For unit-rate Erlang distribution of shape two claims put $q=\lambda /c<1/2$. The adjustment coefficient satisfies $q(2-R)=(1-R{)}^2$, so $1-R=(q+\sqrt{q^2+4q})/2$. In the interior-adjustment prefactor $(1-2q)/(2q(1-R{)}^{-3}-1)$, substituting $q=(1-R{)}^2/(2-R)$ gives the displayed expression. It is positive for $0<R<1$ and tends to one as the loading tends to zero.

An ideal planar alpha-Omega dynamo places poloidal regeneration above an interface and shear-generated toroidal induction below it. With equal constant diffusivities, decay away from the interface and ordinary electromagnetic matching, its surface-wave branch grows when $D>32$ for positive $k,\alpha ,V$. The model separates the two regeneration regions while allowing diffusive coupling. The threshold refers to this model, not every solar interface dynamo or a full isotropic alpha closure.

Write the old nuisance parameter as $\lambda =g(\psi ,\eta )$ with $g_{\eta }\ne 0$. The transformed score functions are $u_{\psi }^{\mathrm{new}}=u_{\psi }+g_{\psi }{u}_{\lambda }$ and $u_{\eta }=g_{\eta }{u}_{\lambda }$. Their expected cross product is $g_{\eta }(I_{\psi \lambda }+g_{\psi }{I}_{\lambda \lambda })$. Setting it to zero gives the displayed ordinary differential equation along fixed-$\eta$ curves, assuming positive nuisance Fisher information. Smooth solutions with distinct initial conditions give a local interest-respecting orthogonal coordinate system.

For a stationary autoregressive process of order one with unit-variance normal distribution marginals and fixed $0<\rho <1$, the Gaussian kernel covariance correction at lag $j$ has the displayed form. The difference of two observations has variance $2(1-{\rho }^j)$, and their characteristic functions reduce the Fourier integral to the difference of two Gaussian integrals. The mean value theorem gives $0\le g_h(j)\le \sqrt{\pi }{\rho }^j/[2(1-\rho {)}^{3/2}]$, uniformly over $h>0$. Thus the integrated variance differs from its independent-sample counterpart by $O(n^{-1})$. The marginal bias is unchanged, so this dependence does not alter the leading optimal bandwidth or the $n^{-4/5}$ mean integrated squared error for a second-order density kernel.

The Rees-Sciama effect is the microwave temperature shift produced by time-dependent gravitational potentials in nonlinear structure. A photon gains energy entering a potential well and loses it leaving; the shifts do not cancel when the well changes during transit. To leading order in its effect on the photon spectrum, this is a frequency-independent thermodynamic temperature shift, unlike the thermal Sunyaev-Zeldovich effect. A nonlinear potential can evolve even during matter domination, when linear growing-mode potentials are constant.

Suppose the instantaneous forward rate field has mean ${\mu }_{s,T}$ and covariance $c(s\wedge r,T,U)$, with deterministic initial curve and suitable integrability and maturity regularity. Its discounted zero-coupon bond prices are martingales exactly when the displayed mean identity holds. PutThe discounted bond price is $\exp [-A(s,T)-Y(s,T)]$, where $Y$ is the integrated Gaussian forward-rate process. Its conditional Gaussian exponential expectation gives the equivalent identity $A(s,T)-A(0,T)=v(s,T)/2$. Differentiating this in maturity gives the displayed restriction. Conversely, integrating that restriction on the diagonal and along the current curve recovers the half-variance identity, by symmetry of the covariance over the maturity square. The Gaussian exponential martingale with deterministic variance then verifies the full conditional martingale property. This formulation does not require a time derivative of $c$.

For a positive integer $k$, in the real Heisenberg group with multiplication $(x,y,z)(x^{\prime },y^{\prime },z^{\prime })=(x+x^{\prime },y+y^{\prime },z+z^{\prime }+x{y}^{\prime })$, the displayed set is a discrete cocompact subgroup. Writing $a=(k,0,0)$, $b=(0,k,0)$, $c=(0,0,k)$ gives $[a,b]=c^k$ and $c$ central. Hence its abelianization is ${\mathbb{Z}}^2\oplus \mathbb{Z}/k$. The right quotient maps to ${\mathbb{R}}^2/k{\mathbb{Z}}^2$ by its first two coordinates; each fibre is the central circle $\mathbb{R}/k\mathbb{Z}$.

For $R>0$, a spherical ultraviolet cutoff gives the following exact examples of the one-loop critical-mass subtraction integral:The areas of the unit spheres in dimensions three, four and five are $4\pi$, $2{\pi }^2$ and $8{\pi }^2/3$. Dividing by $(2\pi {)}^D$ reduces the integrals to constants times ${\int }_0^{\Lambda }dp/(p^2+R)$, ${\int }_0^{\Lambda }pdp/(p^2+R)$ and ${\int }_0^{\Lambda }{p}^{2}dp/(p^2+R)$ respectively. Elementary integration proves the formulas. As $R\to 0$, the first has a power divergence, the second a logarithmic divergence, and the third a finite limit. These examples make the ordinary upper critical dimension four visible without treating a self-consistent one-loop equation as an exact interacting critical exponent prediction.

The sensitivity function is the normalized variance of a fitted regression mean. Its average under the design is the number $p$ of model coefficients, since $\int dd\xi =\mathrm{tr}(M^{-1}M)=p$. The directional derivative of $\log \det M$ towards a point mass at $x$ is $d(x,\xi )-p$.

For a zero-one Boolean function, the flip-probability influence is ${\beta }_i=4{\sum }_{S\ni i}\hat{f}(S{)}^2$. Apply Beckner's inequality with noise parameter $\sqrt{\delta }$ to the discrete derivative of a Boolean function, whose nonzero magnitude is $1/2$. Its $L^{1+\delta }$ norm squared is ${\beta }_i^{2/(1+\delta )}/4$. Summing over coordinates gives the displayed weighted Fourier weight bound, a useful intermediate step in influence lower bounds.

For a plane curve defined by a homogeneous polynomial $F$ of degree $d\ge 2$, assume the field characteristic is different from two and $d-1$ is nonzero in the field. A smooth point of a variety is a flex exactly when the determinant of the homogeneous Hessian matrix vanishes there, provided its tangent line is not a component. Put the point at $[0:0:1]$ with tangent $y=0$ and $F_y(p)\ne 0$. The homogeneous identities give $F_{xz}(p)=F_{zz}(p)=0$ and $F_{yz}(p)=(d-1)F_y(p)$, so the determinant is $-(d-1{)}^2{F}_y(p{)}^2{F}_{xx}(p)$. In the completed local ring, write the curve as a formal power series $y=h(x)$. Its linear coefficient is zero and its quadratic coefficient is $-F_{xx}(p)/(2F_y(p))$. Vanishing of that coefficient is exactly intersection multiplicity at least three with the tangent, proving the criterion in these positive characteristics as well as in characteristic zero.

For a smooth homogeneous cubic in characteristic zero, its plane cubic flexes are precisely its intersections with the determinant of its Hessian matrix. Choose coordinates with a given smooth point $(0:0:1)$ and its tangent $y=0$. Up to a nonzero coefficient, the cubic starts as $y{z}^2+a{x}^{2}z+bxyz+c{y}^{2}z$ plus cubic terms in $x,y$. Its tangent contact has order three exactly when $a=0$, and the Hessian determinant at the point is a nonzero constant times $a$. This proves the criterion.

A Pt100 has nominal electrical resistance $100\Omega$ at $0{}^{\circ }\mathrm{C}$.^[ref] Here $0{}^{\circ }\mathrm{C}$ is the reference for the nominal element designation; it is distinct from the $0.01{}^{\circ }\mathrm{C}$ water triple point used in SPRT calibration. The following Pt100 local linear model deliberately stipulates one approximate response for calculations.

Use the binary finite-set encoding $e_k$ and Cantor code $⟨k,t⟩$. Contracting $(\lambda x.P^n{)}^{m+1}{Q}^p$ gives the substituted body with its root index lowered by taking the minimum with $\min (m,n)$, replacing each occurrence $x^r$ by a copy of $Q$ with root index $\min (m,p,r)$. A zero-indexed abstraction instead substitutes $\perp$ and gives the body root index zero. Indices strictly budget the argument information that can be used by application. Finite-witness inequalities for the pairing code prove that indexed reduction increases the interpretation, although it can decrease the underlying Böhm tree.

An increasing event on a finite product of identically distributed coordinates is transitive if a group action preserving the event is transitive on the coordinate set. Its influences are all equal. Transitivity concerns coordinates, and does not assert transitivity on successful configurations.

For a sequence of nontrivial increasing events on finite product measures, write $p_x$ for the parameter at which the success probability is $x$. An additive sharp threshold means that $p_{1-\epsilon }-p_{\epsilon }\to 0$ for every fixed $0<\epsilon <1/2$. When the threshold parameter itself tends to zero, relative sharpness instead compares this window with $p_{1/2}$; additive sharpness alone is then a weaker assertion. The Friedgut-Kalai sharp threshold theorem provides an explicit additive bound for transitive increasing events.

Concurvity is the nonlinear analogue of multicollinearity: a function of one predictor is exactly or approximately expressible as a sum of functions of other predictors. Exact concurvity can make an additive decomposition nonunique; approximate concurvity makes components difficult to estimate stably. A unique fitted sum need not imply unique individual components.

// Target: probability-and-statistics.bigb

In a dilute, well-mixed wall line plume, let $b$ be its width, $w$ its upward velocity, $\varphi \ll 1$ its ice volume fraction, and $e>0$ its one-edge entrainment coefficient. Define $g_i=g({\rho }_l-{\rho }_i)/{\rho }_l$ from the liquid and ice mass densities. The mixture's reduced gravity is approximately $g_i\varphi$, so $g_i$ itself is a material density-contrast scale, not the changing mixture reduced gravity.

Neglect crystal slip, deposition and other buoyancy contributions. If the ice-bearing wall-plume thermodynamic invariant supplies a constant positive ice flux $I_0=\varphi bw$, then volume flux $Q=bw$ and kinematic momentum flux $M=b{w}^2$ obeyA constant-velocity solution has $b=eZ$ with $Z$ measured from its plume virtual origin. Substitution givesIts width grows linearly and ice fraction falls inversely with height. Two exposed entraining edges replace $e$ by $2e$. The formal origin lies outside the dilute model because the ice fraction diverges there; finite-source conditions require matching away from that origin.

Let $X$ be a nonempty positive-dimensional compact complex submanifold of Complex projective space. On a positive-dimensional connected component choose a hyperplane meeting, but not containing, that component. Its linear equation gives a holomorphic section of ${\mathcal{O}}_X(1)$ with a zero and with a nonzero value. For $m>0$, its $m$th power cannot be a section of a trivial holomorphic line bundle, since such sections are constant on the compact connected component. Negative powers are also nontrivial by duality. Every positive-dimensional compact projective component meets every hyperplane: otherwise its affine coordinate functions would all be constant.

The First Chern class of the hyperplane line bundle. On Complex projective space it generates degree-two integral cohomology, and on ${\mathbb{CP}}^n$ its top power evaluates as one on the complex-oriented fundamental class.