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If $f$ is jointly continuous on a compact rectangle $J\times [a,b]$, then the displayed parameter integral is continuous on $J$. For $t_n\to t$, pointwise continuity gives $f(t_n,x)\to f(t,x)$. The extreme value theorem bounds $|f|$ by a constant $M$ on the rectangle, and $M$ is integrable on the finite interval $[a,b]$. The dominated convergence theorem gives $F(t_n)\to F(t)$.

The same conclusion holds on an unbounded parameter interval if $f$ is jointly continuous there: use a compact rectangle around each fixed parameter value. A global bound over the whole parameter range is unnecessary.

For real $a$, the geometric-series expansion of $1/(e^x-1)$ gives an integrable series after multiplication by $\sin (ax)$: each absolute integral is bounded by $|a|{\int }_0^{\infty }x{e}^{-nx}dx=|a|/n^2$. Termwise integration is therefore justified and gives the convergent series ${\sum }_{n\ge 1}a/(a^2+n^2)$.

Use of heat to reduce viable microorganisms. Temperature, exposure time and the material’s thermal behavior affect the result, so a hot surface does not by itself establish the treatment achieved its intended endpoint.

A random variable whose values lie in the integers with probability one. Its probability mass function is $p(j)=\mathbb{P}(X=j)$ for $j\in \mathbb{Z}$. If it is nonnegative, its probability generating function is $G(s)={\sum }_{j\ge 0}p(j)s^j$ wherever this expectation is finite.

For $0<\theta <1$, this probability distribution on positive integers has mass $\mathbb{P}(X=k)={\theta }^k/(k[-\log (1-\theta )])$. The Taylor identity ${\sum }_{k\ge 1}{\theta }^k/k=-\log (1-\theta )$ proves normalization. Its expectation is $-\theta /((1-\theta )\log (1-\theta ))$, by summing the geometric series after multiplying the mass by $k$.

For an independent sample, the maximum-likelihood estimator solves the equation making this model mean equal to the sample mean. The model mean is strictly increasing: set $t=-\log (1-\theta )$ to obtain $(e^t-1)/t$, whose derivative is positive for $t>0$. Hence a sample mean greater than one determines a unique interior estimate. Pooling several categories changes the likelihood and generally changes this estimate; the ungrouped sample mean is no longer recoverable from the pooled category counts alone.

A directed cycle in an edge-weighted graph whose total weight is negative. Repeating it makes walk costs arbitrarily small. For a particular shortest-path distance $d(s,t)$ to be affected, the cycle must be reachable from $s$ and must have a route to $t$; an unrelated component does not change that distance.

Among paths of minimum total weight, choose the fewest edges. Store a lexicographic order pair $(d,h)$ and extend it across weight $w$ as $(d+w,h+1)$. For positive edge weights, every extension strictly increases its first component, so Dijkstra's algorithm settles these pairs correctly when its priority queue and relaxation both compare the full pair. Equal total weight then selects the smaller hop count; arbitrary vertex-name tie-breaking does not express this objective.

Process a digit-array integer representation from low to high positions. The integer division supplies $q=\lfloor a_i/b\rfloor$ and $0\le r<b$. Replacing the current digit by $r$ and adding $q$ to the next position preserves the represented value. Append zero storage when a positive carry leaves the allocated array. The processed prefix remains normalized and the value stays fixed.

This also works with signed temporary digits when the represented value is nonnegative. A negative carry cannot leave the final allocated position: the normalized lower digits represent a value less than that position's next radix power, which could not offset that negative carry. Positive carries eventually stop beyond the value's finite radix expansion. Thus the usual nonnegative-digit algorithm admits this useful extension without interpreting truncated division as floor division.

When the radix is $10^k$, print the highest nonzero normalized digit without leading zeros, then every lower digit as exactly $k$ decimal characters. Internal zero blocks must be retained. With base $1000$, the low-first digits $[4,0,12]$ print as 12 followed by 000 followed by 004, giving 12000004. An all-zero representation prints a single zero, independent of its allocated length.

A deterministic finite automaton whose automaton state is a pair of automaton states of two given deterministic finite automata, updated componentwise by each input symbol. Choosing accepting pairs by a Boolean function on component acceptance realizes intersection, union, or set difference of the recognized regular languages.

In the Leibniz formula for a determinant of a block diagonal matrix, a term is zero unless its permutation maps every block's row indices onto that same block's column indices. The surviving permutations are precisely independent permutations within the blocks; their signs and products of entries factor over the blocks. Summing therefore gives the product of the individual determinants. Applied to $tI-T$, this also proves that the characteristic polynomial of a block diagonal linear operator is the product of the characteristic polynomials of its blocks.

Foundation implies that no descending membership sequence exists. Conversely, with dependent choice, failure of foundation gives a nonempty set in which every element contains another element of that set. Successively selecting such members produces a descending membership sequence. Thus the two conditions are equivalent with dependent choice; full choice is sufficient.

For a solvable group, its derived length is the least nonnegative integer $r$ at which the derived series reaches the trivial group, with $G^{(0)}=G$ and $G^{(r+1)}$ the commutator subgroup of $G^{(r)}$. A nontrivial abelian group has derived length one; a nonabelian group with nontrivial abelian commutator subgroup has derived length two.

For $t$ in a compact positive interval $[a,b]$, split the integration region at one. Near zero the smallest exponent gives the displayed bound, whose integral is $1/a^2$. Above one the largest exponent gives the bound, integrable by exponential decay. A single integrable function therefore dominates all parameter derivatives locally. Dominated differentiation under the integral sign gives ${\Gamma }^{\prime }(t)={\int }_0^{\infty }{e}^{-x}{x}^{t-1}\log xdx$. The same argument with higher powers of $|\log x|$ gives all higher derivatives.

The right derivative is the displayed one-sided limit, when it exists as a finite number. It describes differentiation at the left endpoint of a closed interval without requiring the function to be defined to the left. An ordinary derivative at an interior point requires matching left and right derivatives. For example, the absolute value function has right derivative one at zero and left derivative minus one, so it has no ordinary derivative there.

A purely imaginary root of $\lambda =-e^{-\lambda \tau }$ has frequency one and first appears at $\tau =\pi /2$. Roots cross with positive real derivative $1/(1+{\tau }_c^2)$; all roots lie in the left half-plane for smaller positive delay, since a hypothetical root $a+ib$ with $a\ge 0$ would require both $|b|\tau \ge \pi /2$ and $|b|\le 1$. A small-amplitude expansion $u=1+a\cos (\omega t)+a^2[\cos (2\omega t)/5+\sin (2\omega t)/10]+\cdots$ yields $\omega =1-3a^2/20+\cdots$ and $\tau =\pi /2+(3\pi -2)a^2/40+\cdots$, verifying a nondegenerate Hopf bifurcation. The limiting period is $2\pi =4{\tau }_c$.

For a twice continuously differentiable deformation map, each row of its cofactor matrix has zero divergence in reference coordinates. In three dimensions write $(\mathrm{cof}Dy{)}_{iA}=\frac{1}{2}{\epsilon }_{ijk}{\epsilon }_{ABC}{y}_{j,B}{y}_{k,C}$. Applying ${\partial }_A$ gives terms in which a symmetric pair of second partial derivatives contracts with an antisymmetric Levi-Civita symbol, so each term vanishes. This identity is the cancellation that makes the Piola-Kirchhoff stress tensor transform force balance correctly between configurations.

Two constant deformation gradients are rank-one connected when their difference is an outer product $a\otimes N$. For a nonzero jump its matrix rank is one; equality $A=B$ is the trivial compatible case. The vector $N$ specifies a reference interface normal, and $a$ is the jump amplitude. The Hadamard jump condition identifies this algebraic condition with continuity of an affine deformation across a planar interface. The normal can be normalized by rescaling $a$ without changing the product.

In the usual first-order Hilbert system, logical implication discharge requires the appropriate generalization restriction: a variable generalized in the derivation must not be free in the assumption being discharged. Equivalently use a proof system whose generalization rule already forbids variables free in open assumptions. One can first discharge a temporary assumption and then generalize if the quantified variable is not free in the remaining assumptions. Ignoring this side condition would allow deriving universal conclusions from a single arbitrary instance as an open hypothesis.

Conditional modus ponens derives $A\to C$ from $A\to B$ and $A\to (B\to C)$. It is the step which translates a modus ponens rule inference when discharging a common assumption. In the case-elimination Hilbert calculus, the third scheme first gives $A\to (\neg B\to C)$; the second scheme combines the two conditionals to give $A\to C$.