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A real Lévy process with continuous paths is Brownian motion with deterministic linear drift: $X_t=bt+\sigma W_t$. Continuity forces the jump measure in the Lévy–Khintchine formula to vanish, leaving characteristic function $\exp (t(ibu-{\sigma }^2{u}^2/2))$. If its law is also invariant under $X_t\mapsto {\lambda }^{-1}{X}_{{\lambda }^{2}t}$ for every $\lambda >0$, then $b=0$. The zero-variance case is included.

For a compact hyperbolic surface, the Selberg trace formula equates a sum over Laplace-Beltrami operator eigenvalues to an area term and a sum over primitive closed geodesics and their iterates. It proves that the Laplacian spectrum and the unmarked length spectrum, with multiplicities, determine each other.

A nonzero Lefschetz coincidence number forces a coincidence of the two maps. If their paired map misses the diagonal, the diagonal Thom class restricts to zero on its image and its pullback vanishes, making the coincidence number zero. The argument works for continuous maps and requires neither isolated coincidences nor transversality.

For $f:M\to N$ between closed connected oriented manifolds of equal dimension, cyclic invariance of trace and $f^{!}{f}^{\ast }=(\mathrm{deg}f)\mathrm{id}$ give the formula. Thus if both the degree and the target Euler characteristic are nonzero, $f$ coincides somewhere with every map homotopic to it, by the Lefschetz coincidence theorem. For target ${\mathbb{CP}}^{2k}$ the Euler characteristic is $2k+1$.

If $1\le p,r\le \infty$, $0\le \theta \le 1$ and $1/q=\theta /p+(1-\theta )/r$, then the displayed bound follows by applying the Holder inequality to $|u{|}^{\theta q}|u{|}^{(1-\theta )q}$. Interpret the endpoint cases using the essential supremum. In particular convergence in Lp space and boundedness in a larger-exponent Lp space imply convergence at intermediate exponents.

On a measure space of finite measure, a bounded measurable function satisfies the displayed limit, where the right side is the essential supremum of its absolute value. The upper estimate is $\Vert f{\Vert }_p\le \mu (X{)}^{1/p}\Vert f{\Vert }_{\infty }$. For every $a<\Vert f{\Vert }_{\infty }$, the set where $|f|>a$ has positive measure, giving the reverse limiting estimate. For a continuous function on an open Euclidean domain, the essential supremum equals the ordinary supremum. Finiteness of the measure is part of this statement.

For $f\in L^1(\mathbb{T},m)$ with $m$ normalized Lebesgue measure, translation on the circle group is continuous in the L1 norm: $\Vert f(\cdot +t)-f{\Vert }_1\to 0$ as $t\to 0$ modulo one. Approximate $f$ in the L1 norm by a continuous function, use its uniform continuity, and use translation invariance to preserve the approximation error. In particular, $m(B△(B-t))\to 0$ for every measurable $B$.

For $1\le p<\infty$ and $f\in L^p({\mathbb{R}}^d)$, translations converge to $f$ in the Lp space norm. First prove this for continuous compactly supported functions, using uniform continuity and a common bounded support. Their density in Lp space, together with the norm-preserving translation map and the triangle inequality, proves the assertion for arbitrary $f$. It implies convergence of averages over shrinking intervals, a basic approximate identity argument. The corresponding assertion fails for general $L^{\infty }$ functions, such as an indicator function with a jump.

A measurable function belongs to $L_{\mathrm{loc}}^2(U)$ when its squared absolute value has finite integral on every compact subset of $U$. Functions are identified if they agree almost everywhere. This is a local integrability condition, not a specified global Hilbert space norm. For example, $x$ belongs to $L_{\mathrm{loc}}^2(\mathbb{R})$ but its mean-square averages on expanding intervals diverge.

The integral average of an integrable function over a measurable set $E$ of finite positive measure is $⟨v{⟩}_E=|E{|}^{-1}{\int }_{E}v$. Normalization removes the volume factor when comparing estimates on balls of different radii.

A real or complex measurable function is integrable for a measure $\mu$ when its absolute value has finite integral. For a random variable, this is integrability with respect to its probability measure, and makes its expectation finite.

For a bounded linear operator $K:U\to V$ between Hilbert spaces, a least-squares solution for $f$ exists exactly when $P_{\overline{\mathcal{R}(K)}}f\in \mathcal{R}(K)$, equivalently $f\in \mathcal{R}(K)\oplus \mathcal{R}(K{)}^{\perp }$. Indeed the orthogonal projection onto the closure uniquely minimizes distance to that closed space, and it can be realized by $Ku$ exactly under this condition. Thus all data admit least-squares solutions exactly when the range is closed.

If, for every $\eta >0$, the smooth numbers up to $T$ with all prime factors at most $T^{\eta }$ have density at least $\kappa (\eta )>0$, then for fixed $\epsilon >0$ there are only $O_{\epsilon }(1)$ primes $p\le N$ with $n(p)>N^{\epsilon }$. Apply the variance form of the large sieve to the indicator function of the $N^{\epsilon }$-smooth numbers up to $N^2$. For every exceptional prime, all these numbers are nonzero quadratic residues, so half its nonzero residue classes are empty. Their variance contribution is at least one third of the square of the total count. The large sieve bounds the sum of contributions by a constant times $N^2$ times that count. The density assumption with $\eta =\epsilon /2$ finishes the argument.

The Cauchy-Schwarz inequality gives $\Vert P-Q{\Vert }_{\mathrm{TV}}=\frac{1}{2}{\mathbb{E}}_Q|dP/dQ-1|\le \frac{1}{2}\sqrt{{\chi }^2(P\Vert Q)}$. Every statistical hypothesis testing rule has sum of its Type I error and Type II error at least $1-\Vert P-Q{\Vert }_{\mathrm{TV}}$, proving the displayed bound. A mixture model of alternatives yields a lower bound on the worst alternative error because a maximum dominates an average.

For a subterminal object classified by $u$, the open local operator is $p\mapsto (u\Rightarrow p)$. Its dense monos are exactly those whose image contains the pullback of that subterminal. It is complementary to the corresponding closed local operator.

For a subterminal object classified by $u$, the closed local operator is $p\mapsto u\vee p$. Its closed monos contain the pullback of that subterminal, hence are exactly the monos dense for the corresponding open local operator.

For a well-mixed lava lake of depth $H$ and a thin conductive crust of thickness $a$, put $\Delta T=T_m-T_s$, and use the convective flux $K(\overline{T}-T_m)$ with constant heat transfer coefficient $K>0$. The Stefan condition and liquid heat balance give, to leading order in $a/H$,Here latent heat $L_f$ is per unit mass and $k=\rho c_p\kappa$. Crust heating and changes of liquid depth are neglected. The linear conductive profile uses a quasi-steady approximation; thinness alone does not justify neglecting crust thermal storage. During initial growth, a small Stefan number supplies a sufficient rapid-adjustment regime. Define $h=a/H$, $\theta =(\overline{T}-T_m)/\Delta T$, and $\tau =\kappa t/H^2$. Scaling by the thermal diffusion time $H^2/\kappa$ and $\Delta T$ gives $h^{\prime }=S(1/h-B\theta )$ and ${\theta }^{\prime }=R-B\theta$, where $B=KH/k$, $R=Q{H}^2/(c_p\kappa \Delta T)$, and $S=c_p\Delta T/L_f$ is the Stefan number. Thus $\theta =R/B+({\theta }_0-R/B)e^{-B\tau }$ exactly. The reduction applies while $a/H\ll 1$, the conductive profile is approximately linear, and the thermal convection closure remains valid; it does not model complete freezing of a deep lake.

Given a finite set $F\subset {\mathbb{Z}}^r$ of exponent vectors, there is $w\in {\mathbb{Z}}^r$ for which the integers $w\cdot m$, $m\in F$, are pairwise distinct. For $r=2$, take $w=(1,N)$ with $N$ larger than every first-coordinate difference in $F$. In higher dimensions take $w=(1,N,\dots ,N^{r-1})$ with $N$ sufficiently large. This proves that a nonzero Laurent polynomial can be detected by a substitution $X_i=T^{w_i}$ without cancellation of its distinct terms.

Every unital $k$-subalgebra $A\subseteq k[T,T^{-1}]$ is a Noetherian ring. If $A\subseteq k[T]$ or $k[T^{-1}]$, a nonconstant element $f\in A$ makes that ambient polynomial ring a finitely generated module over $k[f]$. Otherwise $A$ contains an $f$ with both positive and negative exponents, and both $T$ and $T^{-1}$ are integral elements over $k[f]$. In either case $A$ is a submodule of a finite module over the Noetherian ring $k[f]$, and every ideal of $A$ is finitely generated over $k[f]$, hence over $A$.

A nonzero quotient of an $r$-variable Laurent polynomial ring is finite over an embedded Laurent polynomial ring in $m\le r$ variables. Given a nonzero Laurent relation, choose $N$ so that weights $(N,N^2,\dots ,N^{r-1},1)$ separate its finite exponent support. The displayed invertible monomial substitution makes its extreme $Y_r$ coefficients units in the remaining Laurent variables. Normalize the relation to a monic polynomial with unit constant term; both $Y_r$ and $Y_r^{-1}$ are integral over the image of the $(r-1)$-variable subring. Induction and transitivity of finite extensions prove the assertion, over finite fields as well as infinite fields. An infinite-dimensional quotient has $m\ge 1$.