Diversification of multipotential postmitotic mouse retinal ganglion cell precursors into discrete types

The genesis of broad neuronal classes from multipotential neural progenitor cells has been extensively studied, but less is known about the diversification of a single neuronal class into multiple types. We used single-cell RNA-seq to study how newly-born (postmitotic) mouse retinal ganglion cell (RGC) precursors diversify into ~45 discrete types. Computational analysis provides evidence that RGC type identity is not specified at mitotic exit, but acquired by gradual, asynchronous fate restriction of postmitotic multipotential precursors. Some types are not identifiable until a week after they are generated. Immature RGCs may be specified to project ipsilaterally or contralaterally to the rest of the brain before their type identity has been determined. Optimal transport inference identifies groups of RGC precursors with largely non-overlapping fates, distinguished by selectively expressed transcription factors that could act as fate determinants. Our study provides a framework for investigating the molecular diversification of discrete types within a neuronal class.

Coordinate-wise transformation of probability distributions to achieve a Stein-type identity

It is shown that for any given multi-dimensional probability distribution with regularity conditions, there exists a unique coordinate-wise transformation such that the transformed distribution satisfies a Stein-type identity. A sufficient condition for the existence is referred to as copositivity of distributions. The proof is based on an energy minimization problem over a totally geodesic subset of the Wasserstein space. The result is considered as an alternative to Sklar’s theorem regarding copulas, and is also interpreted as a generalization of a diagonal scaling theorem. The Stein-type identity is applied to a rating problem of multivariate data. A numerical procedure for piece-wise uniform densities is provided. Some open problems are also discussed.

Some New Identities for the Generalized Fibonacci Polynomials by the Q(x) Matrix

We obtain some new identities for the generalized Fibonacci polynomial by a new approach, namely, the Q(x) matrix. These identities including the Cassini type identity and Honsberger type formula can be applied to some polynomial sequences such as Fibonacci polynomials, Lucas polynomials, Pell polynomials, Pell-Lucas polynomials and so on, which generalize the previous results in references.

Another generalization of Euler’s arithmetic function and Menon’s identity

We define the k-dimensional generalized Euler function $$\varphi _k(n)$$ φ k ( n ) as the number of ordered k-tuples $$(a_1,\ldots ,a_k)\in {\mathbb {N}}^k$$ ( a 1 , … , a k ) ∈ N k such that $$1\le a_1,\ldots ,a_k\le n$$ 1 ≤ a 1 , … , a k ≤ n and both the product $$a_1\cdots a_k$$ a 1 ⋯ a k and the sum $$a_1+\cdots +a_k$$ a 1 + ⋯ + a k are prime to n. We investigate some of the properties of the function $$\varphi _k(n)$$ φ k ( n ) , and obtain a corresponding Menon-type identity.