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David E. Dobbs; Gabriel Picavet; Martine Picavet-L’Hermitte(2014)
Let R⊂S be an extension of commutative rings, with X an indeterminate, such that the extension RX⊂SX of Nagata rings has FIP (i.e., SX has only finitely many RX-subalgebras). Then, the number of RX-subalgebras of SX equals the number of R-subalgebras of S. In fact, the function from the set of R-subalgebras of S to the set of RX-subalgebras of SX given by T ↦TX is an order-isomorphism.
The main purpose of this paper is to prove the following
conclusion: let p be a prime large enough and let k be a fixed positive integer with 2k|p−1. Then for any finite field Fp and any element 0≠c∈Fp, there exist three generators x, y, and z∈Fp such that xkyk+ykzk+xkzk=c.
Two m×n matrices A,B over a commutative ring R are equivalent
in case there are invertible matrices P, Q over R with B=PAQ. While any m×n matrix over a principle ideal domain
can be diagonalized, the same is not true for Dedekind domains. The first author and T. J. Ford
introduced a coarser equivalence relation on matrices called homotopy and showed any m×n matrix
over a Dedekind domain is homotopic to a direct sum of 1×2 matrices. In this article give,
necessary and sufficie...
The Lucas sequence is defined by:
L0=2,L1=1,Ln=Ln−1+Ln−2 for n≥2. Let
V(n), r(n) denote respectively the number of partitions of n into parts, distinct parts from {Ln}. We
develop formulas that facilitate the computation of V(n) and r(n).
We consider the question of existence of trigonometric series universal in weighted Lμ1[0, 2π] spaces with respect to rearrangements and in usual sense.
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