Solving Non-homogeneous Nested Recursions Using Trees

The solutions to certain nested recursions, such as Conolly's C(n) = C(n-C(n-1))+C(n-1-C(n-2)), with initial conditions C(1)=1, C(2)=2, have a well-established combinatorial interpretation in terms of counting leaves in an infinite binary tree. This tree-based interpretation, which has a natural generalization to a k-term nested recursion of this type, only applies to homogeneous recursions, and only solves each recursion for one set of initial conditions determined by the tree. In this paper, we extend the tree-based interpretation to solve a non-homogeneous version of the k-term recursion that includes a constant term. To do so we introduce a tree-grafting methodology that inserts copies of a finite tree into the infinite k-ary tree associated with the solution of the corresponding homogeneous k-term recursion. This technique can also be used to solve the given non-homogeneous recursion with various sets of initial conditions.Comment: 14 page

Confirming Two Conjectures of Su and Wang

Two conjectures of Su and Wang (2008) concerning binomial coefficients are proved. For $n\geq k\geq 0$ and $b>a>0$, we show that the finite sequence $C_j=\binom{n+ja}{k+jb}$ is a P\'{o}lya frequency sequence. For $n\geq k\geq 0$ and $a>b>0$, we show that there exists an integer $m\geq 0$ such that the infinite sequence $\binom{n+ja}{k+jb}, j=0, 1,...$, is log-concave for $0\leq j\leq m$ and log-convex for $j\geq m$. The proof of the first result exploits the connection between total positivity and planar networks, while that of the second uses a variation-diminishing property of the Laplace transform.Comment: 8 pages, 1 figure, tentatively accepted by adv. in appl. mat

Technology and the Police

“Technology and the Police” primarily seeks to underline the importance of the role technology plays when it comes to law enforcement. As agents of law enforcement, police are needed to protect and defend society. From the use of DNA to video surveillance and tracking tools, technology helps police officers achieve their goal. Furthermore, technology may also serve as a means of control to make sure police are doing their job effectively. Another purpose of “Technology and the Police” is to show that unlike popular belief, the study of history does not start and ends in the past. It is a continuous process from which many lessons can be learnt and understood, it is timeless. Hence, this thesis also attempts to challenge ideas that history and technology cannot blend together because one represents the past and the other represents the present and the future

Comparative Analysis of the Machinability of Various Titanium Alloys

Titanium alloys have a good strength to weight ratio, amazing corrosion resistance. These alloys are mostly used in the aerospace, biomedical and automotive industries. However, titanium alloys are difficult to machine since they are chemically reactive and have low thermal conductivity. Drilling is an important machining process with titanium alloys since it is used is most titanium applications. The machinability of titanium alloys varies depending on the method the titanium alloy was manufactured. There is a lack of knowledge for drilling additive manufactured titanium and powdered metallurgy titanium. This study will investigate and compare the wear of various drill bits with various manufactured titanium alloys using a constant feed rate and cutting speed. The titanium alloys used include the reference alloy of Ti-6Al-4V, powder metallurgy titanium and additive manufactured titanium. The drill bits used include high speed steel drill bits, three flute carbide drill bits and coated carbide drill bits. Additionally, the torque is recorded to analyze the drilling performance for each experiment. The drill bit tips, the drilled holes, and the drilled in the workpiece, and the drilled chips were studied microscopically to determine the wear rate of the drill bits, the material transfer, the quality of the drilled surfaces, and the ease of chip dejection. The additive manufactured titanium showed promising material properties but the machinability aspect showed that the wear was poor compared to the reference Ti-6Al-4V alloy

My Leadership Journey

Tanny Crane is President and C.E.O. of Crane Group. Crane Group, a 70-year old, privately-held diversified portfolio company, is comprised of five operating businesses as well as an investment company that holds real estate, private equity investments, equities / bond portfolio, and hedge funds.https://digitalcommons.otterbein.edu/leadhour/1033/thumbnail.jp