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## Science, Religion and Henry Gee

Posted in Bad Statistics, Books, Talks and Reviews, Science Politics, The Universe and Stuff with tags , , , , , , , , , on September 23, 2013 by telescoper

Last week a piece appeared on the Grauniad website by Henry Gee who is a Senior Editor at the magazine Nature.  I was prepared to get a bit snarky about the article when I saw the title, as it reminded me of an old  rant about science being just a kind of religion by Simon Jenkins that got me quite annoyed a few years ago. Henry Gee’s article, however, is actually rather more coherent than that and  not really deserving of some of the invective being flung at it.

For example, here’s an excerpt that I almost agree with:

One thing that never gets emphasised enough in science, or in schools, or anywhere else, is that no matter how fancy-schmancy your statistical technique, the output is always a probability level (a P-value), the “significance” of which is left for you to judge – based on nothing more concrete or substantive than a feeling, based on the imponderables of personal or shared experience. Statistics, and therefore science, can only advise on probability – they cannot determine The Truth. And Truth, with a capital T, is forever just beyond one’s grasp.

I’ve made the point on this blog many times that, although statistical reasoning lies at the heart of the scientific method, we don’t do anywhere near enough  to teach students how to use probability properly; nor do scientists do enough to explain the uncertainties in their results to decision makers and the general public.  I also agree with the concluding thought, that science isn’t about absolute truths. Unfortunately, Gee undermines his credibility by equating statistical reasoning with p-values which, in my opinion, are a frequentist aberration that contributes greatly to the public misunderstanding of science. Worse, he even gets the wrong statistics wrong…

But the main thing that bothers me about Gee’s article is that he blames scientists for promulgating the myth of “science-as-religion”. I don’t think that’s fair at all. Most scientists I know are perfectly well aware of the limitations of what they do. It’s really the media that want to portray everything in simple black and white terms. Some scientists play along, of course, as I comment upon below, but most of us are not priests but pragmatatists.

Anyway, this episode gives me the excuse to point out  that I ended a book I wrote in 1998 with a discussion of the image of science as a kind of priesthood which it seems apt to repeat here. The book was about the famous eclipse expedition of 1919 that provided some degree of experimental confirmation of Einstein’s general theory of relativity and which I blogged about at some length last year, on its 90th anniversary.

I decided to post the last few paragraphs here to show that I do think there is a valuable point to be made out of the scientist-as-priest idea. It’s to do with the responsibility scientists have to be honest about the limitations of their research and the uncertainties that surround any new discovery. Science has done great things for humanity, but it is fallible. Too many scientists are too certain about things that are far from proven. This can be damaging to science itself, as well as to the public perception of it. Bandwagons proliferate, stifling original ideas and leading to the construction of self-serving cartels. This is a fertile environment for conspiracy theories to flourish.

To my mind the thing  that really separates science from religion is that science is an investigative process, not a collection of truths. Each answer simply opens up more questions.  The public tends to see science as a collection of “facts” rather than a process of investigation. The scientific method has taught us a great deal about the way our Universe works, not through the exercise of blind faith but through the painstaking interplay of theory, experiment and observation.

This is what I wrote in 1998:

Science does not deal with ‘rights’ and ‘wrongs’. It deals instead with descriptions of reality that are either ‘useful’ or ‘not useful’. Newton’s theory of gravity was not shown to be ‘wrong’ by the eclipse expedition. It was merely shown that there were some phenomena it could not describe, and for which a more sophisticated theory was required. But Newton’s theory still yields perfectly reliable predictions in many situations, including, for example, the timing of total solar eclipses. When a theory is shown to be useful in a wide range of situations, it becomes part of our standard model of the world. But this doesn’t make it true, because we will never know whether future experiments may supersede it. It may well be the case that physical situations will be found where general relativity is supplanted by another theory of gravity. Indeed, physicists already know that Einstein’s theory breaks down when matter is so dense that quantum effects become important. Einstein himself realised that this would probably happen to his theory.

Putting together the material for this book, I was struck by the many parallels between the events of 1919 and coverage of similar topics in the newspapers of 1999. One of the hot topics for the media in January 1999, for example, has been the discovery by an international team of astronomers that distant exploding stars called supernovae are much fainter than had been predicted. To cut a long story short, this means that these objects are thought to be much further away than expected. The inference then is that not only is the Universe expanding, but it is doing so at a faster and faster rate as time passes. In other words, the Universe is accelerating. The only way that modern theories can account for this acceleration is to suggest that there is an additional source of energy pervading the very vacuum of space. These observations therefore hold profound implications for fundamental physics.

As always seems to be the case, the press present these observations as bald facts. As an astrophysicist, I know very well that they are far from unchallenged by the astronomical community. Lively debates about these results occur regularly at scientific meetings, and their status is far from established. In fact, only a year or two ago, precisely the same team was arguing for exactly the opposite conclusion based on their earlier data. But the media don’t seem to like representing science the way it actually is, as an arena in which ideas are vigorously debated and each result is presented with caveats and careful analysis of possible error. They prefer instead to portray scientists as priests, laying down the law without equivocation. The more esoteric the theory, the further it is beyond the grasp of the non-specialist, the more exalted is the priest. It is not that the public want to know – they want not to know but to believe.

Things seem to have been the same in 1919. Although the results from Sobral and Principe had then not received independent confirmation from other experiments, just as the new supernova experiments have not, they were still presented to the public at large as being definitive proof of something very profound. That the eclipse measurements later received confirmation is not the point. This kind of reporting can elevate scientists, at least temporarily, to the priesthood, but does nothing to bridge the ever-widening gap between what scientists do and what the public think they do.

As we enter a new Millennium, science continues to expand into areas still further beyond the comprehension of the general public. Particle physicists want to understand the structure of matter on tinier and tinier scales of length and time. Astronomers want to know how stars, galaxies  and life itself came into being. But not only is the theoretical ambition of science getting bigger. Experimental tests of modern particle theories require methods capable of probing objects a tiny fraction of the size of the nucleus of an atom. With devices such as the Hubble Space Telescope, astronomers can gather light that comes from sources so distant that it has taken most of the age of the Universe to reach us from them. But extending these experimental methods still further will require yet more money to be spent. At the same time that science reaches further and further beyond the general public, the more it relies on their taxes.

Many modern scientists themselves play a dangerous game with the truth, pushing their results one-sidedly into the media as part of the cut-throat battle for a share of scarce research funding. There may be short-term rewards, in grants and TV appearances, but in the long run the impact on the relationship between science and society can only be bad. The public responded to Einstein with unqualified admiration, but Big Science later gave the world nuclear weapons. The distorted image of scientist-as-priest is likely to lead only to alienation and further loss of public respect. Science is not a religion, and should not pretend to be one.

PS. You will note that I was voicing doubts about the interpretation of the early results from supernovae  in 1998 that suggested the universe might be accelerating and that dark energy might be the reason for its behaviour. Although more evidence supporting this interpretation has since emerged from WMAP and other sources, I remain sceptical that we cosmologists are on the right track about this. Don’t get me wrong – I think the standard cosmological model is the best working hypothesis we have _ I just think we’re probably missing some important pieces of the puzzle. I don’t apologise for that. I think sceptical is what a scientist should be.

## Last Week of Term

Posted in Biographical, Education, The Universe and Stuff with tags , , , , , on March 26, 2012 by telescoper

So the glorious weather continues. Unfortunately, unlike most UK universities, we’re not finished for Easter yet; at Cardiff University we only get three weeks for the Easter recess instead of the four that colleagues over the border seem to enjoy.

One of the consequences of this is that the annual National Astronomy Meeting (NAM) often falls in Cardiff term time. This year NAM is taking place in the fine city of Manchester (which, for those of you unfamiliar with British geography, is in the Midlands). Many colleagues in the School of Physics & Astronomy are attending NAM, and most of my research group are either there already or travelling up today. I particularly wish Jo and Ian well when they give their talks; one of the excellent things about NAM is the opportunity it offers for younger researchers to talk about their work to a large audience. Nerve-wracking, no doubt, but invaluable experience.

I’m not going to NAM this year because I have too much to do back here at the ranch, including filling in a few lectures for staff who are away.  I’m always reluctant to cancel lectures during term-time, but in the current spell of good weather I doubt if any students would complain too much! I did a cosmology lecture this morning – only the second I’ve done here – and it the room was uncomfortably stuffy. A few of the students failed to fall asleep, however, so I regard that as a major success.

It’s strange how often good weather coincides with times of great stress for students. I recall that most of my undergraduate examinations took place in glorious sunshine, which seemed to have been laid on by some malevolent being to make us suffer. This week our students have project reports and presentations to worry about and other coursework to finish before term ends, as well as revision for the exams that take place in May; being couped up inside is no fun on days like this and I’m sure they’d prefer it to be raining outside so as not to distract them from the tasks in hand…

It’s so quiet around here today that it occurred to me now would be a good time to stage a Coup d’Etat. Come to thank of it, there’s a Staff Meeting  been called on Wednesday which may well amount to something pretty similar…

Anyway, those of us around today have a nice event this evening to look forward to, a lecture by Lord Rees followed by a nice dinner in Aberdare Hall. Here’s the invitation:

You’ll see that this is organized “in association with The Learned Society for Wales“, which I only just learned about when I saw it on the invitation!

Anyway, the prospect of a slap-up dinner persuaded me to just have a sandwich for lunch. Now that’s eaten methinks I’ll get back to work!

UPDATE: It was indeed a very interesting and entertaining lecture by Lord Rees; here he is, in action, watched by Prof. Disney…

## Gravity and Grace

Posted in Biographical, The Universe and Stuff with tags , , , , , on April 26, 2011 by telescoper

This morning I came across the following quotation, which is translated from the book Le Pesanteur et la Grace (i.e. “Gravity and Grace“), written in 1947 by French philosopher Simone Weil:

Science today must search for a source of inspiration higher than itself or it must perish.

Science offers only three points of interest: 1. technical applications; 2. as a game of chess; 3. as a way to God.

I’m not sure I agree with what is written, and in any case the options don’t seem to me to be mutually exclusive, but a number of things did strike me reading it.

For a start, and for what it’s worth, I do think science has value within itself, so I’m at odds a bit with the initial premise. On the other hand, science is a human activity and it therefore doesn’t stand apart from other thing humans are interested in.

Then there is the extent to which we now all have to pretend that pretty much the only point of interest in science is “1. technical applications”. I don’t believe that’s true, actually, and I’m worried that by continually saying that it is, scientists might be sowing the seeds of their own destruction.

And then there’s “the game of chess”. I’m actually hopeless at chess, but I understand this as representing some form of abstract mental challenge.  If that’s what it does mean, then I’d agree that’s probably what got me interested in science. I’ve always been pathologically interested in puzzles. When I look at galaxies and stars, I don’t tend to gaze at them in awe at their enormity or beauty, I just tend to wonder how they work and what they’re made of. I don’t really mind people having a sense of awe, of course, but there’s a danger that if we take that too far we end up being over-awed which might make us shy away from the biggest questions. To me the Universe is just a great big puzzle, though it’s actually rather a tough one. I’m still stuck on 1 across, in fact…

Finally, we have science as “a way to God”. I find it quite interesting that a Christian philosopher could present science as that, especially when so many of my atheistic colleagues regard science and religion as polar opposites. It seems likely to me that anyone who studies science primarily as a means of finding God is probably in for a disappointment. I’m reminded of a quote  from Thomas à Kempis I learned at school:

The humble knowledge of thyself is a surer way to God than the deepest search after science.

But that’s not to say that science and religion are incompatible with each other. I think they’re basically orthogonal, although in an abstract space with an extremely complicated geometry…

One of the interesting things about working in cosmology is that the big questions are very big indeed, which may be the reason why cosmologists tend to have strong views on matters of religion (and metaphysics in a general sense).  Just take the Templeton Prize, for example. The arguments about this year’s award to Lord (Martin) Rees are still simmering on, but it’s worth remembering that many recent winners of this prize, including John Barrow (my PhD supervisor, in fact) and  George Ellis (former collaborator of mine), are most noted for their work in cosmology. Both are religious: John Barrow is a member of the United Reformed Church, and George Ellis is a Quaker. Martin Rees is an atheist. But their religious views are not in conflict with their research. All are outstanding scientists.

I’ve been thinking a lot over the Easter holiday about religion and science. It’s partly the Templeton prize saga, partly the occasion of Easter itself, and partly the fact that I’ve been reading even more of the poems of R.S. Thomas. In case you didn’t know I was brought up in the (Anglican) Christian tradition, attended Sunday School, sang in the local Church Choir, and was confirmed in the Church of England. When I went to seconday school – the Royal Grammar School, Newcastle – I joined the Christian Union and remained in it for 3-4 years.

Although  I was immersed in Christianity – the Christian Union was vigorously Evangelical – it didn’t really stick and eventually all melted away.   I don’t really remember precisely what it was then that made me turn away from religion, although the sins of the flesh might have had something to do with it…

However, although I became an atheist I’ve never been a particularly devout one. The only thing that I’m really sure about is that I don’t know the answers. Does that make me an agnostic rather than an atheist? I don’t know. Perhaps I could just describe myself as a non-believer? That wouldn’t do either, because we all have to believe in some things in order to function at all. Even science starts with unprovable axioms.

A career in cosmology has given me the opportunity to think about many Big Questions. Why does the Universe have laws? Why is there something rather than nothing? And so on. I’m not much of a philosopher, though, and  I don’t have the answers. I do, however, refuse to take the easy way out by denying that the questions have meaning. Of course it’s not entirely satisfactory having to answer “I don’t know”, but I don’t agree with those of my atheist colleagues who think religion is an easy way out. I’m sure that a thinking Christian has just as many difficult issues to grapple with as a thinking atheist. Not thinking at all is the only really easy way out.

A few years ago I spoke at an interesting meeting in Cambridge entitled God or Multiverse? In fact there’s a picture below of the panel discussion at the end -I’m second from the right:

I thought it was an interesting dialogue, but I have to say that, if anything, it strengthened my non-belief. Prof. Keith Ward argued that the primary motivation for belief in God was the existence of “Good”. I have to admit that I find the Universe as a whole amoral and although humans have done good from time to time they have done evil in at least equal measure. The vast majority of people on this Earth live in poverty, many of them in abject misery. Good is a bad word to describe this state of affairs.

I just can’t accept the idea of a God that is interested in the Universe at the level of human beings. We’re so insignificant on the scale of the cosmos, that it seems very arrogant to me to suppose that it’s really got much to do with us. We appeared somehow, miraculously perhaps, but could disappear just as easily. I doubt the Universe would miss us much.

But I might be wrong.

## Mud Wrestling and Microwaves

Posted in The Universe and Stuff with tags , , , on January 13, 2011 by telescoper

Reading through an interesting blog post about the new results from Planck by the ever-reliable Jonathan Amos (the BBC’s very own “spaceman”), I was reminded of a comment I heard made by Martin Rees (now Lord Rees) many years ago.

The remark concerned the difference between cosmology and astrophysics. Cosmology, said Lord Rees, especially the part of it that concerns the very early Universe, involves abstract mathematical concepts, difficult yet logical reasoning and the ability to see deep things in complicated spatial patterns. In that respect it’s rather like chess. Astrophysics, on the other hand, which is not at all elegant and has so many messy complications that it is sometimes difficult even to work out what is going on or what the rules are, is more like mud wrestling.

The following image, which I borrowed from Jonathan Amos’ piece, explains why I was reminded of this and why some cosmologists are having to abandon chess for mud wrestling, at least for the time being. The picture shows the nine individual frequency maps (spanning the range from 30 GHz to 857 GHz) obtained by Planck.

What we cosmologists really want to see is a pristine map of the cosmic microwave background, the black-body radiation that pervades the entire Universe. It’s black body form means that it would have the same brightness temperature across all frequencies, and would also be statistically homogeneous (i.e. looking roughly the same all across the sky).

What you actually see is a mess. There are strong contributions from the disk of our own Galaxy, some of it extending quite a way above and below the plane of the Milky Way. You can also see complicated residuals produced by the way Planck scans the sky. On top of that there is radiation from individual sources within our Galaxy, other Galaxies and even clusters of Galaxies (which I mentioned a couple of days ago). These “contaminants” constitute valuable raw material for astronomers of various sorts, but for cosmologists they are an unwanted nuisance. Unfortunately, there is no other way to reach the jewels of the CMB than by hacking through this daunting jungle of foregrounds and instrumental artefacts.

Looking at the picture might induce one of two reactions. One would be to assume that there’s no way that all the crud can be removed with sufficient accuracy and precision to do cosmology with what’s left. Another is  to appreciate how well cosmologists have done with previous datasets, especially WMAP, have confidence that they’ll solve the numerous problems associated with the Planck data, but understand why  will take another two years of high-powered data analysis by a very large number of very bright people to extract cosmological results from Planck.

There might be gold at the end of the pipeline, but until then it’s going to be mud, glorious mud…

## Science as a Religion

Posted in Books, Talks and Reviews, Science Politics, The Universe and Stuff with tags , , , , , , , on July 6, 2010 by telescoper

With the reaction to Simon Jenkins’ rant about science being just a kind of religion gradually abating, I suddenly remembered that I ended a book I wrote in 1998 with a discussion of the image of science as a kind of priesthood. The book was about the famous eclipse expedition of 1919 that provided some degree of experimental confirmation of Einstein’s general theory of relativity and which I blogged about at some length last year, on its 90th anniversary.

I decided to post the last few paragraphs here to show that I do think there is a valuable point that Simon Jenkins could have made out of the scientist-as-priest idea. It’s to do with the responsibility scientists have to be honest about the limitations of their research and the uncertainties that surround any new discovery. Science has done great things for humanity, but it is fallible. Too many scientists are too certain about things that are far from proven. This can be damaging to science itself, as well as to the public perception of it. Bandwagons proliferate, stifling original ideas and leading to the construction of self-serving cartels. This is a fertile environment for conspiracy theories to flourish.

To my mind the thing  that really separates science from religion is that science is an investigative process, not a collection of truths. Each answer simply opens up more questions.  The public tends to see science as a collection of “facts” rather than a process of investigation. The scientific method has taught us a great deal about the way our Universe works, not through the exercise of blind faith but through the painstaking interplay of theory, experiment and observation.

This is what I wrote in 1998:

Science does not deal with ‘rights’ and ‘wrongs’. It deals instead with descriptions of reality that are either ‘useful’ or ‘not useful’. Newton’s theory of gravity was not shown to be ‘wrong’ by the eclipse expedition. It was merely shown that there were some phenomena it could not describe, and for which a more sophisticated theory was required. But Newton’s theory still yields perfectly reliable predictions in many situations, including, for example, the timing of total solar eclipses. When a theory is shown to be useful in a wide range of situations, it becomes part of our standard model of the world. But this doesn’t make it true, because we will never know whether future experiments may supersede it. It may well be the case that physical situations will be found where general relativity is supplanted by another theory of gravity. Indeed, physicists already know that Einstein’s theory breaks down when matter is so dense that quantum effects become important. Einstein himself realised that this would probably happen to his theory.

Putting together the material for this book, I was struck by the many parallels between the events of 1919 and coverage of similar topics in the newspapers of 1999. One of the hot topics for the media in January 1999, for example, has been the discovery by an international team of astronomers that distant exploding stars called supernovae are much fainter than had been predicted. To cut a long story short, this means that these objects are thought to be much further away than expected. The inference then is that not only is the Universe expanding, but it is doing so at a faster and faster rate as time passes. In other words, the Universe is accelerating. The only way that modern theories can account for this acceleration is to suggest that there is an additional source of energy pervading the very vacuum of space. These observations therefore hold profound implications for fundamental physics.

As always seems to be the case, the press present these observations as bald facts. As an astrophysicist, I know very well that they are far from unchallenged by the astronomical community. Lively debates about these results occur regularly at scientific meetings, and their status is far from established. In fact, only a year or two ago, precisely the same team was arguing for exactly the opposite conclusion based on their earlier data. But the media don’t seem to like representing science the way it actually is, as an arena in which ideas are vigorously debated and each result is presented with caveats and careful analysis of possible error. They prefer instead to portray scientists as priests, laying down the law without equivocation. The more esoteric the theory, the further it is beyond the grasp of the non-specialist, the more exalted is the priest. It is not that the public want to know – they want not to know but to believe.

Things seem to have been the same in 1919. Although the results from Sobral and Principe had then not received independent confirmation from other experiments, just as the new supernova experiments have not, they were still presented to the public at large as being definitive proof of something very profound. That the eclipse measurements later received confirmation is not the point. This kind of reporting can elevate scientists, at least temporarily, to the priesthood, but does nothing to bridge the ever-widening gap between what scientists do and what the public think they do.

As we enter a new Millennium, science continues to expand into areas still further beyond the comprehension of the general public. Particle physicists want to understand the structure of matter on tinier and tinier scales of length and time. Astronomers want to know how stars, galaxies  and life itself came into being. But not only is the theoretical ambition of science getting bigger. Experimental tests of modern particle theories require methods capable of probing objects a tiny fraction of the size of the nucleus of an atom. With devices such as the Hubble Space Telescope, astronomers can gather light that comes from sources so distant that it has taken most of the age of the Universe to reach us from them. But extending these experimental methods still further will require yet more money to be spent. At the same time that science reaches further and further beyond the general public, the more it relies on their taxes.

Many modern scientists themselves play a dangerous game with the truth, pushing their results one-sidedly into the media as part of the cut-throat battle for a share of scarce research funding. There may be short-term rewards, in grants and TV appearances, but in the long run the impact on the relationship between science and society can only be bad. The public responded to Einstein with unqualified admiration, but Big Science later gave the world nuclear weapons. The distorted image of scientist-as-priest is likely to lead only to alienation and further loss of public respect. Science is not a religion, and should not pretend to be one.

PS. You will note that I was voicing doubts about the interpretation of the early results from supernovae  in 1998 that suggested the universe might be accelerating and that dark energy might be the reason for its behaviour. Although more evidence supporting this interpretation has since emerged from WMAP and other sources, I remain skeptical that we cosmologists are on the right track about this. Don’t get me wrong – I think the standard cosmological model is the best working hypothesis we have _ I just think we’re probably missing some important pieces of the puzzle. I don’t apologise for that. I think skeptical is what a scientist should be.

## Water and Energy

Posted in Biographical, Science Politics, The Universe and Stuff with tags , , , , , on May 25, 2010 by telescoper

I’ve refrained from blogging about the fraught history of my attempts to have a new  gas boiler installed in my house. Today, however, at last I have finally succeed in getting a state-of-the-art high-efficiency condensing contraption fit for the 21st Century, which will hopefully save me a few bob in gas bills over the winter but, more importantly, actually produce hot water for more than a minute or so without switching itself off.

The chaps that did the job for me actually had to test all the radiators too, which meant switching them all up to maximum. It wasn’t quite as hot today as it was yesterday but nevertheless the inside of the house was like a Turkish bath for a while. I therefore sat outside in the Sun for a bit waiting for them to get finished and tidy everything up.

While I was sitting there I got thinking about sustainable energy and so on, and was reminded of a comment Martin Rees made in his Reith Lecture not long ago. Wanting to sound positive about renewable energy he referred to the prospect of generating significant tidal power using a Severn Barrage. Given the local relevance to Cardiff – one of the main ideas is a barrage right across the Severn Estuary from Cardiff to Weston-super-Mare -so he presumably thought he was on safe ground mentioning it. In fact there was a lot of uneasy shuffling in seats at that point and the question session at the end generated some tersely sceptical comments. Many locals are not at all happy about the possible environmental impact of the Severn Barrage. That, and the cost – probably in excess of £20 billion – has to be set against the fact that such a barrage could in principle generate 2GW average power from an entirely renewable source. This would reduce our dependence on fossil fuels and increase our energy security too. The resources probably aren’t available right now given the parlous state of the public finances, but I’m glad that the Welsh Assembly Government is backing serious study of the various options. It may be that it won’t be long before we’re forced to think about it anyway. The Wikipedia page on the various proposals for a Severn Barrage is very comprehensive, so I won’t rehearse the arguments here. In any case, I’m no engineer and can’t comment on the specifics of the technology required to construct, e.g., a tidal-stream generator. However, I have to say that I find the idea pretty compelling, provided ways can be found to mitigate its environmental impact.

For a start it’s instructive to look at turbine-generated power. Wind turbines  are cropping up around the British isles, either individually or in wind farms. A  typical wind turbine can generate about 1MW in favourable weather conditions, but it needs an awful lot of them to produce anything like the power of a conventional power station. They’re also relatively unpredictable so can’t be relied upon on their own for continuous power generation. The power $P$ available from a wind turbine is given roughly by

$P \simeq \frac{1}{2} \epsilon \rho A v^3$

where $v$ is the wind speed, $A$ is the area of the turbine, $\rho$ is the density of air, which is about 1.2 kg per cubic metre, and $\epsilon$ is the efficiency with which the turbine converts the kinetic energy of the air into useable electricity.

The same formula would apply to a turbine placed in water, immediately showing the advantage of tidal power.  For comparable efficiencies and sizes the ratio of power generated in a tidal-stream turbine to a wind turbine would be

$\frac{P_{t}}{P_{w}}\simeq \frac{\rho_{t}}{\rho_{w}} \left( \frac{v_{t}}{v_{w}}\right)^{3}$

The speed of the water in a tidal stream can be comparable to the airspeed in a moderate wind, in which case the term in brackets doesn’t matter and it’s just the ratio of the densities of water and air that counts, and that’s a large number! Of course wind speed can sometimes be larger than the fastest tidal current, but wind turbines don’t work efficiently in such conditions and in any case it isn’t the $v$ which provides the killer factor. The density of sea water is about 1025 kg per cubic metre, a thousand times greater than that of air. To get the same energy output from air as from a tidal stream you would need to have winds blowing steadily ten times the velocity of the stream, which would be about 80 knots for the Severn. More than breezy!

Not all proposals for the Severn Barrage involve tidal stream turbines. Some exploit the gravitational potential energy rather than the kinetic energy of the water by exploiting the vertical rise and fall during a tidal cycle rather than the horizontal flow. The energy to be exploited in, for example, a tidal basin of area $A$  would go as

$E \simeq \frac{1}{2} \epsilon A\rho gh^{2}$

where $h$ is the vertical tidal range, about 8 metres for the Severn Estuary, and $g$ is the acceleration due to gravity. The average power generated would be found by dividing this amount of energy by 12 hours, the time between successive high tides. It remains to be seen whether tidal basin or lagoon based on this principle emerges as competitive.

Another thing that struck me doodling these things on the back of an envelope in the garden is that this sort of thing is what we should be getting physics students to think about. I’m quite ashamed to admit that we don’t…

## A Reith Lecture

Posted in Politics, The Universe and Stuff with tags , , on May 13, 2010 by telescoper

I’m a bit late getting around to blogging today, primarily because I spent the evening at a lecture by Martin Rees. Not just any lecture, but one of the annual series of Reith Lectures that he has been chosen to present this year. This event took place in the splendid Reardon Smith Theatre in the National Museum in Cardiff, and was preceded by a wine reception where we mingled amongst the relics of Welsh prehistory. The audience for the lecture  included academics, politicians, journalists and students and there was a lively question-and-answer session afterwards.

The Reith Lectures were inaugurated in 1948 by the BBC to mark the historic contribution made to public service broadcasting by Sir John (later Lord) Reith, the corporation’s first director-general. John Reith maintained that broadcasting should be a public service which enriches the intellectual and cultural life of the nation. It is in this spirit that the BBC each year invites a leading figure to deliver a series of lectures on radio. The aim is to advance public understanding and debate about significant issues of contemporary interest.

The very first Reith lecturer was the philosopher, Bertrand Russell who spoke on “Authority and the Individual”. Among his successors were Arnold Toynbee (The World and the West, 1952), Robert Oppenheimer (Science and the Common Understanding, 1953) and J.K. Galbraith (The New Industrial State, 1966). More recently, the Reith lectures have been delivered by the Chief Rabbi, Dr Jonathan Sacks (The Persistence of Faith, 1990) and Dr Steve Jones (The Language of the Genes, 1991). Since 2002, the Reith Lectures have been presented as was tonight’s,  by Sue Lawley.

I think this is the first time any of these lectures have been delivered in Cardiff. Martin Rees is, in fact, almost a Welshman himself ,  being born in Ludlow in Shropshire only about a mile the wrong side of the border; since being elevated to the peerage a few years ago, he is now known as Baron Rees of Ludlow. He is, of course, an immensely distinguished astrophysicist (he has been Astronomer Royal since 1995) but now has a broader portfolio of responsibility in the higher echelons of British science as President of the Royal Society.

As well as being an eminent scientist, Martin Rees is also a very fine public speaker, possessing an effortless gravitas that  any politician would die for.  He speaks with great clarity, thoughtfully and to the point, but with an economical use of language. He comes across as not only highly intelligent , which he undoubtedly is, but also deeply humane, another rare combination. Martin Rees was therefore an excellent choice to give the Reith Lectures. I had been looking forward to the evening for months after I got a phone call from Auntie Beeb asking me if I’d like to attend.

His lecture this evening wasn’t about astrophysics, and neither are the others in the series which has the pretty vague overall title Scientific Horizons. This lecture, the second of the series of four, was entitled Surviving the Century,and it concerned the role of science in identifying and possibly counteracting the threats facing humanity over the next few decades. He touched on climate change, renewable energy, and the possibility of nuclear or bio-terrorism. Although he spelled out the dangers in pretty stark terms he nevertheless claimed to be an optimist to the extent that he believed science could find solutions to the most pressing problems facing our planet, but I also sensed he was more of a pessimist as to whether the necessary measures could be implemented owing to socio-economic and political constraints. Science is vital to safeguarding the future of the planet, but it isn’t sufficient. People need to change the way they live their lives.

I won’t say any more about the lecture – or the interesting audience discussion that followed it – because you’ll be able to hear it yourselves on BBC Radio 4. The Lectures will be broadcast at 9am on Radio 4 starting on Tuesday 1st June (Lecture 1, called The Scientific Citizen). The lecture I attended tonight will be broadcast at the same time the following week (8th June). Lectures 3 and 4 will follow on 15th and 22nd June. Of course they will also be available as podcasts from the BBC website. If you want to be informed, enriched and challenged then I recommend you check them out.

## Astronomy Look-alikes, No. 19

Posted in Astronomy Lookalikes with tags , on April 6, 2010 by telescoper

While watching Carry on Screaming on the telly tonight, I was reminded of the similarity in visual appearance possessed by the villainous Orlando (played by Kenneth Williams) and the distinguished Astronomer Royal, President of the Royal Society and Master of Trinity College, Cambridge, and the Universe, Lord Rees of Ludlow (who, I hasten to add, is not at all villainous..).

Lord Martin of Rees

Sir Kenneth of Williams

## Cranks Anonymous

Posted in Biographical, Books, Talks and Reviews, The Universe and Stuff with tags , , , , on September 22, 2009 by telescoper

Sean Carroll, blogger-in-chief at Cosmic Variance, has ventured abroad from his palatial Californian residence and is currently slumming it in a little town called Oxford where he is attending a small conference in celebration of the 70th birthday of George Ellis. In fact he’s been posting regular live commentaries on the proceedings which I’ve been following with great interest. It looks an interesting and unusual meeting because it involves both physicists and philosophers and it is based around a series of debates on topics of current interest. See Sean’s posts here, here and here for expert summaries of the three days of the meeting.

Today’s dispatches included an account of George’s own talk which appears to have involved delivering a polemic against the multiverse, something he has been known to do from time to time. I posted something on it myself, in fact. I don’t think I’m as fundamentally opposed as Geroge to the idea that we might live in a bit of space-time that may belong to some sort of larger collection in which other bits have different properties, but it does bother me how many physicists talk about the multiverse as if it were an established fact. There certainly isn’t any observational evidence that this is true and the theoretical arguments usually advanced are far from rigorous.The multiverse certainly is  a fun thing to think about, I just don’t think it’s really needed.

There is one red herring that regularly floats into arguments about the multiverse, and that concerns testability. Different bits of the multiverse can’t be observed directly by an observer in a particular place, so it is often said that the idea isn’t testable. I don’t think that’s the right way to look at it. If there is a compelling physical theory that can account convincingly for a realised multiverse then that theory really should have other necessary consequences that are testable, otherwise there’s no point. Test the theory in some other way and you test whether the  multiverse emanating from it is sound too.

However, that fairly obvious statement isn’t really the point of this piece. As I was reading Sean’s blog post for today you could have knocked me down with a feather when I saw my name crop up:

Orthodoxy is based on the beliefs held by elites. Consider the story of Peter Coles, who tried to claim back in the 1990’s that the matter density was only 30% of the critical density. He was threatened by a cosmological bigwig, who told him he’d be regarded as a crank if he kept it up. On a related note, we have to admit that even scientists base beliefs on philosophical agendas and rationalize after the fact. That’s often what’s going on when scientists invoke “beauty” as a criterion.

George was actually talking about a paper we co-wrote for Nature in which we went through the different arguments that had been used to estimate the average density of matter in the Universe, tried to weigh up which were the more reliable, and came to the conclusion that the answer was in the range 20 to 40 percent of the critical density. There was a considerable theoretical prejudice at the time, especially from adherents of  inflation, that the density should be very close to the critical value, so we were running against the crowd to some extent. I remember we got quite a lot of press coverage at the time and I was invited to go on Radio 4 to talk about it, so it was an interesting period for me. Working with George was a tremendous experience too.

I won’t name the “bigwig” George referred to, although I will say it was a theorist; it’s more fun for those working in the field to guess for themselves! Opinions among other astronomers and physicists were divided. One prominent observational cosmologist was furious that we had criticized his work (which had yielded a high value of the density). On the other hand, Martin Rees (now “Lord” but then just plain “Sir”) said that he thought we were pushing at an open door and was surprised at the fuss.

Later on, in 1996, we expanded the article into a book in which we covered the ground more deeply but came to the same conclusion as before.  The book and the article it was based on are now both very dated because of the huge advances in observational cosmology over the last decade. However, the intervening years have shown that we were right in our assessment: the standard cosmology has about 30% of the critical density.

Of course there was one major thing we didn’t anticipate which was the discovery in the late 1990s of dark energy which, to be fair, had been suggested by others more prescient than us as early as 1990. You can’t win ‘em all.

So that’s the story of my emergence as a crank, a title to which I’ve tried my utmost to do justice since then. Actually, I would have liked to have had the chance to go to George’s meeting in Oxford, primarily to greet my ertswhile collaborator whom I haven’t seen for ages. But it was invitation-only. I can’t work out whether these days I’m too cranky or not cranky enough to get to go to such things. Looking at the reports of the talks, I rather think it could be the latter.

Now, anyone care to risk the libel laws and guess who Professor BigWig was?

## Budget Boost?

Posted in Science Politics with tags , , , , on April 19, 2009 by telescoper

This Wednesday (22nd April 2009) the Chancellor of the Exchequer, Alistair Darling, will deliver the UK government’s budget for this year. The background is of course the economic recession and the consequent collapse of our public finances. The government will have to borrow an estimated £175 billion over the next year, and it likely that taxes will eventually have to rise considerably to balance the books in the longer term.

Rumours are abounding about what will be in the budget and what won’t. According to today’s Observer, the centrepiece is likely to be a £50 billion scheme to revitalize the housing market.  If this is the case then I think it’s a mistake. Our economy has been run for too long on the basis of money raised from inflated property valuations, and we need to take this opportunity to change to a more sustainable way of running the country. Other schemes that may emerge include a £2 billion scheme to help unemployed young people which is a better idea, but much of it would probably be wasted in bureaucracy rather than doing real good.

My own attention will be focussed on whether there is anything in Alistair Darling’s speech that indicates some help for science, particularly fundamental science like physics and astronomy. In yesterday’s Guardian the Astronomer Royal and President of the Royal Society, Lord Martin Rees argued  for an injection of cash to stimulate science and innovation. About a month ago the BBC reported on efforts by Ministers to convince the treasury of the benefit of a £1 billion stimulus package for science along these lines. However, even if the powers that be listen to this argument (which is, in my view, unlikely), any increase in science funding would not necessarily be directed towards fundamental physics. I think if there isn’t anything for those of us working in astronomy in this budget, then we’re completely screwed.

I believe the funding crisis at the Science & Technology Facilities Council (STFC) was precipitated by a conscious government decision to move funds away from blue skies research and into more applied, technology driven areas.  The 2007 Comprehensive Spending Review was extremely tough on STFC but quite generous to some other agencies.  Moreover, within STFC itself there seems to be a shift from science-driven to technology-driven projects,  signalled by the cancellation of projects such as Clover to save a couple of million, and the allocation of funds to projects such as Moonlite which is devoid of any scientific interest and which could end up costing as much as £150 million over the next five years or so.

The true depth of the ongoing STFC crisis is only gradually becoming apparent. It was bad enough to start with, but has been exacerbated by the fall in value of sterling against the euro since 2007 which has meant that the cost of subscriptions to CERN, ESA and ESO have risen dramatically (by about 40%). These form such a large part of STFC’s expenditure – the CERN subscription alone is £70m out of a total budget of around £800m – that it cannot absorb the increased cost and it is now looking to make swingeing cuts on top of the 25% cut in research grants already implemented.

News emerged last week that STFC has abandoned plans to fund any R&D grants for ESA’s Cosmic Vision programme, and there are dark rumours circulating that it is considering cancelling all astronomy grants this year as well as clawing back money already given to universities in previous rounds. I hope these are not true, but I fear the worst.

Cuts on this scale would be devastating, demoralising, and I honestly think would destroy the United Kingdom as a place to do astronomy. They would also signal a complete breakdown of trust between scientists and the research council that is supposed to support them, if that hadn’t happened already.

Incidentally it is noticeable that STFC hasn’t bothered to report any of these matters publically through its website. Instead, the lead story on the STFC news page is about a visit by Prince Andrew to the Rutherford Appleton Lab. No sign yet, then, of the promised improvement in communication between the STFC Executive and its community.

The way I see it, the urgent issue is not whether we get a stimulus package , but whether we even get the bit of sticking plaster that is needed to  saves physics and astronomy from utter ruin. The cost would be a small fraction of the billions lavished on profligate bankers, but I’m not at all sure that the government either appreciates or cares about the scale of the problem.

Anyway, coincidentally, next week sees the Royal Astronomical Society’s National Astronomy Meeting (NAM), which is this year held jointly with the European Astronomical Society’s JENAM at the University of Hertfordshire. I won’t be going because it has unfortunately been organized in term time apparently because European astronomers refuse to attend meetings in the vacations, at least if they’re in places like Hatfield.  STFC representatives  have been invited; it remains to be seen what, if anything, they will have to say.


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# Graphs, groups, and more: celebrating Brian Alspach’s 80th and Dragan Marušič’s 65th birthdays

from 28 May 2018 to 1 June 2018
Koper
UTC timezone
Home > Timetable > Contribution details

# Characterization of generalized Petersen graphs that are Kronecker covers

• Matjaž KRNC

## Content

The family of generalized Petersen graphs $G(n,k)$, introduced by Coxeter et al. [4] and named by Mark Watkins (1969), is a family of cubic graphs formed by connecting the vertices of a regular polygon to the corresponding vertices of a star polygon. The Kronecker cover $\mathrm{KC}(G)$ of a simple undirected graph $G$ is a a special type of bipartite covering graph of $G$, isomorphic to the direct (tensor) product of $G$ and $K_{2}$. We characterize all the members of generalized Petersen graphs that are Kronecker covers, and describe the structure of their respective quotients. We observe that some of such quotients are again generalized Petersen graphs, and describe all such pairs.

format: plain


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# Matasano Crypto Challenges, Set 2

## Challenge 9 Implement PKCS#7 padding

As the challenge states, “A block cipher transforms a fixed-sized block (usually 8 or 16 bytes) of plaintext into ciphertext. But we almost never want to transform a single block; we encrypt irregularly-sized messages.”

The PKCS#7 padding scheme will append the number of bytes of padding to the end of the block. I use the pwntools pack() function to pack the number of padding bytes.

## Challenge 10 Implement CBC mode

This challenge has us implement CBC mode of block encryption.

In CBC mode, each block of plaintext is XORed with the previous ciphertext block before being encrypted. We need an IV for the first block of plaintext.

In set2_utils, I create a CBC Cipher class that takes in a cipher and an IV. The encrypt() function will split the plaintext into blocks (usually size 16), and then do the encryption:

The decryption takes each block of ciphertext, decrypts it, and XORs with the previous block of ciphertext to recover the plaintext.

I can verify I’ve done this correctly by decrypting 10.txt with a CBC Cipher with IV '\x00'*16.

## Challenge 11 An ECB/CBC detection oracle

This challenge asks us to detect whether we’ve encrypted a text with ECB or CBC, chosen at random.

Recall the properties of ECB vs CBC — ECB will take two identical plaintext blocks and produce two identical ciphertext blocks.

This is as simple as asking the oracle to encrypt a string that contains at least two consecutive blocks of identical characters. If the oracle chooses ECB, the ciphertext will have two adjacent identical blocks as well.

To ensure that we have at least two consecutive blocks of identical characters, we need to input at least 43 bytes. Why? Because the oracle pads the plaintext with 5-10 bytes, so we need to give some offset to ensure our identical plaintext blocks are properly aligned.

R = random_nfix

## Challenge 12 Byte-at-a-time ECB decryption (Simple)

I have an oracle that produces AES-128-ECB(your-string || unknown-string, random-key).

I can find unknown-string with this oracle. The idea is that,

1. First, I need to find the block size of the cipher.
2. Then, assuming I know it’s using ECB, I can find the flag byte-by-byte. How? Since I control my-string, I can ensure each time that the oracle encrypts 15 bytes that I know + one unknown byte. I can then create a table of all possible ciphertexts of the 15 known bytes + 1 unknown byte, and compare the ciphertext the oracle returns to the ciphertexts in my table.

To find the block size, I feed in incrementing offsets to the oracle, until the ciphertext length increases. The size of the increase will be block_size, because of the padding.

Next, I need to get an offset of 15 known bytes to feed into the oracle. The first offset is just 15 filler variables, all A’s.

And I create a table of all possible ciphertexts of A…U.

I then feed the block with the unknown byte to the oracle, padding with the same filler variables as my offset. Perform the table lookup.

At the next iteration, I decrease the number of filler variables by 1, since I have a known byte and want the next byte.

When I have 16 known bytes in this manner, I no longer need filler variables in my offset; I can just use the previous 15 known bytes as my offset. Note that my lookup table can be populated with ciphertexts of 16 flag bytes.

Since I have 16-byte ciphertexts in my lookup table, I need to first align, then get the index of, the 16-byte block-of-interest that I’ll look up in my table.

Stopping after I run out of bytes, I find the answer is

## Challenge 14 Byte-at-a-time ECB decryption with random prefix

In this challenge, a random-length prefix is added to the attacker-controlled string, AES-128-ECB(random-prefix || attacker-controlled || target-bytes, random-key). Thus, I need to know the length of the prefix to be able to conduct the same attack as Challenge 12.

Finding this length is not so hard.

1. First, I can find the block index of the last byte of the prefix with just two calls to the oracle.
2. Then, I find the offset of the last byte of the prefix within the last block.

Because this is ECB mode, the first different block between ciphertext1 and ciphertext2 will be the last block of the prefix.

Now that I know what block the last byte of the prefix is in, find_prefix_block_modulo_offset finds the offset of the last byte of the prefix. I want to align two blocks of identical plaintext, to encrypt to two blocks of identical ciphertext. The amount of offset I use to align will tell me the offset of the prefix.

, with offset_len always being 15 known bytes, as in Challenge 12.

The keys in my lookup table are no longer a fixed-length of blocksize — they include all known bytes now. Here, the len(offset) is not always 15, as in Challenge 12. Also, the length of my lookup table keys is not always blocksize; rather, the length of my keys increases with the number of known bytes. I could have implemented this more similarly to Challenge 12, which would have been less computationally expensive.

How do I produce ciphertext keys for my lookup table now? Here’s what I feed into the oracle:

Calling find_next_byte_with_prefix byte-by-byte, I find the same flag as before.

## Challenge 15 PKCS#7 padding validation

Here’s an easy one. We need only validate that the PKCS7 padding is correct.

I check if the last pad_length bytes of s are equal to the string of byte repeated pad_length times, as it should be in a properly padded string.

## Challenge 16 CBC bitflipping attacks

We have a function that takes user input and url-encodes special characters, and we want to exploit the properties of CBC to allow us to insert the string “;admin=true;” without the “;” and “=” being validated out.

Recall that CBC takes each block of ciphertext and XORs it with the next block of decrypted ciphertext to recover the plaintext.

If I can modify at least two consecutive blocks of ciphertext, I can make the second block decrypt to whatever I want.

I know that \x00 ^ \x59 = ; and \x00 ^ \x61 = '=' from ascii chart.

Done with set2!


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# Polynomial Substitution for only higher terms

I'm dealing with elliptic curves in SAGE. Say the elliptic curve I have is $y^2=x^3+x$ working over GF(43) and I have a polynomial expression in x and y: $$y^2 + xy + 1.$$ I want to be able to substitute $y^2=x^3+x$ to get $(x^3+x)+xy+1$. How do I do this?

edit retag close merge delete

n is missing in polyomial

( 2021-09-24 11:09:54 +0100 )edit

Sort by » oldest newest most voted

What you can do is choose a monomial ordering on the polynomial ring such that $y$ is the largest variable, so that in the multivariate division algorithm computing the remainder after division by $y^2 - (x^3 + x)$ will amount to substituting $y^2 = x^3 + x$:

sage: R.<x,y> = PolynomialRing(GF(43), order='invlex')
sage: (y^2 + x*y + 1).reduce([y^2 - (x^3 + x)])
x*y + x^3 + x + 1

Here you could also choose e.g. R.<y,x> = PolynomialRing(GF(43), order='lex').

more

This works in this example, however I am also dealing with expressions of the form $p(x)y^2+ q(x)$ where $p,q$ are polynomials of large degree. Implementing the above does the opposite; it takes the large powers of $x$ and replaces them with $y$

( 2021-10-05 18:08:03 +0100 )edit

If your polynomial is assumed to be zero, then it is also possible to entirely eliminate $y$ by computing resultant of your polynomial and $y^2-(x^3+x)$ with respect to $y$:

sage: R.<x,y> = PolynomialRing(GF(43))
sage: (y^2+x*y+1).resultant(y^2-(x^3+x),y)
x^6 - x^5 + 2*x^4 + x^3 + x^2 + 2*x + 1

more


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# The numerators in partial fraction decomposition involving repeated factors

So I'm confused about the partial fraction decomposition of fractions that involved repeated factors in the denominator. From what I've been taught, when you are breaking your fraction into partial fractions, the numerator of each fraction has a degree of at least one less than the denominator. However, from what I've seen in textbooks and online the partial fraction decomposition of fraction involving repeated factors in denominators often looks like this for linear factors. The degree of the numerator is only one less than the degree of the repeated factor which is $(x-a)$ rather than than the degree of the entire denominator

$\frac{P(x)}{(x - a)^n} = \frac{A}{(x - a)} + \frac{B}{(x -a)^2} + \frac{C}{(x -a)^3} + \dots$

But if I consider the general rule I was taught. Then shouldn't each partial fraction look like this.

$\frac{P(x)}{(x - a)^n} = \frac{A}{(x - a)} + \frac{Bx+C}{(x -a)^2} + \frac{Dx^2+Ex+F}{(x -a)^3} + \dots$

Why is this so? Is there a proof or explanation for this? because I can't seem to find one online. I've found one proof from Cambridge Mathematics 4 unit Year 12, but I don't understand the proof. So could you explain their proof as well, https://i.gyazo.com/ad52aefdf68e4728985498b48262a1ed.png (the proof) especially the part where it express $P(x)$ as a polynomial in $(x-a)$, what does that mean? and what are they doing when they say let $G(x) = P(x+a)= b_{n-1}x^{n-1} + b_{n-2}x^{n-2} + \dots$

With repeated factors your numerator should one less than the basic factor. Like you wrote:

$$\frac{P(x)}{(x - a)^n} = \frac{A}{(x - a)} + \frac{B}{(x -a)^2} + \frac{C}{(x -a)^3} + \dots$$

This is because the fraction $\frac{Dx^2+Ex+F}{(x-a)^3}$ can be rewritten as:

$$=\frac{Dx^2-2Dax+a^2+2Dax+Ex-Ea-2Da^2+F-a^2+2Da^2+Ea}{(x-a)^3}$$

$$=\frac{Dx^2-2Dax+a^2+2Dax+Ex-Ea-2Da^2+F-a^2+2Da^2+Ea}{(x-a)^3}$$

$$=\frac{D(x-a)^2}{(x-a)^3}+\frac{(2Da+E)(x-a)}{(x-a)^3}+\frac{Ea+F-a^2}{(x-a)^3}$$

$$=\frac{D}{x-a}+\frac{2Da+E}{(x-a)^2}+\frac{F-a^2+2Da^2+Ea}{(x-a)^3}$$

$$=\frac{A}{x-a}+\frac{B}{(x-a)^2}+\frac{C}{(x-a)^3}$$

This will work for any degree of polynomials involved and this final form is preferred as it straight forward to integrate.

Update in response to comment:

That is what your textbook is doing when it says let $G(x)=P(x+a)$ - i.e we take your polynomial and replace $x$ with $x+a$ and expand it. This new polynomial will have particular coefficients labelled $b_i$. I.e. you now have a polynomial like:

$$G(x) = P(x+a)= b_{n-1}x^{n-1} + b_{n-2}x^{n-2} + \dots + b_1x + b_0$$

If we then consider your original polynomial $P(x)$ you have:

$$\frac{P(x)}{(x-a)^n}=\frac{G(x-a)}{(x-a)^n}=\frac{b_{n-1}(x-a)^{n-1} + b_{n-2}(x-a)^{n-2} + \dots +b_1(x-a) +b_0}{(x-a)^n}$$

$$=\frac{b_{n-1}(x-a)^{n-1}}{(x-a)^n} + \frac{b_{n-2}(x-a)^{n-2}}{(x-a)^n} + \cdots + \frac{b_1(x-a)}{(x-a)^n} + \frac{b_0}{(x-a)^n}$$

$$=\frac{b_{n-1}}{(x-a)} + \frac{b_{n-2}}{(x-a)^2} + \cdots + \frac{b_1}{(x-a)^{n-1}} + \frac{b_0}{(x-a)^n}$$

• Ah I see. But how do you know it ill work for all degrees can you explain or proof that, maybe using a repeated factor with the degree n? – D.Pham Apr 22 '17 at 15:10


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# Is there any algorithm can solve the following equations efficiently?

$$x_1 + x_2 + \cdots +x_n = c_1$$ $$\frac{x_1^2}{2} + \frac{x_2^2}{2} + \cdots +\frac{x_n^2}{2} = c_2$$
$$\cdots$$ $$\frac{x_1^n}{n!} + \frac{x_2^n}{n!} + \cdots +\frac{x_n^n}{n!} = c_n$$

$c_1,\cdots,c_n$ are known constants. $x_1 \cdots x_n$ are the unknown variables to solve.

• What have you tried ? BTW, using the word algorithm does not make it Computer Science. Solving such equations seems straight math to me. – babou Nov 24 '14 at 1:30
• I am wondering some iteration techniques, but have no experience. Please let me known what kind of mathematics approaciate, I am not sure if numerical mathematics is suitable for math exchange. If necessary i will post it there. – user26143 Nov 24 '14 at 1:34
• Cross-posted on math.se: math.stackexchange.com/questions/1036219/…. – Yuval Filmus Nov 24 '14 at 6:33

One way, which is numerically probably a bad idea, is to use Newton's identities. They give the value of the elementary symmetric functions given the power sums, and from the elementary symmetric functions you can construct a polynomial whose roots are the answers to your problem. Let $$p_k = k! c_k$$, and let $$e_k$$ be the sum of all $$\binom{n}{k}$$ products of $$k$$ variables out of $$x_1,\ldots,x_n$$. Newton's identities state that $$ke_k = (-1)^{k-1} p_k + \sum_{i=1}^{k-1} (-1)^{i-1} e_{k-i} p_i.$$ Using these identities, you can compute $$e_1,\ldots,e_n$$. Now consider the polynomial $$P = x^n + \sum_{i=1}^n (-1)^i e_i x^{n-i}.$$ The unique solution to your equations are the $$n$$ roots of $$P$$.
As an example, let's do the case $$n = 2$$. We have $$e_1 = p_1$$ and $$2e_2 = -p_2 + e_1 p_1 = p_1^2 - p_2.$$ The polynomial $$P$$ is therefore $$x^2 - p_1 x + \frac{p_1^2-p_2}{2}.$$
For $$n=3$$ we calculate additionally $$3e_3 = p_3 - e_1 p_2 + e_2 p_1 = p_3 - p_1 p_2 + \frac{p_1^2-p_2}{2} p_1 = \frac{2p_3 - 3p_1p_2 + p_1^3}{2}.$$ The polynomial $$P$$ is therefore $$x^3 - p_1 x^2 + \frac{p_1^2-p_2}{2}x - \frac{2p_3 - 3p_1p_2 + p_1^3}{6}.$$
• Sorry for late response, excuse me, why the roots of $P$ give the solutions to the eqns above, i.e. $\sum_i^n x_i^k /k!= c_k, k=1,2,\cdots, n$? – user26143 Jan 10 '15 at 13:54
• If we write $P = (x-x_1)(x-x_2)\cdots(x-x_n)$ then we can read off the coefficients of $x^m$ to deduce that $e_1=x_1+\cdots+x_n$, $e_2=x_1x_2+\cdots+x_{n-1}x_n$, and so on. Therefore the roots $x_1,\ldots,x_n$ of $P$ satisfy these equations on $e_1,\ldots,e_n$. We conclude that the roots of $P$ are solutions of the system of equations $e_1=x_1+\cdots+x_n$, $e_2=x_1x_2+\cdots+x_{n-1}x_n$, ..., $e_n = x_1x_2\cdots x_n$. A priori there could be other solutions, but this is not the case here. – Yuval Filmus Jan 10 '15 at 17:25


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Network of Networks

Speaker

Jianxi Gao, Shanghai Jiao Tong University and Boston University

Time

2012.02.22 10:30-11:30

Venue

520 Pao Yue-Kong Library

Abstract

Complex networks appear in almost every aspect of science and technology. Nearly all network results have been obtained by analyzing isolated networks, but many real-world networks do in fact interact with and depend on other networks. Very recently an analytical framework for studying the percolation properties of interacting networks has been developed. Here we review the analytical framework and the results for connectivity properties for a network of networks’’ (NON) formed by interdependent random networks. The percolation properties of a network of networks differ greatly from those of isolated networks. In particular, networks with broad degree distributions, such as scale free networks, that are robust when analyzed as isolated networks, become vulnerable in a NON. Moreover, in a NON, cascading failures appear due to failure of dependent nodes in other networks. When there is strong interdependent coupling between the networks, the percolation transition is discontinuous (a first-order transition), unlike the well-known continuous second-order transition in single isolated networks. These results will likely be useful in a wide range of disciplines, since no networks is completely independent of other networks.

Brief Introducation to Jianxi Gao

Jianxi Gao is a PH.D candidate student in Automation department in SJTU from 2008, and is a research fellow in Physics department in Boston University. He works with H. E. Stanley from Boston University and Shlomo Havlin from Bar-Ilan University. Jianxi Gao studies the robustness of complex network and collaborative control theory. He has focused primarily on the percolation on network of interdependent networks and the optimum synchronization on self-propelled agent systems. He develops a general analytical framework for studying percolation of n interdependent networks and illustrate the analytical solutions for many distinct examples. Jianxi Gao proposes a system of iterative equations somewhat analogous to Kirchhoff equations for the resistor network in the field of percolation on interdependent networks, which shows his intelligent in finding the new percolation law in complex networks. He has one paper published in Nature Physics (IF>18), one paper published in Physical Review Letters (IF>7), and 4 papers in Physical Review E (IF>2). He was invited to several conferences such as “EU-China workshop on Complexity Science” in Switzerland (2011) and “ APS March Meeting” in Boston (2012). He is the referee of several journals, such as: Europhysics Letter, New Journal of Physics, Physica A, and International Journal of Control etc..


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## The Why of Juvix: Ingredients & Architecture

*Image source: Aditya via Wikimedia Commons, CC-BY-SA 3.0

In addition to protocol development, in the past several months METASTATE has embarked upon a new project: research & development of a novel smart contract language, Juvix.

Juvix is designed to address the problems that we have experienced while trying to write & deploy decentralised applications and that we observe in the ecosystem at large: the difficulty of effective verification, the ceiling of compositional complexity, the illegibility of execution costs, and the lock-in to particular backends.

In order to do so, Juvix draws upon and aims to productionise a deep reservoir of prior academic research in programming language design & type theory which we believe has a high degree of applicability to these problems.

There should be a substantial bar to meet before electing to write a new language. After investigating many simpler approaches and developing distributed ledgers & smart contracts ourselves, we've decided that this bar, for the use-case of smart contracts on public ledgers, is met — there are many unique, fundamentally difficult problems which can be convincingly solved at the language level, but only by designing & engineering a language and compiler stack from scratch.

This post, the second part of a two-part series, enumerates the various theoretical ingredients in Juvix that we think will enable it to meet these challenges, describes the current state of specification and implementation, and provides a list of further resources & instructions should you wish to learn more.

## Key ingredients

• How does Juvix aim to realise these goals?
• Which key ingredients are drawn from the theoretical literature to provide the requisite capabilities?

### Dependent types

The first and most essential ingredient is that of dependent types: a typesystem expressive enough to allow terms written in Juvix to express properties about the computation behaviour of other terms.

For example (not yet in Juvix), the fairness of Tendermint proposer election has been partially verified in Idris, which has the same kind of dependent typesystem:

fairlyProportional :
(idA : ProposerId) -> (idB : ProposerId) ->
(wA : ProposerWeight) -> (wB : ProposerWeight) ->
(pA : ProposerPriority) -> (pB: ProposerPriority) ->
(n : Nat) ->
(wA >= 0 = True) -> (wB >= 0 = True) ->
(abs(pA - pB) <= (wA + wB) = True) ->
((count idA (snd (incrementElectMany n ((idA, wA, pA), (idB, wB, pB)))))
>= ((n * (wA / (wA + wB))) - 1) = True,
(count idA (snd (incrementElectMany n ((idA, wA, pA), (idB, wB, pB)))))
<= ((n * (wA / (wA + wB))) + 1) = True)

incrementElectMany, the proposer-election function, is the term whose behaviour we want to reason about, and inhabiting the type of fairlyProportional with a valid term proves that it behaves in the fashion we intend.

In English, this proof could be read as "a validator, in a sequence of proposer elections where no other power changes take place, proposes no fewer blocks than the total blocks in the epoch multiplied by its fraction of stake less one, and proposes no more blocks than the total blocks in the epoch multiplied by its fraction of stake plus one".

Dependent types as implemented in Juvix satisfy all three dimensions of the lambda cube: terms can depend on types, types can depend on types, and types can depend on terms. In accordance with the Curry-Howard correspondence, types in Juvix can express propositions, or properties of terms, and programs which inhabit those types serve as proofs of the propositions.

There are several different formal verification techniques, and they each have different trade-offs. Dependent types are quite expressive, since the conversion rule is just beta reduction, and allow for property verification in the same language, which removes the need for an error-prone translation between a specification and implementation. They allow for progressive verification — contracts can initially be written "unverified", and proofs can be added over time as the logic of an application-in-development solidifies. We have found that dependent types are elegant and become reasonably intuitive with practice, but more tooling support will undoubtedly be necessary to smooth the learning curve and ease some of the proof bureaucracy, which can be onerous at times.

### Usage quantisation

The second theoretical ingredient Juvix brings to the table, used both to allow developers to
reason even more precisely about the behaviour of their programs and to ensure that using dependent types for formal verification incurs no runtime cost, is usage quantisation, drawn from the Quantitative Type Theory paper and extended by us.

Typing judgements in Juvix are annotated with precise usages:

x_1 \overset{ρ_1}{:} S_1, \dots , x_n \overset{ρ_n}{:} S_n \vdash\ M \overset{σ}{:} T

These usage annotations indicate how many times the subject of the judgement can be "computed with", or used. This is a form of linear types, but much more expressive: usage annotations are numerical and can depend on terms just like types can. Usage annotations are checked at compile-time for correctness. Since proofs of properties need not be computed over and can be constructed in the zero-usage fragment, Juvix can then erase proofs at compile time after checking them, ensuring that property verification creates no runtime overhead. Detailed usage accounting also allows Juvix to avoid garbage collection, optimise stack usage, and safely modify values in-place, in various situations depending on the types and the granularity of usage annotations provided.

### Whole-program optimisation & efficient execution

Juvix takes advantage of the unique trade-offs of the smart contract use case — small source-code sizes and infrequent deployment, which allow for longer compile times and wider optimisation searches — to enable several kinds of whole-program optimisations. Whole-program optimisation encapsulates a collection of techniques & optimisation passes rather than any specific one, drawn from multiple inspirations including GRIN, STOKE and Morte, including compile-time evaluation, inlining, fusion, case simplification, common sub-expression elimination, various dead code elimination checks, and rewriting provably equivalent terms.

For certain backends, Juvix can evaluate lambda calculus terms using the abstract model of interaction nets, as outlined by Lamping, Asperti, and Guerrini and inspired by prior work.

Interaction nets minimise the number of beta reductions by sharing evaluations, perform runtime fusion, support automatic parallelism, and handle large closures efficiently. They come with some overhead, but this can be mitigated with bespoke encoding — generating custom rewrite rules at compile-time — for small, performance-critical functions.

### Resource verification

Leveraging the dependent typechecker already in place, Juvix uses techniques from a 2008 POPL paper to check computational resource consumption of terms without executing them by encapsulating computations in a cost monad, where costs can depend on terms in the usual dependent fashion. This requires some annotations by the developer, but tooling can be improved over time, and cost bounds can be added in a progressive fashion just like proofs of properties.

In the future, if ledgers elect to integrate a limited version of the typechecker directly into their state machines, this resource verification mechanism can be used to eliminate runtime gas metering entirely: each contract would have a cost function, computable depending on arguments to the call, and the gas (exactly computable) could be deducted at the beginning of the function prior to execution. This resource verification model can also easily incorporate different denominations of resources (compute, storage, etc.) by tracking several costs at once.

### Backend parameterisation

Juvix parameterises the type theory & core language over a set of primitive data types and primitive values, which can include native data types such as strings, integers, or sets, and native functions such as addition, subtraction, string concatenation, set membership, etc.

The language & typechecker can then be instantiated over a particular backend which provides concrete sets of primitives and a primitive type-checking relation. Backends are integrated into the language itself as special values, so that the standard library can nicely abstract primitives with different underlying implementations into common typeclasses for numbers, strings, etc. and resolve operators to the correct backend at compile time as instructed by the developer.

This will allow contract authors to write a single contract in the frontend language and target different backends with compiler switches or compile to more efficient upgraded versions of a virtual machine later on, with minimal or no code changes, and also eases the path towards integrated typechecking of applications with components on multiple backends (perhaps multiple ledgers and a zkSNARK circuit, for example).

## Current state of affairs

Engineering a language & compiler stack from scratch is tricky business, and undoubtedly we'll make mistakes and change our mind along the way. In order to avoid following too many false leads, we've taken a specification-first approach with Juvix, trying first to reason through all the thorny problems and write down exactly what we plan to do and how the language will work. Specification-wise, Juvix currently has a detailed language reference, the start of a type theory formalisation in Agda, and detailed plans for future work on resource verification, whole-program optimisation, and other features. Implementation-wise, Juvix has a typechecker for the first version of the type theory, a core language and several intermediate forms, erasure of zero-usage terms, a backend parameterisation system, and the start of two backends: one for Michelson and one for LLVM / WASM (through LLVM).

Juvix is not yet ready for end-user or developer use. We expect further work on the frontend language and an initial public release in Q1/Q2 2020. Contributions & critiques alike are welcome anytime.


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# Angle Bisector

An angle bisector is a line or ray that divides an angle into two congruent angles.

In the figure, the ray $\stackrel{\to }{KM}$ bisects the angle $\angle JKL$.

The angles $\angle JKM$ and $\angle LKM$ are congruent.

So, $m\angle JKM=m\angle LKM$.

Note that any point on the angle bisector is equidistant from the two sides of the angle. Some textbooks call this Angle Bisector Theorem, but this name is usually used for another theorem about angle bisectors in a triangle.

Example :

In the figure, $\stackrel{\to }{BD}$ is an angle bisector. Find the measure of $\angle ABC$.

Since $\stackrel{\to }{BD}$ is an angle bisector,

$m\angle CBD=m\angle ABD$

Therefore,

$m\angle CBD=65°$

$m\angle ABC=m\angle ABD+m\angle CBD$.
$\begin{array}{l}m\angle ABC=65°+65°\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=130°\end{array}$


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# Projectile motion with air resistance proportional to velocity

1. Apr 25, 2010

### dykkms

1. The problem statement, all variables and given/known data
The projectile with initial velocity $$v_0$$ is moving in environment with air resistance, which is linear to projectile's velocity. The task is to prove, that horizontal distance is maximal, if the elevation angle ($$\alpha$$) and the angle of trajectory tangent in place of impact are complementary.

2. Relevant equations
$$F_x = - k.v_x$$
$$F_y = - m.g - k.v_y$$
$$v_x = v.\cos(\varphi)$$
$$v_y = v.\sin(\varphi)$$
$$v_{0x} = v_0.\cos(\alpha)$$
$$v_{0y} = v_0.\sin(\alpha)$$

3. The attempt at a solution
equations in $$x$$ direction:
$$a_x = \frac{F_x}{m} = - \frac{k}{m} v_x$$
$$a_x = \frac{\mathrm{d}v_x}{\mathrm{d}t} = - \frac{k}{m} v_x$$
$$\frac{\mathrm{d}v_x}{v_x} = - \frac{k}{m}\mathrm{d}t \Rightarrow v_x = v_{0x}.e^{-\frac{k}{m}t}$$
after integration:
$$x = -\frac{m}{k} v_{0x} . e^{-\frac{k}{m}t} + \frac{m}{k} v_{0x}$$

equations in $$y$$ direction:
$$a_y = \frac{F_y}{m} = - g - \frac{k}{m} v_y$$
$$a_y = \frac{\mathrm{d}v_y}{\mathrm{d}t} = - g - \frac{k}{m} v_y$$
$$\frac{\mathrm{d}v_y}{- g - \frac{k}{m} v_y} = \mathrm{d}t$$
after integrations and mathematical expressing:
$$v_y = \frac{m}{k}\left(g + \frac{k}{m}v_{0y}\right)e^{-\frac{k}{m}t} - g\frac{m}{k}$$
$$y = - \frac{m^2}{k^2}\left(g+ \frac{k}{m}v_{0y}\right).\left(e^{-\frac{k}{m}t} - 1\right) - g\frac{m}{k}t$$

And here is my problem:
I would like to express from $$y$$ equation the time of the impact $$t_i$$. I know about this moment, that $$y = 0$$. If I am able to write explicit function for $$t_i(\alpha)$$, then I would pass it to $$x$$ function. Afterwards I would differentiate this $$x$$ function according to $$\alpha$$. Then I would find the ideal $$\alpha$$, which I would use to find $$t_i$$ which I would finally use to find impact angle.
Unfortunatelly, I can't see the way to find explicit expression of $$t_i$$ from $$y = 0$$ function.

2. Apr 25, 2010

### gabbagabbahey

Basically, you want to maximize $x(t_i)-x(0)$ subject to the constraint $y(t_i)=0$...that seems like a prime candidate for the mathod of Lagrange multipliers to me

3. Apr 26, 2010

### dykkms

Basically, you say (and I think you are right...), there are two other possibilities:
• from function $$y = 0$$ not to express $$t_i(\alpha)$$, but $$\alpha(t_i)$$ and to use this in function $$x(t_i)$$, which I should maximize
• to use method of Lagrange multipliers

First mentioned method brings me to situation, that I am not able to express $$t_i$$, which would maximize $$x(t_i)$$ function, anyway.
Second, I was thinking about method of Lagrange multipliers before, but I tried to avoid it, because I am trying to help with this homework to my friend, who is first-year student at university (industrial management) and he doesn't know the method yet... But I think, I will have to.

4. Apr 26, 2010

### gabbagabbahey

Hmmm.... if this is supposed to be a 1st year level problem, there may be a simpler method...perhaps energy conservation would make things easier.


text:
# How do you find the vertex of a parabola y=2x^2 -12x+23?

Jul 8, 2015

The easiest way is (probably) to convert the given equation into vertex form.
The vertex is at $\left(3 , 5\right)$

#### Explanation:

Vertex form of a parabola is
$\textcolor{w h i t e}{\text{XXXX}}$$y = m {\left(x - a\right)}^{2} + b$ with its vertex as $\left(a , b\right)$

$y = 2 {x}^{2} - 12 x + 23$
$\textcolor{w h i t e}{\text{XXXX}}$extract the $m$
$y = 2 \left({x}^{2} - 6 x\right) + 23$
$\textcolor{w h i t e}{\text{XXXX}}$complete the square
$y = 2 \left({x}^{2} - 6 x + 9\right) + 23 - 2 \left(9\right)$
$\textcolor{w h i t e}{\text{XXXX}}$simplify into vertex form
$y = 2 {\left(x - 3\right)}^{2} + 5$

Since this is in vertex form, we can read the vertex directly from the equation as $\left(3 , 5\right)$


text:
# The Interaction of HCl and Crystalline Ice Clusters at Cryogenic Temperatures: Evidence of the Molecular Complex

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/29621

Files Size Format View
1995-RE-04.jpg 61.32Kb JPEG image

 Title: The Interaction of HCl and Crystalline Ice Clusters at Cryogenic Temperatures: Evidence of the Molecular Complex Creators: Delzeit, Lance; Devlin, J. Paul Issue Date: 1995 Publisher: Ohio State University Abstract: The addition of HCl to the surface of crystalline ice clusters at cryogenic temperatures produces spectroscopic changes which can be related to the formation of the hydronium ion and the HCl acting as a molecular adsorbate. The hydronium ion is identified from its IR active modes below $2200cm^{-1}$. Evidence for the HCl acting as a molecular adsorbate is its effect on the dangling-hydrogen (d-H) mode and the appearance of a $2500cm^{-1}$ band. Analogous spectra for with the $HBr-H_{2}O$ and the $DCl-D_{2}O$ $systems^{1}$ show two bands relative to the one band in the $2500cm^{-1}$ region due to the $HCl-H_{2}O$ interaction. The single band in the HCl system is attributed to the near overlap of the bands due to the molecular and ionic interaction of the HCl and $H_{2}O$. Reference to matrix $isolation^{2,3}$ and computational $work^{4}$ will show the plausibility of the presence of the molecular complex. Shifting of the d-H band is a common effect for molecularly adsorbed species on the surface of $ice.^{5}$ The shifting of the d-H to a position yet unobserved for common adsorbates gives strong evidence of the HCl acting as a molecular adsorbate. Description: 1. Lance Delzeit, Brad Rowland, and J. Paul Devlin; J. Phys. Chem. 97 10312 (1993). 2. B. S. Ault and G. C. Pimentel; J. Phys. Chem. 77 57 (1973). 3. A. Schriver, et. al.; J. Phys. Chem. 87 2095 (1977). 4. G. Kroes and D. C. Clary; J. Phys. Chem. 96 7079 (1992). 5. Brad Rowland, Mark Fisher, and J. Paul Devlin; J. Chem. Phys. 95 1378 (1991). Author Institution: Oklahoma State University, Stillwater, OK 74078 URI: http://hdl.handle.net/1811/29621 Other Identifiers: 1995-RE-04


text:
• Create Account

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### #ActualAlessio1989

Posted 02 March 2013 - 06:01 PM

And another one thing - now I must use Windows 8 SDK to use new D2D, but here is no D3DX.h, so I must find new analogs for old functions such as D3DXMatrixPerspectiveFovLH and D3DXCreateTextureFromFile

Does someone now which functions I can replace these?

By the way. I have tested you code and it work great after few small corrections regarding CreateWindow function. Thanks a lot for this, but I still dont know whixh functions to use instead of D3DX functions. Could you please help me with that question?

Since D3DX is retired you (must) can use DirectX Tool kit, DirectX Tex and DirectXMath respectively for graphics, texture and math function ad structures.

http://directxtk.codeplex.com/

http://directxtex.codeplex.com/

http://msdn.microsoft.com/en-us/library/windows/desktop/ee418730.aspx

If you still want to use D3DX you must install the June 2010 DXSDK and manually configure the vs project: http://msdn.microsoft.com/en-us/library/windows/desktop/ee663275.aspx

### #2Alessio1989

Posted 02 March 2013 - 06:01 PM

And another one thing - now I must use Windows 8 SDK to use new D2D, but here is no D3DX.h, so I must find new analogs for old functions such as D3DXMatrixPerspectiveFovLH and D3DXCreateTextureFromFile

Does someone now which functions I can replace these?

By the way. I have tested you code and it work great after few small corrections regarding CreateWindow function. Thanks a lot for this, but I still dont know whixh functions to use instead of D3DX functions. Could you please help me with that question?

Since D3DX is retired you (must) can use DirectX Tool kit, DirectX Tex and DirectXMath respectively for graphics, texture and math function ad structures.

http://directxtk.codeplex.com/

http://directxtex.codeplex.com/

http://msdn.microsoft.com/en-us/library/windows/desktop/ee418730.aspx

If you still want to use D3DX you must install the June 2010 DXSDK too and manually configure the vs project: http://msdn.microsoft.com/en-us/library/windows/desktop/ee663275.aspx

### #1Alessio1989

Posted 02 March 2013 - 05:59 PM

And another one thing - now I must use Windows 8 SDK to use new D2D, but here is no D3DX.h, so I must find new analogs for old functions such as D3DXMatrixPerspectiveFovLH and D3DXCreateTextureFromFile

Does someone now which functions I can replace these?

By the way. I have tested you code and it work great after few small corrections regarding CreateWindow function. Thanks a lot for this, but I still don`t know whixh functions to use instead of D3DX functions. Could you please help me with that question?

Since D3DX is dead you can (must) use DirectX Tool kit, DirectX Tex and DirectXMath respectively for graphics, texture and math function ad structures.

http://directxtk.codeplex.com/

http://directxtex.codeplex.com/

If you still want to use D3DX you must install the June 2010 DXSDK too and manually configure the vs project: http://msdn.microsoft.com/en-us/library/windows/desktop/ee663275.aspx

PARTNERS


text:
## Optimal Order Results for a Class of Regularizazion Methodes Using Unbounded Operators

• A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an priori parameter choice strategy, it is shown that the method yields optimal order. Error estimates have also been obtained under stronger assumptions on the the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper includes, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also recent results of Tautenhahn (1996) in the setting Hilbert scales.

$Rev: 13581$


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# Comparing escape velocities on Earth and Mars

1. Apr 2, 2005

### six789

The problem is..
the mass of MArs is 0.107 times Earth's mass and its raduis is 0.532 times Earth's raduis. how does the escape velocity on Mars compare to the velocity on Earth?

i did this..
Vof escape = square root of 2G M(mars)/r(Mars)
Vof escape is proportional to the square root of M(mars)/r(Mars)
Vof Mars/Vof Earth = square root of 0.107x M(earth)/0.532x r(earth)
square root of M(earth)/r(earth)
Vof Mars/Vof Earth = square root of 0.107x M(earth)/0.532x r(earth) *square root of r(earth)/M(earth)
Vof Mars = 0.448*Vof Earth

but the answer in my book is .488 x V of Eath... im not sure if im right or the book is right

Last edited: Apr 2, 2005
2. Apr 2, 2005

### 1123581321

What? Was that a question? I can read minds, but not over the internet.

Fibonacci

3. Apr 2, 2005

### six789

The problem is..
the mass of MArs is 0.107 times Earth's mass and its raduis is 0.532 times Earth's raduis. how does the escape velocity on Mars compare to the velocity on Earth?

i did this..
Vof escape = square root of 2G M(mars)/r(Mars)
Vof escape is proportional to the square root of M(mars)/r(Mars)
Vof Mars/Vof Earth = square root of 0.107x M(earth)/0.532x r(earth)
square root of M(earth)/r(earth)
Vof Mars/Vof Earth = square root of 0.107x M(earth)/0.532x r(earth) *square root of r(earth)/M(earth)
Vof Mars = 0.448*Vof Earth

but the answer in my book is .488 x V of Eath... im not sure if im right or the book is right

4. Apr 2, 2005

### Berislav

Your answer does indeed seem to be correct. The answer in the book is most likely a typo.

5. Apr 2, 2005

### six789

Mars

so u mean im right? check the equation if it is right??

6. Apr 2, 2005

### Berislav

$$\frac{v_m}{v_e}={\sqrt{\frac{M_m R_e}{M_e R_m}}={\sqrt{\frac{0,107 M_e R_e}{0,532 M_e R_e}}$$

This does indeed equal 0,448.

Last edited: Apr 2, 2005
7. Apr 2, 2005

### whozum

$$v_{esc} = \sqrt{\frac{2GM}{R}}$$

Me = 5.98 x 10^24
Re = 6.3 x 10^6m
Mm = 6.4 x 10^23kg
Rm = 3.4 x 10^6m

Earth: $$v_{esc} = \sqrt{\frac{2(6.67 x 10^{-11})(5.98 x 10^{24})}{6.3 x 10^6}} = 11252.73m/s$$

Mars: $$v_{esc} = \sqrt{\frac{2(6.67 x 10^{-11})(6.4 x 10^{23})}{3.4 x 10^6}} = 5011.047m/s$$

$$\frac{v_{mars}}{v_{earth}} = \frac{5011.047}{11252.73} = 0.446$$

Its a matter of significant figures.

8. Apr 2, 2005

### six789

thanks

ok... thanks a lot for the big help...

another problem is... a rocket is launched vertically from earth's surface with a velocity of3.4km/s. How high it go a)from Earth's centre and b) from Earth's surface.
a)r = 2GM
V^2
= 2(6.673x10^-11N m^2/kg^2) (5.98 x10^24kg)
(3400m/s)^2
r=7.0x107m

i dont know what to do with b), i am thinking to add the raduis of the earth, but my answer is wrong... the answer should be 650km.

9. Apr 2, 2005

### six789

whoozum

but the correct answer in my book is 0.488

10. Apr 2, 2005

### six789

ok guys...

another problem is... a rocket is launched vertically from earth's surface with a velocity of3.4km/s. How high it go a)from Earth's centre and b) from Earth's surface.
a)r = 2GM
V^2
= 2(6.673x10^-11N m^2/kg^2) (5.98 x10^24kg)
(3400m/s)^2
r=7.0x107m

i dont know what to do with b), i am thinking to add the raduis of the earth, but my answer is wrong... the answer should be 650km.

11. Apr 2, 2005

### whozum

If its 7 x 10^7 from earth's center, thenthe rocket is closer to the surface than the center.

12. Apr 2, 2005

### six789

so what will i do then if it is closer to the earth's surface, the information is so limited?

13. Apr 2, 2005

### six789

14. Apr 2, 2005

### whozum

Center ----------------- Surface ---------------------------Rocket

If you know the distance from the rocet to the center, and from the surface to the center. How do you find the distance from the rocket to the sufrace?

15. Apr 2, 2005

### six789

ull just have to subtract the distance to the answer.... but i dont know the distance

16. Apr 2, 2005

### six789

whozum, i still dont get it.. wat will i do??

Last edited: Apr 2, 2005
17. Apr 2, 2005

### whozum

You can look up the radius of the earth

It is 6.3 x 10^6 m

18. Apr 2, 2005

### six789

rocket

i know it is... will i do this?

change of raduis = r of rocket from earth - r of earth
7x10^7 - 6.38x10^6
63620000m

my answer is 63620000m... but in the book, the answer is 650km

Last edited: Apr 2, 2005
19. Apr 2, 2005

### whozum

I don't think were using the right equation, either that or the answer in the book is incorrect.

20. Apr 3, 2005

### VietDao29

I dunno how come you use the equation to find the escape velocity here:
$$v_{esc} = \sqrt{\frac{2GM}{R}}$$
Draw a graph of:
$$F_{gra} = \frac{GmM}{r^{2}}$$
The integration of F with respect to r is the work done by the Earth.
So
$$\frac{mv^{2}}{2} = -W_{Earth} = \int{\frac{GmM}{r^{2}}dr}$$
With x-axis is r and y-axis is F
Now you have
$$\frac{v^{2}}{2} = \int^{H}_{r_{0}}{\frac{GM}{r^{2}}dr}$$
$$=-\frac{GM}{H} + \frac{GM}{r_{0}}$$
H is the height from the earth center the setellite will go.
M is Earth's mass (6. 10^24 kg)
m is satellite's mass.
ro is the radius of the Earth (6400 km = 6400000 m)
v is the satellite's initial valocity (3400m/s)