symmetric symmetric

symmetric Sentence Examples

• =0, are non-unitary symmetric functions of the roots of a xn-a l xn 1 a2 x n-2 -...

• 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.

• A skew symmetric determinant has a,.

• skew table is much more symmetric.

• " 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 name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.

• 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.

• 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.

• 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..

• n be permuted, is a rational integral symmetric function of the quantities.

• 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 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 .

• 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.

• 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.

• 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.

• It will be ï¿½ shown later that every rational integral symmetric function is similarly expressible.

• 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.

• The b co-ordinates of any point R on the a ?/' t11®V1 a cycloid are expressible in the form x=a(8-}-sin 0); y=a (I -cos 0), M where the co-ordinate axes are the tangent at the vertex 0 and the axis of the curve, a is the radius of the generating circle, and 0 the angle R'CO, where RR' is parallel to LM and C is the centre of the circle in its symmetric position.

• Mailbox SDSL is the big brother of ADSL, providing symmetric bandwidth up to speeds of 2Mbps.

• The methods of objective 1.4 will be applied to these to obtain results on symmetric chaos.

• Can you state, in general, what property a symmetric cipher needs to have for this to work?

• The proposed authentication system is based on symmetric cryptography to minimize the encryption/decryption overhead.

• dilatation of the aorta is symmetric, commencing at the sinus of Valsalva and predisposing to rupture and dissection.

• Nevertheless the PSII core region is clearly 2-fold symmetric and closely resembles the averaged PSII core dimer top view.

• I like the idea of drinking a good espresso in a radially symmetric mirrored universe.

• Preconditioned conjugate gradients are shown to be extremely effective for all symmetric problems.

• This defines a natural homomorphism of C into the symmetric group of degree n.

• We consider the phenomenon of forced symmetry breaking in a symmetric Hamiltonian system on a symplectic manifold.

• The routine calculates the square symmetric matrix of distances between each atom and every other atom currently selected.

• The Intel and SPARC versions have reliable symmetric multiprocessing.

• This was altered in 1928 to the current spruce Bermudan rig with the symmetric spinnaker being adopted in 1969.

• stabilizer calculation in the symmetric group given by the degree component.

• teardrop fractals are derivable from cyclically symmetric fractals with a central element.

• T and L are symmetric tensors, while S is in general asymmetric.

• The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.

• 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, ...'

• 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,Ã¯¿½Ã¯¿½Ã¯¿½ To obtain particular theorems the quantities a l, a 2, a 3, ...a, n are auxiliaries which are at our entire disposal.

• A skew symmetric determinant has a,.

• 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,.

• 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) Ã¯¿½ Ã¯¿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.

• 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.

• THE Theory Of Symmetric Functions Consider n quantities a l, a 21 a 3, ...

• n be permuted, is a rational integral symmetric function of the quantities.

• +ax n, al, a2, ...an are called the elementary symmetric functions.

• A separation is the symbolic representation of a product of monomial symmetric functions.

• ) j1+j2+j3+..Ã¯¿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, a31Ã¯¿½Ã¯¿½Ã¯¿½; x 1, x2, x31..; X1, X2, X3,...

• in terms of x 1, x2, x3,Ã¯¿½Ã¯¿½ The inverse question is the expression of any monomial symmetric function by means of the power functions (r) = sr. Theorem of Reciprocity.-If Ã¯¿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!

• - " If a symmetric function be symboilized by (AÃ¯¿½v...) and (X1X2X3..Ã¯¿½), (Ã¯¿½i/-12Ã¯¿½3Ã¯¿½Ã¯¿½Ã¯¿½), (v1v2v3...)...

• 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 Ã¯¿½8 m' 2s m Ã¯¿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.Ã¯¿½.) +/103(A1X2.Ã¯¿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.

• a Product of Symmetric Functions.

• - 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-...

• =0, are non-unitary symmetric functions of the roots of a xn-a l xn 1 a2 x n-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.

• 1+Eaix+Esiy+ /al a2x 2 +Malt2xy -Z01023,2+Ã¯¿½Ã¯¿½Ã¯¿½ The most general symmetric function to be considered is E 41 041 8424-3033..Ã¯¿½ .conveniently written in the symbolic form (pigi p2g2 p3go...)Ã¯¿½ Observe that the summation is in regard to the expressions obtained by permuting then suffixes I, 2, 3, ...n.

• 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.

• It will be Ã¯¿½ shown later that every rational integral symmetric function is similarly expressible.

• 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.

• 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.

• Observe that, if we subject any symmetric function the diminishing process, it becomes ao 1 - P2 (p2p3...)Ã¯¿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.

• 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.

• The group of two longnecked gazelles facing a palm tree is of extraordinary refinement, and shows the, artistic consciousness in every part; the symmetric rendering of the palm tree, reduced to fit the scale of the animals, the dainty grace of the smooth gazelles contrasted with the rugged stem, the delicacy of the long flowing curves and the fine indications of the joints, all show a sense of design which has rarely been equalled in the ceaseless repetitions of the tree and supporters motive during every age since.

• 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.

• The b co-ordinates of any point R on the a ?/' t11®V1 a cycloid are expressible in the form x=a(8-}-sin 0); y=a (I -cos 0), M where the co-ordinate axes are the tangent at the vertex 0 and the axis of the curve, a is the radius of the generating circle, and 0 the angle R'CO, where RR' is parallel to LM and C is the centre of the circle in its symmetric position.

• Firstly, the skew table is much more symmetric.

• This was altered in 1928 to the current spruce Bermudan rig with the symmetric spinnaker being adopted in 1969.

• This can be done (albeit not very effectively) by a stabilizer calculation in the symmetric group given by the degree component.

• Symmetric cryptography: Block ciphers, including DES and AES, stream ciphers and modes of operation.

• Teardrop fractals are derivable from cyclically symmetric fractals with a central element.

• T and L are symmetric tensors, while S is in general asymmetric.

• Distance metrics Any measure that we use should be a distance metric (non-negative, symmetric and respecting triangle inequality).

• The affected muscles may be on both sides of the body (symmetric paralysis) but are often on unbalanced parts of the body (asymmetric paralysis).

• Growth inhibition during the first stage produces an undersized fetus with fewer cells, but normal cell size, causing symmetric IUGR.

• Japanese clansmen from as far back as 1185 AD admired butterflies for their duality--humble caterpillar and aristocratic butterfly--and their symmetric appearance.

• Symmetric DSL (SDSL) - While this system won't allow use a phone at the same time you send or receive data, but it provides equal receiving and transmission speeds.

• 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, ...'

• 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,.

• 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.

• ) 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,...

• 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 ...

• - " 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.

• Application to Symmetric Function Multiplication.-An example will explain this.

• 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.