120,939 research outputs found

    Binomial coefficients, Catalan numbers and Lucas quotients

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    Let pp be an odd prime and let a,ma,m be integers with a>0a>0 and m≢0(modp)m \not\equiv0\pmod p. In this paper we determine k=0pa1(2kk+d)/mk\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k mod p2p^2 for d=0,1d=0,1; for example, k=0pa1(2kk)mk(m24mpa)+(m24mpa1)up(m24mp)(modp2),\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2}, where ()(-) is the Jacobi symbol, and {un}n0\{u_n\}_{n\geqslant0} is the Lucas sequence given by u0=0u_0=0, u1=1u_1=1 and un+1=(m2)unun1u_{n+1}=(m-2)u_n-u_{n-1} for n=1,2,3,n=1,2,3,\ldots. As an application, we determine 0<k<pa,kr(modp1)Ck\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k modulo p2p^2 for any integer rr, where CkC_k denotes the Catalan number (2kk)/(k+1)\binom{2k}k/(k+1). We also pose some related conjectures.Comment: 24 pages. Correct few typo

    Generalized linear isotherm regularity equation of state applied to metals

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    A three-parameter equation of state (EOS) without physically incorrect oscillations is proposed based on the generalized Lennard-Jones (GLJ) potential and the approach in developing linear isotherm regularity (LIR) EOS of Parsafar and Mason [J. Phys. Chem., 1994, 49, 3049]. The proposed (GLIR) EOS can include the LIR EOS therein as a special case. The three-parameter GLIR, Parsafar and Mason (PM) [Phys. Rev. B, 1994, 49, 3049], Shanker, Singh and Kushwah (SSK) [Physica B, 1997, 229, 419], Parsafar, Spohr and Patey (PSP) [J. Phys. Chem. B, 2009, 113, 11980], and reformulated PM and SSK EOSs are applied to 30 metallic solids within wide pressure ranges. It is shown that the PM, PMR and PSP EOSs for most solids, and the SSK and SSKR EOSs for several solids, have physically incorrect turning points, and pressure becomes negative at high enough pressure. The GLIR EOS is capable not only of overcoming the problem existing in other five EOSs where the pressure becomes negative at high pressure, but also gives results superior to other EOSs.Comment: 9 pages, 3 figure

    Why Students Drop Out of School: A Review of 25 Years of Research

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    Reviews research on the underlying causes of the high school dropout crisis -- individual and institutional characteristics that predict whether a student is likely to drop out of high school. Discusses student engagement, deviance, and other models
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