Abstract We study foliations ℱ {\mathscr{F}} on projective complete intersection K3 surfaces X ↪ ℙ n {X\hookrightarrow\mathbb{P}^{n}} , where ℱ {\mathscr{F}} has isolated singularities and it is the restriction of a foliation of degree d on ℙ n {\mathbb{P}^{n}} that leaves X invariant. We compute the values of the degrees d for which ℱ {\mathscr{F}} is uniquely determined by its singular scheme.
A primary part in the rational homology cobordism group is generated by manifolds for which the order of the first homology is a power of a given prime. Kim and Livingston proved that the rational homology cobordism group does not equal the sum of primary parts. The current paper defines composite parts, generated by manifolds for which the order of the first homology is a product of powers of primes in a given set. We show that the rational homology cobordism group is not generated by composite parts with bounded number of distinct prime factors. Moreover, we give a necessary and sufficient condition for two sums of composite parts to be equal. We also prove the rational homology cobordism group is not right primary decomposable, answering a question of Cha.
A necessary criterion for injective solutions to the set-theoretical Yang-Baxter equation in terms of q-cycle sets is given, improving Soloviev's syntactic criteria. Supporting evidence is provided that the new criterion is sufficient. Generalizing augmented rack solutions, q-cycle groups are revisited, and a general extension theorem for the underlying solution is established. Close relationships between injective solutions, generic coverings, and extensions of q-cycle groups, are obtained. Several characterizations of generic coverings are given.
Very recently, two new notions of para-linear mappings and weak associative orthonormal bases were introduced in octonionic functional analysis, which have been proved to be powerful in formulating the basic theory, such as the Riesz representation theorem and the Parseval theorem. In this article, we continue exploring more properties of these two concepts and initiate the study of octonionic para-linear isometric operators. Surprisingly, it is proven that the condition of the para-linear operator on a Hilbert octonionic bimodule being an isometric isomorphism is equivalent to it mapping any associative orthonormal basis to a weak associative orthonormal basis, which implies also that an octonionic matrix is an isometry if and only if the system of its row vectors is a weak associative orthonormal basis. Furthermore, we introduce the concept of para-linear partial isometric operators and establish the aforementioned analogue in this new setting. Based on these facts, we can provide naturally a new viewpoint of James questions by modifying the definition of octonionic Stiefel space.
Investigating a recent positive solution of a conjecture of Grunewald and O'Halloran for complex finite-dimensional nilpotent Lie algebras, we find results of existence and uniqueness for the construction of complex nilpotent Lie algebras of arbitrary dimension via pseudobosonic operators. We involve the theory of the deformation of Lie algebras of Gerstenhaber, in order to prove our main results. There is not a generalized version of the Grunewald-O'Halloran Conjecture when we consider pseudoquonic operators, which specialize to pseudobosonic operators, so we prove results of uniqueness for C * {C<^>{*}} -algebras of pseudoquonic operators leveraging on different methods of functional analysis and operator theory.
