We prove the inverse conjecture for the Gowers U s+1 N-norm for all s ≥ 1; this is new for s ≥ 4. More precisely, we establish that if f : N → −1,1 is a function with ${\parallel \mathrm{f}\parallel ...}_{{\mathrm{U}}^{\mathrm{s}+1}\left\mathrm{N}\right}\ge \text{\hspace{0.17em}}\mathrm{\delta }$ , then there is a bounded-complexity s-step nilsequence F(g(n)Γ) that correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.
In 1980. Fisher in Fisher, B.,Results on common fixed points on complete metric spaces, Glasgow Math. J., 21 (1980), 165–167 proved very interesting fixed point result for the pair of maps. In 1996. ...Kada, Suzuki and Takahashi introduced and studied the concept of 𝗐–distance in fixed point theory. In this paper, we generalize Fisher’s result for pair of mappings on metric space to complete metric space with 𝗐–distance. The obtained results do not require the continuity of maps, but more relaxing condition (𝐶; 𝑘). As a corollary we obtain a result of Chatterjea.
Computing a function on natural numbers assumes a fixed notation for natural numbers. Usually, we assume the binary or decimal notation. However, different notations might have different sets of ...effectively (PTIME) computable functions. We compare notations with respect to the set of effectively computable functions under a given notation. This approach gives an inclusion based partial ordering on notations, say ≤PTIME. We address the question whether the binary notation is, in some way, distinguished. We say that a notation σ is dense if the length of a σ-numeral for n is a function bounded by poly(log(n+1)). We show that the binary notation is a maximal element in ≤PTIME among dense notations. We also show that a dense notation σ allows to compute effectively a different set of functions than the binary notation only if σ has short numerals for large numbers.
•Among all dense notations for natural numbers, positional notations allow to compute in PTIME a maximal set of functions.•Any dense notation computing in PTIME all functions computable under the binary notation is PTIME isomorphic with the binary one.•Any dense notation computing in PTIME a function not computable in PTIME under the binary notation codes some numbers with short codes.
Humans can represent number either exactly – using their knowledge of exact numbers as supported by language, or approximately – using their approximate number system (ANS). Adults can map between ...these two systems – they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate (i.e., ANS-to-Word), and can generate an approximation, for example by rapidly tapping, when provided with an exact verbal number (i.e., Word-to-ANS). Here we ask how these mappings are initially formed and whether one mapping direction may become functional before the other during development. In two experiments, we gave 2–5year old children both an ANS-to-Word task, where they had to give a verbal number response to an approximate presentation (i.e., after seeing rapidly flashed dots, or watching rapid hand taps), and a Word-to-ANS task, where they had to generate an approximate response to a verbal number request (i.e., rapidly tapping after hearing a number word). Replicating previous results, children did not successfully generate numerically appropriate verbal responses in the ANS-to-Word task until after 4years of age – well after they had acquired the Cardinality Principle of verbal counting. In contrast, children successfully generated numerically appropriate tapping sequences in the Word-to-ANS task before 4years of age – well before many understood the Cardinality Principle. We further found that the accuracy of the mapping between the ANS and number words, as captured by error rates, continues to develop after this initial formation of the interface. These results suggest that the mapping between the ANS and verbal number representations is not functionally bidirectional in early development, and that the mapping direction from number representations to the ANS is established before the reverse.
Regularity lemmas for stable graphs MALLIARIS, M.; SHELAH, S.
Transactions of the American Mathematical Society,
03/2014, Letnik:
366, Številka:
3
Journal Article
Recenzirano
éééonly essential difficulty, by giving a much stronger version of Szemer>éé
We obtain estimates for Vinogradov's integral that for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the ...conjectured asymptotic formula in Waring's problem holds for sums of s kth powers of natural numbers whenever s ≥ 2k 2 + 2k − 3.
A graph is d-degenerate if all its subgraphs have a vertex of degree at most d. We prove that there exists a constant c such that for all natural numbers d and r, every d-degenerate graph H of ...chromatic number r with $\left|\mathrm{V}\left(\mathrm{H}\right)\right|\ge {2}^{{\mathrm{d}}^{2}{2}^{\mathrm{c}\mathrm{r}}}$ has Ramsey number at most ${2}^{{\mathrm{d}2}^{\mathrm{c}\mathrm{r}}}\left|\mathrm{V}\left(\mathrm{H}\right)\right|$. This solves a conjecture of Burr and Erdős from 1973.
By means of a theoretical development of lecture 3, we provide a characterization of the Goldbach Conjecture in an infinite set of even numbers in terms of gradients of deformed hyperbolas.
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