On degree sets in k-partite graphs

The degree set of a k-partite graph is the set of distinct degrees of its vertices. We prove that every set of non-negative integers is a degree set of some k-partite graph.

Understanding the mechanical limitations of the performance of soft X-ray monochromators at MAX IV laboratory

MAX IV is a fourth-generation, or diffraction-limited, synchrotron light source with a number of state-of-the-art beamlines. The performance of a beamline is, to a high degree, set by the energy resolution it can achieve, which in turn is governed to a large extent by the monochromator. During the design phase of a monochromator, the mechanical requirements must be fully understood and met with margin. During commissioning, the performance must be verified and optimized. In this paper, six soft X-ray monochromators at MAX IV beamlines (Bloch, Veritas, HIPPIE, SPECIES, FinEstBeAMS and SoftiMAX) are examined with a focus on their resolving power, energy range and the time required to change measurement range, as those parameters are dependent on each other. The monochromators have a modern commercial design, planned and developed in close collaboration with the vendors. This paper aims to present the current status of the commissioning at MAX IV with emphasis on elucidating the mechanical limitations on the performance of the monochromators. It contains analysis of the outcome and our approach to achieve fast and high-resolution monochromators.

INFORMATION TECHNOLOGIES IN THE WORK OF A CONTEMPORARY TEACHER: CHALLENGES OF NUSH

Introduction.The article substantiates the importance of transformation of semantic content and modernization of educational tools, taking into account the current trendsin the development of the information socie-ty and the active use of information technologies based on the principles of the new Ukrainian school. The article substantiates the need to introduce innovations and change the emphasis in the priorities of the national edu-cation system, focusing on the disclosure of personal potential, the implementation of an individual educational trajectory and the applied nature of knowledge.Purpose. The purpose of the article: to identify the main directions of using information technologies in pri-mary school education, to analyze the possibilities of modern information technologies available for use in school and University settings, in particular, Google soft-ware and applications, in the context of the formation of key competencies and the concept of new Ukrainian school. Methods. To achieve this goal, we used a set of theo-retical (analysis of scientific literature, pedagogical expe-rience) and empirical (questionnaires, surveys, conversa-tions) research methods. Results. And the capabilities of modern information technologies available for use in school and degree set-tings, in particular Google software and applications. Originality. The author outlines the main directions of using information technologies in teaching younger stu-dents in the context of the formation of key competencies and the concept of a new Ukrainian school. Conclusion. The completed research does not exhaust all aspects of further scientific research and does not claim to be complete and complete solutions to the prob-lem. The most promising areas of research include improv-ing educational and methodological support (creating video lectures, electronic textbooks, taking into account modern requirements for the level of teacher training in the context of the NUS concept), improving forms and methods of teaching.

The solvability comes from a given set of character degrees

Let [Formula: see text] be a finite group and the irreducible character degree set of [Formula: see text] is contained in [Formula: see text], where [Formula: see text], and [Formula: see text] are distinct integers. We show that one of the following statements holds: [Formula: see text] is solvable; [Formula: see text]; or [Formula: see text] for some prime power [Formula: see text].

Tripartite graphs with given degree set

If k ≥ 1, then the global degree set of a k-partite graph G = (V1, V2, . . . , Vk, E) is the set of the distinct degrees of the vertices of G, while if k ≥ 2, then the distributed degree set of G is the family of the k degree sets of the vertices of the parts of G. We propose algorithms to construct bipartite and tripartite graphs with prescribed global and distributed degree sets consisting from arbitrary nonnegative integers. We also present a review of the similar known results on digraphs.

Reconstruction of score sets

The score set of a tournament is defined as the set of its different outdegrees. In 1978 Reid [15] published the conjecture that for any set of nonnegative integers D there exists a tournament T whose degree set is D. Reid proved the conjecture for tournaments containing n = 1, 2, and 3 vertices. In 1986 Hager [4] published a constructive proof of the conjecture for n = 4 and 5 vertices. In 1989 Yao [18] presented an arithmetical proof of the conjecture, but general polynomial construction algorithm is not known. In [6] we described polynomial time algorithms which reconstruct the score sets containing only elements less than 7. In [5] we improved this bound to 9. In this paper we present and analyze new algorithms Hole-Map, Hole-Pairs, Hole-Max, Hole-Shift, Fill-All, Prefix-Deletion, and using them improve the above bound to 12, giving a constructive partial proof of Reid’s conjecture.