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### An explicit formula for the number of tunnels in digital objects [article]

Valentin Brimkov, Angelo Maimone, Giorgio Nordo
2005 arXiv   pre-print
In this paper we obtain a formula for the number of tunnels as a function of the number of the object vertices, pixels, holes, connected components, and 2x2 grid squares.  ...  An important concept in digital geometry for computer imagery is that of tunnel.  ...  that is compact and topologically connected [12, 8] . 8 Final remark In this paper we proposed a formula for the number of tunnels in a digital object.  ...

### Formulas for the number of (n−2)-gaps of binary objects in arbitrary dimension

Valentin E. Brimkov
2009 Discrete Applied Mathematics
Then we obtain an explicit formula for the number of gaps in S of maximal dimension, derive combinatorial relations for digital curves, and discuss possible applications to image analysis of digital surfaces  ...  In this paper we define the notion of a gap in an arbitrary digital binary object S in a digital space of arbitrary dimension.  ...  Acknowledgements The author thanks Reneta Barneva and Davide Moroni for their contribution to a conference version of the paper [17] that was reported at the 13th International Conference on Discrete  ...

### Combinatorial Relations for Digital Pictures [chapter]

Valentin E. Brimkov, Davide Moroni, Reneta Barneva
2006 Lecture Notes in Computer Science
As a main result, we obtain an explicit formula for the number of gaps in S of maximal dimension. We also derive a combinatorial relation for a digital curve.  ...  In this paper we define the notion of gap in an arbitrary digital picture S in a digital space of arbitrary dimension.  ...  Acknowledgements The authors thank the two anonymous referees for a number of useful remarks and suggestions.  ...

### Digital Surface of Revolution with Hand-Drawn Generatrix

Eric Andres, Lydie Richaume, Gaelle Largeteau-Skapin
2017 Journal of Mathematical Imaging and Vision
One can choose the topology of the surface that may have 1-tunnels, 0tunnels or no tunnels with applications in 3D printing for instance. An online tool that illustrates the method is proposed.  ...  In this paper we present a simple method to create general 3D digital surfaces of revolution based on a 2D implicit curve of revolution (therefore not limited to a circle) and a hand-drawn generatrix.  ...  Acknowledgement: This work has been supported by the CPER 2015-2020, NUMERIC Program and FEDER-FSE MODEGA Project of the Nouvelle-Aquitaine Region, France. we would like to thank Aurélie Mourier, http:  ...

### Digital Analytical Geometry: How Do I Define a Digital Analytical Object? [chapter]

Eric Andres
2015 Lecture Notes in Computer Science
We will end with open problems and challenges for the future.  ...  We will then explain how, from simple assumptions about properties a digital object should have, one can build mathematical sound digital objects.  ...  Note that, for an arbitrary structuring element F , it is the reectionF that appears in formula (3).  ...

### Tunneling Neural Perception and Logic Reasoning through Abductive Learning [article]

Wang-Zhou Dai, Qiu-Ling Xu, Yang Yu, Zhi-Hua Zhou
2018 arXiv   pre-print
We present the Neural-Logical Machine as an implementation of this novel learning framework.  ...  Inspired by the way language experts decoded Mayan scripts by joining two abilities in an abductive manner, this paper proposes the abductive learning framework.  ...  The abduced answer ∆ contains hypotheses of the previous two abducibles (i.e., a list of my op rules for the op rules predicate in Formula 6 as the reasoning model ∆ C , and a list of (modified) digit  ...

### Neural-like computing with populations of superparamagnetic basis functions [article]

Alice Mizrahi, Tifenn Hirtzlin, Akio Fukushima, Hitoshi Kubota, Shinji Yuasa, Julie Grollier, Damien Querlioz
2018 arXiv   pre-print
Doing so requires that the population components form a set of basis functions in terms of their response functions to inputs, offering a physical substrate for calculating.  ...  For this purpose, the responses of the nanodevices should be non-linear, and each tuned to different values of the input.  ...  The authors thank Mark Stiles, Pierre Bessière, Jacques Droulez and Nicolas Locatelli for fruitful discussions.  ...

### Analytical model for the approximation of hysteresis loop and its application to the scanning tunneling microscope

Rostislav V. Lapshin
1995 Review of Scientific Instruments
Examples of using the results in a scan unit of a scanning tunneling microscope for compensation of raster distortion are given.  ...  An analysis of the loop approximating function was carried out; the parameters and characteristics of the model were defined - coersitivity, remanent polarization, value of hysteresis, spontaneous polarization  ...  In the case of an unsplit hysteresis loop (a = 0); it is easy to pass from the parametric record over to an explicit function definition As a, b,, b, in Eq. (1) are substituted with the corrected value  ...

### Decomposing Cavities in Digital Volumes into Products of Cycles [chapter]

Ainhoa Berciano, Helena Molina-Abril, Ana Pacheco, Paweł Pilarczyk, Pedro Real
2009 Lecture Notes in Computer Science
The homology of binary 3-dimensional digital images (digital volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities.  ...  In the framework of cubical representation of digital volumes with the topology that corresponds to the 26-connectivity between voxels, we introduce a method for algorithmic computation of a coproduct  ...  We express the simplicial result in terms of cubical 1-chains, and in this way we derive an explicit formula for the cubical version of this diagonal in Section 4.  ...

### Collaborating with an Artist in Digital Geometry [chapter]

Eric Andres, Gaelle Largeteau-Skapin, Aurélie Mourier
2017 Lecture Notes in Computer Science
The artist works with voxel shapes and this led our digital geometry team to develop new shape modeling tools and explore a particular class of unfolding problems.  ...  In this invited paper, we are going to present an ongoing collaboration with a local artist, Aurélie Mourier.  ...  The number in the name of the shape corresponds to its number of voxels (797) and the order of the object among all the shapes having the same number of voxels. Once a shape has been dened, A.  ...

### Connectivity of discrete planes

V BRIMKOV
2004 Theoretical Computer Science
On this basis, we obtain a series of results, in particular, we provide an explicit solution to the problem under certain conditions.  ...  Studying connectivity of discrete objects is a major issue in discrete geometry and topology.  ...  We are grateful to Reinhard Klette for reading the paper and making a number of useful remarks and suggestions, in particular concerning the terminology.  ...

### Connectivity of discrete planes

Valentin E. Brimkov, Reneta P. Barneva
2004 Theoretical Computer Science
On this basis, we obtain a series of results, in particular, we provide an explicit solution to the problem under certain conditions.  ...  Studying connectivity of discrete objects is a major issue in discrete geometry and topology.  ...  We are grateful to Reinhard Klette for reading the paper and making a number of useful remarks and suggestions, in particular concerning the terminology.  ...

### Neural-like computing with populations of superparamagnetic basis functions

Alice Mizrahi, Tifenn Hirtzlin, Akio Fukushima, Hitoshi Kubota, Shinji Yuasa, Julie Grollier, Damien Querlioz
2018 Nature Communications
Doing so requires that the population components form a set of basis functions in terms of their response functions to inputs, offering a physical substrate for computing.  ...  We demonstrate experimentally that a population of nine junctions can implement a basis set of functions, providing the data to achieve, for example, the generation of cursive letters.  ...  The authors thank Mark Stiles, Pierre Bessière, Jacques Droulez, and Nicolas Locatelli for fruitful discussions.  ...

### Turing Machines and Understanding Computational Complexity [article]

P. M. B. Vitanyi (National Research Center for Mathematics and Computer Science in the Netherlands
2012 arXiv   pre-print
We describe the Turing Machine, list some of its many influences on the theory of computation and complexity of computations, and illustrate its importance.  ...  By his technique he exhibits for such a theory an explicit construction of an undecidable statement y that says of itself I am unprovable in T .  ...  The formulas are the objects of interest of the theory and constitute the meaningful statements. With each theory we associate a set of true formulas and a set of provable formulas.  ...

### "Fast Track" Analysis of Small Wind Turbine Blade Performance

Małgorzata Stępień, Michał Kulak, Krzysztof Jóźwik
2020 Energies
The small-scale analysis, performed in Reynolds numbers below 100,000, provided information sufficient for evaluation of the redesigned blade.  ...  The research was expanded by model validation in small-scale wind tunnel tests to provide detailed information on BEM data reliability in comparison to the results of the experiment.  ...  Results of analysis were consistent in terms of rotor geometries performance comparison, giving an explicit indication of the design which should be considered for a full scale prototype.  ...
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