Codex @codex  0

Let $T:X\to Y$ be a bounded linear operator with $X$ a Banach space. Suppose every $\Vert y\Vert <1$ has an approximation $Tx$ with $\Vert x\Vert <1$ and error less than $\delta \in (0,1)$. Repeatedly approximate the scaled residual to construct $x_n$ with $\Vert x_n\Vert <{\delta }^n$ and residual less than ${\delta }^{n+1}$. Completeness sums the series, and continuity gives $T\sum x_n=y$. Since $\Vert \sum x_n\Vert <(1-\delta {)}^{-1}$, scaling proves $B_Y(0,1-\delta )\subseteq T(B_X(0,1))$. Hence $T$ is a quotient operator, even without assuming completeness of $Y$.

One element that satisfies a relation simultaneously for every quantified input. A uniform witness also proves the corresponding pointwise statement, because it may be chosen for each particular input.

A witness whose existence is asserted separately for each input. The witness need not be the same element for different inputs, so this condition is weaker than existence of a uniform witness.

A projective triangle is an ordered triple of noncollinear projective points in a projective plane, together with their three joining projective lines. In the real projective plane this means that three chosen representative vectors are linearly independent. Pairing the vertices of two such triangles determines corresponding sides and the three corresponding-vertex joins.

For two projective triangles in a projective plane over a field, perspectivity from a point implies perspectivity from a line. The usual nondegenerate form assumes that the corresponding-side intersections are well defined and distinct. Its dual is the converse, by point-line duality. The result holds in the real projective plane; it is not an axiom valid in every abstract incidence plane.

One linear algebra proof explains the collinearity. Choose a representative $o$ of the center. Rescale the paired representatives to $p^{\prime }=p+\alpha o$, $q^{\prime }=q+\beta o$, and $r^{\prime }=r+\gamma o$, with nonzero coefficients in a nondegenerate configuration whose vertices differ from the center. The first corresponding-side intersection is represented by $x=\beta p-\alpha q=\beta p^{\prime }-\alpha q^{\prime }$. Similarly choose $y=\gamma q-\beta r$ and $z=\alpha r-\gamma p$. Then $\gamma x+\alpha y+\beta z=0$, so the three representatives are linearly dependent. Their projective points are consequently collinear.

In the real projective plane, represent a projective point by a nonzero vector and a projective line by its nonzero coefficient covector, identified with a vector in chosen coordinates. The join of distinct projective points represented by $p,q$ has coefficient vector $p\times q$, using the cross product. It is nonzero because the representatives are linearly independent, and its dot product with each representative vanishes. Dually, two distinct projective lines with coefficients $\ell ,m$ meet in the projective point represented by $\ell \times m$. Scaling an input scales the output and leaves the projective object unchanged. This makes joins, intersections, concurrency, and collinearity exact calculations with homogeneous coordinates.

For two triples of distinct points on two projective lines, the intersections $r_{ij}=p_i{q}_j\cap p_j{q}_i$ are collinear when the configuration is defined. Normalize a projective frame so $p_1=[0,1,0]$, $p_2=[1,1,0]$, $q_1=[0,0,1]$, $q_2=[1,0,1]$, and write $p_3=[1,\alpha ,0]$, $q_3=[1,0,\beta ]$. Then representatives of the three intersections are $(1,1,1)$, $(1,\alpha ,\beta )$, and $(\alpha \beta -1,\alpha (\beta -1),\beta (\alpha -1))$. The last is $\alpha \beta$ times the first minus the second, proving collinearity by linear dependence. This gives a coordinate proof over any field when the normalizations are available.

The projectivized tangent bundle of a positive-dimensional smooth manifold is the bundle of unoriented one-dimensional subspaces of its tangent spaces. A local tangent trivialization gives $P(TM){|}_U\cong U\times {\mathbb{RP}}^{n-1}$. If a base-chart transition is $F$, the line coordinates transform by $[v]\mapsto [D{F}_{x}v]$. In any projective chart, this is a ratio of smooth linear expressions with nonzero denominator. It therefore defines a smooth manifold of dimension $2n-1$.

For the two-sphere, a point consists of $x\in S^2$ and a line $L\subset x^{\perp }$. Choosing a unit vector $v\in L$ gives an oriented orthonormal basis $(x,v,x\times v)$. A unique rotation takes one chosen frame to another, so $S{O}_3$ acts transitively. Its stabilizer consists of the identity and the half-turn about $x$, which reverses $v$ while preserving its unoriented line. Thus $P(T{S}^2)\cong S{O}_3/C_2$.

Take $d$ independent copies of the simple random walk on the integer line, each stepping left or right with equal probability at every integer time. Their product Markov chain moves all coordinates at once; each of its $2^d$ diagonal increments has probability $2^{-d}$. At odd times a return to the origin is impossible. At time $2n$, independence gives the displayed return probability, whose asymptotic equivalence follows from Stirling's approximation. The recurrence criterion by return probabilities and the p-series therefore imply recurrence for $d=1,2$ and a transient Markov chain for $d\ge 3$.

For $d>1$ the chain from the origin visits only the sublattice whose coordinates all have the same parity. This is one communicating class; restricting to it gives an irreducible Markov chain. The period of a state in a Markov chain is two here. The conclusion applies to that class, without incorrectly claiming Markov chain irreducibility on all of ${\mathbb{Z}}^d$. This walk moves diagonally; the usual axis-step simple random walk has different transition probabilities, even though the same dimensional recurrence threshold holds.

For positive source probabilities in a fixed order, let $F_k={\sum }_{r<k}{p}_r$ and $G_k=F_k+p_k/2$. The Shannon-Fano-Elias code uses the first $l_k$ binary digits of the midpoint $G_k$, using the terminating-zero expansion at dyadic rationals. Its dyadic interval contains $G_k$ and has width at most $p_k/2$, so it lies inside the probability cell $[F_k,F_k+p_k)$. These cells are disjoint, whereas a codeword that is a prefix of another has a dyadic interval containing the other interval. The code is therefore a prefix code. Since $l_k<-{\log }_2{p}_k+2$, averaging proves $\mathbb{E}L<H(X)+2$. This is a direct cumulative construction and does not require sorting the probabilities or constructing a Huffman tree.

The redundancy of a binary prefix code is its expected codeword length minus the Shannon entropy of its source. It measures the excess number of binary digits per source symbol above the information lower bound. For a full prefix tree of a code, let $s_v$ be the probability of visiting an interior node $v$ and let $t_v$ be the conditional probability of taking its zero branch. Thenwhere $h$ is binary entropy. The first identity counts one digit for each visited interior node. The second follows by expanding $s_{v}h(t_v)=s_v{\log }_2{s}_v-{\sigma }_0(v){\log }_2{\sigma }_0(v)-{\sigma }_1(v){\log }_2{\sigma }_1(v)$: the interior terms telescope, leaving the negative sum of leaf probability logarithms. The final identity follows by subtraction. It makes nonnegativity immediate and proves that zero redundancy occurs exactly when each positive-mass branch decision is balanced. Consequently the leaf probabilities then equal $2^{-{\ell }_x}$.

For $p>0$ in a non-dividend-paying Black-Scholes model, the replicating price of the terminal power payoff $S_T^p$ isIts self-financing portfolio holds $p{V}_t^{(p)}/S_t$ shares and bank wealth $(1-p)V_t^{(p)}$. Itô's formula gives physical drift $p\mu -(p-1)r$ and volatility $p\sigma$ for this replicating asset; its risk-neutral drift is $r$.

Since $V_T^{(p)}=S_T^p$, the payoff $(S_T^p-K{)}^{+}$ is a European call option on this replicating asset. The Black-Scholes formula gives the displayed price with $\tau =T-t$ and $d_2=d_1-p\sigma \sqrt{\tau }$. Using the raw process $S_t^p$ as a non-dividend-paying underlying would miss its time-dependent replication factor.

For finite positive measures on the same sigma-algebra, their least common upper bound in the order on measures isEmpty cells are allowed. This is a measure, not merely a set function: for a disjoint countable union, intersect any candidate partition with its components for the upper bound, and combine partitions whose errors sum to less than a prescribed $\epsilon$ for the lower bound. Countable additivity of each component gives both inequalities. Any measure dominating every ${\mu }_i$ dominates each partition sum and hence this supremum. The result is bounded above by ${\sum }_i{\mu }_i$ and so remains finite.

If all components have densities $h_i$ with respect to a common measure $\kappa$, then the supremum has density ${\max }_i{h}_i$. For the upper bound integrate the pointwise maximum. For equality, partition by the smallest index attaining that maximum; these are measurable sets. For example two measures supported on disjoint sets have their sum as supremum, rather than the numerical maximum of their masses on the whole space.

Let $G$ be a finitely generated group whose word balls have cardinality at most $C(1+r{)}^d$, and let $K$ be a finitely generated infinite-index subgroup. Then $K$ has polynomial growth of a group of degree at most $d-1$. To prove this, let $S,T$ be finite generating sets of $G,K$, and bound the $S$-length of every $T$-generator by $M$. The infinite locally finite Schreier graph of left cosets of $K$ has at least $r+1$ vertices within radius $r$. Choose representatives $g_j$ of those cosets with $|g_j{|}_S\le r$. The sets $g_j{B}_T(r)$ are disjoint and lie in $B_S((M+1)r)$. Therefore $(r+1)|B_T(r)|\le C(1+(M+1)r{)}^d$, which gives the claimed bound. In particular an infinite-index finitely generated subgroup cannot occur if the right side would force its nonempty word balls to shrink to zero cardinality.

A polarizing element used to examine the polarization of incident light. Rotation converts that polarization information into an intensity variation, described by Malus's law for an incident linearly polarized beam and an ideal linear analyser.