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# LM317 calculator

- tsp
Last update 21 Oct 2022
3 mins

The LM317 is one of the most common used parts in the history of hobby electronics. It’s a flexible linear adjustable voltage regulator that can convert an input voltage between $3 V$ and $40 V$ into an output voltage in the range of $1.25V$ to $37V$. Note that it’s a linear regulator so excessive potential is simply thermally converted. The larger the difference between input and output voltage is the larger your heat sink needs to be. There even had been 2A $3V$ to $12V$ laboratory power supplies that just consisted of some resistors that could be dialed using a rotary dial, a transformer down to around $24 V$, a bridge rectifier, a few capacitors and a LM317 - mounted of course on a really huge heat sink since in this configuration the power supply will most likely dump most of the power.

The following information is just a summary and mini calculator for my own personal convenience - it does not substitute reading the data sheet of course.

## Voltage regulation: The basic schematic

Usually the LM317 is used in conjunction with:

• An input and output decoupling / bypass capacitor. These are usually ceramic or tantalum capacitors in the range of $0.1 \mu F$ on the input side and $1 \mu F$ on the output side. These prevent oscillations of the regulator
• Two protective diodes. $D_2$ provides a discharge path for $C_{adj}$, $D_1$ provides a discharge path for the output capacitor $C_2$.
• $C_{adj}$ is a ripple rejection mechanism and is basically optional - or should be chosen on the order of $10 \mu F$

The resistor network $R_1$ and $R_2$ selects the output voltage, usually $R_1 = 240 \Omega$. The regulator works by keeping the voltage drop over $R_1$ at a constant $1.25V$ above the adjust input.

[ V_0 = V_{ref} * (1 + \frac{R_2}{R_1}) + I_{adj} * R_2 ]

Usually $I_{adj} \approx 50 \mu A$ and is thus usually neglected from most design calculations. With constant $R_1 = 240\Omega$ one can simply derive:

[ V_{Out} = V_{ref} * (1 + \frac{R_2}{240 \Omega}) + \underbrace{I_{adj} * R_2}_{\approx 0} \\ V_{Out} = V_{ref} * (1 + \frac{R_2}{240 \Omega}) \\ V_{Out} = V_{ref} + V_{ref} * \frac{R_2}{240 \Omega} \\ R_2 = \frac{(V_{Out} - 1.25 V) * 240 \Omega}{1.25 V} ]

Of course it doesn’t get much more complicated in case one does not neglect the $I_{adj}$, usually it’s just not required

[ R_2 = \frac{V_{out} - V_{ref}}{\frac{V_{ref}}{R_1} + I_{adj}} ]

### Resistor calculator

In case JavaScript is enabled in your browser the following calculator is able to calculate one of the three parameters in the input/output section (just leave one field empty):

Input and output parameters
Output voltage: V
$R_1$ Ohm
$R_2$ Ohm

## Current limiting

One can also use the LM317 to provide current limiting. This is pretty simply and also exploits the function that the LM317 just tries to maintain $1.25 V$ voltage difference between it’s output pin and the adjust input. One can simply use an sense resistor - in the following schematic this has been the potentiometer $R_{V1}$:

[ 1.25 V = R_{V1} * I_{out} \\ R_{V1} = \frac{1.25 V}{I_{out}} ]

The idea is really simple - when there is too much current the regulator outputs less voltage, when there is not enough current the regulator increases the output voltage as far as possible. One can use this as current regulator or current limiter. In case the input power supply cannot increase over a given voltage it just caps there - providing an effective active current limiter. Note that in worst case the LM317 has to dump the whole power into thermal energy.

For example to provide a $100 mA$ current limit one would require a $12.5 \Omega$ sense resistor. For a $1 mA$ current limit on the other hand it would be a $1.25 k\Omega$ resistor.


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# Some of Mina Rees's papers

We list below 19 of Mina Rees's papers in chronological order. For each paper we give a short extract. Some are interesting as giving Rees's views while others are of interest since they give information about the development of mathematics and of computers. Some of the papers are "letters to the editor" responding to earlier articles by other authors and it is hard to give extracts without the context in which they were written - we have tried as best we can, but, as for all the extracts, the interested reader should consult the original reference.

1. M Rees, The Federal Computing Machine Program, Science 112 (2921) (1950), 731-736.

Howard Aiken has recently emphasised the desirability of having a relatively simple and cheap machine available to small universities. It was precisely a concern that an adequate machine should be available quite generally and inexpensively to university people that induced ONR two years ago to encourage the California group to proceed with the design and construction of what we have referred to as an "intermediate" computer. Professor Morton, the designer of the CALDIC, has now actually completed the design and taken substantial steps in the building of such a machine. We expect that cheapness, like other components, will develop with new ideas, stepwise and not by command, but Professor Morton's estimate of the cost to reproduce this machine in a university laboratory is under $50,000. ... Some workers in the computing-machine field seem quite pessimistic about the ultimate wide availability of such machines, on the grounds that they are complicated, expensive, and difficult to keep in order. This is a point of view with which I entirely disagree. I fully expect that a competent high-speed computer will very soon be regarded as an important and inevitable part of the research equipment of any university having even the most modest research pretentions. ... There is also an important need to explore more fully the mathematical formulation appropriate to the types of problems that arise frequently in applications. The highly competent staffs at Aberdeen and at Dablgren have been impressed by the extensive difficulty of determining even approximately optimum formulation of their problems and of relying on existing mathematical proofs to assure the convergence of various processes they have had to use. Thus there remains an important creative area for the mathematician in facilitating and implementing the uses of digital machines. 2. M Rees, Digital computers - their nature and use, American Scientist 40 (2) (1952), 328-335. This brief paper is an attempt to present rather general information on the subject of digital computers, without regard to the specific relevance of the current computer development to biological research. Only the beginnings have been made on the mathematisation of biological problems; and since computers rely for their usefulness on the quite explicit mathematical formulation of the problems that they are to solve, it is evident that relatively little of specific application in most fields of biology may be expected in the near future. Although the abacus is frequently thought of as the first digital computer (you will remember that automatic computing devices came in for a lot of kidding not so long ago when the occupation forces in Japan staged a race in calculations between a skilful sergeant with a desk calculator and a Japanese with an abacus), the fact is that the first digital calculator used by man consisted of his ten fingers. In limiting discussion to digital computers, I am excluding from consideration the whole family of analogue machines about which I shall speak only in passing. Let me mention only that analogue machines extend in range of complexity from the simplest slide rule to the elaborate differential analysers which are currently being used in much engineering research in the United States. Usually any single analogue computer is applicable to only a restricted class of problems. For example, the differential analyser was designed to solve systems of ordinary differential equations. Digital computers, on the other hand, are as flexible as arithmetic itself. The accuracy of analogue computers is limited by our capacity to construct flawless mechanisms and by the effects of temperature, air pressure, humidity, and age on such mechanisms. But digital computers can achieve almost any desired degree of accuracy if we are willing to use enough equipment. In discussing computers of digital type I shall try to provide some impression of the answers to the following questions: What are they? How do they operate? Where are they chiefly used, and how? What is the state of the art? How much do they cost? What developments may we expect in the immediate future? 3. M Rees, Modern Mathematics and the Gifted Student, The Mathematics Teacher 46 (6) (1953), 401-406. This paper is concerned with the world that confronts the gifted college student of mathematics upon graduation - a world in which employment opportunities are expanding, largely as a result of new emphases in certain fields of mathematical research since the end of World War II. In this article I will describe some of these fields of research, and I will try to indicate very briefly certain elementary aspects of the current activity in mathematics that might challenge the interest of gifted high school boys and girls. Because I have been asked so frequently about opportunities for women, I shall point out certain mathematical careers in which women have been particularly successful. ... You will see that all this adds up to a recognition that college graduates with mathematical training, and particularly Ph.D.'s with mathematical training, have important roles to play in industry and in government as well as in the universities. It is critical throughout our considerations, however, to recognise that our real contributions as teachers He in the direction of providing real mathematics and not a watered-down version to our students, and in seeing to it that they acquire as broad a base in the sciences and in the humanities as is feasible. At the secondary school level the obligation of the mathematics teacher is particularly acute. I was sitting next to a distinguished psychologist at a luncheon the other day, and I was heartened to hear him remark that the welfare of all the sciences depends critically on an improvement in the mathematics curriculum of our secondary schools. Here is an area where the college teachers and the high-school teachers have a great common undertaking, which it is good to know that the university research workers in mathematics are now keenly appreciating. Though we have by no means solved the problem of producing better mathematics textbooks at the college level, urgent and conscientious work is now going on in this area. I know that your organisation is also heavily committed to working for the improvement of the mathematics curriculum so that high school and college students can learn not only routines but also something of the quality of modern mathematics. We need books written in a way that will appeal to young people, that tell of some of the exciting successes, and that particularly explain some of the unsolved problems. Our students need contact with research people so that they will know that mathematics is alive. I consider it a dreadful failure of our educational process that so many students graduate from college utterly unaware that new and vital results in mathematics are being found daily. Our students need to appreciate the great scope and depth and variety of mathematics, and they need to know that the most effective mathematician is a cultivated person with a broad education outside mathematics. The future is, I am sure, rich in opportunity for well-trained mathematicians; not only for those who carry the torch of devoted and imaginative teaching and for those who experience the rewards of mathematical research, but rich in opportunity also for those who accept the challenge of modern developments in the many phases of applied mathematics and who seek careers in government and industry. 4. M Rees, Computers: 1954, The Scientific Monthly 79 (2) (1954), 118-124. In recent newspaper stories, there have been some miraculous-sounding tales of the uses to which electronic computers have been put. In this article, I shall try to put these accounts into perspective by giving some impression of the state of the art in the technology of construction and in the science of use of digital computers. I shall also try to give a little more detail about some of the applications that have excited public interest and to indicate other directions in which applications are being developed and sought. Let me begin by giving some idea of the number, type, and location of the electronic digital computers now in operation, or soon to be completed, and of the principal characteristics of these machines. There are more than 70 in the United States, somewhat more than half in current operation, and the rest soon to be completed. There are two major companies manufacturing what I shall refer to as large machines in a sense that I shall try to explain, and about a score of other companies manufacturing machines of moderate size. Abroad there are 12 machines in England, five in Germany, three in Japan, and one or two in Belgium, Holland, Sweden, Norway, France, and Australia, as well as Canada. England has two companies that are manufacturing the machines for sale. ... Any realistic approach to the introduction of computers in going business concerns must start with the status quo and proceed by modest steps into the future. Many business concerns will find it a big task to tailor their office procedures to the use of computers. In many cases, the office procedures in use are not known in all their detail to any single member of the organisation, and the first job is to find out exactly what is being done. This is not a trivial job, the early savings are elusive, and it will take real conviction as to the usefulness of the outcome to get the initial study under way. IBM has done a splendid job in smaller but comparable situations in introducing its small business machines; and this company can be expected to do a good job here. Remington Rand also has an excellent staff making analyses of the problems of potential users of their machines. It is clear that in the relatively near future we shall see a notable expansion of business and manufacturing applications of digital machines, including automatic process control and other features of the "automatic factory." 5. M Rees, Mathematics and Federal Support, Science, New Series 119 (3099) (1954), 3A. A question much discussed in scientific circles in Washington and in the universities is the level of Federal support of basic research in the sciences that is appropriate in terms of national policy and adequate in the view of research scientists. As new policies emerge in the Defense Department which seek to apply to all military programs the test of relevance to defense needs, it is not unnatural that particular concern should be felt (and expressed) by the mathematicians of the country about the fate they may expect for mathematics, and particularly for so-called "pure" mathematics. It is still recognised in responsible military circles that a country strong scientifically must be strong mathematically. It is still clear that the total pattern of Federal support must provide adequate financing and broad and wise administration of funds. We are, however, in a period of reformulation of the details of policy. ... Responsibility for initiating and defending policy with respect to the support of mathematics within the three military departments is shared by the mathematicians working in these departments; and they, together with the mathematics staff of the National Science Foundation, are trying in the most serious and intelligent way to use the funds and other resources at their disposal to secure a continuation of the lively and significant development of mathematical research that has characterised the post-war era. The responsible leaders of the mathematical community are giving them needed help in this effort; but the position of mathematics would be greatly strengthened by a clearer presentation by mathematicians of the needs of mathematics. 6. M Rees, New Frontiers for Mathematicians, Pi Mu Epsilon Journal 2 (3) (1955), 122-127. For those who are strongly attracted by pure mathematics, a life devoted to teaching and research will, of course, continue to hold the richest rewards; and the university will continue to be the centre where the great advances in mathematics are made. But mathematics is always handmaiden as well as queen; and for those who are thinking of seeking careers in industry, the picture has changed. Until a few years ago, mathematics had only the same sort of general appeal for such young people as had virtually all subjects in the liberal arts curriculum unless actuarial work was the goal. To-day this picture is quite different. Now, college graduates with a major in mathematics have a choice of several interesting jobs in industry, and leading companies compete for their services. And those who go on for a Ph.D. find themselves, if they have suitable temperament and interest, in the novel position of being in as much demand as engineers and physicists. What kinds of industries are bidding for the services of mathematicians, and what is the nature of the mathematics that is being used? For a young A.B. the principal openings are still associated with the great computing centres that have grown up around the new electronic computers. Many of these jobs are actually involved with the preparation of problems for solution on the machines; but because the machines themselves can handle such interesting problems, our young mathematicians have much more exciting work than their predecessors a few decades ago who worked with desk machines. And there are interesting developments in the direction of using specially trained personnel on the sales forces of the great companies that have huge investments in these machines, and are developing them extensively for business uses. The thing to observe is that there are actually careers in the making. A first step, immediately after college, is to serve on the staff that operates the machines; a later forking of opportunities depends on the interest and further training of the young person, either on the sales force, or in an advanced scientific position, or in an administrative post. The types of problems handled, the constantly new points of view, the utterly unimagined developments that can be expected make this a career worth considering. There are interesting mathematical features of work with these machines. One is the exploitation of the representation of numbers in binary notation - in the engineering design of the machines and in much relevant mathematical analysis. Another is the iterative solution of linear equations. Much work has been done on the practical solution of linear equations in the last few years, and there is much interesting new mathematics connected with the solution of these when there are many of them - at least ten, and preferably twenty-five, or a hundred. The problems involved in insuring that the answer actually has significant figures are non-trivial. Still another question that has been receiving attention is the numerical solution of differential equations by methods suitable for use with electronic computers. Linear programming is another aspect of recent mathematical work that is playing an increasingly large role in industrial developments, often in connection with operations research, The linear programming ideas offer an inviting introduction to some primitive concepts of convex sets, and of game theory. A good introduction to the background that motivated much of the early work in this field is the Leontieff article in the Scientific American, some time ago, on Input-Output Analysis. This is a field in which there are increasingly many jobs for mathematicians, though they will usually need an advanced degree. On operations research teams, a mathematician is a welcome participant, but again a considerable amount of advanced training is probably necessary. In all phases of applied mathematics it is in the construction of conceptual models that the applied mathematician often meets his greatest challenge; and it is here that the decisive contribution of the mathematician as a member of a team of scientists lies. 7. M Rees, Digital Computers, The American Mathematical Monthly 62 (6) (1955), 414-423. Mechanical aids to computation go back at least as far as the abacus, but the distinctive thing about these modern machines is that they are automatically sequenced - they can be given instructions before they begin to operate, which will enable them to perform each tiny step that they must take in order to complete the problem. With modern electro-mechanical desk computers, as with the abacus, each computational step requires direct human intervention; but with the automatically sequenced machine, the human operator must think through, in fullest detail, the many steps that must be taken, and tell the machine about them on the instruction tape. There are advantages and hazards in this fact. With a desk calculator, the speed of the total operation is subject to human physiological limitations. With an automatically sequenced machine, the speed of computation is vastly increased, but the programming and coding must take into account all the acts of memory and judgment that usually go into a desk calculation. This means that a code (an instruction tape), once written, must be subjected to a period of "debugging," consisting of trial runs that show up some (hopefully all) of the omissions and mistakes that have been built into the instructions. In recent tests of an IBM machine and a Remington Rand machine on a weather forecasting problem, though there were other considerations that were decisive in selecting the best available machine for the problem, it was discovered that coding for one machine took about 6 man-weeks, while debugging the code took 15 hours of machine time; coding for the other machine took about 17 man-weeks, and debugging took about 50 machine hours. Since one machine hour on these two machines is roughly equivalent to four or five hundred man-weeks on a desk computer, you will see that the debugging operation is non-trivial. One device that has met, at least partially, the problems inherent in correct coding for high speed computers is the development of the library of sub-routines. For example, after a sine function has been coded and run successfully, whenever a sine function is to be computed as part of another problem, the coder splices in the previously tested sub-routine for this operation. This part of the new code then needs no further debugging. If the disadvantages of extended debugging are great, the advantages of vastly increased speed are greater, so that the new machines have taken their places in our fast moving world. Original applications of these machines were to scientific problems which involved relatively little basic data, but required many thousands, and sometimes many millions of operations on these data. Since little information needed to be fed into the machines, and the answers were short, the original emphasis was on speed within the computing element, with little emphasis on speed in the input and output mechanisms. Business and management problems, on the other hand, ordinarily involve the performance of relatively simple mathematical operations on great masses of data. The answers, too, are sometimes very long. Thus effective utilisation of the speed of electronic computers in the solution of problems of this type requires greatly increased speed in input and output mechanisms. In the recent past, there has been a good deal of emphasis on the solution of these problems, and some of the present input and output equipments have attained impressive speed. Many machines are equipped to type out results on an electrically operated typewriter at about 10 characters per second, but output printers operating 40 to 200 times as fast as this are beginning to find widespread use in business practice. 8. M Rees, Mathematicians in the Market Place, The American Mathematical Monthly 65 (5) (1958), 332-343. We need additional mathematicians not only in industry, but in the universities and in the secondary schools. Our present dilemma in staffing the schools and universities is caused partly by the drift of well-trained mathematicians to industry because of the combined lure of higher salaries and interesting work. The need is clear to train more able mathematicians and to raise academic salaries to a level competitive with those offered by industry. The second point I need not emphasise; but on the first point I should like to expand. Since 1950 we have been producing an average of a little over 200 Ph.D.'s in mathematics a year in the United States (and the average before 1950 was considerably smaller). This is a staggeringly small number to fill the needs that exist. It is imperative that we attract more young people to careers in mathematics. But many able boys and girls for whom teaching is not attractive still have the idea that teaching is the only career to which training in mathematics leads; and the nature of industrial mathematics often is not understood by the teachers who provide the only contact young people have with mathematics. We need to acquaint our young people with the great variety of interesting careers open to mathematicians. Inevitably some who embark on the study of mathematics as a road to a career in industry will find themselves diverted by their enthusiasm for research in pure mathematics - a quid pro quo for those lost to business from the university. It may be worth noting that over one-third of living Americans with Ph.D.'s in mathematics came to mathematics from some other field - the most frequent fields being engineering, chemistry, and physics. One additional point needs mention. The problem is not to steal all the ablest youth of the land from other pursuits but merely to maintain the position of mathematics. Mathematicians themselves have not recognised the strength of their position. The occasion of these observations is not the latest Sputnik in the sky. On the contrary, in considering mathematical training, as in the larger educational universe, the family of Sputniks merely verifies what we knew already. 9. M Rees, The impact of the computer, The Mathematics Teacher 51 (3) (1958), 162-168. My theme is the computer, and its impact upon our changing world. Let me for a moment consider my subject in the perspective of the onward march of mathematical research. Mathematics has been called both the queen and the hand maiden of the sciences; and it is as hand maiden that mathematics usually appears in its association with computers. The computer does not provide the imaginative insights, akin to poetry, which characterise the most significant research in pure mathematics, although it sometimes provides the material on which such insights are based. The computer is a servant, strictly obedient and highly versatile, but it must be instructed, as Claude Shannon has said, in words of one microsyllable. The first commercially built large-scale computer in the United States was put into operation in time for limited application in the preparation of the 1950 census. Since that time, there have been great strides in the development of these "giant brains." Their use has expanded to include applications not only in science and engineering, and in business and industry, but also in many directions not dreamed of by the early users. I shall try in this short paper to give you only a glimpse of some of these directions. What is the impact of the computer? What kinds of problems with solutions in accessible until now have come within reach with its advent? There are several categories. In the first case, the mathematical nature of the solution of the problem was fully understood before the coming of the computer, but the solution could not be found. Here a mathematical model had been formulated, but could not be tested against the real world be cause numerical answers were not avail able. These could not be found either because the time required to obtain a solution was prohibitive, or else because the method to be used had not been developed. In this category is the problem of orbit computation when the need for speed is great. If we are merely interested in computing the path of a heavenly body as it moves on its leisurely way through space, then the older punch-card equipment can be made to do the job, not so well as the new equipment but adequately. But if we want to predict the position of Sputnik II as it races around the world, then powerful and fast modern equipment is needed. In this category, also, is the problem of flood control. Within the past few years the United States Army Corps of Engineers has be come interested in numerical flood pre diction and river regulation, and a study has been completed for them by a group at New York University headed by Professor J J Stoker. This group has studied floods in the Upper Ohio River from Wheeling to Cincinnati, at the junction of the Ohio and the Mississippi Rivers, and in the Kentucky Reservoir. The group has demonstrated that, at least for the junction of the Ohio and Mississippi and for the Kentucky Reservoir, and probably whenever the engineering data are good enough, machine computation will solve the prediction and control problems more cheaply and more satisfactorily than the older engineering methods. The Corps of Engineers has adopted the NYU method as a basis for settling design questions for a reservoir to be built in the Chattahoochee River in Alabama. 10. M Rees, Support of Higher Education by the Federal Government, The American Mathematical Monthly 68 (4) (1961), 371-378. In this article I shall discuss the nature and needs for Federal support as these relate to mathematics. It may be worthwhile to consider briefly certain aspects of the history of these questions. At the end of World War II the urgent need of the Government for new scientific results, sometimes in narrowly delimited fields, and the recognition that the universities of the United States were not planning and were financially unable to embark on a program of basic research in the sciences adequate to provide for these needs, led to the decision to initiate a widespread program of Federal support of university research in the sciences. The first steps in this new program (and we should not forget already existing efforts in the universities like those of the Department of Agriculture) were taken by the Navy through the Office of Naval Research. With the passage of time many other agencies entered the field. The establishment of the National Science Foundation in 1950 gave an added impetus to this development. Mathematics was early recognised as an essential element in any vigorous program for the development of leadership in the sciences, and this recognition included the purest of mathematical disciplines as well as applied mathematics and the development of computers. As more extensive programs of support have emerged the position of mathematics has been maintained, not however without the determined and selfless effort of many leaders in the profession. It is worth noting that the nature of the support needed for mathematicians is different from that needed for experimental scientists. There were two main reasons for federal contributions toward the support of research in the experimental sciences: (1) the need for expensive equipment which in many fields like nuclear physics grew more and more expensive with the passage of time; and (2) the need for results in particular areas of research and for basic advances across a broad front of scientific inquiry. The first of these reasons was not present in mathematics except in computer development, in which mathematicians participated with engineers, physicists and other scientists. Though the criteria for the support of research programs that have been emphasised in the past are primarily the two I have listed, a third criterion, the need for trained personnel, was constantly in the picture. As we assess our situation in mathematics today, this need for trained personnel becomes the critical issue. To the extent that this personnel need is acute in other fields, like the social sciences and certain of the humanities, it may be that programs comparable to those suggested for mathematics should be considered. In recent years the nature of the involvement of higher education in federal programs has changed. Federally supported research programs continue, but the extent to which research people in the arts and sciences are now committed to participation in course content improvement programs at the secondary and even at the elementary level has made new demands on the time of scholars. And in mathematics the Mathematical Association, through the work of the Committee on the Undergraduate Program, has emerged as a significant force in the modernisation of collegiate mathematics courses as well as in the design of radical new plans for the training of teachers to handle all the new curricula. All this takes people. Federal government money for research programs tends to go to a relatively few of the large universities with the effect of strengthening these institutions and increasing their power to attract the ablest scholars. Too often they are attracted away from the smaller liberal arts colleges. Moreover, the teachers at the liberal arts colleges often lack the stimulation and vitality gained from actual participation in research. The National Science Foundation, through its education projects, has come to grips with the problem of providing better subject matter competence to teachers in liberal arts colleges and in secondary schools. 11. M Rees, The Nature of Mathematics, Science, New Series 138 (3536) (1962), 9-12. Some of the most noted mathematicians and philosophers have addressed themselves to a discussion of the nature of mathematics, and I can hope to add very little to the ideas they have expressed and the insights they have given; but I shall attempt to draw together some of their ideas and to view the issues in the perspective that seems to me appropriate to the present state of mathematical scholarship, taking account of the great increases that have been taking place in the body of mathematical learning, and of the changes in viewpoint toward the old and basic knowledge that grow out of deeper understandings brought about by generations of mathematical research. In discussions of this subject we find a sharp difference in the views of able mathematicians. This reflects the concern of some that the trend toward abstraction has gone too far, and the insistence of others that this trend is the essence of the great vitality of present-day mathematics. On one thing, however, mathematicians would probably agree: that there are and have been, at least since the time of Euclid, two antithetical forces at work in mathematics. These may be viewed in the great periods of mathematical development, one of them moving in the direction of "constructive invention, of directing and motivating intuition", the other adhering to the ideal of precision and rigorous proof that made its appearance in Greek mathematics and has been extensively developed during the 19th and 20th centuries. ... And now, as I conclude, let me state the major positions that seem to me to emerge from considerations such as those I have set forth. They are these: - That mathematics is a language which must be learned and that the arsenal of techniques of mathematics must be mastered if we are to speak this language. - That mathematics grows by the addition of new theorems, and that the discovery of new theorems is made sometimes by insights furnished by in- tuition, sometimes by insights provided by abstraction and the identification of patterns. - That the proofs of theorems rely on the logic of their day, but that mathematicians are constantly concerned to find the logic that makes the proofs of needed theorems adequate. - That mathematics is both inductive and deductive, needing, like poetry, persons who are creative and have a sense of the beautiful for its surest progress. - That many of the problems of mathematics come from mathematics itself, but that many more, at least in their earliest genesis, come from the realities of the world in which we live. - That realms conquered by mathematics solely because of their intrinsic interest to mathematicians have provided in the past, and continue to provide, parts of the conceptual framework in which other scientists view their worlds. - That the process of abstraction and axiomatisation has provided simplification and a deep understanding of the body of mathematical results and a powerful tool for conquering new mathematical worlds. 12. M Rees, The nature of mathematics, The Mathematics Teacher 55 (6) (1962), 434-440. In discussions of [the nature of mathematics] we find a sharp difference in the views of able mathematicians. This reflects the concern of some that the trend toward abstraction has gone too far, and the insistence of others that this trend is the essence of the great vitality of present-day mathematics. On one thing, however, mathematicians would probably agree: that there are and have been, at least since the time of Euclid, two antithetical forces at work in mathematics. These may be viewed in the great periods of mathematical development, one of them moving in the direction of "constructive invention, of directing and motivating intuition," the other adhering to the ideal of precision and rigorous proof that made its appearance in Greek mathematics and has been extensively developed during the nineteenth and twentieth centuries. The first position, that the emphasis on abstraction has gone too far, is presented by Courant and Robbins in What Is Mathematics?, though their position is modified by their recognition of the power of the axiomatic method and the deep insights it has made possible. They say, in part: "A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not at tract any intelligent person. It would be a game with definitions, rules and syllogisms, without motivation or goal. The notion that the intellect can create meaningful postulational systems at its whim is a deceptive half-truth. Only under the discipline of responsibility to the organic whole, only guided by intrinsic necessity, can the free mind achieve results of scientific value." The second point of view is represented classically by Bertrand Russell's famous definition of mathematics as the "subject in which we do not know what we are talking about or whether what we say is true." Of this, Marshall Stone has this to say: "A modern mathematician would prefer the positive characterisation of his subject as the study of general abstract systems, each one of which is an edifice built of specified abstract elements and structured by the presence of arbitrary but unambiguously specified relations among them." Stone says in two other passages: "While several important changes have taken place since 1900 in our conception of mathematics or in our points of view concerning it, the one which truly involves a revolution in ideas is the discovery that mathematics is entirely independent of the physical world .... At the same time ... mathematical systems can often usefully serve as models for portions of reality, thus providing the basis for a theoretical analysis of relations observed in the phenomenal world. ... Indeed, it is becoming clearer and clearer every day that mathematics has to be regarded as the corner-stone of all scientific thinking and hence of the intricately articulated technological society we are busily engaged in building." In the history of mathematics the emphasis in research is sometimes on constructive intuition and the acquisition of results without too much concern for the strict demands of logic, sometimes on the insights gained by the identification and study of abstract systems within a care fully designed logical framework. But over the years the body of mathematics moves forward inevitably with growth in both directions. An individual mathematician chooses to work on one frontier or the other, and the emphasis changes from one period to another; but mathematics as a whole and the community of mathematicians have their obligation to the total spectrum. For mathematics is the servant as well as the queen of the sciences, and she weaves a rich fabric of creative theory, which is often inspired by observations in the phenomenal world, but is also often inspired by a creative in sight that recognises identical mathematical structures in dissimilar realisations by stripping the realisations of their substance and concerning itself only with undefined objects and the rules governing their relations. As Von Neumann has said: "It is a relatively good approximation to truth ... that mathematical ideas originate in empirics, although the genealogy is some times long and obscure. But, once they are conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations." With this introduction, it will be useful to consider briefly those episodes in the history of mathematics that play a decisive role in the development and under standing of this dichotomy. The Greeks made fundamental contributions in parts of mathematics other than geometry - in addition to Archimedes' wide-ranging interest in applications, I cite only Euclid in number theory and Eudoxus in analysis. But the failure of the Greeks to develop adequate symbols with which to express many of their ideas made their treatment of these subjects cumbersome. Through Euclid's Elements, however, they contributed to mathematics the ideal of the development of a body of knowledge proved by logical deduction on the basis of a limited number of axioms, a concept that has exercised enormous influence. One of the greatest of Euclid's contributions to geometry was his recognition that the parallel postulate could not be derived from the others. For two thousand years after Euclid, the development of geometry is characterised by attempts to prove the parallel postulate. At last, in the time of Gauss at the beginning of the nineteenth century, the problem was solved. And what a solution! A geometry developed independently in Germany by Gauss, in Hungary by the Bolyais, and in Russia by Lobachevski in which this postulate does not hold, and in which the sum of the angles of a triangle is less than 180º. Interestingly enough, Gauss's impulse was to check to determine whether our physical world (and here he meant only the earth on which we live) was described by Euclidean or by this new non-Euclidean geometry. He found that his instruments were not good enough to discriminate; but it is of some interest to recall that the non-Euclidean geometry developed later by Riemann, in which the sum of the angles of a triangle is greater than 180º, was found by Einstein to provide a satisfactory framework within which to develop his ideas of the physical universe. In passing, it should be noted that the parallel postulate, unlike the others, deals with lines that cannot be described by finite considerations. Infinity early raised difficulties for mathematicians; and the subsequent development of our subject sees infinity introducing new and exciting vistas which, however, are recurrently accompanied by logical problems that have caused an upheaval in mathematical thought. 13. M Rees, A Humane Approach to Population Problems, Science, New Series 173 (3995) (1971), 381. There are many scientists in the United States and in other countries across the world who would argue that we have hardly begun to control the population through the technology that is now at hand and through the political, social, and economic devices available to our world; that the policy of "one mouth, one meal" is a wise, humane, and valid goal for all humanity; that we must persist, as a matter of faith, in seeking to promote the human dignity of all the people of the world. Many have warned that we cannot hope to survive in a world in which the gap in the rates of development of its various peoples continues to expand - that what we need is a determined effort which commits the power and affluence of developed countries to assist in raising the standard of living and the productivity of the undeveloped areas of the world. 14. M Rees, Come, Now, and Let Us Reason Together, Science, New Series 178 (4066) (1972), 1155. There are, then, three principal purposes that the Annual Meeting [of the American Association for the Advancement of Science] seeks to serve: to give in-depth presentations of important scientific and technical areas in such a way that they are of interest to a spectrum of disciplines; to present comprehensive symposiums on central scientific and societal problems whose treatment requires the knowledge and in sights of a number of professions, both scientific and non-scientific; and to present programs designed to increase the public understanding of science and the ability to use science and technology for the promotion of human welfare. Although the principal focus is upon advances in science, approximately one-third of the central program this year (nearly 30 symposiums) deals with science and social issues, including such topics as land-use controls, minorities in science, aggression and violence, prison research, behaviour control, and genetic engineering. Within such a meeting, all reasoned and responsible views have a place. Most programs will incorporate an audience participation portion of significant length in which discussion from the floor is welcomed. Because the AAAS is committed to the open competition of ideas, arrangers of issue-oriented symposiums have been encouraged to include differing or conflicting viewpoints in their programs. In this way, constructive concern and criticism are increasingly represented in the formal program. Thus the AAAS provides a setting in which scientists can devote their energies to major social issues and can attempt, through their special expertise, to contribute to the solution of some of society's problems. A large portion of the meeting may be correctly viewed as a forum for the consideration of public policies involving. science, and for the shaping of the contributions that science can make to the betterment of society. In such a setting, disruption has no place. We welcome participants who use the meeting to bring ideas into confrontation; we condemn acts that deny others the opportunity to present their views or to engage in dissent. As an association, we shall take whatever steps we can to prevent this kind of interference. We believe that the effort to provide the scientific and technical community with the. opportunity for full and free discussion of some of society's most pressing and difficult problems is one of the most important things we can do and is essential for the advancement of science. The participants and others who attend our meetings should be protected from the mischief of disrupted meetings; and the public, who are informed of the proceedings by an able corps of science writers, should not have its right to know interfered with. There will always be a place within the AAAS Annual Meeting for thoughtful and emphatic dissent. There is no place for the activities of a self-selected few who would prevent the views of others from being heard. 15. M Rees, Women's Liberation: What Will We Lose?, The American Scholar 42 (1) (1972-73), 146-147. Like all social problems, this one concerning the status of women is far too complex to be dismissed in a simple statement. ... One of my strong feelings ... is a sense of gratitude (in spite of discomfort and impatience with some details) for the many diverse activities in the movement for women's equality and toward the leaders of the varied groups that have been working for the welfare of women. It is through their efforts, and not through some mystic process of maturation, that equitable treatment for women has reached the status of "an idea whose time has come." This has been accompanied by a great variety of gains for all women: the liberalisation of abortion laws, tenuous though the victory remains; the appointment of many more women to important posts in government and industry; the recent guidelines adopted by the New York State Board of Regents and other educational policy groups urging that schools and colleges take the lead in providing equal opportunity for women and end abuses in jobs, salaries and educational opportunities - all these are manifestations of the fact that much has changed. ... From my own point of view, the outstanding need is close to Ms Hellman's observation that the question is, "Where does dignity lie?" For me the matter of central importance is that our society should be so changed that little girls have the same opportunity for fulfilment in their lives that little boys have. And by fulfilment I mean the fulfilment of their intellectual, personal, emotional and social potential. I think it is important that our mores and all our attitudes and procedures be so changed that girls are encouraged to choose their careers on the basis of their abilities and interests, and not because some activities are stereotyped for girls; that education be open and that girls have the same freedom of choice that boys have; that fictions about the physical or intellectual or emotional limitations of women not be permitted to cloud the issue and introduce irrelevancies; and that girls be brought up in a society that provides reasonable models so that their aspirations are not dampened before they have a chance to flourish. ... In the future a variety of life-styles should be open to women as well as to men. But women will need to acknowledge that, with their husbands, they must make decisions about childbearing and rearing, and arrange their lives so that the toll taken in energy is one that they are prepared and able to bear. M Rees, The Saga of American Universities: The Role of Science, Science, New Series 179 (4068) (1973), 19-23. 23 June 1972 is an important date in the history of higher education in the United States, for it was on that date that the finding of Congress that "the nation's institutions of higher education constitute a national resource which significantly contributes to the security, general welfare, and economy of the United States" was signed into law. In the preceding fiscal year, federal obligations for the support of higher education had amounted to$3.5 billion, over 12 percent of the total expenditure for higher education from all sources; but all these federal funds had been appropriated for purposes other than higher education itself, purposes such as the enhancement of the nation's health, or the improvement of agriculture, or the expansion of educational opportunity. The Education Amendments of 1972, however, express concern about the "financial crisis confronting the nation's post-secondary institutions" and establish a National Commission on the Financing of Postsecondary Education to study the nature and causes of the serious financial distress facing institutions of higher education. It is of particular concern to scientists that, in spite of the substantial sums that flowed to the colleges and universities in fiscal 1971, changing national priorities affecting the support of academic science actually resulted in lower support in 1971 than in 1969 and produced critical problems for our major universities.
...
In the hope of contributing to a better understanding of the present financial crisis, I discuss here certain aspects of post-war categorical support of scientific research in the universities, aspects that I believe suggest some principles which should influence future policy. As Congress studies the needs of higher education, what kinds of programs should we in the scientific and educational communities in order to ensure the health and effectiveness of our educational institutions and to preserve the strength of science? What have we learned in the past 25 years that can assist us in building stronger institutions in the years ahead? Should we favour reliance on categorical grants or on formula-based institutional grants? What role should be played by the federal government, what role by the states? In discussing these questions, I am aware that the assistant commissioner for education has warned that we must expect "some difficult times ahead" for education; and I realise that only the strongest public demand can be expected to clothe the authorisation act of 1972 with budgetary reality. It is my hope that, like Lockheed, the institutions of higher education of the nation will be deemed worth saving.

16. M Rees, The Scientist in Society: Inspiration and Obligation: The scientist today must accept a role more difficult than that of his predecessors, American Scientist 63 (2) (1975), 144-149.

Like most people who are scientifically literate I am deeply grateful that my formal education provided me with enough background to appreciate the excitement of some of the fascinating findings of modern science. At this moment, astrophysics with its pulsars and black holes and highly controversial quasars seems to me the most absorbing, but everyone has his favourite field and his most rewarding pursuit. One question we in the universities cannot avoid is the extent of our obligation to provide all our students, whether or not they expect to become scientists, with the opportunity to find the kind of pleasure in science that will make them feel themselves a part of their time. My reference to feeling "a part of their time" is a comment on the fantastic success science has had. The distinctive shape of our society has been brought about most recently by the advances of science and their application in technology. The upsurge in science and technology has resulted in a greatly deepened understanding of our universe, in economic expansion, in the improvement of health and the lengthening of the individual life span, in the possibility of expanded assistance to other nations, and in the general elevation in the quality of life for large segments of the world's population.

It seems to me that much of traditional humanism has retreated in the face of scientific progress during the twentieth century and that many humanists have lost their zest. They have been disheartened by what they view as too much support for science and by the ebullient self-confidence of scientists. Many of us, on the other hand, may be dismayed when we ask what will happen to science if its prestige dims and there is less money for the types of growth that seem to us to be central. What is the future of science and what in its nature brings it under continuing attack, even as the offensive earlier mounted by dogmatic religion has tapered off? Darwinism is still having its troubles in America as one locality after another challenges its treatment in textbooks, but even greater problems have emerged lately with the coming of a broadly based anti-rationalism.

I have been fond of saying that science has as its goal man's under standing of himself and the world and universe about him. Its distinctive method is based on rationality, and its distinctive results produce an increase in the scope and depth of knowledge. This is a simple kind of statement that gives no insight into deep philosophical questions about the meaning of science or the nature of scientific theories, their generality and elegance, their ability to control and predict. Most of us have given little thought to the question of whether or not our logic is infallible - a question that has been profoundly explored by mathematical logicians.

But we were shaken in the late 1960s when we were assailed by a wave of unreason that seemed to overwhelm us on the campuses of the country. For many of us the most familiar exponent of the suspicion and contempt for conventional rationality that left us numbed was Charles Reich, who advocated the primacy of a direct experience of nature, rather than a disciplined study of the evidence, as the new and powerful way to know the "truth." Clearly, doing or understanding science is not one of the experiences embraced in this mode of acquiring knowledge.

Although Charles Reich seems no longer to be a significant centre of influence, the attack on scientific rationality is now largely spearheaded by a group of possibly more beguiling writers. ...

17. M Rees and A H Livermore, Milestone Legislation for a Metric United States, Science, New Series 191 (4223) (1976), 141.

The President's signing the Metric Conversion Act of 1975 is a milestone in the history of the U.S. measurement policy. The United States is now committed to providing a national program that will make the International Metric System the predominant but not exclusive system of measurement throughout the country.

The use of the metric system has been legal in the United States for more than a century, but only in the last few years has actual usage become wide-spread and increasingly visible.
...
Until now metrication in the United States has been voluntary and uncoordinated. The Metric Conversion Act of 1975 is the congressional response to this absence of coordination and direction. The new law establishes a U.S. Metric Board to coordinate voluntary conversion to the metric system within the next ten years.

18. M Rees, The Mathematical Sciences and World War II, The American Mathematical Monthly 87 (8) (1980), 607-621.

Since the Applied Mathematics Panel represented the largest group of mathematicians organised under government auspices to provide mathematical assistance wherever it was needed during the war, it may be of interest to give a brief overview of the nature of the studies carried on by the Panel from its founding in late 1942 until its dissolution at the end of 1945.

Most AMP studies were concerned with the improvement of the theoretical accuracy of equipment by suitable changes in design or by the best use of existing equipment, particularly in such fields as air warfare. It often happened that a considerable development of basic theory was needed. The following illustrations are taken from the work at New York University, Brown, and Columbia.

At New York University, the work in gas dynamics was principally concerned with the theory of explosions in the air and under water and with aspects of jet and rocket theory. New results were obtained in the study of shock fronts associated with violent disturbances of the sort that result from explosions. A request by the Bureau of Aeronautics for assistance in the design of nozzles for jet motors gave rise to an extended study of gas flow in nozzles and supersonic gas jets. In this field, as in every part of the work of the Applied Mathematics Panel, one result of the work was to provide men (alas, there were not many women) who were broadly and deeply informed in a number of important and difficult fields and who were therefore often called upon as consultants. I have a vivid remembrance of a visit in the company of Richard Courant and Kurt Friedrichs to the rocket work going on at the California Institute of Technology. The Caltech people were having trouble with the launching of their rockets, and they were eager for advice. When I talked about that visit fairly recently with Professor Friedrichs, he was characteristically modest; but when we left Pasadena back in 1944, the Caltech people had new experiments planned, at least partially inspired by suggestions they had received. And the outcome, whether or not significantly affected by Friedrichs's suggestions, was successful.

Because so many questions were raised by wartime agencies about the mathematical aspects of the dynamics of compressible fluids, a Shock Wave Manual was prepared at NYU and published in its first version in 1944 by the Applied Mathematics Panel. It was one of the major documents of continuing mathematical interest to grow out of the Panel's work. Its successor, the book Supersonic Flow and Shock Waves, was published in 1948. Its preface stated:
The present book originates from a report issued in 1944 under the auspices of the Office of Scientific Research and Development. Much material has been added and the original text has been almost entirely rewritten. The book treats basic aspects of the dynamics of compressible fluids in mathematical form; it attempts to present a systematic theory of non-linear wave propagation, particularly in relation to gas dynamics. Written in the form of an advanced text book, it accounts for classical as well as some recent developments, and, as the authors hope, it reflects some progress in the scientific penetration of the subject matter. On the other hand, no attempt has been made to cover the whole field of non-linear wave propagation or to provide summaries of results which could be used as recipes for attacking specific engineering problems ...

Dynamics of compressible fluids, like other subjects in which the non-linear character of the basic equations plays a decisive role, is far from the perfection envisaged by Laplace as the goal of a mathematical theory. Classical mechanics and mathematical physics predict phenomena on the basis of general differential equations and specific boundary and initial conditions. In contrast, the subject of this book largely defies such claims. Important branches of gas dynamics still centre around special types of problems, and general features of connected theory are not always clearly discernible. Nevertheless, the authors have attempted to develop and to emphasise as much as possible such general viewpoints, and they hope that this effort will stimulate further advances in this direction.
After the war, the NYU group continued its interest in a number of the problems worked on during the war with support from all the military services. J J Stoker's studies of water waves, in particular, were continued. And, with the growth of computers, the group greatly expanded its work in fields related to computer applications.

At Brown, the work focused on problems in classical dynamics and the mechanics of deformable media. The mathematical output of the Brown group was substantial; but I think it is worth quoting a paragraph from a letter from William Prager, the head of the Brown group, to Churchill Eisenhart, written in June 1978. He says:
While the Applied Mathematics Group at Brown University worked on numerous problems suggested by the military services, I believe that its essential service to American Mathematics was to help in making Applied Mathematics respectable ... The fact that the Program of Advanced Instruction and Research in Applied Mechanics, the forerunner of Brown's Division of Applied Mathematics, relied heavily on the financial support available under a war preparedness program illustrates the influence of the war on the development of the mathematical sciences in the U.S.
It is certainly true of the post-war programs at Brown and at NYU that they drew great strength from the importance of their work to the war effort and from the interest of the military services in their continuing vitality after the war.

At Harvard, the work in underwater ballistics produced a polished account of the water entry problem and, like all the other projects, it provided a group of expert advisers, in this case for the Navy. Moreover, it gave applied mathematics in the United States an important, newly active participant, Garrett Birkhoff.

The three projects I have thus far mentioned were all concerned with what can be described as classical applied mathematics. The largest of the so-called "Applied Mathematics Groups," the one at Columbia, had a different kind of assignment. For several years, its work was devoted primarily to studies in aerial warfare, the most extensive analyses being devoted to air-to-air gunnery. At the time of its establishment in 1943, this group was headed by E. J. Moulton; during its last year, from the beginning of September 1944 to the end of August 1945, Saunders Mac Lane was its "Technical Representative."

The final summary of the work done by the Applied Mathematics Group at Columbia under the AMP contract, as well as related work done elsewhere in the United States and abroad, was reported in the Summary Technical Report of the Applied Mathematics Panel under the following headings: (1) Aeroballistics - the motion of a projectile from an airborne gun; (2) Theory of deflection shooting; (3) Pursuit curve theory - important because the standard fighter employed guns so fixed in the aircraft as to fire in the direction of flight, and important also in the study of guided missiles that continually change direction under radio, acoustical, or optical guidance unwillingly supplied by the target; (4) The design and characteristics of own-speed sights - devices designed for use in the special case of pursuit curve attack on a defending bomber; (5) Lead computing sights - which assume that the target's track relative to the gun mount is essentially straight over the time of flight of the bullet; (6) The basic theory of a central fire control system; (7) The analytical aspects of experimental programs for testing airborne fire control equipment; (8) New developments, such as stabilisation and radar.

That part of the program of the Applied Mathematics Panel that was concerned with the use of rockets in air warfare was primarily the responsibility of Hassler Whitney, who served as a member of the Applied Mathematics Group at Columbia. He not only integrated the work carried on at Columbia and Northwestern in the general field of fire control for airborne rockets but maintained effective liaison with the work of the Fire Control Division of NDRC in this field and with the activities of many Army and Navy establishments, particularly the Naval Ordnance Test Station at Inyokern, the Dover Army Air Base, the Wright Field Armament Laboratory, the Naval Bureau of Ordnance, and the British Air Commission.

All these studies were concerned with the best use of equipment or with changes in equipment that could be effected in time to be of use in World War II. Two studies in air warfare carried out under AMP auspices came closer to having general tactical scope than did most of the other work done by the Panel. In 1944, the Panel responded to a request from the Army Air Force (AAF) asking for collaboration "in determining the most effective tactical application of the B-29 airplane" by setting up three contracts: one at the University of New Mexico, to carry on large-scale experiments; a second at Mt Wilson Observatory, to carry on small-scale optical studies; and a third at Princeton, to provide mathematical support for the whole undertaking. At Mt Wilson the staff was concerned principally with the defensive strength of single B-29's against fighter attack, and the effectiveness of fighters against B-29's. One indirect result of the optical studies was a set of moving pictures showing the fire-power variation of formations as a fighter circles about them. Warren Weaver reports that, concerning such pictures, the President of the Army Air Forces Board remarked that he "believed these motion pictures gave the best idea to air men as to the relative effect of fire power about a formation yet presented." Certain of these pictures were flown to the Marianas and viewed by General LeMay and by many gunnery officers at the front. The extent to which the claim can be made here for the power of mathematics may be limited, but the study was an effective one.

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# Homework Help: Is the uncountably infinite union of open sets is open?

1. Jan 26, 2010

### Johnson04

This is not a homework problem, just a question from a discussion with my classmates about the Cantor set. The original goal is to prove Cantor set is closed. My earlier attempt is to show the complement of the Cantor set is open. Since when construct the Cantor set each time the sets removed are open, I thought the complement of the Cantor set contains uncountably infinite many open sets, and the union of all of these open sets is still open. So the Cantor set is closed. But first, one of my classmate disagree with this proof. He thought the complement of the Cantor set only contains countably infinite many open sets. Second, he thought only countably infinite union of open sets is open, uncountably infinite union of open sets may not be open. But he could not give me convincing proof. So I posted my question here. Any suggestions are highly appreciated!

2. Jan 26, 2010

### Dick

ANY union of open sets is open, countable or not. That's an easy proof. Try it! Prove it to your classmate.

3. Jan 26, 2010

### Johnson04

Thanks a lot. I see, since any points in a open set are interior points, the points in the any union of the open sets are still interior points, accordingly, the union of the open sets is still open, no matter if the union is finite, countably infinite or uncountably infinite. Well, I was wondering if the complement of the Cantor set consists of uncountably infinitely many open sets?

4. Jan 26, 2010

### Dick

Depends on what you mean. ANY open set can be written as an uncountable union of open sets. You can write it as an uncountable union of copies of itself. On the other hand, it's also true that any open set on the real line can be written as a countable union of disjoint intervals, if that's what you really mean. That's a bit harder to prove. But not super hard.

5. Jan 26, 2010

### Johnson04

I think that was my question: if the complement of the Cantor set on [0,1] is the set of uncountably infinitely many disjoint open sets. I think it is, because the end points of the open sets form the Cantor set. It can be proved that the Cantor set is uncountable set by using the Cantor's diagonal process. So there are uncountably infinitely many open sets in the complement of the Cantor set. Did I miss anything important?

6. Jan 26, 2010

### Dick

Yes. You are missing the deep weirdness of the Cantor set. You can't have an uncountable number of disjoint open intervals on the real line. Each interval contains a rational number which is not in any other interval. Right? That means you can associate each interval with a unique rational number. Also, right? That means the cardinality of intervals is equivalent the cardinality of a subset of the rationals. Also, also, right? Can't be denied so far, right? Hence the intervals are countable. The Cantor set is uncountable. The picture of a string of intervals with single points between them is WRONG. Boo!

7. Jan 26, 2010

### Dick

Look at it this way. If you pick x=1/2, then what's the next rational number that is larger than 1/2? I think that kind of sums the problem up.

8. Jan 27, 2010

### Johnson04

Thanks Dick! I think I see what you mean. Each interval contains a unique rational number, because the set of all rational numbers is dense on real line. On real line there are only at most countably infinite many open intervals. Actually this conclusion make the proof of the Cantor set is closed a little easier.

9. Jan 28, 2010

### Johnson04

Well, I believe I totally understand the related properties of the Cantor set now. First, not all points in Cantor set are endpoints of the removed open sets. Second, the complement of the Cantor set only contains countably infinitely many open sets, that means the Cantor set does contain only countably infinitely many endpoints of the removed open sets (since they only have countably infinitely many endpoints). With this very clear picture, I have got proofs of several properties of the Cantor sets. Thanks!


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### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

Four rods are hinged at their ends to form a convex quadrilateral. Investigate the different shapes that the quadrilateral can take. Be patient this problem may be slow to load.

# Eight Factors Only

##### Age 14 to 16 ShortChallenge Level

Finding common factors and products
Factors of 15 and 21 will also be factors of the number
Products of the prime factors will also be factors of the number

The list already has 8 numbers in it so the number is 105.
1 + 3 + 5 + 7 + 15 + 21 + 35 + 135 = 192

Considering numbers which have 15 and 21 as factors
15 and 21 are both factors of the number we are looking for.
The first few numbers in the 15 times table are 15, 30, 45, 60, 75, 90, 105, 120, ...
The first few numbers in the 21 times table are 21, 42, 63, 84, 105, 126, 147, 168, ...
So 105 is the smallest number that has both 15 and 21 as factors.
The factors of 105 are 1 and 105, 3 and 35, 5 and 21, and 7 and 15. That gives 8 factors, so 105 is the number we were looking for.
The sum of all the factors of 105 is 1 + 105 + 3 + 35 + 5 + 21 + 7 + 15 = 192.

Considering what the other factors of the number must be
15 = 3$\times$5, so 3 and 5 must also be factors of the number.
21 = 3$\times$7, so 7 must also be a factor of the number.
Since 5 and 7 are both factors of the number, 5$\times$7 = 35 must also be a factor of the number.
So 3, 5, 7, 15, 21, 35, 1 and the number itself must all be factors of the number - which makes 8 factors.
So the number must be 3$\times$5$\times$7 = 105, and the sum of all of the factors is 1 + 3 + 5 + 7 + 15 + 21 + 35 + 105 = 192.

Considering the lowest common multiple of 15 and 21
15 and 21 are both factors of the number we are looking for, so the lowest common multiple of 15 and 21 must also be a factor of the number we are looking for.
15 = 3$\times$5 and 21 = 3$\times$7, so the lowest common muliple of 15 and 21 is 3$\times$5$\times$7 = 105.
The factors of 105 are 1 and 105, 3 and 35, 5 and 21, and 7 and 15.
That gives 8 factors, so 105 is the number we were looking for (all other numbers in the 105 times table are larger than 105, and have themself as a factor, so have more than 8 factors).
The sum of all the factors of 105 is 1 + 105 + 3 + 35 + 5 + 21 + 7 + 15 = 192.

You can find more short problems, arranged by curriculum topic, in our short problems collection.


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Average velocity backpacker

1. Sep 9, 2009

triplel777

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

In reaching her destination, a backpacker walks with an average velocity of 1.33 m/s, due west. This average velocity results because she hikes for 6.44 km with an average velocity of 2.60 m/s, due west, turns around, and hikes with an average velocity of 0.425 m/s, due east. How far east did she walk?

2. Relevant equations

3. The attempt at a solution

average V= delta r/delta t. since i dont know the time i found that first using v= d/t. 2.6= 6.44/t. t=2.48. using that i set up the equation saying 1.33= 2.6-0.425/2.48-t_0 and solved for t_0 which got me 0.844 and since we are looking for a distance i said v= d/t so 0.425= d/0.844= 0.3587 km....but its wrong

2. Sep 9, 2009

rl.bhat

If d is the distance traveled due east, time taken will be d/0.425 = 2.35*d s.
The average velocity 1.33 = (6440 - d)/(6440/2.6 + 2.35*d)
Now solve for d.

Last edited: Sep 9, 2009
3. Sep 9, 2009

triplel777

d/0.425 = 2.35*d
it will give me d/d = 5.53 which i dont understand and where does the 2.35 come from?

4. Sep 9, 2009

rl.bhat

it will give me d/d = 5.53
How did you get this one?

5. Sep 9, 2009

triplel777

sorry i wrote the wrong number. well you gave me the equation d/0.425 = 2.35*d so 2.35d*0.425=d : d=0.99875d. so d/d=099875 ??

6. Sep 9, 2009

rl.bhat

No. It is not correct.
Go through #2. again. I have done sum editing.
Solve for d.

7. Sep 9, 2009

triplel777

is this suppose to be a fraction over a fraction?

8. Sep 9, 2009

rl.bhat

No. It is
1.33 = (6440 - d)/(2477 + 2.35d)

Last edited: Sep 9, 2009
9. Sep 11, 2009

triplel777

so
1.33*(2477+2.35d)= 6440-d
3294.41+3.126d= 6440-d
-3145.59+3.126d= -d
? what do i do from there?

10. Sep 11, 2009

rl.bhat

3145.59 = 3.126d + d

11. Sep 11, 2009

triplel777

ok so if i add 3.126d+d then that gives me 4.126d
so 3145.59= 4.126d
3145.59/4.126=d
d= 762.94
but thats wrong...???

12. Sep 11, 2009

rl.bhat

1.33 = (6440 - d)/(2477 + 2.35d)

Rewrite this step.
1.33 = (6440 - d)/(2462 + 2.353d)
Now calculate.

13. Sep 12, 2009

triplel777

1.33*(2462+2.353d)= 6440-d
3274.46+3.129d= 6440-d
-3165.54+3.129d= -d
3165.54= d+ 3.129d
3165.54= 4.129d
d= 766.7
but thats wrong....


text:
Latest Posts

how to find a generalized eigenvector

December 4, 2020 4:18 am Leave your thoughts

We will now need to find the eigenvectors for each of these. u2 = B*u1 u2 = 34 22 -10 -27 and . Since (D tI)(tet) = (e +te t) tet= e 6= 0 and ( D I)et= 0, tet is a generalized eigenvector of order 2 for Dand the eigenvalue 1. Also note that according to the fact above, the two eigenvectors should be linearly independent. NOTE: By "generalized eigenvector," I mean a non-zero vector that can be used to augment the incomplete basis of a so-called defective matrix. Theorem Let Abe a square matrix with real elements. Therefore, a r 1 = 0. We find that B2 ≠ 0, but B3 = 0, so there should be a length 3 chain associated with the eigenvalue λ = 1 . So, let’s do that. We proceed recursively with the same argument and prove that all the a i are equal to zero so that the vectors v The generalized eigenvectors of a matrix are vectors that are used to form a basis together with the eigenvectors of when the latter are not sufficient to form a basis (because the matrix is defective). Generalized Eigenvectors When a matrix has distinct eigenvalues, each eigenvalue has a corresponding eigenvec-tor satisfying [λ1 −A]e = 0 The eigenvector lies in the Generalized Eigenvectors of Square Matrices. The vector ~v 2 in the theorem above is a generalized eigenvector of order 2. To find the eigenvectors we simply plug in each eigenvalue into . Generalized Eigenvectors of Square Matrices. This means that for each , the vectors of lying in is a basis for that subspace.. Choosing the first generalized eigenvector . This turns out to be more involved than the earlier problem of finding a basis for , and an algorithm for finding such a basis will be deferred until Module IV. Generalized eigenspaces November 20, 2019 Contents 1 Introduction 1 2 Polynomials 2 3 Calculating the characteristic polynomial 6 4 Projections 8 5 Generalized eigenvalues 11 6 Eigenpolynomials 16 1 Introduction We’ve seen that sometimes a nice linear transformation T (from a vector Note that a regular eigenvector is a generalized eigenvector of order 1. Find all of the eigenvalues and eigenvectors of A= 2 6 3 4 : The characteristic polynomial is 2 2 +10. u3 = B*u2 u3 = 42 7 -21 -42 by Marco Taboga, PhD. and solve. Its roots are 1 = 1+3i and 2 = 1 = 1 3i: The eigenvector corresponding to 1 is ( 1+i;1). GENERALIZED EIGENVECTORS 5 because (A I) 2r i v r = 0 for i r 2. The smallest such kis the order of the generalized eigenvector. $${\lambda _{\,1}} = - 5$$ : In this case we need to solve the following system. The generalized eigenvalue problem of two symmetric matrices and is to find a scalar and the corresponding vector for the following equation to hold: or in matrix form The eigenvalue and eigenvector matrices and can be found in the following steps. u1 = [1 0 0 0]'; we calculate the further generalized eigenvectors . Generalized eigenvector. Table of Contents. The generalized eigenvalue problem is to find a basis for each generalized eigenspace compatible with this filtration. If is a complex eigenvalue of Awith eigenvector v, then is an eigenvalue of Awith eigenvector v. Example Generalized eigenvector From Wikipedia, the free encyclopedia In linear algebra, for a matrix A, there may not always exist a full set of linearly independent eigenvectors that form a complete basis – a matrix may not be diagonalizable. Generalized Eigenvectors of Square Matrices Fold Unfold. Old Lithuanian Alphabet, Banquet Meatballs Recipe, Blueberry Basil Infused Water, Triple Moisture-rich Herbal Whipped Crème Conditioner & Hair Mask, Birthday Cake Shot, Basil In Ilocano, Praesidia Rifle Accessories, Assignment On Maharashtrian Cuisine, Kawai Ca99 Vs Cs11, Northern College Accounting,


text:
# Setting QProgressBar values based on selection

I have a function which writes a report based on technologies selected by the user via QCheckBoxes. Depending on the number of technologies chosen, the QProgressBar will go through set values until the function reaches completion.

For example:

1 tech: 0 -> 99 -> 100
2 tech: 0 -> 50 -> 99 -> 100
3 tech: 0 -> 33 -> 66 -> 99 -> 100
4 tech: 0 -> 25 -> 50 -> 75 -> 99 -> 100

Regardless of the number of technologies chosen, once the last one has been processed, the progressbar would be set to 99 to do some textual clean up before reaching 100.

At the moment, I have two such technologies (PV and wind turbine) where I manually set the progress values.

How could I achieve the above in a more systematic fashion?

My current code:

def report(self):
# Define progress bar
progressMessageBar = iface.messageBar().createMessage(str('Generating report...'))
progress = QProgressBar()
progress.setMaximum(100)
progress.setAlignment(Qt.AlignLeft | Qt.AlignVCenter)
iface.messageBar().pushWidget(progressMessageBar, iface.messageBar().INFO)
progress.setValue(0)
sleep(5)

# Define Report text browser
display_report = self.report_dockwidget.report_textBrowser
# Technology checkboxes
pv_checkBox = self.pv_generation_dockwidget.returnResults_Checkbox
wind_checkBox = self.wind_generation_dockwidget.returnResults_Checkbox

# For PV
if pv_checkBox.isChecked():
# Do some processing
progress.setValue(50)
if not wind_checkBox.isChecked():
progress.setValue(99)
sleep(5)

# For Wind
if wind_checkBox.isChecked():
# Do some processing
if not pv_checkBox.isChecked():
progress.setValue(50)
else:
progress.setValue(99)
sleep(5)

# Do some cleanup
progressMessageBar.setText('Complete')
progress.setValue(100)
sleep(1)

# Show report
self.report_dockwidget.show()

One idea would be to gather all your techs in a queue and iterate over that, updating the progress as you go.

Since my Python is rather Rust-y (I'm sorry for that pun), here's some pseudocode:

# Setup progress bar et cetera
# Gather techs in a queue
techs = Queue()
if pv_checkBox.isChecked():
if wind_checkBox.isChecked():

# Do the work
while not techs.isEmpty():
progress.setValue(newProgress)

# Progress is now 99
doFinalStuff()
progress.setValue(100)

This will do all the tasks in a FIFO fashion and set the progress as you wanted with the last step always being 99.

Now, it would be a good idea to gather all checkboxes in a group or layout and, by getting and filtering all child elements, add the techs via loop as well.

• Thanks for your answer! Interesting, never used Queue() before so will test this at some point and report back :) – Joseph Dec 5 '17 at 13:58

Thanks to @PhilKiener's answer, I translated his pseudocode with the relevant methods for Python 2.7 which works well:

# Setup progress bar et cetera
# Gather techs in a queue
techs = Queue.Queue()
if pv_checkBox.isChecked():


text:
# nLab category

This page is about the concept in mathematics. For the concept of the same name in philosophy see at category (philosophy).

category theory

# Contents

## Idea

• A category consists of a collection of things and binary relationships (or transitions) between them, such that these relationships can be combined and include the “identity” relationship “is the same as.”

• A category is a quiver (a directed graph with multiple edges) with a rule saying how to compose two edges that fit together to get a new edge. Furthermore, each vertex has an edge starting and ending at that vertex, which acts as an identity for this composition.

• A category is a combinatorial model for a directed space – a “directed homotopy 1-type” in some sense. It has “points”, called objects, and also directed “paths”, or “processes” connecting these points, called morphisms. There is a rule for how to compose paths; and for each object there is an identity path that starts and ends there.

• More precisely, a category consists of a collections of objects and a collection of morphisms. Every morphism has a source object and a target object. If $f$ is a morphism with $x$ as its source and $y$ as its target, we write

$f : x \to y$

and we say that $f$ is a morphism from $x$ to $y$. In a category, we can compose a morphism $g : x \to y$ and a morphism $f : y \to z$ to get a morphism $f \circ g : x \to z$. Composition is associative and satisfies the left and right unit laws.

The example to keep in mind is the category Set, in which the objects are sets and a morphism $f : x \to y$ is a function from the set $x$ to the set $y$. Here composition is the usual composition of functions.

For more background on and context for categories see

## Definitions

There are two broad ways to write down the definition of category; in the usual foundations of mathematics, these two definitions are equivalent. It is good to know both, for several reasons:

• Each introduces its own system of notation, both of which are useful in other parts of category theory, so one should know them.
• One definition generalises quite nicely to the notion of internal category, while the other generalises quite nicely to the notion of enriched category; these are both important concepts.
• When examining alternative foundations, sometimes one definition or the other may be more appropriate; in any case, one will want to examine the question of their equivalence.

The two definitions may be distinguished by whether they use a single collection of all morphisms or several collections of morphisms, a family of collections indexed by pairs of objects.

### With one collection of morphisms

A category $C$ consists of

• a collection (see Size issues) $C_0$ of objects;

• a collection $C_1$ of morphisms (or arrows);

• two functions $s, t\colon C_1 \to C_0$ which assign, to every morphism, its source (or domain) and target (or codomain);

• a partial function $\circ\colon C_1 \times C_1 \to C_1$ which assigns, to any pair of morphisms $f, g$ such that $t(f) = s(g)$, their composite morphism $g \circ f \in C_1$ (also written $g f$ or sometimes $f;g$— see diagrammatic order);

• a function $id\colon C_0 \to C_1$ which assigns to each object $x$ a morphism $id_x$ or $1_x$, the identity morphism on $x$;

• such that the following properties are satisfied:

• source and target are respected by composition: $s(g \circ f) = s(f)$ and $t(g\circ f) = t(g)$;

• source and target are respected by identities: $s(1_x) = x$ and $t(1_x) = x$;

• composition is associative: $(h \circ g)\circ f = h\circ (g \circ f)$ whenever $t(f) = s(g)$ and $t(g) = s(h)$;

• composition satisfies the left and right unit laws: if $s(f) = x$ and $t(f) = y$, then $1_y \circ f = f = f \circ 1_x$.

People also often write $x \in C$ instead of $x \in C_0$ as a short way to indicate that $x$ is an object of $C$. Also, some people write $Ob(C)$ and $Mor(C)$ instead of $C_0$ and $C_1$. One usually writes $f\colon x \to y$ if $f \in C_1$ to state that $s(f) = x$ and $t(f) = y$. Finally, people often write $hom(x,y)$, $hom_C(x,y)$, or $C(x,y)$ for the collection of morphisms $f\colon x \to y$.

If the identity-assigning map and its axiom is omitted, then one speaks of a semicategory.

### With a family of collections of morphisms

A category $C$ consists of

• a collection (see Size issues) $C_0$ of objects;

• for each pair $x,y$ of objects, a collection $C_1(x,y)$ of morphisms from $x$ to $y$;

• for each triple $x,y,z$ of objects, a function $\circ\colon C_1(y,z) \times C_1(x,y) \to C_1(x,z)$ which assigns, to any appropriate pair of morphisms $f, g$, their composite morphism $g \circ f$ (also written $g f$ or sometimes $f;g$— see diagrammatic order);

• for each object $x$, an element $id_x$ or $1_x$ of $C_1(x,x)$, the identity morphism on $x$;

• such that the following properties are satisfied:

• composition is associative: for each quadruple $w,x,y,z$ of objects, if $f \in C_1(y,z)$, $g \in C_1(x,y)$, and $h \in C_1(w,x)$, then $(h \circ g)\circ f = h\circ (g \circ f)$;

• composition satisfies the left and right unit laws: for each pair $x,y$ of objects, if $f \in C_1(x,y)$, then $1_y \circ f = f = f \circ 1_x$.

People also often write $x \in C$ instead of $x \in C_0$ as a short way to indicate that $x$ is an object of $C$. Also, some people write $Ob(C)$ instead of $C_0$ and $hom(x,y)$, $hom_C(x,y)$, or $C(x,y)$ instead of $C_1(x,y)$. One usually writes $f\colon x \to y$ to state that $f \in C_1(x,y)$. Finally, people often write $C_1$ or $Mor(C)$ for the disjoint union $\biguplus_{x \in C_0} \biguplus_{y \in C_0} C_1(x,y)$.

### Size issues

We said a category has a ‘collection’ of objects and ‘collection’(s) of morphisms. A category is said to be small if these collections are all sets — as opposed to proper classes, for example. (The alternatives depend on ones foundations for mathematics.)

Similarly, a category is locally small if $C_1(x,y)$ is a set for every pair of objects $x,y$ in that category. The most common motivating examples of categories are all locally small but not small (unless one restricts their objects in some way).

### Alternative definitions

For some purposes it is useful or necessary to vary the way the ordinary definition of category is expressed. See

The first definition, with a single collection $C_1$ of morphisms, generalises to the notion of internal category; essentially, we define a category internal to (some other category) $D$ as above, with ‘collection’ interpreted as an object of $D$ and ‘function’ interpreted as a morphism of $D$. In particular, a category internal to Set is the same thing as a small category.

### Equivalent definitions

A category is equivalently

### Generalizations

#### Enriched categories

The definition, of a category as a family $C_1(x,y)$ of collections of morphisms, generalises to the notion of enriched category: we define a category enriched over (some other category) $D$ as above, with the collection of objects still a ‘collection’ as before, but with objects of $D$ in place of the collections of morphisms and morphisms of $D$ in place of the various functions. In particular, a category enriched over Set is the same thing as a locally small category.

#### Indexed categories

The notion of indexed category captures the idea of woking “over a base” other than Set.

#### Multicategories

There is a rather obvious generalization of the notion of catgeory where one allows a morphism to go from several objects to a single object. This is called a multicategory or operad. See also polycategory and PROP.

## Examples

There is the beginning of a database of categories listing well-known categories (with links to articles on these categories, if such articles exist) and some of their properties.

The classic example of a category is Set, the category with sets as objects and functions as morphisms, and the usual composition of functions as composition. Here are some other famous examples, which arise as variations on this theme:

Note that in all these cases the morphisms are actually special sorts of functions. these are concrete categories. That need not be the case in general!

These classic examples are the original motivation for the term “category”: all of the above categories encapsulate one “kind of mathematical structure”. These are often called “concrete” categories (that term also has a technical definition that these examples all satisfy). But just as widespread in applications as these categorization examples of categories are are other categories (often “small” ones) which, roughly, model something like states and processes of some system.

• Poset A poset can be thought of as a category with its elements as objects and one morphism in each $hom(x,y)$ if $x$ is less than or equal to $y$, but none otherwise.

• Group A group is just a category where there’s one object and all the morphisms have inverses - we call the morphisms “elements” of the group. This may seem weird, but it’s actually a very useful viewpoint. Here’s another way to say it: A group is a groupoid with a single object.

• Monoid More generally, a monoid is a category with a single object. In fact, this is one way to motivate the concept of categories: categories are the many object version of monoids.

• Groupoid A groupoid is a category in which all morphisms are isomorphisms.

• Quiver A quiver may be identified with the free category on its directed graph. Given a directed graph $G$ with collection of vertices $G_0$ and collection of edges $G_1$, there is the free category $F(G)$ on the graph whose collection of objects coincides with the collection of vertices, and whose collection of morphisms consists of finite sequences of edges in $G_1$ that fit together head-to-tail. The composition operation in this free category is the concatenation of sequences of edges.

• Universal structure A category bearing a structure making it initial (or 2-initial) in some doctrine. Examples include the permutation category as the free symmetric monoidal category generated by a single object, or the simplex category which is initial among monoidal categories equipped with a monoid.

## Basic notions

A homomorphism between categories is a functor.

This way small categories themselves form a category, the category Cat whose objects are small categories and whose morphisms are functors. This naturally enhances to a 2-category whose 2-morphisms are natural transformations between functors.

An equivalence of categories is an equivalence in Cat, hence a pair of functors going back and forth, that are each others inverse up to natural isomorphism.

Weaker than the notion of a pair of functors exhibiting an equivalence is the notion of a pair of adjoint functors.

Other standard operations on categories include

## References

The standard textbook is

Expositions include

A textbook with an eye towards the theory of categories of sheaves and their application in homological algebra is

The definition of categories in the foundations of homotopy type theory (hence via the notion of internal category in an (infinity,1)-category) is discussed in

For more references see category theory.

Revised on March 24, 2014 08:37:15 by Urs Schreiber (89.204.138.56)