Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.

Fuglede-Putnam Theorem have been proved for a considerably large number of class of operators. In this paper by using the spectral theory, we obtain a theoretical and general framework from which Fuglede-Putnam theorem may be promptly established for many classes of operators.

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ. Complete asymptotic expansions will be constructed if that order is constant along γ, and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set γ of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve γ. Complete asymptotic expansions will be constructed if that order is constant along γ, and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested.

As we move into the 21st century, with very rapid mobile communication and access to vast stores of information, we seem to be surrounded by more and more information, with less and less time or ability to digest it. The creation of the automatic summarization was really a genius human solution to solve this complicated problem. However, the application of this solution was too complex. In reality, there are many problems that need to be addressed before the promises of automatic text summarization can be fully realized. Basically, it is necessary to understand how humans summarize the text and then build the system based on that. Yet, individuals are so different in their thinking and interpretation that it is hard to create "gold-standard" summary against which output summaries will be evaluated. In this paper, we will discuss the basic concepts of this topic by giving the most relevant definitions, characterizations, types and the two different approaches of automatic text summarization: extraction and abstraction. Special attention is devoted to the extractive approach. It consists of selecting important sentences and paragraphs from the original text and concatenating them into shorter form. Broadly, the importance of sentences is decided based on statistical features of sentences. This approach avoids any efforts on deep text understanding. It is conceptually simple and easy to implement. KeywordsText summarization; Automatic text summarization; Abstractive approach; Extractive approach; Natural language processing.

As we move into the 21st century, with very rapid mobile communication and access to vast stores of information, we seem to be surrounded by more and more information, with less and less time or ability to digest it. The creation of the automatic summarization was really a genius human solution to solve this complicated problem. However, the application of this solution was too complex. In reality, there are many problems that need to be addressed before the promises of automatic text summarization can be fully realized. Basically, it is necessary to understand how humans summarize the text and then build the system based on that. Yet, individuals are so different in their thinking and interpretation that it is hard to create "gold-standard" summary against which output summaries will be evaluated. In this paper, we will discuss the basic concepts of this topic by giving the most relevant definitions, characterizations, types and the two different approaches of automatic text summarization: extraction and abstraction. Special attention is devoted to the extractive approach. It consists of selecting important sentences and paragraphs from the original text and concatenating them into shorter form. Broadly, the importance of sentences is decided based on statistical features of sentences. This approach avoids any efforts on deep text understanding. It is conceptually simple and easy to implement. KeywordsText summarization; Automatic text summarization; Abstractive approach; Extractive approach; Natural language processing.

As we move into the 21st century, with very rapid mobile communication and access to vast stores of information, we seem to be surrounded by more and more information, with less and less time or ability to digest it. The creation of the automatic summarization was really a genius human solution to solve this complicated problem. However, the application of this solution was too complex. In reality, there are many problems that need to be addressed before the promises of automatic text summarization can be fully realized. Basically, it is necessary to understand how humans summarize the text and then build the system based on that. Yet, individuals are so different in their thinking and interpretation that it is hard to create "gold-standard" summary against which output summaries will be evaluated. In this paper, we will discuss the basic concepts of this topic by giving the most relevant definitions, characterizations, types and the two different approaches of automatic text summarization: extraction and abstraction. Special attention is devoted to the extractive approach. It consists of selecting important sentences and paragraphs from the original text and concatenating them into shorter form. Broadly, the importance of sentences is decided based on statistical features of sentences. This approach avoids any efforts on deep text understanding. It is conceptually simple and easy to implement. KeywordsText summarization; Automatic text summarization; Abstractive approach; Extractive approach; Natural language processing.

As we move into the 21st century, with very rapid mobile communication and access to vast stores of information, we seem to be surrounded by more and more information, with less and less time or ability to digest it. The creation of the automatic summarization was really a genius human solution to solve this complicated problem. However, the application of this solution was too complex. In reality, there are many problems that need to be addressed before the promises of automatic text summarization can be fully realized. Basically, it is necessary to understand how humans summarize the text and then build the system based on that. Yet, individuals are so different in their thinking and interpretation that it is hard to create "gold-standard" summary against which output summaries will be evaluated. In this paper, we will discuss the basic concepts of this topic by giving the most relevant definitions, characterizations, types and the two different approaches of automatic text summarization: extraction and abstraction. Special attention is devoted to the extractive approach. It consists of selecting important sentences and paragraphs from the original text and concatenating them into shorter form. Broadly, the importance of sentences is decided based on statistical features of sentences. This approach avoids any efforts on deep text understanding. It is conceptually simple and easy to implement. KeywordsText summarization; Automatic text summarization; Abstractive approach; Extractive approach; Natural language processing.