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The equilibria are $\boxed{(0,0),(\pm \sqrt{2/3},0)}$. The Jacobian is $J=\left(\begin{array}{cc}0& 1\\ 2-9x^2& -1\end{array}\right)$. At the origin it has eigenvalues $1,-2$ and tangent eigenvectors $(1,1)$ and $(1,-2)$, so the origin is a saddle equilibrium. At either nonzero point the eigenvalues are $\boxed{(-1\pm i\sqrt{15})/2}$, giving stable foci.

The energy-like function $H=y^2/2-x^2+3x^4/4$ has $\dot{H}=-y^2\le 0$. It is coercive at large distance, so trajectories remain bounded; there are no periodic orbits because their energy could not return to its starting value unless $y\equiv 0$. The stable manifold of the saddle equilibrium separates the two attracting basins. Its unstable branches leave the origin and approach the right and left foci, respectively. Their negative energy near the origin prevents crossing back through $x=0$.

For either invariant manifold write $y=h(x)=sx+c{x}^3+O(x^5)$. Odd symmetry rules out even powers. Tangency to the vector field requires $h^{\prime }(x)h(x)=2x-h(x)-3x^3$. Matching coefficients yields $s^2+s-2=0$ and $c(4s+1)=-3$. HenceThese are local graphs, not globally exact cubic curves.

For $\nu =1$, the equilibria are $(0,0)$ and $(\pm 1,0)$. At the origin the eigenvalues are $1,-1$, so it is a saddle for every $\kappa$.

At either nonzero point the Jacobian matrix is $\left(\begin{array}{cc}0& 1\\ -2& \kappa \end{array}\right)$, with eigenvaluesBoth points are attracting for $\kappa <0$ and repelling for $\kappa >0$. For $-2\sqrt{2}<\kappa <0$ they are stable foci; for $\kappa <-2\sqrt{2}$ stable nodes. For $0<\kappa <2\sqrt{2}$ they are unstable foci; for $\kappa >2\sqrt{2}$ unstable nodes. At the two repeated-root boundaries they are degenerate nodes. At $\kappa =0$, linearization has purely imaginary eigenvalues; the conserved energy in the next part shows that they are actual nonlinear centres. Thus the zero-real-part case is resolved by an additional first integral, rather than by the linear test alone.

An equilibrium is asymptotically stable when it is Lyapunov stable and attracts every initial state in some neighbourhood. Lyapunov stability means that sufficiently small initial displacement stays small for all future times; attraction additionally requires convergence to the equilibrium. A Lyapunov function is a differentiable positive-definite scalar function near the equilibrium whose derivative along solutions is nonpositive. Strict negativity, or an appropriate invariant-set argument for a non-strict derivative, gives asymptotic stability.

For $\nu =-1$, equilibria satisfy $y=0$ and $x+x^3=0$, so only $(0,0)$ is real. Its Jacobian matrix is $\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, with eigenvalues $\pm i$ for every $\kappa$. It is a linear centre equilibrium, so linearization alone cannot decide nonlinear attraction.

Use the positive proper energyFor $\kappa <0$, energy sublevel sets are compact and invariant. The zero-derivative set is the union of the axes, but its only invariant subset is the origin: on $x=0,y\ne 0$ one immediately leaves that axis, and on $y=0,x\ne 0$ one has $\dot{y}=-x-x^3\ne 0$. LaSalle's invariance principle therefore proves global asymptotic stability. At $\kappa =0$ the conserved energy levels are closed periodic orbits, giving a nonlinear centre equilibrium. For $\kappa >0$, time reversal together with $y\mapsto -y$ converts the system to the attracting negative-$\kappa$ case, so the origin is a nonlinear repeller and is unstable. Nonzero forward trajectories cannot remain bounded; this does not assert existence for all time if finite-time escape occurs.

Orient the flow so $q>0$, and let subscripts 1 and 2 denote upstream and downstream. The mechanical energy flux is $\rho qB$, where $B=u^2/2+gh$. Inserting the relation for $q^2$ in i gives the hydraulic jump head loss,An admissible jump dissipates mechanical energy, so $\rho q(B_1-B_2)>0$. The denominator is positive, henceThe zero depth-change case is continuous flow. A depth-decreasing discontinuity would require an energy input at the jump; it is not the dissipative hydraulic jump selected by this model.

The star is near the southwestern end of the main fault slip patch, so a much longer part of the finite-source rupture propagates northeastward. For comparable distances, ground conditions and buildings, the northeastern site receives the stronger, shorter direct pulse; the southwestern site receives the weaker, longer direct shaking. This is rupture directivity.^[ref]

For a segment of length $\ell$, rupture velocity $v_r$ and seismic wave speed $c>v_r$, successive arrival times span approximatelyForward arrivals overlap and concentrate energy into a shorter pulse. Backward arrivals spread out. The supplied finite-source rupture is not perfectly unilateral, so the prediction concerns the dominant contribution. Seismic radiation patterns, seismic attenuation, sediment amplification and building response can alter the ranking of actual damage or total coda duration.

Specify the amplitude convention. For incident stress/pressure amplitude one, the normal-incidence seismic reflection coefficient and stress seismic transmission coefficient areAt limestone–sandstone, $R_P=-3/19=-0.1579$ and $T_{\sigma P}=16/19=0.8421$. At sandstone–salt, $R_P=0$ and $T_{\sigma P}=1$. The first return reverses the chosen stress-pulse polarity; the second interface produces no normal-incidence P return.

For a particle-velocity amplitude convention, $R_v=-R$ and $T_v=1-R$, giving $(R_v,T_v)=(0.1579,1.1579)$ and $(0,1)$ respectively. These are amplitude coefficients, not fractions of energy. The lossless reflected and transmitted energy fractions are $\mathcal{R}=R^2$ and $\mathcal{T}=4Z_a{Z}_b/(Z_a+Z_b{)}^2=1-R^2$, which sum to one.

The normal-incidence acoustic impedance is $Z=\rho V_p$, with units $\mathrm{kg}{\mathrm{m}}^{-2}{\mathrm{s}}^{-1}$. ThusDefine the pressure-amplitude reflection coefficient and pressure-amplitude transmission coefficient for incidence downwards from layer $i$ into layer $j$ byHere $p$ denotes compressional normal-stress amplitude; in the water it is the acoustic pressure. Continuity of normal traction and normal particle velocity gives these expressions. They describe each interface separately, before any layer layer reverberation.

For a lossless normal-incidence plane wave, intensity is proportional to $p^2/Z$, soA transmitted pressure amplitude exceeding unity is therefore compatible with conservation of energy. If instead the coefficient is defined for particle velocity measured along the same downward coordinate, $r_v=-r_p$ and $t_v=2Z_i/(Z_i+Z_j)=1-r_p$: the two transmitted velocity coefficients are $0.8000$ and $18/29=0.62069$. Upward incidence reverses the sign of $r_p$, interchanges the transmitted pressure and velocity coefficients, and leaves $R_E,T_E$ unchanged. At normal incidence between these ideal isotropic layers there is no P-to-S seismic mode conversion.

For incidence from cap rock (1) into reservoir (2), use the conventional normal-incidence stress-amplitude reflection coefficientHere acoustic impedance is the P wave impedance; $\rho V_s$ is the corresponding shear-wave impedance. Positive $R$ preserves the chosen stress-wave seismic reflection polarity; negative $R$ reverses it. These are amplitude coefficients, not reflected energy fractions. At normal incidence in the ideal isotropic model there is no P-to-S mode conversion. The cap-rock impedances are

The following are approximate readings of the supplied Figure 1 PNG, checked against the matching source PDF, rather than exact tabulated parameters. At zero gas, use $\rho \simeq 1.75\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, $V_p\simeq 1.75\mathrm{km}{\mathrm{s}}^{-1}$, and $V_s\simeq 0.60\mathrm{km}{\mathrm{s}}^{-1}$; at full gas use $\rho \simeq 1.25\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$, $V_p\simeq 1.15\mathrm{km}{\mathrm{s}}^{-1}$, and $V_s\simeq 0.65\mathrm{km}{\mathrm{s}}^{-1}$. The left-hand $V_p$ endpoint is crowded against the axis and the density curve; readings differing by roughly $0.05$ in the plotted units are reasonable. The resulting coefficients deserve about two significant figures, not the apparent precision of the intermediate arithmetic.

Both end members change seismic reflection polarity, but the P wave change is much larger: approximately $+0.15$ to $-0.22$, compared with $+0.08$ to $-0.05$ for S waves. Thus a calibrated P wave survey is better for distinguishing these two saturation end members and should give a stronger reflection from the fully gas-filled reservoir. A negative P wave reflection relative to the water-filled prediction is a useful gas indicator in this stipulated model. In a real seismic reflection survey, lithology, porosity, effective pressure, and seismic wavelet phase also affect the response, so a bright or reversed reflection alone does not prove the presence of gas.

Mammalian endothermy probably emerged through several interacting changes along the synapsid stem, and its precise onset remains disputed because the fossil proxies measure different aspects of physiology. Endothermy concerns substantial metabolic heat production; homeothermy concerns relatively constant body temperature. A large animal can have relatively stable temperature without a mammalian metabolic rate, while an endothermic animal can display heterothermy. These distinctions prevent a single temperature estimate or anatomical character from being treated as a complete diagnosis.

The physiological problem is a heat balance. In a deliberately simplified steady-state model:where $M$ is metabolic energy turnover, $W$ external work, $G$ effective thermal conductance of an animal, $T_b,T_a$ body and ambient temperature, and $E$ evaporative heat loss. Maintaining a warm body in a cool environment can involve increased metabolism, reduced conductance, behaviour, or a combination. For example, $G=0.1\mathrm{W}{\mathrm{K}}^{-1}$ and a $20\mathrm{K}$ difference imply $2\mathrm{W}$ of non-evaporative heat loss in this illustrative model. The numbers are not a reconstruction of a fossil animal.

Along the synapsid lineage, increased sustained activity would require coordinated oxygen uptake, transport and use, greater food processing and sufficient energy assimilation. More effective locomotor construction, ventilation and circulation could support greater aerobic capacity, while insulation would reduce the cost of retaining the resulting heat. Shivering thermogenesis and later forms of non-shivering thermogenesis can contribute in living mammals. Brown adipose tissue in many living placental mammals cannot simply be assigned to early therapsids; a modern mechanism need not have been the ancestral one. Similarly, an expanded secondary palate or an altered limb posture is compatible with increased performance but is not a direct measurement of resting metabolism.

The time sequence must be reconstructed from several lines of evidence. Early synapsids lack the complete modern mammalian package, whereas various therapsids and cynodonts show changes in growth, feeding and respiratory anatomy before the mammalian crown radiation. Rapidly deposited fibrolamellar bone supports rapid growth, but fast-growing ectotherms and the influence of size and environment prevent a unique inference of endothermy. Growth interruptions can reflect seasonality or resource shortage rather than a categorical physiological boundary. Nasal structures interpreted as respiratory nasal turbinates could reduce water and heat loss during high ventilation, but identifying their precise form and separating respiratory from olfactory roles in fossils is difficult.

Preserved hair supplies direct evidence of insulation in some later fossils. Castorocauda, a Middle Jurassic mammaliaform, preserves integument alongside an aquatic anatomical specialization. ^[ref] This establishes that a fur-bearing covering existed by that time, not when insulation first evolved or whether its original function was thermal. Sensory and protective functions can precede elaboration for heat conservation. Direct preservation is particularly valuable, but its late appearance in the sampled record is not proof of biological absence in older lineages.

Stable isotope analysis has suggested earlier elevated thermometabolism. Comparing oxygen-isotope compositions of therapsid apatite with those of coexisting tetrapods, one study inferred differences in physiology and proposed elevated thermometabolism in several Permo-Triassic therapsid groups. ^[ref] The comparison is stronger than analysing one isolated animal because coexisting taxa constrain aspects of the environment. Nevertheless, body water, drinking source, habitat, evaporation and diagenesis affect the isotopic signal. The proposed middle-to-late Permian origin is an interpretation using a physiological framework, not a thermometer reading of an observed living cynodont.

A contrasting biomechanical approach uses the temperature dependence of endolymph properties and the dimensions of the semicircular canals. Calibration against living animals and application to fossil inner ears supported a relatively abrupt transition around 233 million years ago in the Late Triassic, near the origin of mammaliamorphs. ^[ref] This offers a quantitative prediction from a different anatomical system. However, body size, activity, reconstruction and the adequacy of the living calibration remain relevant. A change in a temperature-sensitive inner-ear proxy need not coincide exactly with the first increment in metabolic heat production or the first insulating covering. The disagreement with earlier isotope-based interpretations should therefore be tested, rather than erased by choosing the preferred date.

There are several plausible selective explanations. The aerobic capacity hypothesis proposes that selection for sustained locomotor performance raised maximum oxygen consumption, with elevated resting metabolism and heat production linked to that change. Comparative measurements of resting and maximum oxygen use underpin this proposal. ^[ref] A major issue is the strength and evolutionary direction of the relationship: high resting expenditure is expensive, and an association among living species does not prove which change led historically. Selection for thermal independence, nocturnal activity or performance in fluctuating climates provides related benefits, but “the climate became colder” is not sufficient unless the energetic costs and available food are considered.

A parental care hypothesis instead emphasizes selection for controlling the developmental environment and supplying offspring. Improved provisioning endurance and more stable incubation or offspring temperatures could favour increased metabolic capacity and heat retention. ^[ref] This gives concrete reproductive benefits that might offset energetic costs, while remaining a hypothesis about historical selective circumstances. The evidence required includes relationships among care, thermal performance and reproductive success, not merely the fact that living mammals both care for young and are endothermic. Activity, parental investment, insulation and environmental variability could reinforce one another, so the explanations need not be mutually exclusive.

The most secure account is a mosaic physiological transition with increasingly effective energy supply, heat production and heat retention. A Late Triassic onset of a more fully mammalian condition is a specific supported hypothesis, while earlier therapsid evidence indicates that parts of the transition may be older. fossil anatomy and chemistry constrain when particular features existed; the selective reasons require functional and comparative tests and remain less directly observable.

A Pauling bond-energy excess subtracts a chosen like-atom baseline from the unlike-atom bond-dissociation energy. For an unequal-input arithmetic example, stipulate $D_{AA}=200$, $D_{BB}=450$ and $D_{AB}=421.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, all with one dissociation convention. The arithmetic mean baseline and excess areThe arithmetic-baseline Pauling scale model therefore gives $|{\chi }_{\mathrm{P}}(A)-{\chi }_{\mathrm{P}}(B)|=\sqrt{96.485/96.485}=1$. The bond-dissociation energy $421.485$ is not itself the excess: subtracting the baseline is essential. All three numbers are stipulated, not experimental values for named chemical elements.

Changing the baseline changes the model. For the same stipulated inputs, a geometric mean baseline would be $\sqrt{200\times 450}=300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, yielding an excess of $121.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and a difference of approximately $1.122$. It is a different construction, even though both start from the same bond-dissociation energies. State which baseline is used before comparing electronegativity differences.

The arithmetic-baseline Pauling scale also has a useful inverse: a difference of $0.5$ corresponds to an excess of $24.12125\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, and a difference of $2$ corresponds to $385.94\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Doubling the electronegativity difference quadruples the modeled excess. Interchanging the endpoint labels changes neither quantity; the square root alone cannot choose the direction of bond polarization.

In The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms (printed pp. 3576–3578, Table II and equation (1)), Linus Pauling organized excess energies for chemical bonds into an approximately additive map of their square roots. He also identified a limitation: his H–F map separation of $2$ predicted an excess of $4.00\mathrm{eV}$, exceeding his table's value by $1.23\mathrm{eV}$. Those are historical values and a historical model discrepancy, not modern recommended inputs.^[ref]

Pauling electronegativity is a relative, dimensionless scale motivated by bond-dissociation energies. The interesting observation is that a chemical bond between unlike chemical atoms can be stronger than a simple baseline inferred from the corresponding like-atom chemical bonds. An empirical excess energy can then be associated with an electronegativity difference. The 1932 paper The Nature of the Chemical Bond. IV. The Energy of Single Bonds and the Relative Electronegativity of Atoms built such a map; its original coordinates used hydrogen as zero, so those coordinates should not be mistaken for the modern tabulated Pauling electronegativities.^[ref]

One arithmetic-baseline version of the Pauling scale relation isHere $D_{AB}$, $D_{AA}$ and $D_{BB}$ are consistently defined bond-dissociation energies per molecule, and the arithmetic mean is the chosen baseline. With molar energies instead, write $\sqrt{{\Delta }_D/(96.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1})}$, approximately $0.102\sqrt{{\Delta }_D/(1\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1})}$. The denominator makes the result dimensionless; the factor $0.102$ is not a unit-independent magic constant. This is an empirical relation with a specified baseline, not a definition valid for every chemical bond.^[ref]

For an explicitly synthetic example, suppose $D_{AA}=D_{BB}=300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and $D_{AB}=396.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ with the same temperature and dissociation convention. Then ${\Delta }_D=96.485\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ and the model gives an electronegativity difference of $1$. Replacing $D_{AB}$ by $300\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ gives zero; a negative excess would make this real square-root model inapplicable, not produce a negative electronegativity difference. These invented inputs illustrate the formula and are not measurements of named chemical substances.

Only differences appear, so adding the same constant to every Pauling electronegativity leaves this relation unchanged. A reference value must fix the origin. The bond-dissociation energies alone also do not tell which endpoint attracts electron density more strongly, because interchanging $A$ and $B$ leaves the formula unchanged. Chemical comparisons fix that direction. Thus Pauling electronegativity is useful as a compact ordering, while the underlying measured energies and approximations remain essential.

Let $p$ be the perpendicular distance from the force centre to the tangent of a regular planar orbit. Conserved physical angular momentum has magnitude $|h|=mpv$, while conserved energy is $E=m{v}^2/2+V(r)$. Eliminating speed gives the displayed equation when $h\ne 0$. For a differentiable polar curve, $p=r^2/\sqrt{r^2+(dr/d\theta {)}^2}$, connecting the geometric tangent distance to the radial energy equation.

For a pendulum whose support slides without horizontal force, the support coordinate is cyclic and total horizontal momentum is conserved. At zero horizontal momentum, eliminating support velocity leaves the displayed angle-dependent effective inertia. The energy is $M_{\mathrm{eff}}{\dot{\varphi }}^2/2-m_{2}gl\cos \varphi$. Fixing the momentum before evaluating motion is distinct from fixing the support position.

The two-layer cell seismic layer transfer matrix has reciprocal eigenvalues because its determinant is one. Writing them as $e^{\pm iKd}$ gives the displayed half-trace relation. At a real frequency, an absolute half-trace exceeding one forces an exponentially growing/decaying pair instead of propagating cell modes. A long finite stack then suppresses transmission in this stop band, without dissipating energy: the missing transmitted flux is reflected. Quarter-wave layers of unequal seismic impedance give half-trace $-\frac{1}{2}(q_1/q_2+q_2/q_1)<-1$.

A superposition of two distinct energy eigenstates gives the expectation value of a time-independent observable an interference term proportional to $e^{-i(E_2-E_1)t/\hslash }$. The displayed beat period applies when that term has nonzero amplitude. A vanishing off-diagonal matrix element, zero state coefficient or equal energies instead gives a constant expectation value.

Define transfer gained by the neutron as $\mathbf{Q}={\mathbf{k}}_f-{\mathbf{k}}_i$. If it is perpendicular to ${\mathbf{k}}_i$, expansion of the free-neutron kinetic energy gives $E_f-E_i={\hslash }^2{Q}^2/(2m_n)$. Absorbing one phonon requires this positive energy to equal $\hslash \omega$. A periodic phonon branch may therefore intersect a nonperiodic recoil parabola in an extended-zone representation. The sign of $\mathbf{Q}$ is opposite to the common sample-transfer convention and must be declared.

A proposed cyclic machine that continuously delivers useful work without the corresponding permitted energy input. A first-kind machine violates conservation of energy; a second-kind machine would turn heat from a single reservoir completely into work in a cycle, contrary to the Kelvin-Planck statement. Continuing ideal motion without withdrawing work is a different claim.

A galaxy cluster whose X-ray-emitting core is a well-studied example of interaction between radio bubbles and cooling gas. density fronts and ripples provide diagnostics of the mechanical energy released by the central active galaxy.^[ref]

A superconducting persistent current is an electric current circulating in a closed path with superconductivity after the external driving electromotive force has ceased. Its associated magnetic field makes the circulation observable without inserting a measuring instrument into the path. This article concerns macroscopic circulation in superconductivity. A finite experiment establishes persistence over its observation window with a stated measurement error, rather than directly observing an infinite lifetime.

The ideal circuit connection is flux conservation in a superconducting loop. In a fixed loop with self-inductance $L$, a single circulating electric current $I$, and externally imposed linked magnetic flux ${\Phi }_{\mathrm{ext}}$, the lumped circuit approximation givesWhen the electrical resistance $R$ vanishes, integrating gives $LI+{\Phi }_{\mathrm{ext}}=\text{constant}$. Removing the external linked magnetic flux therefore changes the circulating electric current; holding it fixed afterwards preserves that electric current. This is a circuit approximation with a specified self-inductance, not a derivation of the microscopic state of superconductivity. For thin paths where the carrier contribution matters, fluxoid and flux quantization in a superconductor supplies the more careful description.

For constant $L>0$, constant effective electrical resistance $R\ge 0$, and no changing external linked magnetic flux, the RL circuit instead predictsThis exponential decay follows by separating $dI/I=-(R/L)dt$. A long relaxation time tests extremely small electrical resistance without trying to resolve an equally tiny steady electric potential difference. The stored magnetic energy is $U=L{I}^2/2$, so the same model gives $dU/dt=-R{I}^2$, exactly the Joule heating loss. The finite energy stored in the loop supplies the magnetic readout; the circulating electric current is not a continuously powered source of useful work.

A coarse-grained description in which quadratic inflaton oscillations have zero mean pressure and lose energy at constant effective decay rate $\Gamma$ to a relativistic component. Adding the equations preserves total energy accounting with the appropriate expansion work. Efficient transfer occurs around $H\sim \Gamma$, but producing particles and achieving thermal equilibrium are different steps. The potential alone does not specify the decay rate or thermalization interactions.