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In linear temporal logic, $\alpha \mathrm{W}\beta$ means $(\alpha \mathrm{U}\beta )\vee G\alpha$. Thus it permits the left operand to hold forever without the right operand ever becoming true, unlike strong until.

An instrument for estimating high temperatures. The physical response and calibration must be specified; thermal expansion, a calorimetric transfer, electrical properties and emitted radiation lead to different methods and different limitations.

An average of a specified temperature record over a stated interval or ensemble. For a time mean, observations must represent elapsed durations: irregularly spaced readings generally require time weighting. The half-sum of a daily maximum and minimum is a conventional daily statistic and need not equal the continuous time mean.

Substitution of acyclic concept definitions into expressions to be tested. Primitive inclusions require care: $A⊑D$ can be represented using a fresh name $A_0$ by $A\equiv A_0\sqcap D$, but is not equivalent to $A\equiv D$.

Rewriting suitable general axioms into definitions or primitive inclusions that can be processed when an atomic label is encountered. For example, $A\sqcap B⊑\perp$ can be absorbed as $A⊑\neg B$.

The quadratic Taylor polynomial of $f(x)=1/(1+x^2)$ at $x=1$ is $p(x)=(x^2-4x+5)/4$. Multiplying $p-f$ by $4(1+x^2)$ gives $(x-1{)}^4$, proving the exact displayed identity. Thus $p\ge f$ for every real $x$, with equality only at one. In particular, a claim that $f>p$ to the right of one has the wrong sign. The Lagrange remainder also proves this direction there, since $f^{\prime \prime \prime }(x)=24x(1-x^2)/(1+x^2{)}^4<0$ for $x>1$. The exact identity is stronger: it controls the sign globally and shows the fourth-order contact at the expansion point.

Dandelion; the 1867 medicinal root article uses the historical name Taraxacum dens leonis.

If the zero set of a differentiable function $g$ is a submanifold, the chain rule always places its tangent space inside $\ker D{g}_a$. Equality follows from the full-rank hypotheses of the regular level set theorem, but may fail for a singular defining function even when the zero set itself is smooth. For $g:\mathbb{R}\to \mathbb{R}$, $g(x)=x^2$, the zero set is the single point $\{0\}$. Its tangent space is zero, while $D{g}_0=0$ has kernel $\mathbb{R}$. One should therefore check the constraint qualification before using the kernel formula.

Two smooth differential systems related by a nonlinear change of dependent variables need not define the same weak solutions. If $U_t+F(U{)}_x=0$, a smooth function $V=g(U)$ satisfies $V_t+g^{\prime }(U)F^{\prime }(U)U_x=0$ for smooth $U$; even when this can be written as another divergence, its Rankine-Hugoniot conditions need not be equivalent to those in $U$. The smooth chain rule does not justify multiplying jump distributions by discontinuous coefficients.

For example the shallow water equations can be written smoothly as $h_t+(hv{)}_x=0$ and $v_t+(h+v^2/2{)}_x=0$. Mechanical mass and momentum conservation instead select $h_t+(hv{)}_x=0$ and $(hv{)}_t+(h{v}^2+h^2/2{)}_x=0$. These systems agree for smooth positive $h$, but their jump laws differ. The states $(h,v)=(2,\sqrt{3}/2)$ and $(1,0)$ satisfy the mass/momentum jump laws with speed $\sqrt{3}$, while the velocity-form jump equation would demand $11/(4\sqrt{3})$. A model for discontinuities must specify the conservative variables or an equivalent admissible regularization, rather than using smooth equivalence alone.

A sampled Boolean circuit that suppresses short input changes using recent samples. A three-sample majority function removes isolated one-sample changes; a stateful variant updates its stored output only after three equal input samples. These are different specifications.

Let $K$ be a nonempty compact space and let $\alpha :K\to \mathbb{R}$ be continuous and nowhere zero. On $C(K)$, define $\Vert f{\Vert }_{\alpha }={\sup }_{x\in K}|\alpha (x)f(x)|$. Multiplication by $\alpha$ is an isometry onto $C(K)$ with its ordinary uniform norm, because division by the continuous nonvanishing $\alpha$ provides the inverse. Thus the weighted space is a Banach space. Equivalently, the extreme value theorem gives $0<m=\min |\alpha |\le \max |\alpha |=M<\infty$, and $m\Vert f{\Vert }_{\infty }\le \Vert f{\Vert }_{\alpha }\le M\Vert f{\Vert }_{\infty }$. A weighted Cauchy sequence is uniformly Cauchy, has a continuous uniform limit, and the upper bound gives convergence also in the weighted norm.

An inclusion-minimal superkey: removing any of its database attributes destroys the superkey property. Minimal means inclusion-minimal, not necessarily smallest in size.

A subset $B$ of a topological space $X$ is locally closed if $B=U\cap F$ for an open set $U$ and a closed set $F$. Equivalently, $B$ is open in its closure $\overline{B}$, or $\overline{B}\setminus B$ is closed in $X$. For the equivalence, $B\subseteq F$ implies $\overline{B}\subseteq F$, so $B=U\cap \overline{B}$. Relative complementation proves the second condition implies the third. Conversely, if $C=\overline{B}\setminus B$ is closed, then $B=(X\setminus C)\cap \overline{B}$ is such an intersection.

Every subring $R$ of the rational numbers containing $\mathbb{Z}$ is a principal ideal domain. If $a/b\in R$ is reduced, Bézout's identity gives integers $u,v$ with $ua+vb=1$, hence $1/b=u(a/b)+v\in R$. Thus every reduced denominator occurring in $R$ is a unit.

For an ideal $I$, write $I\cap \mathbb{Z}=d\mathbb{Z}$. Then $dR\subseteq I$. If $a/b\in I$ is reduced, $a=b(a/b)\in I\cap \mathbb{Z}$, so $a=dc$ for some integer $c$. Since $1/b\in R$, $a/b=d(c/b)\in dR$. Therefore $I=dR$, including the zero ideal. This is a concrete instance of extending and contracting ideals under localization of a ring.

Let $Ay=(p{y}^{\prime }{)}^{\prime }+qy$ have real eigenfunctions $u_n$ obeying $A{u}_n={\lambda }_{n}w{u}_n$ and homogeneous Dirichlet boundary conditions, with $N_n={\int }_a^{b}w{u}_n^2>0$. Suppose $Ay+\mu wy=f$, with possibly nonzero endpoint values. The Lagrange identity for a Sturm-Liouville operator givesDivision by $({\lambda }_n+\mu )N_n$ gives the displayed coefficient whenever the denominator is nonzero. At resonance in a differential equation, equality of the forcing projection and the boundary term is the Fredholm solvability condition. The Sturm-Liouville eigenfunction expansion represents the interior solution in the weighted inner product sense; its zero endpoint terms do not reproduce nonzero boundary traces by pointwise substitution. A smooth lift of the endpoint data reduces the calculation to homogeneous Dirichlet boundary conditions.

A finite directed graph admits an infinite edge-following walk from $s$ exactly when a directed cycle is reachable from $s$. An infinite walk repeats a vertex and therefore contains a reachable cycle; conversely, reach a cycle and repeat it. In a strongly connected component, a cycle exists precisely when the component has at least two vertices or is a singleton with a self-loop. Reachability followed by the two-pass strongly connected component algorithm decides the criterion in $O(|V|+|E|)$ time.

The nonempty set of vertices visited infinitely often by a single infinite walk lies within one strongly connected component. For two such vertices, arbitrarily late visits supply directed walks in both directions. This set need not be the whole component: the walk can keep choosing one cycle and omit other vertices.

A directed graph with black and white edges has a closed walk containing both colours exactly when some strongly connected component has internal edges of both colours. Join the two chosen edges with graph paths inside the strongly connected component. A corresponding claim for a simple cycle is false: differently coloured graph cycles can meet at a single vertex.

For normal Hausdorff $X$, if $\beta X$ has a clopen base, extend a Urysohn's lemma separator of two closed subsets of $X$. Their closures in $\beta X$ are separated; compactness gives a clopen separator, whose trace works in $X$. Conversely separate the traces of the low and high level sets of a continuous $f:\beta X\to [0,1]$ by clopen $U\subseteq X$. The characteristic function of $U$ extends into $\{0,1\}$ on $\beta X$. Density ensures its one level contains the zero level of $f$ and misses the one level. Applying this with $f$ separating a point from a closed set gives a clopen base for $\beta X$.

The log-cosh derivative operator is strictly monotone on $H^1$ but does not have $⟨A(v),v⟩/\Vert v{\Vert }_{H^1}\to \infty$. Constant functions $v=t$ make the ratio approach $|\Omega {|}^{1/2}$. Restricting to $H_0^1$ removes the uncontrolled constants: Poincaré gives $⟨A(v),v⟩\ge \Vert v{\Vert }_{H^1}^2/(1+C_P^2)$, which proves coercivity. Strict monotonicity and large-norm coercivity are distinct properties.

Under the usual $C^2$ integrand regularity, an admissible $C^1$ curve satisfying the weak Euler-Lagrange equation, a coercively positive second variation, and the strengthened Weierstrass condition is a strong local minimizer of a variational integral. Strict excess away from the reference slope gives strict minimality. Coercive positivity means ${\delta }^{2}I(u)[\eta ,\eta ]\ge c\Vert \eta {\Vert }_{W^{1,2}}^2$ for every zero-trace variation and some $c>0$.

The second variation provides a neighboring field of extremals with slope $p(x,y)$. The associated endpoint-invariant Hilbert integral yields $I(v)-I(u)=\int E(x,v,p(x,v),v^{\prime })dx$ for uniformly close admissible $v$. The tube condition makes this integral nonnegative even when $v^{\prime }$ is large, explaining the passage from weak to strong minimality.