List of maths fields I have interest in: 1. Group theory in terms of orbits and actions 2. Zero term algebra and division by zero algebra 3. Integration in the language of abstract algebra and as a functional, symmetry of integrands 4. Optimising proofs given axiomatic systems 5. Category theory 6. Unnatural algebraic structures 7. Patterns in expanding multiplications of polynomials 8. Set of all counterexamples given a proposition 9. Tensor visualisation and intuitions 10. Numerical analysis methods to explore special regions or points of mathematical functions or systems of equations
Is it true that if $R$ is a division ring, then $(R\setminus\{0\},\cdot)$ is cancellative? It feels like that should be obviously true, but I've made mistakes about this sort of stuff in the past, so I like to make sure.
In university I have learnt about the concept of tensors, which are multilinear maps in that it is a map such that it is linear with respect to all its arguments
$$f(a+b+c+d+e+f+g....)=f(a)+f(b)+...+f(g)+...$$
and in physics, it is geometric in that it is independent of basis
In biophysics I l...
Can you guys recommend me books on analysis and linear algebra? Like from calculus 1 to formal integrals and beyond, polynomial bounding and stuff that’s more than calculus.
On linear algebra I’m already using @TedShifrin’s book. I’m trying to learn everything about the first year of uni so I can do scientific research.
@TedShifrin following up on the other day -- So if I'm reading this problem correctly, what I ended up having (and again, I don't think the lecturer himself understands what he's asking, since his matrices are not even well defined) is two sets of orthogonal columns -- a set of column vectors for the even functions -- call their span $F$, and a set of column vectors for the odd functions -- call their span $G$, whence $F \perp G$. The question asks how many even left singular vectors in the SVD decomposition of $A$ (where $A$ is not well defined mind you). What I was trying to do is force i…
Here is an identity about numbers of tuples $(a,b,c)$ with $a,b,c\in [1,n]$ such that $ab=c$: $$|\{(a,b,c)\in [1,n]^3|ab=c\}|=2\sum_{i=1}^{\lfloor\sqrt{n}\rfloor}\Big(\lfloor\frac ni\rfloor-i\Big)+\lfloor\sqrt{n}\rfloor$$
I'm not sure how to get this. Does anyone have any idea about the counting method?
(Next time I saw another palidrome question out here that is boring unlike the 3 starred one, you are out of my view. Cannot kept on being banned by producing a rant) (Of course, I already knew at least 3 users have pressed the ignore button on me, but that will not affect my workings)
now to study the above two questions...
@user193319 Intuitively I can think of a loop that covers that, but it will require the point where it joined to be cover by the loop twice. I am not sure if paths like these are legal
While discussing about prime numbers with other users, I noticed that:
$(1)$ There are very few combinations palindromic prime numbers that have products which are palindromes.
Ex : [2, 3], [2, 30203], [2, 30403]
$(2)$ For the large range I tested on PARI/GP, I noticed that all the palindromi...
sorry for being rude, but ever since a long time ago some homework vampired drained me in my tutoring, I have very low patience for pure computation type questions
Regarding palidromes, similar to primes, I don't know if there exist a way to study them systematically. It seems it suffers the same hard problems as transcendental number theory
and regarding that 3 starred question, I thought it would have been easier to solve since it is the subset of the actual open question of whether there are finite palidrome numbers
chat.stackexchange.com/rooms/92782/… Friends i have created this room so that a beginner or any one on Math SE can discuss question (before posting it ,if he wants to discuss )
Problem: Let $F$ be a field with $\text{char} F = p$ for some prime $p$. Show that if $X^p - X - a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$
Solution in back of the book:
This solution confuses me slightly. Why is $f(X)$ splitting as two monic polynomials? ...
Prove that a sequence of continuous functions $f_n : [0,1] → \Bbb R$ converges uniformly to a continuous function $f$ then $f_n (x_n ) → f(x)$ whenever $x_n → x$.
Fixing $i$ in $f_i(x_n)$, we see by continuity of $f_i$ that $f_i(x_n)\to f_i(x)$ as $x_n\to x$ and now $f_i(x)\to f(x)$ by uniform convergence. Am I correct?
be wary though of the bi-sequence $a_{ij} := \delta_{ij} = \begin{cases}1&i=j \\ 0&i \ne j \end{cases}$ where $\displaystyle\lim_{j \to \infty} a_{ij} = 0$ for every $i$, and $\displaystyle \lim_{i \to \infty} a_{ii} = 1$
One of my connecting flights on the return trip got cancelled by the airline. That’s annoying but not Travelocity’s fault
So I contact Travelocity to get the flight changed to one that will work. There is an obvious fix
But it takes days (via their Facebook IM support) for them to convey that info to me correctly
They first quoted the flight as coming from LGA aka Laguardia NY. probably what they meant was LGW aka Gatwick in London. But that’s exactly the same info as the cancelled flight, and they finally confirmed that it should have been LHR aka Heathrow in London
So I finally confirmed that with them, and asked them to schedule it
They tried, but that was on Friday which is apparently is a holiday for Icelandair and they’ll have to wait until Monday. Again, annoying, but not in their control
So I ask them about it yesterday, and they ask me to be patient
Problem: Let $F$ be a field with $\text{char} F = p$ for some prime $p$. Show that if $X^p - X - a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$
Solution in back of the book:
This solution confuses me slightly. Why is $f(X)$ splitting as two monic polynomials? ...
Consider the following problem 5 below.
I am trying to construct the stated function f. I tried many functions one which sends the matrix of the form below to xy. I also tried to send the matrices of the form below to x + y. None of them work.
Recall a left uniform continuous function f on G i...
Consider the following problem 5 below.
I am trying to construct the stated function f. I tried many functions one which sends the matrix of the form below to xy. I also tried to send the matrices of the form below to x + y. None of them work.
Recall a left uniform continuous function f on G i...
If $p(x,t)=x^n+a_1(t)x^{n-1}+...+a_n(t)$ is a family of parametrized polynomials, where t lies in a compact interval and all the $a_i(t)$ are continuous, then why do the roots of $p(x,t)$ in $x$ for all $t$ form a bounded set?
@Thorgott Because the roots of a polynomial vary continuously with the coefficients. This can be proved, for example, with Rouché's Theorem in complex analysis.
In university I have learnt about the concept of tensors, which are multilinear maps in that it is a map such that it is linear with respect to all its arguments
$$f(a+b+c+d+e+f+g....)=f(a)+f(b)+...+f(g)+...$$
and in physics, it is geometric in that it is independent of basis
In biophysics I l...
While discussing about prime numbers with other users, I noticed that:
$(1)$ There are very few combinations palindromic prime numbers that have products which are palindromes.
Ex : [2, 3], [2, 30203], [2, 30403]
$(2)$ For the large range I tested on PARI/GP, I noticed that all the palindromi...
Problem: Let $F$ be a field with $\text{char} F = p$ for some prime $p$. Show that if $X^p - X - a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$
Solution in back of the book:
This solution confuses me slightly. Why is $f(X)$ splitting as two monic polynomials? ...
Consider the following problem 5 below.
I am trying to construct the stated function f. I tried many functions one which sends the matrix of the form below to xy. I also tried to send the matrices of the form below to x + y. None of them work.
Recall a left uniform continuous function f on G i...
