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The odd reflection of a function on a half-space is $Eu(x^{\prime },x_n)=\mathrm{sgn}(x_n)u(x^{\prime },|x_n|)$. When $u$ has zero trace on the boundary, this extension preserves the relevant weak regularity and often converts a homogeneous boundary problem into an interior problem.

Matsumoto's theorem says that any two reduced expressions for one element of a Coxeter group are connected by a finite sequence of braid moves.

If $w=s_1\cdots s_n$ is a reduced expression and $s$ is simple with $\ell (ws)<\ell (w)$, thenfor some index $j$. The analogous statement holds for multiplication on the left.

Pressure forcing of one rectangular free-surface mode reduces its amplitude toAt $\omega =\sqrt{gk}$, the inviscid undamped response contains $t\sin (\omega t)$ and grows without bound in the linear model.

The Wigner--Seitz cell of a lattice point consists of points at least as close to it as to every other lattice point. Its faces lie on perpendicular bisectors to neighbouring lattice points.

A smooth quadric surface is a smooth two-dimensional quadric hypersurface. Over an algebraically closed field of characteristic other than two, every smooth quadric surface in ${\mathbb{P}}^3$ becomesafter a projective linear transformation. The Segre embedding identifies it with ${\mathbb{P}}^1\times {\mathbb{P}}^1$.

The Segre embedding sendsand realizes ${\mathbb{P}}^m\times {\mathbb{P}}^n$ as a projective variety.

The coordinate projections from a product send $(x,y)$ to $x$ and $y$, respectively. For products of projective varieties, they are morphisms.

An umbilical point of a surface is a point where the two principal curvatures are equal, equivalently where the shape operator is a scalar multiple of the identity.

Attach three 2-cells to $S^1\vee S^1$ along loops representing $a^2$, $b^3$, and $(ab{)}^2$. The resulting group isIndeed, $a\mapsto (12)$ and $b\mapsto (123)$ gives a surjection to $S_3$, while $aba=b^{-1}$ reduces every word to one of $b^i$ or $a{b}^i$, so the presented group has at most six elements.

For a bounded stopping time $T$ and a square-integrable representation $M_n=M_0+{\sum }_{k\le n}{B}_k{\xi }_k$,The cross terms vanish because predictable multiples of distinct independent signs are orthogonal martingale differences.

A cardioid is an epicycloid with one cusp; a standard polar equation is $r=a(1+\cos \theta )$.

For each group $g$, let independent cell counts have Poisson means ${\mu }_{gj}$. Conditional on their total $n_g$, the cells have a multinomial distribution with probabilitiesIncluding a free group main effect in the Poisson log-linear model makes its profiled likelihood for the remaining parameters proportional to the corresponding multinomial likelihood. The two fits therefore give the same fitted proportions and likelihood-ratio comparisons.

If a closed oriented manifold is covered by two open sets and a positive-degree cohomology class restricts to zero on both, then its cap product with the fundamental class vanishes in every degree strictly below the top. A chain proof uses the small simplex theorem and replaces the two restricted cocycles by coboundaries.

If $A$ is the adjacency matrix of a directed covering graph, then $\mathrm{tr}(A^n)$ counts its pointed closed walks of length $n$. For a prime number $p$, subtracting the $\mathrm{tr}(A)$ constant walks and identifying the $p$ cyclic choices of starting point givesprimitive closed itineraries of length $p$.