prods                 package:onion                 R Documentation

_V_a_r_i_o_u_s _p_r_o_d_u_c_t_s _o_f _t_w_o _o_n_i_o_n_i_c _v_e_c_t_o_r_s

_D_e_s_c_r_i_p_t_i_o_n:

     Returns the various inner and outer  products of two onionic
     vectors.

_U_s_a_g_e:

     x %<*>% y
     x %>*<% y
     x %<.>% y
     x %>.<% y
     ## S3 method for class 'onion':
     g.even(x,y)
     ## S3 method for class 'onion':
     g.odd(x,y)
     ## S3 method for class 'onion':
     e.even(x,y)
     ## S3 method for class 'onion':
     e.odd(x,y)

_A_r_g_u_m_e_n_t_s:

       x: Onionic vector

       y: Onionic vector

_D_e_t_a_i_l_s:

     This page documents an attempt at a consistent notation for
     onionic products.  The product used by 'Ops.octonion()'  and
     'Ops.quaternion()' (viz "'*'") is sometimes known as the "Grassman
     product".  There is another product known as the Euclidean product
     defined by E(p,q)=p'q where x' is the conjugate of x.

     Each of these products separates into an "even" and an "odd" part,
     here denoted by functions 'g.even()' and 'g.odd()' for the
     Grassman product, and 'e.even()' and 'e.odd()' for the Euclidean
     product.  These are defined as follows:

        *  'g.even(x,y)=(xy+yx)/2'

        *  'g.odd(x,y)=(xy-yx)/2'

        *  'e.even(x,y)=(x'y+y'x)/2' 

        *  'e.odd(x,y)=(x'y-y'x)/2'

     These functions have an equivalent binary operator.

     The Grassman operators have a "'*'"; they are "'%<*>%'" for the
     even Grassman product and "'%>*<%'" for the odd product.

     The Euclidean operators have a "'.'"; they are "'%<.>%'" for the
     even Euclidean product and "'%>.<%'" for the odd product.

     There is no binary operator for the ordinary Euclidean product (it
     is not defined because there is no natural, consistent notation
     available; and it seems to be rarely needed in practice).  Use
     'Conj(x)*y'.

_A_u_t_h_o_r(_s):

     Robin K. S. Hankin

_E_x_a_m_p_l_e_s:

     Oj %<.>% Oall

