For an interior optimal payoff in a complete two-state market under CRRA utility, $W_u^{-R}/W_d^{-R}=Z_u/Z_d$. If the two positive payoffs differ, taking logarithms gives the displayed coefficient. The multiplier can then be recovered from either state. A positive coefficient requires the price-density and payoff orders to be compatible with decreasing marginal utility.

In a parallel-propagated orthonormal frame, orthogonal geodesic deviation in constant sectional curvature reduces to three scalar equations ${\ddot{S}}^i=\kappa S^i$. For $\kappa >0$, $S=A{e}^{\sqrt{\kappa }\tau }+B{e}^{-\sqrt{\kappa }\tau }$: generic initial data grow exponentially, but the pure decaying mode is a counterexample to universal growth. Initially comoving neighbors have $S=S(0)\cosh (\sqrt{\kappa }\tau )$. For $\kappa <0$, $S=A\cos (\sqrt{-\kappa }\tau )+B\sin (\sqrt{-\kappa }\tau )$, so the deviation is bounded and oscillatory. Simultaneous focusing requires compatible vector initial data and is not automatic for every oscillatory solution.

For an infinite connected locally finite graph with a uniform degree bound, let ${\sigma }_n(v)$ count its length-$n$ self-avoiding walks from $v$. The suprema satisfy ${\sigma }_{m+n}\le {\sigma }_m{\sigma }_n$, so the Fekete lemma applied to their logarithms gives a finite connective constant. Unlike the usual rooted definition on a vertex-transitive graph, this definition explicitly takes the supremum over roots.

In an orthonormal coframe in spacetime, antisymmetry applies after lowering the first index with the Lorentzian frame metric. Thus mixed time–space connection 1-forms are equal under index interchange, while mixed spatial forms are opposite. The same rule holds for curvature 2-forms. Confusing lowered antisymmetry with ordinary mixed-index antisymmetry changes the curvature signs.

For a Jacobi field along a geodesic with nonpositive sectional curvature, the curvature convention with positive round-sphere curvature givesThus $|J{|}^2$ is nonnegative and convex. If it vanishes at both ends of a segment, convexity forces it to vanish everywhere. No nonzero Jacobi field can vanish at both ends, so the segment has no conjugate points. Only curvature of planes containing the geodesic tangent is needed, and completeness is unnecessary.

For inner products linear in the first argument, define ${\ast }_E$ by contraction in $\psi \wedge {\ast }_E\eta =(\psi ,\eta )dV$. It is conjugate-linear and takes $(p,q)$-forms with values in $E$ to $(n-p,n-q)$-forms with values in $E^{\ast }$. With induced dual metrics, ${\ast }_{E^{\ast }}{\ast }_E=(-1{)}^{p+q}$.

A finitely generated cone is a conic hull of a finite set of vectors. Every point admits a representation using linearly independent active generators, by eliminating dependence while keeping coefficients nonnegative.

For $\tilde{g}=e^{2\sigma }g$, the displayed correction is a symmetric tensor, so adding it to $\nabla$ gives a torsion-free connection. The two extra metric-compatibility pairings sum to $2d\sigma (X)g(Y,Z)$; this is exactly the derivative of the conformal factor. Uniqueness of the Levi-Civita connection proves the formula. It also yields the critical-point criterion for equal conformal connections.

On a finite measure space $(\Omega ,\Sigma ,\mu )$, let $f\ge 0$ be integrable and let ${\Sigma }_0\subseteq \Sigma$ be a sub-sigma-algebra. The finite measure $\nu (A)={\int }_{A}fd\mu$ on ${\Sigma }_0$ is absolutely continuous with respect to $\mu {|}_{{\Sigma }_0}$. The Radon-Nikodym theorem produces a ${\Sigma }_0$-measurable density $f_0$ withIt is unique almost everywhere. Positive and negative parts extend the construction to real integrable $f$, and real and imaginary parts handle complex $f$. On a probability space, this is conditional expectation.

The finite-base assumption, or a suitable sigma-finiteness assumption on the restricted measure, must not be omitted. On $\mathbb{N}$ with counting measure, take the trivial sub-sigma-algebra and $f=1_{\{1\}}$. A measurable candidate density is constant. Its integral over the whole space is either zero or infinite, never $\int f=1$. Thus a globally sigma-finite base measure need not remain sigma-finite after restriction to a smaller sigma-algebra.

Let $K_s$ be an increasing uniformly computable finite approximation to the diagonal halting set. For $a<b<c$, colour the triple $0$ if $K_b\upharpoonright a=K_c\upharpoonright a$, and $1$ otherwise. No infinite homogeneous set for a colouring can have colour $1$, since the finitely many bits below its first element eventually stabilize. Every infinite homogeneous set $H$ of colour $0$ computes the diagonal halting set: choose $n<a<b$ in $H$ and read membership of $n$ in $K_b$. Homogeneity and unboundedness make this answer final. Thus the colouring has no infinite computable homogeneous set.

For a matrix with at least two columns of unit Euclidean norm, its mutual coherence is the largest magnitude of an inner product between distinct columns. It measures the strongest pairwise ambiguity of linear measurements. The Cauchy-Schwarz inequality gives $0\le \mu \le 1$. The order-two restricted isometry constant equals the mutual coherence, because every two-column Gram matrix has eigenvalues $1\pm |⟨a_i,a_j⟩|$.

The Laguerre unitary ensemble is the positive-eigenvalue ensemble with joint density proportional to ${\prod }_{i<j}(x_j-x_i{)}^2{\prod }_i{x}_i^a{e}^{-x_i}$, for $a>-1$. integer $a=n-m\ge 0$ is realized by a complex Wishart matrix. The Laguerre projection kernel expresses its correlation functions of a point process as determinants.

The rest-frame field $\psi =\cos \alpha e^{i\sin \alpha t}/[\lambda \cosh (\cos \alpha x)]$ has mass $4M\cos \alpha /{\lambda }^2$ and charge $4[\mathrm{sgn}(\alpha )\pi /2-\alpha ]/{\lambda }^2$, for the generator $\delta \psi =i\psi$ and $\alpha \ne 0$. The conserved phase angle is a collective coordinate with momentum $Q$. Quantization of a periodic soliton coordinate gives integer $Q$ in units $\hslash =1$ and the displayed leading mass formula. On the regular branch $0<|Q|<2\pi /{\lambda }^2$, its concave increasing sine law prevents fragmentation into smaller like-charge states.

The fundamental class of a complex projective line generates $H_2({\mathbb{CP}}^2;\mathbb{Z})$. Two distinct projective lines are homologous and meet at one transverse point. In complex affine coordinates their two complex tangent lines concatenate to the complex orientation of the plane, so the signed intersection is $+1$. The integral intersection form therefore has matrix $(1)$, giving a unimodular intersection pairing. This computes a self-intersection through distinct representatives of the same class.

Removing the completion's poles produces an entire function with $\xi (0)=\xi (1)=1/2$ and $\xi (s)=\xi (1-s)$. Its zeros are the nontrivial zeros of the Riemann zeta function. Its finite order allows Hadamard factorization, while the xi modulus criterion for the Riemann hypothesis characterizes their horizontal location.

In a one-period complete market with deterministic initial holdings, two integrable positive pricing variables assigning the same prices to every traded asset are equal almost surely. Replicate indicator functions to obtain equality of their integrals on every event, then apply this to the event where one density exceeds the other.

Two assets with linearly independent payoff vectors across two positive-probability states span all terminal payoffs. The state-price equations have a unique solution. A positive solution gives the state-price density after division by physical state probabilities. Arrow state prices sum to the riskless discount factor, while normalized Arrow state prices are risk-neutral probabilities; these three quantities must not be confused.

If a bipartite graph with class sizes $m,N$ has at least $\beta mN$ edges, double count pairs consisting of an $s$-set in the first class and a common neighbour in the second. Convexity of binomial coefficients gives some $S$ with $|N(S)|\ge N\left(\genfrac{}{}{0ex}{}{\lfloor \beta m\rfloor }{s}\right)/\left(\genfrac{}{}{0ex}{}{m}{s}\right)$. When $s\le \beta m/2$, the quotient is at least $(\beta /2{)}^s$. This turns density into a large complete bipartite subgraph.

On a closed manifold with a Riemannian metric the positive Laplace-Beltrami operator is self-adjoint with compact resolvent. It has a complete smooth orthonormal eigenbasis, finite-dimensional eigenspaces, nonnegative eigenvalues tending to infinity, and polynomial eigenvalue-counting bounds. Elliptic regularity bounds derivatives of its eigenfunctions polynomially in their eigenvalues.

For $n\ge 3$, count the empty graph once, all one-edge graphs once, and each larger accepted graph under its unique center. The latter weighted contribution is $n(n-2)$, giving $1-\left(\genfrac{}{}{0ex}{}{n}{2}\right)+n(n-2)$. Its nonzero value proves evasiveness by the alternating-sum criterion for decision-tree evasiveness.