Questions tagged [big-picture]
Questions designed to get an overview of a specific subject or body of results or to understand the relations among similar definitions, techniques or concepts appearing in different sub-fields of mathematics. While such questions by their very nature sometimes cannot be made very narrow and focused, it can be helpful to keep in mind that the design of MathOverflow does not make it a good fit for questions that are too broad.
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How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher number sets
How did Ramanujan come up with the Ramanujan summation and is it possible to extend it to higher sets (Everything circled in red is what I'm interested in (+ the Cauchy integral to make it Dedekind ...
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Clique-coclique and uncertainty
The clique-coclique inequality states that for a graph $G$ on $n$ vertices that is either distance-regular or vertex-transitive, the independence number $\alpha(G)$ and the clique number $\omega(G)$ ...
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Can $N!$ be computed in less than $\mathcal{O}(N)$ operations?
The standard algorithm to compute the factorial function $N!$ via repeated multiplications has complexity $\mathcal{O}(N)$, in the model in which each operation costs 1, no matter how many digits the ...
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Can this kind of formal Mereological based interpretation of Set Theory be significant in understanding Set Theory?
To Atomic General Extensional Mereology $\sf AGEM$ add primitives of non-directional linking, which is a ternary relation symbol, and the primitive binary relation "is connected to". The ...
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Meaning of a quantum field given by an operator-valued distribution
I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me.
Let $\mathcal{H}$ be a Hilbert space in which ...
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Do empirical studies have a place in contemporary mathematics research?
I was thinking about the Collatz conjecture a while back (I know, not the healthiest thing to think about). It occurred to me that while I might not be able to prove it true for all positive integers, ...
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What is an important mathematical question?
$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste....
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A geometric interpretation of the fractional Fourier transform
I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18):
Once a week, Feynman led Physics X, where freshman and sophomores could ask ...
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Intuition behind Kleene's “second algebra” $\mathcal{K}_2$
The “second Kleene algebra” $\mathcal{K}_2$ is defined, e.g. here on nLab, or in section 1.4.3 of van Oosten's book Realizability: an Introduction to its Categorical Side (2008), or as example 3.4 of ...
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What is the relationship (if any) between constructivism, finitism and predicativism?
The terms “constructivism”, “finitism” and “predicativism” refer to ideas / currents in the philosophy of mathematics (or loosely defined conditions on a system of logic) that I think I understand ...
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Comparing Selberg and Eichler-Selberg trace formulas
The trace formula of Selberg gives an equality between trace of Hecke operators (a spectral sum) on spaces of Maass forms and sums over closed geodesics mostly. The Eichler-Selberg trace formula, ...
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Any concrete survey on infinite and finite injury method?
I hope to have a historic outline of infinite and finite injury method and their main technical introdution.Any concrete survey on infinite and finite injury method recommended?
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Is there any intuition of why the both, regularized logarithm of zero is $-\gamma$ and the regularized logarithm of Bernoulli umbra is $-\gamma$?
If we take the MacLaurin series for $\ln(x+1)$ and evaluate it at $x=-1$, we will get the Harmonic series with the opposite sign: $-\sum_{k=1}^\infty \frac1x$. Since the regularized sum of the ...
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The advantage of asymmetric objects
We know that it is usually much easier to work with highly symmetry objects, the objects that have many automorphisms like the sphere, Lie groups, complete graph,... But is there any advantage of ...
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Sieve theory through variational principles
Disclaimer: I'm just starting to read Sieve Methods by Halberstam and Richert, so my present knowledge of the subject is close to zero, but it made me wonder if some connection to physics could exist, ...
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Examples of rich theories that started out as an infinite-dimensional inquiry
It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it ...
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Stories where a different definition lead to an inaccurate conclusion/a misunderstanding/etc
The overall question: What are some good examples where a different understanding of terminology or notation caused you to misinterpret a result in a way that was inaccurate? The intent here is of ...
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Breakthroughs in mathematics in 2021
This is somehow a general (and naive) question, but as specialized mathematicians we usually miss important results outside our area of research.
So, generally speaking, which have been important ...
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What are the "hot" topics in mathematical QFT at the time?
I am currently finishing my Master's studies in mathematical physics. One topic which always interested me a lot were modern mathematical approaches to Quantum Field Theory (QFT) as well as the ...
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Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]
It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance,
$0^ \left(
\begin{array}{cc}
\frac{1}{2} &...
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In what circumstances do we typically encounter expressions like $(c/2+1/2)^n \pm(c/2-1/2)^n$?
It attracted my attention that in many areas of mathematics we sometimes encounter expressions of the form $(c/2+1/2)^n \pm(c/2-1/2)^n$, where $c$ is some kind of a known constant.
Split-complex ...
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Motivation for birational geometry
I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
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Why do we have two theorems when one implies the other?
Why do we have two theorems one for the density of $C^{\infty}_c(\mathbb{R}^n)$ in $L^p(\mathbb{R}^n)$ and one for the density of $C^{\infty}_c(\Omega)$ in $L^p(\Omega)$? with $\Omega$ an open subset ...
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The use of computers leading to major mathematical advances II
I would like to ask about recent examples, mainly after 2015, where experimentation by computers or other use of computers has led to major mathematical advances.
This is a continuation of a question ...
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Where do these divergent integrals appear in mathematics and physics?
I have already asked a similar question, albeit far more extensive, but it was criticized and closed for being too extensive and promotional. So, here is a greatly truncated and focused version.
Since ...
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Anomalous phenomena [closed]
What are examples of strikingly anomalous phenomena in mathematics, where just one or a rather small number of cases stand out because they don't fit a general pattern?
This is most interesting when ...
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Differentiation of functions over graphs
In short: There are various ways to define differentiation over a graph. I am trying to get the big picture, like a more complete and structured bestiary.
Definitions.
Let $G=(V,E)$ be a directed ...
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What intuitive meaning "determinant" of a divergency (divergent integral, series, germ, pole or a singularity) can have?
I am working on the algebra of "divergencies", that is, infinite integrals, series, and germs.
So, I decided to construct something similar to the modulus or determinant of a matrix of these ...
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On primes of specified length and bit pattern
Denote $P(n,k)$ to be the number of primes between $2^n$ and $2^{n+1}-1$ having $k$ number of $1$s in its binary expansion between the $n+1$th binary digit and the least which is always $1$ if $n>1$...
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Motivation for $C^*$-algebras
I just gave a presentation on exotic group $C^*$-algebras and someone asked why these are studied. I could answer that they can be used to construct $C^*$-algebras with certain properties. However, I ...
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Comparing the existing formulations of universal algebra and their levels of generality
I am a newcomer to universal algebra and I just read this (very good, IMO) book on the topic:
Adámek, J., Rosický, J., & Vitale, E. M. (2010). Algebraic theories: a categorical introduction to ...
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Nonconvexity and discretization
Edit: Here's a more down-to-earth, and somewhat weakened, but I believe still nontrivial, version of the main theorem.
Prototypical nonconvex spaces are $\ell^p$-spaces for $0<p<1$, say $\ell^p(\...
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Morphism of non-commutative algebras
Disclaimer: this question is a "big picture" one that comes from my personal thoughts in physics. If it doesn't fit this site, please tell me.
While having a walk, I thought a bit about what ...
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Sober spaces vs. spatial frames-a big picture
For any topological space $X$ one can consider the so called frame of all open subsets of $X$ to be denoted by $\mathcal{O}(X)$. If $f:X \to Y$ is continuous taking the inverse image we get the ...
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What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?
First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that ...
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Evidence of optimality of sieve algorithms
Sieve techniques apply to integer factoring and discrete logarithm to provide $2^{O(((\log n)(\log\log n)^2)^{1/3})}$ complexity for $n$ bit factoring and $n$ bit prime discrete logarithm.
The state ...
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Is there a bipartite graph whose determinant corresponds to number of perfect matchings?
Let $M\in\{0,1\}^{n\times n}$ be a square integer matrix. If we consider $M$ as biadjacency of a balanced bipartite graph on $2n$ vertices having $n$ vertices of color $1$ and $n$ vertices of color $2$...
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Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?
Consider the operator $\frac D{e^D-1}$ which we will call "shadow":
$$\frac {D}{e^D-1}f(x)=\frac1{2 \pi }\int_{-\infty }^{+\infty } e^{-iwx}\frac{-iw}{e^{-i w}-1}\int_{-\infty }^{+\infty } e^...
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Is there a quantum analog of Kolmogorov Complexity?
Kolmogorov Complexity (interpreted in terms of shortest program computing a string) and Shannon Entropy are quite similar.
Since there is a quantum entropy is it reasonable to ask if there is quantum ...
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Geometric intuition for $R[x,y]/ (x^2,y^2)$, kinematic second tangent bundle, and Wraith axiom
This is a sort of continuation of this question.
In synthetic differential geometry (SDG), we have $D\subset R$ comprised of the second order nilpotents. The Kock-Lawvere axiom (KL axiom) implies that ...
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What is the motivation for a Frobenius manifold?
A Frobenius manifold is a type of manifolds with extra structure.
The main examples are quantum cohomology (viewed as a space itself), GBV algebras, the ``Saito'' examples arising from singularities (...
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Axiomatic definition of quantum groups
This is a question I've discussed with a lot of mathematicians, and have read some mathematical texts about, and watched some conference talks about: what is, axiomatically, a quantum group?
There are ...
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Each mathematician has only a few tricks
The question "Every mathematician has only a few tricks" originally had approximately the title of my question here, but originally admitted an interpretation asking for a small collection ...
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Every mathematician has only a few tricks
In Gian-Carlo Rota's "Ten lessons I wish I had been taught" he has a section, "Every mathematician has only a few tricks", where he asserts that even mathematicians like Hilbert ...
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Mathematics based only on real numbers [closed]
I'm aware that >90% will outright reject this, so feel free to ignore it. I'd much appreciate those trying to figure out in which way this question (or rather its eventual answer) would make sense.
...
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What are some interesting relationships between pi and phi? [closed]
Phi is the golden mean solution to the 1/x=1+x and pi the transcendental number relating the radius of the circle to its area.
A side note: while there are really interesting series converging to pi, ...
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Analogues over finite fields of certain integers defined multiplicatively in $\mathbb Z$
For any irreducible polynomial $f$ of degree $d$ in finite field $\mathbb F$ we have that $$x^{|\mathbb F|^d}-x\equiv0\bmod f$$ and thus the polynomial $x^{|\mathbb F|^d}-x$ is the analog of factorial ...
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Why is modular forms applicable to packing density bounds from linear programming at $n\in\{8,24\}$?
Sphere packing problem in $\mathbb R^n$ asks for the densest arrangement of non-overlapping spheres within $\mathbb R^n$. It is now know that the problem is solved at $n=8$ and $n=24$ using modular ...
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Why does the Lax pair formalism look so similar to the Hamiltonian equations, and what is the significance of this?
If we have a Lax pair for a system, which we'll call operators $L$ and $B$, then the system
\begin{align*}L\psi&=\lambda\psi\\
\psi_t&=B\psi\end{align*}
has as its integrability condition ...
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Update on "Hopf algebras: their status and pervasiveness" by Hazewinkel
Hazewinkel wrote this article in 2005. Perhaps it's time for an update.
For example, updating item
34: Ordinary differential equations much work has been done on the underlying Hopf algebra (HA) of ...