Iterative quantum computations of HO2 bound states and resonances for J=4 and 5

Bound and resonance states of HO2 have been calculated by both the complex Lanczos homogeneous filter diagonalisation (LHFD) method(1,2) and the real Chebyshev filter diagonalisation method(3,4) for non-zero total angular momentum J = 4 and 5. For bound states, the agreement between the two methods is quite satisfactory; for resonances while the energies are in good agreement, the widths are only in general agreement. The relative performances of the two iterative FD methods have also been discussed in terms of efficiency as well as convergence behaviour for such a computationally challenging problem. A helicity quantum number Ohm assignment (within the helicity conserving approximation) is performed and the results indicate that Coriolis coupling becomes more important as J increases and the helicity conserving approximation is not a good one for the HO2 resonance states.

Subtractive and divisive adaptation in visual motion computations

Models of visual motion processing that introduce priors for low speed through Bayesian computations are sometimes treated with scepticism by empirical researchers because of the convenient way in which parameters of the Bayesian priors have been chosen. Using the effects of motion adaptation on motion perception to illustrate, we show that the Bayesian prior, far from being convenient, may be estimated on-line and therefore represents a useful tool by which visual motion processes may be optimized in order to extract the motion signals commonly encountered in every day experience. The prescription for optimization, when combined with system constraints on the transmission of visual information, may lead to an exaggeration of perceptual bias through the process of adaptation. Our approach extends the Bayesian model of visual motion proposed byWeiss et al. [Weiss Y., Simoncelli, E., & Adelson, E. (2002). Motion illusions as optimal perception Nature Neuroscience, 5:598-604.], in suggesting that perceptual bias reflects a compromise taken by a rational system in the face of uncertain signals and system constraints. © 2007.

Noisy random Boolean formulae:a statistical physics perspective

Properties of computing Boolean circuits composed of noisy logical gates are studied using the statistical physics methodology. A formula-growth model that gives rise to random Boolean functions is mapped onto a spin system, which facilitates the study of their typical behavior in the presence of noise. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding macroscopic phase transitions. The framework is employed for deriving results on error-rates at various function-depths and function sensitivity, and their dependence on the gate-type and noise model used. These are difficult to obtain via the traditional methods used in this field.

On the robustness of random Boolean formulae

Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models. © 2010 IOP Publishing Ltd.

Phase transitions and memory effects in the dynamics of Boolean networks

The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between Boolean networks and general Boolean formulas are identified and results common to both system types are highlighted. © 2012 Copyright Taylor and Francis Group, LLC.

Computing with noise: phase transitions in Boolean formulas

Computing circuits composed of noisy logical gates and their ability to represent arbitrary Boolean functions with a given level of error are investigated within a statistical mechanics setting. Existing bounds on their performance are straightforwardly retrieved, generalized, and identified as the corresponding typical-case phase transitions. Results on error rates, function depth, and sensitivity, and their dependence on the gate-type and noise model used are also obtained.

Dynamics of Boolean networks: an exact solution

The dynamics of Boolean networks (BN) with quenched disorder and thermal noise is studied via the generating functional method. A general formulation, suitable for BN with any distribution of Boolean functions, is developed. It provides exact solutions and insight into the evolution of order parameters and properties of the stationary states, which are inaccessible via existing methodology. We identify cases where the commonly used annealed approximation is valid and others where it breaks down. Broader links between BN and general Boolean formulas are highlighted.

On reliable computation by noisy random Boolean formulas

We study noisy computation in randomly generated k-ary Boolean formulas. We establish bounds on the noise level above which the results of computation by random formulas are not reliable. This bound is saturated by formulas constructed from a single majority-like gate. We show that these gates can be used to compute any Boolean function reliably below the noise bound.

Recognizing Decomposition of a Partial Boolean Function

A hard combinatorial problem is investigated which has useful application in design of discrete devices: the two-block decomposition of a partial Boolean function. The key task is regarded: finding such a weak partition on the set of arguments, at which the considered function can be decomposed. Solving that task is essentially speeded up by the way of preliminary discovering traces of the sought-for partition. Efficient combinatorial operations are used by that, based on parallel execution of operations above adjacent units in the Boolean space.

On Graph-Based Cryptography and Symbolic Computations

We have been investigating the cryptographical properties of in nite families of simple graphs of large girth with the special colouring of vertices during the last 10 years. Such families can be used for the development of cryptographical algorithms (on symmetric or public key modes) and turbocodes in error correction theory. Only few families of simple graphs of large unbounded girth and arbitrarily large degree are known. The paper is devoted to the more general theory of directed graphs of large girth and their cryptographical applications. It contains new explicit algebraic constructions of in finite families of such graphs. We show that they can be used for the implementation of secure and very fast symmetric encryption algorithms. The symbolic computations technique allow us to create a public key mode for the encryption scheme based on algebraic graphs.

Algebraic Computations with Hausdorff Continuous Functions

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006.