50 research outputs found

    Computability Theory

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    Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science

    Degrees of members of ∏01 classes

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    Abstract unable to be displayed accurately. Please see eThesis for full detail

    The Reverse Mathematics of CAC for trees

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    CAC for trees is the statement asserting that any infinite subtree of N<N\mathbb{N}^{<\mathbb{N}} has an infinite path or an infinite antichain. In this paper, we study the computational strength of this theorem from a reverse mathematical viewpoint. We prove that TAC for trees is robust, that is, there exist several characterizations, some of which already appear in the literature, namely, the tree antichain theorem (TCAC) introduced by Conidis, and the statement SHER introduced by Dorais et al. We show that CAC for trees is computationally very weak, in that it admits probabilistic solutions.Comment: 28 page

    On the Nonexistence of a Strong Minimal Pair

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    Two nonzero recursively enumerable (r.e.) degrees a\mathbf{a} and b\mathbf{b} form a strong minimal pair if a∧b=0\mathbf{a} \wedge \mathbf{b}=\mathbf{0} and b∨x≥a\mathbf{b}\vee \mathbf{x}\geq \mathbf{a} for any nonzero r.e. degree x≤a\mathbf{x}\leq \mathbf{a}. We prove that there is no strong minimal pair in the r.e. degrees. Our construction goes beyond the usual 0′′′\mathbf{0}'''-priority arguments and we give some evidence to show that it needs 0(4)\mathbf{0}^{(4)}-priority arguments

    Reducibilities in recursive function theory.

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    Massachusetts Institute of Technology. Dept. of Mathematics. Thesis. 1966. Ph.D.Bibliography: leaves 102-103.Ph.D

    A Hyperimmune Minimal Degree and an ANR 2-Minimal Degree

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    Randomness and Computability

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    This thesis establishes significant new results in the area of algorithmic randomness. These results elucidate the deep relationship between randomness and computability. A number of results focus on randomness for finite strings. Levin introduced two functions which measure the randomness of finite strings. One function is derived from a universal monotone machine and the other function is derived from an optimal computably enumerable semimeasure. Gacs proved that infinitely often, the gap between these two functions exceeds the inverse Ackermann function (applied to string length). This thesis improves this result to show that infinitely often the difference between these two functions exceeds the double logarithm. Another separation result is proved for two different kinds of process machine. Information about the randomness of finite strings can be used as a computational resource. This information is contained in the overgraph. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not truth-table complete. This question is answered in the negative. Related results are also established. This thesis makes advances in the theory of randomness for infinite binary sequences. A variant of process machines is used to characterise computable randomness, Schnorr randomness and weak randomness. This result is extended to give characterisations of these types of randomness using truthtable reducibility. The computable Lipschitz reducibility measures both the relative randomness and the relative computational power of real numbers. It is proved that the computable Lipschitz degrees of computably enumerable sets are not dense. Infinite binary sequences can be regarded as elements of Cantor space. Most research in randomness for Cantor space has been conducted using the uniform measure. However, the study of non-computable measures has led to interesting results. This thesis shows that the two approaches that have been used to define randomness on Cantor space for non-computable measures: that of Reimann and Slaman, along with the uniform test approach first introduced by Levin and also used by Gacs, Hoyrup and Rojas, are equivalent. Levin established the existence of probability measures for which all infinite sequences are random. These measures are termed neutral measures. It is shown that every PA degree computes a neutral measure. Work of Miller is used to show that the set of atoms of a neutral measure is a countable Scott set and in fact any countable Scott set is the set of atoms of some neutral measure. Neutral measures are used to prove new results in computability theory. For example, it is shown that the low computable enumerable sets are precisely the computably enumerable sets bounded by PA degrees strictly below the halting problem. This thesis applies ideas developed in the study of randomness to computability theory by examining indifferent sets for comeager classes in Cantor space. A number of results are proved. For example, it is shown that there exist 1-generic sets that can compute their own indifferent sets

    Studies on some bovine mycoplasmas

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    My studies on bovine mycoplasnas fall under three separate headings:A. ANGPLASMA MYQPI.DES VAR. MCOIDES AND CONTÁ2IOUS BOVINE UROPNEUMONIA. An investigation was undertaken to determine whether the allergic reaction could be developed as a useful diagnostic test for contagious bovine pleuropneumonia. Using antigens obtained from M. mycoides var. n coides two types of reaction were observed, an immediate reaction associated with lipopolysaccharide material and a delayed reaction associated with protein. It was impossible to separate the protein completely from the lipopolysaccharide, which elicited some non-specific reactions, and the test was abandoned.A comparison of various serological tests employed in contagious bovine pleuropneumonia diagnosis revealed that no single test was entirely satisfactory, but the complement fixation test was the best available. Growth inhibition of M. mycoides var. mycoides by serum was examined as a diagnostic test and as a measure of immunity. Two tests were evolved, one on solid medium and the other in liquid medium, and both proved to be of potential value for contagious bovine pleuropneumonia diagnosis, but their value in measuring immunity was not determined. Inhibition was observed, to occur in the absence of agglutination and heat-labile accessory factors.The lipopolysaccharide from M. mycoides var. mycoides, responsible for the immediate allergic reaction, was shown to be serologically related to a galactan extracted from normal bovine lung. This lipopolysaccharide was also shown to be aritigenically related to polysaccharides from a number of different sources including other microorganisms and this relationship was considered to be responsible for non-specific reactions discerned in diagnostic tests.It was suggested that the serological relationship between the lipopolysaccharide of M. Ayssides vay_t_mycoides and the galactan from bovine lung might play a role in the pathogenesis of contagious bovine pleuropneumonia. A preliminary examination of this percept was inconclusive, but it was shown, fortuitously, that immunity could be transferred passively.Electronmicroscopic examination of ItmmixleamaEl. ycoides, treated with specific antiserum, revealed a capsule or slime layer round the organisms. This capsule is probably composed of galactan.A method devised for the production, testing and transport of a modified Ti contagious bovine pleuropneumonia vaccine is described.Chicken embryos, from hens previously inoculated with M. mycoides var. mycoides, were immune to challenge with this organism, suggesting that the chicken embryo might be a suitable host for studying immunity to contagious bovine pleuropneumonia.B. MYCOPLASMAS WITH CALF NEUMONIA AND BOVINE KERATOCON'UNCTIVITIS. Material from pneumonic lungs cultured in special media resulted in the isolation of two species of xaycoplasma hitherto unrecorded from lungs. One was a new species of mycoplasma, subsequently named M. dispar, and the other was T-mycoplasna. Mycoplasmas were isolated from 75% of 65 pneumonic lungs examined; these comprised T-mycoplasmas from 58.5% of lungs, M. dispar from 51% and M. bovirhinis from 23%.The pathogenicity of the T-mycoplasmas was tested by inoculating cultures endobronchially into 3-week-old calves. Gross pneumonic lesions were observed in 14 out of the 16 calves at slaughter, compared with 2 out of 9 that revealed lesions following inoculation of control materials (P<0.01). Certain T-mycoplasmas also produced mastitis in the mammary glands of cows when inoculated via the teat canal. Strains both virulent and avirulent for the bovine mammary gland could be isolated from cattle, but human, canine and simian strains of T-mycoplasmas were avirulent. Bovine T-mycoplasmas were shown to be serologically heterogeneous.T-mycoplasmas, M. bovirhinis, Acholejlasma laidlawii and an unidentifiable species of mycoplasma were isolated from eyes of cattle suffering from keratoconjunctivitis.C. MYCOPLASMATALES VIRUSES Three distinct viruses have been isolated. All infect A. laidlawii producing visible plaques on 'lawns' of this organism. The first virus isolated, named Mycoplasmatales virus - laidlawii 1 (MV-L1), is a small (16 x 90 nm.) rod-shaped DNA containing virus. The second virus, named MV-L2, is roughly spherical, apparently enveloped and about 80 nm, in diameter (range 50-125 nm.). It appears to contain DNA and is serologically distinct from MV-Ll. The third virus (MV-L3), only recently isolated, is polyhedral and uniform in size (about 54 nm, diameter) and is serologically distinct from both MV-Li and MV-L2.PUBLICATIONS IN SUPPORT OF CANDIDATURE: A MYCOPLASMA MYCOIDES VAR. MYCOIDES AND CONTAGIOUS BOVINE PLEUROPNEUMONIA 1. GOURLAY, R. N. (1964). The allergic reaction in contagious bovine pleuropneumonia. Journal of Esylpåratiae2211-22122x 74, 286-299. || 2. & PALMER, R. F. (1965). Further studies on the allergic reaction in contagious bovine pleuropneumonia. Journal of Comparative Pathology 75, 89-95. || 3. & SHIFRINE, M. (1965) . Comparison between methods of antigen preparation and the use of adjuvant in the delayed allergic skin reaction in contagious bovine pleuropneumonia. Journal of Comparative Pathology 75, 375-380. || 4. SHIFRINE, M. & GOURLAY,. R. N. (1965). The immediate type allergic skin reaction in contagious bovine pleuropneumonia. Journal of Comparative Pathology 75, 381-385. || 5. & (1967). Serological relationships between Mycoplasma mycoides and other bacteria. Annals of the New York Academy of Science. 143, 311-324. || 6. GOURLAY, R. N. (1935). Comparison between some diagnostic tests for contagious bovine pleuropneumonia. Journal of Comparative Pathology 75, 97-109. || 7. SHIFRINE M. & GOURLAY, R. N. (1967). Evaluation of diagnostic tests for contagious bovine pleuropneumonia. ER112Iinafpizootic Diseases of Africa 15, 7-10, || 8. DOMERMUTH, C. H. & GOURLAY, R. N. (1967). A solid medium test for measuring growth inhibition and neutralisation of ypyARxçc2LdMcolasxes by immune bovine serum. Journal of General Microbiology 47, 289-294. || 9. GOURLAY, R. N. & DOMERMUTH, C. H. (1967). Growth inhibition and 'neutralisation' of Mycoplasma mycoides by immune bovine serum. Annals of the New York Academ of Science 143, 325-336. || 10. SHIFRINE, M. & GOURLAY, R. N. (1965). Serological relationship between galactans from normal bovine lung and Mycoplasma Nature, London 208, 498-499. || 11. GOURLAY, R. N. & SHIFRINE, M. (1966). Antigenic cross-reactions between the galactan from plasmanasoides and polysaccharides from other sources. Journal of Corn arative Patholoc 76, 417-425. || 12. & BROCKLESBY, D. W. (1967). Preliminary experiments with antigens of Theileria påLya (Theiler, 1904). British Veterinary_lamIlL 123, 533-540. || 13. & SHIFRINE, M. (1966). Passive transfer of immunity and formation of lung lesions following intravenous inoculation of Mycoplasma nix,21.sisa and immune bovine serum. Bulletin of Epiaslotic Diseases of Africa 14, 369-372. || 14. GOURLAY, R. N. & THROWER, K. J. (1968). Morl,hology of Llysgpjau_uarzlxcoides thread-phase growth. Journal of General Microbiology 54, 155-159. || 15. BROWN, R. D., GOURLAY, R. N. & MACLEOD, A. K. (1965). The production of Ti broth culture contagious bovine pleuropneumonia vaccine. Bulletin of E izootic Diseases of Africa 13, 149-155. || 16. GOURLAY, R. N. & MACLEOD, A. K. (1966). Fermentation of glucose by Mys2212sma mycoides and its effect on viability. Bulletin of E izootic Diseases of Africa 14, 373-381. || 17. & SHIFRINE, M. (1966). The use of embryonated chicken eggs for the study of immunity to MYcoplasma Tycoides. Veterinary Record 78, 256-257. || 18. & (1968). The virulence and viability of Mycoplasma mycoides strains in chicken embryos from normal and immunised hens. Research in Veterinary Science 9, 185-186. || B MYCOPLASMAS ASSOCIATED WITH CALF PNEUMONIA AND BOVINE KERATOCONJUNCTIVITIS: 19. GOURLAY, R. N. (1969). Isolation of a mycoplasma-like organism from pneumonic calf lungs. Veterinary Record 84, 229-230. || 20. & LEACH, R. H. (1970). A new mycoplasma species isolated from pneumonic lungs of calves (Mycoplasma dispar, Sp. Nov.) Journal of Medical Microbioloq 3, 111-123. || 21. (1968). The isolation of T-strains of Mycoplasma from pneumonic calf lungs. Research in Veterinar Science 9, 376-378. || 22. & THOMAS, L. H. (1969). The isolation of large colony and T-strain mycoplasmas from cases of bovine kerato-conjunctivitis. Veterinary Record 84, 416-417. || 23. & MACKENZIE, A. & COOPER, J. E. (1970). Studies of the microbiology and pathology of pneumonic lungs of calves. Journal of comparatitive Pathology 80, 575-584. || 24. & THOMAS, L. H. (1970). The experimental production of pneumonia in calves by the endobronchial inoculation of T-mycoplasmas. Journal of Com arative Patholoca 80, 585-594. || 25. HOWARD, C. J. & BROWNLIE, J. (1972). The production of mastitis in cows by the intramammary inoculation of T-mycoplasmas. Journal of Hygiene, Cambridge 70, 511-521. || 26. HOWARD, C. J. & GOURLAY, R. N. (1972). Serology of bovine T-mycoplasmas. British Veterinary Journal 128, XXXVII-XL. || 27. ______, _______ & BROWNLIE, J. (1973). The virulence of T-mycoplasmas, isolated from various animal species, assayed by intramammary inoculation in cattle. journal of Hygiene, Cambridge 71, 163-170. || C MYCOPLASMATALES VIRUSES: 28. GOURLAY, R. N. (1970). Isolation of a virus infecting a strain of MycopIasma laidlawii. Nature, London 225, 1165. || 29. , BRUCE, JUDY & GARWES , D. J. (1971). Characterisation of Mycoplasmatales virus laidlawii 1. Nature New Biolaay 229, 118-119. || 30. (1972). Ultrastructural studies of Mycoplasmatales viruses. MedicALILlIg. and Immunology 157, 172, || 31. BRUCE, J., GOURLAY R. N., HULL, R. & GARWES, D. J. (1972). Ultrastructure of Mycoplasmatales virus - laidlawii 1. Journal of 16, 215-221. || 32. GOURLAY, R. N. & WYLD, SARA G. (1972). Some biological characteristics of Mycoplasmatales virus laidlawii 1. Journal of General Virolgaz 14, 15-23. || 33. (1971). Mycoplasmatales virus - laidlawii 2, a new virus isolated from Acholeplasma laidlawii. Journal of General Virology 12, 65-67. || 34. (1972). Isolation and characterisation of mycoplasma viruses. In PathougaisilyssmlåsmIla Ciba Foundation Symposium. Amsterdam. ASP. 145-156. || 35. 4.11 GARWES, D. J., BRUCF J. & WYLD, S. G. (1973). Further studies on the morphology and composition of Mycoplasmatales virus - laidlawii 45 2. Journal of General Virolooy 18, 127-133. || 36. & WYLD, S. G. (1973). Isolation of Mycoplasmatales virus laidlawii 3, a new virus infecting Acholeolasma laidlawii. Journal of General Virology (in press)
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