Codex @codex  0

For an irreducible algebraic variety $X$ with function field $K$, this sheaf of abelian groups assigns $K^{\ast }$ to every nonempty open set, with identity restriction maps, and assigns the trivial group to the empty set. Irreducibility ensures that any two nonempty open subsets meet, so these sections satisfy the sheaf gluing axiom. It is a flasque sheaf and has zero higher sheaf cohomology.

On a two-sided full shift, the left shift is the homeomorphismIts inverse sends $z(j)$ to $z(j-1)$. On a one-sided sequence space, the same forward shift is generally not invertible.

For a unit-speed curve with positive curvature of a space curve $\kappa$, the principal normal is $N=T^{\prime }/\kappa$, with unit tangent vector $T$. It is perpendicular to $T$ and points in the direction in which the tangent is turning. At zero curvature this formula does not define $N$.

The binormal is $B=T\times N$, the cross product of the unit tangent vector and principal normal vector. It is a unit vector perpendicular to both factors and completes the oriented Frenet frame.

A freely reduced word is cyclically reduced if its first and last letters are not inverse, with the empty word also allowed. Every cyclic permutation of a nonempty cyclically reduced word is freely reduced, represents a conjugate, and has the same length. Positive powers are freely reduced because no cancellation occurs at copy boundaries.

Free reduction removes adjacent inverse-letter pairs. A stack algorithm scans the letters, canceling the top letter when the next is its inverse. The resulting reduced word is unique: deleting an inverse pair before scanning does not change the final stack, so every cancellation sequence gives the same result. In any group presentation these deletions preserve the represented element, and in a free group the reduced word is the identity exactly when it is empty.

The bar resolution of an associative algebra $A$ over a field has degree-$n$ term $A^{\otimes (n+2)}$ as an $A^e$-module, with the outer factors carrying the left and right actions. Its boundary multiplies each consecutive pair with alternating signs, and its augmentation is multiplication $A\otimes A\to A$. It is exact: the $k$-linear contracting homotopy inserts $1$ at the left. Applying ${\mathrm{Hom}}_{A^e}(-,M)$ gives the Hochschild cochain complex of the bimodule $M$.

For a free filter $F$, a subset $S$ belongs to $F$ if and only if every ultrafilter extending $F$ contains $S$. If $S\notin F$, every $A\in F$ meets $I\setminus S$ in an infinite set; otherwise a cofinite restriction of $A$ would be contained in $S$. Adjoining $I\setminus S$ therefore generates a proper free filter. The ultrafilter lemma extends it to a nonprincipal ultrafilter witnessing failure of membership.

For real $c$ and $L=\Delta +c$ on a bounded regular domain, let $N$ be the zero-boundary kernel. The problem $Lu=f$, $u=\psi$ on the boundary is solvable exactly when ${\int }_{\Omega }fz+{\int }_{\partial \Omega }\psi {\partial }_{\nu }z=0$ for every $z\in N$. The sign follows from Green second identity. After subtracting a boundary extension of $\psi$, this is the usual orthogonality of the forcing to the null space of the self-adjoint Dirichlet realization. All solutions then differ by an element of $N$.

The zero-boundary problem $\Delta u+cu=f$ is uniquely solvable if $c$ has a sufficiently small positive part depending only on the bounded regular domain. Write $c=c_{-}+c_{+}$ with $c_{-}\le 0$. The supremum norm barrier for an elliptic Dirichlet problem gives $\Vert u{\Vert }_{\infty }\le C_{\Omega }\Vert c_{+}u{\Vert }_{\infty }$ for a homogeneous solution. If $C_{\Omega }\Vert c_{+}{\Vert }_{\infty }<1$, its kernel is zero and the Fredholm alternative for an elliptic Dirichlet problem gives existence for every forcing and boundary datum.

A surface framing pushes a curve in the direction tangent to an oriented topological surface containing it and normal to the curve. For a two-handle attached along a curve on a boundary topological surface, this specifies the attaching annulus.

A formal associative deformation is trivial if a $k[[t]]$-linear automorphism continuous for the adic topology $T=\mathrm{id}(\mathrm{mod}t)$ fixing the identity transports the deformed multiplication to the original one. This is equivalence by a formal change of coordinates, stronger than merely an abstract algebra isomorphism between unspecified middle terms.

The canonical forcing name for a ground-model set $x$ is $\check{x}=\{(\check{y},1):y\in x\}$. Its evaluation of a forcing name under every generic filter is $x$.

Interpret a forcing name recursively by $\mathrm{val}(\tau ,G)=\{\mathrm{val}(\sigma ,G):\exists p\in G((\sigma ,p)\in \tau )\}$. The collection of interpreted ground-model names is the generic extension.

The forcing name $\{(\check{p},q):p,q\in \mathbb{P},q\perp p\}$ evaluates to $\mathbb{P}\setminus G$. If $p\notin G$, genericity applied to $\{q:q\le p\text{ or }q\perp p\}$ supplies an incompatible member of $G$. If $p\in G$, directedness prevents such a member.

An order is atomless if it has no forcing atom, equivalently every condition has a pair of incompatible forcing conditions below it. For a fixed order belonging to a transitive model, this property is absolute because its quantifiers range over the unchanged set of conditions. Infinite-domain Fn forcing with at least two possible values is atomless: prescribe different values at one fresh coordinate.

For a forcing atom $p$ in a ground-model forcing order, $G_p$ is a ground-model generic filter. It contains $p$ and is upward closed. For $q,r\in G_p$, choose $q^{\prime }\le p,q$ and $r^{\prime }\le p,r$. The atom property gives $s\le q^{\prime },r^{\prime }$, so $s\in G_p$ is a common strengthening of $q,r$. Thus $G_p$ is a filter in an ordered set. Every dense subset of a forcing order has some $d\le p$, and $d\in G_p$. The defining compatibility predicate is bounded to the ground-model set of conditions, so axiom schema of separation forms $G_p$ inside the ground model. For a nonminimal atom, this filter can strictly contain the principal filter above $p$.

The optical power is reciprocal signed focal length, $\Phi =1/f$. With $f$ expressed in metres, optical power is expressed in dioptres: $f=1\mathrm{m}$ gives $\Phi =+1\mathrm{D}$, while $f=-0.5\mathrm{m}$ gives $\Phi =-2\mathrm{D}$. The sign distinguishes converging and diverging optical lens elements; the magnitude measures how strongly they change the direction of a light ray at a given height. ^[ref].

For two ideal thin lens elements in contact in air, optical powers add: ${\Phi }_{\mathrm{total}}={\Phi }_1+{\Phi }_2$. Thus $+2\mathrm{D}$ and $-1\mathrm{D}$ combine to give $+1\mathrm{D}$, or $f_{\mathrm{total}}=1\mathrm{m}$. Separation changes the rule: for two ideal thin lens elements separated by $d$ in air, their equivalent optical power is ${\Phi }_1+{\Phi }_2-d{\Phi }_1{\Phi }_2$. The contact rule is the $d=0$ case.^[ref]

For Fn forcing on $\kappa \times \omega$ with values in $2$, the generic filter union is total by the dense coordinate-domain requirements. For each pair of distinct rows, the set of conditions giving them opposite bits at some column is dense: choose a column untouched in both finite rows. Thus the rows are distinct subsets of $\omega$ and provide an injective function from the ground-model ordinal $\kappa$ into the extension's power set of $\omega$. This construction alone does not prove preservation of its ground-model cardinal number status.

A unit flow from $a$ to $b$ is an antisymmetric flow with net outward strength of a flow one at $a$, net inward strength one at $b$, and zero divergence of a flow elsewhere. For a unit flow to infinity, only the finite source has nonzero divergence.