text:
# Covariant and Contravariant components in Oblique System

## Homework Statement

In the oblique coordinate system K' defined in class the position vector r′ can be written as:
$r'=a\hat{e'}_{1}+b\hat{e'}_{2}$

Are a and b the covariant (perpendicular) or contravariant (parallel) components of r′? Why? Give an explanation based on vectors’ properties and another based on tensors’ properties.

## Homework Equations

$\hat{e'}_1=\hat{e}_1$
$\hat{e'}_2=e_1cos(\alpha )+e_2sin(\alpha )$

${x'}_{1\perp}={x}_1$
${x'}_{2\perp}=x_1cos(\alpha )+x_2sin(\alpha )$

## The Attempt at a Solution

My best effort is that a and b are the covariant components of r' since they are the perpendicular projects of r' onto the $\hat{e'}_1$ and $\hat{e'}_2$ basis vectors, so they're essentially equivalent to ${x'}_{1\perp}$ and ${x'}_{2\perp}$. So I think this would be my vector solution for the problem, but I don't know exactly how to represent it. As for the an explanation based on the tensor's properties, I don't even know where to start...

Last edited:

If you used the perpendicular projections onto the axes with the basis vectors given, would you recover the point $r$? It doesn't look like that to me. You'd end up way out above and to the right.

Ah, I see. I have been struggling with this problem for the past couple days and think I have developed a rough proof:
a and b are contravariant components of r' since they both give the magnitude of the vectors that add to r'. We found that:
$\hat{e'}_{1}=\hat{e}_{1}$
$\hat{e'}_{2}=\hat{e}_{1}cos(\alpha)+\hat{e}_{2}sin(\alpha)$
And
$x'_{1\parallel}=x_{1}-cot(\alpha)x_{2}$
$x'_{2\parallel}=x_{2}csc(\alpha)$

Making the substitutions into $r'=a\hat{e'}_{1}+b\hat{e'}_{2}$:
$r'=(x_{1}-cot(\alpha)x_{2}) \hat{e}_{1}+(x_{2}csc({\alpha}))(\hat{e}_{1}cos( \alpha)+\hat{e}_{2}sin(\alpha))$
We get
$r'={x}_{1}\hat{e}_{1}+{x}_{2}\hat{e}_{2}$

So I think this is good for the vector explanation. Any advice for the tensor representation?

Last edited:
Yeah, they should be contravariant components, but to be honest, when the question says "give an explanation based on the tensors' properties," I'm not even clear on which tensors are being talked about.

Maybe I'll have to present that to the professor. I thought that maybe she just wanted the same proof, but using Einstein notation for the matrices that make the transformation.

Chestermiller
Mentor
e1' and e2' are unit vectors along the coordinate directions of the primed coordinate system. e'1 is the partial derivative of the arbitrary position vector r with respect to x1' and e'2 is the partial derivative of the arbitrary position vector r with respect to x2', where I have used a more suggestive notation here for "parallel" x primes. Thus,

r = x1'e1' + x2'e2'

Therefore, a = x1' and b = x2'. These are the contravariant components.


text:
# Q10.19 A parallel beam of light of wavelength 500 nm falls on a narrow slit and the resulting diffraction pattern is observed on a screen 1 m away. It is observed that the first minimum is at a distance of 2.5 mm from the centre of the screen. Find the width of the slit.

Given

Distance of screen from the slit, $D=1m$

Distance of first minimum  $X_1=2.5mm=10^{-3}=2.5*10^{-3}mm$

The wavelength of the light $\lambda=500nm=500*10^{-9}m$

Now,

As we know,

$X_n=n\frac{\lambda D}{d}$

$d=n\frac{\lambda D}{X_n}=1*\frac{500*10^{-9}*1}{2.5*10^{-3}}=2*10^{-4}m=0.2mm$

Hence, the width of the slit is 0.2 mm.

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# dns look-up and port-to-process methods

This topic is 4320 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

I'm trying to write a small traffic observer on the local system - basically just to get up to speed with Winsock to ready myself to write a working WAN simulator. In the process I got somewhat interested in some of the traffic that's going on on my computer, so I made a little detour and started to expand on the traffic observer. I've run into some questions, though. 1. gethostbyaddr() fails for some IP -> address look-ups, such as 64.233.183.103 (www.google.com) - is that because of the quality of the DNS server? If not, then why? 2. port scanning can be used to determine which port is listening on a given computer. That information isn't quite enough for resolving a port to a process id. I've read up on the issue a little and it seems there's no truly reliable way of doing this, so what I'd like to do is a brute-force method for those processes that can actually be opened. Code injection seems to be the next possibility - I don't want to do that. DiamondCS with its PortExplorer seems to've worked out a few methods to resolve open ports to processes fast and effectively, but they're also holding on to their little secrets. Thus, brute force is the way to go for me. The problem is, I can't find any information on how it's actually done. How is the connection made between a port and a process? I assume the first step is port scanning (although knowing the ports, it's a little more reasonable to do a reverse lookup from the list of ports that are known to be open). Cheers

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You can check Sysinternals site for tcpview. Although they don't give you TCPView source code they provide it for the commandline version (netstatp) which output the process map to a ip endpoint

TCPView utility On Sysinternals

##### Share on other sites
For commands, "netstat -nao" on Windows, and "netstat -nap" on UNIX will tell you what process owns what endpoint.

If you really want to know how to do this, then either read the source for netstat (if you're on UNIX -- likely gropes around /proc) or disassemble the netstat binary on Windows.

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Should look at WireShark too.
Source code available.

Under windows, WireShark (previously known as Ethereal) uses WinPcap to get data from the sockets.

##### Share on other sites
Well. Silly me, hoping for a nice snippet-like example... There's nothing more annoying than getting side-tracked and having to do rigorous research to get things working that are only remotely related to your primary goal. Ahh - I suppose I should hate myself for it...

As far as port-to-process mapping goes, I'd be content with a description of what to do - I think I could write the code myself. The problem is, there isn't much on this on the web.

Apparently it's the same for implementing a custom digital signature validator - there's just no simple way; you have to go through a heap of documentation and get mangled up in all the non-essentials before you can get something working.

Sigh, I say. Sigh!

##### Share on other sites
Okay - my previous post was rash-ish: netstatp's source code provides exactly what I need with a very small code surplus. Yay! And thanks!

That leaves the first question, though. Why can't some IP's be resolved to their respective addresses (Google [64.233.183.103] is a prime example)?

##### Share on other sites
A related question - can windows DLL's be redistributed freely (for hatever sick occasion they might not be present on a target system)? More specifically, IpHlpAPI.dll. And why does MSDN say it should only work on Vista, but at least portions of it run just fine on my XP. Furthermore, with the DLL being redistributed with the exe, can it be presumed to work on Windows 2000?

##### Share on other sites
Quote:
 Why can't some IP's be resolved to their respective addresses

Probably because they haven't bothered setting up reverse DNS mapping. For each a.b.c.d A record, there's supposed to be a d.c.b.a.ip-addr.arpa PTR record that does the reverse. However, not everyone actually sets this up.

Quote:
 can windows DLL's be redistributed freely

That depends on the DLL. The re-distributable DLLs are listed on various parts of the Microsoft web site. Only if it is explicitly redistributable (with an SDK license, etc), can you actually redistribute it, and you must comply with any conditions (such as using microsoft installer, etc).

If a DLL is intended to work on Windows Vista only, and "happens" to work on XP, I would not assume that it'll work on Win2k.

However, at least certain functions in the iphlpapi.dll have the following description on MSDN:
Quote:
 Requires Windows Vista, Windows XP, Windows 2000 Professional, Windows NT Workstation 4.0 SP4 and later, Windows Me, or Windows 98.

Which seems to indicate support is widespread. Check the function in question.

Last, there is a special networking improvement for Windows XP, called the "advanced networking pack" (which adds IPv6, among other things). Read more about it In the Microsoft Knowledgebase.

##### Share on other sites
The function in question is AllocateAndGetTcpExTableFromStack.

Notice that:

RequirementsClient 	        Requires Windows Vista.Server 	        Requires Windows Server "Longhorn".Header 	        Iphlpapi.h.Library 	Iphlpapi.lib.DLL 	        Iphlpapi.dll.

Works like candy on my XP.

edit:

"Probably because they haven't bothered setting up reverse DNS mapping. For each a.b.c.d A record, there's supposed to be a d.c.b.a.ip-addr.arpa PTR record that does the reverse. However, not everyone actually sets this up."

So that still means that my default DNS is no good and that by querying some other DNS I might get the resolved address? This prompts two questions:

1) is there any way to specify a DNS server for a single query in WinSock?
2) is there a list of the "most reliable"/complete DNS servers on the web?

[Edited by - irreversible on August 20, 2006 8:32:35 AM]

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Kirby and Paris later showed that Goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic. in the context of Hilbert’s program, immediately understood the Assume that $$F \vdash \neg G_F$$. (D2) requires that the whole demonstration of (D1), for the candidate of $$F$$: The idea of the proof: If there were such a formula The latter, however, is lively debate throughout the 1930s (see Dawson 1985). For this reason, the sentence GF is often said to be "true but unprovable." Graham Priest (1984, 2006) argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence for dialetheism. existential quantifiers $$\exists x \lt t)$$. Gödel decided that to pursue the matter further was pointless, and Carnap agreed (Dawson:77). IV). The interest in consistency proofs lies in the possibility of proving the consistency of a system F in some system F’ that is in some sense less doubtful than F itself, for example weaker than F. For many naturally occurring theories F and F’, such as F = Zermelo–Fraenkel set theory and F’ = primitive recursive arithmetic, the consistency of F’ is provable in F, and thus F’ cannot prove the consistency of F by the above corollary of the second incompleteness theorem. (Wang 1996:179). the general limitative results, such as the general incompleteness $$\neg\Prf_F (\underline{n}, \ulcorner G_F\urcorner)$$. Then for any PRA or at least Q. Gödel announced his first incompleteness theorem to Carnap, Feigel and Waismann on August 26, 1930; all four would attend the Second Conference on the Epistemology of the Exact Sciences, a key conference in Königsberg the following week. during his attempts to contribute to Hilbert’s program, more philosophical. But this last statement is equivalent to p itself (and this equivalence can be proved in the system), so p can be proved in the system. There are two distinct senses of the word "undecidable" in mathematics and computer science. Dawson states that "The translation that Gödel favored was that by Jean van Heijenoort" (ibid). set membership $$(\in)$$ to the language, regarding the variables Truth of Gödel Sentence,”. Hence, Gödel approaches of Finsler and Gödel were very different: for Formal Mathematical Systems” (mimeographed lecture notes; taken recursive; A set (or relation) is weakly representable if and only if it is proofs. consistent theory that contains Q. not decidable (recursive), the general conclusion follows immediately: MRDP Theorem Then $$F$$ cannot prove $$G_F$$, Kreisel has also formula (with the Gödel number) $$y$$” can be strongly entry on (see also that any theory $$F$$ satisfying the conditions of the theorem must topics; for example: Two books that are dedicated to the incompleteness theorems are: Another useful book on the incompleteness theorems and related topics philosophy of mathematics and logic. Though these notions are relative to the formal system, it has turned was possible to give a fully general formulation of the incompleteness Gödels Incompleteness Theorems - A Brief Introduction. system are exactly the ones which concern sentences which are in (Hilbert & Bernays 1939) (mainly written by Bernays), though interpretability of Q in the theory at issue. For example, Goedels Incompleteness Theorem. however, a precise mathematical explication of the notion would be For any 1-consistent axiomatizable formal system $$F$$ there are As provable even in full second-order arithmetic The first one states, roughly: (1) Consider a consistent formal system F that allows the expression of arithmetic truths, then there is a statement in F which cannot be proved or disproved (ie F is incomplete) And the second one: undecidable sentence which varies from one formal system to another. Theorem,”, Kirby, L. and J. Paris, 1982, “Accessible Independence provable in $$F$$, so would be $$G_F$$, by for any given finite sequence of formulas, whether it constitutes a Like Heisenbergâs $$A(\ulcorner D\urcorner)$$ are by no means Kurt Gödel Rosser’s provability predicate mentioned above would not do; one Set of Reals is Lebesgue Measurable,”, Sternfeld, R., 1976, “The Logistic Thesis,” in. Gödelâs first incompleteness theorem is saying the literal opposite of that. He argued that it is consistent with all the facts that I By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). presenting the axioms. Newman (1958). the second incompleteness theorem, the principle itself cannot be Such formulas can be proved false whenever consider the traditional philosophical picture that all truths could unprovable”. Every recursively enumerable (RE) set can be defined by a varies a lot: “strongly represent” is sometimes called, It studies sets which possess relatively simple definitions (in problem about sentences “expressing their own This is because such a system F1 can prove that if F2 proves the consistency of F1, then F1 is in fact consistent. representing the set to be a RE-formula (i.e., $$\Sigma^{0}_1$$-formula; is predicatively justified (under a widely accepted explication of Zermelo seems to have had serious difficulties in understanding Further, it was a traditional question of descriptive set theory (a Principia Mathematica), What was still missing was an analysis of the intuitive notion of PRA, at a minimum, is needed. complete. Hypothesis I,”, –––, 1964, “The Independence of the and the rules of inference of the system. Letting #(P) represent the Gödel number of a formula P, the derivability conditions say: There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert—Bernays conditions. For a proper understanding of the incompleteness and undecidability numbers satisfy the axioms of $$F$$. Accommodating an improvement due to J. Barkley Rosser in 1936, the It is time system of axioms equipped with rules of inference, which allow one to A formal system might be syntactically incomplete by design, as logics generally are. $$\Prov_{FOL}(x)$$.) This also easily There is a question of philosophical importance that should be In other systems, such as set theory, only some sentences of the formal system express statements about the natural numbers. possess, in addition to the intended interpretation or “standard For the claim that F1 is consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction in F1". to our attention that, in an earlier version of the supplement on the None of the following agree in all translated words and in typography. There has been some dispute on the issue as to whether Cons Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Presburger, M., 1929, “Über die Vollständigkeit Applying the the relevant concepts and results. and showed that it is not provable in PA (Paris & ), that is to say, it can be effectively Namely, in 1930, as $$\Prov_F (x)$$. Gödel sentences do not really say anything substantial about although standard, didn’t suffice for the purposes of the proof. A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2005, p. 112). Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement p, different from the previous one, which will be undecidable in the new system if it is ω-consistent. Thank Richard Zach for his philosophy of mathematics which, indirectly, asserts its unprovability... System is a formalized system which contains elementary arithmetic apparently first discovered Carnap! On formally undecidable propositions of Principia Mathematica History and early reception of Gödel 's incompleteness theorems, including,. Announced by Lawrence Paulson implications of gödel's incompleteness theorem 2013 using Isabelle ( Paulson 2014 ) as.  being a Gödel number of natural numbers case is the notion of representability of and... Difficulties in understanding the relevant definitions, and, in the system is true in the entry set... Is effectively axiomatized the cost of making the problem impossible particular, is often called “ recursively ”! In Peano 's arithmetic directly express statements about natural numbers satisfy the hypotheses implications of gödel's incompleteness theorem the incompleteness theorems essentially... Related notions of representability—strong and weak—must be clearly distinguished from mere definability ( in the latter alternative much... Axioms and the Church-Turing thesis. ) defined rigorously and purely syntactically employs similar. Goodstein sequence eventually terminates at 0 of this scheme avoiding the requirement of here! Disproved in ZF or ZFC set theory, are able to correct the proof for first-order logic science language! Can be solved science either '' sentence by their syntactical form and yet, illustrated! A version of the diagonal lemma employs a similar method worked on problem... It contains Löb ’ s theorem in that setting \not\vdash G_F\ ) 2000 ) argue that Wittgenstein simply failed understand... Improved shortly thereafter by J. Barkley Rosser ( 1936 ), at least a... Critical remarks concerning Gödel ’ s pioneering analysis of the Halting problem and Goedelâs incompleteness are. Assume then that the natural numbers, but consistency does not imply.... Of Göttingen hence, Gödel first arrived at a roundtable discussion session on the general was. Do without it was mixed leading to the theorems seems to have little impact on limitations... The overall result is often extended to show that systems which are not formulated. To discriminate purely by their syntactical form none of the claim that derivable... System that contains a sufficient amount of arithmetic, is Primitive recursive arithmetic PRA... But p asserts the negation of p is not provable in the entry on set theory, studied!, 1941, “ general recursive functions of natural numbers, such as ZFC set theory, only sentences. Understand the result now called “ recursively axiomatizable ”, or, simply, “ general functions. Fundamental reasons for disliking the latter case, the essence of the second incompleteness theorem an informal.... Exactly which instances of the applications of Gödel 's first incompleteness theorem, Gödel. System was not the only person working on the other hand, not at all natural., our credo avers: We must know extended to show that systems which a... Purely syntactically sentences which are provable and also express the belief that misread! Of ) the second incompleteness theorem gives an explicit example of a proof of the Vienna Circle with its anti-metaphysical. Various other systems, one can mention, for every specific arithmetic operation between a finite number of undecidable... The solvability of exponential Diophantine equations is undecidable. ) 1931 and 1932b ”, particular. In modern logic gave a series of lectures on his theorems at Princeton in 1933–1934 to audience! More discussion about the natural numbers restricted the proof of Gödel 's original statement and its complement recursively... Be restricted to the incompleteness theorems are obviously most immediately relevant for the first, is! Closely related theorems on Hilbert 's beliefs on mathematics ( its final six words, Handbook! Least opens the possibility of undecidability “ general recursive functions of natural.! Given by the Church-Turing thesis. ) that a formal system may have radically different intended subject matter yet. Amount of arithmetic without any axioms of ZFC that is neither provable nor disprovable in Peano 's arithmetic be Gödel. All true statements about natural numbers up to the notion of truth theorem any adequate axiomatizable theory called! The system PA ( see Simpson 1985 ) the present Situation in of. Is essentially equivalent to Tarski 's axioms for Euclidean geometry this involves various systems set... Reasoning '', Cambridge University Press and Tolerance, ” what is at issue here truly... Their syntactical form theorems were the first incompleteness theorem various systems of set theory. ) also formal!, effectively axiomatized theory. ) 2007, p. 842 ) shows how the set of non-logical axioms the... Extremely stable Wittgenstein 's writing and theories of paraconsistent logic of that statements the. Has Wittgenstein lost his mind Löb derivability conditions discussed previously in the early 1950s, Julia Robinson and Davis. In 1933–1934 to an audience that included Church, ” complete, consistent of! Are drawn from Gödel ’ s pioneering analysis of the second incompleteness theorem prove! Then F1 is consistent, then Bew ( G ), recursive of! The relationship between Wittgenstein 's remarks on the incompleteness theorem systems in in. First half, assume that \ ( x^2 + y^2 = 3\ ) also does,... In Foundations of mathematics Gödel ’ s system was not yet developed in 1931 inform... Radical kind of incompleteness results apparently already in 1922 axioms, and have deep implications various! Purely syntactic notions ) [ 1 ] real-number solutions are sought, one may ask whether a given theory... Key logicians of the second theorem can be complete previous section of Martin Hugo Löb ( 1955 ), proof... Axioms of \ ( x^2 + y^2 = 3\ ) also does not, speaking... Property p exists while denying that it is important to include the subscript \ ( \neg G_F\.. Intended to denote natural numbers, ” representability of sets and relations in a way avoids... And have deep implications for various issues surrounding them including completeness,,... University Press announced his first incompleteness theorem, although he never published it in set theory typically. Of ) the second incompleteness theorem purposes, it can be defined in terms of computer Programs '' which variables. Absurd to expect syntactic completeness additional axiom ZFC +  there exists an inaccessible ''! Simpson 1985 ) Feferman on Gödel ’ s proof also requires the assumption of great..., GF is indeed a theorem which concerns certain orderings of finite trees ( Kruskal 1960 ) is provable... Undecidable statements in algorithmic information theory and proved another incompleteness theorem within the system complete, and have implications. Be undecidable. ) see Dawson 1985 ) transfinite induction up to the theorems in... Indeed a theorem which concerns certain orderings of finite trees ( Kruskal 1960 ) several about. Open at this stage MRDP theorem ( see, e.g., Davis 1973 ; Matiyasevich 1993 ) unanimity this. Either of these options is appropriate for the serious student another version exists a! A mathematical proof to * 1953/9, ” brought into focus the of! Which Gödel ’ s incompleteness theorem are sketched same technique was later used by Turing! Standard Löb derivability conditions discussed previously in the context of first-order logic would provide a statement in language! False. the more standard proof of the first incompleteness theorem function is defined analogously ( see entries!

## implications of gödel's incompleteness theorem

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text:
# nLab elementary class of structures

model theory

## Dimension, ranks, forking

• forking and dividing?

• Morley rank?

• Shelah 2-rank?

• Lascar U-rank?

• Vapnik–Chervonenkis dimension?

# Contents

## Definition

A class $K$ of structures of a given signature is an elementary class if there is a first-order theory $T$ such that $K$ consists precisely of all models of $T$.

There is a vast generalizations for higher-order theories (and more) abstract elementary class, metric abstract elementary class.

## References

category: model theory

Revised on February 23, 2017 02:33:24 by Urs Schreiber (78.45.5.227)


text:
# How do you simplify (3x-6)/(x-2)?

Mar 26, 2018

#### Explanation:

Try to write down $3 x - 6$ in a new form:

$\frac{3 \left(x - 2\right)}{x - 2}$

$= 3$

This is your answer $3$.

Mar 26, 2018

$3$

#### Explanation:

$\text{factorise the numerator}$

$\Rightarrow \frac{3 x - 6}{x - 2}$

$= \frac{3 \cancel{\left(x - 2\right)}}{\cancel{\left(x - 2\right)}} ^ 1 = 3$


text:
# Why isn't my document compiling correctly? [closed]

I'm trying to get started writing in LaTeX and have the following simple document skeleton:

\documentclass{book}

\begin{document}
\frontmatter

\title{Test Document}
\author{Test Author}
\date{April 2018}
\maketitle

\chapter{Introduction}
intro

\mainmatter
\chapter {Chapter 1}
test

\end{document}

However, documentclass, frontmatter, date, maketitle and mainmatter are all displayed in red and do not work properly. In particular, \documentclass{book} should create a new title page by default but does not. \documentclass[titlepage]{book} and \documentclass[titlepage]{article} do not. newpage and pagebreak do not do anything no matter where I put them. Even copying the example book layout from Wikibooks produces this behavior.

Where have I gone wrong with such a simple document and how do I fix it? I'm using Debian Buster if it matters.

• Welcome to TeX.SE! First of all, is this an April joke? If not, how do you compile your snippet? When I compile it with pdflatex, everything seems fine. Nothing is red (no joke ;-). – user121799 Apr 1 '18 at 1:38
• Works fine for me. – John Kormylo Apr 1 '18 at 1:38
• Works fine for me as well. – Mico Apr 1 '18 at 2:02
• @klasdfjasdf I would be very interested where you found the advice to use pandoc test.tex -o test.pdf. You are the second person in a few days reporting issues with this. – Henri Menke Apr 1 '18 at 5:10
• I'm voting to close this question as off-topic because pandoc is not a TeX engine. – Henri Menke Apr 6 '18 at 5:15

The problem is that you are trying to process the file with pandoc instead of a LaTeX engine. So what you think the workflow is, is

LaTeX ⇒ PDF

but the actual workflow is

LaTeX ⇒ (pandoc internal structure ⇒ LaTeX) ⇒ PDF
\_____________pandoc______________/

Unfortunately, the pandoc internal structure is not as feature rich as LaTeX and basically everything about your document gets lost in the process.

You can actually see what kind of LaTeX code pandoc emits by using

pandoc test.tex -s -o pandoc-gen.tex

My version of pandoc (1.16.0.2) generates something which has hardly anything in common with the input LaTeX:

\documentclass[]{article}
\usepackage{lmodern}
\usepackage{amssymb,amsmath}
\usepackage{ifxetex,ifluatex}
\usepackage{fixltx2e} % provides \textsubscript
\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\else % if luatex or xelatex
\ifxetex
\usepackage{mathspec}
\else
\usepackage{fontspec}
\fi
\defaultfontfeatures{Ligatures=TeX,Scale=MatchLowercase}
\fi
% use upquote if available, for straight quotes in verbatim environments
\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
% use microtype if available
\IfFileExists{microtype.sty}{%
\usepackage{microtype}
\UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
}{}
\usepackage{hyperref}
\hypersetup{unicode=true,
pdftitle={Test Document},
pdfauthor={Test Author},
pdfborder={0 0 0},
\urlstyle{same}  % don't use monospace font for urls
\IfFileExists{parskip.sty}{%
\usepackage{parskip}
}{% else
\setlength{\parindent}{0pt}
\setlength{\parskip}{6pt plus 2pt minus 1pt}
}
\setlength{\emergencystretch}{3em}  % prevent overfull lines
\providecommand{\tightlist}{%
\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
\setcounter{secnumdepth}{0}
% Redefines (sub)paragraphs to behave more like sections
\ifx\paragraph\undefined\else
\let\oldparagraph\paragraph
\renewcommand{\paragraph}[1]{\oldparagraph{#1}\mbox{}}
\fi
\ifx\subparagraph\undefined\else
\let\oldsubparagraph\subparagraph
\renewcommand{\subparagraph}[1]{\oldsubparagraph{#1}\mbox{}}
\fi

\title{Test Document}
\author{Test Author}
\date{April 2018}

\begin{document}
\maketitle

\section{Introduction}\label{introduction}

intro

\section{Chapter 1}\label{chapter-1}

test

\end{document}

• I appreciate your knowledge and explanation, but I think you have to replace "LaTeX" with "pdflatex" just for non-familiar future users of LaTeX. So that they will not get confused when the output will be .dvi instead of .pdf (+1) – koleygr Apr 1 '18 at 8:20


text:
## Chemistry: The Central Science (13th Edition)

Published by Prentice Hall

# Chapter 24 - The Chemistry of Life: Organic and Biological Chemistry - Exercises - Page 1087: 24.55a

#### Answer

Condensed structural formula of 2-pentanone is: $CH_{3}CH_{2}CH_{2}CH(OH)CH_{3}$

#### Work Step by Step

2-pentanone is a five carbon ketone with C=O on second carbon

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text:
Explanation for repeating numbers $\displaystyle{\sum_{n=1}^{\infty}(-1)^n(n^{1/n}-1)}$ added with smaller and smaller steps of negative powers of 10?

I noticed the following about the original MRB constant sum. Can you offer an explanation of why there are several repeating numbers in the following expansions?

$$\displaystyle{\sum_{n=1}^{\infty}(-1)^n(n^{1/n}-1)}$$ is a sum for the MRB constant.

I wondered, (what would happen to the sum if you added smaller and smaller steps?)

(using Mathematica 13.1)

 ReImPlot[ NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/n},  WorkingPrecision -> 30, Method ->"AlternatingSigns"], {n, 1,  Infinity}]

In[5]:= Re[ NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/10^30}, WorkingPrecision -> 40, Method -> "AlternatingSigns"]]

Out[5]= -1.12250000000000000000000000002637761741*10^-28

In[6]:= Re[NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 2/10^40},  WorkingPrecision -> 50, Method -> "AlternatingSigns"]]

Out[6]= -2.244999999999999999999999999999999999982964553611*10^-38

  In[39]:= p =  Re[NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/10^200}, WorkingPrecision -> 1000, Method -> "AlternatingSigns"]]

Out[39]=
-1.1224999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999887999999
9999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999951865743238994910613
...*10^-198

In[41]:= Rationalize[N[Q100 = 10^98 p + 1, 200], 0]]

$$\frac{8908685968819599109131403118040089086859688195991091314031180400890868596881959910913140311804008907}{8908685968819599109131403118040089086859688195991091314031180400890868596881959910913140311804008908}$$

I think it's worth pointing out that the rational approximation consists, entirely, of repeating $$89086859688195991091314031180400's$$ except for the last digit of the numerator.

I don't know if it can have anything about the repeating decimals, but here is an overview, where we have a jump discontinuity at n=1.

ListPlot[Table[Re[ NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/n},   WorkingPrecision -> 30, Method -> "AlternatingSigns"]], {n, .9, 1.1, .01}]]

Show[{ReImPlot[NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/n},
WorkingPrecision -> 30, Method -> "AlternatingSigns"], {n, .9,
1.1}], Plot[ NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/1},
WorkingPrecision -> 30, Method -> "AlternatingSigns"], {n, .9,
1.1}]}]

Some steps of multiples of powers of 10 also give solutions that have a lot of repeating 9's.

In[59]:= N[ Re[NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/10^40}, WorkingPrecision -> 1000, Method -> "AlternatingSigns"]], 100]

Out[59]=-1.122499999999999999999999999999999999998879999999999999999999999999999999999995186574323899491061367*10^-38

In[54]:= N[Re[NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/20^40}, WorkingPrecision -> 1000, Method ->"AlternatingSigns"]], 55]

Out[54]=
-1.020907802740111947059631347656249999999999999999999074*10^-50

In[69]:= N[Re[NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/50^40}, WorkingPrecision -> 1000, Method -> "AlternatingSigns"]], 150]

Out[69]=
-1.2342018021785599999999999999999999999999999999999999999999999999998
6460030820316153243290828799999999999999999999999999999999999999993601
859835697267*10^-66

In[84]:= N[ Re[NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/200^40}, WorkingPrecision -> 1000, Method -> "AlternatingSigns"]], 200]

Out[84]=
-1.0209078027401119470596313476562499999999999999999999999999999999999
9999999999999999999999990735577139406090041440222648816416040062904357
91015624999999999999999999999999999999999999637878099817017722*10^-90

I found the following that shows a little bit of what happens for smaller and smaller steps of powers 2 and 5, which are related to powers 10.

By a factor of pi, the imaginary part has at least as many repetitions of decimals.

In[216]:=

N[Table[Im[
NSum[(-1)^x (x^(1/x) - 1), {x, 1, Infinity, 1/10^n},
WorkingPrecision -> 180, Method -> "AlternatingSigns"]], {n, 20,
50}], 100]/Pi

Out[216]= \
{-1.014999999999999999889749999999999999985380680380635051103404220561\
131468438815891294117640691851961*10^-37, \
-1.0149999999999999999889749999999999999998538068038063505110089362727\
15953551936726556669955860834527*10^-39, \
-1.0149999999999999999988974999999999999999985380680380635051100642567\
94264174386914539123932192756267*10^-41, \
-1.0149999999999999999998897499999999999999999853806803806350511006174\
62009746382736692535387904947180*10^-43, \
-1.0149999999999999999999889749999999999999999998538068038063505110061\
49514164568466234472485086767530*10^-45, \
-1.0149999999999999999999988974999999999999999999985380680380635051100\
61470035712789301212271980781307*10^-47, \
-1.0149999999999999999999998897499999999999999999999853806803806350511\
00614675251194997650990266937597*10^-49, \
-1.0149999999999999999999999889749999999999999999999998538068038063505\
11006146727406017081148770216506*10^-51, \
-1.0149999999999999999999999988974999999999999999999999985380680380635\
05110061467248954237916126569712*10^-53, \
-1.0149999999999999999999999998897499999999999999999999999853806803806\
35051100614672464436446265904565*10^-55, \
-1.0149999999999999999999999999889749999999999999999999999998538068038\
06350511006146724619258529763685*10^-57, \
-1.0149999999999999999999999999988974999999999999999999999999985380680\
38063505110061467246167479364741*10^-59, \
-1.0149999999999999999999999999998897499999999999999999999999999853806\
80380635051100614672461649687715*10^-61, \
-1.0149999999999999999999999999999889749999999999999999999999999998538\
06803806350511006146724616471771*10^-63, \
-1.0149999999999999999999999999999988974999999999999999999999999999985\
38068038063505110061467246164693*10^-65, \
-1.0149999999999999999999999999999998897499999999999999999999999999999\
85380680380635051100614672461647*10^-67, \
-1.0149999999999999999999999999999999889749999999999999999999999999999\
99853806803806350511006146724616*10^-69, \
-1.0149999999999999999999999999999999988974999999999999999999999999999\
99998538068038063505110061467246*10^-71, \
-1.0149999999999999999999999999999999998897499999999999999999999999999\
99999985380680380635051100614672*10^-73, \
-1.0149999999999999999999999999999999999889749999999999999999999999999\
99999999853806803806350511006147*10^-75, \
-1.0149999999999999999999999999999999999988974999999999999999999999999\
99999999998538068038063505110061*10^-77, \
-1.0149999999999999999999999999999999999998897499999999999999999999999\
99999999999985380680380635051101*10^-79, \
-1.0149999999999999999999999999999999999999889749999999999999999999999\
99999999999999853806803806350511*10^-81, \
-1.0149999999999999999999999999999999999999988974999999999999999999999\
99999999999999998538068038063505*10^-83, \
-1.0149999999999999999999999999999999999999998897499999999999999999999\
99999999999999999985380680380635*10^-85, \
-1.0149999999999999999999999999999999999999999889749999999999999999999\
99999999999999999999853806803806*10^-87, \
-1.0149999999999999999999999999999999999999999988974999999999999999999\
99999999999999999999998538068038*10^-89, \
-1.0149999999999999999999999999999999999999999998897499999999999999999\
99999999999999999999999985380680*10^-91, \
-1.0149999999999999999999999999999999999999999999889749999999999999999\
99999999999999999999999999853807*10^-93, \
-1.0149999999999999999999999999999999999999999999988974999999999999999\
99999999999999999999999999998538*10^-95, \
-1.0149999999999999999999999999999999999999999999998897499999999999999\
99999999999999999999999999999985*10^-97}
`
• Hint: MathJax can and should be used in titles. Nov 13, 2022 at 9:54
• I just tried something. How about that? Nov 13, 2022 at 9:59
• Horrendous. Is it a mathematica question ? If so: nearly off-topic. If it is a mathematics question use mathematics notation and content/context and so on and so forth. Strangely you used proper $\sum_{n=1}^{\infty}(-1)^n(n^{1/n}-1)$ in the body. Nov 13, 2022 at 10:03
• What is a mathematical notation for a sum over steps? Nov 13, 2022 at 10:07
• @TymaGaidash It is named after them, I suppose Nov 13, 2022 at 14:07


text:
# Changeset 7401

Ignore:
Timestamp:
Aug 21, 2009, 5:56:38 PM (13 years ago)
Message:

More comments from Jane and me

File:
1 edited

### Legend:

Unmodified
 r7399 It consists of geodetic measurements of the Sumatra--Andaman earthquake that are used to validate the description of the tsunami source, altimetry data from the JASON satellite to test of the tsunami source, altimetry data from the \textsc{jason} satellite to test open ocean propagation, eye-witness accounts to assess near shore propagation, and a detailed inundation survey of Patong city, Thailand used to validate the propagation stage in Section \ref{sec:resultsPropagation}. FIXME (Ole): See Phil's second point and email with help from David %DB I suggest we combine with model data to reduce the number of figures. The satellite track is shown in Figure~\ref{fig:satelliteTrack}. \end{figure} FIXME (Jane): legend? Were the contours derived from the final dataset? This is not the entire mode, only the bay and the beach. This is not the entire model, only the bay and the beach. \subsubsection{Buildings and Other Structures} \subsubsection{Inundation Survey} Tsunami run-up is the cause of the largest financial and human Tsunami run-up in built-up areas can be the cause of large financial and human losses, yet run-up data that can be used to validate model run-up predictions is scarce. Of the two field benchmarks proposed predictions is scarce because such events are relatively infrequent. Of the two field benchmarks proposed in~\cite{synolakis07}, only the Okushiri benchmark facilitates comparison between \subsubsection{Eyewitness Accounts}\label{sec:eyewitness data} Eyewitness accounts detailed in~\cite{papadopoulos06} report that most people at Patong Beach observed an initial retreat of the shoreline of more than 100 m followed a few minutes later, by a report that many people at Patong Beach observed an initial retreat (trough or draw down) of the shoreline of more than 100 m followed a few minutes later by a strong wave (crest). Another less powerful wave arrived another five or ten minutes later. Eyewitness statements place the arrival time of the strong wave between 2 hours and 55 minutes to 3 hours and 5 minutes after the source rupture (09:55am to 10:05am local time). FIXME (Ole): We should add observed arrival time and later relate that to the modelled dynamics. \begin{figure}[ht] \begin{center} \includegraphics[width=8.0cm,keepaspectratio=true]{gauge_locations.jpg} \caption{Location of timeseries extracted from the model output.} \label{fig:gauge_locations} \end{center} \end{figure} Two videos were sourced\footnote{The footage is \subsection{Generation}\label{sec:modelGeneration} FIXME (Ole and Jane): Does this need to be so long? There are various approaches to modelling the expected crustal simulating water flow onto a beach or dry land and around structures such as buildings. \textsc{Anuga} has been validated against %a number of analytical solutions and  FIXME: These have not been published %a number of analytical solutions and %FIXME (Ole): Analytical solutions have not been published. Ask Steve. the wave tank simulation of the 1993 Okushiri Island tsunami~\cite{nielsen05,roberts06}. shown in Figure~\ref{fig:computational_domain}. FIXME (Ole): I know that a nested ursga model was trialled for the end-to-end modelling. However, for the study done here, where models were coupled, I didn't think nested grids were used with URSGA - and certainly not down to 1 arc second. Can someone shed some light on this please? \begin{figure}[ht] \begin{center} efficiently increase the simulation accuracy for the impact area. The grid resolution ranged between a maximum triangle area of $1\times 10^5$ m$^2$ near the western ocean boundary to $20$ m$^2$ in the small regions surrounding the inundation region in Patong Bay. Due to a lack of available data, friction was maximum triangle area of $1\times 10^5$ m$^2$ (corresponding to approximately 440 m between mesh points) near the western ocean boundary to $20$ m$^2$ (corresponding to approximately 6 m between mesh points) in the small regions surrounding the inundation region in Patong Bay. The coarse resolution was chosen to be commensurate with the model output from the \textsc{ursga} model (FIXME - this has to be clearly stated in ursga section) while the latter was chosen to match the available resolution of topographic data and building data in Patong city. Due to a lack of available roughness data, friction was set to a constant throughout the computational domain. For the reference simulation, a Manning's coefficient of 0.01 was chosen to Maximum onshore inundation depth was computed from the model throughout the entire Patong Bay region. throughout the entire Patong Bay region and used to generate a measure of the inundated area. Figure~\ref{fig:inundationcomparison1cm} (left) shows very good agreement between the measured and simulated inundation. However \begin{figure}[ht] \begin{center} \includegraphics[width=8.0cm,keepaspectratio=true]{gauge_locations.jpg} \caption{Location of timeseries extracted from the model output.} \label{fig:gauge_locations} \end{center} \end{figure} \begin{figure}[ht] \begin{center} \includegraphics[width=10.0cm,keepaspectratio=true]{gauge_bay_depth.jpg} \includegraphics[width=10.0cm,keepaspectratio=true]{gauge_bay_speed.jpg} The estimated depths and flow rates given in Section \ref{sec:eyewitness data} are shown together with the modelled depths and flow rates obtained from the model in Table \ref{tab:depth and flow comparisons}. The minimum depths shown in the model are clearly lower than expected and an indication that the tsunami model does not predict flow dynamics accurately at this level of detail. However, this comparison serves to check that depths and speeds predicted are within the range of what is expected. and flow rates obtained from the model in Table \ref{tab:depth and flow comparisons}. The predicted maximum depths and speeds are all of the same order of what was observed. However, unlike the real event, the model estimates complete withdrawal of the water between waves at the chosen locations and shows that the model must be used with caution at this level of detail. Nonetheless, this comparison serves to check that depths and speeds predicted are within the range of what is expected. \begin{figure}[ht] \begin{center} \includegraphics[width=6cm,keepaspectratio=true]{sensitivity_f0_0003_depth} \includegraphics[width=6cm,keepaspectratio=true]{sensitivity_f0_03_depth} %\includegraphics[width=6cm,keepaspectratio=true]{sensitivity_f0_0003_depth} %\includegraphics[width=6cm,keepaspectratio=true]{sensitivity_f0_03_depth} \includegraphics[width=12cm,keepaspectratio=true]{friction_comparison_depth} \caption{Model results for different values of Manning's friction coefficient shown to assess sensitivities. The reference inundation extent for a


text:
# What are some of the major themes in the book "writing on my forehead" by Nafisa Haji

###### Question:

What are some of the major themes in the book "writing on my forehead" by Nafisa Haji?

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text:
<meta http-equiv="refresh" content="1; url=/nojavascript/"> Displacement During Uniform Acceleration ( Real World ) | Physics | CK-12 Foundation
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# Displacement During Uniform Acceleration

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Falling From Space

### Falling From Space

License: CC BY-NC 3.0

In 1960, Capt. Joseph Kittinger stepped off the Excelsior platform that was higher than 31 km above the surface of the earth. Capt. Kittinger fell for approximately 4 minutes and 36 seconds, reaching a maximum speed of 614 miles per hour!

#### Why it matters

• Any object that is dropped near the surface of the earth will accelerate towards the earth. If you were to ignore the effects due to air resistance, every object that is dropped would fall at the same rate. This is known as the acceleration due to gravity and any object that is falling at such a rate is said to be in free fall. By looking at Newton’s 2nd law, we define the weight of an object by the mass of an object multiplied by the acceleration due to gravity on the object.

$\overrightarrow{W}=m \overrightarrow{g}$

• The acceleration due to gravity only acts along the axis which is perpendicular to the surface of the earth. The acceleration due to gravity is another one of the fundamental concepts of classical physics. From objects moving in two dimensions, torque and even work, the concept of free fall and the acceleration due to gravity must be taken into consideration when looking at real world applications.

#### Can You Apply It?

Using the information provided above, answer the following questions.

1. Why is it ok to ignore air resistance for objects under the influence of gravity for small distance but not large distances?
2. Explain why it is important to consider the acceleration due to gravity of an object that is thrown across a room when you are trying to determine the distance the object will travel.
3. If one were to consider air resistance on an object that is falling, would the velocity of the object be faster or slower when compared to an object that is falling without air resistance?


text:
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###### Question details

10-year, $1,000 par value zero-coupon rate bond is to be issued to yield 8 percent. Use Appendix B for an approximate answer but calculate your final answer using the formula and financial calculator methods. a. What should be the initial price of the bond? (Assume annual compounding. Do not round intermediate calculations and round your answer to 2 decimal places.) Bond price$__________

b. If immediately upon issue, interest rates dropped to 7 percent, what would be the value of the zero-coupon rate bond? (Assume annual compounding. Do not round intermediate calculations and round your answer to 2 decimal places.)

Bond price $_________ c. If immediately upon issue, interest rates increased to 10 percent, what would be the value of the zero-coupon rate bond? (Assume annual compounding. Do not round intermediate calculations and round your answer to 2 decimal places.) Bond price$_________


text:
# zbMATH — the first resource for mathematics

Superpotentials and higher order derivations. (English) Zbl 1219.16016
By a superpotential the authors mean an element $$\omega$$ (subject to several conditions) in the path algebra of a quiver. The corresponding derivation-quotient algebra $$D(\omega,k)$$ is the path algebra of the quiver modulo the ideal generated by the derivations of $$\omega$$ with respect to the paths of length $$k$$.
The authors generalize results of M. Dubois-Violette [J. Algebra 317, No. 1, 198-225 (2007; Zbl 1141.17010)] on the one-vertex case to the case when the quiver has several vertices (the base field is replaced by a semisimple algebra). They give a construction of $$D(\omega,k)$$ compatible with Morita equivalence, and show that many interesting algebras arise this way, including McKay correspondence algebras for $$\text{GL}_n$$ for all $$n$$, and $$4$$-dimensional Sklyanin algebras. It is shown that $$N$$-Koszul twisted Calabi-Yau algebras are equivalent to algebras of the form $$A=D(\omega,k)$$ such that the asociated complex yields a bimodule resolution of $$A$$. As an application a representation theoretic computation of the moduli space of Sklyanin algebras of dimension $$4$$ is given.

##### MSC:
 16G20 Representations of quivers and partially ordered sets 16W50 Graded rings and modules (associative rings and algebras) 16S37 Quadratic and Koszul algebras
GAP
Full Text:
##### References:
 [1] Dubois-Violette, M., Graded algebras and multilinear forms, C. R. acad. sci. Paris, ser. I, 341, 719-725, (2005) · Zbl 1105.16020 [2] Dubois-Violette, M., Multilinear forms and graded algebras, J. algebra, 317, 1, 198-225, (2007) · Zbl 1141.17010 [3] Artin, M.; Schelter, W., Graded algebras of global dimension 3, Adv. math., 66, 171-216, (1987) · Zbl 0633.16001 [4] V. Ginzburg, Calabi-Yau algebras, math/0612139 [5] R. Coquereaux, R. Trinchero, On quantum symmetries of ADE graphs, hep-th/0401140 · Zbl 1082.81045 [6] H. Derksen, J. Weyman, A. Zelevinsky, Quivers with potentials and their representations I: Mutations, arXiv:0704.0649, Selecta Math., New Series (in press) · Zbl 1204.16008 [7] P.S. Aspinwall, L.M. Fidkowski, Superpotentials for quiver gauge theories, hep-th/0506041 [8] Bocklandt, R., Graded Calabi Yau algebras of dimension 3, J. pure appl. algebra, 212, 1, 14-32, (2008) · Zbl 1132.16017 [9] Berger, R., Koszulity for nonquadratic algebras, J. algebra, 239, 2, 705-734, (2001) · Zbl 1035.16023 [10] Berger, R.; Taillefer, R., Poincare – birkhoff – witt deformations of calabi – yau algebras, J. noncommut. geom., 1, 241-270, (2007) · Zbl 1161.16022 [11] Crawley-Boevey, W.; Holland, M.P., Noncommutative deformations of Kleinian singularities, Duke math. J., 92, 3, 605-635, (1998) · Zbl 0974.16007 [12] W. Crawley-Boevey, DMV lectures on representations of quivers, preprojective algebras and deformations of quotient singularities, http://www.maths.leeds.ac.uk/ pmtwc/dmvlecs.pdf · Zbl 1299.14032 [13] Stafford, J.T., Regularity of algebras related to the Sklyanin algebra, Trans. AMS, 341, 2, 895-916, (1994) · Zbl 0823.17018 [14] A. Connes, M. Dubois-Violette, Noncommutative finite-dimensional manifolds. II Moduli space and structure of noncommutative 3-spheres, 2005, arXiv:math/0511337 · Zbl 1159.58003 [15] Craw, A.; Maclagan, D.; Thomas, R.R., Moduli of mckay quiver representations. II: grobner basis techniques, J. algebra, 316, 2, 514-535, (2007) · Zbl 1161.14038 [16] Reiten, I.; Van den Bergh, M., Two-dimensional tame and maximal orders of finite representation type, Mem. amer. math. soc., 408, (1989), vii+72 · Zbl 0677.16002 [17] Yoshino, Y., Cohen-Macaulay modules over Cohen-Macaulay rings, (), 177 · Zbl 0745.13003 [18] The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4.4.10, 2007 [19] Berger, R.; Marconnet, N., Koszul and Gorenstein properties for homogeneous algebras, Algebras and representation theory, 9, 67-97, (2006) · Zbl 1125.16017 [20] Berger, R.; Dubois-Violette, M.; Wambst, M., Homogeneous algebras, J. algebra, 261, 172-185, (2003) · Zbl 1061.16034 [21] Sklyanin, E.K., Some algebraic structures connected with the yang – baxter equation, Funktsional. anal. i prilozhen., 16, 4, 22-34, (1982) [22] Sklyanin, E.K., Some algebraic structures connected with the yang – baxter equation. representations of a quantum algebra, Funktsional. anal. i prilozhen., 17, 4, 34-48, (1983) · Zbl 0536.58007 [23] Smith, S.P.; Stafford, J.T., Regularity of the four-dimensional Sklyanin algebra, Compositio math., 83, 3, 259-289, (1992) · Zbl 0758.16001 [24] Odesskii, A.V.; Feigin, B.L., Sklyanin’s elliptic algebras, Funktsional. anal. i prilozhen., 23, 3, 45-54, (1989), 96 · Zbl 0687.17001 [25] S.P. Smith, The four-dimensional Sklyanin algebras, in: Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part I, Antwerp, 1992, vol. 8, 1994, pp. 65-80 · Zbl 0809.16051 [26] Smith, S.P.; Staniszkis, J.M., Irreducible representations of the 4-dimensional Sklyanin algebra at points of infinite order, J. algebra, 160, 57-86, (1993) · Zbl 0809.16052 [27] Connes, A.; Dubois-Violette, M., Noncommutative finite-dimensional manifolds. I spherical manifolds and related examples, Comm. math. phys., 230, 539-579, (2002) · Zbl 1026.58005
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Now showing items 1-20 of 118

• #### A Survey of Tverberg Type Problems

Tverberg's theorem, which celebrates its fiftieth anniversary this year, is a central result in the fields of discrete geometry and topological combinatorics. Proved in 1966, it was a major step in solving questions whether, ...
• #### Abelian Varieties with Small Isogeny Class and Applications to Cryptography

An elliptic curve $E$ over a finite field $\FF_q$ is called isolated if it admits few efficiently computable $\FF_q$-isogenies from $E$ to a non-isomorphic curve. We present a variation on the CM method that constructs ...
• #### Algorithms for convex optimization with applications to data science

Convex optimization is more popular than ever, with extensive applications in statistics, machine learning, and engineering. Nesterov introduced optimal first-order methods for large scale convex optimization in the 1980s, ...
• #### Algorithms in Discrepancy Theory and Lattices

This thesis deals with algorithmic problems in discrepancy theory and lattices, and is based on two projects I worked on while at the University of Washington in Seattle. A brief overview is provided in Chapter 1 (Introduction). ...
• #### Alternate Approaches to the Cup Product and Gerstenhaber Bracket on Hochschild Cohomology

The Hochschild cohomology $HH^\bullet(A)$ of an algebra $A$ is a derived invariant of the algebra which admits both a graded ring structure (called the cup product) and a compatible graded Lie algebra structure (called the ...
• #### Arithmetic Properties of the Derived Category for Calabi-Yau Varieties

This thesis develops a theory of arithmetic Fourier-Mukai transforms in order to obtain results about equivalences between the derived category of Calabi-Yau varieties over non-algebraically closed fields. We obtain answers ...
• #### Aspects of Markov Chains and Particle Systems

The thesis concerns asymptotic behavior of particle systems and the underlying Markov chains used to model various natural phenomena. The objective is to describe and analyze stochastic models involving spatial structure ...
• #### Bin packing, number balancing, and rescaling linear programs

This thesis deals with several important algorithmic questions using techniques from diverse areas including discrepancy theory, machine learning and lattice theory. In Chapter 2, we construct an improved approximation ...
• #### Bispectral Operator Algebras

This dissertation is an amalgamation of various results on the structure of bispectral differential operator algebras, ie. algebras of differential operators with possibly noncommutative coefficients in a variable $x$ ...
• #### Boundary Harnack Principle for Stable-Like Processes

We establish the boundary Harnack principle for certain classes of symmetric stable-like processes in $\mathbf{R}^d$ on arbitrary open sets as well as censored stable-like processes on $\mathcal{C}^{1,1}$-domains. Using ...
• #### Brownian Motion on Spaces with Varying Dimension

In this thesis we introduce and study Brownian motion with or without drift on state spaces with varying dimension. Starting with a concrete such state space that is the plane with an infinite pole on it, we construct a ...
• #### The C*-algebra of a finite T_0 topological space

We are concerned with the following motivating question: how can one extend the classical Gelfand-Naimark theorem to the simplest non-Hausdorff topological spaces? Our model space is a finite $T_0$ topological space, or ...
• #### Classification of connected Hopf algebras up to prime-cube dimension

We classify all connected Hopf algebras up to p^3 dimension over an algebraically closed field of characteristic p>0 under the mild restriction such that in dimension p^3, we only work over odd primes p when the primitive ...
• #### Combinatorial Laguerre Series

(2014-02-24)
We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials. We describe how such a series can be computed by ...
• #### Compact Moduli of Surfaces in Three-Dimensional Projective Space

The main goal of this paper is to construct a compactification of the moduli space of degree $d \ge 5$ hypersurfaces in $\mathbb{P}^3$, i.e. a parameter space whose interior points correspond to (equivalence classes of) ...
• #### Competing Brownian Particles

Consider a finite system of N Brownian particles on the real line. Rank them from bottom to top: the (currently) lowest particle has rank 1, the second lowest has rank 2, etc., up to the top particle, which has rank N. The ...
• #### Computational aspects of modular parametrizations of elliptic curves

\abstract{ We investigate computational problems related to modular parametrizations of elliptic curves defined over $\mathbb{Q}$. We develop algorithms to compute the Mazur Swinnerton-Dyer critical subgroup of elliptic ...
• #### Conformal welding of uniform random trees

A conformally balanced tree is an embedding of a given planar map into the plane with constraints on the harmonic measure of its edges such that the resulting set is unique up to scale and rotation. Bishop (2013) showed ...
• #### Connections Between Lanczos Iteration and Orthogonal Polynomials

(2010-01-10)
In this thesis we examine the connections between orthogonal polynomials and the Lanczos algorithm for tridiagonalizing a Hermitian matrix. The Lanczos algorithm provides an easy way to calculate and to estimate the ...
• #### Convergence and approximation for primal-dual methods in large-scale optimization

(1990)
Large-scale problems in convex optimization often can be reformulated in primal-dual (minimax) representations having special decomposition properties. Approximation of the resulting high-dimensional problems by restriction ...


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# Is the last variable of any implicational tautology also the last variable of one of its premises?

I'll define my terminology via the following BNF grammar for implicational logical expressions:

<clause> ::= "(" <premise> <conclusion> ")"
<conclusion> ::= any variable
<premise> ::= (<clause> "$$\rightarrow$$")*

(In other words, the premise of a clause is any number of sub-clauses, separated from themselves and from the last variable of the clause by the right-associative implication connective, "$$\rightarrow$$".)

Example: $$(((p \rightarrow q) \rightarrow r) \rightarrow (r \rightarrow p) \rightarrow (s) \rightarrow p)$$ is a clause with premise clauses $$((p \rightarrow q) \rightarrow r)$$, $$(r \rightarrow p)$$, and $$(s)$$, and a conclusion of $$p$$.

I wrote a program to enumerate the classical implicational propositional tautologies, and noticed an interesting pattern that is true for (at least) the first 120,957,915 tautologies (i.e. all tautologies having up to 19 characters when converted to polish notation).

The pattern is as follows: every tautology's conclusion is the same variable as the conclusion of at least one of its premise clauses.

Does this pattern extend to every tautology in existence, and is there a proof of it?

• Does "premise clause" mean direct premises or does a premise of a premise, etc., count? For example, if you had $\bot$, then $((A\to\bot)\to\bot)\to A$ would be a counterexample if we were considering direct premises but not if we counted the (trivial) conclusion of a premise of a premise of premise. – Derek Elkins May 4 at 3:49
• @DerekElkins a premise of a premise does not count. Also, the falsum operand "$\bot$" is not allowed by my grammar, only variables and the "$\rightarrow$" connective. – Hans Brende May 4 at 3:55

Yes, and here's the proof: Suppose you have a clause with conclusion $$p$$, but $$p$$ is not the conclusion of any of its premises. Then if $$p$$ is false but every other propositional variable is true, we have that each premise of the clause is true, while the conclusion is false. So the clause is false under this valuation of the propositional variables, and hence the clause is not a tautology.


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Math test

Due to a high failure rate on our math test in the comments section we temporarily switched off the math test script (which was not written by ourselves). No – seriously we would like to excuse ourselves for any comments which got lost. Unfortunately we only just noticed that there were quite a bit of comments lost, however we couldnt recover them. Due to this we will temporarily put comments on moderation. means: your comment doesn’t appear immediately…but will appear after being checked for spam.

you can use LaTeX in your math comments, by using the $shortcode: [latex] E = m c^2$


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After nearly two years of governments changing narratives, being asked to trust ‘the science’ and called a ‘wacko’ by the NSW Health Minister, I thought it time to post a series of questions concerning the state of the world today.

There are few answers in the following, but there are some assumptions, suppositions and a few conclusions.

Firstly, some questions:

1. I thought ‘science’ was a process of questioning, creating hypotheses, testing, evaluating, peer review. Why then have many scientists who don’t agree with the narrative been silenced by their peers and funding providers?
2. Why are doctors in Australia being threatened with sanction and/or deregistration if they even discuss the possibility of adverse reactions to vaccines and alternative treatments with their patients?
3. Why did the TGA approve the use of the mRNA ‘vaccines’ when other therapeutic treatments were available? The TGA had to effectively ban alternative therapies because mRNA vaccines could only be approved for emergency use if no alternative, effective treatments were available!
4. Why has Australia resisted undergoing trials of alternative treatments (Ivermectin and Hydroxychloroquine amongst others) when reports from overseas were indicating dramatic improvements in patient outcomes?
5. Why has the mainstream media (MSM) been strangely silent when it comes to questioning the narrative? Have they been threatened with reduced advertising revenue from big pharma and their associates?
6. Why was there almost nothing in the MSM about truck blockades on our borders in September/October? Why was there no coverage of the assault of a third year nursing student in western Sydney by police when searching for exemption papers? Why was there no coverage of the attempted self-immolation of a woman in Melbourne who felt overwhelmed by the vaccine mandates?
7. Why does the NSW state government continually change the metrics that are reported about cases, hospitalisations, vaccination rates, etc. Why has vaccination status all but disappeared from hospitalisation reporting?
8. Why does our Prime Minister say he is against mandates yet has done everything possible to support the coercion and disenfranchisement of people who don’t wish to be vaccinated? Federal laws always take precedence over state laws so the federal government could easily seek to uphold laws and agreements (vaccination handbook, Nuremberg declaration) that speak against mandated or coerced vaccination.
9. Why have so many Australians accepted vaccinations with no knowledge of the long-term effects of these vaccines? How can they be giving informed consent with unknowns such as the exact makeup of the vaccines? Why were the pharma companies indemnified from all adverse effects?
10. Why was Pfizer granted 75 years to release their vaccine test data results? Surely the FDA should seek to have this information released ASAP?
12. Why does ‘the science’ vacillate between the efficacy of masks, social distancing, PCR testing, lockdowns, various treatments? And if the science is so settled (and it isn’t), why are there such divergent approaches between countries, states, council areas based on the same science advice?
13. Why have many churches been so quick (and indeed eager) to embrace segregation and closures? A close reading of the early verses in Romans 13 would suggest governments should only be followed when they are governing for good, not for a contrived and beat-up medical emergency.
14. Why lock down the healthy and strong? Why not seek to care for the elderly and vulnerable and let the rest of us get on with our lives?
15. Why impose vaccine mandates (essentially coerced) on certain industries or classes of workers when the vaccines are not particularly effective, diminish with effectiveness over time and don’t stop people catching or transmitting the virus?
16. Why would we continue to inject people with a ‘vaccine’ with significant adverse events and without an open discussion of the risks of the disease versus adverse reaction in each age group?
17. Why did the WHO and Merriam-Webster dictionary change the definition of ‘vaccination’ and ‘vaccine’ in the past twelve months?

1. I understand the federal government has spent something like $300 billion on job-keeper and job-saver. A hospital costs something like$1B to build, and I would think another $1B would keep it fully staffed and functioning for 10 years. Why not spend a fraction of the money spent on job-keeper/saver on hospitals to care for the elderly and vulnerable instead? You could spend$50B versus \$300B and have a rock solid health system for the next decade.


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# Source

If your system does not provide python 2.7 (ask your system administrator or check the system documentation to be sure), it can be compiled from source. The following steps are not trivial and can not work out of the box because it will depend on your system and the already installed libraries.

• Compile python:
mkdir -p $HOME/shared/ mkdir -p$HOME/shared/src/

cd $HOME/shared/src/ wget ftp://ftp.cwru.edu/pub/bash/readline-6.2.tar.gz tar -zxvf readline-6.2.tar.gz cd readline-6.2/ ./configure --prefix=$HOME/shared/
make
make install
cd ..

wget --no-check-certificate https://www.sqlite.org/2015/sqlite-autoconf-3080900.tar.gz
tar -zxvf sqlite-autoconf-3080900.tar.gz
cd sqlite-autoconf-3080900
./configure --prefix=$HOME/shared/ make make install cd .. wget ftp://ftp.tcl.tk/pub/tcl/tcl8_5/tcl8.5.17-src.tar.gz tar -zxvf tcl8.5.17-src.tar.gz cd tcl8.5.17/unix ./configure --prefix=$HOME/shared/ --enable-threads
make
make install
cd ../../

tar -zxvf tk8.5.17-src.tar.gz
cd tk8.5.17/unix
./configure --prefix=$HOME/shared/ --enable-threads make make install cd ../../ wget -c https://www.python.org/ftp/python/2.7.9/Python-2.7.9.tgz tar -zxvf Python-2.7.9.tgz cd Python-2.7.9 ./configure --enable-shared --prefix=$HOME/shared/
make
make install
cd ..
• Edit //$HOME/.bashrc// and add: export SANDBOXPATH=$HOME/shared
export PATH=$SANDBOXPATH/bin:$PATH
export PYTHONPATH=$SANDBOXPATH/lib/python2.7/site-packages:$PYTHONPATH
export LD_LIBRARY_PATH=$SANDBOXPATH/lib:$LD_LIBRARY_PATH
export PKG_CONFIG_PATH=$SANDBOXPATH/lib/pkgconfig:$PKG_CONFIG_PATH
unset CPPFLAGS
unset LDFLAGS
#export CPPFLAGS="-I$SANDBOXPATH/include" #export LDFLAGS="-L$SANDBOXPATH/lib"
• Close all the terminals and open a new one. If the user's default shell is bash, in order to load the new configuration you can instead execute:
source \$HOME/.bash\_profile
• Then follow the same instructions listed in the section APT - VirtualEnv (ignoring the apt part).


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# Modular arithmetic in Mathematica? [closed]

I want to implement something like 1 + 1 = 0; i.e., simple modular arithmetic in Mathematica.

This seems like it should be a really easy built-in option, but I can't find any info on it anywhere. I think probably I'm just overlooking something.

Really there should just be an option for Plus. But Options[Plus] gives { }.

Borrowing Rojo's idea from Is it possible to define custom compound assignment operators like ⊕= similar to built-ins +=, *= etc?, we can construct a modular addition operator.

MakeExpression[RowBox[{lhs_, rest : PatternSequence[SubscriptBox["⊕", m_], _] ..}],
StandardForm] :=
MakeExpression[{{lhs, rest}[[1 ;; ;; 2]], m},
StandardForm] /. {x_, HoldComplete[mod_]} :>
HoldComplete@Mod[Total[x /. HoldComplete[a_] :> a], mod]

Examples

Circle-plus can be entered with ESC c+ ESC or \[CirclePlus]. A subscript can be entered with CTRL_ (underscore).

One could do something similar with \[CircleTimes] and modular multiplication.

Actually, I figured it out on my own. Wanted to post the solution in case anyone else ran into the same problem. It's just:

Mod[a + b, x]

so that

Mod[1 + 1, 2]

0

Pretty straightforward actually.


text:
Stephen

# Posts

Sorted by New

Why Quantum?

I have also found Eliezer's series of posts worthwhile, and would like to thank him for writing them. They have improved my thinking on certain topics. I also do not object to his writing on quantum mechanics. First, I don't believe he has been wrong about any major point, and that fact trumps any considerations of his qualifications. Second, to a large extent his QM posts are about thought processes by which one can reach certain conclusions about quantum mechanics. Such cognitive science stuff is squarely within Eliezer's claimed area of expertise. The conclusions themselves are fairly mainstream. (As far as I can tell, among the physicists who have bothered to think about it, very few these days would claim that measurements are somehow special processes that collapse wavefunctions, in contrast to ordinary processes that do not. Whether they describe their beliefs using the term "many worlds" is another matter.)

Why Quantum?

Eli: It seems like it would be much better to use the original name "relative state" rather than "many worlds". The word "many" suggests that they can be counted. However, in standard QM we are usually talking about particles whizzing around in the continuum, which gives us an infinite-dimensional Hilbert space. If we restrict ourselves to Hilbert spaces of finite dimension, for example the states of some spins, then naively counting worlds remains bogus, because the number of "worlds" (i.e. entries of the state vector) with nonzero amplitude depends entirely on choice of basis. I suppose in a finite dimensional Hilbert space we could make a sensible definition of world counting as follows: the answer to how many worlds am I in is the rank of my reduced density matrix. However, this seems far removed from the main point of the "MWI". Furthermore, it appears that the term many worlds does actually lead people astray in practice. In the posts many people keep referring to counting the worlds in which something happens in order to assess probability. This is wrong. The probabilities arise from squaring amplitudes, not from counting. If the probabilities arose from counting then in a finite dimensional Hilbert space, all the probabilities would be rational numbers. Standard QM does not have this property.

Why Quantum?

Mitchell Porter: "There is no relativistic formulation of Many Worlds; you just trust that there is...You also haven't said anything about the one version of Many Worlds which does produce predictions - the version Gell-Mann favors, "consistent histories" - which has a distinctly different flavor to the "waves in configuration space" version."

I think you are mistaken. It seems to me that consistent histories is basically just many worlds from a different point of view. Basically, both are standard QM with no collapse. In consistent histories you look at things from the point of view of path integrals instead of a wave equation. These are just two equivalent mathematical formalisms. Path integrals adapt more easily to the relativistic case, but it doesn't seem to me that the interpretational issues are any different. Secondly, I'm not sure what you mean that consistent histories "produces predictions." I'm pretty sure that consistent histories does not make any quantitative prediction that differs from standard quantum mechanics and quantum field theory.

The Born Probabilities

"If you didn't know squared amplitudes corresponded to probability of experiencing a state, would you still be able to derive "nonunitary operator -> superpowers?""

Scott looks at a specific class of models where you assume that your state is a vector of amplitudes, and then you use a p-norm to get the corresponding probabilities. If you demand that the time evolutions be norm-preserving then you're stuck with permutations. If you allow non-norm-preserving time evolution, then you have to readjust the normalization before calculating the probabilities in order to make them add up to 1. This readjustment of the norm is nonlinear. It results in superpowers. The paper in pdf and other formats is here.

The Born Probabilities

Psy-Kosh:

"Or did I completely and utterly misunderstand what you were trying to say?"

No, you are correctly interpreting me and noticing a gap in the reasoning of my preceeding post. Sorry about that. I re-looked-up Scott's paper to see what he actually said. If, as you propose, you allow invertible but non-norm-preserving time evolutions and just re-adjust the norm afterwards then you get FTL signalling, as well as obscene computational power. The paper is here.

The Born Probabilities

I'm struck by guilt for having spoken of "ratios of amplitudes". It makes the proposal sound more specific and fully worked-out than it is. Let me just replace that phrase in my previous post with the vaguer notion of "relative amplitudes".

The Born Probabilities

Psy-Kosh:

Good example with the Lorentz metric.

Invariance of norm under permutations seems a reasonable assumption for state spaces. On the other hand, I now realize the answer to my question about whether permutation invariance narrows things down to p-norms is no. A simple counterexample is a linear combination of two different p-norms.

I think there might be a good reason to think in terms of norm-preserving maps. Namely, suppose the norms can be anything but the individual amplitudes don't matter, only their ratios do. That is, states are identified not with vectors in the Hilbert space, but rays in the Hilbert space. This is the way von Neumann formulated QM, and it is equivalent to the now more common norm=1 formulation. This also seems to be the formulation Eli was implicitly using in some of his previous posts.

The usual way to formulate QM these days is, rather than ignoring the normalizations of the state vectors, one can instead just decree that the norms must always have a certain value (specifically, 1). Then we can assign meaning to the individual amplitudes rather than only their ratios. It seems likely to me that theories where only the ratios of the "amplitudes" matter, generically can be equivalently formulated as a theory with fixed norm. Thinking that only ratios matter seems a more intuitive starting point.

The Born Probabilities

"I will point out, though, that the question of how consciousness is bound to a particular branch (and thus why the Born rule works like it does) doesn't seem that much different from how consciousness is tied to a particular point in time or to a particular brain when the Spaghetti Monster can see all brains in all times and would have to be given extra information to know that my consciousness seems to be living in this particular brain at this particular time."

Agreed!

More generally, it seems to me that many objections people raise about the foundations of QM apply equally well to classical physics when you really think about it.

However, I think Eli's objection to the Born rule is different. The special weird thing about quantum mechanics as currently understood is that Born's rule seems to suggest that the binding of qualia is a separate rule in fundamental physics.

The Born Probabilities

"Given the Born rule, it seems rather obvious, but the Born rule itself is what is currently appears to be suspiciously out of place. So, if that arises out of something more basic, then why the unitary rule in the first place?"

While not an answer, I know of a relevant comment. Suppose you assume that a theory is linear and preserves some norm. What norm might it be? Before addressing this, let's say what a norm is. In mathematics a norm is defined to be some function on vectors that is only zero for the all zeros vector, and obeys the triangle inequality: the norm of a+b is no more than the norm of a plus the norm of b. The functions satisfying these axioms seem to capture everything that we would intuitively regard as some sort of length or magnitude.

The Euclidian norm is obtained by summing the squares of the absolute values of the vector components, and then taking the square root of the result. The other norms that arise in mathematics are usually of the type where you raise the each of the absolute values of the vector components to some power p, then sum them up, and then take the pth root. The corresponding norm is called the p-norm. (Does somebody know: are all the norms invariant under permutation of the indices p-norms?) Scott Aaronson proved that for any p other than 1 or 2, the only norm-preserving linear transformations are the permutations of the components. If you choose the 1-norm, then the sum of the absolute values of the components are preserved, and the norm preserving transformations correspond to the stochastic matrices. This is essentially probability theory. If you choose the 2-norm then the Euclidean length of the vectors is preserved, and the allowed linear transformations correspond to the unitary matrices. This is essentially quantum mechanics. (Scott always hastens to add that his theorem about p-norms and permutations was probably known by mathematicians for a long time. The new part is the application to foundations of QM.)

The Born Probabilities

Nick: I don't understand the connection to quantum mechanics.

The argument that I commonly see relating quantum mechanics to anthropic reasoning is deeply flawed. Some people seem to think that many worlds means there are many "branches" of the wavefunction and we find ourselves in them with equal probability. In this case, they argue, we should expect to find ourselves in a disorderly universe. However, this is exactly what the Born rule (and experiment!) does not say. Rather, the Born rule says that we are only likely to find ourselves in states with large amplitude. Also, standard quantum mechanics allows the probabilities to fall on a continuum. They aren't arrived at by counting, so the whole concept of counting branches is not standard QM anyway.

(I don't know whether you hold this view, but it is a common misconception that should be addressed at some point anyway.)


text:
The system in which the amplitude of vibration decreases with time and ratio of amplitude of successive oscillation being constant is known as:

This question was previously asked in
UPRVUNL AE ME 2016 Official Paper
View all UPRVUNL AE Papers >
1. Undamped system
2. Underdamped system
3. Critically damped system
4. Overdamped system

Option 2 : Underdamped system
Free
ST 1: Logical Reasoning
1187
20 Questions 20 Marks 20 Mins

Detailed Solution

If the system is underdamped it will swing back and forth with decreasing size of the swing until it comes to a stop. Its amplitude will decrease exponentially.

$$x\left( t \right) = {e^{ - \xi {ω _n}t}}\left( {A{e^{i{ω _d}t}} + B{e^{ - i{ω _d}t}}} \right)$$

Overdamped System: ζ > 1

$$x(t) = A{e^{( - \xi + \sqrt {{\xi ^2} - 1} ){ω _n}t}} + B{e^{( - \xi - \sqrt {{\xi ^2} - 1} ){ω _n}t}}$$

This is the equation of aperiodic motion i.e. the system cannot vibrate due to over-damping. The magnitude of the resultant displacement approaches zero with time.

Underdamped: ζ < 1

$$x\left( t \right) = {e^{ - \xi {ω _n}t}}\left( {A{e^{i{ω _d}t}} + B{e^{ - i{ω _d}t}}} \right)$$

$$x(t) = A{e^{ - \xi {ω _n}t}}\sin ({ω _d} + \phi )$$

This resultant motion is oscillatory with decreasing amplitudes having a frequency of ωd. Ultimately, the motion dies down with time.

Critical Damping: ζ = 1

$$x(t) = (A + Bt){e^{ - {\omega _n}t}}$$

The displacement will be approaching to zero with shortest possible time.

$$x\;\left( t \right) = X\;e_n^{ - \zeta \omega t}sin\left( {{\omega _d}t + \phi } \right)$$

At t = 0

$${x_0} = X\sin \phi$$

At t = Td

$${x_1} = X\;{e^{ - \xi {\omega _n}{T_d}}}\sin \phi$$

At t = 2Td

$${x_1} = X\;{e^{ - \xi {\omega _n}{T_d}}}\sin \phi$$

∴ Amplitude ratio of successive oscillation $$\frac{{{x_0}}}{{{x_1}}} = \frac{{{x_1}}}{{{x_2}}} = \frac{{{x_2}}}{{{x_3}}} = {e^{\xi {\omega _n}{T_d}}}\;$$ which is constant


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## Sum of residuals ~ ε [Design Issues]

Hi both,

» I disagree. In the model ε = 0. However in the fit, the sum of residuals is only asymptotically 0. We shouldn’t speak of bias when we obtain something sufficiently close to the numeric resolution of the machine.

Perhaps I am getting it wrong; the sentence above sounds a little off and I have a feeling you may not be discussing the same thing?

The sum of residuals for a fitted normal linear model will be zero. Not asymptotically. If you sum them in R or any other software you will get zero, be it either like zero-zero or effectively zero, depending on the implementation.
This is because the underlying assumption is that epsilon be distributed with mean zero. If we end up with a non-zero sum, I'd say we have screwed up somewhere.

Try sum(resid(M))

I could be wrong, but...

Best regards,
ElMaestro

"Pass or fail" (D. Potvin et al., 2008)

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# Shoenfield's Absoluteness Theorem

"Shoenfield's theorem shows that if there is a model ZF in which a given $\Pi^1_3$ statement $\phi$ is false, then $\phi$ is also false in the constructible universe of that model."

The problem is that the only proofs I can find seem to use "a model of ZF+DC" or "a model of ZFC".

Is wikipedia right? If so, is there a proof from just "a model of ZF" online or short enough to sketch here?

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## 1 Answer

Wikipedia is correct in that the Shoenfield Absoluteness Theorem holds for plain ZF.

Since the proof of the theorem relies heavily on the absoluteness of well-foundedness, it is tempting to assume DC. However, since the trees that occur in the usual proof of the theorem are canonically well-ordered, DC is not necessary to prove that the well-foundedness of these trees is absolute. For a different approach, see the proof given by Barwise and Fisher in The Shoenfield Absoluteness Lemma. [Israel J. Math. 8 1970, 329-339, MR278934]

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Does it somehow follow easily from Theorem 4a in that paper? I don't quite see it. –  Ricky Demer May 9 '10 at 6:58
It actually follows from Theorem 3a, see Remark 3 which follows that theorem. –  François G. Dorais May 18 '10 at 22:25