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# Books by Ralph Philip Boas Jr

Below we list extracts from reviews of books by Ralph Philip Boas Jr. We list these in chronological order except when there are later editions which we list below the previous editions.

1. Entire functions (1954), by Ralph Philip Boas Jr.
1.1. Review by: A Pfluger.
Mathematical Reviews MR0068627 (16,914f).

The book gives a summary account of the extensive results obtained by mathematical research in the class of functions of exponential type in the last fifteen years. These functions are analytic and of exponential type in an angular space or in the plane (entire functions of exponential type). There is no complete or exclusive treatment of entire functions.

1.2. Review by: Maurice Heins.
Science, New Series 121 (3142) (1955), 390.

The theory of entire functions is a vast, rich chapter in the general theory of functions of a complex variable. Its literature is tremendous. The bibliography of the present monograph, which is concerned principally with the special class of entire functions of exponential type, consists of 17 pages. Intentionally, many topics of the general theory of entire functions, such as the Picard theorem and its developments, are omitted in this monograph. The result of this imposed restriction is a unity of theme and method. The interest of the special class of entire functions of exponential type is, of course, well known. As the author remarks, an account of the applications of functions of exponential type would call for a book by itself. What has been achieved here is a comprehensive account of the modern theory of functions of exponential type. The book is a valuable addition to the existing monographic literature on the theory of functions of a complex variable and is to be warmly recommended to both specialist and student.

1.3. Review by: R W.
Science Progress (1933-) 43 (171) (1955), 525.

Apart from polynomials and rational functions, the simplest kind of analytic functions are uniform functions with one essential singularity. When this is isolated and at infinity we get the class of integral functions. It is natural that the study of such functions should have been developed more completely than almost any other branch of function theory. What is remarkable is that this book on Entire Functions is the first book on integral functions to appear in English since the publication of Valiron's classical work. Professor Boas's book does not claim to be a comprehensive treatise on integral functions. It is largely concerned with the properties of functions of exponential type, that is, functions of at most order one, mean type, regular in the whole plane or in an angular region. This should have been made clear in the title since a number of important topics belonging to the general theory of integral functions have been omitted. ... Although this is a book for experts the introductory chapters are particularly good, and no one who is interested in integral functions should be without

1.4. Review by: Archibald James Macintyre.
The Mathematical Gazette 40 (331) (1956), 70-71.

This book deals with integral functions from a special point of view. Its aim is not unfairly described as the discussion of the validity of the classical interpolation expansions of Newton and Gauss, and their natural generalisations together with the immediate consequences of these expansions. ... The subject matter-apart from a 50 page introduction to the general theory of integral functions - is quite modern. There are very few references in the extensive bibliography outside the period 1940-1954. Nevertheless, the book is really an elementary one, the new ideas being notable for their originality of conception rather than their difficulty or depth. The author has managed to arrange in a systematic way a very large number of individual contributions (many being his own) and in considerable detail at the end of each chapter, an analysis of the relevant sources.

1.5. Review by: J Korevaar.
Bull. Amer. Math. Soc. 62 (1) (1956), 57-62.

This is the first book which is devoted primarily to the theory of analytic functions of exponential type, and it gives a practically complete account of the subject. ... Besides the theory of these functions the book contains a large chapter on entire functions in general, a chapter on the minimum modulus of an entire function, and a chapter on applications of functions of exponential type. The author and his publisher have rendered a great service by making so much material available in a single volume of moderate size and price. ... Besides the theory of these functions the book contains a large chapter on entire functions in general, a chapter on the minimum modulus of an entire function, and a chapter on applications of functions of exponential type. The author and his publisher have rendered a great service by making so much material available in a single volume of moderate size and price.

2. Polynomial expansions of analytic functions (1958), by Ralph Philip Boas Jr and R Creighton Buck.
2.1. Review by: W F Newns.
Mathematical Reviews MR0094466 (20 #984).

The objects of this monograph are to explain a general method, the method of kernel expansion, of expanding analytic functions in series ... and to illustrate the power of the method by working out the details in a large number of important cases.

2.2. Review by: Archibald James Macintyre.
The Mathematical Gazette 43 (345) (1959), 235.

According to the preface the authors have tried to bring about a certain amount of order and completeness and to formulate results and methods in a fashion which will make them more generally accessible. They have succeeded in producing a very compact and readable account of a great deal of recent work and its more classical background. There is a twenty page introduction which reaches a form of generalised Appell polynomials. Proceeding to more special cases they discuss properties and expansion theories of the polynomials of Bernoulli, Laguerre, Hermite, Tchebycheff, Jacobi and many others. A brief but suggestive final chapter on applications to uniqueness problems and functional equations must be mentioned and the authors commended for their provision of a detailed contents list, index, bibliography, and list of special symbols.

2.3. Review by: Casper Goffman.
Amer. Math. Monthly 66 (8) (1959), 740.

This little book differs from most items in the Ergebnisse series in that, rather than being a survey of a field, it is a semi-expository presentation of the authors' approach to a topic. ... While the book is by no means light reading, it should be within the scope of a large segment of the mathematical public. In particular, it should make an excellent seminar topic.

2.4. Review by: John Todd.
Mathematics of Computation 15 (73) (1961), 104-105.

A great part of numerical analysis is concerned with polynomial approximation to analytic functions, and so this booklet appears of immediate interest to the numerical analyst. However, with numerical analysis in its present state, it is more relevant for studies in the general theory of functions of a complex variable or the theory of special functions. ... The material is accessible to those familiar with the classical methods of complex variable theory, and its study by numerical analysts is recommended. It will, for instance, encourage us to get off the real axis, reveal some thought-provoking "bad examples", and show us how mathematics should be written.

3. A primer of real functions (1960), by Ralph Philip Boas Jr.
3.1. Review by: R L Jeffery.
Mathematical Reviews MR0118779 (22 #9550).

The author states his purpose in the following words: "My idea is to go reasonably far in a few directions with a minimum amount of special terminology." One finds that the special symbolism is restricted to those for inclusion, union and intersection. With this meagre equipment the author proves such theorems as the following. A non-empty closed set E in a complete metric space is also a complete metric space. A complete metric space is of the second category. The elements of the space C of continuous functions that have, even at one point, a finite derivative, even on one side, form a set of the first category in C. A continuous function is uniformly continuous on any compact set in its domain. Baire's theorem that a complete metric space is of the second category. ... The text is a valuable contribution to mathematical literature in that it sets forth in simple language and in short space the parts of the real variable theory that are essential to further study in the various fields of mathematics.

3.2. Review by: T M Flett.
The Mathematical Gazette 45 (354) (1961), 356-357.

There is a tendency for the modern mathematician to sit in his ivory tower and to concern himself only with the broad sweep of the view which lies within his gaze - he is rather like the landscape gardeners of the eighteenth century who considered anything less than a whole field as unworthy of their attention. It is good, therefore, to be reminded of the interest and the beauty which can be found in the more intensive cultivation of relatively small domains. In the book under review, we are given some of the results achieved by such an intensive cultivation, in that part of the field of "real variables" which deals particularly with the properties of continuity and differentiability. The book is not a systematic treatise, but is more in the nature of a course of informal lectures; the author has set out to tell readers who have little previous knowledge of the subject some of the results which he has himself found particularly interesting, and he has limited himself to the material that seemed essential for the results he had in mind, together with "as much related material as seemed interesting and not too complicated". His aim has been, too, "to preserve some of the sense of wonder that was associated with the subject in its early days but has now largely been lost". In this last aim Professor Boas has been strikingly successful, and his book presents a fascinating and very readable account of many of the interesting and beautiful results which lie in his chosen field. The sense of wonder is admirably preserved, and the book could well serve to revive the interest in analysis of those students to whom "analysis" at present too often means the proof, in successive annual stages of increasing rigour, of results learnt in outline at school.

3.3. Review by: Fritz Herzog.
J. Amer. Statistical Association 56 (293) (1961), 214-215.

Boas' monograph presents to the beginning student of the subject a good introduction into the concepts and methods usually presented in a course on Real Functions. The treatment is essentially the classical treatment of the subject, which in the reviewer's opinion enhances the value of the book from the expository standpoint. Topological concepts are kept to a minimum, while emphasis is put on those ideas which are at the basis of every rigorous treatment of real analysis. ... The exposition is usually clear and concise. More difficult parts of the book are printed in small type and can be skipped without harm. The book contains a good number of exercises, whose solutions are found at the end of the book. .... It might be doubtful whether, as the author claims, a student with a knowledge of calculus should be able to read this book. There is no doubt, however, that a slightly more advanced student (one who has had advanced calculus) will know a considerable amount of good mathematics after studying this primer.

3.4. Review by: Pasquale Porcelli.
Amer. Math. Monthly 68 (2) (1961), 192-193.

This excellent two-chapter little book treats a variety of specialised subjects in lively manner. The subject matter, for the most part, is selected from the foundations of analysis and ranges from important pathological examples to fundamental theorems and some of their applications. The first chapter presents some basic notions inherent to the foundations of analysis, and the last chapter treats some important properties of various classes of functions. Of particular interest here is the treatment of the linear function and convex functions. The entire presentation is (as the author states in his preface) informal and includes some sprinkled bits of wit and philosophy.

3.5. Review by: I S Gál.
Bull. Amer. Math. Soc. 68 (1) (1962), 10-12.

Long before this book was published Professor Boas told me of his plans to write a new Carus monograph on a less specialized topic than those presented in the last few volumes. He had already decided to write a first introduction to real variable theory and in order to have a guiding line he planned to include everything which is necessary to formulate and prove the following proposition: Suppose the continuous real valued function $f$ of a real variable x has derivatives of all orders everywhere and for each x there is an order n(x) such that the n(x)th derivative of $f$ vanishes at x. Then $f$ is a polynomial function. ... The book is clearly written by a man who knows mathematics, has something to say and is able to communicate with others. Undergraduates and promising high school students could profit a great deal by reading it.

3.6. Review by: Richard Bellman.
SIAM Review 5 (3) (1963), 290.

This is an interesting and well-written introduction to real variable theory. It is highly recommended as a text for undergraduate and graduate courses and for the individual who wishes to forge his own path. Although the writing is leisurely (and a welcome relief from some of the modern pseudo-abstract expositions), a large amount of material is covered in an elegant fashion. A number of the results and proofs will delight even the expert. A very attractive feature is a large set of exercises, together with answers at the end of the book.

4. A primer of real functions (3rd edition) (1982), by Ralph Philip Boas Jr.
4.1. From the Preface.

In preparing this revision I have tried to resist the temptation to insert additional material; but a considerable number of references have been added to the notes.

4.2. Review by: David Tall.
The Mathematical Gazette 67 (442) (1983), 316.

This little book in an earlier edition has graced my shelf for many years, being as a source of personal inspiration and also for material used in university elegant and friendly in mathematical style claiming 'no previous knowledge assumed of the reader but he should have had at least a course in calculus'. It in 1960 and since then has grown in size from 190 to 232 pages, the latest additions considerable number of references added to the notes'. For those readers who have not come across the book in the last two decades, it concerns itself with sets and functions. ... I used it myself when learning about everywhere continuous nowhere differentiable functions which led to me writing a little article on the topic a couple of years ago in the 'Gazette'. I have reason to be thankful for its clear and helpful way of explaining abstruse ideas in analysis and have referred to it in my work on a number of occasions for nitty gritty results invaluable in tying up loose ends. A colleague of mine has used it as the basis of a more advanced analysis course. The fact that it has been reprinted and extended after over twenty years says all there is needed to say of this charming textbook.

4.2. Review by: Nicholas D Kazarinoff.
Amer. Math. Monthly 91 (3) (1984), 213-214.

Professor Boas' book is one that is both classical and modern in content, and in addition it excites the mathematical spirit - if it exists in the reader. Yet, beautiful and pedagogically sound as Professor Boas' book is, perhaps it is not the right book for the next generation. There is no mention of computing or numerical analysis either in it or in many widely used texts. An undergraduate course in real analysis (or in functions of several variables or advanced calculus) ought to serve the needs of the computer scientist and computer oriented mathematical scientist, as well as the future mathematician. For a basic analysis course introducing mathematical rigour and satisfying these needs, the concepts of number and function require reinterpretation with mention of the concrete and discrete and discussion of function evaluation algorithms. The constructive and algorithmic aspects of many other topics need to be taught. For example, the implicit function theorem could be followed by computation of Jacobians and Newton and/or quasi-Newton solution of systems of equations (algebraic or transcendental). Why should the cornerstone of the mathematics curriculum be relaid? The answer is that computer usage, from number crunching to investigation of geometric phenomena via computer graphics, is entering mathematics with strength and speed. Differential geometers, logicians, algebraists, and analysts of many varieties now use the computer. For those not entering mathematical research and teaching careers in universities, but who use mathematics in their work, the computer and computer graphics will be even more important. My preaching now gives way to praise of a respected colleague's work. Both the elderly and the young can enjoy Boas' 'Primer'. There are gems to delight any reader.

5. A primer of real functions (4th edition) (1996), by Ralph Philip Boas Jr.
5.1. From the Preface.

5.2. Review by: Walter J Sanders.
The Mathematics Teacher 90 (6) (1997), 499.

This 300-page fourth edition of Ralph Boas' Carus Monograph Number 13, first published in 1960 by the Mathematical Association of America, is an extension of the original by Harold Boas, son of Ralph Boas. This edition includes a new chapter on Lebesque integration. The Carus Monographs "are in tended for a wide circle of thoughtful people familiar with basic graduate mathematics ... who wish to extend their knowledge without prolonged study of journals." The reader should be aware that the reading can require effort, even for majors. ... I highly recommend this monograph to mathematics majors who wish to review or extend their knowledge of real analysis, and to workers in other fields who wish to learn of the major results of real analysis.

6. Integrability theorems for trigonometric transforms (1967), by Ralph Philip Boas Jr.
6.1. Review by: R A Askey.
Mathematical Reviews MR0219973 (36 #3043).

There has long been interest in the question of the relative sizes of a function and of its Fourier coefficients. Except for L2, where there is Parseval's theorem, it was long ago realized that some auxiliary conditions must be placed on either the function or the coefficients before really satisfactory theorems could be obtained. ... This book is very well written and gives an up-to-the-minute survey; in addition, a number of open questions are given. One thing that these questions show is that we are still short of a number of special series which will provide counterexamples.

7. Invitation to complex analysis (1987), by Ralph Philip Boas Jr.
7.1. From the Preface.

This book is intended as a textbook for a first serious (senior or graduate level) course on complex analysis; but it is also intended to be useful for independent study. It assumes that its readers have had a course in advanced calculus or introductory real analysis (the material in the first part of my A primer of real functions [The Carus Mathematical Monographs, No. 13, Published by The Mathematical Association of America, 1960] is more than enough background).

7.2. Review by: George Piranian.
Amer. Math. Monthly 96 (4) (1989), 376-378.

Boas avoids the folly of an impossible definition by making a modest declaration. In his preliminary statement to students he writes: "Complex analysis was originally developed for its applications; however, the subject now has an independent and active life of its own, with many elegant and even surprising results." The declaration does not characterize complex analysis; but complex analysts know that no reasonable description of their territory could ever have remained satisfactory for more than a quarter century. An understanding of the term complex analysis requires at least peripheral participation. To those who by virtue of necessity or volition refrain from participation we might simply say that complex analysis began as the art of using complex-valued functions in the analysis of various physical problems and that today it is primarily the study, by analysis and synthesis, and with geometric, topological, algebraic, number-theoretic, or other cultural orientations, of complex-valued functions in spaces of one or more complex variables. ... Boas obviously believes in exercises, and he has confidence in the soundness of his pedagogical opinions. ... Boas is a master at conveying central truth without telling everything he knows. Although the book's title does not promise a sit-down dinner, that's what we get. An important aspect of the occasion is that no curriculum committee has caterer's rights. The MONTHLY'S editorial rules forbid recitations from the menu; but I can report that the host has chosen and prepared the food in his own inimitable way. The meal's nutritional balance is above reproach, and each dish presents itself attractively. The herbs come from Boas's private garden, and the beverages have aged in his cellar.

7.3. Review by: Lawrence Zalcman.
Amer. Math. Monthly 97 (3) (1990), 262-266.

Ralph Boas' 'Invitation to Complex Variables' is .. good ... The distillation of an intimate acquaintance with the field that extends over half a century, it is, in my opinion, the best undergraduate text ever written on the subject. The years of thought that must have gone into the making of this book are evident not simply in its organization and choice of topics (both of which are exemplary) but also in many little touches. Boas is the only author I know who approaches Morera's theorem from the proper direction (viz. Green's formula); and even when he comes to prove a result as familiar as the fundamental theorem of algebra, he has something interesting to show us. Again and again, one encounters illuminating discussions of matters ignored entirely by most other authors. Cases in point include the section on changing variables in improper integrals and the discussion of methods for calculating harmonic conjugates. Another nice feature of the book are the notes, which contain well-chosen references to the literature, attributions, pointed remarks, an occasional reminiscence, and some amusing speculations. As Boas observes early on, "complex analysis was originally developed for the sake of its applications." In addition to the usual applications to irrotational flows of incompressible, nonviscous fluids, there are discussions of the Plemelj formulas, Nyquist diagrams, and asymptotic series. But make no mistake about it: this is a mathematics book.

7.4. Review by: Richard Rochberg.
American Scientist 77 (4) (1989), 402.

This is a text for a first course in complex analysis. In addition to the standard topics based on the integral formulas and power series, there is an attractive collection of short discussions of additional topics including intuitive Riemann surfaces, non-Euclidean geometry, and Fourier series. This is a very friendly book. The style is informal, there are many figures, and there are lots of interesting asides - conceptual, historical, and whimsical. Boas's love of the subject comes through clearly and makes the book fun to read. If you are teaching an introductory course in function theory and do not use this book, at least tell your students about it. If you need to learn (or relearn) these things on your own, this book is one of the best sources.

8. Lion hunting & other mathematical pursuits (1995), by Ralph Philip Boas Jr.
8.1. From the Introduction.

Occasionally over the course of time, a person with the ability to do high quality research in mathematics also stands out in other ways, contributing to mathematics and the mathematical community in many areas. Ralph Philip Boas, Jr, was such a person: mathematician, author, editor, teacher, and administrator. This collection of his work represents an assortment of many of his lighter mathematical papers along with verse, stories, anecdotes, and recollections. In addition to highlighting his many different abilities, we hope that this volume will stimulate the reader in the way in which Ralph Philip Boas Jr intended to stimulate the readers of his articles when they were first published. The mathematics remains fresh and the comments on teaching are as cogent today as when he wrote them. The recollections and anecdotes included here, with a few exceptions, have not appeared in print before. These will provide the reader with a personal glimpse into the life of a mathematician who moved in the circles of the great mathematicians and scientists of his day, such as G H Hardy and many others. The stories have tantalizing depth and the verses offer imaginative and insightful views of many aspects of the life of a mathematician.

8.2. Review by: Walter Sanders.
The Mathematics Teacher 88 (7) (1995), 610.

If you are one who believes that all mathematicians are off in a world by themselves and not regular people - read this book. If you enjoy reading light, humorous vignettes about famous people - read this book. If you like the short, humorous poetry of Ogden Nash or the snappy observations of Dorothy Parker - read this book. If you just plain like mathematics - read this book! This tribute to Ralph Philip Boas, a fine mathematician, also shares with the reader a much broader profile of a man who contributed to the mathematics community as editor, teacher, administrator, and insightful humorist. Included are many of his lighter mathematical papers, along with verse, stories, anecdotes, and recollections of his interactions with most of the prominent mathematicians of 1930 to 1985. Perhaps the most important feature of this books is how it subtly makes the reader aware of the nature of mathematics.

8.3. Review by: Leonard Gillman.
Amer. Math. Monthly 103 (7) (1996), 607-609.

The stage is set by a twenty-page autobiographical sketch (taken from 'More Mathematical People', D J Albers et al., Harcourt Brace Jovanovich, 1990). It is a fascinating tour through Boas's school days, college years, and professional life, and includes a charming account of how he met his wife, Mary, as well as photographs of his family and of some of the eminent mathematicians he encountered. The book proper presents enough of a mix to delight readers of many stripes: reprints of a good number of Boas' expository mathematical papers (on Lion Hunting, Infinite Series, The Mean-Value Theorem and Indeterminate Forms, Complex Variables, Inverse Functions, Polynomials, The Teaching of Mathematics); several sections titled Recollections and Verse; Reminiscences by several close friends, students, and son Harold (also a mathematician); Reviews and other miscellany. What I found most engaging were the autobiography and other sections containing Boas' insightful comments about mathematics or about people and their foibles. His prose is simple and direct, and at the same time, elegant and witty ...

Last Updated February 2017


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William Blum's website

On text sorting, the fact that Notepad++ is the best text editor, and 2011 wishes!

UPDATE: Version 1.0.0.2 fixes intermittent crashes. NppColumnSort-1.0.0.2.zip

A few years ago I spent some time looking for the best text editor for Windows. After testing pretty much all the editors available at the time (Notepad2, Notepad++, UltraEdit, UltraPad, TextPad...) I finally reckoned that the most powerful and versatile one was Notepad++. I have been using it since then and I regularly recommend it to other people. I could not convince my father to ditch TextPad though because of its superior text sorting feature. It's true that the standard sorting plugin shipping with Notepad++ is pretty basic. It only lets you sort the file at a given column and by lexicographical order only. TextPad on the other hand lets you define a sort criterion based on up to three user-defined columns. Also for each column the comparison can be either numerical or lexicographical.

The thing is that I am very stubborn: Notepad++ is the best editor for Windows, and so to prove it to my father there were no way round it: I had to implement the missing sorting features for Notepad++!

Because Notepad++ is open-source I was able to start from existing code. It thus took less than a day to implement the missing feature as a plugin! Of course it had to be superior than TextPad's implementation. Hence it lets you define an unlimited number of columns, unlike TexPad which is limited to only three.

I figured out that my father may not be the only one interested by the plugin so here you go: You can download the plugin including sources at NppColumnSort.zip NppColumnSort-1.0.0.2.zip

To install it just copy the plugin DLL (under 'Unicode Release' in the zip file) to the Notepad++ plugin directory (%ProgramFiles(x86)%\Notepad++\plugins if your Windows is 64bit; %ProgramFiles%\Notepad++\plugins if your Windows is 32bit).

I take the opportunity to wish you a great year 2011 and hope that all your problems will be sorted in whichever order you prefer be it lexicographical or numerical!

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# Moderate Algebra Solved QuestionAptitude Discussion

Common Information

A boy is roaming on the roof of his house observes that the insects are there in bunches. He found '$p$' insects on day 1. On day two, the number of insects doubled and he takes out '$q$' insects and kills them. On the third day again, the insects left is doubled and he again takes out '$q$' insects and kill them. On the fourth day also he found the same, insects left is doubled and he takes out '$q$' insects again. He is surprised to find that there are no insects left on his roof.

 Q. Common Information Question: 1/3 If the number of insects on the roof, on day 1 is 63, then the number of insects that they kill on the fourth day is:
 ✖ A. 63 ✖ B. 90 ✔ C. 72 ✖ D. 75

Solution:
Option(C) is correct

Day (1) -> Number of insects $-> p$

Day (2) -> Number of insects $-> 2p−q$

Day (3) -> Number of insects $-> 4p−3q$

Day (4) -> Number of insects $-> 8p−7q$

$\Rightarrow P$ has to multiple of 7 so that $q$ is an integer

If $p=63$

$\Rightarrow q =72$

## (1) Comment(s)

Parag
()

no insects left after last day so

$8p-7q = 0$

$8\times 63 -7q = 0$

$7q = 504$

$q = 72$


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## A Study on the Pointwise Time-Space Estimates of Fundamental Solution for Higher Order Schr\odinger Equation Whose Symbol Function is One Kind of Degenerate Polynomial    [PDF]

Myong-Jin Kim, Jin-Myong Kim
In this paper we investigated the point-wise time-space estimates of the fundamental solution for higher order Schr\"odinger equation. These estimates improve the result of Yao [2] and generalize the one of Cui [1].
View original: http://arxiv.org/abs/1303.4838


text: Name: Gargi Shaw Affiliation: DAA, TIFR Conference ID : ASI2022_617 Title : Spectroscopic modelling of two high-mass X-ray binaries, Cyg X--3 and 4U 1538--522 Authors : Gargi Shaw and Sudip Bhattacharyya Abstract Type: Poster Abstract Category : Stars, ISM and Galaxy Abstract : High-mass X-ray binary (HMXB) systems are co-rotating close binary stellar systems where the compact star is a neutron star or a black hole, and the companion star is a massive star with a mass typically greater than 10 solar mass. HMXBs are sources of strong X-ray radiation. Here we report detailed modelling of X-ray emission lines, Fe XXV (6.700 keV) and Fe XXVI (6.966 keV), from two stellar wind-fed HMXB systems, Cyg X--3 and 4U 1538--522, using the spectral synthesis code CLOUDY. We find that for Cyg X–3 and 4U 1538--522, the inner radius of the ionized gas is at a distance of 10\$^{12.25}\$ cm and 10\$^{10.43}\$ cm, respectively, from the primary star, which is the main source of ionization. The hydrogen number densities of the ionized gas for Cyg X--3 and 4U 1538--522 are \$\sim\$ 10\$^{11.35}\$ cm\$^{-3}\$ and 10\$^{11.99}\$ cm\$^{-3}\$, respectively. The corresponding wind velocities are 2000 km s\$^{-1}\$ and 1500 km s\$^{-1}\$. The respective predicted hydrogen column densities for Cyg X–3 and 4U 1538–522 are \$10^{23.2}\$ cm\$^{-2}\$ and 10\$^{22.25}\$ cm\$^{-2}\$. In addition, we find that the magnetic field affects the strength of the Fe XXV (6.700 keV) and Fe XXVI (6.966 keV) lines through cyclotron cooling. We estimate the most probable strength of the magnetic field for Cyg X–3 and 4U 1538–522, where the Fe XXV and Fe XXVI lines originate, is \$\sim\$ 10\$^{2.5}\$ G.


text:
During the first eight weeks after fertilization, a developing human is known as an embryo. Please select the best answer from

Question:

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(e – 7)(e +5) The heck is the simplified version

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Based on Table N, determine the time required for an original 100.-milligram sample of P-32 to decay until only 25 milligrams of the sample

Based on Table N, determine the time required for an original 100.-milligram sample of P-32 to decay until only 25 milligrams of the sample remain unchanged....

Use the graph to find the solution to the system of equations. write your answer as an ordered pair

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text:
The Gaussian and the Sine

My favorite math problem is the following:

A Gaussian and a sinusoid have very similar shapes. The big difference is that the Gaussian has one “hump”, while the sinusoid has infinitely many. Prove the following formalization of this intuition:

Let $$G(x, \mu)$$ be the standard Gaussian $$e^{-(x-\mu)²/2}$$. Then the infinite sum $\sum_{k\in\mathbb{Z}} G(x, k \pi)$ of shifted Gaussians is just a shifted sinusoid $\alpha \cos(x) + \beta.$

The theorem demanded is not outrageous. The Gaussian is a variant on the exponential function, as is the sinusoid. There are some geometric series to immediately make use of. The theorem seems within reach of some clever algebra.

Nonetheless, it makes sense to test the claim on a computer.

The two curves diverge because I have added up only three shifted Gaussians, not infinitely many of them; the red curve is the sum of Gaussians, and the blue curve is the properly-shifted sinusoid. They clearly match; the error is small enough to dismiss as floating-point artifacts.

The computer affirms what we already suspected: the theorem must be a few clever algebraic manipulations away. But the truth is not so.

In fact, the theorem is false, and the graph above a vicious lie. The graph hides an error term of order 0.0003; to see why, we need to go deeper into the problem.

It’s simplest to talk about equivalence to a sinusoid in frequency space. So, let’s try to understand our sum of Gaussians in the frequency domain.

Our sum $$\sum G(x, k \pi)$$ adds shifted versions of a function together; thus it is a convolution. In this case, it is a convolution of the Gaussian $$G(x, 0)$$ with the Dirac comb $$W(\pi x)$$. To compute the Fourier transform $$t$$: $\mathcal{F} \{ \sum_{k \in \mathbb{Z}} G(x, k \pi) \} = \mathcal{F} \{ G(x, 0) * W(\pi x) \},$ we can use the fact that the Fourier transform of a convolution is the product of Fourier transforms: $t = \mathcal{F} \{ G(x, 0) \} \mathcal{F}\{W(\pi x)\}.$

However, both of these component Fourier transforms are easy to compute; both the Gaussian $$G$$ and the Dirac comb $$W$$ are fixed points of the Fourier transform. So we have $t = G(x, 0) W(2 x).$

$$t$$ is the sum of delta functions, scaled appropriately. There is one delta function at 0, which represents the constant offset; one at $$\pm 2$$, which represent a sinusoid with wavelength $$\pi$$; and then delta functions at every other even integer.

Those delta functions at other even integers are errors in our representation of the sum of Gaussians as a sinusoid. But did we not notice their effect?

The answer is only that those spikes are scaled by very small numbers. How small? Well, $$G(x, 0)$$ falls off very rapidly, so even the spikes at $$\pm4$$ are scaled by $$G(4, 0) = e^{-4^2 / 2} = e^{-8}$$. Thus our error term is many orders of magnitude smaller than the “signal”, which is why we didn’t notice it before.

If you carry out the calculation with a bit more rigor (I dropped a multiple of $$\pi$$ or two), you can easily find the exact size of the error term. If you add a second sinusoid to the expression, the error term is smaller than single-float precision, so your computer can even more confidently swear to the “truth” of the theorem.

By on . Share it—it's CC-BY-SA licensed.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.


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A volcano wipes out the majority of the members of a population of mice. By chance,...

Question:

A volcano wipes out the majority of the members of a population of mice. By chance, the surviving members have, on average, darker coloration than the original population. Coloration is a heritable trait. If the next generation has darker coloration than the original population, which of the following has occurred in this population? All of these have occurred Natural selection Gene flow None of these have occurred Evolution

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text:
# Energy Density vs Total Energy

1. Jan 25, 2007

### cosine

Here is a detail that makes me doubt:

1. The problem statement, all variables and given/known data
I am given the equation for a standing wave with a dependence in 't' and 'z' only... I am told that the wave is propagating in a confined space (2*2*2) m
Q1. Calculate the energy density for n=1,2,3. I didn't have any pb with this.
Q2. Calculate the total energy for n=1,2,3. (!!!)

2. Relevant equations

3. The attempt at a solution
Ok, I can't find anything in my lecture notes about total energy for waves. So I went back to the definition of energy density (h) which the amount of energy per unit of volume.
Therefore if I multiply the volume by h I should get the energy (right?)
The volume here is 8 m^3 BUT since the wave in question is only in 1 direction (z) do I still have to multiply by V or only by L=2m instead (for this particular case)

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jan 26, 2007

### dextercioby

Well, since the integrated function has no "x" and "y" dependence, then the integration wrt them should be trivial, right ?

3. Jan 28, 2007

### cosine

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text:
# Introduction

The notes are for the self-use only and not it’s just a test of how convenient it can be to write here $$\alpha^2$$


text:
# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
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Dynamics of a higher order rational difference equation. (English) Zbl 1158.39301
Summary: We will investigate a nonlinear rational difference equation of higher order. Our concentration is on invariant intervals, periodic character, the character of semicycles and global asymptotic stability of all positive solutions of $$x_{n+1}=\frac{\beta x_n+\gamma x_{n-k}}{Bx_n+Cx_{n-k}}\,,\quad n=0,1,\dots\,.$$ It is worth to mention that our results solve the open problem proposed by {\it H. L. S. Kulenvić} and {\it G. Ladas} in their monograph [Dynamics of second order rational difference equations: with open problems and conjectures, Chapman & Hall/CRC, Boca Raton (2002; Zbl 0981.39011)].

##### MSC:
 39A20 Generalized difference equations
##### Keywords:
recursive sequence; global asymptotic stability
Full Text:
##### References:
 [1] Abu-Saris, R. M.; Devault, R.: Global stability of xn+1=A+xnxn-k. Appl. math. Lett. 16, 173-178 (2003) · Zbl 1049.39002 [2] Amleh, A.; Grove, E.; Ladas, G.; Georgiou, G.: On the recursive sequence xn+1=A+xn-1xn. J. math. Anal. appl. 233, 790-798 (1999) · Zbl 0962.39004 [3] Cunningham, K.; Kulenovic, M. R. S.; Ladas, G.; Valicenti, S. V.: On the recursive sequence $xn+1=\alpha +\beta$xnBxn+Cxn-1. Nonlinear anal. 47, 4603-4614 (2001) · Zbl 1042.39522 [4] Dehghan, M.; Douraki, M.; Douraki, M. J.: Dynamics of a rational difference equation using both theoretical and computational approaches. Appl. math. Comp., 1-20 (2004) · Zbl 1085.39006 [5] Devault, R.; Kosmala, W.; Ladas, G.; Schultz, S. W.: Global behavior of yn+1=p+yn-kqyn+xn-k. Nonlinear anal. 47, 4743-4751 (2001) · Zbl 1042.39523 [6] El-Owaidy, H.; Ahmed, A.; Mousa, M.: On asymptotic behaviour of the difference equation xn+1=A+xn-kxn. Appl. math. Comput. 147, 163-167 (2004) · Zbl 1042.39001 [7] Kosmala, W.; Kulenovic, M. R. S.; Ladas, G.; Teixeira, C. T.: On the recursive sequence yn+1=p+yn-1qyn+xn-1. J. math. Anal. appl. 251, 571-586 (2000) · Zbl 0967.39004 [8] Kulenovic, M. R. S.; Ladas, G.: Dynamics of second order rational difference equations with open problems and conjectures. (2002) [9] Kulenovic, M. R. S.; Ladas, G.; Prokup, N. R.: A rational difference equation. Appl. math. Comput. 41, 671-678 (2001) · Zbl 0985.39017 [10] Elaydi, Saber N.: An introduction to difference equations. (1996) · Zbl 0840.39002 [11] Saleh, M.; Aloqeili, M.: On the rational difference equation xn+1=A+xnxn-k. Appl. math. Comput. 171, 862-869 (2005) · Zbl 1092.39019 [12] Li, W. -T.; Sun, H. -R.: Dynamics a rational difference equations. Appl. math. Comput. 157, 713-727 (2004)


text:
Author

I am Joannes Vermorel, founder at Lokad. I am also an engineer from the Corps des Mines who initially graduated from the ENS.

I have been passionate about computer science, software matters and data mining for almost two decades. (RSS - ATOM)

Meta
Tuesday
Apr102018

A weirder definition of Bitcoin

While attemps have been made in the past to come up with a definition of Bitcoin, I felt that those definitions were somehow failing at capturing the very esssence of Bitcoin, so I decided to roll my own. Enjoy!

A weirder definition of Bitcoin

Abstract: Bitcoin is best characterized as an exceedingly weird virtue-inducing artifact. Attempts at making Bitcoin less weird have only two outcomes: either the attempt fails and Bitcoin just becomes weirder; or the attempt succeeds and this is not Bitcoin anymore. The weirdest part of Bitcoin is probably that its own excess of weirdness can be reliably relied upon, as the author demonstrates by providing corollaries of practical interest out of the present definition. Anecdotally, it also explains why Satoshi Nakamoto opted to remain anonymous, as it is usually frowned upon to conjure exceedingly weird inventions.

Friday
Apr062018

Addressing a few loose angles of Bitcoin

Two weeks ago, I had the unique privilege of meeting not one, but a whole series, of truly remarkable people at Satoshi’s Vision in Tokyo. This list includes Amaury Séchet, Shammah Chancellor, Tomas van der Wansem, and quite a few others. While Bitcoin had gained my interest back in 2011, I never had taken much time to think about the Nakamoto consensus itself. To my defense, running Lokad, my company, was simply capturing my day-to-day interests. Thus, in Tokyo, I came to realize that there were some angles within Bitcoin which were maybe not getting yet the proper solutions they deserve.

Terabyte-sized blocks, which I have contemplating since last December, represent only a narrow angle of Bitcoin - albeit an important one.

Below, I am presenting four articles, redacted upon my return from Tokyo, covering angles which I believe to be relevant for Bitcoin. Those articles are all still early drafts, and I apologize in advance for their sorry state of writing. Once again, running my own company does not give me all the freedom it would take to get those ideas either polished or debunked depending on their merits.

By making those drafts public, I am primarily seeking the constructive feedback of the Bitcoin community at large. Unlike many speakers of the Satoshi’s Conference who can claim almost a decade of experience in those matters, I am only claiming about 10 days or so of experience. So please, take those findings with a grain of salt.

Ansible, practical faster-than-light secure 0-conf transactions for Bitcoin

Abstract: The Ansible is a peer-to-peer pre-consensus signal that, by itself, makes 0-conf transactions secure. The Ansible is a self-fulfilling prophecy because making it so aligns with the economic interests of the miners who have to remain competitive. The Ansible confers two seemingly counterintuitive properties to the Nakamoto consensus. First, securing 0-conf transactions do not require any kind of retaliation against Byzantine miners, thus no change to the Nakamoto consensus. Second, 0-conf transactions can be secured with arbitrarily low latencies on earth, despite the fact that this proposition appears to violate the speed of light. The Ansible perspective clarifies why larger and infrequent blocks are actually highly desirable to secure 0-conf transactions.

Midas, united non-colluding transaction fees for Bitcoin

Abstract: Transaction fees are an integral component of the Bitcoin social contract. They reward miners into playing the long game of Bitcoin, when the monetary mass of Bitcoin will not be growing anymore, not with economic relevance anyway. Through an analysis of Bitcoin looking inward, but even more importantly, looking outward, we demonstrate that soviet economics are required in the short term to set the transaction fees, but that the transition toward market-driven fees should and will happen in the future. The author proposes Midas, a pre-consensus signal intended for transaction fees, which preserves the competition within the Bitcoin mining market. Midas unifies miners through their mutual interest of preserving the security model of Bitcoin which includes microlatent transactions. Midas does not require any change to the Nakamoto consensus.

Tokeda, Viable token-driven metadata within Bitcoin

Abstract: Tokeda addresses both the challenge of viably preserving an unbounded amount of metadata without endangering Bitcoin itself and the challenge of introducing tokens within Bitcoin by weaving the two problems through aligned economic incentives. Tokeda is compatible with stateless wallets (which include SPV wallets) and requires no consensus change. As a token scheme, Tokeda relies on a trust-but-verify security model centered around the issuer. The issuer is trusted with the relay of inbound transactions from users. The issuer takes care of routing the metadata to remedy the lack of such capabilities within the Bitcoin script. As a metadata layer, Tokeda leverages the UTX set (unspent transactions) as a purposefully pruneable key-value store, which is a superset of the UTXO set (unspent transaction outputs). Tokeda creates a market signaling mechanism at the issuer level, to foster an ecosystem of nodes which can selectively persist the metadata in the UTX set depending on the originating issuer. The author argues that Tokeda is an economically superior form of tokens compared to the code-is-law approach adopted by some of the competitors of Bitcoin. The author also argues that Tokeda is a provable way to incentivize miners to foster a token-driven economy backed by Bitcoin, instead of expecting the miners to subsidize tokens operated over Bitcoin.

Sakura, long term UTXO recycling mechanism for Bitcoin

Abstract: The very long term viability, centuries ahead, of Bitcoin depends on preventing the runaway growth of the UTXO set (unspent transaction outputs). Also, the ever shrinking monetary mass of Bitcoin is a complication which hinders economic agents from fully relying on perfectly predictable monetary conditions. Here, we propose Sakura, a long term recycling mechanism to prune “dead” UTXO entries, defined as entries that have remained untouched for 80 years (defined as 4,200,000 blocks). It allows those “dead” entries to re-enter the pool of mining rewards, on top of the normal halving mechanism which normally occurs every 210,000 blocks. Sakura comes with a trigger condition that “dead” UTXO entries should represent more than 50% of the UTXO set. This condition prevents a premature activation of the change of consensus if there is not enough economic gains to justify the change. Sakura proposes an exponential decay mechanism associated to a half-period of roughly 20 years. The paper also presents a discussion to justify why those seemingly arbitrary choices are made.

Thursday
Mar292018

Satoshi's Vision, talk on Terabyte Blocks for Bitcoin

Last week, I was in Tokyo at the Satoshi's Vision conference. I gave a talk about Terabyte Blocks for Bitcoin. Here are the slides. Check the video too, there are some good questions raised at the end of the talk.

Overall, it was a incredible event, tremendously positive for Bitcoin. I am thrilled to see so many hard-working contributors doing their best to address all the challenges that Bitcoin is facing.

Wednesday
Jan242018

Fast 1D convolution with AVX

Convolutions are important in a variety of fields. For example, in deep learning, convolutional layers represent a critical building block for most signal processing: image, sound or both. My company Lokad is also extensively using convolutions as part of its own algebra of distribution.

One technical problem associated to convolutions is that they are slow. The theory tells you that FFT can be used to obtain good asymptotic performance, but FFT isn't typically an option when your signal has only a few dozen data points; but that you still need to process tens of millions of such signals.

It is possible to speed-up convolutions with a GPU, but my latest experients also tells me that a massive speed-up can already be achieved with CPUs as well, by leveraging their now widespread vector instructions. This idea isn't new, and I was inspired by an original post of Henry Gomersall on the very same topic.

I have just released my own fast 1D convolution in C++ which differs substantially from the original one posted by Henry Gomersall. More specifically:

• This implementation works with AVX (rather than SSE), for further performance boost.
• The approach can canonically be upgraded to AVX2 (or even larger vector instruction).
• It delivers full convolutions rather than exact ones (NumPy terminology).

Compared to the naive C#/.NET implementation of convolution - which was my starting point - this implementation delivers a rough 20x speed-up; but it comes at the cost of doing PInvoke from C#/.NET.

Sidenote for .NET enthusiats: AVX intrinsics are coming to .NET. It will soon be possible to write such ultra-optimized code directly from C#/.NET.

Tuesday
Dec262017

Mankind needs fractional satoshis

Update: Dr Craig Wright is pointing out that payment channels are a viable alternative to fractional satoshis. I am not an expert in payment channels, but it would certainly largely help in mitigating the problem discussed below. Then, choosing between on-chain scaling and payment channels boils down in establishing the actual limits of on-chain scaling.

Bitcoin Cash aims at becoming the world currency. As discussed previously, terabyte blocks are needed to achieve this goal. However, the Bitcoin Cash protocol also needs a few changes as well. In this post, I will demonstrate why fractional satoshis are needed for Bitcoin Cash.

In the following, for the sake of concision, Bitcoin always refers to Bitcoin Cash.

Overview of the issue

A satoshi is, presently, the smallest unit of payment that can be sent across the Bitcoin network. There are 100 million satoshis in 1 bitcoin. In particular, the smallest non-zero transaction fee that can be paid is 1 satoshi. Non-zero transaction fees are desirable in order to eliminate spam; however, the original intent behind Bitcoin is clearly to keep those fees vanishingly small as far humans are concerned.

At mankind scale, let’s assume that we have 10 billion humans, and that every human wants to do 50 transactions a day. This might seem a bit high - after all, mankind won’t reach 10 billion humans before 2050, however, good engineering implies safety margins and thinking ahead. I firmly believe that Bitcoin must be engineered to support 10 billion humans and 50 transactions per day per human.

Let’s further assume that those transactions are secured by paying exactly 1 satoshi per transaction (1). The miners collect 1e10 * 50 * 365 / 1e8 ≈ 1.8 millions BCH per year. This amount is huge, about 10% of the total BCH that will ever be in existence (2).

Bitcoin Cash needs to be designed in such a fashion that it is possible for mankind to spend less than 0.001% of its whole monetary supply per year in order to transact freely. Over the lifetime of a human, 100 years, the total transaction fees would remain below 0.1% her or his average monetary capital, which feels about right.

Practical example: let’s assume that my average monetary capital is 100,000€ (just counting cash, not any other asset classes). Over the course of my 100 year’s lifetime, I will pay 0.1% of this amount to cover all my transaction fees, that is, 100€. On average, it’s 1€/year, that is, 0.27 cents per day. We are not far of the 1/10th of a cent per day of my previous analysis.

As such, the current Bitcoin protocol is not tenable at mankind scale. Satoshis are too large, mankind needs fractional satoshis.

14-bit left shift

The Bitcoin transactions are encoded with 64-bits integers. This choice, made back in 2009, remains sound. For the foreseeable future, all CPUs will be 64-bits CPUs. However, the current Bitcoin implementation is wasteful. There are 14 bits that are wasted, as we will see below. Yet, it turns out that those 14 bits are exactly what Bitcoin needs to make transaction fees low enough at mankind scale.

Proposal: 1 bitcoin is redefined as 1,638,400,000,000 naks - nak being the shorthand of Nakamoto - that is 214 nakamoto per satoshi.

Let’s demonstrate why 14 bits makes sense. With 50 bits, it is possible to represent 250 satoshis, that is, about 11 millions BCH. The richest BCH address in existence contains about 400k BCH. It’s unlikely that this address will ever grow 1M BCH, let alone 11M BCH.

Thus, in order to represent even the richest BCH address, the protocol only needs 50 bits. While it may be theoretically possible to accumulate more than 11M BCH on a single address, it’s straightforward to add a rule in the Bitcoin protocol to invalidate any transaction which would try to accumulate more than 11M BCH on a single address, forcing the owner of such a fortune to split her/his fortune over 2 addresses instead.

Now, the protocol is left with 64-50 = 14 bits which are “wasted” if we want to preserve the encoding of transactions inputs and outputs as 64-bits unsigned integers. Re-encoding all amounts in sat as nak only requires a 14-bit shift to the left.

As 214 = 16384, we can revisit our initial back-of-the-envelop calculations with 10 billion humans doing 50 transactions a day. We have 1e10 * 50 * 365 / (16384 * 1e8) = 111.4 BCH paid in fees to the miners per year. This is much better, about 0.0005% of the whole monetary supply paid to the miner per year, that is, 0.05% over the 100 year lifetime of a human.

A non-urgent yet kind-of-urgent change

Fractional satoshi won’t become a problem until about 1% of mankind starts using Bitcoin to pay for everything. However, every single day that passes, there are more software out there which are dependent on the current instance of the Bitcoin protocol. Thus, the Bitcoin ecosystem is accumulating technical debt.

We know that this debt will have to be paid back. Indeed, as demonstrated above, keeping satoshis as the smallest payment unit is not tenable. We also know that this debt comes with compound interests. At present time, fixing this issue will only incur a modest friction in the ecosystem. 10 years from now, if Bitcoin has any measurable degree of success at being a currency, then, it will be a huge mess. Every single piece of Bitcoin-dependent software will be broken by such a change.

Thus, I call to the Bitcoin developers to coordinate in order to introduce fractional satoshis in their mid-term roadmap.

Will I still own the same amount of BCH? Yes, there is zero impact on your current BCH holding. If you have 1 BCH now, you will still have exactly 1 BCH after the change.

Does it change the upper limit on the number of BCH? Technically yes, but in practice, no. This change would push back the date when mining becomes purely fee-funded by 4*14=52 years; and the total amount of extra BCH which will ever be mined will be less than 0.001 BCH. Hardly noticeable.

Nitpicking, why not 18 bits?

I do feel strongly that fractional satoshis are needed, and that 14 bits of extra precision is a minimum. However, if someone has a good reason to motivate a shift beyond 14 bits, maybe up to 18 bits, then, this person might be right. The discussion below is merely opinionated. This is not a demonstration.

The richest BCH address hold about 400k BCH. Thus, technically, it is still possible to adjust the protocol to free up to 18 bits, with a hard-cap at 703k BCH for a single BCH address. However, I do see for potential edge cases in the ecosystem of Bitcoin.

If Bitcoin succeeds, then the world will start implementing accounting packages, ERP, POS, CRM …, where assets are valued in Bitcoins, or rather in naks. Most of those software developers will use int64 integers (signed) to track the valuations of those assets. Why? Just because it’s what naturally comes to mind as a developer if you need large signed integers.

As the non-monetary assets are typically valued more than the monetary assets - eg. for most people, their home is worth more than the cash they have at the bank, the same goes for companies - those accounting books may contain values that exceed 10M BCH. Those situations, arguably rare, would trigger bugs known as numeric overflows.

Through a 14-bit shift, naive financial software implementations would still work up to 5M BCH (beware, signed integers, we lose 1 bit of precision), while a 18-bit shift will cap the maximal amount at 350k BCH, that is, 16 times less. While, it’s only a hunch, my take is that this numeric precision of a 14-bit shift would be sufficient to eliminate all int64 numerical overflows in finance calculations even when dealing with the budget of giant corporations. With, a 18-bit shift, edge cases would remain somewhat possible.

(1) Most Bitcoin wallets that exist today do not let you pay 1 satoshi. Instead, the minimal non-zero payable fee is 1 satoshi per byte. However, this behavior only reflects the implementation of the wallet, not a limitation of the Bitcoin protocol itself.

(2) Technically, there is a limit at 21M BCH, however, experts suspect that a few millions BCH are lost forever. Anecdotal evidence: I personally know one person who has irremediably lost about 100 BCH. Thus, those estimates sound right. In any case, even if those coins where not actually lost, it would not fundamentally change the discussion above.

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# Distinction or rule about $d$ and $\delta$ in differential forms

I have in trouble understanding the differential forms.

For $k$ form $\alpha$ and $l$ form $\beta$ we have \begin{align} \alpha \wedge \beta = (-1)^{kl} \beta \wedge \alpha \end{align} And for differentiation, we have \begin{align} d(\alpha \wedge \beta) = d \alpha \wedge \beta + (-1)^k \alpha \wedge d \beta \end{align}

Apply this for usual Riemannian case, for 1-form spin connection $w$, and two form curvature $R= dw+ w\wedge w$ \begin{align} d R = d^2 w + dw \wedge w - w \wedge dw = d w \wedge w - w \wedge dw \end{align}

Now i want to variation of differential forms. How about variation? is the same rule holds?

In page 10 of lecture note and some computation in https://physics.stackexchange.com/questions/222100/variations-of-actions-of-lie-algebra-valued-differential-forms, it seems they treat $\delta$, following usual Lebiniz rule. $i.e$, \begin{align} \delta R = \delta d w+ \delta w \wedge w + w \wedge \delta w \end{align}

Is their procedure right?

In usual differentiation or variation case, this does not be a big problem, (As far as i known, the role of differentiation and variation are similar whether they treat function or functional) but in terms of differential form. I got confused.

Can you give me some formula for variation $\delta$ acting on $(\alpha \wedge \beta)$?

How about Lie derivatives?

I tried to find some reference related with variation on differential form, but they only treat differentiation. Recommendation of any kinds of references are welcomed

I think what needs to be emphasised is that $d$ and this variational $\delta$ (as opposed to the codifferential $\delta$ that is the adjoint of $d$) are two very different operations, that both happen to be a bit like derivatives:
• Imprecisely, $d$ talks about the variation in the form $\alpha$ near $x$ as we move around on the manifold. A simple, though coarse, analogue is the derivative $d/dx$.
• Whereas $\delta$ is talking about what happens to a function of $\alpha$ if we change $\alpha$ by a small amount; I find the physicists' notation is rather lacking here. The (more precise) analogue is the functional derivative, $DF[\alpha](\phi) = \lim_{h \to 0} (F[\alpha+h\phi]-F[\alpha])/h$.
In particular, there is antisymmetry built into the definition of $d$, but $\delta$, while superficially looking the same, is an operation talking about different sorts of variations in a different place. A more mathematical way to write the variation is to expand $F[\alpha+h\phi]$ to first order in $h$, so, for example, $$(\alpha+h\phi) \wedge (\alpha+h\phi) = \alpha \wedge \alpha + h(\phi \wedge \alpha + \alpha \wedge \phi) + o(h),$$ and subtracting and taking $h \to 0$ gives Leibniz for the variational derivative.


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# bisquick drop dumplings

It gives an indication of whether a turbine or burner will be able to run on an alternative fuel source without tuning or physical modifications. El índice de Wobbe es un parámetro importante cuando se quiere mezclar gases combustibles y el aire (en una reacción de combustión), se controla este índice para asegurar la combustión satisfactoria en un quemador. The Wobbe index should not be used as the sole criteria by which to judge the feasibility of a fuel swap out. formula for wobbe index Posted on October 11, 2020 by 0000372561 00000 n 0000016789 00000 n 0000966065 00000 n A stovetop designed for gas with a Wobbe index number of 50, for example, will provide too much air if supplied with gas possessing a Wobbe index number of 35. I2P Not to sound like a grinch, but the wobbe Index is just a measure of how much heat you get through a hole at constant (and low) pressure. Wobbe Index. �y�T�P��&؄�W&g�D��6��H�rS�N��&��Ĕ��\$v�J�F�U�PrT�������9ʰ��Tl�1��st&R�\gM��'Pۯ~�Fl�9v�Fj�B��hsZuF���L��� ��kB ��q��FIi�Z�|K�b�ey�= V��\#1| �>٬��F�l2��ܛն��hGEF3ɯ�)BB+�3��!�5P6��U \�� 0000015093 00000 n Summary. 0000381931 00000 n 0000014347 00000 n 0000015843 00000 n 0000014096 00000 n the Wobbe Index, Combustion Air Requirement, Calorific value based on a residual oxygen measurement. Community Property Barbri, 0000013646 00000 n Wobbe index for common fuel gases - propane, butane, methane and more Sponsored Links The Wobbe Index indicates the interchange ability of a fuel gas and is the best indicator of similarity between natural gas and a specific propane-air mixture. 0000955696 00000 n 0000396853 00000 n (ăBf����"��]/��f��8b�Q�cH�s��.�ef���#�D;06Q/���cZ�7҃�M���*}c��V�� ��~�0=0� �@��Kӊ��j 0000949656 00000 n We usually give it for a unit mass of the fuel. 0000410690 00000 n 0000019083 00000 n Pressure, specific gravity, and calorific value make up a fuel's potential energy, and vary between families and even among gases of the same type, depending on the supplier. The units are often deliberately omitted to prevent confusion with the actual energy per volume of the fuel gas. Wobbe Index uses to compare the combustion energy of various fuel gasses. Wobbe index and calorific value are measured with a fast response time by maintaining a stoichiometric mixture in a mixer supplied with the gaseous fuel at a constant pressure together with ambient air. cardiac index formula. Wobbe Index From the measurement of CH 4 and C2+, the upper and lower heating value is calculated according to ISO EN6976-2005. The BAGGI BA-WIA is an on-line process analyzer for combustible gases. Methane numbers of some individual component gases are: Wobbe Index and Net CV ... Good agreement was obtained when tested against the UK Safe Wobbe Envelope method for determining interchangeability. X;L1(�忭6�칣?�n:�x}܅�]� ; � 0000018783 00000 n Wobbe Index is: where: SGgas = Specific gravity of Gas. 5% Modified Wobbe Index limit. A continuous gas sample is mixed with dry air at a precisely maintained constant ration, which depends on the BTU range of the gas to be measured. Definitie. body mass index formulas. Available here, 1.”Light my fire (2152952690)”By Michal Osmenda from Brussels, Belgium – Light my fire, (CC BY-SA 2.0) via Commons Wikimedia. For the use of gas fuels having a significant vari-ation in composition or heating value, a per-manent gas chromatograph shall be furnished in the plant’s main gas supply line. It measures in real time the Wobbe Index, Calorific value and specific gravity. 0000013296 00000 n The lower a fuel's Wobbe index, the less energy it releases per volume combusted. Ballad Of Death Valley Imdb. The Wobbe Index is one of the main indicator of fuel gasses interchangeability in the application of gas turbines. Note that the range 26 does not have to be the same as range 24. the Wobbe index was computed according to IS0 6976 from the gas composition. Excessive carbon monoxide formation - Caused by incomplete combustion, if allowed to build up can be harmful or fatal to people. Figure 3 shows the following: 0000956196 00000 n. Airlie Beach Resorts, De Wobbe-index is de verhouding van de verbrandingswaarde of calorische waarde van het gas en van de vierkantswortel uit de relatieve dichtheid van het gas (): = De relatieve dichtheid is de dichtheid van het gas gedeeld door de dichtheid van lucht bij dezelfde temperatuur en druk (namelijk 273,15 kelvin en 101,325 kilopascal).. De Wobbe-index wordt uitgedrukt in dezelfde â¦ This may snuff the flame and damage equipment. This can result in the following negative effects: Where the Wobble index of the replacement fuel is too low the fuel will not deliver enough energy. The results are based on GIIGNL 2013 average LNG composition data, with MWM [11] calculation method for methane number. Over Heating - Higher than expected flame temperature leading to damage to the burner, turbine or equipment being fired. No Limits Racing, Wobbe Indekset bruges især til at indstille forbrændingen af gas i for eksempel gaskomfurer. 0000018037 00000 n > Wobbe index is the factor that affects with heat released from a machine. \text{BTU}/\text{SCF}. A method and system for modulating the Modified Wobbe Index (MWI) of a fuel is provided. profitability index formula. 0000929332 00000 n The below infographic presents the difference between wobbe index and calorific value in tabular form for quick reference. Reactive Components - The presences of reactive components like nitrogen, carbon monoxide and alkenes may cause problems for fuel system components. Wobbe index is the factor that affects with heat released from a machine. El concepto de este índice es el de caudal calorífico que pasa por el orificio de un quemador. \text{MJ}/\text{Sm}^{3}. Any use, including reproduction requires our written permission. 0000932137 00000 n What is Calorific Value %%EOF 0000964949 00000 n Therefore, this is the key difference between wobbe index and calorific value. Wobbe Index uses to compare the combustion energy of various fuel gasses. 0000012846 00000 n 0000015593 00000 n Digital inputs (Start Measurement, Start Calibration, Hold Output signal, Solenoid Valves Control signals), The absolute filter needs a frequent cleaning; therefore it is redundant, o avoid affecting the analyzer operation during the clean-up. Wobbe - Index: The Wobbe - Index (WI) or Wobbe number is an indicator of the interchangeability of fuel gases such as natural gas, liquefied petroleum gas (LPG), and town gas and is frequently defined in the specifications of gas supply and transport utilities. Soot build up - Increased potential for soot to build up due to incomplete combustion. The WIM9900 Wobbe Index analyser offers an unmatched combination of response time, accuracy and availability over a large range of gas compositions without the concerns of flameouts or the need for support gases. In this post, we will see the Wobbe Index and Modified Wobbe Index formula and Wobbe index of commonly used fuel gasses in industries. The instrument is a high speed combustion calorimeter complemented by the sampling system (naphtha removal for blast-furnace gas and coke oven gas). In one embodiment, the Wobbe Index of the low Wobbe Index fuel 11a is increased by mixing a component 22 with a high Wobbe Index into the low Wobbe Index fuel 11a to produce a blend 25, and the blend 25 has a Wobbe Index that falls within a desired range 26. For the use of gas fuels having a significant vari-ation in composition or heating value, a per-manent gas chromatograph shall be furnished in the plantâs main gas supply line. 0000687057 00000 n 2. simpsons diversity index formula. Uconn Women's Hoops, Replaces the first edition, which has been technically revised. 6. LHV and specific gravity readings from the gas chro-matograph are used to regulate the amount of Wobbe Indeks (I W) bruges til at sammenligne brændværdierne mellem forskellige brændbare gasser, såsom naturgas, biogas og bygas.Hvis er øvre brændværdi, også kaldet den kalorimetriske brændværdi, og er specifikke tyngde, så er Wobbe Indekset, , defineret som =. For example, this value indicates whether a turbine will run on an alternative fuel source without any tuning or physical alteration. 2.5 Wobbe index: The superior calorific value on a volumetric basis at specified reference conditions, divided by the square root of the relative density at the same specified metering reference conditions. L'índex de Wobbe és un paràmetre important quan es vol mesclar gasos combustibles i l'aire (en una reacció de combustió), es controla aquest índex per assegurar la combustió satisfactòria en un cremador. The key difference between Wobbe index and Calorific Value is that the Wobbe index indicates the interchangeability of fuel gases whereas the Calorific Value indicates the total amount of heat energy generated when we burn a unit mass of fuel.. 0000017289 00000 n The WIM9900 Wobbe Index analyser offers an unmatched combination of response time, accuracy and availability over a large range of gas compositions without … 0000959277 00000 n Available here  When we consider the fuels containing hydrogen atoms, there are two different calorific values; The gross calorific value includes two values; when we completely burn a fuel containing hydrogen, it converts into steam, then if we cool it down to room temperature, the steam condenses into water. The accuracy of this formula within the range specified is better than 0.02%. The accuracy of this formula within the range specified is better than 0.02%. 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# Co-prime or Not Co-prime?

$\large{\begin{cases} a &=& 100 + 85i \\ b &=& 208 + 39i \\ c &=& 188 + 22i \\ \end{cases} }$

Given that $$a$$, $$b$$, & $$c$$ are Gaussian integers as shown above, if $$d$$ is the greatest common divisor (GCD) of $$a$$, $$b$$, & $$c$$, what is the value of $$│d│^{2}$$?

×


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# The Cardioid as the Envelope of Pencil Curves

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Draw a circle and divide the circumference into and even number of equally spaced points.  In this example I have used 44:

$1,2,3, \dots , 44$

Draw following chords between numbers in such a way that when viewed as a sequemce, the second point of each pair moves twice as fast around the circle.:
$1 \rightarrow 2,\ 2 \rightarrow 4,\ 4 \rightarrow 8, \dots,\ 22 \rightarrow 44$
and
$23 \rightarrow 2,\ 24 \rightarrow 4,\ 25 \rightarrow 8, \dots,\ 44 \rightarrow 44$

and on completion, sketch the curve which owns each of the chords as a tangent:


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# reconstruct¶

Reconstruct mailboxes

## Synopsis¶

reconstruct [ -C config-file ] [ -p partition ] [ -x ] [ -r ]
[ -f ] [ -U ] [ -s ] [ -q ] [ -G ] [ -R ] [ -o ]
[ -O ] [ -M ] mailbox...

reconstruct [ -C config-file ] [ -p partition ] [ -x ] [ -r ]
[ -f ] [ -U ] [ -s ] [ -q ] [ -G ] [ -R ] [ -o ]
[ -O ] [ -M ] [ -u ] users...

reconstruct [ -C config-file ] -m

## Description¶

reconstruct rebuilds one or more IMAP mailboxes. When invoked with the -m switch, it rebuilds the master mailboxes file. It can be used to recover from almost any sort of data corruption.

If reconstruct can find existing header and index files, it attempts to preserve any data in them that is not derivable from the message files themselves. The state reconstruct attempts to preserve includes the flag names, flag state, and internaldate.

reconstruct derives all other information from the message files.

reconstruct reads its configuration options out of the imapd.conf(5) file unless specified otherwise by -C. Any mailbox directory underneath the path specified in the partition-news configuration option is assumed to be in news format.

reconstruct does not adjust the quota usage recorded in any quota root files. After running reconstruct, it is advisable to run quota(8) with the -f switch in order to fix the quota root files.

## Options¶

-C config-file

Use the specified configuration file config-file rather than the default imapd.conf(5).

-p partition

Search for the listed (non-existant) mailboxes on the indicated partition. Create the mailboxes in the database in addition to reconstructing them. (not compatible with the use of wildcards)

-x

When processing a mailbox which is not in the mailbox list (e.g. via the -p or -f options), do not import the metadata from the mailbox, instead create it anew (this specifically affects at least the mailbox’s seen state unique identifier, user flags, and ACL).

-r

Recursively reconstruct all sub-mailboxes of the mailboxes or mailbox prefixes given as arguments.

-f

Examine the filesystem underneath mailbox, adding all directories with a cyrus.header found there as new mailboxes. Useful for restoring mailboxes from backups.

-s

Don’t stat underlying files. This makes reconstruct run faster, at the expense of not noticing some issues (like zero byte files or size mistmatches). “reconstruct -s” should be quite fast.

-q

Emit less verbose information to syslog.

-n

Don’t make any changes. This gives equivalent behaviour to chk_cyrus(8) where problems are reported, but not fixed.

-G

Force re-parsing of the underlying message (checks GUID correctness). Reconstruct with -G should fix all possible individual message issues, including corrupted data files.

-R

Perform a UID upgrade operation on GUID mismatch files. Use this option if you think your index is corrupted rather than your message files, or if all backup attempts have failed and you’re happy to be served the missing files.

-U

Use this option if you have corrupt message files in your spool and have been unable to restore them from backup. This will make the mailbox IOERROR free and fix replication.

WARNING: this deletes corrupt message files for ever - so make sure you’ve exhausted other options first!

-o

Ignore odd files in your mailbox disk directories. Probably useful if you are using some tool which adds additional tracking files.

-O

Delete odd files. This is the opposite of -o.

-M

Prefer mailboxes.db over cyrus.header - will rewrite ACL or uniqueid from the mailboxes.db into the header file rather than the other way around. This feature was introduced in version 3.0.

-V version

Change the cyrus.index minor version to a specific version. This can be useful for upgrades or downgrades. Use a magical version of max to upgrade to the latest available database format version.

-u

-m

NOTE: CURRENTLY UNAVAILABLE

Rebuild the mailboxes file. Use whatever data in the existing mailboxes file it can scavenge, then scans all partitions listed in the imapd.conf(5) file for additional mailboxes.

## Examples¶

reconstruct -r -f tech.support

Recursively reconstruct all mailboxes within the tech.support hierarchy, restoring any directories containing cyrus.header files.
tech.support uid 9634 rediscovered - appending
tech.support uid 9635 rediscovered - appending
tech.support uid 9642 rediscovered - appending
tech.support
tech.support.Archive
tech.support.Spam

reconstruct -r -f tech.support.Archive.2%

Recursively reconstruct all mailboxes within the tech.support.Archive hierarchy with names begining with ‘2’, restoring any directories containing cyrus.header files.
tech.support.Archive.2001
tech.support.Archive.2002
tech.support.Archive.2003
tech.support.Archive.2004
tech.support.Archive.2005
tech.support.Archive.2006
tech.support.Archive.2007
tech.support.Archive.2008
tech.support.Archive.2009
tech.support.Archive.2010
tech.support.Archive.2011
tech.support.Archive.2012
tech.support.Archive.2013

reconstruct -r -f -u jsmith

Recusively reconstruct all mailboxes belonging to jsmith, restoring any directories containing cyrus.header files.
user.jsmith
user.jsmith.Archive
user.jsmith.Drafts
user.jsmith.Lists
user.jsmith.Outbox
user.jsmith.Sent
user.jsmith.Spam
user.jsmith.Trash

## History¶

The options -k (keep flags) and -g (clear GUID) have been deprecated in Cyrus version 2.4.

The -u and -V options were added in Cyrus version 2.5.

The -M option was added in Cyrus version 3.0.

/etc/imapd.conf