general topology – Question about the nearly open set

‎‎‎‎A subset ‎‎$‎S‎$‎ of ‎$‎(X,tau ‎)‎$‎ is said to ‎be:‎

  • $ ‎alpha‎$‎-open ‎in‎$‎(X,tau ‎)‎$‎ ‎if‎ $ S‎ ‎‎subset ‎‎int(cl(int(S)))_{tau}‎$‎‎

  • ‎pre-‎open‎ ‎in ‎$‎(X,tau ‎)‎$‎ ‎if‎ $‎S ‎subset ‎‎‎int(cl(S)‎)‎_{tau}$

  • semi-‎preopen‎ ‎in$‎(X,tau ‎)‎$‎ ‎if‎ $‎S ‎subset‎ ‎‎‎cl(int(cl(S)))_{tau}$

  • ‎regular‎-‎open‎ ‎in‎$‎(X,tau ‎)‎$‎ ‎if‎ $‎S ‎=‎ ‎‎‎int(cl(S)‎)‎_{tau}$‎.‎‎‎

‎The ‎familly ‎of ‎all‎ ‎$ ‎alpha‎$‎-open ‎(Res.‎ ‎pre-‎open ,‎ ‎‎‎semi-‎preopen,‎ ‎regular-‎open)‎ subset of ‎‎$‎(X,tau ‎)‎$‎‎‎ ‎is ‎denoted ‎by‎‎$ ‎tau‎^{‎alpha‎}‎‎ $‎(‎‎Res. $‎PO‎(X,tau), SPO(X,tau), RO(X,tau)‎‎$‎)‎.‎

‎‎‎‎For every space ‎$‎(X, tau ‎)‎$‎ the family $ ‎tau‎^{alpha}‎$‎ forms a topology on ‎‎‎‎$‎X‎‎$‎‎‎ and $tau subset tau^{alpha}$‎.‎‎‎

Let‎‎$ Y ‎‎subseteq X , ‎‎‎Y‎ ‎in‎ SPO(X,‎tau)‎ $‎, ‎which ‎one ‎is ‎correct?‎

  • if ‎‎$‎V in ‎PO‎(X,‎tau)‎$‎, ‎then‎ ‎$ Y ‎cap V‎ ‎in ‎PO(Y, ‎tau‎_{Y})‎‎ $‎.‎
  • ‎if ‎‎$‎V in ‎tau‎^{‎alpha‎}‎‎ ‎$‎, ‎then‎ ‎$ Y ‎cap V‎ ‎in tau‎^{‎alpha‎}‎‎‎(Y,tau‎_{Y})‎‎$‎.‎
  • ‎if ‎‎$‎V in ‎R‎O‎(X,‎tau)‎$‎, ‎then‎ ‎$ Y ‎cap V‎ ‎in ‎RO(Y, ‎tau‎_{Y}) ‎$‎.‎

( $‎tau‎_{Y}$ is the subspace topology of $Y$.)

Question about Merkle path verification

Merkle trees in general are useful in the context of a prover-verifier model.

A prover Peggy wants to prove to verifier Victor that a merkle root R, which Victor knows ahead of time, commits to a tree which includes a specific leaf L.

To do so, Peggy would send the element L well as the Merkle proof (or branch) containing all hashing partners L is combined with to produce R. In your example diagram, that proof consists of Hk, Hij, Hmnop, and Habcdefgh. Victor uses these elements to recompute R from it, and compare it with his pre-existing knowledge of R.

So to answer your question: certainly something has to keep Hk (or the elements that has to it, so it can be recomputed), but it doesn’t need to the same party as the one that does the verification. If there is only one party involved, there is nothing to prove.

Question related to Vector Space, Subspaces and their complements?

Thanks for contributing an answer to Mathematics Stack Exchange!

  • Please be sure to answer the question. Provide details and share your research!

But avoid

  • Asking for help, clarification, or responding to other answers.
  • Making statements based on opinion; back them up with references or personal experience.

Use MathJax to format equations. MathJax reference.

To learn more, see our tips on writing great answers.

pr.probability – Math grade 12 question

There are 16 kids to place into 4 different vans. Van 1 can hold 5 kids, Van 2 can hold 2 kids, Van 3 can hold 6 kids and Van 4 can hold 3 kids. How many ways can the 16 kids be placed in the vans?

soft question – Stages in mathematical perception

Mathematics can be viewed at different levels. But those views has to be attained in the course of time.

In initial days, Mathematics is generally perceived as a collection of facts, algorithms, methods etc., I can call it algorithmic stage of perception.

Next level, according to me, is perceiving Mathematics as a tool for solving problems in other domains. I can call it a tool level of perception. Can also be called as intuition level of perception. Because it needs an Mathematical intuition to use concepts of Mathematics in a particular context.

I can experience only these two levels. Is the second level, the highest? Are there any further levels of perceiving Mathematics?

databases – Basic Sqlite3 Python Question

I have a beginner sqlite3 python question. I have a large function f() that sequentially calls a series of intermediate functions f1(),…,f10(). f() takes about 10 minutes to complete. It is being called simultaneously by multiple (eight) different threads (different arguments for each call). f5() tries to write to an sqlite3 database—the same database for each thread. This write is not intensive and should be very short. Even with a connection timeout=80, there are times when I am getting a OperationalError: database is locked. I am trying to figure why this is happening.

In f5(), I have the following try-except block

try:  
    con = sqlite3.connect()
    do stuff and write
    con.close()
    return (and move on to f6())

except:
    contingency
    (never call con.close())
    return (again move on to f6())

As shown, I realized that I never call con.close() in the except portion of the try-except block. If the connection opens in the try block and moves to the except block before calling con.close(), is this connection remaining open when I move on from f5() to f6(). Or is it automatically deleted once f5() hits a return?

Thanks!

magento2 – Magento 2.3.3 Tax question

I know this is a common issue, but cant seem to find a proper answer anywhere.
I am running a Magento 2.3.3 shop and have my product imported through a plugin and these are imported excluding tax (i cannot change this, my industry suppliers work on excluding tax and this is how the prices are imported).
No matter what tax settings i use the inc VAT prices are always wrong when added to the cart.
For example, i have a product that sells for £8 inc VAT, so the ex VAT price set in the catalog is £6.67. The catalog and product pages display this correct at £8, but when a quantity of 2 is added to that cart it shows at £8.01 each.
I had an order yesterday for:
2 x £8
2 x 17 (these show as £17.01 in the cart)
£5 delivery

Magento works this out at £55.02, not £55. Before tax it displays as £45.85, not £45.83 like it should be.

No customers have yet questioned it, but it is frustrating and only a matter of time until somebody does.

My current tax settings are:
Tax Calculation Method Based On: Total (Have tried unit and row and i would prefer unit)
Catalog Prices: Excluding Tax (has to be this way)
Shipping Prices: Excluding Tax

Discounts are a whole other ball game and dont seem to be ale to do calculate discount after and including tax but this is not my worry for now.

I would like to figure out how i can display, in this example the prices correct with a total of £55 inc VAT. I know this is a rounding issue but dont know how to solve it. All prices are imported at 2 decimal places.

Thanks

Question about queuing theory

Suppose that two types of customers arrive at a queueing system
according to independent Poisson process of rate λ/2 . Both types of customers require
exponentially distributed service times of rate μ. Type 1 customers are always accepted
into the system, but type 2 customers are turned away when the total number of customers
in the system exceeds K.
(a) Sketch the state transition diagram with total number of customers.
(b) Find the steady state PMF.

trigonometry – Trigonometric question combined with questions about Mathematica

I’m new to Mathematica and I am struggling with the Syntax and the functions of this “programming” language.

I’m looking for a way to solve sin2 α cos3 α = tan4 α

But if I try to run it through the Solve[] function I recieve an error.

I’m not looking for hand holding and having someone give me the answer. What I am trying to do is to learn which functions work where and how I successfully can calculate questions like these in Mathematica.