Illini Rhythm Syndicate

Illini Rhythm Syndicate UIUC's Rhythm Games Club - SDVX, CHUNITHM, maimai, Taiko, pop'n, Arcaea, Project SEKAI, etc.

04/18/2026

In linear algebra, a Jordan normal form, also known as a Jordan canonical form, is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them.

Let V be a vector space over a field K. Then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is algebraically closed (for instance, if it is the field of complex numbers). The diagonal entries of the normal form are the eigenvalues (of the operator), and the number of times each eigenvalue occurs is called the algebraic multiplicity of the eigenvalue.

If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.

The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.
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04/15/2026

In thermodynamics, the fugacity of a real gas is an effective partial pressure which replaces the ideal partial pressure in an accurate computation of chemical equilibrium. It is equal to the pressure of an ideal gas which has the same temperature and molar Gibbs free energy (chemical potential) as the real gas.

Fugacities are determined experimentally or estimated from various models such as a Van der Waals gas that are closer to reality than an ideal gas. The real gas pressure and fugacity are related through the dimensionless fugacity coefficient φ=f/P.

For an ideal gas, fugacity and pressure are equal, and so φ = 1. Taken at the same temperature and pressure, the difference between the molar Gibbs free energies of a real gas and the corresponding ideal gas is equal to RT ln φ.

The fugacity is closely related to the thermodynamic activity. For a gas, the activity is simply the fugacity divided by a reference pressure to give a dimensionless quantity. This reference pressure is called the standard state and normally chosen as 1 atmosphere or 1 bar.

Accurate calculations of chemical equilibrium for real gases should use the fugacity rather than the pressure. The thermodynamic condition for chemical equilibrium is that the total chemical potential of reactants is equal to that of products. If the chemical potential of each gas is expressed as a function of fugacity, the equilibrium condition may be transformed into the familiar reaction quotient form (or law of mass action) except that the pressures are replaced by fugacities.

For a condensed phase (liquid or solid) in equilibrium with its v***r phase, the chemical potential is equal to that of the v***r, and therefore the fugacity is equal to the fugacity of the v***r. This fugacity is approximately equal to the v***r pressure when the v***r pressure is not too high.
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04/12/2026

In mathematics, a Green’s function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.

This means that if L is a linear differential operator, then the Green’s function G is the solution of the equation LG= δ, where δ is Dirac’s delta function; the solution of the inhomogeneous problem Ly = f is the convolution, y = (G * f).
By the superposition principle, given a linear ordinary differential equation (ODE), Ly = f, one can first solve LG=δ_s, for each s. If the source is a sum of delta functions, then the solution is a sum of Green’s functions as well due to linearity of L. This means that the integral, viewed as a continuous sum, can reconstruct a wide class of sources, f through the convolution integral. Whenever the integral converges, then the solution to the inhomogeneous equation, Ly = f, is given by y = G * f.

Green’s functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green’s functions are studied largely from the point of view of fundamental solutions instead, which take into account the modern language of the theory of distributions or generalized functions.

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04/09/2026

In relativistic physics, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as “the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space”.

Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. Lorentz covariance has two distinct, but closely related meanings:

A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors. In particular, a Lorentz covariant scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a Lorentz invariant (i.e., they transform under the trivial representation).
An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term invariant here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame; this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame. This condition is a requirement according to the principle of relativity; i.e., all non-gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference.

On manifolds, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors can be Lorentz covariant quantities.

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04/07/2026

In continuum mechanics, the maximum distortion energy criterion (also von Mises yield criterion) states that yielding of a ductile material begins when the second invariant of deviatoric stress J_2 reaches a critical value. It is a part of plasticity theory that mostly applies to ductile materials, such as some metals. Prior to yield, material response can be assumed to be of a linear elastic, nonlinear elastic, or viscoelastic behavior.

In materials science and engineering, the von Mises yield criterion is also formulated in terms of the von Mises stress or equivalent tensile stress, σ_v. This is a scalar value of stress that can be computed from the Cauchy stress tensor. In this case, a material is said to start yielding when the von Mises stress reaches a value known as yield strength, σ_y. The von Mises stress is used to predict yielding of materials under complex loading from the results of uniaxial tensile tests. The von Mises stress satisfies the property where two stress states with equal distortion energy have an equal von Mises stress.
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04/04/2026

In machine learning, reinforcement learning from human feedback (RLHF) is a technique to align an intelligent agent with human preferences. It involves training a reward model to represent preferences, which can then be used to train other models through reinforcement learning.

In classical reinforcement learning, an intelligent agent’s goal is to learn a function that guides its behavior, called a policy. The function is iteratively optimized to increase the reward signal derived from the agent’s task performance. However, explicitly defining a reward function that accurately approximates human preferences is challenging. Therefore, RLHF seeks to train a “reward model” directly from human feedback. The reward model is first trained in a supervised manner to predict if a response to a given prompt is good (high reward) or bad (low reward) based on ranking data collected from human annotators. This model then serves as a reward function to improve an agent’s policy through an optimization algorithm like proximal policy optimization.

04/01/2026

In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean n-spaces. For lower dimensions n = 1, 2, 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n-dimensional volume, n-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by λ(A).
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03/29/2026

Carcinization in biology refers to the phenomenon where over millions of years many different lineages have come together and all created something that resembled crabs because it’s evolutionarily advantageous in hardware recently I have seen a phenomenon where all GPU manufacturers are basically converging on the same architecture for AI GPU despite different parts of that architecture being introduced by different companies at different times. everything is coming together in a fashion where we have a certain number of core clusters, within which some resources are shared by individual multiprocessor units, each multiprocessor unit subgroup comes with tensor cores of some sort, supporting some type of high speed interconnect between GPUs just before DRAM, and supporting some amount of dataflow transfer of data in unicast and multicast within each subgroup nvidia blackwell, rdna 4.0, even intel’s new opencl apis for the b70 released today is trying to replicate this architecture on the battlemage hardware.
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03/27/2026

Noether’s theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether’s second theorem) published by the mathematician Emmy Noether in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system’s behavior can be determined by the principle of least action. This theorem applies to continuous and smooth symmetries of physical space. Noether’s formulation is quite general and has been applied across classical mechanics, high energy physics, and recently statistical mechanics.

Noether’s theorem is used in theoretical physics and the calculus of variations. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian alone (e.g., systems with a Rayleigh dissipation function). In particular, dissipative systems with continuous symmetries need not have a corresponding conservation law.
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03/21/2026

The 1 .4 GHz NXP i.MX 937 quad-core Cortex-A55 microprocessor (MPU) for HMI and Edge AI applications aims to fill the gap between entry-level NXP i.MX 93 SoCs and higher-end parts like the NXP i.MX 952 processor family, while offering pin-to-pin compatibility with the latter.

The i.MX 937 MPU also features a dedicated 667 MHz Arm Cortex -M7 for real-time workloads and a low-power Arm Cortex-M33 core for system management tasks, supports LPDDR4x or LPDDR5 memory, integrates an Arm Mali G310 3D GPU, a VPU for 1080p H.26x video encoding and decoding, and a 2 eTOPS NXP eIQ Neutron NPU for machine learning (ML) acceleration. Since it targets HMI applications, we’ll also find MIPI DSI and LVDS display interfaces, and a 4-lane MIPI CSI camera interface, plus various other I/Os.
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