13
votes
1answer
600 views

How to prove that every polynomial in an infinite family is irreducible over Q?

Consider the bivariate polynomial $$p(X,Y) = X^5 - (2 Y + 1) X^3 - (Y^2 + 2) X^2 + Y (Y-1) X + Y^3.$$ For every integer $y \ge 4$, I conjecture that $p(X,y)$ is irreducible in $\mathbb{Q}[X]$. How can ...
1
vote
0answers
49 views

1d random walk probability of previous n positions

I have the following question. (May be it is very simple, but I cannot find the answer). Suppose I have a 1d random walk on integer numbers with equal pobabilities of unit step in either direction ...
1
vote
1answer
73 views

Weyl algebras $A_n(k)$ as tensor product of the first Weyl algebra

In afew threads I've read that the Weyl algebra $A_{n+1}(k)$ is isomorphic to the $k$-tensor product of $A_n(k)$ with $A_1(k)$, why is this true?
-2
votes
0answers
87 views

Formal definition of function: equality [on hold]

Consider this formal definition of function, which does not require $A$ to be domain. \vspace{0.5cm} \textit{Let $A$, $B$ be sets, then $F\subseteq A \times B$ is said to be a function iff ...
7
votes
1answer
212 views

The moduli space of special Lagrangian submanifolds

Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of ...
-4
votes
0answers
42 views

Mathametical Proof for a series [on hold]

Prove for the series given below z/((1−z)^2) =∑N≥1 N(z^N) Do i need to do this by induction?
2
votes
1answer
144 views

Explicit bound on $\sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$

I'm looking for an explicit bound for $f(x) = \sum_{N\mathfrak p \leq x}\chi(\mathfrak p)\ln(N\mathfrak p)$, where $\chi$ is a Hecke character for a number field $K$ of degree $n$, on the ideals ...
5
votes
1answer
405 views

Has this strengthening of the PNT already been conjectured?

Suppose $f:\mathbb{Z}_{\geq 0}\to\mathbb{Z}_{\geq 0}$ is an arithmetic function that grows slower than the identity map. Has it already been conjectured that, under this general hypotheses, ...
0
votes
0answers
5 views

Bipartite Graph [migrated]

Is there a bipartite graph with the following degrees: 3, 3, 3, 3, 3, 3, 3, 3, 3, 5, 6, 6? I've tried so many different combinations and I don't think there is a way to make a bipartite graph this ...
0
votes
0answers
98 views

Error term in formula for products of necklaces

Let us consider the $\lim_{n\to \infty}\prod_{p=1}^n N(p,a)$, where the number of fixed necklaces of length n composed of a types of beads $N(n,a)$ can be calculated via totient function: ...
3
votes
0answers
42 views

Hopf Lemma for strong solutions

The Hopf lemma still holds for strong solutions? Let $u\in W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)$ with $p>n$ such that $\Delta u + c(x) u\le 0$. If $u(x)>0$ for all $x\in\Omega$ and ...
4
votes
1answer
140 views

Blow-up of $\mathbb{P}^4$ along a quadric surface

Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
2
votes
0answers
45 views

Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite? By a "nice" stack I mean a smooth finite ...
1
vote
0answers
57 views

Do we know a lower bound for the number of critical zeros of the Riemann zeta-function with irrational imaginary part?

If I'm not mistaken, the imaginary parts of the critical zeros of the Riemann Zeta function are conjectured to be linearly independent over $\mathbb{Q}$, but I think we're very far from proving such a ...
0
votes
0answers
31 views

Single element extensions of subspace arrangement over finite field

Matroids have single element extensions found by Crapo[2] to create ground set size $n+1$ matroids from ground set size $n$ matroids. Do subspace arrangements over a finite field $\mathbb{F}_q$ have ...
-1
votes
1answer
99 views

notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals? the motivation for this question is: fractals are very difficult mathematical objects to work with, and many ...
5
votes
1answer
175 views

Can eta invariant be written in terms of topological data?

The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In ...
0
votes
0answers
16 views

norm is invariant under scaling for subspace of wiener space

Consider the subspace H_1 of $C_0(0,\infty)$, where $\phi=\int_0^t\dot{\phi}(s)ds$ and $\int_0^{\infty}{\dot{\phi}}^2ds<\infty$. The transformation is $(T\phi)(t)=t\phi(\frac{1}{t})$. How to show ...
0
votes
0answers
70 views

Eigenvalue of (0-1) matrix [on hold]

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
1
vote
1answer
91 views

On the geometry of roots of a sum of complex linear fractions

I was wondering if there is anything known about the geometry (position) of the roots of a rational map of the form $$R(z):= \sum_{j=1}^{n} \frac{a_j}{z-p_j},$$ where the $a_j$'s are nonzero complex ...
-1
votes
0answers
17 views

Gradient Estimation Using Bicubic Interpolation and Finite Differences [on hold]

Suppose we know the values f(x,y) takes on in a 4x4 grid defined by all pairwise combinations of x={0,1,2,3} and y={0,1,2,3}. Bicubic interpolation using centered differences provides a way of ...
-3
votes
0answers
34 views

complexe integration around a branch point [on hold]

I have this complex integral to which I don't know if it's possible to assign a value: The integral is on a small circle around the origin. The function is 1/((z-1)*sqrt(z)). The fact is that z=0 is ...
4
votes
1answer
142 views

Section of the homology functor on spectra

Consider the (reduced) homology functor $H_*$ from the category of spectra to the category of graded Abelian groups. I wanted to know whether there is a "section" of this functor, i.e., a functor $F$ ...
2
votes
1answer
175 views

What is “tilting” in the context of large deviations?

I have seen references to the "tilting method" in the theory of large deviations. Is there a simple explanation of what this is, exactly?
2
votes
2answers
81 views

Bounds for the fat tail after trimming the mean?

I am interested in the quantity $$f(X,t) = \int_t^\infty\negthinspace x\ p(x)\ dx,$$ where $p$ is a probability distribution for a positive variable $X$. 1) Does this quantity $f(X,t)$ have a name? ...
-1
votes
1answer
91 views

Suppose I know $\int h(t) dt = H(t)$, is there a way to find $\int h(t)^N dt$?

I am trying to find the -1 moments of sum of N geometric random variable, i.e. $E[\frac{1}{\sum_{i=1}^N X_i}]$ Suppose the probability mass function is $f_X(x) = (1 - p)^{x - 1} p$ The moment ...
1
vote
1answer
101 views

Follow up: Is continuity preserved when taking comma object?

Here, I asked wether taking lax pullback preserves continuity, but got no precise answer. However, I have found this recent article by Riehl and Verity which proves something very similar, but I ...
1
vote
1answer
124 views

Question on an arithmetic function with the sieve of Eratosthenes

I want to ask some question related with the sieve of Eratosthenes. The sieve of Eratosthenes: write it as $E_1(x) (=\pi(x)-\pi(\sqrt x)+1)$. Then we have an obvious result $$E_1(x)/x\ln^{-1}x = ...
1
vote
1answer
70 views

Equivalence of star products on two differents Poisson algebras?

Let $A$, $B$ be two commutative and associative $\mathbb k$-algebras and let $A_\hbar:=A[[\hbar]]$, $B_\hbar:=B[[\hbar]]$ be the corresponding ring of formal series. Of sense [Deformation theory and ...
0
votes
0answers
46 views

plotting parametrized algebraic curves near singularities

I have a parametrized algebraic curve: x(t)=A(t)/D(t); y(t)=B(t)/D(t); with A(t) and B(t) being polynomials in t. The curve is solution of a linear system in two unknowns x and y with Cramer's ...
1
vote
1answer
90 views

what's the cohomological dimension of a Stein space?

I want to know the "cohomological dimension" of a Stein manifold. I know that: for $X$ differential manifold and for every sheaf $F$ of abelian groups, I have $H_c^j(X,F)=H^j(X,F)=0$ for ...
3
votes
2answers
136 views

Structure of an intersection of $L^p$-spaces

In what follows, $L^p$ denotes the space of functions from $\mathbb{R}$ to $\mathbb{R}$ such that $\int_{\mathbb{R}} |f(x)|^p\mathrm{d}x < \infty$. I am interested to understand the structure we ...
3
votes
1answer
170 views

What is the group cohomology of the mapping class group of a surface

Let $MCG_g$ be the mapping class group of genus $g$ closed surface. (Say $MCG_1=SL(2,Z)$). I would like to know what is the group cohomology of $MCG_g$ with coefficients in Z, such as ...
3
votes
2answers
270 views

Beautiful constructions in algebraic topology that facilitate one's understanding of homotopy theory [on hold]

There is an army of interesting constructions in AT, and Understanding them are usually very helpful for appreciate the theory underneath. So I would like to invite you to share those examples that ...
1
vote
0answers
60 views

Reference Request: Algebraic Serre's Duality Theorem for Curves

Serre's Duality Theorem is well known and well studied and, as far as I know, there is a "big" algebraic proof for the general case, which is now kind of standard, and can be found in Hartshorne ...
11
votes
1answer
358 views

Normality of $\pi$ in base 16

It seems that in spite of the Bailey–Borwein–Plouffe formula it is still unknown whether $\pi$ is normal in base 16. What are the difficulties in using it for this purpose? In a comment to his answer ...
1
vote
0answers
62 views

Local system over $\mathcal A_{g,[n]}$ with unipotent monodromy

Let $\mathcal A_{g,[n]}$ denote the moduli space of principal polarized abelian varieties with level-[n] structure and $\bar {\mathcal A}_{g,[n]}\supset \mathcal A_{g,[n]}$ a smooth Toroida ...
2
votes
1answer
36 views

Centre manifold theory for a curve of equilibrium points

I am looking for advice concerning a specific situation related to centre manifold theory (compare Perko 2001). The part which is known Let's consider a differential equation in higher-dimensional ...
1
vote
0answers
131 views

Integer solutions of $ z^3 y^2 = x(x-1)(x+1)$

According to a conjecture there are no three consecutive powerful numbers. Necessary condition for this is integer solution of $$ z^3 y^2 = x(x-1)(x+1) \qquad (1) $$ What are integer solutions ...
1
vote
2answers
81 views

Invariant measure of Euler-Maruyama Discretisation of an Ito diffusion

Let $(X_t)_{t \geq 0}$ be a diffusion process with dynamics governed by the stochastic differential equation \begin{equation} dX_t = b(X_t)dt + \sigma(X_t)dW_t, ~~ X_0 = x_0, \end{equation} where ...
0
votes
0answers
79 views

Degree and quasi projective family

Let $V$ be a quasi-projective variety in $\mathbb{P}^{n}\times\mathbb{P}^m$. If $p\in \mathbb{P}^m$, we define the degree of $V_p$ as the degree of its closure in $\mathbb{P}^n$. Question : $\exists ...
-4
votes
0answers
73 views

Where to include contact details in math paper? [on hold]

I recently submitted a paper to a math journal on a prime number patter using latex formatting, but they sent an email back saying that the contact details for the corresponding author should be in ...
1
vote
0answers
52 views

$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$ Let $1\leq p \leq ...
1
vote
2answers
161 views

Can $T$ act trivial in a repn of SL$_2(\mathbb{Z}_N)$?

I am confronted with the following problem: If $\rho : \text{SL}_2(\mathbb{Z}) \to \text{GL}_{\mathbb{C}}(V)$ is a finite dimensional representation such that $\text{ker}(\rho)$ contains the ...
-1
votes
0answers
53 views

On the schur Multiplier of a group [on hold]

Let $G$ be a finite simple group and its Schur multiplier is 2. Is it true that if ${M\over K}\cong G$ and $|K|=2$, then $M\cong {\Bbb Z}_2\times G$ or $M\cong 2.G$, according to the symbols in the ...
1
vote
1answer
139 views

lattice in number field already a fractional ideal?

Let $K=\mathbb{Q}[\alpha]$ where $\alpha$ is integral over $\mathbb{Z}$ such that the Galois hull of $K$ can be embedded in $\mathbb{R}$. Let $S=\mathbb{Z}[\alpha]$. Let $x_1, \ldots , x_n$ be a ...
10
votes
2answers
204 views

Algorithms in hyperbolic groups

I'm stuck in some algorithms in hyperbolic groups, which may be rather simple. Let $G$ be a hyperbolic group given by a finite presentation. It is known that the hyperbolicity constant $\delta$ can ...
2
votes
0answers
31 views

QR-Decomposition of matrix valued function

Suppose I have a matrix valued function $$ F:\mathbb{R}\rightarrow\mathbb{R}^{m\times n},\qquad F(x)=\tilde Q\tilde R+xu_1v_1^T+xu_2v_2^T $$ where $\tilde Q\in\mathbb{R}^{m\times m}$ is orthogonal, ...
1
vote
1answer
102 views

Estimate infinity norm with Lp and W1p norm

Let $p \in [1,\infty)$. Does there exist $C>0$ such that for every $f \in W^{1,p}([0,1],\mathbb{R})$ we have $$\|f\|_{L^\infty}\leq C\|f\|_{L^p}^{1-\frac{1}{p}}\|f\|_{W^{1,p}}^{\frac{1}{p}}?$$ My ...
3
votes
0answers
55 views

Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)

I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the ...

15 30 50 per page