The Journal of the Indian Mathematical Society

R. Vaidyanathaswamy ¹

Affiliations
1 Sri Venkateswara University, Tirupathi, IN

Abstract

The idea of classes of power-residues modulo a prime p (or composite number) is in substance identical with Gauss' idea of the 'periods' of complex ischolar_mains of unity.

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R. Vaidyanathaswamy

Abstract

Let F = Σ a_ij , x_i y_j (i,j = 1, 2, …. n) be a bilinear form in two sets of digredient variables (x₁, x₁ … x ), (y₁, y₂, … y). The matrix | a_ij | of order n may be called the matrix of F, and its rank r, the rank of F.

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R. Vaidyanathaswamy

Abstract

In the paper ' On the Rank of the Double-binary Form' (Proceedings of the London Mathematical Society, Series 3, Vol..94, Part 2, p. 83), the rank of an (m. n) form

a_x^m b_yⁿ, was defined as the rank of the matrix of m + 1 rows and n + I columns, formed by its co-efficients.

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R. Vaidyanathaswamy

Abstract

If a surface V² is immersed in a Euclidean space E³ of three dimensions, then we know from Dupin's Theorem, that at any point of V² the sum of the curvatures of any two perpendicular normal sections is a constant equal to the sum of the principal curvatures.

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R. Vaidyanathaswamy ¹

Affiliations
1 Madras University, IN

Abstract

The main object of this paper is to investigate the curious and mutually correlated properties of three triads of conies connected with a triangle ABC. These are: (l) the three rectangular hyperbolas which pass through the in- and ex-centres I, Ij, I, I3, and have concurrent normals at these points, (2) the three parabolas which are inscribed to ABC so as to have concurrent normals at the points of contact with the sides, and (3) the three parabolas circumscribed to ABC so as to have concurrent normals at these points It is shewn that the centres of the three rectangular hyperbolas (1), and the foci of (he three inscribed parabolas (2) are the same three points p, q, r on the circum-circle ABC, and that these may be obtained parametrically as the ischolar_mains of the binary Jacobian of the triad ABC, and the pair of circular points. Several other remarkable properties connected with the points p q r are obtained.

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R. Vaidyanathaswamy

Abstract

1. Desmic Tetrahedra.

2. The Three Involutoric Cubic Transformations Γ₁, Γ₂, TΓ₃ defined by three Desmic Tetrahedra.

3. The Projection and Intersection-Configuration of the Desmic System.

4. The Geometry of the Fundamental Quartic Curves.

5. The Geometry of the Transformations Γ_p1, Γ_t , on a common invariant quadric Q_t.

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R. Vaidyanathaswamy

Abstract

A quadric in space of three dimensions contains infinitely many straight liner, which fall into two distinct systems such that two lines intersect only when they belong to different systems. That an analogous property holds for the generating planes of a quadric in space of five dimensions was proved from line-geometric considerations by Cayley in 1873, and also independently by the writer.

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R. Vaidyanathaswamy

Abstract

The usual theory of quadratic residues can be extended in all its aspects to polynomials with integral co-efficients modulo p, p being an odd prime. Namely, if f(x), F(x) are two polynomials, which are mutually prime mod. p, we say that f(x) is a quadratic residue of F(x) mod.

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R. Vaidyanathaswamy

Abstract

The Twisted Cubic is a well-known curve and has been much studied both through parametric representation and by synthetic methods. The principal properties of the curve can be obtained straight-way by reducing it to the form

x₁ = t³, x₂ = t², x₃ = t, x₄ = 1.

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R. Vaidyanathaswamy

Abstract

If there exists one self-polar triangle of a conic S inscribed in a second conic S', then there exists an one-fold infinity and S' is out-polar or ex-harmonic to S. On the other hand it is easily seen that there always exists a finite number of self-polar triangles of S with their three vertices situated severally on three assigned conies S₁, S₂, S₃. The object of the present note is to shew that under certain circumstances the conies S₁, S₂, S₃ admit an infinity of such triangles. For the symbolic method which is used throughout, reference may be made to Grace and Young : Algebra of Invariants (Ternary Forms).

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R. Vaidyanathaswamy

Abstract

The set of point-groups which are ischolar_mains of polynomials of the form, f(x) - λΦ(x), [where f(x), Φ(x) are of order n, and λ is an arbitrary parameter], is often called a generalised involution of order n. An examination of the cross ratio of four point-groups of a generalised involution, will reveal the principle upon which depends the expression given by Mr. R. Gopalaswamy (J. I. M. S., Dec 1922: 'Pencils of Conics') for the cross ratio of four conics of a pencil-or, what is the same thing, of four point-pairs of an Involution of order 2.

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R. Vaidyanathaswamy

Abstract

The theory of characteristic (or 'latent') ischolar_mains of a matrix is perfect smooth-sailing when they are all distinct. But when a characteristic ischolar_main a is repeated a certain number of times-say r times, then a little reflection will suffice to show that the properties of the matrix in relation to the characteristic ischolar_main a are not completely determined by a knowledge of the number r alone.

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R. Vaidyanathaswamy , T. Venkatarayudu

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Let A be a commutative ring with unit element and A[x] denote the ring of polynomials in x with coefficients from A. Hilbert's theorem is : -

If every ideal in A has a finite basis, then every ideal in A[x] also has a finite basis.

The proof of this as given by Van der Waerden in his Moderne Algebra Bd. II gives us no information beyond the fact that every ideal in A [x] has a finite basis. If we just reverse the argument in his proof we will be able to give actually a canonical basis for every ideal in A[x] which will be found to be a powerful tool in several applications.

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R. Vaidyanathaswamy

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Various proofs of this theorem have been given, the most recent being from Prof. H. F. Baker. I am concerned here with some simple considerations which throw light on the nature of this theorem, and help incidentally to prove it.

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R. Vaidyanathaswamy

Abstract

In a recent paper I shewed that the classes C₁,C₂,... into which the elements of a group G fall with respect to any group of automorphisms of G, combine among themselves by the group-operation in the sense that each element of any class C_k occurs the same number γ_ij^k of times among the elements which result on combining each element of C_i with each element of C_j by the group-operation. A simple case of this general principle was studied in the paper referred to, namely the classes of a cyclic group with respect to its total group of automorphisms. The present paper is devoted to another simple case of the same principle.

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R. Vaidyanathaswamy

Abstract

We owe to Bernstein the result that every Boolean Algebra can be exhibited as a group, and that all the group-operations R are commutative.

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R. Vaidyanathaswamy

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The author has studied in previous papers the correspondences on a circle related to the pedal line theory of the triangle, by the symbolic methods of the invariant theory of binary forms. Langley's pedal line chain shews however that the method must only be a special case of a uniform theory applicable to the n-point inscribed in a circle, for any value of n.

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R. Vaidyanathaswamy

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The notion of determinant was extended in a satisfactory and general manner to arrays with many suffixes by Rice and Lecat, and independently by the present writer.

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R. Vaidyanathaswamy , B. Ramamurti

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In the paper 'On the rational norm curve II', a one-to-one correspondence has been effected between quadric envelopes Q in a space S_n of n dimensions and linear line complexes L in a space of S_n+1 of (n+1) dimensions in the following manner. Any simplex inscribed in a rational norm curve R_n in S_n may be specified by the binary (n+1)-ic, a_x^n+x, giving the parameters of its vertices.

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R. Vaidyanathaswamy

Abstract

The classification of plane conics into hyperbolas, parabolas and ellipses can, it is well known, be extended to Euclidean-affine space of any number of dimensions. It is the purpose of this note to shew that the classification can be completely described by four numbers which are respectively the rank and signature of the quadric itself, and the rank and signature of its section by the prime at infinity. These four numbers are independent in the sense that no one of them can be unambiguously determined from the others. The description of quadric-types by the values of these four numbers is very convenient, and does not appear to have been stated before. The usual terms, ellipse, hyperbola, parabola, cylinder are used in an extended sense, and serve to characterise affine types.

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