# Search Results

## 1 - 10 of 16 items

• Author: Liang Zhao
Clear All Modify Search  ## Vector Functions and their Differentiation Formulas in 3-dimensional Euclidean Spaces

In this article, we first extend several basic theorems of the operation of vector in 3-dimensional Euclidean spaces. Then three unit vectors: e1, e2, e3 and the definition of vector function in the same spaces are introduced. By dint of unit vector the main operation properties as well as the differentiation formulas of vector function are shown .

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## Gauss Lemma and Law of Quadratic Reciprocity

In this paper, we defined the quadratic residue and proved its fundamental properties on the base of some useful theorems. Then we defined the Legendre symbol and proved its useful theorems , . Finally, Gauss Lemma and Law of Quadratic Reciprocity are proven.

MML identifier: INT 5, version: 7.8.05 4.89.993

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## Linear Congruence Relation and Complete Residue Systems

In this paper, we defined the congruence relation and proved its fundamental properties on the base of some useful theorems. Then we proved the existence of solution and the number of incongruent solution to a linear congruence and the linear congruent equation class, in particular, we proved the Chinese Remainder Theorem. Finally, we defined the complete residue system and proved its fundamental properties.

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## BCI-algebras with Condition (S) and their Properties

In this article we will first investigate the elementary properties of BCI-algebras with condition (S), see . And then we will discuss the three classes of algebras: commutative, positive-implicative and implicative BCK-algebras with condition (S).

MML identifier: BCIALG 4, version: 7.8.09 4.97.1001

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## Several Higher Differentiation Formulas of Special Functions

In this paper, we proved some basic properties of higher differentiation, and higher differentiation formulas of special functions .

MML identifier: HFDIFF 1, version: 7.8.10 4.100.1011

Open access

## Abstract

The aim of this work is to present some aggregation operators with triangular intuitionistic fuzzy numbers and study their desirable properties. Firstly, the score function and the accuracy function of triangular intuitionistic fuzzy number are given, the method for ranking triangular intuitionistic fuzzy numbers are developed. Then, some geometric aggregation operators for aggregating triangular intuitionistic fuzzy numbers are developed, such as triangular intuitionistic fuzzy weighted geometric (TIFWG) operator, the triangular intuitionistic fuzzy ordered weighted geometric (TIFOWG) operator and the triangular intuitionistic fuzzy hybrid geometric (TIFHG) operator. Moreover, an application of the new approach to multi-criteria decision making method was proposed based on the geometric average operator of TIFNs, and the new ranking method for TIFNs is used to rank the alternatives. Finally, an example analysis is given to verify and demonstrate the practicality and effectiveness of the proposed method.

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## Abstract

Bronchobiliary fistula (BBF) is one of the very rare disease. In this report, we described a BBF case. The case was diagnosed by endoscopic retrograde cholangio-pancreatography (ERCP) and percutaneous transhepatic cholangial drainage (PTCD) examinations, and treated properly. From the diagnosis of this BBF case, a patient with cough, biliptysis, fever and pain, should be considered for diagnosis of BBF.

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## Abstract

As the air flow control system has the characteristics of delay and uncertainty, this research designed and achieved a practical air flow control system by using the hydrodynamic theory and the modern control theory. Firstly, the mathematical model of the air flow distribution of the system is analyzed from the hydrodynamics perspective. Then the model of the system is transformed into a lumped parameter state space expression by using the Galerkin method. Finally, the air flow is distributed more evenly through the estimation of the system state and optimal control. The simulation results show that this algorithm has good robustness and anti-interference ability

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## Abstract

The compressive strength and water absorption of cement mortars with different water-binder ratio (0.35, 0.45 and 0.55) and fly ash content (0, 10%, 20% and 30%) under water immersion were investigated, and the correlation between them was further analyzed. The internal microstructure and phase composition of mortar was studied by scanning electron microscopy (SEM) and X-ray diffraction (XRD), respectively. The results show that the inside of mortar mixed with fly ash displayed the loose and porous microstructure. Therefore, the incorporation of fly ash reduced the compressive strength of mortar, especially the early strength, and the strength decreased with the increase of fly ash content, and the water absorption of mortar also increased. There was a linear correlation between the compressive strength and water absorption of mortar with the equation: f c = −3.838β + 62.332, where f c and β represented the compressive strength and water absorption, respectively. Therefore, when the water absorption of mortar immersed in water was measured, its corresponding compressive strength could be preliminarily inferred through this equation, which was of great significance for detecting and identifying the stability and safety of hydraulic structures.

Open access

## Abstract

The fruit fly optimization algorithm (FOA) is a global optimization algorithm inspired by the foraging behavior of a fruit fly swarm. In this study, a novel stochastic fractal model based fruit fly optimization algorithm is proposed for multiobjective optimization. A food source generating method based on a stochastic fractal with an adaptive parameter updating strategy is introduced to improve the convergence performance of the fruit fly optimization algorithm. To deal with multiobjective optimization problems, the Pareto domination concept is integrated into the selection process of fruit fly optimization and a novel multiobjective fruit fly optimization algorithm is then developed. Similarly to most of other multiobjective evolutionary algorithms (MOEAs), an external elitist archive is utilized to preserve the nondominated solutions found so far during the evolution, and a normalized nearest neighbor distance based density estimation strategy is adopted to keep the diversity of the external elitist archive. Eighteen benchmarks are used to test the performance of the stochastic fractal based multiobjective fruit fly optimization algorithm (SFMOFOA). Numerical results show that the SFMOFOA is able to well converge to the Pareto fronts of the test benchmarks with good distributions. Compared with four state-of-the-art methods, namely, the non-dominated sorting generic algorithm (NSGA-II), the strength Pareto evolutionary algorithm (SPEA2), multi-objective particle swarm optimization (MOPSO), and multiobjective self-adaptive differential evolution (MOSADE), the proposed SFMOFOA has better or competitive multiobjective optimization performance.

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