ABSTRACT Let P(x) be the generating function for the primes pn, that is P(x)=^\infty p_nx^n We may also define the invert transform as applied the a generating function G(x) as I[G(x)]={1-G(x)}-1 Let I(n)[G(x)] denote a repeated application of the invert transform n times. Then I^{(2)}[G(x)]={1-({1-G(x)}-1)}-1\\ I^{(3)}[G(x)]={1-({1-({1-G(x)}-1)}-1)}-1 We then have I^{(0)}[P(x)]=2x+3x^2+5x^3+7x^4+11x^5+\cdots\\ I^{(1)}[P(x)]=2x+7x^2+25x^3+88x^4+311x^5+\cdots\\ I^{(2)}[P(x)]=2x+11x^2+61x^3+337x^4+1863x^5+\cdots\\ I^{(3)}[P(x)]=2x+15x^2+113x^3+850x^4+6395x^5+\cdots\\ I^{(4)}[P(x)]=2x+19x^2+181x^3+1723x^4+16403x^5+\cdots\\ we can see that if we take all the coefficients of terms xm, m ∈ [1, ∞) we can generate further sequences S_1=2,2,2,2,2,2,2,\cdots\\ S_2=3,7,11,15,19,\cdots\\ S_3=5,25,61,113,181,\cdots\\ S_4=7,88,337,850,1723, \cdots\\ S_5=11,311,1863,6395,1640, \cdots We can then find the sequence function for each of these sequences, for example S_1(n)=2\\ S_2(n)=4n-1\\ S_3(n)=8n^2-4n+1\\ S_4(n)=16n^3-12n^2+5n-2\\ S_5(n)=32n^4-32n^3+18n^2-10n+3\\ These are then polynomials, such that Sk(1)=pk. We note that the constant terms are A030018, the coefficients of the generating function . For a general polynomial, there are relationships we have S_k(n)=^{k} a_i n^{i-1} \\ S_k(1)=^{k} a_i = p_k In general we may note that for Sk(n) a_k = 2^k\\ a_{k-1} = -(k)2^{k-1} \\ a_{k-2} = (3k+k^2)2^{k-3} \\ a_{k-3} = -}{3}(38k+9k^2+k^3) \\ a_{k-4} = }{3}(378k+179k^2+18k^3+k^4) \\ a_{k-5} = -}{15}(9864k+3030k^2+515k^3+30k^4+k^5) \\ a_{k-6} = }{45}(125640k+90634k^2+12915k^3+1165k^4+45k^5+k^6)\\ a_{k-7} = }{315}(1684080k+2003652k^2+463204k^3+40005k^4+2275k^5+63k^6+k^7)\\ a_{k-8} = }{315}(42089040k+50017932k^2+14438676k^3+1728769k^4+101640k^5+4018k^6+84k^7+k^8)\\ a_{k-9} = }{2835}(415900800k+1431527472k^2+492751916k^3+69663132k^4+5253969k^5+225288k^6+6594k^7+108k^8+k^9)\\ However we must shift the series along to deal with the fact that Sk(n) has k terms, and we end up with a_k = 4\cdot2^{k-2}\\ a_{k-1} = -(k-1)2^{k-3} \\ a_{k-2} = (3(k-2)+(k-2)^2)2^{k-5} \\ a_{k-3} = -}{3}(38(k-3)+9(k-3)^2+(k-3)^3)\\ a_{k-4} = }{3}(378+179(k-4)+18(k-4)^2+(k-4)^3) \\ a_{k-5} = -}{15}(9864(k-5)+3030(k-5)^2+515(k-5)^3+30(k-5)^4+(k-5)^5) \\ a_{k-6} = }{45}(125640(k-6)+90634(k-6)^2+12915(k-6)^3+1165(k-6)^4+45(k-6)^5+(k-6)^6) The we can see that the powers of 2 follow a sequence 0, 1, 3, 4, 7, 8, 10, 11, 15, 16, (18?) which is potentially A005187, the number of ones in the binary expansion of 2n, and the denominators of the convergents of $1/$, we will denote this quantity ξ(n), with ξ(0)=0, ξ(1)=1, ⋯. It then appears that for ak − m the coefficent of the highest power of (k − m) is 1, the coefficent of the next highest power is 3m(m + 1)/2 = 3, 9, 18, 30, 35, 63.... We also see the sequence 1, 1, 1, 3, 3, 15, 45, 315, 315, 2835 is A049606, the largest odd divisor of n!. This is very useful. We can attempt to factor n! out of the coefficients. Then, letting χ(n) be A011371, the number of binary digits in n, we have \kappa(n)=}{n!} and a_k = \kappa(0)\\ a_{k-1} = \kappa(1)k \\ a_{k-2} = \kappa(2)(3k+k^2) \\ a_{k-3} = \kappa(3)(38k+9k^2+k^3) \\ a_{k-4} = \kappa(4)(378k+179k^2+18k^3+k^4) \\ a_{k-5} = \kappa(5)(9864k+3030k^2+515k^3+30k^4+k^5) \\ a_{k-6} = \kappa(6)(125640k+90634k^2+12915k^3+1165k^4+45k^5+k^6)\\ a_{k-7} = \kappa(7)(1684080k+2003652k^2+463204k^3+40005k^4+2275k^5+63k^6+k^7)\\ a_{k-8} = \kappa(8)(42089040k+50017932k^2+14438676k^3+1728769k^4+101640k^5+4018k^6+84k^7+k^8)\\ a_{k-9} = \kappa(9)(415900800k+1431527472k^2+492751916k^3+69663132k^4+5253969k^5+225288k^6+6594k^7+108k^8+k^9)\\ a_{k-10} = \kappa(10)(501863040k+39753346896k^2+17788750740k^3+2992825520k^4+261174375k^5+13782153k^6+451710k^7+10230k^8+135k^9+k^{10})\\ a_{k-11} = \kappa(11)(228247891200k+1120677132960k^2+670268327256k^3+132747091620k^4+13570264070k^5+820667925k^6+32340693k^7+838530k^8+15180k^9+165k^{10}+k^{11})\\ a_{k-12} = \kappa(12)(11086611782400k + 34639931748960k^2+25815979252008k^3+6164459073916k^4+720004626990k^5+50304599015k^6+2260051794k^7+69522783k^8+1464210k^9+21725k^{10}+198k^{11}+k^{12}) \\ a_{k-13} \to () Now we focus on the polynomial aspect in k. We may define a function πn(k) such that a_{k-m}= \kappa(m)\pi_m(k) then we have \pi_0(k)=1\\ \pi_1(k)=k\\ \pi_2(k)=3k+k^2\\ \pi_3(k)=38k+9k^2+k^3\\ \pi_4(k)=378k+179k^2+18k^3+k^4 From the above we can see that \pi_m(k) = k^m + {2}k^{m-1} + {24}k^{m-2} + {16}k^{m-3} + {5760}k^{m-4} \\ + {3840}k^{m-5} + we can then expect a reduction \pi_m(k) = k^m + ^{m-1}c_i\left[^{i} (m+j)\right]\sigma_i(m)k^{m-i} where \sigma_1=3\\ \sigma_2=(125+27m)\\ \sigma_3=(118+125m+9m^2)\\ However after shifting to adjust the sequences we have \pi_m(k) = k^m + {2}k^{m-1} + {24}k^{m-2} + {16}k^{m-3} + {5760}k^{m-4} \\ + {3840}k^{m-5} + {2903040}k^{m-6} + \frac{}{1935360} which can then be re-written as \pi_m(k) = k^m + {2}k^{m-1} + {24}k^{m-2} + {16}k^{m-3} + {5760}k^{m-4} \\ + {3840}k^{m-5} + {2903040}k^{m-6} Now the denominators may be explained by the sequence A053657 d_n=(n)=\prod_p p^{^\infty \lfloor {(p-1)p^k} \rfloor } = 1, 2, 24, 48, 5760, 11520, 2903040, 5806080... Then which can then be re-written as \pi_m(k) = {d_1} + {d_2}k^{m-1} + {d_3}k^{m-2} + {d_4}k^{m-3} + {d_5}k^{m-4} \\ + {d_6}k^{m-5} + {d_7}k^{m-6} we can see the subsequence 71, 213, 19170, 19170, 1811565 is 1, 3, 270, 270, 25515 when divided by 71. The number of 3’s in the lead coeficcient are 1, 3, 3, 5, 6, 8 which could be a number of sequences. The number of 3’s in the second coefficients are 0, 1, 3, 3, 6 which could also be many sequences. Then we have \pi_m(k) = ^{m-1}^{i} (m-j)\right]}{d_{i+1}}\sigma_i(m)k^{m-i} and we focus on the σi(m) polynomials \sigma_0(m)=3 = 3(1)\\ \sigma_1(m)=27m+71 \\ \sigma_2(m)=27m^2+213m-528 = 3 (9 m^2+71 m-176)\\ \sigma_3(m)=1215m^3+19170 m^2-69835 m+191522\\ \sigma_4(m)=729m^4+19170m^3-66945m^2+199686 m-1863360 = 3 (243 m^4+6390 m^3-22315 m^2+66562 m-621120)\\ \sigma_5(m)=45927 m^5+1811565 m^4-3671325 m^3-20584781 m^2-243156858 m-181415824\\ \sigma_6(a)=3(6561 a^6+362313 a^5+273105 a^4-22127777 a^3+15860502 a^2-1815371056 a+12146754816)\\ \sigma_7(a)=885735 a^7+65216340 a^6+316232910 a^5-8968949640 a^4+26098075255 a^3-1410488924700 a^2+16110491218964 a-46896294366576\\ \sigma_8(a)=3(98415 a^8+9316620 a^7+95579190 a^6-2167195968 a^5+5460923335 a^4-506938037980 a^3+7682473899284 a^2-34005560902256 a+42398207462400)\\ \sigma_9(a)= We find that the sequence 0, 3, 27, 27, 1215, 729, 45927, 19683, 885735, 295245, 5845851, 1594323, ... is given by C_m=\left((n)}{n!} \right)3^{m-1}-2(-1)^{m-1}+(-2)^{m-1})}{2}\Gamma where \left((n)}{n!} \right)= 1, 1, 1, 1, 1, 3, 1, 9, 9, 15, 3, 9, 3, 945, 135, 27, 27, 405, 45, 8505, 1701, 66825, 6075, 18225, 6075, 995085, 76545,\\ 3^{m-1}= 1,3,9,27...\\ -2(-1)^{m-1}+(-2)^{m-1})}{2}= 0, 1, 3, 1, 15, 1, 63, 1, 255, 1, 1023, 1, 4095, 1, 16383, 1 \\ \Gamma= 1,1,1,1,1,1,1,1,{17},1,{31},1,1,1,{5461},1,{257}... We can see this is related by {N[tan(x)]}=1,1,1,{17},{31},1,{5461},{257},{73},{1271} Where N[f(x)] is the numerator of a Taylor expanded f. However there is one later coefficient which doesn’t seem to agree. 4097 in the original sequence has in this cot/tan expansion 241 which is 17 times too small. Odd that 17 is the first coefficient, also later in the sequence is the term 32505887 which doesn’t agree with the cot/tan term 61681. However, the former divied by the latter is 17 ⋅ 31, i.e. the product of the first two terms.

MAIN I will work out how to potentially express the series reversion for a general quintic (and possibly the conditions that this fails) using the determinants of matrices. Let us define our quintic polynomial P(x) = a_0 + a_1x + a_2x^2+a_3x^3 + a_4x^4 + a_5x^5 we can treat this is a series and do a series reversion about a₀ giving P^{-1}(x) = {a_1} - {a_1^3} + {a_1^5}+ {a_1^7} + \cdots there is a clear pattern for the denominators and the (x − a₀) term. Can we express this sequence in the form P^{-1}(x) = ^\infty |M_n|}{a_1^{2n+1}} where Mn is an n × n matrix, and | ⋅ | is the determinant. Then we have M₀ as the empty matrix, define the determinant |M₀|=1, then M_1=-[a_2]\\ M_2=- a_1 & a_2\\ 2a_2 & a_3 \\ M_3= - a_1 & a_2 & 0 \\ 0 & a_1 & 5a_2 \\ a_2 & a_3 & a_4 \\ M_4 = - a_1 & a_2 & 2a_3/3 & 0 \\ 0 & 6a_1 & 14a_2 & 18a_3 \\ 0 & 0 & a_1/6 & a_2 \\ a_2 & a_3 & a_4 & a_5 \\ M_5 = - a_1 & a_2 & a_3 & a_4 & 0 \\ 0 & 7a_1 & 14a_2 & 7a_3 & 0 \\ 0 & 0 & a_1 & 3a_2 & a_3 \\ 0& 0 & 0 & a_1 & a_2 \\ a_2 & a_3 & a_4 & a_5 & 0 we could also swap rows and columns being careful about sign, there are likely other (potentially prettier) solutions. If we could work out a general pattern, then we would have a form for the inverse of a general quintic. However, it can be seen that if a₁ → 0, the inverse function becomes ill defined. INVERSE OF LESSER POLYNOMIALS We might quickly want to apply this method to lesser polynomials, for example the linear inverse P_1(x) = a_0 + a_1 x then we have the inverse by series reversion P_^{-1}(x) = {a_1} we can note this is also the first term of the quintic expression above. For the quadratic we have P_2(x) = a_0 + a_1 x + a_2 x^2 and by reversion the sequence P_2^{-1}(x) = {a_1} - {a_1^3} + {a_1^5} - {a_1^7} + \cdots where the coefficients 1, 1, 2, 5, ⋯ are the Catalan numbers, A000108, which gives a form for this expression P_2^{-1}(x) = ^\infty {n!(n+1)!}}{a_1^{2n+1}} = }{2a_2} which clearly resembles the quadratic formula, this however gives little opportunity to yield a determinant structure, it is clear that the a₂n coefficients are arriving from the determinant. Next we may try the cubic equation P_3(x) = a_0 + a_1x + a_2 x^2 + a_3 x^3 with reversion P_3^{-1}(x) = {a_1} - {a_1^3} + {a_1^5} + {a_1^7} + \cdots this is sharing three terms with the quintic equation reversion, it also gives rise to the use of the determinants. The coefficient of a₂n in each term appears to be the Catalan numbers as in P₂−1(x). We can try to derive the series of matricies ₃Mn which describe the cubic pattern, _3M_1=-[a_2]\\ _3M_2=- a_1 & 2a_2\\ a_2 & a_3 \\ _3M_3= - a_1 & a_2 & 0 \\ 0 & a_1 & 5a_2 \\ a_2 & a_3 & 0 which are the same, if not similar. We now note that there seems to always be a coeffient that is the current Catalan number in the matrix, and it is on a₂. We might be better off assiging these coefficients to the lower left corner which is consistently a₂, _3M_1=-[a_2]\\ _3M_2=- a_1 & a_2\\ 2a_2 & a_3 \\ _3M_3= - a_1 & a_2 & 0 \\ 0 & a_1 & a_2 \\ 5a_2 & a_3 & 0 \\ _3M_4= - 3a_1 & a_2 & 2a_3/7 & 0 \\ 0 & a_1 & a_2 & a_3 \\ 0 & 0 & a_1 & a_2\\14a_2 & a_3 & 0 & 0 \\ _3M_5= - 7a_1 & -a_2 & -37a_3/126 & 0 & 0\\0 & 3a_1 & a_2 & 2a_3/7 & 0 \\ 0& 0 & a_1 & a_2 & a_3 \\0 & 0 & 0 & a_1 & a_2\\42a_2 & a_3 & 0 & 0 & 0 GENERAL SERIES We could also attempt this from the infinite limit, and then set the corresponding coefficients to 0 afterwards. P(x) = a_0 + a_1 x + a_2 x^2 + \cdots = ^\infty a_kx^k then we can describe the inverse term by P^{-1}(x) = ^\infty {na_1^n}\left((-1)^{s+t+u+\cdots}{s!t!u!\cdots}\left({a_1}\right)^s\left({a_1}\right)^t\cdots\right)(x-a_0)^n where the sum is over all combinations such that, s + 2t + 3u + ⋯ = n − 1 (all positive integers). [Morse Feschbach 1953] For example, when n = 1, the only combination is for s = t = u = ⋯ = 0. When n = 2, then then only combination is s = 1, for n = 3 we may have either, s = 2, t = 0 or s = 0, t = 1, giving two terms etc. The goal at this stage would be to develop a regular family of n × n matrices that can express these rules through their determinants. We may find that the matrices are made of the sum or product of matrices expressing each term.

INTRODUCTION The Automated Alignment System (AAS) and Back End Active Stabilization of Shear and Tilt (BEASST) together have been proposed to align the beam trains in the MROI and to keep these aligned during the night. A Preliminary Design for the AAS has been developed and some of the subsystems of the AAS have been prototyped. On the other hand, BEASST has been developed to the conceptual design stage only. We propose here a modification to the AAS concept which will allow the existing goals to be met while addressing drifts in alignment during the night more rapidly and robustly. At the same time, some of the constraints on the BEASST system can be relaxed, allowing a higher-performance system to be built more quickly. In the following we recall the original goals and concept for the AAS and outline a proposed modification. The advantages and disadvantages of this modification are then discussed and a way forwards proposed. FUNCTIONAL REQUIREMENTS The functions of the combined AAS and BEASST systems (hereafter collectively referred as “the AAS” since they can be considered to be a single system addressing alignment needs) can be briefly described as: 1. when a new beam train is installed or configured, the AAS must allow for coarse (and possibly manual) alignment of the beam train to allow an alignment beam to pass between the beam combining area (BCA) and the unit telescope mount (UTM) without vignetting; 2. at the beginning of each observing night, the AAS must re-align the beam train so that beams of starlight arriving at the beam combiners meet the tilt and shear error budget; 3. during the night, the AAS must make fine adjustments to the alignment so that the tilt and shear error budget is maintained in the presence of thermal drifts, typically introduced by those optics located in temperature unstable environments; 4. when a new beam train is installed or configured, the AAS must allow for measurement of the “piston” term due to the optical path between the beam combiners and the telescopes, in order to allow for rapid acquisition of stellar fringes; 5. at the beginning of the night, the AAS must allow the pathlength differences between the fringe tracker and the science instrument to be measured so that these can be equalized using the switchyard mirrors.

Let \Pi_N(a,x)= ^N a+x^{p_k} Where pk is the kth prime. Then (\log\Pi_N(a,x) - \log a^N) = {2a}+{3a} -{4a^2} + {5a} + {6a^3} +{7a} -{8a^4}+{9a^3}+}{10a^5}+\cdots It would appear there are a few patterns here. - The power of a in the denominator is the largest proper divisor of the power of x. - The denominator is the power of x. - For prime powers of x the numerator is also the power of x. - For a perfect square or cube power of x, the prime divisor is the numerator. - For a composite power of x the numerator is a polynomial in a, with coefficients of the divisors of x. - After removing the denominator coefficient of the power of x, all raw coefficients present in the above expansion are prime. Focus on the composites x^4 \to -{a^2}\\ x^6 \to {a^3}\\ x^9 \to {a^3}\\ x^{10} \to {a^5}\\ x^{12} \to -{a^6}\\ x^{42} \to }{a^{21}}\\ x^{44} \to -}{a^{22}}\\ x^{30} \to -{a^{15}}\\ x^{66} \to -+11a^{27}}{a^{33}}\\ x^{69} \to }{a^{23}} It seems that for numbers with 3 prime factors, the coefficient of the largest power of a in the numerator is the power of the second largest power of a. Although only a few numbers have been checked so far. It would appear that for a polynomial of the form x^q \to {a^o}\\ xo=q\\ y(o-m)=q\\ z(o-n)=q giving x^q \to {a^{D_m(q)}} (-1)^?pa^{D_m(q)-{p}} where Dm(q) is the largest proper divisor of q. This gives (\log\Pi_N(a,x) - \log a^N) = ^\infty {qa^{D_m(q)}} (-1)^?pa^{D_m(q)-{p}}x^q \\ ^\infty 1+x^{p_k}=\exp\left(^\infty {q} (-1)^?px^q\right) where the pattern for the negative signs is yet to be found. It would appear that if the power of x is divisible by 2 and the power of x divided by 2 is not equal to 1, then the overall sign of the temr in the expansion is negative. Then the polynomials in the numerator must be investigated. It would appear that the polynomial coefficient of 2 is negative if the power of x is of the form 4n + 2, n = 0, 1, 2, 3, .... All other prime divisor polynomial coefficients appear to be positive. This then explains all terms (as far as we know). This gives (\log\Pi_N(a,x) - \log a^N) = ^\infty {qa^{D_m(q)}} (-1)^{q+1}p^*(q)a^{D_m(q)-{p}}x^q \\ ^\infty 1+x^{p_k}=\exp\left(^\infty {q} (-1)^{q+1}p^*(q)x^q\right) where p*(q) means p^*(q)=(-1)^{q/2}\cdot 2,\; p=2\\ p^*(q)=p,\; We may now manipulate this expression ^\infty 1+x^{p_k}=\exp\left(\left[}^\infty {q} (-1)^{q+1}p^*(q)x^q\right] + \left[}^\infty {q} (-1)^{q+1}p^*(q)x^q\right]\right) \\ ^\infty 1+x^{p_k}=\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)\\ ^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}=\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)\\ \log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}\right)=}^\infty {q} (-1)^{q+1}p^*(q)x^q\\ \log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}\right)=}^\infty x^q Where the right hand side is now the ordinary generating function for primes. Then of course we also have \log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty x^q\right)}\right)=}^\infty {q} (-1)^{q+1}p^*(q)x^q Then if we set x = 1, and change the infinite sums to n (checked that the series are generated at the same rate) we should have have the prime counting function \pi(n) = \left.\log\left(^\infty 1+x^{p_k}}{\exp\left(}^\infty {q} (-1)^{q+1}p^*(q)x^q\right)}\right)\right|_{x=1}

ABSTRACT I consider defining transforms to mimic the inverse series transform in structure MAIN If we have an arbitrary series expansion, P(x) = a_0 + a_1 x + a_2 x^2 + \cdots = ^\infty a_kx^k then we can describe the inverse term by P^{-1}(x) = ^\infty {na_1^n}\left((-1)^{s+t+u+\cdots}{s!t!u!\cdots}\left({a_1}\right)^s\left({a_1}\right)^t\cdots\right)(x-a_0)^n [Morse and Feshbach (1953)], where, critically s + 2t + 3u + ⋯ = n − 1, all of s, t, u, ⋯ are positive integers and the sum is then over the combinations to do this. This transform is clearly very important, can we create a similar looking transform, and see if any integer sequences are related by the two? EXAMPLE 1 Again let P(x) = a_0 + a_1 x + a_2 x^2 + \cdots = ^\infty a_kx^k this time define the transformed polynomial as [P](x) = ^\infty \left((sa_1+ta_2+ua_3+\cdots)\right)(x-a_0)^n\\ s+t+u+\cdots=n \\ s\ge t \ge u \ge \cdots we may need to enforce that the higher terms can be 0, the definition can be reworked if this is the case. Then we end up with [P](x) = (a_1)(x-a_0)+(3a_1+a_2)(x-a_0)^2+(6a_1+2a_2+a_3)(x-a_0)^3+(12a_1+5a_2+2a_3+a_4)(x-a_0)^4 + \cdots which looks like it may well have a nice form for certain input ak terms. If we use P(x)=x, then we find that the transform is the generating function for A006128. [x] = {^n(1-x^k)} likewise for P(x)=x², we have the generating function for A096541. For P(x)=x + x² + x³ + x⁴ + ⋯, we get the generating function for A066186. For P(x)=x − x² + x³ − x⁴ + ⋯, we get the generating function for A066897. For P(x)=x + 2x² + 3x³ + 4x⁴ + ⋯, we get the generating function for A066184. For P(x)=x + x³ + x⁵ + ⋯, we get A207381. To generate the generating function for the natural numbers we need to put in the function FN(x)=x − x² − x³ − x⁴ − x⁵ + x⁶ − x⁷ + 3x⁸ + 0x⁹ + ⋯, the rule is not clear. Interestingly if we input P(x)=FN(x)+x, we get the generating function for A225596. To generate the generating function for the prime numbers we need to put in the function FP(x)=2x − 3x² − x³ − x⁵ + x⁶ − 4x⁷ + 3x⁸ + 0x⁹ + ⋯, the rule is not clear. If we input P(x)=FP(x)−FN(x), we do seem to get the generating function for A014689, which makes sense. To get π(n), we can use the coefficients 1, −1, −2, 0, −1, 3, −2, 2, 2, ⋯ CHANGE INEQUALITIES If we force the inequalities to be strict, that is s > t > u > ⋯, then with P(x)=x we have the g.f. of A005895. with P(x)=x + x² + x³ + x⁴ + ⋯, we have the g.f. of A066189. TYPE 2 What about the following transform P(x) = a_0 + a_1 x + a_2 x^2 + \cdots = ^\infty a_kx^k then we can describe the inverse term by T_2[P(x)] = ^\infty \left((-1)^{s+t+u+\cdots}a_0^sa_1^ta_2^ua_3^w\cdots\right)x^n\\ s+2t+3u+\cdots=n-1 Does this generate anything coherent? We get T_2[P(x)] = x - a_0x^2 + (a_0^2 - a_1)x^3 + (a_0a_1 - a_0^3 - a_2)x^4 + (a_0a_2 + a_1^2 - a_3 - a_0^2a_1 + a_0^4)x^5+\cdots with P(x)=1, we will then get T_2[1] = x - x^2 + x^3 - x^4 + x^5 + \cdots with P(x)=1 + x, we will then get T_2[1+x]= x - x^2 + x^5 -x^6 + \cdots = ^\infty x^{4k+1}-x^{4k+2} with P(x)=1 + x², we will then get T_2[1+x]= x-x^2+x^3-2x^4+2x^5-2x^6+3x^7-3x^8+3x^9-4x^{10}+\cdots nicely if we input P(x)=1 + 2x + 3x² + 4x³ + ⋯, then we have T_2\left[{(x-1)^2}\right]= x -x^2 -x^3 -2x^4 +2x^5 -x^6 + 4x^7 -x^8 + 18x^9 -22x^{10} + \cdots\\ T_2\left[{(x-1)^2}\right]= x {1+mx^m} where the bottom equaility is due to A022693. It would also seem that T_2[1+2x]= {1+x+2x^2+2x^3} for T_2[1-x+x^2-x^3+x^4-\cdots] we have the alternating signs of the partition numbers, A000041. POSSIBLE RESULTS Below I list a few significant transformations of simple input generating functions. There appears to be a strong connection to modular forms. T_2\left[{(x-1)^2}\right]= x {1+mx^m}\\ T_2\left[{x-1}\right]={ (1-3x^m)}\\ T_2\left[{1-x}\right]= \eta(t)}{\eta(t^2)}\\ T_2\left[{1+x}\right]= x^{25/24}}{\eta(-\tau)}\\ T_2\left[{x^2-1}\right]=\eta(t^2)}{\eta(t)}\\ T_2\left[{1-x^2}\right]= x^{23/24}\left({16}\right)^{1/24}\\ T_2\left[{1-x^3}\right] = \eta(t)\eta(t^6)}{\eta(t^2)\eta(t^3)}\\ T_2\left[{x^3-1}\right] = \eta(t^3)}{\eta(t)}\\ T_2\left[{x^4-1}\right] = \eta(t^4)}{\eta(t^2)}\\ T_2\left[1+x+x^4-x^5+x^6-x^7+x^9+x^{10}-x^{11}+x^{12}+x^{13}-x^{15}+x^{16}-x^{17}+\cdots\right] = \eta(t)\eta(t^{12})}{\eta(t^2)\eta(t^3)\eta(t^4)}\\ T_2[0 0 -1 1 0 -1 0 2 -1 0 0 0 0 0 -1 4 0 -1] = \eta(t^4)}{\eta(t^3)}\\ T_2\left[ {x^6-1} \right] = \eta(t^6)}{\eta(t^2)}\\ T_2\left[ {x^6-1} \right] = \eta(t^6)}{\eta(t^3)}\\ t^n={2\pi i}\\ \tau^n={2\pi i}\\ m=\lambda\left(t^2\right) with η(q), Dedekind Eta function, λ, is the modular lambda function. HOW DOES THIS COME ABOUT We can observe a similar set of sums, with restricted tuples of indices. f_1(k)=i_1 \to 1,2,3,4,5,6,7,\cdots\\ f_2(k)= i_1i_2 \to 1,2,5,8,14,20,30,\cdots\\ f_3(k)= i_1i_2i_3 \to 1,2,5,10,18,30,49,\cdots reading the : as ’such that’. The generating functions for these (up to offset) are f_1(k) \to ((1-x))^{-1}\\ f_2(k) \to ((1-x)(1-x^2))^{-1}\\ f_3(k) \to ((1-x)(1-x^2)(1-x^3))^{-1}\\ f_3(k) \to ((1-x)(1-x^2)(1-x^3)(1-x^4))^{-1} so we can see f∞(k) will have the generating function f_\infty(k) \to {(x;x)_\infty^2}={\phi(x)^2} with (a; q)k the q-pochammer symbol and ϕ(x) the Euler function. Then f∞(k), will give the number of partitions of k into parts of 2 kinds, A000712. If we change the equalities to less than symbols g_1(k)=i_1 \\ g_2(k)= i_1i_2 \\ g_3(k)= i_1i_2i_3 then it would appear that g∞(k) gives A000713, with generating function g_\infty(k) \to {(1-x)\phi(x)^2}

ABSTRACT Investigate a composition product representation of a set of equations. MAIN Consider the composition product of a matrix of functions on a vector of variables as follows f_{11} & f_{12} \\ f_{21} & f_{22} \odot x_1 \\ x_2 = c_1 \\ c_2 Which is the statement that f_{11}(x_1)+f_{12}(x_2) = c_1 \\ f_{21}(x_1)+f_{22}(x_2) = c_2 and example might be \sin & \cos \\ \cos & -\sin \odot x \\ y = 1 \\ 0 Then \sin(x)+\cos(y)=1\\ \cos(x)-\sin(y)=0 this has solutions x={6}(12\pi n+\pi),\;y={3}(6\pi m + \pi), \; n,m \in \\ x={6}(12\pi n+5\pi),\;y={3}(6\pi m - \pi), \; n,m \in It would be lovely to have the concept of an inverse such that for A \odot v = c\\ v= A^{-1} \odot c Although there clearly exist two families of vectors which “transform” using A to get to c given two solutions. However, picking periodic functions may be the cause of this. There are a few potential answers but one of them is x \\ y = & \beta \\ \gamma & \odot 1 \\ 0 where we require that \beta(0)= {6}\\ \gamma(1)={3}

ABSTRACT An adapted version of the zeta function is defined as θm(s), this is used to explore the sequence of prime numbers.

ABSTRACT Investigate what I percieve to be a Hilbert Space of numbers. I use the concept of a number unit, in analogy to length, area, volume etc. Such that a prime has dimensions of p. Compunds and partitions are visualised.

ABSTRACT Investigate the results of a vector stacking operator in analogy to the matrix multiplication or inner and outer vector products. INTRODUCTION We have vector dot product a&b\cdotc\\d =ac+bd and outer ’dyadic’ of this product by reversing the position of row and column vectors a \\ b \cdot c & d = ac & ad \\ bc & bd So investigate coupling of row and column vectors such that a \\ b \cdot c \\ d = ac \\ ad \\ bc \\ bd or ac \\ bc \\ ad \\ bd Higher matricies could be made with a transform such as M=(v\cdot u)\otimes(v^T\cdot u^T) With this concept in mind let us examine the matrix muliplication operation a & b \\ c & d \cdot e & f \\ g & h = ae+bg & af+bh \\ ce+dg & cf+dh = a&b\cdote\\g & a&b\cdotf\\h \\ c&d\cdote\\g & c&d\cdotf\\h With this concept in mind let us define a similar operation by switching row and column vectors a&b\\c&d\otimes e&f\\g&h= a\\b\cdote&g& a\\b\cdotf&h\\ c\\d\cdote&g& c\\d\cdotf&h = ae & ag & af & ah \\ be & bg & bf & bh \\ ce & cg & cf & ch \\ de & dg & df & dh This appears to be the outer product of two vectors of the form (a b c d) \otimes (e g f h) It can be seen that the trace of each sub matrix is the result of a regular matrix-matrix product! However this differs from the Kronecker product which would yeild a&b\\c&d\otimes e&f\\g&h= a\\b\cdote&g& a\\b\cdotf&h\\ c\\d\cdote&g& c\\d\cdotf&h = ae & af & be & bf \\ ag & ah & bg & bh \\ ce & cf & de & df \\ cg & ch & dg & dh

There will be an abstract

ABSTRACT We introduce the novel concept of a split number line, in which the positive part is as usual and the negative part undergoes an either symmetric or asymmetric bifurcation at zero. An algrbra is made between the two kinds of negative components and positive compoments in the form of a stable triplex. The implications of this create a further abstraction on the concept of rings over number fields, such as to narrow the application of the Frobenius Theorem. TRIPLEXES AND CONCEPT According to [Source: Ben] a stable simplex on a number field cannot be created, implications of Frobenius Theorem leading any real division algebra isomorphic to either ℝ, ℂ or ℍ where, the set of quaternions ℍ are the only non-Abelian algebra. Exclusion of hypercomplex, and therefore non-associative alegbras as we stick to real division algebra only. We form an abstraction on the signality of a number n ∈ N, where N is a ring over real numbers. Instead of the binary +,- number precursors indicating greater or less than zero we allow, +,-,| where + indicated greater than zero and - and | both indicate a number less than zero. There is a component form to these negative numbers, the “full” value is defined to be -n \to a} + b} with a,b positive constants, and $}, }$ basis elements that transform according to some algebraic rule along with the basis element $}$

ABSTRACT I investigate an arithmetic transform where digits in a number can be swapped, calculations performed and then swapped back to give sensible results. The errors in such a process are considered.

ABSTRACT Introduce a fold function and investigate it’s properties. For a given finite barrier or well starting at x_1 and ending at x_2, that it, tending to zero. INTRODUCTION Define a fold function f*(x) as a transform from a defined function f(x) as f_*(x)=^{x_2} f(s) \;ds}{ ^{x} f(s) \;ds} Which should result in a function normalised to the interval [ − 1, 1) ??? what needs doing to make this. This is something like a simple auto-correlation function although only in concept, it’s properties will be quite different. The function should higlight the ’area centre’ of some function, such that the area on the left side is that on the right side. For a given function with an ’area centre’ of 0, the function may mor may not be symetric about the y axis. For simple functions the derivative of the fold function equal to zero will yeild a location for this cetre point. Porbably more realistic to scale down the ∞ to some large value for terminating functions etc. numeric analysis. Moved on to study the transform F_0(x)=2f_{max}\bigg[ ^{x_2} f(s) \;ds}{ ^{x} f(s) \;ds} + ^{x} f(s) \;ds}{ ^{x_2} f(s) \;ds}\bigg]^{-1} \\ F_0(x)=}{f_*(x)+f_*^{-1}(x)} There then appears to be a higher order of interest F_1(x)=\bigg[ ^{x_2} f(s) \;ds}{ ^{x+w} f(s) \;ds} + ^{x+w} f(s) \;ds}{ ^{x_2} f(s) \;ds}\bigg]^{-1} - \bigg[ ^{x_2} f(s) \;ds}{ ^{x-w} f(s) \;ds} + ^{x-w} f(s) \;ds}{ ^{x_2} f(s) \;ds}\bigg]^{-1} where w is a quarter width of the well. Let us turn our attention to the harmonic oscillator potential function given by [Rae] as f(x)={2}m\omega_c^2x^2 The constants will drop out when taking the fold of this potential f_*(x)={ {x^3-x_1^3}+ {x_2^3-x^3} } which is f_*(x)={ (x_2^3-x^3)^2 + (x^3-x_1^3)^2 } becoming reminiscant of the form {k_1^2 +k_2^2} Start Here: We are seeking a functional transformation, a functional on V(x) such that for any given V(x) we can predict the wavefunction of a system. That is, the direct relationship \psi(x)=F(V(x)) We use the folding transforms as candidate functions for F().

ABSTRACT Introduce the concept of Dinary notation, use this to form a divisor matrix and explore its properties. Attempt to develop a Quantum Number Theory on the divisor notation as fermionic states.

ABSTRACT A recreational transform is investigated. It involves taking a vector of binary numbers, extrapolating the vector into a binary matrix, transposing the matrix and repacking the into a different vector which can have a different number of elements for non-square matrices. This allows for sets of transformations from higher to lower dimensional spaces to be noted.

ABSTRACT 13/04/2014 I present a very simple algorithm (visual sieve) for testing if a number is prime. The algorithm was first developed visually in a cellular automation manner. A data map is created from a simple recursive function or an equivalent criterion, from this map it is easy to find if a number is prime from the structure of it’s divisors. By checking from the dense end of the factor list the runtime is minimised. All primes up to 14983 were generated in a few seconds on a laptop.

ABSTRACT An attempt to elucidate a form of recursive integral was made. A link to chaotic systems in bifurcation diagrams was found as the iterative solution to an integral under variation of a general parameter in the integrand. This could propose in some sense the integral having multiple values, with the solution a scaled varient of the logistic map. Approximate piecewise functions are used to map the bifurcation pattern in an attempt to close the form of the recursive integral for small numbers of bifurcations. In the process a potential connection between the Embree-Trefethen constant, Feigenbaum Constant δ, and the closely fitting functional forms of the integral is exhibited. A highly (elliptic hyperbolic?) transformation is defined to move from one bifurcation to the other, a repeated application adds intricate (bound under a value) structure in the region r ∈ [0, 4].

ABSTRACT 02/06/2014, Benedict Irwin. We define fractional creation and annihilation operators and explore the consequences and sub algebras. It is found that the number operator presents itself as an interaction term. The fractional operators are shown to be linked through an Abelian operator.