scholarly journals III. On the orders and genera of quadratic forms containing more than three indeterminates.—Second notice

1868 ◽  
Vol 16 ◽  
pp. 197-208 ◽  
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Royal Society ◽  
Present Communication ◽  
The Matrix ◽  
Primitive Quadratic Form ◽  
Definition Of ◽  
Greatest Common Divisors

The principles upon which quadratic forms are distributed into orders and genera have been indicated in a former notice (Proceedings of the Royal Society, vol. xiii. p. 199). Some further results relating to the same subject are contained in the present communication. I. The Definition of the Orders and Genera. Retaining, with some exceptions to which we shall now direct attention, the notation and nomenclature of the former notice, we represent by f 1 a primitive quadratic form containing nindeterminates, of which the matrix is || A n x n i, j ; by f 2 , f 3 , . . . f n -1 , the fundamental concomitants o f 1 , of which the last is the contravariant. The matrices of these concomitants are the matrices derived from the matrix of f 1 , so that the first coefficients of f 2 , f 3 , .. . f n -1 , are respectively the determinants |A 2 x 2 i, j |, | · A 3 x 3 i, j |,... |A n -1 x n -1 i, j |, taken with their proper signs. The discriminant of f 1 , i. e. the determinant of the matrix |A n x n i, j |, which is supposed to be different from zero, and which is to be taken with its proper sign, is represented by ∇ n . The greatest common divisors of the minors of the orders n - 1, n - 2, . . . 2, 1 in the same matrix are denoted by ∇ n -1 , ∇ n -2 , ∇ 2 , ∇ 1 , of which the last is a unit; we shall presently attribute signs to each of these greatest common divisors.

scholarly journals Theorems Relating to Quadratic Forms and their Discriminant Matrices

1953 ◽  
Vol 10 (1) ◽  
pp. 13-15
Author(s):  
S. Vajda
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Normal Population ◽  
Royal Statistical Society ◽  
Unit Variance ◽  
The Matrix ◽  
Chi Squared ◽  
Image Position ◽  
Latent Roots

In a paper read before the Research Branch of the Royal Statistical Society (Ref. 1, p. 150) the following case was considered:Let the expression be given; introduce, for c, a linear form in and obtainIf the yi are sample values from a normal population with unit variance, then it is known (Ref. 2) that (1) is distributed as where zi varies as chi-squared with one degree of freedom and the li are the latent roots of the matrix of the quadratic form. If these latent roots are f times unity and n—f times zero, then this reduces to a chi-squared distribution with f degrees of freedom.

Quadratic Forms of Type E6, E7 and E8

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Quadratic Form ◽  
Division Algebra ◽  
Quadratic Forms ◽  
Quadratic Extension ◽  
The Other ◽  
Quadratic Space ◽  
Division Algebras ◽  
Arbitrary Characteristic ◽  
Definition Of ◽  
Quaternion Division Algebra

This chapter presents various results about quadratic forms of type E⁶, E₇, and E₈. It first recalls the definition of a quadratic space Λ‎ = (K, L, q) of type Eℓ for ℓ = 6, 7 or 8. If D₁, D₂, and D₃ are division algebras, a quadratic form of type E⁶ can be characterized as the anisotropic sum of two quadratic forms, one similar to the norm of a quaternion division algebra D over K and the other similar to the norm of a separable quadratic extension E/K such that E is a subalgebra of D over K. Also, there exist fields of arbitrary characteristic over which there exist quadratic forms of type E⁶, E₇, and E₈. The chapter also considers a number of propositions regarding quadratic spaces, including anisotropic quadratic spaces, and proves some more special properties of quadratic forms of type E₅, E⁶, E₇, and E₈.

On Gleason's Definition of Quadratic Forms

1981 ◽  
Vol 24 (2) ◽  
pp. 233-236
Author(s):  
T. M. K. Davison
Keyword(s):  
Quadratic Form ◽  
Commutative Ring ◽  
Quadratic Forms ◽  
Definition Of

Suppose R is a commutative ring with identity. Let M be an R -module, and suppose f is a function from M to R. How do we characterize the property that f be a quadratic form?

XII. On the orders and genera of ternary quadratic forms

1867 ◽  
Vol 157 ◽  
pp. 255-298 ◽  
Keyword(s):  
Quadratic Form ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Greatest Common Divisor ◽  
Ternary Quadratic Forms ◽  
The Matrix

Eisenstein, in a Memoir entitled "Neue Theoreme der höheren Arithmetik", has defined the ordinal and generic characters of ternary quadratic forms of an uneven determinant; and, in the case of definite forms, has assigned the weight of any given order or genus. But he has not considered forms of an even determinant, neither has he given any demonstrations of his results. To supply these omissions, and so far to complete the work of Eisenstein, is the object of the present memoir. Art. 2. We represent by f the ternary quadratic form a x 2 + a ' y 2 + a '' z 2 +2 byz +2 b ' xz +2 b '' xy ; . . . . . . . (1) we suppose that f is primitive ( i. e . that the six integral numbers a , a ', a '', b , b ', b " admit of no common divisor other than unity), and that its discriminant is different from zero; this discriminant, or the determinant of the matrix a , b ", b ' b ", a ', b b ', b , a " . . . . . . . (2) we represent by D ; by Ω we denote the greatest common divisor of the minor determinants of the matrix (2); by Ω F the contravariant of f , or the form ( a ' a ''— b 2 ) x 2 + ( a " a — b ' 2 ) y 2 + ( a a '— b '' 2 ) z 2 + 2( b ' b "— ab ) yz + 2( b " b — a ' b ') zx + 2( bb ' — a '' b ") xy ; }. . . . . . . . (3) we shall term F the primitive contravariant of f , and we shall write F = A x 2 + A' y 2 + 2B yz + 2B' xy + 2B'' xy . . . . . . (4)

FINITENESS RESULTS FOR REGULAR DEFINITE TERNARY QUADRATIC FORMS OVER ${\mathbb Q}(\sqrt{5})$

2007 ◽  
Vol 03 (04) ◽  
pp. 541-556 ◽  
Author(s):  
WAI KIU CHAN ◽  
A. G. EARNEST ◽  
MARIA INES ICAZA ◽  
JI YOUNG KIM
Keyword(s):  
Quadratic Form ◽  
Number Field ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Integral Quadratic Form ◽  
Ring Of Integers ◽  
Ternary Quadratic Forms ◽  
Totally Positive ◽  
Finiteness Result

Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over [Formula: see text]. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over [Formula: see text], and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].

De la structure des vaisseaux anglais, considérée dans ses derniers perfectionnements

1833 ◽  
Vol 2 ◽  
pp. 62-63
Keyword(s):  
Royal Society ◽  
Present Communication ◽  
Naval Architecture ◽  
Ship Building ◽  
Oblique Direction ◽  
Rectangular Frame ◽  
Longitudinal Length ◽  
Frame Work ◽  
Academy Of Sciences

Being engaged in collecting materials for a work entitled “A Picture of Naval Architecture in the 18th and 19th Centuries,” the author was induced to visit this country, with a view to become acquainted with the various innovations and improvements lately introduced here in the art of ship-building; and, in the present communication, offers some remarks upon the plans proposed by Mr. Seppings, an account of which has formerly been before the Royal Society, and is printed in their Transactions for 1814. After giving an outline of the fundamental principles upon which Mr. Seppings’s improvements in naval architecture principally depend, and dwelling especially upon the diagonal pieces of timber which he employs to strengthen the usual rectangular frame-work, the author proceeds to state that similar contrivances were long ago suggested and even practised by the French ship-builders, in order to give strength to the general fabric of their vessels. Instead of making the ceiling parallel to the exterior planks, they arranged it in the oblique direction of the diagonals of the parallelograms formed by the timber and the ceiling, in the whole of that part of the ship’s sides between the orlop and limber-strake next the kelson. They then covered this ceiling with riders, as usual, and placed crosspieces between them in the direction of the second diameter of the parallelogram. This system, however, was abandoned in the French navy, on account of its expense, of its diminishing the capacity of the hold, and of the erroneous notion that the longitudinal length of the ship was diminished by the obliquity of the ceiling. In 1755, the Academy of Sciences rewarded M. Chauchot, a naval engineer, for the suggestion of employing oblique for transverse riders; and in 1772, M. Clairon des Lauriers employed diagonal strengtheners in the construction of the frigate l’Oiseau.

scholarly journals REPRESENTATIONS OF ALTERNATIVE CLIFFORD ALGEBRAS OF QUADRATIC FORMS

2014 ◽  
Vol 57 (3) ◽  
pp. 579-590 ◽  
Author(s):  
STACY MARIE MUSGRAVE
Keyword(s):  
Quadratic Form ◽  
Clifford Algebra ◽  
Algebraic Structure ◽  
Quadratic Forms ◽  
Clifford Algebras ◽  
Future Research ◽  
Octonion Algebras

AbstractThis work defines a new algebraic structure, to be called an alternative Clifford algebra associated to a given quadratic form. I explored its representations, particularly concentrating on connections to the well-understood octonion algebras. I finished by suggesting directions for future research.

scholarly journals Infinite Matrix Products and the Representation of the Matrix Gamma Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo
Keyword(s):  
Gamma Function ◽  
Infinite Product ◽  
Infinite Matrix ◽  
Convergence Results ◽  
Matrix Products ◽  
The Matrix ◽  
Definition Of

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

On the existence of integral ternary quadratic forms without automorphs of finite order

2017 ◽  
Vol 26 (14) ◽  
pp. 1750102 ◽  
Author(s):  
José María Montesinos-Amilibia
Keyword(s):  
Quadratic Form ◽  
Finite Order ◽  
Quadratic Forms ◽  
Ternary Quadratic Form ◽  
Ternary Quadratic Forms

An example of an integral ternary quadratic form [Formula: see text] such that its associated orbifold [Formula: see text] is a manifold is given. Hence, the title is proved.

scholarly journals Featural definition of syntactic positions: Evidence from hyper-raising

2018 ◽  
Vol 3 (1) ◽  
pp. 2 ◽  
Author(s):  
Suzana Fong
Keyword(s):  
Phase Problem ◽  
Matrix Clause ◽  
The Matrix ◽  
The Subject ◽  
Embedded Clause ◽  
Definition Of ◽  
Finite Clause

Hyper-raising consists in raising a DP from an embedded finite clause into the matrix clause. HR introduces a phase problem: the embedded clause is finite, which is supposed to be impervious to raising. This can be overcome by postulating A-features at the C of the the embedded clause. They trigger the movement of the subject to [Spec, CP]. Being at the edge of a phase, it is visible to a matrix probe. If successful, this analysis provides support for the claim that syntactic positions are not inherently A or A-bar; they can be defined featurally instead.

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