Stationary turbulent closure via the Hopf functional equation

Exact closed-form solutions are exhibited for the Hopf equation for stationary incompressible 3D Navier-Stokes flow, for the cases of homogeneous forced flow (including a solution with depleted nonlinearity) and inhomogeneous flow with arbitrary boundary conditions. This provides an exact method for computing two- and higher-point moments, given the mean flow.     

Exploiting the diversity of time scales in the immune system: A B-cell antibody model

Using the continuous shape space formalism, we develop an immune system model involving both B lymphocytes and antibody molecules. The binding and cross-linking of receptors on B cells stimulates the cells to divide and, with a lag, to secrete antibody. Using the method of multiple scales, we show how to correctly formulate long-time-scale equations for the population dynamics of B cells, the total antibody concentration, and rate of antibody secretion. We compare our model with previous phenomenological formulations.     

The ground state for sticky disks

It is proven that the ground state of the two-dimensional sticky potential is the triangular lattice.Research supported in part by NSF grant MCS 78-01520-A01.     

Shannon Entropy Loss in Mixed-Radix Conversions

This paper models a translation for base-2 pseudorandom number generators (PRNGs) to mixed-radix uses such as card shuffling. In particular, we explore a shuffler algorithm that relies on a sequence of uniformly distributed random inputs from a mixed-radix domain to implement a Fisher–Yates shuffle that calls for inputs from a base-2 PRNG. Entropy is lost through this mixed-radix conversion, which is assumed to be surjective mapping from a relatively large domain of size 2J to a set of arbitrary size n. Previous research evaluated the Shannon entropy loss of a similar mapping process, but this previous bound ignored the mixed-radix component of the original formulation, focusing only on a fixed n value. In this paper, we calculate a more precise formula that takes into account a variable target domain radix, n, and further derives a tighter bound on the Shannon entropy loss of the surjective map, while demonstrating monotonicity in a decrease in entropy loss based on increased size J of the source domain 2J. Lastly, this formulation is used to specify the optimal parameters to simulate a card-shuffling algorithm with different test PRNGs, validating a concrete use case with quantifiable deviations from maximal entropy, making it suitable to low-power implementation in a casino.     

The linear noise approximation for reaction-diffusion systems on networks

Stochastic reaction-diffusion models can be analytically studied on complex networks using the linear noise approximation. This is illustrated through the use of a specific stochastic model, which displays travelling waves in its deterministic limit. The role of stochastic fluctuations is investigated and shown to drive the emergence of stochastic waves beyond the region of the instability predicted from the deterministic theory. Simulations are performed to test the theoretical results and are analyzed via a generalized Fourier transform algorithm. This transform is defined using the eigenvectors of the discrete Laplacian defined on the network. A peak in the numerical power spectrum of the fluctuations is observed in quantitative agreement with the theoretical predictions.     

PEGylated Luminescent Gold Nanoclusters: Synthesis,Characterization, Bioconjugation,and Application to One‐ and Two‐Photon Cellular Imaging

Biocompatible, near‐infrared luminescent gold nanoclusters (AuNCs) are synthesized directly in water using poly(ethylene glycol)‐dithiolane ligands terminating in either a carboxyl, amine, azide, or methoxy group. The ≈1.5 nm diameter AuNCs fluoresce at ≈820 nm with quantum yields that range from 4–8%, depending on the terminal functional group present, and display average luminescence lifetimes approaching 1.5 μs. The two‐photon absorption (TPA) cross‐section and two‐photon excited fluorescence (TPEF) properties are also measured. Long‐term testing shows the poly(ethylene glycol) stabilized AuNCs maintain colloidal stability in a variety of media ranging from saline to tissue culture growth medium along with tolerating storage of up to 2 years. DNA and dye‐conjugation reactions confirm that the carboxyl, amine, and azide groups can be utilized on the AuNCs for carbodiimide, succinimidyl ester, and Cu^I‐assisted cycloaddition chemistry, respectively. High signal‐to‐noise one‐ and two‐photon cellular imaging is demonstrated. The AuNCs exhibit outstanding photophysical stability during continuous‐extended imaging. Concomitant cellular viability testing shows that the AuNCs also elicit minimal cytotoxicity. Further biological applications for these luminescent nanoclustered materials are discussed.     

Dynamic non-linear optical processes in semiconductors

In this article active non-linear optical effects in semiconductors are reviewed. These processes arise because of the dynamics of excited populations of charge carriers temporarily created in otherwise empty states when a beam of laser radiation is incident on the material. A large number of different effects have been observed in different semiconductor materials recently and the non-linearities cover a wide range of magnitudes both in time-scale and size of non-linearity. The theory and physical concepts relevant to these processes are described and experimental observations using both high power pulsed and low power c.w. lasers are reviewed. Applications in optical bistability, phase conjugation, optical gating and optoelectronic gating are discussed.     

Fastiterative methods for least squares estimations

Least squares estimations have been used extensively in many applications, e.g. system identification and signal prediction. When the stochastic process is stationary, the least squares estimators can be found by solving a Toeplitz or near-Toeplitz matrix system depending on the knowledge of the data statistics. In this paper, we employ the preconditioned conjugate gradient method with circulant preconditioners to solve such systems. Our proposed circulant preconditioners are derived from the spectral property of the given stationary process. In the case where the spectral density functions() of the process is known, we prove that ifs() is a positive continuous function, then the spectrum of the preconditioned system will be clustered around 1 and the method converges superlinearly. However, if the statistics of the process is unknown, then we prove that with probability 1, the spectrum of the preconditioned system is still clustered around 1 provided that large data samples are taken. For finite impulse response (FIR) system identification problems, our numerical results show that annth order least squares estimator can usually be obtained inO(n logn) operations whenO(n) data samples are used. Finally, we remark that our algorithm can be modified to suit the applications of recursive least squares computations with the proper use of sliding window method arising in signal processing applications.Research supported in part by HKRGC grant no. 221600070, ONR contract no. N00014-90-J-1695 and DOE grant no. DE-FG03-87ER25037.     

Riemann-measurable selections

LetX be a Polish space equipped with a σ-finite regular Borel measure μ. IfE is a metric space andF a set-valued function:X → 2 ^E with complete values, and ifF is lower semicontinuous at almost all points ofX, we prove that there exists a Riemann-measurable selections ofF.