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 $\mu \ge 2$.

A Mehrstellen method increases the accuracy of a compact finite difference method discretization by using the differential equation to express its leading truncation terms in terms of the prescribed source. For $\Delta u=f$, the square-grid nine-point finite-difference stencil for the Laplacian satisfies $D_{9}u=\Delta u+h^2{\Delta }^{2}u/12+O(h^4)$. Replacing ${\Delta }^{2}u$ by $\Delta f$ and approximating this source derivative gives $D_{9}U=(I+h^2{D}_5/12)f$, with fourth-order normalized consistency of a numerical method while retaining a compact nine-point matrix. The uncorrected nine-point finite-difference stencil operator is only second-order accurate on general functions.

For $u_t=u_x$, the recurrence $U_m^{n+1}-U_m^{n-1}={\sum }_{j=\pm 1,\pm 2}{a}_j{U}_{m+j}^n$ has fourth-order normalized consistency of a numerical method at fixed $\mu =k/h$ ifTo derive this, match moments $M_r={\sum }_j{a}_j{j}^r$ to the temporal Taylor series: $M_0=M_2=M_4=0$, $M_1=2\mu$, $M_3=2{\mu }^3$. The even conditions force antisymmetry; the two odd conditions then determine the displayed coefficients. The normalized residual begins with $h^4({\mu }^2-1)({\mu }^2-4)u_{xxxxx}/120$. Its amplification polynomial of a multilevel finite difference scheme is $G^2-2iA(\theta )G-1$, where $A=a_1\sin \theta +a_2\sin 2\theta$. At $\mu =1/2$, $|A|\le 3/4$, giving a uniform power bound. At $\mu =3/2$, $A(\pi /3)=19\sqrt{3}/32>1$, and wave packets on the growing polynomial root prove Cauchy linear instability.

Finite difference coefficients are the weights in a linear combination of sampled function values approximating a derivative. Matching polynomial moments determines the weights and approximation order. Forward, backward and centered stencils have different coefficients.

For $u_t=(a{u}_x{)}_x$, this conservative finite difference method uses coefficient samples at edge midpoints. For smooth $a,u$, its spatial residual is $h^2(a{u}_{xxxx}/12+a^{\prime }{u}_{xxx}/6+a^{\prime \prime }{u}_{xx}/8+a^{\prime \prime \prime }{u}_x/24)+O(h^4)$. With zero Dirichlet boundary conditions, $-v^T{L}_{h}v=h^{-2}{\sum }_j{a}_{j+1/2}(v_{j+1}-v_j{)}^2$. If $0<a\le a_{+}$, its eigenvalues lie in $[-4a_{+}/h^2,0]$, giving the mesh-independent forward Euler method bound $\Delta t/h^2\le 1/(2a_{+})$. Constant $a=a_{+}$ proves sharpness as the grid is refined.

Suppose a compact skeleton $Y$ embeds as $e(Y)\subset {\mathbb{R}}^m$. Attach a disk by $a:S^{k-1}\to Y$ and extend its boundary coordinate map radially by $G(ru)=re(a(u))$. Map the disk to $(G(x),(1-|x|)x,1-|x|)$ and $Y$ to $(e(y),0,0)$. The boundary identifications agree; a positive last coordinate recovers each interior point. This defines a continuous injection of the compact quotient into Euclidean space, hence an embedding. Repeating proves Hausdorffness and Euclidean embeddability for every finite CW complex, without assuming triangulability of arbitrary attaching maps.

The finality property of a functor means every required comma category is nonempty and connected. It is the condition defining a final functor; the displayed criterion uses categorical connected components.

On $(0,1{]}^d$ with Lebesgue probability measure, the dyadic filtration is generated by the partitions into half-open cubes of side $2^{-n}$. The conditional expectation of an integrable function is its average on each atom. The increasing partitions generate the Borel sigma-algebra. Step functions on these partitions are dense in $L^1$; approximation and the Doob maximal inequality show that their conditional averages converge in $L^1$ and almost everywhere to the original function.

The $I$-adic topology on a module $M$ has the powers $I^{n}M$ as a neighbourhood basis of zero. It is generated by the I-adic filtration; separatedness means ${\bigcap }_n{I}^{n}M=0$.

An ascending filtration is an increasing chain of additive subgroups, or vector subspaces when a scalar field is fixed. It is exhaustive if their union is the whole object. For a filtered ring, multiplicative compatibility means $F_{i}R{F}_{j}R\subseteq F_{i+j}R$. For an ideal $I$, intersecting each level with $I$ gives the induced filtration. Its associated graded module has components $(I\cap F_{n}R)/(I\cap F_{n-1}R)$.