Now showing items 1-20 of 57

• #### Geodesics and Bounded Harmonic Functions on Infinite Graphs

It is shown there that an infinite connected planar graph with a uniform upper bound on vertex degree and rapidly decreasing Green's function (relative to the simple random walk) has infinitely many pairwise finitely-intersecting geodesic rays starting at each vertex. We then demonstrate the existence of nonconstant bounded harmonic functions on the graph.
• #### Cogrowth of Regular Graphs

Let G be a d-regular graph and T the covering tree of G. We define a cogrowth constant of G in T and express it in terms of the first eigenvalue of the Laplacian on G. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Laplacian on G is zero. Grigorchuk's criterion for amenability of finitely generated groups follows.
• #### Amenability and superharmonic functions

Let G be a countable group and u a symmetric and aperiodic probability measure on G . We show that G is amenable if and only if every positive superharmonic function is nearly constant on certain arbitrarily large subsets of G. We use this to show that if G is amenable, then the Martin boundary of G contains a fixed point. More generally, we show that G is amenable if and only if each member of a certain family of G-spaces contains a fixed point.
• #### On the Commute Time of Random Walks on Graphs

Given a simple random walk on an undirected connected graph, the commute time of the vertices x and y is defined as C(x,y) = ExTy+EyTx. We give a new proof, based on the optional sampling theorem for martingales, of the formula C(x,y) = 1/(Π(y)e(y,x)) in terms of the escape probability e(y,x ) (the probability that once the random walk leaves x, it hits y before it returns to x) and the stationary distribution Π(·). We use this formula for C(x,y) to show that the maximum commute time among all barbell-type graphs in N vertices is attained for the lollipop graph and its value is O[(4N3)/27]
• #### On the spectrum and Martin boundary of homogeneous spaces

Given a conservative, spatially homogeneous Markov process X on an homogeneous spaces χ, we show that if the bottom of the spectrum of the generator of X is zero then the Martin boundary of contains a unique point fixed by the isometry group of χ.
• #### On Iterates of Moebius transformations on fields

Let p be a quadratic polynomial over a splitting field K, and S be the set of zeros of p. We define an associative and commutative binary relation on G ≡ K ∪ {∞ } -S so that every Moebius transformation with fixed point set S is of the form x plus" c for some c. This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If K is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Moebius transformation m to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of m are distinct and the iterates of m converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.
• #### A note on the Zeta Function of a Graph

The number of splanning trees in a finite graph is first expressed as the derivative (at 1) of a determinant and then in terms of a zeta function. This generalizes a result of Hashimoto to non-regular graphs.
• #### Woman-As-Symbol: Intersections of Indian Nationalism, Gender, and Identity

The purpose of this article is to explore the connection between Indian nationalism and gender identity. I provide a critique of Radhakrishnan and Chatterjee's notion of the outer/inner dichotomy of Indian nationalism by stating that religion, in postcolonial India, has emerged as a discursive totality that has subsumed the politics of indigenous or inner identity more so than other rhetoric of caste, tribal, gender, and class. I provide a groundwork for this debate via the writings of Nehru and Gandhi. I conclude, through an analysis of the practices of amniocentesis and Sati, that women and their bodies have been used as representations of the conflicts surrounding national subjectivity.
• #### Associativity of the Secant Method

Iterating a function like 1+1/x gives a sequence which converges to the Golden Mean but does so at a much slower rate than those sequences derived from Newton's method or the secant method. There is, however, a surprising relation between all these sequences. This relation, easily explained by the use of good notation, is generalized by means of Pascal's "Mysterium Hexagrammicum". Throughout, we make contact with many areas of mathematics and physics including abstract groups, calculus, continued fractions, differential equations, elliptic curves, Fibonacci numbers, functional equations, fundamental groups, Lie groups, matrices, Moebius transformations, pi, polynomial approximation, relativity, and resistors.
• #### Assessing Outcomes with Nursing Research Assignments and Citation Analysis of Student Bibliographies

What are the library and information research requirements in a typical undergraduate nursing program? Do distance-learning library services provide undergraduate nursing students with the research materials they require for their academic work? In order to determine how the broad range of reference, instruction, and access services offered by Feinberg Library at Plattsburgh State University of New York, are used by students, the author reviewed selected nursing course syllabi for research requirements and the resulting student research bibliographies as an outcome assessment. The review included 441 bibliographic citations from 78 student research papers from 1998-1999. Results indicated no significant difference between on and off-campus student bibliography citations with regards to currency, format or number of citations. Results also indicated that the reviewed undergraduate nursing research assignments were indeed designed to promote research integration into nursing practice, and that student access to information was sufficient to allow them to complete their academic assignments.
• #### Those Immersed Resurface: A Follow Up with Track 2 Participants of the First Information Literacy Immersion

So what happens to all those great ideas and all that motivation that we get when we attend conferences and professional development opportunities? In the case of the first Track 2 participants of ACRL's Institute for Information Literacy's Immersion program, quite a lot. Two years after the first Immersion program, a follow-up survey pursued this question and found where great ideas and motivation are taking librarians and the institutions they work for.
• #### Reinforcing Information and Technology Literature: The Plattsburg Tip Sheet

This article describes a Plattsburgh State University Library and Information Ser­vices (LIS) faculty workshop on information and technology literacy. The workshop was developed in response to a call for the redraft­ ing and submission for academic curriculum review of all courses intended for General Education approval and credit to meet new college General Education requirements. The focus of this article is on the infor­ mation and technology literacy specifics of the new requirements, the particular style of tip sheet developed for the workshop, and its potential for use by other librarians. The essence of the Plattsburgh Tip Sheet is a practical approach to rethinking lectures, class activities, and assignments to reinforce information and technology literacies.
• #### An Analysis of Executive Functions in Children Referred for Auditory Processing Evaluation

It is well known that attention deficits frequently co-occur with auditory processing disorder yet little is known about the (cognitive) executive functions (EFs) that regulate attention and self-control in children with auditory processing concerns. We present an analysis of EFs in children referred for auditory processing assessment, based on both teacher and parent report forms of a norm-referenced assessment instrument.
• #### On integral Apollonian circle packings

The curvatures of four mutually tangent circles with disjoint interior form what is called a Descartes quadruple. The four smallest curvatures of circles in an Apollonian circle packing form what is called a root Descartes quadruple and, if the curvatures are relatively prime, we say that it is a primitive root quadruple. We prove a conjecture of Mallows by giving a closed formula for the number of primitive root quadruples with minimum curvature -n. An Apollonian circle packing is called strongly integral if every circle has curvature times center a Gaussian integer. The set of all such circle packings for which the center of the largest circle is in the unit square and for which curvature plus curvature times center is congruent to 1 modulo 2 is called the standard super-gasket. These centers are in oneto-one correspondence with the primitive root quadruples and exhibit certain symmetries first conjectured by Mallows. We prove these symmetries; in particular, the centers are symmetric around y = x if n is odd, around x = 1/2 if n is an odd multiple of 2, and around y = 1/2 if n is a multiple of 4.
• #### Not mixing is just as cool

Newton's law of cooling, a staple of the Calculus curriculum, is an empirical law not meant for mathematical proof. However, we show it is mathematically equivalent to the intuitively appealing principle that the average temperature of two cooling objects is equal to the temperature of a single object with initial temperature the average of the other two.
• #### Do Executive Skills or Language Skills Best Predict Social Competence?

Research has shown that children with developmental language impairment are at high risk for social and behavioral problems, although the reasons for this relationship are not entirely clear. Some have proposed that language impairment leads to social and behavioral problems, while others have suggested that there is some other mediating factor. In this study, executive dysfunction was found to be a powerful predictor of social skills and problem behaviors, while language alone was not.
• #### Should Malingering Matter to Speech Language Pathologists?

Malingering, or the intentional feigning of illnesses or disorders for secondary gain, is a financial and legal burden to society. Documented malingered disorders that are of particular interest to speech-language pathologists (SLPs) include malingered dysphagia, stuttering, mutism, dysphonia, language disorders, and cognitive impairment. Unfortunately, little information on malingering is available to SLPs. In this presentation we will introduce SLPs to issues of malingering, and provide information on how to address malingering in the assessment process.
• #### Executive Functions and Their Relationship to Social Skills and Problem Behaviors

This study examined social, behavioral, and executive function characteristics in a group of children referred for auditory and language processing assessment. Teacher ratings of executive functions were compared to parent ratings to determine if teacher's or parent's ratings are better predictors of social skill deficits or problem behaviors. Further, eight executive function skills were examined to determine which are most related to social skills and which are most related to problem behaviors.
• #### On two types of exotic addition

We classify, under reasonable assumptions, all differentiable functions f for which the 'secant method' [xf(y)- yf(x)]/[f(y)- f(x)] is continuous and associative. Further, we classify all differentiable functions for which the similar type of addition xf(y) + yf(x) is associative.