The existence of non trivial zeros off the critical line for a function obtained by analytic continuation of a particular Dirichlet series is studied. Our findings are in contradiction with some results of computations which were present in the field for a long time. In the first part of this note, we illustrate how the approximation errors may have had as effect inexact conclusions and in the second part we prove rigorously our point of view.
This short paper demonstrates a sketch of a generic algorithm that can generate integers written as the sum of N mth positive powers with any N≥2 and m≥1 in two different ways. As it is shown in the paper, such a generic algorithm is NP-hard. One can infer from this result that to generate exact values of the generalized "Taxicab" (m,N,j)– the smallest sum of N mth positive powers expressed in j different ways – is at least as hard as the hardest problems in NP. This implies that except for small N to find the generalized "Taxicab " (m,N,j) would be unfeasible on any computing device.
The concept of semi-compatible and occasionally weakly compatible mappings is used to prove a common fixed point theorem. The theorem thus obtained is a generalization of the result of Cho et al.  in a non-Archimedean Menger PM-space.
The main challenge that faces any researcher in the field of machine learning is determining the quality of an indicator used for measuring the efficiency of classifier techniques. This issue based on Multiple-Criteria Decision Making (MCDM) has not been tackled by any researcher until now. The previous work concerned with a single classical criterion (Accuracy Level) ignoring other important criteria in real-life. This paper presents a novel indicator for measuring the efficiency of classifier techniques. This measure is a global indicator with multi-criteria approach based on the technique for preference by similarity to the ideal solution (TOPSIS). This indicator is characterized by its ability to taking in account all previous criteria. In addition, two novel criteria are created by authors: Learning Efficiency Ratio (LER), and the CPU time efficiency. The classifiers evaluation process includes the classical classifiers: Support Vector Machines (SVM), Multi-layer perceptron (MLP), Gene Expression Programming (GEP), Single Decision Tree (STR), and the techniques that achieved the best results in literature. Inaddition, the latest classifiers: Tropical Collective Machine Learning (TCML), and Dempster-Shafer Collective Machine Learning (DSCML) using the proposed indicator. The comparison is performed using twenty-five standard datasets (benchmarks). The results supported by statistical analysis (T-test) show the efficiency and effectiveness of the proposed global indicator for selecting the best classifier and its ability to measure the classifier efficiency based on multi-criteria. Results promise the optimistic use of the global indicator in the classifiers evaluation process for real-life problems.
In this paper, we consider some available models and then introduce a model which simulates the interaction between ATM protein and DNA damage signal, which motivated biologically. Next we find a Hopf bifurcation for this system. Biologically we find a region for the DNA damage signal and ATM protein where solutions in this region are not of those solutions that DNA healing process occurs on these. In fact entering solutions in this region aren’t biologically appropriate solutions.
Let and where p , q , I are distinct odd primes, I is a primitive root both modulo Pn and .We obtain explicit expressions for all dmn + m + n + 1 I -cyclotomic cosets modulo Pn qm We explicitly determine generating polynomials and enumeration formulas of all self-orthogonal cyclic codes and complementary-dual cyclic codes of length Pn qm fIAs an example, we give all self orthogonal cyclic codes and complementary-dual cyclic codes of length 175 over F3.