Stability - Science topic

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))[1];

# Display results

dfe, endemic_eq, R0;

```

Remember to adapt this code to your specific model and parameters.

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.

Naser Ali

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.

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?

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.

If your simulink model is based on differential and algebraic equations, then you can use 'linmod' MATLAB command to linearize it.

Hi,

I suggest you see the links and attached file on the topic.

Best regards

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).

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

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.

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.

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

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

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.

Hope this makes sense, and addresses your question.

Cheers,

Jorg

Yes I am

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.

https://www.nature.com/articles/srep01865?proof=t%29.

Good luck.

Dear Dr. Douglas Nuttall

Thank you very much for the insightful description.

Regards

Gowhar Meraj

Dear Dr. @Muhammad Ali

Thank you very much.

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:

Dear Alireza Hosseinnajad

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

@Muhammad Kaleem,

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.

Labile H are lost.

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.

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

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.

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

Zeinab Sadooghi

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.

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":

(see attached pdf file)

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

Thanks for the answer and for the reference.

Regards

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

Have a look at

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."

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

Priyabrata Sahoo

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.

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...

And join our research team! You are very welcome!

André

Libor Pekař

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.

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.

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.

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.

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

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:

and

Control of human gait stability through foot placement15J. R. Soc. Interfacehttp://doi.org/10.1098/rsif.2017.0816

Syed Hussain Ali Bokhari ,

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.

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

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

the base position (with ref value at zero), and the base acceleration

Lok Kin Chuen

How did you dilute your samples? Hopefully, not in DI water...

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?

ARTICLES YOU MAY BE INTERESTED IN,

If you are working in the feild of Spinel Ferrites

You can find a lot of shared simulink simulations and Matlab codes here in Matlab File Exchange.

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!

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

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.

Dear Hassan,

Ramadan Kareem!

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

Can you add a reference to the definition you have in mind?

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)

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.

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.

If the system is second order and the cost function is quadratic then It is also possible to solve suitable linear quadratic (LQ) problem.

We are going to release a new version that supports this functionality, expect something like March 2019.

Dear Demilade . . .

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.

recently i have read a paper about this subject which i want to suggest it to you to read it. its about how to design the law for your MRAC.

it might be helpful.

best regards

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.

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.

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.

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

If your equation/system of equations is prone to stiffness or strongly nonlinear the ode 45 solver may not be adapted for some ICs because it uses a fully explicit numerical scheme. This page can help you to chose better adapted solvers: https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html

If none of these solvers works then you may need to write yourself a numerical scheme adapted to your case, probably an implicit or semi-implicit scheme (more stable than explicit schemes) since matlab solvers don't work.

Dear Dr. Cuong Dinh Nguyen

Greating!

You can read the following links that may be useful to answer

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With best wishes