An Explicit Criterion for the Existence of Positive Solutions of the Linear Delayed Equation x

The paper investigates an equation with single delay ẋ(t)=-c(t)x(t-τ(t)), where τ:[t0-r,∞)→(0,r], r>0, t0∈R, and c:[t0-r,∞)→(0,∞) are continuous functions, and the difference t-τ(t) is an increasing function. Its purpose is to derive a new explicit integral criterion for the existence of a positive solution in terms of c and τ. An overview of known relevant criteria is provided, and relevant comparisons are also given

Construction of the General Solution of Planar Linear Discrete Systems with Constant Coefficients and Weak Delay

Planar linear discrete systems with constant coefficients and weak delay are considered. The characteristic equations of such systems are identical with those for the same systems but without delayed terms. In this case, the space of solutions with a given starting dimension is pasted after several steps into a space with dimension less than the starting one. In a sense this situation copies an analogous one known from the theory of linear differential systems with constant coefficients and weak delay when the initially infinite dimensional space of solutions on the initial interval on a reduced interval, turns (after several steps) into a finite dimensional set of solutions. For every possible case, general solutions are constructed and, finally, results on the dimensionality of the space of solutions are deduced

A Final Result on the Oscillation of Solutions of the Linear Discrete Delayed Equation Δ

A linear (k+1)th-order discrete delayed equation Δx(n)=−p(n)x(n−k) where p(n) a positive sequence is considered for n→∞. This equation is known to have a positive solution if the sequence p(n) satisfies an inequality. Our aim is to show that, in the case of the opposite inequality for p(n), all solutions of the equation considered are oscillating for n→∞

New Families of Third-Order Iterative Methods for Finding Multiple Roots

Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong's methods and Neta's method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order

Recent advances in understanding the roles of whole genome duplications in evolution

Ancient whole-genome duplications (WGDs)—paleopolyploidy events—are key to solving Darwin’s ‘abominable mystery’ of how flowering plants evolved and radiated into a rich variety of species. The vertebrates also emerged from their invertebrate ancestors via two WGDs, and genomes of diverse gymnosperm trees, unicellular eukaryotes, invertebrates, fishes, amphibians and even a rodent carry evidence of lineage-specific WGDs. Modern polyploidy is common in eukaryotes, and it can be induced, enabling mechanisms and short-term cost-benefit assessments of polyploidy to be studied experimentally. However, the ancient WGDs can be reconstructed only by comparative genomics: these studies are difficult because the DNA duplicates have been through tens or hundreds of millions of years of gene losses, mutations, and chromosomal rearrangements that culminate in resolution of the polyploid genomes back into diploid ones (rediploidisation). Intriguing asymmetries in patterns of post-WGD gene loss and retention between duplicated sets of chromosomes have been discovered recently, and elaborations of signal transduction systems are lasting legacies from several WGDs. The data imply that simpler signalling pathways in the pre-WGD ancestors were converted via WGDs into multi-stranded parallelised networks. Genetic and biochemical studies in plants, yeasts and vertebrates suggest a paradigm in which different combinations of sister paralogues in the post-WGD regulatory networks are co-regulated under different conditions. In principle, such networks can respond to a wide array of environmental, sensory and hormonal stimuli and integrate them to generate phenotypic variety in cell types and behaviours. Patterns are also being discerned in how the post-WGD signalling networks are reconfigured in human cancers and neurological conditions. It is fascinating to unpick how ancient genomic events impact on complexity, variety and disease in modern life

Oscillation of Solutions of a Linear Second-Order Discrete-Delayed Equation

A linear second-order discrete-delayed equation Δx n −p n x n − 1 with a positive coefficient p is considered for n → ∞. This equation is known to have a positive solution if p fulfils an inequality. The goal of the paper is to show that, in the case of the opposite inequality for p, all solutions of the equation considered are oscillating for n → ∞

Asymptotic Convergence of the Solutions of a Dynamic Equation on Discrete Time Scales

The paper investigates a dynamic equation Δy(tn)=β(tn)[y(tn−j)−y(tn−k)] for n→∞, where k and j are integers such that k>j≥0, on an arbitrary discrete time scale T:={tn} with tn∈ℝ, n∈ℤn0−k∞={n0−k,n0−k+1,…}, n0∈ℕ, tn<tn+1, Δy(tn)=y(tn+1)−y(tn), and limn→∞tn=∞. We assume β:T→(0,∞). It is proved that, for the asymptotic convergence of all solutions, the existence of an increasing and asymptotically convergent solution is sufficient. Therefore, the main attention is paid to the criteria for the existence of an increasing solution asymptotically convergent for n→∞. The results are presented as inequalities for the function β. Examples demonstrate that the criteria obtained are sharp in a sense