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93,175 results · page 3611 of 3,883

Dec 10, 2014·Walisongo Repository (Walisongo State Islamic University)
0 cites
Analisis hukum Islam terhadap alat tukar Bitcoin (studi kasus jual-beli Bitcoin di dunia maya)

Ari Prayogo

Perkembangan teknologi informasi dan didukung pula dengan teknologi komputer yang semakin canggih serta adanya jaringan internet yang menciptakan terkoneksinya perangkat-perangkat komputer baik berbentuk personal komputer maupun super komputer sehingga terciptanya aktivitas bisnis dengan teknologi jaringan internet atau disebut electronic commerce (e-commerce). Maka dari sinilah lahir pula mata uang virtual dunia maya (virtual currency) dengan konsep nilai berdasarkan supply dan demand serta mudah, praktis dan efisien dalam melakukan transaksi perdagangan maupun bisnis lainnya. Namun dampak terbesar dalam transaksi online seperti ini diantaranya adalah penipuan, pencurian dan kerugian yang akan dialami pengguna secara tidak wajar.
\nBerangkat dari masalah di atas ada beberapa permasalahan yang dirumuskan untuk mengetahui analisis jual-beli bitcoin di dunia maya yaitu meliputi bagaimana analisis undang-undang tentang mata uang terhadap alat tukar bitcoin dan prespektif hukum Islam terhadap jual-beli bitcoin di dunia maya. Penyusun memilih tempat penelitian tersebut di website bitcoin exchanger (www.vip.bitcoin.co.id.)
\nDalam menelusuri, menjelaskan dan menyimpulkan objek pembahasan pada skripsi ini. Penyusun menempuh metode jenis penelitian lapangan di dunia maya dengan menggunakan metode observasi partisipasif lengkap dimana peneliti menjadi bagian dalam aktivitas pada sumber data yang diteliti. Untuk mendapatkan informasi secara akurat, aktual dan terpercaya. Penyusun melakukan beberapa metode pengumpulan data yakni menggunakan metode wawancara, observasi dan dokumentasi yang selanjutnya dianalisis menggunakan metode deskriptif kualitatif.
\nDari hasil penelitian ini dapat disimpulkan yakni analisis Undang-Undang No 7 Tahun 2011 Tentang Mata uang, bahwa bitcoin yang dijadikan alat tukar dalam pembayaran di Negara Indonesia ini termasuk melanggar undang-undang karena undang-undang sudah mengatur mengenai alat pembayaran yang sah di Indonesia. Penggunaan bitcoin tidak dijadikan sebagai alat tukar atau hanya sebagai komoditas ini tidak dipermasalahkan secara yuridis karena tidak ada peraturan yang mengatur maupun melarang dari OJK atau Bank Indonesia terhadap bitcoin. Akan tetapi terkait resiko seperti kehilangan dan kerugian itu ditanggung sendiri oleh penggunanya sebagaimana siaran pers “Pernyataan Bank Indonesia Terkait Bitcoin dan Virtual Currency Lainnya” No: 16/ 6 /DKom. Dalam prespektif hukum Islam bitcoin yang dijadikan sebagai alat tukar maupun alat investasi diharamkan. Sebab praktek yang terjadi terdapat unsur gharar dan maisir, serta menghindari kemadharatan yang dapat terjadi pada pengguna.

Open access
Islamic Finance and Communication
SMEs Development and Digital Marketing
Blockchain Technology in Education and Learning
Original source
Dec 10, 2014·International public management review
4 cites
Republic of Moldova: 20 years of Independence and Experiments with Public Finance System

Boris Morozov

Democratic movements, market mechanisms, and decentralization reforms have been at the heart of transformation of former USSR republics. Republic of Moldova was not an exception. The initial argument behind such fascination with these reforms stems from the idea that market mechanisms in an open and democratic society will facilitate the efficient and effective resources’ allocation, compensating for interim distributive inequities and macro-economic challenges. Such a natural experiment has been in progress for more than 20 years in the Republic of Moldova. During these 20 years Moldova experienced democracy in its infancy, democratic return of communism, and once again return to democracy and its core ideas and values. Public administration system along with public finance and budgeting systems experienced comparably abrupt changes in their development. This qualitative study of Moldovan public administration and finance frameworks will provide an evaluative description of these three distinct, but interrelated periods in Moldovan history of independence and development of public administration and public budgeting systems. Such description will allow for evaluation of the public administration and public finance systems’ developments in Moldova as well as reasons of success and failure of these reforms.

Open access
Fiscal Policy and Economic Growth
Original source
Dec 9, 2014·Automation technological and business processes
0 cites
ЦИФРОВАЯ ВАЛЮТА BITCOIN

В. В. Приян

В статье рассматривается цифровая виртуальная интернет валюта Bitcoin, ее общие понятия и участие в мировой экономике

Open access
Military Technology and Strategies
Legal and Regulatory Analysis
Linguistic, Cultural, and Literary Studies
Original source
Dec 8, 2014·ACM SIGMETRICS Performance Evaluation Review
589 cites
Proof of Activity

Iddo Bentov, Charles Lee, Alex Mizrahi, Meni Rosenfeld

We propose a new protocol for a cryptocurrency, that builds upon the Bitcoin protocol by combining its Proof of Work component with a Proof of Stake type of system. Our Proof of Activity protocol offers good security against possibly practical attacks on Bitcoin, and has a relatively low penalty in terms of network communication and storage space.

Cryptography and Data Security
Blockchain Technology Applications and Security
Internet Traffic Analysis and Secure E-voting
Original source
Dec 4, 2014·European Scientific Journal ESJ
1 cites
DECENTRALIZATION PROCESS AND FINANCIAL AUTONOMY OF LOCAL GOVERNMENT UNITS IN ALBANIA

Entela Hoxhaj

The purpose of this paper is to give a general overview on the decentralization process and on the financial autonomy of local government units in Albania. The principal base for the analysis done is the legal framework, accompanied shortly with examples of its implementation in several communes or municipalities. The paper is organised in 3 issues, which treat respectively the organization of local government in Albania, decentralization of functions and responsibilities in favour of the units of local government, and their budget and resources of financing. There has been some difficulties in practice during decentralization process raised out when fulfilling the whole legal framework on exercising exclusive and shared competences of local units of government, because of the dependence by the respective institutions for the implementation of legal acts for exclusive and shared functions in the fields of education, economic support and health care, and by lack of funds of the responsible ministry to transfer the service of water supply. In Albania seems yet early to give right importance to regions, and especially with regard to the aspirations for European integration many challenges raise to the Albanian institutions.

Open access
Government, Law, and Information Management
Fiscal Policies and Political Economy
Regional Development and Policy
Original source
Dec 3, 2014·HAL (Le Centre pour la Communication Scientifique Directe)
0 cites
Modeling and stabilization of sociopolitical networks – application to country coalitions

G. N. Vinogradova

The main subject of this thesis is a paradigm of instability and stabilization in coalition forming among countries as rational actors, presented through a Statistical Physics inspired model.This is an interdisciplinary work involving the fields of applied mathematics and sociophysics, as well as the political applications in the real cases from past and present. Applied to political,economic and social problems, the models can be used to analyze a wide variety of real cases -- international alliances, economic or business alliances, coalitions of political parties, socialnetworks, and organizational structures.In the first part of this thesis we present and analyze the coalition forming and the instability among rational actors coupled with pairwise static historical propensity bonds that have evolvedindependently. Such organization leads to discordant associations into coalitions and the instability as a consequence of decentralized maximization of the individual benefits gained fromjoining or leaving the coalitions. We define Natural Model of coalition forming and address the questions of instability and stabilization among actors possessing different levels of rationality. The framework presented here allows to analytically calculate the optimal and non-optimal stable configurations of actors' coalitions. We then investigate the coalition forming and the stabilization under the influence of externally-set opposing global alliances, which are represented in Global Alliance Model. The stabilization is produced through new cooperations based on the effect of polarization of several distinct interests shared by actors, which generates interest-based propensities and enables a planned coalition forming. We then investigate the effect of dissolution of a global alliancewhich, together with the competing alliance, has previously generated stable coalitions.A special section of the thesis is devoted to investigation and illustration of coalition forming in real historical cases. This part presents the analysis of unstable coalitions in Europe - cycling in the England-Spain-France conflicting triangle and creation of the Italian state, as well as of the remarkable historical cases of the Soviet global alliance collapse, of the recent internal conflict in Syria, and of the "paradoxical stability" in the Eurozone.In the second part of this thesis, we present a simulation of the coalition forming models. The simulation allows to follow graphically the coalition forming processes. We present the methodology used in the simulation, as well as its application in the illustration of coalition forming in the prototypes of real case systems. Given exact propensity values, which is fairly consideredto be the most difficult part of coalition forming modeling, the simulation tool can be used to predict optimal and non-optimal spontaneous stabilizations and globally motivated stabilities inreal cases.An independent part of the thesis is devoted to the subject of viability correction in dynamic network of actors. The model is a finite set of autonomous actors with states that evolveindependently and connected into a network via their connection operators, which evolve independently as well. The network is defined to be viable if a joint evolution satisfies the centralized scarcity constraints set by the environment. In order to restore the viability of these decentralized dynamics, we apply to the method of correction by viability multipliers used in Viability Theory, where the multipliers play the role of decentralizing prices. Standing apart from the main course of the thesis, the subject of viability correction in dynamic network of actors suggests an interesting theoretic dynamical generalization of coalition stabilization in our models inspired from Statistical Physics.

Open access
Opinion Dynamics and Social Influence
Game Theory and Applications
Complex Systems and Time Series Analysis
Original source
Dec 2, 2014·New Journal of Physics
85 cites
Inferring the interplay between network structure and market effects in Bitcoin

Dániel Kondor, István Csabai, János Szüle, Márton Pósfai · 5 authors

A main focus in economics research is understanding the time series of prices of goods and assets. While statistical models using only the properties of the time series itself have been successful in many aspects, we expect to gain a better understanding of the phenomena involved if we can model the underlying system of interacting agents. In this article, we consider the history of Bitcoin, a novel digital currency system, for which the complete list of transactions is available for analysis. Using this dataset, we reconstruct the transaction network between users and analyze changes in the structure of the subgraph induced by the most active users. Our approach is based on the unsupervised identification of important features of the time variation of the network. Applying the widely used method of Principal Component Analysis to the matrix constructed from snapshots of the network at different times, we are able to show how structural changes in the network accompany significant changes in the exchange price of bitcoins.

Open access
2 source records
Complex Systems and Time Series Analysis
Complex Network Analysis Techniques
Stock Market Forecasting Methods
Original source
Dec 1, 2014·SSRN Electronic Journal
18 cites
Chain of a Lifetime: How Blockchain Technology Might Transform Personal Insurance

Michael Mainelli, Chiara Von Gunten

Chain Of A Lifetime: How Blockchain Technology Might Transform Personal Insurance is the outcome of a research project conducted between August and December 2014 which explored how blockchain technology might transform personal insurance and in particular interactions among individuals and insurance companies over time. Blockchain technology's main innovation is an electronic public transaction record of integrity without central authority. Beside cryptocurrencies and distributed payment systems, blockchain applications could include areas of finance where a central, trusted third party has traditionally been used, trade reporting, depository receipts, escrow accounts or trade finance. Blockchains can contain set of documents, record assets and help to manage interconnected devices. Emerging applications, such as smart contracts and decentralised autonomous organisations, might in future also permit blockchains to act as automated agents. The report concludes that blockchain technology could transform the way people manage identities and personal information; blur even further the divide between global and local; influence consumer perception of time; drive honesty and transparency; and, influence consumer perceptions of risk that could change the way insurers support mutualisation. The report highlights how, at the time of writing, most insurance companies do not yet seem ready to experiment with blockchain technology. They find it difficult enough to understand Bitcoin or cryptocurrencies. Non-insurers are more likely to be the first to create insurance or insurance-related applications. Blockchain applications in insurance are likely to start with digital identity systems and management of personal data.

Open access
Blockchain Technology Applications and Security
FinTech, Crowdfunding, Digital Finance
Original source
Dec 1, 2014
6 cites
Privacy and customer segmentation in the smart grid

Lillian J. Ratliff, Roy Dong, Henrik Ohlsson, Álvaro A. Cárdenas · 5 authors

In the electricity grid, networked sensors which record and transmit increasingly high-granularity data are being deployed. In such a setting, privacy concerns are a natural consideration. In order to obtain the consumer's valuation of privacy, we design a screening mechanism consisting of a menu of contracts offered to the energy consumer with varying guarantees of privacy. The screening process is a means to segment customers. Finally, we design insurance contracts using the probability of a privacy breach to be offered by third-party insurance companies.

Smart Grid Security and Resilience
Power Line Communications and Noise
Vehicular Ad Hoc Networks (VANETs)
Original source
Dec 1, 2014·Jurnal Bina Praja
3 cites
Urgensi Peningkatan Kapasitas Pengelolaan Keuangan Daerah di Era Desentralisasi (Dinamika Pengelolaan Pajak Parkir Kecamatan Banguntapan, Kabupaten Bantul Provinsi DI. Yogyakarta)

Bambang Sunaryo, Celly Cicellia

AbstrakPeningkatan kapasitas pengelolaan keuangan daerah menjadi hal yang sangat penting di era desentralisasi. Pengelolaan pajak parkir yang sekilas terlihat sebagai ranah kebijakan prosedural administratif, tidak pernah bisa lepas dari berbagai patologi yang sangat problematik bagi pengelolaan keuangan daerah. Kecamatan Banguntapan dipilih sebagai unit analisis dalam penelitian ini untuk merepresentasikan permasalahan pengelolaan pajak parkir di wilayah sub urban Kabupaten Bantul. Hal ini dikarenakan kharakteristik kewilayahan Bantul sebagai daerah sub urban dapat dilihat dari kharakteristik peri urban yang ada di Kecamatan Banguntapan. Selain itu, Kecamatan Banguntapan merupakan satu-satunya wilayah di Kabupaten Bantul yang memiliki subyek pajak parkir bertarif self assessment dan flat sehingga dinamika problema manajerial keuangan daerah dapat diobservasi dan dianalisis lebih mendalam di Kecamatan Banguntapan ini untuk melihat komparasi 2 sistem pemungutan pajak parkirtersebut. AbstractCapacity building of local finance management becomes important on decentralizations era. Empirically this research is aim to show that the parking tax management at glance seen as an administrative- procedural policy domain cannot release from phenomena on the existence of problematic pathology for regional financial management. District of Bantul Banguntapan characteristic as its suburbs form themain attractionin the selection of research because of the generallocus in sub-urbanarea, began to metamorphose into a parking tax revenues fromregionsthat contribute to the area thoughnot as big financial contribution income tax parking tax in urban areas. The Banguntapan sub district was chosen as analysis unit in this research to represent the issue of parking tax management in sub urban area of Bantul. This was due to the characteristic of Bantul area as sub urban area that can be seen from the characteristics of sub urban in Banguntapan sub district. Moreover, the Banguntapan sub district is the only area in Bantul which has a parking tax subject that the cost is self assessment and flat thus the dynamic of local finance managing problems can be observed and in-depth analyzed in Banguntapan sub district to seek the comparison of those 2 parking tax collection systems.

Open access
2 source records
Economic Growth and Fiscal Policies
Local Governance and Development
Public Administration in Developing Nations
Original source
Dec 1, 2014·Tijdschrift Erfrecht
1 cites
De executeur in een nalatenschap met bitcoins en andere ‘digitale bezittingen’

L.A.G.M. van der Geld

De executeur in een nalatenschap met bitcoins en andere ‘digitale bezittingen’ In deze bijdrage gaat de auteur op zoek naar wat de verplichtingen zijn die de executeur heeft ten aanzien van de onlinenalatenschap. Moet een executeur actief op zoek naar bitcoins en andere digitale bezittingen? Hoever gaat zijn onderzoeksplicht in een onlinenalatenschap? Ook gaat de auteur kort in op de erfbelasting en digitale bezittingen: de executeur moet ook hier aangifte van doen.

Open access
European and International Contract Law
Original source
Dec 1, 2014·EC Tax Review
1 cites
VAT & BITCOIN

S. Capaccioli

No abstract is available for this record.

Corporate Taxation and Avoidance
Original source
Dec 1, 2014
0 cites
Plan de Desarrollo del Turismo Comunitario para el Cantón Pedro Vicente Maldonado, ubicado al Noroeste de la Provincia de Pichincha

Daniel Alberto González Martínez, Lourdes Maricela Guamán Armijos, Jonathan Rodrigo Hidalgo Verdezoto

The canton Pedro Vicente Maldonado is located northwest of the province of Pichincha. This sector is characterized by its temperate climate ranging between 15-20 ° C. It is ideal for the practice of tourism activities quintessential natural region, mainly related to ecotourism. Its cantonal head is the city of Pedro Vicente Maldonado, whose highest authority is the mayor. The Autonomous Decentralized Government is governed under the Organic Code of Territorial Organization and Decentralization (COOTAD). In this county there is great natural and cultural richness, making it an attractive area for tourism. They have not been adequately exploited. No representing an effective boost to the economy of Canton, preventing a better quality of life for its inhabitants. According to statistics provided by the City of Canton, since 2010 there has been increased momentum of tourism growth, is particularly community tourism.

Business, Innovation, and Economy
Diverse Applied Research Studies
Organizational Management and Innovation
Original source
Dec 1, 2014
2 cites
An Alternative Methodology for Authentication and Confidentiality Based on Zero Knowledge Protocols Using Diffie-Hellman Key Exchange

Pratibha Kumari, A. Damodaram

This paper presents a concept for a new method to provide the authentication and confidentiality using zero knowledge protocol and key exchange. Zero knowledge proof protocol is a essential component of cryptography, which in recent years has increasingly popular amongst scholars. Its applications have widened and it has made inroads in several areas including mathematics and network safety and so on. This simple protocol based on zero knowledge proof by which user can prove to the authentication server that he has the password without having to send the password to the server either clear text or in encrypted format. This is a protocol in which the data learned by one party (i.e., The inspector) allow him/her to verify that a statement is true but does not reveal any additional information. In this paper we first discuss about zero-knowledge protocol proof system of knowledge and also key exchange between users and which then is modified into an authentication scheme with secret key exchange for confidentiality. The whole protocol involves mutual identification of two users, exchange of a random common secret key or session key for the verification of public keys.

Cryptography and Data Security
Chaos-based Image/Signal Encryption
Security in Wireless Sensor Networks
Original source
Dec 1, 2014
2 cites
ParaMASK: A Multi-Agent System for the efficient and dynamic adaptation of HPC workloads

Mateusz Guzek, Xavier Besseron, Sébastien Varrette, Grégoire Danoy · 5 authors

The growing parallelism and heterogeneity of modern computing infrastructures such as High Performance Computing (HPC) platforms raises new challenges to their programmers and users. Additional requirements have emerged nowadays, such as minimizing the consumed energy, reducing the utilized system resources, or providing built-in reliability mechanisms. Therefore High Performance Computing (HPC) applications require adaptation mechanisms and then must avoid traditional monolithic centralized approaches in favor of novel autonomous, flexible and decentralized decision systems. In this context, we describe here a dynamic and flexible adaptation scheme based on a Multi-Agent System (MAS) to handle parallel or distributed executions in an HPC environment. More precisely, we model and extend the existing HPC middleware Kaapi to offer the power of the ParaMoise multi-agent organizational framework. Our proposed solution, named ParaMASK, relies on the similarities between ParaMoise workflow-based functional specifications and the Direct Acyclic Graph (DAG) representation of the distributed execution within Kaapi. As a result, ParaMASK permits to analyze and reorganize the scheduling of tasks that compose a program in an autonomous and decentralized way, while additionally handling dynamic adaptations (using task migration to fulfill energy consumption goals for example). The proposed solution was implemented on top of the existing Kaapi middleware and includes an optimized algorithm for the agent coordination. ParaMASK has been validated with a series of experiments on a real computational grid. Experimental results show a good scalability and an exceptional low overhead induced by the approach: less than 1.5% execution time increase with periodic coordinations every 15 seconds on 2662 cores.

Open access
Distributed and Parallel Computing Systems
Distributed systems and fault tolerance
Cloud Computing and Resource Management
Original source
Dec 1, 2014
6 cites
A Cloud-Oriented Services Self-Management Approach Based on Multi-agent System Technique

Fu Hou, Xinjun Mao, Wei Wu, Lu Liu · 5 authors

The services in open and dynamic Cloud Computing environments are often decentralized, evolving and are cross-organized. Efficient management of the Cloud resources for the improvement of the quality and efficiency of the service provisions is still being a continuous challenge in Cloud domain. To this end, this paper proposes a self-management approach for the cloud services, with an autonomous and context-aware management of the resources by employing a number of service agents in the Cloud environment. Such service agents are responsible for organizing the service management through their local interactions in the system, and further leading to the expected emergence of the service organization. Based on this approach, we present a cloud-oriented services self-management framework with suitable mechanisms for service aggregations and service provisions. And, two supporting algorithms are designed to implement the proposed services self-organization process and also the service provision process. We conduct various experiments on public data sets, and the experimental results prove the efficiency of our system in improving the quality of the Cloud services and also the practical adaptability of our system to the ever changing Cloud environment.

Cloud Computing and Resource Management
Software System Performance and Reliability
Peer-to-Peer Network Technologies
Original source
Dec 1, 2014·PubMed
12 cites
The First Stages of Liberalization of Public Hospitals in Iran: Establishment of Autonomous Hospitals and the Barriers.

Nader Markazi Moghaddam, Aidin Aryankhesal, Mohammad Arab

BACKGROUND: Liberalization and decentralization of public sector has been triggered in some developing countries and in Iran by the Ministry of Health and Medical Education (MOHME) that granted autonomy to 54 public hospitals. However, establishment of such a complex organizational reform was rather unsuccessful. We aimed to explore the obstacles and barriers caused such a failure and their mechanisms. METHODS: Using a qualitative approach in 2013, we consulted key informants at the autonomous hospitals and their affiliating universities. Data collection was done within two phases: (i) 276 unstructured questionnaires asking respondents of barriers, and (ii) 23 semi-structured interviews from the first phase's key respondents. The first phase data were analyzed using thematic analysis and the second's by framework approach based on the frame shaped at the first phase. RESULTS: Nine obstacles were recognized including "autonomous hospitals' board composition", "delay in announcing autonomous hospitals' charges by the MOHME", "lack of financing by the committed organizations", "poor follow up for implementation of the reform", "irregular board meetings", "lack of an external overseer", "shortage of full-time physicians", "lack of management stability", and "health insurance organizations' delayed payments". CONCLUSION: The MOHME and insurance organizations did not pay the reform expenses. There were some competing motives as well to slow the reform or to shut it down. The stages of policy formulation and implementation were done separately in Iran, so this big organizational reform encountered serious obstacles.

Open access
Healthcare Systems and Reforms
Global Maternal and Child Health
Healthcare Quality and Management
Original source
Dec 1, 2014·Journal of Computer Science
3 cites
SECURE WIRELESS AD HOC NETWORKS USING ZERO KNOWLEDGE PROOF

Benfano Soewito, Yonathan Marcellinus, Manik Hapsara

A Mobile Ad-hoc Network (MANET) is a group of wireless mobile nodes that dynamically form a network without any pre-established infrastructure or centralized administration, Soewito (2014). Some network hops may be needed to send a packet from one node to another node in the MANET. To do the communication between the nodes, a route has to be selected in the network, therefore it need a routing protocol that manage selection of the route. Selection route in mobile ad-hoc network is not easy because nodes always move so that the topology of network always changed every time. This is a big issue in selection route in mobile ad-hoc network because the route can be broken anytime. Moreover, MANET is more vulnerable than other wireless communication types because every mobile node serves as both the host and the router and forwards packets on behalf of each other. This study presents the analyzing and evaluation several routing algorithms and a novel scheme to build an authentication system by adding the modified zero knowledge proof algorithm to each mobile node in MANET.

Open access
Mobile Ad Hoc Networks
Security in Wireless Sensor Networks
Vehicular Ad Hoc Networks (VANETs)
Original source
Dec 1, 2014
64 cites
Investigating factors behind choosing a cryptocurrency

Aamna Al Shehhi, Mayada Oudah, Zeyar Aung

This paper aims to address two challenging questions: "What are the bases on which online users choose to use and/or mine their cryptocurrency?" and "Which factors strongly affect the coin's popularity and value?" We try to answer these two questions for eight chosen cryptocurrencies by conducting an online survey. Results obtained from the online survey indicate that more than half of the participants believe that the currency name and logo affect the choice of using and/or mining a cryptocurrency. Moreover, the majority of the participants stated the ease of mining, having strong, fun and large community along with anonymity, privacy, and the currency's value, popularity, potential and technology as advantages of the currencies they choose to use and/or mine. These results help in observing the people behavior and predicting how successful and promising a cryptocurrency might become in the future.

Blockchain Technology Applications and Security
Original source
Nov 30, 2014
0 cites
5 Analytic Continuation

Leslie Copley

Our focus in the last chapter was on the construction of an analytic function from a knowledge of its singularities.More often than not, however, we are confronted with the inverse problem: given some knowledge of a function in a restricted region of its domain of holomorphy, determine its singularities.This will be the focus of the present chapter.We have seen repeatedly that one need not know all that much about an analytic function in order to determine its value everywhere in the complex plane or on its Riemann surface.Cauchy's Integral Representation can be viewed as the embodiment of this property and thus far it has provided the key to exploiting it.We are now going to nd out what constitutes a minimal set of information for the determination of an analytic function.The answer is one that is best exploited not by Cauchy's Integral but by one of its consequences, the Taylor series.In so doing, we shall also nd out how to use a representation of a function that is valid in one domain of the complex plane to determine its values at points outside the domain or indeed, at any points where it is holomorphic.Our starting point is the following theorem which, despite its innocuous appearance, is one of the most remarkable results of complex analysis.Theorem: Let f (z) and f (z) be holomorphic in a domain D of the complex plane.If the two functions coincide in any neighbourhood, however small, of a point z in D, or even on a point set with an accumulation point in D, then they coincide throughout D. Proof: The function f (z)f (z) is holomorphic throughout D and has a set of zeros consisting of the points where f (z) and f (z) coincide, with an accumulation point in D. We know that in any domain where it is holomorphic a function either has isolated zeros or it is identically zero.Thus,What this theorem establishes is that a holomorphic function is uniquely determined everywhere within its domain of holomorphy by its behaviour in the neighbourhood of an arbitrary point of that domain.But how can one exploit this remarkable property?Obviously not by means of a Cauchy Integral or dispersion representation or anything else of that ilk as we lack the necessary input information.However, what we do have is precisely the information needed to determine a Taylor series representation.Suppose that we know the value of the function f (z) throughout a neighbourhood of the point z = z which is a point lying within the function's domain of holomorphy, D. This is su cient to permit calculation of the coe cients c = f (z ), c = f (z ), . . ., cm = m! f (m) (z ), . . .

Open access
Algebraic and Geometric Analysis
Holomorphic and Operator Theory
Analytic and geometric function theory
Original source