m koszul artin schelter regular algebras abstract this paper studies the homological determinants and nakayama automorphisms of not necessarily noetherian m koszul twisted calabi yau or equivalently m koszul artin schelter regular algebras dubois violette showed that such an algebra is isomorphic to a derivation quotient algebra d w i for a unique up to scalar multiples twisted superpotential w by definition d w i is the quotient of the tensor algebra tv where v d w i 1 by i w the ideal generated by all i th order left partial derivatives of w the restriction map v is used to identify the group of graded algebra automorphisms of d w i with a subgroup of gl v we show that the homological determinant of a graded algebra automorphism of an m koszul artin schelter regular algebra d w i is given by the formula hdet w m i w it follows from this that the homological determinant of the nakayama automorphism of an m koszul artin schelter regular algebra is 1 as an application we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian artin schelter regular algebras of dimension 3

on the local equatorial characterization of zonoids and intersection bodies abstract in this paper we show that there is no local equatorial characterization of zonoids in odd dimensions this gives a negative answer to the conjecture posed by w weil in 1977 and shows that the local equatorial characterization of zonoids may be given only in even dimensions in addition we prove a similar result for intersection bodies and show that there is no local characterization of these bodies

inequalities between projection functions of convex bodies we obtain injectivity results for certain integral transforms on grassmann manifolds these are used to solve the intermediate cases of a problem of shephard and to give other applications to projection functions of convex bodies of revolution

interaction equations for short and long dispersive waves abstract we show the time local well posedness for a system of nonlinear dispersive equations for the water wave interaction formula it is shown that for any initial data u0 v0 hs r hs 1 2 r s 0 the solution for the above equation uniquely exists in a subset ofc t t hs c t t hs 1 2 and depends continuously on the data by virtue of a special structure of the nonlinear coupling the solution is stable under a singular limiting process

derivatives of the lp cosine transform abstract the l p cosine transform of an even continuous function f c e s n 1 is defined by h x s n 1 x p f d x r n it is shown that if p is not an even integer then all partial derivatives of even order of h x up to order p 1 including p 1 if p is an odd integer exist and are continuous everywhere in r n 0 as a result of the corresponding differentiation formula we show that if f is a positive bounded function and p 1 then h 1 p is a support function of a convex body whose boundary has everywhere positive gauss kronecker curvature

on low regularity of the ostrovsky stepanyams and tsimring equation abstract in this paper we prove by using the fourier restriction norm method that the initial value problem of the ostrovsky stepanyams and tsimring equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r for any s 5 4 which improve our former result in zhao and cui 2009 5

corrigendum to m koszul artin schelter regular algebras j algebra 446 2016 373 399 abstract the definition of the homological determinant in 2 2 of the published paper is not correct despite that all the statements in the paper that involve properties of the homological determinant are correct

kernels with prescribed surface tension mobility for threshold dynamics schemes abstract we show how to construct a convolution kernel that has a desired anisotropic surface tension and desired anisotropic mobility to be used in threshold dynamics schemes for simulating weighted motion by mean curvature of interfaces including networks of them in both two and three dimensions moreover we discuss necessary and sufficient conditions for the positivity of the kernel which in the case of two phase flow ensures that the resulting scheme respects a comparison principle and implies convergence to the viscosity solution of the level set formulation of the flow in particular we show in a barrier type statement that the kernel cannot possibly be positive unless both the mobility and the surface tension satisfy necessary conditions in three dimensions and give a complete characterization among other results is a threshold dynamics scheme that is guaranteed to dissipate a non local approximation to the interfacial energy in the fully anisotropic multiphase setting using the new kernel construction

intersection bodies and generalized cosine transforms the paper is focused on intimate connection between geometric properties of intersection bodies in convex geometry and generalized cosine transforms in harmonic analysis a new concept of intersection body that unifies some known classes of geometric objects is introduced a parallel between trace theorems in function theory restriction onto lower dimensional subspaces of the spherical radon transforms and the generalized cosine transforms and sections of intersection bodies is established new integral formulas for different classes of cosine transforms are obtained and examples of intersection bodies are given we also revisit some known facts in this area and give them new simple proofs

the second mixed projection problem and the projection centroid conjectures abstract we provide partial answers to the open problems 4 5 4 6 of 3 and 12 9 of 12 regarding the classification of fixed points of the second mixed projection operator and iterates of the projection and centroid operators

comparison of volumes of convex bodies in real complex and quaternionic spaces abstract the classical busemann petty problem 1956 asks whether origin symmetric convex bodies in r n with smaller hyperplane central sections necessarily have smaller volumes it is known that the answer is affirmative if n 4 and negative if n 4 the same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning we give unified analysis of this circle of problems in real complex and quaternionic n dimensional spaces all cases are treated simultaneously in particular we show that the busemann petty problem in the quaternionic n dimensional space has an affirmative answer if and only if n 2 the method relies on the properties of cosine transforms on the unit sphere we discuss possible generalizations

local well posedness of the ostrovsky stepanyams and tsimring equation in sobolev spaces of negative indices abstract in this paper we prove by using the method of bilinear estimate in the bourgain type spaces that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 1

on the low regularity of the benney lin equation abstract we consider the low regularity of the benney lin equation u t u u x u x x x u x x u x x x x u x x x x x 0 we established the global well posedness for the initial value problem of benney lin equation in the sobolev spaces h s r for 0 s 2 improving the well posedness result of biagioni and linares h a biaginoi f linares on the benney lin and kawahara equation j math anal appl 211 1997 131 152 for s 2 we also prove some ill posedness issues

inversion of fractional integrals related to the spherical radon transform explicit inversion formulas are obtained for the analytic family of fractional integrals t f x n sn xy 1 f y dyon the unit sphere in rn 1 arbitrary complex andn 2 are considered in the ease 0 the integralt fcoincides with the spherical radon transform for 1 1 3 5 such integrals are known as the blaschke levy representations and arise in convex geometry probability and the banach space theory for 1 3 5 the integralt fis defined by continuity as the spherical convolution with the power logarithmic kernel different inversion methods are discussed

well posedness of the cauchy problem for ostrovsky stepanyams and tsimring equation with low regularity data abstract in this paper we prove that the initial value problem of the ost equation u t u x x x h u x h u x x x u u x 0 x r t 0 where 0 and h denotes the usual hilbert transformation is locally well posed in the sobolev space h s r when s 3 4 and globally well posed in h s r when s 0

on a perturbation of the benjamin ono equation abstract we prove using the fourier restriction norm method that the initial value problem associated to a perturbation of the benjamin ono equation u t u u x h u x x h u x u x x 0 where x r t 0 0 and h denotes the usual hilbert transform is locally well posed in the sobolev space h s r for any s 1 2 and globally well posed in h s r when s 0 for s 1 we also prove some ill posedness issues

convolution kernels and stability of threshold dynamics methods threshold dynamics and its extensions have proven useful in computing interfacial motions in applications as diverse as materials science and machine learning certain desirable properties of the algorithm such as unconditional monotonicity in two phase flows and gradient stability more generally hinge on positivity properties of the convolution kernel and its fourier transform recent developments in the analysis of this class of algorithms indicate that sometimes as in the case of certain anisotropic curvature flows arising in materials science these properties of the convolution kernel cannot be expected other applications such as machine learning would benefit from as great a level of flexibility in choosing the convolution kernel as possible in this paper we establish certain desirable properties of threshold dynamics such as gamma convergence of its associated energy for a substantially wider class of kernels than has been hitherto possible we also present variants of the algorithm that

a positive answer to the busemann petty problem in three dimensions we prove that in e3 the busemann petty problem concerning cen tral sections of centrally symmetric convex bodies has a positive answer together with other results this settles the problem in each dimension

intersection bodies that are not polar zonoids a flat top condition in dimensions four and six abstract we find general geometric conditions on a convex body of revolution k in dimensions four and six so that its intersection body i k is not a polar zonoid we exhibit several examples of intersection bodies which are not polar zonoids

local and equatorial characterization of unit balls of subspaces of lp p 0 and properties of the generalized cosine transform abstract in this paper we show that there is no local equatorial characterization of bodies that embed in l p in odd dimensions for all p not even 0 p however bodies that embed in l p for p odd are local equatorially characterizable provided that the dimension is even but not locally characterizable in general this extends results given in panina 1988 13 goodey and weil 1992 5 and nazarov ryabogin and zvavitch 2008 12 concerning the local equatorial characterization of zonoids and intersection bodies these results were for bodies that embed in l 1 and l 1 respectively

on the cauchy problem for the zakharov system abstract we study the local cauchy problem in time for the zakharov system 1 1 and 1 2 governing langmuir turbulence with initial data u 0 n 0 t n 0 h k h lscr h l 1 in arbitrary space dimension we define a natural notion of criticality according to which the critical values of k l are 2 3 2 2 2 using a method recently developed by bourgain we prove that the zakharov system is locally well posed for a variety of values of k l the results cover the whole subcritical range for 4 for 3 they cover only part of it and the lowest admissible values are k l 1 2 0 for 2 3 and k l 0 1 2 for 1 as a by product of the one dimensional result we prove well posedness of the benney system 1 14 and 1 15 governing the interaction of short and long waves for the same values of k l

skew group algebras of calabi yau algebras abstract the calabi yau property of skew group algebras is discussed it is shown that the skew group algebra a g of a koszul calabi yau algebra a with a finite subgroup g of automorphisms of a is calabi yau if and only if g is a finite subgroup of the special linear group sl a which is defined by means of the homological determinant using the a algebra structure on the yoneda algebra some results in bocklandt et al 2010 bsw are generalized say every connected graded p koszul calabi yau algebra is derived from a superpotential the superpotential for the skew group algebra a g is also constructed

fourier and poisson transformation associated to a semisimple symmetric space soit g un groupe de lie semisimple connexe contenu dans un groupe de lie semisimple simplement connexe dont l algebre de lie est la complexification de celle de g soit une involution non triviale de g et une involution de cartan commutant avec soit h un groupe point fixe de on etudie les representations h spheriques de dimension finie de g contenant un vecteur k fixe ou k est un sous groupe compact maximal stable de g sous representations de dimension finie de le noyau de poisson la distribution u i la transformation de fourier et poisson formules integrales

inversion ofk plane transforms via continuous wavelet transforms the inversion formulae fork plane transforms of functionsf lp rn are obtained in terms of continuous wavelet transforms generated by a wavelet measure the admissibility conditions for a wavelet measure are formulated in terms of the fourier transform of and without using the fourier transform the investigation is based on the wavelet type representations of riesz fractional derivatives

on the well posedness for the generalized ostrovsky stepanyams and tsimring equation abstract in this work we prove that the initial value problem associated with equation u t u xxx h u x h u xxx u k u x 0 x r t 0 where 0 and h denotes the usual hilbert transform is locally well posed in sobolev spaces h s r for s 0 and k 1 2 3 and globally well posed in l 2 r

the cos λ and sin λ transforms as intertwining operators between generalized principal series representations of sl n 1 k abstract in this article we connect topics from convex and integral geometry with well known topics in representation theory of semisimple lie groups by showing that the cos and sin transforms on the grassmann manifolds gr p k su n 1 k s u p k u n 1 p k are standard intertwining operators between certain generalized principal series representations induced from a maximal parabolic subgroup p p of sl n 1 k the index p indicates the dependence of the parabolic on p the general results of knapp and stein 1971 1980 20 21 and vogan and wallach 1990 44 then show that both transforms have meromorphic extension to c and are invertible for generic c furthermore known methods from representation theory combined with a selberg type integral allow us to determine the k spectrum of those operators

on the range of the radon d plane transform and its dual we present direct group theoretic proofs of the range theorem for the radon d plane transform on r the original proof by richter involves extensive use of local coordinate calculations on g d n the grassmann manifold of affine a planes in r we show that moment conditions are not sufficient to describe this range when d n in contrast to the compactly supported case finally we show that the dual rf plane transform maps g g d n surjectively onto f r

a fourier transform approach to christoffel s problem we use fourier transform techniques to provide a new approach to berg s solution of the christoffel problem this leads to an explicit description of berg s spherical kernel and to new regularity properties of the associated integral transform

inversion of k plane transforms and applications in computer tomography the mathematics behind computerized tomography ct is based on the study of the parallel beam transform p and the divergent beam transform d both of these map a function f in mathbb r n into a function defined on the set of all lines in mathbb r n by integrating f along these lines the parallel and divergent k plane transforms are defined in a similar fashion by integration over k planes i e translates of k dimensional subspaces and are also denoted p and d respectively a related transform is the spherical k plane transform s which maps a function f on the sphere s n 1 into its integrals over k dimensional great circlesthis paper discusses the properties of the k plane transforms p d and s and their inverses emphasizing relationships and similarities between the operators and their relation to ct some new results are included most notable are more general conditions under which the inversion formulas hold

lp affine isoperimetric inequalities the lp analogues of the petty projection inequality and the busemannpetty centroid inequality are established an affine isoperimetric inequality compares two functionals associated with convex or more general bodies where the ratio of the functionals is invariant under non degenerate linear transformations these isoperimetric inequalities are more powerful than their better known euclidean relatives

local properties of intertwining operators on the sphere blaschke s original question regarding the local determination of zonoids or projection bodies has been the subject of much research over the years in recent times this research has been extended to include intersection bodies and it has been shown that neither zonoids nor intersection bodies have local characterizations however it has also been proved that both these classes of bodies admit equatorial characterizations in odd dimensions but not in even dimensions the proofs of these results were mostly analytic using properties of associated spherical integral transforms the cosine transform and the radon transform here we elaborate a general principle showing that such local or equatorial characterization problems are equivalent to corresponding support properties of the spherical operators we discuss this within a general framework for intertwining operators on c functions and apply the results to further geometric constructions namely to certain mean section bodies to lq centroid bodies and to k intersection bodies