Volume 22, Issue 2

In this paper, we consider the parabolic magnetic Schrödinger differential operators ∂ t + H ( a, V ) where the nonnegative potential V belongs to the reverse Hölder class. The L p (1 < p < ∞) and weak-type (1,1) estimates are obtained under certain conditions.

Cappell's unitary nilpotent groups are calculated for the integral group ring R = ℤ[ C 2 ] of the cyclic group C 2 of order two. Specifically, they are determined as modules over the Verschiebung algebra 𝒱 using the Connolly–Ranicki isomorphism [Adv. Math. 195: 205–258, 2005] and the Connolly–Davis relations [Geometry and Topology 8: 1043–1078, 2004].

Let ℳ be a W *-algebra with a faithful normal finite trace and think of the unitary group U ℳ endowed with the ultraweak topology. By using the measure topology associated with the aforementioned trace, we prove that U ℳ is a topological group with Lie algebra in the sense of K.H. Hofmann and S.A. Morris (J. Group Theory 8: 115–133, 2005). We discuss the corresponding exponential map and find that it is never locally injective in the case of factors of type II 1 .

We prove integral formulas for closed hypersurfaces in ℂ n +1 , which furnish a relation between elementary symmetric functions in the eigenvalues of the complex Hessian matrix of the defining function and the Levi curvatures of the hypersurface. Then we follow the Reilly approach to prove an isoperimetric inequality. As an application, we obtain the “Soap Bubble Theorem” for star-shaped domains with positive and constant Levi curvatures bounding the classical mean curvature from above.

For a finite group G it was known that the abelian group NK 0 (ℤ G ) is always finitely generated over the Verschiebung algebra. This result inspired Connolly and the author to asked the question whether NK 1 (ℤ G ) is also finitely generated over the Verschiebung algebra. Based on a computation of Weibel we obtain a negative answer to this question by proving that NK 1 (ℤ[ C 2 × C 2 ]) is not finitely generated over the larger Verschiebung and Frobenius algebra. This raises the question of a different finiteness of NK 1 . In the last part we show that the filtration on NK 1 (ℤ G ) which is induced by the degree of a polynomial has the property that the quotients are finite groups.

We extend Hendriks' classification theorem and Turaev's realisation and splitting theorems for PD 3 -complexes to the relative case of PD 3 -pairs. The results for PD 3 -complexes are recovered by restricting the results to the case of PD 3 -pairs with empty boundary. Up to oriented homotopy equivalence, PD 3 -pairs are classified by their fundamental triple consisting of the fundamental group system, the orientation character and the image of the fundamental class under the classifying map. Using the derived module category we provide necessary and sufficient conditions for a given triple to be realised by a PD 3 -pair. The results on classification and realisation yield splitting or decomposition theorems for PD 3 -pairs, that is, conditions under which a given PD 3 -pair decomposes as interior or boundary connected sum of two PD 3 -pairs.

We consider probability distributions which are uniformly distributed on a disjoint union of balls with equal radius. For small enough radius the optimal quantization error is calculated explicitly in terms of the ball centroids. We apply the results to special self-similar measures.

The unipotent subgroup of a finite group of Lie type over a prime field 𝔽 p comes equipped with a natural set of generators; the properties of the Cayley graph associated to this set of generators have been much studied. In the present paper, we show that the diameter of this Cayley graph is bounded above and below by constant multiples of np + n 2 log p , where n is the rank of the associated Lie group. This generalizes the result of [Ellenberg, A sharp diameter bound for an upper triangular matrix group, Harvard University, 1993], which treated the case of SL n (𝔽 p ).

We discuss the Bousfield localization L E X for any spectrum E and any HR -module X , where R is a ring with unit. Due to the splitting property of HR -modules, it is enough to study localizations of Eilenberg–MacLane spectra. Using general results about stable localizations, we give a method to compute the localization L E HG for any spectrum E and any abelian group G , and describe it explicitly when G belongs to certain classes of abelian groups. The results depend basically on the E -acyclicity patterns of H ℚ and H ℤ/ p for each prime p , as in earlier work of Bousfield in the unstable case.

In this article, we study the structure maps of the A ∞ coalgebra structure for the homology H ∗( K ( π, n ); ℤ p ) of an Eilenberg-Mac Lane space K ( π, n ), where π is a finitely generated abelian group, n is a positive integer and p is a prime number different to 2. Using diverse techniques of homological perturbation, we obtain that all the components of Δ i vanish except for Δ i ( p –2)+2 of degree i ( p –2) and i ≥ 0.

Let Γ = G 1 ∗ G 2 ∗ ⋯ ∗ G n be an n -fold free product. For g i ∈ G i and g k ∈ G k , let be the automorphism induced by where means conjugating g k by g j ). Then is a Whitehead automorphism , and the group generated by all is known at the Whitehead automorphism group of Γ , denoted Wh( Γ ). In this paper, we calculate H *(Wh( Γ )). The main tools used are the action of Wh( Γ ) on a space constructed by McCullough-Miller and the equivariant spectral sequence. We also specialize to the case of Γ a free Coxeter group, where further analysis allows us to determine a presentation for the cohomology ring.

Let , n ⩾ 2, be the classical motion group, where SO ( n ) acts on by rotation. In this paper, we identify the dual space , i.e. the set of equivalence classes of irreducible unitary representations of M n , with the quotient space of admissible co-adjoint orbits via a nice geometric parameterization of the two spaces. We show then that this identification is a homeomorphism.