For a monad $(T,\eta ,\mu )$, the free algebra functor sends $X$ to $(TX,{\mu }_X)$ and is a left adjoint to the forgetful functor from the Eilenberg-Moore category. The transpose bijection sends an monad algebra morphism $h:TX\to A$ to $h{\eta }_X$; its inverse sends $k:X\to A$ to $aT(k)$, where $a:TA\to A$ is the algebra structure. The monad and algebra laws make these inverse and natural. The adjunction unit is $\eta$ and the adjunction counit at $(A,a)$ is $a$, so the induced monad has exactly the original endofunctor, unit and multiplication.

Killed planar Brownian motion conditioned to exit at a smooth boundary point $a$ is the Doob h-transform by its positive Poisson kernel, normalized at the starting point. Its generator is $\frac{1}{2}\Delta +\nabla \log h\cdot \nabla$. Conditioning on shrinking boundary arcs yields the same localized laws and specifies a canonical version of an otherwise probability-zero conditioning event.

For positive $n,m$, $\mathrm{Cl}({\mathbb{P}}^n\times {\mathbb{P}}^m)={\mathbb{Z}}^2$, generated freely by the pullbacks of coordinate hyperplanes. Restrict a divisor to the standard affine chart, where unique factorization makes it principal, and subtract this principal Weil divisor to leave only the two boundary hyperplanes. Any principal relation between them would be given by a rational function whose divisor vanishes on the chart; unique factorization then makes that function a constant, proving independence. The same argument gives $\mathrm{Cl}({\mathbb{P}}^r)=\mathbb{Z}$.

An abelian group $A$ is $p$-divisible when multiplication by $p$ is surjective, equivalently $A/pA=0$. This does not mean it has no $p$-torsion: $\mathbb{Q}/\mathbb{Z}$ is divisible and has elements of every finite order. The Kummer cohomology divisibility criterion relates this property to Galois cohomology.

If a degree-$n$ characteristic polynomial has distinct roots of a polynomial ${\lambda }_j$ of multiplicities $m_j$, all homogeneous distribution solutions are the displayed combinations. There are ${\sum }_j{m}_j=n$ independent coefficients. Repeated characteristic roots of a constant-coefficient differential equation account for the polynomial factors multiplying exponentials; conjugate pairs give real sine and cosine combinations when real solutions are desired.

For a positive-order constant-coefficient ordinary differential equation, the kernel of multiplication by a coordinate gives $xP(D)u=0$ exactly when $P(D)u=C{\delta }_0$. Hence every solution is $u=v+CE$, where $P(D)v=0$ and $E$ is any fundamental solution of a linear differential operator. Taking a retarded fundamental solution of a constant-coefficient ordinary differential operator shows that the extra freedom is a single jump in the derivative of order one below the operator order. The coordinate multiplies the already differentiated distribution: this is a different operator from $P(D)(xu)$.

For a smooth function $u$, the distributional derivative of the product of $u$ with the Heaviside function is $D(Hu)=H{u}^{\prime }+u(0){\delta }_0$. Induction gives the displayed formula. It records all boundary sources, including derivatives of the Dirac delta distribution, and determines which initial derivatives must vanish in a retarded fundamental solution.

The Lighthill acoustic analogy for a moving body has a volume acoustic quadrupole, a surface acoustic monopole from acoustic thickness noise, and a surface acoustic dipole from acoustic loading noise. For an impermeable moving surface, with outward normal $n$, normal surface speed $v_n$, and fluid and body normal velocities equal, the surface mass coefficient is ${\rho }_0{v}_n$ and the surface loading is the force exerted on the fluid. Schematically, the density-source equation isThe density perturbation is understood with the chosen interior extension. Permeable-surface versions have additional mass and momentum flux terms. Source approximations must distinguish local thickness radiation from cancellation of its compact net-volume contribution.

The response to a localized perturbation must be followed in space as well as time. Growth at a fixed position is absolute wave-packet instability; growth transported away while the original position decays is convective wave-packet instability. The Briggs-Bers criterion evaluates the relevant complex spatial branches of a dispersion relation, while a real-wavenumber temporal mode test alone does not generally distinguish those behaviors.

For a primitive Dirichlet character whose conductor of a Dirichlet character is $q>1$ and whose character parity is $a$, Poisson summation gives ${\theta }_{\chi }(x)={\epsilon }_{\chi }{x}^{-a-1/2}{\theta }_{\overline{\chi }}(1/x)$, where ${\epsilon }_{\chi }=\tau (\chi )/(i^a\sqrt{q})$. Its rapid decay at both ends yields the entire Mellin representation of the completed Dirichlet L-function. The case of conductor of a Dirichlet character equal to one uses the ordinary Jacobi theta function and a subtracted constant term.

For a primitive nonprincipal Dirichlet character, the theta Mellin integral makes this an entire function satisfying $\Lambda (s,\chi )={\epsilon }_{\chi }\Lambda (1-s,\overline{\chi })$. The gamma poles cancel the trivial zeros of a Dirichlet L-function. For the principal Dirichlet character whose conductor of a Dirichlet character is one the completion is meromorphic, with poles at zero and one.

The inducing primitive Dirichlet character has conductor of a Dirichlet character $q_0$. Increasing the modulus removes Euler factors only at new prime divisors. The correction creates zeros on the imaginary axis; raising the exponents of primes already dividing the conductor of a Dirichlet character changes no Euler factor. Primitive gamma factors continue to use $q_0$. For real-even characters this includes principal primitive conductor of a Dirichlet character one, with its Riemann zeta function pole treated separately.

For a nonprincipal Dirichlet character, the line is zero-free. Euler product positivity for L-function nonvanishing proves this for nonreal characters at every height, and for all characters at nonzero height, where a possibly principal squared-character factor has no pole. The point of height zero is the nonvanishing of nonprincipal Dirichlet L-functions at one. A principal L-function has a pole at one, and no zeros at the other points of this line.

The coefficients of $\zeta (s)L(s,\chi )$ are multiplicative. On $p^k$, their values are $k+1$ when $\chi (p)=1$, alternating one and zero when $\chi (p)=-1$, and one when $\chi (p)=0$. In particular $a_{m^2}\ge 1$. Their logarithmic Euler coefficients $\Lambda (n)(1+\chi (n))$ are also nonnegative, supporting uniqueness of a possible exceptional real Dirichlet zero.

An orthogonal direct sum is a direct sum whose distinct summands are mutually orthogonal. For $v\in V$ and $w\in W$, the Pythagorean identity gives $\Vert v+w{\Vert }^2=\Vert v{\Vert }^2+\Vert w{\Vert }^2$. In a Hilbert space, a closed subspace of a Hilbert space and its orthogonal complement give an orthogonal direct sum of the entire space. The component maps are orthogonal projections and have operator norm one when their ranges are nonzero.

In a Noetherian local ring $(R,\mathfrak{m})$ of dimension $d$, a list $x_1,\dots ,x_d\in \mathfrak{m}$ whose generated ideal has radical $\mathfrak{m}$. Such lists exist, by successively avoiding the finitely many top-dimensional minimal primes. No list shorter than $d$ can generate a maximal-primary ideal, by the Krull height theorem.

The supremum of the lengths of strict chains of nonempty irreducible closed subsets. It is the maximum of the dimensions of the irreducible components. For an affine algebraic set it equals the Krull dimension of its coordinate ring; for a quasi-projective algebraic set it is the supremum of these dimensions over affine open subsets.

The codimension is the difference between the dimension of a manifold and that of its submanifold. For complex submanifolds, complex codimension one means real codimension two.

For the dihedral group of a regular $2n$-gon, $n\ge 2$, use $r^{2n}=s^2=1$ and $srs=r^{-1}$. Its involutions are the central half-turn $r^n$ and all reflections $s{r}^j$. The half-turn has singleton conjugacy class and normal closure $\{1,r^n\}$. Conjugating a reflection changes its exponent by an even integer or negates it, so its conjugacy class is $\{s{r}^{j+2k}:0\le k<n\}$. Products of consecutive reflections generate $r^2$, giving normal closure $⟨r^2,s{r}^j⟩$, of order $2n$ and index two. The two reflection classes are distinguished by exponent parity.

For a differentiable-in-quadratic-mean path, normalization gives $⟨(\sqrt{p_t}-\sqrt{p})/t,\sqrt{p_t}+\sqrt{p}⟩=0$. The two factors converge in L2 space to $g\sqrt{p}/2$ and $2\sqrt{p}$. Continuity of the inner product therefore gives $Pg=0$. This proves score function centering without differentiating the density integral.