ABSTRACT I consider a special type of sum where for two small polynomials (quadratic) f(x) and g(x), treated as a series representation, we define the inverted sum as (f−1(x)+g−1(x))−1. MAIN For now consider quadratics f(x) = a_0 + a_1 x + a_2 x^2 \\ g(x) = a_0 + b_1 x + b_2 x^2 noting the two must share the same constant a₀. Define q = a₁ + b₁Then we have that (f^{-1}(x)+g^{-1}(x))^{-1} = a_0 + {q}x + {q^3}x^2 -{q^5}x^2 + {q^7}x^3 + \cdots however if we set a₂ = b₂ = 1, we get (f^{-1}(x)+g^{-1}(x))^{-1} = a_0 + {q}x + {q^2}x^2 - {q^3}x^3 + {q^4}x^4 + \cdots which appears to be (f^{-1}(x)+g^{-1}(x))^{-1} = a_0 + {q}x + {q^2}x^2 + ^\infty (-1)^k {q^k}x^k where the coefficients ck seem to be A045623, with c₃ = 2, c₄ = 5, ⋯. If c_k= (k+1)2^{k-4}, \;\; k>2 as suggested on the OEIS, then we can write (f^{-1}(x)+g^{-1}(x))^{-1} = a_0 + a_1b_1{q} + (a_1^2-a_1b_1+b_1^2){q^2} + ^\infty (-1)^k(k+1)2^{k-4}(a_1-b_1)^2{q^k} we can let y = x/q and evaluate the sum then we have (f^{-1}(x)+g^{-1}(x))^{-1} = a_0 + a_1b_1y + (a_1^2-a_1b_1+b_1^2)y^2 - {(1+2y)^2}

ABSTRACT I ponder about some braid like-structures. I don’t have the pictures to put in at the moment, so it will look confusing until I make something to draw pictures. They can easily be linked to numbers and code structures. MAIN Define some braid like structures, these are lines with going from left to right, in this generalised sense we will let them twist, fork, merge etc. I had a notation _1S_2 = _2 S_1^{-1} = \\ _2S_1 = _1 S_2^{-1} = \\ T_2 = \\ T^{-1}_2 = \\ I_1 = \\ I_2 = there are some operators, addition is to join the two (in order), such that the right parts of the first object meet the left parts of the second object, for example I_1+I_1 = I_1 \\ _1S_2 + _2S_1 = I_1 \\ T_2 + T^{-1}_2 = I_1 \\ _1S_2 + I_2 = _1S_2 but we also have a stacking operator ∩ (in order top to bottom), such that I_1 \cap I_1 = I_2\\ I_2 + _1S_2 = _1S_2 \cap I_1 we have the rule that we always join the highest possible lines, unless an empty line I₀ is added in the capping sequence I_2 + I_0\cap _1S_2 = I_1 \cap _1S_2 It seems that any kind of flow pattern desired can be chopped up into these small parts, although if we wanted a one to three split, we might need to allow _1S_3 = _1S_2+(I_1\cap _1S_2) Three way permutations may be made for example T_2\cap I_1 + I_1 \cap T_2 = where σabc → bca is a permutation operator that takes the tuple [a, b, c] and maps it to [b, c, a]. PRODUCT We could potentially devise other operations such as a product. This might work as follows _1S_2 \times _1S_2 = _1S_2 + _1S_2\cap _1S_2 PROGRAMMING These kind of diagrams could easily be usedto create a code of sorts. We can assign an operator to the split functions as such _2S_1(+) = a+b = \\ _2S_1(\times) = a\times b = \\ _2S_1(-) = a-b = \\ _2S_1(\%) = a\;\;b = \\ _2S_1(=) = a==b = \\ _2S_1(<) = a<b = \\ for the inverse, we will copy the input into two lines, then a code to take an input x and double it might look like [S_{12}+S_{21}(+)]x=2x where the code is acting as an operator in curly braces. I have switched to the notation S₁₂ and S₂₁ for ease, and may change it above. We may set x to be 1 and see how we might produce other numbers with a ’single’ operation... I will remove the plus for now, and state that it is implied. \{S_{12}+S_{21}\}=2\\ \{S_{12}+S_{12}\cap I_1 + I_1 \cap S_{21} + S_{21}\}=3\\ \{S_{12}+I_1\cap S_{12} + I_1 \cap S_{21} + S_{21}\}=3\\ Note there is a nice set of symmetry. Note there are more than one code to achieve the same result. One could try and interpret the number of different codes that produce each number from 1. Of course, we would have to exclude the infinite number of single lines we could add. There is a nice feature of the code \{S_{12}+I_1\cap S_{12} + I_1 \cap S_{21} + S_{21}\} as it features the code for 2 inside it, let us call the code for 2 A_2, A_2=\{S_{12}+S_{21}\} then we can write \{S_{12}+ I_1 \cap A_2 + S_{21}\}=3 again it is invariant under switching I₁ inside the cap. we will also see that \{S_{12}+ A_2 \cap A_2 + S_{21}\}=4 A_2=\{S_{12}+S_{21}\}=2\\ A_3=\{S_{12}+ I_1 \cap A_2 + S_{21}\}=3\\ A_4=\{S_{12}+ A_2 \cap A_2 + S_{21}\}=4\\ we then see the rules of addition forming \{S_{12}+ A_2 \cap A_3 + S_{21}\} = 5\\ \{S_{12}+ A_3 \cap A_3 + S_{21}\} = 6\\ In effect I₁ = A₁, so the rules hold for smaller examples, these then relate to partitions of integers, there will be a huge number of equivalent codes for larger n. We can consider that prime numbers might struggle to form symmetric g-braids, because symmetry (about the horizontal axis) implies divisibility of through repetition of sub units here. It is possible to draw a vertical line through the center of the braid and it will cut n lines, where the code generates n. We can see an algorithm for factorization checking. Draw n points in a vertical line. One can connect these points rightwards with as many S₂₁ objects as needs be, and the sum will terminate to be n, however, drawing valid modular patterns leftwards is another matter. In light of this, a more sensible definition of the product of two g-braids is the following. Draw the braid such that it is symmetric about the vertical axis, it should always be possible to do this, (perhaps call this the standard form), then make a cut through the vertical axis and detach the two halves, then insert the product braid in-between the two. That way if we have one representation of 3 as \{S_{12}+I_1\cap S_{12} + I_1 \cap S_{21} + S_{21}\}=3 if we times by 2, we will get S_{12}+I_1\cap S_{12} + S_{12}\cap S_{12}\cap S_{12} + where symm. means the symmetric terms. This has a triple capped structure in the middle, but ends up summing to 6, which preserves the product. We can reduce this by factoring in A₂, and the whole expression becomes \{S_{12}+I_1\cap S_{12} + A_2 \cap A_2 \cap A_2 + I_1\cap S_{21}+ S_{21}\}=6 we see the effect was to dump 3, capped A₂’s into the centre of the original string. Because we did this in the mid-point, the 3 will be generalised to n for any product n × m. FACTORISATION If we start with a random valid ’net’ then is there a nice way to rearrange in-situ to find repeatable mid-axis terms? For example, one can add and take, it is possible to shuffle small loops. SYMMETRY BREAKING What if all numbers are not the same. Although two nets may sum to the same number, they are different. Perhaps, we can investigate a number system where all numbers randomly start in a valid configuration (must count degeneracy). Then when combining them, they obey statistics. INFINITE SUMS, NON-INTEGER NETS AND SO ON Consider certain formulae that lead to non-integers.

ABSTRACT If we have a series for a function f(x), we might choose an expansion over ℕ, in the sense that the powers of x are integers. We might even do an expansion over ℤ, allowing negative integer powers. I am thinking about ℝ. EXAMPLE We have 1+x+x^2+x^3+\cdots = {1-x} the spacing between the powers Δ = 1. Then we can shrink Δ, 1+x^{1/2}+x+x^{3/2}+\cdots = {1-} with Δ = 1/2. Then in general ^\infty x^{k/m} = {1-x^{1/m}} then if we weight each term by Δm = 1/m, we have f_m(x)=^\infty \Delta_m x^{k/m} = {m(1-x^{1/m})} and we can take the limit to give f_m(x) = -{\log(x)} which is the same result as \int_0^\infty x^k \; dk = -{\log(x)} FURTHER So if our function had some coefficients, f(x) = ^\infty a_k x^k we would expect them to change into a function for continuous k, f_\infty(x) = \int_0^\infty a(k)x^k\;dk however, we need to modify the function f_m(x) = ^\infty \Delta_m a_{k/m} x^{k/m} for example if f(x) = ^\infty k x^k then f_m(x) = ^\infty \Delta_m {m} x^{k/m} giving f_m(x) = {\log^2(x)} = \int_0^\infty kx^k \;dk to get the result to match the integral. So we have normally that \exp(x) = ^\infty {k!} = ^\infty {\Gamma(k+1)} then we need to make the transition \exp_m(x) = ^\infty \Delta_m}{\Gamma\left({m}+1\right)} = (x^{1/m})}{m} where Eα(z), is the Mittag-Leffler function where we could try an iffy substitution ofr 1/m as 0 giving, $E_{0}(x) = {1-x}$. Then we can take the infinite limit \exp_m(x) = {m(1-x^{1/m})} = -{\log(x)} we have the same result, but I’m not sure whether to trust it. FINITE SUMS There is an interesting relationship when finite polynomials are used to begin with. For example P(x) = x+2x^2+3x^3 = ^3 kx^k we can transform this as P_m(x) = ^{3m} \Delta_m {m}x^{k/m} and take the limit P_m(x) = {\log(x)^2} the same as \int_0^3 kx^k \; dk={\log(x)^2} IS THIS USEFUL Perhaps, I guess if you had an integral \int_0^L f(k)\;dk you could turn it into the sum ^{m\lfloor L \rfloor} \Delta_m f(k/m) and try and perform the sum for the largest m possible. This is just a numerical integration scheme. But it allows use to get convergent functions/approximations quite quickly, with the number of terms/complexity of powers rapidly increasing. Non-integer L do become a problem. Rounding will be needed. Another technique is to simply rescale m to be a non-integer, such that the sum is over integers.

AbstractWhen in the course of human events it is found necessary.Underwater noise pollution from ships is a chronic, global stressor impacting a wide range of marine species. Ambient ocean noise levels nearly doubled each decade from 1963-2007 in low-frequency bands attributed to shipping, inspiring a pledge from the International Maritime Organization to reduce ship noise and a call from the International Whaling Commission for member nations to halve ship noise within a decade. Our analysis of data from 1,582 ships reveals that half of the total power radiated by a modern fleet comes from just 15% of the ships, namely those with source levels above 179 dB re 1 μPa @ 1 m. We present a range of management options for reducing ship noise efficiently, including incentive-based programs, without necessarily regulating the entire fleet.

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 Consider the set of polynomials created by using the Sierpinski triangle as a set of coefficients. Can we find a recursion relation or something similar? MAIN The polynomials will then be the following P_{0}(x)=1\\ P_{1}(x)=1+x^{1}\\ P_{2}(x)=1+x^{2}\\ P_{3}(x)=1+x^{1} +x^{2} +x^{3}\\ P_{4}(x)=1+x^{4}\\ P_{5}(x)=1+x^{1} +x^{4} +x^{5}\\ P_{6}(x)=1+x^{2} +x^{4} +x^{6}\\ P_{7}(x)=1+x^{1} +x^{2} +x^{3} +x^{4} +x^{5} +x^{6} +x^{7}\\ P_{8}(x)=1+x^{8}\\ P_{9}(x)=1+x^{1} +x^{8} +x^{9}\\ P_{10}(x)=1+x^{2} +x^{8} +x^{10}\\ P_{11}(x)=1+x^{1} +x^{2} +x^{3} +x^{8} +x^{9} +x^{10} +x^{11}\\ P_{12}(x)=1+x^{4} +x^{8} +x^{12}\\ P_{13}(x)=1+x^{1} +x^{4} +x^{5} +x^{8} +x^{9} +x^{12} +x^{13}\\ P_{14}(x)=1+x^{2} +x^{4} +x^{6} +x^{8} +x^{10} +x^{12} +x^{14}\\ P_{15}(x)=1+x^{1} +x^{2} +x^{3} +x^{4} +x^{5} +x^{6} +x^{7} +x^{8} +x^{9} +x^{10} +x^{11} +x^{12} +x^{13} +x^{14} +x^{15}\\ P_{16}(x)=1+x^{16}\\ P_{17}(x)=1+x^{1} +x^{16} +x^{17}\\ P_{18}(x)=1+x^{2} +x^{16} +x^{18}\\ P_{19}(x)=1+x^{1} +x^{2} +x^{3} +x^{16} +x^{17} +x^{18} +x^{19} \cdots Some remarks P_n(x)=P_0(x)P_n(x)&n=0+n\\ P_3(x)=P_1(x)P_2(x)&3=1+2\\ P_5(x)=P_1(x)P_4(x)&5=1+4\\ P_6(x)=P_2(x)P_4(x)&6=2+4\\ P_7(x)=P_3(x)P_4(x)&7=3+4\\ P_{11}(x)=P_3(x)P_8(x)&11=3+8\\ P_{15}(x)=P_7(x)P_8(x)&15=7+8\\ P_{2^k}(x)=1+x^{2^k}\\ P_{2^k-1}(x)=^{2^k-1} x^i Some difference equations P_5 +(x^2-x)P_4 = P_6\\ P_1 +(x^2-x)P_0 = P_2\\ P_3 +(x^4-x^2)P_1 = P_5 Some composition equations P_4(x)=P_2(x^2)\\ P_6(x)=P_3(x^2)\\ P_{12}(x)=P_6(x^2)\\ If we take the sums of the powers that are numbers of the form 2k, k ∈ ℕ𝟘, for each polynomial then the powers sum to the index of the polynomial. For example P_{15}(x)=1+x^{1} +x^{2} +x^{3} +x^{4} +x^{5} +x^{6} +x^{7} +x^{8} +x^{9} +x^{10} +x^{11} +x^{12} +x^{13} +x^{14} +x^{15}\\ S[P_{15}(x)]=x^{1} +x^{2} +x^{4} +x^{8}\\ C[S[P_{15}(x)]] = 1+2+4+8 = 15 where S is a sieving transform to remove non-power of two powers and C is a transform converting powers to values. P_{15}(x)=1+x^{1} +x^{2} +x^{3} +x^{4} +x^{5} +x^{6} +x^{7} +x^{8} +x^{9} +x^{10} +x^{11} +x^{12} +x^{13} +x^{14} +x^{15}\\ S^*[P_{15}(x)]=1+x^{3}+x^{5} +x^{6} +x^{7}+x^{9} +x^{10} +x^{11} +x^{12} +x^{13} +x^{14} +x^{15}\\ C[S^*[P_{15}(x)]]=0+3+5+6+7+9+10+11+12+13+14+15 = 105 = 7\cdot 15 where S*[P]=P − S[P] k & S^*[P_k(x)] \\ \hline \\ 0 & 0\\ 1 & 0\\ 2 & 0\\ 3 & 3\\ 4 & 0\\ 5 & 5\\ 6 & 6\\ 7 & 21=3\cdot 7\\ 8 & 0\\ 9 & 9\\ 10 & 10\\ 11 & 33=3\cdot 11\\ 12 & 12\\ 13 & 39=3\cdot13\\ 14 & 42=3\cdot 14\\ 15 & 105 = 7\cdot 15\\ 16 & 0\\ 17 & 17\\ 18 & 18\\ 19 & 57=3\cdot 19 We can see that for powers of 2, S*[P]=0, but for other powers k, it seems to always be a multiple of k. If we define the number of terms (including the 1) of the polynomial as 𝒩(P), then we can see that S[P_k(x)]+S^*[P_k(x)] = (P)k}{2}, It then seems that if for a given k S^*[P_k(x)]=k then k is the sum of two unique powers of 2.

I summarize briefly the ideas behind Grimoire / Grok for the purposes of academic reference. Grimoire / Grok is a memory state saving application that is aimed both at expanding the current working space a mind has available, and also keeping track of a larger number of projects / ideas / factoids, than a mind is capable of by itself. This could be useful either for student studying purposes, for researchers, or (hopefully - further research needed) for mental disabilities such as Alzheimer's / dementia, which limit the brains ability to keep track of and recall certain thoughts or memories. For example, one could couple the application with some sort of unobtrusive heads up display, of which several types exist on the market (current the software is rendered in a browser, so it should work on a hud).Grimoire / Grok has two modes. The "Grimoire" mode is aimed at collecting / preserving / updating / keeping track of a large number of segmented thoughts. Thoughts are organized by topic / item, for example: "calculus" / "stokes theorem" is one possible topic / item pair. Users may navigate to different topic / item notes through a central index, a search bar, or by linking from page to page with links. Thoughts are written in markdown / html / latex / javascript, with the primary mode being Markdown. Thoughts are generally separated by Markdown header, which servers a dual purpose for Grok mode. For a folder structure, thoughts are stored by grimoire / topic / item / (files related to this thought). Typically different Grimoires should correspond to contexts, such as 'work', 'home', 'hobby', 'school', 'research' etc. Topics are split up into subareas of context. For instance, school might have topics such as 'geometry', 'language', 'history,' 'art'. Items then deal with specifics. For example, one might store in the programming / C++ /  some items corresponding to strings, math, and so on. Thoughts, as files and a folder structure are tracked through git  \cite{2009} so that any accidental changes can be reverted, and so that a clear progression of thoughts can be maintained. The author imagines that stronger cryptographic guarantees could be given to memory and mental state through signed git messages, although recently the hash function used in git (sha1) has been shown to be non-collision resistant by Google (citation needed), so some changes to the software would likely be necessary for human lifespan length use. Further types of security guarantees are likely possible (and likely desired, if one is to rely on such software for the integrity of ones thoughts).While the Grimoire mode is aimed at context specific long-term memory recall, Grok mode is aimed at short-term, working memory improvement. Grok mode works by first selecting a subset of topics, and then the program proceeds through each item in the topic. Each item is split up by markdown headers. Each header is asked as a question, while the body below the header corresponds to the answer. The user is expected to actively improve and prune each note throughout the process of using grok mode. The user then decides how well they know each topic, and if they know a certain topic, selecting 'good' will increase the time before it is asked about again, similar to the Pimsleur scheduled memory learning \cite{Pimsleur_1967}. In fact, the application uses a simple progression based on the Fibonacci sequence: items are quizzed again after 1, 1, 2, 3, 5, 8, ... days, assuming they are answered successfully each time. There are other software applications which do spaced repetition learning as well, such as Anki (citation needed).  Grimoire / topic / item structure is exactly the same as in Grimoire mode, so that no additional work needs to be done to create quizzes. In this way, a user can quickly refresh a given topic shortly before it is needed.

Donald Trump promete fazer a América grande novamente. Para seus seguidores isso significa trazer de volta empresas, criar empregos, acabar com o "globalismo" e fazer o país mais seguro. Há chances dessas medidas propostas pelo Presidente dos Estados Unidos Unidos darem certo?

ABSTRACT: We present an L3-type photonic crystal cavity in silicon with a theoretical quality factor of 20.9 million. This highly-optimized design is made by shifting the positions of the holes surrounding the cavity, and was obtained through an automated global optimization procedure.

AbstractThe central complex is a highly conserved insect brain region composed of morphologically stereotyped neurons that arborize in distinctively shaped substructures. The region has been implicated in a wide range of behaviors, including navigation, motor control and sleep, and has been the subject of several modeling studies exploring the underlying circuit mechanisms. Most studies so far have relied on assumptions about connectivity between neurons in the region based on their overlap in light-level microscopic images. Here, we present an extensive functional connectome of Drosophila melanogaster's central complex at cell-type resolution. Using simultaneous optogenetic stimulation and GCaMP recordings and pharmacology, we tested the connectivity between over 70 presynaptic-to-postsynaptic cell-type pairs. The results reveal a range of inputs to the central complex, some of which had not been previously described, and suggest that the central complex has a limited number of output channels. Despite the high degree of recurrence in the circuit, network connectivity appears to be sparser than anticipated from light-level images. Finally, the connectivity matrix we obtained highlights the potentially critical role of a class of bottleneck interneurons of the protocerebral bridge known as the Δ7 neurons. All data is provided for interactive exploration in a website with the capacity to accommodate additional connectivity information as it becomes available.IntroductionPositioned in the middle of the insect brain, the central complex provides a unique opportunity to obtain mechanistic insights into the way brains build and use abstract representations. Studies in a variety of insects, including locusts, dung beetles and monarch butterflies, have used intracellular recordings to chart maps of polarized light E-vectors in substructures of the region \cite{Heinze_2007a,26305929}, and extracellular recordings from the cockroach have found sensory and motor correlates throughout the region \cite{Bender_2010,Guo_2012,Roy_2012}. More recently, experiments in behaving Drosophila have shown that both visual and motor cues can update a fly's internal representation of heading \cite{Seelig_2015}. Independently, neurogenetic studies have used disruptions of the normal physiology of the structure to highlight its involvement in a variety of functions, including motor coordination \cite{Poeck_2008}, visual memory \cite{16452971}, sensory-motor adaptation \cite{Triphan_2010}, and short- and long-term spatial memory \cite{Neuser_2008,Ofstad_2011}. It is likely that these tasks rely on the correct establishment and use of an internal representation of heading. Moreover, the scale of the network —a few thousands neurons in the fly— and the ease of genetic access to individual cell types in Drosophila melanogaster, make this circuit tractable with existing theoretical and experimental methods. Detailed light level anatomy \cite{Hanesch_1989,Lin_2013,Wolff_2015} of a significant fraction of the cell types, along with the availability of tools to genetically target these neurons by type \cite{Wolff_2015}, have given rise to the first mechanistic investigations of how the circuit constructs a stable heading representation \cite{Kim_2017}, and how this representation updates as the animal turns in darkness \cite{Turner_Evans_2017,Green_2017}. Such results and related findings from other insects have also inspired a number of modeling studies aimed at predicting or reproducing physiologically and behaviorally relevant response patterns \cite{kakaria_ring_2017,givon_generating_2017,chang_topographical_2017,Turner_Evans_2017,Cope_2017,Su_2017,Fiore_2017,Kim2017,Stone2017}. Many of these models make assumptions about connectivity within the central complex based on the degree of overlap at the light microscopy level between processes that look bouton-like and those that seem spiny,  which are suggestive of pre- and post-synaptic specializations respectively. We aimed to construct a connectivity map based on functional data, which includes information about whether connections are effectively excitatory or inhibitory. This map will help dissect the function of the central complex by constraining large-scale models and aiding the formulation and testing of new hypotheses. Given the likely number of existing and undiscovered cell types in the central complex,  the diversity of neurotransmitters and receptors they express, the mixture of pre- and post-synaptic specializations in their arbors, and the dense recurrence of the network, we see this map as an initial scaffold, which will allow new information to be incorporated as it becomes available.The quest to obtain circuit diagrams can be dated back to the work of Cajal and Golgi \cite{Ram_n_y_Cajal_1894,Pannese_1999}, who used sparse labeling techniques to reveal circuit architectures. Anatomical methods based on marking a discrete subset of neurons and imaging them with light microscopy have recently been revived in the form of techniques relying on stochastic genetic labeling \cite{Livet_2007,Hampel_2011,Nern_2015,Lee_2001,Chiang_2011} and photoactivatable fluorophores \cite{Patterson_2002,Ruta_2010}. These methods allow the extraction of the detailed anatomy of individual neurons. But even when used in combination with synaptic markers \cite{Nicolai_2010,8229205,Zhang_2002,Fouquet_2009}, such methods do not offer definitive evidence of synaptic connections, as they rely solely on the proximity of putative pre- and post-synaptic compartments. Recently, promising trans-synaptic genetic tagging systems \cite{Talay_2017,Huang_2017} have been developed to address some of these issues. However, none of these approaches provide any insight into the functional properties of potential connections. Despite such shortcomings, light-level imaging constitutes a good starting point by constraining the search for possible connections within large populations of neurons —at the simplest level, if putative pre- and post-synaptic compartments do not overlap in light microscopy images, they cannot be in synaptic contact. More recently, electron microscopy (EM) reconstruction has become the gold standard for connectomics \cite{Briggman_2012,Zheng2017,Schneider_Mizell_2016}. Under ideal conditions, it permits the unambiguous identification of synapses between all neurons in a given volume. As game-changing as this capability is, the technique also suffers from a few limitations. Acquiring, processing and analyzing the data is still time-consuming. As a result, connectomes from EM data are typically based on data from a single animal. In addition, EM does not permit the identification of neurotransmitter types at a given synapse, nor does it detect gap-junctions in invertebrate tissue, at least at present \cite{Zheng_2017}. Finally, it can be challenging to assess the strengths of connections between two neurons, because it is not yet clear whether the number of synapses predicts the functional strength of the connection. Functional methods address some of these drawbacks. Simple measures of activity have been used to assess a form of functional connectivity: regions or neurons whose simultaneously recorded activity is correlated —either spontaneously or during a given task— are deemed connected. This has been used with EEG, fMRI and MEG recordings in humans to establish maps at the brain region level \cite{Salvador_2004,Stam_2004} and with multi-electrode recordings in monkeys and rodents (for example, \citealt{Gerhard_2011}).  Functional connectivity has also been inferred from correlations or graded changes in the response properties of neurons recorded in different animals, usually in cases where the neurons have overlapping arbors when examined with light microscopy. This approach has been employed to suggest polarized light processing pathways in the central complex of the locust and monarch butterfly \cite{Heinze_2009a,Heinze_2014}. However, such functional methods are correlative and do not provide a causal basis for the inferred connectivity.To obtain a causal description of functional connectivity —sometimes termed effective connectivity— one needs to either stimulate one node of the network while recording from another one, or record both at sufficiently high resolution as to detect hallmarks of direct connectivity. The most reliable application of such an idea is paired patch-clamp recording, which identifies connected pairs and their functional properties with a high level of detail \cite{Huang_2010,Yaksi2010,Fişek2014}, but can only be performed at low throughput \cite{Jiang_2015}. In  recent years, the development of optogenetics has expanded the toolkit for simultaneous stimulation and recording experiments \cite{17435752}. In Drosophila, the ease of use of genetic reagents renders such approaches particularly attractive. Combinations of P2X2 and GCaMP \cite{yao_analysis_2012}, P2X2 and patch-clamp recordings \cite{hu_functional_2010}, Channelrhodopsin-2 and patch-clamp \cite{gruntman_integration_2013} and CsChrimson and GCaMP (Hampel et al., 2015; Zhou et al., 2015; Ohyama et al., 2015) have been used in individual studies to investigate a small number of connections. Methods that rely on the genetic expression of calcium indicators to detect potential post-synaptic responses operate at a lower resolution than paired-recordings since they usually establish connectivity between cell types, as defined by the genetic driver lines used, rather than between individual neurons. They are also ambiguous as far as the directness of discovered connections, which are not precluded from being several synapses away from the stimulated neuron (but see Results/Discussion) and are limited by the sensitivity of the calcium sensors used. Despite these shortcomings, these methods constitute a good compromise as they still provides a causal measure of functional connectivity, and at a much higher throughput than double patch recordings. It is also worth noting that the advantages and limitations of these techniques complement those of serial EM reconstructions. We chose to apply this combination of optogenetics and calcium imaging on a large scale by systematically testing genetically defined pairs of central complex cell types, therefore building a large and extensible map of functional connections in the structure at cell-type resolution.Cell types and information flow in the central complex

INTRODUCTION When conducting a meta-analysis that includes previously published data, differences between treatments reported with P-values, least significant differences (LSD), and other statistics provide no direct estimate of the variance. ESTIMATING STANDARD ERROR FROM OTHER SUMMARY STATISTICS (_P_, _LSD_, _MSD_) In the context of the statistical meta-analysis models that we use, overestimates of variance are okay, because this effectively reduces the weight of a study in the overall analysis relative to an exact estimate, but provides more information than either excluding the study or excluding any estimate of uncertainty (though there are limits to this assumption such as ...). Where available, direct estimates of variance are preferred, including Standard Error (SE), sample Standard Deviation (SD), or Mean Squared Error (MSE). SE is usually presented in the format of mean (±SE). MSE is usually presented in a table. When extracting SE or SD from a figure, measure from the mean to the upper or lower bound. This is different than confidence intervals and range statistics (described below), for which the entire range is collected. If MSE, SD, or SE are not provided, it is possible that LSD, MSD, HSD, or CI will be provided. These are range statistics and the most frequently found range statistics include a Confidence Interval (95%CI), Fisher’s Least Significant Difference (LSD), Tukey’s Honestly Significant Difference (HSD), and Minimum Significant Difference (MSD). Fundamentally, these methods calculate a range that indicates whether two means are different or not, and this range uses different approaches to penalize multiple comparisons. The important point is that these are ranges and that we record the entire range. Another type of statistic is a “test statistic”; most frequently there will be an F-value that can be useful, but this should not be recorded if MSE is available. Only if there is no other information available should you record the P-value.

ABSTRACT Angle-resolved (AR) RABBIT measurements offer a high information content measurement scheme, due to the presence of multiple, interfering, ionization channels combined with a phase-sensitive observable in the form of angle and time-resolved photoelectron interferograms. In order to explore the characteristics and potentials of AR-RABBIT, a perturbative 2-photon model is developed; based on this model, example AR-RABBIT results are computed for model and real systems, for a range of RABBIT schemes. These results indicate some of the phenomena to be expected in AR-RABBIT measurements, and suggest various applications of the technique in photoionization metrology.

Jingjing  Liang1*, Thomas W. Crowther2, Nicolas Picard3, Susan Wiser4, Mo Zhou1, Giorgio Alberti5, Ernst-Detlef Schulze6, A. David McGuire7, Fabio Bozzato8, Hans Pretzsch9, Sergio de-Miguel10,11, Alain Paquette12, Bruno Hérault13, Michael Scherer-Lorenzen14, Christopher B. Barrett15, Henry B. Glick16, Geerten M. Hengeveld17,17.5, Gert-Jan Nabuurs17,17.6, Sebastian Pfautsch18, Helder Viana19,20, Alexander C. Vibrans21, Christian Ammer22, Peter Schall22, David Verbyla23, Nadja Tchebakova24, Markus Fischer25,26, James V. Watson1, Han Y.H. Chen27, Xiangdong  Lei28, Mart-Jan Schelhaas17, Huicui Lu29, Damiano Gianelle30,31, Elena I. Parfenova24, Christian Salas32, Eungul Lee33, Boknam Lee34, Hyun Seok Kim34,35,36,37, Helge Bruelheide38,39, David A. Coomes40, Daniel Piotto41, Terry Sunderland42,43, Bernhard Schmid44, Sylvie Gourlet-Fleury45, Bonaventure Sonké46, Rebecca Tavani47, Jun Zhu48,49, Susanne Brandl9,49.5, Jordi Vayreda50,51, Fumiaki Kitahara52, Eric B. Searle27, Victor J. Neldner53, Michael R. Ngugi53, Christopher Baraloto54, Lorenzo Frizzera30, Radomir Bałazy55, Jacek Oleksyn56, Tomasz Zawiła-Niedźwiecki57, Olivier Bouriaud58,58.5, Filippo Bussotti59, Leena Finér60, Bogdan Jaroszewicz61, Tommaso Jucker40, Fernando Valladares62, Andrzej M. Jagodzinski56,63, Pablo L. Peri64,65,66, Christelle Gonmadje46,67,William Marthy68, Timothy O'Brien68, Emanuel H. Martin69, Andrew R. Marshall70,70.5, Francesco Rovero71, Robert  Bitariho72, Pascal A. Niklaus73,74, Patricia Alvarez-Loayza75, Nurdin Chamuya76, Renato Valencia77, Frédéric Mortier78, Verginia Wortel79, Nestor L. Engone-Obiang80, Leandro V. Ferreira81, David E. Odeke82, Rodolfo M. Vasquez83, Simon L. Lewis84,85, Peter B. Reich18,86

AbstractThis paper doesn't present the findings of an experiment. It presents a tool created (and still under active development) to put an specific teaching method, dynamic assessment, into practice through an online dashboard.This paper explores if students felt comfortable with this teahing method, if it helped them take control of their learning and how they felt with this dashboard.Even if the tasks to do where both individual and group tasks, only the individual activities are analyzed. This tool is used for two years, but data presented in this paper are only those collected last year (Fall 2016). Data were anonymised, cleaned and published on Zenodo (10.5281/zenodo.290129).IntroductionIn higher education, most of the time students are evaluated by mid-term and/or final exams. This means that the student's understanding and learning is evaluated on predefined day and that (s)he has to succeed that day. Failing is not allowed, however it could be good to help students learn. The idea was to allow students to fail thanks to a dynamic assessment \cite{sharples_innovating_2014}. Instead of giving student only one bullet, it allows them to fail and learn to improve until they succeed.The Dynamic assessement dashboard (DAD) has been created to assess students dynamically throughout the semester. Giving them the control on their learning (pace, tasks) leads to self-regulation \cite{hattie_visible_2012} and was expected to increase students' motivation. Getting a bonus for completing a set of tasks includes gamification features that supports students' engagement \cite{Hamari_2016}.DAD also includes some gamification mechanics like bonuses \cite{Deterding_2011,muletier_gamification:_2014}. DAD has been created as a personal dashboard. A student can't access another student's dashboard and achievements.DAD is intended to increase students' self-efficacy \cite{Zimmerman_2000} whatever their learning style is. The mix of individual and group activities should help students reach the zone of proximal development as defined by Vygotsky \cite{vygotsky_interaction_1978}.DADThe idea of DAD is born from the combination of the reading of the Open University's Innovating pedagogy 2014 report \cite{sharples_innovating_2014} and the observation of how young children's learning is assessed. The former presents the concept of dynamic assessment to give the learner personalized assessement and the latter is based on simple stamps indicating when a task has been successfully achieved. DAD is an attempt to put that in an online dashboard that displays activities defined by the teacher. All tasks are meant to help students reach the course's objectives. Students choose what to do and when to do it. If the teacher allows it, they can even choose if they want to do it or not.

ABSTRACT I want to note down a few sets of functions from symmetries in the arguments of the Meijer G-function. MAIN We have the Meijer G-function which is often defined by the line integral G^{m,n}_{p,q}\left( a_1, \cdots, a_p \\ b_1,\cdots,b_q \Bigg| z\right)={2\pi i}\int_L ^m \Gamma(b_j-s)^n \Gamma(1-a_j+s)}{^q\Gamma(1-b_j+s)^p \Gamma(a_j-s)}z^s\;ds which is an extension to hypergeometric type functions defined by the Mellin-Barnes integral. Many special functions can be noted down as corner cases of this general function. For example \cos(x) = G^{1,0}_{0,2}\left( \cdot \\ 0,{2} \Bigg| {4}\right)\\ \sin(x) = G^{1,0}_{0,2}\left( \cdot \\ {2},0 \Bigg| {4}\right)\\ \cosh(x) = G^{1,0}_{0,2}\left( \cdot \\ 0,{2} \Bigg| -{4}\right)\\ \sinh(x) = -iG^{1,0}_{0,2}\left( \cdot \\ {2},0 \Bigg| -{4}\right) seeing the difference with sinh, shows that this function may have been defined ’wrong’ (not that there is a wrong), but it breaks the symmetry here. This would pertain to the formula for sinh \sinh x = {2}(e^x - e^{-x}) not having the factor of i in the denominator that sin has \sin x = {2i}(e^{ix}-e^{-ix}) we can consider another group c_2(x) = 2G^{1,0}_{0,2}\left( \cdot \\ 0,{2} \Bigg| {4}\right)\\ s_2(x) = 2G^{1,0}_{0,2}\left( \cdot \\ {2},0 \Bigg| {4}\right)\\ c_3(x) = 4G^{1,0}_{0,2}\left( \cdot \\ 0,{2} \Bigg| {4}\right)\\ s_3(x) = 4G^{1,0}_{0,2}\left( \cdot \\ {2},0 \Bigg| {4}\right) we would normally have the relations {dx} \cos(x) = -\sin(x)\\ {dx} \sin(x) = \cos(x) then these two functions obey {dx} c_2(x) = -x\cos(x)\\ {dx} s_2(x) = x\sin(x)\\ and {dx} c_3(x) = x\cos(x)+x^2\sin(x)\\ {dx} s_3(x) = x\sin(x)-x^2\cos(x)\\ if we label c₁ = cos(x) and s₁ = sin(x) then we have {dx} c_2(x) = -xc_1(x) = -x {dx}s_1(x)\\ {dx} s_2(x) = xs_1(x) = -x {dx}c_1(x) and {dx} c_3(x) = x{dx} s_1(x)+x {dx} s_2(x)\\ {dx} s_3(x) = -x{dx} c_1(x)+x {dx} c_2(x)\\

De tempos em tempos, circula pela Internet a estória motivacional dos “cinco macacos”. E sempre resulta em muitos comentários positivos. Neste início de ano, não foi diferente. Várias postagens, em diferentes redes, lembraram a estória que estimula as pessoas a pensarem diferente do senso comum. Uma espécie de convite ao pensamento crítico.Resumidamente, para quem nunca recebeu um post ou e-mail com essa narrativa, a estória se inicia com o relato de um experimento científico. Um grupo de pesquisadores pendurou um cacho de bananas no teto de uma jaula com uma escada embaixo. Na jaula havia cinco macacos. Quando um dos macacos, após algum tempo observando a situação, subiu na escada para pegar as bananas, todos receberam um jato d’água fria. Passado algum tempo, outro macaco tenta subir na escada e todos novamente são alvo do jato de d’água. Logo, quando um dos macacos demonstra a intenção em subir a escada, os demais o impedem. O experimento segue, um dos macacos é trocado e não há mais jato d’água. Quando o novato tenta subir na escada para pegar as bananas, os quatro que presenciaram a situação anterior o impedem. O novato tenta e novamente é impedido. Os macacos são trocados um a um, e a cena se repete.Ao final do experimento, mesmo sem ter presenciado a situação desagradável inicial, os macacos não tentam mais subir na escada para pegar a banana.Com essa ilustração, o texto quer instigar as pessoas a serem críticas, proativas e inovadoras. A mensagem é: ao continuar a fazer as coisas do jeito que todos fazem você pode estar perdendo oportunidades que só conhecerá se se arriscar.Desde que lançada, a estória tornou-se viral. Apareceu inicialmente em 2011, no blog do escritor Michael Michalko, autor de vários bons textos motivacionais sobre criatividade nos negócios, entre os quais: “Creative Thinkering: putting your imagination to work”.O autor convida o leitor a ter uma visão crítica de si mesmo: será que você não é como um macaco do experimento, aquele que reproduz o mesmo jeito de fazer as coisas sem saber o motivo?Você já se sentiu repreendido pelo grupo quando tentou fazer algo diferente? Provavelmente a grande maioria dos leitores dirá sim a estas perguntas embutidas no texto. Talvez isso explique o sucesso que essa estória faz.Desde que recebi pela primeira essa mensagem (e já foram inúmeras!), chamou-me a atenção a ampla aceitação positiva dessa narrativa. Parece demonstrar que muitas pessoas não querem parecer conformadas e buscam ter um pensamento crítico em relação ao senso comum.Uma postura proativa e inovadora, requer de fato um pensamento crítico e criativo. E pensamento crítico significa rever conceitos pré-estabelecidos.Mas a inovação se faz a partir do acúmulo de conhecimento, não da negação da experiência adquirida como indiretamente sugere a estória dos cinco macacos. Além disso, a inovação depende da capacidade das instituições na criação de um ambiente inovador.A estória dos cinco macacos enfatiza que quem tolhe as iniciativas são os iguais, os colegas de trabalho. Mas na verdade, são as instituições, não os indivíduos, que criam um ambiente favorável ou inibidor da crítica e da diversidade de ideias.Sabe-se que o ponto de partida do pensamento crítico está na problematização da realidade por meio de informações e conhecimento sobre a realidade. O próximo passo consiste em separar, organizar, classificar, hierarquizar os fatos conhecidos. Com base em análise e método, faz-se a proposição de alternativas mais adequadas para o problema inicial. Daí surge a verdadeira inovação.A criatividade que retira soluções do nada é magia. A criatividade que formula soluções a partir da análise da experiência acumulada, está sim gera conhecimento e tem impacto inovador.A narrativa dos cinco macacos induz o leitor a crer que se está diante de uma experiência científica verdadeira. O curioso é que o adjetivo científico deveria significar exatamente o oposto. Um conhecimento científico é aquele obtido por um método demonstrável e passível de ser questionado. Mas é tratado erroneamente como uma afirmação de verdade inquestionável.A “experiência científica” que deu origem à estória dos cinco macacos não existiu. É uma narrativa ficcional criada por Michalko. Supõem-se que tenha sido inspirada pelo experimento (este sim real) do Prof. Gordon Stephenson, do Departamento de Zoologia da Universidade de Wisconsin, publicado em 1966, no artigo: Cultural Acquisition of a Specific Learned Response among Rhesus Monkeys .No artigo do Prof. Stephenson, pares de macacos Rhesus são usados para testar se há transmissão de conhecimento entre essa espécie. Não são cinco macacos, não há banana pendurada no teto, não há jato d’água fria.A pergunta da pesquisa do Professor de Zoologia de Wisconsin era bem mais objetiva: há transmissão de comportamento adquirido entre os animais?No experimento real, os pares eram compostos por um animal condicionado a evitar um alimento (com jatos de ar, não água) e outro não condicionado. Stephenson queria saber se o animal condicionado (ele chamava de “demonstrador”) transmitiria o seu “conhecimento” para aquele não condicionado (que ele denominava de “ingênuo”).Quem ler o artigo do Prof. Gordon Stephenson verá que a conclusão do estudo é bem diferente da conclusão do texto sobre os cinco macacos. Na pesquisa real, em alguns pares, o macaco ingênuo copiou o comportamento do macaco condicionado (como reproduzido na estória dos cinco macacos). Em outros pares não. Houve pares em que se deu o contrário, o animal “ingênuo” acabou influenciando o macaco demonstrador, que passou por cima do seu condicionamento inicial e comeu o alimento (o oposto da estória dos cinco macacos). A narrativa dos cinco macacos não tem relação alguma com a realidade.Na ficção dos cinco macacos, o novato é impedido pelos outros de se aproximar da escada, pois os veteranos sabem, por experiência o que receiam: o jato d’água fria. O macaco novato ignora o alerta e é repreendido pelos demais. O macaco novato se conforma. O leitor se identifica com o macaco novato e lamenta todas as vezes que teve uma iniciativa e foi tolhido. Diante de uma situação análoga, o leitor é incentivado a não se conformar com os veteranos e ir adiante atrás da sua banana.A intenção aqui não é fazer uma crítica ao texto de Michalko, muito menos ao trabalho acadêmico do Prof. Gordon Stephnson. O objetivo é analisar a forma como a mensagem dos cinco macacos é interpretada, apontando exclusivamente a superação individual como caminho para inovação.Para tanto, vamos ver esse experimento imaginativo por outro ângulo. Note que apenas os observadores (os pesquisadores na estória fictícia) sabem que não há mais jato d’água. Os macacos não sabem se haverá ou não jato d’água fria e estão confinados em uma jaula, não têm por onde sair. Logo, para os cinco macacos, a água fria continua a ser uma possibilidade concreta. O risco é real. Quem impõe o risco e incentiva o comportamento não criativo são os empreendedores do experimento, no caso, os cientistas imaginários.A narrativa humaniza os possíveis comportamentos dos cinco macacos. Seguindo essa mesma linha, vamos supor que o novato, ao entrar na jaula, não fosse informado pelos demais sobre o perigo de se aproximar das bananas. Os veteranos sabiam, mas não disseram nada. Seria essa uma atitude racional para o grupo? Certamente, não. Os macacos fictícios, ao socializarem sua experiência, minimizaram o risco existente, representado pelos gestores do experimento que continuavam com a mangueira a postos. Ou seja, para eles, compartilhar a informação era uma forma de minimizar o risco a que todos estavam submetidos.Ao final do experimento, nenhum dos cinco macacos tinha visto de fato o jato d’água. Mas, mesmo assim, por que teriam motivos para desconfiar da informação repassada a eles? Os primeiros macacos de fato receberam a carga desagradável de água fria. Essa era a única informação concreta disponível.Você, se estivesse lá, arriscaria subir a escada, descartando a experiência dos seus companheiros? Se um macaco novato tentasse subir a escada, o que você faria? Incentivaria o novato ou tentaria dissuadi-lo, sabendo que as consequências do ato dele recaria sobre todos?A estória é muito usada para motivar pessoas. O foco recai exclusivamente sobre o comportamento do indivíduo. Mas, na realidade, o problema maior está nas próprias instituições: a jaula, o jato d’água controlado pelos os observadores de fora.Apostar que um indivíduo possa ir contra o senso comum de uma empresa ou instituição não preparada para mudanças é ingênuo.Talvez, os macacos da estória não fossem tão conformados assim. Simplesmente, não confiavam que os gestores de fora da jaula (as instituições) não iriam jogar-lhes água fria. Você confiaria?A estória dos cinco macacos é apenas uma versão pseudocientífica dos velhos adágios “gato escaldado tem medo de água fria” e "macaco velho não põe a mão em cumbuca".Desprezar a experiência acumulada é um erro grave. O pensamento crítico não significa apenas olhar para o lado oposto e fazer o que outros não fazem. Quem aponta defeitos pode ser um crítico no sentido corriqueiro da palavra, mas não significa necessariamente ser uma pessoa com pensamento crítico voltado à inovação. Até porque, nem todos que identificam um problema corretamente, têm uma solução correta para o mesmo problema. Por isso, inovar requer um pensamento crítico coletivo, cujo ponto de partida está na experiência concreta acumulada pelo grupo.O desafio é ir além do conformismo sem cair na ingenuidade voluntarista. A solução está em usar a experiência e o conhecimento da situação, incluindo erros e acertos, como a base sobre a qual se deverá erguer o olhar em busca de novas soluções, bem além do alcance do senso comum. Este é o pensamento crítico inovador.Mas, não basta apenas a motivação pessoal. É necessário que haja liberdade para o pensamento e para a crítica. E essa situação só ocorre em ambientes institucionais que garantem as condições para a livre reflexão sobre a experiência comum adquirida, sem mangueiras reais ou imaginárias apontando para todos.A motivação individual é a base, mas a inovação não é produto da ação individual isolada. Resulta de um ambiente favorável comum. Por isso, é papel das instituições criar as condições coletivas para a inovação e a criatividade, começando por compartilhar experiências, dar confiança e garantir que o pensamento novo, a diversidade de ideias e de opiniões não serão tratados com água fria.

MAIN Say we have a continued fraction function f(x) = {1+{1+{1+{1+\cdots}}}} = {1+} and we wanted to find the derivative without knowing the right hand side. Then if we take the partial convergents to the function f(x)_1 = {1}\\ f(x)_2 = {1+{1}}\\ f(x)_3 = {1+{1+{1}}}\\ f(x)_4 = {1+{1+{1+{1}}}}\\ and take their derivatives (as series expansions). We find that the series expansions all tend to the same series f(x) = 1-2x+6x^2-20x^3+70x^4-252x^5+924x^6-\cdots where the coefficients are the signed central binomial coefficients A000984 a(n) = (-1)^n {n!^2}, \; n=0,1,2,\cdots we have that ^\infty (-1)^n {n!^2} x^n = {} which agrees with the original right hand side of equation 1. This may be useful to work out the closed forms for some continued fractions. If we take the derivative until a known integer sequence appears, we could potentially integrate back to the closed form.

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.

IntroductionInformation Literacy \cite{zurkowski_information_1974} is for decades the playground for teaching librarians. But as the understanding of this concept is limited outside of the libraries, frameworks all around the world \cite{american_library_association_information_2000,bundy_australian_2004,adbu_referentiel_2012,deutscher_bibliotheksverband_standards_2009} including Switzerland \cite{informationskompetenz_referentiel_2011} have been created in order to advertise the meaning and importance of Information Literacy.Aside from the meaning of Information Literacy, the first challenge to face when teaching transferable skills is that everyone feels competent. After all, everyone uses transferable skills everyday. The problem is that these skills are used, but not mastered. Then, the first goal is to turn students from an unconscious incompetent student (they don't know that they don't know) into a conscious incompetent one (they know that they don't know) \cite{allan_no-nonsense_2013}. Once, they realise that they don't know, they feel the need to learn something new to fix this.

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MAIN I want to see if there is a general expression for a ratio of two polynomials {\alpha+\beta x+\gamma x^2 + \delta x^3+\cdots} = ^\infty {\alpha^{n+1}}x^n where Mn are n + 1 by n + 1 matrices. It would appear that the first few matrices could be M_0=a\\ M_1= \alpha & \beta \\ a & b \\ M_2= \alpha & 0 & \beta \\ \beta & \alpha & \gamma \\ b & a & c = 0 & \alpha & \beta \\ \alpha & \beta & \gamma \\ a & b & c where the matrices may have the rows and columns swapped still preserving the determinant. Is there an obvious pattern in these matrices. It seems that we need one extra term from each of the polynomials per new matrix. By seeing the pattern, it seems that we can set M₃ to be M_3 = - 0 & 0 & \alpha & \beta \\ 0 & \alpha & \beta & \gamma \\ \alpha & \beta & \gamma & \delta \\ a & b & c & d noting the minus sign! Then it indeed appears that M_4 = 0 & 0 & 0 & \alpha & \beta \\ 0& 0 & \alpha & \beta & \gamma \\ 0& \alpha & \beta & \gamma & \delta \\ \alpha & \beta & \gamma & \delta & \varepsilon \\ a & b & c & d & e strangely it seems that M₅ follows the same form by also doesn’t have a minus sign. What is the pattern of the minus signs. THE MINUS SIGNS We can see that the pattern becomes clearer. we have from n = 0 the pattern 1, −1, −1, 1, 1, −1, −1, 1, 1, −1, −1, 1, 1, .... We should rephrase the statement to a better notation ^\infty a_k x^k}{^\infty b_k x^k} = ^\infty (-1)^{\lceil {2} \rceil}{b_0^{n+1}}x^n where the (n + 1)×(n + 1) matrices Mn have elements given by M_{n,ij}= \Bigg\{ 0, & j < n+1-i\\ a_{j-1}, & i=n+1\\ b_{i+j-n-1},&

Considere f(x₁, x₂, …, xn) a função a ser minimizada. Define-se avaliar um ponto como calcular o valor da função nesse ponto. 1 - Defina n + 1 pontos iniciais com n dimensões. xi = (xi1,xi2,…,xin), em que 1 ≤ i ≤ n + 1. Ordene e renomeie de forma que f(x₁)<f(x₂)<…<f(xn + 1). 2 - Calcule o centróide xg = (xg1,xg2,…,xgn) dos n pontos com menor avaliação: $\displaystyle x_{gj} \leftarrow {n}^{n + 1}x_{ij}$, 1 ≤ j ≤ n. 3 - Calcule o ponto de reflexão xr = (xr1,xr2,…,xrn): rri ← xgi + α(xgi−x(n + 1)i), 1 ≤ i ≤ n. Avalie esse ponto: f(xr). 4 - Se f(x₁)<f(xr)≤f(xn), então faça x(n + 1)j ← xrj, 1 ≤ j ≤ n. Ordene os pontos por ordem crescente de avaliação e vá para o passo 2. 5 - Se f(xr)≤f(x₁), então calcule o ponto de expansão xe = (xe1,xe2,…,xen): xej ← xrj + β(xrj−xgj), 1 ≤ j ≤ n. Avalie esse ponto: f(xe). 6 - Se f(xe)≤f(xr), então faça x1j ← xej e xij ← x(i − 1)j, 1 ≤ j ≤ n + 1, e vá para o passo 2. Senão, faça x1j ← xrj e xij ← x(i − 1)j, 1 ≤ j ≤ n + 1, e vá para o passo 2. 7 - Se f(xr)>f(xn), então calcule o ponto de contração xc = (xc1,xc2,…,xcn): xcj ← xgj + γ(x(n + 1)j − xgj), 1 ≤ j ≤ n. Avalie esse ponto: f(xc). 8 - Se f(xc)≤f(xn + 1), então faça x(n + 1)j ← xcj, 1 ≤ j ≤ n. Ordene os pontos por ordem crescente de avaliação e vá para o passo 2. 9 - Se f(xc)>f(xn + 1), então realize uma contração ao longo de todas as dimensões em direção ao ponto x₁: xij ← xij + ν(xij − x1j), 2 ≤ i ≤ n + 1, 1 ≤ j ≤ n. Ordene os pontos por ordem crescente de avaliação e vá para o passo 2. Valores recomendados: α = 1, β = 1, γ = 0, 5 e ν = 0, 5.

MAIN If we have some function with an expansion f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + a_4x^4 + \cdots then we could define a transformation from this function to a new function T_{\pi} f(x) = g(x) = a_0 + a_0a_1x + a_0a_1a_2x^2 + a_0a_1a_2a_3x^3 + \cdots this new function has the partial products of the last functions series coefficients. We could envisage many transformations T_\sigma f(x) = a_0 + (a_0+a_1)x + (a_0+a_1+a_2)x^2 + (a_0+a_1+a_2+a_3)x^3 + \cdots \\ T_\delta f(x) = a_0 + (a_1-a_0)x + (a_2-a_1)x^2 + (a_3-a_2)x^3 + \cdots \\ T_! f(x) = {0!} + {1!}x + {2!}x^2 + {3!}x^3 + \cdots a curious one would be the following T_{frac} f(x) = a_0 + \left(a_0+{a_1}\right)x + \left(a_0+{a_1+{a_2}}\right)x^2 + \left(a_0+{a_1+{a_2+{a_3}}}\right)x^3 + \cdots for the function whose coefficients are all one namely f(x) = 1+{1-x} we would then have a sequence whose coefficients are the (slow) convergents to the golden ratio, T_{frac}[1+{1-x}] = 1+ 2x + {2}x^2 + {3}x^3 + {5}x^4 + {8}x^5 + {13}x^6 + {21}x^7 \cdots i.e. the Fibbonacci numbers. We then see the denominator of the next coefficient is the numerator of the last, and the numerator of the next coefficient is the sum of the denominator and numerator of the last. We may define c_f(n) = }{}((1+)^n-(1-)^n) then we have that T_{frac}[1+{1-x}] = ^\infty {c_f(n+1)}x^{n} are there any functions that are eigenfunctions to this transform? We require a set of coefficients such that a_0=a_0\\ a_1=\left(a_0+{a_1}\right)\\ a_2=\left(a_0+{a_1+{a_2}}\right)\\ a_3=\left(a_0+{a_1+{a_2+{a_3}}}\right) and so on. Which is a statement that a_0=a_0\\ a_1={2}\left(a_0\pm\right)\\ a_2=}{a_1} we can arbitrarily set a₀ to 1, and we require a_0=1\\ a_1=\varphi={2}(1+) \\ a_2={2}(1-})\\ a_3=... the begin to get very complicated as algebraic numbers.

MAIN It seems that ^\infty {n^2-1} = {m-1}, \; 1<m\in the main result is that ^\infty {n^2-1} = 2 so then for any other m, we have ^\infty {n^2-1} = ^\infty {n^2-1}}{^{m-1} {n^2-1}} \\ ^\infty {n^2-1} = {^{m-1} {n^2-1}} \\ ^\infty {n^2-1} = 2^{m-1} {n^2} = {m-1} there is something a little beautiful about ^\infty {n^2-1} = 2^{m-1} {n^2} if it is true that ^\infty {n^2-1} = 2 then all the divisors of the terms on the bottom must cancel with the divisors of the terms on the top, except for a single 2. That is {3\cdot8\cdot15\cdot24\cdot35\cdots}={3\cdot(2\cdot2\cdot2)\cdot(3\cdot5)\cdot(2\cdot2\cdot2\cdot3)\cdot(5\cdot7)\cdots} it might be nice to separate into primes and non-primes. If we separate into these two sums normally we would see that 2=}^\infty {p^2-1}}^\infty {q^2-1} and we have a closed form of }^\infty {p^2-1}=\zeta(2)={6} so then we have the complementary }^\infty {q^2-1} = {\pi^2} So then there is a complementary form of the zeta function defined by the Euler product as \xi(s)=} {1-q^{-s}} and we know that \xi(2)={\pi^2} it would appear that \xi(3)=(\pi)}{2}}{\zeta(3)}\\ \xi(4)=(\pi)}{\pi^3}\\ \xi(5)=)\Gamma(2-\sqrt[5]{-1}^2)\Gamma(2+\sqrt[5]{-1}^3)\Gamma(2-\sqrt[5]{-1})^4}{\zeta(5)}\\ \xi(6)=^2(\pi}{2})}{\pi^4} so each of these sums converge separately. We can consider the Wallace product and convert it into a product integral ^\infty {4n^2-1} = {2} then the product integral equivalent is \exp\left( \int_1^\infty\log\left( {4n^2-1} \right) \; dn \right) = }{4} curiously Mathematica states that \exp\left( \int_0^\infty\log\left( {4n^2-1} \right) \; dn \right) = i the reason being \int_0^\infty\log\left( {4n^2-1} \right) \; dn = {2} (branch dependent?). But this reclaims the π/2.