This tag is to be used only when re-tagging highly(!) off-topic questions where none of the actual tags would make sense; all actual tags the questioner has used are removed and something is needed to have some tag, which is enforced by the software, so this tag is used. However note that this tag ...

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1
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2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
4answers
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
1answer
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
3answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
1answer
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
3answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
4answers
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
1answer
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
0answers
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
3answers
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
2answers
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
1answer
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
0answers
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
4answers
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
1answer
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
2answers
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
3answers
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
2answers
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
0answers
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
0answers
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
2answers
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
0answers
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
1answer
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
2answers
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
1answer
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
2answers
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
0answers
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
1answer
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
3answers
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
1answer
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
2answers
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
1answer
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
2answers
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 ...