# Symmetric Sentence Examples

- Now by the theory of
**symmetric**functions, any**symmetric**functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations. - =0, are non-unitary
**symmetric**functions of the roots of a xn-a l xn 1 a2 x n-2 -... - Although many pseudo-
**symmetric**twins are transformable into the simpler form, yet, in some cases, a true polymorph results, the change being indicated, as before, by alterations in scalar (as well as vector) properties. - The theories of determinants and of
**symmetric**functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics. - +amam Expanding the right-hand side by the exponential theorem, and then expressing the
**symmetric**functions of al, a2, ...a m, which arise, in terms of b1, b2, ...' - If al, a2, ...a, n be the roots of f=o, (1, R2, -Ai the roots of 0=o, the condition that some root of 0 =o may qq cause f to vanish is clearly R s, 5 =f (01)f (N2) ï¿½ ï¿½;f (Nn) = 0; so that Rf,q5 is the resultant of f and and expressed as a function of the roots, it is of degree m in each root 13, and of degree n in each root a, and also a
**symmetric**function alike of the roots a and of the roots 1 3; hence, expressed in terms of the coefficients, it is homogeneous and of degree n in the coefficients of f, and homogeneous and of degree m in the coefficients of 4.. - There is no difficulty in expressing the resultant by the method of
**symmetric**functions. - THE Theory Of
**Symmetric**Functions Consider n quantities a l, a 21 a 3, ... - +ax n, al, a2, ...an are called the elementary
**symmetric**functions. - A separation is the symbolic representation of a product of monomial
**symmetric**functions. - In terms of x 1, x2, x3,ï¿½ï¿½ The inverse question is the expression of any monomial
**symmetric**function by means of the power functions (r) = sr. Theorem of Reciprocity.-If ï¿½1 P2 "3 01 Q 2 7 3 Al A 2 A3 X m1 X m2 X m3 ... - The general monomial
**symmetric**function is a P1 a P2 a P3. - (0B) = (e), &c. The binomial coefficients appear, in fact, as
**symmetric**functions, and this is frequently of importance. - The sum of the monomial functions of a given weight is called the homogeneous-product-sum or complete
**symmetric**function of that weight; it is denoted by h.; it is connected with the elementary functions by the formula 1 7r1l7r2!7r3! - The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of
**symmetric**functions symbolized by the k specifications; and that () St =ti ts. - " The
**symmetric**function (m ï¿½8 m' 2s m ï¿½3s ...) whose is 2s 3s partition is a specification of a separation of the function symbolized by (li'l2 2 l3 3 ...) is expressible as a linear function of**symmetric**functions symbolized by separations of (li 1 12 2 13 3 ...) and a**symmetrical**table may be thus formed." - The introduction of the quantity p converts the
**symmetric**function 1 2 3 into (XiX2X3+...) -Hu Al (X 2 A 3 .-) +/l02(X1X3.ï¿½.) +/103(A1X2.ï¿½.) +.... - P operators D upon a monomial
**symmetric**function is clear. - It has been shown (vide " Memoir on
**Symmetric**Functions of the Roots of Systems of Equations," Phil. - - Suppose f to be a product of
**symmetric**functions f i f 2 ...f m . - Application to
**Symmetric**Function Multiplication.-An example will explain this. - Which is satisfied by every
**symmetric**fraction whose partition contains no unit (called by Cayley non-unitary**symmetric**functions), is of particular importance in algebraic theories. - For example, the theory of invariants may be regarded as depending upon the consideration of the
**symmetric**functions of the differences of the roots of the equation aox n - (i) a i x n - 1 + (z) a 2 x n 2 - ... - 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. - =o, are
**symmetric**functions of differences of the roots of aox n - 1!(n)a4xn-1+2!()a2xn-2-... - = 0; and on the other hand that
**symmetric**functions of the differences of the roots of aox n (7)alxn-1+ (z)a2xn-2-... - It is thus possible to study simultaneously all the theories which depend upon operations of the group. Symbolic Representation of
**Symmetric**Functions.-Denote the s 8 s elementar**symmetric**function a s by al a 2 a3 ...at pleasure; then, Y y si,, si,... - Denote by brackets () and []
**symmetric**functions of the quantities p and a respectively. - Being subsequently put equal to a, a non-unitary
**symmetric**function will be produced. **Symmetric**Functions Several Systems Quantities.- The weight of the function is bipartite and consists of the two numbers Ep and Eq; the symbolic expression of the
**symmetric**function is a partition into biparts (multiparts) of the bipartite (multipartite) number Ep, Eq. - All
**symmetric**functions are expressible in terms of the quantities ap g in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit. - DaP4 References For
**Symmetric**Functions.-Albert Girard, In- -vention nouvelle en l'algebre (Amsterdam, 1629); Thomas Waring, Meditationes Algebraicae (London, 1782); Lagrange, de l'acad. - 1852; MacMahon, " Memoirs on a New Theory of
**Symmetric**Functions," American 1 Phil. - 1888-1890; " Memoir on
**Symmetric**Functions of Roots of Systems of Equations," Phil. - Every
**symmetric**function denoted by partitions, not involving the figure unity (say a non-unitary**symmetric**function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz. - It was noted that Stroh considers Method of Stroh.-In the section on "
**Symmetric**Function," (alai +a 2 a 2 +... - Remark, too, that we are in association with non-unitary
**symmetric**functions of two systems of quantities which will be denoted by partitions in brackets ()a, ()b respectively. - Twinning according to the second law can only be explained by reflection across the plane (roi), not by rotation about an axis; chalcopyrite affords an excellent example of this comparatively rare type of
**symmetric**twinning. - Hesse, and centro-
**symmetric**determinants by W. - The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is
**symmetric**about the axis. - B., we obtain by comparison with the middle series the symbolical representation of all
**symmetric**functions in brackets () appertaining to the quantities p i, P2, P3,ï¿½ï¿½ï¿½ To obtain particular theorems the quantities a l, a 2, a 3, ...a, n are auxiliaries which are at our entire disposal. - When a skew
**symmetric**determinant is of even degree it is a perfect square. - A skew determinant is one which is skew
**symmetric**in all respects,. - ) j1+j2+j3+..ï¿½ (J1+ j2 +j3+...-1)!/T1)?1(J2)72 (J 3)/3..., j11j2!j3!... ?.1 for the expression of Za n in terms of products of
**symmetric**functions symbolized by separations of (n 1 1n 2 2n 3 3) Let (n) a, (n) x, (n) X denote the sums of the n th powers of quantities whose elementary**symmetric**functions are a l, a 2, a31ï¿½ï¿½ï¿½; x 1, x2, x31..; X1, X2, X3,... - - " If a
**symmetric**function be symboilized by (Aï¿½v...) and (X1X2X3..ï¿½), (ï¿½i/-12ï¿½3ï¿½ï¿½ï¿½), (v1v2v3...)... - A Product of
**Symmetric**Functions. - It is thus possible to study simultaneously all the theories which depend upon operations of the group. Symbolic Representation of
**Symmetric**Functions.-Denote the s 8 s elementar**symmetric**function a s by al a 2 a3 ...at pleasure; then, Y y si,, si,... - 1+Eaix+Esiy+ /al a2x 2 +Malt2xy -Z01023,2+ï¿½ï¿½ï¿½ The most general
**symmetric**function to be considered is E 41 041 8424-3033..ï¿½ .conveniently written in the symbolic form (pigi p2g2 p3go...)ï¿½ Observe that the summation is in regard to the expressions obtained by permuting then suffixes I, 2, 3, ...n. - Observe that, if we subject any
**symmetric**function the diminishing process, it becomes ao 1 - P2 (p2p3...)ï¿½ Next consider the solutions of 0=o o which are of degree 0 and weight w. - The extraordinary advantage of the transformation of S2 to association with non-unitary
**symmetric**functions is now apparent; for we may take, as representative forms, the**symmetric**functions which are symbolically denoted by the partitions referred to. - The partitions being taken as denoting
**symmetric**functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula. - A skew
**symmetric**determinant has a,.