Related papers: Lebesgue measure zero modulo ideals on the natural…

Let $R = \bigoplus_{n \in \mathbb{N}_{0}} R_{n}$ be a standard graded ring, $M$ be a finite graded $R$-module and $J$ be a homogenous ideal of $R$. In this paper we study the graded structure of the $i$-th local cohomology module of $M$…

The paper establishes several inequalities between cardinal characteristics of the continuum. In particular, it is shown that the partition splitting number is not larger than the uniformity of the meagre ideal; not all sets of reals having…

We study perturbations of non-recurrent parameters in the exponential family. It is shown that the set of such parameters has Lebesgue measure zero. This particularly implies that the set of escaping parameters has Lebesgue measure zero,…

This paper defines Lebesgue measure preserving Thompson's monoid, denoted by $\mathbb{G}$, which is modeled on Thompson's group $\mathbb{F}$ except that the elements of $\mathbb{G}$ are non-invertible. Moreover, it is required that the…

The Levi-Civita field $\mathcal{R}$ is the smallest non-Archimidean ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order. In an earlier paper [Shamseddine-Berz-2003], a…

Let $R$ be a Noetherian ring, $I_1,\ldots,I_r$ be ideals of $R$, and $N\subseteq M$ be finitely generated $R$-modules. Let $S = \bigoplus_{\underline{n} \in \mathbb{N}^r} S_{\underline{n}}$ be a Noetherian standard $\mathbb{N}^r$-graded…

We introduce a non real-valued measure on the definable sets contained in the finite part of a cartesian power of an o-minimal field $R$. The measure takes values in an ordered semiring, the Dedekind completion of a quotient of $R$. We show…

In this paper, we introduce the concept of S-J-ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of J-ideals apply to S-J-ideals and examine…

The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of $\mathbb{R}$ which is Baire (=has the Baire property, BP) or measurable is shift-compact: it contains some subsequence of any null sequence to within translation by…

Suppose $E, F$ are Borel sets in the plane, $\dim_{\mathcal{H}} E>1$, $\dim_{\mathcal{H}} E+\dim_{\mathcal{H}} F>2$, and $F$ has equal Hausdorff and packing dimension. We prove that there exists $y\in F$ such that the pinned distance set…

Let $\epsilon$ be a fixed positive quantity, $m$ be a large integer, $x_j$ denote integer variables. We prove that for any positive integers $N_1,N_2,N_3$ with $N_1N_2N_3>m^{1+\epsilon},$ the set $$ \{x_1x_2x_3 \pmod m: \quad x_j\in [1,N_j]…

Let I be a nonzero proper ideal in a Noetherian integral domain R. In this paper we establish the existence of a finite separable integral extension domain A of R and a positive integer m such that all the Rees integers of IA are equal to…

We show that properties of pairs of finite, positive and regular Borel measures on the complex unit circle such as domination, absolute continuity and singularity can be completely described in terms of containment and intersection of their…

The paper, that continuous some previous work of Sch\"onherr & Schuricht, treats density measures on ${\mathbb R}^n$ that concentrate in any neighborhood of a Lebesgue null set. Such measures are typical for purely finitely additive…

In this paper, we study a class of Borel measures on $\mathbb{R}^n$ that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the…

Let $R$ be a Noetherian local ring. We define the minimal $j$-multiplicity and almost minimal $j$-multiplicity of an arbitrary $R$-ideal on any finite $R$-module. For any ideal $I$ with minimal $j$-multiplicity or almost minimal…

We study the symbolic powers of determinantal ideals of generic, generic symmetric, and Hankel matrices of variables, and of Pfaffians of generic skew-symmetric matrices, in prime characteristic. Specifically, we show that the limit…

R.D.Mauldin asked if every translation invariant $\sigma$-finite Borel measure on $\RR^d$ is a constant multiple of Lebesgue measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on…

We consider modules $M$ over Lie algebroids ${\mathfrak g}_A$ which are of finite type over a local noetherian ring $A$. Using ideals $J\subset A$ such that ${\mathfrak g}_A \cdot J\subset J $ and the length $\ell_{{\mathfrak g}_A}(M/JM)<…

Let $R$ be a Noetherian local ring, $I$ and $J$ two ideals of $R$, $M$ an $R$-module and $s$ and $t$ two integers. We study the relationship between the Bass numbers of $M$ and $H^{i}_{I,J}(M)$. We show that…