Operator Algebras Research Articles

The nonlinear mixed bi-skew lie triple derivations on *-algebras

Let A be a unital *-algebra. In this paper, under some mild conditions on A, it is shown that a map ? : A ? Ais a nonlinear mixed bi-skew Lie triple derivation if and only if ? is an additive*-derivation. As applications, nonlinear mixed bi-skew Lie triple derivations on prime *-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

A lattice theoretical interpretation of generalized deep holes of the Leech lattice vertex operator algebra

Abstract We give a lattice theoretical interpretation of generalized deep holes of the Leech lattice VOA $V_\Lambda $ . We show that a generalized deep hole defines a ‘true’ automorphism invariant deep hole of the Leech lattice. We also show that there is a correspondence between the set of isomorphism classes of holomorphic VOA V of central charge $24$ having non-abelian $V_1$ and the set of equivalence classes of pairs $(\tau , \tilde {\beta })$ satisfying certain conditions, where $\tau \in Co.0$ and $\tilde {\beta }$ is a $\tau $ -invariant deep hole of squared length $2$ . It provides a new combinatorial approach towards the classification of holomorphic VOAs of central charge $24$ . In particular, we give an explanation for an observation of G. Höhn, which relates the weight one Lie algebras of holomorphic VOAs of central charge $24$ to certain codewords associated with the glue codes of Niemeier lattices.

Stability assessment and parametric sensitivity analysis based on extended Gershgorin Theorem for multiple grid-connected converters

• A multiple grid-connected converters is modeled as a medium and high dimensional negative feedback system with a stable open-loop function in unified dq frame, which fills in the limitations of traditional impedance ratios for the small disturbance stability analysis of multiple grid-connected converters system. • In terms of estimating eigenvalues for the multiple grid-connected converters system, a stability assessment method based on extended Gershgorin Theorem is proposed in frequency domain. Moreover, the algebraic form of the criterion is further simplified to quantify the oscillatory stability, which has the lower complexity compared with generalized nyquist criterion. The proposed method is of great potential in analyzing the stability issue for large-scale new energy power systems. • The reshaped eigenvalue estimation area based on extended Gershgorin Theorem is smaller than traditional Gershgorin Theorem, which can reduce its conservatism to improve system stability margin. • Compared to observe the changed trend of system stability by the change of the related parameters, the parameter sensitivity analysis based on equivalent impedance network model is used to quantify the impact factors on stability, which can be given to guide the parameter optimization of the multiple grid-connected converters system. The large-scale integration of power electronic based systems is bringing new oscillatory stability issues, further threatening the safe and stable operation of power system. Early researches are focused on stability analysis of single grid-connected converters to simple networks. Instead, there have been growing interests in studying the small signal stability of multiple grid-connected converters nowadays, whose main method is to apply traditional stability assessment with high complexity and conservatism. To address this problem, a simplified criterion based on extended Gershgorin Theorem to assess the stability of multiple grid-connected converter. The proposed method has less conservatism than tradition Gershgorin Theorem-based criteria, and has the lower computational complexity with simple algebraic operation compared to generalized nyquist criteria. Moreover, the parametric sensitivity analysis is used for analyzing the impact factors to promote system stability based on the proposed criteria. Theoretical results as well as electromagnetic transient simulations have verified the validity. The proposed method has the potential to assess the high-order system, and provides guidance on system parameters optimization based on the sensitivity.

A Shuffle Theorem for Paths Under Any Line

AbstractWe generalize the shuffle theorem and its$(km,kn)$version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the$(km,kn)$Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whosexandyintercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of$\operatorname {\mathrm {GL}}_{l}$characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.

The Summation of Series Based on the Laplace Transformation Method in Mathematics Teaching

Abstract It is more difficult to give Laplace transform directly in a defined form or derive it by Fourier transform in mathematics teaching. The article gives a solution for solving high exponential series sum by using Laplace transform. With the help of Laplace transform, calculus operations can be transformed into complex plane algebra operations. The application of the algorithm to the option hedging strategy verifies the applicability of the algorithm proposed in this article.

New facts related to dilation factorizations of Kronecker products of matrices

<abstract><p>The Kronecker product of two matrices is known as a special algebraic operation of two arbitrary matrices in the computational aspect of matrix theory. This kind of matrix operation has some interesting and striking operation properties, one of which is given by $ (A \otimes B)(C \otimes D) = (AC) \otimes (BD) $ and is often called the mixed-product equality. In view of this equality, the Kronecker product $ A_1 \otimes A_2 $ of any two matrices can be rewritten as the dilation factorization $ A_1 \otimes A_2 = (A_1 \otimes I_{m_2})(I_{n_1} \otimes A_2) $, and the Kronecker product $ A_1 \otimes A_2 \otimes A_3 $ can be rewritten as the dilation factorization $ A_1 \otimes A_2 \otimes A_3 = (A_1\otimes I_{m_2} \otimes I_{m_3})(I_{n_1} \otimes A_2 \otimes I_{m_3})(I_{n_1} \otimes I_{n_2} \otimes A_3) $. In this article, we proposed a series of concrete problems regarding the dilation factorizations of the Kronecker products of two or three matrices, and established a collection of novel and pleasing equalities, inequalities, and formulas for calculating the ranks, dimensions, orthogonal projectors, and ranges related to the dilation factorizations. We also present a diverse range of interesting results on the relationships among the Kronecker products $ I_{m_1} \otimes A_2 \otimes A_3 $, $ A_1 \otimes I_{m_2} \otimes A_3 $ and $ A_1 \otimes A_2 \otimes I_{m_3} $.</p></abstract>

Robot sensors process based on generalized Fermatean normal different aggregation operators framework

<abstract><p>Novel methods for multiple attribute decision-making problems are presented in this paper using Type-Ⅱ Fermatean normal numbers. Type-Ⅱ Fermatean fuzzy sets are developed by further generalizing Fermatean fuzzy sets and neutrosophic sets. The Type-Ⅱ Fermatean fuzzy sets with basic aggregation operators are constructed. The concept of a Type-Ⅱ Fermatean normal number is compatible with both commutative and associative rules. This article presents a new proposal for Type-Ⅱ Fermatean normal weighted averaging, Type-Ⅱ Fermatean normal weighted geometric averaging, Type-Ⅱ generalized Fermatean normal weighted averaging, and Type-Ⅱ generalized Fermatean normal weighted geometric averaging. Furthermore, these operators can be used to develop an algorithm that solves MADM problems. Applications for the Euclidean distance and Hamming distances are discussed. Finally, the sets that arise as a result of their connection to algebraic operations are emphasized in our discourse. Examples of real-world applications of enhanced Hamming distances are presented. A sensor robot's most important components are computer science and machine tool technology. Four factors can be used to evaluate the quality of a robotics system: resolution, sensitivity, error and environment. The best alternative can be determined by comparing expert opinions with the criteria. As a result, the proposed models' outcomes are more precise and closer to integer number $ \delta $. To demonstrate the applicability and validity of the models under consideration, several existing models are compared with the ones that have been proposed.</p></abstract>

On the Concepts of Two- Fold Fuzzy Vector Spaces and Algebraic Modules

The concept of two-fold algebras is generated by merging some special sets such as fuzzy sets, neutrosophic sets or plithogenic sets with various algebraic structures. The main objective of this research paper is to provide a strict mathematical definition and a general study of two-fold fuzzy vector spaces defined over the real numbers and also two-fold fuzzy vector spaces defined over the complex numbers, as well as two-fold fuzzy algebraic modules defined over commutative rings with unity. The elementary properties and algebraic operations of those structures will be explained through many theorems and mathematical proofs.

A Computer Program For The System Of Weak Fuzzy Complex Numbers And Their Arithmetic Operations Using Python

Weak fuzzy complex number system is considered as a generalization of complex numbers by using an algebraic element with fuzzy property. It is believed that this new numerical system will play many important roles in various computer science disciplines, proceeding from this importance this paper aims to introduce for the first time the algebraic operations defined on weak fuzzy complex numbers using Python and Jupyter Notebook. Hence, by building a new class and its functions in Python for this new set, we can now use weak fuzzy complex numbers and the main arithmetic operations on them.

A study of quadratic Diophantine fuzzy sets with structural properties and their application in face mask detection during COVID-19

<abstract><p>During the COVID-19 pandemic, identifying face masks with artificial intelligence was a crucial challenge for decision support systems. To address this challenge, we propose a quadratic Diophantine fuzzy decision-making model to rank artificial intelligence techniques for detecting masks, aiming to prevent the global spread of the disease. Our paper introduces the innovative concept of quadratic Diophantine fuzzy sets (QDFSs), which are advanced tools for modeling the uncertainty inherent in a given phenomenon. We investigate the structural properties of QDFSs and demonstrate that they generalize various fuzzy sets. In addition, we introduce essential algebraic operations, set-theoretical operations, and aggregation operators. Finally, we present a numerical case study that applies our proposed algorithms to select a unique face mask detection method and evaluate the effectiveness of our techniques. Our findings demonstrate the viability of our mask identification methodology during the COVID-19 outbreak.</p></abstract>

An effective treatment of adding-up restrictions in the inference of a general linear model

<abstract><p>This article offers a general procedure of carrying out estimation and inference under a linear statistical model $ {\bf y} = {\bf X} \pmb{\beta} + \pmb{\varepsilon} $ with an adding-up restriction $ {\bf A} {\bf y} = {\bf b} $ to the observed random vector $ {\bf y} $. We first propose an available way of converting the adding-up restrictions to a linear matrix equation for $ \pmb{\beta} $ and a matrix equality for the covariance matrix of the error term $ \pmb{\varepsilon} $, which can help in combining the two model equations in certain consistent form. We then give the derivations and presentations of analytic expressions of the ordinary least-squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of parametric vector $ {\bf K} \pmb{\beta} $ using various analytical algebraic operations of the given vectors and matrices in the model.</p></abstract>

A naive justification of hyperbolic discounting from mental algebraic operations and functional analysis

<abstract><sec><title>Background</title><p>Intertemporal decision-making, which involves making choices between outcomes at different time points, is a fundamental aspect of human behavior. Understanding the underlying mental processes is vital for comprehending the complexities of human decision-making and choice behavior.</p> </sec> <sec><title>Objective</title><p>The main objective of this study is to investigate the interplay of mental processes, specifically cognitive evaluation, subjective valuation, and comparison, in the context of intertemporal decision-making, with a specific focus on understanding the discounting process.</p> </sec> <sec><title>Methodology</title><p>Development of a mathematical representation of the discounting process that incorporates the mental processes associated with intertemporal decision-making.</p> </sec> <sec><title>Result</title><p>Our findings indicate that hyperbolic discounting aligns well with the cognitive processes underlying intertemporal decision-making. Subsequent research will employ qualitative questionnaires to establish the discount function relevant to specific groups, thereby enhancing our comprehension of the discounting process within intertemporal decision-making.</p> </sec></abstract>

Characterizations of matrix equalities involving the sums and products of multiple matrices and their generalized inverse

<abstract><p>It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $ A_1B_1^{g_1}C_1 + A_2B_2^{g_2}C_2 + \cdots + A_kB_k^{g_k}C_{k} = D $ and $ A_1B_1^{g_1}A_2B_2^{g_2} \cdots A_kB_k^{g_k}A_{k+1} = A $, where $ A_1 $, $ A_2 $, $ \ldots $, $ A_{k+1} $, $ C_1 $, $ C_2 $, $ \ldots $, $ C_{k} $ and $ A $ and $ D $ are given, and $ B_1^{g_1} $, $ B_2^{g_2} $, $ \ldots $, $ B_k^{g_k} $ are generalized inverses of matrices $ B_1 $, $ B_2 $, $ \ldots $, $ B_k $. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $ \{1\} $- and $ \{1, 2\} $-generalized inverses of matrices with $ k = 2, 3 $ by using some elementary methods, including a series of explicit rank equalities for block matrices.</p></abstract>

Analisis Kesulitan Siswa Kelas VII dalam Memecahkan Masalah Matematika Materi Operasi Bentuk Aljabar

This study aims to determine the difficulties of junior high school students in solving problems of operating algebraic forms. This research is included in the type of descriptive research with a qualitative approach, with the research design used is a case study. The subjects of this study were 20 students of class VII SMP Negeri 5 Sendana and 4 students were selected using a purposive sampling technique. The results of the analysis show that students' difficulties in solving algebraic operational problems consist of 4 (four) difficulty criteria, namely; 1) Able to solve problems properly and correctly; 2) Be able to understand the problem and be able to identify information on the problem, be able to plan a solution to the problem but make a mistake in implementing the problem-solving plan and re-examine the answers that have been written; 3) Difficulty understanding the problem, difficulty planning problem solving, difficulty implementing problem solving plans and difficulty re-checking answers; 4) Able to understand the problem and identify the information contained in the problem, experience difficulties in problem-solving plans, difficulties in implementing problem-solving plans and re-checking answers. In conclusion, this study resulted in an analysis of the difficulties of class VII students in solving mathematical problems material for algebraic operations in three ability categories, namely; 1) Subjects with high ability category (ST); 2) Subjects with medium ability category (SS) and 3) Subjects with low ability category (SR)
 
 Keywords: Difficulty analysis, Problem solving, Operations of Algebraic Forms

PENGEMBANGAN LKPD DENGAN PENDEKATAN PMRI PADA MATERI OPERASI PERKALIAN ALJABAR MENGGUNAKAN KONTEKS COVID-19

This study aims to produce a worksheet with the PMRI approach on algebraic multiplication operations material using the context of covid 19 which is valid, practical, and effective. This type of research is research and development from Tessmer which focuses on two stages, namely the preliminary stage and the formative evaluation stage which consists of the expert review, one-to-one evaluation, small group, and field test stages. The subjects of this research were class VII students of SMP Negeri 35 Palembang. Data collection techniques used are interviews, questionnaires and tests. From the results of the study, the following conclusions were obtained: (1) The validity of Student Worksheets can be shown bythe validation results with the validators at the expert review stage with an average score result of 3,65 by three validators. (2) The practicality of Student Worksheets can be shown by the results of the student questionnaire at the one-to-one evaluation stage with an average practicality level of 3.56, the small group stage with an average practicality level of 3.47, and the field test stage with an average practicality level of 3.58. (3) The effectiveness of student worksheets can be shown by student learning outcomes in the form of post tests in the small group stage with an average level of effectiveness of 79.33 and also in the field test stage with an average level of effectiveness of 86.93.

A universal algorithm for linear and quadratic programming in the problems of contact deformation of cable-stayed structures with one-way connections

Problems in production activities that require optimization problems are extremely numerous and very diverse. Optimization approaches are most often associated with the search for the best version of a structure or building. Mathematical methods for solving such problems are developing rapidly and are widely used. They actively and productively penetrate into many areas of scientific research, into engineering and design developments, are an important tool for improving design efficiency throughout the entire process of creating structures. The search for the best design solution is reduced to the selection of a set of parameters that provide a stationary value of the objective function. A wide range of extreme problems of practical orientation, as a rule, in mathematical models contains restrictions on design parameters of the equality-inequality type. In general, their set makes up the content of such a section of mathematics as Mathematical Programming. Due to the fact that there is no single solution method, diversity in research approaches has formed, which divides them into groups, classes, etc. Linear programming (LP), as one of the sections, with a linear objective function and constraints is well studied and successfully applied. Methods of solving problems of nonlinear programming, which includes quadratic programming, are more complex. Therefore, the development of convenient computational schemes is relevant. The essence of this work is that the statements of 2 optimization problems are formalized in a single and convenient form of a symmetric matrix dependence, which makes it possible to obtain an effective (in our opinion) algorithm for their implementation. Namely, a unified scheme for solving both LP and KP problems based on matrix algebra operations is proposed. Quadratic programming (QP), as the second section, also has wide possibilities, in particular, it allows considering the practical tasks of calculating VAT in the mechanics of a deformed solid body under the conditions of contact interaction. Such problems, in particular, include cable-stayed structures with one-way connections and span lengths that can reach tens or hundreds of meters. As an example, the behavior of a model cable-stayed span structure under varying wind loads is considered. The given results may be of interest.

A New Fast Logarithm Algorithm Using Advanced Exponent Bit Extraction for Software-Based Ultrasound Imaging Systems

Ultrasound B-mode imaging provides anatomical images of the body with a high resolution and frame rate. Recently, to improve its flexibility, most ultrasound signal and image processing modules in modern ultrasound B-mode imaging systems have been implemented in software. In a software-based B-mode imaging system, an efficient processing technique for calculating a logarithm instruction is required to support its high computational burden. In this paper, we present a new method to efficiently implement a logarithm operation based on exponent bit extraction. In the proposed method, the exponent bit field is first extracted and then some algebraic operations are applied to improve its precision. To evaluate the performance of the proposed method, the peak signal-to-noise ratio (PSNR) and the execution time were measured. The proposed efficient logarithm operation method substantially reduced the execution time, i.e., eight times, compared to direct computation while providing a PSNR of over 50 dB. These results indicate that the proposed efficient logarithm computation method can be used for lowering the computational burden in software-based ultrasound B-mode ultrasound imaging systems while improving or maintaining the image quality.

Generic Gelfand-Tsetlin modules of quantized and classical orthogonal algebras

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard q-deformed enveloping algebra Uq′(son) defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra U(son). In this paper we only consider the case when q is not a root of unity, and q→1 for the classical case. Extending work by Mazorchuk on son, we provide rational matrix coefficients for these infinite-dimensional modules of both Uq′(son) and U(son). We use these modules with rationalized formulas to embed the respective algebras into skew group algebras of shift operators. Casimir elements of Uq′(son) were given by Gavrilik and Iorgov, and we consider the commutative subalgebra Γ⊂Uq′(son) generated by these elements and the corresponding subalgebra Γ1⊂U(son). The images of Γ and Γ1 under their respective embeddings into skew group algebras are equal to invariant algebras under certain group actions. We use these facts to show that Γ is a Harish-Chandra subalgebra of Uq′(son) and Γ1 is a Harish-Chandra subalgebra of U(son).

3D mirror symmetry and the βγ VOA

We study the simplest example of mirror symmetry for 3d [Formula: see text] SUSY gauge theories: the A-twist of a free hypermultiplet and the B-twist of SQED. We particularly focus on the category of line operators in each theory. Using the work of Costello–Gaiotto, we define these categories as appropriate categories of modules for the boundary vertex operator algebras present in each theory. For the A-twist of a free hyper, this will be a certain category of modules for the [Formula: see text] VOA, properly containing the category previously studied by Allen-Wood. Applying the work of Creutzig–Kanade–McRae and Creutzig–McRae–Yang, we show that the category of line operators on the A side possesses the structure of a braided tensor category, extending the result of Allen-Wood. In addition, we prove that there is a braided tensor equivalence between the categories of line operators on the A side and B side, completing a nontrivial check of the 3d mirror symmetry conjecture. We derive explicit fusion rules as a consequence of this equivalence and obtain interesting relations with associated quantum group representations.

Quasi-Banach algebras and Wiener properties for pseudodifferential and generalized metaplectic operators

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik (Ann Inst Fourier 58:2279–2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219–233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrödinger equations, arXiv:2208.00505).