Improvements on overdetermined problems associated to the $ p $-Laplacian

<abstract><p>This work presents some improvements on related papers that investigate certain overdetermined problems associated to elliptic quasilinear operators. Our model operator is the $ p $-Laplacian. Under suitable structural conditions, and assuming that a solution exists, we show that the domain of the problem is a ball centered at the origin. Furthermore we discuss a convenient form of comparison principle for this kind of problems.</p></abstract>

Research on the coordinated optimization operation method of Park Comprehensive Energy System Based on master-slave game

With the transformation of the energy market from the traditional vertical integrated structure to the interactive competitive structure, the distributed characteristics of the energy system become more and more obvious, and the traditional centralized optimization method is difficult to reveal the interaction between the multi-agent. In this paper, a method based on master-slave game is proposed to optimize the operation of park integrated energy system. Firstly, user load model, user benefit model, operator revenue and cost model are established for park integrated energy system. Secondly, the Stackelberg master-slave game model of interactive optimization operation is established, and the peak cutting compensation price is adjusted. Both of them aim at maximizing their own interests until the game equilibrium is achieved. A distributed cooperative optimization model with one master and many slaves is established and solved by the combination of genetic algorithm and quadratic programming. Finally, an example is given to verify the effectiveness of the proposed method.

The effect of English Non-Modal Operators on Proficiency and Performance of Undergraduate Students

This study focuses on the problems posed by the English non-modal operators to the undergraduate level students of Hazara University, Mansehra, Pakistan. The data was collected from hundred students selected through non-random and convenience sampling technique. A proficiency test was used as a tool for data collection. The test was focused on all the uses of non-modal operators. The results show that some of these problems were caused by the intervention of some of grammatical concepts like tense, aspect, back shifting and voice. While some grammatical operations like negation, interrogation and insertion/omission had no role and so were found comparatively easy. These operators when used after wh-word such as when, while, before and if posed difficulty for the subjects. Similarly, different forms such as non-tensed form and uses such as dynamic and non-dynamic of non-modal operators were also problematic for the subjects. The highest frequency of error was found in the use of non-model operator for emphasis and surprise. However, the degree of difficulty posed by non-modal operators in idiomatic expressions was not significant.

The Detectable Subspace for the Friedrichs Model: Applications of Toeplitz Operators and the Riesz–Nevanlinna Factorisation Theorem

We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. In this paper, we choose functions arising as parameters in the Friedrichs model in certain Hardy classes. This allows us to determine the detectable subspace by using the canonical Riesz–Nevanlinna factorisation of the symbol of a related Toeplitz operator.

The Detectable Subspace for the Friedrichs Model

This paper discusses how much information on a Friedrichs model operator can be detected from ‘measurements on the boundary’. We use the framework of boundary triples to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible from boundary measurements’. The Friedrichs model, a finite rank perturbation of the operator of multiplication by the independent variable, is a toy model that is used frequently in the study of perturbation problems. We view the Friedrichs model as a key example for the development of the theory of detectable subspaces, because it is sufficiently simple to allow a precise description of the structure of the detectable subspace in many cases, while still exhibiting a variety of behaviours. The results also demonstrate an interesting interplay between modern complex analysis, such as the theory of Hankel operators, and operator theory.