Online text: openstax.org/books/anatomy-and-physiology-2e/pages/1-5-homeostasis

Cited passages: §1.5, Figure 1.10 and thermoregulatory examples following it; §1.5, Negative Feedback, Figure 1.10: sensor, control center and effector; §1.5, opening paragraphs and Negative Feedback: normal ranges and compensatory responses.

Online text: openstax.org/books/anatomy-and-physiology-2e/pages/17-9-the-endocrine-pancreas

Cited passages: 17.9, delta cells and somatostatin; 17.9, glucagon signaling through cAMP and glycogen breakdown; 17.9, pancreatic islet cell types and regulation of blood glucose.

Online text: openstax.org/books/anatomy-and-physiology-2e/pages/12-2-nervous-tissue

Cited passages: Anatomy and Physiology 2e, §12.2, Parts of a Neuron: nucleus, organelles, processes, branching axon, initial segment and terminals; Anatomy and Physiology 2e, §12.2, Types of Neurons and Figure 12.9: multipolar, bipolar and human pseudo-unipolar cells; sensory projection past the cell body.

Online text: chem.libretexts.org/Bookshelves/Analytical_Chemistry/Supplemental_Modules_(Analytical_Chemistry)/Analytical_Sciences_Digital_Library/Courseware/Analytical_Electrochemistry:_Potentiometry/06_Experiments/01_Potentiometric_Titration_of_an_Unknown_Weak_Monoprotic_Acid

A Green's function is the response of a specified linear differential operator $L$ to a point source, together with specified boundary conditions or causality conditions. In distributional notation, $L_{x}G(x,y)=\delta (x-y)$, where $\delta$ is the Dirac delta function. Integrating this response against a source gives $u(x)=\int G(x,y)f(y)dy$ when the representation is valid.

For $L=-d^2/d{x}^2$ on $0<x<1$ with zero Dirichlet boundary conditions, $G(x,y)=\min (x,y)(1-\max (x,y))$. It is linear on either side of $y$, continuous there, and its first derivative jumps by $-1$, so $-G^{\prime \prime }={\delta }_y$. Changing either the operator sign or the boundary conditions changes the kernel. The example shows how a difficult source is assembled from elementary impulse responses.

Suppose that, as $t\downarrow 0$,and that the corresponding Laplace integral has suitable integrability and growth control away from zero. Watson lemma gives the endpoint expansionas $x\to +\infty$. Each finite truncation of the local amplitude expansion produces the corresponding finite truncation and remainder of the integral expansion.

A function is piecewise continuous on a finite interval if it is continuous between finitely many breakpoints and has finite one-sided limits at those points and at the endpoints. On $[0,\infty )$, require this on every finite interval. A Heaviside step function is an example; a function with an infinite blow-up at an interior point is not. This condition guarantees local integrability while allowing jumps in an input.

Harmonic analysis studies functions through elementary oscillations, generalizing Fourier series and the Fourier transform. On a locally compact Abelian group, the oscillations are continuous unitary characters and Haar measure supplies translation-invariant integration.

The mechanism is visible in $d(e^{i\xi x})/dx=i\xi e^{i\xi x}$: differentiation becomes multiplication by a frequency parameter. A suitable transform can therefore simplify a differential equation, while the inverse transform reconstructs the solution. The subject also asks which transforms, averages and singular integral operators preserve particular function-space bounds. Those estimates control whether the formal decomposition remains valid for rough or limiting data.

A rough path augments a path with compatible higher-order iterated-integral data. The extra data records effects, such as signed area, that are not continuously determined by the first-level path in the uniform topology. A rough differential equation uses this enhancement to recover a continuous solution map for irregular drivers.

The Faraday constant is the electric charge associated with one mole of elementary charges, $F=N_{\mathrm{A}}e\approx 96485.33212\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$. The product of the Avogadro constant and elementary charge is exact under the SI definitions; the displayed decimal is rounded. For ions with signed charge number $z$, the molar electrical contribution to an electrochemical potential is $zF\psi$, where $\psi$ is electric potential. ^[ref].

The Avogadro constant converts amount of substance into a dimensionless entity count: $N=n{N}_{\mathrm{A}}$. Its value is exact under the SI definition of the mole. ^[ref]

Online text: telescope-optics.net/Seidel_aberrations.htm

Cited passages: Amateur Telescope Optics, §3.5.1, opening paragraph: historical classification and equation 5.

Online text: www.telescope-optics.net/coma.htm

Cited passages: Amateur Telescope Optics, §4.2, Figure 41: geometrical spots versus diffraction images; Amateur Telescope Optics, §4.2, classical coma formula after equation 15 and equation 13: pupil-cubic and field-linear dependence; Amateur Telescope Optics, §4.2, opening paragraphs and Figure 39: field coma versus misalignment coma.

For fixed-length messages in ${\mathbb{F}}_q^L$, choose independent uniform field elements $a,b$. Each output is uniform because $b$ masks it. For distinct messages and prescribed output difference $d$, the equation ${\sum }_{j=1}^L(m_j^{\prime }-m_j)a^j=d$ is a nonzero polynomial equation of degree at most $L$, so the root bound for a polynomial gives at most $L$ choices of $a$. The joint output probability is at most $L/q^2$. Thus the family is $(L/q)$-almost strongly universal when $L<q$. Starting the message polynomial at power one matters: using a freely varying message coefficient at power zero would permit a known constant tag shift after observing one tag.

The allele frequency is the proportion of gene copies at a specified genetic locus carrying an allele. Two unrelated copies can have the same observed allele without being identical by descent; under independent sampling their matching probability is the sum of squared allele frequencies. Pedigree autozygosity instead traces copies to the same ancestral copy.