#### Advances in Pure Mathematics

Journal Information

ISSN / EISSN :
21600368 / 21600384

Current Publisher: Scientific Research Publishing, Inc. (10.4236)

Total articles ≅ 642

Archived in

SHERPA/ROMEO

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#### Latest articles in this journal

Advances in Pure Mathematics, Volume 10, pp 39-55; doi:10.4236/apm.2020.101004

**Abstract:**

The traditional estimation of Gaussian mixture model is sensitive to heavy-tailed errors; thus we propose a robust mixture regression model by assuming that the error terms follow a Laplace distribution in this article. And for the variable selection problem in our new robust mixture regression model, we introduce the adaptive sparse group Lasso penalty to achieve sparsity at both the group-level and within-group-level. As numerical experiments show, compared with other alternative methods, our method has better performances in variable selection and parameter estimation. Finally, we apply our proposed method to analyze NBA salary data during the period from 2018 to 2019.

Advances in Pure Mathematics, Volume 10, pp 86-99; doi:10.4236/apm.2020.102006

**Abstract:**

A standard method is proposed to prove strictly that the Riemann Zeta function equation has no non-trivial zeros. The real part and imaginary part of the Riemann Zeta function equation are separated completely. Suppose ξ(s) = ξ1(a,b) + iξ2(a,b) = 0 but ζ(1-s) = ζ1(a,b) + iζ2(a,b) ≠ 0 with s = a + ib at first. By comparing the real part and the imaginary part of Zeta function equation individually, a set of equation about a and b is obtained. It is proved that this equation set only has the solutions of trivial zeros. In order to obtain possible non-trivial zeros, the only way is to suppose that ζ1(a,b) = 0 and ζ2(a,b) = 0. However, by using the compassion method of infinite series, it is proved that ζ1(a,b) ≠ 0 and ζ2(a,b) ≠ 0. So the Riemann Zeta function equation has no non-trivial zeros. The Riemann hypothesis does not hold.

Advances in Pure Mathematics, Volume 10, pp 101-113; doi:10.4236/apm.2020.103007

**Abstract:**

In this paper, we research the regression problem of time series data from heterogeneous populations on the basis of the finite mixture regression model. We propose two finite mixed time-varying regression models to solve this. A regularization method for variable selection of the models is proposed, which is a mixture of the appropriate penalty functions and l2 penalty. A Block-wise minimization maximization (MM) algorithm is used for maximum penalized log quasi-likelihood estimation of these models. The procedure is illustrated by analyzing simulations and with an application to analyze the behavior of urban vehicular traffic of the city of São Paulo in the period from 14 to 18 December 2009, which shows that the proposed models outperform the FMR models.

Advances in Pure Mathematics, Volume 10, pp 57-85; doi:10.4236/apm.2020.102005

**Abstract:**

This paper presents a new scheme of flaw searching in surface modeling based on Euler Characteristic. This scheme can be applied to surface construction or reconstruction in computer. It is referred to as Euler Accompanying Test (EAT) algorithm in this paper. Two propositions in algebraic topology are presented, which are the foundation of the EAT algorithm. As the modeling is the first step for rendering in the animation and visualization, or computer-aided design (CAD) in related applications, the flaws can bring some serious problems in the final image or product, such as an artificial sense in animation rendering or a mistaken product in industry. To verify the EAT progressive procedure, a three-dimensional (3D) stamp model is constructed. The modeling process is accompanied by the EAT procedure. The EAT scheme is verified as the flaws in the stamp model are found and modified.

Advances in Pure Mathematics, Volume 10, pp 12-20; doi:10.4236/apm.2020.101002

**Abstract:**

In recent years, many methods have been used to find the exact solutions of nonlinear partial differential equations. One of them is called the first integral method, which is based on the ring theory of commutative algebra. In this paper, exact travelling wave solutions of the Non-Boussinesq wavepacket model and the (2 + 1)-dimensional Zoomeron equation are studied by using the first integral method. From the solving process and results, the first integral method has the characteristics of simplicity, directness and effectiveness about solving the exact travelling wave solutions of nonlinear partial differential equations. In other words, tedious calculations can be avoided by Maple software; the solutions of more accurate and richer travelling wave solutions are obtained. Therefore, this method is an effective method for solving exact solutions of nonlinear partial differential equations.

Advances in Pure Mathematics, Volume 10, pp 21-38; doi:10.4236/apm.2020.101003

**Abstract:**

There are two classes of continuities, parametric continuities and geometric continuities, which are used to illuminate the smoothness of a composite surface in surface construction and reconstruction in computer graphics (CG) and computer aided design (CAD). A parametric continuity is more stiff than its corresponding geometric continuity of the same order. This paper uncovers the geometric properties of parametric and geometric continuities less than and equal to second order and presents the proofs for the corresponding propositions. These propositions can be applied to the existent or promising schemes of surface construction or reconstruction, which can provide a convincing theory for researchers to establish their schemes in surface construction. Three examples are used in this paper to show the applications of these propositions.

Advances in Pure Mathematics, Volume 10, pp 1-11; doi:10.4236/apm.2020.101001

**Abstract:**

The Meyer-König and Zeller operator is one of the most challenging operators. Sometimes the study of its properties will rely on the weighted approximation by Baskakov operator. In this paper, this relation is extended to complex space; the quantitative estimates and the Voronovskaja type results for analytic functions by complex Meyer-König and Zeller operators were obtained.

Advances in Pure Mathematics, Volume 10, pp 114-124; doi:10.4236/apm.2020.103008

**Abstract:**

Factoring quadratics over Z is a staple of introductory algebra and textbooks tend to create the impression that doable factorizations are fairly common. To the contrary, if coefficients of a general quadratic are selected randomly without restriction, the probability that a factorization exists is zero. We achieve a specific quantification of the probability of factoring quadratics by taking a new approach that considers the absolute size of coefficients to be a parameter n. This restriction allows us to make relative likelihood estimates based on finite sample spaces. Our probability estimates are then conditioned on the size parameter n and the behavior of the conditional estimates may be studied as the parameter is varied. Specifically, we enumerate how many formal factored expressions could possibly correspond to a quadratic for a given size parameter. The conditional probability of factorization as a function of n is just the ratio of this enumeration to the total number of possible quadratics consistent with n. This approach is patterned after the well-known case where factorizations are carried out over a finite field. We review the finite field method as background for our method of dealing with Z [x]. The monic case is developed independently of the general case because it is simpler and the resulting probability estimating formula is more accurate. We conclude with a comparison of our theoretical probability estimates with exact data generated by a computer search for factorable quadratics corresponding to various parameter values.

Advances in Pure Mathematics, Volume 10, pp 125-154; doi:10.4236/apm.2020.103009

**Abstract:**

The most interesting and famous problem that puzzled the mathematicians all around the world is much likely to be the Fermat’s Last Theorem. However, since the Theorem was proposed, people can’t find a way to solve the problem until Andrew Wiles proved the Fermat’s Last Theorem through a very difficult method called Modular elliptic curves in 1995. In this paper, I firstly constructed a geometric method to prove Fermat’s Last Theorem, and in this way we can easily get the conclusion below: If a and b are integer and a = b, n ∈ Q and n > 1, the value of c satisfies the function an + bn = cn that can never be integer; if a, b and c are integer and a ≠ b, n is integer and n > 2, the function an + bn = cn cannot be established.

Advances in Pure Mathematics, Volume 9, pp 15-42; doi:10.4236/apm.2019.91002