# Guide Encyclopaedia of Mathematics: Orbit — Rayleigh Equation

The use of dynamical cores constructed according to the sophisticated version of fluid dynamics discussed here might provide crucial for improving the ability of atmospheric models in representing correctly the global budgets of physically relevant quantities also in the case when forcing and dissipative processes are taken into account. As discussed by Lucarini and Ragone [ ] for the case of energy, this is far from being a trivial task. Like Hamiltonian mechanics, the Nambu approach is a versatile tool for the analysis and simulation of dynamical systems.

Here some possible research directions are outlined. Modular Modeling and Approximations. Decomposition leads to subsystems where the constitutive Casimirs are conserved. Statistical Mechanics. Thus, these conservation laws have a twofold impact: They determine the dynamics in a Nambu bracket and the canonical probability distribution in equilibrium. Dynamics of Casimirs. Casimir functions of a conservative system are ideal observables to characterize the dynamics in the presence of forcing and dissipation. This might prove especially interesting when studying the response of a system to perturbations in the context of the Response theory proposed by Ruelle [ , a, b, ] and recently used in a geophysical context by various authors with promising results [ Eyink et al.

As illuminating example, we mention the recent work of Pelino and Maimone [ ] and Gianfelice et al. We have seen in the previous section that different models of geophysical flows have a specific mathematical structure: they are Hamiltonian systems and have an infinite number of conserved quantities—the Casimirs. The previous section has shown how one could take advantage of these features and construct theoretically rich representation of the dynamics and provide proposals for constructing new numerical codes of GFD flows.

This section goes in the direction of constructing a probabilistic description of GFD flows, basically taking the point of view that due to the large amount of degrees of freedom involved, one can consider the state of the atmosphere and the ocean as random variables. Here we shall review the progress that has been made by using the simplest class of possible probability distributions: the equilibrium distributions depending only on the conserved quantities. However, most of the standard applications of equilibrium statistical mechanics deal with dynamics on a finite dimensional phase space e.

The equations describing the dynamics of geophysical flows violate both these constraints. Several solutions have thus been proposed: they are reviewed briefly in the next sections, going from the main fundamental ideas to selected geophysical applications. Onsager [ ] was the first to understand that the coherent structures and persistent circulations that appear ubiquitously in planetary atmospheres and in the Earth's oceans could be explained on statistical grounds. In particular, the microcanonical probability measure, acting as invariant—i.

It is easily proved that, for a bounded domain, and hence a finite volume phase space, this function reaches a maximum for a given value of the energy. When the temperature is negative, configurations with maximum energy are favored. It satisfies the normalization. This probability density is expected to converge toward its statistical equilibrium: the equilibrium distribution maximizes the statistical entropy.

The point vortex model suffers from a number of limitations inherent to the approach. First of all, when we let the number of vortices tend to infinity the thermodynamic limit , we have to introduce an ad hoc scaling of the Lagrange parameters to retain the organized, negative temperature states. Besides, there is no unique way to approximate a vortex patch by a finite number of vortices. A consequence is also that the area of vorticity patches cannot be conserved in this singular formulation.

Rather than a discretization in physical space, one may consider a finite number of modes in Fourier space, as proposed by Lee [ ] and Kraichnan [ ] in the context of the Euler equations. This dynamics preserves two quadratic quantities: the energy and the enstrophy. Here this condition reads and. Hence, statistical mechanics for the truncated system predicts that when the enstrophy is small enough compared to the energy, we expect the energy to be transferred to the large scales.

In particular, the equilibrium energy spectrum at large scales is shallower than the energy inertial range spectrum. Similarly, the transfer of enstrophy in the corresponding inertial range should be towards the small scales. In particular, they conserve similar quadratic invariants, and the theory can be extended in a straightforward manner [ Holloway , ; Salmon , ].

We will discuss in this section the effect of stratification and beta effect, resulting from the fact that the planetary vorticty depends on the latitude. The corresponding partition function can be computed, and the spectrum studied in the various regimes, with similar results. In particular, negative temperature states are accessible, which correspond to condensation of the energy on the largest horizontal scales and the Fofonoff [ ] solutions mentioned below. Maybe more interestingly, although the various forms of energy kinetic energy K 1 , K 2 in each layer and potential energy P are not individually conserved, we can compute their average value at equilibrium, as Salmon et al.

Alternatively, the standard decomposition in terms of the barotropic and baroclinic modes constructed by taking the average and the difference of the stream functions in the two layers , with their kinetic energies K T and K B , can be used. As Salmon et al. These results have been extended to an arbitrary number of layers and to continuously stratified flows by Merryfield [ ]. The second dominant effect in geophysical flows, in addition to stratification, is rotation. The Coriolis force introduces a linear term in the equations, which does not affect directly the previous analysis of the nonlinear energy transfers: the conserved quantities remain the same and the statistical theory is easily extended by replacing relative vorticity with absolute vorticity.

However, the variation of the Coriolis force with latitude is responsible for the appearance of Rossby waves, which modify the physical interpretation of the predicted cascade of energy. As anticipated by Rhines [ ] and verified numerically [e. This leads to the preferential formation of zonal flows.

As a matter of fact, a downscale transfer of energy is needed in the ocean to feed enhanced vertical mixing [e. It is therefore natural to ask how equilibrium statistical mechanics can help understanding how energy is exchanged by nonlinear interactions between the slow, balanced motions and the fast, wave motions. Errico [ ] first observed a tendency for unforced inviscid flows described by hydrostatic primitive equations to reach an energy equipartition state, in which the energy in the fast wave modes is comparable to that in the slow balanced modes.

The study by Warn [ ], in the context of the shallow water equations, essentially confirms that QG flows are not equilibrium states and that a substantial part of the energy may end up in the fast surface wave modes at statistical equilibrium, implying a direct cascade of energy to the small scales. Bartello [ ] has obtained analytically the equilibrium energy spectrum for the Boussinesq equations neglecting the nonlinear part of potential vorticity , in the presence of rotation, confirming the direct cascade of energy.

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A natural interpretation would be that vortical modes are responsible for the inverse cascade while waves cascade energy downscale simultaneously. Hence, we are interested in the variational problem: 51 or equivalently, 52 The solutions of this variational problem correspond to the most probable states for a given set of conserved quantities. This will automatically ensure that the equilibrium states are nonlinearly stable steady states [ Chavanis , ].

Most of the analytical solutions have been obtained in the linear case, by decomposing the fields on a basis of eigenfunctions of the Laplacian on the domain. The same method was extended to the case of barotropic flows, replacing vorticity by potential vorticity. Taking into account the beta effect, Fofonoff [ ] flows are obtained as statistical equilibria in a rectangular basin [ Naso et al. The relative vorticity is confined to a boundary layer, whose width decreases with the total energy or when the beta effect i. The flow is westward in the interior of the basin, with an eastward compensating flow near the boundaries.

Bouchet and Simonnet [ ] have also considered the role of a small nonlinearity in the relationship for a rectangular domain of aspect ratio close to 1, with periodic boundary conditions, thereby obtaining two topologies for the equilibrium states: dipole and unidirectional flows. Adding a small stochastic forcing generates transitions from one to the other equilibrium.

Still in the continuous case, Venaille et al. As the beta effect increases, barotropization is facilitated, until we enter a regime dominated by waves. It is well known that baroclinic dynamics is hindered by strong rotation [ Holton , ]. Results from equilibrium statistical mechanics have found practical applications in the development of parameterization methods. Such a parameterization has been implemented, tested, and commented in a number of studies [e.

Along similar lines, Kazantsev et al. In the previous sections the focus has been on identifying symmetry properties and conservation laws of GFD flows and relate these to dynamical features and statistical mechanical properties. Neglecting forcing and dissipation has led us to study reversible equations whose statistical properties can be described using equilibrium statistical mechanics. Indeed, this provides the backbone of the properties of GFD flows and are of great relevance for studying more realistic physical conditions.

Nonetheless, at this stage, a reality check is necessary.

In addition to that, spatial gradients in chemical concentrations and temperature as well as their associated internal matter and energy fluxes can be established and maintained for a long time within nonequilibrium systems e. In this and in the next sections we will take such a point of view. The basis of the physical theory of climate was established in a seminal paper by Lorenz [ ], who elucidated how the mechanisms of energy forcing, conversion, and dissipation are related to the general circulation of the atmosphere.

Such instabilities create a negative feedback, as they tend to reduce the temperature gradients they feed upon by favoring the mixing between masses of fluids at different temperatures. Furthermore, in a forced and dissipative system like the Earth's climate, entropy is continuously produced by irreversible processes [ Prigogine , ; de Groot and Mazur , ]. Contributions to the total material entropy production , which is related to the nonradiative irreversible processes [ Goody , ; Kleidon , ], come from the following: dissipation of kinetic energy due to viscous processes, turbulent diffusion of heat and chemical species, irreversible phase transitions associated to various processes relevant for the hydrological cycle, and chemical reactions relevant for the biogeochemistry of the planet.

It is important to note that the study of the climate entropics has been revitalized after Paltridge [ , ] proposed a principle of maximum entropy production MEPP as a constraint on the climate system. While the scientific community disagrees on the validity of such a point of view—see, e. In this paper we will not discuss MEPP and other nonequilibrium variational principles for an updated review, see Dewar et al. We first focus on developing equations describing the energy budget of the climate system. The total specific per unit mass energy of a geophysical fluid is given by the sum of internal, potential, kinetic, and latent energy.

In this formula, we neglect the heat content of the liquid and solid water and the heat associated to the phase transition between solid and liquid water. Similar relationships can be written for the atmosphere, ocean, and land provided that energy fluxes of sensible, latent heat as well as radiative fluxes are taken into account at the surface [ Peixoto and Oort , ].

At observational level nonzero energy balances are found at TOA and at the surface [ Trenberth and Fasullo , ; Wild et al. However, a physically consistent climate model should feature a vanishing net energy balance when its parameters are held fixed and statistical stationarity is eventually obtained. Spurious energy biases may be associated with nonconservation of water in the atmospheric branch of the hydrological cycle [ Liepert and Previdi , ; Liepert and Lo , ] and in the water surface fluxes [ Lucarini et al.

The meridional distribution of the radiative fields at the top of the atmosphere poses a strong constraint on the meridional general circulation [ Stone , ]. As clear from equation 56 , the stationarity condition 59 leads to the following indirect relationship for T T : Atmospheric and oceanic circulations act as responses needed to equilibrate such an imbalance [ Peixoto and Oort , ]. As emphasized by Enderton and Marshall [ ], if one assumes drastic changes in the meridional distributions of planetary albedo differences emerge with respect to Stone's theory.

A comprehensive thermodynamic theory of the climate system that is able to predict the peak location and strength of the meridional transport, the partition between atmosphere and ocean [ Rose and Ferreira , ], and to accommodate the variety of processes contributing to it is still missing. For simplicity, we here refer to T T. There is still not an accurate estimate of such a fundamental quantity for testing the output of climate models, despite the efforts of several authors [ Trenberth and Caron , ; Wunsch , ; Fasullo and Trenberth , ; Trenberth and Fasullo , ; Mayer and Haimberger , ].

The precision of the estimates relies on the knowledge of the boundary fluxes F R , F S , and F L and on the reanalysis data sets. Wunsch [ ], by using measurements of the radiative fluxes at the top of the atmosphere and previous estimates of the oceanic enthalpy transport, gave a range of values of 3. Unfortunately reanalysis data sets are affected by mass and energy conservation e.

Furthermore, these estimates are dependent on the analysis method and the model used; Trenberth and Caron [ ], using other reanalysis data set National Centers for Environmental Prediction , found a value of the maxima 0. The use of numerical climate model does not necessarily help reducing such uncertainties. For an intercomparison of the cloud distribution in different climate models, see Probst et al.

Interesting information emerge when looking at the position of the peaks of the transport. Similarly, the spread among models is small with respect to the position of the peak of T A in both hemispheres and of T O in the Northern Hemispheres, while a larger uncertainty exists in the position of the peak of T O in the Southern Hemisphere. The basic understanding of the maintenance of the atmospheric general circulation was achieved nearly 60 years ago by Lorenz [ , ] through the concepts of available potential energy and atmospheric energy cycle.

The concept of available potential energy, first introduced by Margules [ ] to study storms, is defined as , where T r is the temperature field of the reference state, obtained by an isentropic redistribution of the atmospheric mass so that the isentropic surfaces become horizontal and the mass between the two isentropes remains the same. By its own definition, this state minimizes the total potential energy at constant entropy. Such a definition is somewhat arbitrary, and different definitions lead to different formulations of atmospheric energetics [ Tailleux , ]. For example, the choice of a reference state maximizing entropy at constant energy [ Dutton , ] leads in a natural way to the concept of exergy.

Exergy is the part of the internal energy measuring the departure of the system from its thermodynamic and mechanical equilibrium, i. Lorenz [ ] proposed the following picture of the transformation of energy in the atmosphere. Under hydrostatic approximation one can show that [see, e. In the Lorenz framework one considers the hydrological cycle as a forcing to the atmospheric circulation.

This amounts to separating the budget of the moist static energy and of the part related to the phase changes of water [see Peixoto and Oort , , chapter 13]. The conversion term W can be interpreted as the instantaneous work performed by the system. Stationarity implies that , and therefore, , which is referred to as the intensity of the Lorenz energy cycle. One has to note that the latter can be expressed as the average rate of generation of available potential energy, , where T r is the temperature field of the reference state [ Lorenz , ].

On the other hand, general circulation models feature values of ranging from 2 to 3. Another aspect to be considered is that the intensity of the Lorenz energy cycle is formulated assuming hydrostatic conditions. In the case of the ocean, available potential energy is generated through thermohaline forcings due to the correlation of density inhomogeneities and density forcings e.

In addition to that, mechanical energy enters the ocean through direct transfer of kinetic energy by surface winds and though tidal effects. The understanding of the details of the oceanic Lorenz energy cycle is still at a relatively early stage. Johnson [ ] proposed an interesting construction for further elucidating the idea that the climate can be seen as a heat engine. Therefore, the atmosphere can be interpreted as a heat engine, in which and are the net average heat gain and loss and is the average mechanical work.

The efficiency of the atmospheric heat engine, i. We wish now to emphasize a different aspect of the climate's thermodynamics, namely its irreversibility by the investigation of its material entropy production, i. The entropy budget of the fluid can be rewritten as 66 so that we separate the contribution coming from the absorption of the radiation from other effects related to the other irreversible processes occurring in the fluid. Note that, in the previous formula, we refer to the entropy budget of the whole climate, not of the atmosphere, as done, instead, in the previous section.

The material entropy production rate can be expressed as , i. Therefore, we will limit the discussion to processes occurring in the interior and at the boundaries of the atmosphere. The importance of and in the context of thermodynamic theories of moist convection is extensively discussed in Pauluis and Held [ b ] and Pauluis [ ]. An indirect estimate of 67 can be obtained from the entropy budget for water as discussed in Pauluis and Held [ b ], where is the rate of change of entropy of water and is the neat heating amount of heat per time that the water substance receives from its environment i.

Therefore, it is possible to compute the material entropy production by considering exclusively the heat exchanges and the temperature at which such exchanges take place, thus bypassing the need for looking into the complicated details of phase separation processes. In particular, Woollings and Thuburn [ ] showed that dispersive dynamical cores can lead to negative numerical entropy production. More generally, it has been argued that parameterizations of subgrid turbulent fluxes of heat, water vapor and momentum should conform to the second law of thermodynamics and therefore should lead to locally positive definite entropy production, this being generally not the case [ Gassmann , ].

At steady state, we have that. The postindustrial case corresponds to the first years after the stabilization of the CO 2 in the A1B climate change scenario. Issues related to the effective nonstationarity of the system have been treated as in Lucarini and Ragone [ ]. Comparing with the direct computation [ Pascale et al.

The contribution due to vertical processes is dominant by about 1 order of magnitude with respect to the contribution due to horizontal processes. This suggests that from the point of view of the entropy production, the climate system approximately behaves as a collection of weakly coupled vertical columns where mixing takes place [ Lucarini and Pascale , ]. This fits well with what reported in Lucarini et al. First, the integrand in equation 70 is positive everywhere, indicating that vertical exchange processes lead to irreversible mixing of the fluid properties.

Small deviations can be found over Antarctica and Greenland, where the approximations leading to equation 70 are not necessarily valid see the discussion in Lucarini et al. Overall, the local material entropy production due to vertical processes seems to be a good indicator of the geographical distribution of convective activity: the highest values are observed in the warm pool of the western Pacific and Indian Ocean and in land areas characterized by warm and moist climates, while relatively low values are instead observed in the cold tongue of the eastern Pacific, near western boundary currents, and in the temperate and cold oceans, as well as on deserts and middle and high latitudes of terrestrial areas.

Note that also in this case the role of latent heat releases is fundamental in determining the characteristics of the system, showing how the hydrological cycle is a crucial aspect of thermodynamics of the climate system. The local vertical component of material entropy production increases almost everywhere, with negative anomalies confined to polar regions and to limited areas of the Southern Hemisphere, with very small values.

The positive anomalies are extremely high in the tropical regions, particularly in the eastern and western Pacific Ocean. Note that the pattern of increase does not strictly follow the pattern of the absolute value in the preindustrial case. In particular, the maximum of the increase is located eastward to the maximum of the entropy production in the preindustrial case, a signature of a shifting of the warm pool and a modification of the Walker circulation [ Bayr et al. High values are also found in the Indian Ocean, suggesting an increase of the convective activity connected with the monsoon [ Turner and Annamalai , ; IPCC , ].

Significant local maxima are also observed in the Gulf of Mexico and along the Gulf Stream, and in the Mediterranean Sea. The pattern of increase is correlated to the pattern of variation of the surface temperature only to a minor extent. The reason is that this indicator contains in a synthetic way the information of the change in the surface temperature, in the vertical stability of the atmosphere, and in the intensity of the energy fluxes connected to the vertical processes.

Moreover, the range of variation due to climate change of the local vertical entropy production is rather high if compared to the range of variation of standard fields like surface temperature or pressure. Based on the evidence supported by Hoffman and Schrag [ ] and from numerical models [ Budyko , ; Sellers , ; Ghil , ], it is expected that the Earth is potentially capable of supporting multiple steady states for the same values of some parameters such as, for example, the solar constant.

This is due to the presence of two disjoint strange chaotic attractors. The boundary of the domain in the parametric space where two states are admissible corresponds to the tipping points of the system. The thermodynamical and dynamical properties of the W and SB states are largely different. In the W states, surface temperature is 40—60 K higher than in the corresponding SB state and the hydrological cycle dominates the dynamics.

The SB state is eminently a dry climate, with entropy production mostly due to sensible heat fluxes and dissipation of kinetic energy. The entropy production increases for both states, but for different reasons: the system becomes more irreversible and less efficient in the case of W states, while stronger atmospheric motions lead to stronger dissipation and stronger energy transports in the case of SB states.

The negative feedbacks tend to counteract the differential heating due to the stellar insolation pattern, thus leading the system closer to equilibrium. At the bifurcation point, the negative feedbacks are overcome by the positive feedbacks, so that the system makes a global transition to a new state, where, in turn, the negative feedbacks are more efficient in stabilizing the system [ Boschi et al. It is interesting to study the possibility of constructing empirical relations between different thermodynamical quantities for a variety of parametric configurations.

If one could provide convincing arguments and methods for constructing such relations, it would be possible to express nonequilibrium thermodynamical properties of the system in terms of parameters which are more directly accessible through measurements [ Lucarini et al. A large number of the exoplanets discovered so far are tidally locked to their parental star, experiencing extreme stellar forcing on the dayside where temperature up to K can be reached.

Starlight energy, deposited within the atmosphere at the planet's dayside, is then transported by atmospheric circulation to the nightside. Numerical simulations [ Perna et al. Relating this definition of efficiency with the many different definitions used to characterize global circulations [ Johnson , ; Schubert and Mitchell , ; Perna et al.

The latter symbol is used in the theory of pseudodifferential operators, where it has a different meaning. There will be no risk of misunderstanding in our case. We introduce a partial order in the set of closed semi-bounded forms. By definition, a I b if d[a]3 d[b]and a[z] I b[x]for all z E d[b]. If, in addition, a[. The Riedrichs Extension Let A0 be a symmetric operator in 4. The theory of extensions see Akhiezer and Glazman ; Kato ; Birman and Solomyak answers the question concerning the conditions under which A0 has a self-adjoint extension.

The theory also provides an abstract description of all such extensions. If A0 is semi-bounded, then self-adjoint extensions are certain to exist and we can select one that plays a special role. On completing D A0 in the ao-norm and extending a0 by continuity, we obtain a set d C 4 and a closed positive definite form a defined on d.

It is called the the Fnedrichs extension.

## Global study of a family of cubic Liénard equations

On the other hand, given a, we first construct a bounded self-adjoint operator T in fj. The existence and uniqueness of such h follows from the Riesz theorem on the general form of a linear functional in a Hilbert space. Now let a be an arbitrary lower semi-bounded form.

A is the operator to be associated with a. It satisfies the above relations 1. It suffices to specify the corresponding quadratic form b[z]. The relation 1. The operator T is bounded and self-adjoint in d. Extending the form by continuity, we can obtain a bounded form b[z] on d[a].

It is therefore natural to associate the spectrum of this operator with 1. In applications it is often expedient to go over from the unbounded operator A to the bounded operator T. A simplified terminology is often used in the study of the spectra of operators determined by variational triples. Formula 1. The following important result is a consequence of 1.

Let A and B be semi-bounded self-adjoint operators such that A 5 B. Let us now assume that A is a semi-bounded self-adjointoperator in a space L z X , p. The kernel eA X;z, y of this operator is called the spectral function of A. Then x j 1. This subject is presented in more detail in Maurin , Dunford and Schwartz , Berezanskij , and Berezin and Shubin As opposed to N X ; A ,the spectral function is not a unitarily invariant object, since it requires that the Hilbert space rj be realized as an L2 space.

Differential Expressions and Their Symbols 1. Essential Self-Adjointness Here we shall briefly present some information about compact completely continuous operators in a Hilbert space fj. It is not always easy to verify whether a differential operator is self-adjoint or not. In a number of cases the domain of a self-adjoint differential operator admits an indirect description only.

In addition, the action of the operator on such a domain may not always be given explicitly. In what follows we take L 2 X as our basic Hilbert space. X of order m 2 1 be given in X. It is obvious that 1. It is a necessary condition for L to have self-adjoint realizations. The polynomials in t E an 20 2.

Essential Self-Adjointness are referred to as the symbol or the complete symbol and the principal symbol of the differential expression 2. As opposed to the general uniform ellipticity condition 2. The Maximal and Minimal Operators On many occasions it is also said that L x, It has already been mentioned that, under the conditions 2. Essential Self-Adjointness 2. This condition is necessarily satisfied if Cmin is lower C,,, - c I for c 22 r r 23 2. Essential Self-Adjointness of Elliptic Operators has the largest possible domain. On the other hand, as a rule, only domains D 2 C r X are considered.

The closure of C C r X is called the minimal operator generated by C. One of the simplest cases in which a differential expression has a unique self-adjoint realization is the case of an elliptic operator on a manifold without boundary. Example 2. Let X be a domain in Rn and let C be an elliptic differential expression of order m with constant coefficients. Theorem 2. Then the operator C is essentially self-adjoint on Coo X. The equality Example 2. For u E Dmin C some boundary conditions on d X can also be retained in the general non-elliptic case.

On the other hand, the functions u E Dmax do not, in general, satisfy any boundary conditions. However, if X is an unbounded domain or the coefficients of C are rapidly increasing near d X , then the relation Cu E L 2 X can involve implicit conditions at infinity or on ax that must be satisfied by the functions from D,,,.

It is also said that C is an essentially self-adjoint differential expression on C F X. In particular, this is the case in Example 2. Self-adjoint realizations do not always exist. The theorem can be extended to the case of elliptic differential expressions on bundles.

We shall now discuss the case when C , Cmin, confining ourselves to elliptic differential expressions. If D is the domain of a self-adjoint realization C TO, then it satisfies 2. It follows from 2. From this point of view, any self-adjoint realization of C is determined by the boundary conditions. The above discussion admits localization. Similarly, we can talk of boundary conditions at infinity, etc. Weyl considered this problem as a limit of problems on the expanding system of intervals 0 , l. It follows that the discs Dl contract either to a 'limitpoint' or a 'limit-circle' as 1 4 In the former case the operator does not require any boundary conditions at infinity, but it does in the latter.

We shall present the classic examples of self-adjoint operators defined by a formally self-adjoint differential expression with boundary conditions. The basic boundary value problems for a second-order elliptic differential expression. Let a uniformly elliptic formally self-adjoint differential expression 2. We set 2. The boundary conditions in 2. The operator of the Dirichlet problem for a n equation of order 2r. The operator 2. In particular, the operators 2. The theorem follows from the theory of solubility of elliptic boundary value problems Lions and Magenes ; Hormander The symmetry condition assumed in the theorem implies formal self-adjointness of L and also stipulates certain consistency between L and the boundary conditions the operators 2.

In general, Theorem 2. For instance, if X c R2 is a domain whose boundary contains a corner point, the interior angle being greater than T , then AD is not a self-adjoint operator in this connection, see Example 3. Regular elliptic boundary value problems can also be stated for operators acting in spaces of sections of vector bundles. We shall collect all such operators as well as elliptic differential operators on compact manifolds without boundary under the name of regular elliptic differential operators.

Singular Differential Operators Let us agree to refer to differential operators other than regular elliptic ones as singular operators. An operator may be singular because of various reasons such as the non-compactness of the manifold, violation of the uniform ellipticity condition, non-regularity of the coefficients or the boundary, and the like. From this list it is clear that it is a matter of convention to collect all such operators under one name.

Many singular differential operators have properties similar to those of regular elliptic operators. The following result, Y w! The Schrodinger Operator which can be easily proved by going over to the Fourier transforms, is one of the exceptions. Below we present a typical result based on Theorem 1. Let V be a real-valued potential that satisfies the following conditions: a i f n 2 4, then 26 27 2.

The real-valued within the scope of the self-adjoint theory function V is usually called the potential. In accordance with 2. In order that the evolution operator of this equation be unitary, the realization of C must be a self-adjoint operator. Roughly speaking, the essential self-adjointness of L on CF Wn means that a wave at first localized in space cannot escape to infinity in finite time.

The completeness of a Hamiltonian system with Hamiltonian L x ,E , that is, the existence of a solution of the system for all t E R and any initial data x0,to such that EO 0, serves as the classical analogue of essential self-adjointness. See Reed and Simon , Vol. Only in rare cases can one succeed in giving an exact description of the domain on which 2. For the most part the domain can be studied on the basis of Theorem 1. In other words, under the assumptions of the theorem, the operator 2.

Example 2 8. In particular, this is the case for the Schrodinger operator of an atom with Coulomb interaction between the particles. The assumptions of Theorem 2. For such potentials the domain on which the operator 2. Separation is always connected with a specific regular behaviour of the potential.

As an illustration of the theorem, we consider the following example. Then the operator 2. Firstly, 2. Secondly, it rules out rapid oscillations and other irregularities in the behaviour of V z. Therefore L is essentially self-adjoint on C,M. We remark that the operator 2. There are also separation conditions that admit local irregularities of the potential. The precise definition uses capacity terms. Regarding this point, see Mynbaev and Otelbaev , which also contains further references.

In a more complex situation one fails to give an exact description of the domain of a self-adjoint realization of the Schrodinger operator. Then the investigation of conditions for V under which 2. The theorem below is one of the basic results in this direction. We shall discuss the case when the potential V in 2. In accordance with Sect. There arises the question of whether or not L requires any boundary conditions on F.

The Dirac Operator The Dirac operator in W3, which describes the behaviour of a relativistic particle, serves as another important example. Let crl,. As opposed to the Laplace operator, LO is not semi-bounded, which can be seen immediately on applying the Fourier transform. The essential self-adjointness of C on Cr R3 depends only on the local properties of the potential. For the operator 2. The Dirac operator with scalar Coulomb potential. One can demonstrate Reed and Simon , Vol. The qualitative and quantitative characteristics of the spectrum of a semibounded self-adjoint operator can be well described in terms of its quadratic form.

If the minimal operator exists, is lower semi-bounded, and not essentially self-adjoint, then the method of forms distinguishes one concrete self-adjoint realization of C, namely, the Friedrichs extension of Cmin. The construction of a differential operator from a quadratic form is closely related to the problems of classical variational calculus.

The differential expression L corresponding to the form appears on the left-hand side of the Euler equation. Examples 53 Defining an Operator by a Quadratic Form 3. Examples We shall present a number of typical examples. Example 3. Boundary value problems for a second-order elliptic operator see Example 2. Let X c R" be an arbitrary domain and let 33 Example 3. The operator -A D in a domain with corners. Therefore, if a The quadratic functional 3.

The description of the operator given in 2. The form 3. Analogously, let F c d X be a closed set and let k l X ;F be the closure in H 1 X of the set of functions vanishing in a neighbourhood of F. We assume that the boundary is compact and of class C2. Thus, by the embedding theorem, the boundary integral in 3. In general, under the above assumptions for aij and the domain which are weaker compared with Example 2. Here all the boundary conditions are the natural ones.

The operators of the Dirichlet and Neumann problems f o r degenerate elliptic differential expressions. The form I , r H k X is non-negative and closed. We denote by C,,N the self-adjoint operator in L 2 X that corresponds to this form. The operator C,,D corresponds to the same form 3. The Dirichlet problem for non-elliptic differential expressions with constant coefficients. In general, it is scarcely possible to describe the boundary conditions explicitly unless C is elliptic.

In analogy with Example 3. Generalized Schrodinger operator. Let the 'potential' V satisfy 3. We consider the quadratic functional 3. The corresponding self-adjoint operator A is taken as the realization of the differential expression 2. In the case in question one can give a more or less explicit description of the domain of A Kato; see Reed and Simon , Vol. The functional 3. The operator Op a, can be taken as a realization of the differential expression On multiplying the differential expression 2.

It is only important that the natural domain of 3. It is only necessary that the negative part of 3. The generalized Schrodinger operator with weakly nonsemi-bounded potential. Under these conditions, the functional 3. As before, the corresponding self-adjoint operator in L2 Rn is taken as the realization of the differential expression 3. From 3. This approach enables one to consider potentials with stronger negative local singularities than the approach described in Theorem 1.

It follows from 3. Here d must be a subspace of finite deficiency in H' on which 3. For a reasonable choice of d depending on the 'weight' p in 3. For details, see Birman and Solomyak , The Steklov problem. Weighted Polyharmonic Operator Here we shall consider examples of operators that can be defined by variational triples see Sect. In our examples a[. The Dirichlet problem in a bounded domain. Then 3. Let p E L, X with y defined by 3. The Neumann problem. Let X c Bn be a bounded domain with Lipschitzian boundary and let 3.

The Dirichlet problem in an unbounded domain. The passage to an unbounded X involves two new difficulties. One of them, which is purely technical, is that the boundedness conditions for 3. The point is that k X is not necessarily complete in the metric 3.

Its completion is not always a space of functions. The operator that corresponds to this triple is what we take as the operator of the Dirichlet problem for equation 3. In particular, for the unperturbed Dirac operator that is, for the operator 2. Denote by s C the set of values of C There are two basic sources of differential operators that admit exact computation of the spectrum. One of them consists of invariant operators on Lie groups or homogeneous spaces, in which case the spectral characteristics can be computed by means of representation theory. Special functions and separation of variables constitute the other source which is, in fact, closely related to the former one.

Many examples of the latter kind can be found in standard textbooks on mathematical physics. Examples of operators whose spectrum can be computed explicitly are important because of many reasons. In particular, they make it possible to observe relationships which are sometimes rather difficult to discover in the general situation.

Quite often the analysis of such examples constitutes the first step in proving general assertions - see, for example, the estimates of the spectrum of regular elliptic differential operators in Sect. Here we shall present several examples to be used later on. The examples are completely elementary and their analysis requires no direct reference to representation theory. As a consequence, there are no points of the spectrum to the left of Xo. Since this proof makes little use of the Fourier transform, it can be easily adapted to analyse the following example.

Example 4. The operator C D X see Example 3. We assume that the domain X c R" contains balls of arbitrarily large radius. Then the spectrum of C D X coincides with the closure of the range of C. Indeed, to prove that s C c a C ,it suffices to consider suitable shifts of the functions u k used above. The proof of the inclusion in the opposite direction remains unchanged. Scalar differential operators on R".

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The Fourier transform turns C into the operator of multiplication by the real symbol C 0 and coincides with the closure of the set of values of C c ,where E Rn. In particular, the spectrum of -A. Scalar differential operator on the torus T". The functions exp i j. The eigenvalues are equal to C j , where j E Z". If IC c r The Fourier transform makes it also possible to carry out the spectral analysis of differential operators with constant coefficients defined on vector- W" and on a Torus 4a.

Dafferential operator in the space of vector-valued functions o n a torus. Its symbol C 0 ,where E W",is a Hermitian kx k-matrix. Operators on a Sphere and a Hemisphere into account and eigenvectors of L,the latter being orthonormalized in C k. It follows that L has pure point spectrum; e p L consists of all the eigenvalues Xl j. By an orthogonal coordinate transformation in Rn diagonalizing B , one can reduce the operator to the form 40 4.

In this notation the multiplicity of the eigenvalues is automatically taken into account. Hence it is seen that the 'formal' eigenvalues X 0 for the time being we do not assume that u,v E are the same for the two operators 4. Sometimes this simple argument enables one to find the point spectrum of a differential operator.

Harmonic oscillator. To within normalization, U k are the Hermite functions H k X. It follows that the spectrum of the operator 4. The example below can be reduced to the last one. Many-dimensional harmonic oscillator. Let B be a positive definite n x n -matrix. The operator is self-adjoint which follows, for example, from Theorem 2.

The essential spectrum of this operator coincides with [0,00 see Theorem 6. We denote by lk the operators 4. Since l? Next, by induction, we find that if n - 1 4. Operators on a Sphere and a Hemisphere Example 4. The Laplace-Beltrami operator on the sphere 5'". The spherical functions form a complete system of eigenfunctions of this operator. Here designates the number of q-element combinations out of p elements. If p i Example 4.

As in Example 4. In Sect. Estimates of Eigenvalues The following assertion, which belongs to the general theory of operators, can, as a rule, serve as the basis for establishing that the spectrum is discrete. Lemma 5. Once it is established that the spectrum is discrete, there arise quantitative questions concerned with the estimates of the eigenvalues and eigenfunctions, their asymptotic behaviour, and the like. Here we shall touch upon the simpler, but, nevertheless, important question concerning the estimates.

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Example 5. Regular elliptic operator A see Theorems 2.

The operator -A. For example, the operator I0 is compact for any domain X of finite measure, including any bounded domain. There is also a similar criterion for 2r 5 n, but in this case one has t o consider cubes that may contain a sufficiently small in the sense of capacity portion of the complement of X.

For a rigorous formulation, see Maz'ya By Lemma 5. The latter requires much stronger conditions for X compared with Example 5. For instance, it suffices that X be bounded and its boundary be Lipschitzian. For the precise conditions see Maz'ya It is only required that the corresponding quadratic form a[. In particular, under the assumptions of Example 3. Degenerate second-order elliptic operators see Example 3. Here the problem can be solved by means the embedding theorems for weighted Sobolev classes. The eigenvalues of the operator coincide with C j , where j E Z".

By the uniform ellipticity condition 2. Thus, obviously, C j 2 yoljlm - C for some yo E 0,y and C 2 0. The same is true for the generalized Schrodinger operator from Example 3. The above discussion rests on the following two facts: the L2-norm of u outside a sufficiently large ball is small compared with a[u],and the embedding of d[a] in L2 is compact in a bounded domain. The latter is true, since the potential is locally semibounded. Both conditions can, in fact, be relaxed. Similar criteria for the spectrum to be discrete are also valid for the operator 3. If 2r 5 n, then, even for a semi-bounded potential, a criterion for the spectrum to be discrete can be stated only in terms of capacity see Maz'ya In the non-semi-bounded case 5.

As was demonstrated by Agmon , one can deduce 5. The estimate 5. The subsequent development of estimates of the form 5. Thus, the estimate 5. For the eigenvalues f X j A themselves, an estimate equivalent to 5. For instance, for the Neumann problem the constant in the estimate depends on X it worsens as the properties of the boundary deteriorate. Moreover, the estimate itself is violated for small A, because zero is an eigenvalue of -A. The independence of the estimates of the domain for the Dirichlet problem makes it possible to extend 5. In particular, this includes the assertion that the spectrum of -A pD is discrete for such domains see Example 5.

Estimates of the Spectrum of a Weighted Polyharmonic Operator 5. Estimates of the Spectrum of a Weighted the asymptotic formula. We can see that, under the assumptions of Example 5. Indeed, these conditions imply the very strong estimate 5. We use the variational formulation of the problem, that is, by the spectrum of the Dirichlet or Neumann problem for 5. For these problems one can succeed in obtaining estimates of. Such estimates have important applications in the study of spectral asymptotics by the variational method see as well as in the study of the spectrum of the Schrodinger operator.

The Dirichlet problem f o r equation 5. Under the assumptions adopted in Examples 3. The estimates 5. The Neumann problem f o r equation 5. Under the assumptions of Example 3. However, in this case the constant c, in 5. As in Example 5. From the estimate 5. Theorem 5. The constant c does not depend on p or E 2 0. Rn see 3. In general, the form is unbounded. It follows that the estimate 5.

According to 1. Applying the same formula 1. It remains to apply 5. From 5. Nevertheless, one can easily construct potentials such that the estimate 5. Then the estimates N Theorem 5. Estimates of Eigenfunctions For a fixed x, the spectral function eA 5. Estimates of the Spectrum: Heuristic Approach Theorem 5. Then the estimate d. Then a is semi-bounded and closed on d , and 6. The Schrodinger operator 2. In this connection there arise many new questions concerning the location of the essential spectrum and its spectral characteristics in the first place, one is interested in conditions for the absolute continuity of the spectrum; see Sect.

These classes are most interesting from the viewpoint of applications, and, at the same time, relatively complete results are available for them. Nowadays the theory of the Schrodinger operator with an almost periodic potential is also developing rapidly, however, it is still far from being complete. We also postpone to 57 the presentation of certain facts concerned with the spectral theory of the multiparticle Schrodinger operator. Theorem 6. Let V E Loo,lOcand let 6. Alternative conditions for V that imply 6.

If, in addition, V z L 0 under the assumptions of Theorem 6. It follows that the spectrum of the Schrodinger operator with a non-negative potential that satisfies the assumptions of Theorem 6. The location of the essential spectrum can often be determined on the basis of the following result from abstract perturbation theory see Birman and Solomyak Negative Spectrum of the Schrodinger Operator If A and A0 are introduced by means of quadratic forms, then it is more convenient to use the following corollary of Theorem 6. If V z Corollary. Let a0 be a closed positive definite form in 4.

The hydrogen atom. It proved to be an efficient tool in the study of various problems concerned with the spectrum. We shall illustrate the application of 6. The abstract scheme of such a reduction was developed by Birman , We shall state only the simplest result from Birman The proof can be reduced to comparing the formulae below, which follow directly from 1.

For an operator on a half-axis the presence or absence of a resonance at zero is determined by the boundary condition. There is a resonance for the operator of the Neumann problem the proof is the same as in the case of an axis. The estimates of the most accurate order have been obtained in Egorov and Kondrat'ev In the case of the one-dimensional Schrodinger operator the following simple estimates are valid for the operator on an axis or half-axis with the Neumann condition see Birman : 55 be studied on the basis of Theorem 6.

According to Theorem 6. The latter two conditions ensure that 2. Eigenvalues within the Continuous Spectrum cf. For the operator of the Dirichlet problem on a half-axis the term 1 should be discarded. From Theorem 6. When the number of unknown quantities exceeds the number of equations, the equations will admit of innumerable solutions, and are therefore said to be indeterminate.

Thus if it be required to find two numbers such that their sum be 10, we have two unknown quantities x and y, and only one equation, viz. It is, however, usual, in such questions as this, to restrict values of the numbers sought to positive integers, and therefore, in this case, we can have only these nine solutions,. This branch of analysis was extensively studied by Diophantus, and is sometimes termed the Diophantine Analysis. Indeterminate problems are of different orders, according to the dimensions of the equation which is obtained after all the unknown quantities but two have been eliminated by means of the given equations.

Those of the first order lead always to equations of the form. Before proceeding to the general solution of these equations we will give a numerical example. But since these quantities are required to be positive, it is evident, from the value of y, that z must be either 0 or positive, and from the value of x, that it must be less than 4; hence z may have these four values, 0, 1, 2, 3. In this case the number of solutions is limited. For the determination of the number of solutions the reader is referred to H.

Hall and S. From this last equation we may find values of x and y of this form,. For more advanced treatment of linear indeterminate equations see Combinatorial Analysis. The possibility of rendering the proposed formula a square depends altogether upon the coefficients a, b, c; and there are four cases of the problem, the solution of each of which is connected with some peculiarity in its nature.

Case 1. Case 2. Case 3. Case 4. The application of the preceding general methods of resolution to any particular case is very easy; we shall therefore conclude with a single example. This equation is evidently of such a form as to be resolvable by the method employed in case 1. Cubic equations, like all equations above the first degree, are divided into two classes: they are said to be pure when they contain only one power of the unknown quantity; and adfected when they contain two or more powers of that quantity. Let us now consider such cubic equations as have all their terms, and which are therefore of this form,.

This transformation is a particular case of a general theorem. Thus we have obtained a value of the unknown quantity y, in terms of the known quantities q and r; therefore the equation is resolved. Next to pure biquadratic equations, in respect of easiness of resolution, are such as want the second and fourth terms, and therefore have this form,. When a biquadratic equation has all its terms, its resolution may be always reduced to that of a cubic equation.

There are various methods by which such a reduction may be effected. The following was first given by Leonhard Euler in the Petersburg Commentaries , and afterwards explained more fully in his Elements of Algebra. We have already explained how an equation which is complete in its terms may be transformed into another of the same degree, but which wants the second term; therefore any biquadratic equation may be reduced to this form,. For this purpose let us compare the equations. Hence it follows that the roots of the proposed equation are generally expressed by the formula.

These considerations enable us to determine that four of the eight expressions for the root belong to the case in which q is positive, and the other four to that in which it is negative. Thus the equation. Form this cubic equation. The equation may be numerical ; that is, the coefficients p 1 , p 2 n , Or the equation may be algebraical ; that is, the coefficients are not then restricted to denote, or are not explicitly considered as denoting, numbers.

Postponing all consideration of imaginaries, we take in the first instance the coefficients to be real, and attend only to the real roots if any ; that is, p 1 , p 2 , It is clear that, in general, y is a continuous one-valued function of x, finite for every finite value of x, but becoming infinite when x is infinite; i. The great step was effected by the theorem of J. Sturm —viz. The practical difficulty is when two or more roots are very near to each other.

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Suppose, for instance, that the theorem shows that there are two roots between 0 and 10; by giving to x the values 1, 2, 3, Supposing the separation once effected, the determination of the single real root which lies between the two given limits may be effected to any required degree of approximation either by the processes of W. First as to Horner and Lagrange. The arrangement of the calculations is very elegant, and forms an integral part of the actual method. It is to be observed that after a certain number of decimal places have been obtained, a good many more can be found by a mere division.

It is in the progress tacitly assumed that the roots have been first separated. The method is theoretically very elegant, but the disadvantage is that it gives the result in the form of a continued fraction, which for the most part must ultimately be converted into a decimal. There is one advantage in the method, that a commensurable root that is, a root equal to a rational fraction is found accurately, since, when such root exists, the continued fraction terminates.

Referring to fig. It thus appears that for the proper application of the method we require more than the mere separation of the roots. When this is so, the point C may be taken anywhere on the proper side of X, and within the portion XN of the axis; and the process is then the one already explained. The approximation is in general a very rapid one. The difference is that with Horner the integer part of this quotient is taken as the presumptive value of h, and the figure is verified at each step. With Newton the quotient itself, developed to the proper number of decimal places, is taken as the value of h; if too many decimals are taken, there would be a waste of work; but the error would correct itself at the next step.

Of course the calculation should be conducted without any such waste of work. It will be recollected that the expression number and the correlative epithet numerical were at the outset used in a wide sense, as extending to imaginaries. This extension arises out of the theory of equations by a process analogous to that by which number, in its original most restricted sense of positive integer number, was extended to have the meaning of a real positive or negative magnitude susceptible of continuous variation.

We thus arrive at the extended signification of number as a continuously varying positive or negative magnitude.

Such numbers may be added or subtracted, multiplied or divided one by another, and the result is always a number. Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above. There may very well be, and perhaps are, numbers in a more general sense of the term quaternions are not a case in point, as the ordinary laws of combination are not adhered to , but in order to have to do with such numbers if any we must start with them.

The capital theorem as regards numerical equations thus is, every numerical equation has a numerical root; or for shortness the meaning being as before , every equation has a root. Of course the theorem is the reverse of self-evident, and it requires proof; but provisionally assuming it as true, we derive from it the general theory of numerical equations. As the term root was introduced in the course of an explanation, it will be convenient to give here the formal definition. We have thus the theorem —A numerical equation of the order n has in every case n roots, viz.

If the equation has equal roots, these can in general be determined, and the case is at any rate a special one which may be in the first instance excluded from consideration. If the coefficients p 1 , p 2 , But an equation with real coefficients may have as well imaginary as real roots, and we have further the theorem that for any such equation the imaginary roots enter in pairs, viz. It follows that if the order be odd, there is always an odd number of real roots, and therefore at least one real root.

In the case of an equation with real coefficients, the question of the existence of real roots, and of their separation, has been already considered. This is solved theoretically by means of a theorem of A. Cauchy , viz. It would appear that this proof of the fundamental theorem in its most complete form is in principle identical with the last proof of K. Gauss of the theorem, in the form—A numerical equation of the nth order has always a root. Very little has been done in regard to the calculation of the imaginary roots of an equation by approximation; and the question is not here considered.

The foregoing conclusions apply, viz. And these can be found numerically by the extraction of the square root, and of an nth root, of real numbers, and by the aid of a table of natural sines and cosines. The theory will be resumed further on. As the coefficients of an algebraical equation may be numerical, all which follows in regard to algebraical equations is with, it may be, some few modifications applicable to numerical equations; and hence, concluding for the present this subject, it will be convenient to pass on to algebraical equations.

In either case we have. As already explained, the epithet algebraical is not used in opposition to numerical; an algebraical equation is merely an equation wherein the coefficients are not restricted to denote, or are not explicitly considered as denoting, numbers. But if it be asked what there is beyond numerical equations included in the term algebraical equation, or, again, what is the full extent of the meaning attributed to the term—the latter question at any rate it would be very difficult to answer; as to the former one, it may be said that the coefficients may, for instance, be symbols of operation.

As regards such equations, there is certainly no proof that every equation has a root, or that an equation of the nth order has n roots; nor is it in any wise clear what the precise signification of the statement is. But it is found that the assumption of the existence of the n roots can be made without contradictory results; conclusions derived from it, if they involve the roots, rest on the same ground as the original assumption; but the conclusion may be independent of the roots altogether, and in this case it is undoubtedly valid; the reasoning, although actually conducted by aid of the assumption and, it may be, most easily and elegantly in this manner , is really independent of the assumption.

But the existence of a double root implies a certain relation between the coefficients; the general case is when the roots are all unequal. We have then the theorem that every rational symmetrical function of the roots is a rational function of the coefficients. This is an easy consequence from the less general theorem, every rational and integral symmetrical function of the roots is a rational and integral function of the coefficients.

Hence in each system the root-functions can be determined linearly in terms of the powers and products of the coefficients:. The proof in the case of a rational non-integral function is somewhat more complicated. From the theorem that a rational symmetrical function of the roots is expressible in terms of the coefficients, it at once follows that it is possible to determine an equation of an assignable order having for its roots the several values of any given unsymmetrical function of the roots of the given equation.

For example, in the case of a quartic equation, roots a, b, c, d , it is possible to find an equation having the roots ab, ac, ad, bc, bd, cd being therefore a sextic equation : viz. This is in fact E. The answer is 3, viz. In the case of 4 letters there exist as appears above 3-valued functions: but in the case of 5 letters there does not exist any 3-valued or 4-valued function; and the only 5-valued functions are those which are symmetrical in regard to four of the letters, and can thus be expressed in terms of one letter and of symmetrical functions of all the letters.

These last theorems present themselves in the demonstration of the non-existence of a solution of a quintic equation by radicals. The theory is an extensive and important one, depending on the notions of substitutions and of groups q. Returning to equations, we have the very important theorem that, given the value of any unsymmetrical function of the roots, e.

The a priori ground of this theorem may be illustrated by means of a numerical equation. Suppose for a moment that t 1 , t 2 , t 3 are all known; then the equations being linear in y 1 , y 2 , y 3 these can be expressed rationally in terms of the coefficients and of t 1 , t 2 , t 3 ; that is, y 1 , y 2 , y 3 will be known.

We now consider the question of the algebraical solution of equations, or, more accurately, that of the solution of equations by radicals. It will be observed that the coefficients p, q It does not even follow that in the case of a numerical equation solvable by radicals the algebraical solution gives the numerical solution, but this requires explanation. Consider first a numerical quadric equation with imaginary coefficients.

The case of a numerical cubic equation will be considered presently. It would have been wrong to complete the solution by writing. In the case of a numerical cubic, even when the coefficients are real, substituting their values in the expression. By what precedes there is nothing in this that might not have been expected; the algebraical solution makes the solution depend on the extraction of the cube root of a number, and there was no reason for expecting this to be a real number. It is well known that the case in question is that wherein the three roots of the numerical cubic equation are all real; if the roots are two imaginary, one real, then contrariwise the quantity under the cube root is real; and the algebraical solution gives the numerical one.

But when the quartic is numerical the same thing happens as in the cubic, and the algebraical solution does not in every case give the numerical one. It will be understood from the foregoing explanation as to the quartic how in the next following case, that of the quintic, the question of the solvability by radicals depends on the existence or non-existence of k-valued functions of the five roots a, b, c, d, e ; the fundamental theorem is the one already stated, a rational function of five letters, if it has less than 5, cannot have more than 2 values, that is, there are no 3-valued or 4-valued functions of 5 letters: and by reasoning depending in part upon this theorem, N.

Abel showed that a general quintic equation is not solvable by radicals; and a fortiori the general equation of any order higher than 5 is not solvable by radicals. The general theory of the solvability of an equation by radicals depends fundamentally on A. The formulae for the cubic were obtained by J. Lagrange from a different point of view. For a quartic the formulae present themselves in a somewhat different form, by reason that 4 is not a prime number. It is to be remarked, in regard to the question of solvability by radicals, that not only the coefficients are taken to be arbitrary, but it is assumed that they are represented each by a single letter, or say rather that they are not so expressed in terms of other arbitrary quantities as to make a solution possible.

It is proper to distinguish the cases n prime and n composite; and in the latter case there is a distinction according as the prime factors of n are simple or multiple. The process of solution due to Karl Friedrich Gauss depends essentially on the arrangement of the roots in a certain order, viz. Some interesting developments in regard to the theory were obtained by C. Abel , and since called Abelian equations, viz. Abel applied his theory to the equations which present themselves in the division of the elliptic functions, but not to the modular equations.

But the theory of the algebraical solution of equations in its most complete form was established by Evariste Galois born October , killed in a duel May ; see his collected works, Liouville , t. The fundamental theorem is the Proposition I. Every function of the roots invariable by the substitutions of the group is rationally known. Reciprocally every rationally determinable function of the roots is invariable by the substitutions of the group. And in saying that a function is rationally known, it is meant that its value is expressible rationally in terms of the coefficients and of the adjoint quantities.

For instance in the case of a general equation, the group is simply the system of the 1. But the problem of solution by radicals, instead of being the sole object of the theory, appears as the first link of a long chain of questions relating to the transformation and classification of irrationals. And hence, to determine whether an equation of a given form is solvable by radicals, the course of investigation is to inquire whether, by the successive adjunction of radicals, it is possible to reduce the original group of the equation so as to make it ultimately consist of a single permutation.

The condition in order that an equation of a given prime order n may be solvable by radicals was in this way obtained—in the first instance in the form scarcely intelligible without further explanation that every function of the roots x 1 , x Luther, Crelle , t. Among other results demonstrated or announced by Galois may be mentioned those relating to the modular equations in the theory of elliptic functions; for the transformations of the orders 5, 7, 11, the modular equations of the orders 6, 8, 12 are depressible to the orders 5, 7, 11 respectively; but for the transformation, n a prime number greater than 11, the depression is impossible.

The general theory of Galois in regard to the solution of equations was completed, and some of the demonstrations supplied by E. Betti See also J. Hermite, L. Kronecker and F. Annalen , t. The modern work, reproducing the theories of Galois, and exhibiting the theory of algebraic equations as a whole, is C. The work is divided into four books—book i. A glance through the index will show the vast extent which the theory has assumed, and the form of general conclusions arrived at; thus, in book iii. Burnside and A. Panton, The Theory of Equations 4th ed. Matthews, Algebraic Equations See also the Ency.

Its amount is the correction which must be applied positively or negatively to the mean anomaly in order to obtain the true anomaly. It arises from the ellipticity of the orbit, is zero at pericentre and apocentre, and reaches its greatest amount nearly midway between these points.

See Anomaly and Orbit. It goes through a double period in the course of a year. Its amount varies a fraction of a minute for the same date, from year to year and from one longitude to another, on the same day. The following table shows an average value for any date and for the Greenwich meridian for a number of years, from which the actual value will seldom deviate more than 20 seconds until after Scheuer ; the modern spelling has confused the word with the Lat.

At the British court, equerries are officers attached to the department of the master of the horse, the first of whom is called chief equerry see Household , Royal. According to the older classification this family was taken to include only the forms with tall-crowned teeth, more or less closely allied to the typical genus Equus. The Equidae , in this extended sense, together with the extinct Palaeotheriidae , are indeed now regarded as forming one of four main groups into which the Perissodactyla are divided, the other groups being the Tapiroidea, Rhinocerotoidea and Titanotheriide.

For the horse-group the name Hippoidea is employed. All four groups were closely connected in the Lower Eocene, so that exact definition is almost impossible. In the Hippoidea there is generally the full series of 44 teeth, but the first premolar is often deciduous or wanting in the lower or in both jaws. The incisors are chisel-shaped, and the canines tend to become isolated so as in the now specialized forms to occupy nearly the middle of a longer or shorter gap between the incisors and premolars.

In the upper molars the two outer columns of the primitive tubercular molar coalesce to form an outer wall, from which proceed two crescentic transverse crests; the connexion between the crests and the wall being imperfect or slight, and the crests themselves sometimes tubercular. Each of the lower molars carries two crescentic ridges. The number of toes ranges from four to one in the fore-foot, and from three to one in the hind-foot. The paroccipital, postglenoid and post-tympanic processes of the skull are large, and the latter always distinct.

Normally there are no traces of horn-cores. The calcaneum lacks the facet for the fibula found in the Titanotheroidea. In the earlier Equidae the teeth were short-crowned, with the premolars simpler than the molars; but there is a gradual tendency to an increase in the height of the crowns of the teeth, accompanied by increasing complexity of structure and the filling up of the hollows with cement. Similarly the gap on each side of the canine tooth in each jaw continues to increase in length; while in all the later forms the orbit is surrounded by a ring of bone.

A third modification is the increasing length of limb as well as in general bodily size , accompanied by a gradual reduction in the number of toes from three or four to one. All the existing members of the family, such as the domesticated horse Equus caballus and its wild or half-wild relatives, the asses and the zebras, are included in the typical genus.

In all these the crowns of the cheek-teeth are very tall fig. Each limb terminates in one large toe; the lateral digits being represented by the splint-bones, corresponding to the lateral metacarpals and metatarsals of Hipparion. Not unfrequently, however, the lower ends of the splint-bones carry a small expansion, representing the phalanges.

Remains of horses indistinguishable from E. The ancestor of these Pleistocene horses is probably E. In India a nearly allied species E. In North America species of Equus occur in the Pleistocene and from that continent others reached South America during the same epoch. In the latter country occurs Hippidium , in which the cheek-teeth are shorter and simpler, and the nasal bones very long and slender, with elongated slits at the side. The limbs, especially the cannon-bones, are relatively short, and the splint-bones large. The allied Argentine Onohippidium , which is also Pleistocene, has still longer nasal bones and slits, and a deep double cavity in front of the orbit, part of which probably contained a gland.

Onohippidium is certainly off the direct line of descent of the modern horses, and, on account of the length of the nasals and their slits, the same probably holds good for Hippidium. Species from the Pliocene of Texas and the Upper Miocene Loup Fork of Oregon were at one time assigned to Hippidium , but this is incorrect, that genus being exclusively South American. The name Pliohippus has been applied to species from the same two formations on the supposition that the foot-structure was similar to that of Hippidium , but Mr J. Gidley is of opinion that the lateral digits may have been fully developed.

Apparently there is here some gap in the line of descent of the horse, and it may be suggested that the evolution took place, not as commonly supposed, in North America, but in eastern central Asia, of which the palaeontology is practically unknown; some support is given to this theory by the fact that the earliest species with which we are acquainted occur in northern India. Be this as it may, the next North American representatives of the family constitute the genera Protohippus and Merychippus of the Miocene, in both of which the lateral digits are fully developed and terminate in small though perfect hoofs.

In both the cheek-teeth have moderately tall crowns, and in the first named of the two those of the milk-series are nearly similar to their permanent successors. In Merychippus , on the other hand, the milk-molars have short crowns, without any cement in the hollows, thus resembling the permanent molars of the under-mentioned genus Anchitherium. From the well-known Hipparion , or Hippotherium , typically from the Lower Pliocene of Europe, but also occurring in the corresponding formation in North Africa, Persia, India and China, and represented in the Upper Miocene Loup Fork beds of the United States by species which it has been proposed to separate generically as Neohipparion , we reach small horses which are now generally regarded as a lateral offshoot from the Merychippus type.

The cheek-teeth, which have crowns of moderate height, differ from those of all the foregoing in that the postero-internal pillar the projection on the right-hand top corner of c in fig. The skull, which is relatively short, has a large depression in front of the orbit, commonly supposed to have contained a gland, but this may be doubtful.

In the typical, and also in the North American forms these were complete, although small, lateral toes in both feet fig. If this be so, we have the development of a monodactyle foot in this genus independently of Equus. The foregoing genera constitute the subfamily Equinae , or the Equidae as restricted by the older writers. In all the dentition is of the hypsodont type, with the hollows of the cheek-teeth filled by cement, the premolars molariform, and the first small and generally deciduous.

The orbit is surrounded by a bony ring; the ulna and radius in the fore, and the tibia and fibula in the hind-limb are united, and the feet are of the types described above. Between this subfamily and the second subfamily, Hyracotheriinae , a partial connexion is formed by the North American Upper Miocene genera Desmatippus and Anchippus or Parahippus. The characteristics of the group will be gathered from the remarks on the leading genera; but it may be mentioned that the orbit is open behind, the cheek-teeth are short-crowned and without cement fig.

The longest-known genus and the one containing the largest species is Anchitherium , typically from the Middle Miocene of Europe, but also represented by one species from the Upper Miocene of North America. The European A. The cheek-teeth are of the type shown in a of figs. The summits of the incisors were infolded to a small extent. Nearly allied is the American Mesohippus , ranging from the Lower Miocene to the Lower Oligocene of the United States, of which the earliest species stood only about 18 in.

The incisors were scarcely, if at all, infolded, and there is a rudiment of the fifth metacarpal fig. By some writers all the species of Mesohippus are included in the genus Miohippus , but others consider that the two genera are distinct. Mesohippus and Miohippus are connected with the earliest and most primitive mammal which it is possible to include in the family Equidae by means of Epihippus of the Uinta or Upper Eocene of North America, and Pachynolophus , or Orohippus , of the Middle and Lower Eocene of both halves of the northern hemisphere.

The final stage, or rather the initial stage, in the series is presented by Hyracotherium Protorohippus , a mammal no larger than a fox, common to the Lower Eocene of Europe and North America. The general characteristics of this progenitor of the horses are those given above as distinctive of the group. The cheek-teeth are, however, much simpler than those of Anchitherium ; the transverse crests of the upper molars not being fully connected with the outer wall, while the premolars in the upper jaw are triangular, and thus unlike the molars.

The incisors are small and the canines scarcely enlarged; the latter having a gap on each side in the lower, but only one on their hinder aspect in the upper jaw. The fore-feet have four complete toes fig. The vertebrae are simpler in structure than in Equus. From Hyracotherium , which is closely related to the Eocene representatives of the ancestral stocks of the other three branches of the Perissodactyla, the transition is easy to Phenacodus , the representative of the common ancestor of all the Ungulata.

See also H. Such a feeling of security is necessary both for maintaining any posture, such as standing, or for performing any movement. If this feeling is absent or uncertain, or if there are contradictory sensations, then definite muscular movements are inefficiently or irregularly performed, and the body may stagger or fall. When we stand erect on a firm surface, like a floor, there is a feeling of resistance, due to nervous impulses reaching the brain from the soles of the feet and from the muscles of the limbs and trunk.

In walking or running, these feelings of resistance seem to precede and guide the muscular movements necessary for the next step. If these are absent or perverted or deficient, as is the case in the disease known as locomotor ataxia, then, although there is no loss of the power of voluntary movement, the patient staggers in walking, especially if he is not allowed to look at his feet, or if he is blind-folded. He misses the guiding sensations that come from the limbs; and with a feeling that he is walking on a soft substance, offering little or no resistance, he staggers, and his muscular movements become irregular.

Such a condition maybe artificially brought about by washing the soles of the feet with chloroform or ether. And it has been observed to exist partially after extensive destruction of the skin of the soles of the feet by burns or scalds. This shows that tactile impulses from the skin take a share in generating the guiding sensation. In the disease above mentioned, however, tactile impressions may be nearly normal, but the guiding sensation is weak and inefficient, owing to the absence of impulses from the muscles.

The disease is known to depend on morbid changes in the posterior columns of the spinal cord, by which impulses are not freely transmitted upwards to the brain. These facts point to the existence of impulses coming from the muscles and tendons. It is now known that there exist peculiar spindles, in muscle, and rosettes or coils or loops of nerve fibres in close proximity to tendons.

These are the end organs of the sense. The transmission of impulses gives rise to the muscular sense , and the guiding sensation which precedes co-ordinated muscular movements depends on these impulses. Thus from the limbs streams of nervous impulses pass to the sensorium from the skin and from muscles and tendons; these may or may not arouse consciousness, but they guide or evoke muscular movements of a co-ordinated character, more especially of the limbs.

In animals whose limbs are not adapted for delicate touch nor for the performance of complicated movements, such as some mammals and birds and fishes, the guiding sensations depend largely on the sense of vision. This sense in man, instead of assisting, sometimes disturbs the guiding sensation. It is true that in locomotor ataxia visual sensations may take the place of the tactile and muscular sensations that are inefficient, and the man can walk without staggering if he is allowed to look at the floor, and especially if he is guided by transverse straight lines.

On the other hand, the acrobat on the wire-rope dare not trust his visual sensations in the maintenance of his equilibrium. He keeps his eyes fixed on one point instead of allowing them to wander to objects below him, and his muscular movements are regulated by the impulses that come from the skin and muscles of his limbs.

The feeling of insecurity probably arises from a conception of height, and also from the knowledge that by no muscular movements can a man avoid a catastrophe if he should fall. A bird, on the other hand, depends largely on visual impressions, and it knows by experience that if launched into the air from a height it can fly. Here, probably, is an explanation of the large size of the eyes of birds. Cover the head, as in hooding a falcon, and the bird seems to be deprived of the power of voluntary movement. Little effect will be produced if we attempt to restrain the movements of a cat by covering its eyes.

A fish also is deprived of the power of motion if its eyes are covered. But both in the bird and in the fish tactile and muscular impressions, especially the latter, come into play in the mechanism of equilibrium. In flight the large-winged birds, especially in soaring, can feel the most delicate wind-pressures, both as regards direction and force, and they adapt the position of their body so as to catch the pressure at the most efficient angle.

The same is true of the fish, especially of the flat-fishes. In mammals the sense of equilibrium depends, then, on streams of tactile, muscular and visual impressions pouring in on the sensorium, and calling forth appropriate muscular movements. It has also been suggested that impulses coming from the abdominal viscera may take part in the mechanism.