Question is about a paper “A Block-sorting Lossless Data Compression Algorithm” by M. Burrows and D.J. Wheeler

In the paper A Block-sorting Lossless Data Compression Algorithm by M. Burrows and D.J. Wheeler Link. On page number 5. please describe this line

If the original string $ S$ is of the form $ Z^p$ for some substring $ Z$ and some $ p > 1$ , then the sequence $ T^i[I]$ for $ i = 0,…., N – 1$ will also be of the form $ Z’^p$ for some subsequence $ Z’$ .

Question on preventing k from reducing too quickly during KMV intersection

This question considers KMV, an algorithm that is able to estimate the cardinality (unique item) from a stream of data.

The way it does it is to first map the stream of data to a space that almost guarantee uniform distribution and then only store the minimum k values. To estimate the cardinality it looks at the spacing between the k values and extrapolate the cardinality.

One of the benefit of KMV compare to hyperloglog is that KMV can perform set intersection that create better result than hyperloglog with inclusion exclusion principle

I am looking at a particular implementation of KMV call theta sketch and I am interest in its implementation in intersection

Specifically I am having a hard time understanding how its k size is reducing almost 50% slower than my implementation.

I expect k‘s reduction to match the probably of intersection between the 2 sets, which I do see in my code. However I am seeing k‘s reduction in theta sketch much less prominent.

I look at the code and I think the only major difference is that theta sketch is much more optimized and its bound mechanism is through storing any number less than a certain number (ie. theta) instead of storing the minimum k number. But I still expect that k to reduce as fast as my implementation.

Can someone point me a direction ?

[ Etiquette ] Open Question : Poll, yes or no it is your responsibility as an adult to pay your bills on time?

Seems kind of common sense, but the longer I work with the public, the more I realize it isn’t so common sense to a lot of people. Right now I work in insurance and…yes…you do have to pay to insure your car, house, business, etc. It’s not free. We all know this. I have clients who have had their insurance for months, even years, that forget to make a payment but somehow make it OUR problem and throw a fit if their policy has a late fee, cancel notice, or worse completely cancels due to non-pay. They’ll use every excuse in the book. “I didn’t get my bill in the mail so I didn’t pay” is always my favorite You should still know about when your bills are due. I pay around a dozen bills a month, I know when they are due. And if I am late…guess what? That’s MY problem. Not the company’s problem. Even if I didn’t get a bill. Even if I thought I already paid and turns out I didn’t. So, am I nuts here in assuming as adults we alone are responsible for paying our bills on time? Yes? No? Why is this such a complicated concept to some? **Obviously this doesn’t count for people who have had a situation they absolutely cannot help. I.e. in the hospital or family member death or illness. We always do our best to accommodate for those situations. But interesting enough, those are the people who actually don’t get upset at us. Go figure. Also, for those of you who have been in this boat of grown *** adults getting mad at you or your company for penalties of being late, I am curious to hear your stories. 🙂 Actually, to the anonymous answer, you have a point. But I have been in that predicament. Especially with credit cards because they’re not as important as food, gas, rent, electric…BUT I do not get mad at the company for my negligence. At the end of the day if I am going to own a car insurance is part of the deal.

Notational question about quadratic differentials in Strebel’s book “Quadratic differentials”

In Kurt Strebel’s book “Quadratic Differentials”, in Chapter 2, $ \S4$ , he begins by saying:

“Every analytic function $ \varphi$ is a domain $ G$ of the $ z$ -plane defines, in a natural way, a field of line elements $ dz$ , namely by the requirement that $ \varphi(z)dz^2$ is real and positive. This means of course that $ \text{arg }dz = -\frac{1}{2}\text{arg }\varphi(z)\mod \pi$ , and thus $ dz$ is determined, up to sign, for every $ z$ , where $ \varphi(z)\ne 0,\infty$ . One may then ask for the integral curves of this field of line elements.”

I am having some trouble with the language used here.

Note that I am not a differential geometer by training. My background in differential geometry mostly comes from Voisin’s first book on Hodge theory, Bott-Tu’s “Differential forms in Algebraic Topology”, and a bit of Kobayashi-Nomizu and a few snippets from elsewhere.

This book began its first chapter on background material on Riemann surfaces, and the point of the book seems to be to study the differential geometry of Riemann surfaces. Thus, I’m sure $ G$ must be a domain in $ \mathbb{C}$ , $ z$ is a holomorphic coordinate, and “analytic” probably means “complex analytic”.

Now, normally, for me, a line element should be a differential 1-form. Though, for him, since he says “field of line elements”, I’m assuming he uses “line element” to refer to a cotangent vector at a point, and thus his “field of line elements” should be taken to be a differential 1-form.

Okay, fine, but what would it mean for $ \varphi(z)dz^2$ to be real and positive? In all analogous texts, $ dz^2$ is really short for $ dz\otimes dz$ , ie a holomorphic section of the tensor square of the complex-valued cotangent bundle, but presumably for him he really means $ \varphi(z_0)(dz|_{z_0}\otimes dz|_{z_0})$ as an element of the tensor square of the complex-valued cotangent space at $ z_0\in G$ ? In this case, if we view $ dz = dx + idy$ as a complex-valued differential 1-form, one might interpret his requirement as saying that $ $ \varphi(z_0)(dz|_{z_0}\otimes dz|_{z_0})(X\otimes X) := \varphi(z_0)\cdot dz(X)\cdot dz(X)\in\mathbb{R}_{>0}\qquad\text{for all $ X\in T_{G,z_0}$ }$ $ where $ T_{G,z_0}$ is the (real) tangent space at $ z_0$ , but then this condition will never be satisfied since if it holds for $ X$ , then it will fail for $ iX$ , where $ i$ is the complex involution on $ T_{G,z_0}$ .

But maybe there is hope, as he explains that this means $ \text{arg }dz = -\frac{1}{2}\text{arg }\varphi(z)\mod \pi$ . However, this only confuses me more. For a complex number $ z_0 = r\cdot e^{i\theta}$ with $ r\in\mathbb{R}_{>0},\theta\in\mathbb{R}$ , $ \text{arg }(z_0) := \theta\mod 2\pi$ . But what do they mean by the argument of a differential/cotangent vector? The best I can think of is: Identify $ T_{G,z_0}$ with $ \mathbb{C}$ via $ \frac{\partial }{\partial x}\mapsto 1$ and $ \frac{\partial}{\partial y}\mapsto i$ , and then $ \text{arg }dz$ is how much the $ \mathbb{C}$ -linear map $ dz|_{z_0} : T_{G,z_0} = \mathbb{C}\rightarrow\mathbb{C}$ “rotates” the tangent vector (viewed as an element in $ \mathbb{C}$ ). However, this still does not resolve the previous issue with $ \varphi(z)dz^2$ being “real and positive”.

Lastly, what are “the integral curves of this field of line elements”? Usually one takes integral curves of a vector field. Are the $ dz$ ‘s really tangent vectors?

Counterspell Question [duplicate]

This question already has an answer here:

  • Counterspelling a counterspell 3 answers

Say you are casting fireball at level 5, and an enemy casts counterspell. Can you use your reaction to cast counterspell targeting their counterspell?

Alternatively, can I use counterspell on a later turn to stop an enemy from casting a spell, assuming I’ve cast my level 5 fireball earlier in the same round?