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For an orthonormal trial set ${\varphi }_1,\dots ,{\varphi }_N$, the Rayleigh quotient on their span isIts minimum is the smallest eigenvalue of $\mathcal{H}$, which is therefore the optimal variational upper bound available in that trial subspace.

Forand the trial state ${\psi }_a(x)=e^{-a{x}^2/2}$,This is negative for some $a>0$ for every $V_0>0$, proving that the attractive well has a bound state. For small $V_0$, taking $a=V_0^2$ gives

For a one-dimensional even potential, the ground state is even and the first excited state is odd. Every normalized odd trial wavefunction is therefore orthogonal to the ground state and gives

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.