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Is the following equivalent to the axiom of choice? How to Set Firefox 4 to only store History for a set period of time, i.e. one day? I want to setup Firefox 4 to only store my history for 1 day (or some other set period of time), but I can't seem to find an option to do that in the options dialog, all I can see is "Remember History", "Never Remember History" and "Custom History Settings". "Custom History Settings" only lets me select what to keep, not for how long.  Is there anyway I can do this? Is circular reasoning always a fallacy? Is the 9-speed Sora crankset compatible with 105 11-speed chain? Note: $x_n$ is a sequence which is not necessarily convergent.    The following was my attempt. Since $\lim_{n\to \infty}a_n=a$ then $\limsup_{n\to \infty}a_n=a$ . Also $\sup(a_nx_n)=\sup(a_n)\sup(x_n)$. Therefore, $\limsup_{n\to \infty}a_nx_n=\limsup_{n\to \infty}a_n\limsup_{n\to \infty}x_n=a\limsup_{n\to \infty}x_n$.    I am not sure if this is correct. So could someone please show me how it is done. Thanks Let's have breakfast    Let's have a good breakfast      I've learned that indefinite articles aren't used before uncountable nouns, so why is a used before the adjective "good" here?    Is there an exception to the rules or is it wrong? html - how to remove right margin completely? Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell Prove: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ and $X\sqcup_gD^n$ are homotopy equivalent.   I think it can be proved by showing they are both deformation retracts of $X\sqcup_H(D^n\times I)$ where $H$ is the homotopy between $f$ and $g$.  However, I have difficult in proving that the deformation retracts are continuous map. In fact, I have difficulty in representing a map in quotient spaces like $X\sqcup_fD^n$. I think a map from $X\sqcup_fD^n$ to $W$ can be represented by two maps: $m_1: X\to W$, $m_2: D^n\to W$, where for $x\in S^{n-1}$, $m_1\circ f(x)=m_2\circ i(x)$.  Then I construct the deformation retract this way: $m_1: X\to X$. For $x\in H(S^{n-1},t)$, $m_1(x)=H(S^{n-1},0)$, otherwise $m_1(x)=x$.  $m_2: D^n\times I\to D^n\times {0}$: $m_2((D^n,t))=(D^n,0)$.  It is easy to verify that $m_1$ and $m_2$ define a map from $X\sqcup_H(D^n\times I)$ to $X\sqcup_fD^n$. As long as this is a continous map, obvioulsy then we find a deformation retract. But it seems such a map is not continous? Does there exist a function such that $\lim_{x \to a} f(x) = L$ for all $a \in \mathbb R$ but $f(x)$ is never $L$? How to get a list of monomials of a polynomial without coefficients? Can I use the valid Schengen visa in old passport? Show that if $f$ is integrable on $[a, b]$ then $\lvert f \rvert$ is also integrable. Hint: Show that $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ I have: $$U (P , \lvert f \rvert) \ge U (P , f )$$ and $$ L(P , \lvert f \rvert) \ge L(P,f)$$ Thus, $$U (P , \lvert f \rvert) − L(P , \lvert f \rvert) ≤ U (P , f ) − L(P , f ).$$ Workflow rule order of execution When should you use struct and not class in C#? My conceptual model is that structs are used in times when the item is merely a collection of value types.  A way to logically hold them all together into a cohesive whole.  I came across these rules :   A struct should represent a single value. A struct should have a memory footprint less than 16 bytes. A struct should not be changed after creation.   Do these rules work? What does a struct mean semantically? Spurious whitespace with algpseudocode and noend I'm currently typesetting some pseudocode and was up to now pretty happy with the algorithmicx package and the algpseudocode command set. However, I would now like to use the noend option to omit end lines. That works, but unfortunately introduces additional whitespace (see lines 6 and 7 in the example below). I fiddled around with negative \vspaces but could not come up with a consistent value that works in all cases.    \documentclass{article} \usepackage[noend]{algpseudocode}  \begin{document}  \begin{algorithmic}[1]   \Function{lsolve}{$L$, $b$}     \State $x \gets b$     \For{$j \gets 0$ to $n - 1$}       \State $x_j \gets x_j / l_{jj}$       \ForAll{$i &gt; j$ where $l_{ij} \neq 0$}         \State $x_i \gets x_i - l_{ij} x_j$       \EndFor     \EndFor     \State \Return $x$   \EndFunction \end{algorithmic}  \end{document}   Is there any way to patch algorithmicx or do I have to finetune the pseudocode on a case by case basis? The following problem came up from a though I had while reading:  Let's say we have $M=\mathbb{Z}^n$ and we have another free $\mathbb{Z}$-module, $N$, inside of $M$ also with rank $n$.  We know we can make a matrix $A$ that changes $N$ to its representation in $M$ (ie has as columns the basis $N$ expressed in coordinates coming from the basis of $M$). If the index of $N$ in $M$ is $m$, I am pretty sure just from tinkering that the determinant of $A$ should also be $m$, however, I have not been able to show this.  It is unclear to me how to proceed. I though maybe I could do something related to the points in $\mathbb{Z}^n$ that are within the unit box of $N$, but I cannot really make sense of this.  Thank you for any help or direction. My computer is loud: How do I tell which fan is the issue? On my new computer, my fan(s) is/are plain LOUD! Using SpeedFan, I find that my temps are all under 32 degrees C under my normal use (IDK how accurate that is but the CPU itself says 27 C).  The fan doesn't sound unbalanced and it isn't making sounds in rhythm (Not the usual rrrrrrrRRRRRR... rrrrrrrRRRRRR... rrrrrrrRRRRRR... rrrrrrrRRRRRR), just a constant fan. I am thinking about oiling the fans, but I can't decide if it is the PSU fan or a case fan or the CPU fan. If it's the CPU fan I will most likely just replace that with a quiet fan. How can I tell which fan is making the noise (if not multiple)?  One thing that I am thinking may be the problem is my CPU fan isn't PWM so would swapping that out help? There is no "linear voltage" or etc thing in BIOS so I think it may be running at full speed. Also, I could have too much airflow because it is also making a whistling noise that you can hear close to it that sounds like when I accedentially put part of the side cover over my floor vent to get it out of the way. The type initializer for 'Emgu.CV.CvInvoke' threw an exception If anyone who use Emgu openCv wrapper in .net 3.5 framework,please tell me why this exception appears,Its inner exception is unable to load cxcore.dll. How do I add a .sty file to my MacTeX/TeXShop installation? Is calling a linear-equation a linear-function, misnomer or completely wrong? From my college life, I remember many professors used to call a linear-equation a linear-function, however:  A standard definition of linear function (or linear map) is:  $$f(x+y)=f(x)+f(y),$$ $$f(\alpha x)=\alpha f(x).$$  Where as linear equation is defined as:  $$f(x)=mx+b.$$  So, linear-equation is NOT a linear-function, according to the definitions defined above.  Though, for $b=0$ the linear-equation becomes a linear-function, but it is not true in general.  Question: Is it misnomer to call a linear-equation a linear-function, or it is completely wrong to say that? And linear-equation must be considered strictly as an affine-mapping.