ENCYCLOPAEDIA OF MATHEMATICS Volume 3 ENCYCLOPA EDIA OF MATHEMATICS Managing Editor M. Hazewinkel Scientific Board 1. F. Adamst, S. Albeverio, 1. B. Alblas, S. A. Amitsur, I. 1. Bakeiman, J. W. de Bakker, C. Bardos, H. Bart, H. Bass, A. Bensoussan, M. Bercovier, M. Berger, E. A. Bergshoeff, L. Berkovitz, E. Bertint, F. Beukers, A. Beutelspacher, K.-D. Bierstedt, H. P. Boas, J. Bochnak, H. J. M. Bos, B. L. J. Braaksma, T. P. Branson, D. S. Bridges, A. E. Brouwer, M. G. de Bruin, R. G. Burns, P. Cameron, H. Capel, P. Cartier, C. Cercignani, J. M. C. Clark, Ph. Clement, A. M. Cohen, J. W. Cohen, P. Conrad, H. S. M. Coxeter, R. F. Curtain, M. H. A. Davis, M. V. Dekster, C. Dellacherie, G. van Dijk, H. C. Doets, I. Dolgachev, A. Dress, J. J. Duistermaat, D. van Dulst, H. van Duyn, H. Dym, A. Dynin, M. L. Eaton, W. Eckhaus, 1. Eells, P. van Emde Boas, H. Engl, G. Eskin, G. Ewald, V. I. Fabrikant, A. Fasano, M. Fliess, R. M. Fossum, B. Fuchssteinert, G. B. M. van der Geer, R. D. Gill, V. V. Goldberg, J. de Graaf, J. Grasman, P. A. Griffith, A. W. Grootendorst, L. Gross, P. Gruber, E. 1. Hannan, K. P. Hart, G. Heckman, A. J. Hermans, W. H. Hesselink, C. C. Heyde, K. Hirscht, M. W. Hirsch, K. H. Hofmann, A. T. de Hoop, P. J. van der Houwen, N. M. Hugenholtz, C. B. Huijsmans, 1. R. Isbell, A. Isidori, E. M. de Jager, D. Johnson, P. T. Johnstone, D. Jungnickel, M. A. Kaashoek, V. Kac, W. L. 1. van der Kallen, D. Kanevsky, Y. Kannai, H. Kaul, M. S. Keane, E. A. de Kerf, W. Klingen berg, T. Kloek, J. A. C. Kolk, G. Komen, T. H. Koornwinder, L. Krop, B. Kupershmidt, H. A. Lauwerier, J. van Leeuwen, J. Lennox, H. W. Lenstra Jr., J. K. Lenstra, H. Lenz, M. Levi, J. Lindenstrauss, J. H. van Lint, F. Linton, A. Liulevicius, M. Livshits, W. A. J. Luxemburg, R. M. M. Mattheij, L. G. T. Meertens, P. Mekenkamp, A.R. Meyer, J. van Mill, I. Moerdijk, J. P. Murre, H. Neunzert, G. Y. Nieuwland, G. J. Olsder, B. 0rsted, F. van Oystaeyen, B. Pareigis, K. R. Parthasarathy, I. 1. Piatetskii-Shapiro, H. G. J. Pijls, N. U. Prabhu, G. B. Preston, E. Primrose, A. Ramm, C. M. Ringel, 1. B. T. M. Roerdink, K. W. Roggenkamp, G. Rozenberg, W. Rudin, S. N. M. Ruysenaars, A. Salam, A. Salomaa, 1. P. M. Schalkwijk, C. L. Scheffer, R. Schneider, 1. A. Schouten, A. Schrijver, F. Schurer, I. A. Segal, 1. J. Seidel, A. Shenitzer, V. Snaith, T. A. Springer, 1. H. M. Steenbrink, J. D. Stegeman, F. W. Steutel, P. Stevenhagen, I. Stewart, R. Stong, L. Streit, K. Stromberg, L. G. Suttorp, D. Tabak, F. Takens, R. 1. Takens, N. M. Temme, S. H. Tijs, B. Trakhtenbrot, L. N. Vaserstein, M. L. 1. van de Vel, F. D. Veldkamp, P. M. B. Vitanyi, N. J. Vlaar, H. A. van der Vorst, J. de Vries, F. Waldhausen, B. Wegner, J. J. O. O. Wiegerinck, J. Wiegold, J. C. Willems, J. M. Wills, B. de Wit, S. A. Wouthuysen, S. YuzvinskiY, L. Za1cman, S. I. Zukhovitzkii ENCYCLOPAEDIA OF MATHEMATICS Volume 3 Heaps and Semi-Heaps - Moments, Method of (in Probability Theory) An updated and annotated translation of the Soviet 'Mathematical Encyclopaedia' Springer Science+Business Media, B.V. 1995 This International Edition in 6 volumes is an unabridged reprint of the originall0-volume hardbound library edition. (ISBN originallO-volume set: 1-55608-101-7) ISBN 978-0-7923-2975-6 ISBN 978-1-4899-3793-3 (eBook) DOI 10.1007/978-1-4899-3793-3 All Rights Reserved © 1995 by Springer Science+ Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint of the hardcover 1st edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner _ _ H _ _ lIEAPs AN» SEMI-IIEAPS - Algebras with one ter groups with involution. fi an involution fJ, which is an nary operation that satisfies certain identities. Heaps anti-automorphism, is defined on a semi-group S, then are defined by the identities the ternary operation [818283]=81fJ(.f2)s3 converts S = into a semi-heap. Any semi-heap is isomorphic to a [[XtX2X3]x4XS] [XtX2[X3X4XS)), sub-semi-heap of a semi-group with involution, [4]. [XtXtX2] = X2, [XtX2X2] = x .. References while semi-heaps are defined by the identities (1) PIlCPl!R, H.: '1bcorie dcr Abelschen Gruppen', Math. Z. 10 = = (1924), 16S-187. [[XtX2X3]x4XS] [Xt[X4X3X2]xS] [XtX2[X3X4XS)). (2) BAn, R.: 'Zur EiDfOhrung des Scharbegriffs', J. ReîM Angew. A1l heaps are also semi-heaps. Math. 160 (1929), 199-207. fi, in the set ~Â, B) of a1l one-to-one mappings of a (3) CmtTAINB, J.: '1be tcmary operation (abc)=ab-1c of a group', BuJL Amer. Math. Soc. 49 (1943),869-877. set  into a set B, one defines the ternary operation (4) VAGNER, V.V.: '1be theOIy of gcneralized heaps and of gen that puts an ordered triplet of mappings 4»1,4>2,4>3 into eraIized groups', Mat. Sb. 32, DO. 3 (1953), 54S-632 (in Rus correspondence with the mapping that is the composite sian). of 4»1,4»21,4>3, then ~Â, B) is a heap. Any heap is iso [S) VAGNER, V.V.: 'Poundations of differential geometry and con temporary algebra', in Proc. 4-th AU-Union. Mat. Conţ, VoI. 1, morphic to some heap of one-to-one mappings. fi a ter LaUngrad, 1963, pp. 17-29 (in Russian). nary operation is introduced into an arbitrary group G [S) GLUSItlN, L.M.: ldeoJs of semi-groups, 1, Saratov, I96S, pp. by putting [glg2g3]=glgi1g3, a heap is obtained (the 179-197; 198-228 (in Russian). V.. N. Sw'_Il.Vl heap associated with the given group). The concept of a AMS 1980 Subject Classification: 20N10 heap was introduced in the study of the above ternary operation on an Abelian group [1]. Heaps have been HEAT EQUATION - ef. Thennal-conductaDce equa studied from their abstract aspect [2], [3]. In particular, tiOD. it was shown by R. Baer [2] that if an arbitrary given AMS 1980 Subject Classification: 35K05 element 80 is fixed in a heap S, then the operations defined by the equations 8182 =[818082], 8-1 =[80880] define a group structure on S in which 80 is the unit; HEAvv SPHERE, ME1HOD OF THE - A method the heap associated with this group coincides with the for solving a minimization problem for a differentiable initial heap, while the groups obtained from a heap by function on a Euclidean space Eri. The method is based fixing various elements of it are isomorphic. In other on considering the system of differential equations words, the variety of a1l heaps is equivalent to the variety of a1l groups. d2x +adx + fJX) 2 = 0, 1;;'0, (1) The set $(Â, B) of a1l binary relations (ef. Binary dt2 dt 1+ I (x) I relation) between the elements of two sets  and B is a which describes the movement of a material point over semi-heap with respect to the triple multiplication the surface y = f(x) in an attracting field directed in the [P1P2P3]=P1Pi1P3. The set of a1l invertible partial negative direction of the y-axis under the condition that mappings of  into B is also closed with respect to the the point may not leave the surface and that the attrac If I triple multiplication and is a generalized heap [4], i.e. a tion is proportional to the velocity; (x) is de gra semi-heap with the identities dient of f(x) at a point x and a;;'O is the attraction f = coefficient. Taking into account that I (x) 12 is small [xxx] x, in a neighbourhood of a stationary point, (1) is oft en = [XtXt[X2X2X3)) [X2X2[XtXtX3)), replaced by the system = [[XtX2X2]x3X3] [[XtX3X3]x2X2]· dd2t2x + addxi +f (x) = 0, t;;'O. (2) Generaliz.ed heaps find application in the foundations of differential geometry in the study of coordinate Subject to certain assumptions on f(x) and under the atlases [5]. Heaps are closely connected with semi- initial conditions 1 HEAV Y SPHERE, METHOD OF THE = dx(O) = always the same as that of the other handle-body and is x(O) Xo, dt Xl ca11ed the genus of the Heegaard decomposition. Two it can be shown that the corresponding solution x(t) of Heegaard decompositions of the same manifold M3 are (1) or (2), as t~oo, converges to some stationary point equivalent if the dividing surface (the common boun x. of fix). H !(x) is a convex function, then x. is a dary of the handle-bodies) of one of them can be car minimum point of it. Thus, the method of the heavy ried into that of the other by means of a certain sphere is a particu1ar case of the adju.tment methocl. homeomorphism of the manifold M3• For the numerical solution of (1), or (2), one may, use, References e.g., difference methods. In dependence on the choice [1] lIBEoAAltD, P.: 'Sur l'analyse situs', BulL Soc. Math. France 44 of this difference method, discrete analogues of the (1916), 161-242. Translation of thcsis (in Danisb, 1898). method of the heavy sphere are obtained, inc1uding S. V. Matveev those for functions depending strongly on a few vari AMS 1980 Subject Classification: 57N10 ables, the conjuga~gradient method, etc. (cf. MiDimi zation metbods for functioIw dependiDg strongIy on a IlEEGAAIID DIAGItAM - One of the most common few variables; Conjupte gradients, method of). The methods for representing a c10sed orientable dIree choice of the step of the difference method and of the dimensional manifold. A Heegaard diagram of genus n quantity a strongly influences the rate of convergence consists of two systems of simple c10sed curves on a of the method of the heavy sphere. Instead of (1), (2) c10sed orientable surface F of genus n. The curves of other first- or second-order systems may be used (cf. each system satisfy the following conditions: I) the [ID. In the problem of minimizing a function fix) number of curves in the system is n; 2) the curves of under the restrictions the system are disjoint; 3) after cutting the surface F by gj(X)<O, i=l, ... ,m; gj(X) =0, i=m+l, ... ,s, these curves, a connected surface must result (a sphere with 2n deleted open discs). Heegaard diagrams are the method of the heavy sphere is applied in combina c1ose1y connected with Heegaard decompositions (cf. tion with the method of penalty functions, Lagrange lIeegaard deoomposition): the curves of each system are functions, etc. (cf. [2], [3], Penalty t'unc:tioM, method of; a complete system of meridians (secants of the circles Lagnmge function). of the handles) of one handle-body of the decomposi References tion; the curves of the second system are a complete [1] BAXIIVALOV, N.S.: Numericol metlwds: ana/ysis, algebra, system of meridians of the other handle-body. Two ordi1llllY differentiol equations, Mir, 1977 (translated from the Russian). Heegaard diagrams are ca11ed equivalent if the [2] VA SJL'EV, F.P.: Numeric:ol mt!thods for soIving extremum prob Heegaard decompositions corresponding to them are lems, MOSQOW, 1980 (in Russian). equivalent. It is known, for example, that any two [3] Evrusm!NKo, YU.G.: Numeric:ol optimization techniques, Heegaard diagrams of a three-dimensional sphere are Optimization Software, 1985 (translated from the Russian). F.P. Vasil'ev equivalent if they have the same genus. The genus of a Heegaard diagram can always be increased by taking AMS 1980 Subject Classification: 65K10 instead of the surface F its connected sum with a two dimensional torus and by adding to the curves of the lIEEGAAIm DECOMPOSmON - A representation diagram the meridian and a longitude of this torus. of a c10sed three-dimension manifold as a union of This operation is ca11ed stabilization. Any two Heegaard two three-dimensional submanifolds with a common diagrams of the same manifold are stably equivalent, boundary, each of which is a handle-body (that is, a that is, become equivalent after applying to each of three-dimensional ball with several handles of index I). them several stabilization operations. It was defined by P. Heegaard [I] in 1898. Heegaard For references see Heegaard decomposition. decompositions are one of the most commonly used S. V. Matveev devices in the study of three-dimensional manifolds, although there are other More effective methods for AMS 1980 Subject Classification: 57N10 decomposing three-dimensional manifolds into simple pieces (connected sums, hierarchies). Every c10sed HEIGIff, IN DIOPHANTINE GEOMETRY - A cer three-dimensional manifold has a Heegaard decomposi tain numerical function on the set of solutions of a tion. For the handle-bodies of the decomposition one Diophantine equation (cf. Diopbantine equations). In may take, for example, a regular neighbourhood of the the simplest case of a solution in integers (Xl> ••• ,xn) one-dimensional skeleton of a certain triangulation of of a Diophantine equation, the height is a function of the manifold and the c10sure of its complement. The the solution, and equals max 11 Xi 11. It is encountered in genus (number of handles) of one handle-body is this form in Fermat's method of descent. Let X be a 2 HEIGIIT OF AN IDEAL projective algebraic variety defined over a global field [3] MUMFORD, D.: Abelian varieties, Oxford Univ. Press, 1970. Appendix in Russian translation: Yu.I. Manin; The K. The height is a c1ass of real-valued functions hL(P) Mordell-Weil theorem (in Russian). defined on the set X (K) of rational points P and [4] MANIN, Yu.I.: 'Height of theta points on an Abelian manifold, depending on a morpbism L: x~pn of the variety X their variants and applications', Izv. Akad. Nauk SSSR Sero into the projective space pn• Each function in tbis c1ass Mat. 28 (1964), 1363-1390 (in Russian). [5] MUMFORD, D.: 'A remark on Mordell's conjecture', Amer. J. is also called a height. From the point of view of Math. 87 (1965), 1007-1016. estimating the number of rational points there are no [6] N~RON, A.: 'Quasi-fonctions et hauteurs sur les varietes essential differences between the functions in tbis c1ass: abeliennes', Ann. of Math. (2) 82 (1965), 249-331. A.N. Parshin for any two functions h~ and h~ there exist constants c' >0 and c" >0, such that c' h~ :;;;;:;'h~ :;;;;:;'c" h~. Such func Editorial comments. The notion of height is a major tool tions are ca1led equivalent, and this equivalence is in arithmetic algebraic geometry. It plays an important role denoted (here) as ~. in Faltings' proof of the Tate conjecture on endomorphisms Fundamental properties 01 the height. The function of Abelian varieties over number fields, the Shafarevich con hL(P) is functorial with respect to P, i.e. for any mor jecture that there are only finitely many isomorphism classes pbism 1 : X ~ Y and morpbism L: Y ~pn, of Abelian varieties over a number field over K of given dimension g;;;..1 with good reduction outside a finite set of hf*L(P) ~ hL(j(P», PEX(K). places S of K, and the Morde" conjecture on the finiteness If the morpbisms L, L) and L2 are defined by inverti of the set of rational points X(K) of a smooth curve of genus ble sheaves 2, 2) and 2 2, and if 2=2) ®22, then g;;;..2 over a number field K. Heights also play an important hL~hLJhL,. The set of points PEX(K) of bounded role in Arakelov intersection theory, wh ich via moduli spaces of algebraic curves has also become important in height is finite in the followmg sense: If the basic field string theory in mathematical physics. K is an algebraic number field, the set is finite; if it is an algebraic function field with field of constants k, the References elements of X(K) depend on a finite number of param [A1] FALTINGS, G., ET AL.: Rational points, Vieweg, 1986. eters from the field k; in particular, K is finite if the AMS 1980 Subject Classification: 14K15 I' I field k is finite. Let v run through the set of all norms of K. One may then define the height of a point IlEIGHf, IN ELEMENTARY GEOMETRY - The (xo: ... : xn) of the projective space pn with coordi segment of the perpendicular dropped from a vertex of nates from K as a geometrie figure (a triangle, a pyramid, a cone) on its IIsup I Iv. (*) Xi base or on its extension, and also the length of this seg v I ment. This is weIl defined because of the product formula II I x Iv = 1, x EK. Let X be an arbitrary projective v variety over K and let L be a c10sed imbedding of X into the projective space; the height hL may then be The height of a trapezium, prism, cylinder, spherical obtained by transferring the function (*), using the segment, or a pyramid or a cone truncated parallel to imbedding, to the set X (K). Various projective imbed the base, is the distance between the lower and the dings, corresponding to the same sheaf 2, define upper base. The figure shows the heights h of a trian equivalent functions on X (K). A linear extension yields the desired function hL• The function hL is occasionally gle, of a trapezium and of a truncated cone. BSE-3 replaced by its logarithm - the so-called logarithmic AMS 1980 Subject Classification: 51 N1 O. 51 M25 height. The above estimates may sometimes follow from IlEIGHf OF AN IDEAL - The minimum of the exact equations [3], [4], [5]. There is a variant of the heights of the prime ideals containing the ideal. The height function - the Neron-Tate height - which is height ht(~) 01 a prime ideal ~ in a ring A is the largest defined on Abelian varieties and behaves as a functor number h (or 00 if such a number does not exist) such with respect to the morpbisms of Abelian varieties that there exists achain of different prime ideals preserving the zero point. For the local aspect see [6]. PoCPI C"'CPh=P, The local components of a height constructed there play the role of intersection indices in arithmetic. The co-height coht(~) 01 a prime ideal ~ is defmed as the largest h for wbich there exists achain of prime ideals References [1] WEIL, A.: 'Number-theory and algebraic geometry', in Proc. P = Po C PI c··· C Ph :;6A. Internat. Congress Mathematicians Cambridge, 1950, Amer. In other words, Math. Soc., 1952, pp. 90-100. [2] LANG, S.: Diophantine geometry, Interscience, 1962. ht(p) = dim(A ~), coht(p) = dim(A / p), 3 HEIGHT OF AN IDEAL where dim denotes the dimension of the corresponding representation of interaction, and that they are Krull ring. The height of a prime ideal is equal to the equivalent, is due to the fact that it is not A or '" by codimension of the variety defined by the ideal, while themselves but only the average value of the operators the co-height equals the dimension of this variety. The A in the state '" that must be invariant with respect to height and the co-height of a prime ideal are connected unitary transformations of the type (1) and, conse by the inequality quently, the average value should not depend on the selection of the representation. Differentiation of (1) ht(~)+coht(~) .so;; dimA, with respect to I yields an equation for the operators which becomes an equality if, for example, A is a loca1 AH(/) in the Heisenberg representation that eontains Coben - Macaulay ring. eomplete information on the variation of the state of The prime ideals of height zero are the minima] the quantum system with the time I: prime ideals. The existence of prime ideals of height . aAH(t) one in Noetherian integral domains is established by lh-a-t - = AsH-HAs, the principa/ ideal theorem: The height of a non-zero principal ideal is one (cf. Krull ring). A more general where the operators H and As do not usually eommute. result - Krull's theorem - interconnects the height Named after W. Heisenberg, who introduced it in with the number of generators of the ideal: In a Noeth 1925 in a matrix formulation of quantum mechanies. erlan ring the height of an ideal generated by r ele Y.D. Kukin ments is not larger than r, and conversely: A prime Editorial comments. ideal of height r is the smallest of all prime ideals con References taining some r elements. In particular, any ideal in a [A1] MEHRA, J. and RECHENBERG, H.: The historical developrnent Noetherian ring has finite height; this is not true of the o( quantum theory, 1-4, Springer, 1982. co-height [2]. AMS 1980 Subject Classification: 81 DOS, 81805, References 43A65 [I] KItULL, W.: Primidealketten in allgmreinen Ringbereichen, Berlin-Leipzig, 1928. [2] NAGATA, M.: LocoI rings, Interscience, 1962. HELICAL CALCULVS, screw calculus - A branch of [3] ZAIusKI, O. and SAMum., P.: Commutative algebra, 1, Springer, vector calcu1us dealing with operations on screws, which 1975. are ordered pairs of eollinear vectors (r,.-o) having their [4] SJ!RRE, J.-P.: Aigebre locale. MuJtiplicites, Springer, 1965. orlgins at a eommon point. The vector r is called the Y.I. Danilov screw veclor, the axis defined by this vector is called the AMS 1980 Subject Classification: 16A33, 13C15 screw axis, .-0 is the screw momenl, while the number p in the equation .-0 =pr is said to be the screw parameier. HEINE - BOREL THEOREM on open coverings - Helical calculus deals with the operations of addition Cf. Borel-Lebesgue covering theorem. of screws, multiplication by a number, scalar and vec AMS 1980 Subject Classification: 54030 tor products, etc. In this eontext, the operations of heli cal calculus are reduced to operations on complex vec HEISENBERG REPRESENTATION - One of the tors of the form r+wtJ principal possible equivalent representations (together I I with the Schrödinger representation and the representa where w2 =0; the eomplex number r eWP is said to be tion of interaction, cf. Interaction, representation of) of the complex modulus of the screw; the number 0: + wo:o the dependence of the operators A and the wave func is said to be the complex angle between the screws (0: is tions i' on the time t in quantum mechanies and in the angle between the axes, while 0:0 is the distance quantum field theory. In the Heisenberg representation between them). All formulas of helical calculus are the operators AH depend on I, while the wave functions identical with the formulas of vector calculus if the "'H do not depend on I, and are connected with the modulus of the vector is replaced by the complex corresponding I-independent operators As and 1- modulus of the screw, and the ordinary angle between dependent wave functions "'s(t) in the Schrödinger straight lines is replaced by the eomplex angle. representation by a unitary transformation For instance, the scalar product of two screws is equal to the product of their complex moduli and the = = AH(/) eilH /hAse-itH /h; t/;H eilH /ht/;s(t), (1) eosine of the eomplex angle between them where the Hermitian operator H is the complete Hamil (cos(o:+ wo:o)= coso:-wo:o sino:); the screw product of tonian of the system, which is independent of time. two screws is a screw whose axis is perpendicular to the That it is possible to introduce the Heisenberg axes of the factors; a vector has the direction of the representation, the Schrödinger representation and the vector product of its factors, while the eomplex 4 HEUCOlD modulus is equal to the product of the complex moduli intersect the cylinder axis at a right angle. The length of the screws and the sine of the complex angle of a segment of a helical line between two of its succes between the axes of the factors sive points of intersection with the same generator is (sin( a + wao) = sin a + wao cos a). The correspondence called the turn of the helical line, while the length of the between the formulas of vector analysis and those of corresponding segment on the generator is said to be helical calculus, wbich involve complex scalar functions the pace of the helical line. The parametric equations of and screw functions of screw arguments, is established a conical helicalline are: in a similar manner. = = = x ceml cos t, y ceml sin t, Z ceml cotan a, Helical calculus is employed in mechanies, where where t is the angular parameter on the basic circle for arbitrary displacements of asolid body or an arbitrary the cone, a is the angle between the axis of the cone system of forces acting on a body may be expressed by and some generator, m = sin a / tan 4>, and 4> is the screws [4), and in geometry in the theory of ruled sur angle between the tangent to the helical line and the faces [3), [5). corresponding generator of the cone. The projection of Helical calculus was born in the early nineteenth cen the conical helical line parallel to the axis of the cone tury following the studies of 1. Poinsot, M. Chasles, A. onto the plane perpendicular to the axis of the cone is Möbius, and J. Plücker; the first major treatise was due a logarithmic spiral with pole in the projection of the to R. Ball [I). Helical calculus proper was developed by apex of the cone. The ratio between the curvature and A.P. Kotel'nikov [2). the torsion of a helical line remains constant at all References points. [1) BAll, R.: A treatise on the theory of screws, Dublin, 1876. One distinguishes between right-handed and left (2) KOTEL'NIKOV, A.P.: Screw calcuJus and some applications ofit, etc., Kazan, 1895 (in Russian). handed helical lines, i.e. as the value of the coordinate z (3) BLASCHKE, W.: Vorlesungen über Differentialgeometrie und increases, the rotation of the helical line around the geometrische Grundlagen von Einsteins Relativitätstheorie. Affine axis is clockwise or counter-clockwise. A generalized Differentialgeometrie, 2, Springer, 1923. (4) DIMENTBERG, F.M.: Screw calcuJus and its applications 10 helical line is a line on an arbitrary cylinder wbich mechanics, Moscow, 1965 (in Russian). intersects all generators of the cylinder at a constant (5) ZEiLIGER, D.N.: Complex line geometry, Leningrad-Moscow, angle. A helical line is a special case of a curve of con 1934 (in Russian). A.B.lvanov stant slope. AMS 1980 Subject Classification: 53A45 , 26812 References [1) BYUSHGENS, S.S.: Differential geometry, Moscow-Leningrad, 1940 (in Russian). HEUCAL UNE - A curve in space situated on the (2) BLASCHKE, W.: EinjUhrung in die Differentialgeometrie, surface of a circular cylinder (a cylindrical helical line; Springer, 1950. E. V. Shikin see Fig. 1) or a circular cone (a conical helicalline; see Editorial comments. A helical line is also called a helix. Fig. 2) wbich intersects all generators at the same References angle. [A1] Do CARMO, M.: Differential geornetry of curves and surfaces, Prentice Hall, 1976, p. 26. r z [A2) BERGER, M. and GoSTIAUX, B.: Differential geometry, Springer, 1988 (translated from the French). [A3] BLASCHKE, W. and LEICHTWEISS, K.: Einführung in die Dif ferentialgeometrie, Springer, 1973. AMS 1980 Subject Classification: 53A04 HEucolD - A ruled surface described by a straight y line that rotates at a constant angular rate around a Fig. 1. Fig.2. fixed axis, intersects the axis at a constant angle a, and at the same time becomes gradually displaced at a con The parametric equations of a cylindrical helical line stant rate along tbis axis. are = = = x a cos t, y a sin t, Z ht, where t is proportional to the arc length of the curve and a is the radius of the cylinder. The parallel projec tion of a cylindrical helical line onto a plane parallel to the generators of the cylinder is a sinusoid. The curva ture and the torsion of a cylindrical helicalline are con stant. The principal normals of a cylindrical helical line 5