How to use grep to match a string with spaces in it?

Can anyone help me with this:

I am trying to extract a time stamp, by matching a string. For this am using combination of grep/awk commands. As the sting has 4 spces before [1] and one after, am confused how to work around it. Am new to programming, so need some help here.

IS="$  (grep 'Starting    [1] TaskInit' process.log |  awk '{print $  4}')" echo "$  IS" 

Aim: It should match the string in the process.log file and should print out the time stamp related to that row.

How to convert a file with several spaces into a tab-delimited file?

I had file1:

MSTRG.10807.1   0.494311896511595   MID    0.423993026403461 2.39379412548345   1.99703339651136 MSTRG.10884.1   0.365770947942799   EARLY  1.46416917664929   1.16816186543633   0.689075392478972 MSTRG.10958.1   0.52855355823638    MID    0.885493751836316  2.28463841550375   1.26867555157512 

I used command the sed 's/ /,/g' file1 > file2 and this produced file2:

MSTRG.10807.1,,,0.494311896511595,,,MID,,,,0.423993026403461,,2.39379412548345,,,1.99703339651136 MSTRG.10884.1,,,0.365770947942799,,,EARLY,,1.46416917664929,,,1.16816186543633,,,0.689075392478972 MSTRG.10958.1,,,0.52855355823638,,,,MID,,,,0.885493751836316,,2.28463841550375,,,1.26867555157512 

But I want a tab-delimited file.

Where have you seen an application of “arrangement spaces”?

I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature).

Let $ v_i\in\Bbb R^d,i\in N:=\{1,…,n\}$ be a finite family of points. Let’s call this an arrangement and write $ v$ when referring to all the points at once.

Definition. The arrangement space $ U\subseteq\Bbb R^n$ of $ v$ is the column span of the matrix $ M$ in which the $ v_i$ ‘s are the rows.

The definition is motivated by a recurring idea in geometry, and despite its simplicity has some interesting and non-trivial applications. While I have seen numerous computations that could have been formulated with this concept, I have never seen anyone giving a name to this specific idea.

So I wonder where some of you came across this idea, how you have used it, and where you have already seen it under a different name. I will start to name a few:

  • The arrangement space determines the arrangement up to invertible linear transformations. It is therefore of interest wherever people study linear, affine and convex dependencies between points (point configurations, oriented matroids, …)
  • The arrangement space determines all properties that are determined up to invertible linear transformations. E.g. the dimension of the span of the points, or whether the points are a linear transformation of a rational arrangement or a 01-arrangement.
  • The space of possible $ d$ -dimensional arrangement spaces in $ \Bbb R^n$ is the Grassmannian $ \mathrm{Gr}(n,d)$ . We therefore can parametrize arrangments via the Grassmannian, where distinct points in $ \mathrm{Gr}(n,d)$ describe arrangements that are not related by linear transformations.
  • Arrangement spaces give a natural definition of a form of Gale duality, which I have only seen defined in artificial and technical ways. The idea is as follows: $ $ v\;\;\mapsto \;\;U\;\;\mapsto\;\; U^\bot\;\;\mapsto\;\;\bar v.$ $ Two arrangements are Gale duals of each other, if and only if their arrangement spaces are orthogonal complements of each other.
  • The linear matroid on $ v$ can equivalently be defined by just knowing its arrangement space $ U$ : the independent sets are $ I\subseteq N$ for which $ U^\bot$ has trivial intersection with $ \mathrm{span}\{e_i\mid i\in I\}$ . In this form it is especially easy to prove that the dual matroid is again a linear matroid, namely, it is the linear matroid of the Gale dual of $ v$ (as defined above).
  • Let $ \Gamma\subseteq\mathrm{Sym}(N)$ be a permutation group that acts on $ \Bbb R^n$ via permutation matrices. If the arrangement space of $ v$ is $ \Gamma$ -invariant, then the arrangement expresses certain symmetries prescribed by $ \Gamma$ . The arrangement space can hence be used in the study of symmetric arrangements.
  • If $ G=(N,E)$ is a graph on $ N$ , and the arrangement space of $ v$ is an eigenspace of $ G$ , then the points in $ v$ are in a certain balanced configuration w.r.t. the edges of $ G$ . This has then be applied in graph drawings, stable arrangements from strongly regular graphs, eigenpolytopes, etc.

Note how all these are connected via their common use of the arrangement space. Hence, if you provide an answer, please explain how the arrangment space can be used to simplify what you are doing.

Classify 2-fold covering spaces of three-times punctured plane

Ahoy Mathematicians,

We are preparing for qualifying exams and ran across this question. There were two proposed methods of approaching it.

  1. Recognize that the thrice punctured plane is homotopic to the wedge of 3 circles. Following the example in Hatcher for the 2-fold coverings of the wedge of 2 circles, we construct the 3 connected covering spaces (up to relabeling) of the wedge of three circles. There is also the disconnected covering space of two copies of itself.

  2. Use the lifting correspondence to show that the 2 fold covering spaces of the thrice punctured disk are in one-to-one correspondence with subgroups H of $ \mathbb{Z} * \mathbb{Z} * \mathbb{Z}$ of index 2. There are 3 such groups, giving 3 connected covering spaces.

You’ll notice that the ideas in this agree. The question is one of rigor. Which answer is more accurate? Does Approach 1 actually give us the correct covering spaces, or do we need to think about crossing it with something to cover the surface rather than the 1-mfld? If we use Approach 2, does this answer the question, or do we need to give more of a classification than just the fundamental groups?


Which Shimura varieties admit or don’t admit $p$-adic uniformization by Drinfeld spaces?

$ p$ -adic uniformization is a powerful tool for studying Shimura curves and Shimura varieties. For instance, we know cohomology groups of Drinfeld spaces, so we know some information about the Shimura variety.

But it seems we don’t know all Shimura varieties have such uniformization. Which Shimura varieties admit or don’t admit $ p$ -adic uniformization by (products of) Drinfeld spaces?

Do we have some general sufficient conditions for the existence of such a uniformization? Do we know any explicit example that can’t have such $ p$ -adic uniformization?

The Newton-Raphson method in Banach spaces

I am not sure if this question is too easy for mathoverflow – please tell me to remove this question if it is too e before any downvotes. I have asked this on MSE (Link), but it received only a few comments and no answers.

Note: the screenshot at the bottom is where my question comes from.

This question is quite different from other versions of conditions of convergence of Newton iteration. For example, Kantorovich theorem.

I am now analysing the Newton-Raphson iteration in general Banach spaces $ E,F$ . Let $ x_0\in E$ , and let $ f:B_t(x_0)\to F$ be a differentiable function. ($ B$ denotes an open ball with radius $ t$ .) $ L(E,F)$ is the set of linear mapping from $ E$ to $ F$ .

By definition, $ f$ is differentiable at $ x$ with derivative $ Df_x\in L(E,F)$ (which is a linear functional from $ E$ to $ F$ ) if $ \exists r(h),f(x+h)=f(x)+Df_x(h)+r(h)$ , where $ r(h)/\|h\|\to 0$ as $ h\to 0$ .

To make it simple, I assume that there exist $ s>0$ such that

  • $ \|f(x_0)\|\leq t/(2s)$
  • If $ x,y\in B_t(x_0)$ then $ \|Df_x-Df_y\|\leq 1/(2s)$
  • $ \forall x\in B_t(x_0),\exists J_x\in L(F,E)$ such that $ J_xDf_x=Df_xJ_x=I_E$ and $ \|J_x\|\leq s$ .

Now let’s work on the iteration. Let’s fix $ x\in B_t(x_0)$ . Set $ x_n=x_{n-1}-J_x(f(x_{n-1}))$ . In real analysis course, we often take $ x=x_{n-1}$ , but here I have to fix $ x$ to be anything in $ B_t(x_0)$ . Just assume for a moment that $ \forall x\in B_t(x_0)$ . I will explain why later.

Firstly I have to show that $ x_n$ converges. Now I can use the inequality $ $ \|f(a)-f(b)-T(a-b)\|\leq \|a-b\|\sup_{c\in [a,b]} \|Df_c-T\|, $ $ where $ [a,b]$ is the line segment joining $ a,b$ , and $ T\in L(E,F)$ .

To use this inequality, we define $ g(y)=J_x(f(y))$ , so $ x_n=x_{n-1}-g(x_{n-1})$ , and $ Dg_y=J_xDf_y$ .(The reason why I cannot set $ x=x_{n-1}$ is that if I do it that way, then $ g(y)=J_y(f(y))$ , and I cannot find the derivative of $ g$ in this case.) Since $ x$ is fixed, we can assume there is NO $ x$ dependence in $ g$ . Therefore, $ $ \|x_{n+1}-x_{n}\|=\|f(x_{n})-f(x_{n-1})-(x_{n}-x_{n-1})\|\ \leq \|x_{n}-x_{n-1}\|\sup_{c\in [x_n,x_{n-1}]} \|Dg_c-I\|\=\|x_{n}-x_{n-1}\|\sup_{c\in [x_n,x_{n-1}]} \|J_xDf_c-J_xDf_x\|\ \leq \|x_{n}-x_{n-1}\|\|J_x\|\|Df_c-Df_x\|\ \leq \frac{1}{2} \|x_{n}-x_{n-1}\|. $ $ Also, $ $ \|x_1-x_0\|=\|J_x(f(x_0))\|\leq t/2 $ $ The conclusion is $ \|x_n-x_{n-1}\|\leq t/2^n$ .

My question: is it really OK to let $ x$ be anything fixed in $ B_t(x_0)$ ? Does that really work? If it is wrong, how can I fix it?

To prove that $ f(x_n)$ converges to zero, I feel that I should prove something like $ \|f(x_n)\|\leq t/(2^{n+1}s)$ (Suggested in a book of real analysis). I try to start by considering this: $ $ \|f(x_n)\|\leq \|Df_x\|\|x_{n+1}-x_n\| $ $ but it goes nowhere. From $ \|J_x\|\leq s$ we cannot obtain an upper bound on $ Df_x$ .

So how can I prove $ \|f(x_n)\|\leq t/(2^{n+1}s)$ ?

It should be clear that $ x_n$ is a Cauchy sequence – but it might not converge into $ B_t(x_0)$ – is that a problem?

It is a long question, so if I have made mistakes please point it out.

Please look at the following screenshot if the above is not clear.

Source of my problem: A course in mathematical analysis (screenshot)

Here is a theorem of Kantorovich which is related but not the same.

Two directed colimits same spaces with different inclusions

For any natural number $ n$ , let $ i_{n},j_{n}:X_{n}\rightarrow X_{n+1}$ be a pair of monomorphisms of simplcial sets.

Define $ $ X=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{i_n} X_{n+1}\cdots \} $ $ and $ $ Y=\operatorname*{colim}_n \{\cdots X_n \rightarrow_{j_n} X_{n+1}\cdots \}$ $

Question: I’m looking for an example where $ X$ is not isomorphic to $ Y$ as a simplicial set.

Homeomorphism between LP-Spaces on Metric Spaces and Lp-Spaces on Euclidean Space

Let $ (X,d_X,\mu)$ and $ (Y,d_Y)$ are complete metric spaces and $ \mu$ be doubling; moreover suppose that $ (X,d_X)$ is homeomorphic to a Euclidean space $ E^D$ and $ (Y,d_Y)$ is homemorphic to $ E^n$ .

Fix an arbitrary $ x_0 \in X$ and define the $ L^p$ -space $ L^p(X,Y;\mu)$ are the equivalence class of all measurable functions $ f$ for which $ $ \int_{x \in X} d^p(f(x),f(x_0)) \mu(dx)<\infty. $ $ I know that this is a complete metric space (standard result).

Since $ X$ is homeomorphic to $ E^D$ , then is $ L^p(X,Y;\mu)$ homeomorphic to $ L^p(E^d,E^n;\nu)$ for some measure $ \nu$ ?(probably $ \nu$ should be the push-forward of $ \mu$ along the homeomorphism relating $ (X,d_X,\mu)$ to $ E^D$ ).

In short, is $ L^p(X,Y;\mu)$ homeomorphic to a separable Banach space?

No spaces in command prompt when using tmux

I pulled pytorch/pytorch:1.1.0-cuda10.0-cudnn7.5-runtime from dockerhub, and install tmux and zsh in this docker.

At first, after I start tmux, it display extra space in tmux, I added the following to ~/.zshrc:

export LC_ALL=en_US.UTF-8   export LANG=en_US.UTF-8  

and added the following to .tmux.conf

set -g utf8 set-window-option -g utf8 on 

however, after I start tmux,there is no space in command prompt enter image description here

I bought 3 banners (ad spaces) on 3 forums for $170

Lately, I've been looking for new places to advertise one of my websites, and I thought to give a try on buying banner ads.

I myself am blind to ads and banners, so I didn't have high hopes about them, but I thought I'd give it a try, since my product is about SEO, and I had the opportunity to buy ad spaces on 3 forums about SEO and online marketing.

On May 6 I launched ads (banners 728×90) on all 3 websites.

I spent $ 170,10 on 3 ads. They'll be shown for a month, 10K impressions each….

I bought 3 banners (ad spaces) on 3 forums for $ 170