A Mealy machine is a finite-state machine whose transition and output both depend on the current state and input. It consists of a finite state set $Q$, input and output alphabets $\Sigma ,\Gamma$, an initial state, a transition $\delta :Q\times \Sigma \to Q$, and an output map $\lambda :Q\times \Sigma \to \Gamma$.

For a running parity checker, let states be $0,1$ and put $\delta (q,a)=q+a(\mathrm{mod}2)$, $\lambda (q,a)=q+a(\mathrm{mod}2)$. On input $1101$, it outputs $1001$, the parity after each prefix. Unlike a Moore machine, whose output is attached to a state, the output here is attached to a transition. This model naturally represents synchronous controllers and stream transformations.

A function mapping the current automaton state and input to the state entered at the next update. In a Moore machine, a separate output function depends only on the current state; in a Mealy machine, it may also depend on the current input.

A finite-state automaton recognizes a nonempty formal language exactly when some accepting automaton state is reachable from an initial automaton state. Reachability in a graph decides this in a finite transition graph. For an Epsilon-NFA, include its epsilon transitions; the non-epsilon labels of a successful graph path give an accepted word.

To decide whether a finite-state automaton accepts exactly one distinct word, first obtain some accepted word $w$ by a finite automaton emptiness test. If there is none, answer no. Otherwise construct a finite-state automaton for the accepted formal language minus the singleton $\{w\}$ and test it for emptiness. Answer yes exactly when that difference is empty. Subset construction handles multiple accepting runs without confusing runs with distinct words.

For a deterministic finite automaton with automaton state set $Q$ and input alphabet $\Sigma$, the function $\delta :Q\times \Sigma \to Q$ specifies the next automaton state after reading one input symbol. A nondeterministic finite automaton instead allows a set of possible next automaton states.

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 $y^{\prime }=y$.