Does a base have to be an integer? If not, pi in base(pi) is 10 exactly.
Yep. Writing a real number x in base b (not necessarily an integer) essentially means writing
x = \sum_{n=0}{\infty} a_{n}/b^{n}
for suitable integers a_n. (`Essentially' here means that there may be some non-uniqueness, for example 1.000...=0.999....)
To prove that the real number x is normal to base b, one looks at the ergodic properties the transformation defined on the unit interval [0,1] by
T_b: x \mapsto bx \mod 1
(that is, the function that takes x, multiplies it by b and then takes the fractional part). When
b is an integer, this transformation preserves Lebesgue measure (length) and is ergodic. Birkhoff's Ergodic Theorem then says that Lebesgue almost every point x is such that the sequence \{ T_b^{n}(x) \mid n = 0,1, 2,...} is uniformly distributed mod 1, and such a point x is easily seen to be normal in base b. Now there are only countably many different integer bases and the countable intersection of sets of full Lebesgue measure also has full Lebesgue measure. Hence (Lebesgue) almost every point is normal in every (integer) base.
What about the case when b is not an integer? Well, the map T_b can still be defined, but it no longer preserves Lebesgue measure. It does, however, preserve a measure that is equivalent (in the sense of having the same sets of measure zero) to Lebesgue measure, and it is ergodic with respect to this measure. One can again conclude that for a given base b (not necessaily integral), Lebesgue almost every point x is normal to base b.
However, one cannot then go on to say that almost every point is normal to every (real, non-zero) base b. This is because the intersection of uncountably many sets each of full measure can (and typically will) have zero measure. However, the following is true: take any sequence b_1, b_2,..., then almost every point is normal in each base b_n.
Of course, the set of full measure obtained will depend on the sequence b_1, b_2,....
There's a whole theory based around the ergodic properties of the map T_b for a non-integer b. Such transformations are called \beta-transformations, and there's a vast literature; see Parry, W.
On the $\beta $-expansions of real numbers.
Acta Math. Acad. Sci. Hungar. 11 1960 401--416,
for starters.
There is actually a theory that Pi is one of those special numbers (whose name escapes me -
making this post a little less useful) which is random in all bases.
Such numbers are called normal.
Specifically, a real number x is normal in base b if the frequency with which the digit r occurs in the base b expansion of x is equal to 1/b, for each r \in {0,1,...,b-1}. A real number is normal if it is normal in each (integer) base b.
It's easy to construct (usually artificial) numbers that are normal to any given base. One can also sure (easily - using the Birkhoff Ergodic Theorem) that Lebesgue almost all real numbers are normal. However, there are no known examples of such a number. Thus `most' real numbers are normal (in the sense of `pick a real number at random then with probability one it will be normal'), but no explicit examples are known.
This is all straightforward ergodic theory (actually, `uniform distribution mod 1'), and there's an excellent account in Kuipers & Neidreitter's book, which is sadly out of print.
> perhaps someone should write to those scientists
> who do publish in the publications not meeting
> the demands, to make them aware of the issues.
> I don't think this boycott has the high profile
> in the scientific community we'd like.
Rob Kirby has a list comparing mathematics journal prices. (Although some of the data is somewhat ('97) old now.) In particular, there is an excellent comparison of the (estimated) costs involved of producing the Pacific Journal of Mathematics (published by UCI) and Inventiones Mathematicae (published by Springer-Verlag).
I think most people in the mathematics community are aware of the issues here. However, certainly in the UK, there are far more pressing factors that determine which journals one submits to. Firstly, there's the obvious implications for one's career and promotion prospects: a paper in Annals of Mathematics or Inventiones looks better on a C.V. than a paper in some random journal no-one's heard of. Secondly, in the UK each member of staff in every university department is assessed for the quality of its research output (this is the dreaded Research Assessment Exercise). The ranking of each department (on the slightly bizarre scale of 1,2,3a,3b,4,5,5*) determines the level of research income that the government will provide.
The quality of research is primarily assessed by each member of staff submitting their 4 best published papers over the last 5 years to be peer-reviewed by the RAE panel. They obviously can't read every paper, and so it's important for each department to submit as many papers in as many prestigous journals as possible: too few and your department could be closed down! Each submitted paper has to be put into the correct category: peer-reviewed journal, refereed conference proceedings, non-peer-refereed proceedings, etc. What is interesting is that refereed electronic journals are counted separately to traditional refereed print journals. The RAE panel claims that the two will be treated equally, but given that the panels usually comprise of the more senior figures in the discipline (hence, older and more adverse to innovations like electronic journals), and given the huge sums of money involved, it's regarded as foolish to submit papers to purely electronic journals in case they are perceived as being of lower quality.
I - and many others - would much prefer to publish in electronic journals, or to journals which are published by the academic organisations, but the potential career and financial risks are just too great.
This says that once we know \zeta(s) for Re(s)>1 (and this is the region for which the Dirichlet series \sum_n 1/n^s converges), then we know \seta(s) for Re(s) 1, it follows that \zeta(-2k) must be zero to cancel the pole of the Gamma function.
EULAs and the UK Sale of Goods Act
on
EULA In Games
·
· Score: 2
I've often wondered whether some EULAs contradict the UK's Sale of Goods Act. (The Sale of Goods Act is one of the better bits of consumer legislation in the UK and gives consumers a great deal power. Here is an explanation of the main points.)
For example, the goods must be of `Fit for their purpose'. This means that if you're lead to believe that a product can do something, and it can't, then the retailer is obliged to refund your money. I would imagine that, for example, the packaging for System Shock 2 implies that you will find out what's going on on the ship. Not without downloading the patch you won't! Are the goods fit for their purpose? (This isn't to slag of SS2 - it was just the first example I thought of.)
Another example, the goods must be of `merchantable quality'. What this means is left vague and for the (usually small claims) court to decide. A yogurt containing rotting fruit is clearly not of merchantable quality, but I guess the question `Is Win9x of merchantable quality' would occupy the finest Slashdot minds for days...)
Notice that the Sale of Goods Act applies only between the consumer and the retailer. *NOT* the manufacturer. As I understand it, any contract exists between the retailer and the consumer, not the manufacturer and consumer. Moreover, retailers can't get around the Sale of Goods act by introducing their own terms (although there are some minor cop-outs, for example, you can't complain about a fault if it was made aware to you at the time of purchase. (Although, I can't see Dixons or Comet reading out BugTrak to everyone who wants to buy the upgrade to WinME...))
I should add that IANAL. Also, for examples of fun things to do with the S of G act, see uk.legal.
Firstly, I'm a named contributor in Eric's Treasure Trove (which means I got a freebie copy of the printed version - wheee!). When Eric was first getting involved with CRC press, I remember that he sent me (and the other contributors) a form to sign to transfer copyright. I didn't keep a copy of the form, but I'm almost certain that I assigned copyright over my entries to CRC. Incidentally, Eric told me (and the other contributors) that he would try to negotiate an agreement with CRC by which a web version of the treasure trove could remain on the web - if he hadn't have done this then I would have been unwilling to let my entries be used. Such an agreement between Eric and CRC was reached, because after publication of the printed version, the web version would have certain entries unavailable (on a rotating basis), presumably at the request of CRC.
Going back to who owns the copyright of the individual entries, a lot of entries on the properties of sequences of integers were submitted by Steven Finch of MathSoft. Steven still maintains a website with this material on, so I wonder if CRC will start chasing him? (Maybe he has a separate agreement with CRC, though - I don't know.)
Incidentally, some academic journals in mathematics allow for authors to have an electronic version of their papers on their homepages. The AMS is one example, where you will often see in the copyright notice on a paper `copyright retained by author'. A lot of other journals turn a blind eye. (As you might expect, the copyright notice in the CRC Encyclopedia is the standard `it's ours so hands off' one: no reproducing or transmitting in any form or by any means, electronic or mechanical, etc etc.)
My own feelings are that the best place for the encyclopedia is on the web. Some of the entries are mathematically wrong, and many are misleading. This is not a criticism of Eric, who obviously put a lot of work into the project, it's just a fact that a book containing so much material will contain many many errors. (See the (often extrmely rude) posts from about 5 years ago on sci.math.research complaining about the lack of mathematical precision in the treasure trove!)
Having the treasure trove on the web would and should have allowed the project to grow, both in terms of the accuracy and the number of the entries. Sadly, the only way that such errors could be corrected in the printed version would be for CRC to issue a second edition - something I would imagine Eric is now unlikely to want to get involved in...
Slashdot Mirror
Yep. Writing a real number x in base b (not necessarily an integer) essentially means writing
for suitable integers a_n. (`Essentially' here means that there may be some non-uniqueness, for example 1.000...=0.999...To prove that the real number x is normal to base b, one looks at the ergodic properties the transformation defined on the unit interval [0,1] by
(that is, the function that takes x, multiplies it by b and then takes the fractional part). When b is an integer, this transformation preserves Lebesgue measure (length) and is ergodic. Birkhoff's Ergodic Theorem then says that Lebesgue almost every point x is such that the sequence \{ T_b^{n}(x) \mid n = 0,1, 2,...} is uniformly distributed mod 1, and such a point x is easily seen to be normal in base b. Now there are only countably many different integer bases and the countable intersection of sets of full Lebesgue measure also has full Lebesgue measure. Hence (Lebesgue) almost every point is normal in every (integer) base.What about the case when b is not an integer? Well, the map T_b can still be defined, but it no longer preserves Lebesgue measure. It does, however, preserve a measure that is equivalent (in the sense of having the same sets of measure zero) to Lebesgue measure, and it is ergodic with respect to this measure. One can again conclude that for a given base b (not necessaily integral), Lebesgue almost every point x is normal to base b.
However, one cannot then go on to say that almost every point is normal to every (real, non-zero) base b. This is because the intersection of uncountably many sets each of full measure can (and typically will) have zero measure. However, the following is true: take any sequence b_1, b_2, ..., then almost every point is normal in each base b_n.
Of course, the set of full measure obtained will depend on the sequence b_1, b_2, ....
There's a whole theory based around the ergodic properties of the map T_b for a non-integer b. Such transformations are called \beta-transformations, and there's a vast literature; see Parry, W. On the $\beta $-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 1960 401--416, for starters.
Such numbers are called normal.
Specifically, a real number x is normal in base b if the frequency with which the digit r occurs in the base b expansion of x is equal to 1/b, for each r \in {0,1,...,b-1}. A real number is normal if it is normal in each (integer) base b.
It's easy to construct (usually artificial) numbers that are normal to any given base. One can also sure (easily - using the Birkhoff Ergodic Theorem) that Lebesgue almost all real numbers are normal. However, there are no known examples of such a number. Thus `most' real numbers are normal (in the sense of `pick a real number at random then with probability one it will be normal'), but no explicit examples are known.
This is all straightforward ergodic theory (actually, `uniform distribution mod 1'), and there's an excellent account in Kuipers & Neidreitter's book, which is sadly out of print.
> who do publish in the publications not meeting
> the demands, to make them aware of the issues.
> I don't think this boycott has the high profile
> in the scientific community we'd like.
Rob Kirby has a list comparing mathematics journal prices. (Although some of the data is somewhat ('97) old now.) In particular, there is an excellent comparison of the (estimated) costs involved of producing the Pacific Journal of Mathematics (published by UCI) and Inventiones Mathematicae (published by Springer-Verlag).
I think most people in the mathematics community are aware of the issues here. However, certainly in the UK, there are far more pressing factors that determine which journals one submits to. Firstly, there's the obvious implications for one's career and promotion prospects: a paper in Annals of Mathematics or Inventiones looks better on a C.V. than a paper in some random journal no-one's heard of. Secondly, in the UK each member of staff in every university department is assessed for the quality of its research output (this is the dreaded Research Assessment Exercise). The ranking of each department (on the slightly bizarre scale of 1,2,3a,3b,4,5,5*) determines the level of research income that the government will provide. The quality of research is primarily assessed by each member of staff submitting their 4 best published papers over the last 5 years to be peer-reviewed by the RAE panel. They obviously can't read every paper, and so it's important for each department to submit as many papers in as many prestigous journals as possible: too few and your department could be closed down! Each submitted paper has to be put into the correct category: peer-reviewed journal, refereed conference proceedings, non-peer-refereed proceedings, etc. What is interesting is that refereed electronic journals are counted separately to traditional refereed print journals. The RAE panel claims that the two will be treated equally, but given that the panels usually comprise of the more senior figures in the discipline (hence, older and more adverse to innovations like electronic journals), and given the huge sums of money involved, it's regarded as foolish to submit papers to purely electronic journals in case they are perceived as being of lower quality.
I - and many others - would much prefer to publish in electronic journals, or to journals which are published by the academic organisations, but the potential career and financial risks are just too great.
One can show that \zeta(s) = 0 whenever s = -2k, for any k > 0. (\zeta(0)=-1/2.) This follows from the functional equation, which in pidgin-TeX is:
\zeta(s) = 2^{s} \pi^{s-1} \sin(\pi s/2) \Gamma(1-s) \zeta(1-s).
This says that once we know \zeta(s) for Re(s)>1 (and this is the region for which the Dirichlet series \sum_n 1/n^s converges), then we know \seta(s) for Re(s) 1, it follows that \zeta(-2k) must be zero to cancel the pole of the Gamma function.
For example, the goods must be of `Fit for their purpose'. This means that if you're lead to believe that a product can do something, and it can't, then the retailer is obliged to refund your money. I would imagine that, for example, the packaging for System Shock 2 implies that you will find out what's going on on the ship. Not without downloading the patch you won't! Are the goods fit for their purpose? (This isn't to slag of SS2 - it was just the first example I thought of.)
Another example, the goods must be of `merchantable quality'. What this means is left vague and for the (usually small claims) court to decide. A yogurt containing rotting fruit is clearly not of merchantable quality, but I guess the question `Is Win9x of merchantable quality' would occupy the finest Slashdot minds for days...)
Notice that the Sale of Goods Act applies only between the consumer and the retailer. *NOT* the manufacturer. As I understand it, any contract exists between the retailer and the consumer, not the manufacturer and consumer. Moreover, retailers can't get around the Sale of Goods act by introducing their own terms (although there are some minor cop-outs, for example, you can't complain about a fault if it was made aware to you at the time of purchase. (Although, I can't see Dixons or Comet reading out BugTrak to everyone who wants to buy the upgrade to WinME...))
I should add that IANAL. Also, for examples of fun things to do with the S of G act, see uk.legal.
-- C.
Going back to who owns the copyright of the individual entries, a lot of entries on the properties of sequences of integers were submitted by Steven Finch of MathSoft. Steven still maintains a website with this material on, so I wonder if CRC will start chasing him? (Maybe he has a separate agreement with CRC, though - I don't know.)
Incidentally, some academic journals in mathematics allow for authors to have an electronic version of their papers on their homepages. The AMS is one example, where you will often see in the copyright notice on a paper `copyright retained by author'. A lot of other journals turn a blind eye. (As you might expect, the copyright notice in the CRC Encyclopedia is the standard `it's ours so hands off' one: no reproducing or transmitting in any form or by any means, electronic or mechanical, etc etc.)
My own feelings are that the best place for the encyclopedia is on the web. Some of the entries are mathematically wrong, and many are misleading. This is not a criticism of Eric, who obviously put a lot of work into the project, it's just a fact that a book containing so much material will contain many many errors. (See the (often extrmely rude) posts from about 5 years ago on sci.math.research complaining about the lack of mathematical precision in the treasure trove!) Having the treasure trove on the web would and should have allowed the project to grow, both in terms of the accuracy and the number of the entries. Sadly, the only way that such errors could be corrected in the printed version would be for CRC to issue a second edition - something I would imagine Eric is now unlikely to want to get involved in...