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For an arc-length profile with radius $Y(u)>0$, the equation $K=c$ isThus $Y=A\cos u$ gives $K=1$ locally, while $Y=A\cosh u$ gives $K=-1$ wherever $|Y^{\prime }|<1$. At $u=0$ their mean curvatures are respectivelyso different choices of $A$ give locally noncongruent surfaces with the same constant Gaussian curvature.

The curvature of a connection satisfies $d_{A}F(A)=0$. In a local frame this follows directly by expanding $d(dA+A\wedge A)+A\wedge F-F\wedge A$.

The Legendre transform is the convex function

For $S=\{{\sum }_i{l}_i{a}_i:-L_i/2\le l_i\le L_i/2\}$ and ${\alpha }_i=Q\cdot a_i$,For large $L_i$, each factor is sharply peaked when ${\alpha }_i\in 2\pi \mathbb{Z}$, exactly the condition $Q\in {\Lambda }^{\ast }$.

The critical line is the symmetry line $\Re s=1/2$ inside the critical strip.

A Coxeter element is a product of all simple generators in some order. For a finite irreducible Coxeter group, all Coxeter elements are conjugate.

For involutory generators, the folding condition requires that simple multiplication never preserve length and that simultaneous left and right ascents either combine to a two-step ascent or fold: if both $s_{i}w$ and $w{s}_j$ have length $\ell (w)+1$, then either $\ell (s_{i}w{s}_j)=\ell (w)+2$ or $w=s_{i}w{s}_j$.

For $J\subseteq I$, the standard parabolic subgroup is $W_J=⟨s_j:j\in J⟩$. Its cosets index faces of a fixed type in the Coxeter complex.

For a finite-rank Coxeter system, let $V$ have basis $(e_i{)}_{i\in I}$ and symmetric bilinear formIts geometric representation sends the generator $x_i$ to the reflection

A reduced expression in a Coxeter group has the least possible Coxeter length among expressions in the generators of its Coxeter system.

A reduced expression for $w\in W$ is a product of the fewest possible simple generators representing $w$. The number of factors is its Coxeter length $\ell (w)$.

The Coxeter matrix of a Coxeter system is the symmetric matrix whose entry $m_{ij}$ is the order of $x_i{x}_j$.

A geometric quotient $X\to X/G$ has fibers equal to group orbits and carries precisely the invariant regular functions locally on the quotient.

The Cox ring of a toric variety is the polynomial ring with one variable $x_{\rho }$ for each ray, graded by the divisor class group through $\mathrm{deg}{x}_{\rho }=[D_{\rho }]$.

For a real random vector $X=(X_1,\dots ,X_p{)}^T$ with finite second moments, its covariance matrix collects all pairwise covariances:The diagonal entries are variances. The covariance matrix is a symmetric matrix and a positive semidefinite matrix, because for every real vector $a$,Thus one matrix describes the variance of every linear combination of the coordinates. It is a positive-definite matrix exactly when no nonzero linear combination is constant almost surely. A deterministic linear relation produces a zero eigenvalue.

For example,has unit coordinate variances, covariance $0.8$, and eigenvalues $1.8$ and $0.2$. Its unit eigenvectors $(1,1{)}^T/\sqrt{2}$ and $(1,-1{)}^T/\sqrt{2}$ describe directions of greatest and least projected variance. For a unit vector $a=(\cos \theta ,\sin \theta {)}^T$, direct multiplication gives $a^T\Sigma a=1+0.8\sin (2\theta )$: the variation in the sum direction is nine times that in the difference direction.

The spectral theorem supplies an orthonormal basis of these directions. This connects covariance matrices to principal component analysis: its leading eigenvector identifies the unit-direction projection with the largest variance. For a linear transformation $A$, the transformed random vector has covariance matrix $A\Sigma A^T$, so uncertainties propagate through a linear model by matrix multiplication.

If every $a_j=b_j=1$, then $p_n=F_{n+2}$ and $q_n=F_{n+1}$. Binet's formulashows that the convergents tend to $\varphi$ and gives sharp scaled errors through $F_{n+1}-\varphi F_n={\psi }^n$.

A uniformly continuous function has one input tolerance valid throughout its domain of a function. For $f:D\to \mathbb{R}$, with $D\subseteq \mathbb{R}$, uniform continuity meansChoose the output tolerance $\epsilon$ first, then a positive $\delta$ valid for every pair of function inputs. It may depend on $\epsilon$, the function and its domain of a function, but not on the pair’s location. The absolute values measure input and output distance.^[ref]

Compare the quantifier order with the epsilon-delta definition of continuity:Both lines end with $|x-a|<\delta \Rightarrow |f(x)-f(a)|<\epsilon$. In the first, $\delta$ can depend on the point $a$; in the second, it is chosen before $a$. Thus uniform continuity implies continuity at every point. Each requested $\epsilon$ gets its own shared tolerance.

For an exact example, take the polynomial function $f(x)=x^2$ on the closed interval $D=[0,2]$. For every pair of function inputs in that interval, polynomial factorization givesConsequently $\delta =\epsilon /4$ works everywhere on this domain of a function. With $\epsilon =0.4$, the common tolerance is $\delta =0.1$. Near the right endpoint, for example, $x=1.92$ and $y=2$ have input distance $0.08$ and output distance $4-1.92^2=0.3136<0.4$. The bound is sufficient; it does not claim the largest admissible input window at each point.

On the whole real line, the same polynomial function is a continuous function but fails uniform continuity. Take the sequences $x_n=n$ and $y_n=n+1/n$, for positive integers $n$. ThenFor any proposed $\delta >0$, choose an integer $n>1/\delta$. The inputs are closer than $\delta$, yet their outputs violate the fixed tolerance $\epsilon =1$. The sequences need not themselves have a finite limit: their mutual distance is what tends to zero. This is the obstruction captured by the sequential criterion for uniform continuity. Restricting the domain of a function to $[0,2]$ removes these escaping pairs.

A uniformly continuous function need not be a bounded function. The identity function $f(x)=x$ on the real line has $|f(x)-f(y)|=|x-y|$, so $\delta =\epsilon$ always works, although its image of a function is the unbounded real line. Uniform continuity controls changes between nearby inputs, not the overall size of outputs. Likewise, an unbounded domain of a function does not by itself prevent uniform continuity.

The square root $g(x)=\sqrt{x}$ supplies a different common tolerance on the interval $[0,\infty )$. Assuming $x\ge y\ge 0$, put $u=\sqrt{x}$ and $v=\sqrt{y}$. Since $0\le u-v\le u+v$,Interchanging the inputs when necessary provesThus $\delta ={\epsilon }^2$ guarantees uniform continuity on the entire domain of a function, including zero. For $\epsilon =0.4$, any input distance below $0.16$ suffices. The pairs $(0,0.09)$ and $(4,4.09)$ both have input distance $0.09$; their output distances are respectively $0.3$ and $0.09/(\sqrt{4.09}+2)<0.0225$. The common guarantee accommodates quite different local sensitivities.

This Hölder continuity bound allows a curved tolerance rule. It is weaker than a Lipschitz bound: at zero the output change divided by the input change isNo finite Lipschitz constant works near zero, even though uniform continuity does. A Lipschitz function does have uniform continuity, choosing $\delta =\epsilon /L$ when its Lipschitz bound $L$ is positive; when $L=0$ it is a constant function.^[ref]

For general metric spaces, replace the absolute values by the respective distance functions. The Heine-Cantor theorem gives an existence guarantee: every continuous function on a compact metric space is uniformly continuous. In particular, continuity on a finite closed interval provides a common tolerance, even when no simple formula for it is known.^[ref]

For a short proof outline, fix $\epsilon >0$. Continuity gives each point $a$ an open ball of radius $r_a>0$ within which outputs differ from $f(a)$ by less than $\epsilon /2$. The open balls of half these radii still cover the domain of a function. Compactness supplies a finite subcover. Take $\delta$ to be the smallest half-radius in this finite set. Two inputs closer than $\delta$ lie together in one of the original open balls, so the triangle inequality bounds their output distance by $\epsilon$. The beautiful step is that the compact set turns local choices into finitely many choices, whose minimum stays positive.

A shared tolerance also makes uniformly continuous functions preserve Cauchy sequences: after choosing $\delta$ for $\epsilon$, all sufficiently late input pairs are closer than $\delta$, so all corresponding output pairs are closer than $\epsilon$. Unlike a convergent sequence argument, this does not require an input limit inside the domain of a function. It is why uniform continuity is useful for extending a function toward missing endpoints and for the decay argument below.

A function $f$ on the real line is right-continuous at $t$ when $f(s)\to f(t)$ as $s\downarrow t$.

Two-sided continuity requires both the right and left limits to equal the function value.

The continuity set of a function is the set of points where it is continuous. For a real function $f$, define its oscillation at $x$ byThe sets $\{x:{\omega }_f(x)<\epsilon \}$ are open, and $f$ is continuous at $x$ exactly when ${\omega }_f(x)=0$. Hence its continuity set is the $G_{\delta }$ Borel set

If $f$ is continuous at $x$ and $g$ is continuous at $f(x)$, then $g\circ f$ is continuous at $x$.