## UML class diagrams – how do I draw a class diagram that shows a self-reference? [closed]

So I have the following case. I have an interface A and an implementation AImpl. Now I have another implementation of A, called A2Impl, which references any instance of A besides implementing A already.

So I have now drawn the following UML class diagram:

Would this be correct? How do i draws this correctly?

## $NL^2 = NDSPACE(\log^2n)$ is closed under complement

• From Savitch’s theorem we have $$NL^2 \subseteq L^4$$, which is deterministic and thus closed under complement.
• From Immerman–Szelepcsényi theorem we have $$NL = coNL$$.

Why then $$NL^2 = coNL^2$$

## Characterization of continuous weakly closed 1-forms

Recall that a differential $$k$$-form $$\alpha$$ on a smooth manifold $$M$$ is called weakly closed if

$$\int_M \alpha \wedge d\beta = 0,$$

for all smooth forms $$\beta$$ of degree $$n-k-1$$, where $$n = \dim M$$. My question is:

If $$\alpha$$ is a weakly closed continuous 1-form on a closed manifold $$M$$, can we conclude that $$\alpha$$ is the sum of a smooth closed 1-form and $$d\phi$$, where $$\phi$$ is a $$C^1$$ function on $$M$$?

This appears to follow from the properties of the de Rham regularization operator(s) and associated homotopy operator(s) on forms; see, for instance, Theorem 12.5 in V. Gol’dshtein and M. Troyanov, Sobolev inequalities for differential forms and $$L_{q,p}$$-cohomology, Journal of Geometric Analysis, vol. 6, no. 4, 2006.

Any thoughts on this would be appreciated.

## DSPACE(f(n)) closed under complement

Im thinking that you can create the complementary Language that is an element of DSPACE(f(n)) (f(n) >= log(n)) by adding a step to the algorithm that reverses the answer. By that the function f(n) hasnt changed,therefore showing that DSPACE(f(n)) is closed under complement. I know that most likely my answer is wrong and can someone explain why ?

## A closed form of mean-field equations

Assume that a system at time t, for example number of costumers in a line at time $$t$$ which is denoted by $$q(t)$$, follows a Markov chain with these dynamics (probabilities)

$$P(q(t+\Delta t)-q(t)=1)=\alpha \Delta t$$ $$P(q(t+\Delta t)-q(t)=-1)=\beta\hspace{0.1cm}v(q(t)) \Delta t$$

for some known constant rates $$\alpha$$ and $$\beta$$. where $$v(x)$$ is piecewise function

$$$$\label{v} v(q)=\begin{cases} 0 & q \leq 0 \ \frac{q}{q_{*}} & 0\leq q\leq q_{*}\ 1 &q\geq q_{*} \end{cases}$$$$

Where $$q_*$$ is a constant. Using Dyknin formula we can find that the density, $$E[q(t)]$$=the expected number of customers at time $$t$$, can be obtained from this differential equation: $$$$\frac{d}{dt} E[q(t)]=\alpha-\beta E[v(q(t))] \label{eq:1}$$$$

I was wondering if there is any way to find a closed form above differential equation which means to find an expression for $$E[v(q(t))]$$ in terms of $$E[q(t)]$$.

Some efforts: In above equation replace $$E[v(q(t))] = v(E[q(t)])$$. Clearly this is not a good assumption because $$v(x)$$ is a concave function so according to Jensen’s inequality we have $$E[v(q(t))] \geq v(E[q(t)])$$

So replacing $$E[v(q(t))]$$ by $$v(E[q(t)])$$ may not give us a good estimation of exact solution of above ODE. I’ve done simulations for $$E(q(t))$$= the exact solution of differential equation (non-closed one), and the solution of closed form differential equation where $$E[v(q(t))]$$ is replaced by $$v(E[q(t)])$$ and this solution is not close to the exact solution. There are some other suggestions like using a quadratic equation in form of
$$E[v(q(t))]= a\hspace{0.1cm}E[q(t)]^2+b\hspace{0.1cm}E[q(t)]+ c$$ and then trying to find coefficients $$a$$, $$b$$ and $$c$$, but this effort also fails in some cases! I think any effort should use the nature of function $$v(x)$$ but till this moment I don’t know how to find it. So please let me know if you have any idea.

## Does the isometry group of a closed simple smooth curve in the plane constrain its perimeter^2/area ratio?

Let $$C$$ be a simple closed smooth curve delimitating a bounded domain $$D$$ in the euclidean plane of isometry group $$G$$ and of given area $$A$$. Does the minimal possible ratio $$\dfrac{P^{2}}{A}$$ where $$P$$ is the perimeter hence the total length of $$C$$ decrease when $$G$$ runs over a sequence $$(G_{i})_{i>0}$$ of groups such that $$i implies $$G_{i}$$ is a strict subgroup of $$G_{j}$$?

## Does $\bigcup_{c \ge 1} \mathsf{DTime}(2^{cn})$ closed under polynomial reduction?

It’s well known $$EXP$$ is closed under polynomial reduction. It means $$\bigcup_{c \ge 1} \mathsf{DTime}(2^{c^{n}})$$ is closed under polynomial reduction. But what about $$\bigcup_{c \ge 1} \mathsf{DTime}(2^{cn})$$? Is it also closed under polynomial reduction?

## File structure of object-oriented projects seems cluttered [closed]

In computer science courses at University, assignments written in OO languages such as Java had file systems similar to this:

• TreeNode.java
• BinaryTree.java
• Assignment1.java
• etc. …

In writing some of my own projects, it seems like splitting up each class into a file is very trivial, especially when classes are very small. Are there other design patterns that circumvent having many files lots of small helper classes, or is this pretty much the only standard?

## Remmina VNC server closed connection

I have FreeNas installed on a old computer which is working fine. I have installed Ubuntu 19.04 on a virtual machine on FreeNas. The installation went well but only when I want to type something in the terminal of ubuntu server the keyboard layout is messed up. But this is an issue of the web RealVNC Viewer which I don’t blame Ubuntu for. So I saw on my pc (OS: Ubuntu 18.04) I have Remmina which can also see VNC connections so I gave it a shot. But how many times I try to connect it won’t work (yes I also tried to connect right after the vm is starting up). Every time I get the message ‘VNC server closed connection’. When I power off the VM it Remmina says instantly ‘Can connect to VNC-server’. So it knows when the VM is only and when not. I also tried installing the realVNC software on my Windows 10 system and I could connect right after I start the vm (I gave the exact same arguments to remmina as i did to realVNC). Anyone having and idea what I’m doing wrong? Please note that i’m beginner to this stuff.

Many thanks YaMoef

## Visual representation for higher-order functions [closed]

I’m creating a visual programming language. I base on functional programming paradigm, because I believe that declarative thing should be mapped to visual more easily. I mostly done with conceptual design but there is one thing that I need to make that language full-featured and I still can’t come to the solution: higher-order functions (the functions that take one or more other functions as its arguments and do whatever it wants but mostly apply that functions to some data and do something with their outputs).

The best solution I found is one by Autodesk’s Max Creation Graph (there is also a good article explaining HOF’s). Here is how it looks:
(source: autodesk.com)

Here Map is a higher-order function, stuff in white dashed border is its function argument, orange dot with hint – argument of that “child” function, and the rightmost dark-cyan is the output of “child” function. This solution, however, have some obvious weak points. First of all you need a hand-drawn hint, to figure out where input is. Second – what if you have more than one such unconnected orange dots (e.g. optional arguments)? Or more than one unconnected outputs (e.g. some side-effect functions are used)? And finally, what if “child” function need more then one argument (e.g. for Map it can be element and index so you can not only transform an element of the array but also do something based on it’s position in that array) – it’s kind of ಠ_⊙ what the heck will happen in that situation at all…

So, maybe you have some kind of hints, insights or better examples? Or… fresh eye at least.