Set the dimension of the zero module to $-1$. A nonzero module has dimension zero exactly when it is an Artinian module. Inductively, dimension at most $d$ means that in every descending chain the successive quotients eventually have dimension less than $d$. The least such ordinal is the Krull dimension; the ordinal definition also covers dimensions beyond finite integers. The left dimension of a ring is that of its left regular module. In noncommutative rings this is not defined simply as the length of a chain of two-sided prime ideals.

For a ring $D$, an automorphism $\sigma$ and central $t$, adjoin $X,Y$ with $Xd=\sigma (d)X$, $Yd={\sigma }^{-1}(d)Y$, $YX=t$ and $XY=\sigma (t)$. This construction includes Weyl algebras and rank-one universal enveloping algebras.

A ring is semilocal when its quotient by the Jacobson radical is a semisimple ring that is Artinian. Equivalently, that quotient is a right Artinian ring: it has zero Jacobson radical, and the Artinian structure theorem makes it semisimple. A commutative ring is semilocal exactly when it has finitely many maximal ideals.

A regular element of a ring is a two-sided non-zero-divisor. The two cancellation conditions must both hold in a noncommutative ring. Products of regular elements remain regular.

A ring is right Goldie if it has finite right uniform dimension and satisfies the ascending chain condition on right annihilators. The left-sided definition reverses the multiplication. These two definitions need not be equivalent.

A ring is semiprime when it has no nonzero nilpotent ideals. Equivalently, a two-sided ideal with square zero must be zero. This is weaker than having no nonzero nilpotent elements: a matrix ring over a field is semiprime and can have nilpotent matrices.

A Hahn series over a field $k$ with exponents in an ordered abelian group $\Gamma$ is a formal sum ${\sum }_{\gamma }{a}_{\gamma }{t}^{\gamma }$ whose nonzero exponents form a well-ordered set. Such series form the Hahn series field.

An absolute value on a field is a nonnegative multiplicative function that vanishes only at zero and satisfies the triangle inequality.

A commutative Noetherian ring is Cohen-Macaulay when every localization at a prime ideal has depth of a Noetherian local ring equal to its Krull dimension. A regular local ring is Cohen-Macaulay. If $(R,\mathfrak{m})$ is a Cohen-Macaulay local ring and $x\in \mathfrak{m}$ is a non-zero-divisor, then $R/(x)$ is Cohen-Macaulay and both its depth of a Noetherian local ring and Krull dimension are one less than those of $R$.

The requirement $x\in \mathfrak{m}$ excludes units, whose quotient would be the zero ring. In particular, a hypersurface in a regular ambient variety satisfies the depth condition $(S_2)$ in the Serre criterion for normality.

NADPH is the reduced form of nicotinamide adenine dinucleotide phosphate. It carries electrons used in synthesis; in the Calvin cycle it becomes the oxidized form NADP+, available for reuse by light-dependent reactions. NADPH and ATP serve different roles even though both connect light capture to carbon fixation.

NADP+ is the oxidized form of nicotinamide adenine dinucleotide phosphate. It can accept electrons to form NADPH; the light-dependent reactions supply this conversion during oxygenic photosynthesis.

The radioactive chain converts nickel-56 through cobalt-56 into iron-56, releasing energy that can power supernova light curves. Electron capture and, where allowed, beta-plus decay convert protons into neutrons; gamma rays and charged decay products deposit some of the released energy. Ionization and escape of decay radiation affect the observed heating.

Newton–Cartan theory describes Newtonian mechanics geometrically using absolute time, a degenerate spatial inverse metric, and a compatible torsion-free affine connection. It is not a theory with a nondegenerate Lorentzian metric. Gravitationally accelerated particles are geodesics of that connection, whose curvature measures tidal gravity.

Friction is a contact force tangent to the surfaces. Kinetic friction opposes contact slip; static friction maintains contact without contact slip, including rolling without slipping. The perpendicular contact component is the normal force. These two components answer different questions: how strongly the surfaces press together, and how strongly they resist sliding past one another.

The ideal Coulomb dry-contact model combines a bound on static friction with a sliding law for kinetic friction. Its parameters are the coefficient of static friction and the coefficient of kinetic friction; constant coefficients are an empirical approximation, rather than a consequence of Newton's laws of motion. Surface condition matters, and the static coefficient need not universally exceed the kinetic coefficient. ^[ref]

The model's historical setting was practical machinery: Coulomb's Théorie des machines simples investigated friction and the stiffness of ropes. The Académie's bibliography records its 1781 prize and its appearance in volume 10, pages 161–332, of the Recueil des savants étrangers. ^[ref]

Neglecting vertical acceleration and the vertical driving-force component, the radial normal reaction is $-Mg{h}^{\prime }(r)$. A drive parallel to the horizontal velocity cancels when the radial acceleration equation is combined with $r_{\varphi }/r$ times the azimuthal equation. For a prescribed path $r(\varphi )$ this gives the displayed speed constraint. It describes the projected approximation, not arbitrary trajectories with substantial vertical motion.

// Target: classical-mechanics.bigb

A particle constrained to the graph $z=h(x,y)$ has kinetic energy $m[|\dot{\mathbf{r}}{|}^2+(\nabla h\cdot \dot{\mathbf{r}}{)}^2]/2$ and potential energy $mgh$. For a gently varying graph and velocities for which its geometric kinetic correction is higher order, the leading Euler-Lagrange equations are the displayed planar gradient-force equations. Small slope alone at arbitrarily large speed or curvature does not justify discarding the normal inertial acceleration. For a radial height, the leading planar angular momentum is conserved, allowing an effective potential reduction.

Electroencephalography records differences in electric potential using electrodes, commonly at the scalp. Summed activity in neural tissue contributes to the measured signal, providing millisecond-scale information about changes over time.

An event-related potential compares or averages this activity relative to defined events. Volume conduction and overlapping sources limit direct localization, so a scalp waveform cannot simply be equated with one small neural generator.

Magnetoencephalography measures very small magnetic fields generated by electrical activity in neural tissue. It supplies a time-resolved signal that can constrain when activity changes relative to an event.

Inferring its neural sources is an inverse problem: sensor readings combine contributions from multiple locations and depend on their geometry. The technique therefore does not directly display the activity of one isolated neuron.

A sequence of action potential times, represented mathematically as a point process or a sum of Dirac deltas. Its event counts, intervals and correlations summarize different aspects of neural activity.

Representation by a neuron's firing frequency or the expected spike count in a chosen time window. The window length and averaging convention matter; equal counts do not imply equal temporal patterns.