PHYSICS: BARGMANNAND WIGNER
pairing of G to G' is defined and the products gl g2 (g1, g92 G) all have order two. The group Hn(wr, n; G) is isomorphic with the group of homomorphisms of H5(ir,,n) into G, and H,(r, n) is isomorphic with 7r. Hence Hn(7r, n; 7r) is isomorphic with the group of endomorphisms of 7r; let dn be the element of Hn(7r, n; 7r) which acts as the identity endomorphism of 7r. For n > 2, 70:7rn(X) -rw+1(X) is a homomorphism with values in 2(7rn+l(X)) and therefore a commutative self-pairing of 7rn(X) to 7r,+l(X) can be found such that a. a = 710(a). It then follows that a! (3 always has order two and therefore Sqn_2:Hn(7rn(X), n; 7rn(X)) -H+2(n(X) n; 7rn+1(X)) is defined and is independent of the particular choice of the self-pairing. Using this pairing, we find THEOREM 6. If n > 2, Sqn52(dn) = kn+2. 1 Freudenthal, H., Comp. Math., 5, 299-314 (1937). 2 Fox, R. H., Bull. Am. Math. Soc., 49, abstract 172 (1943). 3 Hurewicz, W., Proc. Akad. Amsterdam, 39, 117-126'(1936). 4 Hopf, H., Comm. Math. Helv., 17, 307-326 (1945). 6 These results depend on Pontrjagin's result that TWn+2(S) = 0 for n 2 3. A proof was outlined in C. R. Acad. Sci. U.R.S.S., 19, 361-363 (1938), but a complete proof has not yet appeared. 6 Eilenberg, S., and MacLane, S., Ann. Math., 46, 489-509 (1945). 7 Cf. ]ilenberg, S., and MacLane, S., Proc. Nat. Acad. Sci., 32, 277-280 (1946) for the case n = 1. 8 Eilenberg, S., and MacLane, S., Ann. Math., 43, 757-831 (1942). 9 Steenrod, N. E., Ibid., 48, 290-320 (1947).
STEREOSCOPIC ACUITY FOR VARIOUS LEVELS OF ILL UMI-
7Dirac, P. A. M., Proc. Roy. Soc., A, 155, 447-459 (1936). 8 The literature on relativistic wave equations is very extensive. Besides the papers quoted in reference 11, we only mention the book by de Broglie, L., Theorie gene'rale des particides a' spin (Paris, 1943), and the following articles which give a systematic account of the subject: Pauli, W., Rev. Mod. Phys., 13, 203-232 (1941); Bhabha, H. J., Rev. Mlod. Phys., 17, 203-209 (1945); Kramers, H. A., Belinfante, F. J., and Lubanski, J. K., Physica, VIII, 597-627 (1941). In this paper, the sum of (14) over all v was postulated; (14a) then has to be added as an independent equation (except for N = 1). Reference 11 uses these equations In the form given by Kramers, Belinfante and Lubanski. 9 One may derive this result in a more elegant way, without specializing the co6rdinate system. For the sake of brevity, we omit this derivation. 10 de Wet, J. S., Phys. Rev., 58, 236-242 (1940), in particular, p. 242. "Wigner, E. P., Z. Physik, (1947).
Kinematic Modelling and Simulation of a 2R Robot Using SolidWorks and Verification by MATLAB / Simulink Regular Paper
The  simulation  of  robot  systems  is  becoming very popular, especially with  the  lowering of  the cost of computers,  and  it  can  be  used  for  layout  evaluation, feasibility studies, presentations with animation and off‐ line programming. The  trajectory  planning  of  redundant manipulators  is  a very  active  area  since  many  tasks  require  special characteristics  to  be  satisfied.  The  importance  of redundant manipulators has  increased over  the  last  two decades  because  of  the  possibility  of  avoiding singularities  as  well  as  obstacles  within  the  course  of motion.  The  angle  that  the  last  link  of  a  2  DOF manipulator makes with the x‐axis is required in order to find the solution for the inverse kinematics problem. This angle  could  be  optimized  with  respect  to  a  given specified  key  factor  (time,  velocity,  torques)  while  the end‐effector  performs  a  chosen  trajectory  (i.e., avoiding an obstacle) in the task space. Modeling  and  simulation  of  robots  could  be  achieved using  either  of  the  following  models:  the  geometrical model (positions, postures), the kinematic model and the dynamic model. To  do  so,  the  modelization  of  a  2‐R  robot  type  is implemented. Our main  tasks  are  comparing  two  robot postures with the same trajectory (path) and for the same length  of  time,  and  establishing  a  computing  code  to obtain the kinematic and dynamic parameters. SolidWorks  and MATLAB/Simulink  softwares  are  used to check the theory and the robot motion simulation. This  could  be  easily  generalized  to  a  3‐R  robot  and possibly therefore to any serial robot (Scara, Puma, etc.). The verification of the obtained results by both softwares allows  us  to  qualitatively  evaluate  and  underline  the validityof  the  chosen  model  and  obtain  the  right conclusions. The results of  the simulations are discussed and an agreement between the two softwares is certainly obtained.