Linear independence of values of $G$-functions

Algebraic Independence of Values of Elliptic Functions

Linear independence of values of Tschakaloff series

In this note we investigate arithmetic properties of values of the Tschakaloff function Tq(z) = P∞ ν=0 q −ν(ν+1)/2zν , |q| > 1. One of the open problems is proving linear independence of the values of Tq(z) with different q. The only result obtained in this direction in [1] is very restrictive. We refer an interested reader to the survey [2] for an account of known linear and algebraic independ...

Criterion for linear independence of functions

where T1, . . . , Tn, S1, . . . , Sn : [a, b] → R. (2) Suppose K 6= 0. Then we may consider each of the systems {T1, . . . , Tn}, {S1, . . . , Sn} linearly independent. Indeed, starting from an expression of kind (1), we consequently reduce the number of items while it is needed. Assume that we need to express the functions (2) in terms of K. In order to do it, we find such points (a proof of e...

Singular values of convex functions of matrices

‎Let $A_{i},B_{i},X_{i},i=1,dots,m,$ be $n$-by-$n$ matrices such that $‎sum_{i=1}^{m}leftvert A_{i}rightvert ^{2}$ and $‎sum_{i=1}^{m}leftvert B_{i}rightvert ^{2}$  are nonzero matrices and each $X_{i}$ is‎ ‎positive semidefinite‎. ‎It is shown that if $f$ is a nonnegative increasing ‎convex function on $left[ 0,infty right) $ satisfying $fleft( 0right)‎ ‎=0 $‎, ‎then  $$‎2s_{j}left( fleft( fra...

On the Algebraic Independence of the Values of E - Functions