Optimization of Adams-type difference formulas in Hilbert space W2(2,1)(0,1)W_2^{(2,1)}(0,1)

In this paper, we consider the problem of constructing new optimal explicit and implicit Adams-type difference formulas for finding an approximate solution to the Cauchy problem for an ordinary differential equation in a Hilbert space. In this work, I minimize the norm of the error functional of the difference formula with respect to the coefficients, we obtain a system of linear algebraic equations for the coefficients of the difference formulas. This system of equations is reduced to a system of equations in convolution and the system of equations is completely solved using a discrete analog of a differential operator $d^2/dx^2-1$. Here we present an algorithm for constructing optimal explicit and implicit difference formulas in a specific Hilbert space. In addition, comparing the Euler method with optimal explicit and implicit difference formulas, numerical experiments are given. Experiments show that the optimal formulas give a good approximation compared to the Euler method

CCD Photometry of Standard Stars at Maidanak Astronomical Observatory in Uzbekstan: Transformations and Comparisons

Observation of standard stars is of crucial importance in stellar photometry. We have studied the standard transformation relations of the UBVRI CCD photometric system at the Maidanak Astronomical Observatory in Uzbekistan. All observations were made with the AZT-22 1.5m telescope, SITe 2k CCD or Fairchild 486 CCD, and standard Bessell UBVRI filters from 2003 August to 2007 September. We observed many standard stars around the celestial equator observed by SAAO astronomers. The atmospheric extinction coefficients, photometric zero points, and time variation of photometric zero points of each night were determined. Secondary extinction coefficients and photometric zero points were very stable, while primary extinction coefficients showed a distinct seasonal variation. We also determined the transformation coefficients for each filter. For B, V, R, and I filters, the transformation to the SAAO standard system could be achieved with a straight line or a combination of two straight lines. However, in the case of the U filter and Fairchild 486 CCD combination, a significant non-linear correction term - related to the size of Balmer jump or the strength of the Balmer lines - of up to 0.08 mags was required. We found that our data matched well the SAAO photometry in V, B-V, V-I, and R-I. But in U-B, the difference in zero point was about 3.6 mmag and the scatter was about 0.02 mag. We attribute the relatively large scatter in U-B to the larger error in U of the SAAO photometry. We confirm the mostly small differences between the SAAO standard UBVRI system and the Landolt standard system. We also attempted to interpret the seasonal variation of the atmospheric extinction coefficients in the context of scattering sources in the earth's atmosphere.Comment: 14 pages, 13 figure

Investigation of the Surface Structure of Si, Ge, GaAs, GaP by the Method of Ion Bombarding When Growing Solid Solutions from a Liquid Phase

The article presents the results of simulation with the method of ion bombardment of the surface of Si, Ge, GaAs, GaP. Theoretically, the impact parameters p(?) of surface components are determined when exposed to Ar+, Ne+, Xe+ ions with an initial energy of 1 keV, at scattering angles in the range from 00

Solution of the Goursat problem for a fourth-order hyperbolic equation with singular coefficients by the method of transmutation operators

In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operato

The New White Flight

White charter school enclaves—defined as charter schools located in school districts that are thirty percent or less white, but that enroll a student body that is fifty percent or greater white— are emerging across the country. The emergence of white charter school enclaves is the result of a sobering and ugly truth: when given a choice, white parents as a collective tend to choose racially segregated, predominately white schools. Empirical research supports this claim. Empirical research also demonstrates that white parents as a collective will make that choice even when presented with the option of a more racially diverse school that is of good academic quality. Despite the connection between collective white parental choice and school segregation, greater choice continues to be injected into the school assignment process. School choice assignment policies, particularly charter schools, are proliferating at a substantial rate. As a result, parental choice rather than systemic design is creating new patterns of racial segregation and inequality in public schools. Yet the Supreme Court’s school desegregation jurisprudence insulates racial segregation in schools ostensibly caused by parental choice rather than systemic design from regulation. Consequently, the new patterns of racial segregation in public schools caused by collective white parental choice largely escapes regulation by courts. This article argues that the time has come to reconsider the legal and normative viability of regulating racial segregation in public schools caused by collective white parental choice. The article makes two important contributions to the legal literature on school desegregation. First, using white charter school enclaves as an example, it documents the ways in which school choice policies are being used to allow whites as a collective to satisfy their preference for segregated predominately white schools. Second, the article sets forth both constitutional and normative arguments for regulating the private choices that result in stark racial segregation patterns in public schools