Narrow Results

In [6], J. Dixmier posed six problems for the Weyl algebra A₁ over a field K of characteristic zero. Problems 3, 5,and 6 were solved respectively by Joseph and Stein [7]; the author [1]; and Joseph [7]. Problems 1, 2, and 4 are still open.

For an arbitrary algebra A, Dixmier's problem 6 is essentially aquestion: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], deg_t(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (e.g., rings of differential operators on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra of a completely solvable algebraic Lie algebra ). This gives an affirmative answer (with a short proof) to an analogue of Dixmier's Problem 6 for certain algebras of small Gelfand–Kirillov dimension, e.g. the ring of differential operators on a smooth irreducible affine curve X, Usl(2), etc. (see [3] for details). In this paper an affirmative answer is given to an analogue of Dixmier's Problem 3 but for the ring .

For any positive integer $k$, let ${{𝒢}}_k$ denote the set of finite groups $G$ such that all Cayley graphs $\mathrm{\text{Cay}}(G,S)$ are integral whenever $\left|S\right|\le k$. Estélyi and Kovács [On groups all of whose undirected Cayley graphs of bounded valency are integral, Electron. J. Combin.21 (2014) #P4.45.] classified ${{𝒢}}_k$ for each $k\ge 4$. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class ${{𝒢}}_3$ is characterized. As an application, the classification of ${{𝒢}}_k$ is obtained again, where $k\ge 4$.

The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings.

We discuss an algebraic problem (Separation and Irreducibility Conjecture) which arises from the study of the nonlinear Schrödinger equation (NLS for short). This problem is about separation and irreducibility (over the ring of integers) of the characteristic polynomials of the graphs, describing blocks of a normal form for the NLS. For the cubic NLS the problem has been completely solved (see [C. Procesi, M. Procesi and B. Van Nguyen, The energy graph of the nonlinear Schrödinger equation, Rend. Lincei Mat. Appl.24(2) (2013) 229–301]), meanwhile for higher degree NLS it is still open, even in small dimensions (see [C. Procesi, The energy graph of the non-linear Schrödinger equation, open problems. Int. J. Algebra Comput.23(4) (2013) 943–962]). In this work, the author will give a partial answer for this problem, in particular, the author will prove Separation and Irreducibility Conjecture for NLS of arbitrary degree on one-dimensional and two-dimensional tori.

The zero-divisor graph $\mathrm{\Gamma }(R)$ of a commutative ring $R$ is the graph whose vertices are the nonzero zero divisors in $R$ and two vertices $x$ and $y$ are adjacent if and only if $xy=0$. We study the adjacency and Laplacian eigenvalues of the zero-divisor graph $\mathrm{\Gamma }(R)$ of a finite commutative von Neumann regular ring $R$. We prove that $\mathrm{\Gamma }(R)$ is a generalized join of its induced subgraphs. Among the $\left|Z{(R)}^{\ast }\right|$ eigenvalues (respectively, Laplacian eigenvalues) of $\mathrm{\Gamma }(R)$, exactly $|B(R)|-2$ are the eigenvalues of a matrix obtained from the adjacency (respectively, Laplacian) matrix of $\mathrm{\Gamma }(B(R))$-the zero-divisor graph of nontrivial idempotents in $R$. We also determine the degree of each vertex in $\mathrm{\Gamma }(R)$, hence the number of edges.