Tensor invariants of generalized Kenmotsu manifolds

In this paper, we study the properties of generalized Kenmotsu manifolds, consider the second-order differential geometric invariants of the Riemannian curvature tensor of generalized Kenmotsu manifolds (by the symmetry properties of the Riemannian geometry tensor). The concept of a tensor spectrum is introduced. Nine invariants are singled out and the geometric meaning of these invariants turning to zero are investigated. The identities characterizing the selected classes are singled out. Also, 9 classes of generalized Kenmotsu manifolds are distinguished, the local structure of 8 classes from the selected ones is obtained.

Genus expansion of matrix models and ћ expansion of KP hierarchy

We study ћ expansion of the KP hierarchy following Takasaki-Takebe [1] considering several examples of matrix model τ-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter ћ are τ-functions of the ћ-KP hierarchy and the expansion in ћ for the ћ-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the ћ-formulation of the KP hierarchy [2, 3] with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of τ-functions is straightforward and algorithmic.

Connection Between A Right Triangle And An Equal Side Triangle

There is some evidence that a right triangle and an equilateral triangle are related. Information about Pythagorean numbers is given. The geometric meaning of the relationship between right triangles and equilateral triangles is shown. The geometric meaning of the relationship between an equilateral triangle and an equilateral triangle is shown.

Hierarchical Linear Disentanglement of Data-Driven Conceptual Spaces

Conceptual spaces are geometric meaning representations in which similar entities are represented by similar vectors. They are widely used in cognitive science, but there has been relatively little work on learning such representations from data. In particular, while standard representation learning methods can be used to induce vector space embeddings from text corpora, these differ from conceptual spaces in two crucial ways. First, the dimensions of a conceptual space correspond to salient semantic features, known as quality dimensions, whereas the dimensions of learned vector space embeddings typically lack any clear interpretation. This has been partially addressed in previous work, which has shown that it is possible to identify directions in learned vector spaces which capture semantic features. Second, conceptual spaces are normally organised into a set of domains, each of which is associated with a separate vector space. In contrast, learned embeddings represent all entities in a single vector space. Our hypothesis in this paper is that such single-space representations are sub-optimal for learning quality dimensions, due to the fact that semantic features are often only relevant to a subset of the entities. We show that this issue can be mitigated by identifying features in a hierarchical fashion. Intuitively, the top-level features split the vector space into different domains, making it possible to subsequently identify domain-specific quality dimensions.