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Equal variables can replace selected free occurrences in a first-order formula if the inserted variable is not captured. Starting with atomic equality replacement, derive symmetry, then prove both replacement directions together by structural induction. Negation uses the reverse induction implication; implication uses both directions; a universal quantifier is handled by instantiation, replacement and generalization with a variable fresh for the equality assumptions. Bound occurrences are not replaced as free ones.

The number of load inputs driven by one output of a logic gate. A large fan-out increases the capacitive load and can increase propagation delay; the allowable number also depends on the logic technology and input characteristics.

A two-input logic gate whose output is one exactly when its two input bits agree. Its Boolean function is $(a\wedge b)\vee (\neg a\wedge \neg b)$, equivalently $(a\vee \neg b)\wedge (\neg a\vee b)$. This follows by considering the two cases in which inputs are equal. It implements logical equivalence of the input truth values.

A deterministic Turing machine with read-only input, $O(\log (n+2))$ work space, and successive write-only output whose contents it cannot inspect. A transducer for a total function halts on every input. It has a polynomial number of internal Turing machine configurations, so a halting run emits only a polynomial number of symbols. The unreadable output position does not affect subsequent transitions.

For canonical positive binary numeral system inputs, a logspace transducer can emit the canonical representation of their sum most significant digit first. Count the operand lengths, compute the final arithmetic carry to determine output length, and then compute each requested result digit by restarting the least-significant-first column recurrence. Each pass uses only a fixed number of $O(\log n)$-bit indices and one arithmetic carry. The binary-addition carry invariant proves correctness. Logspace composition by recomputation consequently permits any L property of the resulting digit string, including being a binary palindrome, to be tested without storing the sum.

For the strictly convex polynomial $f(x)=x^2/2+x^4/4$, applying Newton root-finding iteration to $f^{\prime }(x)=x+x^3$ givesThe unique minimum is at zero. For any sufficiently small nonzero initial point, the errors satisfy $|x_{k+1}|/|x_k{|}^3\to 2$, so the order is cubic, not merely quadratic. This provides a simple counterexample to treating the usual quadratic convergence theorem as an upper bound on possible orders. In general, the quadratic coefficient for Newton optimization near a nondegenerate stationary point is $f^{(3)}(x_{\ast })/(2f^{\prime \prime }(x_{\ast }))$; when it vanishes, a higher order is possible.

The helicoid $H(s,v)=(\sinh s\cos v,\sinh s\sin v,v)$ and catenoid $C(s,v)=(\cosh s\cos v,\cosh s\sin v,s)$ have the same first fundamental form ${\cosh }^{2}s(d{s}^2+d{v}^2)$. Therefore matching equal parameter pairs gives a local isometry. The result is local: a full turn changes the height on the helicoid while the angular coordinate on the catenoid repeats. Equal intrinsic lengths do not require the same embedding into Euclidean space.

Jeff Haby authored forecasting notes that connect the formation of surface condensation and frost with surface cooling, humidity and wind. The notes distinguish ice deposited from vapour from frozen dew.^[ref]

A surface exposed to a clear night sky can become colder than the air measured above it. Forecasting a freezing surface therefore requires its energy balance and moisture conditions, not only a standard-height minimum air temperature.

A linear storage law gives $\kappa Q_{\tau }+Q_x=M(\tau )$ with zero upstream water flux. Integration along characteristic curves yields the displayed boundary-generated solution. For $M=1-\delta \cos \tau$, $J=\tau -\delta \sin \tau$ and $J^{-1}(\theta )=\theta +\delta \sin \theta +O({\delta }^2)$. A fixed-flux contour therefore obeysIts extrema have phase $\tau =\kappa Q/2(\mathrm{mod}\pi )$ when the first-order amplitude is nonzero. The phase lag arises from water storage and downstream propagation, even if the ice thickness is approximately frozen during one fast seasonal cycle.

The instant used to identify an operation's abstract effect in a proof of linearizability. Loading the published pointer can be the linearization point of an immutable snapshot read, with suitable ordering and lifetime guarantees.

In $n$ independent trials with category probabilities $p_1,\dots ,p_m$, let $K_n$ count the distinct categories observed. Write it as the sum of indicator random variables for the categories' appearances. Category $j$ is absent with probability $(1-p_j{)}^n$, so linearity of expectation gives the displayed formula. The appearance indicators need not be independent. For equally likely categories the expression becomes $m(1-(1-1/m{)}^n)$; uneven probabilities can substantially delay observation of rare categories.

For $a_t+(V(s)a{)}_s=0$ with $V>0$, the flux perturbation $b=Va$ solves $b_t+V{b}_s=0$. The travel coordinate transforms it into unit-speed transport. Thus the flux perturbation is constant on characteristics while area amplitude changes as $1/V$. In a supplied power-law river, $A_{\mathrm{eq}}^{\prime }=M/V$, giving travel time $[A_{\mathrm{eq}}(l)-A_{\mathrm{eq}}(0)]/M$. At a zero-speed head, this time can remain integrable even though the amplitude formula is not uniform.

For nodes $a=x_0<\cdots <x_n=b$ and prescribed real node values, let $s$ be the linear spline joining them. Among all absolutely continuous functions $u$ with the same node values and square-integrable derivative, $s$ uniquely minimizes the Dirichlet energy ${\int }_a^b|u^{\prime }{|}^2$.

On each node interval, $s^{\prime }$ is constant and $u-s$ vanishes at both endpoints. The fundamental theorem of calculus therefore gives $\int s^{\prime }(u^{\prime }-s^{\prime })=s^{\prime }[(u-s)(x_{i+1})-(u-s)(x_i)]=0$. Expanding the square and summing proves the displayed energy decomposition. Its final term is nonnegative; equality forces $u^{\prime }=s^{\prime }$ almost everywhere and the endpoint values fix the additive constant, proving uniqueness. In particular, the minimum is ${\sum }_i(u(x_{i+1})-u(x_i){)}^2/(x_{i+1}-x_i)$. If all interior node values are zero and there is at least one interior node, only the two endpoint intervals contribute to this minimum.

For $y^{\prime }+ay=g$ with real $a>0$ and $2\pi$-periodic forcing, comparing Fourier coefficients gives $(a+in){\hat{y}}_n={\hat{g}}_n$. There is no vanishing denominator, including the constant mode. With real cosine and sine coefficients $a_n,b_n$ for $g$ and $A_n,B_n$ for $y$, the equivalent equations giveWhenever termwise differentiation is justified, these formulas construct a periodic particular solution. For a forcing series with coefficients $O(n^{-2})$, the differentiated response coefficients are also summable, which supplies such a justification by uniform convergence. All other solutions add the homogeneous solution $C{e}^{-ax}$. Since this exponential cannot be periodic unless $C=0$, the periodic solution is unique. This is also a simple example of how a linear equation changes the amplitude and phase of each harmonic independently.

A fundamental system for a homogeneous linear ordinary differential equation of order $n$ on an interval is a basis of its vector space of homogeneous solutions. Every solution has a unique expression ${\sum }_{j=1}^n{c}_j{y}_j$ with constant coefficients. Under the usual continuity assumptions and a nonvanishing leading coefficient, a nonzero Wronskian at one point proves that $n$ solutions form such a system: their initial-value vectors are independent and span all possible initial data, and the Picard-Lindelöf theorem supplies uniqueness. For a first-order vector system, the corresponding columns form a fundamental matrix.

A characteristic exponent is a value $\lambda$ for which an exponential mode $e^{\lambda t}$ solves the homogeneous constant-coefficient linear equation or system. Repeated exponents can give polynomial factors multiplying that exponential.

An invertible linear transformation replaces one system of coordinates by another. If $x=Pu$ in a linear differential equation $\dot{x}=Ax$, then $\dot{u}=P^{-1}APu$. The similar matrices $A$ and $P^{-1}AP$ have the same characteristic polynomial and eigenvalues, although their coordinate descriptions of the eigenvectors differ. Such a change can turn a planar linear system with eigenvalues $\alpha \pm i\beta$ into $\dot{u}=\alpha u-\beta v$, $\dot{v}=\beta u+\alpha v$.

A rational-linear map is a linear map when its domain and codomain are regarded as vector spaces over the rational numbers. An additive map on a rational vector subspace of the real numbers is automatically rational-linear: integer homogeneity follows by repeated addition, and division by a positive integer gives rational homogeneity of an additive function.

A real-linear map is a linear map over the real numbers. For a map from the real line to itself, real linearity means multiplication by a fixed real number, its value at one. Rational-linear maps need not be real-linear without an additional regularity condition.

In the one-sorted vector-space language naming each scalar multiplication, a first-order formula reduces by quantifier elimination to finitely many linear equations. An independent pair satisfies only the identically zero equations among them. A dependent pair $(x,ax)$ with nonzero $x$ satisfies the same listed equations if the scalar $a$ avoids their finitely many forbidden ratios, which is possible over an infinite field. Thus no formula defines pair independence. The complete independent-pair type is still consistent in an infinite-dimensional model; it is nonisolated.