The focus of this entire note is on positive integers. I will start by the basic notion of divisibility. We say that a divides b, or a is a divisor of b, or b is a multiple of a, if there exists an integer m such that b = am We also denote this by a | b (a divides b). If a is not a divisor of b, the we write a | b, but we can still talk about division by a with a remainder r. Given any two inte...

We are interested in subsets S of Z which admit simultaneous orderings, that is, sequences {an}n≥0 such that Qn−1 k=0 (an−ak) divides Qn−1 k=0 (x− ak) for every x ∈ S (analogously to n! which divides (m+n)!/m! for every m). In particular, we characterize the polynomials f of degree 2 such that f(N) admits a simultaneous ordering.

We investigate two density questions for Sobolev, Besov and Triebel--Lizorkin spaces on rough sets. Our main results, stated in the simplest Sobolev space setting, are that: (i) an open set $\Omega\subset\mathbb R^n$, $\mathcal{D}(\Omega)$ is dense $\{u\in H^s(\mathbb R^n):{\rm supp}\, u\subset \overline{\Omega}\}$ whenever $\partial\Omega$ has zero Lebesgue measure $\Omega$ "thick" (in sense o...

The Garsia-Haiman modules are doubly graded modules for the symmetric groups, introduced by A. Garsia and M. Haiman [GH] to prove Macdonald’s positivity conjecture [M1]. These modules are defined for partitions of positive integers n, denoted by Dμ. The dimension of Dμ is given by n! whenever μ is a partition of n [H2]. As this fact implies, the Garsia-Haiman modules Dμ are isomorphic to the co...

The Wronski map is a finite, PGL2(C)-equivariant morphism from the Grassmannian Gr(d, n) to a projective space (the projectivization of a vector space of polynomials). We consider the following problem. If Cr ⊂ PGL2(C) is a cyclic subgroup of order r, how may Cr-fixed points are in the fibre of the Wronski map over a Cr-fixed point in the base? In this paper, we compute a general answer in term...

Given an integer matrix M ∈ GLn(R) and a point y ∈ R/Z, consider the set Ẽ(M,y) def = { x ∈ R : y / ∈ {Mkx mod Zn : k ∈ N} } . S.G. Dani showed in 1988 that whenever M is semisimple and y ∈ Q/Z, the set Ẽ(M,y) is winning in the sense of W. Schmidt (a property implying density and full Hausdorff dimension). In this paper we strengthen this result, extending it to arbitrary y ∈ R/Z and M ∈ GLn(R)...

Building on work by Bouc and by Shareshian and Wachs, we provide a toolbox of long exact sequences for the reduced simplicial homology of the matching complex Mn, which is the simplicial complex of matchings in the complete graph Kn. Combining these sequences in different ways, we prove several results about the 3-torsion part of the homology of Mn. First, we demonstrate that there is nonvanish...

Given a commutative unital ring $R$, we show that the finiteness length of group $G$ is bounded above by Borel subgroup rank one $\mathbf{B}_2^\circ(R)=\left( \begin{smallmatrix} * & \\ 0 \end{smallmatrix} \right)\leq\mathrm{SL}_2(R)$ whenever admits certain $R$-representations with metabelian image. Combined results due to Bestvina--Eskin--Wortman and Gandini, this gives new proof (a generaliz...

non-beneficial and harmful weeds are plants that are unwanted, outside their home farms are growing and have the potential to exceed. this study was done in order to screening fungai and isolating nlp phytotoxine from them for selective biocontrol of amaranthus retroflexus and rumes crispus as a dicot, common and chemical herbicide resistance weeds. nlps are effective just on dicot plants. cont...