Abstract: In this paper, we prove and discuss some new $\left( H_{p},weak-L_{p}\right) $ type inequalities of maximal operators of $T$ means with respect to Vilenkin systems with monotone coefficients. We also apply these results to prove a.e. convergence of \ such $T$ means. It is also proved that these results are the best possible in a special sense. As applications, both some well-known and new results are pointed out.

Abstract: This paper is concerned with the oscillatory behavior of first-order linear differential equations with variable non-monotone deviating arguments and nonnegative coefficients. Corresponding differential equations of both delay and advanced type are studied. Sufficient oscillation conditions involving $% \lim \sup $ and $\lim \inf $ are obtained by applying an iterated method. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained conditions over known ones.

Abstract: Present paper deals with a nutrient-phytoplankton-zooplankton model with viral disease in phytoplankton species. We have worked out some threshold conditions for extinction of the species. We have derived out the condition for existence of equilibrium points and their stability. We have explained the basic reproduction number and we analyzed the the community structure by the help of these basic repro- duction number. We also derived the condition of Hobf Bifurcationand persistence of the system. To observe the global dynamics wehave performed the extensive numerical simulations. Our Simulation results shows that viral infection is responsible for limit cycle plankton oscillations. It is also observed that nutrient may play important rolefor controlling diseased induced plankton osciilations. Our numerical studies clearly indicate that viral infection is responsible for zooplank-ton survival as well as plankton oscillation dynamics. It is also implied that there a nutrient threshold is required for plankton oscillations as well as zooplankton survival. Nutrient threshold value is changed bynthe variation of viral infection in phytoplankton population.

Abstract: In this paper, we use the Banach contraction mapping principle and the Krasnoselskii fixed point theorem to obtain the existence and uniqueness of solutions for nonlinear retarded and advanced implicit Hadamard fractional differential equations with nonlocal conditions. The results obtained here extend the work of Benchohra, Bouriah and Henderson ben . Two examples are also given to illustrate the results.

Abstract: In this paper, we have formulated and analyzed a mathematical model describing the dynamics of the phytoplankton producing toxin and the fish population by using an ordinary differential equation system. The phytoplankton population is divided into two groups, namely infected phytoplankton, and susceptible phytoplankton. We aim to analyze the effect of the toxic substance on the fish population. The equilibria stability of the model has been studied locally and globally around the basic reproduction ratio R0. The mathematical analysis of the model shows that the equilibrium without disease is globally asymptotically stable if R0 >1 and the endemic equilibrium is globally asymptotically stable if R0 > 1: Numerical simulations arecarried out to illustrate the feasibility of the theoretical results.

Abstract: This article deals with a mathematical model on cervical cancer dynamics at cellular level to describe the interactions between Cancerous cell, Natural killer cell (NK), Effector T cell and Human Papilloma virus. Our body immune system is capable to kill the cancer cells and Natural killer (NK), Effector T cells, are ultimately responsible for eradicating cervical cancer which is entirely induced by a virus, the Human Papillomavirus (HPV). The qualitative analysis of the important parameters and the stability of each possible steady states is described in this article. The theoretical and numerical outcomes have been supported through experimental data from different literatures. Furthermore, we have used autologous immune enhancement therapy (AIET) as an optimal control strategy to enhance the power of NK cells and Effector T cells which helps to eradicate the cancer more quickly.

Abstract: In this paper, we generalize the Euler method for dynamic equations on general time scales. We give estimates for the local and global truncation errors. We apply the method to specimen examples and provide numerical results to support our theoretical discussion.

Abstract: We prove the existence and uniqueness of global solutions by Faedo-Galerkin method for the Cauchy problem concerning the evolution equation $$ u_{tt} + \Delta^2 u - \Phi( \| abla u \|^2_2 ) \Delta u - \mathrm{div}( a(x) | abla u_t |^{p-2} abla u_t) =0,$$ suggested by the study of plates and beams, where $p \geq 2$ and $\Phi$ is a real function. We also investigate the asymptotic behavior of the solutions, under suitable growth assumptions. We will use for this task an appropriate perturbed energy coupled with multiplier technique.

Abstract: The question of a possible excitation and emergence of fractional type dynamics, as a more realistic framework for understanding emergence of complex systems, directly from a conventional integral order dynamics, in the form a continuous transition or deformation, is of significant interest. Although there have been a lot of activities in nonlinear, fractional or not, dynamical systems, the above question appears yet to be addressed systematically in the current literature. The present work may be considered to be a step forward in this direction. Based on a novel concept of asymptotic duality structure, we present here an extended analytical framework that would provide a scenario for realizing the above stated continuous deformation of integral order dynamics to a local fractional order dynamics on a fractal and fractional space. The related concepts of self dual and strictly dual asymptotics are introduced and there relevance in connection with smooth and nonsmooth deformation of the real line are pointed out. The relationship of the duality structure and renormalization group is examined. The ordinary derivation operator is shown to be invariant under this duality enabled renormalization group transformation, leading thereby to a {\em natural} realization of local fractional type derivative in a fractal space. As an application we discuss linear wave equation in one and two dimensions and show how the underlying integral order wave equation could be deformed and renormalized suitably to yield meaningful results for vibration of a fractal string or wave propagation in a region with fractal boundary.

Abstract: We determine the expressions of the directional derivatives $\xi_{ij}\xi_{pq}$ regarding to the special tangential vector fields $\xi_{ij}$ on the shape space $\Sigma_{m}^{k}$ introduced by D.G. Kendall. The latter helps to associate a geometrical shape to arbitrary subsets of matrices in $\mathbb{R}^{m\times k}$ through the elimination of the effects of basic geometrical transformations. Then, the calculus that we need to perform here manipulates the inner products of the horizontal liftsof the tangential vectors $\xi_{ij}$ since the shape space $\Sigma_{m}^{k}$ coincides with the quotient space of the unit hypersphere modulo specific special orthogonal group.

Abstract: Estimation of Errors by approximating signals (functions) of different classes has been considered by various researchers under different summability means In the present paper, presumably, a new theorem has been established under $(\overline{N},p_{n},q_{n})(E,s)$-product summability mean of conjugate Fourier series of a function of $Lip(\alpha,r)$-class Moreover, the result obtained here is a generalization of several known theorems

Abstract: Search for alternative antibacterial therapies has been intensified in last fifty years This search has become inevitable because many of the well-known antibiotics of yesteryears are no longer effective againstthe resistant pathogens today Bdellovibrio and similar bacterial species (BLO; Bdellovibriolike organisms) capable of predating on other bacteria, specifically gram-negative pathogens arepromising candidates for anti-bacterial therapy However, most experimental studies on predatory behavior of BLOs are carried out in simple lab simulated conditions and even their mathematical modelling also adopted similar simplifications This paper proposes a set of nonlinear stage-structured differential equations under the influence of discreteconstant delay and white noise to describe predatory behavior of BLOs It has been observed by previous authors that even in a simple liquid culture the mature BLO cell growth exhibit some delay and involve a so-called non-virulent phase Our analysis revealed, when the BLOs are exposed to few tens of prey cells in a tissuethe delay parameter & white noise has a great role for dynamical behavior of theBLO population andlogistic growth rate of the preyFurther it represents microbial ecosystem dynamics together with the interactions between ecosystems and cellular as well as environmental noise Here, the local and global stability analysisof the model with and without delay has been discussed The stochastic perturbation analysiswas carried out and qualitatively compared with delay analysis Finally, all three populations viz, mature and immature BLO plus the prey were studied as diffusive species to account

Abstract: In the present work, our thought process is to investigate the effect of harvesting and refugia on the dynamics of a continuous-time tri-trophic food chain model. To peruse these features we have explored the local stability behavior of various equilibrium points. Conditions for Hopf-bifurcation and persistence have been inferred. Extensive numerical simulation work has been performed to reveal the dynamics of the system. Simulation results exhibit the chaotic dynamics of the system when the value of the half-saturation constant is increased. Further, it is established that the chaotic behavior is controlled by increasing the harvesting parameter value. Again, the chaotic behavior is observed to be controlled by increasing the value of the refugia parameter. Thus we infer that harvesting and refugia parameters can be used to restrain the chaotic dynamics of the model system.

Abstract: In this paper, we investigate exponential stabilization problem for a class of linear systems with two continuous variable delays such that they belong to given intervals, but not necessary to be differentiable. By aid of a suitable Lyapunov-Krasovskii functional, Leibniz-Newton’s formula and linear matrix inequalities (LMIs), we derive some new sufficient conditions for the exponential stability of the zero solution of the considered system.

Abstract: The present paper explores a tri-trophic food chain model with alternative food source for prey species and harvesting in prey and intermediate predator. We perform the local dynamical analysis of the model system. We also derive conditions for Turing instability in the diffusive model. To investigate the global dynamics we have performed extensive numerical experiments in the both non-diffusive and diffusive system. We find that harvesting has an important role in preventing chaotic dynamics and sustaining the ecosystem. It is noticed that alternative food also has a stabilizing effect on chaotic dynamics and leads a stable dynamics through chaos, period-doubling and limit cycles.Thus both the harvesting parameter and alternative food may be considered as bio-controlling parameters for controlling chaotic dynamics in three species food chain model. In the diffusive system we observed that the harvesting parameter and alternative food may cause Turing instability.

Abstract: Every countable language which conforms to classical logic is shown to have an extension which conforms to classical logic, and has a definitional theory of truth. That extension has a semantical theory of truth, if every sentence of theobject language is valuated by its meaning either as true or as false. These theories contain both a truth predicate and a non-truth predicate. Theories are equivalent when the sentences of the object language are valuated by their meanings.

Abstract: Algal blooms are a sudden increase in the population of algae in aquatic eco-system The harmful algal blooms can make water toxic and hence can cause sufferings to humans and fish populations So the control of algal bloom is very much desirable In this paper, a four-species food-chain model for the biological control of algal bloom is proposed and analyzed Here we consider algae population, zooplankton, small fish and large fish linked as a food chain The large fish acts as a top predator and small fish & zooplankton as middle predator The algae population is serving as prey The formulation of this mathematical model is based on the assumption that fish population is very much effective in regulating the dynamics of algal bloom and the harvesting of large predatory fish is linked with the growth of algae population The equilibria of the proposed model are found and local stability of these equilibria are discussed in detail Numerical simulation is performed to visualize the impact of harvesting rate constant of the large predatory fish and carrying capacity of algae population on algal blooms

Abstract: The aim of this paper is to study general and abstract control systems with incomplete data Actually, we introduce the notions of averaged no-regret control and its approximation, the averaged low-regret control to get a full characterization for the optimal control via an optimality system As an example, we apply the described theory on an optimal control problem for an equation of vibrating thin plate depending on an uncertainty parameter and with missing initial conditions where optimality systems characterizing the averaged no-regret control and the averaged low-regret control are given