On K4 Of The Gaussian And Eisenstein Integers, 2010 University of Massachusetts - Amherst

#### On K4 Of The Gaussian And Eisenstein Integers, Mathieu Dutour Sikiric, Herbert Gangl, Paul Gunnells, Jonathan Hanke, Achill Schürmann, Dan Yasaki

*Paul Gunnells*

Abstract. In this paper we investigate the structure of the algebraic K-groups K4(Z[i]) and K4(Z[ρ]), where i := √ −1 and ρ := (1 + √ −3)/2. We exploit the close connection between homology groups of GLn(R) for n 6 5 and those of related classifying spaces, then compute the former using Voronoi’s reduction theory of positive definite quadratic and Hermitian forms to produce a very large finite cell complex on which GLn(R) acts. Our main results are (i) K4(Z[i]) is a finite abelian 3-group, and (ii) K4(Z[ρ]) is trivial.

Asymptotic Behavior Of The Finite-Size Magnetization As A Function Of The Speed Of Approach To Criticality, 2010 University of Massachusetts - Amherst

#### Asymptotic Behavior Of The Finite-Size Magnetization As A Function Of The Speed Of Approach To Criticality, Richard S. Ellis, Jonathan Machta, Peter Tak-Hun Otto

*Richard S. Ellis*

The main focus of this paper is to determine whether the thermodynamic magnetization is a physically relevant estimator of the finite-size magnetization. This is done by comparing the asymptotic behaviors of these two quantities along parameter sequences converging to either a second-order point or the tricritical point in the mean-field Blume–Capel model. We show that the thermodynamic magnetization and the finite-size magnetization are asymptotic when the parameter α governing the speed at which the sequence approaches criticality is below a certain threshold α0. However, when α exceeds α0, the thermodynamic magnetization converges to 0 much faster than the finite-size ...

Invariant Weighted Wiener Measures And Almost Sure Global Well-Posedness For The Periodic Derivative Nls, 2010 University of Massachusetts - Amherst

#### Invariant Weighted Wiener Measures And Almost Sure Global Well-Posedness For The Periodic Derivative Nls, Andrea Nahmod, Tadahiro Oh, Luc Rey-Bellet, Gigliola Staffilani

*Andrea Nahmod*

In this paper we construct an invariant weighted Wiener measure associated to the periodic derivative nonlinear Schr\"odinger equation in one dimension and establish global well-posedness for data living in its support. In particular almost surely for data in a Fourier-Lebesgue space ${\mathcal F}L^{s,r}(\T)$ with $s \ge \frac{1}{2}$, $2 < r < 4$, $(s-1)r <-1$ and scaling like $H^{\frac{1}{2}-\epsilon}(\T),$ for small $\epsilon >0$. We also show the invariance of this measure.

Inverse Limits With Upper Semi-Continuous Set Valued Bonding Functions: An Example, 2010 Missouri University of Science and Technology

#### Inverse Limits With Upper Semi-Continuous Set Valued Bonding Functions: An Example, Christopher David Jacobsen

*Masters Theses*

"While there is a wealth of information pertaining to inverse limits with single valued bonding maps, comparatively little is known about inverse limits with upper semi-continuous set valued bonding functions. In order to add somewhat to the communal knowledge on the subject, this paper provides an example of an inverse limit with a single upper semi-continuous set valued bonding function. It is then shown that the space is a continuum, and its structure is examined via its arc components and through various of its properties, such as dimension and decomposability"--Abstract, page iii.

Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, 2010 West Chester University of Pennsylvania

#### Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin

*Mathematics Faculty Publications*

We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.

Some Applications Of A Bailey-Type Transformation, 2010 West Chester University of Pennsylvania

#### Some Applications Of A Bailey-Type Transformation, James Mclaughlin, Peter Zimmer

*Mathematics Faculty Publications*

If k is set equal to aq in the definition of a WP Bailey pair, βn(a, k) = Xn j=0 (k/a)n−j (k)n+j (q)n−j (aq)n+j αj (a, k), this equation reduces to βn = Pn j=0 αj . This seemingly trivial relation connecting the αn’s with the βn’s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for poly-basic hypergeometric series.

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, 2010 West Chester University of Pennsylvania

#### Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills, Peter Zimmer

*Mathematics Faculty Publications*

Lucy Slater used Bailey's 6*Ã*6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10*Ã*10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters ...

An Identity Motivated By An Amazing Identity Of Ramanujan, 2010 West Chester University of Pennsylvania

#### An Identity Motivated By An Amazing Identity Of Ramanujan, James Mclaughlin

*Mathematics Faculty Publications*

Ramanujan stated an identity to the effect that if three sequences {an}, {bn} and {cn} are defined by r1(x) =: ∑∞ n=0 anx n , r2(x) =: ∑∞ n=0 bnx n and r3(x) =: ∑∞ n=0 cnx n (here each ri(x) is a certain rational function in x), then a 3 n + b 3 n − c 3 n = (−1)n , ∀ n ≥ 0. Motivated by this amazing identity, we state and prove a more general identity involving eleven sequences, the new identity being ”more general” in the sense that equality holds not just for the power 3 (as in Ramanujan’s ...

Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, 2010 Northern Illinois University

#### Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi

*Mathematics Faculty Publications*

We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive m-versions of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such m-versions of Rogers Ramanujan-type identities.

General Wp-Bailey Chains, 2010 West Chester University of Pennsylvania

#### General Wp-Bailey Chains, James Mclaughlin, Peter Zimmer

*Mathematics Faculty Publications*

Motivated by a recent paper of Liu and Ma, we describe a number of general WP-Bailey chains. We show that many of the existing WP-Bailey chains (or branches of the WP-Bailey tree), including chains found by Andrews, Warnaar and Liu and Ma, arise as special cases of these general WP-Bailey chains. We exhibit three new branches of the WP-Bailey tree, branches which also follow as special cases of these general WP-Bailey chains. Finally, we describe a number of new transformation formulae for basic hypergeometric series which arise as consequences of these new WP-Bailey chains.

Right Focal Boundary Value Problems For Difference Equations, 2010 Baylor University

#### Right Focal Boundary Value Problems For Difference Equations, Johnny Henderson, Xueyan Liu, Jeffrey W. Lyons, Jeffrey T. Neugebauer

*Mathematics Faculty Articles*

An application is made of a new Avery et al. fixed point theorem of compression and expansion functional type in the spirit of the original fixed point work of Leggett and Williams, to obtain positive solutions of the second order right focal discrete boundary value problem. In the application of the fixed point theorem, neither the entire lower nor entire upper boundary is required to be mapped inward or outward. A nontrivial example is also Provided.

Singular Points Of Real Sextic Curves I, 2010 George Fox University

#### Singular Points Of Real Sextic Curves I, David A. Weinberg, Nicholas J. Willis

*Faculty Publications - Department of Mathematics*

A complete classiﬁcation of the individual types of singular points is given for irreducible real sextic curves. This classiﬁcation is derived by using the computer algebra system Maple. There are 191 types of singular points for real irreducible sextic curves. We clarify that the classiﬁcation is based on computing just enough of the Puiseux expansion to separate the branches. A signiﬁcant portion of the proof consists of a sequence of large symbolic computations that can be done nicely using Maple.

Mathematical Biology At An Undergraduate Liberal Arts College, 2010 Harvey Mudd College

#### Mathematical Biology At An Undergraduate Liberal Arts College, Stephen C. Adolph, Lisette G. De Pillis

*All HMC Faculty Publications and Research*

Since 2002 we have offered an undergraduate major in Mathematical Biology at Harvey Mudd College. The major was developed and is administered jointly by the mathematics and biology faculty. In this paper we describe the major, courses, and faculty and student research and discuss some of the challenges and opportunities we have experienced.

Recognizing Graph Theoretic Properties With Polynomial Ideals, 2010 University of California - Davis

#### Recognizing Graph Theoretic Properties With Polynomial Ideals, Jesus A. De Loera, Christopher J. Hillar, Peter N. Malkin, Mohamed Omar

*All HMC Faculty Publications and Research*

Many hard combinatorial problems can be modeled by a system of polynomial equations. N. Alon coined the term *polynomial method* to describe the use of nonlinear polynomials when solving combinatorial problems. We continue the exploration of the polynomial method and show how the algorithmic theory of polynomial ideals can be used to detect k-colorability, unique Hamiltonicity, and automorphism rigidity of graphs. Our techniques are diverse and involve Nullstellensatz certificates, linear algebra over finite fields, Gröbner bases, toric algebra, convex programming, and real algebraic geometry.

Lifting Module Maps Between Different Noncommutative Domain Algebras, 2010 University of Denver

#### Lifting Module Maps Between Different Noncommutative Domain Algebras, Jonathan Von Stroh

*Electronic Theses and Dissertations*

The classical Carathéodory interpolation problem is the following: let n be a natural number, a_{0}, a_{1}, . . . , a_{N} be complex numbers, and **D** the unit disk. When does there exist an analytic function F : **D** → **C** and complex numbers a_{N+1}, a_{N+2}, . . . such that F(z) = a_{0} + a_{1}z + a_{2}z^{2} + . . . + a_{N}z^{N} + a_{N+1}z^{N+1} + . . . and ||F||_{∞} < 1? In 1967, Sarason used operator theory techniques to give an elegant solution to the Carathéodory interpolation problem. In 1968, Sz.-Nagy and Foias extended Sarason's approach into a commutant lifting theorem. Both the theorem and the technique of the proof have become standard tools in control theory. In particular, the commutant lifting theorem approach lends itself to a wide range of generalizations. This thesis concerns one such generalization.

Arias presented generalizations of the original commutant lifting theorem relating to the full Fock space. Popescu then refined the approach by introducing domain algebras. While Arias and Popescu ...

Methods Of Variational Analysis In Pessimistic Bilevel Programming, 2010 Wayne State University

#### Methods Of Variational Analysis In Pessimistic Bilevel Programming, Samarathunga M. Dassanayaka

*Wayne State University Dissertations*

Bilevel programming problems are of growing interest both from theoretical and practical points of view. These models are used in various applications, such as economic planning, network design, and so on. The purpose of this dissertation is to study the pessimistic (or strong) version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation developed by B. Mordukhovich.

This dissertation begins with analyzing pessimistic bilevel programs, formulation of the problems, literature review, practical application, existence of ...

Asymptotic Properties Of Markov Modulated Sequences With Fast And Slow Time Scales, 2010 Wayne State University

#### Asymptotic Properties Of Markov Modulated Sequences With Fast And Slow Time Scales, Son Luu Nguyen

*Wayne State University Dissertations*

In this dissertation we investigate asymptotic properties of Markov modulated random processes having two-time scales. The model contains a number of mixing sequences modulated by a switching process that is a discrete-time Markov chain. The motivation of our study stems from applications in manufacturing systems, communication networks, and economic systems, in which regime-switching models are used.

This thesis focuses on asymptotic properties of the Markov modulated processes under suitable scaling. Our main effort focuses on obtaining weak convergence and strong approximation results.

Review: Pythagoras’ Revenge: A Mathematical Mystery; The Housekeeper And The Professor, 2010 Oberlin College

#### Review: Pythagoras’ Revenge: A Mathematical Mystery; The Housekeeper And The Professor, Susan Jane Colley

*Faculty & Staff Scholarship*

No abstract provided.

Alternative Technical Efficiency Measures: Skew, Bias, And Scale, 2010 Syracuse University. Center for Policy Research

#### Alternative Technical Efficiency Measures: Skew, Bias, And Scale, William Clinton Horrace, Qu Feng

*Center for Policy Research*

In the fixed-effects stochastic frontier model an efficiency measure relative to the best firm in the sample is universally employed. This paper considers a new measure relative to the worst firm in the sample. We find that estimates of this measure have smaller bias than those of the traditional measure when the sample consists of many firms near the efficient frontier. Moreover, a two-sided measure relative to both the best and the worst firms is proposed. Simulations suggest that the new measures may be preferred depending on the skewness of the inefficiency distribution and the scale of efficiency differences.

Generalized Fourier-Feynman Transforms, Convolution Products, And First Variations On Function Space, 2010 Dankook University

#### Generalized Fourier-Feynman Transforms, Convolution Products, And First Variations On Function Space, Seung Jun Chang, Jae Gil Choi, David Skough

*Faculty Publications, Department of Mathematics*

In this paper we examine the various relationships that exist among the first variation, the convolution product and the Fourier-Feynman transform for functionals of the form *F(x) *= *f*((α_{1}, *x*), . . . , (α* _{n}*,

*x*)) with

*x*in a very general function space

*C*

_{a,b}[0,

*T*].