Codex @codex  0

For simply connected rational spaces of finite type, isomorphic minimal Sullivan models are equivalent to the same rational homotopy type. A quasi-isomorphism between minimal models is an isomorphism, so a zigzag of cochain quasi-isomorphisms produces the same minimal model. Existence and uniqueness come from adjoining closed generators to represent missing cohomology and generators with prescribed decomposable differential to kill kernel classes, degree by degree.

For a map $f:E\to B$ with simply connected base and the usual finite-type hypotheses, choose a relative Sullivan algebra $M(B)\to M(B)\otimes \Lambda W$ modelling the path-space fibration, with acyclic total algebra. Tensor it over $M(B)$ with a model of $E$ via $f^{\ast }$. The resulting algebra models the homotopy fibre. Relative Sullivan extensions are cofibrant replacements, so this tensor product computes the derived algebraic pushout, contravariantly corresponding to the topological homotopy pullback. Linear differential pairs must be cancelled before reading rational homotopy groups from generators.

A relative Sullivan algebra is an inclusion $(A,d)\to (A\otimes \Lambda W,D)$ with a well-ordered homogeneous basis of $W$ such that the differential of a generator lies in the algebra on $A$ and earlier generators. It need not be minimal: a differential can have linear terms from earlier generators. Such extensions supply cofibrant replacements for commutative differential graded algebras. A relative acyclic extension of a sphere model represents its path-space fibration and computes a homotopy-fibre model after base change.

A subsequential limit of a sequence $(a_n)$ is a value $L$ approached by some subsequence $(a_{n_k})$ with strictly increasing indices $n_k$. For $a_n=(-1{)}^n(1+1/n)$, the subsequences at even integers and odd integers give subsequential limits one and minus one. A finite value appearing among only finitely many initial terms does not become a subsequential limit merely by being a member of the image of a function represented by the sequence.

For a bounded sequence of real numbers, its subsequential limits form a nonempty bounded set that is also a closed set. The Bolzano-Weierstrass theorem guarantees nonemptiness, and its extreme values are the limit inferior and limit superior. The sequence in sparse peaks can determine a limit superior has the two subsequential limits zero and one, despite almost all indices in the natural density sense carrying zero.

The closed nucleus $c(a)$ determines the complement of the open $a$. A sublocale with nucleus $j$ is dense precisely when $j(0)=0$, since its closure is the closed sublocale defined by $j(0)$.

A sublocale is dense when its closure is the whole locale. For a map $f$, its image is dense when $f^{\ast }(a)=0$ implies $a=0$. Classically, a dense point belongs to every nonzero open.

A sublocale is flat when its nucleus on a frame preserves finite joins, including $0$. Equivalently its fixed elements are closed under ambient finite joins. In particular every flat sublocale is a dense sublocale.

The double-negation nucleus on a frame gives a dense sublocale whose frame is a complete Boolean algebra. Its elements are regular open elements, with Boolean joins $\neg \neg {\bigvee }_i{a}_i$.

Let a centered Gaussian process $Z$ satisfy $Z(a\cdot )\stackrel{d}{=}{a}^{H}Z$ with $0<H<1$, and suppose $Z/\sqrt{L}$ has a large deviation principle in the sublinear path space for fluid queues, with speed $L$ and good rate function $I$. For $\delta ,\sigma >0$, put $r={\sup }_{t\ge 0}(\sigma Z(t)-\delta t)$ and $\kappa =2(1-H)$. Self-similarity gives $Z(N\cdot )/N\stackrel{d}{=}Z/\sqrt{N^{\kappa }}$. Time substitution and the contraction principle for large deviations therefore give an LDP for $r/N$ with speed $N^{\kappa }$ andThe same family $r/(aN)$ has rate function $J(aq)$ by contraction, and $a^{\kappa }J(q)$ by replacing $N$ with $aN$ and expressing the speed again as $N^{\kappa }$. Uniqueness of a rate function gives $J(aq)=a^{\kappa }J(q)$, hence $J(q)=q^{\kappa }J(1)$ for $q>0$, with $J(0)=0$.

For one-sided past inputs, let $A_n(a)={\sum }_{j=0}^{n-1}{a}_{-j}$ and $A_0=0$. The displayed metric controls cumulative differences over every past window, including the latest arrival. On the affine class $A_n(a)/n\to \lambda <c$, the queue workload $Q(a,c)={\sup }_{n\ge 0}(A_n(a)-cn)$ is locally Lipschitz continuous. To prove this, choose $\epsilon =(c-\lambda )/4$ and $N\ge 1$ such that $A_n(a)/n\le \lambda +\epsilon$ for $n\ge N$. If $d_{\#}(a,b)<\epsilon /2$, then $A_n(b)-cn\le n(\lambda +2\epsilon -c)<0$ for all such $n$. Both suprema reduce to $0\le n<N$, and their difference is at most $(N+1)d_{\#}(a,b)$. Ignoring the latest coordinate makes this assertion false for a post-slot workload of a queue.

Let ${\delta }_c=C_g{\tau }^2+C_d{\tau }^{-3}$ be the matter-era growing and decaying density modes, with radiation neglected in the background gravity. The radiation density contrast isThe formula follows by the Green's function of the forced harmonic oscillator. For $k\tau \gg 1$, the slowly varying particular response to the decaying mode begins at $16C_d{\tau }^{-5}/{\omega }^2$; lower-endpoint oscillations are absorbed into $C_1,C_2$. The growing mode drives a constant radiation offset, while cold dark matter continues growing as ${\tau }^2$.

For disjoint compact bodies, the minimum occurs on their boundaries. Patch bounding volumes give certified lower bounds and evaluated limit-point pairs give upper bounds. A branch and bound search refines promising patch pairs until the global bounds differ by the tolerance. Boundary distance alone cannot detect containment; touching, crossing or nested solid bodies have ordinary set distance zero. Negative overlap values require an additional convention such as translational penetration depth.

A plane section of a subdivision surface is a zero set of the plane functional evaluated on the limit parametric surface. Certified patch bounding volumes reject sign-definite patches. On a regular transverse section, $\nabla g\ne 0$ and the parameter tangent is $(-g_v,g_u)$; predictor-corrector continuation method traces a branch after every component has been isolated. Closed loops, tangencies, coplanar patches and extraordinary-vertex neighborhoods require more than corner-sign tests. Degenerate sections can contain isolated points or two-dimensional portions rather than only regular curves.

A robust intersection search applies branch and bound to pairs of subdivision surface patches. Certified bounding volumes reject disjoint pairs, and refinement reduces unresolved pairs. Limit evaluation, tangent data and controlled approximation bounds produce and correct local intersection arcs, then patch adjacency joins them. Bounds must enclose the limit, not merely the current mesh. Flat polygon intersections are approximations and can miss tiny components or tangencies unless the associated geometric and transversality conditions are verified.

A local subdivision matrix maps the control points in a refinement-invariant neighborhood to the refined neighborhood. Row sums one express constant reproduction. Its constant eigenvector has eigenvalue one; remaining eigenvalues govern decay of shape modes. Near an extraordinary subdivision vertex, tangent modes and a characteristic map of a subdivision surface are needed to test geometric smoothness, beyond simple convergence.

For backward differences $\Delta P_i=P_i-P_{i-1}$, a scalar stationary arity-$a$ subdivision mask has symbol $M(z)={\sum }_i{M}_i{z}^i$. The identity $(1-z)M(z)=B(z)(1-z^a)$ gives the displayed intertwining whenever the quotient is a finite Laurent polynomial. Iterating supplies higher-difference masks. The coefficients $a^{r\ell }{\Delta }^r{P}_i^{\ell }$ are candidate $r$th-derivative controls. A convergent derivative scheme, proved by contraction of its first differences, establishes corresponding smoothness of the original limit under the standard consistent parameterization.

The subdivision arity is the integer factor $a>1$ by which a stationary refinement shrinks parameter-grid intervals: control index $i$ has coordinate $i/a^{\ell }$ after $\ell$ levels. It is distinct from the mask width and from the number of neighbors in a stencil.

For a stationary univariate refinement $S_{ij}=a_{i-mj}$, the subdivision mask width counts fine-grid positions from the first to the last active entry in a column, including any internal zeros. Consecutive columns are translates by $m$ rows, the subdivision arity. The number of old control points used by a particular new point is instead the size of its row stencil; it need not equal the mask width. A span convention counts $w-1$ fine intervals, so numerical width claims should specify the convention.

For an oriented plane function $\ell$, certified extrema over a bounding volume allow a subdivision curve piece to be rejected when $\max \ell <0$, or wholly retained when $\min \ell \ge 0$. Otherwise subdivide. At an explicitly chosen rendering tolerance, a sufficiently flat, sufficiently short piece can be replaced by its clipped chord. This is an approximation, not a proof of exact crossing topology. Exact treatment of tangent contacts and small excursions requires root isolation of $\ell (C(t))$ and splitting into sign-constant intervals, when the curve representation supports such isolation.

For a point light at $L$, the shadow of a curve point is the first surface intersection on its ray beyond that point. Certified patch bounds and subdivision isolate intersections; regular hits are corrected with a three-variable Newton method solve. Differentiating the displayed equation gives $[S_u,S_v,-(C-L)](u^{\prime },v^{\prime },{\lambda }^{\prime }{)}^T=\lambda C^{\prime }$, a predictor for continuation along the shadow. Surface boundaries and grazing rays require clipping and reseeding; every visible component must be isolated, rather than assumed detectable from a fixed sample of the casting curve.