We consider separately radial (with corresponding group ${\mathbb{T}}^n$) and radial (with corresponding group
${\rm U}(n))$ symbols on the projective space ${\mathbb{P}^n({\mathbb{C}})}$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^*$-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the $C^*$-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between ${\mathbb{T}}^n$ and ${\rm U}(n)$., Raul Quiroga-Barranco, Armando Sanchez-Nungaray., and Obsahuje bibliografii
We determine in \mathbb{R}^{n} the form of curves C corresponding to strictly monotone functions as well as the components of affine connections \Delta for which any image of C under a compact-free group Ω of affinities containing the translation group is a geodesic with respect to \Delta . Special attention is paid to the case that Ω contains many dilatations or that C is a curve in \mathbb{R}^{3}. If C is a curve in \mathbb{R}^{3} and Ω is the translation group then we calculate not only the components of the curvature and the Weyl tensor but we also decide when \Delta yields a flat or metrizable space and compute the corresponding metric tensor., Josef Mikeš, Karl Strambach., and Obsahuje seznam literatury
Let (M, g) be a 4-dimensional Einstein Riemannian manifold. At each point p of M, the tangent space admits a so-called Singer-Thorpe basis (ST basis) with respect to the curvature tensor R at p. In this basis, up to standard symmetries and antisymmetries, just 5 components of the curvature tensor R are nonzero. For the space of constant curvature, the group O(4) acts as a transformation group between ST bases at TpM and for the so-called 2-stein curvature tensors, the group Sp(1) ⊂ SO(4) acts as a transformation group between ST bases. In the present work, the complete list of Lie subgroups of SO(4) which act as transformation groups between ST bases for certain classes of Einstein curvature tensors is presented. Special representations of groups SO(2), T2, Sp(1) or U(2) are obtained and the classes of curvature tensors whose transformation group into new ST bases is one of the mentioned groups are determined., Zdeněk Dušek, Hradec Králové., and Obsahuje seznam literatury
We obtain a conditional, under the Generalized Riemann Hypothesis, lower bound on the number of distinct elliptic curves E over a prime finite field F_{p} of p elements, such that the discriminant D(E) of the quadratic number field containing the endomorphism ring of E over F_{p} is small. For almost all primes we also obtain a similar unconditional bound. These lower bounds complement an upper bound of F. Luca and I.E. Shparlinski (2007)., Igor E. Shparlinski., and Obsahuje seznam literatury
The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic ω-α-Bloch space and characterize it in terms of \omega \left( {{{\left( {1 - {{\left| x \right|}^2}} \right)}^\beta }{{\left( {1 - {{\left| y \right|}^2}} \right)}^{\alpha - \beta }}} \right)\left| {\frac{{f\left( x \right) - f\left( y \right)}}{{x - y}}} \right| and \omega \left( {{{\left( {1 - {{\left| x \right|}^2}} \right)}^\beta }{{\left( {1 - {{\left| y \right|}^2}} \right)}^{\alpha - \beta }}} \right)\left| {\frac{{f\left( x \right) - f\left( y \right)}}{{\left| x \right|y - x'}}} \right| where ω is a majorant. Similar results are extended to harmonic little ω-α-Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G.Ren, U.Kähler (2005)., Xi Fu, Bowen Lu., and Obsahuje seznam literatury
Euler's pentagonal number theorem was a spectacular achievement at the time of its discovery, and is still considered to be a beautiful result in number theory and combinatorics. In this paper, we obtain three new finite generalizations of Euler's pentagonal number theorem., Ji-Cai Liu., and Seznam literatury