**1**

vote

**0**answers

139 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

588 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 ...

**7**

votes

**3**answers

983 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 ...

**3**

votes

**1**answer

364 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, ...

**1**

vote

**3**answers

568 views

### Ultrafilters and principal filters [closed]

Can someone give me an example of an ultrafilter which is not principal?

**4**

votes

**1**answer

650 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

203 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

541 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

462 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

283 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

470 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 ...

**3**

votes

**1**answer

327 views

### Topological space with some conditions

Can one give an example of non-compact space $X$ which satisfies the following conditions:
the countable union of compact subsets is relatively compact,
for every closed noncompact subset $A$ of $X$ ...

**1**

vote

**4**answers

1k 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 ...

**3**

votes

**2**answers

813 views

### Are presheaves of constant functions sheaves?

Hey there, I was reading 'An introduction to homological algebra' by Rotman, and on page 279 in the section about sheaves, example 5.64, Rotman gives an example of a constant presheaf $\mathcal{P}$ ...

**2**

votes

**1**answer

333 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 ...

**10**

votes

**1**answer

655 views

### An exponential polynomial with at least one bounded positivity component

In a forthcoming paper on nodal domains of Gaussian random functions, we (I and Misha Sodin) have a statement that is, roughly speaking, the following: if bounded nodal domains are possible at all, ...

**0**

votes

**0**answers

329 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 ...

**-10**

votes

**1**answer

1k views

### Direct product of filters

Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$.
I will denote the principal filter ...

**-8**

votes

**2**answers

1k views

### Filters and intersection of two binary relations

Let $\mathfrak{F}$ is the complete lattice of filters (including the improper filter) on some set, ordered
inverse to set-theoretic inclusion.
I will denote $\left\langle f \right\rangle \mathcal{X} ...

**0**

votes

**1**answer

301 views

### Sufficient conditions for Hausdorffness

Let $(X,\tau)$ be a $T_1$ topological space and $Y\subset X$ a dense subspace which is completely metrizable. Are there any sufficient conditions to ensure that $(X,\tau)$ is Hausdorff using the known ...

**0**

votes

**2**answers

427 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

557 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

262 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

848 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 ...

**2**

votes

**1**answer

293 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

252 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

967 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

2k 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 ...

**11**

votes

**0**answers

546 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

457 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

325 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 $n$ ...

**2**

votes

**0**answers

254 views

### Prime divisors of the difference set

Fix $c\in(0,1)$, and let $N$ be a (large) positive integer. Given a set $A=\{0=a_1<\dots<a_n=N\}$ of density $\alpha:=n/N>c$ with $\gcd(A)=1$, I want to find a prime dividing as few ...

**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

2k 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

407 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 ...

**4**

votes

**1**answer

5k 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

795 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 ...

**0**

votes

**1**answer

319 views

### Triviality of finite fiber bundles [closed]

Hello,
I suspect the following is true and easy but I am unable to prove. Suppose (E, B, π, F) is a fiber bundle, where E,B are compact and F is finite, prove that E is a trivial fiber bundle. Any ...

**1**

vote

**0**answers

311 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 ...

**3**

votes

**1**answer

600 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

2k 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

355 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.
...

**5**

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 ...

**33**

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

478 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 ...

**2**

votes

**1**answer

448 views

### Mandelbrot and “log-derivative”

I am reading Mandelbrot, and stubling upon his use of the limit ("almost a Hölder exponent")
\lim_{\epsilon -> 0} log(f(x+\epsilon) - f(x))/log(\epsilon).
To simplify, lets assume that f is ...

**14**

votes

**2**answers

593 views

### Curve integral of exponent of superharmonic function.

Let $\phi$ be a real smooth superharmonic function on unit disc $D$ in $\mathbb C$; i.e. $\triangle \phi\le 0$.
Then there is a curve $\gamma$ from the center of $D$ to its boundary such that
...

**1**

vote

**1**answer

521 views

### What is the Stirling formula for x(x+1)(x+2)…(x+n-1)?

Let x be a complex number.
What is the Stirling formula for x(x+1)(x+2)...(x+n-1) when n goes to infinity?

**7**

votes

**3**answers

1k views

### Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer.
...

**11**

votes

**2**answers

571 views

### “Probabilistic ultrafilters?”

A naive question.
Let $S$ be a set and let $[0,1]^S$ the set of functions from $S$ to the closed interval $[0,1]$.
Suppose given some function $P \colon [0,1]^S \to [0,1]$ satisfying the following ...