In this paper, we propose a
new approach to the problem of degree reduction of Bézier curves based on the
given endpoint constraints. A differential term is added for the purpose of controlling
the smoothness to a certain extent. Considering the adjustment of second
derivative in curve design, a modified objective function including two parts
is constructed here. One part is a kind of measure of the distance between
original high order Bézier curve and degree-reduced curve. The other part
represents the second derivative of degree-reduced curve. We tackle two kinds
of conditions which are position vector constraint and tangent vector
constraint respectively. The explicit representations of unknown points are
presented. Some examples are illustrated to show the influence of the
differential terms to approximation and smoothness effect.

The marked variation between China and other nations in well-being may be due to the influence of national culture [1]. Considering the dramatically different cultural background between China and western societies, to investigate comprehensively Chinese well-being is important. In this literature review, firstly, we will introduce the understanding of well-being for Chinese people. Secondly, we introduce the concept of beauty, which is an important component of Chinese well-being be neglected by western scholars. Thirdly, the differences between Chinese and western well-being are presented.

Abstract:
The analog electronics is a challenging subject for undergraduate students in electrical engineering, due to the complex combination of many previous subjects, such as linear circuit analysis, signal and system, linear control theory and some sort of mathematics. This paper presents the modeling, analysis and design of the operational amplifier, which is used as benchmark system for analog electronics, for undergraduate studies in electrical engineering. Followed by the introduction of the operation amplifier circuit, the design of feedback network for the operational amplifier using MATLAB is presented. The bandwidth and sensitivity analysis for the feedback control loop are also discussed. In order to enhance the stability margin and dynamic characteristics of the operational amplifier, the lead compensator is designed for the feedback loop by adding capacitive component to the feedback resistive network. The presented analysis and design method of the operational amplifier by using MATLAB/SIMULINK can be highly effective to compliment the classroom teaching for circuit design courses for undergraduate studies in electrical engineering.

Abstract:
The power electronics course is a rather challenging subject for instructors and undergraduate students pursuing Bachelor’s Degree in Electrical Engineering. To enhance teaching effectiveness and motivate self-learning capabilities of the students, this paper presents a pedagogical approach for mathematical modeling and simulation of switching mode DC-DC converters. The Buck and Boost converters are analyzed as benchmark systems to study the power converter modeling methodologies. And a comparative analysis using digital simulation from Matlab/Simulink and ATP/EMTP is presented. A summary of student survey is also presented, which shows a high level of satisfaction. The presented pedagogical approach would be useful for classroom teaching for the power electronics course and similar engineering courses.

Abstract:
It is shown that a bounded quiver algebra having a 2-truncated oriented cycle is of infinite Hochschild homology dimension and global dimension, which generalizes a result of Solotar and Vigu\'{e}-Poirrier to nonlocal ungraded algebras having a 2-truncated oriented cycle of arbitrary length. Therefore, a bounded quiver algebra of finite global dimension has no 2-truncated oriented cycles. Note that the well-known "no loops conjecture", which has been proved to be true already, says that a bounded quiver algebra of finite global dimension has no loops, i.e., truncated oriented cycles of length 1. Moreover, it is shown that a monomial algebra having a truncated oriented cycle is of infinite Hochschild homology dimension and global dimension. Consequently, a monomial algebra of finite global dimension has no truncated oriented cycles.

Abstract:
It is shown that a recollement of derived categories of algebras induces those of tensor product algebras and opposite algebras respectively, which is applied to clarify the relations between recollements of derived categories of algebras and smoothness and Hochschild cohomology of algebras.

Abstract:
Extension conjecture states that if a simple module over an artin algebra has nonzero first self-extension group then it has nonzero i-th self-extension group for infinitely many positive integers i. It is shown by recollement of triangulated categories and differential graded homological algebra approach that extension conjecture is true for finite-dimensional elementary algebras over a field, particularly, for finite-dimensional algebras over an algebraically closed field. Moreover, bimodule approach is introduced to strong no loop conjecture, which provides two new proofs of Igusa-Liu-Paquette theorem.

Abstract:
It is shown that Hattori-Stallings trace induces a homomorphism of abelian groups, called Hattori-Stallings character, from the $K_1$-group of endomorphisms of the perfect derived category of an algebra to its zero-th Hochschild homology, which provides a new proof of Igusa-Liu-Paquette Theorem, i.e., the strong no loop conjecture for finite-dimensional elementary algebras, on the level of complexes. Moreover, the Hattori-Stallings traces of projective bimodules and one-sided projective bimodules are studied, which provides another proof of Igusa-Liu-Paquette Theorem on the level of modules.

Abstract:
Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the Basic-Wild-Rank Conjecture. A covering criterion on the rank of a basic wild algebra is given, which can be effectively applied to verify the Basic-Wild-Rank Conjecture for concrete algebras. It makes all conjectures much reliable.