Binomial coefficients, Catalan numbers and Lucas quotients

Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for any integer $r$, where $C_k$ denotes the Catalan number $\binom{2k}k/(k+1)$. We also pose some related conjectures.Comment: 24 pages. Correct few typo

Generalized linear isotherm regularity equation of state applied to metals

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

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

UV-Photolithography Fabrication of Poly-Ethylene Glycol Hydrogels Encapsulated with Hepatocytes

The development of biomanufacturing technologies particularly, layered manufacturing has advanced cell encapsulation processes in an effort to mimic the cellular microenvironment for invitro studies. This paper illustrates an inexpensive UV-photolithographic method for encapsulation of human hepatocytes in three dimensional structures using poly-ethylene diacrylate (PEGDA) hydrogels as candidate substrates. In order to further develop this technology for layered fabrication, we have quantified the long-term effects of the photo-initiator concentration and UV light exposure on the metabolic rates of encapsulated human hepatocytes under a 21 day study. The photoinitator toxicity was observed immediately after polymerization with no significant cytotoxicity on a long term basis. A cellular viability is examined and reported for the UV photopolymerization process. Cell phenotype maintenance was observed by measuring the amount of urea produced over a 1 week time period. This photo encapsulation process may find use in the fabrication of spatially complex 3D scaffolds for tissue engineering applications, elucidation of the 3D structure-pharmacokinetic response relationship and the fabrication of complex multi-compartment liver tissue analog devices for drug screening applications.Mechanical Engineerin

Precision Extrusion Deposition of Polycaprolactone/Hydroxyapatite Tissue Scaffolds

Freeform fabrication provides an effective process tool to manufacture advanced tissue scaffolds with specific designed properties. Our research focuses on using a novel Precision Extrusion Deposition (PED) process technique to directly fabricate Polycaprolactone (PCL) and composite PCL/ Hydroxyapatite (HA) tissue scaffolds. The scaffold morphology and the mechanical properties were evaluated using SEM and mechanical testing. In vitro biological studies were conducted to investigate the cellular responses of the composite scaffolds. Results and characterizations demonstrate the viability of the PED process as well as the good mechanical property, structural integrity, controlled pore size, pore interconnectivity, and the biological compatibility of the fabricated scaffolds.Mechanical Engineerin

Kinetics of dissociative chemisorption of methane and ethane on Pt(110)-(1X2)

The initial probability of dissociative chemisorption Pr of methane and ethane on the highly corrugated, reconstructed Pt(110)‐(1×2) surface has been measured in a microreactor by counting the number of carbon atoms on the surface following the reaction of methane and ethane on the surface which was held at various constant temperatures between 450 and 900 K during the reaction. Methane dissociatively chemisorbs on the Pt(110)‐(1×2) surface with an apparent activation energy of 14.4 kcal/mol and an apparent preexponential factor of 0.6. Ethane chemisorbs dissociatively with an apparent activation energy of 2.8 kcal/mol and an apparent preexponential factor of 4.7×10^(−3). Kinetic isotope effects were observed for both reactions. The fact that P_r is a strong function of surface temperature implies that the dissociation reactions proceed via a trapping‐mediated mechanism. A model based on a trapping‐mediated mechanism is used to explain the observed kinetic behavior. Kinetic parameters for C–H bond dissociation of the thermally accommodated methane and ethane are extracted from the model