laplacian and diffusional growth a unified theoretical description for symmetrical and parity broken patterns abstract nowadays laplacian growth patterns such as the saffman taylor instability are well understood mainly because of exact results derived by conformal mapping techniques this is not the case for crystal growth processes except for the free needle crystal normally it is thought that the crystal growth process can be understood by an analogy with the equivalent steady laplacian process the results that follow from this analogy however are in contradiction with numerical and experimental results even at very low velocities we present here a way to mimic the diffusion instabilities in a simple way which allow to describe the formation and the dynamics of new patterns we show that the problem of crystal growth in a channel is closely related to the saffman taylor problem in a sector by using this analogy we obtain the family of zero surface tension solutions for small peclet numbers including an asymmetric degree of freedom we study analytically the selection problem and confirm that parity broken solutions originate from the symmetrical solutions through a bifurcation in agreement with recent numerical investigations we obtain scaling laws for the parity broken solutions for a wide range of parameters we extend these results to the infinite geometry and show that parity broken double fingers can exist even in the range of small undercooling with fully isotropic surface tension the scaling law for the selected velocity of double fingering structures is very different from the scaling law of a free dendrite with anisotropy

quantum analogues of richardson varieties in the grassmannian and their toric degeneration abstract in the present paper we are interested in natural quantum analogues of richardson varieties in the type a grassmannians to be more precise the objects that we investigate are quantum analogues of the homogeneous coordinate rings of richardson varieties which appear naturally in the theory of quantum groups our point of view here is geometric we are interested in the regularity properties of these non commutative varieties such as their irreducibility normality cohen macaulayness in the spirit of non commutative algebraic geometry a major step in our approach is to show that these algebras have the structure of an algebra with a straightening law from this it follows that they degenerate to some quantum analogues of toric varieties

3 d flow of a compressible viscous micropolar fluid with spherical symmetry regularity of the solution abstract in this paper we consider the nonstationary 3 d flow of a compressible viscous and heat conducting micropolar fluid that is in the thermodynamical sense perfect and polytropic the fluid domain is a subset of r 3 bounded with two concentric spheres that present the solid thermoinsulated walls the corresponding mathematical model is set up in the lagrangian description we assume that the initial data are spherically symmetric functions and that the initial density and temperature are strictly positive this problem has a unique spherically symmetric generalized solution globally in time here we introduce the holder continuous initial functions and prove that for any t 0 the state function is also holder continuous

global solution to 3d problem of a compressible viscous micropolar fluid with spherical symmetry and a free boundary abstract we consider a nonstationary 3d flow of a compressible viscous heat conducting micropolar fluid which is in the thermodynamical sense perfect and polytropic in the domain bounded with two concentric spheres in r 3 in this paper we establish the existence of a global solution to the free boundary problem defined with the homogeneous boundary conditions for velocity microrotation heat flux on the fixed border and homogeneous boundary conditions for strain microrotation and heat flux on the free boundary we suppose that the initial data are spherically symmetric with positive initial density and temperature having the zero density on the free boundary because of the spherical symmetry the starting three dimensional problem is transformed to the one dimensional problem in lagrangian coordinates in the domain that is a segment the solution to our problem is obtained as a limit of the sequence of approximate solutions derived from suitable semi discrete finite difference approximate systems by using the derived bounded estimates of the approximate solutions and the results of the weak and strong compactness we establish the convergence to the generalized solution of our problem globally in time

selection of saffman taylor fingers in the sector geometry we study numerically and analytically the selection of saffman taylor fingers in cells having the shape of a disk sector for divergent fingers we find results which are qualitatively new compared to the standard linear geometry instead of a discrete set of solutions all converging to a relative width of 1 2 for zero surface tension we find that the fingers disappear below some surface tension this happens by the merging of neighboring branches of the discrete set of solutions this behavior is in quantitative agreement with experimental results

radial fingering in a hele shaw cell a weakly nonlinear analysis abstract the saffman taylor viscous fingering instability occurs when a less viscous fluid displaces a more viscous one between narrowly spaced parallel plates in a hele shaw cell experiments in radial flow geometry form fan like patterns in which fingers of different lengths compete spread and split our weakly nonlinear analysis of the instability predicts these phenomena which are beyond the scope of linear stability theory finger competition arises through enhanced growth of sub harmonic perturbations while spreading and splitting occur through the growth of harmonic modes nonlinear mode coupling enhances the growth of these specific perturbations with appropriate relative phases as we demonstrate through a symmetry analysis of the mode coupling equations we contrast mode coupling in radial flow with rectangular flow geometry

double canonical bases abstract we introduce a new class of bases for quantized universal enveloping algebras u q g and other doubles attached to semisimple and kac moody lie algebras these bases contain dual canonical bases of upper and lower halves of u q g and are invariant under many symmetries including all lusztig s symmetries if g is semisimple it also turns out that a part of a double canonical basis of u q g spans its center and consists of higher casimirs which suggests physical applications

the pointwise estimates of diffusion wave of the compressible micropolar fluids abstract the pointwise estimates for the compressible micropolar fluids in dimension three are given which exhibit generalized huygens principle for the fluid density and fluid momentum as the compressible navier stokes equation while the micro rational momentum behaves like the fluid momentum of the euler equation with damping to circumvent the complexity from 7 7 green s matrix we use the decomposition of fluid part and electromagnetic part for the momentums to study three smaller green s matrices the following from this decomposition is that we have to deal with the new problem that the nonlinear terms contain nonlocal operators we solve it by using the natural match of these new green s functions and the nonlinear terms moreover to derive the different pointwise estimates for different unknown variables such that the estimate of each unknown variable is in agreement with its green s function we develop some new estimates on the nonlinear interplay between different waves

brunn minkowski inequality for multiplicities this is a convex subset of the real vector space p zr in fact it is a convex polytope see br where this polytope is discussed from an algebraic point of view it is known that the total mass of m is a polynomial in m for su ciently large m denote by k its degree and m k m m weak d where d is the lebesgue measure supported on and the density is a piecewise polynomial function we will not use this piecewise polynomiality in this paper recall that a real function f de ned on a convex subset u of a vector space v is called concave if

wavelength selection in hele shaw flows a maximum amplitude criterion as in most interfacial flow problems the standard theoretical procedure to establish wavelength selection in the viscous fingering instability is to maximize the linear growth rate however there are important discrepancies between previous theoretical predictions and existing experimental data in this work we perform a linear stability analysis of the radial hele shaw flow system that takes into account the combined action of viscous normal stresses and wetting effects most importantly we introduce an alternative selection criterion for which the selected wavelength is determined by the maximum of the interfacial perturbation amplitude the effectiveness of such a criterion is substantiated by the significantly improved agreement between theory and experiments

toric degeneration of schubert varieties and gelfand tsetlin polytopes this note constructs the flat toric degeneration of the manifold fln of flags in cn due to gonciulea and lakshmibai transform groups 1 3 1996 215 as an explicit git quotient of the grobner degeneration due to knutson and miller grobner geometry of schubert polynomials ann math 2 to appear this implies that schubert varieties degenerate to reduced unions of toric varieties associated to faces indexed by rc graphs reduced pipe dreams in the gelfand tsetlin polytope our explicit description of the toric degeneration of fln provides a simple explanation of how gelfand tsetlin decompositions for irreducible polynomial representations of gln arise via geometric quantization

canonical bases for the quantum group of type a_r and piecewise linear combinatorics this work was motivated by the following two problems from the classical representation theory both problems make sense for an arbitrary complex semisimple lie algebra but since we shall deal only with the ar case we formulate them in this generality 1 construct a good basis in every irreducible finite dimensional slr 1 module v which materializes the littlewood richardson rule a precise formulation of this problem was given in 3 we shall explain it in more detail a bit later 2 construct a basis in every polynomial representation of glr 1 such that the maximal element w0 of the weyl group sr 1 considered as an element of glr 1 acts on this basis by a permutation up to a sign and explicitly compute this permutation this problem is motivated by recent work by john stembridge 10 and was brought to our attention by his talk at the jerusalem combinatorics conference may 1993

study of q commutations in uq n algebra abstract let g be a semi simple lie algebra with maximal nilpotent subalgebra n let q g and u q n be the quantized enveloping algebras we describe the set of normal elements of u q n we prove that each ideal of u q n can be generated by a normal sequence

mean field diffusion limited aggregation a density model for viscous fingering phenomena we explore a universal density formalism to describe nonequilibrium growth processes specifically the immiscible viscous fingering in hele shaw cells usually referred to as the saffman taylor problem for that we develop an alternative approach to the viscous fingering phenomena whose basic concepts have been recently published in a rapid communication phys rev e 63 045305 r 2001 this approach uses the diffusion limited aggregation dla paradigm as a core we introduce a mean field dla generalization in stochastic and deterministic formulations the stochastic model a quasicontinuum dla simulates monte carlo patterns which demonstrate a striking resemblance to natural hele shaw fingers and for steady state growth regimes follow precisely the saffman taylor analytical solutions in channel and sector configurations the relevant deterministic theory a complete set of differential equations for a time development of density fields is derived from that stochastic model as a principal conclusion we prove an asymptotic equivalency of both the stochastic and deterministic mean field dla formulations to the classic saffman taylor hydrodynamics in terms of an interface evolution

okounkov bodies and seshadri constants abstract okounkov bodies which are closed convex sets defined for big line bundles have rich information on the line bundles on the other hand seshadri constants are invariants which measure the positivity of line bundles in this paper we prove that okounkov bodies give lower bounds of seshadri constants

dynamics of viscous fingers in rotating hele shaw cells with coriolis effects a growing number of experimental and theoretical works have been addressing various aspects of the viscous fingering formation in rotating hele shaw cells however only a few of them consider the influence of coriolis forces the studies including coriolis effects are mostly restricted to the high viscosity contrast limit and rely on either purely linear stability analyses or intensive numerical simulations we approach the problem analytically and use a modified darcy s law including the exact form of the coriolis effects to execute a mode coupling analysis of the system by imposing no restrictions on the viscosity contrast a dimensionless viscosity difference we go beyond linear stages and examine the onset of nonlinearities our results indicate that when coriolis effects are taken into account an interesting interplay between the reynolds number re and a arises this leads to important changes in the stability and morphological features of the emerging interfacial patterns we contrast our mode coupling approach with previous theoretical models proposed in the literature

okounkov bodies and restricted volumes along very general curves abstract given a big divisor d on a normal complex projective variety x we show that the restricted volume of d along a very general complete intersection curve c x can be read off from the okounkov body of d with respect to an admissible flag containing c from this we deduce that if two big divisors d 1 and d 2 on x have the same okounkov body with respect to every admissible flag then d 1 and d 2 are numerically equivalent

optimal time decay of the compressible micropolar fluids abstract this paper primarily studies the large time behavior of solutions to the cauchy problem on the compressible micropolar fluid system which is a generalization of the classical navier stokes system the asymptotic stability of the steady state with the strictly positive constant density the vanishing velocity and micro rotational velocity is established under small perturbation in regular sobolev space moreover it turns out that both the density and the velocity tend time asymptotically to the corresponding equilibrium state with rate 1 t 3 4 in l 2 and the micro rotational velocity also tends to the equilibrium state with the faster rate 1 t 5 4 in l 2 norm the proof is based on the spectrum analysis and time weighted energy estimate

stability of rarefaction waves for 1 d compressible viscous micropolar fluid model abstract in this paper we study the large time behavior of the cauchy problem of the compressible micropolar fluids in one dimensional space according to darcy s law we see from the equation describing the evolution of the microrotation that 0 as t actually if the microstructure of the fluid is not taken into account that is to say the effect of the angular velocity fields of the particle s rotation is omitted i e 0 then the micropolar fluids equations reduce to the classical navier stokes equations therefore the system should tend time asymptotically to the corresponding classical full navier stokes equations we consider the case that the far field of the initial data for the microrotation velocity is zero and the far fields of the initial data for other variables such as the specific volume v velocity u and entropy s are connected by rarefaction waves to the corresponding euler equations in this case we prove the stability of rarefaction waves for this compressible micropolar fluids model compared with the classical navier stokes equations the angular velocity in this model brings both benefit and trouble the benefit lies in the fact that the term v is a damping term which provides extra regularity of while the trouble is brought by the term v 2 which increases the nonlinearity of the system

field of fractions of certain quantum algebras abstract in this paper we show that the field of fractions of certain algebras appearing in the theory of quantum groups are isomorphic to the field of fractions of quantum n space when the parameter q is not a root of 1 more precisely the quantum serre relations of type a n and the relations defining the quantum weyl algebra induce q communtation relations in the field of fractions also we study the problem of separation of these skew fields using the standard embeddings in non commutative series

global okounkov bodies for bott samelson varieties abstract we use the theory of mori dream spaces to prove that the global okounkov body of a bott samelson variety with respect to a natural flag of subvarieties is rational polyhedral as a corollary okounkov bodies of effective line bundles over schubert varieties are shown to be rational polyhedral in particular it follows that the global okounkov body of a flag variety g b is rational polyhedral as an application we show that the asymptotic behaviour of dimensions of weight spaces in section spaces of line bundles is given by the volume of polytopes

invariant prime ideals in quantizations of nilpotent lie algebras de concini kac and procesi defined a family of subalgebras u of a quantized universal enveloping algebra uq g associated to the elements of the corresponding weyl group w they are deformations of the universal enveloping algebras u n adw n where n are the nilradicals of a pair of dual borel subalgebras based on results of gorelik and joseph and an interpretation of u as quantized algebras of functions on schubert cells we construct explicitly the h invariant prime ideals of each u and show that the corresponding poset is isomorphic to w where h is the group of group like elements of uq g moreover for each h prime of u w we construct a generating set in terms of demazure modules related to fundamental representations using results of ramanathan and kempf we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related poisson structures on schubert cells in flag varieties

geometric lifting of the canonical basis and semitoric degenerations of richardson varieties in the sl n case a berenstein and a zelevinsky studied the schutzenberger involution in terms of lusztig s canonical basis 3 we generalize their construction and formulas for any semisimple lie algebra we use for this the geometric lifting of the canonical basis on which an analogue of the schutzenberger involution can be given as an application we construct semitoric degenerations of richardson varieties following a method of p caldero 6

effacement des derivations et spectres premiers des algebres quantiques abstract given any commutative field k and any iterated ore extension r k x1 x2 2 2 xn n n satisfying some suitable assumptions we construct the so called derivative elimination algorithm it consists of a sequence of changes of variables inside the division ring f fract r starting with the indeterminates x1 xn and terminating with new variables t1 tn these new variables generate some quantum affine space r such that f fract r this algorithm induces a natural embedding spec r spec r which satisfies the following property if p is in spec r then fract r p fract r p we study both the derivative elimination algorithm and natural embedding and use them to produce for the general case a common proof of the quantum gelfand kirillov property for the prime homomorphic images of the following quantum algebras u q w g b q w g w w r q g where g denotes any complex semi simple connected simply connected lie group with associated lie algebra g and weyl group w quantum matrices algebras quantum weyl algebras and quantum euclidean respectively symplectic spaces another application will be given in g cauchon j algebra to appear in the general case the prime spectrum of any quantum matrices algebra satisfies the normal separation property

normality of torus orbit closures in g p abstract the purpose of this note is to classify the torus orbit closures in an arbitrary algebraic homogeneous space g p that are toric varieties

weyl group multiple dirichlet series eisenstein series and crystal bases we show that the whittaker coecients of borel eisenstein series on the metaplectic covers of glr 1 can be described as multiple dirichlet series in r complex variables whose coecients are computed by attaching a number theoretic quantity a product of gauss sums to each vertex in a crystal graph these gauss sums depend on string data previously introduced in work of lusztig berenstein and zelevinsky and littelmann these data are the lengths of segments in a path from the given vertex to the vertex of lowest weight depending on a factorization of the long weyl group element into simple reections the coecients may also be described as sums over strict gelfand tsetlin patterns the description is uniform in the degree of the metaplectic cover

invariants in the enveloping algebra of a semisimple lie algebra for the adjoint action of a nilpotent lie subalgebra let be a complex semisimple lie algebra and let resp be its symmetric resp enveloping algebra it is shown in 1 that finite algebras can be realized up to a central extension as the algebra of invariants resp for the adjoint action of a lie subalgebra of for nilpotent we use the taylor lemma to give a description generators and relations for these algebras

3 d flow of a compressible viscous micropolar fluid with spherical symmetry large time behavior of the solution abstract we consider the non stationary 3 d flow of a compressible viscous heat conducting micropolar fluid in the domain that is the subset of r 3 bounded with two concentric spheres that present solid thermo insulated walls under the assumption that the fluid is perfect and polytropic in the thermodynamical sense as well as that the initial density and temperature are strictly positive and that the initial data are sufficiently smooth spherically symmetric functions the corresponding problem with homogeneous boundary data has a unique generalized solution for any time interval 0 t t r in this work we analyze large time behavior of the solution and prove its stabilization

torus invariant prime ideals in quantum matrices totally nonnegative cells and symplectic leaves the algebra of quantum matrices of a given size supports a rational torus action by automorphisms it follows from work of letzter and the first named author that to understand the prime and primitive spectra of this algebra the first step is to understand the prime ideals that are invariant under the torus action in this paper we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size as a corollary we obtain explicit generating sets of quantum minors for the torus invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over mathbb q

stability of contact discontinuity for 1 d compressible viscous micropolar fluid model abstract this paper is concerned with a one dimensional compressible micropolar fluid model with initial data whose behaviors at far fields x are different motivated by the relationship between micropolar fluid model and navier stokes system we can prove that the solutions to the one dimensional compressible micropolar fluid model tend time asymptotically to a viscous contact wave which is constructed from a contact discontinuity solution of the riemann problem on euler system this result is proved by an elaborate elementary energy estimates

global solutions for a one dimensional compressible micropolar fluid model with zero heat conductivity abstract in this study we consider global solutions for the one dimensional compressible micropolar fluid model with zero heat conductivity which is a hyperbolic parabolic system the pressure velocity and angular velocity are dissipative because of the viscosity whereas the entropy is non dissipative due to the absence of heat conductivity compared with the classical navier stokes equations there is an extra angular velocity in the micropolar fluid model which causes both benefits and problems the benefit is due to the fact that the term v is a damping term and it provides extra regularity for and the problem is due to the term v 2 because it increases the nonlinearity of the system the global solutions can be obtained by combining the local existence and a priori estimates if the h 2 norm of the initial perturbation around a constant state is sufficiently small the asymptotic behavior is also determined in this study

okounkov bodies and toric degenerations let vardelta be the okounkov body of a divisor d on a projective variety x we describe a geometric criterion for vardelta to be a lattice polytope and show that in this situation x admits a flat degeneration to the corresponding toric variety this degeneration is functorial in an appropriate sense

vanishing theorem for the cohomology of line bundles on bott samelson varieties abstract we use the grossberg karshon s degeneration of bott samelson varieties to toric varieties and the description of cohomology of line bundles on toric varieties to deduce vanishing results for the cohomology of lines bundles on bott samelson varieties

totally nonnegative cells and matrix poisson varieties we describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices that is the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by a postnikov in order to do this we relate the totally nonnegative cells to torus orbits of symplectic leaves of the poisson varieties of complex matrices in particular we describe the minors that vanish on a torus orbit of symplectic leaves we prove that such families of minors are exactly the admissible families and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices

global behavior for compressible viscous micropolar fluid with spherical symmetry abstract in this paper we prove the regularity and exponential stability of solutions for a viscous micropolar fluid in the domain to be subset of r 3 bounded with two concentric spheres that present the solid thermoinsulated walls some new ideas and more delicate estimates are introduced to prove these results

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

universal attractors for the navier stokes equations of compressible and heat conductive fluid in bounded annular domains in ℝn this paper is concerned with the dynamics for the navier stokes equations for a polytropic viscous heat conductive ideal gas in bounded annular domains n in n n 2 3 one of the important features of this problem is that the metric spaces h 1 and h 2 we work with are two incomplete metric spaces as can be seen from the constraints 0 and u 0 with and u being absolute temperature and specific volume respectively for any constants 1 2 3 4 5 satisfying certain conditions two sequences of closed subspaces h i h i i 1 2 are found and the existence of two maximal universal attractors in h 1 and h 2 is proved

on theq commutations inuq n at roots of one abstract let e be a root of one and g a semisimple lie algebra with triangular decomposition g n h n letu e resp ures e be the nonrestricted resp restricted quantum enveloping algebra of n we prove that fract u eis a quantum weyl field we then give a description of the e center ofu e letufin ebe the finite part ofures e via the drinfeld correspondence theufin e covariant space of a weyl module is e central in case g s l n this enables us to describe this space in terms of semistandard young tableaux

newton okounkov bodies semigroups of integral points graded algebras and intersection theory generalizing the notion of newton polytope we dene the newtonokounkov body respectively for semigroups of integral points graded algebras and linear series on varieties we prove that any semigroup in the lattice z n is asymptotically approximated by the semigroup of all the points in a sublattice and lying in a convex cone applying this we obtain several results we show that for a large class of graded algebras the hilbert functions have polynomial growth and their growth coecients satisfy a brunn minkowski type inequality we prove analogues of the fujita approximation theorem for semigroups of integral points and graded algebras which imply a generalization of this theorem for arbitrary linear series applications to intersection theory include a far reaching generalization of the kushnirenko theorem from newton polytope theory and a new version of the hodge inequality we also give elementary proofs of the alexandrov fenchel inequality in convex geometry and its analogue in algebraic geometry

coexistence of symmetric and parity broken dendrites in a channel abstract we numerically solve the steady state equation and the stability spectrum for solidification in a channel for a large range of parameters stable symmetric and stable parity broken solutions coexist the branches of parity broken solutions are found to originate from the symmetric solutions through standard bifurcations

geometric approach to viscous fingering on a cone we study fluid flow and the formation of viscous fingering patterns on a two dimensional conical background space defined as the conical hele shaw cell we approach the problem geometrically and study how the nontrivial topological structure of the conical cell affects the evolution of the interface separating two viscous fluids we perform a perturbative weakly nonlinear analysis of the problem and derive a mode coupling differential equation which describes fluid fluid interface behaviour our nonlinear study predicts the formation of fingering structures in which fingers of different lengths compete and split at their tips the shape of the emerging patterns show a significant sensitivity to variations in the cell s topological features which can be monitored by changing the cone opening angle we find that for increasingly larger values of the opening angle finger competition is inhibited while finger tip splitting is enhanced

okounkov bodies on projectivizations of rank two toric vector bundles abstract the global okounkov body of a projective variety is a closed convex cone that encodes asymptotic information about every big line bundle on the variety in the case of a rank two toric vector bundle e on a smooth projective toric variety we use its klyachko filtrations to give an explicit description of the global okounkov body of p e in particular we show that this is a rational polyhedral cone and that p e is a mori dream space

systematic weakly nonlinear analysis of radial viscous fingering we present a weakly nonlinear analysis of the interface dynamics in a radial hele shaw cell driven by both injection and rotation we extend the systematic expansion introduced in e alvarez lacalle et al phys rev e 64 016302 2001 to the radial geometry and compute explicitly the first nonlinear contributions we also find the necessary and sufficient condition for the uniform convergence of the nonlinear expansion within this region of convergence the analytical predictions at low orders are compared satisfactorily to exact solutions and numerical integration of the problem this is particularly remarkable in configurations with no counterpart in the channel geometry for which the interplay between injection and rotation allows that condition to be satisfied at all times in the case of the purely centrifugal forcing we demonstrate that nonlinear couplings make the interface more unstable for lower viscosity contrast between the fluids

convex bodies appearing as okounkov bodies of divisors abstract based on the work of okounkov okounkov 1996 15 2003 16 lazarsfeld and musta 2009 13 and kaveh and khovanskii preprint 10 have independently associated a convex body called the okounkov body to a big divisor on a smooth projective variety with respect to a complete flag in this paper we consider the following question what can be said about the set of convex bodies that appear as okounkov bodies we show first that the set of convex bodies appearing as okounkov bodies of big line bundles on smooth projective varieties with respect to admissible flags is countable we then give a complete characterisation of the set of convex bodies that arise as okounkov bodies of r divisors on smooth projective surfaces such okounkov bodies are always polygons satisfying certain combinatorial criteria finally we construct two examples of non polyhedral okounkov bodies in the first one the variety we deal with is fano and the line bundle is ample in the second one we find a mori dream space variety such that under small perturbations of the flag the okounkov body remains non polyhedral

integrable systems toric degenerations and okounkov bodies let x be a smooth projective variety of dimension n over mathbb c equipped with a very ample hermitian line bundle mathcal l in the first part of the paper we show that if there exists a toric degeneration of x satisfying some natural hypotheses which are satisfied in many settings then there exists a surjective continuous map from x to the special fiber x 0 which is a symplectomorphism on an open dense subset u from this we are then able to construct a completely integrable system on x in the sense of symplectic geometry more precisely we construct a collection of real valued functions h 1 ldots h n on x which are continuous on all of x smooth on an open dense subset u of x and pairwise poisson commute on u moreover our integrable system in fact generates a hamiltonian torus action on u in the second part we show that the toric degenerations arising in the theory of newton okounkov bodies satisfy all the hypotheses of the first part of the paper in this case the image of the moment map mu h 1 ldots h n x rightarrow mathbb r n is precisely the newton okounkov body delta delta r v associated to the homogeneous coordinate ring r of x and an appropriate choice of a valuation v on r our main technical tools come from algebraic geometry differential kahler geometry and analysis specifically we use the gradient hamiltonian vector field and a subtle generalization of the famous ojasiewicz gradient inequality for real valued analytic functions since our construction is valid for a large class of projective varieties x this manuscript provides a rich source of new examples of integrable systems we discuss concrete examples including elliptic curves flag varieties of arbitrary connected complex reductive groups spherical varieties and weight varieties

viscous fluid fingering on a negatively curved surface viscous fingering formation in flat hele shaw cells is a classical and widely studied fluid mechanical problem we examine the development of viscous fluid fingering on a two dimensional surface of constant negative gaussian curvature the hyperbolic plane h 2 a perturbative mode coupling formalism is applied to study the influence of the negative surface curvature on the two most important pattern formation mechanisms of the system fingertip splitting and finger competition we also report on a time dependent control strategy placed on the injection rate which is able to minimize viscous fingering growth on h 2

representation theory of p adic groups and canonical bases in this paper we interpret the gindikin karpelevich formula and the casselman shalika formula as sums over kashiwara lusztig s canonical bases generalizing the results of bump and nakasuji 2010 7 to arbitrary split reductive groups we also rewrite formulas for spherical vectors and zonal spherical functions in terms of canonical bases in a subsequent paper kim and lee preprint 14 we will generalize these formulas to p adic affine kac moody groups

transfer results for frobenius extensions abstract we study frobenius extensions which are free filtered by a totally ordered finitely generated abelian group and their free graded counterparts first we show that the frobenius property passes up from a free graded extension to a free filtered extension then also from a free filtered extension to the extension of their rees algebras our main theorem states that under some natural hypotheses a free filtered extension of algebras is frobenius if and only if the associated graded extension is frobenius in the final section we apply this theorem to provide new examples and non examples of frobenius extensions

richardson varieties and equivariant k theory we generalize standard monomial theory smt to intersections of schubert varieties and opposite schubert varieties such varieties are called richardson varieties the aim of this article is to get closer to a geometric interpretation of the standard monomial theory as constructed in p littelmann j amer math soc 11 1998 551 567 in fact the construction given here is very close to the ideas in p lakshmibai c s seshadri j algebra 100 1986 462 557 our methods show that in order to develop a smt for a certain class of subvarieties in g b which includes g b it suffices to have the following three ingredients a basis for h0 g b l compatibility of such a basis with the varieties in the class certain quadratic relations in the monomials in the basis elements an important tool as in p lakshmibai c s seshadri j algebra 100 1986 462 557 will be the construction of nice filtrations of the vanishing ideal of the boundary of the varieties above this provides a direct connection to the equivariant k theory products of classes of structure sheaves with classes of line bundles where the combinatorially defined notion of standardness gets a geometric interpretation

removing the stiffness from interfacial flows with surface tension a new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in two dimensional irrotational and incompressible fluids through the laplace young condition at the interface surface tension introduces high order terms both nonlinear and nonlocal into the dynamics this leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult this new formulation has all the nice properties for time integration methods that are associated with having a linear constant coefficient highest order term that is using this formulation we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in fourier space the approach is based on a boundary integral formulation and applies more generally even to problems beyond the fluid mechanical context here they are applied to computing with high resolution the motion of interfaces in hele shaw flows and the motion of free surfaces in inviscid flows governed by the euler equations one hele shaw computation shows the behavior of an expanding gas bubble over long time as the interface undergoes successive tip splittings and finger competition a second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification in euler flows the computation of a vortex sheet shows its roll up through the kelvin helmholtz instability this motion culminates in the late time self intersection of the interface creating trapped bubbles of fluid this is we believe a type of singularity formation previously unobserved for such flows in 2d finally computations of falling plumes in an unstably stratified boussinesq fluid show a very similar behavior

braid group actions on coideal subalgebras of quantized enveloping algebras abstract we construct braid group actions on coideal subalgebras of quantized enveloping algebras which appear in the theory of quantum symmetric pairs in particular we construct an action of the semidirect product of z n and the classical braid group in n strands on the coideal subalgebra corresponding to the symmetric pair sl 2 n c sp 2 n c this proves a conjecture by molev and ragoucy we expect similar actions to exist for all symmetric simple complex lie algebras the given actions are inspired by lusztig s braid group action on quantized enveloping algebras and are defined explicitly on generators braid group and algebra relations are verified with the help of the package quagroup within the computer algebra program gap

asymptotic stability of a composite wave for the one dimensional compressible micropolar fluid model without viscosity abstract we are concerned with the large time behavior of solutions to the cauchy problem of the one dimensional compressible micropolar fluid model without viscosity where the far field states of the initial data are prescribed to be different if the corresponding riemann problem of the compressible euler system admits a contact discontinuity and two rarefaction waves solutions we show that for such a non viscous model the combination of the viscous contact wave with two rarefaction waves is time asymptotically stable provided that the strength of the composite wave and the initial perturbation are sufficiently small the proof is given by an elementary l 2 energy method

fields of fractions of quantum solvable algebras abstract we introduce the notion of pure q solvable algebra the quantum matrices quantum weyl algebra u q n are the examples it is proved that the skew field of fractions of a pure q solvable algebra r is isomorphic to the skew field of twisted rational functions this is a quantum version of the gelfand kirillov conjecture for solvable algebraic lie algebras

relating the approaches to quantised algebras and quantum groups this paper constructs two representations of the quantum groupu q g by exploiting its quotient structure and the quantum double construction here the quantum group is taken as the dual to the quantised algebrau q g a one parameter deformation of the universal enveloping algebra of the lie algebra g as in drinfel d 6 and jimbo 10 from the two representations the hopf structure of the quantised algebrau q g is reexpressed in a matrix format this is the very structure given by faddeev et al 7 in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated lie group to g

dynamics of finger arrays in a diffusion limited growth model with a drift abstract we propose a phenomenological model of fingering dynamics in the presence of an external drift motivated by recent experiments on quasi two dimensional electrodeposition with hydrodynamic convection we study the dynamical transition between a laplacian growth regime and a weak finger competition regime the model proposed defines a wavelength selection problem of the finger array which is coupled in a nontrivial way to the single finger selection problem in a channel the wavelength selection mechanism proposed is associated to the crossover from a diffusive to a ballistic behavior of the aggregating particles as the coarsening spatial scale of the pattern matches the appropriate diffusion length this yields specific predictions for the selected wavelength the dynamics of the model are studied numerically with boundary integral methods in the quasistatic approximation and complementarily using a biased random walk monte carlo scheme on a lattice results are in qualitative agreement with the theoretical predictions

hamiltonian formulation towards minimization of viscous fluid fingering a variational approach has been recently employed to determine the ideal time dependent injection rate q t that minimizes fingering formation when a fluid is injected in a hele shaw cell filled with another fluid of much greater viscosity however such a calculation is approximate in nature since it has been performed by assuming a high capillary number regime in this work we go one step further and utilize a hamiltonian formulation to obtain an analytical exact solution for q t now valid for arbitrary values of the capillary number moreover this hamiltonian scheme is applied to calculate the corresponding injection rate that minimizes fingering formation in a uniform three dimensional porous media an analysis of the improvement offered by these exact injection rate expressions in comparison with previous approximate results is also provided

semigroups of sl_3 mathbb c tensor product invariants abstract we find presentations for a family of semigroup algebras related to the problem of decomposing sl 3 c tensor products along the way we find new toric degenerations of the grassmannian variety gr 3 c n which are t invariant for t gl n c the diagonal torus

quantum graded algebras with a straightening law and the as cohen macaulay property for quantum determinantal rings and quantum grassmannians abstract we study quantum analogues of quotient varieties namely quantum grassmannians and quantum determinantal rings from the point of view of regularity conditions more precisely we show that these rings are as cohen macaulay and determine which of them are as gorenstein our method is inspired by the one developed by de concini eisenbud and procesi in the commutative case thus we introduce and study the notion of a quantum graded algebra with a straightening law on a partially ordered set showing in particular that among such algebras those whose underlying poset is wonderful are as cohen macaulay then we prove that both quantum grassmannians and quantum determinantal rings are quantum graded algebras with a straightening law on a wonderful poset hence showing that they are as cohen macaulay in this last step we are led to introduce and study to some extent natural quantum analogues of schubert varieties

toric degenerations of grassmannians and schubert varieties from matching field tableaux abstract we study grobner degenerations of grassmannians and the schubert varieties inside them we provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in the sense of sturmfels and zelevinsky in 31 we prove that these ideals are all quadratically generated and they yield a sagbi basis of the plucker algebra this leads to a new family of toric degenerations of grassmannians moreover we apply our results to construct a family of grobner degenerations of schubert varieties inside grassmannians we provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields permutations and semi standard tableaux