Counting permutations by alternating descents

We find the exponential generating function for permutations with all valleys even and all peaks odd, and use it to determine the asymptotics for its coefficients, answering a question posed by Liviu Nicolaescu. The generating function can be expressed as the reciprocal of a sum involving Euler numbers. We give two proofs of the formula. The first uses a system of differential equations. The second proof derives the generating function directly from general permutation enumeration techniques, using noncommutative symmetric functions. The generating function is an "alternating" analogue of David and Barton's generating function for permutations with no increasing runs of length 3 or more. Our general results give further alternating analogues of permutation enumeration formulas, including results of Chebikin and Remmel

Shattering Thresholds for Random Systems of Sets, Words, and Permutations

This paper considers a problem that relates to the theories of covering arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability thresholds. Specifically, we want to find the number of subsets of [n]:={1,2,....,n} we need to randomly select, in a certain probability space, so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to words, we ask for the number of n-letter words on a q-letter alphabet that are needed to shatter all t-subwords of the q^n words of length n. Finally, we explore the number of random permutations of [n] needed to shatter (specializing to t=3), all length 3 permutation patterns in specified positions. We uncover a very sharp zero-one probability threshold for the emergence of such shattering; Talagrand's isoperimetric inequality in product spaces is used as a key tool.Comment: 25 page

Strain-induced energy band gap opening in two-dimensional bilayered silicon film

This work presents a theoretical study of the structural and electronic properties of bilayered silicon films under in-plane biaxial strain/stress using density functional theory. Atomic structures of the two-dimensional silicon films are optimized by using both the local-density approximation and generalized gradient approximation. In the absence of strain/stress, five buckled hexagonal honeycomb structures of the bilayered silicon film have been obtained as local energy minima and their structural stability has been verified. These structures present a Dirac-cone shaped energy band diagram with zero energy band gaps. Applying tensile biaxial strain leads to a reduction of the buckling height. Atomically flat structures with zero bucking height have been observed when the AA-stacking structures are under a critical biaxial strain. Increase of the strain between 10.7% ~ 15.4% results in a band-gap opening with a maximum energy band gap opening of ~168.0 meV obtained when 14.3% strain is applied. Energy band diagram, electron transmission efficiency, and the charge transport property are calculated.Comment: 18 pages, 5 figures, 1 tabl

Consumer Choice of Private Label or National Brand: The case of organic and non-organic milk

We use a two-stage, sample selection model to investigate organic milk purchases using Neilsen’s Homescan data. In the first stage, households decide on a weekly basis to buy mainly organic milk or non-organic milk. Results from this stage show that higher income, better education, having children at home, and several other demographic and marketing variables have a positive effect on organic choice. In the second stage, consumers then choose to buy mainly private label milk or national brand milk conditional on their first-stage choice. Most demographic and marketing variables are found to affect the organic and non-organic private label decision in the same way. However, our results show that a few factors, such as marriage status and children, affect the private label decision differently for organic and non-organic milk customers.organic milk, private label, sample selection, Agribusiness, Consumer/Household Economics, Demand and Price Analysis,

Logarithmic Representability of Integers as k-Sums

A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each element in {0,1,...,kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function phi(n), with lim(phi(n))=infinity, we say that A is a truncated phi(n)-representative k-basis for [n] if for each j in [alpha n, (k-alpha)n] the number of ways that j can be represented as a k-sum of elements of A_{k,n} is Theta(phi(n)). In this paper, we follow tradition and focus on the case phi(n)=log n, and show that a randomly selected set in an appropriate probability space is a truncated log-representative basis with probability that tends to one as n tends to infinity. This result is a finite version of a result proved by Erdos (1956) and extended by Erdos and Tetali (1990).Comment: 18 page

Utilizing blockchain technology for clinical trial optimization

Clinical trials are the cornerstone of treatment discovery because they provide comprehensive scientific evidence on the safety, efficacy, and optimal use of therapeutics. However, current clinical trials are facing multiple challenges such as patient recruitment, data capture, and overall management. There are various causes of patient recruitment challenges such as inefficient advertising models, complex protocols, and distant trial sites. Data inconsistency is the main challenge of the data capture process. Source data verification, a standard method used for data monitoring, is resource-intensive that can cost up to 25 percent of the total budget. The current clinical trial management system market is fragmented and lacks thorough designs with all desired features so that nearly all respondents to management systems from the annual global survey reported dissatisfaction with the current management system. Based on these challenges, disruptive technologies such as blockchain may provide feasible solutions by utilizing its unique features. Blockchain is an open-source distributed ledger technology that was first applied in the financial sector. Its features such as public audibility, data security, immutability, anonymity, and smart contracts are a good fit for the needs of many healthcare applications. However, there are several common challenges of blockchain technology so that most blockchain designs for healthcare applications are still in the early stage of implementation. This dissertation aims at optimizing clinical trials by developing multiple applications using blockchain technology to provide feasible solutions to the current challenges. We will use real-world data to conduct large-scale simulations to evaluate the feasibility and performance of proposed blockchain models for clinical trial applications