Optimization Methods for Solving Systems Ax=B

У роботі обґрунтовано, що ітераційні методи класу x(k+1)= B(k)x(k) + Bk w(k) , Bk є M n*n(R),w(k) є Rn, Bk є R, не є ефективними при розв’язуванні систем Ax=b, b є imA. З погано зумовленими матрицями A є M n*n(R), rankA = n, довільної структури, великих порядків: сповільнюється швидкість збіжності, оскільки наближення при мінімізації норми вектора нев’язки або вектора похибки попадають в область K min – область мінімальних нев’язок; базисні вектори з підпростору Крилова, на яких ґрунтується збіжність методу, сильно зумовлені, похибки обчислень приводять до не монотонності процесу збіжності. Запропонований двоциклічний алгоритм мінімізує похибку обчислень і строго монотонно збігається. Алгоритм заснований на основі базису Крилова Kr r Ar A r m ={ , ,..., m-1 } , r p – нев’язка і системи повних базисів Ke e Ae A e e i i i i i = { , ,..., , m-1 } i n=1 { } i n=1 – одиничний базис. Базис Krm використовується для побудови початкового наближення, базиси {Kei}in=1 – для уточнення напрямного вектора на розв’язок, у заданій (обчисленій) точці x(0) , що гарантує стійкість процесу обчислень. Критерій прийняття наближеного рішення системи стійкий до похибок.The work proved that kind of iterative methods x(k+1)= B(k)x(k) + Bk w(k) , Bk є M n*n(R),w(k) є Rn, Bk є R, are not effective in solving systems, Ax=b, b є imA with ill predefined matrices A є M n*n(R), rankA = n, arbitrary structure, large orders, slowing the rate of convergence as the approach vector regulations while minimizing the residual error vector or fall in the set K min - set of minimum residuals; basis vectors of Krylov subspace on which the convergence method, greatly due, calculation errors do not lead to monotony process of convergence. The proposed algorithm based dvotsyklichnyy which minimizes the error computation and strictly monotonously the same. The algorithm is based on the basis of the Krylov basis Kr r Ar A r m ={ , ,..., m-1 } , r – discrepancy and complete system of bases Ke e Ae A e e i i i i i = { , ,..., , m-1 } i n=1 { } i n=1 – unit basis. The basis Krm used to build the initial approach, bases {Kei}in=1 – to refine the guide on the solution vector in the set (computed) point x(0) that guarantees process stability calculations. Criterion adoption approximate solution of a system resistant to errors

ON CHOOSING RATIONAL SET PARAMETERS OF THE PISTON PNEUMATIC ENGINE WITH VALVE AIR-DISTRIBUTION PERFORMANCE

The results of estimate research with the purpose of choosing the rational values of the set working parameters of the piston pneumatic engine with valve air-distribution and automated shift of opening and closing phases depending on loading, frequency of rotation and operating conditions have been considered

Molecular weight and pH aspects of the efficacy of oligochitosan against methicillin-resistant Staphylococcus aureus (MRSA)

Oligochitosan samples varying in molecular weight (Mw) and having narrow polydispersities were prepared by means of depolymerization of chitosan in hydrochloric acid, and their antibacterial activity against methicillin-resistant Staphylococcus aureus (MRSA) was measured at pH values 5.5-8.0. The antibacterial testing of oligochitosans obtained showed that oligochitosans having Mw in the range of 0.73-20.0 kDa could be used both at slightly acidic and neutral pH values, and that the activity against MRSA remained moderate for oligochitosan samples having Mw about 3-5 kDa even at slightly basic pH values. The self-assembling behavior of oligochitosan macromolecules in the dilute solution at various pH values as a function of chain length was investigated. At first it was shown that oligochitosans formed supramolecular aggregates in dilute solutions below the critical pH value 6.5. Despite the aggregation phenomenon, the formation of nano-sized aggregates did not prevent oligochitosan from demonstrating the bactiostatic activity. © 2011 Elsevier Ltd. All Rights Reserved

Basic System in the Problems of Mathematical Modeling

У статті виведені формули інтерполяції та чисельних квадратур з використанням сіток з вузлами послідовності золотого перерізу. Доведено, що такі сітки мінімізують похибку обчислень, а коефіцієнти інтерполяційного многочлена Лагранжа та квадратурної (кубатурної) формули на його основі є лінійними формами параметра золотого перерізу з цілими раціональними коефіцієнтами. В результаті дослідження, дійшли до висновку, що узагальнені формули золотого перерізу використовують для мінімізацію похибок квадратурних формул. Таким чином можна обґрунтувати побудову оптимізаційних методів на основі послідовностей золотого перерізу.Formulas of interpolation and numerical integration on grids, received on the base of golden ratio, were obtained. It was proved, that these grids have properties of minimizing error of computations and Lagrange coefficients of the polynomial interpolation and quadrature (cubature) forms on the basis thereof are linear forms of the parameter of the golden section with integer (rational) coefficients. The study, concluded that the generalized formula golden section is used to minimize errors of quadrature formulas. So you can justify building optimization techniques based on the sequences of the golden section

The MHD nature of ionospheric wave packets excited by the solar terminator

We obtained the first experimental evidence for the magnetohydrodynamic (MHD) nature of ionospheric medium-scale travelling wave packets (MSTWP). We used data on total electron content (TEC) measurements obtained at the dense Japanese network GPS/GEONET (1220 stations) in 2008-2009. We found that the diurnal, seasonal and spectral MSTWP characteristics are specified by the solar terminator (ST) dynamics. MSTWPs are the chains of narrow-band TEC oscillations with single packet's duration of about 1-2 hours and oscillation periods of 10-20 minutes. Their total duration is about 4--6 hours. The MSTWP spatial structure is characterized by a high degree of anisotropy and coherence at the distance of more than 10 wavelengths. The MSTWP direction of travelling is characterized by a high directivity regardless of seasons. Occurrence rate of daytime MSTWPs is high in winter and during equinoxes. Occurrence rate of nighttime MSTIDs has its peak in summer. These features are consistent with previous MS travelling ionosphere disturbance (TID) statistics obtained from 630-nm airglow imaging observations in Japan. In winter, MSTWPs in the northern hemisphere are observed 3-4 hours after the morning ST passage. In summer, MSTWPs are detected 1.5-2 hours before the evening ST occurrence at the point of observations, at the moment of the evening ST passage in the magneto-conjugate point. Both the high Q-factor of oscillatory system and synchronization of MSTWP occurrence with the solar terminator passage at the point of observations and in the magneto-conjugate area testify the MHD nature of ST-excited MSTWP generation. The obtained results are the first experimental evidence for the hypothesis of the ST-generated ion sound waves.Comment: 12 pages, 3 figure

INCREASE OF INDEXES OF CAR ENGINE OPERATION AT EXPENSE OF FOUR-VALVE CYLINDER HEAD APPLICATION

The results of research of car engine operation index at transition from two-valve to fourvalve cylinder head of MeMЗ-307 car engine are presented