Metrical results on the discrepancy of Halton–Kronecker sequences

By a Halton–Kronecker sequence we mean a sequence in the (s+t)-dimensional unit-cube which is the combination of an s-dimensional Halton-sequence and a t-dimensional Kronecker sequence. The distribution of such sequences was studied for the first time quite recently by Niederreiter. In this paper we obtain metrical results for the discrepancy of Halton–Kronecker sequences which are similar to results for the pure Kronecker sequences obtained by Khintchine and by W.M. Schmidt.     

Lower estimate of Tandori’s norm

The conventional monotonicity hypothesis on the coefficients given in the sharpest lower estimate of the celebrated Tandori-norm is weakened to locally almost monotonicity assumption.     

Non-stretch mappings for a sharp estimate of the Beurling–Ahlfors operator

In this paper we identify certain classes of non-stretch mappings that enjoy a sharp estimate of the Beurling–Ahlfors operator. We first make use of a property of subharmonic functions to prove that the Bañuelos–Wang conjecture and the Iwaniec conjecture are true for a class of mappings that satisfy a quasilinear conjugate Beltrami equation. By utilizing the principal solutions of Beltrami equations, we further explicitly construct some classes of non-stretch mappings for which the Bañuelos–Wang conjecture and the Iwaniec conjecture are true.     

Using rational idempotents to show Turyn’s bound is sharp

Davis, Dillon, and Jedwab all showed the existence of difference sets in groups C_2^r+2×C_2^r{C_{2^{r+2}}\times C_{2^{r}}} . Turyn’s bound had previously shown that abelian 2-groups with higher exponents could not admit difference sets. We give a new construction technique that utilizes character values, rational idempotents, and tiling structures to produce Hadamard difference sets in the group C_2^r+2×C_2^r{C_{2^{r+2}}\times C_{2^{r}}} to replicate the result.     

The number of integer points on Vinogradov’s quadric

An asymptotic formula is obtained for the number of integer solutions of bounded height on Vinogradov’s quadric. Two leading terms are determined, and a strong estimate for the error term is given.     

Pseudo-randomness of van der Corput’s sequences

The aim of this paper is to find main terms of the star D _N ^* and extremal D _N discrepancies of the two dimensional sequence (x _n , x _n+1), n = 0, 1, 2, ..., N − 1, where x _n , n = 0, 1, 2, ..., is the van der Corput sequence. This give a quantitative form of a well-known result that van der Corput sequence is not pseudorandom. This research was supported by the Slovak Academy of Sciences Vega Grant No. 2/7138/27.     

Uniform distribution of generalized Kakutani’s sequences of partitions

We consider a generalized version of Kakutani’s splitting procedure where an arbitrary starting partition π is given and in each step all intervals of maximal length are split into m parts, according to a splitting rule ρ. We give conditions on π and ρ under which the resulting sequence of partitions is uniformly distributed.     

Roth’s theorem in many variables

We prove that if A ? {1, ..., N} has no nontrivial solution to the equation x ₁ + x ₂ + x ₃ + x ₄ + x ₅ = 5y, then \(|A| \ll Ne^{ - c(\log N)^{1/7} } \) , c > 0. In view of the well-known Behrend construction, this estimate is close to best possible.     

Complex Euler’s groups and values of Euler’s function at complex integer Gauss points

The complex Euler group is defined associating to an integer complex number z the multiplicative group of the complex integers residues modulo z, relatively prime to z. This group is calculated for z=(3+0i) ⁿ : it is isomorphic to the product of three cyclic group or orders (8, 3 ⁿ⁻¹ and 3 ⁿ⁻¹).     

Ramanujan’s estimate for the gamma function via monotonicity arguments

The aim of this paper is to extend and refine an approximation formula of the gamma function by Ramanujan.     

A survey of Skolem-type sequences and Rosa’s use of them

Let D be a set of positive integers. A Skolem-type sequence is a sequence of i ∈ D such that every i ∈ D appears exactly twice in the sequence at positions a _i and b _i , and |b _i − a _i | = i. These sequences might contain empty positions, which are filled with null elements. Thoralf A. Skolem defined and studied Skolem sequences in order to generate solutions to Heffter’s difference problems. Later, Skolem sequences were generalized in many ways to suit constructions of different combinatorial designs. Alexander Rosa made the use of these generalizations into a fine art. Here we give a survey of Skolem-type sequences and their applications. Supported by an NSERC Graduate fellowship. This work is in partial fulfillment of an M.Sc.     

A note on Burgess’s estimate

By introducing the Rademacher-Menchov device, we prove “maximal” analogs of principal bounds of character sums. This allows us to present the Burgess method so as to separate the main idea of this method from the technical issues.     

Estimating upper bounds on the limit points of majorizing sequences for Newton’s method

We present new sufficient conditions for the semilocal convergence of Newton’s method to a locally unique solution of an equation in a Banach space setting. Upper bounds on the limit points of majorizing sequences are also given. Numerical examples are provided, where our new results compare favorably to earlier ones such as Argyros (J Math Anal Appl 298:374–397, 2004), Argyros and Hilout (J Comput Appl Math 234:2993-3006, 2010, 2011), Ortega and Rheinboldt (1970) and Potra and Pták (1984).     

The lattice point discrepancy of a body of revolution: Improving the lower bound by Soundararajan’s method

For a convex body in which is invariant under rotations around one coordinate axis and has a smooth boundary of bounded nonzero curvature, the lattice point discrepancy (number of integer points minus volume) of a linearly dilated copy is estimated from below. On the basis of a recent method of K. Soundararajan [16], an -bound is obtained that improves upon all earlier results of this kind.Dedicated to the memory of Professor Erich LamprechtThis revised electronic version of the Abstract includes the formulas that were missing in the previous electronic version published online in September 2004.     

New estimate in Vinogradov’s mean-value theorem

We refine the upper bound for the Vinogradov integral.     

A new sharp approximation for the Gamma function related to Burnside’s formula

In this paper, based on Burnside’s formula, a similar continued fraction approximation of the factorial function and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the Burnside’s formula and the classical Stirling’s series, some numerical computations are given.     

The lattice discrepancy of bodies bounded by a rotating Lamé’s curv

This article provides an asymptotic result for the lattice point discrepancy of the special three-dimensional body for fixed k > 2 and large t. Authors’ addresses: Ekkehard Kr?tzel, Faculty of Mathematics, University of Vienna, Nordbergstra?e 15, 1090 Wien, ?sterreich; Werner Georg Nowak, Department of Integrative Biology, Institute of Mathematics, Universit?t für Bodenkultur Wien, Gregor Mendel-Stra?e 33, 1180 Wien, ?sterreich