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

Biomimetic Design and Fabrication of Interior Architecture of Tissue Scaffolds Using Solid Freeform Fabrication

Modeling, design and fabrication of tissue scaffolds with intricate architecture, porosity and pore size for desired tissue properties presents a challenge in tissue engineering. This paper will present the details of our development in designing and fabrication of the interior architecture of scaffolds using a novel design approach. The Interior Architecture Design (IAD) approach seeks to generate scaffold layered freeform fabrication tool path without forming complicated 3D CAD scaffold models. This involves: applying the principle of layered manufacturing to determine the scaffold individual layered process planes and layered contour; defining the 2D characteristic patterns of the scaffold building blocks (unit cells) to form the Interior Scaffold Pattern; and the generation of process tool path for freeform fabrication of these scaffolds with the specified interior architecture. Feasibility studies applying the IAD algorithm to example models and the generation of fabrication planning instructions will be presented.Mechanical Engineerin

ANALYSIS OF THE WORLD OIL CROPS MARKET

The world of oilseeds, meals, and oils has been evolving. New issues have emerged for researchers and policymakers. The United States is still the largest player, mainly due to its dominant position in soybeans and soybean meals. Nevertheless, the position of the United States has been challenged by several countries. The growth by Brazil and Argentina in the production of soybeans and soybean meal has especially eroded the market share of the United States in recent years. Malaysia and Indonesia have been aggressively marketing their palm oil, intensifying the competition for U.S. soybean oil exports. Other issues include changing trade policies and demand in major importing countries, trade liberalization, and genetically modified crops.oilseeds, soybeans, Brazil, Argentina, trade, Crop Production/Industries,

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