The integrals \(\int_{-\infty}^\infty t_+^{\lambda-1} \phi(t)dt\) and \(\int_0^t(t-s)^{\lambda -1}b(s)ds\) are considered, \(\lambda\neq 0,-1,-2…\), where \(\phi\in C^\infty_0(\mathbb{R})\) and \(0\le b(s)\in L^2_{loc}(\mathbb{R})\). These integrals are defined in this paper for \(\lambda\le 0\), \(\lambda\neq 0,-1,-2,…\), although they diverge classically for \(\lambda\le 0\). Integral equations and inequalities are considered with the kernel \((t-s)^{\lambda -1}_+\).

In this paper we give a new variation of the prime labeling. We call a graph \(G\) with vertex set \(V(G)\) has an odd prime labeling if its vertices can be labeled distinctly from the set \(\big\{1, 3, 5, …,2\big|V(G)\big| -1\big\}\) such that for every edge \(xy\) of \(E(G)\) the labels assigned to the vertices of \(x\) and \(y\) are relatively prime. A graph that admits an odd prime labeling is called an odd prime graph. We give some families of odd prime graphs and give some necessary conditions for a graph to be odd prime. Finally, we conjecture that every prime graph is odd prime graph.

In this study, we define anti complex fuzzy subgroups and normal anti complex fuzzy subgroups under $s$-norms and investigate some of characteristics of them. Later we introduce and study the intersection and composition of them. Next, we define the concept normality between two anti complex fuzzy subgroups by using \(s\)-norms and obtain some properties of them. Finally, we define the image and the inverse image of them under group homomorphisms.

Harvesting has a strong impact on the dynamic evolution of a population subjected to it. In this paper, a fractional-order predator-prey interaction is studied with harvesting affecting both predator and prey populations. Local stability of the coexistence equilibrium point is discussed depending upon the harvesting of prey. Moreover, period-doubling and Neimark-Sacker bifurcations are studied for a wide range of constant harvesting effort of prey.

In this paper, we have established a theorem involving a pair of mappings satisfying a weakly contraction type condition in the context of quasi \(\alpha\)b-metric space and proved the existence and uniqueness of coupled coincidence and coupled common fixed points. The concept of weakly compatibility of the pair of maps is applied to show the uniqueness of coupled common fixed point. This work offers an extension to the published work of Nurwahyu and Aris [1].

In the present paper, we study a differential inequality involving certain differential operator. As a special case of our main result, we obtained certain differential inequalities implying sufficient conditions for meromorphic starlike and meromorphic convex functions of certain order.

In this paper we are interested in the existence of solutions for Navier problem associated with the degenerate nonlinear elliptic equations in the setting of the weighted Sobolev spaces.

In this paper, we study the existence of nontrivial solution for the fractional differential equation of order \(\alpha\) with three point boundary conditions having the following form
$$
D^{\alpha}u(t)=f(t,v(t),D^{\nu}v(t)),\quad t\in(0,T)$$
$$u(0)=0,\quad u(T)=au(\xi),$$
where \(1<\alpha<2\), \(\nu, a>0\), \(\xi\in (0,T)\), \(T^{\alpha-1}+a\xi^{\alpha-1}\neq0\). \(D\) is the standard Riemann-Liouville fractional derivative operator and \(f\in C([0,1]\times\mathbf{R}^{2},\mathbf{R})\). Applying the Leray-Schauder nonlinear alternative we prove the existence of at least one solution. As an application, we also given some examples to illustrate the results obtained.

The critical analysis of the foundations of vector calculus and classical electrodynamics is proposed. Methodological basis of the analysis is the unity of formal logic and rational dialectics. The main results are the following statements: (1) a vector is a property of the motion and of the interaction of material objects, i.e., the concept of a vector is the concept of a physical property. Therefore, the concept of a vector is a general and abstract concept; (2) a vector is depicted in the form of an arrow (i.e., “straight-line segment with arrowhead”) in a real (material) coordinate system. A vector drawn (depicted) in a coordinate system does not have the measure “meter”. Therefore, a vector is a pseudo-geometric figure in a coordinate system. A vector is an imaginary (fictitious) geometric figure; (3) geometrical constructions containing vectors (as pseudo-geometric figures) and vector operations in a coordinate system are fictitious actions; (4) the scalar and vector products of vectors represent absurd because vectors (as abstract concepts, as fictional geometric figures that have different measures) cannot intersect at the material point of the coordinate system; (5) the concepts of gradient, divergence, and rotor as the basic concepts of vector analysis are a consequence of the main mathematical error in the foundations of differential and integral calculus. This error is that the definition of the derivative function contains the inadmissible operation: the division by zero; (6) Maxwell’s equations the main content of classical electrodynamics are based on vector calculus. This is the first blunder in the foundations of electrodynamics. The second blunder is the methodological errors because Maxwell’s equations contradict to the following points: (a) the dialectical definition of the concept of measure; (b) the formal-logical law of identity and the law of lack of contradiction. The logical contradiction is that the left and right sides of the equations do not have identical measures (i.e., the sides do not have identical qualitative determinacy). Thus, vector calculus and classical electrodynamics represent false theories.

In this paper, we study some properties of induced topology by a uniform space generated by a family of ideals of a BCC-algebra. Also, by using Cauchy nets we construct a uniform space which is completion of this space.