Unanswered Questions
49,209 questions with no upvoted or accepted answers
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Bounding $l^0$ norm of random quantity
There are many techniques in high dimensional probability for bounding quantities of the form
$$ \mathbf{E}( \sup_{s \in S} X_s ) $$
where $\{ X_s \}$ are a family of random variables which are not ...
- 455
-1
votes
1
answer
785
views
What is a type $\text{II}_1$ factor von Neumann algebra?
After finding formal definitions in various texts (see, eg, Witten, Notes On Some Entanglement Properties Of Quantum Field Theory, Rev. Mod. Phys. 90, 45003 (2018), doi:10.1103/RevModPhys.90.045003 ...
- 227
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votes
2
answers
638
views
Peano axioms— mathematical induction and other axioms
The Peano axioms of $\Bbb N$ are:
$1 \in \Bbb N$, i.e. $\Bbb N$ is not empty and contains an element denoted by $1$.
Every natural number has a successor, i.e. $\forall n\in\Bbb N, \exists!s(n)\in\...
- 101
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votes
1
answer
236
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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 ...
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-1
votes
1
answer
283
views
Lowerbounding expectation value of binomial tail
I'm trying to find a lower bound for the following expression for $q\ge p$:
$$f(q,p,n) := \sum_{v=0}^n \sum_{k=v}^n \binom{n}{v} \binom{n}{k}q^v(1-q)^{n-v}p^k(1-p)^{n-k}.$$
It can be thought of as the ...
- 702
-1
votes
1
answer
215
views
Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques
Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
- 2,089
-1
votes
1
answer
193
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Limit of the convolution of derivative of Gaussian heat kernel
I'm looking for the following limit:
$$\lim_{\varepsilon\to 0^+}\int_{-\sqrt{\varepsilon}}^{\sqrt{\varepsilon}}\frac{1}{\sqrt{2\pi}\varepsilon^{3/2}}\left(-1+\frac{x^2}{\varepsilon}\right)e^{-\frac{x^...
-1
votes
1
answer
113
views
Approximating expectation of exponential of Wishart matrix
I am trying to obtain an Approximating expectation of exponential of Wishart matrix $X (N,N)$ with $\operatorname{rank} (X)=K$defined as:
\begin{align}
J = E[{e^{{v^H}Xv}}]
\end{align}
where $v$ is $...
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-1
votes
1
answer
246
views
Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators
Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
- 167
-1
votes
1
answer
162
views
A topological space whose closed subsets are locally connected
Let $X$ be a compact $T_0$ topological space such that every closed subset of $X$ is locally connected. Is there any characterization for such a space? I guess $X$ is Noetherian, but I cannot prove ...
-1
votes
1
answer
124
views
Borel summation
If $f(z)=\sum_{n=0}^\infty a_n z^n$ is a formal power series with complex coefficients, then its Borel transform is defined by
$$B(f)(z)=\sum_{n=0}^\infty a_n \tfrac{z^n}{n!}.$$
Suppose that $f$ and ...
- 139
-1
votes
1
answer
98
views
Tuza theorem to prove vizing theorem
The Tuza theorem states that every graph with no cycle congruent to 1 mod $k$ is $k$ colorable. Now, the line graph of any simple graph of maximum degree $d$ is seen to posess the property that it has ...
- 2,089
-1
votes
1
answer
312
views
expectation of upper quantile proportion
(edited considerably following comments)
We have a collection $\boldsymbol{S}$ of $n$ discrete random variables $X_1$, $X_2$, $\dots$, $X_n$ $\overset{\small \text{i.i.d.}}{\small \sim}$ $\mathcal{D}$...
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votes
1
answer
114
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Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values
To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
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-1
votes
1
answer
138
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Integral zeros of the Newton polynomial
I'm trying to understand the following result;
Statement: A newton polynomial of the form
$$a_1 {x\choose c_1}+a_2{x\choose c_2}+a_3{x\choose c_3}+⋯+a_s{x\choose c_s},$$
where $0 ≤c_1<c_2<c_3&...
-1
votes
1
answer
126
views
Is there a name for this family of integral?
This one: $\int_{0}^{\bar{x}}e^{-x^{a}}x^{b}(1-x)^{c}dx,a,b,c\ge0$. When $a=1,c=0,\bar{x}=\infty$ it is the gamma function.
- 101
-1
votes
1
answer
895
views
Restriction of a Cartier divisor
Let $X$ be a surface (so $2$-dimensional proper $k$-scheme)
$D \subset X$ an effective Cartier divisor of $X$ which corresponding to an invertible sheaf $\mathcal{L}=O_X(D)$ and
$C \subset X$ a closed ...
- 6,038
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votes
1
answer
149
views
Perform a univariate integral, involving a Gauss hypergeometric function
This is a follow-up question to the one posed in Compute the two-fold partial integral, where the three-fold full integral is known . (I hope that doing so is viewed as a legitimate step. If not so, I ...
- 723
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votes
1
answer
64
views
square weighted l^2
I am looking the sequence spaces $l^1$ and $$\{(x_k)_k: \|x\|_{sq}^2 := \sum_{k=1}^\infty k^2\cdot x_k^2 < \infty\}. $$
I am particularly interested in relations between their respective norms: It ...
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votes
1
answer
395
views
Odd & even permutations and unit fractions
One more motivated by recent questions of Zhi-Wei Sun.
Let $S_n$ be the group of permutations of $\{1,2,\ldots, n\}$.
Is it true that, for every $n \ge 8$, there is at least one even permutation $\...
- 4,589
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votes
1
answer
2k
views
conditional expectation brownian motion
$B=(B_t,t\in[0,1])$ a standard brownian motion on $[0,1]$.
For $t\in[0,1]$, we define $$\mathcal{F}_t=\sigma(B_s,s\in[0,t]),$$ $$\mathcal{G}_t=\mathcal{F}_t\,\vee\,\sigma(B_1).$$
How can we show $$\...
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votes
1
answer
94
views
How many hyper-rectangle-like objects are intersecting a hyperplane?
Let $A\in \mathbb R^{n\times n},\ b\in \mathbb R^n$ such that $\forall x\in \{-1,1\}^n : Ax\ne b$.
Let us denote: $S=\{x\in\mathbb R^n|Ax=b\}$ ('S' for solution set).
Is $\ \#\Big\{H\in\big\{ \{-1\},...
- 265
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votes
1
answer
76
views
Transforming random variables for having good property?
For arbitrary functions $A$ and $B$ and independent random variables $X$ and $Y$, assume that
\begin{align}
\Omega&\triangleq \{(x,y): A(x,y)=1\},\\
\Lambda&\triangleq \{x: B(x)=1\}.
\end{...
- 287
-1
votes
1
answer
76
views
transformation of two measures on different space
Let $\{e_1,e_2,...,e_n\}=E $ be the standard bases of $\mathbb{R}^n$, and $U\subset\mathbb{R}^n$ be a linear space generated by $\{e_1,e_2,...,e_n\}$.
Let $\Sigma_U$ be the smallest $\sigma-$ field ...
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-1
votes
1
answer
105
views
What type of graph is this? (Edges that are valid / invalid depending on route to node)
I'm trying to model a questionnaire where the flow between questions depends on the answers given in previous questions.
Example. (Node represent questions, edges represent answers).
As you can see ...
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-1
votes
1
answer
64
views
Minimax solution but game has no value
Fix convex sets $\Delta,\Pi$ and let $r: \Pi \times \Delta \in [0,\infty]$ be linear (i.e., concave and convex) in its first parameter for every fixed second parameter.
I'm looking for a situation ...
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-1
votes
1
answer
124
views
Typed Values in Formulas
Question:
are there any "standard" ways of indicating the meaning of numerical values in formulas, resp. general mathematical texts (theorems, proofs, etc.)?
I am especially looking for ...
- 13.2k
-1
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
-1
votes
1
answer
428
views
Bins and colored balls
Consider $n$ color balls. We throw them as follows. For a given ball $i$, randomly choose $k$ bins; create $k$ 'copies' of the ball (i.e., of the same color of the ball $i$); throw a 'copy ball' into ...
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votes
1
answer
99
views
Finitely many subvarieties as divisor
Let $X$ be a smooth projective variety over an algebraically closed field of characteristic $0$ and of dimension $n\geq 10$. Let $(C_i)_{1\leq i\leq N}$ (resp. $(S_i)_{1\leq i\leq N}$) be smooth ...
- 1,463
-1
votes
2
answers
512
views
Deriving the joint distribution of multivariate normal transformed into Bernoulli
Given a covariance matrix $\sum_{ij}$ and a mean vector $\mu$ I have sampled $N$ multivariate normal vectors $Z = (z_1,...z_n)$ My goal is to create a vector of Bernoulli random variables $Y = (y_1,......
-1
votes
1
answer
180
views
Orthogonal polynomials of the second kind
Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $L(x^n)= s_n, n\ge 0.$ Given a positive ...
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-1
votes
1
answer
131
views
Isomorphism of Stable groups formed by different sequences
I have been thinking the relation between two stable groups formed as follow:
1) $GL(2,\mathbb R)\subseteq GL(3,\mathbb R)\subseteq\cdots GL(n,\mathbb R)$,
$G(\infty,\mathbb R)=\cup_{n=2}^{\infty} GL(...
-1
votes
1
answer
37
views
Graph with small automorphism and large isomorphism
Is there a graph family on $n$-vertices such that any graph $G$ in the family have small automorphism group (say $|Aut(G)|\leq n^c$ for some fixed $c>0$) which if $G$ and $H$ are isomorphic then ...
- 13.9k
-1
votes
1
answer
63
views
Idempotent solutions to the implict function theorem other than the identity?
I am interested in the following problem. Assume that an (anti)symmetric function $g:\mathbb{R}^2 \to \mathbb{R}$ satisfies the implicit function theorem. That is, $g(x,y) = \pm g(y,x)$ and $g(x,y)=0$ ...
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-1
votes
1
answer
130
views
Is there an algorithm to find a linear dependence between points on elliptic curves?
Let $E$ be an elliptic curve over a finite field $\mathbb{F}_p$ of characteristic $p$. Let $P,Q\in E(\mathbb{F}_q)$, such that $Q=mP+n\tau(P)$, where $\tau$ is the p-th power of frobenious map and $m$ ...
- 369
-1
votes
1
answer
173
views
For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$
Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
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-1
votes
1
answer
460
views
Orthogonal decomposition of conditional expectations
Suppose I have a random variable $x$ and a set of conditional distributions on $x$. Here is an example where the conditionals are nested:
$$q_1 := E(x|y_1), \quad q_2 := E(x|y_1,y_2),\quad q_3 := E(x|...
- 427
-1
votes
1
answer
139
views
Reference request: automorphism of abelian $p$-groups of rank 2
There is a result saying that if $P$ is an abelian $p$-group of rank $2$ then $$|Aut(P)|=(p-1)^kp^j(p+1)^r,$$
for some $k,j,r$ which depend on the order of $P$.
Moreover, if $P=C_{p^t}\times C_{p^s}$ ...
- 11
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votes
2
answers
441
views
Motivation for weak solution of a PDE (initial condition)
The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ \...
- 1
-1
votes
1
answer
148
views
Analytic extension of the exterior Newtonian potential into the domain
I just want to know for a domain with smooth boundary whether exterior Newtonian potential has singular analytic extension into the domain or into a part of the domain.
Definition of Newtonian ...
-1
votes
1
answer
104
views
For $q$ prime and $\gcd(q, n) = 1$, can $N = qn^2$ be almost perfect?
In general, for $q$ prime and $\gcd(q, n) = 1$, can $N = qn^2$ be almost perfect?
An almost perfect number $N$ is a positive integer satisfying $\sigma(N) = 2N - 1$, where $\sigma(x)$ is the sum of ...
-1
votes
1
answer
293
views
spectrum of a special class of tridiagonal matrices
Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e.,
$$\begin{bmatrix}...
- 15
-1
votes
1
answer
200
views
Collecting terms of a linear expression with nested sums and combinatorics in coefficients
I need to collect the $\Pr(\cdot)$ terms of the following expression:
$\sum_{m=3}^{n}\frac{g_{m}\left( \cdot \right) }{\left( \sqrt{\theta \left(
1-\theta \right) }\right) ^{m}}\left[ \sum_{j=2}^{m-1}...
-1
votes
1
answer
249
views
An extension of the real semiring with multiple degrees of infinity
Is it possible to define an extension of the probability semiring $(\mathbb{R}^+, +, \times, 0, 1)$ such that
Closure $a^* = 1 + a + a^2 + \ldots$ is defined for every element of the semiring, not ...
-1
votes
1
answer
152
views
Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )
Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2.
More generally, is it true that eigenfunctions ...
-1
votes
1
answer
183
views
one-cocycles over finite groups
Let $G$ be a finite group and let $S$ be a subgroup of $G$. Let
$\alpha$ be a one-cocycle $G\to M$ for some $G$-module $M$, and
suppose that $Res_{S}^{G}\alpha=0$. For $g\in G$, and $S^{g}=gSg^{-1}$,
$...
-1
votes
1
answer
2k
views
Variance of euclidean norm of Gaussian vectors
Let $X$ be a Gaussian vector in dimension $n$, with $0$ mean and covariance identity. Let $A$ be a square matrix of size $n$, and $Y = A X$. Let $N$ be the square of $Y$ euclidean norm: $N = \sum Y_i^...
- 11
-1
votes
1
answer
504
views
Dominant morphism of Affine Scheme
Suppose to have $\phi$ a ring morphsim from $A$ to $B$, let $X=SpecA$ , $Y=SpecB$ and $\psi$ the induced morphism of affine schemes. It's true that if $\psi$ dominant than $\phi$ is injective?
- 29
-1
votes
1
answer
493
views
Upper bound on iterations count for power iteration algorithm
I'm stuck trying to get upper bound on iterations count for power iteration algroithm for finding first eigenvalue of adjacency matrix $A$ given tolerance value. I've tried to figure something out ...
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