We study the moduli space of rank 2 instanton sheaves on ℙ3 in terms of representations of a quiver consisting of three vertices and four arrows between two pairs of vertices. Aiming at an alternative compactification for the moduli space of instanton sheaves, we show that for each rank 2 instanton sheaf, there is a stability parameter θ for which the corresponding quiver representation is θ-stable (in the sense of King), and that the space of stability parameters has a non-trivial wall-and-chamber decomposition. Looking more closely at instantons of low charge, we prove that there are stability parameters with respect to which every representation corresponding to a rank 2 instanton sheaf of charge 2 is stable and provide a complete description of the wall-and-chamber decomposition for representation corresponding to a rank 2 instanton sheaf of charge 1.

Let M(u), H(u) be the maximal operator and Hilbert transform along the parabola (t, ut2). For U ⊂ (0, ∞) we consider Lp estimates for the maximal functions sup u∈U|M(u)f| and sup u∈U|H(u)f|, when 1 < p ≤ 2. The parabolas can be replaced by more general non-flat homogeneous curves.

We construct prime amphicheiral knots that have free period 2. This settles an open question raised by the second-named author, who proved that amphicheiral hyperbolic knots cannot admit free periods and that prime amphicheiral knots cannot admit free periods of order > 2.

The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

We describe the connected components of the space $\text {Hom}(\Gamma ,SU(2))$ of homomorphisms for a discrete nilpotent group $\Gamma$. The connected components arising from homomorphisms with non-abelian image turn out to be homeomorphic to $\mathbb {RP}^{3}$. We give explicit calculations when $\Gamma$ is a finitely generated free nilpotent group. In the second part of the paper, we study the filtration $B_{\text {com}} SU(2)=B(2,SU(2))\subset \cdots \subset B(q,SU(2))\subset \cdots$ of the classifying space $BSU(2)$ (introduced by Adem, Cohen and Torres-Giese), showing that for every $q\geq 2$, the inclusions induce a homology isomorphism with coefficients over a ring in which 2 is invertible. Most of the computations are done for $SO(3)$ and $U(2)$ as well.

Let r be an integer with 2 ≤ r ≤ 24 and let pr(n) be defined by $\sum _{n=0}^\infty p_r(n) q^n = \prod _{k=1}^\infty (1-q^k)^r$. In this paper, we provide uniform methods for discovering infinite families of congruences and strange congruences for pr(n) by using some identities on pr(n) due to Newman. As applications, we establish many infinite families of congruences and strange congruences for certain partition functions, such as Andrews's smallest parts function, the coefficients of Ramanujan's ϕ function and p-regular partition functions. For example, we prove that for n ≥ 0,

\[ \textrm{spt}\bigg( \frac{1991n(3n+1) }{2} +83\bigg) \equiv \textrm{spt}\bigg(\frac{1991n(3n+5)}{2} +2074\bigg) \equiv 0\ (\textrm{mod} \ 11), \]

\[ \textrm{spt}\bigg( \frac{143\times 5^{6k} +1 }{24}\bigg)\equiv 2^{k+2} \ (\textrm{mod}\ 11), \]

We address the problem of defining Lyapunov exponents for an expansive homeomorphism f on a compact metric space (X, dist) using similar techniques as those developed in Barreira and Silva [Lyapunov exponents for continuous transformations and dimension theory, Discrete Contin. Dynam. Sys.13 (2005), 469–490]; Kifer [Characteristic exponents of dynamical systems in metric spaces, Ergod. Th. Dynam. Sys.3 (1983), 119–127]. Under certain conditions on the topology of the space X where f acts we obtain that there is a metric D defining the topology of X such that the Lyapunov exponents of f are different from zero with respect to D for every point x ∈ X. We give an example showing that this may not be true with respect to the original metric dist. But expansiveness of f ensures that Lyapunov exponents do not vanish on a Gδ subset of X with respect to any metric defining the topology of X. We define Lyapunov exponents on compact invariant sets of Peano spaces and prove that if the maximal exponent on the compact set is negative then the compact is an attractor.

Let $\mathcal {P}(\mathbf{N})$ be the power set of N. We say that a function $\mu ^\ast : \mathcal {P}(\mathbf{N}) \to \mathbf{R}$ is an upper density if, for all X, Y ⊆ N and h, k ∈ N+, the following hold: (f1) $\mu ^\ast (\mathbf{N}) = 1$; (f2) $\mu ^\ast (X) \le \mu ^\ast (Y)$ if X ⊆ Y; (f3) $\mu ^\ast (X \cup Y) \le \mu ^\ast (X) + \mu ^\ast (Y)$; (f4) $\mu ^\ast (k\cdot X) = ({1}/{k}) \mu ^\ast (X)$, where k · X : = {kx: x ∈ X}; and (f5) $\mu ^\ast (X + h) = \mu ^\ast (X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)–(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that $\mu ^\ast (X)\le 1$ for every X ⊆ N. Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

In this paper, for a positive integer n ≥ 1, we look at the size and prime factors of the iterates of the Ramanujan τ function applied to n.

We show that Ringrose's diagonal ideals are primitive ideals in a nest algebra (subject to the continuum hypothesis). This answers an old question of Lance and provides for the first time concrete descriptions of enough primitive ideals to obtain the Jacobson radical as their intersection. Separately, we provide a standard form for all left ideals of a nest algebra, which leads to insights into the maximal left ideals. In the case of atomic nest algebras, we show how primitive ideals can be categorized by their behaviour on the diagonal and provide concrete examples of all types.

We enumerate the number of isoclinism classes of semi-extraspecial p-groups with derived subgroup of order p2. To do this, we enumerate GL (2, p)-orbits of sets of irreducible, monic polynomials in 𝔽p[x]. Along the way, we also provide a new construction of an infinite family of semi-extraspecial groups as central quotients of Heisenberg groups over local algebras.

In this paper the authors consider the weighted estimates for the Calderón commutator defined by

\mathcal{C}_{m+1, A}(a_1,\ldots,a_{m};f)(x)={\rm p. v.} \displaystyle\int_{\mathbb{R}}\displaystyle\frac{P_2(A; x, y)\prod\nolimits_{j=1}^m(A_j(x)-A_j(y))}{(x-y)^{m+2}}f(y){\rm d}y,

$L^{p_1}(\mathbb {R},w_1) \times \cdots \times L^{p_{m+1}}(\mathbb {R},w_{m+1})$

$L^{p}(\mathbb {R},\nu _{\vec {\kern 1pt w}})$

$\vec {\kern 1pt w}=(w_1, \ldots , w_{m+1})\in A_{\vec {P}}(\mathbb {R}^{m+1})$

$\mathcal {C}_{m+1, A}$

We consider a simple preferential attachment graph process, which begins with a finite graph and in which a new (t + 1)st vertex is added at each subsequent time step t that is connected to each previous vertex u ≤ t with probability du(t)/t, where du(t) is the degree of u at time t. We analyse the graph obtained as the infinite limit of this process, and we show that, as long as the initial finite graph is neither edgeless nor complete, with probability 1 the outcome will be a copy of the Rado graph augmented with a finite number of either isolated or universal vertices.

Can smoothing a single crossing in a diagram for a knot convert it into a diagram of the knot's mirror image? Zeković found such a smoothing for the torus knot T(2, 5), and Moore–Vazquez proved that such smoothings do not exist for other torus knots T(2, m) with m odd and square free. The existence of such a smoothing implies that K # K bounds a Mobius band in B4. We use Casson–Gordon theory to provide new obstructions to the existence of such chiral smoothings. In particular, we remove the constraint that m be square free in the Moore–Vazquez theorem, with the exception of m = 9, which remains an open case. Heegaard Floer theory provides further obstructions; these do not give new information in the case of torus knots of the form T(2, m), but they do provide strong constraints for other families of torus knots. A more general question asks, for each pair of knots K and J, what is the minimum number of smoothings that are required to convert a diagram of K into one for J. The methods presented here can be applied to provide lower bounds on this number.

Let R be a semiprime ring with the extended centroid C and Q the maximal right ring of quotients of R. Set [y, x]1 = [y, x] = yx − xy for x, y ∈ Q and inductively [y, x]k = [[y, x]k−1, x] for k > 1. Suppose that f : R → Q is an additive map satisfying [f(x), x]n = 0 for all x ∈ R, where n is a fixed positive integer. Then it can be shown that there exist λ ∈ C and an additive map μ : R → C such that f(x) = λx + μ(x) for all x ∈ R. This gives the affirmative answer to the unsolved problem of such functional identities initiated by Brešar in 1996.

Let f be a holomorphic self-map of the unit ball in dimension 2, which does not have an interior fixed point. Suppose that f has a Wolff point p with the boundary dilatation coefficient equal to 1 and the non-tangential differential dfp = id. The local behaviours of f near p are quite diverse, and we give a detailed study in many typical cases. As a byproduct, we give a dynamical interpretation of the Burns–Krantz rigidity theorem. Note also that similar results hold on two-dimensional contractible smoothly bounded strongly pseudoconvex domains.

We begin by defining a homoclinic class for homeomorphisms. Then we prove that if a topological homoclinic class Λ associated with an area-preserving homeomorphism f on a surface M is topologically hyperbolic (i.e. has the shadowing and expansiveness properties), then Λ = M and f is an Anosov homeomorphism.

In 1969, Baumslag introduced a family of parafree groups Gi,j which share many properties with the free group of rank 2. The isomorphism problem for the family Gi,j is known to be difficult; a few small partial results have been found so far. In this paper, we compute the twisted Alexander ideals of the groups Gi,j associated with non-abelian representations into $SL(2,{\mathbb Z}_2)$. Using the twisted Alexander ideals, we prove that several pairs of groups among Gi,j are not isomorphic. As a consequence, we solve the isomorphism problem for sub-families containing infinitely many groups Gi,j.

We extend the notions of μ*-sequences and Tjurina numbers of functions to the framework of Bruce–Roberts numbers, that is, to pairs formed by the germ at 0 of a complex analytic variety X ⊆ ℂn and a finitely ${\mathcal R}(X)$-determined analytic function germ f : (ℂn, 0) → (ℂ, 0). We analyze some fundamental properties of these numbers.

In this paper, we show that the numbers of t-stack sortable n-permutations with k − 1 descents satisfy central and local limit theorems for t = 1, 2, n − 1 and n − 2. This result, in particular, gives an affirmative answer to Shapiro's question about the asymptotic normality of the Narayana numbers.