Logarithmic Tables Research Articles

Effects of various hyperopia intervention levels on male college students' gait kinematics.

Background: Hyperopia is a common blurred vision phenomenon that affects postural control in gait; however, current research has focused on the alteration and correction of hyperopia's physiological characteristics, ignoring the effect of hyperopia on gait kinematic characteristics. The effect of hyperopia on the basic form of movement walking is a worthy concern. Objective: To investigate the gait kinematic characteristics of male college students with varying degrees of visual acuity (normal vision, hyperopia 150°, and hyperopia 450°), as well as to provide a theoretical foundation for the effect of visual acuity on gait and fall risk reduction. Methods: Twenty-two male college students with normal visual acuity were chosen. Their vision was tested using a standard visual acuity logarithm table at normal and with 150° and 450° concave lenses. Gait kinematic data were collected under normal vision and hyperopic conditions using the PN3 Pro advanced inertial motion capture system and Axis Studio application program. Results and conclusion: 1. The change of center of gravity in Pre-double support was smaller than normal vision; Late-single support and Late-swing was larger than normal vision; 2. The percentage of the double-leg support decreased; the percentage of the single-leg support and the Late-swing increased; 3. For the joints' range of motion, Trunk flexion and extension range of motion in Pre-single support, Late-double support and Pre-swing smaller than normal visual acuity, and Late-swing larger than normal; hip internal abduction and adduction and internal and external rotation are larger than normal vision in Late-single support; knee and ankle in abduction and adduction direction are larger than normal vision in the swing stage; hip flexion and extension, internal external rotation are larger than normal vision in the swing stage. Hyperopic interventions have an impact on the kinematic characteristics of gait in male college students, mainly in terms of altered balance, increased instability, increased difficulty in maintaining trunk stability, and increased risk of injury.

Zipf’s Law, Benford’s Law, and Pareto Rule

From a basic probabilistic argumentation, the Zipfian distribution and Benford’s law are derived. It is argued that Zipf’s law fits to calculate the rank probabilities of identical indistinguishable objects and that Benford’s distribution fits to calculate the rank probabilities of distinguishable objects. i.e. in the distribution of words in long texts all the words in a given rank are identical, therefore, the rank distribution is Zipfian. In logarithmic tables, the objects with identical 1st digits are distinguishable as there are many different digits in the 2nd, 3rd… places, etc., and therefore the distribution is according to Benford’s law. Pareto 20 - 80 rule is shown to be an outcome of Benford’s distribution as when the number of ranks is about 10 the probability of 20% of the high probability ranks is equal to the probability of the rest of 80% low probability ranks. It is argued that all these distributions, including the central limit theorem, are outcomes of Planck’s law and are the result of the quantization of energy. This argumentation may be considered a physical origin of probability.

Four and Five Figures of Kifilideen (Power of Base 11) and Antikifilideen (Antipower of Base 11) Tables as a Tool For Mathematics Computation

The four-figure of power of base numbers of tables needs to be created for easy synchronization with the existing Logarithm table of power of base 10 which is working on four figures. Although the five-figure of power base numbers of tables are more accurate because it is having less approximation in its establishment. The four-figure table is easy to compute doing utilization because it is having less digits to work with. This paper presents the four and five figures of Kifilideen (Power of base 11) and AntiKifilideen (Antipower of base 11) tables for the computation of mathematical problems. The four and five figures are both reliable to work with. However, there is a tradeoff between easy computation as related to the four-figure table and more accuracy as related to the five-figure table in their utilization.
 

High throughput error correction in information reconciliation for semiconductor superlattice secure key distribution

Semiconductor superlattice secure key distribution (SSL-SKD) has been experimentally demonstrated to be a novel scheme to generate and agree on the identical key in unconditional security just by public channel. The error correction in the information reconciliation procedure is introduced to eliminate the inevitable differences of analog systems in SSL-SKD. Nevertheless, the error correction has been proved to be the performance bottleneck of information reconciliation for high computational complexity. Hence, it determines the final secure key throughput of SSL-SKD. In this paper, different frequently-used error correction codes, including BCH codes, LDPC codes, and Polar codes, are optimized separately to raise the performance, making them usable in practice. Firstly, we perform multi-threading to support multi-codeword decoding for BCH codes and Polar codes and updated value calculation for LDPC codes. Additionally, we construct lookup tables to reduce redundant calculations, such as logarithmic table and antilogarithmic table for finite field computation. Our experimental results reveal that our proposed optimization methods can significantly promote the efficiency of SSL-SKD, and three error correction codes can reach the throughput of Mbps and provide a minimum secure key rate of 99%.

A Common Risk Classification Concept for Safety Related Gas Leaks and Fugitive Emissions?

Gas leaks in the oil and gas industry represent a safety risk as they, if ignited, may result in severe fires and/or explosions. Unignited, they have environmental impacts. This is particularly the case for methane leaks due to a significant Global Warming Potential (GWP). Since gas leak rates may span several orders of magnitude, that is, from leaks associated with potential major accidents to fugitive emissions on the order of 10−6 kg/s, it has been difficult to organize the leaks in an all-inclusive leak categorization model. The motivation for the present study was to develop a simple logarithmic table based on an existing consequence matrix for safety related incidents extended to include non-safety related fugitive emissions. An evaluation sheet was also developed as a guide for immediate risk evaluations when new leaks are identified. The leak rate table and evaluation guide were tested in the field at five land-based oil and gas facilities during Optical Gas Inspection (OGI) campaigns. It is demonstrated how the suggested concept can be used for presenting and analysing detected leaks to assist in Leak Detection and Repair (LDAR) programs. The novel categorization table was proven valuable in prioritizing repair of “super-emitter” components rather than the numerous minor fugitive emissions detected by OGI cameras, which contribute little to the accumulated emissions. The study was limited to five land based oil and gas facilities in Norway. However, as the results regarding leak rate distribution and “super-emitter” contributions mirror studies from other regions, the methodology should be generally applicable. To emphasize environmental impact, it is suggested to include leaking gas GWP in future research on the categorization model, that is, not base prioritization solely on leak rates. Research on OGI campaign frequency is recommended since frequent coarse campaigns may give an improved cost benefit ratio.

Panniculitis in the setting of dermato/rheumatologic diseases.

Panniculitides represent a heterogeneous group of inflammatory diseases that are traditionally considered one of the most difficult challenge for clinicians and pathologists. They may occur in a variety of dermato/rheumatologic diseases and are of particular relevance for clinicians dealing with such pathologies, including immune-mediated/autoimmune and autoinflammatory disorders. In fact, panniculitides can be the initial sign of presentation of a dermato/rheumatologic disease, thereby providing the physician with important clues to the correct diagnosis. Then, panniculitides may serve as an easy-to-access indicator of both systemic involvement and prognostic outcome in dermato/rheumatologic disorders. This review will focus on clinical and histopathological findings of panniculitides in the setting of dermato/rheumatologic disorders and discusses the value of skin biopsies and consequent histopathological examination in the diagnosis of these disorders with the help of a logarithmic table.

Benford's law: A “sleeping beauty” sleeping in the dirty pages of logarithmic tables

Benford's law is an empirical observation, first reported by Simon Newcomb in 1881 and then independently by Frank Benford in 1938: the first significant digits of numbers in large data are often distributed according to a logarithmically decreasing function. Being contrary to intuition, the law was forgotten as a mere curious observation. However, in the last two decades relevant literature has grown exponentially—an evolution typical of “Sleeping Beauties” (SBs) publications that go unnoticed (sleep) for a long time and then suddenly become the center of attention (are awakened). Thus, in the present study, we show that the two papers, Newcomb () and Benford (), Newcomb (, American Journal of Mathematics, 4, 39–40) and Benford (1938, Proc. Am. Phil. Soc., 78, 551–572) papers are clearly SBs. The former was in a deep sleep for 110 years, whereas the latter was in a deep sleep for a comparatively lesser period of 31 years up to 1968, and in a state of less deep sleep for another 27 years, up to 1995. Both SBs were awakened in the year 1995 by Hill (, Statistical Science, 10, 354–363). In so doing, we show that the waking prince (Hill, ) is more often quoted than the SB whom he kissed—in this Benford's law case, wondering whether this is a general effect—to be usefully studied.

On the Reconstruction of Text Phylogeny Trees: Evaluation and Analysis of Textual Relationships.

Over the history of mankind, textual records change. Sometimes due to mistakes during transcription, sometimes on purpose, as a way to rewrite facts and reinterpret history. There are several classical cases, such as the logarithmic tables, and the transmission of antique and medieval scholarship. Today, text documents are largely edited and redistributed on the Web. Articles on news portals and collaborative platforms (such as Wikipedia), source code, posts on social networks, and even scientific publications or literary works are some examples in which textual content can be subject to changes in an evolutionary process. In this scenario, given a set of near-duplicate documents, it is worthwhile to find which one is the original and the history of changes that created the whole set. Such functionality would have immediate applications on news tracking services, detection of plagiarism, textual criticism, and copyright enforcement, for instance. However, this is not an easy task, as textual features pointing to the documents’ evolutionary direction may not be evident and are often dataset dependent. Moreover, side information, such as time stamps, are neither always available nor reliable. In this paper, we propose a framework for reliably reconstructing text phylogeny trees, and seamlessly exploring new approaches on a wide range of scenarios of text reusage. We employ and evaluate distinct combinations of dissimilarity measures and reconstruction strategies within the proposed framework, and evaluate each approach with extensive experiments, including a set of artificial near-duplicate documents with known phylogeny, and from documents collected from Wikipedia, whose modifications were made by Internet users. We also present results from qualitative experiments in two different applications: text plagiarism and reconstruction of evolutionary trees for manuscripts (stemmatology).

The mathematics behind Jost Bürgi's method for calculating sine tables

AbstractJost Bürgi (1552–1632) is not only one of the first persons who set up tables of logarithms for practical use but also invented a method to calculate highly precise tables of the sine function with small increment of the argument, and this with only a modest amount of calculations. Bürgi's manuscript on his method, however, was never published by him and got lost for almost four centuries until Menso Folkerts discovered it, cf. [2]. The article [1] on this manuscript gives a proof for the correctness of this method which, however, relies on the Theorem of Perron‐Frobenius that is over three centuries younger than Bürgi's method. The present contribution discusses how Bürgi may have come to his method and what considerations may have convinced him that it should work. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

HISAB AWAL WAKTU SALAT DALAM KITAB AL-KHULASAH FI AL-AQWAT AL-SYAR’IYYAH BI AL-LUGHARITMIYYAH KARYA MUHAMMAD KHUMAIDI JAZRY

Prayer is the pre-determined-timed obligatory worship. The jurists decide the time of prayer based on the rotation of the earth or natural phenomena. Such concept is still thought to be applicable until today. On the contrary, the astronomers decide the time of prayer based on a more mathematical formula. Since the invention of arithmetic, prayer time decision formula and methods are always evolving until now. Good progress occurs in the formula as well as in the use of tools. There are several methods used by the experts to confirm it. One is the concept of the calculation using logarithm tables. The book that discusses the methods is the book Al-Khulasah fi al-Awqat al-Syar’iyyah bi al-Lugharitmiyyah by Muhammad Khumaidi Jazry. In this book, in addition to making the calculation formula, the author also specifies the use of a table of five decimal logarithms. The results compared with calculations using the ephemeris method do not produce a significant difference. The difference between every prayer time is only in the range of two to four minutes. Kata kunci: Waktu Salat, Tabel Logaritma, Al-Khulasah fi Al-Awqat al-Syar’iyyah bi al-Lugharitmiyyah

Miller, S. J. (ed.): Benford’s law. Theory and applications

Ever since its rediscovery in the late 1930s byFrankA.Benford, the strangedistribution of first digits observed in many data sets has become an increasingly attractive object of investigation in mathematics, statistics, economics, accounting and many other fields. And it is from this universe of applications and inspirations that the 19 chapters of this nicely edited survey are drawn. I particularly liked passages like the one by Frank A. Benford’s grandson (also named Frank A. Benford) on the history of the subject, or personal accounts where authors explain how their interest in Benford’s law arose in the first place. Then there are many chapters on the law’s most popular application, i.e. how Benford’s law can be used to detect fraud in data sets of various sorts, with an in depth treatment given to election outcomes, economic macrodata, or financial statements of all sorts. Although most of these applications have long been known and have also been widely discussed in the media, in particular those concerning fraud in tax returns, it is nice to have them all collected together in one place. More mathematically inclined readers will also appreciate the collection of mathematical explanations for Benford’s law in various probability distributions and stochastic processes. And for statisticians, there is also some discussion on how the significance of deviations from Benford’s law can be best determined. This diversity of authors and approaches almost by necessity engenders considerable variation in mathematical rigour and level of abstraction. It also leads to considerable overlap at various places. For instance, after reading for the umpteenth time that Benford detected his law by studying tables of logarithms, and that the relative frequency of the first digit 1 is equal to log 2, some weariness arises. Because of this diversity, I also very much doubt whether this book is useful as a textbook for advanced statistics courses,

CURIOSIDADES DE LA FISICA Y MATEMATICAS

The story began in 1881 when the mathematician and astronomer Simon Newcomb US noted that the table of logarithms that used for computations showed a progressive wear to pages; the former were considerably more worn than the latter.

Study on Compilation of <i>Bi Li Shu Biao</i>

Bi Li Shu Biao is a logarithm table compiled by Jean Nicolas Smogolenski and Xue Fengzuo in the beginning of Qing dynasty and it has ten thousand common logarithm values of natural number. By analyzing these values, we found that they were obtained through cutting western logarithm values and adopting the method of five homes six into on the basis of reference to the Western logarithmic table. Almost logarithm values is correct, its false rate is only 0.76 percent.

The future of schooling: Children and learning at the edge of chaos

This paper describes the effect that assistive technologies, such as paper, printing, protractors, logarithm tables, computers, and the Internet, have on pedagogy. It reports the results of experiments with self-organising systems in primary education and develops the concept of a self-organised learning environment (SOLE). It then describes how SOLEs operate, and their possible effects on primary education in remote areas, and discusses the implications of the physics of complex systems and their possible connection with self-organised learning amongst children. Finally, it proposes a change in the examination system that would incorporate the Internet and concepts of self-organisation into schooling.

‘And John Napier created logarithms...’

The Mirifici logarithmorum canonis descriptio which Napier published in 1614 tabulates this ‘marvellous measure of ratio-numbers’ to 7 places, giving the ‘Napierian’ logarithms (we would now say) of the sines of angles from 0 to 90, and in a wealth of worked examples displays how these may be applied to reduce to the equivalent additions and subtractions the lengthy multiplications and divisions necessary in problems of plane and spherical trigonometry. The book had a short-lived heyday: one abruptly ended just ten years later when Henry Briggs brought out his 14-place table of common logarithms (these computed by a different and more powerful method).It is well known that the last places in Napier’s table are not to be trusted. And no one after 1624 would dream of constructing logarithms his way (which was not just by computing powers of …). Why then make so much fuss over his discovery if in fact he no more invented our logarithms than Logie Baird the TV cathode-ray tube?

Construction of logarithm tables for Galois Fields

A branch of mathematics commonly used in cryptography is Galois Fields GF(p n ). Two basic operations performed in GF(p n ) are the addition and the multiplication. While the addition is generally easy to compute, the multiplication requires a special treatment. A well-known method to compute the multiplication is based on logarithm and antilogarithm tables. A primitive element of a GF(p n ) is a key part in the construction of such tables, but it is generally hard to find a primitive element for arbitrary values of p and n. This article presents a naive algorithm that can simultaneously find a primitive element of GF(p n ) and construct its corresponding logarithm and antilogarithm tables. The proposed algorithm was tested in GF(p n ) for several values of p and n; the results show a good performance, having an average time of 0.46 seconds to find the first primitive element of a given GF(p n ) for values of n = {2, 3, 4, 5, 8, 12} and prime values p between 2 and 97.

The Silver Lining of Language Loss

Just as zoologists and botanists are concerned about the extinction of biological species, linguists are concerned about the rapid loss of languages [“What's lost when languages are,” A. Pires, Books et al. , 23 April, p. [431][1], review of ([ 1 ][2])]. We should, however, view the situation from the user's standpoint as well. In this modern and connected world, we cannot fully realize our potential unless we understand, speak, and write the language of our school, media, government, place of employment, and economic centers of power. If our mother language does not provide these connections (and most languages do not), one alternative is bilingualism or multilingualism, but many people are not able or willing to attain fluency in more than one language. The problem thus becomes a political one: Either the speakers of such a language maintain the autonomy and capability to promote enduring bilingualism, as the Catalans did, or the language takes the path toward extinction, as is happening to Occitan, Yiddish, and the native languages of North America. We may, as outsiders, deplore such an evolution, but language is a practical tool. We do not advocate returning to the steam engine, the slide rule, or the logarithmic table. The drastic reduction of the number of languages is natural, unavoidable, and—from the viewpoint of communication and integration into a world community—desirable. ![Figure][3] Losing languages. The number of Yiddish speakers is dwindling. CREDIT: WIKIMEDIA COMMONS 1. [↵][4] 1. K. David Harrison , When Languages Die: The Extinction of the World's Languages and the Erosion of Human Knowledge (Oxford Univ. Press, Oxford, 2008). [1]: /lookup/doi/10.1126/science.1163922 [2]: #ref-1 [3]: pending:yes [4]: #xref-ref-1-1 View reference 1 in text

94.15 Napier's table of logarithms

94.15 Napier's table of logarithms

A Real Cloud Computer

Attempts to calculate the weather numerically have a long history. The first effort along these lines took place not in some cutting-edge university or government lab but on what the lone man doing it described as "a heap of hay in a cold rest billet." Lewis Fry Richardson, serving as an ambulance driver during World War I and working with little more than a table of logarithms, made a heroic effort to calculate weather changes across central Europe from first principles way back in 1917. The day he chose to simulate had no particular significance-other than that a crude set of weather-balloon measurements was available to use as a starting point for his many hand calculations. It's no surprise that the results didn't at all match reality. Three decades (and one world war) later, mathematician John von Neumann, a computer pioneer, returned to the problem of calculating the weather, this time with electronic assistance, although the limitations of the late-1940s computer he was using very much restricted his attempt to simulate nature. The phenomenal advances in computing power since von Neumann's time have, however, improved the accuracy of numerical weather forecasting and allowed it to become a routine part of daily life. Will it rain this afternoon? Ask the weatherman, who in turn will consult a computer calculation.

A nomogram for the atmosphere

When our departmental library cleared out some of its old and largely unused books – not always a reflection on their quality: there were two copies of Lamb’s Hydrodynamics – I came across a book by H. J. Allcock and J. Reginald Jones called The Nomogram, a title intriguing enough to warrant a closer look. A nomogram (or nomograph) is a graphical computation chart. Like a slide rule, it works by reading off numbers on graduated scales, but, unlike a slide rule, a nomogram is designed to perform only one calculation or a few related calculations efficiently. Like the slide rule, nomograms have been largely displaced by electronic calculators, but unlike the slide rule, some nomograms are still in use. A web search will reveal several examples. A tephigram is a type of nomogram: it is more than just a graph; it also represents relationships between several variables and can be used for graphical computations such as the determination of atmospheric stability, water vapour mixing ratios or convective available potential energy. In meteorology we often need to convert between pressure, density, temperature and potential temperature. Following Allcock and Jones, I constructed a nomogram to do these computations. We further need a device to lay an ‘index line’ over the nomogram: a ruler or, more practically, an overhead transparency with a single straight line drawn on it. The nomogram itself can be seen in Figure 1. The nomogram contains four graduated scales: for potential temperature, density, pressure and temperature. We can choose any two of these and read off the other two using the ruler: put the ruler against the values of the two chosen variables and read off the other two missing values. But there is more: the temperature scale also shows a saturated vapour pressure scale, and the pressure scale also shows a 1976 US standard atmosphere height scale, so these two quantities can be thrown into the mix as well. For good measure I included a Celsius –Fahrenheit converter, which consists of a separate linear scale with two graduations, the simplest form of nomogram possible. All calculations used are detailed in Box 1. As an example, consider an air parcel at 500 hPa and –10 °C. A ruler crossing these scales at the given pressure and temperature values crosses the density scale between 0.65 and 0.675 kg m and the potential temperature scale just above 320K. The corresponding saturated vapour pressure is between 2.75 and 3 hPa and the corresponding standard height is slightly above 5.5 km. Using this nomogram reveals the great advantage of graphical computation methods: qualitative and quantitative relationships between variables become immediately clear. Not just single point values can be read off; their ranges and gradients also become obvious. We can see where small changes in one variable can lead to big changes in another variable or vice versa. Answers to questions such as ‘how much would the temperature decrease if an air parcel were lifted adiabatically to such-andsuch a pressure level’ can be read off easily as well. The nomogram places a single computation in a continuum of similar computations. Figure 2 is a schematic to get an idea of how the nomogram works. The red index line crosses the three parallel equidistant lines A, B and C. It is now easy to see that for any index line the locations of the crossing points will satisfy p + r = 2q, so in this way we can perform additions. The coefficients can be changed by changing the relative distances. If we now put logarithmic scales on each parallel line it is clear that we can also do multiplications (including fixed powers, depending on the scalings and relative distances of the parallel lines.) The atmospheric nomogram is essentially a scaled and sheared version of this basic design with the added complexity of two simultaneous multiplications where all values fall within specified ranges. I have found my nomogram quite satisfying to use and I have been showing it to students who were intrigued to figure out how, and why, it works: many of today’s students are unaware of logarithmic tables and slide rules. Readers are invited to download an A nomogram for the atmosphere