Simple Roots Research Articles

Basins of attraction for family of Popovski’s methods and their extension to multiple roots

In this paper we revisit Popovski’s family of methods for simple roots. We compare several members using basins of attraction visually and qualitatively by comparing the run-time on several examples, the average number of iterations and the number of divergent points. We chose 5 different members of the family. We also develop an equivalent family of methods for multiple roots and compare several members on six different numerical examples.

Discriminant and Root Trajectories of Characteristic Equation of Fractional Vibration Equation and Their Effects on Solution Components

The impulsive response of the fractional vibration equation z′′(t)+bDtαz(t)+cz(t)=F(t), b>0,c>0,0≤α≤2, is investigated by using the complex path-integral formula of the inverse Laplace transform. Similar to the integer-order case, the roots of the characteristic equation s2+bsα+c=0 must be considered. It is proved that for any b>0, c>0 and α∈(0,1)∪(1,2), the characteristic equation always has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface. Particular attention is paid to the problem as to how the couple conjugated complex roots approach the two roots of the integer case α=1, especially to the two different real roots in the case of b2−4c>0. On the upper-half complex plane, the root s(α) is investigated as a function of order α and with parameters b and c, and so are the argument θ(α), modulus r(α), real part λ(α) and imaginary part ω(α) of the root s(α). For the three cases of the discriminant b2−4c: >0, =0 and <0, variations of the argument and modulus of the roots according to α are clarified, and the trajectories of the roots are simulated. For the case of b2−4c<0, the trajectories of the roots are further clarified according to the change rates of the argument, real part and imaginary part of root s(α) at α=1. The solution components, i.e., the residue contribution and the Hankel integral contribution to the impulsive response, are distinguished for the three cases of the discriminant.

Twisting functors and Gelfand–Tsetlin modules over semisimple Lie algebras

We associate to an arbitrary positive root [Formula: see text] of a complex semisimple finite-dimensional Lie algebra [Formula: see text] a twisting endofunctor [Formula: see text] of the category of [Formula: see text]-modules. We apply this functor to generalized Verma modules in the category [Formula: see text] and construct a family of [Formula: see text]-Gelfand–Tsetlin modules with finite [Formula: see text]-multiplicities, where [Formula: see text] is a commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] generated by a Cartan subalgebra of [Formula: see text] and by the Casimir element of the [Formula: see text]-subalgebra corresponding to the root [Formula: see text]. This covers classical results of Andersen and Stroppel when [Formula: see text] is a simple root and previous results of the authors in the case when [Formula: see text] is a complex simple Lie algebra and [Formula: see text] is the maximal root of [Formula: see text]. The significance of constructed modules is that they are Gelfand–Tsetlin modules with respect to any commutative [Formula: see text]-subalgebra of the universal enveloping algebra of [Formula: see text] containing [Formula: see text]. Using the Beilinson–Bernstein correspondence we give a geometric realization of these modules together with their explicit description. We also identify a tensor subcategory of the category of [Formula: see text]-Gelfand–Tsetlin modules which contains constructed modules as well as the category [Formula: see text].

Simple and robust multiple shoot regeneration and root induction cycle from different explants of Hypericum perforatum L. genotypes

Hypericum perforatum L. commonly known as Saint John’s Wort (SJW) is an economically important medicinal plant known for accumulating its valuable bioactive compounds in a compartmentalized fashion. The dark glands are very rich in hypericin, and translucent glands are filled with hyperforin. The antibiotic properties of the afore mentioned bioactive compounds make it hard to establish tissue regeneration protocols essential to put in place a transformation platform that is required for testing gene function in this challenging species. In this study, we report the establishment of a regeneration and root induction cycle from different types of explants. The regeneration cycle was set up for the continuous supply of roots and leaf explants for downstream transformation experiments. The most effective medium to obtain multiple shoot-buds from node cultures was MS (Murashige and Skoog, Physiol Plant 15:473–497, 1962) medium supplemented with 0.5 mg L−1 6-Benzylaminopurine (BAP) and 0.5 mg L−1 indole-3-butyric acid (IBA). The same combination yielded copious amounts of shoots from root and leaf explants as well. For rooting the elongated shoots, MS medium devoid of plant growth regulators (PGRs) was sufficient. Nevertheless, addition of a low amount of IBA improved the quantity and quality of roots induced. Additionally, the roots obtained on a medium containing IBA readily developed shoot buds.

Cataclysms for Anosov representations

In this paper, we construct cataclysm deformations for theta -Anosov representations into a semisimple non-compact connected real Lie group G with finite center, where theta subset Delta is a subset of the simple roots that is invariant under the opposition involution. These generalize Thurston’s cataclysms on Teichmüller space and Dreyer’s cataclysms for Borel-Anosov representations into mathrm {PSL}(n, mathbb {R}). We express the deformation also in terms of the boundary map. Furthermore, we show that cataclysm deformations are additive and behave well with respect to composing a representation with a group homomorphism. Finally, we show that the deformation is injective for Hitchin representations, but not in general for theta -Anosov representations.

Common values of a class of linear recurrences

Let (an),(bn) be linear recursive sequences of integers with characteristic polynomials A(X),B(X)∈Z[X] respectively. Assume that A(X) has a dominating and simple real root α, while B(X) has a pair of conjugate complex dominating and simple roots β,β̄. Assume further that α,β,α/β and β̄/β are not roots of unity and δ=log|β|/log|α|∈Q. Then there are effectively computable constants c0,c1>0 such that the inequality |an−bm|>|an|1−(c0log2n)/nholds for all n,m∈Z≥02 with max{n,m}>c1. We present c0 explicitly.

General higher—order breathers and rogue waves in the two‐component long‐wave–short‐wave resonance interaction model

AbstractGeneral higher order breather and rogue wave (RW) solutions to the two‐component long wave–short wave resonance interaction (2‐LSRI) model are derived via the bilinear Kadomtsev–Petviashvili hierarchy reduction method and are given in terms of determinants. Under particular parametric conditions, the breather solutions can reduce to homoclinic orbits, or a mixture of breathers and homoclinic orbits. There are three families of RW solutions, which correspond to a simple root, two simple roots, and a double root of an algebraic equation related to the dimension reduction procedure. The first family of RW solutions consists of bounded fundamental RWs, the second family is composed of bounded fundamental RWs coexisting with another fundamental RWs of different a bounded state ( being positive integers), while the third one has fundamental bounded RWs ( being nonnegative integers). The second family can be regarded as the superpositions of the first family, while the third family can be the degenerate case of the first family under particular parameter choices. These diverse RW patterns are illustrated graphically.

Discovering the Roots: Uniform Closure Results for Algebraic Classes Under Factoring

Newton iteration is an almost 350-year-old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all roots simultaneously. In this form, the process yields better circuit complexity in the case when the number of rootsris small but the multiplicities are exponentially large. Our method sets up a linear system inrunknowns and iteratively builds the roots as formal power series. For an algebraic circuit\( f(x_1,\ldots ,x_n) \)of sizes, we prove that each factor has size at most a polynomial insand the degree of the squarefree part off. Consequently, if\( f_1 \)is a\( 2^{\Omega (n)} \)-hard polynomial, then any nonzero multiple\( \prod _{i} f_i^{e_i} \)is equally hard for arbitrary positive\( e_i \)’s, assuming that\( \sum _i\deg (f_i) \)is at most\( 2^{O(n)} \).It is an old open question whether the class of poly(n) size formulas (respectively, algebraic branching programs) is closed under factoring. We show that given a polynomialfof degree\( n^{O(1)} \)and formula (respectively, algebraic branching program) size\( n^{O(\log n)} \), we can find a similar-size formula (respectively, algebraic branching program) factor in randomized poly(\( n^{\log n} \)) time. Consequently, if the determinant requires an\( n^{\Omega (\log n)} \)size formula, then the same can be said about any of its nonzero multiples.In all of our proofs, we exploit the following property of multivariate polynomial factorization. Under a random linear transformation\( \tau \), the polynomial\( f(\tau \overline{x}) \)completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. Therefore, with the help of the strong mathematical characterizations and the ‘allRootsNI’ technique, we make significant progress towards the old open problems; supplementing the vast body of classical results and concepts in algebraic circuit factorization (e.g., [17,51,54,111]).

Minimal bandwidth for {mathbb C}^*-actions on generalized Grassmannians

The bandwidth of a {mathbb C}^*-action of a polarized pair (X, L) is a natural measure of its complexity. In this paper, we study {mathbb C}^*-actions on rational homogeneous spaces, determining which provide minimal bandwidth. We prove that the minimal bandwidth is linked to the smallest coefficient of the fundamental weight, in a base of simple roots, which describes the variety as a marked Dynkin diagram. As a direct application of the results we study the Chow groups of the Cayley plane mathrm{E}_6(6).

Optimization of winter irrigation management for salinized farmland using a coupled model of soil water flow and crop growth

Drip irrigation under film mulch (DIUFM) and subsurface pipe drainage (SPD) are important measures to cope with water shortages and soil salinization in arid areas of northwest China. To investigate the coordinated operation mode of DIUFM and SPD, a new H2DSWAP model was developed based on coupling the HYDRUS-2D with a Soil–Water–Atmosphere–Plant model (SWAP). In the H2DSWAP model, real-time evapotranspiration, simple root growth, and the interaction between crops and soil water and salt are considered to improve the simulation accuracy. The model was calibrated and validated using parameter estimation and uncertainty analysis software (PEST) using field experiment datasets in 2019 and 2020 respectively. Compared with the original HYDRUS-2D model, the simulation accuracy of the H2DSWAP for soil water (root mean square error (RMSE) = 0.011 cm3·cm−3; mean absolute error (MAE) = 0.008 cm3·cm−3; determinant coefficient (R2) = 0.869) and soil salt (RMSE = 0.296 g·kg−1; MAE = 0.231 g·kg−1; R2 = 0.959) contents has been greatly improved. In addition, the simulation of the leaf area index (LAI) and yield also fitted well with field observations. The calibrated model was used to predict salt transport at depths of 0–100 cm and the change in maize yield under DIUFM. The results indicated that maize yield decreased yearly, and soil salinity increased yearly under DIUFM without SPD. Based on the obtained results, the management strategies of winter irrigation (WIR) under different drip irrigation water amounts, namely 600 mm (S1), 540 mm (S2), 480 mm (S3), and 420 mm (S4), were further investigated. Water productivity (WP) and yield were used as evaluation indices. The WIR was carried out every 4–5 years when the drip irrigation amount of the maize field was S1 and S2 was recommended. However, it was recommended to carry out WIR every 2 and 3 years under S3 and S4, respectively. Overall, the H2DSWAP model can be used as a useful tool to guide the operation mode of drip irrigation under mulch and subsurface pipe drainage in saline soils.

Polynomial representations of classical Lie algebras and flag varieties

Recently we have started a program to describe the action of Lie algebras associated with Dynkin-type diagrams on generic Verma modules in terms of polynomial vector fields. In this paper we explain that the results for the classical ABCD series of Lie algebras coincide with the more conventional approach, based on the knowledge of the entire algebra, not only the simple roots. We apply the coset description, starting with a large representation and then reducing it with the help of the algebra, commuting with the original one. The irreducible representations are then obtained by gauge fixing this residual symmetry.

Quantized nilradicals of parabolic subalgebras of and algebras of coinvariants

Let PJ be the standard parabolic subgroup of SLn associated to a subset J of simple roots, and let be the standard Levi decomposition. We let and denote the quantized algebras of regular functions on PJ and LJ , respectively. Following work of the first author, we study the quantum analogue of an induced coaction and the corresponding subalgebra of coinvariants. It was previously shown that the smash product algebra is isomorphic to In view of this, – while it is not a Hopf algebra – can be viewed as a quantum analogue of the coordinate ring In this article we prove that when is nonzero and not a root of unity, is isomorphic to a quantum Schubert cell algebra associated to a certain parabolic element w in the Weyl group of In this setting, the quantum Schubert cell algebra is a q-deformation of the universal enveloping algebra of the nilradical of a parabolic subalgebra of We also compute explicit commutation relations among the Lusztig root vectors for these particular quantized nilradicals and we give explicit algebra isomorphisms from to

Asymptotic solutions of boundary value problem for singularly perturbed system of differential-algebraic equations

This paper deals with the boundary value problem for a singularly perturbed system of differential algebraic equations of the second order. The case of simple roots of the characteristic equation is studied. The sufficient conditions for existence and uniqueness of a solution of the boundary value problem for system of differential algebraic equations are found. Technique of constructing the asymptotic solutions is developed.

Para-orthogonal polynomials on the unit circle generated by Kronecker polynomials

The Kronecker polynomialK(z)K(z)is a finite product of cyclotomic polynomialsCj(z)C_j(z). Any Kronecker polynomialK(z)K(z)of degreeN+1N+1with simple roots on the unit circle generates a finite setΦ0=1\Phi _0=1,Φ1(z)\Phi _1(z), …,ΦN(z)\Phi _N(z)of polynomials (para) orthogonal on the unit circle (POPUC). This set is determined uniquely by the conditionΦN(z)=(N+1)−1K′(z)\Phi _N(z) = (N+1)^{-1} K’(z). Such set can be called the set of Sturmian Kronecker POPUC. We present several new explicit examples of such POPUC. In particular, we define and analyze properties of the Sturmian cyclotomic POPUC generated by the cyclotomic polynomialsCM(z)C_M(z). Expressions of these polynomials strongly depend on the decomposition ofMMinto prime factors.

Ribozyme-mediated CRISPR/Cas9 gene editing in pyrethrum (Tanacetum cinerariifolium) hairy roots using a RNA polymerase II-dependent promoter

BackgroundTraditional CRISPR/Cas9 systems that rely on U6 or U3 snRNA promoters (RNA polymerase III-dependent promoters) can only achieve constitutive gene editing in plants, hampering the functional analysis of specifically expressed genes. Ribozyme-mediated CRISPR/Cas9 systems increase the types of promoters which can be used to transcribe sgRNA. Therefore, such systems allow specific gene editing; for example, transcription of the artificial gene Ribozyme-sgRNA-Ribozyme (RGR) is initiated by an RNA polymerase II-dependent promoter. Genetic transformation is indispensable for editing plant genes. In certain plant species, including pyrethrum, genetic transformation remains challenging to do, limiting the functional verification of novel CRISPR/Cas9 systems. Thus, this study’s aim was to develop a simple Agrobacterium rhizogenes-mediated hairy root transformation system to analyze the function of a ribozyme-mediated CRISPR/Cas9 system in pyrethrum.ResultsA hairy root transformation system for pyrethrum is described, with a mean transformation frequency of 7%. Transgenic hairy roots transformed with the pBI121 vector exhibited significantly increased beta-glucuronidase staining as a visual marker of transgene expression. Further, a ribozyme-based CRISPR/Cas9 vector was constructed to edit the TcEbFS gene, which catalyzes synthesis of the defense-related compound (E)-β-farnesene in pyrethrum. The vector was transferred into the hairy roots of pyrethrum and two stably transformed hairy root transgenic lines obtained. Editing of the TcEbFS gene in the hairy roots was evaluated by gene sequencing, demonstrating that both hairy root transgenic lines had DNA base loss at the editing target site. Gas chromatography–mass spectrometry showed that the (E)-β-farnesene content was significantly decreased in both hairy root transgenic lines compared with the empty vector control group. Altogether, these results show that RGR can be driven by the CaMV35S promoter to realize TcEbFS gene editing in pyrethrum hairy roots.ConclusionAn A. rhizogenes-mediated hairy root transformation and ribozyme-mediated CRISPR/Cas9 gene editing system in pyrethrum was established, thereby facilitating gene editing in specific organs or at a particular developmental stage in future pyrethrum research.

Classification of radial Kerr geodesic motion

We classify radial timelike geodesic motion of the exterior non-extremal Kerr spacetime by performing a taxonomy of inequivalent root structures of the first order radial geodesic equation using a novel compact notation and by implementing the constraints from polar, time and azimuthal motion. Four generic root structures with only simple roots give rise to eight non-generic root structures when either one root becomes coincident with the horizon, one root vanishes or two roots becomes coincident. We derive the explicit phase space of all such root systems in the basis of energy, angular momentum and Carter's constant and classify whether each corresponding radial geodesic motion is allowed or disallowed from existence of polar, time and azimuthal motion. The classification of radial motion within the ergoregion for both positive and negative energies leads to 6 distinguished values of the Kerr angular momentum. The classification of null radial motion and near-horizon extremal Kerr radial motion are obtained as limiting cases and compared with the literature. We explicitly parametrize the separatrix describing root systems with double roots as the union of the following three regions that are described by the same quartic respectively obtained when (1) the pericenter of bound motion becomes a double root; (2) the eccentricity of bound motion becomes zero; (3) the turning point of unbound motion becomes a double root.

An Efficient Root Transformation System for Recalcitrant Vicia sativa.

Common vetch (Vicia sativa) is a multi-purpose legume widely used in pasture and crop rotation systems. Vetch seeds have desirable nutritional characteristics and are often used to feed ruminant animals. Although transcriptomes are available for vetch, problems with genetic transformation and plant regeneration hinder functional gene studies in this legume species. Therefore, the aim of this study was to develop a simple, efficient and rapid hairy root transformation system for common vetch to facilitate functional gene analysis. At first, we infected the hypocotyls of 5-day-old in vitro or in vivo, soil-grown seedlings with Rhizobium rhizogenes K599 using a stabbing method and produced transgenic hairy roots after 24 days at 19 and 50% efficiency, respectively. We later improved the hairy root transformation in vitro by infecting different explants (seedling, hypocotyl-epicotyl, and shoot) with R. rhizogenes. We observed hairy root formation at the highest efficiency in shoot and hypocotyl-epicotyl explants with 100 and 93% efficiency, respectively. In both cases, an average of four hairy roots per explant were obtained, and about 73 and 91% of hairy roots from shoot and hypocotyl-epicotyl, respectively, showed stable expression of a co-transformed marker β-glucuronidase (GUS). In summary, we developed a rapid, highly efficient, hairy root transformation method by using R. rhizogenes on vetch explants, which could facilitate functional gene analysis in common vetch.

Continuum Kac–Moody Algebras

We introduce a new class of infinite-dimensional Lie algebras, which we refer to as continuum Kac-Moody algebras. Their construction is closely related to that of usual Kac-Moody algebras, but they feature a continuum root system with no simple roots. Their Cartan datum encodes the topology of a one-dimensional real space and can be thought of as a generalization of a quiver, where vertices are replaced by connected intervals. For these Lie algebras, we prove an analogue of the Gabber-Kac-Serre theorem, providing a complete set of defining relations featuring only quadratic Serre relations. Moreover, we provide an alternative realization as continuum colimits of symmetric Borcherds-Kac-Moody algebras with at most isotropic simple roots. The approach we follow deeply relies on the more general notion of a semigroup Lie algebra and its structural properties.

The Inner Aggregation Newton’s Method for Solving Nonlinear Equations

This paper deals with the new variants of Newton’s method based on aggregation functions for finding simple real roots of nonlinear equations. Unlike some well-known two step modifications of Newton’s method based on various means, the presented methods use the same means in a different manner to achieve the third convergence order. A new general iterative scheme is analyzed in details and the theoretical results are verified on several test examples from real-life and literature.

АЛГЕБРАИЧЕСКИЕ УРАВНЕНИЯ В УНИТАРНОМ ПРОСТРАНСТВЕ И КРАТЧАЙШЕЕ АЛГЕБРАИЧЕСКОЕ ДОКАЗАТЕЛЬСТВО ОСНОВНОЙ ТЕОРЕМЫ АЛГЕБРЫ

The article deals with the presentation of some results obtained in the study of algebraic equations with complex coefficients from a complex variable in a unitary space. In the orthonormal basis, two vectors are introduced, which are called the vector of root and the vector of coefficients of an algebraic polynomial. With the help of these vectors, an algebraic polynomial is represented as a scalar product of them in an orthonormal basis. The criterion of linear independence of a set of root vectors is formulated and proved. A Theorem is formulated and proved that the maximum number of simple roots of an algebraic polynomial is one less than the dimension of a unitary space. A generalization of Vieta’s Theorem is obtained and new formulas connecting the coefficients of an algebraic polynomial with its roots are derived. The general case of an algebraic polynomial is considered, when some of its coefficients may be equal to zero. Two orthogonal vector systems from combinations of coefficients of an algebraic polynomial are introduced. Their properties have been studied. The shortest algebraic proof of the fundamental Theorem of algebra among the known proofs is given, which does not go beyond the concepts of the algebra of polynomials and uses the scalar product of vectors in a unitary space, as well as one property of a two-place predicate from mathematical logic. The results of this work can also be used in the educational process in the algebra course.