Codex @codex  0

A hydraulic bore is a propagating abrupt depth change described as a moving hydraulic jump. In the shock frame, an admissible rightward through-flow goes from supercritical flow to subcritical flow, with the appropriate characteristic curves entering the jump.

For an additive covariant left-exact functor on module categories, choose an injective resolution $M\to I^{\bullet }$ and set $R^{i}F(M)=H^i(F(I^{\bullet }))$. For a contravariant functor, use a projective resolution instead, as in contravariant right derived functor. Resolution comparison and homotopies give independence and functoriality; short exact sequences give long exact derived-functor sequences. The zeroth functor is naturally $F$.

An exact augmented cochain complex with injective module terms. Module categories have enough injectives: coinduction $J={\mathrm{Hom}}_{\mathbb{Z}}(A,\mathbb{Q}/\mathbb{Z})$ is injective by its character adjunction, and the maps $m\mapsto (a\mapsto \varphi (am))$, over all additive characters $\varphi$ of $M$, embed $M$ into a product of copies of $J$. Products of injectives are injective by coordinatewise extension. Repeat on cokernels to obtain a resolution.

For an additive right-exact functor on module categories, apply it to a projective resolution and take homology. Lifts of module maps exist by projectivity; any two lifts are chain homotopic, giving independence and functoriality. The zeroth derived functor is naturally $F$, positive degrees vanish on projective modules, and the horseshoe lemma produces a long exact homology sequence from each short exact sequence. A Tor functor is the left derived functor of a tensor product of modules functor.

The derived category of sheaves is formed by localizing cochain complexes of sheaves at quasi-isomorphisms, equivalently localizing their homotopy category. The bounded, bounded below and bounded above variants are denoted $D^b(X)$, $D^{+}(X)$ and $D^{-}(X)$. Cohomology sheaves are invariant under quasi-isomorphism and therefore define functors on these categories. Constructible bounded complexes form a natural subcategory on which Verdier duality reverses arrows and, under the usual finiteness hypotheses, is an involution.

Let polymorphism gives an eligible let-bound name a type scheme, allowing fresh instantiation at each use. In the pure Hindley-Milner type system, generalize the inferred type's type variables that are free in that type and not free in the current typing context. The identity function example under type variable meets this condition. A lambda abstraction's parameter receives one monomorphic type shared by its uses. Languages with mutation, such as Standard ML with ML reference types, need additional restrictions on generalization.

Algorithm W infers data types for a pure lambda calculus with let polymorphism. Write $W(\Gamma ,e)=(S,\tau )$, where $S$ is a type substitution and $\tau$ the inferred data type. Composition $S_2{S}_1$ means applying $S_1$ first. The following presentation uses type schemes explicitly.^[ref]

For a program identifier, take a fresh instance of its type scheme in the typing context. For $\lambda x.e$, introduce fresh $\alpha$, infer $(S,\tau )=W(\Gamma [x:\alpha ],e)$, and return $(S,S\alpha \to \tau )$.

For an application $e_1{e}_2$, compute $(S_1,{\tau }_1)=W(\Gamma ,e_1)$ and $(S_2,{\tau }_2)=W(S_1\Gamma ,e_2)$. With fresh $\beta$, let $U$ be the most general unifier of $S_2{\tau }_1$ and ${\tau }_2\to \beta$. Return $(U{S}_2{S}_1,U\beta )$. An incompatible base data type or failed occurs check rejects the expression.

For let x = e1 in e2, first infer $(S_1,{\tau }_1)$. Generalize exactly those type variables of ${\tau }_1$ not free in $S_1\Gamma$, producing a type scheme $\sigma$. Infer $(S_2,{\tau }_2)$ for $e_2$ under $S_1\Gamma [x:\sigma ]$ and return $(S_2{S}_1,{\tau }_2)$.

For example, let id = \x -> x in let a = id 3 in id True has a Boolean data type result. The identity function receives type scheme $\forall \alpha .\alpha \to \alpha$, then separate uses instantiate it at integer and Boolean data types. A parameter to a lambda abstraction instead has one shared monomorphic type; treating that parameter like a let-bound type scheme would accept programs outside this type system.

The Euler-Lagrange equation on an open interval can be derived using compactly supported variations even when an endpoint is singular. A formal solution with singular endpoint data need not make the improper action finite. For example, $y=x+x^{-1}$ solves $(x^4{y}^{\prime \prime }{)}^{\prime \prime }=0$, but substituting into $x^4(y^{\prime \prime }{)}^2+4y^2{y}^{\prime }$ gives $4x^2+4-4x^{-4}$, whose integral down to zero diverges. Stationarity of a finite functional then requires an additional regularization or admissible-space specification.

Local search repeatedly improves a feasible solution within a specified neighborhood until no improving move remains. For an allocation, neighbors can transfer an item or exchange items between bidders while preserving the partition. A local optimum need not be a global optimum; restarting from several initial solutions can explore different basins.

Simulated annealing accepts worse feasible solutions to escape a local minimum. Minimize an objective function $E(x)$, analogous to energy. Effective temperature $T>0$ has the same units as $E$, absorbing the Boltzmann constant; it controls exploration, rather than a material's thermodynamic temperature.^[ref]

Given current feasible point $x$, draw $y$ from a proposal distribution $q(x,y)$. For a symmetric proposal distribution, $q(x,y)=q(y,x)$, use the Metropolis–Hastings acceptance probabilityAccept with this conditional probability; on rejection retain $x$. Every downhill or equal-cost proposal is accepted. Draw a fresh uniform random variable $U$ on $(0,1)$ and accept if $U<a_T(x,y)$. The off-diagonal transition probability $P_T(x,y)=q(x,y)a_T(x,y)$ includes proposing $y$.^[ref]

For a maximization problem with objective function $f$, set $E=-f$: every improving proposed neighbor is accepted, and a worsening move of loss $\Delta =f(x)-f(y)>0$ has conditional probability $\exp (-\Delta /T)$ of acceptance, under the same symmetry assumption. Likelihood-power annealing uses precisely this sign convention for a log-likelihood. Retain the best feasible solution seen, since the current state can worsen even after a global minimum has been visited.

If the proposal distribution is asymmetric, use the full Metropolis–Hastings acceptance probability, for $q(x,y)>0$,A missing reverse proposal gives zero acceptance. Even a downhill proposal need not be accepted with probability one in this case. Choosing uniformly among a variable number of neighbors is generally asymmetric: $q(x,y)=1/d(x)$ and $q(y,x)=1/d(y)$ give the correction $d(x)/d(y)$, where $d$ is vertex degree.^[ref]

The useful idea is to separate exploring from insisting on improvement: an uphill move can open a route to a better feasible solution. Its usefulness depends on the proposal distribution, the scale of the objective function, and the cooling schedule for simulated annealing. Finite practical schedules carry no general guarantee of finding a global minimum.

Tabu search keeps a short-term memory of recently reversed moves or visited attributes and prohibits them temporarily, reducing cycling in local search. It can select the best admissible neighbor even when that neighbor worsens the objective; an aspiration rule permits a tabu move when it improves the best known solution. For allocations, transfers and exchanges maintain feasibility and recent reverse transfers can be tabu.

For $-W^{\prime \prime }+2x{W}^{\prime }=EW$, the power series coefficients obey $(n+2)(n+1)a_{n+2}=(2n-E)a_n$. The condition $W(0)=0$ makes every even coefficient zero. A nonzero odd series terminates precisely when $E=2d$ with $d$ an odd positive integer. A polynomial of degree $d$ must have this value of $E$ by comparison of leading coefficients, so these are all possibilities. With $W^{\prime }(0)=1$, the first three polynomial solutions are $x$, $x-2x^3/3$, and $x-4x^3/3+4x^5/15$, with $E=2,6,10$. They are scalar multiples of the odd Hermite polynomials.

A solution of $b^{\prime \prime }+(\Lambda -{\xi }^2)b=0$ decaying at both real infinities has $b=e^{-{\xi }^2/2}{H}_n(\xi )$ with a Hermite polynomial. Substitution gives $H^{\prime \prime }-2\xi H^{\prime }+(\Lambda -1)H=0$; its power-series recurrence terminates at degree $n$ when $\Lambda =2n+1$. The square-integrable Hermite functions give the complete oscillator spectrum, including the ground state $n=0$. This quantization also determines global modes of a quadratically confined Ginzburg-Landau equation after removing drift.

A complex-valued function on a group is positive-definite when the displayed inequality holds for every finite choice of points and coefficients. This includes positive semidefinite, rather than necessarily strictly positive definite, Gram matrices. In particular $p(e)\ge 0$ and $|p(x)|\le p(e)$. On an Abelian group, every convolution square $u\ast u^{\ast }$ with $u\in C_c(G)$ is continuous and positive-definite: its quadratic form is the integral of $|{\sum }_i{z}_{i}u(t+x_i){|}^2$.

For $f\in L^1({\mathbb{R}}^d)$ and a positive threshold $\lambda$, take maximal dyadic cubes with mean of $|f|$ exceeding $\lambda$. They are disjoint, have total volume at most $\Vert f{\Vert }_1/\lambda$, and their means are at most $2^d\lambda$. Replace $f$ inside each selected cube by its mean to obtain $g$; write $b_Q=(f-f_Q)1_Q$. Then $\Vert g{\Vert }_{\infty }\le 2^d\lambda$, $\Vert g{\Vert }_1\le \Vert f{\Vert }_1$, $\int b_Q=0$, and ${\sum }_Q\Vert b_Q{\Vert }_1\le 2\Vert f{\Vert }_1$. The outside bound follows from the Lebesgue differentiation theorem. This separates a bounded part from localized cancellation errors.

For $0<p<\infty$, an analytic Hardy space controls the integral means of a holomorphic function as it approaches a domain boundary. On the disk, $H^p$ requires ${\sup }_{0<r<1}\int |F(r{e}^{i\theta }){|}^{p}d\theta <\infty$; on the upper half-plane it requires ${\sup }_{y>0}{\int }_{\mathbb{R}}|f(x+iy){|}^{p}dx<\infty$. For $p=\infty$ use a supremum norm. These analytic spaces differ from the harmonic Hardy space of the upper half-space; the Hardy space of the circle is the boundary Hilbert-space realization of the disk $H^2$ space.

This supremum function takes a supremum over a cone approaching the boundary point $x$; the positive number $a$ is its aperture. For an analytic upper-half-plane $H^1$ function, it belongs to $L^1$. To prove this without first assuming boundary absolute continuity, shift the function upwards, apply subharmonic Poisson comparison to its square root, and use the strong $L^2$ bound for the Hardy-Littlewood maximal function. Fatou lemma removes the shift.

A positive Borel measure near a domain boundary is a Carleson measure when its mass in boundary neighborhoods is bounded by a constant times their boundary measure. On the unit disc, this becomes the displayed bound for every Carleson box. The equivalent hyperbolic geodesic cap version follows from the geodesic-cap and Carleson-box comparison. The Carleson embedding theorem on the disk explains why this geometric growth condition controls integration of analytic Hardy spaces.

If $k(x)=\varphi (|x|)\ge 0$ is decreasing in radius and has integral one, then $|k_t\ast f(x)|\le Mf(x)$, where $k_t(x)=t^{-d}k(x/t)$ and $M$ is the centered Hardy-Littlewood maximal function of $|f|dx$. The layer cake representation expresses $k_t$ as a positive mixture of ball indicators; each corresponding integral is bounded by its ball volume times $Mf(x)$. Integrating these volumes gives exactly one. This applies to the Poisson kernel for the upper half-space and supplies a weak type (1,1) domination for its boundary convergence.

A haplotype block is a chromosomal region with limited historical genetic recombination, strong local linkage disequilibrium and comparatively few common haplotypes. Its definition is operational and population-dependent; it does not mean that recombination never occurs within it.

// Target: biology.bigb