The effect of intragranular microstress in A1203-SiC nanocomposites

A theoretical model is established to investigate the intragranular particle residual stress in A1203-SiC nanocom-posites. Using this model, we calculate the average compressive stress on the A1203 grain boundary （GB） and the average tensile stress within A1203 grains caused by SiC nanoparticles. The normal compressive stress strengthens the GB, and the average tensile stress weakens the grains. The model gives a reasonable interpretation of the strength changes of A1203-SiC nanocomposites with the number of SiC particles.     

Double Standard Maps

We investigate the family of double standard maps of the circle onto itself, given by (mod 1), where the parameters a,b are real and 0 ≤ b ≤ 1. Similarly to the well known family of (Arnold) standard maps of the circle, (mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a,b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues that begin at the level b = 0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior. The first author was partially supported by NSF grant DMS 0456526. The second author was supported by FCT Grant SFRH/BD/18631/2004. CMUP is supported by FCT through POCTI and POSI of Quadro Comunitário de apoio III (2000-2006) with FEDER and national funding.     

Semigroup Actions of Expanding Maps

We consider semigroups of Ruelle-expanding maps, parameterized by random walks on the free semigroup, with the aim of examining their complexity and exploring the relation between intrinsic properties of the semigroup action and the thermodynamic formalism of the associated skew-product. In particular, we clarify the connection between the topological entropy of the semigroup action and the growth rate of the periodic points, establish the main properties of the dynamical zeta function of the semigroup action and relate these notions to recent research on annealed and quenched thermodynamic formalism. Meanwhile, we examine how the choice of the random walk in the semigroup unsettles the ergodic properties of the action.     

Observables and Statistical Maps

This article begins with a review of the framework of fuzzy probability theory. The basic structure is given by the -effect algebra of effects (fuzzy events) and the set of probability measures on a measurable space . An observable is defined, where is the value space of X. It is noted that there exists a one-to-one correspondence between states on and elements of and between observables and -morphisms from to . Various combinations of observables are discussed. These include compositions, products, direct products, and mixtures. Fuzzy stochastic processes are introduced and an application to quantum dynamics is considered. Quantum effects are characterized from among a more general class of effects. An alternative definition of a statistical map is given and it is shown that any statistical map has a unique extension to a statistical operator. Finally, various combinations of statistical maps are discussed and their relationships to the corresponding combinations of observables are derived.     

Maps of Matrices and Portrait Maps of the Density Operators of Composite and Noncomposite Systems

We obtain a new inequality for arbitrary Hermitian matrices. We describe particular linear maps called the matrix portrait of arbitrary N × N matrices. The maps are obtained as analogs of partial tracing of density matrices of multipartite qudit systems. The structure of the maps is inspired by “portrait” map of the probability vectors corresponding to the action on the vectors by stochastic matrices containing either unity or zero matrix elements. We obtain new entropic inequalities for arbitrary qudit states including a single qudit and discuss entangled single qudit state. We consider in detail the examples of N = 3 and 4. Also we point out the possible use of entangled states of systems without subsystems (e.g., a single qudit) as a resource for quantum computations.     

Common Limits of Fibonacci Circle Maps

We show that limits for the critical exponent tending to ∞ exist in both critical circle homeomorphism of golden mean rotation number and Fibonacci circle coverings. Moreover, they are the same. The limit map is not analytic at the critical point, which is flat, but has non-trivial complex dynamics.     

Local Critical Perturbations of Unimodal Maps

We introduce a new complete metric in the space of unimodal C ²-maps of the interval, with two maps close if they are close in the C ²-metric and differ only on a small interval containing their critical points. We identify all structurally stable maps in the sense of this metric. They are maps for which either (1) the trajectory of the critical point is attracted to a topologically attracting (at least from one side) periodic orbit, but never falls into this orbit, or (2) the critical point is mapped by some iterate to the interior of an interval consisting entirely of periodic points of the same (minimal) period. We verify the generalized Fatou conjecture for and show that structurally stable maps form an open dense subset of . Partially supported by NSF grant DMS 0456748. Partially supported by NSF grant DMS 0456526.     

Morrey Potentials and Harmonic Maps

This paper discusses trace estimates for Morrey potentials (i.e., Riesz potential integrals of Morrey functions), leading to a consideration of the C ^∞ smoothness of a class of generalized harmonic maps.     

Planar Maps and Continued Fractions

We present an unexpected connection between two map enumeration problems. The first one consists in counting planar maps with a boundary of prescribed length. The second one consists in counting planar maps with two points at a prescribed distance. We show that, in the general class of maps with controlled face degrees, the solution for both problems is actually encoded into the same quantity, respectively via its power series expansion and its continued fraction expansion. We then use known techniques for tackling the first problem in order to solve the second. This novel viewpoint provides a constructive approach for computing the so-called distance-dependent two-point function of general planar maps. We prove and extend some previously predicted exact formulas, which we identify in terms of particular Schur functions.     

Spectral Conditions for Positive Maps

We provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions. This construction generalizes the celebrated Choi example of a map which is positive but not completely positive. It is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements. Such maps provide basic tools to study quantum entanglement in multipartite systems.     

Rational Misiurewicz Maps are Rare

We show that the set of Misiurewicz maps has Lebesgue measure zero in the space of rational functions for any fixed degree d ≥ 2.     

Nonuniformly Expanding 1D Maps

This paper attempts to make accessible a body of ideas surrounding the following result: Typical families of (possibly multi-model) 1-dimensional maps passing through ``Misiurewicz points' have invariant densities for positive measure sets of parameters. The research of both authors is partially supported by grant from the NSF     

Conformal Maps and Integrable Hierarchies

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as “string equations”. The same hierarchy locally solves the 2D inverse potential problem, i.e., reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c= matter. We also introduce a concept of the τ-function for analytic curves. Received: 20 December 1999/ Accepted: 2 March 2000     

Boundary from Bulk Integrability in Three Dimensions: 3D Reflection Maps from Tetrahedron Maps

We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process on the Sierpinski gasket in the presence of a variable speed boundary. We obtain, at the hydrodynamics level, the heat equation evolving on the Sierpinski gasket with either Dirichlet or Neumann boundary conditions, depending on whether the reservoirs are fast or slow. For a particular strength of the boundary dynamics we obtain linear Robin boundary conditions. As for the fluctuations, we prove that, when starting from the stationary measure, namely the product Bernoulli measure in the equilibrium setting, they are governed by Ornstein-Uhlenbeck processes with the respective boundary conditions.

     

Theory of size-selective precipitation

A theory of size-selective precipitation of spherical nanoparticles is presented. The precipitate is assumed to be a random loose packing of spheres with low polydispersity (less than ≈30%). The interparticle interaction is treated in the Derjaguin approximation. The theory is used to calculate the polydispersity in the fractionated particle samples for various problem parameters. Our calculations show that, for typical particle concentrations, fractionation is effective for polydispersities above 20% and ineffective for polydispersities below 5%.     

Distribution of Resonances for Open Quantum Maps

We analyze a simple model of quantum chaotic scattering system, namely the quantized open baker’s map. This model provides a numerical confirmation of the fractal Weyl law for the semiclassical density of quantum resonances. The fractal exponent is related to the dimension of the classical repeller. We also consider a variant of this model, for which the full resonance spectrum can be rigorously computed, and satisfies the fractal Weyl law. For that model, we also compute the shot noise of the conductance through the system, and obtain a value close to the prediction of random matrix theory.