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A singular isothermal sphere with one-dimensional velocity dispersion $\sigma$ has $\rho (r)={\sigma }^2/(2\pi G{r}^2)$, enclosed mass $M(<r)=2{\sigma }^{2}r/G$, and constant circular speed $\sqrt{2}\sigma$.

Stellar dynamics treats stars as gravitating bodies whose collective motions reveal and reshape their environment. In a collisionless stellar system, stars mainly follow the smooth collective gravitational potential; in a sufficiently dense star cluster, accumulated gravitational encounters produce two-body relaxation. These encounters need not involve physical contact between stars. Observers combine astrometry with radial velocities to constrain orbits and infer unseen mass. The stellar orbits at the Galactic centre provide a particularly direct example: following a short-period star around Sagittarius A* measures the compact central mass, rather than estimating it from the light emitted by its surroundings.^[ref]

The tidal radius bounds material that remains gravitationally attached to a satellite in the differential gravitational field of its host.

Chandrasekhar dynamical friction is the drag on a massive body produced by its gravitational wake in a background of lighter particles. Its magnitude is proportional to the body's mass and the background density.

A European contingent claim pays a specified function of market variables at one fixed maturity. Under an equivalent martingale measure, an attainable claim is priced by the discounted conditional expectation of its payoff.

A European put option gives its holder the right, without an obligation, to sell one unit of an underlying financial asset at the strike price $K$ only on the expiry date $T$. For one stock share, its terminal financial payoff is the positive partFor strike price $K=100$, terminal stock prices $80$, $100$, and $120$ give financial payoffs $20$, $0$, and $0$, respectively. At $S_T=80$, selling for $100$ rather than the market price $80$ supplies the difference $20$; at $S_T=120$, the holder can let the right expire. Equality at $S_T=100$ gives zero net exercise value. The European call option example plots both financial payoffs on the same axes.

Combining a European put option with the underlying stock share produces the terminal amount $S_T+(K-S_T{)}^{+}=\max (S_T,K)$. Its floor at the strike price is why a European put option can describe protection against a fall in terminal stock value. The protection has an initial cost and lasts to the specified expiry; the terminal floor alone says nothing about the financial return. Put-call parity replicates this same amount with a European call option and a zero-coupon bond.

Using the put-call parity example's price $P_0=10+100e^{-0.05}-100\approx 5.12294245$, the cost financed for one year at interest rate $r=0.05$ grows to $P_0{e}^{0.05}=100-90e^{0.05}\approx 5.38560133$. The terminal net amount is therefore approximately $14.61439867$ when $S_T=80$, and $-5.38560133$ when $S_T=100$ or $120$. These computations use the unrounded price and distinguish the financial payoff from the financed gain. Unlike an American put option, a European put option supplies no right to exercise before $T$.

A European call option gives its holder the right, without an obligation, to buy one unit of an underlying financial asset at the strike price $K$ on the expiry date $T$. European describes exercise timing, not geography: exercise is allowed only at $T$. Here the underlying is one stock share, $S_T$ is its price at expiry, and the financial payoff is expressed as a cash amount. Physical exercise delivers a share against payment of $K$; immediate resale gives the same net amount under the assumed frictionless market.

The holder exercises only when buying at the strike price is advantageous. Writing $x^{+}=\max (x,0)$ for the positive part, the terminal financial payoff isThe European call option is a reusable building block for claim replication: its financial payoff is zero below the strike price and rises by one unit for each unit of stock price above it. The corner at $K$ lets combinations of European call options construct other payoff shapes, as in static replication on a finite terminal support.

For strike price $K=100$, the following are deterministic calculations conditional on each displayed terminal stock price. They assign no probabilities to the three scenarios. The European put option has the opposite exercise right and the financial payoff $(K-S_T{)}^{+}$.

A financial payoff excludes the initial contract price. If the illustrative European call option costs $C_0=10$, its financial payoff at $S_T=120$ is $20$, and subtracting its initial cost gives $10$. If that cost is instead borrowed for one year at the constant continuously compounded interest rate $r=0.05$, the terminal net amount is $20-10e^{0.05}\approx 9.48728904$. At $S_T=80$ or $100$, the corresponding net amount is $-10e^{0.05}\approx -10.51271096$. Keeping all cash amounts at the same date avoids confusing a financial payoff with a financed gain.

Robert Shiller's Options Markets lecture, Yale ECON 252 (2011) introduces these exercise rights and payoff shapes.

Let $P$ be the state-by-asset matrix of discounted gains. There exists $\theta$ with $Y-P\theta \ge 0$ exactly whenfor every $q\ge 0$ satisfying $P^{T}q=0$. This is the finite-dimensional cone-separation, or Farkas-duality, form of superhedging.

A retarded fundamental solution is supported in a chosen forward region, such as $[0,\infty )$ for an ordinary differential operator.

Every nonzero constant-coefficient linear partial differential operator has a distributional fundamental solution.

For the closed orientable surface of genus $g$,In particular, the sphere has trivial fundamental group and the torus has fundamental group ${\mathbb{Z}}^2$.

Let $X=U\cup V$ be an open cover such that $U$, $V$ and $U\cap V$ are nonempty and path-connected, and choose a base point $x_0\in U\cap V$. The inclusions induce a pushout of groups:The maps from the overlap need not be injective. The amalgamated free product here means the universal group receiving the two inclusion-induced homomorphisms with their overlap relations.

A loop can be subdivided into pieces in $U$ or $V$; connecting their endpoints through the path-connected overlap expresses its class using the two local fundamental groups. Subdividing a homotopy shows that the overlap relations are exactly the identifications needed.

The overlap condition matters. Two open arcs can cover a circle while their intersection has two components. Both arcs have trivial fundamental group, but the circle has fundamental group $\mathbb{Z}$. The displayed single-basepoint formula cannot be applied to that disconnected overlap; a groupoid version keeps the extra component information.

A continuous based map $f:(X,x_0)\to (Y,y_0)$ induces a homomorphism $f_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0)$ by composition of loops with $f$.

Concatenating two based homotopies proves that loop concatenation depends only on based-homotopy classes. Endpoint-fixing reparametrizations provide associativity and the identity laws on classes.

A variation of a function $y$ is a one-parameter family $y+\epsilon \eta$, where $\eta$ satisfies the required boundary conditions.

A Lagrangian is a function or density whose stationary integral encodes an extremization problem. In Lagrangian mechanics its action gives the equations of motion; in geometry it describes geodesics and minimal surfaces.

In Lagrangian, the action integrates such a density over spacetime.

A finite-dimensional optimization Lagrangian combines an objective function with Lagrange multiplier terms for its constraints.

An energy functional assigns an integral energy to a function. Minimization of a suitable energy is the basis of the Dirichlet principle. For the modified Helmholtz equation, a natural example is

A periodic function repeats exactly after a nonzero shift of its input. For $f:\mathbb{R}\to Y$, a positive period of a function $T$ satisfiesOne interval of length $T$ therefore determines the whole function. For another domain of a function $D$, a period must also preserve the domain, $D+T=D$; equality only where two domains happen to overlap is weaker. Periodicity is an exact translation of a function symmetry, rather than merely two similar-looking pieces of its graph.

The sine $\sin (2\pi x)$ has fundamental period one. The function ${\sin }^2(2\pi x)$ repeats after half a unit: the sine changes sign under that shift, but squaring removes the sign. The mixed function $g(x)=2+\sin (2\pi x)+\frac{1}{2}\cos (4\pi x)$ again has fundamental period one. All positive integer multiples of a period are periods, so specifying one period need not identify the shortest repetition.

If the input is time and $T_0$ is the fundamental period, the frequency is $1/T_0$ and the angular frequency is $2\pi /T_0$. Repetition lets a Fourier series describe a cycle using sines and cosines; the period average separates the reference level from the oscillating terms. Strang develops this representation and its Fourier coefficients in §4.1 of Computational Science and Engineering by Gilbert Strang.^[ref]

This connection grew out of a physical problem. In his 1822 book Théorie analytique de la chaleur by Joseph Fourier, Joseph Fourier used periodic functions to extend profiles beyond their initial interval while treating heat conduction. Chapter III, §§233–234 combines even functions and odd functions into sine–cosine expansions and rescales their period.^[ref] A modern Fourier series interpretation still requires the appropriate convergence conditions.

A constant function has every real shift as a period and no fundamental period. Without continuity, even a nonconstant periodic function can lack a smallest positive period; periodic function without a least positive period gives a concrete example. The period group of a function organizes all these possibilities.

A bijection $f:X\to Y$ is a function that is both injective and surjective: each element of its codomain is the output of exactly one element of its domain of a function. The two requirements exclude unused targets and merged inputs respectively. The finite example $f(A)=1$, $f(B)=2$, $f(C)=3$ is a bijection from $\{A,B,C\}$ to $\{1,2,3\}$, whereas adding an unused target $4$ makes that same rule fail surjectivity.

A bijection gives an inverse function by reversing every pairing. Conversely, a two-sided inverse function implies a bijection, as the proof under inverse function shows. This is useful for transporting structures without losing or identifying elements. It also defines equality of cardinality: two sets have equal cardinality exactly when a bijection pairs them. An infinite example is $n\mapsto 2n$ from the integers to the even integers, with inverse $m\mapsto m/2$ on that specified codomain. A proper subset can therefore have the same cardinality as an infinite set. ^[ref].

A real-valued function takes each function input to a real number. Its codomain can be all of $\mathbb{R}$ or a specified subset. Thus $f(x)=x^2$ on the real line is a real-valued function and has image $[0,\infty )$, whether the declared target is that interval or all of $\mathbb{R}$.

The inputs need not be real numbers: the cardinality of a finite set and the squared norm of a vector are also real-valued functions. The adjective describes outputs, while scalar-valued function allows outputs in other fields, notably the complex numbers.