At $b=1$ the Hénon map has unit Jacobian determinant. The upper fixed point is linearly elliptic for $-1<\mu <3$, and its newborn two-cycle is elliptic for $3<\mu <4$. These are neutral multiplier statements, not attraction. The endpoints have double multipliers $+1$ and $-1$. Constant-area scaling also excludes a simple invariant closed curve enclosing positive area when $b>0$ and $b\ne 1$: invariance of its interior would require both $\mathrm{area}(F(D))=b\mathrm{area}(D)$ and $F(D)=D$. Thus the $b=1$ elliptic stability boundary is a conservative degeneracy, not a generic dissipative Neimark-Sacker bifurcation. Arbitrary non-area-preserving perturbations can change this behavior even though generic fold and flip curves persist.

On a square Dirichlet grid with spacing $h=1/J$ and interior indices $1,\dots ,J-1$, the five-point Laplacian is diagonalized by products of the sine modes used in the discrete sine transform. The discrete equation $({\Delta }_h+\lambda I)U=F$ is nonsingular exactly when $\lambda$ is not one of the displayed positive eigenvalues of $-{\Delta }_h$, with $1\le p,q\le J-1$. Values below the first resonance are sufficient but not necessary for invertibility.

Two Green functions of the same Helmholtz equation with the same homogeneous Robin boundary condition satisfy the displayed reciprocity by Green's second identity. It is a bilinear identity without complex conjugation, so it concerns the transpose rather than the Hermitian adjoint. In an exterior problem both kernels use the same radiation convention.

On a projective variety over a number field, a very ample line bundle gives a logarithmic projective height by an embedding. Tensor products correspond to sums by the Segre embedding, and a sufficiently ample twist expresses any line bundle as a difference of very ample ones. This defines its height modulo bounded functions. It is additive and functorial under morphism pullback. An ample bundle has the Northcott theorem finiteness property for bounded height and bounded point degree; a trivial bundle need not. The product formula makes these absolute heights independent of scalar coordinates and of the number field used to compute them.

For a symmetric ample line bundle $L$ on an abelian variety, the canonical height is ${\hat{h}}_L(P)={\lim }_{n\to \infty }{4}^{-n}{h}_L([2^n]P)$. It is quadratic and differs from the associated logarithmic Weil height by a bounded function. The canonical height of an elliptic curve specializes this construction to a one-dimensional abelian variety.

For an algebraic Hecke character, the character on archimedean ideles is a product of integral powers of embeddings and their complex conjugates. In the convention $\psi ((a))=a$ for primary $a$ over an imaginary quadratic field, the ideal character has type $(1,0)$ and the idele character has archimedean component $z^{-1}$. Distinguishing these conventions avoids inverse errors in CM Galois representations.

Suppose $F$ contains the CM field $K$ and all CM endomorphisms are defined over $F$. CM reciprocity compares the torus with its idele-transformed lattice and gives $a(s)\in K^{\times }$ for finite ideles. Then ${\psi }_E(s)=a(s_f)/N_{F/K}(s{)}_{\infty }$. On principal ideles, $a(b_f)=N_{F/K}(b)$, so the character descends to the idèle class group. At a good prime its ideal value reduces to the Frobenius isogeny. If the original field omits $K$, construct the character over $FK$ and induce the corresponding representation down to the original field.

A motivating bench experiment asks whether a hot bath receives more heat than the electrical work supplied to a heat pump. PHYWE's authored operating instructions, section 4, describe a sealed vapor-compression refrigeration apparatus with its cold and hot coils immersed in separate water vessels. The quantitative procedure calls for insulated vessels, stirring to reduce temperature gradients, timed temperature readings, and a wattmeter for the input. It waits about five minutes for operating conditions to settle before measurements and explicitly includes the glass vessel and immersed copper pipe when calculating the heated body's heat capacity. One bath cooling while the other warms makes the origin of the delivered energy observable. ^[ref]

The following analysis extends that procedure to simultaneous measurements of both finite baths. Weigh the water in each vessel; determine the effective heat capacities $C_c$ and $C_h$, including immersed hardware, by a separate known electrical-input calibration over the working temperature range. A matched pair of calibrated thermometers placed in stirred water measures bulk $T_c(t)$ and $T_h(t)$; keeping probes away from the coils avoids mistaking a local contact temperature for the whole bath. Record the room temperature and synchronise the electrical meter and thermometry. Use short intervals after startup, with no boiling or freezing and small enough temperature changes that constant effective heat capacities are a reasonable approximation. These are proposed measurement controls, rather than reported results from the manual.

Over a common interval $[t_0,t_1]$, the uncorrected bath estimates and measured electrical input areHere the wattmeter records real electrical power $P_{\mathrm{el}}$. The integral handles changing input, and the signs make both estimated useful heat quantities positive during heating and cooling. The effective heat capacity includes water mass times its specific heat capacity plus the response of the vessel and immersed hardware. Without that correction, equal temperature changes need not imply equal heat transfers.

Insulation reduces heat leaks but does not establish an isolated system. A warmer bath can lose heat to the room while a colder bath gains it. As a proposed control, measure pump-off drift at matched bath and room temperatures, with the same stirring and insulation. Writing $L_h>0$ for outward hot-bath leak power and $L_c>0$ for inward cold-bath leak power, the bath balances giveIntegrate these rates over the same interval before dividing by $W_{\mathrm{el}}$. For example, in a pump-off hot-bath control, $L_h=-C_{h}d{T}_h/dt$; in the cold-bath control, $L_c=C_{c}d{T}_c/dt$. Check that a stopped machine does not create a different leak path from the running one; otherwise use an independent bath calibration and keep that discrepancy as an uncertainty. Repeat at the same starting temperatures to distinguish drift from a change in COP as the baths separate.

In the illustrated toy model, $C_h=C_c=4500\mathrm{J}{\mathrm{K}}^{-1}$, $P_{\mathrm{el}}=50\mathrm{W}$, ${\dot{Q}}_h=150\mathrm{W}$, ${\dot{Q}}_c=100\mathrm{W}$, $L_h=15\mathrm{W}$, and $L_c=5\mathrm{W}$ are stipulated constants for $180\mathrm{s}$. The slopes are $d{T}_h/dt=0.030\mathrm{K}{\mathrm{s}}^{-1}$ and $d{T}_c/dt=-19/900\mathrm{K}{\mathrm{s}}^{-1}$. Consequently,The raw heating and cooling ratios are $2.7$ and $1.9$, whose difference is $0.8$. Correcting the bath leaks gives $3$ and $2$, whose difference is one. The apparent failure of $Q_h=Q_c+W$ was incomplete measurement of heat, not a violation of the first law of thermodynamics.

Real apparatus can also release motor heat and pipe heat directly to the room, and store internal energy during startup. For a machine boundary that receives electrical work, an additional outward heat transfer $Q_{\mathrm{room}}$ gives $Q_h+Q_{\mathrm{room}}=Q_c+W_{\mathrm{el}}-\Delta U_{\mathrm{machine}}$. The residual $Q_h-Q_c-W_{\mathrm{el}}$ therefore tests the boundary and corrections; do not force it to zero. Include any auxiliary electrical work in the input if quoting whole-apparatus COP, and account separately for stirring work deposited in a bath. Ignoring vessel heat capacity, sensor lag or room leaks can bias the inferred result. These limitations follow from extending the measured energy budget, and the manual's explicit hardware correction illustrates why such boundaries matter.

Finally, these vessels are finite bodies whose temperatures change; they are not fixed thermal reservoirs. Their ratio of integrated useful heat to integrated input work is an interval COP. A Carnot's theorem comparison using simultaneous bath temperatures is a local approximation only when the machine settles much faster than the baths change. An exact reversible comparison for changing bath temperatures instead uses the bath entropy changes, for example $C_h\ln (T_{h,1}/T_{h,0})+C_c\ln (T_{c,1}/T_{c,0})$ for constant heat capacities, negligible net machine entropy change and no room leaks. The experiment's beauty is that two thermometers, known heat capacities and an electrical-energy measurement expose both conservation of energy and the practical gap from reversibility.

For the cyclic two-side energy balance in refrigerator, define the dimensionless cooling and heating coefficients byThis identity applies even to irreversible machines, provided both coefficients refer to the same operating point, work input and boundary with no additional energy transfer or storage. It does not compare two separately operated appliances. Unlike thermal efficiency of a heat engine, a COP can exceed one: its numerator counts heat moved, including energy taken from another region. ^[ref]

To obtain temperature-only bounds, impose stronger assumptions: a cyclic machine exchanges heat only with two ideal thermal reservoirs maintained at uniform, fixed absolute thermodynamic temperatures $0<T_c<T_h$, and receives work through an ideal input carrying no entropy. Include the machine and the heat transfer interfaces in the entropy production accounting. Because its own entropy returns to its initial value each cycle, the Second law of thermodynamics givesCombining this inequality with $Q_h=Q_c+W$ givesEquality requires a reversible thermodynamic process for the entire cycle and its contacts, as in a reversed Carnot cycle. These are Carnot's theorem bounds independent of the working substance. Use kelvin values, not degree Celsius readings, in the numerators; the temperature difference can be expressed in either unit. ^{[ref] [ref]}

The diagram fixes $T_c=280\mathrm{K}$ and sets $T_h=320\mathrm{K}$, so the reversible bounds are $280/40=7$ and $320/40=8$. The stipulated machine with ${\mathrm{COP}}_{\mathrm{r}}=3$ and ${\mathrm{COP}}_{\mathrm{heat}}=4$ satisfies both. Its reservoir entropy production per cycle is $4000/320-3000/280=25/14\mathrm{J}{\mathrm{K}}^{-1}>0$. Increasing the temperature lift $\Delta T=T_h-T_c$ at fixed $T_c$ lowers both bounds. Equivalently, delivering fixed $Q_h$ needs at leastThese expressions reveal why a small temperature lift is attractive. Their divergence as $T_h-T_c\to 0^{+}$ is an ideal reversible limit, not a promise of infinite heating or cooling power. For finite heat transfer rates, the working fluid must be colder than the cold bath and hotter than the hot bath; these contact differences produce entropy and increase the internal temperature lift. Dissipative compression and the pressure-reducing restriction of vapor-compression refrigeration supply further departures from reversibility. Thus bath temperatures set an outer bound, not a prediction of real COP. ^[ref]

Heat-engine calorimetry measures heat transfers so that observed work can be compared with the correct energy input. Measuring motion alone cannot supply the denominator of thermal efficiency.

The institutional Laboratory 16: Stirling Engine describes a modern teaching arrangement: an ethanol lamp drives a Stirling engine coupled to an electrical output device, which supplies a resistor immersed in water in a calorimeter. The prescribed readouts include fuel mass consumed, the pressure-volume loop, electrical readings, and water temperature. The sequence separates fuel input, gas-cycle work, electrical output, and Joule heating of the water. Warming of the water is the final useful output; it is not the engine's rejected $Q_c$. The handout distinguishes warm-up from steady operation and asks students to compare the final warming with direct heating by the lamp. It gives a procedure, not a table of measured efficiencies. Its pp. 1–3 therefore establish an apparatus and measurement design without supplying a new experimental result.

For a proposed calibration, use a known input of electrical work to determine the effective heat capacity $C_{\mathrm{eff}}$ of the water, vessel and immersed parts; stir to make the measured temperature representative. A matched run with no input estimates exchange with the room. For constant $C_{\mathrm{eff}}$ and no phase transition, the resulting model iswhere $Q_{\mathrm{leak},\mathrm{out}}$ is the signed net heat lost to the room during the measurement. Here $E_{\mathrm{in}}$ is supplied electrical work when the resistor lies inside the calibration boundary; for a separate external heat input it is the received heat. These are proposed controls, not controls reported in that handout. Slow cooling, changing material properties and imperfectly matched room exchange limit this inference; the approach to thermal equilibrium and heat capacity matter as much as the thermometer reading.

To measure the actual engine-input $Q_h$, one must instrument its hot-side transfer separately, or close a calibrated energy balance around a specified heater boundary. A heater's electrical work includes changes in its stored internal energy and heat transfer that bypasses the engine. At periodic operation a cold-side flowing-water measurement can instead estimate ${\dot{Q}}_c=\dot{m}{c}_p(T_{\mathrm{out}}-T_{\mathrm{in}})$, if mass flow rate $\dot{m}$ and specific heat capacity at constant pressure $c_p$ are known, with negligible storage and other heat exchange in that measurement boundary. It measures only the rejection captured by that stream. Include all remaining transfers before using the residual to infer ${\dot{Q}}_h=P_{\mathrm{out}}+{\dot{Q}}_c$. This measurement discipline makes a comparison with Carnot's theorem meaningful.

A Stirling engine repeatedly heats and cools a confined gas, whose expansion and compression drive a moving mechanical load. This makes it a particularly accessible example of a heat engine: both the repeating motion and the two regions exchanging heat can be observed.

In the published Demonstration experiments with a Stirling engine by Christopher G Deacon and others, an electrical heater warms one end of a cylinder and flowing water cools the other at $12{}^{\circ }\mathrm{C}$. A moving displacer shuttles gas through copper wool that temporarily stores and returns internal energy. A laser spot traces a closed pressure-volume path on a screen: mechanical motion controls one coordinate, pressure the other. Calibrating both scales converts the enclosed area to work per thermodynamic cycle; multiplying by rotation frequency gives gas-cycle power. At $97\mathrm{W}$ heater input, the paper reports $4.4\mathrm{W}$ gas-cycle power, while a separate Prony brake measures $3.4\mathrm{W}$ shaft output. The cold-side temperature is controlled and two higher heater settings are compared. The gas has a nonuniform temperature, and heater energy also escapes to the room: neither output/input ratio measures $W/Q_h$ for an ideal two-thermal reservoir engine. These observations demonstrate work production and losses, not a direct measurement of the bound in Carnot's theorem. The inspected apparatus, readouts and results are on pp. 180–182, Figures 1–3 and Table 1.

Sensible heat is transferred heat associated with a temperature change within a single thermodynamic phase. By contrast, latent heat can change the phase fraction while the temperature stays fixed during equilibrium phase coexistence. Neither name denotes an additional stored substance: both describe how transferred heat participates in a process.

For fixed mass $m$ and composition, use the heat capacity appropriate to the heating constraint. Under the quasistatic process assumptions in heat capacity, with only pressure-volume work and unchanged bulk kinetic energy and potential energy, fixed-pressure heating givesHere $c_p$ is specific heat capacity at constant pressure; fixed-volume heating instead uses specific heat capacity at constant volume $c_V$. A cooling step has negative $Q_{\mathrm{sens}}$ with the sign convention in heat. A temperature difference has the same numerical value in units of the kelvin or degree Celsius. The integral follows by adding the small transfers $\delta Q=m{c}_p(T)dT$, rather than assuming a constant heat capacity.

As a stylized ice-to-water example for a fixed-composition closed system at fixed pressure $1\mathrm{atm}$, take $m=0.5\mathrm{kg}$, an initial temperature of $-10{}^{\circ }\mathrm{C}$, melting at $0{}^{\circ }\mathrm{C}$, and a final temperature of $20{}^{\circ }\mathrm{C}$. Assume constant specific heat capacity at constant pressure values $c_{\mathrm{ice}}=2.1\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$ and $c_{\mathrm{water}}=4.18\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$, and latent heat of fusion $L_f=334\mathrm{kJ}\mathrm{k}{\mathrm{g}}^{-1}$. These are stipulated, rounded model values; ignore the container and heat losses, retain all the material, and assume only pressure-volume work under the quasistatic process conditions above. The three steps areOnly the first and third terms are sensible heat; the middle term is latent heat. The plateau begins after $10.5\mathrm{kJ}$ and ends after $177.5\mathrm{kJ}$ of total supplied heat. At $94\mathrm{kJ}$ total input, half the mass is liquid water and the temperature remains $0{}^{\circ }\mathrm{C}$. The equilibrium phase transition temperature depends on pressure; this example stipulates a single fixed pressure. ^[ref]

This split makes heating budgets easy to interpret: melting consumes $167/219.3\approx 76.2\%$ of the supplied heat in this model despite causing no temperature rise. Within either single thermodynamic phase, the heating curve slope is $dT/dQ=1/(m{c}_p)$; a larger heat capacity makes the slope shallower. A flat interval represents changing phase fraction, so it cannot be modeled by applying $Q=m{c}_p\Delta T$ across the phase transition. The complementary constructed example in latent heat tracks that phase fraction explicitly.

A Hausdorff space is completely regular if a point $x$ outside a closed set $F$ can be separated from $F$ by a continuous function $u:X\to [0,1]$ with $u(x)=1$ and $u{|}_F=0$. These functions determine the original topology: for any neighborhood $O$ of $x$, choose $F=X\setminus O$ to obtain $x\in \{u>1/2\}\subseteq O$. This is exactly the point-separation property needed for the evaluation embedding into the weak-star dual of bounded continuous functions.

The least ordinal that does not admit an injection into $X$. Its existence, supplied by Hartogs theorem, does not require the axiom of choice.

A bead in a liquid keeps undergoing Brownian motion even when an optical trap holds its average position fixed. For a stationary harmonic optical trap in thermal equilibrium at absolute temperature $T$, the equipartition theorem assigns mean potential energy $k_{B}T/2$ to this one quadratic coordinate. With $k_B$ the Boltzmann constant and $x=0$ at equilibrium,At the same temperature, quadrupling the spring constant divides the position variance by four and the standard deviation by two. The Boltzmann factor gives the complete equilibrium probability density function, a Gaussian distribution:The prefactor makes ${\int }_{-\infty }^{\infty }p(x)dx=1$; its second moment is ${\int }_{-\infty }^{\infty }{x}^{2}p(x)dx=k_{B}T/k$. The probability density function has units of inverse length, and its area gives probability. This density is exact for the ideal quadratic model. In a finite optical trap it approximates the occupied well when thermal excursions remain within the harmonic region and escape is negligible during observation. Driven motion, appreciable nonconservative forces or laser heating need not give this equilibrium density.

Here $\ell =\sqrt{k_{B}T/k_0}$, $q=x/\ell$ and $\kappa =k/k_0$. The plotted density per unit $q$ is $p_q(q)=\ell p(\ell q)=\sqrt{\kappa /(2\pi )}{e}^{-\kappa q^2/2}$. Each curve has total area one; their dimensionless standard deviations are $1$, $1/\sqrt{2}$ and $1/2$. The curves are exact model densities, not noisy experimental data.

Brownian motion becomes a measuring tool: a calibrated position measurement supplies ${\sigma }^2=⟨(x-⟨x⟩{)}^2⟩$, and the known temperature supplies $k=k_{B}T/{\sigma }^2$. Once $k$ is known, a slowly applied constant external force $f$ shifts the mean by $⟨x⟩=f/k$, so $f=k⟨x⟩$ in the same harmonic range. Fluctuations therefore calibrate the conversion from displacement to force. Svoboda and Block used this equipartition theorem method when calibrating their bead traps. ^[ref].

For the example stiffness $0.01pN/nm$ and $T=300\mathrm{K}$, $k_{B}T\approx 4.142\mathrm{pN}\mathrm{nm}$ and $\sigma \approx 20.35\mathrm{nm}$. Using a variance measured in uncalibrated detector units would not determine that physical stiffness. Measurement noise, finite exposure time, drift and insufficient sampling of the correlated Brownian motion can bias the estimate; record positions over many optical trap relaxation times and account for the detector response. The deterministic relaxation curve above describes relaxation without thermal forcing; an individual thermal trajectory continues to fluctuate.

For a bead in a stationary viscous liquid, linear drag often dominates inertia on the observed timescale. Neglecting both inertia and the random thermal force, the deterministic force balance givesHere $\zeta >0$ is the drag coefficient and $\tau$ the optical trap relaxation time: the time to reduce the deterministic displacement by a factor $e$. The bead approaches equilibrium without oscillation in this overdamped model. The approximation requires times long compared with the inertial time $m/\zeta$ and, for separated timescales, $m/\zeta \ll \zeta /k$, where $m$ is the bead's mass. For a no-slip spherical bead of radius $a$ in an unbounded liquid of dynamic viscosity $\eta$, Stokes drag law gives $\zeta =6\pi \eta a$. Nearby walls change that drag coefficient.

For example, $a=0.5\mu \mathrm{m}$, $\eta =10^{-3}\mathrm{Pa}\mathrm{s}$ and $k=0.01pN/nm=10^{-5}N/m$ give $\zeta \approx 9.42\times 10^{-9}\mathrm{N}s/m$ and $\tau \approx 0.942\mathrm{ms}$. A displacement of $100\mathrm{nm}$ gives a restoring force of magnitude $1\mathrm{pN}$, and its deterministic displacement drops to $100/e\approx 36.8\mathrm{nm}$ after one optical trap relaxation time. These are predictions for the stated model, not measured trajectories.

The figure uses $q=x/\ell$, $\kappa =k/k_0$ and $s=t/(\zeta /k_0)$, with arbitrary positive reference length $\ell$ and stiffness $k_0$. Thus $U/(k_0{\ell }^2)=\kappa q^2/2$, $F/(k_0\ell )=-\kappa q$ and, from $q(0)=1$, $q(s)=e^{-\kappa s}$. At fixed drag, greater stiffness gives both a steeper potential energy well and faster relaxation. Thermal Brownian motion is omitted from these deterministic curves.

The disk Poisson integral is the harmonic function with boundary data $u$, given byIt is the real part of the Schwarz integral on the unit disk for real data. The nonnegative kernel has integral one; smooth boundary data extend smoothly to the circle. This disk formula should be distinguished from the half-plane formula.

A locally uniformly bounded sequence of harmonic functions with a pointwise limit converges uniformly on compact subsets, and its limit is harmonic. A radial mollifier of fixed local radius reproduces each harmonic function by the mean value property for harmonic functions. Differentiating that convolution gives uniform interior gradient bounds, hence equicontinuity. A finite covering of a compact set converts pointwise convergence to uniform convergence. Passing to the limit in the radial convolution identity proves smoothness; passing to the limit against the Laplacian of each test function proves harmonicity.

A smooth real function on a complex manifold is pluriharmonic if $\partial \bar{\partial }f=0$. A complex-valued function satisfies the same equation exactly when both real and imaginary parts are pluriharmonic. Restrictions to complex coordinate lines are harmonic functions. On a compact connected complex manifold, choose a maximum of $|f|$ and apply the harmonic maximum principle successively along coordinate lines; the function is constant on a polydisc, then on the manifold by open-and-closed propagation. Without connectedness, constancy is componentwise.

For a $C^2$ harmonic function $u$ and Brownian motion stopped before leaving its domain, Itô formula gives $du(B_t)=\nabla u(B_t)\cdot d{B}_t+\frac{1}{2}\Delta u(B_t)dt$. The drift vanishes. On a compact subdomain, the stopped process is a bounded martingale. Optional stopping theorem at an almost-surely finite exit time therefore gives the boundary mean formula. At the centre of a disk, rotational invariance of planar Brownian motion makes the exit law uniform on the boundary circle. This proves the mean value property for harmonic functions probabilistically.