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For a nondegenerate diagonal quadratic form $q=⟨a_1,\dots ,a_m⟩$, this product of quadratic Hilbert symbols is independent of diagonalization. Over a p-adic field, dimension, determinant square class and this invariant classify the form up to isometry. The product of the local invariants of a globally diagonalized form is one by the Hilbert reciprocity law; real places additionally record signature.

For two increasing events of equal probability $a$ under a product measure, positive association of their complements gives $1-P(A\cup B)\ge (1-a{)}^2$. Hence $a\ge 1-\sqrt{1-P(A\cup B)}$. For $r$ equal-probability increasing events, the same reasoning gives $a\ge 1-(1-P({\bigcup }_j{A}_j){)}^{1/r}$.

For independent integrals $G_1=H,G_2,\dots ,G_r$, the Hamiltonian $K=H+\frac{1}{2}{\sum }_j{G}_j^2$ agrees with $H$ to first order on the common zero level. On a nonempty level $G=c\ne 0$, the vector-field difference is ${\sum }_j{c}_j{X}_{G_j}$ and is nowhere zero by independence of the differentials. Nonemptiness is necessary if one expresses the conclusion as inequality between restricted vector fields.

On a symplectic manifold, use ${\iota }_{X_f}\omega =-df$ and $\{f,g\}=\omega (X_f,X_g)=X_f(g)$ for the Poisson bracket. Cartan's magic formula gives ${\mathcal{L}}_{X_f}\omega =0$. Taking the interior product of a differential form with $[X_f,X_g]$ then yields $-d\{f,g\}$, proving the homomorphism. Its kernel consists of locally constant functions, since $X_f=0$ precisely when $df=0$. On a connected manifold these are the constants. The quotient of smooth functions by this kernel is therefore the Lie algebra of Hamiltonian vector fields. A different contraction sign requires a compatible bracket sign.

For $Q=\hat{p}-isA(x)$ with real $A$ and positive $s$, the factorization gives potential $(s^2{A}^2-s\hslash A^{\prime })/(2m)$ and a nonnegative energy form. Its zero mode is proportional to $\exp [-s\int Adx/\hslash ]$. Taking $A=x$ and optimizing positivity over $s$ gives the Heisenberg uncertainty principle for states with finite second moments.

Every finite soluble group has a Hall subgroup for each prime set. Induct on the group order using an elementary abelian minimal normal subgroup. In the coprime hard case, lift a minimal normal subgroup of the quotient, choose its Sylow subgroup, and apply the Frattini argument. A proper normalizer reduces the order; a normal Sylow subgroup allows induction in its quotient. This proves existence without assuming an independent complement theorem.

The normalized Haar scaling functions and wavelets obeyThis orthogonal two-by-two transformation gives the multiresolution analysis decomposition $V_{j+1}=V_j\oplus W_j$. Iterating it expresses a Haar approximation either through level-$j$ cell averages or through coarse averages and all wavelet details at lower levels. The identities remain valid locally for coefficient integrals of locally integrable functions, without a global $L^2$ assumption.

A greenhouse gas absorbs and emits planetary thermal radiation and therefore contributes to the greenhouse effect. Its climatic impact depends on absorption spectrum, concentration, lifetime and chemical effects. Carbon dioxide, methane and water vapour are important examples.

Evaluating the potential of a razor-thin disk at height $ϵ$ gives a softened horizontal force. Its Fourier multiplier is $e^{-kϵ}$ times the razor-thin disk Poisson kernel. This suppresses short wavelengths while retaining the long-wave potential. A real finite-thickness profile generally has a different full reduction factor.

The parity grading records whether a homogeneous element has even or odd Grassmann degree. Homogeneous elements obey $XY=(-1{)}^{|X||Y|}YX$. An odd differential uses the graded Leibniz rule; this is the sign bookkeeping behind left-acting BRST differential calculations.

A natural number coding a finite syntactic object under a fixed Gödel numbering. For program descriptions it records a function in intension; different codes may denote the same function in extension.

The graph of the factorial function has a fixed-length first-order formula in the language of ordered rings interpreted on the natural numbers. A remainder code records $a_0=1$, $a_n=y$, and $a_{i+1}=(i+1)a_i$ for every $i<n$. Uniqueness of remainders proves $y=n!$; sufficiently large pairwise-coprime moduli and the Chinese remainder theorem supply a code for every input. Formula length is independent of the input variable's value, although substituting a numeral adds its encoding length.

For an onto strategyproof rule on unrestricted strict preference orders, rank-raising monotonicity implies unanimity and respect for unanimous pairwise preference. Promote each alternative pair to the top two positions in all reports. The selected member depends only on each voter's comparison of that pair. These binary choices form a strict transitive social order: a directed three-cycle would contradict the rule's choice when that triple is placed above all other alternatives, because the chosen member must win both of its binary comparisons. The original outcome is the top of this order. This creates a social ordering with pairwise independence without presuming it as an axiom of the original choice rule.

With Hermitian $b_0,c_0$ and $\{b_0,c_0\}=1$, a ket annihilated by $b_0$ has zero bare full-ghost norm. Nonzero amplitudes need FP ghost zero-mode saturation. Low-level Virasoro descendant calculations can instead normalize the highest-weight coefficient and compute $L_n{L}_{-n}|\Omega ⟩$ algebraically. Such normalized contravariant coefficients are not positive-definite Hilbert norms. The pairing convention must be stated before using FP ghost descendants in a norm argument.

Objects are geometric formulas in context modulo provable equivalence. Morphisms are provably total, single-valued geometric relations; composition is existential conjunction. Conjunction and equality give finite limits. Interpreting formulas in a model gives a finite-limit-preserving functor, with the additional continuity condition supplied by the geometric syntactic topology.

A geometric quotient adds geometric sequents over the same signature. Quotients are compared modulo provable equivalence. In a syntactic site the extra sequents become additional covers, giving a subtopos of the original classifying topos. Stronger axioms correspond to a larger site topology and a smaller subtopos.

Let the summands have gamma distribution with shape two and rate $\beta$, and let the independent positive-support geometric distribution count have parameter $p$. Put $a=\beta (1-\sqrt{1-p})$ and $b=\beta (1+\sqrt{1-p})$. The aggregate moment-generating function factors as $ab/((a-t)(b-t))$, so it is the sum of two independent exponential distributions of rates $a,b$. Its probability density function is $ab(e^{-as}-e^{-bs})/(b-a)$ for $s>0$, nonnegative and integrating to one. This contrasts with the geometric sum of exponential variables, which is itself exponential.

A subtopos is a Grothendieck topos embedded by a geometric embedding, considered up to equivalence over the ambient topos. On a site $(\mathcal{C},J)$, subtoposes correspond to Grothendieck topologies $J^{\prime }\supseteq J$. On a classifying topos this is the duality between geometric quotients and subtoposes.

For a positive-integer geometric distribution $P(N=j)=p{q}^{j-1}$ with $q=1-p$, the capped time has masses $p{q}^{j-1}$ for $1\le j<k$ and $q^{k-1}$ at $k$. The last mass includes all later arrivals, not only $N=k$. Its probability generating function and expected value areThe expectation follows either by differentiating the PGF or summing the tail probabilities up to $k$.

For a geodesic variation $F(s,\tau )$, the deviation vector $S=F_{\ast }{\partial }_s$ describes infinitesimal separation at equal parameter values. With $T=F_{\ast }{\partial }_{\tau }$, commuting parameter derivatives give $[S,T]=0$. A torsion-free connection then gives ${\nabla }_{T}S={\nabla }_{S}T$. For an affinely parametrized variation, geodesic deviation states ${\nabla }_T^{2}S=R(T,S)T$ with the convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$.