Rotating frames in special relativity. II. Generalized theory

The transformation theory for rotating frames presented in a previous paper (Strauss, 1974) is generalized by replacing the usual conditionr=R for Rr=Rg( _R ) so thatr is now defined for all values ofR,0R. This generalization does not affect the kinematic transformation {,T}{^(r),{^(r)}}, and the result group structure required by the theoretical constraints previously established, provided the old parameter r (=R) is now identified throughout with eitherr orR; for physical reasons it must be identified withR. The functiong, which cannot be fixed by theoretical constraints, determines the degree of geometrical anisotropy in the rotating planez=const. More specifically, sinceg enters the expression for the ratioC/D (circumference/diameter) its choice corresponds to the choice of a congruence definition for lengths in radial and tangential directions. While on this (purely geometrical) levelg remains undetermined, it can be uniquely determined experimentally on the kinematic level, e.g., by observing in the motion of a free particle. Thus the supremacy of kinematics over geometry is explicated by a further instance. At the same time, special relativity theory (SRT) is shown to belong to the class of theories with theoretically unsolvable problems.     

Existence of solitary waves in higher dimensions

The elliptic equation u=F(u) possesses non-trivial solutions inR ⁿ which are exponentially small at infinity, for a large class of functionsF. Each of them provides a solitary wave of the nonlinear Klein-Gordon equation.This work was supported in part by NSF Grant MCS 75-08827     

Synthesis and stability of reactive salts of dodecafluoro-closo-dodecaborate(2-)

The synthesis and characterization of several salts of the B(12)F(12)(2-) anion are reported. The potassium salt was prepared in 72% recrystallized yield by treating K(2)B(12)H(12) with liquid HF at 70 degrees C for 14 h and 20% F(2)/N(2) in liquid HF at 25 degrees C for 72 h. The CPh(3)(+), N(n-Bu)(4)(+), NH(n-C(12)H(25))(3)(+), NH(4)(+), and Li(+) salts were prepared by metathesis reactions. The [NH(n-C(12)H(25))(3)](2)[B(12)F(12)] salt is soluble in aromatic hydrocarbon solvents. The B(12)F(12)(2-) anion is remarkably stable. The salts Li(2)B(12)F(12) and [NH(4)](2)[B(12)F(12)] were stable when heated to 450 and 480 degrees C, respectively. The B(12)F(12)(2-) anion did not react with 98% H(2)SO(4), 70% HNO(3), 3 M KOH, a 10-fold excess of Ce(NH(4))(2)(NO(3))(6) in aqueous solution, or metallic sodium in THF. In addition, B(12)F(12)(2-) did not react with metallic lithium in a mixture of ethylene carbonate and dimethyl carbonate, was not reduced at 0 V versus Li(+/0) in that solvent, and underwent a quasi-reversible oxidation at 4.9 V versus Li(+/0). The structure of [CPh(3)](2)[B(12)F(12)] was determined by single-crystal X-ray diffraction: tetragonal, space group I4(1)/acd, a = 19.102(2), b = 19.102(2), c = 20.535(3) A, V = 7492.2(2) A(3), Z = 8, T = 173(2) K, R(1) = 0.064. The B(12)F(12)(2-) anion weakly interacts with the two symmetry related CPh(3)(+) cations via F.C contacts of 3.087(2) A, which are very close to the 3.17 A sum of van der Waals radii for these two atoms. Taken together, the data suggest that B(12)F(12)(2-) may be useful as a very robust weakly coordinating anion.     

Top down characterization of larger proteins (45 kDa) by electron capture dissociation mass spectrometry.

The structural characterization of proteins expressed from the genome is a major problem in proteomics. The solution to this problem requires the separation of the protein of interest from a complex mixture, the identification of its DNA-predicted sequence, and the characterization of sequencing errors and posttranslational modifications. For this, the "top down" mass spectrometry (MS) approach, extended by the greatly increased protein fragmentation from electron capture dissociation (ECD), has been applied to characterize proteins involved in the biosynthesis of thiamin, Coenzyme A, and the hydroxylation of proline residues in proteins. With Fourier transform (FT) MS, electrospray ionization (ESI) of a complex mixture from an E. coli cell extract gave 102 accurate molecular weight values (2-30 kDa), but none corresponding to the predicted masses of the four desired enzymes for thiamin biosynthesis (GoxB, ThiS, ThiG, and ThiF). MS/MS of one ion species (representing approximately 1% of the mixture) identified it with the DNA-predicted sequence of ThiS, although the predicted and measured molecular weights were different. Further purification yielded a 2-component mixture whose ECD spectrum characterized both proteins simultaneously as ThiS and ThiG, showing an additional N-terminal Met on the 8 kDa ThiS and removal of an N-terminal Met and Ser from the 27 kDa ThiG. For a second system, the molecular weight of the 45 kDa phosphopantothenoylcysteine synthetase/decarboxylase (CoaBC), an enzyme involved in Coenzyme A biosynthesis, was 131 Da lower than that of the DNA prediction; the ECD spectrum showed that this is due to the removal of the N-terminal Met. For a third system, viral prolyl 4-hydroxylase (26 kDa), ECD showed that multiple molecular ions (+98, +178, etc.) are due to phosphate noncovalent adducts, and MS/MS pinpointed the overall mass discrepancy of 135 Da to removal of the initiation Met (131 Da) and to formation of disulfide bonds (2 x 2 Da) at C32-C49 and C143-C147, although 10 S-S positions were possible. In contrast, "bottom up" proteolysis characterization of the CoaBC and the P4H proteins was relatively unsuccessful. The addition of ECD substantially increases the capabilities of top down FTMS for the detailed structural characterization of large proteins.     

Homomorphisms into N *

We study the properties of continuous homomorphisms from β S into N ^* and from N ^* into N ^* . We show that the image C of N ^* under a continuous homomorphism which does not arise from the continuous extension of a homomorphism mapping N to itself, has the property that C+C is a singleton.