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A multiderivative multistep method uses higher time derivatives as well as values and first derivatives from several time levels. For an autonomous ordinary differential equation $y^{\prime }=f(y)$, the second time derivative is $y^{\prime \prime }=f^{\prime }(y)f(y)$ by the chain rule. Its local truncation error must include the higher-derivative terms. Consequently the usual second-order barrier for A-stable linear multistep methods, which only use first derivatives, does not apply.

The Cauchy principal valuedefines a distribution. Near zero the odd constant contribution cancels, and the remaining numerator is $O(x)$. It satisfies $x\mathrm{pv}(1/x)=1$. Hence all solutions of $xv=1$ are $v=\mathrm{pv}(1/x)+c{\delta }_0$ by the kernel of multiplication by a coordinate.

For smooth $a$ and a point $r$,These follow by applying both sides to a test function and using the Leibniz rule. In particular $(x-r){\delta }_r^{\prime }=-{\delta }_r$.

If a smooth function $a$ on $\mathbb{R}$ has finitely many zeros $r_j$, all simple, then $aw=0$ exactly when $w={\sum }_j{c}_j{\delta }_{r_j}$. Away from the zeros, divide a test function by the nonvanishing $a$ to show $w$ vanishes. Near $r_j$, write $a(x)=(x-r_j)b_j(x)$ with smooth nonzero $b_j$. The kernel of multiplication by a coordinate, after translation and multiplication by $b_j$, says that the local distribution is a multiple of ${\delta }_{r_j}$. A partition of unity gives the global sum. Simplicity of the zeros excludes derivatives of deltas.

Replace each positive power $x_i^a$ in a monomial by $x_i$ and combine equal monomials. The result is a multilinear polynomial agreeing with the original polynomial at every point of $\{0,1{\}}^n$, because $x_i^a=x_i$ there for $a\ge 1$. Its polynomial degree does not increase. This reduction identifies polynomial functions on the Boolean lattice with their unique square-free monomial representations: uniqueness follows by successively evaluating at characteristic vectors of sets in increasing order of size.

On the uniform layer of the Boolean cube ${\Omega }_r$, every multilinear polynomial of polynomial degree at most $s\le r$ is in the span of the degree-$s$ square-free monomials. For $T\subseteq [n]$, $|T|=j\le s$, and $x_T={\prod }_{i\in T}{x}_i$, the identity isAt the characteristic vector of a set $A$ of size $r$, both sides vanish if $T⊈A$. Otherwise exactly $\left(\genfrac{}{}{0ex}{}{r-j}{s-j}\right)$ summands are $1$. Hence the dimension of the restricted polynomial space is at most $\left(\genfrac{}{}{0ex}{}{n}{s}\right)$. No assertion of independence of the spanning monomials is needed, so no condition $r+s\le n$ is required. Unlike ordinary homogenization, this operation preserves degree bounds by using the fixed-weight evaluation domain.

A multilinear map $F:V_1\times \cdots \times V_r\to W$ is a map between vector spaces that is linear in each argument when all other arguments are fixed. For example the evaluation $(\omega ,v)\mapsto \omega (v)$ is bilinear, and the determinant is multilinear in its columns. The universal property of a tensor product replaces this separate linearity by a single linear map from $V_1\otimes \cdots \otimes V_r$.

A transport map from $\mu$ to $\nu$ is a measurable function $T$ satisfying $T_{\#}\mu =\nu$. Its graph defines the transport plan $(\mathrm{Id},T{)}_{\#}\mu$.

A monad is idempotent when its multiplication is invertible. Then $T\eta =\eta T={\mu }^{-1}$. If $(A,a)$ is an algebra for a monad, its unit law and naturality give $a{\eta }_A=1_A$ and ${\eta }_{A}a=Ta{\eta }_{TA}=T(a{\eta }_A)=1_{TA}$. Thus every algebra is isomorphic to a free algebra, and the Kleisli comparison functor into the Eilenberg-Moore category is an equivalence.

Let $M$ be the monoid of order-preserving functions $\mathbb{N}\to \mathbb{N}$, with $\mathbb{N}=\{0,1,\dots \}$, regarded as a one-object category. The endofunctor $T$ fixes that object and sends $f$ to $Tf(0)=0$, $Tf(n+1)=f(n)+1$. The displayed maps give its monad unit and multiplication. The multiplication is not injective, so this is not an idempotent monad. Its only algebra for a monad is $\mu$, because $a\eta =1$ forces $a(n+1)=n$ and monotonicity forces $a(0)=0$. Algebra endomorphisms are exactly the functions fixing zero. The Kleisli comparison functor sends $f$ to the function which is zero at zero and equals $f(n)$ at $n+1$; its inverse sends $h$ to $n\mapsto h(n+1)$. Thus the comparison is an isomorphism of categories even though the monad is not idempotent.

The mass second-moment tensor about a centre of mass is $I_{jk}=\int y_j{y}_{k}dm$. It measures the shape of a mass distribution and appears in tidal torque theory. It differs from the mechanical inertia tensor, which is ${\mathcal{I}}_{jk}=\mathrm{tr}(I){\delta }_{jk}-I_{jk}$. For a uniform ellipsoid of mass $M$ and semiaxes $a_1,a_2,a_3$, its principal values are $M{a}_j^2/5$.

The subset of the unmarked moduli space of Riemann surfaces with hyperbolic systole at least $\epsilon >0$ is compact for fixed genus $g\ge 2$. Use a Bers pants decomposition theorem bound, finitely many topological decomposition types and Dehn twists to place representatives in finitely many compact boxes in Fenchel–Nielsen coordinates.

A nonzero module is indecomposable if it cannot be expressed as a direct sum of two nonzero submodules. The zero module is excluded. A module of finite composition length is indecomposable exactly when its endomorphism ring has no idempotents other than zero and one: an idempotent splits the module into its image and kernel.

A nonzero module is uniform if every pair of nonzero submodules has nonzero intersection. Equivalently, every nonzero submodule is an essential submodule. A right Noetherian domain is uniform as a right module: if $aA\cap bA=0$ for nonzero $a,b$, the right ideals $b^{i}aA$, $i\ge 0$, form an infinite direct sum, contradicting the ascending chain condition.

The modular representation ring is the Grothendieck group of finite-dimensional group representations, with relations $[V]=[U]+[W]$ for every short exact sequence $0\to U\to V\to W\to 0$, and multiplication $[V][W]=[V{\otimes }_{k}W]$ using the tensor product of group representations. The Jordan–Hölder theorem makes the classes of simple modules a free integral basis. Tensoring over a field is exact, so this multiplication is well defined. When $k$ is a splitting field for finite group representations, Brauer characters identify $\mathbb{C}{\otimes }_{\mathbb{Z}}{R}_k(G)$ with the algebra of complex class functions on p-regular elements. The splitting hypothesis is essential: ${\mathbb{F}}_2{C}_3\cong {\mathbb{F}}_2\times {\mathbb{F}}_4$ has two simple modules, although $C_3$ has three $2$-regular conjugacy classes.

Here a weak modular form is a holomorphic function on the complex upper half-plane satisfying the weight-$k$ transformation law, with no growth condition at any cusp of a modular group. This convention differs from weakly holomorphic modular forms, which are required to be meromorphic at the cusps. At level one, translation invariance gives a Laurent expansion on the punctured unit disc, possibly with infinitely many negative powers.

If $N_{1}D\mid N$ and $f\in M_k({\Gamma }_0(N_1))$, then $f(Dz)\in M_k({\Gamma }_0(N))$. Conjugating by $\mathrm{diag}(D,1)$ changes a lower-left matrix entry $c$ to $c/D$, which is divisible by $N_1$. A rational upper-triangular factorization at every cusp preserves boundedness and hence cusp holomorphy.

The Serre derivative sends a level-one weight-$k$ modular form to weight $k+2$. Differentiating $f(-1/z)=z^{k}f(z)$ produces the extra term $k{z}^{k+1}f/(2\pi i)$; the anomalous transformation of the Eisenstein series of weight two cancels it. Translation invariance and the Fourier expansion of a modular form verify the remaining holomorphy conditions. For $k>0$, its constant coefficient is $-k{a}_0(f)/12$, so it is a cusp form exactly when $f$ is.

A modular form on an arbitrary finite-index subgroup $\Gamma \le S{L}_2(\mathbb{Z})$ is a holomorphic function on the complex upper half-plane, invariant under the weight-$k$ slash operator for modular forms, and holomorphic at a cusp at every cusp of a modular group for $\Gamma$. If $\sigma \in S{L}_2(\mathbb{Z})$ maps infinity to a cusp, choose a genuine translation period $h>0$ with $\sigma T^h{\sigma }^{-1}\in \Gamma$. Then $f{|}_k\sigma$ has a convergent series in $e^{2\pi iz/h}$ with nonnegative exponents. A cusp form has zero constant term at every cusp. This definition allows noncongruence subgroups and avoids sign ambiguities in odd weights when a smaller width is defined only modulo the center.

The quotient $\mathbb{H}/\Gamma (2)$ is a genus-zero Riemann surface with three cusps and no elliptic Möbius transformations, hence is biholomorphic to $\mathbb{C}\setminus \{0,1\}$. The integer level-two principal congruence subgroup has projective index six; reduction modulo two gives three cusp classes.