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A market is incomplete if some contingent claim cannot be replicated by a self-financing strategy. In a one-period finite-state market, attainable payoff vectors form the span of the traded assets' terminal payoff vectors. If this span is a proper subspace of the full payoff space, the market is incomplete. For example, two linearly independent asset-payoff vectors on three states leave at least one payoff direction unattainable. Absence of arbitrage only requires an appropriate positive state-price density; it does not imply that every claim is attainable.

Suppose a productive asset pays total consumption ${\delta }_t$, while price-taking agents value their consumption shares $p_t^i=c_t^i/{\delta }_t$ through discounted utility functions. Marginal pricing by a common state-price density ${\xi }_t$ gives $e^{-{\rho }_{i}t}{U}_i^{\prime }(p_t^i)/{\delta }_t=y_i{\xi }_t$. Writing $b(t)={\xi }_t{\delta }_t$, market clearing reduces to the scalar equation ${\sum }_i{I}_i(y_i{e}^{{\rho }_{i}t}b(t))=1$, where $I_i$ is the inverse marginal utility. Its unique positive solution is deterministic. Therefore the consumption shares are deterministic even when dividend levels are random.

With finite fundamental prices, the asset's price is $S_t={\delta }_{t}b(t{)}^{-1}{\int }_t^{\infty }b(s)ds$. Initial budget equations determine the multipliers. Identical preferences and discount rates make the shares constant and equal to initial ownership fractions. The aggregate benchmark is taken as given in the price-taking first-order conditions.

For discounted one-period asset gains $Y$, a dominated martingale measure is a probability measure $Q$ absolutely continuous with respect to the physical probability measure $P$, under which the gains have zero expectation. Its Radon-Nikodym derivative $Z$ satisfies $Z\ge 0$, ${\mathbb{E}}_{P}Z=1$ and ${\mathbb{E}}_P[ZY]=0$. An equivalent martingale measure additionally has $Z>0$ almost surely. Absolute continuity permits some positive-probability physical states to receive zero pricing probability; equivalence does not.

A competitive dividend economy specifies the joint law of total output, an information filtration, utility functions, discount factors, initial ownership shares and trading restrictions. Its derived objects are a common ex-dividend price, individually optimal consumption and share holdings satisfying ${\sum }_j{C}_t^j=d_t$ and ${\sum }_j{\theta }_t^j=1$. Each investor obeys the discrete dividend budget equation and, at an unconstrained interior optimum, marginal utility pricing in a dividend economy. A common price is required; a unique common state-price density or arbitrary contingent transfers require extra market completeness or implementability hypotheses.

A trinomial tree allows three next-step states. For the explicit log-price scheme for local-volatility pricing these are $x-h,x,x+h$, with the displayed weights. They sum to one; if nonnegative they are transition probabilities and make backward pricing a conditional average. Their first two increment moments are $bk$ and $2ak$, matching the diffusion to first order in the time step. Coefficients can depend on the node, so the tree need not have constant transition probabilities.

Credit risk is the risk of loss through a counterparty's inability or failure to meet a financial obligation, or deterioration in its credit quality. A loss model distinguishes credit default frequency from loss given default; shared exposures or systematic factors may produce dependence beyond an independent Poisson model.

Value at risk is the displayed upper loss quantile for a specified horizon and confidence level. It is distinct from a two-sided confidence interval for a model parameter and does not by itself describe the severity of losses exceeding that quantile.

A traded claim or instrument carrying a current price and future payments or resale value. In a one-period financial market, its terminal payoff is a random variable and a portfolio combines payoffs linearly through its asset holdings. A stock, a zero-coupon bond and a bank account are basic examples. Dividends require treating the full gain process rather than only resale prices.

A model specifying traded financial assets, their price and payment processes, available information, and allowed admissible trading strategies. A self-financing portfolio generates wealth using gains on these assets. Absence of arbitrage and market completeness respectively concern whether trading can create a riskless profit from zero capital and whether every specified contingent claim can be replicated.

For initial asset value $V_0>0$, final value $V_1$, and intervening cash distribution $D$, the net financial return is $(V_1-V_0+D)/V_0$. Its gross return is $1+R=(V_1+D)/V_0$. In a one-period portfolio with initial-value weights $w_i$ summing to one, the net return is ${\sum }_i{w}_i{R}_i$. The expected return is its expected value; a realized return need not equal it.

The incubation period is the interval from infection to symptom onset in a person who develops symptoms. Its probability distribution describes onset times, not the full timing of infectiousness. Exposure may only be known to fall in an interval, requiring a corresponding observation model when estimating this distribution.

Quarantine restricts contacts of exposed individuals whose infection status is uncertain, with monitoring to detect illness. It differs from isolation of recognized infected or ill individuals. An incubation period quantile can enter a duration calculation, but assumptions about exposure time, symptom detection and infectiousness are needed before the calculation describes transmission risk.

In a continuous-time epidemic model, a constant recovery rate $\gamma$ gives an exponential distribution of infectious duration with mean $1/\gamma$. Other duration distributions require additional stages or infection-age structure.

A stochastic epidemic model represents infection and recovery as random events. A continuous-time Markov chain can track integer counts, while a stochastic SIR model illustrates random fade-out and finite-population outbreak variability that a deterministic limit does not show.

A macroparasite burden model tracks the distribution of within-host parasite burdens and couples it to transmission or environmental infectious stages. Mean burden alone is insufficient when worm mating, nonlinear fecundity, immunity or host mortality depend on that distribution.

Each undirected cross-group partnership is counted once from each group's contact total. Thus group size times per-person contact rate times cross-group fraction must agree. If one group occupies fraction $p$ while its cross-group per-person rate stays fixed, aggregate cross-group contact is order $p$ and the other group's cross-contact fraction is order $p$. Assigning both per-person rates independently generally breaks this balance.

For a smoothly moving volume whose boundary has normal velocity $v_b\cdot n$, the derivative of an integral equals the local time-change integral plus the boundary-transport flux. The boundary term comes from the thin layer swept by the moving surface during a short time interval. Taking $f=1$ gives $d|V|/dt=\int v_b\cdot ndS$. If the boundary moves with the fluid, this combines with the divergence theorem to give the integral of the material derivative plus $f$ times the velocity divergence.

The magnetic helicity integrates the dot product of a magnetic vector potential and its magnetic field. It measures linkage and twisting of magnetic flux. For a solenoidal vector field $\mathbf{B}$ tangent to the boundary, it is unchanged by the gauge transformation $\mathbf{A}\mapsto \mathbf{A}+\nabla \chi$, since the change is ${\int }_V\nabla \cdot (\chi \mathbf{B})dV={\int }_{\partial V}\chi \mathbf{B}\cdot d\mathbf{S}=0$. If its density satisfies ${\partial }_{t}h=\nabla \cdot (f\mathbf{B}-h\mathbf{u})$ with both vector fields tangent to the boundary of a fixed volume, the divergence theorem gives conservation of its integral. This is an instance of helicity conservation by tangent boundary conditions.

A Gouy balance detects the force exerted by a nonuniform magnetic field on a sample of uniform cross-section extending between high- and low-field regions. TeachSpin's implementation reverses the usual arrangement: a rare-earth permanent-magnet assembly sits on a one-pan electronic balance, while a separate holder brings the sample into the field. The magnet's apparent mass change measures the opposite of the sample's vertical force, by Newton's third law. The described pole faces have diameter $25\mathrm{mm}$ and separation $15\mathrm{mm}$; samples occupy cuvettes of cross-section $(10\mathrm{mm}{)}^2$ and height about $32\mathrm{mm}$. ^[ref]

The same balance calibrates magnetic flux density by measuring the reaction force on the magnet when a wire of known horizontal length $L$ carries known electric current $i$ perpendicular to the field: $|F_z|=iL{B}_0$. The manufacturer describes water being repelled by the high-field region, identifying negative magnetic susceptibility. This observed sign supplies a qualitative test, not a measured Curie's law curve. ^[ref]

For weak reversible linear response, a uniform sample displacing a weakly magnetic surrounding medium has, to leading order, downward forceThis follows by integrating the induced-dipole force density $({\chi }_v-{\chi }_{\mathrm{medium}}){\partial }_z(B_0^2)/(2{\mu }_0)$ along a downward coordinate. $A$ is the cross-sectional area, and the two $B_0$ values are applied magnetic flux densities. The force depends on the endpoint fields; the entire field-gradient profile need not be measured under these assumptions. In the reversed apparatus, $F_{\downarrow }=-g\Delta m_{\mathrm{magnet}}$ after subtracting the holder background, where $g$ is gravitational acceleration. This sign must be reversed before inferring volume magnetic susceptibility.

Useful controls are an empty-holder reading at the same position, a distant-sample baseline, repeated wire-current reversals during field calibration, and stable temperature. They separate holder response, balance drift, force offsets, and thermal changes; these are proposed checks for the stated inference. Finite endpoint fields, nonuniform cross-section, uneven powder packing, stray forces and a demagnetizing field limit the simple formula. If magnetization is nonlinear, the weak-response force involves $A{\int }_{B_{0,\mathrm{top}}}^{B_{0,\mathrm{bottom}}}M(B_0)d{B}_0$ instead of a constant-${\chi }_v$ endpoint square. A balance reading then measures an integrated response, not directly the local differential magnetic susceptibility.

Saturation magnetization is the limiting magnetization when the responding magnetic moments are fully aligned within the stated model or experimental regime. In independent-moment magnetic response, $M_s=n\mu$. Reaching a large magnetization does not imply a large differential magnetic susceptibility: the remaining slope can be almost zero.