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

Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same. The main purpose of this paper is to provide an update to our 2011 survey paper. In particular, we discuss recent uniform estimates in comonotone approximation, mention recent developments and state several open problems in the (co)convex case, and reiterate that co-$q$-monotone approximation with $q\ge 3$ is completely different from comonotone and coconvex cases. Additionally, we show that, for each function $f$ from $\Delta^{(1)}$, the set of all monotone functions on $[-1,1]$, and every $\alpha>0$, we have $$\limsup_{n\to\infty} \inf_{P_n\in\mathbb P_n\cap\Delta^{(1)}} \left\| \frac{n^\alpha(f-P_n)}{\varphi^\alpha} \right\| \le c(\alpha) \limsup_{n\to\infty} \inf_{P_n\in\mathbb P_n} \left\| \frac{n^\alpha(f-P_n)}{\varphi^\alpha} \right\|$$ where $\mathbb P_n$ denotes the set of algebraic polynomials of degree $<n$, $\varphi(x):=\sqrt{1-x^2}$, and $c=c(\alpha)$ depends only on $\alpha$

Dynamics of individual indicators of protein metabolism in the body of broiler chickens on the background of combined stress when included in the diet “Reasil Humic Vet” + “Laktin” and “Reasil Humic Health”

The body of&nbsp; broiler chickens is characterized by a high metabolism, which promotes rapid growth and development, but their performance indicators are largely related to housing conditions, which are known to include a number of technological stressors. With the development of stress in the body of poultry changes the course of metabolic processes, which causes changes in all types of metabolism, including protein. The aim of the study was to determine the changes in individual indicators of protein metabolism in the body of broiler chickens against the background of combined stress when included in the diet “Reasil Humic Vet” + “Laktin” and “Reasil Humic Health”. At 13th day of life, all clinically healthy poultry were exposed to combined stress – revaccination (intranasal Newcastle disease) plus cold stress (for 60 minutes by air conditioning and 5 °C in the vivarium). Material for research was selected for 3 days after the action of the stress factor (stage of anxiety), 13, 20 and 26 days after the action of the stress factor (the resistance stage). The material for the study was blood plasma, which determined the concentration of total protein, as well as the content of albumin, globulin fraction (α1, α2, β, γ). It was found that 3 days after the action of stress in chickens of Control group the development of adaptive reactions is manifested by the stability of the total protein content and redistribution of the fractional composition of plasma globulins in the form of increased albumin and α2-globulins. At different stages of development of the stage of resistance in birds of Control group, the content of total protein decreases by an average of 14.6 % due to the content of albumin by 6.9 % and α2-globulins – by 15.9 %, against the background of increasing α1- and γ-globulins by 23.1 and 33.5 % with the stabilization of individual studied indicators at the final stages of development of the resistance stage. The use in the diet of broilers feed “Reasil Humic Vet”, probiotic feed supplement “Laktin”, feed “Reasil Humic Health” under the influence of complex stress helps to increase the intensity of protein metabolism in poultry with the development of adaptation syndrome, as evidenced by the growth of total protein on average by 37.8 % (P &lt; 0.05) and albumin – by 17.0 % (P &lt; 0.05), which indicates an increase in the intensity of protein-synthesizing properties of the organism. In different periods of stress in the blood of broiler chickens Research groups there is an increase in the ratio of individual protein fractions, especially the content of γ-globulins - an average of 21.3 % (P &lt; 0.05), which indicates an increase in the immune status of their body with a predominance of numerical values in the Research 1 group of poultry