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| Edgar E. Escultura... |
 Posted: Thu Nov 05, 2009 12:21 am |
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Guest
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CLARIFICATION OF THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E. Escultura
Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:
1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.
2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].
3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)
for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.
4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.
The following references include references used in the consolidated paper [8] plus [2] which applies [8]
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
[4] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[5] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[6] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[7] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[8] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[9] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[10] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[11] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[12] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[13] Counterexamples to Fermat’s last theorem, http://users.tpg.com.au/pidro/
[14] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Vishakhapatnam, AP, India
http://users.tpg.com.au/pidro/ |
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| Vindicator2009... |
| Posted: Thu Nov 05, 2009 1:59 am |
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Guest
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On Nov 5, 12:21 pm, "Edgar E. Escultura" <escultu... at (no spam) yahoo.com> wrote:
[quote]CLARIFICATION OF THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E. Escultura
Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding.. Let me, therefore, make the following clarification:
1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.
2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].
3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)
for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.
4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture..
The following references include references used in the consolidated paper [8] plus [2] which applies [8]
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
[4] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[5] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[6] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[7] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[8] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[9] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[10] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[11] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[12] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[13] Counterexamples to Fermat’s last theorem,http://users.tpg.com.au/pidro/
[14] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Vishakhapatnam, AP, Indiahttp://users.tpg.com.au/pidro/
[/quote]
So this is what happens when engineers deal with mathematics and
things way over their heads...
You're an idiot. |
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| Dirk Van de moortel... |
| Posted: Thu Nov 05, 2009 7:05 am |
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Guest
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Vindicator2009 <Vindicator2009 at (no spam) live.com> wrote in message
f23e2574-648b-4383-abb6-3f9ec5132b5e at (no spam) k19g2000yqc.googlegroups.com
[quote]On Nov 5, 12:21 pm, "Edgar E. Escultura" <escultu... at (no spam) yahoo.com> wrote:
CLARIFICATION OF THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E. Escultura
[/quote]
[snip]
[quote]E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Vishakhapatnam, AP, Indiahttp://users.tpg.com.au/pidro/
So this is what happens when engineers deal with mathematics and
things way over their heads...
You're an idiot.
[/quote]
Worse things happen when they turn their attention to special
relativity. You might enjoy visiting sci.physics.relativity sometime.
Dirk Vdm |
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| Baugh... |
| Posted: Thu Nov 05, 2009 9:41 am |
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On Nov 5, 11:59 am, Vindicator2009 <Vindicator2... at (no spam) live.com> wrote:
[quote]On Nov 5, 12:21 pm, "Edgar E.Escultura" <escultu... at (no spam) yahoo.com> wrote:
CLARIFICATION OF THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E.Escultura
Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:
1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.
2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].
3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)
for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.
4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.
The following references include references used in the consolidated paper [8] plus [2] which applies [8]
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
[4] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[5] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[6] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[7] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[8] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[9] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[10] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[11] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May2009, 101-107.
[12] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.),2009, Paris..
[13] Counterexamples to Fermat’s last theorem,http://users.tpg.com..au/pidro/
[14] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E.Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Vishakhapatnam, AP, Indiahttp://users.tpg.com.au/pidro/
So this is what happens when engineers deal with mathematics and
things way over their heads...
You're an idiot.
[/quote]
'Ello 'ello, making such an outrageous and sweeping generalization
suggests the commenter above mighn't be much smarter than the deluded
prof.
First of all, there are many competent engineers who were initially
trained in engineering, but master so much math along the way that
they're able to contribute significantly to the field and become math
professor, for instance look up Mark Davis in the Mathematics
Department at Imperial College, London. Be aware that there plenty of
different kinds of engineers out there with different levels of math
mastery, not all are nuts and bolts engineers!
Secondly, Escultura is certainly no engineer! He is in fact supposed
to be a trained mathematician (in this case very poorly trained) with
a PhD degree in maths from UW-Madison in the 70's under the
supervision of L. C. Young. You'll find many graduate engineering
students who grasp math better than this guy. |
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| spudnik... |
| Posted: Thu Nov 05, 2009 7:04 pm |
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what the Hey?...
EEE's result is just the old "trivial" one,
a triple of (1,0,1), a degenerate pythagorean trigon
(if using "two" for the exponent, n).
he used to couch it in the "nonequality
of 1.0000... and 0.9999...," which notion he seems
to have dropped.
lest ye fugeddabowdit,
http://wlym.com/~animations/fermat/Observations20on%20Diophantus.pdf
thus:
well, that was a _Peter_ Michelson. He and
Smolin are some kind of freaked, that they'd worry
about the idea of the index of refraction, varying
for different kinds of Newtonian "photons;" but,
how can a zero-mass point-particle have a frequency, or
a wavelength?
--McSudan Crusades for carbon credits!?!
http://wlym.com/campaigner/8011.pdf |
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| Edgar E. Escultura... |
| Posted: Thu Nov 05, 2009 7:57 pm |
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"So this is what happens when engineers deal with mathematics and
things way over their heads...
You're an idiot."
Name calling based on wrong information not only reveals intellectual inadequacy and emptiness at the top but also racism.
E. E. Escultura
Research Professor
GVP - V. Lakshmikantham Institute for Advanced Studies
and Departments of Mathematics and Physics
GVP College of Engineering, JNT University |
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| Edgar E. Escultura... |
| Posted: Thu Nov 05, 2009 8:15 pm |
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"Escultura is certainly no engineer! He is in fact supposed to be a trained mathematician (in this case very poorly trained) with a PhD degree in maths from UW-Madison in the 70's under the supervision of L. C. Young. You'll find many graduate engineering
students who grasp math better than this guy."
Just as those who cannot do mathematics comment or philosophise about mathematics, those who cannot rebut a comment makes empty claim about the commenter. E. E. Escultura |
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| Edgar E. Escultura... |
| Posted: Thu Nov 05, 2009 8:27 pm |
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"he used to couch it in the "nonequality
of 1.0000... and 0.9999...," which notion he seems
to have dropped."
Read on and there is a summation of the whole debate regarding 1 and 0.99...; 1 and 0.99... are distinct objects like apple and orange and to write spple = orange is simply nonsense. Go beyond SciMath into the universe of blogs and websites and you'll find the definitive resolution of this issue. E. E. Escultura |
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| Edgar E. Escultura... |
| Posted: Thu Nov 05, 2009 8:44 pm |
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| Just as those who cannot do mathematics comment or philosophise about mathematics or comment about its history, those who cannot rebutt a comment makes empty claims about the commenter. E. E. Escultura |
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| Edgar E. Escultura... |
| Posted: Thu Nov 05, 2009 8:49 pm |
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Read on and you'll find my summation of the debate on 1 and 0.99... and related issues or go beyond SciMath and you'll find definitive resolution of the issue. At any rate, 1 and 0.99... are distinct objects like apple and orange and to write apple = orange is simply nonsense.
E. E. Escultura |
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| Justin Benfield... |
| Posted: Thu Nov 05, 2009 9:49 pm |
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First of all 0.999...=1, provided of course we are speaking about the real numbers and not some alternate set of numbers. There are many ways to prove this, if you agree that 0.333....=1/3, then 3*(1/3)=(3/1)*(1/3)=(3*1)/(1*3)=3/3=1/1=1 and 3*(1/3)=3*0.333....=0.999.... which implies 0.999...=1. 0.999... and 1 are distinct notations but a quantity has many possible ways of being written. Another way to prove 0.999...=1 is to use a summation, from the definition of decimal notation 0.999...=(9/(10^1))+((9/(10^2))+(9/(10^3))+... which can be written as a summation of (9/(10^n)) as n goes from 1 to infinity, it isn't too difficult to evaluate this sum.
FLT states that x,y,z are positive nonzero integers so the come from the set {1,2,3,4,...}, and n is a positive integer greater than 2. Andrew Wiles successfully proved the theorem in 1994-1995. |
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| spudnik... |
| Posted: Fri Nov 06, 2009 9:51 am |
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let's not cite Wiles' proof -- too much work
to "get" that. it's also nice to prove
that "one equals dot-nines," but
it was really created by Stevin's "decimals,"
there-shown as the only ambiguity.
the real joke is that EEE fails to give us a "real one,"
1.0000..., ro compare to a real 0.9999... -- alas.
thus:
what the Hey?...
EEE's result is just the old "trivial" one,
a triple of (1,0,1), a degenerate pythagorean trigon
(if using "two" for the exponent, n).
he used to couch it in the "nonequality
of 1.0000... and 0.9999...," which notion he seems
to have dropped.
lest ye fugeddabowdit,
http://wlym.com/~animations/fermat/Observations20on%20Diophantus.pdf
thus:
well, that was a _Peter_ Michelson. He and
Smolin are some kind of freaked, that they'd worry
about the idea of the index of refraction, varying
for different kinds of Newtonian "photons;" but,
how can a zero-mass point-particle have a frequency, or
a wavelength?
--McSudan Crusades for carbon credits!?!
http://wlym.com/campaigner/8011.pdf |
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| Pubkeybreaker... |
| Posted: Fri Nov 06, 2009 1:23 pm |
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On Nov 5, 6:59 am, Vindicator2009 <Vindicator2... at (no spam) live.com> wrote:
[quote]On Nov 5, 12:21 pm, "Edgar E. Escultura" <escultu... at (no spam) yahoo.com> wrote:
CLARIFICATION OF THE COUNTEREXAMPLES TO FERMAT’S LAST THEOREM
By E. E. Escultura
Although all issues related to the resolution of Fermat’s last theorem have been fully debated worldwide since 1997 and NOTHING had been conceded from my side I have seen at least one post expressing some misunderstanding. Let me, therefore, make the following clarification:
1) The decimal integers N.99… , N = 0, 1, …, are well-defined nonterminating decimals among the new real numbers [8] and are isomorphic to the ordinary integers, i.e., integral parts of the decimals, under the mapping, d* -> 0, N+1 -> N.99… Therefore, the decimal integers are integers [3]. The kernel of this isomorphism is (d*,1) and its image is (0,0.99…). Therefore, (d*)^n = d* since 0^n = 0 and (0.99…)^n = 0.99… since 1^n = 1 for any integer n > 2.
2) From the definition of d* [8], N+1 – d* = N.99… so that N.99… + d* = N+1. Moreover, If N is an integer, then (0.99…)^n = 0.99… and it follows that ((0.99,..)10)^N = (9.99…)10^N, ((0.99,..)10)^N + d* = 10^N, N = 1, 2, … [8].
3) Then the exact solutions of Fermat’s equation are given by the triple (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,
x^n + y^n = z^n, (F)
for n = NT > 2. The counterexamples are exact because the decimal integers and the dark number d* involved in the solution are well-defined and are not approximations.
4) Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false [8]. They are exact solutions, not approximation. One counterexample is, of course, sufficient to disprove a conjecture.
The following references include references used in the consolidated paper [8] plus [2] which applies [8]
References
[1] Benacerraf, P. and Putnam, H. (1985) Philosophy of Mathematics, Cambridge University Press, Cambridge, 52 - 61.
[2] Brania, A., and Sambandham, M., Symbolic Dynamics of the Shift Map in R*, Proc. 5th International
Conference on Dynamic Systems and Applications, 5 (2008), 68–72.
[3] Corporate Mathematical Society of Japan , Kiyosi Itô, Encyclopedic dictionary of mathematics (2nd ed.), MIT Press, Cambridge, MA, 1993
[4] Escultura, E. E. (1997) Exact solutions of Fermat's equation (Definitive resolution of Fermat’s last theorem, 5(2), 227 – 2254.
[5] Escultura, E. E. (2002) The mathematics of the new physics, J. Applied Mathematics and Computations, 130(1), 145 – 169.
[6] Escultura, E. E. (2003) The new mathematics and physics, J. Applied Mathematics and Computation, 138(1), 127 – 149.
[7] Escultura, E. E., The new real number system and discrete computation and calculus, 17 (2009), 59 – 84.
[8] Escultura, E. E., Extending the reach of computation, Applied Mathematics Letters, Applied Mathematics Letters 21(10), 2007, 1074-1081.
[9] Escultura, E. E., The mathematics of the grand unified theory, in press, Nonlinear Analysis, Series A:
Theory, Methods and Applications; online at Science Direct website
[10] Escultura, E. E., The generalized integral as dual of Schwarz distribution, in press, Nonlinear Studies.
[11] Escultura, E. E., Revisiting the hybrid real number system, Nonlinear Analysis, Series C: Hybrid Systems, 3(2) May 2009, 101-107.
[12] Escultura, E. E., Lakshmikantham, V., and Leela, S., The Hybrid Grand Unified Theory, Atlantis (Elsevier Science, Ltd.), 2009, Paris.
[13] Counterexamples to Fermat’s last theorem,http://users.tpg.com..au/pidro/
[14] Kline, M., Mathematics: The Loss of Certainty, Cambridge University Press, 1985.
E. E. Escultura
Research Professor
V. Lakshmikantham Institute for Advanced Studies
GVP College of Engineering, JNT University
Madurawada, Vishakhapatnam, AP, Indiahttp://users.tpg.com.au/pidro/
So this is what happens when engineers deal with mathematics and
things way over their heads...
You're an idiot.- Hide quoted text -
[/quote]
I don't think that he is smart enough to qualify
as an idiot. |
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| Edgar E. Escultura... |
| Posted: Fri Nov 06, 2009 4:38 pm |
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Guest
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| 1 and 0.99... are distinct objects in the real world called decimals. As distinct objects they are like apple and orange and to write apple = orange is certainly nonsense. E. E. Escultura |
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| Edgar E. Escultura... |
| Posted: Fri Nov 06, 2009 4:58 pm |
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Guest
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1/3 is a binary operation that maps the pair of decimals (1,3) to the decimal 0.33...; strictly speaking, 1/3 and 0.33... are not equal; rather, 0.33... is the image of 1/3. Thus, the equation, 1/3 = 0.33... is sloppy mathematics. Moreover, one cannot multiply or add nonterminating decimals because either operation needs the last digit on the right. One can only approximate the reult. Therefore, the equation, 3(1/3) = 0.99..., is incorrect because the left side of the equation is only an approximation of the right side.
BTW, this comment is well informed and so I respond in the same manner. E. E. Escultura |
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