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The pole-subtracted theta integral for the completed zeta function bounds the Riemann xi function on $|s|\le R$ by a polynomial in $R$ times ${\int }_1^{\infty }{e}^{-\pi t}{t}^{(R+1)/2}dt$. The Stirling formula gives logarithmic growth $O(R\log R)$. On the positive real axis, the definition and $\zeta (R)\ge 1$ give $\log \xi (R)=\frac{R}{2}\log R+O(R)$. Thus its order of an entire function is exactly one.

// Target: analytic-number-theory.bigb

List positive ordinates of Nontrivial zeros of the Riemann zeta function in nondecreasing order, including multiplicity. The Riemann–von Mangoldt formula gives $N(T)\sim T\log T/(2\pi )$. For each fixed $\epsilon >0$, evaluate this at $T=(1\pm \epsilon )2\pi n/\log n$ to bracket the $n$th ordinate. Letting $\epsilon$ decrease to zero proves the displayed asymptotic without assuming simple zeros or distinct ordinates.

List positive ordinates of Nontrivial zeros of the Riemann zeta function in nondecreasing order, including multiplicity. If all sufficiently late gaps were at most $c/\log n$, summing would give ${\gamma }_n\le (c+o(1))n/\log n$, since ${\sum }_{j\le n}1/\log j\sim n/\log n$ with the sum starting at two. The Riemann–von Mangoldt formula would then give $n\le N({\gamma }_n)\le (c/(2\pi )+o(1))n$, a contradiction when $c<2\pi$. The same argument applies when only distinct ordinates are listed, because $n\le N({\gamma }_n)$ still holds.

A contact discontinuity in one-dimensional gas dynamics moves with the common fluid velocity $s=u_L=u_R$ and has continuous pressure, while mass density and specific entropy can jump. The mass jump condition $s[\rho ]=[\rho u]$ is automatic with common $u=s$; momentum conservation $s[\rho u]=[\rho u^2+p]$ then requires $[p]=0$. The energy jump condition is also satisfied. Thus a contact is a material interface, not a compression shock. A smooth self-similar positive-density flow cannot spread this zero-speed-relative-to-the-fluid family into a fan: $u=x/t$ on an interval would require both $du/d(x/t)=1$ and, from continuity, $du/d(x/t)=0$.

Two positive uniform ideal-gas states can be joined by one nontrivial rarefaction wave precisely when their specific entropy agrees, $p_L/{\rho }_L^{\gamma }=p_R/{\rho }_R^{\gamma }$, and for one $\sigma \in \{\pm 1\}$ they satisfy $u_R-u_L=2\sigma (c_R-c_L)/(\gamma -1)$ with $\sigma (c_R-c_L)>0$. These equations are the constant-entropy and Riemann invariant conditions of the self-similar ideal-gas rarefaction. They are sufficient because its affine $u,c$ profiles match the two states at ${\xi }_L=u_L+\sigma c_L$ and ${\xi }_R=u_R+\sigma c_R$, and ${\xi }_R>{\xi }_L$. The left family has $c_R<c_L$, while the right family has $c_R>c_L$. The inequality $u_R>u_L$ alone does not impose the two matching equalities.

For a univalent function $f:\Omega \to \mathbb{D}$ with $f(z_0)=0$, suppose $a\ne 0$ is omitted. Apply the disk automorphism $T_a$ so the resulting image omits zero, take a holomorphic square root, then apply the disk automorphism taking the value at $z_0$ to zero. If $\Omega$ is a simply connected domain, the root exists. The resulting normalized univalent map has derivative magnitude multiplied by $(1+|a|)/(2\sqrt{|a|})>1$, since $(1-\sqrt{|a|}{)}^2>0$. Thus a map maximizing the derivative cannot omit an interior disk point.

A rational jump with no zeros or poles on the contour can be split into factors analytic and nonvanishing on the two sides. The exterior factor may carry a power of $z$ recording the winding number. For a circle enclosing $0,\pm 1$, the jump $G=z/(z^2-1)$ has the canonical factors $X_{+}=1$, $X_{-}=z-z^{-1}$. The exterior growth of $X_{-}$ imposes compatibility conditions when a Cauchy transform is required to vanish at infinity.

Assume a regular invertible normalized Riemann-Hilbert problem solution $\mu =I+{\mu }_1/k+{\mu }_2/k^2+\cdots$ with jump tending to the identity and conjugated by $e^{-i(kx+2k^{2}t){\sigma }_3}$. Residuals of its two differential equations have no jump and vanish at infinity, so Liouville theorem gives a Lax pair. Coefficient comparison gives the displayed reconstruction and $A=2Q$; compatibility yields $i{Q}_t-Q_{xx}{\sigma }_3+2Q^3{\sigma }_3=0$. A jump matrix tending to zero instead is inconsistent with the normalization. Recovering the initial jump from a prescribed initial potential still requires direct scattering.

Suppose a scalar canonical factor has exterior growth $X_{-}(z)\sim z^q$ with index $-q<0$, and the original exterior unknown must decay as $O(z^{-1})$. Dividing by the canonical factor gives an additive jump $D_{+}-D_{-}=g$ with exterior decay $O(z^{-q-1})$. The exterior Cauchy integral expands in moments $-{\int }_L{t}^{j}g(t)dt/(2\pi i{z}^{j+1})$. Its first $q$ coefficients must therefore vanish, yielding the displayed solvability conditions. Under this decay there is no nonzero entire polynomial addition. A nonlocal moment appearing in $g$ must additionally agree with the moment of the reconstructed original jump.

Combine columns of the three normalized Lax pair eigenfunctions in their common Volterra analyticity sectors for a half-line NLS Lax pair, dividing by the relevant determinants $a,d,d^{♯},a^{♯}$. The resulting matrix is normalized to the identity at infinity and has jumps on the real and imaginary axes. Connection matrices determine each jump explicitly. Zeros of those determinants require residue conditions or local contour deformations.

Use the bounded columns supplied by Volterra analyticity sectors for reverse-dispersion mKdV and divide the necessary columns by their determinants. This gives a normalized meromorphic matrix on the six-ray contour $\arg k=m\pi /3$. The constant scattering data connection matrices generate its jumps by conjugation with $e^{i(kx-4k^{3}t){\sigma }_3}$. Zeros of column determinants require residue conditions; in a sufficiently small-data, zero-free regime only jumps are needed. The off-diagonal expansion coefficient reconstructs $q=-2i{\lim }_{k\to \infty }k{M}_{12}$ for this sign convention.

The ratio of stable buoyancy destruction to shear production of turbulent kinetic energy. With down-gradient transport and turbulent Prandtl number ${\mathrm{Pr}}_t$, it obeys ${\mathrm{Ri}}_f={\mathrm{Ri}}_g/{\mathrm{Pr}}_t$, rather than automatically equaling the gradient Richardson number. Unstable buoyancy production gives a negative value in this convention.

On the matchings in a graph, a lazy Markov chain proposes adding an edge, deleting an edge, or exchanging an edge for the unique matching edge incident with it. A proposal incident with two matching edges is rejected. Select the proposed edge uniformly and use the Metropolis–Hastings acceptance probability for the displayed weights. Symmetric proposals give detailed balance; deletions connect every state to the empty matching in a graph. The normalizing constant is the matching generating function.

A polynomial regression birth move can introduce a coefficient $z$ while shifting the intercept by $-cz$, with $c$ the sample average of the new monomial. The fitted-value increment is $z(x^{k+1}-c)$, whose sample average vanishes. The Jacobian determinant of the coefficient transformation is one. A reversible-jump Markov chain Monte Carlo acceptance ratio must still include the normalized dimension-dependent prior distributions, the model-order prior distribution, the birth/death selection probabilities and the proposal density for $z$.

For nested regression models, append one coefficient $u$ drawn from density $g_k$ while retaining the variance $v$ and existing coefficients $a$. The dimension-matching map $(a,v,u)\mapsto (a,u,v)$ has absolute Jacobian one. With joint model/parameter target $t_k$, birth-selection probability $b_k$, and reverse death-selection probability $d_{k+1}$, the displayed acceptance probability and its reciprocal death rule enforce detailed balance. Include normalized parameter-prior densities and a model-order prior in $t_k$: constants that depend on model order cannot be dropped from a cross-model ratio.

To minimize $E_j(\theta )=2k_j-2{\ell }_j(\theta )$ over models with different parameter dimensions, sample from a target proportional to $w_j{e}^{-E_j(\theta )/(2T)}$ and cool $T$. A dimension-matching proposal augments the smaller model by auxiliary variables, transforms them bijectively into the larger parameter vector, and includes the proposal probability density function and absolute Jacobian determinant in the Metropolis–Hastings acceptance probability. For a positive Poisson mean $\lambda$ and normal parameters $(\mu ,v)$, one convenient transformation isIts absolute Jacobian determinant is one. If the forward and reverse model-jump selection probabilities are $b$ and $d$, the Poisson-to-normal ratio isThe reverse ratio is its reciprocal at the inverse map. This enforces detailed balance. When the target is proper and the annealing chain explores it adequately, the low-temperature laws concentrate on the minimum Akaike information criterion, rather than the maximum marginal model evidence. Likelihoods compared across models must describe the same observed data with a compatible reference measure.

For $N◃G$ and $\theta \in \mathrm{Irr}(N)$, set ${\theta }^g(n)=\theta (g^{-1}ng)$. Its inertia group is its stabilizer under this conjugation action and contains $N$. Write $\mathrm{Irr}(G\mid \theta )$ for the irreducible characters of $G$ whose restriction to $N$ contains $\theta$.

An extension of $\theta \in \mathrm{Irr}(N)$ from $N◃G$ is a character $\chi$ of $G$ whose restriction is exactly $\theta$. It is necessarily irreducible, because a nontrivial invariant subspace for $G$ would also be one for $N$. An ordinary extension can fail to exist even when $\theta$ is invariant under $G$.

A central composite design combines a factorial portion with axial points $\pm \alpha e_j$ and center runs. For a full $2^d$ factorial design, $d$ quantitative factors and $n_0\ge 1$ center runs, its run count is $2^d+2d+n_0$. The factorial portion supplies main effects and interaction terms, while the axial runs allow the individual squared terms of a full quadratic regression model to be estimated. Additional center replicates estimate pure error.

For two factors and $n_0=5$, this gives $4+4+5=13$ runs. The four corners are $(\pm 1,\pm 1)$, the four axial points are $(\pm \alpha ,0)$ and $(0,\pm \alpha )$, and the remaining five runs repeat $(0,0)$. Choosing $\alpha =\sqrt{2}$ makes this full-factorial design rotatable. Choosing $\alpha =1$ keeps the axial points on the faces and uses only three levels per factor, but does not give that rotatability in dimension at least two.

Circumscribed designs place the axial points outside the original factorial cube; inscribed designs rescale the cube inward to respect physical factor limits. A fractional factorial portion can reduce runs, but must retain sufficient resolution of a fractional factorial design to estimate the desired interactions: axial points have zero cross-products and cannot separate interactions that remain aliased in the factorial portion. The institutional handbook describes augmentation from a full or resolution V factorial portion.^[ref]

A rotatable design for a specified regression model has variance of a fitted regression mean depending on $x$ only through its Euclidean distance from the design center. It concerns prediction precision, not rotational symmetry of the mean response itself.

For a full quadratic regression model in $d\ge 2$ variables, consider all $2^d$ factorial corners, the $2d$ axial points $\pm \alpha e_i$, and center runs giving a nonsingular information matrix of an experimental design. Sign symmetry cancels odd moments and permutation symmetry makes the second coordinate moments equal. Fourth-order rotational balance requires, for distinct $i,j$,The factorial corners contribute $2^d$ to both sums, axial runs contribute $2{\alpha }^4$ only to the first, and center runs contribute zero. Thus $2^d+2{\alpha }^4=3\cdot 2^d$, givingFor $d=2$, $\alpha =\sqrt{2}$; for $d=3$, $\alpha =8^{1/4}$. Center replication can estimate pure error but does not change this fourth-moment equation. If a fractional factorial portion is used, its own moments and estimability must be checked; simply substituting the original factor count into the full-factorial formula is not justified. In one dimension rotational invariance only means reflection symmetry, so the distinct-coordinate fourth-moment equation does not impose the displayed choice.