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.
12,407 questions
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Practical and historical role of Jordan measure
In my earlier questions, the proofs given by Asigan and D.R. showed that the Jordan outer/inner measure of the subgraph $[0,f]$ and the Darboux upper/lower integrals of $f$ are essentially the same ...
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Want an explanation of using Extreme Value theorem to prove Rolle's theorem(solved)
I'm confused about using extreme value theorem here
proof from
https://mathcenter.oxford.emory.edu/site/math111/proofs/rollesTheorem/
Consider the two cases that could occur:
Case 1:
$f(x) = 0$ for ...
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Examples of propositions for which there is $N,N'$ s.t. the proposition is true for $1\leq n\leq N,$ false if $N< n\leq N',$ true if $n>N'?$
I know of examples of "natural" (i.e. not contrived) propositions which are false for the first few, for example, $3,$ values of $n,$ but are true thereafter, for example, for all $n\geq 4.$ ...
- 25.2k
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Series that is known to converge/diverge but for which all these standard tests are inconclusive .
I have noticed that nearly every series I have been asked to analyze its convergence or divergence can be handled by the usual collection of tests: the limit test, Cauchy condensation, the integral ...
- 9,035
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Mathematics for Perspective Drawing
I am an undergraduate math major who likes to draw, and I would like to learn the math behind perspective drawing.
I recently watched this video: Everything about Perspective & Correct ...
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Why do we say "Let $ ABC$ be a triangle"? [closed]
My question is not just about let $ ABC$ be a triangle but rahter about all the mathematical statements where we say "Let some XYZ be PQR"
so why we? I mean even without let or suppose if ...
- 3,240
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Question regarding definitions of non-zero, zero divisors.
I was just wondering if a ring being torsion free is the same as an integral domain. Both definitions seem to state it means you have no non-zero, zero divisors. The only sense I can make of the (...
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Was a vector "an arrow with magnitude and direction" at the beginning, or did the abstraction happen early on? [migrated]
Many people seem to think that the picture of a vector that introductory physics gives (as an arrow with magnitude and direction) came before the more modern, abstract notion that we have today. It is ...
- 335
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Comparing different methods (Cayley transform and graph norm) to get a bounded operator out of an unbounded self-adjoint operator
Given a Hilbert space $\mathcal{H}$, if $T$ is a bounded self adjoint operator according to the spectral theorem this can be rewritten in eigen expansion as
$$
Tx = \sum_k \lambda_k \langle x, \phi_k \...
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How should I write proofs requiring complicated, yet routine, symbolic manipulation?
Often proofs may involve multiple lines of routine symbolic manipulation (e.g. taking derivatives, applying routine identities, or routine algebraic manipulations) which are distracting + tedious.
How ...
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Munkres Topology exercise which apparently does not relate to its chapter
Exercise 31.6 of Munkres "Topology" asks to prove $A=\{x \mid f(x)=g(x)\}$ is closed, when $f,g:X \to Y$ are continuous and $Y$ is Hausdorff.
The proof is straightforward. If $f(x) \ne g(x)$,...
- 2,611
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Can the squeeze theorem be used as part of a proof for the first fundamental theorem of calculus?
Can the squeeze theorem be used as part of a proof for the first fundamental theorem of calculus?
I ask because I was watching an MIT lecture on the proof of the first fundamental theorem of calculus ...
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Is $\log_0(0)$ undefined, indeterminate, or both? [duplicate]
I understand that $\log_1(1)$ is considered an indeterminate form, but the expression $\log_0(0)$ seems even more subtle. Algebraically, it is undefined because a logarithm cannot have a base of zero, ...
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(Hidden) symmetries examples in probability
I am looking for some probability problems in which there is a symmetry that you don't notice in the first place, but by noticing it, you can easily get the answer.
I have two examples in mind, and in ...
- 6,477
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Is there a word for saying that for two members $x$ and $y$ of a set, any true statement will be true if $x$ and $y$ are swapped?
Is there a word for saying that for two members $x$ and $y$ of a set, any true statement will be true if $x$ and $y$ are swapped?
For example, consider the group that you get when you take from the ...
- 1,244
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Understanding the order on $\mathbb{N\times N}$ defined by $(m,n)\leq (p,q)\iff \frac{2m+1}{2^n}\leq\frac{2p+1}{2^q}$. [closed]
I encountered an order relation $\leq$ on the set $\mathbb{N\times N}$ defined by $$(m,n)\leq (p,q)\iff \frac{2m+1}{2^n}\leq\frac{2p+1}{2^q}\text{, for every }(m,n),(p,q)\in \mathbb N\times \mathbb N$$...
- 4,416
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Solution to $\sum_{n=0}^{N-1} h(n) \cos (((\alpha - n)) \omega) = 0$
While studying Finite Impulse Response (FIR) filters, I came across this,
$$\sum_{n=0}^{N-1} h(n) \cos (((\alpha - n)) \omega) = 0$$
Where, (if required)
$h(n)$ are the amplitudes of the filter ...
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Why does choice give us "more" sets whereas foundation gives us "fewer"?
The axiom of foundation is typically described as a "restriction" on the set-theoretic universe. By adopting it, we focus our attention on the hereditarily well-founded slice of the universe:...
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Comparison of some books on functional analysis
I want to compare the books "A course in Functional Analysis + A Course in Operator Theory" by Conway with the books "Fundamentals of Theory of Operator Alegbras Vol I and II" by ...
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Why does the notation for derivative of a function depend on context?
Recently I have been getting caught up in mathematical notation and how hand-wavy & ambiguous it is in practice. Here is an example of something that troubles me:
From what I understand, functions ...
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How to rigorously robustness to outliers and prove that Least Absolute Deviation is more robust than Ordinary Least Square?
I am an engineer who really love math, and recently watched an educational video "Fitting a line WITHOUT using least squares?" where at timestamp 7:10, the presenter demonstrates that Least ...
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650
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What goes wrong when one goes from elliptic PDEs of second order to higher-order ones?
I was speaking with a friend the other day, and he asked me why so many elliptic PDE references (e.g. Gilbarg-Trudinger, Han-Lin) restrict themselves to PDEs of second order.
Annoyingly, I realized I ...
- 1,170
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Classical theorem which cannot be modified to be proven constructively
To which extent constructive mathematics can recover classical mathematics? Is there classical theorem about computable objects and operations which can not be proven constructively?
I know that many ...
- 650
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Wouldn't it be sufficient to consider only wedge products of $1$-covectors?
Why do we consider the wedge product of an element
$f\in A_k(V)$ and an element $g\in A_l(V)$?
Wouldn't it be sufficient to consider only wedge products of $1$-covectors?
After all, any element of $...
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Why are people so okay with stereographic projection mapping "every direction of infinity" to a single point? [closed]
stereographic projection gives a homeomorphism between $S^1$ and the real number line with an added point at infinity, so that whether you journey off in the distance to negative infinity or positive ...
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