Unique Root Research Articles

The general Randić index of trees with given number of pendent vertices

The general Randić index of a graph G is defined as Rα(G)=∑uv∈E(G)(d(u)d(v))α, where d(u) denotes the degree of a vertex u in G and α is a real number. In this paper, we determine the maximum general Randić indices of trees and chemical trees with n vertices and k pendent vertices for 4≤k≤⌊n+23⌋ and α0 ≤ α < 0, where α0≈−0.5122 is the unique non-zero root of the equation 6·4α−20·9α+10·12α−16α+5·24α=0. The corresponding extremal graphs are also characterized.

A generalized Banach–Stone theorem for C0(K,X) spaces via the modulus of convexity of X⁎

Suppose that K and S are locally compact Hausdorff spaces, X is a Banach space and λ is a real number satisfying2λ(1−δX⁎(2λ))<r, where δX⁎ denotes the modulus of convexity of the dual space X⁎ and r is the unique real root of the polynomial p(x)=2x3−x2+8x+20. We prove that if T is an isomorphism from C0(K,X) onto C0(S,X) with‖T‖‖T−1‖≤λ, then K is homeomorphic to S. This improves an earlier generalization of the Banach–Stone theorem via geometric properties of X⁎ due to Jarosz, the case where K and S are compact and r is exchanged by the smaller number 4/3. In the process of the proof we establish a connection between the moduli of convexity of X and X⁎ which has independent interest. Namely, put C=r/4. Then C is approximately 0.41883 and for all Banach space X and ε∈(0,2], we have1−Cε<δX(ε)⟹1−12ε<δX⁎(ε).

Approximating the principal matrix square root using some novel third-order iterative methods

Abstract It is known that the matrix square root has a significant role in linear algebra computations arisen in engineering and sciences. Any matrix with no eigenvalues in R - has a unique square root for which every eigenvalue lies in the open right half-plane. In this research article, a relationship between a scalar root finding method and matrix computations is exploited to derive new iterations to the matrix square root. First, two algorithms that are cubically convergent with conditional stability will be proposed. Then, for solving the stability issue, we will introduce coupled stable schemes that can compute the square root of a matrix with very acceptable accuracy. Furthermore, the convergence and stability properties of the proposed recursions will be analyzed in details. Numerical experiments are included to illustrate the properties of the modified methods.

Exponential growth rates of free and amalgamated products

We prove that there is a gap between \(\sqrt 2 and\left( {1 + \sqrt 5 } \right)/2\) for the exponential growth rate of nontrivial free products. For amalgamated products G = A*CB with ([A: C] − 1)([B: C] − 1) ≥ 2, we show that an exponential growth rate lower than \(\sqrt 2 \) can be achieved. Indeed, there are infinitely many amalgamated products for which the exponential growth rate is equal to ψ ≈ 1.325, where ψ is the unique positive root of the polynomial z3−z−1. One of these groups is \(PGL\left( {2,\mathbb{Z}} \right) \cong \left( {{C_2} \times {C_2}} \right){*_{{C_2}}}{D_6}\). However, under some natural conditions the lower bound can be put up to \(\sqrt 2 \). This answers two questions by Avinoam Mann [The growth of free products, Journal of Algebra 326, no. 1 (2011), 208–217]. We also prove that ψ is a lower bound for the minimal growth rates of a large class of Coxeter groups, including cofinite non-cocompact planar hyperbolic groups, which strengthens a result obtained earlier by William Floyd, who considered only standard Coxeter generators.

Neural reconstruction of bone-eating Osedax spp. (Annelida) and evolution of the siboglinid nervous system.

BackgroundBone-devouring Osedax worms were described over a decade ago from deep-sea whale falls. The gutless females (and in one species also the males) have a unique root system that penetrates the bone and nourishes them via endosymbiotic bacteria. Emerging from the bone is a cylindrical trunk, which is enclosed in a transparent tube, that generally gives rise to a plume of four palps (or tentacles). In most Osedax species, dwarf males gather in harems along the female’s trunk and the nervous system of these microscopic forms has been described in detail. Here, the nervous system of bone-eating Osedax forms are described for the first time, allowing for hypotheses on how the abberant ventral brain and nervous system of Siboglinidae may have evolved from a ganglionated nervous system with a dorsal brain, as seen in most extant annelids.ResultsThe intraepidermal nervous systems of four female Osedax spp. and the bone-eating O. priapus male were reconstructed in detail by a combination of immunocytochemistry, CLSM, histology and TEM. They all showed a simple nervous system composed of an anterior ventral brain, connected with anteriorly directed paired palp and gonoduct nerves, and four main pairs of posteriorly directed longitudinal nerves (2 ventral, 2 ventrolateral, 2 sets of dorso-lateral, 2 dorsal). Transverse peripheral nerves surround the trunk, ovisac and root system. The nervous system of Osedax resembles that of other siboglinids, though possibly presenting additional lateral and dorsal longitudinal nerves. It differs from most Sedentaria in the presence of an intraepidermal ventral brain, rather than a subepidermal dorsal brain, and by having an intraepidermal nerve cord with several plexi and up to three main commissures along the elongated trunk, which may comprise two indistinct segments.ConclusionsOsedax shows closer neuroarchitectural resemblance to Vestimentifera + Sclerolinum (= Monilifera) than to Frenulata. The intraepidermal nervous system with widely separated nerve cords, double brain commissures, double palp nerves and other traits found in Osedax can all be traced to represent ancestral states of Siboglinidae. A broader comparison of the nervous system and body regions across Osedax and other siboglinids allows for a reinterpretation of the anterior body region in the group.

Bounds for eigenvalues of matrix polynomials with applications to scalar polynomials

We first generalize to complex matrix polynomials an improvement of an upper bound by Cauchy on the zeros of complex scalar polynomials. The bound requires the unique positive root of a real scalar polynomial of the same degree as the given complex scalar or matrix polynomial. We then create a recursive procedure to represent a matrix polynomial by another matrix polynomial of larger size, but of lower degree. We apply this procedure to scalar polynomials, and then apply the generalized improved Cauchy bound to their matrix polynomial representation, often obtaining a better bound by solving a real scalar polynomial equation of significantly reduced degree.

Semigroup-theoretic approach to identification of linear diffusion coefficients

Let $X$ be a complex Banach space and $A:\,D(A) \to X$ a quasi-$m$-sectorial operator in $X$. This paper is concerned with the identification of diffusion coefficients $\nu > 0$ in the initial-value problem: \[ (d/dt)u(t) + {\nu}Au(t) = 0, \quad t \in (0,T), \quad u(0) = x \in X, \] with additional condition $\|u(T)\| = \rho$, where $\rho >0$ is known. Except for the additional condition, the solution to the initial-value problem is given by $u(t) := e^{-t\,{\nu}A} x \in C([0,T];X) \cap C^{1}((0,T];X)$. Therefore, the identification of $\nu$ is reduced to solving the equation $\|e^{-{\nu}TA}x\| = \rho$. It will be shown that the unique root $\nu = \nu(x,\rho)$ depends on $(x,\rho)$ locally Lipschitz continuously if the datum $(x,\rho)$ fulfills the restriction $\|x\|> \rho$. This extends those results in Mola [6](2011).

The unique axon trajectory of the accessory nerve is determined by intrinsic properties of the neural tube in the avian embryo

The accessory nerve is a cranial nerve, composed of only motor axons, which control neck muscles. Its axons ascend many segments along the lateral surface of the cervical spinal cord and hindbrain. At the level of the first somite, they pass ventrally through the somitic mesoderm into the periphery. The factors governing the unique root trajectory are unknown. Ablation experiments at the accessory nerve outlet points have shown that somites do not regulate the trajectory of the accessory nerve fibres. Factors from the neural tube that may control the longitudinal pathfinding of the accessory nerve fibres were tested by heterotopic transplantations of an occipital neural tube to the cervical and thoracic level. These transplantations resulted in a typical accessory nerve trajectory in the cervical and thoracic spinal cord. In contrast, cervical neural tube grafts were unable to give rise to the typical accessory nerve root pattern when transplanted to occipital level. Our results show that the formation of the unique axon root pattern of the accessory nerve is an intrinsic property of the neural tube.

Local convergence and a chemical application of derivative free root finding methods with one parameter based on interpolation

We present a local convergence analysis of a derivative free fourth order method with one parameter based on rational interpolation in order to approximate a locally unique root of a function. The method is optimal in the sense of Traub. In earlier studies such as Steffensen (Scand Actuar J 16(1):64–72, 1933) and Zafer et al. (Sci World J, 2015. doi:10.1155/2015/934260) the convergence was based on hypotheses on the third derivative or even higher. We extend the applicability of theses methods using only the first derivative. Moreover, we provide computable radii and error bounds based on Lipschitz constants. Furthermore, the dynamics of this method are studied in order to find the best choice of the parameter in terms of convergence. An application is also presented in this study.

Consistency of stochastic approximation algorithm with quasi-associated random errors

ABSTRACTMany mathematical and physical problems are led to find a root of a real function f. This kind of equation is an inverse problem and it is difficult to solve it. Especially in engineering sciences, the analytical expression of the function f is unknown to the experimenter, but it can be measured at each point xk with M(xk) as expected value and induced error ξk. The aim is to approximate the unique root θ under some assumptions on the function f and errors ξk. We use a stochastic approximation algorithm that constructs a sequence (xk)k ⩾ 1. We establish the almost complete convergence of the sequence (xk)k to the exact root θ by considering the errors (ξk)k quasi-associated and we illustrate the method by numerical examples to show its efficiency.

A hierarchy of Banach spaces with C(K) Calkin algebras

For every well founded tree T having a unique root such that every non-maximal node of it has countable infinitely many immediate successors, we construct a L∞-space XT . We prove that for each such tree T , the Calkin algebra of XT is homomorphic to C(T ), the algebra of continuous functions defined on T , equipped with the usual topology. We use this fact to conclude that for every countable compact metric space K there exists a L∞-space whose Calkin algebra is isomorphic, as a Banach algebra, to C(K).

Asymptotic distribution of the numbers of vertices and arcs of the giant strong component in sparse random digraphs

Two models of a random digraph on n vertices, D(n; Prob(arc) = p) and D(n; number of arcs = m) are studied. In 1990, Karp for D(n;p) and independently T. Luczak for D(n;m = cn) proved that for c > 1, with probability tending to 1, there is an unique strong component of size of order n. Karp showed, in fact, that the giant component has likely size asymptotic to n 2 , where = (c) is the unique positive root of 1 = e c . In this paper we prove that, for both random digraphs, the joint distribution of the number of vertices and number of arcs in the giant strong component is asymptotically Gaussian with the same mean vector n (c), (c) := ( 2 ;c 2 ) and two distinct 2 2 covariance matrices, nB(c) and n B(c) +c( 0 (c)) T ( 0 (c))) . To this end, we introduce and analyze a randomized deletion process which determines the directed (1; 1)-core, the maximal digraph with minimum in-degree and out-degree at least 1. This (1; 1)-core contains all nontrivial strong components. However, we show that the likely numbers of peripheral vertices and arcs in the (1; 1)-core, those outside the largest strong component, are of polylog order, thus dwarfed by anticipated uctuations, on the scale of n 1=2 , of the giant component parameters. By approximating the likely realization of the deletion algorithm with a deterministic trajectory, we obtain our main result via exponential supermartingales and Fourier-based techniques.

Analysis of human dentition from Early Bronze Age: 4000-year-old puzzle.

This paper presents the first ever paleodontological investigation of human remains from an archeological site in Central Europe dating from the Early Bronze Age and attributed to the Strzyzow Culture. It corroborates the knowledge gained from archeological, anthropological and genetical investigations. Our study aimed to assess dental status, dental morphology and dental pathologies as well as tooth wear and enamel hypoplasia based on visual inspection and stereomicroscopic investigation. The research was supported by CBCT imaging to obtain digital images and 3D reconstructions as well as 2D radiographs essential for dental age estimation. All of the 191 teeth discovered showed morphological similarity, with adult teeth showing similar color, shape and size. A maxillary molar presenting with a unique root morphology and a mandibular molar with a rare occlusal surface were found. Both permanent and deciduous dentition presented significant tooth wear. A few specimens displayed signs of dental caries, periapical pathology and antemortem tooth loss. Three individuals exhibited linear enamel hypoplasia. CBCT provided high-quality 2D images useful for dental age estimation by non-destructive methods. Estimated dental age correlated with the age estimated by other anthropological methods. In one case, this was crucial because of insufficient material for anthropological analysis. The presented studies have proved that besides the skeleton, teeth can be used as a fundamental tool in assessing the overall health and living conditions of paleopopulations. It would seem that there is potential for considerable development to be made in the research and investigation of paleodontological material using CBCT.

Fredholm representation of multiparameter Gaussian processes with applications to equivalence in law and series expansions

We show that every multiparameter Gaussian process with integrable variance function admits a Wiener integral representation of Fredholm type with respect to the Brownian sheet. The Fredholm kernel in the representation can be constructed as the unique symmetric square root of the covariance. We analyze the equivalence of multiparameter Gaussian processes by using the Fredholm representation and show how to construct series expansions for multiparameter Gaussian processes by using the Fredholm kernel.

HvEXPB7, a novel β-expansin gene revealed by the root hair transcriptome of Tibetan wild barley, improves root hair growth under drought stress

Tibetan wild barley is a treasure trove of useful genes for crop improvement including abiotic stress tolerance, like drought. Root hair of single-celled structures plays an important role in water and nutrition uptake. Polyethylene-glycol-induced drought stress hydroponic/petri-dish experiments were performed, where root hair morphology and transcriptional characteristics of two contrasting Tibetan wild barley genotypes (drought-tolerant XZ5 and drought-sensitive XZ54) and drought-tolerant cv. Tadmor were compared. Drought-induced root hair growth was only observed in XZ5. Thirty-six drought tolerance-associated genes were identified in XZ5, including 16 genes specifically highly expressed in XZ5 but not Tadmor under drought. The full length cDNA of a novel β-expansin gene (HvEXPB7), being the unique root hair development related gene in the identified genes, was cloned. The sequence comparison indicated that HvEXPB7 carried both DPBB_1 and Pollon_allerg_1 domains. HvEXPB7 is predominantly expressed in roots. Subcellular localization verified that HvEXPB7 is located in the plasma membrane. Barley stripe mosaic virus induced gene silencing (BSMV-VIGS) of HvEXPB7 led to severely suppressed root hairs both under control and drought conditions, and significantly reduced K uptake. These findings highlight and confer the significance of HvEXPB7 in root hair growth under drought stress in XZ5, and provide a novel insight into the genetic basis for drought tolerance in Tibetan wild barley.

Banach function norms via Cauchy polynomials and applications

Let X1,…,Xk be quasinormed spaces with quasinorms | ⋅ |j, j = 1,…,k, respectively. For any f = (f1,⋯,fk) ∈ X1 ×⋯× Xk let ρ(f) be the unique non-negative root of the Cauchy polynomial [Formula: see text]. We prove that ρ(⋅) (which in general cannot be expressed by radicals when k ≥ 5) is a quasinorm on X1 ×⋯× Xk, which we call root quasinorm, and we find a characterization of this quasinorm as limit of ratios of consecutive terms of a linear recurrence relation. If X1,…,Xk are normed, Banach or Banach function spaces, then the same construction gives respectively a normed, Banach or a Banach function space. Norms obtained as roots of polynomials are already known in the framework of the variable Lebesgue spaces, in the case of the exponent simple function with values 1,…,k. We investigate the properties of the root quasinorm and we establish a number of inequalities, which come from a rich literature of the past century.

First 3D reconstruction of the rhizocephalan root system using MicroCT

Parasitic barnacles (Cirripedia: Rhizocephala) are highly specialized parasites of crustaceans. Instead of an alimentary tract for feeding they utilize a system of roots, which infiltrates the body of their hosts to absorb nutrients. Using X-ray micro computer tomography (MicroCT) and computer-aided 3D-reconstruction, we document the spatial organization of this root system, the interna, inside the intact host and also demonstrate its use for morphological examinations of the parasites reproductive part, the externa. This is the first 3D visualization of the unique root system of the Rhizocephala in situ, showing how it is related to the inner organs of the host. We investigated the interna from different parasitic barnacles of the family Peltogastridae, which are parasitic on anomuran crustaceans. Rhizocephalan parasites of pagurid hermit crabs and lithodid crabs were analysed in this study.

Volume entropy for minimal presentations of surface groups in all ranks

We study the volume entropy of a class of presentations (including the classical ones) for all surface groups, called minimal geometric presentations. We rediscover a formula first obtained by Cannon and Wagreich (Math Ann 293(2), 239–257, 1992) with the computation in a non published manuscript by Cannon (The growth of the closed surface groups and the compact hyperbolic coxeter groups, 1980). The result is surprising: an explicit polynomial of degree n, the rank of the group, encodes the volume entropy of all classical presentations of surface groups. The approach we use is completely different. It is based on a dynamical system construction following an idea due to Bowen and Series (Inst Hautes Etudes Sci Publ Math 50, 153–170, 1979) and extended to all geometric presentations in Los (J Topol, 7(1), 120–154, 2013). The result is an explicit formula for the volume entropy of minimal presentations for all surface groups, showing a polynomial dependence in the rank \(n>2\). We prove that for a surface group \(G_n\) of rank n with a classical presentation \(P_n\) the volume entropy is \(\log (\lambda _n)\), where \(\lambda _n\) is the unique real root larger than one of the polynomial $$\begin{aligned} x^{n} - 2(n - 1) \sum _{j=1}^{n-1} x^{j} + 1. \end{aligned}$$

Q-Completions of Free Solvable Groups

A group G is said to be complete if, for any natural n and any element g ∈ G, an equation x n = g is solvable in G. If every such equation in the group has at most one solution, then we say that the condition for uniqueness of root extraction is satisfied. A complete group with unique root extraction can be treated as a ℚ Q-power group since it admits an operation of raising an element to any rational power. Let a group G be embedded in a complete group H with unique root extraction, and let H be generated as a ℚ -group by the set G. Then H is called a ℚ -completion of G. We prove that every m-rigid group G is independently embedded in a complete m-rigid group. Under the specified condition for independence of an embedding, the ℚ -completion of the group G in the class of rigid groups is defined uniquely up to G-isomorphism. It is stated that the centralizer of any element of an independent ℚ -completion of a free solvable group, which does not belong to the last nontrivial member of a rigid series of this completion, is isomorphic to the additive group of a field ℚ of rational numbers.

The plastic number and its generalized polynomial

The polynomial $X^{3}-X-1$ has a unique positive root known as plastic number, which is denoted by $\rho$ and is approximately equal to $1.32471795$. In this note we study the zeroes of the generalized polynomial $X^{k}-\sum_{j=0}^{k-2}X^{j}$ for $k\geq 3$ and prove that its unique positive root $\lambda_{k}$ tends to the golden ratio $\phi=\frac{1+\sqrt{5}}{2}$ as $k \to \infty$. We also derive bounds on $\lambda_{k}$ in terms of Fibonacci numbers.