Glossary

• German word for attempt, approach.
• math A rough guess of solution.

• Circular object with rotational symmetry.

• Circular object with rotational and reflective symmetry.

Function whose domain is a family of subsets of a set, and codomain is the extended real number line.

• It is a series \(\sum t_{n} \), such that \(t_{n}=a_{n+1}-a_{n}\) for some sequence \((a_{n})\).
• It is useful because the partial sum cancels down to only two terms. This cancellation technique is called method of differences, or sometimes telescoping.

• Olympiad level counting (Generating functions) - YouTube
• An expression which its series expansion contains the sequence as coefficients.
• This allows one to make a claim about the global properties of the coefficients.

Set of functions between two fixed sets.

Collection of objects that can be unambiguously defined by a property that all its members share.

A \(n\) dimensional simplex is a convex hull of \(n\) independent—non of them are cohyperplaner—points (or vectors).

• A map \( \varphi\colon X/\sim\, \to Y \) is well-defined if all the elements that are equivalent to \( x\in X \) under the equivalence relation \( \sim \) maps to the same element \( y\in Y \).
• And generally, there is no case in which it is not undefined.

• An object \( A \) is positive definite if there's a sense in which: \[ A > 0\quad \text{(positive)} \] for every cases, besides the trivial case in which \( A = 0\ \text{(definite)} \).
• The semidefiniteness also allows \( A=0 \) for non-trivial cases.
• It is similarly defined for negative definite ( \( < \) ) and negative semidefinite ( \( \le \) ).

A set \( A \) is complete, if every converges within \( A \).

• The adjective for the edge cases.
• The opposite is well-behaved.

• The condition that gives rise to a qualitatively different object from the rest, the degenerate case.

In a normed space, \[ \Vert\mathbf{a}+\mathbf{b}\Vert \leq \Vert \mathbf{a}\Vert +\Vert\mathbf{b}\Vert. \] It can be proved with the law of cosine within the Euclidean space

An element \( x \) of a ring is idempotent, if: \[ x^2 = x \iff \forall n, x^n = x. \]

An element \( x \) of a ring is idempotent, if: \[ \exists n\in\mathbb{N}, x^n = 0. \]

An element \( x \) of a ring is idempotent, if \( x - 1 \) is nilpotent.

Singularity is a point at which a given mathematical object is not defined, or it ceases to be well-behaved in some way.

Such point or object can be described as singular.

Analyze the form of the problem and guess the solution.

• Noticing the pattern.
• Making an educated guess.

• This name I came up with.
• An equation in terms of the components that represents a set of equations one for each index.

• Naming slots with indices

• Einstein Summation Convention, Einstein Summation Notation

There is implicit summations when indices get paired.

Covariant indices are superscripted and contravariant indices are subscripted. Although this is sometimes ignored.

The horizontal position of the index can be used to represent the order in which the tensor takes arguments. Since there is inverse relation between the transformation on the vector and the transformation on the covector, the horizontal rearrangement of indices naturally mean taking inverses. For example, \[ \Lambda_{\mu}{}^{\nu} := (\Lambda^{-1})^{\nu}{}_{\mu}. \]

• Descending Factorial, Falling Sequential Product, Lower Factorial

• Pochhammer Function, Pochhammer Polynomial, Ascending Factorial, Rising Sequential Product, Upper Factorial

These are my notations

It does seem a little too unwieldy but, what do I know? it is a complicated operation in the first place.

This is to alleviate the conflicting and numerous notations:

• \(D_x\), h\(D_\mathbf{x}\), \(\nabla_{\mathbf{x}}\) (with respect to \(x\), in the direction \( \mathbf{x} \)).
• \(\nabla_{\mathbf{A}}\) (act only upon \(\mathbf{A}\)) in vector calculus. See Feynman Subscript Notation.
• \[ \left( \pdv{}{T} \right)_{V}\] in thermodynamics

Fix

• \[ D\bigg|_y \] Fix \(y\) while taking derivatives.
• \[ \frac{\partial}{\partial x}\bigg|_\text{rot}\mathbf{r} \] hold the rotating frame of reference. take it to be invariant.
• \[ \frac{\partial}{\partial P}\!\!\!\!\!\!\mathop{\bigg|}\limits_{n, V, T} \]

Vary

• \[ D\bigg|^y \] Vary only \(y\), that is, take derivative with respect to \(y\).

• \(\nabla^2\) is mathematically not convincing, \(\Delta\) gets confused with the difference, the spacing of \(|\nabla|^2\) looks horrible.
• Therefore, let's use \(\nabla^{\cdot 2}\).

• \(\mathrm{d}^{\wedge n}\mathbf{r}\). Imaging taking the parameters of \(\mathbf{r}\) and making a corresponding volume form out of it.
• This makes the dimension and the variable that the one is integrating with respect to clear.

• The arguments of a function or the additional structures of an object can be omitted when it's clear—suppressed.
• e.g. \(f(x, y) \rightsquigarrow f\) and \((M, d) \rightsquigarrow M\)
• big list - Suggestions for good notation - MathOverflow
• \(\looparrowright\) \looparrowright: Immersion of smooth manifolds
• \(\hookrightarrow\) \hookrightarrow: Is used for embeddings and inclusions.
• \(G \curvearrowright X\) \curvearrowright: Group action of \(G\) on \(X\)
• \(2^X\) : The power set of \(X\).
  • Also for the homset \(\mathrm{hom}(A, B)\): \(B^A\), and set of functors from \(\mathcal{C}\) to \(\mathcal{D}\): \(\mathcal{D}^\mathcal{C}\)
  • This is reasonable since:
    • \[ (A^B)^C = A^{B\times C} \]
    • \[ A^B \times A^C = A^{B\sqcup C} \]
• \(\displaystyle \binom{X}{k}\): Set of \(k\)-subsets of \(X\)
  • Notice \[ 2^X = \bigcup_{k=0}^{|X|} \binom{X}{k} \]
  • Which is analogous to