Description:
Mathematical discussions and pursuits.
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find trig identities
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EXAMPLE Solve, e^(inx) = me^(ix) sin(nx)=msinx cos(nx)=mcosx tan(nx)=tanx nx=(x+kpi), k=1,2,3,..., x(n-1)=kpi x=kpi/(n-1) e^(iknpi/(n-1))=me^(ikpi/(n-1) ) e^in = [m^((n-1)/kpi)]e^i e^(i(n-1)) = m^((n-1)/kpi) e^i = m^(1/kpi)=cos(1)+isin(1) m = e^(ikpi)=cos(kpi)+isin(kpi) k=1,2,3,.. m=-1,1,-1,1,.., isin(1)=m^(1/kpi)-cos(1)... more »
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Forumula for the Roots to the nth Degree Polynomial
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FORMULA FOR THE ROOTS TO THE nth DEGREE POLYNOMIAL a[0] + a[1]x + a[2]x^2 + a[3]x^3 + ... + a[n]x^n = 0 x = { (C^( (j-1)/n ))*S }^(1/(n-j)) j=1,2,3,...,n-1 ALL SUMS ARE FROM p=1 to n C = { 1-(a[0]*SUM{a[p]})/(SUM{(a[p]) ^2}) }^(1/n) D = -a[0]/SUM{a[p]*(C^(p-1))} S = { (1/n)*SUM{(D^(2n-2p))*(C^(2p-2 ))} }^(1/2)... more »
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DTFT versus CFT
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Assume a bandlimited signal xs(t). If we sample that signal in a way satisfying the Nyquist sampling theorem we end up with a discrete sequence xd[n] = xs(n*T), with T being the sampling period. Assume the DTFT of xd[n] is Xd(w), with w being the normalized frequency. Then the CFT of the original signal xs(t) is T * Xd(w*T).... more »
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Prime gaps
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If one forms the raito p/q of all primes p and q is there anything know about the distribution? I plotted a histogram and there is regular gaps between clusters and an obvious enevlope.
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A formula for a sum
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Let n be a positive integer. I need to write F(i,j,n)= -2(j-i)^n + (i-1) sum_{k=0}^{n-1} (j-i)^k (j-i+1)^{n-1-k} in terms of x_1=(i-1), x_2=(i-1)(i-2),...., x_n=(i-1)(i-2)...(i-n). For n=1: F= 3(i-1)-2(j-1) = 3x_1 - 2(j-1) n=2: F= -4x_2 + (6j-9)x_1 - 2(j-1)^2 n=3: F= 5x_3 - 12(j-2)x_2 + 3(3j^2-9j+7)x_1 - 2(j-1)^3.... more »
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Scientists agree on this
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Every probabilistic problem from orthodox mathematics can be reworded and solved by discarding the random variable and replacing it with considerations based on existential indeterminacy and of course conservation. One can compose existent magnitudes with nonexistent magnitudes to obtain existentially indeterminate magnitudes.... more »
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US Thanksgiving on 11/25 future
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I have a relative born on 11/25. Next year 11/25 will be Thanksgiving. What are the future dates 11/25 up to the year 2102? Tim
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simple function approximation
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Assuming it is predicted that when two players compete in a best-of-5 match, the weaker competitor has a probability p of winning the match. The probability of the weaker player winning a best-of-1 match is the solution of x^5 + 5*x^4*(1-x) + 10*x^3*(1-x)^2 = p ; or, more simply, 6*x^5 - 15*x^4 + 10*x^3 - p = 0... more »
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