Moderate Deviations for the overlap parameter in the Hopfield model

We derive moderate deviation principles for the overlap parameter in the Hopfield model of spin glasses and neural networks. If the inverse temperature is different from the critical inverse temperature _c=1 and the number of patterns M(N) satisfies M(N)/N 0, the overlap parameter multiplied by N, 1/2 < < 1, obeys a moderate deviation principle with speed N^1–2 and a quadratic rate function (i.e. the Gaussian limit for = 1/2 remains visible on the moderate deviation scale). At the critical temperature we need to multiply the overlap parameter by N, 1/4 < < 1. If then M(N) satisfies (M(N)⁶ log N M(N)²N⁴ log N)/N 0, the rescaled overlap parameter obeys a moderate deviation principle with speed N^1–4 and a rate function that is basically a fourth power. The random term occurring in the Central Limit theorem for the overlap at _c = 1 is no longer present on a moderate deviation scale. If the scaling is even closer to N^1/4, e.g. if we multiply the overlap parameter by N^1/4 log log N the moderate deviation principle breaks down. The case of variable temperature converging to one is also considered. If _N converges to _c fast enough, i.e. faster than the non-Gaussian rate function persists, whereas for _N converging to one slower than the moderate deviations principle is given by the Gaussian rate. At the borderline the moderate deviation rate function is the one at criticality plus an additional Gaussian term.Research supported by the Volkswagen-Stiftung (RiP-program at Oberwolfach, Germany).Mathematics Subject Classification (2000): 60F10 (primary), 60K35, 82B44, 82D30 (secondary)     

On a semilinear double fractional heat equation driven by fractional Brownian sheet

In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=(\Delta_\alpha+\Delta_\beta)u+\frac{\partial f}{\partial x}(t,x,u)+\frac{\partial^2W}{\partial t\partial x},$$ where $1<\beta<\alpha< 2$, $W(t,x)$ is a fractional Brownian sheet, $\Delta_\theta:=-(-\Delta)^{\theta/2}$ denotes the fractional Lapalacian operator and $f:[0,T]\times \mathbb{R}\times \mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear measurable function. We introduce the existence, uniqueness and H\"older regularity of the solution. As a related question, we consider also a large deviation principle associated with the above equation with a small perturbation via an equivalence relationship between Laplace principle and large deviation principle.     

Bounded stationary reflection II

Bounded stationary reflection at a cardinal λ is the assertion that every stationary subset of λ reflects but there is a stationary subset of λ that does not reflect at arbitrarily high cofinalities. We produce a variety of models in which bounded stationary reflection holds. These include models in which bounded stationary reflection holds at the successor of every singular cardinal $\mu >{\mathrm{?}}_{\omega }$ and models in which bounded stationary reflection holds at ${\mu }^{+}$ but the approachability property fails at μ.     

子系统为S≥3的非线性大系统在近平衡区内的稳定性

本文采用文[1,2]中提出的平衡结构法研究子系统s≥3的非线性大系统在近平衡区内的稳定性;得到了判别这类大系统稳定性的两个一般性准则.     

Particle deposition model for particulate flows at high temperatures in gas turbine components

This study proposes an improved physical model to predict sand deposition at high temperature in gas turbine components. This model differs from its predecessor (Sreedharan and Tafti, 2011) by improving the sticking probability by accounting for the energy losses during particle-wall collision based on our previous work (Singh and Tafti, 2013). This model predicts the probability of sticking based on the critical viscosity approach and collision losses during a particle–wall collision. The current model is novel in the sense that it predicts the sticking probability based on the impact velocity along with the particle temperature. To test the model, deposition from a sand particle laden jet impacting on a flat coupon geometry is computed and the results obtained from the numerical model are compared with experiments (Delimont et al., 2014) conducted at Virginia Tech, on a similar geometry and flow conditions, for jet temperatures of 950 °C, 1000 °C and 1050 °C. Large Eddy Simulations (LES) are used to model the flow field and heat transfer, and sand particles are modeled using a discrete Lagrangian framework. Results quantify the impingement and deposition for 20–40 μm sand particles. The stagnation region of the target coupon is found to experience most of the impingement and deposition. For 950 °C jet temperature, around 5% of the particle impacting the coupon deposit while the deposition for 1000 °C and 1050 °C is 17% and 28%, respectively. In general, the sticking efficiencies calculated from the model show good agreement with the experiments for the temperature range considered.     

Interfacial area concentration in gas–liquid bubbly to churn flow regimes in large diameter pipes

This study performed a survey on existing correlations for interfacial area concentration (IAC) prediction and collected an IAC experimental database of two-phase flows taken under various flow conditions in large diameter pipes. Although some of these existing correlations were developed by partly using the IAC databases taken in the low-void-fraction two-phase flows in large diameter pipes, no correlation can satisfactorily predict the IAC in the two-phase flows changing from bubbly, cap bubbly to churn flow in the collected database of large diameter pipes. So this study presented a systematic way to predict the IAC for the bubbly-to-churn flows in large diameter pipes by categorizing bubbles into two groups (group 1: spherical or distorted bubble, group 2: cap bubble). A correlation was developed to predict the group 1 void fraction by using the void fraction for all bubble. The group 1 bubble IAC and bubble diameter were modeled by using the key parameters such as group 1 void fraction and bubble Reynolds number based on the analysis of Hibiki and Ishii (2001, 2002) using one-dimensional bubble number density and interfacial area transport equations. The correlations of IAC and bubble diameter for group 2 cap bubbles were developed by taking into account the characteristics of the representative bubbles among the group 2 bubbles and the comparison between a newly-derived drift velocity correlation for large diameter pipes and the existing drift velocity correlation of Kataoka and Ishii (1987) for large diameter pipes. The predictions from the newly-developed two-group IAC correlation were compared with the collected experimental data in gas–liquid bubbly to churn flow regimes in large diameter pipes and their mean absolute relative deviations were obtained to be 28.1%, 54.4% and 29.6% for group 1, group 2 and all bubbles respectively.     

Study on imaging method for measuring droplet size in large sprays

Information of droplet size and size distribution lays the basis for investigations of atomization mechanisms and performance optimization. However, the laser diffraction and phase Doppler particle analyzers have difficulty in accurately characterizing sprays with a wide range of droplet sizes and very large droplets, especially if a large number of droplets are aspherical. A method to measure size in such large-droplet sprays based on digital imaging with backward illumination was developed, including an image acquisition system and image process programs. Calibration of the measurement system was performed using a dot calibration target with different dot sizes. An experimental setup was designed and established to characterize spray nozzles under different operation loads, as well as different nozzle arrangements. Results show that the droplet size of sprays ranges from dozens of microns to several millimeters. The superiority of wide load range for such nozzles was indicated by the size-measurement results under half-load to full-load operations. The present study revealed that the image processing technique can be effectively implemented for in-line size measurements of sprays with a wide distribution of droplet size and aspherical droplets, which would be difficult to characterize by other methods.     

Large deflections of a beam subject to three-point bending

In this paper, a solution for the equilibrium configuration of an elastic beam subject to three-point bending is given in terms of Jacobi elliptical functions. General equations are derived, and the domain of the solution is established. Several examples that illustrate a use of the solution are discussed. The obtained numerical results are compared with the results of other authors. An approximation formula by which the beam load is given as a polynomial function of beam deflection is also derived. The range of applicability of the approximation is illustrated by numerical examples.     

Contributions of Computational Aeroacoustics to Jet Noise Research and Prediction

A brief review of recent progress in the field of computational aeroacoustics (CAA) is proposed. This paper is complementary to the previous reviews of Tam [(1995a) “Computational aeroacoustics: issues and methods”, AIAA J. 33(10), 1788–1796], Lele [(1997) “Computational Aeroacoustics: a review”, AIAA Paper 97–0018, 35th Aerospace Sciences Meeting and Exhibit, Reno, Nevada] and Glegg [(1999) “Recent advances aeroacoustics: the influence of computational fluid dynamics”, 6th International Congress on Sound and Vibration, Copenhagen, Danemark, 5–8 July, 43–58] on advances in CAA. After a short introduction concerning the current motivations of jet noise studies, connections between computational fluid dynamics (CFD) and CAA using hybrid approaches are discussed in the first part. The most spectacular advances are probably provided by the direct computation of jet noise, and some recent results are shown in the second part.