On Computable Numbers: Corrections and Critiques

2004 ◽  
Author(s):  
Alan Turing ◽  
Emil Post
Keyword(s):  
Short Note ◽  
Degree Theory ◽  
London Mathematical Society ◽  
Problem Solver ◽  
Symbolic Logic ◽  
Present Position ◽  
Symbol Space ◽  
Important Field ◽  
Formed Part ◽  
The Right

As is not uncommon in work of such complexity, there are a number of mistakes in ‘On Computable Numbers’. Turing corrected some of these in his short note 2.1, published in the Proceedings of the London Mathematical Society a few months after the original paper had appeared. The mathematician Emil L. Post’s critique of ‘On Computable Numbers’ was published in 1947 and formed part of Post’s paper ‘Recursive Unsolvability of a Problem of Thue’. Post is one of the major figures in the development of mathematical logic in the twentieth century, although his work did not gainwide recognition until after his death. (Born in 1897, Post died in the same year as Turing.) By 1936 Post had arrived independently at an analysis of computability substantially similar to Turing’s. Post’s ‘problem solver’ operated in a ‘symbol space’ consisting of ‘a two way infinite sequence of spaces or boxes’. A box admitted ‘of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke’. The problem solver worked in accordance with ‘a fixed unalterable set of directions’ and could perform the following ‘primitive acts’: determine whether the box at present occupied is marked or not; erase any mark in the box that is at present occupied; mark the box that is at present occupied if it is unmarked; move to the box to the right of the present position; move to the box to the left of the present position. Later, Post considerably extended certain of the ideas in Turing’s ‘Systems of Logic Based on Ordinals’, developing the important field now called degree theory. In his draft letter to Church, Turing responded to Post’s remarks concerning ‘Turing convention-machines’. It is doubtful whether Turing ever sent the letter. The approximate time of writing can be inferred from Turing’s opening remarks: Kleene’s review appeared in the issue of the Journal of Symbolic Logic dated September 1947 (12: 90–1) and Turing’s ‘Practical Forms of Type Theory’ appeared in the same journal in June 1948.

Finite combinatory processes—formulation

1936 ◽  
Vol 1 (3) ◽  
pp. 103-105 ◽  
Author(s):  
Emil L. Post
Keyword(s):  
General Problem ◽  
Specific Problem ◽  
Infinite Sequence ◽  
Problem Solver ◽  
Symbolic Logic ◽  
Present Formulation ◽  
Symbol Space ◽  
Gödel’S Theorem ◽  
Unsolvable Problems

The present formulation should prove significant in the development of symbolic logic along the lines of Gödel's theorem on the incompleteness of symbolic logics and Church's results concerning absolutely unsolvable problems.We have in mind a general problem consisting of a class of specific problems. A solution of the general problem will then be one which furnishes an answer to each specific problem.In the following formulation of such a solution two concepts are involved: that of a symbol space in which the work leading from problem to answer is to be carried out, and a fixed unalterable set of directions which will both direct operations in the symbol space and determine the order in which those directions are to be applied.In the present formulation the symbol space is to consist of a two way infinite sequence of spaces or boxes, i.e., ordinally similar to the series of integers …, − 3, − 2, − 1, 0, 1, 2, 3, …. The problem solver or worker is to move and work in this symbol space, being capable of being in, and operating in but one box at a time. And apart from the presence of the worker, a box is to admit of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke.

scholarly journals Foreign Policy of Hungary Towards Ukraine or “European Menu à la Carte”

2018 ◽  
pp. 603-612
Author(s):  
Serhii Esaulov
Keyword(s):  
Foreign Policy ◽  
International Relations ◽  
Law And Order ◽  
Governmental Organizations ◽  
Non Governmental Organizations ◽  
Historical Justice ◽  
The World ◽  
Formed Part ◽  
The Right ◽  
Expert Community

The author raises the issue of settling conflicts around the world and discusses modern attempts to establish law and order. Particular attention is paid to the intricate relations between Hungary and Ukraine. With Russia’s aggression against Ukraine there was ruined a system of international relations, which provided for the rule of law, the right to settle disputes without applying military tools, force or threats. Russia initiated a new precedent of impunity, insolent violation of the fundamental norms of international law, and demonstrated the world how the borders may be redrawn as one sees fit and “bring historical justice”. The author notes that one of the reasons for the escalation of the conflict between Hungary and Ukraine has become the language issue. Still, however pity it is, all attempts of the Ukrainian side to resolve conflict matters have appeared to be vane, since Budapest is reluctant to listen to and consider any arguments of Kyiv, being fully distracted by its demand. It is hard to imagine that in civilized “old” Europe, Germany, for instance, would express claims or even threaten France for the fact that pupils in schools of the French region of Alsace (until 1918, its territory formed part of Germany that attempted to annex it at times of the Second World War) are taught in the official language – French, not in the language of the neighbouring country, even though the Alsatian and German languages are equally spoken there. Unfortunately, Hungary seems not to be ready to follow the example of the Franco-German reconciliation in terms of relations with all neighbours, despite the philosophy of its membership in the EU and NATO. The revenge-seeking attitudes of the Hungarian political establishment regarding the revision of borders according to the Versailles and Yalta systems of international relations are constantly boosted in all directions in the neighbouring countries, where ethnic Hungarians live (Romania, Slovakia, Serbia and Ukraine). The so-called “formula of protecting interests of Russian citizens in Crimea and Donbas” adopted from Putin has apparently laid the foundation for the foreign policy strategy of V. Orban. First, as regards the appeal to make the region of ethnic Hungarians’ residence autonomous and subsequently – the appeal to hold a referendum on separation. The author summarizes that along with the political and diplomatic efforts, a substantial role in easing the tension in relations with Budapest should be played by non-governmental organizations and the expert community though holding forums and scientific conferences aiming at discussing the above-mentioned issues. Keywords: Hungary, conflict, Law on Language, geopolitics, strategies, foreign policy, Ukraine.

The face of energy

2019 ◽  
Vol 59 (3) ◽  
Author(s):  
Peter Bennett
Keyword(s):  
Environmental Sustainability ◽  
Public Perception ◽  
Oil And Gas ◽  
Political Influence ◽  
Oil And Gas Industry ◽  
Problem Solver ◽  
Gas Industry ◽  
The Public ◽  
The Face ◽  
The Right

In this era of technological disruption, when many industries are fighting to stay relevant, the oil and gas industry seems to be stagnant. It is in this environment where public perception of the modern industry is becoming more critical and as younger consumers grow in both number and political influence, their viewpoints will become especially vital to the continued relevance of the industry. The oil and gas industry gives itself high marks for innovation, safety and environmental sustainability, and yet the public opinion in these areas is often portrayed very negatively. We have an image problem. The belief that oil and gas is good for society seems to decline with each younger generation. The public believes the industry is necessary for society, though they still see it as a problem causer, not a problem solver. But support for the industry falls with each generation and millennials are more likely to believe the industry is bad for society and a problem causer. The oil and gas industry needs to communicate and engage with consumers to identify ways to better understand their motivations and concerns. Clearly there is a gap in how the public and executives view the industry and the time to address these perceptions is now. To view the video, click the link on the right.

The Nature of Ususfructus

1946 ◽  
Vol 9 (2) ◽  
pp. 159-170
Author(s):  
Kopel Kagan
Keyword(s):  
Great Difficulty ◽  
Roman Law ◽  
Present Position ◽  
Modern Times ◽  
The Subject ◽  
The One ◽  
Definition Of ◽  
The Right ◽  
Opening Statements ◽  
In Virtue Of

No satisfactory definition of Dominium in Roman Law has yet been achieved. Amongst English writers Austin many years ago found great difficulty in this question while in modern times Professor Buckland has written ‘it is thus difficult to define Dominium precisely.’ Again, Poste, dealing with Gaius' discussion of dominium, says that his opening statements are ‘deplorably confused.’ These examples are enough to indicate the condition, of uncertainty which prevails. In my submission this uncertainty exists mainly because the conception of ususfructus has never yet been explained adequately. Of Possessio it has been said ‘the definition of Possessio to give the results outlined is a matter of great difficulty. No perfectly correct solution may be possible,’ and this statement is generally accepted as a correct assessment of the present position in juristic literature. But here, too, in my opinion, the reason is again connected with usufruct, for the possessio of the usufructuary has not yet been adequately determined. Gaius (2.93) tells us ‘usufructuarius vero usucapere non potest; primuum quod non possidet, sed habet ius utendi et fruendi.’ Ulpian holds that he had possessio in fact (‘Naturaliter videtur possidere is qui usum fructum habet’ D.41.2.12). On this subject Roby says ‘the fructuary was not strictly a possessor and therefore if he was deprived from enjoying he had not a claim to the original interdict de vi but in virtue of his quasi-possessio a special interdict was granted him.’ Austin saw difficulty in the whole problem of possessio. He wrote ‘by Savigny in his treatise on possessio it is remarked that the possessio of a right of usufruct … resembles the possessio of a thing, by the proprietor, or by an adverse possessor exercising rights of property over the thing. And that a disturbance of the one possession resembles the disturbance of the other. Now this must happen for the reason I have already stated:—namely, that the right of usufruct or user, like that of property, is indefinite in point of user. For what is possession (meaning legal possession not mere physical handling of the subject) but the exercise of a right ?’

Global Bifurcation of Periodic Solutions in Symmetric Reversible Second Order Systems with Delays

2021 ◽  
Vol 31 (12) ◽  
pp. 2150180
Author(s):  
Zalman Balanov ◽  
Joseph Burnett ◽  
Wiesław Krawcewicz ◽  
Huafeng Xiao
Keyword(s):  
Periodic Solutions ◽  
Autonomous Systems ◽  
Global Bifurcation ◽  
Degree Theory ◽  
Second Order ◽  
Equivariant Degree ◽  
Commensurate Delays ◽  
Spatio Temporal ◽  
Second Order Systems ◽  
The Right

Global bifurcation and spatio-temporal patterns of periodic solutions (with prescribed period) to second order reversible equivariant autonomous systems with commensurate delays are studied using the Brouwer/Leray–Schauder [Formula: see text]-equivariant degree theory. Here, [Formula: see text] is related to the reversal symmetry combined with the autonomous form of the system, [Formula: see text] reflects extra spacial symmetries of the system, and [Formula: see text] is related to the oddness of the right-hand side. Abstract results are supported by a concrete example with [Formula: see text] — the dihedral group of order 12.

scholarly journals Revolt from the Right: Russia’s Right-Wing Students Between Conservatism and Radicalism

2016 ◽  
Vol 47 (1) ◽  
pp. 32-54
Author(s):  
George Gilbert
Keyword(s):  
Moral Education ◽  
Russian Society ◽  
Right Wing ◽  
Spiritual Renewal ◽  
Late Imperial ◽  
Anti Semitism ◽  
Imperial Period ◽  
Formed Part ◽  
The Russian Empire ◽  
The Right

The article examines the Academist movement between 1900 and 1914 – the student branches of a number of right-wing groups that emerged in the Russian Empire between 1900 and 1905 and endured throughout the late imperial period. It will argue that these groups arose separately from the Russian autocracy, and formed part of an independent, ‘right-wing’ approach to the problems facing Russian society in the late imperial period. It is particularly concerned with the idea, widely present on the right, that the Russian present was in a period of crisis and a more drastic approach to moral and spiritual renewal was needed. It will consider the nature of the Academists’ conceptions of moral education, spiritual renewal of society, and also their violence, anti-Semitism and emergence of an ethno-populist politics. The contention is that the emergence of an independent right-wing movement contributed to the wider instability in the Russian autocracy in the late imperial period.

Options: as Safe as Houses?

1984 ◽  
Vol 43 (1) ◽  
pp. 55-87
Author(s):  
Stephen Tromans
Keyword(s):  
Real Property ◽  
Long Time ◽  
The Law ◽  
Formed Part ◽  
The Right

Options over land have formed part of the law of real property for a long time. Their oldest uses are probably in the context of wills, allowing stated persons to purchase part of the testators” property, and in the law of landlord and tenant, giving the tenant the right to call for a renewal of the term or in some cases to purchase the freehold. Options are still used for both of these purposes, although statutory developments in the law of landlord and tenant have rendered the option to renew of less significance than was once the case. In addition, other uses have been found for options by landowners and conveyancers.

Deciding Optimal Noise Monitoring Sites with Matrix Gray Absolute Relation Degree Theory

2015 ◽  
Vol 14 (04) ◽  
pp. 1550032
Author(s):  
Zhihua Gao ◽  
Yadan Li ◽  
Limin Zhao ◽  
Shuangwei Wang
Keyword(s):  
Noise Level ◽  
Measurement Data ◽  
Degree Theory ◽  
Actual Measurement ◽  
Optimistic Solution ◽  
Noise Measurements ◽  
Noise Data ◽  
The Right ◽  
Theoretical Simulations ◽  
Noise Map

Noise maps are applied to assess noise level in cities all around the world. There are mainly two ways of producing noise maps: one way is producing noise maps through theoretical simulations with the surrounding conditions, such as traffic flow, building distribution, etc.; the other one is calculating noise level with actual measurement data from noise monitors. Currently literature mainly focuses on considering more factors that affect sound traveling during theoretical simulations and interpolation methods in producing noise maps based on measurements of noise. Although many factors were considered during simulation, noise maps have to be calibrated by actual noise measurements. Therefore, the way of obtaining noise data is significant to both producing and calibrating a noise map. However, there is little literature mentioned about rules of deciding the right monitoring sites when placed the specified number of noise sensors and given the deviation of a noise map produced with data from them. In this work, by utilizing matrix Gray Absolute Relation Degree Theory, we calculated the relation degrees between the most precise noise surface and those interpolated with different combinations of noise data with specified number. We found that surfaces plotted with different combinations of noise data produced different relation degrees with the most precise one. Then we decided the least significant one among the total and calculated the corresponding deviation when it was excluded in making a noise surface. Processing the left noise data in the same way, we found out the least significant datum among the left data one by one. With this method, we optimized the noise sensor’s distribution in an area about 2[Formula: see text]km2. And we also calculated the bias of surfaces with the least significant data removed. Our practice provides an optimistic solution to the situation faced by most governments that there is limited financial budget available for noise monitoring, especially in the undeveloped regions.

Arthur W. Apter. On the least strongly compact cardinal. Israel journal of mathematics, vol. 35 (1980), pp. 225–233. - Arthur W. Apter. Measurability and degrees of strong compactness. The journal of symbolic logic, vol. 46 (1981), pp. 249–254. - Arthur W. Apter. A note on strong compactness and supercompactness. Bulletin of the London Mathematical Society, vol. 23 (1991), pp. 113–115. - Arthur W. Apter. On the first n strongly compact cardinals. Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 2229–2235. - Arthur W. Apter and Saharon Shelah. On the strong equality between supercompactness and strong compactness.. Transactions of the American Mathematical Society, vol. 349 (1997), pp. 103–128. - Arthur W. Apter and Saharon Shelah. Menas' result is best possible. Ibid., pp. 2007–2034. - Arthur W. Apter. More on the least strongly compact cardinal. Mathematical logic quarterly, vol. 43 (1997), pp. 427–430. - Arthur W. Apter. Laver indestructibility and the class of compact cardinals. The journal of symbolic logic, vol. 63 (1998), pp. 149–157. - Arthur W. Apter. Patterns of compact cardinals. Annals of pure and applied logic, vol. 89 (1997), pp. 101–115. - Arthur W. Apter and Moti Gitik. The least measurable can be strongly compact and indestructible. The journal of symbolic logic, vol. 63 (1998), pp. 1404–1412.

2000 ◽  
Vol 6 (1) ◽  
pp. 86-89
Author(s):  
James W. Cummings
Keyword(s):  
Israel Journal ◽  
American Mathematical Society ◽  
London Mathematical Society ◽  
Mathematical Society ◽  
Symbolic Logic ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Applied Logic ◽  
Strong Compactness ◽  
Strongly Compact Cardinals

Memoirs: Notes on the Development of Paludina Vivipara, with Special Reference to the Urinogenital Organs and Theories of Gasteropod Torsion

1902 ◽  
Vol s2-46 (181) ◽  
pp. 97-141
Author(s):  
ISABELLA M. DRUMMOND
Keyword(s):  
Special Reference ◽  
Comparative Anatomy ◽  
Vertical Displacement ◽  
The Body ◽  
Negative Evidence ◽  
Present Position ◽  
Forward Movement ◽  
The Right

To sum up, then, theories of Gasteropod torsion may be divided into two classes: a. Those which view the present position of the palleal complex as due to a forward movement along the right side of the body, which resulted from greater growth of the left side of the body than of the right. h. Those which view the present position of the palleal complex as due to a ventral flexion followed by a vertical rotation of the whole visceral hump upon the head. The evidence for the second of these views seems greater than that for the first, in that-- 1. A vertical displacement through 180° of all the organs contained in the visceral hump takes place in the course of ontogeny. 2. There is some evidence, both from comparative anatomy and embryology, for believing that the oesophagus has undergone an actual twist. 3. Monstrosities which retain the palleal complex in a ventral position show a tendency to form an exogastric coil. The innervation of the mantle was shown to be equally difficult to explain on either hypothesis. Also, against the first view was urged the insufficiency of the evidence upon which Bütschli bases his conclusions with regard to zones of unequal growth. With regard to the phylogenetic cause of the vertical twist, embryology can only give negative evidence; while in considering the ontogenetic cause we are thrown back upon unsolved problems of heredity, and must confess our ignorance. In conclusion, I wish to offer most hearty thanks to Professor Weldon, not only for having placed freely at my disposal all the resources of the laboratory, but also for most kind personal aid at all stages of the work. My thanks are also due to Mr. Richard Evans for much help in the technique.

Sign in / Sign up

Export Citation Format

Share Document