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The paper deals with a new fuzzy version of the Lee-Carter (LC) mortality model, in which mortality rates as well as parameters of the LC model are treated as triangular fuzzy numbers. As a starting point, the fuzzy Koissi-Shapiro (KS) approach is recalled. Based on this approach, a new fuzzy mortality model – CNMM – is formulated using orthonormal expansions of the inverse exponential membership functions of the model components. The paper includes numerical findings based on a case study with the use of the new mortality model compared to the results obtained with the standard LC model.

This study focused on students’ inability to correctly use psychometric tests applied for complex numbers in the context of generating an Algebraic abelian group (closure, commutative, associatively, inverse, and identity). This was done using a psychometric test (Kuder and Richardson formula) and APOS Theory after they had been taught the relevant content. The 41 participants were second-year students in the Department of Statistics at the Federal Polytechnic Ile-Oluji. They were given a class test comprising eleven (11) objective questions, and 4 questions relating to group theory were extracted for this analysis. After that, six (6) students were interviewed to gain insight into their perspectives. It was observed that most of the students had difficulty understanding the concept of algebraic groups about complex numbers, founded on not seeing the relevance in the prerequisite knowledge of an algebraic group. Also, if the reliability test of the question is unacceptable, then a good percentage of the participant lacked the mental construction relating to APOS to answer such a question. Keywords: Complex Number, Algebraic group, Prerequisite, Difficulty, Discrimination index, Psychometrics.

本文给出了非均匀复数的定义，非均匀复解析函数的定义和性质，建立了非均匀复解析函数与偏微分方程的关系。 In this paper, we introduce the definitions of heterogeneous complex number and heterogeneous complex function, and then discuss the property of heterogeneous complex function. We establish the relationship between heterogeneous complex analytic function and partial differential equation system.

Network security is a general idea to ensure information transmission over PC and portable systems. Elliptic curve cryptosystems are nowadays widely used in public communication channels for network security. Their security relies upon the complexity of clarifying the elliptic curve discrete alogarithm issue. But, there are several general attacks in them. Elliptic bend number juggling is actualized over complex fields to enhance the security of elliptic curve cryptosystems. This paper starts with the qualities of elliptic curve cryptosystems and their security administrations. At that point we talk about limited field number-crunching and its properties, prime field number-crunching, twofold field math and complex number-crunching, and elliptic bend number-crunching over prime field and parallel field. This paper proposes how to execute the unpredictable number of math under prime field and double field utilizing java BigInteger class. also, we actualize elliptic bend math and elliptic bend cryptosystems utilizing complex numbers over prime field and double field and talk about our trials that got from the usage.

The zoning blocks in the free building order are the special case of the separate, block and adjacent building regulations in the implementation zoning plans. Buildings are established in order to meet human needs in areas within the zoning boundaries by giving such building regulations. Free-order zoning parcels are another type of mostly separate and block orders. In areas where such parcels are located, only the equivalent value, which is the total construction area of the island, is usually given. Based on this precedent value, the residential area of the building is given to the parcel with various methods or approaches in accordance with the relevant law and regulation. With the complex (complex) number approach of the sample plan representations in the study, it was tried to find the tensile measures with the help of the angle, based on the direction with a certain edge. The complex approach, on the other hand, is the process of subtracting the base area on a parcel basis in a certain coordinate system axially depending on the angle and distance parameters along the Y-X axes. In this way, the necessity of giving residential areas to zoning parcels with different identities was examined.

Modern wireless communication systems are facing an increasingly complex electromagnetic environment, which is affected by a variety of noise and interference signals. In the communication process, if the communication party can effectively identify the type of interference signal, it can take corresponding anti-interference measures to avoid or suppress interference to the maximum extent and reduce the damage of interference to communication quality. In recent years, with the continuous development of deep learning and its outstanding performance in image and speech processing, it has been verified that deep learning has extremely strong nonlinear mapping and data expression capabilities, inspiring researchers to apply deep learning to the field of communication anti-jamming, and improving the antijamming capability of the system. In this paper, an algorithm of interference signal recognition based on a complex convolution neural network is proposed. It also introduces the network architecture. Next, six typical interference signals at each level are identified, and the results are analyzed. The last chapter of the paper summarizes the full text. The experimental results show that the recognition ability of the interference signal recognition algorithm proposed in this paper is better than that of the traditional convolutional neural network.

Forecasts of female and male mortality that are conducted independently run the risk of projecting implausible sex differentials and fail to exploit correlations that are known to exist between the sexes. We present a new model for the simultaneous modeling of female and male mortality. The model casts mortality as a complex-valued process where the real and imaginary parts correspond to female and male mortalities, respectively. Calculations proceed similarly to the usual Lee–Carter model, via the singular value decomposition, albeit in complex form. Initial applications suggest that the complex Lee–Carter gives fits that are broadly comparable to independent real fits, while offering the advantage of explicit modeling of the relationship between the sexes. Furthermore, model parameters are informative and easily-interpretable.

This qualitative research, drawing on the theoretical frameworks by Even (1990, 1993) and Sfard (2007), investigated five high school mathematics teachers' geometric interpretations of complex number multiplication along with the roots of unity. The main finding was that mathematics teachers constructed the modulus, the argument, and the conjugate of a complex number along with the roots of unity through a series of discursive transformations without specifying the common terminology. While teachers exhibited a variety of visualizations, each founded in a diversity of approaches on the dynamic geometry software, writing mathematical expressions and equations proved challenging. Construction of roots of unity required teachers' mathematical proficiency—in particular, strategic competence in simultaneously coordinating various interpretations of complex numbers, and representational fluency in analytic geometrical and transformational reasoning.

In this study, the combining strategies are considered to be effective tools to improve the performance of the image quality assessment (IQA) metrics. A new metric is proposed to evaluate the quality of test images of combined degradation and individual degradation. The complex numbers are used to describe the image structure in the proposed method. On the basis of that, the properties of the classical IQA method based on singular value decomposition are analysed. The difference between energy visual map and structure visual map is shown. The complex-number-based approach is different from the classical scalar-based techniques, which are insufficient to describe image structure. The proposed C_SVDQ metric can be considered as a vectorial expansion of structure similarity. In the experiments, an extensive comparison between the proposed C_SVDQ and other IQA metrics on image quality database was performed. Both the overall tests and the individual distortion tests show the superiority of this new approach in IQA.

This criterion by Macha has been repeatedly verified, analyzed and special cases of this criterion have been taken into account, reducing the complex stress to its uniaxial equivalent. Since both normal and shear stresses are vectors, we encounter the mathematical problem of adding these vectors, and the question arises how to consider the obtained equivalent stress, because two perpendicular vectors are added with weighting factors. Therefore, in this work it is proposed to adopt a system of complex numbers. Normal stress is defined here as a real part and shear stress as an imaginary part. As a result, on the basis of the defined complex number and based on pendulum bending and double-sided torsion, after transformations, the expression for equivalent stress proves to be identical to those previously defined on the basis of the concept of Macha.

In this perspective review, we discuss the power of polarized Raman spectroscopy to study optically anisotropic 2D materials, belonging to the orthorhombic, monoclinic and triclinic crystal families. We start by showing that the polarization dependence of the peak intensities is described by the Raman tensor that is unique for each phonon mode and we discuss how to determine the tensor elements from the angle-resolved polarized measurements by analyzing the intensities in both the parallel- and cross-polarized scattering configurations. We present specific examples of orthorhombic black phosphorous and monoclinic 1T'-MoTe2, where the Raman tensors have null elements and their principal axes coincide with the crystallographic ones, following by a discussion on the results for triclinic ReS2 and ReSe2, where the axes of the Raman tensor do not coincide with the crystallographic axes and all elements are non-zero. We show that the Raman tensor elements are, in general, given by complex numbers and that phase differences between tensor elements are needed to describe the experimental results. We also discuss the dependence of the Raman tensors on the excitation laser energy and thickness of the sample within the framework of the quantum model for the Raman intensities. We show that the wavevector dependence of the electron-phonon interaction is essential for explaining the distinct Raman tensor for each phonon mode. Finally, we close with our concluding remarks and perspectives to be explored by angle-resolved polarized Raman spectroscopy in optically anisotropic 2D materials.