A process has conditionally symmetric increments if, for all deterministic $s\le t$ and bounded measurable $g$,Equivalently, the conditional characteristic function of every increment is invariant under changing the sign of its argument. This property is stronger than unconditional symmetry and is different from independent increments. It supplies identities between positive and negative exponential terminal martingales.

Suppose $X,A$ are continuous, start at zero, and $A$ is increasing. If $e^{X-A/2}$ and $e^{-X-A/2}$ are local martingales, their logarithms first recover adaptation of $X,A$ and the semimartingale property of $X$. In the decomposition $X=N+V$, the two Itô drift measures give $dV+(d[N]-dA)/2=0$ and $-dV+(d[N]-dA)/2=0$. Thus $V=0$ and $[X]=A$. All real exponential parameters may be assumed, but these two suffice.

For a contact form on a $(2r+1)$-manifold, $d\alpha {|}_{\ker \alpha }$ is nondegenerate: the contact condition says its top exterior power is nowhere zero on the hyperplanes. This fiberwise symplectic form identifies contact-plane vectors with covectors on that plane and determines the horizontal generator of contact stability.

For CRRA utility of a consumption satisfaction stock, jointly scaling initial portfolio wealth, consumption satisfaction and controls gives homogeneity of degree $1-R$. The value function therefore depends on one dimensionless ratio after factoring out ${\xi }^{1-R}$. The effective consumption coefficient becomes ${\xi }^{-R}[(1-R)v-(x+1)v^{\prime }]$.

If consumption contributes $c(V_{\xi }-V_w)$ to the Hamilton-Jacobi-Bellman equation and has no finite upper bound, finite value requires $V_{\xi }\le V_w$. Strict inequality makes zero consumption optimal locally. Equality describes an active transfer boundary, often reached through singular consumption control.

In singular stochastic control, an unrestricted consumption rate can approximate instantaneous transfers from portfolio wealth into a consumption satisfaction stock. In the relaxed problem a nondecreasing control $C$ records these transfers. A transfer $\Delta C$ decreases portfolio wealth and increases consumption satisfaction by the same amount. Rate controls need not attain the supremum when the relaxed optimum has a jump.

For simple loops surrounding a marked point, containment in two simply connected domains is containment in their intersection component containing that point. Finite deficits therefore determine all finite intersections of avoidance events by inclusion-exclusion. Restrict to finite windows $H_r={\mathbf{L}}_D\setminus {\mathbf{L}}_{rD}$ and use the sigma-finite uniqueness theorem for measures; as $r\downarrow 0$ these windows exhaust the pointed loop space. Finiteness of the windows is essential to this argument.

A previous outcome may affect a subsequent outcome after controlling both measured covariates and persistent unobserved heterogeneity. This is true state dependence. An observed lagged-outcome association alone does not establish it, because omitted individual propensities can create apparent persistence.

The accumulated number of previous successes may enter the conditional mean in a history-dependent logistic regression. Its coefficient exponentiates to the conditional success odds ratio per previous success. For a specified future path, update this cumulative predictor after each outcome before multiplying the conditional probabilities.

Countably many completely Ramsey-null sets can be avoided simultaneously in a stem-preserving Ellentuck topology refinement. Select an increasing sequence $b_j$ with nested infinite remaining tails. At stage $j$, thin the tail to avoid the $j$th set for every stem $s\cup t$, where $t$ ranges over all subsets of the first $j$ selected points. There are only finitely many such stems. Any infinite subset of the final selected sequence uses some such $t$ before stage $j$ and has all remaining points in the thinned tail. Thus it avoids every forbidden set. This explains why an argument checking only the full selected prefix is insufficient.

In the finite-limit setting, a Cartesian comonad is a comonad whose underlying endofunctor preserves finite limits. Its category of coalgebras for a comonad has finite limits created by the forgetful functor. On an elementary topos, equalizer constructions of exponentials in a coalgebra topos and the subobject classifier of a coalgebra topos make that coalgebra category an elementary topos.

In barycentric coordinates with unit primary separation, a small secondary mass fraction $\alpha$ gives distances $(\alpha /3{)}^{1/3}$ from that secondary to the two neighboring Lagrange points at leading order. This is the dimensionless Hill radius. The opposite exterior L3 Lagrange point has coordinate $-1-5\alpha /12+O({\alpha }^2)$.

A simple closed curve in a closed orientable surface has an annular collar neighbourhood, so its collapse is governed by the collapsing a pair theorem. In the long exact sequence in relative homology, the decisive map is $H_1(A;\mathbb{Z})\to H_1({\Sigma }_g;\mathbb{Z})$, sending the circle generator to its homology class. That class is zero for a separating curve and a primitive homology class for a nonseparating curve. The two possibilities lead to homology after collapsing a separating surface curve and homology after collapsing a nonseparating surface curve.

For even $n\ge 2$, attach an $(n+1)$-cell to $S^n\times S^n$ along the diagonal. The cellular chain complex has boundary $(1,1)$ for the new cell. The degree-$n$ cohomology generator restricts to $a-b$, and the top generator restricts to $ab$. Naturality of the cup product therefore gives $u^2=-2v$, since $(a-b{)}^2=-2ab$. Thus the resulting cohomology ring is $\mathbb{Z}[u,v]/(u^2+2v,uv,v^2)$, with $|u|=n$ and $|v|=2n$. This distinguishes the attachment from a wedge of spheres, whose positive-degree products vanish.

A Coleman power series encodes a norm-compatible sequence of local units in a cyclotomic tower by a single integral power series evaluated at ${\zeta }_{p^n}-1$. A logarithmic derivative and trace correction give a measure, relating the local images of cyclotomic units to p-adic L-functions.

The central binomial coefficient satisfies $4^m/(2m+1)\le \left(\genfrac{}{}{0ex}{}{2m}{m}\right)\le 4^m$. Prime factors between $m$ and $2m$ give a dyadic upper bound for the Chebyshev theta function. The prime-power exponents in the factorial quotient are sums of zeros and ones, so the central coefficient gives a linear lower bound for the second Chebyshev function. Higher prime powers contribute only $O(\sqrt{x}\log x)$; hence the prime-counting function has upper and lower bounds of order $x/\log x$.

The invariants $\mu ,\lambda$ of a finitely generated torsion Iwasawa module measure its $p$-power elementary factors and distinguished-polynomial factors. Vanishing of $\mu$ means that the module is finitely generated over ${\mathbb{Z}}_p$, up to finite error. It does not mean that the module is finite: a factor $\Lambda /(T)$ has $\mu =0$ and is ${\mathbb{Z}}_p$.

A character of a finite abelian group is a homomorphism into the complex unit circle. Equivalently it is a linear character; finite order forces each character value to be a root of unity. These are the elements of the character group of a finite abelian group and form the orthogonal Fourier basis of functions on the group.

If $H$ is a subgroup of a finite abelian group, $a\notin H$, and $k$ is the least positive integer with $ka\in H$, then each character of a finite abelian group on $H$ has exactly $k$ extensions to $H+⟨a⟩$. The displayed root choice makes the extension well-defined. Starting with the trivial subgroup and adjoining generators proves that the number of characters equals the group order without using a decomposition into cyclic groups.

A contracting homotopy on a chain complex is a degree-one map $h$ whose displayed identity makes the identity chain-homotopic to zero. It implies that all homology vanishes. The converse is not valid for arbitrary complexes of modules.