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...8 Shortest Co-cycle Bases of Graphs BY HAN REN AND JING HAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 On Involute and Evolute Curves of Spacelike Curve with a Spacelike Principal Normal in Minkowski 3 -Space BY BAHADDIN BUKCU AND MURAT KEMAL KARACAN . . . . . . . . . . . . . . . 27 Notes on the Curves in Lorentzian Plane L2 BY S¨ UHA YILMAZ . ....

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...Math.Soc.13(1963), 743-768.International J.Math. Combin. Vol.1 (2009), 27-37 On Involute and Evolute Curves of Spacelike Curve with a Spaceli... ...can@usak.edu.tr,mkkaracan@yahoo.com Abstract In this study, we have generalized the involute and evolute curves of the spacelike curve α with a spac... ...ike principal normal in Minkowski 3-Space. Firstly, we have shown that, the length between the spacelike curves α and β is constant. Furthermore, the... ...ant. Furthermore, the Frenet frame of the involute curve β has been found as depend on curvatures of the curve α. We have determined the curve α is ... ...n orientation−→O if ω|→− (u) = 0. Otherwise, let ω|→− (u) = (ω1(u), ω2(u), · · · , ωn(u)). The point u is elliptic or O Ohyperbolic d... ... Ohyperbolic determined by the following inductive programming.STEP 1. If ω1(u) < 0, then u is elliptic; otherwise, hyperbolic if ω1(u) > 0... ...otherwise, hyperbolic if ω1(u) > 0;STEP 2. If ω1(u) = ω2(u) = · · · = ωi(u = 0, but ωi+1(u < 0 then u is elliptic; otherwise,hyperbolic if ωi+1(u) > ... ...0 then u is elliptic; otherwise,hyperbolic if ωi+1(u) > 0 for an integer i, 0 ≤ i ≤ n − 1. Denote these elliptic, Euclidean and hyperbolic point set... ... V eu = { u ∈ Rn | u an Euclidean point }, −→ V el = { v ∈ Rn | v an elliptic point }. −→ ...

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.... . . 15 The Forcing Weak Edge Detour Number of a Graph BY A.P.SANTHAKUMARAN AND S.ATHISAYANATHAN . . . . . . . . . . . . . 22 Special Smarandache Curves in the Euclidean Space BY AHMAD T. ALI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 The H-Line Signed Graph of a Signed Graph BY R.RANGARAJAN, M. S. SUBRAMANYA AND P. SIVA KOTA REDDY. . . .37...

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...ph, Ars Combin. (To appear)International J.Math. Combin. Vol.2 (2010), 30-36 Special Smarandache Curves in the Euclidean Space ... ... Email: atali71@yahoo.com Abstract: In this work, we introduce some special Smarandache curves in the Euclidean space. We study Fre... ...variants of a special case. Besides, we illustrate examples of our main results. Key Words: Smarandache Curves, Frenet-Serret Trihedra, Euclidean Sp... ...AMS(2000): 53A04§1. IntroductionIt is safe to report that the many important results in the theory of the curves in E3 wereinitiated by G. Monge; and ... ...dapted the geometrical models to relativistic motion of charged particles.Consequently, the theory of the curves has been one of the most fascinating ... ...ine passing through an exterior point, is kept orvalidated. While in the Riemannian geometry, called elliptic geometry, the fifth Euclideanpostul...

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...two-dimensional heat equation and delves deeply into how this equation can be solved across different geometric domains (square, rectangular, elliptical, and triangular). The solution methods for these geometric regions have been developed based on Green's functions and elliptic functions, and these novel approaches are compared to classical methods. The ...

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.......................................................................... 210 Analysis of Solution of the Two-Dimensional Heat Equation in Terms of Elliptic Functions Nurcan BAYKUŞ SAVAŞANERİL Chapter 10 ........................................................................... 248 Creep Behaviours of the Alabastrine Type Gypsums Under Uniaxial and Tria...

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...are, behavior of alabastrine-type gypsumrectangular, elliptical, and triangular). over time are exami... ... storage systems. The book offers newbased on Green's functions and elliptic insights into the behavior of gypsum infunctio... ... ANALYSIS OF SOLUTION OF THE TWO-DIMENSIONAL HEATEQUATION IN TERMS OF ELLIPTIC FUNCTIONS Nurcan BAYKUŞ SAVAŞANERİLCh... ... x2 y2complex physical processes. In thiscontext, the use of elliptic functions for Similarly, the Dirichlet problem for thesolvi... ...ecifically, Green's This paper presents a comprehensivefunctions and elliptic functions have analysis of various methods for solvingeme... ...ress conditions, creeprecovery develop within balance, curve’s linear section continues tosecondary creep stage (stead... ... tertiary creep stage (Hunsche, 1994;Therefore unit strain–time creep curve Özarslan et al. 2007).reaches near linear view. Whe... ...ndependent%, due to change in the range given = f ( ) curves are given in Table 3.above, gypsum samples tertiary creep ... ...MPa). In given stressTests stages, creep curve could not shift to the tertiary ...

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... clear which Riemann had in mind (see [3]), but today wewould call it either elliptic or spherical geometry. All of these geometries (Euclidean,hyperb... ...liptic or spherical geometry. All of these geometries (Euclidean,hyperbolic, elliptic, and spherical) are homogeneous and isotropic. This is to say th... ...ometry sometimes refers specifically to one of thegeometries just mentioned (elliptic or spherical), it is now most likely to be associatedwith a clas... ...,geometry is studied through curvature, and the basic Euclidean, hyperbolic, elliptic, andspherical geometries are particular constant curvature examp... ...s those possible universes that inhabitantsmight mistake as being Euclidean, elliptic, or hyperbolic. Great insight comes from therealization that the... ...at make congruentvertical angles with the edge.A geodesic in a manifold is a curve that is as straight as possible. Lines in an s-manifoldwill be the ... ...from thelines in the Euclidean plane. An s-line will be any piecewise linear curve that can beconstructed from a line segment lying within one of the ... ...manifold, and in extending indefinitely, we may betraversing the same closed curve an infinite number of times. In addition, we will see thats-lines m... ...d flattened the surface. In the papermodel shown in Figure 11, these s-lines curve, but only in a direction perpendicular to thesurface. In other word...

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... QUANTUM PARADOXES: Comparative Logic and Modern Quantum TheoryTable 1A. Elliptic Parameters of S'. {θ = arcsin(v)}S' a b ... ...distinguish between the trivial and fatal objection.3. GDT Relativity3.1 Elliptic and 4d SpaceWe begin the parametric definition of... ... 0 a1 θ a2 Figure 1. GDT Projective and Elliptic Geometry Theorems a.) Right conic projectio... ... determining S’ ellipse. b.) Cross-section of S’. c.) Geometric construction of elliptic major axis: (a1 < r < a2). ... ...e of theta. In Fig.1c the length of the major axis radius, {a1 ,a2} of the vectorequivalent elliptic bodies defined by v are determined from the Pytha... ... Graph 3. Correlation between Proton O/I Ratio and Mass-Defect Curve [22] Excess ... ... 3. we find the plotted noble proton O/I ratio has notable correspondence to themass defect curve―approximating its pattern of 'middleweight' nuclear ... ...'one big pile is better than two small piles'. At thenuclear scale however, the mass-defect curve indicates the relationship between matter particlevo... ...hotons can be connected to a given κ (space-time) quantum. Thetemperature proportional bell-curve of black-body radiation appears to correlate to this...

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... indeterminate (unclear, vague,ambiguous, unknown) data or curves. 1.a. Neutrosophic Bar Graph: 14 INTRODU... ...third component themaximum indeterminacy (i), and connect them. Weget a 3D curve. 4.b. Neutrosophic 3D Plot for ... ...TATISTICS For examples, the classical function 𝑎(𝑟, 𝑟) = 0represents a curve in the 2D-space, while theneutrosophic function 𝑎𝑁(𝑟, 𝑟) = 0 can ... ... is, instead of acurve in classical statistics, for example:a neutrosophic curve (we can call it a „thick curve”,or „strip curve”), like: ... ...osophic non-linear regressions of seconddegree one mentions the parabolic, elliptic, andhyperbolic regressions. 78 INTRODUC... ... neutrosophic (i.e. havingsome indeterminacy). Instead of one bell-shaped curve, we may havetwo or more bell-shaped curves that have commonand uncomm...

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.... . . . . . . . . . . . . . . . . . . . . . . . . 1045.3 Combinatorial construction of an algebraic curve of genus . . . . . . . . . ... ... of Klein surface is introduced by Alling and Greenleaf [2] in 1971concerned with real algebraic curves, correspondence with that of Riemann su... ...lgebraic curves, correspondence with that of Riemann surfaceconcerned with complex algebraic curves. For introducing this concept, it is need... ...umber of the disks around them. A vertexaround five, six or seven triangular disks is called an elliptic vertex, a Euclid vertexor a hype... ... a graph Γ representing a vertex u by apoint p(u), p(u) 6= p(v) if u 6= v and an edge (u, v) by a curve connecting the pointsp(u) and p(v) on the pl... ...o obtainhomeomorphisms of 2-cells {(x, y)|x2 + y2 < 1, x, y ∈ R} if we remove from S allthe curves used to partite S. There is a classical r... ...alency in {5, 6, 7} as follows: Classical Type: (1) Δ1 = {5 − regular triangular maps} (elliptic); (2) Δ2 = {6 − regular triangular m... ...olic). Smarandache Type: (4) Δ4 = {triangular maps with vertex valency 5 and 6} (euclid-elliptic); (5) Δ5 = {triangular maps with vertex... ... 3 1 χ(M) = ν(M) − ε(M) + φ(M) = ( k − 2)φ(M). If M is elliptic, then k = 5. Whence, χ(M) = φ(M) > 0. ...

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...factoring of numbers with 100digits or more within reasonable time: • the elliptic curve method (ECM) by Lenstra [187], with enhancements by Montgo... ... of numbers with 100digits or more within reasonable time: • the elliptic curve method (ECM) by Lenstra [187], with enhancements by Montgomery [M87... ...size to lie are given in the following table: log p I log n B1 curves time 20 40 1.5 . 10" 100 7 minutes 30 60... ...remaining composites were searched with ECM parameter B1=4OCXX) and200 curves were computed. Therefore, the probability of a remaining factor wit... ... REFERENCES [AM93] A.C.L.Atkin and F.Morain: Elliptic CUnJU and primality proving, Math. Compo 60 (1993) 399-40... ...son-Wesley, 1981 [I.S7] H.W.Lenstza, Jr.: Factoring integers lI1iih elliptic CUnJa, Annala of Mathematics (2) 1:l6 (1987), 649-673 ... ...ag, Berlin, 1993 [M87] Peter L. Montgomery: Speeding the Pollard and Elliptic Curve Methods of Factorizaiion, Math. Comp. 48 (1987), 24... ...n, 1993 [M87] Peter L. Montgomery: Speeding the Pollard and Elliptic Curve Methods of Factorizaiion, Math. Comp. 48 (1987), 243-264 ... ... (1987), 243-264 [M92] Peter L. MontgomeIy: An FFT Ezteruion of the Elliptic Curve Method of FactonzaiWn, Ph.D. dissertation, Mathemat...

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...reek mathematician and astronomer, the first to recognize geodesics on a curved surface as natural analogs of straight lines. Here, in this stud... ...eek mathematician and astronomer, the rst to recognize geodesics on a curved surface as natural analogs of straight lines. The well-known Men... ...ntary principles of geometry are the point, the line, the space, the curve, the plane, the geometrical figures (such as the segment, triangle,... ... time, and obtaining in this way continuous forms of curves, as trajectories of points or curves, in the plane (2D) or... ...tion and real, between circular and hyperbolic, between circular and elliptic, etc. Between the values of numerical eccentricity of s = 0 ... ...1980), 189–196. [4] Selariu, M.E., The definition of the elliptic eccentric with fixed eccenter, A V-a Conf. Nat. ... ... Constr. Masini, Timisoara, 1985, 175–182. [5] Selariu, M.E., Elliptic eccentrics with mobile eccenter, A V-a Conf. Nat. ... ...lly, with unique mathematical entities (line, circle, square,sphere, cube, elliptic integrals, etc). Through the mere expulsion of the pole P and c... ...ions On CIRCLE  Euler circular functions On ELIPSE  Jacobi elliptic functions On SQUARE (rotated with )  Alaci quadratic f...

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...randache ManifoldsClassical Type: (1) Δ1 = {5 − regular triangular maps} (elliptic); (2) Δ2 = {6 − regular triangular maps}(euclidean); (3) Δ3 = {7... ...ache Type: (4) Δ4 = {triangular maps with vertex valency 5 and 6} (euclid-elliptic); (5) Δ5 = {triangular maps with vertex valency 5 and 7} (ellipti... ...ientable or not. Definition 5.3. A point u in a map space (M, µ) is called elliptic, euclidean or hyperbolicif ρ(u)µ(u) < 2π, ρ(u)µ(u) = 2π or ρ(u)µ(u... ... > 2π. Definition 5.4. Let (M, µ) be a map space. An m-line in (M, µ) is a curve with a constantcurvature. Points in (M, µ) are called m-points. We h... ... k-polygon in a map space with each line segment passes throughat most one elliptic or hyperbolic point. If H is the set of elliptic points ... ...hen then the sum of its internal angles is equal to π; i (ii) if 4 is elliptic, then the sum of its internal angles is less than π; (iii) if 4 ... ... − (c) Riemannsurfaces?§6.2 Combinatorial Construction of an Algebraic Curve of Genus A complex plane algebraic curve Cl is a homogeneous equat... ...z with coefficients in C. Thedegree of f (x, y, z) is said the degree of the curve C . For a Riemann surface S, a well-known l... ...holomorphic mapping ϕ : S → P2C such that ϕ(S) is a complex planealgebraic curve and g(S) = (d(ϕ(S)) − 1)(d(ϕ(S)) − 2) . ...

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...factors that have been found by me and others, have beenfound by using the elliptic curve method (ECM) [Le], some have been foundusing the Multiple Po... ...hat have been found by me and others, have beenfound by using the elliptic curve method (ECM) [Le], some have been foundusing the Multiple Polynomial ... ...ors and remaining cofactors have been proven prime or compositeby means of Elliptic Curve Primality Proving (ECPP) [At], or the Adleman-Pomerance-Rume... ...emaining cofactors have been proven prime or compositeby means of Elliptic Curve Primality Proving (ECPP) [At], or the Adleman-Pomerance-Rumely test [... ...d this to 25 digits. Some factors have alreadybeen tried to 25 digits (300 curves with Bl = 50000). For more detail onthe progress check the follow... ...e Number, Ann. Math. 117, 173-206 (1983)[At] A.O. Atkin and F. Morain: Elliptic curves and primality proving, Math. Compo 60, 399-405 (1993)[Co... ... Ann. Math. 117, 173-206 (1983)[At] A.O. Atkin and F. Morain: Elliptic curves and primality proving, Math. Compo 60, 399-405 (1993)[Co] H. Coh... .../rkmath.rikkyo.ac.jp/ubibm/[Le] H.W. Lenstra, Jr: Factoring integers with elliptic curves, Annals of Mathematics (2) 126, 649-673 (1987)[Mo] F. M... ...c curves, Annals of Mathematics (2) 126, 649-673 (1987)[Mo] F. Morain: Elliptic Curve Primaiity Proving, http://pauillac.inria.fr/algo/morain/l...

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...act: A paradoxist Smarandache geometry combines Euclidean, hyperbolic, and elliptic geometry intoone space along with other non-Euclidean behaviors of... ...re for hyperbolic, zero curvature for Euclidean, andpositive curvature for elliptic. In general. the curvature may vmy within a particular Riemannian ... ...an manifold,so it is possible that the geodesics, the straightest possible curves, will behave like the lines of Euclideangeometry in one region and l... ...nes of Euclideangeometry in one region and like the lines of hyperbolic or elliptic geometry in another. We would expect,therefore, to find geometries... ...cewise linearmanifolds topologically, and they have geodesics that exhibit elliptic, hyperbolic. and Euclidean behaviorsimilar to those in Riemannian ... ...and vertices. Lines (geodesics) in the manifold are those piecewise linear curves with the followingproperties. They are straight in the Euclidean sen... ... 5There are five equilateral triangles around an elliptic vertex in an S-manifold. We can take a region aroundan elliptic ve... ...nding of the relatimsbip ~ stated thequestim wether space is curved or not. Gauss never published his revolutiooary ideas OIl noo-Euclid... ...rential geometry. For most purposes, differential geometry is the study of curved surfaces by wayof ideas from calculus. Geometries had thus pia)'ed a...

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...t two different values of N. The interpolating function for that family of curves is given by: a(N ) · e−b(N) L·where a(n) and b(n)... ...arger we choose d the larger will be the standarddeviation, i.e. the trend curve in figure 6 will be flattened and a very large number of tosseswill be ... ... For the same number of tosses theaverage is much closer to π with a trend curve that is closer to the average. Figure 6. ... ...s and algebraicequations. One example is shown in Fig.1, in where the left elliptic curve is transformed to theright circle by changing the metric alo... ...ebraicequations. One example is shown in Fig.1, in where the left elliptic curve is transformed to theright circle by changing the metric along x, y-a... ...ansformed to theright circle by changing the metric along x, y-axes and an elliptic equation x2 + y2 = 1 ... ... suitable metrics to simplify the equations of surfaces and curves inE3. Problem 3.2 Choose suitable metrics to simplify...

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...ss the sea, the lower part of theobject appears cut off by the interposing curved mass of water.The sagacity of Ptolemy enabled him to adduce another ... ...participates inthe horizontal motion, so that though it really describes a curve as anypasser-by would observe, yet it appears to the rider himself me... ...is to say, described uniformly inthose circles which were the only perfect curves.The simplest observations, however, show that the movements of thepl... ...nor even by combinations of circles, but that it could berepresented by an elliptic path. In this path the sun is situated at one ofthose two points i... ...s moved round in such a wayas to keep the string stretched, that beautiful curve known as theellipse is delineated, while the positions of the pins in... ...bservations would not have beensufficiently accurate to have exhibited the elliptic nature of a planetaryorbit which, like that of Venus, differed ver... ...sun withgreater angular velocity than at others. But it so happens that in elliptictracks which differ but little from circles, as is the case with al... ...ase of the comet, is so extremely elongated that the very small part ofthe elliptic path within which the comet is both near enough and brightenough t... ...and in consequence of thesemutual attractions the earth is swayed from the elliptic track which itwould otherwise pursue. In like manner Venus is also...

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...ometry of these paper models corresponds directly to the geometry of smoothly curvedsurfaces, SQ they can be u:sed as an introduction to a study of Ri... ...e paper). The cone and spherealso share a notion of positive curvature and an elliptic geometry. The cone formed by removing a wedge meBsllring f) rad... ...ean away from the cone point, the geodesiccurvature for thi.." circle 8..') a curve on the cone must be the same. That is, fig = ;. 'What is different... ...it should make senile to studentsthat the total rotation for (1 simple closed curve in the plane must always be 211". Since any smalldeformation of th... ...n the plane, it should abo make rscm;e that the totalrotation for a closed curve around the cone point will alwa.ys be 211" minus the angle defect.... ...iemannian (differential) geometry, the differences betweenEuclidean geometry, elliptic geometry, and hyperbolic geometry are understood interms of cur... ... the value of K. Figure 3. Bowl-shaped surfaces have positive Gauss curvature.Elliptic and nyperbolic qeometryThe Euclidean, hyperbolic, and elliptic ... ...t sphere has constant Gauss curvature K+1 (see Figure 4), and a model for the elliptic geometry axioms in Appendix A of[3] can be obtained by identify... ...ngle sum is 180°. It issmaller than this in hyperbolic geometry and larger in elliptic geometry. Inparticular for a triangle with area A and ...

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... in the usedapproximation is given by Jn/mn = (GMr1)1/2n.For elliptical orbits, this relation is given by [4] ... ...which they applied to description of major semi-axesof allowed (discretized) elliptical orbits of the bodies of the planetary system ... ... y, z) and E is the total energy. Negative Eclassically correspond to the elliptic Kepler orbits and the localization property (boundstate) is ... ...n earlystages of the origin of the planets, thus nowadays we already observe elliptical orbits, whichare very close to circular orbits. This procedur... ...from conservation laws 11 The integral in question is elliptic integral [2, 1], whose general form is ... ...ngent space of the ”world of classical worlds” (WCW) just like tangent for a curve gives the first approximation for the curve near a given point. ... ...lained by all known facts. It is our current understanding that only matter curves spacetime, this means that locally the curvature is quitewell defi... ... gravity is geometry. Minimality means thatfied to include galaxies’ rotation curve, and also it is actually the general covariance of General Relativi...

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...EORY of our solar system. - The exponents of the LAWS OF PLANETARY MOTION and the ELLIPTICAL ORBITS of the PLANETS. - The explorers of ... ...er COSMIC BODIES. This simply meant that our PLANETS were actually TRAPPED by the CURVEDSPACE that had been CREATED by the SUN. The CLOSER a PLANE... ... the CARTESIANCOORDINATES) to locate and describe any POINT, LINE or CURVE in hisCOORDINATE PLANE. These included the X-COORDINAT... ...e Y-COORDINATE (or VERTICAL AXIS). In DESCARTES’ ANALYTIC GEOMETRY, every LINE or CURVE represented agiven ALGEBRAIC EQUATION and conversely, ever... ...agiven ALGEBRAIC EQUATION and conversely, every EQUATION represented a givenLINE or CURVE. This meant that with the creation of ANALYTIC GE... ...contributions to the THEORY OF NUMBERS andto the branch of CALCULUS, which dealt with ELLIPTICAL INTEGRALS (ELLIPTICALFUNCTIONS). The study of ELL... ...s a German Mathematician who made fundamentalcontributions in the areas of ELLIPTICAL FUNCTIONS, DYNAMICS, DIFFERENTIALEQUATIONS and NUMBER... ...DRIEN LEGENDRE. JACOBI’S greatest accomplishment was his THEORY OF ELLIPTICALFUNCTIONS (pioneered by LEGENDRE), which he deve... ...nipulations of SIMULTANEOUS EQUATIONS. JACOBI was among the first to apply ELLIPTICAL FUNCTIONS to NUMBERTHEORY. ...

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...tilled water.SPADNS solution and Zirconyl acid solutions were mixed in equal volume.To prepare the calibration curve, 0.221 g of anhydrous sodium fluo... ...er sample was taken and repeated the step. The concentration of F-/l was calculatedafter using the calibration curve.3. Preliminaries: -Def.3.1 Fuzzy ... ...mean, variance, Shannon entropy, reliability measures. We also present the graphical representation of density curves, cumulative distribution functio... ... behavior is exponentially decreasing but starts from a specific point on the y-axis. For𝛼𝑁 > 1, the density curves showing unimodal behavior.Figure... ... NKw distribution is 𝐸𝑁(𝑋) = 1 − (1 − 𝑋𝛼𝑁)𝛼𝑁 (2)We plot CDF curves for some selected values of para... ...this geometry, for example, the sum of the three interior angles of a triangle is less than 180°). III. Elliptic geometry satisfies only Euclid... ...rface of a sphere. It is an example of non-Euclidean geometry. Spherical geometry is the simplest model of elliptical geometry, in which a lin... ...rceived as an attractive gravitational field. Smarandache (2021)said that Riemannian geometry, which is called elliptic geometry, is an antigeometry t...

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...n portion of the plane and threefor any body in space.' But evidently, the curves, surfaces and volumes may be socomplex as to make it necessary to di... ...", Dover Publications Inc., New York, 1970. [9] Theodore Andrea Cook, "The curves of life", Dover Publications Inc., New York, 1979. H. Scholfield,... ...re only five cases which all were confirmed to be primes using theelliptic curve prime factorization program, the largest being term 49: 1357911131517... ....This program was used to factorize the 18 first terms of the sequence. An elliptic curve factorizationprogram, ECMUB, conceived by Y. Kida was adapte... ...gram was used to factorize the 18 first terms of the sequence. An elliptic curve factorizationprogram, ECMUB, conceived by Y. Kida was adapted to gene... ... kDiagram 5. loglo m as a function of k. The upper curve corresponds to the computer count. ... ...e there are only fivecases which all were confirmed to be primes using the elliptic curve prime factorization program. Intable 7 # is the term number...

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.............................. 211 Developed (early 1800’S) his THEORY of ELLIPTICAL FUNCTIONS.KARL T. WEIERSTRASS (1815 - 1897)GERMAN MATHEMATICIAN... ... FRENCH MATHEMATICIAN .. 214 Solved EQUATIONS of the 5TH DEGREE using ELLIPTIC FUNCTIONS. XXXVIIGEORG CANTOR (1845 - 1918)... ... Proposed ( early 1600’s) his THREE LAWS of PLANETARY MOTION and his ELLIPTICAL PLANETARY ORBITS around the SUN, which revolutionized the W... .............................. 310 Developed (1952) a new PERIOD-LUMINOSITY CURVE Studied the ANDROMEDA GALAXY and concluded that it was farther a... ...LUID, LIGHT & DARK, TRANSPARENT & OPAQUE, SOLUBLE & INSOLUBLE, STRAIGHT & CURVED, ORGANIC & INORGANIC, LIVING & DEAD, MALE & FEMALE, GOOD & EVIL, ... ...BEAUTIFUL PLANET EARTH ORBITS OUR FANTASTIC SUNBECAUSE IT IS TRAPPED IN ITS CURVED SPACE/TIME FABRIC WITH AN INCREASING SPACE/TIME AND ENERGY/MATT... ... WITH EACH SUCCESSIVE ORBIT IT GETS CLOSER ANDGOES DEEPER INTO THE POWERFUL CURVED SPACE/TIME FABRIC OF OUR GLORIOUS SUN AND ITS PRODIGIOUS FORCE OF G... ...r solar system. - The exponents of the LAWS OF PLANETARY MOTION and the ELLIPTICAL ORBITS of the PLANETS. - The explorers of our SOLAR ... ...ons to the THEORY OF NUMBERS andto the branch of CALCULUS, which dealt with ELLIPTICAL INTEGRALS (ELLIPTICALFUNCTIONS). The study of ELLIPTICAL IN...