Narrow Results

Sol Golomb was my supervisor, mentor, and long-term collaborator, and played an important role in my life. This article is written according to my talks at various occasions in memory of Sol for describing some experiences when I worked with Sol.

Although noise-based logic shows potential advantages of reduced power dissipation and the ability of large parallel operations with low hardware and time complexity the question still persist: Is randomness really needed out of orthogonality? In this Letter, after some general thermodynamical considerations, we show relevant examples where we compare the computational complexity of logic systems based on orthogonal noise and sinusoidal signals, respectively. The conclusion is that in certain special-purpose applications noise-based logic is exponentially better than its sinusoidal version: Its computational complexity can be exponentially smaller to perform the same task.

Chains of the CME Group Time and Sales E-mini S&P 500 futures tick prices and their a-b-c-d-increments are studied. A discrete probability distribution based on the Hurwitz zeta function and Dirichlet series is suggested for the price increments. The randomness of the ticks is discussed using the notions of typicalness, chaoticness, and stochasticness introduced by Kolmogorov and Uspenskii and developed by predecessors, them, and pupils. They define randomness in terms of the theory of algorithms.

The gist of the paper is the fundamental uncertain nature of all kinds of uncertainties and consequently a critical epistemic review of historical and recent approaches, computational methods, algorithms. The review follows the development of the notion from the beginnings of thinking, via the Aristotelian and Skeptic view, the medieval nominalism and the influential pioneering metaphors of ancient India and Persia to the birth of modern mathematical disciplinary reasoning. Discussing the models of uncertainty, e.g. the statistical, other physical and psychological background we reach a pragmatic model related estimation perspective, a balanced application orientation for different problem areas. Data mining, game theories and recent advances in approximation algorithms are discussed in this spirit of modest reasoning.

An overview is given of recent results concerning systems described by a set of at least two slow dynamical variables. The simplest model contains a relaxing order parameter coupled to the energy density (model C). The effects induced by randomness in such a model are discussed. At the superconducting transition the gauge dependence of the critical dynamics is considered for a model of two coupled relaxation equations.

We put forward a novel quantum secret sharing (QSS) protocol based on round-robin differential-phase-shift scheme (RRDPS), which achieves the distribution of a full secret key to one party and two partial keys to two parties that neither of the two partials alone can decipher the cryptographic message by full key. Making use of the advantages of round-robin differential-phase-shift, the protocol has a high tolerance of bit errors and enjoys a constant amount of privacy amplification and inherent randomness without precise signal disturbance estimation, especially when a finite key effect is considered. In terms of configuration, it is a simple modification, a variable-delay interferometer replacing the one-pulse-delay counterpart, of differential-phase-shift quantum secret sharing(DPS-QSS).

Quantum counterparts of certain classical systems exhibit chaotic spectral statistics of their energy levels; eigenvalues of infinite random matrices model irregular spectra. Eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices admit a gamma distribution approximation, as do the hermitian unitary (GUE) and quaternionic symplectic (GSE) cases. Then chaotic and non-chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for general stochastic systems.

Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events or voids among objects. Here we look at the properties of the Shannon entropy function and calculate its corresponding flow curves, relating them to examples of constrained degeneration from ordered processes. We consider univariate and bivariate gamma, and Weibull distributions since these also include exponential distributions.

The computational complexity of a set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Löf random sets, where we quantify the changes of initial segments. Secondly, we look at c.e. sets, where we quantify the overall amount of changes by obedience to cost functions. Finally, we combine the two settings. The discussions lead to three basic principles on how complexity and changes relate.

The method by using the combination Monte Carlo stochastic finite element method and χ2 test method to study the random method of bulk temperature of gears was put forward. The results showed that when the input random variables obeys normal distribution, the each point of the temperature on the gear also obeys it; The bulk temperature of gears has the feature is that it is constant for the linear system to normal variable input system response; With the mean temperature increasing, the discrete degree of temperature gets greater; The result of the mean temperature of the each point is all greater than the finite element method analysis ,so the results are conservative at the certainty model of the bulk temperature of gears.