On some sequencing problems in finite groups

A finite group is called Z-sequenceable if its non-identity elements can be listed x₁, x₂, …, x_n so that x_ix_i+1 for i = 1, 2, …, n ? 1. Various necessary and sufficient conditions are determined for such sequencings to exist. In particular, it is proved that if n ? 3, then the symmetric group S_n is not Z-sequenceable.     

The spin of prime ideals

Fixing a nontrivial automorphism of a number field K, we associate to ideals in K an invariant (with values in {0,±1}) which we call the spin and for which the associated L-function does not possess Euler products. We are nevertheless able, using the techniques of bilinear forms, to handle spin value distribution over primes, obtaining stronger results than the analogous ones which follow from the technology of L-functions in its current state. The initial application of our theorem is to the arithmetic statistics of Selmer groups of elliptic curves.     

School mathematics and creativity at the elementary and middle-grade levels: how are they related?

In this study, we evaluated students’ creativity, as expressed in the solution methods of three problems for groups of students in different grades. Posing the same problems to students of similar (advanced) mathematical abilities in different grades allowed us to look for possible connections between creativity and mathematical knowledge. The findings indicate that at the elementary school level, the number of solution methods and creativity scores increased with age. The collective methods space of the eighth graders seemed to narrow almost exclusively to algebraic methods, but the increase in the number of solutions was renewed in the ninth grade.     

An inexact-restoration method for nonlinear bilevel programming problems

We present a new algorithm for solving bilevel programming problems without reformulating them as single-level nonlinear programming problems. This strategy allows one to take profit of the structure of the lower level optimization problems without using non-differentiable methods. The algorithm is based on the inexact-restoration technique. Under some assumptions on the problem we prove global convergence to feasible points that satisfy the approximate gradient projection (AGP) optimality condition. Computational experiments are presented that encourage the use of this method for general bilevel problems. This work was supported by PRONEX-Optimization (PRONEX—CNPq/FAPERJ E-26/171.164/2003—APQ1), FAPESP (Grants 06/53768-0 and 05-56773-1) and CNPq.     

Preconditioning by Projectors in the Solution of Contact Problems: A Parallel Implementation

A non-overlapping domain decomposition algorithm of the Neumann–Neumann type for solving contact problems of elasticity is presented. Using the duality theory of convex programming, the discretized problem turns into a quadratic one with equality and bound constraints. The dual problem is modified by orthogonal projectors to the natural coarse space. The resulting problem is solved by an augmented Lagrangian algorithm. The projectors ensure an optimal convergence rate for the solution of the auxiliary linear problems by the preconditioned conjugate gradient method. Relevant aspects on the numerical linear algebra of these problems are presented, together with an efficient parallel implementation of the method.     

Nonlinear Instability in Two Dimensional Ideal Fluids: The Case of a Dominant Eigenvalue

It is proved that any steady 2 dimensional ideal fluid flow is nonlinearly unstable with respect to L ² growth in the velocity, provided there exists an eigenvalue for the linearised Euler equation with Re>. Here is the maximal Lyapunov exponent of the steady flow.     

Nanoparticle Microreactor: Application to Synthesis Of Titania by Thermal Decomposition of Titanium Tetraisopropoxide

The nanoparticle microreactor (NPMR) is a new concept that we have introduced to describe a very small-scale system capable of converting an aerosol precursor to solid particles. The liquid precursor of about 1 µl is injected by a syringe through a septum into a tubular evaporator of 1.0 cm³ in volume with stopcocks at both ends. The evaporator has been preheated by a heating tape to a temperature sufficiently high for vaporization to occur in half a minute. By opening the stopcocks, the vaporized precursor is transported by a carrier gas stream into a quartz tube which is mounted along the axis of a tubular furnace. The nanoparticle aggregates produced in the reactor are sampled by deposition on an electron micrograph grid at the reactor exit. The NPMR was applied first to the synthesis of TiO₂ particles by thermal decomposition of titanium tetraisopropoxide (TTIP) in a nitrogen carrier gas, with TTIP concentrations varying from 1.0 to 7.0 mol% or 2.35×10^–6 to 1.65×10^–5 in TiO₂ volume loading, and decomposition temperatures from 300°C to 1000°C. Studies were made with a 2 mm reaction tube and a 4 mm tube with sheath gas. With the 2 mm tube, a considerable fraction of the TTIP precursor was consumed at the wall by surface reaction, resulting in very small particles. With the 4 mm tube, the primary particle size was comparable to that reported in the literature for steady flow experiments using a 22.2 mm tube. Primary particle sizes ranged from 200 to 400 nm. Depending on TTIP concentration and reactor temperature, the particles exhibited a bimodal size distribution, probably due to a two-stage nucleation. A fourfold increase in the gas flow rate had little effect on particle size, indicating that particle growth ended early, within one-fourth the tube length. Residence time in the reactor was between 0.35 and 1.4 s, and total run time about 1 min. The NPMR has potential for rapid assembly of large databases and is adaptable to combinatorial discovery of nanoparticles with novel properties. Design requirements for an ideal aerosol microreactor are discussed briefly.     

Double exponential sums over thin sets

We estimate double exponential sums of the form

where is of multiplicative order modulo the prime and and are arbitrary subsets of the residue ring modulo . In the special case , our bound is nontrivial for with any fixed 0$">, while if in addition we have it is nontrivial for .

     

On the distribution of the power generator

We present a new method to study the power generator of pseudorandom numbers modulo a Blum integer . This includes as special cases the RSA generator and the Blum-Blum-Shub generator. We prove the uniform distribution of these, provided that the period with fixed 0$"> and, under the same condition, the uniform distribution of a positive proportion of the leftmost and rightmost bits. This sharpens and generalizes previous results which dealt with the RSA generator, provided the period . We apply our results to deduce that the period of the binary sequence of the rightmost bit has exponential length.