Which lattices are rotatable into their scaled copy?

Let $ L=\{\sum_i n_iv_i\mid n_i\in\mathbb Z\}$ be some lattice generated by $ d$ independent vectors $ (v_i)_1^d$ from $ \mathbb R^d$ . Call $ L$ rotatable if for some $ M$ , a scalar multiple of some rotation (orthogonal transformation that is other than the identity), we have $ M(L)\subset L$ . For example, $ \mathbb Z^d$ is rotatable, and so is $ \{n_1(1,0)+n_2(0,\sqrt 2)\}$ , but $ \{n_1(1,0)+n_2(0,\pi)\}$ is not. I’m sure this is some well-known notion with several equivalent definitions, but I couldn’t find anything, so I would be grateful for any pointers/results about them.


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Is Yahoo Changing Their Name to Altaba?

I heard a rumor, but I don't know how trustworthy the source was. I heard Yahoo was signing a deal with some other company. According to this source, the deal excludes Yahoo's 15% stake in Alibaba Group and 35.5% stake in Yahoo! Japan; BUT following the completion of the acquisition, these assets will be retained under the name Altaba, with a new executive team.


Logic to create/update values while maintaining their uniqueness (in a multi-threaded environment)

Assume I have a dumb repository which stores numbers (for sake of this example). It is dumb because it may only create new record, update specified record and list all the existing records – no other logic included.

But I want to implement a business logic which will allow me to store new records safely, knowing that I will not store a duplicate record. Also I want to update existing records knowing that it will not lead to having duplicates.

Let’s start with a creation of new records. The naive approach tells me that I need two steps to add a record:

  • (A) read existing records from repository
  • (B) check if existing records already contain same record as I need to store
  • (C) if existing records are free of duplicates – store record.

Works fine as long as everything runs in a single thread. How about multi-threading?

Another naive approach tells me, that I can introduce the lock around steps A,B,C. I am on a safe side again (let’s ignore all the performance considerations).

Now let’s consider records update. Steps are similar to creation:

  • (A) read all records
  • (B) check if duplicates will occur after update
  • (C) if check is in our favor – update the record.

Problems with multi-threading again? Just use lock again.

But what about the situation when one thread creates a record 200 while another thread updates record 100 to a value of 200. They both pass their checks on steps (B), both perform steps (C)… ant we end up with a duplicate.

It leads us to a conclusion that the whole design is wrong.

But what is the right design?

How to group 50 columns for every row in sas and create a new column for their values?

I have 50 categorical columns which contain numbers and 1 separate column for a unique identifier and 100 rows. I want to create a new dataset out of it which would have 3 columns: Column1 for the unique identifiers; Column2 for the categorical columns grouped for each row; and, Column3 for the numerical values.

What is the right term/theory for prediction of Binary Variables based upon their continuous value?

I am working with a linear programming problem in which we have around 3500 binary variables. Usually IBM’s Cplex takes around 72 hours to get an objective with a gap of around 15-20% with best bound.In the solution, we get around 85-90 binaries which have value of 1 and others are zero. The objective value is around 20 to 30 million. I have created an algorithm in which I am predicting (fixing their values) 35 binaries (with the value of 1) and letting the remaining ones solved through the Cplex. This has reduced the time to get the same objective to around 24 hours (the best bound is slightly compromised). I have tested this approach with the other (same type of problems) and it worked with them also. I call this approach as “Probabilistic Prediction”, but I don’t know what is the standard term for it in mathematics?

Below is the algorithm:

Let y=ContinousObjective(AllBinariesSet); WriteValuesOfTheContinousSolution(); Let count=0;  Let processedbinaries= EmptySet; while (count < 35 ) { Let maxBinary =AllBinariesSet.ExceptWith(processedJourneys).Max();//Having Maximum Value between 0 & 1 (usually lesser than 0.6)             processedJourneys.Add(maxBinary); maxBinary=1; Let z = y; y = ContinousObjective(AllBinariesSet); if (z > y + 50000)                  { //Reset maxBinary maxBinary.LowerBound = 0; maxBinary.UpperBound = 1; y = z; } else { WriteValuesOfTheContinousSolution(); count=count+1; }              } 

According to me, it’s working because the solution matrix is very sparse and there are too many good solutions.