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For independent $N(\mu ,{\mu }^2)$ observations with $\mu >0$, the sufficient statistics are their sum and squared norm. Condition on $a=\sqrt{n}\Vert y\Vert /{\sum }_i{y}_i>0$ and write $v=\hat{\mu }=\Vert y\Vert /(q\sqrt{n})$, where $q=(\sqrt{1+4a^2}+1)/(2a)$. Polar coordinates supply radial measure $\Vert y{\Vert }^{n-1}d\Vert y\Vert$, while the conditional angular factor does not depend on $\mu$. This proves the displayed shape. Its normalization depends on $a,n$ but not $\mu$ after the substitution $v=\mu z$, and agrees with the normalized P-star approximation.

If a regular plane curve has positive curvature $\kappa$ and a particle moves with speed $v=1/\kappa$ in the chosen time units, its elapsed time is $\int \kappa ds$, the total curvature traversed. On a logarithmic spiral $r=a{e}^{b\theta }$, both $ds/d\theta$ and $1/\kappa$ equal $r\sqrt{1+b^2}$. Hence $dt=d\theta$ and every complete turn takes time $2\pi$, regardless of its radius.

Where a regular curve has positive curvature $\kappa$, its radius of curvature is the radius $\rho =1/\kappa$ of its osculating circle. A sharper bend has a smaller radius. At zero curvature there is no finite radius; writing $\rho =\infty$ is a limiting convention rather than an ordinary real reciprocal of zero.

At a point of a twice continuously differentiable regular curve with positive curvature $\kappa$, its osculating circle has radius $1/\kappa$ and center $c=r+N_p/\kappa$, where $N_p$ is the principal normal vector. It matches the point, tangent line and curvature. For a space curve, it lies in the plane spanned by the unit tangent vector and principal normal vector. The parabola example under curvature of a plane curve gives center $(0,1/2)$ and radius $1/2$ at the origin.

In the exterior orthonormal coframe $E^0=\sqrt{V}dt$, $E^1=dr/\sqrt{V}$, $E^2=rd\theta$, $E^3=r\sin \theta d\varphi$, use ${\Omega }^a{}_b=\frac{1}{2}{R}^a{}_{bcd}{E}^c\wedge E^d$. The independent mixed-index curvature 2-forms are ${\Omega }^0{}_1=-V^{\prime \prime }{E}^0\wedge E^1/2$, ${\Omega }^0{}_A=-V^{\prime }{E}^0\wedge E^A/(2r)$, ${\Omega }^1{}_A=-V^{\prime }{E}^1\wedge E^A/(2r)$ for $A=2,3$, and ${\Omega }^2{}_3=(1-V)E^2\wedge E^3/r^2$. The orthonormal Ricci tensor is diagonal with entries $V^{\prime \prime }/2+V^{\prime }/r$, $-V^{\prime \prime }/2-V^{\prime }/r$, and twice $(1-V-r{V}^{\prime })/r^2$. The Einstein field equations with negative cosmological constant then give $V=1-2M/r+r^2/l^2$.

// Target: general-relativity.bigb

A torsion dynamometer compares an applied torque with the restoring torque of a twisted elastic element. Returning the lever to its zero position allows a current-loop torque experiment to hold the test orientation fixed while reading the compensating torque.

A radial current dipole has moment parallel to its displacement from the sphere centre. The Sarvas formula gives identically zero exterior magnetic field for such a source in a spherical volume conductor model, although its electric potential need not vanish. This is a nonuniqueness mechanism for the inverse problem in magnetoencephalography.

For a nonnegative continuous event time with survival function S, hazard function h and H the cumulative hazard, split the expectation at c. The censored contribution is S(c)H(c). The event contribution is ${\int }_0^{c}H(t)f(t)dt=1-S(c)-S(c)H(c)$ by integration by parts, since $dH=hdt$ and $f=hS$. Adding gives the displayed identity. Linearity extends it to the expected observed event count in a cohort, without requiring independence between subjects.

Let $X=\min (T,C)$, $V=1_{\{T\le C\}}$, and suppose $T$ and $C$ are independent, with continuous failure density $f=hS$. Then the Tonelli theorem givesThus $V-H(X)$ has mean zero. The identity also holds conditionally on covariates when censoring is independent conditionally on them. For an estimated hazard, the mean error is controlled by $\mathbb{E}|\hat{H}(X)-H(X)|$, not by pointwise fit alone.

// Target: survival-analysis.bigb

For independent and identically distributed random variables with a moment-generating function finite on an open interval containing zero, the empirical mean satisfies a large deviation principle with Rate function given by the Legendre transform of a cumulant-generating function. The exponential-moment hypothesis is part of this theorem; mere integrability does not guarantee this full large-deviation principle.

For independent identically distributed continuous random variables, let $K(t)=\log \mathbb{E}{e}^{tY}$ be finite near zero. At a mean value $x$ reached by $K^{\prime }(\hat{t})=x$ in the interior of that domain, the displayed leading approximation describes the density of the sum $S_n$. Exponential tilting by $\hat{t}$ puts the sum at its tilted mean; a local normal approximation gives the denominator, and undoing the tilt gives the exponential factor. Under suitable smooth nonlattice regularity, at fixed interior $x$ the exact density is the displayed quantity times $1+O(n^{-1})$. This statistical approximation is an application of the saddle-point approximation to a transform inversion integral. It requires more than the existence of a few ordinary moments.

For the cubic-matter no-scale vacuum with an exponential dilaton, put $x=S+\bar{S}$ and $A=a{e}^{-\alpha S}$. At $C=0$, zero energy requires $G(S)=b+(1+\alpha x)A=0$. On a fixed-imaginary-part slice, $s=\mathrm{Re}S$ gives $dG/ds=\alpha (1-2\alpha s)A$. At the double root $s_0=1/(2\alpha )$, the first derivative vanishes but the second is $-2{\alpha }^{2}A$, so $G=-{\alpha }^{2}A(s-s_0{)}^2+O((s-s_0{)}^3)$. Therefore $V=|G{|}^2/(x{y}^3)$ begins at fourth order. This real scalar has zero quadratic mass but is not an exact flat direction of a scalar potential. The imaginary part of $S$ is still lifted quadratically, and the matter field $C$ also has a quartic leading potential. Exact flat directions instead come from the complex volume modulus $T$ along the zero-energy locus.

Minimize the displayed criterion using a Leave-one-out cross-validation estimate in the second term. Its expectation is the exact mean integrated squared error minus $R(f)$, which does not depend on h. With standard smoothness and appropriate search ranges, the chosen bandwidth is asymptotically optimal. Its relative fluctuations are ordinarily of order $n^{-1/10}$, considerably larger than the estimation error achievable by well-chosen plug-in pilots.

Subtract the diagonal variance contribution from the derivative estimate's squared norm, then insert the result into asymptotic mean integrated squared error. The expected curvature estimate equals $R(f^{\prime \prime })+O(h^2)$, apart from a negligible finite-n term. This makes the resulting criterion asymptotically correct even though it is not an exactly unbiased estimate of integrated risk. Under standard conditions its selector is ratio-consistent with the optimal bandwidth and has relative fluctuation order $n^{-1/10}$.

Nested cross-validation separates selection from assessment using two levels of held-out data. Within each outer training fold, inner cross-validation chooses a model or tuning parameters; the complete selected fitting procedure is then applied to that outer training fold and scored on its untouched outer evaluation fold. This estimates the performance of the model selection procedure, including its choices, rather than reusing its best inner score as an assessment. A final model can then be selected and fitted using all available training data. ^[ref]

Normalize three distinct points on a generalized circle by a Möbius transformation $f$ taking them to $1,0,\infty$. The inverse image of the extended real line is their generalized circle, and $J=f^{-1}\circ c\circ f$, with $c(z)=\overline{z}$, fixes that circle pointwise and squares to the identity. Changing the three normalizing points replaces $f$ by $M\circ f$ with a real-coefficient Möbius map $M$. Since $M$ commutes with conjugation, $J$ is unchanged. Thus this construction depends only on the circle. It is inversion in a circle for an ordinary circle and Euclidean reflection for a straight line, each extended to the Riemann sphere.

The cross-product matrix $A\mathbf{x}=\mathbf{n}\times \mathbf{x}$ obeys $A\mathbf{n}=0$ and $A^2=\mathbf{n}{\mathbf{n}}^T-|\mathbf{n}{|}^{2}I$, by the vector triple product. On the orthogonal complement of a nonzero $\mathbf{n}$, this is $-|\mathbf{n}{|}^{2}I$; hence the two remaining complex eigenvalues are $\pm i|\mathbf{n}|$. In an orthonormal basis adapted to $\mathbf{n}$, $A$ is a planar quarter-turn multiplied by $|\mathbf{n}|$, with zero action along its axis.

In the quasistatic spherical model, ${\eta }_0=a_0/a_c\in (0,1)$. Separating the radius equation gives $t_e=(S/\kappa ){\int }_0^{a_0}{a}^2/(a_c-a)da$, which is finite and evaluates to the displayed expression. Near disappearance, $a^3\sim 3\kappa a_c(t_e-t)/S$. The time diverges logarithmically as the initial radius approaches the critical radius from below.

Assume deterministic discount factors, deterministic risk-neutral default intensity $\lambda$, fixed fractional recovery $\mathcal{R}$, and continuous premium until default or maturity. Survival is $G(t)=\exp (-{\int }_0^t\lambda (u)du)$. Expected discounted protection is $N(1-\mathcal{R})\int B\lambda G$, while premium is $Ns\int BG$. Equating them gives the par spread. Constant intensity gives $s=(1-\mathcal{R})\lambda$. Discrete premiums, default accrual, stochastic recovery and counterparty losses require changing the corresponding cash-flow expectations.

A basket credit contract whose protection payment is triggered by the first specified default among its reference entities. The loss and settlement terms refer to the defaulting name according to the contract. Premium generally stops at that first default or maturity. Pricing requires the joint default-time distribution; separate marginal default probabilities alone do not determine the first-default law.