Related papers: A Class of Continued Radicals
We introduce a necessary and sufficient criterion for determining the existence and the values of ratio limits of complex sequences generated by arbitrary linear recurrences.
In some recent papers, the authors considered regular continued fractions of the form \[ [a_{0};\underbrace{a,...,a}_{m}, \underbrace{a^{2},...,a^{2}}_{m}, \underbrace{a^{3},...,a^{3}}_{m}, ... ], \] where $a_{0} \geq 0$, $a \geq 2$ and $m…
In the first part we associate a periodic sequence to a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved
For integers $m \geq 2$, we study divergent continued fractions whose numerators and denominators in each of the $m$ arithmetic progressions modulo $m$ converge. Special cases give, among other things, an infinite sequence of divergence…
In this note we produce examples of converging sequences of Galois representations, and study some of their properties. Some of the results here are used in the preprint math.NT/0210296.
Combinatorial formulas expressing cyclic rook polynomials and cyclic permanents of rectangular matrices in terms of expansions along rows are presented
In this paper we prove that the gradient ideal of a Morse polynomial is radical. This gives a generic class of polynomials whose gradient ideals are radical. As a consequence we reclaim a previous result that the unconstrained polynomial…
The ring of integers and the discriminant are determined for number fields which are simple radical extensions.
In this paper, we study the complexity of p-adic continued fractions of a rational number, which is the p-adic analogue of the theorem of Lame. We calculate the length of Browkin expansion, and the length of Schneider expansion. Also, some…
In this paper we introduce and study the concept of normality degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be normal. Explicit formulas are obtained for some particular classes of…
We compute the Hilbert coefficients of a graded module with pure resolution and discuss lower and upper bounds for these coefficients for arbitrary graded modules.
We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including $[0;\overline{p-1,1,u(a+2nb)-1,p-1,1,v(a+(2n+1)b)-1 }\,\,]_{n=0}^\infty$, $[0; \overline{c + d m^{n}}]_{n=1}^{\infty}$ and…
In this paper we are interested in the class numbers of a family of real quadratic fields for which the square roots of the discriminants have a known expansion in continued fraction. In particular we prove that $h(D)>1$, with possibly a…
We consider limits over categories of extensions and show how certain well-known functors on the category of groups turn out as such limits. We also discuss higher (or derived) limits over categories of extensions.
It is widely believed that the continued fraction expansion of every irrational algebraic number $\alpha$ either is eventually periodic (and we know that this is the case if and only if $\alpha$ is a quadratic irrational), or it contains…
We present a general introduction to continued fractions, with special consideration to the function fields case. These notes were prepared for a summer class given this year in Beijing at Beihang university.
A polynomial of degree $n$ in two variables is shown to be uniquely determined by its Radon projections taken over $[n/2]+1$ parallel lines in each of the $(2[(n+1)/2]+1)$ equidistant directions along the unit circle.
A systematic method of summing the corrections to the renormalon residue arising from higher order renormalons is discussed.
We look at a class of transcendental real numbers xi which, together with their square, satisfy some extremal property of simultaneous approximation by rational numbers with the same denominator. We give a sufficient condition for such a…
In the present paper, we give sufficient conditions on the elements of the continued fractions $A$ and $B$ that will assure us that the continued fraction $A^B$ is a transcendental number. With the same condition, we establish a…