Operator Algebras Research Articles

A Simple Divide-and-Conquer-based Distributed Method for the Accelerated Failure Time Model

The accelerated failure time (AFT) model is an appealing tool in survival analysis because of its ease of interpretation, but when there is a large volume of data, fitting an AFT model and carrying out the associated inference on one computer can be computationally demanding. This poses a severe limitation for the application of the AFT model in the face of big data. The article addresses this problem by developing a simple distributed method for estimating the parameters of an AFT model based on the divide-and-conquer strategy, which has the dual benefits of statistical efficiency and computational economy. It is an iterative method that involves, for the most part, some rather simple algebraic operations, except for obtaining the initial estimate, which is based on a smoothed approximation of the Gehan estimating equation. Our results show that the proposed method yields estimates that converge after a few iterations and an estimator that is asymptotically as efficient as the benchmark estimator obtained by using the full data in one go. We also develop an associated inference procedure. The merits of the proposed method are demonstrated via an extensive simulation study. The method is applied to a kidney transplantation dataset. Supplementary materials for this article are available online.

Geometric phases characterise operator algebras and missing information

We show how geometric phases may be used to fully describe quantum systems, with or without gravity, by providing knowledge about the geometry and topology of its Hilbert space. We find a direct relation between geometric phases and von Neumann algebras. In particular, we show that a vanishing geometric phase implies the existence of a well-defined trace functional on the algebra. We discuss how this is realised within the AdS/CFT correspondence for the eternal black hole. On the other hand, a non-vanishing geometric phase indicates missing information for a local observer, associated to reference frames covering only parts of the quantum system considered. We illustrate this with several examples, ranging from a single spin in a magnetic field to Virasoro Berry phases and the geometric phase associated to the eternal black hole in AdS spacetime. For the latter, a non-vanishing geometric phase is tied to the presence of a centre in the associated von Neumann algebra.

Signal Processing Application Based on a Hybrid Wavelet Transform to Fault Detection and Identification in Power System

The power system is one of the most susceptible systems to failures, which are most frequently caused by transmission line faults. Transmission line failures account for 85% of all power system malfunctions. However, over the last decade, numerous fault detection methods have been developed to ensure the reliability and stability of power systems. A hybrid detection method based on the idea of redundancy property is presented in this paper. Because the continuous wavelet transform itself does not extract fault features for small defects effectively, the stationary wavelet transform approach is employed to assist in their detection. As a result of its ability to decompose the signal into high- and low-frequency components, undecimated reconstruction by using the algebraic summation operation (ASO) is used. This approach creates redundancy, which is useful for the feature extraction of small defects and makes faulty parts more evident. The numerical value of the redundancy ratio’s contribution to the original signal is approximately equal to 36%. Following this method for redundant signal reconstruction, a continuous wavelet transform is used to extract the fault characteristic significantly easier in the time-scale (frequency) domain. Finally, the suggested technique has been demonstrated to be an efficient fault detection and identification tool for use in power systems. In fact, using this advanced signal processing technique will help with early fault detection, which is mainly about predictive maintenance. This application provides more reliable operation conditions.

АЛГЕБРАИЧЕСКИЙ АЛГОРИТМ СТАТИСТИЧЕСКОЙ ОЦЕНКИ ПАРАМЕТРА БИНОМИАЛЬНОГО РАСПРЕДЕЛЕНИЯ И ПРИМЕР ЕГО ПРИМЕНЕНИЯ В ОДНОЙ ГЛОБАЛЬНОЙ ГЕОИНФОРМАЦИОННОЙ ЗАДАЧЕ ПРИКЛАДНОЙ КЛИМАТОЛОГИИ

The binomial distribution is often used in modeling, including in applied climatology, when assessing the likelihood of adverse or dangerous climate-related impacts. The article outlines an algebraic algorithm for obtaining such estimates based on climate monitoring data or modeling data. Using only algebraic operations in calculations minimizes calculation time, which is important when performing detailed global assessments. As an example of such a calculation, the result of an assessment of the global climatic range of a dangerous insect pest of forest and garden trees and shrubs, the gypsy moth, is given.

A Block Projection Operator in the Algebra of Measurable Operators

A Block Projection Operator in the Algebra of Measurable Operators

Nonlinear A*B+B*A Type Derivations on *-Algebras

Let [see formula in PDF] be a unital *-algebra with the unit [see formula in PDF] and a nontrivial projection [see formula in PDF]. Suppose that [see formula in PDF] satisfies [see formula in PDF] implies [see formula in PDF] and [see formula in PDF] implies [see formula in PDF] In this paper, we prove that [see formula in PDF] is a nonlinear [see formula in PDF] type derivation on [see formula in PDF] if and only if [see formula in PDF] is an additive *-derivation. This is then applied to prime *-algebra, von Neumann algebras with no central summands of type [see formula in PDF] factor von Neumann algebras and standard operator algebras.

Analysis of difficulties in solving high-level thinking ability questions for middle school students

This study aims to determine students' higher order thinking skills (HOTS) and to analyze the difficulties experienced by students in solving HOTS questions at SMP Negeri 10 Lhokseumawe. The study used was descriptive qualitative research which was carried out by class VIII students of SMP Negeri 10 Lhokseumawe. The collection of data used is a test in the form of descriptions with material forms and algebraic operations and interviews. The data analysis technique used follows the Miles and Huberman model with data reduction, data presentation and drawing conclusions. The results showed that the students' high-level thinking skills were still in the low category. This can be seen from the success of each aspect of students' higher order thinking skills, namely the ability to analyze by 70.83%, the ability to evaluate by 12.50% and the ability to create or be creative by 33.33%. The results of the interviews found that students were not familiar with questions in the form of application of concepts or problem-solving, and there were still many students who did not have reinforcement of material concept abilities.

Methods of automation and interpretation of the quality educational tests in distributed virtual learning systems

The subject matter of the article is the development of mathematical and algorithmic support for intellectual tools, which allows you to conduct continuous control of the knowledge of students (subjects of study) objectively and comprehensively. The goal of the work is to create methods for assessing the quality of educational tests and automating such processes. The following tasks were solved in the article: formation of a testing model in a distributed virtual learning environment and a validity assessment model based on the content of sets of test tasks. The following methods used are – algebra of finite predicates and operations, methods of mathematical statistics and methods of intellectual data analysis. The following results were obtained – the principles of intellectual analysis of the value of reliability coefficients, validity coefficients, discriminability coefficient, the difficulty index of the task of assessing the knowledge of subjects of training are formulated. Conclusions: the application of methods of formalization of test evaluation, analysis of software requirements, software development confirm the need to introduce quantitative methods of assessing students' knowledge into educational practice. The introduction of quantitative methods involves the correct setting of control goals, the selection of the measurement object and the selection of measurement tools. The use of pedagogical tests contributes to the effective implementation of all control functions and corresponds to its main principles for solving the problem of assessing the quality of tests, based on distributed virtual learning models and analysis methods. A test example of calculations was developed, with the help of which the efficiency of the proposed methods was researched.

A New Heuristic Tool for Designing Multiple-Valued Logic Systems

In this paper, a brand new heuristic optimization algorithm for multiple-valued logic (MVL) functions and the performance of the implementation will be introduced. In logic synthesis, optimization procedure is a fundamental process in order to design more efficient systems. The proposed algorithm, called MVL-MIN, utilizes Cube Algebra operators. MVL-MIN computes the prime cubes that cover a target cube at a time, instead of extracting all the primes for a given MVL function. MVL-MIN consists of five major procedures, called (a) checking, (b) expansion, (c) elimination, (d) nondisjoint sharp, and (e) prime determination. In order to find a near-minimal cover, three different prime determination functions are developed. The implementation based on the new algorithm is compared with MVSIS. For testing, three sets of test bench are prepared. Each set includes 5 MVL functions. Test results proved that MVL-MIN is able to solve all test files within a fixed allocation of computer resources while MVSIS could not. MVL-MIN achieved 1.9× to 9.7× speedup over MVSIS. In terms of cover size, since current MVL-MIN implementation cannot recognize redundant cubes, MVSIS computed more concise covers for all test benches than MVL-MIN.

Some Generalizations of $$*$$-Lie Derivable Mappings and Their Characterization on Standard Operator Algebras

Some Generalizations of $$*$$-Lie Derivable Mappings and Their Characterization on Standard Operator Algebras

An Efficient and Secure DNA based Image Encryption Technique

In this paper, an efficient and secure DNA based image encryption technique using hyper chaos Lorenz function is proposed. The technique employs SHA-512 hash of gray scale image and primary key to alter the initial seeds of hyper chaos Lorenz function. Initially, chaotic sequences are generated and redesigned. Then, an image is divided into blocks and randomly shuffled according to the primary key. Further perform circular shift operation over the rows of the shuffled image matrix and index based scrambling operation over the columns. After that, perform dynamic encoding and diffusion based DNA algebraic operation over the encoded DNA sequence and DNA based Key sequence using chaotic sequence. Finally, DNA decoding is employed to convert the diffused dynamic matrix into a ciphered image. The dynamicity of the chaotic sequences makes its extraordinary and contributes to the enhanced security and robustness of the technique. Implementation, security and performance analysis demonstrate that the technique has not only an outstanding encryption effect, but also posses large key space, less pixel correlation, low computation time and successfully resilient to the statistical, differential and brute-force attack. Comparative analysis with other references proves its efficiency and practicability for real time applications.

Eris: efficiently measuring discord in multidimensional sources

AbstractData integration is a classical problem in databases, typically decomposed into schema matching, entity matching and data fusion. To solve the latter, it is mostly assumed that ground truth can be determined. However, in general, the data gathering processes in the different sources are imperfect and cannot provide an accurate merging of values. Thus, in the absence of ways to determine ground truth, it is important to at least quantify how far from being internally consistent a dataset is. Hence, we propose definitions of concordant data and define a discordance metric as a way of measuring disagreement to improve decision-making based on trustworthiness. We define the discord measurement problem of numerical attributes in which given a set of uncertain raw observations or aggregate results (such as case/hospitalization/death data relevant to COVID-19) and information on the alignment of different conceptualizations of the same reality (e.g., granularities or units), we wish to assess whether the different sources are concordant, or if not, use the discordance metric to quantify how discordant they are. We also define a set of algebraic operators to describe the alignments of different data sources with correctness guarantees, together with two alternative relational database implementations that reduce the problem to linear or quadratic programming. These are evaluated against both COVID-19 and synthetic data, and our experimental results show that discordance measurement can be performed efficiently in realistic situations.

Algorithm 1037: SuiteSparse:GraphBLAS: Parallel Graph Algorithms in the Language of Sparse Linear Algebra

SuiteSparse:GraphBLAS is a full parallel implementation of the GraphBLAS standard, which defines a set of sparse matrix operations on an extended algebra of semirings using an almost unlimited variety of operators and types. When applied to sparse adjacency matrices, these algebraic operations are equivalent to computations on graphs. A description of the parallel implementation of SuiteSparse:GraphBLAS is given, including its novel parallel algorithms for sparse matrix multiply, addition, element-wise multiply, submatrix extraction and assignment, and the GraphBLAS mask/accumulator operation. Its performance is illustrated by solving the graph problems in the GAP Benchmark and by comparing it with other sparse matrix libraries.

Generalized parametric representation and extension operators

We consider conditions under which mappings of the open Euclidean unit ball of Cn with generalized parametric representation with respect to an appropriately defined locally integrable function A:[0,∞)→L(Cn), where L(Cn) is the algebra of linear operators from Cn into Cn, on an interval [s,∞), s≥0, are extended to mappings of the same type defined in a higher-dimensional ball (with respect to a related operator-valued function) using two classes of extension operators, namely perturbations of the Pfaltzgraff–Suffridge extension operator and modifications of the Roper–Suffridge extension operator introduced by the author. Extensions of Loewner chains of order p∈[1,∞], i.e., non-normalized Loewner chains that satisfy a locally uniform local Lp-continuity condition on their real parameter, using these operators have previously been considered by the author, but, unlike in the case of standard parametric representation, mappings with generalized parametric representation may not be as nicely compatible with Loewner chains. We will consider the ramifications of this. As part of our work, we will see that A-normalized Loewner chains and evolution families will have order p when ‖A(⋅)‖ is locally Lp. In addition, we will completely characterize generalized parametric representation in the open unit disk. This characterization involves families of bounded univalent functions and allows us to show that the modified Roper–Suffridge extension operators extend families of univalent functions bounded by some M>1 to biholomorphic mappings bounded by M under certain conditions as a consequence of our main extension results.

Circular q-Rung Orthopair Fuzzy Set and Its Algebraic Properties

Circular intuitionistic fuzzy sets (CIFS) are a recent extension of intuitionistic fuzzy sets (IFS) that can handle imprecise membership values effectively. However, its representation is limited to the space under the intuitionistic fuzzy interpretation triangle (IFIT). To address this, a new generalization of CIFS called circular q-rung orthopair fuzzy sets (Cq-ROFS) is proposed, extending the IFIT to cover a larger space of imprecision. Several relations and operations, including algebraic operations for Cq-ROFS are proposed. In addition, modal operators and their properties are then investigated.

Cosets from equivariant 𝒲-algebras

The equivariant W \mathcal W -algebra of a simple Lie algebra g \mathfrak {g} is a BRST reduction of the algebra of chiral differential operators on the Lie group of g \mathfrak {g} . We construct a family of vertex algebras A [ g , κ , n ] A[\mathfrak {g}, \kappa , n] as subalgebras of the equivariant W \mathcal W -algebra of g \mathfrak {g} tensored with the integrable affine vertex algebra L n ( g ˇ ) L_n(\check {\mathfrak {g}}) of the Langlands dual Lie algebra g ˇ \check {\mathfrak {g}} at level n ∈ Z > 0 n\in \mathbb {Z}_{>0} . They are conformal extensions of the tensor product of an affine vertex algebra and the principal W \mathcal {W} -algebra whose levels satisfy a specific relation. When g \mathfrak {g} is of type A D E ADE , we identify A [ g , κ , 1 ] A[\mathfrak {g}, \kappa , 1] with the affine vertex algebra V κ − 1 ( g ) ⊗ L 1 ( g ) V^{\kappa -1}(\mathfrak {g}) \otimes L_1(\mathfrak {g}) , giving a new and efficient proof of the coset realization of the principal W \mathcal W -algebras of type A D E ADE .

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˜.