The requirement for solving a polynomial is a means of breaking its symmetry, which in the case of the quintic, is that of the symmetric group S₅. Induced by its five-dimensional linear permutation representation is a three-dimensional projective action. A mapping of complex projective 3-space with this S₅ symmetry can provide the requisite symmetry-breaking tool. The article describes some of the S₅ geometry in CP³ as well as several maps with particularly elegant geometric and dynamical properties. Using a rational map in degree six, it culminates with an explicit algorithm for solving a general quintic. In contrast to the Doyle-McMullen procedure, which involves three 1-dimensional iterations, the present solution employs one 3-dimensional iteration.

We find and prove relationships between Riemann zeta values and central binomial sums. We also investigate alternating binomial sums (also called Apery sums). The study of nonalternating sums leads to an investigation of different types of sums which we call multiple Clausen values. The study of alternating sums leads to a tower of experimental results involving polylogarithms in the golden ratio.

Using a class of polynomials that generalizes Legendre polynomials, we unify previous works of E. A. Rukhadze, A. K. Dubitskas, M. Hata, D. V. and G. V. Chudnovsky about irrationality measures of $\log 2$ and $\pi/\sqrt{3}$

We propose a random graph model which is a special case of sparse random graphs with given degree sequences which satisfy a power law. This model involves only a small number of parameters, called logsize and log-log growth rate. These parameters capture some universal characteristics of massive graphs. From these parameters, various properties of the graph can be derived.

We discussed closed form representations of filter coefficients of wavelets on the real line, half real line and on compact intervals. We show that computer algebra can be applied to perform this task. Moreover, we present a matrix analytical approach that unifies constructions of wavelets on the interval.

Using a result on Riesz factorizations, we show that there are at most 2^N-1 and at least 2^[N/2] distinct Daubechies scaling functions with support in [1-N, N].

Let $\f(X)$ be a cubic polynomial defining a simplest cubic field in the sense of Shanks. We study integral points on elliptic curves of the form $\Y² = f(X)$. We compute the complete list of integral points on these curves for the values of the parameter below 1000. We prove that this list is exhaustive by using the methods of Tzanakis and de Weger, together with bounds on linear forms in elliptic logarithms due to S. David. Finally, we analyze this list and we prove in the general case the phenomena that we have observed. In particular, we find all integral points on the curve when the rank is equal to 1.

An embedded curve is presented which under numerical simulation of the averaged mean curvature flow develops first a loss of embeddedness and then a singularity where the curvature becomes infinite, all in finite time. This leads to the conjecture that not all smooth embedded curves persist for all times under the averaged mean curvature flow.

This paper deals with the computation and classification of 5- and 6-dimensional torsion-free crystallographic groups, known as Bieberbach groups. We describe the basis of an algorithm that decides torsion-freeness of a crystallographic group as well as the triviality of its centre. The computations were done using the computer package CARAT, which handles enumeration, construction, recognition and comparison problems for crystallographic groups up to dimension 6. The complete list of isomorphism types of Bieberbach groups up to dimension 6 can be found online.

For λ $\in$(0, 2) let k(λ) denote the smallest positive value of κ so that the truncated power function φ _{λ, κ} (t) = (1 -- |t|^λ)^κ₊ is positive definite. We give lower and upper estimates of Kuttner's function k(λ) through detailed numerical and symbolic computations, and we show analytically that k((4n+1)/(2n+1)) ≤ 2n+1 for n $\in$ N.

It is known that if the period s(d) of the continued fraction expansion of $\sqrt{d}$ satisfies s(d) ≤ 2, then all Newton's approximants R_n = (1/2)((p_n/q_n) + (dq_n/p_n)) are convergents of $\sqrt{d}$, and moreover R_n = p_2n+1/q_2n+1 for all N ≤ 0. Motivated by this fact we define j = j(d, n) by R_n = p_2n+1+2J/q_2n+1+2j if R_n is a convergent of $\sqrt{d}$}|. The question is how large |j| and b can be. We prove that |j| is unbounded and gie some examples supporting a conjecture that b is unbounded too. We also discuss the magnitude of |j| and be compared with d and s(d).

We give an efficient algorithm for computing relative power integral bases in cubic relative extensions. The problem leads to solving relative Thue equations as described by [Gaál and Pohst 1999] using the enumeration method of [Wildanger 1997]. The article is illustrated by examples of relative cubic extensions of quintic and sextic fields which emphasizes the power of the method. This is the first case that unit equations of 12 unknown exponents are completely solved. The experiences of our computations may be useful for other related calculations, as well.

From the geometric study of the elementary cell of hexagonal circle packings---a flower of 7 circles---the class of conformally symmetric circle packings is defined. Up to Möbius transformations, this class is a three parameter family, that contains the famous Doyle spirals as a special case. The solutions are given explicitly. It is shown that these circle packings can be viewed as descretization s of the quotient of two Airy functions. The online version of this paper contains Java applets that let you experiment with the circle packings directly. The applets are found at http://www-sfb288.math.tu-berlin.de/Publications/online/cscpOnline/Applets.html

We determine the full list of anticanonically embedded quasismooth Fano hypersurfaces in weighted projective 4-spaces. There are 48 infinite series and 4442 sporadic examples. In particular, the Reid-Fletcher list of 95 types of anticanonically embedded quasismooth terminal Fano threefolds in weighted projective 4-spaces is complete. We also prove that many of these Fano hypersurfaces admit a Kähler-Einstein metric, and study the nonexistence of tigers on these Fano 3-folds. Finally, we prove that there are only finitely many families of quasismooth Calabi-Yau hypersurfaces in weighted projective spaces of any given dimension. This implies finiteness for various families of general type hypersurfaces.