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Here $n$ is the crossing count and $k$ is the number of connected components of the planar link diagram, not necessarily the number of link components. The all-$A$ and all-$B$ state counts satisfy $s_A+s_B\le n+2k$, by summing the Turaev surface Euler characteristic bound over projection components. The Kauffman bracket breadth bound and $t=A^{-4}$ give the displayed bound. It is sharp for an unlink diagram with $n=0$, whose Jones polynomial has breadth of a Laurent polynomial $k-1$.

For unforced homogeneous turbulence with isotropic turbulence statistics, put $F=u^{2}f=u^2-S_2/2$ and $S_3=6u^{3}K$. The longitudinal Kármán-Howarth equation is $F_t=r^{-4}{\partial }_r[r^4(S_3/6-\nu S_2^{\prime })]$. Since ${\partial }_t{u}^2=-2ϵ/3$, regularity at zero givesNeglecting the last term in a locally equilibrated small-scale range gives the displayed equation. The Kolmogorov four-fifths law follows where the viscous term is negligible as well.

For an infinite cardinal $\kappa$, the tree property says that every tree of height $\kappa$ with nonempty levels of cardinality less than $\kappa$ has a cofinal branch. In the all-trees-of-size-$\kappa$ convention it also forces regularity: disjoint chains of lengths cofinal in a singular $\kappa$ form a counterexample. It does not, by itself, imply strong inaccessibility. A strong limit cardinal with the tree property satisfies the self-partition relation by applying it to the end-homogeneous coloring tree. At a strongly inaccessible cardinal this is the usual weak-compactness characterization.

For a real scalar field with canonical field normalization, impose $[\dot{\varphi }(t,\mathbf{x}),\varphi (t,\mathbf{y})]=-i{\delta }^3(\mathbf{x}-\mathbf{y})$. The Källén–Lehmann spectral representation also represents its vacuum commutator as the weighted sum of free commutators. The equal-time derivative of each free commutator is $-i{\delta }^3(\mathbf{x}-\mathbf{y})$ independently of mass. Comparison therefore gives the displayed sum rule, whenever the canonical commutator and spectral integral admit this distributional limit. Since the measure is nonnegative, an isolated one-particle weight obeys $0\le Z\le 1$, and is strictly positive if the field couples to that particle. Arbitrary rescaling of the field rescales the total spectral weight, so positivity alone does not imply this normalization or upper bound.

With fixed neighbor positions, independent equal-variance errors and weights $w,(1-w)/2,(1-w)/2$, the displayed variance is minimized at $w=1/3$, with value ${\sigma }^2/3$. Its bias is $w{m}_1+(1-w)(m_2+m_3)/2-m(x)$. Larger nearest-point weight often reduces smoothing bias, but neither the bias nor the variance is universally monotone over $0<w<1$. If the nearest point is the target, the squared bias is $(1-w{)}^2[(m_2+m_3)/2-m(x){]}^2$.

// Target: probability-and-statistics.bigb

Resolve the two crossings in a skein relation using the Kauffman bracket, then correct by the writhe of a link diagram. The other smoothing cancels, leaving the displayed coefficient of the oriented smoothing. This relation and the value one on the unknot determine the Jones polynomial: crossing changes give descending diagrams, which are unlinks, while the smoothing terms have fewer crossings.

For knots, matching bracket smoothing states across a connected sum of knots joins one state circle from each side. Their total circle count is therefore the sum of the two counts minus one. The reduced Kauffman bracket state weights multiply, and writhe of a link diagram adds, proving the formula.

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.