Use of tabular format to insert images
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Nicely force 66 characters per line Sigma algebra and algebra difference An algebra is a collection of subsets closed under finite unions and intersections. A sigma algebra is a collection closed under countable unions and intersections.   Whats the difference between finite and countable unions and intersections? Does "countable" mean it implies there can be infinitely many unions and intersections?    Secondly, I was reading a definition  For an algebra on a set: By De Morgan's law, $A \cap B = (A^c \cup B^c)^c$, thus an algebra is a collection of subsets closed under finite unions and intersections.  What law are they using here to get $A \cap B = (A^c \cup B^c)^c$? I thought de morgan's law was $(A\cap B)^c = A^c \cup B^c$?  Finally, what exactly do they mean by "closed under finite unions and intersections? Linear regression prediction interval The Safari app has disappeared from my iPhone? I can't find the safari icon on my iphone 4?  I have tried resetting home screen layout and that doesn't work.  My restrictions button is 'on' and it asks for a password for this button.  I don't have a password.  Can you help? Thanks, I want to do:  int a = 255;  cout &lt;&lt; a;   and have it show FF in the output, how would I do this? How can change in cost function be positive? In  it says     To make gradient descent work correctly, we need to choose the learning rate η to be small enough that Equation (9) is a good approximation. If we don't, we might end up with $\Delta C&gt;0$, which obviously would not be good! At the same time, we don't want $\eta$ to be too small, since that will make the changes $\Delta v$ tiny, and thus the gradient descent algorithm will work very slowly. In practical implementations, $\eta$ is often varied so that Equation (9) remains a good approximation, but the algorithm isn't too slow. We'll see later how this works.   But just a few paragraphs before we established that $\Delta C\approx−\eta\nabla C⋅\nabla C=−\eta\|\nabla C\|^2$ is obviously always negative (for positive $\eta$). So how can $\Delta C$ be positive if we don't choose a small enough learning rate? What is meant there? Strange behavior of $\lim_{x\to0}\frac{\sin\left(x\sin\left(\frac1x\right)\right)}{x\sin\left(\frac1x\right)}$ Alright, scratch everything below the line. Let me present one cohesive question not marred by repeated edits.  The limit $\lim_{x\to a}f(x)=L$ exists iff for every $\epsilon&gt;0$ there is a $\delta&gt;0$ such that $|f(x)-L|&lt;\epsilon$ when $0&lt;|x-a|&lt;\delta$.  Thus, $\lim_{x\to0}\sin\left(\frac1x\right)$ does not exist because, being that it oscillates infinitely near $0$, there is no $\epsilon,\delta$.  On the other hand, with the limit$$\lim_{x\to0}\frac{\sin\left(x\sin\left(\frac1x\right)\right)}{x\sin\left(\frac1x\right)}\\\lim_{x\to0}x\sin\left(\frac1x\right)=0\\y=x\sin\left(\frac1x\right)\\\lim_{y\to0}\frac{\sin y}{y}=1\\\lim_{x\to0}\frac{\sin\left(x\sin\left(\frac1x\right)\right)}{x\sin\left(\frac1x\right)}=1$$  this proof can be shown. However, since $\sin\left(\frac1x\right)$ oscillates infinitely, by the same definition of limit we used to show the above, the limit does not exist. How do I resolve this discrepancy? Adding CSS Reference on Site Page What is Chromatic Aberration? What is the advantage of using sudo apt-get autoremove over a cleaner app? Is particle superposition reflected in the particle's gravitational footprint? Is it possible to overload the null-coalescing operator for a class in C#?    Say for example I want to return a default value if an instance is null and return the instance if it's not. The code would look like something like this:     return instance ?? new MyClass("Default");     But what if I would like to use the null-coalescing operator to also check if the MyClass.MyValue is set? My text introduces multi-quibt quantum states with the example of a state that can be &quot;factored&quot; into two (non-entangled) substates. It then goes on to suggest that it should be obvious1 that the joint state of two (non-entangled) substates should be the tensor product of the substates: that is, for example, that given a first qubit $$\left|a\right\rangle = \alpha_1\left|0\right\rangle+\alpha_2\left|1\right\rangle$$ and a second qubit $$\left|b\right\rangle = \beta_1\left|0\right\rangle+\beta_2\left|1\right\rangle$$ any non-entangled joint two-qubit state of $\left|a\right\rangle$ and $\left|b\right\rangle$ will be $$\left|a\right\rangle\otimes\left|b\right\rangle = \alpha_1 \beta_1\left|00\right\rangle+\alpha_1\beta_2\left|01\right\rangle+\alpha_2\beta_1\left|10\right\rangle+\alpha_2\beta_2\left|11\right\rangle$$ but it isn't clear to me why this should be the case. It seems to me there is some implicit understanding or interpretation of the coefficients $\alpha_i$ and $\beta_i$ that is used to arrive at this conclusion. It's clear enough why this should be true in a classical case, where the coefficients represent (where normalized, relative) abundance, so that the result follows from simple combinatorics. But what accounts for the assertion that this is true for a quantum system, in which (at least in my text, up to this point) coefficients only have this correspondence by analogy (and a perplexing analogy at that, since they can be complex and negative)? Should it be obvious that independent quantum states are composed by taking the tensor product, or is some additional observation or definition (e.g. of the nature of the coefficients of quantum states) required?  1:  (bottom of p. 18) &quot;so the state of the two qubits must be the product&quot; (emphasis added). How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case. $\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: . Is there a law that you can add or multiply to both sides of an equation? How does Ethernet know how long a frame is? How does MySQL store data $ \int_1^2\int_1^2 \int_1^2 \int_1^2 \frac{x_1+x_2+x_3-x_4}{x_1+x_2+x_3+x_4}dx_1dx_2dx_3dx_4 $ What is the Mediator in this Feynman Diagram $q\bar{q} \rightarrow gg$? Let's consider the t-channel version of the process $q\bar{q} \rightarrow gg$. What is the mediator in this case, a $q$ or a $\bar{q}$? Naively, I would have thought a $q$ (as to have one fermion line), but I am not $100 \%$ sure.