# All Questions

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### I need to know the most active research topic which depend on Real analysis and functional analysis? [on hold]

I need to know the most current topic in pure math with depend mainly on real analysis and functional analysis and not need a good knowledge in algebra and geomtry ? is delay differntial equations is ...
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### Unique Stationary Distribution of A Markov Chain

I have a Markov Chain like $Y_i=\sum_n\pi_{n,i}(Y)Y_n$, i=1,2,3...N. So the Markov chain has N states and the transition matrix depends on the vector $\textbf{Y}$. I am wondering what conditions $\pi$ ...
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### Randomly put $k$ balls in $2n$ circular boxes, pick $n$ consecutive boxes such that the number of balls is minimum!

You are given $2n$ boxes that are arranged circular (you can imagine all boxes are on the edge of a circular table). Then randomly, you put $k$ balls in the boxes such that each box is containing ...
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### Random variable of random variable [on hold]

This is confusing and difficut, but I hope it makes a sence. I am interested in kind of like Random variable of Random variable. This issue might've been mentioned below before. The Probability ...
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### MDAS solution glitch [on hold]

I always get confused with this kind of math problem, How do we solve this problem? Example: 1 * 5 - 10 + 2 = ? Solution 1 [My solution]: Of couse I use MDAS ...
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### Sobolev regularity for systems of elliptic boundary value problems

My question is about Sobolev estimates near the boundary for elliptic systems (equivalently, elliptic boundary-value problems for vector-valued functions). Note, results for the scalar case are ...
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### Riemannian manifolds conformally diffemorphic to sphere [on hold]

let a Riemannian manifold be conformally diffeomorphic to a sphere. Is its scalar curvature constant?
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### Counting tournaments with ties

An improper tournament, or tournament with ties, is a graph in which every pair of nodes is connected by a single uniquely directed edge or by a single undirected edge. There are 1, 2, and 7 improper ...
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### Uncountable divisible groups and the existence of order-preserving ismorphisms of their subsetss

Let $(G,+,0,<)$ be an ordered divisible group of uncountable dimension. Consider the subset $G^{<0}$ of $G$. Question: Are $G$ and $G^{<0}$ isomorphic as ordered sets? Does there exists an ...
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### Whether a given algebra is the algebra of endomorphisms for a vector space

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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### Sub-matrices with a real spectrum

This question arises from the study of PT-symmetric quantum mechanics. Let $A$ be an $n\times n$ complex-valued matrix with a real spectrum. If $A$ is Hermitian, then any sub-matrix corresponding to ...
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### Elementary question: Curvature change under Complexified Gauge Transformation

Forgive me for this elementary question. Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ ...
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### Section of ellipsoids

I am reading some survey about euclidean sections of convex bodies(http://arxiv.org/abs/1110.6401). It is written that any $k$-dimensional ellipsoid easily seen to have a $k/2$-dimensional section ...
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### About consecutive integers covered by arithmetic progressions

Help me please to solve the following problem. There are $n$ arithmetic progressions of the form: $$(2i+1)k + x_i,~~~~ i = 1,\ldots,n, k \geq 0$$ Initial integer terms $x_i \geq 0$ are varying. ...
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### On the automorphism group of binary quadratic forms

This question is a continuation of the following two questions: Discriminants of indefinite integral binary quadratic forms admitting 3 or 6 torsion. On certain solutions of a quadratic form ...
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### Comparison inequalities for Hamiltonian mechanics with convex potential - analogue to Rauch's theorem?

I'm asking about an area (Hamiltonian mechanics) that I don't know at all well; thus, I keep the question somewhat vague. In differential geometry, there are a number of results saying that geodesics ...
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### The composition of a dissipative operator and a positive operator is dissipative?

Consider the following bilinear system on a open and bounded domain $\Omega$ \left\{\begin{array}{r c l} \displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\ y(0) &...
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### Non-lattice Veech groups

I was thinking of Veech surfaces, which are translation surfaces whose stabilizer under the $\mathrm{Sl}_2(\mathbb{R})$ action is a lattice in $\mathrm{Sl}_2(\mathbb{R})$. They seem to have been ...
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### Hodge standard conjecture for étale cohomology

It is known that Hodge standard conjecture is true for étale cohomology for field $k$ in characteristic zero. It means that the following pairing $$(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\rangle$$ ...
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### $L^\infty(\Omega)$-regularity for strongly damped wave equation

I am interested in the following IBVP for the strongly damped wave equation: u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\ u=0 \quad \text{on} \ ...
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### Matrix minimization

Is there an explicit solution to the problem of minimizing $||X-X_0||_F^2+||X^{-1}-Y_0||^2_F$, with respect to matrix X, where $X_0$ and $Y_0$ are given, and all matrices are real $n\times n$ and ...
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### Parabolic characters of subgroups $\Gamma \subset \operatorname{SL}_2(\textbf{Z})$ generated by parabolic and elliptic elements [on hold]

In the paper Generalized Modular Forms from Knopp and Mason, one can read in page $6$: Remark. It is not too hard to prove that a subgroup $\Gamma$ of finite index in $\operatorname{SL}_2(\textbf{Z})$...
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### Algebraic structure on homotopy groups of spheres

It is about a "conjecture" I heard (when I was student). There would exist an algebraic structure on the homotopy groups of spheres such that this algebraic structure would be the free algebraic ...
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### Number of fixed points in Zagier's involution (Fermat's Theorem) [on hold]

Zagier's has found a famous one sentence proof for Fermat's theorem on sums of two squares. It centers on the following involution of the set $S= \lbrace (x,y,z) \in N^3: x^2+4yz=p \rbrace$ having ...
I try to understand the following version of the Kodaira embedding theorem: Let $X$ be a compact Kähler manifold. A line bundle $L$ is positiv if and only if it is ample. I have a problem with the '...