The joint Laplace transforms for killed diffusion occupation times

The approach of Li and Zhou(2014)is adopted to find the Laplace transform of occupation time over interval(0,a)and joint occupation times over semi-infinite intervals(-∞,a)and(b,∞)for a time-homogeneous diffusion process up to an independent exponential time e_(q)for 0<a<b.The results are expressed in terms of solutions to the differential equations associated with the diffusion generator.Applying these results,we obtain explicit expressions on the Laplace transform of occupation time and joint occupation time for Brownian motion with drift.

生成式人工智能与设计流程的关系研究——以Stable Diffusion为例

如今以计算机技术为基石的人工智能正在高速发展,以人工智能为主体的第二次智能自动化革命或将到来。生成式人工智能是对人类思维的模拟,大语言模型与人工智能绘画等技术的迅速发展,使扩散神经网络、对抗神经网络等技术出现在大众视野中。通过对以Stable Diffusion为例的人工智能软件与设计流程的研究,分析生成式人工智能的发展历程与逻辑,探讨这一技术对设计研究的影响。人工智能作为一种工具,其背后决策依然是人的行为。其发展更加清晰的证明了设计过程中人的决策的不可替代性。同时生成式人工智能也体现出技术发展使设计过程回归设计本身这一现象。

Large Deviation Theorem for Empirical Measures of Degenerate Diffusion Processes

A class of multi dimensional degenerate diffusion processes X ε(t) in R r(r≥2) are considered and the asymptotic properties of empirical measures are investigated; here X ε(t) saitisfies the stochastic differential equation dX ε(t)=σ(X ε(t)) d W(t)+B(X ε(t)) d t+ εσ～(X ε(t)) d W(t),ε>0. X ε(t) are small random perturbations of the degenerate diffusion process X(t), which satisfies the stochastic differential equation dX(t)=σ(X(t)) d W(t)+B(X(t)) d t. A large deviation theorem for projection measures ν on R r-n (n<r) of empirical measures μ are proved

ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES

Let X（1） = {X（1）（s）, s ∈ R＋} and X（2） = {X（2）（t）, t ∈ R＋} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X （1） and X （2） are studied. More gener-ally, let E1, E2？（0,∞） and F ？Rd be Borel sets. A necessary condition and a suffcient condition for P{X（1）（E1）∩X（2）（E2）∩F 6=？}〉0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space （R＋× R＋× Rd,ρb）, whereρb is an unsymmetric metric defined in R＋× R＋× Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.

From Nonparametric Density Estimation to Parametric Estimation of Multidimensional Diffusion Processes

The paper deals with the estimation of parameters of multidimensional diffusion processes that are discretely observed. We construct estimator of the parameters based on the minimum Hellinger distance method. This method is based on the minimization of the Hellinger distance between the density of the invariant distribution of the diffusion process and a nonparametric estimator of this density. We give conditions which ensure the existence of an invariant measure that admits density with respect to the Lebesgue measure and the strong mixing property with exponential rate for the Markov process. Under this condition, we define an estimator of the density based on kernel function and study his properties (almost sure convergence and asymptotic normality). After, using the estimator of the density, we construct the minimum Hellinger distance estimator of the parameters of the diffusion process and establish the almost sure convergence and the asymptotic normality of this estimator. To illustrate the properties of the estimator of the parameters, we apply the method to two examples of multidimensional diffusion processes.

Two-Sided First Exit Problem for Jump Diffusion Distribution Processes Having Jumps with a Mixture of Erlang

In this paper, we consider the two-sided first exit problem for jump diffusion processes having jumps with rational Laplace transforms. We investigate the probabilistic property of conditional memorylessness, and drive the joint distribution of the first exit time from an interval and the overshoot over the boundary at the exit time.

Criteria for Regularity of Elliptic Diffusion Processes and It’s Application

The cone condition for x to be tegular for B under the elliptic diffusionprocess was proved. We also gave a necessary and sufficient condition for 0 to be regular for thornunder the elliptic diffusion process.

Insiders' Hedging for Jump Diffusion Processes with Applications to Index Tracking

The hedging problem for insiders is very important in the financial market.The locally risk minimizing hedging was adopted to solve this problem.Since the market was incomplete,the minimal martingale measure was chosen as the equivalent martingale measure.By the F-S decomposition,the expression of the locally risk minimizing strategy was presented.Finally,the local risk minimization was applied to index tracking and its relationship with tracking error variance (TEV)-minimizing strategy was obtained.

Impulse Control Problem of Partially Observed Diffusion Processes

The authors investigate the problem of impulse control of a partially observed diffusion process. The authors study the impulse control of Zakai type equations. The associated value function is characterized as the only viscosity solution of the corresponding quasi-variational inequality. The authors show the optimal cost function for the problem with incomplete information can be approximated by a sequence of value functions of the previous type.

On Optimal Sparse-Control Problems Governed by Jump-Diffusion Processes

A framework for the optimal sparse-control of the probability density function of a jump-diffusion process is presented. This framework is based on the partial integro-differential Fokker-Planck (FP) equation that governs the time evolution of the probability density function of this process. In the stochastic process and, correspondingly, in the FP model the control function enters as a time-dependent coefficient. The objectives of the control are to minimize a discrete-in-time, resp. continuous-in-time, tracking functionals and its L2- and L1-costs, where the latter is considered to promote control sparsity. An efficient proximal scheme for solving these optimal control problems is considered. Results of numerical experiments are presented to validate the theoretical results and the computational effectiveness of the proposed control framework.

A two-step transient liquid phase diffusion bonding process of T91 steels

In this study, a two-step heating process is introduced for transient liquid phase （ TLP） diffusion bonding fo r sound joints with T91 heat resistant steels. At first, a short-time higher temperature heating step is addressed to melt the interlayer, followed by the second step to complete isothermal solidification at a low temperature. The most critical feature of our new method is producing a non-planar interface at the T9／ heat resistant steels joint. We propose a transitional liquid phase bonding of T91 heat resistant steels by this approach. Since joint microstructures have been studied, we tested the tensile strength to assess joint mechanical property. The result indicates that the solidified bond may contain a primary solid-solution, similar composition to the parent metal and free from precipitates. Joint tensile strength of the joint is not lower than parent materials. Joint bend＇s strengths are enhanced due to the higher metal-to-metal junction producing a non-planar bond lines. Nevertheless, the traditional transient liquid phase diffusion bonding produces planar ones. Bonding parameters of new process are 1 260 °C for 0. 5 min and 1 230 °C fo r 4 min.

A GENERAL FORM OF THE INCREMENTS OF A TWO-PARAMETER WIENER PROCESS

In this paper, we consider a general form of the increments for a two-parameter Wiener process. Both the Csorgo-Revesz's increments and a class of the lag increments are the special cases of this general form of increments. Our results imply the theorem that have been given by Csorgo and Revesz (1978), and some of their conditions are removed.

Segregation of Molten Salt on Chromizing in Thermal Diffusion Process

The segregation of thermal diffusion salt bath chromizing process was analyzed. The experimental chromizing ingredients were prepared by the four groups A, B, C, and D. In order to study the segregation status of this case, the cooling molten salt in the crucible was removed by drilling from the heart core of molten salt. The core of molten salt was analyzed by X-ray fluorescence spectroscopy and XRD. Through the analysis, we can conclude that the Cr element deposited in the bottom was 4.51 times than the top. Chloride added to the molten salt will reduce segregation. Meantime we proposed some measures to overcome the segregation problem.

ENTROPY PRODUCTION RATE OF THE MINIMAL DIFFUSION PROCESS

The entropy production rate of stationary minimal diffusion processes with smooth coefficients is calculated. As a byproduct, the continuity of paths of the minimal diffusion processes is discussed, and that the point at infinity is absorbing is proved.

EXPECTED DISCOUNTED PENALTY FUNCTION AT RUIN FOR RISK PROCESS PERTURBED BY DIFFUSION UNDER INTEREST FORCE

In this article, the risk process perturbed by diffusion under interest force is considered, the continuity and twice continuous differentiability for Фδ（u,w） are discussed,the Feller expression and the integro-differential equation satisfied by Фδ （u ,w） are derived. Finally, the decomposition of Фδ（u,w） is discussed, and some properties of each decomposed part of Фδ（u,w） are obtained. The results can be reduced to some ones in Gerber and Landry＇s,Tsai and Willmot＇s, and Wang＇s works by letting parameter δ and （or） a be zero.

An Improved Cooperative Team Spraying Control of a Diffusion Process With a Moving or Static Pollution Source

This paper is concerned with a control problem of a diffusion process with the help of static mesh sensor networks in a certain region of interest and a team of networked mobile actuators carrying chemical neutralizers.The major contribution of this paper can be divided into three parts:the first is the construction of a cyber-physical system framework based on centroidal Voronoi tessellations(CVTs),the second is the convergence analysis of the actuators location,and the last is a novel proportional integral(PI)control method for actuator motion planning and neutralizing control(e.g.,spraying)of a diffusion process with a moving or static pollution source,which is more effective than a proportional(P)control method.An optimal spraying control cost function is constructed.Then,the minimization problem of the spraying amount is addressed.Moreover,a new CVT algorithm based on the novel PI control method,henceforth called PI-CVT algorithm,is introduced together with the convergence analysis of the actuators location via a PI control law.Finally,a modified simulation platform called diffusion-mobile-actuators-sensors-2-dimension-proportional integral derivative(Diff-MAS2D-PID)is illustrated.In addition,a numerical simulation example for the diffusion process is presented to verify the effectiveness of our proposed controllers.

Thiourea crystal growth kinetics,mechanism and process optimization during cooling crystallization

In the cooling crystallization process of thiourea,a significant issue is the excessively wide crystal size distribution(CSD)and the abundance of fine crystals.This investigation delves into the growth kinetics and mechanisms governing thiourea crystals during the cooling crystallization process.The fitting results indicate that the crystal growth rate coefficient,falls within the range of 10^(-7)to 10^(-8)m·s^(-1).Moreover,with decreasing crystallization temperature,the growth process undergoes a transition from diffusion-controlled to surface reaction-controlled,with temperature primarily influencing the surface reaction process and having a limited impact on the diffusion process.Comparing the crystal growth rate,and the diffusion-limited growth rate,at different temperatures,it is observed that the crystal growth process can be broadly divided into two stages.At temperatures above 25℃,1/qd(qd is diffusion control index)approaches 1,indicating the predominance of diffusion control.Conversely,at temperatures below 25℃,1/qd increases rapidly,signifying the dominance of surface reaction control.To address these findings,process optimization was conducted.During the high-temperature phase(35-25℃),agitation was increased to reduce the limitations posed by bulk-phase diffusion in the crystallization process.In the low-temperature phase(25-15℃),agitation was reduced to minimize crystal breakage.The optimized process resulted in a thiourea crystal product with a particle size distribution predominantly ranging from 0.7 to 0.9 mm,accounting for 84%of the total.This study provides valuable insights into resolving the issue of excessive fine crystals in the thiourea crystallization process.

AN ESTIMATE ON THE DISTRIBUTION AND MOMENTS OF THE LAST EXIT TIME OF AN ELLIPTIC DIFFUSION PROCESS

Let Ls be the last exit time from a compact set B of an elliptic diffusion process X. A moderate estimate for the distribution of Ls is obtained, and the sufficient and necessary condition for E^x（L^κB） 〈∞ is proved.

Fabrication of ultrafine Nd-Fe-B powder by a modified reduction-diffusion process

In order to obtain ultrafine Nd-Fe-B powder, a spray-dried precursor was treated by reduction-diffusion (R/D) process. And, unlike the conventional R/D process, calcium reduction that is a crucial step for the formation of Nd2Fe14B was performed without conglomerating the precursor with Ca powder. By adopting this modified process, it is possible to synthesize the hard magnetic Nd2Fe14B at the reaction temperature as low as 850 ℃. The average size of Nd2Fe14B particles that are uniformly distributed in the optimally treated powder was <<1 μm. Most Nd2Fe14B particles were enclosed with thin layers of Nd-rich phase. Typical magnetic properties of such powder without eliminating impurity CaO were iHc=～5.9 kOe, Br=～5.5 kG, and (BH)max=～6 MGOe.

CFD study of flow-diffusion process in Y-shape micromixer

A CFD simulation was carried out to investigate the mixing process in a Y-shape micromixer with the software Fluent 6.3. The definition of the "diffusion angle" is proposed to describe the molecular diffusion process associated with the flow at low Reynolds number. The linear relationship between the diffusion angle and the Peclet number(Pe) is determined by both theoretical analysis and numerical simulation. Moreover, the simulation results reveal that the diffusion angle is only related to the Peclet number whilst it is irrelevant to the changes of Re(Reynolds number) and Sc(Schmidt number). The range of Peclet number and Reynolds number for experimental measurement are also suggested as Pe≤10000 and Re≤10.