inclusion of finite temperature effects in a nonlinear description of strongly interacting many body systems abstract we consider a class of quantum many body systems which are coupled by one and two body interactions following our earlier work on deriving nonlinear field equations of motion for such systems we incorporate the effects of finite temperature through a semiclassical free energy functional we first obtain implicit analytical formulas for the entropy density which lend themselves to simple asymptotic expansions we then study the effects of entropy in both homogeneous and inhomogeneous regimes in the former case critical amplitude rescaling transition temperature shifts and crossover effects are found in the latter case saturation effects are shown to promote the emergence of spatial periodicities in 1d

on the fundamental group of hyperelliptic fibrations and some applications we prove that for a hyperelliptic fibration on a surface of general type with irreducible fibers over a possibly non complete curve the image of the fundamental group of a general fiber in the fundamental group of the surface is finite examples show that the result is optimal as a corollary of this result we prove two conjectures the shafarevich conjecture on holomorphic convexity for the universal cover of these surfaces and a conjecture of nori on the finiteness of the fundamental groups of some surfaces we also prove a striking general result about the multiplicities of multiple fibers of a hyperelliptic fibration on a smooth projective surface

minor identities for quasi determinants and quantum determinants we present several identities involving quasi minors of noncommutative generic matrices these identities are specialized to quantum matrices yieldingq analogues of various classical determinantal formulas

strong q convexity in uniform neighborhoods of subvarieties in coverings of complex spaces the main result is that for any projective compact analytic subset y of dimension q 0 in a reduced complex space x there is a neighborhood of y such that for any covering space upsilon colon widehat x to x in which widehat y equiv upsilon 1 y has no noncompact connected analytic subsets of pure dimension q with only compact irreducible components there exists a c exhaustion function varphi on widehat x which is strongly q convex on widehat omega upsilon 1 omega outside a uniform neighborhood of the q dimensional compact irreducible components of widehat y

exact solutions of the generalized lane emden equation a generalized lane emden equation with indices n is discussed which reduces to the lane emden equation proper for 2 1 1 general properties of the set of solutions of this equation are derived and exact solutions are given these include a singular solution without free integration constant for arbitrary n and for particular relations between n and among the two parameter solutions nonequivalent families of solutions of the same equation are obtained

symmetry reduction for nonlinear relativistically invariant equations symmetry reduction is studied for the relativistically invariant scalar partial differential equation h u u 2 u 0 in n 1 dimensional minkowski space m n 1 the introduction of k symmetry variables 1 k as invariants of a subgroup g of the poincare group p n 1 having generic orbits of codimension k n in m n 1 reduces the equation to a pde in k variables all codimension 1 symmetry variables in m n 1 n arbitrary reducing the equation studied to an ode are found as well as all codimension 2 and 3 variables for the low dimensional cases n 2 3 the type of equation studied includes many cases of physical interest in particular nonlinear klein gordon equations such as the sine gordon equation and hamilton jacobi equations

sur la convexite holomorphe des revetements lineaires reductifs d une variete projective algebrique complexe let x be a smooth connected compact projective variety over we study the shafarevich conjecture concerning holomorphic convexity along the lines of kollar s approach when the fundamental group of x admits large finite dimensionnal representations we prove that given n the topological covering space widetilde x n of a projective algebraic compact complex manifold x corresponding to the intersection of the kernels of all linear reductive representations of 1 x to gln is holomorphically convex in the surface case this is a corollary of a theorem due to katzarkov and ramachandran

discrete groups and holomorphic functions we determine conditions under which the quotient of a kahler hadamard manifold by a torsion free discrete group of isometries admits a big set of nontrivial holomorphic functions we also generalize a theorem of g a margulis on centerless of cocompact discrete groups of automorphisms of a caratheodory hyperbolic manifold by a different approach based on the caratheodory pseudodistance

modeling polymerization of microtubules a semi classical nonlinear field theory approach abstract in this paper for the first time a three dimensional treatment of microtubules polymerization is presented starting from fundamental biochemical reactions during microtubule s assembly and disassembly processes we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in three dimensions we found that the dynamics of a microtubule is mathematically expressed via a cubic quintic nonlinear schrodinger nls equation we show that in 3d a vortex filament a generic solution of the nls equation exhibits linear growth shrinkage in time as well as temporal fluctuations about some mean value which is qualitatively similar to the dynamic instability of microtubules by solving equations numerically we have found spatio temporal patterns consistent with experimental observations

exact solutions of the multidimensional classical φ6 field equations obtained by symmetry reduction the often used 6 model of classical critical phenomena is studied in 3 1 dimensional minkowski and euclidean spaces the euler lagrange equations describing the kinetics of the scalar order parameter are in this case nonlinear klein gordon equations the method of symmetry reduction is systematically applied to derive all the solutions invariant under subgroups with generic orbits of codimension 1 whenever the obtained ordinary differential equations have the painleve property they can be transformed to one of two standard forms these are then solved in terms of elliptic functions or elementary ones this results in a large number of new exact solutions particularly interesting solutions are found in the immediate vicinity of the tricritical point our treatment of the 6 theory is complete only for the four dimensional spaces m 3 1 and e 4 but many of the results are given for the more general cases of m n 1 and e n 1

on the disk theorem we construct an example of a 2 dimensional stein normal space x with one singular point x0 such that x x0 is simply connected and it satisfies the disk condition this answers a question raised by fornaess and narasimhan we also prove that any increasing union of stein open sets contained in a stein space of dimension 2 satisfies the disk condition starting from the above example we exhibit without using deformation theory a new type of 2 dimensional holes which cannot be filled

coherent structures in strongly interacting many body systems ii classical solutions and quantum fluctuations for pt i see ibid vol 22 p 4877 1989 previously i the author considered a second quantised hamiltonian which can be used to model a large number of important strongly interacting many body systems he has shown that the dynamics of such systems can be described exactly using quantum fields the equations of motion derived in this way are highly nonlinear partial differential equations pde he now uses a standard field theoretical approach where initially the fields are treated as classical functions he studies the solutions of these equations which remarkably may be obtained making full use of very recent mathematical discoveries the symmetry reduction method he finds exact solutions of the equations of motion for the classical field for each of the four main cases separately found in the preceding paper these cases may correspond to different physical situations and the last the second order case corresponds to the most general physical situation subsequently he outlines possible quantisation procedures for these fields he then briefly discusses the type of boundary conditions which may be applied to both the classical field and the quantum fluctuations which arise

some q convexity properties of coverings of complex manifolds let c be a nowhere dense compact analytic subset of a 2 dimensional complex manifold x a result of napier is the construction of a neighborhood v of c such that for any covering space open image in new window in which open image in new window is holomorphically convex there is a c exhaustion function on open image in new window which is strictly plurisubharmonic on 1 v away from the compact irreducible components of open image in new window col oiu and v j itu obtained a q convex version of this result for c a q dimensional fiber of a proper surjective submersion x y the goal of this paper is a similar version for cprojective

linear shafarevich conjecture in this paper we settle armatively shafarevich s uniformization conjecture for varieties with linear fundamental groups we prove the strongest to date uniformization result the universal covering space of a complex projective manifold with a linear fundamental group is holomorphically convex the proof is based on both known and newly developed techniques in non abelian hodge theory

algebraic deformations of toric varieties i general constructions abstract we construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry isospectral deformations and noncommutative geometry in braided monoidal categories our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus we develop a sheaf theory using techniques from noncommutative algebraic geometry the cases of projective varieties are studied in detail and several explicit examples are worked out including new noncommutative deformations of grassmann and flag varieties our constructions set up the basic ingredients for a thorough study of instantons on noncommutative toric varieties which will be the subject of the sequel to this paper

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

two remarks on the topology of projective surfaces we will prove two results about the topology of complex projective surfaces the first result says that if the shafarevich conjecture has an affirmative answer in dimension two then the second homotopy group of a smooth projective surface is a torsion free abelian group the second result is that for any 2 dimensional function field k c there is a normal projective simply connected surface with function field k

exact solutions of the multidimensional derivative nonlinear schrodinger equation for many body systems of criticality the authors investigate the problem of strongly interacting many body systems near criticality as recently described in terms of nonlinear dynamics by dixon and tuszynski 1989 they show that in the first order of approximation the equation of motion for the order parameter can be mapped on the derivative nonlinear schrodinger equation and thus can be solved exactly they solve the equations both in the absence and presence of current densities

inversion des matrices de hankel abstract hankel determinants can be viewed as special schur symmetric functions this provides without computations of determinants or matrices most of the algebra of hankel determinants and explains their links with other classical fields

malcev neumann series on the free group and questions of rationality let g be an ordered free group b a basis contained in g the sub monoid spanned by b and k a field we construct a family q n g n n of sub division rings in k m g the division ring consisting of the so called malcev neumann series describe their intersections with k and give some applications concerning rational series in k

convexity properties of coverings of 1 convex surfaces we prove that there exists a 1 convex surface whose universal covering does not satisfy the discrete disk property

the hartogs type extension theorem for meromorphic maps into compact kahler manifolds extension theorems of this type are often called hartogs type extension theorems the above result was conjectured by griffiths see problem 1 in gr who proved it in the special case when d p o i n t and f d x is holomorphic important contributions towards the proof of this conjecture were made by b shiffman y t siu n sibony see sh sil sb3 in particular y t siu proved a thullen type extension theorem for meromorphic maps into compact kfihler manifolds see w 1 the above theorem can be applied to the study of covering maps of zariski open subsets in compact kfihler manifolds recall that a subdomain d of a complex manifold m is called levi pseudoconvex if for any peod there exists a neighbourhood u of p in m such that u c d is stein we say that d covers a complex manifold x if there is a subgroup g c aut d acting properly discontinuously and freely on d with d g x

lie symmetries of a generalised nonlinear schrodinger equation i the symmetry group and its subgroups the symmetry group of the generalised non linear schrodinger equation i psi t delta psi a0 psi a1 mod psi mod 2 psi a2 mod psi mod 4 psi in three space dimensions is shown to be the extended galilei group g 3 for a1a2 not 0 and the galilei similitude group gd 3 including a dilation for a1 oor a2 0 all lie subgroups of g 3 and gd 3 are found they will be used in a subsequent paper to obtain group invariant solutions of the equation

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

classical flag varieties for quantum groups the standard quantum sl n c abstract we suggest a possible programme to associate geometric flag like data to an arbitrary simple quantum group in the spirit of the noncommutative al gebraic geometry developed by artin tate and van den bergh we then carry out this programme for the standard quantum sl n of drinfel d and jimbo where the varieties involved are certain t stable subvarieties of the ordinary flag variety

classification of quadruple galois canonical covers i in this article we classify quadruple galois canonical covers of smooth surfaces of minimal degree the classification shows that they are either non simple cyclic covers or bi double covers if they are bi double then they are all fiber products of double covers we construct examples to show that all the possibilities in the classification do exist there are implications of this classification that include the existence of families with unbounded geometric genus in sharp contrast with triple canonical covers and families with unbounded irregularity in sharp contrast with canonical covers of all other degrees together with the earlier known results on double and triple covers a pattern emerges that motivates some general questions on the existence of higher degree canonical covers some of which are answered in this article

on the fundamental groups of some open rational surfaces if d is an irreducible curve with ordinary nodes in a relatively minimal smooth projective rational surface m then from the solution of zanski s problem in 12 proposition 3 27 it follows that n m d is abelian on the other hand in 15 o zariski has given an example of an irreducible sextic d in ip 2 with six ordinary cusps lying on a conic such that q f 2 d is an infinite group in view of this the following conjecture seems significant

globally well and ill posedness for non elliptic derivative schrodinger equations with small rough data abstract we show that there exists s c 0 such that the cubic quartic non elliptic derivative schrodinger equations with small data in modulation spaces m 2 1 s r n for n 3 n 2 are globally well posed if s s c and ill posed if s s c in 2d cubic case using the gabor frame we get some time global dispersive estimates for the schrodinger semi group in anisotropic lebesgue spaces which include a time global maximal function estimate in the space l x 1 2 l x 2 t by resorting to the smooth effect estimate together with the dispersive estimates in anisotropic lebesgue spaces we show that the cubic hyperbolic derivative nls in 2d has a unique global solution if the initial data in feichtinger segal algebra or in weighted sobolev spaces are sufficiently small

coherent structures in strongly interacting many body systems i derivation of dynamics numerous examples in condensed matter physics possess an effective hamiltonian structure involving two body anharmonic interactions between elementary excitations the effective second quantised hamiltonian used to model these systems is essentially of the same form except for the particulars of the interaction parameters and the statistics of the particles which are interacting the author investigates such systems in the limit of strongly anharmonic interactions which may possibly lead to symmetry breaking he introduces quantum field operators and calculates their equations of motion the interaction coefficients are expanded in a taylor series about the critical mode to second order to illustrate the method but he gives the expansion to infinite order for completeness it is demonstrated that both in first and second order the equations of motion are reduced to a system of highly nonlinear coupled partial differential equations pde for the field operator he shows that the form of the second order equation is exact due to renormalisation theory in the case where the hamiltonian coefficients represent operators in real space which are isotropic and incorporates symmetries and topologies which would appear in an expansion to infinite order this special situation arises in a very wide range of physical examples these equations can be solved exactly using a particular ansatz in conjunction with recent mathematical developments in the field of non linear analysis in particular the method of symmetry reduction for pde provides a complete set of symmetry variables for the type of equations used in the study

a decomposition of solomon s descent algebra abstract a descent class in the symmetric group sn is the collection of permutations with a given descent set it was shown by l solomon j algebra41 1976 255 264 that the product in the group algebra q sn of two descent classes is a linear combination of descent classes thus descent classes generate a subalgebra of q sn we refer to it here as solomon s descent algebra and denote it by n this algebra is not semisimple but it has a faithul representation in terms of upper triangular matrices the main goal of this paper is a decomposition of its multiplicative structure it develops that n acts in a natural way on the so called lie monomials this action has a purely combinatorial description and is a crucial tool in the construction of a complete set of indecomposable representations of n in particular we obtain a natural basis of irreducible orthogonal idempotents indexed by partitions of n for the quotient n n natural bases of nilpotents and idempotents for the subspaces e ne for two arbitrary partitions and are also constructed and the dimensions of these spaces are given a combinatorial interpretation in terms of the so called decreasing factorization of an arbitary word into a product of lyndon words

a hopf algebra approach to inner plethysm abstract we use the hopf algebra structure of the algebra of symmetric functions to study the adams operators of the complex representation rings of symmetric groups and we give new proofs of all of littlewood s formulas for inner plethysm we also study the adams operations for orthogonal and symplectic group characters

a certain class of solutions of the nonlinear wave equation in this paper are investigated some differential geometry methods in the theory of the nonlinear wave equation 2u u u u a special class of solutions is discussed for which u u is constant on each level of the function u it is proved that levels of such solutions form in the space of independent variable s hypersurfaces with all principal curvatures constant the general form of such hypersurfaces is given then it is proved that via the method of characteristics it is possible to construct in principle all the solutions of the discussed class they may be obtained by integration of an ode of second order using a special class of the polynomial functions some new solutions are given for equations v 4av3 3bv2 2cv d v exp v v sin v v cosh v and v sinh v

quantum and quasi plucker coordinates abstract we demonstrate a passage from the quasi plucker coordinates of gelfand and retakh to the quantum plucker coordinates built from q generic matrices in the process we rediscover the defining relations of the quantum grassmannian of taft and towber and provide that algebra with more concrete geometric origins

exponential estimate for the asymptotics of bergman kernels we prove an exponential estimate for the asymptotics of bergman kernels of a positive line bundle under hypotheses of bounded geometry further we give bergman kernel proofs of complex geometry results such as separation of points existence of local coordinates and holomorphic convexity by sections of positive line bundles

on the shafarevich conjecture for genus 2 fibrations we prove a result about fundamental group of a smooth projective surface with ample canonical divisor which admits a genus two fibration as a corollary we prove that the universal cover of such a surface is holomorphically convex this proves the conjecture of shafarevich for such surfaces

convexity properties of coverings of 1 convex manifolds we show the following result let x be a 1 convex manifold a its exceptional set k dima and p y x any covering then y can be exhausted by an increasing sequence y nu nu in mathbb n of smoothly bounded k convex domains

α modulation spaces ii derivative nls abstract the current paper is a continuation of our early work 11 we use the techniques and properties of modulation spaces developed in 11 to study the cauchy problem for the derivative nonlinear schrodinger equations by applying some new type of inhomogeneous linear estimate to control the high low frequency interaction appearing from the nonlinearity we obtain the critical regularity index for the global well posedness with small initial data in modulation spaces for one type of derivative nonlinear schrodinger equations we also study the generalized derivative schrodinger equations

vandermonde and wronskian matrices over division rings in l the first author initiated the study of vandermonde matrices with entries over a division ring k even in the case when k is a field the definition of a vandermonde matrix in l is already more general than that in the classical sense as it takes into account a given endomorphism s of k under the epilogue of l it was further pointed out that the results of that paper remain valid if in the definition of a vandermonde matrix one allows for a given s derivation d and uses the appropriate definition of the power functions in the s d setting such a generalization is worthwhile because in this setting the study of vandermonde matrices over k also encompasses the study of wronskian matrices with respect to d so results on vandermonde matrices may be used to study linear differential equations arising from the operator d more generally vandermonde matrices seem to be an important tool in studying the skew polynomial ring k t s d so a deeper understanding of vandermonde matrices should be important to the study of the arithmetic of a division ring the purpose of this paper is threefold first we shall develop under sec tion 2 the basic facts on skew polynomials in the s d setting define the evaluation of such polynomials and prove the all important product theorem 2 7 second we shall give a general computation of the rank of a vandermonde matrix with respect to s d as in l the computation is first reduced to the case when the elements a a used to build the vandermonde matrix are pairwise s d conjugate in this case the rank of the vandermonde matrix is computed by the dimension of a vector space over a certain division subring of k cf theorem 4 5 last under

quantum deformation of the grassmannian manifold abstract in this paper we work out a deformation of g r n the grassmannian of r subspaces in a vector space of dimension n over a field k of characteristic 0 g r n is deformed as an homogeneous space for sl n k the special linear group of k n this means that k g r n the coordinate ring of g r n gets deformed together with with the coaction of k sl n the coordinate ring of sl n k on it our deformation comes together with a coaction of the corresponding deformation of sl n k at the end we give an explicit presentation of the deformed grassmannian in terms of generators and relations

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

a new class of solutions to a generalized nonlinear schrodinger equation in this paper we compute new classes of symmetry reduction and associated exact solutions of a generalized nonlinear schrodinger equation gnls the generalized terms modelling dispersion and scattering several authors have obtained symmetry reductions of one two and three dimensional nonlinear schrodinger equations in all cases to date reductions have been based on a real new independent variable in this paper we compute reductions in which the new independent variable is complex we seek new reductions from a two dimensional gnls to a pde in two independent variables and also reductions to odes five new classes of reduction are found