For each fixed integer $q\ge 2$, every sufficiently high-degree finite regular graph has an assignment of $q$ colors such that every vertex has a neighbour whose color is its successor modulo $q$. This assignment need not be a proper vertex coloring. If the degree is $r$, independent uniform vertex colors make a given vertex fail with probability $(1-1/q{)}^r$. That event depends on its closed neighbourhood, and has at most $r^2$ dependencies, since only centres within distance two matter. The symmetric Lovász local lemma therefore gives such an assignment whenever $e(r^2+1)(1-1/q{)}^r\le 1$, which holds for all sufficiently large $r$.

Choose one successor neighbour from each vertex. The resulting directed graph has outdegree one and hence contains a directed cycle. Its color increases once modulo $q$ on each step, so its length is divisible by $q$. Thus a nontrivial cycle of length a multiple of $q$ is obtained when $q\ge 3$; for $q=2$ a directed two-cycle may simply traverse one undirected edge twice, so this last cycle inference needs a separate argument if an undirected simple cycle is required.

For a transverse monochromatic electromagnetic wave with angular frequency $\omega \ne 0$, ${\partial }_t^2\mathbf{E}=-{\omega }^2\mathbf{E}$. The displayed electric potential and magnetic vector potential therefore give $-\nabla \Phi -{\partial }_t\mathbf{A}=\mathbf{E}$. Transversality gives $\nabla \cdot \mathbf{A}=0$, so the Lorenz gauge holds. For $\mathbf{E}=E_0(\cos \theta ,\sin \theta ,0)$ and $\theta =\omega t-Kz$, taking a curl yields $\mathbf{B}=(K/\omega )\hat{\mathbf{z}}\times \mathbf{E}$. The vacuum Maxwell equations require $K^2={\omega }^2/c^2$. The field has circular polarization, with $|\mathbf{B}|=|\mathbf{E}|/c$ and perpendicular electric and magnetic directions.

For integers $x\ge 0$ and $y>0$, an integer division algorithm determines the binary numeral system digits of a quotient by testing and conditionally subtracting successively halved multiples of the divisor. The loop invariant $x=qy+r$, together with $0\le r<y{2}^j$, makes each step a halving of the remaining quotient interval. Start with $q=0,r=x$ and $j$ large enough that $x<y{2}^j$. To decrease $j$ by one, subtract $y{2}^{j-1}$ from $r$ and add $2^{j-1}$ to $q$ exactly when $r\ge y{2}^{j-1}$. Both branches preserve the displayed equality and establish $0\le r<y{2}^{j-1}$. When $j=0$, the quotient and remainder conditions hold.

Lengths of formulas logically equivalent to a fixed propositional formula form a nonempty subset of the natural numbers, so a minimum exists. With variables counted as atomic symbols and no truth constants, $(p\to p)$ and $(p\leftrightarrow p)$ are distinct shortest tautologies. A formula with no binary connective is a variable preceded by negations and cannot be a tautology. Thus minimization need not give a canonical representative.

When an equality-like symbol is treated as an ordinary binary predicate, no first-order theory consisting of assertions valid for actual identity can force it to be identity. Given a genuine-equality structure on $A$, replace the domain by $A\times \{0,1\}$ and pull all relations back along the projection $\pi$. Lift functions so that $\pi$ commutes with them. Induction on first-order formulas preserves satisfaction because $\pi$ is surjective. Yet the two copies of each point satisfy the equality-like predicate. Thus actual identity must be part of the semantics, or the resulting equivalence classes must be quotiented out.

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.