An elementary proof of the irrationality of Tschakaloff series

We present a new proof of the irrationality of values of the series $T_q(z)=\sum_{n=0}^\infty z^nq^{-n(n-1)/2}$ in both qualitative and quantitative forms. The proof is based on a hypergeometric construction of rational approximations to $T_q(z)$.Comment: 5 pages, AMSTe

The Moore-Penrose Inverse in Art

Abstract The "Moore-Penrose inverse" of a matrix A corresponds to the (unique) matrix solution X of the system AXA=A, XAX=X, (AX) T =AX, (XA) T =XA. This generalized inverse has many applications, ranging from Gauss' historical prediction for finding Ceres to modern electrical engineering problems. The present paper provides some applications related to art: one about mathematical color theory, and one about curve fitting in architectural drawings or paintings

The generalized inverse of a sum with radical element: Applications

AbstractAn expression for the generalized inverse of a sum with radical element in a ring with unity is generalized to the case of a sum ϕ + η, in an additive category C, of a morphism ϕ with a reflexive Von Neumann regular inverse ϕ(1,2) and a morphism η of C which is such that 1X + ϕ(1,2)η is invertible. Also, if C is an additive category with an involution ∗, ϕ a morphism with Moore-Penrose inverse ϕ†, and η such that 1X + ϕ†η, 1X − λ = 1X − (1X + ϕ†η)−1(1X − ϕ†ϕ)η∗ϕ†∗(1X + η∗ϕ†∗)−1, and 1Y − μ = 1Y − (1Y + ϕ†∗η∗)−1ϕ†∗η∗(1Y − ϕϕ†)(1Y + ηϕ†)−1 are invertible, then ϕ + η − (1Y − ϕϕ†)η(1X + ϕ†η)−1(1X − ϕ†ϕ) has a Moore-Penrose inverse, given by (1X − λ)−1(1X + ϕ†η)− ϕ†(1Y − μ)−1. Relations with results of Wynn, Roth, and Nashed are discussed

Generalized inverses of a sum with a radical element

AbstractGiven a von Neumann regular element a and an element j in the Jacobson radical J(R) of a ring R with unity, a necessary and sufficient condition is given for a + j to be von Neumann regular. Moreover, the Moore-Penrose invertibility and group invertibility of the element a + j are considered. An application is given for matrices over semiprimary SBI rings