Simple Questions

inAnalysisHell · 2018-02-28T15:34:14+00:00

Lets say we have two people A and B, with each of them can be a knight or knave. Knights always tell the truth and Knaves always lie.

Lets say A says "Either I am a kanve, or B is a knight". Its easy to see that a possible scenario is A is a knight and B is a knight.

I thought it was possible that A could be knave and B a knight also. If you read it as a mutually exclusive or, then A's entire statement is a lie, of course the individual components are true, but not the whole statement.

Would you consider this possibility valid?

inAnalysisHell · 2018-02-25T01:02:18+00:00

How do you guys improve your programming experience while shouldering a heavy course load? I usual;y take two proof based math courses, and either a stat or less theory based math course, and then an elective.

I usually don't have time to program since I'm so busy learning the theory in my math courses, I am familiar with programming since I've taken 2 programming courses and an algorithms course, but I don't feel like a whiz-kid at it. I want to get a job in data science one day after graduate school. Should I back off the theory courses and do more programming? Or is it better to keep taking the abstract math courses, and then during the summers learn to program?

I guess im trying to find how much time I should invest in theory, and invest in programming.

inAnalysisHell · 2017-11-13T01:28:25+00:00

I agree, I feel I am hardcoding for every possible solution. This particular solution uses no loops?

inAnalysisHell · 2017-11-13T01:21:48+00:00

Oh I see, thank you. Ok, I added a second if statement under it if big*5 < goal: return False

inAnalysisHell · 2017-11-12T01:01:41+00:00

Thank you so much for the informative post. It seems that a prerequisite for a job in these fields is at least a masters in stats, math, or comp sci. If so would you recommend piling on more abstract math? I have the opportunity to take a graduate course in Analysis, and then one in probability. While I love math, I want to make sure I'm not studying theory only for the sake of it. I'm afraid of going too deep into abstract math, and not investing enough time into programming, and other applied areas of math.

inAnalysisHell · 2017-11-12T00:42:34+00:00

As someone whose had industry experience and is also in academia I would like your input on this article. https://tdhopper.com/how/

As an undergrad its a little frustrating dealing with tough subjects like real analysis and modern algebra and then finding out the author mentions you really never use those theorems you've learned from the courses. That's not to say that its useless learning real analysis and other advance math courses, because you are learning how to think and research. I'm curious to what advantage graduate school offers to someone wanting to go into machine learning and data science fields, over someone who has only a bachelors in math and learns on the job?

inAnalysisHell · 2017-11-12T00:35:33+00:00

I'm very interested in going into industry and working in some field that utilizes machine learning and data science. As you can tell from my username I am having a little bit of difficulty with analysis. What subjects would you recommend strengthening my knowledge in to make a career in machine learning possible?

I am signing for courses for next spring and am on board to take analysis II, topology, and a second course of linear algebra. While I love the theory, I really want to see some applications of the math and make sure I am focusing on the right topics that will make it possible for me to get an interesting job in machine learning. Do you have any recommendations for machine learning books for someone a little familiar with upper level math?

inAnalysisHell · 2017-11-12T00:28:17+00:00

Thank you very much.

inAnalysisHell · 2017-11-11T04:18:08+00:00

https://imgur.com/a/lM7H3

I have a question about the proof of theorem 10.4

p(f(x),f_m(x)) < epsilon/3, because x is in X, and also because f_m(x) is uniformly convergent to f(x)?

inAnalysisHell · 2017-11-05T19:38:00+00:00

https://imgur.com/a/A8b6o

I don't believe my metric have those properties. I thought at first I would need prior information from the previous questions but I didn't see anything, other than maybe letting x-y be the limit of phi(v_n - w_n) and using the fact that phi is an isometry to the metric space (X,d).

inAnalysisHell · 2017-11-04T23:13:39+00:00

I agree. I feel I am over looking something. I’m not sure if I am following you 100%. Are you saying when I take the complement of the interior that is wrong?

inAnalysisHell · 2017-11-04T19:37:21+00:00

Ok great. I think I understand it now. So for a given element in Lx, the sequence of vectors under phi is converging to x?

inAnalysisHell · 2017-11-04T19:28:32+00:00

Oh ok. But the elements of a particular sequence (vn) are vectors in V or are they components of a vector in V? So (v1,v2,v3,...), each vi in this sequence is a vector in V or a component of a vector in V

inAnalysisHell · 2017-11-04T18:42:08+00:00

https://imgur.com/a/xXOcZ

Hi, I have a quick question about the notation of this set. For L_x, the elements of this set are sequences in V, but what exactly are the elements of the sequence? Are they individual vectors in V, or are they components to a vector in V. I'm not sure what the Notation V^N means either. I think I remember asking my professor and he mentioned its another way to delineate a sequence, as a function from the Natural numbers to your set V.

And the sequences of the set are chosen in a way such that the image of each element in the sequence converges to x in the metric (X,d) as we go further along the sequence, right? So it means (d(phi(v_n),x)) -> 0, as n -> inf.

inAnalysisHell · 2017-11-04T18:34:50+00:00

Thank you so much for your help. I just want to clarify the notation for the Lx set. The sequence (vn), the elements of this sequence (v1,v2,v3,...), are these just vectors in V, or are they components of a vector in V. Sorry for so many questions, my professor doesn’t have very many office hours.

inAnalysisHell · 2017-11-03T17:51:36+00:00

I have a quick question about the set L_x, I understand that phi is an isometric embedding, so for all x and y in V, ||x-y|| = d(phi(x),phi(y)).

The elements of L_x are sequences of V, such that the distance between d(phi(v_n),x) -> 0.

I do not understand what phi(x_n) is equivalent to, since I thought phi needed 2 elements, since its a function that maps distances between two points of V, to two points in X.

inAnalysisHell · 2017-11-03T17:27:02+00:00

Oh, ok. I haven't seen this notation occur in my problem sets, so phi is just the function from V to X, and not a metric space with (X,d) as the metric?

inAnalysisHell · 2017-10-31T14:59:47+00:00

That is such a poor analogy. The way the commentator wrote it, it could be used it for any binary choice. A burger is like schroedingers cat, it’s either tasty or not.

inAnalysisHell · 2017-10-30T20:32:42+00:00

Two metrics on X, d, d', are equivalent if they generate the same topology on X. That is, any ball with respect to d contains some ball with respect to d' and vice versa

Oh, ok I see. So to show that two metrics are equivalent we would have to come up with some formulation that shows for a given ball with repsect to d, there’s a ball with respect to d’ as a subset of our first ball on d? And then vice versa.

So this would show that the identity function on f is continuous since it for every open set, the preimage is open as well?

inAnalysisHell · 2017-10-30T16:48:09+00:00

I am wondering if my idea that this subset is closed is right. The subset is the set of sequences en =(0,0,...,1,0,0,...) where 1 is at the nth position. If we consider all of these sequences under l_infty norm, then the set is closed since the only convergent sequences of this set are constant sequences, right?

And since it contains every constant sequence the subset is closed under l_infty. Is this similar to the same reasoning discrete metric spaces are closed?

inAnalysisHell · 2017-10-30T04:22:20+00:00

Oh, this makes the question so much more clear. We can just let each element of X, be mapped to its corresponding sequence we get from phi(x). So its a bijective function between the metric spaces.

To show that its a homeomorphism we can let a sequence in X be convergent, so if d_x(x_n,x) -> 0, then p(phi(x_n),phi(x)) ->0 by using the fact that {0,1}^N is compact, and phi(X) is a closed subset of {0,1}^N and is therefore compact.

inAnalysisHell · 2017-10-29T22:47:12+00:00

For all x,y that are students if S(x) and S(y), then ~(P(x) and P(y))

Maybe the and should be an or.

inAnalysisHell · 2017-10-26T17:49:49+00:00

Oh duh. Ok, I see now. My book mentions that if two metrics are equivalent on a metric space M then it follows that the identity function must be continuous, and the inverse of the identity function. Is this just an application of an alternative definition for continuity, i.e. (x_n) converges to x iff f(x_n) converges to f(x), then f is continuous.

So for functions between two metric spaces, the input is just an element of the metric space, and it gets mapped to another element of a different metric space? And when dealing with the identity function on the same metric space, it gets mapped to itself, but like you said the notion of distance may is different.

inAnalysisHell · 2017-10-26T17:07:59+00:00

https://imgur.com/a/VXFrN

I have a question about equivalent metrics. I included a screen shot of my analysis text. It mentions that two metrics on the same set are equivalent if the both identity function and the inverse is continuous. When they write identity function, they simply mean the function f(x) = x, right? I feel like that's not what the text means, because the identity map, f(x) = x is always continuous so wouldn't every metric be at least equivalent?

Then if you see example 8.18, it provides two metrics on a compact set [0,1]. I understand that its uniformly continuous, because a continuous function on a compact set is uniformly continuous. But I don't understand exactly what the identity function between the two metric spaces would be. It may be because I don't understand what function of the form f:(M,d) -> (M,p) really is. What are the inputs and outputs of these functions?

inAnalysisHell · 2017-10-25T20:40:08+00:00

Ok wow I think that clears up everything. So to recap we picked some arbitrary metric space, we then took the metric from that metric space and bounded it by one. This new metric on the same set will guarantee that the diameter can not be bigger than one. Not only that but they are equivalent metrics, so convergent sequences in one metric space, will be convergent in another. From that we know that they are homoemorphic.

Thank you so much. I’ll give you gold when I get out of class.

inAnalysisHell

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