Frequent Questions
18,060 questions
-1
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
-1
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
-1
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
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
-1
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
-1
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
-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
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
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
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
-1
votes
1
answer
88
views
Interval topology on $(\mathbb{N}^\mathbb{N},\leq^*)$
Given a quasi-ordered set $(Q,\leq)$ the interval topology on $Q$ is generated by
$$\{Q\setminus\downarrow x : x\in Q\} \cup \{Q\setminus\uparrow x : x\in Q\},$$
where $\downarrow x = \{y\in Q: y\leq ...
- 48.9k
-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
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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
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
-1
votes
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
-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
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
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
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
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
-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
1
answer
215
views
Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$ [closed]
I am reading from the book Topics in Galois theory by Serre.
I have the following question ,
take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by
$$\sigma x\;=\;1/(1-x)$$
where $\sigma$ ...
- 591
-2
votes
1
answer
389
views
Bounded metric spaces with non-surjective self-isometry
A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$.
A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
- 48.9k
-2
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
-2
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
-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
-2
votes
2
answers
299
views
Adjunctions between Groupoids and Hilbert spaces
I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the ...
- 1,313
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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 $$1, ...
-2
votes
1
answer
113
views
Does one have $2r_{0}(n)\lesssim k_{0}(n)(\log n)^{1+1/k_{0}(n)}$?
Under Goldbach's conjecture, I'm trying to find an upper bound for $r_{0}(n):=\inf\{r>0,(n-r,n+r)\in\mathbb{P}^{2}\}$ that would generalize Cramer's conjecture.
Denoting by $k_{0}(n)$ the quantity ...
- 6,982
-2
votes
1
answer
96
views
Hadwiger partitions where one block is always a singleton
Let $G=(V,E)$ be a simple, undirected graph.
We call a partition ${\cal P}$ of a non-empty subset of $V$ a Hadwiger partition if
every block (member of ${\cal P}$) is non-empty and connected, and
if ...
- 48.9k
-2
votes
4
answers
2k
views
Why don't quaternions have an overall phase? [closed]
The product of a quaternion multiplied by a real number is a quaternion, but the product of a quaternion multiplied by a complex number is not in general a quaternion. Why are the quaternions defined ...
- 25
-2
votes
1
answer
314
views
Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
- 375
-2
votes
2
answers
764
views
Reduced ring with all non-prime ideals finitely generated
Let $R$ be a reduced ring with all non-prime ideals finitely generated. Then is $R$ Noetherian ? If not, then is it true at least in the local case ?
Without reduced assumption, it is not true even ...
user111492
-2
votes
1
answer
167
views
If we limit matters what ZFC can prove, would that be consistent?
I was thinking about a principle that occurred to me regarding provability in ZFC and truth. The principle outrageously states that: whatever ZFC shows, it is! In other words whatever ZFC can prove ...
- 11.3k
-2
votes
1
answer
217
views
Convergence and roots of alternating periodic infinite series
Let $0<\alpha <1$ and $\beta > 0$. Consider the mapping $$F(\alpha, \beta) = \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}.$$ Can we prove $F(...
- 317
-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
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
345
views
Generalizations of the twin primes conjecture [closed]
This is a question about generalizations of the twin primes conjecture.
I would like to know a counterexample, or a proof, for the following couple of related arithmetical sentences. The first is
...
-3
votes
1
answer
1k
views
Matrix sieve theorem [closed]
I have formulated the following conjecture:
Odd positive integer $ N=6n-1$ is a prime number iff neither of two diophantine equations
$6x^2+(6x−1)y=n$
$6x^2+(6x+1)y=n$
has solution. $x=1,2,3,..y=0,...
- 11
-3
votes
2
answers
450
views
Expected values of two random variables related to a simple urn problem
In an urn there are $u$ balls, $b$ of which are black.
If we perform $n$ trials of one ball at a time with replacement, the probability of the event $E$ to get $n$ times a black ball is $P(E)=\left(\...
- 145
-3
votes
1
answer
531
views
Counter net decidability [closed]
Let one Deterministic Counter Net ($\mathrm{1DCN}$), which is a finite-state automata where every state is complete means all states has transition of all input symbols and their respective weight ...
- 63
-3
votes
0
answers
137
views
Approximation on Dirichlet's arithmetic progression by means of central limit theorem
In this video lecture on
Number theory over function fields taught by Will Sawin
is presented a 'conceptional' reason for error estimation
$\#\{p \in \Bbb P: p =a \ \text{mod} \ N, p <x \}
=\frac{1}...
- 619
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votes
1
answer
117
views
Can stratification be used to internalize functions on models of $\sf Z$?
Suppose $M$ is a model of $\sf Z +\neg AC$ that is externally bijective to an element $k \in M$. Obviously if $j$ denotes such an external bijection, then it cannot be used in Separation within $M$, ...
- 11.3k
-3
votes
1
answer
76
views
Minimal norm problem with linear combination of translation operator to be estimated
Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...
- 115
-3
votes
1
answer
296
views
Can this form of reflection be consistent?
Is this form of reflection consistent?
First I'll begin by clarifying the notation I'm using here:
By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...
- 11.3k
-4
votes
1
answer
199
views
Is Bounding Reflection consistent?
Working in the first order language of set theory.
Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$".
Here a ...
- 11.3k
-4
votes
1
answer
412
views
A topological groupoid structure on a pair $(X,A)$
Assume that $X$ is a compact Hausdorff space and $A\subset X$ is a retract of $X$.
Is there a topological groupoid structure on the topological pair $(X,A)$ where, in the corresponding ...
- 366
-4
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
-4
votes
1
answer
267
views
Is Nested Selection equivalent to AC?
Nested Selection: For every infinite set $G$ of pairwise disjoint infinite sets such that any two distinct elements $x,y$ of $G$ either "$y$ is a set of proper supersets of elements of $x$ and ...
- 11.3k
-4
votes
1
answer
101
views
Regular graph such that $2$ distinct vertices have same neighborhood set [closed]
If $G=(V,E)$ is a simple, undirected graph and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in E\}$.
Is there an integer $k>1$ and a connected $k$-regular graph $G=(V,E)$ such that there are $v\neq ...
- 48.9k