# All Questions

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7 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(...

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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 effective
interest rate in the ...

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20 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 ...

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10 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 ...

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47 views

### Variant Karamata inequality

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 ...

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12 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 $[H,G]$, ...

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23 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 $f=\nabla\times g$...

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39 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 ...

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34 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 $\...

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60 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^...

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30 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 ...

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vote

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26 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 $...

**2**

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48 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 ...

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25 views

### Probability - transformation

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 ...

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votes

**2**answers

101 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 $\lambda =(\lambda_1,\dots,\lambda_n)$ ...

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69 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 ...

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83 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**

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66 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 ...

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24 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 ...

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**0**answers

35 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
$$
(-1)^{k}\frac{\partial^k f}{\partial x_{i_1} \cdots \partial x_{i_k}} \geq 0
$$
...

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92 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 ...

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23 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

**1**answer

37 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 ...

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20 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 \...

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61 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 ...

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75 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 ...

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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!}\...

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59 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
$$[...

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vote

**1**answer

107 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

**1**answer

69 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

**1**answer

130 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 $...

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27 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 ...

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**0**answers

49 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**

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49 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. ...

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**1**answer

88 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 ...

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50 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 ...

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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\} $....

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**1**answer

37 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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**1**answer

97 views

### Minimum separation between $m$ random points on an $n$-dimensional unit sphere

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be
$$
\rho = \min_{i,j\in{...

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47 views

### prove that $\min\{|z_{j} - w_{1}|,|z_{j} - w_{2}|\}\leq 1$ holds [on hold]

Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $$ (z-z_{1})(z-z_{2})+(z-z_{2})(z-z_{3})+(z-z_{3})(z-z_{1})=0$$ ...

**2**

votes

**1**answer

67 views

### On the existence of a square root for a unitary modular tensor category

The centre $Z(\mathcal{C})$ of a fusion category $\mathcal{C}$, is a unitary modular tensor category.
Question: What about the converse, i.e., can we characterize every unitary modular tensor ...

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23 views

### Comparing log functions of CDFs and PDFs (related to order statistics) with non-log functions of the same

Let $f$ and $F$ denote the respective pdf and cdf of a probability
distribution on $\mathbb{R}$. Take any natural $n\geq3$ and any real $a$ and
$c$ such that $a\leq c$, and $\rho\geq0$.
We want to ...

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24 views

### Formulation of constraints

I would like to formulate the following constraints in a tractable form so that I can perform an optimization over the decision variables $A,x_i,y_i$:
$$ A + \sum_{i=1}^N x_i D_i + \beta \big(\sum_{i=...

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4 views

### For which category (if any) are Lie algebras the algebras of a monad? [migrated]

I was reading about monads recently, and it came to me that the purpose of the category of algebras of a monad seems to be to switch to a "representation" which is easier for computations. Soon after ...

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**1**answer

92 views

### Right inverse of the Seiberg-Witten functional

For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e.
$$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...

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**1**answer

75 views

### An inequality of real continuous function with f'>0 and f''>0

I proposed my conjecture as follows:
Let $f(x)$ is a real continuous function on $[m, M]$ and $f'>0, f''>0$ on $[m, M]$, let $m \le x_i \le M$, for $i=1, 2,..., n$. Then
$$\frac{f(x_1)+f(x_2)+...

**2**

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107 views

### What techniques are available for constructing D-modules over smooth projective varieties?

I'm trying to learn about D-modules for computing intersection cohomology but I'm having trouble coming up with explicit constructions of D-modules on projective varieties. Since this is an involved ...

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19 views

### Covering Number of a Positive Semidefinite Cone (Approximate the Objective of a SDP)

I was wondering what the covering number of a positive semidefinite cone is. Consider the semidefinite optimization program
\begin{align}
\max\langle \mathbf{C}, \mathbf{X} \rangle~~\text{subject to}~...

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113 views

### Research topics in Curves and Surfaces [on hold]

I advance that I'm not a mathematician but I'm an undergraduate student of mathematics. In my courses at university I have studied a bit of Differential Geometry, in particoular differential geometry ...

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104 views

### Where can I find basic “computations” of equivariant stable homotopy groups?

I am new to this subject; so please correct me if I will say something wrong or if you don't like my notation. In particular, I don't know whether it is reasonable to consider an infinite group $G$ (...