Shape-preserving projections in low-dimensional settings and the q-monotone case

Let P:X → V be a projection from a real Banach space X onto a subspace V and let S ⊂ X. In this setting, one can ask if S is left invariant under P, i.e., if PS ⊂ S. If V is finite-dimensional and S is a cone with particular structure, then the occurrence of the imbedding PS ⊂ S can be characterized through a geometric description. This characterization relies heavily on the structure of S, or, more specifically, on the structure of the cone S * dual to S. In this paper, we remove the structural assumptions on S * and characterize the cases where PS ⊂ S. We note that the (so-called) q-monotone shape forms a cone that (lacks structure and thus) serves as an application for our characterization.Нехай P:X→V — проекцiя дiйсного банахового простору X на пiдпростiр V i, крiм того, S⊂X. У цiй постановцi виникає питання: чи є S лiвоiнварiантним пiд дiєю P, тобто чи має мiсце вкладення PS⊂S? Якщо пiдпростiр V є скiнченновимiрним, а S є конусом iз певною структурою, то вкладення PS⊂S може бути охарактеризовано шляхом геометричного опису. Ця характеризацiя iстотно залежить вiд структури S, або, точнiше, вiд структури конуса S∗, спряженого до S. У цiй роботi усунено структурнi припущення щодо S∗ i охарактеризовано випадки, у яких PS⊂S. Вiдзначено, що (так звана) q-монотонна форма утворює конус, який (не має структури i тому) може бути використаний для застосування нашої характеризацiї

Shape-preserving projections in low-dimensional settings and the q-monotone case

Let P:X → V be a projection from a real Banach space X onto a subspace V and let S ⊂ X. In this setting, one can ask if S is left invariant under P, i.e., if PS ⊂ S. If V is finite-dimensional and S is a cone with particular structure, then the occurrence of the imbedding PS ⊂ S can be characterized through a geometric description. This characterization relies heavily on the structure of S, or, more specifically, on the structure of the cone S * dual to S. In this paper, we remove the structural assumptions on S * and characterize the cases where PS ⊂ S. We note that the (so-called) q-monotone shape forms a cone that (lacks structure and thus) serves as an application for our characterization.Нехай P:X→V — проекцiя дiйсного банахового простору X на пiдпростiр V i, крiм того, S⊂X. У цiй постановцi виникає питання: чи є S лiвоiнварiантним пiд дiєю P, тобто чи має мiсце вкладення PS⊂S? Якщо пiдпростiр V є скiнченновимiрним, а S є конусом iз певною структурою, то вкладення PS⊂S може бути охарактеризовано шляхом геометричного опису. Ця характеризацiя iстотно залежить вiд структури S, або, точнiше, вiд структури конуса S∗, спряженого до S. У цiй роботi усунено структурнi припущення щодо S∗ i охарактеризовано випадки, у яких PS⊂S. Вiдзначено, що (так звана) q-монотонна форма утворює конус, який (не має структури i тому) може бути використаний для застосування нашої характеризацiї

Hygienic assessment of PM 10 and PM 2.5 contents in the atmosphere and population health risk in zones infleunced by emissions from stationary sources located at industrial enterprises

Our research focused on air contamination with solid particles which occurred in settlements influenced by stationary sources located at enterprises involved in construction materials production. Our goal was to examine concentrations and fractional structure of solid particles and to assess health risks caused by air contamination with fine-dispersed solid particles for population living on territories adjoining to sanitary-hygienic zones of industrial enterprises. The research was conducted with laboratory control techniques, health risk assessment, sanitary-hygienic and statistic techniques. We measured solid particles concentrations in real-time detecting them incessantly, and it allowed us to obtain data on concentrations of fine-dispersed solid particles (10 and 2.5 microns diameter) averaged over 20-minutes period; we also managed to calculate sums of solid particles (dust/aerosol not differentiated in its compound) in the atmosphere in settlements influenced by stationary sources located at in-dustrial enterprises. We analyzed fractional structure of solid particles, performed a hygienic assessment of atmospheric air contamination, and determined population health risks caused by atmospheric air contamination with fine-dispersed particles. The obtained results gave grounds for working out analytical (laboratory) techniques for control over atmospheric air contamination at a border between a residential area and a sanitary-hygienic zone and for hygienic assessment of solid particles content in the air in settlements, both for overall fraction and for particles with aerodynamic diameter 10 microns and 2.5 micron

Coconvex Pointwise Approximation

Assume that a function f ∈ C[−1, 1] changes its convexity at a finite collection Y := {y 1, ... y s} of s points yi ∈ (−1, 1). For each n > N(Y), we construct an algebraic polynomial Pn of degree ≤ n that is coconvex with f, i.e., it changes its convexity at the same points yi as f and |f(x)−Pn(x)| ≤ cω₂ (f, (√(1−x²))/n,x∈[−1,1], where c is an absolute constant, ω₂(f, t) is the second modulus of smoothness of f, and if s = 1, then N(Y) = 1. We also give some counterexamples showing that this estimate cannot be extended to the case of higher smoothness.Нехай функція f ∈ C[−1,1] змінює свою опуклість у скінченному наборі Y := {y₁,...ys} точок yi ∈ (−1,1). Для кожного n > N(Y) будується алгебраїчний многочлен Pn степеня ≤n, який є коопуклим з f, тобто змінює свою опуклість в тих самих точках yi, що й f, а |f(x)−Pn(x)| ≤ cω₂ (f, (√(1−x²))/n,x∈[−1,1], де c — абсолютна стала, ω₂(f,t)—другий модуль неперервності f, і якщо s=1, то N(Y)=1. Наведено також контрприклади, що показують, зокрема, неможливість поширення цієї оцінки для більшої гладкості

Weighted D-T moduli revisited and applied

$f\in L_p[-1,1]\cap C^{r-1}(-1,1)$ , $r\ge1$ , that have an $(r-1)$ st absolutely continuous derivative in $(-1,1)$ and such that $\varphi^rf^{(r)}$ is in $L_p[-1,1]$ , where $\varphi(x)=(1-x^2)^{1/2}$ . These moduli are equivalent to certain weighted D-T moduli, but our definition is more transparent and simpler. In addition, instead of applying these weighted moduli to weighted approximation, which was the purpose of the original D-T moduli, we apply these moduli to obtain Jackson-type estimates on the approximation of functions in $L_p[-1,1]$ (no weight), by means of algebraic polynomials. We also have some inverse theorems that yield characterization of the behavior of the derivatives of the function by means of its degree of approximation

Calculation and Construction of Load Diagrams and Static Characteristics of Multi-Motor Electric Drive System Using Methods of Equivalent Forces and Reduced Moments

This paper includes the calculations and construction of load diagrams and static characteristics of individual mechanisms of a mining combine, which form a combined multi-motor electric drive system, using methods of equivalent efforts and reduced moments. The operating forces of idle and on-load mechanisms are calculated, operation time intervals and a complete working cycle are determined. The load diagrams of mechanisms are constructed and their kinematic diagrams are designed. Based on the calculations made, asynchronous motors with a short-circuit rotor, which have the closest power values and are suitable in terms of voltage and operating mode, were selected from the catalogue. The gear reduction ratio is calculated, static resistance moments and operating speeds to the motor shaft are specified. The static characteristics of the most powerful electric motor of a mining combine - effector - are formulated