Computing the Ball Size of Frequency Permutations under Chebyshev Distance

Let $S_n^\lambda$ be the set of all permutations over the multiset $\{\overbrace{1,...,1}^{\lambda},...,\overbrace{m,...,m}^\lambda\}$ where $n=m\lambda$. A frequency permutation array (FPA) of minimum distance $d$ is a subset of $S_n^\lambda$ in which every two elements have distance at least $d$. FPAs have many applications related to error correcting codes. In coding theory, the Gilbert-Varshamov bound and the sphere-packing bound are derived from the size of balls of certain radii. We propose two efficient algorithms that compute the ball size of frequency permutations under Chebyshev distance. Both methods extend previous known results. The first one runs in $O({2d\lambda \choose d\lambda}^{2.376}\log n)$ time and $O({2d\lambda \choose d\lambda}^{2})$ space. The second one runs in $O({2d\lambda \choose d\lambda}{d\lambda+\lambda\choose \lambda}\frac{n}{\lambda})$ time and $O({2d\lambda \choose d\lambda})$ space. For small constants $\lambda$ and $d$, both are efficient in time and use constant storage space.Comment: Submitted to ISIT 201

Dynamics of Moving Average Rules in a Continuous-time Financial Market Model

Within a continuous-time framework, this paper proposes a stochastic heterogeneous agent model (HAM) of financial markets with time delays to unify various moving average rules used indiscrete-time HAMs. The time delay represents a memory length of a moving average rule indiscrete-time HAMs.Intuitive conditions for the stability of the fundamental price of the deterministic model in terms of agents' behavior parameters and memory length are obtained. It is found that an increase in memory length not only can destabilize the market price, resulting in oscillatory market price characterized by a Hopf bifurcation, but also can stabilize another wise unstable market price, leading to stability switching as the memory length increases. Numerical simulations show that the stochastic model is able to characterize long deviations of the market price from its fundamental price and excess volatility and generate most of the stylized factso bserved in financial markets.asset price; financial market behavior; heterogeneous beliefs; stochastic delay differential equations; stability; bifurcations; stylized facts

Pixel level optical-transfer-function design based on the surface-wave-interferometry aperture

The design of optical transfer function (OTF) is of significant importance for optical information processing in various imaging and vision systems. Typically, OTF design relies on sophisticated bulk optical arrangement in the light path of the optical systems. In this letter, we demonstrate a surface-wave-interferometry aperture (SWIA) that can be directly incorporated onto optical sensors to accomplish OTF design on the pixel level. The whole aperture design is based on the bull’s eye structure. It composes of a central hole (diameter of 300 nm) and periodic groove (period of 560 nm) on a 340 nm thick gold layer. We show, with both simulation and experiment, that different types of optical transfer functions (notch, highpass and lowpass filter) can be achieved by manipulating the interference between the direct transmission of the central hole and the surface wave (SW) component induced from the periodic groove. Pixel level OTF design provides a low-cost, ultra robust, highly compact method for numerous applications such as optofluidic microscopy, wavefront detection, darkfield imaging, and computational photography

The number of ramified covering of a Riemann surface by Riemann surface

Interpreting the number of ramified covering of a Riemann surface by Riemann surfaces as the relative Gromov-Witten invariants and applying a gluing formula, we derive a recursive formula for the number of ramified covering of a Riemann surface by Riemann surface with elementary branch points and prescribed ramification type over a special point.Comment: LaTex, 14 page