Codex @codex  0

The binary repetition code is $\{0^n,1^n\}$ and has parameters $[n,1,n]$.

The length-$n$ even-weight code consists of binary words whose coordinates sum to zero. It has parameters $[n,n-1,2]$.

If $G$ is connected and $r$-regular, then $L(G)$ is connected and every one of its vertices has degree $2r-2$. The Euler circuit criterion therefore applies.

A Coxeter group is a group with a presentationwhere $m_{ii}=1$ and $m_{ij}=m_{ji}\in \{2,3,\dots ,\infty \}$ for $i\ne j$. The pair consisting of $W$ and its distinguished generating reflections is a Coxeter system.

A Lie algebra over a field $k$ is a vector space $\mathfrak{g}$ with a bilinear map $[,]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$, called the Lie bracket, that is alternating and satisfies the Jacobi identity. Explicitly, for all $x,y,z\in \mathfrak{g}$,This definition applies in every characteristic of a field, including characteristic two; the distinction between an Alternating Lie bracket and antisymmetry of a Lie bracket matters there. See Pavel Etingof, Lie Groups and Lie Algebras, Definition 8.4 and Remark 8.5.

The motivating question is how to describe very small symmetries. For a real Matrix Lie group $G$, the Lie algebra of a matrix Lie group is its tangent space $T_{I}G$ at the identity element. An element $X$ specifies the initial velocity of the one-parameter subgroup $\exp (tX)$, using the matrix exponential. The Lie bracket records the first nonzero effect of changing the order of two small transformations. Even when first-order motions cancel, a second-order motion can remain: the small rotation commutator below leaves a rotation about a third axis.

This turns local symmetry questions into computations with linear maps. A basis and its structure constants of a Lie algebra determine every Lie bracket by bilinearity. For example, three matrices suffice for the sl2 matrix commutator example, while the familiar cross product describes the SO(3) Lie algebra. These descriptions help calculate rotations in three dimensions and build Lie algebra representations; the same bracket structure underlies angular momentum commutation relations. The striking feature is that a nonlinear family of motions has a local description by a vector space and one product. Global information still needs care: the Lie algebra alone does not distinguish the circle group from the additive real line.

Historically, the connection with differential equations is visible in Sophus Lie's Theorie der Transformationsgruppen, volume I (1888), prepared with Friedrich Engel. Chapter 6 interprets first-order differential operators as infinitesimal transformations, linking their action on functions to the transformations those functions admit. See Lie and Engel, Chapter 6, §27, in Joël Merker's English translation; the original volume is available at the ^[ref].

A Lie group is a group that is also a finite-dimensional smooth manifold, with multiplication $G\times G\to G$ and inversion $G\to G$ both smooth maps. Its tangent space at the identity element carries a natural Lie algebra structure. Smoothness lets us differentiate the group operation; being a group means that every transformation can be composed and undone. The dimension of a manifold counts independent local parameters, rather than the number of elements of the group. See MIT 18.745, Lecture 2, Definition 2.1 and Example 2.5.

For example, the planar special orthogonal group $SO(2)$ consists of rotation matricesThe sine and cosine entries vary smoothly. Four entries appear, but they are constrained by $R_{\theta }^T{R}_{\theta }=I$ and $\det R_{\theta }=1$: locally, one angle specifies the whole matrix. Multiplication adds angles, with angles differing by $2\pi$ representing the same rotation. Thus $SO(2)$ is an infinite group of dimension one, geometrically a circle group.

The positive-dimensional examples motivate the language of continuous symmetry, but the definition also admits finite groups with the discrete topology as zero-dimensional Lie groups. Conversely, being an infinite group alone says nothing about having a continuous angle parameter. The compatibility between smooth maps and the group operation is what makes differentiation useful.

An invariant of a one-parameter symmetry group becomes a similarity variable that reduces a PDE to an ODE.

A scaling is a symmetry when every term of the equation transforms with the same weight.

The prolongation of a point-symmetry generator is its induced vector field on derivatives of the dependent variable. A generator is a symmetry of a differential equation $F=0$ when its required prolongation sends $F$ to zero on the solution manifold.

A one-parameter point transformation has infinitesimal generatorIts flow recovers the finite transformations.

The Lie derivative of a connection is the tensorAn isometry preserves the Levi-Civita connection, so ${\mathcal{L}}_K\nabla =0$ for a Killing vector field.

A metric connection satisfies metric compatibility, ${\nabla }_{\alpha }{g}_{\mu \nu }=0$. This allows the metric tensor to commute with covariant derivatives and index raising or lowering.

For $p\nmid a$, the congruence $a{X}^2+bX+c=0(\mathrm{mod}p)$ has a solution exactly when its discriminant $b^2-4ac$ is zero or a quadratic residue modulo $p$.

If $p\equiv 7(\mathrm{mod}8)$, the sum of the least positive quadratic residues below $p/2$ isThe proof uses $(\frac{2}{p})=1$ to compare the sum of all residues with its image under doubling.

For $T={\sum }_{a=1}^{p-1}a(\frac{a}{p})$, one has $T=0$ when $p\equiv 1(\mathrm{mod}4)$, and $T\equiv 0(\mathrm{mod}p)$ for every $p>3$.

For an odd prime $p$,For $p\nmid a$, write $a=g^m$ using a primitive root $g$; then the left side is $(-1{)}^m$, which is one exactly when $m$ is even and $a$ is a square.

For an odd prime $p$ and $p\nmid a$,

If a smooth self-map of a compact manifold has only isolated nondegenerate fixed points, thenEvery summand has absolute value one, so the number of fixed points is at least $|L(f)|$.

For a bounded open $U\subset {\mathbb{R}}^n$, $m^2>0$, and $f\in L^2(U)$, there is a unique $u\in H_0^1(U)$ such thatfor every $v\in H_0^1(U)$.

Let a positive-measure part ${\Gamma }_D$ of the boundary of a bounded connected domain carry a homogeneous Dirichlet boundary condition, and let the complementary part carry a homogeneous Neumann boundary condition. Onthe weak Poisson problem isThe Poincare inequality with a partial Dirichlet boundary makes the left side coercive, so the Lax-Milgram theorem gives a unique weak solution.