In &quot;Introduction to Mathematical logic&quot; by Elliot Mendelson , page $50$ and $56$ . I couldn't understand why they sometimes use double subscripts . For example , in page $50$ :  &quot;We shall often indicate that some of the variables $x_{i_1},..., x_{i_k}$  are free variables in a wf $\mathscr B$ by writing $\mathscr B$ as $\mathscr B(x_{i_1},..., x_{i_k} )$ This does not mean that $\mathscr B$ contains these variables as free variables, nor does it mean that $\mathscr B$ does not contain other free variables. This notation is convenient because we can then agree to write as $\mathscr B(t_1,..., t_k)$ the result of substituting in $\mathscr B$ the terms $t_1,..., t_k$ for all free occurrences (if any) of $x_{i_1} , ... , x_{i_k}$ respectively.&quot;  Here , I didn't understand why they used double subscript notation in $x_{i_1},..., x_{i_k}$ instead of single subscript notation like in $\mathscr B(t_1,..., t_k)$ . In page $56$ :  &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 $\mathscr B$  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;  Again , they used Double subscript notation in ${x_j}_1 , {x_j}_2 ,  ... , {x_j}_n$ but also there is this condition $j_1 &lt; j_2 &lt; ... &lt; j_n$ . This just adds to the confusion to me because I don't understand at all why do they use Double subscript in the first place (To the point that it seems completely unnecessary. Now , I am completely new to subjects like first order logic and model theory , so I might struggle sometimes with the significance of some notation which actually has a good technical reason for using.This might be the case for this one as well.Can someone tell me the deep technical reason why we use double subscript sometimes below variables? (It could be possible that there are multiple reasons)
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 have an ArrayList of objects in Java.  The objects have four fields, two of which I'd use to consider the object equal to another.  I'm looking for the most efficient way, given those two fields, to see if the array contains that object.  The wrench is that these classes are generated based on XSD objects, so I can't modify the classes themselves to overwrite the .equals.    Is there any better way than just looping through and manually comparing the two fields for each object and then breaking when found?  That just seems so messy, looking for a better way.  Edit: the ArrayList comes from a SOAP response that is unmarshalled into objects. Selecting the right crank length What is the best way to determine the correct crank length for my height and more specifically my leg length? Performing reverse clip in ArcMap? Is there a way to perform a 'reverse' of the clip function in ArcMap?  I am not exactly sure how to explain this so here is a diagram: How to detect if any specific drive is a hard drive? In C# how do you detect is a specific drive is a Hard Drive, Network Drive, CDRom, or floppy? Subtracting even and odd binomial coefficients? How do you get to the Secret Room in the Tavern? My friend and I noticed that there seems to be an inaccessible room in the Tavern with something behind it. We saw that there was a Treasure Chest seemingly floating somewhere off the map.     In addition, if you walk up to the second floor of the Tavern and position your camera through the wall just right, you can highlight something called the Defender's Forge.   Confirmation of the speculation was confirmed in the notes for the 7.07 patch, which had a note that said the following:     Moved bonus-item crates back into the main section of the Halloween Tavern, out from the Secret Room.   What is this Secret Room? How do I get there? Do I need to beat the campaign first or something else? Is there anything beyond subtle hints that Jack becomes the Face of Boe? A rubber viewfinder cover is connected to my camera strap.  What's it for, and when do I use it?  What happens to a photo if I don't use it? How do I break out of a loop in Perl? I'm trying to use a break statement in a for loop, but since I'm also using strict subs in my Perl code, I'm getting an error saying:     Bareword "break" not allowed while   "strict subs" in use at ./final.pl   line 154.   Is there a workaround for this (besides disabling strict subs)?  My code is formatted as follows:  for my $entry (@array){     if ($string eq "text"){          break;     } } How to illuminate the darkness inside glass objects in Cycles? My model has a clear plastic (or glass) dome with objects inside.  The objects appear dark.  I've tried the glass shader with color 100% RGB, with very small IOR, and using just a refraction shader.    I know Cycles has problems with caustics, but I just want broad illumination to fall clean through the glass.  The light-to-column path can ignore refraction, but is blocked or tremendously attenuated. The camera-column path needs refraction, and seems to work fine anyway.  This demo image offers all optical paths of interest:   light upon object, seen directly light upon object, seen through glass (looks okay) light through glass onto object, seen directly. This is the shadow on the lit side of the column just below the glass tubing.  I expect full illumination, but there is shadow.  There is some light, but it's diffuse light from the floor. light through glass onto object, seen through glass. This     I did at one point try non-progressive rendering, and find extremely coarse light where there shouldn't be shadow, but this is not a practical way to make nice images. I've even tried cranking up certain sampling settings to many thousands, but still get bad noise though less, at the cost of long render times. I  would like to finish my creations sometime before "Star Trek, the Generation After the Next" becomes real life, if you know what I mean.  The next image is an example. I think Transmission in Sampling was 5000, but not sure.    I'm using Cycles in Blender 2.67 on a quad core Linux machine, 16GB RAM. Machine limits have never been a problem for 3D rendering.  Possibly, I screwed up some setting when I made my startup file, and just need to fix it. OTOH maybe this is just a limitation of path tracing, or of the way Cycles does it. If that's the case, what to real-world Blender artists do when they need to produce images on a budget and schedule?  What is the practical real-world fix? Run a command before & after ubuntu apt upgrade & unattended-upgrades I just installed Ubuntu 14.04.3 LTS after installing windows 10 in my new VN7-571G and the wireless is not working. It never had: even during the installation it was not recognized. I run a script to gather the data , which is . lspci -knn | grep Net -A2 output 03:00.0 Network controller [0280]: Qualcomm Atheros Device [168c:003e] (rev 20)         Subsystem: Lite-On Communications Inc Device [11ad:0804]  The OS behaves like it has no wifi card. I do not see wifi settings in network settings I'm not a linux power user, so I do not know what to do to solve the issue. What should I do to make it work? Is there a word for happiness made possible by a tragic situation? This is the profile of the object I'd like to wrap...    Here is the object's profile after being wrapped...    Is it possible to prevent the distortion? The following post here:  did not affect my object. It seemed very promising, maybe I'm missing something?  Note: I have applied the scale to both my object and curve. How to implement "select all" check box in HTML? What are the practical applications of the Taylor Series? I started learning about the  in my calculus class, and although I understand the material well enough, I'm not really sure what actual applications there are for the series.  Question: What are the practical applications of the Taylor Series? Whether it's in a mathematical context, or in real world examples. I would have expected the correlation coefficient to be the same as a regression slope (beta), however having just compared the two, they are different. How do they differ - what different information do they give? I have a dictionary: keys are strings, values are integers.  Example:   stats = {'a':1000, 'b':3000, 'c': 100}   I'd like to get 'b' as an answer, since it's the key with a higher value.  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[    3.336456] iwlwifi 0000:02:00.0: base HW address: 50:eb:71:d7:0b:40 [    3.353553] iwlwifi 0000:02:00.0 wlp2s0: renamed from wlan0 [    4.457920] iwlwifi 0000:02:00.0: Applying debug destination EXTERNAL_DRAM [    4.621843] iwlwifi 0000:02:00.0: FW already configured (0) - re-configuring [    4.629999] iwlwifi 0000:02:00.0: BIOS contains WGDS but no WRDS   Someone have idea why this happened?