Tensor product of operads and iterated loop spaces

A algebraic characterization of an n-fold loop space in terms of its n different 1-fold loop structures is established. This amounts to describing the higher homotopy commutativity for such a space as a strict partial commutativity of the 1-fold loop structures. The tensor product of operads (a special case of the construction for algebraic theories) is ideally suited for this. In particular we show that the operad of little n-cubes C_n is homotopy equivalent to the n-fold tensor product Cⁿ₁, i.e., ‘tensoring these A_∞-structures yields an iterated loop structure’. This is not true for arbitrary A_∞-operads.     

Partial inner product spaces and semi-inner product spaces

A comparison is made between the two objects mentioned in the title. Connections between them are threefold: (i) both are particular instances of dual pairs of locally convex spaces; (ii) many partial inner product spaces consist of chains or lattices of semi-inner product spaces; (iii) the basic structure behind both of them is that of Galois connections. A number of common open problems are described.     

Tensor invariants and homogeneous Riemann spaces

Homogeneous Riemann spaces M = $ {\mathcal p} $ /? with an irreducible isotropy group are considered. On such spaces dimensions are calculated for spaces of $ {\mathcal p} $ -invariant tensor fields of valence 2, 3, 4. In particular, consideration is given to homogeneous spaces whose isotropy groups are given by the diagrams given in Table 8.     

Tensor product q-Bernstein polynomials

An affine de Casteljau type algorithm to compute q-Bernstein Bézier curves is introduced and its intermediate points are obtained explicitly in two ways. Furthermore we define a tensor product patch, based on this algorithm, depending on two parameters. Degree elevation procedure is studied. The matrix representation of tensor product patch is given and we find the transformation matrix between a classical tensor product Bézier patch and a tensor product q-Bernstein Bézier patch. Finally, q-Bernstein polynomials B _n,m (f;x,y) for a function f(x,y), (x,y)∈[0,1]×[0,1] are defined and fundamental properties are discussed. AMS subject classification (2000)  65D17     

Schatten properties,nuclearity and minimality of phase shift invariant spaces

We extend Feichtinger's minimality property on the smallest non-trivial time-frequency shift invariant Banach space, to the quasi-Banach case. Analogous properties are deduced for certain matrix spaces.We use these results to prove that the pseudo-differential operator $\mathrm{Op}(a)$ is a Schatten-q operator from $M^{\infty }$ to $M^p$ and r-nuclear operator from $M^{\infty }$ to $M^r$ when $a\in M^r$ for suitable p, q and r in $(0,\infty ]$.     

Tensor product of partially-additive monoids

Partially-additive monoids (pams) were introduced by Arbib and Manes in order to provide an algebraic semantics for programming languages. In this paper, we prove that the categoryP a m of pams and additive maps is a closed category whose monoids are partially-additive semirings. We follow the tensor product construction of R. Guitart [7] for categories of algebras which generalize the case of modules. Nevertheless, the problem here is more difficult owing to the fact that pams are partial algebras rather than algebras. Thus, we have to make some modifications to Guitart’s approach.     

Tensor product of C-injective modules

The purpose of this paper is to calculate all the character tables of Hecke algebras associated with finite Chevalley groups of exceptional type and their maximal parabolic subgroups when they are commutative. In the case when the groups are of classical type, the character values of Hecke algebras are expressed by using the q-Krawtchouk polynomials and the q-Hahn polynomials (See [10] and [15]). On the other hand, the character tables of commutative Hecke algebras associated with exceptional Weyl groups and their maximal parabolic subgroups are given in [12]. In §1, we discuss the structure of Hecke algebras and in §2, we calculate all the character tables of these commutative Hecke algebras associated with finite Chevalley groups of exceptional type. Although some of them are well known, we include them for completeness     

Tensor products and spaces of vector-valued continuous functions

Quasibarrelled, barrelled and bornological tensor products of locally convex spaces are studied. A device, called the desintegration lemma, is developed for the most difficult case, that of the injective topology. Applications are given to spaces of vector-valued continuous functions.Both authors thank the Volkswagenstiftung and the A. von Humboldt Stiftung. The second author is a research associate of the Belgian National Fund for Scientific Research N.F.W.O.     

Tensor product of quaternion hilbert modules

One of the main problems in the theory of quaternion quantum mechanics has been the construction of a tensor product of quaternion Hilbert modules. A solution to this problem is given by studying the tensor product of quaternion algebras (over the reals) and some of its quotient modules. Real, complex, and (covariant) quaternion scalar products are found in the tensor product spaces. Annihilationcreation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The gauge transformations of a tensor product vector and the gauge fields are studied.On Sabbatical leave from the School of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel. Work supported in part by a fellowship from the Ambrose Monell Foundation.     

Criteria for the nuclearity of spaces of functions of infinitely many variables

The present paper evolves from Berezanskii and Gali (Ukrainian Math. J.24 (4) (1972), 435–464) and Berezanskii, Gali, and Zuk (Soviet. Math. Dokl.13 (2) (1972)), in which, it was shown how one can construct a weighted infinite tensor product H_e,δ = ?_{n = 1:e,δ}^∞ = H_τn of Hilbert spaces H_τn with a given stabilizing sequence δ = (δ_n)_{n = 1}^∞(δ_n > 0). Here a weighted infinite tensor product ?_e = ?_{n = 1,e}^∞?_n of nuclear spaces ?_n is established first. Criteria for nuclearity of the constructed spaces are also given. Some examples of nuclear spaces of functions of infinite many variables K(T^∞) and A(T^∞) are obtained.     

Tensor product and property (b)

Periodica Mathematica Hungarica - A Banach space operator satisfies property (b) if the complement of its essential Weyl approximate point spectrum in its approximate point spectrum is the set of...     

Tensor product stabilization in Kac-Moody algebras

We consider a large class of series of symmetrizable Kac-Moody algebras (generically denoted X_n). This includes the classical series A_n as well as others like E_n whose members are of Indefinite type. The focus is to analyze the behavior of representations in the limit n→∞. Motivated by the classical theory of A_n=sl_n+1C, we consider tensor product decompositions of irreducible highest weight representations of X_n and study how these vary with n. The notion of “double-headed” dominant weights is introduced. For such weights, we show that tensor product decompositions in X_n do stabilize, generalizing the classical results for A_n. The main tool used is Littelmann's celebrated path model. One can also use the stable multiplicities as structure constants to define a multiplication operation on a suitable space. We define this so-called stable representation ring and show that the multiplication operation is associative.     

Tensor product methods for stochastic problems

In the solution of stochastic partial differential equations (SPDEs) the generally already large dimension N of the algebraic system resulting from the spatial part of the problem is blown up by the huge number of degrees of freedom P coming from the stochastic part. The number of degrees of freedom of the full system will be NP, which poses severe demands on memory and processor time. We present a method how to approximate the system by a data-sparse tensor product (based on the Karhunen-Loève decomposition with M terms), which uses only memory in the order of M (N + P), and how to keep this representation also inside the iterative solvers. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)