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Abstract :Denoting these bounds by Nd(n), 0 ? d ? n we prove that Nd(n)/(n + 1)! is a polynomial Pd(n) of degree d with rational coefficients. We give the polynomials for d ? 5 explicitly. The proof uses the fact that these bounds Nd(n) are also the number of d-faces of the Voronoï cell of the weight lattice of the Lie algebra An (it is also the Cayley diagram of the symmetric group Sn+1, which is isomorphic to the Weyl group of An). Each d-face of this cell is a zonotope that can be defined by a symmetry group ~ G d (?), (d-dimensional reflection subgroup of the A n Weyl group. We show that for a given d and n large enough, all such subgroups of A n are represented, and we compute explicitly N(G d (?),n) the number of d-faces of type G d (?) in the Voronoï cell of L = A w n. The final result is obtained by summing over a. That also yields the simple expression Nd(n)=(n+1?d)!S(n+1?d)n+1Nd(n)=(n+1?d)!Sn+1(n+1?d) where the last symbol is the Stirling number of second kind.</text>
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                <text>Constraints on spin rotation parameters due to isospin conservation</text>
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&#13;
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&#13;
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                <text>Rydberg states of atoms and molecules. Basic group theoretical and topological analysis</text>
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                <text>Abstract : Rydberg states of atoms and molecules are studied within the qualitative approach-based primarily on topological and group theoretical analysis. The correspondence between classical and quantum mechanics is explored to apply the results of qualitative (topological) approach to classical mechanics developed by Poincaré, Lyapounov and Smale to quantum problems. The study of the action of the symmetry group of the problems considered on the classical phase space enables us to predict qualitative features of the energy level patterns for quantum Rydberg operators.</text>
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                <text>Static finite-energy solutions of Gauge fields with separated radial variable</text>
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                <text>Abstract : For arbitrary compact gauge group G and real representations of the Higgs fields, we seek static finite-energy solutions for which the radial dependence of the fields is factorized. We find that the gauge fields vanish outside a fixed SO(3) subgroup of G, and that inside SO(3) they reduce to the 't Hooft-Polyakov solution with unit magnetic charge. The Higgs fields may belong to any integer representation of this SO(3).</text>
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