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| In [[additive number theory]], the '''Schnirelmann density''' of a [[sequence]] of numbers is a way to measure how "dense" the sequence is. It is named after [[Russia]]n [[mathematician]] [[L.G. Schnirelmann]], who was the first to study it.<ref name=Schnirelmann1>Schnirelmann, L.G. (1930). "[http://mi.mathnet.ru/eng/umn/y1939/i6/p9 On the additive properties of numbers]", first published in "Proceedings of the Don Polytechnic Institute in Novocherkassk" (in Russian), vol '''XIV''' (1930), pp. 3-27, and reprinted in "Uspekhi Matematicheskikh Nauk" (in Russian), 1939, no. 6, 9–25.</ref><ref name=Schnirelmann2>Schnirelmann, L.G. (1933). First published as "[http://link.springer.com/article/10.1007/BF01448914 Über additive Eigenschaften von Zahlen]" in "Mathematische Annalen" (in German), vol '''107''' (1933), 649-690, and reprinted as "[http://mi.mathnet.ru/eng/umn/y1940/i7/p7 On the additive properties of numbers]" in "Uspekhi Matematicheskikh Nauk" (in Russian), 1940, no. 7, 7–46.</ref>
| | True said, "Paradise lies beneath the feet of the mother" and that heaven has no match and God bless one with only if he is immense lucky. However, if we talk about a materialistic heaven, it is nothing else besides our home and everyone want this place to be nothing less than a luxury accommodation. While talking about construction of a house, few things comes to our mind such as flooring, painting, electronic components, and civil engineering material and resin crafts. In this article, we will be exploring the use of epoxy resin for around mentioned areas.<br><br>Paint:<br><br>Painting is the major part of a house because until a house is not coated with the colors, it will be more likely a barren country side property, throwing impressions of Haunted house. When we are after coating services, Epoxy Resin, is a major component because of below mentioned properties:<br><br>This adhesive (the common name of Epoxy Resin)is blessed with marvelous chemical resistance, specifically alkali<br><br>The Epoxy Resin has no better substitute when it is about film adhesion, heat resistance and electrical insulation.<br><br>This chemical has no match when the question of color retention arises.<br><br>Electrical components:<br><br>Epoxy Resin is blessed with awesome insulation and sealing performance and structural length and that is what makes this curing is widely used for low voltage electrical components like solenoid, contactor coils, high voltage electrical insulation package and dry type transformers as well. In addition to this, it is also preferably used in encapsulation devices, laminated plastic and electronic insulation adhesive.<br><br>Flooring:<br><br>Now comes the turn of flooring and this epoxy resin in an undoubted backbone of seamless flooring, so far. We all know that concrete is an impermeable material and it is strength is unquestionable; this is what makes it durable. If you cherished this article and you would like to receive extra info with regards to [http://wmx6.newsvine.com/_news/2014/06/14/24327888-acogimiento-del-reciclado-de-fibra-de-vidrio-para-mejorar-otros-materiales resina epoxi] kindly check out the web-site. However, when is about flooring, the risk of water and decayed floor due to water, is always there. Any sort of continuous water flow is capable enough of damaging the floor and here, one cannot also deny from water sipping in the roots/foundation and eradicating it thoroughly. We do know that, concrete floors are generally designed with simply design using one tone coloration across the whole floor. Now if somehow, one part of floor breaks or cracks, it makes whole floor look ugly and turning a blind eye to that is something we cannot afford. This is where; The Resin technology comes in offers spotless flooring. |
| | |
| ==Definition==
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| The '''Schnirelmann density''' of a set of [[natural number]]s ''A'' is defined as
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| :<math>\sigma A = \inf_n \frac{A(n)}{n},</math>
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| where ''A''(''n'') denotes the number of elements of ''A'' not exceeding ''n'' and inf is [[infimum]].<ref name=N961912>Nathanson (1996) pp.191–192</ref>
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| The Schnirelmann density is well-defined even if the limit of ''A''(''n'')/''n'' as {{nowrap|''n'' → ∞}} fails to exist (see [[asymptotic density]]).
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| ==Properties==
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| By definition, {{nowrap|0 ≤ ''A''(''n'') ≤ n}} and {{nowrap|''n'' σ''A'' ≤ ''A''(''n'')}} for all ''n'', and therefore {{nowrap|0 ≤ σ''A'' ≤ 1}}, and {{nowrap|σ''A'' {{=}} 1}} if and only if {{nowrap|''A'' {{=}} '''N'''}}. Furthermore,
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| : <math>\sigma A=0 \Rightarrow \forall \epsilon>0\ \exists n\ A(n) < \epsilon n.</math>
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| ===Sensitivity===
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| The Schnirelmann density is sensitive to the first values of a set:
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| : <math>\forall k \ k \notin A \Rightarrow \sigma A \le 1-1/k</math>.
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| In particular,
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| :<math>1 \notin A \Rightarrow \sigma A = 0</math>
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| and
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| :<math>2 \notin A \Rightarrow \sigma A \le \frac{1}{2}.</math>
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| Consequently, the Schnirelmann densities of the even numbers and the odd numbers, which one might expect to agree, are 0 and 1/2 respectively. Schnirelmann and [[Yuri Linnik]] exploited this sensitivity as we shall see.
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| ==Schnirelmann's theorems==
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| If we set <math>\mathfrak{G}^2 = \{k^2\}_{k=1}^{\infty}</math>, then [[Lagrange's four-square theorem]] can be restated as <math> \sigma(\mathfrak{G}^2 \oplus \mathfrak{G}^2 \oplus \mathfrak{G}^2 \oplus \mathfrak{G}^2) = 1</math>. (Here the symbol <math>A\oplus B</math> denotes the [[sumset]] of <math>A\cup\{0\}</math> and <math>B\cup\{0\}</math>.) It is clear that <math> \sigma \mathfrak{G}^2 = 0</math>. In fact, we still have <math> \sigma(\mathfrak{G}^2 \oplus \mathfrak{G}^2) = 0</math>, and one might ask at what point the sumset attains Schnirelmann density 1 and how does it increase. It actually is the case that <math> \sigma(\mathfrak{G}^2 \oplus \mathfrak{G}^2 \oplus \mathfrak{G}^2) = 5/6</math> and one sees that sumsetting <math>\mathfrak{G}^2</math> once again yields a more populous set, namely all of <math>\N</math>. Schnirelmann further succeeded in developing these ideas into the following theorems, aiming towards Additive Number Theory, and proving them to be a novel resource (if not greatly powerful) to attack important problems, such as [[#Waring's problem|Waring's problem]] and [[#Goldbach's conjecture|Goldbach's conjecture]].
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| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Theorem.''' Let <math>A</math> and <math>B</math> be subsets of <math>\N</math>. Then
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| <center>
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| <math>\sigma(A \oplus B) \ge \sigma A + \sigma B - \sigma A \cdot \sigma B.</math>
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| </center>
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| </blockquote>
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| Note that <math>\sigma A + \sigma B - \sigma A \cdot \sigma B = 1 - (1 - \sigma A)(1 - \sigma B)</math>. Inductively, we have the following generalization.
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| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Corollary.''' Let <math>A_i \subseteq \N</math> be a finite family of subsets of <math>\N</math>. Then
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| <center>
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| <math>\sigma(\bigoplus_i A_i) \ge 1 - \prod_{i}(1 - \sigma A_i).</math>
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| </center>
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| </blockquote>
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| The theorem provides the first insights on how sumsets accumulate. It seems unfortunate that its conclusion stops short of showing <math>\sigma</math> being [[superadditive]]. Yet, Schnirelmann provided us with the following results, which sufficed for most of his purpose.
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| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Theorem.''' Let <math>A</math> and <math>B</math> be subsets of <math>\N</math>. If <math>\sigma A + \sigma B \ge 1</math>, then
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| <center> | |
| <math>A \oplus B = \N.</math> | |
| </center>
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| </blockquote>
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| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Theorem.''' (''Schnirelmann'') Let <math>A \subseteq \N</math>. If <math>\sigma A > 0</math> then there exists <math>k</math> such that
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| <center>
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| <math>\bigoplus^k_{i=1} A=\N.</math>
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| </center>
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| </blockquote>
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| ==Additive bases==
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| A subset <math>A \subseteq \N</math> with the property that <math>A \oplus A \oplus \cdots \oplus A = \N</math> for a finite sum, is called an '''additive basis''', and the least number of summands required is called the ''degree'' (sometimes ''order'') of the basis. Thus, the last theorem states that any set with positive Schnirelmann density is an additive basis. In this terminology, the set of squares <math>\mathfrak{G}^2 = \{k^2\}_{k=1}^{\infty}</math> is an additive basis of degree 4.
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| ==Mann's theorem==
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| Historically the theorems above were pointers to the following result, at one time known as the <math>\alpha + \beta</math> hypothesis. It was used by [[Edmund Landau]] and was finally proved by [[Henry Mann]] in 1942.
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| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Theorem.''' {{harv|Mann|1942}} Let <math>A</math> and <math>B</math> be subsets of <math>\N</math>. In case that <math>A \oplus B \ne \N</math>, we still have
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| <center>
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| <math>\sigma(A \oplus B) \ge \sigma A + \sigma B.</math>
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| </center>
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| </blockquote>
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| An analogue of this theorem for lower asymptotic density was obtained by Kneser.<ref>Nathanson (1990) p.397</ref>
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| ==Waring's problem==
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| {{main|Waring's problem}}
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| Let <math> k</math> and <math> N</math> be natural numbers. Let <math> \mathfrak{G}^k = \{i^k\}_{i=1}^\infty</math>. Define <math> r_N^k(n)</math> to be the number of non-negative integral solutions to the equation
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| :<math> x_1^k + x_2^k + \cdots + x_N^k = n\,</math> | |
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| and <math> R_N^k(n)</math> to be the number of non-negative integral solutions to the inequality
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| :<math> 0 \le x_1^k + x_2^k + \cdots + x_N^k \le n,\,</math>
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| in the variables <math> x_i</math>, respectively. Thus <math> R_N^k(n) = \sum_{i=0}^n r_N^k(i)</math>. We have
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| *<math> r_N^k(n)>0 \leftrightarrow n \in N\mathfrak{G}^k, </math>
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| *<math> R_N^k(n) \ge \left(\frac{n}{N}\right)^{\frac{N}{k}}.</math>
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| The volume of the <math>N</math>-dimensional body defined by <math> 0 \le x_1^k + x_2^k + \cdots + x_N^k \le n</math>, is bounded by the volume of the hypercube of size <math> n^{1/k}</math>, hence <math>R_N^k(n) = \sum_{i=0}^n r_N^k(i)= n^{N/k}</math>. The hard part is to show that this bound still works on the average, i.e.,
| |
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| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Lemma.''' (''Linnik'') For all <math>k \in \N</math> there exists <math>N \in \N</math> and a constant <math>c = c(k)</math>, depending only on <math>k</math>, such that for all <math>n \in \N</math>,
| |
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| <center>
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| <math>r_N^k(m) < cn^{\frac{N}{k}-1}</math>
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| </center>
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| for all <math>0 \le m \le n.\,</math>
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| | |
| </blockquote> | |
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| With this at hand, the following theorem can be elegantly proved.
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| | |
| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Theorem.''' For all <math>k</math> there exists <math>N</math> for which <math>\sigma(N\mathfrak{G}^k) > 0</math>.
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| </blockquote>
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| We have thus established the general solution to Waring's Problem:
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| <blockquote style="border: 1px dashed #cccccc; padding: 10px; ">
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| '''Corollary.''' {{harv|Hilbert|1909}} For all <math>k</math> there exists <math>N</math>, depending only on <math>k</math>, such that every positive integer <math>n</math> can be expressed as the sum of at most <math>N</math> many <math>k</math>-th powers.
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| </blockquote> | |
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| ==Schnirelmann's constant==
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| In 1930 Schnirelmann used these ideas in conjunction with the [[Brun sieve]] to prove '''Schnirelmann's theorem''',<ref name=Schnirelmann1/><ref name=Schnirelmann2/> that any [[natural number]] greater than one can be written as the sum of not more than ''C'' [[prime numbers]], where ''C'' is an effectively computable constant:<ref name=N96208>Nathanson (1996) p.208</ref> Schnirelmann obtained ''C'' < 800000.<ref>Gelfond & Linnik (1966) p.136</ref> '''Schnirelmann's constant''' is the lowest number ''C'' with this property.<ref name=N96208/>
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| [[Olivier Ramaré]] showed in {{harv|Ramaré|1995}} that Schnirelmann's constant is at most 7,<ref name=N96208/> improving the earlier upper bound of 19 obtained by [[Hans Riesel]] and [[Robert Charles Vaughan (mathematician)|R. C. Vaughan]].
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| Schnirelmann's constant is at least 3; [[Goldbach's conjecture]] implies that this is the constant's actual value.<ref name=N96208/>
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| ==Essential components==
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| [[Aleksandr Khinchin|Khintchin]] proved that the sequence of squares, though of zero Schnirelmann density, when added to a sequence of Schnirelmann density between 0 and 1, increases the density:
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| : <math>\sigma(A+\mathfrak{G}^2)>\sigma(A)\text{ for }0<\sigma(A)<1.\,</math>
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| This was soon simplified and extended by [[Paul Erdős|Erdős]], who showed, that if ''A'' is any sequence with Schnirelmann density α and ''B'' is an additive basis of order ''k'' then
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| : <math>\sigma(A+B)\geq \alpha+ \frac{\alpha(1-\alpha)}{2k}\,,</math><ref name=Rus177>Ruzsa (2009) p.177</ref>
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| and this was improved by Plünnecke to | |
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| :<math>\sigma(A+B)\geq \alpha^{\frac{1}{1-k}}\ . </math><ref name=Rus179>Ruzsa (2009) p.179</ref>
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| Sequences with this property, of increasing density less than one by addition, were named '''essential components''' by Khintchin. [[Yuri Linnik|Linnik]] showed that an essential component need not be an additive basis<ref>{{cite journal | first=Yu. V. | last=Linnik | authorlink=Yuri Linnik | title=On Erdõs's theorem on the addition of numerical sequences | journal=[[Sbornik: Mathematics|Mat. Sb.]] | volume=10 | year=1942 | pages=67–78 | zbl=0063.03574 }}</ref> as he constructed an essential component that has ''x''<sup>o(1)</sup> elements less than ''x''. More precisely, the sequence has
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| : <math>e^{(\log x)^c}\,</math>
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| elements less than ''x'' for some ''c'' < 1. This was improved by E. Wirsing to
| |
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| : <math>e^{\sqrt{\log x}\log\log x}.\,</math>
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| For a while, it remained an open problem how many elements an essential component must have. Finally, [[Imre Z. Ruzsa|Ruzsa]] determined that an essential component has at least (log ''x'')<sup>''c''</sup> elements up to ''x'', for some ''c'' > 1, and for every ''c'' > 1 there is an essential component which has at most (log ''x'')<sup>''c''</sup> elements up to ''x''.<ref name=Rus184>Ruzsa (2009) p.184</ref>
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| ==References==
| |
| {{reflist}}
| |
| * {{Cite journal | last1=Hilbert | first1=David | author1-link=David Hilbert | title=Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl ''n''ter Potenzen (Waringsches Problem) | doi=10.1007/BF01450405 | mr=1511530 | year=1909 | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=67 | issue=3 | pages=281–300 | ref=harv | postscript=<!--None-->}}
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| * {{cite journal | first=L.G. | last=Schnirelmann | authorlink=Lev Schnirelmann | title=On additive properties of numbers | language=Russian | journal=Ann. Inst. polytechn. Novočerkassk | volume=14 | pages=3–28 | year=1930 | ref=harv | zbl=JFM 56.0892.02 }}
| |
| * {{cite journal | first=L.G. |last=Schnirelmann | authorlink=Lev Schnirelmann | title=Über additive Eigenschaften von Zahlen | language=German | journal=Math. Ann. | volume=107 | pages=649–690 | year=1933 | doi=10.1007/BF01448914 | ref=harv | zbl=0006.10402 }}
| |
| * {{Cite journal |last=Mann |first=Henry B.| authorlink=Henry Mann| title=A proof of the fundamental theorem on the density of sums of sets of positive integers | doi=10.2307/1968807 | mr=0006748 | year=1942 | journal=[[Annals of Mathematics]] | series = Second Series | issn=0003-486X | volume=43 | pages=523–527 | issue=3 | publisher=Annals of Mathematics | ref=harv | postscript=<!--None--> | jstor=1968807 | zbl=0061.07406 }}
| |
| * {{cite book | first1=A.O. | last1=Gelfond | authorlink1=Alexander Gelfond | first2=Yu. V. | last2=Linnik | authorlink2=Yuri Linnik | title=Elementary Methods in Analytic Number Theory | publisher=George Allen & Unwin | year=1966 | editor=L.J. Mordell | editor-link=Louis J. Mordell }}
| |
| *{{cite book |last=Mann |first=Henry B. |authorlink=Henry Mann |title=Addition Theorems: The Addition Theorems of Group Theory and Number Theory
| |
| |publisher=[http://www.krieger-publishing.com/subcats/MathematicsandStatistics/mathematicsandstatistics.html Robert E. Krieger Publishing Company]
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| |location=Huntington, New York |year=1976 |edition=Corrected reprint of 1965 Wiley |isbn=0-88275-418-1 |mr=424744 |ref=harv }}
| |
| * {{cite book | zbl=0722.11007 | last=Nathanson | first=Melvyn B. | chapter=Best possible results on the density of sumsets | pages=395–403 | editor1-last=Berndt | editor1-first=Bruce C. | editor1-link=Bruce C. Berndt | editor2-last=Diamond | editor2-first=Harold G. | editor3-last=Halberstam | editor3-first=Heini | editor3-link=Heini Halberstam | editor4-last=Hildebrand | editor4-first=Adolf | title=Analytic number theory. Proceedings of a conference in honor of Paul T. Bateman, held on April 25-27, 1989, at the University of Illinois, Urbana, IL (USA) | series=Progress in Mathematics | volume=85 | location=Boston | publisher=Birkhäuser | year=1990 | isbn=0-8176-3481-9 | ref=harv }}
| |
| * {{cite journal | first=O. | last=Ramaré | authorlink=Olivier Ramaré | title=On Šnirel'man's constant | journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV | volume=22 | year=1995 | issue=4 | pages=645–706 | url = http://www.numdam.org/item?id=ASNSP_1995_4_22_4_645_0 | accessdate = 2011-03-28 | ref=harv | zbl=0851.11057 }}
| |
| * {{cite book | title=Additive Number Theory: the Classical Bases | volume=164 | series=[[Graduate Texts in Mathematics]] | first=Melvyn B. | last=Nathanson | publisher=[[Springer-Verlag]] | year=1996 | isbn=0-387-94656-X | zbl=0859.11002 }}
| |
| * {{cite book | first=Melvyn B. | last=Nathanson | title=Elementary Methods in Number Theory | volume=195 | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | year=2000 | isbn=0-387-98912-9 | zbl=0953.11002 | pages=359–367 }}
| |
| *{{Cite document | author1-link = Aleksandr Khinchin | last1 = Khinchin | first1 = A. Ya. | title = Three Pearls of Number Theory | publisher = Dover | location = Mineola, NY | date = 1998 | isbn = 978-0-486-40026-6 | ref = harv | postscript = <!--None-->}} Has a proof of Mann's theorem and the Schnirelmann-density proof of Waring's conjecture.
| |
| * {{cite book | first1=Alina Carmen | last1=Cojocaru | first2=M. Ram | last2=Murty | author2-link=M. Ram Murty | title=An introduction to sieve methods and their applications | series=London Mathematical Society Student Texts | volume=66 | publisher=[[Cambridge University Press]] | isbn=0-521-61275-6 | pages=100–105 | year=2005 }}
| |
| * {{cite book | last=Ruzsa | first=Imre Z. | chapter=Sumsets and structure | pages=87–210 | editor1-last=Geroldinger | editor1-first=Alfred | editor2-last=Ruzsa | editor2-first=Imre Z. | others=Elsholtz, C.; Freiman, G.; Hamidoune, Y. O.; Hegyvári, N.; Károlyi, G.; Nathanson, M.; Solymosi, J.; Stanchescu, Y. With a foreword by Javier Cilleruelo, Marc Noy and Oriol Serra (Coordinators of the DocCourse) | title=Combinatorial number theory and additive group theory | series=Advanced Courses in Mathematics CRM Barcelona | location=Basel | publisher=Birkhäuser | year=2009 | isbn=978-3-7643-8961-1 | zbl=1221.11026 | ref=harv }}
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| [[Category:Additive number theory]]
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| [[Category:Mathematical constants]]
| |
True said, "Paradise lies beneath the feet of the mother" and that heaven has no match and God bless one with only if he is immense lucky. However, if we talk about a materialistic heaven, it is nothing else besides our home and everyone want this place to be nothing less than a luxury accommodation. While talking about construction of a house, few things comes to our mind such as flooring, painting, electronic components, and civil engineering material and resin crafts. In this article, we will be exploring the use of epoxy resin for around mentioned areas.
Paint:
Painting is the major part of a house because until a house is not coated with the colors, it will be more likely a barren country side property, throwing impressions of Haunted house. When we are after coating services, Epoxy Resin, is a major component because of below mentioned properties:
This adhesive (the common name of Epoxy Resin)is blessed with marvelous chemical resistance, specifically alkali
The Epoxy Resin has no better substitute when it is about film adhesion, heat resistance and electrical insulation.
This chemical has no match when the question of color retention arises.
Electrical components:
Epoxy Resin is blessed with awesome insulation and sealing performance and structural length and that is what makes this curing is widely used for low voltage electrical components like solenoid, contactor coils, high voltage electrical insulation package and dry type transformers as well. In addition to this, it is also preferably used in encapsulation devices, laminated plastic and electronic insulation adhesive.
Flooring:
Now comes the turn of flooring and this epoxy resin in an undoubted backbone of seamless flooring, so far. We all know that concrete is an impermeable material and it is strength is unquestionable; this is what makes it durable. If you cherished this article and you would like to receive extra info with regards to resina epoxi kindly check out the web-site. However, when is about flooring, the risk of water and decayed floor due to water, is always there. Any sort of continuous water flow is capable enough of damaging the floor and here, one cannot also deny from water sipping in the roots/foundation and eradicating it thoroughly. We do know that, concrete floors are generally designed with simply design using one tone coloration across the whole floor. Now if somehow, one part of floor breaks or cracks, it makes whole floor look ugly and turning a blind eye to that is something we cannot afford. This is where; The Resin technology comes in offers spotless flooring.