It's no secret that Mathematics is an essential part of the study of science, especially Physics. Mathematics and Physics are intimately intertwined, and each field has inspired discovery and developments in the other. As a physicist, I will share some ideas on how educators in the two fields might learn from each other in the context of some entertaining physics demonstrations.

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Abstract: Following in the rich tradition of Instant Insanity and Rubik's Cube, the current popularity of Sudoku puzzles presents a golden opportunity for teachers of mathematics.  A whole new population is potentially motivated to learn about permutations, invariance groups, graph-coloring and Boolean constraint programming.  This talk will overview some of the mathematical prehistory of Sudoku puzzles and explore some of the mathematical issues related to the puzzles.

Biography: Jim received his undergraduate degrees in physics and mathematics from Miami University (in Ohio, not that upstart in Florida!), his masters degree from Cornell University and his Ph.D. from the University of Rochester in probability under the direction of Michael Cranston.  Prior to coming to Ithaca College, Jim taught at Allegheny College as well as the Rochester and Cornell; at Cornell he received the Clark Award for Distinguished Teaching.  At Ithaca College he has served as department chair and interim associate dean.  Jim is also the 2005 recipient of Clarence F. Stephens Distinguished Teaching Award within the Seaway Section. His current major mathematical interests are in probability, mathematical biology and mathematical economics.  His favorite escape from e-mail and voice-mail is going off camping and hiking with his family.

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Abstract: This talk will survey the history, meaning, and significance of the P vs. NP question. Could there be a miracle machine to efficiently perform all the problem solving work of mathematics? To zip through problems of engineering design and discovery? The existence of such a miracle machine is exactly the question of P vs. NP. If P=NP, there would be a miracle machine that could take in any mathematical statement S and output a proof of the statement in time polynomially bounded by the length of the shortest proof of S. In other words, our miracle machine could solve problems in time polynomially related to the length of the answer. Our faith in the "genius" of our own creativity leads us to conjecture that no such magic bullet problem solver exists. Is the conjecture that P<>NP common sense, or our failure of imagination?

P vs. NP stands as one of the unsolved Millennium Challenges to mathematics. In this talk, we will survey the role of P vs. NP in a much broader theory of computational complexity. We will touch on the major "approaches" and why they fall short.

Biography: Steven Rudich has been a Professor of Computer Science at Carnegie Mellon University since 1990. He received his PhD. in Computer Science from the University of California at Berkeley under the advisor ship of Manuel Blum. Professor Rudich's research expertise is in computational complexity theory, the mathematical foundations of cryptography, and in the often surprising interplay between the two areas. He has a special
interest in the history, status, and resolution of the P vs NP question. Professor Rudich is the recipient of Carnegie Mellon's departmental and university-wide awards for teaching excellence, and was chosen by the Mathematical Association of America as Polya Lecturer (2004-2006). He is the director of Andrew's Leap, which is a summer program for gifted high school aged students to gain exposure to all aspects of computer science. Steven Rudich is also a gourmet cook and a professional close-up magician.

Further information on Steven's research can be found on his website:

http://www.cs.cmu.edu/~rudich/

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Abstract: Relaxation oscillations were named and studied by van der Pol in the 1920's. They are periodic orbits of dynamical systems with multiple time-scales.  Geometric and analytic methods have been used to characterize the transitions between slow and fast motion in relaxation oscillations. In early work with these systems, Cartwright and Littlewood discovered chaotic dynamics while studying the forced van der Pol equation. This lecture will survey geometric methods for studying relaxation oscillations, bringing a fresh perspective and new insights to our understanding of the bifurcations that create or destroy these periodic orbits.

Biography: John Guckenheimer earned his PhD from Berkeley in 1970, as a student of Stephen Smale. He has held long term positions at University of California, Santa Cruz (1973-85) and Cornell (1985-present). He served as president of SIAM 1997-98. He is currently Associate Dean of Computing and Information Sciences at Cornell University and Director of Cornell IGERT Program in Nonlinear Systems.  His research interests focus on dynamical systems and their applications. 

Further information on John's research can be found on his website:

http://www.math.cornell.edu/~gucken/

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Anurag Agarwal,  RIT
On Thue-Mahler Quartic Diophantine Impossibilities

Abstract:  In this talk we will explore the intriguing world of Diophantine equations. Why is the world of Diophantine equations so fascinating and also challenging? In particular, I will talk about a family of quartic Diophantine equations $x^4-kx^2y^2+y^4=2^j$ (also known as Thue-Mahler equations), which emerged from Euler's work on concordant forms. I will illustarte the solution techniques for such equations using various number-theoretic tools.

Sanghmitra Agarwal, SUNY Buffalo
The Nature of Pre-service Secondary Mathematics Teachers’ Knowledge of Mathematics for Teaching of Functions

Abstract: Everyone is in agreement with the idea that teacher’s knowledge is one of the most important influences on what is done in classrooms and ultimately on what students learn. However there is no consensus about the critical knowledge a teacher should have to teach effectively in the classroom.

    The concept of a function plays an important role in the mathematics education of pre-service secondary school teachers, and most of them do not seem to have a clear understanding of functions.

    Some of the findings of the study that I will discuss are: (1) pre-service secondary mathematics teachers who have a conceptual knowledge of functions were able to convey the concepts in a more effective manner; (2) they were also able to represent ideas in multiple ways and were able to move from one representation of functions to another flexibly.

Cristina Bacuta, John Best, and R. Bruce Mattingly,  SUNY Cortland
Proofs We Thought Our Students Could Do

Abstract:  This presentation will discuss particular observations we made regarding what we teach students about proofs, and how well they can put it into practice. We will include examples of proofs from discrete mathematics, linear algebra and abstract algebra that our students have found difficult. This work is supported by our PMET mini-grant "Assessing pre-service teachers' abilities to do proofs"

Michael Barbosu,  SUNY Brockport and Karen Wells, Monroe Community College
A Statistical Analysis On a Company's Accounts Payable, Using an Empirical Law

Abstract:  This work is an extension of our previous research on Benford’s Law. We utilize this law to perform a statistical analysis of a company's accounts payable. Performing a digit analysis, we examine digit and number patterns to detect abnormal recurrences of digit combinations or specific numbers. Those findings will be discussed in this paper.

William Basener,  RIT
A Survey of Applied Topology

Abstract:  Over recent decades, researchers in scientific and engineering fields have been using topology as a powerful tool.  In many cases, they have solved problems which would have been very difficult without the aid of topology.  We will present some of these applications in robotics, computer graphics, condensed matter physics, biology, cosmology, chemistry, imaging science, geology, and economics.  Our focus will be on problems where researchers from outside of mathematics have collaborated with topologists to solve practical problems, and where the solution would be difficult or impossible to answer without topology.  We will discuss geometric topology, including index theory, homotopy and homology, from an intuitive point of view, emphasizing the perspective of applied researchers.  The talk will be accessible to anyone without prior knowledge of topology.

David Biddle,  Cornell
Experiences teaching Freshman Writing Seminars in Mathematics

Abstract: As we enter a new phase of mathematics education, a phase that increases the amount of writing mathematics students have to use, the emphasis has been to use writing as a form of communicating and understanding mathematics.  In the Fall of 2003 and the Spring of 2004 I had the unique experience of teaching and designing a Freshman Writing Seminar (FWS) at Cornell with the tantalizing name 'The Dementia of Dimension'.  The distinguishing factor of such a course is that the tables are turned:  the students use mathematical ideas to help improve, expand, and broaden their writing skills.  I will briefly talk about the setup of such a course, give some examples of writing exercises, and show how mathematics and metamathematics leads to 5 different styles of writing.

Bernard Brooks,  RIT
Rumour Propagation on a Small World Network

Abstract: Mathematical models of rumour propagation have traditionally used a 'rumour as epidemic' approach that grossly oversimplifies the spatial and demographic distribution of the people infected with the rumour. A dynamical system modeling rumour propagation in a population represented by a social small world network will be presented. Comparisons will be made to the rumor flow over other network topologies such as regular and random. The transmission probability function of the rumour in any given interaction will be a function of belief in the rumour, the novelty of the rumour and the demographic group status of the transmitter and receiver.

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Katrine Danforth and Jayashree Aiyah, Corning Community College
An Assessment Model for Calculus

Abstract:  As a consequence of SUNY General Education requirements and a recent Middle States Accreditation visit, Corning Community College has been taking a serious look at what we do, why we do it, and how we assess the effectiveness of our work. The presenters will share their recent work assessing CCC's Calculus I course. Two different levels of course objectives, Global Course Objectives and Student Learning Outcomes will be discussed. Different types of assessment will be discussed. This talk is geared specifically for assessing Calculus I, but will be of interest to anyone interested in the assessment of course objectives.

Kevin Dempsey,  Clarkson University
Numerical solution of first kind Volterra integral equations with nonsmooth kernels

Abstract:  A Volterra integral equation (VIE) of the first kind and of convolution type occurs in the uplift dynamics of floating plates. This particular VIE distinguishes itself in three respects: it is not able to be recast as a second-kind equation, the convolution kernel is only once differentiable, and the kernel is zero at the origin. Assumptions made ubiquitously in the literature are not true for this problem and high-order discretization methods are not easily attainable. In this talk, analytical and numerical aspects of this VIE will be discussed.

Pat Halpin, SUNY Oswego
On the Sum of Squares

Abstract: Many of our students take a course in Abstract Algebra and a course in Elementary Number Theory.  Unfortunately they often do not experience the rich and fruitful interplay between these two areas.  This talk will use undergraduate level results from Algebra (Abelian groups and commutative rings) and from Elementary Number Theory (classical results on sums of squares of integers and quadratic residues) to yield some results on sums of squares in certain familiar rings.

Keary Howard et al.,  SUNY Fredonia
Teachers’ Masters Capstone Projects in Secondary and College Mathematics

Abstract:  These three twenty-five minute sessions are highlighted by the presentation of research results from secondary school mathematics teachers completing their capstone Masters projects.  Topics and presenters include:

• Session 1: Homework in Middle and College Mathematics Classrooms
  • Does the Length of Homework Assignments Really Matter? Results from a College General Education Mathematics Class.   Amanda Kelchlin, SUNY Fredonia
  • Does Somebody Have a Case of the Mondays?  Students versus Homework:  When Do They Do It?  Joe Enright, Amherst Catholic Middle School
• Session 2: Organization and Athletics:  Their Effects on Secondary Math Achievement**
  • Organization as Motivation in Mathematics Education:  The Relationship Between Organization and Achievement in High School Mathematics.  Bill Persons, Chautauqua Lake Central School
  • Friday Night Lights:  Math Achievement for In-Season and Non-Season Student Athletes.  John Homich, Silver Creek Central School
• Session 3: Innovation in Math Pedagogy and Projects**
  • Class-Wide Peer Tutoring in the Math Classroom:  Teacher or Tutor - Who Is Really More Effective?  Leeann Mislin, SUNY Fredonia
  • Optimize This:  Precalculus Students’ Solution Methods to Linear Optimization Problems.  Bill Doyle, SUNY Fredonia

Vincenzo Isaia, Hobart and William Smith Colleges
An Efficient Rescaling Algorithm for the Numerical Simulation of Blow-up Phenomena

Abstract: Self-similarity in the intermediate asymptotic behavior for initial value problems is an important facet because it demonstrates a stabilization of the physics at hand. Renormalization group (RG) techniques have been used to capture, via analytic proof and numerical computation, a wide variety of asymptotic behaviors for initial value problems, including anomalous exponents, logarithmic corrections, traveling waves and now  blow-up phenomena. 

    For this talk, an RG based numerical scheme will be presented for determining asymptotic behavior of the initial value problem:

with and .  Although rescaling algorithms have been formulated for such blow-up problems (e.g. Berger and Kohn, 1988), this RG based version is very tidy and efficient without the need for any special numerical procedures, and it captures the logarithmic correction present in the spatial scaling which is not present in other algorithms.  Due to the ease with which the blow-up time can be extracted from this process, a study of the structure of the blow-up time with respect to the equation parameters and initial conditions is possible

Dawn Jones,  SUNY Brockport
Panel Discussion: Activities and Manipulatives in Proof Intensive Classes

Abstract: Many Mathematics Educators, Teachers, Professors, and students have seen and used manipulatives and activities in non-proof intensive classes. In  Mathematics for Elementary Teachers classes there are many manipulatives that are typically used. Now if we consider a class such as College Geometry, Topology, or Abstract Algebra, what manipulatives come to mind? Should we even use manipulatives at that level? Should we spend time in class to do activities with manipulatives or activities at all? In this panel discussion we will touch on these and other related questions. Additionally, the panelists will share some manipulatives that have been used in these and other proof-intensive courses.

Leo Jonker,  Queens University
What can a Mathematics department do to improve the preparation of elementary school teachers?

Abstract: The presentation will reflect on weaknesses in the mathematical preparation of elementary school teachers at a typical Ontario university.  It will reflect on the ingredients necessary for mathematics program that is appropriate and effective, and that attracts the right group of students.   I will  describe work done in the Queen's University Mathematics and Statistics Department to create such a course.  As a key requirement, students in the course present a weekly mathematics enrichment class to grade 7 and 8 students in local schools. 

Isa S. Jubran,  SUNY Cortland
Iterated Function Systems (IFS's)  and  Fractals

Abstract:  The study of fractals arising as attracttors of IFS's became an area of practical importance thanks to Mandelbrot's fundamental insight that many natural objects have some self similarity and Barnsley's insight that it is possible to begin with a shape and determine an IFS whose attractor converges onto that shape .  A number of artificial as well as natural fractals and their possible IFS's will be investigated using java applets which are freely available on the Web.

Marc Laforest, École Polytechnique,  Montréal
Error propagation and cancellation in two schemes for nonlinear conservation laws

Abstract: It is one thing to show that a numerical scheme accurately approximates the solution to a nonlinear hyperbolic conservation law, it is quite another to provide a practical means of estimating that error. We present computable and rigorous error estimators for two closely related schemes : the front-tracking scheme and Glimm's scheme.   In both cases we indicate the mechanisms whereby the error is generated, propagated and sometimes cancelled through interactions with other errors. Numerical results are also presented.

Blair F. Madore and Cheryl Chute Miller, SUNY Potsdam
Research and Teaching; working together

Abstract: Connections between traditional research and teaching are frequently discussed, but often include the notions of “keeping faculty actively engaged in their subject”.  As most traditional research in mathematics is far beyond the scope of undergraduate courses, it can be difficult for a faculty member to see a direct impact on their teaching. Occasionally a topic will allow this connection to be seen very clearly.  In this presentation we will discuss how traditional research by Madore led to teaching ideas for an undergraduate Abstract Algebra course taught by Miller.

James Marengo,  RIT
Fisher Information and the Rao-Cramer Inequality

Abstract:  The Rao-Cramer Inequality was discovered in the !940's. It gives a positive lower bound for the variance of an unbiased estimator of an unknown parameter based on a random sample taken from a population which depends on this parameter . The lower bound is expressed in terms of the Fisher Information. This inequality acts as an uncertainty principle in the sense that it puts a limit on the precision of the estimation. The speaker will give a simple proof of this result after defining the necessary terminology.  We will then look at some examples and we will investigate the issue of whether the bound is always attainable

Claudiu Mihai, Daemen College
Optimization Problems or just Patterns?

Abstract: In this talk I solve several optimization problems in the calculus books by showing that the maximal area of a rectangle with vertices on the side of any triangle always equals one half of the area of the triangle. I also show that the ratio between the largest area of a rectangle with two vertices on a horizontal line and two vertices on a parabola, and the area between the parabola and the horizontal line  is always constant and equals 1/3,and that the rectangle with the maximum area is always obtained by cutting the parabola at a constant height which is 2/3 of the height of the parabola. I show similar results for cylinder inside a cone and cylinder inside a paraboloid

G. Arthur Mihram, Princeton, NJ and Danielle Mihram,  University of Southern California
An Historical Basis for Viewing Our Mathematics’ True Value as Other Than its Content

Abstract:  We propose to make two points:  1. Mathematics is an art, not truly a science.; and 2. The value of mathematics in the tertiary-level  educational curriculum (as well as that of our secondary education!) is not its content, but rather its capability to discipline each student’s mind in preparation for his/her becoming an adult in the nation which each is about to serve. We derive both of these conclusions from an understanding of the Scientific Method, itself a near-algorithmic process of six stages. The value of mathematics in training one to make and to confirm one’s analogies is underscored as well.

Morris Orzech,  Queen’s University
A Teaching Apprenticeship Initiative

Abstract:  I will describe a departmental initiative providing teaching opportunities to interested graduate students for whom an individual course was not possible or appropriate.  Support for the graduate students and faculty mentors were modeled on a Peer Consultation Program being piloted by our university's Centre for Teaching and Learning.  I will describe what we did and why, and what our early experience has been.

Gary Raduns,  Roberts Wesleyan College
Determinants and Zero Divisors in Rings of Matrices

Abstract: In introductory linear algebra courses we typically teach that if  and are square matrices and  then det A = det B = 0.  In stating this, we have either implicitly or explicitly made the assumption that the entries in the matrices are real numbers (or elements of some field-an integral domain suffices).  If A and B are matrices over a ring R then our statement must be modified:  If  and  are square matrices over a commutative ring R with unity then if and only if det A and det B are zero or zero-divisors in R.  This talk explores the question:  If A is a non-zero square matrix over a commutative ring R with unity and det A is a zero-divisor in R, under what conditions is there a square matrix B such that det B is non-zero and AB=0?

Charles Ragozzine, Jr. ,  SUNY Oneonta
A Formula for  Following Euler's Approach

Abstract:  The Riemann zeta function is defined as , for Re(z) > 1. The reader is probably familiar with Euler's method of computing  that compares the Taylor series and Weierstrass product for . By going to the complex plane, this technique can be generalized to find a formula for , where  is a positive

even integer. The approach is algebraic and elementary in nature, using facts about Taylor series, Weierstrass products, roots of unity, and culminates with a formula for finding Bernoulli numbers.

Hossein Shahmohammed,  RIT
Matching, Flow and Chromatic Polynomials in Graph Theory

Abstract:  In this talk we briefly talk about the 3 significant polynomials in Graph Theory and the close relationship among these three polynomials. Many examples, properties and computations of each will be given.

Yishi Wang,  Binghamton University
Normality tests based on Empirical Processes

Abstract:Every statistics textbook in use has questions that start with "Assume the lifetime of light bulbs (people's height, weights or whatever) is normally distributed with mean...". What if this assumption does not hold? Many statistical procedures become invalid. Therefore the first step of any data analysis is testing the normality assumption.
   In this talk we will present the construction of some normality tests based on empirical processes. Two new normality tests will be introduced and we will outline some of their properties.

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Sandra Barbosu, SUNY Brockport
Applications With Maplets

Maple is a computer algebra system that can be used both in teaching and research. Maplets are graphical friendly tools that allow users to explore mathematics by nteracting with this powerful software. In order to illustrate the potential of these tools, I created Maplets that deal with various applications. For this project I considered examples from Algebra, Calculus and Fractal Geometry.

Bonita Bryson, SUNY Potsdam
Friezes and Tilings

We assume that a frieze tiles a wallpaper pattern if and only if the frieze in question has width equal to the length of some translation vector running perpendicular to the direction of translation in the frieze. There are 119 different combinations of patterns and friezes to consider. I will discuss how the composition of the pattern affects the set of friezes which may tile it.

Tom Church, Cornell University
The Abelian Cover of a Surface

Every topological space has an abelian cover associated to it; in the case of a surface S; the abelian cover is another surface S'. It has the useful property that many objects on S, including points, lines, and homeomorphisms, can be "lifted" to S' in a way that brings deeply-buried facts about them to the surface. Unfortunately, although the general theory tells us certain algebraic properties of the abelian cover, in general we have no mental picture of it.

In the particular case of a surface, we can find a model that lets us visualize the abelian cover and even draw parts of it. By translating algebraic objects into pictorial ones, we can perform calculations on them purely topologically. I will explain how to construct this picture, show how various concepts can be extended to it, and, if time permits, give examples of its practical application to the Magnus representation and mapping class groups.

Deana B. Connell, Rochester Institute of Technology
Rumor Flow over Random and Small World Networks

Various dynamical systems modeling rumor transmission on a social network are introduced. The different speeds of rumor flow determined by the time which the rumor has passed completely through the network are compared using the constructs of both a Random Social Network and a Small World Social Network. The structural and statistical differences are presented through graphs using node coloring (not chromatic numbering).

From unique adjacency matrices that define our random networks, we can manipulate the initial conditions in order to obtain a small world adaptation. Interesting comparisons can be made about the networks' clustering coefficients as well as their average path lengths. Significant correlations to consider are those between transmission rates vs. time, connectivity vs. average path length, and connectivity vs. clustering coefficient. A rumor will clearly flow through a small world network faster than in a random network, mainly due to the shortened path lengths from increased connectivity.

Tony Lee, Rochester Institute of Technology
Proving an Inequality Pair Proposed by Erdos

Erdos proposed the following problem in Tomorrow's Math by Ogilvy. Prove that if

((m - 1)/m)^n > 1/2 and ((m - 2)/(m - 1))^n < 1/2

then the following inequalities hold:

(m - 1)^n > (m - 2)^n + (m - 3)^n + ... + 1^n

(m + 1)^n < m^n + (m - 1)^n + ... + 1^n

In this joint work with my advisor Manuel Lopez, we approached the problem using basic techniques in the calculus of finite differences and linear algebra to put the question in a geometric context. Then using properties of convex sets we proved a condition that implies the desired inequalities. To the best of our knowledge, this problem has remained unsolved and has been in obscurity since Ogilvy published it in his book Tomorrow's Math.

Steven Link, SUNY Fredonia
Phantom Phenomena: Audible False-fundamental Tones in Quartet Singing

Where did that extra sound come from? Sometimes in barbershop quartet singing, the audience and the performers will hear five distinct notes instead of just the four that are being sung. Using discoveries of Pythagoras and the scientists of today, we will combine their findings through the use of various multiplicative relationships and sine waves to explain these 'phantom tones.'

Jason Ouderkirk, SUNY Cortland
(GPS)-Global Positioning Systems and Mathematics

A brief view of the mathematics involved in how (GPS) works. Namely trilateration.

Jason Ouderkirk, SUNY Cortland
Hypercube (What is a Hypercube?)

A hands on project of building a hypercube, also ways of calculating distance.

Nathan Reff, Rochester Institute of Technology
Analysis of Convex Shapes Using Tridrafters

In 2000, the Eternity puzzle offered about a 2 million dollar prize for the first solver. This talk will cover the history, rules, and current status of the Eternity puzzle. We will also discuss a simplified set of the Eternity pieces and analyze the shapes that these pieces will fit into.

Candice Rockell, St. John Fisher College
Reliving the Struggles of Fermat's Last Theorem and Developing Related Conjectures

Fermat's Last Theorem is one of the most widely studied theorems in the history of mathematics. There were many brilliant mathematicians that contributed to this theorem. In particular, I will look at its history and the mathematicians involved before 1850. Other interesting components of FLT are the conjectures and ideas that spawned as a result of this theorem. I will speak to these and also present some of my own ideas that were a result of my research.

Angela Tessoni, Nazareth College
Signed, Sealed, Delivered: A Look into Digital Signatures

In a world of that is becoming increasingly dependent on computers, the need for paper documents is becoming obsolete. Almost any document these days can be sent electronically. Now we must be concerned with how to sign these documents and how to keep these signatures secure so that they cannot be duplicated. The study of cryptography is then introduced, specifically the use of digital signatures. People can use digital signature algorithms to attach signatures to electronic documents and keep them secure to the point where the signature cannot be copied and their identity is permanently bound to that document. When describing how a digital signature works, the topics of primality testing and discrete logarithms come up. There is also the question of how secure these signatures are and what types of attacks can be made.

Erik Wallace, Hartwick College
On the Sum and Product of Simple Continued Fractions

We will begin with a brief introduction to continued fractions, and highlight two important theorems for simple continued fractions. We will then present a theorem that is a direct result of continuity. We will observe that addition and multiplication may be thought of as continuous functions of two variables, and use this theorem to define a method for adding and multiplying continued fractions. We will then pose and answer the question "can this sum or product of two simple continued fractions, again be a simple continued fraction?" We will show that the answers are "never" and "sometimes" for multiplication and addition respectively. In conclusion we will suggest extending this technique to other operators, which must also be continuous functions, and rephrasing the problem of adding or multiplying two simple continued fractions in terms of the problem of rewriting an arbitrary continued fraction as a simple continued fraction.

Matthias L. Youngs, SUNY Geneseo
Just One More Transcendental

C is a field extension of Q. What if we had some transcendental number a that is not even in C - thus a is a new transcendental - and we adjoin a to C. Is C(a) over Q isomorphic to C over Q? Maybe not, but we can improve this conjecture. We will slightly modify the conjecture and then find an equivalent statement.