Integer-valued definable functions in ℝan,exp

2021 ◽  
pp. 1-14
Author(s):  
Gareth Jones ◽  
Shi Qiu
Keyword(s):  
Growth Condition ◽  
Dense Subset ◽  
Suitable Sense ◽  
Growth Bound ◽  
Definable Functions ◽  
Positive Integers

We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of [Formula: see text]-definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in [Formula: see text] that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function [Formula: see text], Wilkie showed that it must be eventually equal to a polynomial. Assuming a stronger growth condition, but only assuming that the function takes values sufficiently close to integers at positive integers, we show that the function must eventually be close to a polynomial. In a different variation we show that it suffices to assume that the function takes integer values on a sufficiently dense subset of the positive integers (for instance the primes), again under a stronger growth bound than that in Wilkie’s result.

scholarly journals On ranges of variants of the divisor functions that are dense

2015 ◽  
Vol 11 (06) ◽  
pp. 1905-1912 ◽  
Author(s):  
Colin Defant
Keyword(s):  
Real Number ◽  
Open Problem ◽  
Dense Subset ◽  
Arithmetic Function ◽  
Divisor Functions ◽  
Positive Integers ◽  
Multiplicative Arithmetic ◽  
Multiplicative Arithmetic Function

For a real number t, let st be the multiplicative arithmetic function defined by [Formula: see text] for all primes p and positive integers α. We show that the range of a function s-r is dense in the interval (0, 1] whenever r ∈ (0, 1]. We then find a constant ηA ≈ 1.9011618 and show that if r > 1, then the range of the function s-r is a dense subset of the interval [Formula: see text] if and only if r ≤ ηA. We end with an open problem.

Computability and λ-definability

1937 ◽  
Vol 2 (4) ◽  
pp. 153-163 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Computable Function ◽  
General Recursive Function ◽  
Technical Details ◽  
Intuitive Idea ◽  
Short Space ◽  
Definition Of ◽  
Definable Functions ◽  
Positive Integers ◽  
Computable Functions ◽  
Definable Function

Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene. It is shown that every λ-definable function is computable and that every computable function is general recursive. There is a modified form of λ-definability, known as λ-K-definability, and it turns out to be natural to put the proof that every λ-definable function is computable in the form of a proof that every λ-K-definable function is computable; that every λ-definable function is λ-K-definable is trivial. If these results are taken in conjunction with an already available proof that every general recursive function is λ-definable we shall have the required equivalence of computability with λ-definability and incidentally a new proof of the equivalence of λ-definability and λ-K-definability.A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.

scholarly journals On one type of compactification of positive integers

2015 ◽  
Vol 64 (1) ◽  
pp. 127-131
Author(s):  
Milan Paštéka
Keyword(s):  
Dense Subset ◽  
Positive Integers

Abstract The object of observation is a compact metric ring containing positive integers as dense subset. It is proved that this ring is isomorphic with a ring of reminder classes of polyadic integers.

Stepping in the Shoes of Leaders of Populist Right-Wing Parties

2017 ◽  
Vol 48 (1) ◽  
pp. 40-46 ◽  
Author(s):  
Jolanda Jetten ◽  
Rachel Ryan ◽  
Frank Mols
Keyword(s):  
Economic Growth ◽  
Growth Condition ◽  
Baseline Condition ◽  
Right Wing ◽  
Economic Context

Abstract. What narrative is deemed most compelling to justify anti-immigrant sentiments when a country’s economy is not a cause for concern? We predicted that flourishing economies constrain the viability of realistic threat arguments. We found support for this prediction in an experiment in which participants were asked to take on the role of speechwriter for a leader with an anti-immigrant message (N = 75). As predicted, a greater percentage of realistic threat arguments and fewer symbolic threat arguments were generated in a condition in which the economy was expected to decline than when it was expected to grow or a baseline condition. Perhaps more interesting, in the economic growth condition, the percentage realistic entitlements and symbolic threat arguments generated were higher than when the economy was declining. We conclude that threat narratives to provide a legitimizing discourse for anti-immigrant sentiments are tailored to the economic context.

VLS Growth of Si nanowhiskers on a H-terminated Si{111} surface

1998 ◽  
Vol 536 ◽  
Author(s):  
N. Ozaki ◽  
Y. Ohno ◽  
S. Takeda ◽  
M. Hirata
Keyword(s):  
High Pressure ◽  
Growth Condition ◽  
Growth Mechanism ◽  
Quantum Confinement ◽  
Critical Value ◽  
Extended Defects ◽  
Vapor Liquid Solid ◽  
Minimum Diameter ◽  
Vls Growth ◽  
Vapor Liquid

AbstractWe have grown Si nanowhiskers on a Si{1111} surface via the vapor-liquid-solid (VLS) mechanism. The minimum diameter of the crystalline is 3nm and is close to the critical value for the effect of quantum confinement. We have found that many whiskers grow epitaxially or non-epitaxially on the substrate along the 〈112〉 direction as well as the 〈111〉 direction.In our growth procedure, we first deposited gold on a H-terminated Si{111} surface and prepared the molten catalysts of Au and Si at 500°C. Under the flow of high pressure silane gas, we have succeeded in producing the nanowhiskers without any extended defects. We present the details of the growth condition and discuss the growth mechanism of the nanowhiskers extending along the 〈112〉 direction.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Update on nitrite in animal feed – Results of the studies of the ESST-working group ‘Nitrite in feed’

2016 ◽  
pp. 91-96
Author(s):  
Stefan Frenzel
Keyword(s):  
Growth Condition ◽  
Analytical Methods ◽  
Working Group ◽  
Animal Feed ◽  
Extraction Process ◽  
Study Group ◽  
External Effects ◽  
Complex Behaviour ◽  
Sugar Extraction ◽  
Expert Study

The presentation includes monitoring results of the companies participating in the ESST expert study group on “Nitrite in feed”. It will be obvious that the complex behaviour of nitrite in the sugar extraction process overlaps with external effects such as growth condition of the beet which are not under the control of the process owner. Currently the lack of reliable and validated analytical methods do not allow to comply to the questionable maximum nitrite limit for animal feed materials.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

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