Statistics of the polarization mode dispersion dynamics

We characterize the joint two-time statistics of the polarization-mode dispersion (PMD) vector. Good agreement with experimental PMD measurements taken on an installed system is achieved. The results can be used to obtain the temporal evolution of the average penalty of a system close to an outage condition.     

Biases and Variability from Costly Bayesian Inference

When humans infer underlying probabilities from stochastic observations, they exhibit biases and variability that cannot be explained on the basis of sound, Bayesian manipulations of probability. This is especially salient when beliefs are updated as a function of sequential observations. We introduce a theoretical framework in which biases and variability emerge from a trade-off between Bayesian inference and the cognitive cost of carrying out probabilistic computations. We consider two forms of the cost: a precision cost and an unpredictability cost; these penalize beliefs that are less entropic and less deterministic, respectively. We apply our framework to the case of a Bernoulli variable: the bias of a coin is inferred from a sequence of coin flips. Theoretical predictions are qualitatively different depending on the form of the cost. A precision cost induces overestimation of small probabilities, on average, and a limited memory of past observations, and, consequently, a fluctuating bias. An unpredictability cost induces underestimation of small probabilities and a fixed bias that remains appreciable even for nearly unbiased observations. The case of a fair (equiprobable) coin, however, is singular, with non-trivial and slow fluctuations in the inferred bias. The proposed framework of costly Bayesian inference illustrates the richness of a ‘resource-rational’ (or ‘bounded-rational’) picture of seemingly irrational human cognition.     

New semi-Hamiltonian hierarchy related to integrable magnetic flows on surfaces

We consider magnetic geodesic flows on the two-torus. We prove that the question of existence of polynomial in momenta first integrals on one energy level leads to a semi-Hamiltonian system of quasi-linear equations, i.e. in the hyperbolic regions the system has Riemann invariants and can be written in conservation laws form.     

Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

In recent years, sum–product estimates in Euclidean space and finite fields have received great attention. They can often be interpreted in terms of Erdős type incidence problems involving the distribution of distances, dot products, areas, and so on, which have been studied quite extensively by way of combinatorial and Fourier analytic techniques. We use both kinds of techniques to obtain sharp or near-sharp results on the distribution of volumes (as examples of d-linear homogeneous forms) determined by sufficiently large subsets of vector spaces over finite fields and the associated arithmetic expressions. Arithmetic–combinatorial techniques turn out to be optimal for dimension d≥4 to this end, while for d=3 they have failed to provide us with a result that follows from the analysis of exponential sums. To obtain the latter result we prove a relatively straightforward function version of an incidence results for points and planes previously established in [D. Hart, A. Iosevich, Sums and products in finite fields: An integral geometric viewpoint, in: Radon Transforms, Geometry, and Wavelets, Contemp. Math. 464 (2008); D. Hart, A. Iosevich, D. Koh, M. Rudnev, Averages over hyperplanes, sum–product theory in vector spaces over finite fields and the Erdős–Falconer distance conjecture, arXiv:math/0711.4427, preprint 2007].More specifically, we prove that if E=A××A is a product set in , d≥4, the d-dimensional vector space over a finite field , such that the size |E| of E exceeds (i.e. the size of the generating set A exceeds ) then the set of volumes of d-dimensional parallelepipeds determined by E covers . This result is sharp as can be seen by taking , a prime sub-field of its quadratic extension , with q=p². For in three dimensions, however, we are able to establish the same result only if (i.e., , for some C; in fact, the bound can be justified for a slightly wider class of “Cartesian product-like” sets), and this uses Fourier methods. Yet we do prove a weaker near-optimal result in three dimensions: that the set of volumes generated by a product set E=A×A×A covers a positive proportion of if (so ). Besides, without any assumptions on the structure of E, we show that in three dimensions the set of volumes covers a positive proportion of if |E|≥Cq², which is again sharp up to the constant C, as taking E to be a 2-plane through the origin shows.     

Are Corals Colorful?

Using in situ spectrometry data and visual system modeling, we investigate whether the colors conferred to the reef-building corals by GFP-like proteins would look colorful not only to humans, but also to fish occupying different ecological niches on the reef. Some GFP-like proteins, most notably fluorescent greens and nonfluorescent chromoproteins, indeed generate intense color signals. An unexpected finding was that fluorescent proteins might also make corals appear less colorful to fish, counterbalancing the effect of absorption by the photosynthetic pigments of the endosymbiotic algae, which might be a form of protection against herbivores. We conclude that GFP-determined coloration of corals may be an important factor in visual ecology of the reef fishes.     

Magnetic field effect on the superconducting magnetic gap of Nd1.85Ce0.15CuO4

Inelastic neutron-scattering measurements on the archetypical electron-doped material Nd1.85Ce0.15CuO4 up to a high relative magnetic-field strength, H/H(c2) approximately 50%, reveal a simple linear magnetic-field effect on the superconducting magnetic gap and the absence of field-induced in-gap states. The extrapolated gap-closing field value is consistent with the upper critical field H(c2), and the high-field response resembles that of the paramagnetic normal state.     

Tunnel ionization of molecules and orbital imaging

We study whether tunnel ionization of aligned molecules can be used to map out the electronic structure of the ionizing orbitals. We show that the common view, which associates tunnel ionization rates with the electronic density profile of the ionizing orbital, is not always correct. Using the example of tunnel ionization from the CO(2) molecule, we show how and why the angular structure of the alignment-dependent ionization rate moves with increasing the strength of the electric field. These modifications reflect a general trend for molecules.     

Wirtinger numbers and holomorphic symplectic immersions

For any subvariety of a compact holomorphic symplectic K?hler manifold, we define the symplectic Wirtinger number W(X). We show that W(X) \leqslant 1,W(X) \leqslant 1, and the equality is reached if and only if the subvariety X ì MX \subset M is trianalytic, i.e. compatible with the hyperk?hler structure on M. For a sequence X₁ ? X₂ ? ?X_n ? MX_1 \to X_2 \to \ldots X_n \to M of immersions of simple holomorphic symplectic manifolds, we show that W( X₁ ) \leqslant W( X₂ ) \leqslant ?\leqslant W( X_n ).W\left( {X_1 } \right) \leqslant W\left( {X_2 } \right) \leqslant \ldots \leqslant W\left( {X_n } \right).