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For a prime number $p$, a p-valuation is a separated filtration on a group such that every $x\ne 1$ satisfiesA group equipped with one is called a p-valued group.

An equivariant index theorem expresses the supertrace of a symmetry on the kernel and cokernel of an elliptic operator as an integral of local characteristic data over the symmetry's fixed-point set.

A positive integer is a congruent number when it is the area of a right triangle with rational side lengths. For square-free $D$, this is equivalent to the congruent number elliptic curve having a Mordell-Weil group of positive rank.

Every identity-preserving biholomorphism of $\mathbb{C}/\Lambda$ lifts to multiplication by a nonzero complex number $\zeta$ satisfying $\zeta \Lambda =\Lambda$. Relative to a $\mathbb{Z}$-basis of $\Lambda$, multiplication by $\zeta$ is represented by a unimodular matrix $A\in {\mathrm{GL}}_2(\mathbb{Z})$. Its real determinant is $|\zeta {|}^2$, while complex multiplication preserves orientation, so $\det A=1$ and $|\zeta |=1$. The Cayley-Hamilton theorem givesConsequently $2\mathrm{Re}\zeta =\mathrm{tr}(A)\in \{-2,-1,0,1,2\}$, and $\zeta$ is a root of unity of order $1$, $2$, $3$, $4$, or $6$.

Let ${\pi }_{\Lambda }:\mathbb{C}\to \mathbb{C}/\Lambda$ and ${\pi }_{{\Lambda }^{\prime }}:\mathbb{C}\to \mathbb{C}/{\Lambda }^{\prime }$ be the universal covering maps of two one-dimensional complex tori. Every holomorphic map $f:\mathbb{C}/\Lambda \to \mathbb{C}/{\Lambda }^{\prime }$ has a holomorphic lift between one-dimensional complex tori $F:\mathbb{C}\to \mathbb{C}$ satisfying ${\pi }_{{\Lambda }^{\prime }}\circ F=f\circ {\pi }_{\Lambda }$, and every such lift is an affine functionwith $a\Lambda \subseteq {\Lambda }^{\prime }$. Indeed, for $\lambda \in \Lambda$, the continuous function $F(z+\lambda )-F(z)$ takes values in the discrete set ${\Lambda }^{\prime }$ and is therefore constant. Hence $F^{\prime }$ is $\Lambda$-periodic. It is bounded on a compact fundamental parallelogram and therefore on the complex plane, so the Liouville theorem makes $F^{\prime }$ constant. If $f$ preserves the identity and the lift is chosen with $F(0)=0$, then $b=0$ and the lift is a linear map.

Every holomorphic map $f:\mathbb{C}\to \mathbb{C}/\Lambda$ is constant or surjective. It lifts to an entire function $F:\mathbb{C}\to \mathbb{C}$. If $F$ is nonconstant, the Little Picard theorem says it omits at most one point, while every fiber of $\mathbb{C}\to \mathbb{C}/\Lambda$ is an infinite lattice coset; hence every point of the torus has a preimage under $f$.

Negation is a biholomorphic involution of $E=\mathbb{C}/\Lambda$. Its fixed points satisfy $2z\in \Lambda$, so they are indexed by the four elements of $\frac{1}{2}\Lambda /\Lambda$.

Let $L=a^{ij}(x)D_{ij}$ be uniformly elliptic with continuous coefficients. The coefficient-freezing interior second-derivative estimate extends from smooth functions to the graph norm closure of $L$ on $H^1(U)$: if $u_k\to u$ in $H^1(U)$ and $L{u}_k\to f$ in $L^2(U)$, then $(u_k)$ is Cauchy in $H^2(W)$ for every $W⋐U$. Hence $u\in H_{\mathrm{loc}}^2(U)$ and $Lu=f$. Equivalently, the standard local regularization argument gives the same conclusion for weak solutions.

On $y^2=x^3+a_2{x}^2+a_{4}x+a_6$, let $\lambda$ be the secant slope through distinct points $P,Q$, or the tangent slope when $P=Q$. Then

The degree-$p$ Čech cochain group for an ordered cover is ${\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0}\cap \cdots \cap U_{i_p})$.

For per-capita birth and death rates $B$ and $D$, the rates are $b_n=Bn$ and $d_n=Dn$. If $\mu =\mathbb{E}{N}_t$ and ${\sigma }^2=\mathrm{var}{N}_t$, thenThe expected value sees only the net growth rate $B-D$, whereas the variance also sees the total event rate $B+D$.

Choose a closed one-form $\nu$ on $S^1$ with integral one and let $p:M\times S^1\to M$ be projection. Every rotation-invariant $k$-form is uniquelyThe exterior derivative acts componentwise. Averaging therefore proves thatis the displayed isomorphism.

If $A$ is an ample line bundle on a projective variety and $L$ is any line bundle, then $A^{\otimes m}\otimes L$ is very ample for all sufficiently large integers $m$. Taking $L$ to be trivial also makes $A^{\otimes m}$ very ample.

For the centered second-difference matrix with diagonal $-2$ and off-diagonals $1$, the weighted-Jacobi eigenvalues areThe method converges for every grid size exactly when $0<\omega \le 1$.

Multiplying both $A$ and $b$ by $-1$ leaves $D^{-1}A$ and the weighted Jacobi method unchanged. Thus the same formula applies to the positive second-difference matrix $T_n$ with diagonal $2$ and neighboring entries $-1$, the finite difference form of the one-dimensional Poisson equation with Dirichlet boundary conditions. Its eigenvectors have components $\sin (jk\pi /(n+1))$, and $D^{-1}{T}_n$ has eigenvalues $1-\cos (k\pi /(n+1))$. At $\omega =2/3$, upper-half modes have factors between $-1/3$ and $1/3$, while low-frequency factors remain close to one on fine grids. This explains why the weighted Jacobi method helps a multigrid method by reducing oscillatory errors, even when it is a slow solver on its own.^[ref]

On a genus-two smooth projective curve, the gap set at any point is either $\{1,2\}$ or $\{1,3\}$. The integer $1$ is always a gap on a positive-genus curve. If $2$ is not a gap, then $3$ must be a gap, since pole orders are closed under addition and $2,3$ would otherwise generate every integer at least two.

The Virasoro central extension is the term $c(m^3-m){\delta }_{m+n,0}/12$ in $[L_m,L_n]$. It records the conformal anomaly and commutes with every Virasoro generator.

Suppose that for every $x\in U$ there is a $y\in V$ such that $\mathcal{A}$ is fixed by the smaller compression $C_{U\setminus \{x\},V\setminus \{y\}}$. ThenIndeed, deleting an element outside $U$ from a newly compressed member gives the $UV$-compression of an old shadow member. If the deleted element is $x\in U$, stability under the chosen smaller compression shows that the resulting set already belongs to the old lower shadow. Thusand compression preserves cardinality.

A set family is left-compressed when replacing a member $j$ by a missing smaller element $i<j$ always produces another member. Equivalently, it is fixed by every $\{i\},\{j\}$-compression with $i<j$.

The time-evolution operator maps an initial quantum state to its state at a later time. For a time-independent Hamiltonian operator $H$, it is the unitary operator $U(t)=e^{-iHt/\hslash }$.

More generally, write $|\psi (t)⟩=U(t,t_0)|\psi (t_0)⟩$. The time-evolution operator obeysFor a constant self-adjoint Hamiltonian operator, $U(t,t_0)=\exp [-iH(t-t_0)/\hslash ]$. With zero as the initial time, this gives the group law $U(t+s)=U(t)U(s)$. Successive evolution is reversible; a measurement followed by conditioning on an outcome is a separate operation described by projective measurement.

For a time-dependent Hamiltonian operator, the defining equation is $i\hslash {\partial }_{t}U(t,t_0)=H(t)U(t,t_0)$. In finite dimension, a continuous Hermitian $H(t)$ still gives unitary time evolution, sinceThe solution is a time-ordered exponential, with later factors acting on the left. One may replace it by $\exp [-i{\int }_{t_0}^{t}H(s)ds/\hslash ]$ when $[H(s),H(s^{\prime })]=0$ at all times; independently changing pulse axes generally requires time ordering.^[ref]

For a unitary operator $U$ on a Hilbert space, the Cesàro averagesconverge strongly to the orthogonal projection onto the fixed-point subspace $\ker (U-I)$.