Recursive Signatures and the Signature Left Near-Ring

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1 Recursive Signatures and the Signature Left Near-Ring J. Conrad Oct 2, 0207 Abstract A new characterization of the INVERT transform is given for the class of -beginning sequences. Its properties are canonized by a familiar algorithm. We construct an additive operation from a compatible homomorphism, and explore the immediate consequences of the operation and its ability to streamline identity proving. Then we extend the parameters of the signature function to construct a multiplicative group which is left-distributive over the additive operation, forming the signature left near-ring. Finally, we explore a peculiar algorithm with a surprisingly simple interpretation via the signature function. Contents Introduction 2. The Invert Transform and the Signature Function The Inverse Signature Function F Antidiagonal summation and ω Signature Addition 4 2. Convolution of -beginning sequences Internal Applications Signature Convolution 5 3. Parameterized Antidiagonal Summation Right and Left inverses The Signature (Left!) Near-Ring Signatures in Action 7 4. The Continuum Hypothesis (just kidding) Transforming signatures Conclusion & Afterthoughts 0

2 Introduction. The Invert Transform and the Signature Function In Bernstein & Sloane s Some Canonical Sequences of Integers [2], the Invert transform of a sequence a is the sequence b which satisfies + b n x n = n= () a n x n n= As formal power series over R[[x]] this is simply + bx = ax By this algorithm we may define a function F : D O where all sequences b O take the form + bx (the one-beginning sequences). Though the name Invert is a useful mnemonic for this formula, there is a recursive algorithm which computes this transform more quickly for a sequence d D by F d (0) = F d (x) = (2) F d (x k) d k (3) k= The Invert transform becomes an identity for the entire sequence: F d = dx (4) For simplicity, we accept two assumptions about every sequence d: d x<0 = 0 d + {0} d (5) These rules together form the null postulate. It allows every sequence to be treated as infinite in both directions. This may be optimized in code to reduce unnecessary additions and multiplications, but it provides a consistent upper bound for all sums, and thus greatly simplifies the syntax for various formulas. A sequence adhering to this postulate is said to be null compliant. 2

3 .2 The Inverse Signature Function F We can compute its inverse F : O D by solving for d in terms of F d : F d (x) = From this, substituting F d F d (x k) d k k= x F d (x) = F d (0) d x + F d (x k) d k k= x F d (x) = d x + F d (x k) d k k= x d x = F d (x) F d (x k) d k (5) F d k= for d and increasing the index yields (x) = d x+ k= d x k F d (k ) (7) With this new structure, the notion of the invert transform is less intuitive. For this reason, I have elected to refer to this treatment as the recursive signature function or simply the signature function for short..3 Antidiagonal summation and ω It is well known that summation along the diagonals of Pascal s Triangle yields the Fibonacci numbers. This relationship has been explored in further detail by Hoggatt Jr & Bicknell [3]. In general, we may select a polynomial p and sum along the x-th diagonal to yield F p (x) = p k x k (8) Additionally, we may describe F d as a sum of power series. To do this, we define the series ω = [0, ]. While it is normal to use x to describe position in this way, it is beneficial to use ω to distinguish these as null-compliant. Then the signature function is also computed by F d = (dω) k F d (x) = (dω) k x (9) Finally, we may describe F d (x) without requiring it be piecewise. We may substitute F d (0) = for F 0 to get F d (x) = F 0 (x) + F d (x k) d k (0) k= 3

4 2 Signature Addition 2. Convolution of -beginning sequences Recall that the convolution of two sequences a and b is given by ab x = a x k b k = b x k a k () In the set of -beginning sequences O, convolution is a closed operation. This means that we may describe a homomorphism F a F b = F (a b) (2) for a binary operation : D D D where D is the set of null-compliant (integer) sequences. By (2), we have and thus ( aω)( bω) = aω bω + abω 2 (3) a b = F (F a F b ) = a + b abω (4) To find an inverse, we solve a + b abω = 0. If we factor out b from the latter two terms, we may substitute the reciprocal of F b for ( bω) to find that In addition to this inverse, note that b = af a (5) a F a = a + a nf a = a + n (6) This describes an isomorphism to (integer) addition with identity 0, but may be easily extended to the reals and complex numbers. We may also use Eq. (0) to yield a convenient recursive formula which is amenable to memoization: (F a F b )(x) = F a (x) Internal Applications F a F b (x k) d k (7) The information given by this group can help us quickly solve problems when they are portrayed in terms of their signatures. For example, we have that k= F ( dω) = F ( F d ) = 0 d = df d (8) or conversely F ( + dω) = df d (9) 4

5 Through this, we can quickly solve a more complex problem symbolically without relying on the explicit algorithm: When a =, we have F ( + af b ω) = (af b )F afb = af b afb = af b a (20) F ( + F b ω) = F b (2) but we can substitute F 0 for, and reach the same solution in an albeit roundabout way: F ( + F b F 0 ω) = F b F 0 Fb = F b Fb = F b (22) Another roundabout solution to this form of problem takes advantage of an interesting almost distributive case: Which is used to solve F ( + F a F b ω): a (b c) = a + a b a c (23) F (+F a F b ω) = F a F b Fa = F a (b Fa) = F a+a b a Fa = F a+a b (a+) = F a b (24) This isn t the best way to solve this problem, but it showcases the versatility of this construction. Because the signature function is linear over signature addition, it is easier to say that F ( + F a F b ω) = F ( + F a b ω) = F a b (25) 3 Signature Convolution 3. Parameterized Antidiagonal Summation If we elect to treat each term of antidiagonal summation as the product of itself and, then we can rephrase it in terms of the signature function as d k x k F (k) = F d (x) (26) From this, we can experiment with alternative signatures to. Using 0, for example, yields d k x k F 0 (k) = F 0 (x) (27) And with 2 we get d k x k F 2 (k) = F 2d (x) (28) Already we have multiplicative qualities akin to scalar multiplication, and nullification by the identity of signature addition. By describing this transformation 5

6 as a binary operation : D D D, we can focus on the signature of the solution rather than the entire solution. Thus we define this operation as the satisfaction of F a b (x) = a k x k F b (k) (29) which as a series is F a b = (aω) k F b (k) (30) Finally, a formula for the multiplication itself is given by a b = a (aω) k b k (3) with each value of the sequence given by (a b)(x) = 3.2 Right and Left inverses a k+ x k b k (32) With a bit of manipulation (left as an exercise, but solved in the same way as Eq (5)) we can derive an inverse which computes either a or b in terms of a b. First, we have the left inverse which computes a left operand, denoted \: (a\b) x = x a x (a\b) k+ x k b k b 0 (33) Next, we have the right inverse which computes a right operand, denoted /: (a/b) x = x a x b k+ x k (a/b) k b x+ 0 (34) Because these inverses are different, we can conclude that is not commutative. Furthermore, we may compare it to deconvolution, the inverse of convolution: (a b) x = x a x b x k (a b) k b 0 (35) If we treat convolution as an ansatz, this new operation is amenable to the name signature convolution, which distinguishes its application to sequences rather than numbers. 6

7 3.3 The Signature (Left!) Near-Ring Recall that for the system (D,, 0,, ) to satisfy the near-ring axioms, it must meet the following three conditions: D is a group under the additive operation * D is a semigroup under the multiplicative operation Multiplication distributes on the right XOR left Because of the additivity of signature addition, signature convolution distributes over by definition in Eq. (30): a (aω) k (b c) k = a (aω) k b k a (aω) k c k = a b a c (36) It follows that its right inverse right-distributes: a b d a c e a (b c) = d e (d e)/a = d/a e/a = b c (37) Right-distributivity and left inverse distributivity can be disproven by any number of random test cases. There are conditions where appears to either distribute or commute, but such cases are easily explained via its associativity and shared signature factorization. Take for example [, ] [, 2, 2, ] = [, ] [, ] [, ] = [, 2, 2, ] [, ] (38) Thus signature addition and convolution together form a left near-ring over the integer sequences. Like previous constructions, this can extend to form a left near-field as its inverse commutes and signature convolution forms a group under the rationals, reals, and complex numbers. This may also be enumerated by signature factorization, by observing that by induction a (x) a (y) = a (x+) a (y ) = a (x ) a (y+) (39) where the parenthetical exponents are the signature power of the sequence. Then for x = y = we get a (0) a (0) = a () a ( ) = a ( ) a () (40) 4 Signatures in Action 4. The Continuum Hypothesis (just kidding) Though we have seen the near-ring of signatures used to solve internal problems, it was originally motivated by a peculiar representation of Cantor s diagonal argument as a power series. It begins with an infinite set, whose cardinality is σ, 7

8 which we conjecture to be a complete set of the real numbers. By the diagonal argument, it is always possible to define at least one element which must be real but is not present in the set (in Cantor s case the binary reals are used, but we may use any base (b + ). We add this set of absent real numbers, and include it as the second element of the series. This process may be repeated indefinitely, but by induction will never be complete. I chose to represent this recursively as C(0) = σ C(x) = C(x ) + b C(x ) lim x C(x) < ℵ (4) Next, we take the finite difference of this function, C, and we construct a matrix which satisfies ( M[0, 0] = M[x, y] = n : b x M[x, k] C (k)) = C (x) (42) There is an interesting equivalence between the rows of this matrix and the base-b logarithm of C ; if we state that then it is true that b < σ = log b σ = σ (43) x M[x, k] C (k) = log b (C (x)) (44) We may take this a step further, and specify that this is a linear logarithm-like function. This would allow us to express iteration of this pseudologarithm as matrix powers (assume that each of these matrices is padded with sufficient zeroes to make matrix multiplication valid): (45) 8

9 The shape of the initial matrix is identical to the one exhibited by Mats Granvik and Gary Adamson [], in which they compute the Bell numbers by the powers of a modified Pascal s Triangle. In this case, however, we will add a final step and produce a sequence from the partial sums of the remaining column of M. We may frame this as the sequence P where P (x) = M k [k, 0] (46) which produces P (x) = 2 x. For brevity s sake we will call this the pseudologarithmic series. 4.2 Transforming signatures We may elect to define a signature-based recursive function C d, which is nullcompliant, and compute it as C d (0) = σ C d (x) = C d (x ) + b ( C d (x k) d k ) k= (47) Then the matrix M d is given and its pseudologarithmic series is M d [0, 0] = M d [x, y] = x y d k (48) P d (x) = F d+ (x) (49) More curious still, an antidiagonal summation algorithm exists, which we ll call S, such that S d (x) = M x k d [k, 0] = F dω+ (x) (50) This example hints at the possibility of a broader treatment of signatures, and a deeper relationship with antidiagonal summation of powers. Perhaps there is someone smarter than me who could find a suitable topology which describes the behavior of signatures in significant detail. 9

10 5 Conclusion & Afterthoughts The signature near-ring has not been fully explored, and I will be continuing this work in the future. But I consider the construction complete enough to be publishable. This paper envelops everything I ve learned, including ideas I wouldn t have anticipated, like the possibility to apply it to select matrix multiplication algorithms such as the one derived from Cantor s diagonal argument. While defining, proving, and exploring the signature near-ring is not easily accessible to students, I intend to make it easy to understand, compute its operations, and demonstrate one of the many universes of arithmetic capable of being constructed. Education in the U.S. is streamlined for calculus, with surprisingly little attention paid to arithmetic after completing algebra. The world of abstract algebra could be used to examine the way we teach and the systems we use to teach, but instead seems just a means to an end. I am not a canonical mathematician. I ve learned math independently and as such my understanding of the subject does not lean towards calculus and analysis in the way that a formal education so often does. Thus my attention is not usually focused on the number-theoretical on which professionals are focused. I would like to do so one day, but I would be satisfied to see this available on arxiv, and used to refresh what I believe is a poorly-documented aspect of the Online Encyclopedia of Integer Sequences. I began compiling this information when my vocabulary was not as strong, and even the concept of a formal power series was foreign to me. I started learning what came naturally, and landed in combinatorics and abstract algebra because most of what I was learning revolved around those subjects. I learned the ins and outs of Pascal s Triangle, binomial transformations, and the behavior of summation, and independently learned to derive a lot of qualities of groups, sequences, polynomials, and more. When I discovered the OEIS it became one of my most important tools, and I decided that I would do my best to contribute as an amateur. Mathematics at this level feels all but trivialized - not only by its connotation, but by the explanatory power of vast abstract systems constructed, all of which are constantly synthesizing years of research into cursory theorems and corollaries. It enables us to present the illusion of understanding to the world, and advance intermediate students to analysis and category theory without much thought to the trivialities themselves. Many discuss the constructions of the rational and real numbers which make them unique from the integers, but far fewer seem to stop and think about them. It is easier to generalize and parameterize a problem and make it trivial than it is to stop and think about a solution to the problem. This isn t to blame computers for the neglect of arithmetic; they only provide academics of all sorts the ability to do so as they always wished. Special thanks to: my love, Ann Valor, for believing in me from the start; Dr. Doron Zeilberger, who encouraged me to keep going; Tristen Harr, for learning from and with me; and to my reliable computer, for helping to prove me right. 0

11 References [] Granvik, Mats and Adamson, Gary. The invert transform, Bell numbers, Pascal triangle, permanents, matrix multiplication and matrix inversion, Catalan numbers. The Mobius function Blog. 8 Feb. 20. Web. Retrieved from on 30 Sep [2] Sloane, NJA and Bernstein, M. Some Canonical Sequences of Integers. Mathematical Sciences Research Center. AT&T Bell Laboratories. Retrieved from on 30 Sep [3] Hoggatt, Jr, V. E. and Bicknell, Marjorie. Diagonal Sums of Generalized Pascal Triangles. Web. Retrieved from Scanned/7-4/hoggatt-a.pdf

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