DOCSLIB.ORG

View Front and Back Matter from The

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
View Front and Back Matter from The TRANSACTIONS OF THE American Mathematical Society EDITED BY DAVID RAYMOND CUKTISS LUTHER PFAHLER EISENHART PERCEY F. SMITH WITH THE COOPERATION OF GEORGE D. BIRKHOFF ARTHUR B. COBLE WALL1E A. HURWITZ DUNHAM JACKSON EDWARD KASNER WILLIAM R. LONGLEY CHARLES N. HOORE ROBERT L. MOORE FOREST R. MOULTON ARTHUR RANDH FRANCIS R. 8HARFE ERNEST J. WILCZYNSKI VOLUME 20 1919 PUBLISHED BY THE SOCIETY Lancaster, Pa., awd New Tobk 1919 Reprinted with the permission of The American Mathematical Society Johnson Reprint Corporation 111 Fifth Avenue, New York 3, N. Y. Johnson Reprint Company Limited Berkeley Square House, London, W. 1 First Reprinting, 1963, Johnson Reprint Corporation TABLE OF CONTENTS VOLUME 20, 1919 PAGES Alexander, J. W., of New York, N. Y. Note on two three-dimensional manifolds with the same group.339 Blumberg, H., of Urbana, 111. On convex functions.40 Carmichael, R. D., of Urbana, 111. On a general class of integrals of the form <p{t)g{x+ t)dt.313 Jof Chittenden, E. W., of Iowa City, la. On the limit functions of sequences of continuous functions converging relatively uniformly .... 179 - (with A. D. Pitcher). On the theory of developments of an abstract class in relation to the calcul fonctionnel.213 Dines, L. L., of Saskatoon, Can. Projective transformations in func- tion space.45 Eisenhart, L. P., of Princeton, N. J. Transformations of surfaces applicable to a quadric.323 Glenn, O. E., of Philadelphia, Pa. Modular concomitant scales, with a fundamental system of formal covariants, modulo 3, of the binary quadratic.154 -On a new treatment of theorems of finiteness.203 Green, G. M., of Cambridge, Mass. Memoir on the general theory of surfaces and rectilinear congruences.79 Gronwall, T. H., of Washington, D. C. On the influence of keyways on the stress distribution in cylindrical shafts.234 Hopkins, J. W., of Cambridge, Mass. Some convergent developments associated with irregular boundary conditions.245 Larew, G. A., of Lynchburg, Va. Necessary conditions in the problems of Mayer in the calculus of variations.1 Manning, W. A., of Palo Alto, Cal. On the order of primitive groups (IV).66 Miller, G. A., of Urbana, 111. Groups possessing a small number of sets of conjugate operators.260 Moore, R. L., of Philadelphia, Pa. Concerning a set of postulates for plane analysis situs.169 Pell, A.. J., of Bryn Mawr, Pa. Linear equations with unsymmetric systems of coefficients.23 tv TABLE OF CONTENTS - A general system of linear equations.343 Pitcher, A. D., of Cleveland, 0. (with E. W. Chittenden). On the theory of developments of an abstract class in relation to the calcul fonctionnel.213 Shaepe, F. R., of Ithaca, N. Y. (with V. Snyder). Certain types of involutorial space transformations.185 Snyder, V., of Ithaca, N. Y. (with F. R. Sharpe). Certain types of involutorial space transformations.185 Van Vleck, E. B., of Madison, Wis. On the combination of non-loxo- dromic substitutions.299 Wilczynski, E. J., of Chicago, 111. Line-geometric representations for functions of a complex variable.271 Errata, volume 20.356 TRANSACTIONS OF THE American Mathematical Society Volumes 11-20 1910-1919 Indices CONTENTS PAGE Index by Authors. 5 Index by Subject Matter. 17 Arrangement of the Index by Subject Matter A. Logical Analysis of Mathematical Disciplines. 17 (Cf. F 1.) B. Algebra B 1. Rational Functions. Theory of Equations. Determinants. Sym- metric Functions. 17 B 2. Algebraic Forms. Invariants. (Cf. F 2.). 18 B 3. Linear Associative Algebra. Hypèrcomplex Number Systems. Fields. (Cf. F 1.). 18 B 4. Algebra of Logic. — C. Theory of Numbers. 19 (Cf. E 1, E 2.) D. Analysis D 1. Functions of Real Variables. Point Sets. 19 D 2. Functions of Complex Variables. 21 D 3. Particular Functions. 22 D4. Ordinary Differential Equations. (Cf. D 2, D 3, F 4.). 23 D 5. Partial Differential Equations. 24 D 6. Differential Forms. — D 7. Calculus of Variations. 24 D 8. Difference Equations. 24 D 9. Integral Equations. 25 E. Groups E 1. Discrete Groups in General. (Cf. E 2.). 25 E 2. Linear Groups in Arbitrary or Special Fields. 26 E 3. Continuous Groups. (Cf. E 2, F 1, F 3, G 3.). 26 F. Geometry F 1. Synthetic Geometry. Non-Euclidean Geometry. Geometry of n Dimensions. Analysis Situs. (Cf. D 1, D 2.). 26 3 4 CONTENTS PAGE F 2. Algebraic Geometry. Algebraic Curves and Surfaces. (Cf. B 2, D 3, E 2, F 1, F 4, F 5.). 27 F 3. Geometry of Special Space-Elements. 27 F 4. Differential Geometry. (Cf. D 6.). 27 F 5. Geometric Transformations. Geometry of Motion. (Cf. E 2.).. 29 G. Applied Mathematics G 1. General Mechanics. 30 G 2. Celestial Mechanics. 30 G 3. Elasticity, Hydromechanics, and Sound. 31 G 4. Electricity, Light, and Heat. — i INDEX BY AUTHORS Vol.* pages Alexander, J. W., II, A proof of the invariance of certain constants of analysis situs. 162 148-154 -On the factorization of Cremona plane transformations. 173 295-300 -Note on two three-dimensional manifolds with the same group. 204 339-342 Barrow, D. F., Oriented circles in space. 163 235-258 Bates, W. H., An application of symbolic methods to the treatment of mean curvatures in hyperspace. 12i 19-38 Beck, H. Ein Seitenstück zur Möbius'schen Geometrie der Kreisverwandtschaften. 114 414^148 Bernstein, B. A., A set of four independent postulates for boolean algebras. 17i 50- 52 BiANCHi, L., Concerning singular transformations Bk of surfaces applicable to quadrics. I83 379-401 Bill, E. G., An existence theorem for a problem of the calculus of variations in space. 13i 50- 58 BiRKHOFF, G. D., A simplified treatment of the regular singular' . point. 112 199-202 -General theory of linear difference equations. 122 243-284 -Proof of Poincaré's geometric theorem. 14i 14- 22 -On a simple type of irregular singular point. 144 462-476 -Infinite products of analytic matrices. 173 386-404 -Dynamical systems with two degrees of freedom. I82 199-300 Blichfeldt, H. F., Theorems on simple groups. Hi 1-14 -On the order of linear homogeneous groups (fourth paper) 12i 39- 42 -A new principle in the geometry of numbers, with some applications. 153 227-235 Bliss, G. A. (with M. Mason), Fields of extremals in space. 113 325-340 -A generalization of Weierstrass' preparation theorem for a power series in several variables. 132 133-145 -(with A. L. Underhill), The minimum of a definite integral for unilateral variations in space. 153 291-310 * The numbers in this column indicate for every paper the volume and part of volume of publication, and thus the date of publication, e. g., the first paper in the list was published in April, 1915. In this column references are made also to the Notes and errata published in the concluding numbers of various volumes, e. g., anote to the paper by B. H. Camp appears in the Notes and errata of volume 16. >> 6 INDEX BY AUTHORS [October Vol. Pages -The Weierstrass E-Function for problems of the calculus of variations in space. 154 369-378 -Jacobi's condition for problems in the calculus of variations in parametric form. 172 195-206 -The problem of Mayer with variable end points. 193 305-314 Blumberg, H., On the factorization of expressions of various types. 174 517-544 -On convex functions. 20i 40- 44 Bôcher, M., Applications and generalizations of the concep- tion of adjoint systems. 144 403-420 -Note "On supplementary to the paper certain pairs of transcendental functions whose roots separate each other" 184 519-521 Brink, R. W., A new integral test for the convergence and divergence of infinite series. 192 186-204 Buchanan, D., A new isosceles triangle solution of the three body problem. 163 259-274 Camp, B. H., Singular multiple integrals, with applications to series. 14x 42- 64 -Lebesgue integrals containing a parameter, with applica- fl5i 87-106 tions.*[l64 532 Car Michael, R. D., Linear difference equations and their analytic solutions. 12i 99-134 -On transcendentally transcendental functions. 143 311-319 -On a general class of series of the form Y,cng{x + n). 173 207-232 -On a general class of integrals of the form <p{t)g{x-\-t)dt204 313-322 JoJ Carpenter, A. F., Ruled surfaces whose flecnode curves have plane branches. 164 509-532 Chittenden, E. W., Relatively uniform convergence of se- quences of functions. 152 197-201 -On the equivalence of écart and voisinage. 182 161-166 -(with A. D. Pitcher), On the foundations of the calcul fonctionnel of Fréchet. 19i 66- 78 -On the limit functions of sequences of continuous func- tions converging relatively uniformly. 202 179-184 -(with A. D. Pitcher), On the theory of developments of an abstract class in relation to the calcul fonctionnel.203 213-233 Clements, G. R., Implicit functions defined by equations with vanishing jacobian. 143 325-342 Coble, A. B., An application of Moore's cross-ratio group to the solution of the sextic equation. 123 311-325 -An application of finite geometry to the characteristic theory of the odd and even theta functions. 142 241-276 -Points sets and allied Cremona groups. 162 155-198 1919] INDEX BY AUTHORS 7 Vol. Pages -Points sets and allied Cremona groups (Part II). 173 345-385 -Points sets and allied Cremona groups (Part III). 183 331-372 Cole, F. N., The triad systems of thirteen letters. 14i 1- 5 CooliDge, J. L., The metrical aspect of the line-sphere trans- formation . 12i 43-69 -A study of the circle cross. 142 149-174 -Congruences and complexes of circles. 15i 107-134 Conner, J. R., Multiple correspondences determined by the rational plane quintic curve.
Load more

Recommended publications
  • Arxiv:1406.1948V2 [Math.GM] 18 Dec 2014 Hr = ∆ Where H Ouino 1 Is (1) of Solution the 1
    Solution of Polynomial Equations with Nested Radicals Nikos Bagis Stenimahou 5 Edessa Pellas 58200, Greece [email protected] Abstract In this article we present solutions of arbitrary polynomial equations in nested periodic radicals Keywords: Nested Radicals; Quintic; Sextic; Septic; Equations; Polynomi- als; Solutions; Higher functions; 1 A general type of equations Consider the following general type of equations ax2µ + bxµ + c = xν . (1) Then (1) can be written in the form b 2 ∆2 a xµ + = xν , (2) 2a − 4a where ∆ = √b2 4ac. Hence − 2 µ b ∆ 1 ν x = − + 2 + x (3) 2a r4a a Set now √d x = x1/d, d Q 0 , then ∈ + −{ } 2 µ b ∆ 1 x = − + + xν (4) s 2a 4a2 a arXiv:1406.1948v2 [math.GM] 18 Dec 2014 r hence µ/ν 2 ν b ∆ 1 ν x = − + 2 + x (5) s 2a r4a a Theorem 1. The solution of (1) is µ v 2 µ/ν 2 2 u b v ∆ 1 b ∆ 1 µ/ν b ∆ x = u− + u + v− + v + − + + ... (6) u 2a u4a2 a u 2a u4a2 a s 2a 4a2 u u u u r u u u t t t t 1 Proof. Use relation (4) and repeat it infinite times. Corollary 1. The equation ax6 + bx5 + cx4 + dx3 + ex2 + fx + g = 0 (7) is always solvable with nested radicals. Proof. We know (see [1]) that all sextic equations, of the most general form (7) are equivalent by means of a Tchirnhausen transform y = kx4 + lx3 + mx2 + nx + s (8) to the form 6 2 y + e1y + f1y + g1 = 0 (9) But the last equation is of the form (1) with µ = 1 and ν = 6 and have solution 2 1/6 2 2 f v ∆ 1 f ∆ 1 1/6 f ∆ y = 1 + u 1 v 1 + 1 1 + 1 + ..
    [Show full text]
  • ALGEBRAIC GEOMETRY (1) Compute the Points of Intersection
    ALGEBRAIC GEOMETRY HOMEWORK 5 (1) Compute the points of intersection of the circle x2 +y2 +x−y = 0 with the lemniscate (x2 +y2)2 = x2 −y2 using resultants (can you compute the affine points of intersections directly?), and compute the multiplicities. Explain which coordinate system you choose, how the equations look like, what the resultant is, and what its factors are. (2) Here you will learn how to solve cubics using resultants. Assume you want to solve the cubic equation x3 + ax2 + bx + c = 0. (a) Show that the substitution y = x + a/3 transforms the cubic into y3 + By + C = 0. (b) Now assume that we have to solve x3 + bx + c = 0. Using linear re- lations y = rx ∗ b will not make the linear term disappear (without bringing back the quadratic term). Thus we try to put y = x2 + cx + d and then eliminate x using resultants: type in p = polresultant(x^3+b*x+c,y-x^2-d*x-e,x) polcoeff(p,2,y) polcoeff(p,1,y) (you can find out what polcoeff is doing by typing ?polcoeff). Now we know that we can keep the quadratic term out of the equation by choosing e = 2b/3. With this value of e go back and recompute the resultant; this will now be a cubic in y without quadratic term. The coefficient of the linear term can be made to disappear by solving a quadratic equation (which is called the resolvent of the original cubic). (c) Thus you can solve the cubic by solving the resolvent, using the trans- formations above to turn the cubic into a pure cubic y3 + c, which in turn is easy to solve.
    [Show full text]
  • On the Solution to Nonic Equations
    On the Solution to Nonic Equations By Dr. Raghavendra G. Kulkarni Abstract. This paper presents a novel decomposition tech- nique in which a given nonic equation is decomposed into quartic and quintic polynomials as factors, eventually lead- ing to its solution in radicals. The conditions to be satisfied by the roots and the coefficients of such a solvable nonic equation are derived. Introduction It is well known that the general polynomial equations of degree five and above cannot be solved in radicals, and have to be solved algebraically using symbolic coefficients, such as the use of Bring radicals for solving the general quintic equation, and the use of Kampe de Feriet functions to solve the general sextic equation [1, 2]. There is hardly any literature on the analytical solution to the polynomial equations beyond sixth degree. The solution to certain solvable octic equations is described in a recent paper [4]. This paper presents a method of determining all nine roots of a nonic equation (ninth-degree polynomial equation) in radicals. First, the given nonic equation is decomposed into two factors in a novel fashion. One of these factors is a fourth-degree polynomial, while the other one is a fifth-degree polynomial. When the fourth- degree polynomial factor is equated to zero, the four roots of the given nonic equation are extracted by solving the resultant quar- tic equation using the well-known Ferrari method or by a method given in a recent paper [3]. However, when the fifth-degree polyno- mial factor is equated to zero, the resultant quintic equation is also required to be solved in radicals, since the aim of this paper is to solve the nonic equation completely in radicals.
    [Show full text]
  • 04. Rukhin-Final5a-116-1.Qxd
    Volume 116, Number 1, January-February 2011 Journal of Research of the National Institute of Standards and Technology [J. Res. Natl. Inst. Stand. Technol. 116, 539-556 (2011)] Maximum Likelihood and Restricted Likelihood Solutions in Multiple-Method Studies Volume 116 Number 1 January-February 2011 Andrew L. Rukhin A formulation of the problem of considered to be known an upper bound combining data from several sources is on the between-method variance is National Institute of Standards discussed in terms of random effects obtained. The relationship between and Technology, models. The unknown measurement likelihood equations and moment-type Gaithersburg, MD 20899-8980 precision is assumed not to be the same equations is also discussed. for all methods. We investigate maximum likelihood solutions in this model. By representing the likelihood equations as Key words: DerSimonian-Laird simultaneous polynomial equations, estimator; Groebner basis; [email protected] the exact form of the Groebner basis for heteroscedasticity; interlaboratory studies; their stationary points is derived when iteration scheme; Mandel-Paule algorithm; there are two methods. A parametrization meta-analysis; parametrized solutions; of these solutions which allows their polynomial equations; random effects comparison is suggested. A numerical model. method for solving likelihood equations is outlined, and an alternative to the maximum likelihood method, the restricted Accepted: August 20, 2010 maximum likelihood, is studied. In the situation when methods variances are Available online: http://www.nist.gov/jres 1. Introduction: Meta Analysis and difficult when the unknown measurement precision Interlaboratory Studies varies among methods whose summary results may not seem to conform to the same measured property.
    [Show full text]
  • International Journal of Pure and Applied Mathematics ————————————————————————– Volume 18 No
    International Journal of Pure and Applied Mathematics ————————————————————————– Volume 18 No. 2 2005, 213-222 NEW WAY FOR A TWO-PARAMETER CANONICAL FORM OF SEXTIC EQUATIONS AND ITS SOLVABLE CASES Yoshihiro Mochimaru Department of International Development Engineering Graduate School of Science and Engineering Tokyo Institute of Technology 2-12-1, O-okayama, Meguru-ku, Tokyo 152-8550, JAPAN e-mail: [email protected] Abstract: A new way for a two-parameter canonical form of sextic equations is given, through a Tschirnhausian transform by multiple steps, supplemented with some solvable cases at most in terms of hypergeometric functions. AMS Subject Classification: 11D41 Key Words: sextic equation, Tschirnhausian transform 1. Introduction 6 6 6−n Actually the general sextic equation x + anx = 0 can be solved in nX=1 terms of Kamp´ede F´eriet functions, and it can be reduced to a three-parameter Received: October 21, 2004 c 2005, Academic Publications Ltd. 214 Y. Mochimaru 6 4 2 resolvent of the form y + ay + by + cy + c = 0 by Joubert [3]. Such a type of the sextic equation was treated by Felix Klein and a part was given in [4]. Other contributions to treatment on sextic equations were given e.g. by Cole [2] and Coble [1]. In this paper, a new way leading to a two-parameter canonical form of sextic equations is given. 2. Analysis 2.1. General Let the original given sextic equation be expressed as 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ ∗ x + a x + b x + c x + d x + e x + f = 0. (1) 3 Here the quantity 5a∗ − 18a∗b∗ + 27c∗ corresponding to the polynomial of the left hand side of equation (1) is invariant with respect to the parallel shift x → x+constant.
    [Show full text]
  • Analytic Solutions to Algebraic Equations
    Analytic Solutions to Algebraic Equations Mathematics Tomas Johansson LiTH{MAT{Ex{98{13 Supervisor: Bengt Ove Turesson Examiner: Bengt Ove Turesson LinkÄoping 1998-06-12 CONTENTS 1 Contents 1 Introduction 3 2 Quadratic, cubic, and quartic equations 5 2.1 History . 5 2.2 Solving third and fourth degree equations . 6 2.3 Tschirnhaus transformations . 9 3 Solvability of algebraic equations 13 3.1 History of the quintic . 13 3.2 Galois theory . 16 3.3 Solvable groups . 20 3.4 Solvability by radicals . 22 4 Elliptic functions 25 4.1 General theory . 25 4.2 The Weierstrass } function . 29 4.3 Solving the quintic . 32 4.4 Theta functions . 35 4.5 History of elliptic functions . 37 5 Hermite's and Gordan's solutions to the quintic 39 5.1 Hermite's solution to the quintic . 39 5.2 Gordan's solutions to the quintic . 40 6 Kiepert's algorithm for the quintic 43 6.1 An algorithm for solving the quintic . 43 6.2 Commentary about Kiepert's algorithm . 49 7 Conclusion 51 7.1 Conclusion . 51 A Groups 53 B Permutation groups 54 C Rings 54 D Polynomials 55 References 57 2 CONTENTS 3 1 Introduction In this report, we study how one solves general polynomial equations using only the coef- ¯cients of the polynomial. Almost everyone who has taken some course in algebra knows that Galois proved that it is impossible to solve algebraic equations of degree ¯ve or higher, using only radicals. What is not so generally known is that it is possible to solve such equa- tions in another way, namely using techniques from the theory of elliptic functions.
    [Show full text]
  • Closed-Form Solution of Polynomial Equations
    CLOSED-FORM SOLUTION OF POLYNOMIAL EQUATIONS ALEXANDER KHEYFITS Abstract. Complex analysis is used to find closed-form expres- sions of the roots of a univariate polynomial of any degree as inte- grals of elementary functions. 1. Introduction The roots of any univariate algebraic equation of degree four or less can be expressed explicitly through its coefficients as superpositions of four arithmetic operations and radicals, while for no n 5 there exists a general formula that represents all the n complex roots≥ of every n th degree polynomial through its coefficients by means of the arithmet− ic operations and radicals. The reader can consult, e.g., [4] or [7] for the history and the details of these results. The topic is of interest even today; see, e.g., a comprehensive current monograph of Teo Mora [9] and the very recent papers by Boyd [2] and Nash [10]. The Abel-Galois result did not, of course, stop scientists from search for other means to represent the roots of the quintics and more general equations. Already in 1858, Brioschi, Hermite, and Kronecker indepen- dently expressed the roots of a general quintic through elliptic modular functions; quarter century later Klein published his famous book [8]. Thus, in solving the quintics, mathematicians have followed the stan- arXiv:1804.00698v2 [math.CV] 29 Apr 2018 dard path of extending the existing toolbox. For instance, the equation x2 = 1 cannot be solved in real numbers, but it can be done if we extend− the available numbers to the imaginary numbers; to study the ordinary differential equation x2y′′ + xy′ +(x2 ν2)y = 0, which can- not be solved in elementary functions, the special− (Bessel) functions were introduced and included in the set of possible solutions.
    [Show full text]
  • A Combined Methodology for the Approximate Estimation of the Roots of the General Sextic Polynomial Equation
    A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation Giorgos P. Kouropoulos ( [email protected] ) Technological Educational Institute of Athens https://orcid.org/0000-0003-2033-0034 Method Article Keywords: sextic polynomial, general sextic, hexic polynomial, radical solutions, estimation of roots, approximate estimation, Galois theory, regression analysis, statistics Posted Date: September 8th, 2021 DOI: https://doi.org/10.21203/rs.3.rs-882192/v2 License: This work is licensed under a Creative Commons Attribution 4.0 International License. Read Full License A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation Giorgos P. Kouropoulos BSc in Energy Technology, Technological Educational Institute of Athens E-mail: [email protected] ORCID id: 0000-0003-2033-0034 Abstract In this article we present the methodology, according to which it is possible to derive approximate solutions for the roots of the general sextic polynomial equation as well as some other forms of sextic polynomial equations that normally cannot be solved by radicals; the approximate roots can be expressed in terms of polynomial coefficients. This methodology is a combination of two methods. The first part of the procedure pertains to the reduction of a general sextic equation H(x) to a depressed equation G(y), followed by the determination of solutions by radicals of G(y) which does not include a quintic term, provided that the fixed term of the equation depends on its other coefficients. The second method is a continuation of the first and pertains to the numerical correlation of the roots and the fixed term of a given sextic polynomial P(x) with the radicals and the fixed term of the sextic polynomial Q(x), where the two polynomials P(x) and Q(x) have the same coefficients except for the fixed term which might be different.
    [Show full text]
  • Solution of Polynomial Equations with Nested Radicals
    Solution of Polynomial Equations with Nested Radicals Nikos Bagis Stenimahou 5 Edessa Pellas 58200, Greece [email protected] Abstract In this article we present solutions of certain polynomial equations in periodic nested radicals. Keywords: Nested Radicals; Bring Radicals; Iterations; Quintic; Sextic; Equations; Polynomials; Solutions; Higher functions; 1 Introduction In (1758) Lambert considered the trinomial equation xm + q = x (1) and solve it giving x as a series of q. Euler, write the same equation into, the more consistent and symmetrical form xa xb = (a b)νxa+b, (2) − − using the transformation x x−b and setting m = ab, q = (a b)ν in (1). Euler gave his solution as → − 1 1 xn =1+ nν + n(n + a + b)µ2 + n(n + a +2b)(n +2a + b)+ 2 6 1 + n(n + a +3b)(n +2a +2b)(n +3a + b)ν4+ 24 arXiv:1406.1948v4 [math.GM] 17 Dec 2020 1 + n(n + a +4b)(n +2a +3b)(n +3a +2b)(n +4a + b)ν5 + ... (3) 120 The equation of Lambert and Euler can formulated in the next (see [4] pg.306- 307): Theorem 1. The equation aqxp + xq = 1 (4) 1 admits root x such that ∞ n Γ( n + pk /q)( qa)k xn = { } − , n =1, 2, 3,... (5) q Γ( n + pk /q k + 1)k! =0 Xk { } − where Γ(x) is Euler’s the Gamma function. d 1/d Moreover if someone defines √x := x , where d Q+ 0 , then the solution x of (4) can given in nested radicals: ∈ −{ } q q/p x = 1 aq 1 aq q/p 1 aq q/p√1 .
    [Show full text]
  • Calculus Glossary High School Level
    Calculus Glossary High School Level English - Spanish Updated November 2016 GLOSSARY ENGLISH LANGUAGE ARTS ENGLISH ‐ SPANISH THE STATE EDUCATION DEPARTMENT / THE UNIVERSITY OF THE STATE OF NEW YORK / ALBANY, NY 12234 P‐16 Office of Elementary, Middle, Secondary and Continuing Education and Office of Higher Education Office of Bilingual Education and Foreign Language Studies http://www.emsc.nysed.gov/biling/ THE UNIVERSITY OF THE STATE OF NEW YORK Regents of the University BETTY A. ROSA, Chancellor, B.A., M.S. in Ed., M.S. in Ed., M.Ed., Ed.D. ............ Bronx T. ANDREW BROWN, Vice Chancellor, B.A., J.D. ………………......................................... Syracuse NAN EILEEN MEAD, B.A. ………….................................................................................. Manhattan JOSEPHINE VICTORIA FINN, B.A., J.D. ……………………................................................... Albany BEVERLY L. OUDERKIRK, B.S., M.S. ............................................................................. Saratoga ELIZABETH. S. HAKANSON ……………........................................................................ Onondaga CATHERINE COLLINS, B.S., M.S., Ph.D. ....................................................................... West New York JAMES R. TALLON, R., B.A., M.A. ................................................................................. Binghamton ROGER TILLES, B.A., J.D.................................................................................................. Great Neck JUDITH JOHNSON, B.A., M.A., Ed.D. ...........................................................................
    [Show full text]
  • The Maximum Modulus of a Trigonometric Trinomial
    The maximum modulus of a trigonometric trinomial Stefan Neuwirth Abstract Let Λ be a set of three integers and let CΛ be the space of 2π-periodic functions with spectrum in Λ endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T ∈ CΛ and prove that x is unique unless |T | has an axis of symmetry. This enables us to compute the exposed and the extreme points of the unit ball of CΛ, to describe how the maximum modulus of T varies with respect to the arguments of its Fourier coefficients and to compute the norm of unimodular relative Fourier multipliers on CΛ. We obtain in particular the Sidon constant of Λ. 1 Introduction Let λ1, λ2 and λ3 be three pairwise distinct integers. Let r1, r2 and r3 be three positive real numbers. Given three real numbers t1, t2 and t3, let us consider the trigonometric trinomial i(t1+λ1x) i(t2+λ2x) i(t3+λ3x) T (x)= r1 e + r2 e + r3 e (1) for x R. The λ’s are the frequencies of the trigonometric trinomial T , the r’s are the moduli or ∈ it1 it2 it3 intensities and the t’s the arguments or phases of its Fourier coefficients r1 e , r2 e and r3 e . −1 −e iπ/3 Figure 1: The unit circle, the hypotrochoid H with equation z =4e −i2x +e ix, the segment from 1 to the unique point on H at maximum distance and the segments from e iπ/3 to the two points− on H at maximum distance.
    [Show full text]
  • An Application of Moore's Cross-Ratio Group To
    AN APPLICATIONOF MOORE'SCROSS-RATIO GROUP TO THE SOLUTIONOF THE SEXTICEQUATION* BY ARTHUR B. COBLE f Introduction. By making use of the fact that all the double ratios of n things can be expressed rationally in terms of a properly chosen set of n — 3 double ratios, E. H. Moore\ has developed the "cross-ratio group," C„,, a Cremona group in Sn_3 of order n\ which is isomorphic with the permutation group of n things. H. E. Slaught§ has discussed the (75! in considerable detail. Both C6! and C6, have been listed by S. Kantor. || I have already pointed out^f that Cn, defines a form-problem which I shall denote by PCn,. The solution of PCn, carries with it the solution of the n-ic equation, and I have worked out in detail the application to the quintic equation. After the adjunction of the square root of the discriminant of the n-ic, a new form-problem, PCn]l2, can be enunciated whose underlying group is the even subgroup CnU2 of Cnl. It appears in the particular case, n= 5, that the solu- tion of PC5,I2 can be expressed rationally in terms of the solution of Klein's ^.-problem.** The transition from the one problem to the other is accomplished by a very direct process which is suggested by simple geometric considerations. Explicit formulée are always desirable, and these have been lacking hitherto in the solutions of the quintic equation in terms of the ^4-problem. By intro- ducing PC5!I2 as an intermediate stage I have obtained such formulae from the long known invariant theory of the binary quintic.
    [Show full text]