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

Gabrielâ€™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 âˆž. 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 âˆž:

Volume

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

The total volume of this infinitely long trumpet isÏ€.

Surface Area

To determine the surface area, we first need the functionâ€™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:

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.

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.

Dear MATLAB community,

How can I help my close friend who's bad at math and programming learn MATLAB?

He's a final year chemical engineering student who struggles even to plot two functions on the same graph in his computational fluid dynamics class (there was no prereq for matlab skills).

In his first year, I saw him get dragged through the introductory engineering classes which was his first encounter with MATLAB. Students were taught a few rudimentary programming skills and then were expected to make a code for a 'simple' tic-tac-toe game. It took him hours of blank looks and tutoring to even understand the simplest of boolean operators. He was never able to write a working function without the supervision of a friend or tutor. Needless to say, he was permanently scarred by the experience and swore to avoid using it forever.

After 3 years of avoiding MATLAB, he realised how not knowing it hurt him during his final year project. He had to solve a system of pdes to model the performance of a reactor and practically speaking, MATLAB was the most suitable software at hand. He ended up having to get a friend to help him code the equations in while also having to oversimplify his model.

The weird thing is that: most students from his chemical engineering faculty were not expected or encouraged to use MATLAB, almost all of their prior assignments required no use of MATLAB except that infamous first year course, and most of his peers also avoided using MATLAB and resorted to Excel. It is my understanding that Excel cannot match MATLAB's efficiency and clarity when solving calculus problems so it was not uncommon to see extremely long Excel spreadsheets.

Anyway, my friend is, with the help of a friend's past year MATLAB codes, trying to finish up his computational fluid dynamics assignment that's due soon. He finishes university in 2 weeks time.

Even though he knows that not every engineer has to use MATLAB in the workplace, he somehow wishes he was able to learn MATLAB at his glacial pace. I find it such a pity that he was never able to keep up with the pace of learning that was expected which begs the question: are students who are too slow at learning programming better of in a different field of study?

If you've managed to read to the end of this, thank you so much. I just don't know how to help my friend and I'm hoping some of you might be able to suggest how I can help him be better at it. I believe he has potential but needs special help when it comes to MATLAB.

All helpful and constructive suggestions considered,

Thank You All

While searching the internet for some books on ordinary differential equations, I came across a link that I believe is very useful for all math students and not only. If you are interested in ODEs, it's worth taking the time to study it.

A First Look at Ordinary Differential Equations by Timothy S. Judson is an excellent resource for anyone looking to understand ODEs better. Here's a brief overview of the main topics covered:

- Introduction to ODEs: Basic concepts, definitions, and initial differential equations.
- Methods of Solution:

- Separable equations
- First-order linear equations
- Exact equations
- Transcendental functions

- Applications of ODEs: Practical examples and applications in various scientific fields.
- Systems of ODEs: Analysis and solutions of systems of differential equations.
- Series and Numerical Methods: Use of series and numerical methods for solving ODEs.

This book provides a clear and comprehensive introduction to ODEs, making it suitable for students and new researchers in mathematics. If you're interested, you can explore the book in more detail here: A First Look at Ordinary Differential Equations.

The study of the dynamics of the discrete Klein - Gordon equation (DKG) with friction is given by the equation :

In the above equation, W describes the potential function:

to which every coupled unit adheres. In Eq. (1), the variable $$ is the unknown displacement of the oscillator occupying the n-th position of the lattice, and is the discretization parameter. We denote by h the distance between the oscillators of the lattice. The chain (DKG) contains linear damping with a damping coefficient , whileis the coefficient of the nonlinear cubic term.

For the DKG chain (1), we will consider the problem of initial-boundary values, with initial conditions

and Dirichlet boundary conditions at the boundary points and , that is,

Therefore, when necessary, we will use the short notation for the one-dimensional discrete Laplacian

Now we want to investigate numerically the dynamics of the system (1)-(2)-(3). Our first aim is to conduct a numerical study of the property of Dynamic Stability of the system, which directly depends on the existence and linear stability of the branches of equilibrium points.

For the discussion of numerical results, it is also important to emphasize the role of the parameter . By changing the time variable , we rewrite Eq. (1) in the form

. We consider spatially extended initial conditions of the form: where is the distance of the grid and is the amplitude of the initial condition

We also assume zero initial velocity:

the following graphs for and

% Parameters

L = 200; % Length of the system

K = 99; % Number of spatial points

j = 2; % Mode number

omega_d = 1; % Characteristic frequency

beta = 1; % Nonlinearity parameter

delta = 0.05; % Damping coefficient

% Spatial grid

h = L / (K + 1);

n = linspace(-L/2, L/2, K+2); % Spatial points

N = length(n);

omegaDScaled = h * omega_d;

deltaScaled = h * delta;

% Time parameters

dt = 1; % Time step

tmax = 3000; % Maximum time

tspan = 0:dt:tmax; % Time vector

% Values of amplitude 'a' to iterate over

a_values = [2, 1.95, 1.9, 1.85, 1.82]; % Modify this array as needed

% Differential equation solver function

function dYdt = odefun(~, Y, N, h, omegaDScaled, deltaScaled, beta)

U = Y(1:N);

Udot = Y(N+1:end);

Uddot = zeros(size(U));

% Laplacian (discrete second derivative)

for k = 2:N-1

Uddot(k) = (U(k+1) - 2 * U(k) + U(k-1)) ;

end

% System of equations

dUdt = Udot;

dUdotdt = Uddot - deltaScaled * Udot + omegaDScaled^2 * (U - beta * U.^3);

% Pack derivatives

dYdt = [dUdt; dUdotdt];

end

% Create a figure for subplots

figure;

% Initial plot

a_init = 2; % Example initial amplitude for the initial condition plot

U0_init = a_init * sin((j * pi * h * n) / L); % Initial displacement

U0_init(1) = 0; % Boundary condition at n = 0

U0_init(end) = 0; % Boundary condition at n = K+1

subplot(3, 2, 1);

plot(n, U0_init, 'r.-', 'LineWidth', 1.5, 'MarkerSize', 10); % Line and marker plot

xlabel('$x_n$', 'Interpreter', 'latex');

ylabel('$U_n$', 'Interpreter', 'latex');

title('$t=0$', 'Interpreter', 'latex');

set(gca, 'FontSize', 12, 'FontName', 'Times');

xlim([-L/2 L/2]);

ylim([-3 3]);

grid on;

% Loop through each value of 'a' and generate the plot

for i = 1:length(a_values)

a = a_values(i);

% Initial conditions

U0 = a * sin((j * pi * h * n) / L); % Initial displacement

U0(1) = 0; % Boundary condition at n = 0

U0(end) = 0; % Boundary condition at n = K+1

Udot0 = zeros(size(U0)); % Initial velocity

% Pack initial conditions

Y0 = [U0, Udot0];

% Solve ODE

opts = odeset('RelTol', 1e-5, 'AbsTol', 1e-6);

[t, Y] = ode45(@(t, Y) odefun(t, Y, N, h, omegaDScaled, deltaScaled, beta), tspan, Y0, opts);

% Extract solutions

U = Y(:, 1:N);

Udot = Y(:, N+1:end);

% Plot final displacement profile

subplot(3, 2, i+1);

plot(n, U(end,:), 'b.-', 'LineWidth', 1.5, 'MarkerSize', 10); % Line and marker plot

xlabel('$x_n$', 'Interpreter', 'latex');

ylabel('$U_n$', 'Interpreter', 'latex');

title(['$t=3000$, $a=', num2str(a), '$'], 'Interpreter', 'latex');

set(gca, 'FontSize', 12, 'FontName', 'Times');

xlim([-L/2 L/2]);

ylim([-2 2]);

grid on;

end

% Adjust layout

set(gcf, 'Position', [100, 100, 1200, 900]); % Adjust figure size as needed

Dynamics for the initial condition , , for , for different amplitude values. By reducing the amplitude values, we observe the convergence to equilibrium points of different branches from and the appearance of values for which the solution converges to a non-linear equilibrium point Parameters:

Detection of a stability threshold : For , the initial condition , , converges to a non-linear equilibrium point.

Characteristics for , with corresponding norm where the dynamics appear in the first image of the third row, we observe convergence to a non-linear equilibrium point of branch This has the same norm and the same energy as the previous case but the final state has a completely different profile. This result suggests secondary bifurcations have occurred in branch

By further reducing the amplitude, distinct values of are discerned: 1.9, 1.85, 1.81 for which the initial condition with norms respectively, converges to a non-linear equilibrium point of branch This equilibrium point has norm and energy . The behavior of this equilibrium is illustrated in the third row and in the first image of the third row of Figure 1, and also in the first image of the third row of Figure 2. For all the values between the aforementioned a, the initial condition converges to geometrically different non-linear states of branch as shown in the second image of the first row and the first image of the second row of Figure 2, for amplitudes and respectively.

Refference:

Check out this episode about PIVLab: https://www.buzzsprout.com/2107763/15106425

Join the conversation with William Thielicke, the developer of PIVlab, as he shares insights into the world of particle image velocimetery (PIV) and its applications. Discover how PIV accurately measures fluid velocities, non invasively revolutionising research across the industries. Delve into the development journey of PI lab, including collaborations, key features and future advancements for aerodynamic studies, explore the advanced hardware setups camera technologies, and educational prospects offered by PIVlab, for enhanced fluid velocity measurements. If you are interested in the hardware he speaks of check out the company: Optolution.

In the MATLAB description of the algorithm for Lyapunov exponents, I believe there is ambiguity and misuse.

The lambda(i) in the reference literature signifies the Lyapunov exponent of the entire phase space data after expanding by i time steps, but in the calculation formula provided in the MATLAB help documentation, Y_(i+K) represents the data point at the i-th point in the reconstructed data Y after K steps, and this calculation formula also does not match the calculation code given by MATLAB. I believe there should be some misguidance and misunderstanding here.

According to the symbol regulations in the algorithm description and the MATLAB code, I think the correct formula might be y(i) = 1/dt * 1/N * sum_j( log( ||Y_(j+i) - Y_(j*+i)|| ) )

Let's talk about probability theory in Matlab.

Conditions of the problem - how many more letters do I need to write to the sales department to get an answer?

To get closer to the problem, I need to buy a license under a contract. Maybe sometimes there are responsible employees sitting here who will give me an answer.

Thank you

ðŸ“š New Book Announcement: "Image Processing Recipes in MATLAB" ðŸ“š

I am delighted to share the release of my latest book, "Image Processing Recipes in MATLAB," co-authored by my dear friend and colleague Gustavo Benvenutti Borba.

This 'cookbook' contains 30 practical recipes for image processing, ranging from foundational techniques to recently published algorithms. It serves as a concise and readable reference for quickly and efficiently deploying image processing pipelines in MATLAB.

Gustavo and I are immensely grateful to the MathWorks Book Program for their support. We also want to thank Randi Slack and her fantastic team at CRC Press for their patience, expertise, and professionalism throughout the process.

___________

Are you local to Boston?

Shape the Future of MATLAB: Join MathWorks' UX Night In-Person!

When: June 25th, 6 to 8 PM

Where: MathWorks Campus in Natick, MA

ðŸŒŸ Calling All MATLAB Users! Here's your unique chance to influence the next wave of innovations in MATLAB and engineering software. MathWorks invites you to participate in our special after-hours usability studies. Dive deep into the latest MATLAB features, share your valuable feedback, and help us refine our solutions to better meet your needs.

ðŸš€ This Opportunity Is Not to Be Missed:

- Exclusive Hands-On Experience: Be among the first to explore new MATLAB features and capabilities.
- Voice Your Expertise: Share your insights and suggestions directly with MathWorks developers.
- Learn, Discover, and Grow: Expand your MATLAB knowledge and skills through firsthand experience with unreleased features.
- Network Over Dinner: Enjoy a complimentary dinner with fellow MATLAB enthusiasts and the MathWorks team. It's a perfect opportunity to connect, share experiences, and network after work.
- Earn Rewards: Participants will not only contribute to the advancement of MATLAB but will also be compensated for their time. Plus, enjoy special MathWorks swag as a token of our appreciation!

ðŸ‘‰ Reserve Your Spot Now: Space is limited for these after-hours sessions. If you're passionate about MATLAB and eager to contribute to its development, we'd love to hear from you.

Are you a Simulink user eager to learn how to create apps with App Designer? Or an App Designer enthusiast looking to dive into Simulink?

Don't miss today's article on the Graphics and App Building Blog by @Robert Philbrick! Discover how to build Simulink Apps with App Designer, streamlining control of your simulations!

Hi to all.

I'm trying to learn a bit about trading with cryptovalues. At the moment I'm using Freqtrade (in dry-run mode of course) for automatic trading. The tool is written in python and it allows to create custom strategies in python classes and then run them.

I've written some strategy just to learn how to do, but now I'd like to create some interesting algorithm. I've a matlab license, and I'd like to know what are suggested tollboxes for following work:

- Create a criptocurrency strategy algorythm (for buying and selling some crypto like BTC, ETH etc).
- Backtesting the strategy with historical data (I've a bunch of json files with different timeframes, downloaded with freqtrade from binance).
- Optimize the strategy given some parameters (they can be numeric, like ROI, some kind of enumeration, like "selltype" and so on).
- Convert the strategy algorithm in python, so I can use it with Freqtrade without worrying of manually copying formulas and parameters that's error prone.
- I'd like to write both classic algorithm and some deep neural one, that try to find best strategy with little neural network (they should run on my pc with 32gb of ram and a 3080RTX if it can be gpu accelerated).

What do you suggest?

The study of the dynamics of the discrete Klein - Gordon equation (DKG) with friction is given by the equation :

above equation, W describes the potential function :

The objective of this simulation is to model the dynamics of a segment of DNA under thermal fluctuations with fixed boundaries using a modified discrete Klein-Gordon equation. The model incorporates elasticity, nonlinearity, and damping to provide insights into the mechanical behavior of DNA under various conditions.

% Parameters

numBases = 200; % Number of base pairs, representing a segment of DNA

kappa = 0.1; % Elasticity constant

omegaD = 0.2; % Frequency term

beta = 0.05; % Nonlinearity coefficient

delta = 0.01; % Damping coefficient

- Position: Random initial perturbations between 0.01 and 0.02 to simulate the thermal fluctuations at the start.
- Velocity: All bases start from rest, assuming no initial movement except for the thermal perturbations.

% Random initial perturbations to simulate thermal fluctuations

initialPositions = 0.01 + (0.02-0.01).*rand(numBases,1);

initialVelocities = zeros(numBases,1); % Assuming initial rest state

The simulation uses fixed ends to model the DNA segment being anchored at both ends, which is typical in experimental setups for studying DNA mechanics. The equations of motion for each base are derived from a modified discrete Klein-Gordon equation with the inclusion of damping:

% Define the differential equations

dt = 0.05; % Time step

tmax = 50; % Maximum time

tspan = 0:dt:tmax; % Time vector

x = zeros(numBases, length(tspan)); % Displacement matrix

x(:,1) = initialPositions; % Initial positions

% Velocity-Verlet algorithm for numerical integration

for i = 2:length(tspan)

% Compute acceleration for internal bases

acceleration = zeros(numBases,1);

for n = 2:numBases-1

acceleration(n) = kappa * (x(n+1, i-1) - 2 * x(n, i-1) + x(n-1, i-1)) ...

- delta * initialVelocities(n) - omegaD^2 * (x(n, i-1) - beta * x(n, i-1)^3);

end

% positions for internal bases

x(2:numBases-1, i) = x(2:numBases-1, i-1) + dt * initialVelocities(2:numBases-1) ...

+ 0.5 * dt^2 * acceleration(2:numBases-1);

% velocities using new accelerations

newAcceleration = zeros(numBases,1);

for n = 2:numBases-1

newAcceleration(n) = kappa * (x(n+1, i) - 2 * x(n, i) + x(n-1, i)) ...

- delta * initialVelocities(n) - omegaD^2 * (x(n, i) - beta * x(n, i)^3);

end

initialVelocities(2:numBases-1) = initialVelocities(2:numBases-1) + 0.5 * dt * (acceleration(2:numBases-1) + newAcceleration(2:numBases-1));

end

% Visualization of displacement over time for each base pair

figure;

hold on;

for n = 2:numBases-1

plot(tspan, x(n, :));

end

xlabel('Time');

ylabel('Displacement');

legend(arrayfun(@(n) ['Base ' num2str(n)], 2:numBases-1, 'UniformOutput', false));

title('Displacement of DNA Bases Over Time');

hold off;

The results are visualized using a plot that shows the displacements of each base over time . Key observations from the simulation include :

- Wave Propagation: The initial perturbations lead to wave-like dynamics along the segment, with visible propagation and reflection at the boundaries.
- Damping Effects: The inclusion of damping leads to a gradual reduction in the amplitude of the oscillations, indicating energy dissipation over time.
- Nonlinear Behavior: The nonlinear term influences the response, potentially stabilizing the system against large displacements or leading to complex dynamic patterns.

% 3D plot for displacement

figure;

[X, T] = meshgrid(1:numBases, tspan);

surf(X', T', x);

xlabel('Base Pair');

ylabel('Time');

zlabel('Displacement');

title('3D View of DNA Base Displacements');

colormap('jet');

shading interp;

colorbar; % Adds a color bar to indicate displacement magnitude

% Snapshot visualization at a specific time

snapshotTime = 40; % Desired time for the snapshot

[~, snapshotIndex] = min(abs(tspan - snapshotTime)); % Find closest index

snapshotSolution = x(:, snapshotIndex); % Extract displacement at the snapshot time

% Plotting the snapshot

figure;

stem(1:numBases, snapshotSolution, 'filled'); % Discrete plot using stem

title(sprintf('DNA Model Displacement at t = %d seconds', snapshotTime));

xlabel('Base Pair Index');

ylabel('Displacement');

% Time vector for detailed sampling

tDetailed = 0:0.5:50; % Detailed time steps

% Initialize an empty array to hold the data

data = [];

% Generate the data for 3D plotting

for i = 1:numBases

% Interpolate to get detailed solution data for each base pair

detailedSolution = interp1(tspan, x(i, :), tDetailed);

% Concatenate the current base pair's data to the main data array

data = [data; repmat(i, length(tDetailed), 1), tDetailed', detailedSolution'];

end

% 3D Plot

figure;

scatter3(data(:,1), data(:,2), data(:,3), 10, data(:,3), 'filled');

xlabel('Base Pair');

ylabel('Time');

zlabel('Displacement');

title('3D Plot of DNA Base Pair Displacements Over Time');

colorbar; % Adds a color bar to indicate displacement magnitude

As far as I know, the MATLAB Community (including Matlab Central and Mathworks' official GitHub repository) has always been a vibrant and diverse professional and amateur community of MATLAB users from various fields globally. Being a part of it myself, especially in recent years, I have not only benefited continuously from the community but also tried to give back by helping other users in need.

I am a senior MATLAB user from Shenzhen, China, and I have a deep passion for MATLAB, applying it in various scenarios. Due to the less than ideal job market in my current social environment, I am hoping to find a position for remote support work within the Matlab Community. I wonder if this is realistic. For instance, Mathworks has been open-sourcing many repositories in recent years, especially in the field of deep learning with typical applications across industries. I am eager to use the latest MATLAB features to implement state-of-the-art algorithms. Additionally, I occasionally contribute through GitHub issues and pull requests.

In conclusion, I am looking forward to the opportunity to formally join the Matlab Community in a remote support role, dedicating more energy to giving back to the community and making the world a better place! (If a Mathworks employer can contact me, all the better~)

I created an ellipse visualizer in #MATLAB using App Designer! To read more about it, and how it ties to the recent total solar eclipse, check out my latest blog post:

Github Repo of the app (you can open it on MATLAB Online!):