Mathematical Research Seminar - Archive
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Valuated matroids were introduced by Dress and Wenzel in 1992 as a valuated generalization of matroids. They are a central object in discrete convex analysis, and play important roles in other areas such as mathematical economics and tropical geometry. Finding a constructive characterization, i.e., showing that all valuated matroids can be derived from a simple class by some basic operations has been a natural question proposed in various contexts.Motivated by this, we study the class of R-minor valuated matroids, that includes the indicator functions of matroids, and is closed under operations such as taking minors, duality, and induction by network. Our main result exhibits valuated matroids that are not R-minor, giving a negative answer to a question asked by Frank in 2003. Valuated matroids are inherently related to gross substitute valuations in mathematical economics. By the same token we refute the Matroid Based Valuation Conjecture by Ostrovsky and Paes Leme from 2015, asserting that every gross substitute valuation arises from weighted matroid rank functions by repeated applications of merge and endowment operations.
This is joint work with Georg Loho, Ben Smith, and László Végh.
We are looking forward to meeting you at FAMNIT-MP1.
Our Math Research Seminar will also be broadcasted via Zoom.
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Everyone is welcome and encouraged to attend.
In this talk, first we find the number of idempotents, nilpotents and the zero-divisors of a matrix ring over a finite field F.
Next, given the order of the Jacobson radical of the finite unital ring R, we find the number of units, nilpotents and zero-divisors of R, and give an upper bound for the number of idempotents of R, which is an extension of one of the previously founded results.
Finally, we find the above-mentioned numbers in some matrix rings and quaternion rings.
We are looking forward to meeting at the video-conference.
Join the Zoom Meeting Here.
See you there!
Everyone is welcome and encouraged to attend.
