For P ⊂ 3 a finite volume geodesic polyhedron, with the property that all interior angles between incident faces are of the form π/mij (mij ≥ 2 an integer), there is a naturally associated Coxeter group ΓP. Furthermore, this Coxeter group is a lattice inside the semi-simple Lie group O+(3, 1) = Isom(3), with fundamental domain the original polyhedron P. In this paper, we provide a procedure for computing the lower algebraic K-theory of the integral group ring of such groups ΓP in terms of the geometry of the polyhedron P. As an ingredient in the computation, we explicitly calculate the K−1 and Wh of the groups Dn and Dn × 3, and we also summarize what is known about the 0.

We extend the classical formula of Porteous for blowing-up Chern classes to the case of blow-ups of possibly singular varieties along regularly embedded centers. The proof of this generalization is perhaps conceptually simpler than the standard argument for the nonsingular case, involving Riemann–Roch without denominators. The new approach relies on the explicit computation of an ideal, and a mild generalization of a well-known formula for the normal bundle of a proper transform ([8, B·6·10]).

We also discuss alternative, very short proofs of the standard formula in some cases: an approach relying on the theory of Chern–Schwartz–MacPherson classes (working in characteristic 0), and an argument reducing the formula to a straightforward computation of Chern classes for sheaves of differential 1-forms with logarithmic poles (when the center of the blow-up is a complete intersection).

For C*-algebras A1, A2 the map (I1, I2) → ker(qI1 ⊗ qI2) from Id′(A1) × Id′(A2) into Id′(A1 ⊗minA2) is a homeomorphism onto its image which is dense in the range. Here, for a C*-algebra A, the space of all proper closed two sided ideals endowed with an adequate topology is denoted Id′(A) and qI is the quotient map of A onto A/I. This result is used to show that any continuous function on Prim(A1) × Prim(A2) with values into a T1 topological space can be extended to Prim(A1 ⊗minA2). This enlarges the scope of [7, corollary 3·5] that dealt only with scalar valued functions. A new proof for a result of Archbold [3] about the space of minimal primal ideals of A1 ⊗minA2 is obtained also by using the homeomorphism mentioned above. New proofs of the equivalence of the property (F) of Tomiyama for A1 ⊗minA2 with certain other properties are presented.

Let G be a compact, connected, abelian group with dual group Γ. The set E ⊂ has zero discrete harmonic density (z.d.h.d.) if for every open U ⊂ G and μ ∈ Md(G) there exists ν ∈ Md(U) with = on E. I0 sets in the duals of these groups have z.d.h.d. We give properties of such sets, exhibit non-Sidon sets having z.d.h.d., and prove union theorems. In particular, we prove that unions of I0 sets have z.d.h.d. and provide a new approach to two long-standing problems involving Sidon sets.

We study the squaring operation Sq0 on the dual of the minimal -generators of the Dickson algebra. We show that this squaring operation is isomorphic on its image. We also give vanishing results for this operation in some cases. As a consequence, we prove that the Lannes–Zarati homomorphism vanishes (1) on every element in any finite Sq0-family in except possibly the family initial element, and (2) on almost all known elements in the Ext group. This verifies a part of the algebraic version of the classical conjecture on spherical classes.

We show that when r ≥ 5 is prime, the SO(3) Witten–Reshetikhin–Turaev quantum invariants for three-manifolds at the level r form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.

The radial index of a 1-form on a singular set is a generalization of the classical Poincaré–Hopf index. We consider different classes of closed singular semi-analytic sets in n that contain 0 in their singular locus and we relate the radial index of a 1-form at 0 on these sets to Poincaré–Hopf indices at 0 of vector fields defined on n.

The inverse problem for Schrödinger operators on metric graphs is investigated in the presence of magnetic field. Graphs without loops and with Euler characteristic zero are considered. It is shown that the knowledge of the Titchmarsh–Weyl matrix function (Dirichlet-to-Neumann map) for just two values of the magnetic field allows one to reconstruct the graph and potential on it provided a certain additional no-resonance condition is satisfied.

We apply the averaging theory of first, second and third order to the class of generalized polynomial Liénard differential equations. Our main result shows that for any n, m ≥ 1 there are differential equations of the form ẍ + f(x)ẋ + g(x) = 0, with f and g polynomials of degree n and m respectively, having at least [(n + m − 1)/2] limit cycles, where [·] denotes the integer part function.