Cellular generators

The aim of this paper is twofold. On the one hand, we show that the kernel of the Bousfield periodization functor is cellularly generated by a space , i.e., we construct a space such that the smallest closed class containing is exactly . On the other hand, we show that the partial order is a complete lattice, where if . Finally, as a corollary we obtain Bousfield's theorem, which states that )$"> is a complete lattice, where A$"> if .

     

Turbulent chaos

A new definition of a chaotic invariant set for a continuous semi-flow in a metric space is given. This definition generalizes the known definition of Devaney and allows us to take into account a specific feature arising in a noncompact and infinite-dimensional case, the so-called turbulent chaos.     

New theory of spatio-temporal chaos in nonlinear systems of partial differential equations

It is shown that there exists one universal scenario of transition to chaos in three considered systems of nonlinear partial differential equations of diffusion type. It is the subharmonic cascade of bifurcations of two-dimensional tori along one or both frequencies. So, it is shown that the new universal Feigenbaum-Sharkovskii-Magnitskii theory of dynamical chaos in nonlinear differential equations is applicable also for nonlinear partial differential equations of diffusion type. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Varieties without minimal generators

While many familiar varieties have a minimal varietal generator, i.e., a regular projective finitely presentable regular generator such that none of its retracts is a regular generator, and even a unique one, we present (a) a variety having no minimal varietal generator at all and (b) a variety having two non-isomorphic minimal varietal generators. Moreover we demonstrate that the same effects can happen with respect to a weaker notion of minimality and are common even in module categories. Received April 7, 1999; accepted in final form July 10, 2000.     

Interaction and program generators

We consider problems connected with the automatic synthesis of programs. We propose a functional approach to the organization of program generators. We give algorithms for extracting programs from the reduction of object terms and the organization of dialog for specification of free variables. Translated fromDinamicheskie Sistemy, No. 13, 1994, pp. 103–106.     

On generators and cogenerators

We consider the existence of generators and cogenerators in several categories. In many cases the non-existence of cogenerators results from the construction of arbitrarily high dimensional simple (restricted) Lie algebras. The existence of generators in the Hopf algebra categories results from the existence of generators in a certain coalgebra category and the construction of a free Hopf algebra on a coalgebra.     

Devaney’s chaos implies distributional chaos in a sequence

Let X be a complete metric space without isolated points, and let f:X→X be a continuous map. In this paper we prove that if f is transitive and has a periodic point of period p, then f is distributionally chaotic in a sequence. Particularly, chaos in the sense of Devaney is stronger than distributional chaos in a sequence.     

Structurization of chaos

Vortex cascades of instabilities forming a core are studied. Large-scale linear waves in a fluctuating medium are described.     

Minimal generators for symmetric ideals

Let be the polynomial ring in infinitely many indeterminates over a field , and let be the symmetric group of . The group acts naturally on , and this in turn gives the structure of a module over the group ring . A recent theorem of Aschenbrenner and Hillar states that the module is Noetherian. We address whether submodules of can have any number of minimal generators, answering this question positively.

     

Designing optimum cost function generators

The ultimate aim of design is optimization. Not much has been reported in the literature on the cost optimization of four-bar function generators. An example problem is solved to illustrate the application of the method.     

Pseudorandom generators for space-bounded computation

Pseudorandom generators are constructed which convertO(SlogR) truly random bits toR bits that appear random to any algorithm that runs inSPACE(S). In particular, any randomized polynomial time algorithm that runs in spaceS can be simulated using onlyO(Slogn) random bits. An application of these generators is an explicit construction of universal traversal sequences (for arbitrary graphs) of lengthn ^O(logn).The generators constructed are technically stronger than just appearing random to spacebounded machines, and have several other applications. In particular, applications are given for deterministic amplification (i.e. reducing the probability of error of randomized algorithms), as well as generalizations of it.This work was done in the Laboratory for Computer Science, MIT, supported by NSF 865727-CCR and ARO DALL03-86-K-017     

Predicting nonlinear pseudorandom number generators

Let be a prime and let and be elements of the finite field of elements. The inversive congruential generator (ICG) is a sequence of pseudorandom numbers defined by the relation . We show that if sufficiently many of the most significant bits of several consecutive values of the ICG are given, one can recover the initial value (even in the case where the coefficients and are not known). We also obtain similar results for the quadratic congruential generator (QCG), , where . This suggests that for cryptographic applications ICG and QCG should be used with great care. Our results are somewhat similar to those known for the linear congruential generator (LCG), , but they apply only to much longer bit strings. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for LCG.

     

Bifurcation continuation,chaos and chaos control in nonlinear Bloch system

A detailed analysis is undertaken to explore the stability and bifurcation pattern of the nonlinear Bloch equation known to govern the dynamics of an ensemble of spins, controlling the basic process of nuclear magnetic resonance. After the initial analysis of the parameter space and stability region identification, we utilize the MATCONT package to analyze the detailed bifurcation scenario as the two important physical parameters γ (the normalized gain) and c (the phase of the feedback field) are varied. A variety of patterns are revealed not studied ever before. Next we explore the structure of the chaotic attractor and how the identification of unstable periodic orbit (UPO) can be utilized to control the onset of chaos.     

Run distribution innonlinear binary generators

In this work, an easy method of computing the exact number of runs of ones and zerosin the sequence obtained from LFSR-based nonlinear generators has been developed. The procedure is based on the decomposition of the generating function in global minterms. If the obtained results are not in the expected range, then the sequence generator must be rejected for cryptographic purposes.     

Invariant generators for generalized distributions

The existence of invariant generators for distributions satisfying a compatibility condition with the symmetry algebra is proved.