Codex @codex  0

For a figure-eight contour around $1$ anticlockwise and $-1$ clockwise, start $(t^2-1{)}^z$ at $t=0$ with argument $-\pi$. Shrinking the two loops gives the displayed identity. If the interval integral instead uses $(t^2-1{)}^z=e^{-i\pi z}(1-t^2{)}^z$, the corresponding identity includes the extra factor $e^{i\pi z}$. Omitting this phase changes the value, rather than merely the locations of the meromorphic continuation's poles.

If the derivative characteristic polynomial $\sigma$ of a linear multistep method has a simple root outside the unit disk and keeps its degree in the limit, then its absolute stability region is bounded. Otherwise a stable sequence with $|z|\to \infty$ would give amplification polynomials $\sigma -\rho /z$ tending coefficientwise to $\sigma$. A root near the exterior root would persist by continuity, contradicting the root condition for a multistep method. This argument excludes stability at every sufficiently large complex $z$, not only along the negative real axis.

An amplification root is a root of a polynomial governing a geometric mode $y_n={\xi }^n$ of a numerical recurrence on the Dahlquist test equation. Its modulus measures damping or growth per step. A repeated root produces additional modes such as $n{\xi }^n$; therefore a unit-circle root can be permitted by a bounded-root condition only when it is simple. The principal root approximates $e^z$ near zero, while additional roots can describe parasitic amplification roots.

In the notation of the toric basepoint-free criterion, the divisor is ample exactly when $⟨m_{\sigma },v_{\rho }⟩>-a_{\rho }$ for each ray outside $\sigma$. Equivalently, the normal fan of its divisor polytope is the original fan.

// Target: geometry-and-topology.bigb

A reduced alternating link diagram is an alternating link diagram with no nugatory crossing. Its Tait graph and dual have no loops, so it is an adequate link diagram. If connected, its Jones polynomial breadth equals its crossing count.

// Target: module-theory.bigb

For r colours, k letters and d variable blocks, set $P_i={\sum }_{j<i}{L}_j$ and $L_i=r^{k^{P_i+d-i}}$. Process blocks from right to left. In block i, the $L_i+1$ words $a^t{b}^{L_i-t}$ have colour profiles indexed by all $k^{P_i}$ earlier words and $k^{d-i}$ later variable assignments. Two profiles agree, giving a nonempty variable interval insensitive to exchanging a and b. Previously obtained insensitivity holds for all earlier words and survives their restriction. This constructs the combinatorial subspace required by the alphabet insensitivity lemma.

Massive stars produce prompt enrichment through core-collapse supernovae, whereas a Type Ia supernova follows binary stellar evolution and contributes substantial iron after a distribution of delays. A stellar population formed rapidly before much delayed enrichment can have a larger alpha-element to iron ratio than a stellar population formed over a prolonged interval. Relative abundance also depends on the initial mass function, stellar yields, and gas exchange; this is a chemical clock with model assumptions.

An alpha-2 adrenergic receptor is an adrenergic receptor commonly coupled to inhibitory Gi-family signalling. It can reduce adenylyl cyclase activity and alter ion-channel regulation. Presynaptic receptors can participate in feedback control of transmitter release, while receptors in other tissues have additional roles. Its effects therefore differ from the common Gq-linked signalling of an alpha-1 adrenergic receptor. ^[ref]

An alpha-1 adrenergic receptor is an adrenergic receptor commonly coupled to Gq-family signalling. Activation can stimulate phospholipase C, generating inositol trisphosphate and diacylglycerol, with subsequent changes in intracellular calcium and kinase activity. In smooth muscle, this can contribute to contraction. Coupling describes a common mechanism rather than a guarantee of an identical response in every cellular context. ^[ref]

If $\Omega ={\theta }_1\wedge \cdots \wedge {\theta }_n$ and $d\Omega =0$, differentiating ${\theta }_j\wedge \Omega =0$ gives $d{\theta }_j\wedge \Omega =0$. Only the $(0,2)$ component of $d{\theta }_j$ can survive this wedge product; multiplication by $\Omega$ is injective on that component. Thus no $d{\theta }_j$ has a $(0,2)$ part. The bracket and differential-form criteria for integrability make the induced almost complex structure integrable. Local decomposability is part of the hypothesis and is not implied solely by nonvanishing of $\Omega \wedge \overline{\Omega }$.

The group $I{S}_i^{\bar{p}}(X)$ consists of finite singular chains whose nonzero simplices are allowable singular simplices and whose boundary, after cancellation, also consists of allowable simplices. Since ${\partial }^2=0$, these groups form a chain complex. Its homology is intersection homology.

The Alfvén Mach number compares a characteristic flow speed with the Alfvén speed $v_A=B/\sqrt{{\mu }_0\rho }$. In an axisymmetric magnetohydrodynamic wind, the relevant poloidal quantity is $M_{Ap}=|{\mathbf{u}}_p|\sqrt{{\mu }_0\rho }/|{\mathbf{B}}_p|$. With magnetohydrodynamic mass loading $\rho {\mathbf{u}}_p=\kappa {\mathbf{B}}_p$, its square is ${\mu }_0{\kappa }^2/\rho$. The Alfvén surface is the locus $M_{Ap}=1$, and a smooth crossing requires the corresponding Alfvén-surface regularity condition for an axisymmetric wind. The total-speed and poloidal-speed definitions should be distinguished when there is substantial rotation or a strong toroidal magnetic field.

For a subset $S$ of the nonzero countable limit ordinals, let $B_{\alpha }=1+\mathbb{Q}$ when $\alpha \in S$ and $B_{\alpha }=\mathbb{Q}$ otherwise. The order sum is an aleph-one-like linear order whose nonempty proper initial segments are $\mathbb{Q}$ or $\mathbb{Q}+1$. At the limit cut before block $\delta$, the complementary final segment has a least point exactly when $\delta \in S$. Club agreement of countable filtrations makes this condition invariant modulo a nonstationary difference. Disjoint stationary subsets of omega-one consequently supply $2^{{\aleph }_1}$ nonisomorphic orders.

For a normal linear model with known error variance ${\sigma }^2$, the scaled Akaike criterion differs by a model-independent constant from

Starting from a fitted model, compare admissible single-term additions and deletions within specified lower and upper scopes, retaining hierarchical lower-order terms when an interaction requires them. Select a change reducing the Akaike information criterion and repeat until no admissible change improves it. For a Gaussian normal linear model with $N$ observations and $k$ fitted mean coefficients, the criterion used by R's linear-model stepwise search is $N\log (\mathrm{RSS}/N)+2k$, up to constants common to the models. The search is greedy and need not find the best model in the whole scope. Ordinary coefficient tests in the selected model do not account for this model selection. The software arguments are documented in the MASS documentation.

For a strongly convex rational cone $\sigma$, the group generated by $S={\sigma }^{\vee }\cap M$ is $M$. Inverting its finitely many monomial generators gives $k[M]$, so $\mathrm{Spec}k[M]$ is a dense open algebraic torus in $\mathrm{Spec}k[S]$. The map ${\chi }^m\mapsto {\chi }^m\otimes {\chi }^m$ extends torus multiplication to the entire variety.

// Target: geometry-and-topology.bigb

For $r>1$ and $\gcd (r,q)=1$, the quotient of ${\mathbb{A}}^2$ by $(x,y)\mapsto (\zeta x,{\zeta }^{q}y)$ is a cyclic quotient surface singularity. Its toric model uses the first-quadrant cone in ${\mathbb{Z}}^2+\mathbb{Z}(1,q)/r$. Conversely, every singular full-dimensional affine toric surface over an algebraically closed field of characteristic zero has this form. Smooth cones and cones with a torus factor require separate treatment.

// Target: geometry-and-topology.bigb

An arbitrary real affine equation has steady states exactly when $\mathrm{rank}A=\mathrm{rank}[A\mid -c]$. Its solution set is $s_{\ast }+\ker A$. For a finite-dimensional time-independent Lindblad equation, physical stationary states always exist by time averaging trajectories. If $A$ is invertible the steady state is unique. Conversely, a nonzero traceless Hermitian stationary direction can be decomposed into the difference of two equally weighted density operators; time averaging their trajectories yields two distinct stationary density operators. Thus singularity of the physical Bloch matrix also obstructs physical uniqueness.

For a two-level density operator, rates ${\gamma }_{12}$ from level one to two and ${\gamma }_{21}$ in the reverse direction give ${\dot{p}}_1=-{\gamma }_{12}{p}_1+{\gamma }_{21}{p}_2$. Their sum sets the longitudinal population relaxation rate, and their ratio fixes the stationary population imbalance. If both rates vanish, populations are conserved.

Decay of the off-diagonal coherence of a two-level density operator. Population relaxation contributes half its total rate; additional dephasing contributes ${\gamma }_{\varphi }\ge 0$ in the usual completely positive two-level model. Thus transverse relaxation should not automatically be identified with pure dephasing alone.