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Open problems in compressed sensing

What are the main open problems in compressed sensing? I am interested in theoretical and in numerical angles as well.
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How can i be distinguished from -i? [migrated]

Mathematicians designate one solution to x^2 = -1 as i and the other as -i. Would anybody notice if we switched their identities? Any polynomial p(x) with a complex root will also have its conjugate ...
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Sets of squares representing all squares up to $n^2$

Let $S_n=\{1,2,\ldots,n\}$ be natural numbers up to $n$. Say that a subset $S \subseteq S_n$ square-represents $S_n^2$ if every square $1^2,2^2,\ldots,n^2$ can be represented by adding or subtracting ...
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Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize, ...
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Prime labelling of graphs

A prime labeling of a graph is an injective function f: V(G) -> {1, 2, ..., |V(G)|} such that for every pair of adjacent vertices ...
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Model of function of 2 random variables [on hold]

In my model W = f(E, K). f is a complex function (several operations on E and K). for any W, infinity pairs of (E, K) exist that satisfy f. E and K are between [0, +oo] I have observations for W ...
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On a continuous extension of a linear 2nd order PDE

Consider an elliptic (hyperbolic) equation $A(x,y) u_{xx} + 2B(x,y) u_{xy} + C(x,y) u_{yy} = 0$ in a bounded open plane set $D$, with real-valued functions $A$, $B$, and $C$. Is it true that at least ...
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How the modular theory of von Neumann algebras, deal with generating C*-algebras?

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Suppose the existence of a bicyclic ...
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Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
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Murray–von Neumann equivalence on C$^*$-algebra and von Neumann algebra

Let $H$ be a separable infinite dimensional Hilbert space, $M \subset B(H)$ a von Neumann algebra and $A \subset M$ a separable $C^*$-algebra such that $A''=M$. Let $p,q \in A$ be (Murray–von ...
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What's so special about $1$-categories?

I have been pretty thoroughly convinced for some time now that, when thinking about mathematics, one really should be thinking 'categorically', that is, one should always be thinking of the morphisms ...
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Arithmetic functions associated with Hurwitz Zeta function raised to arbitrary complex powers, $\zeta(s,q)^z$ for $q \in \mathbb{N}$?

If $\zeta(s)$ is the Riemann Zeta function, then $\zeta(n)^z$, with $z \in \mathbb{C}$, $\Re(s)>1$, can be represented as $$\zeta(s)^z=\sum_{n=1}^\infty \frac{d_z(n)}{n^{-s}}$$ where $d_z(n)$ ...
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What happens to the angles of an isosceles triangle if one vertex is at infinity? [migrated]

My son and I were trying to decide whether an isosceles triangle can ever have 90 degree base angles. I would argue that if the two equal length sides are both infinitely long, they must have 90 ...
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variants of ramification groups - need terminology and sources

I've asked this question in several more elementary forums, and haven't get any answer. So I presume this is not so an elementary question. Let $L/K$ be a Galois extension, and $w$ be a valuation of ...
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Bounded dg algebra vs unbounded dg algebras

1)Let $Cd_{\geq 0}ga$ be the category of non negatively commutative cochain dg algebra over a field $\Bbbk$ of charachteristic zero. Let $w\: : \: Cd_{\geq 0}ga\to dg_{\geq 0}Mod$ be the forgethfull ...
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Application of the Riemann hypothesis and the ABC conjecture to independence results

In Old Home Page of Andreas Weiermann Andreas Weiermann has stated the following: Quite recently I submitted a preprint about an application of the Riemann hypothesis and the ABC conjecture to ...
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Let $a_{1},\dots,a_{n}$ be positive natural numbers ($n>2$) such that $a_{i}\neq a_{j}$ if $i\neq j$. I want to prove that $$\left\lvert \left\{ p \text{ prime} \; : \; p \mid \sum_{i=1}^n ... 0answers 68 views this sequence A_{n} have recursive relations? Let$$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$I want find the A_{n} recursive relations,such as following form ... 0answers 68 views metric geometry,geometric measure theory in riemannian geometry [on hold] I am interested in knowing the role played by Alexandroff geometry,geometry measure theory in riemannian geometry. I know 2 examples like proof of positive mass conjecture used geometric measure ... 1answer 119 views How to solve this differential equation with an infinite sum? I would like to find solutions of the following differential equation:  \sum_{1}^{\infty} a_n f(nx) + f''(x)+ x^2 f(x)=\lambda f(x) For example in space of function from \mathbb R^* to \mathbb ... 2answers 463 views Where do Set Theory and Number Theory meet together? As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are ZFC-provable if and only if they are consistent with ZFC, it's because of absoluteness of essence of ... 1answer 182 views Is there a Riemann existence theorem for orbifolds? For smooth algebraic varieties X over \mathbb{C}, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of X and finite unramified ... 0answers 27 views Floquet-Bloch solutions of the quasiperiodic Schrödinger equation I am concerned about the Schrödinger equation -x''(t)+q(t)x(t)=Ex(t). Here, the potential q is real and quasiperiodic with frequency vector \omega. That is, we let T^d be the d-dimensional ... 0answers 14 views how to efficiently compute the mean function for non-homogeneous poisson process? Suppose that I know all intensity functions lambda(t) during given period [0,t], how can I compute the mean function m(t) for non-homogeneous Poisson process? Basically, m(t) in the integral of ... 0answers 39 views Intersection of ordinary subspaces at different primes Choose two distinct primes \ell and \ell', and embeddings \iota_\ell : \overline{\mathbb Q} \to \overline{\mathbb Q_\ell}, \iota_{\ell'} : \overline{\mathbb Q} \to \overline{\mathbb ... 1answer 124 views Strong Morita equivalence and representation theory In the context of pure algebra we say that two algebras (in general: rings) A,B are Morita equivalence when there are bimodules _AP_B,_BQ_A such that P \otimes_B Q \cong _AA_A and Q \otimes_A P ... 1answer 79 views Eigenvectors of a symmetric positive definite Toeplitz matrix I wish to efficiently compute the eigenvectors of an n x n symmetric positive definite Toeplitz matrix K. A full eigendecomposition would be even better. Although I assumed this would be a well ... 1answer 175 views How many k-subsets of the integers {1,…,n} sum to N? Given the set of integers S = \{1,..n\}, how many subsets of S with k elements sum to N\in \mathbb Z? 3answers 455 views Complete resolutions of GCH Let's say that a "complete resolution of GCH" is a definable class function F: \operatorname{Ord}\longrightarrow \operatorname{ Ord} such that 2^{\aleph_\alpha} = \aleph_{F(\alpha)} for all ... 1answer 76 views Simple Isogeny Question I'm looking for a reference of an isogeny fact that I've used many times but am having a hard time proving formally. One can define the degree of an isogeny as the degree of extension fields of the ... 0answers 36 views transition matrix [on hold] Gene mutation. Suppose a gene in a chromosome is of type A or type B. Assume that the probability that a gene of type A will mutate of type B in one generation is 10-4 and that a gene of ... 1answer 145 views Nonperiodic points of homeomorphisms of a ball Suppose B is a d-dimensional ball (for some d \geq 1) and T is a homeomorphism from B to itself. Suppose also that T is not of finite order (that is, for no n \geq 1 is it the case that ... 0answers 41 views Does an arbitrary product of f and f^\dagger belong to a universal enveloping algebra of the Heisenberg algebra? The Heisenberg algebra is essentially the canonical commutation relations (CCR) for bosons [f,f^\dagger]=1. f is called an annihilation operator in physics (f^\dagger creation operator). ... 0answers 35 views Existence of solution to weak form of linear equation with boundary integral (parabolic PDE) Let W(0,T) := \{ u \in L^2(0,T;H^{\frac 12}(\partial\Omega)) \mid u_t \in L^2(0,T;H^{-\frac{1}{2}}(\partial\Omega))\}. Let \gamma and \xi denote the trace map and its right inverse. Does there ... 1answer 196 views Example of a Frobenius algebra that is not projective over a Frobenius subalgebra I'd like to know an example of a Frobenius algebra A, with a subalgebra B that is itself a Frobenius algebra, such that A is not projective as a left B-module. I don't require any ... 1answer 906 views The letters of the word “ART” Are there only a finite number of connected topological spaces X (up to homeomorphism) with the property that X has an open subset U such that U and X-U are homeomorphic to \mathbb{R}? I ... 0answers 40 views Linear algebra over principal rings 1 [on hold] If N is a left-idea of ring R and R is a left R-module, then submodule N is a direct sum if and only if N has a right unit. 1answer 740 views Joyal's letter to Grothendieck Mostly out of curiosity: Where do I find Joyal's letter to Grothendieck in which he defines a model structure on simplicial sheaves? The question was already asked in this MO post, but that ... 1answer 153 views journal to submit mathematic books' review it has been asked to me to write a review on a book about the history of mathematics in Italy between the two world wars. The book is a non-technical one. I would like to know which journal accepts ... 0answers 122 views On matrix rank inequality Let A be a \{0,1\} square matrix. Let J be all 1 matrix. Let \bar{A}=J-A. Is it possible for rk_+(A)\geq c\cdot rk_+(\bar{A})^d-1 and rk_+(\bar{A})\geq c\cdot rk_+({A})^d-1 for some ... 1answer 193 views Can ergodic theory help to prove ergodicity of general Markov chain? I am a beginner in ergodic theory. I have read some lecture notes(such as this and this) about it in hope that I could find something which helps to prove the ergodicity of some Markov chain taking ... 2answers 192 views Real algebraic solution Suppose a system of polynomial equations with rational coefficients has a real solution. Does necessarily there exists a real solution with algebraic coordinates? What about the simplest case of one ... 3answers 183 views Injective map between two schemes Assuem we have a finite surjective map between two irreducible, separated schemes, f:X \rightarrow Y, and for a dense open U \subset Y and for any y \in U, |X_y| =1, then can we say f is ... 1answer 75 views Dual connections for Information Geometry In information Geometry, there is a definition of dual connection, which is: two affine connections \nabla and \nabla^* are called dual if they satisfied ... 1answer 116 views Prove that \sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}) [on hold] Let a,b\in\mathbb{Z}, and f\in C^2([a,b]) such that |f''(t)|\asymp \lambda for a\le t\le b. Prove that$$\sum_{a<n\le b}\{f(n)\}=\frac{1}{2}(b-a)+O(\lambda^{1/3}(b-a)+\lambda^{-1/2}). ...
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There are early successes of the moduli theory - the construction and compactification of the moduli spaces of curves $\overline{\mathcal{M}}_g$ . I want to study about the moduli of algebraic ...
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Finding an example for [on hold]

Let $\varphi$ be a periodic function s.t. at zero and every integer points it is equal to 1. Moreover it's equal to one in at least one point between each integer. Can we have two distinct density ...
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invariant measures of the expanding maps on the circle

I would be very happy to know about original references for the following results; For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1) (1) There exist ...
We have a homogeneous graph $G = (V,E)$ with a function $f:V\rightarrow \mathbb{R}$. We define the following modulus: $\displaystyle \omega(s) = \sup\{f(x)-f(y) \ | \ |x-y|=s \}$ and wish to lower ...