For $n\ge 3$, a full star graph requires every nonincident edge to be certified absent, and fixing those bits suffices. Rejection is certified by at most three present edges with empty common intersection; a triangle needs all three. Hence query certificate complexity is $\max \{3,(\genfrac{}{}{0ex}{}{n-1}{2})\}=\Theta (n^2)$ for this property.

If $G:\mathcal{C}\to \mathcal{D}$ preserves small categorical limits, the projection $(X\downarrow G)\to \mathcal{C}$ creates the small categorical limits available in $\mathcal{C}$. A categorical cone of arrows $X\to G{A}_j$ induces a unique arrow $X\to G(\lim A_j)$ by preservation; this equips the underlying categorical limit with its comma category structure. The universal property proves that every categorical cone factorization preserves that structure.

The slice over $A$ has morphisms $x:X\to A$ as objects and commuting triangles $yu=x$ as arrows $u:(X,x)\to (Y,y)$. Its terminal object is $1_A$, regardless of whether $\mathcal{C}$ has a terminal object. It is the comma category $(1_{\mathcal{C}}\downarrow A)$, viewing $A$ as a functor from the terminal category.

The dimensional matter power spectrum contains a factor $k^{n_s}$, rather than $k^{n_s+3}$. The cosmological transfer function tends to one below the matter-radiation equality scale and to $\ln (k/k_{\mathrm{eq}})/(k/k_{\mathrm{eq}}{)}^2$ above it. The smooth spectrum therefore scales as $k^{n_s}$ at small wavenumber and $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$ at large wavenumber. The logarithmic factor records slow cold-matter growth during radiation domination. Multiplying by $k^3$ instead produces dimensionless matter power.

Suppose $1\to X\to \mathcal{G}\to \Gamma \to 1$ is an exact sequence of pro-p groups, with $X$ abelian and $\Gamma \cong {\mathbb{Z}}_p$. Conjugation gives $X$ a compact Galois module structure. The closed commutator subgroup is $(\gamma -1)X$: its image is closed by compactness, and after quotienting by it, a lift of $\gamma$ centralizes $X$ and topologically generates the remaining quotient. Hence $0\to X_{\Gamma }\to {\mathcal{G}}^{\mathrm{ab}}\to \Gamma \to 0$ is exact.

An antiperiodic free fermion field with inverse propagator ${\partial }_{\tau }+\xi$ has a thermally normalized determinant $1+e^{-\beta \xi }$. The factor counts empty and occupied states of one level. Its logarithmic Matsubara sum is meaningful only with a regularization fixing the determinant normalization; an unqualified infinite product is not convergent.

A CNOT gate can correlate either the original control's computational label with a computational pointer, or the original target's sign-basis label with a sign-basis pointer. For the second interpretation prepare the original control in $|+⟩$ and read it in the Hadamard basis. Its two pointer states become $|+⟩$ and $|-⟩$, measuring $X$ on the original target. An apparatus role therefore depends on the ready state and readout as well as the measurement interaction.

The design effect is a sampling variance under a complex design divided by the corresponding simple-random-sampling variance. For independent equal-size clusters of size $m$ with exchangeable intraclass correlation $\rho$, a common mean-estimation approximation is $1+(m-1)\rho$. It expresses the loss of effective sample size from positive correlation; finite-population and unequal-cluster-size designs require their own calculation.

For a regular uncountable cardinal $\lambda$ and $B:\lambda \to [\lambda {]}^{\omega }$, the limit ordinals closed under all earlier values of $B$ form a club set. Countably iterate bounds for ${\bigcup }_{\alpha <\gamma }B(\alpha )$ to obtain unboundedly many closure points. Regularity keeps each bound and the countable supremum below $\lambda$. Limits of closure points remain closure points.

For an irreducible variety which is an affine variety of dimension $d$ and a nonzero nonunit regular function $f$, each irreducible component of $V(f)$ has dimension $d-1$. Choose a closed point on just the component in question. In its local ring $R$, $\sqrt{(f)}$ is that component's prime ideal. A system of parameters of $R/(f)$, lifted and supplemented by $f$, generates a maximal-primary ideal; the Krull height theorem gives $\dim R/(f)\ge d-1$. Extending any chain above $(f)$ by the zero prime of the domain gives the reverse inequality. This proof does not identify dimension with transcendence degree.

A linear map $T:E\to F$ between Fréchet spaces is continuous if its graph of a linear operator is closed in $E\times F$. The closed graph is itself a Fréchet space. Its first-coordinate projection is a continuous linear bijection onto $E$, so the open mapping theorem for Fréchet spaces makes the inverse continuous. Compose that inverse with the second-coordinate projection to obtain $T$. Completeness of both original spaces is part of the theorem.

The nonnegative integer laws whose probabilities satisfy this recurrence for all $n\ge 1$. Their probability generating function solves $(1-az)G^{\prime }(z)=(a+b)G(z)$. This is the $(a,b,0)$ count class used in the Panjer aggregate algorithm; the count recurrence itself should be distinguished from that aggregate recursion. It includes the Poisson distribution, binomial distribution, and negative binomial distribution with suitable parameters.

The circuit value problem restricted to AND and NOT gates remains P-complete. Replace each OR by $\neg (\neg x\wedge \neg y)$, using a constant-size gadget. Gate indices and gadget positions can be emitted in logarithmic space, and assignments are preserved. Constants, when needed, can be represented by fixed-valued input nodes.

OR together one minterm for each accepted length-$n$ input. Each minterm is an AND of the appropriate Boolean literals. There are at most $2^n$ minterms, and their negated input wires can be shared, giving $O(n{2}^n)$ bounded-fan-in gates. This is a nonuniform existence statement, not an algorithm for determining an undecidable truth table.

A cyclic chain of distinct circles between two disjoint circle boundaries, each tangent to both boundaries and its two neighbours. In concentric coordinates with inner radius $\rho$ and outer radius $R$, the chain circles have radius $r=(R-\rho )/2$ and centre distance $d=(R+\rho )/2$. A simple disjoint chain with $n\ge 3$ closes when $r/d=\sin (\pi /n)$.

For spin-projective measurements on a two-qubit state, let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ be the eigenvalues of $T^{T}T$, where $T$ is its Pauli correlation tensor. Bob's sum and difference of unit axes are orthogonal and have lengths $2\cos \varphi$ and $2\sin \varphi$. Maximizing Alice's two axes aligns them with the corresponding images under $T$. Maximizing $\varphi$ then gives $2\sqrt{\Vert T\mathbf{u}{\Vert }^2+\Vert T\mathbf{v}{\Vert }^2}$ for orthonormal $\mathbf{u},\mathbf{v}$. The maximum of this quadratic-form sum is ${\lambda }_1+{\lambda }_2$, by diagonalizing $T^{T}T$ and selecting its top two eigenvectors. Thus the displayed expression is both an upper bound and attainable. The same maximum applies to the form with the sum of two absolute values, since local outcome signs can select their positive branches.

For fixed $p$ and $\gamma >0$, Janson inequality bounds the failure probability for a given large vertex subset by $\exp (-c|U{|}^2/(\log n{)}^4)$. A union bound over all at most $2^n$ subsets still tends to zero. This uniform statement is essential because vertex sets left by a greedy colouring procedure depend on the graph; one cannot assume each adaptively chosen remainder is an independent fresh random graph.

A Dirichlet character with even parity satisfies $\chi (-n)=\chi (n)$, so its extension to integer arguments is an even function.

A Dirichlet character with odd parity satisfies $\chi (-n)=-\chi (n)$, so its extension to integer arguments is an odd function.

For any group, take the group-algebra image on the direct sum of two finite-dimensional complex semisimple representations. Its radical annihilates their simple summands and hence the faithful direct sum, so the image algebra is semisimple. Equality of traces on group elements gives equality on their linear span. Traces on primitive central blocks then recover every simple multiplicity, proving that equal characters imply isomorphism.