boundary regularity on the parabolic monge ampere equation abstract this paper establishes boundary estimates for solutions to the parabolic monge ampere equation u t det d 2 u 1 n f x t

aubry mather sets for finite twist maps of a cylinder and semilinear duffing equations in this paper based on a generalized version of the aubry mather theorem about twist maps of an infinite cylinder without the usual infinite twist assumption we establish the existence of certain quasiperiodic solutions and unlinked periodic solutions in some semilinear duffing equations x g x p t where p t is a continuous periodic function of period 1 and the semilinearity means that 0 g x x k

gauss curvature flow the fate of the rolling stones we prove firey s 1974 conjecture that convex surfaces moving by their gauss curvature become spherical as they contract to points

kumjian pask algebras of locally convex higher rank graphs the kumjian pask algebra of a higher rank graph generalises the leavitt path algebra of a directed graph we extend the definition of kumjian pask algebra to row finite higher rank graphs with sources which satisfy a local convexity condition after proving versions of the graded uniqueness theorem and the cuntz krieger uniqueness theorem we study the kumjian pask algebra of rank 2 bratteli diagrams by studying certain finite subgraphs which are locally convex we show that the desourcification procedure of farthing and webster yields a row finite higher rank graph without sources such that the kumjian pask algebras of and are morita equivalent we then use the morita equivalence to study the ideal structure of the kumjian pask algebra of by pulling the appropriate results across the equivalence

kumjian pask algebras of finitely aligned higher rank graphs abstract we extend the definition of kumjian pask algebras to include algebras associated to finitely aligned higher rank graphs we show that these kumjian pask algebras are universally defined and have a graded uniqueness theorem we also prove the cuntz krieger uniqueness theorem to do this we use a groupoid approach as a consequence of the graded uniqueness theorem we show that every kumjian pask algebra is isomorphic to the steinberg algebra associated to its boundary path groupoid we then use steinberg algebra results to prove the cuntz krieger uniqueness theorem and also to characterize simplicity and basic simplicity

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

the logarithmic minkowski problem in analogy with the classical minkowski problem necessary and sufficient conditions are given to assure that a given measure on the unit sphere is the cone volume measure of the unit ball of a finite dimensional banach space

free boundary regularity on the focusing problem for the gauss curvature flow with flat sides we consider the motion of a compact weakly convex two dimensional surface of revolution sigma under the gauss curvature flow we assume that the initial surface has a flat side and as a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate at points where the curvature is zero expressing the strictly convex part of the surface near the interface as the graph of a function z f r t we show that if at ti me t 0 g sqrt f vanishes linearly at the flat side then g r t will become smooth up to the interface for t 0 and it will remain smooth up to the focusing time t of the flat side we also show that at the focusing time of the flat side the function g is of class c 1 beta for all beta and no better than c 1 2 over 5 this implies that at the focusing time the surface sigma is of class c 2 beta for all beta and no better than c 2 2 over 5 in the case of the evolution monge ampere equation we find the exact self similar profile of the function g at its focusing time

the dual minkowski problem for symmetric convex bodies abstract the dual minkowski problem for even data asks what are the necessary and sufficient conditions on a prescribed even measure on the unit sphere for it to be the q th dual curvature measure of an origin symmetric convex body in r n a full solution to this is given when 1 q n the necessary and sufficient conditions turn out to be an explicit measure concentration condition to obtain the results a variational approach is used where the functional is the sum of a dual quermassintegral and an entropy integral the proof requires two crucial estimates the first is an estimate of the entropy integral which is obtained by using a spherical partition the second is a sharp estimate of the dual quermassintegrals for a carefully chosen barrier convex body

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

c 1 α regularity of solutions to parabolic monge ampere equations we study interior c 1 alpha regularity of viscosity solutions of the parabolic monge amp ere equation u t b x t big det d 2 u big p with exponent p 0 and with coefficients b which are bounded and measurable we show that when p is less than the critical power 1 over n 2 then solutions become instantly c 1 alpha in the interior also we prove the same result for any power p0 at those points where either the solution separates from the initial data or where the initial data is c 1 beta

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

new lp affine isoperimetric inequalities we prove new lp affine isoperimetric inequalities for all p 8 1 we establish for all p n a duality formula which shows that lp affine surface area of a convex body k equals affine surface area of the polar body k

variational discretization of axisymmetric curvature flows we present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail although numerical methods for geometric flows have been used frequently in axisymmetric settings numerical analysis results so far are rare in this paper we present stability equidistribution existence and uniqueness results for the introduced approximations numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one dimensional problem has to be solved the good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries

a groupoid approach to discrete inverse semigroup algebras abstract let k be a commutative ring with unit and s an inverse semigroup we show that the semigroup algebra ks can be described as a convolution algebra of functions on the universal etale groupoid associated to s by paterson this result is a simultaneous generalization of the author s earlier work on finite inverse semigroups and paterson s theorem for the universal c algebra it provides a convenient topological framework for understanding the structure of ks including the center and when it has a unit in this theory the role of gelfand duality is replaced by stone duality using this approach we construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups generalizing the case of an inverse semigroup with finitely many idempotents more generally we describe the irreducible representations of an inverse semigroup s that can be induced from associated groups as precisely those satisfying a certain finiteness condition this finiteness condition is satisfied for instance by all representations of an inverse semigroup whose image contains a primitive idempotent

a flow of conformally balanced metrics with kahler fixed points while the anomaly flow was originally motivated by string theory its zero slope case is potentially of considerable interest in non kahler geometry as it is a flow of conformally balanced metrics whose stationary points are precisely kahler metrics we establish its convergence on kahler manifolds for suitable initial data we also discuss its relation to some current problems in complex geometry

subharmonic solutions of second order subquadratic hamiltonian systems with potential changing sign abstract with the aid of some symplectic transformations the existence of subharmonic solutions of the second order hamiltonian system v x t x is considered where the potential v is subquadratic in x as x goes to infinity and can change sign with respect to t

on the uniqueness of lp minkowski problems the constant p curvature case in r3 abstract we study the c 4 smooth convex bodies k r n 1 satisfying k x u x 1 p where x s n k is the gauss curvature of k u is the support function of k and p is a constant in the case of n 2 either when p 1 0 or when p 0 1 in addition to a pinching condition we show that k must be the unit ball this partially answers a conjecture of lutwak yang and zhang about the uniqueness of the l p minkowski problem in r 3 moreover we give an explicit pinching constant depending only on p when p 0 1

the stability of the equilibrium of a conservative system abstract in this paper we consider the conservative system formula where a t is a continuous and 1 periodic function in the time t e t x is also 1 periodic in the time t and dominated by the power x 2 n 2 in a neighborhood of x 0 a sufficient and necessary condition for the stability in the liapunov sense of the trivial solution x 0 is given

α gauss curvature flows with flat sides abstract in this paper we study the deformation of the 2 dimensional convex surfaces in r 3 whose speed at a point on the surface is proportional to power of positive part of gauss curvature first for 1 2 1 we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with c 1 1 estimate before the collapsing time if the initial surface is only convex moreover we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while we also show the interface between the flat side and the strictly convex side of the surface remains smooth on 0 t t 0 under certain necessary regularity and non degeneracy initial conditions where t 0 is the vanishing time of the flat side

periodic and subharmonic solutions of second order hamiltonian systems abstract some solvability conditions of periodic solutions and subharmonic solutions are obtained for a class of new superquadratic non autonomous second order hamiltonian systems by the minimax methods in critical point theory

worn stones with flat sides all time regularity of the interface we study the all time regularity of the free boundary problem associated to the deformation of a compact weakly convex surface in 3 with a flat side by its gaussian curvature we show that under certain necessary regularity and non degeneracy initial conditions the interface separating the flat from the strictly convex side remains smooth on 0 t tc up to the vanishing time tc of the flat side

on graded irreducible representations of leavitt path algebras using the e algebraic branching systems various graded irreducible representations of a leavitt path k algebra l of a directed graph e are constructed the concept of a laurent vertex is introduced and it is shown that the minimal graded left ideals of l are generated by the laurent vertices or the line points leading to a detailed description of the graded socle of l following this a complete characterization is obtained of the leavitt path algebras over which every graded irreducible representation is finitely presented a useful result is that the irreducible representation v p induced by infinite paths tail equivalent to an infinite path p we call this a chen simple module is graded if and only if p is an irrational path we also show that every one sided ideal of l is graded if and only if the graph e contains no cycles supplementing the theorem of one of the co authors that every leavitt path algebra l is graded von neumann regular we show that l is graded self injective if and only if l is a graded semi simple algebra made up of matrix rings of arbitrary size over the field k or the graded field k xn x n where n n

anisotropic flow of convex hypersurfaces by the square root of the scalar curvature abstract we show the existence of a smooth solution for the flow deformed by the square root of the scalar curvature multiplied by a positive anisotropic factor when the strictly convex initial hypersurface in euclidean space is suitably pinched we also prove the convergence of rescaled surfaces to a smooth limit manifold which is a round sphere for a general case in dimension two it is shown that with a volume preserving rescaling the limit profile satisfies a soliton equation

periodic solutions of second order superquadratic hamiltonian systems with potential changing sign ii abstract we consider the periodic solutions of the second order hamiltonian system x x h t v x with v being positive and superquadratic at infinity h being c 1 2 periodic sign changing and all zeros being simple some existence and multiplicity results of periodic solutions are given

on the lp dual minkowski problem abstract the l p dual curvature measure was introduced by lutwak yang zhang in an attempt to unify the l p brunn minkowski theory and the dual brunn minkowski theory the characterization problem for l p dual curvature measure called the l p dual minkowski problem is a fundamental problem in this unifying theory the l p dual minkowski problem contains the l p minkowski problem and the dual minkowski problem two major problems in modern convex geometry that remain open in general in this paper existence results on the l p dual minkowski problem in the weak sense will be provided moreover existence and uniqueness of the solution in the smooth category will also be demonstrated

subharmonic solutions for nonautonomous sublinear second order hamiltonian systems some existence theorems are obtained for subharmonic solutions of nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

on a prioric 1 α andw 2 p estimates for a parabolic monge ampere equation in the gauss curvature flows this paper establishes holder estimates of du and l p estimates of d 2 u for solutions u to the parabolic monge ampere equation au t det d 2 u 1 n

on the planar dual minkowski problem abstract in this paper we resolve the planar dual minkowski problem proposed by huang et al 2016 31 for all positive indices without any symmetry assumption more precisely given any q 0 and function f on s 1 bounded by two positive constants we show that there exists a convex body in the plane containing the origin in its interior whose dual curvature measure c q has density f in particular if f is smooth then is also smooth

symplectic transformations and periodic solutions of hamiltonian systems abstract symplectic transformations with a kind of homogeneity are introduced which enable us to give a unified approach to some existence problems on the periodic solutions for the first and second order hamiltonian systems

periodic and subharmonic solutions of a class of subquadratic second order hamiltonian systems some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

the theory of prime ideals of leavitt path algebras over arbitrary graphs given an arbitrary graph e and a field k the prime ideals as well as the primitive ideals of the leavitt path algebra lk e are completely characterized in terms of their generators the stratification of the prime spectrum of lk e is indicated with information on its individual stratum necessary and sufficient conditions are given on the graph e under which every prime ideal of the leavitt path algebra lk e is primitive leavitt path algebras with krull dimension zero are characterized and those with various prescribed krull dimension are constructed the minimal prime ideals of lk e are described in terms of the graphical properties of e and using this complete descriptions of the height one as well as the co height one prime ideals of lk e are given

flow by powers of the gauss curvature abstract we prove that convex hypersurfaces in r n 1 contracting under the flow by any power 1 n 2 of the gauss curvature converge after rescaling to fixed volume to a limit which is a smooth uniformly convex self similar contracting solution of the flow under additional central symmetry of the initial body we prove that the limit is the round sphere for 1

the lp minkowski problem for polytopes for 0 p 1 abstract necessary and sufficient conditions are given for the existence of solutions to the discrete l p minkowski problem for the critical case where 0 p 1

convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature we consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity we prove that the ratio of circumradius to inradius is bounded by a function of the circumradius with limit 1 at zero we apply this result to the motion of hypersurfaces by arbitrary speeds which are smooth homogeneous functions of the principal curvatures of degree greater than one for smooth strictly convex initial hypersurfaces with ratio of principal curvatures sufficiently close to one at each point we prove that solutions remain smooth and strictly convex and become spherical in shape while contracting to points in finite time recently several papers have considered the flow of convex hypersurfaces by speeds which are homogeneous functions of the principal curvatures of degree 1 under suitable pinching conditions on the curvature of the initial hypersurface the aim is to prove that solutions become spherical as they contract to points this behaviour has been established for a wide range of flows where the speed is homogeneous of degree 1 in the principal curvatures including the mean curvature flow h1 the flow by nth root of gauss curvature ch1 square root of scalar curvature ch2 and a large family of other speeds a1 a3 a4 the first such result with degree of homogeneity higher than 1 was due to ben chow ch1 and concerned flow by powers of the gauss curvature he proved that flow by k with 1 n produces a spherical limiting shape provided the initial hypersurface is sufficiently pinched i n the sense that hij c hgij later such results were proved by schulze s3 for powers of the mean curvature by alessandroni and sinestrari for powers of the scalar curvature a as and by cabezas rivas and sinestrari crs for normalized flows by powers of elementary symmetric functions a feature of the results mentioned above is that the flows all have some divergence structure a point which is used crucially in deriving sufficient regularity of solutions to deduce the existence of a smooth limiting hypersurface in ch1 the divergence structure was used to deduce pinching of the principal curvatures using integral estimates in a manner similar to h1 in s3 a and as the curvature pinching was proved using maximum principle arguments but the divergence structure was still needed because higher regularity of solutions was established using results for divergence form porous medium equations to date we are not aware of any work which provides holder continuity for solutions of porous medium equations in non divergence form without assuming regularity of the coefficients in this paper we provide a geometric estimate which circumvents this difficulty we prove in section 3 that if the ratios of principal curvatures at each point are bounded in terms of the maximum principal curvature by a function which approaches one at infinity then the ratio of circumradius to inradius is as close to one as desired if the circumradius is sufficiently small using