# Invariant Sentence Examples

invariant
• This 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.

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

• Remark.-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.

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

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

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

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

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

• Moreover, 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.

• The 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.

• If 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.

• From 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.

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

• Since (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.

• The 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.

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

• We 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.

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

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

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

• Invariant 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.

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

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

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

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

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

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

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

• The 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.

• A 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'.

• The 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.

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

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

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

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

• When 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.

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

• It 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.

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

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

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

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

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

• In 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.

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

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