Thomas Antony

CubeCL is a Rust crate that is used for writing GPU programs in pure Rust, which is compatible with a wide range of GPU platforms.

The inner/dot product for Euclidean vectors is defined as: $$ \langle u, v \rangle = \sum_{i} u_i v_i $$ Following on from how we defined a norm for square integrable functions, we can similarly define an inner product for such functions in $\mathbb{L}^2$:

We know that functions can be treated as infinite dimensional vectors.

Functions can be treated as infinite dimensional vectors.

An inner product space1 is a real or complex vector space that has an operation called an “inner product” defined for it.

A Hilbert space is an inner product space that is also “complete” w.r.t the inner product - i.e. no sequence of elements in the set converges to an element outside the set.

A volumetric scene represenation is a data structure or algorithmic representation of a scene - typically associated with computer vision.

Volumetric rendering is an algorithm used for rendering radiance fields such as [[NeRFs]] and [[3D Gaussian Splats]]12.

Git worktrees allow you to check out multiple branches simultaneously without re-cloning the repository multiple times 1.

Ingredients Water - 131ml 2% Ultra-Filtered Milk - 93ml All Purpose Flour - 320g Red Star Instant Yeast - 8 g Diamond Crystal salt - 6g Granulated Sugar - 15g Vital Wheat Gluten (Anthony’s) - 1.0 tbsp Kerrygold Unsalted Butter - 30g Diastatic malt (Hoosier

The 1-2-3 Rule For an IFR flight, you do not have to file an alternative if the following conditions are forecast at ETA +/- 1 hour: At least 2000 ft ceiling At least 3 SM visibility Alternate Minimums In order to qualify as an alternate, an airport must…

Cold Start (First start of the day or Oil temp < 100F) Mixture: Full Rich Power Lever: Full Forward Fuel Pump - PRIME (2-3 seconds), then BOOST Power Lever: Open 1/4 inch Start Oil pressure check, Amp check Lean for Max RPM Hot Start (Oil temp > 100F) If…

In the event of an unexplained loss of manifold pressure in an SR22T, the engine reverts to operating like a naturally aspirated engine.

A Schmitt trigger is a Comparator with hysteris.

SINDy1 is an algorithm for identifying a nonlinear dynamic model from data.

A fragment shader is a program that is part of a rendering pipeline1 which takes as input a fragment (generated by rasterizing the outputs of a vertex shader) and outputs a pixel color value and depth. 1 Fragment - OpenGL Wiki 2 Fragment Shader - OpenGL W…

A vertex shader is a program that runs on the GPU that is one of the stages in a “rendering pipeline” (e.g. the OpenGL rendering pipelline1).

If you want to skip to the code, you can find it here. 3D gaussian splats have been all the rage since they were published.

If you want to skip to the code, you can find it here. 3D gaussian splats have been all the rage since they were published.

Here are some rough notes on the “Hyena Hierarchy” architecture described in the paper 1.

This paper1 describes a method for augmenting an existing language model with external memory to improve its performance without requiring any extra training.

The following command sends the 4-byte packet ‘\x01\x02\x03\x04’ to 127.0.0.1 on port 2222 echo -n -e '\x01\x02\x03\x04' > /dev/udp/127.0.0.1/2222

Troubleshooting If you get EXC_BAD_INSTRUCTION on M1 Mac, downgrade openssl from 3.1.0 to 3.0.7 (and Python to 3.10.9)

Train a GPT-like model to “understand langauge”.

Plasma, also called the “fourth state of matter”, consists of a collection of ions and electrons that is quasi-neutral and exhibits collective behavior.

I have been experimenting with OpenAI’s ChatGPT since it came out a couple of days ago.

3 Hamiltonian Mechanics Numerical experiments are just what their name implies: experiments.

2.12 Nonsingular Coordinates and Quaternions The Euler angles provide a convenient way to parameterize the orientation of a rigid body.

2.11 Spin Orbit Coupling The rotation of planets and natural satellites is affected by the gravitational forces from other celestial bodies.

Exercise 2.13: Bicycle Wheel a.

Exercise 2.12: Derivation of Euler Angle Kinematics Fill in the details of the derivation of equation (2.73).

Exercise 2.11: Conservation of Angular Momentum Fill in the details of the argument that Noether’s theorem implies that vector angular momentum is conserved by the motion of the free rigid body.

2.9 Euler’s Equations For a free rigid body we have seen that the components of the angular momentum on the principal axes comprise a self-contained dynamical system: the variation of the principal axis components depends only on the principal axis compon…

2.8.2 Qualitative Features of Free Rigid Body Motion The evolution of the components of the angular momentum on the principal axes has a remarkable property.

2.8.1 Computing the Motion of Free Rigid Bodies Lagrange’s equations for the motion of a free rigid body in terms of Euler angles are quite disgusting, so we will not show them here.

Exercise 2.10: Uniformly accelerated rigid body Show that a rigid body subject to a uniform acceleration rotates as a free rigid body, while the center of mass has a parabolic trajectory.

2.8 Motion of a Free Rigid Body The kinetic energy of a rigid body, expressed in suitable generalized coordinates, is a Lagrangian for its motion.

Exercise 2.9: Euler Angles It is not immediately obvious that all orientations can be represented in terms of the Euler angles.

2.7 Euler Angles Euler Angles are a set of generalized coordinates for describing the orientation of a rigid body.

Exercise 2.5: Principal moments of inertia For each of the configurations described below find the principal moments of inertia with respect to the center of mass, and find the corresponding principal axes. a.

Exercise 2.8: Rotational angular momentum Verify that expression in Eq. (2.50) for the components of the rotational angular momentum Eq. (2.49) in terms of the inertia tensor is correct. $$ \vec{L} = \sum_\alpha m_\alpha \vec{\xi}_\alpha \times \left(\vec…

2.6 Vector Angular Momentum The vector angular momentum of a particle is the cross product of its position vector and its linear momentum vector.

1.8.3 Central Forces in Three Dimensions Consider the motion of a particle in a potential field $V(r)$ in three dimensions using spherical coordinates, $r, \theta, \varphi$, where $\theta$ is the colatitude and $\varphi$ is the longitude.

Exercise 2.5: A constraint on the moments of inertia Show that the sum of any two of the moments of inertia is greater than or equal to the third moment of inertia.

Exercise 2.4: Jupiter a.

Exercise 2.3: Some useful moments of inertia Show that the moments of inertia of the following objects are as given: a.

2.5 Principal Moments of Inertia In the previous section, we showed how the inertia tensor transforms under a rotation: $$ \mathbf{I}' = \mathbf{R}^{\mathscr{T}} \mathbf{I} \mathbf{R} \tag{2.35} $$ This transformation can be used to show that there are sp…

Exercise 2.2: Steiner’s theorem Let I be the moment of inertia of a body with respect to some given line through the center of mass.

2.4 Inertia Tensor The representation of kinetic energy in terms of the inertia tensor involved a rectangular coordinate frame with basis vectors, $\hat{e}$.

Seam carving is an algorithm that can be used to resize an image while removing the least important pixels from it.