Related papers: Equipartition of a segment

We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…

We show that for every integer $m > 0$, there is an ordinary abelian variety over ${\mathbb F}_2$ that has exactly $m$ rational points.

We introduce an interesting method of proving separable reduction theorems - the method of elementary submodels. We are studying whether it is true that a set (function) has given property if and only if it has this property with respect to…

A positive integer n is said to be perfect if sigma(n)=2n, where sigma denotes the sum of the divisors of n. In this article, we show that if n is an even perfect number, then any integer m<=n is expressed as a sum of some of divisors of n.

We prove that for any finite-dimensional differential graded algebra with separable semisimple part the category of perfect modules is equivalent to a full subcategory of the category of perfect complexes on a smooth projective scheme with…

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.

Let $f \in \mathbb{R}[x]$ be a polynomial with real coefficients. We say that $f$ is eventually non-negative if $f^m$ has non-negative coefficients for all sufficiently large $m \in \mathbb{N}$. In this short note, we give a classification…

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for…

In this paper we solve in the positive the question of whether any finite set of integers, containing the zero, is the mapping degree set between two oriented closed connected manifolds of the same dimension. We extend this question to the…

After introducing the partially separable concept, we proved the equivalence between the partial separability of a given $m$-partite subsystem with $m$ qubits and the purity of states of this $m$-partite subsystem for a pure state in…

We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of derived categories and even marked filtrations, where one descends not…

New splitting theorems in a semi-Riemannian manifold which admits an irrotational vector field (not necessarily a gradient) with some suitable properties are obtained. According to the extras hypothesis assumed on the vector field, we can…

A triangle decomposition of a graph is a partition of its edges into triangles. A fractional triangle decomposition of a graph is an assignment of a non-negative weight to each of its triangles such that the sum of the weights of the…

For a positive integer $m$, a finite set of integers is said to be equidistributed modulo $m$ if the set contains an equal number of elements in each congruence class modulo $m$. In this paper, we consider the problem of determining when…

A class theorem is presented and proved: the complex Fourier transforms of a certain class of exponential functions have all their zeros on the real line. A class of basis functions is first considered, and the class is then extended via…

A subset $M$ of a continuum $X$ is called a \textit{meager composant} if $M$ is maximal with respect to the property that every two of its points are contained in a nowhere dense subcontinuum of $X$. Motivated by questions of Bellamy,…

Let $ f $ be a real-valued function on a compact subset in $ \mathbb{R}^n $. We show how to decide if $ f $ extends to a nonnegative and $ C^1 $ function on $ \mathbb{R}^n $. There has been no known result for nonnegative $ C^m $ extension…

We consider the set of power functions defined on the set of positive real number, and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative…

In this short note, we give basic enumerative results on colored integer partitions.

Let F be a field complete for a real valuation. It is a standard result in valuation theory that a finite extension of F admits a valuation basis if and only if it is without defect. We show that even otherwise, one can construct bases in…