Hausdorff dimension and conformal measures of Feigenbaum Julia sets

We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon', there exist many Feigenbaum Julia sets whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent is equal to the hyperbolic dimension . Moreover, if , then . In the stationary case, the last statement can be reversed: if , then . We also give a new construction of conformal measures on that implies that they exist for any , and analyze their scaling and dissipativity/conservativity properties.

     

Optimal investment with two-factor uncertainty

This paper presents a real options model to value the option to invest in a project contingent on two stochastic factors. A general sensitivity analysis is conducted highlighting the importance of the variance and correlation between the two variables. A higher correlation is shown to increase always the values of the trigger, the active project and the option. The impact of uncertainty is more complex and depends on the assumption about which variables adjust and the correlation between the variables and the market.     

The Dirichlet ring and unconditional bases in L 2[0, 2π]

It is observed that the Dirichlet ring admits a representation in an infinite-dimensional matrix algebra. The resulting matrices are subsequently used in the construction of nonorthogonal Riesz bases in a separable Hilbert space. This framework enables custom design of a plethora of bases with interesting features. Remarkably, the representation of signals in any one of these bases is numerically implementable via fast algorithms.     

Splitting Formulas for Tutte Polynomials

We present two splitting formulas for calculating the Tutte polynomial of a matroid. The first one is for a generalized parallel connection across a 3-point line of two matroids and the second one is applicable to a 3-sum of two matroids. An important tool used is the bipointed Tutte polynomial of a matroid, an extension of the pointed Tutte polynomial introduced by Brylawski.     

Harmonic and Superharmonic Majorants on the Disk

A proof is given to show that a positive function on the unitdisk admits a harmonic majorant if and only if it has a certainexplicit upper envelope that admits a superharmonic majorant.The (logarithmic) Lipschitz regularity of this superharmonicmajorant is discussed. 2000 Mathematics Subject Classification31A05.     

Free vibrations of thin annular plates made from functionally graded material

Subject of the consideration is thin annular plate made of a two-phase functionally graded composte. The plate has periodically inhomogeneous microstructure slowly varying in space: the λ-periodic structure along circular coordinate, but smoothly graded apparent (averaged) properties in the perpendicular, radial direction. The aim of the contribution is to derive and apply a deterministic macroscopic model describing the free vibrations of this plate. Modeling procedure is based on tolerance averaging technique. We received, equations system with smooth coefficients. We made numerical solution of this problem, using finite difference method, and analyze influence of material proportion and microstructure size on first frequency of free vibrations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Statistical Properties of Unimodal Maps

We consider typical analytic unimodal maps which possess a chaotic attractor. Our main result is an explicit combinatorial formula for the exponents of periodic orbits. Since the exponents of periodic orbits form a complete set of smooth invariants, the smooth structure is completely determined by purely topological data (“typical rigidity”), which is quite unexpected in this setting. It implies in particular that the lamination structure of spaces of analytic unimodal maps (obtained by the partition into topological conjugacy classes, see [ALM]) is not transversely absolutely continuous. As an intermediate step in the proof of the formula, we show that the distribution of the critical orbit is described by the physical measure supported in the chaotic attractor.     

Large free sets in powers of universal algebras

We prove that for each universal algebra ${(A, \mathcal{A})}$ of cardinality ${|A| \geq 2}$ and infinite set X of cardinality ${|X| \geq | \mathcal{A}|}$ , the X-th power ${(A^{X}, \mathcal{A}^{X})}$ of the algebra ${(A, \mathcal{A})}$ contains a free subset ${\mathcal{F} \subset A^{X}}$ of cardinality ${|\mathcal{F}| = 2^{|X|}}$ . This generalizes the classical Fichtenholtz–Kantorovitch–Hausdorff result on the existence of an independent family ${\mathcal{I} \subset \mathcal{P}(X)}$ of cardinality ${|\mathcal{I}| = |\mathcal{P}(X)|}$ in the Boolean algebra ${\mathcal{P}(X)}$ of subsets of an infinite set X.     

Observation of geometric phases for mixed states using NMR interferometry

Examples of geometric phases abound in many areas of physics. They offer both fundamental insights into many physical phenomena and lead to interesting practical implementations. One of them, as indicated recently, might be an inherently fault-tolerant quantum computation. This, however, requires one to deal with geometric phases in the presence of noise and interactions between different physical subsystems. Despite the wealth of literature on the subject of geometric phases very little is known about this very important case. Here we report the first experimental study of geometric phases for mixed quantum states. We show how different they are from the well-understood, noiseless, pure-state case.