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In &quot;Introduction to Mathematical logic&quot; by Elliot Mendelson , page $59$ . I couldn't understand the basic preliminary notations for the rigorous definition of Satisfiability. For example , at a line they said, &quot;Moreover, instead of talking about the $n$-tuples of objects that satisfy a wf that has $n$ free variables, it is much more convenient from a technical standpoint to deal uniformly with denumerable sequences. What we have in mind is that a denumerable sequence $s = (s_1, s_2 , s_3 , ... )$ is to be thought of as satisfying a wf $\beta$ that has ${x_j}_1 , {x_j}_2 ,  ... , {x_j}_n$ as free variables (where $j_1 &lt; j_2 &lt; ... &lt; j_n$ ) if the $n$-tuple $\langle {s_j}_1 , {s_j}_2 ,  ... , {s_j}_n \rangle$ satisfies $\mathscr B$ in the usual sense.&quot; I didn't quite understand why they used ${s_j}_1$ and ${x_j}_1$ instead of $s_1$ and $x_1$ . Maybe there is a critical detail I am missing out? Also , Another passage for the rigorous definition: &quot;Let $M$ be an interpretation of a language $\mathscr L$ and let $D$ be the domain of $M$. Let $\sum$ be the set of all denumerable sequences of elements of $D$. For a wf $\mathscr B$ of $\mathscr L$ , we shall define what it means for a sequences $s= (s_1, s_2, ... )$ in $\mathscr L$ to satisfy $\mathscr B$ in $M$. As a preliminary step, for a given s in $\sum$ we shall define a functions $s^*$ that assigns to each term $t$ of $\mathscr L$ an element $s^{*}(t)$ in $D$.&quot; They did say that for each $s = (s_1, s_2, ... )$ , there is a defined function $s^*$ , but even after further reading the text afterwards , I still did not understand how exactly $s^*$ is defined from $s$ . Can someone tell me where the gap in my understanding here is? I've read in several places that one motivation for category theory was to be able to give precise meaning to statements like, "finite dimensional vector spaces are canonically isomorphic to their double duals; they are isomorphic to their duals as well, but not canonically."  I've finally sat down to work through this, and -  Okay, yes, it is easy to see that the "canonical isomorphism" from $V$ to $V^{**}$ is a functor that has a natural isomorphism (in the sense of category theory) to the identity functor.  Also, I see that there is no way that the functor $V\mapsto V^*$ could have a natural isomorphism to the identity functor, because it is contravariant whereas the identity functor is covariant. My question amounts to:     Is contravariance the whole problem?   To elaborate:   I was initially disappointed by the realization that the definition of natural isomorphism doesn't apply to a pair of functors one of which is covariant and the other contravariant, because I was hoping that the lack of a canonical isomorphism $V\rightarrow V^*$ would feel more like a theorem as opposed to an artifact of the inapplicability of a definition.  Then I tried to create a definition of a natural transformation from a covariant functor $F:\mathscr{A}\rightarrow\mathscr{B}$ to a contravariant functor $G:\mathscr{A}\rightarrow\mathscr{B}$. It seems to me that this definition should be that all objects $A\in\mathscr{A}$ get a morphism $m_A:F(A)\rightarrow G(A)$ such that for all morphisms $f:A\rightarrow A'$ of $\mathscr{A}$, the following diagram (in $\mathscr{B}$) commutes:  $$\require{AMScd}\begin{CD} F(A) @&gt;m_A&gt;&gt; G(A)\\ @VF(f)VV @AAG(f)A\\ F(A') @&gt;&gt;m_{A'}&gt; G(A') \end{CD}$$  This is much more stringent a demand on the $m_A$ than the typical definition of a natural transformation.  Indeed, it is asking that $m_A=G(f)\circ m_{A'}\circ F(f)$, regardless of how $f$ or $A'$ may vary. Taking $\mathscr{A}=\mathscr{B}=\text{f.d.Vec}_k$, $F$ the identity functor and $G$ the dualizing functor, it is clear that this definition can never be satisfied unless $m_V$ is the zero map for all $V\in\text{f.d.Vec}_k$ (because take $f$ to be the zero map). In particular, it cannot be satisfied if $m_V$ is required to be an isomorphism.     Is this the right way to understand (categorically) why there is no natural isomorphism $V\rightarrow V^*$?   As an aside, are there any interesting cases of some kind of analog (the above definition or another) of natural transformations from covariant to contravariant functors?  Note: I have read a number of math.SE answers regarding why $V^*$ is not naturally isomorphic to $V$. None that I have found are addressed to what I'm asking here, which is about how categories make the question and answer precise. ( was closest.) Hence my question here. Assume you are given two objects whose exact locations are unknown, but are distributed according to normal distributions with known parameters (e.g. $a \sim N(m, s)$ and $b \sim N(v, t))$. We can assume these are both bivariate normals, such that the positions are described by a distribution over $(x,y)$ coordinates (i.e. $m$ and $v$ are vectors containing the expected $(x,y)$ coordinates for $a$ and $b$ respectively). We will also assume the objects are independent.  Does anyone know if the distribution of the squared Euclidean distance between these two objects is a known parametric distribution? Or how to derive the PDF / CDF for this function analytically? Mail vs. mail what is the difference What is this business about mail and Mail? [On a CentOS box] Why mail does not work but won't return an error either while Mail works fine with the exact same syntax?  Mail is basically a link to mail, isn't it?  Sorry, this was a mixup, apparently (for historical reasons?!) there is a mail and a Mail  Nowadays it looks like they both point to the same thing, on my machine:  sr-linux&gt; ls -l `which mail` -rwxr-xr-x    1 root     mail        66492 Jun 24  2001 /bin/mail sr-linux&gt; ls -l `which Mail` lrwxrwxrwx    1 root     root           14 Nov 19  2001 /usr/bin/Mail -&gt; ../../bin/mail How do I start skype in the system tray? What reputation do you need on Stack Overflow to receive an invite to Careers? I couldn't find much on the new Lucene-based search, and don't know much about the engine in general, so I'm asking this question.  Does anybody have a description of how search results are ranked?  Does the rank calculation include:   rep of post's author? views? unique referers? PageRank of referers? rank of questions that link to the post? whether a question is closed? vote count? answer count? How does the linux kernel know device major and minor numbers? As far as I know, the kernel detects hardware, adds information to sysfs creates a device in /dev and then generates a udev event. My question is, do device drivers do all of this or it is the kernel itself? If drivers do it, then they would know the device major and minor number to create the file in devtmps.   I also understand (maybe wrongly) that it is udev who calls modprobe to load the drivers that handle a device.   I was told that they do so with the contents of the MODALIAS uevent environment variable, thus loading all modules whose aliases match this string (the correct driver). In short: what are the steps of hardware detection in linux and WHO performs each step: the kernel itself, the driver inside the kernel or udev in userspace. Thanks. How to remove selected commit log entries from a Git repository while keeping their changes? I would like to remove selected commit log entries from a linear commit tree, so that the entries do not show in the commit log.  My commit tree looks something like:  R--A--B--C--D--E--HEAD   I would like to remove the B and C entries so that they do not show in the commit log, but changes from A to D should be preserved. Maybe by introducing a single commit, so that B and C become BC and the tree looks like.  R--A--BC--D--E--HEAD   Or, ideally, after A comes D directly. D' representing changes from A to B, B to C and C to D.  R--A--D'--E--HEAD   Is this possible? if yes, how?  This is a fairly new project so has no branches as of now, hence no merges as well. How do I migrate my Keychain's "Local Items" to a new Mac? Can colliding gravitational waves create a black hole? LWC Navigation: "this[f.NavigationMixin.Navigate] is not a function" When attempting to use the Lightning navigation service in my component's base class (.js), I get the following error when functionally testing in the UI:  this[f.NavigationMixin.Navigate] is not a function  I am calling the NavigationMixin function provided by the lightning/navigation module. I have imported the module: import { NavigationMixin } from 'lightning/navigation';  And am calling the function as below: this[NavigationMixin.Navigate]({     type: 'standard__recordPage',     attributes: {         recordId: this.org.Id,         objectApiName: 'Account',         actionName: 'view'     } }); I try to create a common extent for rasters in QGIS 2.99. Here is my script:  import processing  processing.runalg("script:unifyextentandresolution",                   "/qgis_data/rasters/Image2.tif;                   /qgis_data/rasters/Image1.tif",                  -9999,"/qgis_data/rasters",True)   However, processing.runalg is being used for QGIS 2.99. I need it for QGIS 2.99 Problem in notation: $(\vec{v}\cdot\vec{\nabla})\vec{v}$ versus $L_v(\vec{v})$ Does "it is entirely possible that" require the subjunctive form of the verb following "that"? Math Induction Proof: $(1+\frac1n)^n < n$ So I have to prove:      For each natural number greater than or equal to 3,   $$(1+\frac1n)^n&lt;n$$   My work: Basis step: $n=3$ $$\left(1+\frac13\right)^3&lt;3$$ $$\left(\frac43\right)^3&lt;3$$ $$\left(\frac{64}{27}\right)&lt;3$$ which is true.  Now the inductive step, assume $P(k)=\left(1+\frac1k\right)^k&lt;k$ to be true and prove $P(k+1)=\left(1+\frac1{k+1}\right)^{k+1}&lt;k+1$.   This is where I am stuck because usually you add or multiply by $k+1$ or some similar term. Is it valid to have more than one question mark in a URL? Writing conditional (if/then) statements into ArcGIS Field Calculator using Python Parser? I have gotten stuck on seemingly simple ArcGIS field calculator python expression.  I obtained a great answer to a another similar question a few years ago: . Now I need to populate a new field based on 6 ranges of values (bedrock depth).  My attempt at writing this expression is below:    I just need to populate the empty field "bedrockdp" using the ranges in the toc called brockdepmin.     Pre-Logic Script Code:  def !GRIDCODE! (value):     if value &gt;0 and value &lt; 50:         return "0 - 50 cm from the surface"     else:         return "n/a"   bedrockdep =  GRIDCODE (!bedrockdep!) Extended Euclidean Algorithm: backward vs. forward Remote start up a machine that is in the shut down state? Possible Duplicate:          May I know how can one remote start up a machine that is in a shut down state?