Hensel lifting improves a solution of a polynomial congruence modulo a prime power to one modulo a higher power. For $f\in \mathbb{Z}[X]$, suppose $f(a)\equiv 0(\mathrm{mod}{p}^k)$ and $f^{\prime }(a)\not\equiv 0(\mathrm{mod}p)$. A lift $a+t{p}^k$ satisfiesThe invertible derivative therefore determines a unique $t$ modulo $p$.

For $f(X)=X^2-2$, the root $3$ modulo seven lifts to $10$ modulo $49$, since $3+7=10$ and $10^2-2=98$. Repetition gives a root in the p-adic integers. The simple-root hypothesis ensures uniqueness; when the derivative vanishes modulo $p$, there may be multiple lifts or none.

Heliconius melpomene is a tropical butterfly used in research on wing-pattern evolution and speciation. Genome comparisons link mimicry-pattern variation with exchange of genetic material among related species.^[ref]

Heliconius cydno is a butterfly closely related to Heliconius melpomene. Differences in habitat use, wing pattern and mating behavior make their comparison informative about reproductive isolation despite some exchange of genetic material.^[ref]

Heliconius erato is a tropical butterfly with geographically variable wing patterns. Resemblance to defended butterflies in the same region makes it a useful system for studying Müllerian mimicry and the relationship between local selection and inherited pattern variation.

The classical Heisenberg model assigns a three-component unit vector to each site and couples pairs by their dot product. Positive $J$ favors parallel alignment. With no applied magnetic field or anisotropy, simultaneous rotations of all vectors preserve the energy, so an ordered state chooses a direction and exhibits spontaneous symmetry breaking. The vector order parameter is the spatially averaged orientation. A quartic Landau free energy describes its mean-field onset near a continuous transition; fluctuations can change the critical exponents.^[ref]

The XXZ spin chain has equal transverse exchange interactions and a possibly different longitudinal coupling. For $J>0$ the transverse coupling is antiferromagnetic, while the sign and size of $\Delta$ determine the longitudinal preference. A periodic even chain is bipartite. Its continuous symmetry is rotation about $z$ for generic $\Delta$, enhanced to full spin rotation at $\Delta =1$.

For $H=(\Delta /4){\mathbf{\sigma }}_1\cdot {\mathbf{\sigma }}_2-({\mu }_{0}B/2)(g_1{\sigma }_{1z}+g_2{\sigma }_{2z})$, the parallel-spin energies are $\Delta /4\mp {\mu }_{0}B(g_1+g_2)/2$. The zero-projection block has energies $-\Delta /4\pm \sqrt{{\Delta }^2+[{\mu }_{0}B(g_1-g_2){]}^2}/2$. Signed moment conventions must be fixed before assigning a physical opposite-charge case.

Writing the unit symmetry axis as $n$ and axial spin momentum as $s=I_3{\omega }_3$, angular momentum is $L=I_{1}n\times \dot{n}+sn$. Taking its derivative and applying gravitational torque gives $I_{1}n\times \ddot{n}+s\dot{n}=-mg\ell n\times \hat{z}$.

A lumped sample of heat capacity $C$ coupled by thermal conductance $G$ to a bath has the displayed linearized energy balance. For constant internal power, the bath-to-sample temperature response is $1/(1+i\omega C/G)$. Insulation and heat capacity suppress rapid bath fluctuations but have unity static bath-to-sample gain. A steady power change shifts the sample by $\delta P/G$, so improved isolation increases self-heating sensitivity and equilibration time. The lumped approximation requires internal equilibration faster than the variations being modeled.

thermal insulation reduces unwanted heat exchange. Conduction through supports, convection in gas and thermal radiation require different treatments. vacuum suppresses gas transport but leaves solid conduction and radiation; lowering thermal conductance also increases the temperature rise caused by a fixed internal heat load.

Thermal power generated per unit volume. Its integral over a region is the region's total heat generation rate.

For a square of side $\ell$ initially at constant temperature $T_0$, held at zero boundary temperature for $t>0$, the heat equation $T_t=\kappa \Delta T$ with $\kappa >0$ has solutionThe double Fourier sine series of the initial constant gives the coefficients. The spatial modes are Dirichlet Laplacian eigenfunctions, so separation of variables gives their decay rates. At late times the $(1,1)$ mode dominates, giving the displayed profile; the next nonzero decay exponent is $10\kappa {\pi }^2/{\ell }^2$. At early times the series approaches the constant at each interior point and in L2 space, while the boundary remains at zero. An instantaneous change to the boundary condition therefore need not produce a solution continuous at the initial boundary in space-time.

On $0<x<L$ with zero-temperature endpoints, let the initial temperature rise linearly from zero to height $H$ at $L/2$ and then fall linearly to zero. Integration by parts on the two halves gives Fourier sine series coefficients $8H\sin (n\pi /2)/({\pi }^2{n}^2)$. Under the heat equation $T_t=D{T}_{xx}$ with $D>0$, each coefficient decays by $e^{-D(n\pi /L{)}^{2}t}$, giving the displayed solution. Only odd spatial modes occur; absolute coefficient summability proves uniform convergence for $t\ge 0$ via the Weierstrass M-test. This spatial profile also occurs in a midpoint-plucked fixed-end string, where the time factors oscillate instead of decaying.

For positive thermal diffusivity $\nu$, thermal conductivity $k$ and length $L$, subtract the stationary solution $w(x)=Hx/k$. The remainder satisfies the heat equation with a homogeneous Dirichlet boundary condition at zero and a homogeneous Neumann boundary condition at $L$. The mixed Dirichlet-Neumann modes on an interval are $\sin (k_{n}x)$, where $k_n=(n+1/2)\pi /L$. Thus for an initial temperature profile $u_0$,For zero initial temperature, $b_n=-2H(-1{)}^n/(kL{k}_n^2)$. The series gives the initial profile in the usual Fourier series sense and a smooth solution for $t>0$. If $H\ne 0$, the zero initial profile does not satisfy the prescribed endpoint derivative at $t=0$: the temperature can be continuous there without its spatial derivative extending continuously to that corner. The sign $k{u}_x(L,t)=H>0$ describes incoming heat flux, since the outward conductive heat flux is $-k{u}_x$.

A sinusoidal surface temperature applied to a homogeneous semi-infinite conductor produces a periodic depth profile with penetration length $\delta =\sqrt{2\kappa /(\omega C)}$. It follows by substituting a complex exponential into $C{\partial }_{t}T=\kappa {\partial }_z^{2}T$ and taking the root that decays at depth. A depth $z$ attenuates the amplitude by $e^{-z/\delta }$ and delays the phase by $z/\delta$. This is diffusion with phase delay, rather than a nondispersive travelling wave.

Integrating energy conservation over a fixed material volume gives storage equal to incoming conductive heat flux plus internally generated heat. Here $c_{\mathrm{vol}}$ is volumetric heat capacity, with units $\mathrm{J}{\mathrm{m}}^{-3}{\mathrm{K}}^{-1}$. For a mass-specific heat capacity, the density factor must be included. Constant isotropic conductivity reduces the divergence to $k{\nabla }^{2}T$; variable conductivity must remain inside the divergence.

For density $F=\alpha (T-T_c)M^2/2+b{M}^4/4-{\mu }_{0}HM$, the zero-field ordered minimum is $-{\alpha }^2(T-T_c{)}^2/(4b)$. Its constant-field heat-capacity density jump is $\Delta c_H=T_c{\alpha }^2/(2b)$. In an Arrott plot, let physical slope be $S={\mu }_0/b$ and horizontal intercept be $X_0(T)=a(T)/{\mu }_0$. ThenThis makes numerical sensitivity explicit: uncertainty in the intercept slope is squared. A volumetric free energy determines a volumetric heat-capacity jump; total or molar values need further sample information.

For independent particles with interconverting state families $i$, the equilibrium partition function is a sum $q={\sum }_i{q}_i$ and the family probabilities are $x_i=q_i/q$. Let $u_i$ be the conditional mean one-particle energy in family $i$ and $c_i=d{u}_i/dT$. The total canonical energy variance separates into within-family and between-family contributions, givingThe final positive term is absent if family fractions are held fixed. A frozen ortho-para mixture and an equilibrating mixture can therefore have different heat capacities at the same instantaneous composition.

For an ideal diatomic gas with intact molecules and all translational, rotational and harmonic vibrational degrees of freedom active, the classical molar heat capacity is $7R/2$. Quantum freeze-out of vibrations and then rotations produces approximate $5R/2$ and $3R/2$ regimes. The model excludes dissociation, condensation and electronic excitation.

The nonretarded attractive normal pressure obtained by differentiating the half-space interaction energy per area: $P=-A_H/(6\pi d^3)$. Total force additionally requires the interacting area.