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59 views

Demystifying the tensor product

It seems to me, through my mathematical immaturity, that the tensor product seems to beg for more well-definition. I am working in vector spaces (so we always have a free module) and here is what my ...
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93 views
+50

Algebraic Geometry Project ideas related to Computer Science

I am a Computer Science Undergrad student with an interest towards Algebraic Geometry.I have just recently started and am currently reading Miles Reids' Undergraduate Algebraic Geometry(I have read ...
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47 views

Can Stochastic Integration be Further Generalized?

Is the idea of stochastic integration to accept convergence towards the stochastic integrals in probability instead of almost surely (pathwise)? I.e. to accept a weaker form of convergence for the ...
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30 views

Ask for some advice in proving Calculus problems

Recently, I have done some exercise about single variable calculus. Most of them are solved by theorems like Lagrange/Cauchy mean value theorem, Rolle theorem, and Taylor series. And most of the time ...
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32 views

What category of math do “theoretical” jigsaws fall into?

Imagine if like to investigate the structure of solutions made from various puzzle pieces. What area of math is this? A good analogy is how Rubix cube solutions are related to group theory.
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124 views

How does commutative and/or differential algebra think about total derivatives?

If we apply the "operator" $\frac{d}{dx}$ to the polynomial $xy$, we get the expression $y+x\frac{dy}{dx}.$ (Source: high school.) Thinking of $xy$ as an element of the polynomial ring $\mathbb{R}[x,y]...
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113 views

Manifold in Milnors Morse Theory

While reading "Morse Theory" by Milnor, I noticed that certain arguments would not work, if the considered manifolds have nonempty boundary. Example: Proof of 3.5 I could not find the definition ...
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159 views

What Constitutes a Pattern

Mathematics is often referred to as the "study of patterns." What I'm wondering is whether there is somehow a technical way to describe a pattern. For the length of this question let's assume that we'...
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248 views

Good starting point for learning noncommutative geometry?

Currently, I am attempting to learn noncommutative geometry. My interests mostly lie on the boundaries of pure mathematics and theoretical physics, so I am not only interested in the mathematical ...
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52 views

Generalizing convexity of sets

Let $X$ be a subset of some Euclidean space. We say that $X$ is convex if for any two points $p$, $q$ in $X$, the line segment joining $p$ and $q$ is also in $X$. But what if we loosen this definition ...
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156 views

Solving a question by using special products (Students debate to Teacher)

So today,we got back our exam papers,and we found a question marked wrongly and teacher said that it is wrong.We all students do NOT believe this.So here is what happened. Before reading the next ...
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113 views

Algorithm for finding “fact families”

My friend's 3rd grader encountered the following question regarding "fact families" on her math homework: I was in 3rd grade sometime in the 1980s, so I don't believe I ever encountered this term ...
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75 views

Why should the open mapping theorem be expected?

Soft question alert. I want to know why to expect the open mapping theorem to be true. My thoughts: I know that one nice consequence of the OMT could be thought of as the universal property of ...
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108 views

Apparent Arbitrariness in Mathematics

Something about definitions in mathematics has always interested – confused? - me, I call it “arbitrariness in Mathematics” - it's a bad name, but I don't know a better one. Let me explain: 1st - ...
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77 views

For cardinals, if $\mathfrak{a}\ne\mathfrak{b}$ then $2^\mathfrak{a}\ne 2^\mathfrak{b}$

In the usual ZF (or ZFC) set theory, let $\mathfrak{a}$ and $\mathfrak{b}$ be cardinal numbers. Is it correct that one can neither prove nor disprove the statement: $$\mathfrak{a}\ne\mathfrak{b} \...
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143 views

From algebraic master degree to algebraic geomery Phd

I am a foreign master student in algebra at the final year. I'm familar with categorical algebra and have interesting in algebraic geomery and number theory. I have learned some knowledge about scheme ...
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81 views

Numbers Made From Concatenating Prime Factorizations

I came across the following curious problem while playing around with my calculator. Take any positive integer $n$; for this example we'll use $216$. Create a sequence as follows: Factor $n$ into ...
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83 views

Math opportunities for a graduate

Are there any opportunities for young mathematicians who have finished their undergraduate education and are taking a year before going to graduate school, but want to do something math related? To ...
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99 views

Do Natural transformations make 'God given' precise?

Often in mathematics, one encounters certain structures in which there are natural choices to be made. For example, there could be a 'god given' map relating two mathematical objects, in the sense ...
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167 views

Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have $$d(x,...
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189 views

Sufficient conditions for closed infinite pasting lemma

It's well known that the pasting lemma for infinitely many closed sets is false. It's reasonably easy to cook up examples such that for $X = \bigcup X_i$ with $X_i$ closed in $X$ such that $\left. f\...
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158 views

Reference Request: Prereqs for Lecture Notes on “Abstract Linear Algebra”

I just found this set of lecture notes on linear algebra which seems to go over several things I've been wondering about as I study linear algebra. Unfortunately there are very few exercises in the ...
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66 views

Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point ...
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268 views

Note-taking software which is capable of representing mathematical relation

After quite a few studying experiences (both alone and following a class, studying topology, algebra, analysis, measure theory, differential geometry) I have acquired a lot of notes that I have either ...
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172 views

Is there a way to figure out the minimum number of participants or maximum number of rounds in my tournament style?

I just finished hosting a Euchre tournament at work that was meant to get people to meet other people in the company. This is the third time I've hosted this type of tournament. The first two times, ...
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162 views

Intuition about structures in Galois Theory.

Background Often in mathematics I find we can ask "why" something is true. Of course, this is not a well defined question. However, it usually prompts answers that includes words like "intuitively..."...
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95 views

Making modular representation theory and cohomology 'compelling' and 'accesible'

I'm currently putting together an application for a dissertation completion fellowship offered through my university. A part of the application includes 500-1000 words describing my dissertation. ...
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151 views

definition of determinant in Artin

In Artin, the discussion on determinant starts from the standard recursive expansion by minors. Artin defines determinant as a function $\delta$ from a square matrix to a real number. Then Artin lists ...
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102 views

Is there a proper etiquette for asking an author for their (mathematical) software?

This may not be the correct place to ask such a question. I have read a mathematical paper on multiclass total variation clustering. I wish to use the algorithm in the contents to compare with another ...
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300 views

Difference between dynamic system and complex system

I know this might not be an easy question, I've already read the wikipedia page, and there is an interesting view: Therefore, the main difference between chaotic systems and complex systems is ...
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220 views

Examples of good phone interview quetions.

My school asked me to prepare some students for maths questions in a phone interview. But this is my first time doing this kind of job and I never had this kind of interview myself so I am a little ...
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421 views

Paul Erdős Joke.

I was watching the great documentary "$N$ is a Number" and in it Erdős tells a joke where he writes: PGOM LD AD LD CD Which means poor great old man, living dead, archeological discovery, legally ...
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1k views

which branch of maths studies Standard Logical Matrices

In classical logic you have truthtables like: & | T | F ---|---|--- T | T | F F | F | F In many valued logic you have tables like: (this one is ...
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320 views

Logical consequence of Euclid's theorem

Are there any far reaching non-trivial consequences of Euclid's infinitude of primes where theorems make use of it? Wikipedia does not have the list of applications of this theorem, rather modern ...
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939 views

Publication date for Michael Spivak - Physics for Mathematicians II?

I bought the book "Physics for Mathematicians I" by Michael Spivak (http://www.amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/0914098322), have worked through quite some chapters and ...
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271 views

Convex Hulls vs Shrink Wrap

I was recently explaining to a friend what the convex hull of a set of points is using the analogy of an elastic band around a set of nails hammered into a board. I was about to say that we can ...
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352 views

Differential Forms

I just had a general question about differential forms. Background. Let $f = f_k$ and define $f^{k-1}$ on $I^{k-1}$ by $$f_{k-1}(x_1, \dots, x_{k-1})= \int_{a_k}^{b_k} f_{k}(x_1, \dots, x_{k-1}, x_k) ...
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41 views

Is there a schsim between classical logic and categorical logic?

I've been trying to learn a little bit more about the foundations of mathematics, and it has strike me that there seems to be two competing points of view about what the foundations should be. While ...
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69 views

Topology on $\mathcal{C}(X,Y)$ to work with homotopy.

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
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66 views

Concrete Mathematics: Not taught how to solve problems.

I've read and studied the first 3 paragraphs from Concrete Mathematics and now I've made the first couple of problems. When I got to the actual homework problems I was lost because I tried to solve ...
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96 views

Notebooks used by Paul Erdős

Paul Erdős was known for living out of two half-full (or half-empty) suitcases; one had a few clothes and the other had mathematical papers. Some of these papers were probably referring to his ...
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35 views

(Soft Question) Formal name for something one is taking the limit of

When you are taking integral you have an integrand. When you are taking a sum you have a summand. Is there an analog of this for limits? How should one refer to $f(x)$ in the expression $\lim\limits_{...
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75 views

Functions with “ugly” inverses

Inspired by this post: I was amazed to see that (at least according to wolframalpha) the inverse of such a nice and simple function as $f(x)=x^3+x$ is: $$ f^{-1}(x) = \sqrt[3]{\frac{2}{3( \sqrt{81x^2+...
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52 views

Is there a dictionary for math notation?

As a non-mathematician who loves the elegance of mathematics, I'm often confused about certain syntax I see. For example, $2+2$ is "2 plus 2" obviously. But things get more complicated when, for ...
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97 views

Can the tehniques of higher level mathematics solve most of Olympiad level math problems through straighforward applications?

Working through many Olympiad math problems(pre-undergrad) I've found that simple applications of undergrad math will solve many of them. Does this trend go on? Can it be that Putnam problems are ...
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121 views

Why is recursion theory suffering from terminological bloat?

Several questions on MSE in recent months and most recently this one have made me feel that recursion theory is suffering from terminology bloat. Why have so many synonyms for "recursive" and "...
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81 views

Mathematics Wallpapers

I know that this sounds very silly. But I don't know where else to ask. Is there a good free site for mathematics wallpapers , pictures etc ? Most of the time it is very difficult to find exact ...
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41 views

What are some concrete examples of how Graph Theory can be applied to Finite State Machines?

Conceptually, Graph Theory is not very hard to understand, even its terminology is very intuitive, however, it appears to have a very wide range of applications, one of which is the field of Finite ...
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57 views

Why Navier-Stokes solutions are expected to be $C^\infty$?

I am curious, why the Millennium problem about Navier-Stokes is about smooth ($C^\infty$) not just $C^1$ solutions?
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Universal geometry

Is there something analog as to what universal algebra is for algebra for geometry? Some formalism incorporating both analytical geometry, differential geometry and algebraic topology. Or what comes ...