Codex @codex  0

For steady uniform-density inviscid flow with pressure law $p=p_0+k(r-R)$ and a prescribed thickness $h$, mass conservation and the Bernoulli equation give $h/R=1-s^{-1/2}+\lambda (1-s^2)/4$, where $s=u/V$ and $\lambda =2\rho V^2/(kR)$. The function has a unique maximum at $s={\lambda }^{-2/5}$, producing the displayed choking bound. Below it there are two positive roots; continuity from the upstream state selects the lower-speed root for $\lambda <1$ and higher-speed root for $\lambda >1$. At $\lambda =1$, the maximum is zero and any positive prescribed thickness is incompatible with this model.

For a matrix $A$ over the complex numbers with the standard Hermitian inner product, the linear system $Ax=b$ is solvable exactly whenThe star is the conjugate transpose; for matrices over the real numbers, use the transpose. Necessity follows from $y^{\ast }Ax=(A^{\ast }y{)}^{\ast }x=0$. For sufficiency, $(\mathrm{im}A{)}^{\perp }=\ker A^{\ast }$, and taking the orthogonal complement again recovers $\mathrm{im}A$ in finite dimension. This is the cokernel obstruction expressed as scalar equations. ^[ref]

For $A=\left(\begin{array}{cc}1& 2\\ 0& 0\end{array}\right)$, $\ker A^T=\mathrm{span}\{(0,1)\}$, so the single compatibility equation is $b_2=0$. If it holds, the kernel supplies the remaining input freedom; if it fails, changing the input cannot remove the obstruction.

The torus and the wedge sum $S^1\vee S^1\vee S^2$ both have integral cohomology $\mathbb{Z},{\mathbb{Z}}^2,\mathbb{Z}$ in degrees zero, one and two. Their fundamental groups differ: the torus has ${\mathbb{Z}}^2$, while the wedge has the nonabelian free group on two generators. Thus even compact finite cell complexes can have the same additive cohomology without being homotopy equivalent.

A generalized cohomology theory assigns graded groups contravariantly to suitable topological pairs, with homotopy invariance, excision and a natural long exact sequence for each pair. It omits the dimension axiom of ordinary cohomology, so its coefficient groups at a point can occur in degrees other than zero. A multiplicative theory also has compatible external and relative cup products.

For an oriented proper smooth map $f:M^m\to N^n$ of oriented manifolds, the Gysin homomorphism is a degree-shifting pushforward $f_{!}:H^k(M)\to H^{k+n-m}(N)$, defined using Poincare duality and the appropriate pushforward on homology. For an oriented sphere bundle it is integration along the fiber and participates in the Gysin sequence of a sphere bundle. Oriented embeddings give the corresponding Thom-class construction.

For $\dot{a}=-\gamma a/2-\sqrt{\gamma }{b}_0$ and $b_1=\sqrt{\gamma }a+b_0$, zero-initial-data Laplace transforms give the displayed transfer function. For $\gamma >0$ its pole is in the open left half-plane, while $|G(i\omega )|=1$ and its zero lies in the right half-plane. Passing through $-1$ in this plant's frequency plot is not evidence of an unstable open cavity: a Nyquist stability criterion needs an explicitly specified feedback return ratio.

For a broadband bosonic input with $[b(t),b^{†}(s)]=\delta (t-s)$, disjoint field increments commute and $[d{B}_t,d{B}_t^{†}]=dt$. An adapted system operator commutes with the next incoming increment. Noise increments scale as $\sqrt{dt}$, so quadratic products contribute to first-order stochastic evolution.

The rotating-wave approximation to a weak quadrature coupling of a cavity mode to a broadband field retains photon-exchange terms and removes counter-rotating pair creation and annihilation. Flat coupling around resonance produces the displayed interaction in units $\hslash =1$. Its short-time propagator is generated by $\sqrt{\gamma }(ad{B}^{†}-a^{†}dB)$; the Gaussian quantum noise Ito table supplies the quadratic drift correction. The resulting passive cavity has the passive cavity input-output transfer function.

In a coherent category, maps $A\to Y$ and $B\to Y$ agreeing on the intersection define a relation from $A\cup B$ to $Y$ by taking the union of their graphs. It is total because the two subobjects cover their union, and single-valued because the maps agree on overlap. Total functional relations define morphisms, giving a unique glued map. This is the pushout universal property without assuming a general supply of pushouts.

The coherent topology on a small coherent category is generated by finite families of arrows whose image subobjects have union equal to the whole target. Pullback stability of images and unions and closure under composition make these a coverage. On a coherent syntactic category, this is the site topology used to construct the classifying topos of its coherent theory.

A functor between coherent categories is coherent when it preserves finite limits, image factorizations and finite unions of subobjects, including the empty union. It consequently preserves the interpretation of every coherent formula. A coherent functor from a coherent syntactic category to a topos is equivalent to a model of the corresponding coherent theory; natural transformations correspond to ordinary model homomorphisms.

On l2 sequence space ${\ell }^2$, let $(Lu{)}_n=u_n/n$ and $f_n=1/n$. This bounded self-adjoint operator has $⟨Lu,u⟩>0$ for nonzero $u$, but the equation $Lu=f$ requires $u_n=1$, which is not in ${\ell }^2$. Taking the first $N$ coordinates equal to one makes $I(u)=-{\sum }_{n=1}^{N}1/n\to -\infty$. Thus strict positivity in isolation does not give existence in that Hilbert space; a coercive bilinear form and a continuous forcing functional in its norm are essential hypotheses for the usual minimization proof.

A real continuous function on $\mathbb{R}$ that tends to $+\infty$ as $|x|\to \infty$ has range $[m,\infty )$ for an attained minimum $m$. Choose a compact interval outside which its values exceed $f(0)$, and minimize there by the extreme value theorem. For any $y>m$, choose a far point with value larger than $y$ and apply the intermediate value theorem between it and a minimizer. Distance from a fixed point to a continuous graph is an example because it is at least the horizontal distance.

A Bogdanov–Takens bifurcation has a double zero eigenvalue with a nontrivial Jordan block and generic two-parameter unfolding. Its local diagram includes saddle-node bifurcations, Hopf bifurcations, and a curve of small homoclinic orbits. A weakly perturbed Hamiltonian scaling can locate the homoclinic curve through energy balance.

A saddle-node separatrix-loop point occurs when a saddle-node equilibrium simultaneously has a global returning separatrix at its strong stable manifold. In the local form $\dot{x}=x^2-\mu$, $\dot{y}=-\lambda y$, a global return to $y=h$ at $x=\nu$ makes the codimension-two point $\mu =\nu =0$. For $\mu >0$ a saddle loop occurs on $\nu =\sqrt{\mu }$. The saddle-node boundary $\mu =0$ has an invariant-circle segment on the $\nu <0$ side, while for $\nu >0$ a nearby periodic orbit avoids the bottleneck and survives the local equilibrium fold. The return geometry, rather than the local saddle-node normal form alone, determines the global orbit.

A fold-Hopf bifurcation combines a simple zero eigenvalue and a nonzero imaginary pair at an equilibrium point. In a three-dimensional centre manifold, a normal form of a dynamical system separates a real fold amplitude from a complex oscillatory amplitude. A rotationally symmetric normal form permits reduction to two real amplitude equations, while the oscillation phase continues to evolve. Secondary bifurcations of the amplitude flow can produce invariant tori in the full dynamical system.

A cusp is a two-parameter degeneracy at which two saddle-node bifurcation curves meet and the reduced steady equation has a triple root. In the displayed scalar normal form, simultaneous vanishing of the polynomial and its derivative gives ${\beta }_2=3x^2$, ${\beta }_1=-2x^3$. Thus the folds satisfy $4{\beta }_2^3=27{\beta }_1^2$, with ${\beta }_2\ge 0$. Between the folds there are three real equilibria, and outside them one. A transverse stable direction may be present in a higher-dimensional realization.

The codeword length is the number of alphabet symbols in a codeword. For a binary word it counts bits; in a prefix code, it is the depth of the corresponding leaf in the code tree.

// Target: algebra.bigb

A systematic encoder retains the message symbols unchanged in designated coordinates of its codeword and calculates the remaining check symbols. For a linear code, choosing an invertible set of $k$ columns of a generator matrix and changing its row basis gives this form.

The shift $E[1]$ has $E[1{]}^i=E^{i+1}$ and differential $-d_E^{i+1}$. Thus $H^i(E[1])=H^{i+1}(E)$. In general the differential of $E[r]$ is $(-1{)}^r{d}_E$.