In this article we derived an important example of the inconsistent countable set in second order ZFC (ZFC2) with the full second-order semantics. Main results: (i) ¬Con(ZFC2), (ii) let k be an inaccessible cardinal and Hk is a set of all sets having hereditary size less then k, then ¬Con(ZFC + (V = Hk)).
This paper presents an automatic diagnosis model of erythemato-squamous diseases. The proposed model consists of two stages. In the first stage, two filter based feature selection methods, namely rough set using Johnson's algorithm and ranked features for feature selection of erythemato-squamous diseases are employed to select the optimal feature subset from the original feature set for dimensionality reduction in order to further improve the diagnostic accuracy. Next, for the sake of comparison, the diagnoses decisions are made by four different classification algorithms: k-nearest neighbors, Naive Bayesian classifier, linear discriminant analysis and decision tree. Experimental results show that the accuracies of the four base classifiers using ranked features outperformed those using rough set with Johnson's algorithm and the base classifiers without using feature selection. Using erythemato-squamous diseases dataset taken from UCI (University of California at Irvine) machine learning database. The accuracies of these four classifiers using ranked features on test sets (50% of the dataset) are 97.21, 98.32, 96.09, and 98.32, respectively. Therefore, we can conclude that the ranked features method is very promising in detection of erythemato-squamous diseases compared to the rough set using Johnson's algorithm and also compared favorably with previously reported results. This tool enables doctors to differentiate six types of erythemato-squamous diseases using clinical and histopathological parameters obtained from a patient.
It is well known that the Cauchy problem for elliptic partial dierential equations is ill-posed. The question, which arises, how a priori knowledge about solutions can bring about stability? A parabolic transform is dened to discuss the stability of the Cauchy problem for some stochastic partial dierential equations under a priori knowledge about solutions. With the help of the parabolic transform, existence results are established for general linear and nonlinear stochastic partial dierential equations, without any restrictions on the characteristic forms. Many physical and engineering problems in areas like seismology, geophysics and biology require the solutions of ill-posed problems. The Cauchy problem for general stochastic dierential equations has many dierent important applications with amazing range.
In this paper category of multisets (Mul) is presented and various operations of category of sets such as points, products, sums, etc., were introduced in Mul and related results were proved. It is further shown that the product of two objects is unique as is the case for the sum, and the terminal object helps in separating arbitrary arrows. Thus, similar to category of sets if two arrows agree on points, they are the same arrows.
Limit is a basic concept of calculus. However, according to the updated definition, the limit of periodic function at infinity is not in existence. This conclusion of description does not suit with the periodic phenomenon. For example, the temperature on earth is changed periodically every year since the birth of the earth (viewed as t=0 ). Today (viewed as t →∞ ) the temperature on earth is continuing. Continuation means that the limit exists. In this paper, a new definition of limit of periodic function and periodic g-contractive mapping at infinity is defined by the value of its initial point based on transformation of variables. Similar definition is made for g- contractive ratio of periodic g-contractive mapping with k-related fixed points. These definitions can be used to describe the k-polar problems and calculation the limit of combinations of periodic functions at infinity. Furthermore, the new definition on contractive ratio of periodic iterative g-contractive mapping at infinity can help us to find the constant G and improves the application of the periodic iterative g-contractive mapping theorem.