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The purpose of the present article is to obtain class number formulas for imaginary pure quartic number fields by means of class field theory and complex multiplication. This is a particular case of the quite general situation, which has been treated in the author’s thesis [8]. On the other side, this paper may be considered as a “quantitative “ supplement of Parry’s work [9] on the same subject, which is more of a “qualitative” character.

The term “simplest” field has been used to describe certain totally real, cyclic number fields of degrees 2, 3, 4, 5, 6, and 8. For each of these degrees, the fields are defined by a one-parameter family of polynomials with constant term ±1. The regulator of these “simplest” fields is small in an asymptotic sense: in consequence, the class number of these fields tends to be large.

The authors present a simple approximate analytical expression for the energy eigenvalues E(d) of the pure quartic oscillation in d

The Riemann-Roch problem for divisors on a smooth surface in ℙ3 is studied. This problem is solved for some smooth quartic surfaces which are called Mori quartics; as a consequence the Hilbert function of any integral curve on a Mori quartic is determined.

The two-loop β-functions for the scalar quartic couplings are computed in a general renormalizable quantum field theory with scalar, spin - 1 2 , and (vector) gauge fields associated with a general gauge group G, using dimensional regularization and modified minimal subtraction (−MS). A more explicit form is given for the two-loop β-function of the quartic coupling of the Higgs doublet in the minimal QCD electroweak theory based on SU(3) × SU(2) × U(1).

The expansion of built-up area in western Kobe is quantitatively analysed and mapped through trend surfaces. Two trend surface techniques, 3-variable and 4-variable trend surfaces, are fitted to the distribution of gross floor index calculated by census tract-wise.First, by using 3-variable trend surface analysis, six cross-section surfaces are mapped at five years intervals from 1950 to 1975 (Fig. 2 a-f). Between 1955 and 1960, we find remarkable expansion of the built-up area into the southwestern suburbs, which is more adequately presented through quartic surface. From 1960, therefore, we change the degree of trend from cubic to quartic. Each map of isopleth lines lucidly explains the regional trend of expansion.Secondly, 4-variable trend surface analysis, which unites time and space coordinates, is applied in order to consider the spatial-temporal process of expansion. Time slices over the study periods present almost same results as in case of 3-variable trend surface fitting (Fig. 4 a-f).Regional trends are largely composed of the following three elements; the westward principal expansion from the high density core area which involves maximum in isopleth pattern, the northern mountain parts which shape minimum and work as hindrance to the advance of built-up area, and the recent construction of large-scale public housing estates which are found in southwestern parts and form newly emerged maximum.Residuals found on the 4-variable trend surface are carefully considered from both spatial and temporal series. Various factors affect the functional characteristics of dwelling agglomeration in each census tract. The following factors, which are considered as significant to interpret the deviation from general trend patterns, are pointed out.(1) Relative rapidity and delay in expansion of built-up area with comparison to the neighbouring tracts, which particularly explain residual occurences in temporal series.(2) The differences in land use and residential use. Especially the high ratio of industrial area is a factor for negative residuals, and the presense of commercial area is related to positive residuals. As for residential use, high ratios of row houses and dwelling houses combined with other uses, distribution of which coincides with densely inhabited area, account for positive residuals.(3) Moutains and relief energy as physical barriers against the expansion. This factor, however, gradually loses its hidrance effects through the progress of civil engineering technology.(4) The development of large-scale public housing estates, which brings about the drastic changes in residual values and keep the values high and stable hereafter.Among above factors, factor (2) is considered as most important. The effects of factor (2) on residual distribution are confirmed through principal components regression analysis (Tab. 1. and 2).Two suggestive conclusions are obtained from the present article.(1) We can safely say that the temporal change such as the expansion of built-up area may well be examined in a spatial-temporal system.(2) Spatial autocorrelation problem has been curiously considered to be removed as a noise. However, spatial autocorrelation in residual distribution patterns should be positively incorporated into spatial models, for it seems to have possibility to offer new angle to interpret spatial patterns. It will open the way to overcome the limitation of trend surface analysis as a simple procedure for areal smoothing and to construct the more strict spatial model which would lead to further refining of spatial forecasting.

In this paper we prove that a field Q ( 3 d ) Q{(^3}\sqrt d ) is euclidean with respect to the ordinary norm if and only if d = 2 , 3 d = 2,3 or 10. We also prove that certain fields of the form Q ( 4 − d ) , d > 0 Q{(^4}\sqrt { - d} ),d > 0 , are or are not euclidean.

This paper develops and explains the construction of a piecewise quartic space curve that interpolates positional, tangent, and curvature data. The construction is local and explicit; that is, it does not involve the solution of equations. If only positional data are known, then the tangent and curvature data can he derived by simple local default rules. Another option is to reduce the degree of the curve or minimize the a-norm of its derivative by solving a diagonally dominant, banded, linear system.

For integers r, s, t, u define the recursion A ( n + 4 ) = r A ( n + 3 ) − s A ( n + 2 ) + t A ( n + 1 ) − u A ( n ) A(n + 4) = rA(n + 3) - sA(n + 2) + tA(n + 1) - uA(n) where the initial conditions are set up in such a way that A ( n ) = α n + β n + γ n + δ n A(n) = {\alpha ^n} + {\beta ^n} + {\gamma ^n} + {\delta ^n} where α , β , γ , δ \alpha ,\beta ,\gamma ,\delta are the roots of the associated polynomial f ( x ) = x 4 − r x 3 + s x 2 − t x + u f(x) = {x^4} - r{x^3} + s{x^2} - tx + u In this paper a detailed deterministic procedure using the A ( n ) A(n) for finding how f ( x ) f(x) splits modulo a prime integer p is given. This gives for p not dividing the discriminant of f ( x ) f(x) the splitting of p in the field obtained by adjoining a root of f ( x ) f(x) to the rational numbers. There is an interesting connection between the results here for reciprocal polynomials and some work of D. Shanks.

This brief paper derives via quartic spline function a new consistency recurrence relation connecting the quartic spline function values at equidistant knots and the corresponding values of the second derivatives. It is shown how this consistency relation may be used in an algorithm for computing quartic spline approximations to the solution and its higher derivatives for a linear two-point boundary value problem. Some numerical evidence is also included to demonstrate the practical usefulness of the algorithm.

This paper gives a computation of the irregularity of the Fano surface of lines on the double cover branched in a quartic. A tangent bundle theorem is proved for?, from which it follows that determines uniquely. It is shown that the Abel-Jacobi map is an isogeny. Bibliography: 8 titles.

1. The equation of the scroll R of tangents of the curve Γ of intersection of two quadrics in general position in [3] was found by Cayley in 1850 ((l), p. 50). Cayley also considered special positions of the quadrics but we shall not be concerned with these. was reproduced in Salmon's treatise ((8), p. 191) and Salmon, on this same page, provides an equation in covariant form and so applicable to pairs of quadrics whether in canonical form, or in general position, or not.

Some of Watson's quartic centrifugal distortion parameters have the undesirable property that they become infinite when two rotational constants are equal. A proposal is made for a redefinition of the parameters in such a way that they stay well defined in the symmetric top limit.

SynopsisThe group of a harmonic elliptic quartic has exceptional action on the edges of the self-polar tetrahedron, singling out one pair of opposite edges. Geometrical explanations are given. One concerns the possession by each of these two edges of a certain harmonic tetrad constructible in three ways. The same constructions yield, for a general quartic, three distinct tetrads on each edge of its tetrahedron, with a related fourth. The cases of exceptional overlap of these tetrads are examined: they occur for quartics of moduli 0, ∞ and — 32/49.

For a classical system of interacting particles with on-site cubic or quartic anharmonicity explicit analytic solutions of the d'Alembert equation are obtained in the form of kinks in the presence of dissipation (viscous or Rayleigh) and a constant force. These kinks will be asymptotically stable in the case of quartic anharmonicity and unstable in the case cubic anharmonicity.

The singularities of the eigenvalues of the quartic anharmonic oscillator hamiltonian are studied numerically as a function of the quartic coupling parameter. Their location is found to be consistent with previous WKB calculations but the identification of which levels are singular is found to differ.

The paper seeks to elucidate the geometry of a simple engineering mechanism, comprising four bars smoothly jointed together to form a movable quadrilateral with one fixed side. The configurations of this mechanism correspond to the points of an elliptic curve, to which is associated interesting geometry and Morse theory. By appropriate projection, this curve yields the 2-parameter family of plane curves described by points rigidly attached to the side of the quadrilateral opposite the fixed side: the geometry of the general projection is related to the configuration of lines on a Segre quartic surface.

In this paper we study the intermediate Jacobian J3(X) of a double covering X of P3 branched at a smooth quartic which does not contain projective lines. We prove an analogue of the Riemann theorem for the Poincare divisor of the intermediate Jacobian J3(X), the global Torelli theorem for X, and the nonrationality of X. Bibliography: 13 titles.

We establish the dimension of the space of $C^1 $ bivariate piecewise quartic polynomials defined on a triangulation of a connected polygonal domain. Our approach is to construct a minimal determining set and an associated explicit basis for the space. For general triangulations, the minimal determining set must be defined globally.

Imaginary quadratic fields with class groups that have C(4) as a subgroup are analyzed in depth, and the units of associated dihedral quartic fields are thereby evaluated using Epstein zeta functions. Emphasis is placed on extreme examples such as Q(√−3502) which is probably the last case having an even discriminant and C(4) × C(4) as its class group. These extreme examples lead to very remarkable approximations and series for π such as π= 6 3502 log (2π)+7.37×10 −82 where u is the product of four, rather simple, quartic units. The approximations and series relate to Baker's theory of linear forms in logarithms and to certain modular identities. Related topics are discussed briefly.

SynopsisLet F(X, Y, Z) be a non-singular quadratic form with rational coefficients. The curve EF(x2, y2, z2) = 0 is of genus 3. A procedure is described for deciding whether there is an effective divisor on E of degree 3 defined over the rationals. There is such a divisor if and only if there is a point on E defined over some algebraic number field of odd degree. An example is constructed for which there is no such divisor although (i) there are points on E defined over all p-adic fields and over the reals and (ii) there are infinitely many rational points on each of the three curves F(X, y2, z2) = 0, F(x2, Y, z2) = 0 and F(x2, y2, Z) = 0.

The paper is concerned with a curve F, the complete intersection of a quadric with a quartic surface, that admits a group of self-projectivities isomorphic to the symmetric group of degree 5. Every generator of the quadric is, as shown at the end of the paper, cut by F equianharmonically. F has 80 stalls, points where its osculating plane is stationary; they are of two kinds, 60 to be labelled ∑, the other 20 Ω. F also has inflections at 24 points which compose a figure encountered on earlier occasions. A search is made for tritangent planes of F of which, when reckoned according to proper multiplicity, there must be 2048. Among them are 60 all of whose three contacts are ∑ while a further 120 each involve a single ∑ among their contacts and 420 each involve a single Ω.

Explicit conditions are given for a cyclic quartic field to have a relative integral basis over its unique quadratic subfield. Throughout this paper, K denotes a cyclic quartic extension of the rational number field Q. By Theorem 1 of [3] we know that K can be expressed uniquely in the form

We investigate the energy eigenvalue spectra in reggeon quantum mechanics when the hamiltonian contains (non-hermitian) cubic, symmetric quartic and asymmetric quartic interactions. We describe two new methods for finding eigenvalues numerically. When the asymmetric quartic coupling is zero, the energy eigenvalues cross the vacuum state sequentially as predicted by Bronzan as long as r ≲0.7. For r ⩾0.7 the energy eigenvalues above the ground state pinch together pairwise above the energy axis, leaving the ground state to oscillate about the vacuum. The addition of an asymmetric quartic term appears to dilute the effects produced by increasing the cubic coupling strength. Quantitative graphs of the functional dependence of the eigenvalues on the parameter are given. An alternative derivation of Bronzan's formula for vanishing eigenvalues is given in the appendix.

We study the curvature dependence of the quartic self-coupling constant in a λφ4 theory in the background of a static closed Einstein universe. Contrary to what happens in flat space-time at finite temperature, we find that X always increases with the curvature the symmetry being either broken or not at the classical level.

The effect of enlarging the class of admissible movable singularities in Painleve analysis to include all rational algebraic branch points is examined for a class of quartic polynomial potentials. Eight homogeneous quartic potentials are found in addition to the seven integrable cases given by weak Painleve analysis. They are examined using various numerical techniques and Ziglin’s theorem. Only one of them remains a candidate for integrability, which indicates that, in general, most rational algebraic branch points are incompatible with integrability. Movable logarithmic singularities also appear to be inconsistent with integrability. However, the remaining potentials are either regular or nearly regular for the energies examined, and are therefore still of interest for numerical purposes. The corresponding surfaces of section are found to be particularly simple in structure and belong to a small number of topologically distinct classes. Their stable and unstable periodic orbit structures are examined to provide information about their regularity and for use in Ziglin’s theorem. There appears to be a correlation between the resonances of the stable periodic orbits and the order of the corresponding movable singularities.

It is shown that the Henon-Heiles and two coupled quartic anharmonic oscillator systems possess non-trivial generalised Lie symmetries for specific sets of parametric values, for which second integrals of motion can also be constructed directly using Noether's theorem, thereby establishing their complete integrability.

A procedure is presented for solving quartic equations. It requires finding a single real root of an associated sixth‐order equation, but it can be shown that such roots must exist and simple bounds suitable for the application of the bisection method obtained.

The author studies birational automorphisms of a hypersurface of degree 4 in the four-dimensional projective space over an algebraically closed field having a unique ordinary double point of general type. It is shown that the group of birational automorphisms of a quartic is a semidirect product of a normal subgroup of finite index freely generated by 25 birational involutions and the (finite) group of biregular (projective) automorphisms of the quartic. Such a quartic is not rational.The proof is based on the techniques of maximal singularities due to V. A. Iskovskih and Yu. I. Manin.Bibliography: 7 titles.

We investigate three-dimensional hamiltonian systems with quartic potentials that are even in x, y, and z. We use the constraint that reductions to two dimensions should belong to the known integrale systems. In this way we obtain a classification that is complete to the extent that the two-dimensional results are complete. Two new integrable systems are obtained and their constants of motion explicitly given.

The congruence relation modulo 5 between the class numbers of the real and imaginary quartic subfields of the extension of a quadratic number field obtained by adjoining a fifth root of unity is studied. 1. Introduction. The aim of this note is to study the congruence relation modulo 5 between the class numbers of the real and imaginary quartic subfields of the extension of a quadratic number field obtained by adjoining a fifth root of unity. Our result gives different proofs of the main theorems of Parry (6, 7). Let m be a square free rational integer prime to 5, f a primitive fifth root of unity and Q the field of rational numbers. Let K = Q({m , f)> which is of degree 8 over Q and has the following subfields

Each covariant locus of a rational plane curve contributes its share toward the covariant and invariant tlleorST of rational curves of higher order by reason of the fact that osculantst of rational curves play the same role in regard to rational curves as polars do in regard to binary forms. Especially is this true of covariant points and covariant lines of the rational curve of order n (whicll we shall call Rn) because corresponding to each covariant line or point of all Rn is a series of covariant rational curves of any Rk, where D > n; the parametric equations of these covariant rational curves are easy to obtain. In tlle first section of this paper a new kind of covariant loclls is defined for Rn and illustrated by important loci associated witll tlle R4. Section II is devoted to combinant curves of the R5. The theory of combinant curves of the R5 in the plane corresponds to the invariant theory of the R6 in space; tllis correspondence is pointed out and illustrated. Finally a sufficient number of combinants and combillative sl-iCS of two line sections of tlle plane R) have been foulld to form the algebraic basis of an analytic treatment of combinant curves of the Ro in the plane, which is shown to be identical with the illvariant theory of the Ro in space.

A tetrad of Möbius tetrahedra consists of a set of 4 mutually inscribed and therefore circumscribed tetrahedra whose 16 vertices and 16 faces form a Kummer's 166 configuration (5; 11; 12; 21). As pointed out by the refere, fundamental to all work on the 166 figure are the 10 quadrics, called fundamental for the associated Kummer's quartic surface (13). To every quadric F correspond a matrix scheme of the 16 points or planes, arranged in 4 rows or columns, such that the 8 Rosenhain tetrahedra (7) formed of the rows and columns are all self-polar for F. The rows form one and the columns another tetrad of Mobius tetrahedra. Nine new schemes can be derived from one such scheme to make the total ten as explained by Baker (3, p. 133) leading to 80 Rosenhain tetrahedra in all. The 16 nodes (5; 8) or tropes of the Rummer's quartic are the 16 common elements of the 10 schemes such that the nodes and tropes are poles and polars for any one of the 10 quadrics. Each trope touches the quartic along a singular conic through the 6 points of the figure lying therein. The lines tangent to the surface at its each node N generate a quadric cone which is enveloped by the 6 tropes through N (12).