Checking the zero and scale of a differential pressure transducer against equal port pressures and independently known pressure differences. The dynamic pressure measurement discussion gives a liquid-column check.

For a statistically homogeneous vector random field, its spectral tensor is the Fourier transform of the componentwise two-point correlation $C_{ij}(\mathbf{r})=\mathbb{E}[B_i(\mathbf{x}+\mathbf{r})B_j(\mathbf{x})]$. It distributes cross-correlations between components over wavevectors. With $\hat{B}(\mathbf{k})=\int B(\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{r}}{d}^{d}r$, homogeneity gives $\mathbb{E}[{\hat{B}}_i(\mathbf{k}){\hat{B}}_j({\mathbf{k}}^{\prime })]=(2\pi {)}^d\delta (\mathbf{k}+{\mathbf{k}}^{\prime }){\hat{C}}_{ij}(\mathbf{k})$, understood as an identity of distributions for an infinite stationary field.

A power-law distribution uses a power law for a probability mass function or a probability density on a specified range. For a continuous model with $p_X(x)=C{x}^{-\beta }$ on $x\ge x_{\min }>0$ and zero below it, the total integral is finite exactly when $\beta >1$. Normalization then requires $C=(\beta -1)x_{\min }^{\beta -1}$, giving a Pareto distribution with shape parameter $\alpha =\beta -1$. Discrete models instead normalize a sum. The support and normalizing coefficient are part of the model, and a power law for its tail probability uses a different exponent from its probability density.

A Dirichlet form on a real $L^2$ space is a densely defined closed nonnegative symmetric bilinear form with the Markov contraction property. In the real case, truncating $u$ to $(0\vee u)\wedge 1$ keeps it in the form domain and cannot increase $\mathcal{E}(u,u)$. The Dirichlet form of a Markov chain is a discrete energy example; Laplacian energy on a region supplies a continuous example.

For unit channel width, incompressible flow towards a corner withdrawal point has $\psi =0$ on the lower wall and $\psi =-m$ on the upper and left walls, with $\psi \to -my$ far downstream. For irrotational flow, separation of variables gives the displayed harmonic correction: the Fourier sine coefficients of $-m(1-y)$ are $-2m/(n\pi )$. Near the corner, $\psi \sim -2m\theta /\pi$ and radial velocity is $-2m/(\pi r)$, whose integrated inward flux across a quarter-circle is $m$.

For $x,z\in {\mathbb{R}}^d$, a polynomial kernel is the positive-semidefinite kernel $k(x,z)=(x^{T}z+c{)}^r$, where $r$ is a positive integer and $c\ge 0$. The case $c=0$ uses monomials of total degree $r$; $c>0$ also includes lower degrees. The feature map weights matter: listing each distinct monomial once with coefficient one generally gives a different inner product.^[ref]

For the homogeneous quadratic case in two dimensions, take the scaled quadratic feature mapThe inner product identity is explicit:The factor $\sqrt{2}$ accounts for the two equal cross terms in the square. Thus an inner product of three derived coordinates can be obtained from an inner product of the two input coordinates followed by squaring. The kernel trick uses this equality inside an algorithm; the feature map explains which nonlinear functions the algorithm can represent.

Cell-cycle activation feedback is the reinforcement of Cdc2 activation through Cdc25 and suppression of Wee1. Its role is to help make transitions in the cell cycle decisive; the cell-cycle hysteresis experiment tests the resulting states rather than inferring them solely from circuit arrows.

Saturating positive feedback limits the reinforcing rate as activity increases. Here is an explicitly synthetic ordinary differential equation for normalized nonnegative input $u$, response $x$, and return strength $\beta$:The return is an increasing, nonproportional function of $x$, while removal is proportional to $x$. Time and activities are dimensionless variables. At $x=0$ the vector field points into the nonnegative domain, and $dx/dt\le u+\beta -x$ prevents unbounded growth there. This makes the toy useful for asking about a bounded switch; it supplies no molecular mechanism for the chosen rate law.

At an equilibrium point, $f=0$. For a simple equilibrium point, the sign of the derivativedetermines local asymptotic stability. If $\beta =1$, the maximum reinforcing slope is $3\sqrt{3}/8<1$, so $f$ decreases strictly: every $u\ge 0$ has one asymptotically stable equilibrium point despite positive feedback.

Three equilibrium points become possible only if the reinforcing slope exceeds removal's slope somewhere. Its maximum is $3\sqrt{3}\beta /8$, at $x=1/\sqrt{3}$. Consequently this toy requiresfor two distinct folds. This is a necessary threshold for a bistability interval in this toy, not a universal biochemical number. At a fold, both $f=0$ and $f_x=0$, givingFor $\beta =1.8$, the folds are at $(x,u)\simeq (0.350003,0.153563)$ and $(0.887240,0.094405)$. Between these input values there are two asymptotically stable equilibrium points separated by an unstable one. At $u=0.12$, the three activities are approximately $0.171331$, $0.621230$ and $1.127439$; the low and high ones are stable.

A sufficiently slow increasing sweep follows the low branch to $u_{\mathrm{on}}\simeq 0.153563$ and then jumps upward. A decreasing sweep follows the high branch to $u_{\mathrm{off}}\simeq 0.094405$ and then jumps downward: hysteresis. These are limiting quasistatic thresholds, with relaxation between input changes and negligible perturbations that could cause earlier switching. Finite sweep rates and fluctuations can alter observed thresholds. A switchlike response with only one stable equilibrium point does not have this equilibrium hysteresis mechanism.

A generalized eigenstate of the position operator is labeled by its position eigenvalue. In a continuous position coordinate the normalization is distributional: $⟨q|q^{\prime }⟩=\delta (q-q^{\prime })$ and $\int dq|q⟩⟨q|=1$. Individual position eigenstates are not normalizable Hilbert-space vectors. The Dirac delta distribution normalization makes $⟨q_f|U|q_i⟩$ the integral kernel of a time-evolution operator, such as the free-particle propagator.

A porous medium is a solid framework containing connected voids in which a fluid can reside and flow. Its porosity is the void fraction; its permeability of a porous medium measures the ease of pressure-driven flow through the connected pores. The averaged Darcy flux is measured per total cross-sectional area, whereas the pore velocity is that flux divided by porosity. Not every material with voids has connected permeable pores.

A sharp advected temperature transition in a fluid-filled porous medium. If fluid and solid equilibrate locally in temperature, their effective heat capacity per total volume is $C_{\mathrm{eff}}=\varphi C_f+(1-\varphi )C_s$, while the advected heat flux is $C_{f}UT$ with $U$ the Darcy velocity. Neglecting conduction gives $C_{\mathrm{eff}}{T}_t+C_{f}U{T}_x=0$, so the thermal front speed is $\Gamma U$ with $\Gamma =C_f/C_{\mathrm{eff}}$. The material pore velocity is $U/\varphi$, so thermal retardation relative to that speed is $\Gamma \varphi$. Temperature-dependent viscosity couples thermal-front perturbations back to the Darcy flux.

For a fixed pressure difference across two noncommunicating layers, an activated viscous polymer segment of length ${\ell }_i$ adds hydraulic resistance in series with the remaining fluid. The Darcy flux is the displayed expression when polymer raises dynamic viscosity by a factor $\lambda$. Tracking particle entry time determines activation and exit, while ${\dot{X}}_i=q_i/\varphi$ gives each displacement front. Preferentially occupying the more permeable layer shifts the relative inflow toward the other layer, but cannot increase that layer's absolute flux at fixed pressure. It can delay its breakthrough and leave original reservoir fluid flowing after the unmodified reservoir would be fully swept.

An aquifer is a permeable geological formation containing and transmitting groundwater. A confined aquifer lies beneath a confining layer; an unconfined aquifer instead has a free water table.

The logistic growth equation applies the logistic differential equation to population dynamics. For positive growth rate $r$ and carrying capacity $K$, an initial condition $0<N(0)<K$ gives $N(t)=K/[1+b{e}^{-rt}]$, where $b=K/N(0)-1$. Its per-capita growth falls linearly with abundance, and a trajectory starting below $K/2$ reaches an inflection point when $N=K/2$. The exact solutions of a differential equation and their diagrams are developed under logistic differential equation. A translated and scaled logistic function has the same form. In nonlinear regression for biological size, an analogous saturation curve need not imply that circumference itself obeys a mechanistic population law.

Two populations grow at the same per-capita rate and each loses members at a rate proportional to their product. On the invariant diagonal $x=y=z$, the logistic differential equation gives $z(t)=\lambda /[\mu +(\lambda -\mu )e^{-\lambda t}]$ for $z(0)=1$ and positive rates. The positive symmetric equilibrium $(\lambda /\mu ,\lambda /\mu )$ is stable along that diagonal but unstable transversely: $(x-y{)}^{\prime }=\lambda (x-y)$, while the Jacobian matrix has eigenvalues $-\lambda ,+\lambda$. Thus an exactly symmetric deterministic solution can relax toward coexistence while a small imbalance grows. A corresponding density-dependent Markov jump process has a controlled fluid limit on each fixed finite interval, rather than uniform control for all time.

The Nicholson-Bailey model describes seasonal host-parasitoid interaction. Hosts surviving random attacks reproduce, while parasitoid offspring are proportional to attacked hosts. For $r>1$, its positive equilibrium is $Y_{\ast }=(\log r)/a$, $X_{\ast }=r{Y}_{\ast }/[c(r-1)]$. Counting attacked rather than surviving hosts makes its predator reproduction law different from the annual pulse-breeding predator-prey model.

A carrying capacity is an abundance scale at which a specified environment's net population growth vanishes. In the logistic growth equation, positive unharvested populations approach $K$. It depends on the species, resources and environment rather than being an immutable universal population limit.

The useful distinction is between a growth rate and a sustainable abundance. In a logistic differential equation, $r$ controls how quickly a population responds, while the carrying capacity $K$ sets the positive equilibrium point. Increasing $r$ with $K$ fixed accelerates approach without raising that equilibrium. Increasing $K$ changes the destination. The logistic differential equation diagrams show populations approaching the same carrying capacity from below and above.

In ecological use, the carrying capacity describes what a particular environment can support under specified conditions. Nutrient supply, available space, and accumulated waste can change those conditions; it is not a permanent property of a species alone. A population temporarily above a carrying capacity is possible. In the simplest constant-$K$ logistic growth equation, its net growth becomes negative; real population dynamics can include delays, seasonal variation, and changing resources that the basic mathematical model omits. ^[ref].

Cellulose is a polysaccharide of glucose units whose chains form strong assemblies in plant cell walls. It illustrates a structural role for carbohydrates. See Alberts et al., Molecular Biology of the Cell: the plant cell wall.

A polynomial identity is equality of formal polynomials: every corresponding polynomial coefficient agrees. For example, $(X+Y{)}^2=X^2+2XY+Y^2$ holds in the polynomial ring over every commutative ring. Polynomial evaluation preserves such an identity, so it gives an equality of values at every allowed substitution.

The converse requires a hypothesis on the coefficient field. Over an infinite field, a polynomial that vanishes at every point is zero: use the Lagrange root bound over a field in one variable, and induction on the number of variables. Over the finite field ${\mathbb{F}}_q$, the nonzero formal polynomial $X^q-X$ induces the zero polynomial function. Thus equality of polynomial functions on a finite domain need not be a formal polynomial identity. Coefficient comparison, as used with the binomial theorem, concerns formal polynomial identities.

A quadratic polynomial in one variable over a field has polynomial degree two and the displayed form. Its roots of a polynomial satisfy a quadratic equation. The polynomial and its associated polynomial function are distinct objects: over a finite field, different polynomials can define the same function.