Physical Sciences and Mathematics Commons^™

Elementary Differential Equations With Boundary Value Problems, William F. Trench

William F. Trench

Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation. If your syllabus includes Chapter 10 (Linear Systems of Differential Equations), your students should have some prepa- ration in linear algebra. In writing this book I have been guided by the these principles: An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student’s place, and have chosen to err on the side of too much detail rather than not enough. …

Elementary Differential Equations, William F. Trench

William F. Trench

Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation. If your syllabus includes Chapter 10 (Linear Systems of Differential Equations), your students should have some preparation in linear algebra. In writing this book I have been guided by the these principles: An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student’s place, and have chosen to err on the side of too much detail rather than not enough. An …

Student Solutions Manual For Elementary Differential Equations And Elementary Differential Equations With Boundary Value Problems, William F. Trench

William F. Trench

No abstract provided.

Introduction To Real Analysis, William F. Trench

William F. Trench

This is a text for a two-term course in introductory real analysis for junior or senior math- ematics majors and science students with a serious interest in mathematics. Prospective educators or mathematically gifted high school students can also benefit from the mathe- matical maturity that can be gained from an introductory real analysis course. The book is designed to fill the gaps left in the development of calculus as it is usually presented in an elementary course, and to provide the background required for insight into more advanced courses in pure and applied mathematics. The standard elementary calcu- lus sequence …

Finite Symmetries Of S^4, Weimin Chen Chen, Slawomir Kwasik, Reinhard Shultz

Weimin Chen

This paper discusses topological and locally linear actions of finite groups on S4. Local linearity of the orientation preserving actions on S4 forces the group to be a subgroup of SO(5). On the other hand, orientation reversing topological actions of “exotic” groups G (i.e. G 6⊂ O(5)) on S4 are constructed, and local linearity and stable smoothability of the actions are studied.

Modeling The Progress And Retention Of International Students Using Markov Chains, Lucas Gagne

Lucas Gagne

Motivation For Achievement And Attitudes Toward Mathematics Instruction In A Required Calculus Course At The Norwegian University Of Science And Technology, Donna Sundre, Carol Barry, Vidar Gynnild, Erin Tangen Ostgard

Donna L. Sundre

This study from the Norwegian University of Science and Technology (NTNU) examines students’ learning goals and attitudes toward mathematics in a first-year calculus course in undergraduate engineering education. Achievement motivation research using the Achievement Goal Questionnaire (AGQ) is advanced from current literature with two additions: (1) a course specific context using introductory college calculus students, and (2) participation of Norwegian students. Pre- and posttest measures of attitudes indicate that students do change learning goals over time, unfortunately opposite to the instructors’ aspirations. A significant increase in “Mastery Avoidance” and “Work Avoidance” was accompanied with a drop in “Mastery Approach” and …

Ropelength Criticality, Jason Cantarella, Joseph H.G. Fu, Robert B. Kusner, John M. Sullivan

Robert Kusner

The ropelength problem asks for the minimum-length configuration of a knotted diameter-one tube embedded in Euclidean three-space. The core curve of such a tube is called a tight knot, and its length is a knot invariant measuring complexity. In terms of the core curve, the thickness constraint has two parts: an upper bound on curvature and a self-contact condition.

We give a set of necessary and sufficient conditions for criticality with respect to this constraint, based on a version of the Kuhn–Tucker theorem that we established in previous work. The key technical difficulty is to compute the derivative of thickness …

On The Data Of Images, Lori Ziegelmeier

Lori Beth Ziegelmeier

No abstract provided.

A Mean-Field Analogue Of The Hong-Ou-Mandel Experiment With Bright Solitons, Zhi-Yuan Sun, Panos Kevrekidis, Peter Kruger

Panos Kevrekidis

In the present work, we theoretically propose and numerically illustrate a mean-field analog of the Hong-Ou-Mandel experiment with bright solitons. More specifically, we scatter two solitons off of each other (in our setup, the bright solitons play the role of a classical analog to the quantum photons of the original experiment), while the role of the beam splitter is played by a repulsive Gaussian barrier. In our classical scenario, distinguishability of the particles yields, as expected, a 0.5 split mass on either side. Nevertheless, for very slight deviations from the completely symmetric scenario, a near-perfect transmission can be constructed instead, …

An Unnoticed Consequence Of Szego's Distribution Theorem, William F. Trench

William F. Trench

No abstract provided.

Making The Journey From A Traditional Model To An Online Model, Jenny Shrensker, Nazire Koc

Jenny Shrensker

Crystal Graphs, Tokuyama's Theorem, And The Gindikin-Karpelevic Formula For G2, Holley Friedlander, Louis Gaudet, Paul E. Gunnells

Paul Gunnells

We conjecture a deformation of the Weyl character formula for type G2 in the spirit of Tokuyama’s formula for type A . Using our conjecture, we prove a combinatorial version of the Gindikin–Karpelevič formula for G2 , in the spirit of Bump–Nakasuji’s formula for type A .

A Bijective Proof Of A Factorization Formula For Specialized Macdonald Polynomials, Nicholas A. Loehr, Elizabeth Niese

Elizabeth Niese

Let μ and ν = (ν 1, . . . , ν k ) be partitions such that μ is obtained from ν by adding m parts of sizer. Descouens and Morita proved algebraically that the modified Macdonald polynomials H~μ(X;q,t) satisfy the identity H~μ=H~νH~(rm) when the parameter t is specialize to an mth root of unity. Descouens, Morita, and Numata proved this formula bijectively when r ≤ ν k and r∈{1,2}. This note gives a bijective proof of the formula for all r ≤ ν k .

New Combinatorial Formulations Of The Shuffle Conjecture, Nicholas A. Loehr, Elizabeth Niese

Elizabeth Niese

The shuffle conjecture (due to Haglund, Haiman, Loehr, Remmel, and Ulyanov) provides a combinatorial formula for the Frobenius series of the diagonal harmonics module DHn, which is the symmetric function∇(en). This formula is a sum over all labeled Dyck paths of terms built from combinatorial statistics called area, dinv, and IDes. We provide three new combinatorial formulations of the shuffle conjecture based on other statistics on labeled paths, parking functions, and related objects. Each such reformulation arises by introducing an appropriate new definition of the inverse descent set. Analogous results are proved for the higher-order shuffle conjecture involving ∇m(en). We …

A New Recursion For Three-Column Combinatorial Macdonald Polynomials, Elizabeth Niese

Elizabeth Niese

The Hilbert series of the Garsia–Haiman module Mμ can be described combinatorially as the generating function of certain fillings of the Ferrers diagram of μ where μ is an integer partition of n . Since there are n ! fillings that generate , it is desirable to find recursions to reduce the number of fillings that need to be considered when computing combinatorially. In this paper, we present a combinatorial recursion for the case where μ is an n by 3 rectangle. This allows us to reduce the number of fillings under consideration from (3n)! to (3n)!/(3!nn!).

Outage Constrained Robust Secure Transmission For Miso Wiretap Channels, Shuai Ma, Mingyi Hong, Engin Song, Xiangfeng Wang, Dechun Sun

Mingyi Hong

In this paper, we consider the robust secure beam-former design for multiple-input-single-output wiretap channels. Assuming that the eavesdroppers' channels are only partially available at the transmitter, we seek to maximize the secrecy rate under the transmit power and the secrecy rate outage probability constraint. The outage probability constraint requires that the secrecy rate exceed certain thresholds with high probability. Therefore, including such constraint in the design naturally ensures the desired robustness. Unfortunately, the presence of the probabilistic constraints makes the problem nonconvex and, hence, difficult to solve. In this paper, we investigate the outage probability constrained secrecy rate maximization problem …

Network Analysis Of World Trade Using The Baci-Cepii Dataset, Luca De Benedictis, Silvia Nenci, Gianluca Santoni, Lucia Tajoli, Claudio Vicarelli

Luca De Benedictis

In this paper we explore the BACI-CEPII database using Network Analysis. Starting from the visualization of the World Trade Network, we then define and describe the topology of the network, both in its binary version and in its weighted version, calculating and discussing some of the commonly used network’s statistics. We finally discuss some specific topic that can be studied using Network Analysis and International Trade data, both at the aggregated and sectoral level. The analysis is done using multiple software (Stata, R, and Pajek). The scripts to replicate part of the analysis are included in the appendix, and can …

Computing, Symbols And Math, Stephen M. Watt

Stephen M. Watt

No abstract provided.

Kazhdan-Lusztig Cells In Planar Hyperbolic Coxeter Groups And Automata, Mikhail V. Belolipetsky, Paul E. Gunnells, Richard A. Scott

Paul Gunnells

Let C be a one- or two-sided Kazhdan–Lusztig cell in a Coxeter group (W, S), and let Red(C) be the set of reduced expressions of all w ∈ C, regarded as a language over the alphabet S. Casselman has conjectured that Red(C) is regular. In this paper, we give a conjectural description of the cells when W is the group corresponding to a hyperbolic polygon, and show that our conjectures imply Casselman's.

Composite Dilation Wavelets With High Degrees, Tian-Xiao He

Tian-Xiao He

No abstract provided.

Asymptotic Expansions And Computation Of Generalized Stirling Numbers And Generalized Stirling Functions, Tian-Xiao He

Tian-Xiao He

Here presented is a unified approach to Stirling numbers and their generalizations as well as generalized Stirling functions by using generalized factorial functions, k-Gamma functions, and generalized divided difference. Previous well-known extensions of Stirling numbers due to Riordan, Carlitz, Howard, Charalambides-Koutras, Gould-Hopper, Hsu-Shiue, Tsylova Todorov, Ahuja-Enneking, and Stirling functions introduced by Butzer and Hauss, Butzer, Kilbas, and Trujilloet and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations and generating functions are discussed. Some asymptotic expansions for the generalized Stirling functions and generalized Stirling numbers are established. …

A Study Of Flipping Vs. Not Flipping In Applied Calculus, Lori Ziegelmeier

Lori Beth Ziegelmeier

No abstract provided.

The Strauss Conjecture On Kerr Black Hole Backgrounds, Hans Lindblad, Jason Metcalfe, Christopher D. Sogge, Mihai H. Tohaneanu, Chengo Wang

Mihai H. Tohaneanu

We examine solutions to semilinear wave equations on black hole backgrounds and give a proof of an analog of the Strauss conjecture on the Schwarzschild and Kerr, with small angular momentum, black hole backgrounds. The key estimates are a class of weighted Strichartz estimates, which are used near infinity where the metrics can be viewed as small perturbations of the Minkowski metric, and a localized energy estimate on the black hole background, which handles the behavior in the remaining compact set.

Reasoning & Proof In The Hs Common Core, Laurie O. Cavey

Laurie O. Cavey

No abstract provided.

English Translation (From Latin) Of "A Solution Of A Most Difficult Problem, In Which The Two Forms Aaxx + Bbyy Et Aayy + Bbxx Must Be Rendered Into Squares" By L. Euler, Christopher Goff

Christopher Goff

A Covariance Inequality With A Non-Monotone Function, Martin Egozcue

Martin Egozcue

No abstract provided.

The Pitman Inequality For Exchangeable Random Vectors, J. Behboodian, Naveen Bansal, Gholamhossein Hamedani, Hans Volkmer

Naveen Bansal

In this short article the following inequality called the “Pitman inequality” is proved for the exchangeable random vector (X₁,X₂,…,X_n)(X1,X2,…,Xn) without the assumption of continuity and symmetry for each component X_iXi:

P(|1n∑i=1nXi|≤|∑i=1nαiXi|)≥12 ,

where allαi≥0 are special weights with∑i=1nαi=1.

Empirical Bayes And Hierarchical Bayes Estimation Of Skew Normal Populations, Naveen K. Bansal, Mehdi Maadooliat, Xiaowei Wang

Naveen Bansal

We develop empirical and hierarchical Bayesian methodologies for the skew normal populations through the EM algorithm and the Gibbs sampler. A general concept of skewness to the normal distribution is considered throughout. Motivations are given for considering the skew normal population in applications, and an example is presented to demonstrate why the skew normal distribution is more applicable than the normal distribution for certain applications.

Randomized Detection Of Extraneous Factors, Manfred Minimair

Manfred Minimair

A projection operator of a system of parametric polynomials is a polynomial in the coefficients of the system that vanishes if the system has a common root. The projection operator is a multiple of the resultant of the system, and the factors of the projection operator that are not contained in the resultant are called extraneous factors. The main contribution of this work is to provide a randomized algorithm to check whether a factor is extraneous, which is an important task in applications. A lower bound for the success probability is determined which can be set arbitrarily close to one. …