Highest scored questions
159,029 questions
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2
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Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumptions? [closed]
I am interested in asking the following question:
What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation ...
- 6,132
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votes
3
answers
670
views
Remarkable articles about the distribution of prime numbers that were written by contemporary physicists [closed]
I would like to ask about if you know papers containing remarkable achievements that were written by contemporary physicists concerning the distribution of prime numbers (or closely related, maybe the ...
-4
votes
3
answers
942
views
Is the given expression, monotonically increasing or decreasing with increasing x?
Is the given expression, monotonically increasing or decreasing with increasing x?
$\frac{1}{x \log(x)} \sum_{i=1}^{\infty} \frac{\mu(i)}{i} x^{1/i}$
EDIT: This is the derivative of the prime ...
- 731
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votes
2
answers
228
views
An elementary-looking integral inequality
This might seem a bit easy but I still like to ask it for pedagogical reasons.
QUESTION. Is this inequality true for non-negative integers $n$?
$$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
- 43.2k
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votes
1
answer
813
views
How i show this beautiful inequality :$\frac{x^n}{x^m+y^m}+\frac{y^n}{y^m+z^m}+\frac{z^n}{z^m+x^m}\geq \frac{3} {2}(\frac{1}{\sqrt{3}})^{n-m}$? [closed]
This question accross to this question from SE which there some answers but they r n't
enough to me hop to see MO what can they say about it .
let $m,n$ be integers, show that if $ n>m\geq 0 $ :
...
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1
answer
442
views
Are spectra determined by their homotopy groups?
A famous theorem of Whitehead essentially states that spaces are determined by their homotopy groups. Is this true for spectra too?, i.e,
$$
\text{question: is a spectrum $E$ determined by its ...
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votes
2
answers
2k
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what part of using vieta's formulas violates quintic non-solvability? [closed]
You can write the n roots of an n degree polynomial in terms of its n coefficients, i.e., "Vieta's" formulas.
You can solve this system of nonlinear equations using Newton's method and the Jacobian. ...
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votes
1
answer
2k
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Is this equivalent to Goldbach's conjecture?
As one can easily prove https://math.stackexchange.com/questions/20564/sums-of-square-free-numbers-is-this-conjecture-equivalent-to-goldbachs-conjec every integer greater than $1$ is a sum of two ...
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votes
2
answers
421
views
Would it be simpler, pedagogically speaking, if textbook writers introduced root systems as an example of a quandle?
I could never, for the life of me, recall the definition of a root system in Lie theory. It probably doesn't help that I've never taken a course on Lie Theory - the algebra, or the groups, or the ...
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votes
1
answer
605
views
Multiplicative Persistence - Highest persistence found? [closed]
tried to ask on the math reddit but got deleted due to my account being new.
Is the record for highest multiplicative persistence found still 11? As I may have just found a number with persistence of ...
- 21
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votes
2
answers
6k
views
Factorizing polynomials of several variables (in a different perespective)
I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^...
-4
votes
1
answer
882
views
Are there infinitely many $n$ such that $n!-1$ and $n!+1$ are prime numbers? [closed]
Are there infinitely many $n$ such that $n!−1$ and $n!+1$ are prime numbers?
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votes
2
answers
950
views
What is a coordinate less definition of differentiable manifolds
![enter image description here]
From Clifford algebra to geometric calculus by d. Hestenes
https://en.wikipedia.org/wiki/Universal_geometric_algebra
The attempt above is to have the base manifold ...
- 147
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votes
3
answers
524
views
Relation between elliptic curve and Fermat's last thereom
I am looking for a elaborate explanation how the elliptic curve $E (a, b) := y^2=x(x-a)(x-b)$ is associated with the solution of $a^n+b^n=c^n$.
In 1969 Hellegouarch performed the elliptic curves $E (a,...
-4
votes
2
answers
272
views
Has nontrivial solution in positive integers of a diophantine equation: $x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$ [closed]
Has nontrivial solution in positive integers of a diophantine equation as follows ?
$$x_1^2+x_2^2+x_3^2+x_4^2=y_1^2+y_2^2+y_3^2+y_4^2$$
Where trivial solutions are $x_i=y_j$.
Can you send me any ...
- 477
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votes
1
answer
614
views
What is an oracle, really? [closed]
Regarding oracles, might this be a reasonable description of their inner workings (this from Hartley Rogers, Jr.'s text, Theory of Recursive Functions and Effective Computability)?
Why should I ask ...
- 6,132
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votes
1
answer
365
views
Are there major research areas in math? Or is it a lot of individual efforts? [closed]
In physics, for example, dark matter is a major research area now. And there are specific parts of that trending. Is there anything similar in math? Is there something the majority of mathematicians ...
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votes
1
answer
583
views
What's the minimum amount of knowledge to start doing research? [closed]
There are cases in which you have too much knowledge of something to do anything interesting ,and cases in which a lack of experience with a problem (and the prejudices about it) helps someone solve ...
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votes
2
answers
1k
views
Riemann Siegel function and gamma function
I ask about an idea to prove this formula:
$Γ(1/2-iβ)=((\sqrt{π})/(\sqrt{\coshπβ}))\exp(-i(2ϑ(β)+βln2π+\arctan(\tanh(1/2)πβ)))$
where $ϑ(β)$ is the Riemann Siegel function.
- 1,197
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2
answers
2k
views
How do you calculate/prove the length of n, the number of non-repeating digits preceeding a periodic sequence of a fractional repeating decimal [closed]
Is there a way to calculate the number of non-repeating digits that precede the periodic repeating portion of a decimal expansion? For example:
1/6 = 0.1666.... (there is 1 non repeating digit) **(...
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votes
2
answers
785
views
Spectral sequence [closed]
what is Koszul resolution? what is its role played in the computation of spectral sequence?
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1
answer
310
views
Limit of recursion relation
Consider the sequence of functions $\{F_n\}_{n\in \mathbb{N}}$, where each $F_n$ is defined on $\{0,...,n\}$ by recurrence of the following form: $$ F_n(0)=3 \textrm{
and }F_n(k)=\frac{1}{k^2}+\frac{\...
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votes
1
answer
201
views
Does the equation $x^k+y^k-z^k-w^k=3\ (k>3)$ have a solution over $\mathbb N$?
Clearly,
$$3=0^2+2^2-1^2-0^2\ \ \mbox{and}\ \ 3=4^3 +4^3-5^3-0^3.$$
Question. Let $k>3$ be an integer. Does the equation
$$ x^k+y^k-z^k-w^k=3\quad \ (x,y,z,w\in\mathbb N=\{0,1,2,\ldots\})\tag{1}$$
...
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votes
2
answers
256
views
Reconstructing a graph from the multiset of degrees
Suppose $G, H$ are finite, simple, undirected graphs and there is a bijection between the vertex sets $\varphi:V(G) \to V(H)$ such that for all $v\in V$ we have $$\text{deg}_G(v) = \deg_H(\varphi(v)).$...
- 48.9k
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votes
1
answer
465
views
Asymptotic formula for $\prod_{p\leq x} (1-p^{-1})$ [closed]
Does there exists a good asymptotic formula for
$$A(x) := \prod_{p\leq x}(1-\frac 1p).$$
By using a heuristic argument one can guess:
$$A(x) \sim \frac{1}{2\,\mathrm{ln}(x)}.$$
Here is the argument:...
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1
answer
710
views
Lie algebraic Grassmannian
Assume that $L$ is a Lie algebra structure on $\mathbb{R}^{n}$, and $1<k<n$ is given.
We define $Gr(k,n)_{L}$, the space of all $k$ dimensional Lie subalgebra of $(\mathbb{R}^{n}, L)$.
For ...
- 366
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votes
1
answer
636
views
Solution to $ \sum (-1)^k \binom{n}{k} \alpha_k = b_n$? [closed]
Is there anyone can tell me any information about the integer solution to the combinatotial equation
$$
\sum (-1)^k \binom{n}{k} \alpha_k = b_n
$$
(all variables are integers)?
For example,
suppose ...
- 39
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votes
1
answer
328
views
Does a coarser topology lead to a non-Hausdorff topological manifold? [closed]
Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold?
NOTE: PLEASE avoid the ...
- 612
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votes
2
answers
530
views
Inverse square-law as a positive definite kernel?
Newtons law for gravity states that:
$$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$
The function :
$$k(x,y):=\exp(-| x-y|^2)$$
is known to be a positive definite function, called the RBF-kernel.
It ...
- 3,074
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votes
1
answer
571
views
What is the proof for any non trivial zero? [closed]
There are many known nontrivial zeros of the Riemann Zeta function, but I have never seen proof that any of them actually resolve to zero. The trivial zeros make sense because there is a more ...
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votes
1
answer
229
views
A generalization Bertrand's postulate [closed]
Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$?
When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question ...
- 1,776
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votes
1
answer
224
views
Do monoid homomorphisms from $X^X$ to a group factor through $\text{Sym}(X)$? [closed]
Let $X$ be a set and let $(X^X,\circ)$ denote the monoid of all maps $f: X\to X$, together with composition. Let $(\text{Sym}(X),\circ)$ be the group of all bijections from $X$ to itself.
Does there ...
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votes
1
answer
600
views
Is SOC known to imply the Grand Riemann Hypothesis? [closed]
I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...
- 6,982
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votes
2
answers
173
views
Why exactly is Simpson's rule better than the Trapezoidal rule? [closed]
I am reading up on numerical integration and have trouble to really understand why or rather in what sense Simpson's rule is better than the Trapezoidal rule in general. There is a lot of stuff ...
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votes
2
answers
399
views
Two equivalent statements about formulas projected onto an Ultrafilter
Question 1:
In the same language, let $ X $ be a nonempty set, and let $ \{ (\forall x_{x(i)} f(i)) \ | \ i \in X \} $ be a set of formulas. We use $ x(i) $ to denote the index of the variable on ...
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1
answer
213
views
Is the proof in "On Hilbert’s 8th Problem" published on Brazilian Journal of Probability and Statistics correct? [closed]
The article can be freely accessed here. The proof is only five pages. I am quite in doubt.
A new version (2021) of that paper can be found here.
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votes
2
answers
462
views
Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism?
Consider the definition of measurable cardinal (this definition was found in Neil Barton's paper, "Large cardinals and the iterative conception of set"):
Definition 8. A cardinal $\kappa$ ...
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votes
2
answers
187
views
Is $x=\frac{1}{2}$ the solution of this equation $\zeta(2)= 1+{{{{x}^{x}}^{x}}^{x}}^{\cdots } $?
I would like to study the irrationality of ${{{{x}^{x}}^{x}}^{x}}^{\cdots } $
for $x=\frac{1}{2} $ using the irrationality of $\zeta(2)$ .
Some computations in wolfram alpha show to me that :
$${...
user99666
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votes
1
answer
79
views
probably an easy inequality [closed]
Show that for any positive $a,b,c$ and any $\alpha \geq 65/66$ we have
$$\alpha\left( \frac{1}{a b} + \frac{1}{(b+c)(a+b+c)} + \frac{1}{(a+b+c)(a+b)} + \frac{1}{b c}\right)+ \frac{1}{(b+c)^2} + \...
-4
votes
1
answer
238
views
What exactly is wrong with this statement (Lucas-Penrose fallacy)? [closed]
Statement
"For every computer system, there is a sentence which is undecidable for the computer, but the human sees that it is true, therefore proving the sentence via some non-algorithmic method."
...
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votes
2
answers
233
views
If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$? [closed]
If $~(c - b) ^ 2 + 3cb = a^3~$ has nonzero integer solutions, then $~(a,c) \gt 1~$ or $~(b,c) \gt 1$?
I think this is true, how to prove this?
- 1
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votes
1
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2k
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A question on the number of subgroups of symmetric groups
Let $G$ be a finite group. Several recent papers (see e.g. http://www.jstor.org/discover/10.2307/2695441) deal with the following notion: $G$ is called a group with perfect order subsets or briefly, a ...
-4
votes
1
answer
468
views
Symplectic forms and 1-forms [closed]
Suppose we have a real symplectic manifold $(M,\omega)$. Under what conditions can we find a global 1-form $\alpha$ such that $\omega = \alpha \wedge\alpha$?
Obviously there are some simple ...
- 1,025
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votes
3
answers
8k
views
What is information-theoretic lower bound? [closed]
Hi all,
Please tell me what is information-theoretic lower bound. what does it really means
Thank you
- 1
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votes
1
answer
163
views
What are all the complex structures on $\mathbb{R}^2$ which live inside $\mathrm{SL}_2(\mathbb{Z})$? [closed]
By "complex structure" I am referring to 2x2 matrices which square to $-\mathrm{Id}_2$. I need to know those with integer entries and determinant equal to 1.
Thank you
- 3
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votes
1
answer
550
views
Amount of mathematical knowledge required for starting Ph.D. in pure mathematics [closed]
How much mathematics should one know before starting a Ph.D. program in pure mathematics? For example what topics one must understand well to pursue a Ph.D. in US University in Number Theory (...
- 310
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votes
1
answer
149
views
Hilbert’s third problem and what a polyhedron is [closed]
What is the definition of a polyhedron used by Hilbert’s third problem?
- 2,773
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votes
1
answer
143
views
Conjugacy classes of $PSL_2(11)$ and $PGL_2(11)$ in $Aut(HN)$
How many conjugacy classes each of $PSL_2(11)$ and $PGL_2(11)$ subgroups are contained in the automorphism group of the Harada-Norton group?
- 2,773
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votes
1
answer
190
views
Does Rankin-Selberg convolution preserve primitivity?
Call $L$-function any element of an L-rig (see Are there infinitely many L-rigs? for a definition). Suppose $F$ and $G$ are two primitive L-functions. Is $F\otimes G$ itself primitive? If yes, does ...
- 6,982
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votes
1
answer
180
views
Is the underlying set of every renormalization group countable and finite？ [closed]
Is the underlying set of every renormalization group countable and finite？
Suppose A is a renormalization group, and the elements of it compose of the set B. Is B the set countable and finite?
- 3,104