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Collection of "wait a minute!" facts
A couple of months ago, my Professor of Analysis brought to our attention the idea that sometimes we apply standard rules without checking if we can. To convince us it was the case, he suggested to...
View ArticleProperties (algebraic) of the "pro-sum". [duplicate]
On the set $\mathbb{R}$, we define the following operation which is called the "pro-sum" of two real numbers :For all $a, b \in \mathbb{R},\ a \circledast b = (a \times b) + (a+b)$, where $\times$ and...
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How to explicitly construct flat polynomial approximation to the exponential...
I have been reading the paper “Learning quantum Hamiltonians at any temperature in polynomial time”. This paper seems to be very interesting and has appeared in Quanta Magazine 2024 for "2024's Biggest...
View ArticleHow can I make sure I never forget.
I am currently refreshing my Elementary Algebra using Schaum's Outlines. I find them useful as they are choc full of exercises (Which I now realize is the only way to master algebra). I am worried...
View ArticleA mathematical object that can be an element of a matrix.
I read the definition of matrix on wikipedia. However, if you read this content, you can see the following phrase.In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers,...
View ArticleEquivalent to Riemann Hypothesis
Through last number theory, I did learn that Riemann hypothesis is equivalent to the following inequality : $|\pi(x)-\operatorname{Li}(x)| \leq \sqrt{x} \log(x)$ where $\operatorname{Li}(x)$ is the...
View ArticleIs the question of Lebesgue measurability of projective sets “leaning” one...
A lot of independence results (Continuum Hypothesis, existence of Suslin trees, etc.) can seem (to my mind anyway) a bit “esoteric”, i.e. not impacting in an obvious way to “normal” mathematics (areas...
View ArticleBooks with SAGE portions
I recently finished working through Adventures in Group Theory and really appreciated the use of SageMath it employs. I considered myself moderately proficient with Sage, but I found working through...
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Why does packing exactly 992 circles in a square behave exceptionally?
It is not so surprising that the problem of Circle packing in a square is a chaotic and often-unpredictable problem. However, after looking over the data on hydra.nat.uni-magdeburg.de, we find quite...
View ArticlePrime factorization and eigendecomposition [closed]
There seems to be an analogy between (a) the role of prime numbers relative to integers and multiplication and (b) the role of eigenpairs relative to vector spaces and matrix multiplication. Wondering...
View ArticleSignificance & Relevance of the Metric Differential for Maps into Metric Spaces.
I came across a generalization of the derivative for a map $ f: \mathbb{R}^n \to X $, where $ (X, d) $ is a metric space.The Metric Differential of $ f $ in the direction of $ v \in \mathbb{R}^n $ at $...
View ArticleCan partial derivatives help solve systems of elementary equations?
Let's say I have two systems of two equations,$x\ y = P(x,y)$ and $y+x = Q(x,y)$, where $P$ and $Q$ are rational functions.If $P$ and $Q$ are simple enough, this might be solvable via traditional...
View ArticleNeed clarification on Fredholm alternative
Let $A$ be an m by n matrix with, say, real cofficients. Then we have a linear transformation:$A: \mathbb{R}^n \rightarrow \mathbb{R}^m$It follows from (Gilbert Strangs) fundamental theorem of linear...
View ArticleStudy suggestions [closed]
I am a 16 year old high school student with too much free time. For the past two years I have been teaching myself university level maths in my free time. I have studied the basic first year subjects...
View ArticleStarting point for learning etale cohomology
I am currently interested in learning etale cohomology. My background is what you could find in Hartshorne (mostly in chapter II and III) and I also have strong intuition from algebraic topology. I...
View ArticleConic Sections formulae using Complex Numbers
I have been preparing for the JEE Examination here in India and have been studying Complex Numbers for the past few days. One of the topics which falls under Complex Numbers is their application in...
View ArticleIrrational numbers in reality
I have a square tile which measures 1 metre by 1 metre, by the Pythagorean identity the diagonal from one corner to another will be $\sqrt 2$ metres. However $\sqrt 2$ is an irrational number, could...
View ArticleNumber theory book recommendation.
I'm currently trained in Algebra, Calculus and Statistics in high school level. Basically, I've no knowledge at all in number theory as this subject is not taken seriously in my country.I was thinking...
View ArticleConnected rings and ideals
In what follows, a ring will denote a commutative ring with unity.I was doing some problems from a Commutative Algebra class and I came across the following problem/idea. It is a common practice to...
View ArticleMinimal Prerequesites to Study Section 3.1 (Cohomology Groups) in Hatcher
I want to take Algebraic Topology next year, and in the meantime, I thought it would be an excellent opportunity to listen to a topology reading group organized by some older students. So far, I have...
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