When the set of roots for a semi-simple Lie algehra have beenselected appropriately, the scalar product of the weights of the elementaryrepresentation can be calculated.

When the set of roots for a semi-simple Lie algebra have been selected appropriately,the scalar product of the weights of the elementary representation can be calculated.

In this paper by using cartan matrix,a method for calculating the scalar product of the weights of the elementary representation for semi- simple Lie algebra is presented,and from this,the eigen value of the second Casimir operator is worked out.

In a group of orthogonal vectors,a vector’s standard product is 1 with itself while 0 with others. With this property,watermark information is spread spectrum by vector from orthogonal vectors,then multiple watermark information are linear compounded and embedded as a single information,watermark information can be detected anytime without impacted by other embedded watermark.

Applying the character of rotation matrices, it is not difficult to obtain the recurrence formulas of direction cosines of Cartesian unit vectors, to calculate the scalar products and triple products of these unit vectors, and to derive the 6th constraint equation.

In this paper by using cartan matrix, a method for calculating thescalar product of the weights of the elementary representation for semi-simple Lie algebra is presented, and from this, the eigen value of the aecondCasimir operator is worked out.

A unified electromagnetic field vector is formed in the 3 d space. Its equations,and the relations between square of the field vector and energy,momentum are given. It is found that the dot product of the unified field vector with itself in the 3 d space is a relativistic invariant.

We introduce a family of linear differential operators ${\cal K}^n =(-i)^nP_n^{\cal M}(i\frac{d}{dt})$, called the chromatic derivatives associated with M, which are orthonormal with respect to a suitably defined scalar product.

Define the scalar product in H1 to be (f(x), g(x))1=

In this paper we prove an "explicit formula" relating the scalar product of the Hecke series of two fields of algebraic numbers to the zeta-function composites of these fields.

Construction of Banach spaces using a generalization of the scalar product

For such a space we introduce, as in the case of the usual scalar product, a series of concepts, and prove, as an example a theorem concerning orthogonal expansion.

It is shown that there is a similar identity when the inner product is replaced by an indefinite quadratic formq and h is a Л-harmonic distribution, where Л is the differential operator canonically associated toq.

We present a method for constructing wavelet bases on the unit sphere S2 of R3, using the radial projection and an inner product associated to a convex polyhedron having the origin inside.

A modified Bates and Watts geometric framework is proposed for quasi-likelihood nonlinear models in Euclidean inner product space.

We introduce the notion of inner product with sign-sensitive weight and construct systems of nonsymmetrically orthonormalized polynomials.

Estimates of polynomials orthogonal with respect to the Legendre-Sobolev inner product

The wind fields are taken from the 1998-2004 NCEP/NCAR Reanalysis data with better spatial resolution (1° × 1°) than the standard product, which are publicly available on the Internet.

Lack of bioequivalence of a generic mefloquine tablet with the standard product

In particular, we show that a simple formula for updating the pricing vector can be used with some advantage in the standard product form simplex algorithm and with very considerable advantage in two recent developments: P.M.J.

Certain uses of the analysis of variance with standard product specifications

In this paper firms may deviate from standard product specifications by investing in flexible production.

An input-ouptut equation of the general spatial 7R mechanism is derived in this paper by using the method in [1] and applying the rotation matrices. The result is the same as [2], but the process of derivation is simpler. Applying the character of rotation matrices, it is not difficult to obtain the recurrence formulas of direction cosines of Cartesian unit vectors, to calculate the scalar products and triple products of these unit vectors, and to derive the 6th constraint equation. Moreover, an algorithm, which...

An input-ouptut equation of the general spatial 7R mechanism is derived in this paper by using the method in [1] and applying the rotation matrices. The result is the same as [2], but the process of derivation is simpler. Applying the character of rotation matrices, it is not difficult to obtain the recurrence formulas of direction cosines of Cartesian unit vectors, to calculate the scalar products and triple products of these unit vectors, and to derive the 6th constraint equation. Moreover, an algorithm, which consists of successive applications of row transformation and expansion based on Laplace's Theorem, is given to evaluate the 16X16 determinant according to its characteristic, so that the evaluation is much simplified.

The simple roots for any semi-simple Lie algebra can be representedby the weights of appropriate elementary representation. Conversely, theweights for the elementary representation can also be repeesented by simpleroots. When the set of roots for a semi-simple Lie algehra have beenselected appropriately, the scalar product of the weights of the elementaryrepresentation can be calculated. In this paper by using cartan matrix, a method for calculating thescalar product of the weights of the elementary representation...

The simple roots for any semi-simple Lie algebra can be representedby the weights of appropriate elementary representation. Conversely, theweights for the elementary representation can also be repeesented by simpleroots. When the set of roots for a semi-simple Lie algehra have beenselected appropriately, the scalar product of the weights of the elementaryrepresentation can be calculated. In this paper by using cartan matrix, a method for calculating thescalar product of the weights of the elementary representation for semi-simple Lie algebra is presented, and from this, the eigen value of the aecondCasimir operator is worked out.

The simple roots for any semi-simple Lie algebra can be represented by the weights of appropriate elementary representation.Conversely,the weights for the elementary representation can also be represented by simple roots.When the set of roots for a semi-simple Lie algebra have been selected appropriately,the scalar product of the weights of the elementary representation can be calculated. In this paper by using cartan matrix,a method for calculating the scalar product of the weights of the elementary representation...

The simple roots for any semi-simple Lie algebra can be represented by the weights of appropriate elementary representation.Conversely,the weights for the elementary representation can also be represented by simple roots.When the set of roots for a semi-simple Lie algebra have been selected appropriately,the scalar product of the weights of the elementary representation can be calculated. In this paper by using cartan matrix,a method for calculating the scalar product of the weights of the elementary representation for semi- simple Lie algebra is presented,and from this,the eigen value of the second Casimir operator is worked out.