A set $N$ is completely Ramsey-null when every Ellentuck topology neighborhood $[s,A]$ has a refinement $[s,B]$ disjoint from $N$, with the same finite stem $s$. Assuming open Ellentuck sets are completely Ramsey, every star-nowhere dense set is completely Ramsey-null: apply the open-set property to the dense complement of its closure. The Ellentuck meagre-set fusion lemma shows that countable unions retain this avoidance property.

The three-uniform tetrahedron is the hypergraph on four vertices with all four possible triple hyperedges. It is the complete uniform hypergraph $K_4^3$, and differs from its geometric realization as a tetrahedron. Its edge pattern encodes four-term arithmetic progressions.

For a straight shearing-sheet encounter, $V=S|x|$, and the outer and inner longitudinal changes have opposite signs. The complete-flyby gravitational impulse gives $\chi =2$. Some simplified prescribed kick models use a different coefficient; carrying it explicitly makes the corresponding one-sided impulse torque on a disk and migration rates unambiguous. This coefficient does not assert an exact solution of the full rotating Hill encounter.

Let $A$ be a finite-dimensional semisimple algebra over an algebraically closed field $k$, acting faithfully on a finite-dimensional vector space $T$. Write its isotypic decomposition as $T={⨁}_i{S}_i{\otimes }_k{M}_i$. The image of $A$ acts by the full matrix algebra ${\mathrm{End}}_k(S_i)$ on each simple factor, while its commutant of an operator algebra acts by ${\mathrm{End}}_k(M_i)$ on the multiplicity factors. Taking the commutant twice therefore recovers the original image. This is the algebraic mechanism behind Schur–Weyl duality.

The splitting assumption matters for this matrix-factor description. Over a general field, simple factors can involve division algebras, and multiplicity spaces and endomorphisms must be taken over those division algebras. For example, the real algebra of quaternions acting on itself by left multiplication has the opposite quaternion algebra as its commutant, rather than an ordinary full real matrix algebra.

Three pairwise ultraparallel hyperbolic lines may have disjoint common perpendiculars or coincident ones. In the upper-half-plane model, three semicircles of radius $1/2$ centred at $-5/2,0,5/2$ have mutually disjoint common perpendiculars. Three concentric semicircles instead have the same vertical common perpendicular. Thus pairwise ultraparallelism alone does not determine the incidence of their perpendiculars.

For a proper hereditary graph property with unbounded orders, the colouring number determines the leading labelled speed. The lower bound comes from one contained clique-independent partition class. For the upper bound, forbid one induced graph from each next-size partition type and use the induced regularity template lemma. Its intermediate-pair graph is clique-free, and sparse or almost-complete pairs have negligible entropy. Bounded-order properties have colouring number zero and are outside this formula.

On the projective line, an infinite direct sum of copies of $\mathcal{O}(-1)$ is a quasi-coherent sheaf but not a coherent sheaf: at any point its residue-vector-space dimension is infinite. Its global sections vanish because each summand has no sections, and sections commute with direct sums for the finite two-chart equalizer. Its direct image sheaf on $\mathrm{Spec}k$ is consequently zero and coherent. This shows that a nonaffine direct image can lose information needed to detect coherence.

A field of characteristic $p$ satisfies ${\mathrm{cd}}_p(K)\le 1$. The Artin-Schreier sequence and vanishing of positive-degree cohomology of the additive separable-closure module yield the bound. It concerns discrete $p$-primary modules; it does not assert that the $p$-primary Brauer group vanishes.

For a pro-p group $P$, vanishing of $H^{n+1}(P,{\mathbb{F}}_p)$ implies vanishing in that degree for every discrete $p$-primary module. Finite such modules have composition factors equal to the trivial module ${\mathbb{F}}_p$, so the long exact sequence gives the result by induction. General modules are unions of finite stable submodules, and continuous cohomology commutes with filtered unions. Dimension shifting gives ${\mathrm{cd}}_p(P)\le n$. The converse is immediate from the definition.

A coherent sequent true in all set models of a coherent theory is coherently derivable. A generic underivable sequent remains false after the Boolean cover by a generic quasi-closed subtopos. Slice over its nonzero Boolean counterexample to show classical consistency of the theory with a named tuple satisfying the antecedent and negated consequent. The Henkin construction then gives a set-valued countermodel. This argument does not assume that every Boolean topos has points.

For unit Euclidean norm columns, cumulative coherence measures the largest sum of $k$ correlations with one other column. Its domain is $0\le k\le N-1$, with ${\mu }_1(0)=0$ by the empty-sum convention. It is nondecreasing, ${\mu }_1(1)=\mu$ for $N\ge 2$, and ${\mu }_1(k)\le k\mu$, where $\mu$ is the mutual coherence. Keeping a sum of actual correlations can be substantially sharper than replacing each summand by the largest one.

The excess measures a coalition's complaint against an allocation: a positive excess means its own attainable value exceeds its assigned total. The core of a cooperative game requires every excess to be nonpositive at an efficient allocation. The nucleolus lexicographically minimizes the decreasingly sorted excess vector over imputations.

For a co-computably enumerable equivalence relation, the least element of each equivalence class forms a complete transversal of a set family. Membership is semidecidable: run the finitely many inequivalence tests against smaller numbers in parallel and accept when all succeed. Infinitely many classes are needed only to make the resulting transversal infinite.

For an infinite cardinal $\kappa$, the square principle supplies club sets $C_{\zeta }\subseteq \zeta$ at limit $\zeta <{\kappa }^{+}$, with order type at most $\kappa$, coherent at limit points: $C_{\gamma }=C_{\zeta }\cap \gamma$ whenever $\gamma$ is a limit point of $C_{\zeta }$. The bound precludes a single club set threading the entire sequence. For regular uncountable $\kappa$, coherence together with the weaker-looking size clause $\mathrm{cf}(\zeta )<\kappa \Rightarrow |C_{\zeta }|<\kappa$ implies the order-type bound. At $\kappa =\omega$, that clause alone is vacuous and cannot replace the bound.

Geometric theories are Morita-equivalent when their classifying toposes are equivalent. This gives equivalent internal model categories pseudonaturally in every Grothendieck topos. A common classifying topos transports intrinsic invariants between different presentations; agreement of set-valued model categories alone is insufficient. The duality between geometric quotients and subtoposes transfers geometric theory extensions along such an equivalence.

Each geometric quotient theory is classified by the corresponding subtopos of the original classifying topos. Extra axioms impose extra covers on the geometric syntactic topology. Conversely additional definable covers give the corresponding deductively closed quotient. Under Morita equivalence of geometric theories, transporting a subtopos produces a corresponding quotient of the other theory, with equivalent classifiers. Stronger axioms correspond to smaller subtoposes under inclusion.

In a one-period complete market, an invertible payoff Gram matrix identifies the unique holdings replicating each claim. The pricing weight is $Z=P_0^T{Q}^{-1}{P}_1$ and the random portfolio kernel is $W=Q^{-1}{P}_1$. Positive definiteness gives uniqueness of holdings; positivity of $Z$ requires an additional no-arbitrage condition.

The quotient metric of circumference one uses the shorter distance around the circle: $\Vert x\Vert ={\min }_{m\in \mathbb{Z}}|\tilde{x}-m|$. It also measures separation across the identified endpoints. This is the metric used by circular spacing in the analytic large sieve inequality.

Church numeral successor, addition, multiplication, and exponentiation are represented by $\lambda nfx.f(nfx)$, $\lambda mnfx.mf(nfx)$, $\lambda mnfx.m(nf)x$, and $\lambda mnfx.(nm)fx$. The last term takes the base first and the exponent second. Its outer abstractions preserve the usual numeral normal form at exponent zero, implementing $m^0=1$, including $0^0=1$. At base type $A$, exponentiation uses an exponent at Church type $C_{A\to A}$ and a base at $C_A=(A\to A)\to A\to A$.

For an integrable response with continuous strictly increasing distribution function $F$, the population risk for the check loss has derivative $F(q)-\tau$. Bounded difference quotients justify differentiation by dominated convergence. Integrating the derivative between the unique quantile and another location proves strict optimality. This proves identifiability without requiring a positive density.