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October 2012

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[math-fun] A causes B causes A
by Eric Angelini 03 Oct '12

03 Oct '12

Hello Math-fun, I've received this interesting press release yesterday and would like to share it with you today (or the contrary?) Best, É. http://www.eurekalert.org/pub_releases/2012-10/uov-qcr100212.php

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Re: [math-fun] is your Hurwitz # unlisted?
by Bill Gosper 01 Oct '12

01 Oct '12

BG> By uncontrived I meant not explicitly constructed as a continued fraction. NeilB>Just my two cents:http://ttk.pte.hu/mii/html/pannonica/index_elemei/mp17-1/mp17-1-091-110.pdf claims that (1;2,4,8,16...) is equal to 1+Sum[1/((-1)^n*2^(1 + n)^2*QPochhammer[4, 4, n]), {n, 0, Infinity}]/ Sum[1/((-1)^n*2^n^2*QPochhammer[4, 4, n]), {n, 0, Infinity}] (The source, Takao Komatsu's "Hurwitz and Tasoev Continued Fractions with Long Period" cites another paper, "On Hurwitzian and Tasoev's Continued Fractions", by the same author, but which I can't seem to find) --Neil Bickford On Thu, Sep 27, 2012 at 10:33 AM, Allan Wechsler <acwacw(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=acwacw%40gmail.com>> wrote:>> If x is Hurwitz then so is x/2, so, uh ...>> I think that by "contrived" Gosper means "with cf terms explicitly chosen> to be non-periodic". An example would be (1;2,4,8,16...). I'm assuming> that there is not a single such example that anybody knows a "closed form"> for; the closed form thereof would answer Gosper's query.>> On Thu, Sep 27, 2012 at 12:54 PM, meekerdb <meekerdb(a)verizon.net <http://gosper.org/webmail/src/compose.php?send_to=meekerdb%40verizon.net>> wrote:>> > Is the smallest contrived Hurwitz, thereby uncontrived? :-)> >> > Brent> >> >> > On 9/27/2012 5:38 AM, Bill Gosper wrote:> >> >> Can someone name a single (uncontrived) constant (e.g., π, e^3, 2^(1/3),> >> parity number,...)> >> that is provably nonHurwitz? And if Hurwitz, is not homographically> >> equivalent to a single-> >> mover, linear?> >>> >> Boy, are we ignorant.> >> --rwg YOW! This paper has the general "single-mover" formula, and also has double movers! --rwg Out[584]= 1 Tanh[-------] Sqrt[2] ------------- Sqrt[2] In[585]:= hursigs2[%584, 99, 1] During evaluation of In[585]:= 0.006964 Out[585]= {1, 0 1 } 1 0 2 {3 + 4 k, 10 + 8 k} In[586]:= hursigs2[%584, 99, 2] During evaluation of In[586]:= 0.035164 Out[586]= 0 2 {3, 1 1 8 {1, 1, 8 k, 1, 1, 1 + 4 k, 20 + 32 k, 3 + 4 k}} 0 2 1 0 8 {1, 1, 3 + 4 k, 36 + 32 k, 5 + 4 k, 1, 1, 12 + 8 k} 0 1 2 0 2 {1 + 4 k, 6 + 8 k} In[587]:= hursigs2[%585, 99, 3] During evaluation of In[587]:= 0.082445 Out[587]= 0 3 1 2 14 {1, 2, 8 k, 2, 1, 4 k, 2, 1, 2 + 8 k, 1, 2, 2 + 4 k, 54 + 72 k, 3 + 4 k} {4, } 0 3 1 1 14 {1, 2, 3 + 4 k, 2, 1, 8 + 8 k, 45 + 36 k, 11 + 8 k, 2, 1, 5 + 4 k, 1, 2, 13 + 8 k} 0 3 1 0 10 {1, 3 + 4 k, 78 + 72 k, 5 + 4 k, 102 + 72 k, 6 + 4 k, 2, 1, 13 + 8 k, 2} 0 1 3 0 10 {1, 2, 4 k, 1, 2, 3 + 8 k, 21 + 36 k, 6 + 8 k, 33 + 36 k, 8 + 8 k} In[588]:= hursigs2[%585, 99, 4] During evaluation of In[588]:= 0.087023 Out[588]= 0 4 {7, 1 3 8 {1, 3, 4 k, 3, 1, 2 k, 40 + 64 k, 1 + 2 k} } 0 4 1 2 12 {1, 1, 2 k, 3, 1, 1 + 4 k, 1, 3, 1 + 2 k, 1, 1, 17 + 16 k} 0 4 1 1 12 {1, 1, 1 + 2 k, 3, 1, 3 + 4 k, 1, 3, 2 + 2 k, 1, 1, 25 + 16 k} 0 4 1 0 8 {1, 3, 2 + 4 k, 3, 1, 1 + 2 k, 72 + 64 k, 2 + 2 k} 0 2 2 1 8 {1, 1, 4 k, 12 + 32 k, 2 + 4 k, 1, 1, 6 + 8 k} 0 2 2 0 2 {3 + 4 k, 10 + 8 k} 0 1 4 0 8 {1, 1, 2 + 4 k, 28 + 32 k, 4 + 4 k, 1, 1, 10 + 8 k}

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