lifespan theorem for constrained surface diffusion flows we consider closed immersed hypersurfaces in mathbb r 3 and mathbb r 4 evolving by a class of constrained surface diffusion flows our result similar to earlier results for the willmore flow gives both a positive lower bound on the time for which a smooth solution exists and a small upper bound on a power of the total curvature during this time by phrasing the theorem in terms of the concentration of curvature in the initial surface our result holds for very general initial data and has applications to further development in asymptotic analysis for these flows

noncommutative ricci flow in a matrix geometry we study noncommutative ricci flow in a finite dimensional representation of a noncommutative torus it is shown that the flow exists and converges to the flat metric we also consider the evolution of entropy and a definition of scalar curvature in terms of the ricci flow

sextic anharmonic oscillators and orthogonal polynomials under certain constraints on the parameters a b and c it is known that schrodinger s equation d2 dx2 ax6 bx4 cx2 e a 0 with the sextic anharmonic oscillator potential is exactly solvable in this paper we show that the exact wavefunction is the generating function for a set of orthogonal polynomials p t n x in the energy variable e some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi exact solvability we also prove that this set of orthogonal polynomials can be reduced by means of a simple scaling transformation to a remarkable class of orthogonal polynomials pn e p 0 n e recently discovered by bender and dunne

quantal problems with partial algebraization of the spectrum we discuss a new class of spectral problems discovered recently which occupies an intermediate position between the exactly solvable problems e g harmonic oscillator and all others the problems belonging to this class are distinguished by the fact that a part of the eigenvalues and eigenfunctions can be found algebraically but not the whole spectrum the reason explaining the existence of the quasi exactly solvable problems is a hidden dynamical symmetry present in the hamiltonian for one dimensional motion this hidden symmetry issl 2 r it is shown that other groups lead to a partial algebraization in multidimensional quantal problems in particular sl 2 r sl 2 r so 3 andsl 3 r are relevant to two dimensional motion inducing a class of quasi exactly solvable two dimensional hamiltonians typically they correspond to systems in a curved space but sometimes the curvature turns out to be zero graded algebras open the possibility of constructing quasi exactlysolvable hamiltonians acting on multicomponent wave functions for example with a non minimal superextension ofsl 2 r we get a hamiltonian describing the motion of a spinor particle

parabolic monge ampere equations on riemannian manifolds on a compact riemannian manifold m g we consider the parabolic monge ampere equation t x t logdet g x hess x t detg x x t f x x 0 phiv 0 x here is a real parameter andf 0 m r are smooth functions we show existence of for all timestindependent of if 0 then t t converges exponentially towards a solution of the stationary problem ast in the special case 0 f 0 one has 0 and we determine the convergence rate t in thel2 norm more precisely

deforming three manifolds with positive scalar curvature in this paper we prove that the moduli space of metrics with positive scalar curvature of an orientable compact three manifold is path connected the proof uses the ricci flow with surgery the conformal method and the connected sum construction of gromov and lawson the work of perelman on hamilton s ricci flow is fundamental as one of the applications we prove the path connectedness of the space of trace free asymptotically flat solutions to the vacuum einstein constraint equations on r 3

automatic histogram threshold using fuzzy measures in this paper an automatic histogram threshold approach based on a fuzziness measure is presented this work is an improvement of an existing method using fuzzy logic concepts the problems involved in finding the minimum of a criterion function are avoided similarity between gray levels is the key to find an optimal threshold two initial regions of gray levels located at the boundaries of the histogram are defined then using an index of fuzziness a similarity process is started to find the threshold point a significant contrast between objects and background is assumed previous histogram equalization is used in small contrast images no prior knowledge of the image is required

perelman s entropy functional at type i singularities of the ricci flow we study blow ups around fixed points at type i singularities of the ricci flow on closed manifolds using perelman s w functional first we give an alternative proof of the result obtained by naber 18 and enders muller topping 8 that blow up limits are non flat gradient shrinking ricci solitons our second and main result relates a limit w density at a type i singular point to the entropy of the limit gradient shrinking soliton obtained by blowing up at this point in particular we show that no entropy is lost at infinity during the blow up process

collapsing irreducible 3 manifolds with nontrivial fundamental group let m be a closed orientable irreducible non simply connected 3 manifold we prove that if m admits a sequence of riemannian metrics which volume collapses and whose sectional curvature is locally controlled then m is a graph manifold this is the last step in perelman s proof of thurston s geometrisation conjecture

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

a modified kahler ricci flow in this note we study a kahler ricci flow modified from the classic version in the non degenerate case strong convergence at infinite time is achieved the main focus should be on degenerate case where some partial results are presented

two loop renormalization group flow for lorentzian manifolds we study two loop renormalization group flow rg 2 on lorentzian three manifolds the fixed points of flow are analysed on a three dimensional lorentzian manifold and we present a classification of the solitons that evolves only by homotheties furthermore we prove the existence of a lorentzian rg 2 cigar soliton of constant positive curvature

measuring fuzzy uncertainty first this paper reviews several well known measures of fuzziness for discrete fuzzy sets then new multiplicative and additive classes are defined we show that each class satisfies five well known axioms for fuzziness measures and demonstrate that several existing measures are relatives of these classes the multiplicative class is based on nonnegative monotone increasing concave functions the additive class requires only nonnegative concave functions some relationships between several existing and the new measures are established and some new properties are derived the relative merits and drawbacks of different measures for applications are discussed a weighted fuzzy entropy which is flexible enough to incorporate subjectiveness in the measure of fuzziness is also introduced finally we comment on the construction of measures that may assess all of the uncertainties associated with a physical system

image segmentation by histogram thresholding using fuzzy sets methods for histogram thresholding based on the minimization of a threshold dependent criterion function might not work well for images having multimodal histograms we propose an approach to threshold the histogram according to the similarity between gray levels such a similarity is assessed through a fuzzy measure in this way we overcome the local minima that affect most of the conventional methods the experimental results demonstrate the effectiveness of the proposed approach for both bimodal and multimodal histograms

coupling of brownian motions and perelman s l functional abstract we show that on a manifold whose riemannian metric evolves under backwards ricci flow two brownian motions can be coupled in such a way that their normalized l distance is a supermartingale as a corollary we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non decreasing function and the normalized l distance in particular it provides a new proof of a recent result of topping p topping l optimal transportation for ricci flow j reine angew math 636 2009 93 122

on some polynomial potentials in d dimensions the d dimensional schrodinger s equation is analyzed with regard to the existence of exact solutions for polynomial potentials under certain conditions on the interaction parameters we show that the polynomial potentials v8 r k 18 krk 8 0 and v10 r k 110 krk 10 0 are exactly solvable by examining the polynomial solutions of certain linear differential equations with polynomial coefficients the necessary and sufficient conditions for the existence of these exact solutions are discussed finding accurate solutions for arbitrary values of the potential parameters using the asymptotic iteration method is also presented

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

curvature contraction of convex hypersurfaces by nonsmooth speeds the research of the second author was supported by an australian postgraduate award the research of the first third fourth and fifth authors was supported by discovery project grant dp120100097 of the australian research council

complete gradient shrinking ricci solitons with pinched curvature we prove that any n dimensional complete gradient shrinking ricci soliton with pinched weyl curvature is a finite quotient of mathbb r n mathbb r times mathbb s n 1 or mathbb s n in particular we do not need to assume the metric to be locally conformally flat

pt symmetric quantum mechanics this paper proposes to broaden the canonical formulation of quantum mechanics ordinarily one imposes the condition h h on the hamiltonian where represents the mathematical operation of complex conjugation and matrix transposition this conventional hermiticity condition is sufficient to ensure that the hamiltonian h has a real spectrum however replacing this mathematical condition by the weaker and more physical requirement h h where represents combined parity reflection and time reversal pt one obtains new classes of complex hamiltonians whose spectra are still real and positive this generalization of hermiticity is investigated using a complex deformation h p2 x2 ix e of the harmonic oscillator hamiltonian where e is a real parameter the system exhibits two phases when e 0 the energy spectrum of h is real and positive as a consequence of pt symmetry however when 1 e 0 the spectrum contains an infinite number of complex eigenvalues and a finite number of real positive eigenvalues b

the generalized ricci flow for three dimensional manifolds with one killing vector we consider three dimensional 3d flow equations inspired by the renormalization group rg equations of string theory with a three dimensional target space by modifying the flow equations to include a u 1 gauge field and adding carefully chosen de turck terms we are able to extend recent two dimensional results of bakas to the case of a 3d riemannian metric with one killing vector in particular we show that the rg flow with de turck terms can be reduced to two equations the continual toda flow solved by bakas plus its linearizaton we find exact solutions which flow to homogeneous but not always isotropic geometries

ricci soliton homogeneous nilmanifolds we study a notion weakening the einstein condition on a left invariant riemannian metric g on a nilpotent lie groupn we consider those metrics satisfying ric g ci d for some c in mathbb r and some derivationd of the lie algebra mathfrak n ofn where ric g denotes the ricci operator of n g this condition is equivalent to the metric g to be a ricci soliton we prove that a ricci soliton left invariant metric on n is unique up to isometry and scaling the following characterization is also given n g is a ricci soliton if and only if n g admits a metric standard solvable extension whose corresponding standard solvmanifold s tilde g is einstein this gives several families of new examples of ricci solitons by a variational approach we furthermore show that the ricci soliton homogeneous nilmanifolds n g are precisely the critical points of a natural functional defined on a vector space which contains all the homogeneous nilmanifolds of a given dimension as a real algebraic set

the second twisted betti number and the convergence of collapsing riemannian manifolds let mi overset d gh longrightarrow x denote a sequence of n manifolds converging to a compact metric space x in the gromov hausdorff topology such that the sectional curvature is bounded in absolute value and dim x n we prove the following stability result if the fundamental groups of mi are torsion groups of uniformly bounded exponents and the second twisted betti numbers of mi vanish then there is a manifold m and a sequence of diffeomorphisms from m to a subsequence of mi such that the distance functions of the pullback metrics converge to a pseudo metric in c0 norm furthermore m admits a foliation with leaves diffeomorphic to flat manifolds not necessarily compact such that a vector is tangent to a leaf if and only if its norm converges to zero with respect to the pullback metrics these results lead to a few interesting applications

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

convergence of kahler ricci flow in this paper we prove a theorem on convergence of kahler ricci flow on a compact kahler manifold which admits a kahler ricci soliton

wave character of metrics and hyperbolic geometric flow in this paper we illustrate the wave character of the metrics and curvatures of manifolds and introduce a new understanding tool the hyperbolic geometric flow this kind of flow is new and very natural to understand certain wave phenomena in the nature as well as the geometry of manifolds it possesses many interesting properties from both mathematics and physics several applications of this method have been found

investigations on fuzzy thresholding based on fuzzy clustering thresholding the problem of pixel classification is attempted here using fuzzy clustering algorithms the segmented regions are fuzzy subsets with soft partitions characterizing the region boundaries the validity of the assumptions and thresholding schemes are investigated in the presence of distinct region proportions the hard k means and fuzzy c means algorithms have been found useful when object and background regions are well balanced fuzzy thresholding is also formulated as extraction of normal densities to provide optimal partitions regional imbalances in gray distributions are taken care of in region normalized histograms

conformal deformations of integral pinched 3 manifolds abstract in this paper we prove that under an explicit integral pinching assumption between the l 2 norm of the ricci curvature and the l 2 norm of the scalar curvature a closed 3 manifold with positive scalar curvature admits a conformal metric of positive ricci curvature in particular using a result of hamilton this implies that the manifold is diffeomorphic to a quotient of s 3 the proof of the main result of the paper is based on ideas developed in an article by m gursky and j viaclovsky

flow of pinched convex hypersurfaces by powers of curvature functions in hyperbolic space abstract this paper concerns closed h convex hypersurfaces of dimension n 2 in the hyperbolic space h n 1 of constant sectional curvature evolving in direction of its normal vector where the speed equals a power 1 of a curvature function f which is monotone symmetric homogeneous of degree 1 it is shown that if the initial h convex hypersurface is pinched then this is maintained under the flow and the hypersurfaces shrink to a round point in h n 1 in finite time as a consequence when rescaling appropriately the evolving hypersurfaces converge smoothly and exponentially to the unit sphere of r n 1

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

ricci flow on quasi projective manifolds we consider the kahler ricci flow on complete finite volume metrics that live on the complement of a divisor in a compact kahler manifold x assuming certain spatial asymptotics on the initial metric we compute the singularity time in terms of cohomological data on x we also give a sufficient condition for the singularity if there is one to be type ii

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

inverse mean curvature flow with forced term abstract in this paper we study the evolution of convex hypersurfaces by inverse mean curvature minus an external force field c we prove that the flow will preserve the convexity for any c if c 1 h on the initial surface we prove that the flow will expand the hypersurface for all time and after rescaling the hypersurface will converge to a sphere if c 1 h on the initial surface we show that the maximum existence time of the flow is finite and the hypersurface will contract to a point when approaching the final time

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

image segmentation based on fuzzy connectedness using dynamic weights traditional segmentation techniques do not quite meet the challenges posed by inherently fuzzy medical images image segmentation based on fuzzy connectedness addresses this problem by attempting to capture both closeness based on characteristic intensity and hanging togetherness based on intensity homogeneity of image elements to the target object this paper presents a modification and extension of previously published image segmentation algorithms based on fuzzy connectedness which is computed as a linear combination of an object feature based and a homogeneity based component using fixed weights we provide a method called fuzzy connectedness using dynamic weights dyw to introduce directional sensitivity to the homogeneity based component and to dynamically adjust the linear weights in the functional form of fuzzy connectedness dynamic computation of the weights relieves the user of the exhaustive search process to find the best combination of weights suited to a particular application this is critical in applications such as analysis of cardiac cine magnetic resonance mr images where the optimal combination of affinity component weights can vary for each slice each phase and each subject in spite of data being acquired from the same mr scanner with identical protocols we present selected results of applying dyw to segment phantom images and actual mr computed tomography and infrared data the accuracy of dyw is assessed by comparing it to two different formulations of fuzzy connectedness our method consistently achieves accuracy of more than 99 15 for a range of image complexities contrast 5 65 noise to contrast ratio of 6 18 and bias field of four types with maximum gain factor of up to 10

real analyticity of solutions of hamilton s equation il existe des constantes positives cn et sn dependant seulement de la dimension n de la variete m telles que pour une metrique initiale g 0 avec r g 0 1 l equation de hamilton a une solution jusqu a t sn et l estimation suivante est vraie pour la metrique solution en t o t sn k 0 t k k 1 12 k r 2 c n

a quantum exactly solvable nonlinear oscillator related to the isotonic oscillator a nonpolynomial one dimensional quantum potential representing an oscillator which can be considered as placed in the middle between the harmonic oscillator and the isotonic oscillator harmonic oscillator with a centripetal barrier is studied first the general case that depends on a parameter a is considered and then a particular case is studied with great detail it is proven that it is schr dinger solvable and then the wavefunctions n and the energies en of the bound states are explicitly obtained finally it is proven that the solutions determine a family of orthogonal polynomials related to the hermite polynomials and such that i every is a linear combination of three hermite polynomials and ii they are orthogonal with respect to a new measure obtained by modifying the classic hermite measure

space of state vectors in symmetric quantum mechanics the space of states of symmetric quantum mechanics is examined the requirement that eigenstates with different eigenvalues must be orthogonal leads to the conclusion that eigenfunctions belong to a space with an indefinite metric self consistent expressions for the probability amplitude and the average value of operator are suggested further specification of the space of state vectors yields a superselection rule redefining the notion of the superposition principle an expression for the probability current density satisfying the equation of continuity and vanishing for the bound state is proposed

new li yau hamilton inequalities for the ricci flow via the space time approach we prove li yau hamilton inequalties that extend hamilton s matrix inequality for solutions of the ricci flow with nonnegative curvature operators to obtain our extensions we apply the space time formalism of s c chu and one of the authors to solutions of the ricci flow modified by a cosmological constant then we adjoin to the ricci flow the evolution of a 1 form and a 2 form flowing by a system of heat type equations by a rescaling argument the inequalities we obtain in this manner yield new inequalities which are reminiscent of the linear trace inequality of hamilton and one of the authors

the calabi flow on kahler surfaces with bounded sobolev constant i we consider the formation of singularities along the calabi flow by assuming the uniformly bounded sobolev constants on kahler surfaces we prove that if curvature tensor is not uniformly bounded then one can form a singular model called deepest bubble such deepest bubble has to be a scalar flat ale kahler metric in certain kahler classes on toric fano surfaces the sobolev constants are a priori bounded along the calabi flow with small calabi energy we can also show in certain cases no deepest bubble can form along the flow it follows that the curvature tensor is uniformly bounded and the flow exists for all time and converges to an extremal metric subsequently to illustrate our results more clearly we focus on an example on mathbb cp 2 blown up three points at generic position our result also implies existence of constant scalar curvature metrics on mathbb cp 2 blown up three points at generic position in the kahler classes where the exceptional divisors have the same area

mean curvature flow of pinched submanifolds to spheres the evolution of hypersurfaces by their mean curvature has been studied by many authors since the appearance of gerhard huisken s seminal paper hu1 more recently mean curvature flow of higher codimension submanifolds has also received attention in this paper we prove a result analogous to that of hu1 for submanifolds of any codimension let f0 n rn k be a smooth immersion of a compact manifold the mean curvature flow with initial condition f0 is a smooth family of immersions f 0 t rn k satisfying

new quasi exactly solvable sextic polynomial potentials a hamiltonian is said to be quasi exactly solvable qes if some of the energy levels and the corresponding eigenfunctions can be calculated exactly and in a closed form an entirely new class of qes hamiltonians having sextic polynomial potentials is constructed these new hamiltonians are different from the sextic qes hamiltonians in the literature because their eigenfunctions obey symmetry rather than hermitian boundary conditions these new hamiltonians present a novel problem that is not encountered when the hamiltonian is hermitian it is necessary to distinguish between the parametric region of unbroken symmetry in which all of the eigenvalues are real and the region of broken symmetry in which some of the eigenvalues are complex the precise location of the boundary between these two regions is determined numerically using the extrapolation techniques and analytically using the wkb analysis

on the positive mass theorem for manifolds with corners we study the positive mass theorem for certain non smooth metrics following p miao s work our approach is to smooth the metric using the ricci flow as well as improving some previous results on the behaviour of the adm mass under the ricci flow we extend the analysis of the zero mass case to higher dimensions

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

lifespan theorem for simple constrained surface diffusion flows abstract we consider closed immersed hypersurfaces in r 3 and r 4 evolving by a special class of constrained surface diffusion flows this class of constrained flows includes the classical surface diffusion flow in this paper we present a lifespan theorem for these flows which gives a positive lower bound on the time for which a smooth solution exists and a small upper bound on the total curvature during this time the hypothesis of the theorem is that the surface is not already singular in terms of concentration of curvature this turns out to be a deep property of the initial manifold as the lower bound on maximal time obtained depends precisely upon the concentration of curvature of the initial manifold in l 2 for m 2 immersed in r 3 and additionally on the concentration in l 3 for m 3 immersed in r 4 this is stronger than a previous result on a different class of constrained surface diffusion flows as here we obtain an improved lower bound on maximal time a better estimate during this period and eliminate any assumption on the area of the evolving hypersurface