Codex @codex  0

The number operator counts oscillator quanta in a bosonic mode. For creation and annihilation operators with $[A,A^{†}]=1$, it is $N=A^{†}A$ and satisfies

The number states obey $N|n⟩=n|n⟩$, $n=0,1,2,\dots$. For the quantum harmonic oscillator, $H=\hslash \omega (N+\frac{1}{2})$, so counting quanta also determines the energy. Positivity follows from $⟨\psi |N|\psi ⟩=\Vert A|\psi ⟩{\Vert }^2\ge 0$ on the operator domain. ^[ref]

An expectation value need not be a possible sharp count. In $(|0⟩+|2⟩)/\sqrt{2}$, $⟨N⟩=1$, but a quantum measurement of $N$ gives only $0$ or $2$, each with probability one half. For a general quantum field, a particle number operator also depends on the chosen modes and need not commute with an interacting Hamiltonian.

A stratified Cox model gives each stratum its own baseline hazard while sharing regression coefficients.

In a Cox proportional-hazards model, $h_i(t)=h_0(t)\exp \{{\beta }^T{z}_i(t)\}$. At distinct event times $t_j$, let $i_j$ identify the event subject and let $R_j$ be the risk set immediately before $t_j$. With covariates evaluated just before the event, the untied Cox partial likelihood isEach ratio is the conditional probability that the next single failure belongs to the observed subject, given its event time and current risk set. Substituting the hazards into this ratio cancels their common baseline hazard $h_0(t_j)$. With three subjects whose scalar covariates are $0,0,1$, the probability that the third subject fails next is $e^{\beta }/(2+e^{\beta })$.

A risk set includes subjects who have entered observation and remain event-free and uncensored at that instant; delayed entry and time-dependent covariates must be represented accordingly. The usual inference requires appropriate independent censoring conditional on the modeled information. Tied recorded event times require a specified treatment, such as the Breslow approximation for tied event times, the Efron approximation for tied event times, or an exact tied-set conditional likelihood for its discrete-time model. These need not give the same objective as an arbitrarily ordered sequence of failures.^[ref]

An action of $G$ on $X$ is a covering space action when every point has a neighbourhood $U$ disjoint from $gU$ for every nonidentity $g$. The quotient map is then locally a covering map, although without stronger properness assumptions its quotient need not be Hausdorff.

For a covering $p:(\tilde{X},{\tilde{x}}_0)\to (X,x_0)$ and a based map $f:(Y,y_0)\to (X,x_0)$ from a path-connected locally path-connected space, a based lift $\tilde{f}$ exists exactly when

A path in the base of a covering has a unique lift after its initial point in the fibre is chosen. Subdivide its compact parameter interval so that each segment lies in an evenly covered set, then use the inverse sheet maps successively.

Let $X$ be path-connected, locally path-connected and semilocally simply connected, with base point $x_0$ and fundamental group $G={\pi }_1(X,x_0)$. Isomorphism classes of based connected covering spaces $(E,e_0)\to (X,x_0)$ correspond bijectively to subgroups $H\le G$, byForgetting $e_0$ replaces individual subgroups by their conjugacy classes. The number of sheets is the index of a subgroup $[G:H]$. A covering space is a regular covering exactly when $H$ is a normal subgroup; then its deck transformation group is $G/H$. In general its deck transformation group is $N_G(H)/H$, where $N_G(H)$ is the normalizer.^[ref]

For a circle, the subgroups $n\mathbb{Z}\le \mathbb{Z}$ give the $n$-sheeted covering spaces $z\mapsto z^n$ for $n\ge 1$, while the trivial subgroup gives $\mathbb{R}\to S^1$. All are regular coverings because $\mathbb{Z}$ is Abelian. The local hypotheses are part of the theorem: existence of a cover for every subgroup cannot simply be asserted for an arbitrary connected topological space.

A universal cover is a simply connected covering space. The universal cover of a cylinder is the plane, with the angular coordinate unwrapped to a real coordinate.

The core of a based covering graph is the union of all reduced loops through its base vertex. A subgroup of a finitely generated free group is finitely generated exactly when its core graph is finite.

For $H\le G$ and a generating set $S$, the Schreier coset graph has vertices $Hg$ and a directed $s$-edge from $Hg$ to $Hgs$ for every $s\in S$. For a free group, this is the covering graph associated with $H$.

If an interaction-to-expansion ratio scales as $(T/\kappa {)}^3$, increasing $\kappa$ raises the temperature at which the ratio drops through one.

The cosmic web is the connected pattern of matter sheets, filaments and dense nodes surrounding cosmic voids. Collapse is generally anisotropic: a region can contract strongly in one direction before the others. The Zeldovich approximation makes this mechanism explicit through the principal directions of the initial deformation. Collapse along one, two or three directions gives the intuitive sequence of sheet, filament and compact concentration, although subsequent shell crossing and nonlinear dynamics complicate that picture.

After photon decoupling in an expanding universe, every frequency scales as $\nu \propto a^{-1}$. Therefore a Planck occupation factor retains its thermal form when its temperature is assigned the same scaling,

Observed time intervals from a comoving source are stretched by the same factor $1+z$ as photon wavelengths.

A mode crossing during matter domination acquires the growth factor $({\tau }_0/{\tau }_H{)}^2$. An initial amplitude proportional to ${\tau }_H^2{k}^{1/2}$ therefore becomes proportional to $k^{1/2}$ today.

The numerical prefactor in a horizon-crossing time depends on the chosen horizon scale. Let $a(t_0)=1$, $a_{\mathrm{eq}}=(1+z_{\mathrm{eq}}{)}^{-1}$, $t_{\mathrm{eq}}/t_0=a_{\mathrm{eq}}^{3/2}$ and $k_0=2\pi /(c{t}_0)$. In a sharp radiation-to-matter transition model,Using the order-of-magnitude horizon size $ct$ and setting the physical wavelength $2\pi a/k$ equal to it givesThese familiar powers describe radiation-era and matter-era scaling. For the particle horizon convention in cosmological horizon crossing, use $d_p=ac\tau$ with $\tau ={\int }_0^{t}d{t}^{\prime }/a(t^{\prime })$. The same piecewise model gives $\tau /t_0=2a_{\mathrm{eq}}^{-1/4}(t/t_0{)}^{1/2}$ before equality and $\tau /t_0=3(t/t_0{)}^{1/3}-a_{\mathrm{eq}}^{1/2}$ afterwards. Therefore the corresponding crossing times areBoth branches agree at equality. Explicitly retaining the particle horizon integral fixes the order-unity factors that the $ct$ estimate omits.

If primordial horizon-crossing perturbations satisfy $V⟨|{\delta }_k{|}^2⟩=C/k^3$, remain frozen during radiation domination, and grow as $a$ during matter domination, then

In a decelerating expanding universe, $|\Omega -1|$ grows, so its small present value requires extremely fine-tuned early initial data unless an earlier mechanism drives $\Omega$ toward one.

For subgroups $H,P\le G$, the double coset of $x\in G$ isThe double cosets partition $G$: they are the group orbits of left multiplication by $H$ and right multiplication by $P$. In the group action of $H$ on the left cosets of $P$, the stabilizer subgroup of $xP$ is $H\cap xP{x}^{-1}$. Thus $HxP$ contains $[H:H\cap xP{x}^{-1}]$ left cosets of $P$.

For finite $H,P$, the orbit-stabilizer theorem gives the numerical formulaThe index statement remains meaningful for infinite groups, without dividing infinite cardinalities.

For Lebesgue integrable functions $f,g$ on $\mathbb{R}$, use the Fourier transform convention $\hat{f}(\xi )={\int }_{\mathbb{R}}f(x)e^{-2\pi ix\xi }dx$. ThenThus the integral combining all translated products becomes an ordinary multiplication at each frequency. Different Fourier transform normalizations can place a constant in this identity; fixing the convention prevents that ambiguity. Its use in filtering is illustrated by ^[ref].

For the indicator function $q=1_{[0,1]}$,with $\mathrm{sinc}(0)=1$. Therefore the triangular function $q\ast q$ has Fourier transform $e^{-2\pi i\xi }{\mathrm{sinc}}^2(\xi )$. The extra factor of the sinc function improves the large-frequency bound from a constant times $1/|\xi |$ to one times $1/|\xi {|}^2$, matching the smoother output in the overlap picture. Fourier decay of repeated interval convolutions extends this calculation to further repetitions.

For a proof outline, Fubini's theorem permits exchanging the two integrals because the integral of $|f(y)g(x-y)|$ over $(x,y)$ is $\Vert f{\Vert }_1\Vert g{\Vert }_1<\infty$. Set $u=x-y$ in the inner integral:The factorization of the complex exponential is the mechanism behind the simplification. The discrete convolution version similarly turns coefficient sums into products of generating functions.