Logarithmic Measure and the Decibel

1954 ◽  
Vol 26 (3) ◽  
pp. 449-450 ◽  
Author(s):  
William M. Hall
Keyword(s):  
Logarithmic Measure

A Law of Work-Hardening

1948 ◽  
Vol 15 (3) ◽  
pp. 265-273
Author(s):  
A. M. Freudenthal ◽  
M. Reiner
Keyword(s):  
Mild Steel ◽  
Work Hardening ◽  
Wire Drawing ◽  
Total Work ◽  
Exponential Functions ◽  
Recoverable Strain ◽  
Annealed State ◽  
Logarithmic Measure ◽  
Work Of Deformation ◽  
Crystalline Metal

Abstract Based on the “blocking” theory of the strength of a poly-crystalline metal, a law of work-hardening is derived and checked experimentally on mild steel deformed by wire-drawing up to a deformation of 4.6 in the logarithmic measure. The law correlates the recoverable strain work with the total work of deformation in a series of exponential functions, the number of which corresponds to the number of sizes of crystal grains present in the annealed state.

A logarithmic measure of image enhancement

2006 ◽  
Author(s):  
Eric Wharton ◽  
Sos Agaian ◽  
Karen Panetta
Keyword(s):  
Image Enhancement ◽  
Logarithmic Measure

scholarly journals Properties of strong ground motion earthquakes*

1955 ◽  
Vol 45 (3) ◽  
pp. 197-218
Author(s):  
George W. Housner
Keyword(s):  
Strong Ground Motion ◽  
Wave Train ◽  
Square Root ◽  
Earthquake Fault ◽  
Relative Slip ◽  
Logarithmic Measure ◽  
Upper Limits ◽  
Slip Area ◽  
Sudden Release ◽  
Strong Ground

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.

A centripetal-pump effect in air

1957 ◽  
Vol 240 (1221) ◽  
pp. 173-188 ◽  
Keyword(s):  
Shear Modulus ◽  
Finite Time ◽  
Vertical Axis ◽  
Rheological Equation ◽  
Logarithmic Measure ◽  
High Speeds ◽  
State Of Stress ◽  
Stress Strain Relation ◽  
Outer Atmosphere ◽  
Elastic Shear

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.

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

1979 ◽  
Vol 46 (1) ◽  
pp. 78-82 ◽  
Author(s):  
L. Anand
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Finite Strain ◽  
Large Deformations ◽  
Strain Energy Function ◽  
Strain Measure ◽  
Logarithmic Measure ◽  
Infinitesimal Strain ◽  
Isotropic Elasticity ◽  
Good Agreement

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.

scholarly journals Fundamentals of the logarithmic measure for revealing multimodal diffusion

2021 ◽  
Author(s):  
Benjamin A. Dalton ◽  
Ivo F. Sbalzarini ◽  
Itsuo Hanasaki
Keyword(s):  
Logarithmic Measure

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

2020 ◽  
Vol 54 (2) ◽  
pp. 135-145
Author(s):  
A. O. Kuryliak ◽  
O. B. Skaskiv ◽  
S. I. Panchuk
Keyword(s):  
Entire Function ◽  
Power Series ◽  
Entire Functions ◽  
Counting Function ◽  
Type Inequality ◽  
Continuous Functions ◽  
Maximal Term ◽  
Logarithmic Measure ◽  
Circular Domains

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.

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