Abstract:
A new modeling of heating and evaporation of fuel droplets and ignition of a fuel vapour/ air mixture in continuous form is suggested. The size distribution of fuel droplets is assumed to be continuous and found from the solution of the kinetic equation for the probability density function (PDF). The semi-transparency of droplets, the difference between the gas temperature and the external temperature are take into account. The model represent in dimensionless from, and the dynamics of the system is present in term of the dynamics of a multi-scale, singularly perturbed system (SPS)

Abstract:
The process of making complex and controversial decisions, that is, dealing with moral or ethical dilemmas, have intrigued people and inspired writers from time immemorial. Dilemmas give both color and depth to characters in good literary works. But beyond literary fiction, dilemmas occupy society in every day issues such as in introducing legislation or solving current political problems. One example of a current political dilemma is how to deal with Iran’s quest for nuclear weapons. If it were possible to assess and quantify each of the alternative solutions for a given problem, the process of decision making would be much easier. If a problem involves only two optional solutions, game theory techniques can be used. However, real life problems are usually multi-unit, multi-optional problems, as in Iran

Abstract:
The dynamics of a subdiffusive continuous time random walker in an inhomogeneous environment is analyzed. In each microscopic jump, a random time is drawn from a waiting time probability density function (WT-PDF) that decays as a power law: phi(t;k)~k/(1+kt)^(1+beta), 0 beta;, mu=beta, but when 1-gamma

Abstract:
We present a general framework for computing two-dimensional Voronoi diagrams of different classes of sites under various distance functions. The framework is sufficiently general to support diagrams embedded on a family of two-dimensional parametric surfaces in $R^3$. The computation of the diagrams is carried out through the construction of envelopes of surfaces in 3-space provided by CGAL (the Computational Geometry Algorithm Library). The construction of the envelopes follows a divide-and-conquer approach. A straightforward application of the divide-and-conquer approach for computing Voronoi diagrams yields algorithms that are inefficient in the worst case. We prove that through randomization the expected running time becomes near-optimal in the worst case. We show how to employ our framework to realize various types of Voronoi diagrams with different properties by providing implementations for a vast collection of commonly used Voronoi diagrams. We also show how to apply the new framework and other existing tools from CGAL to compute minimum-width annuli of sets of disks, which requires the computation of two Voronoi diagrams of two different types, and of the overlay of the two diagrams. We do not assume general position. Namely, we handle degenerate input, and produce exact results.

Abstract:
Models that explain the economical and political realities of nowadays societies should help all the world's citizens. Yet, the last four years showed that the current models are missing. Here we develop a dynamical society-deciders model showing that the long lasting economical stress can be solved when increasing fairness in nations. fairness is computed for each nation using indicators from economy and politics. Rather than austerity versus spending, the dynamical model suggests that solving crises in western societies is possible with regulations that reduce the stability of the deciders, while shifting wealth in the direction of the people. This shall increase the dynamics among socio-economic classes, further increasing fairness.

Abstract:
Deducing an underlying multi-substate on-off kinetic scheme (KS) from the statistical properties of a two-state trajectory is the aim from many experiments in biophysics and chemistry, such as, ion channel recordings, enzymatic activity and structural dynamics of bio-molecules. Doing so is almost always impossible, as the mapping of a KS into a two-state trajectory leads to the loss of information about the KS (almost always). Here, we present the optimal way to solve this problem. It is based on unique forms of reduced dimensions (RD). RD forms are on-off networks with connections only between substates of different states, where the connections can have multi-exponential waiting time probability density functions (WT-PDFs). A RD form has the simplest toplogy that can reproduce a given data. In theory, only a single RD form can be constructed from the full data (hence its uniqueness), still this task is not easy when dealing with finite data. For doing so, a toolbox made of known statistical methods in data analysis and new statistical methods and numerical algorithms develped for this problem is presented. Our toolbox is self-contained: it builds a mechanism based only on the information it extracts from the data. The implementation of the toolbox on the data is fast. The toolbox is automated and is available for academic research upon electronic request.

Abstract:
Renewal-anomalous-heterogeneous files are solved. A simple file is made of Brownian hard spheres that diffuse stochastically in an effective 1D channel. Generally, Brownian files are heterogeneous: the spheres' diffusion coefficients are distributed and the initial spheres' density is non-uniform. In renewal-anomalous files, the distribution of waiting times for individual jumps is exponential as in Brownian files, yet obeys: {\psi}_{\alpha} (t)~t^(-1-{\alpha}), 0<{\alpha}<1. The file is renewal as all the particles attempt to jump at the same time. It is shown that the mean square displacement (MSD) in a renewal-anomalous-heterogeneous file, , obeys, ~[_{nrml}]^{\alpha}, where _{nrml} is the MSD in the corresponding Brownian file. This scaling is an outcome of an exact relation (derived here) connecting probability density functions of Brownian files and renewal-anomalous files. It is also shown that non-renewal-anomalous files are slower than the corresponding renewal ones.

Abstract:
Normal dynamics in a quasi-one-dimensional channel of length L (\to\infty) of N hard spheres are analyzed. The spheres are heterogeneous: each has a diffusion coefficient D that is drawn from a probability density function (PDF), W D^(-{\gamma}), for small D, where 0\leq{\gamma}<1. The initial spheres' density {\rho} is non-uniform and scales with the distance (from the origin) l as, {\rho} l^(-a), 0\leqa\leq1. An approximation for the N-particle PDF for this problem is derived. From this solution, scaling law analysis and numerical simulations, we show here that the mean square displacement for a particle in such a system obeys, ~t^(1-{\gamma})/(2c-{\gamma}), where c=1/(1+a). The PDF of the tagged particle is Gaussian in position. Generalizations of these results are considered.

Abstract:
In numerous systems in biophysics and related fields, scientists measure (with very smart methods) individual molecules (e.g. biopolymers (proteins, DNA, RNA, etc), nano - crystals, ion channels), aiming at finding a model from the data. But the noise is not solved accurately in not so few cases and this may lead to misleading models. Here, we solve the noise. We consider two state photon trajectories from any on off kinetic scheme (KS): the process emitting photons with a rate {\gamma}on when it is in the on state, and emitting with a rate {\gamma}off when it is in the off state. We develop a filter that removes the noise resulting in clean data also in cases where binning fails. The filter is a numerical algorithm with various new statistical treatments. It is based on a new general likelihood function developed here, with observable dependent form. The filter can solve the noise, in the detectable region of the rate space: that is, we also find a region where the data is "too" noisy. Consistency tests will find the region's type from the data. If the data is ruled "too noisy", binning obviously fails, and one should apply simpler methods on the raw data and realizing that the extracted information is partial. We show that not applying the filter while cleaning results in erroneous rates. This filter (with minor adjustments) can solve the noise in any discrete state trajectories, yet extensions are needed in "tackling" the noise from other data, e.g. continuous data and FRET data. The filter developed here is complementary with our previous projects in this field, where we have solved clean two state data with the development of reduced dimensions forms (RDFs): only the combined procedures enabling building the most accurate model from noisy trajectories from single molecules