linear algebra – Condition for a finite sequence to be a basis of a subspace of the dual space

Let $E$ be a vector space and $(x_i^*)_{1leq ileq p}$ a family of elements of $E^*$. Write $G’$ for the subspace generated by the $x^*_i$ and let $F$ be the orthogonal of $G’$ in $E$. Assume the $x^*_i$ are linearly independent. We want to show the existence of a family $(x_i)_{1leq ileq p}$ of elements of $E$ such that $langle x_i,x^*_jrangle=delta_{ij}$ for all $i,j$. There author says that

$E/F$ and $G’$ can each be canonically identified with the dual of
the other; if the family $(x^*_i)$ is free, there is in $E/F$ a basis
$(dot{x}_i)$ the dual of $(x^*_i)$ and every representative system of
the classes $dot{x}_i$ has the required properties.

Let $pi:Erightarrow E/F$ be the canonical linear surjection. Then $^tpi:(E/F)^*rightarrow G’$ is an isomorphism (the transpose of $pi$). This shows that we can identity $(E/F)^*$ and $G’$. But how should we identify $G’^*$ and $E/F$? Taking the transpose again gives another isomorphism $^{tt}pi:(G’)^*rightarrow(E/F)^{**}$. This means that $(E/F)^{**}$ is finite dimensional and so the canonical mapping $c_{E/F}:E/Frightarrow (E/F)^{**}$ is an isomorphism. Does the author mean that
$$dot{x}_i:=c_{E/F}^{-1}(^{tt}pi(x^{**}_i)),$$
where $(x^{**}_i)$ is the dual basis of $(x^{*}_i)$?

ag.algebraic geometry – On the Dimension of the Dual Variety of a Singular Hypersurface

I was primarily interested in the following question. Let $ngeq 3$, and let $Xsubset mathbb{P}^n$ be a degree $d$ hypersurface. Assume that its singularity locus $S$ (with reduced structure) is irreducible and smooth of dimension $k$. Is it true that
$$
dim S^vee >dim X^vee ?
$$

The statement is true for quadratic hypersurfaces and hypersurfaces with isolated singularities. I am wondering if in general this holds.

I was reading the book Discriminants, Resultants, and Multidimensional Determinants by Gelfand-Kapranov-Zelevinksy. In the first chapter they introduce the Katz dimension theorem, which computes the dimension of the dual varieties via local coordinates. But I found it difficult to use, since it’s quite hard to write down the local coordinates. Is there any comments on the Katz dimension theorem, especially on how to use it ?

dual boot – Ubuntu MBR issue

I have a Lenovo ThinkPad e14 gen 2 AMD which came with 250 gb WD NVME ssd. I decided to install Ubuntu 18.04 in dual boot with windows in a separate drive. So, I bought a second ssd (Kingston 250gb NVME) to install Ubuntu 18.04 on.

I did the following steps before installing Ubuntu:

  • Disable Bitlocker
  • Disable fast startup
  • Disable secure boot in BIOS

I then created a live USB using balena etcher and this Ubuntu iso image

From the boot menu I chose the usb and proceeded with the installation normally.

In the disk partition part, I left the first ssd labelled /dev/nvme0n1 (Windows drive) untouched and to the new ssd (/dev/nvme1n1) I made the following partitions.

  • efi – 512 mb Primary – Beginning
  • swap – 16000 mb Primary – Beginning
  • / – 60000 mb Primary – Beginning – /ext4
  • /home – 173547 mb Primary – Beginning – /ext4

For the device for bootloader installation I picked the new drive (/dev/nvme1n1)
I proceeded with the installation and rebooted and removed the usb.
I intended to install grub in the efi partition.
After rebooting I was presented with the grub menu and I realized that the new ssd was partitioned as MBR, because of which I think the bootloader was in MBR and not efi partition.
I can boot into both windows and Ubuntu but the times are not syncing.
I’m not sure if what I have done is correct.

My question is:

  • Why did Ubuntu install as MBR and not gpt even though my laptop is uefi and windows is uefi?
  • Why does the efi partition in ssd 2 show up as local disk D in windows 10?
  • Will Ubuntu in proper efi mode solve the time sync issue?
  • If I need to reinstall Ubuntu how can I do it safely?
  • What change should I make during installation or the bios to install Ubuntu correctly(UEFI with correct bootloader)?

ssd1 windows partition
ssd2_ubuntu_partitions
partitions
partition_on_windows
thinkpad_boot_menu

Can’t get grub2 to install and be able to dual boot windows and ubuntu from same SSD

I had Elementary OS running for awhile alongside my W10, but I wanted to switch to Kubuntu while keeping my W10. Something went wrong in installation, so I went ahead and installed ubuntu 20.04 instead. Now, I only have legacy boot options and cannot use grub2 as UEFI to be able to boot into either Ubuntu 20.04 or W10.

How can I get around this issue? I have four partitions, one of which is windows. I have heard it’s a bad idea to install grub into a partition. I think a similar question has been asked before, but apologies since I am rather new to linux. Thank you.

Best,
George

usability – Order of list boxes in a dual list box

A dual list box is a component-based on two list boxes placed together that allows adding, removing, and ordering items.

According to Kaley from the nngrounp: “The Listbox on the left holds available options and the Listbox on the right represents selected items. The Add button moves an item from the available list to the selected list and the Remove button moves a selected option back to the available list, to deselect it. Users can also move options up and down to reorder elements in the list

My concern is the following, due to the normal order of reading of a user, it seems more natural to place the selected list box on the left instead of the right. Because the page scanning is faster when you only want to review while defining is the opposite.

enter image description here

So, my question is:

Does it matter the order of the list boxes? or it’s just to be consistent using always the same order because users will learn the pattern reading the labels?

dual boot – systemd-coredump files

I was using a shared home partition (NTFS) between Windows 10 and Ubuntu 20.10 on my hp elitebook. for about a good 3 months I had no issues, then today I went to copy a file while in Ubuntu in the shared partition and found that “root” had taken ownership. i’ve shared a partition before without incident, and found that it’s kind of a “no-no” to do. so, I decided it was a good time to repartition anyway, and using a Ubuntu 20.10 Live USB I set out and used my HDD docking station to copy everything from the NTFS shared partition to a EXT4 partition, called “myfiles”. i did everything one folder/directory at a time (i only copied folders that’d be found in /home/user; nothing else). i let every instance of each copy go without being stopped and without doing anything else on my laptop (no multitasking, no internet browsing, nothing, i just let it sit there and copy). when everything was done, i restarted my laptop, everything turned on and mounted as it should – but when I went to the copied partition, “myfiles”, i was told i didn’t have permission to access or to see the partition because it was now “systemd-coredump.” 1) i don’t know what this means. 2) what did do that caused this? so i can avoid it again. 3) how do i fix this?

tourist visas – Hypothetical Dual Citizen Passport Scenario for Covid 19 Travel

This question might be a little complicated, so I’ve made a reference chart to refer to throughout.

Reference Legend:

  1. Country A – Have Citizenship
  2. Country B – Have Citizenship
  3. Country
    C – Tourist, currently staying
  4. Country D – Intended destination as a tourist. No current visa, but normally issues visas upon arrival. However due to Covid-19, is currently not issuing these visas upon arrival at this time
    for people from countries A or B.

Assumption:
My goal is to enter Country D when it resumes visa upon arrival issuance for citizens of either country A or B. Due to the coronavirus pandemic however country D is no longer issuing these visas upon arrival and is also restricting travel from counties A & B (but not country C with the appropriate visa, which is immaterial to me since I’m just a tourist in C anyway so couldn’t avail of this).

Full Question:
Let’s say I have dual nationality/passports from countries A and B, but am living as a tourist currently in country C. Ideally I’d like to travel to country D (my fiance lives there) as a tourist directly from country C , but country D is currently banning tourists from both countries A and B due to the coronavirus pandemic.

Let’s say in the foreseeable future that country D allows tourist travel and visa upon arrival issuance for country A but not from country B (still banned). This would mean I have citizenship of both a banned and allowed country for the purposes of being issued a tourist visa upon arrival in country D. Would it be advisable to travel directly from country C to country D using my passport from country A (not banned) at the immigration for country D when I arrive there? For relevance, let’s assume I have previously entered this country with my passport from Country B (banned) so they may well have a record of me being of that banned nationality anyway.

In this hypothetical scenario, should I also declare to the immigration at country D that I’m a citizen of the banned country B upon arrival, but will be using my passport from country A which is not banned, and that I have in fact come directly from country C (which let’s assume isn’t banned either)?

I’d just like to confirm whether this would be an appropriate hypothetical course of action, particularly for a dual national living as a tourist in yet another country (C), that ultimately wants to reside as a tourist in yet another country (D), in the times of this pandemic where some countries may re-open tourist entry to certain countries (such as country A) but not others (such as country B).

Would there also be any potential issue coming from country C instead of country A directly on my journey to country D, provided country C isn’t banned? Ie: Would it be likely for the immigration to care that I was merely a tourist in C where I came from, provided I can still produce a passport from country A?

I realize the answer may depend upon the actual country that country D is, but I’m just curious in terms of “generally” whether this course of action would be appropriate.

Thank you, I hope this question is understandable.

macbook pro – Upgrading from sierra to catalina while maintaining dual boot with kali

Yesterday I installed Kali linux in macbook air having macos (sierra).today I downloaded Catalina through appstore . It downloads Catalina file around 7gb and to install it reboots
MAGIC
While rebooting it enters into Linux grub option.
Then I tried to reboot again with option button
It show Macintosh disk and I clicked it.then
Normal sierra opened
What about Catalina os . I have files of Catalina

dual boot – Crash:systemd-journald:failed to write entry. ignoring input/output. systemd-journald:failed to rotate /var/log/journal/ read only file system

I have a Hp laptop, dual boot, running windows 10,and Ubuntu 18.04LTS.
Recently I have been facing issue, that Ubuntu suddenly crashes and shows error of this type Link of the image and another time this type of error link of another image.

I am a newbie and don’t know what is causing this error, still I have a speculation that might be my HDD is causing this, my laptop has become very slow, its 3+ years old and recently ran the SMART Test, the test result was OK, and bad sector count was 25168. Please help.