There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Table Of Integrals Series And Products

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N →∞, for the number of integer points n ∈ ℤ d ∩ K for which the integers ψ 1 (n), ... , ψ t (n) are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ 1 , ... , ψ t are affinely related; this excludes the important "binary" cases such as the twin prime or Goldbach conjectures, but does allow one to count "nondegenerate" configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowers-norm conjecture (GI(s)) and the Mobius and nilsequences conjecture (MN(s)), where s ∈ {1,2,...} is the complexity of the system and measures the extent to which the forms ψ i depend on each other. The case s = 0 is somewhat degenerate, and follows from the prime number theorem in APs. Roughly speaking, the inverse Gowers-norm conjecture GI(s) asserts the Gowers U s+1 -norm of a function f: [N] → [-1, 1] is large if and only if f correlates with an s-step nilsequence, while the Mobius and nilsequences conjecture MN(s) asserts that the Mobius function μ is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity. These conjectures have long been known to be true for s = 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s = 2 in two papers of the authors. Thus our results in the case of complexity s ≤ 2 are unconditional. In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 ≤ N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.

https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n3-p08-p.pdf

Linear equations in primes

Die grundlagen der arithmetik: by G. Frege. English translation by J. L. Austin. 119 pages, 14 × 22 cm. Breslau, Verlag von Wilhelm Koebner, 1884, and New York, Philosophical Library, 1950. Price, $4.75

Results on algorithmic problems on strings that are given in a compressed form via straightline programs are surveyed. A straight-line program is a context-free grammar that generates exactly one string. In this way, exponential compression rates can be achieved. Among others, we study pattern matching for compressed strings, membership problems for compressed strings in various kinds of formal languages, and the problem of querying compressed strings. Applications in combinatorial group theory and computational topology and to the solution of word equations are discussed as well. Finally, extensions to compressed trees and pictures are considered.

https://www.informatik.uni-leipzig.de/~lohrey/12-SURVEY.pdf

Algorithmics on SLP-compressed strings: A survey

Decidability of the theory of the natural integers with the cantor pairing function and the successor

Given two strings (a text t of length n and a pattern p) and a natural number w, window subsequence problems consist in deciding whether p occurs as a subsequence of t and/or finding the number of size (at most) w windows of text t which contain pattern p as a subsequence, i.e. the letters of pattern p occur in the text window, in the same order as in p, but not necessarily consecutively (they may be interleaved with other letters). We are searching for subsequences in a text which is compressed using Lempel-Ziv-like compression algorithms, without decompressing the text, and we would like our algorithms to be almost optimal, in the sense that they run in time O(m) where m is the size of the compressed text. The pattern is uncompressed (because the compression algorithms are evolutive: various occurrences of a same pattern look different in the text).

/pdf/window-subsequence-problems-for-compressed-texts-35ysno164x.pdf

Window subsequence problems for compressed texts

Given two strings, text t of length n , and pattern p = p 1 …p k of length k , and given a natural number w , the subsequence matching problem consists in finding the number of size w windows of text t which contain pattern p as a subsequence, i.e. the letters p 1 ,…,p k occur in the window, in the same order as in p , but not necessarily consecutively (they may be interleaved with other letters). Subsequence matching is used for finding frequent patterns and association rules in databases. We generalize the Knuth–Morris–Pratt (KMP) pattern matching algorithm; we define a non-conventional kind of RAM, the MP-RAMs which model more closely the microprocessor operations; we design an O (n) on-line algorithm for solving the subsequence matching problem on MP-RAMs.

Window-accumulated subsequence matching problem is linear

On arithmetical first-order theories allowing encoding and decoding of lists

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Papers

Decidability of the theory of the natural integers with the cantor pairing function and the successor

Window subsequence problems for compressed texts

Window-accumulated subsequence matching problem is linear

On arithmetical first-order theories allowing encoding and decoding of lists

Window-accumulated subsequence matching problem is linear