The mutual intensity and its equivalent phase-space representations quantify an optical field's state of coherence and are important tools in the study of light propagation and dynamics, but they can only be estimated indirectly from measurements through a process called coherence retrieval, otherwise known as phase-space tomography. As practical considerations often rule out the availability of a complete set of measurements, coherence retrieval is usually a challenging high-dimensional ill-posed inverse problem. In this paper, we propose a trace-regularized optimization model for coherence retrieval and a provably convergent adaptive accelerated proximal gradient algorithm for solving the resulting problem. Applying our model and algorithm to both simulated and experimental data, we demonstrate an improvement in reconstruction quality over previous models as well as an increase in convergence speed compared to existing first-order methods.

We give exact formulas for the growth of the ideal Aλ for λ a quadratic element of the algebra of Boolean functions over the Galois field GF(2). That is, we calculate dim A_k λ where A_k is the subspace of elements of degree less than or equal to k. These results clarify some of the assertions made in the article of Yang, Chen and Courtois [22,23] concerning the efficiency of the XL algorithm.

Recent studies have shown that the entropy production rate for the master equation consists of two nonnegative terms: the adiabatic and non-adiabatic parts, where the non-adiabatic part is also known as the dissipation rate of a Boltzmann-Shannon relative entropy. In this paper, we provide some nonzero lower bounds for the relative entropy, the entropy production rate, and its adiabatic and non-adiabatic parts. These nonzero lower bounds not only reveal some novel dissipative properties for general nonequilibrium processes which are much stronger than the second law of thermodynamics, but also impose some new constraints on thermodynamic constitutive relations. Moreover, we also provide a mathematical application of these nonzero lower bounds by studying the long-time behavior of the master equation. Extensions to the Tsallis statistics are also discussed, including the nonzero lower bounds for the Tsallis-type relative entropy and its dissipation rate.

We prove that a random map drawn from any class \mathfrak{C} of polynomial maps from (F_q)^n to (F_q)^{n+r} that is (i) totally random in the affine terms, and (ii) has a negligible chance of being not strongly one-way, provides a secure PRNG (hence a secure stream cipher) for any q. Plausible choices for are semi-sparse (i.e., the affine terms are truly random) systems and other systems that are easy to evaluate from a small (compared to a generic map) number of parameters. To our knowledge, there are no other positive results for provable security of specialized polynomial systems, in particular sparse ones (which are natural candidates to investigate for speed). We can build a family of provably secure stream ciphers a rough implementation of which at the same security level can be more than twice faster than an optimized QUAD (and any other provably secure stream ciphers proposed so far), and uses much less storage. This may also help build faster provably secure hashes. We also examine the effects of recent results on specialization on security, e.g., Aumasson-Meier (ICISC 2007), which precludes Merkle-Damgaard compression using polynomials systems {uniformly very sparse in every degree} from being universally collision-free. We conclude that our ideas are consistent with and complements these new results. We think that we can build secure primitives based on specialized (versus generic) polynomial maps which are more efficient.

Schrodinger operators with strongly singular interactions, typically supported by a set of Lebesgue measure zero, eg, by a nite or countable set of points, appeared in quantum mechanics from the very beginning~cf., eg,[35, 28]. Their use had essentially two motivations. First of all, they were expected to provide a reasonable description in the physical situations when the real interaction has a very small range comparing to the other characteristic lengths of the problem, for instance, to the wavelength of the scattered particles. On the other hand, the fact that the interaction support is small and outside it a free solution of the corresponding Schrodinger equation may be used often makes such models explicitly solvable. Until the beginning of the sixties, however, the occasional use of these point interactions remained purely formal. Only after the work of Berezin and Faddeev [11] did it become clear they can be