Some Experimental Results on the Frobenius Problem

Some Experimental Results on the Frobenius Problem

Some results on the symmetric doubly stochastic inverse eigenvalue problem

‎The symmetric doubly stochastic inverse eigenvalue problem (hereafter SDIEP) is to determine the necessary and sufficient conditions for an $n$-tuple $sigma=(1,lambda_{2},lambda_{3},ldots,lambda_{n})in mathbb{R}^{n}$ with $|lambda_{i}|leq 1,~i=1,2,ldots,n$‎, ‎to be the spectrum of an $ntimes n$ symmetric doubly stochastic matrix $A$‎. ‎If there exists an $ntimes n$ symmetric doubly stochastic ...

The Frobenius problem on lattices

It is widely known that if p and q are relatively prime positive integers then (a) the set of linear combinations of p and q with nonnegative integer coefficients includes all integers greater than pq − p − q, (b) exactly half the integers between 0 and pq − p − q belong to this set and (c) an integer m belongs to this set if and only if pq−p−q−m does not. A multidimensional version of statemen...

The Frobenius Coin Problem Upper Bounds on The Frobenius Number

In its simplest form, the coin problem is this: what is the largest positive amount of money that cannot be obtained using two coins of specified distinct denominations? For example, using coins of 2 units and 3 units, it is easy so see that every amount greater than or equal to 2 can be obtained, but 1 cannot be obtained. Using coins of 2 units and 5 units, every amount greater than or equal t...

Some results on the Oberwolfach problem

The well-known Oberwolfach problem is to show that it is possible to 2-factorize Kn (n odd) or Kn less a 1-factor (n even) into predetermined 2-factors, all isomorphic to each other; a few exceptional cases where it is not possible are known. In this paper we introduce a completely new technique which enables us to show that there is a solution when each 2-factor consists of k r-cycles and one ...