The map $h$ is a homeomorphism from the real line to its image with the subspace topology: its continuous inverse is $h^{-1}(x,y)=\log \sqrt{x^2+y^2}$. Different parameters have different radii, so winding does not destroy injectivity. The origin is an accumulation point of the spiral but is not a point of this space.

Here $\psi ={\Gamma }^{\prime }/\Gamma$ is the digamma function, $B=\log 2+\frac{1}{2}\log \pi -1-\gamma /2$, and $\gamma$ is the Euler--Mascheroni constant. The sum runs over Nontrivial zeros of the Riemann zeta function with multiplicities. Each displayed paired summand is $O_s(|\rho {|}^{-2})$, so it converges locally uniformly away from zeros. Differentiate the genus-one Hadamard factorization of the Riemann xi function and use the Gamma function recurrence. Unlike an unpaired reciprocal-zero sum, this expression does not require an unstated summation convention.

For $c>0$ and a centered normal distribution prior, this conditional log-posterior has second derivative $-c{e}^{\alpha }-s^{-2}<0$. It is strictly concave but does not belong to a normal distribution or gamma distribution family. A random-walk Metropolis algorithm, or a locally scaled proposal around its unique mode with the full proposal correction, can update it. Gaussian conjugacy is unavailable because the Poisson distribution likelihood contains an exponential rate.

For independent Poisson cell counts in stratified two-by-two contingency tables, the model $\log {\mu }_{abj}={\gamma }_j+{\alpha }_{aj}+{\beta }_{bj}$ allows both margins to vary by stratum but excludes within-stratum association. Its fitted means are the displayed products of margins, because the Poisson regression score equations match those margins. Every stratum's odds ratio is one. With positive nondegenerate margins, $J$ strata give $3J$ parameters and $J$ residual degrees of freedom. A common association adds one parameter; stratum-specific associations saturate the tables.

Independent Poisson distributions with a separate nuisance intercept in every stratum become multinomial distributions after conditioning on their totals. Profiling the intercept gives $e^{{\hat{\gamma }}_s}=N_s/{\sum }_l{e}^{x_s^T{\beta }_l}$. Substitution produces the same parameter-dependent log-likelihood as the conditional multinomial model. Both approaches fit ${\hat{\mu }}_{sl}=N_s{\hat{p}}_{sl}$. This exact equivalence differs from approximating a single binomial event count by a rare-event Poisson distribution.

A compact-open subgroup is both a compact subset and an open subgroup. Its coset space is discrete, and compact subsets meet only finitely many cosets. Finite intersections of compact-open subgroups are again compact-open, making them useful common stabilizers for finitely many vectors.

Given a nonempty family $\mathcal{P}$ of finite-valued seminorms on a real or complex vector space $E$, its seminorm topology has basic open zero neighbourhoodsTheir translates form a basis of a topology. Only finite intersections are required: an infinite intersection might impose a stronger topology. Equivalently, put $q(x)={\max }_i{p}_i(x)/{\epsilon }_i$; then $q$ is a seminorm and $U=\{q<1\}$. See Günther Hörmann, Advanced Functional Analysis, section 4, neighbourhood systems from seminorms for the general construction.

Each defining seminorm is continuous, since $|p(x)-p(y)|\le p(x-y)$. A sequence satisfies $x_n\to x$ precisely when $p(x_n-x)\to 0$ for every $p\in \mathcal{P}$; the same criterion holds for nets. Finitely many measurements can each meet a prescribed tolerance after a common index.

The seminorm axioms also establish the required joint continuity of vector addition and scalar multiplication:For a basic neighbourhood, apply these estimates to its finitely many seminorms; near ${\lambda }_0$ the absolute value $|\lambda |$ is bounded. This gives a locally convex space even when some directions are invisible to every measurement.

A compact identity neighbourhood $K$ produces a symmetric compact neighbourhood $S=K\cup K^{-1}\cup \{e\}$. The subgroup ${\bigcup }_{n\ge 1}{S}^n$ is open because it contains an identity neighbourhood, and is sigma-compact because each finite product $S^n$ is compact. This construction does not require the original group itself to be sigma-compact.

The ring $k\times k$, for a field $k$, has two prime ideals, $0\times k$ and $k\times 0$. Their localizations at a prime ideal are both isomorphic to $k$. Nevertheless $(1,0)(0,1)=0$ is a product of nonzero elements, so $A$ is not an integral domain. Nonzero orthogonal idempotents separate the two components of its spectrum of a commutative ring.

A sublocale is determined by a nucleus on a frame $j$ on $\mathcal{O}(X)$. Its frame consists of the fixed elements of $j$, with inherited meets and regularized joins. Increasing the nucleus makes the sublocale smaller.

A locale is compact when every directed family of opens whose join is $1$ contains $1$. Equivalently every open cover has a finite subcover. The truth-valued test $a\mapsto [a=1]$ is a preframe homomorphism exactly when the locale is compact.

A locale is Hausdorff when its diagonal into its localic square is a closed sublocale. Its complement is the join of rectangles $a\times b$ with $a\wedge b=0$. This is stronger than the underlying space being a Hausdorff space, because the localic square need not be spatial.

A locale is totally unordered when $f^{\ast }\le g^{\ast }$ for two maps into it implies $f=g$, for every common domain. This tests generalized points rather than just ordinary points. Every Hausdorff locale has this property: comparable maps annihilate the complement of the diagonal and hence factor through the diagonal.

An open, or overt, locale has a positivity map $\mathrm{Pos}:\mathcal{O}(X)\to \Omega$ left adjoint to the scalar inverse image ${!}^{\ast }:\Omega \to \mathcal{O}(X)$, satisfying Frobenius reciprocity. Thus $a\le {!}^{\ast }\mathrm{Pos}(a)$ and positivity preserves arbitrary joins. Classically every locale is overt, with positivity meaning $a\ne 0$.

A locale is extremally disconnected when the closure of every open is open. Equivalently its regular open elements are complemented, or double negation preserves binary joins, or its Booleanization of a locale is a flat sublocale.

This relation says that the closure of the open $u$ is contained in $v$. A regular locale has $a={\bigvee }_{u\prec a}u$ for every $a$. If $u\prec a$, then $\neg \neg u\le a$ and $\neg \neg u\prec a$.

Under classical logic a positive locale is connected when it cannot be written as the union of two disjoint nonzero opens. Equivalently its only complemented opens are $0$ and $1$. An open sublocale uses this criterion relative to its own top element. Positivity of the whole locale excludes the empty locale. This definition is stated with its logical convention; replacing positivity by negated nonemptiness is not a constructive step.

For a local volatility model $d{S}_t=S_t(rdt+\sigma (S_t)d{W}_t^Q)$ with bank account $B_t=B_0{e}^{rt}$, a nonnegative classical solution $V$ ofdefines a possible price of a European contingent claim. Indeed, Itô formula gives $d(V(t,S_t)/B_t)=B_t^{-1}\sigma (S_t)S_t{V}_S(t,S_t)d{W}_t^Q$, a local martingale. Thus the same equivalent local martingale measure works for the market augmented by this price. For an admissible trading strategy whose discounted wealth is bounded below, the discounted gains are a local martingale bounded below, hence a supermartingale by localization and the Fatou lemma. A zero-cost terminal gain that is nonnegative and strictly positive with positive probability would then contradict the supermartingale expectation inequality. This proves the absence of arbitrage for such strategies. If $V$ is bounded on the finite time interval, its discounted value is a bounded martingale and consequently $V(t,S_t)=B_t{\mathbb{E}}_Q[g(S_T)/B_T\mid {\mathcal{F}}_t]$. Nonnegativity alone guarantees the local martingale calculation, not this latter equality for every possible solution.

In the zero-interest local volatility model $d{S}_t=\sqrt{S_t}d{W}_t^Q$, the risk-neutral pricing value of the European contingent claim $e^{v{S}_T}$, for $v>0$ and $v(T-t)<2$, isThe compound Poisson transition law of a driftless square-root diffusion proves finiteness and this conditional-expectation formula. Alternatively, substituting $V(t,S)=e^{a(t)S+b(t)}$ into the pricing equation for a local volatility model gives $a^{\prime }+a^2/2=0$ and $b^{\prime }=0$, with terminal values $a(T)=v,b(T)=0$. Hence $a(t)=v/[1-v(T-t)/2]$ and $b(t)=0$. The conditional-expectation calculation verifies that this formal solution is the price of the unbounded claim. The delta hedge holds $a(t)V$ stocks and $(1-a(t)S_t)V/B_t$ bank units.

Under a product unitary operator $U_A\otimes U_B$, the reduced density matrix of $A$ becomes $U_A{\rho }_A{U}_A^{†}$. The unitary on $B$ cancels under the partial trace. Unitary conjugation preserves eigenvalues, so the squared Schmidt coefficients of a bipartite pure state are unchanged. Conversely, two normalized bipartite pure states with the same squared Schmidt coefficients, including multiplicities and zero padding, have Schmidt decompositions carried into each other by local unitaries mapping their Schmidt bases. Thus reduced-state spectra completely classify bipartite pure states up to local unitary operations.