Let $N\sim \mathrm{Pois}(\lambda )$ be independent of independent and identically distributed random variables $(Y_i,Z_i)$ with finite second moments. Conditioning on $N$ gives covariance $N\mathrm{Cov}(Y_1,Z_1)$ and conditional expectations $N\mathbb{E}{Y}_1,N\mathbb{E}{Z}_1$. Law of total covariance and $\mathrm{Var}N=\lambda$ yield the formula. For per-claim excess of loss reinsurance payouts $Y=\min (X,M)$ and $Z=(X-M{)}_{+}$, it becomes $\lambda M\mathbb{E}[(X-M{)}_{+}]$. The insurer and reinsurer aggregates share their claim count and are generally dependent.

If a Poisson distribution count $M$ of clusters has mean $\nu$, and each cluster independently contains a Poisson number of claims of mean $\lambda$, then the total count is a compound Poisson distribution with the displayed probability generating function. Conditioning gives $\mathbb{E}N=\nu \lambda$ and $\mathrm{Var}N=\nu \lambda (1+\lambda )$, so it is not Poisson for positive parameters. With independent severities of mean $\mu$ and variance ${\sigma }^2$, the aggregate payment has mean $\nu \lambda \mu$ and variance $\nu [\lambda ({\sigma }^2+{\mu }^2)+{\lambda }^2{\mu }^2]$. independent validity thinning replaces the per-cluster count mean by $\lambda (1-p)$ and multiplies expected payment by $1-p$.

If independent summands have exponential distribution of mean $\mu$ and the independent count has a zero-based geometric distribution of parameter $p$, their random sum of independent claims has an atom $p$ at zero and an exponential positive component of rate $p/\mu$. Its Laplace transform of a nonnegative random variable is $p/[1-(1-p)/(1+\mu s)]=p+(1-p)p/(p+\mu s)$. Thus $\mathbb{P}(S>x)=(1-p)e^{-px/\mu }$ for $x>0$. Unlike the geometric sum of exponential variables with a positive count, this law is not purely exponential.

For independent factories with accident probability $p$, and independent Poisson distribution claim counts of mean $\lambda$ per accident, the accident count is binomial and each accident supplies a cluster. The random-sum transform identity gives the displayed probability generating function. In particular the no-claim chance is $(1-p+p{e}^{-\lambda }{)}^m$; a factory with an accident can still have zero claims. The binomial accident law has Panjer parameters $a=-p/(1-p)$ and $b=(m+1)p/(1-p)$.

For an even-rank real vector bundle $\xi$, a stable complex structure is a complex structure on $\xi \oplus {\mathbb{R}}^{2s}$, considered up to adding trivial complex bundles and homotopy. This gives a stable complex class but does not necessarily make $\xi$ itself a complex vector bundle. Stable Chern classes therefore need not recover the unstable Euler class: $T{S}^4\oplus {\mathbb{R}}^2$ can be made trivial complex, while $e(T{S}^4)$ evaluates to $2$.

This is a smoothly varying positive-definite Hermitian form on each fiber, without requiring a holomorphic base or bundle. A partition of unity glues local positive metrics into a global one. Applying smooth Gram-Schmidt to local frames then makes the transitions unitary, reducing the structure group of a vector bundle to $U(k)$. For a line bundle its unit vectors form a principal bundle for the circle group.

For a crack on the real axis under applied shear traction $p$, write $G=\mu F^{\prime }$ for the additional complex stress potential for antiplane shear derivative. Reflection of the induced displacement gives $G^{-}=\overline{G^{+}}$. Traction-free crack faces therefore require $G^{+}+G^{-}=-2p$; intact portions have $G^{+}=G^{-}$. This is a scalar Riemann-Hilbert problem. Endpoint regularity, single-valued displacement and the absence of additional remote loads fix its homogeneous terms.

Substituting the free complex scalar field modes into $i\int ({\varphi }^{†}\dot{\varphi }-{\dot{\varphi }}^{†}\varphi )d^{3}x$ gives the formally ordered expression $Q_{\mathrm{bare}}=\int d{\Pi }_p(a^{†}a-b{b}^{†})$. Its divergent vacuum constant is removed by normal ordering, equivalently by choosing $Q|0⟩=0$. The resulting complex scalar charge operator has $[Q,a^{†}]=a^{†}$ and $[Q,b^{†}]=-b^{†}$. Subtracting the constant changes neither its action on fields nor current conservation, but it is essential when identifying the vacuum charge and the charges of one-particle states.

Every scalar holomorphic function on a compact connected complex manifold is constant by the maximum modulus principle. Apply this to each entry of the holomorphic Adjoint representation of a Lie group; its value is constantly the identity. Differentiation makes the complex Lie algebra abelian, hence connectedness makes the group abelian. The holomorphic Lie exponential map identifies it with ${\mathbb{C}}^n$ modulo a discrete subgroup. Compactness forces that subgroup to span the underlying real vector space, giving a full-rank Euclidean lattice and a complex torus.

For real $c_1<\cdots <c_n$ and $C=\mathrm{diag}(c_1,\dots ,c_n)$, the critical points are the coordinate projections $P_I$, indexed by $k$-subsets $I=\{i_1<\cdots <i_k\}$. Indeed $df([A,P])=\mathrm{tr}([P,C]A)$, which vanishes for every skew-Hermitian $A$ exactly when $[P,C]=0$. In the graph chart at $P_I$,Every coefficient is nonzero, so the Hessian matrix is nondegenerate. Each pair $j<i$ supplies two negative real directions, giving the displayed Morse index. The function is a Morse function even if distinct critical points happen to have equal critical values.

This operation reflects a holomorphic function across the real axis while preserving holomorphic dependence on the new argument. A transform of real data with real kernel coefficients satisfies $F^{♯}=F$. In a polygonal global relation, complex rotation factors are conjugated as well; an explicit prefactor $i$ changes sign.

A Dedekind cut can be represented by a nonempty proper subset $A$ of the rational numbers satisfying two conditions: if $q\in A$ and $r<q$ is rational, then $r\in A$; and $A$ has no maximum. The complementary set contains all the rational values on the other side of the boundary. Using the convention that the lower set never includes its boundary avoids two representations of a rational boundary.

The set $\{q\in \mathbb{Q}:q<0\}$ represents zero. The set $\{q\in \mathbb{Q}:q<0\text{ or }{q}^2<2\}$ represents $\sqrt{2}$, even though no rational number occupies that boundary by the irrationality of the square root of two. Thus a Dedekind cut describes an irrational number entirely using rational comparisons. Ordering these cuts by the subset relation and defining their arithmetic gives a construction of the real numbers. ^[ref].

Let E be the even positive integers, and take a finite-symmetric-difference parity colouring $\epsilon$ of $[E{]}^{\omega }$. The family $\{\{1\}\cup X:X\in [E{]}^{\omega },\epsilon (X)=0\}$ is Ramsey even under the every-ground-set convention, since every infinite ground set can be thinned to omit 1. It is not completely Ramsey: inside $[\{1\},E]$, every infinite tail has two subfamilies of opposite parity. Its supporting neighbourhood is closed and nowhere dense in the ordinary topology on infinite subsets, but open in the Ellentuck topology.

Choose ordered minimal topological generators $a_1,\dots ,a_d$ of a uniform pro-p group for odd $p$, put $b_i=a_i-1$, and choose $1/(p-1)<s<1$. Ordered monomials $b^{\alpha }=b_1^{{\alpha }_1}\cdots b_d^{{\alpha }_d}$ give power-series coordinates in the integral completed group algebra. The displayed Non-Archimedean absolute value filtration gives a complete normed ${\mathbb{Q}}_p$-algebra after completion of the rational group algebra. The power relation raises valuation by one, and interchanging $b_i,b_j$ adds terms of valuation at least $1+s>2s$, so multiplication does not decrease total valuation. The p-adic logarithm and p-adic exponential are inverse on the relevant small balls. In particular $\log G$ is a faithful realization of the Lie algebra of a uniform pro-p group, useful for verifying its intrinsic limit operations and the Baker--Campbell--Hausdorff formula without assuming the correspondence in advance.

On ${\mathbb{R}}^2$, the vector fields $X=y{\partial }_x$ and $Y=(x^2/2){\partial }_y$ are complete, since their local flows are $(x+ty,y)$ and $(x,y+t{x}^2/2)$. Their Lie bracket of vector fields is $-(x^2/2){\partial }_x+xy{\partial }_y$. Its integral curve of a vector field starting at $(2,0)$ is $(2/(1+t),0)$, which cannot continue through $t=-1$. The sum has an integral curve of a vector fieldthat cannot continue through $t=2\sqrt{3}$. Thus neither operation preserves complete vector fields on a general noncompact smooth manifold. This contrasts with a compact manifold without boundary, where every smooth vector field is complete.

An $r$-uniform hypergraph is strongly saturated with respect to a complete uniform hypergraph of order $r+t$ when every missing edge immediately completes such a clique. Existing cliques are allowed. All $r$-sets meeting a fixed $t$-set give an example: the witness for a missing edge is its union with that fixed set. It has $\left(\genfrac{}{}{0ex}{}{n}{r}\right)-\left(\genfrac{}{}{0ex}{}{n-t}{r}\right)$ edges, and the exterior-power quotient bound for strong hypergraph saturation proves optimality.

A partially ordered set is directed-complete if every directed set in it has a least upper bound. A complete lattice is directed-complete. Conversely, directed completeness together with all finite joins, including the empty join, gives arbitrary joins: take the directed set of joins of finite subsets.

For a Banach space $E$ satisfying $E\cong ({⨁}_{n\ge 1}E{)}_p$, a complemented subspace $W$ is absorbed by $E$: write $E\cong W\oplus V$, expand the countable sum, and absorb one additional copy of $W$ into the countably many $W$ summands. Hence $E\oplus W\cong E$. If $Z$ is complemented in $E$ and contains a complemented copy of $E$, write $Z\cong E\oplus W$ with $W$ complemented in $E$; the preceding argument gives $Z\cong E$. The decompositions are Banach space isomorphisms, not merely algebraic decompositions.

If a counting process $N$ has predictable conditional intensity $\lambda$, subtracting its compensator of a counting process gives the associated martingale $M$. Conditional expected event increments and compensator increments agree, so their difference is conditionally centered. With the necessary integrability, $M$ is a mean-zero martingale; for a simple no-ties process its predictable quadratic variation is ${\int }_0^t\lambda (u)du$. A predictable weighted martingale integral has mean zero and variance determined by the square of its weights. In survival analysis, $\lambda =Yh$ produces the event-minus-expected representation used to derive the Nelson–Aalen estimator.

The metric construction of a compatible almost complex structure is continuous in the auxiliary Riemannian metric, and recovers $J$ when applied to $g_J=\omega (\cdot ,J\cdot )$. Interpolating $g_J$ linearly to a fixed auxiliary metric therefore contracts the space of compatible almost complex structures. In particular it is nonempty and path connected.