All Questions
5,076 questions with no upvoted or accepted answers
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123
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Isolated points of the spectra of self-adjoint operators on Hilbert spaces
Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$.
I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
0
votes
0
answers
230
views
Ramanujan's infinite sum for pi
Ramanujan's famous pi formula states that
\begin{equation}
\frac{1}{\pi}=\frac{2\sqrt{2}}{99^2}\sum_{k=0}^{\infty}\frac{(4k)!}{k!^4}\frac{26390k+1103}{396^{4k}}
\end{equation}
How can one prove this?...
- 45
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votes
0
answers
101
views
Equation $u_t - u_{tx} - u_{xx} = 0$
Consider the following heat equation with a perturbation given by a second order mixed derivative:
$$u_t - u_{tx} - u_{xx} = 0$$
Does this equation have a name? How can one prove a wellposedness ...
- 839
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answers
140
views
Graph theory: Closed neighourhoods and generalized clustering coefficients
The neighbourhood of node $v$ in graph $G$ is the subgraph of $G$ induced by all vertices adjacent to $v$.
The number of edges between neighbours divided by the number of pairs of neighbours is ...
- 9,730
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0
answers
142
views
Counting special paths on a certain rectangle integer grid (binary matrix)
Crossposting from MSE after getting no answers. The bounty on the MSE question is still open, but not for long. Be advised that the comments of the MSE question regard an obsolete version, and that ...
- 149
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votes
0
answers
295
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Reference: Irreducible components of the Steinberg variety are conormal bundles
The (partial) Springer resolution is defined as a map $\mu: T^*\mathcal{F} \to \mathcal{N}$, where $\mathcal{F}$ is the partial flag variety consisting of $n$-step partial flags of $\mathbb{C}^d$, and ...
- 159
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votes
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answers
186
views
Generalization of elementary symmetric polynomials
The elementary symmetric polynomials (ESPs) are defined as -
\begin{align*}
E_{1}^{1}
&= X_1,
\\
E_{1}^{2}
&= X_1 + X_2,
\\
E_{2}^{2}
&= X_1 X_2,
\\
E_{2}^{3}
&= X_1 X_2 + X_1 X_3 + ...
- 47
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votes
0
answers
113
views
Viewing limit as a map
Question: Let $X$ be a set of functions from $\mathbb{R}$ to itself. Consider the subset $X_0$ of the sequences $(f_n)_{n=1}^{\infty}\in X^{\mathbb{N}}$ for which
$$
f_{\infty}(x) = \lim\limits_{n \...
- 5,405
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votes
0
answers
127
views
mean curvature for codimension $>1$?
The mean curvature of a hypersurface in a Riemannian manifold is defined to be the trace of the second fundamental form. I was curious, does the notion of mean curvature generalise to higher ...
- 3,625
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votes
0
answers
30
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Maximum nonintersecting interval pick
This surely has been solved in the context of scheduling already! (Shall I ask on some computer SE instead?)
Assume we have a set of closed "intervals" on $\mathbb Z$ ($\mathbb R$ isn't ...
- 4,829
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votes
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109
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The role of a combination of Eneström-Kakeya and Gauss-Lucas theorems: reference request or soft question, asking for this combination as tool
In past days I was trying to create problems or direct applications invoking Eneström—Kakeya and Gauss-Lucas theorems for certain arithmetic functions that I know from analytic number theory. These ...
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answers
83
views
Need reference of books which deals with ideal theory of tensor product
Is there any book which deals with Ideal theory of tensor product of $C^{\ast}-$ algebras
- 1,115
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votes
0
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84
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Prerequisites/Preparation for understanding a research paper - global solutions to Einstein field in Bondi Coordinates
I would like to read this paper:
João L. Costa, Filipe C. Mena, Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi ...
- 101
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answers
91
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Image of Frobenius element under irreducible representation is diagonalizable
Let $K/ \mathbb Q$ be a Galois extension, and $\rho$ be an irreducible representation of the Galois group $Gal(K/ \mathbb Q)$. Consider an integer prime $p$ which doesn't ramify in $K$, and let $\...
- 739
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177
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Status of the $n$ conjecture and, as secondary question or reference request, what about a transfer method for this conjecture $n>3$
The n conjecture is a generalization of the abc conjecture. What is the current status of the $n$ conjecture? See also [1]
Question 1. Can you tell us what about the current status of the $n$ ...
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answers
226
views
On Prime Numbers which can be Norms of an Integral Ideal of a Number Field
We know that since the ring $\mathbb Z [i]$ of Gaussian integers is a Principal Ideal Domain, the only integer primes which can norms of some ideal of $\mathbb Z [i]$ are those which can be expressed ...
- 739
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answers
155
views
On the equation that involves the Dedekind psi function $\psi(x)=n$ with unique solution $x$, for a fixed integer $n\geq 1$
The Dedekind psi function is defined for a positive integer $m>1$ as
$$\psi(m)=m\prod_{\substack{p\mid m\\p\text{ prime}}}\left(1+\frac{1}{p}\right)\tag{1}$$
with the definition $\psi(1)=1$. See ...
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answers
141
views
What is the distribution of the norm of the multivariate $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d?$
Let $X \sim \mathcal{N}(\mu, \Sigma) \in \mathbb{R}^d$ follow a multivariate normal distribution. Then what's the distribution (PDF, CDF etc.) of $X?$ When $\mu = 0, \Sigma = I_d,$ we know that $||X||...
- 1,467
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votes
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answers
154
views
Use of this space of very rapidly decreasing continuous functions
Let $C_n$ denote the subspace of continuous function on $[0,\infty)$ supported on $[n,n+1]$. Denote the $\ell^p$-direct sum Banach space
$$
V_p
:=
\left\{
f \in C([0,\infty)):\,
\sum_{n=1}^{\infty} ...
- 5,405
0
votes
0
answers
79
views
What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?
What do Sylow 2-subgroups look like for Schur covering groups of finite simple groups?
Are there any references in which we can find the stucture of Sylow 2-subgroups of Schur covering groups of ...
- 271
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votes
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answers
180
views
When is $\phi(a^n+b^n+c^n)=0\mod n$?
A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
user147204
0
votes
0
answers
56
views
Reference about Relation between Probabilistic Turing Machine and Turing Machine of every hierarchy
What are the relation between Probabilistic Turing Machine and Turing Machine of every hierarchy, for instance, are the Probabilistic PDA and NPDA equivalent? the Probabilistic LBA and LBA equivalent?...
- 3,104
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votes
0
answers
36
views
Non asymptotic error bound for non parametric estamation $f(x)=\mathbb{E}[Y|X=x]$
I am considering the following model:
$(X_i,Y_i)_{i=1}^n$ are iid random pairs with $(X_i,Y_i)\in[0,1]^2$. Let $f(x)=\mathbb{E}[Y|X=x]$. Consider an estimate $\hat{f}_n$ of $f$.
Under some hypothesis ...
- 1
0
votes
0
answers
121
views
Commensurability of arithmetic, irreducible, nonuniform lattices
Let $n \in \mathbb{Z}_{\geq 2}$ be arbitrary. Let $r_1$ and $r_2$ be arbitrary elements of $\mathbb{Z}_{\geq 0}$ that satisfy $r_1 + r_2 > 0.$ Let $G := {\rm SL}_n(\mathbb{R})^{r_1} \times {\rm SL}...
- 255
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66
views
Generalized compact open topology?
Let $X,Y$ be topological spaces. The compact-open topology on $C(X,Y)$ is generated by the sub-basic open sets
$$
\left\{U_{K,O}: \mbox{ K is compact in X and O is open in Y}\right\}\\
U_{K,O}:=\...
- 5,405
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votes
0
answers
45
views
Notation of $P^+$-families - bibliography searching
have you ever met with notation of $P^+$-families in other papers than Iian B. Smythe "A local Ramsey theory for block sequences" and his phd?
Thank you in advance
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votes
0
answers
142
views
English translation Gaston Darboux paper works
I'm looking for an English translation of Gaston Darboux's thesis:
Sur les équations aux dérivées partielles du second ordre
and
Mémoire sur la théorie des fonctions discontinues
Does anyone know ...
- 101
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votes
0
answers
116
views
Distribution of the $k$-th largest eigenvalue of in the sample covariance matrix?
Let us assume we've a rectangular data matrix $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$, where the $x_i \in \mathbb{R}^{p \times 1}$ are iid column vectors. I'm not assuming here that the ...
- 1,467
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votes
0
answers
79
views
Associativity and truncatedness of a $\mathbb N$-filtered object
$\require{AMScd}$Consider the monoid monad: $S : Set \to Set$ sends a set $X$ to the free monoid on $X$:
$$
SX := \coprod_{n\in\mathbb N} X^n
$$ the empty sequence $()\in X^0$ is the identity element ...
- 13.6k
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votes
0
answers
145
views
“Chapman-Kolmogorov”-convolution vs. smoothness
Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...
- 1,461
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votes
0
answers
94
views
Neat expresion for an anti-symmetric matrix
Fix a column vector $\pmb{v}$ and consider the cross product $\pmb{v}^T\times\pmb{x}^T$ for any column vector $\pmb{x}\in\mathbb{R}^3$. One can write
$$\pmb{v}^T\times\pmb{x}^T=A(\pmb{v})\pmb{x}$$
for ...
- 43.2k
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votes
0
answers
135
views
Reference for discrete Laplacian on $\mathbb{Z}$
For $x\in \mathbb{R}^\mathbb{Z}$, let the discrete Laplacian be defined as
\begin{align*}
(\Delta x)_k = 2x_k-x_{k+1}-x_{k-1}.
\end{align*}
I am looking for good references about its spectrum (or ...
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votes
0
answers
84
views
Relation between two matrices associated with a positive definite function
Let $f:\mathbb{R}^N \to \mathbb{R}$ be a positive definite function. Let $$g(h) = \int_{\mathbb{R}^N}f(x)f(h-x)\mathrm{d}x$$ Due to Bochner's and Parseval's theorems, $g$ is also a positive definite ...
- 698
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votes
0
answers
43
views
Reference request: Ito formula for function $G(t, x)$ when $G$ depend on $\omega$
There is proved Lemma in book : Let the function $G(t,x)$ is defined when $t\in [0,T], x\in(-\infty,\infty)$, $G$ has continuous derivative with respect to $t$ and twice continuously diferentiable ...
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206
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Any good references on the decay rate of Legendre coefficient?
Let $P_n:[-1,1]\rightarrow\mathbb{R}$ be the $n$-th Legendre Polynomial. and, let
$$a_n:=\int_{-1}^1 f(t) P_n(t)\, dt$$
for some $f:[-1,1]\rightarrow\mathbb{R}$.
Are there any good references on the ...
- 469
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votes
0
answers
46
views
Generalizing CIT-groups to odd case
A CIT-group is a group such that the centralizer of any involution is a 2-subgroup. The structure of these groups is known from the works of Suzuki and others.
Here is my question: has the odd case ...
- 307
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votes
0
answers
53
views
Which knots in the Rolfsen knot table are (quasi) positive braid knots?
A knot is called a positive braid knot if it can be presented as the closure of a positive braid. A knot is called a quasi-positive braid knot if it can be presented as the closure of a braid which is ...
- 1,430
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votes
0
answers
163
views
Gaps between primes - what to read?
I did a little search about available literature on gaps between primes, and, although I was confident in thinking that books about that topic are very rare, I really wasn´t sure that I will find ...
user147968
0
votes
0
answers
54
views
On $L^\infty$ norm of solutions to time dependent differential equations
I am new to the theory of differential equations and weak solutions.
I am looking for references regarding the analysis of the $L^\infty$ norm of weak solutions to linear second order time ...
- 597
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votes
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answers
132
views
Reference request: mathematical expectation of a random object in a topological space
Recently I got interested in the following question: what does a mathematical expectation look like for a random object taking values in a topological space?
This turns out to be a difficult ...
- 984
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answers
62
views
Can we have relations of the same type of their composing ordered pairs?
Is it possible to define a binary function $P$ in the language of set theory that obeys the characteristic property of ordered pairs and such that for any two sets $A,B$, for any definable relation $R$...
- 11.3k
0
votes
0
answers
137
views
Ask for some percolation reference textbook
I try to learn Bernoulli percolation recently. Could anyone provide some lecture notes or textbooks to enter this field? Thanks.
- 288
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answers
186
views
Subset of reals associated to pairs of matrices in $\mathrm{SL}(2,\mathbb{R})$
Let $\Gamma$ be a subgroup of $\mathrm{SL}(2,\mathbb{R})$. I would like to ask if there is any research on the following set:
$$\Gamma*\Gamma:=\bigg\{\dfrac{(a+b)(a'+b')}{(c+d)(c'+d')}\bigg|\begin{...
- 333
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votes
0
answers
89
views
Asymptotics of perturbations of polynomial systems
Disclaimer: the following question is taken from math.SE. It relates to perturbation theory, and I'm interested in references (if any) that relate to the following problem:
Suppose we are given a ...
- 101
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votes
0
answers
133
views
What about an alternative formulation for different prime constellations in the spirit of Suzuki's theorem for twin primes?
It is known that the twin prime conjecture is a special case of the $k$-tuple conjecture. See if you want the article with title k-Tuple Conjecture from the encyclopedia Wolfram MathWorld.
On the ...
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votes
0
answers
23
views
A linear map satisfying the given property
Let $A$ and $B$ be two Banach algebras such that $B$ is a Banach $A$-bimodue and $T:A\rightarrow B$ a linear map satisfying
$T(aa')=aT(a')+T(a)a'+T(a)T(a')$ for all $a,a'\in A$.
If the algerba ...
- 167
0
votes
0
answers
83
views
Is it possible to get a conjecture similar to Mandl's conjecture for a different arithmetic function of number theory, mainly related to primes?
I'm curious to know if are in the literature analogous conjectures to the conjecture due to Mandl, I ask about these analogous conjectures for different sequences playing an important role in number ...
0
votes
0
answers
86
views
Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
0
votes
0
answers
141
views
Research Request for a Paper of A.M. Leontovich
I am looking for a digital copy of a the English version of the paper "The Number of Mappings of Graphs, Ordering of Graphs, and Muirhead's Theorem" by A.M. Leontovich. The math.ru link to the paper ...
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votes
0
answers
121
views
Reference request: smooth affine curves are planar
Let $X\rightarrow\mathrm{Spec}\:\mathbb{C}$ be an affine smooth morphism of relative dimension$\leq 1$. What is a reference for the fact that there exists a $\mathbb{C}$-locally closed immersion $X\...
user142076