In the Black-Scholes model, take $B_t=e^{-\rho (T-t)}$ as a zero-coupon bond numéraire. Holding the fixed stock value $\theta B_t$ requires $g_t=\theta B_t/S_t$ shares. The self-financing conditions for smooth stock and bond holdings giveunits of the bond. Since $V_t=g_t{S}_t+h_t{B}_t$, this yields the displayed wealth. Under the risk-neutral measure, $V_t/B_t=w_0/B_0+\theta \sigma W_t^Q$: the discounted gain is a constant multiple of Brownian motion.

For the orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^4$, define ${\eta }_{ij}^a={ϵ}_{aij}$, ${\eta }_{i4}^a={\delta }_{ai}$, and extend antisymmetrically. The three forms $\frac{1}{2}{\eta }_{\mu \nu }^{a}d{x}^{\mu }\wedge d{x}^{\nu }$ are self-dual. Reversing the sign of the components involving index four gives the anti-self-dual symbols $\bar{\eta }$. They identify the two three-dimensional eigenspaces of the Hodge star on differential two-forms and give compact formulas for the BPST instanton.

For a uniform elastic filament clamped below and free above, its own weight gives filament tension $T(z)=-\lambda g(L-z)$. With filament bending modulus $B=EI$, the static slope $u$ satisfies $B{u}^{\prime \prime }+\lambda g(L-z)u=0$, $u(0)=0$, and $u^{\prime }(L)=0$. Setting $\eta =(2/3)\sqrt{\lambda g/B}(L-z{)}^{3/2}$ gives $u={\eta }^{1/3}{J}_{-1/3}(\eta )$ up to amplitude; the $J_{1/3}$ branch violates the zero-moment condition. The first positive zero $j_{-1/3,1}$ therefore yields $\lambda g{L}_c^3/B\simeq 7.83735$. This is a self-buckling of a vertical rod threshold, not a constant-end-load Euler threshold.

A densely defined semibounded symmetric operator on a Hilbert space has a canonical semibounded extension that is a self-adjoint operator obtained by closing its quadratic form. After shifting the lower bound to zero, complete the initial form domain in the norm $(\Vert u{\Vert }^2+q[u]{)}^{1/2}$; the associated closed form determines the Friedrichs extension. For the positive Laplace-Beltrami operator on a polygonal surface, this specifies the finite-energy realization at corners. Different choices of the initial form domain impose Dirichlet boundary conditions, Neumann boundary conditions, or mixed boundary conditions; the extension does not choose the boundary condition independently of that domain.

Let $\ddot{q}=Fq$ on a Hilbert space, with a time-independent self-adjoint operator $F$ that is a bounded linear operator, and conserved quadratic energy $E=\Vert \dot{q}{\Vert }^2/2-⟨q,Fq⟩/2$. If $⟨q_0,F{q}_0⟩>0$, set $\dot{q}(0)=\alpha q_0$ with ${\alpha }^2=⟨q_0,F{q}_0⟩/\Vert q_0{\Vert }^2$ and $\alpha >0$. This gives $E=0$. For $N=\Vert q{\Vert }^2$, $N^{\prime }=2\mathrm{Re}⟨q,\dot{q}⟩$ and $N^{\prime \prime }=4\Vert \dot{q}{\Vert }^2$, soby the Cauchy-Schwarz inequality. Since $(\ln N{)}^{\prime }(0)=2\alpha$, $N(t)\ge N(0)e^{2\alpha t}$ for $t\ge 0$. Positivity of $N$ follows from $N^{\prime }(0)>0$ and $N^{\prime \prime }\ge 0$. Thus a negative direction of the restoring energy $W=-⟨q,Fq⟩/2$ yields an exponentially growing solution without a normal-mode expansion or a variational spectral theorem. If $W$ is positive definite, conserved $E$ instead controls the energy norm; a bound in the original Hilbert space norm additionally follows from a coercive lower bound on $W$. For an unbounded self-adjoint operator, the same estimate applies to sufficiently regular solutions for which the energy identity is justified; existence for arbitrary initial data must be established separately.

Variation of the small-slope bending energy of an elastic filament gives endpoint work $A[h^{(2)}\delta h^{(1)}-h^{(3)}\delta h]$. Arbitrary endpoint displacement and slope variations force the bending moment and transverse force to vanish. These are $h^{(2)}=h^{(3)}=0$. They define a self-adjoint nonnegative fourth-order bending operator on a finite interval; its kernel contains rigid translation and tilt.

A response model and a dropout model share latent random effects. Conditional on those effects and observed predictors, the two processes may be independent, yet marginally dropout is missing not at random because the shared effects also predict unseen outcomes. Joint likelihood fitting integrates the product of both models over the shared effects; integrating the outcome model alone is insufficient. This is an explicit identifying model, and its latent-dependence assumptions should be assessed by sensitivity analysis.

If the displayed fiber bundle had a section $s$, its projection $p$ would satisfy $s^{\ast }{p}^{\ast }=1$. The cohomology ring of complex projective space is $\mathbb{Z}[h]/(h^4)$ with $|h|=2$. For a generator $u\in H^4(S^4;\mathbb{Z})$, write $p^{\ast }u=m{h}^2$. But $s^{\ast }h=0$ since $H^2(S^4;\mathbb{Z})=0$, giving $u=s^{\ast }{p}^{\ast }u=0$. This contradiction proves that no section exists, without needing the value of $m$.

For a nonnegative random variable $N$ with positive expected value and finite second moment, write $N=N{1}_{\{N>0\}}$. The Cauchy-Schwarz inequality giveswhich proves the bound in second moment method. It is the zero-threshold case of the Paley-Zygmund inequality. Equality occurs when $N$ is a fixed positive value on the event $N>0$ and zero elsewhere, almost surely. For $N=10B$ with $B$ a Bernoulli random variable of parameter $1/10$, $(\mathbb{E}N{)}^2/\mathbb{E}{N}^2=1/10$ equals the exact positivity probability.

The variance identity $\mathbb{E}{N}^2=(\mathbb{E}N{)}^2+\mathrm{Var}(N)$ turns the lower bound into a statement about variance relative to the squared mean. A small ratio forces positivity to be likely. A positive expected value alone does not do so: keeping $\mathbb{E}N=1$ while taking $N=1/q$ with probability $q$ and zero otherwise makes the positivity probability as small as any chosen $q>0$.

Let $C^a{}_{bcd}={\partial }_d{\Gamma }_{bc}^a-{\partial }_c{\Gamma }_{bd}^a-{\Gamma }_{ec}^a{\Gamma }_{bd}^e+{\Gamma }_{ed}^a{\Gamma }_{bc}^e$, with all other index positions obtained by the metric tensor. This convention is the negative of the convention $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$. The Ricci identity is $k_{a;bc}-k_{a;cb}=C_a{}^d{}_{bc}{k}_d$, with $k_{a;bc}={\nabla }_c{\nabla }_b{k}_a$. For a Killing vector, the Killing equation makes this second derivative antisymmetric in $a,b$. Applying the commutator three times gives $2k_{a;bc}=(C_a{}^d{}_{bc}+C_c{}^d{}_{ba}-C_b{}^d{}_{ac})k_d=-2C_{abc}{}^d{k}_d$, where the last step uses the first Bianchi identity. Switching the sign of the Riemann curvature tensor switches the sign in this identity; moving a raised index by ordinary metric contraction does not switch it.

Mechanical redistribution of sea ice changes thickness-class area through ridging, rafting and lead formation. In the conservative thickness-distribution equation, local volume conservation gives $\int h\psi (h)dh=0$; area normalization including open water gives $\int \psi (h)dh=\nabla \cdot v$ when thermodynamic boundary fluxes are accounted for consistently.

// Target: geophysics.bigb

For critical planar percolation in a conformal quadrilateral, Cardy's boundary-crossing probability is $I_{\theta }(1/3,1/3)$, where $\theta \in (0,1)$ is its conformal cross-ratio and $I$ is the regularized incomplete beta function. This conformally invariant crossing law is associated with Schramm–Loewner evolution at parameter six.

The conformally invariant critical percolation crossing probability can be expressed in an appropriate ordered boundary cross ratio $\theta \in [0,1]$ asFor the corresponding SLE exploration at parameter six, it arises from the beta integral for strict ordering of coupled Bessel lifetimes with $a=1/3$. In the upper-half-plane configuration $0<x<y$, the strict-swallowing probability is this function at $\theta =1-x/y$. Exchanging the two relevant boundary arcs exchanges the probability with its complement. This is a crossing probability, not the probability of visiting a fixed boundary point.

A pole of the continued scattering matrix associated with outgoing boundary conditions and an unstable trapped state. A pole below the real energy axis gives time dependence $e^{-i{E}_{R}t/\hslash }{e}^{-\Gamma t/(2\hslash )}$, so the probability lifetime is $\hslash /\Gamma$. Bound-state poles instead lie on the positive imaginary wavenumber axis and produce square-integrable decaying spatial waves.

Hold a stable input law and service rate of a queue fixed while the buffer size of a queue grows. Under interior exponential-moment and root hypotheses, the Cramér-Lundberg workload exponent gives an exponential tail linear in buffer size. A likely overflow accumulates during a lookback horizon proportional to the buffer size.

Scale independent source count, service and buffer by the same factor $L$. For lookback horizon $n$, the source-average cumulative input has rate function $I_n$, and reaching buffer level $b$ requires input above $b+cn$. Optimizing this horizon gives the displayed exponent under uniform control of the infinite stationary supremum. For independent slots, $I_n(x)=n{I}_A(x/n)$; discrete time keeps $n$ an integer. Temporal dependence instead enters through the cumulative-input cumulant-generating function.

For a bufferless queue with Poisson mean input $L\lambda$, place the service margin at $C{L}^{(1+\beta )/2}$, $0<\beta <1$. It exceeds the ordinary square-root fluctuation scale but is sublinear in total input. The Poisson moderate deviation principle gives the displayed overflow exponent. This interpolates between a central-limit safety margin and a fixed per-source excess capacity.

For fixed independent and identically distributed arrivals of mean $\mu$ and positive finite variance ${\sigma }^2$, put service at $\mu +\epsilon d$. Rescaling net input by space $\epsilon$ and time ${\epsilon }^{-2}$ gives Brownian motion with drift $-d$ by the functional central limit theorem. The continuous Skorokhod reflection map on the half-line gives the displayed transient limit, with a correspondingly scaled initial workload. A stationary approximation additionally requires justification of limit interchange. When valid, the stationary law of negatively drifted reflected Brownian motion predicts $\mathbb{P}(Q_{\epsilon }>b/\epsilon )\approx e^{-2db/{\sigma }^2}$.

Apply the vector triple product identity with first vector $b\times c$. Its contraction with $c$ vanishes, so $(b\times c)\times (c\times a)=[a\cdot (b\times c)]c$. Dotting with $a\times b$ gives the square of the scalar triple product. This is also the Gram determinant of $a,b,c$, hence is nonnegative even when the vectors are linearly dependent.

The cross product of two cross products sharing their first vector is parallel to that vector, with scalar factor equal to the scalar triple product. In suffix notation, use ${ϵ}_{ijk}{ϵ}_{kpq}={\delta }_{ip}{\delta }_{jq}-{\delta }_{iq}{\delta }_{jp}$. The component of the double cross product becomes $A_i{ϵ}_{q\ell m}{C}_q{A}_{\ell }{B}_m-C_i{ϵ}_{p\ell m}{A}_p{A}_{\ell }{B}_m$. The second term vanishes by antisymmetry of the Levi-Civita symbol; the first equals $A_i(A\cdot (B\times C))$, proving the identity even when the vectors are dependent.

The Brans-Dicke theory adds a dynamical scalar field $\Phi$ to a metric tensor theory of gravity, with constant dimensionless coupling $\omega$. In a $(+---)$ signature and the reversed-curvature convention used here, the displayed Lagrangian is supplemented by matter and integrated against the metric tensor volume density with prefactor $1/(16\pi G)$. Its scalar multiplier of the Ricci scalar lets the curvature coupling vary in spacetime. Both the metric tensor and scalar are varied; the scalar equation of Brans-Dicke theory follows by eliminating curvature between the scalar equation and the trace of the metric tensor equation. The original construction is described in Brans and Dicke's 1961 article.