let $r$ be a ring with unity. the undirected nilpotent graph of $r$, denoted by $gamma_n(r)$, is a graph with vertex set ~$z_n(r)^* = {0neq x in r | xy in n(r) for some y in r^*}$, and two distinct vertices $x$ and $y$ are adjacent if and only if $xy in n(r)$, or equivalently, $yx in n(r)$, where $n(r)$ denoted the nilpotent elements of $r$. recently, it has been proved that if $r$ is a left ar...

Remark 1. We quickly recall a couple of definitions Let DivQ(X) := Div(X)⊗Q. On smooth projective surfaces all Q-Weil divisors are also Q-Cartier, hence we can write Q-Cartier every divisor D as a finite sum ∑ xiCi, where the Ci are distinct integral curves and xi ∈ Q. A divisor D is called effective (or sometimes positive) if xi ≥ 0 ∀i. If D · C ≥ 0 for all integral curves C then D we be calle...

AdigraphC is called a zero divisor if there exist non-isomorphic digraphs A and B for which A×C ∼= B ×C , where the operation is the direct product. In other words,C being a zero divisormeans that cancellation property A×C ∼= B×C ⇒ A ∼= B fails. Lovász proved that C is a zero divisor if and only if it admits a homomorphism into a disjoint union of directed cycles of prime lengths.Thus any digra...

When we speak about a function field of one variable over a field K, we mean a finitely generated regular extension F of K of transcendence degree 1. We briefly recall the definitions of the main objects attached to F/K and their properties. See the books [Che51] or [Sti93] for details. A more comprehensive survey can be found [FrJ08, Sections 3.1-3.2]. A K-place of F is a place φ: F → K̃ ∪ {∞} ...

We give explicit bounds on sums of $d(n)^2$ and $d_4(n)$, where $d(n)$ is the number divisors $n$ $d_4(n)$ ways writing as a product four numbers. In doing so we make slight improvement upper bound for class numbers quartic fields.

Рассмотрен абстрактный автомат, осуществляющий деление на 13(4) в четверичной системе счисления. Автомат является типичным представителем семейства делителей, отличающихся отсутствием общих характерных признаков. Вследствие чего почти каждый делитель источником оригинальных особенностей. Примером такого делителя предлагаемый для рассмотрения делитель. Некоторые его особенности представляют необ...