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invariant

from the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2.

150This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions.

81F(a ' a ' a, ...a) =r A F(ao, a1, a2,ï¿½ï¿½ï¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.

70This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.

60Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.

60Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, o.

601 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).

601 A2B' Where The Denominator Factors Indicate The Forms Themselves, Their Jacobian, The Invariant Of The Quadratic And Their Resultant; Connected, As Shown By The Numerator, By A Syzygy Of Degreesorder (2, 2; 2).

60When the latter invariant, but not the former, vanishes, the system reduces to a single force.

69respectively: then the result arrived at above from the logarithmic expansion may be written (n)a(n x) = (n)x, exhibiting (n) $ as an invariant of the transformation given by the expressions of X1, X2, X3...

50Such a symbolic product, if its does not vanish identically, denotes an invariant or a covariant, according as factors az, bz, cz,...

50If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.

502 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.

50The invariant theory then existing was classified by them as appertaining to " finite continuous groups."

40By the x process of Aronhold we can form the invariant S for the cubic ay+XH:, and then the coefficient of X is the second invariant T.

40then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - ï¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.

40a property independent of the particular axes of co-ordinates used in the representation of the curve by its equation; for instance, the curve may have a node, and in order to this, a relation, say A = o, must exist between the coefficients of the equation; supposing the axes of co-ordinates altered, so that the equation becomes u' = o, and writing A' = o for the relation between the new coefficients, then the relations A = o, A' = o, as two different expressions of the same geometrical property, must each of them imply the other; this can only be the case when A, A' are functions differing only by a constant factor, or say, when A is an invariant of u.

40By the x process of Aronhold we can form the invariant S for the cubic ay+XH:, and then the coefficient of X is the second invariant T.

40This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axe~.

44A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

30It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.

30The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system.

31The vanishing of this invariant is the condition for equal roots.

33Then if j, J be the original and transformed forms of an invariant J= (a1)wj, w being the weight of the invariant.

20The fourth shows that every term of the invariant is of the same weight.

20Moreover, if we add the first to the fourth we obtain aj 2w ak = 7 1=6, j, =0j, where 0 is the degree of the invariant; this shows, as we have before observed, that for an invariant w= - n0.

20If 0 be the degree of an invariant j - aj aj a; oj =a ° a a o +al aa l +...

20x x x To form an invariant or covariant we have merely to form a product of factors of two kinds, viz.

20If now the nti c denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples.

20Siace E2 + if + ~1, or ef, is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that XE + un + v~ is also an absolute invariant.

20Moreover, if we add the first to the fourth we obtain aj 2w ak = 7 1=6, j, =0j, where 0 is the degree of the invariant; this shows, as we have before observed, that for an invariant w= - n0.

20x x x To form an invariant or covariant we have merely to form a product of factors of two kinds, viz.

20From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics.

10in addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.

10In either case (AB) =A 1 B 2 -A 2 B 1 = (A/2)(ab); and, from the definition, (ab) possesses the invariant property.

10Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0.

10(ab)i(ac)j(bc)k..., that the symbolic product (ab)i(ac)j(bc)k..., possesses the invariant property.

10may be always viewed as a simultaneous invariant of a number of different linear forms a x, x, c x, ....

10may be a simultaneous invariant of a number of different forms az', bx 2, cx 3, ..., where n1, n 2, n3, ...

10=n3, 'If' the forms a:, b:, cy 7 ...be identical the symbols are alternative, and provided that the form does not vanish it denotes an invariant of the single form ay.

10The linear invariant a s is such that, when equated to zero, it determines the lines ax as harmonically conjugate to the lines xx; or, in other words, it is the condition that may denote lines at right angles.

10In either case (AB) =A 1 B 2 -A 2 B 1 = (A/2)(ab); and, from the definition, (ab) possesses the invariant property.

10=n3, 'If' the forms a:, b:, cy 7 ...be identical the symbols are alternative, and provided that the form does not vanish it denotes an invariant of the single form ay.

10If the forms be all linear and different, the function is an invariant, viz.

11possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...

11- We have seen that (ab) is a simultaneous invariant of the two different linear forms a x, bx, and we observe that (ab) is equivalent to where f =a x, 4)=b.

11If the forms be all linear and different, the function is an invariant, viz.

11When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.

14When the latter invariant, but not the former, vanishes, the displacement is equivalent to a pure rotation.

14A single linear form has, in fact, no invariant.

00(ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant.

00= wj, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant.

003 is absolutely unaltered by transformation, and is termed the absolute invariant.

002 - 9 m 2 (1 3 m 2)) 2 we have a cubic equation for determining m 2 as a function of the absolute invariant.

00We can see that (abc)a x b x c x is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant.

00The complete system consists of the form itself and this invariant.

00Operating with 5l-xidxlwe find S2C 0 =o; that is to say, C ° satisfies one of the two partial differential equations satisfied by an invariant.

00Unlike the other descriptors the chain code histogram is not a rotation invariant descriptor.

00This was the first known result on a topological invariant.

00Invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariant Theory of Finite Groups This introductory lecture will be concerned with polynomial invariants of finite groups which come from a linear group action.

00invariant torus: A torus where any trajectories on its surface do not change, despite any change in control space.

00invariant subspaces on which it is possible to calculate the action of the matrix.

00invariant subspace is a famous long-standing open problem.

00invariant manifold.

00invariant latitude.

00invariant Sections in the license notice of the combined work.

00QFT is relativistically invariant in a way which is not possible in QM.

00The project should develop the small amount of topology needed to understand what a knot invariant is.

00The filtered images are analyzed and rotation invariant features extracted at each pixel.

00Safety is related to the concept of a loop invariant.

00You wouldn't be able to create a scale invariant picture.

00invariant under one-to-one transformations of the co-ordinates.

00Opposite to an unstable manifold, both are types of invariant manifold.

00pyridine amplificant is invariant.

00rotation invariant ' texture classification schemes can fail when the 3D textures are rotated.

00The question whether every Hilbert space operator has a non-trivial invariant subspace is a famous long-standing open problem.

00Those solutions belong (or asymptotically tend) to a certain invariant linear subspace - cluster manifold.

00invariant torus: A torus where any trajectories on its surface do not change, despite any change in control space.

00This arose from the study by Felix Klein and Sophus Lie of a new theory of groups of substitutions; it was shown that there exists an invariant theory connected with every group of linear substitutions.

00The invariant theory then existing was classified by them as appertaining to " finite continuous groups."

00This expression of R shows that, as will afterwards appear, the resultant is a simultaneous invariant of the two forms.

00respectively: then the result arrived at above from the logarithmic expansion may be written (n)a(n x) = (n)x, exhibiting (n) $ as an invariant of the transformation given by the expressions of X1, X2, X3...

00F(a ' a ' a, ...a) =r A F(ao, a1, a2,Ã¯¿½Ã¯¿½Ã¯¿½an), 0 1 2 n the function F(ao, al, a2,...an) is then said to be an invariant of the quantic gud linear transformation.

00From these formulae we derive two important relations, dp4 = or the function F, on the right which multiplies r, is said to be a simultaneous invariant or covariant of the system of quantics.

00in addition, and transform each pair to a new pair by substitutions, having the same coefficients a ll, a12, a 21, a 22 and arrive at functions of the original coefficients and variables (of one or more quantics) which possess the abovedefinied invariant property.

00We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients.

00Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being aob l -a l b 0.

00A single linear form has, in fact, no invariant.

00When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance.

00(ab)i(ac)j(bc)k..., that the symbolic product (ab)i(ac)j(bc)k..., possesses the invariant property.

00may be always viewed as a simultaneous invariant of a number of different linear forms a x, x, c x, ....

00may be a simultaneous invariant of a number of different forms az', bx 2, cx 3, ..., where n1, n 2, n3, ...

00possess the invariant property, and we may write (AB) i (AC)'(BC) k ...A P E B C...

00possesses the invariant property.

00It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical.

00(ab)(ac)bxcx = - (ab)(bc)axcx = 2(ab)c x {(ac)bx-(bc)axi = 1(ab)2ci; so that the covariant of the quadratic on the left is half the product of the quadratic itself and its only invariant.

00The operation of taking the polar results in a symbolic product, and the repetition of the process in regard to new cogredient sets of variables results in symbolic forms. It is therefore an invariant process.

00Moreover, its operation upon any invariant form produces an invariant form.

00- We have seen that (ab) is a simultaneous invariant of the two different linear forms a x, bx, and we observe that (ab) is equivalent to where f =a x, 4)=b.

00Then if j, J be the original and transformed forms of an invariant J= (a1)wj, w being the weight of the invariant.

00= wj, aa 1 aa 2 a a 3 the complete system of equations satisfied by an invariant.

00The fourth shows that every term of the invariant is of the same weight.

00The second and third are those upon the solution of which the theory of the invariant may be said to depend.

00In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.

00If 0 be the degree of an invariant j - aj aj a; oj =a Ã‚° a a o +al aa l +...

00A leading proposition states that, if an invariant of Xax and i ubi be considered as a form in the variables X and, u, and an invariant of the latter be taken, the result will be a combinant of cif and b1'.

00A binary form of order n contains n independent constants, three of which by linear transformation can be given determinate values; the remaining n-3 coefficients, together with the determinant of transformation, give us n -2 parameters, and in consequence one relation must exist between any n - I invariants of the form, and fixing upon n-2 invariants every other invariant is a rational function of its members.

00The vanishing of this invariant is the condition for equal roots.

003 is absolutely unaltered by transformation, and is termed the absolute invariant.

002 - 9 m 2 (1 3 m 2)) 2 we have a cubic equation for determining m 2 as a function of the absolute invariant.

00Now, when C = o, clearly (see ante) R 2 j = 6 2 p where p = S +2 B a; and Gordan then proves the relation 6R 4 .f = B65Ã¯¿½5B64p - 4A2p5, which is Bring's form of quintic at which we can always arrive, by linear transformation, whenever the invariant C vanishes.

00Remark.-The invariant C is a numerical multiple of the resultant of the covariants i and j, and if C = o, p is the common factor of i and j.

00Such a symbolic product, if its does not vanish identically, denotes an invariant or a covariant, according as factors az, bz, cz,...

00We can see that (abc)a x b x c x is not a covariant, because it vanishes identically, the interchange of a and b changing its sign instead of leaving it unchanged; but (abc) 2 is an invariant.

00If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.

00The complete system consists of the form itself and this invariant.

00The simplest invariant is S = (abc) (abd) (acd) (bcd) cf degree 4, which for the canonical form of Hesse is m(1 -m 3); its vanishing indicates that the form is expressible as a sum of three cubes.

00Operating with 5l-xidxlwe find S2C 0 =o; that is to say, C Ã‚° satisfies one of the two partial differential equations satisfied by an invariant.

00from the invariant a2 -2a 1 a 3 -2aoa4 of the quartic the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2.

00Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the quartic invariant of degree order 4, o.

00The linear transformation replaces points on lines through the origin by corresponding points on projectively corresponding lines through the origin; it therefore replaces a pencil of lines by another pencil, which corresponds projectively, and harmonic and other properties of pencils which are unaltered by linear transformation we may expect to find indicated in the invariant system.

00If now the nti c denote a given pencil of lines, an invariant is the criterion of the pencil possessing some particular property which is independent alike of the axes and of the multiples, and a covariant expresses that the pencil of lines which it denotes is a fixed pencil whatever be the axes or the multiples.

002 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.

00then of course (AB) = (ab) the fundamental fact which appertains to the theory of the general linear substitution; now here we have additional and equally fundamental facts; for since A i = Xa i +,ia2, A2= - Ã¯¿½ay + X a2, AA =A?-}-A2= (X2 +M 2)(a i+ a z) =aa; A B =AjBi+A2B2= (X2 +, U2)(albi+a2b2) =ab; (XA) = X i A2 - X2 Ai = (Ax i + /-Lx2) (- /-jai + Xa2) - (- / J.x i '+' Axe) (X a i +%Ga^2) = (X2 +, u 2) (x a - = showing that, in the present theory, a a, a b, and (xa) possess the invariant property.

00It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.

00The linear invariant a s is such that, when equated to zero, it determines the lines ax as harmonically conjugate to the lines xx; or, in other words, it is the condition that may denote lines at right angles.

00We know that this x2 is an invariant; i.e.

00Siace E2 + if + ~1, or ef, is necessarily an absolute invariant for all transformations of the (rectangular) co-ordinate axes, we infer that XE + un + v~ is also an absolute invariant.

00When the latter invariant, but not the former, vanishes, the system reduces to a single force.

00This expression gives the work done by a given wrench when the body receives a given infinitely small twist; it must of course be an absolute invariant for all transformations of rectangular axe~.

00a property independent of the particular axes of co-ordinates used in the representation of the curve by its equation; for instance, the curve may have a node, and in order to this, a relation, say A = o, must exist between the coefficients of the equation; supposing the axes of co-ordinates altered, so that the equation becomes u' = o, and writing A' = o for the relation between the new coefficients, then the relations A = o, A' = o, as two different expressions of the same geometrical property, must each of them imply the other; this can only be the case when A, A' are functions differing only by a constant factor, or say, when A is an invariant of u.

00The attachment locant " 4 " in each pyridine amplificant is invariant.

00Existing ' rotation invariant ' texture classification schemes can fail when the 3D textures are rotated.

00Those solutions belong (or asymptotically tend) to a certain invariant linear subspace - cluster manifold.

00This is the famous and still open " invariant subspace problem " for operators on a Hilbert space.

00We examine our invariant measure approximation in more detail, and include encouraging numerical examples for the HÃ©non system and a nonlinear torus map.

00We cannot, however, say that it is an invariant unless it is expressible in terms of the real coefficients.

01When either of the forms is of an order higher than the first (ab), as not being expressible in terms of the actual coefficients of the forms, is not an invariant and has no significance.

01possesses the invariant property.

01It is always an invariant or covariant appertaining to a number of different linear forms, and as before it may vanish if two such linear forms be identical.

01Moreover, its operation upon any invariant form produces an invariant form.

01The second and third are those upon the solution of which the theory of the invariant may be said to depend.

01We know that this x2 is an invariant; i.e.

01The Aronhold process, given by the operation a as between any two of the forms, causes such an invariant to vanish.

02The Aronhold process, given by the operation a as between any two of the forms, causes such an invariant to vanish.

02In particular, when the product denotes an invariant we may transform each of the symbols a, b,...to x in succession, and take the sum of the resultant products; we thus obtain a covariant which is called the first evectant of the original invariant.

03

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