• The Mathematical Education
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• v.16 no.2
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• pp.22-26
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• 1978

It is shown that a map having an extension to an open map between the Alex-androff base compactifications of its domain and range has a unique such extension. J.S. Wasileski has introduced the Alexandroff base compactifications of Hausdorff spaces endowed with Alexandroff bases. We introduce a definition of morphism between such spaces to obtain a category which we denote by ABC. We prove that the Alexandroff base compactification on objects can be extended to a functor on ABC and that the compact objects give an epireflective subcategory of ABC. For each topological space X there exists a completely regular space $\alpha$X and a surjective continuous function $\alpha$$_{x}$ : Xlongrightarrow$\alpha$X such that for each completely regular space Z and g$\in$C (X, Z) there exists a unique g$\in$C($\alpha$X, 2) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\alpha$$_{x}$, $\alpha$X) is called a completely regularization of X. Let TOP be the category of topological spaces and continuous functions and let CREG be the category of completely regular spaces and continuous functions. The functor $\alpha$ : TOPlongrightarrowCREG is a completely regular reflection functor. For each topological space X there exists a compact Hausdorff space $\beta$X and a dense continuous function $\beta$x : Xlongrightarrow$\beta$X such that for each compact Hausdorff space K and g$\in$C (X, K) there exists a uniqueg$\in$C($\beta$X, K) with g=g$^{\circ}$$\beta$$_{x}$. Such a pair ($\beta$$_{x}$, $\beta$X) is called a Stone-Cech compactification of X. Let COMPT$_2$ be the category of compact Hausdorff spaces and continuous functions. The functor $\beta$ : TOPlongrightarrowCOMPT$_2$ is a compact reflection functor. For each topological space X there exists a realcompact space (equation omitted) and a dense continuous function (equation omitted) such that for each realcompact space Z and g$\in$C(X, 2) there exists a unique g$\in$C (equation omitted) with g=g$^{\circ}$(equation omitted). Such a pair (equation omitted) is called a Hewitt's realcompactification of X. Let RCOM be the category of realcompact spaces and continuous functions. The functor (equation omitted) : TOPlongrightarrowRCOM is a realcompact refection functor. In [2], D. Harris established the existence of a category of spaces and maps on which the Wallman compactification is an epirefiective functor. H. L. Bentley and S. A. Naimpally [1] generalized the result of Harris concerning the functorial properties of the Wallman compactification of a T$_1$-space. J. S. Wasileski [5] constructed a new compactification called Alexandroff base compactification. In order to fix our notations and for the sake of convenience. we begin with recalling reflection and Alexandroff base compactification.

• The Korean Journal of Air & Space Law and Policy
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• v.1
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• pp.223-243
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• 1989

The practical application of modern space science and technology have resulted in many actual and potential gains of mankind. These successes have conditioned and increased the need for a viable space law regime and the challenge of space has ultimately led to the formation of an international legal regime for space. Space law is no longer a primitive law. It is a modern law. Yet, in its stages of growth, it has not reached the condition of perfection. Therefore, under the existing state of thing, we could carefully say that the space law is one of the most newest fields of jurisprudence despite the fact that no one has so far defined it perfectly. However, if space law can be a true jurisprudential entity, it must be definable. In defining the space law, first of all, the grasp of it's nature iis inevitable. Although space law encompasses many tenets and facets of other legal discriplines, its principal nature is public international law, because space law affects and effects law relating intercourse among nations. Since early 1960s when mankind was first able to flight and stay in outer space, the necessity to control and administrate the space activities of human beings has growingly increased. The leading law-formulating agency to this purpose is the United Nation's ad hoc Committee on Peaceful Uses of Outer Space("COPUOS"). COPUOS gave direction to public international space law by establishing the 1963 Declaration of Legal Principles Governing the Activities of the States in the Exploration and Use of Outer Space("1963 Declaration"). The 1963 Declaration is very foundation of the five international multilateral treaties that were established successively after the 1963 Declaration. The five treaties are as follows: 1) The Treaty on Principles Governing the Activities of States in the Exploration and Use of Outer Space including Moon and other Celestial Bodies, 1967. 2) The Agreement on the Rescue of Astronauts, the Return of Astronauts, and the Return of Objects Launched into Outer Space, 1968. 3) The Convention on International Liability for Damage Caused by Space Objects, 1972. 4) The Convention on Registration of Objects Launched into Outer Space, 1974. 5) The Agreement Governing Activities of States on the Moon and Other Celestial Bodies: Moon Treaty, 1979. The other face of space law is it's commercial aspect. Space is no longer the sole domination of governments. Many private enterprise have already moved directly or indirectly into space activities in the parts such as telecommunications and space manufacturing. Since space law as the public international law has already advanced in accordance with the developments of space science and technology, there left only a few areas untouched in this field of law. Therefore the possibility of rapid growth of space law is expected in the parts of commerical space law, as it is, at this time, in a nascent state. The resources of the space environment are also commercially both valuable and important since the resources include the tangible natural resources to be found on the moon and other celestial bodies. Other space-based resources are solar energy, geostationary and geosynchronous orbital positions, radio frequencies, area possibly suited to human habitations, all areas and materials lending themselves to scientific research and inquiry. Remote sensing, space manufacturing and space transportation services are also another potential areas in which commercial. endeavors of Mankind can be carried out. In this regard, space insurance is also one of the most important devices allowing mankind to proceed with commercial space venture. Thus, knowlege of how space insurance came into existence and what it covers is necessary to understand the legal issues peculiar to space law. As a conclusion the writer emphasized the international cooperation of all nations in space activities of mankind, because space commerce, by its nature, will give rise many legal issues of international scope and concern. Important national and world-community interests would be served over time through the acceptance of new international agreements relating to remote sencing, direct television broadcasting, the use of nuclear power sources in space, the regularization of the activities of space transportation systems. standards respecting contamination and pollution, and a practical boundary between outer space and air space. If space activity regulation does not move beyond the national level, the peaceful exploration of space for all mankind will not be realized. For the efficient regulation on private and governmental space activities, the creation of an international space agency, similar to the International Civil Aviation Organization but modified to meet the needs of space technology, will be required. But prior to creation of an international organization, it will be necessary to establish, at national level, the Office of Air and Space Bureau, which will administrate liscence liscence application process, safety review and sale of launch equipment, and will carry out launch service.

• Journal of Intelligence and Information Systems
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• v.29 no.3
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• pp.167-183
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• 2023

In the recent field of recommendation systems, various studies have been conducted to model sparse data effectively. Among these, GLocal-K(Global and Local Kernels for Recommender Systems) is a research endeavor combining global and local kernels to provide personalized recommendations by considering global data patterns and individual user characteristics. However, due to its utilization of kernel tricks, GLocal-K exhibits diminished performance on highly sparse data and struggles to offer recommendations for new users or items due to the absence of side information. In this paper, to address these limitations of GLocal-K, we propose the GEase-K (Global and EASE kernels for Recommender Systems) model, incorporating the EASE(Embarrassingly Shallow Autoencoders for Sparse Data) model and leveraging side information. Initially, we substitute EASE for the local kernel in GLocal-K to enhance recommendation performance on highly sparse data. EASE, functioning as a simple linear operational structure, is an autoencoder that performs highly on extremely sparse data through regularization and learning item similarity. Additionally, we utilize side information to alleviate the cold-start problem. We enhance the understanding of user-item similarities by employing a conditional autoencoder structure during the training process to incorporate side information. In conclusion, GEase-K demonstrates resilience in highly sparse data and cold-start situations by combining linear and nonlinear structures and utilizing side information. Experimental results show that GEase-K outperforms GLocal-K based on the RMSE and MAE metrics on the highly sparse GoodReads and ModCloth datasets. Furthermore, in cold-start experiments divided into four groups using the GoodReads and ModCloth datasets, GEase-K denotes superior performance compared to GLocal-K.