Questions tagged [soft-question]
For questions whose answers can't be objectively evaluated as correct or incorrect, but which are still relevant to this site. Please be specific about what you are after.
11,840
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Is there a theory of isometrically embedded polyhedra on manifolds?
There is a book that is called embeddings in manifolds that studies topological embeddings and how they relate to each other (by homeomorphisms).
I was wondering if there is a study of isometrically ...
- 11
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How much to relearn before diving into Algebraic Number Theory?
I got my bachelors in mathematics in 2020 and haven't really touched anything related since. I could hardly get a passing grade now if I tackled the exams and HWs that I was able to solve before with ...
- 209
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Topic ideas for a 30-minute presentation in Analysis? [closed]
I have just completed the first two years of my math Bachelor's and I have completed Calculus 1 and 2 very successfully. This year, we have to do a presentation of 30 minutes (with 10 minutes of Q&...
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Book recommendation or sources that help me to improve my mental arithmetic skill and mental mathematics skill
I am interested in learning how to perform arithmetic operations (such as addition, subtraction, multiplication, division, calculating many steps on my head ,etc.) faster and more accurately in my ...
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Is this equivalence between two ways of expressing the coordinates of a rotated point useful?
Using " matrix by vector" multiplication, one can establish that the coordinates of the image ($P'$) of a point $P=(a,b)$ under a rotation by $\alpha$ radians ( counterclockwise) are :
$$ \...
- 1,597
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Best Chalkboard Paint? [closed]
I know mathematicians have strong opinions about chalk, so I was wondering if anyone has a strong preference for chalkboard paint? I have a chalkboard that is very glossy and even Hagoromo doesn’t ...
- 389
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Number of conjugacy classes of integral matrices with irreducible characteristic polynomials and ideal classes
Let $f\in \mathbb{Z}[X]$ be an irreducible monic polynomial of degree $n$.
A well-known theorem of Latimer and McDuffee states that their is a bijection between the following sets:
the set of $\...
- 14k
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Sequential characterization of limits in $\mathbb{R}^n$
Sequential characterization of limits:
Let $I$ be an open interval, $a \in I$, and $f:I \to \mathbb{R}$, then:
$\\$
$$\lim_{x \to a}f(x) = L \text{ if and only if }\lim_{n \to \infty}f(x_n) = L \text{ ...
- 1,188
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The book you'll need when someone came up with a random car plate number
I believe I've seen this book in a bookstore somewhere: it was a directory of integers, in ascending order, annotated with why each integer is interesting in certain aspects. e.g. the smallest (...
- 755
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How can I "compress" a known homogenous coordinate affine transformation?
Having worked with geometry I am aware of homogenous coordinates and affine transformations. With linear algebra we can express it as
$$\begin{bmatrix}x_1\\1\end{bmatrix} = \begin{bmatrix}R&T\\0&...
- 25.3k
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Where and how should I focus myself in my self-studies?
During the last few years of secondary school I found myself developing somewhat of an infatuation for the field of Maths, and particularly for the immense beauty that pure Maths gives way to. This ...
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Looking for the interesting usages of linearity of expectation [duplicate]
I am going to give a presentation about the basics of probability and for that, I would like to include some interesting (or even better mind-blowing) examples of how linearity of expectation can help ...
- 5,544
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Do Hoffman and Kunze seem to forgo easy "if and only if" statements?
For those who have read Hoffman and Kunze, am I imagining that they seem to have this really bad habit of stating "one-way" implications when it is literally no more work to state (and show) ...
- 1,143
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$\mu_{x}(A):=\langle P_Ax,x\rangle$ is also called Spectral measure?
As I know the spectral measure of a linear possibly unbounded self adjoint operator $T$ on a seperable Hilbert space is a projection valued measure defined by its measurable functional calculus $A\...
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Ways to check a homogeneous space $G/H$ is Riemannian homogeneous
When a homogeneous space $G/H$ is given, a differential geometer may ask
Is this homogeneous space also Riemannian homogenous?
Now many sources I read interpret this question as whether this space ...
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Is elementary number theory still being researched? [closed]
I was always under the impression that elementary number theory (where the techniques required to solve a problem don't require beyond high school level concepts) is not an active area of research and ...
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Notation to disambiguate substitution from multiplication of polynomials
Substitution of an element of a $K$-algebra into a polynomial from $K[X]$ for $X$ is notationally indistinguishable from multiplication. For example: if $P,Q\in K[X]$, is $P(Q - 1)$ the product of $P$...
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Platonic graphs in spaces having genus > 0
In the book "Introduction to Graph Theory", Richard J Trudeau dedicates a chapter on g-platonic graphs (platonic graphs in topological spaces with genus > 0).
My question is, what are ...
- 140
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Einstein summation notation and multi indices [closed]
Consider the Einstein summation notation $A^{ij}B_{ij}$. I know that this summation takes over a each pair of indices $i,j$ and there is no confusion if they are tensors and the indexing sets are ...
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Why is the abstract functorial definition of the tangent bundle not widely accepted?
The following quote from page 595 of Spivak's Calculus exemplifies my viewpoint on definitions:
It is an important part of a mathematical education to follow a construction of the real numbers in ...
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Why is it that $1-1+1-1+... \not = \frac{1}{2}$? [closed]
I am having a hard time understanding what is faulty with Euler's argument that $$1-1+1-1+... = \frac{1}{2}$$ using the geometric series summation formula. Is there a philosophical reason we reject ...
- 349
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Why the study of spanning tree is important in graph theory?
A spanning tree of a graph is a subgraph obtained by deleting only edges of the graph and which is also a tree.
Why does one study "spanning tree" in graph theory?
What are "spanning ...
- 4,955
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What means to say that a result depends on a theorem?
In everyday talk we say things like: we must use the fundamental theorem of calculus to calculate this integral; we must use the results of analysis to proof the fundamental theorem of algebra; we ...
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How to properly do detrending for the multidimensional Fourier transform?
Detrending as described here is a useful technique used as a preprocessor for the Fast Fourier Transform.
However how can it best be applied in a multidimensional case?
Own work:
What I have tried ...
- 25.3k
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1
answer
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Looking for properties of a pseudo-logarithm where you add all the (prime factor$\cdot$exponent) terms in an integer's prime factorization?
If, for positive integer $n=p_1^{q_1}p_2^{q_2}\dots p_k^{q_k}$ with the right side the prime factorization of $n$, we define $J(n)$ as $p_1q_1+p_2q_2+\dots+p_kq_k$, does $J(n)$ have any generally &...
- 1,725
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Is expression and its result is the same thing?
So, $\frac{1}{2}$ and $0.5$ are just two different ways to address the same object which is rational number $\frac{1}{2}$.
What about more complex expressions? Like $\{a, b\} \cap \{b, c\}$ is just ...
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Why do two equivalent systems of equations produce a different result?
I've been reading a book about material and energy balance when I faced a confusing problem when solving a system of equations. I came to notice some new concepts (that were already there, but didn't ...
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Why are local submersions and local immersions not studied in Lee's Introduction to Smooth Manifolds?
I'm reading through Lee's Introduction to Smooth Manifolds and wondered why local submersions or local immersions are not studied.
Let $M$ and $N$ be smooth manifolds (with or without boundary) and ...
- 4,710
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Does second order cone programming harder to solve than geometric programming?
As an Electrical Engineer, I have been studying convex optimization for a while. During my study, I see that most textbook claim that both second order cone programming and geometric programming can ...
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What is the difference between rigorous and formal?
This is a bit of a soft question, but is there a difference between rigorous and formal, and if so, what is it? I have seen those terms used interchangeably, and I use them interchangeably myself, but ...
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Frechet differentiability, RNP, and Asplundness (soft question)
I'm looking for references which provide good surveys and histories of results related to Gateaux and Frechet derivatives and their relationship to Asplundness and RNP. Specifically, any papers or ...
user520024
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integral of inverse powers of distance
I have encountered some integrals of this form
$\int d^n y \ \frac{1}{\Pi_{i=1}^m|x_i-y|^{p_i}}$
where $x_i,y \in \mathbb{R}^n$ and $|x-y|$ is the usual Cartesian distance. Also $\Sigma_{i=1}^m p_i&...
- 303
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How do topologies capture the notion of space? A concrete example?
When trying to make sense of locales and comparing to usual topologies, I realized I have no idea how topologies relate to my everyday intuition of space. To make the question simpler, I'll restrict ...
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Is there a rigorous textbook on step by step development for coming up with the equations of motion of classical dynamical systems?
I was trying to find some references for modelling the equations of motion of a simple dynamical system (say a pendulum on a moving mass) when I realized that the very vast majority of the material ...
- 11.1k
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Books like Béla Bollobás' The Art of Mathematics: Coffee Time in Memphis
I have been looking at The Art of Mathematics: Coffee Time in Memphis and The Art of Mathematics - Take Two: Tea Time in Cambridge by Béla Bollobás, and am really enjoying these books. I like the ...
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How far does this analogy go between smooth manifolds and sufficiently nice topological spaces?
In the last few months, I have read parts of the elementary theory of smooth manifolds and also more about general topology. One point made in a number of smooth manifold texts is that there are ...
- 3,143
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Why study probabilities in deterministic dynamical systems?
When studying hyperbolic dynamics and ergodic theory, one often come with probabilities (measures that give to the total space measure 1) even in deterministic systems. Do these have some intuitive ...
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Is labeling a numerical quantity "high," as opposed to "large," metaphorical, conventional, or technical? [migrated]
Why do we say "high frequency" and "high IQ" instead of large frequency or large IQ? Are these spatial metaphors? Are they just examples of language conventions with no technical ...
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Problem solving strategies for Measure theory (soft question)
I'm a beginner in learning Measure theory. I would like to ask are there some common problem solving tricks in Measure theory.
For example, usually when I'm doing a measure theory problem, once I ...
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Do techniques in contest math differ from those used in solving advanced (but *not research level*) course problems?
How does contest math differ from challenging course/textbook/test problems (not research math)?
There are several good posts on math.SE describing how contest math differs from research math. Most ...
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Is this notation a little bit ambigious? [duplicate]
We are all used to seeing $y=f(x)$ where we wish to plot the function $f$ on the $xy$ plane. We can differentiate both sides, $\frac{d}{dx}(y) = \frac{d}{dx}(f(x))$ to get the good old $\frac{dy}{dx} =...
- 1,081
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Does our perception of what $a$ is in these two expression's change?
Take the polynomial $f(x) = x^2 + ax + 2$ we might think of $a$ as being a paramter (arbitrary constant) and $x$ being a variable. This means, we choose a value of $a$ then we vary $x$ to get the ...
- 1,081
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Can I Start Studying Probability and Statistics After Algebra $1$, Geometry, Algebra $2$, and Linear Algebra?
I have a question regarding my mathematical background and its readiness for studying Probability and Statistics. I have completed Algebra $1$, Geometry, Algebra $2$, and Linear Algebra (half done).
...
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Difference between arbitrary constant and free variable
Take the following mathematical sentence $f(x) = ax^2 + bx + c$, before I learnt about first-order-logic I would say, $x$ is a variable and $a,b,c$ are all arbitrary constants. Now after learning ...
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Why are infinite-dimensional vector spaces usually equipped with additional structure?
In a first course in linear algebra, it is common for instructors to mostly restrict their attention to finite-dimensional vector spaces. These vector spaces are usually not assumed to be equipped ...
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Comparing coefficients gone wrong. How do we know it's allowed?
Disclaimer: This is a problem I am currently working through. I do not wish for the solution, but rather a good detailed explanation about the topic of comparing coefficients, and why in my case (...
- 1,081
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57
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Problem (exercise) books in commutative algebra [duplicate]
I am looking for problem books in commutative algebra. I could find a book named Exercises in Classical Ring Theory by T Y Lam. This book contains problems about non commutative rings also, but I need ...
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Help me ask this confused question about the divergence of the harmonic series
I've struggled for a couple of weeks to figure out the question I really want to ask, but without any success, so I'm just putting out this rather confused question out in hopes that someone can help ...
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Video lecture for charts on modular curves
I'm reading Diamond and Shurman's A First Course in Modular Forms, and I just can't seem to wrap my head around the construction of charts at elliptic points on a modular curve, and the proof that the ...
- 1,521
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Does this gradient-adjusting variant of the FFT have a name?
Consider the case of calculating an FFT of a not-quite-1-periodic function $$ t\to f(t) , t \in [0,1] , f(0) \neq f(1) $$ by transforming it into a periodic function :
$$f \to \hat f = f(t) - g(t) = f(...
- 25.3k