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Turing degrees of isomorphism types of geometric objects [article]

Wesley Calvert, Valentina Harizanov, Alexandra Shlapentokh
2011 arXiv   pre-print
In particular, we show that any Turing degree may occur as the least degree of an isomorphic copy of a structure of these kinds. We also show that these structures may fail to have a least degree.  ...  We initiate the computability-theoretic study of ringed spaces and schemes.  ...  Then for any Turing degree d, there is a member of C whose isomorphism type has Turing degree d. There is also a member of C whose isomorphism type has no Turing degree.  ... 
arXiv:1106.2763v2 fatcat:d72o3ztqjjemzonwkvjclr2skq

A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

RUSSELL MILLER, BJORN POONEN, HANS SCHOUTENS, ALEXANDRA SHLAPENTOKH
2018 Journal of Symbolic Logic (JSL)  
AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields.  ...  a countable field ${\cal F}$ of arbitrary characteristic with the same essential computable-model-theoretic properties as ${\cal S}$.  ...  The degree spectrum DgSp A (R) of a relation R on a computable structure A is the set of all Turing degrees of images of R under isomorphisms f from A onto computable structures B.  ... 
doi:10.1017/jsl.2017.50 fatcat:juoiknhs3bbvta5mfydwvcop3e

Automorphism Groups of Substructure Lattices of Vector Spaces in Computable Algebra [chapter]

Rumen Dimitrov, Valentina Harizanov, Andrei Morozov
2016 Lecture Notes in Computer Science  
Shlapentokh, "Turing degrees of isomorphism types of geometric objects," Computability 3 (2014), pp. 105-134. 54. E. Fokina, V. Harizanov, and A.  ...  Harizanov, "Groups with orderings of arbitrary algorithmic complexity," 22 pages, accepted for publication in: Sets and Computations (IMS, National University of Singapore), R. Dilip, S.D.  ...  Frederick Nelson, A Geometric Approach to Rations of π/3-Congruent Numbers, Department of Mathematics, PhD, Howard University, 2011. 9.  ... 
doi:10.1007/978-3-319-40189-8_26 fatcat:2p5s2kf5xzfwfdf7zakfx43mpy

Page 1927 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
For these types of struc- Schoen, R. [Schoen, Richard M.] (1-STF; Stanford, CA); tures one has isomorphisms between the cotangent bundle and Wolfson, J. (1-MIS; East Lansing, M1) the tangent bundle.  ...  Edith Padron (E-LALAM-FM;; La Laguna) Lagrangian submanifolds have been important geometric objects of study in symplectic geometry since its birth.  ... 

Page 2015 of Mathematical Reviews Vol. , Issue 95d [page]

1995 Mathematical Reviews  
The proof requires only to show that a subvariety S (of degree d) of Y of codimension one is a complete intersection of Y (of degree m) and a hypersurface of P”.  ...  of general type with p, > 0 is base-point-free.  ... 

Spaces in mathematics

Boris Tsirelson
2018 WikiJournal of Science  
1][details 1] A space consists of selected mathematical objects that are treated as points, and selected relationships between these points.  ...  While modern mathematics use many types of spaces, such as Euclidean spaces, linear spaces, topological spaces, Hilbert spaces, or probability spaces, it does not define the notion of "space" itself. [  ...  A general definition of "structure", proposed by Bourbaki [2] , embraces all common types of spaces, provides a general definition of isomorphism, and justifies the transfer of properties between isomorphic  ... 
doi:10.15347/wjs/2018.002 fatcat:azjykkza4ffmrbson6jnsso7nm

Page 6211 of Mathematical Reviews Vol. , Issue 97J [page]

1997 Mathematical Reviews  
Fur- ther, he describes some geometrical objects invariant with respect to such isomorphisms. Josef Janyska (Brno) 97}:53025 53B21 53B20 Zhao, Pei Biao Projectively recurrent Riemannian spaces.  ...  (RS-NZNV; Nizhnii Novgorod) Trajectory isomorphisms of quasigeodesic flows of second degree and their invariants. (Russian) Izv. Vyssh. Uchebn. Zaved.  ... 

A Modern Point of View on Anomalies [article]

Samuel Monnier
2019 arXiv   pre-print
We review the concept of anomaly field theory, namely the fact that the anomalies of a d-dimensional field theory can be encoded in a d+1-dimensional field theory functor.  ...  We also sketch the status of global anomaly cancellation checks in string theory. This paper is based on a talk given at the Durham Symposium 'Higher Structures in M-theory' in August 2018.  ...  I would like to thank the organizers of the Durham Symposium "Higher Structures in M-theory" for the opportunity to present this work. Key words.  ... 
arXiv:1903.02828v1 fatcat:winn5t4d7nalboje5qgoi6jmgy

Page 4683 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
The authors use the so-called Miwa coordinates to reformulate the key objects of the (algebro-geometric) theory of the KP hierarchy, such as tau-function, wave function, bilinear identity and addition  ...  This is a nice survey of old and new results related to Arnol’d’s strange duality for 14 isomorphism classes of exceptional uni- modal surface singularities.  ... 

Page 644 of Mathematical Reviews Vol. , Issue 88b [page]

1988 Mathematical Reviews  
the global fine structure of &,-categorical struc- tures.  ...  The first part of the paper concerns the proof of the following re- sults: (a) (Fourth degree bound) If S C R’ is semialgebraically de- scribable by polynomials fj, ---, fm € R[x1,---,xX,] of total degree  ... 

Page 4496 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
Differential characters can be identified with equivalence classes of various kinds of geometric objects. For example H?  ...  If I*(G) is the space of invariant polynomials of degree k on the Lie algebra of G then the Weil homomorphism /*(G) — H**(BG,R) is an isomorphism.  ... 

The Digital and the Real Universe. Foundations of Natural Philosophy and Computational Physics

Klaus Mainzer
2019 Philosophies  
This example underlines that modern physics, in the tradition of Newton's Principia Mathematica Philosophiae Naturalis, is a further development of natural philosophy with the rigorous methods of mathematics  ...  Computational Foundations of Mathematics, Science, Technology, and Philosophy."  ...  Mathematical theories of nature are isomorphic to mathematical structures of different degrees and, by that, embedded in the categorical framework of mathematics.  ... 
doi:10.3390/philosophies4010003 fatcat:mgv373e4sbhgreenikab6am5qa

Simple structures with complex symmetry

V. Harizanov, R. Miller, A. S. Morozov
2010 Algebra and Logic  
We define the automorphism spectrum of a computable structure M, a measurement of the complexity of the symmetries of M, and prove that certain sets of Turing degrees can be realized as automorphism spectra  ...  , while certain others cannot. of degrees ≥ T d, then d is hyperarithmetical.  ...  this correspondence preserves the Turing degree of any isomorphism of degree ≥ deg(A).  ... 
doi:10.1007/s10469-010-9079-4 fatcat:6ynpw7jymjehnodcfv2cm7am2y

Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry [article]

Alexei Kanel-Belov, Alexei Chilikov, Ilya Ivanov-Pogodaev, Sergey Malev, Eugeny Plotkin, Jie-Tai Yu, Wenchao Zhang
2020 arXiv   pre-print
In Section 2.1 we describe the geometric equivalence, the elementary equivalence, and the isotypicity of algebras.  ...  In this setting Plotkin's problem on the structure of automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of categories is pretty natural and important.  ...  Two algebras are isotypic if the sets of realizable types over H1 and H2 coincide. So, by some abuse of language these algebras have the same logic of types.  ... 
arXiv:2008.09788v2 fatcat:b6v5vptirrgk5fa6rvckojevja

Searching for pharmacophoric patterns in databases of three-dimensional chemical structures

Peter Willett
1995 Journal of Molecular Recognition  
of the struc- tures passing the screening and geometric searches) but they were also substantially faster in execution.  ...  Inspection of these hits using molecular graphics software demonstrates a substantial degree of 3D overlap with the target struc- ture that would be difficult, if not impossible, to achieve with a simple  ... 
doi:10.1002/jmr.300080503 pmid:8619950 fatcat:zrw25m53obdyjp5posqipup26a
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