Problem of Optimal Control of the Turbulent Boundary Layer on a Permeable Surface in Supersonic Gas Flow

The problem of constructing the law of distribution of the normal component of the velocity of blowing to the turbulent boundary layer at supersonic flow velocities which ensure the minimum convective heat flow transmitted from the boundary layer to the surface is considered. The power of the control system calculated with regard to Darcy’s law of flow through a porous medium acts as the isoperimetric condition. The problemis solved using the Dorodnitsyn generalized integral relations. The numerical experiments carried out in the case of flow past a sphere showed the effectiveness of the optimal blowing laws as compared with the uniform law, namely, the gain in the minimized functional reaches 31.82%.     

Estimation of Kloosterman sums with primes and its application

Suppose that p is a large prime. In this paper, we prove that, for any natural number N < p the following estimate holds: $$ \left. {\mathop {\max }\limits_{\left( {a,p} \right) = 1} } \right|\left. {\sum\limits_{q \leqslant N} {e^{{{2\pi iaq*} \mathord{\left/ {\vphantom {{2\pi iaq*} p}} \right. \kern-\nulldelimiterspace} p}} } } \right| \leqslant \left( {N^{{{15} \mathord{\left/ {\vphantom {{15} {16}}} \right. \kern-\nulldelimiterspace} {16}}} + N^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} p^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} } \right)p^{0\left( 1 \right)} , $$ where q is a prime and q* is the least natural number satisfying the congruence qq* ≡ 1 (modp). This estimate implies the following statement: if p > N > p ^16/17+? , where ? > 0, and if we have λ ? 0 (modp), then the number J of solutions of the congruence $$ q_1 \left( {q_2 + q_3 } \right) \equiv \lambda \left( {\bmod p} \right) $$ for the primes q ₁, q ₂, q ₃ ≤ N can be expressed as $$ J = \frac{{\pi \left( N \right)^3 }} {p}\left( {1 + O\left( {p^{ - \delta } } \right)} \right), \delta = \delta \left( \varepsilon \right) > 0. $$ This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition p > N > p ^38/39+? was required.     

Concentration of Points on Two and Three Dimensional Modular Hyperbolas and Applications

Let p be a large prime number, K, L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I ₂(M; K, L) of solutions of the congruence

New estimates of double trigonometric sums with exponential functions

We establish a new bound for the exponential sum

Azocalix[4]arene-Rhodamine Supramolecular Hypoxia-Sensitive Systems: A Search for the Best Calixarene Hosts and Rhodamine Guests

A potential hypoxia-sensitive system host-guest complex of three calixarenes (including two with four anionic carboxyl and sulphonate azo fragments on the upper rim and a newly synthesized bis-azo adduct of calixarene in the cone configuration with azo fragments on the lower rim with the most widespread cationic and zwitterionic rhodamine dyes (123, 6G and B)) was studied using UV-VIS spectrometry and fluorescence as well as 1D and 2D NMR techniques. It was found that all three calixarenes form a complex with rhodamine dyes with a 1:1 composition. The association constants of calixarene-dye complexes with sulfonate calixarenes, especially in the case of tetra-anionic calixarene, turned out to be higher compared with carboxyl calixarene due to the more intense electrostatic interactions. For the first time using an HRESI MS technique, it was shown that the treatment of rhodamine 6G and 123 with sodium dithionite (SDT) produces a non-fluorescent leuco form of the dye, and only rhodamine B can be used with SDT without the occurrence of a side reduction. Moreover, it was identified that in addition to the reduction in the azo groups, SDT causes partial cleavage of the aryl ether bonds. The found features of SDT should be taken into account when SDT is used as an azoreductase mimic.     

On the concentration of points of polynomial maps and applications

For a polynomial ${f\in{\mathbb {F}}_p[X]}$ , we obtain upper bounds on the number of points (x, f (x)) modulo a prime p which belong to an arbitrary square with the side length H. Our results in particular are based on the Vinogradov mean value theorem. Using these estimates we obtain results on the expansion of orbits in dynamical systems generated by nonlinear polynomials and we obtain an asymptotic formula for the number of visible points on the curve ${f(x)\equiv y\, ({\rm mod}\, p)}$ , where ${f\in{\mathbb {F}}_p[X]}$ is a polynomial of degree d?≥ 2. We also use some recent results and techniques from arithmetic combinatorics to study the values (x, f (x)) in more general sets.     

Synthesis of cell-permeable stapled BH3 peptide-based Mcl-1 inhibitors containing simple aryl and vinylaryl cross-linkers

We report the synthesis of a series of distance-matching aryl and vinylaryl cross-linkers for constructing stapled peptides containing cysteines at i,i+7 positions. Langevin dynamics simulation studies helped to classify these cross-linkers into two categories: the rigid cross-linkers with narrower S–S distance distribution and the flexible cross-linkers with wider S–S distance distribution. The stapled Noxa BH3 peptides with the flexible distance-matching cross-linkers gave the highest degree of helicity as well as the most potent inhibitory activity against Mcl-1. However, the stapled peptides with the highest hydrophobicity showed the most efficient cellular uptake. Together, this work illustrates the divergent nature of binding affinity and cellular uptake, and the vital importance of choosing appropriate cross-linkers in constructing stapled peptides with the drug-like properties.     

Thermodynamic Efficiency of Interactions in Self-Organizing Systems

The emergence of global order in complex systems with locally interacting components is most striking at criticality, where small changes in control parameters result in a sudden global reorganization. We study the thermodynamic efficiency of interactions in self-organizing systems, which quantifies the change in the system’s order per unit of work carried out on (or extracted from) the system. We analytically derive the thermodynamic efficiency of interactions for the case of quasi-static variations of control parameters in the exactly solvable Curie–Weiss (fully connected) Ising model, and demonstrate that this quantity diverges at the critical point of a second-order phase transition. This divergence is shown for quasi-static perturbations in both control parameters—the external field and the coupling strength. Our analysis formalizes an intuitive understanding of thermodynamic efficiency across diverse self-organizing dynamics in physical, biological, and social domains.