Search

Given Banach space operators A, B, let δ A , B denote the generalised derivation δ (X) = (L A - R B) (X) = A X - X B and ▵ A , B the length two elementary operator ▵ A , B (X) = (I - L A R B) (X) = X - A X B . This note considers the structure of m-symmetric operators δ ▵ A 1 , B 1 , ▵ A 2 , B 2 m (I) = (L ▵ A 1 , B 1 - R ▵ A 2 , B 2 ) m (I) = 0 . It is seen that there exist scalars λ i ∈ σ a (B 1) , 1 ≤ i ≤ 2 , such that δ λ 1 A 1 , λ 2 A 2 m (I) = 0 . Translated to Hilbert space operators A and B this implies that if δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 , then there exists λ ¯ ∈ σ a (B ∗) such that δ (λ A) ∗ , λ A m (I) = 0 = δ λ ¯ B , λ B ∗ m (I) . We prove that the operator δ ▵ A ∗ , B ∗ , ▵ A , B m is compact if and only if (i) there exists a real number α and finite sequnces (i) { a j } j = 1 n ⊆ σ (A) , { b j } j = 1 n ⊆ σ (B) such that a j b j = 1 - α , 1 ≤ j ≤ n ; (ii) decompositions ⊕ j = 1 n H j and ⊕ j = 1 n H J of H such that ⊕ j = 1 n (A - a j I) | H j and ⊕ j = 1 n (B - b j I) | H j are nilpotent. If δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 implies δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 , then A and B satisfy a (Putnam-Fuglede type) commutativity theorem; conversely, a sufficient condition for δ ▵ A ∗ , B ∗ , ▵ A , B m (I) = 0 to imply δ ▵ A ∗ , B ∗ , ▵ A , B (I) = 0 is that λ A and λ ¯ B satisfy the commutativity property for scalars lambda ¯ ∈ σ a (B ∗) . An analogous result is seen to hold for the operators ▵ δ A ∗ , B ∗ , δ A , B m and ▵ δ A ∗ , B ∗ , δ A , B m (I) . Perturbation by commuting nilpotents is considered. [ABSTRACT FROM AUTHOR]

Let sl (2) ⋉ h n , n ≥ 1 , be the Galilean Lie algebra over a field of characteristic zero, here h n is the Heisenberg Lie algebra of dimension 2 n + 1 , and sl (2) acts on h n so that, sl (2) -modules, h n ≃ V (2 n - 1) ⊕ V (0) (here V(k) denotes the irreducible sl (2) -module of highest weight k). In this paper, we study the tensor product of two uniserial representations of sl (2) ⋉ h n . We obtain the sl (2) -module structure of the socle of V ⊗ W and we describe the space of intertwining operators Hom sl (2) ⋉ h n (V , W) , where V and W are uniserial representations of sl (2) ⋉ h n . The structure of the radical of V ⊗ W follows from that of the socle of V ∗ ⊗ W ∗ . The result is subtle and shows how difficult is to obtain the whole socle series of arbitrary tensor products of uniserials. In contrast to the serial associative case, our results for sl (2) ⋉ h n reveal that these tensor products are far from being a direct sum of uniserials; in particular, there are cases in which the tensor product of two uniserial (sl (2) ⋉ h n) -modules is indecomposable but not uniserial. Recall that a foundational result of T. Nakayama states that every finitely generated module over a serial associative algebra is a direct sum of uniserial modules. This article extends a previous work in which we obtained the corresponding results for the Lie algebra sl (2) ⋉ a m where a m is the abelian Lie algebra of dimension m + 1 and sl (2) acts so that a m ≃ V (m) as sl (2) -modules. [ABSTRACT FROM AUTHOR]

This paper characterize important concepts such as the Riesz bases and pair of dual frames for tensor product of Hilbert spaces and introduce some special properties of pair of dual frames in these spaces. We show that Riesz bases and pair of dual frames in H ⊗ K are preserved under invertible and unitary operators on H . Finally, We present a representation for Hilbert–Schmidt operators by a pair of dual frames. [ABSTRACT FROM AUTHOR]

We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin's condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra A and a compact space X, let A be a Banach A-valued function algebra on X and let A be the subalgebra of A consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra A and the associated algebras A and A. That A inherits a certain regularity condition P to A and A is the easy part of the problem. We investigate the converse and show that, under certain conditions, A receives P form A and A. The results apply to tensor products of commutative Banach algebras as they are included in the class of vector-valued function algebras. [ABSTRACT FROM AUTHOR]

The notion of Cauchy dual for left-invertible covariant representations was studied by Trivedi and Veerabathiran. Using the Moore-Penrose inverse, we extend this notion for the covariant representations having closed range and explore several useful properties. We obtain a Wold-type decomposition for regular completely bounded covariant representation whose Moore-Penrose inverse is regular. Also, we discuss an example related to the non-commutative bilateral weighted shift. We prove that the Cauchy dual of the concave covariant representation (σ , V) modulo N (V ~) is hyponormal modulo N (V ~) . [ABSTRACT FROM AUTHOR]

A m-quasi class A k , denoted as M ∈ Q (A k , m) , is defined as an operator M acting on a complex Hilbert space H , provided that the following condition holds true: M ∗ m | M k + 1 | 2 k + 1 M m ≥ M ∗ m | M | 2 M m , where both k and m are positive integers. In this research, we unveil fundamental structural characteristics of these operators. Leveraging these findings, we can establish that P is self-adjoint if P represents the Riesz idempotent associated with the isolated point λ in the spectrum of M ∈ Q (A k , m) . Additionally, we provide a necessary and sufficient condition for M ⊗ N to belong to Q A k when both M and N are non-zero. [ABSTRACT FROM AUTHOR]

Using the Wold–von Neumann decomposition for the isometric covariant representations due to Muhly and Solel, we prove an explicit representation of the commutant of a doubly commuting pure isometric representation of the product system over N 0 k. As an application we study a complete characterization of invariant subspaces for a doubly commuting pure isometric representation of the product system. This provides us a complete set of isomorphic invariants. Finally, we classify a large class of an isometric covariant representations of the product system. [ABSTRACT FROM AUTHOR]

We study projective and injective tensor products of Banach L 0 -modules over a σ -finite measure space. En route, we extend to Banach L 0 -modules several technical tools of independent interest, such as quotient operators, summable families, and Schauder bases. [ABSTRACT FROM AUTHOR]

We introduce a simple algorithm that efficiently computes tensor products of Pauli matrices. This is done by tailoring the calculations to this specific case, which allows to avoid unnecessary calculations. The strength of this strategy is benchmarked against state-of-the-art techniques, showing a remarkable acceleration. As a side product, we provide an optimized method for one key calculus in quantum simulations: the Pauli basis decomposition of Hamiltonians. [ABSTRACT FROM AUTHOR]

In this work, we study non-rough norms in L(X, Y), the space of bounded linear operators between Banach spaces X and Y. We prove that L(X, Y) has non-rough norm if and only if X ∗ and Y have non-rough norm. We show that the injective tensor product X ⊗ ^ ε Y has non-rough norm if and only if both X and Y have non-rough norm. We also give an example to show that non-rough norms are not stable under projective tensor product. We also study a related concept namely the small diameter properties in the context of L (X , Y) ∗ . These results lead to a discussion on stability of the small diameter properties for projective and injective tensor product spaces. [ABSTRACT FROM AUTHOR]

For a standard graded ring R of dimension ≥ 2 over a perfect field of characteristic p > 0 and a homogeneous ideal I of finite colength, the HK density function of R with respect to I is a compactly supported continuous function f R , I : [ 0 , ∞) ⟶ [ 0 , ∞) , whose integration yields the HK multiplicity e HK (R , I) . Here we answer a question of V. Trivedi about the Hilbert–Kunz density function of the tensor product of standard graded rings and show that it is the convolution of the Hilbert–Kunz density function of the factor rings. Using Fourier transform, as a corollary, we get that HK multiplicity of the tensor product of rings is a product of the HK multiplicity of the factor rings. We compute the Fourier transform of the HK density function of a projective curve. [ABSTRACT FROM AUTHOR]

We introduce the algebras satisfying the (ℬ , n) condition. If Λ, Γ are algebras satisfying the (ℬ , n), (ℰ , m) condition, respectively, we give a construction of (m+n)-almost split sequences in some subcategories (B ⊗ E) (i 0 , j 0) of mod(Λ ⊗ Γ) by tensor products and mapping cones. Moreover, we prove that the tensor product algebra Λ ⊗ Γ satisfies the ((B ⊗ E) (i 0 , j 0) , n + m) condition for some integers i0, j0; this construction unifies and extends the work of A. Pasquali (2017), (2019). [ABSTRACT FROM AUTHOR]

In this paper, we characterize when, for any infinite cardinal α , the Fremlin tensor product of two Archimedean Riesz spaces (see Fremlin in Am J Math 94:777–798, 1972) is Dedekind α -complete. We also provide an example of an ideal I in an Archimedean Riesz space E such that the Fremlin tensor product of I with itself is not an ideal in the Fremlin tensor product of E with itself. [ABSTRACT FROM AUTHOR]

Explaining the foundation of cognitive abilities in the processing of information by neural systems has been in the beginnings of biophysics since McCulloch and Pitts pioneered work within the biophysics school of Chicago in the 1940s and the interdisciplinary cybernetists meetings in the 1950s, inseparable from the birth of computing and artificial intelligence. Since then, neural network models have traveled a long path, both in the biophysical and the computational disciplines. The biological, neurocomputational aspect reached its representational maturity with the Distributed Associative Memory models developed in the early 70 s. In this framework, the inclusion of signal-signal multiplication within neural network models was presented as a necessity to provide matrix associative memories with adaptive, context-sensitive associations, while greatly enhancing their computational capabilities. In this review, we show that several of the most successful neural network models use a form of multiplication of signals. We present several classical models that included such kind of multiplication and the computational reasons for the inclusion. We then turn to the different proposals about the possible biophysical implementation that underlies these computational capacities. We pinpoint the important ideas put forth by different theoretical models using a tensor product representation and show that these models endow memories with the context-dependent adaptive capabilities necessary to allow for evolutionary adaptation to changing and unpredictable environments. Finally, we show how the powerful abilities of contemporary computationally deep-learning models, inspired in neural networks, also depend on multiplications, and discuss some perspectives in view of the wide panorama unfolded. The computational relevance of multiplications calls for the development of new avenues of research that uncover the mechanisms our nervous system uses to achieve multiplication. [ABSTRACT FROM AUTHOR]

We consider the family P of n-tuples P consisting of polynomials P 1 , ... , P n with nonnegative coefficients, which satisfy ∂ i P j (0) = δ i , j , i , j = 1 , ... , n. With any such P, we associate a Reinhardt domain ▵ P n that we call the generalized Hartogs triangle. We are particularly interested in the choices P a = (P 1 , a , ... , P n , a) , a ⩾ 0 , where P j , a (z) = z j + a ∏ k = 1 n z k , j = 1 , ... , n. The generalized Hartogs triangle associated with P a is given by: ▵ a n = { z ∈ C × C ∗ n - 1 : | z j | 2 < | z j + 1 | 2 (1 - a | z 1 | 2) , j = 1 , ... , n - 1 , | z n | 2 + a | z 1 | 2 < 1 }. The domain ▵ 0 2 is the Hartogs triangle. Unlike most domains relevant to the multi-variable operator theory, the domain ▵ P n , n ⩾ 2 , is never polynomially convex. However, ▵ P n is always holomorphically convex. With any P ∈ P and m ∈ N n , we associate a positive semi-definite kernel K P , m on ▵ P n. This, combined with the Moore's theorem, yields a reproducing kernel Hilbert space H m 2 (▵ P n) of holomorphic functions on ▵ P n. We study the space H m 2 (▵ P n) and the multiplication n-tuple M z acting on H m 2 (▵ P n). It turns out that M z is never rationally cyclic, but H m 2 (▵ P n) admits an orthonormal basis consisting of rational functions on ▵ P n. Although the dimension of the joint kernel of M z ∗ - λ is constant of value 1 for every λ ∈ ▵ P n , it has jump discontinuity at the serious singularity 0 of the boundary of ▵ P n with the value equal to ∞. We capitalize on the notion of joint subnormality to define a Hardy space H 2 (▵ 0 n) on the n-dimensional Hartogs triangle ▵ 0 n. This in turn gives an analog of the von Neumann's inequality for ▵ 0 n. [ABSTRACT FROM AUTHOR]

The two well-known numerical radius inequalities for the tensor product A ⊗ B acting on H ⊗ K , where A and B are bounded linear operators defined on complex Hilbert spaces H and K , respectively are 1 2 ‖ A ‖ ‖ B ‖ ≤ w (A ⊗ B) ≤ ‖ A ‖ ‖ B ‖ and w (A) w (B) ≤ w (A ⊗ B) ≤ min { w (A) ‖ B ‖ , w (B) ‖ A ‖ }. In this article, we develop new lower and upper bounds for the numerical radius w (A ⊗ B) of the tensor product A ⊗ B and study the equality conditions for those bounds. [ABSTRACT FROM AUTHOR]

Let E be a Banach space. For a topological space X, let C b (X , E) be the space of all bounded continuous E-valued functions on X, and let C K (X , E) be the subspace of C b (X , E) consisting of all functions having a pre-compact image in E. We show that C K (X , E) is isometrically isomorphic to the injective tensor product C b (X) ⊗ ^ ε E , and that C b (X , E) = C b (X) ⊗ ^ ε E if and only if E is finite dimensional. Next, we consider the space Lip(X, E) of E-valued Lipschitz operators on a metric space (X, d) and its subspace LipK(X, E) of Lipschitz compact operators. Utilizing the results on C b (X , E) , we prove that LipK(X, E) is isometrically isomorphic to a tensor product Lip (X) ⊗ ^ α E , and that Lip (X , E) = Lip (X) ⊗ ^ α E if and only if E is finite dimensional. Finally, we consider the space D1(X, E) of continuously differentiable functions on a perfect compact plane set X and show that, under certain conditions, D1(X, E) is isometrically isomorphic to a tensor product D 1 (X) ⊗ ^ β E . [ABSTRACT FROM AUTHOR]

Tootkaboni and Vahed introduced the notion of some large sets near idempotent along with some combinatorial properties. We characterize when the finite Cartesian product of central sets near idempotent is central near idempotent. Moreover, we provide a partial characterization for the infinite Cartesian product of the same. We then study the abundance of some large sets near idempotent. We also investigate the effect of tensor product near zero. Finally, as an application we provide a characterization of members of polynomials (with constant term 0) evaluated at idempotents in a near zero semigroup. [ABSTRACT FROM AUTHOR]

In signal processing and image science, capturing the phase of measurements is often difficult. In such cases, one needs to reconstruct signal/image from intensity measurements, that is, perform phase retrieval. Like phase retrieval, norm retrieval restores the norm of a signal from its intensity measurements. In the last few decades, phase retrieval and norm retrieval have become active research areas. In this paper, we study phase retrieval sequences and norm retrieval sequences in the tensor product of Hilbert spaces. We explore various methods of construction of phase (norm) retrieval sequences in tensor product spaces by means of tensor product of vectors and tensor product of operators. [ABSTRACT FROM AUTHOR]

The approximation properties of the Aldaz–Kounchev–Render (AKR) operators where investigated in several papers. We improve some existing quantitative results concerning these approximation properties. Moreover, we describe classes of functions for which these operators approximate better than the classical Bernstein operators and classes of functions for which Bernstein operators approximate better than AKR operators. The new results, in particular involving monotonic convergence and Voronovskaja type formulas, are then extended to the bivariate case on the square [ 0 , 1 ] 2 and compared with other existing results. Several numerical examples, illustrating the relevance and supporting the theoretical findings, are presented. [ABSTRACT FROM AUTHOR]

Let X be a reflexive Banach space with an unconditional basis and Y be any Banach space. We show that X ⊗ ^ Y has Pelczyński’s property (V) if and only if Y has Pelczyński’s property (V) and every bounded linear operator from X to Y ∗ is compact. We also show that if Y is reflexive then K (X , Y) has Pelczyński’s property (V). As an application, we provide new examples of non-reflexive Banach spaces with Pelczyński’s property (V). [ABSTRACT FROM AUTHOR]

The starting point of this work is the fact that the class of evolution algebras over a fixed field is closed under tensor product. We prove that, under certain conditions, the tensor product is an evolution algebra if and only if every factor is an evolution algebra. Another issue arises about the inheritance of properties from the tensor product to the factors and conversely. For instance, nondegeneracy, irreducibility, perfectness and simplicity are investigated. The four-dimensional case is illustrative and useful to contrast conjectures, so we achieve a complete classification of four-dimensional perfect evolution algebras emerging as tensor product of two-dimensional ones. We find that there are four-dimensional evolution algebras that are the tensor product of two nonevolution algebras. [ABSTRACT FROM AUTHOR]

Commuting pairs of algebraic structures on a set have been studied by several authors and may be described equivalently as algebras for the tensor product of Lawvere theories, or more basically as certain bifunctors that here we call bifold algebras. The much less studied notion of commutant for Lawvere theories was first introduced by Wraith and generalizes the notion of centralizer clone in universal algebra. Working in the general setting of enriched algebraic theories for a system of arities, we study the interaction of the concepts of bifold algebra and commutant. We show that the notion of commutant arises via a universal construction in a two-sided fibration of bifold algebras over various theories. On this basis, we study special classes of bifold algebras that are related to commutants, introducing the notions of commutant bifold algebra and balanced bifold algebra. We establish several adjunctions and equivalences among these categories of bifold algebras and related categories of algebras over various theories, including commutative, contracommutative, saturated, and balanced algebras. We also survey and develop examples of commutant bifold algebras, including examples that employ Pontryagin duality and a theorem of Ehrenfeucht and Łoś on reflexive abelian groups. Along the way, we develop a functorial treatment of fundamental aspects of bifold algebras and commutants, including tensor products of theories and the equivalence of bifold algebras and commuting pairs of algebras. Because we work relative to a (possibly large) system of arities in a closed category V , our main results are applicable to arbitrary V -monads on a finitely complete V , the enriched theories of Borceux and Day, the enriched Lawvere theories of Power relative to a regular cardinal, and other notions of algebraic theory. [ABSTRACT FROM AUTHOR]

Tensor decomposition is widely used to reduce the amount of model parameters and release the pressure on resource-constrained devices. However, exiting studies on tensor decomposition need tensor rank selection, which introduce a large number of additional hyper-parameters and extensive combinatorial searches on tensor ranks. In this paper, we present a new tensor product (TP) based linear layer that can replace the original convolution layer, fully-connected (FC) layer, and vector FC layer in capsule networks without tensor rank selection. Specifically, tensor-matrix product and tensor-vector product are used to the original matrix multiplication. Tensor-matrix product and tensor-outer product are used to replace element product operation. Tensor-outer product is for convolution. These tensor product operations are made up of numerous factors and have fewer parameters. The experimental results show that, when compared to the tensor decomposition algorithm, our algorithm can compress the parameter amount several times while maintaining accuracy without fine-tuning and tensor rank selection. [ABSTRACT FROM AUTHOR]

In this paper, we study irreducible non-weight modules over the mirror Heisenberg-Virasoro algebra D , including Whittaker modules, U (ℂ d 0) -free modules and their tensor products. More precisely, we give the necessary and sufficient conditions for the Whittaker modules to be irreducible. We determine all the D -module structures on U (ℂ d 0) , and find the necessary and sufficient conditions for these modules to be irreducible. At last, we determine the necessary and sufficient conditions for the tensor products of Whittaker modules and U (ℂ d 0) -free modules to be irreducible, and obtain that any two such tensor products are isomorphic if and only if the corresponding Whittaker modules and U (ℂ d 0) -free modules are isomorphic. These lead to many new irreducible non-weight modules over D . [ABSTRACT FROM AUTHOR]

Twisted generalized Weyl algebras (TGWAs) A(R,σ,t) are defined over a base ring R by parameters σ and t, where σ is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and σ, there is a natural algebra map A (R , σ , t t ′) → A (R , σ , t) ⊗ R A (R , σ , t ′) . This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,σ,t). We give presentations of these Grothendieck rings for n = 1,2, when R = ℂ [ z ] . As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over s l 2 is a tensor product of two Weyl algebra modules. [ABSTRACT FROM AUTHOR]

This paper is concerned with studying a new class of multivariable commuting operators know as left (m , C) -invertible p-tuples and related to a given conjugation transformation C on a Hilbert space Y . Some structural properties of some members of this class are given. [ABSTRACT FROM AUTHOR]

For two given graphs, we construct an augmentation graph. We obtain an injective algebra homomorphism from the Leavitt path algebra of the augmentation graph to the tensor product of the Leavitt path algebras of the given graphs, and give a necessary and sufficient condition on the surjectivity of the homomorphism. This gives rise to a realization of the tensor product of two Leavitt path algebras as a Leavitt path algebra. [ABSTRACT FROM AUTHOR]

The performance of a wide area damping controller (WADC) can be affected considerably if a photovoltaic (PV) power plant is integrated with the power system. A WADC using state feedback control law is proposed in this paper to eliminate the frequency oscillation of interconnected power system integrated with solar PV cell. Output power of the solar PV cell varies with irradiation. This variation in power degrades the performance of the WADC. Communication delay, which is always present in case of wide area measurement system (WAMS) also has a detrimental effect on the effectiveness of WADC. Another non-linear phenomenon which is always present in the WADC is the saturation of the actuator which reduces the effectiveness of the controller. The damping controller developed in this paper contemplates the effect of saturation of the actuator, time varying delay and changing operating condition using linear matrix inequality (LMI) criterion. The changing operating conditions are considered using a linear parameter varying (LPV) modelling of the system. After developing the LPV model of the system, the model is converted into a tensor product (TP) model. A supplementary damping controller (SDC) is considered in this paper as the WADC. The actuator of the damping controller is a thyristor controlled series capacitor (TCSC). A modified two area 11-bus power system with PV power plant and modified IEEE 39-bus power system with PV power plant is used as a test system to understand the usefulness of the designed controller. [ABSTRACT FROM AUTHOR]

A dynamic phasor estimation technique based on the tensor product is presented to enhance the accuracy and robustness of phasor measurement units (PMUs) in the smart electric grid. The performance of the proposed tensor product-based dynamic phasor estimation algorithm (TDPE) is demonstrated with various test signals, i.e., DC decaying, noise, harmonics, power swings, frequency ramps, abrupt change, off-nominal frequency events as per Standard IEEE C37.118. 1a-2014 to represent peculiar dynamic scenarios of the modern power system. The simulation results reveal that the performance factor, i.e., total vector error (%TVE) of the proposed TDPE approach, is below 3 % during various simulated dynamic environments. Thus, the proposed tensor-based dynamic phasor estimation methodology can be effective to extract the phasors of voltage/current signals during peculiar dynamics in the smart electric grids and enhance the several applications based on phasor measurement units (PMUs) information for the stable and secure power system. [ABSTRACT FROM AUTHOR]

The primary goal of this article is devoted to the study of fractal bases and fractal frames for L 2 (I × J) , the collection of all square integrable functions on the rectangle I × J . The fractal function when recognized as an internal binary operation paved way for the construction of right and left partial fractal convolution operators on L 2 (I) , for any real compact interval I. The aforementioned operators defined on one dimensional space have been applied to obtain operators on the space L 2 (I × J) by the identification of L 2 (I × J) with the tensor product space L 2 (I) ⊗ L 2 (J) . In this paper, we establish properties of this bounded linear operator which eventually helps to prove that L 2 (I × J) admits Bessel sequences, Riesz bases and frames consisting of products of fractal (self-referential) functions in a nice way. [ABSTRACT FROM AUTHOR]

In this paper we study the cancellation law for the tensor product in the category Sup of complete lattices and join-preserving maps. First, we investigate the tensor product of generalized power-set lattices. Based on which, we prove that the cancellation law for the tensor product has a close relation to that for the cartesian product of posets, and give a class of complete lattices which do not satisfy the cancellation law for the tensor product. Then, we also investigate the cancellation law for particular subclasses of complete lattices. [ABSTRACT FROM AUTHOR]

In a recent paper, Dave Benson and Peter Symonds defined a new invariant γG(M) for a finite dimensional module M of a finite group G which attempts to quantify how close a module is to being projective. In this paper, we determine this invariant for permutation modules of the symmetric group corresponding to two-part partitions using tools from representation theory and combinatorics. [ABSTRACT FROM AUTHOR]

Solving linear systems of equations is a fundamental problem in mathematics. When the linear system is so large that it cannot be loaded into memory at once, iterative methods such as the randomized Kaczmarz method excel. Here, we extend the randomized Kaczmarz method to solve multi-linear (tensor) systems under the tensor–tensor t-product. We present convergence guarantees for tensor randomized Kaczmarz in two ways: using the classical matrix randomized Kaczmarz analysis and taking advantage of the tensor–tensor t-product structure. We demonstrate experimentally that the tensor randomized Kaczmarz method converges faster than traditional randomized Kaczmarz applied to a naively matricized version of the linear system. In addition, we draw connections between the proposed algorithm and a previously known extension of the randomized Kaczmarz algorithm for matrix linear systems. [ABSTRACT FROM AUTHOR]

In this paper, we consider the frameworks for quantum-classical and classical-quantum cryptographic translation by (Nimbe et al. Int. J. Theor. Phys. 60, 793–818 2021). The high or low level framework for quantum-classical cryptographic translation has three major phases with the framework for quantum-classical conversion being the intermediary phase. The high or low level framework for classical-quantum cryptographic translation also has three major phases with the framework for classical-quantum conversion being the intermediary phase. Mathematical steps are presented for these conversion frameworks and are shown to be implementable. Implementations of these intermediary phases are presented using c++ programming language. [ABSTRACT FROM AUTHOR]

In this work we introduce a three-party quantum conference where the involved parties initially hold asymmetric quantum information which are finally shared amongst themselves by the execution of a quantum communication protocol. The three conferencing parties along with a controller are connected amongst themselves through a multipartite quantum entanglement which is used as quantum resource in the protocol. The role of the controller is confined to a final action without which the conference can not be performed. The present work is an instance of applications of multipartite entangled resources in communication problems. [ABSTRACT FROM AUTHOR]

Multi-strut platform is widely used for precise instrument vibration isolation. In this paper, a Newton-Euler based 6-DOF 12-strut platform model is proposed. Nonlinearity of platform dynamic component is derived by establishing polytopic linear parameter varying (LPV) system. To guarantee the linearization accuracy of LPV system while reducing model elements to a real-time computing level. Tensor product(TP) model transformation and truncated high-order singular value decomposition(HOSVD) are used to decompose LPV system high-order tensor into unique principle basis. Then low-rank approximation of system is implemented by discarding minor singular basis vectors, for the sake of minimizing storage space and computing complexity. And then the parameter varying system is represented by convex combination of discretized system vertexes. So quadratic regulator method can be applied to vertex linear time-invariant subsystem controller to construct the global controller. Performance of the proposed multi-strut platform is demonstrated through hardware in loop simulation. [ABSTRACT FROM AUTHOR]

Let F be an infinite field of characteristic different from two and E be the unitary Grassmann algebra of an infinite dimensional F-vector space L. Denote by Egr an arbitrary ℤ 2 -grading on E such that the subspace L is homogeneous. We consider Egr ⊗ E⊗n as a (ℤ 2 × ℤ 2 n) -graded algebra, where the grading on E is supposed to be the canonical one, and we find its graded ideal of identities. [ABSTRACT FROM AUTHOR]

Some representations of states of quantum systems are discussed, and their equivalence is proved. In particular, an approach going back to L.D. Landau in which the density operator is constructed using a reduction of a pure state of a quantum system described by the tensor product of suitable Hilbert spaces is presented. Under these assumptions, changes in the states of subsystems of a quantum system caused by experiments are investigated. [ABSTRACT FROM AUTHOR]

The aim of this research work is to define a new class of multialgebras that are left(right) operators and fuzzy operators and related properties are investigated. They are a generalization of semihypergroups and semigroups. With this concept we can connect a non-empty set to a semihypergroup. Also, the concepts of the lower and upper approximations of left(right) operators are introduced and some results are obtained. [ABSTRACT FROM AUTHOR]

For an even graph G and non-negative integers r and s, the pair (r , s) is an admissible pair if 4 r + 8 s = | E (G) |. If G admits a decomposition into r copies of C 4 , the cycle of length four, and s copies of C 8 , the cycle of length eight, for every admissible pair (r , s) , then G has a { C 4 r , C 8 s } -decomposition. In this paper, a necessary and sufficient condition is obtained for the existence of a { C 4 r , C 8 s } -decomposition of the complete k-partite graph K a 1 , a 2 , ⋯ , a k , where k ≥ 3. Further, a characterization is obtained for the graph K m × K n , where × denotes the tensor product of graphs, to admit a { C 4 r , C 8 s } -decomposition. [ABSTRACT FROM AUTHOR]

This is an expository paper on tensor products where the standard approaches for constructing concrete instances of algebraic tensor products of linear spaces, via quotient spaces or via linear maps of bilinear maps, are reviewed by reducing them to different but isomorphic interpretations of an abstract notion, viz. the universal property, which is based on a pair of axioms. [ABSTRACT FROM AUTHOR]

Assume that X is a reflexive Banach space with an unconditional basis. In this paper, we show that the Banach space K(X) of compact operators on X has Pełczyński's property V. [ABSTRACT FROM AUTHOR]

The autocorrelation function of a stochastic process is one of the essential mathematical tools in the description of variability that is successfully applied in many scientific fields such as signal processing or financial time series analysis and forecasting. The aim of the paper is to provide an analysis of fuzzy transform of higher degree applied to stochastic processes with a focus on its autocorrelation function. We introduce a higher-degree fuzzy transform of a stochastic process and investigate its basic properties. Further, we introduce a higher-degree fuzzy transform of bivariate functions in the tensor product of polynomial spaces and demonstrate its approximation ability. In addition, we show that the bivariate higher-degree fuzzy transform of multiplicative separable functions is a product of univariate fuzzy transform of the respective functions, which is a desirable property in the processing of functions of higher dimensions. The obtained results are used to demonstrate an interesting identity between the autocorrelation function of the fuzzy transform of a stochastic process and the bivariate fuzzy transform of the autocorrelation function of the stochastic process. This identity provides two ways to determine the autocorrelation function of the fuzzy transform of a stochastic process, which can be useful, especially in a situation where its direct calculation becomes complex or even impossible, for example, the calculation of a sample autocorrelation function. [ABSTRACT FROM AUTHOR]

A finite element cochain complex on Cartesian meshes of any dimension based on the H 1 -inner product is introduced. It yields H 1 -conforming finite element spaces with exterior derivatives in H 1 . We use a tensor product construction to obtain L 2 -stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of arbitrary order. [ABSTRACT FROM AUTHOR]

In this paper we prove that the ℓ -group tensor product of two Archimedean f-rings is again an f-ring. We will use this result to characterize multiplicative ℓ -bimorphisms between unital f-rings. [ABSTRACT FROM AUTHOR]

Let A be a right noetherian algebra over a field k. If the base field extension A ⊗kK remains right noetherian for all extension fields K of k, then A is called stably right noetherian over k. We develop an inductive method to show that certain algebras of finite Gelfand-Kirillov dimension are stably noetherian, using critical composition series. We use this to characterize which algebras satisfying a polynomial identity are stably noetherian. The method also applies to many ℕ -graded rings of finite global dimension; in particular, we see that a noetherian Artin-Schelter regular algebra must be stably noetherian. In addition, we study more general variations of the stably noetherian property where the field extensions are restricted to those of a certain type, for instance purely transcendental extensions. [ABSTRACT FROM AUTHOR]

We study the tensor product decomposition of the split real quantum group U q q ~ (sl (2 , R)) from the perspective of finite-dimensional representation theory of compact quantum groups. It is known that the class of positive representations of U q q ~ (sl (2 , R)) is closed under taking tensor product. In this paper, we show that one can derive the corresponding Hilbert space decomposition, given explicitly by quantum dilogarithm transformations, from the Clebsch–Gordan coefficients of the tensor product decomposition of finite-dimensional representations of the compact quantum group U q (sl 2) by solving certain functional equations arising from analytic continuation and using normalization arising from tensor products of canonical basis. [ABSTRACT FROM AUTHOR]

Biquadratic tensors play a central role in many areas of science. Examples include elastic tensor and Eshelby tensor in solid mechanics, and Riemannian curvature tensor in relativity theory. The singular values and spectral norm of a general third order tensor are the square roots of the M-eigenvalues and spectral norm of a biquadratic tensor, respectively. The tensor product operation is closed for biquadratic tensors. All of these motivate us to study biquadratic tensors, biquadratic decomposition, and norms of biquadratic tensors. We show that the spectral norm and nuclear norm for a biquadratic tensor may be computed by using its biquadratic structure. Then, either the number of variables is reduced, or the feasible region can be reduced. We show constructively that for a biquadratic tensor, a biquadratic rank-one decomposition always exists, and show that the biquadratic rank of a biquadratic tensor is preserved under an independent biquadratic Tucker decomposition. We present a lower bound and an upper bound of the nuclear norm of a biquadratic tensor. Finally, we define invertible biquadratic tensors, and present a lower bound for the product of the nuclear norms of an invertible biquadratic tensor and its inverse, and a lower bound for the product of the nuclear norm of an invertible biquadratic tensor, and the spectral norm of its inverse. [ABSTRACT FROM AUTHOR]