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The only zeros of $f$, modulo the reciprocal lattice, are the two Dirac valleys in graphene,At these momenta ${\sum }_j\sin ({\mathbf{K}}_{\tau }\cdot {\mathbf{b}}_j)=\tau 3\sqrt{3}/2$. Thus the two Dirac mass coefficients, measured in energy units, are $m_{\tau }=M-\tau \mu$, with $\mu =3\sqrt{3}{t}_2\sin \varphi$. For $\mathbf{k}={\mathbf{K}}_{\tau }+\mathbf{q}$,The two Dirac cones have opposite orientation, encoded by $\tau$. A scalar sublattice offset $M$ gives equal-sign Dirac masses; imaginary second-neighbour hopping at $M=0$ gives opposite-sign Dirac masses. Both mechanisms open valley gapsAn observed band gap alone therefore cannot establish a Chern insulator. In particular, the gapless nearest-neighbour model of graphene and its sublattice-mass deformation should not be confused with the topologically nontrivial Haldane model.

A band gap closes when all three components of $\mathbf{d}$ vanish. Since $d_x=d_y=0$ only at the two valleys, the phase boundaries are exactly $M=\pm 3\sqrt{3}{t}_2\sin \varphi$. Usually one valley becomes massless; at $M=0$, $\sin \varphi =0$, both do. The occupied-band Chern number is undefined at every such touching, even if a numerical momentum grid misses its location.

A hypothetical Van der Waals equation model illustrates positive and negative Joule-Thomson coefficient regions and a finite throttle that first warms, then cools. Use the molar convention already distinguished in Van der Waals equation:Here $v_{\mathrm{m}}$ is molar volume, $R$ the molar gas constant, $u_{\mathrm{m}}$ internal energy per mole, and $C_{V,\mathrm{m}}$ the stipulated constant molar heat capacity at fixed volume. The second equation is an additional caloric assumption: an equation of state alone does not specify the temperature dependence of internal energy. It satisfies the equilibrium identity $(\partial u_{\mathrm{m}}/\partial v_{\mathrm{m}}{)}_T=T(\partial p/\partial T{)}_{v_{\mathrm{m}}}-p=a_{\mathrm{m}}/v_{\mathrm{m}}^2$. The arbitrary energy reference is set to zero.

Define dimensionless variables $r=v_{\mathrm{m}}/b_{\mathrm{m}}$, $\theta =R{b}_{\mathrm{m}}T/a_{\mathrm{m}}$, $\pi =b_{\mathrm{m}}^{2}p/a_{\mathrm{m}}$ and $\eta =b_{\mathrm{m}}{h}_{\mathrm{m}}/a_{\mathrm{m}}$, where $h_{\mathrm{m}}$ is molar enthalpy. These differ from the critical-value scaling in Van der Waals law of corresponding states. The throttle calculation reduces toThus a prescribed inlet fixes $\eta$, and solving the first and last equations at the outlet pressure gives the final temperature. For an inlet $({\theta }_1,{\pi }_1)=(0.8,0.12)$ and outlet ${\pi }_2=0.025$, the model gives ${\theta }_2\simeq 0.752287$, a decrease of $0.047713$. The inlet $(2.2,0.12)$ instead gives ${\theta }_2\simeq 2.205786$, an increase of $0.005786$. The purple inlet $(1.45,0.30)$ initially warms but ends at ${\theta }_2\simeq 1.437585$ for the same outlet pressure. These are reduced toy values, not kelvin readings or experimental evidence. No physical gas parameters were fitted.

For a short inversion derivation, implicit differentiation of the Van der Waals equation at fixed pressure givesInsert this into $T(\partial v_{\mathrm{m}}/\partial T{)}_p=v_{\mathrm{m}}$ and substitute back into the equation of state. The Joule-Thomson inversion curve isPositive inversion pressure requires $r>3/2$. Its maximum pressure is $\pi =1/3$ at $r=3$, while the upper dilute limit is ${\theta }_{\mathrm{inv}}\to 2$. Using Van der Waals critical point values, the latter is $T_{\mathrm{inv},\max }/T_c=27/4$. The inversion locus depends on the equation of state; the chosen molar heat capacity additionally determines how much temperature changes along each finite throttle.^[ref]

The displayed region has $\theta >8/27$, above the model's critical temperature, and uses only $r>1$, $p>0$, $(\partial p/\partial v_{\mathrm{m}}{)}_T<0$ and positive $C_{p,\mathrm{m}}$. Subcritical calculations require attention to phase coexistence, metastability and the Maxwell construction; an unstable homogeneous continuation is not an equilibrium throttle endpoint. Even on a stable branch, this mean-field theory with constant $C_{V,\mathrm{m}}$ is a qualitative model, not an accurate inversion map for a particular real gas. Phase transitions, changing composition, appreciable heat leakage or bulk kinetic energy changes require a different property calculation or steady-flow energy balance. At phase coexistence, the smooth single-thermodynamic phase coefficient formula cannot replace a balance involving the phase fraction.

The variational method estimates the lowest energy of a Quantum Hamiltonian by choosing a trial wavefunction, evaluating its energy, and optimizing the trial parameters. It uses the quantum variational principle, the Rayleigh-Ritz variational principle expressed for quantum states. Let $H$ be a self-adjoint operator bounded below, and define $E_0=\inf \sigma (H)$, the bottom of its spectrum. Every normalized admissible trial quantum state satisfiesWhen $E_0$ is an energy eigenvalue, it is the ground-state energy; equality holds precisely for a quantum state in the ground-state subspace. Otherwise the bottom of the spectrum need not be attained by any normalizable wavefunction.

Admissible means finite form energy. Choose a real $b\le E_0$ and use the shifted quadratic form of a positive quantum Hamiltonian:The form domain $Q(H)$ is independent of this choice of $b$ and can be larger than the operator domain $D(H)$. For $\psi \in D(H)$, $E_{\psi }=⟨\psi ,H\psi ⟩$ is the usual expectation value; for nonzero unnormalized trials, divide $q_H[\psi ]$ by $\Vert \psi {\Vert }^2$ to get the form version of the Rayleigh quotient. See the quantum variational principle for the general domain justification.

Thus each trial energy is an upper bound on $E_0$, and minimizing within a trial family improves that upper bound. It is not a lower bound on $E_0$. The Gaussian variational energy for the harmonic oscillator reaches $E_0=1/2$ because its family contains an exact ground state. The exact variational solution in a two-level system makes the equality condition visible as vanishing excited-state probability amplitude.

The method is useful when solving the Schrödinger equation exactly is difficult but integrals of a trial wavefunction are tractable. Enlarging a trial family cannot increase its optimal energy, although an optimum need not be attained. The beauty is that even an imperfect trial gives a rigorous bound, provided its form energy is computed exactly: numerical integration errors require separate control. The Gaussian variational estimate for a solvable sextic potential provides an existing example beyond a quantum harmonic oscillator.

A short proof explains the direction. With a complete orthonormal eigenbasis of energy eigenstates, write $\psi ={\sum }_n{c}_n{\varphi }_n$, where $H{\varphi }_n=E_n{\varphi }_n$ and ${\sum }_n|c_n{|}^2=1$. ThenThis is a weighted mean of nonnegative energy differences; equality requires every occupied energy eigenstate to have energy $E_0$. ^[ref]. For continuous spectrum, the spectral theorem replaces the sum by an integral against the scalar spectral measure of $\psi$, with the same nonnegative integrand. This proves the bound without assuming an orthonormal eigenbasis exists.

The historical setting predates quantum mechanics: Rayleigh's The Theory of Sound, volume I, appeared in 1877, and Ritz's paper on solving variational problems of mathematical physics appeared in volume 135 of Crelle's journal, catalogued by its publisher as 1909, pp. 1–61. These are original works associated with the Rayleigh-Ritz method; the dates here identify the publications rather than assert priority. ^[ref]; ^[ref].

Change in vascular volume per change in distending pressure; it describes storage rather than an independent source of energy.

In local continuously turbulent scaling at $z/{\ell }_O\gg 1$, dependence on the distance to the wall disappears at leading order. The velocity profile is linear and the buoyancy frequency is constant. Local energy balance gives $ϵ=(c_m-1)|B_0|$ and ${\mathrm{Ri}}_f=1/c_m$, with positive dissipation requiring $c_m>1$. The coefficients depend on a closure, and this regime should not be imposed on fully collapsed turbulence.

Victor S. L'vov is a researcher in theoretical physics whose work includes turbulence, wave turbulence and nonlinear dynamical systems. His Weizmann Institute of Science research homepage identifies wave and hydrodynamic turbulence as central interests.^[ref]

In 1998 he and collaborators introduced an improved shell model of turbulence designed to reduce spurious correlations and oscillations between model scales. A shell model of turbulence replaces a spatially resolved velocity field by amplitudes $u_n$, which are complex numbers attached to wavenumbers $k_n=k_0{\lambda }^n$, $\lambda >1$. Nonlinear interactions transfer energy between shells, while forcing and viscosity supply and dissipate it. With $\lambda =2$, ten shell steps cover a factor of $1024$ in wavenumber: a small collection of variables can investigate an energy cascade spanning widely separated scales.^[ref]

This research makes a useful distinction between reproducing scale statistics and reproducing the full spatial geometry of a turbulent flow. The paper tests scaling and shell correlations in a deliberately reduced dynamical system; that success does not make it an exact replacement for the Navier-Stokes equations in arbitrary apparatus.

Victor Weisskopf developed a statistical treatment of particle emission from highly excited heavy atomic nuclei. His 1937 paper compares the process with evaporation: the excitation energy is shared among many internal quantum states, and the available final states constrain which particles can escape with which energies. This connects nuclear level density with emission probabilities and approximately thermal-looking outgoing spectra. Nuclear evaporation supplies a way to predict the aggregate outcome of a complicated nuclear reaction without following every individual internal trajectory.^[ref]

A Virasoro character records the graded dimensions of a positive-energy Virasoro algebra module. A state of weight $h+N$ contributes $q^{h+N-c/24}$, and its multiplicity is the dimension of that energy eigenspace. The shift accounts for the Casimir energy of a two-dimensional conformal field theory. For a Virasoro Verma module, the character is $q^{h-c/24}{\prod }_{n\ge 1}(1-q^n{)}^{-1}$, by counting the ordered descendant basis.

For an isolated quasistatic cluster with near-zero escape energies, the bound negative energy is approximately fixed. The virial theorem and gravitational radius then imply $R_g\propto M^2$. A homologous mean density scales as $M^{-5}$ and increases as mass evaporates. Tidal limitation or finite positive escape energies modifies this simple model.

For a sufficiently regular solution of the mass-critical focusing nonlinear Schrödinger equation with finite variance, the variance and its first derivative areA second differentiation and integration by parts give the virial identityThese constants use Laplacian coefficient one in the evolution equation. Energy conservation makes the second derivative constant. If the energy is negative, the resulting quadratic expression for $I(t)$ eventually becomes negative, contradicting its definition as a nonnegative integral; this proves negative-energy blowup for the mass-critical focusing nonlinear Schrödinger equation under the finite-variance hypotheses.

The weighted moment condition is additional to an H1 norm bound. A localized virial identity replaces the unbounded weight by a smooth cutoff when only tail control is available.

For fixed cloud mass $M$ and isothermal sound speed $c_s$, approximate the binding energy by $-3G{M}^2/(5r)$. The boundary-pressure virial theorem gives $A=3M{c}_s^2/(4\pi )$ and $B=3G{M}^2/(20\pi )$. The required external pressure peaks at $r_c=4B/(3A)$. A homologous radial displacement is restoring on the branch $d{p}_{\mathrm{ext}}/dr<0$, or $r>r_c$; the smaller-radius positive-pressure branch is unstable. This is a one-radius virial model, not the exact hydrostatic isothermal-sphere sequence.

For a homologous, isolated collapsing sphere, write its Newtonian gravitational potential energy as $U=-aG{M}^2/R$. At spherical-collapse turnaround the coherent kinetic energy vanishes. Conserving total energy and applying the virial theorem to the final equilibrium gives $U_{\mathrm{ta}}=U_{\mathrm{vir}}/2$ and hence $R_{\mathrm{vir}}=R_{\mathrm{ta}}/2$. If the gravitational structure coefficient changes, the ratio is $a_{\mathrm{vir}}/(2a_{\mathrm{ta}})$ instead. Mass loss, surface pressure, external tides and energy exchange also change this estimate.

The virial theorem relates the average value of a function along a trajectory to the forces that bind it. For an isolated system of particles interacting through Newtonian gravitational potential energy, it gives $2⟨K⟩+⟨U⟩=0$: the average kinetic energy is half the magnitude of the average negative potential energy. An eccentric Kepler orbit makes the distinction vivid: kinetic energy and potential energy vary throughout the orbit, although their sum is constant. The relation concerns their time averages, rather than their values at every instant. It underlies the virial mass estimate and the negative heat capacity of a virialized gravitational system.

Use constant positive particle masses $m_i$, positions ${\mathbf{r}}_i$, mechanical momenta ${\mathbf{p}}_i=m_i{\dot{\mathbf{r}}}_i$, and total kinetic energy $K={\sum }_i{\mathbf{p}}_i^2/(2m_i)$. The auxiliary quantity $G(t)$ is a sum of dot products, not the gravitational constant, which we denote here by $G_N$:This finite-$T$ identity is exact wherever Newton's second law holds along a regular trajectory; ${\mathbf{F}}_i$ must include every force. The endpoint term is what a finite observation window leaves behind. For the infinite-time theorem, suppose the averages on the left exist and $(G(T)-G(0))/T\to 0$. A sufficient condition is uniformly bounded positions and velocities for finitely many particles. More generally $G(T)=o(T)$ suffices, even without those individual bounds. Bounded positions alone do not supply a bound on momenta, especially near a singular potential energy; nor does a small endpoint term prove thermodynamic equilibrium. Periodic motion with $G(T)=G(0)$ gives the identity without an endpoint term at that finite $T$. For an isolated system, use an inertial centre of mass frame so translational drift does not contribute bulk kinetic energy.^[ref]

For a differentiable potential energy that is a homogeneous function of degree $q$, choose its additive constant so that the scaling law really holds. With conservative forces, the Euler theorem for homogeneous functions givesThus a harmonic oscillator has $q=2$ and $⟨K⟩=⟨U⟩$. An isolated system with Newtonian gravitational potential energy hasEach interacting pair occurs once, and $U=0$ at infinite separation. Here the total energy $E=K+U$ is conserved. An external field or a mixture of different scaling powers requires its own force term; one cannot assign the whole potential energy a single degree $q$. The virial theorem for a homogeneous pair potential states the corresponding general energy relation.

Consider the exact two-body problem with masses $m_1,m_2$. In the inertial centre of mass frame, its internal kinetic energy is $K=m|\dot{\mathbf{r}}{|}^2/2$, where $\mathbf{r}$ is their relative position, $m=m_1{m}_2/(m_1+m_2)$ is the reduced mass, and $U=-\kappa /r$ with $\kappa =G_N{m}_1{m}_2$. Take a noncollision elliptic Kepler orbit with semi-major axis $a$ and orbital eccentricity $0<e<1$. Write $\eta$ for eccentric anomaly, to avoid confusing it with total energy $E$. With mean motion $n=\sqrt{\kappa /(m{a}^3)}$ and orbital period $P=2\pi /n$, Kepler's equation and the orbit geometry giveThe time origin is periapsis. Since $dt=(1-e\cos \eta )d\eta /n$, the time average of $1/r$ over one orbital period is exactly $1/a$. Therefore $⟨K{⟩}_P=\kappa /(2a)$ and $⟨U{⟩}_P=-\kappa /a$, independently of orbital eccentricity. Equal steps in eccentric anomaly are not equal time steps; unweighted averaging of a drawing of the ellipse gives the wrong answer.

The original dimensionless example in both diagrams has $m=a=\kappa =1$ and $e=0.6$. At periapsis, $r=0.4$, $K=2$, $U=-2.5$; at apoapsis, $r=1.6$, $K=0.125$, $U=-0.625$. Both have $E=-0.5$, but $2K+U$ is respectively $1.5$ and $-0.375$. Only the orbital period averages are $⟨K{⟩}_P=0.5$ and $⟨U{⟩}_P=-1$. The instantaneous zero of $2K+U$ occurs at $r=a$, not at every point of the orbit.

The endpoint term can also be evaluated exactly: $G=m\mathbf{r}\cdot \dot{\mathbf{r}}=mn{a}^{2}e\sin \eta$. Starting at periapsis, a window ending at eccentric anomaly $\eta$ hasAt $T=P/4$, Kepler's equation gives $\eta \simeq 2.09133$ and a positive residual $\simeq 0.3314\kappa /a$. At $T=P/2$ or $P$, $G$ returns to zero and the residual vanishes exactly. The lower panel shows the running residual and its endpoint expression coinciding; the shaded quarter-orbital period window explains why a short observation need not satisfy the infinite-time theorem.

A hopping process through an intermediate configuration of different unperturbed energy, contributing to a low-energy effective description through perturbation theory. If an intermediate state costs $\Delta >0$, second-order nondegenerate perturbation theory gives a contribution $-|⟨a|T|0⟩{|}^2/\Delta$. Its amplitude is of order $t/\Delta$ and its energy correction of order $-t^2/\Delta$; a degenerate low-energy sector requires the corresponding degenerate perturbation theory before identifying the ground state.

An intermediate contribution to a second-order transition amplitude need not be an occupied, on-shell stationary atomic state. Both time orderings must be summed with their energy denominators. Large detuning can suppress real intermediate occupation while preserving a coherent two-photon coupling. Near resonance, finite linewidth, saturation and sequential population transfer must be treated explicitly.

For small kinematic viscosity $\nu$, the mean viscous dissipation per unit horizontal area is $2\rho \nu g{k}^2|A{|}^2$. Since the mean wave energy is $\rho g|A{|}^2/2$, the amplitude obeys

Viscous drag is resistance to relative motion arising from viscosity. It transfers momentum between a moving object and its surrounding fluid, with mechanical energy dissipated as heat.

For a rigid sphere of radius $a$ moving at speed $U$ through an unbounded Newtonian fluid with dynamic viscosity $\mu$, Stokes law gives drag magnitude $F=6\pi \mu aU$ in the creeping-flow limit. The force opposes relative motion. This linear law requires a small Reynolds number; a quadratic drag law appropriate to other flow regimes cannot be substituted without checking the assumptions.

Vitamin B3 is the nutritional category including nicotinic acid and nicotinamide, compounds the body can use to make nicotinamide adenine dinucleotide and nicotinamide adenine dinucleotide phosphate. These coenzymes participate in metabolism, including transfer of reducing equivalents between reactions. Adequate dietary supply supports a reusable biochemical system rather than providing the bulk of the body's fuel.^[ref]

For example, an oxidized coenzyme can accept reducing equivalents in one reaction and transfer them in another. The coenzyme is regenerated during the cycle, so its importance is not measured by how much of it is burned for energy. Humans can also derive niacin equivalents from tryptophan; severe inadequacy is associated with pellagra.^[ref]

Vladimir Bulović's organic-optoelectronics course connects molecular excited states with practical light-emitting and photovoltaic devices. Its treatment follows excitons from their creation through diffusion and energy transfer, then asks whether that excitation produces light or collected electric charge. This distinction explains why efficient absorption alone does not guarantee an efficient solar cell or organic light-emitting diode. Comparing molecular energy levels, interfaces and transport makes device behavior understandable from the processes that compete inside the material.^[ref]

MIT lists the instructor of Organic Optoelectronics lecture notes by Vladimir Bulović as Vladimir Bulovic.