For a self-financing portfolio earning both running reward $e^{-\rho t}{W}^{\gamma }$ and terminal reward $W_T^{\gamma }$, $0<\gamma <1$, with no consumption withdrawals, Hamilton-Jacobi-Bellman equation gives $J(W,t)=A(t)W^{\gamma }$ and the displayed constant risky fraction. The coefficient satisfies $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, where $\kappa =\gamma r+\gamma (\mu -r{)}^2/(2(1-\gamma ){\sigma }^2)$. Its positive solution is $e^{\kappa (T-t)}+{\int }_t^T{e}^{\kappa (s-t)-\rho s}ds$. Risk aversion is $1-\gamma$; increasing $\gamma$ raises risky investment and can require borrowing.

For Gaussian terminal prices with positive definite covariance $V$, set $M=(\gamma V{)}^{-1}$ and $\eta =(S_0^{T}M\mu -w_0)/(S_0^{T}M{S}_0)$, assuming $S_0\ne 0$. The normal moment-generating function turns expected exponential utility into a strictly concave mean-minus-variance objective. Its free optimum has cash $S_0^{T}M{S}_0(1+r-\eta )$. If this is positive, keep the free optimum; otherwise the cash constraint binds and the multiplier becomes $\eta$. Thus zero optimal cash is equivalent to $\eta \ge 1+r$, including equality.

With positive welfare weights $a_j$, maximize ${\sum }_j{a}_j{\beta }_j^t[-e^{-{\gamma }_j{c}_j}/{\gamma }_j]$ subject to ${\sum }_j{c}_j=d$. Interior first-order conditions give $a_j{\beta }_j^t{e}^{-{\gamma }_j{c}_j}=\nu$. Put $\Gamma =({\sum }_j{\gamma }_j^{-1}{)}^{-1}$, $\log \bar{\beta }=\Gamma {\sum }_j{\gamma }_j^{-1}\log {\beta }_j$ and $\log A=\Gamma {\sum }_j{\gamma }_j^{-1}\log a_j$. ThenIf consumption must be nonnegative, the interior formula is replaced by $c_j=\max \{0,(\log a_j+t\log {\beta }_j-\log \nu )/{\gamma }_j\}$, with $\nu$ determined by resource clearing. These are Pareto allocations; realizing them as a competitive equilibrium with one productive asset requires feasible financing. For common ${\beta }_j=\beta$ and initial ownership $\Gamma /{\gamma }_j$, constant holdings and $C_t^j=(\Gamma /{\gamma }_j)d_t$ give such an equilibrium whenever fundamental prices and the utility sums are finite.

For exponential utility $(1-e^{-ax})/a$ on unrestricted terminal portfolio wealth in the Black-Scholes model, the optimal dollar holding in the stock is the displayed deterministic amount. Set $\vartheta =(\mu -\rho )/\sigma$ and let $W^Q=W+\vartheta t$ be the Brownian motion under the risk-neutral measure. The terminal optimizer is $X^{\ast }=w_0{e}^{\rho T}+\vartheta W_T^Q/a$. Its marginal utility is proportional to the state-price density, and the concave supporting-tangent inequality proves global optimality under the state-price budget constraint. The expected utility at the optimum is $(1-\exp (-a{w}_0{e}^{\rho T}-{\vartheta }^{2}T/2))/a$. A nonnegative-wealth constraint changes this optimizer.

For unrestricted final portfolio wealth in an $n$-period binomial market, write $R$ for the bank growth factor, $a>0$ for absolute risk aversion, and $Z=dQ/dP$ for the binomial-market probability density. The displayed optimizer has the state-price budget constraint ${\mathbb{E}}_Q{H}^{\ast }=R^n{w}_0$. Its exponential utility derivative satisfies $e^{-a{H}^{\ast }}=\lambda Z$ for a constant $\lambda >0$. The concave supporting-tangent inequality bounds any competing expected utility by this value, because the expected linear error is $\lambda {\mathbb{E}}_Q(H-H^{\ast })=0$. Strict concavity proves uniqueness, and market completeness supplies replication. The formula is altered by a constraint requiring nonnegative final wealth. With physical up probability $p$ and risk-neutral probability $q$, ${\mathbb{E}}_Q\log Z=n[q\log (q/p)+(1-q)\log ((1-q)/(1-p))]$.

A nonzero uniform equilibrium of the conserved-field real amplitude equation obeys $\mu =R^2-\alpha R$. Fixed zero mean excludes the constant $B$ perturbation. Its homogeneous amplitude eigenvalue is $-r$. For a nonzero Fourier wavenumber $k$, the stability matrix has trace $-r-(1+\sigma )k^2$ and determinant $\sigma k^2(k^2+r-2\delta R^2)$. The displayed conditions are necessary and sufficient for strict linear stability on a period $2L$. In an infinite-domain limit the nonzero-mode condition becomes $r\ge 2\delta R^2$; a long-wave instability need not occur in a sufficiently short domain.

For $0<\delta <1$ and $\alpha \ne 0$, a broad localized plateau in the conserved-field real amplitude equation has effective parameter $\nu =\mu -\delta ⟨A^2⟩$ and stationary equation $A^{\prime \prime }+\nu A+\alpha A^2-(1-\delta )A^3=0$. Equal endpoint values of its first integral give $R=2\alpha /[3(1-\delta )]$, $\nu =-2{\alpha }^2/[9(1-\delta )]$. If the plateau occupies fraction $h$ of a long periodic interval, $⟨A^2⟩\simeq h{R}^2$ supplies the displayed relation. It applies when both the plateau and exterior are long relative to the front width; $0<h<1$ gives its leading existence interval.

For equal heat extraction, the density increase caused by brine rejection can exceed that caused by cooling near the freezing point because the thermal density response is weak. Using conservation of salt during freezing, the ratio is $R=(\partial \rho /\partial S)S_0{f}_i/[(1-f_i)|\partial \rho /\partial T|\Delta T]$, with $f_i=c_{p,w}\Delta T/L_f$. This is a parcel comparison, not a guarantee that freezing overcomes any ambient density stratification.

Let liquid mass $M$ freeze incrementally, and let each new ice increment retain salinity $rS$ with constant $0\le r\le 1$, where $S$ is the instantaneous well-mixed liquid salinity. Conservation gives $d(MS)=rSdM$, hence $dS/S=(r-1)dM/M$. Integrating yields $S/S_0=(M/M_0{)}^{r-1}$. For frozen mass fraction $f$, $\Delta S=S_0[(1-f{)}^{-(1-r)}-1]\simeq (1-r)S_{0}f$. Specifying a fixed bulk ice salinity instead produces a different, though first-order compatible, finite-mass balance.

Basic radiance is radiance divided by the square of the refractive index. For radiance $L$ at a fixed wavelength,Lossless propagation conserves reduced radiance along each light ray; $L$ itself is conserved only when the endpoint refractive indices agree. A larger irradiance at a smaller image can result from admitting more directions without increasing reduced radiance. Uniform wave absorption can lower radiant flux while leaving the occupied geometric etendue unchanged; an opaque aperture removes part of that occupied set. ^[ref]

Place two equal point masses of $1\mathrm{kg}$ each on opposite sides of a fixed origin, at equal radius $r$, and let them share a signed angular velocity $\omega =\dot{\theta }$ in a plane. Their total mass is $M=2\mathrm{kg}$ and their moment of inertia about the perpendicular axis is $I=M{r}^2$. Radial pulling forces have zero torque about that origin, soTake an initial radius $r_i=1\mathrm{m}$ and angular velocity ${\omega }_i=1rad/s$, and a final radius $r_f=0.5\mathrm{m}$. ThenandThese are endpoint states with ${\dot{r}}_i={\dot{r}}_f=0$. During the pull, the full kinetic energy also contains radial motion: $K=M{\dot{r}}^2/2+K_{\mathrm{rot}}$. Assume the pulling apparatus supplies all net work to the two point masses; the work-energy theorem then fixes its net work at $+3\mathrm{J}$, including any radial acceleration and braking. Each mass spirals inward during the pull, allowing its inward force to do positive work. Conservation of angular momentum therefore predicts the final angular velocity without requiring kinetic energy conservation. For an apparatus included in a larger system, its stored energy supplies the increase consistently with conservation of energy. OpenStax uses the same mechanism to explain a skater pulling her arms inward. ^[ref]

Every nontrivial oriented knot is a finite connected sum of knots of prime knots, uniquely up to permutation and oriented link isotopy. Existence follows by induction from additivity of Seifert genus. For uniqueness, compare complete decomposing sphere systems for a knot and remove their intersection circles by the prime-ball exchange lemma. Once a prime ball is disjoint from both systems, its factor can be removed from both; induction then matches all factors.

For $d=4-ϵ$, write the bare field as ${\varphi }_0=Z^{1/2}\varphi$ and the renormalized source coupling as ${\mu }^{-ϵ/2}h\varphi$. In bare variables this is $h_0{\varphi }_0$, with the displayed source. At fixed bare parameters, the source running is $\mu d\log h/d\mu =ϵ/2+\frac{1}{2}\hat{\beta }{\partial }_{\lambda }\log Z$. Normalizing the generating functional by its zero-source value cancels source-independent vacuum diagrams and the constant measure factor from field rescaling. It does not remove source-dependent ultraviolet divergences or infrared divergences; the former require the ordinary field, mass and interaction counterterms.

For a normalized Minkowski source functional $Z=e^{iW}$ with vanishing one-point function, $C_n=i^{1-n}{\delta }^{n}W/\delta J^n$ andDifferentiating $e^{iW}$ assigns the four labels to connected blocks. Blocks of size one vanish, leaving one block of four or one of the three pair partitions. Thus $G_4=i{W}_4-\sum W_2{W}_2$ in this source convention.

// Target: perturbative-quantum-field-theory.bigb

Two objects of a category lie in the same connected component if a finite zigzag of morphisms, with either orientation allowed, joins them. This is an equivalence relation; its classes give full subcategories, each a connected category. For a small category, the collection of components is a set denoted ${\pi }_0(\mathcal{C})$. A functor induces a function on components. Empty categories have no components.

For a proper subgroup $B$ of a finite group $G$, the number $m$ of its conjugates is $[G:N_G(B)]\le [G:B]$, because its normalizer contains $B$. All conjugates contain the identity, so their union has size at most $1+m(|B|-1)\le |G|-[G:B]+1<|G|$. Thus some element is not conjugate to any element of $B$. Finiteness is essential to this cardinal counting argument.

Along a nonconstant affinely parametrized geodesic, the multiplicity of a conjugate point at positive time $t_0$ is the dimension of the space of Jacobi fields vanishing at both endpoints. The tangential component of such a Jacobi field is affine and vanishes at both endpoints, hence is zero. Uniqueness for the Jacobi field equation embeds this space into the $(n-1)$-dimensional normal space by $J\mapsto D_{t}J(0)$. Thus its dimension is at most $n-1$.

For a proper cone $K$, suppose a point satisfies $Ax-b\in \mathrm{int}K$ and $p_{\ast }$ is finite. This generalized Slater condition gives strong duality and attainment of the conic dual problem's optimum. It does not by itself assert primal attainment.^[ref]

The two worked examples have strict primal points: $(u,v)=(2,2)$ has slack $(2,2,1)$ in the interior of the nonnegative orthant, and $(t,z)=(2,0)$ has slack $(2,1,0)$ in the interior of the second-order cone. Both optima were already certified by explicit feasible pairs. Slater's condition explains why a matching optimal dual variable can be expected under these hypotheses; checking a supplied zero-gap pair needs only weak duality.

Changing cone geometry preserves the bound mechanism. For a positive semidefinite cone, the dual cone pairing is the Frobenius inner product of two matrices. This connects the same certificate idea to semidefinite programming, while the central path uses strictly interior primal and dual points to approach vanishing gap.

In 1995, Lieven Vandenberghe and Stephen Boyd presented a primal-dual interior-point method for convex optimization with matrix inequalities. Their algorithmic study included exploiting problem structure and allowing incomplete conjugate gradient method solves within an iteration. This illustrates how the primal-dual viewpoint guides computation as well as checking a final answer.^[ref]

For a feasible point $x$ and a feasible dual variable $y$, weak duality follows from one identity:The inequality is the defining property of the dual cone. Every feasible pair therefore encloses the optimal objective function value:Both optima are finite when such a pair exists. Consequently,These are absolute cost bounds, rather than bounds on distance to an optimizing vector. Zero gap certifies both optima immediately. A small positive gap bounds the remaining improvement even if no strong duality theorem has been invoked.

For the nonnegative-orthant optimality certificate example, $x=(2.9,0.2{)}^T$ is feasible and costs $3.3$. Pairing it with $y=(0,1,1{)}^T$ gives $g=0.3$, so its cost exceeds the optimum by at most $0.3$. Here the optimum is known to be $3$, and the bound is exact. Pairing the same $x$ with the weaker $r=1/2$ certificate gives $g=1.8$: a valid but less informative guarantee.

The identity assumes exact primal and dual feasibility. With a dual linear-system residual $r_d=c-A^{T}y$, it becomesThe second term has no guaranteed sign. Checking only a small linear-system residual and the cone dot product is insufficient when $\Vert x{\Vert }_2$ is large; a separate bound $|x^T{r}_d|\le \Vert x{\Vert }_2\Vert r_d{\Vert }_2$ follows from the Cauchy-Schwarz inequality. Approximate cone membership likewise needs a quantified correction before claiming a certificate.

For an exact arithmetic example, minimize $x$ subject to $x\ge 10^9$. The point $x=10^9$ is optimal. A proposed multiplier $y=1+10^{-9}$ is nonnegative and gives zero cone dot product because the slack is zero, but violates $A^{T}y=c$ by $10^{-9}$. Its purported lower bound is $10^9+1$, exceeding the optimum. The missing term $x^T{r}_d=-1$ explains the failure.

A curved feasible set can have the same certificate structure. ConsiderWrite $x=(t,z{)}^T$ and use the second-order cone $K=\{(s_0,s_1,s_2):s_0\ge \sqrt{s_1^2+s_2^2}\}$. In this coordinate ordering,Self-duality of a second-order cone makes $K^{\ast }=K$. The equality $A^{T}y=c$ forces $y_0=1$ and $y_2=0$, leaving $-1\le y_1\le 1$. Its dual objective function is $b^{T}y=-y_1$, maximized at $y=(1,-1,0{)}^T$. Thus every feasible point satisfies $t\ge 1$, and $(t,z)=(1,0)$ attains that bound: $\boxed{p_{\ast }=d_{\ast }=1}$.

The zero dot product is $(1,1,0)\cdot (1,-1,0)=0$. Individual coordinate products need not be nonnegative here: the cancellation $1-1=0$ occurs across coordinates. The valid complementary slackness condition is cone-level orthogonality, rather than the coordinatewise interpretation of the nonnegative orthant.