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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.
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- 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!):

The latest release is pretty much upon us. Official annoucements will be coming soon and the eagle-eyed among you will have started to notice some things shifting around on the MathWorks website as we ready for this.

The pre-release has been available for a while. Maybe you've played with it? I have...I've even been quietly using it to write some of my latest blog posts...and I have several queued up for publication after MathWorks officially drops the release.

At the time of writing, this page points to the pre-release highlights. Prerelease Release Highlights - MATLAB & Simulink (mathworks.com)

What excites you about this release? why?

I found this link posted on Reddit.

https://workhunty.com/job-blog/where-is-the-best-place-to-be-a-programmer/Matlab/

I'm excited to share some valuable resources that I've found to be incredibly helpful for anyone looking to enhance their MATLAB skills. Whether you're just starting out, studying as a student, or are a seasoned professional, these guides and books offer a wealth of information to aid in your learning journey.

These materials are freely available and can be a great addition to your learning resources. They cover a wide range of topics and are designed to help users at all levels to improve their proficiency in MATLAB.

Happy learning and I hope you find these resources as useful as I have!

Let S be the closed surface composed of the hemisphere and the base Let be the electric field defined by . Find the electric flux through S. (Hint: Divide S into two parts and calculate ).

% Define the limits of integration for the hemisphere S1

theta_lim = [-pi/2, pi/2];

phi_lim = [0, pi/2];

% Perform the double integration over the spherical surface of the hemisphere S1

% Define the electric flux function for the hemisphere S1

flux_function_S1 = @(theta, phi) 2 * sin(phi);

electric_flux_S1 = integral2(flux_function_S1, theta_lim(1), theta_lim(2), phi_lim(1), phi_lim(2));

% For the base of the hemisphere S2, the electric flux is 0 since the electric

% field has no z-component at the base

electric_flux_S2 = 0;

% Calculate the total electric flux through the closed surface S

total_electric_flux = electric_flux_S1 + electric_flux_S2;

% Display the flux calculations

disp(['Electric flux through the hemisphere S1: ', num2str(electric_flux_S1)]);

disp(['Electric flux through the base of the hemisphere S2: ', num2str(electric_flux_S2)]);

disp(['Total electric flux through the closed surface S: ', num2str(total_electric_flux)]);

% Parameters for the plot

radius = 1; % Radius of the hemisphere

% Create a meshgrid for theta and phi for the plot

[theta, phi] = meshgrid(linspace(theta_lim(1), theta_lim(2), 20), linspace(phi_lim(1), phi_lim(2), 20));

% Calculate Cartesian coordinates for the points on the hemisphere

x = radius * sin(phi) .* cos(theta);

y = radius * sin(phi) .* sin(theta);

z = radius * cos(phi);

% Define the electric field components

Ex = 2 * x;

Ey = 2 * y;

Ez = 2 * z;

% Plot the hemisphere

figure;

surf(x, y, z, 'FaceAlpha', 0.5, 'EdgeColor', 'none');

hold on;

% Plot the electric field vectors

quiver3(x, y, z, Ex, Ey, Ez, 'r');

% Plot the base of the hemisphere

[x_base, y_base] = meshgrid(linspace(-radius, radius, 20), linspace(-radius, radius, 20));

z_base = zeros(size(x_base));

surf(x_base, y_base, z_base, 'FaceColor', 'cyan', 'FaceAlpha', 0.3);

% Additional plot settings

colormap('cool');

axis equal;

grid on;

xlabel('X');

ylabel('Y');

zlabel('Z');

title('Hemisphere and Electric Field');

Although, I think I will only get to see a partial eclipse (April 8th!) from where I am at in the U.S. I will always have MATLAB to make my own solar eclipse. Just as good as the real thing.

Code (found on the @MATLAB instagram)

a=716;

v=255;

X=linspace(-10,10,a);

[~,r]=cart2pol(X,X');

colormap(gray.*[1 .78 .3]);

[t,g]=cart2pol(X+2.6,X'+1.4);

image(rescale(-1*(2*sin(t*10)+60*g.^.2),0,v))

hold on

h=exp(-(r-3)).*abs(ifft2(r.^-1.8.*cos(7*rand(a))));

h(r<3)=0;

image(v*ones(a),'AlphaData',rescale(h,0,1))

camva(3.8)