Science topic

# Stability - Science topic

Explore the latest questions and answers in Stability, and find Stability experts.
Questions related to Stability
Question
I am working on an infectious disease of 10 to 15 compartment. I need I need maple code to solve the DFE,ENDEMIC,BASIC REPRODUCTION NUMBER AND ALSO TO PERFORM SENSITIVITY ANALYSIS.
Dear friend Kayode Bolaji
I don't have direct access to specific codes or Maple scripts, but I can guide you Kayode Bolaji on how you might approach solving stability, finding the disease-free equilibrium (DFE), endemic equilibrium, and basic reproduction number in Maple.
**1. Define the Model Equations:**
Define your system of differential equations that models the infectious disease. You Kayode Bolaji can use standard compartmental models like the SIR model or SEIR model, depending on your needs.
**2. Find Disease-Free Equilibrium (DFE):**
Set the right-hand side of your differential equations to zero to find the steady-state or equilibrium points. For the DFE, this means setting the infection compartments to zero.
**3. Find Endemic Equilibrium:**
To find the endemic equilibrium, set the right-hand side of your differential equations to zero and solve for the non-zero steady-state values.
**4. Calculate Basic Reproduction Number ($$R_0$$):**
The basic reproduction number is typically calculated as the spectral radius of the next-generation matrix. For a basic SIR model, $$R_0$$ is often calculated as the dominant eigenvalue of the matrix associated with the model.
**5. Perform Sensitivity Analysis:**
You Kayode Bolaji can perform sensitivity analysis by perturbing parameters and observing the effect on relevant outputs. Sensitivity of equilibrium points and basic reproduction number to parameters can be explored by varying the parameter values and observing the changes.
**6. Use Maple to Solve:**
Maple provides a range of functions for solving differential equations, finding eigenvalues, and performing algebraic manipulations. For example, you Kayode Bolaji can use the dsolve function to solve your differential equations and LinearAlgebra package for eigenvalues.
Here is a simplified example code using Maple syntax. Assume S, I, and R are compartments for susceptible, infected, and recovered individuals:
maple
restart;
# Define your system of differential equations
eq1 := diff(S(t), t) = -beta * S(t) * I(t);
eq2 := diff(I(t), t) = beta * S(t) * I(t) - gamma * I(t);
eq3 := diff(R(t), t) = gamma * I(t);
# Define parameters
beta := 0.1; # infection rate
gamma := 0.05; # recovery rate
# Find Disease-Free Equilibrium
dfe := solve([eq1 = 0, eq2 = 0, eq3 = 0], [S(t), I(t), R(t)]);
# Find Endemic Equilibrium
endemic_eq := solve([eq1 = 0, eq2 = 0, eq3 = 0], [S(t), I(t), R(t)], explicit);
# Calculate Basic Reproduction Number (R0)
next_gen_matrix := LinearAlgebra:-JacobianMatrix([eq1, eq2, eq3], [S(t), I(t), R(t)]);
R0 := abs(LinearAlgebra:-Eigenvalues(next_gen_matrix));
# Display results
dfe, endemic_eq, R0;

Keep in mind that for more complex models with 10 to 15 compartments, the equations and their solutions will be more intricate, and the approach might involve numerical methods. Ensure that you Kayode Bolaji have a clear understanding of the model you're working with and verify your results with domain-specific literature or experts.
Question
Long time a go, I came across a journal paper that describes the effect of vial or container diameter on the dispersion stability of nanofluid. It mentioned that, if three vials had the same height but with different diameters the dispersed particles behaviour in the suspension will be different. Technically, the authors fabricated their nanofluids in these different diameters containers and monitored their physical stability, where they found that there is link between the container diameter and the suspension stability.
I was wondering if you have come across this article and probably share it with me because I cannot seem to find it anywhere. Also, if their are any similar articles that discusses the container geometry effect on the nanofluids stability or thermophysical properties then kindly share them with me.
Thank you very much in advance.
Naser
The dependence of the diameter of nanoparticles and the stability of their dispersion is the basis of nanotechnology and is studied in colloid chemistry. This academic discipline has long been created and is the fundamental basis of nanotechnology. The smaller the nanoparticles (the smaller their diameter), the greater the specific surface area of the nanoparticles. The larger the specific surface, the greater the Gibbs energy of the surface of the dispersed system, the more unstable it is. Our nature is such that any system tends to spontaneously reduce the Gibbs energy of the surface, which means that nanoparticles will spontaneously aggregate, increase in diameter and settle. A system with small particles is more unstable than a system with a large diameter. Read the collection of terms and concepts of colloidal chemistry.
Question
I am conducting k-means clustering analysis. My clusters separate very well. Anova, effect size and discriminant analysis results also support this separation. But I can't get efficient results from methods such as cluster stability index, jaccard coefficient, rand coefficient. My clusters are not stable. What can I do? Is it a necessary step clusters to be stable? Can you reccomend me a scientific arcticle or book to understand this well and reference it?
Thank you sincerely for your detailed answer! I have two more questions, I would be very grateful if you could answer them.
1. The Coefficient of Variation (CV) values of my clusters are quite good. The clusters are quite consistent within themselves. I work in the field of social sciences and when I examine the clusters crosswise with other variables, I get meaningful results. Discriminant analysis also indicates that my clusters are classified in a meaningful way, 90% correctly. Are these included in the internal stability factors? It that enough for validity and reliability?
2. I have tried Gaussian Mixture Models (GMM) instead of K-means, I think this algorithm more flexible. But there are even fewer resources on this topic and I am a social scientist not a computer scientist. Can you suggest me some resources on GMM that can help me interpret the validity and reliability of the resulting clusters?
Question
Hello Everyone,
I am trying to do a stability analysis of a gravity dam using sap2000. what is the best way to model the soil structure interaction using this software (The soil is rigid rock)? also, is there a possibility to model the concrete joints in sap2000?
Thank you
Modeling soil-structure interaction (SSI) in SAP2000 involves considering the interaction between the foundation and the surrounding soil. There are several methods that can be used to model SSI in SAP2000, including:
Finite Element Method (FEM): In this method, the soil is modeled using finite elements and the foundation is modeled using beam or plate elements. The interaction between the soil and the foundation is represented by interface elements that connect the soil and foundation elements.
Substructuring Method: In this method, the soil is modeled as a separate substructure and the foundation is modeled as another substructure. The two substructures are connected using interface elements that represent the interaction between the soil and foundation.
Equivalent Linearization Method: This method involves linearizing the non-linear soil behavior by dividing it into several linear segments. The foundation is modeled using linear elements and the interaction between the soil and foundation is represented by spring elements that connect the foundation to the soil.
To model SSI in SAP2000, the following steps can be followed:
Define the geometry and material properties of the soil and foundation.
Define the type of SSI method to be used (FEM, Substructuring, Equivalent Linearization).
Assign interface elements that represent the interaction between the soil and foundation.
Define the boundary conditions for the soil and foundation.
Apply loads to the structure and analyze the response.
Interpret the results and evaluate the performance of the foundation under SSI.
It is important to note that modeling SSI in SAP2000 can be complex and requires expertise in both structural engineering and geotechnical engineering. Therefore, it is recommended to consult with a qualified engineer or seek specialized training before attempting to model SSI in SAP2000.
Question
Hello all,
I have a simulink model for an autonomous microgrid which has no input and no output and I I would like to get the system poles at a certain operating point using the control system design tool in MATLAB. However when I start this tool with linearization, it demands specifying system inputs and outputs (There are currently no linearization I/Os marked on the model) to get the A B C D matrices which is not applicable in my model... So, how could I employ this tool to get the system modes ?
If your simulink model is based on differential and algebraic equations, then you can use 'linmod' MATLAB command to linearize it.
Question
This book, for example, deals with the dynamics of ordinary differential equations, is there anything similar to stochastic differential equations? Especially for infinite dimensional systems.
 Wiggins S, Golubitsky M. Introduction to applied nonlinear dynamical systems and chaos. New York: Springer, 2003.
Hi,
I suggest you see the links and attached file on the topic.
Best regards
Question
I prepared polymer blend of low and ultra high molecular weight Polyolefins. I need to know which techniques should I use to know if this polymer blends are stable over time and no phase separation will be happened over time. Is there any test may be high temperature aging or other accelerated test that I can conduct to see if there is any phase separation happens or they are stable over time. Also, which characterization method will be useful to characterize the phase separation of blends.
I believe that you should prove that they are compatible in the first place. Polyolefin compounds are rarely compatible in the classical sense. You should carefully analyse the DSC and XRD properties after various ambient storage times. Higher temperature annealing may change the crystalline properties (recrystallization and the like).
Question
Like MPFP, important sampling etc.
You should perform Monte Carlo simulations with at least 10,000 iterations at worst process corner and temperature.
For HSNM and RSNM: FS corner and 125 'C
For WSNM: SF corner and -40 'C
Question
NormFinder and RefFinder are algorithms designed to detect the stability of the reference genes in gene expression analyses.
NormFinder is an algorithm for identifying the optimal normalization gene among a set of candidates.
RefFinder is a user-friendly web-based comprehensive tool developed for evaluating and screening reference genes from extensive experimental datasets. It integrates the currently available major computational programs (geNorm, Normfinder, BestKeeper, and the comparative Delta-Ct method) to compare and rank the tested candidate reference genes. Based on the rankings from each program, It assigns an appropriate weight to an individual gene and calculated the geometric mean of their weights for the overall final ranking.
Question
Most dams are designed with traffic lanes that could be used by regular vehicules or mobile cranes for example. Why isn't this type of loading considered in the stability analysis of dams?
Thank you.
Hi Ilham.
It is of course an error not considering traffic loads. However, a typical distribute load will be of 10 to 20 KPa acting on the width of the road (7 to 10 m wide, normally), which is very small, when compared to the selfweight of the dam. Therefore, this load probably is often neglected for overall stability.
Question
In literature, many researchers recommend prob sonication for stability but in my observations bath sonication gives better stability to hybrid nanofluids. So is there any influence of the sonication method on stability? Does it have anything to do with the frequency of sonication?
and which method is best by your recomendations
Abdul Rehman One experiment I would recommend is taking 18 Meg ohm or DI water and subjecting it to a ultrasonic probe and measuring the conductivity over a period of time. You may be surprised by the results. Both probe and bath sonication have their issues. It all depends what you want to accomplish. See the following webinar (registration required):
Ultrasound, cavitation, and the singing kettle
Stability may not be related to sonication - the 3 steps in making a stable dispersion are wetting, separation (sonication usually in the wet) and stabilization (preventing recombination of separated particles). See:
PST & BDAS - An acronym approach to laser diffraction method development https://www.malvernpanalytical.com/en/learn/events-and-training/webinars/W190815PST
Question
I have some mutants that have produced atypical melt curves via just standard DSF with Sypro orange then nanoDSF looking at intrinsic fluorescence. Ideas of further experiments that could be used to probe this further and elucidate details of stability and what's going on with these proteins?
Dear @Gemma..
and what about the CBD? with sypro orange? Ir is show a normal slope?
From the 222 and CS6 spectra, that start with an high fluorescence alreraDY at low temperatures, it seems that your mutants are at least partially unfolded.
You can see an example of the DSF curve of a protein unfolded when trated with DTT at the minute 2:30 of the following video
pubblished on my blog, ProteoCool
I think that you can verify it in some alternative ways to confirm it:
1) check in SEC chromatography if they have different aggregation state
2) Run an 1D NMR spectra (unfolded proteins show lower peak dispertion)
3) Check the stability of the samples under concentration (If the protein is unfolded will tend to precipitate at lower concentration)
best regards
Manuele
Question
I am currently focusing on 3D geomechanical modeling. And in the future, I want to extend it to a 4D model. During my recent studies, I realized that most of the 4D geomechanical modeling that has been done has not properly updated the elastic properties such as Young's modulus, bulk modulus, Poisson ratio, etc. If a 3D static model is extended to a dynamic model, or a two-way or one-way coupling is performed, it is necessary to consider all material behaviors in a time-dependent manner. Please share if you have useful information in this regard or if you have a suggestion, I would be grateful if you could comment.
Hi Erfan Rahimi , in time-dependent 3D geomechanical simulation ("4D" coupled flow and geomechanical simulations) there a many interdependencies between material properties (e.g., elastic properties, strength properties, porosity/ permeability, fluid properties) and simulated fields (stress, strain, pore pressure).
You have to think carefully when including additional interdependencies (or vice versa NOT including these interdependencies), whether they add (i) a lot of additional complexity, (ii) create a lot of additional insight, (iii) create complexity without creating insight, (iv) the error incurred by failing to include the interdependency. As you correctly point out, updating elastic properties in the overburden is NOT commonly for coupled flow and geomechanical modelling. This is one of the cases where you make the simulations a lot more complex, without adding a lot of insight. Elastic properties due to stress/strain changes in the overburden change by less than a percent from their initial value - and updating the elastic properties will affect the simulated stress field by an amount which is insignificant compared to our ability to calibrate the stress field. In your picture, you show a loop which includes updating of velocities for 4D seismic attribute generation. Here updating the velocities (even by less than 1%) results in something we can observe in field data in the form of time-lapse timeshifts. In a similar manner, if permeabilities are stress dependent in a significant manner and neglecting to include this coupling will create a large error, it is customary to include this coupling. Another example of coupling which is sometimes, but not always, used is to use non-linear stress-strain relationships in the reservoir, if significant compaction occurs and the reservoir rock will experience irreversible compaction.
In summary, keep models as simple as possible, and add complexity if there is a good reason. Do not fall into the trap of making models "complex" for the sake of complexity. Complex models are harder to interpret, and don't necessarily provide more insight. There is an unfortunate tendency of assuming that "complex" models are "better" models. They sometimes are, and often are not.
Cheers,
Jorg
Question
Dear researchers
As we know, recently a new type of derivatives have been introduced which depend two parameters such as fractional order and fractal dimension. These derivatives are called fractal-fractional derivatives and divided into three categories with respect to kernel: power-law type kernel, exponential decay-type kernel, generalized Mittag-Leffler type kernel.
The power and accuracy of these operators in simulations have motivated many researchers for using them in modeling of different diseases and processes.
Is there any researchers working on these operators for working on equilibrium points, sensitivity analysis and local and global stability?
Thank you very much.
Best regards
Yes I am
Question
Hello, can someone help me in understanding how the variation in angle of attack effect the features of the transonic flow? I have read books on transonic flow, but they explain the effect of variation in mach number with transonic flow. How does tansonic features over a wing evolve with angle of attack ? What constitutes a good transonic wing ?
Hello dear Vinay, in transonic flow we have a complex mixing of subsonic, sonic and weakly supersonic behaviour, angle of attack variation induces unstable choc waves in appearance and position, that create unsteady boundary layer separation.
the common soulution is to use supercritic airfoils, weakly curved in extrados and slightly more curved in intrados, that's divide strong choc wave in many little weaker choc waves with less noise effect.
I try to find you good references and book titles.
Good luck.
Question
Hello,
Would you have an idea on how to model stable regular networks in a phase field model ?
For instance a square network like in the joined picture (the picture was made artificially) ? I know that the Swift-Hohenberg equation allows to create some regularity but I would like to control more the size of the network and the distance between the branches, and maybe the type of network.
Question
I am looking for alkali stable radical scavengers.
I have a glycol solution of 80:20 ethyl diglycol:butyl glycol.
And want to add 10% KOH (so a waterless alkaline).
I have tried spiking the glycol solution with up to 2% BHT (butylated hydroxy toluene) before adding the KOH. As BHT is commonly used as antioxidant (radical scavenger)
However, it still seems unstable.
I was wondering if there are any Alkali stable radical scavengers or anything to supress discoloration and further (aldol) condensation of the present glycols.
Question
Dears all
I would like to know what are the relationships among watershed health, watershed sustainability and watershed stability!
Thanks
Dear Dr. Douglas Nuttall
Thank you very much for the insightful description.
Regards
Gowhar Meraj
Question
Dear researchers
As you know, nowadays optimal control is one of the important criteria in studying the qualitative properties of a given fractional dynamical system. What is the best source and paper for introducing optimal control analysis for fractional dynamical models?
Thank you very much.
Question
Any suggestions on analytical and numerical solutions? Any available codes to solve Mathieu equation available other than in math work websites ?
The Mathieu equation has periodic solutions and therefore it is possible to use finite difference approximations to obtain an equivalent matrix eigenvalue problem. The matrix will of tridiagonal kind with an extra two coefficients, one top right and the other bottom left when second order accurate differences are used.
Also see:
Question
hello dear colleagues
I was working on a problem where I decide to merge sliding mode controller with homogeneous one. I was planning to define the sliding manifold based on homogeneous system of integrators.
Has anybody tried it out? are there any advantages?
best regards
You are welcome. In my opinion, I think such papers have not yet been contributed, because the topic is brand-new and needs more research contributions to be recently developed. You may be the one who contributes the first paper on this field and with respect to the finite-time stabilization, compare such controllers together, in terms of advantages and disadvantages.
Regards
Question
I am trying to evaluate MWCNT nanofluid (NF) stability by measuring absorbance by UV vis, however since the concentration I am interested in leads to an opaque solution, I get absorbance higher than the max recommended for UV measurements (I get around 5). Is there is any why to come around this without changing concentration? , would decreasing cuvette path length solve the problem?
Many thanks for the spectra from the reference. It looks like their spectra from the films spun out from their solutions. Your spectra are from solutions. They should be different if it is true. You could measure the spectra from your thin fims from your solutions to compare with. This is also important information for your research too. Please make sure that you use UV grade substrates.
Question
concentration: 1and 2 %w
Solvent: deionized water
Storage time: 1 or a few days
Also a layer of mold can be see on it.
Labile H are lost.
Question
The Van der Pol oscillator can be give in state model form as follows:
dx/dt = y
dy/dt = mu (1 - x^2) y - x,
where mu is a scalar parameter.
When mu = 0, the Van der Pol oscillator has simple harmonic motion. Its behavior is well-known.
When mu > 0, the Van der Pol oscillator has a stable limit cycle (with Hopf bifurcation).
While we can show the existence of a stable limit cycle with a MATLAB / SCILAB plot with some initial conditions and some positive value for mu like mu = 0.1 or 0.5 (for simulation), I like to know if there is a smart analytical proof (without any simulation) showing the existence of a limit cycle.
Specifically I like to know - is there any energy function V having time-derivative equal to zero along the trajectories of Van der Pol oscillator? Is there some smart calculation showing the existence of a stable limit cycle..
I am interested in knowing this - your help on my query is most welcome. Thanks!
The following steps are helpful for finding the dynamics of the system:
1. Check the stability of equilibrium by the Jacobian method.
2. Find the parametric conditions of stability/unstability.
3. Check what type of Hopf bifurcation (subcritical/supercritical) is there.
4. If the Hopf bifurcation is supercritical, then there is a stable limit cycle.
5. You can also solve the system numerically in Matlab/Mathematica and plot the limit cycle.
Question
I have a real stable system. However, when I try to reconstruct the state-space matrices of my system by using the subspace identification, it resulted in an unstable A matrix where its eigenvalues are located outside the unit circle.
I know that there are some ways to forced the A matrix to be stable. But it tends to give us a biased result since the stability is forced, not naturally identify as a stable system.
During the modeling process, it may be important to consider the relations of (input and output variables) , the governing differential modeling principles (including simplifications and linearity assumptions, steady-state conditions…), and after the model is obtained, see the properties of the eigenvalues of the system martrix (sign of roots, complex roots, if roots are bounded or not…), and or the poles of the transfer function T may be seen.
Given some system with matrix A find the characteristics polynomial of A.
Apply the “ Routh-Hurwitz” stability criterion on the polynomial.
(1) Negative eigenvalues lead to stable system
(2) Positive eigenvalues lead to unstable ones.
(3) Complex eigenvalues (may) tell us oscillatory conditions….
There may be other cases of considerations...
Thanks
Question
Please suggest if any specific software is used.
I am in a holidays, away from the lab computer to check the options of origin, but anyane who have the experience in using it can provide help. Sorry for being unable to help at this time period.
Question
Hi everyone,
As known, the buoyancy frequency is defined as N^2=g/rho*drho/dz. Normally, the positive or negative characteristics of N^2 is the same as the gradient of water density (drho/dz), depended on the water temperature vertical profiles. In summer, the temperature of surface water is always higher than the lower zone, so drho/dz is positive. However, in autumn and winter, the surface temperature decreases, the temperature of surface water becomes lower than the lower zone, so drho/dz should be negative. The paradox is N^2 is a square number and should be always positive. I'd like to know how to explain such paradox of negative density gradient (drho/dz) and always positive buoyancy frequency (N^2)? Thanks a lot.
Dear Pr. Carton,
Thank u very much for your clear answer, I understand your explanation well. In actuality, I am numerically solving a lake mixing model of methane, and I need to know the value of N2 to calculate the turbulence diffusivity (D), the equation as follows,
dC/dt = -Dd2C/dz2-r; r is the oxidation
In some articles, D is defined as D=8.94e-4[m^2/d]/(N2[1/s^2])^0.43. For this model, N2 cannot be a negative here. I don't know whether it is the problem of this model or sth. else. Thanks again.
Sincerely
Enze
Question
Hello
Why do we see the deposition of nanoparticles by adding different surfactants (cationic, anionic and non-ionic) with different concentrations to cobalt ferrite nanofluids?
Cobalt ferrite nanoparticles were synthesized by co-precipitation method.
I offer you our method for the synthesis of cobalt ferrite nanoparticles in a system of straight micelles. Its advantage: stability of dispersion, the formation of self-organized uniformly distributed nanoparticles as a result of a quantum-thermal phase transition.
Question
In the case of water splitting the water oxidation (OER) is the bottleneck for the process and because of the harsh conditions of the WO reaction, molecular catalysts with high TOFs are not stable in the matrix and quickly decompose. On the other hand heterogeneous catalysts are more stable but they often do not have high TOF quantities. Now what is your opinion about the future catalyst type? Will the ideal catalysts fabricated by turning the molecular catalysts to heterogeneous catalysts ( by loading the molecular catalysts on the solid surfaces ) or it will be from the single atom heterogeneous catalyst type?
Dear Sina Safavi thank you for initiating this interesting RG discussion. Although we are inorganic chemists, I'm not an expert in this field. However, I agree with Yurii V Geletii in that there is no such thing as a true catalyst. In this context I suggest that you have a look at the following relevant article which has been published Open Access. It is stated there that future catalysts should be "low-cost and earth-abundant":
Earth-Abundant Electrocatalysts for Water Splitting: Current and Future Directions
(see attached pdf file)
Question
Hi! I am trying to find some studies (or be educated) on the stability of the following groups in highly acidic (pH ~ 1-3) aqueous solutions of either an oxidizing acid (sulfuric or nitric) or reducing acid (HBr):
-Polycarbonate diol
-Polyacrylate (say, HEA or IBOA)
I have a UV curable coating with the above components (polyurethane prepolymer dissolved in an acrylate monomer) that will be exposed to various acidic solutions for up to 30 days at RT.
I am trying to determine where any reactions would take place and understand if the polymer would degrade. I don't think it will based off the backbone, but I am curious to learn the reasons why / why not.
Thanks!
Dear Michael Bay, hydrolytic hydrolysis is the problem. To overcome this problem, superhydrophobic surfaces are the most followed solution. Please check the following documents. My Regards
Question
Dear all,
I am trying to couple a simulation with two codes, say code A and code B (different discretizations). For code A we solve simply the conduction equation (solid), while in B we solve the NS + temperature convection/diffusion (fluid).
The coupling is explicit, while the time scheme for each code is implicit.
at each iteration, code A gives code B the heat flux to impose at the boundary of domain B, while code B gives code A the temperature to impose at the boundary of domain A. So in short, the coupling is done via a Neumann/Dirichlet BCs.
I am facing stability issues that don't allow me to finish correctly the transient solution.
Let us focus only on domain A (code A). When code A receives the temperature values from code B to apply a Dirichlet BC, the values are stocked in a layer of cells; so called the ghost cells. We denote these values as T_wall.
Now, we need to compute the heat flux at the boundary to give to code B. There are two possibilities to compute such a flux :
1- use T_wall and one inner temperature value
2- use 2 or 3 inner temperature values to calculate the flux, and then extrapolate it to the boundary.
I am getting serious instabilities if I calculate the flux via the first approach (T_wall and 1 inner temperature value). Otherwise, with the second approach (only inner temperature values), the simulation is stable and a steady-state is reached.
Can I have please your opinion on this subject. Have you ever faced such kind of problems ?
Is it really that computing the flux with the second approach leads to a stable solution, but to a non-consevative scheme ?
Best regards
Elie Saikali
Thanks for the answer and for the reference.
Regards
Question
I need to measure the CCT from rotor angle graph >> is it possible ? HOW?
it will be highly appreciated, if you advice any method to calculate the CCT
This is a trial an error process in a multi.machine system, first of all you need to know a clearing time for a stable operating condition and another time for an unstable condition, then you can try another time from such an interval. Should the resulting condition is unstable you have to reduce the clearing time and run the simulation again, in this way you are reducing the interval until you reach the CCT
Question
I need guidance on the analytical model and calculation for shape (cross-section) optimization of Euler Bernoulli cantilever column under buckling with a point load. It may be using variational methods, with the use of Rayleigh Quotient.
For it, I found a similar case in "Haftka, Raphael & Gurdal, Zafer & Kamat, Manohar. Elements of structural optimization. 2nd revised ed." but with different boundary conditions, I have attached the scans of the procedure used in the text for reference.
Question
Dear all,
I am trying to couple a simulation with two codes, say code A and code B (different discretizations). For code A we solve simply the conduction equation (solid), while in B we solve the NS + temperature convection/diffusion (fluid).
The coupling is explicit, while the time scheme for each code is implicit.
at each iteration, code A gives code B the heat flux to impose at the boundary of domain B, while code B gives code A the temperature to impose at the boundary of domain A. So in short, the coupling is done via a Neumann/Dirichlet BCs.
I am facing stability issues that don't allow me to finish correctly the transient solution.
Let us focus only on domain A (code A). When code A receives the temperature values from code B to apply a Dirichlet BC, the values are stocked in a layer of cells; so called the ghost cells. We denote these values as T_wall.
Now, we need to compute the heat flux at the boundary to give to code B. There are two possibilities to compute such a flux :
1- use T_wall and one inner temperature value
2- use 2 or 3 inner temperature values to calculate the flux, and then extrapolate it to the boundary.
I am getting serious instabilities if I calculate the flux via the first approach (T_wall and 1 inner temperature value). Otherwise, with the second approach (only inner temperature values), the simulation is stable and a steady-state is reached.
Can I have please your opinion on this subject. Have you ever faced such kind of problems ?
Is it really true that computing the flux with the second approach leads to a stable solution, but to a non-consevative scheme ?
Best regards
Elie Saikali
The stability condition is Fo Bi < 1 (CAST3M club presentation in 2019 at the CAST3M website by myself)... and serious references on the question :
Giles, M. B. "Stability analysis of numerical interface conditions in fluid–structure thermal analysis." International journal for numerical methods in fluids 25.4 (1997): 421-436.
Errera, Marc Paul, and Florent Duchaine, 2015. "STABLE AND FAST NUMERICAL SCHEMES FOR CONJUGATE HEAT TRANSFER."
Question
I have a question regarding STB in cadence should I set the operating point with STB analysis or not important? also, another question when I get the loop gain I found it below 0 dB what does it mean does that mean that there is no oscillation and the system is stable? … Read more
Dear Hagar,
All you need is to apply the Barkhausen criterion” for sustained oscillations.
It is so that the negative feedback amplifier will be stable so long mag. AB is smaller at -180 degrees phase shift and it may oscillate if mag. AB is greater than one at -180 degrees phase shift.
Accordingly, so the amplifier at the top figure is stable. While it will be unstable in the bottom figure. The probable point of oscillations is slightly to the left of M2.
You can test the stability by an input impulse, if the amplifier is stable the impulse will decay and it will grow to grow if the amplifier is unstable.
This means that you let the amplifier function as an oscillator.
Yes what do you said is okay.
But I see in the top figure, the amplifier shows zero response at specific frequencies.
Best wishes
Question
I have the water dispersion of WO3 and Ni(OH)2. Both the dispersions containing 2D nanosheets are stable for more than a month. Dispersions of the bulk materials are not stable though. There are no other additives/surfactants in the dispersion. Measured Zeta potential value is -2mV. How the dispersion is stable then? Is it because of the less mass of the nanosheet compared to bulk Or any other reason?
I do not believe you measured zeta potential (ZP) but some other property (e.g. electrophoretic or acoustic mobility) and you are using some/many assumptions in a conversion to ZP. This distinction is important because at higher concentrations, you will have particle-particle interactions and hence reduced mobility (which may be 'transformed' to a 'ZP' closer to 0 mV).
Why do you believe that ZP is a measure of 'stability'? [Whatever you mean by that] -echoing the comments of John Francis Miller .
Stability is conferred through charge (erroneously called electrostatic) or steric means. ZP is irrelevant to steric stability as the slipping plane is in the protective polymer 'coat'. Both stabilization mechanisms involve additives or change in pH. Hence DI water is about the worst material to dilute any system as it washes out/dilutes any stabilizing agents. The 'mother liquor' is the correct dilution medium.
Your comment 'bulk material is heavy and has higher chance of aggregation' is completely illogical. A density difference corresponds to settling propensity.
Pedantically I also cannot understand 'Dispersions of the bulk materials are not stable though. There are no other additives/surfactants in the dispersion. Measured Zeta potential value is -2mV'. You talk of 'dispersions' and 'bulk materials' (both plural) yet only have one single 'zeta potential' measurement (- 2 mV) - without indicating pH or other important parameters such as concentration. I would expect many 'ZP' measurements.
Question
We know that these elements are non-carbide formers. Therefore, C is more likely to be present in the solution which should make the austenite more stable. By this logic, adding these elements should make austenite stable up to an even lower temperature.
Dear Arka,
Good afternoon!
I always try to keep in mind that C is an interstitial alloying element while Al, Co and Si are substitutional alloying elements n Fe based alloys (like steels).
That means that Al, Co and Si will "take the place" of Fe atoms in the crystals, differently than C, B and N that will be in the interstitial positions in both austenite and ferrite crystals...
In Fe alloys there are only 4 gammagenic alloying elements, as follows: Ni, Mn, C and N. These 4 chemical elements are, ALWAYS, austenite formers in Fe alloys.
Si is a strong alphagenic element. It means it is a ferrite former.
I would as well keep in mind that both Al and Si have strong affinity with oxygen, been considered deoxidant elements in Fe based alloys.
What I can guarantee to you is that Co will always shift the TTT curve to the left... But I do not agree that Al and Si will do the same.
Co is a very interesting alloying element in Fe alloys. Keep in mind that Co is neighbour of Fe in the periodic table! I would tell you that the important effect of reduction of stacking-fault energy (SFE) that Co creates in Fe based alloys may be the answer to this particular effect that Co will always shift the TTT curve to the left... But it is something that is not 100% explained yet.
Very interesting question... Thank you!
I hope to help you. If it makes sense to you, I invite you to read some of my papers...
André
Question
I am still unsure about the relationship between BIBO and Lyapunov stability of simple undelayed LTI SISO systems.
Basic facts:
1) The system is STABLE if it has all system poles (eigenvalues) in the open left-half plane (LHP) or even single poles on the imaginary axis.
2) The system is ASYMPTOTICALLY or EXPONENTIALLY STABLE if it has all system poles (eigenvalues) in the open left-half plane (LHP).
3) The system is BIBO STABLE if it has all system poles (eigenvalues) in the open left-half plane. Or, the system is BIBO STABLE if its impulse function is absolutely integrable (i.e., it is L1-stable).
4) Btw., it is a fact that LTI SISO systems with DELAYS can have infinitely many poles in the LHP except for the complex infinity. Such systems are EXPONENTIALLY stable but they can/cannot be ASYMPTOTICALLY, Hinf or BIBO stable. Here, moreover, BIBO implies Hinf stability.
Notes:
- Some authors consider BIBO stability as a feature of the TRANSFER FUNCTION, not the SYSTEM itself. That is, there may exist unstable modes that cannot be seen at the output in the system. Therefore, every asymptotically Lyapunov stable system is BIBO, not vice-versa.
- I found also the idea in the literature that BIBO is stronger than asymptotical Lyapunov stability – however, I mean that this is incorrect.
Could anyone clearly explain me whether it exist any general relationship (inclusion, implication,…) between BIBO and (asymptotic) Lyapunov stability for SISO LTI delay-free systems, please?
Dear Libor,
I don’t know if you eventually found the answer to your question. I shall give my answer below.
The output y(t) of a system has two components: the free component y_L(t) generated by the initial conditions and the forced component y_F(t) generated by the input signal.
The forced component is the convolution between the causal impulse response h_C(t) (the inverse Laplace of the transfer function H(s)) and the input signal u(t) and has two sub-components: the transitory component y_T(t) related to the system’s structure (influenced by the signs of the poles) and the permanent component y_P(t) related to the input signal (influenced by the input’s poles).
If the linear system is exponential stable, then the free- and transitory components of the output response vanish as the time increase sufficiently, i.e. y_L(t) -> 0 and y_T(t) -> 0 as t-> infinity. Thus, the only component which remains is y_P(t) and moreover, for exponentially stabile systems the permanent component is of the same shape as the input signal as t-> infinity.
BIBO-stability reflects the input-output stability and it is evaluated on the transfer function, H(s). On the state space model one evaluates Lyapunov, asymptotic, exponential and other kind of stability properties.
Consequently:
- If the linear system is internally Lyapunov exponential stable, then for bounded inputs the output responses are bounded, i.e. THE SYSTEM IS BIBO-STABLE;
- If the linear system is internally unstable, but it is also non-minimal such that all the uncontrollable or the unobservable eigenvalues are those with positive sign (within the open right complex half -plane, inducing instability), then THE SYSTEM IS BIBO-STABLE. Explanation: taking into account that the transfer function obtained from the state-space model is always the irreducible, the unstable roots of the characteristic polynomial (poles of H(s)) will be simplified by the similar zeros of the nominator of H(s); consequently, H(s) will have only negative poles which means that the transitory component vanish as time tends to infinity and for bounded inputs the permanent component of the output response will be also bounded as determined by the poles of the input (the Laplace transform of u(t)).
- Conversely, if the transfer function has all its poles with negative real part, i.e. the system is BIBO stable, we cannot know if it is also Lyapunov exponentially stable (the characteristic polynomial can have uncontrollable or unobservable eigenvalues with positive real parts). From a stable transfer function one obtains a minimal state realization which is thus exponentially stable, but this will not reflect the internal state of the system which can be evaluated only on the system mathematical model of differential equations – i.e. the equations which result directly from the mathematical modeling process.
- A transfer function with all its poles in the open left complex half-plane has the impulse response absolutely integrable, i.e. it is L1-stable and vice-versa.
Question
I would like to store biological samples containing biotinylated proteins at -80 °C for several weeks, before continuing protein extraction and streptavidin affinity purification. I was wondering whether the biotin-protein bond is stable for long periods at -80 °C? Thank you for your advice.
The biotinylation is unlikely to be a problem, if your protein itself is stable under your conditions. Most proteins are, some are not (especially in complex mixtures where proteases may be present).
As Kiran-Kumar Shivaiah pointed out, freeze-thaw cycles can be a problem. Apart from his solution to freeze the sample in aliquots you may also consider storage at -20 °C in a glycerol-containing solution. At this temperature, solutions between 50 and 75% glycerol remain liquid, the eutectic is 66.7% with a freezing point of -46.5 °C. I have used this method to store labelled secondary antibodies and found it reliable and convenient.
Question
If an operating region is divided into a number of sectors and if it is shown that the system is stable in equilibrium point corresponding to each of the sectors, can it be said that the system is asymptotically stable in the entire operating region consisting of each of these sectors? Is there any theorem that can be cited to reinforce the premise?
Biswajit Debnath Please send me your problem I will try to check the complete stability and get back to you. I got through the concepts from measure theory book and relate to mathematical control on my own. I am sorry I don't have any references to send you. It is my understanding only.
Question
I'm using the Turbiscan Lab Instrument to assess shelf life and stability properties of my dispersions at different temperatures.
However, once analysing my data, I can only export the graphs into a pdf format.
I couldn’t find a method helping me access my raw data.
Are you aware of any tricks on the software to attain my raw data as a table enabling me to compare and analyse them?
The simplest method is to manually select and copy the scan data to an excel file. Inside the main Turbiscan software window, you can hold the 'ctrl' button, then left click on the time stamp of the scans you want to copy (usually on the right hand side of the screen.)
Then, right click, select 'copy', then paste them into an excel file. The result will be something like in the attached file. The first column is the height, then transmission, then backscattering data.
From there, I recommend using a Jupyter Notebook (with Python and Pandas) for plotting the data.
Question
I am working on postural stability during static standing position in condition of using mobile phone and without mobile.
I need literature an suggestion for methods.
Thnak you Dariusz...
and what you thinking on longitudinal monitoring of large group of participates via mobile app and using of data for science?
Dragan
Question
I'm currently do my research about stability for gait analysis by using Lyapunov Exponent. Can I calculate the Lyapunov exponent based on the ankle angle and ankle moment? Currently I'm still confusing with the the code steps on how to calculate the Lyapunov exponent. Can someone guide me. Thank you so much.
Yes, you can practically use any kinematic time series as long its long enough (at least 100 cycles per recording). For more info on the process the following paper is very detailed:
• Bruijn Sjoerd M.
and
• van Dieën Jaap H.
Control of human gait stability through foot placement15J. R. Soc. Interfacehttp://doi.org/10.1098/rsif.2017.0816
Question
I have seen unit cell designs that only perform at normal incidence while there are some designs that perform for a wide range of incidence angles. what is the reason or explanation behind this. what makes them angularly stable.
In a word, resonance.
Phase on the surface may be related to the phase of an incident plane wave via phase-matching principles and the constitutive relations between wave vector components. So then, observing a large range of incident angles is equivalent to observing a large range of wave vector (k) values on the surface.
If the phase incurred across the metasurface element has a strong frequency dependence, the metasurface will tend to only operate over a small range of angles, which correspond to the frequency/frequencies of operation of the surface mode. Alternatively, if the phase across the metasurface has a weak frequency dependence, it will tend to operate over a wide range of incident angles.
We can visualize these modes on a dispersion (k-Beta or w-k) diagram. Modes which have a strong frequency dependence vs angle will exist over a large portion of these diagrams with some k-dependent slope. Resonant modes, however, will exist as pretty much straight lines -- indicating that over all possible k values, the frequency band stays pretty much constant.
See Yang and Rahmat-Samii's book, "Electromagnetic Bandgap Structures in Antenna Engineering" (ISBN: 9780521889919), Chapter 2, for a more detailed description.
Question
As MRT express the decay rate (average time that biochar can persist from decaying) and this rate decreases with time. Therefore, the incubation period becomes pivotal for the MRT calculation because shorter duration may lead to higher estimated mineralization rate and shorter MRT. So, can we consider other means of getting justifiable stability period?
please suggest a methodology for above.
Dear Shiv,
you may check following publications to get some ideas on how to proceed:
Enders et al 2012 10.1016/j.biortech.2012.03.022
Spokas 2010 10.4155/cmt.10.32
Kuzyakov et al 2014 10.1016/j.soilbio.2013.12
Wang et al 2016 10.1111/gcbb.12266
Harvey et al 2012 10.1021/es2040398
and more...
BR
Nils
Question
In a diverse panel of genotypes, why some genotypes have high stability in certain traits (i.e. flowering time and yield) across multiple environments? In other words, why certain genotypes have consistent or similar phynotypic performance across different environments? Why other genotypes have low stability? Thanks for your help in advance!
Thanks Joel Ivan Cohen for the good answer to Jiang, its true some phenotypic traits are much heritable across genotypes or varieties and are influenced by the environmental interactions. Take an example of corn crop (similar genotype) tested across different zones that do pose different day length, it will mature at different days, and give different yield even when you provide same rates of fertilizers, water and other agronomic operations like seeding days, seeding rates, timely weeding. The cumulative degree growing days (CDGD) influence on growth stages, and so do to timed operations.
Testing the genotype across environments help you to reveal the hidden traits and also to test performance, and you can be able to recommend same genotype to different growing zones.@Cohen Ivan
Question
What if the pendulum angle set-point of a inverted pendulum (IP) is not zero? Have you ever tried it?
If the result is the same with the zero set-point , Does it mean that the system is controllable but not accessible?
the base position (with ref value at zero), and the base acceleration
Question
As attached in the picture, my emulsion contains Tween80 or food emulsifier occasionally. They both show very opposite but large magnitude of zeta potential. Can i consider this as stable emulsion? Or does anybody know how i could improve.
Tried concentration from 4% to 30% oil
How did you dilute your samples? Hopefully, not in DI water...
Question
I'm currently looking into the existing methods of estimating the lightship weight. In particular the estmation of outfitting weight is of interest to me, since it is very dependend on the type of vessel. Ultimately I would like to find out if it is possible to parameterize the outfitting weight for different vessel types.
I would very much like to hear your suggestions on literatur etc. or just your opinions.
Dear İlkay Özer Erselcan,
Thank you for your input. I will check those books out.
Does anyone have any practical information on how estimation and control of wheights (aka wheight management) is handled at a shipyard?
Question
I want to use NiCo2O4 as a substrate in 0.5 M H2SO4 solution. Just wondering will it be chemically stable in strong acidic solutions?
Question
Hey everyone,
I'm reading the "Stability- and performance-robustness tradeoffs: MIMO mixed.." and I need to simulate everything.
I would like to know if there is a space/place where the Matlab code has been shared, so that I can simulate the reported results as I need.
Is the paper code available anywhere?
If anyone could help, I would be immensely grateful!
You can find a lot of shared simulink simulations and Matlab codes here in Matlab File Exchange.
Question
Hi all.
I am looking for clarifications on the hexahedral mixed displacement-pressure elements.
1.) What exactly is the Taylor-Hood hexahedral element? Is it hex element with 20 nodes for displacement and 8 nodes for pressure or 27 nodes for displacement and 8 nodes for pressure? (Considering continuous interpolation for pressure.)
2.) Are both 20/8 or 27/8 elements inf-sup stable? If not, which one is inf-sup stable and which is not? Is there any good reference on this topic?
Thanks.
I would recommend Section 4.5.2.3 of the following book:
@book{reddy2010finite, title={The finite element method in heat transfer and fluid dynamics}, author={Reddy, Junuthula Narasimha and Gartling, David K}, year={2010}, publisher={CRC press}}
Here, it is said that both Q_2 Q_1 elements with 20 or 27 nodes for the velocities (for fluid dynamics) are LBB-stable.
Taylor-Hood elements often refer to [P_k]^d P_{k-1} or [Q_k]^d Q_{k-1} elements, where k is the polynomial order and d is the dimension.
But there are also other element types that are LBB-stable, e.g., [P_1-bubble]^2 P_1. See for example:
@article{suli2013brief, title={A brief excursion into the mathematical theory of mixed finite element methods}, author={S{\"u}li, ENDRE}, journal={Lecture Notes, University of Oxford}, pages={24--29}, year={2013}}
Hope that helps!
Question
If I have a MIMO system with a number of subsystems with moderate amount of coupling sensitivity and if those individual subsystems are BIBO stable then under what conditions the whole system can be considered to be BIBO stable ? Is there any necessary and sufficient conditions that needs to be satisfied ? An answer from the control theoretic point of view would be most appreciated.
Biswajit,
In that case you are trying out some very old concepts afresh !
Not very relevant today perhaps (that's just my opinion, of course !), because control concepts themselves are a lot more mature and evolved today !!
Have you checked out total stability concepts, for instance ?
Cheers !
-Sanjay
Question
We know that the balancing of load and generation cannot be exactly achieved in a power system. This means that the swing equation cannot be in the steady-state conditions. So in such conditions are the frequencies of a small generating unit and a large generating unit equal?
Yes, while operating large system, as experienced with proper time stamping measurements done, though average frequency for an integrated synchronous system has been found to be more or less same at each bus under 'so-called' steady-state operating condition (as inherently load changes from time-to-time and so the generation to match) due to sub-transient / transient condition (resulting from disturbances, faults, etc.) variations are not uncommon with coherency and inertia playing significant role.
Question
PEDOT:PSS is hygroscopic in nature and absorbs moisture from sorrounding environment. One more problem with PEDOT:PSS is it's acidic nature which affects ITO/HTL interface.These problems lead to degradation of perovskite material and hence stability of PSCs reduced. Now question arises, what are the best solutions to enhance the stability of PEDOT:PSS based PSCs?
Dear Hassan,
I think the solution is in using a metal oxide substitute for the PEDOT: PSS organic hole conducting material. It is found that NiOx is one of the suitable metal oxide hole conductors. For more information please follow the link: https://www.materialstoday.com/energy/news/niox-perovskite-solar-cells-promise-stability/
Best wishesd
Question
The independence property is defined by Shelah. I am looking for other versions or generalization of the definition in which some circumstances are changed.
Main reference (p.316, Def. 4.1):
Shelah, S. "Stability, the f.c.p. and superstability." Ann. Math. Logic 3, 271–362 (1971)
Another reference: (First Def. in introduction)
Gurevich, Yuri, and Peter H. Schmitt. "The theory of ordered abelian groups does not have the independence property." Transactions of the American Mathematical Society 284.1 (1984): 171-182.
Can you add a reference to the definition you have in mind?
Question
Generally it is and industry practice to conduct the accelerated shelf life study of foods products at a temperature of 40+/-5 degree Celsius and 70+/-5% RH. Is there any authentic document which we can refer for this?
Question
Spectral methods are one of the good tools to find the solution of the PDEs,
I have a parabolic PDE like
dP/dt=f(P, dP/dx1, dP/dx2, d^2P/dx1^2,..)
where f could be
f=a(x1,x2)*dP/dx1 +b (x1,x2)* d^P/dx2+c(x1,x2)*P
I formulated a 2D PDE, but the dimensions could be higher than 2.
a formal way in spectral method to solve the PDE is to solve the eigenvalue problem for operator f like:
f(phi_j)=lambda_j * phi_j
where phi_j is eigenfunctions (preferably orthogonal) and lamda_j are corresponding eigenvalues.
this way we can find the solution of parabolic PDE as well as analyze the stability of the solution based on the eigenvalues.
I should note that the boundary condition might be in Dirichlet , Neumman or Robbin form.
is there any general method to solve this eigenvalue problem and find the eigenfunctions and eignevalues of this general operator?
Unfortunately the things are much more complex. The Lax theorem is valid for linear schemes but if you look to a proof of convergence for the numerical solution of non-linear equations this theorem cannot guarantee convergence.
For a non-linear formulation, Lax and Wendroff theorem states that if a conservative scheme converges it does towards a weak solution. That means we must previously ensure that convergence exists. (see the texbook of Leveque, page 239)
Question
I need literature how set-up for markers for 3D analysis of upper body stability with Vicon ( 9 cameras stereophotogrammetric system). Suggestion and advice please.
Hi Dragan, E. Jaspers and E. Butler have both created an upper limb model to be applied in Vicon. The protocol developed by E. Jaspers has an open-source software toolbox (with matlab) which can be found online.
Question
I am trying to test some algebraic theory using many different Runge-Kutta schemes, with varying order, number of stages, stability, etc. Does anybody have a reference with a coherent list of lots of established RK schemes and their stability functions? I have pieced together a good number from various books and literature, but an encyclopedia of sorts would be very useful.
Ben
John Butcher developed the stability theory for Runge-Kutta methods and suspect his book(s) have the Butcher tableau's for many orders and cases.
And it does not require Dr. Dodd's pills like certain matrices.
Question
Hi,
Let us say that a plant model of the following form is given:
\dot{q} = A*q + B*u,
y = C*q.
The output of the model (y) is its position (x) and velocity (v).
Using this output, I am going to construct a PI controller (u) of the following form:
u = K1*x + K2*v.
Then, the task is to find the optimal gains (K1) and (K2) such that some given cost function is minimized.
Without any constraints, this problem can be solved easily using the MATLAB command 'fminsearch' or 'fminunc.'
But I'd like to add a stability constraint such that the closed-loop system with PI control is always stable.
Mathematically, it is realized by constraining the eigenvalues of the closed-loop system in the left-hand plane.
But I'm wondering how to add this constraint using MATLAB commands.
I tried to use the command 'eig' in the constraint function (nonlcon) of 'fmincon' but MATLAB says that command is not allowed to be used in the fmincon constraint function.
Any comments will be greatly appreciated.
Thank you.
If the system is second order and the cost function is quadratic then It is also possible to solve suitable linear quadratic (LQ) problem.
Question
The bifurcation curve generated in MatCont is displayed in the same color(blue). The stability of the equilibrium or limit cycle can only be tracked in the layout of "numeric" in "Window" by Eigenvalues or Multipliers. Does anyone know how to display the stability of the curve graphically by color or any other ways(e.g. line style)?
We are going to release a new version that supports this functionality, expect something like March 2019.
Question
I am trying to use Model Reference Adaptive Control (MRAC) based on Lyapunov's rule on a system. My question is: should the reference model be assumed or is there a systematic procedure for determining its parameters in relation to the dynamism of the system in question.
Adapting/adjusting the controller parameters for such cases (where plant parameters were not accurately known) is called adaptive control. so as it can be infer, you should have an acceptable approximation of your dynamic system to tune the adaptive control parameters.
best regards
Question
In a simplest case imagine we have a continuous finite-dimensional dynamic system described by and ODE
x_dot=f(x) (1) ,
Is it possible to prove the asymptotic stability of (1) by investigating the discrete time version of (1)
x(k+1)=F(x(k)) (2)
((2) might be written by RK4 discretization, Euler first order descretization or any other descretization scheme)
if it is possible to do that, what about infinite dimensional system resulting from space descritization of PDEs?
Dear Ali,
your question is interesting. In general, the discrete version will depend on two fundamental choices: the discretization scheme used (there are several) and the discretization time. Notice that it is typicaly not sufficient just to replace t for k and \dot{x(t)} with x(k+1). The stability of discretized models will usually depend on the discretization time. This feature is obviously beyond the continuous-time counterpart.
Question
in most cases for continuous time stochastic systems which are modeled by SDE, the Lyapunov stability conditions can guarantee the stochastic stability of the system,
another definition In stochastic literature is detailed balance which guarantee the convergence of the probability of the states of SDE to a stationary probability density.
I want to know which one is more strong stability condition?
With Markov chains having a detailed balance distribution is a stronger condition then having a stationary distribution (that you can check by Lyapunov theory).
Every detailed balance distribution is stationary, but not all stationary distributions are detailed balanced. The proof is very easy. Detail balance means there exist a distribution pi(k) such that
pi(k)q_kj =pi(j)q_jk,
where q_jk is any non-diagonal entry of the generator matrix Q.
Now if on both sides you take the sum over all the k that are different from j you get that stationarity follows (pi Q=0).
You can check it on Durret's book Essentials of stochastic processes, page 130
I'm not sure what happens with diffusions, but it shuold be the same.
Question
In the theory of the stability of the differential operators, one could prove the stability results based on spectra of an operator, (all eigenvalues must be negative for example).
one problem with the above method is that not all linear operators are self-adjoint (for examples operators in convection diffusion form) and their corresponding eigenvalue problem can not be solved analytically, hence spectra of the operator can not be calculated analytically. On the other hand there is a definition related to spectra, which is called pseudo-spectra, that somehow evaluates the approximated spectrum , even for non self-adjoint operators.
I want to know is it possible to establish stability results for a differential operator based on pseudo-spectra?
The question you ask is a very broad one. Since differential operators are unbounded - the definition of the on which one defines the operator is important. When one goes to infinite dimensions, there might not even be a point spectrum (eigenvalues) - the spectrum may be only the continuous spectrum and residual spectrum. If the operator has a self joint extension or a normal extension then the problem becomes more tractable but for non-normal operators the problem is quite complex.
At one time a class of operators "spectral operators" were defined, see Dunford and Schwartz, "Linear Operators", vol III, to address the issues on non-normal operators. Proving a non-normal operator - even an ordinary differential operator - was not a trivial task. The goal of this effort was to expand the concept of Jordan Canonical form to first bounded linear operators on a Hilbert/Banach space and then to unbounded linear operators, e.g., differential operators. For compact operators or operators with compact resolvents - that has been solved. For bounded normal and unitary operators, that has been solved (spectral theorem). For unbounded operators with a compact resolvent, that has been pretty much solved (by using the spectral theorem the spectral theorem). For unbounded self adjoint operators, there is also a spectral theorem. However, for non-normal bounded operators and non-self adjoint unbounded operators - there is no general result on a canonical form.
In these problems stability questions be quite problem specific.
Question
Is there any scientific data avaibale on the AAV vector stability in 15-20% Tween 20 solution? Also any information on the AAV9 aggregation behaviour at 6E+13 concentration range?
Hi Navneet，an AAV at the titer of 6E+13 would not aggregate, we can produce as high as 1E+15, no aggregation was observed.
You could find more on the website: www.genemedi.net/i/aav-packaging
Question
Could there be nonlinear ODEs that could not even be solved numerically?
I am working on a third-order nonlinear problem which is giving accurate results for a given set of initial conditions (ICs) but for other ICs the matlab ODE solver (checked 4-5 algorithms) do not converge.