For $A\subseteq {\mathbb{R}}^2$ and a nonzero direction $\mathbf{b}$, its affine section is the inverse image of $A$ under the line parametrization $t\mapsto \mathbf{a}+t\mathbf{b}$. The resulting set is a subset of the parameter line, so changing a parametrization can change the section by translation or scaling. The four real parameters can be encoded by a single real using real-number pairing by separated digits. Thus no fixed planar set can realize every real subset as a section, by Cantor's theorem.

The two coordinate axes divide a Cartesian coordinate system in the plane into four open quadrants, distinguished by the signs of their coordinates. The positive quadrant is the first quadrant.

For a strictly bifunctional $AA+BB$ feed, label the limiting functional groups $A$, set $0<r=A_0/B_0\le 1$, and define $p=J/A_0$ as in functional-group conversion in step growth. The initial molecule count is $N_0=(A_0+B_0)/2$, and each intermolecular join consumes one $A$ and one $B$. ThusThis version of the Carothers equation uses limiting-group conversion, not the fraction of all groups reacted.^[ref]

As an original finite example, mix $95$ $AA$ monomer molecules with $100$ $BB$ monomer molecules. Then $(N_0,A_0,B_0)=(195,190,200)$ and $r=0.95$. At $p=0.90$, $171$ joins leave $24$ molecules, giving $X_n=195/24=8.125$. At $p=1$, $190$ joins leave five chains and ten unreacted $B$ groups; every chain has two $B$ ends, and $X_n=39$. Continuing to wait cannot remove the missing-$A$ constraint within this model.

For $r<1$, complete limiting-group conversion gives the ceiling $X_{n,\max }=(1+r)/(1-r)$. Reaching a target $X_n=K>1$ therefore requires $r\ge (K-1)/(K+1)$, even before considering incomplete conversion. For $K=200$, $r\ge 199/201\approx 0.99005$. Equality requires $p=1$; any lower conversion demands closer balance. The imbalance $1-r$ is measured relative to the initial number of $B$ groups, not relative to the total number of groups. A monofunctional impurity changes the initial groups-per-molecule count, so it cannot be inserted into this strictly bifunctional formula by retaining an unexamined $r$.

On a connected manifold, a smooth bracket-generating family of vector fields connects every pair of points by a piecewise smooth horizontal curve. For a free step-N nilpotent Lie group, this says that every group element is the path signature of a sufficiently regular path. In the planar step-two case, a closed rectangle prescribes the signed area and a following straight segment prescribes the displacement.

A boundary neighborhood over an arc $I$ of the unit circle, with radial depth equal to its normalized angular length $m(I)$. Changing that depth by any fixed positive factor gives an equivalent Carleson measure condition. Large arcs and central mass can be handled separately by finitely many fixed neighborhoods covering the unit disc.

A positive Borel measure on the unit disc satisfies $\int |f|d\mu \le N\Vert f{\Vert }_1$ for every H1 space on the unit disk function exactly when it is a Carleson measure, with comparable optimal constants. The normalized squared Cauchy kernel tests necessity. For sufficiency, the boundary shadow of every point with $|f(z)|>t$ lies inside a component arc of $\{Nf>t\}$. The geodesic-cap and Carleson-box comparison bounds the measure of each shadow region by its component length, giving $\mu \{|f|>t\}\le C\Vert \mu {\Vert }_{\mathrm{Carl}}m\{Nf>t\}$. The layer cake representation and H1 non-tangential maximal inequality on the disk give the embedding. The broader kernel and covering mechanism is discussed in Terence Tao's harmonic-analysis notes, Exercise 2 after Lemma 4.1; the analytic endpoint here uses the fractional-power argument just given.

Zhu, Miao, Xu and Cao's 2007 experiment drove a triangular metal frame electromagnetically with a sinusoidal generator. In a roughly $3\mathrm{cm}$-deep, temperature-regulated pool, it produced nearly plane capillary waves. A $632.8\mathrm{nm}$ laser illuminated the surface obliquely; the moving ripples acted as a reflection diffraction grating. One split beam monitored laser stability, while a CCD recorded diffraction orders on a screen $6.76\mathrm{m}$ away. An air-supported optical table reduced ambient vibration. ^[ref]

The frequency sweep was $100$–$290\mathrm{Hz}$ in $10\mathrm{Hz}$ steps, using distilled water and kerosene at $20{}^{\circ }\mathrm{C}$. For each setting they averaged at least three measurements of first-order spot separation. Known optical geometry and the grating equation yielded $k=2\pi /\lambda$; generator readings yielded $\omega =2\pi f$. Agreement with the predicted log–log slope $3/2$ tests the capillary law, and its intercept gives surface tension if mass density is known. The authors identify surface contamination as a major error source; noncontact optical detection reduces additional disturbance. ^[ref]

The optical measurement can be understood from the grating equation. Let ${\lambda }_L$ be the laser wavelength, $\alpha$ the incident grazing angle above the mean surface, and ${\beta }_m$ the outgoing grazing angle of diffraction order $m$, with the sign of $m$ chosen so thatThus calibrated spot positions determine outgoing angles, which determine the ripple wavelength $\lambda$, independently of the mechanical dispersion relation. This optical geometry treats the surface as a reflection diffraction grating at the laser wavelength. ^[ref]

The inferred surface tension in the full deep-water model isThis formula makes useful experimental controls explicit. Measure mass density at the sample's temperature; check depth with $kh$; keep the driven wave amplitude small enough that $ka\ll 1$; and distinguish $f$ from angular frequency $\omega$. For the stipulated parameters, the $100$–$290\mathrm{Hz}$ window corresponds to calculated wavelengths about $3.62$–$1.76\mathrm{mm}$ and $kh\simeq 52$–$107$ at $h=30\mathrm{mm}$. Depth corrections are negligible here, but the gravity contribution is about $4.5\%$–$1.1\%$ of the capillary contribution. These are ideal predictions, not digitized results from the experiment.

Omitting gravity would infer ${\gamma }_{\mathrm{cap}}=\rho {\omega }^2/k^3=\gamma +\rho g/k^2$, biasing surface tension upward. In the capillary limit, the differential estimate $d\gamma /\gamma \simeq d\rho /\rho +2d\omega /\omega -3dk/k$ also shows why careful spot-spacing calibration matters. A clean surface and weak viscosity are modeling assumptions: appreciable damping, surfactant stresses or large wave amplitudes call for additional physics. The experiment tests the short-wave regime; its drive window does not establish the crossover or the long-wave surface gravity wave limit.

A gravity-capillary wave has restoring contributions from both gravity and surface tension. Consider an initially resting, horizontally unbounded liquid layer of constant depth $h$, with uniform mass density $\rho$, incompressible flow, inviscid flow and irrotational flow. The upper gas exerts constant pressure and has negligible inertia. Assume a clean interface with constant surface tension $\gamma$: no surfactant stresses, imposed current or wind. The wave amplitude $a$ must satisfy $ka\ll 1$ and $a\ll h$, so linearized boundary conditions apply. Deep water means $kh\gg 1$, rather than a particular depth in metres. ^[ref]

Let $z=0$ be the undisturbed free surface, with liquid below it. Potential flow gives a velocity potential $\varphi$ with $\mathbf{u}=\nabla \varphi$, and incompressible flow gives the Laplace equation. The linearized kinematic boundary condition for a free-surface graph and the Unsteady Bernoulli equation, including gravity and the Young–Laplace equation, giveHere $\eta (x,t)$ is the interface displacement; the signed small-slope pressure jump is $p_{\mathrm{liquid}}-p_{\mathrm{gas}}=-\gamma {\eta }_{xx}$. A wave crest has ${\eta }_{xx}<0$, so its capillary pressure is positive. Choose a right-travelling Fourier mode of positive wavenumber $k$ and angular frequency $\omega$:The factor $e^{kz}$ decays into the deep liquid, satisfies the Laplace equation, and makes ${\varphi }_z={\eta }_t$ at $z=0$. Substituting in the dynamic boundary condition leavesThis derivation displays the two restoring contributions with the same sign. Dimensional analysis also checks the result: $\gamma /\rho$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-2}$, so both $gk$ and $(\gamma /\rho )k^3$ have units ${\mathrm{s}}^{-2}$. ^[ref]

At finite depth, a flat impermeable bottom at $z=-h$ requires ${\varphi }_z=0$. Replacing $e^{kz}$ by the normalized vertical factor $\cosh (k(z+h))/\cosh (kh)$ makes the surface ratio ${\varphi }_z/\varphi =k\tanh (kh)$. Eliminating the mode amplitudes then givesThus the same model supplies its own depth check: use the deep-water formula only when $\tanh (kh)$ is sufficiently close to one for the required accuracy. For example, $kh=3$ reduces $\omega$ by about $0.248\%$ relative to deep water, while $kh=1$ reduces it by about $12.7\%$. These percentages are calculations from this factor. ^[ref]

In deep water the phase velocity magnitude follows a wave crest, while the group velocity magnitude follows the wave envelope of a narrow wave packet. Differentiating the positive-frequency dispersion relation givesFor $k{\ell }_c\ll 1$, the surface gravity wave limit has $c_g\simeq c_p/2$. For $k{\ell }_c\gg 1$, the deep-water capillary-wave dispersion relation has $c_g\simeq 3c_p/2$: a wave crest slips backward relative to the faster wave envelope, even while both advance. Neither speed is the oscillating fluid element's own velocity. ^[ref]

There is a particularly neat coincidence at the crossover. Differentiate $c_p^2=g/k+\gamma k/\rho$: its unique minimum is at $k_c=\sqrt{\rho g/\gamma }=1/{\ell }_c$, exactly where the restoring terms are equal. ThereforeFor the stipulated parameters in capillary length, the minimum is $0.231\mathrm{m}{\mathrm{s}}^{-1}$. At $\lambda =3\mathrm{mm}$, direct substitution gives $f=\omega /(2\pi )\simeq 131\mathrm{Hz}$, $c_p\simeq 0.394\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.580\mathrm{m}{\mathrm{s}}^{-1}$. This short wavelength is capillary dominated, but its group velocity is still slightly below the pure-capillary ratio $3c_p/2$ because gravity has not vanished.

On $\{0,1{\}}^{\mathbb{N}}$ this sum is a metric, since each summand obeys the triangle inequality and a zero sum forces coordinatewise equality. A prefix cylinder set fixing the first $n$ coordinates contains the open ball of radius $2^{-n}$ about each of its points and has diameter at most $2^{-n}$. These bounds show that the metric topology equals the prefix cylinder set topology, the usual Cantor space topology.

For a spatial potential $\mathbf{A}$ with $\mathbf{E}=-\dot{\mathbf{A}}-\nabla A_0$, the Maxwell Lagrangian gives $\mathbf{\Pi }=-\mathbf{E}$ and ${\Pi }^0=0$. Its Hamiltonian is $\int [({\mathbf{\Pi }}^2+{\mathbf{B}}^2)/2+A_0\nabla \cdot \mathbf{\Pi }],d^{3}x$. Preserving the temporal momentum first-class constraint gives the Gauss law constraint in gauge theory. In free space, Coulomb gauge and decay at spatial infinity set $A_0=0$ and leave only transverse modes. The reduced Dirac bracket uses the projector ${\delta }_{ij}-k_i{k}_j/|\mathbf{k}{|}^2$, so quantization gives the transverse equal-time commutator and the positive Hamiltonian displayed above.

For the affine action $S=-\frac{m}{2}\int g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }d\tau$ with signature $(-+++)$, differentiating with respect to velocity gives the displayed canonical momentum. Future-oriented proper time in a conformally flat metric yields $P_i^c=-a\gamma m{v}_i$, whereas the conventional future-directed mechanical momentum has $q_i=a\gamma m{v}_i$. Changing the action's overall sign interchanges these canonical labels but leaves its classical geodesics unchanged. Both labels are conserved by spatial translation invariance, and either gives the same physical momentum magnitude proportional to $a^{-1}$. The sign should be stated when comparing a worldline canonical covector with a positive-frequency field-mode label.

These bounded-unitary relations encode a pair of canonical commutation relations without multiplying unbounded position and momentum operators. In a fixed convention on $L^2(\mathbb{R})$, $W(x,y)f(t)=e^{ixy+2iyt}f(t+x)$ and $a\wedge b=x_1{y}_2-y_1{x}_2$. Strong continuity is part of a regular Weyl representation. Different nonzero choices of the central normalization rescale the canonical coordinates.

A calorimeter infers transferred heat from a measured change, such as a temperature change and a calibrated heat capacity. The heat-engine calorimetry example requires accounting for the vessel and heat transfer to the surroundings as well as the water.

A practical receiver has an inner vessel, insulation or an outer shield, a lid, a thermometer and a stirrer. Its participating heat capacity includes the immersed parts that follow the water's temperature. Insulation reduces heat transfer to the room; stirring makes a single reading more representative of the contents. These serve different purposes: a well-insulated vessel can still have substantial heat capacity, and a uniform temperature does not imply zero external heat transfer.

The University of Toronto's Heat Capacity (University of Toronto laboratory handout) describes a copper inner vessel, outer shield, thermometer and heated metal block. Weigh the block and receiving water, heat the block in a bath, then transfer it quickly into enough water to cover it. The observed temperature rises, reaches a maximum, then falls. The inferred specific heat capacity comes from the block's cooling and the warming of both water and vessel. In its cooling exercise the handout deliberately exposes the inner vessel to room air, recommends readings every $15\mathrm{s}$, and uses the later common cooling to estimate the earlier loss. Starting below room temperature reduces exchange without guaranteeing cancellation. The handout notes limited time and thermometry for small corrections; it supplies no dataset.^[ref]

For a proposed modern version, independently perform thermometer calibration and calorimeter calibration, record bath and room temperatures, and keep immersion depth, lid position and stirring rate fixed. Ensure the block actually reaches bath temperature; minimize and account for adhering hot bath water. Compare repeated transfers with different delays and a reference sample of known specific heat capacity. A resting run tests environmental heat transfer, while a stirred run with no added sample checks mechanical input. These are proposed controls, not claimed observations from the handout. Sensor delay, changing heat capacity, incomplete thermal equilibrium and material escaping the boundary limit the resulting estimate.

Mixing calorimetry determines heat capacities by bringing bodies at different temperatures into thermal contact. The striking feature is that the final temperature is a weighted arithmetic mean with heat capacities as weights, rather than weights given by mass. The vessel participates in the exchange even though it is not the sample being studied.

For a hot nonreacting sample of mass $m_x$, initial temperature $T_x$ and unknown specific heat capacity at constant pressure $c_x$, let receiving water have mass $m_w$, initial temperature $T_w$ and known specific heat capacity at constant pressure $c_w$. Let the vessel and immersed parts start at $T_w$ and have combined heat capacity $C_{\mathrm{cal}}$, excluding that water. Assume a common final temperature $T_f$, approximately constant pressure and heat capacities, no phase transition, no chemical reaction and negligible external nonexpansion work. For signed outward heat loss $Q_{\mathrm{loss},\mathrm{out}}$, the combined enthalpy balance isHenceThe loss is positive if the room receives net heat, negative if the room supplies it. If the vessel starts at a different temperature $T_{\mathrm{cal},i}$, replace its warming term by $C_{\mathrm{cal}}(T_f-T_{\mathrm{cal},i})$; do not assume the vessel shared the water's initial temperature. For variable material properties, replace each $mc\Delta T$ by the corresponding integral $m\int c_p(T)dT$ and each vessel term by $\int C_{\mathrm{cal}}(T)dT$. The inferred constant $c_x$ then represents an average over the sample's cooling interval.^[ref]

For a synthetic no-loss example, mix $0.100\mathrm{kg}$ of hot water at $80{}^{\circ }\mathrm{C}$ with $0.200\mathrm{kg}$ of cold water at $20{}^{\circ }\mathrm{C}$ in a vessel initially at $20{}^{\circ }\mathrm{C}$. Stipulate $c_w=4180\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ and $C_{\mathrm{cal}}=100\mathrm{J}{\mathrm{K}}^{-1}$. The enthalpy balance givesThe hot water releases $17.337\mathrm{kJ}$: the cold water receives $15.485\mathrm{kJ}$ and the vessel $1.852\mathrm{kJ}$. Omitting the vessel predicts $40{}^{\circ }\mathrm{C}$, despite perfect insulation. These are computed numbers from stipulated properties, not laboratory observations.

For an unknown-sample example using the same cold water and vessel, stipulate $m_x=0.100\mathrm{kg}$, $T_x=80{}^{\circ }\mathrm{C}$ and $c_x=450\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. With no loss, $T_f\approx 22.752{}^{\circ }\mathrm{C}$. Inverting the full balance returns $450\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$; ignoring vessel storage returns about $402\mathrm{J}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$. With a positive outward loss, ignoring that loss also biases the inferred $c_x$ downward under these hot-sample conditions. A small water temperature rise makes sensor offsets and loss estimates relatively influential; propagation of uncertainty must include the measured temperatures, masses and calibrated vessel term.

A calibration is a closed real differential form whose comass is at most one. Restricting it to any oriented unit tangent plane therefore gives at most that plane's volume. A calibrated submanifold attains equality everywhere, and calibration implies volume minimization among homologous competitors.

An oriented submanifold is calibrated by a calibration when its tangent planes attain the comass bound at every point. Closedness then compares its volume with homologous competitors by Stokes theorem.

A generalized calibration bounds the full energy of a supersymmetric brane, including its potential or gauge couplings. For a membrane, compatible spinor bilinears obey $d\Omega ={\iota }_{K}F$. In a stationary gauge ${\mathcal{L}}_{K}A=0$, Cartan's magic formula proves $d(\Omega +{\iota }_{K}A)=0$. The closed charge form then supplies the energy bound. This extends ordinary calibration from volume alone to charged branes, as developed in Topological charges for branes in M-theory.

A real-valued function $f:D\to \mathbb{R}$, with domain of a function $D\subseteq \mathbb{R}$ in the usual order of the real numbers, is a monotonic function if it is either a nondecreasing function throughout $D$ or a nonincreasing function throughout $D$. The same direction must work for every pair $x<y$ in $D$. Being a strictly increasing function on one part and a strictly decreasing function on another does not make the whole function monotone: for example, $x^2$ on the real line changes direction at zero. A constant function is both a nondecreasing function and a nonincreasing function.

The word "increasing" is ambiguous across textbooks: some allow equal outputs and some require unequal outputs. Use nondecreasing function for the weak inequality and strictly increasing function for the strict inequality; use nonincreasing function and strictly decreasing function in the other direction. Neither continuity nor differentiability is part of the definition of a monotonic function.

Strict monotonicity gives injectivity, so an equation $f(x)=b$ has at most one solution and an inverse function exists on the image of a function. Weak monotonicity permits repeated outputs and jumps. In probability theory, a cumulative distribution function is a nondecreasing function; its quantile function uses a generalized inverse to handle those flat parts and jumps. Monotonicity alone does not guarantee continuity, even for a strictly increasing function.

Taylor's theorem expresses a sufficiently differentiable function as a finite Taylor polynomial plus a Taylor remainder. One convenient version assumes $f$ is $n+1$ times continuously differentiable on the segment between $a$ and $x$. Thenfor some intermediate point $\xi$, when $x\ne a$. Thus a bound $|f^{(n+1)}|\le M$ on that segment gives $|R_{n,a}(x)|\le M|x-a{|}^{n+1}/(n+1)!$. The intermediate point usually need not be computed: bounding the derivative throughout the segment is enough. This is why local approximation can yield reliable numerical bounds. A finite Taylor polynomial does not require an infinite Taylor series to represent the function.

Historically, the Royal Society holds Brook Taylor's letter of 26 July 1712 to John Machin. Its catalogue connects his work on Kepler's second law with the expansion later published in the 1715 Methodus incrementorum directa et inversa. This is a documented route from an orbital calculation to a general approximation method, rather than a claim that every earlier use of such expansions began with Taylor. See Royal Society archive, letter MS/82/2.

The floor function and ceiling function turn a real number into directed integer bounds. They answer two different questions: which integer is the largest one that does not exceed $x$, and which is the smallest one that is not less than $x$?Both outputs are integers. These are exact mathematical operations; they need not choose the closest integer, and they do not refer to roundoff error in floating-point arithmetic.

For example,The fractional part is $\{x\}=x-\lfloor x\rfloor$. The negative example matters: discarding the digits after the decimal point would give $-2$, which is truncation toward zero, rather than the floor function's value $-3$. At an integer, the floor function and ceiling function agree and the fractional part vanishes.

The usefulness is a bridge from continuous quantities to discrete decisions. An integer count in a closed interval uses both bounds. A ceiling count for fixed-size batches counts how many containers or pages are needed; a floor function counts how many complete groups fit. In Euclidean division, the same bounds specify the quotient, while the fractional part records the scaled remainder. A small amount of notation removes many cases that would otherwise have to be stated separately.

The reflection identityfollows by reflecting the ordered integers across zero: the largest one below $-x$ becomes the smallest one above $x$. For every integer $m$, translation gives $\lfloor x+m\rfloor =\lfloor x\rfloor +m$ and $\lceil x+m\rceil =\lceil x\rceil +m$. These formulas turn upper bounds into lower bounds and move examples between neighboring intervals.

For a historical example of the notation in use, Kenneth E. Iverson's 1962 conference paper A Programming Language lists the floor function, ceiling function and residue operation alongside the ordinary arithmetic operations. It states the two defining inequalities above, then extends operations component by component to vectors and matrices. This shows their role as tools for describing algorithms, rather than merely conventions for writing decimal numbers. ^[ref]