from real affine geometry to complex geometry we construct from a real ane manifold with singularities a tropical manifold a degeneration of calabi yau manifolds this solves a fundamental problem in mirror symmetry furthermore a striking feature of our approach is that it yields an explicit and canonical order by order description of the degeneration via families of tropical trees this gives complete control of the b model side of mirror symmetry in terms of tropical geometry for example we expect that our deformation parameter is a canonical coordinate and expect period calculations to be expressible in terms of tropical curves we anticipate this will lead to a proof of mirror symmetry via tropical methods

on the lie symmetry algebra of a general ordinary differential equation we give a description of the lie symmetry algebra of a general ordinary differential equation which clarifies certain issues in its practical use in particular we present examples of symmetry algebras for well known equations which illustrate features of this algebra

boundary conditions for integrable equations the problem of constructing boundary conditions for nonlinear equations compatible with higher symmetries is considered in particular this problem is discussed for the sine gordon jiber shabat liouville and kdv equations new results are obtained for the last two ones the boundary condition for the kdv contains two arbitrary constants the substitution maps it onto the boundary condition with linear dependence on t for the potentiated kdv

non classical symmetry reduction example of the boussinesq equation a symmetry of an equation will leave the set of all solutions invariant a conditional symmetry will leave only a subset of solutions defined by some differential condition invariant the authors show how a specific class of conditional symmetries can be used to reduce a partial differential equation to an ordinary one in particular for the boussinesq equation these conditional symmetries together with the ordinary ones provide all possible reductions to ordinary differential equations a group theoretical explanation of the recently obtained new reductions is provided

conditional symmetry of a porous medium equation abstract we obtain an exhaustive description of conditional symmetries admitted by a nonlinear porous medium equation in one spatial dimension it is proved that solving the determining equations for the conditional symmetries either yields classical lie symmetries or is equivalent to solving the porous medium equation itself the second possibility is shown to be typical for an arbitrary evolution type partial differential equation in one spatial variable we show that this very property is responsible for the well known phenomenon when using non classical symmetries yield exact solutions which prove to be invariant ones

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

semitoric integrable systems on symplectic 4 manifolds let m be a symplectic 4 manifold a semitoric integrable system on m is a pair of smooth functions j h c m r for which j generates a hamiltonian s 1 action and the poisson brackets j h vanish we shall introduce new global symplec tic invariants for these systems some of these invariants encode topological or geometric aspects while others encode analytical information about the singularities and how they stand with respect to the system our goal is to prove that a semitoric system is completely determined by the invariants we introduce

semiclassical inverse spectral theory for singularities of focus focus type we prove assuming that the bohr sommerfeld rules hold that the joint spectrum near a focus focus singular value of a quantum integrable system determines the classical lagrangian foliation around the full focus focus leaf the result applies for instance to pseudodifferential operators on cotangent bundles and berezin toeplitz operators on prequantizable compact symplectic manifolds

improved kinetic renormalisation group approach to diffusion limited aggregation an improved kinetic renormalisation group approach to diffusion limited aggregation is presented this approach is based on the growth process itself and accounts for the dispersity in the growth probabilities it yields the multifractal spectrum d q with better values at smaller q on the 2d square lattice the authors find df 1 694 for the fractal dimensions of the cluster and that of its interface and d 1 1 01 for the information dimension the former agrees with the simulation results df approximately 1 70 and the latter compares very well with the exact value d 1 1

constructing integrable systems of semitoric type let m be a connected symplectic 4 manifold a semitoric integrable system on m essentially consists of a pair of independent real valued smooth functions j and h on m for which j generates a hamiltonian circle action under which h is invariant in this paper we give a general method to construct starting from a collection of five ingredients a symplectic 4 manifold equipped a semitoric integrable system then we show that every semitoric integrable system on a symplectic 4 manifold is obtained in this fashion in conjunction with the uniqueness theorem proved recently by the authors this gives a classification of semitoric integrable systems on 4 manifolds in terms of five invariants

initial value problems for evolutionary partial differential equations and higher order conditional symmetries we suggest a new approach to the problem of dimensional reduction of initial boundary value problems for evolution equations in one spatial variable the approach is based on higher order generalized conditional symmetries of the equations involved it is shown that reducibility of an initial value problem for an evolution equation to a cauchy problem for a system of ordinary differential equations can be fully characterized in terms of conditional symmetries which leave invariant the equation in question we also give some examples of the solution of initial value problems for second and third order nonlinear differential equations by reduction by their conditional symmetries we give a systematic classification of general second order partial differential equations admitting second order conditional symmetries based on lie s classification of invariant second order ordinary differential equations this yields five classes of principally new initial value problems for nonlinear evolution equations which admit no lie symmetries and are reducible via second order conditional symmetries

a precise definition of reduction of partial differential equations we give a comprehensive analysis of interrelations between the basic concepts of the modern theory of symmetry classical and non classical reductions of partial differential equations using the introduced definition of reduction of differential equations we establish equivalence of the non classical conditional symmetry and direct ansatz approaches to reduction of partial differential equations as an illustration we give an example of non classical reduction of the nonlinear wave equation in 1 3 dimensions the conditional symmetry approach when applied to the equation in question yields a number of non lie reductions which are far reaching generalizations of the well known symmetry reductions of the nonlinear wave equations

semiclassical quantization and spectral limits of ħ pseudodifferential and berezin toeplitz operators we introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system this gives a quick alternative solution to the isospectrality problem for quantum toric systems if the operators are uniformly bounded the convergence is uniform analogous results hold for non commuting operators

mirror symmetry via logarithmic degeneration data i this paper lays the foundations of a program to study mirror symmetry by studying the log structures of illusie fontaine and kato on degenerations of calabi yau manifolds the basic idea is that one can associate to certain sorts of degenerations of calabi yau manifolds a log calabi yau space which is a log structure on the degenerate fibre the log cy space captures essentially all the information of the degeneration and hence all mirror statements for the large complex structure limit given by the degeneration can already be derived from the log cy space in this paper we begin by discussing affine manifolds with singularities given such an affine manifold along with a polyhedral decomposition we show how to construct a scheme consisting of a union of toric varieties in certain non degenerate cases we can also construct log structures on these schemes conversely given certain sorts of degenerations one can build an affine manifold with singularities structure on the dual intersection complex of the degeneration mirror symmetry is then obtained as a discrete legendre transform on these affine manifolds thus providing an algebro geometrization of the strominger yau zaslow conjecture the deepest result of this paper shows an isomorphism between log complex moduli of a log cy space and log kahler moduli of its mirror

inverse spectral theory for semiclassical jaynes cummings systems quantum semitoric systems form a large class of quantum hamiltonian integrable systems with circular symmetry which has received great attention in the past decade they include systems of high interest to physicists and mathematicians such as the jaynes cummings model 1963 which describes a two level atom interacting with a quantized mode of an optical cavity and more generally the so called systems of jaynes cummings type in this paper we consider the joint spectrum of a pair of commuting semiclassical operators forming a quantum integrable system of jaynes cummings type we prove assuming the bohr sommerfeld rules hold that if the joint spectrum of two of these systems coincide up to mathcal o hbar 2 then the systems are isomorphic

symmetries of linear ordinary differential equations we discuss the lie symmetry approach to homogeneous linear ordinary differential equations in an attempt to connect it with the algebraic theory of such equations in particular we pay attention to the fields of functions over which the symmetry vector fields are defined and by defining a noncharacteristic lie subalgebra of the symmetry algebra are able to establish a general description of all continuous symmetries we use this description to rederive a classical result on differential extensions for second order equations

convexity properties of the moment mapping ii be its associated momen t mapping see w for definitions the set x is the union of co adjoint orbits the main result of this paper is a description of the orbit structure of this set to describe this result we first note that the coadjoint orbits in q can be parametr ized as follows let t be a caf tan subalgebra of g and let b be a positive definite g invariant bilinear form on by means of b we get a g equivariant identification q let t be the subspace of q corresponding to t then every co adjoint orbit intersects i in a weyl group orbit so there is a one one correspondence between co adjoint orbits in q and weyl group orbits in t to get an even sharper parametrization let t be a weyl chamber in then every weyl group orbit intersects t in a single point so t is a modul i space for the co adjoint orbits two orbits in x are said to be of the same type if they are isomorphic as homogeneous g spaces since x is compact there are at most a finite number of different types of orbits see 1113 let h be a closed subgroup of t of codimension r and let cit be the set of all orbits in x of type g h

fast community detection based on sector edge aggregation metric model in hyperbolic space by studying the edge aggregation characteristic of nodes in hyperbolic space sector edge aggregation metric seam model is proposed and theoretically proved in this paper in hyperbolic disk seam model determines the minimum angular range of a sector which possesses the maximal edge aggregation of nodes the set of nodes within such sector has dense internal links which corresponds with the characteristic of community structure based on seam model we propose a fast community detection algorithm called greedy optimization modularity algorithm goma which employs greedy optimization strategy and hyperbolic coordinates goma firstly divides initial communities according to the quantitative results of sector edge aggregation given by seam and the nodes hyperbolic coordinates then based on greedy optimization strategy only merges the two angular neighboring communities in hyperbolic disk to optimize the network modularity function and consequently obtains high quality community detection the strategies of initial community partition and merger in hyperbolic space greatly improve the speed of searching the most optimal modularity experimental results indicate that goma is able to detect out high quality community structure in synthetic and real networks and performs better when applied to the large scale and dense networks with strong clustering

constructing conditionally integrable evolution systems in 1 1 dimensions a generalization of invariant modules approach given a generalized lie backlund vector field satisfying certain nondegeneracy assumptions we explicitly describe all 1 1 dimensional evolution systems that admit this vector field as a generalized conditional symmetry the connection with the theory of symmetries of systems of odes and with the theory of invariant modules is discussed

moment polytopes for symplectic manifolds with monodromy abstract a natural way of generalising hamiltonian toric manifolds is to permit the presence of generic isolated singularities for the moment map for a class of such almost toric 4 manifolds which admits a hamiltonian s 1 action we show that one can associate a group of convex polygons that generalise the celebrated moment polytopes of atiyah guillemin sternberg as an application we derive a duistermaat heckman formula demonstrating a strong effect of the possible monodromy of the underlying integrable system

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant