arXiv subjects
Publications and source records attributed to Pace P. Nielsen.
We provide a new, elementary, algebraic proof of the fundamental theorem of algebra. Furthermore, the method recovers a more general version of the theorem recently obtained by Joseph Shipman. The key idea is to understand extension fields in which a polynomial gains a factor of a given degree.
An algebraic framework in which to study infinite sums is proposed, complementing and augmenting the usual topological tools. The framework subsumes numerous examples in the literature. It is developed using many varied examples, with a particular emphasis on infinitizing the usual group and ring axioms. Comparing these examples reveals that a few key algebraic properties play a crucial role in the behaviors of different forms of infinite summation. Special attention is given to associativity, which is particularly difficult to properly infinitize. In that context, there is an important technique called the Eilenberg-Mazur swindle that is studied and greatly generalized. Some special properties are singled out as potential axioms. Interactions between these potential axioms are analyzed, and numerous results explore how to impose new axioms while retaining old ones. In some cases the axioms classify or categorize a given example. Surprisingly, such axiomatizations are very concise, relying on only a handful of natural conditions. These investigations reveal more precisely the part that topology plays in the formation of infinite sums. Special attention is given to the methods of partial summation and unconditional summation. In the opposite direction, it is proved that from the infinite sums alone one can create a refined topology, lying between the original topology and its sequential coreflection. Another especially interesting application of these ideas is the construction of new algebraic quotient structures that retain the ability to handle infinite summation.
We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.
We construct a Euclidean domain with no multiplicative Euclidean norm to a compatibly well-ordered monoid, and hence with no multiplicative Euclidean norm to $\mathbb{R}$ (under its usual order). A key step in the proof is showing that the UFD property is preserved when adjoining a free factorization.
Levels of cancellativity in commutative monoids $M$, determined by stable rank values in $\mathbb{Z}_{> 0} \cup \{\infty\}$ for elements of $M$, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb{Z}_{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid $M$, the possible stable rank values in archimedean components of $M$ are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.
We provide two new formulations of the separativity problem. First, it is known that separativity (and strong separativity) in von Neumann regular (and exchange) rings is tightly connected to unit-regularity of certain kinds of elements. By refining this information, we characterize separative regular rings in terms of a special type of inner inverse operation, which is defined via a single identity. This shows that the separative regular rings form a subvariety of the regular rings. The separativity problem reduces to the question of whether every element of the form $(1-aa')bac(1-a'a)$ in a regular ring is unit-regular, where $a'$ is an inner inverse for $a$. Second, it is known that separativity in exchange rings is equivalent to regular matrices over corner rings being reducible to diagonal matrices via elementary row and column operations. We show that for $2\times 2$ invertible matrices, four elementary operations are sufficient, and in general also necessary. Dropping the invertibility hypothesis, but specializing to separative regular rings, we show that three elementary row operations together with three elementary column operations are sufficient, and again in general also necessary. The separativity problem can subsequently be reframed in terms of the explicit number of operations needed to diagonally reduce.
We very slightly improve the leading constant of the (currently best) proven asymptotic zero-free region of the Riemann zeta-function, by using an easy improvement to a trigonometric polynomial.
We parameterize solutions to the equality $Φ_3(x)=Φ_3(a_1)Φ_3(a_2)\cdotsΦ_3(a_n)$ when each $Φ_3(a_i)$ is prime. Our focus is on the special cases when $n=2,3,4$, as this analysis simplifies and extends bounds on the total number of prime factors of an odd perfect number.
We prove that for every indecomposable ordinal there exists a (transfinitely valued) Euclidean domain whose minimal Euclidean norm is of that order type. Conversely, any such norm must have indecomposable type, and so we completely characterize the norm complexity of Euclidean domains. Modifying this construction, we also find a finitely valued Euclidean domain with no multiplicative integer valued norm.
We prove that every distinct covering system has a modulus divisible by either 2 or 3.
Vaughan Pratt has introduced objects consisting of pairs $(A,W)$ where $A$ is a set and $W$ a set of subsets of $A,$ such that (i) $W$ contains $\emptyset$ and $A,$ (ii) if $C$ is a subset of $A\times A$ such that for every $a\in A,$ both $\{b\mid (a,b)\in C\}$ and $\{b\mid (b,a)\in C\}$ are members of $W$ (a "crossword" with all "rows" and "columns" in $W),$ then $\{b\mid (b,b)\in C\}$ (the "diagonal word") also belongs to $W,$ and (iii) for all distinct $a,b\in A,$ the set $W$ has an element which contains $a$ but not $b.$ He has asked whether for every $A,$ the only such $W$ is the set of all subsets of $A.$ We answer that question in the negative. We also obtain several positive results, in particular, a positive answer to the above question if $W$ is closed under complementation. We obtain partial results on whether there can exist counterexamples to Pratt's question with $W$ countable.
We construct an example of a unit-regular ring which is not strongly clean, answering an open question of Nicholson. We also characterize clean matrices with a zero column, and this allows us to describe an interesting connection between unit-regular elements and clean elements. It is also proven that given an element $a$ in a ring $R$, if $a,a^2,\ldots, a^k$ are all regular elements in $R$ (for some $k\geq 1$), then there exists $w\in R$ such that $a^{i}w^{i}a^{i}=a^{i}$ for $1\leq i\leq k$, and a similar statement holds for unit-regular elements. The paper ends with a large number of examples elucidating further connections (and disconnections) between cleanliness, regularity, and unit-regularity.
Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation for the minors of minor matrices in terms of the weights of families of paths in a network.
An odd perfect number, N, is shown to have at least nine distinct prime factors. If 3 does not divide N, then N must have at least twelve distinct prime divisors. The proof ultimately avoids previous computational results for odd perfect numbers.
We show that a suitable ring with a ``nice'' topology, in which convergent limits of units are units, is an \aleph_0-exchange ring. We generalize the argument to show that a semi-regular ring, R, with a ``nice'' topology, is a full exchange ring. Putting these results in the language of modules, we show that a cohopfian module with finite exchange has countable exchange. Also, all modules with Dedekind-finite, semi-regular endomorphism rings are full exchange modules.