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The Duhamel formula for Hamiltonian transport, phase space preservation and Minkowski integral inequality give the displayed bound. A time-independent source in an Lp space yields finite space-time norm on every bounded interval; it need not yield integrability on infinite time.

If $f_0=0$ and a nonzero time-independent source is invariant along the Hamiltonian flow, then the Duhamel formula for Hamiltonian transport gives $f_t=th$. For the nonzero-frequency oscillator, $h=e^{-H_{\omega }}$ is smooth and belongs to all finite Lp spaces, yet its resulting solution has infinite space-time norm on $(0,\infty )$.

The Hamiltonian $(v^2-x^2)/2$ gives $\dot{x}=v$, $\dot{v}=x$. Its characteristic flow map is multiplication by $A_t$, with $A_t{A}_s=A_{t+s}$ and determinant one. The backward map is $A_{-t}$. This expanding-contracting linear flow preserves phase-space Lebesgue measure and every finite-$p$ integral of a transported density, despite stretching its level sets.

With the displayed signed Lie-Poisson bracket and evolution $\dot{F}=\{F,H\}$, the equations are ${\dot{x}}_1=2(C-B)x_2{x}_3$, ${\dot{x}}_2=2(A-C)x_3{x}_1$, and ${\dot{x}}_3=2(B-A)x_1{x}_2$. The energy and the Casimir function of a Poisson manifold $x_1^2+x_2^2+x_3^2$ are conserved. For $A=1/(2I_1)$, $B=1/(2I_2)$, $C=1/(2I_3)$ with positive inertias, these are the Euler equations for a torque-free rigid body. Reversing the bracket while fixing $H$ reverses time orientation.

If $H$ is $C^2$ and its spatial Hessian matrix is uniformly bounded on every finite time interval, its Hamiltonian vector field is globally Lipschitz continuous in phase space with at most linear growth. The Picard-Lindelöf theorem and Gronwall inequality give a unique Hamiltonian flow for all finite forward and backward times. Bounds on ${\nabla }_{z}H(t,0)$ follow from time continuity on compact intervals.

For homogeneous Hamiltonian Liouville equation transport with autonomous $H$, compact initial support remains in the fixed energy sublevel determined by its maximum initial energy. If these sublevels are compact, this proves uniform compact support without solving the flow. The harmonic energy $H_{\omega }=(|v{|}^2+{\omega }^2|x{|}^2)/2$ has this property exactly when $\omega \ne 0$; at zero frequency spatial free streaming can escape every compact set.

For $H_{\omega }=(|v{|}^2+{\omega }^2|x{|}^2)/2$, the displayed linear Hamiltonian flow solves $\dot{X}=V$, $\dot{V}=-{\omega }^{2}X$. Its inverse is $S_{-t}$, its determinant is one and it preserves $H_{\omega }$. The zero-frequency limit is the shear $(x,v)\mapsto (x+tv,v)$.

For a real scalar field with Lagrangian density $\mathcal{L}={\dot{\varphi }}^2/2-|\nabla \varphi {|}^2/2-V(\varphi )$, the canonical momentum is $\pi =\dot{\varphi }$. Its Hamiltonian density is the sum of momentum energy, gradient energy and potential energy. Hamilton's equations recover the Euler-Lagrange equation.

A real linear functional $g$ on a vector subspace, dominated by a sublinear function $p$, can be extended across one new vector $v$ by setting $\tilde{g}(m+tv)=g(m)+tc$. The displayed lower and upper bounds are compatible because $g(m+n)\le p(m-v)+p(n+v)$. A value between them preserves domination for both signs of $t$. Chain unions and Zorn's lemma then yield the full Hahn-Banach theorem.

If $Y$ is a closed linear subspace of a Hausdorff locally convex space and $x_0\notin Y$, choose a continuous seminorm $q$ with $q(x_0-y)\ge 1$ for all $y\in Y$, using a basic neighborhood disjoint from $Y$. The linear functional $g(y+t{x}_0)=t$ is bounded by $q$. The Hahn-Banach theorem extends it to a continuous linear functional with the displayed values. This is the linear-subspace form of separation, sharper than merely distinguishing the two sets.

A futility boundary specifies an interim result below which a clinical trial stops without rejecting the null hypothesis. Removing only potential rejection paths cannot increase the type I error. With a finite boundary and a nonsingular joint normal test-statistic distribution, it strictly reduces that error relative to the corresponding fixed-final-boundary test.

A matrix coefficient of a finite-dimensional group representation is a scalar function obtained by applying a linear functional to the translate of a fixed vector. In a chosen basis these functions include $x\mapsto \rho (x{)}_{ij}$. For a finite group, the Schur orthogonality relations make the scaled coefficients of all inequivalent unitary irreducible representations an orthonormal basis of its scalar functions.

An intertwiner between group representations $\rho$ on $V$ and $\sigma$ on $W$ is a linear map $A:V\to W$ satisfying $A\rho (x)=\sigma (x)A$ for every group element. The Schur lemma controls these maps between irreducible representations.

The regular group representation of a finite group over the complex numbers contains each irreducible representation with multiplicity equal to its dimension. Consequently the sum of the squared dimensions is the group order. Once a list of distinct irreducibles exhausts this sum, it is complete.

For a group representation of $G$ and a subgroup $H$, restriction keeps the vector space and lets only the elements of $H$ act. It is denoted ${\mathrm{Res}}_H^{G}V$.

A finite group presentation has finitely many generators and finitely many relators. It defines the quotient of the free group on $X$ by the normal closure of $R$. Free products of finitely presented groups and HNN extensions along explicitly finitely generated associated subgroups have finite presentations obtained by adjoining finitely many generators and relations.

For a group presentation $\mathcal{P}=⟨X\mid R⟩$ with finitely many generators and a prime number $p$, defineThis uses the unshifted convention. Identity relators have zero weight, and a divergent sum gives value $-\infty$. Some literature subtracts one instead; inequalities must be shifted accordingly. Taking roots in the free group matters: roots appearing only after passing to the quotient do not change the relator weight. The p-rank of a group bounds this quantity above, and the index-p rewriting bound for p-deficiency gives the infinitude criterion p-deficiency at least one implies infinitude.

Joint squared-error nodewise regression with this penalty groups the two coefficients of each unordered edge. Its norm is nonsmooth only when the whole pair is zero, encouraging edge-wise sparsity. A fitted edge is present when its pair norm is positive. The penalty does not require equal directional coefficients, two nonzero entries in every selected group, or a symmetric positive definite precision matrix.

Cayley theorem embeds a finite group of order $n$ into $S_n$. Extend $\sigma \in S_n$ to $n+2$ letters and multiply it by the disjoint transposition $(n+1n+2)$ exactly when $\sigma$ is odd. Additivity of parity makes this an injective group homomorphism into the alternating group $A_{n+2}$. Thus every finite group is isomorphic to a subgroup of some alternating group.

For a finite group, the usual integral notion asks for a class $\Delta \in H^n(G;\mathbb{Z})$ whose cup product gives the displayed isomorphisms for every coefficient module $M$ and every $i>0$. The analogous claim for just trivial coefficients over a fixed field is weaker and should be stated as such. A free cohomologically orientation-preserving sphere action supplies that field-coefficient version through periodic group cohomology from a free sphere action.