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Acme

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  1. Noting again the OEIS error in listing {1,2,3,4,5} with the set of non-polygonal numbers, I want to point out that while 3 is technically the second triangular number (and 1 is the first), it is proper to exclude them and all n= 1 or 2 values for all polygonals of side s. Otherwise, all numbers would be trivially polygonal. OEIS does properly show this restriction but does not follow it with the listing. Just so, and following the restriction, it can be proved that all multiples of 3 are polygonal and so no non-polygonal numbers are multiples of 3. Here's my proof.

     

     

    [math]F=frac{1}{2}(n^2s-2n^2-ns+4n)[/MATH]

     

    let n = 3

     

    let s = {2,3,4,5...} set of integers >=2

     

    [math]F=frac{1}{2}(3^2s-2*3^2-3s+4*3)[/MATH]

     

    [math]F=frac{1}{2}(9s-18-3s+12)[/MATH]

     

    [math]F=frac{1}{2}(6s-6)[/MATH]

     

    [math]F=(3s-3)[/MATH]

     

    [math]F=3(s-1)[/MATH]

     

    Δ, all multiples of 3 are figurate numbers

     

    Δ, no non-figurate numbers are multiples of 3

     

    (sorry for the latex errors; little help?)

  2. Just dropping in to drop off some eye-candy. The design is mine, the execution is computer generated by fella_2 and recolored by hand by moi. Not sure if spiral arrays are original to me, but I came up with the idea independently. The array begins in the middle, spirals out clockwise, and sequential cells represent the natural numbers. In this particular view the black cells are polygonal numbers and white not. Thanks for having a taste & hoping it's as fun to eat as it was to cook. :)

  3.  

     

    Excellent!! Thank you!! This has been a long-standing question for me/us. Here is my friend's version of a solution and another fella's variation for comparison. I will pass this all on to them as they came about the solutions independently. Coinikydinkily, the variant example I'm giving uses the same number 325 as the example on page 259 in the track you gave. I will be looking at the geometric analysis a bit more as we came at things only algebraically from the generalized expression for polygonals.

     

    fella 1: I'd handle the problem this way:

     

    F = 1/2((n^2)*s) - 2*n^2 - n*s + 4*n)

    147 = 1/2((n^2)*s) - 2*n^2 - n*s + 4*n)

    294 = ((n^2)*s) - 2*n^2 - n*s + 4*n)

     

    294 factorises as 1,2,3,7,7. One of those factors must be n

     

    checking n=1: 294 = s - 2 - s + 4 = 2: Clearly n=1 doesn't work

    checking n=2: 294 = 4s - 8 - 2s + 8 =2s: Gives s=147. A trivial result: n=2 goes in steps of 1, generating

     

    every number.

    checking n=3: 294 = 9s - 18 - 3s + 12 = 6s - 6: Gives 6s=300, s=50

    checking n=4: 294 = 49s - 98 -7s + 28 = 4ss - 70: Gives 42s=364, =8.6666666666666666666666666666667 which is

     

    fractional and therefore not a solution.

     

    answer: n=3, s=50.

     

    fella 2: alternate method given

     

    factors of 325*2

    650 2, 5, 10, 13, 25, 26, 50, 65, 130, 325

     

    650/5 = 130 -2 = 128/4 = 32

     

    650/10 = 65 -2 = 63/9 = 7 + 2 = 9

     

    650/13 = 50 -2 = 48/12 = 4 + 2 = 6

     

    650/25 = 26 -2 = 24/24 = 1 + 2 = 3

     

    basically, if a number, multiplied by 2 divides n, and then after division by n and subtracting 2, is

     

    divisible by n-1, then it's figurate. in short 2*F = n*((n-1)*a +2)

    which gon it is is determined after dividing by n-1. just add two the result, with n being the term.

     

    Fella 2 has also been kind enough to do some programming for me and set me up with portable Python so that I could generate sequential extensive lists of the polygonals which I have done on the interval 6 to 1000000000. In that interval I found the greatest "multiplicity" to be 17 at the number 879207616 with the following solutions [n,s]. [4, 146534604, 7, 41867031, 14, 9661624, 16, 7326732, 28, 2325948, 31, 1890771, 56, 570916, 58, 531888, 118, 127368, 236, 31708, 248, 28708, 406, 10696, 496, 7164, 518, 6568, 1711, 603, 2146, 384, 7192, 36]

     

    This is all rather incidental to my interest in the set of numbers which are not polygonal and perhaps I could impose on you further for any information/knowledge you might have of that set. ? The OEIS listing of the set gives credit for the set listing to a fella named Beiler but Beiler's cited work we have checked and found it does not give that set at all. (OEIS actually gives the set incorrectly in listing {1,2,3,4,5} as members in spite of the restrictions n & s >=3.)

     

    Anyway, this is a recreational pursuit of mine that keeps me off the streets and I sincerely appreciate the help and interest. If you do have an interest in the non-polygonal set, I have independent work o'plenty to share. Thanks again. :)

  4. Fundamentals of Diophantine Geometry by Serge Lang would be a good place to start.

     

    Thanks. I see the volume goes for $92.00 as an e-book so it could be some time before I could purchase it as I am retired & on a small fixed income. Google books gives only a few excerpts and they do not address my question. I doubt my local community library has it but I will check. My community has no universities either, though we do have a community college which I can check too.

     

    Do you know for a fact the solution(s) I seek are in the book? Do you have the book and if so are you willing to check it for me? Do you know of a solution or if one exists?

     

    So as not to appear too "dodgy" I should say that I do know a solution though it is not mine. It can be written in a short paragraph and does not involve any complex mathematics at all. What I want to get at is if there are any other extant solutions and if so who wrote them and when and how they compare to the solution I have.

     

    You [all] of course understand my reticence to believe Fermat's claim and I beg everyone's forebearance in not giving the solution I have for the time being. In part I want to avoid presenting a bias and while I do have permission to use it I would want to touch base with the author again before posting it here. (Assuming anyone is interested in such a thing.)

     

     

    Thanks again. :)

  5. THe OP is anything but specific. Keep in mind that this poster apparently believes in the ultra-crackpot nonsense of the expanding earth "theory": http://www.sciencefo...g-mass-resolved.

    How did I miss that!? I was taking the original question at face value. My mistake & thanks for the heads-up. Marking topic ultra-super-dooper crackpot nonsense. And I thought I had joined a science forum. D'oh.

  6. I understand there are exceptions but for the most part my model is a good description of reality... Just like sea ice, a huge ice berg can ride up over sea ice and sea ice can ride up over layers of it's self... In the ocean this process is driven by under water currents on the earth it is driven by moving molten basalt deep in the earth.

     

    In my opinion, the ice analogy is a poor model for plate tectonics. The OP is quite specific and i found all the replies lacking so I gave specific examples of subduction under oceans. Then too, not all spreading centers are oceanic, e.g. the East African Rift and the molten basalt is a result of the hot asthensophere rising and decompressing so the basalt is not the driver. The production of any igneous rock is more of a chemical matter than it is mechanical.

  7. The thing to remember is that continental plates are like icebergs floating in the sea... granite bergs floating in a sea of basalt... the molten basalt wells up in the middle of oceans along the mid oceanic ridge, this basalt spreads out and then subducts under the lighter granite continents...

     

    Again, not all subduction zones involve continental plates. For example, the Mariana islands are ocean volcanoes along a line where the Pacific Plate is subducting under the Mariana plate. Neither of these plates is continental.

     

    Mariana Plate

    The Mariana Plate is a small tectonic plate located west of the Mariana Trench and forms the basement of the Mariana Islands. It is separated from the Philippine Sea Plate by a long divergent boundary with numerous transform fault offsets. The boundary between the Mariana and the Pacific Plate to the east is a subduction zone with the Pacific Plate subducting beneath the Mariana....

     

     

    Mariana_Plate_map-fr.png

     

     

     

    Cross_section_of_mariana_trench.jpg

  8. Why doesn't it take place in the middle of the oceans if there are no rules governing how plates move?

     

     

    subduction does occur in the [middle of] oceans.

     

    Plate Tectonics: Island Arcs

    ...Examples of island arcs created in this way are the Aleutians, the Kuriles, Japan, the Ryukyus, and the Philippines, found clustered around the northern and western borders of the Pacific Plate like a necklace. There are other island arcs to the south (Indonesia and the Solomon's), and although scientists are still puzzled by the exact origin of these southern island arcs, plate subduction is the suspected architect.

    ...

  9. snip...

    And an add-on question, why is bacteria allowed in the digestive tract? Does the body recognize the bacteria as 'self', or is the digestive tract somehow not covered by the immune system?

     

    I don't know the answers to the snipped bit, but I do know something on the add on question in regard to bacteria in the human gut . I hope a wiki article will suffice; if not there are other links there to more in depth expositions. :

     

     

    Gut flora

    Gut flora consists of microorganisms that live in the digestive tracts of animals and is the largest reservoir of human flora. In this context, gut is synonymous with intestinal, and flora with microbiota and microflora.The human body, consisting of about 100 trillion cells, carries about ten times as many microorganisms in the intestines.[1][2][3][4] The metabolic activities performed by these bacteria resemble those of an organ, leading some to liken gut bacteria to a "forgotten" organ.[5] It is estimated that these gut flora have around 100 times as many genes in aggregate as there are in the human genome.[6]

     

    Bacteria make up most of the flora in the colon[7] and up to 60% of the dry mass of feces.[2] Somewhere between 300[2] and 1000 different species live in the gut,[3] with most estimates at about 500.[4][5][8] However, it is probable that 99% of the bacteria come from about 30 or 40 species.[9] Fungi and protozoa also make up a part of the gut flora, but little is known about their activities.

     

    Research suggests that the relationship between gut flora and humans is not merely commensal (a non-harmful coexistence), but rather a symbiotic relationship. ...

  10. ...Given that there are animals that are definately not aware, a duck for instance. This would automatically mean we can create a linear graph, unaware at the begining and humans (conceted I know but what else) at the finnish.

     

    My question is this, what type of animal is just aware. My thinking is perhaps an octopus it's a problem solveing animal but has to relearn the problem however many times it's presented.

     

    In his book I Am A strange Loop, Douglas Hofstader draws a downward pointing cone as a graph for self-awareness. the lower on the cone the less "soul" [say self awareness] a creature has. he labels the unit value "hunekers" after James Huneker who wrote a line containing the phrase "small souled men". Hofstadter rates dogs as having fewer hunekers than people, but more than mosquitoes. one has to make their own assesment about where each creature lies.

     

    the graph reads the same in a mirror, which would definitely please Hofstader even if he did not intend it. ;)

  11. i'm talking about incest between close relatives, eg, father/daughter, mother/son, brother/sister

     

    if it doesn't necessarily lead to birth defects, then what is the probability of it leading to defects?

     

    i don't have a degree or any other qualification in biology so if you can answer this can you put it in simple language please. thanks

     

    No, inbreeding does not necessarily lead to defects. Here is a chart from Wikipedia of some probabilities.

     

    The inbreeding is computed as a percentage of chances for two alleles to be identical by descent. This percentage is called "inbreeding coefficient". There are several methods to compute this percentage, the two main ways are the path method[10] and the tabular method.[11][unreliable source?]

     

    Typical inbreeding coefficient percentages are as follows, assuming no previous inbreeding between any parents:

     

    • Father/daughter, mother/son or brother/sister → 25%
    • Grandfather/granddaughter or grandmother/grandson → 12.5%
    • Half-brother/half-sister → 12.5%
    • Uncle/niece or aunt/nephew → 12.5%
    • Great-grandfather/great-granddaughter or great-grandmother/great-grandson → 6.25%
    • Half-uncle/niece or half-aunt/nephew → 6.25%
    • First cousins → 6.25%
    • First cousins once removed or half-first cousins → 3.125%
    • Second cousins or first cousins twice removed → 1.5625%
    • Second cousins once removed or half-second cousins → 0.78125%
    • Third cousins or second cousins twice removed → 0.390625 %
    • Third cousins once removed or half-third cousins → 0.195 %

    Inbreeding

  12. I-racked my brain and I thought of Conway's Life. Hope that's not off the topic but I hope it does stimulate questions. For me the garden in the Pentateuch is no more or less interesting than any other ancient myth, but I do find them mildly interesting. What will we think of next?

     

    In a cellular automaton, a Garden of Eden configuration is a configuration that cannot appear on the lattice after one time step, no matter what the initial configuration. In other words, these are the configurations with no predecessors.They resemble the concept of the Garden of Eden in Abrahamic religions, which was created out of nowhere, hence the name. According to Moore (1962), this name was coined by John Tukey in the 1950s.

     

    A Garden of Eden is a configuration of the whole lattice (usually a one- or two-dimensional infinite square lattice). Each Garden of Eden configuration contains at least one finite pattern (an assignment of states to a finite subset of the cells) that has no predecessor regardless of how the surrounding cells are filled. Such a pattern is called an orphan. Alternatively, an orphan is a finite pattern such that each configuration containing that pattern is a Garden of Eden.

    http://en.wikipedia....lular_automaton)

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