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For an odd-dimensional sphere $S^{2m+1}$, only ${\pi }_{2m+1}(S^{2m+1})\otimes \mathbb{Q}$ is nonzero. For an even-dimensional sphere $S^{2m}$, the nonzero rational homotopy groups occur in degrees $2m$ and $4m-1$, and both are isomorphic to $\mathbb{Q}$.

The Chow group is $A_k(X)=Z_k(X)/{\mathrm{Rat}}_k(X)$, the group of $k$-dimensional algebraic cycles modulo rational equivalence.

For $a_n=n!/n^n$,Thus $\sum n!/n^n$ converges and $n/(n!{)}^{1/n}\to e$.

For a positive sequence, if $a_{n+1}/a_n\to L\ge 0$, thenFor $L>0$ this follows by taking logarithms and applying Cesàro averaging; the case $L=0$ follows from an eventual geometric upper bound.

The inertia group of a finite Galois extension of local fields is the kernel of the action of its Galois group on the residue field. In lower numbering it is the zeroth ramification group $G_0$.

Suppose $x_n\rightharpoonup x$ in a separable Hilbert space with Hilbertian basis $(e_i)$. Then $x_n\to x$ in norm exactly whenThe condition prevents norm from escaping to successively higher basis coordinates.

Mutual absolute continuity is equivalent to the Radon--Nikodym derivative being finite and strictly positive almost everywhere.

If the ordinary power series ${\sum }_{n\ge 0}{b}_n{w}^n$ has radius $R\in (0,\infty )$, thenconverges for $\mathrm{Re}z<\log R$ and diverges for $\mathrm{Re}z>\log R$. Behavior on the boundary depends on the coefficients.

If $L=\mathrm{limsup}{a}_n^{1/n}$ for positive coefficients, then the coefficients $a_n^2$ have root limsup $L^2$. Coefficients $a_n^{a_n}$ instead have root termsso their radius depends on the growth of $a_n\log a_n$ relative to $n$.

For the power series ${\sum }_{n\ge 0}{a}_n(z-c{)}^n$, the Cauchy-Hadamard theorem giveswith $R=\infty$ when the limit superior is zero, and $R=0$ when it is infinite. Away from the centre, the root test explains the formula: the limit superior of the term roots at displacement $z-c$ is $|z-c|/R$, interpreted as zero for $R=\infty$ and infinite for $R=0$. Values below one give absolute convergence; values above one imply that the terms fail to tend to zero. A value of one leaves the boundary question open. At the centre the series converges regardless of $R$.

For $a_{2n}=1$ and $a_{2n+1}=0$, the coefficient roots alternate between one and zero, so their ordinary limit does not exist. Their limit superior is nevertheless one, giving $R=1$ for ${\sum }_{n\ge 0}{z}^{2n}=1/(1-z^2)$ inside the unit disk. A ratio of consecutive coefficients would be undefined at the zero coefficients; the root-limsup formula handles the gap directly.

In the first part of his 1892 study of functions given by Taylor series, Jacques Hadamard introduced an upper-limit formulation for sequences and used coefficient roots to determine the radius of convergence. Sections 4–6 separate finite positive, zero, and infinite root-growth cases; the later parts study singularities and continuation. See Hadamard, Essai sur l'étude des fonctions données par leur développement de Taylor, Journal de mathématiques pures et appliquées, fourth series 8 (1892), pp. 107–109. This original text explains the coefficient-growth question without implying that the study of power series began there.

For intensity $\lambda$ in ${\mathbb{R}}^d$, let $R_k$ be the distance to the $k$th closest point and put $\rho =\lambda v_d$. Then $R_k^d$ is the $k$th arrival time of a rate-$\rho$ Poisson process, so

The endomorphism ring ${\mathrm{End}}_R(M)$ consists of the $R$-module homomorphisms $M\to M$, with pointwise addition and composition as multiplication.

For every $\epsilon ,p,r,\delta$, there is $T$ such that every Boolean function has a set $J$ with $|J|\le T$ for which a ${\mu }_p$-random restriction on $J$ leaves an $(\epsilon ,p,r)$-quasirandom function with probability at least $1-\delta$.

A geodesic metric space has the bottleneck property if every path joining the endpoints of a geodesic passes within one uniform distance of that geodesic's midpoint. A geodesic metric space is a quasi-tree exactly when it has this property.

If $U(s)=e^{-isQ/\hslash }$ and $U(s{)}^{†}HU(s)=H$, differentiation at $s=0$ givesIn the Heisenberg picture,

A 't Hooft anomaly is an obstruction to gauging a global symmetry. It is preserved by renormalization-group flow and constrains possible infrared phases.

The chiral anomaly is the quantum nonconservation of a classically conserved axial current in a gauge-field background.

A gauge anomaly destroys a gauge redundancy needed to remove unphysical states and makes the quantum gauge theory inconsistent unless the anomaly cancels.

Polar coordinates give ${\mathbb{C}}^{\ast }\cong S^1\times \mathbb{R}$ after taking logarithmic radius.

The vector fields $(cx+d){\partial }_x-2cu{\partial }_u$ generateTheir second prolongations multiply the differential equation by $-4c$, so they are infinitesimal Lie symmetries.