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1. CMB 2012 (vol 56 pp. 570)
Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups In this paper we give a lower bound
with respect to block length
for the trace of non-elliptic conjugacy classes
of the Hecke groups.
One consequence of our bound
is that there are finitely many
conjugacy classes of a given trace in any Hecke group.
We show that another consequence of our bound
is that
class numbers are finite for
related hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms.
We give canonical class representatives
and calculate class numbers
for some classes of hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms.
Keywords:Hecke groups, conjugacy class, quadratic forms Categories:11F06, 11E16, 11A55 |
2. CMB 2011 (vol 56 pp. 258)
The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field Dufresnoy and Pisot characterized the smallest
Pisot number of degree $n \geq 3$ by giving explicitly its minimal
polynomial. In this paper, we translate Dufresnoy and Pisot's
result to the Laurent series case.
The
aim of this paper is to prove that the minimal polynomial
of the smallest Pisot element (SPE) of degree $n$ in the field of
formal power series over a finite field
is given by $P(Y)=Y^{n}-\alpha XY^{n-1}-\alpha^n,$ where $\alpha$
is the least element of the finite field $\mathbb{F}_{q}\backslash\{0\}$
(as a finite total ordered set). We prove that the sequence of
SPEs of degree $n$ is decreasing and converges to $\alpha X.$
Finally, we show how to obtain explicit continued fraction
expansion of the smallest Pisot element over a finite field.
Keywords:Pisot element, continued fraction, Laurent series, finite fields Categories:11A55, 11D45, 11D72, 11J61, 11J66 |
3. CMB 2011 (vol 55 pp. 774)
Pell Equations: Non-Principal Lagrange Criteria and Central Norms We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any non-square integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of
congruence conditions modulo $D$.
Keywords:Pell's equation, continued fractions, central norms Categories:11D09, 11A55, 11R11, 11R29 |
4. CMB 2005 (vol 48 pp. 121)
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ |
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$ We look at the simple continued fraction expansion of $\sqrt{D}$
for any $D=2^hc $ where $c>1$ is odd with a goal of
determining necessary and
sufficient conditions for the central norm (as determined by
the infrastructure of the underlying real quadratic order therein) to be
$2^h$. At the end of the paper, we also address the case where $D=c$
is odd and the central norm of $\sqrt{D}$ is equal to $2$.
Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fields Categories:11A55, 11D09, 11R11 |
5. CMB 2004 (vol 47 pp. 12)
On Newton's Method and Rational Approximations to Quadratic Irrationals In 1988 Rieger exhibited a differentiable function having a zero at
the golden ratio\break
$(-1+\sqrt5)/2$ for which when Newton's method for approximating
roots is applied with an initial value $x_0=0$, all approximates
are so-called ``best rational approximates''---in this case, of the
form $F_{2n}/F_{2n+1}$, where $F_n$ denotes the $n$-th Fibonacci
number. Recently this observation was extended by Komatsu to the
class of all quadratic irrationals whose continued fraction
expansions have period length $2$. Here we generalize these
observations by producing an analogous result for all quadratic
irrationals and thus provide an explanation for these phenomena.
Categories:11A55, 11B37 |
6. CMB 2002 (vol 45 pp. 428)
Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ The purpose of this article is to provide criteria for the
simultaneous solvability of the Diophantine equations $X^2 - DY^2 =
c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in
\mathbb{N}$ is not a perfect square. This continues work in
\cite{me}--\cite{alfnme}.
Keywords:continued fractions, Diophantine equations, fundamental units, simultaneous solutions Categories:11A55, 11R11, 11D09 |
7. CMB 1998 (vol 41 pp. 86)
On \lowercase{$q$}-exponential functions for \lowercase{$|q| =1$} We discuss the $q$-exponential functions $e_q$, $E_q$ for $q$ on the unit
circle, especially their continuity in $q$, and analogues of the limit
relation $ \lim_{q\rightarrow 1}e_q((1-q)z)=e^z$.
Keywords:$q$-series, $q$-exponentials Categories:33D05, 11A55, 11K70 |