Let $X_0^{star}(k,n,s)$ denote the sum of all multiple zeta-star values of weight $k$, depth $n$ and height $s$. Kaneko and Ohno conjecture that for any positive integers $m,n,s$ with $m,ngeqslant s$, the difference $(-1)^mX_0^{star}(m+n+1,n+1,s)-(-1
)^nX_0^{star}(m+n+1,m+1,s)$ can be expressed as a polynomial of zeta values with rational coefficients. We give a proof of this conjecture in this paper.
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
We study the self-gravitating stars with a linear equation of state, $P=a rho$, in AdS space, where $a$ is a constant parameter. There exists a critical dimension, beyond which the stars are always stable with any central energy density; below which
there exists a maximal mass configuration for a certain central energy density and when the central energy density continues to increase, the configuration becomes unstable. We find that the critical dimension depends on the parameter $a$, it runs from $d=11.1429$ to 10.1291 as $a$ varies from $a=0$ to 1. The lowest integer dimension for a dynamically stable self-gravitating configuration should be $d=12$ for any $a in [0,1]$ rather than $d=11$, the latter is the case of self-gravitating radiation configurations in AdS space.
In this paper we investigate the equilibrium self-gravitating radiation in higher dimensional, plane symmetric anti-de Sitter space. We find that there exist essential differences from the spherically symmetric case: In each dimension ($dgeq 4$), the
re are maximal mass (density), maximal entropy (density) and maximal temperature configurations, they do not appear at the same central energy density; the oscillation behavior appearing in the spherically symmetric case, does not happen in this case; and the mass (density), as a function of the central energy density, increases first and reaches its maximum at a certain central energy density and then decreases monotonically in $ 4le d le 7$, while in $d geq 8$, besides the maximum, the mass (density) of the equilibrium configuration has a minimum: the mass (density) first increases and reaches its maximum, then decreases to its minimum and then increases to its asymptotic value monotonically. The reason causing the difference is discussed.
