Tensor Product Of Spaces Research Articles

Numerical solution to a linear equation with tensor product structure

SummaryWe consider the numerical solution of a c‐stable linear equation in the tensor product space , arising from a discretized elliptic partial differential equation in . Utilizing the stability, we produce an equivalent d‐stable generalized Stein‐like equation, which can be solved iteratively. For large‐scale problems defined by sparse and structured matrices, the methods can be modified for further efficiency, producing algorithms of computational complexity, under appropriate assumptions (with ns being the flop count for solving a linear system associated with ). Illustrative numerical examples will be presented.

A new method for reconstruction of cross-sections using Tucker decomposition

The full representation of a d-variate function requires exponentially storage size as a function of dimension d and high computational cost. In order to reduce these complexities, function approximation methods (called reconstruction in our context) are proposed, such as: interpolation, approximation, etc. The traditional interpolation model like the multilinear one, has this dimensionality problem. To deal with this problem, we propose a new model based on the Tucker format – a low-rank tensor approximation method, called here the Tucker decomposition. The Tucker decomposition is built as a tensor product of one-dimensional spaces where their one-variate basis functions are constructed by an extension of the Karhunen–Loève decomposition into high-dimensional space. Using this technique, we can acquire, direction by direction, the most important information of the function and convert it into a small number of basis functions. Hence, the approximation for a given function needs less data than that of the multilinear model. Results of a test case on the neutron cross-section reconstruction demonstrate that the Tucker decomposition achieves a better accuracy while using less data than the multilinear interpolation.

Boundary Hilbert spaces and trace operators

We discuss the introduction of boundary Hilbert spaces for a class of physical systems for which it is not possible to factor their state spaces as tensor products of Hilbert spaces naturally associated to their boundaries and bulks, respectively. In order to do this we make use of the so called trace operators that play a relevant role in the analysis of PDEs in bounded regions. By taking advantage of these operators and some functorial aspects of the construction of Fock spaces, we will show how to obtain quantum dynamics at the boundaries defined in appropriate Hilbert spaces associated with them.

HL-homotopy of handlebody-links and Milnor's invariants

A handlebody-link is a disjoint union of embeddings of handlebodies in S3 and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this paper, we construct a family of invariants for HL-homotopy classes of general handlebody-links, by using Milnor's μ‾-invariants. Moreover, we give a bijection between the set of HL-homotopy classes of almost trivial handlebody-links and tensor product space modulo some general linear actions, especially for 3- or more component handlebody-links. Through this bijection we construct invariants of HL-homotopy classes which can be used to distinguish the classes.

Derivation of matrix product states for the Heisenberg spin chain with open boundary conditions.

Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-1/2) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.

Covariance structure of parabolic stochastic partial differential equations with multiplicative Lévy noise

The characterization of the covariance function of the solution process to a stochastic partial differential equation is considered in the parabolic case with multiplicative Lévy noise of affine type. For the second moment of the mild solution, a well-posed deterministic space–time variational problem posed on projective and injective tensor product spaces is derived, which subsequently leads to a deterministic equation for the covariance function.

Octahedral norms in tensor products of Banach spaces

We continue the investigation of the behaviour of octahedral norms in tensor products of Banach spaces. First, we will prove the existence of a Banach space Y such that the injective tensor products l1⊗^εY and L1⊗^εY both fail to have an octahedral norm, which solves two open problems from the literature. Secondly, we will show that in the presence of the metric approximation property octahedrality is preserved from a non-reflexive L-embedded Banach space taking projective tensor products with an arbitrary Banach space.

A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen–Loeve expansion of Gaussian random fields. We develop, validate, and discuss an algorithm based on the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor-product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals, allowing the implementation of a fast and efficient algorithm that provides considerable reduction in CPU time, when compared with classical Galerkin methods. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We illustrate the numerical solution of a diffusion problem with random input data with the present method. In this problem, accuracy was retained after dropping the coefficients below a threshold value that was numerically determined. A similar method with scaling functions rather than wavelet functions does not need a discrete wavelet transform and leads to eigenvalue systems with better conditioning but lower sparsity.

ℓ1-norm and entanglement in screening out braiding from Yang–Baxter equation associated with [formula omitted] parafermion

The relationships between quantum entangled states and braid matrices have been well studied in recent years. However, most of the results are based on qubits. In this paper, we investigate the applications of 2-qutrit entanglement in the braiding associated with Z3 parafermion. The 2-qutrit entangled state |Ψ(θ)〉, generated by the action of the localized unitary solution R˘(θ) of YBE on 2-qutrit natural basis, achieves its maximal ℓ1-norm and maximal von Neumann entropy simultaneously at θ=π/3. Meanwhile, at θ=π/3, the solutions of YBE reduces braid matrices, which implies the role of ℓ1-norm and entropy plays in determining real physical quantities. On the other hand, we give a new realization of 4-anyon topological basis by qutrit entangled states, then the 9×9 localized braid representation in 4-qutrit tensor product space (C3)⊗4 is reduced to Jones representation of braiding in the 4-anyon topological basis. Hence, we conclude that the entangled states are powerful tools in analysing the characteristics of braiding and R˘-matrix.

Some fixed point theorems in 2-Banach spaces and 2-normed tensor product spaces

Some fixed point theorems in 2-Banach spaces and 2-normed tensor product spaces

Additive decomposability preservers and related results on tensor products of matrices

In this paper we characterize additive maps between tensor spaces that send decomposable tensors to decomposable tensors. As an application, we classify all additive maps from tensor products of spaces of rectangular matrices to spaces of rectangular matrices which do not increase the rank of tensor product of rank one matrices.

Error Estimates for Multivariate Regression on Discretized Function Spaces

In this paper, we will discuss the discretization error for the regression setting and derive error bounds relying on the approximation properties of the discretized space. Furthermore, we will point out how the sampling error and the discretization error interact and how they can be balanced appropriately. We will present two examples based on tensor product spaces (sparse grids, hyperbolic crosses) which provide a suitable approach in the case of large sample sets in moderate dimensions.

Projective tensor product of proto-quantum spaces

A proto-quantum space is a (general) matricially normed space in the sense of Effros and Ruan presented in a ‘matrix-free’ language. We show that these spaces have a special (projective) tensor product possessing the universal property with respect to com

Tensor-splitting properties of $n$-inverse pairs of operators

In this paper we study n-inverse pairs of operators on the tensor product of Banach spaces. In particular we show that an n-inverse pair of elementary tensors of operators on the tensor product of two Banach spaces can arise only from l- and m-inverse pairs of operators on the individual spaces. This gives a converse to a result of Duggal and Muller, and proves a conjecture of the second named author. Our proof uses techniques from algebraic geometry, which generalize to other relations among operators in a tensor product. We apply this theory to obtain results for n-symmetries in a tensor product as well.

Concerning the Szlenk index

We discuss pruning and coloring lemmas on regular families. We discuss several applications of these lemmas to computing the Szlenk index of certain $w^*$ compact subsets of the dual of a separable Banach space. Applications include estimates of the Szlenk index of Minkowski sums, infinite direct sums of separable Banach spaces, constant reduction, and three space properties. We also consider using regular families to construct Banach spaces with prescribed Szlenk index. As a consequence, we give a characterization of which countable ordinals occur as the Szlenk index of a Banach space, prove the optimality of a previous universality result, and compute the Szlenk index of the injective tensor product of separable Banach spaces.

Factorization of (q,p)-summing polynomials through Lorentz spaces

We present a vector valued duality between factorable (q,p)-summing polynomials and (q,p)-summing linear operators on symmetric tensor products of Banach spaces. Several applications are provided. First, we prove a polynomial characterization of cotype of Banach spaces. We also give a variant of Pisier's factorization through Lorentz spaces of factorable (q,p)-summing polynomials from C(K)-spaces. Finally, we show a coincidence result for (q,p)-concave polynomials.

The Szlenk power type and tensor products of Banach spaces

We prove a formula for the Szlenk power type of the injective tensor product of Banach spaces with Szlenk index at most ω \omega . We also show that the Szlenk power type as well as summability of the Szlenk index are separably determined, and we extend some of our recent results concerning direct sums.

Diagonals of injective tensor products of Banach lattices with bases

Let E be a Banach lattice with a 1-unconditional basis \(\{e_i: i \in \mathbb {N}\}\). Denote by \(\Delta (\check{\otimes }_{n,\epsilon }E)\) (resp. \(\Delta (\check{\otimes }_{n,s,\epsilon }E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach space tensor product, and denote by \(\Delta (\check{\otimes }_{n,|\epsilon |}E)\) (resp. \(\Delta (\check{\otimes }_{n,s,|\epsilon |}E)\)) the main diagonal space of the n-fold full (resp. symmetric) injective Banach lattice tensor product. We show that these four main diagonal spaces are pairwise isometrically isomorphic. We also show that the tensor diagonal \(\{e_i\otimes \cdots \otimes e_i: i \in \mathbb {N}\}\) is a 1-unconditional basic sequence in both \(\check{\otimes }_{n,\epsilon }E\) and \(\check{\otimes }_{n,s,\epsilon }E\).

SOME FIXED POINT THEOREMS ON THE SUM AND PRODUCT OF OPERATORS IN TENSOR PRODUCT SPACES

Let X and Y be Banach spaces and P and Q be two subsets of X and Y respectively. Let T1 : X ⊗ Y → X and T2 : X ⊗ Y → Y be two mappings and S be a self mapping on P⊗Q. Using T1 and T2 we define a self mapping T on X⊗Y. Different conditions under which T +TS+S has a fixed point in P ⊗Q are established here. Analogous results are also established taking the pair (T1,T2) as (k,k / ) contraction mappings. Again considering X ⊗ Y as a reflexive Banach space. We derive the conditions for 1 m (T + ST + S), m > 2, m ∈ N, for having a fixed point in P ⊗ Q. Some iteration schemes converging to a fixed point of T + ST + S in P ⊗ Q are also presented here.

⁎-freeness in finite tensor products

In this paper, we consider the following question and variants thereof: given D:=(a1;i⊗⋯⊗aK;i:i∈I), a collection of elementary tensor non-commutative random variables in the tensor product of probability spaces (A1⊗⋯⊗AK,φ1⊗⋯⊗φK), when is D ⁎-free? (See Section 1.2 for a precise formulation of this problem.)Settling whether or not freeness occurs in tensor products is a recurring problem in operator algebras, and the following two examples provide a natural motivation for the above question:(A)If (a1;i:i∈I) is a ⁎-free family of Haar unitary variables and ak,i are arbitrary unitary variables for k≥2, then the ⁎-freeness persists at the level of the tensor product D.(B)A converse of (A) holds true if all variables ak;i are group-like elements (see Corollary 1.7 of Proposition 1.6).It is therefore natural to seek to understand the extent to which such simple characterizations hold true in more general cases. While our results fall short of a complete characterization, we make notable steps toward identifying necessary and sufficient conditions for the freeness of D. For example, we show that under evident assumptions, if more than one family (ak,i:i∈I) contains non-unitary variables, then the tensor family fails to be ⁎-free (see Theorem 1.8 (1)).