Frequent Questions
17,981 questions
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The existence of a maximal “cross-sectional” filter on the Boolean algebra of measurable subsets of [0, 1] modulo almost everywhere equivalence
Let $\mathcal{B}([0, 1])$ be the Boolean algebra of measurable subsets of $[0, 1]$ modulo almost everywhere equivalence, i.e., two measurable sets which differ only by a Lebesgue null set are ...
- 2,830
-1
votes
1
answer
570
views
Is Selberg's eigenvalue conjecture related to RH?
I took a quick glance on a survey paper about superzeta functions where one considers a pair $\rho\leftrightarrow 1-\rho$ of non trivial zeroes of the Riemann zeta function. The assumption of RH, i.e $...
- 6,982
-1
votes
1
answer
258
views
A number theoretical identity of exponential sum
I try to understand a number theoretical identity used by
Jan-Christoph Schlage-Puchta in this answer.
He defined the function
$$S(\alpha)=\sum_{n\leq N}\Lambda(n) e(n\alpha)$$
where $\Lambda(n)$ is ...
- 6,038
-1
votes
2
answers
961
views
Alternative characterization of homotopy equivalence
Using the formalism of model categories its possible define the concept of homotopy as done here.
If we take as model category $\mathbf{Top}$ having homotopy-equivalence as weak-equivalence and ...
- 3,349
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votes
1
answer
369
views
Would this go to 0 [closed]
Let $t_{m}$ be the sup of the sum of the pairwise distances
between any $2m$ points in the unit disk. Does $t_{m}/m^{2}$ go to
$0$ as $m\rightarrow\infty$?
- 1
-1
votes
2
answers
946
views
co spanning tree
Hi,
Does anyone know that what is co spanning tree. If there are some good answers then it would be really good to have an example also.
Thanks
- 99
-1
votes
2
answers
263
views
Selection problem in a collection of non-empty sets
Is there a set $X\neq\emptyset$ and a collection ${\cal F}\subseteq {\cal P}(X)\setminus\{\emptyset\}$ of non-empty subsets of $X$ with the following properties?
$a\in {\cal F} \implies |a|\geq 2$,
$...
- 48.9k
-1
votes
1
answer
395
views
Odd & even permutations and unit fractions
One more motivated by recent questions of Zhi-Wei Sun.
Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$.
Is it true that, for every $n \ge 8$, there is at least one even permutation $\...
- 4,589
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votes
1
answer
1k
views
Publication Of 50 pages [closed]
Does anyone know of a research journal in mathematics that is willing to publish 50 pages of peer-review research? I would like to submit research that explores how to develop predicate models for ...
-1
votes
1
answer
370
views
What's the probability of two independent events in time domain?
Suppose there are two independent events A and B. The probability that A or ...
- 29
-1
votes
1
answer
208
views
Does this function belong to $L^2(\mathbb{D})$?
Edit: After the answer of Prof. Eremenko to the previous version, I realized that a weaker assumption works for the main motivation of this post. so I revise the question.
The unit ...
- 366
-1
votes
2
answers
368
views
Is this expression always irrational? [closed]
Is it right that
$$\sqrt[a]{2^{2^n}+1}$$
for every $$a>1,n \in \mathbb N $$
is always irrational?
- 21
-1
votes
1
answer
378
views
Marriages in infinite bipartite graphs with many neighbors
Let $A,B\neq \emptyset$ be disjoint and suppose $G = (A\cup B, E)$ is bipartite where for all $e\in E$ we have $e\cap A \neq \emptyset\neq e\cap B$. For $a\in A$ we set $N_G(a) = \{b\in B: (\exists e\...
- 48.9k
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votes
1
answer
143
views
Does transforming a periodic function imply periodicity
Let $f(x,y)$ be a periodic function for every fixed $y = \beta$ with respect to $x$ in the domain $x\in \mathbb{R}$ and consider this transform of $f$:
\begin{equation}
f^\star(\alpha,\beta ) = \sum_{...
-1
votes
1
answer
519
views
Poisson kernel is the Cauchy distribution, reference?
Let $d = 2$, and consider the domain $D = \mathbb{H}$, the upper half-plane. Can someone give me a reference to a proof that the Poisson kernel is the Cauchy distribution?
-1
votes
1
answer
74
views
Invariant ergodic measure Volterra operator
Define the Volterra operator $V$ on $C_0([0,1])\triangleq \{g \in C([0,1]):g(0)=0\}$ by
$$
f \mapsto \int_0^{\cdot} f(s)ds.
$$
Is there an example of an ergodic and $V$-invariant Borel probability ...
- 5,405
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votes
3
answers
312
views
Binomial Coefficients sum [closed]
Any idea on whether or not $$\sum_{i = 0}^b(-1)^i\binom{b}{i}\binom{a+b-i-1}{a-i}$$
has a closed formula on $a$ and $b$ (and on what it is, in case it does)?
It is supposed that $b \le a$.
- 369
-1
votes
1
answer
827
views
Is it possible to derive the rules of set theory as transfers from the pure finite set world, and can we extend this further?
Informally the idea of this question is about whether the rules of set theory can be derived as a transfer of some rules from the hereditarily finite set realm, and whether this transfer principle ...
- 11.3k
-1
votes
1
answer
261
views
Is Proper Class Choice equivalent to Global Choice?
Working in "MK-Regularity-Limitation of Size + Replacement for sets", call it the Base theory, let's coin the following axiom:
Axiom of Super-Choice:$$\forall \ relation \ R \ \exists F \...
- 11.3k
-1
votes
1
answer
204
views
Cauchy reduction formula with measure (a variation)
The Cauchy reduction formula conveniently compresses $n$ integrations of a function $F(x)$ into a single integral. Here I am interested in reducing the following "curved-space" ...
- 141
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votes
2
answers
409
views
$X$ is Polish and $N$ is countable. Is $N^X$ Polish? [closed]
$X$ is a separable, completely metrizable topological space equipped with its sigma algebra of Borel sets. $N$ is a countable space.
$X^N$ is the collection of all mappings from $N$ to $X$. It is ...
- 263
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1
answer
1k
views
A question about independent set in regular graphs
Suppose that $G$ is a simple $r$-regular graph with $n$ vertices.
We say $H$ is a dominating set for $T$, if
for every vertex $v\in T$, we have $v\in H$ or there is a vertex $u\in H$ such that $vu\in ...
- 385
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votes
1
answer
507
views
loop homology product for oriented compact manifolds with boundary
This is my first steeps in string topology and please forgive the basic level of my questions: I reformulate my question
Chas and Sullivan define the loop homology product for closed (=compact with ...
- 189
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votes
3
answers
637
views
What is the consistency strength of Z+ Accessibility?
Informally the axiom schema of accessibility states that for each unary function $F$ that is definable over the whole universe of discourse "in the language of set theory", like the powerset function $...
- 11.3k
-1
votes
14
answers
2k
views
Priming for the primes [closed]
I have to confess that most often my eyes begin to glaze over when someone starts discussing the prime numbers. However, my ears have perked up at times over the primes--maybe first when I learned of ...
-1
votes
1
answer
179
views
Categories that admit all finite products but not all finite coproducts
What are examples for categories that admit all finite products but not all finite coproducts?
(See also this question: Categories that admit all products but not all coproducts .)
-1
votes
1
answer
512
views
Functions of several variables over finite fields [closed]
For a finite field $F$ any function $f\colon F\to F$ is given by a polynomial. My question is what happens when we are given a function of two or more variables? Is this necessarily a polynomial ...
- 2,583
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votes
1
answer
236
views
Natural candidates for sub-half-exponential which limit to half-exponential function from below
There are no closed form candidates for half-exponential functions "Closed-form" functions with half-exponential growth.
However sub-half-exponentials (functions whose composition grows ...
- 1,826
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votes
1
answer
163
views
Alternate property of H^2(T, Z) [closed]
Let us take $T = S^1_X \times S^1_Y$, which is a torus, where the former $S^1_X$ has the coordinate $X$, whereas the latter $S^1_Y$ has $Y$. If we consider the generator $dX \wedge dY \in H^2(T, {\Bbb ...
- 563
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votes
1
answer
402
views
Ubiquity beyond infinity, transitive closure and the recursion theorem?
I am considering a Principle of Ubiquity, expressed as follows - for a class theory where precisely the elements are sets - with the aid of set abstracts:
For $\alpha(y,z)$ a first order condition so ...
- 3,574
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votes
1
answer
679
views
Naturally definable sets of natural numbers (2): Can the circle be broken?
(follow-up to: Naturally definable sets of natural numbers)
Every formula $\Psi(x)$ in the first-order language of Peano arithmetic defines a set of natural numbers. Some of these sets are finite, ...
- 9,720
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votes
1
answer
573
views
is the existence of an inaccessible cardinal stronger than just CON(ZFC)? [closed]
is it even stronger than that ZFC has a transtitive model?
- 21
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votes
1
answer
342
views
Finite or polynomial number integral points clarification on Coppersmith's theorems (possibility of ellipse counter example?)
Coppersmith states if $f(x,y)$ is an irreducible bivariate with total degree $\delta$ then if he can list all roots $(X,Y)$ of the polynomial in $\mathsf{poly}(\log D,\delta)$ time if the roots ...
- 13.9k
-1
votes
2
answers
225
views
Precise asymptotic of diophantine approximation
I know that every irrational number $\xi$ can be approximated by rational numbers in such a way that
$$
\left| \xi - \frac{p}{q} \right| \leq \frac{c}{q^2}
$$
for infinitely many choices of $p$ and $q$...
- 459
-1
votes
1
answer
280
views
A question on assigning finite values to divergent sums involving expression of primes
We know the following:
$$\gamma=\lim_{n\to\infty }\left(\sum_{k=1}^n\frac{1}{k}-\ln(n)\right).$$
This could be a good candidate for renormalized sum of $\left(\sum_{k=1}^{\infty}\frac{1}{k}\right)$.
...
- 149
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votes
1
answer
521
views
Generalized Leibniz rule [closed]
Suppose on a manifold $M$ we have a differential operator $D$ of order 1 from the smooth sections $C^\infty(M, E)$ to smooth sections $C^\infty(M, F)$, where $E$ and $F$ are vector bundles of rank $n$ ...
-1
votes
1
answer
315
views
Bounds for the number of points on projective hyperelliptic curves over finite fields
Let $C$ be projective hyperelliptic curve over finite field $K$.
What are bounds for the number of points $\#C(K)$?
The Hasse-Weil bound requires smooth curves, and hyperelliptic curves are
not smooth ...
- 25.4k
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votes
1
answer
312
views
expectation of upper quantile proportion
(edited considerably following comments)
We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
- 121
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votes
1
answer
406
views
Topological properties of complex valued Riemann sum limit curve and a particular integral inequality
I am studying under what conditions the following integral inequality would hold ($a$ real, $a>0$):
$$ \int_{-\infty} ^{\infty} \frac{f(ix)}{a\pm ix}dx\ = 0 \ \ \ \ \Rightarrow \ \ \ \int_{-\...
- 362
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votes
1
answer
218
views
Function on quadratic numbers
Let $\mathbb{N}$ denote the set of the positive integers. We consider the following function $f:\mathbb{N}\times \mathbb{N}\to \mathbb{Q}$: $$f(a,b)=\frac{a^2+b^2}{1+ab} \text{ for all } a,b\in\mathbb{...
- 48.9k
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votes
1
answer
138
views
Identifying two non-adjacent vertices and the effect on the Hadwiger number [closed]
Suppose $G$ is a finite simple graph. Let $h(G)$ denote the Hadwiger number of $G$; that is, the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$.
What is an example of a graph $G_0=(V_0, ...
- 48.9k
-2
votes
1
answer
155
views
Does MK prove internally that there are more proper classes than sets?
Is the following provable in MK?
$\not \exists S: \\ \text{ } \\1. \ \ \forall s \in S \exists a,b (s=\langle a,b \rangle) \\ \text{ } \\2. \ \ \forall x (set(x) \to \exists! X (\neg set(X) \land \{...
- 11.3k
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votes
1
answer
1k
views
Derivative of log determinant [closed]
Let $x_i \in\mathbb{R}^d$ and $a_i\in [0,1]$ for $i = 1,\dots,k$. How to compute the following derivative?
$$
\frac{d}{da_j}\log \det\left(\sum_{i = 1}^k a_ix_ix_i^\top\right).
$$
- 245
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votes
1
answer
402
views
collective slide-hosting for Mathematics [closed]
Has anyone considered using SlideShare to host slides from talks? In much the same way arXiv hosts papers.
Truth be told, the slides are often much easier to absorb than the papers.
Sometimes I will ...
- 22.8k
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votes
2
answers
764
views
Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
user111492
-2
votes
1
answer
154
views
Is there any pseudoprime that pass this test above tested range, or any prime that does not show these ending patterns?
if the recurrence sequence is defined by the following foormula, $d_{n + 3} = 3d_{n + 2} - d_{n + 1} - 2d_n$ where $d_1 = 1, d_2 = 3$ and $ d_3 = 7$, this produce the following complex sequence $$1, ...
-2
votes
1
answer
192
views
Can we have consistent theories stating opposing provability statements that are non-standardly coded?
I want to coin a notion of "strong provability", to be defined as:
$S$ is strongly provable in $T$ if and only if there is a Gödel code of its proof in $T$ that is strictly smaller than any ...
- 11.3k
-2
votes
1
answer
167
views
If we limit matters what ZFC can prove, would that be consistent?
I was thinking about a principle that occurred to me regarding provability in ZFC and truth. The principle outrageously states that: whatever ZFC shows, it is! In other words whatever ZFC can prove ...
- 11.3k
-2
votes
1
answer
203
views
Holomorphic maps on $\mathbb{R}^{n}$ (for n not necessarily even)
Edit according to the comment of user36931 I remove the "motivation" from the previous version and I add an statement to the first question
We consider the following two classes of smooth maps on $...
- 366
-2
votes
4
answers
2k
views
Why don't quaternions have an overall phase? [closed]
The product of a quaternion multiplied by a real number is a quaternion, but the product of a quaternion multiplied by a complex number is not in general a quaternion. Why are the quaternions defined ...
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