Search

Cervical cancer is a very prevalent disease among women all over the world. Cervical cancer can form in the cervix cells found in the lower uterus. Women all over the world are at death risk as a result of this type of cancer. Cervical cancer has seven stages: normal intermediate, normal superficial, columnar, mild dysplasia, moderate dysplasia, severe dysplasia, and carcinoma in situ. Doctors in hospitals find it difficult to recognise cancer cells as it is challenging to view a nucleus through the naked eye. A normal cell's nucleus is smaller than an abnormal cell's nucleus. It is possible to calculate the size of the abnormal nucleus with the naked eye in order to assess the stages of cervical cancer. A tool for identifying and quantifying Pap smear cell images to detect cervical cancer has recently been suggested by several researchers. This method has the potential to increase detection and classification precision, resulting in improved results with balanced data and samples. A comprehensive study of nucleus detection cervical cancer classification techniques was conducted in this paper. As a result of the findings, the function database, detection and classification process, and device performance were all investigated for further evaluation. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence, types, and the cardinality of the integral solutions of the Diophantine equation x4 + y4 = 2kz3, for k > 1. The objective of this paper is to develop methods to be used in finding all integer solutions to this equation. Results of the study show the existence of infinitely many integral solutions to this type of Diophantine equation for both cases, x = y and x ≠ y. For the case when x=y, the form of the solutions is given by (a, b, c) = (2k−1n3, 2k-1n3, 2k−1n4) when 1 ≤ k < 5, and (a, b, c) = (2k−1−3tn3, 2k−1−3tn3, 2k−1−4tn4), for t ≤ k − 1 4 when k ≥ 5. Meanwhile, for the case when x ≠ y, the form of solutions is given by (a, b, c) = (2kun2, 2kvn2, 2kn3) or (a, b, c)= (2kdu, 2kdv, 2kdn), depending on the value of k. The main result obtained is a formulation of the generalized method to find all the solutions for this type of Diophantine equation. [ABSTRACT FROM AUTHOR]

In this note, the homotopy analysis method (HAM) is applied as a tool for solving the system of non-linear Fredholm-Volterra integral equations. The generalized chain rule is implemented for differentiation of the non-linear kernel functions with many variables, and the non-linear problem is reduced into a sequence of known non-linear integral equa- tions. The proposed method is compared with the homotopy perturbation method (HPM) given in Yusufogli [20] and Biazar and Ghazyini [21]. Numerical results reveal that the proposed method is very effective with high accuracy and dominated the HPM. [ABSTRACT FROM AUTHOR]

In recent years, considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of physics and engineering. Many essential phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, and material science are well described by differential equations of fractional order (Podlubny [8]). In certain circumstances, many exact solutions for linear fractional differential equation had been found. However, in general, there exists no method that yields an exact solution for nonlinear fractional differential equations. In this note, we consider the following nonlinear mixed Volterra-Fredholm integro-differential equation (VFIDEs) of fractional order. c D 0 + α u (t) φ (t) + λ ∫ 0 t ∫ 0 T k (x , s) F (u (s)) d x d s , (1) u (i) (0) = b i , i = 0 , ... , n − 1 , n − 1 < α ≤ n , (2) where t ∈ Ω = [0; T], k : Ω × Ω → ℝ, ϕ : Ω → ℝ are known functions, F : C(Ω, ℝ) → ℝ is nonlinear function, and u(s) is unknown function to be determined, bi (i = 0,..., n − 1) and λ are constants, c D 0 + α is the Caputo fractional derivative of order α. At first, Eq. (1)-(2) is written into operator form and applied standard homotopy analysis method (Liao, 1992) to reduce it into a sequence of known integral equation problems. Solving the latter equations step by step using the truncated Taylor series, an approximate solution is obtained. Description of the method and discretization of the problems together with the control convergence parameter are thoroughly studied. In addition, one example is illustrated to show the accuracy and validity of the proposed approach. Numerical results reveal that the proposed method is highly accurate and quickly approaches the exact solution when the number of iteration increases. [ABSTRACT FROM AUTHOR]