# Invariants sentence example

invariants
• Under the general heading "Analysis" occur the subheadings "Foundations of Analysis," with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; "Theory of Functions of Complex Variables," with the topics functions of one variable and of several variables; "Algebraic Functions and their Integrals," with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; "Other Special Functions," with the topics Euler's, Legendre's, Bessel's and automorphic functions; "Differential Equations," with the topics existence theorems, methods of solution, general theory; "Differential Forms and Differential Invariants," with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; "Analytical Methods connected with Physical Subjects," with the topics harmonic analysis, Fourier's series, the differential equations of applied mathematics, Dirichlet's problem; "Difference Equations and Functional Equations," with the topics recurring series, solution of equations of finite differences and functional equations.
• Also such ideas as those of invariants, groups and of form, have modified the entire science.
• The investigation of the properties of these functions, as well for a single form as for a simultaneous set of forms, and as well for one as for many series of variables, is included in the theory of invariants.
• The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies.
• The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions.
• Occasionally the word " invariants " includes covariants; when this is so it will be implied by the text.
• This notion is fundamental in the present theory because we will find that one of the most valuable artifices for finding invariants of a single quantic is first to find simultaneous invariants of several different quantics, and subsequently to make all the quantics identical.
• For the quartic (ab) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.
• It will be shown later that all invariants, single or simultaneous, are expressible in terms of symbolic products.
• All the forms obtained are invariants in regard to linear transformations, in accordance with the same scheme of substitutions, of the several sets of variables.
• An instantaneous deduction from the relation w= 2 n0 is that forms of uneven orders possess only invariants of even degree in the coefficients.
• The invariants in question are invariants qud linear transformation of the forms themselves as well as qud linear transformation of the variables.
• The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.
• Further, in the case of invariants, we write A= (1, i') 2 and take three new forms B = (i, T) 2; C = (r, r`) 2; R = (/y).
• Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.
• - (aa) and then expresses the coefficients, on the right, in terms of the fundamental invariants.
• Hence, solving the cubic, R 2 j = (S -m i a) (S - m 2 a) (S - m3a) wherein m 1 m2, m 3 are invariants.
• The discriminant is the resultant of ax and ax and of degree 8 in the coefficients; since it is a rational and integral function of the fundamental invariants it is expressible as a linear function of A 2 and B; it is independent of C, and is therefore unaltered when C vanishes; we may therefore take f in the canonical form 6R 4 f = BS5+5BS4p-4A2p5.
• Hesse's canonical form shows at once that there cannot be more than two independent invariants; for if there were three we could, by elimination of the modulus of transformation, obtain two functions of the coefficients equal to functions of m, and thus, by elimination of m, obtain a relation between the coefficients, showing them not to be independent, which is contrary to the hypothesis.
• For an algebraic solution the invariants must fulfil certain conditions.
• When a z and the invariants B and C all vanish, either A or j must vanish; in the former case j is a perfect cube, its Hessian vanishing, and further f contains j as a factor; in the latter case, if p x, ax be the linear factors of i, f can be expressed as (pa) 5 f =cip2+c2ay; if both A and j vanish i also vanishes identically, and so also does f.
• Besides the invariants and covariants, hitherto studied, there are others which appertain to particular cases of the general linear substitution.
• Thus what have been called seminvariants are not all of them invariants for the general substitution, but are invariants for the particular substitution xl = X11 + J-s12, X 2 = 112 Again, in plane geometry, the most general equations of substitution which change from old axes inclined at w to new axes inclined at w' =13 - a, and inclined at angles a, l3 to the old axis of x, without change of origin, are x-sin(wa)X+sin(w -/3)Y sin w sin ' _sin ax y sin w a transformation of modulus sin w' sin w' The theory of invariants originated in the discussion, by George Boole, of this system so important in geometry.
• Of the quadratic axe+2bxy+cy2, he discovered the two invariants ac-b 2, a-2b cos w+c, and it may be verified that, if the transformed of the quadratic be AX2=2BXY+CY2, sin w 2 AC -B 2 =) (ac-b2), sin w A-2B cos w'+C = (sin w'1 2(a - 2bcosw+c).
• 2 cos w xy+y 2 = X 2 +2 cos w'XY+Y2, from which it appears that the Boolian invariants of axe+2bxy-y2 are nothing more than the full invariants of the simultaneous quadratics ax2+2bxy+y2, x 2 +2 cos coxy+y2, the word invariant including here covariant.
• Since +xZ=x x we have six types of symbolic factors which may be used to form invariants and covariants, viz.
• For instance, there are the symbols A, D, E used in the calculus of finite differences; Aronhold's symbolical method in the calculus of invariants; and the like.
• In the course of the ensuing ten years he published a large amount of original work, much of it dealing with the theory of invariants, which marked him as one of the foremost mathematicians of the time.
• At Woolwich he remained until 1870, and although he was not a great success as an elementary teacher, that period of his life was very rich in mathematical work, which included remarkable advances in the theory of the partition of numbers and further contributions to that of invariants, together with an important research which yielded a proof, hitherto lacking, of Newton's rule for the discovery of imaginary roots for algebraical equations up to and including the fifth degree.
• The Hessian A has just been spoken of as a covariant of the form u; the notion of invariants and covariants belongs rather to the form u than to the curve u=o represented by means of this form; and the theory may be very briefly referred to.