Jordan (Super)Coalgebras and Lie (Super)Coalgebras

We discuss the question of local finite dimensionality of Jordan supercoalgebras. We establish a connection between Jordan and Lie supercoalgebras which is analogous to the Kantor–Koecher–Tits construction for ordinary Jordan superalgebras. We exhibit an example of a Jordan supercoalgebra which is not locally finite-dimensional. Show that, for a Jordan supercoalgebra (J,) with a dual algebra J ^*, there exists a Lie supercoalgebra (L ^c (J), _L ) whose dual algebra (L ^c (J))^* is the Lie KKT-superalgebra for the Jordan superalgebra J ^*. It is well known that some Jordan coalgebra J ⁰ can be constructed from an arbitrary Jordan algebra J. We find necessary and sufficient conditions for the coalgebra (L ^c (J ⁰),_L) to be isomorphic to the coalgebra (Loc(L _in (J)⁰), _L ⁰), where L _in (J) is the adjoint Lie KKT-algebra for the Jordan algebra J.     

Projections in Krein spaces

Let (H,J) be a Krein space with selfadjoint involution J. Starting with a canonical representation of a J-selfadjoint projection, J-projection in short, as the sum of a J-positive projection and a J-negative one we study in detail the structure of a regular subspace, that is, the range of a J-projection. We treat the problem when the sum of two regular subspaces is again regular. We also treat the problem when the closure of the range of the product of a J-contraction and a J-expansion becomes regular.     

Positive Twistor Bundle of a Kähler Surface

The aim of this paper is to characterize Kähler surfaces in terms oftheir positive twistor bundle. We prove that an oriented four-dimensionalRiemannian manifold (M, g) admits a complex structure J compatible with the orientation and such that (M, g, J is a Kähler manifold ifand only if the positive twistor bundle (Z ⁺(M), g _c ) admits a verticalKilling vector field.     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Shuffles of permutations and the Kronecker product

The Kronecker product of two homogeneous symmetric polynomialsP ₁,P ₂ is defined by means of the Frobenius map by the formulaP ₁oP ₂=F(F ⁻¹ P ₁)(F ⁻¹ P ₂). WhenP ₁ andP ₂ are the Schur functionsS _I ,S _J then the resulting productS _I oS _J is the Frobenius characteristic of the tensor product of the two representations corresponding to the diagramsI andJ. Taking the scalar product ofS _I oS _J with a third Schur functionsS _K gives the so called Kronecker coefficientc _I,J,K =<S _I oS _J ,S _K >. In recent work lascoux [7] and Gessel [3] have given what appear to be two separate combinatorial interpretations for thec _I,J,K in terms of some classes of permutations. In Lascoux's workI andJ are restricted to be hooks and in Gessel's both have to be zigzag partitions. In this paper we give a general result relating shuffles of permutations and Kronecker products. This leads us to a combinatorial interpretation of <S _I oS _J ,S _K > forS _I a product of homogeneous symmetric functions andJ, K unrestricted skew shapes. We also show how Gessel's and Lascoux's results are related and show how they can be derived from a special case of our result. Work supported by NSF grant at the University of California, San Diego.     

Beurling-Malliavin theory for Toeplitz kernels

We consider the family of Toeplitz operators T_J[`(S)]^aT_{J\bar{S}^{a}} acting in the Hardy space H ² in the upper halfplane; J and S are given meromorphic inner functions, and a is a real parameter. In the case where the argument of S has a power law type behavior on the real line, we compute the critical value

Weak convergence using higher-order cumulants

Denote byc _j (F) thejth cumulant (or semi-invariant) of the distribution functionF. We say thatF is specified by its higher-order cumulants if it is the unique distribution functionG having the following property: there exists a positive integerJ such thatc _j (G)=c _j (F) forj=1,2 andjJ. Let (F _n n1) be a sequence of distribution functions, and suppose that there existsJ such thatc _j (F _n )c _j (F) asn, forj=1,2 andjJ. It is proved thatF _n F so long asF is specified by its higher-order cumulants. It is an open problem to characterize the family of distributions which are specified by their higher-order cumulants.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

Choosing ther-dimension for the FCV family of clustering algorithms

A strategy is given for selecting the dimensionr of the linear variety which is used to define the criterion functionalJ _vrm and which determines the shape of the data clusters detected by the correspondingc-Varieties (FCV) clustering algorithms.     

On the existence of a crepant resolution of some moduli spaces of sheaves on an abelian surface

Let J be an abelian surface with a generic ample line bundle . For n≥1, the moduli space M_J(2,0,2n) of (1)-semistable sheaves F of rank 2 with Chern classes c₁(F)=0, c₂(F)=2n is a singular projective variety, endowed with a holomorphic symplectic structure on the smooth locus. In this paper, we show that there does not exist a crepant resolution of M_J(2,0,2n) for n≥2. This certainly implies that there is no symplectic desingularization of M_J(2,0,2n) for n≥2. Jaeyoo Choy was partially supported by KRF 2003-070-C00001 Young-Hoon Kiem was partially supported by a KOSEF grant R01-2003-000-11634-0.     

On local ergodic convergence of semi-groups and additive processes

We prove the local ergodic theorem inL _∞: Let {T _t}_t>0 be a strongly continuous semi-group of positive operators onL ₁. IfT _t is continuous at 0, then ɛ⁻¹∫ ₀ ^F T ₁ ^* f(x)dt→T ₀ ^* f(x) a.e., for everyf∈L _∞. The technique shows how to obtain theL _p local ergodic theorems from theL ₁-contraction case. It applies also to differentiation ofL p additive processes. Then-dimensional case, which is new, is proved by reduction to then-dimensionalL ₁-contraction case, solved by M. Akcoglu and A. del Junco. Research carried out during a sabbatical leave at the Ohio State University.     

Asymptotic Distribution of Estimated Affinity between Multiparameter Exponential Families

Let F ₁,...,F _J be the distributions of J independent multiparameter exponential families, and _J (F ₁,...,F _J ) denote the affinity between F ₁,...,F _J . We consider the problem of estimating _J on the basis of independent random samples from these distributions. Subject to some mild regularity conditions, we derive the asymptotic distribution of the maximum likelihood estimator of _J . Applications to hypothesis testing and discriminant analysis are discussed, and an example is provided.     

A global Newton method for the zeros of cylinder functions

The zeros of cylinder functions C _u (x)=cos α, J _u (x) - sin α, Y _u(x) coincide with those of the ratios H _u (x)=C _u (x)/C _u-1 (x) except, perhaps, at x = 0. We show monotonicity properties of H u(x) and f _u (x) = x ^2v-1 H u(x) and their derivatives for x > 0. We then build a Newton-Raphson iterative method based on the monotonic function f u(x) which is shown to be convergent, for any real values of u and α and any starting value x ₀ > 0, to an sth positive root c ,s of C _u (x) = 0, s being such that c _,s and x0 belong to the same interval (c _u-1 ,s', c _{u -1} ,s'+1]. We also show applications of the method. In particular, taking advantage of the fact that the ratio H _u (x) for first kind Bessel functions J u(x) can be evaluated by using a continued fraction, a very simple algorithm is built; it becomes especially efficient for low values of u and s and it allows the evaluation of the real zeros for arbitrary orders u, positive or negative. This revised version was published online in August 2006 with corrections to the Cover Date.     

Kurosh–Amitsur Right Jacobson Radicals of Type-1 and 2 for Right Near-Rings

Near-rings considered are right near-rings. Let ν ∈ {1, 2}. J ^r _ν , the right Jacobson radical of type-ν, was introduced for near-rings by the first and second authors. In this paper properties of these radicals J ^r _ν are studied. It is shown that J ^r _ν is a Kurosh-Amitsur radical (KA-radical) in the variety of all near-rings R in which the constant part R _c of R is an ideal of R. Thus, unlike the left Jacobson radical of type-1 of near-rings, J ^r ₁ is a KA-radical in the class of all zero-symmetric near-rings. J ^r _ν is not s-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings. Received: April 1, 2007. Revised: July 11, 2007.     

Magnetic bubble refraction and quasibreathers in inhomogeneous antiferromagnets

We study the dynamics of magnetic bubble solitons in a two-dimensional isotropic antiferromagnetic spin lattice in the case where the exchange integral J(x, y) is position dependent. In the near-continuum regime, this system is described by the relativistic O(3) sigma model on a space-time with a spatially inhomogeneous metric determined by J. We use the geodesic approximation to describe the low-energy soliton dynamics in this system: the n-soliton motion is approximated by geodesic motion in the moduli space M _n of static n-solitons equipped with the L ² metric γ. We obtain explicit formulas for γ for various natural choices of J(x, y). Based on these, we show that single soliton trajectories are refracted with J⁻¹ being analogous to the refractive index and that this refraction effect allows constructing simple bubble lenses and bubble guides. We consider the case where J has a disk inhomogeneity (with the value J ₊ outside a disk and J ₋ < J ₊ inside) in detail. We argue that for sufficiently large J ₊/J ₋, this type of antiferromagnet supports approximate quasibreathers: two or more coincident bubbles confined within the disk spin internally while their shape oscillates with a generically incommensurate period. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 152, No. 1, pp. 191–208, July, 2007.     

Projective Summands in Tensor Products of Simple Modules of Finite Dimensional Hopf Algebras

Abstract

Let H be a finite dimensional Hopf algebra over a field k. We show that H contains a unique maximal Hopf ideal J _w (H) contained in J(H), the Jacobson radical of H. We give various characterizations of J _w (H), for example J _w (H) = Ann _H ((H/J(H))^?n ) for all large enough n. The smallest positive integer n with this property is denoted by l _w (H). We prove that l _w (H) equals the smallest number n such that (H/J(H))^?n contains every projective indecomposable H/J _w (H)-module as a direct summand. This also equals the minimal n such that the tensor product of n suitable simple H-modules contains the projective cover of the trivial H/J _w (H)-module as a direct summand. We define projective homomorphisms between H-modules, which are used to obtain various reciprocity laws for tensor products of simple H-modules and their projective indecomposable direct summands. We also discuss some consequences of our general results in case H = kG is a group algebra of a finite group G and k is a field of characteristic p.     

Embedding of Jordan Copairs into Lie Coalgebras

Let (V, Δ) be a Jordan copair over a field Φ and let V? be its dual pair. Then there exists a Lie coalgebra (L ^c (V), Δ _L ) whose dual algebra (L ^c (V))? is the Kantor–Koecher–Tits construction for the pair V?. If Φ is a field of characteristic other than 2 or 3 then the Lie coalgebra (L ^c (J), Δ _L ) is locally finite-dimensional. As a corollary we derive that Jordan copairs over fields of characteristic other than 2 or 3 are locally finite-dimensional.     

On semi-infinite systems of linear inequalities

The ordered fieldR(M) consists of the realsR with a transcendentalM adjoined, which is larger than any realr ∈R. Given any semi-infinite matrix (s.i.m.) interpreted as linear inequalities:u ^tP_i≧c _i, ∀ _i ∈I, an arbitrary index set, it is also shown that the following are equivalent. (1) For every finiteJ ⊆I the systemu ^tP_i≧c _i,i ∈J is consistent, and (2) the s.i.m. has a solutionu ∈R(M) ⁿ. Some consequences for “duality gaps” are also given. These results were obtained as part of the activities of the Management Science Research Group and School of Urban and Public Affairs, Carnegie-Mellon University.     

Codimensions of products and of intersections of verbally primeT-ideals

By Kemer’s theory [9],T idealsJ ₁ ∪…∪J _r andJ ₁ …J _r, where eachJ _i is verbally prime, are of fundamental importance in the theory of P.I. algebras. We calculate, approximately and asymptotically, the codimensions of suchT-ideals, thereby extending the corresponding results about matrix algebras. In all such cases, the exponential growth of the codimensions is calculated; in particular, it is always an integer. Partially supported by NSF grant DMS 9303230. Partially supported by NSF grant DMS 9101488.