Volume 28, Issue 2

We give some explicit conditions under which the singular values of Δ-quotients generate ring class fields over imaginary quadratic fields.

Let p be a fixed prime number. The main purpose of this paper is to introduce irreducible p -local compact groups and to describe all p -local compact groups of rank 1 in terms of this concept.

Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra 𝔫 such that 𝔫=𝔞⊕𝔟⊕𝔷${\mathfrak {n=a\oplus b\oplus z}}$, [𝔞,𝔟]⊆𝔷${[ \mathfrak {a},\mathfrak {b}] \subseteq \mathfrak {z}}$, the algebras 𝔞,𝔟,𝔷${\mathfrak {a},\mathfrak {b,z}}$ are abelian, 𝔞=ℝ- span {X1,X2,...,Xd}${\mathfrak {a}=\mathbb {R}\mathrm {\hbox{-}span}\lbrace X_{1},X_{2},\ldots ,X_{d}\rbrace }$, and 𝔟=ℝ- span {Y1,Y2,...,Yd}${\mathfrak {b}=\mathbb {R}\mathrm {\hbox{-}span}\lbrace Y_{1},Y_{2},\ldots ,Y_{d}\rbrace }$. Also, we assume that det[[Xi,Yj]]1≤i,j≤d${\det [ [ X_{i} ,Y_{j}] ] _{1\le i,j\le d}}$ is a non-vanishing homogeneous polynomial in the unknowns Z1,...,Zn-2d${Z_{1},\ldots ,Z_{n-2d}}$ where {Z1,...,Zn-2d}${\lbrace Z_{1},\ldots ,Z_{n-2d}\rbrace }$ is a basis for the center of the Lie algebra. Using well-known facts from time-frequency analysis, we provide some precise sufficient conditions for the existence of sampling spaces with the interpolation property, with respect to some discrete subset of N . The result obtained in this work can be seen as a direct application of time-frequency analysis to the theory of nilpotent Lie groups. Several explicit examples are computed. This work is a generalization of recent results obtained for the Heisenberg group by Currey and Mayeli in [Rocky Mountain J. Math. 42 (2012), no. 4, 1135–1151].

Let G be a real reductive algebraic group with maximal compact subgroup K , and let F r be a rank r free group. We show that the space of closed orbits in Hom( F r , G )/ G admits a strong deformation retraction to the orbit space Hom( F r , K )/ K . In particular, all such spaces have the same homotopy type. We compute the Poincaré polynomials of these spaces for some low rank groups G , such as Sp(4,ℝ) and U(2,2). We also compare these real moduli spaces to the real points of the corresponding complex moduli spaces, and describe the geometry of many examples.

This paper discusses topological and locally linear actions of finite groups on S 4 . Local linearity of the orientation preserving actions on S 4 forces the group to be a subgroup of SO(5). On the other hand, orientation reversing topological actions of “exotic” groups G (i.e., G ¬⊂ O(5)) on S 4 are constructed, and local linearity and stable smoothability of the actions are studied.

Minkowski's second theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of an o -symmetric convex body in terms of its successive minima. In this paper we study analogs of this theorem from two different points of view: either relaxing the symmetry condition, assuming that the centroid of the body lies at the origin, or replacing the volume functional by the surface area.

A one-relator group with lower central series of length ω 2 is constructed. This answers a problem of G. Baumslag.

We construct a finitely-presented group such that its Vogel–Levine localization is not transfinitely nilpotent. This answers a problem of J. P. Levine.

A time inhomogeneous generalized Mehler semigroup on a real separable Hilbert space ℍ is defined through p s , t f ( x ) = ∫ ℍ f ( U ( t , s ) x + y )μ t , s ( d y ), s , t ∈ ℝ, t ≥ s , x ∈ ℍ, for every bounded measurable function f on ℍ, where ( U ( t , s )) t ≥ s is an evolution family of bounded operators on ℍ and (μ t , s ) t ≥ s is a family of probability measures on (ℍ,ℬ(ℍ)) satisfying the following time inhomogeneous skew convolution equations: μ t , s = μ t , r * (μ r , s ∘ U ( t , r ) -1 ), t ≥ r ≥ s . This kind of semigroups typically arise as the “transition semigroups” of non-autonomous (possibly non-continuous) Ornstein–Uhlenbeck processes driven by some proper additive process. Suppose that μ t , s converges weakly to δ 0 as t ↓ s or s ↑ t . We show that μ t , s has further weak continuity properties in t and s . As a consequence, we prove that for every t ≥ s , μ t , s is infinitely divisible. Natural stochastic processes associated with (μ t , s ) t ≥ s are constructed and are applied to get probabilistic proofs for the weak continuity and infinite divisibility. Then we analyze the structure, existence and uniqueness of the corresponding evolution systems of measures (= space-time invariant measures) of ( p s , t ) t ≥ s . We also establish a dimension free Harnack inequality for ( p s , t ) t ≥ s and present some of its applications.

We describe explicitly the Auslander–Reiten translation in the category of bounded complexes of finitely generated maximal Cohen–Macaulay modules, 𝐂 b (CM R ), over a commutative local Cohen–Macaulay ring R with a canonical module ω. Then the Auslander–Reiten formula is generalized for complexes in 𝐂 b (CM R ) and we prove the existence theorem of Auslander–Reiten sequences. As an application of our results, we investigate the existence of Auslander–Reiten triangles in the category of perfect complexes as a full triangulated subcategory of 𝐃 b (mod R ).

We prove that, for N /( N -2) < p < ∞, α > 2, the operator L u = (1 + | x | 2 ) α/2 Tr( A ( x ) D 2 u ) = (1 + | x | 2 ) α/2 ∑ i , j =1,..., N a ij ( x ) D ij u admits realizations generating analytic semigroups in L p (ℝ N ). Moreover, for α < N / p ', we provide an explicit description of the domain.

We give an upper bound for the exponential sum ∑ m =1,..., M exp(2 i π f ( m )) where f is a real-valued function whose fourth derivative has the order of magnitude λ > 0 small. Van der Corput's classical bound, in terms of M and λ only, involves the exponent 1/14. We show how this exponent may be replaced by any θ < 1/12 without further hypotheses. The proof uses a recent result by Wooley on the cubic Vinogradov system.