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For $c>0$, the backward causal cone at $(x_{\ast },t_{\ast })$ isIts boundaries are characteristic lines for the one-dimensional wave equation; its base on $t=0$ is the initial-data domain of dependence. The forward cone from $(s,0)$ is $t\ge 0$, $|x-s|\le ct$. Here the cone is set by the model's wave speed, which may be acoustic or a string wave speed; it does not assert propagation at the speed of light.

A finite p-group is powerful if $[G,G]\le G^p$ for odd $p$, or $[G,G]\le G^4$ for $p=2$. Here powers denote generated subgroups. The same definition applies to pro-p groups using closed power and commutator subgroups. Its Frattini subgroup is $G^p$. For odd $p$, successive power layers admit linear correction of roots, which makes every element of $G^p$ an actual $p$th power.

The displayed descending filtration combines powers and group commutators. In a pro-p group, take the closure of the generated product at each step. For a powerful p-group these layers are $P_i(G)=G^{p^{i-1}}$, with iterated closed power subgroups in the infinite case. The successive quotients are elementary abelian p-groups.

A p-subgroup is a subgroup which is a finite p-group, for a specified prime $p$. Every p-subgroup lies in a Sylow subgroup.

A Frobenius group has a nontrivial proper subgroup $H$ satisfying $H\cap H^g=\{1\}$ for every $g\notin H$, with $H^g=g^{-1}Hg$. Such $H$ is a Frobenius complement. The Frobenius kernel theorem turns the identity together with elements outside all conjugates of $H$ into a normal subgroup $N$, giving $G=N⋊H$.

A finite group is monomial if every irreducible character is a monomial character. Finite metabelian groups have this property.

A finite group is ambivalent when every element is conjugate to its inverse. Since every complex character satisfies $\chi (g^{-1})=\overline{\chi (g)}$, an ambivalent group has real values for every irreducible character. Conversely, if every irreducible value is real, irreducible characters separate conjugacy classes and force every $g$ to be conjugate to $g^{-1}$.

In finite-group character theory, an elementary group is a group that is p-elementary group for some prime $p$. Such groups are solvable and provide the subgroup tests in Brauer's characterization of characters. This does not mean elementary abelian group.

For a set $\pi$ of primes, a pi-group is a finite group whose order has no prime divisors outside $\pi$. The trivial group qualifies for every $\pi$.

The p-core is the largest normal p-subgroup of a finite group. The product of two normal p-subgroups is again a normal p-subgroup, so this subgroup exists and is characteristic.

The p-prime core is the largest normal subgroup whose order is coprime to $p$. Products of normal subgroups of order coprime to $p$ again have order coprime to $p$, so it exists and is characteristic.

A rectangular lock of length ${\ell }_0$ and depth $h_0$ has rarefaction head speed $-c_0$, where $c_0=\sqrt{g^{\prime }{h}_0}$. It reaches a closed rear wall at $t_r={\ell }_0/c_0$. The first reflected positive characteristic curve follows $x={\ell }_0+2c_{0}t-3c_0{t}_r^{2/3}{t}^{1/3}$ inside the undisturbed rarefaction, then crosses the uniform shelf and reaches the front at the displayed time. This marks the end of the initial constant-speed front stage in the ideal fixed-front-Froude model.

For $A=\sqrt{\pi /8}$, define real tail functions by $Q+iP=e^{-i{\lambda }^2}{\int }_{\lambda }^{\infty }{e}^{i{t}^2}dt$. Then $C=A+P\sin {\lambda }^2-Q\cos {\lambda }^2$ and $S=A-P\cos {\lambda }^2-Q\sin {\lambda }^2$. Repeated integration by parts giveswhere $(-1)!!=1$. These are asymptotic expansions, rather than convergent infinite sums at a fixed argument.

On a periodic uniform grid, shifting an index in the discrete inner product gives the displayed skew-adjoint identity for the centered first difference. It also gives $⟨U,D_{xx}V{⟩}_h=-⟨D_{+}U,D_{+}V{⟩}_h$. With finite boundaries there are additional boundary terms unless the prescribed endpoint traces eliminate them. These identities are discrete counterparts of integration by parts and prove energy estimates for finite difference methods.

An arbitrary $L^2$ initial function has no well-defined nodal point samples. Cell averages or a bounded projection to the discrete approximation space supply grid data independently of its pointwise representative. Jensen's inequality bounds the cell-average discrete L2 norm by the continuous L2 norm. Combining consistent bounded initialization, stability, and density of smooth data can extend convergence to rough initial functions, without claiming their nonexistent classical derivatives.

For $u_t+c{u}_x=0$ on a uniform grid, the update $U_m^{n+1}=(U_{m-1}^n+U_{m+1}^n)/2-\nu (U_{m+1}^n-U_{m-1}^n)/2$ has symbol $\cos \theta -i\nu \sin \theta$, where $\nu =c\Delta t/h$. Its squared modulus is $1+({\nu }^2-1){\sin }^2\theta$, giving contraction exactly for $|\nu |\le 1$. Stability is separate from the dissipative and dispersive approximation errors.

A three-point implicit advection stencil can match translated data through cubic Taylor expansion while retaining a rational Fourier amplification symbol. For the coefficient family $a=\mu (1+\mu )/6$, $b=(2-\mu )(1+\mu )/3$, $c=(2-\mu )(1-\mu )/6$, $d=(2-\mu )/3$, $e=(1+\mu )/3$, its factor is $G=(d+e{e}^{i\theta })/(a{e}^{-i\theta }+b+c{e}^{i\theta })$. Direct subtraction gives $|D{|}^2-|N{|}^2=\mu (\mu -2)(\mu -1)(\mu +1)(1-\cos \theta {)}^2/9$. The displayed ranges are the contraction ranges in the discrete L2 norm, with a nonvanishing denominator there. For forward time and positive speed parameter, retain the nonnegative ranges. For fixed nonzero nonexceptional Courant number, the normalized truncation error is third order; at $\mu =-1,0,1,2$ the update is the corresponding exact grid translation.

On a rectangular grid with zero Dirichlet boundary conditions, the nine-point Poisson equation matrix has diagonal $10/3$, axial off-diagonal entries $-2/3$, and diagonal-neighbor entries $-1/6$. Its quadratic form is the sum of squared differences over internal links, weighted by $2/3$ or $1/6$, plus the weighted squares of values linked to the fixed zero boundary. The connected axial grid forces a zero quadratic form to have all entries zero. This proves positive-definite matrix in every ordering. The Jacobi method converges because its nonnegative iteration matrix has row sums at most one, with a strict deficit at boundary-adjacent vertices; a hypothetical unit-modulus eigenvector would propagate its maximum modulus to such a deficit row, which is impossible.

With zero endpoints, the centered first-difference matrix has opposite off-diagonal entries and is a skew-adjoint operator in the mesh-weighted Euclidean inner product. Adding a real multiple of it to the negative symmetric second-difference matrix does not change the real energy form. The semidiscrete convection-diffusion solution therefore obeys a mesh-independent energy estimate, even when its drift is large compared with its diffusion coefficient.

Consider a one-step stencil with new-level coefficients $a=\mu (1+\mu )/2$, $b=(1+\mu )(2-\mu )$ and $c=(1-\mu )(2-\mu )/2$ at offsets $-1,0,1$, and old-level coefficients $d=2-\mu$, $e=1+\mu$ at offsets $0,1$. For $u_t=u_x$, its exact-solution residual starts at fourth degree with coefficient $\mu (\mu -2)(\mu -1)(\mu +1)/24$. Thus its normalized local truncation error is third order at fixed nonzero Courant number, except at the exact-shift values $-1,1,2$; zero gives the zero-step identity. For its Fourier symbol, $|D{|}^2-|N{|}^2=\mu (\mu -2)(\mu -1)(\mu +1)(1-\cos \theta {)}^2$. Uniformly invertible stable steps occur for $\mu \le -1$, $0\le \mu \le 1$ or $\mu \ge 2$.