International Journal of Mathematics and Mathematical Sciences

We present upper bounds on the diameter of bipartite and triangle-free graphs with prescribed edge connectivity with respect to order and size. All bounds presented in this paper are asymptotically sharp.

We introduce a mathematical model that shows the interaction dynamics between the uninfected and the infected cancer cell populations with oncolytic viruses for the benign and the malignant cancer cases. There are two important parameters in our model that represent the malignancy level of the cancer cells and the efficacy of the therapy. The parameters play an important role to determine the possibility to have successful therapy for cancer. Our model is based on the predator-prey model with logistic growth, functional response, and the saturation effect that show the possibility for the virus to be deactivated and blocked by the human immune system after they reach a certain value. In this paper, we consider the appearance of the Hopf bifurcation on the system to characterize the treatment response based on the malignancy effect of the disease. We employ numerical bifurcation analysis when the value of the malignancy parameter is varied to understand the dynamics of the system.

In this paper, we introduce a new concept of -admissible almost type -contraction and prove some fixed point results for this new class of contractions in the context of complete metric spaces. The presented results generalize and unify several existing results in the literature.

Let denote an algebraically closed field with a characteristic not two. Fix an integer ; let , , and be the equitable basis of over . Let denote an irreducible -module with dimension ; let . In this paper, we show that if each of the pairs , , and acts on as a Leonard pair, then these pairs are of Krawtchouk type. Moreover, is a linear combination of 1, , , and .

We propose reduced Collatz conjecture and prove that it is equivalent to Collatz conjecture but more primitive due to reduced dynamics. We study reduced dynamics (that consists of occurred computations from any starting integer to the first integer less than it) because it is the component of original dynamics (from any starting integer to 1). Reduced dynamics is denoted as a sequence of “I” that represents ()/2 and “O” that represents . Here, and are combined together because is always even and thus followed by . We discover and prove two key properties on reduced dynamics: (1) Reduced dynamics is invertible. That is, given reduced dynamics, a residue class that presents such reduced dynamics can be computed directly by our derived formula. (2) Reduced dynamics can be constructed algorithmically, instead of by computing concrete and step by step. We discover the sufficient and necessary condition that guarantees a sequence consisting of “I” and “O” to be a reduced dynamics. Counting from the beginning of a sequence, if and only if the count of over the count of is larger than ln3/ln2, reduced dynamics will be obtained (i.e., current integer will be less than starting integer).

The purpose of this study is to see whether it is possible to eradicate the disease theoretically using mathematical modeling with the aid of numerical simulation when disease occurs in a population by implementing adequate preventive measures. For this, we consider a mathematical model for the transmission dynamics of cholera and its preventive measure as one cohort of individuals, namely, a protected cohort in addition to susceptible, infected, and recovered cohorts of individuals including the concentration of Vibrio cholerae in the contaminated aquatic reservoir with small modifications. We calculate the basic reproduction number, , and investigate the existence and stability of equilibria. The model possessed forward bifurcation. Moreover, we compute the sensitivity indices of each parameter in relating to of the model. Numerical simulations are carried out to validate our theoretical results. The result indicates that the disease dies out in areas with adequate preventive measures and widespread and kills more people in areas with the inadequate preventive measures.