In this paper it is shown that if the volume sum ∑r = 1∞ Ψ(r) converges for a monotonic function Ψ then the set of points (x, z, w) ∈ ℝ × ℂ × ℚp which simultaneously satisfy the inequalities |P(x)| ≤ H−v1 Ψλ1(H), |P(z)| ≤ H−v2 Ψλ2(H) and |P(w)|p ≤ H−v3 Ψλ3(H) with v1 + 2v2 + v3 = n − 3 and λ1 + 2λ2 + λ3 = 1 for infinitely many integer polynomials P has measure zero.

We obtain a number of new bounds for exponential sums of the type S(χ, f) = ∑x = 1p−1 χ(x) ep(f(x)), with p a prime, f(x) = ∑i = 1raixki, ai, ki ∈ ℤ, 1 ≤ i ≤ r and χ a multiplicative character (mod p). The bounds refine earlier Mordell-type estimates and are particularly effective for polynomials in which a certain number of the ki have a large gcd with p − 1. For instance, if f(x) = ∑i = 1maixki + g(xd) with d|(p − 1) then . If f(x) = axk + h(xd) with d|(p − 1) and (k, p − 1) = 1 then , and if f(x) = axk + bx−k + h(xd) with d|(p − 1) and (k, p − 1) = 1 then .

Let I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits also exist.

As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

We expand the notion of core to cl-core for Nakayama closures cl. In the characteristic p > 0 setting, when cl is the tight closure, denoted by *, we give some examples of ideals when the core and the *-core differ. We note that *-core(I) = core(I), if I is an ideal in a one-dimensional domain with infinite residue field or if I is an ideal generated by a system of parameters in any Noetherian ring. More generally, we show the same result in a Cohen–Macaulay normal local domain with infinite perfect residue field, if the analytic spread, ℓ, is equal to the *-spread and I is Gℓ and weakly-(ℓ − 1)-residually S2. This last is dependent on our result that generalizes the notion of general minimal reductions to general minimal *-reductions. We also determine that the *-core of a tightly closed ideal in certain one-dimensional semigroup rings is tightly closed and therefore integrally closed.

By a covering presentation of a 3-manifold, we mean a labelled link (i.e., a link with a monodromy representation), which presents the 3-manifold as the simple 4-fold covering space of the 3-sphere branched along the link with the given monodromy. It is known that two labelled links present a homeomorphic 3-manifold if and only if they are related by a finite sequence of some local moves. This paper presents a method for constructing topological invariants of 3-manifolds based on their covering presentations. The proof of the topological invariance is shown by verifying the invariance under the local moves. As an example of such invariants, we present the Dijkgraaf–Witten invariant of 3-manifolds.

In this paper, we introduce two new invariants that are closely related to Milnor's curvature-torsion invariant. The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points. This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties. The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor's curvature-torsion invariant is 6π.

Let K be a convex body which is (i) symmetric with respect to each of the coordinate hyperplanes and (ii) in isotropic position. We prove that most linear functionals acting on K exhibit super-Gaussian tail behavior. Using known facts about the mean-width of such bodies, we then deduce strong lower bounds for the volume of certain caps. We also prove a converse statement. Namely, if an arbitrary isotropic convex body (not necessarily satisfying (i)) exhibits similar cap-behavior, then one can bound its mean-width.

We introduce new formulations of aperiodicity and cofinality for finitely aligned higher-rank graphs Λ, and prove that C*(Λ) is simple if and only if Λ is aperiodic and cofinal. The main advantage of our versions of aperiodicity and cofinality over existing ones is that ours are stated in terms of finite paths. To prove our main result, we first characterise each of aperiodicity and cofinality of Λ in terms of the ideal structure of C*(Λ). In an appendix we show how our new cofinality condition simplifies in a number of special cases which have been treated previously in the literature; even in these settings our results are new.

Internal diffusion-limited aggregation is a growth model based on random walk in ℤd. We study how the shape of the aggregate depends on the law of the underlying walk, focusing on a family of walks in ℤ2 for which the limiting shape is a diamond. Certain of these walks—those with a directional bias toward the origin—have at most logarithmic fluctuations around the limiting shape. This contrasts with the simple random walk, where the limiting shape is a disk and the best known bound on the fluctuations, due to Lawler, is a power law. Our walks enjoy a uniform layering property which simplifies many of the proofs.

Let H be a Hamiltonian, e ∈ H(M) ⊂ ℝ and ƐH, e a connected component of H−1({e}) without singularities. A Hamiltonian system, say a triple (H, e, ƐH, e), is Anosov if ƐH, e is uniformly hyperbolic. The Hamiltonian system (H, e, ƐH, e) is a Hamiltonian star system if all the closed orbits of ƐH, e are hyperbolic and the same holds for a connected component of −1({ẽ}), close to ƐH, e, for any Hamiltonian , in some C2-neighbourhood of H, and ẽ in some neighbourhood of e.

In this paper we show that a Hamiltonian star system, defined on a four-dimensional symplectic manifold, is Anosov. We also prove the stability conjecture for Hamiltonian systems on a four-dimensional symplectic manifold. Moreover, we prove the openness and the structural stability of Anosov Hamiltonian systems defined on a 2d-dimensional manifold, d ≥ 2.