The problem of the derivation of hydrodynamics from the Boltzmann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions. We review a few instances where such hydrodynamic manifo
lds were found analytically both as the result of summation of the Chapman--Enskog asymptotic expansion and by the direct solution of the invariance equation. These model cases, comprising Grads moment systems, both linear and nonlinear, are studied in depth in order to gain understanding of what can be expected for the Boltzmann equation. Particularly, the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short-wave limit and the viscosity modification at high divergence of the flow velocity are indicated as severe obstacles to the resolution of Hilberts 6th Problem. Furthermore, we review the derivation of the approximate hydrodynamic manifold for the Boltzmann equation using Newtons iteration and avoiding smallness parameters, and compare this to the exact solutions. Additionally, we discuss the problem of projection of the Boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts. Finally, a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the Boltzmann equation in the future.
Principal component analysis (PCA) is an important tool in exploring data. The conventional approach to PCA leads to a solution which favours the structures with large variances. This is sensitive to outliers and could obfuscate interesting underlyin
g structures. One of the equivalent definitions of PCA is that it seeks the subspaces that maximize the sum of squared pairwise distances between data projections. This definition opens up more flexibility in the analysis of principal components which is useful in enhancing PCA. In this paper we introduce scales into PCA by maximizing only the sum of pairwise distances between projections for pairs of datapoints with distances within a chosen interval of values [l,u]. The resulting principal component decompositions in Multiscale PCA depend on point (l,u) on the plane and for each point we define projectors onto principal components. Cluster analysis of these projectors reveals the structures in the data at various scales. Each structure is described by the eigenvectors at the medoid point of the cluster which represent the structure. We also use the distortion of projections as a criterion for choosing an appropriate scale especially for data with outliers. This method was tested on both artificial distribution of data and real data. For data with multiscale structures, the method was able to reveal the different structures of the data and also to reduce the effect of outliers in the principal component analysis.
This case study tests the possibility of prediction for success (or winner) components of four stock & shares market indices in a time period of three years from 02-Jul-2009 to 29-Jun-2012.We compare their performance ain two time frames: initial fra
me three months at the beginning (02/06/2009-30/09/2009) and the final three month frame (02/04/2012-29/06/2012). To label the components, average price ratio between two time frames in descending order is computed. The average price ratio is defined as the ratio between the mean prices of the beginning and final time period. The winner components are referred to the top one third of total components in the same order as average price ratio it means the mean price of final time period is relatively higher than the beginning time period. The loser components are referred to the last one third of total components in the same order as they have higher mean prices of beginning time period. We analyse, is there any information about the winner-looser separation in the initial fragments of the daily closing prices log-returns time series. The Leave-One-Out Cross-Validation with k-NN algorithm is applied on the daily log-return of components using a distance and proximity in the experiment. By looking at the error analysis, it shows that for HANGSENG and DAX index, there are clear signs of possibility to evaluate the probability of long-term success. The correlation distance matrix histograms and 2-D/3-D elastic maps generated from ViDaExpert show that the winner components are closer to each other and winner/loser components are separable on elastic maps for HANGSENG and DAX index while for the negative possibility indices, there is no sign of separation.
There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types of principa
l curves and principal trees, and so on. For each type of approximators the measure of the approximator complexity was developed too. These measures are necessary to find the balance between accuracy and complexity and to define the optimal approximations of a given type. We propose a measure of complexity (geometrical complexity) which is applicable to approximators of several types and which allows comparing data approximations of different types.
We study systems with finite number of states $A_i$ ($i=1,..., n$), which obey the first order kinetics (master equation) without detailed balance. For any nonzero complex eigenvalue $lambda$ we prove the inequality $frac{|Im lambda |}{|Re lambda |}
leq cotfrac{pi}{n}$. This bound is sharp and it becomes an equality for an eigenvalue of a simple irreversible cycle $A_1 to A_2 to... to A_n to A_1$ with equal rate constants of all transitions. Therefore, the simple cycle with the equal rate constants has the slowest decay of the oscillations among all first order kinetic systems with the same number of states.
For many real physico-chemical complex systems detailed mechanism includes both reversible and irreversible reactions. Such systems are typical in homogeneous combustion and heterogeneous catalytic oxidation. Most complex enzyme reactions include irr
eversible steps. The classical thermodynamics has no limit for irreversible reactions whereas the kinetic equations may have such a limit. We represent the systems with irreversible reactions as the limits of the fully reversible systems when some of the equilibrium concentrations tend to zero. The structure of the limit reaction system crucially depends on the relative rates of this tendency to zero. We study the dynamics of the limit system and describe its limit behavior as $t to infty$. If the reversible systems obey the principle of detailed balance then the limit system with some irreversible reactions must satisfy the {em extended principle of detailed balance}. It is formulated and proven in the form of two conditions: (i) the reversible part satisfies the principle of detailed balance and (ii) the convex hull of the stoichiometric vectors of the irreversible reactions does not intersect the linear span of the stoichiometric vectors of the reversible reactions. These conditions imply the existence of the global Lyapunov functionals and alow an algebraic description of the limit behavior. The thermodynamic theory of the irreversible limit of reversible reactions is illustrated by the analysis of hydrogen combustion.
We consider continuous--time Markov kinetics with a finite number of states and a given positive equilibrium distribution P*. For an arbitrary probability distribution $P$ we study the possible right hand sides, dP/dt, of the Kolmogorov (master) equa
tions. We describe the cone of possible values of the velocity, dP/dt, as a function of P and P*. We prove that, surprisingly, these cones coincide for the class of all Markov processes with equilibrium P* and for the reversible Markov processes with detailed balance at this equilibrium. Therefore, for an arbitrary probability distribution $P$ and a general system there exists a system with detailed balance and the same equilibrium that has the same velocity dP/dt at point P. The set of Lyapunov functions for the reversible Markov processes coincides with the set of Lyapunov functions for general Markov kinetics. The results are extended to nonlinear systems with the generalized mass action law.
MicroRNAs can affect the protein translation using nine mechanistically different mechanisms, including repression of initiation and degradation of the transcript. There is a hot debate in the current literature about which mechanism and in which sit
uations has a dominant role in living cells. The worst, same experimental systems dealing with the same pairs of mRNA and miRNA can provide ambiguous evidences about which is the actual mechanism of translation repression observed in the experiment. We start with reviewing the current knowledge of various mechanisms of miRNA action and suggest that mathematical modeling can help resolving some of the controversial interpretations. We describe three simple mathematical models of miRNA translation that can be used as tools in interpreting the experimental data on the dynamics of protein synthesis. The most complex model developed by us includes all known mechanisms of miRNA action. It allowed us to study possible dynamical patterns corresponding to different miRNA-mediated mechanisms of translation repression and to suggest concrete recipes on determining the dominant mechanism of miRNA action in the form of kinetic signatures. Using computational experiments and systematizing existing evidences from the literature, we justify a hypothesis about co-existence of distinct miRNA-mediated mechanisms of translation repression. The actually observed mechanism will be that acting on or changing the limiting place of the translation process. The limiting place can vary from one experimental setting to another. This model explains the majority of existing controversies reported.
In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl Pearson invented principal component analysis in 1901 and fou
nd lines and planes of closest fit to system of points. The famous k-means algorithm solves the approximation problem too, but by finite sets instead of lines and planes. This chapter gives a brief practical introduction into the methods of construction of general principal objects, i.e. objects embedded in the middle of the multidimensional data set. As a basis, the unifying framework of mean squared distance approximation of finite datasets is selected. Principal graphs and manifolds are constructed as generalisations of principal components and k-means principal points. For this purpose, the family of expectation/maximisation algorithms with nearest generalisations is presented. Construction of principal graphs with controlled complexity is based on the graph grammar approach.
The concept of the limiting step is extended to the asymptotology of multiscale reaction networks. Complete theory for linear networks with well separated reaction rate constants is developed. We present algorithms for explicit approximations of eige
nvalues and eigenvectors of kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms is demonstrated on simple examples. Application of algorithms to nonlinear systems is discussed.
