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Ginseng is a well-known medicinal plant used in traditional Oriental medicine. In recent decades, ginseng root has gained popularity as a dietary supplement in the United States. Ginseng has also been commonly used in Oriental medicine to treat diabetes-like conditions. The present review discusses the research on the anti-diabetic effects of ginseng and the possible mechanisms of its anti-diabetic actions.

Let h₁, h₂,… be a sequence of elements in a free group and let H be the subgroup they generate. Let H′ be the subgroup generated by w₁, w₂, …, where each w_i is a word in h_i and possibly other h_j, such that the associated directed graph has the finite paths property. We show that rank H′≥ rank H. As a corollary we get that , where is the subgroup generated by the roots of the elements in H. If H₀ is finitely generated and the sequence of subgroups H₀, H₁, H₂, … satisfies then the sequence stabilizes, i.e. for some m, H_i=H_i+1 for every i≥ m. When applied to systems of equations in free groups, we give conditions on a transformation of the system such that the maximal rank of a solution (the inner rank) does not increase. In particular, we show that if in "Lyndon equation" the exponents a_i satisfy gcd(a₁,…,a_n)≠1 then the inner rank is ⌊ n/2⌋. The proofs are mostly elementary.

When considering affine tropical geometry, one often works over the max-plus algebra (or its supertropical analog), which, lacking negation, is a semifield (respectively, $\nu$-semifield) rather than a field. One needs to utilize congruences rather than ideals, leading to a considerably more complicated theory. In his dissertation, the first author exploited the multiplicative structure of an idempotent semifield, which is a lattice ordered group, in place of the additive structure, in order to apply the extensive theory of chains of homomorphisms of groups. Reworking his dissertation, starting with a semifield${}^{†}$$F$, we pass to the semifield${}^{†}$$F({\lambda }_1,\dots ,{\lambda }_n)$ of fractions of the polynomial semiring${}^{†}$, for which there already exists a well developed theory of kernels, which are normal convex subgroups of $F({\lambda }_1,\dots ,{\lambda }_n)$; the parallel of the zero set now is the $1$-set, the set of vectors on which a given rational function takes the value 1. These notions are refined in supertropical algebra to $\nu$-kernels (Definition 4.1.4) and $1^{\nu }$-sets, which take the place of tropical varieties viewed as sets of common ghost roots of polynomials. The $\nu$-kernels corresponding to tropical hypersurfaces are the $1^{\nu }$-sets of what we call “corner internal rational functions,” and we describe $\nu$-kernels corresponding to “usual” tropical geometry as $\nu$-kernels which are “corner-internal” and “regular.” This yields an explicit description of tropical affine varieties in terms of various classes of $\nu$-kernels. The literature contains many tropical versions of Hilbert’s celebrated Nullstellensatz, which lies at the foundation of algebraic geometry. The approach in this paper is via a correspondence between $1^{\nu }$-sets and a class of $\nu$-kernels of the rational $\nu$-semifield${}^{†}$ called polars, originating from the theory of lattice-ordered groups. When $F$ is the supertropical max-plus algebra of the reals, this correspondence becomes simpler and more applicable when restricted to principal$\nu$-kernels, intersected with the $\nu$-kernel generated by $F$. For our main application, we develop algebraic notions such as composition series and convexity degree, leading to a dimension theory which is catenary, and a tropical version of the Jordan–Hölder theorem for the relevant class of $\nu$-kernels.

The centralizer of a group element g is the group of elements commuting with g. Powers and, if existent, roots of g are contained in the centralizer. We call a centralizer trivial if it consists of the integer powers of g, only.

In the group of torus homeomorphisms, we study the centralizer of torus diffeomorphisms of hyperbolic Anosov type. As a result, the centralizer can be calculated, isomorphically, in the much smaller group of affine torus automorphisms. In particular, Anosov diffeomorphisms of the 2-torus with a unique fixed point possess trivial centralizer, up to a trivial involution.

The identification of individual reactors from data on reactor cascades, for example in chemical engineering, is one source of motivation for the study of roots of diffeomorphisms. Another possible source is the study of commuting diffeomorphisms in finite-dimensional spatially or spatio-temporally chaotic systems.

We consider a natural generalization of the concept of order of an element in a group $G$: an element $g\in G$ is said to have order $k$ in a subgroup $H$ (respectively, in a coset $Hu$) of $G$ if $k$ is the first strictly positive integer such that $g^k\in H$ (respectively, $g^k\in Hu$). We study this notion and its algorithmic properties in the realm of free groups and some related families.

Both positive and negative (algorithmic) results emerge in this setting. On the positive side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e. the set of elements of a given order) with respect to finitely generated subgroups are always computable in free and free times free-abelian groups. On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even not computably enumerable sets of natural numbers. Also, we take advantage of Mikhailova’s construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.