Bulletin of Mathematical Sciences

The celebrated Diamond Lemma of Bergman gives an effectively verifiable criterion of uniqueness of normal forms for term rewriting in associative algebras. We revisit that result in the context of deformation theory and homotopical algebra; this leads to a new proof using multiplicative free resolutions. Specifically, our main result states that every such resolution of an algebra with monomial relations gives rise to its own Diamond Lemma, where Bergman’s condition of “resolvable ambiguities” is precisely the first nontrivial component of the Maurer–Cartan equation in the corresponding tangent complex. The same approach works for many other algebraic structures, emphasizing the relevance of computing resolutions of algebras with monomial relations.

In this paper, we deal with fractional $p$-Laplace Kirchhoff equations with exponential growth of the form

A new class of nonlocal multipoint boundary value problems involving a dual hybrid system of nonlinear Riemann–Liouville-type $q$-fractional differential equations is studied in this paper. Existence and uniqueness results for the given problem are derived by applying the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle. Examples are presented for illustrating the obtained results. The work established in this paper is a useful contribution to the existing literature on $q$-fractional differential equations. Some interesting special cases are also discussed.

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi-homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

In this paper, we investigate the following fractional Nirenberg problem:

Our purpose is to generalize some recent comparison principles for operators driven by $p$-Laplacian to a wide class of quasilinear equations including $(p,q)$-Laplacian. It turns out, in particular, that adding a $q$-Laplacian to $p$-Laplacian allows to weaken the assumptions needed on the Hamiltonian of lower order terms. The results are specialized in the case that the Hamiltonian has at most polynomial growth in the gradient with coefficients depending on $x$ and $u$.