THE IMPACT OF TOXICANTS IN THE MARINE THREE ECOLOGICAL FOOD-CHAIN ENVIRONMENT: A MATHEMATICAL APPROACH

To explore the impact of toxicants on a marine ecological food chain system consisting of three species, this work develops and analyzes a non-linear mathematical model. The model consists of five state variables: phytoplankton, zooplankton, fish, environmental toxicant, and organismal toxicant. We have incorporated the Monod-Haldane functional response as a predation function for each species. Using the Jacobian matrix, the stability analysis was conducted, and necessary constraints were obtained for the system's local and global stability. Hopf bifurcation analysis was performed for carrying capacity ($K$) and the rate of decrease in the growth rate of phytoplankton due to the presence of toxicants ($r_1$). Also, phase portraits are presented for different parameters of the model. In addition, numerical simulations are executed using MATLAB to prove theoretical findings and explore the impact of parameter variation on ecological species behavior

STABILITY OF GENERAL QUADRATIC EULER–LAGRANGE FUNCTIONAL EQUATIONS IN MODULAR SPACES: A FIXED POINT APPROACH

In this paper, we establish a result on the  Hyers–Ulam–Rassias stability of the Euler–Lagrange functional equation. The work presented here is in the framework of modular spaces. We obtain our results by applying a fixed point theorem. Moreover, we do not use the $\Delta_\alpha$-condition of modular spaces in the proofs of our theorems, which introduces additional complications in establishing stability. We also provide some corollaries and an illustrative example. Apart from its main objective of obtaining a stability result, the present paper also demonstrates how fixed point methods are applicable in modular spaces

TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS

In the present paper, we investigate 2-local linear operators on vector spaces. Sufficient conditions are obtained for the linearity of a 2-local linear operator on a finite-dimensional vector space. To do this, families of matrices of a certain type are selected and it is proved that every 2-local linear operator generated by these families is a linear operator. Based on these results we prove that each 2-local derivation of a finite-dimensional null-filiform Zinbiel algebra is a derivation. Also, we develop a method of construction of 2-local linear operators which are not linear operators. To this end, we select matrices of a certain type and using these matrices we construct a 2-local linear operator. If these matrices are distinct, then the 2-local linear operator constructed using these matrices is not a linear operator. Applying this method we prove that each finite-dimensional filiform Zinbiel algebra has a 2-local derivation that is not a derivation. We also prove that each finite-dimensional naturally graded quasi-filiform Leibniz algebras of type I has a 2-local automorphism that is not an automorphism

TOPOLOGIES ON THE FUNCTION SPACE YXY^X WITH VALUES IN A TOPOLOGICAL GROUP

Let $Y^X$ denote the set of all functions from $X$ to $Y$. When $Y$ is a topological space, various topologies can be defined on $Y^X$. In this paper, we study these topologies within the framework of function spaces. To characterize different topologies and their properties, we employ generalized open sets in the topological space $Y$. This approach also applies to the set of all continuous functions from $X$ to $Y$, denoted by $C(X,Y)$, particularly when $Y$ is a topological group. In investigating various topologies on both $Y^X$ and $C(X,Y)$, the concept of limit points plays a crucial role. The notion of a topological ideal provides a useful tool for defining limit points in such spaces. Thus, we utilize topological ideals to study the properties and consequences for function spaces and topological groups

ATTRACTION SETS IN ATTAINABILITY PROBLEMS WITH ASYMPTOTIC-TYPE CONSTRAINTS

In control theory, the problem of constructing and investigating attainability domains is very important. However, under perturbations of constraints, this problem lacks stability. It is useful to single out the case when the constraints are relaxed. In this case, greater opportunities arise in terms of attainability, and often a useful effect can be observed even under slight relaxation of the constraints. This situation is analogous to the duality gap in convex programming. Very often, it is not possible to specify in advance how much relaxation of the constraints will occur. Therefore, attention is focused on the limit of the attainability domains under unrestricted tightening of the relaxed conditions. As a result, a certain attainability problem with asymptotic-type constraints arises. This problem formulation can be significantly generalized. Namely, we do not consider any unperturbed conditions at all and instead pose asymptotic-type constraints directly by means of a nonempty family of sets in the space of ordinary controls. Moreover, not only the case of control problems can be considered. In this general formulation, an analogue of the limit of attainability domains naturally appears as the relaxed conditions are infinitely tightened. For asymptotic constraints of this kind, we introduce solutions which are, at the conceptual level, similar to the approximate solutions of J.Warga, but we use filters or directedness, and not just sequences of ordinary solutions (controls). We investigate the most general attainability problem, in which asymptotic-type constraints can be generated by any nonempty family of sets in the ordinary solution space. It is shown, however, that the most practically interesting case is realized by filters, and the role of ultrafilters is noted as well. The action of constraints is associated with sets and elements of attraction. Furthermore, some properties of the family of all attraction sets are investigated.

A PAIR OF FOUR-ELEMENT HORIZONTAL GENERATING SETS OF A PARTITION LATTICE

Let $\lfloor x \rfloor$ and $\lceil x\rceil$ denote the lower integer part and the upper integer part of a real number $x$, respectively. Our main goal is to construct four partitions of a finite set $A$ with $n\geq 7$ elements such that each of the four partitions has exactly $\lceil n/2\rceil$  blocks and any other partition of $A$ can be obtained from the given four by forming joins and meets in a finite number of steps. We do the same with $\lceil n/2\rceil-1$ instead of $\lceil n/2\rceil$, too. To situate the paper within lattice theory, recall that the partition lattice $\mathrm{Eq}(A)$ of a set $A$ consists of all partitions (equivalently, of all equivalence relations) of $A$. For a natural number $n$, $[n]$ and $\mathrm{Eq}(n)$ will stand for $\{1,2,\dots,n\}$ and $\mathrm{Eq}([n])$, respectively. In 1975, Heinrich Strietz proved that, for any natural number $n\geq 3$, $\mathrm{Eq} (n)$ has a four-element generating set; half a dozen papers have been devoted to four-element generating sets of partition lattices since then. We give a simple proof of his just-mentioned result. We call a generating set $X$ of $\mathrm{Eq}(n)$ horizontal if each member of $X$ has the same height, denoted by $h(X)$, in $\mathrm{Eq} (n)$; no such generating sets have been known previously. We prove that for each natural number $n\ge 4$, $\mathrm{Eq}(n)$ has two four-element horizontal generating sets $X$ and $Y$ such that $h(Y)=h(X) +1$; for $n\geq 7$, $h(X)= \lfloor n/2 \rfloor$

ASYMPTOTIC BEHAVIOR FOR THE LOTKA–VOLTERRA EQUATION WITH DISPLACEMENTS AND DIFFUSION

In this paper, we consider the Lotka–Volterra equation with displacements and diffusion, that is transport-diffusion system describing the evolution of prey and predator populations with their displacements and their diffusion, in a periodic domain in $\mathbb{R}$. It is shown that the solution to this equation and its logarithm are globally bounded, and that, when the solution converges to the stationary solution in mean value, it converges uniformly with respect to the time variable as well as the space variable. These results are obtained by using $L^2$-estimate of the well-known Lyapunov functional, and, in particular, an estimate of the point-wise growth of the solution by means of the application of the fundamental solution of the heat equation

A STUDY ON PERFECT ITALIAN DOMINATION OF GRAPHS AND THEIR COMPLEMENTS

Perfect Italian Domination is a type of vertex domination  which can also be viewed as a graph labelling problem. The vertices of a graph $G$ are labelled by 0, 1 or 2 in such a way that a vertex labelled 0 should have a neighbourhood with exactly two vertices in it labelled 1 each or with exactly one vertex labelled 2. The remaining vertices in the neighbourhood of the vertex labelled 0 should be all 0's. The minimum sum of all labels of the graph G  satisfying these conditions is called its Perfect Italian domination number. We study the behaviour of graph complements and how the Perfect Italian Domination number varies between a graph and its complement. The Nordhaus–Gaddum type inequalities in the Perfect Italian Domination number are also discussed

A REMARK AND AN IMPROVED VERSION ON RECENT RESULTS CONCERNING RATIONAL FUNCTIONS

This paper extends as a lemma an auxiliary result obtained by Singh and Chanam. Using it, we prove a refinement of the Turán-type inequality for rational functions obtained recently by Akhter et al.  Next, using examples, we discuss the result of Mir et al

STATISTICAL CONVERGENCE IN TOPOLOGICAL SPACE CONTROLLED BY MODULUS FUNCTION

The notion of $f$-statistical convergence in topological space, which is actually a statistical convergence's generalization under the influence of unbounded modulus function is presented and explored in this paper. This provides as an intermediate between statistical and typical convergence. We also present many counterexamples to highlight the distinctions among several related topological features. Lastly, this paper is concerned with the notions of $s^{f}$-limit point and $s^{f}$-cluster point for a unbounded modulus function $f$