The homogeneous Dirichlet boundary condition kernel for $u_t=u_{xx}+\alpha u_x$ on $x>0$. Here $H(r,t)=e^{-r^2/(4t)}/\sqrt{4\pi t}$ is the heat kernel. The exponential reflection weight makes $K_{\alpha }(0,y,t)=0$. Gaussian decay makes its initial-data integral convergent even when the exponentially weighted initial data are not integrable separately.

For a continuously differentiable periodic kernel, integration by parts and the Riemann-Lebesgue lemma give $c_n=o(1/|n|)$. Nonperiodic data $e^{\alpha x}$ with $\alpha \notin i\mathbb{Z}$ have Fourier coefficients asymptotic to a nonzero multiple of $1/n$. Their inverse coefficients $y_n/c_n$ therefore fail square summability, and there is no Hilbert-space inverse or least-squares minimizer. Periodic exponential data $\alpha =im$ instead give the single-mode solution $e^{imx}/c_m$.

If entry is uniform on calendar interval $[{\tau }_a,{\tau }_b]$ and follow-up stops at ${\tau }_c$, the censoring duration is uniform on $[{\tau }_c-{\tau }_b,{\tau }_c-{\tau }_a]$. Its hazard function diverges at the longest possible follow-up. It is independent censoring when entry date is independent of prognosis after the required conditioning.

A densely defined operator $T$ on a Hilbert space is symmetric whenEquivalently, $T$ agrees with its adjoint operator on $D(T)$, while $D(T^{\ast })$ may be larger. A self-adjoint operator additionally requires equality of these operator domains.

For fixed-composition monatomic ideal gas plus equilibrium blackbody radiation, let $\mathcal{R}=k/(\mu H)$ and $\beta =P_g/P$. The mass-specific heats are $c_P=\mathcal{R}(32-24\beta -3{\beta }^2)/(2{\beta }^2)$ and $c_V=\mathcal{R}(24-21\beta )/(2\beta )$. The specific-heat ratio is ${\Gamma }_1/\beta$, not generally any one of the stellar adiabatic exponents. Differentiate at constant total pressure rather than at constant $\beta$.

A nonzero finitely generated subgroup of the additive rational numbers is an infinite cyclic group. Put the generators over a common positive denominator. Their integer numerators generate the ideal given by their greatest common divisor, and Bezout identity expresses that divisor as an integer combination of them. Dividing back by the denominator gives a single generator.

Consider an infinite planar membrane of areal mass density $m_s>0$ and tension $\tau \ge 0$, separating identical inviscid fluids of mass density $\rho$ and speed of sound $c$. For a plane component with time factor $e^{-i\omega t}$, tangential wavenumber $\kappa$ and positive normal wavenumber $q=\sqrt{{\omega }^2/c^2-{\kappa }^2}$, the fluid's normal acoustic impedance is $Z=\rho \omega /q>0$.

The membrane displacement $\eta$ has normal velocity $v=-i\omega \eta$, and its tension-mass balance gives the pressure jumpThe same normal velocity occurs on both sides. For pressure reflection and transmission amplitudes $r,t$ relative to the incident pressure, velocity continuity gives $t=1-r$ and the pressure jump gives $2r=Z_{s}t/Z$. HenceSince $Z_s$ is imaginary, $|r{|}^2+|t{|}^2=1$: the membrane stores and returns energy but has no mean dissipation. If its inertial and tension terms cancel, $Z_s=0$ and this component is fully transmitted. These formulas assume a propagating incident component; evanescent components do not have the same positive incident-power interpretation.

Under the conditional-variance and overshoot assumptions of the absorption-time bound from conditional variance and overshoot, $\mathbb{P}(T>n)\le 2\tau /(\sigma \sqrt{n})$. Bound the probability by $\mathbb{P}(T_R<\infty )+\mathbb{E}(T\wedge T_R)/n$ and optimize $1/R+{\tau }^{2}R/({\sigma }^{2}n)$. If the resulting bound is below one, its minimizing threshold is greater than one; otherwise the probability bound is trivial.

If $M$ is a transitive model of ZF, its internally computed $L_{\alpha }$ is the actual $L_{\alpha }$ for each ordinal $\alpha \in M$. At a successor stage use absoluteness of satisfaction for the same set structure, formula codes and parameters; at a limit stage take the union of the already identical earlier stages. Choice is not required.

In the model $d{r}_t=r_t(r_t-1)(dt+d{W}_t)$ under the pricing measure, with $0\le r_t\le 1$, the discounted affine expression $D_t[A(T-t)+B(T-t)r_t]$ has zero drift when $A^{\prime }=0$ and $B^{\prime }=-A-B$. Terminal conditions $A(0)=1$, $B(0)=0$ yield the displayed unit bond price. The expression is bounded, so the bounded local martingale criterion justifies pricing by conditional expectation.

Suppose random continuous real functions have unique zeros. If pointwise convergence in probability gives opposite strict limiting signs at the two ends of every sufficiently small interval about the target, the intermediate value theorem traps their zeros in those intervals with probability tending to one. The limiting function need not be continuous and the random functions need not be monotone.

After a scalar vacuum expectation value, a Yukawa matrix produces a bilinear $-{\bar{f}}_L{M}_f{f}_R+h.c.$. Independent unitary left/right rotations diagonalize $M_f$ with nonnegative masses. Extra scalar-dependent operators can make the physical scalar coupling matrix fail to diagonalize with the mass matrix.

A pseudoscalar Yukawa interaction couples a Dirac field bilinear containing the chirality matrix to a spin-zero field. In the standard mostly-minus gamma matrix convention with ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the usual Dirac adjoint, $ig\bar{\psi }{\gamma }^5\psi \varphi$ is Hermitian for real $g$ and real $\varphi$. If $\varphi$ is a pseudoscalar, the interaction also preserves parity symmetry in quantum field theory; if $\varphi$ is parity even, it breaks that symmetry. Moving the factor $i$ into the definition of ${\gamma }^5$ changes the displayed vertex coefficient but not the interaction. Each vertex has one bosonic leg and two oppositely oriented fermion legs, and preserves Dirac fermion number conservation.

The Standard Model lepton doublet $L=({\nu }_L,e_L{)}^T$ has hypercharge $-1/2$, the electron singlet $e_R$ has $-1$, and the Higgs doublet has $1/2$. Therefore $-y_e\bar{L}\varphi e_R+h.c.$ is gauge-invariant even though a bare ${\bar{e}}_L{e}_R$ mass is not. The vacuum expectation value generates a Dirac mass $|y_e|v/\sqrt{2}$. In unitary gauge, the Higgs boson coupling is $-(m_e/v)h\bar{e}e$ after rephasing the electron.

Drift driven by the Yarkovsky effect can carry a body out of a belt or into an unstable orbital region. A useful two-limit removal law is $t_y=BD+C/D$, with slow escape at both large and small diameters.

For a compact simple gauge group, use $B=-\kappa$ on its algebra and an averaged positive invariant Hermitian form $h$ on any supplied finite-dimensional complex representation. With anti-Hermitian connection and signature $(+---)$, $\mathcal{L}=-B({\mathcal{F}}_{\mu \nu },{\mathcal{F}}^{\mu \nu })/(4g^2)+h(D_{\mu }\Phi ,D^{\mu }\Phi )-V(\Phi )$ is gauge-invariant whenever $V$ is invariant. A real scalar representation instead has a symmetric form and kinetic prefactor $1/2$.

Stationarity of the Yang-Mills action in the absence of sources gives $D_{\mu }{F}^{\mu \nu }=0$, using the adjoint gauge covariant derivative. This dynamical equation differs from the geometric gauge-theory Bianchi identity.

For the convention $D_{\mu }={\partial }_{\mu }+g{A}_{\mu }$, a gauge group transformation $\Phi \mapsto R\Phi$ acts on the gauge potential by $A_{\mu }\mapsto R{A}_{\mu }{R}^{-1}-g^{-1}({\partial }_{\mu }R)R^{-1}$. The gauge field strength transforms by conjugation. This makes $D_{\mu }\Phi$ transform in the same gauge group representation as $\Phi$.

The self-dual Yang-Mills equations require a gauge curvature two-form to have positive Hodge star operator eigenvalue on an oriented Riemannian four-manifold. By the Bianchi identity, they imply the full second-order Yang-Mills equations. Smooth finite-action solutions saturate the Yang-Mills instanton Bogomolny bound. Reversing orientation exchanges this equation with the Anti-self-dual Yang-Mills equations.

Both the self-dual Yang-Mills equations and the Anti-self-dual Yang-Mills equations imply the second-order Yang-Mills equations because the Bianchi identity holds for every principal connection. Finite action is additionally required to call the solution a Yang-Mills instanton; the implication itself is a local differential identity.