Operator Algebras Research Articles

Nonlinear Skew Lie-Type Derivations on ∗-Algebra

Let A be a unital ∗-algebra over the complex fields C. For any H1,H2∈A, a product [H1,H2]•=H1H2−H2H1* is called the skew Lie product. In this article, it is shown that if a map ξ : A→A (not necessarily linear) satisfies ξ(Pn(H1,H2,…,Hn))=∑i=1nPn(H1,…,Hi−1,ξ(Hi),Hi+1,…,Hn)(n≥3) for all H1,H2,…,Hn∈A, then ξ is additive. Moreover, if ξ(ie2) is self-adjoint, then ξ is ∗-derivation. As applications, we apply our main result to some special classes of unital ∗-algebras such as prime ∗-algebra, standard operator algebra, factor von Neumann algebra, and von Neumann algebra with no central summands of type I1.

The Ring of Conway Numbers in Mizar

Summary Conway’s introduction to algebraic operations on surreal numbers with a rather simple definition. However, he combines recursion with Conway’s induction on surreal numbers, more formally he combines transfinite induction-recursion with the properties of proper classes, which is diffcult to introduce formally. This article represents a further step in our ongoing e orts to investigate the possibilities offered by Mizar with Tarski-Grothendieck set theory [4] to introduce the algebraic structure of Conway numbers and to prove their ring character.

Modular Models of Monoids with Operations

Inspired by algebraic effects and the principle of notions of computations as monoids, we study a categorical framework for equational theories and models of monoids equipped with operations. The framework covers not only algebraic operations but also scoped and variable-binding operations. Appealingly, in this framework both theories and models can be modularly composed. Technically, a general monoid-theory correspondence is shown, saying that the category of theories of algebraic operations is equivalent to the category of monoids. Moreover, more complex forms of operations can be coreflected into algebraic operations, in a way that preserves initial algebras. On models, we introduce modular models of a theory, which can interpret abstract syntax in the presence of other operations. We show constructions of modular models (i) from monoid transformers, (ii) from free algebras, (iii) by composition, and (iv) in symmetric monoidal categories.

Restricted modules and associated vertex algebras of extended Heisenberg-Virasoro algebra

In this paper, a family of infinite dimensional Lie algebras L˜ is introduced and investigated, called the extended Heisenberg-Virasoro algebra, denoted by L˜. These Lie algebras are related to the N=2 superconformal algebra and the Bershadsky-Polyakov algebra. We study restricted modules and associated vertex algebras of the Lie algebra L˜. More precisely, we construct its associated vertex (operator) algebras VL˜(ℓ123,0), and show that the category of vertex algebra VL˜(ℓ123,0)-modules is equivalent to the category of restricted L˜-modules of level ℓ123. Then we give uniform constructions of simple restricted L˜-modules. Also, we present several equivalent characterizations of simple restricted modules over L˜.

Chaotic Color Image Encryption Based on Eight-Base DNA-Level Permutation and Diffusion.

Images, as a crucial information carrier in the era of big data, are constantly generated, stored, and transmitted. Determining how to guarantee the security of images is a hot topic in the information security community. Image encryption is a simple and direct approach for this purpose. In order to cope with this issue, we propose a novel scheme based on eight-base DNA-level permutation and diffusion, termed as EDPD, for color image encryption in this paper. The proposed EDPD integrates secure hash algorithm-512 (SHA-512), a four-dimensional hyperchaotic system, and eight-base DNA-level permutation and diffusion that conducts on one-dimensional sequences and three-dimensional cubes. To be more specific, the EDPD has four main stages. First, four initial values for the proposed chaotic system are generated from plaintext color images using SHA-512, and a four-dimensional hyperchaotic system is constructed using the initial values and control parameters. Second, a hyperchaotic sequence is generated from the four-dimensional hyperchaotic system for consequent encryption operations. Third, multiple permutation and diffusion operations are conducted on different dimensions with dynamic eight-base DNA-level encoding and algebraic operation rules determined via the hyperchaotic sequence. Finally, DNA decoding is performed in order to obtain the cipher images. Experimental results from some common testing images verify that the EDPD has excellent performance in color image encryption and can resist various attacks.

A Survey of Accelerating Parallel Sparse Linear Algebra

Sparse linear algebra includes the fundamental and important operations in various large-scale scientific computing and real-world applications. There exists performance bottleneck for sparse linear algebra since it mainly contains the memory-bound computations with low arithmetic intensity. How to improve its performance has increasingly become a focus of research efforts. Using parallel computing techniques to accelerate sparse linear algebra is currently the most popular method, while facing various challenges, e.g., large-scale data brings difficulties in storage, and the sparsity of data leads to irregular memory accesses and parallel load imbalance. Therefore, this article provides a comprehensive overview on acceleration of sparse linear algebra operations using parallel computing platforms, where we focus on four main classifications: sparse matrix-vector multiplication (SpMV), sparse matrix-sparse vector multiplication (SpMSpV), sparse general matrix-matrix multiplication (SpGEMM), and sparse tensor algebra. The takeaways from this article include the following: understanding the challenges of accelerating linear sparse algebra on various hardware platforms; understanding how structured data sparsity can improve storage efficiency; understanding how to optimize parallel load balance; understanding how to improve the efficiency of memory accesses; understanding how do the adaptive frameworks automatically select the optimal algorithms; and understanding recent design trends for acceleration of parallel sparse linear algebra.

Highly efficient secure linear algebra for private machine learning classifications over malicious clients in the post-quantum world

Over the last decade there has a been widespread usage of Machine Learning (ML) classifiers in cases such accurate disease diagnosis at clinics, credit card fraud detection in banks, cyber-attacks prevention of computer systems in different industries, etc. However, privacy and security concerns and law regulations have been an obstacle to the usage of ML classifiers. To this end, this paper addresses the scenario where a server has a private trained ML model, and one or more clients have private queries that they wish to classify using the server's model. During the process, the server learns nothing, while the clients learn only their final classifications and nothing else. Several ML classification algorithms, such as Deep Neural Networks, Support Vector Machines, Logistic Regression, different flavors of Naïve Bayes, etc., can be expressed in terms of linear algebra operations. To this end, initially, as building blocks, several novel secure linear algebra operations are proposed. On top of them novel secure ML classification algorithms are proposed for the aforementioned classifiers under strict security, privacy and efficiency constraints and their security is proven under the semi-honest model. Since the used underlying cryptographic primitives are shown to be resilient to quantum computer attacks, the proposed algorithms are also suitable for the post-quantum world. Furthermore, the proposed algorithms are non-interactive and, based on where the bulk of the operations are done, they have the flexibility to be server or client centric. Theoretical analysis and extensive experimental evaluations over benchmark datasets show that the proposed secure linear algebra operations, hence the secure ML algorithms build on top of them, outperform the state-of-the-art schemes in terms of computation and communication costs as well as on security and privacy characteristics. Moreover, and to the best of the authors’ knowledge, for the first time in literature the security of the proposed algorithms is proven when dealing with multiple malicious clients during classifications.

Hybrid Koopman C∗ –formalism and the hybrid quantum–classical master equation *

Abstract Based on Koopman formalism for classical statistical mechanics, we propose a formalism to define hybrid quantum–classical dynamical systems by defining (outer) automorphisms of the C ∗ algebra of hybrid operators and realizing them as linear transformations on the space of hybrid states. These hybrid states are represented as density matrices on the Hilbert space obtained from the hybrid C ∗ –algebra by the GNS construction. We also classify all possible dynamical systems which are unitary and obtain the possible hybrid Hamiltonian operators.

Asymptotic States and S-Matrix Operator in de Sitter Ambient Space Formalism

Within the de Sitter ambient space framework, the two different bases of the one-particle Hilbert space of the de Sitter group algebra are presented for the scalar case. Using field operator algebra and its Fock space construction in this formalism, we discuss the existence of asymptotic states in de Sitter QFT under an extension of the adiabatic hypothesis and prove the Fock space completeness theorem for the massive scalar field. We define the quantum state in the limit of future and past infinity on the de Sitter hyperboloid in an observer-independent way. These results allow us to examine the existence of the S-matrix operator for de Sitter QFT in ambient space formalism, a question which is usually obscure in spacetime with a cosmological event horizon for a specific observer. Some similarities and differences between QFT in Minkowski and de Sitter spaces are discussed.

An Extended Type-2 Fuzzy Relational Database Model for Aggregate and Grouping Operations

This paper introduces an extended type-2 fuzzy relational database model (ET-2FRDB) for aggregate and grouping operations that can represent and query uncertain and imprecise information in the real world applications. In ET-2FRDB, each fuzzy relation is represented by a type-2 fuzzy set whose membership degree of each tuple is a fuzzy number on [0, 1], the fuzzy aggregate functions, the fuzzy relational algebraic, aggregate and grouping operations are defined as extensions of those in the classical relational database model thereby the membership degree of tuples associated by using the minimum and maximum of fuzzy numbers. Also, some properties of the fuzzy relational algebraic, aggregate and grouping operations in ET-2FRDB are formulated and proven.

Asymptotically entropy-conservative and kinetic-energy preserving numerical fluxes for compressible Euler equations

This paper proposes a hierarchy of numerical fluxes for the compressible flow equations which are kinetic-energy and pressure equilibrium preserving and asymptotically entropy conservative, i.e., they are able to arbitrarily reduce the numerical error on entropy production due to the spatial discretization. The fluxes are based on the use of the harmonic mean for internal energy and only use algebraic operations, making them less computationally expensive than the entropy-conserving fluxes based on the logarithmic mean. The use of the geometric mean is also explored and identified to be well-suited to reduce errors on entropy evolution. Results of numerical tests confirmed the theoretical predictions and the entropy-conserving capabilities of a selection of schemes have been compared.

Remote data possession checking scheme with supporting efficient group user authority management for shared cloud data

Under the shared big data environment, the existing shared data auditing schemes rarely consider the authorization management of group users. Thus, in this paper, we propose a novel remote shared data checking possession scheme that support group authority management. To implement group user rights management, we firstly introduce a trusted entity group manager. To assist the group manager in authorization management, we formalize a new algebraic structure operator named authorization invisible authenticator (AIA). Meanwhile, we provide a basic AIA scheme for general security scenarios and a standard AIA scheme suitable for high-level security scenarios respectively. The standard AIA scheme can fully meet the needs of the group manager to safely perform rights management work. It is composed of the User Information Table (UIT) and the basic AIA scheme, which has higher security and is applicable to a wider range of scenarios. By distributing AIA through the standard AIA scheme, the group manager can easily carry out authority management, including enrolling, revoking, updating. After solving the problem of authorization management, the detailed design of the scheme based on identity-based encryption (IBE) is given. Furthermore, the security analysis and performance evaluation demonstrate that the scheme is safe and feasible.

Typology and Modeling Profiles of Moving-Class Process-Entity Interactions. Framework for Multimodal Route Network

Configuration of routes is a rather comprehensive, necessary and at the same time self-sufficient component of an integrated process that models the displacement of objects. This article is focused on the development of the multimodal route network model that leverages on only the pertinent suite of source abstract object types, which preliminary differentiate according to the open architecture of categorized types. The main emphasis is on the detection of relevant associations, which take place into the transportation domain being modeled, and constraints on its populations. One of these restrictions is the base segment constraint, which guarantees correct instances of the specialization segment relationship. The model contains formal expressions for verifying data reliability in terms of segments specialization, coherence and sequence order of the latter in routes. Another group of constraints encompasses the terminal pool restrictions that comprise constraints of the sequence order, non-redundancy, and unique combinations of source and destination route points. All terminal constraints have been formalized by means of the relational algebra operators. Such decision is capable of supporting integrity constraints by using triggers or stored procedures of database management systems. We have also exploited the universal structural primitive for the multilevel representation of entity lists, as well as have demonstrated the role and function of dedicated datasets under the forming of links between entities.

The implicit rigid tube model predictive control

A computationally efficient reformulation of the rigid tube model predictive control is developed. A unique feature of the derived formulation is the utilization of the implicit set representations. This novel formulation does not require any set algebraic operations to be performed explicitly, and its implementation requires merely the use of the standard optimization solvers.

On topological indices and entropies of diamond structure

AbstractA topological index is a numerical parameter that represents the molecular structure of a compound based on its graph‐theoretical properties. It provides a means to describe the structure‐activity relationship, as well as the physicochemical properties of the compound. Topological indices are used in various scientific fields, including chemistry, biology, and computer science, to study and predict the behavior of molecules. In this study, we concentrate on the calculation of M‐polynomials and entropy for the allotropic form of carbon, diamond. Algebraic operations are applied to these polynomials to obtain degree‐dependent topological invariants.

Classification of Consistent Systems of Handlebody Group Representations

Abstract The classifying spaces of handlebody groups form a modular operad. Algebras over the handlebody operad yield systems of representations of handlebody groups that are compatible with gluing. We prove that algebras over the modular operad of handlebodies with values in an arbitrary symmetric monoidal bicategory $\mathcal{M}$ (we introduce for these the name ansular functor) are equivalent to self-dual balanced braided algebras in $\mathcal{M}$. After specialization to a linear framework, this proves that consistent systems of handlebody group representations on finite-dimensional vector spaces are equivalent to ribbon Grothendieck-Verdier categories in the sense of Boyarchenko-Drinfeld. Additionally, it produces a concrete formula for the vector space assigned to an arbitrary handlebody in terms of a generalization of Lyubashenko’s coend. Our main result can be used to obtain an ansular functor from vertex operator algebras subject to mild finiteness conditions. This includes examples of vertex operator algebras whose representation category has a non-exact monoidal product.

Hopf algebroids from non-commutative bundles

We present two classes of examples of Hopf algebroids associated with non-commutative principal bundles. The first comes from deforming the principal bundle while leaving unchanged the structure Hopf algebra. The second is related to deforming a quantum homogeneous space; this needs a careful deformation of the structure Hopf algebra in order to preserve the compatibilities between the Hopf algebra operations.

Vertex operator algebras and topologically twisted Chern-Simons-matter theories

We consider several topologically twisted Chern-Simons-matter theories and propose boundary VOAs whose module categories should model the category of line operators of the 3d bulk. Our main examples come from the topological A and B twists of the exotic mathcal{N} = 4 Chern-Simons-matter theories of Gaiotto-Witten, but we show that there is a topological “A-twist” for a much larger class of mathcal{N} ≠ 4 theories. We illustrate a particular example of this new class of theories that admits the p = 2 singlet VOA mathfrak{M} (2) on its boundary and comment on its relation to the ψ → ∞ limit of the Gaiotto-Rapčák corner VOA Y1,1,0[ψ].

Nonlinear bi-skew Jordan derivations in prime ∗-rings

Let [Formula: see text] be a unital prime ∗-ring containing a nontrivial symmetric idempotent and let [Formula: see text] be the maximal symmetric ring of quotients of [Formula: see text]. Using the technique of Peirce decomposition and the theory of functional identities, we prove that a map [Formula: see text] satisfies [Formula: see text] for all [Formula: see text] if and only if [Formula: see text] is an additive ∗-derivation unless [Formula: see text] and [Formula: see text]. As an application, we shall characterize such maps in different operator algebras.

The Lie algebra of the lowest transitively differential group of degree three *

We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous polynomial coefficients of degree up to and including three. (2) The generator of the algebra must include the translation operators for all the variables . (3) The Lie algebra is the smallest indecomposable Lie algebra satisfying (1) and (2). It turns out to be a 39-dimensional Lie algebra in six variables (k = 6) and the construction of this algebra is also the simplest possible case in the general construction of the Lie algebras of the transitively differential groups introduced by Guillemin and Sternberg in 1964 involving the coefficients of degree 3. Those algebras and various subalgebras have similarities with algebras related to different applications in physics such as the Schrödinger, Conformal and Galilei algebras with and without central extension. The paper is devoted to the presentation of the structure and different decompositions of the Lie algebra under investigation. It is also devoted to the presentation of relevant Lie subalgebras and the construction of their Casimir invariants using different methods. We rely, in particular, on differential operator realizations, symbolic computation packages, the Berezin bracket and virtual copies of the Lie algebras.