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Kernel (algebra) by

Ciro Santilli 37 Updated 2025-07-16

Product of group subsets by

Ciro Santilli 37 Updated 2025-07-16

Subgroup by

Ciro Santilli 37 Updated 2025-07-16

Domain of an elliptic curve by

Ciro Santilli 37 Updated 2025-07-16

Largest known ranks of an elliptic curve over the rational numbers by

Ciro Santilli 37 Updated 2025-07-16

web.math.pmf.unizg.hr/~duje/tors/rankhist.html gives a list with Elkies (2006) on top with:

y^{2} + x y + y = x^{3} - x^{2} - 20067762415575526585033208209338542750930230312178956502 x + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429

TODO why this non standard formulation?

Fraction by

Ciro Santilli 37 Updated 2025-07-16

Notes on Elliptic Curves (II) by BSD by

Ciro Santilli 37 Updated 2025-07-16

The paper that states the BSD conjecture.

Likely paywalled at: www.degruyter.com/document/doi/10.1515/crll.1965.218.79/html. One illegal upload at: virtualmath1.stanford.edu/~conrad/BSDseminar/refs/BSDorigin.pdf.

Limit (mathematics) by

Ciro Santilli 37 Updated 2025-07-16

The fundamental concept of calculus!

The reason why the epsilon delta definition is so venerated is that it fits directly into well known methods of the formalization of mathematics, making the notion completely precise.

Total derivative by

Ciro Santilli 37 Updated 2025-07-16

The total derivative of a function assigns for every point of the domain a linear map with same domain, which is the best linear approximation to the function value around this point, i.e. the tangent plane.

E.g. in 1D:

T o t a l d er i v a t i v e = D [f (x_{0})] (x) = f (x_{0}) + \frac{\partial f}{\partial x} (x_{0}) \times x

and in 2D:

D [f (x_{0}, y_{0})] (x, y) = f (x_{0}, y_{0}) + \frac{\partial f}{\partial x} (x_{0}, y_{0}) \times x + \frac{\partial f}{\partial y} (x_{0}, y_{0}) \times y

(2)

Carleson's theorem by

Ciro Santilli 37 Updated 2025-07-16

The Fourier series of an

L^{2}

function (i.e. the function generated from the infinite sum of weighted sines) converges to the function pointwise almost everywhere.

The theorem also seems to hold (maybe trivially given the transform result) for the Fourier series (TODO if trivially, why trivially).

Only proved in 1966, and known to be a hard result without any known simple proof.

This theorem of course implies that Fourier basis is complete for

L^{2}

, as it explicitly constructs a decomposition into the Fourier basis for every single function.

TODO vs Riesz-Fischer theorem. Is this just a stronger pointwise result, while Riesz-Fischer is about norms only?

One of the many fourier inversion theorems.

Orthonormality by

Ciro Santilli 37 Updated 2025-07-16

Coordinate chart by

Ciro Santilli 37 Updated 2025-07-16

Inner product by

Ciro Santilli 37 Updated 2025-07-16

Appears to be analogous to the dot product, but also defined for infinite dimensions.

Norm induced by an inner product by

Ciro Santilli 37 Updated 2025-07-16

An inner product

x \cdot y

induces a norm with:

∣ x ∣ = < x, x >

Torus by

Ciro Santilli 37 Updated 2025-07-16

Riemann zeta function by

Ciro Santilli 37 Updated 2025-07-16

Tensor by

Ciro Santilli 37 Updated 2025-07-16

A multilinear form with a domain that looks like:

V^{m} \times V *^{n} \to R

where

V *

is the dual space.

Because a tensor is a multilinear form, it can be fully specified by how it act on all combinations of basis sets, which can be done in terms of components. We refer to each component as:

T_{i_{1} \dots i_{m}}^{j_{1} \dots j_{n}} = T (e_{i_{1}}, \dots, e_{i_{m}}, e^{j_{1}}, \dots, e^{j_{m}})

(2)

where we remember that the raised indices refer dual vector.

Some examples:

Explain it properly bibliography: