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On a closed manifold with a Riemannian metric, the minimum of a smooth heat solution cannot fall below its initial minimum. At a spatial minimum, $\Delta u\le 0$ for the positive Laplace-Beltrami operator. Adding a small increasing function of time and considering the first crossing gives the assertion. Applying the result to nonnegative initial functions shows that the Riemannian heat kernel is nonnegative.

A Fourier mode has multiplier $e^{D{k}^{2}t}$, amplifying arbitrarily small high-frequency data. For a fixed positive time $t_0$, initial modes $e^{-D{n}^2{t}_0/2}\sin (nx)$ tend to zero in $L^2$ while their values at $t_0$ grow without bound. This violates continuous dependence on initial data and gives an ill-posed problem in ordinary unweighted spaces. An analytic-data restriction or physical short-scale regularization changes that conclusion.

Classical heat equation solutions with identical continuous initial data and a uniform Gaussian spatial growth bound on each finite time interval are unique. For the zero-data difference, choose $b>a$ and compare on a short slab with $\epsilon {\Phi }_b$, where ${\Phi }_b=(1-4bt{)}^{-n/2}\exp (b|x{|}^2/(1-4bt))$ solves the equation for $t<1/(4b)$. Its faster spatial growth controls the lateral boundary of large cylinders. The heat equation maximum principle bounds both signs of the difference by $\epsilon {\Phi }_b$ inside. Increasing the cylinder radius and then sending $\epsilon$ to zero proves uniqueness on that slab; repeating slabs covers the finite interval.

At ordered possible failure times $t_j$, the discrete hazard is $h_j=\mathbb{P}(T=t_j\mid T>t_{j-1})$. Its survival function satisfies $S(t_j)=S(t_{j-1})(1-h_j)$, and hence $S(t)={\prod }_{t_j\le t}(1-h_j)$. Estimating $h_j$ by the observed number of failures divided by the risk set size under independent censoring yields the Kaplan–Meier estimator. Unlike a continuous-time hazard rate, a discrete hazard is a probability between zero and one.

The core disk of a $k$-handle attachment $D^k\times D^{n-k}$ is $D^k\times \{0\}$. Together with its cap in the preceding stage, it often represents a homology class; for a surgery trace, capped two-handle cores give the basis of its surgery linking matrix.

On binary words of length $n$, normalized Hamming distance is the fraction of differing coordinates. Unlike unscaled Hamming distance, it makes the Hamming cubes a Lévy family under uniform probability measures.

A machine's halting times admit a total computable function as a bound on all inputs in its domain if and only if that domain is a computable set. A bound decides the domain by finite simulation; conversely, a domain decider allows returning zero outside the domain and the exactly simulated halting time inside it. In particular every machine computing a total computable function has an exact computable running-time function, while nontotal machines may or may not have total bounds.

The usual half-level NS claim that all physical vector polarizations have nonnegative norm for $a\le 1/2$ assumes nonzero particle momentum. At zero momentum and limiting intercept, all positive-mode constraints and transversality allow a timelike polarization with negative norm, and $A\sim A+\kappa p$ removes nothing. Excluding this exceptional momentum restores the ordinary null-vector argument; retaining it makes the unrestricted elementary criterion strict $a<1/2$.

Both the ensemble position density transported by the guidance equation and the squared wave amplitude satisfy the same probability continuity equation. Subject to existence and uniqueness of that transport, an initial Born rule distribution stays a Born rule distribution. This is a preservation statement, not a derivation that every initial ensemble is already in equilibrium.

If the order of a finite abelian group $\Delta$ is invertible in the coefficient ring and all its character values belong to that ring, its character idempotents are orthogonal and sum to one. They decompose any module into character components. In a cyclotomic tower with odd $p$, the Teichmüller character provides these components integrally over ${\mathbb{Z}}_p$.

In a cycle square, each triple of consecutive vertices is a clique. A proper three-colouring therefore repeats with period three and closes exactly when the length is divisible by three. For all other cycle lengths at least four colours are needed; the five-cycle square is a complete five-vertex graph.

If every graph component has at most $(k+1)n/[k(2k+1)]$ vertices, assigning whole graph components to two sides gives an empty graph cut with the displayed bound. To prove it, maximize the smaller side's weight $S\le n/2$. If $S<kn/(2k+1)$, every component on the larger side has weight at least $n-2S$, because moving a smaller one would improve the smaller side. There are at least $k+1$ such components, since their total exceeds $k$ times the permitted maximum weight. Thus $n-S\ge (k+1)(n-2S)$, a contradiction. The proof also works for arbitrary positive weights.

A family of positively many $s$-cliques forces a logarithmic balanced graph blow-up, which can also contain a matching of that many members of the family. To induct on $s$, prune faces with few extensions, apply induction to the remaining $(s-1)$-faces, and use a matching of those faces as one side of a bipartite incidence graph. The common neighbourhood from bipartite density estimate selects logarithmically many disjoint faces with polynomially many common extension vertices.

Normalize the conjugated cube average by the number of integer cubes whose vertices all lie in the interval. Equivalently, extend $f$ by zero to a cyclic group of order greater than $8N$ and divide its Gowers U3 norm by that of the interval's indicator function. A quadratic phase then has norm one. Omitting the denominator gives a different but uniformly equivalent interval normalization.

For functions bounded by one on a cyclic group of prime order greater than three, three applications of Cauchy-Schwarz bound the four-term arithmetic progression average by the smallest Gowers U3 norm of its factors. Zero extension yields the corresponding interval bound with an absolute factor. This makes Gowers U3 norm a measure of uniformity relevant to four-term arithmetic progression counts.

For a nonnegative all-integer row $x_B+{\sum }_j{a}_j{x}_j=b$ with $a_j>0$, the Gomory fractional cut requires ${\sum }_j\{a_j\}{x}_j\ge \{b\}$. The original row gives ${\sum }_j{a}_j{x}_j\le b$. If a constant $\kappa$ satisfies $\{a_j\}\le \kappa a_j$ for every $j$ and $\kappa b<\{b\}$, these inequalities contradict the cut. A single row therefore certifies integer infeasibility. Slacks may be used as integer variables when the original constraint coefficients and right sides are integral and the decision variables are integral.

Curvature depresses the equilibrium melting temperature of solid convex into liquid. Here $\mathcal{K}$ is the sum of principal curvatures, equal to twice a common convention for mean curvature, and $T_m$ is absolute temperature. The coefficient uses solid-liquid surface energy, mass density and latent heat. This stabilizes fine corrugations of a solidification front.

Fix $k\ge 1$, ${\gamma }_0>1/(k+1)$, and ${\gamma }_0\le \gamma \le 1-\omega (n)/\log n$ with $\omega (n)\to \infty$. A binomial random graph has one giant component of the displayed order and all other graph components are tree components of orders at most $k$, with high probability. The expected value and variance of the isolated vertex count concentrate it around $n^{1-\gamma }\to \infty$. A Cayley formula upper bound for connected sets excludes component sizes $k+1$ through a small fixed fraction of $n$; the empty-graph cut bound excludes the remaining sizes up to $n/2$. An extra-edge count excludes cyclic small graph components. The expected value of the number of vertices in small nonisolated tree components is $O(nd{e}^{-2d})=o(n{e}^{-d})$, where $d=np$. The Markov inequality completes the count of vertices outside the unique large graph component.

Suppose $\epsilon =o(1)$ and $\epsilon \ge n^{-1/6}$. The breadth-first exploration of a binomial random graph has deterministic drift $F(t)=\epsilon t-t^2/(2n)+O({\epsilon }^{3}n+\epsilon )$ up to $3\epsilon n$ steps. Its weighted martingale has variance $O(\epsilon n)$. For $h=\sqrt{\epsilon }+(n{\epsilon }^3{)}^{-1/8}$, the Doob L2 maximal inequality makes its maximum smaller than $h{\epsilon }^{2}n/6$ with high probability. Then $A_t>0$ throughout $[h\epsilon n,(2-h)\epsilon n]$, giving a graph component of order $(2-o(1))\epsilon n$.

For the upper expectation bound, let $K=\lceil {\epsilon }^{-3}\rceil$. The dominating binomial branching process has branching survival probability $(2+o(1))\epsilon$ by the binomial branching survival correction. Its branching process conditioned on extinction has mean $1-\epsilon +O({\epsilon }^2+\epsilon /n)$ and total-progeny expected value $O(1/\epsilon )$. Therefore $\mathbb{E}{N}_{\ge K}\le n\rho +O(n/(\epsilon K))$. Since $L_1\le K+N_{\ge K}$ and $K=o(\epsilon n)$, this yields the matching upper bound. Controlling rare large components is necessary to conclude an expected value asymptotic from a typical-size statement.

A geometric morphism is surjective when its inverse-image functor is faithful, equivalently conservative for geometric inverse images. Reflection of isomorphisms of subobjects already proves faithfulness by applying the inverse image to equalizers of parallel arrows. The surjection-embedding factorization of a geometric morphism separates this property from a fully faithful direct image.