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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.

Hydrostatic pressure continuity across both interfaces fixes the partition of current thickness. The modified current pressure is $\Delta \rho gh$, with $\Delta \rho =({\rho }_0-{\rho }_1)({\rho }_2-{\rho }_0)/({\rho }_2-{\rho }_1)$. The equivalent radial body force is $-\Delta \rho g{h}_r{\mathbf{e}}_r$.

The gravity force per interfacial area is $\Delta \rho g{H}^2/R$, while ambient viscous traction is $\mu u/R$. Their balance gives the radial velocity scale independently of current viscosity within the appropriate plug-flow window.

When vertical diffusion equation controls depth, $H\sim \sqrt{Dt}$, while conserved excess solute mass gives $\Delta \rho R^{2}H\sim \Delta {\rho }_0{V}_0$. Ambient viscous resistance then leads to a $t^{1/2}$ spreading law. The enriched-fluid volume grows, so the fixed-volume $t^{1/5}$ law does not apply.

The instantaneous continuously compounded interest rate. The continuous-time bank account grows at this rate; it equals the instantaneous forward rate at current maturity.

The continuously compounded rate inferred for an infinitesimal investment interval at future maturity $T$, as seen at time $t$. For a unit-face-value zero-coupon bond, $P(t,T)=\exp (-{\int }_t^{T}f(t,u)du)$ and the short rate is $f(t,t)$.

A multiplicative factor converting a value into units of a chosen initial account. With the continuous-time bank account $B$, $D_t=1/B_t$; discounted traded asset prices are martingales under the corresponding equivalent martingale measure.

A model evolving the entire instantaneous forward rate curve. In the displayed one-factor risk-neutral measure dynamics the drift restriction makes every suitably integrable discounted zero-coupon bond price a martingale. The multi-factor form replaces the product of volatilities by their dot product.