For a finite set of primes choose $0<g(p)<1$, put $k(p)=g(p)/(1-g(p))$, and let $W={\prod }_p(1+k(p))$. Give each squarefree integer $d$ on these primes probability $k(d)/W$. Inclusion of each prime is represented by a Bernoulli random variable with parameter $g(p)$; these are independent random variables, so $\mathbb{E}[\log d]={\sum }_{p}g(p)\log p$. If this is at most $\theta \log L$ with $0<\theta <1$, the Markov inequality givesThis turns a full Euler product into a lower bound for the truncated normalizing sum of a Selberg sieve.

Writing $q=u+iv$ in the free Schrodinger equation gives $u_t=-v_{xx}$ and $v_t=u_{xx}$, hence $u_{tt}+u_{xxxx}=0$. To encode initial velocity $u_1$, choose the decaying primitive $v_0(x)=-{\int }_x^{\infty }(s-x)u_1(s)ds$. A prescribed endpoint curvature becomes the time derivative of the imaginary boundary trace. This turns the normalized Euler-Bernoulli beam equation into a complex second-order boundary problem.

For a real vector $v$, the square of its Euclidean norm is $\Vert v{\Vert }_2^2=v\cdot v={\sum }_j{v}_j^2$. Unlike the norm itself it is a quadratic function, so expanding sums directly exposes diagonal and cross terms. This is useful for the mean-square displacement of an isotropic planar random walk.

If a strongly additive arithmetic function $g$ is supported on primes up to $y$, its $j$th uniform central moment differs from the independent Bernoulli random variables model with probabilities $1/p$ by $O(j{y}^j({\sum }_{p\le y}|g(p)|{)}^j/N)$. Expansion of divisibility indicator functions reduces the comparison to $\lfloor N/d\rfloor /N=1/d+O(1/N)$.

For a group action on $X$ and a contractible free $G$-space $EG$, the diagonal orbit space $EG{\times }_{G}X$ maps to the classifying space $BG$ with fibre $X$. It defines equivariant cohomology. When $X\to X/G$ is a principal $G$-bundle, the map $EG{\times }_{G}X\to X/G$ has contractible fibre $EG$ and gives a homotopy equivalence for spaces of CW type. Freeness without the bundle hypotheses should not be substituted for this assertion for arbitrary topological groups.

A perfect fluid in general relativity stellar model in which pressure depends only on energy density, $p=p(\rho )$. A positive derivative makes this relation locally invertible and turns outward-decreasing pressure into outward-decreasing density. This general relation need not have the constant ratio $p/\rho$ used for a constant cosmological fluid parameter.

The completion ${\mathcal{E}}_{\Lambda }(s)={\pi }^{-s}\Gamma (s)E_{\Lambda }(s)$ satisfies ${\mathcal{E}}_{\Lambda }(s)=m(\Lambda {)}^{-1}{\mathcal{E}}_{{\Lambda }^{\vee }}(n/2-s)$. Its simple poles at zero and $n/2$ have residues $-1$ and $m(\Lambda {)}^{-1}$. The pole-subtracted theta integral for an Epstein zeta function gives both the continuation and this dual lattice symmetry.

For finite-alphabet independent and identically distributed random variables, fixed $0\le ϵ<1$ and $\delta >0$, every epsilon-sufficient set $S_n$ of length-$n$ words satisfies $|S_n|\ge (1-ϵ)2^{n(H-\delta )}/2$ for sufficiently large $n$. Intersect $S_n$ with a typical set whose excluded mass is at most $(1-ϵ)/2$. The intersection retains that much probability and each word has probability at most $2^{-n(H-\delta )}$.

In a Cartesian frame rotating with an epicyclic guiding center, a small planar oscillation obeys $x=X\cos (\kappa t+\chi )$ and $y=-(2\Omega X/\kappa )\sin (\kappa t+\chi )$. The ellipse is elongated azimuthally when the radial epicyclic frequency satisfies $\kappa <2\Omega$. Conservation of angular momentum makes the motion around the guiding centre retrograde.

The azimuthal advance over one complete small radial oscillation is $\Delta \psi =2\pi \Omega /\kappa$. The oscillatory azimuthal correction averages to zero at first order. This angle compares successive radial turning points of the same type and describes apsidal precession when it differs from a full revolution.

A small planar velocity kick on a stellar circular orbit produces a local epicyclic motion in the source's rotating reference frame. With outward $x$, forward $y$, and initial displacements zero, Hill equations give $x/(\gamma a)=2\cos \theta (1-\cos nt)+\sin \theta \sin nt$ and $y/(\gamma a)=\cos \theta (4\sin nt-3nt)+2\sin \theta (\cos nt-1)$. Tangential kicks combine an epicycle with Keplerian shear; a radial kick gives a closed leading-order epicyclic ellipse.

The reduced positive definite triples are $(1,0,11)$, $(2,2,6)$ and $(3,\pm 2,4)$. The imprimitive middle form represents only even numbers; the two last forms have identical value sets after reversing one variable.

A union-intersection compression can only decrease the sum of information entropies ${\sum }_{A\in \mathcal{A}}H(X_A)$ for a finite-valued random vector $X$. Each elementary step follows from entropy submodularity; iteration proves the assertion with multiset multiplicities. Uniform coordinate multiplicity gives Shearer inequality by compressing to repeated full sets and empty sets.

For an extensive equilibrium system, the thermodynamic Euler relation gives $E=TS-PV+\mu N$. At zero chemical potential, divide by $V$ to obtain $s=(\rho +P)/T$. Equivalently, $dP=(\rho +P)dT/T$ makesThe first law of thermodynamics identifies this as $dS$; the thermodynamic Euler relation fixes the entropy normalization. For adiabatic expansion satisfying the cosmological perfect-fluid continuity equation, the numerator on the right vanishes, proving cosmological entropy conservation.

For $E(\gamma )=\frac{1}{2}{\int }_0^1|\dot{\gamma }{|}^{2}dt$ and variation vector field $V$, the first variation is $[⟨V,\dot{\gamma }⟩{]}_0^1-{\int }_0^1⟨V,D_t\dot{\gamma }⟩dt$. For fixed endpoints the boundary term vanishes, so the critical points of the energy are exactly affinely parametrized geodesics.

The centered process ${\mathcal{C}}_n(u)=\sqrt{n}({\phi }_n(u)-\phi (u))$ describes fluctuations of the empirical characteristic function. At a fixed frequency, its real and imaginary parts satisfy a multivariate central limit theorem. A process-level limit requires further control over the frequency index.

For a classical solution of $\Delta u=f$ on a half-ball with zero data on its flat face, a tangential Caccioppoli inequality controls every second partial derivative having at least one tangential index. The equation recovers the remaining second normal derivative. The resulting Sobolev norm estimate has a dimension-only constant for the fixed radii one and two. Nonzero data are handled by homogenization of Dirichlet boundary data, adding the $W^{2,2}$ norm of the supplied boundary extension. No condition is imposed on the curved face because the smooth cutoff function vanishes there.

For a smooth uniformly elliptic homogeneous equation away from other boundaries, an interior gradient estimate on a ball of radius comparable to the distance $s$ from a flat side gives $|\nabla u|\le C{s}^{-1}{\sup }_{B_{cs}}|u|$. If $u$ extends continuously with value zero on that side, $s|\nabla u|\to 0$ locally along it. This justifies vanishing boundary terms such as $u_x\sin (nx)$ in a distributional Fourier sine projection.

For $\Delta v=\mathrm{div}F+G$ and $U\subset \subset V$, a cutoff function energy estimate gives $\Vert v{\Vert }_{H^1(U)}\le C(\Vert v{\Vert }_{L^2(V)}+\Vert F{\Vert }_{L^2(V)}+\Vert G{\Vert }_{L^2(V)})$. One derivative of forcing is permitted because integration by parts places it on the test function. This estimate allows a mollified distributional solution initially in $L^2$ to gain its first weak derivative.

Suppose $u\in L_{\mathrm{loc}}^2$ solvesas a distribution, with $b,c$ and $f$ smooth on the open domain. The divergence-forcing interior H1 estimate first gives $u\in H_{\mathrm{loc}}^1$. If $u\in H_{\mathrm{loc}}^m$, smooth multiplication puts $\mathrm{div}(bu)+cu$ in $H_{\mathrm{loc}}^{m-1}$, and the smooth forcing belongs there too. Interior Poisson regularity then yields $u\in H_{\mathrm{loc}}^{m+1}$. Iteration and the Sobolev embedding theorem give a smooth representative.^[ref]

The regularity of $f$ is part of this argument. With only $f\in H_{\mathrm{loc}}^r$ for a nonnegative integer $r$, the same process gives $u\in H_{\mathrm{loc}}^{r+2}$, rather than unlimited smoothness. Smooth coefficients cannot remove a rough forcing term from the right-hand side.