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Journal articles on the topic 'Computable topology'

Author: Grafiati
Published: 5 June 2025
Last updated: 1 August 2025

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1

Weihrauch, Klaus, and Tanja Grubba. "Elementary Computable Topology." JUCS - Journal of Universal Computer Science 15, no. (6) (2009): 1381–422. https://doi.org/10.3217/jucs-015-06-1381.

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We revise and extend the foundation of computable topology in the framework of Type-2 theory of effectivity, TTE, where continuity and computability on finite and infinite sequences of symbols are defined canonically and transferred to abstract sets by means of notations and representations. We start from a computable topological space, which is a T0-space with a notation of a base such that intersection is computable, and define a number of multi-representations of the points and of the open, the closed and the compact sets and study their properties and relations. We study computability of b
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2

Weihrauch, Klaus. "Computable Separation in Topology, from T0 to T2." JUCS - Journal of Universal Computer Science 16, no. (18) (2010): 2733–53. https://doi.org/10.3217/jucs-016-18-2733.

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This article continues the study of computable elementary topology started in [Weihrauch and Grubba 2009]. For computable topological spaces we introduce a number of computable versions of the topological separation axioms T0, T1 and T2. The axioms form an implication chain with many equivalences. By counterexamples we show that most of the remaining implications are proper. In particular, it turns out that computable T1 is equivalent to computable T2 and that for spaces without isolated points the hierarchy collapses, that is, the weakest computable T0 axiom WCT0 is equivalent to the stronges
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3

Brattka, Vasco. "Some Notes on Fine Computability." JUCS - Journal of Universal Computer Science 8, no. (3) (2002): 382–95. https://doi.org/10.3217/jucs-008-03-0382.

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A metric defined by Fine induces a topology on the unit interval which is strictly stronger than the ordinary Euclidean topology and which has some interesting applications in Walsh analysis. We investigate computability properties of a corresponding Fine representation of the real numbers and we construct a structure which characterizes this representation. Moreover, we introduce a general class of Fine computable functions and we compare this class with the class of locally uniformly Fine computable functions defined by Mori. Both classes of functions include all ordinary computable function
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4

KOROVINA, MARGARITA, and OLEG KUDINOV. "Computable elements and functions in effectively enumerable topological spaces." Mathematical Structures in Computer Science 27, no. 8 (2016): 1466–94. http://dx.doi.org/10.1017/s0960129516000141.

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This paper is a part of the ongoing program of analysing the complexity of various problems in computable analysis in terms of the complexity of the associated index sets. In the framework of effectively enumerable topological spaces, we investigate the following question: given an effectively enumerable topological space whether there exists a computable numbering of all its computable elements. We present a natural sufficient condition on the family of basic neighbourhoods of computable elements that guarantees the existence of a principal computable numbering. We show that weakly-effective
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Lu, Hong, and Klaus Weihrauch. "Computable Riesz Representation for Locally Compact Hausdorff Spaces." JUCS - Journal of Universal Computer Science 14, no. (6) (2008): 845–60. https://doi.org/10.3217/jucs-014-06-0845.

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By the Riesz Representation Theorem for locally compact Hausdorff spaces, for every positive linear functional I on K(X) there is a measure μ such that I(f) =∫ f dμ where K(X) is the set of continuous real functions with compact support on the locally compact Hausdorff space X. In this article we prove a uniformly computable version of this theorem for computably locally compact computable Hausdorff spaces X. We introduce a representation of the positive linear functionals I on K(X) and a representation of the Borel measures on X and prove that for every such functional I a measure μ can be co
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6

Kalantari, Iraj, and Larry Welch. "On Turing degrees of points in computable topology." MLQ 54, no. 5 (2008): 470–82. http://dx.doi.org/10.1002/malq.200710062.

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Kalantari, Iraj, and Larry Welch. "On degree-preserving homeomorphisms between trees in computable topology." Archive for Mathematical Logic 46, no. 7-8 (2007): 679–93. http://dx.doi.org/10.1007/s00153-007-0056-0.

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8

SCHRÖDER, MATTHIAS. "The sequential topology on is not regular." Mathematical Structures in Computer Science 19, no. 5 (2009): 943–57. http://dx.doi.org/10.1017/s0960129509990065.

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The compact-open topology on the set of continuous functionals from the Baire space to the natural numbers is well known to be zero-dimensional. We prove that the closely related sequential topology on this set is not even regular. The sequential topology arises naturally as the topology carried by the exponential formed in various cartesian closed categories of topological spaces. Moreover, we give an example of an effectively open subset of that violates regularity. The topological properties of are known to be closely related to an open problem in Computable Analysis. We also show that the
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9

Brattka, Vasco. "Effective representations of the space of linear bounded operators." Applied General Topology 4, no. 1 (2003): 115. http://dx.doi.org/10.4995/agt.2003.2014.

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<p>Representations of topological spaces by infinite sequences of symbols are used in computable analysis to describe computations in topological spaces with the help of Turing machines. From the computer science point of view such representations can be considered as data structures of topological spaces. Formally, a representation of a topological space is a surjective mapping from Cantor space onto the corresponding space. Typically, one is interested in admissible, i.e. topologically well-behaved representations which are continuous and characterized by a certain maximality condition
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10

Heath, Philip R. "Two, more readily computable, equivariant Nielsen numbers II." Topology and its Applications 156, no. 2 (2008): 186–204. http://dx.doi.org/10.1016/j.topol.2008.06.007.

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11

Friedl, Stefan, Joan Licata, Stephan Tillmann, and Paula Truöl. "MATRIX-MFO Tandem Workshop: Invariants in Low-Dimensional Topology: Combinatorics, Geometry, and Computation." Oberwolfach Reports 21, no. 3 (2025): 2511–50. https://doi.org/10.4171/owr/2024/43.

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The MATRIX-MFO tandem workshop addressed several research questions in low-dimensional topology and related areas. The format of the workshop consisted primarily of discussion sessions focusing on computable invariants of colored spatial graphs, random knots, small clasp number knots, algebraic models for Poincaré duality complexes in dimension 4, and the topological volume of the three-torus.
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12

BAUER, ANDREJ, and PAUL TAYLOR. "The Dedekind reals in abstract Stone duality." Mathematical Structures in Computer Science 19, no. 4 (2009): 757–838. http://dx.doi.org/10.1017/s0960129509007695.

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Abstract Stone Duality (ASD) is a direct axiomatisation of general topology, in contrast to the traditional and all other contemporary approaches, which rely on a prior notion of discrete set, type or object of a topos.ASD reconciles mathematical and computational viewpoints, providing an inherently computable calculus that does not sacrifice key properties of real analysis such as compactness of the closed interval. Previous theories of recursive analysis failed to do this because they were based on points; ASD succeeds because, like locale theory and formal topology, it is founded on the alg
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13

Bridges, Douglas, and Luminiţa Vîţă. "Strong and Uniform Continuity – the Uniform Space Case." LMS Journal of Computation and Mathematics 6 (2003): 326–34. http://dx.doi.org/10.1112/s1461157000000486.

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AbstractIt is proved, within the constructive theory of apartness spaces, that a strongly continuous mapping from a totally bounded uniform space with a countable base of entourages to a uniform space is uniformly continuous. This lifts a result of Ishihara and Schuster from metric to uniform apartness spaces. The paper is part of a systematic development of computable topology using apartness as the fundamental notion.
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14

Brattka, Vasco, and Guido Gherardi. "Effective Choice and Boundedness Principles in Computable Analysis." Bulletin of Symbolic Logic 17, no. 1 (2011): 73–117. http://dx.doi.org/10.2178/bsl/1294186663.

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AbstractIn this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this purpose theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice,
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15

Schröder, Matthias. "A Note on Closed Subsets in Quasi-zero-dimensional Qcb-spaces." JUCS - Journal of Universal Computer Science 16, no. (18) (2010): 2711–32. https://doi.org/10.3217/jucs-016-18-2711.

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We introduce the notion of quasi-zero-dimensionality as a substitute for the notion of zero-dimensionality, motivated by the fact that the latter behaves badly in the realm of qcb-spaces. We prove that the category QZ of quasi-zero-dimensional qcblt;sub>0lt;/sub>-spaces is cartesian closed. Prominent examples of spaces in QZ are the spaces of the Kleene-Kreisel continuous functionals equipped with the respective sequential topology. Moreover, we characterise some types of closed subsets of QZ-spaces in terms of their ability to allow extendability of continuous functions. These results a
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16

Simson, Daniel. "The Euler characteristic and Euler defect for comodules over Euler coalgebras." Journal of K-Theory 7, no. 1 (2009): 91–113. http://dx.doi.org/10.1017/is009010019jkt081.

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AbstractLet K be a field. We study a class of left C-comodules over a basic left Euler coalgebra C by means of the Euler ℤ-bilinear form associated to C, the Euler characteristic χC(M,N) of left C-comodules M,N, and the defect ∂C(M,N) ∊ ℤ associated to any computable Euler pair (M,N) of left C-comodules. We show that (lgthM, lgthN) = χC(M,N) + ∂C(M,N), for any computable Euler pair (M,N) of comodules over a left Euler coalgebra C. One of the main results of the paper asserts that the defect ∂C(M,N) is zero and (lgthM,lgthN) = χC(M,N), if the comodules M,N are finite-dimensional.
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17

Montalbán, Antonio. "Computable Linearizations of Well-partial-orderings." Order 24, no. 1 (2007): 39–48. http://dx.doi.org/10.1007/s11083-007-9058-0.

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18

Stoltenberg-Hansen, V., and J. V. Tucker. "Complete local rings as domains." Journal of Symbolic Logic 53, no. 2 (1988): 603–24. http://dx.doi.org/10.1017/s0022481200028498.

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Contents: Introduction. §1: Computable rings and modules. §2: Ideal membership relation. §3: Effective structured domains. §4: Completion of a local ring as a domain. §5: The recursive completion. Epilogue. References.Introduction. Completion is an important general mathematical device. Often, but not always, a completion takes the following form. Let A be a topological algebraic structure whose topology is derived from a metric. For A, a topological algebra  and an embedding i: A →  are constructed such that  is a complete metric space in which A is densely embedded by i. The long list of
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19

Bailor, Maximillian H., Xiaoyan Sun, and Hashim M. Al-Hashimi. "Topology Links RNA Secondary Structure with Global Conformation, Dynamics, and Adaptation." Science 327, no. 5962 (2010): 202–6. http://dx.doi.org/10.1126/science.1181085.

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Thermodynamic rules that link RNA sequences to secondary structure are well established, but the link between secondary structure and three-dimensional global conformation remains poorly understood. We constructed comprehensive three-dimensional maps depicting the orientation of A-form helices across RNA junctions in the Protein Data Bank and rationalized our findings with modeling and nuclear magnetic resonance spectroscopy. We show that the secondary structures of junctions encode readily computable topological constraints that accurately predict the three-dimensional orientation of helices
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20

Heath, Philip R. "Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads." Topology and its Applications 156, no. 2 (2008): 165–85. http://dx.doi.org/10.1016/j.topol.2008.06.006.

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21

Parad�s, Jaume, Llu�s Bibiloni, and Pelegr� Viader. "On actually computable bijections between ? and Q +." Order 13, no. 4 (1996): 369–77. http://dx.doi.org/10.1007/bf00405596.

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22

Gao, Dayou, Jiaxi Huang, Tianchang Liu, and Tiantian Liu. "There is no algorithm to determine if the homotopy group is trivial or not for all compact spaces." Theoretical and Natural Science 53, no. 1 (2024): 236–41. http://dx.doi.org/10.54254/2753-8818/53/20240222.

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Abstract. Constructive mathematics involves numbers and objects that can be constructed and computed through algorithmic methods. In this paper we prove that there is no algorithm to determine if the homotopy group _n is trivial or not for all constructive compact spaces. In the beginning we introduce constructive mathematics and topology theory. Then we introduce constructive compact spaces to describe the compact space in algorithmic way, and we calculate certain homotopy groups. Finally, we use an unextendible partially computable function to prove that there is no algorithm that can always
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23

Hofmann, Martin. "An Application of Category-Theoretic Semantics to the Characterisation of Complexity Classes Using Higher-Order Function Algebras." Bulletin of Symbolic Logic 3, no. 4 (1997): 469–86. http://dx.doi.org/10.2307/421100.

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AbstractWe use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PVω defines exactly the PTIME-iunctions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types.By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with -induction over PVω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellanto
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24

BOBROWSKI, OMER, and MATTHEW STROM BORMAN. "EULER INTEGRATION OF GAUSSIAN RANDOM FIELDS AND PERSISTENT HOMOLOGY." Journal of Topology and Analysis 04, no. 01 (2012): 49–70. http://dx.doi.org/10.1142/s1793525312500057.

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In this paper we extend the notion of the Euler characteristic to persistent homology and give the relationship between the Euler integral of a function and the Euler characteristic of the function's persistent homology. We then proceed to compute the expected Euler integral of a Gaussian random field using the Gaussian kinematic formula and obtain a simple closed form expression. This results in the first explicitly computable mean of a quantitative descriptor for the persistent homology of a Gaussian random field.
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25

Brandts, Jan H., and Ricardo Reis da Silva. "Computable eigenvalue bounds for rank-k perturbations." Linear Algebra and its Applications 432, no. 12 (2010): 3100–3116. http://dx.doi.org/10.1016/j.laa.2010.02.010.

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26

CHANG, EE-CHIEN, SUNG WOO CHOI, DO YONG KWON, HYUNGJU PARK, and CHEE K. YAP. "SHORTEST PATH AMIDST DISC OBSTACLES IS COMPUTABLE." International Journal of Computational Geometry & Applications 16, no. 05n06 (2006): 567–90. http://dx.doi.org/10.1142/s0218195906002191.

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An open question in Exact Geometric Computation is whether there are transcendental computations that can be made "geometrically exact". Perhaps the simplest such problem in computational geometry is that of computing the shortest obstacle-avoiding path between two points p,q in the plane, where the obstacles are a collection of n discs. This problem can be solved in O(n2 log n) time in the Real RAM model, but nothing was known about its computability in the standard (Turing) model of computation. We first give a direct proof of the Turing-computability of this problem, provided the radii of t
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27

GAROUFALIDIS, STAVROS, and XINYU SUN. "THE NON-COMMUTATIVE A-POLYNOMIAL OF TWIST KNOTS." Journal of Knot Theory and Its Ramifications 19, no. 12 (2010): 1571–95. http://dx.doi.org/10.1142/s021821651000856x.

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The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative A-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form [Formula: see text] given a recursion relation for [Formula: see text] and the hypergeometric kernel c(n, k). As an application of our method, we explicitly compute the non-commutative A-polynomial for twist knots with -15 and 15 crossings. The non-commutative A-polynomial
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28

Yue, Yunguang, Fengchun Lei, Xingwu Liu, and Jie Wu. "Asynchronous Computability Theorem in Arbitrary Solo Models." Mathematics 8, no. 5 (2020): 757. http://dx.doi.org/10.3390/math8050757.

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In this paper, we establish the asynchronous computability theorem in d-solo system by borrowing concepts from combinatorial topology, in which we state a necessary and sufficient conditions for a task to be wait-free computable in that system. Intuitively, a d-solo system allows as many d processes to access it as if each were running solo, namely, without detecting communication from any peer. As an application, we completely characterize the solvability of the input-less tasks in such systems. This characterization also leads to a hardness classification of these tasks according to whether
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29

Fujikoshi, Yasunori. "Computable error bounds for asymptotic approximations of the quadratic discriminant function." Hiroshima Mathematical Journal 50, no. 3 (2020): 313–24. http://dx.doi.org/10.32917/hmj/1607396491.

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30

Hunt, Sebastian, David Sands, and Sandro Stucki. "Reconciling Shannon and Scott with a Lattice of Computable Information." Proceedings of the ACM on Programming Languages 7, POPL (2023): 1987–2016. http://dx.doi.org/10.1145/3571740.

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This paper proposes a reconciliation of two different theories of information. The first, originally proposed in a lesser-known work by Claude Shannon (some five years after the publication of his celebrated quantitative theory of communication), describes how the information content of channels can be described qualitatively , but still abstractly, in terms of information elements , where information elements can be viewed as equivalence relations over the data source domain. Shannon showed that these elements have a partial ordering, expressing when one information element is more informativ
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31

Zenil, Hector, Narsis Kiani, and Jesper Tegnér. "Symmetry and Correspondence of Algorithmic Complexity over Geometric, Spatial and Topological Representations." Entropy 20, no. 7 (2018): 534. http://dx.doi.org/10.3390/e20070534.

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We introduce a definition of algorithmic symmetry in the context of geometric and spatial complexity able to capture mathematical aspects of different objects using as a case study polyominoes and polyhedral graphs. We review, study and apply a method for approximating the algorithmic complexity (also known as Kolmogorov–Chaitin complexity) of graphs and networks based on the concept of Algorithmic Probability (AP). AP is a concept (and method) capable of recursively enumerate all properties of computable (causal) nature beyond statistical regularities. We explore the connections of algorithmi
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32

Tadokoro, Yuuki. "Pointed harmonic volume and its relation to the extended Johnson homomorphism." Journal of Topology and Analysis 12, no. 01 (2018): 87–103. http://dx.doi.org/10.1142/s1793525319500407.

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The period for a compact Riemann surface, defined by the integral of differential 1-forms, is a classical complex analytic invariant, strongly related to the complex structure of the surface. In this paper, we treat another complex analytic invariant called the pointed harmonic volume. As a natural extension of the period defined using Chen’s iterated integrals, it captures more detailed information of the complex structure. It is also one of a few explicitly computable examples of complex analytic invariants. We obtain its new value for a certain pointed hyperelliptic curve. An application of
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33

HENCKELL, KARSTEN, STUART W. MARGOLIS, JEAN-ERIC PIN, and JOHN RHODES. "ASH'S TYPE II THEOREM, PROFINITE TOPOLOGY AND MALCEV PRODUCTS: PART I." International Journal of Algebra and Computation 01, no. 04 (1991): 411–36. http://dx.doi.org/10.1142/s0218196791000298.

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This paper is concerned with the many deep and far reaching consequences of Ash's positive solution of the type II conjecture for finite monoids. After reviewing the statement and history of the problem, we show how it can be used to decide if a finite monoid is in the variety generated by the Malcev product of a given variety and the variety of groups. Many interesting varieties of finite monoids have such a description including the variety generated by inverse monoids, orthodox monoids and solid monoids. A fascinating case is that of block groups. A block group is a monoid such that every e
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34

Castro-Rivera, Jorge, María Lucila Morales-Rodríguez, Nelson Rangel-Valdez, Claudia Gómez-Santillán, and Luciano Aguilera-Vázquez. "Modeling Preferences through Personality and Satisfaction to Guide the Decision Making of a Virtual Agent." Axioms 11, no. 5 (2022): 232. http://dx.doi.org/10.3390/axioms11050232.

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Satisfaction is relevant for decision makers (DM, Decision Makers). Satisfaction is the feeling produced in individuals by executing actions to satisfy their needs, for example, the payment of debts, jobs, or academic achievements, and the acquisition of goods or services. In the satisfaction literature, some theories model the satisfaction of individuals from job and customer approaches. However, considering personality elements to influence satisfaction and define preferences in strategies that optimize decision making provides the unique characteristics of a DM. These characteristics favor
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35

Fujikoshi, Yasunori. "Computable error bounds for asymptotic expansions of the hypergeometric function ${}_1F_1$ of matrix argument and their applications." Hiroshima Mathematical Journal 37, no. 1 (2007): 13–23. http://dx.doi.org/10.32917/hmj/1176324092.

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36

Valenti, Manlio. "A journey through computability, topology and analysis." Bulletin of Symbolic Logic 28, no. 2 (2022): 266–67. http://dx.doi.org/10.1017/bsl.2022.13.

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AbstractThis thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and (effective) descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy.We first analyze the strength of the open and clopen Ramsey theorems. Since there is
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Feng, Xinxin, and Sirui Li. "A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals." Axioms 11, no. 10 (2022): 495. http://dx.doi.org/10.3390/axioms11100495.

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The Doi–Onsager molecular theory is capable of providing a rather accurate description of the local behavior of molecules; however, its computation is extremely time-consuming, since some higher-dimensional variables are typically involved. Therefore, establishing a computable reduced model that can capture essential physical properties is an important issue. In this work, we derived a reduced Q-tensor hydrodynamic theory that described smectic phases with density variations from the Doi–Onsager molecular theory using the Bingham closure approximation. The coefficients in the tensor model were
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38

Zhang, Chi, John P. Huelsenbeck, and Fredrik Ronquist. "Using Parsimony-Guided Tree Proposals to Accelerate Convergence in Bayesian Phylogenetic Inference." Systematic Biology 69, no. 5 (2020): 1016–32. http://dx.doi.org/10.1093/sysbio/syaa002.

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Abstract Sampling across tree space is one of the major challenges in Bayesian phylogenetic inference using Markov chain Monte Carlo (MCMC) algorithms. Standard MCMC tree moves consider small random perturbations of the topology, and select from candidate trees at random or based on the distance between the old and new topologies. MCMC algorithms using such moves tend to get trapped in tree space, making them slow in finding the globally most probable trees (known as “convergence”) and in estimating the correct proportions of the different types of them (known as “mixing”). Here, we introduce
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39

Filom, Khashayar, Roozbeh Farhoodi, and Konrad Paul Kording. "On PDE Characterization of Smooth Hierarchical Functions Computed by Neural Networks." Neural Computation 33, no. 12 (2021): 3204–63. http://dx.doi.org/10.1162/neco_a_01441.

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Abstract Neural networks are versatile tools for computation, having the ability to approximate a broad range of functions. An important problem in the theory of deep neural networks is expressivity; that is, we want to understand the functions that are computable by a given network. We study real, infinitely differentiable (smooth) hierarchical functions implemented by feedforward neural networks via composing simpler functions in two cases: (1) each constituent function of the composition has fewer inputs than the resulting function and (2) constituent functions are in the more specific yet
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Yasugi, Mariko, Yoshiki Tsujii, and Takakazu Mori. "Sequential Computability of a Function. Effective Fine Space and Limiting Recursion." JUCS - Journal of Universal Computer Science 11, no. (12) (2005): 2179–91. https://doi.org/10.3217/jucs-011-12-2179.

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We consider real sequences in I = [0, 1) and real functions on I. It is first shown that, as for real sequences from I, R-computability (computability with respect to the Euclidean topology) implies " weak Fine-computability." Using this result, we show that " Fine­sequential computability" and " -sequential computability" are equivalent for effectively locally Fine-continuous functions as well as for Fine-continuous functions.
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Bhupati, Chokara, and Kodanda Rama Jammalamadaka Sastry. "Hybrid models for computing fault tolerance of IoT networks." TELKOMNIKA (Telecommunication, Computing, Electronics and Control) 21, no. 2 (2023): 333–45. https://doi.org/10.12928/telkomnika.v21i2.22429.

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Many Internet of Things (IoT) - based networks are being built to develop applications spanning multiple domains. Many small to large devices connected in various ways increases the risk of IoT networks failing. Small devices in the devices layer frequently fail due to their small size and high usage. Intermittent failures of the IoT networks lead to catastrophes at times. The IoT systems must be designed to be fault-tolerant. Fault tolerance of IoT networks must be computable so that the same can be considered while designing IoT networks. However, the computation of fault tolerance of IoT ne
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42

Dale, Anders M., and Martin I. Sereno. "Improved Localizadon of Cortical Activity by Combining EEG and MEG with MRI Cortical Surface Reconstruction: A Linear Approach." Journal of Cognitive Neuroscience 5, no. 2 (1993): 162–76. http://dx.doi.org/10.1162/jocn.1993.5.2.162.

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We describe a comprehensive linear approach to the problem of imaging brain activity with high temporal as well as spatial resolution based on combining EEG and MEG data with anatomical constraints derived from MRI images. The "inverse problem" of estimating the distribution of dipole strengths over the cortical surface is highly underdetermined, even given closely spaced EEG and MEG recordings. We have obtained much better solutions to this problem by explicitly incorporating both local cortical orientation as well as spatial covariance of sources and sensors into our formulation. An explicit
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43

Soare, Robert I. "Computability Theory and Differential Geometry." Bulletin of Symbolic Logic 10, no. 4 (2004): 457–86. http://dx.doi.org/10.2178/bsl/1102083758.

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Abstract. LetMbe a smooth, compact manifold of dimensionn≥ 5 and sectional curvature ∣K∣ ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics onMmodulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machineTe,eϵ ω, there is a sequence (uniformly effective ine) of homologyn-sphereswhich are also hypersurfaces, such thatis diffeomorphic to the standardn-sphereSn(denoted)iffTehalts on inputk, and in this case the connected sum, so, and
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44

ELLIS, J. L., G. KEDEM, T. C. LYERLY, et al. "THE RAYCASTING ENGINE AND RAY REPRESENTATIONS: A TECHNICAL SUMMARY." International Journal of Computational Geometry & Applications 01, no. 04 (1991): 347–80. http://dx.doi.org/10.1142/s0218195991000256.

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Solid modeling is computationally intensive. Thus far its use in industry has been limited mainly to simple parts and simple applications, and this is not likely to change much until 'massive' computing power can be made available at an affordable cost. The RayCasting Engine is one specialized source of 'massive' computing power for solid modeling, and it is but the simplest member of a potentially large family of 'classification computers'. The RayCasting Engine (RCE) is a highly parallel, custom-VLSI computer that classifies grids of parallel lines against solids represented in CSG. The sets
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45

CAMASSA, ROBERTO, RICHARD M. McLAUGHLIN, and LONGHUA ZHAO. "Lagrangian blocking in highly viscous shear flows past a sphere." Journal of Fluid Mechanics 669 (February 16, 2011): 120–66. http://dx.doi.org/10.1017/s0022112010004933.

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An analytical and computational study of Lagrangian trajectories for linear shear flow past a sphere or spheroid at low Reynolds numbers is presented. Using the exact solutions available for the fluid flow in this geometry, we discover and analyse blocking phenomena, local bifurcation structures and their influence on dynamical effects arising in the fluid particle paths. In particular, building on the work by Chwang & Wu, who established an intriguing blocking phenomenon in two-dimensional flows, whereby a cylinder placed in a linear shear prevents an unbounded region of upstream fluid fr
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46

XU, YATAO, and TANJA GRUBBA. "On computably locally compact Hausdorff spaces." Mathematical Structures in Computer Science 19, no. 1 (2009): 101–17. http://dx.doi.org/10.1017/s0960129508007366.

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Locally compact Hausdorff spaces generalise Euclidean spaces and metric spaces from ‘metric’ to ‘topology’. But does the effectivity on the latter (Brattka and Weihrauch 1999; Weihrauch 2000) still hold for the former? In fact, some results will be totally changed. This paper provides a complete investigation of a specific kind of space – computably locally compact Hausdorff spaces. First we characterise this type of effective space, and then study computability on closed and compact subsets of them. We use the framework of the representation approach, TTE, where continuity and computability o
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47

BOI, LUCIANO. "IDEAS OF GEOMETRIZATION, GEOMETRIC INVARIANTS OF LOW-DIMENSIONAL MANIFOLDS, AND TOPOLOGICAL QUANTUM FIELD THEORIES." International Journal of Geometric Methods in Modern Physics 06, no. 05 (2009): 701–57. http://dx.doi.org/10.1142/s0219887809003783.

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The aim of the first part of this paper is to make some reflections on the role of geometrical and topological concepts in the developments of theoretical physics, especially in gauge theory and string theory, and we show the great significance of these concepts for a better understanding of the dynamics of physics. We will claim that physical phenomena essentially emerge from the geometrical and topological structure of space–time. The attempts to solve one of the central problems in 20th theoretical physics, i.e. how to combine gravity and the other forces into an unitary theoretical explana
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NEHANIV, CHRYSTOPHER LEV. "ALGEBRAIC CONNECTIVITY." International Journal of Algebra and Computation 01, no. 04 (1991): 445–71. http://dx.doi.org/10.1142/s0218196791000316.

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Let [Formula: see text] be a type of algebra in the sense of universal algebra. By defining singular simplices in algebras and emulating singular [co] homology, we introduce for each variety, pseudo-variety, and divisional class V of type [Formula: see text], a homology and cohomology theory which measure the V-connectivity of type-[Formula: see text] algebras. Intuitively, if we were to think of an algebra as a space and subalgebras which lie in V as simplices, then V-connectivity describes the failure of subalgebras to lie in V, i.e., it describes the "holes" in this space. These [co]homolog
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49

Korovina, Margarita, and Oleg Kudinov. "The Rice-Shapiro theorem in Computable Topology." Logical Methods in Computer Science Volume 13, Issue 4 (December 29, 2017). https://doi.org/10.23638/lmcs-13(4:30)2017.

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We provide requirements on effectively enumerable topological spaces which guarantee that the Rice-Shapiro theorem holds for the computable elements of these spaces. We show that the relaxation of these requirements leads to the classes of effectively enumerable topological spaces where the Rice-Shapiro theorem does not hold. We propose two constructions that generate effectively enumerable topological spaces with particular properties from wn--families and computable trees without computable infinite paths. Using them we propose examples that give a flavor of this class.
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50

Steinberg, Florian. "Complexity theory for spaces of integrable functions." Logical Methods in Computer Science Volume 13, Issue 3 (September 12, 2017). https://doi.org/10.23638/lmcs-13(3:21)2017.

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This paper investigates second-order representations in the sense of Kawamura and Cook for spaces of integrable functions that regularly show up in analysis. It builds upon prior work about the space of continuous functions on the unit interval: Kawamura and Cook introduced a representation inducing the right complexity classes and proved that it is the weakest second-order representation such that evaluation is polynomial-time computable. The first part of this paper provides a similar representation for the space of integrable functions on a bounded subset of Euclidean space: The weakest rep
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