Let H denote a two dimensional qubit Hilbert Space

  1. Explain what is meant by a spanning set for an n dimensional Hilbert space. What distinctions can be made between a spanning set, a basis and an orthonormal basis?

  2. Hilbert spaces may be partitioned according to the equivalence classes that its vectors may be perceived as belonging to. Explain what is mean by an equivalence class and how such a partitioning may naturally be realised in a Hilbert space. Describe, with examples, how a density operator may also be seen as an equivalence class representing a range of different possible ensembles. (

Is it safe to delete logcat.txt files or the whole psysinfo folder to make more space on my Android phone?

I have an android phone and the space on the sdcard is getting scarce… (I don’t understand how android functions – just a simple user.) In the following directory: mnt > sdcard

There is a psysinfo folder, which is 3.02 GB … And in this folder there are subfolders like log6, log5 and so on. In these subfolders, there are a few logcat.txt files and three of them are like 1GB large.

Will my android phone function OK if I just delete these files and free some space (3.02 GB)? I view the folders with ‘ES File Explorer’ and am thinking of deleting these files using the same app. This app says the above folder and logcat.txt files are the largest in my phone. My Android version is 4.1.2.

I searched online to find what the [ …> psysinfo ] folder in Android is, but I couldn’t find any info about it and then searched about deleting logcat.txt files but the info available seems to be for folks like developers. I couldn’t really understand it.


handle space in filename for imagemagic command

I am trying to resize image based on file name and size :

import subprocess import os  os.environ.setdefault('DJANGO_SETTINGS_MODULE', 'hiren.settings')  from hiren.settings import BASE_DIR   def resize(image_file, size):     os.chdir(BASE_DIR + '/convert/')     file_name = os.path.splitext(os.path.basename(image_file))[0]     file_ext = os.path.splitext(os.path.basename(image_file))[1]      for i in size:         cmd = ['convert', image_file, '-resize', i, file_name + i + file_ext]         # subprocess.check_call(f'convert {image_file} -resize {i} {file_name + i + file_ext}', shell=True)         subprocess.check_call(cmd, shell=True)   resize(BASE_DIR + '/convert/x/images 2.jpeg', ['308x412', '400x400']) 

then here is errors :

Traceback (most recent call last):   File "/home/../image.py", line 26, in <module>     resize(BASE_DIR + '/convert/x/images_2.jpeg', ['308x412', '400x400'])   File "/home/.../image.py", line 23, in resize     subprocess.check_call(cmd, shell=True)   File "/usr/lib/python3.6/subprocess.py", line 311, in check_call     raise CalledProcessError(retcode, cmd) Version: ImageMagick 6.9.7-4 Q16 x86_64 20170114 http://www.imagemagick.org subprocess.CalledProcessError: Command '['convert', '/home/../convert/x/images_2.jpeg', '-resize', '308x412', 'images_2308x412.jpeg']' returned non-zero exit status 1. Copyright: © 1999-2017 ImageMagick Studio LLC License: http://www.imagemagick.org/script/license.php Features: Cipher DPC Modules OpenMP  Delegates (built-in): bzlib djvu fftw fontconfig freetype jbig jng jpeg lcms lqr ltdl lzma openexr pangocairo png tiff wmf x xml zlib Usage: convert-im6.q16 [options ...] file [ [options ...] file ...] [options ...] file 

I am using python 3.6. Now how can I handle empty space or space less file name ?

Bookthickness of covering space

A book embedding of a graph G consists of placing the vertices of G on a spine and assigning edges of the graph to pages so that edges in the same page do not cross each other. The page number is a measure of the quality of a book embedding which is the minimum number of pages in which the graph G can be embedded.

If a graph $ G$ is a covering graph of graph $ B$ , is there any relation between their pagenumber?

I think the covering graph is more complicated than the basis graph. So does $ pn(G)\geq pn(B)$ hold in general?

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?

Why we need size of the page table should be some fraction of virtual address space

The Page Table should have all virtual page number which are in its logical address space, Why it’s the case?

  1. Is it because we want to access Page Table entry fast just like an array where key is virtual page number i.e. constant time?


  1. Is it due to structure of the process? (I mean Our program uses whole logical space; In general at address 0 we have Code and at address Max we have stack which is variable. Which means can point to any address of logical address space)