Placing the two link diagrams in disjoint disks keeps all state circles separate. The reduced Kauffman bracket therefore acquires one extra loop factor $-A^2-A^{-2}$. Correcting writhe of a link diagram and setting $t=A^{-4}$ gives the displayed formula. It concerns a split link, not an arbitrary union of linked components.

For a nonempty link with $\ell$ components, the Jones polynomial skein relation at $t=1$ makes a crossing change preserve the value. Crossing changes turn the link into an unlink with the same number of components. Repeated Jones polynomial of a split union gives the formula. In particular the Jones polynomial is never the zero polynomial.

Reflecting a link diagram interchanges its two bracket smoothings and negates its writhe of a link diagram. This replaces $A$ by $A^{-1}$ in the normalized Kauffman bracket, hence $t$ by $t^{-1}$. A knot whose Jones polynomial differs from its reciprocal cannot be an amphichiral knot.

Use the Conway-normalized Alexander polynomial. Choosing $t^{1/2}=i$ in the Jones polynomial skein relation gives $V_{L_{+}}(-1)-V_{L_{-}}(-1)=-2i{V}_{L_0}(-1)$. The quantities $(-1{)}^{\ell -1}{\nabla }_L(2i)$ obey the same relation, because smoothing changes the component count by one. Their unlink values agree, so crossing-change induction proves equality for every link. For a knot, the sign is one and ${\nabla }_K(2i)={\Delta }_K(-1)$. This proves the signed identity, not only equality of knot determinants.

This asks whether the Jones polynomial equal to one characterizes the unknot among knots. A zero breadth of a Laurent polynomial makes $V_K$ a monomial; the identities $V_K(1)=1$ and $V_K^{\prime }(1)=0$ then force that monomial to be one. Thus using Jones breadth as a strictly positive complexity for nontrivial knots requires this detection property. To prove the derivative identity, differentiate the Jones polynomial skein relation at one for a crossing change in a knot. Both crossed diagrams have value one, while the smoothed two-component link has value minus two by Jones polynomial evaluation at one, so $V_{K_{+}}^{\prime }(1)=V_{K_{-}}^{\prime }(1)$. Crossing changes reduce to the unknot, whose derivative is zero. The corresponding Alexander polynomial detection property is false, as nontrivial untwisted Whitehead doubles demonstrate. The Jones question is discussed in research on closed four-strand braids.

The meet of two elements of a lattice is their greatest lower bound. For set families ordered by inclusion, it is their intersection whenever that intersection lies in the specified lattice.

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.