What is the mental model for (modular) lattices?
When we do computations* in abstract commutative rings, we use a mental model all the time: the arithmetic of integers. After the foundations are developed, we don't really care about the specific ring...
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Easy (graphic) interpretation of the third isomorphism theorem of groups
I came across these slides today and this diagram is shown when the third isomorphism theorem was presented.Could somebody please provide an intuitive way to explain the picture? Why do the green boxes...
View ArticleTranspose method for finding a basis for the row space
My linear algebra textbook outlines a method of finding a basis for the row space of a matrix by finding a basis for the column space of its transpose. Is there any point of using this method? It seems...
View ArticleIn order theory, is there a term for a set that upper bounds everything...
Suppose we have a partial order $(X, \leq)$, and for some element $x\in X$ we find a set $S\subseteq X$ whereNo two elements of $S$ are comparable, so for any $s,t\in S$ we have $s\leq t$ if and only...
View ArticleWhy n-Dimension? [closed]
In the real world, all objects, including ourselves, exist in three dimensions. Even no 4D object exists. However, in math we often work with n-dimensional spaces (might be infinity), which do not...
View Articleundecidable problems in Euclidean geometry
What is (are) the most "elementary" question(s) one could ask in Euclidean geometry with all its postulates/axioms including the "5th" that is (are) known to be undecidable. The "5th" is undecidable by...
View ArticleWhy are modular lattices important?
A lattice $(L,\leq)$ is said to be modular when$$(\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$where $\vee$ is the join operation, and $\wedge$ is the meet...
View ArticleTerminology for different notions of directional derivatives
I need to differentiate between two versions of directional derivatives of functions between finite-dimensional affine spaces :as the total derivative/differential applied to a fixed vector,as the...
View ArticleAlternative proof that base angles of an isosceles triangle are equal
The "classic textbook proof" of equality of base angles of an isosceles triangles which I studied in my school days is as follows:Let $\Delta ABC$ be a triangle with $AB = AC$ and let $D$ be the mid...
View ArticleWhich kind of non-linear component is required for a differential equation in...
Which kind of non-linear component is required for a differential equation in order to show frequency mixing?I going to focus the question on the electromagnetic wave equation as an example, by I am...
View ArticleAn example of a solution to a differential equation having a singularity (not...
Looking for examples about ordinary differential equations whose solutions are not everywhere defined (have a singularity), like a blown up on some finite time.I want to remember the exact definition...
View ArticleThermodynamics for math majors
I'm about to wrap a course in partial differential equations. We've discussed the heat/wave equations and introductory Fourier Analysis.I'd like to do some reading into the field of thermodynamics....
View ArticleVarious Ways to Prove the Half-Angle Formulae for Sine and Cosine
I was pondering about the different methods by which the half-angle identities for sine and cosine can be proved.$$\left|\sin\left(\frac{x}2\right)\right|=\sqrt{\frac{1-\cos...
View ArticleInteresting Applications of the $n^\text{th}$ Roots of Unity
I was recently looking at a trigonometry textbook for revision and I found that the identities$$\cos x+\cos\left(x+\frac{2\pi}3\right)+\cos\left(x+\frac{4\pi}3\right)=0$$$$\sin...
View ArticleConsequences of finite many twin primes?
Suppose , it turns out that the number of twin primes is finite (this is very unlikely, but let us assume it). Which consequences would such a result have for number theory ?To be more concrete : Which...
View ArticleStruggling to retain mathematical knowledge as a self-learner.
I’m a self-learner who studies mathematics because of a deep passion for the subject. However, I’ve been facing a persistent issue that has started to seriously affect both my progress and motivation....
View ArticleConfusion about a holonomy-related construction
Motivation: We start by posing a simple to state geometric optimization question: What is the largest surface of revolution with constant positive Gaussian curvature that can be embedded in...
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Approximate permutation block diagonalization for correlation matrices.
I'm working with correlation matrices and I'd like to find such permutation of variables that the correlation matrix is 'most block diagonal' (cf. picture below). How does one go about doing this? Is...
View ArticleWhy is Willans formula for prime's not considered a valid formula?
So, I was watching this video on Willan's formula by Eric Rowland, and it was mentioned that this formula does not count for generating primes, as it is extremely hard and time consuming to compute. I...
View ArticleIs $(-\infty,\infty)$ is an open or a closed interval?
Working in the reals, not the extended reals, I recently encountered several mathematicians, including my lecturer whose area of interest is topology, independently refer to $(-\infty,\infty),$...
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