scholarly journals Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients

2010 ◽  
Vol 467 (2130) ◽  
pp. 1563-1576 ◽  
Author(s):  
Martin Hutzenthaler ◽  
Arnulf Jentzen ◽  
Peter E. Kloeden
Keyword(s):  
Exact Solution ◽  
Finite Time ◽  
Weak Sense ◽  
Mean Square ◽  
Euler’S Method ◽  
Lipschitz Continuous ◽  
The Difference ◽  
And Diffusion ◽  
Weak Divergence ◽  
Time Point

The stochastic Euler scheme is known to converge to the exact solution of a stochastic differential equation (SDE) with globally Lipschitz continuous drift and diffusion coefficients. Recent results extend this convergence to coefficients that grow, at most, linearly. For superlinearly growing coefficients, finite-time convergence in the strong mean-square sense remains. In this article, we answer this question to the negative and prove, for a large class of SDEs with non-globally Lipschitz continuous coefficients, that Euler’s approximation converges neither in the strong mean-square sense nor in the numerically weak sense to the exact solution at a finite time point. Even worse, the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean-square sense and in the numerically weak sense.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

scholarly journals Linear Structural Trends and Multi-Phase Intergrowths in Helvine-Group Minerals, (Zn,Fe,Mn)8[Be6Si6O24]S2

Minerals ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 325
Author(s):  
Sytle Antao
Keyword(s):  
Cell Parameter ◽  
Structural Parameters ◽  
Unit Cell ◽  
Fluid Composition ◽  
Ternary Solid Solution ◽  
X Ray Diffraction ◽  
X Ray ◽  
The Difference ◽  
Multi Phase ◽  
And Diffusion

Synchrotron high-resolution powder X-ray diffraction (HRPXRD) and Rietveld structure refinements were used to examine the crystal structure of single phases and intergrowths (either two or three phases) in 13 samples of the helvine-group minerals, (Zn,Fe,Mn)8[Be6Si6O24]S2. The helvine structure was refined in the cubic space group P4¯3n. For the intergrowths, simultaneous refinements were carried out for each phase. The structural parameters for each phase in an intergrowth are only slightly different from each other. Each phase in an intergrowth has well-defined unit-cell and structural parameters that are significantly different from the three endmembers and these do not represent exsolution or immiscibility gaps in the ternary solid-solution series. The reason for the intergrowths in the helvine-group minerals is not clear considering the similar radii, identical charge, and diffusion among the interstitial M cations (Zn2+, Fe2+, and Mn2+) that are characteristic of elongated tetrahedral coordination. The difference between the radii of Zn2+ and Mn2+ cations is 10%. Depending on the availability of the M cations, intergrowths may occur as the temperature, pressure, fugacity fS2, and fluid composition change on crystallization. The Be–Si atoms are fully ordered. The Be–O and Si–O distances are nearly constant. Several structural parameters (Be–O–Si bridging angle, M–O, M–S, average <M–O/S>[4] distances, and TO4 rotational angles) vary linearly with the a unit-cell parameter across the series because of the size of the M cation.

scholarly journals Exponential Mean-Square Stability of Numerical Solutions to Stochastic Differential Equations

2003 ◽  
Vol 6 ◽  
pp. 297-313 ◽  
Author(s):  
Desmond J. Higham ◽  
Xuerong Mao ◽  
Andrew M. Stuart
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Finite Time ◽  
Mean Square ◽  
Convergence Conditions ◽  
Finite Time Convergence ◽  
Mean Square Stability ◽  
Exponential Mean Square Stability ◽  
Positive Results

AbstractPositive results are proved here about the ability of numerical simulations to reproduce the exponential mean-square stability of stochastic differential equations (SDEs). The first set of results applies under finite-time convergence conditions on the numerical method. Under these conditions, the exponential mean-square stability of the SDE and that of the method (for sufficiently small step sizes) are shown to be equivalent, and the corresponding second-moment Lyapunov exponent bounds can be taken to be arbitrarily close. The required finite-time convergence conditions hold for the class of stochastic theta methods on globally Lipschitz problems. It is then shown that exponential mean-square stability for non-globally Lipschitz SDEs is not inherited, in general, by numerical methods. However, for a class of SDEs that satisfy a one-sided Lipschitz condition, positive results are obtained for two implicit methods. These results highlight the fact that for long-time simulation on nonlinear SDEs, the choice of numerical method can be crucial.

Bidirectional saturable transport of LHRH across the blood-brain barrier

1991 ◽  
Vol 261 (3) ◽  
pp. E312-E318 ◽  
Author(s):  
C. M. Barrera ◽  
A. J. Kastin ◽  
M. B. Fasold ◽  
W. A. Banks
Keyword(s):  
Blood Brain Barrier ◽  
Ovarian Function ◽  
Brain Barrier ◽  
Multiple Time ◽  
Blood Brain ◽  
Permeability Surface ◽  
Cerebrovascular Permeability ◽  
The Difference ◽  
Single Time Point ◽  
Time Point

Systemic administration of luteinizing hormone-releasing hormone (LHRH) in rats has been found to influence behavior independently of pituitary or ovarian function. A previous study has shown that LHRH can cross the blood-brain barrier in one direction, but it was not known whether this was due to a saturable transport system. The rate of entry of 125I-labeled LHRH from blood to brain was determined by two different single-pass methods of carotid perfusion. The first, a multiple time point method, measures Ki from the slope of the linear regression when brain-to-blood ratios of radioiodinated LHRH are plotted against time. Saturable transport was determined by the difference between the Ki of rats perfused with 125I-LHRH (12.51 X 10(-3) mg.g-1.min-1) vs. rats perfused with 125I-LHRH and unlabeled LHRH (10 nmol/ml; 2.20 X 10(-3) ml.g-1.min-1). The inhibition by the unlabeled peptide was statistically significant (P less than 0.001). The second method, a single time point technique, measures the cerebrovascular permeability-surface area coefficient (PA). Saturable transport was determined in rats by the competition of unlabeled LHRH with 125I-LHRH. The PA value for 125I-LHRH (20.00 X 10(-3) ml.g-1.min-1) was significantly greater (P less than 0.05) than for 125I-LHRH with the addition of 10 nmol/ml unlabeled LHRH (4.14 X 10(-3) ml.g-1.min-1). Saturable transport of LHRH from brain to blood in mice was also determined.(ABSTRACT TRUNCATED AT 250 WORDS)

scholarly journals On an extension of the stochastic linearization

1993 ◽  
Vol 15 (4) ◽  
pp. 1-6
Author(s):  
Di Paola Mario ◽  
Nguyen Dong Anh
Keyword(s):  
Nonlinear Systems ◽  
Nonlinear Equation ◽  
Random Excitation ◽  
Linearization Method ◽  
Mean Square ◽  
Fundamental Idea ◽  
Stochastic Linearization ◽  
The Mean ◽  
The Difference

Stochastic linearization method is one of the most useful tools for analysis of nonlinear systems under random excitation. The fundamental idea of the classical stochastic linearization consists in replacing the original nonlinear equation by a linear one in such a way that the difference between two equations is minimized in the mean square value. In this paper a new version of the stochastic linearization is proposed. It is shown that for two nonlinear systems considered the new version gives good results for both the weak and strong nonlinearities.

scholarly journals INFLUENCE OF DIFFERENT VERSIONS ARTIFICIAL INSEMINATION OF GILTS OF THEIR TRAITS OF THE REPRODUCTIVE ABILITY CONSOLIDATION

2018 ◽  
Vol 56 ◽  
pp. 130-135
Author(s):  
M. V. Tserenyuk ◽  
O. M. Tserenyuk
Keyword(s):  
Artificial Insemination ◽  
Reproductive Capacity ◽  
Mean Square ◽  
Mean Square Deviation ◽  
Reproductive Ability ◽  
Coefficient Of Variability ◽  
Current Instruction ◽  
The Mean ◽  
The Difference ◽  
The Impact

In addition to assessing the absolute performance of animals, the impact of certain technological approaches, breeding influence and other external organized factors on the consolidation of pigs groups on certain grounds should be evaluated. Breeding consolidation is a desirable breeding process that is realized through more motivated consolidation of intra-breeding structural units for maintaining a significant level of intergroup differentiation and variability. If the issue relates to the rearing young animals that are introduced into the herd, this is the most urgent issue that needs to be monitored. The purpose of the research is to determine the consociality of the main signs of reproductive ability of the gilts, which are checked at different rates of artificial insemination of sows of the main herd. The research was carried out at the Shubs`ke farm in the Bogoduhivsky District of Kharkiv Oblast at the pig farm for the pure breeding of the Welsh breed pigs (breeding reproductive status from 2014). In order to evaluate the optimal multiplicity (from single to triple), a gilts with a total of 30 heads was selected. The main indicators of reproductive ability of gilts (multiplicity, weight of nest at birth and excommunication and keeping piglets before weaning on day 28) were evaluated. Recalculation on the weight of the litter at weaning at the weight of the nest at extermination in 60 days was carried out in accordance with the current instruction on the boning of pigs. The results of researches were worked out according to traditional methods by the method of variation statistics. Consolidation of individual indicators of reproductive capacity of gilts was calculated to the total number of evaluated animals. In the previous stage of the research, it was found that the use of multiple insemination of gilts compared to single insemination is positively reflected at the level of their reproductive ability. A decrease in the percentage of non-productive inseminants in animal groups has been established. Triple insemination of gilts allows for 1.14 piglets to be obtained more compared to single insemination of the same pigs (p < 0.01). Also, for three times insemination of gilts, in comparison with single insemination, larger nest for weaning were obtained (by 14.24 kg, converted to 60th day p < 0.01). At the same time, the reduction in the keeping of piglets to weaning for increase frequency of insemination of gilts. According to the multiplicity of gilts, the most consolidated level of symptoms per group was obtained for single insemination. Increasing the multiplicity of insemination negatively affected the consolidation of the multiplicity of the piglets (as for determining the coefficients of phenotypic consolidation due to the mean square deviation, so, and for determining the coefficient of variability). The least consolidated in terms of multiplicity were uterus for double insemination. It should also be noted that there is little difference, in general, on the indicators of the theonotypical consolidation of the multiplicity of piglets between different groups of animals. So this difference between the most contrasting groups was 0.447 points for determining the coefficients due to the mean square deviation and 0.397 points for the determination due to the coefficient of variability, respectively. By weight of the same litters at birth, the most consolidated group was gilts, which was inseminated twice. The least consolidated are trimmed three times, respectively. The difference as a whole, according to the indicators of the theonotype consolidation of the mass of the litter’s piglets at birth, between different groups of pigs was even smaller. The difference between the most contrasting groups was 0.270 points for determining the coefficients due to the mean square deviation and 0.260 points for the determination due to the coefficient of variability, respectively. On the basis of the mass of the litter at weaning, the most consolidated level of symptoms was obtained in the group of gilts for triple insemination. The least consolidated group, respectively, was gilts for single insemination. At the same time, on the basis of the weight of the litter at weaning, the slightest differences were obtained after the consolidation of the signs between different groups of pigs. The difference between the most contrasting groups was 0.173 points for determining the coefficients due to the mean square deviation and 0.248 points for the determination due to the coefficient of variation, respectively. At different rates of insemination of gilts, there are no significant differences between the groups between the groups in terms of the consolidation of signs of reproductive ability. Also, there was no clear effect on increasing the multiplicity of insemination of gilts to the consolidation of their signs of reproductive qualities.

Diuretic activities of fayiren and coix seed in normal rats

2020 ◽  
Author(s):  
Tao Hong ◽  
Hao Chen ◽  
Li Liu ◽  
Weiqi Liu ◽  
Longxue Li ◽  
...  
Keyword(s):  
Low Dose ◽  
Urine Volume ◽  
Positive Control ◽  
Total Output ◽  
High Dose ◽  
Model Group ◽  
Urine Ph ◽  
Coix Seed ◽  
The Difference ◽  
Time Point

Abstract Background Coix seed is a traditional Chinese medicine with pharmacological effects and food efficacy. It is well-know and widely used in China and other Asian countries. Fayiren is a new artillery material processed from Coix Seed. This is the first study of the diuretic activities of Fayiren and coix seed in normal rats. Methods Forty-eight rats were randomly divided into 6 groups (8 rats/group) according to body mass and urine volume stratification, namely model (physiological saline) group, positive control (furosemide 20 mg/kg) group, coix seed (1666 mg/kg) group, low-dose of Fayiren group (166 mg/kg) and high-dose Fayiren group (1666 mg/kg). Rats were fasted for 16 h before the experiment, with normal water supply. Immediately after administration, each group of rat was placed in a metabolic cage, and urine was collected once after every hour for a total of 6 times. The urine volume at each time point for collection was measured, and then totaled. After the 6 h recording procedure, the treatments were administered once a day and the urine was collected at the 24th, 48th, 72th, 96th, 110th, 134th, and 158th hours from the start respectively. The urine pH and Na +, K +, Ca 2+, Cl - concentrations at each time point of each group of rats were measured. The contents of ALD, ANP, ADH, Na+-K+-ATPase, the gene expression and protein levels of AQP1, AQP2 and AQP3 in urine of rats in each group were determined. SPSS22.0 statistical software was used to analyze the experimental data. One way ANOVA and multiple comparisons between LSD groups were performed. The difference is considered significant When p < 0.05 or p < 0.01. Results After the administration of Fayiren and coix seed, compared with normal rats, the total output of urine in the high-dose coix seed & Fayiren group and the positive control group both increased; In addition, the contents of Na+ and Cl-in urine in these groups increased as well. However, the content of K+ did not increase in all groups. Contents of ALD and ADH was decreased and increased contents of ANP observed in urine in both high-dose and low-dose Fayiren groups with significant differences from model group (P < 0.05). Coix seed significantly reduced ADH and increased ANP levels compared with model group (P < 0.05). There was no obvious change in the urine pH content in urine of all rats. All the aquaporin (AQP1, AQP2, and AQP3) contents were decreased comparing to the model group, the coix seed group, as well as the Fayiren group. The effect of coix seed was particularly notable on AQP2, and the difference was significant against the Furosemide group (P < 0.05). By contrast, Fayiren showed greater effects on AQP1 and AQP3, with statistically significant difference comparing to the model group (P༜0.05). Conclusions This study will provide experimental evidence for explaining the different mechanisms of diuretic effects in natural medicine use of coix seed and artillery material processed from Coix Seed. The Fayiren may affect the reabsorption of water in the kidney through mechanisms related to increased ANP secretion,decreased ALD and ADH secretion in urine, and it significantly affect the expression of aquaporin 1 and aquaporin 3. However, coix seed may increase ANP but decreased ADH secretion, further affecting the expression of aquaporin2 and exert diuretic effect.

Stochastic storage networks: stationarity and the feedforward case

1997 ◽  
Vol 34 (02) ◽  
pp. 498-507 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Total Variation ◽  
Finite Time ◽  
Initial Conditions ◽  
Stability Conditions ◽  
Input Process ◽  
Stationary Increments ◽  
Stationary Version ◽  
Storage Network ◽  
The Difference ◽  
Storage Networks

We show that for a certain storage network the backward content process is increasing, and when the net input process has stationary increments then, under natural stability conditions, the content process has a stationary version under which the cumulative lost capacities have stationary increments. Moreover, for the feedforward case, we show that under some minimal conditions, two content processes with net input processes which differ only by initial conditions can be coupled in finite time and that the difference of two content processes vanishes in the limit if the difference of the net input processes monotonically approaches a constant. As a consequence, it is shown that for the natural stability conditions, when the net input process has stationary increments, the distribution of the content process converges in total variation to a proper limit, independent of initial conditions.

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