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- 1. v1 Oct.28.2017 Satoshi Egi Rakuten Institute of Technology Rakuten, Inc.
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- 3. 3 My aim is to create a language that can represent directly all algorithms that can be discovered. Currently, my biggest challenge is to improve Egison in order to represent directly calculations that appear in mathematical physics.
- 4. 4 Pattern-matching against the wider range of data types. Customizable symbolic computation using Egison pattern-matching. Tensor index notation in programming.
- 5. 5 https://www.egison.org/math
- 6. 6 https://www.egison.org/math
- 7. 7 The theory for investigating a curved space. It is easy for us in the 3 dimensional space to recognize a 2 dimensional curved surface is curved. How about our world? Is it curved for who can recognize the higher dimensional space? https://commons.wikimedia.org/wiki/File:Trian gles_(spherical_geometry).jpg
- 8. 8 The theory explains the gravity as the curve of 4 dimensional time-space. https://commons.wikimedia.org/wiki/File:Spacetime_lattice_analogy.svg
- 9. 9 In our recognition, the space-time is flat.
- 10. 10 But, according to the theory, the space-time is curved.
- 11. 11 And, things move straight in this space-time.
- 12. 12 In our recognition, it looks like things are falling.
- 13. 13 Formulas in Riemannian geometry are represented with partial derivative operator and tensor index notation. We can represent both of them concisely in Egison!
- 14. 14 Egison program that represents the above formula Formula of Riemann curvature tensor ~: superscript _: subscript
- 15. 15 Pattern-matching against the wider range of data types. Customizable symbolic computation using Egison pattern-matching. Tensor index notation in programming.
- 16. 16 Pattern-matching against the wider range of data types. Customizable symbolic computation using Egison pattern-matching. Tensor index notation in programming.
- 17. 17 Pattern Body Matcher Target
- 18. 18 are patterns.
- 19. 19 Pattern-matching against the wider range of data types. Customizable symbolic computation using Egison pattern-matching. Tensor index notation in programming.
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- 21. 21 We can apply Egison pattern-matching against math expressions. Math expressions are a multiset of terms. Terms are a multiset of factors. Therefore, Egison pattern-matching is very useful to handle them.
- 22. 22 Pattern-matching against the wider range of data types. Customizable symbolic computation using Egison pattern-matching. Tensor index notation in programming.
- 23. 23 Tensor index notation is a popular notation in the field of mathematics and physics.
- 24. 24 We can control the way for multiplying vectors by indices.
- 25. 25 We can use tensor index notation to multiply any order of tensors. The tensor index notation is necessary to represent the multiplication of order tensors higher than matrices.
- 26. 26 In Egison method, we can apply directly both “∂/∂” and “.” functions to tensors. Egison program that represents the above formula Formula of Riemann curvature tensor ~: superscript _: subscript
- 27. 27 Wolfram program that represents the above formula Egison program that represents the above formula Formula of Riemann curvature tensor
- 28. 28https://arxiv.org/abs/1702.06343
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- 30. 30 https://commons.wikimedia.org/wiki/File:T riangles_(spherical_geometry).jpg
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- 37. 37 https://commons.wikimedia.org/wiki/File:Spacetime_lattice_analogy.svg
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- 39. 39 Currently, I am working to represent differential forms, exterior derivative, and Hodge operator directly in Egison. If we realize that, there are the wide range of application, e.g. mathematics, physics, and computer simulation.