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A finite-dimensional associative algebra is semisimple when its left regular module is a direct sum of simple modules. Equivalently, every left ideal is a direct summand. In particular, a semisimple algebra has no nonzero square-zero left ideal: if $L^2=0$ and a module projection $A\to L$ exists, its value $e$ at $1$ satisfies $e\in L$ and $e^2=e$, forcing $e=0$.

A left ideal $L$ of an associative algebra $A$ is a vector subspace satisfying $AL\subseteq L$. Equivalently, it is a submodule of the left regular $A$-module.

The group algebra $k[G]$ is the vector space with basis $G$ and multiplication obtained by extending the group law $k$-bilinearly.

A commutative Noetherian ring is Artinian if and only if every prime ideal is maximal. Equivalently, it is Artinian exactly when its Krull dimension is zero.

The Condorcet paradox is a cycle in collective pairwise preferences even though each individual voter's preference is transitive.

A set is in ${\Pi }_2$ when membership can be written with one universal block followed by one existential block over a computable predicate. Equivalently, in the word-function formulation, there is a partial computable $f$ such that

A set $A$ is in ${\Sigma }_1$ exactly when it is the domain of a partial computable function. Equivalently, there is a computable predicate $R$ such that

A ${\Sigma }_1$ formula is equivalent to $\exists \mathbf{y}\delta$ with $\delta$ bounded. A ${\Pi }_1$ formula is equivalent to $\forall \mathbf{y}\delta$ with $\delta$ bounded.

If $g$ is holomorphic and nowhere zero on $0<|z|<r$, the residue of its logarithmic derivative is an integer:where $C$ circles zero once counterclockwise. It is the winding number of $g(C)$ about zero. Removing $z^k$ makes every period of the remaining logarithmic derivative vanish, so $z^{-k}g$ has a holomorphic logarithm on the punctured disk.

This does not make that derivative removable at zero. The counterexample $g(z)=e^{1/z}$ has $g^{\prime }/g=-1/z^2$ and residue zero, but still has a double pole in its logarithmic derivative. Its holomorphic logarithm $1/z$ is defined on the punctured disk, not across the missing point.

If $g$ additionally extends meromorphically across zero, write $g=z^{k}h$ with $h$ holomorphic and nonzero there. Then $g^{\prime }/g=k/z+h^{\prime }/h$, and the derivative after removing $z^k$ really is removable. In that stronger setting, $k$ is the zero order, or the negative of the pole order.

If $f,g$ are holomorphic near the closure of a bounded domain and $|g|<|f|$ on its boundary, then $f$ and $f+g$ have the same number of interior zeros, counted with multiplicity. Apply the argument principle to the zero-free boundary homotopy $f+tg$.

A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.

An anyon condenses at a boundary when its string operator may terminate there without leaving an excitation, so the boundary can absorb or emit that anyon.

Two particle types are mutual semions when taking either once around the other multiplies the state by $-1$.

If $\psi =z{e}^{-r^2/(2a^2)}$ in two dimensions, then $L_z\psi =\hslash \psi$ and $L_z{\psi }^{\ast }=-\hslash {\psi }^{\ast }$. Orthogonality therefore makes the angular-momentum expectation of $\alpha \psi +\beta {\psi }^{\ast }$ equal to $(|\alpha {|}^2-|\beta {|}^2)\hslash$ for normalized coefficients.

Areal velocity measures how fast a position vector from a chosen origin sweeps signed swept area. It connects a geometric picture of motion to angular momentum: a long position vector with slow transverse motion can sweep the same area per unit time as a short one with fast transverse motion. Constancy requires a dynamical condition, given by the torque criterion for constant areal velocity.

Choose a fixed origin $O$ in an inertial frame and a fixed plane through $O$, with constant unit vector $\mathbf{n}$ normal to it. For a differentiable trajectory in that plane, let $\mathbf{r}(t)$ be position relative to $O$ and $\mathbf{v}=\dot{\mathbf{r}}$ its velocity. Positive swept area follows the right-hand rule about $\mathbf{n}$. In positively oriented Cartesian coordinates in the plane, and in plane polar coordinates with angle $\theta$ in radians,Here the dot product selects the normal component of the cross product. Motion away from or towards $O$ alone contributes no area. The signed specific angular momentum component $h_n=\mathbf{n}\cdot (\mathbf{r}\times \mathbf{v})$ obeysfor a constant mass $m$. If $\mathbf{n}$ is chosen along a nonzero conserved specific angular momentum vector, then $h_n=h=|\mathbf{r}\times \mathbf{v}|$ and the familiar positive formula is $\dot{A}=h/2$. Reversing the orientation changes the sign; the instantaneous unsigned swept-area rate is $|\dot{A}|$.

Torque measures a force's turning effect about a specified origin. The geometry in the angular momentum diagram applies with force replacing momentum: transverse forces give positive or negative torque, while a radial force gives zero. A longer perpendicular lever arm makes the same force more effective, explaining why pushing near the end of a handle is useful.

For a force $\mathbf{F}$ applied at position $\mathbf{r}$ from a fixed origin $O$ in an inertial frame, its torque, or moment of force, isHere $F=|\mathbf{F}|$, $\varphi$ is the angle between position and force, and $\ell =r\sin \varphi$ is the perpendicular lever arm. Direction follows the right-hand rule; in the $xy$ plane ${\tau }_z=x{F}_y-y{F}_x$, positive counterclockwise as viewed from $+z$. The International System of Units unit is $\mathrm{N}\mathrm{m}=\mathrm{kg}{\mathrm{m}}^2/{\mathrm{s}}^2$. Although torque and energy have the same physical dimensions, torque is an axial vector and energy is a scalar; their roles differ.

For $\mathbf{r}=(2,0,0)\mathrm{m}$ and $\mathbf{F}=(0,3,0)\mathrm{N}$, ${\tau }_z=+6\mathrm{N}\mathrm{m}$. Applying the same force at $(1,0,0)\mathrm{m}$ gives $+3\mathrm{N}\mathrm{m}$; reversing it gives a negative torque; directing it along $\mathbf{r}$ gives zero. More generally, sum ${\mathbf{r}}_i\times {\mathbf{F}}_i$ to find the total torque about one common origin.

For a constant-mass point mass, the product rule and Newton's second law show why net torque changes angular momentum:The first cross product vanishes because its factors are parallel. For a collection with canceling internal torques, the remaining sum is external torque, giving conservation of angular momentum when it vanishes.^[ref]

A nonzero torque can turn the direction of angular momentum as well as change its magnitude. Indeed, the dot product gives $d|{\mathbf{L}}_O{|}^2/dt=2{\mathbf{L}}_O\cdot {\mathbf{\tau }}_O$: a perpendicular torque changes direction without instantaneously changing the magnitude. This is the geometry behind gyroscope precession. For a rapidly spinning symmetric top with a fixed tip, centre of mass distance $d$ from the tip, mass $M$, and spin-dominated angular momentum magnitude $L\simeq I\omega$, the slow steady precession approximation gives $\Omega \simeq Mgd/L$, where $g$ is gravitational acceleration. ^[ref] This is an approximation with $\Omega \ll \omega$; the uniform precession of a heavy symmetric top retains contributions omitted by the spin-dominated estimate.

For $-1<\alpha <1$,with value one at $\alpha =0$ by continuity. A keyhole contour around the positive axis gives a jump factor $1-e^{2\pi i\alpha }$, while the double pole at $-1$ has residue $-\alpha e^{\pi i\alpha }$.

An American put has exercise payoff $(K-S_n{)}^{+}$. In a finite binomial market, backward option pricing compares that payoff with the discounted risk-neutral continuation value at every node.

One coefficient form of the Combinatorial Nullstellensatz says that if $\mathrm{deg}f\le d_1+\cdots +d_n$ and $[x_1^{d_1}\cdots x_n^{d_n}]f\ne 0$, then for every choice of sets $A_i$ with $|A_i|=d_i+1$ there is a point $(a_1,\dots ,a_n)\in A_1\times \cdots \times A_n$ at which $f$ does not vanish.

Sampling the normalized Fourier expansion of $f:{\mathbb{F}}_p^n\to \mathbb{C}$ gives an average of $O(q/{ϵ}^2)$ phase-adjusted characters whose $L^q$ error is at most $ϵ\Vert \hat{f}{\Vert }_1$. The common kernel of those characters is therefore a low-codimension subspace of almost periods of $f$.