2015-05-20

Phase space Feynman path integrals of higher order parabolic type (Chair. Tepper L. Gill)We give a general class of functionals for which the phase space path integrals of higher order parabolic type have a mathematically rigorous meaning. For any functional belonging to our class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the endpoint of position paths and to the starting point of momentum paths. Our class of functionals is rich because it is closed under addition and multiplication. The interchange of the order with the integration with respect to time and the interchange of the order with a limit hold in the phase space path integrals.

A change of scale formula for present time-independent conditional expectations of cylinder functions on an analogue of Wiener space (Chair. Tepper L. Gill)

Let $C[0,t]$ denote a generalized Wiener space, the space of real-valued continuous functions on the interval $[0,t]$ and define a random vector $Z_n : C[0,t]\to\mathbb R^n$ by $Z_n(x)=(\int_0^{t_1}h(s) dx(s),\cdots,\int_0^{t_n}h(s) dx(s))$, where $0 < t_1 < \cdots < t_n < t$ is a partition of $[0,t]$ and $h\in L_2[0,t]$ with $h\neq 0$ a.e.
In this paper we will introduce a simple formula for a generalized conditional Wiener integral on $C[0,t]$ with the conditioning function $Z_n$ and
then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x) = f(\int_0^t e(s)dx(s))$ for $x\in C[0,t]$, where $f\in L_p (\mathbb R)$ $(1\le p\le\infty)$ and $e$ is a unit element in $L_2[0,t]$.
Finally we express the generalized analytic conditional Feynman integralof $F$ as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of $e$ and the conditioning function $Z_n$ does not contain the present position of the generalized Wiener paths.

Complex order modified Bessel functions integrals (Chair. Hyun Jae Yoo)

The proof of new properties and inversion formulas of the modified **Kontorovitch-Lebedev** integral transforms is developed. The **Parseval** equations are proved and sufficient conditions for them are found. Some new representations of these transforms are justified. The inequalities which give estimations of these kernels - the real and imaginary parts of the modified Bessel functions of the second kind $ ReK_{1/2+i\tau}(x) $ and $ ImK_{1/2+i\tau}(x) $ in whole domain of the variable's alteration $x$ and $\tau$ are obtained.
The applications of **Lebedev** type integral transforms and dual integral equations to the solution of some mixed boundary value problems in the wedge domains are presented. The numerical solution of the **Helmholtz** equation in the wedge domain is performed.
The calculation of some characteristics of dipole and quadrypole radiation is conducted. The evaluation of some asymptotic characteristics of vacuum polarization in **Schwarzschild** spacetime is carried out.

The Bishop-Phelps-Bollob ́as theorem for operators from $L_1(\mu)$ to Banach spaces with the Radon Nikody ́m property

Let $Y$ be a Banach space and $(\Omega, \Sigma, \mu)$ be a $\sigma$-finite measure space, where $\Sigma$ is an infinite $\sigma$-algebra of measurable subsets of $\Omega$. We show that if the couple $(L_1(\mu), Y)$ has the Bishop-Phelps-Bollob\'as property for operators, then $Y$ has the \emph{AHSP}. Further, for a Banach space $Y$ with the Radon-Nykod\'ym property, we prove that the couple $(L_1(\mu), Y)$ has the Bishop-Phelps-Bollob\'as property for operators if and only if $Y$ has the \emph{AHSP}.

Analogue of Wiener space in the space of sequences real numbers (Chair. Naoto Kumano-go)

Let $T>0$ be given. Let $(C[0,T], m_\phi)$ be the analogue of Wiener measure space, associated with the Borel probability measure $\phi$ on $\mathbb R$, which is the extended concept of the concrete Wiener measure space. Let $(L_2[0,T],\overset \sim \omega)$ be the centered Gaussian measure space with the correlation operator $(-\frac{d^2}{dx^2})^{-1}$ and let $(l_2,\overset \sim m)$ be the abstract Wiener space, where $l_2$ is the space of all square summable sequences with the norm $\|\langle c_n\rangle\|=\sqrt{\sum_{n=0}^\infty c_n^2}$. Let $U$ be the space of all sequence $\langle c_n \rangle $ in $l_2$ such that the limit
\begin{eqnarray*}
\lim_{m\to\infty}\frac{1}{m+1}\sum_{n=0}^m\sum_{k=}^nc_k\cos\frac{k\pi t}{T}
\end{eqnarray*}
converges uniformly on $[0,T]$ and give a set function $m$ such that for Borel subset $G$ of $l_2$, $m(U\cap P_0^{-1}\circ P_0(G))=\overset \sim m(P_0^{-1}\circ P_0(G))$ where $P_0:l_2\to l_2$ is a function with $P_0(\langle c_0,c_1,c_2, \cdots\rangle )=\langle c_1,c_2,c_3,\cdots \rangle$.
The goal of this note is to study the relationship between the measures $m_\phi$, $\overset \sim \omega$, $\overset \sim m$ and $m$.

Transforms in quantum white noise theory (Chair. Naoto Kumano-go)

Starting with the integral transform by Y. J. Lee on abstract Wiener space, we introduce a general transform of white noise operators which includes Fourier-Gauss and Fourier-Mehler transforms, Bogoliubov transform and quantum Girsanov transform.
Also, we study the characterizations of the general transforms in terms of quantum white noise derivatives.

Shifting for Fourier-Feynman transform and convolution (Chair. Sonia Mazzucchi)

Time and frequency shifting properties for Fourier transform on Euclidean space are useful when apply Fourier transform. We introduce some of important properties relavant to shifting and computational rules for Fourier-Feynman transform and convolution of functionals in a Banach algebra ${\mathcal S}$ on classical Wiener space. Cameron and Storvick's translation theorem can be obtained as a corollary of our result.

2015-05-21

Infinite dimensional integration of oscillatory type and applications (Chair. Juri M. Rappoport)In this talk I shall describe the theory and the main applications of infinite dimensional integration technique of oscillatory type. I shall show that the full scope of these techniques goes beyond the definition of Feynman path integrals for the representation of the solution of the Schr ̈odinger equation, as they can be applied to the construction of generalized Feynman-Kac-type formulae for a more general class of partial differential equations.

A segmentation model for colour images using level set method (Chair. Juri M. Rappoport)

A segmentation model for human detection is developed, which aims to explicitly delineate the figure of a single subject as initial work.
The modelling consists of two processes of segmentation: coarse foreground extraction and refinement using a level set propagation. A level set method is based on variational calculus and has been widely used in many application fields. The level set method makes it very easy to follow shapes that change topology, for example when a shape splits in two, develops holes, or the reverse of these operations.
In our model, we develop a level set formulation for colour images. The formulation is, in particular, for CIE Lab space of colour representation, where both colour information and grey intensity are used. The coarse human shape is employed as an initial surface of the level set propagation and used as a constraint by narrow band marching in the procedure of optimisation. The resulting segmentation is bound close to human shape and delineates the human boundary. The newly developed level set model demonstrates a reasonable performance in human detection.

Change of scale formulas for function space integrals (Chair. Teuk Seob Song)

It has long been known that Wiener measure and Wiener measurability behave badly under the change of scale transformation and under translations. Cameron and Storvick expressed the analytic Feynman integral on classical Wiener space as a limit of Wiener integrals. In doing so, they discovered nice change of scale formulas for Wiener integrals on classical Wiener space. After that, Yoo and Skoug extended these results to an abstract Wiener space. Moreover, Yoo, Song, Kim, Kim, Chang and Cho established change of scale formulas for Wiener integrals and function space integrals of various functionals.
In this talk, we survey change of scale formulas for Wiener integrals and function space integrals of various functionals.

Quantization on spaces with symmetries, and applications (Chair. Il Yoo)

In this talk we will give an overview of recent research on pseudo-differential operators on spaces with addition structures: for example on compact or nilpotent Lie groups, or spaces equipped with a fixed system of eigenfunctions for a given operator. We will make an overview of results and applications in related areas such as harmonic analysis or the theory of partial differential equations.

Standing waves for the pseudo-relativistic nonlinear Schrodinger equations (Chair. Il Yoo)

In this talk, we discuss about standing wave solutions for the pseudo-relativistic nonlinear Schrodinger equations
\begin{equation}\label{sprnse}
(1) \sqrt{-c^2\Delta +m^2c^4}u -mc^2u +\mu u = |u|^{p-2}u \quad \text{ in } \mathbb{R}^n,\, n \geq 2,
\end{equation}
which are one of relativistic version of nonlinear Schrodinger equations.
We will deal with the existence and qualitative properties of ground state solutions and their non-relativistic limit.

One-dimensional three-state quantum walks: weak limits and localization (Chair. Vu Kim Tuan)

We investigate one-dimensional three-state quantum walks. We find a formula for the moments of the weak limit distribution via a vacuum expectation of powers of a self-adjoint operator. We use this formula to fully characterize the localization of three-state quantum walks in one-dimension. The localization is also characterized by investing the eigenvectors of the evolution operator for the quantum walk. As a byproduct we clarify the concepts of localization differently used in the literature. We also study the continuous part of the limit distribution. For typical examples we show that the continuous part is the same kind as that of two-state quantum walks. We provide with explicit expressions for the density of the weak limits of some three-state quantum walks. Joint work with Chul Ki Ko and Etsuo Segawa.

Stochastic differential equations with applications (Chair. Vu Kim Tuan)

Classical mathematical modeling is largely concerned with the derivation and use of ordinary and partial differential equations in the modeling of natural phenomena.
Traditionally, these differential equations are deterministic by which we mean that their solutions are determined in the value by knowledge of initial and boundary conditions. On the other hand, a stochastic differential equation (SDE) is a differential equation with a solution which is influenced by initial and boundary conditions, but not predetermined by them. It is clear that any solution of the SDE must involve some randomness, i.e., we can only hope to be able to say something about the probability distribution of the solutions. The SDEs find applications in many disciplines including finance, economics, physics, biology, and medicine. In this presentation, we deal with mathematical background on SDEs derived by the Wiener process and describe how the concepts, methods, and results in these SDEs can be applied to give a rigorous mathematical model of finance. Also, we construct and implement numerical methods for solving SDEs using MATLAB.
Keywords: Stochastic differential equations, Wiener process, Stochas- tic integration, Financial mathematics

2015-05-22

Lebesgue measure on Banach spaces (Chair. Michael Ruzhansky)Historically, when constructing measures on ${\mathbb{R}^\infty }$, the topology defines an open set ${\mathcal{O}}$ by: ${\mathcal{O}} = \prod _{k = 1}^n{O_k} \times \prod _{k = n + 1}^\infty \mathbb{R}$ (cylinder sets). Any attempt to construct Lebesgue measure using this definition is impossible. If set ${I_0} = \left[ { - \tfrac{1}{2},\;\tfrac{1}{2}} \right]$ and $I = \prod _{k = 1}^\infty {I_0}$, then any natural version of Lebesgue measure must satisfy ${\lambda _\infty }\left[ I \right] = 1$. Thus, another approach is to define the topology using open sets of the form ${\mathcal{O}} = \prod _{k = 1}^n{O_k} \times \prod _{k = n + 1}^\infty {I_0}$ (box sets). The purpose of this talk is to discuss the construction of $\lambda_\infty$ in such a way as to provide a definition of $\lambda_{\mathcal B}$ (Lebesgue measure) for every Banach space $\mathcal B$ with a Schauder (or S-basis). This will allow us to constructively define Schwartz spaces, distributions and the Fourier transforms for every reflexive Banach space with and S-Basis. As an application, we define a universal version of Gaussian measure and use it to provide a direct, constructive solution to the heat equation on Hilbert space. If time permits, we will discuss the natural relationship between our theory and the foundations for path integrals.

Global well-posedness of the Chemotaxis-Navier-Stokes equations in two dimensions (Chair. Michael Ruzhansky)

We consider two dimensional Keller-Segel equations coupled with the Navier-Stokes equations.
\begin{equation}\label{KSNS} \left\{
\begin{array}{ll}
\partial_t n + u \cdot \nabla n - \Delta n= -\nabla\cdot (\chi (c) n \nabla c),\\
\vspace{-3mm}\\
\partial_t c + u \cdot \nabla c-\Delta c =-k(c) n,\\
\vspace{-3mm}\\
\partial_t u +(u \cdot \nabla)u -\Delta u +\nabla p=-n \nabla
\phi,\quad
\nabla \cdot u=0 %\qquad t \in (0,\, T], \qquad x \in \R^d
\end{array}
\right. \quad\mbox{ in }\,\, (x,t)\in \mathbb R^2\times (0,\, T),
\end{equation}
where $c \ge 0$ , $n\ge 0 $, $u$ and $p$ denotes the oxygen concentration, cell concentration, fluid velocity, and scalar pressure, respectively. Assuming that the chemotactic sensitivity and oxygen consumption rate are nondecreasing and differentiable, we prove that
there is no blow-up in a finite time for solutions with large initial data to chemotaxis-Navier-Stokes equations in two dimensions. In addition, temporal decays of solutions are shown, as time tends to infinity.
References
[1] M. Chae, K. Kang and J. Lee, On Existence of the smooth solutions to the Coupled Chemotaxis-Fluid Equations, Discrete Cont. Dyn. Syst. A, 33(6); 2271–2297, 2013.
[2] M. Chae, K. Kang and J. Lee, Global existence and temporal decay in Keller-Segel models coupled to fiuid equations, Comm. Partial Diff. Equations, 39; 1205-1235, 2014.
[3] M. Chae, K. Kang and J. Lee, Asymptotic behavior of solutions for an aerotaxis model
coupled to fluid equations, submitted for publication.
[4] I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler, and R.
E. Goldstein, Bacterial swimming and oxygen transport near contact lines, PNAS, 102(7); 2277–2282, 2005.

Convolution semigroups of completely positive instruments (Chair. Byoung Soo Kim)

We introduce a notion of a (covariant) CP-instrument on a $C^*$-algebra, and construct a (covariant) Stinespring type representation
and a (covariant) Naimark-Paschke type representation for a (covariant) CP-instrument on a $C^*$-algebra. The minimality of the representation associated to a (covariant) CP-instrument is established. We study some properties of CP-instruments on $C^*$-algebras.
Using probability operators and Fourier transforms of CP-instruments on von Neumann algebras, we give necessary and sufficient conditions for various covariant properties. We discuss a convolution semigroup of covariant CP-instruments and a semigroup of probability operators associated with CP-instruments on von Neumann algebras.

Sampling in Paley-Wiener and Hardy spaces (Chair. Jaeseong Heo)

Let $f(t)\in L_2(R)$ and $\hat{f}(\lambda)$ be its Fourier transform. In signal processing $f$ is called a (finite-energy) signal, and $\hat{f}$ is the frequency content of the signal $f$.
Signal $f$ is bandlimited (Paley-Wiener function) with the bandwidth $T$ if $f$ has no frequency higher than $T$. Most of
signals like human voices, are bandlimited.
The fundamental of digital signal processing is the Shannon sampling formula
$$ f(t) =\sum_{n=-\infty}^{\infty}
f\left(\frac{n\pi}{T}\right)\frac{\sin (Tt-n\pi)}{Tt-n\pi},\quad
$$ that allows to recover a bandlimited signal with bandwidth $T$ from its samples at equidistant points spaced $\frac{\pi}{T}$ apart. The sampling rate $\frac{T}{\pi}$ per second is called the Nyquist sampling rate. This is the minimum rate at which the signal needs to be sampled in order to reconstruct it exactly. All available sampling formulae, such as Shannon, Paley-Wiener, and Kramer, need a priori the information of the bandwidth, in order to set the sampling rate.
Now suppose that our receiver gets a signal from an unknown source. So no information about the bandwidth of the signal is known. What conditions should we put on a set of sampling points so that we can recover a bandlimited signal $f$ with unknown bandwidth, and how do we recover the signal?
A solution of this problem is based on the Gelfand-Levitan theory for inverse spectral problems and Kramer's theorem. The connection between the two is that given a sequence of points satisfying certain conditions, we can construct a self-adjoint Sturm-Liouville problem such that these points are precisely its eigenvalues. Using eigenfunctions of this Sturm-Liouville problem, Kramer's theorem then allows us to employ the spectral theorem as a sampling formula.
We also obtain new sampling formulas for functions in the Hardy space $\mathcal{H}^{2}\left( \mathbb{R}_{+}^{2}\right) $. As a consequence, a new inverse formula for the Laplace transform and a new series representation for the Riemann zeta function in the half-plane $\Re(s)>\frac{1}{2}$ are obtained. Upper bounds for the truncation and amplitude errors for the sampling formulae are also provided.

Dirichlet boundary value problems in elliptic and parabolic partial differential equation (Chair. Jaeseong Heo)

We consider Dirichlet boundary value problems for second order linear elliptic or parabolic operator in divergence and non-divergence forms.
For the well known Laplace operator, many results are known and studied including the existence and uniqueness of solutions.
In this talk, we present a boundary value problem in a certain non-smooth domain, which is called (A)-measure domain including time varying domain in the parabolic case. For the proof, we use so called a growth lemma, which is known by E. Landis, M. Safonov, and N. Krylov.
The results of this talks are joint work with M. Safonov, D. Kim, and H. Dong.

Block design based scheduling for neighbor discovery protocol in wireless sensor networks (Chair. Kun Sik Ryu)

In this talk, we discuss neighbor discovery protocol in wireless sensor networks. In the combinatorial design theory, Balanced Incomplete Block Design(BIBD) is widely applied wake up scheduling techniques in the wireless sensor networks. We propose a new scheduling method based on the BIBD.

Conditional integral transforms and convolution products on function space (Chair. Kun Sik Ryu)

In this talk we study various results on conditional integral transforms and convolution products on the function space $K[0,T]$, for some conditioning functions. Also we consider very general conditioning functions $X_\alpha(x)$ which need not depend upon the values of $x$ in $C_0[0,T]$ at only finitely many points in $(0,T]$.