We study correlation bounds under pairwise independent distributions for functions with no large Fourier coefficients. Functions in which all Fourier coefficients are bounded by δ are called δ-uniform. The search for such bounds is motivated by their potential applicability to hardness of approximation, derandomization, and additive combinatorics.

In our main result we show that $\operatorname{\mathbb {E}}[f_{1}(X_{1}^{1},\ldots,X_{1}^{n}) \ldots f_{k}(X_{k}^{1},\ldots,X_{k}^{n})]$ is close to 0 under the following assumptions:

• the vectors $\{ (X_{1}^{j},\ldots,X_{k}^{j}) : 1 \leq j \leq n\}$ are independent identically distributed, and for each j the vector $(X_{1}^{j},\ldots,X_{k}^{j})$ has a pairwise independent distribution;

• the functions f_i are uniform;

• the functions f_i are of low degree.

We compare our result with recent results by the second author for low influence functions and to recent results in additive combinatorics using the Gowers norm. Our proofs extend some techniques from the theory of hypercontractivity to a multilinear setup.

The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle π in the middle points of a certain pair of its skew edges.

We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$, β>0, and λ∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not β-porous in scale λⁿ for n from a set with positive density amongst natural numbers.

We show almost sure simplicity of eigenvalues for several models of Anderson-type random Schrödinger operators, extending methods introduced by Simon for the discrete Anderson model. These methods work throughout the spectrum and are not restricted to the localization regime. We establish general criteria for the simplicity of eigenvalues which can be interpreted as separately excluding the absence of local and global symmetries, respectively. The criteria are applied to Anderson models with matrix-valued potential as well as with single-site potentials supported on a finite box.

We calculate the local Fourier transforms for connections on the formal punctured disk, reproducing the results of J. Fang and C. Sabbah using a different method. Our method is similar to Fang’s, but more direct.

We will introduce a quantity which measures the singularity of a plurisubharmonic function φ relative to another plurisubharmonic function ψ, at a point a. We denote this quantity by ν_{a, ψ}(φ). It can be seen as a generalization of the classical Lelong number in a natural way: if ψ=(n−1)log| ⋅ −a|, where n is the dimension of the set where φ is defined, then ν_{a, ψ}(φ) coincides with the classical Lelong number of φ at the point a. The main theorem of this article says that the upper level sets of our generalized Lelong number, i.e. the sets of the form {z: ν_{z, ψ}(φ)≥c} where c>0, are in fact analytic sets, provided that the weightψ satisfies some additional conditions.

Recently, Haghighi, Terai, Yassemi, and Zaare-Nahandi introduced the notion of a sequentially (S_r) simplicial complex. This notion gives a generalization of two properties for simplicial complexes: being sequentially Cohen–Macaulay and satisfying Serre’s condition (S_r). Let Δ be a (d−1)-dimensional simplicial complex with Γ(Δ) as its algebraic shifting. Also let (h_{i, j}(Δ))_{0≤j≤i≤d} be the h-triangle of Δ and (h_{i, j}(Γ(Δ)))_{0≤j≤i≤d} be the h-triangle of Γ(Δ). In this paper, it is shown that for a Δ being sequentially (S_r) and for every i and j with 0≤j≤i≤r−1, the equality h_{i, j}(Δ)=h_{i, j}(Γ(Δ)) holds true.

Let X⊂V be a closed embedding, with V∖X nonsingular. We define a constructible function ψ_{X, V} on X, agreeing with Verdier’s specialization of the constant function 1_V when X is the zero-locus of a function on V. Our definition is given in terms of an embedded resolution of X; the independence of the choice of resolution is obtained as a consequence of the weak factorization theorem of Abramovich–Karu–Matsuki–Włodarczyk. The main property of ψ_{X, V} is a compatibility with the specialization of the Chern class of the complement V∖X. With the definition adopted here, this is an easy consequence of standard intersection theory. It recovers Verdier’s result when X is the zero-locus of a function on V.

Our definition has a straightforward counterpart Ψ_{X, V} in a motivic group. The function ψ_{X, V} and the corresponding Chern class c_SM(ψ_{X, V}) and motivic aspect Ψ_{X, V} all have natural ‘monodromy’ decompositions, for any X⊂V as above.

The definition also yields an expression for Kai Behrend’s constructible function when applied to (the singularity subscheme of) the zero-locus of a function on V.

Let C and C′ be two smooth self-transverse immersions of S¹ into ℝ². Both C and C′ subdivide the plane into a number of disks and one unbounded component. An isotopy of the plane which takes C to C′ induces a one-to-one correspondence between the disks of C and C′. An obvious necessary condition for there to exist an area-preserving isotopy of the plane taking C to C′ is that there exists an isotopy for which the area of every disk of C equals that of the corresponding disk of C′. In this paper we show that this is also a sufficient condition.