The best bounds in Wallis' inequality

For all natural numbers , let denote a double factorial. Then

The constants and are the best possible. From this, the well-known Wallis' inequality is improved.

     

Projective surfaces with many skew lines

We give an example of a smooth surface of degree that contains pairwise disjoint lines. In particular, our example shows that the degree in Miyaoka's bound is sharp.

     

A note on exponential decay properties of ground states for quasilinear elliptic equations

We give an explicit formula for exponential decay properties of ground states for a class of quasilinear elliptic equations in the whole space .

     

On the -torsion of elliptic curves and elliptic surfaces in characteristic

We study the extension generated by the -coordinates of the -torsion points of an elliptic curve over a function field of characteristic . If is a non-isotrivial elliptic surface in characteristic with a -torsion section, then for 11$"> our results imply restrictions on the genus, the gonality, and the -rank of the base curve , whereas for such a surface can be constructed over any base curve . We also describe explicitly all occurring in the cases where the surface is rational or or the base curve is rational, elliptic or hyperelliptic.

     

Upper and lower bounds of temporal and spatial decays for the Navier-Stokes equations

We obtain the lower bounds of the temporal-spatial decays for weak solutions of the Navier-Stokes equations

     

Coding of Gibbs function flows from nucleotide pools

The biochemical machinery of living systems obeys kinetic laws, but is driven by Gibbs function flows. Both the kinetic and thermodynamic aspects of Gibbs gain, transmission, and utilization are considered. An information-theoretic approach is used to find conditions under which the kinetics encodes the associated Gibbs function flow with the lowest possible error.     

Thermodynamical proof of the Gibbs formula for elementary quantum systems

An elementary derivation is given of the formula for the thermal equilibrium states of quantum systems that can be described in finite-dimensional Hilbert spaces. The three assumptions made, Passivity, Structural Stability, and Consistency, have phenomenological interpretations. Except at zero temperature, Structural Stability follows already from Passivity and a weak form of Consistency.     

Spontaneous translation symmetry breakdown for quantum lattice systems

We study the set of equilibrium states for quantum lattice states in the presence of a translation symmetry of the model. We derive a characterization of the spontaneous breaking of this symmetry, i.e., the decomposition of an invariant equilibrium state into a mixture of noninvariant equilibrium states, in terms of the separability in mean energy of these states for a class of perturbed dynamics.