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.

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What should I do to make up for my negligence as an undergrad?

This is a long one so please bear with me. I am not happy with where I am as a math student at the moment. So far, most of my courses have been conducted entirely online and the importance and ...
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1 answer
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Notation: minimum for all (sub) elements

Have an element list \Delta, these Elements have sub values, \alpha, \beta, ...
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1 answer
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Notation for smallest value greater than a number in a sorted set

In a finite set, is there a concept/notation for the smallest value larger than a particular element? For example, I have a sorted set as $ A = \{a_1, a_2, ..., a_k \} $ where $ a_2 > a_1, a_3 >...
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3 votes
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Learning Math from Old Hard Books

Does any one have a syllabus for how math was taught in UK in the 1800s? For example, a curriculum from the time that made use of the following book (as an example), Toddhunters Treatise on ...
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About Exercise 3.B.18 on p.68 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. Is this modification to 3.B.18 good or not?

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. Exercise 3.B.18 on p.68 Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists a surjective linear ...
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2 votes
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Pseudo-periodicity of analytic self-maps of the upper half-plane

I have a couple of questions, in increasing order of softness: Consider an analytic map of the upper-half plane into itself $f: \mathbb{H}\to\mathbb{H}$. When this function is $1$-periodic, i.e., $f(z+...
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Is there a probabilistic concept or theory for infinitesimal logarithmic product interpretation of integral?

If we have a number of independent events in probability, we can calculate it's likelihood : $$\prod_{\forall i} p_{i}$$ We can also consider ( where $H$ is the Heaviside step function ) $$\int L(t) H(...
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Could you please answer this short survey (30 seconds) about guessing the weight of a chair from an image? [closed]

We are a group of engineering students who are performing a statistical survey and we would like your help! Could you please answer this short survey (30 seconds) about guessing the weight of a chair ...
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Undecidibily of the Word Problem for Groups and First Order Logic

I am trying to derive the undecidability of the Word Problem for an arbitrary group, let's call this Problem $WP(G)$. It is clear to me that FOL and higher are undecidable and I want to reduce the $WP(...
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Intuition behind the notation of differential operators $\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial \overline z}$.

I am a graduate student of Mathematics.In this semester I am studying complex analysis.Stein Shakarchi's complex analysis book defines differential operators $\frac{\partial}{\partial z}$ and $\frac{\...
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2 answers
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Utility of the coordinate free definition of the derivative on manifolds.

Preface: I am not an expert on the topic of smooth manifolds, nor do I have the perspective gained from knowing many theorems proven on smooth manifolds. Please try to look at the problem from the ...
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1 vote
1 answer
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About Exercise 3.D.13 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 3.69 Suppose $V$ is finite-dimensional and $T\in\mathcal{L}(V)$. Then the following are equivalent: (a) $T$ is ...
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1 vote
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Proving a uniform bound given that $0<A-f\leq B$

I've been working in the following problem. The problem itself is quite large and involves many steps, but the point is that I'm in a step where I have a certain open set $\Omega \subseteq \mathbb R^n$...
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If $𝑓∘𝑔∘ℎ=𝑓 ∧ 𝑔∘ℎ∘𝑓=𝑔$ then must $ℎ∘𝑓∘𝑔=ℎ$?

If not, then What can be said of each $𝑓,𝑔,ℎ$ and are there any simpy-definable minimal conditions imposable upon one or more of the indexable functions that would ensure this symmetric closure? ...
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1 vote
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Getting better at Geometry - soft question

I am a high school student in the UK (year 12 UK, grade 11 US. I am very interested in maths and so have been doing some STEP papers (2 and 3) in my spare time. I have become reasonably proficient at ...
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1 answer
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Can Hilbert's Hotel be explained by a difference between ordinal numbers and cardinal numbers

In taking a philosophy of maths course I have been very curious about the notion of infinity, and whether or not it is paradoxical. One thing I have frequently thought is that "infinity" as ...
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What software can I use to graph a solid of revolution between 2 functions?

What software is used to graph solids of revolution between 2 functions like the ones below?
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0 answers
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Examples of Substructures that "do not know they are that substructure"

Just learned $\mathbb{L}\vDash \mathbb{V}=\mathbb{L}$ and was warned that this property is not obvious with the counterexample mentioned being $HOD$. I can think of a few examples of definable ...
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1 vote
2 answers
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Intuition for fibers over a point in $\operatorname{Spec}(A)$

This is a rather soft question, but I would like to see what I get. I'm currently reading/working through Atiyah Macdonald, and I just did exercise 21 of Chapter 3. I won't repeat the entire exercise ...
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3 votes
1 answer
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A Reference From Andrej Bauer's Recent Talk on Countable Reals

Andrej Bauer gave a talk today in the topos institute colloquium (video here) announcing a proof that the dedekind reals can be countable in the absence of LEM and CC. At roughly the 27 minute mark, ...
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4 votes
1 answer
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Practical uses of $p$ norms for $p\notin \{1,2,\infty\}$?

We all love normed spaces, but it seems like the $1$-, $2$-, and $\infty$-norms get the lion's share of the love. That's not admittedly not without good reason, but what of the other unsung norms with ...
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1 vote
1 answer
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Integrating Dirac delta distribution from $0$ to $1$.

Let $\delta$ be rigorously defined as a generalized function (lim of a function). I am guessing that $\int_{-1}^0\delta(x)d x=\int_0^1\delta(x)d x=\frac{1}{2}$? Also, let $E$ denote a set contains 1/3 ...
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6 votes
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What are some good sources for probability heuristics?

I've studied rigorous probability theory at university but I find myself struggling to solve questions in probability as quickly as some of my peers. I think that what I try to do is 'translate' the ...
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6 votes
1 answer
124 views

What is expected from a Master's thesis, especially contrasted with a Bachelor's or PhD thesis? [closed]

I'm thinking of enrolling in a Master's degree in Mathematics and it culminates with a thesis rather than being purely course-based. I am wondering what is expected of students writing a Master's ...
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4 votes
1 answer
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A Collection of Bogus Proofs

Hello M.S.E. people, This question is just for fun, don't take it seriously :). We have all encountered Bogus Proofs, which seem logical and reasonable, but they prove some claims which are completely ...
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PUCT Analoge for Adversarial Bandits

Many people are familiar with PUCT, the multi-armed bandits algorithm that produces good results (logarithmic regret) in the stochastic regime that utilizes 'predictions' of the best arm. This ...
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7 votes
1 answer
146 views

Does "Entropy" explain why the Normal Distribution is so "Popular"?

Recently, I have learned about the Principle of Maximum Entropy with regards to Probability Distribution - in particular, when certain "information" (i.e. constraints) is available about ...
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variational methods of semilinear elliptic problem with critical sobolev index

There are many exitence results of semilinear elliptic problem with critical sobolev index, for example, the Brezis-Nirenberg problem: $$-\Delta u =\lambda u+u|u|^{2^{*}-2}.$$ However, it seems all ...
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Vectors as arrows

Given some vector space, can we take vectors as arrows to form a category? I mean, I am not thinking the vectors spaces as objects, and linear transformations between them as arrows. In the monoid ...
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4 votes
0 answers
28 views

Understanding compatibility of PDE

THEOREM $1$: The equations $$f(x, y, z, p, q)=0\qquad (1)$$ and $$g(x, y, z, p, q)=0\qquad(2)$$ are compatible on a domain $D$ if (i) $J=\frac{\partial(f, g)}{\partial(p, q)}=\left|\begin{array}{ll}f_{...
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2 votes
0 answers
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Proofs and Types: Girard's remarks on Theoretical Computing

In the first chapter of Girard's Proofs and Types (1989) one finds the following remarks: Theoretical Computing is not yet a science. Many basic concepts have not been clarified, and current work in ...
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1 answer
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How to reduce a straight line - of known equation and passing by two rotating points - to a line segment : which condition should be imposed on $x$?

Let point $P=(cos(\alpha), sin (\alpha))$ and point $Q = (cos(\alpha+ \pi), sin (\alpha+\pi))$ be two points moving on a circle ( of center $(0,0)$ and of radius $1$). The straignt line passing ...
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About Exercise 5.B.20 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. Exercise 5.B.20: Suppose $V$ is a finite-dimensional complex vector space and $T\in\mathcal{L}(V)$. Prove that $T$ has ...
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Intermediate Linear Algebra Text

I have been struggling to find an intermediate-level linear algebra textbook. I have tried Strang's Introduction to Linear Algebra and found it convoluted. Additionally, some of the books like Axler's ...
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1 answer
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Soft Question: Introductory books on Algebraic Number Theory

I am a high school student in Britain. I have recently studied elementary number theory and absolutely loved it. I got as far as to prove results like quadratic reciprocity, Fermat's sum of two ...
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1 vote
0 answers
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Should I still pursue math? [closed]

I'm a sophomore studying physics in the US. I have always liked math and physics, and as a high school student I've always aced in both subjects, in fact they are the only subjects that I get good ...
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8 votes
0 answers
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Externalizing Statements about the Effective Topos (or: what is it good for?)

Say we prove some theorem constructively (by which I mean, without choice and without LEM). Then it's true inside an arbitrary topos, and in the case of sheaf topoi we can externalize our theorem into ...
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terminology : Big difference between basic real analysis and basic calculus over reals? [duplicate]

Im a bit confused by terminology: What is the difference ( if any ) between Basic Real Analysis without measure theory. and Basic Real Calculus ( calculus over the reals , not complex analysis nor ...
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2 answers
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Let $h$ be such that $h(x)=\int_a^x f'(t)dt$. For which value of $a$ is function $h$ identical to function $f$?

An informal reasoning had led me to the ( erroneous) conviction that: "the integral of the differential $f'(x)dx$ is identical to $f(x)$, because the (infinite) sum of the infinitesimal linear ...
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-2 votes
2 answers
61 views

How much time should I spend on a problem when preparing for a timed exam? [closed]

I'm preparing for a university entrance exam that has different subjects (like Biology, Mathematics, Physics, Chemistry ,...). In the Math section, there are $30$ problems for $47$ minutes ($94$ secs ...
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1 vote
2 answers
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Is the boundary of singleton set in $T_1$ space is empty?

The singleton set in $T_1 $ topological space is closed so, I wonder how i use this information to prove that boundary of this set is empty.
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When to use Remark and Note

I am writing notes for some topics of Math in details. I am confused about one thing that when do we use word 'Remark' and 'Note'. I referred some texts.. but couldn't figure out the exact difference.
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2 votes
4 answers
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What breaks down when generalising from sets to classes?

I have been familiarising myself recently with some basic definitions in category theory, where we work with classes of objects and morphisms, as opposed to just sets. It feels like I am long overdue ...
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25 votes
5 answers
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Are there are any inherent mathematical reasons some proofs are difficult?

This is not a complaint about my proofs course being difficult, or how I can learn to prove things better, as all other questions of this flavour on Google seem to be. I am asking in a purely ...
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1 answer
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Analytic works versus synthetic works in mathematical research

With the purpose to clarify my ideas about terminology I would like to ask in Mathematics Stack Exchange what is the difference of two verbal expressions: when a professional mathematician understands ...
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-3 votes
0 answers
58 views

Why do we split the Number System into Even and Odd? [closed]

I started thinking about this whilst learning that the difference between a Baryon and a Meson is their possession of an odd or even number of quarks; I couldn't think of the name for the odd-even ...
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11 votes
7 answers
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The vertices of a tetrahedron lie on a sphere

I am struggling a bit with the following (elementary) question: How to prove that every regular tetrahedron admits a circumsphere, i.e. there exist a sphere on which all four vertices lie. I would ...
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2 votes
1 answer
49 views

How to graph the elasticity function ( knowing the - linear-demand function and the price function )? What goes wrong in my Desmos graph?

My goal is to visualize the graph of the elasticity function for a linear demand curve . The problem I face is that the elasticity function graph I came up with looks unfamiliar. I suppose my formula ...
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0 answers
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Does any weak form of compactness can force the conclusion of the closed graph theorem to be true?

Closed Graph Theorem (Topological Version , necessary condition ) : $X,Y$ be two topological space where $ Y $ is a compact Hausdorff space and $f:X\to Y$ be a map with $G_f=\{(x,f(x)):x\in X\}\subset ...
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0 votes
1 answer
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What does a definite integral from a function to function represent?

Everyone knows that integrating a function $f(x)$ from a to b can be represented by the area under $f(x)$ graph from a to b. But If we are integrating lets say $f(y)$ from $h(x)$ to $g(x)$ what will ...
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