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For $(a(x,x/ϵ)y^{\prime }{)}^{\prime }=f(x,x/ϵ)$ with positive one-periodic fast coefficient, a periodic two-scale expansion has leading term independent of the fast variable. The first corrector satisfies $y_{1X}=C(x)/a-y_0^{\prime }$. Its zero-mean derivative requires $C=\hat{a}{y}_0^{\prime }$. Averaging the next equation gives $C^{\prime }={\int }_0^{1}fdX$. Thus the leading equation has the local harmonic-mean transport coefficient and the arithmetic-mean source. Positivity is what makes the secular drift incompatible with periodicity.

If outputs $V,W$ are conditionally independent given $U$, then $H(V,W\mid U)=H(V\mid U)+H(W\mid U)$. Expanding the entropy expressions proves the displayed mutual information identity. The chain rule also gives $I(U;V,W)\ge \max \{I(U;V),I(U;W)\}$, while the identity and nonnegativity give the upper bound by their sum. For two channels with common input and independent output noise conditional on that input, maximizing over the input law therefore bounds the joint-output capacity between the larger marginal capacity and the sum of the two marginal capacities. Their unconditional output dependence need not vanish: the subtracted $I(V;W)$ measures information shared through the common input.

Let $Z_s>0$ be a martingale with $Z_0=1$ and let $dQ=Z_{T}dP$ on a finite horizon. Its restriction to ${\mathcal{F}}_s$ has Radon-Nikodym derivative $Z_s$. If $Y$ is Q-integrable and $U={\mathbb{E}}_Q[Y\mid {\mathcal{F}}_s]$, then for every $A\in {\mathcal{F}}_s$,Also $Z_{s}U$ is P-integrable because ${\mathbb{E}}_P[Z_s|U|]={\mathbb{E}}_Q|U|<\infty$. The defining uniqueness of conditional expectation gives the displayed formula. In particular, if $N_s{Z}_s$ is a P-martingale and $N_s$ is Q-integrable, conditioning first at an intermediate time proves $N_s$ is a Q-martingale.

Let a probability space have countable measurable partitions of positive-measure cells, each partition refining its predecessor. An exceeding point has a first level where its cell average of $|f|$ exceeds $t$. The cells at those first crossings are disjoint: intersecting cells at different levels are nested, and an earlier exceeding cell prevents any of its descendants from being a first crossing. They cover exactly the maximal superlevel set. Summing their strict average inequalities gives the displayed weak type (1,1) estimate with constant one. The same proof works on a general measure space when every relevant cell has positive finite measure.

If $X_n\to X$ almost surely and $|X_n|\le H$ for integrable $H$, conditional expectations converge almost surely and in $L^1$. Conditional Fatou lemma applied to $H+X_n$ and $H-X_n$ bounds the conditional liminf and limsup by the same limit. For $L^1$ convergence, the contraction estimate $\mathbb{E}|\mathbb{E}(X_n-X\mid \mathcal{G})|\le \mathbb{E}|X_n-X|$ and dominated convergence theorem finish the proof. This also permits taking an unbounded measurable factor out of a conditional expectation by clipping it, provided the original product is integrable.

A secondary umbel within a compound umbel.

Every two finite composition series of a module have the same multiset of simple composition factors, up to isomorphism and order. One proves this by induction on the length: if the first simple submodules differ, their intersection is zero, and the quotient by their sum identifies the remaining common comparison problem; if they coincide, pass directly to the quotient. The theorem makes the multiplicity of a simple module independent of a chosen composition series. It does not assert that the module is a direct sum of its composition factors.

The only nontrivial proper normal subgroups of this direct product of groups are its two factors. Indeed a normal subgroup meeting $S\times 1$ nontrivially contains it by the simple group property. Otherwise each of its commutators with $S\times 1$ lies in that trivial intersection. Its first coordinates therefore lie in $Z(S)=1$, so it is contained in $1\times C_p$. Both factors are simple and each quotient by one factor is isomorphic to the other. Hence precisely two composition series occur, one through each factor. This is compatible with the Jordan–Hölder theorem, since the two series have the same simple factors in opposite orders. For $S=A_5,p=2$, the group has order $120$.

The hodograph plane records velocity rather than position. For a two-dimensional potential flow, the complex coordinate $q=u-iv$ is the complex velocity. A straight impermeable wall fixes the velocity direction and hence maps to a radial line in this plane. A constant-speed free streamline maps to a circle centered at zero. Thus a free-boundary problem in physical space can become a fixed geometric domain in velocity space. Recovering the physical map then uses $dz/dw=1/q(w)$, wherever $q$ is nonzero.

At zero, $[f(h)-f(0)]/h\to 1$, so the complex derivative exists. At a nonzero point, write $r=|z|$. The Cauchy-Riemann equations for $x/(1+r)$ and $y/(1+r)$ force $x^2=y^2$ and $xy=0$, impossible when $r>0$. Hence the function is complex differentiable only at zero and is holomorphic on no nonempty open subset. A complex derivative at one point gives no neighborhood of complex differentiability.

A nonprincipal complete type has no first-order consistent first-order formula implying all its members. In a complete countable first-order theory such a partial type can be omitted in a countable first-order model by the omitting types theorem. A partial type realized by a parameter-free definable tuple is principal.

A complete one-type is a complete type in one free variable. It chooses exactly one of each first-order formula and its negation consistently with the first-order theory and the named parameters.

A complete two-type is a complete type in two free variables. Such a partial type describes both individual properties and relations between a pair of elements.

For the linear program maximizing $c^{T}x$ subject to $Ax=b$, $x\ge 0$, let $x^{\ast }$ be optimal and let $y^{\ast }$ be an optimal dual linear program solution. Complementary slackness makes $A_j^T{y}^{\ast }=c_j$ on every positive coordinate of $x^{\ast }$. Any nonnegative $\hat{x}$ supported on those coordinates therefore hasThe same $y^{\ast }$ is dual feasible when the right-hand side is changed to $\hat{b}=A\hat{x}$. Equality of feasible objectives and weak duality prove optimality of $\hat{x}$ for that modified problem.

There is also a direct primal proof. If $z\ge 0$ had $Az=A\hat{x}$ and $c^{T}z>c^T\hat{x}$, then $x^{\ast }+\epsilon (z-\hat{x})$ would remain feasible and nonnegative for sufficiently small $\epsilon >0$. On zero coordinates of $x^{\ast }$, support inclusion makes its perturbation nonnegative; on positive coordinates, smallness preserves positivity. Its objective would improve, contradicting optimality of $x^{\ast }$.

For a monic polynomial $f(t)=t^d+c_{d-1}{t}^{d-1}+\cdots +c_0$, use the companion matrix convention with ones on the subdiagonal and last column $(-c_0,\dots ,-c_{d-1}{)}^{\mathsf{T}}$. In $tI-C(f)$, the cofactors of the last-column entries, from top to bottom, are $1,t,\dots ,t^{d-1}$. This follows by expanding their bidiagonal minors: the subdiagonal factors $-1$ cancel the cofactor signs. Laplace expansion along that column therefore gives $c_0+c_{1}t+\cdots +c_{d-2}{t}^{d-2}+(t+c_{d-1})t^{d-1}=f(t)$. The degree-one case is the same identity for the one-by-one matrix $(-c_0)$.

Suppose a first-order theory characterized equivalence relations with exactly one infinite class. Name two inequivalent points and require each named class to have at least $n$ elements for every positive integer $n$. Every finite subset is satisfied with one infinite class and a sufficiently large finite class. Compactness theorem would produce a first-order model with two infinite classes, a contradiction.

In a Hausdorff space, a point $x$ outside a compact subset $K$ can be separated from all of $K$ by disjoint open sets. Choose disjoint open neighbourhoods $U_k$ of $k$ and $V_k$ of $x$ for each $k\in K$. Finitely many $U_k$ cover $K$ by compactness. Their union and the matching finite intersection of the $V_k$ give the displayed separation. If $K$ is empty, choose $U=\varnothing$, $V=X$. Thus $X\setminus K$ is open, proving that $K$ is closed.

A closed filter base in a compact space has nonempty intersection. If the space is Hausdorff and every base member is connected, the intersection is connected. Otherwise normality separates its two closed components by disjoint open sets. Compactness of the complement supplies finitely many base members whose intersection lies in that union; directedness then gives a single base member meeting both sides, contradicting its connectedness.

The elapsed time between a change at a logic gate's input and the corresponding settled change at its output, under specified loading and transition conditions. Dependent gate delays add along a path. The combinational critical path gives the longest such input-to-output path; reconvergent paths of unequal delay can produce a logic hazard even when the final Boolean function value does not change.

A logic hazard is a transient output error caused by unequal propagation delays along paths in a Boolean circuit. A static hazard temporarily changes an output that should remain constant; a dynamic hazard introduces extra transitions when the output should change once. A steady-state truth table records final values and therefore does not describe these timing effects.

For example, implement $F=ab+\bar{a}c$ with AND gates, an OR gate and a NOT gate. With $b=c=1$, changing $a$ should leave $F=1$. But if the first product falls before the second rises, both OR inputs can momentarily be zero. Adding the redundant term $bc$ gives $F=ab+\bar{a}c+bc$: it does not change the Boolean function, by the consensus theorem, and remains one throughout that single-input transition. This argument assumes the other inputs remain fixed and the circuit begins settled. Simultaneous input changes and more general implementations require their own timing analysis.