Generalized interval exchanges and the 2–3 conjecture

We introduce the notion of a generalized interval exchange induced by a measurable k-partition of [0,1). can be viewed as the corresponding restriction of a nondecreasing function on ℝ with . A is called λ-dense if λ(A _i ∩(a, b))>0 for each i and any 0≤ a< b≤1. We show that the 2–3 Furstenberg conjecture is invalid if and only if there are 2 and 3 λ-dense partitions A and B of [0,1), such that . We give necessary and sufficient conditions for this equality to hold. We show that for each integer m≥2, such that 3∤2m+1, there exist 2 and 3 non λ-dense partitions A and B of [0,1), corresponding to the interval exchanges on 2m intervals, for which and commute.     

Emergence and control of multiphase nonlinear waves by synchronization

Large amplitude multiphase solutions of the periodic Korteweg-de Vries equation are excited and controlled by a small forcing. The approach uses passage through an ensemble of resonances and subsequent multiphase self-locking of the system with eikonal-type perturbations. The synchronization of each phase in the Korteweg-de Vries wave is robust, provided the corresponding driving amplitude exceeds a threshold.     

Estimates of the spectral radius of graphs

We give lower and upper bounds on the spectral radius of directed Eulerian multigraphs and undirected multigraphs.     

Richtungskorrelation für Kernresonanzfluoreszenz und Lebensdauer des 412 keV-Niveaus von198Hg

Using the method of thermal Doppler-broadening of the emission line, the effective differential cross sections for nuclear resonance scattering for 412 keV photons by¹⁹⁸Hg has been measured for well defined scattering angles of 120° and 160°. The results indicate a partial mean lifetime for this energy level of τ_γ=(5·12±0·23)·10^?11 sec. This result differs appreciably from the values reported by other workers. Possible reasons for these discrepancies are discussed. The ratio of the differential cross sections for resonance scattering at angles of 120° and 160° is consistent with a 0-2-0 transition.     

Variation of Grassman powers and spectra

The operator norm of the derivative of the map which takes a finite-dimensional linear operator to its kth Grassman power (the kth compound) is evaluated. This leads to a bound for the distance between the Grassman powers of two operators. As an important application, a bound for the distance between the eigenvalues of two operators is obtained.     

Inclusive and exclusive preequilibrium spectra in a model of independently interacting excitons

Starting from a previous ansatz ofindependently interacting excitons the more refined model INCLUSIVE INDEX is developed yieldinginclusive preequilibrium proton and neutron spectra. It is shown that three stages are sufficient to describe single- and multi-nucleon emission to all orders of practical importance. By use of recurrence relations the loss of particles and energy due to emission from previous stages is fully accounted for. In order to predict the population of specific residual nuclei the precompound process is reformulated in terms ofexclusive emission spectra. This intricate statistical problem is solved in the EXCLUSIVE INDEX model by algorithms representing the average probability for a specified number of particles not to be emitted but to reach the next stage of nuclear relaxation. Differential excitation probabilities can be calculated for those six nuclei which are populated by the emission of none, one, or two preequilibrium nucleons in two stages via alltogether 15 principal paths. Typical features of the INCLUSIVE and EXCLUSIVE INDEX model are discussed. The strength of one- and two-nucleon emission is compared with HYBRID model predictions.     

Analysis of inclusive nucleon spectra fromp- andα-induced reactions in the frame of the INDEX model,and implications for stopped π−-induced reactions

The previously published INDEX model is tested for nucleon spectra fromp- andα-induced reactions. The results of two alternative versions, the INCLUSIVE INDEX model and the EXCLUSIVE INDEX model, quite well agree with the data. It is found that in the INCLUSIVE INDEX model three preequilibrium stages are sufficient to describe single- and multi-nucleon emission. The model provides an useful first order estimate of the influence of the finite Fermi energy on particle spectra. This effect is very strong for nucleon induced reactions while forα-induced reactions it can be neglected. The deduced mean-free-path multiplier corroborates the long stated discrepancy between models in which excitons interact independently or not. Using preequilibrium parameters similar to those found for nucleon induced reactions the important branching ratio of contributingnp andpp pairs in stoppedπ ^?-absorption can be determined by INDEX model calculations. Deduced values from publishedn- andp-spectra agree reasonably well with those of other experimental analyses but deviate significantly from microscopic model predictions.     

Sparsity and non-Euclidean embeddings

We present a relation between sparsity and non-Euclidean isomorphic embeddings. We introduce a general restricted isomorphism property and show how it enables one to construct embeddings of ? _p ⁿ , p > 0, into various types of Banach or quasi-Banach spaces. In particular, for 0 < r < p < 2 with r ≤ 1, we construct a family of operators that embed ? _p ⁿ into $\ell _r^{(1 + \eta )n}$ , with sharp polynomial bounds in η > 0.     

Random embedding of $${\ell_p^n}$$ into $${\ell_r^N}$$

For any 0 < p < 2 and any natural numbers N > n, we give an explicit definition of a random operator \({S : \ell_p^n \to \mathbb{R}^N}\) such that for every 0 < r < p < 2 with r ≤ 1, the operator \({S_r = S : \ell_p^n \to \ell_r^N}\) satisfies with overwhelming probability that \({\|S_r\| \, \|(S_r)_{| {\rm Im}\, S}^{-1}\| \le C(p,r)^{n/(N-n)}}\), where C(p, r) > 0 is a real number depending only on p and r. One of the main tools that we develop is a new type of multidimensional Esseen inequality for studying small ball probabilities.