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For a geodesic in the metric $d{u}^2+Y(u{)}^{2}d{v}^2$, the cyclic coordinate $v$ givesEquivalently, the geodesic meets parallels according to Clairaut's relation.

A circular cylinder of radius $R$ has principal curvatures $0$ and $\pm 1/R$, according to orientation. Hence $K=0$ and $|H|=1/(2R)$.

This is the circular generating-curve case of a Cylinder; the curvature formulas here refer to its lateral surface.

For $X(u,v)=(Y(u)\cos v,Y(u)\sin v,Z(u))$ with $Y^{\prime 2}+Z^{\prime 2}=1$, the induced metric is

For the surfaces generated by $(e^u,0,u)$ and $(\cosh s,0,s)$,The diffeomorphism $u=\log \sinh s$ matches these curvatures, although comparison of the first fundamental forms shows that no local isometry exists.

Forthe area density after integrating over $\theta$ is $2\pi \varphi \sqrt{1+{\varphi }^{\prime 2}}$. If every height interval has area $2\pi r$ times its length, continuity forcesWhere $0<\varphi <r$, the sign of ${\varphi }^{\prime }$ is constant and $\sqrt{r^2-{\varphi }^2}$ has derivative $\pm 1$. Hence the profile lies on a circle of radius $r$.

A catenoid is the surface of revolution obtained by rotating a catenary about its directrix. One conformal parametrization is

A banded-unlink diagram consists of an unlink $U$ and disjoint bands whose simultaneous surgery produces another unlink. Capping the lower and upper unlinks by disks and tracing the band surgeries constructs a surface in $S^4$.

A small-amplitude deep-water wave with surface elevation $\eta =A{e}^{i(kx-\omega t)}$ has dispersion relation ${\omega }^2=gk$ and a velocity potential that decays as $e^{kz}$ below the surface.

A superfield is a function on superspace. Its finite Taylor expansion in Grassmann coordinates packages component fields into a supersymmetry representation.

Active-galactic-nucleus feedback transfers radiative or mechanical energy and momentum from black-hole accretion to surrounding galactic gas.

The Polonyi model uses a canonical Kähler potential and a superpotential linear in one chiral superfield to obtain spontaneous supersymmetry breaking.

A prefix sum stores the accumulated sum through a specified position in a finite sequence or array. For inputs $a_1,\dots ,a_n$, include the empty sum $P_0=0$ and setFor the original input $(4,-2,5,-1,3,-4)$, the prefix array including $P_0$ is $(0,4,2,7,6,9,5)$. A negative entry makes the accumulated sum decrease; a prefix sum does not have to be monotone. Conversely, the original entry is recovered by subtraction: $a_m=P_m-P_{m-1}$.

The useful cancellation is that any contiguous block has sumBoth prefixes include the entries before $\ell$; subtracting cancels those entries. In the example, the sum from positions two through four is $-2+5-1=2$, recovered as $P_4-P_1=6-4$. The arithmetic mean of that block is $2/3$, obtained by dividing its sum by its length. Prefix sums therefore make many repeated questions about an unchanged array cheap: reuse accumulated totals instead of resumming every block.

The sequential recurrence builds all $n+1$ prefixes in $O(n)$ time complexity and storage; each block query needs two reads and one subtraction, hence $O(1)$ time complexity. These counts assume constant-cost arithmetic and access to array entries. If one input changes, all later stored prefixes change, so this simple representation is most useful when queries outnumber updates. For integer inputs, sufficient numeric capacity must be retained; with floating-point arithmetic, subtracting large nearby prefixes can lose relative accuracy for a small block total. The exact mathematical identity does not eliminate representation error.

The apparently sequential recurrence also motivates a prefix scan: associativity permits a tree of partial calculations that yields all accumulated results in parallel. Guy E. Blelloch's 1990 technical report systematically develops this operation and its applications; its authored chapter version explains the two tree sweeps used to obtain all prefixes. ^[ref]. ^[ref].

Every finite graph $G$ has a three-colourable subgraph with at least $2e(G)/3$ edges. Give vertices three independent uniform colours and retain edges whose endpoints have different colours. Each edge survives with probability $2/3$, so the expected number surviving is $2e(G)/3$.

The subgraph induced by a vertex set $W$ contains every edge of the original graph whose two endpoints lie in $W$.

For a regular self-adjoint Sturm-Liouville problem, eigenfunctions are orthogonal in the weighted inner productand the coefficient of $y_n$ is $⟨f,y_n{⟩}_w/⟨y_n,y_n{⟩}_w$.

A Dirichlet eigenvalue is an eigenvalue for an eigenfunction constrained to vanish at the endpoints of its interval or on the boundary of its domain.

In a two-parameter Gaussian simple linear regression centered at $\bar{x}$, the intercept estimator is $\bar{Y}$. With residual standard deviation $s$,which gives the usual two-sided interval from the corresponding quantiles.

An element belongs to $\mathrm{acl}(A)$ when it lies in a finite set definable with parameters from $A$.

Let $N=pq$ for distinct odd prime numbers $p,q$ with $\gcd (p-1,q-1)=10$. Exactly $100$ unit bases make $N$ a Fermat pseudoprime, and exactly $50$ of these make it a strong pseudoprime. The Fermat bases form $C_{10}\times C_{10}$ under the Chinese remainder theorem for unit groups. If $N-1=2^{s}m$, then $5\mid m$ and $m$ is odd, so the $m$th-power map sends five elements of each $C_{10}$ to $1$ and five to $-1$. The strong condition requires equal signs in both components, giving $5^2+5^2=50$.

A two-dimensional conformal field theory is invariant under local holomorphic and antiholomorphic conformal transformations.