Sheaf Theory

@KTHuniversity | @ASMLcompany | H^n(X, F)

[Sheaf Theory @BrewMasterNull]

@aVanishingCycle i'll pay extra for the one with the night sky but no stars

[Ria @UnRialist]

mathematicians who are into literature and philosophy are precious

[Mathematica @mathemetica]

Why can a heat pump deliver more heat than the work supplied while no engine converts every bit of heat into work?

Engines, heat pumps and refrigerators share the same energy balance but measure success differently.

- Engine: absorbs Qₕ, produces W → η = W / Qₕ.
- Heat pump: work W delivers Qₕ → ε = Qₕ / W (often >1).
- Refrigerator: work W extracts Q_c → ε = Q_c / W.

This math runs car engines, efficient heating and food cooling. Energy conservation is universal; the goal sets the figure of merit.

[Antonio Lupetti @antoniolupetti]

"Masters Math Lectures and Assignments" by Richard G. Klotz is a free collection of university mathematics lecture notes and problem sets.

The website has material on Calculus I–III, Vector Analysis, Differential Equations, Linear Algebra, Analysis, Abstract Algebra, Complex Analysis, Differential Geometry, Calculus on Manifolds, and Real Analysis. Each course has lecture notes and assignments available as downloadable PDFs.

The notes explain both the theory and the methods used to solve common problems, including derivatives, integrals, differential equations, linear systems, vector spaces, and other topics.

This is a useful resource for university students, for those starting a degree in mathematics, engineering, physics, or computer science, and for anyone who needs to review or study undergraduate mathematics.

https://t.co/U1UboAfPUd

[Antonio Lupetti @antoniolupetti]

"Algebra, Topology, Differential Calculus, and Optimization Theory for Computer Science and Machine Learning" is an excellent free university-level textbook on the mathematical foundations of computer science and machine learning.

Across more than 1,000 pages, it covers groups, rings, fields, vector spaces, bases, linear maps, matrices, direct sums, determinants, Gaussian elimination, LU and Cholesky decomposition, echelon forms, iterative methods for solving linear systems, differential calculus, topology, convexity, optimization, and many other topics.

It is a rigorous, broad, and modern reference with a clear focus on the mathematics behind AI, machine learning, and computer science.

https://t.co/ykCMrlmGCg

[Sheaf Theory @BrewMasterNull]

@mathemetica Lmao @AssumeTheAxiom did this during our lunch break… while eating Haribo

[Mathelirium @mathelirium]

A Black Hole Can Be Dragged Through Spacetime

A Black Hole does not have to sit still.

Einstein’s equations allow a stranger possibility... a Black Hole that accelerates.

This is the C-metric, an exact solution of General Relativity describing an accelerating Black Hole.

The Black Hole accelerates because the geometry of Spacetime says it must.

The C-metric is one of those exact solutions that reminds us how wild General Relativity really is. Gravity is not just attraction, and Spacetime is not just a stage.

Sometimes the stage itself contains the force.

#Astrophysics #GeneralRelativity #Einstein #CMetric #BlackHole #CosmicStrings #Spacetime #Cosmology #Mathematics #Physics

[Mathematica @mathemetica]

Intricate hydrodynamic patterns emerge when fluids interact with curved manifold geometries.

The upper panel demonstrates the exterior calculus formulation of continuum mechanics on manifolds through a three-dimensional radial manifold shape, a curved surface patch with tangent vectors, and coordinate representations of the tangent and cotangent spaces.

The lower panel shows computed hydrodynamic flow responses on complex manifold shapes, with applied forces inducing vortices and saddle points in the flow.

It is used to simulate fluid behavior on curved surfaces in studies of soft materials and biological interfaces.