If λ = (λ1,…, λm) where λ1,…,λm are nonnegative integers with λ1 ≥…≥ λm, then λ is a partition of |λ| = λ1 + …+λm, and we write λ⊢ |λ|. The non-zero λi's are the parts of λ, so λ1 is the largest part, and ℓ(λ) is the number of parts of λ. Two partitions with the same parts, so they differ only in number of zeros, are the same. The set of all partitions, including the partition of 0 (with 0 parts) is denoted by The conjugate of λ, denoted by , is the partition (μ1,…, μk), in which μi is the number of λ's that are ≥i , for i = 1,…, k, where k=λ1.

Several results in the existing literature establish Euclidean density theorems of the following strong type. These results claim that every set of positive upper Banach density in the Euclidean space of an appropriate dimension contains isometric copies of all sufficiently large elements of a prescribed family of finite point configurations. So far, all results of this type discussed linear isotropic dilates of a fixed point configuration. In this paper, we initiate the study of analogous density theorems for families of point configurations generated by anisotropic dilations, i.e., families with power-type dependence on a single parameter interpreted as their size. More specifically, we prove nonisotropic power-type generalizations of a result by Bourgain on vertices of a simplex, a result by Lyall and Magyar on vertices of a rectangular box, and a result on distance trees, which is a particular case of the treatise of distance graphs by Lyall and Magyar. Another source of motivation for this paper is providing additional evidence for the versatility of the approach stemming from the work of Cook, Magyar, and Pramanik and its modification used recently by Durcik and the present author. Finally, yet another purpose of this paper is to single out anisotropic multilinear singular integral operators associated with the above combinatorial problems, as they are interesting on their own.

In the paper we formulate a Covering Property Axiom, $\text{CP}{{\text{A}}_{\text{prism}}}$, which holds in the iterated perfect set model, and show that it implies the following facts, of which (a) and (b) are the generalizations of results of J. Steprāns.

(a) There exists a family $\mathcal{F}$ of less than continuum many ${{C}^{1}}$ functions from $\mathbb{R}$ to $\mathbb{R}$ such that ${{\mathbb{R}}^{2}}$ is covered by functions from $\mathcal{F}$, in the sense that for every $\left\langle x,\,y \right\rangle \,\in \,{{\mathbb{R}}^{2}}$${{\mathbb{R}}^{2}}$ there exists an $f\,\in \,\mathcal{F}$ such that either $f\left( x \right)\,=\,y$ or $f\left( y \right)\,=\,x$.

(b) For every Borel function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many “${{C}^{1}}$” functions (i.e., differentiable functions with continuous derivatives, where derivative can be infinite) whose graphs cover the graph of $f$.

(c) For every $n\,>\,0$ and a ${{D}^{n}}$ function $f:\,\mathbb{R}\,\to \,\mathbb{R}$ there exists a family $\mathcal{F}$ of less than continuum many ${{C}^{n}}$ functions whose graphs cover the graph of $f$.

We also provide the examples showing that in the above properties the smoothness conditions are the best possible. Parts (b), (c), and the examples are closely related to work of A. Olevskiî.

We give mild sufficient conditions on a nonlinear functional to have eigenvalues. These results are intended for the study of boundary value problems for semilinear elliptic equations.

Let L be an orthomodular poset. A positive measure ξ on L is said to be weakly purely finitely additive if the zero measure is the only completely additive measure majorized by ξ. It was shown in [15] that, in an arbitrary orthomodular poset L, every positive measures μ is the sum v + ξ of a positive completely additive measure v and a weakly purely finitely additive measure ξ. We give sufficient conditions for this Yosida-Hewitt-type decomposition to be unique.

A positive measure λ on L is said to be filtering if every non-zero element p in L majorizes a non-zero element q on which λ vanishes. A filtering measure is weakly purely finitely additive. Filtering measures play a mediator role throughout these investigations since some of the aforementioned conditions are given in terms of these.

The results obtained here are then viewed in the context of Boolean lattices and applied to lattices of idempotents of non-associative JBW-algebras.

This paper has been motivated by earlier work of the first two authors (see [3] ), where distinct Nielsen classes of generating systems for a Fuchsian group have been established and, in the case of odd and pairwise relative prime exponents π(i),classified. As a consequence they could distinguish nonisotopic Heegaard decompositions of Seifert fibred 3-manifolds. In proving that these decompositions are actually non-homeomorphic (see [3], Section 2), they investigated the question whether the different Nielsen classes of generating systems for G remain distinct, if one passes over to the weaker notion of “Nielsen equivalence up to automorphisms” (see [12], p. 3.5, 4.11 a-c): this means that the automorphisms of G are added to the Nielsen equivalence relations on the generators.

Let $\mathbb{G}$ be a step-two nilpotent group of $\text{H}$-type with Lie algebra $\mathfrak{G}\,=\,V\,\oplus \,\text{t}$. We define a class of vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ on $\mathbb{G}$ depending on a real parameter $k\,\ge \,1$, and we consider the corresponding $p$-Laplacian operator ${{L}_{p,\,k}}u\,=\,di{{v}_{X}}\left( {{\left| {{\nabla }_{X}}u \right|}^{p-2}}{{\nabla }_{X}}u \right)$. For $k\,=\,1$ the vector fields $X\,=\,\left\{ {{X}_{j}} \right\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb{G}$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator ${{L}_{p,\,k}}$ and as an application, we get a Hardy type inequality associated with $X$.

For amenable Lie groups of NC-type the heat kernel satisfies pt ~ t-a. We find the exact value of a ≥ 0.

Let $(Y,\,T)$ be a minimal suspension flow built over a dynamical system $(X,\,S)$ and with (strictly positive, continuous) ceiling function $f:\,X\,\to \,\mathbb{R}$. We show that the eigenvalues of $(Y,\,T)$ are contained in the range of a trace on the ${{K}_{0}}$-group of $(X,\,S)$. Moreover, a trace gives an order isomorphism of a subgroup of ${{K}_{0}}\left( C(X)\,{{\rtimes }_{S}}\,\mathbb{Z} \right)$ with the group of eigenvalues of $(Y,\,T)$. Using this result, we relate the values of $t$ for which the time-$t$ map on the minimal suspension flow is minimal with the $K$-theory of the base of this suspension.

For a smooth strongly convex Minkowski norm $F:\mathbb {R}^n \to \mathbb {R}_{\geq 0}$ , we study isometries of the Hessian metric corresponding to the function $E=\tfrac 12F^2$ . Under the additional assumption that F is invariant with respect to the standard action of $SO(k)\times SO(n-k)$ , we prove a conjecture of Laugwitz stated in 1965. Furthermore, we describe all isometries between such Hessian metrics, and prove Landsberg Unicorn Conjecture for Finsler manifolds of dimension $n\ge 3$ such that at every point the corresponding Minkowski norm has a linear $SO(k)\times SO(n-k)$ -symmetry.

1. Introduction. If m and n are two integers chosen at random, the probability that they are relatively prime (2, p. 267) is 6π-2. This result may still hold when m and n are functionally related. Thus, Watson (3) recently proved that for α irrational, the positive integers n for which (n, [αn]) = 1, have density 6π-2. A different proof of a slightly more general result was given by Estermann (1).

For fixed $\alpha \in [0,1]$ , consider the set $S_{\alpha ,N}$ of dilated squares $\alpha , 4\alpha , 9\alpha , \dots , N^2\alpha \, $ modulo $1$ . Rudnick and Sarnak conjectured that, for Lebesgue, almost all such $\alpha $ the gap-distribution of $S_{\alpha ,N}$ is consistent with the Poisson model (in the limit as N tends to infinity). In this paper, we prove a new estimate for the triple correlations associated with this problem, establishing an asymptotic expression for the third moment of the number of elements of $S_{\alpha ,N}$ in a random interval of length $L/N$ , provided that $L> N^{1/4+\varepsilon }$ . The threshold of $\tfrac {1}{4}$ is substantially smaller than the threshold of $\tfrac {1}{2}$ (which is the threshold that would be given by a naïve discrepancy estimate).

Unlike the theory of pair correlations, rather little is known about triple correlations of the dilations $(\alpha a_n \, \text {mod } 1)_{n=1}^{\infty } $ for a nonlacunary sequence $(a_n)_{n=1}^{\infty } $ of increasing integers. This is partially due to the fact that the second moment of the triple correlation function is difficult to control, and thus standard techniques involving variance bounds are not applicable. We circumvent this impasse by using an argument inspired by works of Rudnick, Sarnak, and Zaharescu, and Heath-Brown, which connects the triple correlation function to some modular counting problems.

In Appendix B, we comment on the relationship between discrepancy and correlation functions, answering a question of Steinerberger.

Let V be a paracompact real analytic manifold of dimension n ≥ 1. Following the terminology of the theory of distributions of Schwartz (4), is the linear space of infinitely differentiable functions with compact support in V with the appropriate inductive limit topology, is the Frechet space of infinitely differentiable functions on V, is the dual space of consisting of the distributions on V, the dual space of consisting of the distributions with compact support on V. Let 𝒰(V) be the linear space of real analytic functions on V.

In this paper we consider anti-commutative algebras with an invariant form, that is, an algebra A over a field F such that

and A possesses a symmetric bilinear form f(x, y) such that

Lie and Malcev algebras (2, 3) are examples of such algebras and we shall consider generalizations of these algebras obtained by introducing commutation, x o y = xy — yx, as a new multiplicative operation in the non-commutative Jordan algebras of (1).

It is assumed that all topological spaces discussed in this paper are T1 spaces. A function ƒ mapping a topological space X into itself is a closed function if ƒ[H] is closed for each closed subset H of S. The semigroup, under composition, of all closed functions mapping X into X is denoted by Γ(X). These were among the semigroups under consideration in (4).

Introduction. Let G denote the multiplicative group of matrices

where a, b, c, d are integers and ad — bc = 1. G is one of the well-known modular groups. Let G0(n) denote the subgroup of G characterized by c ≡ 0 (mod n), where n is a positive integer. In this note we determine the normalizer of G0(n) in G, denoted by .

Une structure de contact sur une variété M est un champ d'éléments de contact tangents Σ dont le système caractéristique Δ est régulier. Lorsque codim Σ = 1 une telle structure est définie localement par une forme de Pfaff ω ou encore par l'équation ω = 0. Le théorème classique de Darboux nous donne un critère d'équivalence de deux Pfaffiens ω et ω′ à l'aide de l'invariant classe ω. Cet invariant est cependant trop fort pour déterminer l'équivalence des structures de contact Σ et Σ′ définies par ω = 0 et ω′ = 0 respectivement. D'une façon précise, les structures Σ et Σ′ peuvent être équivalentes sans que les Pfaffiens ω et ω′ le soient. Dans cette note nous considérons un invariant plus faible, classe Σ, attaché à une structure de contact Σ et nous montrons qu'il détermine l'équivalence en codimension 1 (cf. corollaire 3).

In [1] the category of relations in an abelian category was studied. A relation from A to B appeared as an equivalence class of pairs of -morphisms (ϕ, ψ),

Moreover, this equivalence class may be written as the composite , where is canonically embedded in and is the image of Γ in the canonical involution on . Every equivalence class has an essentially unique minimal representative (ϕ0,ψ0), characterized by the property that

where

and if and only if the square

is exact.

A summation method defined by the linear transformation

will be called an l-l method if ∑|yr| < ∞ whenever ∑|xk| < ∞; if in addition we have ∑yr = ∑xk whenever ∑|xk| < ∞ we shall say the method is absolutely regular. (It should be observed that we are dealing with series-to-series methods, not sequence-to-sequence as usual.)

Throughout this note all groups are abelian, written additively. We refer to Kurosh (8; 9) for notation, terminology and theorems used without reference. We recall the notion of a serving subgroup (or pure subgroup) of a group . This is a subgroup in which for every natural number n every equation nx = s, s ∊ can be solved provided that it can be solved in . If is torsion-free, “linearly closed” subgroups coincide with serving subgroups and is a serving subgroup if and only if / is torsionfree.