For the three-dimensional discrete paraboloid over an odd prime field, the surface transform away from zero has magnitude at most $p^{-1}$, by completing squares and evaluating quadratic Gauss sums. The adjoint-composition convolution kernel is the transform of the surface measure. Subtract its unit mass at the origin. Convolution by the remainder has ${\ell }^1\to {\ell }^{\infty }$ norm at most $p^{-1}$ and ${\ell }^2\to {\ell }^2$ norm at most $p$. The Riesz-Thorin interpolation theorem gives ${\ell }^{4/3}\to {\ell }^4$ norm at most one; adding the identity kernel gives at most two. The adjoint-composition identity and duality therefore give the displayed extension constant for normalized surface measure and ambient counting measure. This argument works for either quadratic character of minus one.

If $i^2=-1$ in the prime field, the line $\ell =\{(t,it,0):t\in {\mathbb{F}}_p\}$ lies in the three-dimensional discrete paraboloid. Its indicator has normalized surface $L^2$ norm $p^{-1/2}$. Its extension is $p^{-1}$ on the two-dimensional annihilator plane and zero elsewhere. Thus its ambient ${\ell }^q$ norm is $p^{2/q-1}$, proving the displayed lower bound, which diverges for $q<4$.

An incoming morphism into the image of a specified object has exactly one lift with that specified codomain. Both its source object and its lifted morphism are determined. Applying uniqueness to identity and composite morphisms gives the functoriality of lifting.

Zero padding extends a finite sequence by zero entries to a specified length. For input lengths $r,s$, their full discrete convolution has length at most $r+s-1$. Padding both inputs to $N\ge r+s-1$ allows circular convolution to represent this full result without folded entries.

Circular convolution of two length-$N$ sequences uses modular arithmetic for their indices. Extending both inputs periodically gives the displayed formula. Equivalently, fold a finite discrete convolution into residue classes modulo $N$ by adding entries whose indices are congruent. The discrete Fourier transform of the circular convolution is the product of the two discrete Fourier transforms, with the usual unnormalized forward transform and $1/N$ inverse convention.

Preserving the modified contents of a dirty memory page in its backing storage. The reference model waits for successful page writeback before reusing the victim's page frame; failure cannot authorize losing its modifications. A reference bit records recent access, while the dirty flag records a write obligation: reading can set the first without setting the second. Buffer cache writeback concerns the analogous obligation for a buffered device block.

For the positive Dirichlet Laplacian $A$ on a bounded smooth domain, with normalized eigenfunctions $e_j$ and positive eigenvalues ${\mu }_j$, define $A^{\alpha }u={\sum }_j{\mu }_j^{\alpha }(u,e_j)e_j$ on the vectors for which ${\sum }_j{\mu }_j^{2\alpha }|(u,e_j){|}^2<\infty$. The resulting Hilbert space embeds in the Sobolev space $H^{2\alpha }$, with the boundary conditions appropriate to the exponent. In particular $H_{\alpha }$ embeds in $C^{0,\beta }$ whenever $0<\beta <\min (1,2\alpha -n/2)$. High powers impose further boundary compatibility; ordinary smoothness alone does not imply membership in every fractional domain.

If ${\beta }_1,{\beta }_2,{\beta }_3$ are the affine barycentric coordinates of an equilateral triangle of side $\ell$, then $h={\prod }_j\sin (\pi {\beta }_j)=\frac{1}{4}{\sum }_j\sin (2\pi {\beta }_j)$ vanishes on the boundary and satisfies $\Delta h=-16{\pi }^{2}h/(3{\ell }^2)$. Thus the problem $\Delta q-4\lambda q=0$ with zero Dirichlet boundary data has distinct solutions zero and $h$ at the displayed negative parameter. Their normal traces differ, demonstrating failure of a uniquely determined Dirichlet-to-Neumann map at resonance.

For a primitive nonprincipal Dirichlet character of conductor $q$ and parity $a$, the completed Dirichlet L-function is entire of order one. Its genus-one Hadamard factorization gives the displayed logarithmic derivative, with $B_{\chi }$ the logarithmic derivative of the completion at zero and zeros counted with multiplicity. Combined summands converge locally normally away from zeros. For an imprimitive character induced by ${\chi }^{\ast }$, add ${\sum }_{p\mid Q,p\nmid q}{\chi }^{\ast }(p)\log p/(p^s-{\chi }^{\ast }(p))$ for the removed Euler factors. The primitive principal case is the global partial-fraction expansion of the zeta logarithmic derivative.

// Target: algebraic-number-theory.bigb

For real $s>1$, Orthogonality of Dirichlet characters gives $\log {\mathcal{P}}_q(s)=\phi (q){\sum }_{p\nmid q}{\sum }_{j\ge 1,p^j\equiv 1(\mathrm{mod}q)}1/(j{p}^{js})\ge 0$. Thus ${\mathcal{P}}_q(s)\ge 1$. Only the principal Dirichlet character contributes a pole at one, and that pole is simple. A zero of a nonreal Dirichlet L-function at one brings a zero of its conjugate too; their two zeros overpower that pole and would make the product tend to zero. This proves nonvanishing for nonreal Dirichlet characters and is a useful complement to arguments for real Dirichlet characters.

For $D_N(t)=\sin ((N+1/2)t)/\sin (t/2)$, define ${\Lambda }_N=(2\pi {)}^{-1}{\int }_{-\pi }^{\pi }|D_N|$. Evenness and $\sin (t/2)\le t/2$ on $(0,\pi ]$ give ${\Lambda }_N\ge (2/\pi ){\int }_0^{(N+1/2)\pi }|\sin v|/vdv$. The first $N$ complete sine periods of length $\pi$ have absolute sine integral two, and on the $j$th such interval $1/v\ge 1/(j\pi )$. This proves the displayed unbounded lower bound. The real functional $f\mapsto S_N(f,0)$ on continuous functions has norm exactly ${\Lambda }_N$: the upper bound is its kernel integral, and the continuous functions $D_N/\sqrt{D_N^2+{\delta }^2}$ approach that bound as $\delta \downarrow 0$ by dominated convergence.

The Dirichlet hyperbola method gives $D(x)=2{\sum }_{a\le r}\lfloor x/a\rfloor -r^2$, where $r=\lfloor \sqrt{x}\rfloor$. The harmonic number estimate $H_r=\log r+\gamma +O(1/r)$, with $\gamma$ the Euler's constant, yields $D(x)=2x(\log r+\gamma )-r^2+O(x/r+r)$. Substitution of $r=\sqrt{x}+O(1)$ gives the displayed asymptotic expansion. Comparison of ${\sum }_{n\le x}\log n$ with its integral consequently gives ${\sum }_{n\le N}(\tau (n)-\log n-2\gamma )=O(\sqrt{N})$ for integer $N$.

The direct divisor summatory function formula evaluates $N$ floor divisions. Its balanced Dirichlet hyperbola method formula evaluates $r=\lfloor \sqrt{N}\rfloor$ floor divisions and reuses their sum for the other strip, followed by doubling and square-overlap subtraction. Thus it uses $O(\sqrt{N})$ arithmetic operations, with big O notation, in a model that counts each division, addition and multiplication as one operation. At $N=10^6$, the division counts for these two formulas are $10^6$ and $10^3$.

For a general Dirichlet convolution, the asymmetric formula has $\lfloor A\rfloor +\lfloor B\rfloor$ outer terms; evaluating $F$ and $G$ has an additional cost. Under $AB=N$, equal thresholds minimize $A+B$, but an asymmetric split can be useful if the two partial sums have different evaluation costs. The balanced divisor summatory function gains the further reuse because its two arithmetic functions are identical.

These are arithmetic operation counts, rather than counts of elementary operations on binary digits: the integers grow with $N$, and division of longer integers costs more in that model of computational complexity theory. Computing $r$ also requires an exact integer square root, or checking and correcting a numerical square root until $r^2\le N<(r+1{)}^2$ holds.

A balanced Dirichlet hyperbola method split takes real thresholds $A=B=\sqrt{N}$. An asymmetric split takes $1\le A\le N$ and $B=N/A$. Every lattice point with $ab\le N$ then has $a\le A$ or $b\le B$, and every point in their overlap rectangle is allowed because $AB=N$. For arithmetic functions $f,g$, define their partial sums by $F(t)={\sum }_{a\le t}f(a)$ and $G(t)={\sum }_{b\le t}g(b)$, always over positive integers. The same inclusion-exclusion principle, now with weight $f(a)g(b)$, yieldsNo multiplicativity assumption is needed: this is a finite rearrangement of Dirichlet convolution. The constant-one arithmetic function in both positions recovers the divisor summatory function. For $N=12$, $A=2$, $B=6$ givesIt uses two columns and six rows, with a twelve-point overlap.

For general positive real thresholds, $AB\ge N$ is sufficient for coverage, since $a>A$ and $b>B$ would imply $ab>AB\ge N$. If $AB>N$, however, the overlap can include points above the hyperbola. The correct subtraction is then ${\sum }_{a\le A,b\le B,ab\le N}f(a)g(b)$; the product $F(A)G(B)$ is justified only when the whole integer rectangle lies below it. For example, $N=12$, $A=B=4$ covers every allowed pair, but its rectangle contains the forbidden pair $(4,4)$, so subtracting all sixteen points would give $34$ instead of $35$.

Conversely, the balanced integer thresholds $r,r$ have $r^2\le N$ and still cover every allowed pair because $(r+1{)}^2>N$. More generally, positive integer thresholds $U,V$ give coverage exactly when $(U+1)(V+1)>N$, and their overlap is a full rectangle exactly when $UV\le N$. These discrete conditions explain why the square-root identity works even when $r^2<N$.

The divisor summatory function adds the divisor function up to a cutoff $x\ge 1$. It counts all ordered pairs of positive integers whose product is at most $x$. For $N=12$, the twelve individual divisor function values areCounting columns of lattice points instead gives $12+6+4+3+2+2+1+1+1+1+1+1=35$. The Dirichlet hyperbola method needs only the first three column counts:The arithmetic work of the Dirichlet hyperbola method is much smaller even though the answer is identical.

This is useful for elementary averages: individual divisor function values fluctuate, but their summation has the regular behavior given by the divisor summatory asymptotic. Dividing that estimate by a positive integer $N$ giveswhere $\log$ is the natural logarithm, $\gamma$ is the Euler--Mascheroni constant, and $O$ uses big O notation. This describes an average, rather than an estimate for every individual divisor function value. ^[ref]

Consider determinant-one matrices $\left(\begin{array}{cc}u& v\\ \overline{v}& \overline{u}\end{array}\right)$ over Gaussian integers with $u+v\equiv 1(\mathrm{mod}2\mathbb{Z}[i])$, modulo their common sign. They act on the Poincare disc model by fractional linear maps. Their origin-centered Dirichlet domain isFor a nonidentity matrix set $N=|v{|}^2\ge 1$ and $uv=m+in$. The group orbit point is $(m+in)/(N+1)$ and its bisector inequality is $2(mx+ny)\le N(1+x^2+y^2)$. Since $m^2+n^2=N(N+1)$, integrality gives $|m|,|n|\le N$, so the four inequalities defining $D$ imply all of them. They are necessary because the matrices with $u=1\pm i$, $v=\pm i$ realize the four combinations $m,n=\pm 1$. The four sides are arcs of the unit-radius circles centered at $\pm 1\pm i$, and the ideal vertices are $1,i,-1,-i$.

For the directly forced cubic amplitude equation, a saddle-node bifurcation requires ${\Lambda }^2=1/s-1/(4s^4)$ and $\mu =s+1/(2s^2)$. The right side has its unique maximum $3/4$ at $s=1$, giving a cusp bifurcation at $(\Lambda ,\mu ,s)=(\sqrt{3}/2,3/2,1)$. The stability matrix has trace $2(\mu -2s)$ and determinant $(\mu -3s)(\mu -s)+{\Lambda }^2$. A Hopf bifurcation requires $\mu =2s$, $s(s^2+{\Lambda }^2)=1$ and ${\Lambda }^2-s^2>0$. These imply $\Lambda >2^{-1/3}$; equality is a double-zero linear degeneracy rather than an ordinary Hopf bifurcation.

For a linear imposed temperature $T=T_E+Gz$, local liquidus equilibrium gives $C=-(T_E+Gz)/m$ in the mushy layer. With equal densities, pure solid, pulling velocity $-V$, and diffusion only in liquid volume fraction $q=1-\varphi$, steady solute flux is $-VqC-Dq{C}_z=-V{C}_0$. The exterior concentration is $C_0+(C_i-C_0)e^{-V(z-h_0)/D}$. Marginal equilibrium at a mush-liquid boundary gives $C_i-C_0=GD/(mV)$ and hence the displayed thickness. In the mush, $q[C+(D/V)C_z]=C_0$, so $\varphi =(h_0-z)/(m{C}_0/G+h_0-z)$. A positive thickness requires the thermal interval to exceed the diffusive length $D/V$.

For pure solid rejecting all solute, pulling speed $V>0$, solutal diffusivity $D$ and finite solute inventory per unit area $N$, a stationary planar front has $C(z)=C_I{e}^{-Vz/D}$ in its liquid half-space. The advection-diffusion equation gives this profile, and ${\int }_0^{\infty }Cdz=N$ fixes $C_I=VN/D$. A liquidus $T_I=-m{C}_I$ fixes its position in an imposed temperature gradient. The pure-solid branch requires $C_I<C_E$, where $C_E$ is the eutectic composition.

The frozen-temperature approximation prescribes the external thermal field during interface perturbations, rather than solving a perturbation heat equation. An interface displacement samples that prescribed field at its new location. Local liquidus and Gibbs-Thomson effect conditions still apply, and the solute field may retain full time-dependent diffusion.