ABSTRACT I comment on the entropy of waves by using the Shannon Entropy on the density of standing wave solutions to an infinite box and harmonic oscillator potentials, finding closed form results. The standing wave box entropies do not change with excited state, the harmonic oscillator entropies do change for excited states. INTRODUCTION The Shannon Entropy is a functional H[f(x)]=-\int f(x)f(x) \;dx which can be written in a similar form H[f(x)]=\int f^{-f(x)}(x)\;dx Consider a wave in a box of length π, from 0 to pi which has a spatial wavefunction \psi_n(x)=(nx) Then if we take the density \rho_n(x)=^2(nx) we can evaluate the corresponding entropy of these states. We find very quickly that \int_0^\pi \;^{-2^2(nx)}(nx)\;dx={2}((4)-1) for any value of n! That is the entropy of each state is the same, independant of the level of excitation. When adjusting for waves of any amplitude α we get s_n=-\int_0^\pi \alpha^2\sin^2(nx)\ln \alpha^2\sin^2(nx) \; dx =\pi\alpha^2\left(\left({\alpha}\right)-{2}\right) HARMONIC OSCILLATOR We can analyse the entropy of the harmonic oscillator with wavefunctions \psi_n(x) = {}\left({\pi\hbar}\right)^{1/4}\exp\left({-{2\hbar}}\right)H_n\left({\hbar}}\right), \;\;n=0,1,2,\cdots we then set {\hbar}=1 and analyse the wavefunction integral \sigma_0=-^\infty {\sqrt[4]{\pi}}e^{-x^2/2}\left({\sqrt[4]{\pi}}e^{-x^2/2}\right)\;dx = -(2+\ln(\pi))}{2} and the density integral s_0=-^\infty {}e^{-x^2}\left({}e^{-x^2}\right)\;dx = {2}(\ln(\pi)+1) With added constants we find s_0=-^\infty {}e^{-bx^2}\left({}e^{-bx^2}\right)\;dx = {2}(\ln({a^2})+1) Then with a^2=b={\hbar} s_0= {2}(\ln({m\omega})+1) Then for the next integrals with H₁(x)=2x we have s_1 = }{4} -^\infty e^{-x^2}x^2\ln(x^2) \; dx - ^\infty e^{-x^2}x^2\ln(A) \; dx Using inverse symbolic methods thanks to RIES and ISC, we find from accurate numerical integration \sigma_1 = }{\sqrt[4]{\pi}}^{\infty} xe^{-x^2/2}(}{\sqrt[4]{\pi}}xe^{-x^2/2})\;dx = \sqrt[4]{\pi^3}i and s_1 = {}^{\infty} x^2e^{-x^2}({}x^2e^{-x^2})\;dx = \gamma + (2) + {2}((\pi)-1) where γ is the Euler-Mascheroni constant. When solving for the next integrals we have H₂(x)=4x² − 2, and additional complexity. We find by laking a log expansion some progress can be made to finding a closed form for the total entropy of the density state. We have s_2=-^{\infty}(16x^4-16x^2+4)}{8}\left(^\infty {n}\left((16x^4-16x^2+4)}{8} -1 \right)^n \right)\;dx Note down each term in the converging sequence 1-{64\pi^{1/2}}\\ {2}+{54\pi}-{64\pi^{1/2}}\\ {3}-{3145728\pi^{3/2}}+2{54\pi}-{64\pi^{1/2}}\\ {4}+{1562500\pi^2}-3{3145728\pi^{3/2}}+3{54\pi}-{64\pi^{1/2}}\\ {5}-{2866544640\pi^{5/2}}+4{1562500\pi^2}-6{3145728\pi^{3/2}}+4{54\pi}-{64\pi^{1/2}}\\ \vdots\\ {n} + ^{n-1} n-1\\ka_k}{b_k\pi^{(k+1)/2}}\\ s_2=^\infty{n} + ^{n-1} n-1\\ka_k}{b_k\pi^{(k+1)/2}} We have some higher terms from Mathematica {83047723206\pi^3}\\ {\pi^{7/2}} It seems clear that a factor of 8n + 1 must be removed from the denominator in the sequence. 1-{2^2\cdot8^2\pi^{1/2}}\\ {2}+{3^3\cdot8^3\pi}-{2^2\cdot8^2\pi^{1/2}}\\ {3}-{3\cdot4^4\cdot8^4\pi^{3/2}}+2{3^3\cdot8^3\pi}-{2^2\cdot8^2\pi^{1/2}}\\ {4}+{5^3\cdot8^5\cdot5^5\pi^2}-3{3\cdot4^4\cdot8^4\pi^{3/2}}+3{3^3\cdot8^3\pi}-{2^2\cdot8^2\pi^{1/2}}\\ {5}-{15\cdot8^6\cdot6^6\pi^{5/2}}+4{1562500\pi^2}-6{3145728\pi^{3/2}}+4{54\pi}-{2^2\cdot8^2\pi^{1/2}}\\ \vdots\\ s_2=^\infty{n} + ^{n-1} n-1\\ka_k}{8^{(k+2)}b_k(k+2)^{(2k+5)}\pi^{(k+1)/2}}

ABSTRACT Knowledge of genetic cause in neurodevelopmental disorders can highlight molecular and cellular processes critical for typical development. Furthermore, the relative homogeneity of neurodevelopmental disorders of known genetic origin allows the researcher to establish the subsequent neurobiological processes that mediate cognitive and behavioural outcomes. The current study investigated white matter structural connectivity in a group of individuals with intellectual disability due to mutations in _ZDHHC9_. In addition to shared cause of cognitive impairment, these individuals have a shared cognitive profile, involving oro-motor control difficulties and expressive language impairment. Analysis of structural network properties using graph theory measures showed global reductions in mean clustering coefficient and efficiency in the _ZDHHC9_ group, with maximal differences in frontal and parietal areas. Regional variation in clustering coefficient across cortical regions in _ZDHHC9_ mutation cases was significantly associated with known pattern of expression of _ZDHHC9_ in the normal adult human brain. The results demonstrate that a mutation in a single gene impacts upon white matter organisation across the whole-brain, but also shows regionally specific effects, according to variation in gene expression. Furthermore, these regionally specific patterns may link to specific developmental mechanisms, and correspond to specific cognitive deficits.

This blog post describes the rationale and motivation behind the LIBRARY OF WORDS, a digital collection of pages filled with every possible combination of 320 words.

INTRODUCTION I begin to investigate the concept of a growing concatenate prime. Let us define the notation a\dagger b = a|a+1|...|b-1|b, for b > a, where, as in the last few articles | indicates a digitwise concatenation, therefore to give a few examples 5 \dagger 7 = 567 \\ 3 \dagger 8 = 345678 \\ 13 \dagger 15 = 131415 \\ 1 \dagger 1 = 1 The reason for trying this is primarily out of curiosity. However, by fixing the ’form’ of numbers (for example in the d;n notation), we have already shown it is possible to extract information for very large numbers in previous work. Now, it is clear that the numeric interpretation of such digits strings will grow in magnitude extremely fast, a number such as 100 † 120 will be 60 digits long. It is also clear that the numbers made in this fashion are base dependent. A good idea at first may be to tabulate conceptual intervals such as n † n + 1 {|r|r r|} \hline n & n \dagger n+1 & \Delta\\ \hline -3 & -32 & - \\ -2 & -21 & 11 \\ -1 & -10 & 11 \\ 0 & 1 & 11 \\ 1 & 12 & 11 \\ 2 & 23 & 11 \\ ... & ... & 11 \\ 7 & 78 & 11 \\ 8 & 89 & 11 \\ 9 & 910 & 821 \\ 10 & 1011 & 101 \\ 11 & 1112 & 101 \\ ... & ... & 101 \\ 18 & 1819 & 101 \\ 19 & 1920 & 101 \\ 20 & 2021 & 101 \\ ... & ... & 101 \\ 98 & 9899 & 101 \\ 99 & 99100 & 89201 \\ 100 & 100101 & 1001 \\ \hline We see here the special numbers 11, 101, 1001 forming in the differences. These numbers and their concatenates, 10101, 1001001, ... form the repdigit basis. But there are the other numbers 821, 89201, 8992001, which are made by 910 − 89, 99100 − 9899, 9991000 − 998999, .... These two sets of numbers can be described compactly then by 1|0;k|1 , \;\;\; k \in ^0 \\ 8|9;k|2|0;k|1, \;\;\; k \in ^0 By arranging this family of numbers in a table of divisors we can look for patterns, for example: {| c | l |} \hline k & Factors \;\; of \;\; 8|9;k|2|0;k|1 \in \\ \hline 0 & 821 \\ 1 & 7 \times 12743 \\ 2 & 29×149×2081 \\ 3 & 899920001 \\ 4 & 139×2239×289181 \\ 5 & 12041×747445561 \\ 6 & 31×29032255483871 \\ 7 & 7×12857142742857143 \\ 8 & 5939×259681×5835646939 \\ 9 & 31×120167×241599258215113 \\ 10 & 19×4736842105221052631579 \\ 11 & 2078217726601×4330633833401 \\ 12 & 899999999999920000000000001 \\ 13 & 7×13151×977655148440631347969193 \\ 14 & 19×131×709×2539×329641069×6093507681211 \\ 15 & 647×8762041×158757023470472699268463 \\ 16 & 5386471×3336989011711×5007067047060521 \\ 17 & 5386471×3336989011711×5007067047060521 \\ 18 & 199×1699×1119415751×93500544191×25432584853061 \\ 19 & 7×47×273556231003039513675379939209726443769 \\ 20 & 59×152542372881355932203254237288135593220339 \\ 21 & 31×21391×131117953566049×10351125295707468458809169 \\ 22 & 421×22621×2701121747698887529×3498681867987224789609 \\ 23 & 199×977×5661367×8176615206505410031316643554451717361 \\ 24 & 31×2408264179×12055263005477825957505994169206911982949 \\ 25 & ...\\ 40 & 89999999999999999999999999999999999999999200000000000000000000000000000000000000001 Primes could occur on the following sequences... A034956 , A052482, A120304, A007993, A053043, A032093, Check 3, 12, 40, 70... Interesting that we have the sequences {|c|c|} \hline k & \\ \hline 1 & 7 \times 12743 & 1|274|3\\ 7 & 7 \times 12857142742857143 & 1|285714|274|285714|3 \\ 13 & 7 \times 12857142857142742857142857143 & 1|285714|285714|274|285714|285714|3 \\ \hline We can state that the integer 7 provides a mapping 8|9;k|2|0;k|1 \;\; [\to_7] \;\; 1|285714;n|274|285714;n|3, \\ \forall k=1+6n, \;\; n \in ^0 We note that 6 is the period of the repdigit in the right hand concatenate, but also that {7} = 39., with period 6, and the order of the sequence is jumbled somewhat. Also note that 12743 and 821 are prime, (that is the k = 0 and n = 0 cases). It may be a coincidence that 13 the outermost pairing on the right hand side is 3 times 39. {|c|c|} \hline k & \\ \hline 3 & \\ 6 & 31 \times 2903225|5483871 \\ 9 & 31 \times 2903225|806193|5483871 \\ 12 & \\ 15 & - \\ 18 & - \\ 21 & 31×2903225|806|451612903225|548387096774|193|5483871 \\ 24 & 31×2903225|806|451612903225|806|193|548387096774|193|5483871\\ 27 & - \\ 30 & - \\ 33 & - \\ 36 & 31×2903225|806|451612903225|806|451612903225|548387096774|193|548387096774|193|5483871 \\ 39 & 31×2903225|806|451612903225|806|451612903225|806|193|548387096774|193|548387096774|193|5483871 \in \\ \hline {31} = 2903225|806451612903225;n|548387096774193;n|5483871\\ {31} = 2903225|806451612903225;n|806193|548387096774193;n|5483871\\ Now a strange/fantastic result is that in the number 2903225|806451612903225548387096774193|5483871 each digit 0 − 9 appears exactly three times in the 30 digit central partition! The numbers 806 and 193 are ’9 conjugate’, their sum is 999. The number is likely to split between the double 5 in the remaining 24 digits in the centre, as it did on the k = 6.

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 a recreational maths transform. We turn a number into its prime representation, then turn that into a digit string, and repeat. There may exist numbers other than primes that are invarient under such a transformation, others will grow tending to infinity.

ABSTRACT I work towards a formula to predict 2n for any n. I don’t expect to complete it, but I have found the last four digits for any n ∈ ℕ. Update: I have found a relationship between the sequences that in theory should allow the calculation of any digit of 2n by storing only 4 digits... Update II: The sequence fails for some numbers as it cycles in a non prime manner, numbers would either need to be stored, or a prime cycling sequence discovered which seems unlikely.

ABSTRACT I investigate the use of the notation d; n, meaning repeat the digit d, n times. The convergence of rational expressions and the primality of larger sequences are explored. A way of ruling some non-trivial numbers out in log₁₀(digits) is found for prime searches. The relationship of numbers written in such notation and thier prime decompositions is investigated.

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 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 (B.W.J Irwin, S.Wilkinson) We investigate the concept of a derivative with respect to 1 via a standard trick to solve integrals, it appears to have a well defined form which is closely linked to q-calculus/ the q-derivative. INTRODUCTION When confronted with an integral of the form ^{\infty} e^{-ax^2} \;dx = {a}} for a > 0. There is a standard trick to take the derivative with respect to the parameter a such that {da}^{\infty} e^{-ax^2} \;dx = {da}{a}} \\ ^{\infty} x^2e^{-ax^2} \;dx = {2}{a^3}} for a > 0. However, one could have introduced the parameter a if it was not there, and set it to one at the end to generate the seemingly arbitrary step. This of course required knowledge of where a would appear on the right hand side, this knowledge creates the information that a factor of −1/2 is added. However one can ask the question, is it possible to define a derivative with respect to 1. Such as to preserve the logic of \big({da}e^{ax}=xe^{ax}\big) = \big( {d}e^{x}=xe^{x} \big) One can create a formal definition which yeilds such a result that {d}={\delta} In this manner {d}=}{\delta}={\delta}=xe^x We compare the fact that ex is the eigenfunction that produces an eigenvalue of 1 with application of the d/dx operator, with x being the eigenfunction that produces an eigenvalue of 1 with application of the d/d𝕀 operator. Another interesting ’derivative’ {d}sin(x) = {\delta}(sin(x) - (x - \delta x - {3!} + {3!} = {5!} - {5!} -O((1-\delta)x^7))) \\ {d}sin(x) = {\delta}(sin(x) - sin(x) -\delta x( -1 + {2!}-{4!}...) \\ {d}sin(x) = xcos(x) Likewise we can find d/d𝕀cos(x)= − xsin(x). Which is making the results look suspiciously like x times the ’regular derivative’. Just going by the rule that we times the factor (1 − δ) to any occurence of x in the function would leave the derivative of a constant as 0 still, (which would indeed be x times the regular derivative). However, the first integral shown, the right hand side had the derivative taken, this scenario needs to be protected or contradication will occur! We now wish to perform {d}^{\infty} e^{-x^2} \;dx ={d} Now instead of using an a parameter, we may replace the derivative with the ’regular derivative’ times x ^{\infty} x(-2xe^{-x^2}) \;dx ={d} We see the factor of 2 has presented itself via a new method! Rather than the square root index it has now been generated by the exponential function, however, to keep consistency, if this is true it must be the case that {d} =- Such a result is very curious... Another alternative is that only the modulus of the ’regular derivatiive’ is taken and the derivative of $$ is constant, or that this result is a conincidence... We can attempt a new integral and postulate the result... Try {d}^{\infty} e^{-x^4} \;dx ={d} 2\Gamma({4}) I would have no idea where to plop an a on the right hand side if we brought one into the exponent, it may well be on the bottom but I don’t know at this stage... However using our theorem that we can replace x times the regular derivative on the left hand side and flip the sign of the constant right we would have ^{\infty} x^4e^{-x^4} \;dx ={2}\Gamma({4}) Our postulated result. Amazingly enough it is correct! (when verified elsewhere). We can try another result, potentially as \int_0^\infty {e^x-1} dx = {15} We could differentiate the integrand and times by x, then flip the sign and gain the result \int_0^\infty {(e^x-1)^2} dx = -{15} But this would then imply {(e^x-1)}=-1 Which doesn’t appear to be true... In theory for any function which ends up as a polynomial, the differentiation and multiplication with x should leave the form unchanged except for a numeric factor, which will drop out in some circumstances. A simpler set of examples with the normal integral on the left, and the answer on the right {|c|c|} \hline Eqn & d/d \\ \hline \int_0^\pi sin(x)dx =2 & \int_0^\pi xcos(x)dx = -2 \\ \int_0^\infty e^{-x} dx =1 & \int_0^\infty -xe^{-x} dx = -1 \\ \int_0^\infty x^{z-1}e^{-x} dx= \Gamma(z) & \int_0^\infty x^{z-1}e^{-x}(-x+z-1) dx= -\Gamma(z) \\ \int_0^\infty x^{z-1}e^{-x}(-x+z-1) dx= -\Gamma(z) & \int_0^\infty x^{z-1}e^{-x} (x+x^2+(-1+z)^2-2xz)dx=\Gamma(z) \\ ^{\infty} 1 dx = \infty & ^{\infty} -1 dx = -\infty \\ \hline Above we have used (or discovered depending on your point of view!) that (z − 1)Γ(z)−Γ(z + 1)= − Γ(z). I.e (n − 1)(n − 1)! − n!= − (n − 1)!, which allows us to discover the recursive formula n!=n(n − 1)!. Of course, it also applies to the complex functionality etc. Then reused this again an used/discovered that (1 − 2z)Γ(z + 1)+Γ(z + 2)+(z − 1)²Γ(z)=Γ(z)... This might not be so obvious and upon rearranging elucidates that n!=(2n-3)[(n-1)!]+(4n-3-n^2)[(n-2)!] Altohugh if one knows the inner workings of the gamma function this is perhaps obvious, we could potentially using this method, discover identities for horrendously complicated functions without knowing what they were... I will demonstrate below later on. This derivative has explicity operated on functions of x, which makes it still in some sense “with respect to x”. One could imagine multiple operations on multidimensional functions... Powerful ideas, think about \int_0^\infty {x} \;dx = {2} We could potentially have a situation where we know that \int_0^\infty cos(x) \;dx - \int_0^\infty {x} \;dx = -{2} Which would give a specific value of 0 for the cos integral. However, it is uncertain that one can trust this method. Some contradictory examples to it working are {|c|c|} \hline Eqn & d/d \\ \hline \int_0^\pi cos(x)dx =0 & \int_0^\pi xsin(x)dx = \pi \\ \int_0^\pi x^2 dx =\pi^3/3 & \int_0^\pi 2x^2dx = 2\pi^3/3 \\ \int_0^\pi x^2 + 4x + 2 dx = {3}(6+6\pi+\pi^2) & \int_0^\pi 2x^2 + 4x dx = {3}(6+\pi^2) \\ \hline If it is such that the integral cannot equal 0, it may be that d0/d𝕀 can be any number/ is not defined. In the last example information was lost equation to 2π, i.e the integral of the constant term in te polynomial, this would infer that the handling of a constant in the left hand side was done incorrectly with the rule of “use the regular derivative times x”. We must revisit the formal definition of the derivative. In fact we should probably use our rule that constant terms are made negative. Infinite limits and functions that terminante at one of the limits seem to work well, this might mean there is a problem with scaling the limits...

INVERSE For a 3x3 non-singular matrix A with a determinant |A| defined by A= a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} we can calculate the inverse as A^{-1}={|A|} a_{22} & a_{23} \\ a_{32} & a_{33} & a_{13} & a_{12} \\ a_{33} & a_{32} & a_{12} & a_{13} \\ a_{22} & a_{23} \\ a_{23} & a_{21} \\ a_{33} & a_{31} & a_{11} & a_{13} \\ a_{31} & a_{33} & a_{13} & a_{11} \\ a_{23} & a_{21} \\ a_{21} & a_{22} \\ a_{31} & a_{32} & a_{12} & a_{11} \\ a_{32} & a_{31} & a_{11} & a_{12} \\ a_{21} & a_{22} If we define an operation that is RCRij(A):=Rij, remove the ith column and the jth row of A, then this is expressible as A^{-1}= |R_{11}| & |R_{12}J| & |R_{13}| \\ |R_{21}J| & |R_{22}| & |R_{23}J| \\ |R_{31}| & |R_{32}J| & |R_{33}| Where J= 0 & 1 \\ 1 & 0 Clearly the condition for a right multiplication of the J matrix being i + j = odd. Alternatively |A|(A^{-1})_{ij}=|R_{ij}J^{(i+j-1)}|=|J^{(i+j-1)R_{ij}}| This likely works because for any 2x2 matrix |A|= − |AJ₂|= − |J₂A|=|J₂AJ₂|. This property |A|= − |AJ₂| also appears to hold for a 3x3 matrix. Extrapolating backwards for a two by two matrix we get the correct formula on the proviso we define J₁ ≡ −1. This makes some sense, as for any JnJn = I and JnJnA = A. We can further extrapolate to the inverse of a 1x1 matrix A = A₁₁, taking the R₁₁ element to be the zero matrix, the determinant of this matrix is 1 and the reciprocal of the determinant of A is then just the reciprocal of A₁₁, which again is the inverse of the 1x1 matrix. Proof J₁ = −1: For any Jn, n > 1, |Jn|= − 1 as Jn is defined to be an antidiagonal matrix. |AB|=|A||B|. Therefore |AJn|= − |A|. If one extrapolates to the case n = 1, For the above to remain true, J₁ = −1 OTHER CONCEPT Looking at the same formula we can define four 3x3 matrices, top left, top right, bottom left, bottom right A^{TL}= a_{22} & a_{13} & a_{12} \\ a_{23} & a_{11} & a_{13} \\ a_{21} & a_{12} & a_{11} \\ A^{TR}= a_{23} & a_{12} & a_{13} \\ a_{21} & a_{13} & a_{11} \\ a_{22} & a_{11} & a_{12} \\ A^{BL}= a_{32} & a_{33} & a_{22} \\ a_{33} & a_{31} & a_{23} \\ a_{31} & a_{32} & a_{21} \\ A^{BR}= a_{33} & a_{32} & a_{23} \\ a_{31} & a_{33} & a_{21} \\ a_{32} & a_{31} & a_{22} These matrices are constructed from the rows of the original matrix as such, if R_i is the ith row of the original matrix, and P_n is an operator which cycles that row forward n times we have A^{TL}= R_2^TP_2 & R_1^TP_1 & R_1^TP_2 =A^{r(211)}_{p(212)}\\ A^{TR}= R_2^TP_1 & R_1^TP_2 & R_1^TP_1 =A^{r(211)}_{p(121)}\\ A^{BL}= R_3^TP_2 & R_3^TP_1 & R_2^TP_2 =A^{r(332)}_{p(212)}\\ A^{BR}= R_3^TP_1 & R_3^TP_2 & R_2^TP_1 =A^{r(332)}_{p(121)} These matricies are then such that {|A|}(A^{TL} \circ A^{BR} - A^{TR} \circ A^{BL}) = A^{-1} Where ∘ is the Hadamard product or element-wise product.

PRIMES Consider the infinite table {c | c c c c c c c c} k & 1 & 2 & 3 & 4 & 5 & 6 & 7 & \dots \\ \hline P_k & 2 & 3 & 5 & 7 & 11 & 13 & 17 & \dots\\ S^1_k & 2 & 1 & 2 & 2 & 4 & 2 & 4 & \dots\\ S^2_k & 2 & -1 & 1 & 0 & 2 & -2 & 2 & \dots\\ S^3_k & 2 & -3 & 2 & -1 & 2 & -4 & 4 & \dots\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & Where Pk is the sequence of prime numbers and Sk¹ is the difference of terms in Pk, Sk² is the difference of terms in Sk¹ etc. One can define a set of generating functions for each column such that for a given generating function Gn(an)(z), when z = 0, Gn = Pn, the, nth coeficcient of the nth power of z is the nth prime. The functions are then G_1={z-1} = 2+2z+2z^2+2z^3 \dots \\ G_2={(z-1)^2} = 3+1z-1z^2-3z^3 \dots\\ G_3={(z-1)^3} = 5+2z+1z^2+2z^4 \dots\\ G_4={(z-1)^4} \\ G_5={(z-1)^5} \\ \vdots Here one can see that the coefficients of the resulting polynomial match the table columns. It seems a better idea to tabulate the coefficients of this new set of functions. {c | c c c c c c c} & c & z & z^2 & z^3 & z^4 & z^5 & z^6 & z^7 & z^8 & z^9\\ \hline G_1(z-1)^1 & -2 \\ G_2(z-1)^2 & 3 & -5 \\ G_3(z-1)^3 & -5 & 13 & -10 \\ G_4(z-1)^4 & 7 & -26 & 34 & -17 \\ G_5(z-1)^5 & -11 & 51 & -92 & 78 & -28 \\ G_6(z-1)^6 & 13 & -76 & 181 & -222 & 143 & -41\\ G_7(z-1)^7 & -17 & 115 & -331 & 521 & -477 & 245 & -58\\ G_8(z-1)^8 & 19 & -150 & 514 & -996 & 1186 & -876 & 378 & -77 \\ G_9(z-1)^9 & -23 & 203 & -794 & 1802 & -2606 & 2474 & -1520 & 562 & -100 \\ G_{10}(z-1)^{10} & 29 & -284 & 1247 & -3230 & 5456 & -6260 & 4910 & -2564 & 823 & -129 \\ By construction the first column must be the alternating sign prime number sequence such that when z = 0, the equation reduces to the primes. We can attempt to find the sequences however for the diagonals of this table! The main diagonal appears to be negative the sequence OEIS A007504, which is the sum of the first few primes. This is achieved also by extending table1 back by one extra row. Looking a the next diagonal down, this appears to be the partial sum of the sequence OEIS A117495, which is the product of a prime with the number of primes less than itself. That is pnπ(pn − 1) when π(x) is defined as the number of primes smaller or equal to x. The next diagonal was found to be another sum over a prime product with a different element in the pi function. To re create the table above we can add piecewise terms. M_{ij}=-\delta^i_j^jP_k+\delta^i_{j+1}^jP_{k+1}\pi(P_{k+1}-1) -\delta^i_{j+2}^j P_{k+2}\pi(P_{k(k+1)/2})+\dots If we let the Mij → Mi, 1 then this is just the alternating sequence of primes!, The sums disappear. M_{i1}=(-1)^iP_i=-\delta^i_1P_1+\delta^i_{2}P_{2}\pi(P_{2}-1) -\delta^i_{3}P_{3}\pi(P_{2}-1)+\dots However this is a statement of the obvious! It implies consitency with what was done so far. Further diagonal were found, the key being the polynomial in k in the index of the prime number inside the π function the next terms in the series were found to be +\delta^i_{j+3}^j P_{k+3}\pi(P_{k(k+1)(k+2)/6}) Can assume a form of polynomial (which one is better ?/same thing) Q(k)={(m+1)!} \\ Q_n(k)={n!}^n(k+i-1) Also, removing the unecesarry pi function, we can simply state the number of primes below or equal to the kth prime is k. I.e π(Pk)=k. Then we have M_{ij}=^j -\delta^i_jP_{k+0}Q_0+\delta^i_{j+1}P_{k+1}Q_1-\delta^i_{j+2}P_{k+2}Q_2+ \dots Which can be written as an ’infinite’ sum M_{ij}=-^{\infty}^j (-1)^{l}\delta^i_{j+l}P_{k+l}Q_l(k) \\ M_{ij}=-^{\infty}^j \bigg[ }{l!}\delta^i_{j+l}P_{k+l}^l(k+m-1) \bigg ] But it is not really infinite, as the kronecker deltas will all be zero at some l and greater, so this can be expressed as M_{ij}=-^j \bigg[ }{(i-j)!}P_{k+i-j}^{i-j}(k+m-1) \bigg ] We can now express the coeficcients of a generating function’s polynomial directly for example G_4={(z-1)^4}=z^3+M_{43}z^2+M_{42}z+M_{41}}{(z-1)^4} For any general column generating function of the original table we than have G_q=^q M_{qr}z^{r-1}}{(z-1)^q} which is G_q={(z-1)^q}^q ^r \bigg[ }{(q-r)!}P_{k+q-r}^{q-r}(k+m-1) \bigg ]z^{r-1} We should then reclaim the fact that when z = 0 the series Gq will be that of the prime numbers. The only non zero term will be when r = 1 P_q={(-1)^q}\bigg[ }{(q-1)!}P_{q}^{q-1}(m) \bigg ] Which is indeed reclaimed!

ABSTRACT Under matrix multiplication, different shaped matrices (diagonal, upper/lower triangular and anti diagonal variants etc.) form various other shapes as a result. For example in LU decomposition, a lower and upper matrix form a square matrix. I attempt to search for a group and explore this concept a little. INTRODUCTION We can draw a Cayley table under the matrix multiplication, however this is a non-commutative operation so we do not expect the table to be symmetric. Let us define, s as a square positive matrix (all elements greater than zero). d and a as diagonal and anti diagonal positive matrices respectively, then nw, ne, sw and se (compass directions) as triangular positive matrices around the respective corner. //(Examples for explicit) { c| c c c c c c c} & s & sw & ne & se & nw & d & a \\ \hline s & s & s & s & s & s & s & s \\ sw & s & sw & s & se & s & sw & se \\ ne & s & s & ne & s & nw & ne & nw \\ se & s & s & se & s & sw & se & sw \\ nw & s & nw & s & ne & s & nw & ne \\ d & s & sw & ne & se & nw & d & a \\ a & s & nw & se & ne & sw & a & d \\ It is clear from this table that the element s acts like 0 as when it is multiplied to anything it creates itself. Then the element d acts like 1 as when it is multiplied to anything it leaves it the same, (identity). In some sense a is then like −1 as it switches some property of a given element, a left multiplication of a changes sX to nX and nX to sX, where X is e or w. a right multiplication of a changes Xw to Xe and Xe to Xw, where X is n or s. With this in mind a left and right multiplication of some element with a will change both parts, for example, (a)(se)(a)=(nw). There cannot be an inverse for every operation when we insist that the matrices are positive, thus they cannot form a group.

Vectors, instead of N vectors of size d, why not d vectors of size N. That is, $,,$, store all the x, y, z coordinates for all particles. Does there exist a set of operations for these constructs that give the same answers for vector problems?

PROBABILITY DISTRIBUTION The end result of this project will be a series of volumes sampled from an unknown probability distribution. We want to determine what that distibution will be. We will have a functional form for that distribution P(V|θ) where θ are the unknown parameters of the distibution. In the past we’ve used a generalized Gaussian. P(V|θ) is the biased distribution, so it will be the generalized gaussian times V. This is mostly taken from http://en.wikipedia.org/wiki/Distribution_fitting There are several ways to determine what the values of θ should be. None of which involve histogramming. maximum likelihood method http://en.wikipedia.org/wiki/Maximum_likelihood The likelihood is the joint probability distribution of all the observations given the set of parameters θ L(V_1, ..., V_N | \theta) = ^{N} P(V_i | \theta) If we maximize this (actually the log of this) we get the maximum likelihood estimate for θ. regression on the cumulative distribution This seems fairly ststraightforward. I suspect it reduces to maximum likelihood methods moment fitting We record a number of moments of the observed variables, e.g. ⟨V⟩, ⟨V²⟩, etc. We then fit analytical moments of P(V|θ) to the observed moments. The advantage of this is that you don’t need to save every data point, just the running averages. The downside is obviously that you’re only fitting a few parameters. This might be good if you have too many observations to realistically record them all. Bayesian updates http://en.wikipedia.org/wiki/Bayesian_inference#Parametric_formulation With this method you start with some prior distribution over the parameters P(θ) and apply Bayes’ theorem iteratively to update the distribution given the additional knowledge (the fact of the observation). P(\theta | V_1) = { P(V_1)} The denomenator is simply the integral of the numerator over all possible values of θ (kind of like the partition function). After applying all the updates we have P(\theta | V_1, \dots, V_N ) = { P(V_1, \dots, V_N)} where L is the likelihood as defined earlier. To find the best parameters θ we then maximize the log probability. \log P(\theta | V_1, \dots, V_N ) = \log L(V_1, ..., V_N | \theta) + \log P(\theta) + The advantage of the Bayesian method is that it allows you to naturally incorporate prior information about the distribution of θ. At a minimum this allows you to place bounds, e.g. if some parameters must be positive. It also allows you to incorporate prior beliefs. If you think θ₁ should be a small parameter you can make P(θ)∼exp(−αθ₁²) We can also use the principle of maximum entropy to obtain the prior probability distribution. This would be the prior that corresponds to the least assumptions on our part (the least informative prior). I suspect this would result in the flat distribution, but I’m not sure. Bishop §2.4.3 discusses non-informative priors. He says that a non-informative prior should reflect symmetries of the probability distribution. For example, the mean μ of a Gaussian distribution is translationally invariant, so the prior on μ should be as well, so the non-informative prior is the flat distribution. The lenth scale L of a Gaussian distribution is scale invariant, so the non-informative prior should be ∼1/L. We might be able to formulate the requirement that P(V)→0 for large V in a more flexible way using priors rather than the blunt way of specifing the class functional form of P(V|θ). Another advantage of the Bayesian method is that it gives you a full probability distribution over θ. Which is a lot more information than a simple maximum likelihood estimate. (I’m not sure if the likelihood function can be thought of as a probability distribution. I suspect that the Bayesian probability simply reduces to the Likelihood if the prior P(θ) is flat. But I can’t say for sure.) bayesian model comparison Bishop §3.4 talks about Bayesian model comparison. In our situation, different models would correspond to different functional forms for P(V|θi) with θi being parameters of model i (Mi). If we define the data D = {V₁, ..., VN} then p(M_i | D) \propto p(M_i) p(D|M_i) where p(Mi) is the prior over p(Mi) which we will assume is flat. p(D|Mi) is known as the model evidence. It is also sometime called the marginal likelihood because it can be viewed as the likelihiood function over the space of models, in which the parameters of the models have been marginalized out (integrated out). The ratio of model evidences for two models is known as the Bayes factor. Choosing the model with the highest evidence is called model selection. Even better is to combine multiple models (mixture distribution) p(V| D) = \sum_i p(V|D, M_i) p(M_i | D) If the prior (p(θ)) is improper (not normalizable) then the evidence is not defined. It may however, still be possible to compare different models by taking the ratio. non-parametric formulation If we want to approximate the probability density without specifying a functional form we can use density estimation http://en.wikipedia.org/wiki/Density_estimation (also see Bishop §2.5), especially kernal density estimation http://en.wikipedia.org/wiki/Kernel_density_estimation. This is a bit like histogramming with smoothing, but you end up with an analytic form for the density at the end. This can easily be done using http://scikit-learn.org/stable/modules/density.html

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 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