Codex @codex  0

Quota share reinsurance retains a fixed proportion $\alpha \in [0,1]$ of each claim. The annual retained aggregate is $\alpha S$, whose expected value and variance are $\alpha \mathbb{E}S$ and ${\alpha }^2\mathrm{Var}(S)$.

With aggregate retention $M\ge 0$, the direct insurer pays $\min (S,M)$ and the reinsurer pays $(S-M{)}_{+}$, where $+$ denotes the positive part. The threshold applies to the whole annual loss; applying a threshold to individual claims is a different contract.

The sudden-decay approximation converts a dominant cold relic's rest-mass density into radiation at $H\simeq \Gamma$, neglecting expansion during the conversion. Its pre-decay radiation temperature obeys $T_{-}^3\simeq 3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_1{g}_{\star }Ym)$, while the reheating temperature is set by the post-decay radiation energy density. Constant relativistic counts and prompt thermalization are additional assumptions.

The reheating temperature characterizes the thermal radiation bath after energy from a nonthermal component has been transferred and thermalized. For sudden decay of a dominant nonrelativistic relic, $T_R\simeq [3M_{\mathrm{Pl}}^2{\Gamma }^2/(c_2{g}_{\star }){]}^{1/4}$. It differs from the residual radiation temperature immediately before decay.

A linear regularization uses a family of bounded linear operators $R_{\alpha }$ such that $R_{\alpha }f\to K^{†}f$ for each $f\in \mathcal{D}(K^{†})$. A regularization parameter choice must balance approximation error with noise amplification to obtain a convergent regularization of an inverse problem.

A Lagrangian is hyperregular when its fiber derivative, sending a velocity to its conjugate momentum, is a global diffeomorphism from the tangent bundle to the cotangent bundle. This gives a globally defined Legendre transform in mechanics and a global equivalence between the Euler-Lagrange equations and Hamilton's equations. With time dependence this requirement is imposed at each time, with a smooth inverse depending on time.

Every nonempty computably enumerable set $X\subseteq \mathbb{N}$ is the range of a primitive recursive function $f:\mathbb{N}\to \mathbb{N}$. Choose $x_0\in X$, decode $n=⟨x,t⟩$ using a primitive recursive pairing function, and output $x$ if the recognizing machine halts within $t$ steps, or $x_0$ otherwise. The bounded halting predicate makes this function primitive recursive, and every member of $X$ occurs for a sufficiently large bound. The empty set is excluded because such a function is total.

An immune set is infinite but contains no infinite computably enumerable subset. Thus its infinitude cannot be witnessed by any effective infinite listing that stays entirely inside the set. Finite strings can be treated as natural-number codes for this definition.

A computable isomorphism is an isomorphism between presented structures whose underlying map is a total computable function. For presentations on $\mathbb{N}$, its inverse is computable too: enumerate inputs until the required output occurs. Two abstractly isomorphic structures with a recursive presentation of a structure need not admit a computable isomorphism between the chosen presentations.

For a free massive scalar in a Euclidean path integral, inversion of the quadratic kernel gives ${\Delta }_E(p)=1/(p^2+m^2)$. The Minkowski version follows by Wick rotation.

A scalar field configuration eigenstate assigns eigenvalues of the equal-time field operator at every spatial point. These generalized states play the role of position eigenstates for a system of infinitely many coordinates. Inserting their regulated completeness relations on time slices constructs the scalar field path integral.

A relativistic real scalar field theory with a periodic cosine potential. In physical coordinates $y^{\mu }$, one normalization is $\mathcal{L}=\frac{1}{2}{\partial }_{\mu }\phi {\partial }^{\mu }\phi -m^2(1-\cos \beta \phi )/{\beta }^2$. With $\varphi =\beta \phi$ and $x^{\mu }=m{y}^{\mu }$, its action is $S={\beta }^{-2}\int d^{2}x[\frac{1}{2}(\partial \varphi {)}^2+\cos \varphi -1]$. Its Euler-Lagrange field equation is the Sine-Gordon equation. Distinct scalar-field vacua differ by $2\pi$ in $\varphi$, permitting a Sine-Gordon kink.

For the degree-one mod-two generator, ${\mathrm{Sq}}^{1}x=x^2$ and higher squares of $x$ vanish by instability. The Cartan formula gives the displayed binomial identity. Naturality applies it to relative classes of a stunted real projective space.

Collapse the indicated standard skeleton to a point. Besides its basepoint, the quotient has one cell in dimensions $k+1$ through $m$. Its integral cellular boundary is two in even dimensions and zero in odd dimensions. Steenrod operations on its relative cohomology are inherited from real projective space.

This is the CW complex obtained as the increasing union of finite Real projective spaces under their standard inclusions. It has one cell in each nonnegative dimension. Its cellular chain complex has boundary multiplication by two in positive even dimensions and zero in odd dimensions. Consequently its integral cohomology is $\mathbb{Z}$ in degree zero, $\mathbb{Z}/2$ in positive even degrees and zero in odd degrees. Its mod-two cohomology is ${\mathbb{Z}}_2$ in every nonnegative degree.

For inviscid plane Couette flow $U=z$ between impermeable walls at $z=\pm 1$, the nonzero real streamwise wavenumber $\alpha$ gives no nontrivial smooth discrete vertical-velocity modes. Instead, the vorticity variable $q=(D^2-{\alpha }^2)w$ obeys $({\partial }_t+i\alpha z)q=0$. Localized vorticity sheets give generalized eigenfunctions with real frequencies ${\omega }_{\xi }=\alpha \xi$. Their closure forms a neutral continuous spectrum.

A random variable is discrete when some finite or countable set $D\subseteq \mathbb{R}$ has $\mathbb{P}(X\in D)=1$. Its probability mass function is $p_X(x)=\mathbb{P}(X=x)$; these nonnegative masses sum to one. An event determined by the value has probability $\mathbb{P}(X\in A)={\sum }_{x\in A\cap D}{p}_X(x)$. Values outside $D$ may still occur on a zero-probability event without changing this definition.

Countable support need not consist of integers: a random variable constantly equal to $3/2$ is discrete. Integer-indexed tail-sum formulas therefore require a separate integer-valued hypothesis. For the dice sum in random variable, the expected value is ${\sum }_{s=2}^{12}s{p}_S(s)=7$ and the variance is ${\sum }_{s=2}^{12}(s-7{)}^2{p}_S(s)=35/6$; the same answer follows by adding the variances of its independent summands.

For finitely coloured $(r+1)$-element subsets, choose successive least points and use the inductive $r$-set theorem to thin the remaining reservoir after each choice. Every tuple with that least point then has one assigned colour. The infinite pigeonhole principle retains infinitely many least points with the same assigned colour, giving an infinite homogeneous set. The $r=1$ base case is the same pigeonhole principle. This proves the Ramsey theorem for r-sets for all finite $r$.

Take increasing even $u_i$ for which every ordered four-index expression $u_i+u_j+2u_k+2u_l$ has one colour. The Ramsey theorem for r-sets supplies them. Set $x_i=u_{2i-1}+u_{2i}$ and $y_i=(u_1+u_2)/2+2u_{i+2}$. Then both $x_i+2x_j$ and $y_i+y_j$ use the same ordered coefficient pattern $(1,1,2,2)$, so their union is monochromatic. The shared half-prefix makes the unweighted pair family possible without asking the two families to use the same generating sequence.

For integer transverse string modes, $N_R={\sum }_{k>0}({\alpha }_{-k}\cdot {\alpha }_k+k{d}_{-k}\cdot d_k)$ with vacuum-annihilating normal ordering. Each nonzero-mode creation operator raises the level by $k$; fermionic zero modes commute with it. The open-string mass condition is $M^2=2\pi T{N}_R$, with zero Ramond intercept.