Related papers: A Note on the Manickam-Mikl\'os-Singhi Conjecture …

Motivated by work of Zhang from the early `90s, Medvedev and Scanlon formulated the following conjecture. Let $K$ be an algebraically closed field of characteristic $0$ and let $X$ be a quasiprojective variety defined over $K$ endowed with…

A vector space is commonly defined as a set that satisfies several conditions related to addition and scalar multiplication. However, for beginners, it may be hard to immediately grasp the essence of these conditions. There are probably a…

We provide a formula and a generating function for the dimension of the vector space of relative symmetric polynomials of D_n for all its irreducible one-dimensional and two-dimensional representations which were defined originally by M.…

We prove a conjecture of Ross concerning the value distribution of $\dim S_2^{\rm new}(\Gamma_0(N))$ for $N\in\mathbb{N}$, as well as analogous results for general weight $k\in2\mathbb{N}$ and the full and twist-minimal spaces…

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher dimensional theories of gravity. We extend the results shown by R. Bartnik and G. Fodor for the 3-dimensional case in…

In previous work, the authors established a generalized version of Schmidt's subspace theorem for closed subschemes in general position in terms of Seshadri constants. We extend our theorem to weighted sums involving closed subschemes in…

Let $m,n$ be positive integers. In this short note we prove that the set of all continuous and surjective functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ contains (excluding the 0 function) a $\mathfrak{c}$-dimensional vector space.…

We prove the following results solving a problem raised in [Y. Caro, R. Yuster, On zero-sum and almost zero-sum subgraphs over $\mathbb{Z}$, Graphs Combin. 32 (2016), 49--63]. For a positive integer $m\geq 2$, $m\neq 4$, there are…

Motivated by super-Yang-Mills theory on a Calabi-Yau 4-fold, Nekrasov and Piazzalunga have assigned weights to $r$-tuples of solid partitions and conjectured a formula for their weighted generating function. We define $K$-theoretic virtual…

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of ``positive space'' and its rational powers. Positive spaces are 1-dimensional ``semi-vector…

The well-known Lvov-Kaplansky conjecture states that the image of a multilinear polynomial $f$ evaluated on $n\times n$ matrices is a vector space. A weaker version of this conjecture, known as the Mesyan conjecture, states that if $m=deg(…

A function from $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2^n}$ is $k$th order sum-free if the sum of its values over each $k$-dimensional $\mathbb{F}_2$-affine subspace is nonzero. It is conjectured that for $n$ odd and prime,…

The vector space of all polynomial functions of degree $k$ on a box of dimension $n$ is of dimension ${n \choose k}$. A consequence of this fact is that a function can be approximated on vertices of the box using other vertices to higher…

Huang, McKinnon, and Satriano conjectured that if $v \in \mathbb{R}^n$ has distinct coordinates and $n \geq 3$, then a hyperplane through the origin other than $\sum_i x_i = 0$ contains at most $2\lfloor n/2 \rfloor (n-2)!$ of the vectors…

In this work we study arrangements of $k$-dimensional subspaces $V_1,\ldots,V_n \subset \mathbb{C}^\ell$. Our main result shows that, if every pair $V_{a},V_b$ of subspaces is contained in a dependent triple (a triple $V_{a},V_b,V_c$…

The maximal dimension of commutative subspaces of $M_n(\mathbb{C})$ is known. So is the structure of such a subspace when the maximal dimension is achieved. We consider extensions of these results and ask the following natural questions: If…

The complex conjecture of Stefan Banach states that if V is a Banach space over the complex numbers where for some n, 1<n<dim(V), all of its n-dimensional subspaces are isometric, then V is a Hilbert space. Mikhail Gromov proved it for n…

If an Fq-linear set LU in a projective space is defined by a vector subspace U which is linear over a proper superfield of Fq, then all of its points have weight at least 2. It is known that the converse of this statement holds for linear…

We investigate integral representation of vector-valued function spaces, i.e., of subspaces $H\subset C(K,E)$, where $K$ is a compact space and $E$ is a (real or complex) Banach space. We point out that there are two possible ways of…

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of…