Consider an experiment with d+1 possible outcomes, d of which occur with probabilities $x₁,...,x_d$. If we consider a large number of independent occurrences of this experiment, the probability of any event in the resulting space is a polynomial in $x₁,...,x_d$. We characterize those polynomials which arise as the probability of such an event. We use this to characterize those x⃗ for which the measure resulting from an infinite sequence of such trials is good in the sense of Akin.

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a unique factorization domain of characteristic p > 0. One of these conditions involves Jacobians while the other some properties of factors. In the case m = n this extends the known theorem of Nousiainen, and we obtain a new formulation of the Jacobian conjecture in positive characteristic.

We present a class of counterexamples to the Cancellation Problem over arbitrary commutative rings, using non-free stably free modules and locally nilpotent derivations.

We study the multiplicative lattices $L$ which satisfy the condition $a=(a:(a:b\left)\right)(a:b)$ for all $a,b\in L$. Call them sharp lattices. We prove that every totally ordered sharp lattice is isomorphic to the ideal lattice of a valuation domain with value group $\mathbb{Z}$ or $ℝ$. A sharp lattice $L$ localized at its maximal elements are totally ordered sharp lattices. The converse is true if $L$ has finite character.

Butler groups formed by factoring a completely decomposable group by a rank one group have been studied extensively. We call such groups, bracket groups. We study bracket modules over integral domains. In particular, we are interested in when any bracket $R$-module is $R$ tensor a bracket group.

We describe a cluster algebra algorithm for calculating $q$-characters of Kirillov–Reshetikhin modules for any untwisted quantum affine algebra $U_q\left(\widehat{𝔤}\right)$. This yields a geometric $q$-character formula for tensor products of Kirillov–Reshetikhin modules. When $𝔤$ is of type $A,D,E$, this formula extends Nakajima’s formula for $q$-characters of standard modules in terms of homology of graded quiver varieties.

We define various ring sequential convergences on $\mathbb{Z}$ and $\mathbb{Q}$. We describe their properties and properties of their convergence completions. In particular, we define a convergence ${𝕃}_1$ on $\mathbb{Z}$ by means of a nonprincipal ultrafilter on the positive prime numbers such that the underlying set of the completion is the ultraproduct of the prime finite fields $\mathbb{Z}/\left(p\right)$. Further, we show that $(\mathbb{Z},{𝕃}_1^{*})$ is sequentially precompact but fails to be strongly sequentially precompact; this solves a problem posed by D. Dikranjan.

Let F=X-H:$k^n$ → $k^n$ be a polynomial map with H homogeneous of degree 3 and nilpotent Jacobian matrix J(H). Let G=(G1,...,Gn) be the formal inverse of F. Bass, Connell and Wright proved in [1] that the homogeneous component of $G_i$ of degree 2d+1 can be expressed as $G_i^{\left(d\right)}={\sum }_T\alpha {\left(T\right)}^{-1}{\sigma }_i\left(T\right)$, where T varies over rooted trees with d vertices, α(T)=CardAut(T) and ${\sigma }_i\left(T\right)$ is a polynomial defined by (1) below. The Jacobian Conjecture states that, in our situation, $F$ is an automorphism or, equivalently, $G_i^{\left(d\right)}$ is zero for sufficiently large d....

Let F = X + H be a cubic homogeneous polynomial automorphism from ${ℂ}^n$ to ${ℂ}^n$. Let $p$ be the nilpotence index of the Jacobian matrix JH. It was conjectured by Drużkowski and Rusek in [4] that $deg{F}^{-1}\le 3^{p-1}$. We show that the conjecture is true if n ≤ 4 and false if n ≥ 5.