Bulletin of the Malaysian Mathematical Sciences Society

Given a positive integer t and a graph F, the goal is to assign a subset of the color set $$\{1,2,\ldots ,t\}$$ to every vertex of F such that every vertex with the empty set assigned has all t colors in its neighborhood. Such an assignment is called the t-rainbow dominating function ( $$t\mathrm{RDF}$$ ) of the graph F. A $$t\mathrm{RDF}$$ is independent ( $$It\mathrm{RDF}$$ ) if vertices assigned with non-empty sets are pairwise non-adjacent. The weight of a $$t\mathrm{RDF}$$ g of a graph F is the value $$w(g) =\sum _{v \in V(F)}|g(v)|$$ . The independent t-rainbow domination number $$i_{rt}(F)$$ is the minimum weight over all $$It\mathrm{RDF}$$ s of F. In this article, it is proved that the independent t-rainbow domination problem is NP-complete even if the input graph is restricted to a bipartite graph or a planar graph, and the results of the study provide some bounds for the independent t-rainbow domination number of any graph for a positive integer t. Moreover, the exact values and bounds of the independent t-rainbow domination numbers of some Petersen graphs and torus graphs are given.

Let G be a graph of order n. The binding number $$\text{ bind }(G)$$ of G is $$\min \{\frac{|N_{G}(X)|}{|X|}\mid \emptyset \ne X\subseteq V(G)\,\, \text{ and }\,\, N_{G}(X)\ne V(G)\}$$. Throughout this article, some sufficient conditions about neighborhood union and $$\text{ bind }(G)$$ for a graph G to be fractional covered are obtained. Moreover, some graphs to verify that the results are best possible in a certain sense are gotten.

In this paper, we study the Berezin number inequalities by using the transform $$C_{\alpha ,\beta }\left( A\right) $$ on reproducing kernel Hilbert spaces (RKHS). Moreover, we give Grüss-type inequalities for selfadjoint operators in RKHS.

In this paper, we consider the Cauchy problem to the 2D generalized tropical climate model. We establish a new blow-up criterion only in terms of the first baroclinic mode of the vector velocity in Besov spaces with negative indices.

In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if $$x\in {\mathcal {H}}$$ has a disk orbit under $$T$$ that is somewhere dense in $${\mathcal {H}}$$ , then the disk orbit of $$x$$ under $$T$$ need not be everywhere dense in $${\mathcal {H}}$$ . We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space $${\mathcal {H}}$$ over the field of complex numbers if and only if $$\dim ({\mathcal {H}})=1$$ or $$\dim ({\mathcal {H}})=\infty $$ . Finally, we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

The aim of this paper is to introduce a Dunkl generalization of the operators including two-variable Hermite polynomials which are defined by Krech and to investigate approximating properties for these operators by means of the classical modulus of continuity, second modulus of continuity and Peetre’s K-functional.

Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor networks. In this paper, we study the notion of excess for woven frames and prove that any two frames in a separable Hilbert space that are woven have the same excess. We also show that every frame with a large class of duals is woven provided that its redundant elements have small enough norm. Also, we try to transfer the woven property from frames to their duals and vice versa. Finally, we look at which perturbations of dual frames preserve the woven property; moreover, it is shown that under some conditions, the canonical Parseval frame of two woven frames is also woven.

In this paper, we propose a bioeconomic predator–prey model with Michaelis–Menten type prey harvesting and general functional response which is described by a differential–algebraic system. Using the local parameterization, we derive an equivalent parametric system and investigate its dynamics in terms of local stability and Hopf bifurcation. The economic profit is chosen as a bifurcation parameter to prove the occurrence of Hopf bifurcation phenomenon in the neighborhood of the interior equilibrium. Moreover, we calculate the first Lyapunov coefficient to study the direction of Hopf bifurcation and the stability of bifurcated periodic solutions based on the normal form theory. Numerical simulations are carried out to demonstrate the analytical results.