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Many nondeterminism measures for finite automata have been studied in the literature. The tree width of an NFA (nondeterministic finite automaton) counts the number of leaves of computation trees as a function of input length. The trace of an NFA is defined in terms of the largest product of the degrees of nondeterministic choices in computations on inputs of given length. Branching is the corresponding best case measure based on the product of nondeterministic choices in the computation that minimizes this value. We establish upper and lower bounds for the trace of an NFA in terms of its tree width. We give a tight bound for the size blow-up of determinizing an NFA with finite trace. Also we show that the trace of any NFA either is bounded by a constant or grows exponentially.

A new methodology was introduced to investigate the pattern formation in time series systems due to their viscoelastic behavior. Four stochastic processes, uniform distribution, normal distribution, Nasdaq-100 stock market index, and a melody were studied within this context. The time series data were converted into vectorial forms in a scattering diagram. The sequential vectors can be split into its in-line (or conservative) and out-of-line (or dissipative) components. Thus, one can define the storage and loss modulus for conservative, and dissipative components, respectively. Instead of using the geometric Brownian equation which involves Wiener noise term, the changes were taken into consideration at every step by introducing “lethargy” concept and the deviation from it. Thus, the mathematics is somehow simplified, and the dynamical behavior of time series systems can be elucidated at every step of change. The viscoelastic behavior of time series systems reveals patterns of the viscoelastic parameters such as storage and loss modulus, and also of thermodynamic work-like and heat-like properties. Besides, there occur some minima and maxima in the distribution of the angles between the sequential vectors in the scattering diagram. The same is true for the change of entropy of the system.

A novel metasurface is proposed that aims to generate underwater acoustic waves with various functions by only one actuator. Each metasurface unit consists of an air cavity sandwiched on one side by a vibration plate and connecting rubber supports. By properly selecting the ratio of the plate to unit lengths, a phase shift of $\pi$ can be attained to constitute a binary coding metasurface. Three demonstrations, including focusing, branching and self-bending waves, are chosen to validate the functionality of the design. The design is also shown to work over a wide frequency range by changing the ratio. In addition, the design is extremely compact, with the thickness only about 1/100 of the target wavelength. Compared with commonly used phased array transducers that are utilized to generate underwater acoustic waves, this design offers has the advantage of needing only a single actuator as opposed to needing a lumped electrical control system.

Branching random walks on multidimensional lattice with heavy tails and a constant branching rate are considered. It is shown that under these conditions (heavy tails and constant rate), the front propagates exponentially fast, but the particles inside of the front are distributed very non-uniformly. The particles exhibit intermittent behavior in a large part of the region behind the front (i.e. the particles are concentrated only in very sparse spots there). The zone of non-intermittency (were particles are distributed relatively uniformly) extends with a power rate. This rate is found.

Human solid tumors are believed to be caused by a sequence of genetic abnormalities arising in normal and premalignant cells. The understanding of these sequences is important for improving cancer treatment. Models for the occurrence of the abnormalities include linear structure and a recently proposed tree-based structure. We will describe the oncogenetic tree model and an efficient algorithm for its estimation. We also discuss methods for estimating the reliability and goodness-of-fit of this reconstruction. An R package “Oncotree” implementing the described methodology is available from the authors.

Photonuclear reaction yields were studies at the range of giant dipole resonance for medium-mass targets and the probabilities for population of different final states were deduced and analyzed. In addition to the expected statistical trends expressed in the influence of the reaction threshold and of the product spin, there are observed the microstructure manifestations due to the individual level schemes of the products. The internal status of nucleons and their single-particle orbital momentum also make influence onto the reaction yield. The effects are isolated on the scale from tens % to orders of magnitude for the probability and branching ratio in reactions at modest energy.