For a regular parametric curve $F(u)$ traversed in increasing $u$ with positive speed $v(F(u))$, its travel time isInvert this strictly increasing function to sample at equal time intervals, or integrate $\dot{u}=v(F(u))/\Vert F^{\prime }(u)\Vert$. A zero of the speed requires separate analysis: whether travel time diverges depends on the integrability of this reciprocal-speed expression, and the scalar speed alone does not determine the direction after a turning event.

If a continuous map $f:S^d\to S^d$ has no fixed point, the formula $H_t(x)=((1-t)f(x)-tx)/\Vert (1-t)f(x)-tx\Vert$ is a homotopy from $f$ to the antipodal map. The denominator could vanish only at $t=1/2$ and $f(x)=x$, which is excluded. The homotopy invariance of mapping degree therefore gives $\mathrm{deg}f=(-1{)}^{d+1}$.

The historical interpretation fills the negative-energy electron states of the Dirac equation and uses the Pauli exclusion principle to prevent additional electrons from falling into them. Removing an electron of energy $-E$ and charge $-e$ gives a hole of excitation energy $E$ and charge $+e$, interpreted as a positron. A quantum field theory instead treats negative-frequency field coefficients as independent antiparticle creation operators in a Fock space, with positive excitation energies after normal ordering. No literal infinite material sea is needed. Bosonic antiparticles also exist, although a filled-sea exclusion argument cannot stabilize bosonic negative-energy levels.

Antimatter consists of antiparticles, whose rest masses and spins match those of the corresponding particles while additive internal conserved charges are reversed. A Positron, for example, has the electron's rest mass and opposite electric charge. Both have positive physical energy.

When a particle meets its antiparticle, annihilation can convert the pair into other particles while preserving total energy, momentum and applicable conserved charges. Electron-positron annihilation into two photons is a simple example. Producing a few antiparticles in energetic collisions does not establish a large cosmological antimatter population: the production process and the subsequent abundance are separate questions. The antiparticle description is built into relativistic quantum field theory, rather than requiring negative measured mass or energy.

Particle-antiparticle annihilation converts a particle and its antiparticle into other particles, conserving energy and applicable quantum numbers. Baryon-antibaryon annihilation commonly produces mesons whose subsequent decays can generate radiation.

On a connected open interval, if $F^{\prime }=g$, every antiderivative of $g$ has the form $F+C$, where $C$ is a constant function. Indeed, two solutions satisfy $(G-F{)}^{\prime }=0$: the zero derivative on a connected open set principle makes their difference constant. In one dimension, the mean value theorem proves this on any closed subinterval. ^[ref]

An initial condition or boundary condition can determine $C$. If a particular antiderivative is $F_0$ and $F(x_0)=y_0$, thenFor $g(x)=x^2$, $F_0(x)=x^3/3$ and $F(1)=2$, this gives $C=5/3$. Without that reference value, choosing $C=0$ simply selects one member of the family.

The freedom has a physical meaning in a simple kinematic model. A velocity $v(t)=at$, with constant acceleration $a$, determines position only as $x(t)=a{t}^2/2+C$. The initial condition $x(0)=x_0$ selects $C=x_0$; changing the starting position does not change the prescribed velocity.

For the differential equation $T^{\prime }=-B/r^2$ on $r>0$, the antiderivative is $T=B/r+C$. The boundary condition $T(R_s)=T_s$ gives $C=T_s-B/R_s$. Discarding the constant of integration therefore needs a specified condition or an approximation that quantifies its effect. On a domain with several connected components, the freedom is larger: antiderivatives on disconnected domains require a separate constant on each component.

Anti-self-dual gauge curvature obeys $F=-\ast F$ in oriented Euclidean four-space. In the corresponding two-spinor convention it has vanishing symmetric primed curvature spinor. A matrix-valued two-form which is both self-dual and anti-self-dual is zero, because adding the two equations gives $2F=0$.

A global anomaly obstructs invariance of a quantum theory under a topologically nontrivial symmetry transformation even when infinitesimal local anomaly tests vanish. The Witten SU(2) anomaly is one example; global gravitational anomalies are another class.

For $\xi \ne 0$ set $y=\xi /(2|\xi {|}^2)$, and take $R>r\ge 4|y|$. Translation by $y$ reverses the oscillatory exponential's sign. Comparing the original annulus with its translate gives twice the integral as an integrable kernel difference plus boundary-shell errors. The shell errors are bounded by $C_{d}A|y|(r^{-1}+R^{-1})$. The displayed estimate follows by enlarging the constant. Its right-hand side tends to zero as $r\to \infty$, proving convergence of the Fourier integral's improper far tail without absolute integrability of $K$ itself.

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.