Young, cheap tuo cha showdown!

toucha5Can a cheap pu'erh be good? Is it possible to buy cheap tea and age it in to something amazing? Most say "no" --but that won't put me off from trying. The first step is to taste young teas so that one can better understand the aging process.

I put three teas (all inexpensive young pu'erh) in to three teapots without looking much then mixed them up. Who will win the blind tea taste test?



Tea #1

Astringency, Dryness, nice Hui gan, rather sweet and slightly sour like white grapes. This one needed a longer steep than the other two. Nice back of throat feeling. The leaves are big and plump and young.


Tea #2

Heavy mouth-feel, Astringency, Dryness, some Hui gan... I swear this has this flavor like older tea or shu... it's not strong but its there. Steeps dark very quickly. Thick. This is easiest to drink in some ways.


Tea #3

Astringency, yet wonderful Hui gan, thick, flavors lasts, just a little biter also smoky? why is raw puerh smoky? Love the after-taste.

I think in their current young state #2 is the most "drinkable" -- yet it isn't my favorite. I like #3 a bit more ... in fact I like #2 least since it isn't really what I expect from sheng. I suspect it is blended. Not that that is "bad" I just like things that are less contrived.

How will these age? Damned if I know. I have two of each tuocha, one to nibble on and one to keep for a long time. I don't have an special storage arrangements, by my in NYC apartment I never use AC-- so I'm not too worried. In any case, if they start to dry out I'll seek new arrangements. But, If I had to guess,I'd guess that #1 will end up too sweet and plain with age, #2 won't really age much since it's so unstable, and #3 has a shot at be good in a few years.


Let's see what happens!

And now....

The Reveal

Xia Guan Yellow Ribbon

Xia Guan Yellow Ribbon Tuo Cha 2010 100g Raw ($9)

Jia Ji Tuo Cha * Menghai Dayi Pu-erh Tea

Jia Ji Tuo Cha * Menghai Dayi Pu-erh Tea 2009 100g Raw ($7)   RUNNER-UP

Mystery Tea From ebay

"2012yr Yunnan Wild Old Tea Pu'er Cake tea 100g" Mystery Tea From ebay ($6) WINNER!


And there you have it!


Almost Cauchy Sequence, unfinished business.

I just remembered a question from beginning analysis that my professor mentioned in class:

"Find a sequence A that is not Cauchy, yet any consecutive terms A_n, A_{n+1} will become arbitrarily close."

It was one of those questions that he mentioned in class for us to do on our own, but I never quite got to it. I was thinking about convergence and complete metric spaces today (and about unfinished business, as I've made a lot of my own since I started teaching.) And the question came back to me.

We seek a sequence, A, where \forall \; \epsilon > 0 \quad \exists N \in \mathbb{N} \;  \ni \; \forall n > N \quad |A_n - A_{n+1}| < \epsilon and \exists \epsilon > 0 \; \ni  \quad \forall  N \in \mathbb{N} \quad \exists m,n > N \quad |A_m - A_n| > \epsilon. (Writing the negation of what it means for a sequence to be Cauchy was a little more tricky than I thought...)

The natural log function is the most obvious answer. For large vales of n \ln (n) - \ln (n+1) is small. That is, \displaystyle \lim_{n \rightarrow \infty} \ln \frac{n}{n+1} = 0. This is true becuase after taking EXP of both sides we have: \displaystyle \lim_{n \rightarrow \infty}  \frac{n}{n+1} = 1

Next, it seems obvious that the natural log can't be Cauchy, but just to be certian if we let \epsilon = \frac{1}{2} then, \forall N \in \mathbb{N} if we choose some n > N and let m= ne (so, n,m > N) then

| \ln m - \ln n | = |\ln \frac{m}{n}| = |\ln \frac{ne}{n}| = 1 > \frac{1}{2} = \epsilon.

NOT Cauchy at all!

Ok, I imagine my professor was thinking of the natural log... but, what other functions also have this property? \sqrt[n]{x}? So, I guess this is really asking when \displaystyle \lim_{x \rightarrow \infty} f(x) - f(x+1) = 0 while \displaystyle \lim_{x \rightarrow \infty} f(x) = \pm \infty. It is necesairy but not sufficent for f'(x) > 0 \; \forall x and f''(x) < 0 \; \forall x , both will have the x-axis as a horizontal asymptote.

Nice to have finally gotten around to doing this problem! I think it will be good practice to find more unfinished business in my notes.

Using LaTeX on the iPad: A review of the best and worst apps for writing mathematics.

LaTeX  is a powerful markup language that allows you to generate beautiful equations like this:

\left\{\begin{matrix} n \\ k \end{matrix}\right\} =\sum_{j=1}^k (-1)^{k-j} \frac{j^{n-1}}{(j-1)!(k-j)!} =\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k \choose j} j^n


Without cumbersome "equation editors." You simply type the TeX code for the equation and it appears. (And it looks as good as what you'd find in any research paper since LaTeX is what most publishers use for their typesetting.)

Naturally, if you do math and if you have an iPad you want to find ways to use LaTeX on your iPad. As it turns out there are a few great options, though each leaves a little more to be desired. At this time there is no full LaTeX engine that runs on the iPad that will let you use every typesetting and graphics command that LaTeX offers. But there are a few work-arounds that give you access to that full engine. And there are a myriad of small apps that let you typeset equations. (I often forget that there is far more to LaTeX than just writing equations!) So, here is the round up:


MathBot ★★★½☆

Overall: A bare-bones equation editor that shows your LaTeX equations in real-time as you edit. Most of the important keys for writing mathematics are handy, but, annoyingly, the numbers (0,1, 2, 3,...) are on a a separate screen so typing equations can be quite tedious.  Once you have your equation ready you can export it as an image to email, to the photo album or to other apps.  It also has an "export as TeX" option that lets you move the code in to another editing environment as text. (Though copy/past would work just as well for this, so I don't really see the point.) If you are new to LaTeX you'll find the short-cut keys for most of the basic math symbols  helpful, and the way the keyboard shifts from one view to another is smooth and delightful, fitting well within the iPad user interface.  There are no folders and it is slightly cumbersome to rename files. This app is really for people who are just looking to write one or two quick equations to paste in to an email or document. Naturally, it has no full LaTeX engine and can't make pdfs or postscript files.

Ease of Use: Short-cut buttons are great for beginners. Real-time editing gives instant feedback. However, switching screens to type numbers is annoying.
Customization: Only one font size. Few options for advanced users.
Aesthetics and Flow: Simple clean design, intuitive sliding shortcuts on the keyboard. Classy, no distractions. Designed for the iPad. Uses the full screen.
Price: Free!



Tex Touch ★★★½☆

Overall: Tex Touch is the only LaTeX app (available right now) that will allow you to type documents of any size in LaTeX and generate pdfs as if you were using your desktop... sounds amazing right? Well, the catch is that you ARE using your desktop! The app uses Dropbox (an excellent, free, "cloud" style remote hard-drive) to send .tex files to your desktop computer where a second application installed (and always running) will pick up the file and compile your pdf. Hence, you can create and edit exams, research papers and even books from your iPad. But, as the review implies, the instillation and set-up process, along with the complexity of the program may be too much for some. Once running, it works like a charm and I use it often to revise my exams, (though typing an entire exam in LaTeX on the iPad is perhaps too much even for an enthusiast like myself.) I give it high marks for being the only "real" LaTeX application. But, I and many others, look forward to the day when we can compile our pdfs directly on the iPad without an internet connection, and without leaving a helper application running on our home computer.

Ease of Use: Hard to set up. Not for novices. Requires multiple applications on multiple computers and an account with Dropbox. However, if you were able to install TeX on your iMac you should have no trouble getting this set up. Once running it works like a charm. (Though you must remember to turn the helper application on or you're out of luck.)
Customization: You have access to everything LaTeX can do including things like the picture environment, that lets you use LaTeX to draw graphics on the fly. If it's on your home computer you'll have access to it with TeX Touch.
Aesthetics and Flow: Well designed for editing .tex files, though it would be vastly improved if the code were color coded.
Price: $9.99



LaTeX Help ★★☆☆☆

Overall: This app is a dictionary of common LaTeX commands. They are organized by type (Greek, operations, relations, etc...) Where this app really falls short is in its lack of any kind of search function. For example, \aleph, is listed under "more > misc." and was quite hard to find.  This app is designed for iPhone only, so iPad users will still have to put up with scrolling through the list on a tiny screen with a big black background. I do not recommend this app for iPad. It makes more sense to simply download one of the pdfs containing LaTeX symbols then use your favorite reader to scroll and surf through the document. The pdf will be more functional since you will be able to search! However it is free, so from that stand point, one can't complain. It could also be convenient for iPhone users to have at their side while writing TeX on their desktop.

Ease of Use: It's easy to use, but without a search function often unhelpful.
Aesthetics and Flow: The interface is not bad, it's fun to scroll through all of the symbols. No clutter, easy to read.
Price: Free!

TeX Equation ★★★★½

Overall: If you only get one LaTeX app for your iPad this should be it. It is a bare-bones, yet highly functional application that lets you type TeX mixed with regular text and generate images that are perfect for emails. It includes customizable macros that allow you to speed up your coding by simplifying repetitive character strings. You can set the font size and color. The keyboard is well thought out, including letters, common latex characters and numbers on the same screen so most equations are easy to type without switching. You can cut and paste the images in to email and other applications or save them to your pictures folder.  I find this app handy for conversing with students via email, though I do long for an email client with real LaTeX integration.  The developer runs a forum and is very responsive to user feedback.

Ease of Use: Very easy great keyboard design. Recommend for those familiar with LaTeX as it lacks short-cut keys.
You can change the font size and color and create custom macros for faster coding.
Aesthetics and Flow: A sign of good interface design is when you don't notice the interface. No short-cut keys cluttering the screen. All the keys you need where you want them on one screen.
Price: $6.99

Formula ★★☆☆☆

Overall: I want to like this app, but there are so many little infuriating things about it that I find it unusable. First the interface design: Much of your screen is taken up with cheesy greek columns, it feels like a bad CD rom from the 90s. But, once you get past that, there are a few nice touches: equations appear in real time, copious short-cuts will make this app appealing to the LaTeX beginner.  However, whoever designed it did not test it for writing actual equations. Most  equations contain both numbers and letters, but on the main keyboard you are given access to only a numeric key pad. Try writing something simple like:  x= \frac{-b \pm \sqrt{b^2-4ac}}{2a} and you will end up switching keyboards 7 times! It's very tedious. The app is packed with useless flashy features like voice recognition. In theory you say the name of the symbol and the application will provide the correct short-cut keys... To say this feature "dosen't work" is an understatement. It doesn't work and it is an incredible waste of time!  I have decided that this app is not for people who do math, but rather for people who would like to look like they are doing math.  It's very serious and impressive to look at, and nothing says 'mathematician' like loudly shouting Greek letters into your iPad's built-in microphone. You will be pleased with this app until you try to use it to get any kind of work done. Stay away.

Ease of Use: Looks fancy but poorly thought out.
Customization: You have no control over font size, there are no maros.
Aesthetics and Flow: 1990s CD ROM look. Feels junky.
Price: $3.99