regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

the monge ampere equation and its link to optimal transportation we survey old and new regularity theory for the monge ampere equation show its connection to optimal transportation and describe the regularity properties of a general class of monge ampere type equations arising in that context

nearly round spheres look convex we prove that a riemannian manifold m g close enough to the round sphere in the c 4 topology has uniformly convex injectivity domains so m appears uniformly convex in any exponential chart the proof is based on the ma trudinger wang nonlocal curvature tensor which originates from the regularity theory of optimal transport

a perturbation argument for a monge ampere type equation arising in optimal transportation we prove some interior regularity results for potential functions of optimal transportation problems with power costs the main point is that our problem is equivalent to a new optimal transportation problem whose cost function is a sufficiently small perturbation of the quadratic cost but it does not satisfy the well known condition a 3 guaranteeing regularity the proof consists in a perturbation argument from the standard monge ampere equation in order to obtain first interior c1 1 estimates for the potential and second interior holder estimates for second derivatives in particular we take a close look at the geometry of optimal transportation when the cost function is close to quadratic in order to understand how the equation degenerates near the boundary

a note on canonical forms for matrix congruence abstract canonical forms for matrix congruence for general matrices are exhibited as an easy consequence of results presented in a recent paper by r c thompson on pencils of symmetric and skew symmetric matrices we also show how to exhibit standard equivalence class representatives in the orbits of the semiorthogonal groups acting naturally on skew symmetric forms

the classification of 4 dimensional leibniz algebras abstract this paper is a contribution to the development of the non associative algebras theory more precisely this work deals with the classification of the complex 4 dimensional leibniz algebras note that the classification of 4 dimensional nilpotent complex leibniz algebras was obtained in albeverio et al 2006 1 therefore we will only consider non nilpotent case in this work

canonical matrices of isometric operators on indefinite inner product spaces abstract we give canonical matrices of a pair a b consisting of a nondegenerate form b and a linear operator a satisfying b ax ay b x y on a vector space over f in the following cases f is an algebraically closed field of characteristic different from 2 or a real closed field and b is symmetric or skew symmetric f is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field and b is hermitian or skew hermitian with respect to any nonidentity involution on f these classification problems are wild if b may be degenerate we use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver this method was described in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 3 1988 481 501

canonical forms for mixed symmetric skewsymmetric quaternion matrix pencils canonical forms are described for pairs of quaternionic matrices or equivalently matrix pencils where one matrix is symmetric and the other matrix is skewsymmetric under strict equivalence and symmetry respecting congruence the symmetry is understood in the sense of a fixed involutory antiautomorphism of the skew field of the real quaternions the involutory antiautomorphism is assumed to be nonstandard i e other than the quaternionic conjugation some applications are developed such as canonical forms for quaternionic matrices under symmetry respecting congruence and canonical forms for matrices that are skewsymmetric with respect to a nondegenerate symmetric or skewsymmetric quaternion valued inner product

structure of isometry group of bilinear spaces abstract we describe the structure of the isometry group g of a finite dimensional bilinear space over an algebraically closed field of characteristic not two if the space has no indecomposable degenerate orthogonal summands of odd dimension it admits a canonical orthogonal decomposition into primary components and g is isomorphic to the direct product of the isometry groups of the primary components each of the latter groups is shown to be isomorphic to the centralizer in some classical group of a nilpotent element in the lie algebra of that group in the general case the description of g is more complicated we show that g is a semidirect product of a normal unipotent subgroup k with another subgroup which in its turn is a direct product of a group of the type described in the previous paragraph and another group h which we can describe explicitly the group h has a levi decomposition whose levi factor is a direct product of several general linear groups of various degrees we obtain simple formulae for the dimensions of h and k

fourth order problems with nonlinear boundary conditions we develop a new method of lower and upper solutions for a fourth order nonlinear boundary value problem where the differential equation has dependence on all lower order derivatives our boundary conditions are nonlinear we will assume the functions that define the nonlinear boundary conditions are either monotone or nonmonotone as a result we obtain existence principles which improve recent results in the literature

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

characterizing nilpotent leibniz algebras by a new bound on their second homologies abstract let l be a finite dimensional nilpotent leibniz algebra such that dim l n and dim l 2 m 0 in this paper we prove dim h l 2 l n m 2 n m m 2 where h l 2 l is the second leibniz homology of l as a consequence for a non abelian nilpotent leibniz algebra l we find that s l n 1 2 1 dim h l 2 l 0 furthermore we determine all finite dimensional nilpotent leibniz algebras with s l less than or equal to three

self adjoint operators and pairs of hermitian forms over the quaternions abstract we classify self adjoint operators and pairs of hermitian forms over the real quaternions by providing canonical matrix representations in the preliminaries we discuss the jordan canonical form theorem for quaternionic linear endomorphisms

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

a structure theorem for leibniz homology abstract presented is a structure theorem for the leibniz homology hl of an abelian extension of a simple real lie algebra g as applications results are stated for affine extensions of the classical lie algebras sl n r so n r and sp n r furthermore hl h is calculated when h is the lie algebra of the poincare group as well as the lie algebra of the affine lorentz group the structure theorem identifies all of these in terms of g invariants

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

periodic oscillations for a nonlinear suspension bridge model abstract we look for time periodic solutions of the suspension bridge equation lazer and mckenna showed that for a certain configuration of the parameters one may expect the existence of large amplitude periodic solutions having the same period as the forcing term we prove the existence of large scale subharmonic solutions

nonlinear models of suspension bridges nonlinear variational equations describing one type of suspension bridges are proposed and studied the variational equations describe the behaviour of road bed main cables and cable stays the road bed is described by two functions connected with vertical and horizontal deformation of any cross section the main cable is considered to be perfectly flexible and inextensible the cable stays only resist tensile forces the variational equations are derived from the principle of minimum potential energy the existence of solution is based on the brouwer fixed point theorem the local uniqueness and continuous dependence on the data represented by gravitational forces acting on the road bed are studied the local results are based on the implicit function theorem for banach spaces a certain stability criterion for suspension bridges is formulated and this criterion indicates how to influence the stability of suspension bridges

non commutative hopf algebra of formal diffeomorphisms abstract this paper deals with two hopf algebras which are the non commutative analogues of two different groups of formal power series the first group is the set of invertible series with the group law being multiplication of series while the second is the set of formal diffeomorphisms with the group law being a composition of series the motivation to introduce these hopf algebras comes from the study of formal series with non commutative coefficients invertible series with non commutative coefficients still form a group and we interpret the corresponding new non commutative hopf algebra as an alternative to the natural hopf algebra given by the co ordinate ring of the group which has the advantage of being functorial in the algebra of coefficients for the formal diffeomorphisms with non commutative coefficients this interpretation fails because in this case the composition is not associative anymore however we show that for the dual non commutative algebra there exists a natural co associative co product defining a non commutative hopf algebra moreover we give an explicit formula for the antipode which represents a non commutative version of the lagrange inversion formula and we show that its coefficients are related to planar binary trees then we extend these results to the semi direct co product of the previous hopf algebras and to series in several variables finally we show how the non commutative hopf algebras of formal series are related to some renormalization hopf algebras which are combinatorial hopf algebras motivated by the renormalization procedure in quantum field theory and to the renormalization functor given by the double tensor algebra on a bi algebra

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

initial boundary value problem for the nonlinear string beam system this paper deals with a nonlinear string beam system describing the torsional vertical oscillations of a suspension bridge we consider the initial boundary value problem and study the existence and uniqueness question we assume time independent right hand sides but allow quite general nonlinear terms using the faedo galerkin method we prove the existence of a unique solution on an arbitrary large time interval

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

some cohomologically rigid solvable leibniz algebras abstract in this paper we describe solvable leibniz algebras whose quotient algebra by one dimensional ideal is a lie algebra with rank equal to the length of the characteristic sequence of its nilpotent radical we prove that such leibniz algebra is unique and centerless also it is proved that the first and the second cohomology groups of the algebra with coefficients in the adjoint representation is trivial

mathematical analysis of dynamic models of suspension bridges in this paper we present some mathematical analysis of dynamic pde models of suspension bridges as proposed by lazer and mckenna our results are illustrated by numerical simulation with physical interpretation

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

lie algebras and bilinear forms in characteristic 2 abstract all bilinear forms defined on a finite dimensional vector space over an algebraically closed field of characteristic 2 whose associated lie algebra is reductive are determined

classification of linear mappings between indefinite inner product spaces abstract let a u v be a linear mapping between vector spaces u and v over a field or skew field f with symmetric or skew symmetric or hermitian forms b u u f and c v v f we classify the triples a b c if f is r or c or the skew field of quaternions h we also classify the triples a b c up to classification of symmetric forms and hermitian forms if the characteristic of f is not 2

regularity of optimal maps on the sphere the quadratic cost and the reflector antenna building on the results of ma et al in arch rational mech anal 177 2 151 183 2005 and of the author loeper in acta math to appear we study two problems of optimal transportation on the sphere the first corresponds to the cost function d2 x y where d is the riemannian distance of the round sphere the second corresponds to the cost function log x y known as the reflector antenna problem we show that in both cases the cost sectional curvature is uniformly positive and establish the geometrical properties so that the results of loeper in acta math to appear and ma et al in arch rational mech anal 177 2 151 183 2005 can apply global smooth solutions exist for arbitrary smooth positive data and optimal maps are holder continuous under weak assumptions on the data

topological classification of sesquilinear forms reduction to the nonsingular case abstract two sesquilinear forms c m c m c and c n c n c are called topologically equivalent if there exists a homeomorphism c m c n i e a continuous bijection whose inverse is also a continuous bijection such that x y x y for all x y c m r a horn and v v sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix a that is a direct sum s a s r j n 1 j n p in which s and r are nonsingular and each j n i is the n i by n i singular jordan block in this paper we prove that and are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands j n i and replacement of r c r r by a nonsingular matrix r c r r such that r and r are the matrices of topologically equivalent forms c r c r c analogous results for bilinear forms over c and over r are also obtained

a localization property of viscosity solutions to the monge ampere equation and their strict convexity the purpose of this note is to show a localization property of convex viscosity solutions to the monge ampere inequality 0 1 2 n are strictly convex

partial w 2 p regularity for optimal transport maps abstract we prove that in the optimal transportation problem with general costs and positive continuous densities the potential function is always of class w l o c 2 p for any p 1 outside of a closed singular set of measure zero we also establish global w 2 p estimates when the cost is a small perturbation of the quadratic cost the latter result is new even when the cost is exactly the quadratic cost

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

leibniz algebras with small derived ideal abstract a nilpotent leibniz algebra l is called extra special if dim z l dim l 2 1 in this paper the structure the second homology and the capability of extra special leibniz algebras are completely determined

holder continuity and injectivity of optimal maps consider transportation of one distribution of mass onto another chosen to optimize the total expected cost where cost per unit mass transported from x to y is given by a smooth function c x y if the source density f x is bounded away from zero and infinity in an open region u subset mathbf r n and the target density f y is bounded away from zero and infinity on its support overline v subset mathbf r n which is strongly c convex with respect to u and the transportation cost c satisfies the mathbf a3 rm w condition of trudinger and wang ann sc norm super pisa cl sci 5 8 1 143 174 2009 we deduce the local holder continuity and injectivity of the optimal map inside u so that the associated potential u belongs to c 1 alpha loc u here the exponent 0 depends only on the dimension and the bounds on the densities but not on c our result provides a crucial step in the low interior regularity setting in a sequel figalli et al j eur math soc jems 1131 1166 2013 we use it to establish regularity of optimal maps with respect to the riemannian distance squared on arbitrary products of spheres three key tools are introduced in the present paper namely we first find a transformation that under mathbf a3 rm w makes c convex functions level set convex as was also obtained independently from us by liu calc var partial diff eq 34 435 451 2009 we then derive new alexandrov type estimates for the level set convex c convex functions and a topological lemma showing that optimal maps do not mix the interior with the boundary this topological lemma which does not require mathbf a3 rm w is needed by figalli and loeper calc var partial diff eq 35 537 550 2009 to conclude the continuity of optimal maps in two dimensions in higher dimensions if the densities f are holder continuous our result permits continuous differentiability of the map inside u in fact c 2 alpha loc regularity of the associated potential to be deduced from the work of liu et al comm partial diff eq 35 1 165 184 2010

boundary e regularity in optimal transportation the second author has been partially supported by nsf grant dms 1262411 this material is based upon work supported by the national science foundation under grant no 0932078 000

on strict convexity and continuous differentiability of potential functions in optimal transportation this note concerns the relationship between conditions on cost functions and domains the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings in particular we prove that if the cost function satisfies the condition a3 introduced in our previous work with xinan ma the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function our results extend by different and more direct proof similar results of loeper proved by approximation from our earlier work on regularity

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

canonical forms for complex matrix congruence and congruence abstract canonical forms for congruence and congruence of square complex matrices were given by horn and sergeichuk in linear algebra appl 389 2004 347 353 based on sergeichuk s paper math ussr izvestiya 31 3 1988 481 501 which employed the theory of representations of quivers with involution we use standard methods of matrix analysis to prove directly that these forms are canonical our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms we use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew symmetric matrices as well as canonical forms for simultaneous congruence of pairs of complex hermitian matrices

wildness for tensors abstract in representation theory a classification problem is called wild if it contains the problem of classifying matrix pairs up to simultaneous similarity the latter problem is considered hopeless it contains the problem of classifying an arbitrary finite system of vector spaces and linear mappings between them we prove that an analogous universal problem in the theory of tensors of order at most 3 over an arbitrary field is the problem of classifying three dimensional arrays up to equivalence transformations a i j k i 1 m j 1 n k 1 t i j k a i j k u i i v j j w k k i 1 m j 1 n k 1 t in which u i i v j j w k k are nonsingular matrices this problem contains the problem of classifying an arbitrary system of tensors of order at most three

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159