Universality of the double scaling limit in random matrix models

We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painlevé equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann‐Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann‐Hilbert problem is done with the Deift‐Zhou steepest‐descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. © 2005 Wiley Periodicals, Inc.     

DG method for numerical pricing of multi-asset Asian options—The case of options with floating strike

Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two assets. Further, we focus only on one subclass—Asian options with floating strike—and introduce the concept of the dimensionality reduction with respect to the payoff leading to PDE with two spatial variables. Then the numerical option pricing scheme arising from the discontinuous Galerkin method is developed and some theoretical results are also mentioned. Finally, the aforementioned model is supplemented with numerical results on real market data.     

On the impact of conditional expectation estimators in portfolio theory

In this paper, we investigate the implications for portfolio theory of using conditional expectation estimators. First, we focus on the approximation of the conditional expectation within large-scale portfolio selection problems. In this context, we propose a new consistent multivariate kernel estimator to approximate the conditional expectation and it optimizes the bandwidth selection of kernel-type estimators. Second, we deal with the portfolio selection problem from the point of view of different non-satiable investors, namely risk-averse and risk-seeker investors. In particular, using a well-known ordering classification, we first identify different definitions of returns based on the investors preferences. Finally, for each problem, we examine several admissible portfolio optimization problems applied to the US stock market. The proposed empirical analysis allows us to evaluate the impact of the conditional expectation estimators in portfolio theory.     

Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations

We first prove the existence and regularity of the trajectory attractor for a three-dimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.     

Abelian Cayley Graphs of Given Degree and Diameter 2 and 3

Let CC _d,k be the largest possible number of vertices in a cyclic Cayley graph of degree d and diameter k, and let AC _d,k be the largest order in an Abelian Cayley graph for given d and k. We show that \({CC_{d,2} \geq \frac{13}{36} (d + 2)(d -4)}\) for any d= 6p?2 where p is a prime such that \({p \neq 13}\) , \({p \not\equiv 1}\) (mod 13), and \({AC_{d,3} \geq \frac{9}{128} (d + 3)^2(d - 5)}\) for d = 8q?3 where q is a prime power.     

L ∞ Algebra Representations

It is shown that the direct sum of an L algebra and a module over it has a natural L algebra structure.     

A Sharp Threshold for a Modified Bootstrap Percolation with Recovery

Bootstrap percolation is a type of cellular automaton on graphs, introduced as a simple model of the dynamics of ferromagnetism. Vertices in a graph can be in one of two states: ‘healthy’ or ‘infected’ and from an initial configuration of states, healthy vertices become infected by local rules. While the usual bootstrap processes are monotone in the sets of infected vertices, in this paper, a modification is examined in which infected vertices can return to a healthy state. Vertices are initially infected independently at random and the central question is whether all vertices eventually become infected. The model examined here is such a process on a square grid for which healthy vertices with at least two infected neighbours become infected and infected vertices with no infected neighbours become healthy. Sharp thresholds are given for the critical probability of initial infections for all vertices eventually to become infected.     

Calibration method at the N K‐edge using interstitial nitrogen gas in solid‐state nitrogen‐containing inorganic compounds

The standard method of soft X‐ray beamline calibration at the N K‐edge uses the ν = 0 peak transition of gas‐phase N₂. Interstitial N₂ gas trapped or formed within widely available solid‐state ammonium‐ and amine‐containing salts can be used for this purpose, bypassing gas‐phase measurements. Evidence from non‐nitrogen‐containing compounds (KH₂PO₄) and from He‐purged ammonium salts suggest that production of N₂ gas is through beam‐induced decomposition. Compounds with nitrate or nitrite as anions produce coincident features and are not suitable for this calibration method.     

Surface Gap Soliton Ground States for the Nonlinear Schr?dinger Equation

We consider the nonlinear Schr?dinger equation

Multiscale multimodal imaging with multiphoton microscopy and optical coherence tomography

A multiscale multiphoton microscopy (MPM) and optical coherence tomography (OCT) system has been developed using a sub-10 fs Ti:sapphire laser. The system performs cross-sectional OCT imaging over millimeter field-of-view and en-face high-resolution MPM imaging with submicrometer resolution from the same sample location. With fish cornea, we have demonstrated cross-sectional imaging of cornea tissue layers using OCT, and the zoom-in imaging of cells and collagen fibers in each layer using MPM. The multiscale MPM/OCT system shows the potential of a rapid coarse scan to search for abnormal regions and the subsequent fine zoom-in imaging for diagnosis.