Abstract:
We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^\prime: H\to H$, the germ $(F^\prime-id)$ is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with $\kappa=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form `identity plus a 1-smoothing analytic map'.

Abstract:
The multiple gamma function $\Gamma_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the $\Gamma_n$ function and their applications to summation of series and infinite products.

Abstract:
In this paper a new method is introduced for studying time series of complex systems. This method is based on using the concept of entropy and Jensen-Shannon divergence. In this paper this method is applied to time series of billiard system and heart signals. By this method, we can diagnose the healthy and unhealthy heart and also chaotic billiards from non chaotic systems . The method can also be applied to other time series.

Abstract:
Any quasismooth function f(x) in a finite interval [0,x0], which has only a finite number of finite discontinuities and has only a finite number of extremes, can be approximated by a uniformly convergent Fourier series and a correction function. The correction function consists of algebraic polynomials and Heaviside step functions and is required by the aperiodicity at the endpoints (i.e., f(0)≠f(x0)) and the finite discontinuities in between. The uniformly convergent Fourier series and the correction function are collectively referred to as the corrected Fourier series. We prove that in order for the mth derivative of the Fourier series to be uniformly convergent, the order of the polynomial need not exceed (m

Abstract:
The natural logarithm can be represented by an infinite series that converges for all positive real values of the variable, and which makes concavity patently obvious. Concavity of the natural logarithm is known to imply, among other things, the fundamental inequality between the arithmetic and geometric mean.

Abstract:
We present a new framework to detect various types of variable objects within massive astronomical time-series data. Assuming that the dominant population of objects is non-variable, we find outliers from this population by using a non-parametric Bayesian clustering algorithm based on an infinite GaussianMixtureModel (GMM) and the Dirichlet Process. The algorithm extracts information from a given dataset, which is described by six variability indices. The GMM uses those variability indices to recover clusters that are described by six-dimensional multivariate Gaussian distributions, allowing our approach to consider the sampling pattern of time-series data, systematic biases, the number of data points for each light curve, and photometric quality. Using the Northern Sky Variability Survey data, we test our approach and prove that the infinite GMM is useful at detecting variable objects, while providing statistical inference estimation that suppresses false detection. The proposed approach will be effective in the exploration of future surveys such as GAIA, Pan-Starrs, and LSST, which will produce massive time-series data.

Abstract:
Linear Dynamical System (LDS) is an elegant mathematical framework for modeling and learning multivariate time series. However, in general, it is difficult to set the dimension of its hidden state space. A small number of hidden states may not be able to model the complexities of a time series, while a large number of hidden states can lead to overfitting. In this paper, we study methods that impose an $\ell_1$ regularization on the transition matrix of an LDS model to alleviate the problem of choosing the optimal number of hidden states. We incorporate a generalized gradient descent method into the Maximum a Posteriori (MAP) framework and use Expectation Maximization (EM) to iteratively achieve sparsity on the transition matrix of an LDS model. We show that our Sparse Linear Dynamical System (SLDS) improves the predictive performance when compared to ordinary LDS on a multivariate clinical time series dataset.

Abstract:
Suffcient conditions, necessary conditions for faster convergent infinite series, faster -convergent infinite series are studied. The faster convergence of infinite series of Kummer's type is proved.

Abstract:
The purpose of this paper is to present simple and general algebraic methods for describing series connections in quantum networks. These methods build on and generalize existing methods for series (or cascade) connections by allowing for more general interfaces, and by introducing an efficient algebraic tool, the series product. We also introduce another product, which we call the concatenation product, that is useful for assembling and representing systems without necessarily having connections. We show how the concatenation and series products can be used to describe feedforward and feedback networks. A selection of examples from the quantum control literature are analyzed to illustrate the utility of our network modeling methodology.

Abstract:
We prove that any collection which tiles the positive integers must contain one of two types of sub-collections. We then use this result to prove a variation of the Ratio Test for convergence of series. This version of the Ratio Test shows the convergence of certain series for which the Root Test (which is known to be more powerful than the conventional Ratio Test) fails. This version of the Ratio Test is also used to prove a version of the Banach Contraction Principle for self-maps of a complete metric space.