c# – What are the benefits of multiple projects over namespaces in a solution?

I have been tasked to set up a guide for a microservice solution structure and find myself reflecting on why exactly I do things the way I do them.
I am struggling to find an answer to the question when exactly I want to add a new project to my solution in a microservice context.

Some Microsoft articles suggest you should do so (see first picture): https://docs.microsoft.com/de-de/dotnet/architecture/microservices/microservice-ddd-cqrs-patterns/ddd-oriented-microservice
while there were discussions going somewhat into detail on this topic already, which have a different result:
https://stackoverflow.com/questions/8678251/what-are-the-benefits-of-multiple-projects-and-one-solution#:~:text=If%20you%20only%20ever%20have,things%20like%20sharing%20source%20files.

It is obvious that you need to organize your code inside a solution. So there are two options: projects and namespaces. The question is: If there is no technical reason to create multiple project files – like reusability across solutions, individual deployment – why not simply organize your code with namespaces in a folder structure in your service project?

The separation of frontend and backend is separated from this. I would create different solutions for frontend and backend, unless its a very small application.

As far as I know DDD is just demanding for a logical separation of application/domain/infrastructure layers. And there is an often undervalued Clean Code Priciple: KISS – Keep it simple, stupid (make it as easy as possible, but not easier).

To be on the same page as to what a solution using multiple projects and a solution using namespaces for those projects is:

Multiple Project Solution:

Ordering.sln
|-Ordering.Infrastructure.csproj
||-MyInfrastructureClass.cs
|-Ordering.Domain.csproj
||-Order.cs
||-OrderItem.cs
|-Ordering.Application.csproj
||-CreateNewOrder.cs
|-Ordering.Service.csproj
||-Controller
|||-OrderController.cs
|-Ordering.Tests.csproj

Namespace Solution:

Ordering.sln
|-Ordering.Service.csproj
||-Application
|||-CreateNewOrder.cs
||-Controller
|||-OrderController.cs
||-Domain
|||-Order.cs
|||-OrderItem.cs
||-Infrastructure
|||-MyInfrastructureClass.cs

Pro “New Project”:

  • It forces you to get your dependencies right, as long as you place your classes in the right projects.
  • Smaller/Leaner service project
  • Assemblies are reusable

Pro Namespaces:

  • No need to deploy more Assemblies than necessary
  • Smaller/Leaner solution itself
  • Less complexity in the solution

The only real benefit of the multi-project solution seems to be that the threshold for messing up dependencies is lower in the namespace solution.

My conclusion so far is:

  1. The namespace solution is better, if code reviews are done properly to make sure nobody messes up the architecture of the solution.
  2. Make new projects in the solution only if there is a technical reason for it. Such as reusing an assembly.

So my question is: What do I miss here?
I did quite some research before and found people doing it because of DDD or just to organize their code. From Microsofts code snippets it almost seems like it does not matter. Sometimes its done one way, sometimes another way. But those examples are showing something the article explained, so since the purpose is not to show a good solution structure this may be the reason why.

fa.functional analysis – Smooth dependence of solution to elliptic pde depending on parameter

I have a question in mind but let me generalize it slightly.
Suppose I am looking at some pde like
$$-Delta u + t u = f(u)$$ in $B_1$ (here $u=u(x)$) with $u=0$ on $ partial B_1$ where $B_1$ is the unit ball centred at the origin in $R^N$. For concreteness lets assume $f(u)=u^p$ where $p>1$. Lets assume I can show for all $t>0$ there is a bounded positive radial solution of the given pde.

My goal is to prove that for $t=1$ the solution is nondegenerate; meaning the kernel of the linearized operator is trivial. Now since this equation is not exactly the one I have in mind
I don’t want to prove directly that the solution is nondegenerate since this might not extend to my case.
Using some other tricks I believe I can show that the solution for $t=1$ is nondegerate provided these solutions indexed by $t$ are sufficiently smooth in $t$. I believe the usual method to prove smoothness in $t$ is to use the implicit function theorem (or something close) but of course I can’t do that here since I am really trying to prove one of the hypothesis. Any comments would be greatly appreciated.

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linear algebra – Homogenous Equation with Complex Vector Solution: Converting to Real Functions

Solving the system
$$
begin{array}{l}frac{d x}{d t}=6 x-y \ frac{d y}{d t}=5 x+4 yend{array}
$$

we get $lambda_{1}=5+2 i, lambda_{2}=5-2 i$ eigenvalues. So eigenvectors and corresponding solutions is:

$$
mathbf{K}_{1}=left(begin{array}{c}1 \ 1-2 iend{array}right), quad mathbf{X}_{1}=left(begin{array}{c}1 \ 1-2 iend{array}right) e^{(5+2 i) t}
$$

$$
mathbf{K}_{2}=left(begin{array}{c}1 \ 1+2 iend{array}right), quad mathbf{X}_{2}=left(begin{array}{c}1 \ 1+2 iend{array}right) e^{(5-2 i) t}
$$

Thus general solution:
$$
mathbf{X}=c_{1}left(begin{array}{c}1 \ 1-2 iend{array}right) e^{(5+2 i) t}+c_{2}left(begin{array}{c}1 \ 1+2 iend{array}right) e^{(5-2 i) t}
$$

So converting the above solution to a real function via Euler’s:
$$
begin{array}{l}e^{(5+2 i) t}=e^{5 t} e^{2 t i}=e^{5 t}(cos 2 t+i sin 2 t) \ e^{(5-2 i) t}=e^{5 t} e^{-2 t i}=e^{5 t}(cos 2 t-i sin 2 t)end{array}
$$

My text states to collect terms and replace $ c_1 + c_2 $ by $ C_1 $ and $ (c_1 – c_2)i $ by $ C_2 $ the solution becoming $mathbf{X}=C_{1} mathbf{X}_{1}+C_{2} mathbf{X}_{2}$ where
$$
mathbf{X}_{1}=left(left(begin{array}{l}1 \ 1end{array}right) cos 2 t-left(begin{array}{r}0 \ -2end{array}right) sin 2 tright) e^{5 t}
$$

$$
mathbf{X}_{2}=left(left(begin{array}{r}0 \ -2end{array}right) cos 2 t+left(begin{array}{l}1 \ 1end{array}right) sin 2 tright) e^{5 t}
$$

I seem to be making a mistake somewhere and don’t get to the same simplification. I’m looking for some of the steps to the books specification. I’ve tried a few times by multiplying the general solution across rows, fully expanding outside of its matrix form.

An Integral equation solution

$$displaystyle int_{1}^{infty}frac{(1-(x-(x)))(x^{1-a}-x^a)cos (ln x) }{x^2}mathrm{d}x$$=0 for $0<a<1$, prove that $a=frac{1}2$.

$(.)$ denotes Greatest integer function.
Note that a=1/2 satisfies the above equation.

differential equations – Symbolic solution to periodic boundary conditions

Is it possible to get an analytical solution to the following problem? If so, how?

$$a w^{(1,0)}(x,t)+b w^{(0,1)}(x,t)=p(x)+g w(x,t)$$
$$I.C.: w(x,0)=0 | B.C.: w(0,t)=w(1,t) | x in left(0, 1right)$$

Clear("Global`*")

a = 1;
b = v;
g = -Subscript((Lambda), i);
p = Subscript((Beta), i) Subscript((Psi), 0)*Sin(Pi*x);

ic = w(x, 0) == 0;
bc = w(0, t) == w(1, t);

pde = a*D(w(x, t), x) + b*D(w(x, t), t) == g*w(x, t) + p

DSolve({pde, ic, bc}, w(x, t), {x, t})

$$text{DSolve}left(left{psi _0 beta _i sin (pi x)=lambda _i w(x,t)+v w^{(1,0)}(x,t)+w^{(0,1)}(x,t),w(x,0)=0,w(0,t)=w(1,t)right},w(x,t),{x,t}right)$$


As you can see, in the current formulation, DSolve is unable to solve this problem with the periodic boundary specified. I have tried to provide some guidance to the solution such as shown below and failed. I would expect the solution to have two solutions, 1) $t > x/v$ and 2) $t <= x/v$ representing the propagation of the wave. Hence, I also tried “helping” there as well to no avail.

Element(x, {0, 1});
Simplify(DSolve(...),{0<=x<=1});
Assuming({0<=x<=1}, DSolve(...));
Simplify(DSolve(...),{0<=x<=1,t>x/v});
Assuming({0<=x<=1,t>x/v}, DSolve(...));

Note 1: I had a previous question related to simple Mathematica things which are now more understood and I have moved beyond that and spent a good deal of time solving some other problems to build experience, but now I am back to this one with the periodic boundary addition.

Note 2: This post has me worried that periodic boundary conditions are not possible to be solved symbolically as they were specifically not called out…

Damaged iMac Display – Any Solution for Grey Vertical Column Lines?

After replacing a Fusion Drive with an SSD, it must have damaged my display. Now, there is a column of grey on the left-hand side. I double-checked the cables, and they are snug. Probably damaged beyond repair.

Is there any way to modify the Menu Bar at the top to move it right, past the grey bar, so that all the menu items are visible?

The iMac is still usable, I just use the other 2/3rd of the screen and use a lot of keyboard shortcuts. But I often can’t see the menu and need to select from it.

27″ iMac late 2012 running Catalina

enter image description here

simplifying expressions – Trouble verifying a solution to a differential equation

I am reading a physics book which has discussed one approach towards solving the differential equation

$$
frac{d^2 x(t)}{dt^2} = Cx(t)
$$

as follows:

enter image description here

Using Mathematica to solve the equation. I tried using Mathematica to solve the equation as follows:

In(4):= DSolve(x''(t) == C x(t), x(t), t)

Out(4)= {{x(t) -> E^(Sqrt(c) t) C(1) + E^(-Sqrt(c) t) C(2)}}

But it gives a different (and I assume more general/proper) solution than the one offered by the book.

Question: How can I verify that the solution offered in the physics book has a form which matches the solution offered by Mathematica?

Attempt: Trying

Solve(E^(Sqrt(C) t) C(1) + E^(-Sqrt(C) t) C(2) == 
  C(1) Cos(Sqrt(-C) t) + C(2) Sin(Sqrt(-C) t), C)

yields

enter image description here

Confusion with Regards to General and Particular Solution Terminology in Differential Equations

I have been reading R. Kent Nagle’s Fundamental’s of Differential Equations textbook and I’m really confused as to the meaning of the terms of “Particular Solution” and “General Solution”, specifically as they change from being used in a first-order equation to a second-order equation.

So in a first-order differential equation, your answer should have a “C” somewhere resulting from integrating somewhere. This would be called your general solution because you haven’t specified any initial conditions. If you did, and you incorporated that information into your solution, then it would be called your particular solution. Alright, so far so good.

But when I started learning about second-order differential equations, I got really confused because I understood that if you solve a homogeneous second-order linear differential equation with no initial condition information then you would get a general solution with two “C”s because of its second order. This made sense, but when I learned about the Method of Undetermined Coefficients and the Variation of Parameters, I learned that the general solution to a non-homogeneous second-order linear differential equation involved a particular solution AND the general solution to the homogeneous diff eq. But I don’t see why or how this could make sense. Does this occur because the terms particular and general are redefined for second-order diff EQs? I’m overall just very confused about the terminology. General clarification would be greatly appreciated.

Thanks and have a good one.

set theory – I’m looking for a conjecture on a possible solution in an infinite set

Recently, I came across a conjecture by two mathematicians about something along the lines of “if an infinite set contains a solution modulus an ever increasing power, then a finite solution should exist in the set even though the set appears to only contain ever increasing numbers”. I’m winging it here. It’s something along these lines. Any help would be greatly appreciated.

I checked my browser history since it was less than a month ago, but not a single hit came up.

Thanks,

Bassam