Bitlyan-Gol'dberg type inequality for entire functions and diagonal maximal term

In the article is obtained an analogue of Wiman-Bitlyan-Gol'dberg type inequality for entire $f\colon\mathbb{C}^p\to \mathbb{C}$ from the class $\mathcal{E}^{p}(\lambda)$ of functions represented by gap power series of the form$$f(z)=\sum\limits_{k=0}^{+\infty} P_k(z),\quadz\in\mathbb{C}^p.$$Here $P_0(z)\equiv a_{0}\in\mathbb{C},$ $P_k(z)=\sum_{\|n\|=\lambda_k} a_{n}z^{n}$ is homogeneouspolynomial of degree $\lambda_k\in\mathbb{Z}_+,$ ànd $ 0=\lambda_0<\lambda_k\uparrow +\infty$\ $(1\leq k\uparrow +\infty ),$$\lambda=(\lambda_k)$.\ We consider the exhaustion of thespace\ $\mathbb{C}^{p}$\by the system $(\mathbf{G}_{r})_{r\geq 0}$ of a bounded complete multiple-circular domains $\mathbf{G}_{r}$with the center at the point $\mathbf{0}=(0,\ldots,0)\in \mathbb{C}^{p}$. Define $M(r,f)=\max\{|f(z)|\colon z\in\overline{G}_r\}$, $\mu(r,f)=\max\{|P_k(z))|\colon z\in\overline{G}_r\}$.Let $\mathcal{L}$ be the class of positive continuous functions $\psi\colon \mathbb{R}_{+}\to\mathbb{R}_{+}$ such that $\int_{0}^{+\infty}\frac{dx}{\psi(x)}<+\infty$, $n(t)=\sum_{\lambda_k\leq t}1$ counting function of the sequence $(\lambda_k)$ for $t\geq 0$. The following statement is proved:{\it If a sequence $\lambda=(\lambda_{k})$ satisfy the condition\begin{equation*}(\exists p_1\in (0,+\infty))(\exists t_0>0)(\forall t\geq t_0)\colon\quad n(t+\sqrt{\psi(t)})-n(t-\sqrt{\psi(t)})\leq t^{p_1}\end{equation*}for some function $\psi\in \mathcal{L}$,then for every entire function $f\in\mathcal{E}^{p}(\lambda)$, $p\geq 2$ and for any$\varepsilon>0$ there exist a constant $C=C(\varepsilon, f)>0$ and a set $E=E(\varepsilon, f)\subset [1,+\infty)$ of finite logarithmic measure such that the inequality\begin{equation*}M(r, f)\leq C m(r,f)(\ln m(r, f))^{p_1}(\ln\ln m(r, f))^{p_1+\varepsilon}\end{equation*}holds for all $ r\in[1,+\infty]\setminus E$.}The obtained inequality is sharp in general.At $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and the Bitlyan-Gol'dberg inequality (1959) it follows. In the case $\lambda_k\equiv k$, $p=2$ we have $p_1=1/2+\varepsilon$ and from obtained statement we get the assertion on the Bitlyan-Gol'dberg inequality (1959), and at $p=1$ about the classical Wiman inequality it follows.

On H. Hencky’s Approximate Strain-Energy Function for Moderate Deformations

It is shown that the classical strain-energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain-energy function is replaced by the Hencky or logarithmic measure of finite strain.

A centripetal-pump effect in air

An instrumental arrangement is described which constitutes a centripetal air-pump. It consists of a hollow cylinder closed at the top and open at the bottom which can be brought into rotation at high speeds about a vertical axis. When a receptacle filled with a heavy oil is lifted until the cylinder is partially immersed, the oil does not wet the cylinder, but an air-gap is maintained between the walls of the cylinder and the oil. It can be seen that air is pumped from the outer atmosphere through this gap into the cylinder. This presupposes that the air must be in a state of stress which includes elastic cross-stresses. It is shown that these stresses result from a stress-strain relation in which the strain is defined in Hencky’s logarithmic measure. This confirms Maxwell’s theory that air is an elasticoviscous material possessing an elastic shear modulus, and therefore a finite time of relaxation. A rheological equation for air is proposed accordingly.

Properties of strong ground motion earthquakes*

Summary The analysis given here considers that an earthquake fault is formed by the superposition of a large number of incremental shear dislocations the sudden release of which produces the earthquake. It is postulated that during an earthquake the incremental dislocations are released in such a way that the average slip is proportional to the square root of the area of slip, and that the probability of release of individual incremental dislocations is such that the probability of a total slip area A is inversely proportional to A. With these two postulates a frequency distribution of earthquakes is derived that agrees with observed data; the Richter magnitude is shown to be essentially a logarithmic measure of the average slip on a fault; and an expression is derived for the energy released by an earthquake that agrees with that derived from consideration of the energy carried in a wave train. Expressions are derived also for the areas of slip during earthquakes, the maximum relative slip, and the average annual, over-all shearing distortion of the state of California and these are in satisfactory agreement with observed behavior. It is assumed that an accelerogram is formed by the superposition of a large number of elemental acceleration pulses random in time. It is shown that this agrees with recorded accelerograms, and an accelerogram composed in this fashion is shown to have the characteristics of actual recorded accelerograms. It is also shown that the maximum ground accelerations in the vicinity of the center of the fault, so far as they are dependent upon the size of the slip area, have essentially reached their upper limits for shocks with areas of slip approximately equal to that associated with the El Centro earthquake of 1940.