In this paper we present a new criterion on characterization of real inner product spaces concerning the Euler-Lagrange type identity

$$\|r_2x_1-r_1x_2\|^2 + \|r_1x_1+r_2x_2\|^2=(r_1^2+r_2^2)(\|x_1\|^2+\|x_2\|^2)\,.$$

Reflection of a shock from a solid wedge is a classical problem in gas dynamics. Depending on the parameters either a regular or a irregular (Mach-type) reflection results. We construct regular reflection as an exact self-similar solution for potential flow. For some upstream Mach numbers $M_I$ and isentropic coefficients $\gamma$, a solution exists for all wedge angles $\theta$ allowed by the sonic criterion. This demonstrates that, at least for potential flow, weaker criteria are false.

We introduce and analyse linear systems of hyperbolic partial differential equations that model the replacement and hoarding of currency. The goal is to deduce the hoarding behavior from observations of circulating bills. The large time asymptotics of the models is identified in all cases. The mathematical analysis is novel, partly because of nonstandard boundary conditions. To identify parameters we suggest themeasurement of the age histogram of notes, the rate of growth, and the retard in wear of notes due to hoarding. In our models that suffices to identify all but one quantity.

We find the largest convex minorant of the function

\begin{equation*} F\left( x,y\right) =ax^{2}+xy+by^{2} \end{equation*}

where $a,b$ are positive constants and $x\geq 0,\ y\geq 0$. We explain how the problem is closely connected with finding the ground state Thomas-Fermi electron density for a spin polarized quantum mechanical system with the Fermi-Amaldi correction.

In this paper, we make extensive use of the well-known Krasnoselskii fixed point theorem to obtain the existence of square-mean almost periodic solutions to some classes of hyperbolic stochastic evolution equations with infinite delay. Next, the existence of square-mean almost periodic solutions to not only the heat equation but also to a boundary value problem with infinite delay arising in control systems are studied.

The generalized Hyers–Ulam stability for the Pexiderized Cauchy and Jensen’s equations on restricted domains is investigated. As an application we study an asymptotic behavior of additive mappings.