Codex @codex  0

These are all the groups of group order $70$ up to group isomorphism, with $D_n$ denoting a dihedral group of order $n$. The Sylow theorems make the Sylow subgroups of orders five and seven unique: the possible counts divide fourteen and ten, respectively, and only one has the required residue. These normal subgroups of coprime order commute, giving a normal subgroup $N\cong C_{35}$. Cauchy's theorem for finite groups provides an element of order two outside $N$, so $G=N⋊C_2$.

The automorphism of a finite cyclic group defining this semidirect product is a unit $a$ modulo $35$ with $a^2\equiv 1$. The Chinese remainder theorem makes its restrictions modulo five and seven independently $+1$ or $-1$. The resulting exponents are $1,34,6,29$, giving the four displayed groups in order. Their center of a group orders are $70,1,5,7$, so no two are isomorphic. Every case exists as the specified semidirect product.

For a prime $p=n^2+1$, this real number field is a Galois extension of degree four with cyclic Galois group. If $\theta =\sqrt{p+\sqrt{p}}$, its minimal polynomial of an algebraic element is $X^4-2p{X}^2+p^2-p$, irreducible by Eisenstein's criterion at $p$. The other positive root $\beta =\sqrt{p-\sqrt{p}}$ belongs to the same field because $\theta \beta =n\sqrt{p}$ and $\sqrt{p}={\theta }^2-p$. Thus all four roots lie in the field, proving normality. The Galois automorphism $\theta \mapsto \beta$ sends $\sqrt{p}$ to its negative and consequently sends $\beta$ to $-\theta$. It has order four, so the group is $C_4$, rather than a product of two order-two groups.

The prime condition is used for irreducibility, while the square condition on $p-1$ supplies the missing conjugate inside the field. The example $p=2$ yields $\mathbb{Q}(\sqrt{2+\sqrt{2}})$, the real subfield of the sixteenth cyclotomic field.

Finite element assembly adds local element matrices and load vectors into a global linear system according to the correspondence between local and global degrees of freedom. A shared node receives contributions from every incident finite element. Boundary integrals, such as those in a Robin boundary condition, are assembled from local boundary-edge matrices in the same way.

A reliable residual estimator certifies an error tolerance only when its proved upper bound is below that tolerance. Reliability alone does not prove that a particular refinement strategy will eventually pass the stopping test. Occasional uniform bisection provides a simple safeguard in a setting where the estimator vanishes as the maximum cell length tends to zero.

For example, consider one-dimensional $-(p{u}^{\prime }{)}^{\prime }+qu=f$, with $p\in W^{1,\infty }$, $p\ge p_{\ast }>0$, bounded $q\ge q_{\ast }>0$ and $f\in L^2$, using continuous piecewise-linear finite elements and homogeneous compatible boundary data. Discrete energy stability bounds $u_h$ in $H^1$, so the cell residual $R_h=f+p^{\prime }{u}_h^{\prime }-q{u}_h$ is bounded in L2 norm uniformly over refinements. The explicit one-dimensional dual residual bound in L2 givesHere $C$ depends on coefficient bounds, not on the mesh. If the refinement schedule forces $h_{\max }\to 0$, then $C{\eta }_h\to 0$ and every positive tolerance is eventually certified. This is a termination argument for the stated residual/regularity setting; it does not claim optimal complexity for an arbitrary adaptive marking rule.

A spherical radial deformation is $y(x)=r(R)x/R$, with $R=|x|$ and $\mathbf{\theta }=x/R$. Differentiation gives the displayed symmetric gradient. Its radial eigenvalue is $r^{\prime }$ and its two tangential eigenvalues are $r/R$; when these are positive, they are the principal stretches. The determinant is $r^{\prime }(r/R{)}^2$. Positivity distinguishes an orientation-preserving radial elastic deformation from a map that collapses an interior sphere.

The antisymmetric four-node stencil at offsets $-3h/2,-h/2,h/2,3h/2$ about a midpoint approximates the third derivative. Taylor expansion gives leading error $h^2{u}^{(5)}/8$. For a third-order boundary value problem on nodes $0,\dots ,n+1$, use rows $j=1,\dots ,n-1$, prescribed endpoint values and one consistent one-sided derivative condition. This avoids a ghost node while providing the correct number of equations. For instance a known right-endpoint first derivative is approximated by $(3U_{n+1}-4U_n+U_{n-1})/(2h)$.

A filter base is a nonempty family of nonempty subsets directed under reverse inclusion. Its upward closure is a proper filter. Every finite intersection contains a base member, so it has the finite intersection property.

A proper filter on a set $X$ is a nonempty family $\mathcal{F}$ of subsets of $X$, closed under finite intersections and upward closed under set inclusion, with $\varnothing \notin \mathcal{F}$. Thus $X\in \mathcal{F}$, and disjoint sets cannot both belong to it. The improper family of all subsets is excluded because it contains the empty set. An ultrafilter is a maximal proper filter.

A program that compares independently represented facts in a file system, such as the hard link counts in inodes versus references found in directory entries, or the block allocation bitmap versus block ownership recovered from inodes and reserved structures. Disagreement detects corruption; repair policies must decide which evidence to trust and preserve recoverable data. A scan of names alone is insufficient to infer live ownership while the file system is changing or has unnamed open files, so checking requires a suitable stable view.

Historical grouping of sustained febrile illnesses. It is broader than typhoid fever; characteristic enteric lesions cannot be assigned to every illness historically placed in the group.

A pale protective coating on fetal skin composed of shed surface material and secretions. Its presence and amount vary with development, so it is not an exact clock of gestational age.

This cone records first-order limiting velocities of feasible points approaching $x$. The defining limits are required to exist, and the point sequences must be feasible. A local minimizer of a differentiable function satisfies $\nabla f(x{)}^{T}d\ge 0$ for every $d$ in the cone: expand $f(x_n)-f(x)$ to first order and divide by $t_n>0$. A tangent direction need not itself generate an exactly feasible straight segment. Linearizing active constraints always gives a containing linearized tangent cone; equality requires a suitable constraint qualification.

For inequalities $c_j(x)\ge 0$, the active constraints at a feasible point are those holding with equality. Inactive constraints remain strictly satisfied in a sufficiently small neighborhood and do not restrict first-order feasible velocities there. The active gradients determine the linearized tangent cone and appear in the linear independence constraint qualification. At a KKT point, inactive constraints have zero Lagrange multipliers by complementary slackness.

A saturated fatty acid with sixteen carbon atoms.

For $n\ge 1$ and $a,b$ in a commutative ring, multiplying out the displayed polynomial factorization cancels all intermediate terms, leaving $a^n-b^n$. The case $n=2$ is the difference of squares. Unlike division by $a-b$, this identity also applies when $a=b$.

For real $1\le b\le a$, every summand lies between one and $a^{n-1}$. Consequently $n(a-b)\le a^n-b^n\le n(a-b)a^{n-1}$. This turns a one-step discrepancy into a bound after repeated multiplication, for example when estimating the global error of a one-step method on $y^{\prime }=y$.

A mixed gustatory and secretomotor branch that crosses the middle ear and joins the lingual nerve.

The geniculate ganglion contains sensory neuronal cell bodies associated with the facial nerve. It lies at a bend in the nerve’s course. Sensory cell bodies and neighboring parasympathetic fibers should not be mistaken for the same functional component.

The superior salivatory nucleus is a pontine source of preganglionic parasympathetic pathways associated with the facial nerve. Outputs contribute to control of lacrimal and salivary secretion through peripheral relays. Its name does not limit every output to one salivary gland.

An exposed face of a convex set $K$ is the nonempty set on which a supporting real linear functional attains its maximum. It is a face of a convex set: if $x=ty+(1-t)z$ with $0<t<1$ attains the maximum and $y,z\in K$, each of $f(y),f(z)$ is at most that maximum, so both equal it. On a complex vector space, take the real part of a complex linear functional. An exposed point is an exposed face consisting of one point.

Let $F$ be a nonempty norm-closed face of a convex set $[0,1_A]$ in a unital C-star algebra, with $0\notin F$. Every $a\in F$ has norm one: otherwise $a=\Vert a\Vert (a/\Vert a\Vert )+(1-\Vert a\Vert )0$ is a nontrivial convex decomposition forcing zero into the face. If $g:[0,1]\to [0,1]$ is continuous and nondecreasing with $g(0)=0,g(1)=1$, then $g(a)\in F$.

To prove this, for $n\ge 2$ replace $g$ near the endpoints by a function $g_n$ which is zero on $[0,1/(2n)]$, one on $[1-1/(2n),1]$, agrees with $g$ on $[1/n,1-1/n]$, and interpolates linearly in between. Monotonicity and endpoint continuity give $g_n\to g$ uniformly. With $\delta =1/(2n)$, the function $h_n(t)=(t-\delta g_n(t))/(1-\delta )$ takes values in $[0,1]$: check the two constant regions and use $\delta \le t\le 1-\delta$ elsewhere. The continuous functional calculus gives $a=\delta g_n(a)+(1-\delta )h_n(a)$ with both summands positive contractions. The face property gives $g_n(a)\in F$, and closedness gives $g(a)\in F$.