For a nonnegative symmetric unit-integral kernel for density estimation with $0<{\mu }_2(K)=\int u^{2}K(u)du<\infty$ and $R(K)=\int K^2<\infty$, the canonical scaling satisfies $R(K_c)={\mu }_2(K_c{)}^2$. Since $R(K_c)=R(K)/c$ and ${\mu }_2(K_c)=c^2{\mu }_2(K)$, the unique scale is $c=[R(K)/{\mu }_2(K{)}^2{]}^{1/5}$. Consequently its asymptotic mean integrated squared error is $C(K)[(nh{)}^{-1}+h^{4}R(f^{\prime \prime })/4]$, where $C(K)=R(K{)}^{4/5}{\mu }_2(K{)}^{2/5}$ is invariant under kernel rescaling. The leading optimal smoothing bandwidth becomes $[nR(f^{\prime \prime }){]}^{-1/5}$, independent of the kernel shape. Unit second moment is a different normalization and does not have this separation property.

For a second-order kernel density estimator, if $f(x)>0$ and $f^{\prime \prime }(x)\ne 0$, the leading pointwise asymptotic mean squared error is $R(K)f(x)/(nh)+{\mu }_2(K{)}^2{f}^{\prime \prime }(x{)}^2{h}^4/4$. Differentiation gives the displayed interior optimum. At a zero of $f$, the leading variance term vanishes and this two-term approximation has no positive interior minimizer; at a zero of $f^{\prime \prime }$, a higher-order bias expansion is needed.

For independent observations with common density f, the kernel density estimator has mean $K_h\ast f$. Its pointwise variance is $n^{-1}[(K_h^2\ast f)-(K_h\ast f{)}^2]$. Integrate the variance and add integrated squared bias to get the displayed exact mean integrated squared error, assuming the integrals are finite. Since $R(K_h)=R(K)/h$, this also separates the smoothing bias from the stochastic variance.

Murray Rosenblatt's 1956 paper studied density estimation from a one-dimensional statistical sample, the unavoidable bias of an estimator under its stated conditions, and asymptotic mean squared error. It is an early foundation of kernel density estimation. ^[ref]. Emanuel Parzen's 1962 paper developed statistical consistency and asymptotic normality for probability density function estimates and studied estimation of the mode. ^[ref].

The authors' interests help explain the connection with smoothing. In a conversation recorded in 2006 and published in 2009, Rosenblatt recalled that his density estimation work grew out of estimation of the time-series spectral density and emphasized the connection with a smoothed histogram. This is his retrospective account of the motivation. ^[ref]. Parzen's original paper lists Stanford University as his affiliation; its opening frames the problem as recovering both a probability density function and its mode from independent and identically distributed random variables. ^[ref].

Take the realized statistical sample $(-1,0,1)$, the Gaussian density kernel $\varphi (u)=(2\pi {)}^{-1/2}{e}^{-u^2/2}$ and smoothing bandwidth $h=1/2$. ThenAt $x=0$, the three contributions are $0.03599398$, $0.26596152$, and $0.03599398$, because $\varphi (0)=0.39894228$ and $\varphi (2)=e^{-2}/\sqrt{2\pi }=0.05399097$. Their sum isThis is a probability density function value, not the probability of observing exactly zero. Each of the three component curves has area $1/3$, despite having a different height at zero.

Let $\Phi$ be the cumulative distribution function of the standard normal distribution. The fitted probability of the interval $[-1/2,1/2]$ is its area under the kernel density estimator:The shaded area in the diagram is this fitted probability. Its agreement to a few digits with $1/3$ is incidental: the empirical measure gives this interval mass $1/3$ exactly, whereas kernel density estimation redistributes that mass into neighboring intervals.

For an axisymmetric thin disc supplied at $R_0$ with the local Keplerian rotation specific angular momentum, the displayed source augments the Keplerian viscous diffusion equation. The radial velocity remains $v_R=-3(\Sigma R^{1/2}{)}^{-1}{\partial }_R(\nu \Sigma R^{1/2})$: subtracting local specific angular momentum times conservation of mass from conservation of angular momentum cancels the source term. The normalization of the Dirac delta function makes the integrated supplied mass per unit time equal to $\dot{M}(t)$.

The relative skein module fixes finitely many framed endpoints on the boundary of a 3-manifold and uses properly embedded framed tangles with those endpoints. Local Kauffman bracket relations are imposed away from the boundary. Gluing compatible endpoint sets composes tangles; a rectangle cylinder with n endpoints on each horizontal side gives the Temperley-Lieb diagram algebra.

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.