Highest scored questions
159,032 questions
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Count arrangements with pairs of attacking kings [closed]
I have a $1\times n$ chessboard and $2$ pairs of kings in it. Both components of each pair of kings must be adjacent in the chessboard, that is, they must be attacking.
Now, I want to calculate the ...
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votes
1
answer
211
views
Can a Polish space have two different topologies?
Let $X$ be a Polish space with the compatible metric being $d_1$. So $(X,d_1)$ is a separable complete metric space, and the topology is generated by $d_1$.
Can there be a metric $d_2$ such that $(X,...
- 291
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votes
1
answer
168
views
Is there a summation method where the divergent series $S = U_0+U_1+U_2+\dots$ converges to a finite value(V2)?
I have a question regarding this question here.
is-there-a-summation-method-where-the-divergent-series
if I set $ x+2=c/c-v$ , will I have
$U_n = M\left(c-\frac{c}{n+2}\right)-M\left(c-\frac{c}{n+1}\...
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votes
1
answer
325
views
Is the category of $Z^* $ algebra equivalent to the category of $C^*$ algebras
A $Z^*$ algebra is a $C^*$ algebra whose all positive elements are zero divisor.
They form a category with usual structures.
Question. Is this category equivalent to the category of $C^*$ algebras?
...
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votes
1
answer
208
views
can the square of a polynomial be written as a sum of squares of polynomials with only even degree terms?
Let $f$ be a polynomial with real coefficients in several indeterminates $x_1, \dots, x_n$. Suppose that
$$ f = g^2 $$
for some polynomial $g$.
Is it true that we can find polynomials $h_1, \dots, h_m$...
- 331
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votes
1
answer
241
views
Can we have a hybrid comprehension between Z and NF?
Hybrid Comprehension: if $\phi,\varphi$ are formulas in which $x$ doesn't occur, and $\varphi$ is stratified; then: $$ \forall A \exists x \forall y \, (y \in x \leftrightarrow \varphi \land [wf(A) \...
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1
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81
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Can you do boolean and of 1 and a number less than 1? [closed]
I am reading
imenez, J., Echevarria, J.I., Sousa, T. and Gutierrez, D. (2012), SMAA: Enhanced Subpixel Morphological Antialiasing Computer Graphics Forum, 31: 355-364. https://doi.org/10.1111/j.1467-...
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1
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218
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All group structures on a set with cardinality $\aleph_0$
Assume we consider the additive group $(\mathbb{Z}, 0, +)$. I am wondering what other group structures are there with neutral element 0 fixed? Is there a way to classify them or find them all?
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1
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111
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Are mantissas of irrationals provably unique, at a given precision? [closed]
Many thanks to all responders!
Is there any research as to the uniqueness of mantissas of irrationals? It's easy to see that the mantissa of the square root of 5 (0.236067977...) and the mantissa of ...
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votes
1
answer
2k
views
Bounding sum of square roots in function of the sum value [closed]
Knowing the value of $S=\sum_{k=1}^n s_k$ with $s_k\geq 0$,
is it possible to obtain an upper bound on $\sum_{k=1}^n\sqrt{s_k}$ better than $n \times \sqrt{\max_{1\leq k\leq n} s_k}$ ?
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votes
1
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123
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Are the first 4 statistical moments independent? [closed]
Are the first 4 statistical moments independent? Is there a mathematical demonstration that can show independence one from each other? Can the value of one moment influence the value of another? If so,...
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votes
2
answers
156
views
Getting almost certainty from uncountably many low-probability events
Let $(\Omega,\Sigma,\mathbb{P})$ be a complete probability space, $B\subseteq X$ be a non-empty Borel subset of a polish space $X$, $A$ be an uncountable indexing set, and $\{X_{\alpha,n}\}_{a \in A, ...
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1
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282
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A Bonse's inequality for semiprimes, with a good mathematical content
A semiprime $s$ is a positive integer that is the product of two prime numbers, see Semiprine the encyclopedia Wikipedia. A well-known inequality, with applications, that involves prime numbers is the ...
-3
votes
1
answer
127
views
Hyper-simultaneous equation [closed]
I am new here and I just want to ask if the following system has a general solution:
If a, b and c are given such that:
$$
x + y + z = a
$$
$$
x^2 + y^2 + z^2 = b
$$
$$
x^8 + y^8 + z^8 = c
$$
Is ...
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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(\...
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votes
1
answer
1k
views
Maximum chromatic number of a $k$-regular graph [closed]
Let $c_k$ be the maximum chromatic number that a $k$-regular graph can have. What is $\lim\sup_{k\to\infty}\frac{c_k}{k}$?
- 48.9k
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1
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392
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A generalization of Chebyshev's sum inequality
From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...
- 1,776
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1
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239
views
SAT problem in Gödel numbering [closed]
I am working on a cryptography project and I have come up with this problem.
Let's say I have a boolean expression L with $k$ variables $A_{1},..., A_{k}$. Let's assume this boolean expression is ...
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votes
1
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964
views
On the maximal ideal m of the formal power series ring [closed]
Let $A \colon= K[[X_1,X_2,\ldots,X_{\infty}]]$ be the formal power series ring with infinitely many variables over a field $K$. We can represent it also by the following manner$\colon$
\begin{equation*...
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1
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366
views
Connected homogeneous graphs [closed]
Let's call a simple, undirected graph $G=(V,E)$ homogeneous if for every $v,w\in V$ there is a graph isomorphism $\varphi:G\to G$ such that $\varphi(v)=w$.
It is clear that every finite homogeneous ...
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2
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2k
views
What is the number of self-inverse permutations on a set of cardinality $N$?
Given a function (aka 'permutation') $f:A \rightarrow A$, where $A$ is a finite set such that $|A| = N$, we call it a self-inverse if $f(f(x)) = x$. The sequence of how many such functions exist for ...
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votes
1
answer
155
views
Characterization of some finite cyclic groups [closed]
Given a finite cyclic group $G$ with order $n=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}$, where $p_i$s are distinct prime numbers $n_i>1$ for all $i$. Let $H$ be any abelian group. Assume that Aut$(G)≅$...
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2
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606
views
The number of totatives to the nth primorial, in an interval shorter than the nth primorial
(The notation of this question will be improved over the next few days, sorry for the lack of clarity at the moment.)
Can, and if so when can, we determine the amount of natural numbers which are ...
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votes
1
answer
330
views
Loop space of manifold [closed]
Question A: The free loop space of a manifold is also a manifold?
Question B: The free loop space of an algebraic variety is also a algebraic variety ?
Are these questions asked or answered anywhere ...
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1
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232
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A problem that involves matrix and Lorentz Transformation [closed]
To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes.
$1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G A^T=A^...
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1
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249
views
A general question on nonnegative integer sequence [closed]
Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$.
Let $B=\mathbb Z_{\ge 0}-A$.
Define $\ 2A= \{a+b : a \in A,\ b \in A\}$.
Define $\ 2B=\{a+b : a \in B,\ b \in B\}$.
Then the set ...
- 359
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votes
1
answer
336
views
adjacency matrix of random geometric graphs [closed]
Consider a graph with N nodes. All nodes are distributed as a Poisson point process with density of λ in a L*L area. There is an edge between two nodes if and only if the distance between them is less ...
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votes
1
answer
1k
views
relation between smash product and suspension
let $S$ be the d-sphere.
we know that $S \wedge X = \Sigma^d X$ the $d$-fold suspension of $X$.
what can we say about
$(S\times S) \wedge (S\times S)$ in terms of suspension?
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votes
1
answer
2k
views
Eliminating redundant linear constraints? [closed]
I have an NxN matrix of linear constraints that is not of full rank. In other words, some of the constraints are linear combinations of other constraints. The "standard" linear algebra tools (...
- 159
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votes
1
answer
960
views
how to formalize a notion of symmetric set difference probability? [closed]
I saw in a paper an argument, that seems simple. (Let $\triangle$ be the triangle operator of symmetric set difference between two sets)
It states that if $P(A \triangle B) < \epsilon$, for some ...
-3
votes
1
answer
590
views
A problem regarding definition of p-norm [closed]
Let ${\bf x}=(x_1,...,x_n)$, the p-norm of x is $(|x_1|^p+...+|x_n|^p)^{1/p}$. If one of the components of x is 0, there will be exponential of the form $0^p$. If p is an irrational, $x^p$ is only ...
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votes
1
answer
440
views
Conditional expectation [closed]
Given E[v|X=x]=g[x] and the pdf of X (f[x]), how to calculate E[v|x>=x0]? The pdf of V or the joint pdf of V,X are unknown. My guess is that this problem has no solution.
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votes
2
answers
314
views
Dispensing with the notion of infinity for the sake of coverings [closed]
Instead of taking a one to one correspondence meaning each set has the same number of elements. why not use the concept of coverings of topology? The irrational numbers covers the whole numbers but ...
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votes
2
answers
851
views
Cycle Length of the Positive Powers of Two Mod Powers of Ten [closed]
I want to prove that the positive powers of two, mod 10m, cycle with period 4*5m-1. It's simple to prove that the powers of FIVE cycle with this period (2 is a primitive root mod powers of five), but ...
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votes
1
answer
107
views
Knot group of mirror image [closed]
Are the knot group and the knot group of its mirror image isomorphic?
And,How about the case of knotted surfaces?
- 1
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votes
1
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174
views
why $L=\{\langle M\rangle\mid M \text{ is a TM that accepts all even number}\} \notin \text{RE}$ [closed]
$L=\{\langle M\rangle \mid M \text{ is a TM that accepts all even number}\}$
hello everyone I anderstennd why $L\in \text{coRE} $ b but I don't understand why $ L\notin \text{RE}$
I Have proved that $ ...
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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
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votes
1
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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
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votes
1
answer
375
views
On Haar measure and Spherical measure [closed]
Let $d$-dimensional complex sphere be
$$\{(c_1,\cdots,c_{d})\sum_{i=1}^{d} |c_i|^2=1.\}$$
We can define the Haar measure on this sphere by regarding the unitary group $U(d)$.
We can regard the $d$-...
- 1,503
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votes
1
answer
81
views
What is the asymptotic behavior of the Levy distribution $P (x)$ when the independent variable $x$ approaches $0$ [closed]
What is the asymptotic behavior of the Levy distribution
$$P(x)=\frac{1}{\pi}\int_{0}^\infty \exp(-\gamma q^\alpha)\cos qx\,dq$$
when the independent variable $x$ approaches $0$?
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votes
1
answer
189
views
Propositional logic without rules of inference and assumptions (except MP) [closed]
I was wondering whether it would be possible to do propositional logic without any rules of inference and assumptions (except modus ponens).
I have the following axioms:
$ p \to (q \to p) $
$ (p \to (...
-3
votes
1
answer
167
views
Is there a simple function similar to exp? [closed]
As far as I know exp have such properties:
$f'(x) >0$
$f''(x) >0$
$\lim_{x \to -\infty}f(x)=0$
$\lim_{x \to +\infty}f(x)=\infty$
$f(x)f(-x)=1$
Let's say f(x) comply such rules.
The closest I ...
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votes
1
answer
117
views
Combinatorial meaning of a reduced fraction in a simple probability problem?
A routine exercise for undergraduates says: Given that the number of successes in $20$ independent Bernoulli trials was $8,$ what is the conditional probability that exactly $3$ of those $8$ successes ...
-3
votes
1
answer
101
views
Order-embeddability of ${\frak b}$ and ${\frak d}$ in $\mathbb{R}$ [duplicate]
The starting point of this question is the observation that in ${\sf (ZFC)}$, all ordinals $\alpha < \omega_1$ can be order-embedded in $\mathbb{R}$.
Let $\omega^\omega$ denote the set of all ...
- 48.9k
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votes
1
answer
144
views
How to find the limit of this recursion
When analyzing an algorithm's utility, I've encountered the following recursion, where $n$ is a parameter:
$ a_k=a_{k-1}+1+(1-1/n)^{a_{k-1}}$
I am interested in the limit of $a_n$ as $n\rightarrow\...
- 11
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votes
1
answer
222
views
What is the basis for the quantifier notation? [closed]
The symbols $\forall, \exists$ are the ones officially used to denote universal and existential quantifiers respectively. I understand that the choice of $\exists$ was made by Peano, while of $\forall$...
- 11.3k
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votes
1
answer
156
views
How to prove this equation? [closed]
How to prove this equation:
\begin{align}\sum_{k=1}^{n}\cos ^{2m+1} \frac{(2k-1)\pi}{2n+1} =\frac{1}{2}\end{align}
where $k,m,n\in \mathbb{N}^*$.
-3
votes
1
answer
219
views
Exact sequence of sheaves that generates an exact sequence of Abelian groups [closed]
Let $X$ be a smooth manifold of dimension $n > 1$. Let us denote by $\underline{\mathbb{S}}^{1}$ the sheaf of the smooth functions over circle, $C^{\infty}$ the sheaf of the smooth functions over $\...
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votes
2
answers
608
views
Rational points on the elliptic curve $y^2 = x^{3} - t^{2}z^3$
What are the rational points on the elliptic curve $y^2 = x^3 - t^{2}z^3$ ? I seem not to find any besides the trivial ones whereby $txyz=0$ or $x= \pm z$.
ADDENDUM 1. I have just noticed that if $z^3 ...
- 1,019
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votes
1
answer
237
views
L. Gegenbauer's proof of Infinitude of Primes [closed]
I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that
L. Gegenbauer proved Infinitude of Primes by ...
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