0
votes
0answers
5 views

Can this equality be possible for any nonzero real $b$?

Please may you kindly assist me on this integration exercise: Suppose for some fixed real $a, b$ with $a \neq 0$, we have: $$\int_1^N f(x)\sin(a\log \sqrt x)x^b \mathrm{d}x = \int_1^N f(x)\sin(a\log \...
1
vote
1answer
30 views

Homotopy type of some lattices with top and bottom removed

The only reaction to this question on math.SE was 21 views in 4 days, so I decided to repost it here. I am not changing anything. There was an interesting question on MO which OP removed by some ...
0
votes
1answer
27 views

GCD for two Cullen numbers

The $n$'th Cullen number is $C_n = n\cdot2^n+1$. If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
0
votes
0answers
11 views

Cluster algebra structure compatible with Poisson brackets

Let $X$ be a Poisson variety. There is a concept "cluster algebra structure compatible with Poisson structure" introduced in the paper. Suppose that we construct a maximal independent set of ...
0
votes
0answers
13 views

Mersenne number with small Carmichael function

Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:...
-2
votes
0answers
17 views

I can't derive the integrating factor of this first order linear Equation [on hold]

I can't derive the integrating factor of this first order linear Equation (x2 - y2 - y) dx - (x2 - y2 - x) dy = O. the answer is: integrating factor = 1/(x2 - y2)
0
votes
0answers
25 views

Jordan curve in $C^2$

Can we find a Jordan curve $\gamma$ in $\mathbf{C}^2$ of class $C^1$ such that the projection to the first coordinate plane divides the plane into infinite components of connectivity.
0
votes
0answers
18 views

Asymptotic Growth of Markov Chain

I asked the following question one week ago at math.stackexchange but didn't receive a response, so I want to give it here another try: I'm interested in the following problem: We have got a time-...
1
vote
0answers
17 views

Is there a full-rank map with connected graph and simply connected image that is not injective?

I want to find a continuously differentiable function $F:X\to Y$, where $X\subseteq\mathbb{R}^n$, $Y\subseteq\mathbb{R}^m$ are open ($n\le m$) with ${\rm rk}\, \frac{\partial F}{\partial x}(x) = ...
0
votes
0answers
64 views

The uses of the polar topology in topological vector spaces

The polar topology originates from the $S$-topology and is used in duality pairs. Due to the connection between the original topology and the weak topology, we can rephrase the original topology in ...
0
votes
0answers
8 views

Connectedness of the space of symplectic embeddings into a higher dimensional manifold

Suppose $M$ and $N$ are symplectic manifolds, $N$ is compact, $\dim_{\mathbb R}(N) \leq \dim_{\mathbb R}(M) -4$. Suppose there are embeddings $f_i:N \to M$, $i=0,1$ such that $f_i^*\omega_M$ is non-...
8
votes
1answer
349 views

What should I cite for the Poincaré conjecture?

I'm writing a paper that, rather unexpectedly, needs the Poincaré conjecture for one of the results. (The paper has almost nothing to do with differential geometry!) The conjecture was famously ...
0
votes
0answers
26 views

$L^2$ bound deduce exponential decay

For the cylindrical $$X\times\mathbb R,$$ where $X$ is a closed manifold, we give a product metric $g^{TX}\times dt^2$ on it. Q: If function defined on it is $L^2$ or $L^2_1$ bound, i.e. $f\in L^2(...
-4
votes
0answers
11 views

Exam FM Study Material [on hold]

This is a math question from an Exam FM study textbook that I've been looking over that I need an explanation for: A fund is earning 5% simple interest. Calculate the e ffective interest rate in the ...
4
votes
1answer
60 views

On certain solutions of a quadratic form equation

This is a continuation of this question: A class of quadratic equations Let $f(x,y) = ax^2 + bxy + cy^2$ be an irreducible and indefinite binary quadratic form. Consider the equation $$\displaystyle ...
0
votes
0answers
11 views

$H$ self-adjoint with mass gap, $P≥0,Ω∈D(P),H+λP$ self-adjoint $⟹$ for $λ$ small, $H+λP$ has gap?

Suppose $H$ is a self-adjoint operator on a Hilbert space having a simple isolated least eigenvalue $0$ with gap $1$ ( $H\Omega = 0$, $\Vert \Omega\Vert = 1$ ), $P$ is a non-negative symmetric ...
0
votes
0answers
74 views

Variant Karamata inequality [on hold]

We observe function $f(x)=x^2$ in the figure as following We can see that: $\frac{f(x_1)+f(x_2)}{2}-f(\frac{x_1+x_2}{2}) \ge \frac{f(y_1)+f(y_2)}{2}-f(\frac{y_1+y_2}{2}) \\$ when $AB \ge CD$ when ...
0
votes
0answers
19 views

What are the rank 3 boolean intervals [H,G], with G simple group?

The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$. The lattice $B_{3}$ is the following: Question: What are the rank $3$ boolean intervals of the form $[H,G]$, ...
2
votes
0answers
38 views

Generalization of de Rham cohomology, or cohomology for non-smooth case

Let $\Omega\subseteq \mathbb{R}^{3}$ be a simply connected domain and $f:\Omega\to \mathbb{R}^{3}$ be a vector field on $f$. If $f$ is smooth vector field and $\nabla\cdot f=0$, then $f=\nabla\times g$...
0
votes
0answers
60 views

What is a universal tree?

I came across some slides talking about the Hrushovski construction. One of the examples was the construction of a "universal tree". I was curious because the collection of finite trees does not ...
5
votes
0answers
47 views

Integral basis of an extension of complete fields

Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$. Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure $\...
5
votes
0answers
76 views

When is a given polynomial a square of another polynomial?

I meet a problem in which I hope to show a special polynomial is not a square of another polynomial. More precisely, let's consider the polynomial $f(x):= 1-x+2bx^n-2bx^{n+1}-b^2x^{2n-1}+2b^2x^{2n}-b^...
4
votes
0answers
39 views

Coming up with a represenation for sum of functions in the Fourier algebra

This is my first overflow question, so let me apologize in advance if this belongs on http://math.stackexchange.com. Let $G$ be a discrete group. Let $\lambda:G\to B(\ell^{2}(G))$ be the left ...
1
vote
0answers
28 views

Non-negative Polynomials from Polynomial Ideal?

Related to the thread Nonnegativity conditions for a polynomial in two variables? but on more than two variables. Suppose a probability vector $p$ belongs to a compact polytope where for each entry $...
3
votes
0answers
57 views

Automorphisms of unipotent groups

I start with a hopelessly broad question: what is known about the structure of the automorphism group of a (smooth, connected) unipotent group (over a field), and particularly about the structure of ...
-1
votes
0answers
26 views

Probability - transformation [on hold]

I've just come across this derivation (it's only a fragment I'm interested in): $$ \int p(x | \theta) \frac{\nabla_\theta p(x | \theta)}{p(x | \theta)} f(x) dx = \int p(x | \theta) \nabla_\theta \log ...
7
votes
2answers
113 views

An identity related to partitions into $n$ parts and Schur polynomials

While working with Schur polynomials I found what seems like a nice identity, and I wonder if it has a simple proof. Notation: Suppose $d,n\in\mathbb{N}$, and $\lambda =(\lambda_1,\dots,\lambda_n)$ ...
-2
votes
0answers
72 views

Where to publish paper in mathematical physiology? [on hold]

Recently I discovered one applied area of Mathematics, and that is Mathematical Physiology. I would like to know mathematical journal where it can be publish something from that area of research. This ...
2
votes
0answers
93 views

Another quesion about J.H. Conway's Surreal Numbers

Let CF be Conway's real closed field of Surreal Numbers and let ACF be the algebraic closure of CF. Is there an Extension E of ZFC that provides for the existence of "proper classes", in which ACF can ...
3
votes
1answer
102 views

Euler characteristic of a surface in $\mathbb{R}^3.$

Suppose I have a (smooth) surface in $\mathbb{R}^3,$ given as (a component of) a real algebraic hypersurface. Is there a good algorithm (assuming, for example, we can compute intersections with lines ...
2
votes
0answers
27 views

A List-Like Frobenius Monad

Has anyone ever seen a Monad that is very much like the List Monad but is also a co-monad, and hence a Frobenius monad. I was reading a paper that, I think, suggested zipper as an example. I think ...
-1
votes
0answers
38 views

Closure property of completely monotone functions [on hold]

A $C^{\infty}$ function $f(x_1, \dots, x_n)$ defined on $(0,\infty)^n$ is said to be completely monotone if $$ (-1)^{k}\frac{\partial^k f}{\partial x_{i_1} \cdots \partial x_{i_k}} \geq 0 $$ ...
1
vote
0answers
93 views

The Lie algebra of Harmonic functions

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...
-1
votes
0answers
25 views

Relation between Independent variables in an Equation [on hold]

Description: We define index as an indicator, sign, or measure of something. Let, $A_{i}$ is an index, that measures the benefits of choosing a network station $i$ among other existing network ...
3
votes
1answer
51 views

Restricted Lie algebras with a $p$-nilpotent basis

Let $L$ be a finite-dimensional restricted Lie algebra over a field of characteristic $p>0$. An element $x$ of $L$ is called $p$-nilpotent if $x^{[p]^k}=0$ for some positive integer $k$. If $L$ is ...
2
votes
0answers
22 views

continuous injective extension of a map defined on a hemisphere

Let $S^2 = \{x\in \mathbf R^3\colon |x|=1\}$ be the unit sphere, $S^2_+ = S^2 \cap \{x_3 \ge 0\}$ be the upper hemisphere and $S^1 = S^2 \cap \{x_3 = 0\}$ be the unit circle. Let $u\colon S^2_+\to \...
3
votes
0answers
67 views

How can I describe explicitely a nonsingular model of the elliptic surface?

Consider the surface $\mathcal{E} = Q_1 \cap Q_2 \subset \mathbb{P}_k^3\times \mathbb{P}_k^1$ with homogenous coordinates $x$, $x^\prime$, $y$, $z$ and $t$, $s$ respectively and a field $k$ of even ...
8
votes
0answers
80 views

Trouble with Stable Equivariant Profinite Homotopy Theory

I've heard that there are some problems in developing a good formalism for stable equivariant homotopy theory (either from the spectral mackey functors perspective or from the orthogonal spectra ...
-1
votes
0answers
12 views

Rigorious formulation of approximation of integral as area of a square and its radius of convergence [migrated]

We know that the taylor expansion of $$\left(\int_{t}^{t+\Delta t}a(t')dt'\right) = a(t)\Delta t + \frac{1}{2}\frac{da}{dt}(t) \Delta t^2 + \frac{1}{3!}\frac{d^2a}{dt^2}(t) \Delta t^3 + \frac{1}{4!}\...
0
votes
0answers
71 views

Join of $G$-CW-Complexes

I want to understand the CW-structure on the join of $G$- CW complexes for my master's thesis. Let $G$ be a discrete group and $X$ and $Y$ $G$-CW-complexes. Furtheremore, let $X*Y$ denote the join $$[...
1
vote
1answer
111 views

Lift Lie group action on a small neighborhood

Suppose a manifold $M$ admits a smooth Lie Group action $G$, and $N$ is a closed sub-manifold of $M$ such that $G$ action freely on $N$. Q: Why in a small neighborhood of $N$, $G$ also action ...
3
votes
1answer
74 views

Is there a known criterion for a compact complex analytic space to be projective?

It is known when a compact complex analytic space $X$ is the analytification of a complex projective variety? If $X$ is a manifold, then Kodaira's embedding theorem and Chow's theorem says that $X$ is ...
5
votes
1answer
134 views

A two-point inequality

Let $M(p,q) = (2p-\sqrt{p^{2}+q^{2}})\sqrt{p+\sqrt{p^{2}+q^{2}}}$ and set $B(t) = M(x+t, \sqrt{t^{2}+(y+bt)^{2}})$. Given any real $x,y,b$ is it true that $\varphi(t) = B(t)+B(-t)$ is decreasing in $...
0
votes
0answers
28 views

interpolation inequalities and embeddings

When Z is an interpolation space between two Banach spaces X and Y (say real / complex method), we have a norm inequality $$ \| x \|_Z \le C \| x \|_X^\theta \| x \|_Y^{1-\theta} $$ My question is ...
4
votes
0answers
52 views

Is this map representable? what is the fiber?

Consider the following map of stacks: Let $S$ be the stack whose $A$ points are diagrams of the form \begin{array}{ccccc} {} & {} & U & {} \\ {} & {} & \downarrow & \searrow \\...
-1
votes
0answers
51 views

Applications of computing the averages of arithmetical functions

I often read many papers in which the authors compute the average of certain arithmetical functions like 'On the distribution of the Euler function of shifted smooth numbers' of Shparlinski and al. ...
1
vote
1answer
90 views

Cluster algebra structure on the coordinate ring of $Mat_3$

Let $Mat_3$ be the set of all 3 by 3 matrices. I have some questions on the cluster algebra structure on the coordinate ring of $Mat_3$. We use $\Delta_{j_1\ldots j_n}^{i_1\ldots i_n}$ to denote the ...
3
votes
0answers
52 views

Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature

Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...
4
votes
0answers
53 views

Lie subalgebras of $\chi^{\infty}(M)$ of codimension $n=dim M$

For a connected $n$ manifold $M$, the Lie algebra of all smooth vector fields is denoted by $\chi^{\infty}(M)$. For a pointe $p\in M$ we define $L_{p}=\{X\in \chi^{\infty}(M)\mid X(p)=0\} $....
1
vote
1answer
38 views

Fredholm operator and automorphism of unit disk

Recently, I came across the following question while studying Fredholm operator. Recall an operator $S$ on a Hilbert space $\mathcal H$ is said to be Fredholm if $Range(S)$ is closed along with both $...

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