The join of two elements of a lattice is their least upper bound. In a lattice of closed set families, it is the closure of their union.

For nonzero slip parameter with $|\alpha |<1$, scale shear rate as $x=\tau \sqrt{1-{\alpha }^2}\dot{\gamma }$ and stress as $S=x[b+1/(1+x^2)]$, where $b={\mu }_0/({\alpha }^2{G}_0\tau )$. The derivative is $b+(1-x^2)/(1+x^2{)}^2$ and its least value is $b-1/8$, attained at $x^2=3$. A decreasing homogeneous constitutive segment therefore exists exactly under the displayed condition. It permits different shear rates at the same stress and motivates shear banding; the local curve alone does not select a band-interface stress.

A Poisson distribution with mean $E\lambda$ and positive known exposure has Fisher information $E/\lambda$, so its Jeffreys prior kernel is ${\lambda }^{-1/2}$. The proper shape-rate gamma distribution $\mathrm{Gamma}(1/2,\epsilon )$ has kernel ${\lambda }^{-1/2}{e}^{-\epsilon \lambda }$, approximating the improper kernel where $\epsilon \lambda$ is small. Its posterior is $\mathrm{Gamma}(y+1/2,E+\epsilon )$, converging to $\mathrm{Gamma}(y+1/2,E)$. There is no normalized Jeffreys prior to which the proper priors converge weakly; this is kernel approximation and posterior convergence.

For a phase-mixed one-dimensional collisionless stellar system, $E=v_z^2/2+\psi (z)$ is an integral of motion and $f_z=F(E)$. Integrating both signs of velocity gives $\nu (\psi )=\sqrt{2}{\int }_{\psi }^{\infty }F(E)(E-\psi {)}^{-1/2}dE$. Applying the Abel transform a second time gives ${\int }_E^{\infty }\nu (\psi )(\psi -E{)}^{-1/2}d\psi =\pi \sqrt{2}{\int }_E^{\infty }F(t)dt$; differentiating proves the inverse. Sufficient convergence, regularity, and correct escape-energy boundary conditions are essential. An arbitrary tracer-density/potential pair is physical only if the reconstructed phase-space distribution function is nonnegative. Applying this inversion to a galactic disk requires a locally separable vertical dynamics or a justified approximation to it, and control of radial mixing and the tilted velocity ellipsoid.

For a steady axisymmetric stellar population with no mean vertical motion, the vertical Jeans equation balances the gradient of the vertical random-motion stress and the radial gradient of the tilted velocity ellipsoid against the vertical Newtonian gravitational field. Neglecting the mixed moment gives ${\partial }_z(n{\sigma }_z^2)=-n{\partial }_z\Phi$. A tracer density $n$ need not equal the density sourcing the gravitational potential.

The Abel map of an algebraic curve sends an effective Cartier divisor $D$ to its line bundle ${\mathcal{O}}_C(D)$. Its scheme fibre over a line bundle $L$ is the projective space of one-dimensional subspaces of $H^0(C,L)$. This follows in families: a relative effective Cartier divisor in the fixed class determines a nowhere fibre-zero section of $L$ up to a line bundle from the parameter space, equivalently a line subbundle of $H^0(C,L)\otimes {\mathcal{O}}_T$. Thus the scheme fibre is smooth, not merely reduced on its points. The dimensions can nevertheless jump, so the Abel map of an algebraic curve need not be a smooth morphism.

The theta divisor consists of degree-$g-1$ line bundles with a nonzero global section. It is the irreducible image of the symmetric product of a curve $C^{(g-1)}$, with its reduced Cartier divisor structure. The determinant description of the theta divisor supplies equations, multiplicities and pullbacks in families.

For a smooth connected projective curve $C$ with chosen basepoint $p$, the Abel map of an algebraic curve $i_p:C\to J(C)$ sends $q$ to $[q-p]$. Every morphism $g:C\to A$ to an abelian variety with $g(p)=0$ factors uniquely through a homomorphism $J(C)\to A$. This universal property converts a curve embedded in an abelian variety into a homomorphism from its Jacobian.

A smooth projective curve over the complex numbers is determined up to isomorphism by its Jacobian variety together with its canonical principal polarization of a complex torus. For genus at least three, the branch locus of the theta Gauss map recovers the canonical curve by projective biduality theorem, or recovers the canonical rational normal curve and the branch points of a hyperelliptic curve. In genus two the theta divisor itself is the curve; in genus one the Jacobian variety is the curve after choosing an origin. The polarization is essential data in this statement.

The identity holds for $|q|<1$, with both sides locally uniformly convergent. In the character theta proof of the eta product, pair the terms indexed by $n$ and $-n$ and then select $n=6r+1$; their Dirichlet character signs are $(-1{)}^r$. Removing $q^{1/24}$ gives the displayed product identity.

The odd Dirichlet character modulo four has ${\chi }_4(2r+1)=(-1{)}^r$. The series $B=\frac{1}{2}{\sum }_{n}n{\chi }_4(n)q^{n^2/8}$ is therefore $q^{1/8}$ times the displayed sum. The character theta proof of the eta product gives $B={\eta }^3$, proving the identity.

The dissimilarity is the number of mismatched presences divided by the number of coordinates present in at least one profile. It differs from normalized Hamming distance because coordinates absent from both profiles do not enter its denominator. Define the distance of two empty sets to be zero.

For a regular nonconstant J-holomorphic sphere in a $2n$-dimensional target, the parametrized Fredholm index is $2n+2c_1(A)$. Removing the six-dimensional Möbius transformation reparametrization group gives the displayed local moduli dimension. The quotient is a manifold for a simple J-holomorphic sphere and may be an orbifold for a multiple cover with finite stabilizer. In dimension four the formula is $2c_1(A)-2$.

For a compact symplectic manifold, smooth convergence of tame almost complex structures and a uniform area bound give a subsequence of J-holomorphic spheres converging, after reparametrization, to a finite tree of J-holomorphic spheres, with convergence smooth away from finitely many bubbling points. The total homology class and symplectic area are preserved. Constant components may be added for stability. In a spherical class which cannot split into two positive-area spherical classes, there is only one nonconstant component.

A p-adic measure gives the compatible finite-quotient elements ${\sum }_{gU}\mu (gU)[gU]\in I[G/U]$. The transition maps sum coefficients over fibers. Conversely compatible coefficients define values on cosets, then all clopen sets, and hence integration. Thus the inverse limit Iwasawa algebra is the ring of measures, with multiplication given by convolution of p-adic measures. Completeness of $I$ suffices even when its residue field is infinite.

For an L2-bounded continuous martingale and ordered possibly infinite stopping times $\sigma \le \tau$, interpret the terminal stopped value as $M_{\infty }$. The Doob L2 maximal inequality bounds the expected squared path supremum. Localization of $M^2-[M]$ then gives $\mathbb{E}[M{]}_{\infty }<\infty$, so this square-minus-bracket local martingale has integrable supremum and is uniformly integrable. Apply the optional stopping theorem to it and to $M$. Expanding $(M_{\tau }-M_{\sigma }{)}^2$ makes the cross term vanish conditionally, proving the identity. Bounded ${\mathcal{F}}_{\sigma }$-measurable factors can be multiplied into either side.

The Cauchy-Schwarz inequality and Jensen inequality give the displayed estimate, with $C=(\mathbb{E}|N{|}^{-1/2}{)}^{1/2}<\infty$. No independence between $R_{\epsilon }$ and $B_{\epsilon }$ is required. If $R_{\epsilon }={\int }_0^{\epsilon }(H_s-H_0)d{B}_s$ for a bounded continuous adapted process, the Itô isometry and the dominated convergence theorem make the right side tend to zero. Time shifting proves recovery of $H_t$ in probability from the ratio of a short stochastic integral to its Brownian increment.

For a zero-starting L2-bounded continuous martingale $M$, the space $L^2(M)$ consists of predictable $H$ with $\mathbb{E}{\int }_0^{\infty }{H}_s^{2}d[M{]}_s<\infty$, identifying processes equal almost everywhere for the quadratic-variation measure. The finite-horizon version stops the integral at $T$. The Itô integral is the isometric extension from elementary predictable integrands.

// Target: probability-and-statistics.bigb

For deterministic square-integrable $h$, approximate by step functions. Their Itô integrals are sums of independent normal Brownian increments. The Itô isometry gives convergence in $L^2$, and the variances converge to $\int h^2$; convergence of characteristic functions proves the normal limit. Each linear combination of finitely many such integrals is another deterministic stochastic integral, so they are jointly Gaussian.

For deterministic $X_0=x$ and bounded continuous drift and volatility, the initial right derivative of the mean is $b(x)$ and that of the variance is $\sigma (x{)}^2$. The stochastic integral has mean zero and second moment ${\int }_0^t\mathbb{E}\sigma (X_s{)}^{2}ds$; the bounded drift contributes only higher-order terms to the variance.

// Target: probability-and-statistics.bigb