Attach an $(i+1)$-cell to $S^i$ by a degree-$n$ map, with $i\ge 1$ and $n\ge 2$. The integral cellular differential is multiplication by $n$, while its modulo-$n$ reduction is zero. Lifting the degree-$i$ generator modulo $n^2$ and applying the differential gives $n$, so the Bockstein homomorphism sends that generator to the degree-$(i+1)$ generator. This supplies a nonzero example in every positive degree.

A real-valued function $f:D\to \mathbb{R}$ is a nondecreasing function if every pair $x<y$ in its domain of a function satisfies $f(x)\le f(y)$. Flat intervals are allowed, unlike for a strictly increasing function. For example,is a continuous nondecreasing function, but it repeats its output on each flat interval.

A nondecreasing function can also be discontinuous. The step function $F(x)=0$ for $x<0$ and $F(x)=1$ for $x\ge 0$ has a jump discontinuity at zero without ever decreasing. It is the cumulative distribution function of a random variable that equals zero with probability one; quantile functions handle such jumps through a generalized inverse. Nondecreasing allocation functions permit critical-value payments in single-parameter mechanisms.

For positive summable weights $w_j$ at points $q_j$, the strict cumulative function $F(x)={\sum }_{q_j<x}{w}_j$ is nondecreasing and left-continuous. A finite-head/tail argument proves left-continuity, and its right jump at an atom is the mass there. This strict convention differs at atoms from the usual right-continuous cumulative distribution function.

In the uniform-label monotone coupling of Bernoulli percolation, let $I_p$ mean that the origin belongs to an infinite percolation cluster, and let $M=\inf \{p:I_p\}$. The occurrence parameters form an upper interval. Therefore $\{M<p\}={\bigcup }_{r<p,r\in \mathbb{Q}}{I}_r$ and $\theta (p-)=\mathbb{P}(M<p)$. Since $\{M<p\}\subseteq I_p\subseteq \{M\le p\}$, the jump is $\theta (p)-\theta (p-)=\mathbb{P}(I_p\cap \{M=p\})$. Occurrence at the infimum is not presumed.

If an increasing event $A$ depends on $m$ sites or edges, the uniform-label monotone coupling of Bernoulli percolation and the union bound give $0\le {\mathbb{P}}_q(A)-{\mathbb{P}}_p(A)\le m(q-p)$ for $p\le q$. A change of the event requires a coordinate with label in $(p,q]$. For bond percolation from the origin to the boundary of $[-n,n{]}^2$, first-hit graph paths use the $4n(2n+1)$ internal edges, so the same count bounds the parameter sensitivity.

A monomorphism $m$ is extremal if every factorization $m=he$ with $e$ an epimorphism forces $e$ to be an isomorphism. A strong monomorphism has this property by its lifting property. In a complete category that is a well-powered category, intersecting all strong subobjects containing a given morphism yields an epimorphism–strong-monomorphism factorization; applying it to an extremal monomorphism proves that the extremal monomorphism is strong. The factorization definition and lifting definition should be kept distinct until the needed ambient hypotheses are supplied.

For $V=\lambda {\varphi }^n$ on $\varphi >0$, slow-roll approximation gives $ϵ\simeq n^2{M}_{\mathrm{Pl}}^2/(2{\varphi }^2)$ and $\eta \simeq 2n{M}_{\mathrm{Pl}}^2/{\varphi }^2$. The slow-roll e-fold count integrates to $N\simeq ({\varphi }_N^2-{\varphi }_e^2)/(2n{M}_{\mathrm{Pl}}^2)$. For $n=2,3,4$, the loose order-one conditions $ϵ\le 1$ and $|\eta |\le 1$ imply ${\varphi }^2\ge 2n{M}_{\mathrm{Pl}}^2$ in this approximation. Genuine slow roll requires a much stronger hierarchy. A prescribed endpoint ${\varphi }_e^2=2n{M}_{\mathrm{Pl}}^2$ gives ${\varphi }_N^2=2n(N+1)M_{\mathrm{Pl}}^2$ andThe actual end criterion for accelerated expansion is $ϵ=1$, whose leading potential estimate instead gives ${\varphi }_e^2\simeq n^2{M}_{\mathrm{Pl}}^2/2$. The prescribed endpoint is a convention, not an exact slow-roll or inflation boundary.

A monoidal functor whose tensor and unit comparison maps are isomorphisms.

A lax monoidal functor $\phi$ and opmonoidal functor $\psi$ on the same underlying functor, satisfying ${\psi }_{A\otimes B,C}{\phi }_{A,B\otimes C}=({\phi }_{A,B}\otimes 1)(1\otimes {\psi }_{B,C})$ and ${\psi }_{A,B\otimes C}{\phi }_{A\otimes B,C}=(1\otimes {\phi }_{B,C})({\psi }_{A,B}\otimes 1)$ in coherent notation. It transports dual pairs in a monoidal category and Frobenius monoids.

A morphism $f:A\to B$ satisfying $f{m}_A=m_B(f\otimes f)$ and $f{j}_A=j_B$.

An object that is both a monoid object and a comonoid in a braided monoidal category, with $\Delta ,\epsilon$ preserving multiplication and unit. The product on its tensor square uses the ambient braiding.

A monoid object $A$ with a morphism $\epsilon :A\to I$ such that $\epsilon m$ and a suitable coevaluation make $(A,A)$ a dual pair in a monoidal category. In a category of vector spaces this is a Frobenius algebra. This condition alone is not the usual separability condition.

A probability distribution is moment-determinate if its moments uniquely specify it among probability distributions. A moment-generating function finite in a neighbourhood of zero is a sufficient condition. The standard normal distribution is moment-determinate.

The fourth moment is $\mathbb{E}[X^4]$. Its finiteness is equivalent to square integrability of $X^2$. For centered variables, dividing it by the squared variance gives kurtosis.

For a modulus of a number field $\mathfrak{c}={\mathfrak{c}}_0{\mathfrak{c}}_{\infty }$, the multiplicity at a finite place is the exponent of its prime ideal in ${\mathfrak{c}}_0$. At a real Archimedean place it is one if that place occurs in ${\mathfrak{c}}_{\infty }$ and zero otherwise. Complex places have multiplicity zero. These effective modulus multiplicities encode congruences and positivity requirements in the ray class group.

Changing the coefficient ring of a module along a ring homomorphism can mean restriction of scalars, extension of scalars, or coextension of scalars. The underlying module may be retained with a smaller ring action, or replaced by a tensor or Hom construction with the new ring.

A module is finitely presented when it admits the displayed exact sequence for finite integers $r,s$. Thus it has finitely many generators and finitely many relations. On a Noetherian ring, every finitely generated module has a finite presentation.

For a finite-index inclusion of modular groups and a form with trivial multiplier, multiply its slash operator for modular forms translates over left coset representatives. Right multiplication permutes the factors, producing a form of weight $k[{\Gamma }^{\prime }:\Gamma ]$ for the larger group. modular cusp cusp widths determine the orders contributed by the factors; the product is nonzero if the original form is nonzero.

On a square, a normalized modified Helmholtz adjoint plane wave with positive normal growth toward one side has magnitude one there, magnitude $e^{-2\eta }$ on the opposite side, and an adjacent-side magnitude integral bounded by $1/\eta$. Thus large tangential spectral frequencies suppress remote-side normal-trace coefficients. Paired sine collocation of square modified Helmholtz global relations further cancels adjacent-side unknown traces exactly.

A càdlàg modification of a stochastic process is a modification of a stochastic process whose paths are càdlàg on one event of probability one. Equality almost surely at every separately fixed time does not imply indistinguishability of stochastic processes when the original process lacks path regularity. A stochastic process starting at zero, with independent increments, stationary increments, and stochastic continuity, has a càdlàg modification and is a Lévy process under the intrinsic definition.