The will be linear transofmation $T:V \rightarrow V$   The formula says that $\dim V=\dim(\operatorname{Ker}(T))+\dim(\operatorname{Im}(T))$  and $\operatorname{Ker}(T)\subseteq V$ and $\operatorname{Im}(T) \subseteq V$  Does it implies that $V=\operatorname{Ker}(T)\oplus \operatorname{Im}(T)$ iff $T$ is a homomorphism?
For a vector space $V$ and a linear operator $f : V \to V$, under what conditions does $V = \ker(f) \oplus \operatorname{im}(f)$?  Is it always true, or only in special cases?  Edit: $V$ is finite dimensional.
I use Chrome as my default browser. This includes my browser for work.  We have a trac setup that uses folder paths (see example) for the different projects  Example:   main.employertracsite.com/projects/projectname   I want the ability to manually sort the results that are displayed in the auto-complete.  Chrome displays auto-complete based on the usage history of a web address. Which means that a project that isn't my main focus is displayed in the url instead of the one that I've been using more lately.  Example:   main.employertracsite.com/projects/alliance is the project that I'm not using all the time anymore and is showing up in auto-complete main.employertracsite.com/projects/horde is the project that I am using more often now and want to show up in the auto-complete   I don't want to delete the alliance from my history, but I do want the ability to be able to manually adjust which project shows up in auto-complete. This will be more usable for me in the future because I'm constantly on different projects full-time and they each have their own area.  I also do not want to delete my history. I managed to successfully publish my old bachelor thesis in a good quality national journal in my country. However, I am thinking of retracting the article due to certain circumstances. First, I felt really embarrassed by the low quality research methodology used in my bachelor thesis. I only used simple correlation in my research. After it was published, I realized that the journal editor team forgot to include a reference in that article because they were rushing me throughout the editing process. I informed them about their mistake and it seems that they have done nothing to fix the problem. I became a bit annoyed because the editor team has never made this problem in the past.  I felt that it would be better for me to simply ask for retraction to solve this issue, but I am a bit afraid of negative consequences in the future. Is it okay to do this? I am really interested of entering the academic world in the future. I am trying to figure out how to add a Lightning Web Component to a Lightning tab.  I have it on the Home Page, but I would like it to be on its own tab.  I feel like there should be an option to expose it to the Lightning tabs.  When I am in the Tabs section, I don't see my LWC listed. Conjugacy Class in a matrix group with columns of standard basis I want to find the conjugacy classes of this group: $$G=\{(a_1,a_2,a_3):a_i=\pm e_j\}\subset GL_3(\mathbb C)$$ I know that $S_3$ is one of the subgroups of this group, so I can check which elements of $S_3$ continues to be conjugate in our group, I know that in $Z(G)$ there is two elements $I,-I$. I know that I have to check the diagonal matrices on which ones are conjugate, but I don't know how to finish and find the others conjugacy class of this group, and I don't Know how to find the number of the elements of the conjugacy class. I'll be thankful for help Debian : Verify disk encryption password for already mounted system So the problem is that account 1 and account 2 bought a game together, but account 1 isn't used anymore. Can we transfer the game license from account 1 to account 3?   The game was bought on account 1 and account 2 activated account 1 as primary. In ZF set theory the axiom of regularity (also called axiom of foundation) says that:     In all nonempty sets x there is an element y such that x∩y=∅   As I been told that  of the axiom is to prohibit constructions as $x\in x$, but is this a possible strategy?  Given a binary predicate $R$, written in infix notation, there are unitary predicates $p$ such that:  $\qquad \exists y \forall x: x R y \leftrightarrow p(x)$.  But there also are some convincing counterexamples:   $  p(x) \leftrightarrow \neg(x R x)$ $  p(x) \leftrightarrow \neg(x R f(x))$, where $f$ is any surjection $  p(x) \leftrightarrow$ all chains $x_1 R x, \: x_2 R x_1, \: x_3 R x_2,  \ldots \:$ are finite.   I think there is a major mistake to try to avoid certain paradoxical loops. First, there are no guarantees that every possibility is prohibited. And second, it seems pathological to make constructions that avoid certain proof-techniques (it's like the attitude of a shyster).  So, what protection comes from this axiom? Historically, has it anything to do with the Russell-predicate $x\notin x$?  See also Turn off text message (SMS) sync to Exchange email account I recently started using a Samsung Galaxy S4 and one of the features in a synchronization between text messages and email accounts.  When a text message comes in it throws a copy into my Exchange email accounts that are one the phone. I want to turn this feature off but cannot find a setting for it anywhere.  Any ideas? Is there a word to describe someone who tends to disagree with others only to upset them? Show that $\dim H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m)) = {n + m \choose n}$ if $m \geq 0$, and $0$ if $m &lt; 0$. This statement came up in an algebraic geometry text with no explanation provided, and I'm trying to understand why it's true.   Thoughts so far:   Since $H^0(\mathbb{P}^n, \mathcal{O}_{\mathbb{P}^n}(m))$ is the vector space of regular sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ over $\mathbb{P}^n$, we can try to find a basis. Since the regular sections are the degree $m$ polynomials in $x_0, \dots, x_n$, it seems like the collection of degree $m$ monomials in $x_0, \dots, x_n$ should work. But I don't see how there are ${n + m \choose n}$ of these. Is it just a basic counting argument that I'm struggling with?  On a side note, I think I understand what global sections of $\mathcal{O}_{\mathbb{P}^n}(m)$ look like, but how can we describe the regular sections over a distinguished open set $D(f)$? Is it something like degree $m$ polynomials with denominator $f$? How to properly handle collision in a component-based game? Trying to wrap my head around the ways to properly handle collision in a game designed around components.  I see many examples have some sort of PhysicsComponent that gets added to the entity's list of components but the actual implementation is confusing me.  For this to work, the PhysicsComponent would need access to the world around it. This doesn't make intuitive sense to me. Shouldn't a component be unaware of not only its container (the entity), but its container's container (the world)?  To me, it sounds like the level or scene should maintain a list of these entities and every game update, loop through the entities to determine which collide.  My question is firstly, whether or not this is good design, and secondly, how to determine which entities can collide. I suppose solid entities could implement an empty IRigidBody interface so the level could determine which entities in the list support collision. But is this breaking the component design?  Instead, should they contain an empty RigidBody component? This may actually be better because it may not always be empty and this approach is more future-proof. The only problem with this is the complexity. The scene would have to loop through not only every entity, but also every entity's components to determine if it had this RigidBody component.  Thirdly, when they do collide both entities should be informed somehow and I am unsure on how to accomplish this either.  Let's say that both entities contained a HealthComponent and when they collided both their healths would be decreased by some arbitrary value, 5. I suppose it would be the scene's responsibility to handle this when it detects a collision between two entities?  But then is the scene responsible for too much? I could see this possibly getting out of hand and becoming unwieldy when the scene is responsible for many things entities should not (?) have access to.  Edit: Question updated with more details. Comment hint not disappearing when I start to type comment If I click "add comment", the comment hint (placeholder; "Request clarification or add information.") is not a hint, but editable text. If I delete the text and click outside of the comment then the hint appears as it should and when I click back into the comment box the hint disappears as it should.   I am on Internet Explorer 10.0.7.  Am I just oblivious and this is how it has always been (having to delete the comment hint), or is this a bug? How to calculate $f(x)$ in $f(f(x)) = e^x$? How would I calculate the power series of $f(x)$ if $f(f(x)) = e^x$?  Is there a faster-converging method than power series for fractional iteration/functional square roots? "Cannot find this file" error when deploying from Visual Studio 2010 I'm getting an error in Visual Studio 2010 after pressing F5 to Debug:     Error occurred in deployment step 'Add   Solution': Error: Cannot find this   file specified in the manifest file:   G2_SiteDefinition_List_Instances_Feature\ListInstance1\Elements.xml   But elements.xml file is there with the ListIntance1 folder. What am I missing here? Obfuscation is one way, but it can't protect from breaking the piracy protection security of the application. How do I make sure that the application is not tampered with, and how do I make sure that the registration mechanism can't be reverse engineered?  Also it is possible to convert a C# application to native code, and  is too costly.  C# provides lot of features, and is the ideal language for my code, so writing the whole codebase again in C++ is out of the question.  Secure certificates can be easily removed from the signed assemblies in .NET. So I'm going to be starting a Job here in a few weeks, I've been doing QA for about 6 years now. Pretty much your standard Test Plans/Testing with a little normal SysAdmin stuff that sometimes comes with the task. I've had some good ideas implemented in companies, as far as tracking requirements/test plans/issues differently, and they seem successful. However, this new position will be me laying down the foundation for QA. Right now things are done in what seems to be as needed as far as QA goes, requirements are gathered and sometimes kept in different locations, same with test plans (which are done by developers right now). So my job will be sort of centralizing this, or developing a Flexible way of QA'ing the Software (Web/Mobile development). So it seems the challenge will be centralizing everything, while creating test plans/ or ways of testing similar requirements that is flexible enough to span projects. I've had a few ideas, however, all my previous test work was windows based (So test plans in TFS or Microsoft Test Manager). I wanted to get some ideas from everyone here, or anyone that has started a QA department from scratch. I was also going to see if there were any books on the subject. My thoughts were mainly keeping projects separated for issue tracking (not sure of the software we will use yet). And keeping test plans for common I guess &quot;actions/elements&quot; that are repeated between different customers and executing those. Eventually maybe using selenium to save those &quot;common&quot; elements as test scripts. But I've never set something up like this before solo, so I've been doing quite a bit of reading. Any books/suggestions would be appreciated! Get TEXT value of a CLOB OID in Postgresql How to find basis of intersection of 2 vector spaces with given basis? Suppose we have vector spaces $\mathbb W$ and $\mathbb V$.  A given basis for $\mathbb W$ is $\{(1, 1, 0, -1), (0, 1, 3, 1)\}$, and a given basis for $\mathbb V$ is $\{(1, 2, 2, -2), (0, 1, 2, -1)\}$.  How to find a basis for $\mathbb W\cap\mathbb V$? Show that $\langle 2,x \rangle$ is not a principal ideal in $\mathbb Z [x]$ Hi I don't know how to show that $\langle 2,x \rangle$ is not principal and the definition of a principal ideal is unclear to me. I need help on this, please.   The ring that I am talking about is $\mathbb{Z}[x]$ so $\langle 2,x \rangle$ refers to $2g(x) + xf(x)$ where $g(x)$, $f(x)$ belongs to $\mathbb{Z}[x]$. How does matter know to obey physical laws? First of all, you may say that, this is a "why" question and in physics we don't ask such a question and instead we ask about "how". However, I want to discuss this with you.  We model the behavior of matter by physical laws, for example by quantum mechanics laws. I want to know how the matter, say electron, knows that it must obey such a law. What forces the electron to obey quantum mechanics laws?  If you consider a simulation of the universe in a computer, in the view of someone  simulated in that computer the physical laws are just there. They may not know what keeps  in check the simulated matter to obey physical laws, but in our perspective those simulated physical laws are there because we implemented them as zero and ones in a hard disk. If we wanted, we could change those physical laws and in the eye of the simulated person everything would be still fine.  Is there a scientific attempt to discuss this issue? so that I can read about it.