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9
votes
0answers
145 views

On the infinity of $\{p\in \mathbb {N}:\exists n\in\mathbb{N}~p| \left \lfloor{r^n}\right \rfloor\}$

I've already asked this same question on MSE here, but didn't get much help, so I will try on this site as well. For which $r\in\mathbb{R}$ is the set $\mathscr{P}_r=\{p \in \mathbb{P}:\ (\exists n\...
10
votes
1answer
218 views

On certain order-automorphisms of the rationals

Consider the rationals $\mathbb{Q}$ with the usual order $\leq$. Now let $A$ be a subset of $\mathbb{Q}$, such that foreseen with the induced order $\leq$, $(A,\leq)$ is a dense linear order. ...
-1
votes
0answers
40 views

On standard form [duplicate]

If $(M, \varphi)$ is a vN algebra in standard form in the GNS space $L^{2}(M,\varphi)$ and $P$ is a projection in $M$. What is the standard form of $PMP$? Here $\varphi$ is faithful normal state in $...
5
votes
0answers
109 views
+50

Weaker versions of Gandy ordinals

Gostanian's paper "The next admissible ordinal" (see https://www.sciencedirect.com/science/article/pii/0003484379900251 ), is concerned with the supremum of the $\alpha$-recursive ordinals for various ...
1
vote
0answers
105 views

For non-noether $A$, there exists a complex $L^\bullet$ such that for all $M$ $H^i(X, \mathscr{F} \otimes_A M) \cong h^i(L^\bullet \otimes_A M)$

Let $A$ be a ring, $X \to A$ a proper morphism of finite presentation, and $\mathscr{F}$ be a quasi-coherent module on $X$ of locally finitely presented, which is flat over $A$. Then there exists a ...
4
votes
0answers
86 views

When are extensions of algebraically good groups algebraically good?

Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...
1
vote
0answers
41 views

Lifting of Contact isotopies on a symplectization

Let $\mathbb{R}^{2n}\times\mathbb{R}=\mathbb{R}^n\times\mathbb{R}^n\times\mathbb{R}=\{(q,p,z)|{q},{p}\in\mathbb{R}^n,z\in\mathbb{R}\}$ be a contact manifold with the standard contact form $\alpha=pdq+...
-3
votes
2answers
203 views

What Differential Equations we still don't know how to solve despite being proven to have a solution? [closed]

It seems that we cannot solve all differential equations that bump up into nature, and here I mean finding a general solution (not just numerical approximation and without including the cases where no ...
0
votes
1answer
61 views

flow, stable manifold and tangent

Given vector field $f: \mathbb{R}^2 \to \mathbb{R}^2$, with $f(0)=0$ ODE: $\dot{x}=f(x)$ generates a flow $\Phi^{t}$. so $\Phi^{t}(0)=0$ for all $t \in \mathbb{R}$ So time-one map $\Phi^1$ is diffeo....
2
votes
0answers
69 views

Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
4
votes
1answer
366 views

Are there infinite many two sided prime numbers?

A prime number $p=\overline{a_na_{n-1}\ldots a_1a_0}$ is called a two sided prime number if its reverse representation $q=\overline{a_0a_1\ldots a_{n-1}a_n}$ is a prime number too. Are there ...
9
votes
0answers
186 views

On the status of some conjectures mentioned/used in Harish Chandra's 1970 lecture notes

In van Dijk's notes of Harish Chandra's lectures on harmonic analysis, several conjectures are mentioned throughout, such as in Part 1, section 4 of van Dijk's notes Conjecture I : Let $\omega$ be ...
8
votes
1answer
330 views

Set of points with a unique closest point in a compact set

Let $K\subset\mathbb{R}^n$ be any compact set. Let $\operatorname{Unp}(K)$ be the set of points in $$ \operatorname{Unp}(K)=\{x\in\mathbb{R}^n\setminus K:\, \exists ! y\in K \ \ |x-y|=d(x,K)\}. $$ ...
6
votes
0answers
165 views

Example of a tensor triangulated category with two different monoidal t-structures?

What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures? While I'm at it: is there an example of a tensor ...
4
votes
1answer
83 views

Expected supremum of normalised random walk

Let $X^i\in \mathbb R^d$ be iid. random variables for $i=1$ to $n$. Assume $\mathbb E[X^i]=0$ and the covariance matrix $\mathbb C[X^i] = \mathbb E[X^iX^{iT}] = I$ is the identity matrix. Define $S^k=...
0
votes
0answers
30 views

Optimization of an integral functional when the multiplier rule yields no useful information

Let $(E,\mathcal E,\lambda)$ be a measure space $\mu\ll\lambda$ be a probability measure on $(E,\mathcal E)$ $p\in[1,\infty)$ $k\in\mathbb N$ $f:E\times\mathbb R^k\times\mathbb R^k\to[0,\infty)$ such ...
5
votes
1answer
145 views

Anti-holomorphic involutions of a complex linear algebraic group

Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$. Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$. Let $\sigma$ be an anti-...
2
votes
1answer
60 views

Index and congruence subgroup from scaling variables of Jacobi form

Let $J_{k,m}(N)$ be the space of Jacobi forms of weight $k$, index $m$, and congruence subgroup $\Gamma_{0}(N) \rtimes \mathbb{Z}^{2}$. I do not believe it is relevant here to specify what type of ...
1
vote
1answer
97 views

On existence of certain operators in von Neumann algebra

Let $M\subset B(\mathcal{H})$ be an infinite dimensional vN algebra in standard form. Fix $\xi\neq 0 \in \mathcal{H}$, does there exist $M\ni x_{\xi}\neq I$ such that $x_{\xi}(\xi)=\xi$?
5
votes
1answer
453 views

Are the coefficients of certain product of Rogers-Ramanujan Continued Fraction non-negative?

Let $$R(q) = \cfrac{q^{1/5}}{1 + \cfrac{q}{1 + \cfrac{q^{2}}{1 + \cfrac{q^{3}}{1 + \cdots}}}}$$ The following equality is famous: $$\cfrac{q^{1/5}}{R(q)} = \prod_{k>0} \cfrac{(1-q^{5k-2})(1-q^{...
4
votes
2answers
110 views

Understanding equiprobable trinomial identity

With $f(x_1,x_2,x_3,x_1+x_2+x_3;\,1/3,1/3,1/3):= \frac{(x_1+x_2+x_3)!}{x_1!\,x_2!\,x_3!\, 3^{x_1+x_2+x_3}}$ denoting the probability mass function of the equiprobable trinomial distribution as in wiki/...
3
votes
1answer
147 views

Sum-product estimate in finite fields

There is a paper by Bourgain, Katz and Tao Bourgain, Jean; Katz, N.; Tao, Terence C., A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14, No. 1, 27-57 (2004). ZBL1145....
3
votes
0answers
134 views

Conventions for Riemann curvature tensor

I am aware of two conventions for the Riemann curvature tensor, namely the expression $$\langle\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,W\rangle$$ is either declared to be $R(X,Y,Z,W)$ or $...
0
votes
1answer
104 views

Notion of module over commutative post lie algebra

Let $ (S, \{. \}, [.]) $ be an algebra over a vector space endowed with two bilinear maps $ \{. \}, [.] : S \times S \rightarrow S$ and satisfying some compatibility conditions. In example, if S is ...
2
votes
1answer
98 views

subelliptic Sobolev compact embedding theorem

Consider the smooth vector fields $X=(X_1,X_2,...,X_m)$ defined in a open bounded set $\Omega\in R^n$. And the non-isotropic dimension is $Q$ (see https://arxiv.org/pdf/1502.06332.pdf page 398) In the ...
7
votes
0answers
138 views

Meaning of Elliptic Irregular Primes

The Bernoulli numbers are defined by the equation $$ \frac{t}{e^t-1}=\sum_k b_k \frac{t^k}{k!}. $$ A prime number $p$ is irregular if it divides the numerator of one of the even Bernoulli numbers up ...
7
votes
1answer
225 views

Kinematic formula for Euler characteristic

Is there a formula for $\int \chi(K \cap gL) \: dg$ (where $\chi$ is Euler characteristic) analogous to the kinematic formula for $\int \mu(K \cap gL) \: dg$ (where $\mu$ is Lebesgue measure)? In both ...
0
votes
0answers
34 views

Can we have a stratified theory equivalent to NFU + Infinity + choice with atmost failure of Extensionality?

It is known in NFU + Infinity + Choice, that we can partition the set $U$ of all Ur-elements (empty objects other than the empty set) such that each piece is as big as the set $V$ of all objects, and ...
1
vote
0answers
61 views

Probabilistic lower bound on largest singular value of matrices

I have a distribution $\mathcal{D}$ that spits out vectors in $\{-1, 1\}^N$. Suppose I have a sample of $H$ of these vectors which I arrange into a matrix $M$ of the form $H \times N$. Consider the ...
2
votes
0answers
89 views

View Dirichlet character as a character of Galois group

In Jaclyn Lang's article "On the image of the Galois representation associated to non-CM Hida family" section 2, the Dirichlet character $\chi$ module $N$ is also viewed as a character $\chi\colon\...
-1
votes
1answer
75 views

Coupling argument involved in the contracting and mixing properties of the Glauber dynamics for an Ising model

While doing a research work, I had to read about the Glauber dynamics for an Ising model. A wonderful account on this is given in the book Markov Chains and Mixing Times by Levin, Peres and Wilmer. ...
6
votes
1answer
204 views

Guessing the number of other $1$'s in a binary sequence

I have posed the following question on math.stackexchange.com but have not received an answer. So I would like to seek experts' opinion here. Consider the set of all binary sequence of length $n+1$, $...
0
votes
0answers
14 views

Quantiles of the Q values of an unknown MDP

Consider an MDP with $n$ states, $k$ actions, and discount factor $\gamma \in [0,1)$. We are uncertain of its reward function $R \in \mathbb{R}^{n \times k}$ and transition function $T \in \mathbb{R}^{...
4
votes
1answer
226 views

Is it possible to use Feynman diagrams to represent a dot product $a \cdot b$?

Feynman diagrams are topological entities, but they describe linear operators It has been observed that Feynman diagrams are in particular string diagrams (morphisms in monoidal categories)) in ...
1
vote
0answers
56 views

Existence of fundamental solution of fractional laplacian

Does the fundamental solution of $$(-\Delta)^sF(x, y)+ V(x) F(x, y)= \delta_{y}(x) \text{ in } \mathbb R^N $$ exist. Here $V(x)$ is a positive smooth non-constant bounded function which satisfy $V(x)\...
2
votes
1answer
120 views

Representative in ideal class group coprime to the conductor

Working in an order $\mathcal{O}$ in an imaginary quadratic field $K = \mathbb{Q}(\sqrt{d})$ and given an invertible ideal $\mathfrak{a}\subseteq \mathcal{O}$, I would like to produce another integral ...
1
vote
1answer
102 views

Exponential upper bounds for sums of martingale differences

Let $(X_{i})_{i\geq 1}$ be a sequence of centered real-valued martingale-differences with respect to some filtration $(\mathcal{F}_{i})_{i \geq 1}$. Define $S_{n} = \sum_{i=1}^{n}X_{i}$ and $\Sigma^{2}...
3
votes
1answer
113 views

Commensurator of a subgroup of matrices

Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...
5
votes
0answers
96 views

Is there an equivariant simplicial deformation retract of Teichmüller space?

Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...
0
votes
0answers
21 views

Name for subset selecting matchings

Tutte and also Lovasz and Plummer reduce the calculation of (optimal) f-factors in graph to non-bipartite matching via replacing each vertex with a $K_{f,\delta}$, refered to as a 'gadget' whose ...
0
votes
1answer
117 views

A question about entire functions of order 1

Suppose $f:\mathbb C \to \mathbb C$ is an entire function on the complex plane of order $1$. Additionally, suppose that: $$ \forall\, c \in \mathbb R, \quad \lim_{t \to \pm \infty} \, f(t+ic) =0.$$ ...
2
votes
0answers
118 views

Counting special metrics on finite fields

Define a Galois coding norm of degree n as a map $|\space| : \Bbb F_{2^n}\rightarrow {\Bbb Z}$ with the following properties : (I) $(\Bbb F_{2^n},|\space|)$ is a self-orthogonal code ; i.e. $(x,y)\...
4
votes
2answers
367 views

Convex hull in a discrete space [closed]

I know some algorithms which compute the convex hull in a continuous space. Are there efficient algorithms to compute it in a discrete domain? For example in 3D discrete space, given the blue points, ...
3
votes
0answers
305 views

Is $\sum_{k=1}^{n} \cos(k^ {1/3} )$ bounded by a constant M? [migrated]

I understand what $\sum_{k=1}^{n} \cos(k)$ is bounded by a constant, but I don't have any idea how to prove if $\sum_{k=1}^{n} \cos(k^ {1/3} )$ is bounded or not.
0
votes
0answers
53 views

Sum of i.i.d discrete random vectors, restricted sum of multinomial

We are given a random vector of dimension $n$ which takes a value one with the probability $p_{1,1}$ in the first coordinate, value two with the probability $p_{1,2}$ in the first coordinate and so ...
2
votes
1answer
87 views

Finding all unitary representations of the connected Poincaré group

I am studying representation theory of Lie groups and its combination to theoretical physics, and I am concerned about the following. Is there an exhaustive way to find all unitary representations of ...
3
votes
1answer
75 views

4-polytopes with only one kind of regular facet

Is there a neat way to show (or a reference that already proves) that the 4-cube is the only convex 4-polytope in which all facets are regular 3-cubes? the 24-cell is the only convex 4-polytope in ...
3
votes
0answers
160 views

Does Spec functor sends pushouts of rings into pullbacks of sets?

This question was posted here on StackExchange and it's still without an answer. Let $A$ be a commutative ring and $B,C$ be two commutative $A$-algebras. Consider the pushout square of ring ...
5
votes
1answer
119 views

Translated version of a Caratheodory article

This excellent introduction to Compressive Sensing cites a couple of (seemingly) interesting Caratheodory papers from 1907-1911. These are: [46] C. Caratheodory. Uber den Variabilitätsbereich der ...
6
votes
1answer
254 views

Is min exponents of three positive integers $n$, $n+1$ and $n+2$ $=1$ true or false?

Given a positive integer $P>1$, let its prime factorization be written $$P=p_1^{a_1}p_2^{a_2}p_3^{a_3}\cdots p_k^{a_k}$$ Define the functions $h(P)$ by $h(1)=1$ and $h(P)=\min(a_1, a_2,\ldots,a_k)$...

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