The closed points of $\mathbb{A}^2_\mathbb{R}$ do not correspond to ordered...
In Eisenbud & Harris's The Geometry of Schemes, after a discussion of the closed points in $\mathbb{A}^1_\mathbb{R}$ and in $\mathbb{A}^2_\mathbb{R}$, the authors note thatThe closed points in...
View ArticleModel probabilistic time evolution from vector field distribution
Given an axis-aligned grid${}^1$$x_1 , \ldots , x_n \in \mathbb{R}^2$. Let $\mu \in \mathbb{R}^{2n}$ and $\Sigma \in \mathbb{R}^{2n\times 2n}$ be the means and covariance matrix and assume we have a...
View ArticleWhat is so interesting about the Armijo-Rule or the Wolfe-Conditions for...
Right now I am taking a course on nonlinear optimization where we currently talk about step size rules(Armijo-Rule and Wolfe-Conditions).I also had a course 1 year ago about statistical machine...
View ArticleA polynomial recurrence relation, what does it solve?
A normal recurrence relation looks like so >$$0 = \sum_{k=0}^N c_{k}a_{n-k}$$Often by convention we choose $c_0 = -1$ and rewrite it to arrive at the probably more well-known expression >$$a_n =...
View Articlewhy don't we have more imaginary numbers? [duplicate]
the imaginary number i $ \sqrt{-1} $ was invented in 1545. the only real world application of this I can really think of is in computing which was obviously invented much after 1545 meaning that $...
View ArticleWhat is the importance of the Collatz conjecture?
I have been fascinated by the Collatz problem since I first heard about it in high school.Take any natural number $n$. If $n$ is even, divide it by $2$ to get $n / 2$, if $n$ is odd multiply it by $3$...
View ArticleOptimization of totally ordered set valued function.
I am familiar with the meaning of optimizing a function $f : \Omega \to \mathbb{R+}$. However I was just wondering if there's some theory of math explaining how to optimize mapping from $f : \Omega \to...
View Article"So That" vs. "Such That"
In definitions and exercises, I notice that "so that" and "such that" are seemingly used interchangeably. Are they in fact interchangeable, or is one more appropriate for a specific context?Note:...
View ArticleThe need for functions, which traverse the rectangular grid.
Let $[N]$ denote the set $\{1,2, \ .. . \ ,N\}$ . For any $n,m \in \mathbb N$ what are some functions which traverse the grid $[n]\times[m]$, hitting each element exactly once? In particular, it would...
View ArticleTrying to understand orthogonality of boundary conditions for functionals of...
A question I had whilst reading section 15 of Fomin's "Calculus of Variations" (great book btw!!)The General Question:Among all smooth curves whose end points $p_0$,$p_1$ lie between two curves...
View ArticleIs it a convention that the word "where" following a mathematical formula...
I am not a native english speaker. I learnt about defining and non-defining relative clauses from english grammar books. Grammar books tell me not to use commas in defining relative clauses, so I don't...
View ArticleAre there other usual ways of justifying the passage of differentiation into...
I am recently reading Evan's Partial Differential Equations.In the book, the author sometimes passage differentiation into integrals (i.e. $\frac{\text{d}(\int_{\Omega}...
View ArticleWhy do they say the PNT wasn't proven until Hadamard?
I've been working my way through the MIT OCW course on number theory, and the lecture on the PNT states:However it was not until a century later that the prime number theorem was independently proved...
View ArticleIs there a website that has all the special functions?
There are a lot of special functions, and I wonder if there is a website that collects all of them, similar to how the Encyclopedia of Triangle Centers collects information on triangle centers.Another...
View ArticleHow much notation should there be in a formal proof? Is there a general...
I am writing a conference paper in formal language theory with an involved proof and I've found myself struggling with notation. In particular, I don't know when to favor notation and when to favor...
View ArticleWhy does learning theory study generalization bounds?
Disclaimer: I know that mathematics needs no external motivation to be developed, and that such view is (in the long term) helpful even for applications. Nonetheless, I believe it is crucial for...
View ArticleBook recommendation of complex analysis in several variables
I want to learn complex analysis. I would like a book that teaches about complex analysis in several variables without assuming that the reader knows complex analysis in one variable.The only book I...
View ArticleWhy do sometimes care for where vectors originate from and sometimes not? and...
When I did linear algebra in high-school, it wasn't of much importance where the vectors originated from and for me this is a really hard concept to grasp. It's like no matter where the two vectors are...
View ArticleDo we have complete understanding of $\mathbb N$?
We have some understanding of natural numbers. We have PA theory and we believe that $\mathbb N$ is one of the PA models. But PA can't prove some statements about $\mathbb N$ even though they are true...
View ArticleWhy do limits work? [closed]
Started basics of calculus as a high school student , and Limits already confuse me . Can somebody explain it to me why limits actually work .If the value of x tends to infinity ,which is undefined...
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