In this paper, the authors consider the inverse piston problem for the systemof one-dimensional isentropic flow and obtain that, under suitable conditions, the pistonvelocity can be uniquely determined by the initial state of the gas on the right side of thepiston and the position of the forward shock.

A linear system arising from a polynomial problem in the approximation theoryis studied, and the necessary and sufficient conditions for existence and uniqueness of itssolutions are presented. Together with a class of determinant identities, the resulting theoryis used to determine the unique solution to the polynomial problem. Some homogeneouspolynomial identities as well as results on the structure of related polynomial ideals arejust by-products.

We prove the Murphy and Cohen's conjecture that the maximum number of collisions of n + 1 elastic particles moving freely on a line is $\frac{{n{\left( {n + 1} \right)}}}{2}$ if no interior particle has mass less than the arithmetic mean of the masses of its immediate neighbors. In fact, we prove the stronger result that, for the same conclusion, the condition that no interior particle has mass less than the geometric mean, rather than the arithmetic mean, of the masses of its immediate neighbors suffices.

Let ℚ³ be the common conformal compactification space of the Lorentzian space forms $\mathbb{R}^{3}_{1} ,\mathbb{S}^{3}_{1} \;{\text{and}}\;\mathbb{H}^{3}_{1} $. We study the conformal geometry of space-like surfaces in ℚ³. It is shown that any conformal CMC-surface in ℚ³ must be conformally equivalent to a constant mean curvature surface in $\mathbb{R}^{3}_{1} ,\mathbb{S}^{3}_{1} \;{\text{and}}\;\mathbb{H}^{3}_{1} $. We also show that if x : M → ℚ³ is a space-like Willmore surface whose conformal metric g has constant curvature K, then either K = −1 and x is conformally equivalent to a minimal surface in $\mathbb{R}^{3}_{1}$, or K = 0 and x is conformally equivalent to the surface $\mathbb{H}^{1} {\left( {\frac{1}{{{\sqrt 2 }}}} \right)} \times \mathbb{H}^{1} {\left( {\frac{1}{{{\sqrt 2 }}}} \right)}\;{\text{in}}\;\mathbb{H}^{3}_{1} .$

In this paper, the existence and smoothness of the collision local time areproved for two independent fractional Brownian motions, through L ² convergence andChaos expansion. Furthermore, the regularity of the collision local time process is studied.

Let M be an invariant subspace of $H^{2}_{v} $. It is shown that for each f ∈ M ^⊥, f can be analytically extended across ∂${\partial \mathbb{B}_{d} } \mathord{\left/ {\vphantom {{\partial \mathbb{B}_{d} } {\sigma {\left( {S_{{z_{1} }} , \cdots ,S_{{z_{d} }} } \right)}}}} \right. \kern-\nulldelimiterspace} {\sigma {\left( {S_{{z_{1} }} , \cdots ,S_{{z_{d} }} } \right)}}.$

The authors consider the problem: −div(p∇u) = u ^q−1 +λu, u > 0 in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝ ⁿ, n ≥ 3, $p:\ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega } \to \mathbb{R}$ is a given positive weight such that $p \in H^{1} {\left( \Omega \right)} \cap C{\left( {\ifmmode\expandafter\bar\else\expandafter\=\fi{\Omega }} \right)},\lambda $ is a real constant and $q = \frac{{2n}}{{n - 2}}$, and study the effect of the behavior of p near its minima and the impact of the geometry of domain on the existence of solutions for the above problem.

In this paper, the author proves the Hyers–Ulam–Rassias stability of homo-morphisms in quasi-Banach algebras. This is used to investigate isomorphisms betweenquasi-Banach algebras.

To increase the hydrodynamic performance in different machine elements, ase.g. journal bearings and thrust bearings, during lubrication it is important to understandthe influence of surface roughness. In this connection one encounters homogenization ofthe incompressible Reynolds equation, where the roughness of the lubricated surface isassumed to be periodic. This problem has recently been studied in more engineering-oriented papers by using the formal method of multiple scale expansion. In this paper, werigorously prove both homogenization and corrector results by using two-scale convergence,which may be regarded as a justification of the formal multiple scale expansion methoddescribed above. Moreover, some numerical illustrations and results are presented.