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For a homogeneous phase function $\Phi$ and finite-order symbol class amplitude $a$, the singular support of $I_{\Phi }(a)$ lies in the spatial projection of the closed conic amplitude support intersected with ${\nabla }_{\theta }\Phi =0$. Away from that set, the frequency gradient is uniformly nonzero on compact spatial sets and supported directions. Repeated frequency integration by parts lowers the amplitude order until every desired spatial derivative is absolutely integrable. If the ordinary amplitude support is already conic, it gives the same bound.

This potential subtracts integrated resource prices from weighted logarithmic utility. Continuous nonnegative nondecreasing prices make it strictly concave on positive rates. If every route meets a resource with a nonzero price function, its superlevel sets are compact and its maximizer is unique. The rate adjustment ${\dot{x}}_r={\kappa }_r{x}_r{\partial }_{r}U$ increases it strictly away from equilibrium.

For strictly increasing continuous supply functions $q_j$ with $q_j(0)=0$, this potential is strictly convex on prices with positive route-price sums. Its minimum is interior for every used resource and gives the unique positive-price equilibrium. Multiplicative price dynamics can also have boundary equilibria of multiplicative price dynamics if zero prices are permitted.

In ${\dot{\mu }}_j={\kappa }_j{\mu }_j(y_j-q_j({\mu }_j))$, a zero price remains zero even if resource demand exceeds supply. Such a boundary equilibrium need not minimize the dual congestion potential; uniqueness of its positive minimizer does not imply uniqueness on the nonnegative orthant.

For route prices $p=A^T\mu$, logarithmic users demand $x_r=w_r/p_r$, while resource prices adjust asThis model raises prices under excess demand and lowers them under spare capacity. Positive capacities and nonempty routes are standard well-definedness assumptions. Positive initial prices approach the dual optimum; zero prices are invariant and can create boundary equilibria of multiplicative price dynamics that overload a resource.

A resource congestion price signals scarcity at a capacity-constrained link. Under multiplicative resource-price dynamics, its increase indicates excess demand and its decrease indicates spare capacity. A route pays the sum of these prices, its route congestion price. A zero resource congestion price at an optimum satisfies complementary slackness with possible spare capacity.

A scale is a conformal bijection sending a marked starting boundary point to zero and target boundary point to infinity in the complex upper half-plane. A curve is SLE in that scale when its image is ordinary SLE using the half-plane-capacity parameterization of that image.

A positive travelling wave $f$ on a uniform background $f_0$ has first integral $c{f}^{\prime 2}/(2f^2)+V(f)=V(f_0)$, with $V(f)=\ln f+c/f+(f_0^2-c{f}_0)/(2f^2)$. Equating the crest and background potentials gives the displayed speed relation for crest ratio $\alpha >1$. The elevation-wave speed exceeds $2f_0$ and approaches that long-wave speed as $\alpha \to 1$. The formula requires the positive-area branch, rather than continuation through $f=0$.

At zero growth rate put $P=\psi +iq\theta$, $q=\sqrt{R}$. The coupled Darcy equations become $\Delta P+iq{P}_x=0$. Substituting $P=e^{-iq(x-1/2)/2}F$ gives $\Delta F+(q^2/4)F=0$ with homogeneous Dirichlet boundary conditions. The square Dirichlet Laplacian eigenvalues give $q^2=4{\pi }^2(m^2+n^2)$, $m,n\ge 1$, whose minimum is $8{\pi }^2$. For $S=\sin \pi x\sin \pi z$, $a=q_c/2$, the complex multiples $F=S,iS$ give real pairs $(S\cos (a\xi ),-S\sin (a\xi )/q_c)$ and $(S\sin (a\xi ),S\cos (a\xi )/q_c)$, $\xi =x-1/2$. Their reflection parities are opposite, yielding two independent physical modes.

For $q=q_c+ϵq_1$, ${\theta }_t=ϵ\lambda \theta$, the first-order equations are $\Delta {\psi }_1-q_c^2{\theta }_{1x}=2q_c{q}_1{\theta }_{0x}$ and ${\psi }_{1x}+\Delta {\theta }_1=\lambda {\theta }_0$. Multiply them by ${\psi }_0,q_c^2{\theta }_0$ and integrate. Integration by parts and the leading equations cancel the homogeneous correction terms, giving $2q_c{q}_1⟨{\psi }_0{\theta }_{0x}⟩+q_c^2\lambda ⟨{\theta }_0^2⟩=0$. Since $⟨{\psi }_0{\theta }_{0x}⟩=-⟨|\nabla {\theta }_0{|}^2⟩$, the displayed title formula follows. The reciprocal norm ratio is incompatible with this solvability condition. Reflection parity diagonalizes the first-order splitting of the two marginal modes.

For two independent stage means $X,Y\sim N(\delta ,s^2)$ and continuation $X\ge c$, let $S=(X+Y)/2$ and $v=s/\sqrt{2}$. The second-stage mean is conditionally unbiased. The Rao-Blackwell theorem gives $E(Y\mid S,X\ge c)=S-v\lambda ((c-S)/v)$, using $X\mid S\sim N(S,v^2)$ and truncation. This is a conditionally unbiased estimator using both stages through their combined sample mean.

The essential supremum over the base of the conditional L2 norm. It may be infinite for a globally square-integrable function. For probability fibers, $\Vert f{\Vert }_2\le \Vert f{\Vert }_{2,\infty \mid Y}$; the uniform norm controls all fibers outside a null set rather than only their average.

For invertible probability systems linked by a factor map, invariance and uniqueness of disintegration identify the pushforward of each conditional measure with the conditional measure on the transformed fiber. The displayed norm identity follows for all integer times on a common full-measure set. It transports uniform conditional error bounds along whole orbits.

Let $A,B$ be indicator random variables with conditional independence given a latent variable $T$, and suppose $\mathbb{E}[A\mid T]=\mathbb{E}[B\mid T]=g(T)$. ThenFor a short proof, conditional independence gives $\mathbb{E}[AB\mid T]=g(T{)}^2$. The law of total expectation gives $\mathbb{E}[AB]=\mathbb{E}[g(T{)}^2]$ and $\mathbb{E}[A]=\mathbb{E}[B]=\mathbb{E}[g(T)]$. Subtract their product to obtain the claimed variance. Since $0\le g(T)\le 1$, all the required expected values exist.

In the two-environment example of conditional independence, $g(T)$ is $0.1$ or $0.9$ with equal probabilities. Hence $\mathbb{E}[g(T)]=0.5$ and $\mathrm{Var}(g(T))=((0.1-0.5{)}^2+(0.9-0.5{)}^2)/2=0.16$. The two trials have zero conditional covariance in either environment, yet their unconditional covariance is positive because their success probabilities rise and fall together.

More generally, square-integrable conditionally independent random variables satisfyby the law of total covariance. Equal conditional means reduce this to the displayed variance; unequal conditional means need not give a nonnegative covariance. The equal-mean result explains correlated component evolutionary states in a coeval binary population without correlation between initial masses: variation in a shared environment can supply the entire covariance.

At unit jump rate, this is $X_t={\sum }_{k=1}^{N_t}{\xi }_k$, where $N$ is a rate-one Poisson process and the independent marks are uniform signs. Equivalently it is the difference of two independent rate-$1/2$ Poisson processes. It is a Lévy process with characteristic function $\exp \{t(\cos \theta -1)\}$, mean zero, variance $t$, and Lévy characteristic exponent $1-\cos \theta$ in the negative-exponent convention. A general total jump rate rescales time.

The conjugate bundle has the same underlying real bundle and the opposite complex scalar action. Its transitions are the complex conjugates of the original transitions. For a holomorphic bundle on $X$, these transitions are smooth and generally antiholomorphic on $X$; it is not automatically a holomorphic bundle there. It is naturally holomorphic over the conjugate complex manifold instead.

For real $x,y$, the complex exponential function has $e^{x+iy}=e^x(\cos y+i\sin y)$. Since the trigonometric factor lies on the unit circle,Thus the real part of the exponent controls its modulus, while the imaginary part controls its direction on the complex plane.

Put $a=n/2$, $m=m(\Lambda )$ and $A_{\Lambda }(s)={\int }_1^{\infty }({\Theta }_{\Lambda }(it)-1)t^{s-1}dt$, which is entire. Splitting the Mellin transform at one and using the lattice theta functional equation gives ${\mathcal{E}}_{\Lambda }(s)=A_{\Lambda }(s)+m^{-1}{A}_{{\Lambda }^{\vee }}(a-s)+m^{-1}/(s-a)-1/s$. The two rational terms explicitly retain the contributions of the zero lattice vector.

Let $c_i\ge 0$ sum to one, with no subset sum equal to $1/2$, and let independent Bernoulli random variables $Z_i$ all have parameter $p\ge 1/2$. The subsets of weight greater than $1/2$ form an intersecting self-dual set family. The complementary-layer bound for biased measure proves $\mathbb{P}({\sum }_i{c}_i{Z}_i\ge 1/2)\ge p$. Nonnegative weights ensure that disjoint sets cannot both have weight greater than $1/2$; the no-tie condition supplies self-duality.

Let $\delta =\hat{\beta }-{\beta }^0$, $s=|S|>0$ and $\varphi >0$. On the score event $\Vert X^T\eta {\Vert }_{\infty }/n\le c\lambda$ with $0<c<1$, the prediction inequality from Lasso stationarity gives $Q+(1-c)\lambda \Vert {\delta }_{S^c}{\Vert }_1\le (1+c)\lambda \Vert {\delta }_S{\Vert }_1$. If the Compatibility condition for the Lasso with constant $\varphi$ holds on this cone, then $Q+(1-c)\lambda \Vert {\delta }_{S^c}{\Vert }_1\le (1+c{)}^2{\lambda }^{2}s/{\varphi }^2$. Here $Q=\Vert X\delta {\Vert }_2^2/n$ and $S$ is the true support of a vector.