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Let ℚsymm be the compositum of all symmetric extensions of ℚ, i.e., the finite Galois extensions with Galois group isomorphic to Sn for some positive integer n, and let ℤsymm be the ring of integers inside ℚsymm. Then, TH(ℤsymm) is primitive recursively decidable.

We study the variation of the[...]-Selmer groups of the elliptic curves y 2 = x 3 - Dx under quartic twists by square-free integers. We obtain a complete description of the distribution of the size of this group when the integer D is constrained to lie in a family for which the relative Tamagawa number of the isogeny[...]is fixed.

The paper systematically studies the contiguous relations of terminating 4φ3 -series. Utilizing several relations for the Ωλ,µ -series, we formulate specific identities for the terminating q-series with two integer parameters. Their limiting cases are presented in this paper. Additionally, the previous proof methods for the extension of the adjacent relation of terminating q-series are summarized and analysed.

This paper proposes the concept of envelope numbers for irrational numbers, divides envelope numbers into upper envelope numbers and lower envelope numbers, and proves several important properties of such numbers. Firstly, the paper gives the uniform distribution theorem of irrational integer multiples. By proving several lemmas of upper envelope numbers and lower envelope numbers, it is proved that there are countless upper envelope numbers and lower envelope numbers of irrational numbers. At the same time, it is proved that the sum of the maximum upper envelope number and lower envelope number that does not exceed a given positive integer is also an envelope number.

This study investigates the positive integer solutions of an equation involving the Smarandache function. The equation is given by tφ ( n ) = φ 2 ( n ) + S ( n 16 ), where φ ( n ) represents the Euler function, φ e ( n ) represents the generalized Euler function with e as a positive integer, and S ( n ) represents the Smarandache function. The solutions of this equation are discussed, and it is proven that the equation only has positive integer solutions when t = 1, 2, 3, 6, 7, 9, 12, 17, 18, 19. Furthermore, all positive integer solutions of the equation are provided.

In this work, we extend the regularization framework from Kronqvist et al. (Math Program 180(1):285–310, 2020) by incorporating several new regularization functions and develop a regularized single-tree search method for solving convex mixed-integer nonlinear programming (MINLP) problems. We propose a set of regularization functions based on distance metrics and Lagrangean approximations, used in the projection problem for finding new integer combinations to be used within the Outer-Approximation (OA) method. The new approach, called Regularized Outer-Approximation (ROA), has been implemented as part of the open-source Mixed-integer nonlinear decomposition toolbox for Pyomo—MindtPy. We compare the OA method with seven regularization function alternatives for ROA. Moreover, we extend the LP/NLP Branch and Bound method proposed by Quesada and Grossmann (Comput Chem Eng 16(10–11):937–947, 1992) to include regularization in an algorithm denoted RLP/NLP. We provide convergence guarantees for both ROA and RLP/NLP. Finally, we perform an extensive computational experiment considering all convex MINLP problems in the benchmark library MINLPLib. The computational results show clear advantages of using regularization combined with the OA method.

For nonzero coprime integers a and b , a positive integer is said to be good with respect to a and b if there exists a positive integer k such that divides a k + b k . Since the early 1990s, the notion of good integers has attracted considerable attention from researchers. This continued interest stems from both their elegant number‐theoretic structure and their noteworthy applications across several branches in mathematics, with coding theory being among the most prominent areas where they play a crucial role. This paper provides a comprehensive review of good integers, emphasizing both their theoretical foundations and their practical implications. We first revisit the fundamental number‐theoretic properties of good integers and present their characterizations in a systematic manner. The exposition is enriched with well‐structured algorithms and illustrative diagrams that facilitate their computation and classification. Subsequently, we explore applications of good integers in the study of algebraic coding theory. In particular, special emphasis is placed on their roles in the characterization, construction, and enumeration of self‐dual cyclic codes as well as complementary dual cyclic codes. Several examples are provided to demonstrate the applicability of the theory. This review not only consolidates existing results but also highlights the unifying role of good integers in bridging number theory and coding theory.

Cloud manufacturing is an emerging service-oriented manufacturing paradigm that integrates and manages distributed manufacturing resources through which complex manufacturing demands with a high degree of customization can be fulfilled. The process of service selection optimization and scheduling (SSOS) is an important issue for practical implementation of cloud manufacturing. In this paper, we propose new mixed-integer programming (MIP) models for solving the SSOS problem with basic composition structures (i.e., sequential, parallel, loop, and selective). Through incorporation of the proposed MIP models, the SSOS with a mixed composition structure can be tackled. As transportation is indispensable in cloud manufacturing environment, the models also optimize routing decisions within a given hybrid hub-and-spoke transportation network in which the central decision is to optimally determine whether a shipment between a pair of distributed manufacturing resources is routed directly or using hub facilities. Unlike the majority of previous research undertaken in cloud manufacturing, it is assumed that manufacturing resources are not continuously available for processing but the start time and end time of their occupancy interval are known in advance. The performance of the proposed models is evaluated through solving different scenarios in the SSOS. Moreover, in order to examine the robustness of the results, a series of sensitivity analysis are conducted on key parameters. The outcomes of this study demonstrate that the consideration of transportation and availability not only can change the results of the SSOS significantly, but also is necessary for obtaining more realistic solutions. The results also show that routing within a hybrid hub-and-spoke transportation network, compared with a pure hub-and-spoke network or a pure direct network, leads to more flexibility and has advantage of cost and time saving. The level of saving depends on the value of discount factor for decreasing transportation cost between hub facilities.

In this paper, we examine a mixed integer linear programming reformulation for mixed integer bilinear problems where each bilinearterm involves the product of a nonnegative integer variable and a nonnegative continuous variable. This reformulation is obtained by first replacing a general integer variable with its binary expansion and then using McCormick envelopes to linearize the resulting product of continuous and binary variables. We present the convex hull of the underlying mixed integer linear set. The effectiveness of this reformulation and associated facet-defining inequalities are computationally evaluated on five classes of instances. [PUBLICATION ABSTRACT]