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# Algebraic Number Theory - Science topic

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Questions related to Algebraic Number Theory
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The formula for sin(a)sin(b) is a very well know highschool formula. But is there a more general version for the product of m sine function?
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Why is a Proof to Fermat's Last Theorem so Important?
I have been observing an obsession in mathematicians. logicians and number theorists with providing a "Proof for Fermat's Last Theorem". Many intend to publish these papers in peer reviewed journal. Publishing your findings is good but the problem is that a lot of the papers aimed at providing a proof for Fermat's Last Theorem are erroneous and the authors don't seem to realize that.
So
Why is the Proof of Fermat's Last Theorem so much important that a huge chunk of mathematicians are obsessed with providing the proof and failing miserably?
What are the practical application's of this theorem?
Note: I am not against the theorem or the research that is going on the theorem but it seems to be an addiction. That is why I thought of asking this question.
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Muneeb Faiq , the situation has changed and there should be no fear, when staying before the FLT.
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In general in mathematics we work with groups that includ numbers, generally we work with constants.
So for exempel: For any x from R we can choose any element from R we will find it a constant, the same thing for any complex number, for any z from C we will find it a constant.
My question say: Can we find a group or groups that includ variabels not constants for exemple, we named G a group for any x from G we will find an x' it's not a constant it's another variabel ?
and if it exisit can you give me exempels ?
Thank you !
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Many proposals for solving RH have been suggested, but has it been splved? What do you
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Interesting question for possible future discussions.
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What lessons and topics are prerequisites for algebraic number theory and analytic number theory?
Please tell me the exact topic of each lesson.
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It would help if you studied advanced abstract algebra, topology, mathematical analysis besides the introductory courses in general number theory.
Regards
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In the 19 hundreds, many though that an explicit formula for the partition function was never going to be found. In 2011, finally, an explicit formula for the partition function was discovered.
For that reason, I am fascinated by how close do mathematicians think we are currently to discovering an explicit formula for prime numbers.
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I was wondering is there any
• model theory of number theory ,hence are there model theorists working in number theory
• the development of arithmatic geometry ,does it have anything to do with questions in logic;and is there any group studying this interaction.
• Anyone is welcome and up for collaboration
• I am interested in finding interaction between algerraic and arithmatic number theory with logic,and to study it to answer logical questions about Arithmatic
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As far as I know, in the entire history of mankind, only two philosophers have seriously dealt with logic, this is Aristotle and Hegel. Of these, only Hegel did mathematics. Nobody else dealt with this problem.
Sincerely, Alexander
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Dear fellow researchers,
In your expert opinion, do you think the Riemann Hypothesis is true? The first billion zeros have been computed and they all verify this hypothesis. However, we have previously seen patterns hold until a very large number than break (There is a conjecture that holds for n<10^40). Do you think there is any reason to believe that it might be false?
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If anyone proposes a theory and no one can make it wrong, at least it is not wrong, but it remains a mystery, and that is why it is called a mystery .. This does not mean that the theory is wrong.
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How many more years do you predict it will take before the Riemann Hypothesis is solved?
Do you think we are close or does it seem that we are still very far?
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I agree with you. Although year after year we are getting closer. A few year ago, it was proven that at least 40% of zeros have to be on the critical line. So a lot of progress has been done. Many conjecture have also been presented, which if proven, they would imply the Riemann Hypothesis. So have gotten some important result, however, we have not quite solved it yet.
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Does Hilbert-Polya Conjecture Infer Riemann Hypothesis by the means of spectral theory? Can anyone describe the procedure
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Stam Nicolis , you said that
" However it's not necessary for the Hilbert-Polya conjecture to be true, in order for the Riemann Hypothesis to be true. "
Why is that?
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I find that multiple sums are a very interesting concept. However, an even more interesting concept for me is the sum of such sums (the sum of multiple sums). The sum of multiple sums can be turned into a simple product by the formula in the first attached image.
What mathematical applications could this formula have? Where could it be useful?
I have found 2 interesting applications:
1- sum of multiple zeta values (see image 2).
2- sum of multiple power sums (see image 3).
Could you suggest any applications for these 2 particular cases? Could you suggest additional particular cases that would be of interest to mathematicians or physicists?
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Why are you asking?
Please respond when you read this.
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• We know ,euclidean algorthm is feasible in the set of integers.
• Taking motivation from this,we define an Euclidean Domain(E.D.) as follows:
R is an E.D. if it is a domain,andwe have a map d:R->non-zero integers,
such that, for given a,b(non-zero),
there exists q,r with
a=bq+r and d(q)<d(r).
• Now,this d function ,for any abstract ED,is the counterpart of |.| function,in case of intgers,
we have ,a=bq+r with,0 \le r < |b|.
• Now the question is:
[] It turns out (q,r) that exists for a given (a,b) in case of intgers ,is unique .
[A simple proof would be:if another (q',r') exists for same (a,b),
if q'=q,we have r'=r,and we are done.
and if,q' and q are distinct,bq+r=bq'+r' implies r' and r differ by multiplies of b,thus if 0 \le r < |b| holds,it is clear it won't hold for r',and thus we can never have (q',r') and (q,r) distinct,and we are done.]
[] But,one now would ask,is it true as well that for given (a,b) in any E.D. the existing (q,r) would be unique ? If yes,we need a proof,and clearly,the same proof does not work,as d(.) is much more generalized than |.| .Or,we need a counter eg!
__________________________________________________________________________________
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Let K={|z|<=1} be the closed unit circle (a connected compact) in C, R=O(K) be the ring of holomorphic functions f on K with d(f)=#{zeroes of f in K}, R is a ED.
Let a=a(z)=z^2+2z+1, b=b(z)=z (hence d(a)=2, d(b)=1). Let h=h(z)=sin(z)/z, h(0)=1.
Then
1) z^2+2z+1=(z+2)z+1, q=z+2, r=1 (hence d(r)=0)
2) z^2+2z+1=(z+2+h)z+1-zh, q'=z+2+h, r'=1-zh=1-sin(z) (hence d(r')=0,
since sin(z)=1 iff z=\pi/2 +2\pi*k, k\in Z).
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Does anyone know of applications of multiple sums of a sequence?
I know of the Multiple Zeta values (which is a multiple sum of 1/N^s). This has multiple applications in quantum physics, QED, QCD, connection between knot theory and quantum physics, ...
Does anyone know of potential applications for this more general form which is a general multiple sums? I have written an article about it and about its applications including partition identities, polynomial identities. I wanted to know if anyone know of applications outside mathematics or additional applications in math.
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Thank you for your recommendations.
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Do you think the "New Whole Numbers Classification" exactly describes the organization of set N ?
Whole numbers are subdivided into these two categories:
- ultimates: an ultimate number not admits any non-trivial divisor (whole number) being less than it.
- non-ultimates: a non-ultimate number admits at least one non-trivial divisor (whole number) being less than it.
Non-ultimate numbers are subdivided into these two categories:
- raiseds: a raised number is a non-ultimate number, power of an ultimate number.
- composites: a composite number is a non-ultimate and not raised number admitting at least two different divisors.
Composite numbers are subdivided into these two categories:
- pure composites: a pure composite number is a non-ultimate and not raised number admitting no raised number as divisor.
- mixed composites: a mixed composite number is a non-ultimate and not raised number admitting at least a raised number as divisor.
From the paper:
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very interesting....
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Dears, please help me.
Could you prove that for any big number N there exists a composite Fermat number F_k such
that F_k > N?
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Fermat numbers are odd numbers.
I think you are confused with Fibonacci numbers where F3n even numbers.
Best wishes
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I am a bachelor's student in mathematics.
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Hello Amirali: I am physician and for that I studied math deeply. One day the teacher tell us that all maths are algebra and analisys. I dont want to generalize because there are branches of math that we dont study in physics. And we dont have to proof math conjectures. But Physician have a great instinct to solve problems with the general math. Study deeply Algebra and Analisys was on the best things I did in my live. FIrst is to understand very good these general branches and after you could focus in what you want.
Best regards
Hugo
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I have a fairly large matrix (250*250) in symbolic form in MATLAB. Matrix is square invertible with size multiple of 2. I have to calculate the first two elements of the first two rows of its inverse (i.e. first 2x2 block). Matrix is very large and in symbolic form ,so, Matlab is not able to calculate the whole inverse due to time limitations. I have tried guassian elimination, LU factorization, block wise inverse technique. I have also tried the simple method of cofactors and determinent. In all the cases the problem is the same: very long time in the range of hours. Can anyone suggest some technique?
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Hi, perhaps another solution is to solve two linear systems of the form A.x=b where b is set to (1,0,0...0) and (0,1,0,0....0) respectively, just to obtain the first two columns of the inverse of A.
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We fix a number field K=Q(a) of degree 4 that is a CM-field. For an order O, if I(O) is the monoid of fractional ideals of O and P(O) is the subgroup of principal fractional ideals, we define the Ideal Class Monoid ICM(O) as the quotient I(O)/P(O).
Given two orders O\subset O', I define the ratio r(O,O')=#ICM(O')/#ICM(O).
Set R=Z[a,\bar{a}], where \bar{a} is the complex conjugate of a, and R'=R[S(a)/p], where S is a polynomia with integer coefficients and p a prime number.
I am interested about the value of r(R,R'), or if we have some inequality as for example r(R,R')=<1/p. Do you have some ideas?
Obs: there is a bijection between ICM(Z[a]) and the conjugacy classes of matrices M_4(Z) with characteristic polynomial the minimal polynomial of a, but I dont know how that can be useful.
Thanks in advance.
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There is a paper by Bushnell and Reiner on zeta functions of orders which you may find of interest.  You can find a free copy at:
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Is there an algorithm to find the Lengths of period of continued fraction sqrt((an)^2+4a)) whith a,n are odd integer.And can we find the continued fraction of the previous number.
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If a<0 then the period is apparently 12.
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is ther any solution of the diophantine equation :
x^2-((a*n)^2+4*a)*(y^2) =4*m,-4*m
more spesific : is it true that the equation have integer solution then m>a ?
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class number  " h" :is the order of Ideal class group.
Ideal class group :In Z[sqrt(n)] the fractional ideal F is a group ,and the principal ideal P is a subgroup in it so we can take the quotient F/P.
real quadratic field: an extension of the rationals Q by the root of an irreducible quadratic equation over the rationals Q(sqrt((an)2+4a)).
the equation is :   x2- ((an)2+4a)y2 = 4m [or -4m]  with  :  a: odd integer ,n : odd integer.
m: integer .
the relation of "class number" and "real quadratic field" with the integer problem eich i put in question is: " h=2" then the equation represent a norm of ideal so we have an integer solution . now if I know when the equation have an integer solution i can determine the real quadratic fields
the solution wich i need "or wich I need the condition for exist it or not exist "should be (x,y) with " y" don't equal zero.
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I have solved many quadratic,cubic, biquadratic, quintic, sextic, heptic and mth degree diophantine equations. I wish to know about the applications  in real life as well as in other fields.
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If you look for "applications in real life" of diophantine equations, the first striking example coming to mind is indeed cryptography with « public key » using elliptic curves, as pointed out by Rogier Brussee. An elliptic curve is a plane irreducible projective cubic. It is miraculously endowed with a natural abelian group structure : given 2 points P,Q on the curve, a simple geometric construction allows to associate to them a 3rd point, denoted  S = P + Q. Given a point P, the sum of d copies of P is written dP. The so called « discrete log » problem on an elliptic curve consists in finding d when knowing P and dP (see any textbook on the subject). It can be quite difficult depending on the choice of the elliptic curve. The front cover of any recent « biometrical » passport in Europe and in the USA is equipped with a chip which contains detailed information on the owner, encrypted via an elliptic curve. If this is not « real life »…
If you accept to include in « real life » the real world of physics, then there is another striking –  perhaps less well known - application based on diophantine approximation. Instead of defining this subject in a general way, let us consider a few particular cases.Typically, diophantine results come in the form of certain finiteness statements, e.g. that certain equations have only a finite number of rational or integral solutions. Consider for example the diophantine equation x^3 – 2y^3 = n (a given integer). By factorizing the LHS inside the decomposition field of the polynomial X^3 – 2,  one can easily show that any possible solution (x, y) must satisfy C /⎹ y^3 ⎹ >=⎹  (x/y – cubic root 2)⎹  , where C is a constant independent of x and y. The problem is thus reduced to approximating irrational numbers by rationals in a certain way. The celebrated theorem of Thue-Siegel-Roth asserts that for any  algebraic number a of degree >= 2, for any e > 0 , the inequality ⎹  p/q - a⎹  >= 1/ q^{2+e}  must hold, with the exception of finitely many rationals p/q where p, q are integers and q > 0.
This shows readily that our diophantine example above admits only a finite number of solutions. Actually, using diophantine approximation (before Roth’s result), Siegel had shown that any affine curve of positive genus defined over a number field contains only a finite number of integral points. In fact, most of the important known results in diophantine equations are consequences of the TSR theorem (including Faltings’ second proof of the Mordell conjecture). Now, where does the physical world come in ? An unexpected close relationship between diophantine approximation and dynamic systems has been revealed by Siegel (1942). When studying the stability of the solar system, one comes across a problem of « small divisors (denominators) » associated with the near resonance of planetary frequencies, which could result in the divergence of the solution expressed as a series. A little bit more mathematically, the dynamical properties of a rotation of angle alpha on a circle of length L are radically different according as the parameter alpha / L is rational or not : in the first case the orbits are all periodic, in the second, they are all dense. Siegel’s idea was to impose a suitable diophantine inequality on the frequencies to ensure that they are not too close to resonance. For more mathematical details, see e.g. Rodrigo A. Pérez, « A brief but historic article of Siegel », Notices of AMS, 58 (4), 2011, 558-567.
Applications to other math. domains have been evoked by Rogier Brussee, especially the « Langlands program », which aims to build bridges between 3 islands : 1) Arithmetic varieties (i.e. diophantine equations) and their L-functions   2) Representations of the absolute  Galois group of Q and their L-functions  3) Representations associated to automorphic forms and their L-functions. Will these fundamental (as opposed to applied) developments one day find their way into the real world ? Without discussing the distinction which you seem to make between the real world and the world of ideas (this would bring us back at least to Plato !), a reasonable attitude could be : fundamental science, especially in math., is primarily concerned with knowledge ; the applicable part of it is small, but  this part plays a prominent role in our control of the physical world ; no searcher or scholar can predict what fundamental discovery could be applied one day, and for what purpose. In these times when money tends to go only to applications, what do you think of this fable : it was in the 19-th century ; « deciders» in various governments had decided to pump money and positions into a big plan of research : how to improve lighting by improving the candle ; then a neglected somebody invented the electric bulb.
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A subexponential algorithm for elliptic curves over F2n ?
The main question is whether or not the strategies described in the elliptic curve discrete logarithm can lead to a general subexponential discrete logarithm algorithm for elliptic curves in characteristic 2.
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For small characteristics field like F2^n there are better algorithms quasi-polynomial and so on, much better than the otherwise sub exponential ones (and they do not translate to the general case). A good survey on the state of the art, see:
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If we are in unison inre to the aforementioned ,then the cosmic speed is one-third the speed of light for particles of masses greater than 1kg. Please verify my publications in regard to the same.
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"For the question assumes the consensualist ought to generally agree with the questioner."
Can we be clear: you are not actually asking a question, you are asking for people to agree with you. This is antithetical to the scientific method where models are rigorously tested both by argument and by experiment. If your theory has merit it ought to stand up to a few basic questions. If it can't, it can't ever be anything more than an opinion. If you don't want people to debate your theory with you, you are on the wrong site. Just to be clear.
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Please can someone give me a reference on the distribution of Pisot numbers on the real line.
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Communication protocol  with  high  level Scurry  type   supported   distribution
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In imaginary quadratic fields we have:
* ELL(O_K) : = {elliptic curves E/C with End(E) ∼= O_K}/{isomorphism over C}
∼= {lattices L with End(L) ∼= OK}/{homothety}∼=ideal class group CL(K)
* #CL(K)=#ELL(O_K)
this notation at the papper :A SUMMARY OF THE CM THEORY OF ELLIPTIC CURVES
JAYCE GETZ
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As you say, the theory of "complex multiplication" (looking at modular invariants of elliptic curves with CM) solves the problem of determining the maximal abelian extension of an imaginary quadratic field. But as yet no analogous theory is known for real quadratic fields, although class field theory affords  a complete theoretical (but not explicit) description of the abelian extensions of a given number field. You can  take the measure of our ignorance by  listening to the short negative answer given by Serre at the end of his talk on abelian Galois theory at the occasion of Galois' bicentenary (the talk can be found on You Tube, I think).
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Since it is difficult to write mathematical formulae please consider the attached file.
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${F_{{5^k}n}}(q) \equiv 0\bmod {[{5^k}]_q}$ is equivalent with ${F_{{5^k}}}(q) \equiv 0\bmod {[{5^k}]_q}$
because ${q^{{5^k} + n}} \equiv {q^{{5^k}}}\bmod {[{5^k}]_q}$ and therefore  ${F_{{5^k}(n + 1)}}(q) \equiv {F_{{5^k}n}}(q){F_{{5^k}n + 1}}(q) \equiv 0\bmod {[{5^k}]_q}.$
Therefore it suffices to show that
${F_{{5^k}}}(q) \equiv 0\bmod {[{5^k}]_q}.$
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Factoriangular numbers (denoted by Ftn) are formed by adding corresponding factorials and triangular numbers, that is Ftn = n! + Tn, where Tn is the nth triangular number. The closed form of the exponential generating function of the sequence of factoriangular numbers can be easily derived from the exponential generating functions of n! and of Tn (see the link below). How about the closed form of the ordinary generating function of such sequence?
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Use the Laplace transform
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We define factoriangular numbers as sum of corresponding factorials and triangular numbers, that is, n! + n(n+1)/2, n is a natural number. We listed the first few factoriangular numbers: 2, 5, 12, 34, 135, 741, 5068, 40356, 362925, 3628855, 39916866, 479001678, 6227020891, 87178291305, 1307674368120 and notice that each is either a deficient or an abundant number. Is there any such number that is a perfect number? Or, how can we prove that no factoriangular number is a perfect number?
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It is easy to show that n!+n(n+1)/2 is never an even perfect number. It is known that every even perfect number has form 2p-1(2p-1), where p and 2p-1 are primes. If n!+n(n+1)/2=2p-1(2p-1), then n(n+1)<2p(2p-1), and so n<2p-1. If n is odd then n|(n!+n(n+1)/2), hence n|(2p-1). Since
2p-1 is a prime, we have n=1, but in this case n!+n(n+1)/2 is not a perfect number. If n is even and 2ǁ n, then 2k-1 ǁ (n!+n(n+1)/2), hence
k-1=p-1 and k=p. But 2k≤n<2p, contradiction.
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I have been using DLS to characterize my Pt nanoparticles as well as AFM and TEM, my AFM and TEM results match up but the intensity distribution doesn't that's why I'm using the number distribution but I would like to know how does the software do the conversion.
Thanks
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Hi,
Thank you for your responses.
Yes I know that DLS will give me a hydrodynamic diameter, however I watch a webinar of the same company (Malvern) explaining how to understand the different distributions (Volume, Number and intensity).
Unfortunately, they explain and they don't go through the calculations on how they do that, the software itself has an algorithm that does the conversion automatically. That's what I want to know, what is the fundamental of the conversion.
Thanks
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Digital root as successive sum of the digit of a positive integer.
Where does the digital root came from?
Is there bibliography about this history of mathematics?
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Thanks Danny
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Dear searchers,
The set of all algebraic numbers is countable. I wish if someone can help by giving the smallest gap between algebraic numbers in general, or between the conjugates in particular, or if that gap depends on the degree or the wight or the height of the minimal polynomial of them.
Recall that the conjugates are the roots (zeros) of a minimal polynomial
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According to what is expressed in the previous answers, I think that the smallest gap does not exist because the gap tends to zero when n tends to infinity.
There is something more interesting and is related with the primes:  which is the lower boundary of P (k + 1) - P (k) being both primes when k tends to infinity?
This is my answer:
Dirichlet, demonstrated that:
For any two positive coprime integers a  and b  , there are infinite primes of the form  a+bm, where  n is a non-negative integer (n=1,2,... ). In other words, there are infinite primes which are congruent to a mod b . The numbers of the form  a+bn is an arithmetic progression.
Actually, Dirichlet checks a result somewhat more interesting than the previous claim, since he demonstrated that:
∑_(p=a mod b) [lnp/p]→∞
Which implies that there are infinite primes p≡a mod b.
Therefore
H_1:=〖lim⁡inf〗┬(k→∞)⁡(p_(k+1)-p_k )=2
Becasue according with Dirichlet's theorem:
p_(k+1) when k→∞)=[(a+bn when n→∞)] and
p_(k) when k→∞)=[(b+bn when n→∞)]
Then the smallest gap will be a-b=2, since primes.2 are odd.
This solved the conjecture of twin primes. Zhang came to 246 in  2014 by other form.
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Is a bracket-preserving map between Lie algebras necessarily Linear? Usually, it is assumed to be linear in advance and the second condition of bracket-preservation males it a Lie homomorphism, by definition.
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The answer is NO. Take the a trivial Lie algebra L, that is just a vector space L (over an arbitrary field) with the product [x,y]=0 for all x,y in L. Let a  f:L-->L a non-linear mapping satisfying f(0)=0.
Then [f(x),f(y)]=0=f(0)=f([x,y]) for all x,y in L. No "aritmetic" helps there.
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For the degree 2 or three with the third conjugate positive it is easy.
key words: Pisot number, strong Pisot number, minimal polynomial.
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a strong PV number is a PV number such that the second conjugate µ (for example) is positive < 1 and the remaining conjugates are of the absolute value <µ
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Let a and b be two reals such that -b<a<0<b<1, does a+b+ab always positive?
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Dear Hanifa,
The first answer is NO. Take e.g  a = b = 1/2.
The second is NO, e.g. b = 0.9 , a = - 0.8
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How to find the total number of normal subgroups for symmetric group Sand An and is there any relation between the normal subgroups of two?
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Dear Rafiq,
As it said by Wiwat and Peter, for n=3 or n ≥5, the alternating group A_n is simple ( there are no normal subgroups) . For n≥5, A_n the only normal subgroup of S_n. You can see the link of wiwat or see the book: Introduction à l'algèbre (K. Kostrikin), you will find all details. For n<5, just use Sylow theorems and with information of S_n and A_n, it is so easy to find them, for example for S_4, you have A_4 and if there is an other normal subgroup, then it will be of order 2 or 4 or 8, because by Sylow theorems, there are 4 subgroups of order 3, in fact you can find them: <(123)>, <(124)>, <(134)>, <(234)>. So  there is no normal subgroup of order 3 in S_4. Now, you have to compute for 2, 4 or 8. Bon courage.
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how can i prove the following statement? In other words, how can i prove existence of the following norm?
Let K be any totally disconnected local field. Then there is an integer q=pr, where p is a fixed prime element of K and r is a positive integer, and a norm ∣⋅∣ on K such that for all x∈K we have ∣x∣≥0 and for each x∈K other than 0, we get ∣x∣=qk for some integer k.
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I would like to also recommend chapter 2 ("Global Fields") of the classical book 3Algebraic Number Theory", Cassels & Fhröhlich ed., Acad. Press 1969, which gives a compact and complete account of the subject. Concerning your specific question, see especially §7, p.50
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An infinite-countable set of infinite-tuples of 0 and 1 is countable or uncountable?
Examples:
(0,1,0,0,........)
(0,1,1,0,1,0,0.....)
Note
The number of 1 is finite for each tuple.
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Let us write the X-tuples as 01000..., 0110100... (those from your Examples) etc. Consider the real numbers w from the interval [0,1] written in the form w= 0.[X-tuple].
Since the number of 1 in an X-tuple is finite, the X-tuple can be considered as a diadic form of a rational number. Therefore, each number w obtained in the described manner is rational.
Conclusion: The considered set of X-tuples is countable.
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Since every reduced binary quadratic form with discriminant d corresponds to one class of ideals in the quadratic fields Q(d^(1/2))with d>0 , Is there an effective way to determine all reduced binary quadratic form with discriminant d or at least can we determine the number of reduced binary quadratic form with discriminant d ?
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Look at
1]:Stark,H.M. A complete determination of the complex quadratic ...elds of
class number one.M ich:M ath:J:14(1967); 1 27:
[2]:Birch,J.B.Weber’s Class Invariants.M athematika 16(1969); 283 294:
[3]:Schertz,R."Complex Multiplication",2010.
[4]:Zagier,
also, Pilar Bayer work is very interesting
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How can I calculate the radius of convergence of Pulita pi-exponential?
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Dear Mokhfi,
In the following paper, the authors give an explicit algorithm to calculate what you desire. I have only found references, such as the following article, in french. I hope you find it helpful.
Cheers,
Ana Paula
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I am looking for the properties of Jacobson Radical of modules over noncommutative rings.
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Thanks a lot.
With best regards
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I want to determine the number of points on E in terms of æ and ß. For example:
For Koblitz curves one can compute this cardinality using a Lucas sequence but I am working in Elliptic curves over F_{2n} (not necessary Koblitz curves) given by the equation Y2 + XY = X3 + æX2 + ß where æ and ß are elements in F_{2n} and Tr(æ) = 1.
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Yes, there is a relation between the order of the group of points, the size of the base field (check Hasse's condition) and the curve coefficients. You can use a point counting algorithm like Schoof-Elkies-Atkin (SEA) to determine the order. This algorithm is usually already implemented in software like Sage or MAGMA.
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Does Gaussian include group theory? Are there any limitations?
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Yes. Group theory is very useful in understanding vibrational spectra, as follows:
- Group theory can also help analyze and predict vibrational spectra, both IR and Raman.
Group theory can predict which vibration is IR or Raman active and which is not.
Group theory can predict maximum numbers of active vibrations
Group theory can predict if IR and Raman vibrations will coincide
Group theory can help assign the the measured vibrational spectra, and explain all shoulders and double bands existing in gaseous sample spectra.
Unlike Quantum Theory, Group Theory can not predict positions of the bands. You need to make spectral measurements and then assign them with group theory.  This is a limitation indeed.
Best wishes
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In 1850 P.L. Chebyshev and in 1932  P. Erdos proved that [n,2n] interval contains a prime. Bachraoui proved the case k = 2 in 2006. k=3 case proved by Andy Loo in 2011.
From the other hand Legendre’s Conjecture states that for any n, there is a prime in [n2, (n+1)2]. For instance between 132 (=169) and 142 (=196) there are five primes (173, 179, 181, 191, and 193).
This interests me and I formulated two different questions. I hope that  the problems are interesting for someone else
1. whether there are k and n such that [kn,(k+1)n] contains odd number of primes, while [(k+1)n,(k+2)n] contains even number of primes with interval [(k+2)n,(k+3)n] having odd number of primes?
2. whether there are A, B and n such that [An,Bn] contains odd number of primes, while [An+1,Bn+1] contains even number of primes and at the same time intervals [(A+1)n, (B+1)n] and [(A+1)n+1, (B+1)n+1] have even and odd number of primes, respectively?
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Recently  Kevin Ford, Ben Green, Sergei Konyagin, James Maynard, and Terence Tao have produced a paper that proves a lower bound for the maximal gap
G(X) := \sup_{p_n, p_{n+1} \leq X} p_{n+1}-p_n
between primes bounded by X of the form
G(X) \geq c \log X \frac{\log \log X \log\log\log\log X}{\log\log\log X}
for some small constant c.
see: http://arxiv.org/abs/1412.5029 and the as always excellent blog post by Terrence Tao
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I have a set of data from my experiment (120 instances with 1458 attributes), but these data is in complex numbers.
How can I represent a complex number into another real number? I found that complex number can be represented by using polar coordinates (r and theta), however what I need is a single number, so that I can further process the data using Weka.
Alternatively, is there any equation that can take 2 numbers and produce a single unique number?
Thank you very much in advance.
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Dear Fajri Pratama
You look before the world mathematical community as the module on the complex plane with impotentny argument (forgive me for sad humour). Dear Fajri Pratama find to yourself the mistress or present to your girl or to the wife a box of chocolate truffles. Only then before you the world of natural and imaginary numbers, the main thing, in this question strong argument will open in all beauty.
Alexander
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Thanks.
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1.As for the post of colleague R.C.Mittal, I guess that the topic starter needs not the basic facts about $p$-adic numbers the colleague addresses him to. I even guess that the topic starter is aware of advanced monographs on $p$-adic analysis like the classical Mahler's monograph "$p$-adic numbers and their functions" ore more recent books by Schikhof "Ultrametric calculus",
by Gouvea "Arithmetic of $p$-adic modular forms", by Gouvea "$p$-adic numbers", by Vladimirov et. al. "$p$-adic mathematical physics", several books of Khrennikov on applications of the $p$-adic theory to various disciplines, etc., etc. There are numerous books and numerous papers on the subject, but I guess that the topic starter needs something more definite. My two cents follows.
2. The most known continuous mapping of $p$-adic numbers into reals is the Monna map which is defined as follows: Represent a $p$-adic number $z$ in a canonical form (this representation is unique) $z=\sum_{i=-k}^\infty\alpha_ip^i$ where $k$ is a non-negative (real) integer, $\alpha_i\in\{0,1,\ldots,p-1\}$. The Monna map puts into the correspondence to $z$ the real number $mon(z)=\sum_{i=-k}^\infty\alpha_ip^{-i-1}$. It is clear that the most interesting part is the action of the Monna map on the $p$-adic integers which are represented as $\sum_{i=0}^\infty\alpha_ip^i$: Contrasting to real numbers, the "fractional part" of a $p$-adic number always has a finite canonical representation while the "integral part" not. So further i will speak only of mappings of the space of $p$-adic integers $\mathbb Z_p$ into reals. The space $\mathbb Z_p$ is a unit ball w.r.t. $p$-adic metric and is a sort of analog of a real unit interval. The Monna mapping $mon$ is a continuos mapping of $\mathbb Z_p$ into $[0,1]$ w.r.t the $p$-adic metric on $\mathbb Z_p$ and standard real metric on $[0,1]$. Moreover, $mon$ is a measure-preserving mapping w.r.t the probability Haar measure on $\mathbb Z_p$ and Lebesgue measure on $[0,1]$.
3. The  idea of the construction the Monna map is based on can be extended as follows: Take a real number $\beta>1$ such that the integral part of $\beta$ is $p-1$ and to every $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ put into the correspondence the real number $\sum_{i=0}^\infty\alpha_i\alpha_i\beta^{-i-1}$. In this case we deal with the so-called $\beta$-representations which were introduced by Parry and Renyi and now are rather popular research area, see. e.g. numerous papers of Frougny, Sakarovitch, Nikita Sidorov, and many others (too long to list all of them, sorry).
4. There are numerous generalizations for the case when $\beta$ is negative or even a complex number, and they are also related to real (or complex) representations of infinite words that correspond to $p$-adic numbers and have tight relations to automata theory, cf. e.g. the monograph by Lothaire "Algebraic combinatorics on words". The basic idea is that to a $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ there corresponds a (left-) infinite word $\ldots\alpha_2\alpha_1\alpha_0$ over a $p$-symbol alphabet; the word may be treated as a representation of a real number in some radix system (e.g., as a $\beta$-representation) and then one can study what reals (or sets of reals) obtained this way can be recognized by finite automata, or transformed by automata, etc, c.f., e.g., the monograph by Allouche and Shallitt "Automatic sequences". From this point of view, real representations of $p$-adic numbers give some information of behavior of automata and of the corresponding discrete dynamical system.
5. As automata map infinite words to infinite words, the automata mappings can be considered as mappings of $p$-adic integers to $p$-adic integers: Actually every automaton mapping is a continuous (w.r.t. the $p$-adic metric) function defined on the space of $p$-adic integers $\mathbb Z_p$ and valuated in $\mathbb Z_p$. Moreover, this mapping is 1-Lipschitz (=satisfies the Lipschtz condition with a constant 1) w.r.t. the $p$-adic metric and vice versa, every 1-Lipschitz mapping $\mathbb Z_p\to\mathbb Z_p$ can be performed by an automaton. From this view, real representations of $p$-adic numbers can visualize the automaton mapping (or the evolution of a discrete dynamical system). But being used for such a visualization, both the Monna map and $\beta$-representations can only give an information about short-time behavior of a system since they put the earliest inputted/outputted symbols to most significant positions in real representations. To study long-term behavior we need maps that act somehow in inverse direction. These "maps" are not maps in a strict sense of the word since put into the correspondence to a single $p$-adic integer some set (maybe, even infinite) of real numbers rather than a single number but these "mappings" can also be judged as 'continuous representations' in some meaning.
6. Namely, given a $p$-adic integer $z=\sum_{i=0}^\infty\alpha_ip^i$ we put into the correspondence to $z$ the set of all accumulation points of the sequence $p^{-k}\sum_{i=0}^{k-1}\alpha_ip^i$, $i=0,1,ldots$. From this view, the dynamics of an automaton map (or of a discrete system) can be represented as a real dynamics on the unit real square or, which is more natural, on the 2-dimensional torus. The representations of this can discover import ant peculiarities in discrete dynamics and are now under research, see e.g.
the monograph by Anashin (me) and Khrennikov "Applied algebraic dynamics" (deGryuter, 2009) and my papers on the theme, e.g. "The non-Archimedean theory of discrete systems" Math. Comp. Sci. 2012 (the paper can be found here on ReserchGate or in arxiv.org).
I am not sure that all these things are really that ones the topic starter is interested in but at least they are related to the question under the discussion. Sorry for the lengthy text, but I tried to be as concrete as possible.
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Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?
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I am not sure whether a closed-form expression for your sum can be found, as this sum appears to be a special case of q-hypergeometric function, see e.g. the link below.
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I am looking for a reference giving the decomposition of a prime p in the maximal real subfield of a cyclotomic number field Q(zeta_m). Something along these lines: If p is a prime not dividing m, then let f be the smallest positive integer such that p to the f is congruent to 1 or -1 modulo m. Write f.g=phi(m)/2. Then (p) splits into g different prime ideals of norm f (plus what happens if p divides m).
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Dear Pieter,
I think you can find your answer in the book of Daniel A. Marcus, "Number Fields", Springer Universitex, 19977, chapter 4, exercise 12, p.118 (see also ex. 14)
Best, T. NQD
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I´m currently working on my master's degree thesis and I would like to know some interesting topics on this subject.
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Here is an old paper of mine that studies Gauss sums over finite rings rather than fields
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Is this true or false?
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Thanks Jeremy
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What is the method to decide if a given number (integer) is going to have an integer square-root or not (without actually computing the square-root)?
For example, assume I have a method M that does this. Then it should behave like below:
M(16) should return true (since sq-root(16)==4 which is integer)
M(17) should return false (since sq-root(17) is not an integer)
and M should not actually compute the square-root to decide this.
Any literature or info of this?
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There are 3 main solutions to this problem.
1. Simple to implement is binary search (as already mentioned). Given a pair of values (L,U) such that L^2 < n < U^2 (where n is the number to be tested), we compute ((L+U)/2)^2 and deduce that the solution is contained in an interval of half size. This is polynomial-time (unlike some of the other solutions already stated).
2. Optimal complexity is to use Newton interation. See Chapter 9 of the book by von zur Gathen and Gerhard. In particular, Section 9.5 talks of a 3-adic Newton iteration to compute integer square roots. This leads to a running time of O( M(n)) = O( \log(n) \log\log(n) \log\log\log(n)) the same as integer multiplication. But this is harder to implement.
3. A Monte-Carlo method is to reduce n modulo small primes p and compute the Legendre symbol (n/p) (which will always be 0 or 1 if n is an integer square). One would expect to test around log(n) primes to have good confidence that n is a square. I do not know a reference for a theoretical analysis of this algorithm.
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I am looking for a complete reference on introduction and basic definitions on Selmer group and Shafarevich-Tate group. Any suggestions would be welcomed.
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Thank you Andrej.
I have seen all these books, but they focused on the elliptic curves. I am looking for a reference which deals with Abelian varieties in general. It doesn't seem as there is any book on that.
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P ≡ 1 (mod 48),
2^(p-1)/4 ≡ −1 (mod p), and
6^(p-1)/4 ≡ 1 (mod p).
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Humm the best person to answer this question is Pieter Moree : he is on this site.
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I need to know if there is such that relation in the case of real quadratic fields.
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There is a notion of (divisor) class group 'Cl(X) ' associated to algebraic varieties X--affine and projective. (I will define this notion below for elliptic curves which are projective.). Now, if A is a Dedekind domain, (e.g. the ring of integers in a number field F (where F could be a quadratic field you are interested in) Cl(Spec(A)) coincides with the ideal class group of A as defined in Algebraic Number Theory. So, this gives you the connection between affine varieties and ideal class group. (Spec A is affine.)
I will now define Cl(X) for an elliptic curve X. . A divisor D of X is a linear combination of points x_i with coefficients k_i in Z(the integers).. The degree of D (deg D) is the sum of the k_i's. The divisors of X form a group Div(X)--the free Abelian group (Z-module) generated by the x_i's . A principal divisor is a divisor D such that the degree of D = 0.The principal divisors (P(X)) form a subgroup of Div (X) and the quotient group Div(X)/P(X) is denoted by Cl(X). Now, there is homomorphism deg: Cl(X) --> Z whose kernel is denoted by Cl^o(X). Cl^o(X) has the following interesting property:
There is a one-one correspondence between the points of x and the elements of the group Cl^o(X). Moreover by using this one-one correspondence, one can transfer the group law from Cl^o(X) to X itself.
Elliptic curves is a very active area of research probably because the curves have various deep ramifications. For example, an elliptic curve is an Abelian variety of dimension 1 i.e an irreducible projective algebraic group; it is also a curve of genus 1. etc.
for further information you may look at the following books:
1) Basic Algebraic Geometry--Varieties in Projective spaces by I. R. Shavarevich.
Springer-Verlag.
2) Algebraic Geometry-- R. Hartshorne. Springer-Verlag.
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I had began by examining multiplication modulo n with n being a primorial. The idea was to find a tool to study the differences between prime numbers. To my surprise I found I could prove the twin prime conjecture and even see how to generalize it further. This is because I could show there exists a linerar algebra of Euler products that can count gap patterns in each set. I found why the Hardy Littlewood constant needs to be multiplied by 2 and I found better, more natural formulas for gap pairs. In fact I can now describe all gap behaviors from 2-28 exactly.
The next step, the subtle step, was I needed to extend the PNT. I have to assume a single concept, the PNT operates everywhere with the log at P being the mean of a set and for each P there is a new set of possibilities and a new mean. But it must be random everywhere with a local mean.
From there RH just "fell out", and the result is more surprising than expected. In fact it took me a few months to reconcile Von Koch's famous "if RH than proof" because the answer was somewhat tighter than he found. I show by construction the worst possible error between the log integral and the prime counting function at any x is strictly less than the log integral at root x. That is, the actual worst error is (ln(P))^-3/2 smaller than Von Koch's ln(P) root (P).
It's 30 pages of what I hope is some of the most interesting math you'll read this week.
I'm confident it's correct because the paper is not a proof per se but a framework that proves many conjectures, not just RH. Also, over the last 2 years I have about 30 different programs I wrote amounting to about 100k lines of code that all support this work. I hope every idea has been tested at least a few different ways.
So I ask the local community here to please look at my paper or watch the presentation and if possible find the mistakes or validate the ideas.
This is also actually a test of crowd sourcing as I submitted the paper for peer review 24h ago. How fast can this crowd settle the validity of the paper against 3 editors. The editors promise an answer in 12 weeks, can RG crowd sourcing improve on this?
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Yup Patrick Sole......absolutely....I just shared this paper as it seems to be interesting....
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My question is: Is this paper published in a journal?
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If it is correct, Inventiones or Annals would have been better choices...