[en] When reparamentrizing time in a single-particle system we show that the largest time symmetry is SO(2,1). We explicitly present the general systems which possess such a symmetry. (orig.)

[en] We study the configuration space of the Tomboulis SO(3)xZ(2) formulation with periodic boundary conditions. The dynamical variables are constrained by the required coincidence of Z(2) and SO(3) monopoles. Furthermore, there is an additional constraint coming from the boundary conditions. We propose an update algorithm that satisfies the constraints and is straightforward to implement. We further prove that it reaches all configurations. We show that configurations that are topologically stable due to boundary conditions cannot be reached from the identity configuration; they have zero weight. We measure gauge invariant vortex counters for ''thin,'' ''thick,'' and ''hybrid'' vortex sheets. We also measure projection vortex counters defined in the maximal center gauge for comparison. (c) 2000 The American Physical Society

[en] The reformulation of Lieb's entropy conjecture, in the frame of the harmonic analysis on the SO(3) group, makes it evident that the exact value of the classical entropy of a pure quantum state, which belongs to the Hilbert space Hsub(J) of a (2J+1) - dimensional, unitary, irreducible representation Usub(J) of the SO(3) group, depends only on the parameters which characterize the orbits of Usub(J) in Hsub(J). In the case J = 1 we give the exact analytic dependence of the classical entropy of a quantum state on the parameter which characterizes the orbits and as a consequence we obtain a verification of Lieb's entropy conjecture. We verify this conjecture also for any value of J for the states of the canonical basis of Hsub(J). A natural generalization of Lieb's entropy conjecture, which is a new ''phenomenon'' in the harmonic analysis on SO(3), is discussed in the case J = 1. (author)

[en] Subgroups of the special orthogonal group SO(3, R) that are generated by two rotations of finite order about axes that are also separated by an angle of finite order are sometimes called generalised dihedral groups. They appear as orientation groups in the theory of tilings of Euclidean 3-space. As our main result, we classify the generalised dihedral subgroups of SO(3, Q) as the finite subgroups of SO(3, Q) except the alternating group A₄.

[en] In this paper, the authors discuss N = 3 super Riemann surfaces. When SO(3) modular parameters are souls (nilpotent parts), we obtain the general supermodular transformation on N = 3 supertorus. In this case, SO(3) moduli are dependent parameters which are proportional to even and odd moduli (τ and εⁱ) through the constants

[en] The representation theory of the SO(3) invariant superconformal algebra is discussed. The necessary and sufficient condition of unitarity, the Kac determinants, the character formulae for unitary representations and their modular transformation laws are obtained under some assumptions

[en] In the presence of texture, the concept of X-ray elastic constants as well as sin²Ψ law is inapplicable and the X-ray stress factors connecting average strain and stress have to be used. The SO(3) vector parameterization with smart composition law proved to be a powerful tool for handling transformations between reference systems used in this approach. Decomposition of the 4-th rank elastic constant tensor on the SO(3) irreducible representations allows one to highlight the symmetry properties and to separate isotropic and anisotropic parts. Use of the vector parameterization and irreducible representations for this group enables to obtain transparent analytical expressions for the X-ray stress factors in case of textures described by preferred spherical/fiber components. (authors)

[en] We consider a product of two finite order quantum $SU(2)$-gates U ₁, U ₂ and ask when $U_1\cdot <!--\; ???\; -->U_2$ has an infinite order. Using the fact that $SU(2)$ is a double cover of $SO(3)$ we actually study the product $O(\mathrm{\gamma <!--\; ??\; -->},{\stackrel{\to <!--\; ???\; -->}{k}}_{12})$ of two rotations $O(\mathrm{\varphi <!--\; ??\; -->},{\stackrel{\to <!--\; ???\; -->}{k}}_1)\in <!--\; ???\; -->SO(3)$ and $O(\mathrm{\varphi <!--\; ??\; -->},{\stackrel{\to <!--\; ???\; -->}{k}}_2)\in <!--\; ???\; -->SO(3)$ about axes ${\stackrel{\to <!--\; ???\; -->}{k}}_1$, ${\stackrel{\to <!--\; ???\; -->}{k}}_2\in <!--\; ???\; -->{\mathbb{R}}^3$. In particular, we focus on the case when ${\stackrel{\to <!--\; ???\; -->}{k}}_1\cdot <!--\; ???\; -->{\stackrel{\to <!--\; ???\; -->}{k}}_2=0$, and ${\mathrm{\varphi <!--\; ??\; -->}}_1=\mathrm{\varphi <!--\; ??\; -->}={\mathrm{\varphi <!--\; ??\; -->}}_2$ are rational multiples of π and show that γ is not a rational multiple of π unless $\mathrm{\varphi <!--\; ??\; -->}\in <!--\; ???\; -->\{\frac{k\mathrm{\pi <!--\; ??\; -->}}{2}:k\in <!--\; ???\; -->\mathbb{Z}\}$. The proof presented in this paper boils down to finding all pairs $\mathrm{\gamma <!--\; ??\; -->},\mathrm{\varphi <!--\; ??\; -->}\in <!--\; ???\; -->\{a\mathrm{\pi <!--\; ??\; -->}:a\in <!--\; ???\; -->\mathbb{Q}\}$ that are solutions of $\cos <!--\; ???\; -->\frac{\mathrm{\gamma <!--\; ??\; -->}}{2}={\cos }^2<!--\; ???\; -->\frac{\mathrm{\varphi <!--\; ??\; -->}}{2}$. (paper)

[en] The internal symmetry of the 8-component real Dirac field has been investigated in matrix approach. The classical formulation of the theory has the internal symmetry group SO(3,2). The quantum formulation has the internal symmetry group SO(3). (authors)

[en] The isomorphism between Drinfeld’s new realization and the FRT realization is proved for the Yangian algebra Y(so₃) using Gauss decomposition. (paper)