At a point where $\varphi (b)$ is finite, this is an affine plane supporting the epigraph from below: $\varphi (u)\ge \varphi (b)+{\lambda }^T(u-b)$ for every $u$. A nonzero coefficient of the vertical coordinate permits normalization to the displayed graph form. For a constrained value function, the slope is exactly an attained multiplier in the Strong Lagrangian property. When the value function is differentiable at an interior point of its finite domain, a supporting slope equals its gradient: apply the supporting inequality along both positive and negative multiples of each coordinate direction and take limits.

Let annual losses be $L={\sum }_{i=1}^N{X}_i$ where $N\sim \mathrm{Poisson}(\lambda )$ is independent of an independent identically distributed sequence of nonnegative loss given default amounts of mean $\mu$ and variance ${\sigma }^2$. Conditional expectation and variance give $\mathbb{E}L=\lambda \mu$ and $\mathrm{Var}L=\lambda ({\sigma }^2+{\mu }^2)$. A many-event normal approximation yields the displayed quantile; the actual loss distribution has a zero atom and depends on the entire severity law. For rare events or extreme tails, the moment-based normal approximation need not be accurate.

Electron pair geometry describes the arrangement of all electron domains in the VSEPR model, including lone pairs. Molecular geometry instead describes the arrangement of chemical atoms. Four domains have a tetrahedral ideal arrangement, but keeping only the bonded neighbours gives the different molecular geometries of methane, Ammonia and water.

An electron domain is a region of electron density counted around a central chemical atom in the VSEPR model. One lone pair counts as one domain; a bond to one neighbour counts as one domain even when its bond order is two or three. Domain count is therefore different from simply counting electron pairs.

In the two-component spinor convention with primed symmetric gauge curvature self-dual, an Abelian Maxwell field is anti-self-dual when its primed symmetric gauge curvature two-component spinor vanishes. For a locally exact field $F=dA$, the Bianchi identity gives $dF=0$; the duality relation then also gives $d\ast F=0$. Thus it obeys the source-free Maxwell equations. In complexified Lorentzian signature the Hodge star eigenvalues are $\pm i$; the two-component spinor condition specifies which component is meant without imposing a Euclidean eigenvalue on that signature.

// Destination: general-relativity.bigb

The expected utility hypothesis ranks uncertain outcomes by the expected value of a utility function. When these preference assumptions hold, expected utility maximization selects an admissible outcome with greatest expected utility.

Expected utility is the expected value of a utility function applied to a random outcome, using the decision-maker's probabilities. For a finite lottery, it is ${\sum }_j{p}_{j}u({\omega }_j)$. It is the expectation of utility, which generally differs from utility of the expected outcome.

For a risk-neutral bidder with fixed valuation $v$, winning indicator $I$ and possibly random payment $P$, quasilinear utility gives $vI-P$ and henceIf losing bidders pay nothing, this can be written $\mathrm{Pr}(I=1)\mathbb{E}[v-P\mid I=1]$ when the winning probability is positive. It cannot be used without that payment assumption: in an all-pay auction, a deterministic bid $b$ is paid even after losing, so the expectation is $v\mathrm{Pr}(I=1)-b$.

The additional utility function obtained per additional unit of consumption or portfolio wealth. For a differentiable utility function it is its derivative; for a concave utility function it is nonincreasing. In a complete-market allocation, equating marginal utility to a multiplier times the state-price density compares the utility benefit of a payoff with its financing cost. The concave supporting-tangent inequality makes this comparison a global optimality certificate rather than only a local first-order condition.

For increasing expected utility preferences, an investor is risk-averse when a sure mean is at least as desirable as the corresponding lottery. A concave utility function gives this preference by the Jensen inequality; a strictly concave utility gives strict preference for nondegenerate risks. Local risk aversion is measured by $-U^{\prime \prime }/U^{\prime }$ and its relative counterpart $-W{U}^{\prime \prime }/U^{\prime }$.

For increasing convex expected utility preferences, a decision-maker weakly prefers a lottery to its sure mean, provided the outcome and utility expectations are defined: the Jensen inequality gives $U(\mathbb{E}W)\le \mathbb{E}U(W)$. Strict preference for every nondegenerate lottery requires strict convexity over its relevant outcome range. An affine utility is convex too, but gives equality and risk neutrality.

For example, on nonnegative wealth, $U(w)=w^2$ assigns the equal-probability lottery $0$ or $2$ expected utility $2$, above the utility $1$ of its sure mean. Where $U^{\prime }>0$ and $U^{\prime \prime }>0$ at positive wealth, the relative risk aversion coefficient $-W{U}^{\prime \prime }/U^{\prime }$ is negative. This local curvature statement should be distinguished from strict preference when a utility has affine segments.

Start with a well-founded universe of sets over an infinite set $A$ of urelements. The full symmetric group of $A$ acts recursively on sets. Retain the objects all of whose descendants have a finite support in a permutation action. This class satisfies set-extensionality, foundation, infinity, pairing, union, separation, replacement and its internal power set axiom. A definable functional image has the finite union of the supports of its domain and parameters as a support; uniqueness makes the image invariant. The family of two-element subsets of $A$ has no choice function: a transposition of two atoms outside the function's finite support fixes their pair and moves either possible choice. For untyped membership with extensionality merely restricted to nonempty objects, atoms are also vacuous subsets; add all atoms to each internal power-set object.

For a slender tube of length $L$ supplied from a large reservoir with excess pressure $P$, negligible gravity and entrance inertia, the Unsteady Bernoulli equation gives $L\dot{U}+U^2/2=P/\rho$. A stationary initial liquid column has the displayed speed and volume flux $\pi a^{2}U$ for tube radius $a$. The inertial term controls the early linear acceleration; the kinetic-energy term sets the terminal inviscid speed. This is a leading slender-tube model while the reservoir remains large and the tube full.

A spherical balloon feeding an inviscid uniform tube plug, with entrance excess pressure $2\gamma /R$, satisfies $L\dot{u}=2\gamma /(\rho R)$ and $4\pi R^2\dot{R}=-\pi a^{2}u$. Eliminating $u$ yields the displayed equation. Its first integral is $R^4{\dot{R}}^2+K{R}^2=C$. From rest at radius $R_0$, $C=K{R}_0^2$; integration gives $t=R_0^2{K}^{-1/2}(\pi /4-\theta /2+\sin (2\theta )/4)$, where $\theta =\arcsin (R/R_0)$. The ideal emptying time is $\pi R_0^2/(4\sqrt{K})$. Independently prescribed initial flow changes $C$ and therefore this time.

A locally uniform convergence limit of univalent functions on a domain is either constant or univalent. For a nonconstant limit $f$, suppose distinct $x,y$ satisfy $f(x)=f(y)$. Choose a small closed disk around $x$ excluding $y$ with $f-f(y)$ nonzero on its boundary. The Rouche theorem then forces $f_j-f_j(y)$ to have a zero in that disk for large $j$, contradicting injectivity of $f_j$. This supplies the injectivity step in an extremal proof of the Riemann mapping theorem without assuming it as a special lemma.

For an injective holomorphic function $f$ on an open subset of $\mathbb{C}$, $f^{\prime }(z)$ is nowhere zero. If $f(z)-f(z_0)=(z-z_0{)}^{m}h(z)$ with $h(z_0)\ne 0$, choose a small circle where both $h$ and $mh+(z-z_0)h^{\prime }$ have no zeros. The argument principle counts $m$ zeros of $f(z)-f(z_0)-w$ inside that circle for sufficiently small $w\ne 0$: deforming $w$ to zero cannot change the count. All these zeros are simple, since $f^{\prime }$ only vanishes at $z_0$ inside the circle and $z_0$ is not a zero when $w\ne 0$. Injectivity forces $m=1$, and hence $f^{\prime }(z_0)\ne 0$.

A normalized univalent function is a holomorphic function on the unit disc which is injective and has the displayed normalization. Its Taylor series begins $z+a_2{z}^2+\cdots$. Normalization makes coefficient bounds and image-size estimates independent of translation and dilation; the Koebe quarter theorem is the sharp universal image bound.

For a univalent exterior map $G(\zeta )=\zeta +b_0+{\sum }_{n\ge 1}{b}_n{\zeta }^{-n}$, the image of $|\zeta |=R>1$ is a positively oriented Jordan curve. Green's theorem gives its enclosed area as $\pi (R^2-{\sum }_{n\ge 1}n|b_n{|}^2{R}^{-2n})$. Nonnegativity and $R\downarrow 1$ give the displayed bound. Laurent-series differentiation and integration on a circle of radius greater than one are justified by uniform convergence there.

For a compact group, the matrix coefficients of its finite-dimensional irreducible unitary representations span a dense subspace of $C(G)$ and form a complete orthogonal decomposition of $L^2(G)$ with Haar measure. If $K$ is a closed subgroup and $W$ is a finite-dimensional unitary $K$-representation, the space of L2 sections of the associated homogeneous bundle has multiplicity $\dim {\mathrm{Hom}}_K(E{|}_K,W)$ for each irreducible $G$-representation $E$. This is the compact-group form of Frobenius reciprocity and determines homogeneous-bundle multiplicities from isotropy weights.

A strongly continuous unitary representation of the circle group splits as an orthogonal sum of character subspaces. The projections are $P_n={\int }_{\mathbb{T}}{z}^{-n}{U}_{z}dz$. Character orthogonality makes them mutually orthogonal, and Fejér sum shows their ranges span the represented Hilbert space. The self-adjoint generator has integer eigenvalues, with domain given by ${\sum }_n{n}^2\Vert P_n\xi {\Vert }^2<\infty$.

A strongly continuous unitary representation of the circle has integer spectral weights: $H$ is the orthogonal sum of spaces on which $U_z=z^m$. Positive energy means that these weights are bounded below, equivalently its self-adjoint generator has spectrum bounded below. Multiplication by an integer character shifts the lower bound to zero. A unitary operator with a nonzero rotation weight cannot exist in its operator algebra: repeated application of it or its inverse would move a nonzero vector to arbitrarily low energy.