Limiting Distribution and Mixing Time

2013 ◽  
pp. 121-143
Author(s):  
Renato Portugal
Keyword(s):  
Mixing Time ◽  
Limiting Distribution

scholarly journals Rate of Convergence of the Short Cycle Distribution in Random Regular Graphs Generated by Pegging

10.37236/133 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Pu Gao ◽  
Nicholas Wormald
Keyword(s):  
Rate Of Convergence ◽  
Total Variation ◽  
Upper Bound ◽  
Mixing Time ◽  
Limiting Distribution ◽  
Total Variation Distance ◽  
Regular Graphs ◽  
Short Cycle ◽  
Variation Distance

The pegging algorithm is a method of generating large random regular graphs beginning with small ones. The $\epsilon$-mixing time of the distribution of short cycle counts of these random regular graphs is the time at which the distribution reaches and maintains total variation distance at most $\epsilon$ from its limiting distribution. We show that this $\epsilon$-mixing time is not $o(\epsilon^{-1})$. This demonstrates that the upper bound $O(\epsilon^{-1})$ proved recently by the authors is essentially tight.

Novel Coupling Agents for Silica-filled Rubbers with Superior Processing Safety and Improved Hysteresis of the Vulcanizates

2010 ◽  
Vol 38 (1) ◽  
pp. 80-98 ◽  
Author(s):  
M. Gerster ◽  
C. Fagouri ◽  
E. Peregi
Keyword(s):  
Mixing Time ◽  
Cost Savings ◽  
Coupling Agents ◽  
Tire Industry ◽  
Selective Functionalization ◽  
Current State ◽  
Rubber Compounds ◽  
Volatile Organic ◽  
Filled Rubbers ◽  
Ethanol Removal

Abstract One challenge facing green tire technology is to achieve good silica hydrophobation/dispersion within the polymer matrix without a detrimental increase in the rubber compound’s viscosity during compounding. This phenomenon is well known to be induced by premature and unwanted coupling and/or crosslinking of the traditional coupling agents. The current state-of-the-art polysulfides silanes, bis(3-triethoxysilylpropyl)tetrasulfide and to a lesser extent bis(3-triethoxysilylpropyl)disulfide (“Product Application—VP Si 75/VP X 75-S in the Rubber Industry,” Degussa Hüls Report No. PA 723.1E), need to be carefully incorporated with careful temperature control during the rubber compounding to prevent this “scorchy” behavior. This paper will present novel monofunctional silanes which are suited for preparing highly silica-loaded rubber compounds of superior processability, while applying fewer mixing passes, thereby reducing mixing times which can lead to improved productivity and cost savings. Additionally, these safer coupling agents can be processed at higher temperatures which can, again, lead to reduced mixing time and better ethanol removal thereby improving the tire’s physical properties and reducing the volatile organic compounds generated during the tire’s use. The rubber compounds produced using these monofunctional silanes are characterized by lower Mooney viscosity and improved processability. Advantageously, within these novel chemical classes of coupling agents, selective functionalization of the silanes allows production of tailor-made coupling agents which can respond to the specific requirements of the tire industry (Vilgis, T. A. and Heinrich, G., “Die Physic des Autoreifens,” Physikalische Blätter, Vol. 57, 2001, pp. 1–7).

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

Water Modeling on Circulating Flow and Mixing Time in a Ruhrstahl–Heraeus Vacuum Degasser

2021 ◽  
pp. 2000608
Author(s):  
Chang Liu ◽  
Lifeng Zhang ◽  
Fei Li ◽  
Kaiyu Peng ◽  
Fenggang Liu ◽  
...  
Keyword(s):  
Mixing Time ◽  
Water Modeling ◽  
Vacuum Degasser

scholarly journals Effect of Aspect Ratio on the Mixing Performance in the Kenics Static Mixer

Processes ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 464
Author(s):  
Xingren Jiang ◽  
Ning Yang ◽  
Rijie Wang
Keyword(s):  
Pharmaceutical Industry ◽  
Aspect Ratio ◽  
Flow Reactor ◽  
Mixing Time ◽  
Continuous Process ◽  
Scale Up ◽  
Influence Factors ◽  
Continuous Manufacturing ◽  
Static Mixer ◽  
Mixing Performance

Continuous manufacturing has received increasing interest because of the advantages of intrinsic safety and enhanced mass transfer in the pharmaceutical industry. However, the difficulty for scale-up has limited the application of continuous manufacturing for a long time. Recently, the tubular flow reactor equipped with the Kenics static mixer appears to be a solution for the continuous process scale-up. Although many influence factors on the mixing performance in the Kenics static mixer have been investigated, little research has been carried out on the aspect ratio. In this study, we used the coefficient of variation as the mixing evaluation index to investigate the effect of the aspect ratio (0.2–2) on the Kenics static mixer’s mixing performance. The results indicate that a low aspect ratio helps obtain a shorter mixing time and mixer length. This study suggests that adjusting the aspect ratio of the Kenics static mixer can be a new strategy for the scale-up of a continuous process in the pharmaceutical industry.

scholarly journals Effect of Energy Degeneracy on the Transition Time for a Series of Metastable States

2021 ◽  
Vol 184 (1) ◽  
Author(s):  
Gianmarco Bet ◽  
Vanessa Jacquier ◽  
Francesca R. Nardi
Keyword(s):  
Metastable State ◽  
Stochastic Dynamics ◽  
Transition Probabilities ◽  
Mixing Time ◽  
Stable State ◽  
Temperature Limit ◽  
Metastable States ◽  
Transition Time ◽  
Sharp Estimates ◽  
Model Independent

AbstractWe consider the problem of metastability for stochastic dynamics with exponentially small transition probabilities in the low temperature limit. We generalize previous model-independent results in several directions. First, we give an estimate of the mixing time of the dynamics in terms of the maximal stability level. Second, assuming the dynamics is reversible, we give an estimate of the associated spectral gap. Third, we give precise asymptotics for the expected transition time from any metastable state to the stable state using potential-theoretic techniques. We do this in a general reversible setting where two or more metastable states are allowed and some of them may even be degenerate. This generalizes previous results that hold for a series of only two metastable states. We then focus on a specific Probabilistic Cellular Automata (PCA) with configuration space $${\mathcal {X}}=\{-1,+1\}^\varLambda $$ X = { - 1 , + 1 } Λ where $$\varLambda \subset {\mathbb {Z}}^2$$ Λ ⊂ Z 2 is a finite box with periodic boundary conditions. We apply our model-independent results to find sharp estimates for the expected transition time from any metastable state in $$\{\underline{-1}, {\underline{c}}^o,{\underline{c}}^e\}$$ { - 1 ̲ , c ̲ o , c ̲ e } to the stable state $$\underline{+1}$$ + 1 ̲ . Here $${\underline{c}}^o,{\underline{c}}^e$$ c ̲ o , c ̲ e denote the odd and the even chessboard respectively. To do this, we identify rigorously the metastable states by giving explicit upper bounds on the stability level of every other configuration. We rely on these estimates to prove a recurrence property of the dynamics, which is a cornerstone of the pathwise approach to metastability.

scholarly journals Formation conditions and mechanical properties of aggregates produced in tephra–water–snow flows

2020 ◽  
Vol 72 (1) ◽  
Author(s):  
Hirofumi Niiya ◽  
Kenichi Oda ◽  
Daisuke Tsuji ◽  
Hiroaki Katsuragi
Keyword(s):  
Mechanical Properties ◽  
Mixing Time ◽  
Testing Machine ◽  
Small Scale ◽  
Scaling Analysis ◽  
Compression Tests ◽  
Ice Slurry ◽  
Experimental Conditions ◽  
Formation Conditions ◽  
Snow And Ice

Abstract The formation of aggregates consisting of snow, water, and tephra has been reported in small-scale experiments on three-phase flows containing tephra, water, and snow, representing lahars triggered by snowmelt. Such aggregates reduce the mobility of mud flow. However, the formation mechanism of such aggregates under various conditions has not been investigated. To elucidate the formation conditions and mechanical properties of the aggregates, we performed mixing experiments with materials on a rotating table and compression tests on the resulting aggregates with a universal testing machine in a low-temperature room at $$0\,^{\circ }\text {C}$$ 0 ∘ C . From experiments with varying component ratios of the mixture and tephra diameter, the following results were obtained: (i) the aggregate grew rapidly and reached maturity after a mixing time of 5 min; (ii) the mass of aggregates increased with snow concentration, exhibiting an approximately linear relationship; (iii) single aggregates with large mass formed at lower and higher tephra concentrations, whereas multiple aggregates with smaller mass were observed at intermediate concentrations; (iv) the shape of the aggregate satisfied the similarity law for an ellipsoid; (v) the compressive mechanical behavior could be modeled by an empirical nonlinear model. The obtained mechanical properties of the aggregates were independent of the experimental conditions; (vi) scaling analysis based on the Reynolds number and the strength of the aggregates showed that the aggregates cannot form in ice-slurry lahars. Our findings suggest that low-speed lahars containing snow and ice are likely to generate aggregates, but snow and ice in the ice-slurry lahars are dispersed without such aggregates.

scholarly journals The Study of Carbon Recovery from Electrolysis Aluminum Carbon Dust by Froth Flotation

Metals ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 145
Author(s):  
Hesong Li ◽  
Jiaoru Wang ◽  
Wenyuan Hou ◽  
Mao Li ◽  
Benjun Cheng ◽  
...  
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Mixing Time ◽  
Froth Flotation ◽  
Aluminum Electrolysis ◽  
Carbon Powder ◽  
Molten Salt Electrolysis ◽  
Carbon Recovery ◽  
Aluminum Smelting

A large amount of carbon dust is generated in the process of aluminum smelting by molten salt electrolysis. The carbon dust is solid hazardous waste but contains a large quantity of recyclable components such as carbon and fluoride. How to recycle carbon dust more effectively is a challenge in the aluminum electrolysis field. In this study, X-ray diffraction, scanning electron microscope, and other methods were used to analyze the phase composition of electrolytic aluminum carbon dust. The effects of particle size distribution of carbon dust, impeller speed, reagent addition, mixing time, and flotation time on the flotation recovery of carbon dust were studied. The optimal flotation conditions were obtained and the flotation products were analyzed. The results show that the optimal particle size distribution is 70% of particles below 200 mesh, corresponding to a grinding time of 11 min. The optimum speed of the flotation machine was to be between 1600 and 1800 r/min with the best slurry concentration of 20–30% and 5 min mixing time, and the collector kerosene was suitable for adding in batches. Under the above conditions, the recovered carbon powder with a carbon content of 75.6% was obtained, and the carbon recovery rate was 86.9%.

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