Tensor Product Of Spaces Research Articles

К тензорному анализу решения задачи реализации билинейной системы второго порядка с опозданием

Определены аналитические условия (необходимые/достаточные) решения задачи дифференциальной реализации континуального пучка управляемых траекторных кривых в классе билинейных неавтономных обычных дифференциальных уравнений (с опозданием и без) второго порядка в материальном сепарабельном гильбертовом пространстве. Эта задача относится к типу обратных задач для аддитивной комбинации нестационарных линейных и билинейных операторов эволюционных уравнений в бесконечномерном гильбертовом пространстве. Метамовой данной теории служат конструкции тензорных произведений гильбертовых пространств, структуры решеток с ортодополнением и функциональный аппарат нелинейного оператора Релея-Ритца. При этом показано, что при конечном пучке траекторий наличие свойств типа сублинейности данного оператора позволяет получить достаточные условия для существования таких реализаций. Попутно обосновываются тополого-метрические условия непрерывности проектизации нелинейного функционального опера-тора Релея–Ритца с вычислением фундаментальной группы его образа. Полученные результаты побуждают к развитию теории нелинейной структурной идентификации полилинейных дифференциальных моделей высших порядков (например, для моделирования многоканальных нейроимплантов типа Neuralink).

A direct mixed–enriched Galerkin method on quadrilaterals for two-phase Darcy flow

We develop a locally conservative, finite element method for simulation of two-phase flow on quadrilateral meshes that minimize the number of degrees of freedom (DoFs) subject to accuracy requirements and the DoF continuity constraints. We use a mixed finite element method (MFEM) for the flow problem and an enriched Galerkin method (EG) for the transport, stabilized with an entropy viscosity. Standard elements for MFEM lose accuracy on quadrilaterals, so we use the newly developed AC elements which have our desired properties. Standard tensor product spaces used in EG have many excess DoFs, so we would like to use the minimal DoF serendipity elements. However, the standard elements lose accuracy on quadrilaterals, so we use the newly developed direct serendipity elements. We use the Hoteit-Firoozabadi formulation, which requires a capillary flux. We compute this in a novel way that does not break down when one of the saturations degenerate to its residual value. Extension to three dimensions is described. Numerical tests show that accurate results are obtained.

Quantum-Theoretic Modeling in Computer Science

We work out a quantum-theoretic model in complex Hilbert space of a recently performed test on co-occurrencies of two concepts and their combination in retrieval processes on specific corpuses of documents. The test violated the Clauser-Horne-Shimony-Holt version of the Bell inequalities ('CHSH inequality'), thus indicating the presence of entanglement between the combined concepts. We make use of a recently elaborated 'entanglement scheme' and represent the collected data in the tensor product of Hilbert spaces of the individual concepts, showing that the identified violation is due to the occurrence of a strong form of entanglement, involving both states and measurements and reflecting the meaning connection between the component concepts. These results provide a significant confirmation of the presence of quantum structures in corpuses of documents, like it is the case for the entanglement identified in human cognition.

Approximation in FEM, DG and IGA: a theoretical comparison

In this paper we compare the approximation properties of degree p spline spaces with different numbers of continuous derivatives. We prove that, for a given space dimension, $$\mathcal {C}^{p-1}$$ splines provide better a priori error bounds for the approximation of functions in $$H^{p+1}(0,1)$$ . Our result holds for all practically interesting cases when comparing $$\mathcal {C}^{p-1}$$ splines with $$\mathcal {C}^{-1}$$ (discontinuous) splines. When comparing $$\mathcal {C}^{p-1}$$ splines with $$\mathcal {C}^{0}$$ splines our proof covers almost all cases for $$p\ge 3$$ , but we can not conclude anything for $$p=2$$ . The results are generalized to the approximation of functions in $$H^{q+1}(0,1)$$ for $$q<p$$ , to broken Sobolev spaces and to tensor product spaces.

Iterative low-rank approximation solvers for the extension method for fractional diffusion

We consider the numerical method for fractional diffusion problems which is based on an extension to a mixed boundary value problem for a local operator in a higher dimensional space. We observe that, when this problem is discretized using tensor product spaces as is commonly done, the solution can be very well approximated by low-rank tensors; we provide some analysis to support this observation. This fact motivates us to apply iterative low-rank approximation algorithms in order to efficiently solve this extended problem. In particular, we employ a recently proposed greedy Tucker approximation method as well as a more classical greedy rank one update method. Throughout, all objects of interest are kept in suitable low-rank approximations, which dramatically reduces the required amount of memory compared to the full formulation of the extended problem.Our approach can be used for general, non-structured space discretizations. If the space discretization itself has tensor product structure, we can further decompose the problem in order to deal with even lower-dimensional objects. We also note that the approach can be directly applied to higher-order discretizations both in space and the extended variable.A further contribution of our work is a rank one version of a diagonal preconditioner which can mitigate the severe ill-conditioning of the extended problem arising due to mesh grading and singularity of the coefficient. This preconditioner can be realized using a simple pre- and postprocessing step and does not affect the computational effort of our algorithm.In several numerical examples, we demonstrate the convergence behavior of the proposed methods. In particular, the Tucker approximation approach requires only a few iterations in order to reach the discretization error in all tested settings.

Completability and optimal factorization norms in tensor products of Banach function spaces

Given $$\sigma $$-finite measure spaces $$(\Omega _1,\Sigma _1, \mu _1)$$ and $$(\Omega _2,\Sigma _2,\mu _2)$$, we consider Banach spaces $$X_1(\mu _1)$$ and $$X_2(\mu _2)$$, consisting of $$L^0 (\mu _1)$$ and $$L^0 (\mu _2)$$ measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product $$X_1(\mu _1) \otimes _\pi X_2(\mu _2)$$ is continuously included in the metric space of measurable functions $$L^0(\mu _1 \otimes \mu _2)$$. In particular, we prove that the elements of the completion of the projective tensor product of $$L^p$$-spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally show that given a bounded linear operator $$T:X_1(\mu _1) \otimes _\pi X_2(\mu _2) \rightarrow E$$ (where E is a Banach space), a norm can be found for T to be bounded, which is ‘minimal’ with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.

The joint distribution of the marginals of multipartite random quantum states

We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quantum information theory.

Hamiltonian models of interacting fermion fields in quantum field theory

We consider Hamiltonian models representing an arbitrary number of spin 1 / 2 fermion quantum fields interacting through arbitrary processes of creation or annihilation of particles. The fields may be massive or massless. The interaction form factors are supposed to satisfy some regularity conditions in both position and momentum space. Without any restriction on the strength of the interaction, we prove that the Hamiltonian identifies to a self-adjoint operator on a tensor product of antisymmetric Fock spaces and we establish the existence of a ground state. Our results rely on new interpolated \(N_\tau \) estimates. They apply to models arising from the Fermi theory of weak interactions, with ultraviolet and spatial cutoffs.

Sequential product spaces are Jordan algebras

We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem, these spaces therefore correspond to Euclidean Jordan algebras. We remark on the significance of this result in the context of reconstructions of quantum theory. In particular, we show that sequential product spaces must be C*-algebras when their vector space tensor product is also a sequential product space (in the parlance of operational theories, when the space “allows a local composite”). We also show that sequential product spaces in infinite dimension correspond to JB-algebras when a few additional conditions are satisfied. Finally, we remark on how changing the axioms of the sequential product might lead to a new characterization of homogeneous cones.

Embeddings for infinite-dimensional integration and [formula omitted]-approximation with increasing smoothness

We study integration and L2-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

Remotal sets in tensor product spaces and ε-remotality

Remotal sets in tensor product spaces and ε-remotality

On the Differential Realization of a Second-Order Bilinear System in a Hilbert Space

We study the necessary and sufficient conditions for the existence of a nonlinear differential realization of a continuous infinite-dimensional behaviorist system in the class of nonstationary second-order bilinear ordinary differential (in particular, hyperbolic) equations in a separable Hilbert space. The obtained conditions rely upon the tensor products of Hilbert spaces. In passing, we analytically justify some topological-metrical continuity conditions for the projectivization of the Rayleigh—Ritz operator with the calculation of the fundamental group of its image.

A relation for the Jones–Wenzl projector and tensor space representations of the Temperley–Lieb algebra

ABSTRACT A relation for the Jones–Wenzl projector is proven. It has the following consequence for representations of the Temperley–Lieb algebra on tensor product spaces: if such a representation is built from a Hermitian matrix T of rank r such that , then either and or . For the latter class of representations, new examples are found. This includes explicit examples for r=2,3,4 and any (with one exception) and a solution for n=r+1 with arbitrary r.

Injective tensor products of tree spaces

We study tensor products on tree spaces; in particular, we give necessary and sufficient conditions for the $n$-fold injective tensor product of tree spaces to contain a copy of $\ell _1$.

Remotal sets in tensor product and function spaces

Remotal sets in tensor product and function spaces

A new duality via the Haagerup tensor product

We initiate the study of a new notion of duality defined with respect to the module Haagerup tensor product. This notion not only recovers the standard operator space dual for Hilbert C ⁎ -modules, it also captures quantum group duality in a fundamental way. We compute the so-called Haagerup dual for various operator algebras arising from ℓ p spaces. In particular, we show that the dual of ℓ 1 under any operator space structure is min ⁡ ℓ ∞ . In the setting of abstract harmonic analysis we generalize a result of Varopolous by showing that C ( G ) is an operator algebra under convolution for any compact Kac algebra G . We then prove that the corresponding Haagerup dual C ( G ) h = ℓ ∞ ( G ˆ ) , whenever G ˆ is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.

A simple counting argument of the irreducible representations of $$\mathsf {SU}(N)$$ on mixed product spaces

It is a well known result that the number of irreducible representations of SU(N) on a tensor product containing k factors of a vector space V is given by the number of involutions in the symmetric group on k letters. In this paper, we present an alternative proof for this fact using a basis of projection and transition operators of the algebra of invariants of SU(N). This proof easily generalizes to the irreducible representations of SU(N) on mixed tensor product spaces (consisting of factors of V as well as its dual space). This implies that the number of irreducible representation of SU(N) on such a space remains unchanged if one exchanges factors V for its dual and vice versa, as long as the total number of factors remains unchanged.

Operator space tensor products and inductive limits

We investigate whether the λ -tensor product of operator spaces which includes the operator space projective, Haagerup and Schur tensor products, and the λ -tensor product of operator systems commute with inductive limits, in the sense that if ( X n , ϕ n ) is an inductive system of operator spaces (or operator systems) with an inductive limit X , and if Y is any operator space (or operator system), then whether ( X n ⨂ λ Y , ϕ n ⊗ id ) becomes an inductive system, and if so, whether the inductive limit of this system is same as X ⨂ λ Y , where ⨂ λ denotes the λ -tensor product of operator spaces (or operator systems). We also discuss about the bidual of such an inductive limit and obtain embeddings lim → ( X n ⁎ ⁎ ⨂ λ Y ⁎ ⁎ ) ⊆ ( lim → X n ⨂ λ Y ) ⁎ ⁎ under certain restrictions.

The Algebraic Structure of Quantity Calculus

Abstract The algebraic structure underlying the quantity calculus is defined axiomatically as an algebraic fiber bundle, that is, a base structure which is a free Abelian group together with fibers which are one dimensional vector spaces, all of them bound by algebraic restrictions. Subspaces, tensor product, and quotient spaces are considered, as well as homomorphisms to end with a classification theorem of these structures. The new structure provides an axiomatic foundation of quantity calculus which is centered on the concept of dimension, rather than on the concept of unit, which is regarded as secondary, and uses only integer exponents of the dimensions.

The Gibbs Paradox.

The Gibbs Paradox is essentially a set of open questions as to how sameness of gases or fluids (or masses, more generally) are to be treated in thermodynamics and statistical mechanics. They have a variety of answers, some restricted to quantum theory (there is no classical solution), some to classical theory (the quantum case is different). The solution offered here applies to both in equal measure, and is based on the concept of particle indistinguishability (in the classical case, Gibbs’ notion of ‘generic phase’). Correctly understood, it is the elimination of sequence position as a labelling device, where sequences enter at the level of the tensor (or Cartesian) product of one-particle state spaces. In both cases it amounts to passing to the quotient space under permutations. ‘Distinguishability’, in the sense in which it is usually used in classical statistical mechanics, is a mathematically convenient, but physically muddled, fiction.