# Tagged Questions

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### Semi-algebraicness of cells involved in integrals of semi-algebraic functions

Background: In "Stability under integration of sums of products of real globally subanalytic functions and their logarithms", by R. Cluckers and D.J. Miller, it is shown that the integral of a ...
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### Can a Riemannian metric be analytic in non-analytically different coordinates?

Suppose I have two coordinates on the same (subset of a) Riemannian manifold. If the metric tensor is analytic in both coordinates, is the change of variables between them necessarily analytic? In ...
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### local analyticity of volumes of slices of semi-algebraic sets

I would like a reference and/or a simple proof using well-known results of the following, which I think is true. (If it's false, I'd like to know that as well of course -- and ideally a way to modify ...
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### Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
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### Separating the points of projective spaces with real-analytic functions

Is there an easy way to separate the points of $\Bbb C \Bbb P^n$ or $\Bbb R \Bbb P^n$ (viewed as real-analytic manifolds) with real-analytic functions? If two points lie in a coordinate patch where a ...
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### Complements of images of complex analytic sets

It is known that the complement of an analytic set is connected. In general, the complement of a proper complex analytic set in a connected complex manifold is an arcwise connected dense open set. My ...
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### analytic vector bundles

Let $E$ be a real analytic vector bundle on an analytic manifold $M$. Assume that $E$, as a smooth vector bundle, is a trivial bundle. Is $E$ a trivial analytic vector bundle? I need to the ...
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### I don't understand behavior of this integral, help!

In an answer to a question I needed the following integral: $$f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt;$$ it represented deviation from modularity of some other function. However I noticed ...
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### Comparison of two infinite dimensional Lie Algebras

Is there an example of a real analytic (compact) manifold $M$ such that the following two lie algebras are isomorphic Lie algebras: $\chi^{\infty}(M)$, the Lie algebra of all smooth vector ...
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### hyperfunctions and analytic duals

Let $A(\mathbb R^n)$ be the real analytic functions and $\mathscr B(\mathbb R^n)$ the hyperfunctions, dual to $A(\mathbb R^n)$. Further let $W\subset \mathbb C$ be a cone in the complex plane with ...
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### Derivation on real analytic manifolds

Let $M$ be a real analytic manifold. By $C^{\omega}(M)$ we mean the algebra of all analytic functions from $M$ to $\mathbb{R}$. Assume that $D$ is a derivation on $C^{\omega}(M)$ . Is there a ...
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### Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$? (Possibly by perturbing a rotation in the real-analytic topology?)
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### Is a function with nowhere vanishing derivatives analytic?

My question is the following: Let $f\in C^\infty(a,b)$, such that $f^{(n)}(x)\ne 0$, for every $n\in\mathbb N$, and every $x\in (a,b)$. Does that imply that $f$ is real analytic? EDIT. According to a ...
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### If $f$ is $C^{\infty}$ and $f^2$ is analytic, then $f$ is analytic

Assume that $f:\mathbb{C}^n\rightarrow \mathbb{C}$ is a $C^{\infty}$ function such that $f^2$ is (complex) analytic. Then one can show that $f$ is analytic. (Note: Liviu Nicolaescu and Alexandre ...
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### Surjective entire functions without critical points

It is easy to construct surjective locally univalent holomorphic functions $f: {\mathbb D}\to {\mathbb C}$, where ${\mathbb D}$ is the open unit disk. I am pretty sure that the answer to the ...
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### Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix

Consider a matrix function $A(x)$, analytically depending on single parameter $x$. Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$. The question is whether we can ...
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### An extension of Morera's Theorem

Morera's Theorem states that If $f$ is continuous in a region $D$ and satisfies $\oint_{\gamma} f = 0$ for any closed curve $\gamma$ in $D$, then $f$ is analytic in $D$. I have two questions: ...
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### If $f(x)+f(2x)$ is quasianalytic, is $f(x)$ necessarily quasianalytic?

Assume that $f\in C^{\infty}$ and that $M_n$ is a sequence such that $$\sum_{n=0}^{\infty}\frac{M_n}{(n+1)M_{n+1}}=\infty$$ and for certain compact neighborhood of the origin $U$ of $\mathbb{R}$, ...
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### Inequality for certain analytic functions

Suppose that $f(z)$ is analytic for $\Re(z)>0$ and it is positive on the real axis. Suppose that for some $y_0$ we have $$\left| {f\left( {x + iy} \right)} \right| \le f\left( x \right)$$ for any ...
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### analytic solution to elliptic PDE in R^n

I am looking for (minimal) conditions, which guarantee that the problem Lu = 0 in R^n, where L is a second-order (uniformly) elliptic operator with analytic coefficients, has a unique global ...
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### construct a power series with infinitely many zeros in the complex plane, bounded coefficients???

Hi all. I want to construct a power series $F(z)=\sum_{n=0}^\infty c_nz^n$ centered at zero and with finite radius of convergence in the complex plane, and which has infinitely many zeros (in its ...
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### Univalent functions with non-negative coefficients

Is anything non-trivial known about univalent functions with non-negative coefficients? Let $U$ be the unit disc, and $f$ a univalent (=injective) holomorphic function, $f(0)=0$, $f'(0)>0$. There ...
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### Does Physics need non-analytic smooth functions?

Observing the behaviour of a few physicists "in nature", I had the impression that among the mathematical tools they use a lot (along with possibly much more sofisticated maths, of course), there is ...
Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic*. Is it possible a solution $f$, which is non-analytic at any point of the ...