Approximate inverse limits and (m,n)-dimensions

This paper introduces shape boundary regions in descriptive proximity forms of CW (Closure-finite Weak) spaces as a source of amiable fixed subsets as well as almost amiable fixed subsets of descriptive proximally continuous (dpc) maps. A dpc map is an extension of an Efremovič-Smirnov proximally continuous (pc) map introduced during the early-1950s by V.A. Efremovič and Yu.M. Smirnov. Amiable fixed sets and the Betti numbers of their free Abelian group representations are derived from dpc's relative to the description of the boundary region of the sets. Almost amiable fixed sets are derived from dpc's by relaxing the matching description requirement for the descriptive closeness of the sets. This relaxed form of amiable fixed sets works well for applications in which closeness of fixed sets is approximate rather than exact. A number of examples of amiable fixed sets are given in terms of wide ribbons. A bi-product of this work is a variation of the Jordan curve theorem and a fixed cell complex theorem, which is an extension of the Brouwer fixed point theorem.

Dehn functions of finitely presented metabelian groups

In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

Exploring the Beginnings of Algebraic K-Theory

According to Atiyah, K-theory is that part of linear algebra that studies additive or abelian properties (e.g. the determinant). Because linear algebra, and its extensions to linear analysis, is ubiquitous in mathematics, K-theory has turned out to be useful and relevant in most branches of mathematics. Let R be a ring. One defines K0(R) as the free abelian group whose basis are the finitely generated projective R-modules with the added relation P ⊕ Q = P + Q. The purpose of this thesis is to study simple settings of the K-theory for rings and to provide a sequence of examples of rings where the associated K-groups K0(R) get progressively more complicated. We start with R being a field or a principle ideal domain and end with R being a polynomial ring on two variables over a non-commutative division ring.

Stability of Maximum Functional Equation and Some Properties of Groups

In this research paper, we deal with the problem of determining the function χ:G→R, which is the solution to the maximum functional equation (MFE) max{χ(xy),χ(xy−1)}=χ(x)χ(y), when the domain is a discretely normed abelian group or any arbitrary group G. We also analyse the stability of the maximum functional equation max{χ(xy),χ(xy−1)}=χ(x)+χ(y) and its solutions for the function χ:G→R, where G be any group and also investigate the connection of the stability with commutators and free abelian group K that can be embedded into a group G.

More on bases of uncountable free abelian groups

We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) subset of κ, then there is a κ-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [7], we give a direct construction.

More on bases of uncountable free abelian groups

We extend results found by Greenberg, Turetsky, and Westrick in [7] and investigate effective properties of bases of uncountable free abelian groups. Assuming V = L, we show that if κ is a regular uncountable cardinal and X is a ∆11(Lκ) subset of κ, then there is a κ-computable free abelian group whose bases cannot be effectively computed by X. Unlike in [7], we give a direct construction.

Some new results about a conjecture by Brian Alspach

In this paper, we consider the following conjecture, proposed by Brian Alspach, concerning partial sums in finite cyclic groups: given a subset A of $$\mathbb {Z}_n{\setminus } \{0\}$$ Z n \ { 0 } of size k such that $$\sum _{z\in A} z\not = 0$$ ∑ z ∈ A z ≠ 0 , it is possible to find an ordering $$(a_1,\ldots ,a_k)$$ ( a 1 , … , a k ) of the elements of A such that the partial sums $$s_i=\sum _{j=1}^i a_j$$ s i = ∑ j = 1 i a j , $$i=1,\ldots ,k$$ i = 1 , … , k , are nonzero and pairwise distinct. This conjecture is known to be true for subsets of size $$k\le 11$$ k ≤ 11 in cyclic groups of prime order. Here, we extend this result to any torsion-free abelian group and, as a consequence, we provide an asymptotic result in $$\mathbb {Z}_n$$ Z n . We also consider a related conjecture, originally proposed by Ronald Graham: given a subset A of $$\mathbb {Z}_p{\setminus }\{0\}$$ Z p \ { 0 } , where p is a prime, there exists an ordering of the elements of A such that the partial sums are all distinct. Working with the methods developed by Hicks, Ollis, and Schmitt, based on Alon’s combinatorial Nullstellensatz, we prove the validity of this conjecture for subsets A of size 12.

Spaces of knotted circles and exotic smooth structures

Suppose that $N_1$ and $N_2$ are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_1)$ and $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N_2),$ have the same homotopy $(2n-7)$ -type. In the four-dimensional case, this means that the spaces of smooth knots in homeomorphic $4$ -manifolds have sets $\pi _0$ of components that are in bijection, and the corresponding path components have the same fundamental groups $\pi _1$ . The result about $\pi _0$ is well-known and elementary, but the result about $\pi _1$ appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie–Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie–Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on $\pi _2 \operatorname {\mathrm {Emb}}(\mathrm {S}^1, N)$ . We use our model to show that for every choice of basepoint, each of the homotopy groups, $\pi _1$ and $\pi _2,$ of $ \operatorname {\mathrm {Emb}}(\mathrm {S}^1, \mathrm {S}^1\times \mathrm {S}^3)$ contains an infinitely generated free abelian group.

Free abelian group actions on normal projective varieties: submaximal dynamical rank case

Let X be a normal projective variety of dimension n and G an abelian group of automorphisms such that all elements of $G\setminus \{\operatorname {id}\}$ are of positive entropy. Dinh and Sibony showed that G is actually free abelian of rank $\le n - 1$ . The maximal rank case has been well understood by De-Qi Zhang. We aim to characterize the pair $(X, G)$ such that $\operatorname {rank} G = n - 2$ .

Planar pure braids on six strands

The group of planar (or flat) pure braids on [Formula: see text] strands, also known as the pure twin group, is the fundamental group of the configuration space [Formula: see text] of [Formula: see text] labeled points in [Formula: see text] no three of which coincide. The planar pure braid groups on 3, 4 and 5 strands are free. In this note, we describe the planar pure braid group on 6 strands: it is a free product of the free group on 71 generators and 20 copies of the free abelian group of rank two.