electromigration process in the strip geometry is investigated analytically the void is assumed to travel steadily along the axis of symmetry of the metal strip when its length is large compared to the width of the metal ribbon a strong similarity with the classical saffman taylor viscous fingering is pointed out it allows us to adapt the hodograph method to predict both the width and the growth velocity at fixed applied current in the absence of surface tension the void shape is unique and has a relative width close to 2 3 a result in strong contradiction with the viscous fingering in the same geometry the propagation of symmetric bubbles is also investigated and shows that the set of solutions is less rich than the flow problem the addition of surface tension appears non standard in the area of pattern formation it does not allow large void propagation suppressing the unique infinite solution as for the bubbles the numerically computed shapes with surface tension appear rather circular this paper shows that the modification of a single boundary condition the dirichlet boundary condition is enough to strongly modify our intuition about pattern formation

phase field model for hele shaw flows with arbitrary viscosity contrast i theoretical approach we present a phase field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular hele shaw cell with asymptotic matching techniques we check the model to yield the right hele shaw equations in the sharp interface limit and compute the corrections to these equations to first order in the interface thickness we also compute the effect of such corrections on the linear dispersion relation of the planar interface we discuss in detail the conditions on the interface thickness to control the accuracy and convergence of the phase field model to the limiting hele shaw dynamics in particular the convergence appears to be slower for high viscosity contrasts

shape selection of saffman taylor fingers among all problems of pattern selection the one posed by the saffman taylor finger is yet unsolved although numerics and experiments indicate the very simple property that the relative width of the finger tends to 1 2 in the low surface tension limit we explain this by performing an expansion beyond all orders leading to the formulation of a nonlinear eigenvalue problem with a discrete set of solutions in particular we predict that the relative width of the finger tends to 1 2 as the surface tension to the power 2 3 pacs numbers 47 10 g 03 40 gc 68 10 m

an extremal mathcal n 2 superconformal field theory here we provide an example of an extremal chiral mathcal n 2 superconformal field theory at c 24 the construction is based on a mathbb z 2 orbifold of the theory associated to the a 1 24 niemeier lattice the statespace is governed by representations of the sporadic group m 23

calculus of twisted vertex operators starting from an arbitrary isometry of an arbitrary even lattice twisted and shifted vertex operators are introduced under commutators these operators provide realizations of twisted affine lie algebras this construction generalizing a number of known ones is based on a self contained calculus

categorical representation of locally noetherian log schemes abstract in this paper we show that for a certain fairly general class of log schemes the structure of the log scheme may be recovered entirely from the purely categorical structure of a certain associated category of log schemes of finite type over the given log scheme this result is motivated partly by grothendieck s anabelian philosophy and partly by general philosophical considerations concerning the importance of categories as a foundation for mathematics

realization of the level one standard _ 2 1 modules in this paper we study the level one standard or irreducible integrable highest weight modules for the affine symplectic lie algebras in particular we give concrete realizations of all level one standard modules for the affine symplectic lie algebras of even rank introduction in recent years explicit realizations of nontrivial representations of affine lie algebras have attracted the attention of many researchers because of their surprising connections with different areas of mathematics and physics among these realizations the principal realizations and the untwisted homogeneous realizations have proved to be especially useful in 1978 lepowsky and wilson 15 constructed the basic modules level one standard modules i e irreducible integrable highest weight modules for a in the principal realization in 1 o this construction was generalized to the level one standard modules for all simply laced affine lie algebras frenkel and kac 5 and segal 26 have given explicit construction of these modules in the untwisted realization the constructions of these modules in the general realization are given in 9 12 since then some higher level standard modules for certain simply laced affine lie algebras have also been constructed e g 14 17 18 19 21 22 25 however less is known regarding explicit constructions of standard modules for nonsimply laced affine lie algebras that is affine lie algebras of type b c f and g in the last section of 10 kac kazhdan lepowsky and wilson constructed reducible level one modules for the non simply laced affine lie algebras in the principal realization by taking fixed points of dynkin diagraminduced automorphisms recently in the untwisted realization the analogous representations again reducible have been constructed in 3 and 6 the level one standard modules for the affine orthogonal lie algebras b 1 have been n explicitly realized in 4 8 13 and 20 from different view points in 20 mandia has constructed the level one standard modules for the affine lie algebras f 1 and g 1 in 23 the author gave explicit constructions of the level one standard modules for the affine symplectic lie algebras of rank 3 and 4 received by the editors july 13 1989 presented to the 847th meeting of the ams pheonix arizona january 14 1989 1980 mathematics subject classification 1985 revision primary 17b65 1990 american mathematical society 0002 9947 90 1 00 25 per page

a construction of the level 3 modules for the affine lie algebra and a new combinatorial identity of the rogers ramanujan type amer math soc 348 1996 pp 481 501

orbifolds of lattice vertex algebras under an isometry of order two abstract every isometry of a positive definite even lattice q can be lifted to an automorphism of the lattice vertex algebra v q an important problem in vertex algebra theory and conformal field theory is to classify the representations of the invariant subalgebra v q of v q known as an orbifold in the case when is an isometry of q of order two we classify the irreducible modules of the orbifold vertex algebra v q and identify them as submodules of twisted or untwisted v q modules the examples where q is a root lattice and is a dynkin diagram automorphism are presented in detail

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

automorphisms of niemeier lattices for miyamoto s mathbb z _3 orbifold construction we classify up to conjugation all automorphisms of niemeier lattices to which we can apply miyamoto s orbifold construction using this classification we prove that the voas obtained in miyamoto symmetries integrable systems and representations springer proceedings in mathematics and statistics vol 40 springer london pp 319 344 2013 and sagaki and shimakura trans am math soc to appear are all of holomorphic non lattice voas which we can obtain by applying the mathbb z 3 orbifold construction to a niemeier lattice and its automorphism

graphene lattice field theory and symmetries borrowing ideas from tight binding model we propose a board class of lattice field models that are classified by non simply laced lie algebras in the case of an 1 su n series we show that the couplings between the quantum states living at the first nearest neighbor sites of the lattice lsun are governed by the complex fundamental representations n and n of su n and the second nearest neighbor interactions are described by its adjoint n n the lattice models associated with the leading su 2 su 3 and su 4 cases are explicitly studied and their fermionic field realizations are given it is also shown that the su 2 and su 3 models describe the electronic properties of the acetylene chain and the graphene respectively it is established as well that the energy dispersion of the first nearest neighbor couplings is completely determined by the an roots through the typical dependence n 2 roots cosk with k the wave vector other features such as the so 2n extension and other application

moving boundary approximation for curved streamer ionization fronts solvability analysis the minimal density model for negative streamer ionization fronts is investigated an earlier moving boundary approximation for this model consisted of a kinetic undercooling type boundary condition in a laplacian growth problem of hele shaw type here we derive a curvature correction to the moving boundary approximation that resembles surface tension the calculation is based on solvability analysis with unconventional features namely there are three relevant zero modes of the adjoint operator one of them diverging furthermore the inner outer matching ahead of the front must be performed on a line rather than on an extended region and the whole calculation can be performed analytically the analysis reveals a relation between the fields ahead and behind a slowly evolving curved front the curvature and the generated conductivity this relation forces us to give up the ideal conductivity approximation and we suggest to replace it by a charge neutrality approximation this implies that the electric potential in the streamer interior is no longer constant but solves a laplace equation this leads to a muskat type problem

additional symmetries of generalized integrable hierarchies the non isospectral symmetries of a general class of integrable hierarchies are found by generalizing the galilean and scaling symmetries of the korteweg de vries equation and its hierarchy the symmetries arise in a very natural way from the semi direct product structure of the virasoro algebra and the affine kac moody algebra underlying the construction of the hierarchy in particular the generators of the symmetries are shown to satisfy a subalgebra of the virasoro algebra when a tau function formalism is available the infinitesimal symmetries act directly on the tau functions as moments of virasoro currents some comments are made regarding the role of the non isospectral symmetries and the form of the string equations in matrix model formulations of quantum gravity in two dimensions and related systems

on the period index problem in light of the section conjecture period and index of a curve x k over a p adic local field k such that the fundamental group pi 1 x k admits a splitting are shown to be powers of p as a consequence examples of curves over number fields are constructed where having sections is obstructed locally at a p adic place hence the section conjecture holds for these curves as there are neither sections nor rational points

extensions of tensor products of zp orbifold models of the lattice vertex operator algebra v2ap 1 abstract let p be an odd prime and let be an order p automorphism of v 2 a p 1 which is a lift of a p cycle in the weyl group weyl a p 1 s p we study a certain extension v of a tensor product of finitely many copies of the orbifold model v 2 a p 1 and give a criterion for v that every irreducible v module is a simple current

a moving boundary model motivated by electric breakdown ii initial value problem abstract an interfacial approximation of the streamer stage in the evolution of sparks and lightning can be formulated as a laplacian growth model regularized by a kinetic undercooling boundary condition using this model we study both the linearized and the full nonlinear evolution of small perturbations of a uniformly translating circle within the linear approximation analytical and numerical results show that perturbations are advected to the back of the circle where they decay an initially analytic interface stays analytic for all finite times but singularities from outside the physical region approach the interface for t which results in some anomalous relaxation at the back of the circle for the nonlinear evolution numerical results indicate that the circle is the asymptotic attractor for small perturbations but larger perturbations may lead to branching we also present results for more general initial shapes which demonstrate that regularization by kinetic undercooling cannot guarantee smooth interfaces globally in time

on twisted representations of vertex algebras abstract in this paper we develop a formalism for working with representations of vertex and conformal algebras by generalized fields formal power series involving non integer powers of the variable the main application of our technique is the construction of a large family of representations for the vertex superalgebra v corresponding to an integer lattice for an automorphism v v coming from a finite order automorphism we find the conditions for existence of twisted modules of v we show that the category of twisted representations of v is semisimple with finitely many isomorphism classes of simple objects

nonabelian orbifolds and the boson fermion correspondence for a positive integerl divisible by 8 there is a bosonic holomorphic vertex operator algebra voa v gamma l associated to the spin lattice l for a broad class of finite groupsg of automorphisms of v gamma l we prove the existence and uniqueness of irreducibleg twisted v gamma l modules and establish the modular invariance of the partition functionsz g h for commuting elements ing in particular for any finite group there are infinitely many holomorphic voas admittingg for which these properties hold the proof is facilitated by a boson fermion correspondence which gives a voa isomorphism between v gamma l and a certain fermionic construction and which extends work of frenkel and others

cartan determinants and shapovalov forms we compute the determinant of the gram matrix of the shapovalov form on weight spaces of the basic representation of an affine kac moody algebra of ade type possibly twisted as a consequence we obtain explicit formulae for the determinants of the cartan matrices of p blocks of the symmetric group and its double cover and of the associated hecke algebras at roots of unity

the pro p hom form of the birational anabelian conjecture we prove a pro p hom form of the birational anabelian conjecture for function fields over sub p adic fields our starting point is the corresponding theorem of mochizuki in the case of transcendence degree 1

twist vertex operators for twisted modules abstract we introduce and study twist vertex operators for a lower bounded generalized twisted modules for a grading restricted vertex super algebra we prove the duality property weak associativity a jacobi identity a generalized commutator formula generalized weak commutativity and convergence and commutativity for products of more than two operators involving twist vertex operators these properties of twist vertex operators play an important role in the author s recent general direct and explicit construction of lower bounded generalized twisted modules

elements of the annihilating ideal of a standard module the construction of affine lie algebras by means of vertex operators has proved to be a remarkably fruitful line of research by motivating being motivated by and linking together a disparate range of sectors of mathematics and physics vertex operators entered mathematics through the work of j lepowsky and r l wilson who gave an explicit construction of the affne algebra a l lwl their vertex operators were similar to ones used by physicists in the string model later on i frenkel v kac fk and g segal s found representations of afhne algebras using untwisted vertex operators a large amount of very interesting research has followed stimulated by this observation and by the connection between classical combinatorial identities and the weyl kac character formula k lm lpl 2 lw2 4 ma mil 21 a deeper understanding of vertex operators has been reached more recently as a result of work by borcherds b who investigated general vertex operator algebras and by frenkel lepowsky and a meurman in a series of papers culminating in the recent book flm in particular these three authors showed that the monster finite simple group is the automorphism group of a certain vertex operator algebra thus giving a natural realization of this largest among sporadic groups this also established a surprising connection between the monster and two dimensional conformal quantum field theory which is the physicists analogue of the theory of vertex operator algebras cf bpz dl fhl fj tk t zf flm

on mathbb z _p z p orbifold constructions of the moonshine vertex operator algebra for p 3 5 7 13 we consider a mathbb z p orbifold construction of the moonshine vertex operator algebra v natural we show that the vertex operator algebra obtained by the mathbb z p orbifold construction on the leech lattice vertex operator algebra v lambda and a lift of a fixed point free isometry of order p is isomorphic to the moonshine vertex operator algebra v natural we also describe the relationship between those mathbb z p orbifold constructions and the mathbb z 2 orbifold construction in a uniform manner in appendix we give a characterization of the moonshine vertex operator algebra v natural by the existence of an orthogonal pair of ising vectors

shifted vertex operator algebras and g elliptic systems shifted vertex operator algebras and superalgebras are defined both in bosonic and in fermionic pictures their isomorphism boson fermion correspondence is shown partition functions of bosonic shifted vertex operator algebras are realized analytically and are interpreted as string path integrals over elliptic curves their modular properties follow and elliptic systems are constructed based on fermionic shifted vertex operator algebras

twisted vertex representations of quantum affine algebras recent interests in quantum groups are stimulated by their marvelous relations with quantum yang baxter equations conformal field theory invariants of links and knots and q hypergeometric series besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q deformations or quantum analogues of known structures quantum groups were first defined by drinfeld 2 and jimbo 9 also see 4 as a q deformation of the universal enveloping algebras of the kacmoody algebras in the work of trigonometric solutions of yang baxter equations in the same spirit it was shown in 13 14 that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable kac moody algebras and those of the corresponding quantum groups where both spaces have the same dimension in the case of generic q i e q is not a root of unity moreover one can be very explicit in the case of quantum gl n to write down the irreducible highest weight representations quantum affine algebras are the quantum groups associated to affine lie algebras following drinfeld s realization 3 of q analog of loop algebras the vertex representation of untwisted simply laced quantum affine algebras was constructed in frenkel jing 6 which is a q deformation of frenkel kac 7 and segal 15 construction in the theory of affine lie algebras subsequently the same was done for the quantum affine algebra of type b in 1 in the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the dynkin diagram which generalizes certain important cases in the ordinary twisted vertex operator calculus 5

a moving boundary problem motivated by electric breakdown i spectrum of linear perturbations abstract an interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a laplacian growth model regularized by a kinetic undercooling boundary condition we study the linear stability of uniformly translating circles that solve the problem in two dimensions in a space of smooth perturbations of the circular shape the stability operator is found to have a pure point spectrum except for the eigenvalue 0 0 for infinitesimal translations all eigenvalues are shown to have negative real part therefore perturbations decay exponentially in time we calculate the spectrum through a combination of asymptotic and series evaluation in the limit of vanishing regularization parameter all eigenvalues are found to approach zero in a singular fashion and this asymptotic behavior is worked out in detail a consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations both the linear and the nonlinear initial value problem are considered in a second paper

hirota equations associated with simply laced affine lie algebras the cuspidal class e6 of e6 1 in this paper we derive hirota equations associated with the simply laced affine lie algebras g 1 where g is one of the simply laced complex lie algebras an dn e6 e7 or e8 defined by finite order automorphisms of g which we call lepowsky automorphisms in particular we investigate the hirota equations for lepowsky automorphisms of e6 defined by the cuspidal class e6 of the weyl group w e6 of e6 we also investigate the relationship between the lepowsky automorphisms of the simply laced complex lie algebras g and the conjugate canonical automorphisms defined by kac this analysis is applied to identify the canonical automorphisms for the cuspidal class e6 of e6