I have started to use semantic linefeeds for the purposes of version control, as suggested by e.g. , which I found through the answer:   It seems quite reasonable to put footnotes on their own line, but this inserts an ugly space before the footnote mark. Can this be dealt with in any easy fashion?  MWE:  \documentclass{article}  \begin{document}  This is how footnotes should work.\footnote{My mark is stuck where it should be!}  Attempting semantic linefeeds, the footnotes seem to float away.  \footnote{Can anyone make my mark stick to the previous character?}  \end{document}   This seems to ask something similar, but has not been answered:
Footnote citations in biblatex ignore spacing before the citation command; how can I have the same behavior for all footnotes in a document?  Here's a MWE demonstrating the issue:  \documentclass{article} \usepackage[style=verbose-trad2]{biblatex} \addbibresource{biblatex-examples.bib} \begin{document} Spacing is fine before citations     \autocite{kant:kpv}  but not before footnotes    \footnote{A footnote \autocite{kant:ku}}. \end{document}
If $h : Y \to X$ is a covering map and $Y$ is connected, then the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$. In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition   Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (x = \gamma(0))$, there exists a unique lift $\hat{\gamma} : [0,1] \to Y$ of $\gamma$ with $\hat{\gamma}(0) = y$. Furthermore, for any homotopy $\gamma_t (t \in [0,1])$ of $\gamma$ with $\gamma_t = \gamma(0)$ and $\gamma_t(1) = \gamma(1)$, there exists a unique lift of $\hat{\gamma_t}$ such that $\hat{\gamma_t}$ is the homotopy of $\hat{\gamma}$ with $\hat{\gamma_t}(0) = \hat{\gamma}(0)$ and $\hat{\gamma_t}(1) = \hat{\gamma}(1)$.  The author then follows with; "In the following, we assume that any covering space is connected. By the preceding proposition, the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$."   I am not sure why this results is true. This is my attempt at explaining it to myself. Take two different $x_1, x_2$ and their fibers $h^{-1}(x_1), h^{-1}(x_2)$ such that $y_1 \in h^{-1}(x_1)$ and $y_2 \in h^{-1}(x_2)$. Take two paths $\gamma_1, \gamma_2$ such that $\gamma_1(0) = x_1$ and $\gamma_2(0) = x_2$. Then we get two lifts $\hat{\gamma_1}, \hat{\gamma_2}$ with $\hat{\gamma_1}(0) = y_1$ and $\hat{\gamma_2}(0) = y_2$. Now, since our covering space is connected we can continuously deform $\hat{\gamma_1}$ into $\hat{\gamma_2}$ and conclude that every elements in $h^{-1}(x_1)$ is also in $h^{-1}(x_2)$ and vice versa. I feel that this is wrong but cannot figure out the right way to see this. Any help would be appreciated. Can you switch to the other Unarmored Defense by multiclassing? PHB p164 states:      If you already have the Unarmored Defense feature, you can't gain it again from another class.   So if I start as a Monk and gain the UD(Wis) feature, and then at next level up, take a level in Barbarian, can I switch to the alternate UD(Con)? Or am I stuck with UD(Wis), even if I take 19 levels in Barbarian?  I'm unclear if the rule is stating this or stating that you can't use both at the same time, which I know. How should I act if I discover a user plagiarizing fellow users' posts? What if I find someone plagiarizing external resources without attribution?  See also: How to read such figures with % as "up (by) 120%" and "increased (by) 120%". Do they mean the same? How do I reference the only element in a singleton set? You can turn any element $n$ into a singleton set by adding braces, $\{n\}$. Is there an inverse to this operation, such that if I know the set is a singleton set I can easily reference its element?  For example, say I have the set $A = \{5,6,7,8\}$, and then some process that iteratively eliminates all but one element of $A$, and then I want to see what 10 plus the resulting element is, how could I write that?  The only way I can think of is by referencing the first element by its index, e.g., $10 + A_1$. Is there another way to do it though? In my case the elements of $A$ are already indices, so I would have to index an index, like $10 + x_{A_1}$. Which is not the worst thing in the world, I was just wondering if there is a better way. Sorting an array of an array of objects in PHP by key value Basically I have a setup like the following:  Array (  [0] =&gt; Array ( [0] =&gt; stdClass Object ( [nid] =&gt; 1 [title] =&gt; title1 [uid] =&gt; 1 [parent] =&gt; 0 [weight] =&gt; -15 [name] =&gt; name1 [value] =&gt; 0 )                [1] =&gt; stdClass Object ( [nid] =&gt; 2 [title] =&gt; title2 [uid] =&gt; 1 [parent] =&gt; 0 [weight] =&gt; -7 [name] =&gt; name2 [value] =&gt; 100 )                [2] =&gt; stdClass Object ( [nid] =&gt; 3 [title] =&gt; title3 [uid] =&gt; 2 [parent] =&gt; 0 [weight] =&gt; -1 [name] =&gt; name3 [value] =&gt; 0 )                [3] =&gt; stdClass Object ( [nid] =&gt; 4 [title] =&gt; title4 [uid] =&gt; 2 [parent] =&gt; 0 [weight] =&gt; 1 [name] =&gt; name4 [value] =&gt; 80 )               )   )   What I need is a way to sort all the arrays inside the parent array by the [value] key in the Object. I've been trying for about 2 days now with usort and different methods but I just can't seem to get my head around it. The [value] key will range anywhere from 0 to 100 and I need all of the arrays sorted in decreasing order (IE: 100 down to 0). Login not redirecting back to page where login link was clicked if account created I have notice some sites have the same form for login, register and forgotten password.  Below the form you have 3 buttons    i forgot my password register log me in   What are the pros and cons on using this approach and which version would you go for. I'm quite new to neural networks and currently I'm trying to train a non convolutional neural network on the MNIST data set. I'm observing some behaviour I don't quite understand.   This is the code written with keras as the library:  # the data, split between train and test sets (x_train, y_train), (x_test, y_test) = fashion_mnist.load_data()  y_train = keras.utils.to_categorical(y_train, 10) y_test = keras.utils.to_categorical(y_test, 10) model = Sequential()  early_stopper = EarlyStopping(patience=3)  model.add(Flatten(input_shape=(28,28))) model.add(Dense(128, activation="relu")) model.add(Dense(128, activation="relu")) model.add(Dense(10, activation="softmax")) model.compile(loss="categorical_crossentropy", optimizer="adam", metrics=["accuracy"])  model.fit(x_train, y_train, epochs=20, validation_data=(x_test, y_test), callbacks=[early_stopper])   This gives me a validation_acc of around 20%. The funny thing is, when I change the activation functions to "sigmoid", the loss function to "mean_squared_error" and the optimizer to "sgd" my performance improves to around 85% after 50 epochs.   Having read  I wonder what's the reason for the bad performance of the network I presented in the code. ReLU, cross-entropy and a dynamic optimizer like Adam all seem to improve on the idea of a very vanilla neural network with stochastic gradient optimization, mean squared error as loss and sigmoid activation functions. Yet I get a really bad performance and if I increase the number of nodes in the hidden layers I often get a network that doesn't learn at all.   EDIT: I figured out it has something to do with me not normalizing the input to values between 0 and 1 ... but why is this the problem? How can I prevent iPhoto from opening everytime I plug my iPad/iPhone? How can I prevent iPhoto/Photos/Image Capture from launching everytime I plug in my iPad/iPhone? Why is the content of /run/resolvconf/interface/NetworkManager changed every time I reboot? I have a simple question answer to which would help me more deeply understand the concept of (non)commutative structures. Let's take for example (our teacher's definition of) a ring:  Let $R\neq \emptyset$ be a set, let $\oplus:A\times A \to A$ and $\bullet :A\times A \to A$ be binary operations. Moreover, let $(R, \oplus)$ be a commutative group, $(R, \bullet)$ be a monoid and following property holds for all $a, b, c\in R$: $$a\bullet(b\oplus c) = (a\bullet b)\oplus(a \bullet c)$$ $$(b\oplus c)\bullet a = (b\bullet a)\oplus(c \bullet a)$$ Then ordered triple $\mathbf R = (R, \oplus, \bullet \mathbf)$ is called a (unitary) ring.  Moreover, we call ring $\mathbf R$ commutative iff $(R, \bullet)$ is a commutative monoid. Commutativity of a ring is always a matter of its multiplicative operation because the additive operation is always assumed to be commutative.  Could anyone explain me the bold part? Why do we even in non-commutative rings (and fields etc.) assume the addition to be always commutative? Is there some serious reason? Would it make any trouble? Or studying of structures with non-commutative addition just doesn't give us anything new so we can take addition as commutative simply because of our comfort? How well will WCF scale to a large number of client users? Does anyone have any experience with how well web services build with Microsoft's WCF will scale to a large number of users?  The level I'm thinking of is in the region of 1000+ client users connecting to a collection of WCF services providing the business logic for our application, and these talking to a database - similar to a traditional 3-tier architecture.   Are there any particular gotchas that have slowed down performance, or any design lessons learnt that have enabled this level of scalability? Where did they go?   Where do they went?   Can you please advise which of above forms is correct?  I'm pretty sure that first one is fine but wondering about second one.   Can I use a second form as well or it's incorrect? $\mathbb{Q}^{\times}$ is the group of rational number without $0$ under multiplication, and $\mathbb{Z}^{n}$ is the free abelian group of rank $n$. Show that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$. I tried to assume that there is an isomorphism and to get a contradiction.. didn't succeed Creating point features with exact coordinates in QGIS What is a "year" in Westeros? Android App initial release date When you are viewing an app in Google Play store, it displays the following : Updated, Size, Installs, Current Version, Requires Android, and Rating.  I am trying to do some research and figure out the initial release / publish dates of different apps. Is there a way to get this information from Google Play, or perhaps a website with this information? Why does QgsSpatialIndex.nearestNeighbor(point, 1) sometimes return 2 lines? What is the difference between "likelihood" and "probability"?