If time is like space, why entropy always increases with time  I should not be able to distinguish whether I am going forwards or backwards in time.   As entropy always increase with time. I can distinguish past from future in a four dimensional space-time.  Edit:  It looks like, it is an unresolved problem  There are some solutions  but nothing generally acceptable
From what I understand, entropy is a concept defined by the experimentalist due to his ignorance of the exact microstate of a system. To say the number of accessible microstates $W$ of the universe is constantly increasing is nothing more than saying 'ignorance begets ignorance'.  I have often encountered the argument of ever increasing entropy for the presence of an inherent time asymmetry, most prominently in the works of Penrose. It just doesn't seem to make sense.  Let us imagine alien beings who are experiencing time in reverse. For them, increasing ignorance is in the direction of our decreasing ignorance. So how can perpetual increase in entropy indicate an 'arrow of time'?  One possible explanation (I thought of) for this argument was the fact that there may never be a mechanism to reduce ignorance. I will pose this as a question:  An observer determines the number of possible microstates of system+observer to be $W_0$ at time $t_0$. After improving his measurements, can he (at a later time) measure $W'$ (where $W_0&gt;W'$) as the number of possible microstates? Assume that all the microstates are equally probable in this case.  Unless the above in false in general, how else can anyone claim that entropy reveals an arrow of time?  EDIT: I am performing this discussion for isolated systems (the universe or it's parts if relevant). The means I propose to reduce entropy in the present, is by redefinition of macroscopic variables and of microscopic models used to count the number of distinguishable micro-states (that obviously yield the same macro-state). This is essentially an argument against the robustness of entropy via redefinition.
Hidden features of Python When is the past perfect exactly needed? Suppose we are in $\mathbb R^n$. What can we say about the sign of $\Delta u(\vec x)$ if u($\vec x$) has a local max/min at $\vec x$? I've tried looking at the reverse of the second partial derivative test and it seems to suggest that $\Delta u(\vec x) \leq 0$ if $\vec x$ is a min and $\Delta u(\vec x) \geq 0$ if $\vec x$ is a max. However, I'm not convinced of this result. Can somebody point me towards a statement (theorem, etc.) involving the above. I appreciate the help. why normalize and the definition of directional derivative I'm not understanding how to solve this problem.  I think the problem lies in the fact that I don't understand why normalize and the definition of directional derivative... It's in Portuguese. The question would be: Determine the directional derivative of f in P in direction of vector u: Why can't the Others go around the Wall at sea? As explained in a  earlier , the Wall does not extend all the way to the sea on the East, and the castle there is essentially undefended anyway. Thus, wildlings -- and Stannis' army -- were able to go around it undetected.  We know that the Others and the wights they raise can't cross through the Wall, and I think we can assume that it's built to stand up to any attacks they make trying to bring it down. But there does not appear to be any reason they can't just go around like anyone else.  Possibly related: in the TV Show, we actually see the attempted evacuation of Hardhome, and the Night King allows Jon Snow and his human army and civilians to flee back to their boats to safety, even after raising a massive army of wights. You could interpret this as merely a psychological ploy, but it may be a hint that the dead can't cross the water either.  Has there been any indication (books or show) that the Others/White Walkers and their army of wights are blocked by open water, or are somehow tied to the land of the Frozen North, or that there's any other obstacle keeping them from just sailing/swimming around the Wall? Wrong math symbols with XeLaTeX and LuaTeX after TexLive update I just updated my TeX live installation (tlmgr update --all) and now I get some wrong symbols in math mode if I use XeLaTeX or LuaTeX. This MWE for example  \documentclass{standalone} \begin{document} $. + , - / $ \end{document}   produces the following output:    I suppose it is a bug in the new Latex2e version from 2016/02/01. Any ideas how to fix this?    EDIT:  Adding \usepackage{fontspec} produces a different output. But it is still not perfect. In Ubuntu 17.10 ubuntu upgraded Nautilus (File Browser) to version 3.26.0. New nautilus is not using file-roller anymore, it has  to an integrated compression mechanism.     This makes it complicated to create encrypted archives and/or different types compression archives that file-roller does provide.  How to make nautilus use file-roller as before?  Related question: When proving $\bar x$ and $S^2$ are independent in my noted it says that "functions of independent quantities are independent ". Can someone tell me how functions of independent quantities are independent happen?  Also let $X_1,X_2,X_3,\dots,X_n$ be a random sample.And suppose we want to estimate parameter $\theta$ using $T(x)$ as an estimator. In this I have a expression as $E\{ [  T(X)-E(T(X))][E(T(x)-\theta)]\}$.  I want to know if I can say that in $[T(X)-E(T(X))]$ since $E(T(X)$ (expected value is a one particular fixed value) and $T(X)$ are independent of one another. Also the expression $[E(T(x)-\theta)]$ is independent of $X$.  Hence the two expressions $[T(X)-E(T(X))]$ and $[E(T(x)-\theta)]$  are independent of one another. So that I can use if $a,b$ are independent $E(ab)=E(a)E(b)$ form I am trying to prove the matrix norm equivalence for norms 1, 2, $\infty$ and Frobenius.  I have managed to solve find the constants for $||.||_{1}$ and  $||.||_{2}$ but I cannot see how to continue if I want to prove the following:  $||A||_{\infty} \leq \sqrt{n}||A||_{2}$   and the ones for Frobenius:  $||A||_{F} \leq \sqrt{n}||A||_{1}$  $||A||_{F} \leq \sqrt{n} ||A||_{2}$  Any help is appreciated. Thank you for your time I need to package my Python application, its dependencies and Python into a single MSI installer. The end result should desirably be:   Python is installed in the standard location the package and its dependencies are installed in a separate directory (possibly site-packages) the installation directory should contain the Python uncompressed and a standalone executable is not required How can I form a word like "quadruple" for any number I want? I'm not sure what these are called, but how can I form a word like "quadruple" for any number I want? Like 5× as much is quintuple, what is 31× as much or 147× as much? I want to know how they are formed so I can make my own.  Similarly, how do I construct the prefixes, such as unicycle, bicycle, tricycle, quadricycle? I figured out 12 is duodeci-, but how can I form any prefix like this? Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$? How can we evaluate this series  $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$  Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how to find this sum?  Isn't there a simple solution which doesn't involve Stirling numbers? If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$ I am working on some past prelim problems and one of them is the following: If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$   Update: Following Jackie Chong's advice, I am attempting this problem with the dominated convergence theorem. Since $f\in L^1(\mathbb{R})$, $f$ can be approximated from below by a sequence of continuous functions $f_n$. Then, I believe I can say that, for all $n$ sufficiently large, $|f_n(x+\frac{1}{n})-f_n(x)|\le 2|f(x)|$. (I'm not sure of this part thought). Then $2|f(x)|$ is integrable and hence a dominating function for $|f_n(x+\frac{1}{n})-f_n(x)|$. So by the dominated convergence theorem we can take the limit inside $\lim_{n\rightarrow \infty}\int|f_n(x+\frac{1}{n})-f(x)|dx=\int \lim_{n\rightarrow \infty}|f_n(x+\frac{1}{n})-f(x)|dx=0$ Is this on the right track?  Another approach that I thought may work, but which I'm less certain of is the following: If $f$ can be approximated by $C^1$ functions (I have no real justification for this, so please correct me if its wrong), then for each $n$ and each $x$ there is some $y(x,n)\in B_{1/n}(x)$ such that $|f(x+\frac{1}{n}-f(x)|=\frac{1}{n}|f'(y(x,n)|$ but from here I don't see how to proceed, because just because $|f|$ is integrable, I can't justify $|f'|$ being integrable, but if there is some way to fix this approach, hints would be appreciated.   Another thing I was thinking was to first approximate $f$ from below by continuous functions $f_n$ then to define $g_{m,n}=|f_n(x+\frac{1}{m})-f_n(x)| \chi_{[-m,m]}$. Then, for each $n$, the function $g_{m,n}$ is continuous on its support, and since it support is compact, it is uniformly continuous there. Can this be used to justify bringing the limit $\lim_{n\rightarrow \infty}$ inside the integral? Then apply the monotone convergence theorem? I know this idea is probably even more sketchy than my previous one, but if it can be fixed please let me know :)  Thanks! If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples? In the Law of Reflection, the angle of incidence is equal to angle of reflection. Why is this true? This is clearly true experimentally, but how does one prove this true mathematically? Today I upgraded from Ubuntu 12.04 to Ubuntu 13.10. I am not able to watch any videos in Amazon Prime Instant Video. I am getting this message:   An error occurred and your player could not be updated.   This is likely because your Flash Player or Browser needs to be updated. This update is required to play back this video. Most simple code to populate JTable from ResultSet I'm doing some web scraping and sites frequently use HTML entities to represent non ascii characters. Does Python have a utility that takes a string with HTML entities and returns a unicode type?  For example:  I get back:  &amp;#x01ce;   which represents an "ǎ" with a tone mark.  In binary, this is represented as the 16 bit 01ce.  I want to convert the html entity into the value  u'\u01ce' I installed Ubuntu 15.04 on the VM and I have downloaded Eclipse Luna. When I try to install it, this error occurs:      An error has occurred.   See the log file   /home/master/.eclipse/1473617060_linux_gtk_x86_64/configuration/1442224462970.log   Can you help me to fix  it? Thanks Do we still need swap partitions on desktops? I have read , but it was asked in the context of servers not desktops or notebooks.  If a good amount of RAM is installed on a notebook, say 8 GB, do we still need a swap space?