# All Questions

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### Completely positive maps-equivalent definition

The most usual definition of the completely positive map (c.p.) between two C*-algebras (say, unital) is the following: $\sigma: A \to B$ should satisfy $\sigma(1)=1$ and for each $n \in \mathbb{N}$ ...
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### Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...
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### Finding Distance with Missing Coordinate Set [on hold]

I have a line that runs through the orgin with a slope of 2, with a distance of 5 from the orgin what are the coordinates and how did you solve? I did some searching online and only found lectures on ...
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### Relation between long exact sequences and Derived functors

I know that if i have a short exact sequence of chain complexes $$0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$$ then i can extend it to long exact sequence of homology groups as ...
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### Invariant definition of the space of symbols on a vector bundle (pseudo-differential operators)

Normally, in the context of pseudo-differential operators, a symbol on a vector bundle $E$ is defined as a smooth function on $E$ which in each trivializing chart fulfills the usual symbol estimates ...
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### impossibility and mode of convergence

I have $\mathrm{Pr}(a>b)=0$ with $b\in[0,\infty]$, then can i have $a<0$? If this is true, which type of convergence is this?
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### A question about ordinal definable real numbers

If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is added to it as a new axiom? "There exists a denumerably ...
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### Convergence of empirical random variable [on hold]

Let $X$ be a RV on the real line, of probability measure $P_X$, and let $\{X_n; n=1,...,N\}$ be an iid sample from $P_X$. Let $\hat X_N$ be the RV that samples from $\{X_n; n=1,...,N\}$. I.e. its ...
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### Quotes from Connes

I found the following remark by Connes HERE: "the main quality of the homotopy type of a manifold is to satisfy Poincare duality not only in ordinary homology but in K-homology with the Fredholm ...
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### Homotopy type of a locally contractible compact

Does a locally contractible compact space have the homotopy type of a finite CW complex? (I think it probably does, but I need a reference anyway.) EDIT: My intuition was wrong [to see why, read ...
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### Ideal classes fixed by the Galois group

Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...
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### Dimension of Commutator Space

For each $n\times n$ matrix $A$ with real entries the set $$C(A)=\{X\in M_n(\mathbb{R}): AX=XA\}$$ is obviously a linear subspace of $M_n(\mathbb{R})$. Can we recognize the dimension of this ...
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### Are the quaternions not uncountably categorical?

Boris Zilber has argued that the field of the complex numbers is "logically perfect". For one thing, the theory of an algebraically closed field of characteristic zero is uncountably categorical: it ...
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### Approximating rational values in ]0,1[ by a sum or difference of unit fractions

Let $U=\{\frac{1}{n}: n\in\mathbb{N}\} \cup \{-\frac{1}{n}: n\in\mathbb{N}\}$ be the set of positive and negative unit fractions. Are there positive integers $m<n \in \mathbb{N}$, such that for ...
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### Elliptic curves with trace of Frobenius values always congruent to 0 modulo 2

Let $E/\mathbb{Q}$ be an elliptic curve and suppose that the trace of Frobenius values are such that $a_{p}(E) \equiv 0 \pmod{2}$ for all odd primes avoiding the conductor. Is it the case that $E$ ...
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### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every nonempty connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at ...
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### Can Suslin lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way? Let $\mathcal{C}$ be a class of ...
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### Bound for an integral in complex analysis

Define $S^1:=\{z\in \mathbb C: |z|=1\}$ and suppose $f(z)=\frac{1-\overline{\lambda} z}{\lambda-z}$ for some $\lambda \in \mathbb C, |\lambda|<1.$ Let $z_0$ be the fixed point of $f$ in $S^1$ and ...
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### How to Express Undirected Integration

Is there an agreed way of expressing undirected integration in formulas? my idea of doing so would be to use the absolute value of the differential $$\int_a^b f(x)|dx| = \int_b^a f(x)|dx|$$ but I ...
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### Is there an example of an SDE that has no weak solution?

So far I couldn't find an example for an SDE for which there exists no weak solution. Do you know one?
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### Coloring graph such that the coloring classes are not maximal independent sets

Let $G$ be a (finite or infinite) simple graph. We let $\mathrm{Ind}(G)$ be the collection of independent sets. For any cardinal $\kappa$ and coloring map $\chi: G\to \kappa$ we have ...
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### Conditional probabilities in epidemic model

I was contemplating an epidemic model where infection and recovery rates are determined by links. Here node $i$ is infected first and recovers at a rate $\mu_i$. For all other nodes, the recovery is ...
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### What arrangement of unit cubes minimizes surface area?

Question A. How does one arrange $n$ unit cubes to minimize surface area? Question B. How does one arrange $n$ unit cubes to form a rectangular prism of minimal surface area? Various curricular ...
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### Tensor product of certain Sobolev spaces on non-compact manifolds

Let $M$ be a non-compact Riemannian manifold of bounded geometry (i.e., its injectivity radius is uniformly positive and the curvature tensor and all its covariant derivatives are bounded in ...
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### Probability of matching under cyclic permutations

In A conjecture about the entropy of matrix vector products I asked a conjecture relating to the entropy of a matrix-vector product. This conjecture is as yet unproven. domotorp then made another ...
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### Image of the typenorm contains the squares

I am having a look at the paper Explicit CM-theory for level 2-structures on abelian surfaces by Bröker, Gruenewald and Lauter, and there is an argument which I don't understand. The main reason being ...
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### Coloring summands of given n-partition with given weights of colors

Let $\lambda$ and $\sigma$ be n-partitions $\lambda_1+\lambda_2+\cdots+\lambda_l=n$ and $\sigma_1+\sigma_2+\cdots+\sigma_s=n$ Then let $M_{\lambda \sigma}$ be the number of ways to colour blocks of ...
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### Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations. Why was there the necessity of singling ...
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### Limits of functions with converging zeros

What can one say about the derivatives of a smooth function of several variables that is a limit of smooth functions with converging zeros? More precisely, suppose that $f_i: R^n \to R^m$ is a ...
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### What are TQFTs that are multiplicative under connected sums? Do bordisms with connected sum as monoidal product exist?

In general, one extracts a manifold invariant from a TQFT by interpreting the closed manifold as a bordism from the empty set to the empty set. The TQFT sends this bordism to a homomorphism of the ...
I was looking at approximation in the forlmula of Products of necklaces: $n \to \infty$ we have $\prod_{p=1}^n N(p,a) \approx \frac {a^n} {n!}$ $\prod_{p=1}^n \frac {1-a^p} {1-a}$. (The number of ...
Let $M$ denote a Riemannian manifold, $\gamma$ a geodesic of $M$ defined on $\mathbb{R}$. Let $t_{0} \in \mathbb{R}$ and $(\alpha,\beta) \in \mathbb{R}^{2}$. I define the reparametrized geodesic ...