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This work, investigates the asymptotics for Euler’s q -exponential E q ( z ), Ramanujan’s function A q ( z ), Jackson’s q -Bessel function J v (2) ( z ; q ), the Stieltjes–Wigert orthogonal polynomials S n ( x ; q ) and q -Laguerre polynomials L n (α) ( x ; q ) as q approaches 1.
Let α( n ) denote the Fourier coefficients of cusp forms or the number of divisors of n . Estimates of the type are shown, uniformly in q ≤ X . The methods can be extended to other arithmetic functions, for example, the number of representations of n as a sum of two squares or k -free numbers....
It is proved that every positive operator R on a Banach lattice E dominated by a strictly singular operator T : E → E satisfies that the R 4 is strictly singular. Moreover, if E is order continuous then the R 2 is already strictly singular.
We prove analogues for reductive algebraic groups of some results for finite groups due to Knörr and Robinson from ‘Some remarks on a conjecture of Alperin’, J. London Math. Soc (2) 39 (1989), 48–60, which play a central rôle in their reformulation of Alperin's conjecture for finite groups.
We show that for every orthogonally additive scalar n -homogeneous polynomial P on a C*-algebra A there exists ϕ in A * satisfying P ( x )=ϕ ( x n ), for each element x in A . The vector-valued analogue follows as a corollary.
Let X be a compact quotient of the product of the real Heisenberg group H 4 m +1 of dimension 4 m + 1 and the three-dimensional real Euclidean space R 3 . A left-invariant hypercomplex structure on H 4 m +1 × R 3 descends onto the compact quotient X . The space X is a hyperholomorphic fibration...
We describe a double construction, which associates a symmetric associative algebra to a bialgebra. We show how a block of a finite group with cyclic defect can be realised via this double construction, after a felicitous choice of bialgebra.
We look at plane curve diagrams ( f ,α), which are given by a plane curve multigerm α : (ℝ, S ) → ℝ 2 and a function on it f :(ℝ, S ) → ℝ. We obtain a classification of all such diagrams, where α has 𝒜 e -codimension ≤ 2 and f has finite order. Then we define an equivalence...
Let α 1 , …,α r ∈ R be ‘not very well approximable’, for example, Q -linearly independent real algebraic numbers. Then there are infinitely many positive square-free integers n such that ‖ n α i ‖ << n −(2/3 r )+ϵ (1 ≤ i ≤ r ), where ‖·‖ denotes distance to the nearest...
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