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Fermat and Space

  1. Feb 1, 2004 #1
    x^n + y^n = z^n

    x^3 + y^3 = (x+y)[x^2-xy+y^2]

    A more general identity for primes p, and numbers, n:

    A*B means A times B

    A/B means A divided by B

    + and - , we all know...

    follow the order of operations:


    x^p + y^p

    equals

    (x+y)*[(x+y)^(p-1) + [{(x^p+y^p)/(x+y)} - (x+y)^(p-1)] ]

    for all p and n >= 1

    Question:

    Let T be a metric space with distance function r(x,y) expressing the definitive predication that involves T with the real numbers, R. Therefore the juxtaposition of left and right hemispheres resonates in perfect accordance with the proposition that T and R are embedded simultaneously in the full structure of manifold M. Ergo we pass on to an enlargement *M of M, whereby the non-standard metric space is diffeomorphism invariant.

    So if f(x) is a homeomorphism from T onto S, then for every point p in T, does f(u(p) = u(f(p)) ?


    A metric space is a set of points such that for every pair of points, there is a nonnegative real number called their distance that is symmetric, and satisfies the triangle inequality, which states that the sum of the measures of any two sides of any triangle is greater than the measure of the third side. Space is then a tranformation[invariant]. Two objects with relative velocity will have a relative measure that transforms into the other. In effect, the separation does not exist in an extrinsic sense. ABC = BCA = CAB

    Utilizing the generalized equation:

    x^3 + y^3 = (x+y)*[(x+y)^2 - 3xy]

    x^5 + y^5 = (x+y)*[(x+y)^4 -5x^3 y -5(xy)^2 -5x y^3 ]

    x^7 + y^7 = (x+y)*[(x+y)^6 -7yx^5 -14x^4 y^2 -21(xy)^3 -14x^2 y^4 - 7xy^5 ]

    In general, x^p + y^p = (x+y)*[(x+y)^(p-1) - p*f(x,y) ]
     
  2. jcsd
  3. Feb 2, 2004 #2

    HallsofIvy

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    Well, I don't know. You didn't tell us what "u" is.

    "relative velocity"? How did "velocity" get into a discussion of metric spaces? In any case, I have no idea what you mean by "ABC = BCA= CAB" because you have not defined ABC, BCA, or CAB.

    What are x and y? They are not points in the metric space you mentioned before because such operations are not defined in a general metric space. They certainly aren't numbers that equation simply isn't true for numbers.
     
  4. Feb 2, 2004 #3

    matt grime

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    Erm, what the hell are half the words you use meant to mean?

    Definitive predication that involves T with the real numbers????


    What's S?

    Why does your metric space have a ring structure?

    Where did M come from? why must R embed into it? How does a general metric space embed into some manifold?
     
  5. Feb 2, 2004 #4
    what precisly are you trying to make a point of here?

    your title has Fermat in it however, no where do you use Fermat's last theorem in anything, you only use the function form.
     
  6. Feb 3, 2004 #5
    Very interesting:

    http://www.space.com/scienceastronomy/time_theory_030806.html
    quote:
    --------------------------------------------------------------------------------
    "There isn't a precise instant underlying an object's motion," he said. "And as its position is constantly changing over time -- and as such, never determined -- it also doesn't have a determined position at any time."
    --------------------------------------------------------------------------------
    Heisenberg uncertainty: DxDp >= hbar/2 As the observed expansion of the universe continues, and, as the mean temperature of the universe continues to approach absolute zero, could the universe transform into a condition analogous to a "Bose Einstein condensate"?
    At the foundation of classical logic is the law of excluded middle,
    A|~A|A V ~A
    T_|F||__T
    F_|T||__T
    This forms an invariance principle which is a symmetry
    T|F = F|T = T

    John Nash gives a most excellent equation for spacetime:


    http://www.math.princeton.edu/jfnj/texts_and_graphics/Equation_an_Interesting/note2
    http://www.math.princeton.edu/jfnj/texts_and_graphics/Equation_an_Interesting/equation.gif
    http://www.math.princeton.edu/jfnj/texts_and_graphics/


    Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness[qualia]. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "red" still describes the set. The distributive identity[attribute] "natural number" or "real number" describes an entire collection of individual objects.

    The alephs can be set into a one to one correspondence with a proper subset of of themselves. The "infinite" Cantorian alephs are really distributive[qualia].

    Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it[wave function collapse-condensation into specific localization].

    Symmetry is analogous to a generalized form of self evident truth, and it is a distributive attribute via the laws of nature, being distributed over the entire system called universe. A stratification of Cantorian alephs with varying degrees of complexity. Less complexity = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called Universe and gives it "identity".

    The highest symmetry is a distributive mathematical identity[also a total unknown but possibly analogous to a state of "nothingness"]. This fact is reflected in part, by the conservation laws.

    So an unbound-infinite-potentia and a constrained-finite-bound-actuality, are somehow different yet the same. The difference and sameness relation is a duality. Freedom(higher symmetry) and constraint-complexity-organizational structure(lesser symmetry) form a relation that can be described by an invariance principle.

    On a flat Euclidean surface, the three angles of a triangle sum to 180 degrees. On the curved surface of a sphere, the three angles add up to more than 180 degrees. On the hyperbolic surface of a saddle they sum to less than 180 degrees. The three types of surface are not equivalent.

    There is a rotational invariance for a triangle, that seems to hold for the three types of surface though.

    ABC = BCA = CAB

    CBA = BAC = ACB


    [ abstract representation]--->[semantic mapping]--->[represented system]


    An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

    A duality.

    A point contains an infinite expanse of space and time?

    Could it be, that the "absolute" infinity, is actually a dimensionless point? Or more correctly, an "infinitesimal"?

    Universe? = Zero?


    On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.

    Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.

    According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."

    A point can be defined as an "infinitesimal". The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system.

    Waves are then abstract distributions and particles are convergent "concrete" localizations.

    Quantum mechanics leads us to the realization that all matter-energy can be explained in terms of "waves". In a confined region(i.e. a closed universe or a black hole) the waves exists as STANDING WAVES In a closed system, the entropy never decreases.

    The analogy with black holes is an interesting one but if there is nothing outside the universe, then it cannot be radiating energy outside itself as black holes are explained to be. So the amount of information i.e. "quantum states" in the universe is increasing. We see it as entropy, but to an information processor with huge computational capabilities, it is compressible information.

    Quantum field theory calculations where imaginary time is periodic, with period 1/T are equivalent to statistical mechanics calculations where the temperature is T. The periodic waveforms that are opposed yet "in phase" would be at standing wave resonance, giving the action.

    Periodicity is a symmetry. Rotate into the complex plane and we have
    real numbers on the horizonal axis and imaginary numbers on the
    vertical axis. So a periodic function could exist with periodicity
    along both the imaginary AND the real axis. Such functions would have
    amazing symmetries. Functions that remain unchanged, when the complex
    variable "z" is changed.

    f(z)---->f(az+b/cz+d)

    Where the elements a,b,c,d, are arranged as a matrix, forming an
    algebraic group. An infinite number of possible variations that
    commute with each other as the function f, is invariant under group
    transformations. These functions are known as "automorphic forms".

    Topologically speaking, the wormhole transformations must be
    invariant with regards to time travel. In other words, by traveling
    backwards in time, we "complete" the future, and no paradoxes are
    created.

    So when spacetime tears and a wormhole is created, it must obey
    certain transformative rules, which probably appear to be
    discontinuities from a "3-D" perspective, but really, these
    transformations are continuous!

    So the number of holes[genus] on the surface of space, determine
    whether there exist an infinite, or finite, number of solutions to
    the universal equations?
     
  7. Feb 3, 2004 #6
    ok, now I get it.

    since X,Y, and Z are irrational numbers according to Fermat's Theorem, then at any given time, you can not be measured at a single point in space.

    that makes sense.
     
  8. Feb 3, 2004 #7

    matt grime

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    wow,russell, that must count as the longest and most off topic reply in the history of everything ever in forums.

    Nash, quantum, qualia, GR, infinity, completeness.

    Shame none of them bears any relation to the criticisms of your post.
     
  9. Feb 4, 2004 #8
    [?] The "equation simply isn't true for numbers"???

    x^3 + y^3 = (x+y)*[(x+y)^2 - 3*x*y ]

    3^3 + 4^3 = (3+4)*[(3+4)^2 - 3*3*4 ]

    91 = 7*[49 - 36]

    91 = 7*13

    x^3 + y^3 = (x+y)*[(x+y)^2 -3xy] is true for numbers

    The metric space has distance function r(x,y), definitively characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles.

    We move on to functions of a complex variable in the 2-D Euclidean space Z, utilizing the algebraic structure of complex numbers, where points of Z may be regarded as pairs of real numbers, R. Formal statements concerning Z, are also expressable as statements about R.

    Complex-valued functions e.g. w = f(z), can also be represented as binary, or quaternary relations, these sets of complex valued polynomials then have elements that map Z into Z. Every complex valued polynomial subset p(z), has natural number coefficients a_k, or the "kth" coefficient of p(z)..

    So, for non-zero polynomials, we take into account the number of zeros OF the polynomial O[p(z)] with O ranging over the natural numbers such that
    O[p(z), R ] = v, where v is a finite natural number.

    Thus if G is a metric group and E is a topological group, such, that an open neighborhood U, of the identity is a metric space, with a compliant metric, specifiable to the topology of U, then the distance r(a,b) between any two points (a,b) in U, then in terms of the distance u(E) = J, consists of the points a of U with the fact that r(a,E) is an infinitesimal.
     
  10. Feb 4, 2004 #9
    fermat's theorem is true

    it is true that there are no rational solutions for x and y and z.

    that means that there are irrational solutions for x or y or z. (aint logic grand)

    any way, there are real solutions, however at least one must be irrational.
     
  11. Feb 4, 2004 #10

    matt grime

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    "The metric space has distance function r(x,y), definitively characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles."

    Would you care to elaborate? Let X be some abstract space (possibly even containing a proper subclass) with the trivial metric (r(x,y) =0 if x=y 1 otherwise.) How does one embed this in a manifold? which manifold, what do you mean by embed? I mean as X is not a manifold, you don't mean the usual embedding (or immersion or submersions) of differential geometry. As the metric only takes exactly two values, how do you relate it to all of R? Must every manifold contain an *embedded* copy of R? I can't remember the explicit definititions for embeddings, you see - I was never very good at memorizing all those things, you know, an immersion that's not this is not the other.
     
  12. Feb 8, 2004 #11
    this is pure insanity . or is it pure genius ?? i leave the details of the proof of either to the reader .
     
  13. Feb 8, 2004 #12
    If f(x) is a homeomorphism from T onto S, and for every point p in T, f(U(p)) = U(f(p)), and the monad is invariant under standard topological transformations, with the caveat that the definition also comprizes a type of dynamic situation sematics, where concepts, such as "proper set", "ordinal" and "cardinal" are relativised to context, taking care of paradox at all levels via symmetry, or an invariant many-valued logic, and the "top[set of all sets]", would naturally not exist, of course, since there is nothing outside the universe. So it becomes an infinite chain or composition of ever more inclusive situated sets expressing an interesting informational -topological dynamic.


    Outside of "Total Existence"[TE] there is nothing. This is an irrefutable fact. Or we could say that there IS not an "outside" of Total Existence.

    Therefore the largest possible set does not exist, where "does not exist" is equivalent to "nothing".

    Nothing contains everything, as Russell's paradox so eloquently puts it.

    Before the beginning there is nothing. After the end, there is nothing.

    ERGO

    Alpha = Omega.
     
  14. Feb 8, 2004 #13
    if alpha is 0 and U is omega, where U is the universal set, then
    {0}=1~U where ~ is 'hemitopy'.

    ergo
    alpha = unity = omega (depending on what you mean by =)
     
  15. Feb 8, 2004 #14

    matt grime

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    I hope your last post was tongue in cheek, Phoenixtoth.


    And, Russell, are you going to explain anything ever or just keep posting wildly unrealistic statements that are lacking any meaningful content?
     
  16. Feb 8, 2004 #15
    yeah, it was. there are about 50 million things wrong with what i wrote but i was mimicking russel's style for a second.
     
  17. Feb 8, 2004 #16
    If space is *quantized* yet also continuous, then it too, has the property called "wave-particle" duality. If space consists of indivisible units, then a measurement of space means that Fermat's last theorem holds, for it.

    According to the Pythagorean theorem:

    x^2 + y^2 = z^2

    All possible integer solutions are then rerpresented as:

    [a^2 - b^2]^2 + [2ab]^2 = [a^2 + b^2]^2

    a^4 -2(ab)^2 + b^4 + 4(ab)^2 =

    a^4 + 2(ab)^2 + b^4 = [a^2 + b^2]^2




    all odd numbers can be represented as:

    [a^2 - b^2] or Z^p - Y^p

    if Y is an "even" natural n and Z is odd, same for a and b .

    Fermat's last theorem, for integers a,b,Z,Y,p:

    [a^2 - b^2]^p + Y^p = Z^p

    [a^2 - b^2]^p = Z^p - Y^p

    a^2 - b^2 = [Z^p - Y^p]^[1/p]

    When Z^p - Y^p is a prime number, it cannot have an integer root.

    a^2 - b^2 is not an integer, for [Z^p - Y^p]^[1/p] , for a,b,Z,Y,p, unless p = 2.


    To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This is the axiom that leads to Russell's paradox. For if we let the condition S(x) be: not(x element of x), then B = {x in A such that x is not in x}. Is B a member of B? If it is, then it isn't; and if it isn't, then it is. Therefore B cannot be in A, meaning that nothing contains everything.

    This means that relativity holds in the "topological" sense and T-duality is correct.

    Quantum entities are described as probability distributions, which are attributes of an underlying phase space, where the properties-attributes such as "spin" and "charge" are not the attributes of individual particles, but they are universally distributive entities, being the attributes of a "coherent wave function". It is this wave-distribution property that then "decoheres" into the ostensible "wave function collapse", as waves become localized particles that are "in phase" creating standing-spherical-wave resonances, which are condensations of space itself. The continual collapse-condensation of space into matter-energy is the continual "change", i.e. the property called "time". The spherical waves, or probability distributions are represented by the Schrodinger wave function, "psi".


    The information density of the universal system must be increasing. The increase of information density is analogous to a pressure gradient.

    [density 1]--->[density 2]--->[density 3]---> ... --->[density n]


    [<-[->[<-[-><-]->]<-]->]

    Intersecting wavefronts = increasing density of spacelike slices

    As the wavefronts intersect, it becomes a mathematical computation:

    2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n


    According to conventional theories, the surface area of the horizon surrounding a black hole, measures its entropy, where entropy is defined as a measure of the number of internal states that the black hole can be in without looking different to an outside observer, who must measure only mass, rotation, and charge. Another theory states that the maximum entropy of any closed region of space can never exceed one quarter of the area of the circumscribing surface, with the entropy being the measure of the total information contained by the system.

    S' = S_m + A/4

    So the "black hole" theorists came to realize that the information associated with all phenomena in the three dimensional world, can be stored on a two dimensional boundary, analogous to the storing of a holographic image.

    Since entropy can also be defined as the number of states, that particles can be in within within a region of space, and the entropy of the universe must always increase, the next logical step is to realize that the spacetime density, i.e. the information encoded within a circumscribed region of space, must be increasing in the thermodynamic direction of time.

    Of course, thermodynamic entropy is popularly described as the disorder or "randomness" in a physical system. In 1877, the physicist Ludwig Boltzmann defined entropy more precisely. He defined it in terms of the number of distinct microscopic states that the particles in a system can be configured, while still looking like the same macroscopic system. For example, a system such as a gas cloud, one would count the ways that the individual gas molecules could be distributed, and moving.

    In1948, mathematician Claude E. Shannon, introduced today's most widely used measure of information content: entropy. The Shannon entropy of a message is the number of binary digits, i.e. "bits" needed to encode it. While the structure, quality, or value, of the information in Shannon entropy may be an unknown, the quantity of information can be known. Shannon entropy and thermodynamic entropy are equivalent.

    The universal laws of nature are explained in terms of symmetry. The completed infinities, mathematician Georg Cantor's infinite sets, could be explained as cardinal identities, akin to "qualia" [Universally distributed attributes] from which finite subsets, and elements of subsets [quantum decoherence-wave function collapse] can be derived.

    Completed infinities, called "alephs" are distributive in nature, similar to the way that a set of "red" objects has the distributive property of redness[qualia]. Properties, or "attributes" like red are numbers in the sense that they interact algebraically according to the laws of Boolean algebra. Take one object away from the set of red objects and the distributive number "red" still describes the set. The distributive identity[attribute] "natural number" or "real number" describes an entire collection of individual objects.

    The alephs can be set into a one to one correspondence with a proper subset of of themselves. The "infinite" Cantorian alephs are really distributive[qualia].

    Yet, if we have a finite set of 7 objects, the cardinal number 7 does not really distribute over its individual subsets. Take anything away from the set and the number 7 ceases to describe it[wave function collapse-condensation into specific localization].

    Symmetry is analogous to a generalized form of self evident truth, and it is a distributive attribute via the laws of nature, being distributed over the entire system called universe. A stratification of Cantorian alephs with varying degrees of complexity. Less complexity = greater symmetry = higher infinity-alephs. So the highest aleph, the "absolute-infinity" distributes over the entire set called Universe and gives it "identity".

    The highest symmetry is a distributive mathematical identity[also a total unknown but possibly analogous to a state of "nothingness"]. This fact is reflected in part, by the conservation laws.

    So an unbound-infinite-potentia and a constrained-finite-bound-actuality, are somehow different yet the same. The difference and sameness relation is a duality. Freedom(higher symmetry) and constraint-complexity-organizational structure(lesser symmetry) form a relation that can be described by an invariance principle.

    On a flat Euclidean surface, the three angles of a triangle sum to 180 degrees. On the curved surface of a sphere, the three angles add up to more than 180 degrees. On the hyperbolic surface of a saddle they sum to less than 180 degrees. The three types of surface are not equivalent.

    There is a rotational invariance for a triangle, that seems to hold for the three types of surface though.

    ABC = BCA = CAB

    CBA = BAC = ACB

    According to Einstein, and the CTMU of Langan, www.ctmu.org , "space and time are modes by which we think, and not conditions in which we live". Space becomes abstract, a relation that is perceptual and "mental", where distance interval between two points becomes a mental perception.

    [ abstract representation]--->[semantic mapping]--->[represented system]


    An abstract representation is exactly that, "abstract". It is not a space, or time, but is instead a product of consciousness, or a mental construct. Topologically it is equivalent to a "point". The abstract description contains the concrete topology. Likewise, the concrete contains the abstract.

    A duality.

    A point contains an infinite expanse of space and time?

    Could it be, that the "absolute" infinity, is actually a dimensionless point? Or more correctly, an "infinitesimal"?

    Universe? = Zero?


    On one level of stratification, two photons are separate. On another level, of stratification, the photons have zero separation.

    Instantaneous communication between two objects, separated by a distance interval, is equivalent to zero separation[zero boundary] between the two objects.

    According to the book "Gravitation", chapter 15, geometry of spacetime gives instructions to matter telling matter to follow the straightest path, which is a geodesic. Matter in turn, tells spacetime geometry how to curve in such a way, as to guarantee the conservation of momentum and energy. The Einstein tensor[geometric feature-description] is also conserved in this relationship between matter and the spacetime geometry. Eli Cartan's "boundary of a boundary equals zero."

    A point can be defined as an "infinitesimal". The Topological spaces are defined as being diffeomorphism invariant. Intersecting cotangent bundles[manifolds] are the set of all possible configurations of a system, i.e. they describe the phase space of the system.
     
  18. Feb 8, 2004 #17
    i have neither the time nor the inclination nor the desire to refute your points. let's just take one:

    incorrect. an infinitesimal is bigger than a point yet smaller than every nondegenerate circle. it is correct to say it "Can" be defined that way, but no one usually does.
     
  19. Feb 8, 2004 #18
    http://www.fact-index.com/h/hy/hyperreal_number.html

     
  20. Feb 8, 2004 #19
    it's 'trival' to call 0 an infinitesimal. the hard part is to show that there are nonzero infinitesimals. now what would your quote become if one replaced 'infinitesimal' with 0? what would it say?
     
  21. Feb 8, 2004 #20
    In Newton's calculus 0/0 is replaced with:

    infinitesimal/infinitesimal.

    A point CAN be defined as an infinitesimal ;)
     
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