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If $Q_m\sim {\chi }_m^2$, then $(Q_m-m)/\sqrt{2m}$ converges in distribution to $N(0,1)$.

Light-cone gauge uses residual conformal transformations to make the target-space coordinate $X^{+}$ linear in worldsheet time. The Virasoro constraints then determine $X^{-}$ from the transverse fields.

This is the string-worldsheet realization of light cone gauge.

If an independent superposition of a Poisson process and a non-arithmetic finite-mean renewal process is itself a renewal process, then its first interarrival is exponential. Equating the limiting excess law computed as a renewal excess with the minimum of the Poisson and second-process excesses yields an integral equation whose survival-function solutions are exponential.

If $f$ is holomorphic on $0<|z|<R$ andthen the singularity at zero is removable. The Laurent coefficient formula and Cauchy--Schwarz give $|a_{-k}|=O(r^k)$ for each $k\ge 1$, so every principal-part coefficient vanishes as $r\downarrow 0$.

A holomorphic function tending to a finite limit at infinity becomes holomorphic at zero after reciprocal substitution.

A holomorphic function bounded on a punctured neighbourhood extends holomorphically across the puncture.

This criterion characterizes a removable singularity of a holomorphic function; the criterion is a theorem about the singularity type.

This is the zero-boundary special case of the Rellich-Kondrachov theorem. For every bounded open $\Omega \subset {\mathbb{R}}^d$, the inclusion $H_0^1(\Omega )\hookrightarrow L^2(\Omega )$ is compact.

Translations of one nonzero compactly supported smooth function have equal $W^{1,p}$ norms. On an unbounded domain they can have pairwise disjoint supports, so their mutual $L^p$ distances stay fixed and no subsequence converges in $L^p$.

By Maschke's theorem, the complex regular representation of a finite group is a direct sum of irreducibles. Its kernel is the intersection of their kernels. If the group is simple, each kernel is either trivial or the whole group; faithfulness of the regular representation therefore forces at least one irreducible constituent to be faithful.

For every $\epsilon >0$ and $m_0$ there are $M,n_0$ such that every graph on at least $n_0$ vertices has a partitionwhere $m_0\le m\le M$, $|V_0|\le \epsilon |V|$, the other parts have equal size, and all but at most $\epsilon m^2$ pairs $(V_i,V_j)$ are $\epsilon$-regular.

If a finite right-linear grammar generates an infinite language, some variable is accessible, looping, and terminable. It is the repeated variable on a sufficiently long accepting derivation.

A variable $A$ is terminable when $A{\Rightarrow }^{\ast }w$ for some terminal word $w$.

A variable $A$ is looping when $A{\Rightarrow }^{\ast }wA$ for some terminal word $w$ through a nonempty derivation.

A variable $A$ is accessible when $S{\Rightarrow }^{\ast }wA$ for some terminal word $w$.

Every sentential form reachable from the start variable of a right-linear grammar is either a terminal word $w$ or $wA$, where $A$ is the unique variable and occurs at the right end.

A root datum is a quadruple $(X^{\ast },\Phi ,X_{\ast },{\Phi }^{\vee })$ of dual lattices, roots, and coroots with the natural perfect pairing and reflection axioms. A reductive group with maximal torus $T$ has $X^{\ast }=X^{\ast }(T)$ and $X_{\ast }=X_{\ast }(T)$.

A Borel subgroup is a maximal closed connected solvable subgroup of an affine algebraic group.

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