Сборник заданий межрегиональной олимпиады школьников «Высшая проба». Физика. Электроника. Информатика
We calculate characteristic polynomials of operators explicitly represented as polynomials of rank $1$ operators. Applications of the results obtained include a generalization of the Forman--Kenyon's formula for a determinant of the graph Laplacian and also provide its level $2$ analog involving summation over triangulated nodal surfaces with boundary.
Reinterpretation of approaches of John Locke, Immanuel Kant and Charles Peirce helps to single out three integral organons of cognition. One is mathematics, or cognition of measure and art of all kind of measurement. Another is morphology or cognition of forms and art of arranging shapes and configurations. One more is semiotics or cognition of meanings and art of their transfer. It is demonstrated that all three organons vary and provide specific fields of knowledge and areas of research. Current versions and varieties of disciplinary manifestations of organons are reviewed. Mathematics is the most developed complex of scientific disciplines. Morphology is a constellation of a number of assorted and fully independent disciplines. Semiotics is rifted by a gap between rough outline of general or «pure» semiotics (Morris) and a nebula of unevenly elaborated semiologies of various sorts – that of languages, literatures, cinema, heraldry, race discrimination or ideological manipulations. Analysis of political discourses and speech acts can contribute to integration of common area of semiotic research.
The introductory article clarifies the title of the current issue of «METHOD» and explicates the purpose of the entire publication. It explains slight but telling differences between the Russian, English and German phrasings that expound the meaning of the title and purpose of the yearbook. Subtle but indicative differences between languages and modes of speech and thought highlight a major issue of knowledge transfer. The yearbook departs from knowledge transfer to a more incentive issue of convergence and divergence of cognitive skills. Introduction focuses on transdisciplinary organons. They derive from our basic cognitive abilities. The initial one is the faculty to tell relative degrees of our sensations (bigger - smaller, warmer - colder etc.) and then to rate sizes of things and intensity of processes. The following one is pattern recognition or our ability to single out certain ‘rated’ entities from their environment. The subsequent one is our capacity to assign meaning to the ‘recognized’ figures and forms of the world around. It further supplements with the gift to use words and images to grasp sense and to convey it. Each of the three fundamental cognitive abilities diverge into further generations of abundant skills and proficiencies. Elaborate methods of scientific research outreach to thresholds of our knowledge. Right there they intertwine with each other. Interdisciplinary linkages develop. Transdisciplinary prospects loom. We conceive imminent convergence of our methodological skills into three transdisciplinary organons congenial to the three cognitive abilities. The first one is metretics or the higher technique of measurement and calculus. It resides in mathematical and statistical studies. The next one is morphetics or the expertise of exploring forms, shapes and figures. It resides in all kinds of morphological, comparative, configurative and evolutionary research. The last one is semiotics or the art of processing sense and reference. It resides in still budding semiologies, cognitive arts and still rudimentary humanities.
The paper discusses in detail the scale of translation of primary points scored by school graduates in the unified state exam in mathematics, used from 2013 to the present time. Based on the analysis of the dynamics of these scales, a conclusion is made about the annual increase in the "average" 100-point result, as well as the presence of a significant increase in the final grade compared with the linear scale. Additionally, the authors describe the effect of reducing the value of primary points as they approach the maximum.