**1**

vote

**2**answers

261 views

### Abelian subgroups of ball quotient

Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can ...

**0**

votes

**1**answer

1k views

### Fourier transform of a differential operator

I have a differential operator defined by its Fourier transform:
$\left(\alpha k_x^2 + \beta k_y^2 + \gamma k_x k_y \right)^\delta \hspace{20pt} \alpha,\beta,\gamma,\delta \in \mathbb{R}$
I don't ...

**22**

votes

**1**answer

2k views

### Why do we use $\epsilon$ and $\delta$?

My understanding (from a talk by Rob Bradley) is that Cauchy is responsible for
the now-standard $\epsilon{-}\delta$ formulation of calculus, introduced in his
1821 Cours d’analyse. Although perhaps ...

**4**

votes

**1**answer

423 views

### Localizability of differential operators a la Grothendieck

Hello,
Maybe this question is trivial, so sorry
Let $A$ be a (comm. with 1) $k$-algebra, where $k$ is a ring (comm. with 1).
Then we can define the module of differential operators $D^{\leq n} ...

**10**

votes

**1**answer

573 views

### A problem concerning $L^2([0,1]\times[0,1])$

Trying to solve a conjecture in differential geometry, I am leaded to the following problem (which may seem weird to a analyst). I wonder if anyone know some techniques that happen to solve it.
Let ...

**9**

votes

**4**answers

891 views

### Coincidences amongst classifying spaces and Eilenberg Mac-Lane spaces

Given that $$\mathbb{R}P^{\infty} = B O(1) = K(\widehat{O(1)}, 1)$$ $$\mathbb{C} P^{\infty} = B U(1) = K( \widehat{U(1)}, 2)$$ is there any way to make sense of $$\mathbb{H}P^{\infty} = B Sp(1)$$ in a ...

**2**

votes

**1**answer

123 views

### Spline fit with bounded derivations

How can I do a Spline Fit with bounds on some derivations?
Problem
Given:
Set of data points $t_k, x_k$
Set of nodes $n_i$
order $D$ of the spline (in my case $D=5$)
lower and upper bounds ...

**0**

votes

**3**answers

703 views

### How to isolate $f(x)$ in $f(x+a)=f(x)+a\times g(x)$?

$a \in \mathbb{R}$
$f:\mathbb{R} \rightarrow \mathbb{R}$
$g:\mathbb{R} \rightarrow \mathbb{R}$
For generic functions $f$ and $g$, how isolate $f(x)$ in the equation below?
$f(x+a)=f(x)+a\times ...

**0**

votes

**1**answer

784 views

### How to determine whether a multivariate function is bounded or not

Suppose there is a function $f:\mathbb{R}_+^n\mapsto \mathbb{R}$. Are there any systematic ways to determine whether the range of $f$ is bounded or not?
For example, there is a function ...

**2**

votes

**2**answers

567 views

### Power function inequality

Let $x$ and $p$ be real numbers with $x \ge 1$ and $p \ge 2$ . Show that $(x - 1)(x + 1)^{p - 1} \ge x^p - 1$ .
I recently discovered this result. I am sure it is known, but it is new to me. It is ...

**0**

votes

**0**answers

196 views

### Properties of morphisms induced by divisors on curves

There are a few properties from Hartshorne IV on curves that I am trying to verify. Let $D$ be an effective divisor on a curve (integral scheme of dimension 1, proper over $k$, regular) $X$, $\dim ...

**2**

votes

**1**answer

482 views

### A singular value inequality

Let $s_1,s_2: \mathbb{R}^{2\times 2} \mapsto \mathbb{R}_+$,
$s_{1}\left(\cdot\right)\ge s_{2}\left(\cdot\right)\ge 0$, be the
singular values of a $2\times2$ matrix. Is it true that
...

**1**

vote

**1**answer

348 views

### two polynomials

If $p,q:\mathbb R^3\rightarrow\mathbb R$ are two polynomials, such that $\{p=0\}\cap\{q=0\}$ is two-dimensional, does it follow that $p$ and $q$ have a common factor? (I believe it does.) How to prove ...

**1**

vote

**0**answers

134 views

### Morphisms, inducing isomorphisms on global functions

Let $f:X\to Y$ be a surjective morphism of complex varieties with $Y$ affine. Assume that every fiber of this morphism has the property that all the global functions are constant.
What else do I need ...

**5**

votes

**1**answer

562 views

### Convex Analytic or linear algebraic proof that a certain psd matrix is a sum of rank 1 psd matrices

Can you prove the following using techniques from convex analysis or linear algebra? I was originally seeking an elementary proof, but I think it is better to broaden the scope for this bounty ...

**6**

votes

**3**answers

833 views

### Is the function $e^{x^2/2} \Phi(x)$ monotone increasing?

Hello,
Here is an interesting problem. It looks elementary, but it has taken me some efforts without solving it. Let
$$
h(x) = e^{x^2/2} \Phi(x),\qquad \text{with}\quad \Phi(x):=\int_{-\infty}^x ...

**2**

votes

**1**answer

321 views

### A Bessel integral

Today I came across the integral
$\int_a^\infty e^{-bx} I_n(x) dx$
where $I_n$ is the modified Bessel function of the first kind. There is a solution for $a=0$, provided in Gradshteyn and Ryzhik, ...

**4**

votes

**1**answer

525 views

### Harmonic functions on the plane

I have a question regarding harmonic maps from all of ${\Bbb R}^2$ into a domain in ${\Bbb R}^2$. Before stating my question in full generality, let me ask a special case of the question first. Is it ...

**1**

vote

**1**answer

149 views

### Integrating with sub-level sets

This is a simple question, and I'm sure it was a homework assignment at some point (assuming it's true) but it's one that I'm puzzled over. Suppose I have a compact domain $D \subset \mathbb{R}^n$ ...

**3**

votes

**1**answer

512 views

### moduli of vector bundles on a surface

Let $S$ be a smooth projective surface with an ample divisor $X\subset S$. Consider the
moduli stack of vector bundles $F$ on $S$ such that
1) $c_1(F)=0$
2) $c_2(F)=n$
3) The restriction of $F$ to ...

**3**

votes

**1**answer

359 views

### Connecting points on a variety by the image of a nonsingular curve

In Hartshorne's proof of a result of Igusa (see III, 9.13 of Hartshorne) he claims without proof that any two closed points on a variety can be connected by the image of a nonsingular curve, or by a ...

**1**

vote

**1**answer

270 views

### method for getting function from power series/perturbation series

is there any definite method or algorithm,software to get exact function or expression from series.e.g we get series solution of differential equation and we want exact expression rather than ...

**5**

votes

**2**answers

451 views

### Some weird “system” of inequalities in nonnegative integers.

Suppose I have a bunch of nonnegative integers $(a_{ijkl})_{1 \leq i \leq j \leq k \leq l \leq 17}$ such that for all 17-tuples nonnegative integers $w_t$ (for $1 \leq t \leq 17$) we have that ...

**1**

vote

**4**answers

982 views

### Approximation to the ratio of a Gaussian CDF to PDF

Johnstone and Silverman (2005) claimed that for large x
$\frac{1-\Phi(x)}{\phi(x)} \approx \frac{1}{x}$
where $\Phi(x)$ and $\phi(x)$ are the CDF and PDF for a normal random variable.
I was able ...

**2**

votes

**1**answer

307 views

### Complexity of computing derivatives

Sorry if this is too simple. This is my first question here.
Suppose $f : R^n \to R$ is a differentiable function. Say that we can compute in $T$ arithmetic operations the value $f(x)$ at any point ...

**0**

votes

**0**answers

299 views

### Systems of linear octonionic equations

Is there theory of determinants, rank of matrices and systems of linear equations with octonionic coefficients? Does anybody could indicate references? I want to know mainly does there exist a ...

**4**

votes

**3**answers

791 views

### Computing the q-series of the j-invariant

It is a fundamental fact, often quoted these days in its connection with Monstrous Moonshine, that the q-expansion (i.e., the Laurent expansion in a neighborhood of $\tau = i\infty$) of the ...

**0**

votes

**2**answers

374 views

### Formalization of n-ary functions

Hi there. I've been doing some thinking lately (oh-no!) about function definitions. Specifically, I'm considering functions with multiple parameters.
Now, I'm familiar with "the usual" definition in ...

**0**

votes

**1**answer

480 views

### Finding two bezier control points given three points

My apologies if this is asked in the wrong spot, I believe that this problem has a fairly simple solution... but it is beyond me. Given three points (A,B,C) drawn ...

**0**

votes

**0**answers

250 views

### Is there an intuition for the exterior calculus identity dd = 0

Considering an exterior derivative $d$, which may be discretized as the transpose of the boundary operator on a simplicial mesh, I have read and seen that $dd = 0$, but I have not seen an intuitive ...

**11**

votes

**4**answers

801 views

### Applications of full integral weight modular forms in elementary number theory

Except for Eisenstein series having the divisor functions as their Fourier coefficients, is there any other full integral weight modular form (of some level, preferably full) having arithmetic ...

**1**

vote

**1**answer

270 views

### Presentation of finite modules with null annihilator

Let $R$ be a noetherian local ring and let $M$ be a finite $R$-module. Assume that the annihilator of $M$ is zero. Consider a minimal presentation of M as follows: ...

**1**

vote

**2**answers

233 views

### Singularity at left endpoint for variational calculus problem

Hi all, first, I'd like to apologize if the term "singularity" is being misused. I have the following integral:
$\int _{0}^{\pi/2} \sqrt{ r \left( x \right) ^{2}
+ \left( {\frac {d}{dx}} r \left( x ...

**5**

votes

**3**answers

655 views

### Volume of Minkowski sum of a ball and an ellipsoid

Is there a simple way to calculate/estimate the volume of Minkowski sum of an n-dimensional unit ball and an n-dimensional ellipsoid? Even a simple ellipsoid like $\frac{x_1^2}{a^2} + x_2^2 + \ldots + ...

**31**

votes

**2**answers

1k views

### Gently falling functions

I wonder if it is possible to characterize the class of
gently falling functions, which I would like to define
as follows.
Let $g(x)$ be a $C^2$ function defined on an interval
$R \subseteq ...

**10**

votes

**0**answers

441 views

### Can the similarity between the Riesz representation theorem and the Yoneda embedding lemma be given a formal undergirding?

For example, by viewing Hilbert spaces as enriched categories in some fashion? (I suppose the same idea of considering the inner product of a Hilbert space as a generalized Hom-set has also been ...

**0**

votes

**0**answers

313 views

### covariant derivative complex manifold

Assume we have $X$ a complex manifold and $Y = Y^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ and $Z = Z^{\alpha} \frac{\partial}{\partial z^{\alpha}}$ two vector fields on $X$. Let $\nabla$ be the ...

**5**

votes

**2**answers

315 views

### Distributing points with respect to a concave function

Suppose I have a concave function defined on the unit interval such that $f(0) = f(1) = 0$ and $\int_0^1 f(t) dt = \alpha$, where $\alpha$ is "small" (say $0.01$ or thereabouts). Say I distribute $n$ ...

**6**

votes

**0**answers

1k views

### Why is mechanical differentiation so hard to get right?

This question is related to this question on differentiation/integration which asks why differentiation is mechanical but integration is an art. The answers given all make a huge assumption: that one ...

**2**

votes

**1**answer

1k views

### complex gradient of a function

Let $M$ be a complex $n-$dim manifold and $u : M \rightarrow \mathbb{R}$ be some smooth function. On $M$ assume that we have a Kaehler metric $h$. How is the complex gradient vectorfield defined with ...

**0**

votes

**2**answers

353 views

### What function smoothly interpolates between the identity and exponential (or log and identity) functions? [closed]

Or rather, what function can be parametrized with some value t in [0,1] such that f(x, t= 0) = x, f(x, t = 1) = e^x, and f(x, 0 < t < 1) is a principled interpolation between those two, kind of ...

**0**

votes

**1**answer

3k views

### What's difference between 'functional' and 'function'? [closed]

Hi, I want to know difference that between 'functional' and 'function'.
Of course, in Wikipedia, http://en.wikipedia.org/wiki/Functional_(mathematics), there is many texts.
But what's the simple ...

**0**

votes

**2**answers

731 views

### Functions defined as infinite products

Are there standard references on infinite products of rational functions and their convergence properties? I'd appreciate information on finite products too!
The original motivation for this is the ...

**1**

vote

**0**answers

301 views

### Some help with differential operators

Hello
I am trying to follow the derivation of the Generalized Cornish Fisher Expansion from this paper.
I dont see how the author goes from equation (21) to (22).
For one to be able to do this ...

**2**

votes

**1**answer

526 views

### spectral theorem for infinite-dimensional matrices

Keller and Ochsenius (1995) has a spectral theorem for finite-dimensional symmetric matrices over the field of formal power series with real coefficients $\mathbf{R}((t))$ (they actually have a more ...

**1**

vote

**3**answers

1k views

### How do you calculate the solid angle of a rectangular, axis aligned section of a surface defined by a two dimensional function?

I have $f(x,y) = \frac{1}{2} (1 - x^2 - y^2)$, which is a paraboloid centered around the origin (plot).
Now I want to calculate the solid angle (with the origin as the viewpoint) of the surface area ...

**2**

votes

**1**answer

329 views

### Regularization of Zygmund functions

Dear community.
I would like to derive a "good" estimate on $\frac{d}{dt}f_\epsilon(t)$, where $f_\epsilon$ is a regularization of a Zygmund-continuous function $f$, i.e.
...

**4**

votes

**2**answers

1k views

### Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising

For a image denoising problem (below):
http://www.bekirdizdaroglu.com/ceng/Downloads/ISCE10.pdf
the author has a functional E defined
$E(u) = \int\int_\Omega F \\ d\Omega$
which he wants to ...

**30**

votes

**1**answer

1k views

### Mr. G.P.K.'s questions [closed]

WARNING: An acquaintance of mine, Mr.Goosepond Prhklstr Kratchinabritchisitch, has requested permission to post his questions under my username. When I asked him why he didn't do it under his own ...

**2**

votes

**2**answers

459 views

### Is there a continuous function on $f:\mathbb{R} \rightarrow \mathbb{R}$ with uncountably infinite turning points?

I was thinking about the statement "if f is continuous on the interval I, there is not necessarily an interval J in I on which f is monotone." and this led me to the question "does there exist a ...