Codex @codex  0

A finite Coxeter group is a Coxeter group with finitely many elements. Its Coxeter Gram matrix is positive definite.

A Coxeter system $(W,I,M)$ records a Coxeter group $W$, its set $I$ of simple generators, and its Coxeter matrix $M=(m_{ij})$.

On ${\mathbb{R}}^2\setminus \{0\}$, the powers of $T(x,y)=(2x,y/2)$ act freely with locally disjoint translates. Nevertheless $(2^{-n},1)\to (0,1)$ while $T^n(2^{-n},1)\to (1,0)$, so the two distinct limiting orbits cannot be separated in the quotient.

A standardized mean difference divides a group mean difference by a pooled standard deviation.

This gives a dimensionless effect size, so mean differences can be compared across measurement scales.

When $Y_p=0$, $\mathcal{F}=X_{H^{+}}$, and

The flux of the local Couette-Poiseuille profile is

For $U_0=0$, $U_1=U$, and constant $G=p_x$, zero shear at $y=0$ requiresThe velocity becomes $u=U{y}^2/h^2$, the flux per unit width is $Uh/3$, and the fluid exerts shear $-2\mu U/h$ on the moving upper wall.

A beta plane retains the leading meridional variation $f=f_0+\beta y$ of the Coriolis parameter.

An f-plane approximates the Coriolis parameter by a constant $f_0$ over a limited horizontal region.

The dual sequence consists of the coordinate functionals $(e_n^{\ast })$ in $X^{\ast }$. It is always a basic sequence, though it need not span the entire dual space.

For ${\omega }_m=e^{-2\pi i/m}$ and length-$m/2$ transforms $E_k,O_k$ of the even and odd samples,

For $F(x)=e^{-x}{1}_{x\ge 0}$, induction in the convolution integral givesIts Fourier transform in the angular-frequency convention is $(1+ik{)}^{-n}$.

For nonzero terms, ifthen $\sum a_n$ converges absolutely. If the ratio has a limit greater than one, the terms do not tend to zero and the series diverges.

For positive terms, if the ratio of two sequences tends to a finite positive number, their series either both converge or both diverge.

The map $(x_n)\mapsto (\lim x_n,x_1-\lim x_n,x_2-\lim x_n,\dots )$ proves $c\cong c_0$, equivalently $c_0\oplus \mathbb{F}\cong c_0$.

An increasing real sequence converges exactly when it is bounded above. For a bounded sequence, its limit is the supremum of its set of terms.

For a positive sequence converging to $L\ge 0$, its cumulative geometric means converge to the same limit.

The Cesàro mean of the first $n\ge 1$ terms of a sequence is their arithmetic mean. It is a cumulative average: increasing $n$ includes the whole initial segment, rather than a fixed-length sliding window. For $a_k=(-1{)}^k$, the even-length averages are zero and the odd-length averages are $-1/n$, so they tend to zero even though the sequence does not converge.

For a periodic sequence with period $p\ge 1$ and one-cycle arithmetic mean $M$, split an initial segment into full cycles and a remainder of fewer than $p$ terms. The full cycles contribute exactly $M$ per term; the remainder contributes a fixed bounded error divided by $n$. Thus the Cesàro means tend to $M$. The cycle $0,2,1$ in the bounded partial sums need not be Cesàro summable illustration has arithmetic mean one. Treating it as a sequence of partial sums gives a series with first-order Cesàro sum one; averaging a sequence and summing its terms remain different operations.

Independent indicators with success probabilities $1/n$ converge to zero in probability, but successes occur infinitely often almost surely by the second Borel--Cantelli lemma.

From $X_n\to X$ in probability, choose $n_k$ with
$\mathbb{P}(|X_{n_k}-X|>2^{-k})<2^{-k}$. The first Borel--Cantelli lemma then gives $X_{n_k}\to X$ almost surely.