A primary state is a simultaneous eigenvector of $L_0,{\tilde{L}}_0$ with eigenvalues $h,\tilde{h}$ that is annihilated by all positive modes of both Virasoro algebras. Negative modes generate descendants. The state–operator correspondence identifies it with a primary operator inserted at the origin.

If $V$ satisfies the pricing equation for a local volatility model, hold $h_t^S=V_S(t,S_t)$ stocks and $h_t^B=[V(t,S_t)-S_t{V}_S(t,S_t)]/B_t$ units of the bank account. Their value is $V(t,S_t)$. The pricing equation for a local volatility model and Itô formula giveThus this is a self-financing strategy, with terminal value $g(S_T)$. It is a delta hedge, and nonnegative $V$ makes its wealth an admissible trading strategy. The stock and bank wealth amounts are $S_t{V}_S$ and $V-S_t{V}_S$, rather than the asset-unit holdings.

With $x=\log S$ and $u=e^{r(T-t)}V(e^x,t)$, a local volatility pricing equation becomes the displayed backward equation. Central finite differences and the explicit Euler method give $u_i^{j-1}=u_i^j+k[a_i^j{D}_{xx}{u}_i^j+b_i^j{D}_x{u}_i^j]$. Nonnegative weights require $k{\sigma }_i^2/h^2\le 1$ and $|b_i|h\le {\sigma }_i^2$. Under these conditions the update is a convex combination, yielding maximum-norm stability. With smooth coefficients, positive diffusion, fixed positive remaining maturity and suitable consistent boundary treatment, its ordinary error is $O(k+h^2)$. Volatility bounds alone do not imply all this regularity or a Greek error expansion.

A prequantum line bundle is a Hermitian line bundle over a symplectic manifold with unitary connection on a vector bundle of the displayed curvature. With $\omega =-d\theta$, a local connection is $\nabla =d-i\theta /\hslash$. Reversing the Hamiltonian-vector-field convention or passing to the dual line bundle reverses the corresponding curvature convention. Integrality of $\omega /(2\pi \hslash )$ on closed two-cycles is the topological existence condition.

For ${\iota }_{X_f}\omega =df$ and $F_{\nabla }=i\omega /\hslash$, the curvature commutator and $[X_f,X_g]=-X_{\{f,g\}}$ give $[\hat{f},\hat{g}]=i\hslash \widehat{\{f,g\}}$. In cotangent coordinates, $\theta =pdq$ gives $\hat{f}=-i\hslash X_f+f-\theta (X_f)$. Restriction to a vertical polarization yields $\hat{q}=q$ and $\hat{p}=-i\hslash {\partial }_q$.

A premeasure on an algebra of sets extends to a measure on the generated sigma-algebra. The extension is unique when the premeasure is sigma-finite.

For a commutative monoid object $(M,\ast ,e)$ in preframes, the indicated subset is a frame. Its bottom is $e$, its binary join is $\ast$, and its directed joins and finite meets are inherited. Every homomorphism from a frame regarded as a join monoid factors uniquely through this subset. This gives a coreflective subcategory.

A prefix scan computes successive accumulated results for an associative operation $\oplus$ while keeping input order fixed. A scan based on addition produces prefix sums; one based on matrix multiplication produces successive ordered matrix products. Associativity allows regrouping, while commutativity is unnecessary because the order stays fixed. If an initial empty prefix is included, its value is the identity element for the chosen operation.

The ability to compute accumulated results does not by itself allow subtraction of two prefixes. For example, two accumulated minima cannot reveal a block minimum: the lists $(1,7,8)$ and $(1,2,3)$ both have successive minima $(1,1,1)$, but the minimum of positions two through three is seven for one and two for the other. Prefix sums support block recovery because addition has additive inverses; an arbitrary associative operation need not have that property.

The lengths of every decipherable binary code satisfyConversely, any positive integer lengths satisfying this bound can be realized by a prefix code.

The prefix tree of a code has vertices corresponding to prefixes of codewords, with one edge for each next symbol. The root represents the empty word. In a prefix code, complete codewords are leaves, and their depths are their codeword lengths.

// Target: computer-science.bigb

For a success index $J$ with geometric distribution of parameter $q=2^{-n}$, write $J-1=2^{n}Q+R$, $0\le R<2^n$. Encode $Q$ by $Q$ ones followed by zero, then encode $R$ in exactly $n$ bits. This prefix code has length $n+Q+1$. Since $\mathbb{P}(Q\ge k)=(1-q{)}^{k{2}^n}$, its expected length is the displayed expression, at most $n+e/(e-1)$. Thus a very rare success can be communicated in approximately the logarithm of its inverse probability, rather than one bit per trial.

// Target: quantum-theory.bigb

On $\Omega =(0,\infty )$, observe only $X_t(\omega )=1_{\{t\le \omega \}}$. Before and at the observed lifetime, the natural filtration cannot distinguish any two surviving outcomes. The predictable sigma-algebra on positive times therefore consists exactly of sets whose trace on $D=\{\omega \ge t\}$ is $D\cap (\Omega \times T)$ for a Borel set $T$, and whose trace on $D^c$ is arbitrary product-Borel. To prove sufficiency, $D$ is predictable because $X$ is left-continuous and adapted; on $D^c$, insert a rational $s$ with $\omega <s<t$ and use the predictable rectangles $(E\cap (0,s))\times (s,\infty )$.

A predictable rectangle has the displayed form for $0\le s<t$. Its indicator is a left-continuous adapted process, and these rectangles generate the predictable sigma-algebra on positive times. If the time-zero slice is included, also use $B\times \{0\}$ with $B\in {\mathcal{F}}_0$.

A normal subgroup $N$ is powerfully embedded in a finite p-group or pro-p group $G$ if $[N,G]\le N^p$ for odd $p$, or $[N,G]\le N^4$ for $p=2$; use closed generated subgroups in the pro-p group case. The condition implies that $N$ is a powerful p-group. For odd $p$, both $[N,G]$ and $N^p$ remain powerfully embedded.

For odd $p$, set $P_0=G$ and $P_{i+1}=P_i^p$. Powerful embedding gives $[P_i,G]\le P_{i+1}$. Modulo $P_{i+2}$, commutators with $P_i$ are central and have exponent dividing $p$, so $(xy{)}^p\equiv x^p{y}^p$ when $y\in P_i$. Also the power map $P_i/P_{i+1}\to P_{i+1}/P_{i+2}$ is a surjective homomorphism. First find a root modulo $P_2$, then multiply it by successive corrections in $P_i$. The process terminates for a finite p-group, and converges for a pro-p group with separated power filtration. This proves that the generated power subgroup consists of actual powers.

For odd $p$, a subgroup $H$ of a finite powerful p-group $G$ satisfies $d(H)\le d(G)$. Here $d$ is the dimension of the Frattini quotient. Set $K=H\cap G^p$, $V=G/G^p$, $W=G^p/G^{p^2}$, and $A=H{G}^p/G^p$. The onto power map $\theta :V\to W$ sends $A$ into the image of $H^p$ in $K/\Phi (K)$. Since $\Phi (K)\le G^{p^2}$, this image has dimension at least $\dim \theta (A)$. Also $\Phi (K)\le \Phi (H)\le K$. Induction on $|G|$, applied to the powerful subgroup $G^p$, gives $d(K)\le \dim W$. ConsequentlyThe last inequality is rank-nullity theorem.

For an odd prime number $p$, a finite p-group that is a product of $d(G)$ cyclic subgroups is a powerful p-group. Indeed, $H=G/G^p$ has exponent of a finite group $p$ and the same minimal generator number $d$. Its cyclic factorization gives $|H|\le p^d$, while its Frattini quotient has order $p^d$. Thus $\Phi (H)=1$, $H$ is an elementary abelian group, and $[G,G]\subseteq G^p$. The condition on the number of factors is essential: the nonabelian Heisenberg group over ${\mathbb{F}}_p$ has exponent $p$ and is a product of three cyclic subgroups, although its minimal generator number is two.

The Baker--Campbell--Hausdorff formula converges integrally on a powerful Lie lattice. A degree-$m$ Lie term has bracket divisibility $p^{ϵ(m-1)}$; the standard integral coefficient bound loses at most $\lfloor (m-1)/(p-1)\rfloor$ powers of $p$. The remaining valuation tends to infinity. Associativity follows from the formal exponential identity, the identity is zero, the inverse is $-x$, and $x^{\ast n}=nx$. The resulting pro-p group is a uniform pro-p group, with power subgroups $p^{n}L$.

On the set underlying a uniform pro-p group, define scalar multiplication by $\lambda x=x^{\lambda }$, and define addition and the Lie bracket byRoots are the unique roots in the appropriate power layers. The limits give a powerful Lie lattice. In a suitable complete normed group algebra, the p-adic logarithm identifies these operations with ordinary addition and the associative-algebra commutator.

For a unit vector $u$ spanning the target eigenspace and a normalized iterate $x_k$, defineThis is the norm of the component in the orthogonal complement of $u$, equivalently the sine of the acute angle to its line. It ignores the arbitrary sign or complex unit factor of an eigenvector. Under power iteration convergence for a diagonalizable matrix, ${\epsilon }_k=O({\gamma }^k)$ in Big O notation. The orthogonal projection defines how we measure direction; it does not assume that the eigenbasis of $A$ is orthogonal. In computation, the projection form avoids subtracting nearly equal numbers inside the square root.