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A matrix game is a finite two-player zero-sum game whose entry $A_{ij}$ is the row player's payoff when the players choose row $i$ and column $j$.

The union of the three coordinate axes in ${\mathbb{A}}^3$ and a union of three distinct lines through the origin in ${\mathbb{A}}^2$ are not isomorphic. Their origins are intrinsically the unique points common to all three irreducible components, but their Zariski tangent spaces there have dimensions three and two respectively.

Forin characteristic other than two,The variety has dimension two. Its tangent dimension is two when both coordinate pairs are nonzero, three when exactly one pair vanishes, and four at the origin.

Since $T_{(x,y)}(X\times Y)=T_{x}X\oplus T_{y}Y$, if $Y$ is smooth thenIn particular, dimensions add.

The smooth locus of a variety over a field $k$ is the open subset where its structure morphism to $\mathrm{Spec}k$ is smooth. Over a perfect field, these are precisely the points with regular local rings, and for a reduced finite-type variety the locus is dense.^[ref]

At a closed point over an algebraically closed field, the test isThe left side is the embedding dimension, and equality is the regular local ring criterion. If $X$ is pure of dimension $d$, its closed-point local dimensions are all $d$, so the familiar test is $\dim T_{p}X=d$. On a mixed-dimensional variety the local dimension must be retained. A point on a disjoint smooth one-dimensional component remains smooth even when another component has dimension two.

Over an imperfect field, a regular local ring need not give a smooth point over that field; geometric regularity, rather than regularity alone, is the relevant condition.

For a nonempty irreducible affine variety $X$ over an algebraically closed field,Every closed-point local ring has Krull dimension $\dim X$, and its embedding dimension is at least that value. Equality holds precisely when the local ring is regular, hence at a smooth point of a variety. The density of the smooth locus ensures the minimum is attained. The same statement holds for a pure-dimensional reduced variety; components of different dimensions require local dimension instead.^[ref]

For example, $X=V(xz,yz)\subset {\mathbb{A}}^3$ is the union of the plane $z=0$ and the vertical line $x=y=0$. It has dimension two, but its Zariski tangent space at $(0,0,1)$ has dimension one. That point is smooth on its one-dimensional component. Thus neither the global minimum formula nor testing all points against one global dimension works for a mixed-dimensional variety.

For a polynomial $f$, the distinguished open set $D(f)$ is the locus on which $f$ does not vanish.

A Bingham plastic has $\dot{\gamma }=0$ for $|\tau |<{\tau }_c$ and $\tau ={\tau }_c\mathrm{sgn}\dot{\gamma }+\mu \dot{\gamma }$ after yield.

For an $SU(2)$ connection under a standard trace convention,Its sign depends on orientation and on whether the instanton is self-dual or anti-self-dual.

The persistence length is the decay length of tangent correlations. A common three-dimensional convention is $L_p=A/(k_{B}T)$; a strictly planar chain has tangent-correlation length $2A/(k_{B}T)$.

Power describes how quickly energy is transferred. For mechanical work, the instantaneous power supplied by a force isThe SI unit is $\mathrm{W}=J/s=\mathrm{N}m/s$, distinct from the velocity unit $m/s$. The italic symbol $W$ for work and the upright unit $\mathrm{W}$ for power mean different things. Two processes can transfer the same energy while requiring different powers: doing $12\mathrm{J}$ of work in $2\mathrm{s}$ requires an average $6\mathrm{W}$, whereas doing it in $4\mathrm{s}$ requires an average $3\mathrm{W}$. ^[ref]

For a point mass of constant mass $m$ observed in an inertial frame, the net work along its actual path equals its change in kinetic energy:The middle expression is a line integral; the dot product selects the component of force along the displacement. Positive net work increases speed, negative net work decreases it, and zero net work leaves the endpoint speeds equal. Equal endpoint speeds do not imply equal velocities: a perpendicular force can continually change direction while doing zero work. ^[ref]

The bridge to Newton's second law is short: take the dot product of $m\dot{\mathbf{v}}={\mathbf{F}}_{\mathrm{net}}$ with velocity. The chain rule gives $d(m|\mathbf{v}{|}^2/2)/dt={\mathbf{F}}_{\mathrm{net}}\cdot \mathbf{v}$, equating the rate of change of kinetic energy to instantaneous power. Integrating gives the theorem. No conservative force assumption entered this argument; friction also contributes to the net work.

For the harmonic oscillator with spring stiffness $k=2N/m$, release a mass $m=0.5\mathrm{kg}$ from rest at $x=0.3\mathrm{m}$. The spring exerts $F=-kx$. Its work before the first passage through $x=0$ isThus $m{v}^2/2=0.09\mathrm{J}$ gives speed $0.6m/s$ there. The velocity is negative on this passage because the position is decreasing. The energy calculation alone determines its magnitude, so the direction still requires information about the motion.

For a time-independent potential energy $V$, the conservative force contributes work $-\Delta V$. Separating all other work as $W_{\mathrm{other}}$ gives $\Delta (T+V)=W_{\mathrm{other}}$. Consequently conservation of mechanical energy requires zero net other work; the work-energy theorem itself does not require that conservation. With damping, mechanical energy decreases even though the theorem remains valid. ^[ref]

For two simple turning points $a(E)<b(E)$ surrounding one classically allowed well,

If $\gamma :[0,1]\to \mathbb{C}\setminus \{0\}$ is continuous and closed, choose a continuous lift $\theta$ such thatThen $\mathrm{wind}(\gamma ,0)=\theta (1)-\theta (0)\in \mathbb{Z}$. This agrees with the contour-integral definition when the path is piecewise smooth.

Every conformally flat metric in dimension at least four has identically zero Weyl tensor. The Weyl tensor is the obstruction to local conformal flatness in dimensions at least four.

Every contraction of the Weyl tensor vanishes. Contracting its first and third indices directly cancels the Ricci and scalar-curvature terms; antisymmetry in each pair and symmetry under exchanging the two pairs reduce all other contractions to this one or make them vanish immediately.

Supersymmetry fixes scalar and fermion couplings so that bosonic and fermionic loop contributions to scalar masses have equal quadratic ultraviolet parts with opposite signs.

The divisor class group is the group of Weil divisors modulo principal divisors of rational functions.

A prime Weil divisor on a Noetherian integral scheme $X$ is an integral closed subscheme of codimension one. Its generic point $\eta$ has a one-dimensional local ring ${\mathcal{O}}_{X,\eta }$.

If $X$ is a normal variety, this local ring is a one-dimensional Noetherian normal domain, hence a discrete valuation ring. More generally it suffices that this local ring be regular. A uniformizer $\pi$ then writes every nonzero rational function as $u{\pi }^m$ with $u$ a unit, giving its integer order along the divisor. This is the valuation used in the principal Weil divisor construction.^[ref]

The codimension-one definition alone does not imply regularity. On the cusp $\mathrm{Spec}k[t^2,t^3]$, the closed point $(t^2,t^3)$ is a prime Weil divisor. Its local ring has a maximal ideal requiring two generators, as its quotient by the square has basis $t^2,t^3$. It is therefore not a discrete valuation ring. Passing to the normalization $k[t]$ supplies a different local ring with uniformizer $t$; normality is a substantive hypothesis, not part of the definition of every prime divisor.

Pitt's theorem implies in particular that every bounded operator $c_0\to {\ell }^1$ is compact. Thus two nonreflexive Banach spaces can have only weakly compact operators between them.