If $X_{\lambda }$ has a Poisson distribution, then $\mathbb{P}(X_{\lambda }\le \lfloor \lambda \rfloor )\to 1/2$ as $\lambda \to \infty$. The Poisson central limit theorem standardizes by mean $\lambda$ and variance $\lambda$; the cutoff $(\lfloor \lambda \rfloor -\lambda )/\sqrt{\lambda }$ tends to zero. Squeezing between fixed nearby cutoffs and using continuity of the standard normal distribution function gives the limit.

Dualize a noncollinear finite point set in the real projective plane to projective lines. Their intersection vertices and consecutive line segments form an embedded graph. A vertex incident to $k$ lines has graph degree $2k$ and corresponds to a primal line containing $k$ points. Ordinary lines correspond to degree-four vertices. The faces are polygons with at least three sides.

For a probability measure-preserving system, every measurable $B$ of positive measure and every integer $k\ge 2$ admit an integer $n\ge 1$ with $\mu ({\bigcap }_{j=0}^{k-1}{T}^{-jn}B)>0$. The sets are preimages; neither invertibility nor an ergodic transformation is required. The Furstenberg correspondence principle converts this simultaneous return into arithmetic progressions and yields the Szemerédi theorem. The case $k=2$ follows from the Poincaré recurrence theorem.

For a map between closed oriented manifolds of the same dimension, Poincare duality turns homological pushforward into a degree-preserving map on cohomology. It is adjoint to pullback under the duality pairing: $⟨f^{!}a⌣b,[N]⟩=⟨a⌣f^{\ast }b,[M]⟩$. If $f$ has degree $d$, nondegeneracy gives $f^{!}{f}^{\ast }=d\mathrm{id}$. Maps of different dimensions require the corresponding degree shift.

For a closed oriented manifold of dimension $4r+2$, Poincare duality pairs all rational Betti numbers except the middle one, making their total contribution to the Euler characteristic even. The middle cohomology degree is odd, so its nondegenerate Poincare duality pairing is an alternating bilinear form and has even dimension. Hence the Euler characteristic is even. Nonorientable manifolds can have odd Euler characteristic, as Real projective space of even dimension demonstrates.

For positive source fluxes and positive constant buoyancy flux, the plume flux-balance invariant has a finite constant while $Q$ grows without bound. Dividing by $Q^2$ makes the source imbalance vanish relatively. The balances then give $Q\sim (6\alpha /5)(9\alpha F/10{)}^{1/3}{z}^{5/3}$ and $M\sim (9\alpha F/10{)}^{2/3}{z}^{4/3}$. This attraction concerns relative similarity and does not assert that different finite sources share the same virtual origin.

A projective line $L=0$ is contained in the zero locus of the nonzero cubic $L^3$. The union of three lines is given by their product, and a conic together with a line is likewise a plane cubic. Reducibility and repeated factors are permitted in polynomial coverings used in incidence geometry; a covering by cubics need not consist of smooth irreducible curves.

The surface density of a disk associated with a planar isotropic distribution is $S(\Phi )=2\pi {\int }_{\Phi }^{\infty }F(E)dE$. Differentiation gives $F(E)=-(2\pi {)}^{-1}dS/d\Phi {|}_{\Phi =E}$. A vanishing upper-limit boundary value recovers the original integral; monotonic decrease of $S(\Phi )$ ensures a nonnegative distribution.

The radial martingale $H=\int XdX+YdY$ and the planar Brownian stochastic area $Z=\int YdX-XdY$ have common bracket $A=\int (X^2+Y^2)ds$ and zero cross variation, because their integrand vectors $(X,Y)$ and $(Y,-X)$ are orthogonal. The Itô formula also gives $X_t^2+Y_t^2=2H_t+2t$.

On level-one cusp forms, determinant-normalized Hecke operators satisfy $⟨T_{n}f,g⟩=⟨f,T_{n}g⟩$. Unfolding their finite correspondences transfers a matrix to its inverse; its adjugate has the same determinant and reverses the correspondence. Since these operators commute, the cusp space has a common orthonormal basis of Hecke eigenforms, with real eigenvalues.

LetThe Fourier coefficients of $\theta$ are absolutely summable, so $a$ belongs to the Wiener algebra, as does $a^{-1}$. The identity $\theta (2t)=\theta (t)+\theta (t+\pi )$ implies $a(2t)=a(t)a(t+\pi )$. Starting with a compactly supported Daubechies wavelet having $p\ge 3$ vanishing moments, the periodic phase change of a scaling function gives $m(t)=a(t+\pi )m_0(t)$, while the canonical high-pass construction leaves $\hat{\psi }(2t)=e^{-it}\overline{m_0(t+\pi )}{\hat{\phi }}_0(t)$ unchanged.

At a dyadic point $t_0=2\pi k/2^j$, the terms of the difference quotient with $n\ge j$ and $2^n|h|\le 1$ each contribute $1+O((2^{n}h{)}^2)$. Their number tends to infinity, their total error is bounded, and the remaining tail contributes a bounded amount. Hence $(\theta (t_0+h)-\theta (t_0))/h={\log }_2(1/|h|)+O(1)$, so neither $\theta$ nor $a$ has a finite derivative there. These points are dense. Away from the isolated zero of $m_0$ near $\pi$, multiplication by its nonzero smooth value cannot remove this nondifferentiability. Thus the new MRA low-pass filter is not differentiable throughout any neighborhood of $\pi$, despite integrability of the scaling function and unchanged vanishing moments. Ordinary higher derivatives at $\pi$ cannot be inferred, although the corresponding Peano zero survives.

A penalized least-squares estimator balances squared error with a nonnegative penalty $J$. The Lasso uses the coefficient ${\ell }_1$ norm; graph total variation denoising uses the ${\ell }_1$ norm of endpoint differences. The choice of normalization of the squared loss changes the numerical tuning parameter.

With signature $(+---)$, ${ϵ}_{0123}=+1$, $J_3=M^{12}$ and the lower-index definition of $W_{\mu }$, a rest momentum $(m,0,0,0)$ gives $W_0=0$ and $W_3=-m{J}_3$ on that momentum eigenspace. A Hermitian spin projection $j_3$ consequently gives eigenvalues $0$ and $-m{j}_3$. Raising the third index changes its sign.

With signature $(+---)$ and ${ϵ}_{0123}=+1$, a momentum $(k,0,0,k)$ and helicity $j_3$ give lower components $W_0=k{j}_3$ and $W_3=-k{j}_3$. They obey $W_{\mu }{P}^{\mu }=k(W_0+W_3)=0$. These longitudinal identities require only the momentum and rotation-projection eigenvalues, not a statement about transverse little-group translations.

A periodic corrugation plus a uniform tilt. For the Adler phase equation, $V(\theta )=-\omega \theta -ϵ\cos \theta$ gives force $-V^{\prime }=\omega -ϵ\sin \theta$ and $V(\theta +2\pi )-V(\theta )=-2\pi \omega$. It has wells when $|\omega |<ϵ$ and no extrema when $|\omega |>ϵ$. It is a potential on the unwrapped phase, not a single-valued periodic potential on the circle.

In unit Plummer model scales, $F=3Q^{3/2}/(\sqrt{2}{\pi }^3)$ for $Q=E-L^2/2>0$ gives $\rho =3{\psi }^3/[4\pi (1+r^2)]=3{\psi }^5/(4\pi )$. It reproduces the isotropic Plummer density but has anisotropy $r^2/(1+r^2)$ and different second velocity moments.