Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, "shape" refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function (for definition, see Section 4). It is rather well known that it is possible to approximate a function by algebraic polynomials that preserve its shape (i.e., the Weierstrass approximation theorem is valid for SPA). At the same time, the degree of SPA is much worse than the degree of best unconstrained approximation in some cases, and it is "about the same" in others. Numerous results quantifying this difference in degrees of SPA and unconstrained approximation have been obtained in recent years, and the main purpose of this article is to provide a "bird's-eye view" on this area, and discuss various approaches used. In particular, we present results on the validity and invalidity of uniform and pointwise estimates in terms of various moduli of smoothness. We compare various constrained and unconstrained approximation spaces as well as orders of unconstrained and shape preserving approximation of particular functions, etc. There are quite a few interesting phenomena and several open questions.Comment: 51 pages, 49 tables, survey, published in Surveys in Approximation Theory, 6 (2011), 24-7

Nearly Coconvex Approximation

* Part of this work was done while the second author was on a visit at Tel Aviv University in March 2001Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials, and by piecewise polynomials, which are nearly coconvex with it, namely, polynomials and piecewise polynomials that preserve the convexity of f except perhaps in some small neighborhoods of the points where f changes its convexity. We obtain Jackson type estimates and summarize the positive and negative results in a truth-table as we have previously done for nearly comonotone approximation

Особливості формування творчого підходу в студентів вищої школи економічного профілю у процесі вивчення іноземної мови (The creative attitude formation peculiarities of students in higher economic academic institutions in the foreign) language studying process

У статті розглядаються особливості формування творчого підходу в студентів вищої школи економічного профілю у процесі вивчення іноземної мови. (In the article the peculiarities of creative approach formation of students in higher economic educational institutions in the foreign language studying process are described. Economic disciplines and learning foreign languages nowadays have occupied a leading role not only in economic universities and economic faculties and also in almost all other universities. Professional formation of students in higher education depends on the level of their cognitive properties, such as: perseverance, self-control, emotional stability. A huge influence on the professional development of students in higher education institutions produce their professional orientation, educational objectives and significance of their own activity. Upgraded educational process of higher school of economics should be based on the study of the students’ perception in studying of economic information and foreign languages and levels of development of the students’ creative approach to professional knowledge. This path involves the use of student interest in learning knowledge through the development of creative approach to the process and focus in the right direction of his work. In the development and application of appropriate educational and correctional techniques can organize such an impact on student activities, which will be directed «inside – outside», and namely to improve the adaptability of a student and his teamwork in academic interest group through the student to the educational process. Consider the basic aspects of student creativity and enhance its creative approach to the study of economic sciences, foreign languages and the educational process. Creativity of students in the mastery of professional knowledge in the process of learning a foreign language can be seen as a conscious desire to their deep understanding, evaluation and application in both the academic and professional activities. The aim of the article is to analyze the importance of students’ creativity formation in the process of obtaining professional knowledge in the study of foreign languages to facilitate the reorientation of the educational system of functional-executive basis for content-search and creativity; the factors of the students’ educational information perception of economic subjects and foreign languages, which is based on economic education. From the perception of educational information point of view it is essential to analyze the level of the students’ creative attitude development to professional knowledge, which is especially important to ensure such educational and training activities that will make the learning process interesting for students, according to their goals and wishes and add encouragement in the choice of future profession.

Хірургічне лікування гнійного низхідного медіастиніту, ускладненого сепсисом

Хірургічне лікування гнійного низхідного медіастиніту,ускладненого сепсисо

Legal interpretation of the constitution and the laws of Ukraine by the constitutional court of Ukraine

У статті досліджено повноваження Конституційного Суду України у сфері тлумачення Конституції і законів України. Проаналізовано поняття, види, суб’єкти та об’єкти тлумачення. Автором визначено правову природу рішень Конституційного Суду України щодо офіційного тлумачення Конституції та законів України.The article deals with the investigation of the powers of the Constitutional Court of Ukraine in the interpretation of the Constitution and the laws of Ukraine. The notion, types, subjects and objects of the interpretation have been analyzed. The legal nature of the decisions of the Constitutional Court of Ukraine as to the legal interpretation of the Constitution and the laws of Ukraine is determined