The action $(x,y)\mapsto (\zeta x,{\zeta }^{-1}y)$ has invariant monomials exactly when their two exponents differ by a multiple of $n$. Every such monomial is a product of $x^n,y^n,xy$. Reducing by $uv=w^n$ leaves the monomials $w^j$, $u^i{w}^j$ and $v^i{w}^j$ with $i\ge 1$; their images are distinct monomials in $x,y$, proving that this is the only relation. The map $(x,y)\mapsto (x^n,y^n,xy)$ is finite and its fibres are the finite-group orbits, so the hypersurface is the geometric quotient.

// Destination: ringed-space.bigb

A properly discontinuous group action by isometries of a finitely generated group on a complete $n$-dimensional Riemannian manifold of nonnegative Ricci curvature yields the displayed group growth bound. Distinct orbit points are uniformly separated, while a word of length $m$ moves a basepoint by at most $Dm$. Pack disjoint equal-volume small balls about the orbit points into $B(p,Dm+\epsilon )$ and apply the Bishop-Gromov inequality. A finite point stabilizer contributes its order to the bound; freeness is unnecessary.

A finitely generated group of isometries acting properly discontinuously on a complete connected $n$-dimensional Riemannian manifold with nonnegative Ricci curvature has polynomial growth of a group of degree at most $n$. Choose $\rho >0$ so distinct points of an orbit are separated by more than $2\rho$. Let $L$ be the largest displacement of the base point by a generator. The radius-$m$ word metric ball maps into the metric ball of radius $Lm$, with fibers of size the finite stabilizer $|{\Gamma }_p|$. Disjoint radius-$\rho$ balls around these orbit points have equal volume and fit inside the ball of radius $Lm+\rho$. The Bishop-Gromov inequality bounds their number by $(1+Lm/\rho {)}^n$, giving the displayed estimate.

A tree on $P$ is a subset of ${\bigcup }_{n\ge 0}{P}^n$ closed under initial segments. It is closed when each finite level is closed in the corresponding product topology, and well-founded when it has no infinite branch. A countable-basis approximation tree, with nested sets of shrinking diameter, reduces a closed well-founded tree on a Polish space to a countable well-founded tree and bounds its height by a countable ordinal.

Fix a lattice, a bilinear polarization and Hodge numbers. The period domain consists of the Hodge filtrations with these dimensions satisfying orthogonality, opposition and positivity. It is an open subset of its compact dual of a period domain, described algebraically by the dimensions and orthogonality conditions alone.

Tensor the prequantum bundle with a square root of the canonical determinant bundle of a polarization, when such a square root exists. The half-form contributes to the state pairing and to the observable action. It improves coordinate invariance and accounts for familiar Maslov and oscillator zero-point shifts. Its existence is additional global data, not automatic from prequantum integrality.

The pairing compares state spaces belonging to two polarizations using their prequantum sections and half-forms. When it defines a suitable identification or projection, a classical Hamiltonian flow can transport a state to its transported polarization and the pairing can return it to the original state space. Differentiating this construction supplies operators for some observables that do not directly preserve the original polarization. Existence, domains and unitarity require additional hypotheses; it is not a universal canonical equivalence of all polarizations.

A sphere of conductivity $a_1$ in a host $a_0$ under a uniform incident gradient $E$ has interior gradient $3a_{0}E/(a_1+2a_0)$. Outside a sphere of radius $R$, its potential correction is $-R^3(a_1-a_0)(E\cdot x)/[(a_1+2a_0)|x{|}^3]$. Its polarization is $\beta E$ inside and zero outside. Temperature and normal-flux continuity determine these coefficients.

The exciting field excludes an inclusion's own singular dipole field. Other inclusions and periodic images supply the regular correction at its centre $y$. The angular mean of a self-dipole gradient on a concentric sphere vanishes, while the mean-value property recovers the regular field at the centre. Consequently a spherical average can recover the exciting field, even though the total pointwise field is not constant on that sphere.

Two Poisson structures are compatible when every linear combination ${\Pi }_1+\lambda {\Pi }_2$ is a Poisson structure. This family is a Poisson pencil. In terms of the Schouten-Nijenhuis bracket, compatibility means $[{\Pi }_1,{\Pi }_2]=0$, in addition to the two individual Jacobi identities. Constant Poisson bivectors are compatible. Over a real manifold, complex parameter values refer to the complexification of its function algebra.

For field functionals, a Poisson operator $J$ is an operator whose displayed pairing of variational derivatives defines a Poisson bracket. It must be skew under the integral pairing and satisfy the functional Jacobi identity. A constant skew differential operator, such as $J={\partial }_x$ with periodic or vanishing boundary conditions, gives the simplest examples. The Hamiltonian evolution is $u_t=J(\delta H/\delta u)$.

For independent samples of $n$ observations from Poisson distributions with positive means $a$ and $b$, the maximum-likelihood estimator of $\nu =\log (b/a)$ is the displayed difference when both totals are positive. Its asymptotic variance is $(a^{-1}+b^{-1})/n$, obtained by the delta method or by inverting the two-parameter Fisher information matrix. A zero group total makes its log-mean estimate a boundary value; finite normal intervals require nonzero totals.

If $C_i\sim \mathrm{Poisson}(T_i{\lambda }_i)$ counts events over $T_i$ days, then the daily average $C_i/T_i$ has mean ${\lambda }_i$ and variance ${\lambda }_i/T_i$. It is not itself a Poisson variable unless $T_i=1$. Known observation periods allow count modelling with a log-exposure offset; otherwise Quasi-Poisson regression may be used as a working mean-variance model, with uncertainty qualified by the missing exposure information.

For a continuous predictor $x$ and a categorical factor, an additive Poisson regression with log link has $\log {\mu }_j(x)=a_j+bx$. The log-mean curves are parallel and ${\mu }_j(x)/{\mu }_k(x)=e^{a_j-a_k}$ is constant in $x$. Adding a predictor-by-factor interaction gives $\log {\mu }_j(x)=a_j+b_{j}x$, so this ratio becomes $e^{a_j-a_k+(b_j-b_k)x}$. With $J$ factor levels, the common-slope restriction has $J-1$ independent constraints. A likelihood-ratio test compares the two deviances with an asymptotic chi-squared distribution on $J-1$ degrees of freedom. Evidence against common slopes establishes varying effects on the log scale; it does not by itself determine their signs or the ordering of the mean curves.

For independent aggregate Poisson distributions with means $n\lambda$ and $n\lambda e^{\beta }$, conditioning on the total eliminates $\lambda$. Independent Poisson conditioning gives the displayed binomial distribution. Its conditional likelihood is proportional to $e^{\beta Y_1}(1+e^{\beta }{)}^{-m}$, also the profile likelihood after maximizing over the unrestricted positive baseline rate. Unequal exposures replace the binomial success probability by $n_1{e}^{\beta }/(n_0+n_1{e}^{\beta })$.

For independent Poisson distributions with log means $Z\beta +o$, where $o$ is a known offset, differentiation of the log-likelihood gives the score function $Z^T(y-\mu )$. An interior maximum-likelihood estimator therefore satisfies the displayed equations. An intercept and categorical indicator columns make fitted count totals equal observed count totals in each represented factor level. These equations match counts, rather than unweighted averages of rates when exposures differ.

// Target: probability-and-statistics.bigb

In Poisson regression with log means $a+\beta x_i$, the Fisher information matrix is $\left(\begin{array}{cc}{S}_0& S_1\\ S_1& S_2\end{array}\right)$, where $S_j={\sum }_i{x}_i^j{\mu }_i$. Inverting this matrix gives slope variance $S_0/(S_0{S}_2-S_1^2)$. Writing ${\bar{x}}_{\mu }=S_1/S_0$ reduces it to the displayed reciprocal weighted sum of squares. Thus a common intercept consumes information about the overall rate, while slope information comes from the spread of the covariates under the mean-count weights. Constant covariates make the two parameters unidentifiable.

For a full-column-rank design $X$ and positive fitted means ${\mu }_i=\exp (x_i^T\beta )$, the log-likelihood has Hessian $-X^T\mathrm{diag}({\mu }_i)X$, which is negative definite. Thus any finite stationary point is the unique maximum-likelihood estimator. Full rank does not guarantee existence: with an intercept and all counts zero, the supremum is approached as the intercept tends to $-\infty$. If every observed count is positive, a full-rank design does give a finite maximum: each function $y_i{\eta }_i-e^{{\eta }_i}$ tends to $-\infty$ in either tail and is bounded above, and $\Vert X\beta \Vert \to \infty$ as $\Vert \beta \Vert \to \infty$, so the negative log-likelihood is coercive.

Suppose independent counts have Poisson distributions with means $p_{i1}{\lambda }_i$ and $p_{i2}{\lambda }_{i}r$, where the known exposures are positive and $r>0$ is a shared rate ratio. Put $t_i=y_{i1}+y_{i2}$. Maximizing over each baseline givesZero-total strata contribute no information about $r$; their baseline maximum is at zero in the parameter-space closure. For every positive-total stratum its fitted second-count fraction increases with $r$, so its contribution to the score function decreases. Explicitly,whenever some $t_i>0$. The limiting score function values are ${\sum }_i{y}_{i2}$ and $-{\sum }_i{y}_{i1}$. If both totals are positive, there is therefore exactly one finite positive maximum-likelihood estimator. A zero total in one condition instead gives a boundary supremum at $r=0$ or $r=\infty$; two zero totals leave $r$ unidentifiable.

The geometric series gives $Q_r(t)=2{\sum }_{n\ge 1}{r}^n\sin nt$. Its convolution multiplier is $-i\mathrm{sgn}(n)r^{|n|}$ for nonzero frequencies and zero at frequency zero. Since the series is uniformly absolutely convergent for $r<1$, these identities hold for every Lebesgue integrable function by termwise integration. If all negative Fourier coefficients vanish, $Q_r\ast f=-i(P_r\ast f-\hat{f}(0))$.