In this paper, we study a truncated Toeplitz operator on finite dimensional spaces and give a characterization of normal matrices on such spaces.

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In particular, we find some conditions for truncated Toeplitz operators on finite dimensional spaces to be normal and provide several examples of such operators. Finally, we consider symbols of normal truncated Toeplitz operators on finite dimensional spaces. We get a theorem which shows the existence of the nontrivial weak periodic solution for the Hamiltonian system with singular potential nonlinearity.

We obtain this result by using variational method, critical point theory for indefinite functional. We establish local regular solutions in time and give some characterizations on parameters and initial data for global solutions and blow-up in a finite time. We also prove that there does not exist finite time self-similar solution of the backward type.

In addition, when the solution blows up, we give estimates for the blow-up time and provide blow-up rate. Finally, we show some numerical experiment which illustrate the main results.

## CiteSeerX — Weighted homogeneous singularities and rational homology disk smoothings

Secondly, I'm concerned with the existence and uniqueness of global solutions and uniform decay of the total energy for the system associated with Kirchhoff type wave equation with some boundaries. Lastly, I give some numerical results for a special example. The local sound wave impact by the velocity vector corner singularity occurred at the non-convex edge is propagated into the region along the streamline emanated from the non-convex edge and its derivatives blow up across the streamline. We establish the global-in-time existence of weak solutions and bounded weak solutions depending on some conditions of parameters such as chemotatic sensitivity and consumption rate of oxygen for certain range of diffusive exponents of cell density in two and three dimensions.

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We discusss backward extensions of quadratically hyponormal weighted shifts in this talk. This generalizes our previous result to arbitrary dimension. The key idea is to use a connection between distances with dot-products of two sets on spheres. Kim and Y. Then, we give some rigidity properties of foliations of CMC weighted hypersurfaces in a smooth metric measure space.

Moreover, we assert another non-existence Hopf hypersurface with semi-parallel structure Jacobi operator. Also, we introduce a multi-variable polynomial invariant for virtual links by using the parity of classical crossings, which refines the index polynomial. As consequences, we give some properties of our invariant, and raise some examples. For each of them there corresponds a real toric variety.

### Weighted homogeneous singularities and rational homology disk smoothings (2009)

In this talk, we try to present a formula for the Betti numbers of the real toric variety, generalizing the work of A. Henderson computing the rational cohomology of the real Coxeter toric variety of type A. This operation generalizes the rational blow-down operation of Fintushel-Stern for other configurations, and therefore extends Symington's result about symplectic rational blow-downs. Like a fan is assoicated to a Delzant polytope, a multi-fan introduced by Hattori-Masuda can be associated to an oriented origami template.

We also characterize products of even dimensional spheres which can be toric origami manifolds. As applications, we study exponential convergence rates for weighted additive convolution sums of probability measures corresponding to the free additive convolution. The Black-Scholes model is based on Normal distribution and the model's main drawback is fat-tail problem.

Because of fat-tail problem, the implied volatility surface is shaped skew, smirk or other forms.

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To correct model's drawback, we propose predicted cumulative distribution using hidden Markov Model and simulate the underlying price paths using Monte Carlo method. In the results, our model reflect current implied volatility surface of market satisfactorily. We approach this problem in two different ways. In the first approach, we first prove the large deviation principle for finite dimensional distributions, and then use it to establish the sample path large deviation principle. For this approach, a more careful examination of the affine transform formula is required.

The second approach exploits the exponential martingale method of Donati-Martin et al. We provide an application to importance sampling of affine diffusion models. We introduce a notion of quantum white noise derivatives of white noise operators. Then every super operator admits a series expansion in terms of quantum white noise derivatives. In this work, we consider the problem of recovering missing samples from multi-channel oversampling in a shift-invariant space.

We find conditions under which any finite or infinite number of missing samples can be recovered. We formulate an intermediate function with the fixed free boundary that has Lipschitz character near optimal exercise boundary. Employing it, we can easily determine the optimal exercise boundary by solving a quadratic equation in time-recursive way. We also present several numerical results which illustrate comparison to other methods.

## Rational blow-downs and smoothings of surface singularities

To validate the performance of the proposed new method, we compare the numerical results with those from Monte Carlo simulation in terms of computational cost and accuracy. Pricing an interest rate option in such a multi-factor term structure model involves evaluating a multi-dimensional integration of the underlying claim's payoff on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state variables as a hyperplane tangent to the maximum probability point on the boundary and simplify the multi-dimensional integration into an analytic form.

The maximum probability point can be found using the gradient descent method. We show that the quality of our method is superior to the result of previous studies by directly comparing them to the exact price obtained from the numerical integration.

The method provides more accurate option value with rapid convergence. The standard binomial tree method for Asian option evaluates option prices at each node of the tree by setting representative averages. Our algorithm redefines representative averages by considering an interval about each node along the stock price axis and then evaluates option prices by computing the representative averages over each interval.

We apply the method to the discrete and continuous monitored Asian options. We establish the threshold dynamics of their permanence and the extinction of the predator. Furthermore, we also give sufficient conditions for the global attractiveness and the local stability of the semi-trivial and coexistence equilibria. The model is an alternative way to approximate the sharp interface models such as mean curvature flow, volume-preserving mean curvature flow, Mullins-Sekerka flow and surface diffusion flow.

In this talk, we observe the motions driven by the phase--field model and how they evolve. If algorithms are well fit with the GPU's architecture, the computational time can be remarkably reduced.

## Rational blowdowns and smoothings of surface singularities (2008)(en)(41s)

In this work, we study an operator splitting method for solving the heat and Black-Scholes equations. The governing equation is implicitly discretized in time, based on the finite difference method, and the operator splitting method leads to a tridiagonal matrix system. We discuss a parallel algorithm, which is called a parallel cyclic reduction method, to solve this matrix system by using GPUs. Although the governing equation is harmonic, the fact that the impinging jet has free boundaries makes the problem highly non-linear. In this paper, we propose an efficient algorithm to find the free boundary of the asymmetric jet.

Based on the boundary integral equations, our algorithm can find the gradient toward the solution from an assumed free boundary. Particularly, by considering the far-field behavior of solution, our algorithm can deal with the entire computational domain instead of truncated one. In the numerical results, the force exerted on the wall by the pressure and numerical calculations of diverse examples are shown. Those data are shown to be useful for getting new insights into the structure or into a certain phenomena on the structure.

For an efficient way of dealing with the data, there have been several approaches to construct wavelets on a graph. We investigate the pros and cons of the graph wavelets. The Sage Grapher that we have developed can be used as a supplementary, pedagogical tool for various math courses. Graphs from Sage Grapher can be easily saved on the server so students can review and utilize them at their convenience.