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Expand each factor as a geometric series indexed by a nonnegative matrix. The RSK correspondence preserves its row and column contents, changing the sum to pairs of Semistandard Young tableaux of the same shape and proving the displayed formal identity. Its kernel is also ${\sum }_{\lambda }{h}_{\lambda }(x)m_{\lambda }(y)={\sum }_{\lambda }{p}_{\lambda }(x)p_{\lambda }(y)/z_{\lambda }$. These are dual-basis kernels for the Hall inner product of symmetric functions, so the Schur expansion proves orthonormality. Applying the symmetric-function involution in one alphabet gives the dual identity ${\prod }_{i,j}(1+x_i{y}_j)={\sum }_{\lambda }{s}_{\lambda }(x)s_{{\lambda }^{\prime }}(y)$.

Relative survival compares observed survival in a diseased cohort with expected survival derived from a suitable reference population. A useful individual model adds a known background mortality hazard to a common excess hazard. Reference mortality can depend on age, sex and calendar time. The comparison avoids classifying each death by cause, but its interpretation depends on the reference population being appropriate. An excess-survival ratio is not automatically the effect of a causal intervention removing one cause of death.

Survival data describe the time $T$ from a specified origin to an event, together with covariates and the observation mechanism. Under right censoring, the observed exit time is $Y=\min (T,C)$ and the event indicator is $\delta =1_{\{T\le C\}}$. With left truncation, an entry time $E$ is also recorded and subjects are observed only if $T>E$. Recording the time origin, entry, exit and event indicator is essential for constructing valid risk sets.

A cure model mixes an individual class with identically zero hazard function and a susceptible class with survivor function $S_{\ast }$. For $0\le \pi <1$, its representation as a proportional frailty model with $\mathbb{E}U=1$ is $U=0$ with probability $\pi$ and $U=1/(1-\pi )$ otherwise, with baseline hazard $h_0=(1-\pi )h_{\ast }$. The frailty law is a mixture of two Dirac measures, rather than an ordinary density. Its Laplace transform evaluated at $H_0$ gives the displayed survivor function. The limiting survival equals $\pi$ when $S_{\ast }(t)\to 0$; otherwise the plateau also includes susceptible individuals who never experience the event.

For a broad, thin oil lens below a flat ice-water interface, define the effective interfacial energy $\Gamma ={\gamma }_{io}+{\gamma }_{ow}-{\gamma }_{iw}$ from the relevant surface tensions. At fixed oil volume $\mathcal{V}=Ah$, the energy is $E=\Gamma \mathcal{V}/h+\frac{1}{2}({\rho }_w-{\rho }_o)g\mathcal{V}h$. Setting its derivative to zero gives $h^2=2\Gamma /[({\rho }_w-{\rho }_o)g]$ for $\Gamma >0$, and the positive second derivative proves a minimum. A contact angle measured through the oil gives $\Gamma ={\gamma }_{ow}(1-\cos \vartheta )$. Oil-ice tension alone does not determine this effective quantity. Complete wetting or rough underside pockets invalidate this finite, flat-lens approximation. With densities $1025$ and $850\mathrm{kg}{\mathrm{m}}^{-3}$, a lens below $1\mathrm{cm}$ requires $0<\Gamma <0.08584\mathrm{N}{\mathrm{m}}^{-1}$.

The Bond number compares hydrostatic stress over a length $\ell$ with capillary stress. For a gas bubble of radius $a$ and negligible density, $\mathrm{Bo}=\rho g{a}^2/\gamma$. Small Bond number is the usual sufficient criterion for capillarity to resist gravity-scale deformations. For a clean spherical bubble with negligible internal dynamic viscosity settling in creeping flow, the capillary number is $\mathrm{Ca}=\mu U/\gamma =\mathrm{Bo}/3$. This last relation uses the clean-bubble terminal velocity; it does not apply at arbitrary Reynolds number or to an interface immobilized by surfactants.

Wetting describes how a liquid maintains contact with or spreads over a solid in the presence of another phase, usually a gas. The competition among solid-liquid, solid-gas and liquid-gas interfacial free energies determines the equilibrium contact angle. For an ideal smooth homogeneous surface, Young's contact-angle equation giveswith $\theta$ measured through the liquid.

A small water drop that spreads over clean glass but beads on a water-repelling surface provides a simple comparison. A low contact angle indicates stronger spreading within this partial-wetting description; complete spreading requires considering the associated interfacial free-energy change. Surface roughness, contamination and contact-angle hysteresis can make the observed angle depend on preparation and whether the contact line is advancing or receding, so one photograph need not determine an equilibrium material constant.

For a liquid interface with constant surface tension $\gamma$, the mechanical surface energy is the energy associated with its surface area $A$:The additive constant $C$ does not affect the work needed to change surface area. For several interfaces with the same surface tension, $A$ means their total surface area; the two faces of a thin soap film must both be counted. ^[ref]

With assumed $\gamma =0.072\mathrm{N}{\mathrm{m}}^{-1}$, creating $1\mathrm{c}{\mathrm{m}}^2=10^{-4}{\mathrm{m}}^2$ of one interface requires $7.2\mu \mathrm{J}$ of reversible work. In the sliding-rim example illustrated under surface tension, moving a $20\mathrm{mm}$ rim outwards by $1\mathrm{mm}$ creates $4.0\times 10^{-5}{\mathrm{m}}^2$ across the two faces and requires $2.88\mu \mathrm{J}$. Independently, multiplying its $2.88\mathrm{mN}$ holding force by the displacement gives the same work.

This makes the preference for a round isolated droplet precise. At fixed volume $V$, the three-dimensional isoperimetric inequality giveswith equality for a sphere. Thus a sphere minimizes surface energy among sufficiently regular closed shapes enclosing the same volume, when surface tension is uniform and other energy contributions are negligible. This is an application of the isoperimetric inequality, rather than a proof of it. Walls, wetting, and gravity can change the shape by adding constraints or other energy contributions. ^[ref]

The differential rule $d{E}_{\mathrm{s}}=\gamma dA$ integrates directly to $E_{\mathrm{s}}=\gamma A+C$ because surface tension is constant. At fixed liquid volume, decreasing surface area can drive viscous fluid flow; the released surface energy can supply viscous dissipation. This mechanical bookkeeping is useful for predicting which deformations can release energy. If surface tension depends on temperature or composition, distinguish this mechanical area-work relation from the complete interfacial thermodynamic free energy and internal energy.

Dynamical supersymmetry breaking occurs when quantum dynamics produces a vacuum that does not preserve supersymmetry. An asymptotically free hidden sector can generate the displayed exponentially small scale through dimensional transmutation. This supplies a possible origin of a hierarchy, but strong coupling by itself does not prove breaking: the vacuum equations and dynamics must exclude a supersymmetric vacuum.

// Target: supersymmetry.bigb

For $d$ fermionic creation operators with the canonical anticommutation relations, the Hilbert space $L^2({\mathbb{R}}^d)\otimes {\Lambda }^{\bullet }{\mathbb{C}}^d$ splits into ${\mathcal{H}}_n=L^2({\mathbb{R}}^d)\otimes {\Lambda }^n{\mathbb{C}}^d$, $0\le n\le d$. The fermion number operator $N={\sum }_a{\psi }_a^{†}{\psi }_a$ has value $n$ on ${\mathcal{H}}_n$; its internal multiplicity is $\left(\genfrac{}{}{0ex}{}{d}{n}\right)$. An odd supercharge $Q=q+q^{†}$, with $q$ raising $n$ and $q^2=0$, gives $H=\{q,q^{†}\}$ and preserves these sectors.

On an eigenspace with $E>0$, $Q/\sqrt{E}$ is an odd involution pairing bosonic and fermionic states. More precisely,gives an orthogonal decomposition of each state into the images of $q$ and $q^{†}$: their inner product vanishes because $(q^{†}{)}^2=0$. If $u\in \mathrm{im}{q}^{†}$, then $q^{†}u=0$, $q^{†}qu=Eu$ and $\Vert qu{\Vert }^2=E\Vert u{\Vert }^2$. Thus $q/\sqrt{E}$ pairs such a state in degree $n$ with one in degree $n+1$, and $q^{†}/\sqrt{E}$ is the inverse. States with zero energy are killed by both operators and may remain unpaired. The pairing applies to positive spectral subspaces as well; discreteness is not implied.

For a real smooth function $\chi$ on Euclidean space, put $A_a={\partial }_a-{\partial }_a\chi$ and $q={\sum }_a{\psi }_a^{†}{A}_a$. The adjoint operator for the usual $L^2$ inner product is $A_a^{†}=-{\partial }_a-{\partial }_a\chi$. Since $[A_a,A_b]=0$, the canonical anticommutation relations give $q^2=0$. Expanding $H=\{q,q^{†}\}$ and using $[A_a,A_b^{†}]=-2{\partial }_a{\partial }_b\chi$ givesIn the empty fermionic Fock space sector the last term vanishes; in the filled sector ${\psi }_a^{†}{\psi }_b={\delta }_{ab}$. Consequently the two scalar Hamiltonian operators are $H_0=-\Delta +|\nabla \chi {|}^2+\Delta \chi$ and $H_d=-\Delta +|\nabla \chi {|}^2-\Delta \chi$. On the compatible self-adjoint operator domain the Hamiltonian operator is nonnegative because $⟨v,Hv⟩=\Vert qv{\Vert }^2+\Vert q^{†}v{\Vert }^2$.

With a well-defined regulated supertrace, positive-energy states of supersymmetric quantum mechanics are paired by an odd invertible charge. Their opposite fermion parities cancel, leaving $I=n_B^0-n_F^0$, independent of $\beta$. A nonzero index proves the existence of a supersymmetric vacuum; zero does not prove its absence. Discrete-spectrum or suitable infrared conditions matter, because continuum thresholds and states escaping to infinity can invalidate naive trace manipulations or deformation invariance.

The candidate zero modes are $e^{-{\int }^{x}w}$ in the minus sector and $e^{+{\int }^{x}w}$ in the plus sector; this follows from the positive squared-norm expressions for their energy. Their membership in the Hilbert space determines whether supersymmetry is unbroken. For $w=x$ only the first is normalizable. For $w=x^2+a$, $a>0$, neither is normalizable, while both partner potentials grow quartically and have discrete spectra. Thus their strictly positive lowest energy is paired and supersymmetry is broken, with zero Witten index.

In four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory, Seiberg–Witten theory determines exact low-energy quantities from the geometry of the vacuum parameter space. For rank one, the central charge in supersymmetry is proportional to $n_{e}a+n_m{a}_D$, with integral electric and magnetic charges and $a_D$ the dual scalar period. The BPS bound in supersymmetry fixes the mass of a short state by this charge combination. Its vanishing identifies loci where a magnetic monopole or dyon can become massless, even when the original electric description is strongly coupled.

Minimal ten-dimensional supersymmetric gauge theory contains a gauge field and an adjoint Majorana-Weyl spinor. They have eight propagating bosonic and eight propagating fermionic states per gauge generator and sixteen real supercharges. A toroidal zero-mode dimensional reduction to four dimensions gives one gauge field, six real adjoint scalar fields, and four adjoint Weyl spinors: a four-dimensional N=4 super Yang-Mills theory. The internal field strength $F_{mn}=-ig[X_m,X_n]$ gives the nonnegative scalar potential $V=\frac{g^2}{4}{\sum }_{m,n}\Vert [X_m,X_n]{\Vert }^2$, where the norm uses a positive invariant inner product on the Lie algebra. Its minima are commuting tuples of adjoint scalar fields.

A classical supersymmetric mechanical model takes its coordinates in a Grassmann algebra: the position is even and the fermionic coordinates are odd. Time differentiation is even, while variation with respect to an odd coordinate uses Grassmann derivatives. For example, moving an odd variation to the left and integrating by parts gives $\delta \int \psi \dot{\psi }dt=2\int \delta \psi \dot{\psi }dt$.

For the real variant $L={\dot{x}}^2/2+U^2/2-{\psi }_1{\dot{\psi }}_1/2+{\psi }_2{\dot{\psi }}_2/2+U^{\prime }{\psi }_1{\psi }_2$, the Euler-Lagrange equations areThe conserved even energy is $E={\dot{x}}^2/2-U^2/2-U^{\prime }{\psi }_1{\psi }_2$ and the odd supercharges are $Q_1=\dot{x}{\psi }_1-U{\psi }_2$, $Q_2=\dot{x}{\psi }_2-U{\psi }_1$. For example, ${\dot{Q}}_i=U^{\prime \prime }{\psi }_1{\psi }_2{\psi }_i=0$, since each odd element squares to zero. Also $d({\psi }_1{\psi }_2)/dt=U^{\prime }({\psi }_2^2+{\psi }_1^2)=0$, which proves $\dot{E}=0$.

The gauge kinetic function is a symmetric holomorphic function of chiral superfields multiplying the gauge field-strength F-term. Its real part controls gauge kinetic coefficients and its imaginary part controls topological-angle terms. Its Wilsonian perturbative correction is constrained by one-loop exactness of the holomorphic gauge kinetic function; canonical and one-particle-irreducible gauge couplings require separate normalization and infrared analysis.

// Target: supersymmetry.bigb

A two-derivative four-dimensional $\mathcal{N}=1$ gauge action is specified by a real gauge-invariant Kähler potential, a gauge-invariant holomorphic superpotential, and a holomorphic symmetric gauge kinetic function. A constant Fayet–Iliopoulos term is additionally allowed for an Abelian gauge factor. Renormalizability restricts the matter kinetic function to a quadratic form, the superpotential to degree at most three, and the gauge kinetic function to a constant invariant tensor.

Minimal four-dimensional supergravity couples a graviton to one Majorana gravitino. In the normalization $\delta e_{\mu }{}^a=2\kappa \overline{ϵ}{\gamma }^a{\psi }_{\mu }$, $\delta {\psi }_{\mu }={\nabla }_{\mu }ϵ/\kappa$, its action through quadratic fermion order is $\int e[R/(2{\kappa }^2)-2{\bar{\psi }}_{\mu }{\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\psi }_{\rho }]$. Rescaling the gravitino and parameter by two gives the usual canonical kinetic coefficient $-1/2$. The curvature identity ${\gamma }^{\mu \nu \rho }{\nabla }_{\nu }{\nabla }_{\rho }ϵ=G^{\mu \nu }{\gamma }_{\nu }ϵ/2$ cancels the Einstein-Hilbert variation against the Rarita-Schwinger variation. The omitted cubic-fermion variations require the quartic-fermion completion.

A Killing spinor is a nonzero supersymmetry parameter that makes the fermionic transformations vanish on a bosonic background. In undeformed minimal four-dimensional supergravity it is parallel, ${\nabla }_{\mu }ϵ=0$; in its cosmological deformation the equation uses ${\mathcal{D}}_{\mu }={\nabla }_{\mu }+(m/2){\gamma }_{\mu }$. Its commuting Dirac current is a Killing vector, since the spinor equation makes the symmetric covariant derivative of the current vanish. General supergravity theories also impose the vanishing of matter-fermion transformations.