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In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra ${\cal G}_2$. We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram., 41pages, many figures

If the traditional deep learning framework needs to support a new operator, it usually needs to be highly optimized by experts or hardware vendors to be usable in practice, which is inefficient. The deep learning compiler has proved to be an effective solution to this problem, but it still suffers from unbearably long overall optimization time. In this paper, aiming at the XGBoost cost model in Ansor, we train a cost model based on LightGBM algorithm, which accelerates the optimization time without compromising the accuracy. Experimentation with real hardware shows that our algorithm provides 1.8× speed up in optimization over XGBoost, while also improving inference time of the deep networks by 6.1 %.

The paper discusses the possibilities of constructing vectorized algorithms for solving problems of continuum mechanics, as well as the specialties of their software implementation in the MATLAB. These algorithms, on the one hand, widely use MATLAB functions designed to treat vectors and sparse matrixes, and on the other hand, are distinguished by high efficiency and counting speed.

In this paper Pythagorean fuzzy rough set and Pythagorean fuzzy rough topological spaces are defined for the connected space. Then, the properties of connectedness are discussed with examples.

In the utilization of C language, it often appears that the function calls itself directly or indirectly, that is, the function calls itself recursively. The recursive solution of the same kind of problem can be expressed by recursion, and the specific problem can be reduced by recursion. Based on this, this paper first analyses the concept and function of C language recursion, then studies the process description of recursion algorithm, and finally gives the implementation strategy of C language recursion algorithm.

This paper introduces the idea of £-Smarandache pseudo fuzzy ideal of a pseudo BH–algebra. Some suggestions and examples are given to comprehend the features of this idea and to decide the connected aspects among them with the £-Smarandache pseudo ideal of a pseudo BH–algebra.

Since cryptography relied on mathematical theories and applications of computer science, mathematical algorithms are built with high difficulty. In this paper, we make an improvement and modification NTRU cryptosystem called NTRSH by using a new tripternion algebra and changing the mathematical structure for public and private keys, as well as for text encryption and decryption to obtain more and higher security.

Previously, the Dubrovin method was used to construct spectral curves of multiphase solutions of such integrable models of nonlinear optics as vector and derived nonlinear Schrödinger equations. The method of constructing spectral curves of multiphase solutions of the scalar nonlinear Schrödinger equation was based on replacing matrix differential operator from the Lax pair by the scalar one. The present paper shows that Dubrovin’s method gives the same results. At the same time, it has the advantage that it consists in a simpler construction of multiphase solutions expressed in terms of elementary functions.

NTRU is an open-source public key cryptosystem that encrypts and decrypts data using lattice-based cryptography. It is more resistant to attacks than other common public-key cryptosystems, and its performance has been demonstrated to be significantly higher. Any cryptosystem that uses non-commutative computations in its encryption and decryption processes becomes a highly lattice attack-resistant system as a result of Shamir’s conclusion. This paper presents a new multidimensional public-key cryptosystem which is a non-commutative variant of the NTRU called QOTRU. It operates in the subalgebra of octonion algebra called Qu-octonion subalgebra with a new mathematical structure of two public keys and five private keys. This new structure has enhanced the public key system to be more secure and complex.

In Information Systems (IS), the search for a good design is still a relevant issue. In IS, two sub-systems coexist, applications and data. To articulate these sub-systems, some authors opt for the CRUD matrix. Similarly, Axiomatic Design (AD) theory studies how functional requirements are related to design parameters using a design matrix. In this paper, the application sub-system corresponds to the functional requirements, the data sub-system to the design parameters, and the CRUD matrix to the design matrix. The goal of the CRUD matrix is to maintain the independence of its items to minimize the information content of the design. Similarly, AD uses the design matrix to define a good design. This work aims to develop a theory to create object-oriented elements based on the CRUD matrix aligned with the business strategy.

The paper reveals the development theory of modified Beta Stancu operators. Incorporate Q derivatives and get critical moments from another Q-Beta generator. There is also a simple module inference for Q operators. Primary moments, Voronovskaya method asymptotic process, and error calculations applying these characteristics are produced for these operators. We used Q integral properties to verify the recurrence theorem for Q analog moments of the updated Operators. We still have an asymptotic method. After the day, We examined all such Q operators’ updating to help estimates.