The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.

For initially well-separated Kerr black holes with no orbital angular momentum, exact axisymmetry gives $J_f=J_1+J_2$. Put $X=(A_1+A_2)/(8\pi )$. The black-hole area theorem and the irreducible mass identity implyIndeed, $X\ge |J_1|+|J_2|\ge |J_f|$, and $\frac{1}{2}(Y+J_f^2/Y)$ is nondecreasing for $Y\ge X$. Opposite spin signs lower $J_f^2$ at fixed initial areas and hence raise the allowed radiated-energy bound; this need not be achievable in an actual merger.

For an irrational skew shift, the averages of every continuous function $g$ converge uniformly in the starting point to $\int gd{m}_2$, with $m_2$ normalized Lebesgue measure. Nonconstant Fourier basis characters have either linear or quadratic phases. Linear phases are bounded geometric series; for quadratic phases the Van der Corput inequality for finite scalar sequences reduces to linear correlations of irrational frequency, uniformly in the starting point. Approximation by trigonometric polynomials proves the assertion and identifies every invariant Borel probability measure as $m_2$.

An atmosphere that absorbs and reemits thermal radiation can reduce outgoing energy at a given surface temperature, requiring a warmer surface to balance absorbed stellar energy. The planetary equilibrium temperature refers to the radiation budget, not the resulting surface temperature. Water, carbon dioxide and clouds can alter this balance.

The Hazel model is a smooth stratified shear flow whose base velocity has a hyperbolic-tangent profile and whose squared buoyancy frequency has a hyperbolic-secant-squared profile. The equal-width nondimensional example is $U=\tanh z$ and $N^2=J{\mathrm{sech}}^{2}z$. For $J\ge 0$, its gradient Richardson number is $J{\cosh }^{2}z$, with minimum $J$.

This three-layer shear flow has $U=\Delta Uz/h$ and density ${\rho }_0$ between $z=\pm h/2$. Above and below, the velocity is the corresponding constant $\pm \Delta U/2$, and the mass density is ${\rho }_0\mp \Delta \rho /2$. The density drops and vorticity jumps coincide at both interfaces. Its hydrodynamic stability involves coupled gravity-vorticity interface waves, unlike a profile with uniform shear extending through all three layers.

Let $G_a$ be the Dirichlet Green function of $D^2-a^2$ on $[-1,1]$, with $a=|\alpha |$:It is continuous and has derivative jump one, so $(D^2-a^2)G_a=\delta (z-\xi )$. The Dirac delta multiplication identity gives $(z-\xi )\delta (z-\xi )=0$, proving it is a generalized eigenfunction of the inviscid Couette continuous spectrum. Integrating $G_a(z,\xi )q_0(\xi )e^{-i\alpha \xi t}$ reconstructs the evolving vertical velocity from its initial vorticity.

For $r=n-1$, the Wang sequence gives an additive splittingThe kernel is free, so the splitting exists as groups. The map in even degree is zero; the map in odd degree is multiplication by minus two.

The base class $t$ and coordinate classes $x_i$ all have degree one. Pullback of the mod-two intersection pairing of the Klein bottle gives $x_i^2=t{x}_i$. The Leray-Hirsch theorem gives the basis consisting of square-free products $x_I$ and $t{x}_I$. Thus the additive dimensions match those of a torus, but the degree-one squares differ for $n\ge 2$.

For fixed dimensions $(d_1,d_2,d_3)$, a pair $V_1\stackrel{f}{\to }{V}_2\stackrel{g}{\to }{V}_3$ is classified up to vertexwise change of basis by its three ranks $r,s,t$. The multiplicities of its interval representations of an equioriented three-vertex quiver are $m_{13}=t$, $m_{12}=r-t$, $m_{23}=s-t$, $m_{11}=d_1-r$, $m_{22}=d_2-r-s+t$ and $m_{33}=d_3-s$. Nonnegativity is precisely the existence condition. For dimensions $(2,2,2)$ there are ten orbits, of which $(2,2,2)$ is open; its boundary has nine orbits. The two cases $(1,1,0)$ and $(1,1,1)$ show why the individual arrow ranks are insufficient.

On a compact oriented smooth manifold $N$ of dimension four without boundary, set $Q([\alpha ],[\beta ])={\int }_N\alpha \wedge \beta$ for closed real two-forms. The Stokes theorem makes the expression independent of representatives, and the even degrees make it symmetric. It is the real de Rham cohomology version of the intersection form. The Hodge decomposition theorem and Hodge star operator express it as $Q(h,k)=(h,\ast k{)}_{L^2}$ on harmonic representatives.

A family of words in $[k{]}^n$ is intersecting if any two words agree in some coordinate. For positive $n,k$, its maximum size is $k^{n-1}$: simultaneous cyclic shifts partition the words into classes in which distinct words never agree, and fixing one coordinate attains the resulting bound.

For odd $n$ and $w\ge 1$, an intersecting family contains at most one set from each complementary pair in the Boolean lattice. Choosing the larger member in every pair gives the displayed bound and is feasible, because all sets of size greater than $n/2$ intersect. For $w>1$ it is the unique maximizer; for $w=1$ other maximizing families can exist.

For an intersecting family of $r$-sets with $r\ge 1$, its lower shadow has at least as many members as the family. Apply elementary set shifts and then the ballot reflection injection to obtain this bound.

A set family is two-intersecting if $|A\cap B|\ge 2$ for all of its members, including $A=B$. In particular its members all have at least two elements. This is stronger than being an intersecting family.

For an intersecting family of $r$-sets, some element $x$ is avoided by at most $(r-1{)}^2(\genfrac{}{}{0ex}{}{n-2}{r-2})$ members. If two members meet exactly at $x$, every member avoiding $x$ contains one element from each of their disjoint remainders. If no such pair exists, apply the pair-cover bound for a two-intersecting uniform family.

A pairwise intersecting family of two-element sets is contained in a star graph or in the three edges of a triangle in a graph. If two members are $\{a,b\},\{a,c\}$ and a third avoids $a$, the third must be $\{b,c\}$. Every member meeting all three belongs to that triangle.

For a reflexive pair $f,g:A\rightrightarrows B$ with section $r$ in a preadditive category, each hom-set diagram defines a groupoid. Arrows $a,b:C\to A$ compose when $ga=fb$, by $b\circ a=a+b-rga$. Identity at $x:C\to B$ is $rx$, and the inverse of $a$ is $rfa+rga-a$. Bilinearity proves the endpoint, identity and associativity laws and compatibility with precomposition in $C$. In the hom-set formulation this does not assume that the composable-arrow pullback exists.

A convex three-times differentiable barrier $F$ is self-concordant whenA barrier of parameter $\nu$ additionally satisfies $|DF(x)[h]|\le \sqrt{\nu D^{2}F(x)[h,h]}$ and diverges at the domain boundary. Logarithmically homogeneous cone barriers satisfy $F(tx)=F(x)-\nu \log t$. The orthant barrier $-{\sum }_i\log x_i$ has parameter equal to the dimension, while the Lorentz-cone barrier $-\log (t^2-\Vert z{\Vert }^2)$ on $t>\Vert z\Vert$ has parameter two. Their Hessians define Dikin ellipsoids and control interior Newton steps.

An auxiliary conic optimization problem searches for strict feasibility before following a central path. Given $e\in \mathrm{int}K$, minimizing $t$ subject to $Ax-b+te\in K$ and $t\ge -1$ has a strict feasible start for sufficiently large $t$. Any feasible solution with $t<0$ certifies $Ax-b\in \mathrm{int}K$. If the original problem is strictly feasible, a small negative $t$ is feasible. Equality constraints and dual feasibility require their corresponding auxiliary procedures.