At a lossless vascular junction, pressure continuity and volumetric flow rate balance give $1+r=t$ and $Y_1(1-r)=Y_{d}t$, where $Y_d$ is the sum of daughter characteristic admittances of an arterial wave. Eliminating transmission amplitude $t$ gives the displayed reflection coefficient. A decrease in total area yields positive reflection if parent and daughter wave speeds are equal; differing stiffnesses require comparison of the admittances rather than areas alone.

A concrete laboratory route to an Arrhenius plot is to infer a rate constant at each controlled temperature from a changing electrical conductivity. The Technical University of Munich's Activation Energy of a First-Order Reaction supplies a teaching protocol, rather than a published set of measured rate constants.

The manual follows tertiary amylbromide conversion, which produces conducting ions. A measuring cell connects to a thermostat; an immersed electrical conductivity probe also measures solution temperature, and a magnetic stirrer mixes it. Use $80\mathrm{mL}$ of $80\%$ by volume ethanol in water, equilibrate, then add $0.5\mathrm{mL}$ of tertiary amylbromide. Record the baseline before addition and set time zero at addition. Repeat with fresh mixtures at $25$, $30$, and $35{}^{\circ }\mathrm{C}$, each within $\pm 0.5{}^{\circ }\mathrm{C}$; record for $30$ minutes at the first two temperatures and until completion at the last. Clean the cell between runs and keep the added volume fixed. The completed $35{}^{\circ }\mathrm{C}$ mixture supplies endpoint electrical conductivity measurements at the other temperatures: one endpoint reading cannot be used unchanged at every temperature.^[ref]

To see what is inferred from the readout, let ${\sigma }_0$, $\sigma (t)$, and ${\sigma }_{\infty }$ be the baseline, current, and completed-mixture electrical conductivities at the same temperature. Assume their baseline-subtracted changes are proportional to converted concentration. Combining that measurement model with a first-order reaction givesThus the electrical conductivity increases toward an endpoint, while the remaining reactant fraction decreases. Fit $-\ln f$ against time to obtain $k$; then fit $\ln (k/k_{\mathrm{ref}})$ against $1/T$ in kelvin to estimate an apparent activation energy. This baseline-aware derivation states the measurement assumptions explicitly; it does not infer a rate constant from the steepness of a raw conductivity trace alone.

The controls distinguish a faster chemical reaction from a larger instrumental response. Measure the actual solution temperature, retain the same solvent composition and initial reactant concentration, and compare endpoint and baseline at that temperature. As further diagnostic checks, repeat runs and inspect regression residuals for departure from a first-order reaction. These checks follow from the model: electrical conductivity also depends on how ions move, a baseline drift can mimic conversion, and a mixing delay can distort early points. Near completion, a small endpoint error produces a large relative error in $f$ and its natural logarithm. A successful fit supports an apparent rate model for this solvent and interval; it does not directly image a transition state or identify a unique reaction mechanism. No measured $k$ or $E_a$ is supplied here.

The Modified Arrhenius equation allows a power of thermodynamic temperature in the pre-exponential factor:Here $A_{\ast }$, $E_0$, $n$, and $T_{\ast }$ are constant model parameters. Using the dimensionless quantity $T/T_{\ast }$ leaves $A_{\ast }$ with the same units as $k$; in the equivalent convention $B{T}^n\exp [-E_0/(RT)]$, $B$ instead has units of $k$ multiplied by ${\mathrm{K}}^{-n}$. The power exponent $n$ here is not an order of reaction. The IUPAC definition uses that equivalent $B{T}^n$ convention.^[ref]

Applying the apparent activation energy definition yields the original short calculationFor the orange synthetic Arrhenius plot, $n=2$, $T_{\ast }=300\mathrm{K}$, and $E_0=60\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Its pre-exponential factor changes by $(310/300{)}^2\approx 1.06778$ from $300$ to $310\mathrm{K}$; combined with the exponential function, $k(310\mathrm{K})\approx 0.231991{\mathrm{s}}^{-1}$, compared with $0.217266{\mathrm{s}}^{-1}$ for the constant-pre-exponential factor model. The diagram uses those models only to reveal the difference between an exponent parameter and a slope-derived activation energy.

An Arrhenius plot places $\ln (k/k_{\mathrm{ref}})$ against $1/T$, using a fixed reference rate constant with the same units as $k$. Under the constant-parameter Arrhenius equation,The slope is $-E_a/R$ and the intercept is $\ln (A/k_{\mathrm{ref}})$. Changing the reference units shifts the intercept but leaves the slope unchanged. If the horizontal coordinate is displayed in $10^{-3}{\mathrm{K}}^{-1}$, its numerical slope must be rescaled before multiplying by $-R$. A temperature-dependent pre-exponential factor or activation energy can make the plot curved.

The pre-exponential factor multiplies the exponential function in the Arrhenius equation. It has the same units as the rate constant, so for a first-order reaction both have units ${\mathrm{s}}^{-1}$. Its physical interpretation depends on the reaction mechanism; an assumed constant $A$ should not be identified with a universal collision frequency. Comparing activation energies alone does not rank absolute rate constants when the pre-exponential factors differ. See the pre-exponential factor.

Suppose a monic real quartic polynomial has no nonnegative real zero and its real part on the imaginary axis is everywhere positive. For sufficiently large $R$, its image of the quarter-circle arc from $R$ to $iR$, closed by the straight chord from $p(iR)$ to $p(R)$, has winding number one: compare with $z^4$ on the arc and its constant image on the chord. The rest of the actual sector boundary maps into the right half-plane, so its replacement by that chord preserves the winding number. The argument principle then counts exactly one root in the open first quadrant, including multiplicity.

If $p(z(t))=t$ and $p^{\prime }(z)\ne 0$, implicit differentiation gives $z^{\prime }(t)=1/p^{\prime }(z)$. At a crossing of the unit circle, the displayed radial derivative determines whether the root enters or leaves the disk as the real parameter increases. Root counts are constant between such boundary crossings, including multiplicities. At the crossing parameter itself, boundary roots are excluded from an open-disk count. This complements the winding number form of the argument principle and handles a self-intersecting image contour without guessing its enclosed regions.

A finite list of function values does not determine the winding number of the intervening image contour. For example, sample $\gamma (t)=e^{it}$ only at $t_j=2\pi j/4$, $j=0,1,2,3$. The polynomials $f(z)=z$ and $g(z)=z^5$ agree at all four sampled points, because $\gamma (t_j{)}^4=1$. Joining those values by straight segments draws the same square for both. Yet their true winding numbers are one and five, and their enclosed zero multiplicities differ by four.

Refining samples can help resolve a given example, but an asserted root count needs a bound controlling what happens between them. One sufficient certificate is to enclose the image of each boundary subarc and its interpolating segment in a convex set that excludes zero. Straight interpolation then gives a homotopy avoiding zero, so the polygon and true image contour have the same winding number. A plot that merely looks closed and avoids zero at its sampled points is not that certificate.

The boundary can recover more than a count. Under the hypotheses of the argument principle, let $h$ be a holomorphic function on a neighborhood of the contour and its interior. If the enclosed zeros of a function are $a_j$ with zero multiplicities $m_j$ and the enclosed poles are $b_k$ of orders $n_k$, thenMultiplying the local logarithmic derivative by $h$ changes its residue from $m_j$ to $m_{j}h(a_j)$, or from $-n_k$ to $-n_{k}h(b_k)$. The residue theorem then gives the identity. Choosing $h=1$ recovers the argument principle; choosing $h(z)=z$ gives the sum of the enclosed zero locations minus the sum of the enclosed pole locations, both counted with zero multiplicities or pole orders.^[ref]

For the rational function in argument principle for a rational function, the $h(z)=z$ integral is $2(1/4)-(-1/2)=1$. Adding the known pole location $-1/2$ back gives a zero-location sum of $1/2$. Dividing by the zero count two gives the arithmetic mean of the enclosed zeros, with multiplicity, as $1/4$. Choosing $h(z)=z^k$ similarly gives the difference between the power-sum symmetric polynomials evaluated on the enclosed zero and pole locations, repeating each location according to its multiplicity or order. These identities let boundary information constrain both the number and aggregate locations of enclosed zeros of a function.

Consider the rational functionInside the unit circle are a double zero of a function at $1/4$ and a simple pole at $-1/2$. The simple zero of a function at $2$ is outside. Thus $N=2$, $P=1$, and the winding number is one. The factorization checks the count exactly; the plot illustrates how the boundary records it.

For the concentric circle $|z|=r$, the winding number is zero for $0<r<1/4$, two for $1/4<r<1/2$, one for $1/2<r<2$, and two for $r>2$. At $r=1/4$ or $r=2$, the boundary contains a zero of a function; at $r=1/2$, it contains a pole. The theorem does not apply at those three radii. Moving a boundary changes the count only when it crosses an object being counted; homotopy invariance of winding number describes the unchanged count between crossings.

This triangular planar map has diagonal Jacobian matrix entries one and determinant one, for differentiable $f$. Its inverse subtracts $f(v)$, so it is an area-preserving map. The analogous vertical map $(u,v)\mapsto (u,v+g(u))$ has the same property. Their composition preserves area even when the shifts are nonlinear, as in the logarithmic area-preserving seasonal predator-prey map.

A freely floating body at rest has no supporting contact or applied holding force. Neglecting surface tension and the buoyancy of air, its upward buoyancy balances its downward force from gravity:For a body of uniform mass density ${\rho }_{\mathrm{body}}$ and total volume $V_{\mathrm{body}}$, its mass is $m={\rho }_{\mathrm{body}}{V}_{\mathrm{body}}$. Substitution into Archimedes' principle givesPartial immersion is possible when ${\rho }_{\mathrm{body}}<{\rho }_f$. A nonuniform body uses its mean mass density $m/V_{\mathrm{body}}$ instead; for a hollow body this volume must count the enclosed space that excludes water. This explains how the amount of displaced water can support a vessel whose walls alone are denser than water. ^[ref]

For the $600\mathrm{kg}{\mathrm{m}}^{-3}$ cube in water of mass density $1000\mathrm{kg}{\mathrm{m}}^{-3}$, the displaced fraction is $600/1000=0.60$: sixty percent of its volume lies below the surface. With a horizontal top and bottom, the immersed depth $d$ satisfies $V_{\mathrm{displaced}}=Ad$, soIts remaining height is $0.040\mathrm{m}$, and the buoyancy is $1000\times 9.81\times 0.000600=5.886\mathrm{N}$, exactly balancing its downward force from gravity.

The law of one price requires traded assets or portfolios with the same terminal payoff to have the same earlier price. In a model where every discounted traded price is a true martingale under a common equivalent measure, it follows by taking the conditional expectation of that terminal payoff. With only local martingale prices and a lower-bound restriction on admissible trading strategies, it can fail: the seemingly profitable short position may have unbounded possible losses before maturity.

Work in discounted units with a unit bank account and constant asset-gain dynamics $dS=\mu dt+\sigma dW$. Here $\mu$ is the vector of excess dollar drifts, not an unadjusted physical return. If a holdings vector $v$ satisfiesfinance its initial stock cost through the bank. The resulting zero-cost self-financing portfolio has deterministic wealth $t{v}^T\mu >0$ for $t>0$, giving an arbitrage.

For $A=\sigma {\sigma }^T$, absence of such a direction is equivalent to $\mu \in \mathrm{Range}A$. Indeed, $\ker A=\ker {\sigma }^T$ and $\mathrm{Range}A=(\ker A{)}^{\perp }$. If $\mu$ has a nonzero projection onto that kernel, taking $v$ to be that projection gives $v^T\mu =\Vert v{\Vert }^2>0$.

Scaling this deterministic gain makes any unbounded increasing terminal utility function unbounded on a positive horizon. A singular volatility matrix therefore requires this drift-compatibility check before a finite investment value is asserted. With a nonzero bank rate in undiscounted units, the financing return must first be subtracted.

Arbitrage pricing theory relates expected returns to systematic factor loadings and their risk premiums. With idiosyncratic noise, a diversification or asymptotic argument needs additional assumptions. With no idiosyncratic noise and spanned factors, the formula is an exact consequence of law of one price: portfolios matching factor exposures can differ only by a constant payoff, which absence of arbitrage forces to be zero.

For a minimization problem with positive optimum, this convention measures the relative excess $(\mathrm{ALG}-\mathrm{OPT})/\mathrm{OPT}$. An epsilon-approximation always returns a feasible solution with excess at most $\epsilon$. A multiplicative-ratio convention instead reports $1+\epsilon$. State the convention explicitly rather than treating these two numbers as interchangeable.

For a nonzero reference value $x$, the relative error is the absolute error divided by the reference's absolute value:It is dimensionless and expresses the size of the error compared with the quantity being approximated. Approximating $1/3$ by $0.33$ has absolute error $1/300$ and relative error $1/100=1\%$. The definition is unavailable when the reference is zero; use an absolute error or a separately stated scale in that case.

The absolute error is the absolute value of the approximation error, $|x_{\mathrm{approx}}-x|$, measured in the same units as the numerical reference $x$. For example, approximating $1/3$ by $0.33$ gives absolute error $|0.33-1/3|=1/300$.

A distinguished annular indentation limit has ${\lambda }^{-2}=O(ϵ)$, $\lambda St=O(ϵ)$ and $Re\gg \lambda {ϵ}^{-3}$. Define $\beta =\lambda St/ϵ$, $\sigma =({\lambda }^2ϵ{)}^{-1}{\int }_0^1{u}_0^2/(R+y)dy$, ${\alpha }_1={\gamma }_0/R+{\gamma }_1/(R+1)$ and ${\alpha }_2={\gamma }_0^2/R^2-{\gamma }_1^2/(R+1{)}^2$. Radial inertia supplies the pressure difference $\sigma A_{xx}$ between the walls. The axial momentum limits and the moving-wall condition supply its derivative, giving the displayed forced KdV equation. This reduction assumes the viscous boundary layers remain thin and their displaced flux is smaller than $O({ϵ}^2)$.

The annual pulse-breeding predator-prey model is an area-preserving map in logarithmic population coordinates: $u^{\prime }=u+\log r-\kappa e^v$, then $v^{\prime }=v+\log s+\log (1+b{e}^{u^{\prime }})$. Each step is a shear of determinant one. At a positive fixed point, the ordinary-population Jacobian matrix has determinant one and trace $2-(1-s)\log r$. Eigenvalues lie on the unit circle for $0<(1-s)\log r<4$, but this linear neutrality does not prove nonlinear stability at every resonance. Area preservation excludes an isolated attracting positive periodic cycle.