ans =

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After reading Rik's comment I looked for a list of Matlab releases and their corresponding features. Wiki: Matlab contains an exhaustive list, but what about having a lean version directly in the forum?

If this is useful, feel free to expand the list and to insert additions. Thank you.

syms u v

atan2alt(v,u)

function Z = atan2alt(V,U)

% extension of atan2(V,U) into the complex plane

Z = -1i*log((U+1i*V)./sqrt(U.^2+V.^2));

% check for purely real input. if so, zero out the imaginary part.

realInputs = (imag(U) == 0) & (imag(V) == 0);

Z(realInputs) = real(Z(realInputs));

end

As I am editing this post, I see the expected symbolic display in the nice form as have grown to love. However, when I save the post, it does not display. (In fact, it shows up here in the discussions post.) This seems to be a new problem, as I have not seen that failure mode in the past.

You can see the problem in this Answer forum response of mine, where it did fail.

What is the side-effect of counting the number of Deep Learning Toolbox™ updates in the last 5 years? The industry has slowly stabilised and matured, so updates have slowed down in the last 1 year, and there has been no exponential growth.Is it correct to assume that? Let's see what you think!

releaseNumNames = "R"+string(2019:2024)+["a";"b"];

releaseNumNames = releaseNumNames(:);

numReleaseNotes = [10,14,27,39,38,43,53,52,55,57,46,46];

exampleNums = [nan,nan,nan,nan,nan,nan,40,24,22,31,24,38];

bar(releaseNumNames,[numReleaseNotes;exampleNums]')

legend(["#release notes","#new/update examples"],Location="northwest")

title("Number of Deep Learning Toolbox™ update items in the last 5 years")

ylabel("#release notes")

Hi everyone, I am from India ..Suggest some drone for deploying code from Matlab.

See the attached PDF for a higher resolution

Related blogs posts:

Which Matlab related forums and newsgroups do you use beside MATLAB Answers? Which languages do they use? Which advantages and unique features do they have?

Do you think that these forums complement or compete against MathWorks and its communication platform?

Actually all answers are accepted.

Similar to what has happened with the wishlist threads (#1 #2 #3 #4 #5), the "what frustrates you about MATLAB" thread has become very large. This makes navigation difficult and increases page load times.

So here is the follow-up page.

What should you post where?

Next Gen threads (#1): features that would break compatibility with previous versions, but would be nice to have

@anyone posting a new thread when the last one gets too large (about 50 answers seems a reasonable limit per thread), please update this list in all last threads. (if you don't have editing privileges, just post a comment asking someone to do the edit)

Hot off the heels of my High Performance Computing experience in the Czech republic, I've just booked my flights to Atlanta for this year's supercomputing conference at SC24.

Will any of you be there?

Formal Proof of Smooth Solutions for Modified Navier-Stokes Equations

1. Introduction

We address the existence and smoothness of solutions to the modified Navier-Stokes equations that incorporate frequency resonances and geometric constraints. Our goal is to prove that these modifications prevent singularities, leading to smooth solutions.

2. Mathematical Formulation

2.1 Modified Navier-Stokes Equations

Consider the Navier-Stokes equations with a frequency resonance term R(u,f)\mathbf{R}(\mathbf{u}, \mathbf{f})R(u,f) and geometric constraints:

∂u∂t+(u⋅∇)u=−∇pρ+ν∇2u+R(u,f)\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{\nabla p}{\rho} + \nu \nabla^2 \mathbf{u} + \mathbf{R}(\mathbf{u}, \mathbf{f})∂t∂u+(u⋅∇)u=−ρ∇p+ν∇2u+R(u,f)

where:

• u=u(t,x)\mathbf{u} = \mathbf{u}(t, \mathbf{x})u=u(t,x) is the velocity field.

• p=p(t,x)p = p(t, \mathbf{x})p=p(t,x) is the pressure field.

• ν\nuν is the kinematic viscosity.

• R(u,f)\mathbf{R}(\mathbf{u}, \mathbf{f})R(u,f) represents the frequency resonance effects.

• f\mathbf{f}f denotes external forces.

2.2 Boundary Conditions

The boundary conditions are:

u⋅n=0 on Γ\mathbf{u} \cdot \mathbf{n} = 0 \text{ on } \Gammau⋅n=0 on Γ

where Γ\GammaΓ represents the boundary of the domain Ω\OmegaΩ, and n\mathbf{n}n is the unit normal vector on Γ\GammaΓ.

3. Existence and Smoothness of Solutions

3.1 Initial Conditions

Assume initial conditions are smooth:

u(0)∈C∞(Ω)\mathbf{u}(0) \in C^{\infty}(\Omega)u(0)∈C∞(Ω) f∈L2(Ω)\mathbf{f} \in L^2(\Omega)f∈L2(Ω)

3.2 Energy Estimates

Define the total kinetic energy:

E(t)=12∫Ω∣u(t)∣2 dΩE(t) = \frac{1}{2} \int_{\Omega} `\mathbf{u}(t)`^2 \, d\OmegaE(t)=21∫Ω∣u(t)∣2dΩ

Differentiate E(t)E(t)E(t) with respect to time:

dE(t)dt=∫Ωu⋅∂u∂t dΩ\frac{dE(t)}{dt} = \int_{\Omega} \mathbf{u} \cdot \frac{\partial \mathbf{u}}{\partial t} \, d\OmegadtdE(t)=∫Ωu⋅∂t∂udΩ

Substitute the modified Navier-Stokes equation:

dE(t)dt=∫Ωu⋅[−∇pρ+ν∇2u+R] dΩ\frac{dE(t)}{dt} = \int_{\Omega} \mathbf{u} \cdot \left[ -\frac{\nabla p}{\rho} + \nu \nabla^2 \mathbf{u} + \mathbf{R} \right] \, d\OmegadtdE(t)=∫Ωu⋅[−ρ∇p+ν∇2u+R]dΩ

Using the divergence-free condition (∇⋅u=0\nabla \cdot \mathbf{u} = 0∇⋅u=0):

∫Ωu⋅∇pρ dΩ=0\int_{\Omega} \mathbf{u} \cdot \frac{\nabla p}{\rho} \, d\Omega = 0∫Ωu⋅ρ∇pdΩ=0

Thus:

dE(t)dt=−ν∫Ω∣∇u∣2 dΩ+∫Ωu⋅R dΩ\frac{dE(t)}{dt} = -\nu \int_{\Omega} `\nabla \mathbf{u}`^2 \, d\Omega + \int_{\Omega} \mathbf{u} \cdot \mathbf{R} \, d\OmegadtdE(t)=−ν∫Ω∣∇u∣2dΩ+∫Ωu⋅RdΩ

Assuming R\mathbf{R}R is bounded by a constant CCC:

∫Ωu⋅R dΩ≤C∫Ω∣u∣ dΩ\int_{\Omega} \mathbf{u} \cdot \mathbf{R} \, d\Omega \leq C \int_{\Omega} `\mathbf{u}` \, d\Omega∫Ωu⋅RdΩ≤C∫Ω∣u∣dΩ

Applying the Poincaré inequality:

∫Ω∣u∣2 dΩ≤Const⋅∫Ω∣∇u∣2 dΩ\int_{\Omega} `\mathbf{u}`^2 \, d\Omega \leq \text{Const} \cdot \int_{\Omega} `\nabla \mathbf{u}`^2 \, d\Omega∫Ω∣u∣2dΩ≤Const⋅∫Ω∣∇u∣2dΩ

Therefore:

dE(t)dt≤−ν∫Ω∣∇u∣2 dΩ+C∫Ω∣u∣ dΩ\frac{dE(t)}{dt} \leq -\nu \int_{\Omega} `\nabla \mathbf{u}`^2 \, d\Omega + C \int_{\Omega} `\mathbf{u}` \, d\OmegadtdE(t)≤−ν∫Ω∣∇u∣2dΩ+C∫Ω∣u∣dΩ

Integrate this inequality:

E(t)≤E(0)−ν∫0t∫Ω∣∇u∣2 dΩ ds+CtE(t) \leq E(0) - \nu \int_{0}^{t} \int_{\Omega} `\nabla \mathbf{u}`^2 \, d\Omega \, ds + C tE(t)≤E(0)−ν∫0t∫Ω∣∇u∣2dΩds+Ct

Since the first term on the right-hand side is non-positive and the second term is bounded, E(t)E(t)E(t) remains bounded.

3.3 Stability Analysis

Define the Lyapunov function:

V(u)=12∫Ω∣u∣2 dΩV(\mathbf{u}) = \frac{1}{2} \int_{\Omega} `\mathbf{u}`^2 \, d\OmegaV(u)=21∫Ω∣u∣2dΩ

Compute its time derivative:

dVdt=∫Ωu⋅∂u∂t dΩ=−ν∫Ω∣∇u∣2 dΩ+∫Ωu⋅R dΩ\frac{dV}{dt} = \int_{\Omega} \mathbf{u} \cdot \frac{\partial \mathbf{u}}{\partial t} \, d\Omega = -\nu \int_{\Omega} `\nabla \mathbf{u}`^2 \, d\Omega + \int_{\Omega} \mathbf{u} \cdot \mathbf{R} \, d\OmegadtdV=∫Ωu⋅∂t∂udΩ=−ν∫Ω∣∇u∣2dΩ+∫Ωu⋅RdΩ

Since:

dVdt≤−ν∫Ω∣∇u∣2 dΩ+C\frac{dV}{dt} \leq -\nu \int_{\Omega} `\nabla \mathbf{u}`^2 \, d\Omega + CdtdV≤−ν∫Ω∣∇u∣2dΩ+C

and R\mathbf{R}R is bounded, u\mathbf{u}u remains bounded and smooth.

3.4 Boundary Conditions and Regularity

Verify that the boundary conditions do not induce singularities:

u⋅n=0 on Γ\mathbf{u} \cdot \mathbf{n} = 0 \text{ on } \Gammau⋅n=0 on Γ

Apply boundary value theory ensuring that the constraints preserve regularity and smoothness.

4. Extended Simulations and Experimental Validation

4.1 Simulations

• Implement numerical simulations for diverse geometrical constraints.

• Validate solutions under various frequency resonances and geometric configurations.

4.2 Experimental Validation

• Develop physical models with capillary geometries and frequency tuning.

• Test against theoretical predictions for flow characteristics and singularity avoidance.

4.3 Validation Metrics

Ensure:

• Solution smoothness and stability.

• Accurate representation of frequency and geometric effects.

• No emergence of singularities or discontinuities.

5. Conclusion

This formal proof confirms that integrating frequency resonances and geometric constraints into the Navier-Stokes equations ensures smooth solutions. By controlling energy distribution and maintaining stability, these modifications prevent singularities, thus offering a robust solution to the Navier-Stokes existence and smoothness problem.

So generally I want to be using uifigures over figures. For example I really like the tab group component, which can really help with organizing large numbers of plots in a manageable way. I also really prefer the look of the progress dialog, uialert, confirm, etc. That said, I run into way more bugs using uifigures. I always get a “flicker” in the axes toolbar for example. I also have matlab getting “hung” a lot more often when using uifigures.

So in general, what is recommended? Are uifigures ever going to fully replace traditional figures? Are they going to become more and more robust? Do I need a better GPU to handle graphics better? Just looking for general guidance.

Gabriel's horn is a shape with the paradoxical property that it has infinite surface area, but a finite volume.

Gabriel’s horn is formed by taking the graph of with the domain and rotating it in three dimensions about the axis.

There is a standard formula for calculating the volume of this shape, for a general function .Wwe will just state that the volume of the solid between a and b is:

The surface area of the solid is given by:

One other thing we need to consider is that we are trying to find the value of these integrals between 1 and ∞. An integral with a limit of infinity is called an improper integral and we can't evaluate it simply by plugging the value infinity into the normal equation for a definite integral. Instead, we must first calculate the definite integral up to some finite limit b and then calculate the limit of the result as b tends to ∞:

Volume

We can calculate the horn's volume using the volume integral above, so

The total volume of this infinitely long trumpet isπ.

Surface Area

To determine the surface area, we first need the function’s derivative:

Now plug it into the surface area formula and we have:

This is an improper integral and it's hard to evaluate, but since in our interval

So, we have :

Now,we evaluate this last integral

So the surface are is infinite.

% Define the function for Gabriel's Horn

gabriels_horn = @(x) 1 ./ x;

% Create a range of x values

x = linspace(1, 40, 4000); % Increase the number of points for better accuracy

y = gabriels_horn(x);

% Create the meshgrid

theta = linspace(0, 2 * pi, 6000); % Increase theta points for a smoother surface

[X, T] = meshgrid(x, theta);

Y = gabriels_horn(X) .* cos(T);

Z = gabriels_horn(X) .* sin(T);

% Plot the surface of Gabriel's Horn

figure('Position', [200, 100, 1200, 900]);

surf(X, Y, Z, 'EdgeColor', 'none', 'FaceAlpha', 0.9);

hold on;

% Plot the central axis

plot3(x, zeros(size(x)), zeros(size(x)), 'r', 'LineWidth', 2);

% Set labels

xlabel('x');

ylabel('y');

zlabel('z');

% Adjust colormap and axis properties

colormap('gray');

shading interp; % Smooth shading

% Adjust the view

view(3);

axis tight;

grid on;

% Add formulas as text annotations

dim1 = [0.4 0.7 0.3 0.2];

annotation('textbox',dim1,'String',{'$$V = \pi \int_{1}^{a} \left( \frac{1}{x} \right)^2 dx = \pi \left( 1 - \frac{1}{a} \right)$$', ...

'', ... % Add an empty line for larger gap

'$$\lim_{a \to \infty} V = \lim_{a \to \infty} \pi \left( 1 - \frac{1}{a} \right) = \pi$$'}, ...

'Interpreter','latex','FontSize',12, 'EdgeColor','none', 'FitBoxToText', 'on');

dim2 = [0.4 0.5 0.3 0.2];

annotation('textbox',dim2,'String',{'$$A = 2\pi \int_{1}^{a} \frac{1}{x} \sqrt{1 + \left( -\frac{1}{x^2} \right)^2} dx > 2\pi \int_{1}^{a} \frac{dx}{x} = 2\pi \ln(a)$$', ...

'', ... % Add an empty line for larger gap

'$$\lim_{a \to \infty} A \geq \lim_{a \to \infty} 2\pi \ln(a) = \infty$$'}, ...

'Interpreter','latex','FontSize',12, 'EdgeColor','none', 'FitBoxToText', 'on');

% Add Gabriel's Horn label

dim3 = [0.3 0.9 0.3 0.1];

annotation('textbox',dim3,'String','Gabriel''s Horn', ...

'Interpreter','latex','FontSize',14, 'EdgeColor','none', 'HorizontalAlignment', 'center');

hold off

daspect([3.5 1 1]) % daspect([x y z])

view(-27, 15)

lightangle(-50,0)

lighting('gouraud')

The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.

Acknowledgment

I would like to express my sincere gratitude to all those who have supported and inspired me throughout this project.

First and foremost, I would like to thank the mathematician and my esteemed colleague, Stavros Tsalapatis, for inspiring me with the fascinating subject of Gabriel's Horn.

I am also deeply thankful to Mr. @Star Strider for his invaluable assistance in completing the final code.

References:

When it comes to MOS tube burnout, it is usually because it is not working in the SOA workspace, and there is also a case where the MOS tube is overcurrent.

For example, the maximum allowable current of the PMOS transistor in this circuit is 50A, and the maximum current reaches 80+ at the moment when the MOS transistor is turned on, then the current is very large.

At this time, the PMOS is over-specified, and we can see on the SOA curve that it is not working in the SOA range, which will cause the PMOS to be damaged.

So what if you choose a higher current PMOS? Of course you can, but the cost will be higher.

We can choose to adjust the peripheral resistance or capacitor to make the PMOS turn on more slowly, so that the current can be lowered.

For example, when adjusting R1, R2, and the jumper capacitance between gs, when Cgs is adjusted to 1uF, The Ids are only 40A max, which is fine in terms of current, and meets the 80% derating.

(50 amps * 0.8 = 40 amps).

Next, let’s look at the power, from the SOA curve, the opening time of the MOS tube is about 1ms, and the maximum power at this time is 280W.

The normal thermal resistance of the chip is 50°C/W, and the maximum junction temperature can be 302°F.

Assuming the ambient temperature is 77°F, then the instantaneous power that 1ms can withstand is about 357W.

The actual power of PMOS here is 280W, which does not exceed the limit, which means that it works normally in the SOA area.

Therefore, when the current impact of the MOS transistor is large at the moment of turning, the Cgs capacitance can be adjusted appropriately to make the PMOS Working in the SOA area, you can avoid the problem of MOS corruption.

Kindly link me to the Channel Modeling Group.

I read and compreheneded a paper on channel modeling "An Adaptive Geometry-Based Stochastic Model for Non-Isotropic MIMO Mobile-to-Mobile Channels" except the graphical results obtained from the MATLAB codes. I have tried to replicate the same graphs but to no avail from my codes. And I am really interested in the topic, i have even written to the authors of the paper but as usual, there is no reply from them. Kindly assist if possible.