An Extension of Sharkovsky’s Theorem to Periodic Difference Equations

We present an extension of Sharkovsky’s Theorem and its converse to periodic difference equations. In addition, we provide a simple method for constructing a p-periodic difference equation having an r-periodic geometric cycle with or without stability properties

The stability of collocation methods for VIDEs of second order

Simplest results presented here are the stability criteria of collocation methods for the second-order Volterra integro differential equation (VIDE) by polynomial spline functions. The polynomial spline collocation method is stable if all eigenvalues of a matrix are in the unit disk and all eigenvalues with |λ|=1 belong to a 1×1 Jordan block. Also many other conditions are derived depending upon the choice of collocation parameters used in the solution procedure

The stability of collocation methods for higher-order Volterra integro-differential equations

The numerical stability of the polynomial spline collocation method for general Volterra integro-differential equation is being considered. The convergence and stability of the newmethod are given and the efficiency of the newmethod is illustrated by examples. We also proved the conjecture suggested by Danciu in 1997 on the stability of the polynomial spline collocation method for the higher-order integro-differential equations. 1

Limit cycles for successive projections onto hyperplanes in Rn

AbstractIn this paper we consider successive orthogonal projections onto m hyperplanes in Rn, where m ⩾ 2 and n ⩾ 2. A limit cycle is defined to be a sequence of points formed by projecting onto each of the hyperplanes once in a prescribed order, with the last projection giving the starting point. Several examples, including triangles, quadrilaterals, regular polygons, and arbitrary collections of lines in R2, are solved for the limit cycle. Limit cycles are found in various ways, including by a limiting process and by solving an mn × mn linear system of equations. The latter approach will produce all the limit cycles for an arbitrary ordered set of m hyperplanes in Rn

Unravelling the synergistic interaction of Thrips tabaci and newly recorded, Thrips parvispinus with Alternaria porri (Ellis.) Cif., inciting onion purple blotch

Onion purple blotch is the most indispensable foliar disease of crop and has become a major concern for farmers and research fraternity. An attempt to investigate the role of injury in parasitism by Alternaria porri indicated that disease incidence and severity enhance considerably with injury. Thrips injured plants inoculated with A. porri presented 100% incidence and 52–72% severity while mechanically injured plants inoculated with A. porri showed 60–70% incidence and 28–34% severity. The uninjured plants showed considerably less disease incidence (30–40%) and severity (10–16%). Injured inoculated plants presented reduced leaf length and leaf area while the leaf diameter remained unaffected. The lesion number, lesion length and size was substantially enhanced with concomitant infestation of pest and pathogen. Thrips tabaci injury led to more pronounced symptoms of purple blotch compared to Thrips parvispinus injury. There was substantial decrease in photosynthetic rate and chlorophyll content with stress imposed on plant whilst the relative stress injury was enhanced. The induction of injury and inoculation of A. porri had an impact on the concentration of total phenolics, total soluble sugars, total proteins and hydrogen peroxide in onion leaves. A. porri combined with injury caused a more pronounced decrease in total soluble sugars and total protein content while enhancement in total phenolics and hydrogen peroxide content compared to uninjured plants. The dynamic nature of morpho-physiological and biochemical changes owing to stress conditions imposed on onion plant adds an extra layer of complexity in understanding the onion plant physiology and their ability to work out in response to challenging environment conditions

Data_Sheet_1_Unravelling the synergistic interaction of Thrips tabaci and newly recorded, Thrips parvispinus with Alternaria porri (Ellis.) Cif., inciting onion purple blotch.docx

Onion purple blotch is the most indispensable foliar disease of crop and has become a major concern for farmers and research fraternity. An attempt to investigate the role of injury in parasitism by Alternaria porri indicated that disease incidence and severity enhance considerably with injury. Thrips injured plants inoculated with A. porri presented 100% incidence and 52–72% severity while mechanically injured plants inoculated with A. porri showed 60–70% incidence and 28–34% severity. The uninjured plants showed considerably less disease incidence (30–40%) and severity (10–16%). Injured inoculated plants presented reduced leaf length and leaf area while the leaf diameter remained unaffected. The lesion number, lesion length and size was substantially enhanced with concomitant infestation of pest and pathogen. Thrips tabaci injury led to more pronounced symptoms of purple blotch compared to Thrips parvispinus injury. There was substantial decrease in photosynthetic rate and chlorophyll content with stress imposed on plant whilst the relative stress injury was enhanced. The induction of injury and inoculation of A. porri had an impact on the concentration of total phenolics, total soluble sugars, total proteins and hydrogen peroxide in onion leaves. A. porri combined with injury caused a more pronounced decrease in total soluble sugars and total protein content while enhancement in total phenolics and hydrogen peroxide content compared to uninjured plants. The dynamic nature of morpho-physiological and biochemical changes owing to stress conditions imposed on onion plant adds an extra layer of complexity in understanding the onion plant physiology and their ability to work out in response to challenging environment conditions.</p