Common Logarithms Research Articles

A Proposal for Rescaling Units

Several physical quantities are thought of as having “obvious” upper or lower bounds in their magnitudes. Temperature, for example, cannot go below absolute zero, the magnitude of any velocity is at most c, the speed of light, and time cannot go back past the Big Bang. In life outside the laboratory, we never get very close to any of these bounds, so they pose no problem. However, researchers in cryogenics, cosmology, and high-energy physics work at values that are perceived to be close to these limits. We read of temperatures one one-hundredth of a degree above absolute zero (that is, 0.01 K); of velocities at 99.95% of the speed of light; and of the state of the cosmos at 10−8 seconds after the Big Bang. In these cases, the standard system of units is inconvenient. Getting three orders of magnitude closer to 0 K, or to the time of the Big Bang, or to the speed of light, should not be obscured by the appearance that the change is merely an infinitesimal improvement.A change of units that maps zero to minus infinity and, in the case of velocity, maps c to plus infinity makes it easier to appreciate improvements that get closer to these bounds by orders of magnitude. This principle has been in use for a long time by engineers who measure changes in intensity in decibels (dB), where an increase or decrease by n orders of magnitude is a change of +10n dB or −10n dB, respectively.For physical quantities like time and temperature, the new units may simply be taken as the common logarithms of the old units. Using T to represent the new unit of temperature, we set 1 K = 0 T with (10 n ) K = n T for negative as well as positive values for n. Thus, 0.01 K = −2 T. More generally, x K = (log10 x) T, for any positive real number x. A similar approach applies to time since the Big Bang, where “x seconds after the Big Bang” becomes (log10 x) U, where U is our new, logarithmic measure of time.For velocity magnitudes v, with 0 ≤ v ≤ c, we can rescale to V = tan(πv/2c), with 0 ≤ V ≤ + ∞. At υ=1/2c,, this gives V=tan(π/4)=1..For nanotechnology researchers, the lower limit of zero for weight, length, and so forth can be moved to minus infinity by the same logarithmic technique used above for time and temperature. Names and symbols for these new units should be recommended by appropriate standards committees for each of the research areas involved.© 2001 American Institute of Physics.

Design of a unified arithmetic processor based on redundant constant-factor CORDIC with merged scaling operation

An arithmetic processor is designed based on redundant constant-factor implementation of the coordinate rotation digital computer (CORDIC) algorithm with three different modes: circular, hyperbolic and linear. Both CORDIC types (angle calculation and vector rotation) are considered in this unified processor that is capable of computing a wide variety of arithmetic and elementary functions including: multiplication, division, some common trigonometric functions, natural logarithms, square roots, vector norm and phase. Furthermore, by merging the scaling operation with the regular CORDIC iterations, the processor based on folded (iterative) CORDIC architecture reduces by about 1/4 the total number of iterations in one complete CORDIC operation.

Statistics notes. Logarithms.

Logarithms (or logs for short) are much used in statistics. We often analyse the logs of measurements rather than the measurements themselves, and some widely used methods of analysis, such as logistic and Cox regression, produce coefficients on a logarithmic scale. Here we shall give a brief summary of the properties of logarithms which make them so useful. We shall start with logarithms to base 10. These are the common logarithms formerly widely used to do calculations for which we now use calculators and computers. The log to base 10 of a number …

Does Morphology Predict Ecology? Hypothesis Testing within a Freshwater Stream Fish Assemblage

in gut analyses; and (c) 34 head, fin, and body measurements. Ecological data were arcsine transformed, while morphological data were converted first into common logarithms, then into canonical variate scores and scores from a sheared principal components analysis. A taxonomic matrix based on accepted phylogenetic relationships amongst species of the assemblage was also constructed. Microhabitat, trophic, morphological, and taxonomic matrices were then compared against one another using pair-wise Mantel tests, matrix correlation analyses, and three-way Mantel tests. In analyses across families, two-way Mantel tests revealed that taxonomy and body shape (based on sheared principal component scores) were each significantly related to one another, and to the trophic matrix. Yet, three-way Mantel comparisons indicated that body shapes of constituent species were significantly correlated only with taxonomy, and not with diet. To circumvent the confounding effects of taxonomy, the largest family in the study (Cyprinidae, 8 species represented) was then evaluated separately. The same analyses were performed as before, using trophic, microhabitat, and morphological matrices adjusted for numbers of species. In these analyses, morphology (represented by canonical variate and sheared principal component scores) was significantly related to the microhabitat matrix, and to no others. Previous field studies have indicated that cyprinids segregate along habitat rather than trophic dimensions. Ecomorphological analyses thus appear to be a valid assay for structure within fish assemblages only when limited to within-family comparisons (where the effects of history appear less significant). When many different families are involved in the analysis, historical influences predominate.

Quantitative analysis of human immunodeficiency virus type-1 DNA in asymptomatic carriers using the polymerase chain reaction

A method for detecting human immunodeficiency virus type 1 (HIV-1) provirus DNA in lymphocytes with improved sensitivity and reproducibility was developed using the polymerase chain reaction(PCR). Amplified HIV-1 DNA was hybridized with a 32P-labeled probe and quantitated with a beta-scanner after electrophoresis. A linear relationship was obtained between the common logarithms of the counts detected and the number of HIV-1 DNA copies applied to the PCR. Detectability was from 3 copies/10 5 lymphocytes, and linearity was maintained from 10 to 10 3 copies. HIV-1 DNA was detected in all 9 asymptomatic carriers tested (18 to 2,857 copies/10 5 CD4+ T lymphocytes). The viral burden was inversely related to the CD4+ lymphocyte count, suggesting that quantitation of provirus levels may serve as a predictor of progress in early HIV infection.

Helium isotope ratio and heat discharge rate in the Hokkaido island, Northeast Japan.

We have collected 21 gas samples of various kinds such as volcanic gas, hot spring gas and CH4-rich natural gas in the Hokkaido Island. Helium and neon in the samples were purified in a vacuum line and the 3He/4He and 4He/20Ne ratios were measured by a mass spectrometer. Observed 3He/4He and 4He/20Ne ratios vary significantly from 0.242 × 10-6 to 9.58 × 10-6 and from 0.53 to 96, respectively. There is no correlation between the 3He/4He and 4He/20Ne ratios. Taking into account present data as well as published ones, geographical distribution of 3He/4He ratios indicates a clear contrast, high in the volcanic arc and low in the forearc region. The distribution pattern agrees well with the data of Northeast Japan (Sano and Wakita, 1985) and those of the North Island of New Zealand (Sano et al., 1987). Higher 3He/4He ratios than air in the volcanic arc region can be well explained by the presence of an uprising magma. Lower 3He/4He ratio in the forearc region is attributed to the radiogenic helium accumulated in crustal rocks. A positive correlation is found between the common logarithms of 3He/4He ratio and heat discharge rate of hot springs in the area. The tendency can be explained by a combination of magma aging and the dilution of magmatic fluids by local groundwater.

Epidemiological Immunological Studies on Leprosy in Okinawa

Lepromin reactions of leprosy patients in the mainland and Okinawa of Japan were examined by using the standard (160 million bacilli/ml) and diluted lepromins, for the pur-pose of establishing practical use of the latter and understanding immunological character-istics of leprosy in Okinawa. The size of reactions expressed by mm scale were con-verted by common logarithms for statistical evaluation, because the distribution curves shown in Fig. 1-3 were approximately equal to a distorted shape of normal distribution. The results were as follow.1. Both Fernandez' and Mitsuda's reactions to the lepromins diluted in 1:4 or more were significantly weaker than those to the standard, as demonstrated by the percentage of completely negative (0-1mm) reactions (Tables 1 and 2) and by the difference between two reactions in both forearms of the same patient (Tables 7 and 8).2. Average size of reactions of the patients in Okinawa was significantly larger than those in the mainland, irrespective of dilution degree of the lepromins (Tables 4 and 5).3. A correlation coefficient was significant between the Fernandez' and Mitsuda's reactions to each of the standard, 1:4 and 1:8 diluted lepromins (Table 9).4. Normal human, autoclaved skin suspension caused the cutaneous response resembling Fernandez' reaction in a significant number of the patients, but the most of their responses disappeared completely after 4 weeks and a mean size of reactions in a few cases was significantly smaller than that of Mitsuda's reactions (Tables 3, 6 and 7).5. Significant fall in the percentage of completely negative reactions and significant increase in the mean of 2mm or larger sizes of ractions were observed in the patients with lepromatous leprosy, according to three stages which were classified as progressive, retrogressive and quiescent in this order (Table 10).6. A cut-off point of positive Fernandez' reactions was proposed to be 8mm, according to different criteria for positive reactions in lepromatous cases as well as in non-lepro-matous. On the other hand, Mitsuda's reactions should rather be recorded by the size of reaction which could be used as an indicator of cell-mediated immunity against Mycobacterium leprae.

Proton and Metal Complexation by Water‐soluble Ligands Extracted from Anaerobically Digested Sewage Sludge

AbstractA water‐soluble extract (WSE) of an anaerobically‐digested sewage sludge containing 19.5 mol m−3 (19.5 mM) total soluble carbon (CTS) was acidified to pH 3 and titrated with NaOH to pH 11 in a 50 mol m−3 (50 mM) NaClO4 background at 25°C. Conditional protonation constants were determined from the formation function computed from the titration data. The common logarithms of the average protonation constants for functional groups titrated between pH 3 and 11 were 4.2, 7.2, and 9.6. Trace metal‐WSE complexes were chromatograpbed on a molecular size exclusion gel. Copper formed the more stable complexes with the ligands in the WSE when compared to Ni or Cd. Cadmium was present predominately as the free metal ion or in inorganic complexes. Adsorption of Cu by montmorillonite was significantly reduced in the presence of the WSE. Since most of the organic carbon present in the WSE remained in solution during the adsorption experiments, the primary factor controlling Cu solubility was concluded to be the formation of soluble organo‐Cu complexes. A “mixture model” was developed to simulate qualitatively the proton binding of the inorganic and organic components of the WSE. This model included organic and inorganic ligands whose relative concentrations were previously analytically determined for the WSE. The “mixture model” ligands were employed to model Ca and Cd speciation for gel filtration and adsorption experiments. The “mixture model” qualitatively explained the observed speciation in both experiments.

Appendix E - REMOVED: Common Logarithms

This chapter has been retracted. Please see Elsevier Policy on Article Retraction (http://www.elsevier.com/locate/withdrawalpolicy). This chapter has been removed as the rights have been returned to the author.

GROWTH PATTERNS OF MUSCLE, FAT AND BONE, AND CARCASS COMPOSITION OF DOUBLE MUSCLED AND NORMAL CATTLE

Eighteen Double Muscled (DM), 18 Beef Synthetic (SY) and 18 Hereford (HE) bulls were serially slaughtered from approximately 250 to 800 kg liveweight to determine the influence of maturity type and "double muscling" on tissue growth patterns and compositional differences which were examined by using the allometric equation Y = aXb after transformation to common logarithms (log10). As liveweight (LW) increased, the proportions of total side fat (TSF) and subcutaneous fat (SCF) increased, the proportions of total side muscle (TSM) and intermuscular fat (IMF) remained relatively constant and the proportion of total side bone (TSB) decreased. Relative to LW, DM had a similar growth rate for TSM and lower growth rates for TSF, SCF and IMF than HE and SY. All breed types had similar growth rates for TSB on LW. Relative to dissected side weight (DSW) and TSB, the DM bulls had a higher growth rate for TSM than did the HE and S Y bulls. Differences existed between DM and the more normal genotypes in carcass composition. DM was characterized by a high muscle:liveweight ratio, a high proportion of muscle in the carcass, high muscle:bone and muscle:fat ratios and a low proportion of fat compared to the more normal genotypes. Objective criteria to discriminate between HE and DM were SCF, muscle:bone and TSMTMF ratios, and between SY and DM were TSB and muscle:bone ratio. Key words: Beef bulls (young), growth patterns, carcass composition, double muscling, breed type

Appendix E - WITHDRAWN: Common Logarithms

This Book/Chapter has been withdrawn as the rights have been returned to the author

Wroński's Canons of Logarithms

As everyone knows, the electronic computer and the hand calculator have made tables of logarithms all but obsolete as practical aids to calculation. The history of these tables, however, glistens with examples of remarkable diligence and ingenuity. There may be no more striking example of cleverness in this area than Wrofiski's canons of logarithms. This does not seem too strong a claim to make for a scheme that made it possible to put a seven-place table of common logarithms, in readable type, on a single page less than 17 cm by 22 cm in size.

湖沼最深部底泥の有機物含量と湖盆形態との関係

Based on the analysis of the surface layer (0-3cm) of bottom sediments at the deepest points of 10 lakes in Hokkaido and on the data by other investigators on the organic matter of 10 other Japanese lakes, the relationship between morphometric features and organic matter contents of bottom sediments of lakes was investigated. Concentrations of total nitrogen, organic carbon and ignition loss of the bottom sediments were reversely correlated with the common logarithms of maximum depths in deep (over 30 m in maximum depth) natural lakes. The concentration of organic matter was also related with the ratio of the length of shore line to area except for artificially polluted lakes, man-made lakes, and very large or small lakes. Hence, organic matter of bottom sediments was suggested to be primarily influenced by the morphometric features of lakes. Organic matter of bottom sediments at the deepest points of 20 Japanese lakes consisted of total nitrogen 0.07-0.81%, organic carbon 0.21-7.05% and ignition loss 6-24%.

Edmund Gunter (1581–1626)

The year 1981 marks the fourth centenary of the birth of Edmund Gunter who, during his relatively short life, played a dominant role in England in the advancement of navigation. Born in Hertfordshire, of Welsh parentage, Gunter was educated at Westminster School before entering Christchurch College, Oxford, where he graduated BA in 1603; MA in 1606; and BD in 1615, in which year he was presented to the living at St George's, Southwark.As early as 1603 Gunter had written an account of a ‘New Projection of the Sphere’ which was circulated in manuscript among some of his mathematical acquaintances. This appears to have gained for him the friendship of Henry Briggs, the first Gresham Professor of Geometry and a brilliant mathematician best known for having been first to suggest a table of logarithms to base 10 – ‘common logarithms’, as they are now known.

Trace Metal Complexation by Fulvic Acid Extracted from Sewage Sludge: I. Determination of Stability Constants and Linear Correlation Analysis

AbstractThe complexes formed between either Cd2+ or Pb2+ and fulvic acid extracted from sewage sludge mixtures were investigated by potentiometric titration of K/H fulvate with either Cd(ClO4)2 or Pb(ClO4)2 at pH 5.0 and 25°C, in an ionic medium of 0.1M KClO4. Formation functions were calculated from the titration data and analyzed by the Scatchard plot method. The results were similar to those obtained previously for sludge‐fulvic acid complexes of Ca2+ and Cu2+, in that two classes of complexes were found for both Cd2+ and Pb2+. The common logarithms of the measured conditional stability constants for the two types of complexes with sludge‐derived fulvic acid were hypothesized to be linearly correlated with the Misono softness parameter, which is a quantitative measure of the tendency to covalent bonding on the part of a metal cation. The resulting linear correlation equations had r2‐values ≥ 0.99 and, therefore, were used to predict conditional stability constants for complexes of sludge‐derived fulvic acid with Mg2+, Mn2+, Fe2+, Ni2+, and Zn2+, which cannot be investigated by the method of potentiometric titration.

Treatment of Heterogeneity of Variances in Hydrocyanic Acid Data1

AbstractHomogeneity of variance is a basic assumption of the analysis of variance. Hydrocyanic acid potential (HCN‐p) of samples of sorghum (Sorghum bicolor (L.) Moench) plants may vary from nil to the maximum produced by the genotype‐treatment combination, resulting in heterogeneous variances. The purpose of the present study was to further elucidate heterogeneity of variance in HCN‐p data and to employ appropriate treatment of the heterogeneity.‘Greenleaf’ and ‘Piper’ sudangrass (Sorghum sudanese (Piper) Stapf.) and ‘Suhi‐l’, a sorghum ✕ sudangrass hybrid were subjected to five N treatments: (a) no N applied, (b) 67.2 kg/ha at planting, (c) 33.6 kg/ha at planting and 33.6 kg/ha after second harvest, (d) 134.4 kg/ha at planting, and (e) 33.6 kg/ha at planting and 33.6 kg/ha after first, second, and third harvests. When the plants reached a height of 75 cm, they were cut at a height of 20 cm, resulting in four harvests. The cultivar, N, and harvest treatments were analyzed for heterogeneity of variances.For the natural data the treatment means and variances were positively associated and the variances for several treatment comparisons were heterogeneous. Transforming the natural data to common logarithms failed to eliminate the evidence of heterogeneity of variances. Since the data did not meet the assumption of homogeneity of variance, an approximate test of equality of means based upon the assumption of heterogeneity of variance was recommended for the HCN‐p data.Heterogeneity of variances decreases the accuracy of statistical comparisons of means. Since HCN‐p is converted to highly toxic hydrocyanic acid by ruminant animals, accuracy of statistical decisions is critical. Researchers should examine HCN‐p data for evidence of heterogeneity of variances and, if present, employ the approximate test which includes provisions for heterogeneity of variances.

THE DETERMINATION OF DEBYE CHARACTERISTIC TEMPERATURES OF CRYSTALS FROM X-RAY DIFFRACTION INTENSITIES

The methods of determining Debye characteristic temperatures from X-ray diffraction intensities for the case of homogeneous and isotropic crystals have been fully discussed.It is proposed that if the common logarithms of the ration of the calculated intensities to observed intensities log (Icalc/Iobs) of all diffraction lines are plotted against sin2θ, a straight line should be obtained, the slope of which gives 2Bloge/λ2, where B is a physical quantity to be determined contained in the Debye factor e(-2Bsin2θ/λ2) in the intensity expression, λ being the wave length of the radiation used. In the Debye theory of specific heats, B may be expressed as (6h2T/MkΘD2){Φ(x) + x/4}, where h and k represent Planck constant and Boltzmann constant respectively, M is the mass of the atom or of the group of atoms situated at the lattice points, T is the absolute temperature at the time of taking Debye-Scherrer photographs, and ΘD is the Debye characteristic temperature. X = ΘD/T, and φ(x) is a function of x, given in the original Debye theory. It is seen that if we let G=BMkT/6h2, then φ(x)+x/4=Gx2 Having obtained B, G in this equation is a measurable number, and solution of the equation may be performed graphically. By making Y1=Gx2 and Y2=φ(x)+x/4, the plotting of these two equations should give two curves, the intersection of which should give x which determines the characteristic temperature at that temperature.It is pointed out that owing to the fact that ΘD itself is a function of temperature, the method proposed affords a possibility of determining Debye temperatures at required temperatures.

Cattle, Control of Face and Horn Flies on Dairy Cattle With Ectrin And Rabon Ear Tags, 1979

Abstract Plastic ear tags containing 8.0% wt/wt Ectrtn (fenvalerate) were placed 1/head on 17 dairy cows. Tagging was done on June 29 and the cattle maintained on pasture. On July 2, 18 similar animals were treated by placing 13.7% wt/wt Rabon (stirofos) tags in each ear. An additional 24 head were treated on July 5 by affixing a single Rabon tag per head. A separate herd consisting of ca. 20 head was maintained as an untreated control. All animals were pastured within 1 km of one another. Assess-ment of treatment efficacy was made at weekly intervals via inspection of 10 head selected at random from each herd. Face flies were counted on the entire face, and horn flies on a single side of each animal examined. Data thus obtained was converted to common logarithms prior to analysis; however, true averages are reported. The test was terminated when herd management based on lactation forced intermixing of experimental groups.

Application of Digital Fourier Analysis in Processing Dynamic Aerodynamic Heating Measurements

APRACTICAL application of a digital Fourier analysis (fast Fourier transform) in processing fluctuating heattransfer data was utilized in a wind-tunnel test program at Mach number 8. A 4-deg, half-angle cone instrumented with conventional heat gages was oscillated in pitch at an amplitude of 2 deg and frequencies up to 15 Hz. The gages had a time constant of approximately 120 ms, which was not sufficiently responsive to follow the motion, and the Fourier analysis was used to correct the gage outputs. These measurements were used to map the location of the boundarylayer transition front. The purpose of this paper is to describe in some detail the technique used in processing these transient heat-transfer measurements. A more detailed presentation may be found in Ref. 1. The results indicate that the Fourier analysis was a very powerful method for correcting dynamic heat-transfer measurements. Contents The procedure for analyzing transient sensor data signals which are produced by undefined input signals is well documented. Any transient signal can be viewed in the frequency domain as a spectrum (amplitude distribution of the real and imaginary terms as a function of frequency), and the fast Fourier transform (FFT) algorithm provides the numerical procedure needed to transform digitally recorded transient data from the time domain into the frequency domain. Brigham2 has suggested that the FFT, in many respects, is analogous to using natural (or common) logarithms as a means of substituting simple arithmetic operations such as addition and subtraction for multiplication and division. In the frequency domain, certain numerical operations on a signal such as convolutions, correlations, energy content, and discrete digital filtering can be performed quite simply. In particular, the FFT can be used to determine the response or transfer function of a linear measuring system based on the distorted output of a sensor along with the known calibration pulse which caused the sensor response. A computer calculates the Fourier transforms of both the input and output time histories and ratios the two transforms to

Proton Binding in Fulvic Acid Extracted From Sewage Sludge‐soil Mixtures

AbstractAqueous solutions of two fulvic acids (FA) extracted from sludge‐soil mixtures and converted to K‐fulvates were titrated potentiometrically at 25°C with HCl in ionic media of 0.1M and 0.2M KCl. Data were obtained for a range of concentrations of FA, although during a titration experiment both FA concentration and ionic strength were rigorously held constant. A simple model was employed to interpret the formation functions calculated using the titration data. This model ascribes the acidic properties of a FA solution to an assembly of hypothetical macromolecules (“mean fulvic acid units”) that contain several distinct classes of functional group and are able to associate through hydrogen bonding. Conditional protonation constants were computed for each class of acidic functional group that protonates in the pH range 3–11. The common logarithms of the average protonation constants at 25°C were found to be 3.9, 6.5, and 9.4 for the three classes of weakly acidic functional groups studied. These values were independent (within experimental error) of FA composition and concentration, and of ionic strength in the range investigated.