Narrow Results

Micro-PIXE and inductively coupled plasma mass spectrometry (ICP-MS) were applied to elemental distribution analyses in plant root apex which is composed of various types of tissues and cells in different developmental stages. ICP-MS was so sensitive that a large number of elements including Na, Mg, P, S, K, Ca, Mn, Fe, Cu, Zn, Se, Rb, Sr and Cs could be determined quantitatively. These fourteen elements included almost all the essential elements for plant growth. Only a rough estimation, however, could be obtained by ICP-MS for the elemental distribution at the tissue level, by analyzing sections from the root apex. On the other hand, micro-PIXE was effective for detailed mappings of elemental distributions. The images of elemental distributions were obtained for Na, Mg, P, S, K, Ca, Mn, Fe and Zn, corresponding to the microscopic images of the root structures. The localizations of P, K and Zn in some tissues were observed by the mappings. These results indicated that micro-PIXE and ICP-MS have different, but complementary abilities for the investigation of elemental distributions in plant tissues.

The dual resolution graphs of rational triple point (RTP) singularities can be seen as a generalization of Dynkin diagrams. In this work, we study the triple root systems corresponding to those diagrams. We determine the number of roots for each RTP singularity, and show that for each root we obtain a linear free divisor. Furthermore, we deduce that linear free divisors defined by rational triple quivers with roots in the corresponding triple root systems satisfy the global logarithmic comparison theorem. We also discuss a generalization of these results to the class of rational singularities with almost reduced Artin cycle.

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.

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.

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.

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.

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.

We introduce the class of split Lie algebras of order 3 as the natural generalization of split Lie superalgebras and split Lie algebras. By means of connections of roots, we show that such a split Lie algebra of order 3 is of the form $L={𝒰}+({\sum }_j{I}_j)$ with ${𝒰}$ a linear subspace of $H$ and any $I_j$ a well-described (split) ideal of $L$ satisfying $[I_{j,\bar{0}},I_k]=\{I_{j,ī},I_{k,ī},L_{ī}\}=0$, with $ī\in \{\bar{1},\bar{2}\}$, if $j\ne k$. Additionally, under certain conditions, the (split) simplicity of the algebra is characterized in terms of the connections of nonzero roots, and a second Wedderburn type theorem for the class of split Lie algebras of order 3 (asserting that $L$ is the direct sum of the family of its (split) simple ideals) is stated.

In this paper, we consider roots of multivariate polynomials over a finite grid. When given information on the leading monomial with respect to a fixed monomial ordering, the footprint bound [Footprints or generalized Bezout’s theorem, IEEE Trans. Inform. Theory46(2) (2000) 635–641, On (or in) Dick Blahut’s ‘footprint’, Codes$,$ Curves Signals (1998) 3–9] provides us with an upper bound on the number of roots, and this bound is sharp in that it can always be attained by trivial polynomials being a constant times a product of an appropriate combination of terms consisting of a variable minus a constant. In contrast to the one variable case, there are multivariate polynomials attaining the footprint bound being not of the above form. This even includes irreducible polynomials. The purpose of the paper is to determine a large class of polynomials for which only the mentioned trivial polynomials can attain the bound, implying that to search for other polynomials with the maximal number of roots one must look outside this class.

Phytohormone auxin is the main regulator of plant growth and development. Nonuniform auxin distribution in plant tissue sets positional information, which determines morphogenesis. Auxin is transported in tissue by means of diffusion and active transport through the cell membrane. There is a number of auxin carriers performing its influx into a cell (AUX\LAX family) or efflux from a cell (PIN, PGP families). The paper presents mathematical models for auxin transport in vascular tissues of Arabidopsis thaliana L.root tip, namely protophloem and protoxylem. Tissue specificity of auxin active transport was considered in these models. There is PIN-mediated auxin efflux in both protoxylem and protophloem, but AUX1-mediated influx exists only in protophloem. Optimal values of parameters were adjusted for model solutions to fit the experimentally observed auxin distributions in the root tip. Based on simulation results we predicted that shoot-derived auxin flow to protophloem is lower than one to protoxylem, and the efficiency of PIN-mediated auxin transport in protophloem is higher than in protoxylem. In summary, our simulation showed that despite the same auxin distribution pattern, provascular tissues in the root tip differ in dynamics of auxin transport.

For a nilpotent group $G$ without $\pi$-torsion, and $x$, $y\in G$, if $x^n=y^n$ for a $\pi$-number $n$, then $x=y$; if $x^m{y}^n=y^n{x}^m$ for $\pi$-numbers $m$, $n$, then $xy=yx$. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let $m$ be a given positive integer and $A$ a complex square matrix satisfying that (i) all eigenvalues of $A$ are nonnegative, and (ii) $\mathrm{rank}{A}^2=\mathrm{rank}A$; then $A$ has a unique $m$-th root $X$ with $\mathrm{rank}{X}^2=\mathrm{rank}X$, all eigenvalues of $X$ are nonnegative, and moreover there is a polynomial $f(\lambda )$ with $X=f(A)$. In addition, let $A$ and $B$ be complex $n\times n$ matrices with all eigenvalues nonnegative, and $\mathrm{rank}{A}^2=\mathrm{rank}A$, $\mathrm{rank}{B}^2=\mathrm{rank}B$; then (i) $A=B$ when $A^r=B^r$ for some positive integer $r$, and (ii) $AB=BA$ when $A^s{B}^t=B^t{A}^s$ for two positive integers $s$ and $t$.

Let ${\mathscr{H}}(n,r)$ be the set of the connected $r$-uniform linear hypergraphs with $n$ vertices, where $n,r\ge 2$. The matching polynomial of a hypergraph ${\mathscr{H}}$ is denoted by $\phi ({\mathscr{H}},x)$, where ${\mathscr{H}}\in {\mathscr{H}}(n,r)$. Several properties on the roots of $\phi ({\mathscr{H}},x)$ are derived. We establish different expressions for $\phi ({\mathscr{H}},x)$, such as a higher-order differential formula and an integral formula. A new concept is also introduced for the weighted matching polynomials of weighted hypergraphs, for which several basic calculating formulas are obtained. Based on the results we obtained, some formulas for counting the number of perfect matchings of ${\mathscr{H}}$ are directly derived. Finally, for the weighted matching polynomials of the weighted loose paths and the weighted loose cycles, we not only obtain their recursive formulas, but also deduce their specific expression formulas.