On continuity of Guo Wuwen function

We show that the functions g and gs introduced by Guo Wuwen in [4] are continuous and semialgebraic. We use this fact to prove that the set Nn of ordered n-tuples of real numbers, realizable by nonnegative matrices, is a closed set

Comparison Between Simulated and Observational Results of Galaxy Formation for Large Scale Structures

The Millennium simulation is the largest numerical simulation of how minor fluctuations in the density of the universe’s dark matter distribution are amplified by gravity to develop into the large scale structures(LSS) and galaxy clusters seen today(Springel et al. 2005). Although the simulations have been compared with the astronomical observations of the local universe, the simulations have not been widely compared with high redshift, early universe observations. In our study we compare the simulation data(Wang et al. 2008; Guo et al. 2008(in preparation)) for the first time with observations from the COSMOS survey(Scoville et al. 2006). Three quantities are proposed to characterize the structures and the structures distribution, namely the percent area occupied by LSS at each redshift, the average area of LSS and the shapes as characterized by the square root of the area divided by the circumference. We calculate these quantities for both the observations and the simulations, and quantify discrepancies between the existing simulations and observations. In particular, the simulations exhibit earlier development of dense structures than is seen in the observational data

On 2-adic orders of some binomial sums

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$\sum_{k=-n}^n\binom{2n}{n-k}k^{2r}$$ is divisible by $2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.Comment: 6 page

The Eulerian distribution on the involutions of the hyperoctahedral group is unimodal

The Eulerian distribution on the involutions of the symmetric group is unimodal, as shown by Guo and Zeng. In this paper we prove that the Eulerian distribution on the involutions of the hyperoctahedral group, when viewed as a colored permutation group, is unimodal in a similar way and we compute its generating function, using signed quasisymmetric functions.Comment: 11 pages, zero figure

On spectrum of irrationality exponents of Mahler numbers

We consider Mahler functions $f(z)$ which solve the functional equation $f(z) = \frac{A(z)}{B(z)} f(z^d)$ where $\frac{A(z)}{B(z)}\in \mathbb{Q}(z)$ and $d\ge 2$ is integer. We prove that for any integer $b$ with $|b|\ge 2$ either $f(b)$ is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent for $f(b)$ as soon as the continued fraction for the corresponding Mahler function is known. This improves the result of Bugeaud, Han, Wei and Yao where only an upper bound for the irrationality exponent was provided