I have an exercise that involves this set: $2^\mathbb{N}$, but I don't know what set it is. I know that the elements of the set are functions whose domain is $\mathbb{N}$, but no idea which type of functions they are.  Thanks in advance.
I am not very well versed on set theory or syntax, but I thought I knew the basics.  However, in a book about databases I am reading now, the author uses $2^x$ to signify "a set of $x$."  For example, $2^{\text{dogs}}$ is a set of $\text {dogs}$, etc.    The author never really explained this or why he does it, I just picked up the meaning from context.    I am not sure why the exponent operator is used, nor am I sure what the number $2$ has to do with it.  The sets being represented are NOT powers of $2$ (in size)... they come in all sizes.  Is this a valid notation?  I have not seen it anywhere before...
Inspecting standard container (std::map) contents with gdb Supposing to have something like this:  #include &lt;map&gt; int main(){     std::map&lt;int,int&gt; m;     m[1] = 2;     m[2] = 4;     return 0; }   I would like to be able to inspect the contents of the map running the program from gdb. If I try using the subscript operator I get:    (gdb) p m[1] Attempt to take address of value not located in memory.   Using the find method does not yield better results:  (gdb) p m.find(1) Cannot evaluate function -- may be inlined   Is there a way to accomplish this? Reading order for Dragonlance? I am currently using 19.04 and I got "No new release found for 19.10" Organic yet accurate modeling with the golden spiral Foreword  First off, let me say that I am profoundly amazed at the orderliness in nature. If you haven't heard about the Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, 13, ...), you should check it out.  This mathematical concept of the Fibonacci numbers can be shaped into a set of squares, where the lengths of the sides of the squares follow the Fibonacci sequence:     This is sometimes called a golden ratio. It can be formed into a (golden) spiral. (a.k.a. logarithmic spiral):      Application  I am looking for ways on how to utilize this concept in Blender. Primarily in organic modeling. The golden spiral is found all throughout nature. From flower seed heads, sea shells, and ram horns to hurricanes and spiral galaxies.  My question is: how can I model based off of one or multiple golden spirals? Specifically, I'm interested in modeling a sunflower head like the one in the following image. I'm mostly interested in the point of origin of the individual seeds (not worried about the shape). Does Blender have any features related to this? I am trying to print a List of Appendices (LoA) for my thesis, separate from the Table of Contents (ToC).   I wish for my ToC to look like this, with depth up to subsections (2?):  CONTENTS  CHAPTER   I. Introduction......1    1.1 Section title...1     1.1.1 Subsection...1  Appendices ..... 2   and for an LoA on a new page, with depth 0, thus:  APPENDICES  APPENDIX A. Proof of Theorem 1...1 B. Proof of Theorem 2...1   I am trying to achieve this using the report class, in a way that allows me to type appendices using the hierarchy that I obtain from the chapter, section commands. This is as opposed to defining a custom myappendix command into which I insert my appendix content. (I realize that I only need the titles from the appendix chapters for the LoA, but I'd like for the appendix content to have such structure.)   For instance, this approach would solve my problem:   \chapter{Introduction} \section{Section title} \subsection{Subsection} % \begin{appendices} is also acceptable \appendix   % Some settings are changed now \chapter{Proof of Theorem 1} \chapter{Proof of Theorem 2} % \end{appendices} if environment is begun   whereas the following approach would not:  \chapter{Introduction} \section{Section title} \subsection{Subsection} \myappendix{Proof of Theorem 1} \myappendix{Proof of Theorem 2}   I realize several questions have been asked regarding this before, but none of them quite solved my problem. Notably,  defines a custom appendix command;  defines two custom commands to maintain some hierarchy, but I will have to do more work to handle the inclusion of another layer (subsection) into the appendix.    is the closest to what I want: it does give me a separate LoA, but it also adds appendices to ToC and I can't quite figure out how to suppress the entires in ToC. The moment I use   \addcontentsline{toc}{\setcounter{tocdepth}{-1}}   the line   \xpatchcmd{\@chapter}{\addcontentsline{toc}}{\addcontentsline{toa}}{}{}%   fails because I am not adding anything to the ToC anymore.   My approach: Since I was using titlesec and tocloft packages anyway, I tried to do the following:   Stop ToC once appendix command is issued Start a new counter for appendices using tocloft Using the &lt;after-code&gt; option of titlesec, add contents line after every chapter.    My MWE   \documentclass[a4paper]{report} \usepackage{lipsum} % for arbitrary text  \usepackage{tocloft,etoolbox,titlesec} \newcommand{\listappendixname}{List of Appendices} \newlistof{appendix}{apc}{\listappendixname}  \makeatletter \g@addto@macro{\appendix}{     \clearpage     \addcontentsline{toc}{chapter}{Appendices}     \addtocontents{toc}{\protect\setcounter{tocdepth}{-1}}     \titleformat      {\chapter}      [display]     {\huge\bfseries\filcenter}      {\Large \MakeUppercase{\@chapapp} \thechapter}      {1.5ex}{}     {\refstepcounter{appendix} \addcontentsline{apc}{appendix}{\protect{\@chapapp \thechapter}}} } \makeatother  \title{Title} \author{Author}  \begin{document} \maketitle \tableofcontents \clearpage \listofappendix  \chapter{Introduction} \lipsum[1] \section{Section} \lipsum[2] \subsection{Subsection} \lipsum[3]  \appendix  \chapter{Proof of Theorem One} \lipsum[4] \chapter{Proof of Theorem Two} \lipsum[5]  \end{document}   produces the following LoA:  List of Appendices  Appendix...... 5   It doesn't get the Appendix title or number, and fails after the first one.  Could you please help me obtain a separate LoA in a way that I can still use LaTeX's chapter, section, ... hierarchy? (I am okay with using different packages, the book class, or any approach other than mine.) How can I press "Insert" on a keyboard without that key? Various \textcolor within lstlisting? Why is \[ ... \] preferable to $$ ... $$? I've heard that in the study of finite fields, and other concepts related to finite fields, mathematicians have found a sort of gap: there are various results and things that seem like they correspond to a field with one element, $\mathbf{F}_1$, even though there is no such field. Of course, there is an object which seems similar to a field with one element: the zero ring, $\mathbf{Z}_1$, defined as having one element, $0$, obeying the equations $$0 + 0 = 0 - 0 = 0 \cdot 0 = 1 = 0.$$ The zero ring seems to behave like a field in every respect, except for the fact that it fails to satisfy $0 \ne 1$. (But the importance of the axiom $0 \ne 1$ isn't clear to me.) However, in any case, there seems to be consensus that the zero ring is not the field with one element. People say that the zero ring &quot;does not behave like a finite field&quot; () or that it &quot;does not have the features that mathematicians need&quot; (). I'm not familiar enough with algebraic geometry to understand the properties that $\mathbf{F}_1$ is expected to have. Is there an elementary, undergraduate-level explanation of  why it seems like a field with one element &quot;ought to exist&quot; in the first place, what properties we expect it to have and why, and how the zero ring fails to satisfy these properties?  Or do I need to study algebraic geometry if I want to have an understanding of any of this? (A side question: do we have a good answer to the question of whether the single element of $\mathbf{F}_1$ ought to be $0$, $1$, both, or neither?) What's the standard "roadmap" to learning quantum physics? I'm really interested in quantum physics and would like to learn more. However, I don't know where to start and in what order I should learn things. So, ideally I'm looking for some sort of roadmap of what to learn. What physics topics do I need to know to start learning about quantum mechanics? (In addition to the mathematical topics mentioned at )  My current knowledge is mostly popular science stuff, like tv shows on Discovery Science and National Geographic Channel. So I have a basic understanding of some basic principals. There's also a recent lecture from Brian Cox that I have watched which gave a bit more in-depth information. Although we can create or update fieldsets using salesforce UI but Is there any way to insert or update FieldSets using apex code (using metadata or API calls) ? Is Angular Momentum truly fundamental? This may seem like a slightly trite question, but it is one that has long intrigued me.  Since I formally learned classical (Newtonian) mechanics, it has often struck me that angular momentum (and generally rotational dynamics) can be fully derived from normal (linear) momentum and dynamics. Simply by considering circular motion of a point mass and introducing new quantities, it seems one can describe and explain angular momentum fully without any new postulates. In this sense, I am lead to believe only ordinary momentum and dynamics are fundamental to mechanics, with rotational stuff effectively being a corollary.  Then at a later point I learned quantum mechanics. Alright, so orbital angular momentum does not really disturb my picture of the origin/fundamentality, but when we consider the concept of spin, this introduces a problem in this proposed (philosophical) understanding. Spin is apparently intrinsic angular momentum; that is, it applies to a point particle. Something can possess angular momentum that is not actually moving/rotating - a concept that does not exist in classical mechanics! Does this imply that angular momentum is in fact a fundamental quantity, intrinsic to the universe in some sense?  It somewhat bothers me that that fundamental particles such as electrons and quarks can possess their own angular momentum (spin), when otherwise angular momentum/rotational dynamics would fall out quite naturally from normal (linear) mechanics. There are of course some fringe theories that propose that even these so-called fundamental particles are composite, but at the moment physicists widely accept the concept of intrinsic angular momentum. In any case, can this dilemma be resolved, or do we simply have to extend our framework of fundamental quantities? Ubuntu 14.04 System Logging Configuring dpkg stops processing when calling an HTTP request to Adobe Flash Player "Battery Health" in macOS Catalina 10.15.5? I have just installed the macOS Catalina 10.15.5 update on my 11" Mid 2013 MacBook Air. The update is working fine.  The MacBook Air specifications on the Wikipedia page says that this model has a lithium-ion battery, 'tho I haven't unscrewed the case to look. But the new Battery Health controls in System Preferences → Energy Saver don't appear.  Not a lithium-ion battery? Or perhaps other differences in this model prevent Battery Health from showing up? Something else? How many non congruent trapeziums are possible with the side lengths from {1,3,4,5,6} I need to find number of mutually non  congruent trapeziums with side lengths from {1,3,4,5,6}.   I know for a trapezium with a,c as non parallel sides the condition for my question is that  $$|a-c|&lt;b+d&lt;a+c$$  But i cant apply it to count the cases. Can someone help in how to get the number of cases? Green is the most common circuit-board color because it has become an industry standard.  What I'm interested in is How the traditional "PCB Green" become a standard in the first place?   Were there any interesting historical reasons for the initial choice, or was it just a product of what one particularly successful company was doing that became the de-facto standard? Under which assumptions a regression can be interpreted causally? First, don't panic. Yes, there are many similar question on this site. But I believe none gives a conclusive answer to the question below. Please bear with me.  Consider a data generation process $\text{D}_X(x_1, ... , x_n|\theta)$, where $\text{D}_X(\cdot)$ is a joint density function, with $n$ variables and parameter set $\theta$. It is well known that a regression of the form $x_n = f(x_1, ... , x_{n-1}|\theta)$ is estimating a conditional mean of the joint distribution, namely, $\text{E}(x_n|x_1,...,x_{n-1})$. In the specific case of a linear regression, we have something like $$ x_n = \theta_0 + \theta_1 x_1 + ... + \theta_{n-1}x_{n-1} + \epsilon $$ The question is: under which assumptions of the DGP $\text{D}_X(\cdot)$ can we infer the regression (linear or not) represents a causal relationship? It is well known that experimental data does allow for such interpretation. For what I can read elsewhere, it seems the condition required on the DGP is exogeneity: $$ \text{E}(x_1, ... x_{n-1}|\epsilon) = 0$$ The nature of randomisation involved in experimental data ensures the above is the case. The story goes then to argue why observational data normally fails in achieving such condition, reasons including omitted variable bias, reverse causality, self-selection, measurement errors, and so on. I am however uncertain about this condition. It seems too weak to encompass all potential arguments against regression implying causality. Hence my question above. UPDATE: I am not assuming any causal structure within my DGP. I am assuming that the DGP is complete, in the sense that there must be some causality (an ontological position it could well be debated), and all relevant variables are included. The key is to identify the set of assumptions which ensure me causality goes from certain variables to the other, without assuming from the outset such direction of causality.  Many similar posts on the site spend time mentioning why correlation does not imply causation, without providing hard arguments about when it does. That is the case, for instance, of  very popular post. Additionally, in the  on the site about the topic, the accepted answer gives the very vague answer:  Expose all hidden variables and you have causation.  I do not know how to translate that to my question above. Neither is the second most upvoted answer. And so on. That is why I believe this post does not have an answer elsewhere. Regionalism or just bad English? I've encountered a particular type of writing occasionally and it being, derp, in writing, it's hard to tell whether there's an accent behind it. The English used seems to me to be simply incorrect, but I can't help but feel that the following features are part of a dialect or a regional style of English. One of the most glaring (and grating) features is the substitution of &quot;saw&quot; with &quot;seen&quot;.  He seen him at the shop the other day. I seen that you were looking for... etc.  The use of &quot;away (to)&quot; is also peculiar, if not incorrect.  He was away to open the door when... He was away to get what he had asked for. etc.  I haven't come across these forms very often, and even then it was accompanied by otherwise mostly correct English, which is what makes me wonder what the hell. This is more intellectual curiosity than anything else. How to reset the Xorg / xserver? Xserver keeps crashing on my general user account both in Unity 3D and 2D. It doesn't do it on the guest account.  That's why I want to reset / rollback everything xserver related, hoping this is going to fix it.  Most answers on this involve sudo dpkg-reconfigure xserver-xorg. However, this does nothing for me (generates no output) and I read in other forums, that it's deprecated.  So, what's the current way to reset all xserver / xorg options; what's the alternative to sudo dpkg-reconfigure xserver-xorg? X crashes during work, not before login or so. And the effect of a crash is, that I get thrown out of the session, not a freeze or machine crash.