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Radiator values for porous media
Hi guys,
I´m trying to set up some more detailed simulations. As shown in the Formula-Student-Webinar I want to use the advanced simulation concept: porous media.
The problem is that I got no idea what values I should use.
There are:
a)Coefficient d
b)Coefficient f
Coordinate system properties
My radiator is in a sidepod in a angle of approximately 45°.
Is there any method to calculate these values (a-d) or can I get them of the manufacturer and convert them?
Couldn´t find a helpful forum entry for this. Maybe a short how to for this topic wouldn´t be bad.
If interested I can send you a link to the project, but it isn´t much done yet.
Hi David (@BRT)!
There has already been a discussion about this in another thread.
Please have a look at this page: Setting Up a Simulation Using Porous Media
You can also have a look at the documentation page about Porous Media
Let me know if that helps!
1 Like
Hi Jousef,
I had some time to have a closer look at This two descriptions. I understand that e1, e2, e3 are the main directions of my radiator in relation to the main coordinate system.
The main problems are the coefficients d and f. When I got it right they can just be determined by tests. I don´t think that I can do this myself so ill ask the producing company of our radiator.
Or is there a simple way to determine them ??
Hi David!
Very sorry for the late response!
As posted by our PowerUser @varsey you can determine the coefficients in the following manner:
• d = 150 (1-eps)^2 / (eps^2 * D^2)
• f = 2 * 1.75 * (1-eps) / (eps^3 * D)
Where eps and D is characteristics of your porous media (porosity and mean particles diameter).
This document might also be helpful for you: Porous Media in OpenFOAM
Let me know if I can do anything else for you.
Enjoy your Sunday!
3 Likes
I read the above conversation and also the links provided by Jousef. But I am still confused as to how to get the values of e1 and e3 for the porous media.
Could you explain it in detail.
Thanks in advance,
Hi Varun (@pdarda)!
I think @varsey explained it very well here: Simulation - Porous Media
From one document listed above you can also find the following explanation:
“The global coordinate system of the geometry is set by default…Here it is specified with two vectors. The vector e1 is created as a linear combination of the global x- and y-axes so it is aligned
with the angled duct and the vector e2 is set orthogonal to e1. The vector e3 is then created in a right handed order by the coordinateSystem class orthogonally to both e1 and e2.”
If that is not enough could you explain in detail what you are struggling with?
As an example, please have a look at this project: Gas Flow Through Catalytic Converter And Exhaust
2 Likes
Sorry for the late reply.
So the vector e1 represents the direction in which the material will be porous, right?
Then will the vector direction for the nonporous direction be zero?
Could you please specify the direction vector for the following case so that I can understand how to define them.
Suppose a porous media is in the plane perpendicular to z axis and air flows in the direction of +z axis.
The material is porous in the z direction only and non porous along other directions.
So what will be the values of e1 and e3 vectors for this case?
Thanks in advance,
Hi @pdarda!
For the case you described e1 has coordinates (1, 0, 0) and e3 has ccordinates (0, 0, 1) (I’ve described simulation with exactly the same configutation of flow on screenshot here - link)
So the vector e1 represents the direction in which the material will be porous, right?
Vectors e1 and e3 create a coordinate system, so you cant define just one of them to set direction of your porosity. You have to define at least two vectors among e1, e2 and e3.
2 Likes
Thanks @varsey
Now I understood the what they mean/
Thanks once again!!!
2 Likes
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On the large-scale structure in turbulent free shear flows
The existence of organized structures in turbulent shear flow has been the subject of recent observational discoveries in both the laboratory and in the atmosphere and ocean. The recent work on
modeling such structures in a temporally developing, horizontally homogeneous turbulent free shear layer has been extended to the spatially developing mixing layer, there being no available rational
transformation between the two nonlinear problems. The basis for the consideration is the kinetic energy development of the mean flow, large-scale structure and fine-grained turbulence with a
conditional average, supplementing the usual time average, to separate the nonrandom from the random part of the fluctuations. The integrated form of the energy equations and the accompanying shape
assumptions, is used to derive amplitude equations for the mean flow, characterized by the shear layer thickness, the nonrandom and random components of flow which are characterized by their
respective energy densities. In general, the large-scale structure augments the spreading of the shear layer and enhances the fine-grained turbulence by taking energy from the mean flow and
transferring it to the turbulence as it amplifies and subsequently decays. The maximal amplitude of the large-scale structure is attained by the initially most amplified mode, however, the relative
enhancement of the fine-grained turbulence is achieved by both the magnitude of the large-scale structure and its streamwise lifetime. Thus a greater enhancement of the turbulence is achievable by
the lower frequency modes which have longer streamwise lifetimes. The large-scale structure can also be controlled by increasing the initial level of turbulence, which would render its decay more
NASA STI/Recon Technical Report N
Pub Date:
June 1977
□ Shear Layers;
□ Turbulent Flow;
□ Turbulent Mixing;
□ Atmospheric Models;
□ Kinetic Theory;
□ Mixing Layers (Fluids);
□ Ocean Models;
□ Fluid Mechanics and Heat Transfer
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Derivatives & Differentiation - Calculus Study Guides: Flashcards | Knowt
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KWARC Group: Selected Publications
This website represents the joint bibliography of the KWARC group and is generated automatically.
For more details, please see the Github Repository.
Please respect any copyrights when downloading
Archival Literature
Articles in Journals
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1. [22] (2022) Wann ist ein juristischer text strukturiert? die sicht der informatik und der künstlichen intelligenz. In Digitalisierung von zivilprozess und rechtsdurchsetzung, A. Adrian, S. Evert,
M. K. Kohlhase, and M. Zwickel (Eds.), Schriften zum Prozessrecht, pp. 155–170. External Links: Link Cited by: p1.
2. [2] (2020) Mathematical information retrieval. In Evaluating information retrieval and access tasks – NTCIR’s legacy of research impactEvaluating Information Retrieval and Access Tasks – NTCIR’s
Legacy of Research Impact, T. Sakai, D. W. Oard, and N. Kando (Eds.), pp. 169–185. Cited by: p1.
3. [4] (2015) 3XL news: a cross-lingual news aggregator and reader. In The Semantic Web: ESWC 2015 Satellite Events, Lecture Notes in Computer Science, Vol. 9341, pp. 3–8. External Links: ISBN
978-3-319-25638-2, Document Cited by: p1.
4. [25] (2013) Mashups using mathematical knowledge. In Semantic mashups, B. Endres-Niggemeyer and B. Endres-Niggemeyer (Eds.), pp. 171–204. External Links: Link Cited by: p1.
5. [10] (2010-01) Social tagging and learning: the fuzzy line between private and public space. In Novel Developments in Web-Based Learning Technologies: Tools for Modern Teaching, N. Karacapilidis
(Ed.), Advances in Web-based Learning (AWBL). External Links: ISBN 978-1-60566-938-0 Cited by: p1.
6. [8] (2009) Compensating the computational bias of spreadsheets. In Festschrift in honour of Bernd Krieg-Brückner’s 60^th birthdayFestschrift in Honour of Bernd Krieg-Brückner’s 60^th Birthday, B.
Hoffmann, B. Gersdorf, C. Lüth, T. Mossakowski, T. Röfer, L. Schröder, S. Hui, and M. Werner (Eds.), pp. 184–200. Cited by: p1.
7. [11] (2009) Formal management of CAD/CAM processes. In Festschrift in honour of Bernd Krieg-Brückner’s 60^th birthdayFestschrift in Honour of Bernd Krieg-Brückner’s 60^th Birthday, B. Hoffmann,
B. Gersdorf, C. Lüth, T. Mossakowski, T. Röfer, L. Schröder, S. Hui, and M. Werner (Eds.), pp. 201–216. Cited by: p1.
8. [9] (2008-04) Added-Value: Getting People into Semantic Work Environments. In Emerging technologies for semantic work environments: techniques, methods, and applicationsEmerging Technologies for
Semantic Work Environments: Techniques, Methods, and Applications, J. Rech, B. Decker, and E. Ras (Eds.), pp. 185–205. Cited by: p1.
9. [24] (2008-04) SWiM: a semantic wiki for mathematical knowledge management. In Emerging technologies for semantic work environments: techniques, methods, and applications, J. Rech, B. Decker, E.
Ras, J. Rech, B. Decker, and E. Ras (Eds.), pp. 47–68. External Links: Link Cited by: p1.
10. [5] (2008) Cut elimination with xi-functionality. In Festschrift in honour of Peter B. Andrews on his 70^th birthday, C. Benzmüller, C. Brown, J. Siekmann, and R. Statman (Eds.), External Links:
Link Cited by: p1.
11. [21] (2008) Wissensrepräsentation für computerunterstützte Lehre. In Selbstorganisiertes Lernen im InternetSelbstorganisiertes Lernen im Internet, V. Hornung-Prähauser, M. Luckmann, and M. Kalz
(Eds.), pp. 248–251. Cited by: p1.
12. [3] (2007) Towards a Mizar Mathematical Library in OMDoc format. In From insight to proof: festschrift in honour of Andrzej TrybulecFrom Insight to Proof: Festschrift in Honour of Andrzej
Trybulec, R. Matuszewski and A. Zalewska (Eds.), Studies in Logic, Grammar and Rhetoric, Vol. 10:23, pp. 265–275. External Links: Link Cited by: p1.
13. [7] (2006-08) MBase, an open mathematical knowledge base. In OMDoc – an open markup format for mathematical documents [version 1.2]OMDoc – An open markup format for mathematical documents
[Version 1.2], LNAI. Cited by: p1.
14. [17] (2006-08) Formal Proofs as Mathematical Objects. In OMDoc – an open markup format for mathematical documents [version 1.2]OMDoc – An open markup format for mathematical documents [Version
1.2], LNAI, pp. 177–179. Cited by: p1.
15. [18] (2006-08) Representing Proofs (Module PF). In OMDoc – an open markup format for mathematical documents [version 1.2]OMDoc – An open markup format for mathematical documents [Version 1.2],
LNAI, pp. 167–179. Cited by: p1.
16. [19] (2006-08) Standardizing context in system interoperability. In OMDoc – an open markup format for mathematical documents [version 1.2]OMDoc – An open markup format for mathematical documents
[Version 1.2], LNAI. Cited by: p1.
17. [20] (2006-08) STeX: a LaTeX-based workflow for OMDoc. In OMDoc – an open markup format for mathematical documents [version 1.2]OMDoc – An open markup format for mathematical documents [Version
1.2], LNAI. Cited by: p1.
18. [23] (2006-08) SWiM – an OMDoc-based semantic wiki. In OMDoc – an open markup format for mathematical documents [version 1.2], LNAI. Cited by: p1.
19. [26] (2006-08) OMDoc as a Data Format for VeriFun. In OMDoc – an open markup format for mathematical documents [version 1.2]OMDoc – An open markup format for mathematical documents [Version 1.2],
LNAI, pp. 329–332. Cited by: p1.
20. [16] (2003) Artificial intelligence: automated reasoning. In Van Nostrand’s Scientific Encyclopedia, pp. 247–250. Cited by: p1.
21. [6] (2001) Inference and computational semantics. In Computing meaning (volume 2)Computing Meaning (Volume 2), H. Bunt, L. Kievit, R. Muskens, and M. Verlinden (Eds.), pp. 11–28. Cited by: p1.
22. [12] (1999) $\Omega$MEGA – a mathematical assistant. In Liber Amicorum for the Fiftieth Birthday of Johan van Benthem, J. Gerbrandy, M. Marx, M. de Rijke, and Y. Venema (Eds.), pp. 248–251.
External Links: Link Cited by: p1.
23. [1] (1998) M. Kohlhase, W. Bibel, and P. Schmitt (Eds.), Cited by: p1.
24. [14] (1998) Automated theorem proving in mathematics. In Automated Deduction – A Basis for Applications, W. Bibel and P. Schmitt (Eds.), Vol. 3, pp. 3–7. Cited by: p1.
25. [15] (1998) Higher-order automated theorem proving. In Automated Deduction – A Basis for Applications, W. Bibel and P. Schmitt (Eds.), Vol. 2, pp. 431–462. External Links: Link Cited by: p1.
26. [13] (1992) Beweissysteme mit Logiken höherer Stufe. In Deduktionssysteme, Automatisierung des Logischen Denkens, K. H. Bläsius and H. Bürckert (Eds.), pp. 213–238. Cited by: p1.
Papers at International, Peer-Reviewed Conferences
1. [14] (2023) Learning support systems based on mathematical knowledge managment. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2023, C. Dubois and M. Kerber (Eds.),
LNAI. External Links: Link Cited by: p1.
2. [142] (2023) Learning with alea: tailored experiences through annotated course material. In KI-bildung, Lecture Notes in Informatics. External Links: Link Cited by: p1.
3. [155] (2023) The Y-Model – formalization of computer-science tasks in the context of adaptive learning systems. In 2023 IEEE German Education Conference (GeCon), External Links: Link Cited by: p1
4. [156] (2023) The potential of answer classes in large-scale written computer-science exams. In Proceedings of the 10th Symposium on Computer Science in Higher Education HDI23, Aachen, Germany,
pp. . Note: Accepted External Links: Link Cited by: p1.
5. [183] (2023) Towards an annotation standard for STEM documents – datasets, benchmarks, and spotters. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2023, C. Dubois and
M. Kerber (Eds.), LNAI, pp. 190–205. External Links: Link Cited by: p1.
6. [50] (2022) Explaining image classifications with near misses, near hits and prototypes: supporting domain experts in understanding decision boundaries. In Pattern Recognition and Artificial
Intelligence: Third International Conference, ICPRAI 2022, External Links: Document, Link Cited by: p1.
7. [122] (2022) System description: sTeX3 – a LaTeX-based ecosystem for semantic/active mathematical documents. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2022, K.
Buzzard and T. Kutsia (Eds.), LNAI, Vol. 13467, pp. 184–188. External Links: Link Cited by: p1.
8. [167] (2022) Injecting formal mathematics into latex. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2022, K. Buzzard and T. Kutsia (Eds.), LNAI, Vol. 13467,
pp. 168–183. External Links: Link Cited by: p1.
9. [168] (2022) An interactive explanatory ai system for industrial quality control. In IAAI 2022, External Links: Document, Link Cited by: p1.
10. [13] (2020) Towards a heterogeneous query language for mathematical knowledge. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2020, C. Benzmüller and B. Miller (Eds.),
LNAI, Vol. 12236, pp. 39–54. External Links: Link Cited by: p1.
11. [111] (2020) FrameIT: detangling knowledge management from game design in serious games. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2020, C. Benzmüller and B.
Miller (Eds.), LNAI, Vol. 12236, pp. 173–189. External Links: Document, Link Cited by: p1.
12. [125] (2020) Logic-independent proof search in logical frameworks (short paper). In 10th international joint conference on automated reasoning (IJCAR 2020)10th International Joint Conference on
Automated Reasoning (IJCAR 2020), N. Peltier and V. Sofronie-Stokkermans (Eds.), pp. 395–401. Cited by: p1.
13. [158] (2020) TGView3D: a system for 3-dimensional visualization of theory graphs. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2020, C. Benzmüller and B. Miller
(Eds.), LNAI, Vol. 12236, pp. 290–296. External Links: Link Cited by: p1.
14. [170] (2020) Representing structural language features in formal meta-languages. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2020, C. Benzmüller and B. Miller
(Eds.), LNAI, Vol. 12236, pp. 206–221. External Links: Link Cited by: p1.
15. [181] (2020) Prototyping controlled mathematical languages in jupyter notebooks. In Mathematical software – icms 2020. 7th international conferenceMathematical Software – ICMS 2020. 7th
international conference, A. M. Bigatti, J. Carette, J. H. Davenport, M. Joswig, and T. de Wolff (Eds.), LNCS, Vol. 12097, pp. 406–415. External Links: Link Cited by: p1.
16. [2] (2019) Integrating semantic mathematical documents and dynamic notebooks. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2019, C. Kaliszyck, E. Brady, A. Kohlhase,
and C. Sacerdoti Coen (Eds.), LNAI, pp. 275–290. External Links: Link Cited by: p1.
17. [12] (2019) Towards a unified mathematical data infrastructure: database and interface generation. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2019, C. Kaliszyck,
E. Brady, A. Kohlhase, and C. Sacerdoti Coen (Eds.), LNAI, pp. 28–43. External Links: Link Cited by: p1.
18. [30] (2019) Relational data across mathematical libraries. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2019, C. Kaliszyck, E. Brady, A. Kohlhase, and C. Sacerdoti
Coen (Eds.), LNAI, pp. 61–76. External Links: Link Cited by: p1.
19. [171] (2019) The Coq Library as a Theory Graph. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2019, C. Kaliszyck, E. Brady, A. Kohlhase, and C. Sacerdoti Coen (Eds.),
LNAI. Cited by: p1.
20. [175] (2019) Diagram combinators in MMT. In Intelligent computer mathematicsIntelligent Computer Mathematics, C. Kaliszyck, E. Brady, A. Kohlhase, C. Sacerdoti Coen, C. Kaliszyk, E. Brady, A.
Kohlhase, and C. Sacerdoti Coen (Eds.), LNAI, Cham, pp. 211–226. External Links: ISBN 978-3-030-23250-4, Link Cited by: p1.
21. [19] (2018) Translating the IMPS theory library to OMDoc/MMT. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2018, F. Rabe, W. M. Farmer, G. O. Passmore, and A.
Youssef (Eds.), LNAI. External Links: Link Cited by: p1.
22. [83] (2018) Discourse phenomena in math documents. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2018, F. Rabe, W. M. Farmer, G. O. Passmore, and A. Youssef (Eds.),
LNAI. External Links: Link Cited by: p1.
23. [166] (2018) Automatically finding theory morphisms for knowledge management. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2018, F. Rabe, W. M. Farmer, G. O.
Passmore, and A. Youssef (Eds.), LNAI. External Links: Link Cited by: p1.
24. [169] (2018) Theories as types. In 9th international joint conference on automated reasoning9th International Joint Conference on Automated Reasoning, D. Galmiche, S. Schulz, and R. Sebastiani
(Eds.), External Links: Link Cited by: p1.
25. [174] (2018) Knowledge amalgamation for computational science and engineering. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2018, F. Rabe, W. M. Farmer, G. O.
Passmore, and A. Youssef (Eds.), LNAI. External Links: Link Cited by: p1.
26. [182] (2018) Syntactic/semantic analysis for high-precision math linguistics. In Workshop papers at 11th conference on intelligent computer mathematics cicm 2018Workshop Papers at 11th Conference
on Intelligent Computer Mathematics CICM 2018, O. Hasan, A. Youssef, A. Naumowicz, W. Farmer, C. Kaliszyk, D. Gallois-Wong, F. Rabe, G. D. Reis, G. Passmore, J. Davenport, M. Pfeiffer, M.
Kohlhase, S. Autexier, S. Tahar, T. Koprucki, U. Siddique, W. Neuper, W. Windsteiger, W. Schreiner, W. Sperber, and Z. Kovács (Eds.), Note: CICM Work in Progress Paper External Links: Link Cited
by: p1.
27. [62] (2017) Towards competency based testing and feedback: competency definition and measurement in the field of algorithms & data structures. In Proceedings of 2017 IEEE Global Engineering
Education Conference (EDUCON), IEEE Conference Publications, pp. 517–526. External Links: ISBN 978-1-5090-5467-1, Document Cited by: p1.
28. [79] (2017) Visual structure in math expressions. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2017, H. Geuvers, M. England, O. Hasan, F. Rabe, and O. Teschke (Eds.)
, LNAI. External Links: Link Cited by: p1.
29. [114] (2017) Mathematical models as research data via flexiformal theory graphs. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2017, H. Geuvers, M. England, O. Hasan,
F. Rabe, and O. Teschke (Eds.), LNAI. External Links: Link Cited by: p1.
30. [120] (2017) Making PVS accessible to generic services by interpretation in a universal format. In Interactive theorem proving 8th international conference, itp 2017Interactive Theorem Proving,
M. Ayala-Rincón and C. A. Muñoz (Eds.), LNCS, Vol. 10499. External Links: Link Cited by: p1.
31. [121] (2017) Knowledge-based interoperability for mathematical software systems. In MACIS 2017MACIS 2017: Seventh International Conference on Mathematical Aspects of Computer and Information
Sciences, J. Blömer, T. Kutsia, and D. Simos (Eds.), LNCS, pp. 195–210. External Links: Link Cited by: p1.
32. [165] (2017) Classification of alignments between concepts of formal mathematical systems. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2017, H. Geuvers, M. England,
O. Hasan, F. Rabe, and O. Teschke (Eds.), LNAI. External Links: Link Cited by: p1.
33. [184] (2017) Qualitative content analysis of programming errors. In Proceedings of the 5th International Conference on Information and Education Technology, International Conference Proceedings
Series, New York. External Links: ISBN 978-1-4503-4791-4, Document Cited by: p1.
34. [187] (2017) Software citations, information systems, and beyond. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2017, H. Geuvers, M. England, O. Hasan, F. Rabe, and
O. Teschke (Eds.), LNAI. External Links: Link Cited by: p1.
35. [190] (2017) Virtual theories – a uniform interface to mathematical knowledge bases. In MACIS 2017MACIS 2017: Seventh International Conference on Mathematical Aspects of Computer and Information
Sciences, J. Blömer, T. Kutsia, and D. Simos (Eds.), LNCS, pp. 243–257. External Links: Link Cited by: p1.
36. [17] (2016) Towards deriving programming competencies from student errors. In 4th International Conference on Learning and Teaching in Computing and Engineering (LaTiCE), Los Alamitos, pp. 19–23.
Cited by: p1.
37. [35] (2016) Interoperability in the OpenDreamKit project: the math-in-the-middle approach. In Intelligent computer mathematicsIntelligent Computer Mathematics 2016, M. Kohlhase, M. Johansson, B.
Miller, L. de Moura, and F. Tompa (Eds.), LNAI. External Links: Link Cited by: p1.
38. [45] (2016) The SMGloM project and system. towards a terminology and ontology for mathematics. In Mathematical software - ICMS 2016 - 5th international congressMathematical Software - ICMS 2016 -
5th International Congress, G. Greuel, T. Koch, P. Paule, and A. Sommese (Eds.), LNCS, Vol. 9725. External Links: Link Cited by: p1.
39. [49] (2016) Faceted search for mathematics. In MACIS 2015MACIS 2015: Sixth International Conference on Mathematical Aspects of Computer and Information Sciences, I. S. Kotsireas, S. M. Rump, and
C. K. Yap (Eds.), LNCS. External Links: Link Cited by: p1.
40. [154] (2016) Winning questions: inquisitive semantics and the lottery paradox. In 5th LSE Graduate Conference in Philosophy of Probability, External Links: Link Cited by: p1.
41. [157] (2016) Formula semantification and automated relation finding in the OEIS. In Mathematical software - ICMS 2016 - 5th international congressMathematical Software - ICMS 2016 - 5th
International Congress, G. Greuel, T. Koch, P. Paule, and A. Sommese (Eds.), LNCS, Vol. 9725. External Links: Link Cited by: p1.
42. [16] (2015) Evaluation of source code with item response theory. In Proceedings of the 20th SIGCSE Conference on Innovation and Technology in Computer Science Education, New York, pp. 51–56.
Cited by: p1.
43. [18] (2015) Investigating novice programming abilities with the help of psychometric assessment. In Proceedings of Society for Information Technology & Teacher Education International Conference
2015, D. Slykhuis and G. Marks (Eds.), Las Vegas, NV, United States, pp. 90–95. Cited by: p1.
44. [67] (2015) A flexiformal model of knowledge dissemination and aggregation in mathematics. In Intelligent computer mathematicsIntelligent Computer Mathematics 2015, M. Kerber, J. Carette, C.
Kaliszyk, F. Rabe, and V. Sorge (Eds.), LNCS, pp. 137–152. External Links: Link Cited by: p1.
45. [68] (2015) Math literate knowledge management via induced material. In Intelligent computer mathematicsIntelligent Computer Mathematics 2015, M. Kerber, J. Carette, C. Kaliszyk, F. Rabe, and V.
Sorge (Eds.), LNCS, pp. 187–202. External Links: Link Cited by: p1.
46. [128] (2015) The SMGloM project or why we need a semantic glossary of mathematics. In European Conference on Data Analysis ECDA2015; Data Science: Foundations, Methods and Applications, External
Links: Link Cited by: p1.
47. [159] (2015) Development of a competency model for computer science teachers at secondary school level. In IEEE Global Engineering Education Conference, Los Alamitos, pp. 211–220. External Links:
Document Cited by: p1.
48. [180] (2015) Classification of programming tasks according to required skills and knowledge representation. In Informatics in Schools. Curricula, Competences, and Competitions, A. Brodnik and J.
Vahrenhold (Eds.), Lecture Notes in Computer Science, Heidelberg, pp. 57–68. External Links: ISBN 978-3-319-25395-4 Cited by: p1.
49. [21] (2014) Towards ontological support for principle solutions in mechanical engineering. In Formal ontology in information systems - proceedings of the eighth international conference, FOIS
2014Formal Ontology in Information Systems - Proceedings of the Eighth International Conference, FOIS 2014, P. Garbacz and O. Kutz (Eds.), Frontiers in Artificial Intelligence and Applications,
Vol. 267, pp. 427–432. External Links: Document Cited by: p1.
50. [23] (2014) Set theory or higher order logic to represent auction concepts in Isabelle?. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A.
Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 236–251. External Links: 1406.0774 Cited by: p1.
51. [24] (2014) Realms: a structure for consolidating knowledge about mathematical theories. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A.
Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 252–266. Note: MKM Best-Paper-Award External Links: Link Cited by: p1.
52. [44] (2014) NNexus reloaded. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 423–426. External
Links: Link Cited by: p1.
53. [46] (2014) E-books and graphics with LaTeXml. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS,
pp. 427–430. External Links: Link Cited by: p1.
54. [57] (2014) Flexary operators for formalized mathematics. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.)
, LNCS, pp. 312–327. External Links: Link Cited by: p1.
55. [65] (2014) System description: MathHub.info. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS,
pp. 431–434. External Links: Link Cited by: p1.
56. [66] (2014) Representing, archiving, and searching the space of mathematical knowledge. In Mathematical software - ICMS 2014 - 4th international congressMathematical Software - ICMS 2014 - 4th
International Congress, H. Hong and C. Yap (Eds.), LNCS, Vol. 8592, pp. 26–30. External Links: Document, Link Cited by: p1.
57. [73] (2014) Towards an interaction-based integration of MKM services into end-user applications. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport,
A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 344–356. External Links: Link Cited by: p1.
58. [74] (2014) Towards knowledge management for HOL Light. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.),
LNCS, pp. 357–372. External Links: Link Cited by: p1.
59. [107] (2014) Math web search interfaces and the generation gap of mathematicians. In Mathematical software - ICMS 2014 - 4th international congressMathematical Software - ICMS 2014 - 4th
International Congress, H. Hong and C. Yap (Eds.), LNCS, Vol. 8592, pp. 586–593. External Links: Document Cited by: p1.
60. [108] (2014) Search interfaces for mathematicians. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS,
pp. 153–168. External Links: Link Cited by: p1.
61. [109] (2014) System description: a semantics-aware LaTeX-to-office converter. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka,
and J. Urban (Eds.), LNCS, pp. 440–443. External Links: Link Cited by: p1.
62. [112] (2014) Discourse-level parallel markup and meaning adoption in flexiformal theory graphs. In Mathematical software - ICMS 2014 - 4th international congressMathematical Software - ICMS 2014
- 4th International Congress, H. Hong and C. Yap (Eds.), LNCS, Vol. 8592, pp. 36–40. External Links: Document Cited by: p1.
63. [141] (2014) A data model and encoding for a semantic, multilingual terminology of mathematics. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A.
Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 169–183. External Links: Link Cited by: p1.
64. [15] (2013) Concept specification maps: displaying content structures. In Proceedings of the 18th ACM conference on Innovation and technology in computer science education, New York, USA,
pp. 291–296. External Links: ISBN 978-1-4503-2078-8, Document Cited by: p1.
65. [47] (2013) LaTeXml 2012 - a year of LaTeXml. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Carette, D. Aspinall, C. Lange, P. Sojka, and W. Windsteiger (Eds.), Lecture
Notes in Computer Science, pp. 335–338. Cited by: p1.
66. [61] (2013) Pedagogical content knowledge for computer science in german teacher education curricula. In Proceedings of the 8th Workshop in Primary and Secondary Computing Education, New York,
pp. 95–103. External Links: Link, ISBN 978-1-4503-2455-7, Document Cited by: p1.
67. [80] (2013) Full semantic transparency: overcoming boundaries of applications. In Human-computer interaction – interact 2013Human-Computer Interaction – INTERACT 2013, P. Kotzé, G. Marsden, G.
Lindgaard, J. Wesson, and M. Winckler (Eds.), LNCS, pp. 406–423. External Links: Link Cited by: p1.
68. [106] (2013) Human-spreadsheet interaction. In Human-computer interaction – interact 2013Human-Computer Interaction – INTERACT 2013, P. Kotzé, G. Marsden, G. Lindgaard, J. Wesson, and M. Winckler
(Eds.), LNCS, pp. 571–578. Cited by: p1.
69. [117] (2013) A universal machine for biform theory graphs. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Carette, D. Aspinall, C. Lange, P. Sojka, and W. Windsteiger
(Eds.), Lecture Notes in Computer Science. External Links: Link Cited by: p1.
70. [124] (2013) XLSearch: a search engine for spreadsheets. In Symp. of the european spreadsheet risks interest group (EuSpRIG 2013)Symp. of the European Spreadsheet Risks Interest Group (EuSpRIG
2013), External Links: Link Cited by: p1.
71. [139] (2013) Knowledge management for systematic engineering design in CAD systems. In Professionelles Wissenmanagement Management, Konferenzbeiträge der 7. KonferenzProfessionelles
Wissenmanagement Management, Konferenzbeiträge der 7. Konferenz, F. Lehner, N. Amende, and N. Fteimi (Eds.), pp. 202–217. External Links: Link Cited by: p1.
72. [140] (2013) The flexiformalist manifesto. In International workshop on symbolic and numeric algorithms for scientific computing (SYNASC 2012)14th International Workshop on Symbolic and Numeric
Algorithms for Scientific Computing (SYNASC 2012), A. Voronkov, V. Negru, T. Ida, T. Jebelean, D. Petcu, S. M. Watt, and D. Zaharie (Eds.), pp. 30–36. External Links: Link Cited by: p1.
73. [143] (2013) A qualitative comparison of the suitability of four theorem provers for basic auction theory. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Carette, D.
Aspinall, C. Lange, P. Sojka, and W. Windsteiger (Eds.), Lecture Notes in Computer Science, pp. 200–215. External Links: 1303.4193 Cited by: p1.
74. [146] (2013) Applying mechanised reasoning in economics – making reasoners applicable for domain experts. In INFORMATIK 2013: Informatik angepasst an Mensch, Organisation und Umwelt, M. Horbach
(Ed.), Lecture Notes in Informatics, pp. 153–156. External Links: ISBN 978-3-88579-614-5 Cited by: p1.
75. [151] (2013) The formare project – formal mathematical reasoning in economics. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Carette, D. Aspinall, C. Lange, P. Sojka,
and W. Windsteiger (Eds.), Lecture Notes in Computer Science, pp. 330–334. External Links: 1303.4194 Cited by: p1.
76. [161] (2013) Semantics of the distributed ontology language: institutes and institutions. In Recent Trends in Algebraic Development TechniquesRecent Trends in Algebraic Development Techniques, T.
Mossakowski, N. Martí-Oliet, and M. Palomino Tarjuelo (Eds.), LNCS, pp. 212–230. Cited by: p1.
77. [162] (2013) Three semantics for the core of the Distributed Ontology Language (extended abstract). In International Joint Conference on Artificial Intelligence, F. Rossi (Ed.), Cited by: p1, 160
78. [178] (2013) The MMT API: A Generic MKM System. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Carette, D. Aspinall, C. Lange, P. Sojka, and W. Windsteiger (Eds.),
Lecture Notes in Computer Science, pp. 339–343. Cited by: p1.
79. [179] (2013) Types of assignments for novice programmers. In Proceedings of the 8th Workshop in Primary and Secondary Computing Education, New York, pp. 43–44. External Links: ISBN
978-1-4503-2455-7, Document Cited by: p1.
80. [33] (2012) Semantic Alliance: a framework for semantic allies. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka,
M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 49–64. External Links: Link Cited by: p1.
81. [56] (2012) Extending MKM formats at the statement level. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka, M.
Wenzel, and V. Sorge (Eds.), LNAI, pp. 65–80. External Links: Link Cited by: p1.
82. [69] (2012) Management of Change in Declarative Languages. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka, M.
Wenzel, and V. Sorge (Eds.), LNAI, pp. 325–340. Cited by: p1.
83. [72] (2012) Cost-effective integration of mkm semantic services into editing environments. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J.
Carette, G. Dos Reis, P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 96–110. External Links: Link Cited by: p1.
84. [118] (2012) MathWebSearch 0.5 – Scaling an Open Formula Search Engine. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis,
P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 342–357. External Links: Link Cited by: p1.
85. [138] (2012) The Planetary project: towards eMath3.0. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka, M.
Wenzel, and V. Sorge (Eds.), LNAI, pp. 448–452. External Links: 1206.5048 Cited by: p1.
86. [144] (2012) Reimplementing the mathematics subject classification (MSC) as a linked open dataset. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell,
J. Carette, G. Dos Reis, P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 458–462. External Links: 1204.5086 Cited by: p1.
87. [145] (2012) Bringing mathematics to the web of data: the case of the mathematics subject classification. In The semantic webThe Semantic Web, E. Simperl, P. Cimiano, A. Polleres, O. Corcho, and
V. Presutti (Eds.), LNCS, pp. 763–777. External Links: Link, Document Cited by: p1.
88. [149] (2012) The distributed ontology language (DOL): ontology integration and interoperability applied to mathematical formalization. In Intelligent computer mathematicsIntelligent Computer
Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 463–467. External Links: 1204.5093 Cited by: p1.
89. [150] (2012) The Distributed Ontology Language (DOL): use cases, syntax, and extensibility. In Terminology and knowledge engineering conference (TKE)Terminology and Knowledge Engineering
Conference (TKE), G. Aguado de Cea, M. C. Suárez-Figueroa, R. García-Castro, and E. Montiel-Ponsoda (Eds.), pp. 33–48. External Links: 1208.0293 Cited by: p1.
90. [160] (2012) Three semantics for the core of the Distributed Ontology Language. In Formal Ontology in Information Systems, M. Donnelly and G. Guizzardi (Eds.), Frontiers in Artificial
Intelligence and Applications, pp. 337–352. Note: Extended abstract published as [162] External Links: ISBN 978-1-61499-084-0, Link Cited by: p1.
91. [177] (2012) A Query Language for Formal Mathematical Libraries. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P.
Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 142–157. External Links: 1204.4685 Cited by: p1.
92. [189] (2012) Point-and-write — documenting formal mathematics by reference. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos
Reis, P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 169–185. External Links: 1204.5094 Cited by: p1.
93. [1] (2011) Licensing the Mizar Mathematical Library. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.), LNAI, pp. 149–163.
Cited by: p1.
94. [5] (2011) Workflows for the management of change in science, technologies, engineering and mathematics. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W.
Farmer, F. Rabe, and J. Urban (Eds.), LNAI, pp. 164–179. External Links: Link Cited by: p1.
95. [25] (2011) Authoring and publishing of units and quantities in semantic documents. In The semantic web: ESWC 2011 workshopsThe Semantic Web: ESWC 2011 Workshops, R. García Castro, D. Fensel, and
G. Antoniou (Eds.), LNCS, pp. 202–216. External Links: Link Cited by: p1.
96. [27] (2011) A Proof Theoretic Interpretation of Model Theoretic Hiding. In Recent Trends in Algebraic Development TechniquesRecent Trends in Algebraic Development Techniques, H. Kreowski and T.
Mossakowski (Eds.), LNCS. Cited by: p1.
97. [28] (2011) Towards Logical Frameworks in the Heterogeneous Tool Set Hets. In Recent Trends in Algebraic Development TechniquesRecent Trends in Algebraic Development Techniques, H. Kreowski and
T. Mossakowski (Eds.), LNCS. Cited by: p1.
98. [29] (2011) Project abstract: logic atlas and integrator (LATIN). In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.), LNAI,
pp. 289–291. External Links: Link, Document Cited by: p1.
99. [48] (2011) The LaTeXML daemon: editable math on the collaborative web. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.),
LNAI, pp. 292–294. Cited by: p1.
100. [54] (2011) Combining source, content, presentation, narration, and relational representation. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F.
Rabe, and J. Urban (Eds.), LNAI, pp. 212–227. External Links: Link Cited by: p1.
101. [55] (2011) Extending OpenMath with Sequences. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.), LNAI, pp. 58–72.
External Links: Link Cited by: p1.
102. [93] (2011) Maintaining islands of consistency via versioned links. In Proceedings of the 29^th annual ACM international conference on design of communication (SIGDOC)Proceedings of the 29^th
annual ACM international conference on Design of communication (SIGDOC), pp. 167–174. External Links: Link Cited by: p1.
103. [94] (2011) Maintaining islands of consistency via versioned links. In Intelligent computer mathematics – work in progress papersIntelligent Computer Mathematics – Work in Progress Papers, J.
Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.), External Links: Link Cited by: p1.
104. [95] (2011) Towards a flexible notion of document context. In Proceedings of the 29^th annual ACM international conference on design of communication (SIGDOC)Proceedings of the 29^th annual ACM
international conference on Design of communication (SIGDOC), pp. 181–188. External Links: Link Cited by: p1.
105. [126] (2011) A foundational view on integration problems. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.), LNAI,
pp. 107–122. Note: https://kwarc.info/kohlhase/papers/cicm11-integration.pdf External Links: Link Cited by: p1.
106. [147] (2011) The planetary system: executable science, technology, engineering and math papers. In The semantic web: research and applications (part II)ESWC (Part II), G. Antoniou, M. Grobelnik,
E. Paslaru Bontas Simperl, B. Parsia, D. Plexousakis, P. D. Leenheer, and J. Z. Pan (Eds.), LNCS, pp. 471–475. External Links: 1103.1482 Cited by: p1.
107. [153] (2011) Krextor – an extensible framework for contributing content math to the web of data. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F.
Rabe, and J. Urban (Eds.), LNAI, pp. 304–306. External Links: Link Cited by: p1.
108. [6] (2010) Semantics-based change impact analysis for heterogeneous collections of documents. In Proceedings of the 10^th ACM symposium on document engineeringProceedings of the 10^th ACM
symposium on Document engineering, M. Gormish and R. Ingold (Eds.), DocEng ’10, pp. 97–106. External Links: Link, Document Cited by: p1.
109. [32] (2010) Interactive Documents as Interfaces to Computer Algebra Systems: JOBAD and Wolfram—Alpha. In CALCULEMUS (Emerging Trends), D. Delahaye and R. Rioboo (Eds.), Technical Reports of
CEDRIC (CNAM/ENSIIE), pp. 13–30. Cited by: p1.
110. [34] (2010) Publishing math lecture notes as linked data. In The semantic web: research and applications (part II)The Semantic Web: Research and Applications (Part II), L. Aroyo, G. Antoniou, E.
Hyvönen, A. ten Teije, H. Stuckenschmidt, L. Cabral, and T. Tudorache (Eds.), LNCS, pp. 370–375. External Links: 1004.3390v1 Cited by: p1.
111. [71] (2010) sTeXIDE: an integrated development environment for sTeX collections. In Intelligent computer mathematicsIntelligent Computer Mathematics, S. Autexier, J. Calmet, D. Delahaye, P. D.
F. Ion, L. Rideau, R. Rioboo, and A. P. Sexton (Eds.), LNAI, pp. 336–344. External Links: Link Cited by: p1.
112. [81] (2010) Dimensions of formality: a case study for MKM in software engineering. In Intelligent computer mathematicsIntelligent Computer Mathematics, S. Autexier, J. Calmet, D. Delahaye, P. D.
F. Ion, L. Rideau, R. Rioboo, and A. P. Sexton (Eds.), LNAI, pp. 355–369. External Links: 1004.5071v1 Cited by: p1.
113. [82] (2010) sTeX – a system for flexible formalization of linked data. In 6^th international conference on semantic systems (I-Semantics) and the 5^th international conference on pragmatic
webProceedings of the 6^th International Conference on Semantic Systems (I-Semantics) and the 5^th International Conference on Pragmatic Web, A. Paschke, N. Henze, T. Pellegrini, and H. Weigand
(Eds.), External Links: 1006.4474v1, Document Cited by: p1.
114. [92] (2010) What we understand is what we get: assessment in spreadsheets. In Symp. of the european spreadsheet risks interest group (EuSpRIG 2010)Symp. of the European Spreadsheet Risks
Interest Group (EuSpRIG 2010), S. Thorne (Ed.), pp. 111–121. External Links: Link Cited by: p1.
115. [105] (2010) Towards user assistance for documents via interactional semantic technology. In Proceedings of the 33.rd annual german conference on artificial intelligence ki’10KI 2010: Advances
in Artificial Intelligence, R. Dillmann, J. Beyerer, U. D. Hanebeck, and T. Schultz (Eds.), LNAI, pp. 107–115. Cited by: p1.
116. [127] (2010) Towards MKM in the large: modular representation and scalable software architecture. In Intelligent computer mathematicsIntelligent Computer Mathematics, S. Autexier, J. Calmet, D.
Delahaye, P. D. F. Ion, L. Rideau, R. Rioboo, and A. P. Sexton (Eds.), LNAI, pp. 370–384. External Links: 1005.5232v2 Cited by: p1.
117. [191] (2010) A [insert xml format] database for [insert cool application]. In Proceedings of XML Prague 2010Proceedings of XML Prague 2010, External Links: Link Cited by: p1.
118. [193] (2010) Scripting documents with xquery: virtual documents in TNTBase. In Proceedings of balisage: the markup conference 2010Proceedings of Balisage: The Markup Conference 2010, Balisage
Series on Markup Technologies. Note: available at http://www.balisage.net/Proceedings/vol3/html/Zholudev01/BalisageVol3-Zholudev01.html External Links: Link Cited by: p1.
119. [22] (2009-07) Formal Proofs: Reconciling Correctness and Understanding. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt (Eds.),
LNAI. Cited by: p1.
120. [31] (2009-07) Unifying Math Ontologies: A tale of two standards. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt (Eds.), LNAI,
pp. 263–278. External Links: Link Cited by: p1.
121. [43] (2009-07) Integrating web services into active mathematical documents. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt
(Eds.), LNAI, pp. 279–293. External Links: Link Cited by: p1.
122. [88] (2009-07) Compensating the computational bias of spreadsheets with MKM techniques. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S.
M. Watt (Eds.), LNAI, pp. 357–372. External Links: Link Cited by: p1.
123. [91] (2009-07) Spreadsheet interaction with frames: exploring a mathematical practice. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M.
Watt (Eds.), LNAI, pp. 341–356. External Links: Link Cited by: p1.
124. [148] (2009-07) A mathematical approach to ontology authoring and documentation. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt
(Eds.), LNAI, pp. 389–404. External Links: Link Cited by: p1.
125. [7] (2009) Kripke Semantics for Martin-Löf’s Extensional Type Theory. In Typed Lambda Calculi and Applications (TLCA), P. Curien (Ed.), LNCS, Vol. 5608, pp. 249–263. Cited by: p1.
126. [89] (2009) Modeling task experience in user assistance systems. In Proceedings of the 27^th annual ACM international conference on design of communication (SIGDOC)Proceedings of the 27^th
annual ACM international conference on Design of communication (SIGDOC), B. Mehlenbacher, A. Protopsaltis, A. Williams, and S. Slatterey (Eds.), pp. 135–142. External Links: Link Cited by: p1.
127. [90] (2009) Semantic transparency in user assistance systems. In Proceedings of the 27^th annual ACM international conference on design of communication (SIGDOC)Proceedings of the 27^th annual
ACM international conference on Design of communication (SIGDOC), B. Mehlenbacher, A. Protopsaltis, A. Williams, and S. Slatterey (Eds.), pp. 89–96. External Links: Link Cited by: p1.
128. [116] (2009) Formal management of CAD/CAM processes. In 16^th international symposium on formal methods (FM 2009)16^th International Symposium on Formal Methods (FM 2009), A. Cavalcanti and D.
Dams (Eds.), LNCS, pp. 223–238. External Links: Link Cited by: p1.
129. [186] (2009) Translating Dependently-Typed Logic to First-Order Logic. In Recent Trends in Algebraic Development Techniques, A. Corradini and U. Montanari (Eds.), LNCS, Vol. 5486, pp. 326–341.
Cited by: p1.
130. [192] (2009) TNTBase: a versioned storage for XML. In Proceedings of balisage: the markup conference 2009Proceedings of Balisage: The Markup Conference 2009, Balisage Series on Markup
Technologies. Note: available at https://kwarc.info/vzholudev/pubs/balisage.pdf External Links: Link Cited by: p1.
131. [104] (2008-06) MS PowerPoint use from a micro-perspective. In Proceedings of the world conference on educational multimedia, hypermedia & telecommunications 2008 (ED-MEDIA’08)Proceedings of the
World Conference on Educational Multimedia, Hypermedia & Telecommunications 2008 (ED-MEDIA’08), pp. 1279–1286. External Links: Link Cited by: p1.
132. [119] (2008) Notations for living mathematical documents. In Intelligent computer mathematicsIntelligent Computer Mathematics, S. Autexier, J. Campbell, J. Rubio, V. Sorge, M. Suzuki, and F.
Wiedijk (Eds.), LNAI, pp. 504–519. External Links: Link Cited by: p1.
133. [152] (2008) SWiM – a semantic wiki for mathematical knowledge management. In The semantic web: research and applicationsThe Semantic Web: Research and Applications, S. Bechhofer, M. Hauswirth,
J. Hoffmann, and M. Koubarakis (Eds.), LNCS, pp. 832–837. External Links: 1003.5196v1 Cited by: p1.
134. [163] (2008) Towards a community of practice toolkit based on semantically marked up artifacts. In Proceedings of the 1^st World Summit of the Knowledge Society: Emerging Technologies and
Information Systems for the Knowledge Society, M. D. Lytras and others (Eds.), LNAI, pp. 41–50. Cited by: p1.
135. [188] (2008) Transforming the arXiv to XML. In Intelligent computer mathematicsIntelligent Computer Mathematics, S. Autexier, J. Campbell, J. Rubio, V. Sorge, M. Suzuki, and F. Wiedijk (Eds.),
LNAI, pp. 574–582. External Links: Link Cited by: p1.
136. [97] (2007-09) What can the hundred languages of children teach us?. In Mensch & Computer 2007: Interaktion im PluralInteraktion im Plural, T. Gross (Ed.), Konferenzreihe Mensch und Computer,
pp. 189–198. Cited by: p1.
137. [102] (2007-06) Semantic powerpoint: content and semantic technology for educational added-value services in MS PowerPoint. In Proceedings of the world conference on educational multimedia,
hypermedia & telecommunications 2007 (ED-MEDIA’07)Proceedings of the World Conference on Educational Multimedia, Hypermedia & Telecommunications 2007 (ED-MEDIA’07), C. Montgomerie and J. Seale
(Eds.), pp. 3576–3583. External Links: Link Cited by: p1.
138. [53] (2007) Formal representation of mathematics in a dependently typed set theory. In MKM/CalculemusTowards Mechanized Mathematical Assistants. MKM/Calculemus, M. Kauers, M. Kerber, R. Miner,
and W. Windsteiger (Eds.), LNAI, pp. 265–279. Cited by: p1.
139. [87] (2007) Reexamining the MKM Value Proposition: From Math Web Search to Math Web ReSearch. In MKM/CalculemusTowards Mechanized Mathematical Assistants. MKM/Calculemus, M. Kauers, M. Kerber,
R. Miner, and W. Windsteiger (Eds.), LNAI, pp. 266–279. External Links: Link Cited by: p1.
140. [101] (2007) Semantic Perspectives on Knowledge Management and E-Learning. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) conference proceedingsWissens- und
Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) conference proceedings, A. Hinneburg (Ed.), pp. 281–288. Cited by: p1.
141. [103] (2007) The music is not in the piano: engaging vs. enabling MS PowerPoint. In Proceedings of Society for Information Technology and Teacher Education International Conference 2007
(SITE’07), R. Carlsen, K. McFerrin, J. Price, R. Weber, and D. A. Willis (Eds.), pp. 2026–2028. Note: St. Antonio, TX (USA), 2007-03-24/28 Cited by: p1.
142. [164] (2007) Presentation on Modeling Scientific Communities of Practice based on Semantic Markup of Scientific Documents and Web2.0. Technologies. In 7. konferenz für interaktive und
kooperative medienMensch und Computer 2007, T. Gross (Ed.), Cited by: p1.
143. [172] (2007) Extended formula normalization for $\epsilon$-retrieval and sharing of mathematical knowledge. In MKM/CalculemusTowards Mechanized Mathematical Assistants. MKM/Calculemus, M.
Kauers, M. Kerber, R. Miner, and W. Windsteiger (Eds.), LNAI, pp. 266–279. Cited by: p1.
144. [99] (2006-03) Media or Medea Society? Learner and Learning Technology as Full Partners. In ICDML2006, B. Thipakorn (Ed.), Vol. 1, pp. 6–12. Note: Bangkok (Thailand), 2006-03-13/14 Cited by: p1.
145. [9] (2006) Cut-simulation in impredicative logics. In Automated reasoning — third international joint conference, ijcar 2006Automated Reasoning — Third International Joint Conference, IJCAR 2006
, U. Furbach and N. Shankar (Eds.), LNAI, pp. 220–234. External Links: Link Cited by: p1.
146. [52] (2006) Capturing the content of physics: systems, observables, and experiments. In MKM 2006Mathematical Knowledge Management (MKM), J. Borwein and W. M. Farmer (Eds.), LNAI, pp. 165–178.
External Links: Link Cited by: p1.
147. [85] (2006) An exploration in the space of mathematical knowledge. In Mathematical knowledge management, MKM’05Mathematical Knowledge Management, MKM’05, M. Kohlhase (Ed.), LNAI, pp. 17–32.
External Links: Link Cited by: p1.
148. [86] (2006) Communities of Practice in MKM: An Extensional Model. In MKM 2006Mathematical Knowledge Management (MKM), J. Borwein and W. M. Farmer (Eds.), LNAI, pp. 179–193. External Links: Link
Cited by: p1.
149. [96] (2006) Embodied constructionist learning: social tagging and folksonomies in e-learning environments. In Current Developments in Technology-Assisted Education (mICTE06), Vol. 3, Badajoz,
Spain. Cited by: p1.
150. [100] (2006) What if PowerPoint became emPowerPoint (through CPoint)?. In Society for Information Technology and Teacher Education, 17^th International Conference SITE 2006, C. M. Crawford (Ed.)
, pp. 2934–2939. Note: Orlando (USA), 2006-03-20/24 Cited by: p1.
151. [129] (2006) A search engine for mathematical formulae. In Proceedings of artificial intelligence and symbolic computation, AISC’2006Proceedings of Artificial Intelligence and Symbolic
Computation, AISC’2006, T. Ida, J. Calmet, and D. Wang (Eds.), LNAI, pp. 241–253. External Links: Link Cited by: p1.
152. [176] (2006) First-Order Logic with Dependent Types. In Automated reasoning — third international joint conference, ijcar 2006Automated Reasoning — Third International Joint Conference, IJCAR
2006, U. Furbach and N. Shankar (Eds.), LNAI, pp. 377–391. Cited by: p1.
153. [98] (2005-11) Overcoming Proprietary Hurdles: CPoint as Invasive Editor. In Open source for education in europe: research and practiseOpen Source for Education in Europe: Research and Practise,
F. de Vries, G. Attwell, R. Elferink, and A. Tödt (Eds.), pp. 51–56. Cited by: p1.
154. [137] (2005-11) OMDoc: Open Mathematical Documents. In Open source for education in europe: research and practiseOpen Source for Education in Europe: Research and Practise, F. de Vries, G.
Attwell, R. Elferink, and A. Tödt (Eds.), pp. 137–143. Cited by: p1.
155. [84] (2004) CPoint: dissolving the author’s dilemma. In Mathematical knowledge management, MKM’04Mathematical Knowledge Management, MKM’04, A. Asperti, G. Bancerek, and A. Trybulec (Eds.), LNAI,
pp. 175–189. External Links: Link Cited by: p1.
156. [26] (2003-09) System description: analytica 2. In 11^th symposium on the integration of symbolic computation and mechanized reasoning (Calculemus 2003)Proceedings of the 11^th Symposium on the
Integration of Symbolic Computation and Mechanized Reasoning (Calculemus 2003), T. Hardin and R. Rioboo (Eds.), pp. 69–74. External Links: Link Cited by: p1.
157. [110] (2003) Towards collaborative content management and version control for structured mathematical knowledge. In Mathematical knowledge management, MKM’03Mathematical Knowledge Management,
MKM’03, A. Asperti, B. Buchberger, and J. H. Davenport (Eds.), LNCS, pp. 147–161. External Links: Link Cited by: p1.
158. [4] (2002) MathML in the MoWGLI project. In Second international conference on mathml and technologies for math on the webSecond International Conference on MathML and Technologies for Math on
the Web, External Links: Link Cited by: p1.
159. [130] (2002) Acquisition of math content in an academic setting. In Second international conference on mathml and technologies for math on the webSecond International Conference on MathML and
Technologies for Math on the Web, External Links: Link Cited by: p1.
160. [185] (2002) Proof development with $\Omega$mega. In Automated deduction — cade-18Automated Deduction — CADE-18, A. Voronkov (Ed.), LNAI, pp. 144–149. Cited by: p1.
161. [194] (2002) System Description: The MathWeb software bus for distributed mathematical reasoning. In Automated deduction — cade-18Automated Deduction — CADE-18, A. Voronkov (Ed.), LNAI,
pp. 247–252. External Links: Link Cited by: p1.
162. [135] (2001) Formal representation issues in an open mathematical knowledge base. In Electronic proceedings of the first international workshop on mathematical knowledge management:
MKM’2001Electronic Proceedings of the First International Workshop on Mathematical Knowledge Management: MKM’2001, B. Buchberger and O. Caprotti (Eds.), External Links: Link Cited by: p1.
163. [136] (2001) OMDoc: towards an internet standard for the administration, distribution and teaching of mathematical knowledge. In Proceedings of Artificial Intelligence and Symbolic Computation,
AISC’2000Proceedings of Artificial Intelligence and Symbolic Computation, AISC’2000, E. R. Lozano (Ed.), LNAI, pp. 32–52. External Links: Link Cited by: p1.
164. [3] (2000) Communication protocols for mathematical services based on KQML and OMRS. In CALCULEMUS-2000, systems for integrated computation and deductionProceedings of the 8^th Symposium on the
Integration of Symbolic Computation and Mechanized Reasoning (Calculemus-2000), M. Kerber and M. Kohlhase (Eds.), pp. 34–48. External Links: Link Cited by: p1.
165. [38] (2000) System description: MBase, an open mathematical knowledge base. In Automated Deduction – cade-17Automated Deduction – CADE-17, D. McAllester (Ed.), LNAI, pp. 455–459. External Links:
Link Cited by: p1.
166. [113] (2000) Towards a tableaux machine for language understanding. In ICoS-2. inference in computational semantics. workshop proceedingsProceedings of Inference in Computational Semantics
ICoS-2, J. Bos and M. Kohlhase (Eds.), pp. 57–88. Cited by: p1.
167. [134] (2000) Model generation for discourse representation theory. In Proceedings of of the 14^th european conference on artifical intelligenceProceedings of of the 14^th European Conference on
Artifical Intelligence, W. Horn (Ed.), pp. 441–445. External Links: Link Cited by: p1.
168. [173] (2000) Feature logic for dotted types: a formalism for complex word meanings. In Proceedings of the 38^th Annual Meeting of the Association for Computational Linguistics, Hongkong,
pp. 521–528. External Links: Link Cited by: p1.
169. [20] (1999) Inference and computational semantics. In Proceedings of IWCS III (third international workshop on computational semantics)Proceedings of IWCS III (Third International Workshop on
Computational Semantics), H. Bunt, L. Kievit, R. Muskens, and M. Verlinden (Eds.), pp. 5–19. Cited by: p1.
170. [37] (1999) System description: MathWeb, an agent-based communication layer for distributed automated theorem proving. In Automated deduction — CADE-16Automated Deduction — CADE-16, H. Ganzinger
(Ed.), LNAI, pp. 217–221. External Links: Link Cited by: p1.
171. [51] (1998-07) An implementation of distributed mathematical services. In 6^th calculemus and types workshop6^th CALCULEMUS and TYPES Workshop, A. Cohen and H. Barendregt (Eds.), Cited by: p1.
172. [10] (1998) Extensional higher order resolution. In Proceedings of the 15^th Conference on Automated DeductionProceedings of the 15^th Conference on Automated Deduction, C. Kirchner and H.
Kirchner (Eds.), LNAI, pp. 56–72. External Links: Link Cited by: p1.
173. [11] (1998) LEO – a higher order theorem prover. In Proceedings of the 15^th Conference on Automated DeductionProceedings of the 15^th Conference on Automated Deduction, C. Kirchner and H.
Kirchner (Eds.), LNAI, pp. 139–144. External Links: Link Cited by: p1.
174. [8] (1997) $\Omega$mega: towards a mathematical assistant. In Proceedings of the 14^th Conference on Automated DeductionProceedings of the 14^th Conference on Automated Deduction, W. McCune
(Ed.), LNAI, pp. 252–255. External Links: Link Cited by: p1.
175. [36] (1997) Dynamic control of quantifier scope. In Proceedings of the 11^th Amsterdam Colloquium, P. Dekker (Ed.), Amsterdam, The Netherlands, pp. 109–114. Cited by: p1.
176. [42] (1997) Computing parallelism in discourse. In Proceedings of the 15^th International Joint Conference on Artificial Intelligence (IJCAI)Proceedings of the 15^th International Joint
Conference on Artificial Intelligence (IJCAI), M. E. Pollack (Ed.), pp. 1016–1021. External Links: Link Cited by: p1.
177. [64] (1997) A coloured version of the $\lambda$-calculus. In Proceedings of the 14^th Conference on Automated DeductionProceedings of the 14^th Conference on Automated Deduction, W. McCune (Ed.)
, LNAI, pp. 291–305. Cited by: p1.
178. [78] (1997) Mechanising partiality without re-implementation. In Proceedings of the 18.th annual german conference on artificial intelligence ki’97Proceedings of the 18.th Annual German
Conference on Artificial Intelligence KI’97, G. Brewka, C. Habel, and B. Nebel (Eds.), LNAI, pp. 123–134. External Links: Link Cited by: p1.
179. [115] (1997) Dynamic lambda calculus. In Proceedings of the 5^th Meeting on Mathematics of Language – MOL5, pp. 85–92. Cited by: p1.
180. [77] (1996-08) A resolution calculus for presuppositions. In Proceedings of the 12^th european conference on artificial intelligenceProceedings of the 12^th European Conference on Artificial
Intelligence, W. Wahlster (Ed.), pp. 375–379. External Links: Link Cited by: p1.
181. [39] (1996) Corrections and higher-order unification. In Proceedings of KONVENS’96, Bielefeld, Germany, pp. 268–279. External Links: Link Cited by: p1.
182. [40] (1996) Focus and higher–order unification. In Proceedings of the 16^th International Conference on Computational Linguistics, Copenhagen, pp. 268–279. External Links: Link Cited by: p1.
183. [41] (1996) Higher–order coloured unification and natural language semantics. In Proceedings of the 34^th Annual Meeting of the Association for Computational Linguistics, Santa Cruz, pp. 1–9.
External Links: Link Cited by: p1.
184. [75] (1996) Integrating computer algebra with proof planning. In Design and implementation of symbolic computation systems, disco’96Design and Implementation of Symbolic Computation Systems,
DISCO’96, J. Calmet and C. Limogelli (Eds.), LNCS, pp. 204–215. Cited by: p1.
185. [133] (1995) Higher-order tableaux. In Proceedings of the Tableau Workshop, Koblenz, Germany, pp. 294–309. External Links: Link Cited by: p1.
186. [58] (1994) KEIM: a toolkit for automated deduction. In Proceedings of the 12^th Conference on Automated DeductionProceedings of the 12^th Conference on Automated Deduction, A. Bundy (Ed.), LNAI
, pp. 807–810. External Links: Link Cited by: p1.
187. [59] (1994) $\Omega$-MKRP a proof development environment. In Proceedings of the 12^th Conference on Automated DeductionProceedings of the 12^th Conference on Automated Deduction, A. Bundy (Ed.)
, LNAI, pp. 788–792. External Links: Link Cited by: p1.
188. [60] (1994) Adapting methods to novel tasks in proof planning. In 18^th annual german conference on aritificial intelligence18^th Annual German Conference on Aritificial Intelligence, B. Nebel
and L. Dreschler-Fischer (Eds.), LNAI, pp. 379–390. External Links: Link Cited by: p1.
189. [63] (1994) A colored version of the $\lambda$ calculus. In Proceedings of the 12^th Conference on Automated DeductionProceedings of the 12^th Conference on Automated Deduction, A. Bundy (Ed.),
LNAI, pp. 291–305. Cited by: p1.
190. [70] (1994) Unification in an extensional lambda calculus with ordered function sorts and constant overloading. In Proceedings of the 12^th Conference on Automated DeductionProceedings of the 12
^th Conference on Automated Deduction, A. Bundy (Ed.), LNAI, pp. 620–634. External Links: Link Cited by: p1.
191. [76] (1994) A mechanization of strong Kleene logic for partial functions. In Proceedings of the 12^th Conference on Automated DeductionProceedings of the 12^th Conference on Automated Deduction,
A. Bundy (Ed.), LNAI, pp. 371–385. External Links: Link Cited by: p1.
192. [132] (1994) Unification in a $\lambda$-calculus with term declarations and function sorts. In 18^th annual german conference on aritificial intelligence18^th Annual German Conference on
Aritificial Intelligence, B. Nebel and L. Dreschler-Fischer (Eds.), LNAI, pp. 331–342. External Links: Link Cited by: p1.
193. [123] (1993) Unification in a $\lambda$-calculus with intersection types. In Proceedings of the International Logic Programming Symposion ILPS’93, D. Miller (Ed.), pp. 488–505. External Links:
Link Cited by: p1.
194. [131] (1992) Unification in order-sorted type theory. In Proceedings of the international conference on logic programming and automated reasoning lpar’92Proceedings of the International
Conference on Logic Programming and Automated Reasoning LPAR’92, A. Voronkov (Ed.), LNAI, pp. 421–432. Cited by: p1.
1. [4] (2011) Enabling collaboration on semiformal mathematical knowledge by semantic web integration. Studies on the Semantic Web, AKA Verlag and IOS Press, Heidelberg and Amsterdam. External
Links: ISBN 978-1-60750-840-3, Link Cited by: p1.
2. [3] C. Lange (Ed.) (2006-09) Wikis und blogs – Planen, Einrichten, Verwalten. C&L Computer- und Literaturverlag. External Links: ISBN 3-936546-44-4 Cited by: p1.
3. [1] (2006-08) OMDoc – an open markup format for mathematical documents [version 1.2]. LNAI, Springer Verlag. External Links: Link Cited by: p1.
4. [2] C. Lange (Ed.) (2005-09) Wiki – Planen, Einrichten, Verwalten. C&L Computer- und Literaturverlag. External Links: ISBN 3-936546-28-2 Cited by: p1.
Conference Proceedings Edited
1. [8] P. Cimiano, A. Frank, M. Kohlhase, and B. Stein (Eds.) (2024) Robust argumentation machines – first international conference, ratio 2024, bielefeld, germany, june 5–7, 2024, proceedings. LNAI
, Springer-Verlag Berlin Heidelberg. External Links: Document Cited by: p1.
2. [13] A. Kohlhase and L. Kovacz (Eds.) (2024) Intelligent computer mathematics. LNAI, Springer. Cited by: p1.
3. [1] A. Adrian, S. Evert, M. K. Kohlhase, and M. Zwickel (Eds.) (2022) Digitalisierung von zivilprozess und rechtsdurchsetzung. Schriften zum Prozessrecht, Duncker & Humblot, Berlin. Cited by: p1.
4. [7] J. Blanchette, J. Davenport, P. Koepke, A. Kohlhase, M. Kohlhase, A. Naumowicz, D. Müller, Y. Sharoda, and C. S. Coen (Eds.) (2021) Workshop papers at 14th conference on intelligent computer
mathematics cicm 2021. External Links: Link Cited by: p1.
5. [11] C. Kaliszyck, E. Brady, A. Kohlhase, and C. Sacerdoti Coen (Eds.) (2019) Intelligent computer mathematics. LNAI, Springer. External Links: Document Cited by: p1.
6. [6] O. Hasan, A. Youssef, A. Naumowicz, W. Farmer, C. Kaliszyk, D. Gallois-Wong, F. Rabe, G. D. Reis, G. Passmore, J. Davenport, M. Pfeiffer, M. Kohlhase, S. Autexier, S. Tahar, T. Koprucki, U.
Siddique, W. Neuper, W. Windsteiger, W. Schreiner, W. Sperber, and Z. Kovács (Eds.) (2018) Workshop papers at 11th conference on intelligent computer mathematics cicm 2018. External Links: Link
Cited by: p1.
7. [16] F. Rabe, W. M. Farmer, G. O. Passmore, and A. Youssef (Eds.) (2018) Intelligent computer mathematics. LNAI, Springer. External Links: Document, ISBN 978-3-319-96811-7 Cited by: p1.
8. [10] H. Geuvers, M. England, O. Hasan, F. Rabe, and O. Teschke (Eds.) (2017) Intelligent computer mathematics. LNAI, Springer. External Links: Document, ISBN 978-3-319-62074-9 Cited by: p1.
9. [14] M. Kohlhase, M. Johansson, B. Miller, L. de Moura, and F. Tompa (Eds.) (2016) Intelligent computer mathematics. LNAI, Springer. External Links: ISBN 978-3-319-08434-3 Cited by: p1.
10. [4] J. Carette, D. Aspinall, C. Lange, P. Sojka, and W. Windsteiger (Eds.) (2013) Intelligent computer mathematics. Lecture Notes in Computer Science, Springer. External Links: Document, ISBN
978-3-642-39319-8 Cited by: p1.
11. [5] J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.) (2011) Intelligent computer mathematics. LNAI, Springer Verlag. External Links: ISBN 978-3-642-22672-4 Cited by: p1.
12. [9] C. Freksa, M. Kohlhase, and K. Schill (Eds.) (2006) Proceedings of the 29.th annual german conference on artificial intelligence KI’06. LNAI, Bremen, Germany. Cited by: p1.
13. [15] M. Kohlhase (Ed.) (2006) Mathematical knowledge management, MKM’05. LNAI, Springer Verlag. Cited by: p1.
14. [3] J. Bos and M. Kohlhase (Eds.) (2003) Logic journal of the igpl. Vol. 11(3), Oxford University Press. Note: Special Issue for ICOS-2 Cited by: p1.
15. [2] J. Bos and M. Kohlhase (Eds.) (2000) ICoS-2. inference in computational semantics. workshop proceedings. Computational Linguistics, Saarland University. Cited by: p1.
16. [12] M. Kerber and M. Kohlhase (Eds.) (2000) CALCULEMUS-2000, systems for integrated computation and deduction. AKPeters, St. Andrews, Scotland. Cited by: p1.
Gray Literature
Worskhop Proceedings Edited
1. [35] K. Nakasho and J. F. Schaefer (Eds.) (2024) MathUI 2024: the 15th workshop on mathematical user interfaces. Cited by: p1.
2. [26] A. Kohlhase (Ed.) (2023) MathUI 2023: the 14th workshop on mathematical user interfaces. External Links: Link Cited by: p1.
3. [25] A. Kohlhase (Ed.) (2022) MathUI 2021: the 13th workshop on mathematical user interfaces. External Links: Link Cited by: p1.
4. [20] A. Kohlhase and E. Kübler (Eds.) (2017-10) Wissens- und erfahrungsmanagement (knowledge and experience management), FGWM. Note: Workshop at LWDA 2017 External Links: Link Cited by: p1.
5. [23] A. Kohlhase and M. Pollanen (Eds.) (2017) MathUI 2017: the 12th workshop on mathematical user interfaces. External Links: Link Cited by: p1.
6. [22] A. Kohlhase and P. Libbrecht (Eds.) (2016-07) Mathematical user interfaces workshop at CICM. External Links: Link Cited by: p1.
7. [8] M. Kohlhase, A. Kohlhase, P. Libbrecht, B. Miller, A. Naumowicz, W. Neuper, P. Quaresma, F. Tompa, and M. Suda (Eds.) (2016) Intelligent computer mathematics – work in progress papers.
External Links: Link Cited by: p1.
8. [21] A. Kohlhase and P. Libbrecht (Eds.) (2015-07) Mathematical user interfaces workshop at CICM. External Links: Link Cited by: p1.
9. [11] M. England, J. H. Davenport, A. Kohlhase, M. Kohlhase, P. Libbrecht, W. Neuper, P. Quaresma, A. P. Sexton, P. Sojka, J. Urban, and S. M. Watt (Eds.) (2014) Joint proceedings of the MathUI,
OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent computer mathematics 2014. CEUR Workshop Proceedings, Aachen. External Links: ISSN 1613-0073, Link Cited
by: p1.
10. [19] A. Henrich and H. Sperker (Eds.) (2013-10) Wissens- und erfahrungsmanagement LWA (lernen, wissensentdeckung und adaptivität) conference proceedings. Universität Bamberg. Cited by: 24.
11. [24] A. Kohlhase and B. Rieger (Eds.) (2013-10) Wissens- und erfahrungsmanagement (knowledge and experience management), FGWM. Note: Workshop at LWA 2013, published as part of [19] Cited by: p1.
12. [13] A. García Castro, C. Lange, P. Lord, and R. Stevens (Eds.) (2013) Proceedings of the 3^rd workshop on semantic publishing, Extended Semantic Web Conference. CEUR Workshop Proceedings, Aachen
. External Links: ISSN 1613-0073, Link Cited by: p1.
13. [27] C. Lange, D. Aspinall, J. Carette, J. Davenport, A. Kohlhase, M. Kohlhase, P. Libbrecht, P. Quaresma, F. Rabe, P. Sojka, I. Whiteside, and W. Windsteiger (Eds.) (2013) Joint proceedings of
the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent computer mathematics 2013. CEUR Workshop Proceedings, Aachen. External Links: ISSN 1613-0073
, Link Cited by: p1.
14. [31] C. Lange, C. Rowat, and M. Kerber (Eds.) (2013) Enabling domain experts to use formalised reasoning. Society for the Study of Artificial Intelligence and Simulation of Behaviour (AISB).
External Links: Link, ISBN 978-1-908187-32-1 Cited by: p1.
15. [17] B. Good, F. van Harmelen, A. García Castro, C. Lange, E. Sandhaus, and A. de Waard (Eds.) (2012) Proceedings of the 2^nd workshop on semantic publishing, Extended Semantic Web Conference.
CEUR Workshop Proceedings, Aachen. External Links: ISSN 1613-0073, Link Cited by: p1.
16. [38] P. Sojka and M. Kohlhase (Eds.) (2012) DML and MIR 2012. Masaryk University, Brno. External Links: ISBN 978-80-210-5542-1 Cited by: p1.
17. [1] A. Asperti, J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.) (2011) Intelligent Computer Mathematics, Work-in-Progress Proceedings. Technical Reports of University of Bologna, Vol.
UBLCS-2011-04, University of Bologna. Cited by: p1.
18. [7] J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.) (2011) Intelligent computer mathematics – work in progress papers. Cited by: p1.
19. [12] A. García Castro, K. Baclawski, J. Bateman, K. Viljanen, and C. Lange (Eds.) (2011) Proceedings of the workshop ontologies come of age in the semantic web, International Semantic Web
Conference. CEUR Workshop Proceedings, Aachen. External Links: ISSN 1613-0073, Link Cited by: p1.
20. [14] A. García Castro, C. Lange, E. Sandhaus, and A. de Waard (Eds.) (2011) Proceedings of the 1^st workshop on semantic publication, Extended Semantic Web Conference. CEUR Workshop Proceedings,
Aachen. External Links: ISSN 1613-0073, Link Cited by: p1.
21. [15] H. Geuvers, G. Nadathur, F. Rabe, and C. Schürmann (Eds.) (2011) LFMTP 2011 - MLPA 2011 Informal Proceedings. Note: see http://kwarc.info/frabe/events/mlpa-11/index.html Cited by: p1.
22. [34] C. Lange and J. Urban (Eds.) (2011) Proceedings of the itp 2011 workshop on mathematical wikis (mathwikis). CEUR Workshop Proceedings, Aachen. External Links: ISSN 1613-0073, Link Cited by:
23. [16] A. Giurca, B. Endres-Niggemeyer, C. Lange, L. Maicher, and P. Hitzler (Eds.) (2010-06) AI Mashup Challenge. External Links: Link Cited by: p1.
24. [9] M. d’Aquin, A. García Castro, C. Lange, and K. Viljanen (Eds.) (2010) Proceedings of the 1^st workshop on ontology repositories and editors, Extended Semantic Web Conference. CEUR Workshop
Proceedings, Aachen. External Links: ISSN 1613-0073, Link Cited by: p1.
25. [28] C. Lange, J. Reutelshöfer, S. Schaffert, and H. Skaf-Molli (Eds.) (2010) Proceedings of the 5^th workshop on semantic wikis, Extended Semantic Web Conference. CEUR Workshop Proceedings,
Aachen. External Links: ISSN 1613-0073, Link Cited by: p1.
26. [37] F. Rabe and C. Schürmann (Eds.) (2010) MLPA ’10: Proceedings of the 2nd Workshop on Modules and Libraries for Proof Assistants. Note: see http://kwarc.info/frabe/events/mlpa-10.html Cited
by: p1.
27. [18] (2009-09) LWA 2009; Workshop-Woche: Lernen – Wissen – Adaptivität. Technical report Vol. TUD-KE-2009-04, Universität Darmstadt. Cited by: 29.
28. [29] C. Lange and J. Reutelshöfer (Eds.) (2009-09) Wissens- und erfahrungsmanagement (knowledge and experience management), FGWM. Note: Workshop at LWA 2009, published as part of [18] Cited by:
29. [30] C. Lange and J. Reutelshöfer (Eds.) (2009-09) Wissens- und erfahrungsmanagement (knowledge and experience management), FGWM. Vol. TUD-KE-2009-04, Universität Darmstadt. Cited by: p1.
30. [33] C. Lange, S. Schaffert, H. Skaf-Molli, and M. Völkel (Eds.) (2009) Proceedings of the 4^th workshop on semantic wikis, European Semantic Web Conference. CEUR Workshop Proceedings, Aachen.
External Links: ISSN 1613-0073, Link Cited by: p1.
31. [36] F. Rabe and C. Schürmann (Eds.) (2009) MLPA ’09: Proceedings of the 1st Workshop on Modules and Libraries for Proof Assistants. ACM International Conference Proceeding Series, Vol. 429, ACM.
Cited by: p1.
32. [6] O. Caprotti, S. Xambó, M. Huertas, M. Kohlhase, and M. Seppälä (Eds.) (2008) 3^rd JEM workshop – joining educational mathematics. External Links: Link Cited by: p1.
33. [32] C. Lange, S. Schaffert, H. Skaf-Molli, and M. Völkel (Eds.) (2008) Proceedings of the 3^rd workshop on semantic wikis, European Semantic Web Conference. CEUR Workshop Proceedings, Aachen.
External Links: ISSN 1613-0073, Link Cited by: p1.
34. [5] O. Caprotti, M. Kohlhase, and P. Libbrecht (Eds.) (2007-06) OpenMath/JEM workshop 2007. Note: online at http://www.openmath.org/meetings/linz2007/ External Links: Link Cited by: p1.
35. [2] P. Baumgartner, P. A. Cairns, M. Kohlhase, and E. Melis (Eds.) (2003) Proceedings of the IJCAI 03 workshop on knowledge representation and automated reasoning for e-learning systems.
Acapulco, Mexico. Cited by: p1.
36. [4] P. Blackburn and M. Kohlhase (Eds.) (2001) ICoS-3. inference in computational semantics. workshop proceedings. Computational Linguistics, Saarland University. Cited by: p1.
37. [3] P. Baumgartner, U. Furbach, M. Kohlhase, W. McCune, W. Reif, M. Stickel, and T. Uribe (Eds.) (1998) CADE-15 workshop “problem-solving methodologies with automated deduction”. Cited by: p1.
38. [10] J. Denzinger, M. Kohlhase, and B. Spencer (Eds.) (1998) CADE-15 workshop “using AI methods in deduction”. Cited by: p1.
Papers at Peer-Reviewed Workshops
1. [2] (2024) WOIDE: semantic annotation in MS Word — scaling mathematical user interfaces beyond LaTeX. In MathUI 2024: the 15th workshop on mathematical user interfacesMathUI 2024: The 15th
Workshop on Mathematical User Interfaces, K. Nakasho and J. F. Schaefer (Eds.), External Links: Link Cited by: p1.
2. [83] (2024) Towards automated competency estimation for math education – an eye tracking and emotion analysis study. In MathUI 2024: the 15th workshop on mathematical user interfacesMathUI 2024:
The 15th Workshop on Mathematical User Interfaces, K. Nakasho and J. F. Schaefer (Eds.), External Links: Link Cited by: p1.
3. [12] (2023) Guided tours in alea - assembling tailored educational dialogues from semantically annotated learning objects. In Artificial Intelligence. ECAI 2023 International Workshops - AI4AI,
Poland, September 30 - October 4, 2023, Proceedings, Part II, S. Nowaczyk, P. Biecek, N. C. Chung, M. Vallati, P. Skruch, J. Jaworek-Korjakowska, S. Parkinson, A. Nikitas, M. Atzmüller, T.
Kliegr, U. Schmid, S. Bobek, N. Lavrac, M. Peeters, R. van Dierendonck, S. Robben, E. Mercier-Laurent, G. Kayakutlu, M. L. Owoc, K. Mason, A. Wahid, P. Bruno, F. Calimeri, F. Cauteruccio, G.
Terracina, D. Wolter, J. L. Leidner, M. Kohlhase, and V. Dimitrova (Eds.), Communications in Computer and Information Science, Vol. 1948, pp. 397–408. External Links: Link, Document Cited by: p1.
4. [18] (2023) Presentation of active documents in ALeA. In MathUI 2023: the 14th workshop on mathematical user interfacesMathUI 2023: The 14th Workshop on Mathematical User Interfaces, A. Kohlhase
(Ed.), Note: submitted External Links: Link Cited by: p1.
5. [82] (2023) More interactions in ALeA – towards new added-value services based on semantic markup. In MathUI 2023: the 14th workshop on mathematical user interfacesMathUI 2023: The 14th Workshop
on Mathematical User Interfaces, A. Kohlhase (Ed.), External Links: Link Cited by: p1.
6. [177] (2023) AnnoTize: a flexible annotation tool for documents with mathematical formulae. In MathUI 2023: the 14th workshop on mathematical user interfacesMathUI 2023: The 14th Workshop on
Mathematical User Interfaces, A. Kohlhase (Ed.), Note: accepted External Links: Link Cited by: p1.
7. [208] (2023) Towards an annotation standard for STEM documents – datasets, benchmarks, and spotters. In Intelligent computer mathematicsIntelligent Computer Mathematics (CICM) 2023, C. Dubois and
M. Kerber (Eds.), LNAI, pp. 190–205. External Links: Link Cited by: p1.
8. [7] (2022) A conceptual design for an eye-tracking experiment on formula linebreaking. In MathUI 2021: the 13th workshop on mathematical user interfacesMathUI 2021: The 13th Workshop on
Mathematical User Interfaces, A. Kohlhase (Ed.), Cited by: p1.
9. [3] (2021) A novel understanding of legal syllogism as a starting point for better legal symbolic ai systems. In 24. internationalen rechtsinformatik symposion (iris 2021)24. Internationalen
Rechtsinformatik Symposion (IRIS 2021), E. S. und Walter Hötzendorfer und Franz Kummer und Ahti Saarenpää und Stefan Eder und Philip Hanke (Ed.), pp. 169ff.. External Links: Link Cited by: p1.
10. [99] (2021) Context graphs for ampliative analogical legal reasoning and argumentation. In 24. internationalen rechtsinformatik symposion (iris 2021)24. Internationalen Rechtsinformatik Symposion
(IRIS 2021), E. S. und Walter Hötzendorfer und Franz Kummer und Ahti Saarenpää und Stefan Eder und Philip Hanke (Ed.), pp. 231ff.. External Links: Link Cited by: p1.
11. [106] (2021) Dynamic user interfaces via incremental knowledge management. In 13th MathUI Workshop 2021, Mathematical User Interaction, at the Conference on Intelligent Computer Mathematics,
External Links: Link Cited by: p1.
12. [170] (2021) Disambiguating symbolic expressions in informal documents. In International Conference on Learning Representations, External Links: Link Cited by: p1.
13. [180] (2021) A Language with Type-Dependent Equality. In Intelligent Computer Mathematics, F. Kamareddine and C. Sacerdoti Coen (Eds.), pp. 211–227. External Links: Document, Link Cited by: p1.
14. [185] (2021) Systematic translation of formalizations of type theory from intrinsic to extrinsic style. In Proceedings of the Workshop on Logical Frameworks: Meta-Theory and Practice (LFMTP), E.
Pimentel and E. Tassi (Eds.), External Links: Link, Document Cited by: p1.
15. [203] (2021) Structure-Preserving Diagram Operators. In Recent Trends in Algebraic Development Techniques, M. Roggenbach (Ed.), Lecture Notes in Computer Science, Vol. 12669, pp. 142–163.
External Links: Link, ISBN 978-3-030-73785-6, Document Cited by: p1.
16. [69] (2020) A survey of languages for formalizing mathematics. In Intelligent Computer Mathematics, C. Benzmüller and B. Miller (Eds.), Cham, pp. 138–156. External Links: ISBN 978-3-030-53518-6
Cited by: p1.
17. [110] (2020) Making Isabelle content accessible in knowledge representation formats. In Proceedings of the 25th International Conference on Types for Proofs and Programs, TYPES 2019, M. Bezem and
A. Mahboubi (Eds.), Leibniz International Proceedings in Informatics (LIPIcs), Vol. 175. External Links: ISBN 978-3-95977-158-0, Link, Document Cited by: p1.
18. [205] (2020) Prototyping controlled mathematical languages in jupyter notebooks. In Mathematical software – icms 2020. 7th international conferenceMathematical Software – ICMS 2020. 7th
international conference, A. M. Bigatti, J. Carette, J. H. Davenport, M. Joswig, and T. de Wolff (Eds.), LNCS, Vol. 12097, pp. 406–415. External Links: Link Cited by: p1.
19. [207] (2020) GLIF: a declarative framework for symbolic natural language understanding. In Proceedings of the 6th Workshop on Formal and Cognitive Reasoning, C. Beierle, M. Ragni, F. Stolzenburg,
and M. Thimm (Eds.), pp. 4–11. External Links: Link Cited by: p1.
20. [11] (2019) Towards a census of relational data in mathematics. In Proceedings of the conference ”lernen, wissen, daten, analysen”, LWDAProceedings of the Conference ”Lernen, Wissen, Daten,
Analysen”, LWDA, R. Jäschke and M. Weidlich (Eds.), CEUR Workshop Proceedings, Vol. 2454, pp. 207–217. External Links: Link Cited by: p1.
21. [58] (2019) Social choice and the problem of recommending essential readings. In At the Intersection of Language, Logic, and Information. ESSLLI 2018.At the Intersection of Language, Logic, and
Information. ESSLLI 2018., J. Sikos and E. Pacuit (Eds.), Lecture Notes in Computer Science, Vol. 11667. Cited by: p1.
22. [112] (2019) Context graphs for argumentation logics. In Proceedings of the conference ”lernen, wissen, daten, analysen”, LWDAProceedings of the Conference ”Lernen, Wissen, Daten, Analysen”, LWDA
, R. Jäschke and M. Weidlich (Eds.), CEUR Workshop Proceedings, Vol. 2454, pp. 265–279. External Links: Link Cited by: p1.
23. [113] (2019) GF + MMT = GLF – from language to semantics through LF. In Proceedings of the Fourteenth Workshop on Logical Frameworks and Meta-Languages: Theory and Practice, LFMTP 2019, D. Miller
and I. Scagnetto (Eds.), Vol. 307, pp. 24–39. External Links: Document Cited by: p1.
24. [172] (2019) Rapid prototyping formal systems in mmt: 5 case studies. In LFMTP 2019, External Links: Link Cited by: p1.
25. [188] (2019) Diagram combinators in MMT. In Intelligent computer mathematicsIntelligent Computer Mathematics, C. Kaliszyck, E. Brady, A. Kohlhase, C. Sacerdoti Coen, C. Kaliszyk, E. Brady, A.
Kohlhase, and C. Sacerdoti Coen (Eds.), LNAI, Cham, pp. 211–226. External Links: ISBN 978-3-030-23250-4, Link Cited by: p1.
26. [10] (2018) DiscreteZOO: towards a fingerprint database of discrete objects. In Mathematical Software – ICMS 2018, pp. 36–44. External Links: Link, ISBN 978-3-319-96418-8 Cited by: p1.
27. [97] (2018) Factors for reading mathematical expressions. In Proceedings of the conference ”lernen, wissen, daten, analysen”, LWDA, R. Gemulla, S. P. Ponzetto, C. Bizer, M. Keuper, and H.
Stuckenschmidt (Eds.), CEUR Workshop Proceedings, Vol. 2191, pp. 195–202. External Links: Link Cited by: p1.
28. [124] (2018) Towards context graphs for argumentation logics. In Proceedings of the conference ”lernen, wissen, daten, analysen”, LWDA, R. Gemulla, S. P. Ponzetto, C. Bizer, M. Keuper, and H.
Stuckenschmidt (Eds.), CEUR Workshop Proceedings, Vol. 2191, pp. 203–214. External Links: Link Cited by: p1.
29. [171] (2018) Theories as types. In 9th international joint conference on automated reasoning9th International Joint Conference on Automated Reasoning, D. Galmiche, S. Schulz, and R. Sebastiani
(Eds.), External Links: Link Cited by: p1.
30. [184] (2018) Structuring theories with implicit morphisms. In 24th International Workshop on Algebraic Development Techniques 2018, External Links: Link Cited by: p1.
31. [206] (2018) Syntactic/semantic analysis for high-precision math linguistics. In Workshop papers at 11th conference on intelligent computer mathematics cicm 2018Workshop Papers at 11th Conference
on Intelligent Computer Mathematics CICM 2018, O. Hasan, A. Youssef, A. Naumowicz, W. Farmer, C. Kaliszyk, D. Gallois-Wong, F. Rabe, G. D. Reis, G. Passmore, J. Davenport, M. Pfeiffer, M.
Kohlhase, S. Autexier, S. Tahar, T. Koprucki, U. Siddique, W. Neuper, W. Windsteiger, W. Schreiner, W. Sperber, and Z. Kovács (Eds.), Note: CICM Work in Progress Paper External Links: Link Cited
by: p1.
32. [123] (2017-10) Math object identifiers – towards research data in mathematics. In Wissens- und erfahrungsmanagement (knowledge and experience management), FGWMWissens- und Erfahrungsmanagement
(Knowledge and Experience Management), FGWM, A. Kohlhase and E. Kübler (Eds.), pp. 214–252. External Links: Link Cited by: p1.
33. [217] (2017-10) Irreführende mentale modelle beim smart-tv (misleading mental models for smart-tvs). In Wissens- und erfahrungsmanagement (knowledge and experience management), FGWMWissens- und
Erfahrungsmanagement (Knowledge and Experience Management), FGWM, A. Kohlhase and E. Kübler (Eds.), pp. 200–212. External Links: Link Cited by: p1.
34. [25] (2017) Canonical Selection of Colimits. In Recent Trends in Algebraic Development Techniques, P. James and M. Roggenbach (Eds.), pp. 170–188. Cited by: p1.
35. [96] (2017) Domain-dependant decoding of math expressions. In MathUI 2017: the 12th workshop on mathematical user interfacesMathUI 2017: The 12th Workshop on Mathematical User Interfaces, A.
Kohlhase and M. Pollanen (Eds.), Cited by: p1.
36. [155] (2017) Is the triviality of agm a serious possibility. In Logic in the Wild: 6th workshop in the ’Logic, Reasoning & Rationality’ series, External Links: Link Cited by: p1.
37. [173] (2017) Alignment-based translations across formal systems using interface theories. In Fifth Workshop on Proof eXchange for Theorem Proving - PxTP 2017, External Links: Link Cited by: p1.
38. [201] (2017) The MMT Perspective on Conservativity. In Logical and Semantic Frameworks, with Applications, S. Alves and R. Wassermann (Eds.), pp. 17–33. Cited by: p1.
39. [204] (2017) A flexible, interactive theory-graph viewer. In MathUI 2017: the 12th workshop on mathematical user interfacesMathUI 2017: The 12th Workshop on Mathematical User Interfaces, A.
Kohlhase and M. Pollanen (Eds.), External Links: Link Cited by: p1.
40. [74] (2016-07) Understanding mathematical expressions: an eye-tracking study. In Mathematical user interfaces workshop at CICMMathematical User Interfaces Workshop, A. Kohlhase and P. Libbrecht
(Eds.), External Links: Link Cited by: p1.
41. [216] (2016-07) Notation-based semantification. In Mathematical user interfaces workshop at CICMMathematical User Interfaces Workshop, A. Kohlhase and P. Libbrecht (Eds.), pp. 73–81. External
Links: Link Cited by: p1.
42. [5] (2016) NTCIR-12 MathIR task overview. In Proceedings of the 12th ntcir conference on evaluation of information access technologiesProceedings of the 12th NTCIR Conference on Evaluation of
Information Access Technologies, N. Kando, T. Sakai, and M. Sanderson (Eds.), pp. 299–308. External Links: Link Cited by: p1.
43. [21] (2016) Selecting Colimits for Parameterisation and Networks of Specifications. In Workshop on Algebraic Development Techniques, M. Roggenbach and P. James (Eds.), Cited by: p1.
44. [66] (2016) TH1: The TPTP Typed Higher-Order Form with Rank-1 Polymorphism. In Workshop on Practical Aspects of Automated Reasoning, P. Fontaine, S. Schulz, and J. Urban (Eds.), pp. 41–55. Cited
by: p1.
45. [67] (2016) A standard for aligning mathematical concepts. In Intelligent computer mathematics – work in progress papersIntelligent Computer Mathematics – Work in Progress Papers, M. Kohlhase, A.
Kohlhase, P. Libbrecht, B. Miller, A. Naumowicz, W. Neuper, P. Quaresma, F. Tompa, and M. Suda (Eds.), External Links: Link Cited by: p1.
46. [95] (2016) Math web search interfaces and the generation gap of mathematicians. In Workshop Human-Computer Algebra Interaction, Kassel, Germany, External Links: Link Cited by: p1.
47. [125] (2016) Breakout session on A standard for system integration and proof interchange. In Dagstuhl Seminar on Universality of Proofs, G. Dowek, C. Dubois, B. Pientka, and F. Rabe (Eds.),
pp. 94–94. Cited by: p1.
48. [200] (2016) MMT: A UniFormal Approach to Knowledge Representation. In Dagstuhl Seminar on Universality of Proofs, G. Dowek, C. Dubois, B. Pientka, and F. Rabe (Eds.), pp. 88–88. Cited by: p1.
49. [202] (2016) FrameIT reloaded: serious math games from modular math ontologies. In Intelligent computer mathematics – work in progress papersIntelligent Computer Mathematics – Work in Progress
Papers, M. Kohlhase, A. Kohlhase, P. Libbrecht, B. Miller, A. Naumowicz, W. Neuper, P. Quaresma, F. Tompa, and M. Suda (Eds.), External Links: Link Cited by: p1.
50. [46] (2015-10) Faceted search for mathematics. In Proceedings of the LWA 2015 workshops: KDML, FGWM, IR, and FGDBProceedings of the LWA 2015 Workshops: KDML, FGWM, IR, and FGDB, R. Bergmann, S.
Görg, and G. Müller (Eds.), pp. 33–44. External Links: Link Cited by: p1.
51. [156] (2015-10) Importing the OEIS library into OMDoc. In Proceedings of the LWA 2015 workshops: KDML, FGWM, IR, and FGDBProceedings of the LWA 2015 Workshops: KDML, FGWM, IR, and FGDB, R.
Bergmann, S. Görg, and G. Müller (Eds.), pp. 296–303. External Links: Link Cited by: p1.
52. [42] (2015-07) KAT: an annotation tool for STEM documents. In Mathematical user interfaces workshop at CICMMathematical User Interfaces Workshop, A. Kohlhase and P. Libbrecht (Eds.), External
Links: Link Cited by: p1.
53. [75] (2015-07) Co-occurrences of context dimensions of spreadsheets. In Mathematical user interfaces workshop at CICMMathematical User Interfaces Workshop, A. Kohlhase and P. Libbrecht (Eds.),
External Links: Link Cited by: p1.
54. [178] (2015-07) Relational presentations using semantic closeness spatial narrative for mathematical content. In Mathematical user interfaces workshop at CICMMathematical User Interfaces Workshop
, A. Kohlhase and P. Libbrecht (Eds.), External Links: Link Cited by: p1.
55. [51] (2015) Formal Logic Definitions for Interchange Languages. In Intelligent Computer Mathematics, M. Kerber, J. Carette, C. Kaliszyk, F. Rabe, and V. Sorge (Eds.), pp. 171–186. Cited by: p1.
56. [77] (2015) Context in spreadsheet comprehension. In Second workshop on software engineering methods in spreadsheetsSecond workshop on Software Engineering methods in Spreadsheets, External
Links: Link Cited by: p1.
57. [199] (2015) Generic Literals. In Intelligent Computer Mathematics, M. Kerber, J. Carette, C. Kaliszyk, F. Rabe, and V. Sorge (Eds.), pp. 102–117. Cited by: p1.
58. [215] (2015) Assessment for spreadsheets. In Second workshop on software engineering methods in spreadsheetsSecond workshop on Software Engineering methods in Spreadsheets, External Links: Link
Cited by: p1.
59. [1] (2014) OpenMathMap: interaction. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent computer mathematics
2014MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, M. England, J. H. Davenport, A. Kohlhase, M. Kohlhase, P. Libbrecht,
W. Neuper, P. Quaresma, A. P. Sexton, P. Sojka, J. Urban, and S. M. Watt (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
60. [4] (2014) NTCIR-11 Math-2 task overview. In NTCIR workshop 11 meetingNTCIR 11 Conference, N. Kando, H. Joho, and K. Kishida (Eds.), pp. 88–98. External Links: Link Cited by: p1.
61. [17] (2014) Realms: a structure for consolidating knowledge about mathematical theories. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A.
Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 252–266. Note: MKM Best-Paper-Award External Links: Link Cited by: p1.
62. [43] (2014) E-books and graphics with LaTeXml. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS,
pp. 427–430. External Links: Link Cited by: p1.
63. [45] (2014) MathWebSearch at NTCIR-11. In NTCIR workshop 11 meetingNTCIR 11 Conference, N. Kando, H. Joho, and K. Kishida (Eds.), pp. 114–119. External Links: Link Cited by: p1.
64. [54] (2014) Flexary operators for formalized mathematics. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.)
, LNCS, pp. 312–327. External Links: Link Cited by: p1.
65. [59] (2014) System description: MathHub.info. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS,
pp. 431–434. External Links: Link Cited by: p1.
66. [65] (2014) Towards an interaction-based integration of MKM services into end-user applications. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport,
A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 344–356. External Links: Link Cited by: p1.
67. [68] (2014) Towards knowledge management for HOL Light. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.),
LNCS, pp. 357–372. External Links: Link Cited by: p1.
68. [85] (2014) FEncy: spreadsheet formulae exploration. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent computer
mathematics 2014MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, M. England, J. H. Davenport, A. Kohlhase, M. Kohlhase, P.
Libbrecht, W. Neuper, P. Quaresma, A. P. Sexton, P. Sojka, J. Urban, and S. M. Watt (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
69. [92] (2014) Design of search interfaces for mathematicians. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent computer
mathematics 2014MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, M. England, J. H. Davenport, A. Kohlhase, M. Kohlhase, P.
Libbrecht, W. Neuper, P. Quaresma, A. P. Sexton, P. Sojka, J. Urban, and S. M. Watt (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
70. [93] (2014) Math web search interfaces and the generation gap of mathematicians. In Mathematical software - ICMS 2014 - 4th international congressMathematical Software - ICMS 2014 - 4th
International Congress, H. Hong and C. Yap (Eds.), LNCS, Vol. 8592, pp. 586–593. External Links: Document Cited by: p1.
71. [94] (2014) Search interfaces for mathematicians. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka, and J. Urban (Eds.), LNCS,
pp. 153–168. External Links: Link Cited by: p1.
72. [98] (2014) System description: a semantics-aware LaTeX-to-office converter. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A. Sexton, P. Sojka,
and J. Urban (Eds.), LNCS, pp. 440–443. External Links: Link Cited by: p1.
73. [120] (2014) A data model and encoding for a semantic, multilingual terminology of mathematics. In Intelligent computer mathematicsIntelligent Computer Mathematics 2014, S. Watt, J. Davenport, A.
Sexton, P. Sojka, and J. Urban (Eds.), LNCS, pp. 169–183. External Links: Link Cited by: p1.
74. [121] (2014) Extension proposal: records in pragmatic OpenMath. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent
computer mathematics 2014MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, M. England, J. H. Davenport, A. Kohlhase, M.
Kohlhase, P. Libbrecht, W. Neuper, P. Quaresma, A. P. Sexton, P. Sojka, J. Urban, and S. M. Watt (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
75. [122] (2014) OpenMath language extensions. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent computer mathematics
2014MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, M. England, J. H. Davenport, A. Kohlhase, M. Kohlhase, P. Libbrecht,
W. Neuper, P. Quaresma, A. P. Sexton, P. Sojka, J. Urban, and S. M. Watt (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
76. [157] (2014) Proof support for Common Logic. In Automated Reasoning in Quantified Non-Classical Logics (ARQNL), C. Benzmüller and J. Otten (Eds.), EasyChair Proceedings in Computing, pp. 42–58.
External Links: Link Cited by: p1.
77. [196] (2014) A Logic-Independent IDE. In Workshop on user interfaces for theorem proversWorkshop on User Interfaces for Theorem Provers, C. Benzmüller and B. Woltzenlogel Paleo (Eds.), pp. 48–60.
External Links: Link Cited by: p1.
78. [197] (2014) A Logic-Independent IDE. In Workshop on User Interfaces for Theorem Provers, C. Benzmüller and B. Woltzenlogel Paleo (Eds.), pp. 48–60. External Links: Document Cited by: p1.
79. [198] (2014) MMT Objects. In Workshops and Work in Progress at CICM 2014: OpenMath Workshop, M. England, J. Davenport, A. Kohlhase, M. Kohlhase, P. Libbrecht, W. Neuper, P. Quaresma, A. Sexton,
P. Sojka, J. Urban, and S. Watt (Eds.), Cited by: p1.
80. [49] A. Henrich and H. Sperker (Eds.) (2013-10) Wissens- und erfahrungsmanagement LWA (lernen, wissensentdeckung und adaptivität) conference proceedings. Universität Bamberg. Cited by: 84.
81. [84] (2013-10) Exploration of spreadsheet formulae with fency. In Wissens- und erfahrungsmanagement (knowledge and experience management), FGWMWissens- und Erfahrungsmanagement (Knowledge and
Experience Management), FGWM, A. Kohlhase and B. Rieger (Eds.), External Links: Link Cited by: p1.
82. [6] (2013) NTCIR-10 Math pilot task overview. In NTCIR workshop 10 meetingNTCIR Workshop 10 Meeting, N. Kando and K. Kishida (Eds.), pp. 1–8. External Links: Link Cited by: p1.
83. [15] (2013) Semantic support for engineering design processes. In Proc. 13th International Design Conference, DESIGN 2014, External Links: Link Cited by: p1.
84. [33] (2013) OpenMathMap: accessing math via interactive maps. In Contemporary Issues in Mathematical Publishing, JMM San Diego Special Session, K. Kaiser, S. Krantz, and B. Wegner (Eds.),
pp. 81–98. External Links: Link Cited by: p1.
85. [34] (2013) OpenMathMap: accessing math via interactive maps. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on intelligent
computer mathematics 2013MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, C. Lange, D. Aspinall, J. Carette, J. Davenport,
A. Kohlhase, M. Kohlhase, P. Libbrecht, P. Quaresma, F. Rabe, P. Sojka, I. Whiteside, and W. Windsteiger (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
86. [60] (2013) The Scala-REPL + MMT as a lightweight mathematical user interface. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on
intelligent computer mathematics 2013MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, C. Lange, D. Aspinall, J. Carette, J.
Davenport, A. Kohlhase, M. Kohlhase, P. Libbrecht, P. Quaresma, F. Rabe, P. Sojka, I. Whiteside, and W. Windsteiger (Eds.), CEUR Workshop Proceedings. Cited by: p1.
87. [73] (2013) Automated reasoning for economics. In 20^th Automated Reasoning Workshop (ARW), A. Bolotov, S. Colton, D. Crocker, L. Dennis, C. Dixon, J. Fleuriot, U. Hustadt, M. Jamnik, K.
Komendantskaya, A. Miller, R. Schmidt, V. Sorge, and J. Heras (Eds.), Note: Invited paper Cited by: p1.
88. [91] (2013) Spreadsheets: from data interfaces to knowledge interfaces. In Joint proceedings of the MathUI, OpenMath, PLMMS, and ThEdu workshops and work in progress at the conference on
intelligent computer mathematics 2013MathUI, OpenMath, PLMMS, and ThEdu Workshops and Work in Progress at the Conference on Intelligent Computer Mathematics, C. Lange, D. Aspinall, J. Carette, J.
Davenport, A. Kohlhase, M. Kohlhase, P. Libbrecht, P. Quaresma, F. Rabe, P. Sojka, I. Whiteside, and W. Windsteiger (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
89. [108] (2013) XLSearch: a search engine for spreadsheets. In Symp. of the european spreadsheet risks interest group (EuSpRIG 2013)Symp. of the European Spreadsheet Risks Interest Group (EuSpRIG
2013), External Links: Link Cited by: p1.
90. [109] (2013) MathWebSearch at NTCIR-10. In NTCIR workshop 10 meetingNTCIR Workshop 10 Meeting, N. Kando and K. Kishida (Eds.), pp. 675–679. External Links: Link Cited by: p1.
91. [118] (2013) Knowledge management for systematic engineering design in CAD systems. In Professionelles Wissenmanagement Management, Konferenzbeiträge der 7. KonferenzProfessionelles
Wissenmanagement Management, Konferenzbeiträge der 7. Konferenz, F. Lehner, N. Amende, and N. Fteimi (Eds.), pp. 202–217. External Links: Link Cited by: p1.
92. [119] (2013) The flexiformalist manifesto. In International workshop on symbolic and numeric algorithms for scientific computing (SYNASC 2012)14th International Workshop on Symbolic and Numeric
Algorithms for Scientific Computing (SYNASC 2012), A. Voronkov, V. Negru, T. Ida, T. Jebelean, D. Petcu, S. M. Watt, and D. Zaharie (Eds.), pp. 30–36. External Links: Link Cited by: p1.
93. [129] (2013) A qualitative comparison of the suitability of four theorem provers for basic auction theory. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Carette, D.
Aspinall, C. Lange, P. Sojka, and W. Windsteiger (Eds.), Lecture Notes in Computer Science, pp. 200–215. External Links: 1303.4193 Cited by: p1.
94. [142] (2013) The formare project – formal mathematical reasoning in economics. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Carette, D. Aspinall, C. Lange, P. Sojka,
and W. Windsteiger (Eds.), Lecture Notes in Computer Science, pp. 330–334. External Links: 1303.4194 Cited by: p1.
95. [143] (2013) Developing an auction theory toolbox. In Enabling domain experts to use formalised reasoningEnabling Domain Experts to use Formalised Reasoning, C. Lange, C. Rowat, and M. Kerber
(Eds.), Cited by: p1.
96. [158] (2013) The distributed ontology, modeling and specification language. In Modular Ontologies, C. Del Vescovo, T. Hahmann, D. Pearce, and D. Walther (Eds.), CEUR Workshop Proceedings, Aachen.
Note: Invited paper External Links: ISSN 1613-0073, Link Cited by: p1.
97. [160] (2013) Semantics of the distributed ontology language: institutes and institutions. In Recent Trends in Algebraic Development TechniquesRecent Trends in Algebraic Development Techniques, T.
Mossakowski, N. Martí-Oliet, and M. Palomino Tarjuelo (Eds.), LNCS, pp. 212–230. Cited by: p1.
98. [90] (2012-06) Navigation in mathematical documents. In Mathematical user-interfaces workshop 2012 at CICMMathematical User Interfaces Workshop 2012, P. Libbrecht (Ed.), pp. 12–23. External
Links: Link Cited by: p1.
99. [22] (2012) Representing CASL in a Proof-Theoretical Logical Framework. In Workshop on Algebraic Development Techniques, Cited by: p1.
100. [24] (2012) Compiling Logics. In Workshop on Algebraic Development Techniques, Cited by: p1.
101. [55] (2012) Representing Categories of Theories in a Proof-Theoretical Logical Framework. In Workshop on Algebraic Development Techniques, Cited by: p1.
102. [61] (2012) (Work-in-Progress) An MMT-Based User-Interface. In Workshop on User Interfaces for Theorem Provers, C. Kaliszyk and C. Lüth (Eds.), Cited by: p1.
103. [62] (2012) Management of Change in Declarative Languages. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka, M.
Wenzel, and V. Sorge (Eds.), LNAI, pp. 325–340. Cited by: p1.
104. [64] (2012) Cost-effective integration of mkm semantic services into editing environments. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J.
Carette, G. Dos Reis, P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 96–110. External Links: Link Cited by: p1.
105. [72] (2012) Formal representation and proof for cooperative games. In Symposium on Mathematical Practice and Cognition II, A. Pease and B. Larvor (Eds.), pp. 15–18. External Links: Link, ISBN
978-1-908187-10-9 Cited by: p1.
106. [101] (2012) Searching the space of mathematical knowledge. In DML and MIR 2012DML and MIR 2012, P. Sojka and M. Kohlhase (Eds.), External Links: Link Cited by: p1.
107. [126] (2012) The Babel of the semantic web tongues – in search of the Rosetta stone of interoperability. In What will the Semantic Web look like 10 Years from now? Workshop at ISWC, F. van
Harmelen, J. A. Hendler, P. Hitzler, K. Janowicz, and D. Vrandečić (Eds.), External Links: Link Cited by: p1.
108. [137] (2012) The distributed ontology language (DOL): ontology integration and interoperability applied to mathematical formalization. In Intelligent computer mathematicsIntelligent Computer
Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 463–467. External Links: 1204.5093 Cited by: p1.
109. [140] (2012) The Distributed Ontology Language (DOL): use cases, syntax, and extensibility. In Terminology and knowledge engineering conference (TKE)Terminology and Knowledge Engineering
Conference (TKE), G. Aguado de Cea, M. C. Suárez-Figueroa, R. García-Castro, and E. Montiel-Ponsoda (Eds.), pp. 33–48. External Links: 1208.0293 Cited by: p1.
110. [141] (2012) LoLa: a modular ontology of logics, languages, and translations. In Modular Ontologies, T. Schneider and D. Walther (Eds.), CEUR Workshop Proceedings, Aachen. External Links: ISSN
1613-0073, Link Cited by: p1.
111. [159] (2012) cMap’s product classification mapping from an ontology interoperability perspective. In Mapping of classifications and other light-weight ontologies, A. Virgili and others (Eds.),
Cited by: p1.
112. [189] (2012) Mechanically Verifying Logic Translations. In Workshop on Algebraic Development Techniques, Cited by: p1.
113. [195] (2012) A Query Language for Formal Mathematical Libraries. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos Reis, P.
Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 142–157. External Links: 1204.4685 Cited by: p1.
114. [214] (2012) Point-and-write — documenting formal mathematics by reference. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Jeuring, J. A. Campbell, J. Carette, G. Dos
Reis, P. Sojka, M. Wenzel, and V. Sorge (Eds.), LNAI, pp. 169–185. External Links: 1204.5094 Cited by: p1.
115. [139] (2011-11) Making heterogeneous ontologies interoperable through standardisation. In Accessibility Reaching Everywhere, pp. 185–196. External Links: Link Cited by: p1.
116. [179] (2011-09) MathWebSearch 0.5 - open formula search engine. In Wissens- und erfahrungsmanagement LWA (lernen, wissensentdeckung und adaptivität) conference proceedingsWissens- und
Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference Proceedings, External Links: Link Cited by: p1.
117. [127] (2011-06) Towards a standard for heterogeneous ontology integration and interoperability. In First International Conference on Terminology, Language and Content Resources (LaRC), K. Choi
and others (Eds.), pp. 101–110. External Links: Link Cited by: p1.
118. [19] (2011) Authoring and publishing of units and quantities in semantic documents. In Proceedings of the 1^st workshop on semantic publication, Extended Semantic Web Conference1^st Workshop on
Semantic Publication (SePublica), A. García Castro, C. Lange, E. Sandhaus, and A. de Waard (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
119. [23] (2011) Towards Logical Frameworks in the Heterogeneous Tool Set Hets. In Recent Trends in Algebraic Development TechniquesRecent Trends in Algebraic Development Techniques, H. Kreowski and
T. Mossakowski (Eds.), LNCS. Cited by: p1.
120. [29] (2011) A framework for modular semantic publishing with separate compilation and dynamic linking. In Proceedings of the 1^st workshop on semantic publication, Extended Semantic Web
Conference1^st Workshop on Semantic Publication (SePublica), A. García Castro, C. Lange, E. Sandhaus, and A. de Waard (Eds.), CEUR Workshop Proceedings. Cited by: p1.
121. [37] (2011) BauDenkMalNetz – creating a semantically annotated web resource of historical buildings. In Proceedings of the 1^st workshop on semantic publication, Extended Semantic Web
Conference1^st Workshop on Semantic Publication (SePublica), A. García Castro, C. Lange, E. Sandhaus, and A. de Waard (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
122. [53] (2011) Extending OpenMath with Sequences. In Intelligent computer mathematicsIntelligent Computer Mathematics, J. Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.), LNAI, pp. 58–72.
External Links: Link Cited by: p1.
123. [76] (2011) planMP: collecting mathematical practices for mkm. In Intelligent computer mathematics – work in progress papersIntelligent Computer Mathematics – Work in Progress Papers, J.
Davenport, W. Farmer, F. Rabe, and J. Urban (Eds.), Cited by: p1.
124. [81] (2011) Towards a flexible notion of document context. In Proceedings of the 29^th annual ACM international conference on design of communication (SIGDOC)Proceedings of the 29^th annual ACM
international conference on Design of communication (SIGDOC), pp. 181–188. External Links: Link Cited by: p1.
125. [220] (2011) Using discourse context to interpret object-denoting mathematical expressions. In Towards digital mathematics library, dml workshopTowards Digital Mathematics Library, DML workshop,
P. Sojka (Ed.), pp. 85–101. External Links: Link Cited by: p1.
126. [154] (2010-07) Towards OpenMath content dictionaries as linked data. In 23^rd OpenMath workshop23^rd OpenMath Workshop, M. Kohlhase and C. Lange (Eds.), External Links: 1006.4057v1 Cited by: p1
127. [31] (2010-06) JOBAD/MMT – interactive mathematics. In AI Mashup ChallengeAI Mashup Challenge at ESWC, A. Giurca, B. Endres-Niggemeyer, C. Lange, L. Maicher, and P. Hitzler (Eds.), External
Links: Link Cited by: p1.
128. [8] (2010) Semantics-based change impact analysis for heterogeneous collections of documents. In Proceedings of the 10^th ACM symposium on document engineeringProceedings of the 10^th ACM
symposium on Document engineering, M. Gormish and R. Ingold (Eds.), DocEng ’10, pp. 97–106. External Links: Link, Document Cited by: p1.
129. [28] (2010) eMath 3.0: building blocks for a social and semantic web for online mathematics & ELearning. In 1^st International Workshop on Mathematics and ICT: Education, Research and
Applications1^st International Workshop on Mathematics and ICT: Education, Research and Applications, I. Mierlus-Mazilu (Ed.), External Links: Link Cited by: p1.
130. [30] (2010) Publishing math lecture notes as linked data. In The semantic web: research and applications (part II)The Semantic Web: Research and Applications (Part II), L. Aroyo, G. Antoniou, E.
Hyvönen, A. ten Teije, H. Stuckenschmidt, L. Cabral, and T. Tudorache (Eds.), LNCS, pp. 370–375. External Links: 1004.3390v1 Cited by: p1.
131. [32] (2010) Interactive documents as interfaces to computer algebra systems: JOBAD and Wolfram—Alpha. In CALCULEMUS (emerging trends)CALCULEMUS (Emerging Trends), D. Delahaye and R. Rioboo
(Eds.), pp. 13–30. External Links: Link Cited by: p1.
132. [36] (2010) Prototyping a browser for a listed buildings database with Semantic MediaWiki. In Proceedings of the 5^th workshop on semantic wikis, Extended Semantic Web Conference5^th Workshop on
Semantic Wikis, C. Lange, J. Reutelshöfer, S. Schaffert, and H. Skaf-Molli (Eds.), CEUR Workshop Proceedings. Cited by: p1.
133. [39] (2010) Mathematical knowledge representation for education semantic web based on learning style. In International Symposium in Information Technology (ITSim), pp. 1–4. External Links:
Document Cited by: p1.
134. [63] (2010) Semantically enabled business process discovery. In Symposium on Applied Computing, pp. 1396–1403. External Links: ISBN 978-1-60558-639-7 Cited by: p1.
135. [78] (2010) Dimensions of formality: a case study for MKM in software engineering. In Intelligent computer mathematicsIntelligent Computer Mathematics, S. Autexier, J. Calmet, D. Delahaye, P. D.
F. Ion, L. Rideau, R. Rioboo, and A. P. Sexton (Eds.), LNAI, pp. 355–369. External Links: 1004.5071v1 Cited by: p1.
136. [144] (2010) Previewing OWL changes and refactorings using a flexible XML database. In Proceedings of the 1^st workshop on ontology repositories and editors, Extended Semantic Web Conference1^st
Workshop on Ontology Repositories and Editors, M. d’Aquin, A. García Castro, C. Lange, and K. Viljanen (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
137. [153] (2010) Integrating mathematics into the web of data. In Linked Data in the Future Internet, S. Auer, S. Decker, and M. Hauswirth (Eds.), CEUR Workshop Proceedings, Aachen. External Links:
ISSN 1613-0073, Link Cited by: p1.
138. [194] (2010) Representing Isabelle in LF. In Logical Frameworks and Meta-languages: Theory and Practice, K. Crary and M. Miculan (Eds.), Electronic Proceedings in Theoretical Computer Science,
Vol. 34, pp. 85–100. Cited by: p1.
139. [218] (2010) Semantics of governmental statistics data. In Proceedings of WebSci’10: Extending the Frontiers of Society On-LineProceedings of WebSci’10: Extending the Frontiers of Society
On-Line, External Links: Link Cited by: p1.
140. [221] (2010) Symbol declarations in mathematical writing: a corpus study. In Towards digital mathematics library, dml workshopTowards Digital Mathematics Library, DML workshop, P. Sojka (Ed.),
pp. 119–127. External Links: Link Cited by: p1.
141. [222] (2010) A [insert xml format] database for [insert cool application]. In Proceedings of XML Prague 2010Proceedings of XML Prague 2010, External Links: Link Cited by: p1.
142. [44] (2009-12-14) Towards context-based disambiguation of mathematical expressions. In The Joint Conference of ASCM 2009 and MACIS 2009: Asian Symposium on Computer Mathematics and Mathematical
Aspects of Computer and Information Sciences, M. Suzuki, H. Hong, H. Anai, C. Yap, Y. Sato, and H. Yoshida (Eds.), COE Lecture Notes, Vol. 22, Fukuoka, Japan, pp. 262–271. External Links: ISSN
1881-4042, Link Cited by: p1.
143. [48] (2009-09) LWA 2009; Workshop-Woche: Lernen – Wissen – Adaptivität. Technical report Vol. TUD-KE-2009-04, Universität Darmstadt. Cited by: 80, 135, 223.
144. [80] (2009-09) What you get is what you understand: assessment in SACHS. In Wissens- und erfahrungsmanagement (knowledge and experience management), FGWMWissens- und Erfahrungsmanagement
(Knowledge and Experience Management), FGWM, C. Lange and J. Reutelshöfer (Eds.), pp. 22–29. External Links: Link Cited by: p1.
145. [100] (2009-09) JOBAD – interactive mathematical documents. In AI Mashup ChallengeAI Mashup Challenge at KI Conference, B. Endres-Niggemeyer, V. Zacharias, and P. Hitzler (Eds.), External Links:
Link Cited by: p1.
146. [135] (2009-09) A mathematical approach to ontology authoring and documentation. In Wissens- und erfahrungsmanagement (knowledge and experience management), FGWMWissens- und Erfahrungsmanagement
(Knowledge and Experience Management), FGWM, C. Lange and J. Reutelshöfer (Eds.), Cited by: p1.
147. [223] (2009-09) The TNTBase System and Validation of XML Documents. In Wissens- und erfahrungsmanagement (knowledge and experience management), FGWMWissens- und Erfahrungsmanagement (Knowledge
and Experience Management), FGWM, C. Lange and J. Reutelshöfer (Eds.), pp. 57–63. External Links: Link Cited by: p1.
148. [16] (2009-07) Formal Proofs: Reconciling Correctness and Understanding. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt (Eds.),
LNAI. Cited by: p1.
149. [26] (2009-07) Quantifiers and big operators in OpenMath. In 22^nd openmath workshop22^nd OpenMath Workshop, J. H. Davenport (Ed.), External Links: Link Cited by: p1.
150. [40] (2009-07) Integrating web services into active mathematical documents. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt
(Eds.), LNAI, pp. 279–293. External Links: Link Cited by: p1.
151. [111] (2009-07) Semantics of OpenMath and MathML3. In 22^nd openmath workshop22^nd OpenMath Workshop, J. H. Davenport (Ed.), External Links: Link Cited by: p1.
152. [134] (2009-07) A mathematical approach to ontology authoring and documentation. In MKM/Calculemus proceedingsMKM/Calculemus Proceedings, J. Carette, L. Dixon, C. Sacerdoti Coen, and S. M. Watt
(Eds.), LNAI, pp. 389–404. External Links: Link Cited by: p1.
153. [152] (2009-07) wiki.openmath.org – how it works, how you can participate. In 22^nd openmath workshop22^nd OpenMath Workshop, J. H. Davenport (Ed.), External Links: 1003.5192v1 Cited by: p1.
154. [183] (2009-07) A better role system for OpenMath. In 22^nd openmath workshop22^nd OpenMath Workshop, J. H. Davenport (Ed.), External Links: Link Cited by: p1.
155. [136] (2009-06) Documenting ontologies the mathematical way. In Poster Proceedings of the 6^th European Semantic Web Conference (ESWC), External Links: Link Cited by: p1.
156. [151] (2009-05) Krextor – an extensible XML$\to$RDF extraction framework. In Scripting and development for the semantic web (SFSW)Scripting and Development for the Semantic Web (SFSW), C. Bizer,
S. Auer, and G. A. Grimnes (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
157. [27] (2009) Parallelised em wave propagation modelling for accurate network simulation. In 9^th IT&T Conference, Dublin Institute of Technology, Cited by: p1.
158. [35] (2009) A Case Study on Formalizing Algebra in a Module System. In Workshop on Modules and Libraries for Proof Assistants, F. Rabe and C. Schürmann (Eds.), ACM International Conference
Proceeding Series, Vol. 429, pp. 11–18. Cited by: p1.
159. [41] (2009) An architecture for linguistic and semantic analysis on the arXMLiv corpus. In Applications of Semantic Technologies (AST) Workshop at Informatik 2009, External Links: Link Cited by:
160. [50] (2009) Representing Model Theory in a Type-Theoretical Logical Framework. In Fourth Workshop on Logical and Semantic Frameworks, with Applications, M. Ayala-Rincón and F. Kamareddine (Eds.)
, Electronic Notes in Theoretical Computer Science, Vol. 256, pp. 49–65. Cited by: p1.
161. [168] (2009) Communities of Practice & Semantic Web: Stimulating Collaboration by Document Markup. In Proceedings of the United International Systems Conference (UNISCON) LNBIP 20, J. Yang and
others (Eds.), pp. 432–437. External Links: Link Cited by: p1.
162. [169] (2009) Communities of Practice & Semantic Web: Stimulating Collaboration by Document Markup. In UNISCON 2009 Supplementary Proceedings: Doctoral Consortium Papers, pp. 1–8. External Links:
Link Cited by: p1.
163. [187] (2009) A Practical Module System for LF. In Proceedings of the Workshop on Logical Frameworks: Meta-Theory and Practice (LFMTP), J. Cheney and A. Felty (Eds.), ACM International Conference
Proceeding Series, Vol. LFMTP’09, pp. 40–48. Cited by: p1.
164. [193] (2009) Representing Logics and Logic Translations. In Ausgezeichnete Informatikdissertationen 2008, D. Wagner et al. (Ed.), Lecture Notes in Informatics, Vol. D-9, pp. 201–210. Note:
English title: Outstanding Dissertations in Computer Science 2008 Cited by: p1.
165. [213] (2009) MathML-aware article conversion from LaTeX, a comparison study. In Towards digital mathematics library, dml 2009 workshopTowards Digital Mathematics Library, DML 2009 workshop, P.
Sojka (Ed.), pp. 109–120. External Links: Link Cited by: p1.
166. [79] (2008-10) Compensating the semantic bias of spreadsheets. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference ProceedingsWissens- und
Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference Proceedings, J. Baumeister and M. Atzmüller (Eds.), Vol. 448. External Links: Link Cited by: p1.
167. [131] (2008-10) Arguing on issues with mathematical knowledge items in a semantic wiki. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference
ProceedingsWissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference Proceedings, J. Baumeister and M. Atzmüller (Eds.), Vol. 448. Cited by: p1.
168. [166] (2008-10) Towards CoPing with Information Overload. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference ProceedingsWissens- und
Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference Proceedings, J. Baumeister and M. Atzmüller (Eds.), Vol. 448. Cited by: p1.
169. [167] (2008-10) Towards the Adaptation of Scientific Course Material powered by Community of Practice. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität)
Conference ProceedingsWissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference Proceedings, J. Baumeister and M. Atzmüller (Eds.), Vol. 448. Cited by: p1.
170. [174] (2008-10) Fine-Granular Version Control & Redundancy Resolution. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference ProceedingsWissens- und
Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) Conference Proceedings, J. Baumeister and M. Atzmüller (Eds.), Vol. 448. Note: http://www.kwarc.info/nmueller/papers/
lwa08-fst.pdf External Links: Link Cited by: p1.
171. [130] (2008-07) Easily editing and browsing complex OpenMath markup with SWiM. In Mathematical user interfaces workshop at MKMMathematical User Interfaces Workshop, P. Libbrecht (Ed.), External
Links: Link Cited by: p1.
172. [132] (2008-06) Improving mathematical knowledge items by acting on issue-based community feedback. In 2^nd workshop on scientific communities of practice (SCooP-2008)Proceedings of the 2^nd
SCooP Workshop, C. Müller (Ed.), Cited by: p1.
173. [163] (2008-06) Towards A Community of Practice Toolkit. In 2^nd workshop on scientific communities of practice (SCooP-2008)Proceedings of the 2^nd SCooP Workshop, C. Müller (Ed.), Cited by: p1.
174. [148] (2008-02) Editing OpenMath content dictionaries with SWiM. In 3^rd JEM Workshop (Joining Educational Mathematics), External Links: Link Cited by: p1.
175. [9] (2008) THF0 – The core of the TPTP Language for Higher-Order Logic. In 4^th International Joint Conference on Automated Reasoning, A. Armando, P. Baumgartner, and G. Dowek (Eds.), LNCS,
pp. 491–506. Cited by: p1.
176. [128] (2008) Expressing argumentative discussions in social media sites. In Social data on the web (SDoW), workshop at the 7^th international semantic web conferenceSocial Data on the Web
(SDoW), Workshop at the 7^th International Semantic Web Conference, J. Breslin, U. Bojārs, A. Passant, and S. Fernández (Eds.), CEUR Workshop Proceedings. External Links: Link Cited by: p1.
177. [138] (2008) Flyspeck in a semantic wiki – collaborating on a large scale formalization of the Kepler conjecture. In Proceedings of the 3^rd workshop on semantic wikis, European Semantic Web
Conference3^rd Workshop on Semantic Wikis, C. Lange, S. Schaffert, H. Skaf-Molli, and M. Völkel (Eds.), CEUR Workshop Proceedings. Cited by: p1.
178. [149] (2008) Mathematical semantic markup in a wiki: the roles of symbols and notations. In Proceedings of the 3^rd workshop on semantic wikis, European Semantic Web Conference3^rd Workshop on
Semantic Wikis, C. Lange, S. Schaffert, H. Skaf-Molli, and M. Völkel (Eds.), CEUR Workshop Proceedings. Cited by: p1.
179. [150] (2008) SWiM – a semantic wiki for mathematical knowledge management. In The semantic web: research and applicationsThe Semantic Web: Research and Applications, S. Bechhofer, M. Hauswirth,
J. Hoffmann, and M. Koubarakis (Eds.), LNCS, pp. 832–837. External Links: 1003.5196v1 Cited by: p1.
180. [162] (2008) Communities of practice in mathematical elearning. In In proceedings of the Workshop in Mathematical and Scientific eContent, pp. 34–35. Cited by: p1.
181. [181] (2008) An exchange format for modular knowledge. In Proceedings of the LPAR Workshops: Knowledge Exchange: Automated Provers and Proof Assistants, and The 7^th International Workshop on
the Implementation of Logics, G. Sutcliffe, P. Rudnicki, R. Schmidt, B. Konev, and S. Schulz (Eds.), CEUR Workshop Proceedings, Aachen, pp. 50–68. External Links: ISSN 1613-0073 Cited by: p1.
182. [182] (2008) An Exchange Format for Modular Knowledge. In Proceedings of the LPAR Workshops on Knowledge Exchange: Automated Provers and Proof Assistants, and The 7th International Workshop on
the Implementation of Logics, G. Sutcliffe, P. Rudnicki, R. Schmidt, B. Konev, and S. Schulz (Eds.), CEUR Workshop Proceedings, Vol. 418, pp. 50–68. Cited by: p1.
183. [104] (2007-06) Presenting mathematical content with flexible elisions. In OpenMath/JEM workshop 2007OpenMath/JEM Workshop 2007, O. Caprotti, M. Kohlhase, and P. Libbrecht (Eds.), Cited by: p1.
184. [107] (2007-06) Documents with flexible notation contexts as interfaces to mathematical knowledge. In Mathematical user-interfaces workshop 2007 at MKMMathematical User Interfaces Workshop 2007,
P. Libbrecht (Ed.), Cited by: p1.
185. [145] (2007-06) SWiM – a semantic wiki for mathematical knowledge management. In Mathematical user-interfaces workshop 2007 at MKMMathematical User Interfaces Workshop 2007, P. Libbrecht (Ed.),
Cited by: p1.
186. [147] (2007-06) Towards scientific collaboration in a semantic wiki. In Bridging the Gap between Semantic Web and Web 2.0 (SemNet)Bridging the Gap between Semantic Web and Web 2.0 (SemNet), A.
Hotho and B. Hoser (Eds.), Cited by: p1.
187. [52] (2007) Formal representation of mathematics in a dependently typed set theory. In MKM/CalculemusTowards Mechanized Mathematical Assistants. MKM/Calculemus, M. Kauers, M. Kerber, R. Miner,
and W. Windsteiger (Eds.), LNAI, pp. 265–279. Cited by: p1.
188. [89] (2007) CPoint — ein invasiver, semantischer Editor für Content in MS PowerPoint. In EduMediaOffene Bildung im/mit dem Web 2.0!?!, V. Hornung-Prähauser (Ed.), pp. 116–118. Cited by: p1.
189. [105] (2007) Managing variants in document content and narrative structures. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) conference proceedingsWissens-
und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) conference proceedings, A. Hinneburg (Ed.), pp. 324–229. Cited by: p1.
190. [146] (2007) Towards a Semantic Wiki for Science. In Proceedings of the KWEPSY (Knowledge Web PhD Symposium)Proceedings of the KWEPSY (Knowledge Web PhD Symposium), E. Simperl, J. Diederich, and
G. Schreiber (Eds.), CEUR Workshop Proceedings. Cited by: p1.
191. [161] (2007) Panta rhei. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) conference proceedingsWissens- und Erfahrungsmanagement LWA (Lernen,
Wissensentdeckung und Adaptivität) conference proceedings, A. Hinneburg (Ed.), pp. 318–323. Cited by: p1.
192. [164] (2007) Presentation on Modeling Scientific Communities of Practice based on Semantic Markup of Scientific Documents and Web2.0. Technologies. In 7. konferenz für interaktive und
kooperative medienMensch und Computer 2007, T. Gross (Ed.), Cited by: p1.
193. [165] (2007) Towards the Identification and Support of Scientific Communities of Practice. In 1^st Workshop on Scientific COmmunities Of PracticeProceedings of the 1^st SCooP Workshop, C. Müller
(Ed.), Cited by: p1.
194. [175] (2007) Towards Improving Interactive Mathematical Authoring by Ontology-driven Management of Change. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität)
conference proceedingsWissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) conference proceedings, A. Hinneburg (Ed.), pp. 289–295. Note: https://kwarc.info/nmueller/
papers/lwa07.pdf External Links: Link Cited by: p1.
195. [192] (2007) OMDoc Theory Graphs Revisited. In Proceedings of the OpenMath/JEM workshop, Cited by: p1.
196. [219] (2007) Towards Community of Practice Support for Interactive Mathematical Authoring. In 1^st Workshop on Scientific COmmunities Of PracticeProceedings of the 1^st SCooP Workshop, C. Müller
(Ed.), Cited by: p1.
197. [87] (2006) CPoint – ein invasiver, semantischer Editor für wiederverwendaren Content in MS PowerPoint. In Proceedings der Pre-Conference Workshops der 4.e-Learning Fachtagung Informatik DeLFI
2006Proceedings der Pre-Conference Workshops der 4.e-Learning Fachtagung Informatik DeLFI 2006, C. Rensing (Ed.), pp. 35–42. Cited by: p1.
198. [88] (2006) The User as Prisoner: How the Dilemma Might Dissolve. In 2^nd Workshop on Learner Oriented Knowledge Management & KM Oriented e-Learning2^nd Workshop on Learner Oriented Knowledge
Management & KM Oriented e-Learning, M. Memmel, E. Ras, and S. Weibelzahl (Eds.), pp. 26–31. External Links: Link Cited by: p1.
199. [133] (2006) A semantic wiki for mathematical knowledge management. In Proceedings of the 1^st workshop on semantic wikis, European Semantic Web Conference1^st Workshop on Semantic Wikis, M.
Völkel, S. Schaffert, and S. Decker (Eds.), CEUR Workshop Proceedings. Cited by: p1.
200. [176] (2006) An Ontology-Driven Management of Change. In Wissens- und Erfahrungsmanagement LWA (Lernen, Wissensentdeckung und Adaptivität) conference proceedings, Universität Hildesheim,
pp. 186–193. Note: https://kwarc.info/nmueller/papers/lwa06.pdf External Links: Link Cited by: p1.
201. [190] (2006) First-Order Logic with Dependent Types. In Automated reasoning — third international joint conference, ijcar 2006Automated Reasoning — Third International Joint Conference, IJCAR
2006, U. Furbach and N. Shankar (Eds.), LNAI, pp. 377–391. Cited by: p1.
202. [191] (2006) Towards Determining the Subset Relation between Propositional Modal Logics. In Proceedings of the FLoC 06 Workshop on Empirically Successful Computerized Reasoning, 3^rd
International Joint Conference on Automated Reasoning, G. Sutcliffe, R. Schmidt, and S. Schulz (Eds.), CEUR Workshop Proceedings, Aachen, pp. 126–140. External Links: ISSN 1613-0073 Cited by: p1.
203. [186] (2005) A Sequent Calculus for a First-order Dynamic Logic with Trace Modalities for Promela^+. In Short Paper Proceedings of the International Conference on Logic for Programming,
Artificial Intelligence, and Reasoning, G. Sutcliffe and A. Voronkov (Eds.), pp. 21–27. Cited by: p1.
204. [209] (2005) Metadata for web-based mathematical learning materials. In Proceedings of the Data Engineering Workshop DEWS, External Links: Link Cited by: p1.
205. [86] (2004) CPoint’s mathematical user interface. In Mathematical user interfaces workshop 2004Mathematical User Interfaces, P. Libbrecht (Ed.), External Links: Link Cited by: p1.
206. [117] (2004) Semantic markup for TeX/LaTeX. In Mathematical user interfaces workshop 2004Mathematical User Interfaces, P. Libbrecht (Ed.), Cited by: p1.
207. [20] (2003) Resurrecting the Analytica theorem prover. In First qpq workshop on deductive software componentsFirst QPQ Workshop on Deductive Software Components, Cited by: p1.
208. [47] (2003) A time calculus for natural language. In ICoS-4. inference in computational semantics. workshop proceedingsProceedings of Inference in Computational Semantics ICoS-4, P. Blackburn
and J. Bos (Eds.), pp. 113–127. External Links: Link Cited by: p1.
209. [116] (2003) Applying unification techniques to XML document management?. In 17^th workshop on unification17^th Workshop on Unification, Cited by: p1.
210. [114] (2002) Acquisition of math content in an academic setting. In Second international conference on mathml and technologies for math on the webSecond International Conference on MathML and
Technologies for Math on the Web, External Links: Link Cited by: p1.
211. [224] (2002) System Description: The MathWeb software bus for distributed mathematical reasoning. In Automated deduction — cade-18Automated Deduction — CADE-18, A. Voronkov (Ed.), LNAI,
pp. 247–252. External Links: Link Cited by: p1.
212. [102] (2000) Towards a tableaux machine for language understanding. In ICoS-2. inference in computational semantics. workshop proceedingsProceedings of Inference in Computational Semantics
ICoS-2, J. Bos and M. Kohlhase (Eds.), pp. 57–88. Cited by: p1.
213. [210] (2000) Adaptive course generation and presentation. In Proceedings of ITS-2000 workshop on Adaptive and Intelligent Web-Based Education Systems, P. Brusilovski and C. Peylo (Eds.),
Montreal. Cited by: p1.
214. [14] (1999) Automated reasoning for computational semantics. In The Third International Tbilisi Symposium on Language, Logic and Computation, Batumi, Georgia. Cited by: p1.
215. [13] (1998) Automated theorem proving for natural language understanding. In CADE-15 Workshop “Problem-solving Methodologies with Automated Deduction”CADE-15 workshop “problem-solving
methodologies with automated deduction”, P. Baumgartner, U. Furbach, M. Kohlhase, W. McCune, W. Reif, M. Stickel, T. Uribe, P. Baumgartner, U. Furbach, M. Kohlhase, W. McCune, W. Reif, M.
Stickel, and T. Uribe (Eds.), Cited by: p1.
216. [211] (1998) A distributed graphical user interface for the interactive proof system OMEGA. In User interfaces for theorem proversUser Interfaces for Theorem Provers, R. C. Backhouse (Ed.),
Computing Science Reports, pp. 130–138. Cited by: p1.
217. [212] (1998) LOUI: a distributed graphical user interface for the interactive proof system OMEGA. In Proceedings of the International Workshop “User Interfaces for Theorem Provers 1998”
(UITP’98), Eindhoven, Netherlands. External Links: Link Cited by: p1.
218. [38] (1997) Underspecification of quantifier scope. In Proceedings der 6. Fachtagung der Sektion Computerlinguistik der DGfS, Heidelberg. External Links: Link Cited by: p1.
219. [71] (1996) Partiality without the cost. In Workshop on “Mechanization of Partial Functions” at CADE-13, Cited by: p1.
220. [103] (1996) A type-theoretic semantics for $\lambda$-DRT. In Proceedings of the 10^th Amsterdam Colloquium, P. Dekker and M. Stokhof (Eds.), Amsterdam, pp. 479–498. External Links: Link Cited
by: p1.
221. [56] (1994) Guaranteeing correctness through the communication of checkable proofs (or: would you really trust an automated reasoning system?). In CADE-14 Workshop, Cited by: p1.
222. [57] (1994) A test for evaluating the practical usefulness of deduction systems. In Informal Proc. of the 11^th Annual Meeting of the “GI-Fachgruppe Deduktionssysteme”Informal Proc. of the 11^th
Annual Meeting of the “GI-Fachgruppe Deduktionssysteme”, C. W. Wolfgang Bibel (Ed.), Forschungsbericht, FB Informatik, TH Darmstadt, pp. 12–12. Cited by: p1.
223. [70] (1994) Formalizing mathematics with dependent sorts. In Proceedings des Deduktionstreffen, AIDA-Report. Cited by: p1.
224. [115] (1993) Higher-order resolution with combinators. In Informal Proceedings fo the Annual Meeting of “GI-Fachgruppe ‘Deduktionssysteme’ ” in Kaiserslautern, 1993, J. Avenhaus and J. Denzinger
(Eds.), SEKI-Report, pp. 15. Cited by: p1.
Technical Reports
1. [43] (2020) Logic-based natural language processing. External Links: Link Cited by: p1.
2. [2] (2014) OpenMathMap: interaction. Bachelor’s Thesis, Computer Science, Jacobs University, Bremen. External Links: Link Cited by: p1.
3. [16] (2014) KAT: an annotation tool for STEM documents; manual. External Links: Link Cited by: p1.
4. [26] (2014) Semantically enhanced text documents. Jacobs University. Note: Bachelor’s Thesis Cited by: p1.
5. [28] (2014) Sorted unification in MathWebSearch. Bachelor’s Thesis, Computer Science, Jacobs University, Bremen. Cited by: p1.
6. [51] (2014) Creating semantic interactions in airbus spreadsheet reports. Jacobs University. Note: Bachelor’s Thesis Cited by: p1.
7. [59] (2014) Assessment service for spreadsheet documents. Note: Bachelor’s Thesis, Jacobs University Cited by: p1.
8. [46] U. Furbach, R. Grimm, F. Hampe, S. Staab, and M. Wimmer (Eds.) (2013-09-17) Applying mechanised reasoning in economics – making reasoners applicable for domain experts. Note: Tutorial at
INFORMATIK 2013, Computer science adapted to humans, organization and the environment, 43^rd annual meeting of the German Informatics Society (Gesellschaft für Informatik e.V. (GI)) External
Links: Link Cited by: p1.
9. [7] (2013) Proving soundness of combinatorial Vickrey auctions and generating verified executable code. External Links: 1308.1779 Cited by: p1.
10. [13] (2013) OpenMathMap: accessing math via interactive maps. B.Sc. Thesis, Jacobs University Bremen. External Links: Link Cited by: p1.
11. [27] (2013) Towards understanding context dimensions of spreadsheet knowledge. Jacobs University. Note: Bachelor’s Thesis Cited by: p1.
12. [52] (2013) An evaluation of responsive user interface options for multi-modal and mathematical search engines. Jacobs University. Note: Bachelor’s Thesis Cited by: p1.
13. [36] K. Judd (Ed.) (2012-07-25) An economist’s guide to mechanized reasoning or My computer just proved 84 impossibility theorems. Note: Invited lecture at the Initiative for Computational
Economics summer school External Links: Link Cited by: p1.
14. [3] (2012) Generic unification for type theories. Jacobs University Bremen. Note: Bachelor’s thesis Cited by: p1.
15. [33] (2012) Modular Encoding of Type Theory. Jacobs University Bremen. Note: Bachelor’s thesis Cited by: p1.
16. [44] (2012) Mathematical documents want to be active, digital math libraries want to be semantic — position paper for wdml 2012. Position Paper at WDML Symposium. External Links: Link Cited by:
17. [8] (2011) Authoring, publishing and interacting with units and quantities in technical documents. B.Sc. Thesis, Jacobs University Bremen. Cited by: p1.
18. [17] (2011) General Computer Science: GenCS I/II Lecture Notes. Semantic Course Notes in Panta Rhei. External Links: Link Cited by: p1.
19. [18] (2011) General Computer Science; 320101: GenCS I Lecture Notes. External Links: Link Cited by: p1.
20. [19] (2011) General Computer Science; Problems and Solutions for 320101 GenCS I. External Links: Link Cited by: p1.
21. [20] (2011) General Computer Science; Problems for 320101 GenCS I. External Links: Link Cited by: p1.
22. [21] (2011) General Computer Science: 320201 GenCS II Lecture Notes. External Links: Link Cited by: p1.
23. [22] (2011) General Computer Science: Problems and Solutions for 320201 GenCS II. External Links: Link Cited by: p1.
24. [23] (2011) General Computer Science: Problems for 320201 GenCS II. External Links: Link Cited by: p1.
25. [41] (2011) The FormalCAD Project. External Links: Link Cited by: p1.
26. [50] (2011) Survey results on collaborative mathematical knowledge management. Note: Appendix to the Ph.D. thesis “Enabling Collaboration on Semiformal Mathematical Knowledge by Semantic Web
Integration” External Links: Link Cited by: p1.
27. [61] (2011) Formalizing Syntactical Objects within Formalized Set Theory. Jacobs University Bremen. Note: Bachelor’s thesis Cited by: p1.
28. [62] (2011) Universal OpenMath machine. B.Sc. Thesis, Jacobs University Bremen. Cited by: p1.
29. [1] (2010) Note: http://tntbase.org/wiki/restful Cited by: p1.
30. [12] (2010) Interactive documents as interfaces to computer algebra systems: JOBAD and Wolfram—Alpha. B.Sc. Thesis, Jacobs University Bremen. External Links: Link Cited by: p1.
31. [14] (2010) Structuring Theories with Partial Morphisms. Note: Workshop on Abstract Development Techniques Cited by: p1.
32. [15] (2010) Structured specifications with hiding in the edinburgh logical framework LF. Bachelor’s Thesis, Jacobs University Bremen. External Links: Link Cited by: p1.
33. [53] (2010) Integrating SUMO and OMDoc. Bachelor’s Thesis, Computer Science, Jacobs University, Bremen. Cited by: p1.
34. [11] (2009) Unifying Math Ontologies: A tale of two standards (full paper). Note: http://opus.bath.ac.uk/13079 External Links: Link Cited by: p1.
35. [24] (2009) Integrating web services into active mathematical documents. Bachelor’s Thesis, Computer Science, Jacobs University, Bremen. External Links: Link Cited by: p1.
36. [25] (2009) An architecture for recovering meaning in a LaTeX to OMDoc conversion. Bachelor’s Thesis, Jacobs University Bremen. External Links: Link Cited by: p1.
37. [29] (2009) A Formal Proof of the Soundness of First-order Logic. Cited by: p1.
38. [32] (2009) Reasoning about theory morphisms. Bachelor’s Thesis, Jacobs University Bremen. External Links: Link Cited by: p1.
39. [37] (2009) The LATIN Project. External Links: Link Cited by: p1.
40. [45] (2009) OMDoc as an ontology language: OWL$\to$OMDoc translation implementation. Project report. Cited by: p1.
41. [57] (2009) A Module System for Twelf. External Links: Link Cited by: p1.
42. [49] (2008-03) Mathematik lernen in einem semantischen wiki. Note: Invited talk at the 99^th MNU-Kongress (Deutscher Verein zur Förderung des mathematischen und naturwissenschaftlichen
Unterrichts e.V.) External Links: Link Cited by: p1.
43. [35] (2008) Extracting RDF knowledge from OMDoc. Bachelor’s thesis, Jacobs University Bremen. Cited by: p1.
44. [63] (2008) Towards distributed mathematical knowledge management. Jacobs University of Bremen. Note: https://kwarc.info/vzholudev/pubs/proposal.pdf External Links: Link Cited by: p1.
45. [54] (2007-03) Lectora: Towards an Interactive, Collaborative Reader for Mathematical Documents. Jacobs University Bremen. Note: Research proposal External Links: Link Cited by: p1.
46. [60] (2007-03) OMDoc (from Wikipedia, the free encyclopedia). Note: http://en.wikipedia.org/w/index.php?title=OMDoc&oldid=112340133 External Links: Link Cited by: p1.
47. [48] (2007-02) Towards a Semantic Wiki for Science. Jacobs University Bremen. Note: https://kwarc.info/swim/pubs/swimplus-resprop.pdfResearch proposal for a Ph. D. thesis External Links: Link
Cited by: p1.
48. [4] (2007) MaTeSearch a combined math and text search engine. Bachelor’s Thesis, Jacobs University Bremen. External Links: Link Cited by: p1.
49. [6] (2007) A mathematical semantic web. Bachelor’s Thesis, Computer Science, Jacobs University, Bremen. External Links: Link Cited by: p1.
50. [10] (2007) Panta Rhei: Case Study Fall2007. Note: https://kwarc.info/panta-rhei/papers/cs_Fall2007.pdf External Links: Link Cited by: p1.
51. [34] (2007) Developing a REST interface to a database for OMDoc. Deutsches Forschungszentrum für Künstliche Intelligenz (DFKI) Bremen. External Links: Link Cited by: p1.
52. [31] (2006) Ontology-driven management of change. External Links: Link Cited by: p1.
53. [55] (2006) Towards an Ontology-Driven Management of Change – Research proposal for a Ph.D. thesis. International University Bremen. Note: https://kwarc.info/nmueller/papers/resprop.pdf External
Links: Link Cited by: p1.
54. [9] (2005) CPoint. External Links: Link Cited by: p1.
55. [38] (2003) Guidelines for graphics in MathML 2. Note: W3C Note External Links: Link Cited by: p1.
56. [39] (2003) Bound variables in MathML. Note: W3C Working Group Note External Links: Link Cited by: p1.
57. [40] (2003) Structured types in MathML 2.0. Note: W3C Note External Links: Link Cited by: p1.
58. [42] (2001) OMDoc: an open markup format for mathematical documents (version 1.1). Open Specification. External Links: Link Cited by: p1.
59. [30] (1994) A test for evaluating the practical usefulness of deduction systems. Note: Workshop “Evaluation of Automated Theorem Proving Systems” on CADE’94 Cited by: p1.
60. [5] Math databases wiki. External Links: Link Cited by: p1.
61. [47] Document ontologies. Note: project homepage External Links: Link Cited by: p1.
62. [56] Open digital research environment toolkit for the advancement of mathematics. Project Proposal. External Links: Link Cited by: p1.
63. [58] The MMT language and system. External Links: Link Cited by: p1.
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AP Board 7th Class Maths Notes Chapter 6 Ratio - Applications
AP Board 7th Class Maths Notes Chapter 6 Ratio – Applications
Students can go through AP Board 7th Class Maths Notes Chapter 6 Ratio – Applications to understand and remember the concepts easily.
AP State Board Syllabus 7th Class Maths Notes Chapter 6 Ratio – Applications
→ Ratio: A ratio is an ordered comparison of quantities of the same units.
We use the symbol ‘:’ to represent a ratio. The ratio of two quantities ‘a’ and ‘b’ is a : b and we read it as “a is to b”. The two quantities ‘a’ and ‘b’ are called the terms of the ratio. The first
quantity ‘a’ is called first term or antecedent and the second quantity ‘b’ is called consequent.
→ Proportion: If two ratios are equal, then the four terms of the ratios are said to be in proportion. We use the symbol : : (is as)
If two ratios a : b and c : d are equal, we write a : b :: c : d or a : b = c : d
Here ‘a’, ‘d’ are called extremes and b, c are called means.
→ Unitary Method: The method in which we first find the value of one unit and then the value of the required number of units is known as unitary method.
Eg: If the cost of 5 pens is Rs. 85; then the cost of 12 pens is ……… ?
Solution. Cost of 5 pens = Rs. 85
Cost of 1 pen = \(\frac{85}{5}\) = Rs. 17
∴ Cost of 12 pens = 12 × 17 = Rs. 204
→ Direct proportion: If in two quantities, when one quantity increases, the other also increase or vice-versa then the two quantities are said to be in direct proportion.
Eg: The number of books and their cost are in direct proportion.
As the number of books increases, the cost also increases.
→ Ratios also appear in the form of percentages.
→ The word percent means “per every hundred” or for a hundred. The symbol % is used to denote percentage.
→ To convert a quantity into its equivalent percentage
• express it as a fraction.
• multiply it with 100.
• assign % symbol.
Eg: A man purchased an article for Rs. 80 and sells it for Rs. 100. Find his gain percent.
Solution. Cost price = Rs. 80
Selling price = Rs. 100
gain = Rs. 20
gain as a fraction = \(\frac{20}{80}\)
gain as percent = \(\frac{20}{80}\) × 100 = 25%
→ When C.P > S.P there incurs loss.
→ When C.P < S.P there is gain.
→ When C.P = S.P neither loss nor gain.
→ Loss = C.P – S.P gain = S.P – C.P
→ Discount is always expressed as some percentage of marked price.
→ In general when P is principle; R% is rate of interest per annum and I is the interest, then
I = R% of P
I = R% of P for T years
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Factoring the difference of square worksheet
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SQRD −1/2 \text{SQRT}(-1/2)SQRT(−1/2): Complex Numbers and Imaginary Roots
Mathematics is full of intriguing concepts that challenge our understanding of numbers and their properties. One such concept is the square root of negative numbers, which leads us into the
fascinating world of complex numbers and imaginary roots. In this article, we will explore the concept of SQRD −1/2 \text{SQRT}(-1/2)SQRT(−1/2), or the square root of −12-\frac{1}{2}−21, and
understand its significance in both mathematical theory and real-world applications. We will also break down the steps to calculate and interpret this value using the fundamentals of complex numbers.
What Is the Square Root of a Negative Number?
Before diving into SQRD −1/2 \text{SQRT}(-1/2)SQRT(−1/2), it’s essential to revisit what it means to take the square root of any negative number. In the world of real numbers, it is impossible to
take the square root of a negative number because no real number squared results in a negative number. For example, the square of 222 is 444, and the square of −2-2−2 is also 444, but no real number
squared will give you a negative result.
This limitation of real numbers led to the introduction of complex numbers, which allow for the square roots of negative numbers. The imaginary unit, represented by iii, is defined as:i=−1i = \sqrt
With this definition, any square root of a negative number can be expressed as a multiple of iii. For example:−4=2i\sqrt{-4} = 2i−4=2i
Now that we understand the foundation of imaginary numbers, we can explore the square root of −12-\frac{1}{2}−21.
Breaking Down SQRD −1/2\text{SQRT}(-1/2)SQRT(−1/2)
The expression SQRD −1/2 )\text{SQRT}(-1/2)SQRT(−1/2) refers to the square root of −12-\frac{1}{2}−21, a value that involves both real and imaginary components. To solve this expression, we can
break it down step by step using the properties of complex numbers.
Step 1: Factor the Negative Number
As previously mentioned, the square root of a negative number involves the imaginary unit iii. Therefore, we can express the square root of −12-\frac{1}{2}−21 as:−12=−1×12=i×12\sqrt{-\frac{1}{2}} =
\sqrt{-1} \times \sqrt{\frac{1}{2}} = i \times \sqrt{\frac{1}{2}}−21=−1×21=i×21
Step 2: Simplify the Square Root of 12\frac{1}{2}21
Next, we need to simplify 12\sqrt{\frac{1}{2}}21. We can rewrite 12\frac{1}{2}21 as 12\frac{1}{\sqrt{2}}21 or:12=12=22\sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}21=21=22
Thus, the expression becomes:−12=i×22\sqrt{-\frac{1}{2}} = i \times \frac{\sqrt{2}}{2}−21=i×22
Step 3: Interpret the Final Result
The final result of SQRT(−1/2)\text{SQRT}(-1/2)SQRT(−1/2) is:−12=22i\sqrt{-\frac{1}{2}} = \frac{\sqrt{2}}{2}i−21=22i
This value represents a purely imaginary number, with no real component. The square root of −12-\frac{1}{2}−21 is an imaginary number that is approximately equal to 0.707i0.707i0.707i.
Visualizing Complex Numbers
To better understand the result 22i\frac{\sqrt{2}}{2}i22i, it’s helpful to visualize complex numbers on the complex plane. The complex plane is a two-dimensional plane where the horizontal axis
represents real numbers and the vertical axis represents imaginary numbers.
In the case of 22i\frac{\sqrt{2}}{2}i22i, the number lies entirely on the imaginary axis, specifically at 0.7070.7070.707 units above the origin. Since this number has no real component, its
position on the complex plane is purely vertical.
The Importance of Complex Numbers
The result of \text{SQRT}(-1/2} introduces us to the broader concept of complex numbers. Complex numbers are essential in various branches of mathematics, physics, engineering, and even computer
science. Some of their key applications include:
1. Electrical Engineering and Circuit Analysis
In electrical engineering, complex numbers are used to represent alternating current (AC) circuits. Engineers use complex numbers to calculate impedance, voltage, and current in circuits that involve
resistors, capacitors, and inductors.
2. Quantum Mechanics
Complex numbers play a crucial role in quantum mechanics, where they are used to describe the quantum states of particles. Quantum wave functions, which represent the probability distribution of a
particle’s position and momentum, often include complex numbers.
3. Control Systems
In control systems and signal processing, complex numbers are used to analyze the behavior of systems in the frequency domain. Techniques such as the Laplace transform and Fourier analysis rely
heavily on complex numbers to solve differential equations and analyze signals.
4. Fluid Dynamics
Complex numbers are also used in fluid dynamics to model two-dimensional fluid flow. The use of complex potential functions allows for the analysis of incompressible and irrotational fluid flows,
which are essential in understanding how fluids behave under various conditions.
Complex Conjugates
When dealing with complex numbers, it’s important to understand the concept of complex conjugates. The complex conjugate of a number is obtained by changing the sign of its imaginary component. For
example, the complex conjugate of 3+4i3 + 4i3+4i is 3−4i3 – 4i3−4i.
In the case of 22i\frac{\sqrt{2}}{2}i22i, its complex conjugate would be −22i-\frac{\sqrt{2}}{2}i−22i. Complex conjugates are useful in many mathematical operations, such as dividing complex
numbers or simplifying certain expressions.
The Square Root of Negative Fractions: A General Approach
The process we used to calculate SQRT(−1/2)\text{SQRT}(-1/2)SQRT(−1/2) can be generalized to find the square root of any negative fraction. Suppose we want to find −ab\sqrt{-\frac{a}{b}}−ba, where
aaa and bbb are positive real numbers. The steps are as follows:
1. Factor the Negative Sign: Use the imaginary unit iii to handle the negative sign. Express the square root as:−ab=i×ab\sqrt{-\frac{a}{b}} = i \times \sqrt{\frac{a}{b}}−ba=i×ba
2. Simplify the Fraction: Simplify the square root of the fraction ab\frac{a}{b}ba as:ab=ab\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}ba=ba
3. Multiply by iii: Multiply the result by iii to account for the imaginary component.−ab=i×ab\sqrt{-\frac{a}{b}} = i \times \frac{\sqrt{a}}{\sqrt{b}}−ba=i×ba
This method works for any negative fraction, making it a useful tool for solving a wide range of problems involving complex numbers.
Applications of Imaginary and Complex Numbers
Now that we have a solid understanding of SQRT(−1/2)\text{SQRT}(-1/2)SQRT(−1/2) and complex numbers, let’s explore some additional real-world applications where imaginary and complex numbers are
1. Signal Processing
In signal processing, complex numbers are used to represent and analyze sinusoidal signals. The Fourier transform, a powerful tool in signal analysis, converts time-domain signals into
frequency-domain representations using complex numbers. This allows engineers to study the behavior of signals across different frequencies and make adjustments in communications, audio processing,
and more.
2. Fractals and Chaos Theory
Complex numbers are central to the study of fractals, particularly the famous Mandelbrot set. Fractals are self-similar patterns that exhibit intricate structures at both large and small scales. The
Mandelbrot set is generated using complex numbers and iterative functions, revealing a beautiful and infinitely complex pattern that has fascinated mathematicians and artists alike – SQRD −1/2.
3. Electrical Impedance
In AC circuits, impedance is a measure of how much a circuit resists the flow of alternating current. Impedance is often represented as a complex number, with the real part representing resistance
and the imaginary part representing reactance. Complex numbers allow engineers to calculate impedance and optimize the design of electrical circuits.
4. Control Theory and Stability Analysis
In control theory, complex numbers are used to analyze the stability of dynamic systems. The roots of a system’s characteristic equation, known as poles, are often complex numbers. The location of
these poles on the complex plane helps engineers determine whether a system is stable, unstable, or marginally stable.
The square root of negative numbers, such as SQRD −1/2\text{SQRT}(-1/2)SQRT(−1/2), introduces us to the world of complex numbers and imaginary units. While the concept of imaginary numbers may
initially seem abstract, these mathematical tools have a wide range of practical applications in fields such as engineering, physics, and signal processing.
By understanding how to calculate −12\sqrt{-\frac{1}{2}}−21 and similar expressions, we gain insight into the behavior of complex systems and how they can be analyzed mathematically. Whether you’re
studying quantum mechanics, designing electrical circuits, or simply exploring the beauty of fractals, complex numbers offer a powerful way to model and solve complex problems.
1. What is the value of SQRD −1/2\text{SQRT}(-1/2)SQRT(−1/2)?
The square root of −12-\frac{1}{2}−21 is 22i\frac{\sqrt{2}}{2}i22i, which is approximately 0.707i0.707i0.707i, a purely imaginary number.
2. How do you calculate the square root of a negative fraction?
To calculate the square root of a negative fraction, factor out the negative sign using the imaginary unit iii, then simplify the square root of the positive fraction.
3. What is the significance of complex numbers in mathematics?
Complex numbers extend the real number system to allow for the square roots of negative numbers. They are crucial in various fields such as electrical engineering, quantum mechanics, and signal
4. What is a complex conjugate?
A complex conjugate is obtained by changing the sign of the imaginary component of a complex number. For example, the complex conjugate of 3+4i3 + 4i3+4i is 3−4i3 – 4i3−4i.
5. How are complex numbers used in engineering?
Complex numbers are used in engineering to analyze AC circuits, calculate impedance, model control systems, and study signal processing.
6. Can you graph complex numbers?
Yes, complex numbers can be graphed on the complex plane, where the horizontal axis represents real numbers and the vertical axis represents imaginary numbers.
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Seed, table's system | Studio Irvine
Seed, table’s system
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A cylindrical bucket 32cm high and with a radius of base, 18cm is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed, if the height of the conical heap is 24cm. Find the radius and slant height of the heap.
Hint: In order to solve this problem we need to compare the volume of sand in a cylindrical bucket with Volume of sand in a conical heap. Doing this will solve your problem and will give you the
right answer.
Complete step-by-step answer:
It is given to us that,
Height $\left( {{h_1}} \right)$ of cylindrical bucket= 32cm
Radius $\left( {{r_1}} \right)$ of circular end of bucket= 18cm
Height $\left( {{h_2}} \right)$ of conical heap= 24cm
Let the radius of the circular end of the conical heap be ${r_2}$ .
Therefore, the Volume of sand in the cylindrical bucket that will be equal to the volume of sand in the conical heap.
Now, Volume of sand in the cylindrical bucket = Volume of sand in conical heap
And hence on putting the values we have,
\Rightarrow \pi \times {r_1}^2 \times {h_1} = \dfrac{1}{3}\pi \times {r_2}^2 \times {h_2} \\
Now on putting the given values in the above equation, we have
$ \Rightarrow \pi \times {\left( {18} \right)^2} \times 32 = \dfrac{1}{3}\pi \times {\left( {{r_2}} \right)^2} \times 24$
And hence on doing the simplification, we have
\Rightarrow {\left( {{r_2}} \right)^2} = \dfrac{{3 \times {{18}^2} \times 32}}{{24}} = {18^2} \times 4 \\
{\text{And hence ,}} \\
\Rightarrow {{\text{r}}_2} = 18 \times 2 = 36cm \\
Now, Slant height=$\sqrt {{r_2}^2 + {h_2}^2} = \sqrt {{{12}^2}({3^2} + {2^2})} = 12\sqrt {13} \;cm$
Therefore the radius and the slant height of the conical heap is 36cm and $12\sqrt {13} \;cm$ respectively.
Note: This question is based on mensuration and hence first of all we have to compare the Volume of sand in the cylindrical bucket with Volume of sand in conical heap and with the help of that we can
find the value of radius of the circular end of the conical heap and later on slant height. Proceeding like this will take you towards the right answer.
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225 grams of Boiled Chickpeas in Milliliters • Recipe equivalences
225 grams of Boiled Chickpeas in Milliliters
How many Milliliters are 225 grams of Boiled Chickpeas?
225 grams of Boiled Chickpeas in Milliliters is approximately equal to 321 milliliters.
That is, if in a cooking recipe you need to know what the equivalent of 225 grams of Boiled Chickpeas measure in Ml, the exact equivalence would be 320.51282, so in rounded form it is approximately
321 milliliters.
Is this equivalence of 225 grams to Milliliters the same for other ingredients?
It should be noted that depending on the ingredient to be measured, the equivalence of Grams to Milliliters will be different. That is, the rule of equivalence of Grams of Boiled Chickpeas in Ml is
applicable only for this ingredient, for other cooking ingredients there are other rules of equivalence.
Please note that this website is merely informative and that its purpose is to try to inform about the approximate equivalent values to estimate the weight of the products that can be used in a
cooking recipe, such as Boiled Chickpeas, for example. In order to have an exact measurement, it is recommended to use a scale.
In the case of not having an accessible weighing scale and we need to know the equivalence of 225 grams of Boiled Chickpeas in Milliliters, a very approximate answer will be 321 milliliters.
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Product Substitution Effect: 2011 and 2009 Basket Updates
Waruna Wimalaratne
Statistics Canada, Consumer Prices Division
The Canadian Consumer Price Index (CPI) is a fixed-basket Laspeyres-type index for which the weights in the basket are periodically updated. On March 27, 2013, the 2009 weighting pattern was replaced
with one from 2011. This marked the first time in the Canadian CPI’s history that weights were updated at a two-year interval.
Due to its use of fixed weights, a Laspeyres-type index is typically subject to an upward bias brought on by product substitutions made by consumers. This arises in a fixed quantity index when
consumers change their purchasing behaviour in response to relative price changes. For example, if the price of chicken increases substantially between basket updates, consumers may opt away from
chicken and substitute other meats such as beef. In cases such as these, a fixed-quantity Laspeyres-type price index cannot correctly reflect this expenditure change until basket weights are updated.
This can lead to an overstatement of the importance of changes in the price of chicken in the index and, hence, an upward bias.
One aspect of the CPI Enhancement Initiative, a five-year project to improve the quality of the CPI, is to take into account, as rapidly as possible, changes in consumer behaviours and therefore
minimize the substitution effect. This is achieved by updating basket weights at more frequent two-year intervals, instead of at the four-year interval which has been the most recent practice.
A basket update provides the opportunity to measure the magnitude of the effect of product substitutions in consumer purchases.
A straightforward method of estimating this effect is to measure the difference between the Laspeyres and Fisher indexes^1. The Fisher index formula is the geometric mean of the base period-weighted
Laspeyres price index and the current period-weighted Paasche price index. This incorporates, in a symmetrical and balanced manner, weight information from both the beginning and ending periods for
which data on consumers’ purchasing patterns are available. This ensures a more representative reflection of spending patterns and effectively avoids the bias issue noted above.
It should be noted that, despite having this desirable feature, constructing a CPI using a Fisher (or any other symmetrically-weighted) index is not feasible in a timely monthly production
environment because of the long lags in obtaining current-period weights. However, it can be calculated retrospectively and with a lag when new weight information is obtained.
It follows that the Fisher index, once calculated, can be used as a reference or benchmark, representing the path the CPI would have taken had there been no substitution effects. The magnitude of
substitution effects over a given period can be measured as the difference between the fixed-weighted Laspeyres index and the symmetrically-weighted Fisher index.
The effect of product substitutions was estimated as part of the 2011 basket update by calculating the relevant indexes using weights from 2009 and 2011 at the published class level for Canada as a
whole. The results are shown in the following table, with those obtained from a comparison of 2005 and 2009 spending patterns for reference purposes:
Calculation of the Product Substitution Effect
All-items CPI: Laspeyres index value All-items CPI: Paasche index value All-items CPI: Fisher index value Implied annual upward substitution effect^1
2005 – 2009^2 106.42 104.72 105.57 0.20
2009 – 2011^3 104.82 104.51 104.66 0.07
1. The implied annual upward rate of substitution is measured by the difference between the Laspeyres and the Fisher indexes, expressed as the growth rate per annum. Using the table index values to
calculate the implied annual upward substitution effect may not equal the implied annual upward substitution figure in the table due to rounding.
2. Note that some adjustments were made to the 2009 basket in order to align with that from 2005, mainly due to the addition of two published classes in 2009.http://www23.statcan.gc.ca/imdb-bmdi/
3. In this analysis, the homeowners’ replacement cost component was excluded from both periods because it is the only product that is not an out-of-pocket expense, but rather is an imputed
expenditure value. Moreover, its price movement is imputed from the New Housing Price Index (NHPI), and therefore we should not expect any meaningful interaction between changes in prices and changes
in quantities.
The results show a product substitution effect of 0.07% per year between 2009 and 2011. This is significantly less than the average annual upward bias of 0.20% that was observed between the 2009 and
2005 baskets. This difference underlines the benefits of more frequent basket updates. The impact of using a less representative set of weights is minimized by its more frequent replacement.
A more detailed analysis of the component parts of the CPI revealed that the two largest contributors to the divergence between the Laspeyres and Paasche indexes were gasoline and women’s clothing.
These two products exhibited large price change and quantity shifts in the opposite direction. From 2009 to 2011, gasoline prices increased by 31.0%, while the quantities purchased decreased by 9.2%.
Gasoline is relatively inelastic, with a price elasticity of demand of -0.36 from 2009 to 2011. Over the same period, women’s clothing prices decreased by 7.7% whereas quantities increased by 24.8%,
with a price elasticity of -2.75.^2
1. White, Alan G. “Measurement Biases in Consumer Price Indexes.” International Statistical Review 67.3 (1999): 301-325.
2. Due to rounding, quality and seasonal adjustments, the published indexes may differ from internal data. From 2009 to 2011, the published CANSIM series indicates gasoline has increased 30.9% and
women’s clothing prices have decreased by 7.5%. The primary source of expenditures for the CPI is the Survey of Household Spending (SHS). The SHS collects expenditures (price*quantity). The
quantity change was derived by the following method: [1-((1/(P2009Q2009)(P2011Q2011)*(P2011/P2009))]. The price elasticity of demand was calculated using the midpoint method. By using average
prices and quantities one avoids the value of elasticity being dependent upon whether a price change reflects a price increase or decrease.
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American Mathematical Society
Integrals occurring in problems of molecular structure
HTML articles powered by AMS MathViewer
by A. Dalgarno PDF
Math. Comp. 8 (1954), 203-212 Request permission
J. Slater, “Atomic shielding constants,” Phys. Rev., v. 36, 1930, p. 57-64. Y. Sugiura, “Über die Eigenschaften des Wasserstoff Moleküls in Grundzustande,” Zeit. Phys., v. 45, 1927, p. 484-492.
C. Zener & V. Guillemin, “The $B$-state of the hydrogen molecule,” Phys. Rev., v. 34, 1929, p. 999-1009. J. H. Bartlett, “Orbital valency,” Phys. Rev., v. 37, 1931, p. 507-531. N. Rosen,
“Calculation of interactions between atoms with $s$-electrons,” Phys. Rev., v. 38, 1931, p. 255-276; “The normal state of the hydrogen molecule,” Phys. Rev., v. 38, 1931, p. 2099-2114. H. M.
James, “Wave-mechanical treatment of the ${\operatorname {Li} _2}$ molecule,” Jn. Chem. Phys., v. 2, 1934, p. 794-810. A. S. Coolidge, “A quantum mechanics treatment of the water molecule,” Phys.
Rev., v. 42, 1932, p. 189-209. E. W. Hobson, Spherical and Ellipsoidal Harmonics. Cambridge University Press, London, 1931. (a) A. L. Sklar, “The near ultraviolet absorption of substituted
benzenes,” Jn. Chem. Phys., v. 7, 1939, p. 984-993. (b) M. Goeppert Mayer & K. J. McCallum, “Calculation of the absorption spectrum of Wurster’s salts,” Rev. Mod. Phys., v. 14, 1942, p. 248-258.
(c) R. S. Mulliken, “Quelques aspects de la théorie des orbitals moléculaires,” Journ. Chim. Phys., v. 46, 1949, p. 497-542, 675-713. (d) J. F. Mulligan, “LCAO self-consistent field calculation
of the ground state of carbon dioxide,” Jn. Chem. Phys., v. 19, 1951, p. 347-362. (e) S. L. Altmann, “$\pi - \sigma$ electronic states in molecules I. The Hückel approximation,” Roy. Soc. London,
Proc., v. 210A, 1952, p. 327-343. (f) E. Scrocco & O. Salvetti, “Studio dell’ integrale di scambio e di ricerca di formule approximate per alcuni integrali che si incontrano nei problemi di
struttura molecolare,” La Ricerca Scientifica, v. 23, 1953, p. 98-107. (g) C. R. Mueller & H. Eyring, “The overlap average and central field approximations,” Jn. Chem. Phys., v. 19, 1951, p.
934-938. (h) H. Preuss, “Abschätzung für Zweizentren-integrale,” Zeit. f. Naturforschung, v. 8a, 1953, p. 270-272. (i) C. R. Mueller, “Semilocalised orbitals IV. Relationship of $n$-center
orbitals, two-center orbitals, and bond properties,” Jn. Chem. Phys., v. 20, p. 1600-1604. J. F. Mulligan—reference (12d). K. Rüdenberg—reference (2). Quantum-mechanical methods in valence theory
, Sept. 1951, Nat. Acad. Sci. and Office of Naval Research, New York.
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Additional Information
• © Copyright 1954 American Mathematical Society
• Journal: Math. Comp. 8 (1954), 203-212
• MSC: Primary 65.0X
• DOI: https://doi.org/10.1090/S0025-5718-1954-0064473-9
• MathSciNet review: 0064473
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Review and correction - Custom Scholars
Review and correction
Please review this answer and if any corrections needed please fix it
Accounting Report
Question 1: Incremental Cash Flows
A) Calculations:
1.1. Initial Investment
This is gotten by summing the costs as follows: Initial Investment=Cost of Bicycles
+Cost of Metal Shelves + Design and Printing
Substitute the values:
(2×€1,500) + €4,000 + €15,000 + €3,200
Collect the like terms together:
= €25,200
1.2. Annual Cash Flows
Sales Revenue−Cost of Goods Sold (COGS)−Delivery Guys’ Salaries−Net Working Capital Inv
Year 1
(750× €20 × (1−0.75)) − (750 × €20 × 0.75) − €16,000 − (0.12 × Year 1 Sales)
= €5,350
Year 2
(1,000× €20 × (1−0.75)) −(1,000 × €20 × 0.75) − €16,000 − (0.12 × Year 2 Sales)
= €8,900
Year 3
(1,200× €20× (1−0.75)) − (1,200× €20×0.75) − €16,000− (0.12×Year 3 Sales)
= €12,450
1.3. Salvage Value
Gotten by:
=Salvage value of Bicycles + Salvage value of Metal Shelves
Substitute the values:
= (2 × €400) + 0
1.4. Net Cash Flow:
Annual Cash Flows + Salvage Value
Year 1
€5,350 −€25,200 + €800
Year 2
€8,900 − €16,000 + €800
Year 3
€12,450 − €16,000 + €800
= −€2,750
B) Paragraph of Justification:
All cash flows identified are incremental to the project. The initial investment and annual cash
flows are directly associated with the implementation and operation of the delivery service. The
salvage value represents the recovery of some initial investment at the end of the project. Each
cash flow is crucial in assessing the project’s financial viability.
Question 2: NPV, IRR, and Payback Period
A) Calculations:
2.1. NPV:
NPV = ∑ (Net Cash Flow (1 + Cost of Capital)
NPV = −€19,050/ (1+Cost of Capital) + (−€6,300/ (1+Cost of Capital)2 + (−€2,750/
(1+Cost of Capital) ^3 + €800/ (1+Cost of Capital) ^3
2.2. IRR:
where NPV equals zero
2.3. Payback Period:
Cumulative Cash Flow1=−€19,050
Cumulative Cash Flow2=Cumulative Cash Flow1+(−€6,300)
Cumulative Cash Flow3 = Cumulative Cash Flow2 +(−€2,750 + €800)
= −€27,300
Now, let’s determine the Payback Period:
Payback Period=Number of years before cumulative cash inflows cover the initial investment
The initial investment is covered sometime during Year 2. To find the exact Payback Period:
Payback Period = 1+(Remaining InvestmentCash Flow at the Start of Year 3)
Remaining Investment =−(Cumulative Cash Flow2)
Payback Period=1+ (−(−€25,350) −€2,750)
Payback Period≈1+9.21818
Payback Period≈10.22
B) Paragraph of Justification/Thought Process:
The NPV is a key indicator of the project’s profitability, with a positive value indicating a viable
investment. A higher IRR signifies a better return on investment, and a shorter payback period
implies quicker recovery of the initial investment. In this case, Manuel should consider the NPV,
IRR, and payback period collectively to make an informed decision.
Question 3: Cost of Capital
A) Calculation:
3.1. Weighted Average Cost of Capital (WACC):
WACC = (Cost of Debt × Weight of Debt) + (Cost of Equity × Weight of Equity)
Cost of Debt = 7%
Weight of Debt=0.35
Cost of Equity = Risk-Free Rate + (Beta × Market Risk Premium)
Risk-Free Rate = 2.746% (3-year bonos yield)
Market Risk Premium=5.8%
Beta= 1.3
Cost of Equity = 2.746% + (1.3 × 5.8%)
WACC = (Cost of Debt × Weight of Debt) + (Cost of Equity × Weight of Equity)
WACC= (0.07× 0.35) + (0.10526 × 0.65)
WACC= 0.0245 +0.0682
B) Justification
The WACC represents the overall cost of capital, considering both debt and equity. By using the
current market yields for debt and incorporating the risk premium and beta for equity, the
WACC provides an accurate estimate of the cost of capital for the project. Manuel can use this
WACC as a reliable benchmark for evaluating the project’s financial feasibility.
| Year | Net Cash Flow (€) |
| -19,050
| 5,350
| 8,900
| 12,450
Bicycle price in €
Metal Shelves
Total initial costs
Employee salary
Net working capital rate
Bicycle selling price each
Initial investment
Books selling rate
tax rate
interest rate
total cost
Costs (€)
NPV, IRR, and Cash payment period Calculations
Cash flow
cumulative cash flow
Cash payment period
monthly Books sold
Delivery system
Annual books sold
Sales price (€)
Annual Books sold instore
sales price (€)
Annual revenue (€)
Total annual revenue
total annual costs
Net Working Capital Investment
Cost of Capital
Cost of Debt
Weight of Debt
Risk free rate
Market Risk Premium
Cost of Equity
Weight of Equity
Annual revenue (€)
Costs of books sold
Net cash flows
Assignment A3.1 – Making a Business Decision
Assignment Description
In this final assignment, you are asked to answer several questions regarding a potential
business idea (described below). Apply all that you have learnt throughout the course to
answer the questions – and good luck!
A small bookstore on Gran Via, one of the main arteries through Madrid and a famous
shopping street, has come up with the idea of delivering books to visitors to Madrid who
stay in local hotels. Given the central location of the bookstore, the books would be
delivered to tourists staying in nearby hotels within less than 30 minutes.
The idea for the delivery service originally came from five MBA students at IE Business
School who conducted a market study into the potential need for such a service. From
person experience, the students knew that it is easy to forget one’s current book at
home while travelling. The five MBA students spent last summer interviewing local
tourists about the desirability of such a service. Manuel Munoz, the bookstore owner,
sponsored this market study with €1,250. The market study also revealed that Amazon
were planning to set up a similar delivery service in central Madrid. Due to various
administrative delays, Amazon would only be able to start its delivery service in exactly 3
years from now. It would then immediately price Manuel’s bookstore out of the market.
The bookstore would need to buy two electrical bicycles for the delivery guys at a price
of €1,500 each. The two bicycles would be fully depreciated in a straight line over the
three years. They could be sold for €400 each at the end of the project. The bookstore
would also need to use an empty room at the back of the store as a storeroom. If the
bookstore were not to go ahead with the delivery service, the empty room could be
rented out at an annual rent of €2,000. The storage room would require heavy-duty
metal shelves at a total cost of €4,000. These metal shelves would have to be fixed to
the walls and would therefore be difficult to remove once fixed. Hence, Manuel assumes
that the shelves will not have a resale value. He also assumes that they will not be
If going ahead, Manuel wants to advertise the new delivery service by distributing
leaflets to nearby hotels. He has been quoted a fee of €15,000 for the design of various
promotional items and another €3,200 for the printing of sufficiently enough leaflets and
other promotional items (such as free keyholders) to last for the next three years.
The two delivery guys needed for the delivery service would be paid €16,000 p.a.
However, Manuel expects to relocate an existing employee who already works for the
bookstore to the delivery service. This employee currently earns €15,000 p.a.
The new delivery service would generate sales of 750 books, 1,000 books and 1,200
books each month during the first year, the second year and the third year, respectively.
The average sales price of the books would be €20. The cost of the books sold is 75% of
Omantel – Concealed
their sales price to the customers. Manuel expects that the number of books sold instore
would drop by 25% of the number of books sold via the delivery service. The average
sales price per book for the eroded sales is €15 and the cost of the books sold is 80%.
The bookstore pays 30% of tax on its profits.
Manuel is somewhat confused about his cost of capital. He knows that he should not
rely on the bookstore’s historic cost of debt. He therefore phoned up his bank manager
who quoted him an interest rate of 7% if he were to take out a loan today. Following
some extensive research on the internet, he is pretty sure that the equity beta of his
bookstore is roughly 1.3. He asked one of his regular customers, a finance professor at
IE, what risk premium he should use for the market premium. The professor suggested to
use a market risk premium of 5.8% p.a.
However, Manuel does not know what rate to use for the risk-free rate. He looked up
the yields on various debt securities issued by the Spanish government and he found the
Debt security
Current yield
3-month bills
1-year bills
3-year bonos
5-year bonos
10-year bonos
Note: The Spanish word for bond is “bono”.
The bookstore is currently financed by 35% of debt and 65% of equity. The new project
is expected to require the same capital structure or mix of debt and equity as the entire
Net working capital of 12% of the annual sales revenue would need to be in place at the
start of each year. The net working capital would be fully recovered at the end of the
Omantel – Concealed
Manuel does not feel very confident about estimating his cost of capital. Would you be
able to help him? What value would you suggest for the cost of capital? Please answer
each of the following questions:
1. Determine the incremental cash flows for this project. Clearly explain why a given
cash flow is incremental or not.
2. Compute the NPV, IRR and payback period for the project. Should Manuel go ahead
with this project?
3. Manuel does not feel very confident about his estimated cost of capital. Would you
be able to reassure him? If yes, how?
Assignment Structure
Your assignment will be an Excel spreadsheet cleary showing your answers to the
questions below (three main questions, with each question subdivided into parts).
For Question 1 (25% of your grade):
A. Determine the incremental cash flows for this project – calculate each.
B. Clearly explain why a given cash flow is incremental or not – write your thought
process/justification in another cell (to the right of your calculations for each selected
cash flow).
For Question 2 (50% of your grade):
A. Compute the NPV, IRR and payback period for the project – calculate each.
B. Should Manuel go ahead with this project? Write your recommendation (paragraph,
using a cell in the Excel sheet and clearly label it as your answer to Question 2B) with
your explanation for what your suggestion is for Manuel as far as going ahead with
the project.
For Question 3 (25% of your grade):
A. Manuel does not feel very confident about his estimated cost of capital – calculate
the cost of capital (show your calculation).
B. Would you be able to help him? What would be your suggested cost of capital? Write
your recommendation (paragraph, using a cell in the Excel sheet and clearly label it as
your answer to Question 3B) explaining your thought process.
Assignment Format
Your assignment should be handed in as an Excel sheet containing your calculations and
answers to the questions (as listed above):
Omantel – Concealed
Question 1: A) calculations; B) paragraph of justification for each cash flow selected.
Question 2: A) calculation of NPV, IRR and payback period for the project; B)
paragraph of justification/thought process.
Question 3: A) calculation of cost of capital; B) paragraph of justification/thought
Your assignment will be graded as follows:
Correct identification of each
incremental cash flow and providing
a thorough explanation for each on
why it is or is not an incremental
cash flow
25 possible points
COMPUTING NPV, IRR, AND
Correctly computing each value and
providing an appropriate
recommendation for Manuel on
whether to accept the project or not
50 possible points
Correctly calculating the cost of
capital and providing a thorough
explanation of why the cost of capital
is appropriate for the project.
25 possible points
Considerations for Completing this Assignment
Omantel – Concealed
This is individual work.
Your assignment should be submitted in Excel format. Please make sure you include
both your calculations and the text explanation within the Excel sheet, clearly labeling
your text explanations as “Question 1 Response,” “Question 2b Response,” etc. (where
This assessment will be graded by your Learning Facilitator and will count as 60% of
your grade.
Use the rubric in the Grading section (above) as a guide to understand how you will be
Please label your assignment with the following format: as CXXFinanceAXXLastNameFirstName.
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MathWorks Logo, Part One. Why Is It L Shaped?
MathWorks is the only company in the world whose logo satisfies a partial differential equation. Why is the region for this equation shaped like a capital letter L?
Wave Equation
The wave equation describes how a disturbance travels through matter. If the units are chosen so that the wave propagation speed is equal to one, the amplitude of a wave satisfies $$ {{\partial^2 u}
\over {\partial t^2}} = \triangle u $$ The $\triangle$ denotes Laplace's operator $$ \triangle = {\partial^2 \over {\partial x^2}} + {\partial^2 \over {\partial y^2}} $$
Initial and Boundary Values
Geometry plays a crucial role here. Initial values of the amplitude and velocity of the wave are prescribed on a certain region. Values of the amplitude or its normal derivative are also prescribed
on the boundary of the region. If the wave vanishes outside the region, these boundary values are zero.
Eigenvalues and Eigenfunctions
Separating out periodic time behavior leads to solutions of the form $$ u(x,y,t) = \cos{(\sqrt{\lambda}\,t)} v(x,y) $$ The functions $v(x,y)$ also depend upon $\lambda$ and the region. They satisfy
the differential equation $$ \triangle v + \lambda v = 0 $$ and are zero on the boundary of the region. The quantities $\lambda$ that lead to nonzero solutions are the
, and the corresponding functions $v(x,y)$ are the
. They are determined by the physical properties of the medium and the geometry of the region. The square roots of the eigenvalues are resonant frequencies. A periodic external driving force at one
of these frequencies generates an unboundedly strong response in the medium. Any solution of the wave equation can be expressed as a linear combination of these eigenfunctions. The coefficients in
the linear combination are obtained from the initial conditions.
One Dimension
In one dimension, the eigenvalues and eigenfunctions are easily determined. The simplest example is a violin string, held fixed at the ends of an interval. The eigenfunctions are trig functions. $$
v_k(x) = \sin{(k x)} $$ If the length of the interval is $\pi$, the eigenvalues are determined by the boundary condition, $v_k(k \pi) = 0$. Hence, $k$ must be an integer and $$ \lambda_k = k^2 $$ If
the initial condition is expanded in a Fourier sine series, $$ u(x,0) = \sum_k a_k \sin{(k x)} $$ (And the initial velocity is zero), then the solution to the wave equation is $$ u(x,t) = \sum_k a_k
\cos{(\sqrt{\lambda_k}\,t)} v_k(x) $$ Here are graphs of the first nine eigenfunctions in one dimension. The corresponding eigenvalues are the squares of integers.
eigenvals = (1:9).^2
eigenvals =
A Square
The simplest region in two dimensions is a square. The eigenfunctions are again trig functions. $$ v_{k,j}(x,y) = \sin{(k x)}\,\sin{(j y)} $$ If the sides have length $\pi$, the boundary conditions
imply that $k$ and $j$ must be integers. Here are the first nine eigenvalues and eigenfunctions.
[k,j] = meshgrid(1:3); e = k.^2+j.^2; eigenvals = e(:)'
eigenvals =
A Circular Disc
If the region is a circular disc, we switch to polar coordinates, $r$ and $\theta$. Trig functions are replaced by Bessel functions. The eigenfunctions become $$ v_{k,j}(r,\theta) = B_j(\mu_k r) \,\
sin{(j \theta)} $$ where $B_j$ is the $j$ -th order Bessel function and $\mu_k = \sqrt{\lambda_k}$. To find the eigenvalues we need to have the eigenfunctions vanish on the boundary of the disc. If
the radius is one, we require $$ B_j(\mu_k) = 0 $$ In other words, we need to compute zeros of Bessel functions. Here are the first nine eigenvalues and eigenfunctions of the circular disc. The
violin string has become a tambourine.
eigenvals = [bjzeros(0,3) bjzeros(1,3) bjzeros(2,3)].^2
eigenvals =
Columns 1 through 7
5.7832 30.4713 74.8870 14.6820 49.2185 103.4995 26.3746
Columns 8 through 9
70.8500 135.0207
A Circular Sector
Replace the full circular disc by a three-quarter circular sector. The angle at the origin is $270^\circ$ or $\frac{3}{2}\pi$ radians. We can make our eigenfunctions adapt to this angle. Take $$ v_
{k,j}(r,\theta) = B_{\alpha_j}(\mu_k r) \,\sin{(\alpha_j \theta)} $$ with $$ \alpha_j = \frac{2}{3} j $$ and fractional order Bessel functions. The eigenfunctions satisfy their differential equation
and also satisfy the boundary conditions on both sides of the angle. $$ v_{k,j}(r,\theta) = 0 $$ at $\theta = 0$ and at $\theta = \frac{3}{2}\pi$. By finding the zeros of the Bessel functions we can
also have the eigenfunctions satisfy the boundary conditions on the outer circular portion of the boundary. Here are the first nine eigenvalues and eigenfunctions of the three-quarter circular
eigenvals = [bjzeros(2/3,3) bjzeros(4/3,3) bjzeros(6/3,3)].^2
eigenvals =
Columns 1 through 7
11.3947 42.6442 93.6362 18.2785 56.1131 113.6860 26.3746
Columns 8 through 9
70.8500 135.0207
The L-shaped Membrane
For all the regions we have discussed so far it is possible to express the eigenvalues as zeros of analytic functions. For the interval and the square, the eigenvalues are integers, which are the
zeros of $\sin{\pi x}$. For the circular disc and sector, the eigenvalues are zeros of Bessel functions. Once we have the eigenvalues, it is easy to compute the eigenfunctions using sines and Bessel
functions. So, an L-shaped region formed from three unit squares is interesting for at least two reasons. It is the simplest geometry for which solutions to the wave equation cannot be expressed
analytically; numerical computation is necessary. Furthermore, the 270 degree nonconvex corner causes a singularity in the solution. Mathematically, the gradient of the first eigenfunction is
unbounded near the corner. Physically, a membrane stretched over such a region would rip at the corner. This singularity limits the accuracy of finite difference methods with uniform grids. I used
the L-shaped region as the primary example in my doctoral thesis fifty years ago. MathWorks has adopted a modified surface plot of the first eigenfunction as the company logo. I am going to devote a
series of blog posts to the L. Here are the first nine eigenvalues and eigenfunctions, computed by a function from
Numerical Computing with MATLAB
, which I will discuss in a future posting. Compare these eigenfunctions with the ones from the circular sector, which has the same reentrant corner and resulting singularity.
for k = 1:9
[~,eigenvals(k)] = membranetx(k);
eigenvals =
Columns 1 through 7
9.6397 15.1973 19.7392 29.5215 31.9126 41.4745 44.9485
Columns 8 through 9
49.3480 49.3480
Microwave Waveguide
Simple model problems involving waves on an L-shaped region include an L-shaped membrane, or L-shaped tambourine, and a beach towel blowing in the wind, constrained by a picnic basket on one fourth
of the towel. A more practical example involves ridged microwave waveguides. One such device is a waveguide-to-coax adapter. The active region is the channel with the H-shaped cross section visible
at the end of the adapter. The ridges increase the bandwidth of the guide at the expense of higher attenuation and lower power-handling capability. Symmetry of the H about the dotted lines shown in
the contour plot of the electric field implies that only one quarter of the domain needs to be considered and that the resulting geometry is our L-shaped region. The boundary conditions are different
than our membrane problem, but the differential equation and the solution techniques are the same. The photo is courtesy of Advanced Technical Materials, Inc. See their website,
>, for lots of devices like this.
Published with MATLAB® R2014b
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English Deutsch
Info Letter No. 46 December 1997
Copyright by HEXAGON Software 1997
WL1+ Quick View
The life expectancy of roller bearings has been added to the WL1+'s Quick Output.
WL1+ Safety Margin
Until now WL1+ has calculated the safety margins in relation to alternating load due to bending, tension/pressure and torsion. In the new version you can choose whether torque is alternating,
pulsating or resting.
WL1+ calculates the reference stress in relation to Sigma bw. The result is one value for the safety margin: S = Sigma bw / Sigma v.
WL1+ Load Type
You can select whether the load should be alternating, pulsating or static for torque and axial forces. Bending stress (from shearing force, path load and bending moments) are always alternating due
to the rotating shaft, even when the load is purely static. The load frequency is equal to the rotational speed of the shaft. With the help of the reference stress, the general state of stress is
converted to a plane stress. The safety margin is calculated from:
S = sigma bw / Sigma v
Dependent on the strength theory:
Sigma v = SQRT ( sigma² + ( alpha0 * phi * tau )²)
The different stress states can be taken into account with the
correction value alpha0:
Sigma bw
alpha0= ------------
phi * tau
Whereby when tau occurs with alternating torque tau w is used;
for pulsating torque tau sch is used; and for static load tauf F
is used. phi is dependent upon the strength theory (1, 2 or
Example: A shaft made of St50 is stressed by the bending moment
Mb and the torque Mt. How great is the correction value ã0 in
accordance with the shear strain energy theory when:
a) bending occurs alternating, torsion resting;
b) both occur alternating;
c) bending occurs resting, torsion occurs alternating?
The limit stress can be taken from the WST1 material data base:
sigmabF=450N/mm² sigmabW=250N/mm²
tauF=180N/mm² tauW=150N/mm².
With phi=SQRT3
a) alpha0 = sigmabW/(phi tauF) = 250 / ( SQRT3 * 180 ) = 0.8
b) alpha0 = sigmabW/(phi tauW) = 250 / ( SQRT3 * 150 ) = 0.96 ÷ 1
c) alpha0 = sigmabW/(phi tauF) = 450 / ( SQRT3 * 150 ) = 1.7
The correction value ã0 is ascertained by WL1+ when a material is selected from the data base and the strength values are known.
WL1+ Tension and Bending Stress
To allow for overloading of tension stress onto the reference stress, the tension stress on the bending-alternating stress portion of the reference stress is converted as follows:
sigmaz alternating: Sigma vb = Sigma b + Sigma z * Sigma bw / Sigma w
sigmaz pulsating: Sigma vb = Sigma b + Sigma z * Sigma bw / Sigma sch
sigmaz resting: Sigma vb = Sigma b + Sigma z * Sigma bw / Re
WL1+ Crit.Rotat.Speed for Triple Bearing
Now it is also possible to calculate the resonance speed for bending vibrations for triple bearing shafts. Since this drastically increases calculation time, it is possible to switch off the
calculation for bending critical rotational speed, under "Edit->Calculation". The rotational speed for torsional fatigue is independent of the number of bearing positions and can therefore always be
FED8 - Spring Energy
A diagram showing the curve for spring energy is now included in the FED8 software for calculation of torsion bars. For alternatingly loaded torsion bars, the displacement angle goes from -alpha to
+alpha. The maximum spring operation from -ãn to +ãn is then Mn times ãn (Index n = usable displacement angle for shearing stress tau n = tau perm).
FED2 - Index kn
For the extension spring calculation, the Index n is output as usable spring path for static load, for which the shearing stress tau n = tau zul. For dynamic load, the corrected shearing stresses
(tau k = tau * k) are used for tau k1 and tau k2. The corrected shearing stresses tau k1 and tau k2 are used for positions 1 and 2 in the printout, for Positions 0 and n the values tau 0 and tau n
are used. In the new version, tau kn = tau n * k are also displayed, this makes it clearer why tau k2 can be greater than tau n.
Spring length travel mm Spring force N tau N/mm² S
L0= 27.47ñ0.8 F0= 8.47 tau0 = 86
L1= 39.98 s1= 12.51 F1= 45.62ñ5.38 tauk1= 559 1.39
sh= 10.00 Fh= 29.69 taukh= 364 1.09
L2= 49.98 s2= 22.51 F2= 75.31ñ5.83 tauk2= 923 0.84
Lkn= 45.96 skn= 18.49 Fkn= 63.38 taukn= 777 1.00
Ln= 50.41 sn= 22.94 Fn= 76.58 taun = 777 1.00
(S = tau zul. / tau y)
Sigma perm./ Sigma q2 = 0.68 Sigma hperm./ Sigma bh = 0.67
FED2 - Pre-Stress Force
When only one spring length and matching spring force is available, then F1=0, L1=L0 and sh=s2 are entered for the pressure spring calculation. In the tension spring calculation this is different
because the pre-stress force with which the coils lie over one another must first be overcome. This means that during dimensioning and pre-dimensioning the pre-stress force F0, must be entered for F1
and not F1=0.
10 Year Anniversary for HEXAGON Software
Although HEXAGON GmbH wasn't founded until 1990, we had already developed our Tolerance Program TOL1 in 1987. Parallel to the Further Informatic Degree, our Engineer Office Ruoss, was already writing
software for mechanical engineering designers for use on 8086-PCs with 4,77 MHz and 256 kB memory, with 360 kB disk drives without hard drives. At the time this was highly advanced compared to the
previous computers such as the Sinclair ZX81 with only 1 kB of memory! The development tool Turbo Pascal 3.0 was contained in a 40 kB file (TURBO.COM) which included the compiler and program editor.
This meant that a 360 kB disk included the development tool, the source code and finished program! In comparison: Borland Delphi 3.0, the successor to Turbo Pascal requires 100 MB memory on the hard
drive, this is an increase by a factor 2500! After our Tolerance Program, TOL1 followed the toothing program ZAR1 and the pressure spring program FED1. The highlight in our programs was the
introduction of graphic representation of the tooth form in ZAR1, and the spring drawing in FED1, later followed the feature for output to DXF or IGES files via the CAD interface. At the time, a CAD
interface for calculation programs was something new, and was only available with HEXAGON software. At the time, most users stated that they rarely used the CAD interface, today, this is a standard
part of any calculation program. For data bases with material values and DIN parts, the "old" DBF format from dBase 3+ is used. This means that the files can be editted and worked on with any data
base or spreadsheet program (dBase, Access, FoxPro, Excel, Quattro Pro, StarOffice). After our Tolerance program, Toothing program and Pressure Spring program followed both of the graphic conversion
programs, HPGL Manager and DXF Manager. Both of which were and still are useful for internal program development (generation of source code for DXF drawings). Then followed the programs for tension
and spiral spring calculation, shaft calculation, bolted joints, pressed fits and shaft-hub joints. Along side this, we continually improved and updated our user interfaces.
Since 1992 we have been able to offer a UNIX version of the HPGL Manager and ZARXE, for DEC and SUN work stations. Although Unix version were sought after in 1990-92, this has now dropped off so that
no further development in the Unix area is carried on. Much more successful has been the conversion to Windows. The first HEXAGON Windows version was the HPGL Manager, available in June 1993. All
programs were converted after this to Windows. Today, 95% of our orders are for Windows versions. In 1997 we brought out new programs, FED8 for calculation of torsion bars, and TR1 for girder
calculation. New and future oriented in all of our programs is the feature, as an alternative to printing out on a printer, the possibility for creating HTML files which can be read by Netscape or
Internet Explorer.
WN2 - Dedendum
The tooth dedendum, hfP, or tool addendum haP0, wer until now set a 0.55*m. According to DIN 5480 this is the factor for broach toothing manufacture. However, a factor of 0.6*m is recommended for
hobbing manufacture, and a factor of 0.65*m is recommended for wälzstossen. With the new WN2 version you can enter the dedendum factor freely. I would like to thank Mr Canovas of Moog, Böblingen,
Germany for pointing this out.
FED1+ Quick View
The output for the manufacturing adjustment has been moved down one line. This avoids the text in the second column being written over.
FED3+ Clamping Angle
With twists springs, the clamping angle delta is in the opposite direction to the spring angle alpah. This is often misunderstood, and was displayed falsely in the auxiliary picture for the
application example.
Analogue to pressure spring with L = L0 - s applies to the torsion spring delta = delta0 - alpha
FED3+ Clamping
The program allows you so choose between fixed clamping and supported spiral. For supported spiral, the lever arm for the center axis must also be entered. The program then calculates the deflection
of the spiral and the resulting additional angle displacement ß. For fixed clamped spirals the spring is held directly on the outer diameter. When the spring ends are bent axially or inwards, you
should also select "fixed clamping". A new auxiliary picture is provided for clamping.
Windows Versions for NT
All Windows version since 1996 can be installed without problem under Windows NT. The only limitation is that you can still only use the 8.3 format for file names. This means 8 characters for the
file name, and three characters for the extension.
Don't use blanks in file names!
Please don't use blanks in file names. Although Windows 95/NT and old DOS versions allow this. This can result in your not being able to load files when changing to a newer version. If this has
already happened, install the old DOS version again, load all your files with blanks in the file name and rename and save them under new names.
Seminar Schedule 1998
We are offering the following seminars in 1998: toothing and gearing calculation, spring calculation, working design calculation and DIN ISO 9001 certification. If you are interested in taking part
in one of our seminars, please contact us to obtain our Seminar Schedule. We now offer a seminar on "Vibration Activation and Noise Behaviour in Gears". This seminar, of ever increasing importance
for minimizing gear noise, is held by Dr. Müller.
Seminar: Vibration and Noise Behaviour in Gears
This seminar is a sensible follow-on, providing the newest information about noise and toothing, and noise behaviour in gears, and provides important information on preventing noise development
during gearing pre-dimensioning. The topics covered are listed below:
Basic knowledge in vibration activation and vibration behaviour
Dynamic behaviour of toothing gears; determination of datum rotational speed N, as well as dynamic factor Kv.
Vibration behaviour of the whole gear: Influence of the whole system "gear" including drive and work machines, also accounting for shaft bending.
Measures for influencing the vibration and noise activation by spur gears; toothing geometry, toothing deviation and toothing adjustment, tooth flank play, etc.
Influence parameters for bevel gears
Special occurence of vibration: Getriebeasseln
Relationship between noise activation and noise radiation of gears.
Measures for influencing solid borne noise conduction and noise radiation: Housing design.
Selected question about measuring technology
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Extended Hermite-Hadamard $(H-H)$ and Fejer's inequalities based on $(h_1,h_2,s)$-convex functions
[1] M.U. Awan, Some new classes of convex functions and inequalities, Miskolc Math. Notes 19(1) (2018) 77–94.
[2] G. Cristescu, L. Lupsa and L. Lupsa, Non-connected convexities and applications, Springer Science & Business Media, 2002.
[3] S.S. Dragomir and R. Agarwal, Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett. 11(5) (1998) 91–95.
[4] S.S. Dragomir, J. Pecaric and L.E. Persson, Some inequalities of Hadamard type, Soochow J. Math. 21(3) (1995) 335–341.
[5] M.T. Hakiki and A. Wibowo, Hermite-Hadamard-Fej`er type inequalities for s-convex functions in the second sense via Riemann-Liouville fractional integral, J. Phys. Conf. Ser. 1442(1) (2020)
[6] C.Y. Jung, G. Farid, H. Yasmeen, Y.P. Lv and J. Pecaric, Refinements of some fractional integral inequalities for refined (α, h-m)-convex function, Adv. Diff. Equ. 2021(1) (2021) 1–18.
[7] U.S. Kirmaci, Inequalities for differentiable mappings and applications to special means of real numbers and to midpoint formula, Appl. Math. Comput. 147(1) (2004) 137–146.
[8] P. Liu, M.B. Khan, M.A. Noor and K.I. Noor, New Hermite–Hadamard and Jensen inequalities for log-s-convex fuzzy-interval-valued functions in the second sense, Complex Intell. Syst. (2021) 1–15.
[9] N. Mehreen and M. Anwar, Hermite-Hadamard type inequalities via exponentially (p, h)-convex functions, IEEE Access 8 (2020) 37589–37595.
[10] M.V. Mihai, M.A. Noor, K.I. Noor and M.U. Awan, Some integral inequalities for harmonic h-convex functions involving hypergeometric functions, Appl. Math. Comput. 252 (2015) 257–262.
[11] M.A. Noor, K.I. Noor and M.U. Awan, Integral inequalities for some new classes of convex functions, Amer. J. Appl. Math. 3(3–1) (2015) 1–5.
[12] M.A. Noor, K.I. Noor and M.U. Awan, Generalized convexity and integral inequalities, Appl. Math. Info. Sci. 9(1) (2015).
[13] M.A. Noor, K.I. Noor and M.U. Awan, Integral inequalities for coordinated harmonically convex functions, Complex Variab. Ellipt. Equ. 60(6) (2015) 776–786.
[14] M.A. Noor, K.I. Noor, M.U. Awan and S. Costache, Some integral inequalities for harmonically h-convex functions, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 77(1) (2015) 5–16.
[15] M.Z. Sarikaya, E. Set, H. Yaldiz and N. Ba¸sak, Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities, Math. Comput. Model. 57(9-10) (2013) 2403–2407.
[16] S. Varosanec, On h-convexity, J. Math. Anal. Appl. 326(1) (2007) 303–311.
[17] S. Yasin, M. Misiran and Z. Omar, Hermite-Hadamard and Fejer’s type inequalities for product of different convex functions, Design Engin. 2021(7) (2021) 5550–5560.
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(zk-learning) Deriving functional commitment families
I just finished the second lecture of the Zero Knowledge Proofs MOOC titled Overview of Modern SNARK Constructions (yes I know, I’m two lectures behind schedule. Don’t remind me!!!).
In the lecture Dr. Boneh introduces four important functional commitment families used for building SNARKs
1. A univariate polynomial of at most degree $d$ such that we can open the committed polynomial at a point $x$.
2. A Multilinear polynomial in $d$ variables such that we can open the committed polynomial at a point $x_1, …, x_d$.
3. A vector $\vec{u}$ of size $d$ such that we can open the committed vector at element $u_i$.
4. An inner products on vector $\vec{u}$ of size $d$ such that we can open the committed vector at the inner product of $\vec{u}$ and an input vector $\vec{v}$.
He mentions in passing that any one of these four function families can be built from any of the other four, but leaves it at that. I was left wondering… how?
Like a lot of things professors say, the details are left as an exercise to the reader ;). In this article I’d like to present a special case of that statement: given multlinear polynomials we can
use it to build commitments for any of the other three families.
One by one let’s see how that’s done.
Univariate Polynomials
Recall the definition of a univariate polynomial commitment. We commit to a polynomial up to degree $d$ in a single variable $x$, and want to open it at arbitrary points.
One way to achieve this with a multilinear polynomial in $k$ variables is to “reduce” it to a single variable, and then map its coefficients to the coefficients in the unvariate polynomial.
Here’s what I mean.
The univariate polynomial is of the form
\[f(x) = a_0 + a_1x + ... \> + a_dx^d\]
Now I’m going to be clever and write the multilinear polynomial as follows, collecting each sum of terms with the same number of variables under a single coefficient
\[F(X) = b_0 + b_1(x_0 + ... \> + x_d) + b_2(x_0x_1 + ... \> + x_{d-1}x_d) + ... \> + b_dx_1...x_d\]
What if we substitute $x$ for every $x_i$ in the input vector $X$?
\[F(\begin{bmatrix}x_0 = x\\...\\x_d=x\end{bmatrix}) = b_0 + b_1(x + ... \> + x) + b_2(x + ... \> x) + ... + \> b_dx \\ = b_0 + \binom{d}{1}b_1x + \binom{d}{2}b_2x^2 + ... \> + \binom{d}{d}b_dx^2\]
We’ve reduced it to a polynomial in a single variable! Now we can map the coefficients of $F$ to the coefficients of $f$
\[a_0 = b_0 \\ a_1 = \binom{d}{1}b_1 \Rightarrow b_1 = \frac{a_1}{\binom{d}{1}} \\ ...\]
The general formula is
\[b_i = \frac{a_i}{\binom{d}{i}}\]
Thus, committting $F$ with coefficients $b_i = \frac{a_i}{\binom{d}{i}}$ and opening it at $X = [x,…,x]$ is just like committing $f$ with coefficients $a_0, …, a_d$ and opening it at $x$.
This one is easier than the previous one. In a vector commitment we commit to a vector $\vec{u}$ and open it at one of its elements: $f_{\vec{u}}(i) = u_i$.
We can implement this with a simple multivariate polynomial of the form
\[F(X) = u_1x_1 + ... + u_dx_d\]
For input $i$ to $f_{\vec{u}}$, we evaulate $F(X)$ such that $x_i = 1$ and $x_j = 0, j \neq i$ for every element in $X$.
\[F(X) = u_1x_1 + ... + u_dx_d \Rightarrow \\ F(X) = 0u_1 + ... + 1u_i + ... + 0u_d \Rightarrow \\ F(X) = u_i\]
Thus, committing to
\[F(X) = \sum_{i=1}^{d}{u_ix_i}\]
and opening it at
\[X = \begin{bmatrix}x_0 = 0\\...\\x_i = 1\\..\\x_d=0\end{bmatrix}\]
is equivalent to committing to $f_{\vec{u}}$ and opening at $i$.
Inner Products
In an inner product commitment we commit to a vector $\vec{u}$ and open it by evaluating its dot product with an input vector $\vec{v}: f_{\vec{u}}(\vec{v}) = \vec{u}\cdot\vec{v}$.
This one is very similar to the vector committment. The multivariate polynomial stays the same. But instead of evaluating it at a point where all but one $x_i$ is non-zero, we substitute $v_i = x_i$
for all $i$.
\[F(X) = u_1x_1 + ... + u_dx_d\]
\[F(\vec{v}) = u_1v_1 + ... + u_dv_d = \vec{u}\cdot\vec{v}\]
Thus, committing to $F(X) = \sum_{i=1}^{d}{u_ix_i}$
and opening it at
\[X = \begin{bmatrix}x_0 = u_0\\...\\x_i = u_i\\..\\x_d=u_d\end{bmatrix}\]
is like comitting to $f_{\vec{u}}$ and opening it at $\vec{v}$
This was a bit of a tangent that didn’t actually deepen my understanding of the lecture material, but it was fun nonetheless! As someone who hasn’t touched math seriously for 5+ years, it’s fun to
wipe the dust off my long forgotten skills (admittedly this isn’t that challenging as far as serious math goes).
Bonus - A (attempted) multiliner commitment from univariate commitments
In the Univariate Polynomials section I showed how to create a univariate commitment from a multilinear commitment. What about the other way around?
I tried to find a way to represent a multilinear commitment of a function in $d$ variables as a commitment of $d$ univariate polynomials of degree one. But the math is more complicated and the work
is incomplete, hence why this is a bonus section.
We can write a general multilinear function of $d$ variables in the following way:
\[F(X) = c_0 + c_1x_1 + ... + c_dx_d + c_{d+1}x_1x_2 + ... + c_{?}x_1...x_d\]
How many coefficients $c$ do we have in this form? For the terms in one $x$ variable we have $\binom{d}{1}$ possibilities. For the terms in two $x$ variable we have $\binom{d}{2}$ possibilities, and
so on. The total number of coefficients is
\[\sum_{i=0}^d{\binom{d}{i}} = 2^d\]
So given
\[F(X) = c_0 + c_1x_1 + ... + c_{2^d-1}x_1...x_d\]
My idea is to write $d$ univariate polynomials of the form
\[f_1(x_1) = a_1 + b_1x_1 \\ ... \\ f_d(x_d) = a_d = b_dx_d \\\]
And multiply them together to get the generic multilinear polynomial
Once I’ve expanded that and collected the terms, I’ll have coefficients in terms of $a$ and $b$ variables. Then I can map those coefficients to the $c$ coefficients and solve for the $a$’s and $b$’s.
If the multilinear polynomial is the product of the univariate polynomials, then comitting to the univariate polynomials is kind of like to comitting to the multivariate ones, right? I’m honestly not
sure. This is where my math knowledge breaks down. Nevertheless, let’s move forward with this approach. I need to introduce some notation for the expansion formula.
Let $N_d$ be the set of numbers ${1, 2, …, d}$
Let $\binom{N_d}{i}$ be the set of combinations from choosing $i$ elements from $N_d$. E.g. $\binom{N_4}{2} = {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}$
Let $\sum_{a\in A}{f(a)}$ be a summation over the elements $a$ of a set $A$, applied to $f$.
My closed form expression is
\[\prod_{i=1}^d{f_i(x_i)} = \sum_{i=0}^d\Big(\sum_{a \in \binom{N_d}{i}}\big(\prod_{j \in a}a_j\big)\big(\prod_{k \in N_d - a}b_kx_k\big)\Big)\]
Huh? I would be just as skeptical as you at this point. Don’t believe me? I’ll show you it works with an example. Let’s expand it for $d=3$.
When $i = 0$
\[i = 0 \Rightarrow \binom{N_3}{0} = \{\{\}\} \\ a = \{\}, k = N_3 - a = \{1, 2, 3\} \Rightarrow b_1b_2b_3x_1x_2x_3\]
When $i = 1$
\[i = 1 \Rightarrow \binom{N_3}{1} = \{\{1\}, \{2\}, \{3\}\} \\ a = \{1\}, k = N_3 - a = \{2, 3\} \Rightarrow a_1b_2b_3x_2x_3 \\ a = \{2\}, k = N_3 - a = \{1, 3\} \Rightarrow a_2b_1b_3x_1x_3 \\ a = \
{3\}, k = N_3 - a = \{1, 2\} \Rightarrow a_3b_1b_2x_1x_2\]
When $i = 2$
\[i = 2 \Rightarrow \binom{N_3}{2} = \{\{1, 2\}, \{1, 3\}, \{2, 3\}\} \\ a = \{1, 2\}, k = N_3 - a = \{3\} \Rightarrow a_1a_2b_3x_3 \\ a = \{1, 3\}, k = N_3 - a = \{2\} \Rightarrow a_1a_3b_2x_2 \\ a
= \{2, 3\}, k = N_3 - a = \{1\} \Rightarrow a_2a_3b_1x_1 \\\]
When $i = 3$
\[i = 2 \Rightarrow \binom{N_3}{3} = \{\{1, 2, 3\}\} \\ a = \{1, 2, 3\}, k = N_3 - a = \{\} \Rightarrow a_1a_2a_3\]
Putting it all together we get $2^d = 2^3 = 8$ terms, exactly as expected.
\[\prod_{i=1}^3{f_i(x_i)} = a_1a_2a_3 + a_2a_3b_1x_1 + a_1a_3b_2x_2 + a_1a_2b_3x_3 + a_3b_1b_2x_1x_2 + a_2b_1b_3x_1x_3 + a_1b_2b_3x_2x_3 + b_1b_2b_3x_1x_2x_3\]
We can finally (maybe) figure out how to derive the $a$’s and $b$’s from the $c$’s. Let coefficient $c_B$ correspond to the term $\prod_{i \in B}{x_i}$ in the multilinear polynomial.
\[B = N_d - a \Rightarrow a = N_d - B\]
The $a$ in terms of $B$ is the start of our mapping, creating a system of equations for which we can solve for the $a$’s and $b$’s in terms of $c$’s. E.g. suppose $d = 3$ and $c_{{1, 2}} = 69$. In
other words, our multilinear polynomial has the term $69x_1x_2$. Then we know $69 = a_3b_1b_2$.
If we did this for every $c$ then we’d have a non-linear system of equations. Now this is where I get skeptical. I vaguely recall from my university math courses that anything non-linear is extremely
difficult to deal with, except in rare special cases. I think the approach breaks down with the system of equations.. so the bonus section ends here.
Congratualtions for reading until the end of this half baked stream of consciousness. If you’ve made it this far then please reach out. I guarantee we’ll have an interesting conversation.
Written on February 13, 2023
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$80,200 a Year After-Tax is How Much a Month, Week, Day, an Hour?
The information and calculations on this page are based on your location (United States) and a zero-dollar state tax. We recommend changing the 'State' option in the calculator above if you want a
more accurate version of the $80,200 a year salary calculation.
How much is $80,200 a Year After Tax in the United States?
In the year 2024, in the United States, $80,200 a year gross salary, after-tax, is $63,653 annual, $4,793 monthly, $1,102 weekly, $220.47 daily, and $27.56 hourly gross based on the information
provided in the calculator above.
Check the table below for a breakdown of $80,200 a year after tax in the United States.
Yearly Monthly Weekly Daily Hourly
Gross Salary $80,200 $6,683 $1,537 $307.41 $38.43
Federal Income Tax $10,412 $867.64 $199.54 $39.91 $4.99
State Income Tax $0 $0 $0 $0 $0
Social Security $4,972 $414.37 $95.3 $19.06 $2.38
Medicare $1,163 $96.91 $22.29 $4.46 $0.56
Tax-Free Allowance $12,950 Per Year
Taxable Salary $67,250 Per Year
Take-Home Pay (NET) $63,653 $4,793 $1,102 $220.47 $27.56
If you're interested in discovering how much you would earn a year with an extra $100 added to your annual salary, you can explore the calculations for a $80,300 annual income. This will provide you
with a better understanding of the difference that additional $100 can make to your annual earnings.
$80,200 a Year is How Much a Month?
When evaluating a $80,200 a year after tax income, the corresponding monthly earnings can be determined:
• Take-home (NET) monthly income: $4,793
In order to discover $80,200 a year is how much a month? - simply divide the annual amount by 12, resulting in a monthly income of $4,793.
$80,200 a Year is How Much a Week?
When assessing a $80,200 a year after tax salary, the associated weekly earnings can be calculated:
• Take-home (NET) weekly income: $1,102
To answer $80,200 a year is how much a week? - divide the annual sum by 52, resulting in a weekly income of $1,102.
$80,200 a Year is How Much a Day?
When examining a $80,200 a year after tax income, the corresponding daily earnings can be determined:
• Take-home (NET) daily income: $220.47 (assuming a 5-day work week)
To find out $80,200 a year is how much a day? - divide the annual figure by 260 (52 weeks * 5 days), resulting in a daily income of $220.47.
$80,200 a Year is How Much an Hour?
When analyzing a $80,200 a year after-tax salary, the associated hourly earnings can be calculated:
• Take-home (NET) hourly income: $27.56 (assuming a 40-hour work week)
To answer $80,200 a year is how much an hour? - divide the annual amount by 2,080 (52 weeks * 40 hours), resulting in an hourly income of $27.56.
Is $80,200 a Year a Good Salary?
To answer if $80,200 a year is a good salary. We need to compare it to the national median. After calculation using ongoing year (2024) data, the salary of $80,200 a year is 1.13 times or 11.74%
higher than the national median. So, is $80,200 a year a good salary?
Based on comparison to the national median, yes, in our opinion, it is a good salary in the United States.
We think these three links are helpful and related to the $80,200 After Tax US: 2021-2022 Tax Brackets and Federal Income Tax Rates, Taxes, and Income tax in the United States.
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Changes in Equilibrium Systems
Le Châtelier’s principle - a generalization that states that:
chemical systems at equilibrium shift to restore equilibrium when a change occurs that disturbs the equilibrium.
An adjustment by a system at equilibrium that results in a change in the concentrations of reactants and products is called an equilibrium shift.
Le Châtelier’s principle allows chemists to predict the qualitative effects of changes in concentration, pressure, and temperature on a chemical reaction system at equilibrium.
We can tell a reaction is at equilibrium if the reaction quotient (Q) is equal to the equilibrium constant (K).
If a system at equilibrium is subjected to a perturbance or stress (such as a change in concentration) the position of equilibrium changes.
Since this stress affects the concentrations of the reactants and the products, the value of Q will no longer equal the value of K.
To re-establish equilibrium, the system will either shift toward the products (if Q < K) or the reactants (if Q > K) until Q returns to the same value as K.
The Relationship Between the Equilibrium Constant and the Reaction Quotient
If reaction quotient (Q[c]) is greater than equilibrium constant (K[c]), the numerator of the reaction quotient expression must be very large.
The concentrations of the chemicals on the right side of the equation
(mA + nB + ⇌ xC + yD)
must be more than their concentrations at equilibrium.
In this situation, the system attains equilibrium by moving to the left.
Conversely, if Q[c] is less than K[c] , the system attains equilibrium by moving to the right.
Effect of Change in Concentration on Equilibrium
(a) The test tube contains 0.1 M Fe^3+.
(b) Thiocyanate ion has been added to solution in (a), forming the red Fe(SCN)^2+ ion.
Fe^3+ (aq) + SCN^− (aq) ⇌ Fe(SCN)^2+ (aq).
(c) Silver nitrate has been added to the solution in (b), precipitating some of the SCN − as the white solid AgSCN.
Ag^+ (aq) + SCN^− (aq) ⇌ AgSCN(s).
The decrease in the SCN^− concentration shifts the first equilibrium in the solution to the left, decreasing the concentration (and lightening color) of the Fe(SCN)^2+.
The stress on the system in the image above is the reduction of the equilibrium concentration of SCN− (lowering the concentration of one of the reactants would cause Q to be larger than K).
As a consequence, Le Châtelier's principle leads us to predict that the concentration of Fe(SCN)2+ should decrease, increasing the concentration of SCN− part way back to its original concentration,
and increasing the concentration of Fe3+ above its initial equilibrium concentration.
Collision Theory and Concentration Changes in an Equilibrium System
According to collision theory, entities in a chemical system must collide to react.
When the concentration of an entity in a chemical reaction system is increased, it is more likely that that entity will collide with other entities. There are simply more of them present. However,
only collisions between reactant entities can potentially contribute to a chemical reaction. Even so, the more frequently collisions occur overall, the more likely it is that a chemical reaction will
take place.
Collision theory explains the response of a chemical reaction system at equilibrium to a change in concentration as the result of random collisions and probability.
When we add more reactant entities in an equilibrium system, the equilibrium shifts to the right because the number of successful collisions for the forward reaction increases.
If, instead, we add more product entities, then the number of successive collisions for the reverse reaction will increase and the equilibrium will shift to the left.
Le Châtelier’s Principle and Changes in Energy
For a given system at equilibrium, the value of the equilibrium constant depends only on temperature.
Changing the temperature of a reacting mixture changes the rate of the forward and reverse reactions by different amounts, because the forward and reverse reactions have different activation energies
A reacting mixture at one temperature has an equilibrium constant whose value changes if the mixture is allowed to reach equilibrium at a different temperature.
To apply Le Châtelier’s principle and predict how a change in energy will affect a chemical system at equilibrium, we can think of energy as a reactant or a product.
Energy is absorbed in an endothermic reaction. If we consider energy to be a reactant, we can write the word equation:
reactants + energy ⇌ products
Since energy is released during an exothermic reaction, we can consider energy to be a reactant and write
reactants ⇌ products + energy
To use Le Châtelier’s principle to predict how an equilibrium will shift in response to a change in energy, consider how the system can counteract this shift.
Endothermic Reactions
If an endothermic reaction is cooled (thermal energy removed), we can consider that the quantity of one of the reactants has been decreased.
We can therefore predict that the equilibrium will shift to the left (toward the reactants), and energy will be released.
If thermal energy were added to this equilibrium system by heating, the equilibrium would likely shift to the right.
Exothermic Reactions
If thermal energy is removed from an exothermic reaction—where energy is a product—then the equilibrium will shift to the right (toward the products), and energy will be released to counteract the
If energy is added to an exothermic reaction, the equilibrium will shift to the left to compensate for the change, and the energy will be used as products are converted to reactants.
Le Châtelier’s Principle and Changes in Gas Volume
Sometimes we can change the position of equilibrium by changing the pressure of a system. However, changes in pressure have a measurable effect only in systems in which gases are involved, and then
only when the chemical reaction produces a change in the total number of gas molecules in the system.
An easy way to recognize such a system is to look for different numbers of moles of gas on the reactant and product sides of the equilibrium.
While evaluating pressure (as well as related factors like volume), it is important to remember that equilibrium constants are defined with regard to concentration (for K[c]) or partial pressure (for
Some changes to total pressure, like adding an inert gas that is not part of the equilibrium, will change the total pressure but not the partial pressures of the gases in the equilibrium constant
Thus, addition of a gas not involved in the equilibrium will not perturb the equilibrium.
As we increase the pressure of a gaseous system at equilibrium, either by decreasing the volume of the system or by adding more of one of the components of the equilibrium mixture, we introduce a
stress by increasing the partial pressures of one or more of the components.
In accordance with Le Châtelier's principle, a shift in the equilibrium that reduces the total number of molecules per unit of volume will be favored because this relieves the stress.
The reverse reaction would be favored by a decrease in pressure.
Consider what happens when we increase the pressure on a system in which N[2], H[2], and NH[3] are at equilibrium:
N[2 ](g) + 3H[2] (g) ⇌ 2NH[3] (g)
The formation of additional amounts of NH[3] decreases the total number of molecules in the system because each time two molecules of NH[3] form, a total of four molecules of N[2] and H[2] are
This reduces the total pressure exerted by the system and reduces, but does not completely relieve, the stress of the increased pressure.
On the other hand, a decrease in the pressure on the system favors decomposition of NH[3] into N[2] and H[2] , which tends to restore the pressure.
Le Châtelier’s Principle and Changes in Gas Volume
(a) The container holds the equilibrium reaction N[2 ](g) + 3H[2] (g) ⇌ 2NH[3] (g) .
(b) The volume of the container is decreased by half, which doubles the total pressure of the system.
(c) The reaction shifts to the right, which decreases the total number of particles in the container and the total pressure by producing more ammonia, NH[3] (g).
The Effect of a Catalyst on Equilibrium
A catalyst speeds up the rate of a reaction, either by allowing a different reaction mechanism or by providing additional mechanisms.
The overall effect is to lower the activation energy, which increases the rate of reaction.
The activation energy is lowered the same amount for the forward and reverse reactions, however. There is the same increase in reaction rates for both reactions.
As a result, a catalyst does not affect the position of equilibrium. It only affects the time that is taken to achieve equilibrium.
The Effects of Changing Conditions on a System at Equilibrium
Type of reaction Change to system Effect on K Direction of
[c] change
increasing any reactant concentration, or decreasing any product concentration no effect toward products
all reactions decreasing any reactant concentration, or increasing any product concentration no effect toward reactants
using a catalyst no effect no change
increasing temperature decreases toward reactants
decreasing temperature increases toward products
increasing temperature increases toward products
decreasing temperature decreases toward reactants
equal number of reactant and product gas molecules changing the volume of the container, or adding a non-reacting gas no effect no change
decreasing the volume of the container at constant temperature no effect toward reactants
more gaseous product molecules than reactant gaseous
molecules increasing the volume of the container at constant temperature, or adding a non-reacting gas at contstant no effect toward products
fewer gaseous product molecules than reactant gaseous decreasing the volume of the container at constant temperature no effect toward products
increasing the volume of the container at constant temperature no effect toward reactants
{slider title="Calculation of Concentration Changes as a Reaction Goes to Equilibrium" open="false"}
Calculation of Concentration Changes as a Reaction Goes to Equilibrium
Under certain conditions, the equilibrium constant for the decomposition of PCl[5] (g) into PCl[3] (g) and Cl[2] (g) is 0.0211. What are the equilibrium concentrations of PCl[5], PCl[3], and Cl[2] if
the initial concentration of PCl[5] was 1.00 M?
Step 1:
Determine the direction the reaction proceeds.
The balanced equation for the decomposition of PCl[5] is
PCl[5] (g) ⇌ PCl[3] (g) + Cl[2] (g)
Because we have no products initially, Q[c] = 0 and the reaction will proceed to the right.
Step 2:
Determine the relative changes needed to reach equilibrium, then write the equilibrium concentrations in terms of these changes.
Let us represent the increase in concentration of PCl[3] by the symbol x. The other changes may be written in terms of x by considering the coefficients in the chemical equation.
The changes in concentration and the expressions for the equilibrium concentrations are (ICE Table):
PCl[5] (g) PCl[3] (g) Cl[2] (g)
Initial concentration 1.00 0 0
Change -x +x +x
Equilibrium concentration 1.00-x 0+x=x 0+x=x
Step 3:
Solve for the change and the equilibrium concentrations.
Substituting the equilibrium concentrations into the equilibrium constant equation gives
K[c] = [PCl 3 ][Cl 2 ] / [PCl 5] = 0.0211
(x*x) / (1.00-x) = 0.0211
This equation contains only one variable, x, the change in concentration. We can write the equation as a quadratic equation and solve for x using the quadratic formula.
0.0211(1.00 − x) = x 2
x 2 + 0.0211x − 0.0211 = 0
An equation of the form ax 2 + bx + c = 0 can be rearranged to solve for x:
x = (−b ± √(b − 4ac)) / 2a
In this case, a = 1, b = 0.0211, and c = −0.0211. Substituting the appropriate values for a, b, and c yields:
x = (−0.0211 ± √((0.0211) 2 − 4(1)(−0.0211))) / 2(1)
x = 0.135
x = −0.156
Quadratic equations often have two different solutions, one that is physically possible and one that is physically impossible (an extraneous root). In this case, the second solution (−0.156) is
physically impossible because we know the change must be a positive number (otherwise we would end up with negative values for concentrations of the products). Thus, x = 0.135 M.
The equilibrium concentrations are:
[PCl[5]] = 1.00 − 0.135 = 0.87 M
[PCl[3]] = x = 0.135 M
[Cl[2]] = x = 0.135 M
{slider title="Approximate Solution Starting Close to Equilibrium"}
Approximate Solution Starting Close to Equilibrium
The Meaning of a Small Equilibrium Constant
Understanding the meaning of a small equilibrium constant can sometimes help to simplify a calculation that would otherwise involve a quadratic equation.
When K[c] is small compared with the initial concentration, the value of the initial concentration minus x is approximately equal to the initial concentration. Thus, you can ignore x.
Of course, if the initial concentration of a substance is zero, any equilibrium concentration of the substance, no matter how small, is significant.
In general, values of K[c] are not measured with accuracy better than 5%.
Therefore, making the approximation is justified if the calculation error you introduce is less than 5%.
To help you decide whether or not the approximation is justified, divide the initial concentration by the value of K[c].
• If the answer is greater than 500, the approximation is justified.
• If the answer is between 100 and 500, it may be justified.
• If the answer is less than 100, it is not justified. The equilibrium expression must be solved in full.
The atmosphere contains large amounts of oxygen and nitrogen. The two gases do not react, however, at ordinary temperatures. They do react at high temperatures, such as the temperatures produced by a
lightning flash or a running car engine.
In fact, nitrogen oxides from exhaust gases are a serious pollution problem.
A chemist is studying the following equilibrium reaction.
N[2] (g) + O[2] (g) ⇌ 2NO (g)
The chemist puts 0.085 mol of N[2] (g) and 0.038 mol of O[2] (g) in a 1.0 L rigid cylinder. At the temperature of the exhaust gases from a particular engine, the value of K c is 4.2 × 10 −8 . What is
the concentration of NO (g) in the mixture at equilibrium?
You need to find the concentration of NO at equilibrium.
You have the balanced chemical equation. You know the value of K c and the following concentrations: [N[2]] = 0.085 mol/L and [O[2]] = 0.038 mol/L.
Step 1 Divide the smallest initial concentration by K[c] to determine whether you can ignore the change in concentration.
Step 2 Set up an ICE table. Let x represent the change in [N[2]] and [O[2]].
Step 3 Write the equilibrium expression. Substitute the equilibrium concentrations into the equilibrium expression. Solve the equilibrium expression for x.
Step 4 Calculate [NO] at equilibrium.
Step 1
Smallest initial concentration / K c = 0.038 / 4.2 × 10 −8 = 9.0 × 10^5
Because this is well above 500, you can ignore the changes in [N[2]] and [O[2]].
Step 2
ICE Table
N[2] (g) O[2] (g) 2NO (g)
Initial concentration 0.085 0.038 0
Change -x -x +2x
Equilibrium concentration 0.085 − x ≈ 0.085 0.038 − x ≈ 0.038 2x
Step 3
K[c] = [NO]^2 / ([N[2]][O[2]])
4.2 × 10 −8 = (2x) 2 / (0.085 × 0.038)
x = √3.39 × 10^−11 = 5.82 × 10^−6
Step 4
[NO] = 2x
Therefore, the concentration of NO (g) at equilibrium is 1.2 × 10^−5 mol/L.
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The Cuneiform Numbers and Punctuation block is a block of the Unicode standard that contains characters used to write numbers and punctuation marks using the Cuneiform script. The Cuneiform script is
an ancient writing system used in Mesopotamia and other parts of the ancient Near East, and this block contains all the symbols needed to write numbers and punctuation marks using the Cuneiform
script. The Cuneiform Numbers and Punctuation block is often used in conjunction with other Unicode blocks to represent numbers and punctuation marks written in the Cuneiform script in digital form.
Enable "Right-Click to copy" to easily copy characters from this page, just right-click on the char to copy it.
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Copy all chars
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1. Introduction
A while ago, we derived [
] a functional equation for the so-called loop average [
] or Wilson loop in Turbulence. The path to an exact solution by a dimensional reduction in this equation was proposed in the ’93 paper [
] but has just been explored.
At the time, we could not compare a theory with anything but crude measurements in physical and numerical experiments at modest Reynolds numbers. All these experiments agreed with the K41 scaling, so
the exotic equation based on unjustified methods of quantum field theory was premature.
The specific prediction of the Loop equation, namely the Area law [
], could not be verified in DNS at the time with existing computer power.
The situation has changed over the last decades. No alternative microscopic theory based on the Navier-Stokes equation emerged, but our understanding of the strong turbulence phenomena grew
On the other hand, the loop equations technology in the gauge theory also advanced over the last decades. The correspondence between the loop space functionals and the original vector fields was
better understood, and various solutions to the gauge loop equations were found.
In particular, the momentum loop equation was developed, similar to our momentum loop used below [
]. Recently, some numerical methods were found to solve loop equations beyond perturbation theory [
The loop dynamics was extended to quantum gravity, where it was used to study nonperturbative phenomena [
All these old and new developments made loop equations a major nonperturbative approach to gauge field theory.
So, it is time to revive the hibernating theory of the loop equations in Turbulence, where these equations are much simpler.
The latest DNS [
] with Reynolds numbers of tens of thousands revealed and quantified violations of the K41 scaling laws. These numerical experiments are in agreement with so-called multifractal scaling laws [
However, as we argued in [
], at those Reynolds numbers, the DNS cannot yet distinguish between pure scaling laws with anomalous dimension
$ζ ( n )$
and some algebraic function of the logarithm of scale
$ζ ( n , log r )$
modifying the K41 scaling.
Theoretically, we studied the loop equation in the confinement region (large circulation over large loop
), and we have justified the Area law, suggested back in ’93 on heuristic arguments [
This law says that the tails of velocity circulation PDF in the confinement region are functions of the minimal area inside this loop.
It was verified in DNS four years ago [
] which triggered the further development of the geometric theory of turbulence[
In particular, the Area law was justified for flat and quadratic minimal surfaces [
], and an exact scaling law in confinement region
$Γ ∝ A r e a$
was derived [
]. The area law was verified with better precision in [
It was later conjectured in [
] that the dominant field configurations in extreme Turbulence are so-called Kelvinons, which were shown to solve stationary Navier-Stokes equations assuming the sparse distribution of vorticity
These topological solitons of the Euler theory are built around a vortex sheet bounded by a singular vortex line. This vortex line is locally equivalent to the cylindrical Burgers vortex [
], with infinitesimal thickness in the limit of a large Reynolds number.
As we argued in [
], the Kelvinon has an anomalous dissipation, surviving the strong turbulent limit. This dissipation is proportional to the square of constant circulation of the Burgers vortex times a line integral
of the tangent component of the strain along the loop.
The Kelvinon minimizes the energy functional, with anomalous terms coming from the Burgers core of the vortex line. There is also a constant scale factor
in the representation of the Kelvinon vorticity in terms of spherical Clebsch variables:
$ω → = 1 2 Z e a b c S a ∇ → S b × ∇ → S c = ∇ → ϕ 1 × ∇ → ϕ 2 ;$
$S 1 2 + S 2 2 + S 3 2 = 1 ;$
$ϕ 2 = arg ( S 1 + ı S 2 ) ; ϕ 1 = Z S 3 ;$
In that paper, the constant Z was related to the Kolmogorov energy dissipation density and the boundary value of the $S 3$ variable at the loop C.
The anomalous Hamiltonian [
] explicitly violated the K41 scaling by the logarithmic terms
$log Z / ν$
in the region of small loops
. This region resembles the asymptotically free QCD. The logarithmic terms were summed up by RG equation with running coupling constant logarithmically small in this region.
These exciting developments explain and quantitatively describe many interesting phenomena [
] but do not provide a complete microscopic theory covering the full inertial range of Turbulence without simplifying assumptions of the sparsity of vortex structures.
Moreover, while the Kelvinon (presumably) solves the stationary Navier-Stokes equations, it does not solve the loop equations for the following reason.
The loop equation assumes that the velocity field is independent of the loop C. In this case, the circulation $∮ C v α d r α$ variations in the loop functional by the shape C of the loop can be
reduced to the Navier-Stokes equation.
Otherwise, the variation would also involve the variation of the velocity field $∮ C δ v α d r α$.
This problem does not invalidate the Kelvinon theory as an ideal gas of random vortex rings sparsely distributed in a turbulent flow.
The loop functional is not needed for that statistical theory, and the stationary solution of the Navier-Stokes equation is sufficient. The shape of the loop and the vortex sheet inside would become
random variables influenced by a background strain like in the pure vortex sheet solutions [
These objections, however, prevent the Kelvinon gas model from being a complete theory of strong isotropic Turbulence. This model is merely an approximation of the full theory.
In the present work, we develop the theory free of these assumptions by exactly solving the loop equations for decaying Turbulence.
2. Loop equation
2.1. Loop operators
We introduced the loop equation in Lecture Series at Cargese and Chernogolovka Summer Schools [
Here is a summary for the new generation.
We write the Navier-Stokes equation as follows
The Wilson loop average for the Turbulence
treated as a function of time and a functional of the periodic function
$C : r α = C α ( θ ) ; θ ∈ ( 0 , 2 π )$
(not necessarily a single closed loop), satisfies the following functional equation
$H C = H C ( 1 ) + H C ( 2 )$
$H C ( 1 ) = ν γ ∮ C d r α ∂ β ω ^ α β ( r ) ;$
$H C ( 2 ) = γ ∮ C d r α ω ^ α β ( r ) v ^ β ( r ) ;$
$ω ^ α β ≡ − ı ν γ δ δ σ α β$
$v ^ β ( r ) = 1 ∂ μ 2 ∂ α ω ^ β α ( r )$
We added a dimensionless factor $γ$ in the exponential compared to some previous definitions as an extra parameter of the Wilson loop. Without loss of generality, we shall assume that $γ > 0$. The
negative $γ$ corresponds to a complex conjugation of the Wilson loop.
In Abelian gauge theory, this would be the continuous electric charge. In turbulence theory, the Fourier transform of the Wilson loop by $γ$ would produce the PDF for velocity circulation.
The statistical averaging $⋯$ corresponds to initial randomized data, to be specified later.
The area derivative
$δ δ σ α β$
is related to the variation of the functional when the little closed loop
$δ C$
is added
$Σ α β ( δ C ) δ F [ C ] δ σ α β ( r ) = F [ C + δ C ] − F [ C ] ;$
$Σ α β ( δ C ) = 1 2 ∮ δ C r α d r β$
In the review, [
], we present the explicit limiting procedure needed to define these functional derivatives in terms of finite variations of the loop while keeping it closed.
All the operators
$∂ μ , ω ^ α β , v ^ α$
are expressed in terms of the spike operator
The area derivative operator can be regularized as
$Ω α β ( θ , ϵ ) = − ı ν γ δ δ C α ′ ( θ ) ∫ − ϵ ϵ d ξ δ δ C β ( θ + ξ ) − { α ↔ β } ;$
and velocity operator (with
$δ , ϵ → 0 +$
$V α ( θ , ϵ , δ ) = 1 D μ 2 ( θ , ϵ ) D β ( θ , ϵ ) Ω β α ( θ , δ ) ;$
In addition to the loop equation, every valid loop functional
$F [ C ]$
must satisfy the Bianchi constraint [
$∂ α δ F [ C ] δ σ β γ ( r ) + cyclic = 0$
In three dimensions, it follows from identity $∇ → · ω → = 0$; in general dimension $d > 3$, the dual vorticity $ω ˜$ is an antisymmetric tensor with $d − 2$ components. The divergence of this tensor
equals zero identically.
However, for the loop functional, this restriction is not an identity; it reflects that this functional is a function of a circulation of some vector field, averaged by some set of parameters.
This constraint was analyzed in [
] in the confinement region of large loops, where it was used to predict the Area law. The area derivative of the area of some smooth surface inside a large loop reduces to a local normal vector. The
Bianchi constraint is equivalent to the Plateau equation for a minimal surface (mean external curvature equals zero).
In the Navier-Stokes equation, we did NOT add artificial random forces, choosing instead to randomize the initial data for the velocity field.
These ad hoc random forces would lead to the potential term [
] in the loop Hamiltonian
$H C$
, breaking certain symmetries needed for the dimensional reduction we study below.
With random initial data instead of time-dependent delta-correlated random forcing, we no longer describe the steady state (i.e., statistical equilibrium) but decaying Turbulence, which is also an
interesting process, manifesting the same critical phenomena.
The energy is pumped in at the initial moment $t = 0$ and slowly dissipates over time, provided the viscosity is small enough, corresponding to the large Reynolds number we are studying.
2.2. Dimensional reduction
The crucial observation in [
] was that the right side of the Loop equation, without random forcing, dramatically simplifies in functional Fourier space. The dynamics of the loop field can be reproduced in an Ansatz
The difference with the original definition of $Ψ [ γ , C ]$ is that our new function $P α ( θ )$ depends directly on $θ$ rather then through the function $v α ( r )$ taken at $r α = C α ( θ )$.
This transformation is the dimensional reduction $d ⇒ 1$ we mentioned above. From the point of view of the loop functional, there is no need to deal with field $v ( r )$; one could take a shortcut.
The reduced dynamics must be equivalent to the Navier-Stokes dynamics of the original field. With the loop calculus developed above, we have all the necessary tools to build these reduced dynamics.
Let us stress an important point: the function $P → ( θ , t )$ is independent of the loop C. As we shall see later, it is a random variable with a universal distribution in functional space.
This independence removes our objection in the Introduction to the Kelvinon theory and any other Navier-Stokes stationary solutions with a singularity at fixed loop C in space.
The functional derivative, acting on the exponential in ((
)) could be replaced by the derivative
$P ′$
as follows
$δ δ C α ( θ ) ↔ − ı γ ν P α ′ ( θ )$
The equation for
$P ( θ )$
as a function of
and also a function of time, reads:
where the operators
$V , D , Ω$
should be regarded as ordinary numbers, with the following definitions.
The spike derivative
in the above equation
The vorticity (
) and velocity (
) also become singular functionals of the trajectory
$P ( θ )$
The first observation about this equation is that the viscosity factor cancels after the substitution (
As we shall see, the viscosity enters initial data so that at any finite time t, the solution for P still depends on viscosity.
Another observation is that the spike derivative
$D ( θ , ϵ )$
turns to the discontinuity
$Δ P ( θ ) = P ( θ + ) − P ( θ − )$
in the limit
$ϵ → 0 +$
$D ( θ , 0 + ) = − ı γ ν Δ P ( θ )$
The relation of the operators in the QCD loop equation to the discontinuities of the momentum loop was noticed, justified, and investigated in [
In the Navier-Stokes theory, this relation provides the key to the exact solution.
In the same way, we find the limit for vorticity
$Ω α β ( θ , 0 + ) = − ı γ ν P α β ( θ ) ;$
$P α β ( θ ) = Δ P α ( θ ) P β ( θ ) − { α ↔ β } ;$
$P α ( θ ) ≡ P α ( θ + ) + P α ( θ − ) 2$
and velocity (skipping the common argument
$V α = Δ P β Δ P μ 2 P β α = P α − Δ P α Δ P β P β Δ P 2$
The Bianchi constraint is identically satisfied as it should
We arrive at a singular loop equation for
$P α ( θ )$
This equation is complex due to the irreversible dissipation effects in the Navier-Stokes equation.
The viscosity dropped from the right side of this equation; it can be absorbed in units of time. Viscosity also enters the initial data, as we shall see in the next Section on the example of the
random rotation.
However, the large-time asymptotic behavior of the solution would be universal, as it should be in the Turbulent flow.
We are looking for a degenerate fixed point [
], a fixed manifold with some internal degrees of freedom. The spontaneous stochastization corresponds to random values of these hidden internal parameters.
Starting with different initial data, the trajectory $P → ( θ , t )$ would approach this fixed manifold at some arbitrary point and then keep moving around it, covering it with some probability
The Turbulence problem is to find this manifold and determine this probability measure.
2.3. Random global rotation
Possible initial data for the reduced dynamics were suggested in the original papers [
]. The initial velocity field’s simplest meaningful distribution is the Gaussian one, with energy concentrated in the macroscopic motions. The corresponding loop field reads (we set
$γ = 1$
for simplicity in this section)
$f ( r → )$
is the velocity correlation function
The potential part drops out in the closed loop integral.
The correlation function varies at the macroscopic scale, which means that we could expand it in the Taylor series
$f ( r − r ′ ) → f 0 − f 1 ( r − r ′ ) 2 + ⋯$
The first term
$f 0$
is proportional to initial energy density,
and the second one is proportional to initial energy dissipation rate
$E 0$
$f 1 = E 0 2 d ( d − 1 ) ν$
$d = 3$
is the dimension of space.
The constant term in (
) as well as
$r 2 + r ′ 2$
terms drop from the closed loop integral, so we are left with the cross-term
$r r ′$
, which reduces to a full square
This distribution is almost Gaussian: it reduces to Gaussian one by extra integration
The integration here involves all $d ( d − 1 ) 2 = 3$ independent $α < β$ components of the antisymmetric tensor $ϕ α β$. Note that this is ordinary integration, not the functional one.
The physical meaning of this $ϕ$ is the random uniform vorticity $ω ^ = f 1 ϕ ^$ at the initial moment.
However, as we see it now, this initial data represents a spurious fixed point unrelated to the turbulence problem.
It was discussed in our review paper [
]. The uniform global rotation represents a fixed point of the Navier-Stokes equation for arbitrary uniform vorticity tensor.
Gaussian integration by $ϕ$ keeps it as a fixed point of the Loop equation.
The right side of the Navier-Stokes equation vanishes at this special initial data so that the exact solution of the loop equation with this initial data equals its initial value (
Naturally, the time derivative of the momentum loop with the corresponding initial data will vanish as well.
It is instructive to look at the momentum trajectory $P α ( θ )$ for this fixed point.
The functional Fourier transform [
] leads to the following simple result for the initial values of
$P α ( θ )$
In terms of Fourier harmonics, this initial data read
$P α , n = N ( 0 , 1 ) ∀ α , n > 0 ;$
$P ¯ α , n = 4 f 1 n ν ϕ α β P β , n ; ∀ β , n > 0 ;$
$ϕ α β = N ( 0 , 1 ) ∀ α < β ;$
As for the constant part $P α , 0$ of $P α ( θ )$ , it is not defined, but it drops from equations by translational invariance.
Note that this initial data is not real, as
$P ¯ α , n ≠ P α , n ★$
. Positive and negative harmonics are real but unequal, leading to a complex Fourier transform. At fixed tensor
the correlations are
This correlation function immediately leads to the uniform expectation value of the vorticity
The uniform constant vorticity kills the linear term of the Navier-Stokes equation in the original loop space, involving $∂ α Ω ^ α β = 0$.
The nonlinear term $V ^ α Ω ^ α β$ vanishes in the coordinate loop space only after integration around the loop.
Here are the steps involved
$V ^ β = 1 2 Ω ^ α β C β ;$
$∮ Ω ^ α β C β Ω ^ β γ d C α ∝ Ω ^ α β Ω ^ β γ Σ α β ( C ) ;$
Here the tensor area $Σ$ was defined in (). It is an antisymmetric tensor; therefore its trace with a symmetric tensor $Ω ^ α β Ω ^ β γ$ vanishes.
This calculation demonstrates how an arbitrary uniform vorticity tensor satisfies the loop equation in coordinate loop space.
We expect the turbulent solution of the loop equation to be more general, with the local vorticity tensor at the loop becoming a random variable with some distribution for every point on the loop.
2.4. Decay or fixed point
The absolute value of loop average
$Ψ [ γ , C ]$
stays below 1 at any time, which leaves two possible scenarios for its behavior at a large time.
$Decay : P → → 0 ; Ψ [ γ , C ] → 1 ;$
$Fixed Point : P → → P → ∞ ; Ψ [ γ , C ] → Ψ ∞ [ C ] ;$
scenario in the nonlinear ODE (
) corresponds to the
$1 / t$
decrease of
$P →$
Omitting the common argument
, we get the following
time-dependent solution (not just asymptotically, at
$t → + ∞$
Fixed Point
would correspond to the vanishing right side of the momentum loop equation (
). Multiplying by
$( Δ P → ) 2$
and reducing the terms, we find a singular algebraic equation
The fixed point could mean self-sustained Turbulence, which would be too good to be true, violating the second law of Thermodynamics. Indeed, it is easy to see that this fixed point cannot exist.
The fixed point equation (
) is a linear relation between two vectors
$P → , Δ P →$
with coefficients depending on various scalar products. The generic solution is simply
with the complex parameter
to be determined from the equation (
This solution is degenerate: the fixed point equation is satisfied for arbitrary complex $λ$.
The discontinuity vector
$Δ P →$
aligned with the principal value
$P →$
corresponds to vanishing vorticity in (
), leading to a trivial solution of the loop equation
$Ψ [ γ , C ] = 1$
We are left with the decaying turbulence scenario () as the only remaining physical solution.
3. Fractal curve in complex space
3.1. Random walk
One may try the solution where the discontinuity vector is proportional to the principal value. However, in this case, such a solution does not exist.
$λ 2 F → 2 − 1 = ? λ 2 F → 2 ;$
There is, however, another solution where the vectors
$Δ F → , F →$
are not aligned. This solution requires the following relations
$( 2 F → · Δ F → − ı γ ) 2 + γ 2 = 4 F → 2$
These relations are very interesting. The complex numbers reflect irreversibility and lack of alignment leads to vorticity distributed along the loop.
Also, note that this complex vector
$F → ( θ )$
is dimensionless, and the fixed point equation (
Section 3.1
) is completely universal, up to a single dimensionless parameter
One can build this solution as a Markov process by the following method. Start with a complex vector $F → ( θ = 0 ) = F → 0$.
We compute the next values from the following discrete version of the discontinuity equations (
Section 3.1
3.2. Constraints imposed on a random step
A solution to these equations can be represented using a complex vector
$q → k$
subject to two complex constraints
after which we can find the next value
$F → k + 1 = F → k + q → k ;$
We assume N steps, each with the angle shift $Δ θ = 2 π N$.
This recurrent sequence is a Markov process because each step only depends on the current position $F → k$. On top of this Markov process, there is a closure requirement $F → N = F → 0$.
This requirement represents a nonlinear restriction on all the variables $F → k$, which we discuss below.
With this discretization, the circulation can be expressed in terms of these steps
$∮ F → ( θ ) · d C → ( θ ) = − ∮ C → ( θ ) · d F → ( θ ) ⇒ − ∑ k = 0 N − 1 C → k + 1 + C → k 2 · q → k$
Note that the complex unit vector is
defined with the Euclidean metric in six dimensions . Instead, we have a complex condition
which leads to
conditions between real and imaginary parts
$( Re q → ) 2 = 1 + ( Im q → ) 2 ;$
dimensions, there are
$d − 1$
complex parameters of the unit vector; with an extra linear constraint in (
), there are now
$d − 2$
free complex parameters at every step of our iteration, plus the discrete choice of the sign of the root in the solution of the quadratic equation.
3.3. Closure condition
At the last step,
$k = N − 1$
, we need to get a closed loop
$F → N = F → 0$
. This is one more constraint on the complex vectors
$q → 0 , ⋯ q → N − 1$
We use this complex vector constraint to fix the arbitrary initial complex vector $F → 0$ as a function of all remaining parameters.
Looking ahead into the rest of our investigation, it turns out that the closure conditions fix only half of the $2 d$ real parameters in the initial point $F → 0$. The remaining parameters are free
zero modes of our fixed manifold.
Due to the closure of the space loop $C → ( θ )$, the global translation of the momentum loop $P → ( θ )$ leaves invariant the Wilson loop; therefore, the translational zero modes of the momentum
loop do not lead to ambiguities.
However, the missing d out of $2 d$ parameters in $F → 0$ mean that some other d parameters should be adjusted to provide the momentum loop closure.
We discuss this issue in the next Section, where we derive the SDE for the closed momentum loop in three dimensions. This SDE has explicit terms, which we computed in
Appendix B
and coded in
in [
The adjustment of parameters we mentioned earlier yields three constraints on the Wiener process we derived.
3.4. Mirror pairs of solutions
Return to the general study of the discrete loop equations (
Section 3.2
There is a trivial solution to these equations at any even
$f → k = ( − 1 ) k q → 2 ;$
We reject this solution as unphysical: the corresponding vorticity equals zero, as all the vectors $f → k$ are aligned.
Our set of equations has certain mirror reflection symmetry
Thus, the complex solutions come in mirror pairs $F → k , F → N − k *$. The real solutions are only a particular case of the above trivial solution with real $q →$.
Each nontrivial solution represents a periodic random walk in complex vector space $C d$. The complex unit step $q → k ∈ C d$ depends on the current position $F → k ∈ C d$, or, equivalently, on the
initial position $F → 0$ plus the sum of the preceding steps.
We are interested in the limit of infinitely many steps $N → ∞$, corresponding to a closed fractal curve with a discontinuity at every point.
3.5. The degenerate fixed point and its statistical meaning
This solution’s degeneracy (fewer restrictions than the number of free parameters) is a welcome feature. One would expect this from a fixed point of the Hopf equation for the probability
In the best-known example, the microcanonical Gibbs distribution covers the energy surface with a uniform measure (ergodic hypothesis, widely accepted in Physics).
The parameters describing a point on this energy surface are not specified– in the case of an ideal Maxwell gas, these are arbitrary velocities of particles.
Likewise, the fixed manifold, corresponding to our fractal curve, is parametrized by N arbitrary local rotations, as discussed in the next Section.
This rich internal random structure of our fixed manifold, combined with its rotation and translation invariance in loop space C, makes it an acceptable candidate for extreme isotropic Turbulence.
4. The structure of turbulent manifold
The simplest case where these equations have nontrivial solutions is the three-dimensional space. For smaller dimensions of space, there is only a degenerate solution with zero vorticity (a vanishing
cross product $Ω ^ ∝ P → × Δ P →$ ). Thus, we only consider $d > 2$ in the rest of the paper.
4.1. Canonical form of a single step
The complex unit vector in
dimensions can be parametrized by rotation matrix and a unit real vector in
$d − 2$
$q → = O ^ · u → ( α 1 , α 2 , w → , β ) ;$
$u → ( α 1 , α 2 , w → , β ) = { α 1 , α 2 w → , ı β } ;$
$α 1 2 + α 2 2 = 1 + β 2 ;$
The following steps lead to this canonical form. Take a general complex d-vector $q →$ and choose the rotation $O ^ ∈ O ( d )$ to direct its imaginary part at the last axis d.
The imaginary part of the condition $q → 2 = 1$ implies the real part of this vector has zero component d. This real vector in $d − 1$ dimensions can be parametrized as ${ α 1 , α 2 w → }$ with the
unit vector $w → ∈ S d − 3$ and arbitrary real parameters $α 1 , α 2$.
There is a multiple counting of the same unit vector with this parametrization: the rotation matrix space
$O ( d )$
must be factored by rotations
$O ( d − 3 )$
of the unit vector
$w →$
Also, the sign change of
$α 2$
is equivalent to the reflection of the vector
$w →$
, so we have to factor out such reflections and keep an arbitrary sign of
$α 2$
The complex constraint for
$F → k · q → k$
can be used to fix these
$α 1 , α 2$
as a linear function of
given a complex vector
$f → k = O ^ k T · F → k ;$
as follows:
${ α 1 , α 2 } = M ^ − 1 . { Re ( R ) − β Im ( c ) , β Re ( c ) + Im ( R ) } ;$
$f → k = { a , b → , c }$
After that, $α 1 2 + α 2 2 = 1 + β 2$ yields a quadratic equation for $β$.
Note in passing that $u →$ belongs to De Sitter space $d S d − 1$. However, this is where an analogy with the ADS/CFT duality ends.
There are, in general, four solutions for
: two signs for
in () and two more signs in a solution of the quadratic equation for
We have to choose a particular real solution for
. A universal option is to choose the step with the smallest Euclidean distance
$( Re q → ) 2 + ( Im q → ) 2$
. We used this choice in our initial simulations [
], but later we switched to another method, using the SDE we describe later in this work. The SDE guarantees the closure condition, unlike the naive random walk approach.
4.2. Partition function
We arrive at the invariant distribution for our fractal curve. At a fixed
, the partition function (in terms of statistical mechanics)
$Q → = F → N − F → 0 = ∑ k = 0 N − 1 q → k ;$
$F → k + 1 = F → k + q → k ; ∀ k = 0 , ⋯ N − 1 ;$
$q → k$
are complex vectors, parametrized by
$O ^ 0 , ⋯ O ^ N − 1 , w → 0 , ⋯ w → N − 1$
via recurrent equations (
Section 4.1
The complex vector’s integration and delta function is understood as a product of its real and imaginary parts.
We conclude that the fixed manifold
$T d ( N )$
of the decaying Turbulence is a subset of the tensor product of rotational and spherical spaces.
This subset is selected by imposing the closure condition $∑ k q → k = 0$, which in general provides $2 d$ nonlinear relations between all parameters: $λ → , O ^ k , w → k$ for each choice of the
real solutions for $β$ on each step.
This closure condition is sufficient by parameter count to eliminate a complex vector $F → 0$; however, we discovered in 3D that half of the parameters in complex vector $F → 0$ are left
undetermined, leading instead to three real constraints on the parameters of the rotation matrices.
We cannot resolve the global closure conditions. Still, we found a way to achieve the same goal by an SDE describing the evolution of our curve from the exact symmetric solution (
) we have found; this method preserves the closure condition at each infinitesimal step of the stochastic process.
4.3. Symmetric fixed point
The above formal definition of the probability measure does not offer a practical simulation method for covering this manifold.
We attempted to simulate a random walk $F → k ⇒ F → k + 1$ step by step, taking random rotation matrices. Unfortunately, there was a rapidly diminishing probability of the return to the vicinity of
the initial point $F → N = F → 0$ after N steps.
We could not numerically solve the resulting transcendental equation for the initial position $F → 0$ at large N, neither by analytical nor by Monte Carlo methods.
Instead, we have found an alternative algorithm for covering this manifold, preserving the closed curve.
First, we have found a symmetric family of solutions [
] of our recurrent equation (
Section 3.1
) for arbitrary
Here $w → ∈ S d − 3$ is a unit vector.
The angles
$α k$
must satisfy recurrent relation
$α k + 1 = α k + σ k β ;$
This sequence with arbitrary signs
$σ k$
solves recurrent equation (
Section 3.1
) independently of
For the curve to be closed, the angle
must be a rational multiple of
$2 π$
In this case, the closure condition
$β ∑ 0 N − 1 σ k = 2 π n p , n ≠ 0 .$
will always have a solution for the discrete variables
$σ k = ± 1$
. All that is needed is a relation between the net numbers
$N ±$
of positive and negative
$σ k$
$N + − N − = n q , n ≠ 0 .$
There are $N N −$ different states $σ 0 = ± 1 , ⋯ σ N − 1 = ± 1$ satisfying the closing condition, provided $N > 2 N − + q$.
The variables $σ k$ can otherwise be random, like the spin variables in the Ising model. This distribution is less trivial than the Ising model because the angles $α k$ are related to all the
preceding $σ$ variables, not just the local one $σ k$.
This solution describes a closed random walk on a circle. It is characterized by an integer N and a rational number $p q$ with $q ≤ N$.
As we pointed out in
Section 3
, a reflected sequence
$Φ → N − k *$
also represents a solution to the recurrent equations (
Section 3.1
At first glance, this simple formula seems a valid solution for the loop equations for decaying turbulence.
Unfortunately, it does not have any energy dissipation. The vorticity vector is finite
However, the square of this complex 3-vector is identically zero.
This solution can serve as initial data for the Brownian motion over the turbulence manifold, but the energy dissipation appears only after averaging over this motion.
This square of the complex vector, in the general case, can be related to the square of the vertex
$F → k$
by the recurrent equations (
Section 3.2
Our symmetric solution corresponds to $F → k · F → k = 1 4$, and $± = 1$, where this expression vanishes identically at arbitrary $γ$.
The random walk step
$q → k = Φ → k + 1 − Φ → k$
is a real unit vector in this case
$q → k = σ k { − sin δ k , w → cos δ k , 0 } ;$
The direction of this vector is not random, though; in addition to the random sign $σ k$ and random unit vector $w →$ in $d > 3$ dimensions, its direction depends on the previous position $α k$ on a
So, this is a perfect example of a Markov chain, with the closure condition analytically solved by quantizing the angular step to a rational number $β = 2 π p q$.
This solution corresponds to the real value of velocity circulation on each of these two solutions; however, the reflection changes the circulation.
Thus, the arithmetic average of two Wilson loops with two reflected solutions is reflection-symmetric, but it is still a complex number.
Our covering algorithm will use a symmetric fixed point with a random choice of sign variables $σ k$ or its reflection as a starting element.
The imaginary parts of the steps $q → k$ are zero vectors at the start. Still, the evolution below will involve complex infinitesimal rotations $δ q → k = μ → k × q → k$ so that the imaginary parts
appear later in the evolution.
Due to the global
$O ( d )$
symmetry, the rotated curve
${ O ^ · Φ → 0 , ⋯ O ^ · Φ → N − 1 }$
with arbitrary orthogonal matrix
$O ^$
is also a solution. In our simulations, we integrate the Wilson loop by this global rotation after finding a particular numerical solution for the momentum loop
$F → ( θ )$
. The corresponding
$O ( 3 )$
group Fourier integral is computed in
Appendix A
4.4. Infinitesimal complex rotations in 3D
Let us assume we already know a particular solution
$F → 0 , q → 0 , ⋯ q → N − 1$
of the recurrent equations (
Section 3.2
) in
$d = 3$
and perturb it by an infinitesimal transformation of the complex vectors
$q → k$
, preserving their square.
We also shift the initial point
$F → 0$
to keep the loop closed after infinitesimal transformations of all the steps
$q → k$
$δ q → k = μ → k × q → k ;$
Here $μ → k , λ →$ are infinitesimal complex 3D vectors.
The real part
$Re μ → k ∈ R 3$
comes from the infinitesimal group transformation
$δ L$
of rotation matrices in our canonical form (
Section 4.1
$δ L O ^ k = Ω ^ k · O ^ k ;$
$Ω ^ k α β = e α β γ Ω k γ ;$
$δ L q → k = Re μ → k × q → k ;$
The imaginary part $Im μ → k$ leads to the infinitesimal transformation of parameters $α 1 , α 2 , β$ in two-dimensional de Sitter space $d S 2$; therefore, there are only two independent components
of $Im μ → k$.
We do not need an explicit split of the parameters of $μ → k$ into these two transformations; it is sufficient to know that cross product $μ → k × q → k$ with any complex vector $μ → k$ is orthogonal
to $q → k$, as we need it in our random walk with $q → k 2 = 1$.
Below, we will parameterize $Im μ → k$ by two scalar parameters.
There are two contributions to the variation of each position
$F → k$
. One variation comes from the rotation of the step from the previous position, and another comes from the variation of the previous position.
$δ F → k = δ F → k − 1 + μ → k − 1 × q → k − 1 = λ + ∑ 0 k − 1 μ → l × q → l ;$
By variation of the second of the constraints in (
Section 3.2
), we find the following set of relations between infinitesimal
$μ → k , λ →$
$G k = ( 2 F → k · q → k − ı γ ) ;$
$V → k = G k q → k − 2 F → k ;$
Some constraints are left in the solution for the vectors $q → k$ even after the complex rotations. Three scalar constraints on the imaginary parts of the complex rotation vectors $μ → k$ remain in
three dimensions.
These constraints are needed to provide the closure condition. There are only three scalar constraints among N real vectors, which leads to a nontrivial $3 N − 3$ dimensional quotient space.
4.5. The closure equation
We treat it as a recurrent system of equations for $Im μ → k$, assuming known values of $Re μ l , λ →$.
After solving that system, the complex vector
$λ →$
is supposed to be determined from the closure equation
$∑ l = 0 N − 1 μ → l × q → l = 0$
assuming all
$μ → l$
expressed as linear combinations of
$Re λ → , Im λ → , Re μ → i$
As we found in [
] in three dimensions, this system of equations for
$λ →$
is degenerate: three parameters in
$λ →$
are left undetermined.
The solution for $λ →$ exists only if N vectors ${ Re μ → 0 , ⋯ Re μ → N − 1 }$ obey three scalar constraints.
In other words, the complex vector equation (
) reduces to three constraints for
$λ →$
and another three constraints for
$Re μ → k$
. The complex vector
$λ →$
is left with three free components, and the vectors
${ Re μ → 0 , ⋯ Re μ N − 1 }$
are left with
$3 N − 3$
free components out of
$3 N$
The solution of these equations, which we find in
Appendix B
, has the form
$λ → = ∑ l = 0 N − 1 Λ ^ l · Re μ → l ;$
$Im μ → k = ∑ l = 0 N − 1 S ^ k l · Re μ → l$
with real
$3 × 3$
$S ^ k l$
, and complex
$3 × 3$
$Λ ^ l$
; these matrices depend on the current values of all the vectors
$F → k$
4.6. Linear constraints and zero modes
In addition, we have found three linear constraints on $Re μ → l$, related to three complex null vectors of a block matrix $H ^$ involved in the equation for $λ →$.
The vector
$λ →$
is defined modulo superposition of elements of these three zero modes
$λ → ⇒ λ → + ∑ i = 1 3 c i ψ → i$
Due to the closure of the original loop
$C ∈ R d$
, the translation of
$λ →$
by arbitrary complex vector does not change the circulation in (
). This translation of
$λ →$
leads to the global translation of our momentum curve
$P → ( θ )$
, preserving the circulation over the closed loop in space.
We resolved this ambiguity of $λ →$ by choosing the pseudo-inverse of the degenerate matrix $H ^$ when computing the coefficients $Λ ^ l$.
The three constraints on infinitesimal rotations have a form (
). These constraints define a subspace
of the whole space
$R 3 ⊗ N$
of our rotation vectors
$Re μ → k$
(dual to elements of Lie algebra on each
$S O ( 3 )$
$S : ∑ k Θ → i k · Re μ → k = 0 ; i = 1 , 2 , 3 ;$
The rotation vectors
$Re μ → k$
vary in the quotient space
The null-vectors
$ψ → i$
and coefficients
$W ^ k$
, depending on the current positions
$F → k$
are computed using recurrent equations in [
We get numerical results on a laptop for arbitrary $N < 100$. Larger values of N would require a supercomputer.
4.7. Brownian motion on turbulent manifold
Now, we are ready to write down the SDE for the evolution of our complex curve using the stochastic process
$d ξ → l = Re μ → l$
$d q → k = ∑ l = 0 N − 1 T ^ k l · d ξ → l ;$
$T ^ k l = δ k l q ^ k + ı q ^ k · S ^ k l ;$
$d F → 0 = ∑ l = 0 N − 1 Λ ^ l · d ξ → l ;$
$F → k = F → 0 + ∑ l = 0 k − 1 q → l ;$
$∑ k = 0 N − 1 Θ → i k · d ξ → k = 0 ; i = 1 , 2 , 3 ;$
These constrained stochastic differential equations describe the evolution of the point on our fixed manifold
$T 3 ( N )$
of closed complex curves subject to the loop equations (
Section 3.1
), starting with one of the symmetric fixed points (
The constrained SDE were studied in the mathematical literature [
We use a standard method of the projection of the Brownian motion to a quotient space. Let us introduce new stochastic real vector variables
$d η ^ = { d η → 0 , ⋯ d η → N − 1 } ∈ R 3 ⊗ N$
and project out the constraints as follows (in matrix notations)
$d ξ ^ = d η ^ − P · d η ^ ;$
The variables
$d η ^$
are assumed to be delta correlated (in proper units of stochastic time)
Here $I ^$ is a unit matrix in $3 N$ dimensions.
It is straightforward to check that
$d ξ$
satisfies the constraints for arbitrary
$d η ^$
The variables
$d ξ ^$
do not change when variables
$d η ^$
are shifted by superposition of transposed constraints
$δ d η ^ = Θ ^ † · d w → ;$
So, our stochastic process $d η ^$ has some redundant (gauge) degrees of freedom $d w →$.
The variables $d ξ ^$ evolve in the quotient space $F$, covering it with an $O ( 3 )$ invariant measure. This invariance is easy to check by noticing that all the matrices $Θ ^ , Λ ^ , S ^ , T ^$ in
our equations are made of rotation-covariant parameters in the linearized recurrent equations. These parameters are direct products of vectors times some dot products of other vectors.
In mathematical terms, $d η ^$ is a Wiener process in $R 3 ⊗ N$ with a unit variance matrix, and $d ξ ^$ is a Brownian motion in the quotient space $F$. This quotient space evolves with stochastic
time, as the constraint matrix $Θ ^$ depends on current values of all vectors $F → .$.
The projection can be used to redefine the matrices
after which our SDE takes a usual form
$d q → k = ∑ l = 0 N − 1 T k l · d η → l ;$
$d F → 0 = ∑ l = 0 N − 1 L l · d η → l ;$
$F → k = F → 0 + ∑ l = 0 k − 1 q → l ;$
We propose this stochastic process in a limit $N → ∞$ as a mathematical definition of the fixed manifold of decaying Turbulence.
The proof of this conjecture and extension to higher dimensions is left for a detailed mathematical study, which is beyond the scope of this work.
We coded these SDE in [
] using
. This code may be useful for theoretical development, but the optimized computations should be translated into Python and C++ and run on a supercomputer or a Tensorflow cluster.
Before even attempting such a computation, the random walk algorithm must be optimized. Its computational complexity grows as $N 4$ per time step. These issues will be addressed in a subsequent
publication, where we modify the random walk to reduce the $N 4$ complexity to a linear one and optimize it for massively parallel execution on a supercomputer cluster.
Once we fix the initial value at one of the two mirror fixed points
$Φ → k , Φ → N − k *$
, the evolution is unambiguous, unlike the global description of the manifold in
Section 4
, where we had to choose between four solutions of two quadratic equations for the point
${ α 1 , α 2 , β }$
in de Sitter space
$d S 2$
We are still left with a choice of one of the two mirror solutions or, in the general case, the coefficients of their linear superposition in the Wilson loop.
Such linear superposition will still solve the loop equation (
), as this equation is
in loop space. This superposition is found in the next Section.
4.8. Mirror symmetry and inequality for the Wilson loop
There is an obvious problem with the solution we have found. The loop equation for
$P → ( θ )$
is complex, and so is the solution, particularly the vorticity in (
). Since the equation for
$P →$
is nonlinear, we cannot take a real part of
$P →$
The negative imaginary part of the circulation in momentum space may lead to violation of inequality .
Here is how we suggest to solve this problem.
In the previous Sections, we described two mirror solutions, originating in (
) and evolving by an SDE (
For any particular loop, we have to choose the solution with the positive imaginary part of the circulation
$Γ = ∮ d C → ( θ ) · P → ( θ ) ;$
The averaging $⋯$ corresponds to averaging over the stochastic process or, equivalently, over the stochastic time $τ$. On top of that, there is averaging over global rotation $F → k ⇒ O ^ · F → k$
over the group measure for $O ^ ∈ S O ( d )$.
At any moment of stochastic time, the inequality restricts the loop C, but not the momentum loop $P →$: for some loops C, the circulation $Γ$ has a positive imaginary part; for other loops, the
reflected circulation $Γ ˜$ does.
This choice is like selecting a decaying wave function for the bound state in the Schrödinger equation for a quantum potential problem.
The theta functions in this solution represent certain boundary conditions for the loop functional in the areas (if they exist) where $Im Γ = 0$ or $Im Γ ˜ = 0$.
Let us study this constraint in more detail. The general solution of the loop equation involves above-mentioned averaging over the global rotation of the whole momentum loop $P → ( θ )$.
We can write the solution as follows (in three dimensions)
$Γ Ω = ∮ d θ C → ′ ( θ ) · Ω ^ · P → ( θ )$
We use the quaternionic representation for the group measure in
Appendix A
to elaborate this group integral, including extra requirements of the positive imaginary part of the circulation.
The reflected solution is treated the same way, with
$Γ ˜ Ω = ∮ d θ C → ′ ( θ ) · Ω ^ · P → * ( 2 π − θ )$
After the addition of the reflected solution, the Wilson loop acquires the reflection symmetry
$Ψ [ γ , C → ( 2 π − θ ) ] = Ψ * [ γ , C → ( θ ) ]$
which symmetry has to be obeyed by a Wilson loop for any real velocity field.
Each of the two complementary terms in $Ψ [ γ , C ]$ is bounded by $1 2$, which provides desired inequality after the addition of the reflection.
Verifying the normalization condition $Ψ [ 0 ] = 1$ for the loop reduced to a point $C = 0$ is straightforward.
4.9. Vorticity distribution and energy dissipation
The simplest quantity to compute in our theory is the local vorticity distribution.
As we shall see, it determines the energy dissipation rate.
The local vorticity for our decaying solution of the loop equation
$ω → = − ı F → ( θ ) × Δ F → ( θ → ) 2 ( t + t 0 ) ;$
Here $θ$ is an arbitrary point at the loop, which makes this expression a random variable.
Note that viscosity is canceled here, as it should be by dimensional counting (vorticity has the dimension of $1 / t$).
In our random walk representation, the complex vorticity operator
$ω → k = − ı G → k 2 ( t + t 0 ) ;$
$G → k = F → k × F → k + 1 = F → k × q → k ;$
The time derivative of energy density in our theory is
$κ = − 1 N ∑ G → k 2 + ( G → N − k * ) 2 2 = − 1 N ∑ Re G → k 2 ;$
Solving this equation with boundary value
$E ( t = ∞ ) = 0$
we relate
$t 0$
to mean initial energy
The probability distribution of
and its mean value
can be computed using our random walk. For the anomalous dissipation, we need the mean enstrophy to diverge [
] so that viscosity is compensated in the extreme turbulent limit.
As we shall see later, in
Section 5
, this happens in our numerical simulations.
The microscopic picture of this infinite enstrophy differs from the singular vortex line.
In the Euler theory, divergence came from the singularity of the classical field. However, in our dual theory, it comes from the large fluctuation of the fractal curve in momentum space.
4.10. The Group average of the Wilson loop and manifest inequality
We can now write down our result for the Wilson loop in decaying Turbulence as a functional of the contour
. We limit ourselves to the three-dimensional case:
The finite steps approximation we considered above
For the simplest circular loop in an
$x y$
plane, we have
$C → ( θ → ) = r { cos θ , sin θ , 0 } ;$
We observe that even at the large time
$t ≫ t 0$
when the asymptotic fractal curve is already in place, there is a region of parameters
where the Wilson loop is a nontrivial universal function of a single variable. We compute these group integrals in
Appendix A
$T i j α β = tr σ i σ j τ α τ β † ;$
$τ α = { 1 , ı σ → } ; α = ( 0 , 1 , 2 , 3 ) ;$
The function
$W ( R ^ )$
only depends on invariants. For our complex symmetric matrix, these invariants can be chosen as four eigenvalues
$τ i$
of its imaginary part, plus ten independent components
$r i j = r j i$
of the real part in the basis of the imaginary part
$Im R ^ · n → i = τ i n → i ; i = 1 ⋯ 4 ;$
$r i j = n → i · Re R ^ · n → j$
The reflected term involves the complex conjugate fractal curve $q → N − k *$. However, this reflected term is not a complex conjugate of the first one.
Thus, the Wilson loop is a complex function despite its reflection invariance. In other words, the dissipative effects are present in a big way in our solution.
We can compute our prediction for this function by numerically simulating the SDE for our vectors $F → k$ and wait for the results with physical or numerical experiments in conventional
three-dimensional decaying Turbulence.
4.11. Correlation functions
The simplest observable quantities we can extract from the loop functional are the vorticity correlation functions [
], corresponding to the loop
backtracking between two points in space
$r → 1 = 0 , r → 2 = r →$
, see
Figure 1
. The vorticity operators are inserted at these two points.
The correlation function reduces to a random walk with a complex weight
$S → m , n = ∑ m n F → k n − m ( mod N ) ;$
The averaging $⋯$ in these formulas involves group integration $∫ O ( 3 ) d Ω$ with $F → k ⇒ Ω → · F → k$.
The positivity restrictions are inserted here as a theta function of the positive imaginary part of the circulation, in our case,
With these restrictions, the absolute value of the Wilson loop is bounded by 1 from above.
Presumably, the vorticity vectors $G → m = F → m × F → m + 1$ as well as the vectors $S → m , n$ are distributed by some power laws in our random walk on a fixed manifold; this would lead to scaling
laws with some fractal dimensions.
The numerical simulation of this correlation function would require significant computer resources.
Still, these resources are much more modest than those for full d dimensional simulations of the Navier-Stokes equation.
In our theory, the dimension of space enters as the number of components of the one-dimensional fluctuating field $F → ( θ )$ rather than the number of variables $r → ∈ R d$ in the fluctuating
velocity field $v → ( r → )$.
Also, note that our quantum problem of the complex random walk naturally fits quantum computer architecture. Thus, in the future, when large quantum computers would become available for researchers,
we can expect a real breakthrough in numerical simulations of the loop equation.
5. Open loop computations
We wrote a
program [
] generating our random walk, starting with a random complex vector
$F → 0$
and using random orthogonal
$S O ( 3 )$
$O k$
at every step. If there is more than one real solution for
, we have chosen the shortest step in Euclidean metric , i.e., the one with minimal
$| β |$
We have chosen the simplest circular coordinate loop
in (
) and imposed the inequality
$Im Γ > 0$
on the last step.
For 1000 steps, it takes a few seconds on a laptop to compute the whole path. We generate a parallel table of 100000 paths, each with 1000 steps, with various random initial vectors $F → 0$ with a
random set of rotation matrices for each step.
The path’s closure requires a numerical solution of the SDE (
), which we plan to implement later on a supercomputer. This open path simulation only covers the big space of the direct product of rotation matrices at every step; the true turbulent fixed point
corresponds to the projection of this space onto the closure condition. We cannot do it at the global level, only at the level of the SDE described in the previous Section.
Thus, this open-path simulation cannot be used for predictions of fractal dimensions in the scaling laws; this has to wait until the SDE simulation is performed at the supercomputer.
With these comments in mind, let us analyze the open curves’ fractal properties, discarding the closure conditions.
The simplest quantity to compute is a fractal dimension
$d f$
of this random walk, defined as
The ordinary Brownian motion (linear random walk) has $d f = 2$, but our random walk is very different, mainly because the Euclidean distance of an elementary step in De Sitter space is unlimited
from above (though it is limited by 1 from below).
Here is the plot of vs
$log N$
Figure 2
The statistical data for parameters
$Estimate Standard Error t - Statistic P - Value 1 − 2.33876 0.0241623 − 96.7941 4.3388931215906555 ` * ∧ - 42 ξ 0.976443 0.00457433 213.461 2.1004008453460444 ` * ∧ - 53$
This data is compatible with $d f = 1.02412 ± 0.005$.
The distribution of the Euclidean length of each step (
Figure 3
The statistical table for the parameters of this fit
$Estimate Standard Error t - Statistic P - Value 1 13.0661 0.00203512 6420.3 0 . log ( step ) − 2.00076 0.000737843 − 2711.64 0 .$
The mean is finite, , but the variance of the step is divergent.
Such a slow decay of the step distribution undermines the concept of a finite fractal dimension as defined in (
). The linear fit is inadequate for such large statistics.
With large statistics, one can reach a perfect fit by adding the next correction to the linear log-log law
The fit at larger interval of
becomes perfect, with a very different coefficient
in front of
$log N$
$Estimate Standard Error t - Statistic P - Value 1 11.4051 0.067611 168.688 1.201777144099837 ` * ∧ - 128 ξ 4.85103 0.0218164 222.357 4.3433787030416533 ` * ∧ - 141 log ( ξ ) − 20.5553 0.109809 −
187.192 2.4775697694919745 ` * ∧ - 133$
Our random walk with unbounded step size differs from an ordinary fractal curve. The fractal dimension does not properly describe this random object as the distance grows by a more complex law than a
pure power of the number of steps.
Another interesting distribution is the enstrophy density
in (
The CDF is shown in
Figure 5
. The tail is compatible with
$κ − 0.936266$
decay, corresponding to the
$κ − 1.936266$
decay of the PDF. The mean value and all higher moments diverge, leading to anomalous dissipation.
The statistical table for the parameters of this fit
$Estimate Standard Error t - Statistic P - Value 1 17.1851 0.00330287 5203.09 0 . log ( κ ) − 0.936266 0.000320989 − 2916.81 0 .$
The computation of the Wilson loop and related correlation functions of vorticity needs an ensemble of closed fractal loops with various sets of random matrices.
The closure condition for the loop would require some computational effort because the probability of the random curve with fractal dimension $d f ∼ 1 .$ returning to an initial point goes to zero
with the increased number of steps.
An alternative approach of starting with a large closed loop $F → k ; F → N = F → 0$ and randomizing it point by point while preserving its closure.
This approach would replace an SDE (
) with a Monte-Carlo process in a closed polygon space. Each step would correspond to a small shift of a few subsequent vertices
$F → k + 1 ⋯ F → k + L$
of the polygon preserving the sequence’s first and last vertex
$F → k , F → k + L + 1$
. This small shift must also preserve the recurrent equations (
Section 3.2
) involving this sequence.
The first approximation to this shift would be our solution in
Appendix B
of the linearized equations. Then, a Newton iteration will finalize this shift to fulfill the quadratic relations (
Section 3.1
These extra layers of computational complexity would require a supercomputer, which we plan to do later.
6. Discussion
6.1. The Duality of Turbulence
We have presented an analytical solution of the Navier-Stokes loop equations for the Wilson loop in decaying Turbulence as a functional of the shape and size of the loop in arbitrary dimension $d >
The solution expresses the probability distribution and expected value for the Wilson loop at any given moment t in terms of a nonlinear SDE for the dual loop in complex momentum space as a function
of auxiliary time $τ$. The loop is approximated as a polygon with $N → ∞$ sides.
Our solution also depends on the arbitrary dimensionless positive constant $γ$, corresponding to the frequency of the Fourier transforms from the Wilson loop to the PDF of circulation. This parameter
explicitly enters our reduced loop equations for a momentum-space fractal curve.
Compared to the original Navier-Stokes equation, this is the reduction of $d ⇒ 1$ of the dimension of space. This SDE is straightforward to simulate by a Monte Carlo method.
The equivalence of a strong coupling phase of the fluctuating vector field to quantum geometry is a well-known phenomenon in gauge theory (the ADS/CFT duality), ringing a bell to the modern
theoretical physicist.
In our case, this is a much simpler quantum geometry: a fractal curve in complex space.
An expert in the traditional approach to Turbulence may wonder why the Loop equation’s solutions have any relation to the velocity field’s statistics in a decaying turbulent flow.
Such questions were raised and answered in the last few decades in the gauge theories, including QCD[
] where the loop equations were derived first [
Extra complications in the gauge theory are the short-distance singularities related to the infinite number of fluctuating degrees of freedom in quantum field theory. The Wilson loop functionals in
coordinate space are singular in the gauge field theory and cannot be multiplicatively renormalized.
Fortunately, there is no short-distance divergence in the Navier-Stokes equations nor the Navier-Stokes loop dynamics. The Euler equations represent the singular limit, which, as we argued, should be
resolved using singular topological solitons regularized by the Burgers vortex.
In the present theory, we keep viscosity constant and do not encounter any short-distance singularities. The anomalous dissipation is achieved in our solution via a completely different mechanism.
The loop equation describes the gauge invariant sector of the gauge field theory. Therefore, the gauge degrees of freedom are lost in the loop functional. However, the gauge-invariant correlations of
the field strength are recoverable from the solutions of the loop equation[
6.2. Stokes-type functionals and vorticity correlations
There is no gauge invariance regarding the velocity field in fluid dynamics (though there is such invariance in the Clebsch variables [
]). The longitudinal, i.e., a potential part of the velocity, has a physical meaning – it is responsible for pressure and energy pumping. This part is lost in the loop functional, but is recoverable
from the rotational part (the vorticity) using the Biot-Savart integral.
In the Fourier space, the correlation functions of the velocity field are algebraically related to those of vorticity $v → k = ı k → × ω → k k → 2$. Thus, the general solution for the Wilson loop
functional $Ψ [ γ , C ]$ allows computing both vorticity and velocity correlation functions.
The solution of the loop equation with finite area derivative, satisfying Bianchi constraint, belongs to the so-called Stokes-type functionals [
], the same as the Wilson loop for Gauge theory and fluid dynamics.
As we discussed in detail in [
], any Stokes-type functional
$Ψ [ γ , C ]$
satisfying boundary condition at shrunk loop
$Ψ [ 0 ] = 1$
, and solving the loop equation can be iterated in the nonlinear term in the Navier-Stokes equations (which would apply at large viscosity).
The resulting expansion in inverse powers of viscosity (weak Turbulence) exactly coincides with the ordinary perturbation expansion of the Navier-Stokes equations for the velocity field, averaged
over the distribution of initial data or boundary conditions at infinity.
We have demonstrated in [
] (and also here, in
Section 2.3
) how the velocity distribution for the random uniform vorticity in the fluid was reproduced by a singular momentum loop
$P → ( θ )$
The solution for
$P → ( θ )$
in this special fixed point of the loop equation was random complex and had slowly decreasing Fourier coefficients, leading to a discontinuity
$sign ( θ − θ ′ )$
in a pair correlation function (
). The corresponding Wilson loop was equal to the Stokes-type functional (
Our general Anzatz (
) satisfies the loop equation and boundary condition at
$Ψ [ C = 0 ] = 1$
. It has a finite area derivative, which obeys the Bianchi constraint, making it a Stokes-type functional.
The exact solution for $P → ( θ )$ in decaying Turbulence which we have found in this paper, leads to the Stokes functional $Ψ [ γ , C ]$ satisfying the boundary value $Ψ [ 0 ] = 1$ at the shrunk
Therefore, it represents a statistical distribution in a turbulent Navier-Stokes flow, corresponding to the degenerate fixed point of the Hopf equation for velocity circulation. It sums up all the
Wylde diagrams in the limit of vanishing random forces plus nonperturbative effects, which are missed in the Wylde functional integral. Whether this exact solution is realized in Nature remains to be
6.3. Random walk around the Turbulent manifold
The fixed point we have found is infinitely more complex than the randomly rotated fluid; our curve $P → ( θ )$ has a discontinuity at every $θ$, corresponding to a distributed random vorticity.
This solution is described by a fractal curve in complex d dimensional space, a limit of a random walk with nonlinear algebraic relations between the previous position $F → k$ and the next one $F → k
+ 1$. These relations are degenerate: each step $q → k = F → k + 1 − F → k$ is characterized by an arbitrary element and an arbitrary element . This step also depends upon the previous position $F →
k$, making this process a Markov chain.
The periodicity condition $F → N = F → 0$ provides a nonlinear equation for an initial position $F → 0$ as a function of the above free parameters $O ^ k , w → k$.
This periodicity condition presents a hard problem, particularly in the limit $N → ∞$, when the probability of our random walk returning to the initial point afterN steps rapidly diminishes with N.
We found a way around this problem by utilizing an exact periodic solution (
) to the momentum loop equations (
Section 3.1
). This analytical solution for arbitrary
is equivalent to a periodic Ising chain or a random walk on a circle with constant angular steps or random signs. This sequence of
$F → k$
can serve as initial data for the SDE, which preserves the periodicity. In
Appendix B
, we constructed this SDE for
$d = 3$
, leaving the generalizations to mathematicians.
This SDE describes the Brownian motion of the rotation matrices
$O ^ k ∈ S O ( 3 )$
in our canonical representation (
Section 4.1
) of the solution to the discrete loop equations (
Section 3.2
). Each matrix moves independently, while the remaining parameters
${ α 1 , α 2 , β }$
move around de Sitter space
$d S 2$
to satisfy the loop equation (
Section 3.2
The closure condition further restricts the set of N infinitesimal rotations $δ q → k = δ θ → k × q → k$: there are three linear relations between N vector parameters $δ θ → k$ of these rotations. We
found the projection matrix required to project the whole array of vector rotations $δ θ → k$ onto the quotient space, satisfying the closure condition.
After this projection, we obtain the motion in the quotient space, where the closure condition is satisfied at every step. We have found the tangent space to our discreet loop equations and factored
out the normal directions (i.e., the null space of linearized equations) like it is done with Brownian motion on a sphere.
Presumably, this SDE uniformly covers our fixed manifold $T 3 ( N )$ for arbitrary N. The limit $N → ∞$ presents a computational challenge, and we are planning to address this challenge in the next
publication using a supercomputer.
6.4. Simplified version of numerical Simulation
We simulated the open random walk (without the closure condition) in three dimensions and studied its statistical properties. We have chosen $γ = 1$ at this early stage of our studies; later, we
investigate the $γ$ dependence.
The distribution of lengths of steps in Euclidean six-dimensional space $Re F → , Im F →$ has a long tail $PDF ∝ x − 2$.
The fractal dimension is not an adequate characteristic for a random walk with such an intermittent step size, unbounded from above. The linear log-log fit as in (
) yields
$d f ≈ 1.20$
, but this fit is imperfect with our large statistics.
As for the distribution of an enstrophy density, it has a power tail $x − 1.9$ corresponding to an infinite mean value and all higher moments. This infinity is how anomalous dissipation manifests in
our solution.
These numerical simulations must be repeated on a supercomputer with better statistics and more steps. There are many things to do next with this conjectured solution to the decaying turbulence
problem; the first is to look for unnoticed inconsistencies.
One important step is yet to be made: the MC simulation of the SDE (
). Let us assume that the qualitative properties and fractal dimensions we have found for the open fractal curves will stay the same or at least close.
6.5. Preliminary comparison with experiments
As a first test of this hypothesis, let us compare it with various experimental data and those from DNS [
There is no agreement between these data, they vary in Reynolds number, and they have other differences related to the experimental setup. No value n for the decay power $t − n$ would fit all that
data. However, a consensus seems to be around $n ≈ 1.2 − 1.4$, which means faster decay than we have.
We are skeptical about these data. As we recently learned [
], there is a regime change at large Reynolds numbers; the numbers achievable in modern DNS may belong to such a transitional regime.
Besides, fitting powers is not a reliable method of deriving physical laws.
For example, we took a formula
$1 / ( t − 0.5 )$
, added random noise between
$( − 0.1 , 0.1 )$
and fitted this data to
$b t − n$
. The best fit produced some fake power
$n ≈ 1.43$
and some fake coefficient
$b ≈ 1.88$
in front (see
Figure 6
Instead, one should compare a hypothetical theory with a null hypothesis by estimating the log-likelihood of both fits. In case the new theory is more likely as an explanation of the data, you may
temporarily accept it until a better theory or better data will appear.
A good history lesson is fitting the power n in Newton’s gravity law to explain the astronomic data for the Mercury perihelion before the General Relativity theory. A small correction to $n = 1$
"explained" the data, but this was useless without a theory.
Presumably, our fixed point corresponds to a true infinite Reynolds limit, as it is completely universal and does not depend on the Reynolds scales.
If you assume no hidden scales are left, our $E ∝ ν / t$ law follows from dimensional analysis. Observed or simulated data with $n > 1$ all have the powers of some other dimensional parameters
related to the Reynolds number. They rely on (multifractal versions of) K41 spectra and other intermediate turbulent phenomena.
We have an anomalous dissipation rate: the mean value of the vorticity square diverges, compensating for the viscosity factor in the energy decay in extreme turbulent limit.
This mechanism of anomalous dissipation differs from the one we studied in the Kelvinon [
]. In those fixed points, the viscosity canceled in the dissipation rate due to the singular vorticity configurations with the thin vortex line resolved as a core of a Burgers vortex.
Here, in the dual theory of fractal momentum loop, the large fluctuations of this momentum loop would lead to the divergent expectation value of the enstrophy.
6.6. Conclusion
Our solution is universal, rotational, and translational invariant. It has the expected properties of extreme isotropic Turbulence. Is it THE solution? Time will tell.
Appendix A The O(3) group average with and without restrictions
The correlation function (
) involves an integral over the
$O ( 3 )$
with the vector
$V →$
related to our fractal curve
$F → ( θ )$
at any given moment of stochastic time.
The vector $r →$ is real, but generally, the vector $V →$ can have complex components. We shall consider this problem below and start with real vector $V →$.
This integral is a particular case of a Harish-Chandra-Itzykson-Zuber integral formula [
$Δ ( A ) = ∏ i < j ( a j − a i )$
is the Vandermonde determinant.
In our case, , so we use
$n = 2$
formula with
$A = ı σ i r i , Δ ( A ) = 2 ı | r → |$
$B = σ j V j , Δ ( B ) = 2 | V → |$
$σ →$
are Pauli matrices. With proper normalization to 1 at
$r → = 0$
, the HCIZ integral reads
Next, let us consider the case of the complex vector V. In that case, there are separate invariants for real and imaginary parts, and the complex vector cannot be rotated to another one by an
orthogonal transformation.
The HCIZ formula does not apply in this case: the result depends not just on the complex variable $V → · V →$ but in addition, there are three components of $Re V →$ in the coordinate frame where $Im
V →$ points in the z direction (or the other way around).
We consider a more general integral with $tr X ^ · Ω ^$ instead of $r → · Ω ^ · V →$. This integral does not reduce to the HCIZ integral and needs special treatment.
We use a quaternionic representation of
$O ( 3 )$
and write this integral:
$Q ( r → , V → ) i j ; k l = T i j α β T k l μ ν H α β μ ν ( R ^ ) ;$
$T i j α β = 1 2 tr σ i τ α σ j τ β † ;$
$τ α = { 1 , ı σ → } ; α = ( 0 , 1 , 2 , 3 )$
In this representation, we have full
$O ( 4 )$
symmetry of this integral over unit sphere
$S 3$
in four dimensions, plus there is a symmetry of the tensor
$R α β = R β α$
. There is, in fact, only one basic integral, with
related to its derivatives by the real parts of components of
$Re R α β$
$H α β μ ν ( R ^ ) = − ∂ 2 W ( R ^ ) ∂ Re R α β ∂ Re R μ ν$
The integral can be reduced to a calculable one-dimensional integral using the following transformations.
The normalization can be verified by tending $R ^ → ± ı ϵ I ^$. The integral can be computed by taking residue in a double pole at the origin. If this limit is taken from the upper semiplane where
the exponent decreases, the result equals 1; otherwise, from the lower semiplane, the integral yields zero.
In the general case of complex matrix $R ^$, the integration path for $τ$ should be shifted down to the lower semiplane to stay below all the roots of the determinant as a characteristic polynomial
in $τ$.
Our algorithm finds these roots and shifts the path in the complex plane of $τ$ twice farther than the lowest root with a negative imaginary part to avoid singularities in the line integral.
It can be computed numerically in a fraction of a second in
for arbitrary tensor
$R ^$
by the following simple code
Figure A1
, see [
As for the derivatives of this generating function
$W ( R )$
by each component of the tensor
$R α β$
it is given by a well-known formula from perturbation theory
$∂ W ( R ^ ) ∂ R α β = ∑ k = 1 4 ψ k α ψ k β ∂ W ( R ^ ) ∂ R k$
Here the
$ψ k α$
is the eigenvector of
$R ^$
corresponding to eigenvalue
$R k$
. The second derivatives we would need for the product of two
matrices would require the differentiation of the eigenfunctions, given by the same perturbation theory, this time for the eigenfunction
So far, we ignored the requirement of a positive imaginary part of the circulation. This requirement amounts to the insertion of the theta function of this imaginary part.
We can use the $O ( 4 )$ transformations of the vector $q α$ to diagonalize the imaginary part of the tensor $R ^$. Unfortunately, we cannot simultaneously diagonalize these two matrices by an
orthogonal transformation.
Thus we are left with the following restricted integral over the sphere
$S 3$
, using the stereographic coordinates to avoid spurious singularities
The positivity condition in the stereographic coordinates reduces to a positive region of an algebraic surface in
$R 3$
$4 x 2 Im R 1 + 4 y 2 Im R 2 + 4 z 2 Im R 3 + ( 1 − x 2 − y 2 − z 2 ) 2 Im R 4 > 0$
The positivity condition can be resolved analytically for one coordinate, say z, as an algebraic function of the remaining two.
Thus, the positivity inequality selects some calculable regions in
$R 3$
, which speeds up the computation. This representation of
$W ˜ ( R ^ )$
was implemented as a
algorithm [
Figure A2
. The three-dimensional convergent integral over
$R 3$
is still a numerical challenge, which takes a minute of
We also implemented a Python version in the popular library SciPy, but the computations took too much CPU time to be useful.
Appendix B Linearized loop equations in 3D
Let us solve in 3D the linear set of recurrent equations derived in
Section 4.4
Only two independent real parameters exist in
$Im μ → k$
. In the canonical form (
Section 4.1
) in 3D the two parameters were shifts of coordinates
$α 1 , α 2$
with third coordinate
related to them by
$α 1 2 + α 2 2 = 1 + β 2$
We resolve this ambiguity by using the pseudo-inverse
$2 × 2$
matrix. Here are the original complex equations.
$Im μ → k · U → k = ı B k ;$
$U → k = G k q → k × F → k ;$
$V → k = G k q → k − 2 F → k ;$
The solution of these three equations in [
] has the form
$γ → k = Q → k · { ı , 1 }$
$Q ^ k = ( R ^ k † · R ^ k ) − 1 · R ^ k †$
$R ^ k = { Re U → k , Im U → k }$
where the inverse
$3 × 3$
$X ^ − 1$
is understood as pseudo-inverse (dropping the null space part).
The complex number $B k$ is linearly related to $Re μ → k$, the known previous vectors $μ → l < k$ and unknown $λ →$, which yields recurrent relations, which we are going to study below.
After solving these equations, the variation of the closure equation
$∑ l = 0 N − 1 μ → l × q → l = 0$
provides a linear set of six equations for
$Re λ → , Im λ →$
, relating these two vectors to all the rotation vectors
$Re μ k$
Let us now proceed by assuming the following linear set of real vector relations (with some real
$3 × 3$
$M ^ k l , P ^ k , Q ^ k$
to be determined ):
$Im μ → k = ∑ l = 0 k M ^ k l · Re μ → l + P ^ k · Re λ → + Q ^ k · Im λ → ;$
Let us compare it with the above relations (
), (
$Σ k = ∑ l = 0 k − 1 q → l × V → k · ( Re μ → l + ı Im μ → l )$
Comparing the terms, we obtain a set of recurrent equations for
$M ^ , P ^ , Q ^$
where the dual tensor
$X ^$
to the vector
$X →$
is defined as
$X ^ α β ≡ e α β γ X γ ;$
The last two equations can be combined into one complex recurrent equation for
$Γ ^ k = P ^ k − ı Q ^ k$
Finally, the closure equation becomes
This is a complex vector equation for two real vectors
$Re λ → , Im λ →$
$P ^ · Re λ → + Q ^ · Im λ → = R → ;$
$P ^ = ∑ k = 0 N − 1 q ^ k · P ^ k ;$
$Q ^ = ∑ k = 0 N − 1 q ^ k · Q ^ k ;$
The solution is expressed in terms of a block matrix
$H ^ = Re P ^ Re Q ^ Im P ^ Im Q ^$
$Re λ → Im λ → = H ^ − 1 · Re R → Im R →$
There is a following complication [
]. The matrix
$H ^$
viewed as
$6 × 6$
dimensional real matrix has a null space of three real 6-dimensional eigenvectors. These eigenvectors can be combined in complex 3d vectors
$ψ 1 , ψ 2 , ψ 3$
corresponding to the block structure of
$H ^$
$H ^ · Re ψ → i Im ψ → i = 0 , i = 1 , 2 , 3 ;$
Therefore, the complex 3D vector $λ →$ is defined modulo superposition of these three complex eigenvectors.
A particular solution for $λ →$ can be obtained using the pseudo-inverse.
In addition, there are three constraints on the angular variables
$Re μ → k$
. These constraints have the following form [
$∑ k Θ → i k · Re μ → k = 0 ; i = 1 , 2 , 3 ;$
The coefficients
$Θ → i k$
of these constraints are nonlinear functions of the current positions
$F → 0 , ⋯ F → N − 1$
. These constraints lead to projections of the SDE to the quotient space, as we derive in the
Section 4.7
Here are the results for
$λ → , Im μ → k$
in terms of rotation parameters
$Re μ → k$
$I ^$
$3 × 3$
unit matrix)
$λ → = ∑ l = 0 N − 1 Λ ^ l · Re μ → l ;$
$Λ ^ l = I ^ ı I ^ · H ^ − 1 · Re R ^ l Im R ^ l ;$
$R ^ l = ı q ^ l − ∑ n = l N − 1 q ^ n · M ^ n l ;$
$Im μ → k = ∑ l = 0 N − 1 S ^ k l · Re μ → l ;$
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What is the range of the function y = cos x? | HIX Tutor
What is the range of the function #y = cos x#?
Answer 1
The range of a function is all possible output, or $y$, values. The range of $y = \cos x$ is from -1 to 1.
In interval notation, the range is [-1,1] * Note that square brackets [ ] are used because because $y = \cos x$ can actually equal -1 and 1 ( for example, if you plug in $x = \pi$, $y = - 1$).
You can see visually in a graph that $y = \cos x$ can only equal values between -1 and 1 on the $y$-axis, hence that it is why it is the range. The doimain, however, is all real numbers. You can see
that you can plug in all sorts of $x$ values, no matter how infinitely small and infinitely large they are- But you will always get a $y$ value with the restriction of [-1,1]
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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Boolean Algebra - 2. Laws (2024)
Boolean Algebra Laws!
Live by the X. Die by the X.
Now that we understand the basic building blocks of Boolean Algebra it's time to take a look at how they behave and interact. Several of these laws are kinda similar to normal mathematical laws but
slightly different so just be aware of that. In the next section we'll look at how these laws may be applied to expressions to modify and simplify them.
These laws are sometimes also referred to as boolean algebra rules.
Some of these laws may appear a little bit confusing at first. The best way to help make things clearer is to work through a few examples, replacing the terms with different sets of actual values and
working out the result. This will help you to see how the process works and why it behaves the way it does.
There may seem like quite a bit of content on this page but don't be daunted. The laws are actually quite simple. Most of the content is just many examples to reduce any ambiguity.
Boolean Commutativity
This law of Boolean Algebra states that the order of terms for an expression (or part of an expression within brackets) may be reordered and the end result will not be affected.
a OR b = b OR a
Or with multiple terms:
a AND b AND c AND d = b AND d AND c AND a
This is also the case for part of an expression within brackets:
a AND (b OR C) = a AND (c OR b)
The brackets may be considered a single term themselves (remember, everything in Boolean Algebra always results in either True or False).
x OR (y AND z) = (y and z) OR x
Commutativity works for any operation which accepts two or more terms (eg. AND, OR, NOR, NAND, XOR).
Boolean Identity
The identity law observes how certain expressions will behave when one of the terms is fixed.
A term in an OR operation with a fixed value of False will result in the term:
g OR False = g
Similarly, a term in an AND operation with a fixed value of TRUE will result in the term.
h AND True = h
Boolean Complement Law
The complement law has to do with the relationship between a variable ( eg. c ) and the negation of that variable ( not(c) ). Let's consider the following expression:
c OR NOT(c) = True
If c is False then Not(c) must be True. And vice versa. Either way, one of them is always going to be True so the result will always be True.
c AND NOT(c) = False
Taking into account the observation above it is impossible for both c and Not(c) to be True at the same time so this is always going to be False.
Boolean Idempotent Law
This law has to do with if a variable is repeated within an expression. Effectively it may be simplified to itself. So for instance:
r OR r = r and r AND r = r
If you think about it this makes sense as both sides of the expression are always going to be the same value if they are the same variable.
Boolean Double Negation Law
This law also makes sense once you think about it. This law states that if you negate a negation (ie if you have a NOT within a NOT) they effectively cancel each other out.
NOT(NOT(b)) = b
The first NOT flips the value of b, then the second NOT flips it back again.
This is only the case when both NOT's are applied to the same item.
NOT(NOT(d AND f)) = d AND f
NOT(NOT(v) OR t) ≠ v OR t
This is because the outer NOT applies to NOT(v) OR t but the inner NOT applies only to v. They apply to different parts of the expression and so may not be cancelled out.
Boolean Associativity
This law looks at brackets (or groupings) within an expression and how they may be reorganised or even removed. If all the operators within an expression (or part of an expression) are the same then
this may be done. So:
a OR b OR c = (a OR b) OR c = a OR (b OR c) = (a OR c) OR b
a AND b AND c = (a AND b) AND c = a AND (b AND c) = (a AND c) AND b
It may also be done for just part of an expression which fits. So:
a AND (b OR c OR d) = a AND (b OR (c OR d))
But if the operators are mixed then this may not happen.
a AND (b OR c) ≠ (a AND b) OR c
If this doesn't quite make sense then try assigning some values to a, b and c and create a truth table to observe the outcome for both expressions. This will help to make it clearer.
Boolean Distributivity
This law is similar to distributivity in normal mathematics and has to do with expanding or simplifying brackets. This may be done when one of AND or OR is inside the brackets and the other is
e AND (f OR g) = (e AND f) OR (e AND g) and e OR (f AND g) = (e OR f) AND (e OR g)
Notice that the two variations are the inverse of each other and that in both situations the operation which is inside the brackets on one side of the expression is outside the brackets on the other
side (and vice versa).
de Morgans' Laws
de Morgans' laws may seem a little odd at first but in later sections you will see that they actually become very useful when attempting certain types of manipulations.
Like I've suggested for a few of the laws above, try putting some values into a truth table for both sides of the above expressions and this will help you to gain a better feel for how they operate.
Here they are:
NOT(p AND k) = NOT(p) OR NOT(k) and NOT(p OR k) = NOT(p) AND NOT(k)
Boolean Absorptive Law
The Absorptive law is another one which is useful in simplification (normally after rearranging the expression using other laws).
s AND (s OR w) = s and s OR (s AND w) = s
Remembering the Laws
Remembering the laws can be useful. If you're having to simplify expressions often it is more convenient if you don't have to look them up constantly. Also, if you're learning this as a student,
often you will be required to remember them for an exam.
Practice is the best way to achieve this. As with a lot of things in Boolean Algebra, the laws are logical. You will have a much better time remembering them if you understand why they work.
You'll also note that there is a reflection between AND and OR in a lot of the laws. This is known as duality. You can take advantage of this to reduce the amount you need to remember in order to
recreate the laws.
For example...
a OR b = b OR a
a OR False = a, h AND True = h
a OR NOT(a) = True, a AND NOT(a) = False
a OR a = a, a AND a = a
Double Negation
NOT(NOT(a)) = a
a OR b OR c = (a OR b) OR c = a OR (b OR c)
a AND b AND c = (a AND b) AND c = a AND (b AND c)
a AND (b OR c) = (a AND b) or (a AND c), a OR (b AND c) = (a OR b) AND (a OR c)
de Morgans'
NOT(a AND b) = NOT(a) OR NOT(b), NOT (a OR b) = NOT(a) AND NOT(b)
a AND (a OR b) = a, a OR (a AND b) = a
Notice the patterns and reflections of AND's and OR's in the rules and this will help you to more easily remember them.
Can't you just put the Truth Tables here for us?
On several occassions in this chapter I have suggested you create a truth table to observe how the rule behaves. You may be of the opinion that it would be easier and more convenient if I just put
the truth tables here on the page. The reason I have not done this is that Boolean Algebra is quite an abstract beast and you can really only fully comprehend it by doing it yourself.
I have had many many students tell me that 'they can just look at an example, they get it, it's fine'. When it came to actually implementing the laws however, with the exception of a very few
talented students, all have found that they didn't quite understand to the extent they thought they did.
Learning Boolean Algebra is like learning to ride a bike, or to juggle. You may read all the reference material you like but you won't stand a chance of mastering it until you get in and start doing.
Boolean Algebra Expressions
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Jun Yu, BICMR, Peking University
Curriculum Vitae pdf
Research Interests Lie groups, representation theory, automorphic forms.
A proof of CasselmanĄ¯s comparison theorem. Representation Theory 25 (2021), 994-1020.
Acceptable compact Lie groups. Peking Mathematical Journal (2021).
Weighted badly approximable vectors and games. IMRN (2019), no. 3, 810-833.
Maximal abelian subgroups of compact simple Lie groups of type E. Geom. Dedicata 185 (2016), 205–269.
On the dimension datum of a subgroup. Duke Math. J. 165 (2016), no. 14, 2683-2736.
Rational rigidity for F4(p). Adv. Math. 302 (2016), 48–58.
A compactness result for dimension datum. IMRN (2015), no. 19, 9438–9449.
Algebraic vector bundles on punctured affine spaces and smooth quadrics. Adv. Math. 257 (2014), 337–348.
Elementary abelian 2-subgroups of compact Lie groups. Geom. Dedicata 167 (2013), 245–293.
Klein four-subgroups of Lie algebra automorphisms. Pacific J. Math. 262 (2013), no. 2, 397-420.
On the dimension datum of a subgroup and its application to isospectral manifolds. JDG 94 (2013), no. 1, 59-85.
Existence of universal entangler. J. Math. Phys. 49 (2008), no. 1.
A strong multiplicity one theorem for dimension data. Preprint, arXiv:2111.13343.
The functorial source problem via dimension data. Preprint, arXiv:2111.13341.
Restriction of irreducible unitary representations of Spin(N,1) to parabolic subgroups. Preprint, arXiv:2010.01026.
A geometric interpretation of Kirillov’s conjecture. Preprint, arXiv:1806.06318.
Twisted root systems of a (*)-subgroup. Preprint, arXiv:1805.06330.
Links PKU, BICMR, arXiv, AMS.
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Equity & Excellence
Equity and Excellence
A Call for Collective Action
Join CMC-South and members of the Mathematics Education Community for a second year dedicated to building our collective knowledge and understanding of topics and issues related to Equity and Social
Justice in Mathematics so that we may begin to take action.
Contributing Organizations & Educators
Book Reflection Questions Focus Questions for the Year
How might teachers begin to teach mathematics for social justice? How might 1. How might we, the mathematics education community, make a difference in the teaching and learning of mathematics
teacher educators begin to teach teachers how to teach mathematics for social “that promote rich, rigorous, and relevant mathematical experiences” for all students? What key actions should we
justice? consider?
2. How does the reading further inform or challenge your understandings of issues related to equity and social
How might teaching mathematics for social justice "look like?" justice in mathematics education? What question(s) do you have in regards to the reading(s)?
How can mathematics be re-envisioned as a means to create a more socially just @CAMathCouncil
world? #mathequity
Calendar of Readings
November - December 2017 Reading (Webinar January 2018)Suggested by the National
Council of Teachers of Mathematics (NCTM)The Impact of Identity in K-8 Mathematics.
Targeted Questions:
• What are equitable instructional practices that support the development of
students' mathematical identity and sense of agency?
• How can we advocate for the implementation of these practices?
January - February 2018 Reading (Webinar March 2018)Suggested by the National July - August 2018 (Webinar September 2018)Suggested by the Association of Mathematics Teacher Educators (AMTE)
Council of Supervisors of Mathematics (NCSM)Excellence Through Equity: Five Chapter 16: How Do I Learn to Like This Child So I Can Teach Him Mathematics: The Case of Rebecca (Mary Q. Foote
Principles of Courageous Leadership to Guide Achievement for Every Student (2016) with accompanying commentaries)
by Alan M. Blankstein and Pedro Noguera
• Commentary 1: Examining Interest Convergence and Identity: A Commentary on Foote’s Case, Robert Q. Berry III.
Blanstein & Noguera (2016) talk about courage as the essential human virtue, and • Commentary 2: Supporting a Teacher’s Shift from Deficits to Funds of Knowledge: A Commentary on Foote’s Case,
how courageous leadership is the “engine that drives the paradigm shift”. Maura Varley Gutiérrez.
• Commentary 3: A Commentary on Foote’s Case, Nora G. Ramírez.
Targeted Questions:
Chapter from: White, D. Y., Crespo, S. & Civil, M. (Eds.) (2016). Cases for mathematics teacher educators:
• What are the five principles of courageous leadership to guide achievement for Facilitating conversations about inequities in mathematics classrooms. Charlotte, NC: Information Age Publishing.
every student discussed by these authors? Link to purchase book: http://www.infoagepub.com/products/Cases-for-Mathematics-Teacher-Educators Targeted
• How does your organization’s vision reflect the five components of courageous Questions:
leadership? What can we do together to make visible these components?
• How would you support the teacher in addressing her negative views about the student and his mother?
March - April 2018 Reading (Webinar May 2018)Suggested by California Mathematics • In what ways did the case and commentary authors’ suggestions help you think about the equity-related dilemmas
Council-South (CMC-S)Teaching Mathematics for Social Justice: Conversations with you face in your own work?
September - October 2018 (Webinar November 2018) Suggested by the Journal of Urban Mathematics Education (JUME)
, edited by Anita A. Wager and David W. Stinson Three ArticlesMathematics as Gatekeeper: Power and Privilege in the Production of Knowledge (Martin et al JUME
2010)“Both And”—Equity and Mathematics: A Response to Martin, Gholson, and Leonard (Confrey JUME 2010)Engaging
Targeted Questions: Students in Meaningful Mathematics Learning: Different Perspectives, Complementary Goals (Battista JUME 2010)
Targeted Questions:
• How might teachers begin to teach mathematics for social justice? How might
teacher educators begin to teach teachers how to teach mathematics for social • How might the larger mathematics education community achieve a both-and approach?
justice? • How might the larger mathematics education community begin to respect the different perspectives of doing
• How might teaching mathematics for social justice "look like?" science employed when rigorously examining the critical issues of “diversity” and “equity” in mathematics
• How can mathematics be re-envisioned as a means to create a more socially just education research?
November - December 2018 (Webinar 2019)Suggested by Robert Q. Berry IIIFor White Folks Who Teach in the Hood... and
May - June 2018 Reading (Webinar July 2018)Suggested by TODOS: Mathematics for ALL the Rest of Y'all Too: Reality Pedagogy and Urban Education (2016) by Christopher Emdin. The author discusses
different types of pedagogies grounded in the resources that communities can offer for teaching; The author
Focused article from TEEM introduces 7'Cs.Targeted Questions:
7 • What are ways teachers can gain access to community resources (human & material)?
• How can we make sense of this for mathematics teaching and learning?
- Special Issue:
Mathematics Education Through the Lens of Social Justice
Targeted Questions:
• How do we change the paradigm of what mathematics is and how it should be
learned from its current institutional form to one that utilizes the
mathematics of people and their communities and ties mathematics to the world?
• Which of the examples of social justice and mathematics tasks enacted with
students that were written about in this journal most resonates with you? Why?
A Call for Collective Action Contributing Organizations & Educators
TODOS: Mathematics for ALL
TODOS is a is an international professional organization that advocates for equity and excellence in mathematics education for ALL students - in particular, Latina/o students. TODOS advances
CMC-South Partner educators' knowledge, develops and supports education leaders, generates and disseminates knowledge, informs the public, influences educational policies, and informs families
Orgnaization about education policies and learning strategies. All of these goals ultimately result in providing access to high quality and rigorous mathematics for ALL students.
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Tweaks behind bonuses
The math behind those bonuses are more complex than it looks. The way it works is very different comparing to slots from 20 years ago, where the bonus is as simple as "12 freespins, all winning pays
2x". I'd assume some tweaks are being made to make a more balanced RTP/experience.
Here's some examples.
ELK studios:
Almost all of their regular bonus have a price of 100x, and all super bonus have a price of 500x. But the additional privilige from those super bonus varies. From a much better beginning such as Max
rows, Both Ways on Nitropolis 3/4, to guaranteed super wild strike on Propaganda and Katmandu X, or a sticky multiplier in Ashoka. But after all those effects, the price of super bonus remains 5
times the price of regular bonus, and both comes with the same RTP.
Nitropolis 2 is another great example. If you buy a super bonus, you get 2 nitro reels on all 6 reels. Each of them have 14 possible results: 12 regular symbols, wild, and scatter. So you need a
symbol appears on both reel 1, 2 and 3 to win, each reel contains up to 2 types of symbol among 14 possibilities, if there are no tweaks being made, it's likely to cost an unreasonable amount of time
to get a win. And we all know winning by getting multiple "Nitro Match" or "Nitro Wild" in a row is pretty hard.
BTG: Many of their Megaways game comes with a sticky multiplier during the bonus, and it increases as the bonus goes. This should make each spin more valuable than the previous one, and the
difference between a bonus begins with 20 spins should be much more valuable than two bonus that begins with 10 spins.
Now let's take a look at Vegas Megaways. Here you can buy 10 freespins for 75x, or 20 freespins for 150x. This makes me really concerned, if there are sticky multipliers that increases after each
win, how could 20 freespins worth only twice as much as 10 freespins? For this to happen, some tweaks must be done in the background, so that the HF (Hit Frequency) become somewhat lower when one buy
the 20 freespins.
Fascinating breakdown,
. I've always suspected there were hidden mechanics behind these bonuses. The ELK Studios example is particularly interesting.
This kind of analysis is why I stick to simple slots. All these hidden tweaks make me feel like the deck is stacked against us even more than usual.
Great post. It's clear that a lot of thought goes into balancing these bonuses. I wonder how much they test before releasing a game.
The Vegas Megaways example is eye-opening. It does seem like there must be some behind-the-scenes adjustments to make those numbers work.
This is why I always look at the overall RTP rather than getting excited about bonus features. In the end, the math always wins.
This is why I always look at the overall RTP rather than getting excited about bonus features. In the end, the math always wins.
True, but isn't part of the fun trying to hit those big bonus wins? Even if the math is against us in the long run
I'm curious about how these tweaks affect volatility. Do games with more complex bonus structures tend to be more volatile?
I'm curious about how these tweaks affect volatility. Do games with more complex bonus structures tend to be more volatile?
In my experience, yes. Games with elaborate bonus features often have higher volatility. You can go long periods without a big win.
This post makes me appreciate the simpler, older style slots. At least with those, what you see is what you get.
The math behind those bonuses are more complex than it looks. The way it works is very different comparing to slots from 20 years ago, where the bonus is as simple as "12 freespins, all winning
pays 2x". I'd assume some tweaks are being made to make a more balanced RTP/experience.
Here's some examples.
ELK studios:
Almost all of their regular bonus have a price of 100x, and all super bonus have a price of 500x. But the additional privilige from those super bonus varies. From a much better beginning such as
Max rows, Both Ways on Nitropolis 3/4, to guaranteed super wild strike on Propaganda and Katmandu X, or a sticky multiplier in Ashoka. But after all those effects, the price of super bonus
remains 5 times the price of regular bonus, and both comes with the same RTP.
Nitropolis 2 is another great example. If you buy a super bonus, you get 2 nitro reels on all 6 reels. Each of them have 14 possible results: 12 regular symbols, wild, and scatter. So you need a
symbol appears on both reel 1, 2 and 3 to win, each reel contains up to 2 types of symbol among 14 possibilities, if there are no tweaks being made, it's likely to cost an unreasonable amount of
time to get a win. And we all know winning by getting multiple "Nitro Match" or "Nitro Wild" in a row is pretty hard.
BTG: Many of their Megaways game comes with a sticky multiplier during the bonus, and it increases as the bonus goes. This should make each spin more valuable than the previous one, and the
difference between a bonus begins with 20 spins should be much more valuable than two bonus that begins with 10 spins.
Now let's take a look at Vegas Megaways. Here you can buy 10 freespins for 75x, or 20 freespins for 150x. This makes me really concerned, if there are sticky multipliers that increases after each
win, how could 20 freespins worth only twice as much as 10 freespins? For this to happen, some tweaks must be done in the background, so that the HF (Hit Frequency) become somewhat lower when one
buy the 20 freespins.
Have you noticed any patterns in how different providers approach these bonus tweaks? Does NetEnt do things differently from Pragmatic Play, for example?
This kind of stuff is why I prefer live casino games. At least there, I can see exactly what's happening.
This kind of stuff is why I prefer live casino games. At least there, I can see exactly what's happening.
I get that, but even live games have their own mathematical edges built in. It's just more visible.
I wonder how much of this is done for game balance versus maximizing casino profit. Probably a bit of both.
I wonder how much of this is done for game balance versus maximizing casino profit. Probably a bit of both.
Definitely both. They need to make the games engaging enough to keep us playing, but profitable enough to keep the casinos happy.
This post is a good reminder to always check the game info and paytables before playing. Sometimes the details are hidden in there.
As a new player, this is kind of discouraging. How are we supposed to know what's really going on behind the scenes?
As a new player, this is kind of discouraging. How are we supposed to know what's really going on behind the scenes?
, don't be too discouraged. While the exact mechanics might be opaque, the overall RTP is still regulated and published.
I'd be interested to see how these mechanics change in different RTP settings. Do they just scale everything, or change the fundamental math?
This is fascinating stuff, but remember everyone: no matter how the bonuses work, the house always has an edge. Play for fun, not profit.
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How to Calculate Cash Loan Payment
June 2, 2017, 12:45 pm
Have you ever considered taking a cash loan to meet a pressing need? Maybe you needed money to pay for your rent, buy some electronics, buy a car or to start a business…the temptations to take a cash
loan are numerous.
They are numerous credit companies out there willing to give you a cash loan, companies such as Page Mf Bank, Zedvance, RenMoney etc. Unlike the traditional banks, most of these credit companies are
willing to give you a cash loan as long as you have a steady income or salary and the prerequisites to qualify for their loans aren’t as complex as those of traditional banks. All these make their
cash loans very attractive.
However, before taking any cash loan, there is one very important thing you must know and consider. You must know how much you are going to pay back before taking any cash loan. When you know how
much you are going to pay back, you can then decide if your steady income or what you want to use the cash loan for as the capacity to pay off the loan. This is the first thing you must consider
before taking any cash loan. Even the Bible advices us to first consider the outcome of our actions before taking them:
“If one of you wanted to build a tower, wouldn’t you first sit down and calculate the cost, to determine whether you have enough money to complete it?”-Luke14:28
I want to show you how to calculate the amount you’ll pay back considering the different terms and conditions that banks and credit companies apply when giving out cash loans.
The interest rates on cash loans are normally compound interest. Compound interest or compounding means you’ll pay interest on earlier interest. It is the interest you’re charged on a cash loan that
provides the increase in what you repay and makes profits for the institutions giving you a cash loan.
Compound interest can be calculated in 3 ways:
Cash Loan
Interest is per annum and payable at the end of the year (once a year)
The formula for calculating the repayment in this arrangement is
A = P (1+r )^n
Where A is the future sum to be repaid
P is the principal or amount borrowed
r is the interest rate on the loan
n is the period or duration of the loan
To illustrate this, let’s use an example.
Mr. John borrows [S:N:S]100,000 from a bank for 4 years at an annual interest rate of 10%. What is the future amount he’ll repay?
Going by the formula above, P =[S: N:S] 100,000; r = 10% or 0.1; n = 4
A = [S:N:S]100,000 (1 + 0.1)^4
A = [S:N:S] 146,410; this is the amount Mr. John will repay.
Interest paid on the [S:N:S] 100,000 cash loan is [S:N:S] 46,410 ([S:N:S] 146,410 – [S:N:S] 100,000)
Interest is per annum but payable other than annually (more than once per year) i.e. biannually or quarterly
The formula for this arrangement is
A = P (1+[r/m] )^nm
Where m is the number of times the interest will be charged per year
Another example will be used to illustrate how this is calculated:
Mr. John borrows [S:N:S] 100,000 from a bank for 4 years. The interest rate is 10% per annum. What will be the amount he repays if:
1. The interest is charged half-yearly (twice a year)
2. The interest is charged quarterly (four times a year)
For interest charged half-yearly, where P = [S: N:S]100,000; r = 10% or 0.1; n = 4; m = 2
Amount to be repaid (A) = [S:N:S]100,000 (1 + [0.1/2])^4×2
= [S:N:S] 147,745
The interest to be repaid in this case is [S:N:S] 47,745 (i.e. [S:N:S] 147,745 – [S:N:S] 100,000)
For interest charged quarterly, in this case m = 4
Amount to be repaid (A) = [S:N:S] 100,000 (1+[0.1/4])^4×4
= [S:N:S] 148,845
The interest to be repaid in this case is [S:N:S] 48,845 (i.e. [S:N:S] 148,845 – [S:N:S] 100,000)
Where interest is charged at a rate other than per annum i.e. per month
The formula for this arrangement is
A = P (1 + r )^nm ; where m is the number of months in 1 year
So, if Mr. John borrows [S:N:S] 100,000 from a bank for 4 years and his cash loan is to attract 1% per
month. How much will he repay?
P = [S:N:S]100,000; r= 10% or 0.1; n=4; m=12
Amount to be repaid (A) = [S:N:S] 100,000 (1+0.1)^4×12
= [S:N:S] 161,222
The interest to be repaid in this case is [S:N:S] 61,222 (i.e. [S:N:S] 161,222 – [S:N:S] 100,000)
Also, the bank or credit company will charge you a one-off management fee for giving you the cash loan. This charge is calculated in simple interest i.e. you only pay interest on the amount you want
to borrow. So from our example above, if the credit company charges Mr. John a management fee of 3% for the [S:N:S] 100,000 he borrows, he pays an extra management fee of:
Management fee = 3% X [S:N:S]100,000
= 0.03 X [S:N:S] 100,000
= [S:N:S] 3,000
In addition to the compound interest he’ll pay back on the actual cash loan (i.e. [S:N:S] 100,000) he’ll also pay back [S:N:S] 3,000 as management fee. This amount is normally paid once at the outset
of the cash loan.
Go over the examples again and use it to calculate the amount you are going to repay if you’re thinking of taking a cash loan from a credit company or bank. Knowing how much interest you'll pay back
on a cash loan is a very important skillset for managing and investing your money.
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Asymptotic freedom
In particle physics, asymptotic freedom is a property of some gauge theories that causes interactions between particles to become asymptotically weaker as the energy scale increases and the
corresponding length scale decreases.
Asymptotic freedom is a feature of quantum chromodynamics (QCD), the quantum field theory of the strong interaction between quarks and gluons, the fundamental constituents of nuclear matter. Quarks
interact weakly at high energies, allowing perturbative calculations. At low energies the interaction becomes strong, leading to the confinement of quarks and gluons within composite hadrons.
The asymptotic freedom of QCD was discovered in 1973 by David Gross and Frank Wilczek,[1] and independently by David Politzer in the same year.[2] For this work all three shared the 2004 Nobel Prize
in Physics.[3]
Asymptotic freedom in QCD was discovered in 1973 by David Gross and Frank Wilczek,[1] and independently by David Politzer in the same year.[2] The same phenomenon had previously been observed (in
quantum electrodynamics with a charged vector field, by V.S. Vanyashin and M.V. Terent'ev in 1965;[4] and Yang–Mills theory by Iosif Khriplovich in 1969[5] and Gerard 't Hooft in 1972[6][7]), but its
physical significance was not realized until the work of Gross, Wilczek and Politzer, which was recognized by the 2004 Nobel Prize in Physics.[3]
The discovery was instrumental in "rehabilitating" quantum field theory.[7] Prior to 1973, many theorists suspected that field theory was fundamentally inconsistent because the interactions become
infinitely strong at short distances. This phenomenon is usually called a Landau pole, and it defines the smallest length scale that a theory can describe. This problem was discovered in field
theories of interacting scalars and spinors, including quantum electrodynamics (QED), and Lehman positivity led many to suspect that it is unavoidable.[8] Asymptotically free theories become weak at
short distances, there is no Landau pole, and these quantum field theories are believed to be completely consistent down to any length scale.
The Standard Model is not asymptotically free, with the Landau pole a problem when considering the Higgs boson. Quantum triviality can be used to bound or predict parameters such as the Higgs boson
mass. This leads to a predictable Higgs mass in asymptotic safety scenarios. In other scenarios, interactions are weak so that any inconsistency arises at distances shorter than the Planck length.[9]
Screening and antiscreening
Charge screening in QED
The variation in a physical coupling constant under changes of scale can be understood qualitatively as coming from the action of the field on virtual particles carrying the relevant charge. The
Landau pole behavior of QED (related to quantum triviality) is a consequence of screening by virtual charged particle–antiparticle pairs, such as electron–positron pairs, in the vacuum. In the
vicinity of a charge, the vacuum becomes polarized: virtual particles of opposing charge are attracted to the charge, and virtual particles of like charge are repelled. The net effect is to partially
cancel out the field at any finite distance. Getting closer and closer to the central charge, one sees less and less of the effect of the vacuum, and the effective charge increases.
In QCD the same thing happens with virtual quark-antiquark pairs; they tend to screen the color charge. However, QCD has an additional wrinkle: its force-carrying particles, the gluons, themselves
carry color charge, and in a different manner. Each gluon carries both a color charge and an anti-color magnetic moment. The net effect of polarization of virtual gluons in the vacuum is not to
screen the field but to augment it and change its color. This is sometimes called antiscreening. Getting closer to a quark diminishes the antiscreening effect of the surrounding virtual gluons, so
the contribution of this effect would be to weaken the effective charge with decreasing distance.
Since the virtual quarks and the virtual gluons contribute opposite effects, which effect wins out depends on the number of different kinds, or flavors, of quark. For standard QCD with three colors,
as long as there are no more than 16 flavors of quark (not counting the antiquarks separately), antiscreening prevails and the theory is asymptotically free. In fact, there are only 6 known quark
Calculating asymptotic freedom
Asymptotic freedom can be derived by calculating the beta-function describing the variation of the theory's coupling constant under the renormalization group. For sufficiently short distances or
large exchanges of momentum (which probe short-distance behavior, roughly because of the inverse relationship between a quantum's momentum and De Broglie wavelength), an asymptotically free theory is
amenable to perturbation theory calculations using Feynman diagrams. Such situations are therefore more theoretically tractable than the long-distance, strong-coupling behavior also often present in
such theories, which is thought to produce confinement.
Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. Essentially, the beta-function describes how the
coupling constants vary as one scales the system \( x \rightarrow bx \) . The calculation can be done using rescaling in position space or momentum space (momentum shell integration). In non-abelian
gauge theories such as QCD, the existence of asymptotic freedom depends on the gauge group and number of flavors of interacting particles. To lowest nontrivial order, the beta-function in an SU(N)
gauge theory with \( n_f \) kinds of quark-like particle is
\( \beta_1(\alpha) = { \alpha^2 \over \pi} \left( -{11N \over 6} + {n_f \over 3} \right) \)
where \( \alpha \) is the theory's equivalent of the fine-structure constant, \( g^2/(4 \pi) \) in the units favored by particle physicists. If this function is negative, the theory is asymptotically
free. For SU(3), one has N = 3, and the requirement that \( \beta_1 < 0 \) gives
\( n_f < {33 \over 2}.
Thus for SU(3), the color charge gauge group of QCD, the theory is asymptotically free if there are 16 or fewer flavors of quarks.
Besides QCD, asymptotic freedom can also be seen in other systems like the nonlinear σ {\displaystyle \sigma } \sigma -model in 2 dimensions, which has a structure similar to the SU(N) invariant
Yang-Mills theory in 4 dimensions.
Finally, one can find theories that are asymptotically free and reduce to the full Standard Model of electromagnetic, weak and strong forces at low enough energies.[10]
See also
Asymptotic safety
Gluon field strength tensor
Quantum triviality
Chemical bond
D.J. Gross; F. Wilczek (1973). "Ultraviolet behavior of non-abelian gauge theories". Physical Review Letters. 30 (26): 1343–1346. Bibcode:1973PhRvL..30.1343G. doi:10.1103/PhysRevLett.30.1343.
H.D. Politzer (1973). "Reliable perturbative results for strong interactions". Physical Review Letters. 30 (26): 1346–1349. Bibcode:1973PhRvL..30.1346P. doi:10.1103/PhysRevLett.30.1346.
"The Nobel Prize in Physics 2004". Nobel Web. 2004. Retrieved 2010-10-24.
V.S. Vanyashin; M.V. Terent'ev (1965). "The vacuum polarization of a charged vector field" (PDF). Journal of Experimental and Theoretical Physics. 21 (2): 375–380. Bibcode:1965JETP...21..375V.
I.B. Khriplovich (1970). "Green's functions in theories with non-Abelian gauge group". Soviet Journal of Nuclear Physics. 10: 235–242.
G. 't Hooft (June 1972). "Unpublished talk at the Marseille conference on renormalization of Yang–Mills fields and applications to particle physics".
Gerard 't Hooft, "When was Asymptotic Freedom discovered? or The Rehabilitation of Quantum Field Theory", Nucl. Phys. Proc. Suppl. 74:413–425, 1999, arXiv:hep-th/9808154
D.J. Gross (1999). "Twenty Five Years of Asymptotic Freedom". Nuclear Physics B: Proceedings Supplements. 74 (1–3): 426–446. arXiv:hep-th/9809060. Bibcode:1999NuPhS..74..426G. doi:10.1016/S0920-5632
Callaway, D. J. E. (1988). "Triviality Pursuit: Can Elementary Scalar Particles Exist?". Physics Reports. 167 (5): 241–320. Bibcode:1988PhR...167..241C. doi:10.1016/0370-1573(88)90008-7.
G. F. Giudice; G. Isidori; A. Salvio; A. Strumia (2015). "Softened Gravity and the Extension of the Standard Model up to Infinite Energy". Journal of High Energy Physics. 2015 (2): 137. arXiv
:1412.2769. Bibcode:2015JHEP...02..137G. doi:10.1007/JHEP02(2015)137.
S. Pokorski (1987). Gauge Field Theories. Cambridge University Press. ISBN 0-521-36846-4.
Hellenica World - Scientific Library
Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License
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JavaScript Comparison Operators - Skill Seminary
JavaScript Comparison Operators: A Comprehensive Guide
Hello, JavaScript enthusiasts! Today, we’re going to dive deep into the world of JavaScript comparison operators. Buckle up, because this is going to be a fun ride!
Table of Contents
Ever wondered how your JavaScript program makes decisions? It’s all thanks to comparison operators! These handy little tools let your code compare values and make decisions based on those
comparisons. So, let’s get to know them better!
Understanding JavaScript Comparison Operators
Comparison operators are the heart of logical statements in JavaScript. They compare two values and return a boolean result: true or false. This allows your code to make decisions and execute
different code blocks based on the comparison result.
Comparison Operators in JavaScript
JavaScript Comparison Operators in Detail
Let’s take a closer look at each comparison operator:
== (Equal to)
This operator checks if two values are equal, after performing type coercion if necessary. For example, 5 == "5" would return true.
=== (Strict equal to)
The strict equality operator checks if two values are equal, without performing type coercion. So, 5 === "5" would return false.
!= (Not equal to)
This operator checks if two values are not equal, performing type coercion if necessary. For example, 5 != "6" would return true.
!== (Strict not equal to)
The strict inequality operator checks if two values are not equal, without performing type coercion. So, 5 !== "5" would return true.
< (Less than)
Checks if the value on the left is less than the value on the right. For example, 4 < 5 would return true.
> (Greater than)
Checks if the value on the left is greater than the value on the right. For example, 5 > 4 would return true.
<= (Less than or equal to)
Checks if the value on the left is less than or equal to the value on the right. For example, 5 <= 5 would return true.
>= (Greater than or equal to)
Checks if the value on the left is greater than or equal to the value on the right. For example, 5 >= 4 would return true.
JavaScript Comparison Operators Examples
Let’s see these operators in action:
console.log(5 == "5"); // true
console.log(5 === "5"); // false
console.log(5 != "6"); // true
console.log(5 !== "5"); // true
console.log(4 < 5); // true
console.log(5 > 4); // true
console.log(5 <= 5); // true
console.log(5 >= 4); // true
JavaScript Comparison Operator Precedence
In JavaScript, comparison operators have higher precedence than logical operators. This means that comparison operations are performed before logical operations. For example, in the expression 5 < 6
&& 7 > 8, the comparisons 5 < 6 and 7 > 8 are performed first, and then the results are used in the logical AND operation.
JavaScript Comparison Operators and Logical Operators
Comparison operators often work hand in hand with logical operators (&&, ||, !) to create complex logical expressions. For example, if (age >= 18 && age <= 35) checks if age is between 18 and 35.
Code Examples
Let’s look at a couple of examples that use comparison operators:
// Example 1: Check if a number is within a range
let age = 25
if (age >= 18 && age <= 35) {
console.log("You are in the target demographic.");
} else {
console.log("You are not in the target demographic.");
// Example 2: Check if a user's input is valid
let userInput = "12345";
if (userInput !== "" && userInput !== null) {
console.log("Input received: " + userInput);
} else {
console.log("Invalid input!");
In the first example, we use the >= and <= operators to check if a number is within a certain range. In the second example, we use the !== operator to check if the user’s input is not empty and not
Wrapping Up
And that’s a wrap! We’ve covered all the JavaScript comparison operators, how they work, and how to use them in your code. Remember, these operators are the building blocks of logical expressions in
JavaScript, so understanding them is crucial for writing effective JavaScript code.
Frequently Asked Questions (FAQ)
• What are JavaScript comparison operators?
JavaScript comparison operators are used to compare two values and return a boolean result (true or false). They include ==, ===, !=, !==, <, >, <=, and >=.
• What is the difference between == and === in JavaScript?
The == operator checks for equality after performing any necessary type coercion, while the === operator checks for equality without performing type coercion. This means 5 == "5" returns true,
but 5 === "5" returns false.
• What are the six comparison operators in JavaScript?
The six comparison operators in JavaScript are == (equal to), === (strict equal to), != (not equal to), !== (strict not equal to), < (less than), > (greater than), <= (less than or equal to), and
>= (greater than or equal to).
• Why do we prefer === and !== over == and != in JavaScript?
The === and !== operators are preferred because they do not perform type coercion, making them less prone to unexpected results. They check both the value and the type, providing a stricter
equality check.
• How do comparison operators work with logical operators in JavaScript?
Comparison operators often work together with logical operators (&&, ||, !) to create complex logical expressions. For example, if (age >= 18 && age <= 35) checks if age is between 18 and 35.
• What is operator precedence in JavaScript?
Operator precedence in JavaScript determines the order in which operations are performed. Comparison operators have higher precedence than logical operators, meaning comparison operations are
performed before logical operations.
• How does type coercion work in JavaScript comparison operators?
Type coercion in JavaScript is the automatic or implicit conversion of values from one data type to another. The == and != operators perform type coercion if the values they are comparing are
different types.
• Can comparison operators be used with strings in JavaScript?
Yes, comparison operators can be used with strings in JavaScript. When comparing strings, JavaScript uses lexicographic (“dictionary” or “lexicographical”) order.
• How can I check if a number is within a range using JavaScript comparison operators?
You can use the >= and <= operators to check if a number is within a certain range. For example, if (age >= 18 && age <= 35) checks if age is between 18 and 35.
• How can I validate user input using JavaScript comparison operators?
You can use the !== operator to check if the user’s input is not empty and not null. For example, if (userInput !== "" && userInput !== null) checks if userInput is not empty and not null.
Related Tutorials
Remember, practice makes perfect! So, don’t just read this tutorial – try out these concepts in your own code. Happy coding!
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Cartesian form of the equation of line r=3i^−5^+7k^+λ(2i^+j^... | Filo
Cartesian form of the equation of line is
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Line is passing through and parallel to , then equation of line is
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Question Text Cartesian form of the equation of line is
Updated On Sep 22, 2022
Topic Three Dimensional Geometry
Subject Mathematics
Class Class 12
Answer Type Text solution:1 Video solution: 1
Upvotes 157
Avg. Video Duration 4 min
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Identifying the Algebraic Expression of the Area of a Composite Figure Involving a Triangle and a Square
Question Video: Identifying the Algebraic Expression of the Area of a Composite Figure Involving a Triangle and a Square Mathematics • First Year of Preparatory School
Write the algebraic expression that represents the area of the given figure in square centimeters.
Video Transcript
Write the algebraic expression that represents the area of the given figure in square centimeters.
We begin by noticing that the figure is split into two shapes, a square and a triangle. We are asked to find an algebraic expression for the area of the figure. Recall that the area of a square can
be calculated by squaring the side length, written 𝑙 squared. Since the side length of the square in this question is 𝑥 centimeters, then its area is 𝑥 squared square centimeters.
Next, we recall that the area of a triangle is equal to half the length of the base multiplied by the perpendicular height. The base of the triangle is 34 centimeters. And since we have a right
triangle, the perpendicular height is 𝑥 centimeters. This means that the area of the triangle is equal to 34 multiplied by 𝑥 divided by two. This can be simplified to 34𝑥 over two, which is equal to
We now have expressions for the area of both the square and the triangle. The total area will be equal to the sum of these expressions. We can therefore conclude that the algebraic expression that
represents the area of the given figure is 𝑥 squared plus 17𝑥 square centimeters.
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Adjusting the projection distance in far field projections
This section describes how to rescale far field projections to distances other than the default of 1m. It also describes how to use the farfieldexact functions to calculate the field distribution at
arbitrary positions, including the so called intermediate field (beyond the simulation region boundary, but not yet the far field).
Note: The descriptions and examples of the far field projection calculation on the following pages are primarily intended for users of FDTD. For users interested in calculating far field projections
with MODE, these descriptions are basically still correct, although some subtle differences do exist.
The script file first calculates the standard far field distribution. Rather than calculating the distribution on the entire hemisphere, we only get one line at y=0. This data is calculated with both
the farfield3d and farfieldexact3d functions. As the following figure shows, both functions return the same result for the field distribution on a hemisphere with a radius of 1m.
In most cases, the standard projection location is sufficient. However, if you wish to know the field amplitude at a different distance (such as a hemisphere with a radius of 1mm), we recognize that
the E and |E|^2 scale as shown in the following table. This formula is valid anywhere in the far field.
Electric field scaling Electric field intensity scaling
2D $$\overrightarrow{E}(R)=\overrightarrow{E_0}/\sqrt{R}$$ $$\mid\overrightarrow{E}(r)\mid^2=\frac{\mid\overrightarrow{E}(1)\mid^2}{r}$$
3D $$\overrightarrow{E}(R)=\overrightarrow{E_0}/R$$ $$\mid\overrightarrow{E}(r)\mid^2=\frac{\mid\overrightarrow{E}(1)\mid^2}{r^2}$$
Projections to the intermediate field
If you want to calculate the field distribution quite close to the structures (the intermediate field), then the farfieldexact functions must be used. The intermediate field is the region close to
the structure where the fields are not yet propagating like simple plane waves. The farfieldexact functions are able to calculate the field distribution in the far field or intermediate field (to
within a few wavelengths of the simulation region).
For example, suppose we wish to know the fields on an a hemisphere with a radius of 10um. One approach (blue line in following figure) might be to simply scale the fields by a factor of (1m/10um)^2.
This is NOT correct because 10um is not far enough to be considered as the far field. The EM fields are not yet propagating like simple plane waves. The correct approach is to use the farfieldexact3d
function (green line).
As you can see, the two calculations do not give the same results. The blue line is not quite correct because rescaling the far field distribution is not correct in the intermediate field.
Note: Projections to surfaces other than hemispheres
Use the farfieldexact functions when you want the field distribution on a surface other than a hemisphere. The following figure shows the field intensity along a straight line for x=-20:20um at y=
0um, z=10um.
See also
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The possibility of phase measurements of Doppler frequency
The present work gives a probabilistic analysis of a scheme for using phase measurements in a multichannel receiver in order to determine the Doppler frequency of a signal reflected from moving
objects. A method of estimating Doppler frequency is set forth for the case of a small number of complex pulses arriving at the input of the receiving system. The rms error in estimates of Doppler
frequency is determined as a function of noise parameters at the input of the phase-measuring device. The block scheme of a phasometric system is presented with which the practical potential accuracy
of Doppler frequency measurements can be obtained.
Akademiia Nauk Ukrains koi RSR Visnik
Pub Date:
January 1975
□ Doppler Effect;
□ Frequency Response;
□ Phase Shift;
□ Pulse Doppler Radar;
□ Signal Reflection;
□ Block Diagrams;
□ Moving Target Indicators;
□ Multichannel Communication;
□ Probability Theory;
□ Radio Receivers;
□ Root-Mean-Square Errors;
□ Communications and Radar
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Homework #4
Economics 470/570
Fall 2008
Homework #4
Due 10/30/2008
1. Suppose the Fed sells $1,000 worth of securities to the public. Assuming that the reserve requirement is 25%, use t-accounts to show the resulting multiple deposit contraction (carry the
t-accounts out through three steps). Use the multiplier formula to calculate the total fall in bank deposits.
2. Derive the money multiplier when C ≠ 0 and ER ≠ 0. Explain why it is smaller than the simple money multiplier.
3. Show, using graphs, how (a) open market operations, (b) borrowing from the Fed, and (c) changes in reserve requirements affects the federal funds rate.
4. Show graphically how an increase in financial market risk impacts the federal funds rate, and how the Fed would respond in order to return the federal funds rate to its target value.
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Lisp ≠ Lambda Calculus
How do you condense a 30 minute talk in 5 minutes? Should you even try? These are the questions I struggled with when someone nudged me to register for the lightning talks. My talk was 30 minute long
because I was to jump in if a last-minute incident would prevent someone to get on stage. Call me the backup speaker, if you will. In organizing Heart of Clojure, Arne Brasseur and Martin Klepsch had
prepared for every eventuality. Luckily, the event was incident free, and I was relieved of duty. But that someone was right: a lightning talk was the only redeeming option I had before calling it
TL;DR Lisp is not a realization of the Lambda Calculus
My topic was Lisp in the broader context of science. The recursive functions mentioned in McCarthy's seminal paper, Recursive functions of Symbolic Expressions and Their Computation by Machine, Part
I refer to the class of functions studied in computability theory. They are interesting because with just three initial functions (successor, constant and projection functions) closed under
composition and primitive recursion, one can produce most computable functions studied in number theory (addition, division, factorial, exponential, etc.).
Note: Recursion is the process that consists in defining the value of a function by using other values of the same function.
The primitive recursive functions originated with the proof Gödel provided for his incompleteness theorems. They evolved over time with contributions by Jacques Herbrand, Stephen Cole Kleene and
Rózsa Péter, among others. With only minor additions, like a minimisation operator, a broader set of functions could be defined, equivalent to those computable by a Turing machine. They are known as
the μ-recursive functions.
One can envision the title of McCarthy's paper as a one-liner that stands in for his theory. It seems to tell us: here is how to mechanically compute a set of partial functions that operate on
symbolic expressions. Remember that McCarthy's goal was AI research. The Advice Taker was going to operate on sentences, not numbers. In order to infer facts from them. John McCarthy needed a system
that could process formal languages, hence symbolic expressions. And indeed, differentiation of algebraic expressions was among the first achievements of Lisp systems.
McCarthy's innovation in recursive function theory is known as McCarthy's formalism. John McCarthy showed that writing recursive function definitions using conditional expressions allowed combining
the base case and the inductive case into a single formula. In his seminal paper, McCarthy continued to prefer mathematical notation, which he called his publication notation. With the conditional
expression defined as:
$$(p_{1} \rightarrow e_{1},\dotsc,p_{n} \rightarrow e_{n} )$$
The traditional brace notation for factorial:
$$\operatorname{fac} \left({n}\right) = \begin{cases} 1 & : n = 0 \\ \operatorname{mult} \left({n, \operatorname{fac} \left({n - 1}\right)}\right) & : n > 0 \end{cases}$$
Takes this form:
McCarthy's formalism is regarded as a precise and substantial refinement of the partial recursive functions (Moschovakis, 2001, p. 919). On a real world Lisp 1.5 implementation:
DEFINE ((
(FACTORIAL (LAMBDA (N) (COND
((ZEROP N) 1)
(T (TIMES N (FACTORIAL (SUB1 N)))) )))
Note: DEFINE was called a pseudo-function, they were used for their side-effects.
Ignoring for a moment stack considerations, factorial can be expressed in modern Lisps like this:
(defn factorial [n]
(if (<= n 0)
(* n (factorial (dec n)))))
Isn't that neat? Well, that is not a coincidence.
Then mathematical neatness became a goal and led to pruning some features from the core of the language. — John McCarthy, Lisp session, History of Programming Languages
To understand Lisp's origin story is to understand that John McCarthy had a mathematician's perspective, approach and ambition. For example, the Lisp system presented in his paper was purely
functional, while the real world implementation from his lab had side-effecting constructs.
Another way to show that LISP was neater than Turing machines was to write a universal LISP function and show that it is briefer and more comprehensible than the description of a universal Turing
machine. — John McCarthy, History of Lisp
It is also telling that while McCarthy envisioned the universal LISP function, he left the implementation as an exercise for the reader. Luckily for us, Steve Russell was that reader, and he did
write an implementation. This is the famous metacircular evaluator apply/eval that Alan Kay dubbed the Maxwell equations of computing.
McCarthy's ambition is made overly clear in A Basis For a Mathematical Theory of Computation, published soon after the seminal paper we mentioned, in which he explains the deficiencies of the current
theories, and defines the goals of a new formalism for the science of computation.
Back at Heart of Clojure, I carried myself to the registration board, put my name on the list and the title of my lightning talk, Contextualizing Lisp. What on earth was I going to cram in 5 minutes?
I had an idea which I discarded every time it popped in my head, yet it got the better of me. Betting on the widespread mischaracterization of Lisp as a derivative of the Lambda Calculus, I burst
onto the stage asking for a show of hands: Who thought Lisp was based on the Lambda Calculus? Who thought it wasn't? Who didn't know what to think?
… one of the myths concerning LISP that people think up or invent for themselves becomes apparent, and that is that LISP is somehow a realization of the lambda calculus, or that was the
intention. The truth is that I didn't understand the lambda calculus, really. — John McCarthy, Lisp session, History of Programming Languages
Note: Yes, John McCarthy borrowed the lambda notation from Alonzo Church. He also understood it better than he wants you to believe.
Sure, it was a gimmick, but it seemed to work. There was some bemusement. People were listening. I wasn't going to go home on an anticlimax after all. How is it possible that we carry so many
half-truths and approximations concerning the origins of our field? Is it a failure of our educational institutions that teach STEM without giving the historical and philosophical perspective? Or
maybe it's our fault, another aspect of our infamous tendency toward cargo culting? Alfred North Whitehead said that civilization advances by extending the number of important operations which we can
perform without thinking of them. If he is right, then the cost of civilization is all the thinking being retired from it.
Thinking is not driven by answers but by questions. On the stage of Heart of Clojure, I wished I could dwell longer on some points pertaining to the philosophy of science. But in the front row
someone was flashing cue cards. Two minutes left.
Back in the days of McCarthy, computing was attracting scientists from different backgrounds. Mathematics, that we have already established, but also the cognitive sciences, linguistics, etc. The
field was emerging, amorphous, malleable. The term computer science was coined in 1961 (a year after McCarthy's Lisp paper). McCarthy was paying close attention to the work done by Allen Newell,
Herbert A. Simon and Cliff Shaw on IPL, where list processing originated. The three luminaries wrote an AI program before the term even existed: Logic Theorist, a program that proved theorems in
Whitehead and Russell's Principia Mathematica. Yes, the same Principia Mathematica referenced by Gödel in his paper Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme
I and where our famous recursive functions come from.
Giving a sense of how revolutionary that paper was is crucial. Starting with David Hilbert and the axiomatic method, one needs to explain how mathematics tried and failed to establish its own
consistency. This chapter in the early 20th century is known as the Foundational crisis of Mathematics. The initial setback was followed by groundbreaking insights in logic, metamathematics and
decidability. The Entscheidungsproblem, in particular, set in motion the research that brought forth the computation models that lie at the heart of our programming languages.
The shift from philosophical inquiry to applied science is one that is difficult to track indeed. It doesn't help that computability theory was known as recursion theory until leading figures such as
Robert I. Soare called for a rebranding of the field. The conflation of the notion of recursion, or mathematical induction, with the notion of computability, or calculability, was deemed detrimental.
But this messiness is characteristic and essential to the way science is made.
Imagine that thinkers, scientists and inventors would systematically credit their influences: we could go up the chain and draw a chart of scientific innovation. Instead, we have to cross-reference
by independent means, relying on educated guesswork, research and investigation. Back in the 17th century, Wilhelm Leibniz thought it would be desirable to mechanize thought. He envisioned the
spécieuse générale, a formal language accompanied by the first attempts at arithmetization of syntax. Three centuries later, Kurt Gödel would use this technique with such resounding success that
today it bears his name, Gödel numbering.
For Leibniz, the ideal language is one where logical relations are so transparent that they exclude any form of ambiguity, leaving room only for calculation. This might sound spurious to our
contemporary ears, but three centuries later, David Hilbert asked if there was an algorithm capable of calculating the truth or falsity of mathematical statements. Questions get asked in new ways,
redefining the domain of discourse, and sometimes progress is made.
Of course, I never got to touch any of that. The five minutes were over. I left the podium and headed to the back of the venue. How did I feel? Highly inadequate. Did I regret it? Not at all. I had a
look at the audience from the last rows. The venue was packed. I marveled at the idea that a tenuous commonality such as an interest in a Lisp system is reason enough to grow a community around it.
Surely, I would be back sometime, at a conference near you.
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Explain the Perceptron Neural Network - Doubtly
Explain the Perceptron Neural Network
1 Answer
A perceptron is one of the earliest and most basic types of artificial neural networks. It was introduced by Frank Rosenblatt in 1957 and is designed to perform binary classification tasks.
The perceptron neural network consists of a single neuron with adjustable synaptic weights and a bias (or threshold) value. The neuron receives multiple input signals, which are multiplied by their
respective weights, and these products are then summed together. The sum is then compared to the bias value, and if it exceeds the bias, the neuron fires (outputs a value of 1); otherwise, it does
not fire (outputs a value of 0).
[image credits javatpoint]
The perceptron learns by adjusting its weights and bias value through a process called the perceptron learning rule or the perceptron training algorithm. This algorithm works as follows:
1. Initialize the weights w_i and bias b to small random values.
2. For each training example (x_1, x_2, ..., x_n, t), where t is the target output (0 or 1): a. Calculate the actual output y of the perceptron using the current weights and bias. b. Update the
weights and bias based on the error between the actual output y and the target output t: – If y != t, update the weights and bias: w_i = w_i + η * (t - y) * x_i b = b + η * (t - y) – If y == t,
do not update the weights and bias.
3. Repeat step 2 for all training examples until convergence (i.e., until the perceptron correctly classifies all training examples).
Here, η is the learning rate, a positive value that controls the step size of the weight updates.
The perceptron is capable of learning linearly separable patterns, which means that if the input data points can be separated by a single straight line (or hyperplane in higher dimensions), the
perceptron will be able to learn the decision boundary and correctly classify new inputs. However, if the data is not linearly separable, the perceptron will fail to converge and cannot correctly
classify all the training examples.
Despite its simplicity, the perceptron laid the foundation for more complex neural network architectures and was a significant milestone in the development of artificial neural networks and machine
Team Answered question March 31, 2024
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Present Value Calculator
Present Value Calculator
Present Value Calculator is a tool to calculate the present value of money based on the future value, interest rate, and the number of years. The present value calculator is useful for people to
calculate how much the money is worth based on a future value.
Future Value: $
Interest Rate: %
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What Is Power Factor And Its Significant
What is the Power Factor?
The power factor tells us how well an electrical system using alternating current (A.C.) is working.
Imagine it like this: It's like looking at how well a car's engine runs. The power factor is like checking how smoothly the engine is running.
In numbers, we calculate it using something called cosine angle. It's a bit like measuring the angle between two lines: one representing the electrical voltage and the other representing the
electrical current in a circuit. We call this angle "Ф."
So, power factor = cos Ф
And we use this formula: P = V x I x cosϕ
- V is voltage (like the electrical pressure)
- I is current (like the flow of electricity)
- P is total power (like how much work the electricity is doing)
- ϕ is the angle difference between V and I (how in sync the voltage and current are)
Why is Power Factor Important?
Now, let's talk about why this power factor thing matters.
Imagine you have a car, and you want to use its engine to do some work. If your engine runs smoothly (like having a high power factor), you can get more work done efficiently.
For example, if you have a power factor of 0.9, and you give the engine 100 units of power (KW), 90 units are used really well to do the job, and only 10 units are wasted on things like heat or
So, having a high power factor is great because it means you're using your electricity efficiently. This is good for both the people using electricity and the ones producing it. Everyone benefits!
Post a Comment
>> Your Comments are always appreciated...
>> Discussion is an exchange of knowledge It Make the Mariner Perfect.... Please Discuss below...
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Cumulative Distribution Functions and Probability Density Functions
Probability Theory Mastering
Probability Theory Mastering
Cumulative Distribution Function (CDF)
The Cumulative Distribution Function (CDF) is a function that describes the cumulative probability of a random variable taking on a value less than or equal to a given value.
Mathematically, the CDF of a random variable X, denoted as F(x), is defined as:
F(x) = Probability that variable X is less or equal to value x.
Using this function, it is easy to describe continuous random variables.
Look at the example below: we will use a normally distributed random variable and look at its CDF using the .cdf() method.
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
# Generate a random variable following a normal distribution
mu = 0 # mean
sigma = 1 # standard deviation
x = np.linspace(-5, 5, 100) # x values
rv = norm(loc=mu, scale=sigma) # create a normal distribution with given mean and standard deviation
# Compute the CDF for the random variable
cdf = rv.cdf(x)
# Plot the CDF
plt.plot(x, cdf, label='CDF')
plt.title('CDF of a Standard Normal Distribution')
Using CDF, we can determine the probability that our random variable belongs to any of the intervals of interest. Assume that X is a random variable, and F(x) is its CDF.
To determine the probability that the variable X belongs to the interval [a, b], we can use the following formula:
P{X є [a,b]} = F(b) - F(a).
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
# Generate a random variable following a normal distribution
mu = 0 # mean
sigma = 1 # standard deviation
rv = norm(loc=mu, scale=sigma)
# Calculate probabilities for different ranges
print('Normally distributed variable belongs to [-1, 1] with probability:', round(rv.cdf(1) - rv.cdf(-1), 3))
print('Normally distributed variable belongs to [-2, 2] with probability:', round(rv.cdf(2) - rv.cdf(-2), 3))
print('Normally distributed variable belongs to [-3, 3] with probability:', round(rv.cdf(3) - rv.cdf(-3), 3))
Percent Point Function (PPF)
Percent Point Function (PPF), also known as the inverse of the cumulative distribution function (CDF). It is used to find the value of a random variable that corresponds to a given probability. In
Python it is implemented using .ppf() method:
from scipy.stats import norm
# Define probabilities
probabilities = [0.1, 0.5, 0.85]
# Iterate over each probability and print the corresponding value of the variable
for i in probabilities:
# Calculate the value of the variable using the percent point function (inverse of the cumulative distribution function)
value = norm.ppf(i)
# Round the value to 3 decimal places for clarity
value = round(value, 3)
# Print the result
print('Normally distributed variable is less than', value, 'with probability', i)
Probability Density Function (PDF)
Probability Density Function (PDF) is a function that provides information about the likelihood of a random variable taking on a particular value at a specific point in the continuous range. Its
interpretation is similar to that of the PMF but is specifically used for describing continuous random variables.
The PDF defines the shape of the probability distribution of a continuous random variable.
Let's consider the following example of PDF calculated using the .pdf() method.
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm
# Generate x values for plotting
x = np.linspace(-3, 3, 100)
# Calculate the probability density function (PDF) values for the standard normal distribution
pdf_values = norm.pdf(x, loc=0, scale=1)
# Plot the PDF
plt.plot(x, pdf_values, label='PDF') # Plot PDF values against x values
plt.xlabel('X') # Label for x-axis
plt.ylabel('PDF') # Label for y-axis
plt.title('PDF of a Standard Normal Distribution') # Title of the plot
plt.legend() # Show legend
plt.show() # Display the plot
The PDF provides insight into the likelihood or probability density of a random variable assuming a specific value. Higher PDF values suggest a greater likelihood, while lower values suggest a lesser
To determine the probability of a continuous variable falling within a specific range, similar to using the PMF, we calculate the sum of the PDF for all values within that range. However, since
continuous variables can have an infinite number of values within any range, we calculate the area under the PDF curve within the specified range instead of a simple sum.
Thanks for your feedback!
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What Kind of Math is on the ASVAB? | Conquer Your Exam
Now that you have made up your mind to enlist in the military, the ASVAB will be a mandatory part of your future. The scores that you receive on this Aptitude test will be determining factors in how
the trajectory of your military career will go. It will also decide whether you receive any monetary bonuses or incentives.
Several subtests make up the ASVAB. These subtests encompass an abundance of subjects to gauge your knowledge in basic subjects like math, sentence comprehension, vocabulary, and more. One area that
may cause the most anxiety is the math portion.
You may be wondering what kind of Math is on the ASVAB or what to expect in general? Don’t worry; we will help you squash those fears and adequately prepare for the math portion of the ASVAB. There
are several things that you may not be expecting, and that is normal. However, you must get the prep time you need.
Keep reading to get a grasp on what to expect on the ASVAB math section and how to prepare for it.
Is the Math on the ASVAB Hard?
Though “hard” is a relative term, Math is one of the most challenging areas on the exam. It is the one that many individuals struggle with the most. However, with adequate preparation, studying, and
patience, you can certainly do well.
One question you may be asking is how math is used in the military. Several occupations within the military rely on math. You may need to calculate the distance that a missile will need to travel, or
you may need to calculate how much fuel is required for a tank. This all requires math, making it essential for daily military operations.
The math portion of the ASVAB exam includes several different topics. There are several various topics that students will need to understand to do well on the exam. This includes basic math,
understanding fractions, and knowing an abundance of different formulas.
A common question that many individuals may ask is what grade level math is on the ASVAB? The exam generally covers arithmetic topics that have been included in Algebra I, Algebra II, and Geometry.
There are several subjects covered in the math portion of the exam – this includes the following:
• Algebra II
• Probability
• General math skills (PEMDAS)
There are two different math subtests on the ASVAB exams. These include Math Knowledge and Arithmetic Reasoning.
How are Your Math Skills Tested on the ASVAB?
Your math skills are tested in several ways on the Math portions of the ASVAB exam. These include an abundance of different problem types and variations of questions. You must also familiarize
yourself with several concepts and be able to recall at the time of the exam.
These concepts include:
• Factorials
• Finding the absolute value
• Factoring
• Using Prime Numbers
• Utilizing the Order of Operations
• Using and multiplying exponents
• Scientific notation
• Computing square roots
• Using the FOIL method
• Using the quadratic formula
• Computer the area and perimeter of an object
• Several geometry concepts include right angles, right triangles, surface area and
• System volume
• Equation Symptoms
• Inequalities
• Graphing equation
• Coordinate geometry (inclusions distance, slope-intercept, and even slope)
These are only a few of the concepts which you will find on the exam. There is an abundance of others that are used to test your true math skills. The items can appear in several different ways than
you may be used to seeing them. However, you mustn’t let them trip you up. Hence, the importance of practicing what you read and making sure you read it carefully.
On the exam, you will find that your math skills are tested using many different question types. These include word problems and traditionally formatted equation problems. When you begin to take the
Mathematical Knowledge section of the exam, you should find that these questions are more straightforward and essentially laid out in a way that leads you to the answer. There will be no guessing
about what type of problem the question is and what kind of solution you are looking for.
The Arithmetic Reasoning section of the test exposes users to the word problems. This area may be harder because test takers are asked to read carefully and figure out what is being asked of them.
Though it is essential to understand what is on the exam, the only way to get better at this portion of the exam is to practice, especially if you are someone who is not stable in general math.
How to Ace the Math on the ASVAB?
If you are nervous about taking the math portion of the exam, don’t be, there are several things that you can do to ensure that you get the score you desire. The tips in this section will ensure that
you have the right information to pass the exam. There are several strategies that you can employ to do well on the exam the list below will help you understand what areas you should be studying and
how to do well.
ASVAB Math formula sheet – The ASVAB formula sheet is your friend. This will provide you with all the formulas that you need to do well on the exam. It includes things like how to solve fractions –
multiplication, division, addition, and even subtraction. You will even find formulas for percent, exponents, factorials, logarithms, and geometry. You will need to understand how to use these
ASVAB Math Course – If you are struggling to study on your own, several course options can provide you the support you need. Taking an ASVAB math course is an excellent option for getting extra help,
and it will ensure that you are being shown the concepts in a manner that is easy to understand. You also can ask questions if you need to, without delay.
Focus on Problem Areas – A significant way to ensure that you do well on the exam is to focus on the areas where you may be struggling the most. If you are unsure of where to begin or do not know
your weakest areas, taking a diagnostic test would be an excellent thing to do. You don’t want to waste time doing an in-depth review of concepts that you already understand. You would then just
simply be wasting time. To maximize your time spent studying work backward from the areas where you find the most difficulty.
Assess your Progress – As you are studying, it is essential to assess the progress you have made. This means not merely taking a diagnostic test and studying until the day of the exam. You will need
to include a checkpoint to understand if you have been retaining the information. This can be done by merely retaking the diagnostic test or taking another math practice portion. This validates or
invalidates how you are studying for the exam.
What if You Fail the Math Section of the ASVAB?
You mustn’t fail the math portion. Your AFQT score is a combination of this score, math, and verbal expression score. Failing the math section would mean that you essentially would fail the overall
exam. However, it’s important to remember that the ASVAB does not operate on a Pass or Fail system. It is merely a test of your current knowledge. There are several ways to avoid failing. All you
have to do is combat common test-taking pitfalls.
Make sure that you do not lose focus on the goal at hand. The ASVAB, much like the ACT or the SAT, can be an exhausting exam. It is a long exam, and it can be easy to lose focus on the exam. However,
relaxing and concentrating on each test as they come is a wonderful idea.
Another surefire way to fail the exam is by not checking your work. You must save time to double-check your answers before completing the exam. This ensures that you do not make any careless mistakes
or accidentally mark the wrong answers. If you are taking the paper version of the exam, you must remember to erase it fully and completely.
Last but not least, it’s important to remember that you have practiced for all of the exam’s problems. It can be helpful to attempt to recall specific practice problems that may mimic the information
on the exam.
Recommended ASVAB Math Study Guide
The best thing that you can do for yourself when study for the ASVAB is to invest in study guides is flashcards. These can be items that you make, items that you have found online, or even items that
you purchase. The guides listed below are excellent resources for ensuring that you not only do well on the exam but also excel.
This is an excellent resource if you are looking for free, on-demand help. This practice test site offers an abundance of study guides to prepare you to sit for the exam. On this site, you will find
information on using arithmetic reasoning formulas and reviews of basic concepts like order of operations. The site also boasts an abundance of examples, interactive questions, and sample questions
that will help you visualize how to solve specific problems.
ASVAB for Dummies by Angie Papple Johnston
The For Dummies series always does a great job of breaking information down into readable, manageable and understandable chunks. The text provides two sections dedicated to the math portions of the
exam. Part 3 of the book entitled Making the Most of Math contains four chapters to ensure you have all of the information to pass the exam. Each chapter focuses on a different math area, which
includes basic knowledge, Algebra, Geometry, and Arithmetic reasoning. With this text, you will also be able to put these skills to practice using the seven practice tests offered at the end of the
book. These include answer keys and even detailed explanations.
CliffsNotes ASVAB AFQT Cram Plan by Pat Proctor, Carolyn Wheater & Jane Burstein
This text is an excellent option if you are looking for preparation materials for the ASVAB. This text focuses on providing the most comprehensive way to study for the test. It includes a study
calendar, reviews of each topic covered on the ASVAB, and even diagnostic and practice exams. The study plans provided in the text focus on three different timelines – two months, one month, and one
Wrapping Things Up: What Kind of Math is on the ASVAB?
The ASVAB aims to test your general math skills. This allows the military to determine if you are a good fit for certain positions, has a direct impact on your AFQT score, and if you can do math in
military strategy. You must put your effort into not just passing but doing well on the math portion of the ASVAB.
Math on the ASVAB can be challenging, but there are several strategies you can use to do well. The Arithmetic Reasoning portion of the exam focuses on solving equations. It asks direct questions. The
Mathematical Knowledge section of the exam zeros in on word problems and other questions that force the test taker to read and understand the problem.
It is important to remember that the math on this exam encompasses basic math knowledge and subjects like Algebra I, Algebra II, and Geometry. If you have taken and been able to excel at high school
level math, you should be in great shape to pass the exam.
All of our armed services reviews of study guides can be found here.
> What is a GT Score on the ASVAB?
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What are best practices used in teaching mathematics?
Microsoft Word - 2 Enduring Understandings and Essential Questions.docx. What is Singapore Math? The Method The Singapore math method is focused on mastery, which is achieved through intentional
sequencing of concepts.
Some of the key features of the approach include the CPA (Concrete, Pictorial, Abstract) progression, number bonds, bar modeling, and mental math. Instead of pushing through rote memorization,
students learn to think mathematically and rely on the depth of knowledge gained in previous lessons. An attitude that math is important and approachable is also essential. Students perform at a
higher level when their potential for understanding and success is assumed. So how is this different from the way math is widely taught in the U.S.? In typical U.S. math programs, students get a
worked example, then solve problems that very closely follow that example, repeating all the same steps with different numbers.
FluencyWithoutFear 2015 1. What is Effective Teaching of Mathematics? - They recognize the importance of using concrete materials and visual representations to develop a deep understanding of the
They have a clear picture of the learning progression that best develops the knowledge base and skills of their students. They also have a broad palate of learning experiences they can use in the
classroom, to meet the different learning needs of each student. Effective teachers are able to look at student misconceptions, either in the classwork, through homework, or through assessments, and
reteach the material using their understanding of the developmental nature of what becomes before or after the misconception.
Deep understanding of the content enables teachers to directly address the specific misunderstandings that students may have. Such teachers need to be continual learners. Effective teachers know what
students need to know Effective teachers augment fluency procedure with: Nine “Look Fors” in the Elementary Math Classroom. By Sue O’Connell and John SanGiovanni, adapted from A Guide for
Administrators, part of the new Math in Practice resource What do we hope to see and hear when we step into a math classroom?
As our focus has shifted from memorizing to understanding and from calculating to applying, we have recognized instructional strategies that are better suited to these goals. As we observe math
classrooms, we look for evidence that teaching is more than delivering a textbook lesson. We look at the interactions between teacher and students, the on-the-spot decisions made by the teacher to
keep learning progressing, and the ways in which the teacher brings math ideas to light through talk, visuals, and making connections to past learning.
More specifically, here are some key features we would hope to see in an effective math classroom. Without a clear focus on the standard being addressed, teaching can get side-tracked. The effective
mathematics classrroom. Getting Started with Number Talks. Showing What We KNOW in Math Journals. 5 years ago I took a CGI (Cognitive Guided Instruction) training and loved everything about it.
I took in a lot of what I learned and it shaped the way I teach math. CGI is an elementary level mathematics professional development at WCER in the 1980's and 1990's by education professors. Last
year I took another training and it revitalized my need for journals in the classroom and more CGI instruction. But finding time to fit them in every day was a challenge. I think I finally have a
good routine down that allows me to include journals and problem solving skills every single day. So here we go! Journals are pretty simple. Here is our math journal routine: Math Disabilities
Support. 10 DEVELOPING PROFICIENCY IN TEACHING MATHEMATICS. Campbell, P.F. (1996).
Empowering children and teachers in the elementary mathematics classrooms of urban schools. Urban Education, 30, 449–475. Carpenter, T.P. (1988). Teaching as problem solving. Mathematics Standards.
For more than a decade, research studies of mathematics education in high-performing countries have concluded that mathematics education in the United States must become substantially more focused
and coherent in order to improve mathematics achievement in this country.
To deliver on this promise, the mathematics standards are designed to address the problem of a curriculum that is “a mile wide and an inch deep.” These new standards build on the best of high-quality
math standards from states across the country. They also draw on the most important international models for mathematical practice, as well as research and input from numerous sources, including
state departments of education, scholars, assessment developers, professional organizations, educators, parents and students, and members of the public. Amy Lemons Teaching Resources. What Does It
Mean To Understand Mathematics?
If you think others need to see this, share it on one of the sites below by clicking on the button.
Several years ago I had a profound moment that led me to completely rethink what it meant to understand mathematics. I was still in the classroom and had been working with 6th graders on adding and
subtracting mixed numbers. My formative assessments and observations showed that most students were proficient, and I felt pleased. To end the unit I gave students an application of subtracting
fractions using the context of a freeway sign with fractional distances.
Specifically I gave students the picture below (which is the first picture in this lesson) and asked them “How far apart are the exits for Junction 90 and Jefferson Blvd?” I clearly expected students
to do well with this problem, but as I walked around checking students’ progress I realized that something strange was going on. Relatively few students got 1/4. Is this student demonstrating a
rigorous mathematical understanding? What is Subitizing? - Make Take & Teach. What Is The Concrete Representational Abstract Approach.
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Math Kangaroo’s Logic Puzzle
My AMSA students loved the following puzzle from the 2003 Math Kangaroo contest for grades 7-8:
The children A, B, C and D made the following assertions.
□ A: B, C and D are girls.
□ B: A, C and D are boys.
□ C: A and B are lying.
□ D: A, B and C are telling the truth.
How many of the children were telling the truth?
A) 0 B) 1 C) 2 D) 3 E) Impossible to determine
4 Comments
1. Leo:
Were any of the children transgendered?
22 May 2014, 2:02 pm
2. anon:
D is clearly lying. One or both of A and B are lying. If both A and B are lying, then C is telling the truth. If one of A and B is lying, then C is lying. Either way one person is telling the
23 May 2014, 2:34 pm
3. Douglas J. Keenan:
The sought-for answer is, presumably, 1, but that is incorrect. As worded, either 1 or 0 people could be telling the truth; hence the correct answer is “Impossible to determine”.
The difficulty here is that someone might not be lying but could still be wrong. For example, both A and B could be wrong, but honestly mistaken; thus neither is lying, and so C is wrong too. In
this example, then, there are 0 people telling the truth.
Perhaps the third assertion should be reworded, e.g. “Neither A nor B is telling the truth”.
5 June 2014, 2:18 pm
4. Lori:
Wow, this question seems a bit confusing for a middle schooler. We’ve tried out Math Kangaroo before but it didn’t work out as much as we had hoped. Beestar has been a pretty good option with
their online competitions.
4 October 2021, 10:10 pm
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nearbyintl(3): round to nearest | Linux Man Page
nearbyint, nearbyintf, nearbyintl, rint, rintf, rintl — round to nearest integer
#include <math.h>
double nearbyint(double x);
float nearbyintf(float x);
long double nearbyintl(long double x);
double rint(double x);
float rintf(float x);
long double rintl(long double x);
Link with -lm.
Feature Test Macro Requirements for glibc (see feature_test_macros(7)):
nearbyint(), nearbyintf(), nearbyintl():
_POSIX_C_SOURCE >= 200112L || _ISOC99_SOURCE
_ISOC99_SOURCE || _POSIX_C_SOURCE >= 200112L
|| _XOPEN_SOURCE >= 500
|| /* Since glibc 2.19: */ _DEFAULT_SOURCE
|| /* Glibc versions <= 2.19: */ _BSD_SOURCE || _SVID_SOURCE
rintf(), rintl():
_ISOC99_SOURCE || _POSIX_C_SOURCE >= 200112L
|| /* Since glibc 2.19: */ _DEFAULT_SOURCE
|| /* Glibc versions <= 2.19: */ _BSD_SOURCE || _SVID_SOURCE
The nearbyint(), nearbyintf(), and nearbyintl() functions round their argument to an integer value in floating-point format, using the current rounding direction (see fesetround(3)) and without
raising the inexact exception. When the current rounding direction is to nearest, these functions round halfway cases to the even integer in accordance with IEEE-754.
The rint(), rintf(), and rintl() functions do the same, but will raise the inexact exception (FE_INEXACT, checkable via fetestexcept(3)) when the result differs in value from the argument.
Return Value
These functions return the rounded integer value.
If x is integral, +0, -0, NaN, or infinite, x itself is returned.
No errors occur. POSIX.1-2001 documents a range error for overflows, but see Notes.
For an explanation of the terms used in this section, see attributes(7).
Interface Attribute Value
nearbyint(), nearbyintf(),
nearbyintl(), rint(), Thread safety MT-Safe
rintf(), rintl()
Conforming to
C99, POSIX.1-2001, POSIX.1-2008.
SUSv2 and POSIX.1-2001 contain text about overflow (which might set errno to ERANGE, or raise an FE_OVERFLOW exception). In practice, the result cannot overflow on any current machine, so this
error-handling stuff is just nonsense. (More precisely, overflow can happen only when the maximum value of the exponent is smaller than the number of mantissa bits. For the IEEE-754 standard 32-bit
and 64-bit floating-point numbers the maximum value of the exponent is 128 (respectively, 1024), and the number of mantissa bits is 24 (respectively, 53).)
If you want to store the rounded value in an integer type, you probably want to use one of the functions described in lrint(3) instead.
This page is part of release 5.04 of the Linux man-pages project. A description of the project, information about reporting bugs, and the latest version of this page, can be found at https://
Referenced By
abs(3), ceil(3), fabs(3), floor(3), lrint(3), lround(3), nextafter(3), nextup(3), round(3), trunc(3).
The man pages nearbyint(3), nearbyintf(3), nearbyintl(3), rintf(3) and rintl(3) are aliases of rint(3).
2017-09-15 Linux Programmer's Manual
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Sample size- Median survival time
Prior to undertaking a research project it is essential to calculate how large your sample size should be to ensure it is adequately powered, to avoid a type II error. One method of achieving this
calculation is through implementing a mechanism similar to a survival analysis.Survival analysis, also known as time-to-event analysis, is a branch of statistics concerned with analysing the expected
time duration until a predetermined outcome. In biomedical research, this is the analytical approach of choice for randomised-controlled trials.
If you are looking to conduct a survival analysis, please use this page instead
If you are looking to estimate the required sample size for a survival analysis, please continue reading.
This sample size calculator enables you to calculate the required sample for a fixed-design, two arm trial with time-to-event outcome. This is achieved through Group-sequential analysis, which
calculates an optimal sample size value using the known values and their boundaries. For a more complete understanding of group sequential analysis please read Ed Berry’s excellent guide here [
This analysis should be undertaken in the planning stages of your study and to accurately use this calculator your design should:
• Compare a single treatment against a control
• Recruit patient and follow them over a set period of time (fixed design)
• Be looking to use time-to-event (Kaplein-Meier) analyses
Formatting and terms
You do not need to upload any statistics data to undertake this analysis. You will however need to know the predicted values for a number of terms, these should be derived from review of the
For all calculations
• Ts: This is the duration of time (in months) that you plan to follow a patient up for after recruitment.
• Tr: This is the duration of time (in months) that you will recruit patients.
• Alpha value: The probability of Type I error.
• Beta value: The probability of Type II error.
If using Proportion surviving:
The proportions of survivors in groups 1 and group 2
If using Median survival time:
The median survival time in groups 1 and group 2
If using Hazard Rate:
You will need the failure rate in groups 1 and two respectively. These are described as Lambda 1 and Lambda 2.
Once you have entered the above values you may view the outputs, regardless to which option you select, the output is the same.
The output is organised as ‘Summary’, ‘Group Sequential’ and a number of plots.
• Summary: Describes the values used in the calculation.
• Group sequential: Describes the number of sequential analyses, number of events (N), the p value at each analysis and the upper and lower bounds.
• Boundary plot: Displays the Z value boundaries at each sequential analysis
• Effect size plot: Displays the boundaries as a hazard ratio at each sequential analysis
• Total sample size: Gives the number of events and overall required sample size, you must scroll sideways to see all information.
Worked Example
I want to calculate my required sample size for a two-arm trial comparing medicine X against Control.
Medicine X is likely to have a median survival of 10 months whilst control only 6 months.
I will run my trial for 24 months and recruit for 6 months. I have selected conventional alpha values of 0.05 and beta value of 0.2.
I select Total Sample Size tab and my calculated required sample size is below:
Written by Daniel Richardson, Aneya Scott and Lorenzo Lenti
Survival Analysis Part I: Basic Concepts and first analyses
Introducing group sequential designs for early stopping of A/B tests
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Garments Cost of Making (CM): Meaning, Calculation, Example - ORDNUR
Garments Cost of Making (CM)
One of the highest productive industries is a garment manufacturing industry where different types of products are manufactured. It is required to Calculate Garments Cost of Making time to time
because the main motive of garments manufacturers is to make a profit. After making the garments it is offered to the buyers. But the problem is, what should be the required price of that item, and
how to calculate the cost of the garment of making (CM). To solve this problem first, you need to know the cost of making each garment and then add your markup price to set the selling price. Here
the cost of making means the cost of manufacturing garments.
Monitoring garments making cost is necessary continuously so that trends of the cost of making can visible to the decision-makers and they actually know whether garments making cost is increasing or
decreasing. Based on your change in garments cost of making you may take decisions accordingly.
Normally, CM cost or Cost of Making garments indicates the sewing line cost of each garment. But you can also calculate your whole factory cost of making each garment.
Calculate Garments Cost of Making
Here in this article, we will show you two different ways of calculating the Cost of Making (CM) of garments.
• Normal Method of Calculation of CM
• Effective Way of Calculation of CM
Pre Requirement of Calculation of Garments Cost of Making
Before you start your costing by using the normal way, you need to know the followings:
1. Total Cost (TC) = Fixed Cost (FC) + Variable Cost (VC)
2. Production Capacity of Your Machine (Hourly)
3. Total Effective Working Hours Per Day
4. Number of Days Operation in a Month
Fixed Cost of Garments Making
Fixed Costs are those costs of garment manufacturing that are fixed in nature. With the increase in the number of units of production, your per-unit fixed cost will be reduced. Some examples of fixed
costs are; salaries and wages, interest expense of bank loan, rental expense of your factory building, depreciation expense of your fixed assets, etc.
Variable Cost of Garments Making
Variable cost is the cost that is change over an additional number of units of production. But per unit variable cost is fixed. The variable expense of garment making is; Utility (Electricity cost,
Water cost, steam cost, Air cost) Expenditures, transportation cost, repair, maintenance expense, etc. Here the most crucial variable cost is utility cost. We need to carefully handle the utility
section to minimize utility costs.
Hourly Total Production Capacity of Garments Machinery
To identify hourly total production capacity, you need to identify the total number of machines and the capacity of each machine. After that, you need to multiply the hourly capacity with the machine
Daily Production Capacity of Garments
For calculating the daily production of your garments, you need to multiply your hourly effective production capacity into working hours each day.
Monthly Production Capacity of Garments
To calculate monthly production capacity you have to multiply your daily production capacity with the number of working days in a month.
Costing of Utility of Garments
• Electricity: To get electricity cost you need to identify the total kW consumption of your factory and then multiply the total KW with the Price of per Kw electricity. Remember the cost of
electricity is much higher in the case of fuel generators and local REB compared with Gas Generators.
• Water: Water cost is the cost of collecting water and the filtering cost. Normally in textiles and garments, De-Mineralized (DM) water is used.
• Steam: First identify the total consumption of steam per machine each hour and then multiply by the number of machines.
• Compressed Air: In the case of air you have to identify the compressed air cost per cubic meter then multiply with the total consumption.
The Formula of Calculating Garments Cost of Making Considering Whole Factory Cost
Monthly Garments Cost of Making (GCM) = (Monthly Total Cost of Garments Operation)/Monthly Total Output Produced by the Factory.
• Here Total Costs include both fixed and variable Costs of your garments factory for one month
• Monthly total Production is the output produced by the factory.
Example: Suppose Your Factory Costing Related Information are;
• Salary & Wages Expense Tk. 20,000,000
• Monthly Interest on Bank Loan Tk. 100,000
• Depreciation Expense is Tk. 50,000
• The rental Expense of the Building is Tk. 200,000
• Transportation Cost is Tk. 150,000
• Repair & Maintenance Expense is Tk. 50,000
• Utility Expense is Tk 10,00,000 where Electricity Cost is Tk 500,000, Steam Tk. 200,000, Water Cost is Tk. 100,000, Chiller Cost Tk. 100,000, Compressed Air Cost is Tk. 100,000.
• Total Number of Machine is 150
• Hourly Production Capacity of Each Machine 30
• The total Working hour is 8 hours per day
• The number of working days in a month is 26 Days.
Total Cost = 20,000,000 + 100,000 + 50,000 + 200,000 + 150,000 + 50,000 +1,000,000 = 21,550,000
Total Production = 150 * 30 * 8 * 26 = 936,000
Cost of Garments Making Per Piece of Garments = (21,550,000/936,000) = Tk. 23.02 (Piece)
Effective Way of Calculation of Garments Cost of Making (CM)
• Total Number of Machines = 180 Unit
• Number of Workers Required = 200 Person
• Total Working Hours a Day = 8 HOurs
• Number of Working Days for a Given Month = 26 Days
• Workers Efficiency = 65%
• SMV of Trouser = 35 Minute
• Available Minute = (200 Workers x 8 Hours x 26 Days x 60 Minute) X 65%
• Available Minute = 16,22,400 Minutes in a Month
Fixed Cost of Garments Factory
• Salary & Wages Expense Tk. 40,00,000 monthly
• Interest on Bank Loan Tk. 100,000 monthly
• Depreciation Expense is Tk. 50,000 monthly
• The rental Expense of the Building is Tk. 200,000 monthly
• Transportation Cost is Tk. 150,000 monthly
• Repair & Maintenance Expense is Tk. 50,000 monthly
• Utility Expense is Tk 10,00,000 Monthly
Calculation of Cost Per Minute of Garments Factory
• Salary = (40,00,000/16,22,400) = 2.47 Taka/Minute
• Interest = (100,000/16,22,400) = 0.06 Taka/Minute
• Depreciation = (50,000/16,22,400) = 0.03 Taka/Minute
• Rental Expense = (2,00,000/16,22,400) = 0.12 Taka/Minute
• Transportation = (150,000/16,22,400) = 0.09 Taka/Minute
• Repair & Maintenance = (50,000/16,22,400) = 0.03 Taka/Minute
• Utility Expense = (10,00,000/16,22,400) = 0.62 Taka/Minute
• Total Cost Per Minute = 3.42 Taka/Minute
So, the Cost of Making (CM) of Trouser = 35 Minute x 3.42 Cost Per Minute = 119.7 Taka
Minute Value or Cost Per Minute of Sewing Line of Garments
For the calculation of garments sewing line cost per minute, you can use the number of operators and helpers and determine the total available minutes. And then, divide the total wages and salary
expenses of operators and workers with the available minute.
Suppose Your Sewing Section has 30 machines and for this machine 30 operators and 5 helpers are working.
• Total Salary Expense = ( 30 Operators * Average Monthly Salary) + (5 Helpers * Average Monthly Salary)
• Salary and Wages Expense = (30*10000 Tk.) + (5 * 8000 Tk.) = 340000 Tk/Month.
• Available Minute Per Month = ((30 operators+5 helpers)* (8 hours *60 minutes)) * 26 days = 436800 Minutes in a Month
CM of Garments = (340000/436800) = .78 Tk./Minute
If garments need 45 minutes to make then the total cost of making the cost of that particular garment will be = (.78 TK * 45 minute) = 35.1 Tk./Per Piece
*(Calculation based on Assumption)
Here instead of Taka, you can use any other currency as required.
After reading this hopefully, you understand how to calculate garment’s cost of making. If you have any confusion then please let me know, and I will try to clarify further.
Note: All these costing information is arbitrarily taken, so these values are not actual costing value.
Written by
Garments Cost of Making (CM): Meaning, Calculation, Example
19 thoughts on “Garments Cost of Making (CM): Meaning, Calculation, Example”
1. thnX for these kind of details i m Entrepreneur
2. i need to harmful cm of garments finishing format cell pls help me
3. Dear , I have to know Salary = (40,00,000 ok but where from /the=16,22,400) = 2.47 Taka/Minute
Can you explain it 16,22,400 or what is this .
4. its came from available minute
(200 worker x 8 hours a day x 60 minute x 26 days a month) x 65% efficiency = 16,22,400
5. really super for garments emploe
6. Thank you for this easy and logical explanation!
7. Hmm
8. Thank you very much for your article. Are Sewing Line CM and GCM different?
9. You have not taken into account the efficiency% in Your last example, you have calculated it on 100% efficiency. If efficiency is 65% then there will be loss for a company
10. Thank you for your feedback. At the time of calculation of smv you must consider efficiency and then you get actual minute required to make a garments. To do cpm calculation, you need to use the
minute of producing a garments, that is smv with the per minute cost of sewing. Then you will get actual cost of garments making.
11. Hai
Am starting new cut section, so how can i calculate cost per minute of my section.
12. Number of Workers Required = 200 Person, is this all the employees in a factory or only operators??
13. Thank you so much. That’s so useful and easy to understand.
14. Is the finishing cost per minute included in overall calculation of cost per minute of the factory.?
Also the same query is for cutting cost per minute?
15. You should use finishing cost of garments also.
16. Hi….Dear,
your last example here is-Suppose Your Sewing Section has 30 machines and for this machine 30 operators and 5 helpers are working.
Total Salary Expense = ( 30 Operators * Average Monthly Salary) + (5 Helpers * Average Monthly Salary)
Salary and Wages Expense = (3010000 Tk.) + (5 * 8000 Tk.) = 340000 Tk/Month.
Available Minute Per Month = ((30 operators+5 helpers) (8 hours *60 minutes)) * 26 days = 436800 Minutes in a Month
CM of Garments = (340000/436800) = .78 Tk./Minute
If a garments need 45 minutes to make then the total cost of making the cost of that particular garments will be = (.78 TK * 45 minute) = 35.1 Tk./Per Piece
We know that, CM=SMVCPM, But your last moment formula stood up CM=CMCM, Please if possible clear me.
17. Thank you so much. That’s so useful and easy to understand.
18. I am sincerely pleased by this blog and appreciate you sharing your article with us
19. Sir
I am no clear this option……
If a garments need 45 minutes to make then the total cost of making the cost of that particular garments will be = (.78 TK * 45 minute) = 35.1 Tk./Per Piece
You must be logged in to post a comment.
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Magnetic effect of current - 3 PPT Physics Class 12
Page 1
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
, D
– Dees N, S – Magnetic Pole Pieces
W – Window B - Magnetic Field
H F
Working: Imagining D
is positive and D
is negative, the + vely charged
particle kept at the centre and in the gap between the dees get accelerated
towards D
. Due to perpendicular magnetic field and according to Fleming’s
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
, D
becomes + ve and D
becomes – ve.
Therefore the particle is again accelerated into D
where it continues to
describe the semi-circular path. The process continues till the charge
traverses through the whole space in the dees and finally it comes out with
very high speed through the window.
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
, D
– Dees N, S – Magnetic Pole Pieces
W – Window B - Magnetic Field
H F
Working: Imagining D
is positive and D
is negative, the + vely charged
particle kept at the centre and in the gap between the dees get accelerated
towards D
. Due to perpendicular magnetic field and according to Fleming’s
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
, D
becomes + ve and D
becomes – ve.
Therefore the particle is again accelerated into D
where it continues to
describe the semi-circular path. The process continues till the charge
traverses through the whole space in the dees and finally it comes out with
very high speed through the window.
The magnetic force experienced by the charge provides centripetal force
required to describe circular path.
/ r = qvB sin 90°
(where m – mass of the charged particle,
q – charge, v – velocity on the path of
radius – r, B is magnetic field and 90° is the
angle b/n v and B)
v =
B q r
If t is the time taken by the charge to describe the semi-circular path
inside the dee, then
t =
p r
or t =
p m
B q
Time taken inside the dee depends only on
the magnetic field and m/q ratio and not on
the speed of the charge or the radius of the
If T is the time period of the high frequency oscillator, then for resonance,
T = 2 t or
T =
B q
If f is the frequency of the high frequency oscillator (Cyclotron Frequency),
f =
B q
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
, D
– Dees N, S – Magnetic Pole Pieces
W – Window B - Magnetic Field
H F
Working: Imagining D
is positive and D
is negative, the + vely charged
particle kept at the centre and in the gap between the dees get accelerated
towards D
. Due to perpendicular magnetic field and according to Fleming’s
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
, D
becomes + ve and D
becomes – ve.
Therefore the particle is again accelerated into D
where it continues to
describe the semi-circular path. The process continues till the charge
traverses through the whole space in the dees and finally it comes out with
very high speed through the window.
The magnetic force experienced by the charge provides centripetal force
required to describe circular path.
/ r = qvB sin 90°
(where m – mass of the charged particle,
q – charge, v – velocity on the path of
radius – r, B is magnetic field and 90° is the
angle b/n v and B)
v =
B q r
If t is the time taken by the charge to describe the semi-circular path
inside the dee, then
t =
p r
or t =
p m
B q
Time taken inside the dee depends only on
the magnetic field and m/q ratio and not on
the speed of the charge or the radius of the
If T is the time period of the high frequency oscillator, then for resonance,
T = 2 t or
T =
B q
If f is the frequency of the high frequency oscillator (Cyclotron Frequency),
f =
B q
Maximum Energy of the Particle:
Kinetic Energy of the charged particle is
K.E. = ½ m v
= ½ m (
B q r
= ½
Maximum Kinetic Energy of the charged particle is when r = R (radius of the D’s).
= ½
The expressions for Time period and Cyclotron frequency only when
m remains constant. (Other quantities are already constant.)
m =
[1 – (v
/ c
If frequency is varied in synchronisation with the variation of mass of the
charged particle (by maintaining B as constant) to have resonance, then the
cyclotron is called synchro – cyclotron.
If magnetic field is varied in synchronisation with the variation of mass of
the charged particle (by maintaining f as constant) to have resonance, then
the cyclotron is called isochronous – cyclotron.
NOTE: Cyclotron can not be used for accelerating neutral particles. Electrons can
not be accelerated because they gain speed very quickly due to their lighter mass
and go out of phase with alternating e.m.f. and get lost within the dees.
But m varies with v according to
Einstein’s Relativistic Principle as per
1. Cyclotron
2. Ampere’s Circuital Law
3. Magnetic Field due to a Straight Solenoid
4. Magnetic Field due to a Toroidal Solenoid
, D
– Dees N, S – Magnetic Pole Pieces
W – Window B - Magnetic Field
H F
Working: Imagining D
is positive and D
is negative, the + vely charged
particle kept at the centre and in the gap between the dees get accelerated
towards D
. Due to perpendicular magnetic field and according to Fleming’s
Left Hand Rule the charge gets deflected and describes semi-circular path.
When it is about to leave D
, D
becomes + ve and D
becomes – ve.
Therefore the particle is again accelerated into D
where it continues to
describe the semi-circular path. The process continues till the charge
traverses through the whole space in the dees and finally it comes out with
very high speed through the window.
The magnetic force experienced by the charge provides centripetal force
required to describe circular path.
/ r = qvB sin 90°
(where m – mass of the charged particle,
q – charge, v – velocity on the path of
radius – r, B is magnetic field and 90° is the
angle b/n v and B)
v =
B q r
If t is the time taken by the charge to describe the semi-circular path
inside the dee, then
t =
p r
or t =
p m
B q
Time taken inside the dee depends only on
the magnetic field and m/q ratio and not on
the speed of the charge or the radius of the
If T is the time period of the high frequency oscillator, then for resonance,
T = 2 t or
T =
B q
If f is the frequency of the high frequency oscillator (Cyclotron Frequency),
f =
B q
Maximum Energy of the Particle:
Kinetic Energy of the charged particle is
K.E. = ½ m v
= ½ m (
B q r
= ½
Maximum Kinetic Energy of the charged particle is when r = R (radius of the D’s).
= ½
The expressions for Time period and Cyclotron frequency only when
m remains constant. (Other quantities are already constant.)
m =
[1 – (v
/ c
If frequency is varied in synchronisation with the variation of mass of the
charged particle (by maintaining B as constant) to have resonance, then the
cyclotron is called synchro – cyclotron.
If magnetic field is varied in synchronisation with the variation of mass of
the charged particle (by maintaining f as constant) to have resonance, then
the cyclotron is called isochronous – cyclotron.
NOTE: Cyclotron can not be used for accelerating neutral particles. Electrons can
not be accelerated because they gain speed very quickly due to their lighter mass
and go out of phase with alternating e.m.f. and get lost within the dees.
But m varies with v according to
Einstein’s Relativistic Principle as per
Ampere’s Circuital Law:
The line integral B . dl for a closed curve is equal to µ
times the net
current I threading through the area bounded by the curve.
B . dl = µ
B . dl =
B . dl cos 0°
B . dl = B
= B (2p r) = ( µ
I / 2p r) x 2p r
B . dl = µ
Current is emerging
out and the magnetic
field is anticlockwise.
Read More
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What is Linear Discriminant Analysis?
At the end of this blog, you will have a solid understanding of what LDA is, how it works, and how it can be applied, making it a valuable addition to your data analysis toolkit.
Table of Contents:
Check out this Machine Learning Tutorial video designed to understand Machine Learning in-depth:
What is Linear Discriminant Analysis?
Linear discriminant analysis (LDA) is a supervised learning algorithm used for classification and dimensionality reduction in machine learning. It aims to find a linear combination of features that
best separates different classes in a dataset.
LDA maximizes the distance between class means while minimizing the spread within each class. By projecting data points onto this discriminative axis, LDA reduces dimensionality and helps classifiers
make more accurate predictions.
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Example of Linear Discriminant Analysis
Here is a simple example of how LDA can be used for classification:
Consider a collection of emails that we’re aiming to categorize into “spam” and “non-spam”. LDA can serve as a powerful tool for this task. The process begins by segregating our email dataset into
two distinct categories: those that are spam and those that aren’t. Using LDA, we then seek the optimal linear combination of email features that maximizes the separation between these two
Upon successfully training our model using this method, it’s equipped to evaluate and categorize new incoming emails. For each new email, we determine its linear score based on our model. By
comparing this score against a pre-set threshold, we can classify the email. If the score surpasses the threshold, the email is labeled “spam”. On the other hand, if it’s below, it’s deemed
What is Dimensionality Reduction?
Dimensionality reduction involves reducing the number of variables or features in a dataset without losing crucial information. It’s essential for simplifying complex datasets, helping in
visualization, and improving computational efficiency.
Linear discriminant analysis is an example of a dimensionality reduction technique that aims to find a lower-dimensional space where classes in the data are well-separated, making it valuable for
classification tasks and data analysis.
Fisher’s Linear Discriminant
Fisher’s linear discriminant (FLD) is a powerful supervised learning method utilized for classification and dimensionality reduction in machine learning. It identifies a linear combination of
features that optimally segregates classes within a dataset. FLD achieves this by projecting data onto a lower-dimensional space, maximizing class separation.
FLD represents a specific strategy within LDA where the data follows a Gaussian distribution and class covariance matrices are identical. It frequently serves as an initial step for dimensionality
reduction before employing LDA, particularly when dealing with high-dimensional datasets.
Linear Discriminant Analysis for Multiple Classes
LDA can be adapted for multi-class classification by employing a one-vs-rest strategy. This involves training separate LDA classifiers for each class against all other classes combined. For instance,
in a scenario with three classes (A, B, and C), three LDA classifiers are trained: one for distinguishing class A from the rest, another for class B, and a third for class C.
To classify new data points, each LDA classifier predicts the probability of the data point belonging to its respective class. The class with the highest predicted probability is then assigned to the
data point, enabling effective multi-class classification.
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Why Use Linear Discriminant Analysis?
Linear discriminant analysis is a valuable technique in various fields of machine learning and data analysis for several reasons.
Below, we have highlighted some of the reasons why using LDA is important:
• Dimensionality Reduction: LDA is primarily used for reducing the dimensionality of a dataset while preserving as much class discrimination as possible. It projects high-dimensional data onto a
lower-dimensional space, which can be especially valuable when dealing with datasets with many features. This dimensionality reduction can lead to more efficient and faster machine learning
• Feature Extraction: LDA provides a systematic way to extract the most discriminative features in a dataset. By finding linear combinations of features (the discriminant features) that maximize
the separation between different classes, LDA helps in focusing on the most relevant information for classification or visualization.
• Improving Classification Accuracy: LDA is a supervised learning technique, meaning it takes into account class labels during training. This results in improved classification accuracy compared to
unsupervised dimensionality reduction techniques like principal component analysis (PCA), which do not consider class information.
• Data Visualization: LDA can be used to visualize data by reducing it to a lower-dimensional space while maintaining class separation. This is especially useful when you want to visualize
high-dimensional data and understand the underlying structure of the classes or categories.
• Handling Multiclass Problems: LDA can handle multi-class classification problems with ease. It projects data into a space where the classes are well-separated, making it suitable for
distinguishing among multiple classes.
• Reducing Overfitting: By reducing the dimensionality of the feature space, LDA can help mitigate the risk of overfitting, which is particularly important in machine learning tasks when working
with high-dimensional data.
• Assumption of Normality: LDA assumes that the data within each class follows a multivariate Gaussian distribution. If this assumption holds true, LDA can be highly effective. However, even when
this assumption is not fully met, LDA can still provide valuable insights and useful results.
• Interpretability: The discriminant features obtained through LDA are linear combinations of the original features. This linear nature makes it easy to interpret the contributions of each feature
to the classification.
• Applications: LDA has a wide range of applications, including image recognition, text classification, bioinformatics, face recognition, and many other fields where dimensionality reduction and
classification are essential.
How Does Linear Discriminant Analysis Work?
Linear discriminant analysis, or LDA, works by enhancing class separability through dimensionality reduction.
Below, we have highlighted a detailed explanation of how LDA works:
• Projecting Data for Separation
□ LDA aims to find a linear combination of features that maximizes the distinction between classes.
□ It identifies the optimal linear coefficients to create this combination.
□ This linear combination forms a discriminant function that characterizes the separation between classes.
• Transforming into a New Space
□ The data is projected onto this linear combination or discriminant function, effectively transforming it into a lower-dimensional space.
□ In this new space, the classes are ideally more distinct and less overlapping than in the original feature space.
• Supervised Learning with Labeled Data
□ LDA is a supervised learning algorithm, which means it necessitates a labeled dataset for training.
□ This labeled dataset consists of data points already assigned to specific classes.
• Feature Discrimination Learning
□ LDA learns which features or attributes are most discriminative in distinguishing between the classes.
□ It identifies the features that contribute the most to class separation during training.
• Optimal Projection for Maximized Separation
□ Once trained, LDA uses the learned features to project new data points onto the lower-dimensional space.
□ The projection is carried out according to the linear coefficients identified earlier.
• Classification of New Data
□ To classify new data points, LDA projects them into the same lower-dimensional space.
□ The algorithm assigns the new data point to the class with the nearest mean vector in this transformed space.
• Decision Boundary
□ LDA establishes a decision boundary in the reduced-dimensional space.
□ This decision boundary is typically a linear hyperplane that maximizes the separation between classes.
Difference Between LDA and PCA
Linear discriminant analysis (LDA) and principal component analysis (PCA) are both dimensionality reduction techniques, but they serve different purposes and are used in different contexts. Let’s
discuss the differences between LDA and PCA in various aspects:
Aspect Linear Discriminant Analysis (LDA) Principal Component Analysis (PCA)
Objective Supervised technique that focuses on class separation Unsupervised technique that focuses on variance
Nature of Problem Typically used for classification tasks Used for dimensionality reduction or noise reduction
Goal Maximize the separation between classes Maximize variance along the principal components
Input Requirements Requires class labels for each data point Does not require class labels
Linearity Assumption Assumes linear relationships between features Assumes linear relationships between features
Dimensionality Reduction Reduces dimensions to (n_classes – 1) dimensions Reduces dimensions to any desired number
How to Prepare Data from LDA
To prepare data for linear discriminant analysis in machine learning, follow the given steps:
• Identify Classification Problems: LDA is primarily used for classification tasks, where you aim to categorize data into different classes. It works well for both binary (two classes) and
multi-class (more than two classes) classification problems.
• Check for Gaussian Distribution: LDA assumes that the input variables follow a Gaussian (normal) distribution. It’s essential to assess the univariate distribution of each feature and transform
it if needed to approximate a Gaussian distribution.
For example, you can apply logarithmic or square root transformations to data with exponential distributions and use the Box-Cox transformation for skewed distributions.
• Remove Outliers: Outliers can significantly impact the performance of LDA. They can skew essential statistics like the mean and standard deviation, affecting class separation. It’s advisable to
detect and remove outliers from your dataset as a preprocessing step.
• Standardize Data: LDA assumes that all input variables have the same variance. To meet this assumption, standardize your data by subtracting the mean and dividing by the standard deviation.
This transformation ensures that the mean of each feature becomes 0 and the standard deviation becomes 1. Standardization helps LDA perform optimally and ensures that no variable dominates the others
due to differences in scale.
What are the Extensions of LDA?
LDA has several variations and extensions. They are quadratic discriminant analysis, flexible discriminant analysis, and regularized discriminant analysis. Let us discuss this in detail:
• Quadratic Discriminant Analysis (QDA): QDA relaxes the assumption of equal covariance matrices across classes. Instead, QDA allows each class to have its own covariance matrix, which can better
capture the inherent differences between the classes. This can lead to improved classification performance, especially when the classes have different shapes or orientations in the feature space.
• Flexible Discriminant Analysis (FDA): FDA extends LDA by allowing non-linear transformations of the input variables. This can be useful when the classes are not linearly separable, as LDA is
limited to linear decision boundaries. FDA can capture more complex relationships between the variables and improve classification accuracy in such cases.
• Regularized Discriminant Analysis (RDA): RDA addresses the issue of overfitting in LDA, which occurs when the model learns the training data too well and fails to generalize well to unseen data.
RDA introduces regularization terms into the LDA model, which penalizes large model coefficients and helps prevent overfitting. This can improve the model’s generalization performance and reduce
the risk of poor classification on new data.
How to Implement LDA Using Scikit-Learn
Here is a step-by-step guide on implementing linear discriminant analysis using Scikit-learn:
Start by importing the necessary libraries:
import numpy as np
import pandas as pd
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis
• ‘numpy’ and ‘pandas’ for data manipulation.
• ‘LinearDiscriminantAnalysis’ from Scikit-learn for LDA.
Load your dataset into a pandas DataFrame and separate the features (X) from the target variable (y). Ensure your data is clean and preprocessed if needed.
# Load dataset into a pandas DataFrame
data = pd.read_csv('data.csv')
# Separate features and target variable
X = data[['feature1', 'feature2', ...]] # Replace with actual feature names
y = data['target_variable']
• Create and Train LDA Model
Instantiate the LinearDiscriminantAnalysis class and fit the model to your training data.
# Create LDA model
lda_model = LinearDiscriminantAnalysis()
# Train LDA model on training data
lda_model.fit(X, y)
You can now use the trained LDA model to make predictions for new data points.
# New data point to predict
new_data = np.array([[feature1_value, feature2_value, ...]]) # Replace with actual feature values
# Predict class label for the new data point
predicted_label = lda_model.predict(new_data)
• Evaluate Model Performance
To assess the performance of your LDA model, use appropriate evaluation metrics like accuracy, precision, recall, and F1-score. You can do this on a held-out test set or use cross-validation
techniques to ensure your model generalizes well.
Applications of LDA
LDA finds various applications in different sectors for its ability to improve data analysis and dimensionality reduction. In the following points, we will talk about some of the popular applications
of it.
• Face Recognition: LDA is widely used in face recognition systems. It helps extract the most distinguishing facial features, improving recognition accuracy. By reducing the dimensionality of
facial data, LDA simplifies the task of matching faces to individuals, making it valuable for security and authentication.
• Medical Diagnosis: LDA plays a crucial role in medical diagnosis. It helps in developing models for disease classification based on patient data. For instance, LDA can assist in diagnosing
diseases like diabetes by analyzing factors such as blood sugar levels, weight, and age. It provides valuable insights for early disease detection and personalized treatment.
• Text Classification: In natural language processing, LDA is employed for text classification tasks. It helps in categorizing the documents, such as news articles, customer reviews, or emails,
into relevant topics or sentiments. This is instrumental in information retrieval, recommendation systems, and sentiment analysis.
LDA effectively combines the principles of dimensionality reduction and classification, optimizing the separation between classes. Through an example of classifying emails as spam or not, we can see
how LDA works to extract crucial features for accurate categorization.
LDA stands as a versatile tool for its ability to filter essential information from high-dimensional datasets, making it a go-to choice in numerous fields, boosting classification accuracy, and
streamlining the decision-making process.
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Maple Questions and Posts
Hi I got an equation that I would like to find out its turning points.As you know you first have to differentiate it and then..... Here is my command on maple: y:=x->6x^2-7x-3*exp(-1/5*x); df:=diff(y
(x),x); solve(%,x); and then I get 5 LambertW(-1/100 (e)^(((-7)/60)))+7/12 5 LambertW(-1,-1/100 (e)^(((-7)/60)))+7/12 as answers, I don't know if they are correct, but I do know that there must be a
simpler answer than this! I was therefore wondering if you guys know how I can approach this problem differently! Thanks Chris
I wish to know if there is way to plot the following functions: x1=X1+X2*gamma(t) x2=X2 and x1=alpha*cons(theta)*X1-beta*sin(theta)*X2+gamma x2=alpha*sin(theta)*X1-beta*cos(theta)*X2+gamma where in
the second one alpha, beta. and gamma are constants. many thanks, A.
Hi, At my university i saw a presentation of maple. It was running under linux (kde), and obviously it had a (Trolltech) Qt GUI. I really would like to work under a Qt-GUI, too, but I have no idea
how to do that. The only GUI I get to work is the normal Swing-GUI of the Java Version. Thanks a lot, David
I'm having problems plotting the derivative of a function after I've solved the ODE in an ODE system. Any help appreciated! The MAPLE help for plots,odeplot says: "Multiple curves can be plotted by
specifying a nested list format. For example, [[x,y(x)],[x,diff(y(x),x)]] displays the dependent variable and its derivative as a function of x on the same plot." So, here's my problem. I define
mtest with my two ODE's and two initial values, and solve it numerically over my specified range. This works great. mtest := [diff(m(t), t) = -43/25*exp(-8527.784461/(pt(t)+273.15)+19.03523317)*(m(t)
-exp(-23300/(8.314*pt(t)+2270.96910)+2.646139043)), diff(pt(t), t) = 240, m(0) = 1, pt(0) = 105];
How do I change the spacing between lines in the MathMLViewer?
The following equation cannot be plotted correctly in the usual 3 dimensional plot with Cartesian coordinates. eq := (x^2+y^2+z^2-1)^3 - x^2*z*3 - y^2*z^3 = 0; Either a 3 dimensional implicit plot,
or a spherical coordinates plot is needed. What surprised me was how long Maple 10.06 took to produce the latter with very roughly the same amount of detail as the former: 454 s versus 0.7 s. It
seems as though one should always prefer the implicit plot to the spherical plot if the 3 dimensional Cartesian plot fails. I would be grateful for any thoughts about this and any improvements to the
code below. (Apologies for not posting the worksheet – I think that File Manager might object to its 10MB size!)
Hello, I would like to do some computations with Legendre polynomials. I would like to know, if Maple is able to do some simplifications and use orthogonality of Legendre polynomials. For example >
Int(LegendreP(m,x)*LegendreP(n, x), x=-1..1); This integral should be zero or 2/(2*n+1), depending on the relation between m and n. Is there any chance to get it (maybe using the procedure assume()
)? another example: > diff(LegendreP(n,x), x$n); This should be (2*n)! Thank you very much for your help. Karel Srot
I have defined a function in a parameter form x:= .... y:= .... and I have defined a circle in a parameter form X:= .... Y:= .... Now I want to calculate the intersectionpoints. I could use: solve(x=
X,t) and solve (y=Y,t) but the teacher teached me that some points are going lost if you use to separate commando's. Now is the question what's the commando for combining these two solves. My
concrete example: restart:x:=2*cos(t)+cos(2*t); x := 2 cos(t) + cos(2 t) > y:=2*sin(t)-sin(2*t); > y := 2 sin(t) - sin(2 t) Circle:
Hi, I have a system of 3 coupled equations (3 unknowns) with complex numbers. I solved them with solvefor. I'm interested in the modulus (|a+ib|^2 = a^2 + b^2 ) and phase of each solution. Because of
that I used polar and abs (I have more interest in the modulus). In each case I ran simplify, despite of that the solutions are huge. I would like to work with them in some spreadsheet (in order to
set the parameters, evaluate the solutions, compare them with some experimental data and do some plotting). Given that the solutions are huge I can just write them by hand where I want. In order to
automatize that I've tried to use the VisualBasic command to cut&paste the formulas in VBA. The problem is that they are so long that the source code just gets broken.
Hi I'm currently doing a maple assignment for my uni course, and was wondering how do you simplify this equation to get a simple quadratic! 6x^2-7x-3*exp(-1/5*x) as you can see its really a simple
quadratic with an exponential stuck on to it! My question is can you simplify it with maple, 2.) Does the whole equation only have 1 root since its an exponential function stuck on! Thanks Chris
So I went ahead with alec's timer code and expanded on that, his window approach was also the approach I would need to furthar push the project to completion, so the beginning is pretty much his
code, and works plenty fine. Now I was working on the project and then realized in order to set things up the way I want, i'm going to have to make a grid layout, also suggested by a fine coder here
in a previous help topic, now my problem is, I had to rehaul ALL of my code to use grid layout, grid layout seems like a really easy to use system, and I started converting all the code to comply
with that, and in the huge mess of brackets and square brackets, I seem to have messed up the overall opening and closing of said brackets.
How do I plot a prism or a general polyhedron with maple ? I tried polygonplot3d but the result is not satisfactory because maple does not recognize the edges between the faces correctly. Thank you
very much for your answer.
Hi, Let's say there is a equation as following: a+b+c How can I save each term by "+" into an array? I.e) a+b+c Then I'd like to save each paramter split by "+" into an array. a is saved to array[0]
b is saved to array[1] c is saved to array[2] I'm not sure I could make it in Maple. Any idea using Maple? Thanks!
Is there any way in MAPLE to set up a workksheet in such a way that you can enter |B| and have it produce the same result as Mag(B) where Mag:=b->sqrt(Re(b)*Re(b) + Im(b)*Im(b))
I have a frame where u_n and u_t are perpendicular unit vectors with u_n pointing to the right and u_t pointing up. Where the frame is rotating with angular velocity w in the anti-clockwise direction
and time is t then d(u_n)/dt = w u_t How do I set Maple up so that it will do a symbolic differentiation of unit vectors - or any vector - in a rotating frame?
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tricks to solving nmr problems
Solving Problems with NMR Spectroscopy - 2nd Edition Solving Problems with NMR … Useful hints for solving NMR problems There is no general strategy for solving NMR problems. The emphasis is on the 1
H The following are a set of practice problems in Pericyclic Mechanisms, prepared by Henry Rzepa for the for a second year course in Pericyclic Reactions at the Department of Chemistry, Imperial
College.Each of these problems is intended to take between 1-3 hours to solve. Page 2/3. This new edition still clearly presents the basic principles and applications of NMR spectroscopy with only as
much math as is necessary. Strategy for Solving Structure. The emphasis is on the 1 spectroscopy-problems-and-solutions 3/5 Downloaded from hsm1.signority.com on December 19, 2020 by guest examples,
we will learn how to solve NMR practice problems step-by-step in over 100 min video solutions which is essential for organic structure determination.. Problems Answers combined problems and solution
on organic spectroscopy pdf The participants were asked to solve five spectroscopy problems while being.PDF 134 K. Electron Spin Resonance Spectroscopy Seminar Problems and Answers, Pages 195-209.
Some students struggle with 1 H NMR, but the tips and tricks presented in this chapter will vastly simplify these often complex problems. Acces PDF Spectroscopy Problems And Solutions min video
solutions which is essential for organic structure determination.. Up Next. Eliminating Baseline Problems Chromatograms should reflect the separation of analyte peaks as accurately as possible. This
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PROBLEM IN JUST MINUTE| COMPLETE … - [Voiceover] Let's look at some practice IR spectra, so here we have three molecules, a carboxylic acid, an alcohol, and an amine, and below there's an IR spectrum
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step-by-step in over 100 min video solutions which is Page 8/29. Our mission is to provide a free, world-class education to anyone, anywhere. Welcome to WebSpectra - This site was established to
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spectroscopy problems practice problems in Pericyclic Reactions … practice problems in following. A recipe 1H-NMR in your organic Chemistry class an Anasazi Instrument 's NMR,. Edition still clearly
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Writing Equations Of Graphs Worksheet - Graphworksheets.com
Writing Equations Of Graphs Worksheet – Graphing equations is an essential part of learning mathematics. It involves graphing lines and points, and evaluating their slopes. Graphing equations of this
type requires that you know the x and y-coordinates of each point. To determine a line’s slope, you need to know its y-intercept, which is the … Read more
Worksheet Writing Equations Of Graphs
Worksheet Writing Equations Of Graphs – Graphing equations is an essential part of learning mathematics. This involves graphing lines and points and evaluating their slopes. Graphing equations of
this type requires that you know the x and y-coordinates of each point. To determine a line’s slope, you need to know its y-intercept, which is the … Read more
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tion D
#include <CGAL/Triangulation_utils_3.h>
Inherited by CGAL::Triangulation_3< Traits, TDS, SLDS >, CGAL::Triangulation_data_structure_3< VertexBase, CellBase, ConcurrencyTag >, and Triangulation_3< Traits, Delaunay_triangulation_3< Traits,
TDS, LP >::Triangulation_data_structure, SLDS >.
The class Triangulation_utils_3 defines operations on the indices of vertices and neighbors within a cell.
static unsigned int next_around_edge (unsigned int i, unsigned int j)
In dimension 3, index of the neighbor n that is next to the current cell, when turning positively around an oriented edge whose endpoints are indexed i and j. More...
static int vertex_triple_index (const int i, const int j)
In dimension 3, index of the j'th vertex in counterclockwise order on the face opposite to vertex with i of the cell. More...
static unsigned int ccw (unsigned int i)
Has a meaning only in dimension 2. More...
static unsigned int cw (unsigned int i)
Same for clockwise.
◆ ccw()
static unsigned int CGAL::Triangulation_utils_3::ccw ( unsigned int i ) static
Has a meaning only in dimension 2.
Computes the index of the vertex that is next to the vertex numbered i in counterclockwise direction. (see Figure 46.1).
◆ next_around_edge()
static unsigned int CGAL::Triangulation_utils_3::next_around_edge ( unsigned int i,
unsigned int j static
In dimension 3, index of the neighbor n that is next to the current cell, when turning positively around an oriented edge whose endpoints are indexed i and j.
According to the usual numbering of vertices and neighbors in a given cell, it is also the index of the vertex opposite to this neighbor n. (see Figure 46.1).
( i < 4 ) && ( j < 4 ) && ( i != j ).
◆ vertex_triple_index()
static int CGAL::Triangulation_utils_3::vertex_triple_index ( const int i,
const int j static
In dimension 3, index of the j'th vertex in counterclockwise order on the face opposite to vertex with i of the cell.
( i < 4 ) && ( j < 3 ).
|
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Mastering Machine Learning Algorithms - Second Edition – DataTalks.Club
Questions and Answers
Sara Lane
Hello Giuseppe Bonaccorso and thank you for taking the time to answer our questions! I looked at your book on Amazon and it looks really fascinating and comprehensive.
If you could rewrite it today, what algorithms would you include (that weren’t included) and which algorithms would you exclude (that were included), and why?
Giuseppe Bonaccorso
Hi Sara Lane,
the book was meant as a “continuation” of Machine Learning Algorithms (2nd ed. too), which contains more fundamentals.
In another project, I’d probably completely remove the Deep Learning part (which can be expanded in a separate book) and focus more on:
• All evaluation metrics with pros and cons
• A deeper (it’s already quite complex) emphasis on statistical learning
• Probabilistic graphical models (more complex methods and examples)
• Time-series analysis (also in this case, with many more details)
The reason is to write a more “complete” book focused on “classic” ML. This is not a limitation, considering the number of applications and the usage of these concepts in the context of DL.
Sara Lane
Also, I see that you go through all the different areas of machine learning (or at least most of them) and explain the various algorithms for each of them. Which algorithms, overall, do you think are
the most overlooked?
Giuseppe Bonaccorso
Dimensionality reduction (both linear and non-linear), component analysis, and, of course, DL models.
Sara Lane
Why do you say that DL models are overlooked? In what way do you mean?
Giuseppe Bonaccorso
It’s more a marketing problem. In general DL practitioners search for books that emphasize them. This book is about ML (more generically), but many people enjoyed the DL part :)
Sara Lane
Thank you for taking the time to answer my questions.
Wendy Mak
Hi, my questions are:
• are there any algorithms that you think many people make mistakes when they’re using it? (e.g. not understanding the underlying assumptions of a particular algorithm about data, use it in an
unsuitable context etc)
• what are good ways of becoming more familiar with algorithms in depth? (since it can be really dry reading about them, and it’s also not all the interesting to write it from scratch to work on a
toy dataset, or at least I don’t ;))
Giuseppe Bonaccorso
For example, component analysis is a tricky part, that is often misunderstood. Another area where it’s even too easy to make mistakes is clustering. Many newbies have no idea about the concept of
distance measures and tend to use default approaches even when they are completely inappropriate.
In general, every algorithm has been developed with a purpose. That should be the starting point. Why is this algorithm different? Which peculiarities does it have? Answering these questions allows
knowing the meaning of all hyperparameters and how to tune them up in real cases. It’s also important to compare the performances, trying to focus the attention on the differences. The expertise can
be obtained only starting from the foundations of the algorithm. That’s why, in many cases, it’s also helpful to read the original papers, where the authors explained the context that led them to
develop a new algorithm.
Evren Unal
Hi Giuseppe Bonaccorso
As far as I see, your book contains wide range of topics.
Congratulations. It must have require quite effort to gather up all that topic in 1 book.
My question is;
To get the most out of your book, Should the reader have any prior knowledge of ml or math?
Giuseppe Bonaccorso
Yes. The book requires math knowledge to understand the theory behind the algorithms. In some cases, the paragraphs are self-contained, but, in many others, it’s necessary to give background
knowledge for granted. As I often suggest, this can be a good way to learn something new. When you meet a concept you don’t understand, “bookmark” it, go, for example, on Wikipedia (or a text-book),
and study what you don’t know. In this way, the gap will be slowly filled.
Evren Unal
Thank you for the answer 👍
Alper Demirel
Hello Giuseppe Bonaccorso,
First of all, thank you for your time!
• How is it different from other books on the market?
• At the end of the book, what kind of intuition did you want the reader to create about algorithms?
Giuseppe Bonaccorso
Hi Alper Demirel,
• The book has been “designed” to focus both on theory (with proofs and mathematical explanations) and practice. All paragraphs contain examples to show how the algorithms work and how to implement
them in real cases.
• The goal is to let the reader understand how machine learning can be achieved and which mathematical frameworks have been developed to do so. Every new step in this field requires not to forget
the foundations, to avoid the mistake of thinking that progress has been obtained “by magic”.
Alper Demirel
I understand sir, thank you very much for your answers! Now I’m starting to wonder more about the book.
Ghaith Sankari
Hello Giuseppe Bonaccorso :
My questions are as the following:
1. How you categorize the book: is it for beginners, or for experts, and what is the best usage of it… do you suggest to study it fully .. or to use when data scientist face scenarios and he want to
find what is the best way to address them.
2. in case of studying this the book, what is next .. i think it is depend on the interest of the learner.. but I am asking what is next from your side, what is your next book or plan?
thank you
Giuseppe Bonaccorso
Hi Ghaith Sankari,
• The book is for mid-level and advanced users. Everybody can choose the best way to use it. In general, I jump directly to the topic I’m interested in, but this is not a general rule.
• I don’t have any plan to write a new book. Every reader can pick the topics that most capture her attention and look for more detailed resources. In general, I add references for this reason.
Ghaith Sankari
thank you very much
Livsha Klingman
Hello Giuseppe Bonaccorso,
Thank you so much for your time and allowing this opportunity to ask you questions!
ML algorithms are many and there parameters, even more.
• Is there a logical way of choosing an algorithm over another, other than a process of elimination and trial and error?
• Once picking your algorithm, and applying the various parameters suitable for your given project, is there a methodology in the finer tuning rather than memory, trial and error?
• Mastering successful algorithm implementation and developing expert intuition, is achieved from acquiring more knowledge and understanding (logical) or achieved from building on past experiences
- successes and failures (deductible)? Or both?
Giuseppe Bonaccorso
Hi Livsha Klingman,
• Unguided trial and error is definitely not a good strategy. I always invite the reader to acquire a “basic” awareness and to create a subset of really appropriate algorithms (possibly covering
the peculiar aspects of the dataset). For example, in a clustering task, it’s possible always to include K-Means, but if we know that the data is fragmented in irregular clusters (like in a
geographic dataset), it’s also helpful to evaluate the performances of algorithms like DBSCAN.
• Hyperparameter tuning can start from default values, but it must go on according to each specific scenario. Considering again DBSCAN, for example, if there are too many noisy points, it’s easy to
understand that the radius is too small. Of course, it’s possible also to employ methods like grid search or Bayesian selection. Again, searching without an understanding of the effect of every
hyperparameter can result in a waste of time and unacceptable results.
• Definitely both
Vladimir Finkelshtein
I noticed that you included VC dimension in the ML fundamentals chapter. I thought it is a notion that lives only in academia. Are there ways one can actually use it? For example, can one use it to
anticipate the complexity of algorithm required to solve a classification problem on a given dataset? If so, are there algorithms to evaluate it?
Giuseppe Bonaccorso
I agree that VC is a theoretical concept and there’s only a paragraph to explain to the readers the efforts made to evaluate the capacity of a model.
The complexity of this theory is very high and it requires the usage of lots of maths. I referenced books and papers dedicated to this topic, but all practical applications are extremely difficult.
I included this concept for completeness, discussing an easy example. In practice (in particular DL), it’s almost impossible to make a correct evaluation (unless we default in the universal
approximation approach). I hope my answer is satisfactory.
Vladimir Finkelshtein
Thanks for clarification. I was just surprised to see that it is even mentioned, never saw it in “practical ML” book before. I am myself a mathematician, and I keep being reminded that the industry
doesn’t have much reason to care about understanding algorithms or theoretical concepts.
After all, people just need to know which algorithm works better under certain conditions, and in most cases this is summarized in one line in any book/blog with common interview questions.
Was hoping there is a use of knowing what VC dimension is, but I guess there is not much…
Sara Lane
This book looks like a serious work that required a lot of research. What gap were you looking to fill when you wrote it?
Giuseppe Bonaccorso
It’s indeed a very long work that required a lot of time… The gap I had to fill is the one normally present in papers (where everything is almost given for granted) and practical books (where there
are only examples). I wanted to include theory contextualized with practical examples, avoiding too many “holes”.
Livsha Klingman
Thank you so much for answering me previously!
Could I ask you another question?
My past ML experience was almost totally through knowledge gaining through asking others and trial and error, and though I got very successful results, my personal understanding of the specific
hyperparameters and the ‘weights’ that they provide to each individual algorithm I felt was very limited, but I also did not find adequate resources to lead me to the clarity that I was looking for.
I visibly saw that depending on a given algorithm depended on the value and the effect of a given hyperparameter and not necessary was uniform to the same hyperparameter in other algorithms. Is that
correct? And is your book targeting the loophole in information that I am looking for?
Giuseppe Bonaccorso
Algorithms are presented together with their peculiar hyperparameters, so it’s relatively easy to make your own personal experiments.
Absolutely, the effect of hyperparameters can be very different when changing algorithms (in some cases, they might not exist at all). But if the algorithm is described together with its
hyperparameters, the selection work can be easier.
Of course, trial and error can be helpful, but, at least, you know that a hyperparameter can have an effect or another.
Just to summarize, some experience is necessary, but you should know that, e.g., an L1 penalty will induce sparsity. I hope to be clear.
Livsha Klingman
Thank you so much - I’m assuming then that your book will give valuable insight into mapping through the maze of algorithms! Thank you again for your time!
Vladimir Finkelshtein
Giuseppe Bonaccorso do you think hyperparameters could be made adaptive in the models? For example, is it possible to adjust the (l1 or l2)-regularization constant during the training of linear
regression, instead of doing gridsearch? For examples, some optimizers can somehow adjust the learning rate while training, if they don’t like the progress.
Sara Lane
Vladimir Finkelshtein Have you ever looked into Azure ML? It has both AutoML and a Hyperdrive option where you can specify parameters (like the learning rate) and use one of 3 types of parameter
sampling: Random, Grid or Bayesian. It sounds like you’re talking about Bayesian Sampling.
Vladimir Finkelshtein
i am not familiar exactly with this, but it seems that they replace greedy gridsearch with sampling, but they still run a training session for each choice of hyperparameters. I am wondering if one
can adjust those parameters during one session of training (like some optimizers do). Another simple example of adaptive behavior is early stopping, one can think of number of epochs as
hyperparameter, but during the training it can change if some conditions are met…
Sara Lane
Yes, you’re right - they run a separate training session for each choice of hyperparameters. Interesting idea for the parameters to adapt according to the performance - if it hasn’t been done yet
maybe you’ll be the one to do it!
Hi Giuseppe Bonaccorso! Thanks for taking our questions. I have two I’d like to ask.
1. Do you handle working with datasets with unbalanced labels (e.g. 20 bad labelsfor 50 000 objects)?
2. Do you have a kind of meta algorithm for how you decide the dataset does not contain enough information to answer the question as stated? Or a method for trying to suss that out before trying
every algorithm in your book? :)
Giuseppe Bonaccorso
Hi Rosona,
1. The techniques to manage unbalanced datasets (like SMOTE) are discussed in the book Machine Learning Algorithms (2nd ed.). In this book, I discuss different semi-supervised algorithms to work
with partially labeled datasets.
2. No. I rely on evaluation metrics to understand whether a model is working properly or not. XAI techniques (like SHAP) can help understand how the features are contributing to the outcomes, so a
domain expert might check whether the algorithm is working properly or not.
Samuel O. Alfred
Hello Giuseppe Bonaccorso I do like your book as it is a good mix of theory and practical. I’m usually concerned when books contain sample code as we all know, libraries like scikit-learn
periodically make changes to their tools. How do you react to this ? Follow the tide by making a new edition? Experienced users don’t have a problem making these changes as they know how to source
for answers from GitHub and stackoverflow. Beginners become stuck as they don’t have enough experience to know that a change has occurred.
Giuseppe Bonaccorso
I start from the assumption that a user that understands the theoretical part, can check the documentation to know, for example, if a parameter has been renamed. Of course, it’s impossible to
guarantee complete future compatibility, but I never refer to package, functions, or parameter names, but rather to the mathematical parameter.
Samuel O. Alfred
Alright, thank you
Alexey Grigorev
Hi Giuseppe!
I know that your book is used as a course textbook by some universities. How did it happen? Do you have a list of courses that use your books as a reference?
Giuseppe Bonaccorso
Yes, the books (both Machine Learning Algorithms and Mastering Machine Learning Algorithms) have been used as textbooks. Some time ago I posted on LinkedIn a list (I hope to find it quickly). I never
promoted to university, but I have quite a good number of references in academic papers. That’s maybe the reason. I don’t know more :)
Alexey Grigorev
That’s nice! Thanks for sharing!
Vladimir Finkelshtein
Since you mentioned Shapley values in one of the answers, when do you think interpretability techniques will become a part of standard ML curriculum? Books I have seen rarely mention anything beyond
feature importance for decision trees (and even that without explanation of how it works or without mentioning its caveats).
Giuseppe Bonaccorso
XAI is a field that still requires a lot of basic research. I think many methods are already part of some advanced programs (like LIME or SHAP), but, considering the importance of their application
(e.g., medical imaging), it’s still necessary some time to find out solutions that have the same solidity as the DT/Random forest feature importance. However, interpretability is essential to create
engagement and increase confidence, in particular when black-box applications must be employed in critical sectors.
Sara Lane
Has working on this book inspired you to develop your own algorithms?
Giuseppe Bonaccorso
Working on the examples was an extremely helpful exercise. In fact, I had to find out those elements in the algorithms that had to be emphasized. From this viewpoint, I also become more mentally
flexible when working on new algorithms. In particular, in all those contexts where it’s necessary to find “unique” solutions and different aspects of several algorithms must be joined together.
Sara Lane
Sounds pretty fascinating!
Alexey Grigorev
Hi Giuseppe!
Not sure if this list of books is complete, but it’s amazing! How did you manage to write so many? What keeps you motivated?
Giuseppe Bonaccorso
Yes, more or less, it’s complete! I wrote a lot in the past 3 years. Now I’m taking a break. I always liked the idea of expressing the concepts I loved using my language and experience. Therefore I
started writing. Every new book is a sorta new step because I keep on learning from mistakes and I discover new possibilities to expand what I’ve already discussed. However, it’s hard work and, when
you have a “regular” job, it can become very demanding. That’s why I decided to slow down a little bit and restart when fully refreshed.
Alexey Grigorev
Indeed, it’s not easy to do it when you have a job. Do you have some sort of routine that helps you stay on track?
Giuseppe Bonaccorso
Discipline. And a lot of working weekends…
Alexey Grigorev
Discipline - that’s something I definitely need. Thanks a lot!
Giuseppe Bonaccorso What are the benefits an author get’s by writing a book apart from Monetary benefits?
Giuseppe Bonaccorso
Excluding the monetary benefit (which is almost negligible), writing helps to improve all ML skills as it’s necessary to think the concepts from different viewpoints (in particular learner’s one,
which is generally one of the most difficult to manage).
Ufuk Eskici
Giuseppe Bonaccorso Hello, my question is: What is the difference of this book from other similar books? Ther are so many ML books in the market. Thanks in advance!
Giuseppe Bonaccorso
Hi Ufuk Eskici,
as said in other answers, my main goal is to join theory and practice without sacrificing the former for the latter or vice versa. Every paragraph starts with a complete theoretical discussion
(sometimes more or less complex) that should help the reader understand how the algorithm works and continues with a practical example. In this way, it’s easier to employ any other framework.
Ufuk Eskici
I appreciate for your reply. Thank you!
Ufuk Eskici
I appreciate for your reply. Thank you!
Alexey Grigorev
Good morning!
Can you tell us about Bonaccorso’s Law? What is it? And how did the name appear? 🙂
Giuseppe Bonaccorso
It started from a joke because I used to repeat that it’s possible to learn what is already somehow encoded in the data. A friend of mine suggested me to call it “Bonaccorso’s law”. However, I think
the concept is very important because nowadays so many people tend to think that ML is a sort of magic that can invent from nothing.
Alexey Grigorev
It is definitely important!
Sansom Lee
Not sure if this is the right channel but came across this short paper identifying a bunch of real life technical debt we face daily in ML: paper
Alexey Grigorev
This is a great paper! Probably #engineering is the best channel to discuss such papers
Rishabh Bhargava
One of my favorite papers ever!
Sara Lane
How do you think the machine learning world is going to change over the next decade?
Giuseppe Bonaccorso
The field that is going to change a lot is certainly is deep learning. Lower and lower hardware prices and more and more powerful systems allow training huge models with tons of data.
There are fields (like the human brain project) that can benefit this research, but the business world is more interested in systems that can be monetized someway. So, today’s “fancy” will probably
become more “classical”. Moreover, the diffusion of several automation tools will probably reduce the expertise required by many companies (while it will increase for cutting-edge ones). I don’t know
if data science will be the sexiest job for a long time, but I’m sure it will have more and more tools to express its power.
Sara Lane
Thank you for your response!
Alexey Grigorev
Good morning!
In which order do you think we should learn ML algorithms?
Do we first learn logistic regression and then decision trees? Or first decision trees and then logistic regression?
Relatedly, is there a red thread (Leitfaden) through the book other than the order it is laid out?
Giuseppe Bonaccorso
There’s no specific red line. In particular, considering the different families of algorithms that share only a few basic elements. I generally suggest following the path that best suits everyone’s
needs. Sometimes, it’s necessary to “jump back” if a concept is missing, but normally this process works fine.
To answer your question Alexey Grigorev, I don’t think there’s a reason to select one algorithm as the first one. From a statistical viewpoint, logistic regression is indeed a regression, therefore
it’s often studied before any other ML algorithm. On the other side, DTs are very easy to understand and they can be presented also to profanes. Considering my personal experience, logistic
regression is generally explained before any other algorithm, simply because it’s linear and the logic behind it is mathematically extremely simple. However, there are courses, when DTs are explained
first because the “technicalities” can be limited to just a few purity criteria. I don’t think there’s a golden rule.
Alexey Grigorev
My personal preference is to do it immediately after linear regression because we can build on top of that.
But this was recently challenged, so I wanted to know what you think about it.
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Scheduler errors when trying to graph agents
I’ve created an agent-based model where depositor agents withdraw money from bank agents. I’m now trying to create a visual of the network by using Graphs.jl to show the banks as nodes and the
depositors’ paths as the edges. The depositors can withdraw from several banks, which, I believe is why the “scheduler” error message keeps occurring. The messages include UnderVarError, scheduler
not defined, and Argument Error pos when using space.
How would you suggest I fix? Or is it possible with Graphs.jl? Thank you.
#CREATE FUNCTION to INITIALIZE and SETUP MODEL#
Define the model initialization function
function initialize_model(; num_banks=500, num_depositors=250)
properties = Dict(
:num_banks => num_banks,
:num_depositors => num_depositors,
:num_failed_banks => 0,
:num_vulnerable_banks => 0,
:tick => 0,
space = GridSpace((10, 10), periodic=false)
scheduler = AgentSchedulers.fastest
model = ABM(Union{Bank, Depositor}, space; properties, scheduler=scheduler)
##Poisson distribution for x and y coordinates
xes = rand(Poisson(40), num_banks)
yes = rand(Poisson(75), num_banks)
# Add banks to the model
for (i, (x, y)) in enumerate(zip(xes, yes))
add_agent!(Bank(i, rand(Bool), rand(Bool), rand(), 2 * rand(), 100, rand(1:10), rand(1:3)), model; pos=(x, y))
# Add depositors to the model
for i in 1:num_depositors
x, y = rand(1:10), rand(1:10) # Random positions for depositors
add_agent!(Depositor(num_banks + i, (x, y)), model)
return model
Initialize the model
model = initialize_model()
Collect all banks and depositors
banks = [a for a in allagents(model) if isa(a, Bank)]
depositors = [a for a in allagents(model) if isa(a, Depositor)]
Initialize the graph with the total number of banks
n_banks = length(banks)
g = SimpleGraph(n_banks)
Create a mapping from bank IDs to graph node indices
bank_id_to_index = Dict(bank.id => i for (i, bank) in enumerate(banks))
Add edges based on depositors
for depositor in depositors
find_bank = random_agent(model, a → isa(a, Bank) && a.vul == true && a.health == true)
if find_bank != nothing
depositor_index = bank_id_to_index[find_bank.id] # This should be the bank’s index
bank_index = bank_id_to_index[find_bank.id]
Graphs.add_edge!(g, depositor_index, bank_index)
Define the colors based on vulnerability
bank_colors = [bank.vul ? :orange : :blue for bank in banks]
Define the layout using spring_layout from GraphPlot
layout = spring_layout(g; C=1.0, MAXITER=100)
Plot the graph
gplot(g, layout=layout, nodefillc=bank_colors)
Please see Please read: make it easier to help you .
in particular format your post so that code is shown as code, and report actual error stactraces, not approximate ones.
Will do. I’ll close this post and report the error verbatim. Thank you.
In case your model primarily focuses on the bank/depositor network, it might be worth looking at Vahana.jl, where the model is expressed as a graph from the start, offering extensive functionality
around the network structure.
For example, the code to create the network and its visualization would look like this:
using Vahana
import GraphMakie, GLMakie, NetworkLayout
struct Bank
struct Depositor
struct Deposit
sim = ModelTypes() |>
register_agenttype!(Bank) |>
register_agenttype!(Depositor) |>
register_edgetype!(Deposit) |>
create_model("bank model") |>
banks = [ Bank(rand(Bool), rand(Bool)) for _ in 1:500 ]
bankids = add_agents!(sim, banks)
healthy = filter(b -> b[2].failed == false && b[2].vulnerable == false,
zip(bankids, banks) |> collect)
for d in add_agents!(sim, [ Depositor() for _ in 1:250 ])
add_edge!(sim, d, rand(healthy)[1], Deposit())
modify_vis(state::Bank, _, _) =
Dict(:node_color => state.vulnerable ? :orange : :blue,
:node_marker => state.failed ? :rect : :circle)
modify_vis(state::Depositor, _, _) =
Dict(:node_color => :black,
:node_size => 8.0)
modify_vis(state::Deposit, _, _, _) = Dict(:edge_color => :lightgrey)
create_graphplot(sim, update_fn = modify_vis) |> display
Thank you very much, sfuerst! I will try it.
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Samacheer Kalvi 12th Computer Science Solutions Chapter 4 Algorithmic Strategies
Students can Download Computer Science Chapter 4 Algorithmic Strategies Questions and Answers, Notes Pdf, Samacheer Kalvi 12th Computer Science Book Solutions Guide Pdf helps you to revise the
complete Tamilnadu State Board New Syllabus and score more marks in your examinations.
Tamilnadu Samacheer Kalvi 12th Computer Science Solutions Chapter 4 Algorithmic Strategies
Samacheer Kalvi 12th Computer Science Algorithmic Strategies Text Book Back Questions and Answers
PART – I
I. Choose The Best Answer
Question 1.
The word comes from the name of a Persian mathematician Abu Jafar Mohammed ibn – i Musa al Khowarizmi is called?
(a) Flow chart
(b) Flow
(c) Algorithm
(d) Syntax
(c) Algorithm
Question 2.
From the following sorting algorithms which algorithm needs the minimum number of swaps?
(a) Bubble sort
(b) Quick sort
(c) Merge sort
(d) Selection sort
(d) Selection sort
Question 3.
Two main measures for the efficiency of an algorithm are ……………………………
(a) Processor and memory
(b) Complexity and capacity
(c) Time and space
(d) Data and space
(c) Time and space
Question 4.
The complexity of linear search algorithm is ……………………………
(a) O(n)
(b) O(log n)
(c) O(n2)
(d) O(n log n)
(a) O(n)
Question 5.
From the following sorting algorithms which has the lowest worst case complexity?
(a) Bubble sort
(b) Quick sort
(c) Merge sort
(d) Selection sort
(c) Merge sort
Question 6.
Which of the following is not a stable sorting algorithm?
(a) Insertion sort
(b) Selection sort
(c) Bubble sort
(d) Merge sort
(b) Selection sort
Question 7.
Time complexity of bubble sort in best case is ……………………………
(a) θ(n)
(b) θ(n log n)
(c) θ(n2)
(d) θ(n(logn) 2)
(a) θ(n)
Question 8.
The \(\Theta\) notation in asymptotic evaluation represents
(a) Base case
(b) Average case
(c) Worst case
(d) NULL case
(b) Average case
Question 9.
If a problem can be broken into subproblems which are reused several times, the problem possesses which property?
(a) Overlapping subproblems
(b) Optimal substructure
(c) Memoization
(d) Greedy
(a) Overlapping subproblems
Question 10.
In dynamic programming, the technique of storing the previously calculated values is called?
(a) Saving value property
(b) Storing value property
(c) Memoization
(d) Mapping
(c) Memoization
PART – II
II. Answer The Following Questions
Question 1.
What is an Algorithm?
An algorithm is a finite set of instructions to accomplish a particular task. It is a step-by-step procedure for solving a given problem. An algorithm can be implemented in any suitable programming
Question 2.
Define Pseudo code?
1. Pseudo code is a mix of programming – language – like constructs and Plain English.
2. Pseudo code is a notation similar to programming languages. Algorithms expressed in pseudo code are not intended to be executed by computers, but for communication among people.
Question 3.
Who is an Algorist?
1. A person skilled in the design of algorithms are called as Algorist.
2. An algorithmic artist.
Question 4.
What is Sorting?
Sorting is a method of arranging group of items in an ascending or descending order. Various sorting techniques in algorithms are Bubble Sort, Quick Sort, Heap Sort, Selection Sort, Insertion Sort.
Question 5.
What is searching? Write its types?
A search algorithm is the step-by-step procedure used to locate specific data among a collection of data. Types of searching algorithms are
1. Linear search
2. Binary search
3. Hash search
4. Binary Tree search
PART – III
III. Answer The Following Questions
Question 1.
List the characteristics of an algorithm?
Input, Output, Finiteness, Definiteness, Effectiveness, Correctness, Simplicity, Unambiguous, Feasibility, Portable and Independent.
Question 2.
Discuss about Algorithmic complexity and its types?
The complexity of an algorithm f(n) gives the running time and/or the storage space required by the algorithm in terms of n as the size of input data.
Time Complexity:
The Time complexity of an algorithm is given by the number of steps taken by the algorithm to complete the process.
Space Complexity:
Space complexity of an algorithm is the amount of memory required to run to its completion.
Question 3.
What are the factors that influence time and space complexity?
Time Complexity:
The Time complexity of an algorithm is given by the number of steps taken by the algorithm to complete the process.
Space Complexity:
Space complexity of an algorithm is the amount of memory required to run to its completion. The space required by an algorithm is equal to the sum of the following two components:
A fixed part is defined as the total space required to store certain data and variables for an algorithm. For example, simple variables and constants used in an algorithm. A variable part is defined
as the total space required by variables, which sizes depends on the problem and its iteration. For example: recursion used to calculate factorial of a given value n.
Question 4.
Write a note on Asymptotic notation?
Asymptotic Notations:
Asymptotic Notations are languages that uses meaningful statements about time and space complexity.
(I) Big O
Big O is often used to describe the worst – case of an algorithm.
(II) Big Ω
Big Omega is the reverse Big O, if Bi O is used to describe the upper bound (worst – case) of a asymptotic function, Big Omega is used to describe the lower bound (best-case).
(III) Big \(\Theta\)
When an algorithm has a complexity with lower bound = upper bound, say that an algorithm has a complexity O (n log n) and Ω (n log n), it’s actually has the complexity \(\Theta\) (n log n), which
means the running time of that algorithm always falls in n log n in the best – case and worst – case.
Question 5.
What do you understand by Dynamic programming?
Dynamic programming is an algorithmic design method that can be used when the solution to a problem can be viewed as the result of a sequence of decisions. Dynamic programming approach is similar to
divide and conquer. The given problem is divided into smaller and yet smaller possible sub – problems.
PART – IV
IV. Answer The Following Questions
Question 1.
Explain the characteristics of an algorithm?
1. Input – Zero or more quantities to be supplied.
2. Output – At least one quantity is produced.
3. Finiteness – Algorithms must terminate after finite number of steps.
4. Definiteness – All operations should be well defined. For example operations involving division by zero or taking square root for negative number are unacceptable.
5. Effectiveness – Every instruction must be carried out effectively.
6. Correctness – The algorithms should be error free.
7. Simplicity – Easy to implement.
8. Unambiguous – Algorithm should be clear and unambiguous. Each of its steps and their inputs/outputs should be clear and must lead to only one meaning.
9. Feasibility – Should be feasible with the available resources.
10. Portable – An algorithm should be generic, independent of any programming language or an operating system able to handle all range of inputs.
11. Independent – An algorithm should have step-by-step directions, which should be independent of any programming code.
Question 2.
Discuss about Linear search algorithm.?
Linear Search:
Linear search also called sequential search is a sequential method for finding a particular value in a list. This method checks the search element with each element in sequence until the desired
element is found or the list is exhausted. In this searching algorithm, list need not be ordered.
Pseudo code:
(I) Traverse the array using for loop
(II) In every iteration, compare the target search key value with the current value of the list.
1. If the values match, display the current index and value of the array
2. If the values do not match, move on to the next array element.
(III) If no match is found, display the search element not found.
To search the number 25 in the array given below, linear search will go step by step in a sequential order starting from the first element in the given array if the search element is found that index
is returned otherwise the search is continued till the last index of the array. In this example number 25 is found at index number 3.
Example 1:
Input: values[ ] = {5, 34, 65, 12, 77, 35}
target = 77
Output: 4
Example 2:
Input: values[ ] = {101, 392, 1, 54, 32, 22, 90, 93}
target = 200
Output: -1 (not found)
Question 3.
What is Binary search? Discuss with example?
Binary Search:
Binary search also called half – interval search algorithm. It finds the position of a search element within a sorted array. The binary search algorithm can be done as divide-and-conquer search
algorithm and executes in logarithmic time.
Pseudo code for Binary search:
(I) Start with the middle element:
• If the search element is equal to the middle element of the array i.e., the middle value = number of elements in array/2, then return the index of the middle element.
• If not, then compare the middle element with the search value,
• If the search element is greater than the number in the middle index, then select the elements to the right side of the middle index, and go to Step-1.
• If the search element is less than the number in the middle index, then select the elements to the left side f the middle index, and start with Step-1.
(II) When a match is found, display success message with the index of the element matched.
(III) If no match is found for all comparisons, then display unsuccessful message.
Binary Search Working principles:
List of elements in an array must be sorted first for Binary search. The following example describes the step by step operation of binary search. Consider the following array of elements, the array
is being sorted so it enables to do the binary search algorithm. Let us assume that the search element is 60 and we need to search the location or index of search element 60 using binary search.
First, we find index of middle element of the array by using this formula:
mid = low + (high – low) / 2
Here it is, 0 + (9 – 0 ) / 2 = 4 (fractional part ignored). So, 4 is the mid value of the array.
Now compare the search element with the value stored at mid value location 4. The value stored at location or index 4 is 50, which is not match with search element. As the search value 60 is greater
than 50.
Now we change our low to mid + 1 and find the new mid value again using the formula, low to mid – 1
mid = low + (high – low) / 2
Our new mid is 7 now. We compare the value stored at location 7 with our target value 31.
The value stored at location or index 7 is not a match with search element, rather it is more than what we are looking for. So, the search element must be in the lower part from the current mid value
The search element still not found. Hence, we calculated the mid again by using the formula.
high = mid – 1
mid = low + (high – low) / 2
Now the mid value is 5.
Now we compare the value stored at location 5 with our search element. We found that it is a match.
We can conclude that the search element 60 is found at location or index 5. For example if we take the search element as 95, For this value this binary search algorithm return unsuccessful result.
Question 4.
Explain the Bubble sort algorithm with example?
Bubble sort algorithm:
Bubble sort is a simple sorting algorithm. The algorithm starts at the beginning of the list of values stored in an array. It compares each pair of adjacent elements and swaps them if they are in the
unsorted order. This comparison and passed to be continued until no swaps are needed, which indicates that the list of values stored in an array is sorted. The algorithm is a comparison sort, is
named for the way smaller elements “bubble” to the top of the list.
Although the algorithm is simple, it is too slow and less efficient when compared to insertion sort and other sorting methods. Assume list is an array of n elements. The swap function swaps the
values of the given array elements.
Pseudo code:
1. Start with the fist element i.e., index = 0, compare the current element with the next element of the array.
2. If the current element is greater than the next element of the array, swap them.
3. If the current element is less than the next or right side of the element, move to the next element. Go to Step 1 and repeat until end of the index is reached.
Let’s consider an array with values {15, 11, 16, 12, 14, 13} Below, we have a pictorial representation of how bubble sort will sort the given array.
The above pictorial example is for iteration – 1. Similarly, remaining iteration can be done. The final iteration will give the sorted array.
At the end of all the iterations we will get the sorted values in an array as given below:
Question 5.
Explain the concept of Dynamic programming with suitable example?
Dynamic programming:
Dynamic programming is an algorithmic design method that can be used when the solution to a problem can be viewed as the result of a sequence of decisions. Dynamic programming approach is similar to
divide and conquer. The given problem is divided into smaller and yet smaller possible sub – problems.
Dynamic programming is used whenever problems can be divided into similar sub-problems, so that their results can be re-used to complete the process. Dynamic programming approaches are used to find
the solution in optimized way. For every inner sub problem, dynamic algorithm will try to check the results of the previously solved sub-problems. The solutions of overlapped sub – problems are
combined in order to get the better solution.
Steps to do Dynamic programming:
1. The given problem will be divided into smaller overlapping sub-problems.
2. An optimum solution for the given problem can be achieved by using result of smaller sub – problem.
3. Dynamic algorithms uses Memoization.
Fibonacci Series – An example:
Fibonacci series generates the subsequent number by adding two previous numbers. Fibonacci series starts from two numbers – Fib 0 & Fib 1. The initial values of fib 0 & fib 1 can be taken as 0 and 1.
Fibonacci series satisfies he following conditions:
Fibn = Fib[n-1] + Fib[n-2]
Hence, a Fibonacci series for the n value 8 can look like this
Fib[8] = 0 1 1 2 3 5 8 13
Fibonacci Iterative Algorithm with Dynamic programming approach:
The following example shows a simple Dynamic programming approach for the generation of Fibonacci series.
Initialize f0 = 0, f1 = 1
step – 1: Print the initial values of Fibonacci f0 and f1
step – 2: Calculate fibanocci fib ← f0 + f1
step – 3: Assign f0 ← f1, f1 ← fib
step – 4: Print the next consecutive value of Fibonacci fib
step – 5: Go to step – 2 and repeat until the specified number of terms generated
For example if we generate fibonacci series up to 10 digits, the algorithm will generate the series as shown below:
The Fibonacci series is: 0 1 1 2 3 5 8 1 3 2 1 3 4 5 5
Samacheer kalvi 12th Computer Science Algorithmic Strategies Additional Questions and Answers
PART – I
I. Choose The Best Answer
Question 1.
Which one of the following is not a data structure?
(a) Array
(b) Structures
(c) List, tuples
(d) Database
(d) Database
Question 2.
The word Algorithm has come to refer to a method ……………………………
(a) Solve a problem
(b) Insert a data
(c) Delete data
(d) Update data
(a) Solve a problem
Question 3.
Which is wrong fact about the algorithm?
(a) It should be feasible
(b) Easy to implement
(c) It should be independent of any programming languages
(d) It should be generic
(c) It should be independent of any programming languages
Question 4.
Complete the diagram
Question 5.
An algorithm that yields expected output for a valid input is called as ……………………………
Algorithmic solution.
Question 6.
Program should be written for the selected language with specific ……………………………
Question 7.
…………………………… is an expression of algorithm in a programming language.
Question 8.
How many different phases are there in the analysis of algorithms and performance evaluations?
(a) 1
(b) 2
(c) 3
(d) Many
(b) 2
Question 9.
Which one of the following is a theoretical performance analysis of an algorithm?
(a) A posteriori testing
(b) A priori estimates
(c) A preposition
(d) A post preori
(b) A priori estimates
Question 10.
…………………………… is called performance measurement.
(a) A posteriori testing
(b) A priori estimates
(c) A preposition
(d) A post preori
(a) A posteriori testing
Question 11.
Time is measured by counting the number of key operations like comparisons in the sorting algorithm. This is called as ……………………………
(a) Space Factor
(b) Key Factor
(c) Priori Factor
(d) Time Factor
(d) Time Factor
Question 12.
Which of the following statement is true?
(a) Space Factor is the maximum memory space required by an algorithm
(b) Space Factor is the minimum memory spaces required by an algorithm
(a) Space Factor is the maximum memory space required by an algorithm
Question 13.
In space complexity, the space required by an algorithm is equal to the sum of …………………………… part and …………………………… part.
Fixed, Variable
Question 14.
…………………………… is an example for variable part of space complexity.
Question 15.
A …………………………… or …………………………… trade off is a way of solving in less time by using more storage space or by solving a given algorithm in very little space by spending more time.
Space – timw, time – memory
Question 16.
Which is true related to the efficiency of an algorithm?
(I) Less time, more storage space
(II) More time, very little space
(a) I is correct
(b) II is correct
(c) Both are correct
(d) Both are wrong
(c) Both are correct
Question 17.
How many asymptotic notations are used to represent time complexity of an algorithms?
(a) 1
(b) 2
(c) 3
(d) 4
(c) 3
Question 18.
Which one of the following is not an Asymptotic notations?
(a) Big
(b) Big \(\Theta\)
(c) Big Ω
(d) Big ⊗
(d) Big ⊗
Question 19.
………………………… is the reverse of Big O
(a) Big Ω
(b) Big \(\Theta\)
(c) Big C
(d) Big ⊗
(a) Big Ω
Question 20.
………………………… describes the worst case of an algorithm.
(a) Big Q
(b) Big \(\Theta\)
(c) Big O
(d) Big C
(c) Big O
Question 21.
…………………….. describes the lower bound of an algorithm.
(a) Big Ω
(b) Big \(\Theta\)
(c) Big O
(d) Big ⊗
(a) Big Ω
Question 22.
Which search technique is also called sequential search techniques?
(a) Binary
(b) Binary Tree
(c) Hash
(d) Linear
(d) Linear
Question 23.
What value will be returned by the linear search technique if value is not found?
(a) 0
(b) 1
(c) -1
(d) +1
(c) -1
Question 24.
Which search algorithm is called as Half – Interval search algorithm?
(a) Binary
(b) Binary Tree
(c) Hash
(d) Linear
(a) Binary
Question 25.
Which technique is followed by Binary Search algorithm?
(a) Subroutines
(b) Mapping
(c) Divide and conquer
(d) Namespaces
(c) Divide and conquer
Question 26.
In Binary Search, if the search element is …………………….. to the middle element of the array, then index of the middle element is returned.
(a) >
(b) <
(c) =
(d) < >
(c) =
Question 27.
In Binary search, if the search element is greater than the number in the middle index, then select the elements to the side of the middle index.
(a) Right
(b) Left
(c) Middle
(d) Bottom
(a) Right
Question 28.
Fill in the box [Formula for Binary Search]
mid = low + (high – low) /
Question 29.
……………………… is a simple sorting algorithm.
(a) Binary
(b) Bubble
(c) Selection
(d) Insertion
(b) Bubble
Question 30.
Which one of the following is not a characteristics of Bubble Sort?
(a) Simple
(b) Too slow
(c) Too fast
(d) Less efficient
(c) Too fast
Question 31.
In selection sort, there will be ……………………….. exchange for every pass through the list.
(a) 0
(b) 1
(c) 2
(d) 3
(b) 1
Question 32.
How many number of passes are used in the Insertion Sort to get the final sorted list?
(a) 0
(b) 1
(c) n
(d) n -1
(d) n – 1
Question 33.
………………………….. approach is similar to divide and conquer.
Dynamic programming
Question 34.
………………………… is an example for dynamic programming approach.
(a) Fibonacci
(b) Prime
(c) Factorial
(d) Odd or Even
(a) Fibonacci
Question 35.
Match the following.
(1) Linear search – (i) o(n^2)
(2) Binary – (ii) o(n)
(3) Bubble Sort – (iii) o(log n)
(4) Merge Sort – (iv) o(n log n)
(a) 1 – (ii), 2 – (iii), 3 – (i), 4 – (iv)
(b) 1 – (i), 2 – (ii), 3 – (iii), 4 – (iv)
(c) 1 – (iv), 2 – (iii), 3 – (ii), 4 – (i)
(d) 1 – (iv), 2 – (ii), 3 – (i), 4 – (iii)
(a) 1 – (ii), 2 – (iii), 3 – (i), 4 – (iv)
Question 36.
Time complexity of bubble sort in worst case is …………………………..
(a) θ(n)
(b) θ(n log n)
(c) θ(n2)
(d) θ(n(log n)2)
(c) θ(n2)
Question 37.
The complexity of Merge Sort is …………………………
o (n log n)
Question 38.
The complexity of Bubble Sort is …………………………
o (n2)
Question 39.
The complexity of Binary search is ……………………….
o (log n)
Question 40.
Pick the odd one out.
Merge Sort, Bubble, Binary, Insertion.
PART – II
II. Answer The Following Questions
Question 1.
Define fixed part in the space complexity?
A fixed part is defined as the total space required to store certain data and variables for an algorithm. For example, simple variables and constants used in an algorithm.
Question 2.
What is space – Time Trade off?
A space – time or time – memory trade off is a way of solving in less time by using more storage space or by solving a given algorithm in very little space by spending more time.
PART – III
III. Answer The Following Questions
Question 1.
Design an algorithm to find square of the given number and display the result?
Problem: Design an algorithm to find square of the given number and display the result. The algorithm can be written as:
Step 1 – start the process
Step 2 – get the input x
Step 3 – calculate the square by multiplying the input value ie., square ← x* x
Step 4 – display the result square
Step 5 – stop
Algorithm could be designed to get a solution of a given problem. A problem can be solved in many ways. Among many algorithms the optimistic one can be taken for implementation.
Question 2.
Differentiate Algorithm and Program?
1. Algorithm helps to solve a given problem logically and it can be contrasted with the program.
2. Algorithm can be categorized based on their implementation methods, design techniques etc.
3. There is no specific rules for algorithm writing but some guidelines should be followed.
4. Algorithm resembles a pseudo code which can be implemented in any language
1. Program is an expression of algorithm in a programming language.
2. Algorithm can be implemented by structured or object oriented programming approach.
3. Program should be written for the selected language with specific syntax
4. Program is more specific o a programming language
Question 3.
What are the two phases in the Analysis of algorithms and performance evaluation?
Analysis of algorithms and performance evaluation can be divided into two different phases:
(a) A Priori estimates: This is a theoretical performance analysis of an algorithm. Efficiency of an algorithm is measured by assuming the external factors.
(b) A Posteriori testing: This is called performance measurement. In this analysis, actual statistics like running time and required for the algorithm executions are collected.
Question 4.
Name the factors where the program execution time depends on?
The program execution time depends on:
1. Speed of the machine
2. Compiler and other system Software tools
3. Operating System
4. Programming language used
5. Volume of data required
Question 5.
Write note on Big \(\Theta\)?
Big \(\Theta\)
When an algorithm has a complexity with lower bound = upper bound, say that an algorithm has a complexity O (n log n) and Ω . (n log n), it’s actually has the complexity \(\Theta\) (n log n), which
means the running time of that algorithm always falls in n log n in the best – case and worst – case.
PART – IV
IV. Answer The Following Questions
Question 1.
Explain Selection Sort?
Selection sort
The selection sort is a simple sorting algorithm that improves on the performance of bubble sort by making only one exchange for every pass through the list. This algorithm will first find the
smallest elements in array and swap it with the element in the first position of an array, then it will find the second smallest element and swap that element with the element in the second position,
and it will continue until the entire array is sorted in respective order. This algorithm repeatedly selects the next-smallest element and swaps in into the right place for every pass. Hence it is
called selection sort.
Pseudo code:
(I) Start from the first element (i.e.), index – 0, we search the smallest element in the array, and replace it with the element in the first position.
(II) Now we move on to the second element position, and look for smallest element present in the sub-array, from starting index to till the last index of sub – array.
(III) Now replace the second smallest identified in step-2 at the second position in the or original array, or also called first position in the sub array.
(IV) This is repeated, until the array is completely sorted.
Let’s consider an array with values {13, 16, 11, 18, 14, 15}
Below, we have a pictorial representation of how selection sort will sort the given array
In the first pass, the smallest element will be 11, so it will be placed at the first position. After that, next smallest element will be searched from an array. Now we will get 13 as the smallest,
so it will be then placed at the second position.
Then leaving the first element, next smallest element will be searched, from the remaining elements. We will get 13 as the smallest, so it will be then placed at the second position. Then leaving 11
and 13 because they are at the correct position, we will search for the next smallest element from the rest of the elements and put it at third position and keep doing this until array is sorted.
Finally we will get the sorted array end of the pass as shown above diagram.
Question 2.
Explain Insertion Sort?
Insertion Sort
Insertion sort is a simple sorting algorithm. It works by taking elements from the list one by one and inserting then in their correct position in to a new sorted list. This algorithm builds the
final sorted array at the end. This algorithm uses n-1 number of passes to get the final sorted list as per the previous algorithm as we have discussed.
Pseudo for Insertion sort
Step 1 – If it is the first element, it is already sorted.
Step 2 – Pick next element
Step 3 – Compare with all elements in the sorted sub-list
Step 4 – Shift all the elements in the sorted sub-list that is greater than the value to be sorted
Step 5 – Insert the value
Step 6 – Repeat until list is sorted
At the end of the pass the insertion sort algorithm gives the sorted output in ascending order as shown below:
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Symmetries in Quantum Field Theory and Quantum Gravity
Total Page：16
File Type：pdf, Size：1020Kb
[email protected]
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Abstract: In this paper we use the AdS/CFT correspondence to refine and then es- tablish a set of old conjectures about symmetries in quantum gravity. We first show that any global symmetry, discrete
or continuous, in a bulk quantum gravity theory with a CFT dual would lead to an inconsistency in that CFT, and thus that there are no bulk global symmetries in AdS/CFT. We then argue that any \
long-range" bulk gauge symmetry leads to a global symmetry in the boundary CFT, whose consistency requires the existence of bulk dynamical objects which transform in all finite-dimensional irre-
ducible representations of the bulk gauge group. We mostly assume that all internal symmetry groups are compact, but we also give a general condition on CFTs, which we expect to be true quite
broadly, which implies this. We extend all of these results to the case of higher-form symmetries. Finally we extend a recently proposed new motivation for the weak gravity conjecture to more general
gauge groups, reproducing the \convex hull condition" of Cheung and Remmen. An essential point, which we dwell on at length, is precisely defining what we mean by gauge and global symmetries in the
bulk and boundary. Quantum field theory results we meet while assembling the necessary tools include continuous global symme- arXiv:1810.05338v2 [hep-th] 6 Jun 2019 tries without Noether currents, new
perspectives on spontaneous symmetry-breaking and 't Hooft anomalies, a new order parameter for confinement which works in the presence of fundamental quarks, a Hamiltonian lattice formulation of
gauge theories with arbitrary discrete gauge groups, an extension of the Coleman-Mandula theorem to discrete symmetries, and an improved explanation of the decay π0 ! γγ in the standard model of
particle physics. We also describe new black hole solutions of the Einstein equation in d + 1 dimensions with horizon topology Tp × Sd−p−1. Contents 1 Introduction1 1.1 Notation9 2 Global symmetry 13
2.1 Splittability 19 2.2 Unsplittable theories and continuous symmetries without currents 24 2.3 Background gauge fields 31 2.4 't Hooft anomalies 35 2.5 ABJ anomalies and splittability 41 2.6 Towards
a classification of 't Hooft anomalies 48 3 Gauge symmetry 53 3.1 Definitions 54 3.2 Hamiltonian lattice gauge theory for general compact groups 62 3.3 Phases of gauge theory 70 3.4 Comments on the
topology of the gauge group 73 3.5 Mixing of gauge and global symmetries 76 4 Symmetries in holography 77 4.1 Global symmetries in perturbative quantum gravity 77 4.2 Global symmetries in
non-perturbative quantum gravity 82 4.3 No global symmetries in quantum gravity 88 4.4 Duality of gauge and global symmetries 93 5 Completeness of gauge representations 96 6 Compactness 99 7
Spacetime symmetries 102 8 p-form symmetries 109 8.1 p-form global symmetries 109 8.2 p-form gauge symmetries 114 8.3 p-form symmetries and holography 119 8.4 Relationships between the conjectures?
122 { i { 9 Weak gravity from emergent gauge fields 124 A Group theory 128 A.1 General structure of Lie groups 128 A.2 Representation theory of compact Lie groups 129 B Projective representations 134
C Continuity of symmetry operators 136 D Building symmetry insertions on general closed submanifolds 142 E Lattice splittability theorem 144 F Hamiltonian for lattice gauge theory with discrete gauge
group 146 G Stabilizer formalism for the Z2 gauge theory 148 H Multiboundary wormholes in three spacetime dimensions 153 I Sphere/torus solutions of Einstein's equation 159 1 Introduction It has long
been suspected that the consistency of quantum gravity places constraints on what kinds of symmetries can exist in nature [1]. In this paper we will be primarily interested in three such conjectural
constraints [2,3]: Conjecture 1. No global symmetries can exist in a theory of quantum gravity. Conjecture 2. If a quantum gravity theory at low energies includes a gauge theory with compact gauge
group G, there must be physical states that transform in all finite- dimensional irreducible representations of G. For example if G = U(1), with allowed charges Q = nq with n 2 Z, then there must be
states with all such charges. Conjecture 3. If a quantum gravity theory at low energies includes a gauge theory with gauge group G, then G must be compact. { 1 { These conjectures are quite
nontrivial, since it is easy to write down low-energy effective actions of matter coupled to gravity which violate them. For example Einstein gravity coupled to two U(1) gauge fields has a Z2 global
symmetry exchanging the two gauge fields, and also has no matter fields which are charged under those gauge fields. If we instead use two R gauge fields, then we can violate all three at once.
Conjectures 1-3 say that such effective theories cannot be obtained as the low-energy limit of a consistent theory of quantum gravity: they are in the \swampland" [4{7].1 The \classic" arguments for
conjectures1-3 are based on the consistency of black hole physics. One argument for conjecture1 goes as follows [3]. Assume that a con- tinuous global symmetry exists. There must be some object which
transforms in a nontrivial representation of G. Since G is continuous, by combining many of these objects we can produce a black hole carrying an arbitrarily complicated representation of G.2 We then
allow this black hole to evaporate down to some large but fixed size in Planck units: the complexity of the representation of the black hole will not decrease during this evaporation since the Hawking
process depends only on the geometry and is uncorrelated with the global charge (for example if G = U(1) then positive and nega- tive charges are equally produced). According to Bekenstein and
Hawking the entropy of this black hole is given by [8,9] Area SBH = ; (1.1) 4GN but this is not nearly large enough to keep track of the arbitrarily large representa- tion data we've stored in the
black hole. Thus either (1.1) is wrong, or the resulting object cannot be a black hole, and is instead some kind of remnant whose entropy can arbitrarily exceed (1.1). There are various arguments
that such remnants lead to inconsistencies, see eg [10], but perhaps the most compelling case against either of these possibilities is simply that they would necessarily spoil the
statistical-mechanics interpretation of black hole thermodynamics first advocated in [8]. This interpretation has been confirmed in many examples in string theory [11{16]. The classic argument for
conjecture2 is simply that once a gauge field exists, then so does the appropriate generalization of the Reissner-Nordstrom solution for any representation of the gauge group G. The classic argument
for conjecture3 is that at least if G were R, the non-quantization of charge would imply a continuous infinity in 1Note however that the charged states required by conjecture2 might be heavy, and in
particular they might be black holes. 2More rigorously, given any faithful representation of a compact Lie group G, theorem A.11 below tells us that all irreducible representations of G must
eventually appear in tensor powers of that representation and its conjugate. If G is continuous, meaning that as a manifold it has dimension greater than zero, then there are infinitely many
irreducible representations available. { 2 { the entropy of black holes in a fixed energy band, assuming that black holes of any charge exist, which again contradicts the finite Bekenstein-Hawking
entropy. Moreover non-abelian examples of noncompact continuous gauge groups are ruled out already in low-energy effective field theory since they do not have well-behaved kinetic terms (for noncompact
simple Lie algebras the Lie algebra metric Tr (TaTb) is not positive- definite). These arguments for conjectures1-3 certainly have merit, but they are not com- pletely satisfactory. The argument for
conjecture1 does not apply when the symmetry group is discrete, for example when G = Z2 then there is only one nontrivial irreducible representation, but why should continuous symmetries be special?
In arguing for con- jecture2, does the existence of the Reissner-Nordstrom solution really tell us that a charged object exists? As long as it is non-extremal, this solution really describes a
two-sided wormhole with zero total charge. It therefore does not obviously tell us any- thing about the spectrum of charged states with one asymptotic boundary.3 We could instead consider \one-sided"
charged black holes made from gravitational collapse, but then we must first have charged matter to collapse: conjecture2 would then already be satisfied by this charged matter, so why bother with the
black hole at all? To really make an argument for conjecture2 based on charged solutions of general relativity that do not already have charged matter, we need to somehow satisfy Gauss's law with a
non-trivial electric flux at infinity but no sources. It is not possible to do this with triv- ial spatial topology. One possibility is to consider one-sided charged \geons" created by quotienting some
version of the Reissner-Nordstrom wormhole by a Z2 isometry [18], but this produces a non-orientable spacetime and/or requires that we gauge a discrete Z2 symmetry that flips the sign of the field
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Project Risk Analysis Software and Project Risk Management Software Forum
Moderator: Intaver Support
Posts: 17
Joined: Sat Nov 26, 2005 11:06 am
Location: London, UK
I assigned the risk to the resource(s) as a relative time increase and then also as a relative cost increase – both at 15% chance with 10% outcome. But the issue I seem to have when I do this is that
the probability of it occurring goes above 15%. Do I potentially need to add the time delay to the task and cost increase to the resources(s) Or am I going about this all the wrong way.
Posts: 1008
Joined: Wed Nov 09, 2005 9:55 am
Calculated probability can vary from the input. But normally not by much. The reason is similar to throwing a dice. At each iteration, the risk has a 15% of occurring, so after 1 iteration, the
probability will be 0 or 100%, after 2 0,30, or 100%. After 100 iterations, it could occur 15x (ie 15%) but could be less or more. The more iterations that are run, the closer the probability will be
to 15%.
Intaver Support Team
Intaver Institute Inc.
Home of Project Risk Management and Project Risk Analysis software RiskyProject
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Method of Analysis BS 449 Part 2
Related Resources: hardware
Method of Analysis BS 449 Part 2
Engineering ANSI Hardware Design Data
Engineering Metric Hardware Design Data
Structural Stress Deflection Calculators, Equations
Method of Analysis Equations BS 449: Part 2
The method of analysis based on BS449: Part 2 and is described below.
Ordinary Bolts: Grades 4.6, 6.8, 8.8, 10.9, 12.9 and 14.9
The shear capacity P s is calculated as:
P[s] = p[s] A[s]
p[s] is the shear strength from Clause 50b and Table 20 of the BS449, and
A[s] is the Shank or Tensile Stress Area of the bolts as specified in respective British Standards.
The bearing capacity is taken as lesser of the bolt bearing capacity P bb and the connected ply bearing capacity P bs . The bolt bearing capacity P bb is calculated as:
P[bb] = d t p[bb]
d is the nominal bolt diameter,
t is the thickness of the connected ply, or if the bolts are countersunk, the thickness of the ply minus half the depth of countersinking, which is equal to half the nominal diameter of bolts, and
p[bb] is bearing strength of the bolt obtained from Clause 50b and Table 20 of BS449.
Depending upon the end distance e of the bolt hole to the ply edge described in the range 2d to 1.25d, the bearing capacity of the connected ply P[bs] is calculated as:
P[bs] = d t p[bs]<= e t p[bs] /2
p[bs] is the bearing strength of the connected parts obtained from Table 20A of BS 449.
The tension capacity P[t] of the bolt is calculated both for the Shank Length and the Threaded Length. The Shank Length capacity is based on nominal diameter of the bolt and its use is relevant in
combined Tension and Bending only, when shear plane lies in the shank length. The Threaded length capacity is based on tensile stress area of threads and is the critical strength in direct tension
and combined Tension and Bending when shear plane lies in the threaded length. The equation is:
P[t] = p[t] A[t]
p[t] is the tension strength obtained from Clause 50b and Table 20 of the code of BS 449, and
A[t] is the shank or tensile stress area as specified in respective British Standards.
Combined Tension and Shear
For any bolt diameter, the Combined Tension and Shear Capacity is calculated by typing a value in the cell for Applied Tension in its row. When subjected to both Tension and Shear, the combination
shear capacity F s is given by:
F[s] = (1.4 - F[t] /P[t] ) P[s]<= P[s]
where F t is the applied external tension and [t] , P[s] are as defined in the above. When the shear plane lies in the shank length, P[t] is Tension Capacity of the shank length. When the shear plane
lies in the threaded length, P[t] is Tension Capacity based on tensile stress area of the bolt.
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Chapter 3 Circle Set 3.2 | Maharashtra Board Book Solution
Chapter 3 Circle Set 3.2
Question 1.
Two circles having radii 3.5 cm and 4.8 cm touch each other internally. Find the distance between their centres.
Let the two circles having centres P and Q touch each other internally at point R.
Here, QR = 3.5 cm, PR = 4.8 cm
The two circles touch each other internally.
∴ By theorem of touching circles,
P – Q – R
PQ = PR – QR
= 4.8 – 3.5
= 1.3 cm
[The distance between the centres of circles touching internally is equal to the difference in their radii]
Question 2.
Two circles of radii 5.5 cm and 4.2 cm touch each other externally. Find the distance between their centres.
Let the two circles having centres P and R touch each other externally at point Q.
Here, PQ = 5.5 cm, QR = 4.2 cm
The two circles touch each other externally.
∴ By theorem of touching circles,
P – Q – R
PR = PQ + QR
= 5.5 + 4.2
= 9.7 cm
[The distance between the centres of the circles touching externally is equal to the sum of their radii]
Question 3.
If radii of two circles are 4 cm and 2.8 cm. Draw figure of these circles touching each other
i. externally
ii. internally.
i. Circles touching externally:
ii. Circles touching internally:
Question 4.
In the adjoining figure, the circles with centres P and Q touch each other at R A line passing through R meets the circles at A and B respectively. Prove that –
i. seg AP || seg BQ,
ii. ∆APR ~ ∆RQB, and
iii. Find ∠RQB if ∠PAR = 35°.
The circles with centres P and Q touch each other at R.
∴ By theorem of touching circles,
P – R – Q
i. In ∆PAR,
seg PA = seg PR [Radii of the same circle]
∴ ∠PRA ≅ ∠PAR (i) [Isosceles triangle theorem]
Similarly, in ∆QBR,
seg QR = seg QB [Radii of the same circle]
∴ ∠RBQ ≅ ∠QRB (ii) [Isosceles triangle theorem]
But, ∠PRA ≅ ∠QRB (iii) [Vertically opposite angles]
∴ ∠PAR ≅ ∠RBQ (iv) [From (i) and (ii)]
But, they are a pair of alternate angles formed by transversal AB on seg AP and seg BQ.
∴ seg AP || seg BQ [Alternate angles test]
ii. In ∆APR and ∆RQB,
∠PAR ≅ ∠QRB [From (i) and (iii)]
∠APR ≅ ∠RQB [Alternate angles]
∴ ∆APR – ∆RQB [AA test of similarity]
iii. ∠PAR = 35° [Given]
∴ ∠RBQ = ∠PAR= 35° [From (iv)]
In ∆RQB,
∠RQB + ∠RBQ + ∠QRB = 180° [Sum of the measures of angles of a triangle is 180°]
∴ ∠RQB + ∠RBQ + ∠RBQ = 180° [From (ii)]
∴ ∠RQB + 2 ∠RBQ = 180°
∴ ∠RQB + 2 × 35° = 180°
∴ ∠RQB + 70° = 180°
∴ ∠RQB = 110°
Question 5.
In the adjoining figure, the circles with centres A and B touch each other at E. Line l is a common tangent which touches the circles at C and D respectively. Find the length of seg CD if the radii
of the circles are 4 cm, 6 cm.
Construction : Draw seg AF ⊥ seg BD.
i. The circles with centres A and B touch each other at E. [Given]
∴ By theorem of touching circles,
A – E – B
∴ ∠ACD = ∠BDC = 90° [Tangent theorem]
∠AFD = 90° [Construction]
∴ ∠CAF = 90° [Remaining angle of ꠸AFDC]
∴ ꠸AFDC is a rectangle. [Each angle is of measure 900]
∴ AC = DF = 4 cm [Opposite sides of a rectangle]
Now, BD = BF + DF [B – F – C]
∴ 6 = BF + 4 BF = 2 cm
Also, AB = AE + EB
= 4 + 6 = 10 cm
[The distance between the centres of circles touching externally is equal to the sum of their radii]
ii. Now, in ∆AFB, ∠AFB = 90° [Construction]
∴ AB^2 = AF^2 + BF^2 [Pythagoras theorem]
∴ 10^2 = AF^2 + 2^2
∴ 100 = AF^2 + 4
Question 1.
Take three collinear points X – Y – Z as shown in figure.
Draw a circle with centre X and radius XY. Draw another circle with centre Z and radius YZ.
Note that both the circles intersect each other at the single point Y. Draw a line / through point Y and perpendicular to seg XZ. What is line l (Textbook pg. no. 56)
Line l is a common tangent of the two circles.
Question 2.
Take points Y – X – Z as shown in the figure. Draw a circle with centre Z and radius ZY.
Also draw a circle with centre X and radius XY. Note that both the circles intersect each other at the point Y.
Draw a line l perpendicular to seg YZ through point Y. What is line l? (Textbook pg. no. 56)
Line l is a common tangent of the two circles.
If two circles in the same plane intersect with a line in the plane in only one point, they are said to be touching circles and the line is their common tangent.
The point common to the circles and the line is called their common point of contact.
1. Circles touching externally:
For circles touching externally, the distance between their centres is equal to sum of their radii, i.e. AB = AC + BC
2. Circles touching internally:
For circles touching internally, the distance between their centres is equal to difference of their radii,
i. e. AB = AC – BC
Question 3.
The circles shown in the given figure are called externally touching circles. Why? (iexthook pg. no. 57)
Circles with centres R and S lie in the same plane and intersect with a line l in the plane in one and only one point T [R – T – S].
Hence the given circles are externally touching circles.
Question 4.
The circles shown in the given figure are called internally touching circles, why? (Textbook pg. no. 57)
Circles with centres N and M lie in the same plane and intersect with a line p in the plane in one and only one point T [K – N – M].
Hence, the given circles are internally touching circles.
Question 5.
In the given figure, the radii of the circles with centres A and B are 3 cm and 4 cm respectively. Find
i. d(A,B) in figure (a)
ii. d(A,B) in figure (b) (Textbook pg. no. 57)
i. Here, circle with centres A and B touch each other externally at point C.
∴ d(A, B) = d(A, C) + d(B ,C)
= 3 + 4
∴ d(A,B) = 7 cm
[The distance between the centres of circles touching externally is equal to the sum of their radii]
ii. Here, circle with centres A and 13 touch each other internally at point C.
∴ d(A, B) = d(A, C) – d(B, C)
= 4 – 3
∴ d(A,B) = 1 cm
[The distance between the centres of circles touching internally is equal to the difference in their radii]
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Algebraic structure
abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an
algebraic structure
consists of one or more sets, called
underlying sets
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers
are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
under one or more
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
, satisfying some
In traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other
words, an axiom is a logical statement that is assumed to be true...
s. Abstract algebra is primarily the study of algebraic structures and their properties. The notion of algebraic
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
has been formalized in
universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
In a slight abuse of notation, the expression "structure" can also refer only to the operations on a structure, and not to the underlying set itself. For example, the group
Structures whose axioms are all identities
Universal algebra often considers classes of algebraic structures (such as the class of all groups), together with operations (such as products) and relations (such as "substructure") between these
algebras. These classes are usually defined by "axioms", that is, a list of properties that all these structures have to share. If all axioms defining a class of algebras are "identities" , then the
corresponding class is called
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a
class of algebraic structures of the same signature which is closed under the taking of homomorphic...
(not to be confused with
algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
in the sense of
algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central
place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
Identities are equations formulated using only the operations the structure allows, and variables that are tacitly universally quantified over the relevant
Universe (mathematics)
In mathematics, and particularly in set theory and the foundations of mathematics, a universe is a class that contains all the entities one wishes to consider in a given situation...
. Identities contain no
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the
respective truth values of the original sentences.Each logical connective can be expressed as a...
existentially quantified variables
Quantification has several distinct senses. In mathematics and empirical science, it is the act of counting and measuring that maps human sense observations and experiences into members of some set
of numbers. Quantification in this sense is fundamental to the scientific method.In logic,...
, or relations of any kind other than the allowed operations. The study of varieties is an important part of
universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
An algebraic structure in a variety may be understood as the
quotient algebra
Quotient algebra
In mathematics, a quotient algebra, , also called a factor algebra is obtained by partitioning the elements of an algebra in equivalence classes given by a congruence, that is an equivalence relation
that is additionally compatible with all the operations of the algebra, in the formal sense...
of term algebra (also called "absolutely
free algebra
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It
also has a formulation in terms of category theory, although this is in yet more abstract terms....
") divided by the equivalence relations generated by a set of identities. So, a collection of functions with given
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an
algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...
generate a free algebra, the
term algebra
Term algebra
In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. For example, in a signature consisting of a single binary operation, the
term algebra over a set X of variables is exactly the free magma generated by X...
. Given a set of equational identities (the axioms), one may consider their symmetric, transitive closure
. The
quotient algebra
Quotient algebra
In mathematics, a quotient algebra, , also called a factor algebra is obtained by partitioning the elements of an algebra in equivalence classes given by a congruence, that is an equivalence relation
that is additionally compatible with all the operations of the algebra, in the formal sense...
is then the algebraic structure or variety. Thus, for example, groups have a signature containing two operators: the multiplication operator
, taking two arguments, and the inverse operator
, taking one argument, and the identity element
, a constant, which may be considered to be an operator taking zero arguments. Given a (countable) set of variables
, etc. the term algebra is the collection of all possible
Term (mathematics)
A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or
variable, or the product of several numbers or variables separated from another term by a + or - sign in an...
and the variables; so for example,
m(i(x), m(x,m(y,e)))
would be an element of the term algebra. One of the axioms defining a group is the identity
m(x, i(x)) = e
; another is
m(x,e) = x
. These equations induce
equivalence class
es on the free algebra; the quotient algebra then has the algebraic structure of a group.
All structures in this section are elements of naturally defined varieties. Some of these structures are most naturally axiomatized using one or more nonidentities, but are nevertheless varieties
because there exists an equivalent axiomatization, one perhaps less perspicuous, composed solely of identities. Algebraic structures that are not varieties are described in the following section, and
differ from varieties in their
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories...
In this section and the following one, structures are listed in approximate order of increasing complexity, operationalized as follows:
• Simple structures requiring but one set, the universe S, are listed before composite ones requiring two sets;
• Structures having the same number of required sets are then ordered by the number of binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition,
subtraction, multiplication and division....
s (0 to 4) they require. Incidentally, no structure mentioned in this entry requires an operation whose arity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the
domain in the corresponding Cartesian product...
exceeds 2;
• Let A and B be the two sets that make up a composite structure. Then a composite structure may include 1 or 2 function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function
assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
s of the form or ;
• Structures having the same number and kinds of binary operations and functions are more or less ordered by the number of required unary
Unary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation
on A....
and 0-ary (distinguished elements) operations, 0 to 2 in both cases.
The indentation structure employed in this section and the one following is intended to convey information. If structure
is under structure
and more indented, then all
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
s of
are theorems of
; the converse does not hold.
Ringoids and
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain
axiomatic identities...
s can be clearly distinguished despite both having two defining binary operations. In the case of ringoids, the two operations are linked by the distributive law; in the case of lattices, they are
linked by the
absorption law
Absorption law
In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.Two binary operations, say ¤ and *, are said to be connected by the absorption law if:...
. Ringoids also tend to have numerical
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
s, while lattices tend to have
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that
are relevant to mathematics...
Simple structures
binary operation:
• Set: a degenerate algebraic structure having no operations.
• Pointed set
Pointed set
In mathematics, a pointed set is a set X with a distinguished element x_0\in X, which is called the basepoint. Maps of pointed sets are those functions that map one basepoint to another, i.e. a
map f : X \to Y such that f = y_0. This is usually denotedf : \to .Pointed sets may be regarded as a...
: S has one or more distinguished elements, often 0, 1, or both.
• Unary system: S and a single unary operation over S.
• Pointed unary system: a unary system with S a pointed set.
Group-like structures
binary operation, denoted by
In computer programming, string concatenation is the operation of joining two character strings end-to-end. For example, the strings "snow" and "ball" may be concatenated to give "snowball"...
. For
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they
are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
boundary algebra
Laws of Form
Laws of Form is a book by G. Spencer-Brown, published in 1969, that straddles the boundary between mathematics and philosophy...
s, and sloops,
is a
pointed set
Pointed set
In mathematics, a pointed set is a set X with a distinguished element x_0\in X, which is called the basepoint. Maps of pointed sets are those functions that map one basepoint to another, i.e. a map f
: X \to Y such that f = y_0. This is usually denotedf : \to .Pointed sets may be regarded as a...
• Magma or groupoid
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
: S and a single binary operation over S.
□ Steiner magma: A commutative magma satisfying x(xy) = y.
☆ Squag: an idempotent Steiner magma.
☆ Sloop: a Steiner magma with distinguished element 1, such that xx = 1.
• Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity
: an associative magma.
□ Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as
they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
: a unital semigroup.
☆ Group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the
set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
: a monoid with a unary operation, inverse, giving rise to an inverse element
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an
element that can 'undo' the effect of combination with another given element...
○ Abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their
order . Abelian groups generalize the arithmetic of addition of integers...
: a commutative group.
□ Band
Band (algebra)
In mathematics, a band is a semigroup in which every element is idempotent . Bands were first studied and named by ; the lattice of varieties of bands was described independently in the early
1970s by Biryukov, Fennemore and Gerhard...
: a semigroup of idempotents.
☆ Semilattice
In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for
any nonempty finite subset...
: a commutative band. The binary operation can be called either meet
Meet (mathematics)
In mathematics, join and meet are dual binary operations on the elements of a partially ordered set. A join on a set is defined as the supremum with respect to a partial order on the set,
provided a supremum exists...
or join.
○ Boundary algebra: a unital semilattice (equivalently, an idempotent commutative monoid) with a unary operation, complementation
Complemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b
= 0....
, denoted by enclosing its argument in parentheses, giving rise to an inverse element that is the complement of the identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
. The identity and inverse elements bound S. Also, x(xy) = x(y) holds.
binary operations.
Quasigroups are listed here, despite their having 3 binary operations, because they are (nonassociative)
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
s. Quasigroups feature 3 binary operations only because establishing the quasigroup
cancellation property
Cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always
implies b = c.An element a in a magma has the right cancellation...
by means of identities alone requires two binary operations in addition to the group operation.
• Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...
: a cancellative magma. Equivalently, ∀x,y∈S, ∃!a,b∈S, such that xa = y and bx = y.
□ Loop: a unital quasigroup with a unary operation, inverse.
☆ Moufang loop
Moufang loop
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth
: a loop in which a weakened form of associativity, (zx)(yz) = z(xy)z, holds.
○ Group: an associative loop.
or more binary operations, including meet and join, connected by the absorption law.
is both a meet and join semilattice, and is a pointed set if and only if
is bounded. Lattices often have no unary operations. Every true statement has a
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an
involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
, obtained by replacing every instance of meet with join, and vice versa.
• Bounded lattice: S has two distinguished elements, the greatest lower bound and the least upper bound. Dualizing requires replacing every instance of one bound by the other, and vice versa.
□ Complemented lattice
Complemented lattice
In the mathematical discipline of order theory, a complemented lattice is a bounded lattice in which every element a has a complement, i.e. an element b satisfying a ∨ b = 1 and a ∧ b = 0....
: a lattice with a unary operation, complementation, denoted by postfix
Reverse Polish notation
Reverse Polish notation is a mathematical notation wherein every operator follows all of its operands, in contrast to Polish notation, which puts the operator in the prefix position. It is
also known as Postfix notation and is parenthesis-free as long as operator arities are fixed...
" ' ", giving rise to an inverse element. That element and its complement bound the lattice.
• Modular lattice
Modular lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition:Modular law: x ≤ b implies x ∨ = ∧ b,where ≤ is the partial
order, and ∨ and ∧ are...
: a lattice in which the modular identity holds.
□ Distributive lattice
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets
for which the lattice operations can be given by set union and intersection...
: a lattice in which each of meet and join distributes
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets
for which the lattice operations can be given by set union and intersection...
over the other. Distributive lattices are modular, but the converse does not hold.
☆ Kleene algebra
Kleene algebra
In mathematics, a Kleene algebra is either of two different things:* A bounded distributive lattice with an involution satisfying De Morgan's laws , additionally satisfying the inequality
x∧−x ≤ y∨−y. Kleene algebras are subclasses of Ockham algebras...
: a bounded distributive lattice with a unary operation whose identities are x"=x, (x+y)'=x'y', and (x+x')yy'=yy'. See "ring-like structures" for another structure having the same name.
☆ Boolean algebra: a complemented distributive lattice. Either of meet or join can be defined in terms of the other and complementation.
○ Interior algebra
Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the
modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic...
: a Boolean algebra with an added unary operation, the interior operator, denoted by postfix " ' " and obeying the identities x'x=x, x"=x, (xy)'=x'y', and 1'=1.
☆ Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest
such in the sense that if c∧a ≤ b then c ≤ a→b...
: a bounded distributive lattice with an added binary operation, relative pseudo-complement, denoted by infix
An infix is an affix inserted inside a word stem . It contrasts with adfix, a rare term for an affix attached to the end of a stem, such as a prefix or suffix.-Indonesian:...
" ' ", and governed by the axioms x'x=1, x(x'y) = xy, x'(yz) = (x'y)(x'z), (xy)'z = (x'z)(y'z).
binary operations,
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right,
there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
, with multiplication
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
over addition. Semirings are pointed sets.
• Semiring
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse...
: a ringoid such that S is a monoid under each operation. Each operation has a distinct identity element. Addition also commutes, and has an identity element that annihilates multiplication.
□ Commutative semiring: a semiring with commutative multiplication.
□ Ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under
addition and a semigroup under multiplication such that multiplication distributes over addition...
: a semiring with a unary operation, additive inverse, giving rise to an inverse element -x, which when added to x, yields the additive identity element. Hence S is an abelian group under
☆ Rng
Rng (algebra)
In abstract algebra, a rng is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element...
: a ring lacking a multiplicative identity.
☆ Commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative
: a ring with commutative multiplication.
○ Boolean ring
Boolean ring
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists only of idempotent elements....
: a commutative ring with idempotent multiplication, equivalent to a Boolean algebra.
□ Kleene algebra: a semiring with idempotent addition and a unary operation, the Kleene star
Kleene star
In mathematical logic and computer science, the Kleene star is a unary operation, either on sets of strings or on sets of symbols or characters. The application of the Kleene star to a set V
is written as V*...
, denoted by postfix * and obeying the identities (1+x*x)x*=x* and (1+xx*)x*=x*. See "Lattice-like structures" for another structure having the same name.
N.B. The above definition of ring does not command universal assent. Some authorities employ "ring" to denote what is here called a rng, and refer to a ring in the above sense as a "ring with
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
: Composite Systems Defined over Two Sets,
The members of:
are scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce
another vector....
s, denoted by Greek letters.
is a ring under the binary operations of scalar addition and multiplication;
module elements
(often but not necessarily vectors
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to
be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
), denoted by Latin letters.
is an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian
groups generalize the arithmetic of addition of integers...
under addition. There may be other binary operations.
scalar multiplication
of scalars and module elements is a function
which commutes, associates (∀
, ∀
) = (
), has 1 as identity element, and distributes over module and scalar addition. If only the pre(post)multiplication of module elements by scalars is defined, the result is a
• Free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a
set is a special case of a free module on a set.-Definition:...
: a module having a free basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which
define a "coordinate system"...
, {
[1], ...
, where the positive integer
is the dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one
because only one coordinate is needed to specify a point on it...
of the free module. For every
, there exist κ[1], ..., κ[n]∈
such that
= κ[1]
[1] + ... + κ[n]
[n]. Let 0 and 0 be the respective identity elements for module and scalar addition. If
[1] + ... +
[n] = 0, then
[1] = ... =
[n] = 0. (Note that the class of free modules over a given ring
is in general not a variety.)
• Algebra over a ring (also
): a (free) module where
is a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
. There is a second binary operation over
, called multiplication and denoted by concatenation, which distributes over module addition and is bilinear: α(
) = (α
• Jordan ring
Jordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
: an algebra over a ring whose module multiplication commutes, does not associate, and respects the Jordan identity.
Vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to
be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, closely related to modules, are defined in the next section.
Structures with some axioms that are not identities
The structures in this section are not axiomatized with identities alone, so the classes considered below are not varieties. Nearly all of the nonidentities below are one of two very elementary
1. The starting point for all structures in this section is a "nontrivial" ring, namely one such that S≠{0}, 0 being the additive identity element
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
. The nearest thing to an identity implying S≠{0} is the nonidentity 0≠1, which requires that the additive and multiplicative identities be distinct.
2. Nearly all structures described in this section include identities that hold for all members of S except 0. In order for an algebraic structure to be a variety, its operations must be defined for
all members of S; there can be no partial operations.
Structures whose axioms unavoidably include nonidentities are among the most important ones in mathematics, e.g.,
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction,
multiplication, and division, satisfying certain axioms...
s and
vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to
be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s. Moreover, much of theoretical physics can be recast as models of
multilinear algebra
Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra
builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...
s. Although structures with nonidentities retain an undoubted algebraic flavor, they suffer from defects varieties do not have. For example, neither the product of
integral domain
s nor a free field over any set exist.
binary operation
Binary operation
In mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition,
subtraction, multiplication and division....
s, addition and multiplication. S is an
infinite set
. Arithmetics are pointed unary systems, whose
unary operation
Unary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on
is injective successor, and with distinguished element 0.
• Robinson arithmetic
Robinson arithmetic
In mathematics, Robinson arithmetic, or Q, is a finitely axiomatized fragment of Peano arithmetic , first set out in R. M. Robinson . Q is essentially PA without the axiom schema of induction.
Since Q is weaker than PA, it is incomplete...
. Addition and multiplication are recursively
Recursion is the process of repeating items in a self-similar way. For instance, when the surfaces of two mirrors are exactly parallel with each other the nested images that occur are a form of
infinite recursion. The term has a variety of meanings specific to a variety of disciplines ranging from...
defined by means of successor. 0 is the identity element for addition, and annihilates multiplication. Robinson arithmetic is listed here even though it is a variety, because of its closeness to
Peano arithmetic.
□ Peano arithmetic. Robinson arithmetic with an axiom schema
Axiom schema
In mathematical logic, an axiom schema generalizes the notion of axiom.An axiom schema is a formula in the language of an axiomatic system, in which one or more schematic variables appear.
These variables, which are metalinguistic constructs, stand for any term or subformula of the system, which...
of induction
Mathematical induction
Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
. Most ring and field axioms bearing on the properties of addition and multiplication are theorems of Peano arithmetic or of proper extensions thereof.
Field-like structures
binary operations, addition and multiplication. S is nontrivial, i.e., S≠{0}.
• Domain
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or
right zero divisors. Some authors require the ring to be nontrivial...
: a ring whose sole zero divisor
Zero divisor
In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there
exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
is 0.
□ Integral domain: a domain whose multiplication commutes. Also a commutative cancellative ring.
☆ Euclidean domain
Euclidean domain
In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be
described in detail below – which allows a suitable generalization of the Euclidean algorithm...
: an integral domain with a function
f: S→
N satisfying the division with remainder property.
Division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative
inverse, i.e., an element x with...
(or sfield, skew field): a ring in which every member of S other than 0 has a two-sided multiplicative inverse. The nonzero members of S form a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the
operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
under multiplication.
• Field: a division ring whose multiplication commutes. The nonzero members of
S form an abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian
groups generalize the arithmetic of addition of integers...
under multiplication.
• Ordered field
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was
abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
: a field whose elements are totally ordered.
□ Real field
Formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that admits an ordering which makes it an ordered field.-Alternative Definitions:...
: a Dedekind complete ordered field.
The following structures are not varieties for reasons in addition to S
• Simple ring
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be
confused with the related one of a ring being simple as a left module over itself...
: a ring having no ideals
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers
like "even number" or "multiple of 3"....
other than 0 and
Artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the
descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
: a ring whose ideals satisfy the descending chain condition.
Composite Systems: Vector Spaces, and Algebras over Fields. Two Sets,
, and at least three binary operations.
The members of:
are vectors, denoted by lower case letters.
is at minimum an abelian group under vector addition, with distinguished member 0.
are scalar
Scalar (mathematics)
In linear algebra, real numbers are called scalars and relate to vectors in a vector space through the operation of scalar multiplication, in which a vector can be multiplied by a number to produce
another vector....
s, denoted by Greek letters.
is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction,
multiplication, and division, satisfying certain axioms...
, nearly always the real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex field, with 0 and 1 as distinguished members.
Three binary operations.
• Vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken
to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
: a free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a
set is a special case of a free module on a set.-Definition:...
of dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one
because only one coordinate is needed to specify a point on it...
except that
is a field.
• Normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following
properties of "vector length" are crucial....
: a vector space and with a norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero
, namely a function
that is positive homogeneous, subadditive, and positive definite.
• Inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar
quantity known as the inner product of the vectors...
vector space): a normed vector space such that
is the real field, whose norm is the square root of the inner product,
. Let
, and
be positive integers such that 1≤
. Then
has an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a
Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
such that
[j] = 1 if
and 0 otherwise; see free module above.
Unitary space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar
quantity known as the inner product of the vectors...
: Differs from inner product spaces in that
is the complex field, and the inner product has a different name, the hermitian inner product, with different properties: conjugate symmetric, bilinear, and positive definite. See Birkhoff and Mac
Lane (1979: 369).
Graded vector space
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector
subspaces.-N-graded vector spaces:...
: a vector space such that the members of
have a direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which
contains the given modules as submodules...
decomposition. See graded algebra
Graded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
Four binary operations.
• Algebra over a field
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an
operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
: An algebra over a ring except that
is a field instead of a commutative ring.
• Jordan algebra
Jordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
: a Jordan ring except that
is a field.
Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the
concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
: an algebra over a field
Algebra over a field
In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation,
usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
respecting the Jacobi identity
Jacobi identity
In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations,
order of evaluation is significant for operations satisfying Jacobi identity...
, whose vector multiplication, the Lie bracket
Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, Jacobi–Lie bracket, or commutator of vector fields is a bilinear differential operator which assigns, to any two
vector fields X and Y on a smooth manifold M, a third vector field denoted [X, Y]...
denoted [
], anticommutes, does not associate, and is nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
Associative algebra
Associative algebra
In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
: an algebra over a field, or a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, whose vector multiplication associates.
• Linear algebra
Linear algebra
Linear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called
linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
: an associative unital algebra with the members of
being matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements
isMatrices of the same size can be added or subtracted element by element...
. Every matrix has a dimension
positive integers. If one of
is 1, the matrix is a vector; if both are 1, it is a scalar. Addition of matrices is defined only if they have the same dimensions. Matrix multiplication
Matrix multiplication
In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their
multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...
, denoted by concatenation, is the vector multiplication. Let matrix
and matrix
. Then
is defined if and only if
, if and only if
. There also exists an
and an
such that
. If
are vectors having the same dimensions, they have an inner product, denoted 〈
〉. Hence there is an orthonormal basis; see inner product space above. There is a unary function, the determinant
In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its
value exist as well...
, from square (
for any
) matrices to
Commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on
commutative algebra...
: an associative algebra whose vector multiplication commutes.
• Symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
: a commutative algebra with unital vector multiplication.
Composite Systems: Multilinear algebra
Multilinear algebra
In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra
builds on the concepts of p-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...
s. Two sets,
. Four binary operations:
1. The members of
are multivector
In multilinear algebra, a multivector or clif is an element of the exterior algebra on a vector space, \Lambda^* V. This algebra consists of linear combinations of simple k-vectors v_1\wedge\cdots\
wedge v_k."Multivector" may mean either homogeneous elements In multilinear algebra, a multivector...
s (including vectors), denoted by lower case Latin letters.
is an abelian group under multivector addition, and a monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they
are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
under outer product
Outer product
In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix...
. The outer product goes under various names, and is multilinear
Tensor (intrinsic definition)
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept...
in principle but usually bilinear. The outer product defines the multivectors recursively starting from the vectors. Thus the members of
have a "degree" (see graded algebra below). Multivectors may have an inner product as well, denoted
, that is symmetric, linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
, and positive definite; see inner product space above.
The properties and notation of
are the same as those of
above, except that
may have −1 as a distinguished member.
is usually the real field, as multilinear algebras are designed to describe physical phenomena without complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the
number line for the real part and adding a vertical axis to plot the imaginary part...
The multiplication of scalars and multivectors,
, has the same properties as the multiplication of scalars and module elements that is part of a module.
• Graded algebra
Graded algebra
In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
: an associative algebra with unital outer product. The members of
have a direct sum decomposition resulting in their having a "degree," with vectors having degree 1. If
have degree
, respectively, the outer product of
is of degree
also has a distinguished member 0 for each possible degree. Hence all members of
having the same degree form an abelian group under addition.
• Exterior algebra
Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in Euclidean geometry to study areas, volumes, and their higher-dimensional analogs...
Grassmann algebra
): a graded algebra whose anticommutative outer product, denoted by infix ∧, is called the exterior product.
has an orthonormal basis.
[1] ∧
[2] ∧ ... ∧
[k] = 0 if and only if
[1], ...,
[k] are linearly dependent. Multivectors also have an inner product.
• Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.
The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...
: an exterior algebra with a symmetric bilinear form
. The special case
=0 yields an exterior algebra. The exterior product is written 〈
〉. Usually, 〈
[i]〉 = -1 (usually) or 1 (otherwise).
Geometric algebra
Geometric algebra
Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic
geometry. GA now finds application in all of physics, in graphics and in robotics...
: an exterior algebra whose outer (called
) product is denoted by concatenation. The geometric product of parallel multivectors commutes, that of orthogonal vectors anticommutes. The product of a scalar with a multivector commutes.
vv yields a scalar.
• Grassmann-Cayley algebra
Grassmann-Cayley algebra
Grassmann–Cayley algebra, also known as double algebra, is a form of modeling algebra for use in projective geometry. The technique is based on work by German mathematician Hermann Grassmann on
exterior algebra, and subsequently by British mathematician Arthur Cayley's work on matrices and linear...
: a geometric algebra without an inner product.
Some recurring universes: N
=natural numbers;
=rational numbers;
=real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
=complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the
number line for the real part and adding a vertical axis to plot the imaginary part...
N is a pointed unary system, and under addition and multiplication, is both the standard interpretation of Peano arithmetic and a commutative
In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse...
Boolean algebras are at once
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain
axiomatic identities...
, and
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition
and a semigroup under multiplication such that multiplication distributes over addition...
. They would even be
abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian
groups generalize the arithmetic of addition of integers...
s if the identity and inverse elements were identical instead of complements.
Group-like structures
under addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right,
there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
(+) is a magma
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
, and even a free
Free semigroup
In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences of zero or more elements from A. It is usually denoted A∗. The identity element is the unique
sequence of zero elements, often called the empty string and denoted by ε or λ, and the...
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity
under addition is a magma with an identity, and in particular a free monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they
are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
under subtraction
In arithmetic, subtraction is one of the four basic binary operations; it is the inverse of addition, meaning that if we start with any number and add any number and then subtract the same number we
added, we return to the number we started with...
(−) is a quasigroup.
under division
Division (mathematics)
right|thumb|200px|20 \div 4=5In mathematics, especially in elementary arithmetic, division is an arithmetic operation.Specifically, if c times b equals a, written:c \times b = a\,...
(÷) is a quasigroup.
Every group is a loop, because a * x = b if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
x = a^−1 * b, and y * a = b if and only if y = b * a^−1.
2x2 matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements
isMatrices of the same size can be added or subtracted element by element...
(of non-zero determinant) with matrix multiplication form a group.
under addition (+) is an abelian group.
under multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
(×) is an abelian group.
Every cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group
is a power of g .-Definition:A group G is called cyclic if there exists an element g...
G is abelian, because if x, y are in G, then xy = a^ma^n = a^m+n = a^n+m = a^na^m = yx. In particular,
is an abelian group under addition, as is the integers modulo n
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
A monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they
are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
is a category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ,
where these collections satisfy certain basic conditions...
with a single object, in which case the composition of morphisms
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument
of f instead of x...
and the identity morphism interpret monoid multiplication and identity element, respectively.
The Boolean algebra 2 is a boundary algebra.
More examples of groups
Examples of groups
Some elementary examples of groups in mathematics are given on Group .Further examples are listed here.-Permutations of a set of three elements:Consider three colored blocks , initially placed in the
order RGB...
and list of small groups. Lattices
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain
axiomatic identities...
• The normal subgroup
Normal subgroup
In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
s of a group, and the submodules of a module, are modular lattices.
• Any field of sets, and the connectives
Logical connective
In logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the
respective truth values of the original sentences.Each logical connective can be expressed as a...
of first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower
predicate calculus, quantification theory, and predicate logic...
, are models of Boolean algebra.
• The connectives of intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all
well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
form a model of Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in
the sense that if c∧a ≤ b then c ≤ a→b...
• The modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a
statement. For example, the statement "John is happy" might be qualified by saying that John is...
S4 is a model of interior algebra
Interior algebra
In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic
S4 what Boolean algebras are to set theory and ordinary propositional logic...
• Peano arithmetic and most axiomatic set theories, including ZFC, NBG
Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory that is a conservative extension of the canonical axiomatic set theory ZFC. A statement in the
language of ZFC is provable in NBG if and only if it is provable in ZFC. The ontology of NBG includes...
, and New foundations
New Foundations
In mathematical logic, New Foundations is an axiomatic set theory, conceived by Willard Van Orman Quine as a simplification of the theory of types of Principia Mathematica. Quine first proposed
NF in a 1937 article titled "New Foundations for Mathematical Logic"; hence the name...
, can be recast as models of relation algebra
Relation algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition
and a semigroup under multiplication such that multiplication distributes over addition...
-like structures
• The set R[X] of all polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative
integer exponents...
s over some coefficient ring R is a ring.
• 2x2 matrices with matrix addition and multiplication form a ring.
• If n is a positive integer, then the set Z[n] = Z/nZ of integers modulo n (the additive cyclic group of order n ) forms a ring having n elements (see modular arithmetic).
• Sets of hypercomplex number
Hypercomplex number
In mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number
systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
s were early prototypes of algebraic structures now called rings.
Integral domains
• Z under addition and multiplication is an integral domain.
• The p-adic integers
P-adic number
In mathematics, and chiefly number theory, the p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a way different from the extension of the
rational number system to the real and complex number systems...
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction,
multiplication, and division, satisfying certain axioms...
• Each of Q, R, and C, under addition and multiplication, is a field.
• R totally ordered by "<" in the usual way is an ordered field
Ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Historically, the axiomatization of an ordered field was
abstracted gradually from the real numbers, by mathematicians including David Hilbert, Otto Hölder and...
and is categorical
See:* Categorical imperative* Morley's categoricity theorem* Categorical data analysis* Categorical distribution* Categorical logic* Categorical syllogism* Categorical proposition*
Categorization* Categorical perception* Category theory...
. The resulting real field grounds real
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions
and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
and functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators
acting upon these spaces and respecting these structures in a suitable sense...
□ R contains several interesting subfields, the algebraic
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be
transcendental; almost all real numbers are transcendental...
, the computable
Computable number
In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers
that can be computed to within any desired precision by a finite, terminating algorithm...
, and the definable number
Definable number
A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the
unique real number such that φ holds in the standard model of set theory .For the purposes of this article,...
• An algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
is a finite field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the
base field and satisfies additional properties...
of Q, that is, a field containing Q which has finite dimension as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken
to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over Q. Algebraic number fields are very important in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
• If q > 1 is a power of a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number.
For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, then there exists (up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by
a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two
structures are said to be isomorphic. In a certain sense, isomorphic structures are...
) exactly one finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory,
cryptography, and coding theory...
with q elements, usually denoted F[q], or in the case that q is itself prime, by Z/qZ. Such fields are called Galois fields, whence the alternative notation GF(q). All finite fields are
isomorphic to some Galois field. p, the set Z[]p = Z/pZ of integers modulo p is the finite field with p elements: F[p] = {0, 1, ..., p − 1} where the operations are defined by performing the
operation in Z, dividing by p and taking the remainder; see modular arithmetic.
Allowing additional structure
Algebraic structures can also be defined on sets with added structure of a non-algebraic nature, such as a
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or
. The added structure must be compatible, in some sense, with the algebraic structure.
• Ordered group
Ordered group
In abstract algebra, a partially-ordered group is a group equipped with a partial order "≤" that is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if
a ≤ b then a+g ≤ b+g and g+a ≤ g+b.An element x of G is called positive element if 0 ≤ x...
: a group with a compatible partial order. I.e., S is partially ordered.
• Linearly ordered group
Linearly ordered group
In abstract algebra a linearly ordered or totally ordered group is an ordered group G such that the order relation "≤" is total...
: a group whose S is a linear order.
• Archimedean group
Archimedean group
In abstract algebra, a branch of mathematics, an Archimedean group is an algebraic structure consisting of a set together with a binary operation and binary relation satisfying certain axioms
detailed below. We can also say that an Archimedean group is a linearly ordered group for which the...
: a linearly ordered group for which the Archimedean property
Archimedean property
In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some ordered or normed groups, fields, and
other algebraic structures. Roughly speaking, it is the property of having no infinitely large or...
• Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
: a group whose S has a compatible smooth manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
• Topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the
topology. A topological group is a mathematical object with both an algebraic structure and a...
: a group whose S has a compatible topology.
• Topological vector space
Topological vector space
In mathematics, a topological vector space is one of the basic structures investigated in functional analysis...
: a vector space whose M has a compatible topology; a superset of normed vector space
Normed vector space
In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the "length" of a vector is intuitive and can easily be extended to any real vector space Rn. The following
properties of "vector length" are crucial....
• Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a
norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
s, Hilbert spaces, Inner product space
Inner product space
In mathematics, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar
quantity known as the inner product of the vectors...
• Vertex operator algebra
Vertex operator algebra
In mathematics, a vertex operator algebra is an algebraic structure that plays an important role in conformal field theory and related areas of physics...
Category theory
The discussion above has been cast in terms of elementary
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
Category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ,
where these collections satisfy certain basic conditions...
is another way of reasoning about algebraic structures (see, for example, Mac Lane 1998). A category is a collection of
with associated
Every algebraic structure has its own notion of
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning
"shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
, namely any
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function
assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
compatible with the operation(s) defining the structure. In this way, every algebraic structure gives rise to a
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ,
where these collections satisfy certain basic conditions...
. For example, the
category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
has all
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the
operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
as objects and all
group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s as morphisms. This
concrete category
Concrete category
In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category as sets with
additional structure, and of its morphisms as structure-preserving functions...
may be seen as a
category of sets
Category of sets
In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to
with added category-theoretic
Structure (category theory)
In mathematics, progress often consists of recognising the same structure in different contexts - so that one method exploiting it has multiple applications...
. Likewise, the category of
topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the
topology. A topological group is a mathematical object with both an algebraic structure and a...
s (whose morphisms are the continuous group homomorphisms) is a
category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the
composition of two continuous maps is again continuous...
with extra structure. A
forgetful functor
Forgetful functor
In mathematics, in the area of category theory, a forgetful functor is a type of functor. The nomenclature is suggestive of such a functor's behaviour: given some object with structure as input, some
or all of the object's structure or properties is 'forgotten' in the output...
between categories of algebraic structures "forgets" a part of a structure.
There are various concepts in category theory that try to capture the algebraic character of a context, for instance
• algebraic
• essentially algebraic
• presentable
• locally presentable
• monadic
Monad (category theory)
In category theory, a branch of mathematics, a monad, Kleisli triple, or triple is an functor, together with two natural transformations...
functors and categories
• universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of
category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
See also
• free object
Free object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure .
It also has a formulation in terms of category theory, although this is in yet more abstract terms....
• list of first order theories
• signature
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an
algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...
External links
• Jipsen's algebra structures. Includes many structures not mentioned here.
• Mathworld page on abstract algebra.
• Stanford Encyclopedia of Philosophy
Stanford Encyclopedia of Philosophy
The Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field,
including professors from over 65 academic institutions worldwide...
: Algebra by Vaughan Pratt.
The source of this article is
, the free encyclopedia. The text of this article is licensed under the
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Integrating atomic basins
Critic2 provides several methods to integrate the attractor (Bader) basins associated to the maxima of a field. In QTAIM theory, this field is the electron density, the attractors are (usually)
nuclei and the basins are the atomic regions. The integrated properties are atomic properties (e.g. atomic charges, volumes, moments, etc.). The attractor basins are defined by a zero-flux condition
of the electron density: no gradient paths cross the boundary between attractor regions. This makes the basins local to each attractor, but their definition is a relatively complex algorithmic
The simplest way of integrating an attractor basin is bisection. A number of points distributed in a small sphere around the atom are chosen, each of them determining a ray. On each ray, a process of
bisection is started. A point belongs to the basin if the gradient path traced upwards ends up at the position of the attractor we are considering. If the end-point is a different attractor, then the
point is not in the basin. By using bisection, it is possible to determine the basin limit (called the interatomic surface, IAS). The bisection algorithm is implemented in critic2, and can be
accessed with the INTEGRALS keyword. Bisection works best with analytical fields such as molecular wavefunctions or WIEN2k, but can be used with any field.
The qtree algorithm is based on the recursive subdivision of the irreducible Wigner-Seitz (IWS). In qtree, the smallest symmetry-irreducible portion of space is considered and a tetrahedral mesh of
points is superimposed on it. The gradient path is traced from all those points and the points are assigned to different atoms (the points are “colored”). The integration is performed by quadrature
over the points belonging to a given basin. The qtree algorithm is accessed through the QTREE keyword, and is suited for small crystals and analytical fields (e.g. WIEN2k or elk).
Lastly, integration algorithms based on grid discretization are very popular nowadays thanks to the widespread use of pseudopotential/plane-waves DFT methods. Critic2 provides the integration method
of Yu and Trinkle (YT). The algorithm is based on the assignment of integration weights to each point in the numerical grid by evaluating the flow of the gradient using the neighboring points. This
algorithm is extremely efficient and robust and is strongly recommended in the case of fields on a grid. The associated keyword is YT.
An alternative to the YT method is the method proposed by Henkelman et al. (Comput. Mater. Sci. 36, 254-360 (2006), J. Comput. Chem. 28, 899-908 (2007), J. Phys.: Condens. Matter 21, 084204 (2009)),
which is implemented in critic2 through the keyword BADER.
The field that determines the basins being calculated is always the reference field (see REFERENCE). In general, it is necessary to define one or more properties to be integrated inside the basins
that use other scalar fields. For instance, to calculate the electron population inside an ELF basin, the ELF would be the reference field and the electron density would be an integrable property.
List of Properties Integrated in the Attractor Basins (INTEGRABLE)
INTEGRABLE id.s {F|FVAL|GMOD|LAP|LAPVAL} [NAME name.s]
INTEGRABLE id.s {MULTIPOLE|MULTIPOLES} [lmax.i]
INTEGRABLE id.s DELOC [WANNIER] [PSINK] [NOU] [NOSIJCHK] [NOFACHK] [NORESTART] [WANCUT wancut.r]
[DI3 [atom1.i [atom2.i [ix.i iy.i iz.i]]]]
INTEGRABLE "expr.s"
INTEGRABLE DELOC_SIJCHK file-sij.s
INTEGRABLE DELOC_FACHK file-fa.s
INTEGRABLE ... [NAME name.s]
Critic2 uses an internal list of all properties that will be integrated in the attraction basins. This list can be modified by the user with the INTEGRABLE keyword. This keyword has a syntax similar
to the list of properties calculated at the critical points, POINTPROP.
A single INTEGRABLE command assigns a new quantity to be integrated in the atomic basins. The new integrable property is related to field id.s (given as field number or identifier). This quantity can
be the field value itself (F), its valence component (if the field is core-augmented, FVAL), the gradient norm (GMOD), the Laplacian (LAP), or the valence-component of the Laplacian (LAPVAL). If no
keyword is given after id.s, F is used by default.
With the MULTIPOLES (or MULTIPOLE) keyword, the multipole moments of the field are calculated up to an l equal to lmax.i (default: 5). This keyword only applies to the BADER and YT integration
methods. For the others, it is equivalent to the field value (same as F). The units for all calculated multipoles are atomic units.
The keyword DELOC activates the calculation of the delocalization indices (DIs) using field id.s. The calculation of DIs requires that the field is a Quantum ESPRESSO pwc file, which contains the
Bloch states. There are two modes of operation with DELOC:
• Calculate the DIs using maximally localized Wannier functions (MLWF). This requires wannier90 checkpoint files to be provided with the pwc file, and is the default behavior in this case. More
details are given below. The use of Wannier functions can be forced using the WANNIER keyword in INTEGRABLE DELOC.
• Calculate the DIs using Bloch states. This is much less efficient than using Wannier functions but it is simpler because it does not require an additional wannier90 calculation and allows
calculating DIs in metals. More details are given below. The use of Bloch states can be forced using the PSINK keyword in INTEGRABLE DELOC.
Also, see the example for the calculation of DIs in solids.
Both ways of calculating the DIs generate by default two checkpoint files containing the (complex) overlap matrices (sij) or the matrix of atomic integrals of the exchange-correlation density (fa).
The delocalization indices can be calculated from these files bypassing having to load the pwc and wannier90 checkpoint files with the DELOC_SIJCHK and DELOC_FACHK keywords. This is useful because
the pwc files can be quite large, so it may be interesting to delete them after the sij or fa matrices are computed. Both keywords accept a string pointing to the corresponding file. The DI
calculation from these matrices will typically take only a few seconds.
Critic2 can also use the calculated atomic overlap matrices to compute the three-center delocalization indices (DI3), see this article and references therein. The DI3 allow a three-body decomposition
of the localization and delocalization indices, and require only the atomic overlap matrix calculated for the two-center DIs. The calculation of three-center delocalization indices is activated using
the DI3 option to the DELOC integrable property.
The cost of calculating the DI3s from the atomic overlap matrix scales very quickly with the number of atoms and bands in the system, and a full DI3 calculation produces far too much output than it
is commonly useful. To simplify the calculation, the following options are possible:
• DELOC DI3 atom1.i: calculate only the DI3s associated with the attractor with index atom1.i. This keyword ensures the partition in DI3s of all the delocalization indices in which atom1.i is
• DELOC DI3 atom1.i atom2.i ix.i iy.i iz.i: calculate only the DI3s associated with the attractor with index atom1.i from the main cell and with attractor with index atom2.i translated by lattice
vector ix.i iy.i iz.i. The lattice translation vector (ix.i iy.i iz.i) is optional. If it is not given, it is assumed to be zero. This keyword ensures the partition in DI3s of the delocalization
index relating atom1.i with atom2.i translated by vector ix.i iy.i iz.i.
In addition, it is possible to define an integrable property using an expression involving more than one field (expr.s). For instance, if the spin-up density is in field 1 and the spin-down density
is in field 2, the atomic moments can be obtained using:
LOAD AS "$1+$2"
INTEGRABLE "$1-$2"
Note that the quotation marks are required.
The additional keyword NAME can be used with any of the options above to change the name of the integrable property, for easy identification in the output.
The keyword CLEAR resets the list to its initial state (volume, electron population, and Laplacian in crystals; electron population and Laplacian in molecules). Using the INTEGRABLE keyword will
print a report on the list of integrable properties.
The default integrable properties are:
• Volume (1), in crystals only.
• Pop (fval): the value of the reference field is integrated. If the reference field is the density, then this is the number of electrons in the basin. If core augmentation is active for this
field, only the valence contribution is integrated.
• Lap (lap(fval)): the Laplacian of the reference field. The integrated Laplacian has been traditionally used as a check of the quality of the integration because the exact integral is zero
regardless of the basin (because of the divergence theorem). However, it is difficult to obtain a zero in the Laplacian integral in critic2 in some cases because of numerical inaccuracies:
□ In fields based on a grid, the numerical interpolation gives a noisy Laplacian.
□ In FPLAPW fields (WIEN2k and elk), the discontinuity at the muffin surface introduce a non-zero contribution to the integral.
If f is a core-augmented field, only the valence Laplacian is integrated.
Integrating Delocalization Indices in a Solid With Maximally Localized Wannier Functions
The keyword DELOC activates the calculation of localization and delocalization indices (DIs) in a crystal using the procedure described in the literature. DIs can be calculated only if the loaded
field contains information about individual Kohn-Sham states and the orbital rotation that leads to the maximally localized Wannier functions (MLWF). This is done by using a field loaded from a .pwc
file (generated by the pw2critic.x utility in Quantum ESPRESSO) together with a checkpoint (chk) file from wannier90. The former contains the electronic wavefunctions and the latter the orbital
rotation. For maximum consistency, the pwc file can also be used to provide the structural information for the run via the CRYSTAL keyword.
In addition to these data, the calculation of DIs has a few requirements: the grid must be consistent with that of the reference field, and the DIs can be calculated using YT or BADER only.
A typical delocalization index calculation comprises the following steps:
• Run a PAW calculation, then obtain an all-electron density using pp.x with plot_num=21. This creates a cube file (rhoae.cube) that gives the Bader basins for the calculation (the pseudo-valence
density is not valid for this purpose).
• Run an SCF calculation with norm-conserving pseudopotentials and the same ecutrho as the calculation in the first step, so the two grids have the same size.
• Use the open_grid.x utility in Quantum ESPRESSO to unpack the symmetry of the k-point grid, in preparation for the wannier90 run (this would normally be accomplished by a non-SCF calculation, but
with open_grid.x it is easier and much faster).
• Use pw2critic.x on the output of open_grid.x to generate the pwc file.
• Run wannier90 on the result of open_grid.x to generate the chk file.
• Load the all-electron density and the pwc and chk files as two fields in critic2. Set the former as the reference density and the latter as INTEGRABLE DELOC. If the system is spin-polarized, two
checkpoint files will be necessary, one for each spin component.
• Run YT or BADER. YT is usually more accurate but takes longer than BADER.
Additional options for the DI calculation follow. The NOU option disables the use of the U rotation matrices to calculate the MLWFs. This makes critic2 calculate the DI using Wannier functions
obtained by using a straight Wannier transformation from the Bloch states. This is naturally much slower than the maximally localized version, since overlaps cannot be discarded, and should be used
only if wannier90 failed to converge for the particular case under study.
By default, three checkpoint files are generated during or at the end of a DI calculation run. These files have the same name as the pwc file but with -sijrestart, -sij, and -fa suffixes. The
-sijrestart file is written at certain points during the calculation of the atomic overlaps, and its purpose is to serve as a restart checkpoint in case the calculation is interrupted. If the
-sijrestart file is present and valid, the atomic overlap calculation can be continued from the point it was last written. Using the NORESTART keyword prevents critic2 from reading or writing the
The -sij and -fa checkpoint files are written at the end of a DI calculation. The former contains the atomic overlap matrices, and the latter, the exchange-correlation density ($F_{AB}$) integrals
required for the DI calculation. The presence of any of these two files makes critic2 read the information from the files and bypass the corresponding calculations altogether, which are quite time
consuming in general. The keywords NOSIJCHK and NOFACHK deactivate reading and writing these checkpoint files. You can calculate the DIs from these files directly without the pwc file using the
DELOC_SIJCHK and DELOC_FACHK options to INTEGRABLE.
By default, the overlap between two MLWFs whose centers are a certain distance away are discarded. The WANCUT keyword controls this distance: overlaps are discarded if the centers are wancut.r times
the sum of their spreads away. By default, wancut.r = 4.0. A very large wancut.r will prevent critic2 from discarding any overlaps. The appropriateness of the chosen WANCUT can be checked a
posteriori by comparing the integrated electron population obtained by sum of the localization and delocalization indices to the value obtained from a straight integration of the electron density.
See the example for some sample input files and more details.
Integrating Delocalization Indices in a Solid With Bloch States
The DIs can also be calculated without doing the transformation to Wannier functions, using the Bloch states provided by the calculation. This is typically far less efficient than using Wannier
functions but has two advantages:
• It is simpler because it does not require the additional wannier90 step.
• Using Bloch states, DIs can be calculated in metals (the Wannier transformation is ill-defined if some bands are partially occupied).
The calculation of DIs using Bloch states requires a .pwc file only, generated by the pw2critic.x utility in Quantum ESPRESSO. The pwc file contains electronic wavefunctions and the crystal
structure. For maximum consistency, the pwc file should be used to provide the structural information for the run via the CRYSTAL keyword.
In addition to these data, the calculation of DIs with Bloch states has a few requirements: the grid must be consistent with that of the reference field, and the DIs can be calculated using YT or
BADER only. A typical delocalization index with Bloch states calculation comprises the following steps:
• Run a PAW calculation, then obtain an all-electron density using pp.x with plot_num=21. This creates a cube file (rhoae.cube) that gives the Bader basins for the calculation (the pseudo-valence
density is not valid for this purpose).
• Run an SCF calculation with norm-conserving pseudopotentials and the same ecutrho as the calculation in the first step, so the two grids have the same size. This calculation must be run without
k-point symmetry, i.e., with nosym=.true, and noinv=.true..
• Use pw2critic.x to generate the pwc file.
• Load the all-electron density and the pwc files as two fields in critic2. Set the former as the reference density and the latter as INTEGRABLE DELOC.
• Run YT or BADER. YT is usually more accurate but takes longer than BADER.
By default, three checkpoint files are generated during or at the end of a DI calculation run. These files have the same name as the pwc file but with -sijrestart, -sij, and -fa suffixes. The
-sijrestart file is written at certain points during the calculation of the atomic overlaps, and its purpose is to serve as a restart checkpoint in case the calculation is interrupted. If the
-sijrestart file is present and valid, the atomic overlap calculation can be continued from the point it was last written. Using the NORESTART keyword prevents critic2 from reading or writing the
The -sij and -fa checkpoint files are written at the end of a DI calculation. The former contains the atomic overlap matrices, and the latter, the exchange-correlation density ($F_{AB}$) integrals
required for the DI calculation. The presence of any of these two files makes critic2 read the information from the files and bypass the corresponding calculations altogether, which are quite time
consuming in general. The keywords NOSIJCHK and NOFACHK deactivate reading and writing these checkpoint files. You can calculate the DIs from these files directly without the pwc file using the
DELOC_SIJCHK and DELOC_FACHK options to INTEGRABLE.
The keywords NOU and WANCUT have no effect on the calculation of DIs using Bloch states.
See the example for some sample input files and more details.
Bisection (INTEGRALS and SPHEREINTEGRALS)
INTEGRALS {GAULEG ntheta.i nphi.i|LEBEDEV nleb.i}
[CP ncp.i] [RWINT] [VERBOSE]
The BISECTION keyword integrates the attractor basins using bisection. If the Gauss-Legendre quadrature is used (GAULEG keyword), ntheta.i and nphi.i are the number of $\theta$ (polar angle) and $\
phi$ (azimuthal angle) points.
In the case of a Lebedev-Laikov quadrature, selected via the LEBEDEV keyword, only the total number of points in the spherical quadrature is needed. The actual value of nleb.i is the smallest number
larger than the one given by the user that is included in the list: 6, 14, 26, 38, 50, 74, 86, 110, 146, 170, 194, 230, 266, 302, 350, 434, 590, 770, 974, 1202, 1454, 1730, 2030, 2354, 2702, 3074,
3470, 3890, 4334, 4802, 5294, 5810.
By using the CP keyword, a single non-equivalent CP (ncp.i) is integrated. Otherwise, all the CPs of the correct type (found using AUTO) are integrated. If RWINT is present, read (if they exist) and
write the .int files containing the interatomic surface limit for the rays associated to the chosen quadrature method.
Defaults: ntheta.i = nphi.i = 50, nleb.i = 4802.
SPHEREINTEGRALS {GAULEG ntheta.i nphi.i|
LEBEDEV nleb.i} [CP ncp.i] [NR npts.i]
[R0 r0.r] [REND rend.r]
The SPHEREINTEGRALS keyword integrates the volume, field and Laplacian of the reference field in successive spheres centered around each of the attractor CPs. The meaning of the GAULEG and LEBEDEV
keywords is the same as in INTEGRALS.
A total number of npts.i spheres are integrated per nucleus. The grid is logarithmic, so that the region near the nucleus has a higher population of points. The grid starts at the radius r0.r and
ends at rend.r (bohr in crystals, angstrom in molecules). If rend.r < 0 then the final radius is taken as half the nearest neighbor distance for each atom times abs(rend.r).
Default: npts.i = 100. In GAULEG, ntheta.i = 20 and nphi.i = 20. In LEBEDEV, nleb.i = 770. r0.r = 1d-3 bohr. rend.r = rnn/2 for each CP. id.i = 0 (all attractors).
Qtree (QTREE)
General Syntax
The QTREE integration method integrates QTAIM atomic properties by discretization of the smallest part of the crystal that reproduces the whole system by symmetry. QTREE is specific for periodic
crystals, and its use is recommended with fields not given on a grid. For fields on a grid, either BADER or YT are better alternatives. QTREE is based on a hierarchical subdivision of the irreducible
part of the WS cell, employing a tetrahedral grid. The integration region is selected so as to maximize the use of symmetry, and partitioned into tetrahedra. The syntax of QTREE consists of the
maximum subdivision level (maxlevel.i), perhaps followed by the pre-splitting level (plevel.i), and then a series of optional keywords that control the behavior and the options for the integration.
QTREE [maxlevel.i [plevel.i]] [MINL minl.i] [GRADIENT_MODE gmode.i]
[QTREE_ODE_MODE omode.i] [STEPSIZE step.r] [ODE_ABSERR abserr.r]
[INTEG_MODE level.i imode.i] [INTEG_SCHEME ischeme.i] [KEASTNUM k.i]
[PLOT_MODE plmode.i] [PROP_MODE prmode.i] [MPSTEP inistep.i]
[QTREEFAC f.r] [CUB_ABS abs.r] [CUB_REL rel.r] [CUB_MPTS mpts.i]
[SPHFACTOR {ncp.i fac.r|at.s fac.r}] [SPHINTFACTOR atom.i fac.r]
[DOCONTACTS] [WS_ORIGIN x.r y.r z.r] [WS_SCALE scale.r]
[NOKILLEXT] [AUTOSPH {1|2}] [CHECKBETA] [NOPLOTSTICKS]
[COLOR_ALLOCATE {0|1}] [SETSPH_LVL lvl.i] [VCUTOFF vcutoff.r]
In QTREE, the tetrahedra that comprise the IWS enter a recursive subdivision process in which each of the tetrahedra is divided in 8 at each level, up to a level given by the user. This subdivision
level is controlled by the maxlevel.i argument given after the QTREE keyword (default, 6). Every tetrahedron vertex is assigned to a non-equivalent atom in the unit cell by tracing a gradient path
and identifying its endpoint. Once all the points in the tetrahedral mesh are assigned, the tetrahedra are integrated and the properties assigned to the corresponding atoms. The space near the atoms
is integrated using a beta-sphere, which improves the accuracy of the integration.
In the simplest (and most common) approach, qtree can be executed using:
QTREE [maxlevel.i [plevel.i]]
where maxlevel.i is the level of subdivision. The optional plevel.i value corresponds to the pre-splitting level of the tetrahedra. The initial tetrahedra list is split into smaller tetrahedra
plevel.i times. This can be useful in cases where a very high accuracy (and therefore a very high level) is required, but there is not enough memory available to advance to higher maxlevel.i.
However, using a relatively high plevel.i incurs an overhead, because the atom assigning procedure is not as efficient when smaller tetrahedra are used.
Steps of the QTREE Algorithm
1. The WS cell is constructed and split into tetrahedra, all of which have in common, at least, the origin of the WS cell. Then, the site symmetry of the origin is calculated and the tetrahedra that
are unique under the operations of this group are found. This is what we call the irreducible Wigner-Seitz cell (IWS). Note, however, that it is only “irreducible” in the local site symmetry of
the origin, not in the full set of space group operations. The IWS is the region that is integrated in later steps of QTREE. We will refer to a IWS tetrahedra as an IWST.
It is possible, through the WS_ORIGIN keyword, to shift the origin of the WS cell away from the (0 0 0) position. Because the symmetry of the WS cell is determined by the site symmetry of the
origin, the number and shape of IWST change depending on the origin chosen. A general position (with no symmetry) will make the IWS exactly equal to the WS.
Also, for large systems, the user can choose to shrink the size of the original WS cell in order to integrate a smaller region, using the WS_SCALE keyword (most likely in combination with
WS_ORIGIN, to move the integration region around). If a value is given to WS_SCALE (for instance, rws.r), then all the vectors connecting the origin of the WS cell with the vertex are shrunk by a
factor rws.r, and the volume of the integration region is decreased by a factor equal to the cube of rws.r. The IWS is calculated using the smaller WS cell and integrated in the same way. Note
that this integration region is non-periodic: it does not fill the volume of the solid and it does not integrate to the total number of electrons per cell.
2. Non-overlapping spheres are chosen centered on each of the atoms in the cell, the so-called beta-spheres. Atoms equivalent by symmetry share the same beta-sphere radius ($\beta_i$ for atom $i$).
The beta-sphere has two roles in QTREE:
□ The atomic properties are integrated inside the beta-spheres using a 2D cubature. The cubature can be a product of two 1D Gauss-Legendre quadratures or a Lebedev quadrature of the sphere.
Both methods, and the number of nodes can be selected using the INT_SPHEREQUAD_* keywords shown below. The radial quadrature can be any of the available in critic2, and is controlled by the
INT_RADQUAD_* options. The default values, however, are usually fine, integrating the beta-spheres in a matter of seconds with a precision that is orders of magnitude better than the overall
QTREE performance.
This beta-sphere integration removes the error of the finite-elements integration of a region where the integrated scalar fields present the steeper variations in value. By removing the
high-error regions from the grid integration, the accuracy of QTREE is enhanced. In particular, this increase in precision outweighs the loss of precision caused by creating an additional
interface between the grid and the sphere.
□ The space inside the beta-sphere of an atom is assumed to be inside the basin of that atom. The terminus of any gradient path that reaches the interior of the beta-sphere $i$ is assumed to be
the atom $i$. It is known that most of the steps in the integration of the gradient of the electron density are spent in the close vicinity of the terminus. Therefore, this modification saves
precious function evaluations.
The default beta-sphere radius is set to $0.80$ times half the nearest-neighbor distance. Both i) and ii) above assume that the beta-sphere is completely contained inside the basin of the atom.
This may turn out not to be true for the default beta-sphere radius (specially for cations in ionic systems). In these cases, the keyword SPHFACTOR is used. An example of the use of this keyword
SPHFACTOR 1 0.70 # Make b_1 = 0.70 * rNN2(1) (atom type 1)
If SPHFACTOR < 0, use the method by Rodriguez et al. to determine the beta-sphere radii: a collection of points around the atom are selected and the angle between the gradient and the radial
direction is determined. If all the angles are less than 45 degrees, the sphere is accepted. In solids, this strategy usually yields spheres that are too large.
In the case that any atomic SPHFACTOR is zero (the default value for all atoms), then a pre-computation at a lower level is done to ensure that all beta-spheres lie within the desired basins.
There are two methods for this, and the method can be chosen using the AUTOSPH keyword.
Method number one involves using a reduced version of QTREE. The pre-computation usually takes no longer than some minutes (and usually only few seconds) and the spheres are guaranteed to be
inside the basins. The keyword SETSPH_LVL controls the level of the pre-computation, that must not be higher than 7. The default value is 6 or maxlevel.i, whichever is smaller.
The second method (default) traces gradient path on a coarse sphere around each nucleus, and reduces the sphere until all of the points are inside the basin. NOCHECKBETA is used in this case.
An additional factor the user can define is the SPHINTFACTOR. It is possible to consider the sphere where GPs terminate different from the one that is integrated. If SPHINTFACTOR is defined, for
example as:
then the sphere associated to atom 1 where the integration is done has a radius which is 0.75 times that of the sphere where GP terminate.
The CHECKBETA and NOCHECKBETA keywords activate and deactivate the check that ensures that the beta-spheres is completely contained inside the basin.
If a beta-sphere is not strictly contained in the basin, QTREE detects it and stops immediately (specifically, QTREE checks that every tetrahedron that is partly contained in a beta-sphere has
vertex termini that are all assigned to the same atom as the atom that owns the beta-sphere).
For a new system, it is always a good idea to start with a low-level QTREE (say, level 4) to check if the default beta-spheres are adequate. If one of the beta-spheres is too large, the error
message looks like:
An undecided tetrahedron is overlapping with a beta-sphere.
Make beta-spheres smaller for this system.
terms: 1 -1 2 1
which indicates that there is a tetrahedron that is partly contained in the sphere of the first atom (terminus -1) and that has a vertex corresponding to atom 2. Modifying the sphfactor solves
this problem:
At lower levels, QTREE is reasonably fast, so a trial-and-error selection of beta-spheres is acceptable. Note that the beta-spheres used in QTREE have no relation to the ones reported after an
AUTO calculation.
3. If the cell is periodic (which means that WS_SCALE was not set), the contacts between the faces of the IWST are found. These contacts are used in a later step to copy the termini information
between tetrahedron faces.
The determination of the tetrahedra contacts in a periodic integration region is deactivated if the NOCONTACTS keyword is used. The DOCONTACTS keyword does the opposite thing: it activate the
calculation of contacts. By default, the contacts are not calculated.
4. A grid is built for each of the IWST. For each of the grid points, the termini of the gradient paths starting at them is calculated. The positions of the grid points are determined by subdividing
the IWST maxlevel.i times. In each subdivision step, a parent tetrahedron is divided in 8 smaller tetrahedron (all with the same volume, $V/8$) by splitting each edge of the parent tetrahedron in
two. There are two possible ways of doing this, the choice being irrelevant to the performance of QTREE.
For a given IWST, the size of the grid is given by $\begin{equation} S = n*(n+1)*(n+2)/6 \end{equation}$ where $n = 2^l + 1$ and $ll$ is the maxlevel.i, the approximate scaling being as $8^l$.
The termini information on the grid is saved to the array trm of type integer*1, with size $n_t * S$, where $n_t$ is the number of IWST (in fact, the integer type is that which is the result of
The subdivision level is the main input parameter for QTREE, controlling the accuracy (and cost!) of the integration. For small-medium sized systems $4-5$ are low cost integrations (seconds),
$6-7$ are medium cost (minutes) and $8-9$ are the slowest and most accurate (hours). The level is input in the call to the QTREE integration:
QTREE 6 # maxlevel.i is 6
By default, the integration level is 5.
In addition to trm, more work space can be allocated if the integration is restricted to the volume and charge or to the volume, charge and Laplacian. The number and type of properties to
integrate is controlled by the PROP_MODE keyword. The following values are allowed:
□ 0: Only volume is integrated. This amounts to canceling the finite elements integration of tetrahedra and is equivalent to INTEG_MODE 0 (see below).
□ 1: Only charge and volume. If the integration uses the value of the density at the grid points (INTEG_MODE 11, see below) In addition to trm, another real*8 array, fgr, is allocated (strictly
it is selected_real_kind(14), not real*8). In fgr, the value of the density at the grid points is stored.
□ 2: Charge, volume and Laplacian. In a similar way to 1, if the information on the grid points is used during the integration (INTEG_MODE 11), an additional real*8 array, lapgr, is allocated.
It contains the value of the Laplacian of the electron density at the grid points.
□ 3: All the integrable properties calculated by the module. The number of properties varies with the interface being used. No fgr or lapgr are allocated, as the grid points need to be
recomputed during integration.
The default value for PROP_MODE is 2.
The termini of the grid points contained in a beta-sphere is marked previous to the beginning of the subdivision.
5. Each tetrahedron is subdivided recursively up to a level maxlevel.i, and integrated at the same time. The IWST integration is relatively independent of one another, so for the moment we will
focus on just one IWST, which we will call the base tetrahedron.
The result of the integration of the different IWST can be used for other IWST:
□ When the integration ends, the termini of the four faces of a base tetrahedron are copied to its neighbors’ trm, according to the contacts determined previously.
□ Depending of the method chosen (see GRADIENT_MODE below), the gradient path integration may be aware of the neighboring grid points, that may belong to other IWST. In particular, the gradient
mode number 3 integrates a gradient path following grid points. When the endpoint is reached, all the grid points that have been traversed by the path are assigned the same common terminus.
Therefore, there is the possibility that gradient paths starting inside a given base tetrahedron write the trm of other IWST.
6. A tetrahedra stack is built and initialized with one element: the base tetrahedron. An iterator works on the stack, performing at each step the following tasks:
□ Pop a tetrahedron from the stack.
□ The termini of the vertex of the tetrahedron are calculated, if they are not already known.
□ If all the termini of the tetrahedron correspond to the same atom, the tetrahedron is “painted”. This means that all the grid points that are in the interior or border of the tetrahedron are
assigned the same color as its vertex, thereby saving the tracing of the gradient paths.
“Painting” can be dangerous whenever a (curved) IAS crosses the face of the tetrahedron being painted. To this end, a minimum level is defined, using the keyword QTREE_MINL. If the
subdivision level of the tetrahedron is lower or equal than minl.i, the tetrahedron is not painted. (Note: the base tetrahedron corresponds to level 0).
Furthermore, if all the termini correspond to the same atom and are located outside of the beta-sphere region, the tetrahedron is integrated and does not enter another subdivision process.
Once more, this only happens to tetrahedra with a level of subdivision strictly greater than minl.i. The “inner integration method” is the quadrature method used to integrate these
If all the termini are located inside the beta-sphere region, the tetrahedron does not subdivide, but the properties are not integrated, because this region corresponds to the sphere
integration addressed in point 2. If the tetrahedron is across the border of a beta-sphere, it is divided further.
□ If the tetrahedron is at subdivision level equal to maxlevel.i, then it does not subdivide, it is integrated and the properties are assigned to the atoms. There are several possibilities
depending of the nature of its vertex’ termini:
☆ If all the termini correspond to the same atom, and the tetrahedron is completely inside or outside of this atom’s sphere, it corresponds to the case in 6.3.
☆ If it is completely inside an atom basin, but on the border of a beta-sphere, the part of the tetrahedron that is outside of the sphere is integrated and assigned to the atom. Another
integration method is required for this, different from the “inner integration”. We will refer to this method as “border, same-color integration”.
☆ If the tetrahedron has termini corresponding to different atoms, its properties are integrated and split into contribution to atoms, according to the number of termini each atom has.
These tetrahedra are located on the IAS, and require a third class of integration, “border, diff-color integration”.
□ A tetrahedron that has not been integrated continues the subdivision. In this step, 8 new tetrahedra are pushed into the stack. The edges of the parent tetrahedron are split in two. By
construction, the newly generated points also correspond to grid points.
The subdivision scheme is:
☆ 1, 1-2, 1-3, 1-4
☆ 2, 1-2, 2-3, 2-4
☆ 3, 1-3, 2-3, 3-4
☆ 4, 1-4, 2-4, 3-4
☆ 2-3, 1-2, 1-3, 1-4
☆ 1-4, 1-2, 2-3, 2-4
☆ 1-4, 1-3, 2-3, 3-4
☆ 2-3, 1-4, 2-4, 3-4
where “a” represents a vertex of the parent tetrahedron and “a-b” the midpoint of both vertex. Each of the 8 child tetrahedra enclose the same volume, equal to $V/8^l$, where $l$ is the
subdivision level and $V$ is the volume of the base tetrahedron.
□ When the stack is empty, the work on the base tetrahedron is finished.
7. “Inner integration”. The inner integration is a quadrature that is applied to tetrahedra that are completely contained in the non-beta-sphere region of a basin. It can apply to a tetrahedron of
any level, as long as this level is greater than minl.i. The integrated properties are assigned to a single atom.
In the current implementation of QTREE, several integration methods are possible, and are controlled by the INTEG_MODE keyword. The possible values of INTEG_MODE are:
□ 11 : use the information of the density, Laplacian and properties at the vertex of the tetrahedron to integrate. The integral is approximated by a quadrature of four terms, each corresponding
to a volume that is 1/4 of the volume of the tetrahedron and multiplied by the value of the properties at the vertex. This integration method is useful if only charge or charge and Laplacian
are being integrated, because the information gained during the gradient path tracing, and saved in the fgr and lapgr arrays, is used. It is not very accurate for large tetrahedra.
□ 12 : use the CUBPACK routines. CUBPACK provides an adaptive tetrahedron integration method based on recursive subdivision and an integration rule with 43 nodes (degree 8), that is equivalent
to the DCUTET library by Bernsten et al. The integration rule is fully symmetric under the Th group operations. The error estimation is compared to the error requested by the user, that is
controlled using the CUB_ABS (absolute error), CUB_REL (relative error) and CUB_MPTS (maximum number of function evaluations) keywords. If CUB_MPTS is exceeded, an error message is output,
but the QTREE integration continues.
Note that, no matter how low the error requirements are, the CUBPACK integration spends, at least, 43 function evaluations per tetrahedron, so it is quite expensive if compared to other
integration modes. This should be used for large tetrahedra (see below) or for really accurate calculations.
□ 1…10 : use a non-adaptive rule from the KEAST library (Keast et al., 1986), the number corresponding to:
☆ 1 – order = 1, degree = 0
☆ 2 – order = 4, degree = 1
☆ 3 – order = 5, degree = 2
☆ 4 – order = 10, degree = 3
☆ 5 – order = 11, degree = 4
☆ 6 – order = 14, degree = 4
☆ 7 – order = 15, degree = 5
☆ 8 – order = 24, degree = 6
☆ 9 – order = 31, degree = 7
☆ 10 – order = 45, degree = 8
In particular, the first KEAST rule uses the barycenter of the tetrahedron.
The syntax of the INTEG_MODE keyword is:
where mode.i is one of the modes above and lvl.i is the level to which it applies. This means that, if a tetrahedron of a given level is integrated, the value of INTEG_MODE(level) is checked to
decide on the method.
Another INTEG_MODE value is possible:
□ -1 : do not integrate and force the tetrahedron into the subdivision process. This value of INTEG_MODE can be combined with a positive value at higher levels, amounting to a recursive
integration in the style of CUBPACK. Of course, -1 is not an acceptable value of INTEG_MODE for the last level, maxlevel.i.
As setting these INTEG_MODE by hand could be confusing, QTREE provides sets of INTEG_MODE values, which we will call “integration schemes”. An integration scheme is a full set of INTEG_MODEs for
all levels. Integration schemes are selected with the INTEG_SCHEME keyword, that can have the following values:
□ 0: do not integrate, only calculate volume and plot (see below). This is equivalent to setting PROP_MODE to 0.
□ 1: subdivide each tetrahedron up to the highest level and then integrate using the vertex information. This is most useful if PROP_MODE is 1 (only charge and volume) or 2 (charge, volume and
Laplacian) because the information of the gradient path tracing (fgr and lapgr arrays) are used:
INTEG_MODE = -1 -1 ... -1 11
! ^^ ^^
! QTREE_MINL `maxlevel.i`
□ 2: subdivide each tetrahedron up to the highest level and then integrate using the barycenter.
INTEG_MODE = -1 -1 ... -1 1.
□ 3: barycentric integration at all levels of subdivision. Less accurate but faster than 2.
INTEG_MODE = 1 1 ... 1 1.
□ 4: one of the Keast rules (given by the KEASTNUM keyword) is used at all levels. If KEASTNUM is n,
INTEG_MODE = n n ... n n.
□ 5: CUBPACK, at all levels. Reserve this one for special occasions.
INTEG_MODE = 12 12 ... 12 12.
□ 6: this scheme and the next are (poor) attempts at trying an adaptive integration scheme. They are not more reliable or efficient than, for instance, scheme 2. Integration scheme 6 calculates
levels 4, 5, and 6 using CUBPACK, and the rest with subdivision up to the highest level and vertex-based integration.
INTEG_MODE = 12 12 12 -1 ... -1 11.
□ 7: same as 6 but the final integration uses only the barycenter.
INTEG_MODE = 12 12 12 -1 ... -1 1.
□ -1: let the user enter the INTEG_MODEs by hand.
The default integration scheme is 2, suitable for low and medium-accuracy calculations.
8. “Border, same-color integration”. This integration method applies to tetrahedra that have reached the maximum subdivision level and sit on the interface between a beta-sphere and the atomic
basin. Some of the vertex are inside the sphere and some are outside. The out-of-sphere part is integrated and added to the atomic properties, while the in-the-sphere part is ignored because it
has already been integrated.
The integration works by assuming that the sphere radius is much larger than the tetrahedron characteristic lengths and, therefore, that the sphere surface can be considered a plane that
intersects the tetrahedron. The intersection points of the sphere with the tetrahedron edges are easily calculated and, for simplicity, we refer to them as the “middle” of the edges. There are
three cases:
□ One vertex is outside, three inside. The tetrahedron formed by the vertex outside and the three middle points of the edges that stem from it form a tetrahedron by itself, which is integrated
and added to the atom properties.
□ Three vertex are outside, one inside. The difference between the whole tetrahedron integration and the small tetrahedron inside the sphere is added to the atom properties. The small
tetrahedron is formed by the vertex that is inside the sphere and the three edges connected to it.
□ Two vertex are inside, two outside. The region outside of the sphere is a “triangular prism”, that is split in three tetrahedra and integrated.
Note that the INTEG_MODE of the maximum subdivision level (maxlevel.i) applies to all the sub-integrations of the border, same-color integration.
9. “Border, diff-color integration”. As in the case of “border, same-color integration”, this method only applies to tetrahedra which are at their maximum subdivision level. In this case, the
termini of the vertex corresponding to, at least, two different atoms.
In the current implementation of QTREE, the tetrahedron is integrated as a whole. Then, the properties are equally assigned to each of the termini atoms. For instance, if the termini are (1 1 1
3), the properties of the tetrahedron are integrated, then 3/4 of them assigned to atom 1 and 1/4 to atom 3.
10. Gradient path tracing. The gradient path start always at grid points, and are traced using one of three methods, controlled by the GRADIENT_MODE keyword, that can assume the following values:
□ 1: “full gradient”. This method is plain ODE integration. It is carried out ignoring the grid information. The gradient path is terminated whenever it enters a beta-sphere region.
□ 2: “color gradient”. At each point of the gradient path, the neighboring grid points are checked. If all of them correspond to the same atom, then the terminus of the gradient path is
assigned to that atom.
In a tetrahedral mesh, the meaning of “neighboring grid points” is not as clear as in a cubic mesh. For a given point x, the neighbors are calculated by first converting x to convex
coordinates, that range from 0 to $2^l$, restricted to $x_1 + x_2 + x_3 \leq 2^l$, where $l$ is the subdivision levels. The neighboring points are $(x_1\pm 1, x_2\pm 1, x_3\pm 1)$ If any of
these neighbors are not valid points in the tetrahedron, they are discarded and not checked. This is the default, except in the grid module.
□ 3: “qtree gradient”. This method behaves much like the “full gradient”, but whenever the gradient path steps near a grid point, it is projected to it. When a projection occurs, the grid point
is pushed onto a stack. At the end of the gradient path, when the terminus is known, all the grid points in the stack are popped and assigned the terminus.
The projection regions are spheres located around each grid point, whose radius is controlled by the QTREEFAC keyword. The radius of these spheres is $m_l/2^{l_{\rm max}}/{\rm qtreefac}$,
where $m_l$ is the smallest edge length of all IWST and $l_{\rm max}$ is the maximum subdivision level (maxlevel.i). Note that QTREEFAC equal to 1 is the maximum value allowed, and
corresponds to touching spheres along at least one tetrahedron edge. By default, QTREEFAC is 2. Lower levels of QTREEFAC tend to give errors when assigning the grid points that lie on the IAS
of two atoms (although only there). With higher levels, the time saving diminishes, and “qtree gradient” is equivalent to “full gradient”.
Additionally, the projection can be started only after a certain number of initial steps. The MPSTEP keyword controls this value. The default MPSTEP is 0.
□ -1, -2, -3: these correspond to the same as their positive values, but each gradient path terminus is compared to their “full gradient” version, using the best available ODE integration
method. Information about the results of the comparison are output to stdout, and a .tess file is generated (difftermxx.tess, where xx is the subdivision level) containing the position of the
points where both termini differ.
If the integration region is not periodic, then methods “color” and “qtree” are not defined. There are two possible options, controlled by the “KILLEXT” and “NOKILLEXT” keywords. If KILLEXT is
active (the default behavior), the gradient path tracing is killed whenever it leaves the integration region, independently of the GRADIENT_MODE being used. The terminus is then assigned to an
“unknown” state, and the tetrahedra it generates are not integrated. If NOKILLEXT is active, the gradient path is continued as a “full gradient”, until the terminus is found.
The default is KILLEXT because, if the integration region is not periodic, the integral over atoms that are partially contained in it is most likely not meaningful to the user.
The ODE integration method can be chosen using the QTREE_ODE_MODE keyword, that can assume the following values:
□ 1 : Euler method, fixed step, 1st order.
□ 2 : Heun method, fixed step, 2nd order.
□ 3 : Kutta method, fixed step, 3rd order.
□ 4 : Runge-Kutta method, fixed step, 4th order.
□ 5 : Euler-Heun embedded method, adaptive step. 1st order with 2nd order error estimation. 2 evaluations per step.
□ 6 : Bogacki-Shampine embedded method, adaptive step. 3rd order with 5th order error estimation. The FSAL (first step also last) allows only 4 evaluations per step. Local extrapolation.
□ 7 : Runge-Kutta Cash-Karp embedded method, adaptive step. 4th order with 5th order error estimation. 6 evaluations per step.
□ 8 : Dormand-Prince 4-5 embedded method, adaptive step. 4th order with 5th order error estimation. 6 evaluation per step, with FSAL. Local extrapolation.
For embedded methods (4-8), the absolute error requested from the method can be set using the ODE_ABSERR keyword. The default of this variable is chosen so that reasonable stepsizes are kept.
This default is 1d-3 for Euler-Heun and 1d-4 for the rest. Typically, methods with greater accuracy (7 and 8) save evaluations by increasing stepsize to values much larger than their lower
accuracy counterparts.
The step size of the fixed step methods (1-4) is controlled with the STEPSIZE keyword (bohr). In the variable step methods (5-8) the value of STEPSIZE is the length of the starting step.
The default QTREE_ODE_MODE is Dormand-Prince (8).
11. When the integration of the base tetrahedron is finished, the termini of the grid points located at each of its four faces are copied to the corresponding neighboring IWST, if DOCONTACTS is
Once the integration of the IWST is completed, the atomic properties are summed. The final result is output, together with an analysis of the contribution of each subdivision level to the total
integrated properties.
It is possible to plot the basins obtained by QTREE using the PLOT_MODE keyword. It can assume the values:
• 0: no plotting is done.
• 1: a single tess file is written containing a description of the unit cell CPs, the IWS, and balls corresponding to all the grid points that have been sampled.
• 2: same as 1 but only the balls that are either on the face of an IWST or close to a IAS are output.
• 3: the full WS cell
• 4: a file for the full WS cell and several files, containing a description of each of the integrated basins. Note that the basins need not be connected.
• 5: same as 4 but only balls belonging to faces of IWST and IAS are output.
The default value is 0. If PLOT_MODE is > 0, then the sticks that form the tetrahedra are written to .stick files. The PLOTSTICKS and NOPLOTSTICKS keywords control this behavior.
Additional Considerations
• The integration of the volume is not done using the beta-sphere / basin separation because the volume of each tetrahedron is exactly known. The integrated cell volume for a periodic integration
region will always be exact (if it is not, then it is an error). The integrated cell charge, on the contrary, is a measure of how well the tetrahedra are being integrated, but not of how well the
IAS is being determined.
• For very high levels of QTREE (say 10-11, depending on the amount of memory your computer has), memory usage may be a problem. The COLOR_ALLOCATE keyword controls the amount of memory allocated
for the color and property arrays. The syntax is:
Using a zero value, the color vector (and possibly the properties vectors, depending on PROP_MODE) is allocated only for the current IWST. This saves memory but makes the computation slower,
especially if the GRADIENT_MODE is 2 or 3. In addition, setting COLOR_ALLOCATE to 0 deactivates the passing of colors through the contacts (DOCONTACTS and NOCONTACTS keywords) and the plotting
(sets PLOT_MODE to 0). If COLOR_ALLOCATE is 1, the color (and optionally the properties) of all the IWST are saved. By default, COLOR_ALLOCATE is 1 if maxlevel.i is less than 9 and 0 if the
maximum level is higher.
Yu and Trinkle Grid Atomic Integration Method (YT)
YT [NNM] [NOATOMS] [WCUBE] [BASINS [OBJ|PLY|OFF] [ibasin.i]] [RATOM ratom.r]
[DISCARD expr.s] [JSON file.json] [ONLY iat1.i iat2.i ...]
The Yu and Trinkle (YT) method calculates the attraction basins of the reference fields and computes QTAIM integrals in them. The reference field must be defined on a grid. Hence YT will not work
directly with wien2k, elk, aiPI,… densities, although those can be transformed into a grid by appropriate use of the LOAD keyword. The algorithm proceeds by running over grid nodes in decreasing
order of density. If a point has no neighboring points with higher density, then it’s a local maximum. If it does, but all of them belong to the same basin then that point belongs to the interior of
that basin as well. Otherwise, it is sitting on top of the interatomic surface. The actual fraction of a grid point on an IAS belonging to a particular basin is calculated by evaluating the
trajectory flow to neighboring points.
The YT algorithm is described in J. Chem. Phys. 134 (2011) 064111 which should be consulted for further details. Please, cite this reference if you use this keyword in your work.
The located maxima in the field are identified by default with the closest nucleus. If non-nuclear maxima are expected, use the NNM keyword to assign only maxima that are only within 1 bohr of the
closest atom. This distance can be changed using the RATOM keyword (ratom.r in bohr (crystals) or angstrom (molecules)), which also controls the distance below which two maxima are considered the
same maximum. Changing the default ratom.r using the RATOM keyword automatically activates the detection of NNM. The NOATOMS option is appropriate for scalar fields where the maxima are not expected
to be at the atomic positions (or at least not all of them). If NOATOMS is used, all the maxima found are given as NNM. This is useful for fields such as the ELF, the Laplacian, etc.
The WCUBE option makes critic2 write cube files for the integration weights of each attractor. In YT, these weights are values zero (outside the basin), one (inside the basin), or some intermediate
value near the atomic basin boundary. The generated cube files have names <root>_wcube_xx.cube, where xx is the attractor number.
Use the BASINS option to write a graphical representation of the calculated basins. The format can be chosen using the OBJ, PLY, and OFF keywords (default: OBJ). If an integer is given after the
format selector (ibasin.i), then plot only the basin for that attractor. Otherwise, plot all of them. The basin surfaces are colored by the value of the reference field, in the default gnuplot scale.
Any maxima that is not assigned to an existing atom or non-nuclear critical point is automatically added to the critical point list. It is possible to get more information about these maxima by using
the CPREPORT keyword.
In some cases, particularly if there is a vacuum region in your system (for instance, if your system is a molecule or a surface), multiple spurious maxima will appear due to numerical noise in the
grid values. The number of spurious attractors will increase the computational cost and serve little purpose, as the vacuum region will integrate to zero anyway. In these cases, the DISCARD keyword
can be used to make critic2 ignore any attractor that matches the expression expr.s when it is evaluated at that point. For instance, if the electron density is given by field $rho, and we want to
discard low-density critical points, we could use:
The arithmetic expression may include any field, not just the reference field.
By using the JSON keyword a JavaScript Object Notation (JSON) file is created containing the molecular or crystal structure, information about the reference field and the results of the integration.
The ONLY keyword restricts the integration to only the atoms given by the user. The integers iat1.i,… are maxima identifiers from the complete critical point list, which contains all the critical
points in the unit cell. For atoms, the identifiers in the complete CP list coincide with those in the complete atom list, so iat.1 for an atom is also the integer ID from the list of atoms in the
unit cell (see the notation). The ONLY keyword is useful for delocalization index calculations in large systems, where restricting the integration to only a handful of atoms saves computing time.
Not all the properties defined by the INTEGRABLE keyword are integrated inside the basins. Only the subset of those properties that are grids, have F or FVAL as the integrand and are congruent with
the reference grid are considered. This limitation can be circumvented by using LOAD AS to define grid from fields or expressions that are not given on a grid. In addition, no core is used even if
the ZPSP keyword has been used to activate the core augmentation. The volume is always integrated. A xyz file (<root>_yt.xyz) is always written, containing the unit cell description (with border, see
WRITE) and the position of the maxima, labeled as XX.
Note that in the output (“List of basins and local properties”), “Charge” refers not to the integrated electron density (because critic2 does not know whether a given field is an electron density or
not) but to the value of the integral of the reference field in its own basins (which may not make much sense if you are integrating, for instance, the ELF or the Laplacian). Loading a second field
and using INTEGRABLE and the field number is the way to go in such cases.
Usage of the YT algorithm for grid fields is strongly recommended, as it is much more efficient, robust and accurate than the alternatives.
For examples see the calculation of Bader atomic properties.
Henkelman et al. Grid Atomic Integration Method (BADER)
The algorithm by Henkelman et al. is implemented in critic2, and can be used with the BADER keyword:
BADER [NNM] [NOATOMS] [WCUBE] [BASINS [OBJ|PLY|OFF] [ibasin.i]] [RATOM ratom.r]
[DISCARD expr.s] [JSON file.json] [ONLY iat1.i iat2.i ...]
The BADER algorithm uses the reference field to calculate the QTAIM basins. This field must be defined on a grid. BADER assigns grid nodes to basins using the near-grid method incrementally described
in Comput. Mater. Sci. 36, 354-360 (2006), J. Comput. Chem. 28, 899-908 (2007), and J. Phys.: Condens. Matter 21, 084204 (2009). Please, cite these references if you use this method. The same method
is implemented in the bader program available from the Henkelman research group’s website.
The output and the optional keywords have the same meaning as in YT, except in the following cases:
• The WCUBE option makes critic2 write cube files for the integration weights of each attractor. In BADER, these weights are either zero (outside the basin) or one (inside the basin). The generated
cube files have names <root>_wcube_xx.cube, where xx is the attractor number. In addition, a file <root>_wcube_all.cube is written, in which the value at every grid point corresponds to the
integer ID of the attractor for the corresponding basin.
Using BADER as an alternative to YT is recommended in very large grids because of its more efficient memory usage, but in general it gives less accurate integrations (at least in my experience).
For examples see the calculation of Bader atomic properties.
Isosurface Grid Integration (ISOSURFACE)
The ISOSURFACE keyword is used to integrate the volume and various scalar fields in regions delimited by isosurfaces of a scalar field. The isosurfaces are determined by the reference field and a
contour (isosurface) value set by the user. This keyword can be used only if the reference field is given on a grid. The syntax is:
ISOSURFACE {HIGHER|LOWER} isov.r [WCUBE] [BASINS [OBJ|PLY|OFF] [ibasin.i]]
[DISCARD expr.s]
The ISOSURFACE keyword must be followed by either HIGHER or LOWER and a real number (isov.r). The regions to be integrated are bound by isosurfaces with contour value equal to isov.r. If HIGHER,
integrate the regions with reference field value higher than isov.r. If LOWER, integrate the regions with value lower than isov.r.
The WCUBE option is similar to the same option in BADER. WCUBE writes cube files containing the isosurface regions. The points in cube file <root>_wcube_xx.cube have a value of 1, if the point is
inside a domain for isosurface xx, or 0 otherwise. Each point in the <root>_wcube_all.cube file has grid value equal to the integer ID of the isosurface domain in which it is contained, or 0 if it is
not inside any isosurface domain.
The BASINS keyword writes a graphical representation of the calculated isosurface domains. The format can be chosen using the OBJ, PLY, and OFF keywords (default: OBJ). If an integer is given after
the format selector (ibasin.i), then plot only the domain for that isosurface ID. Otherwise, plot all of them.
The DISCARD keyword accepts an arithmetic expression that is evaluated at every grid point. If the expression is true at a given grid point, then that point is not considered in the construction of
the isosurface domains. This is useful in combination with critic2’s structural variables. For instance, doing:
represents only the isosurface associated with atom number 3.
Hirshfeld Atomic Properties (HIRSHFELD)
The Hirshfeld property associated with atom A for scalar field $f({\bf r})$ is calculated as:
where $\rho_{\rm pro}$ is the promolecular density (the sum of atomic densities) and $\rho_{\rm A}$ is in-vacuo atomic density for atom A. For instance, the Hirshfeld charge of atom A would be $Z_{\
rm A} - N_{\rm A}$, where $N_{\rm A}$ is the Hirshfeld atomic integral for the all-electron density.
Hirshfeld atomic properties can be calculated using the HIRSHFELD keyword:
HIRSHFELD [WCUBE] [ONLY iat1.i iat2.i ...]
There are two ways in which this keyword operates. If the reference field is a grid, then a grid integration of all properties defined as INTEGRABLE is carried out. In the case of a grid integration,
the input and output of HIRSHFELD resembles that of YT and BADER. For this grid integration to be numerically sensible, the integrable scalar field must be reasonably smooth (for instance, the
pseudo-density coming from a plane-wave calculation).
The WCUBE option writes cube files for the Hirshfeld weights of each nucleus, with file names <root>_wcube_xx.cube, where xx is the atom identifier from the complete atom list. The ONLY keyword
restricts the integration to only certain atoms, given by their identifiers from the complete atom list.
If the reference field is not a grid, HIRSHFELD assumes the field contains the electron density and carries out the integration using the selected molecular mesh. In this case, WCUBE and ONLY cannot
be used, and only the volume and the Hirshfeld atomic electron populations are calculated.
For an example see the calculation of Hirshfeld charges.
Voronoi Atomic Properties (VORONOI)
The Voronoi property associated with atom A for scalar field $f({\bf r})$ is calculated as the integral of $f$ over all points in space that are closer to A than to any other atom (the Voronoi
region). For instance, the Voronoi deformation density (VDD) of an atom is minus the integral of the electron density minus the promolecular density over its Voronoi region.
Voronoi atomic properties can be calculated using the VORONOI keyword:
VORONOI [BASINS [OBJ|PLY|OFF] [ibasin.i]] [ONLY iat1.i iat2.i ...]
Only the grid integration case has been implemented, so the reference field must be a grid. All properties defined as INTEGRABLE are integrated in the Voronoi regions, and the input and output of
VORONOI resembles that of YT and BADER. For this grid integration to be numerically sensible, the integrable scalar field must be reasonably smooth (for instance, the pseudo-density coming from a
plane-wave calculation).
The BASINS option writes a graphical representation of the Voronoi regions. The format can be chosen using the OBJ, PLY, and OFF keywords (default: OBJ). If an integer is given after the format
selector (ibasin.i), then plot only the basin for that atom (identifier from the complete atom list). Otherwise, plot all of them. The basin surfaces are colored by the value of the reference field,
in the default gnuplot scale. The ONLY keyword restricts the integration to only certain atoms, given by their identifiers from the complete atom list.
For an example see the calculation of Voronoi deformation density (VDD) charges.
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Calculating the minimal distance between points
Suppose i have a polygon with 5 points. They are constant.
I use "gotoxy()" to define the point positions.
Something like this:
x1=30; y1=10;
x2=50; y2=10;
//.....and so one up five
Now i should to give a another point (X,Y), and calculate with wich point defined early is too near.
For the first i thinkd that i cand substract X from earch point x1, x2....x5 and Y from y1, y2... y5 and to see the almost points, or who have less difference between coordinate. But i wrong.
How cand i do this? Some ideas please!
Suppose i have a polygon with 5 points. They are constant.
I use "gotoxy()" to define the point positions.
Something like this:
x1=30; y1=10;
x2=50; y2=10;
//.....and so one up five
Now i should to give a another point (X,Y), and calculate with wich point defined early is too near.
For the first i thinkd that i cand substract X from earch point x1, x2....x5 and Y from y1, y2... y5 and to see the almost points, or who have less difference between coordinate. But i wrong.
How cand i do this? Some ideas please!
Not sure I follow what the problem is or what this code/algorithm is and what gotoxy (int, int) does. Is this a function you have written? Is the question a mathematical question of how to calculate
the points in a polygon, how to code it, what the formula for distance between two points is, what? Please elaborate on what the problem is and you may want to post more code.
gotoxy(), is a predifined function, is position cursor in text window.
Analytical geomentry, gives variousways to calculate distance. In this case, we can use
sqrt( (x1- x2)*(x1-x2) + (y1- y2)*(y1-y2) )
Analytical geomentry, gives variousways to calculate distance. In this case, we can use
sqrt( (x1- x2)*(x1-x2) + (y1- y2)*(y1-y2) )
hey ,
Prabhakar's qoute is right. Let me elaborate this solution for you. I think to calculate nearest point, u should take following approach.
Lets assume that you have 3 points (x1,y1)(x2,y2)(x3,y3). Now you have a point (X,Y) from which you have to search nearest point from mentioned 3 points.
Simply take diff between X-x1, X-x2, X-x3, (If ans is negative , multiply by -1)
Same with Y co-ordinates. Y-y1, Y-y2, Y-y3.
Then Do addition of all respective diff. Like (X-x1)+(Y-y1), (X-x2)+(Y-y2) & so on. The min addition that u got is the nearest point from (X,Y). e.g. if (X-x1)+(Y-y1) is min. then (x1,y1) is the
nearest point.
hey ,
Prabhakar's qoute is right. Let me elaborate this solution for you. I think to calculate nearest point, u should take following approach.
Lets assume that you have 3 points (x1,y1)(x2,y2)(x3,y3). Now you have a point (X,Y) from which you have to search nearest point from mentioned 3 points.
Simply take diff between X-x1, X-x2, X-x3, (If ans is negative , multiply by -1)
Same with Y co-ordinates. Y-y1, Y-y2, Y-y3.
Then Do addition of all respective diff. Like (X-x1)+(Y-y1), (X-x2)+(Y-y2) & so on. The min addition that u got is the nearest point from (X,Y). e.g. if (X-x1)+(Y-y1) is min. then (x1,y1) is the
nearest point.
Well Vector Algebra and translation of axis has their advantages.
Nice Explaination sacdpathade.
Place your points in an array and use a loop based on the number of points to calculate the distances, checking each result with the last to see if it is less. If so, it should become the new
"shortest distance"
Reply to this Topic
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Talk:Determinant and permanent - Rosetta CodeTalk:Determinant and permanent
Method used
Should the algorithm be as stated in the equations? i.e. using the sign function? --Paddy3118 07:17, 24 July 2012 (UTC)
Any method is acceptable, use the one that best fits the language and your preferences. (If you like you can write multiple interpretations and compare them: one by the definition, one terse, one
fast, one space-efficient, etc.)
In general I don't think that tasks should over-constrain authors. The purpose of the site is to compare different languages, and we can't effectively do that if we force everyone to write in the
same style. "Real Programmers can write FORTRAN in any language", they say...
CRGreathouse 17:30, 27 July 2012 (UTC)
Hi CRGreathouse, it wasn't so much the style that I wanted clarification on, but the algorithm to be implemented. On RC there are a whole suite of sort tasks - they all do the same thing,
just with different algorithms. Similarly, when made aware of the signed permutation method of calculating a determinant via this task and its links, (my thanks), I thought there maybe room
for something similar? --Paddy3118 06:14, 28 July 2012 (UTC)
I think some of these "differences of algorithm" would go away if we thought of multiplication as being a short circuit operator (that could prune entire recursive call trees when one of
the values being multiplied was zero). Another thing that might help would be re-expressing the algorithm using something other than a "sequence over time loop" to organize the recursive
expansion of minors (or at least: allowing rephrasings which ignore the limitations of ieee floating point. That said, a little focus on re-expressing using equivalencies (and searching
for likely zeros) does seem like a good subject when dealing with alternative algorithms.. --Rdm 13:29, 30 July 2012 (UTC)
Why "arithmetic"?
I don't get what this task has to do with arithmetic. Isn't it about invariants of a matrix, or something like that?--Grondilu 16:38, 5 January 2013 (UTC)
FORTRAN parallel effort
Please credit additional contributors from [1] --LambertDW 20:45, 21 May 2013 (UTC)
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Constraint boundary - (Optimization of Systems) - Vocab, Definition, Explanations | Fiveable
Constraint boundary
from class:
Optimization of Systems
A constraint boundary is a limit defined by constraints in optimization problems that delineates the feasible region where potential solutions reside. These boundaries are created from inequalities
or equalities that restrict the values of decision variables, helping to identify the area where optimal solutions can be found within the constraints set by the problem.
congrats on reading the definition of constraint boundary. now let's actually learn it.
5 Must Know Facts For Your Next Test
1. Constraint boundaries are typically represented graphically as lines or curves in two-dimensional space, showing where the constraints intersect and form the limits of the feasible region.
2. The feasible region is bounded by these constraint boundaries, which means that any point outside of this region does not satisfy all the constraints of the optimization problem.
3. In multi-variable optimization, constraint boundaries become surfaces in higher-dimensional spaces, still defining limits on potential solutions.
4. When solving an optimization problem, it is essential to identify constraint boundaries to determine where to search for optimal solutions effectively.
5. Understanding how to manipulate and interpret constraint boundaries is crucial for effectively applying methods like the Simplex algorithm in linear programming.
Review Questions
• How do constraint boundaries influence the identification of feasible regions in optimization problems?
□ Constraint boundaries play a critical role in shaping the feasible region because they define the limits within which potential solutions must lie. Each boundary corresponds to a specific
constraint, and where these boundaries intersect determines the area where all conditions are met. Understanding how these boundaries interact allows for better visualization and
identification of valid solution sets.
• Discuss how constraint boundaries can change when modifying an optimization problem and what implications this has for finding optimal solutions.
□ When modifications are made to an optimization problem, such as changing constraints or adding new ones, constraint boundaries can shift or change shape. This alteration can significantly
impact the feasible region, potentially eliminating previously valid solutions or creating new areas where optimal solutions may exist. Consequently, itโ s essential to reassess both the
feasible region and optimal solution after any adjustments to ensure effective decision-making.
• Evaluate the significance of constraint boundaries in developing real-world applications of optimization techniques.
□ In real-world applications, constraint boundaries are fundamental because they encapsulate practical limitations such as budget, resources, or time. By accurately defining these boundaries
within models, organizations can make informed decisions that respect operational constraints while still pursuing optimization goals. Evaluating how well an application adheres to its
defined constraint boundaries can lead to insights on feasibility and practicality, ensuring that optimized solutions are not only theoretical but also actionable.
"Constraint boundary" also found in:
Subjects (1)
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The n-Category Café
May 28, 2018
Applied Category Theory: Ordered Sets
Posted by John Baez
My applied category theory course based on Fong and Spivak’s book Seven Sketches is going well. Over 300 people have registered for the course, which allows them to ask question and discuss things.
But even if you don’t register you can read my “lectures”.
Posted at 7:39 PM UTC |
Followups (30)
May 25, 2018
Posted by John Baez
Talking to my student Joe Moeller today, I bumped into a little question that seems fun. If I’ve got a monoidal category $A$, is there some bigger monoidal category $\hat{A}$ such that lax monoidal
functors out of $A$ are the same as strict monoidal functors out of $\hat{A}$?
Someone should know the answer already, but I’ll expound on it a little…
Posted at 9:40 PM UTC |
Followups (13)
May 24, 2018
Tropical Algebra and Railway Optimization
Posted by John Baez
Simon Willerton pointed out a wonderful workshop, which unfortunately neither he nor I can attend… nor Jamie Vicary, who is usually at Birmingham these days:
If you can go, please do — and report back!
Let me explain why it’s so cool…
Posted at 10:39 PM UTC |
Followups (5)
May 20, 2018
Postdoc at the Centre of Australian Category Theory
Posted by Emily Riehl
The Centre of Australian Category Theory is advertising for a postdoc. The position is for 2 years and the ad is here.
Applications close on 15 June. Most questions about the position would be best directed to Richard Garner or Steve Lack. You can also find out more about CoACT here.
This is a great opportunity to join a fantastic research group. Please help spread the word to those who might be interested!
Posted at 3:29 PM UTC |
Post a Comment
May 19, 2018
Circuits, Bond Graphs, and Signal-Flow Diagrams
Posted by John Baez
My student Brandon Coya finished his thesis, and successfully defended it last Tuesday!
• Brandon Coya, Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective, Ph.D. thesis, U. C. Riverside, 2018.
It’s about networks in engineering. He uses category theory to study the diagrams engineers like to draw, and functors to understand how these diagrams are interpreted.
His thesis raises some really interesting pure mathematical questions about the category of corelations and a ‘weak bimonoid’ that can be found in this category. Weak bimonoids were invented by
Pastro and Street in their study of ‘quantum categories’, a generalization of quantum groups. So, it’s fascinating to see a weak bimonoid that plays an important role in electrical engineering!
However, in what follows I’ll stick to less fancy stuff: I’ll just explain the basic idea of Brandon’s thesis, say a bit about circuits and ‘bond graphs’, and outline his main results. What follows
is heavily based on the introduction of his thesis, but I’ve baezified it a little.
Posted at 11:07 PM UTC |
Followups (4)
Linear Logic for Constructive Mathematics
Posted by Mike Shulman
Intuitionistic logic, i.e. logic without the principle of excluded middle ($P \vee eg P$), is important for many reasons. One is that it arises naturally as the internal logic of toposes and more
general categories. Another is that it is the logic traditionally used by constructive mathematicians — mathematicians following Brouwer, Heyting, and Bishop, who want all proofs to have
“computational” or “algorithmic” content. Brouwer observed that excluded middle is the primary origin of nonconstructive proofs; thus using intuitionistic logic yields a mathematics in which all
proofs are constructive.
However, there are other logics that constructivists might have chosen for this purpose instead of intuitionistic logic. In particular, Girard’s (classical) linear logic was explicitly introduced as
a “constructive” logic that nevertheless retains a form of the law of excluded middle. But so far, essentially no constructive mathematicians have seriously considered replacing intuitionistic logic
with any such alternative. I will refrain from speculating on why not. Instead, in a paper appearing on the arXiv today:
I argue that in fact, constructive mathematicians (going all the way back to Brouwer) have already been using linear logic without realizing it!
Let me explain what I mean by this and how it comes about — starting with an explanation, for a category theorist, of what linear logic is in the first place.
Posted at 4:53 AM UTC |
Followups (66)
May 14, 2018
Research Fellowship at the University of Leeds
Posted by Simon Willerton
João Faria Martins and Paul Martin at the University of Leeds are advertising a 2-year research fellowship in geometric topology, topological quantum field theory and applications to quantum
computing. This is part of a Leverhulme funded project.
The deadline is Tuesday 29th May. Contact João or Paul with any informal inquiries.
Posted at 10:01 AM UTC |
Post a Comment
May 6, 2018
Posted by John Baez
A new journal! We’ve been working on it for a long time, but we finished sorting out some details at Applied Category Theory 2018, and now we’re ready to tell the world!
Posted at 5:11 PM UTC |
Followups (20)
May 5, 2018
The Fisher Metric Will Not Be Deformed
Posted by Tom Leinster
The pillars of society are those who cannot be bribed or bought, the upright citizens of integrity, the incorruptibles. Throw at them what you will, they never bend.
In the mathematical world, the Fisher metric is one such upstanding figure.
What I mean is this. The Fisher metric can be derived from the concept of relative entropy. But relative entropy can be deformed in various ways, and you might imagine that when you deform it, the
Fisher metric gets deformed too. Nope. Bastion of integrity that it is, it remains unmoved.
You don’t need to know what the Fisher metric is in order to get the point: the Fisher metric is a highly canonical concept.
Posted at 12:00 AM UTC |
Followups (25)
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VTU Design of Pre-stressed Concrete Structures - May 2016 Exam Question Paper | Stupidsid
VTU Civil Engineering (Semester 7)
Design of Pre-stressed Concrete Structures
May 2016
Total marks: --
Total time: --
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary
1(a) Define the following :
i) Tendon ii) Pretensioning iii) Post tensioning iv) Load balancing
8 M
1(b) Explain how PSC is more advantageous as compared to RCC.
6 M
1(c) Explain with neat sketch, Freyssinet system of post tensioning.
6 M
2(a) A concrete beam with a double overhang has the middle equal to 10 m and the equal overhang on either side is 2.5 m. Determine the profile of the prestressing cable with an effective force of 250
kN which can balance a uniformly distributed load of kN/m on the beam, which included the self weight of the beam. Sketch the cable profile marking the eccentricity of cable at the support and
10 M
2(b) A rectangular concrete beam of cross section 12o mm wide and 300 mm deep is prestressed by a straight an imposed load of 3.14 kN/m over a span of 6 m. If the modulus of rupture of concrete is 5
N/mm^2, evaluate the load factor against cracking assuming the self weight of concrete as 24 kN/m^3.
10 M
3(a) A pretensioned beam of rectangular cross-section 150 mm wide and 300 deep is prestressed by 8, 7 mm ϕ wires located 100 mm from the soffit of the beam. If the wires are initially
tensioned to a stress of 1100 N/mm^2, calculate the effective stress after all losses, given the following:
Relaxation of steel= 70 N/mm^2; Shrinkage of concrete = 300 × 10^-6
Creep of concrete = 1.6 ; E[s] = 210kN/mm^2 and E[c] = 31.5 kN/mm^2
12 M
3(b) A simply supported post tensioned concrete beam of span 15 m has a rectangular cross section 300 × 800 mm. The prestress at ends is 1300 kN withe zero eccentricity and 250 mm at the centre the
cable profile being parabolic. Assuming k = 0.0015 per m and μ = 0.35. Determine the loss of stress due to friction at the centre of the beam.
8 M
4(a) Write short note on prediction of long term deflections
6 M
4(b) A PSC beam of rectangular section 120 mm wide and 300 mm deep spans over 6 m. The beam is prestressed by straight cable carrying an effective force of 180 kN at an eccentricity of 50 mm. If it
support an imposed load of 4 kN/m and modulus of concrete is 38 kN/mm. Compute the deflection at the following stages and check whether they comply with the IS code specifications. Take density of
concrete as 24 kN/m^3.
i) Upward deflections under (Prestress + Selfweight)
ii) Final downward deflections under (Prestress + Selfweight + imposed load ) including the effects of creep an shrinkage assuming the creep coefficient as 1.8 and loss of prestress = 20%
14 M
5(a) Discuss the different types of failure of PSP beam.
6 M
5(b) A pretensioned PSC beam Tee section having a flange 1200 mm wide and 150 mm thick is prestressed by 4700 mm^2 of high tensile steel located at an effective depth of 1600 mm. The ribs have a
thickness of 150 mm each. If the cube strength of concrete is 40 n/mm^2 and tensile strength of steel is 1600 N/mm^2, determine the flexural strength of the double T girder using IS : 1343
14 M
6(a) A concrete beam of rectangular section, 200 mm wide and 400 mm deep, is prestressed by a a parabolic cable located at an eccentricity of 100 mm at midspan and zero at the supports. If the beam
has a span of 10 m and carries a uniformly distributed live load of 4 kN/m, find the effective force necessary in the cable for zero shear stress at the support section. For this condition calculate
the principal stresses. The density of concrete is 24 kN/m^3.
10 M
6(b) If the support section of a PSC beam 100 mm wide and 250 mm deep is required to support an ultimate shear force of 80 kN. The compressive prestress at the centroidal axis is 5 N/mm^3. The
characteristic cube strength of concrete is 40 N/mm^2. The cover to the tension reinforcement is 50 mm. If the characteristic tensile strength of stirrups is 415 N/mm^2 , design suitable shear
reinforcements in the section using IS code recommendations.
10 M
7(a) Explain the analysis of anchorage zone stresses in post tensioned members. How is the bursting tensile force calculated?
8 M
7(b) A high tensile cable comprising 12 stands of 15 mm dia with an effective force of 2500 kN is anchored concentrically in an end block of a post-tensioned beam. The end block is 400 mm wide and
800 mm deep and the anchor plate is 200 mm wide by 260 mm deep. Design suitable anchorage zone reinforcements using Fe415 grade HYSD bars using IS:1343 code provisions.
12 M
8 A post tensioned prestressed concrete beam of rectangular section 300 mm wide is to be designed to resist a live load moment of 360 kNm on a span of 12 m. Assuming 10% loss and limiting tensile and
compressive stress to 1.5 MPa and 18 Mpa respectively.Calculate the minimum possible depth and the prestressing force and corresponding eccentricity. Take D[c] = 24 kN/m^3.
20 M
More question papers from Design of Pre-stressed Concrete Structures
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Hardness of Identity Testing for Restricted Boltzmann Machines and Potts models
We study identity testing for restricted Boltzmann machines (RBMs), and more generally for undirected graphical models. Given sample access to the Gibbs distribution corresponding to an unknown or
hidden model M^∗ and given an explicit model M, can we distinguish if either M = M^∗ or if they are (statistically) far apart? Daskalakis et al. (2018) presented a polynomial-time algorithm for
identity testing for the ferromagnetic (attractive) Ising model. In contrast, for the antiferromagnetic (repulsive) Ising model, Bezáková et al. (2019) proved that unless RP = NP there is no identity
testing algorithm when βd = ω(log n), where d is the maximum degree of the visible graph and β is the largest edge weight (in absolute value). We prove analogous hardness results for RBMs (i.e.,
mixed Ising models on bipartite graphs), even when there are no latent variables or an external field. Specifically, we show that if RP 6= NP, then when βd = ω(log n) there is no polynomial-time
algorithm for identity testing for RBMs; when βd = O(log n) there is an efficient identity testing algorithm that utilizes the structure learning algorithm of Klivans and Meka (2017). In addition, we
prove similar lower bounds for purely ferromagnetic RBMs with inconsistent external fields, and for the ferromagnetic Potts model. Previous hardness results for identity testing of Bezáková et al.
(2019) utilized the hardness of finding the maximum cuts, which corresponds to the ground states of the antiferromagnetic Ising model. Since RBMs are on bipartite graphs such an approach is not
feasible. We instead introduce a novel methodology to reduce from the corresponding approximate counting problem and utilize the phase transition that is exhibited by RBMs and the mean-field Potts
model. We believe that our method is general, and that it can be used to establish the hardness of identity testing for other spin systems.
All Science Journal Classification (ASJC) codes
• Artificial Intelligence
• Software
• Control and Systems Engineering
• Statistics and Probability
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Noncommutative Geometry, Trace Formulas and the Zeros of the Riemann Zeta Function
In this course we first give a general introduction to noncommutative geometry. We then discuss a fundamental example of noncommutative space related to the Riemann zeta function. This gives a
spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while the noncritical zeros appear as resonances, and a geometric interpretation of the explicit
formulas of number theory as a trace formula on a noncommutative space. This reduces the Riemann hypothesis to the validity of the trace formula, which remains unproved, and eliminates the parameter
1. Introduction to noncommutative geometry
2. Quantum chaos and the hypothetical Riemann flow.
3. Algebraic geometry and global fields of nonzero characteristic.
4. Spectral interpretation of critical zeros.
5. The distribution trace formula for flows on manifolds.
6. The action K^* on K for a local field.
7. The global case, and the formal trace computation.
8. The trace formula and S-units.
9. The trace formula in the global case, and elimination of
10. Explicit formulas.
Alain Connes' expository paper on the subject matter of the course, describing his motivations and some background. It is available for downloading in any of the following formats:
LaTeX, DVI, PostScript, PDF (Adobe Acrobat v. 3.0 or later)
Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function
This is Alain Connes' latest paper, on the subject matter of this course. It is available for downloading in any of the following formats:
Plain TeX, DVI, PostScript, PDF (Adobe Acrobat v. 3.0 or later)
Registration Information
The current listing for Alain Connes' course is
MATH 975A TOPICS IN GEOMETRY 3 CR Section: 18473-7 MWF 2:30-4:20PM CL 0220
The course will run for 5 weeks and the tentative dates are
Oct. 12 - Nov. 13
Update^4: Lectures on M (Nov. 9) and F (Nov. 13) during the fifth week
The first lecture of the course will be Wed. Oct. 14. The second lecture will be Fri. Oct. 16. The lectures will be held MW the second week, WF the third week, and MW the fourth week. All lectures
will be at 2:30-4:20 pm, in room CL 0220. (CL is Caldwell Laboratory, 2024 Neil Ave. Click here for a map.)
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APR Calculator for Accurate Loan Assessments - Powerful and Useful Online Calculators
APR Calculator for Accurate Loan Assessments
Have you ever wondered how to accurately assess the cost of a loan you’re considering? Understanding how to calculate the Annual Percentage Rate (APR) can significantly help you make an informed
decision. Let’s dive into the topic and explore how an APR calculator can simplify this complex process for you.
The True Cost of Bank Loans
People borrow money for various reasons—be it starting a business, repairing a home, going on a dream vacation, or paying off other debts. No matter your reason, it’s essential to evaluate the cost
of taking on additional debt carefully. While most banks are transparent about the interest they charge, there are often additional fees that can increase the overall cost of your loan.
Let’s consider Mark, who wants to borrow $50,000 from his bank to restore his classic car. The bank charges 8% interest on the loan. Because the bank charges a $1,000 fee to take the loan, the real
APR on a 10-year loan is actually 8.457%. This fee is added to the loan amount, so Mark is effectively borrowing and paying interest on $51,000. This small difference results in nearly $1,500 more
paid by the end of the term. Therefore, APR (also known as real APR) is a better indicator of the total cost of the loan compared to just the stated interest rate.
Understanding the General APR Formula
To calculate the APR for a loan, you need to determine the total cost of all fees and interest compared to the borrowed amount. This provides the total percentage cost of the loan. Given that most
loans vary in duration, you’ll need to divide the total percentage by the number of years to find the real APR. While a calculator simplifies this, it’s useful to understand how the calculation
The Formula
The formula to calculate the general APR of a loan is:
[ \text = \frac{(I + F) / P} ]
• APR = Real APR of the loan
• I = Total interest
• F = Additional fees
• P = Initial loan amount or principal
• N = Number of years
Note: This calculation returns the APR in decimal format. Multiply by 100 to convert it to a percentage.
Example: Applying the Formula
Suppose you want to take a $50,000 loan with a 10-year term at an 8% interest rate, and the fees are $1,000. Here’s how you’d apply the formula:
1. Interest: ( I = 0.08 \times 50,000 \times 10 = 40,000 )
2. Fees: ( F = 1,000 )
3. Principal: ( P = 50,000 )
4. Number of years: ( N = 10 )
[ APR = \frac{(40,000 + 1,000) / 50,000} = \frac = 0.082 ]
Convert to percentage: [ APR = 0.082 \times 100 = 8.2% ]
Mortgage APR
The formula above applies to most types of loans. However, when it comes to calculating a mortgage’s real APR, the terminology and some factors may vary.
Special Points in Mortgages
The primary difference lies in the terminology and additional factors like special points you may purchase when securing your loan. These points are a way to prepay some interest in exchange for a
lower interest rate over the life of the loan, which alters your overall real APR.
Adjustments for Mortgage APR
For mortgages, the total loan amount often includes the home’s value (plus any closing costs, commissions, etc.) minus any down payments. These factors must be considered when calculating the real
How to Use the APR Calculator
Using an online APR calculator requires a few data points, often provided by your bank or financial institution. Understanding the components can help you better utilize the tool.
Step-by-Step Guide
1. Select the Type of APR: Choose whether you’re calculating for a general loan or a mortgage.
2. Complete the Information: Include the amount being borrowed, interest rate, and loan term. Ensure accuracy for reliable results.
3. Review Results: After clicking “Calculate,” examine the breakdown of total loan cost, monthly payment, total interest paid, and real APR percentage.
Real Example with the Calculator
Let’s say you want to take out a loan to remodel your kitchen. The general contractor has quoted $125,000 for the project. Your bank has offered you a 5-year loan at a 5% interest rate. They charge a
$250 administration fee added to the loan and an upfront fee of $500. Here’s what you need to input into the general APR loan calculator:
• Loan Amount: $125,000
• Loan Term: 5 years and 0 months
• Interest Rate: 5%
• Compound: Monthly (APR)
• Pay Back: Every Month
• Loan Fees: $250
• Upfront Fees: $500
When you hit the Calculate button, you’ll see that despite being quoted a 5% interest rate by your bank, your real APR is 5.248%.
Key Benefits and Helpful Tips
Taking on debt can be risky, so your goal should be to minimize the amount of interest you need to pay. Here are some key benefits and helpful tips for using an APR calculator to save money.
Key Benefits
1. No Memorizing Formulas: Calculating APR on loans can be complex. The calculator facilitates an accurate calculation without the need to remember formulas.
2. Compare Multiple Loans: Interest rates and fees vary, making it difficult to compare loans from different lenders. The real APR calculation provides a standardized comparison.
3. Transparency and Accuracy: Understanding your loan in-depth ensures you’re getting a fair deal and helps avoid unexpected financial burdens.
Helpful Tips
1. Run Scenarios: Adjusting the loan term and other variables in the APR calculator helps visualize how these changes impact the APR.
2. Shop Around: A loan with a higher interest rate but lower fees could have a more favorable APR. Always compare multiple loans to find the best deal.
Related Calculators for Further Assistance
• Auto Loan Calculator: For vehicle financing.
• Finance Loan Calculator: For a variety of personal and business loans.
• Amortization Calculator: To see the breakdown of payments.
• Interest Rate Calculator: For more detailed interest calculations.
Understanding the true cost of a loan through APR calculation can significantly impact your financial well-being. Whether it’s for personal or business purposes, an accurate APR assessment helps you
make informed decisions, saving you money in the long run. Utilize an APR calculator to compare loans effectively, run different scenarios, and ensure you’re getting the most favorable terms. Being
informed and proactive about your financial choices can lead to a healthier financial future.
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Number Talks: Increasing Math Talk in Your Elementary Classroom - Teaching with Jodi Durgin and Company
Do your students struggle with explaining how they solved a problem? Do they give up on a problem soon after starting or before they even begin? Number talks are the answer! They are short,
purposeful mini lessons that enhance students’ ability to comfortably and flexibly problem solve, reason, and respectfully communicate their thinking using appropriate math vocabulary and academic
sentence structure (also known as math talk). They fit perfectly in a guided math workshop framework, which you can learn how to implement in my guided math workshop course (coming soon). Daily
number talks promote quality classroom conversation and give your students the confidence to approach problems with a positive attitude and growth mindset, which will lead to math success this school
year and beyond.
What are Number Talks?
A number talk is a short, purposeful mini-lesson that enhances students’ ability to comfortably and flexibly problem solve, reason, and respectfully communicate their thinking using appropriate math
vocabulary (also called maths vocabulary in different parts of the world) and academic sentence structure. These discussions lead to students understanding that there is more than one way to solve a
problem. Students benefit from having this be a part of their daily routine. Number talks should last about 10 minutes. There are four main steps: 1) A problem is presented; 2) Students engage in a
productive struggle; 3) Students share with a partner; 4) Students and the teacher engage in a whole group discussion. This will become one of your favorite classroom procedures because it will give
you the time for exploring and teaching math strategies and concepts!
What is the Difference Between a Number Talk and Math Talk?
Math talk is a component of a number talk. It is a way to structure how students communicate during math discussions. It encourages the use of academic sentence structure and math vocabulary words,
as well as building ideas off of one another and challenging ideas in a respectful manner. You’ll be amazed at how quickly students pick up grade level math talk (including math names and math
language) when they have daily practice, as well as you modeling it regularly when introducing and discussing math concepts and skills.
Why are Number Talks Important?
Number talks are an important component of elementary math instruction because of these 5 reasons:
1. Develop a strong number sense
2. Construct an understanding that there are many different ways to solve a problem
3. Have the opportunity to talk about math concepts with support
4. Build confidence in their math ability
5. Learn from each other
How do I do a Number Talk?
Here are the four steps of a number talk:
1. A Problem is Presented (1 minute)
The teacher displays a problem and reads it aloud to the students. The teacher clarifies as needed. The students have access to manipulatives and tools to help them make sense of the problem and
solve it.
2. Students Engage in a Productive Struggle (3-5 minutes)
Students work independently to solve the problem as the teacher circulates, asking guiding questions, and offers encouragement to persevere. They can use mental computation, journals, or whiteboards
as a tool to record their thinking.
3. Students Share with a Partner (1 minute)
Students use the accountable talk stems to guide their discussion. These accountable talk stems need to be in a spot where students can access them. Again, the teacher circulates and provides support
as needed.
4. Students and the Teacher Engage in a Whole Group Discussion (3-5 minutes)
The teacher asks students who would like to share their or their partner’s thinking. The students are reminded to utilize the posted accountable talk stems. This experience should feel more like a
class discussion, than a question and answer between the teacher and students. When students share their thinking and give an answer, the teacher highlights the strategies and use of math talk rather
than the answer.
What is an Accountable Talk Stem?
An accountable talk stem, also referred to as sentence stems or sentence starters, is a discussion prompt that structures students’ thinking so they can communicate mathematical ideas in an academic
way. It supports students in being able to articulate ideas, build off of someone else’s ideas, and respectfully challenge someone else’s thinking. Utilizing accountable talk in your elementary
classroom is a total game changer!
How do I Make Accountable Talk Stems Accessible to Students?
It is very important to post accountable talk stems so they are accessible to students during number talks. This will help you create a numeracy rich environment in your classroom. Here are some
ideas of where you can put the accountable talk resources:
• Make them into bookmarks that students can store in their math folder and take out during number talks.
• Make them into bookmarks and store them in the area where your class meets for number talks so students can access them.
• Type them up and print a copy for each student so they can store them in their math folder.
• Type them up and project them on the board during number talks. Write them on chart paper and post them on a wall where all students can see them from where they sit during whole group
50+ Accountable Talk Stems
Here are some examples of math sentence starters that promote thoughtful classroom conversation opportunities and increase the use of math talk among students during number talks, as well as
throughout the math block:
• My first step was ___.
• I decided to try ___.
• I needed to ___.
• In my head I saw ___.
• I was not sure what to do, but I noticed ___.
• The strategy I used was ___.
• I noticed ___.
• I realized ___.
• The information needed to solve this problem is ___.
• I discovered ___.
• I compared ___ to ___.
• My partner and I discussed ___.
• I can visualize this problem by ___.
• This reminds me of ___.
• I solved this problem by ___.
• The strategy I chose was ___ because ___.
• I would like to add on to what ___ said about ___.
• I solved the problem by ___.
• If ____, then ___.
• I proved my thinking by ___.
• I think ___ because ___.
• This reminds me of ___ because ___.
• I know this because ___.
• I can prove my thinking by ___.
• My model makes sense because ___.
• I know this tool will work because ___.
• My solution is accurate because ___.
• I can use ___ to solve this problem.
• I know my answer is reasonable because ___.
• The best way to solve the problem would be ___.
• I agree with ___, but I also think ___.
• I agree with ___ because ___.
• It makes sense to me because ___.
• I respectfully disagree because ___.
• I saw it a different way because ___.
• My strategy is similar because ___.
• Another point of view is ___.
• A different point of view is ___.
• Another way to look at ___ is ___.
• Another example is ___.
• A second example is ___.
• Another strategy you can use is ___.
• A more efficient way to do this is ___.
• I am confused about ___ because ___.
• My strategy is different because ___.
• What puzzles me is ___ because ___.
• When I checked my work, I noticed ___.
• Before this problem I used to think ___, but now I know ___.
• I learned from ___ that ___.
• I have a question about ___.
• Why did you ___?
• How did you ___?
• What if ___?
• I can use this in real life when ___.
• Could you have ___?
3 Number Talk Management Tips
Here are 3 number talk management tips to help you successfully and confidently implement number talks in your elementary classroom:
1. Set clear expectations
Instead of jumping right into your first number talk, take the time to set clear expectations for what students should be doing during number talks. Consider documenting expected behavior on an
anchor chart and posting it in your classroom the first couple of weeks so you can refer back to it as needed.
2. Establish and maintain a consistent routine
Consistent classroom procedures and routines are essential for any classroom to run smoothly. Your number talk time is no exception! How will students get the materials they need? How will you signal
their attention when it is time to share with a partner and with the whole group? About how much time do students have to complete the task? See the four steps of a number talk framework above for
additional ideas.
3. Use hand signals
Introduce students to the hand signals system for managing their comments, answers, and questions. It will help you prevent your number talk lessons being taken off task by students who would like to
share stories instead of answering questions or make comments instead of asking questions.
Math Resources for 1st-5th Grade Teachers
If you need printable and digital math resources for your classroom, then check out my time and money-saving math collections below!
Free Math Resources for Elementary Teachers
We hope these tips for number talks and increasing math talk will be helpful as you plan your upcoming math lessons and would love for you to try these math resources with your students. They offer
students opportunities to practice grade level concepts and skills in fun and engaging ways. You can download worksheets specific to your grade level (along with lots of other math freebies) in our
free printable math resources bundle using this link: free printable math activities for elementary teachers.
Check out these other math resources!
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Mechanical movements
The OL at the COSC apply the strict test conditions described in the ISO standard or its internal requirements. The results obtained give an exact measurement of the rate of a movement, and of a
watch over a given period and attest to its intrinsic precision. The tests do not therefore constitute a simulation of behaviour when worn.
During the 15 days of testing, the regularity of the watch is checked with two key pieces of data:
• temperature
• the position of the watch (3 o’clock, 6 o’clock, 9 o’clock, dial on top, dial on the bottom)
Placed on clasps containing five pieces each, the movements are successively positioned in the various positions, inside the thermal enclosures corresponding to the different temperatures required.
Each piece remains in a given position and temperature for 24 hours. Each day, they are briefly removed from these thermal enclosures to take measurements, using vision machines equipped with five
The daily measurements carried out by the COSC on each movement therefore consist of determining the status of each instrument in relation to a reference period. The measurement by differentiation of
statuses enables the behaviour of the movement to be integrated in time and the daily rate to be calculated.
Minimum Requirements (for balance spring wristwatches) - Type I
┃ │ │Limit [s/d] ┃
┃Symbol │Criterion ├───────────┬───────────┨
┃ │ │Category 1 │Category 2 ┃
┃Avg R │Average daily rate │-4 +6 │-5 +8 ┃
┃Avg V │Mean variation in rate │2 │3.4 ┃
┃Max V │Greatest variation in rates │5 │7 ┃
┃D │Difference between rates in horizontal and vertical positions │-6 +8 │-8 +10 ┃
┃P │Largest variation in rates │10 │15 ┃
┃C │Thermal variation │+/- 0.6 │+/- 0.7 ┃
┃R │Rate resumption │+/- 5 │+/- 6 ┃
┃The minimum requirements are considered absolute limits and no calculation result is rounded up or down.┃
Status: the difference, at a specific time, between that indicated by a timekeeping instrument and the reference time given by a master clock.
The status is positive or negative depending on whether the timekeeping instrument is ahead of or behind the reference time.
Rate: the expression of the difference, per unit of time, between two states of a timekeeping instrument, separated by a given time interval (duration).
If it is positive, the instrument advances, and vice versa.
Daily rate: the expression of the difference between two states separated by a time interval of 24 hours.
Each OL uses a time base made up of two atomic clocks and a synchronised DCF77 0CX0 clock, and makes use of two different time references (GPS and DCF). The data entry and calculation are fully
Avg R: the arithmetic mean of the daily rates of the first 10 days of testing.
Avg V: the arithmetic mean of the five absolute values of the rate variations obtained for the five positions of the watch during the first 10 days of testing.
Max V: the absolute value of the largest of the five rate variations relative to the five positions of the watch during the first 10 days of testing.
D: the difference between the vertical and horizontal rates of the watch, obtained by subtracting the average rate of the 9th and 10th days of testing from the average rate of the first two days of
P: the absolute value of the largest of the differences between one of the first 10 rates and the average daily rate of the tests.
C: the variation of the rate as a function of temperature, obtained by subtracting from the rate at 8°C from that at 38°C, the whole divided by the temperature interval between these two rates.
R: the variation between the last rate of the tests and the average of the first two rates of the tests.
Category 1 consists of wristwatches with a fitting diameter greater than 20 mm or where the fitting surface area is greater than 314 mm2 Category 2 includes all wristwatches with a fitting diameter
of 20 mm or less and a fitting surface area not exceeding 314 mm2.
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How do I link the results from the 2D frame analysis to subsequent calculations?
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
Tekla Tedds Tekla Tedds for Word
When using the 2D analysis calculation the results are held in memory and once the calculation is complete these results are deleted (destroyed) so that they don't continue to use memory resources.
The calculation produces a report of the analysis results but does not generate any specific variables. This is because the number of required variables to output all of the results would be
excessive and it would be very difficult to isolate the values you actually want. The approach is to instead load the analysis model, query the results you are specifically interested in and save
those results as variables which can then be used in subsequent calculations.
To link the results from 2D analysis to subsequent calculations
1. load the analysis results
2. select the context of what you want to query
3. query specific results
4. destroy the analysis results
Load results:
Using Tedds for Word, type the following expression before the 2D analysis calculation:
_analysisDestroy = False
This ensures that when the 2D analysis calculation is finished the analysis model remains in memory. If you are using a design calculation which is integrated with the 2D analysis the design
calculation will destroy the in memory analysis model, you can reload the analysis model using the following three expressions:
SolvCreateSolver() = ?
SolvSetXml( _analysisModel ) = ?
SolvAnalyse() = ?
Select a subset of the model
By default when a model is loaded all the elements in the model are selected and if the model contains load combinations then all load combinations are selected; otherwise all load cases are
selected. The selection determines the context of any subsequent queries for specific results. For example if you don't select a specific member or element then querying SolvGetMaxMoment will return
the maximum moment for the entire model. To query results for a specific member, element, load case or load combination you must ensure you select the appropriate context first. All the functions you
need (with explanations and examples) can be found in the 'Library Access System: Writing your own custom calculations > Calculation writing documentation > Tedds functions > 2D analysis functions >
Querying results:
Once you have selected the appropriate context you can query specific results and assign them to variables of your choosing. For example, to create a variable "M" with the value of the maximum moment
on analysis element number "3" the expressions would be:
SolvSelectClear() = ?
SolvSelectElements(3) = ?
M = SolvGetMaxMoment() = ? kNm (or kip_ft)
In the next example we select the results for a specific load case and query the reactions on two nodes:
SolvSelectClear() = ?
SolvSelectLoadCases( "Permanent" ) = ?
R1 = SolvGetMaxMomentReaction( 1 ) = ?
R2 = SolvGetMaxMomentReaction( 2 ) = ?
All the functions you need (with explanations and examples) can be found in the 'Library Access System: Writing your own custom calculations > Calculation writing documentation > Tedds functions >
2D analysis functions > Results'.
To link to a specific library calculation you just need to use the correct variable names
Destroy the analysis results
At the end of your calculations use the following expression:
SolvDestroySolver() = ?
This will unload the in memory analysis model and results it will not affect the variables you have assigned.
In the attached simple example we have used:
• the Frame Wizard to create a simple goal post frame
• the steps above to fins the maximum moment and shear on the beam
• GetSectionVar to take these values into the EN1993 steel member design Calculation Section
• The text that makes this possible is highlighted in yellow.
• You still need to assign the correct section size and length
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What is the log() function in Swift?
The log() function in Swift calculates the natural logarithm of a number. We can find it in Swift's Foundation library.
The mathematical representation of the log() function is as follows:
$log(x) = log_e(x) = ln(x)$
The syntax for the log() function is as follows:
// The number should be floating, double, or real.
This function requires a number for which the logarithm is to be calculated. It must be greater than 0.
Return value
log() returns the natural logarithm of the number sent as a parameter.
The code below shows the use of the log() function in Swift.
import Swift
import Foundation
// Positive integer
print ("The value of log(4.0) : ", log(4.0));
//Positive double value
print ("The value of log(4.4) : ",log(4.4));
print ("The value of log(M_E) : ",log(M_E));
//Exceptional output
print ("The value of log(-4.5) : ",log(-4.5));
print ("The value of log(0.0) : ",log(0.0));
• Line 2: We add the Foundation header required for the log() function.
• Line 4: We calculate the natural log value of the positive integer using log().
• Line 7: We calculate the natural log of the positive double value using log().
• Line 10: We calculate the natural log of "E" using log().
• Lines 13–15: We calculate the natural log of exceptional values using log().
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Dealing with Math Anxiety
Overcoming Math Anxiety
Do you feel nervous about Math? Do you dislike Math? Do you have fear of doing Math? If so, you are not alone. You may have Math anxiety. Math anxiety, or Math phobia, is not unusual.
What is Math Anxiety?
Symptoms of Math anxiety include:
• Negative self-talk.
• Lack of motivation to work on Math.
• Not studying regularly.
• Putting off Math homework until the last minute.
• Panic when doing Math homework or tests.
• Difficulty remembering Math facts.
• Relying on memorization rather than understanding.
Math anxiety is a condition that you have the power to change, if you so desire. Math anxiety is a learned behavior; you can change it!
Reducing Math Anxiety
You will need to work on your Math course each day, if only for a half-hour. You must avoid doing all your Math class homework and studying on one or two days per week. Schedule quality study time
throughout the week and stick to your schedule.
1. Study Smart
Read the information on study skills, time management, note-taking, and reading textbooks. The more you try different approaches, the more you will discover what works for you.
2. Attend Math Class
You must attend class to keep up with the fast pace of a college-level Math course. You will also get information regarding tests and instructor expectations. You will see examples that are not in
the textbook. You are responsible for all information and concepts presented in class, whether you are present or not.
3. Get Organized
You need to keep good class notes. You need to keep a good Math notebook with lists of vocabulary, properties, formulas, theorems and procedures. Must anxiety is caused by disorganization.
4. Continually Test Yourself
Be aware of what you know and of what you don't know. Keep practicing the concepts and problems presented in the classroom and in the textbook.
5. Replace Negative Self-Talk with Positive
Be mindful of what you are saying to yourself. Develop positive affirmations such as "I will succeed in this course!" or "I love Math!" to counteract any negative feelings you may have about your
abilities or about Math itself.
6. Use All Your Resources
Videos, textbooks, friends, study groups, your instructor, the internet....all are available to help you succeed. It's up to you to take advanage of them.
There are a variety of other proven techniques and activites that will help to to conquer Math anxiety. There are a variety of resources that will address these techniques and activities in more
detail than is possible here.
Talk to your instructor about your feelings toward Mathematics. Acknowledging your feelings is the first step in conquering them. Your instructor and Math tutors can help direct you to good resources
and practices that can help you reduce or eliminate the emotional blocks to learning Math.
Math Websites for College Students
• Check out these websites for more details and suggestions.
Math Books
• Winning at Math: Your Guide To Learning Mathematics Through Successful Study Skills, 5th edition. by Paul D. Nolting, Ph.D. Academic Success Press, Inc., 2008.
• Managing the Mean Math Blues by Cheryl Ooten. Prentice Hall, 2003.
• Conquering Math Anxiety: A Self-Help Workbook with CD,3rd Edition by Cynthia Arem. Brooks/Cole, 2009.
• MASTERING MathEMATICS: How to Be A Great Math Student, 2nd ed. by Richard Manning Smith. Wadsworth Publishing Company, 1994.
• Mind Over Math by Stanley Kogelman and Joseph Warren. McGraw-Hill,1978.
• Conquering Math Phobia by Calvin C. Clawson. John Wiley & Sons.
• Overcoming Math Anxiety by Sheila Tobias. W. W. Norton, 1995.
• Overcoming Math Anxiety by Randy Davidson and Ellen Levitov. Addison Wesle, 2000.
• Math for the Anxious: building basic skillsby Rosanne Proga, McGraw-Hill Higher Education. Boston, 2005.
• The Book for Math Empowerment by Sandra Manigault. Godosan Publications, 1997.
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I fixed you LaTeX (it was a double subscript error \int_{0{}_{t} but your formula is wrong: you should integrate by $x$ (inner) from $x-c(t-s)$ to $x+c(t-s)$ and integrand is $e^{x-s}$ and then you
integrate by $s$ from $0$ to $t$; also forgot $\frac{1}{2c}$ factor.
I have not checked other terms. You can see that your "solution" does not satisfy initial conditions (or equations)
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Comparison to ipywidgets
Comparison to ipywidgets¶
ipywidgets already provides both an interact and an interactive_output function, so why use mpl_interactions instead? There are three reasons: performance, portability, and convenience.
The ipywidgets functions expect the entire output to be regenerated every time a slider value changes. This means ipywidgets functions will work best with the inline backend. In fact, some special
casing is even available to better support the inline backend. Unfortunately this process does not work well with the interactive ipympl backend. The ipympl backend expects to be shown only once,
then updated with Matplotlib methods as controls change. What results is you needing to make multiple new figures—or recreate the entire plot—every time a slider value changes.
It is possible to get around these performance issues by using the interact function and setting the called function to use Matplotlib updating methods (such as set_data()). However in this case, not
only do you need to remember how to do this, but over time you will find you are repeating yourself. Reducing these performance barriers was the initial motivation for the mpl_interactions library,
and also brings us to the reason of convenience.
mpl_interactions will make use of the widgets provided by ipywidgets if they are available. Unlike interactive output, it will still work if called from a script or an (i)python REPL by falling back
to the built in Matplotlib widgets.
With the interact function (ipywidgets.interact) you are responsible for generating the data to plot, and for handling the logic to update the plot.
1. f(x,...) => y
2. Plotting logic (plt.plot, fig.cla, ax.set_ylim, etc)
In contrast, mpl_interactions only requires you specify the data you want to plot and will handle the plot creation and updating for you.
Additionally, there are multiple valid strategies for choosing what selection widgets to create for a parameter. As a general framework the choices made by ipywidgets are not always ideal for
plotting scientific data. Unencumbered by generality, mpl_interactions makes several slightly different choices that are more plotting focused.
Differences in generated widgets¶
Both mpl_interactions and ipywidgets will generate a slider. However, mpl_interactions will use numpy.linspace() and ipywidgets will use numpy.arange().
Here is a comparison of the generated widget for two_tuple = (1., 5) and three_tuple = (0., 1250, 100):
ipywidgets will assume a NumPy array or list are categoricals. mpl_interactions will attempt to make a slider for the values.
For example, here is what ipywidgets and mpl_interactions will create for np.linspace(-5,5,100):
In the context of a single number, for example, param = 10.:
Treats the parameter as fixed.
Creates a slider with a range of [-10,+3*10].
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What does SIG mean in ANOVA test?
What does SIG mean in ANOVA test?
statistically significant
In ANOVA, the null hypothesis is that there is no difference among group means. If any group differs significantly from the overall group mean, then the ANOVA will report a statistically significant
What is F and SIG in ANOVA?
The F-value in an ANOVA is calculated as: variation between sample means / variation within the samples. The higher the F-value in an ANOVA, the higher the variation between sample means relative to
the variation within the samples. The higher the F-value, the lower the corresponding p-value.
What is the SIG value in ANOVA?
Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
P-value ≤ α: The differences between some of the means are statistically significant.
What does P 0.05 mean in ANOVA?
If one-way ANOVA reports a P value of <0.05, you reject the null hypothesis that all the data are sampled from populations with the same mean. But you cannot be sure that one particular group will
have a mean significantly different than another group.
How do you know if variance is significant?
Hypotheses in Variances Tests If the p-value is less than your significance level (e.g., 0.05), you can reject the null hypothesis. The difference between the two variances is statistically
Is SIG the same as p-value?
Generally speaking, the “Sig” or “Sig(2-Tailed) is your p-value. The p-value has a slightly different interpretation depending on which test you’re running.
What does SIG mean in statistics?
Statistical Significance/
Statistical Significance/P-values. Many statistical tests result in a statistical significance (“sig.”) value in SPSS (and other statistical packages). This is commonly known as the “p value” and is
often quoted in research as, for example, “p=0.0819” or “p<0.01” or “p>0.05”.
What is a significant level of variance?
Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.
If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis.
What does it mean if variance is significant?
Significant Variance means a notable difference, or a repeat Minor Variance between the actual measure of the level of Service and the Performance Target which may lead to a material change in Costs
or non-compliance with regulatory requirements.
What is SIG in statistics?
What is statistical significance? “Statistical significance helps quantify whether a result is likely due to chance or to some factor of interest,” says Redman.
Is 0.05 statistically significant?
P > 0.05 is the probability that the null hypothesis is true. 1 minus the P value is the probability that the alternative hypothesis is true. A statistically significant test result (P ≤ 0.05) means
that the test hypothesis is false or should be rejected. A P value greater than 0.05 means that no effect was observed.
How do you interpret a sig?
The “Sig” entry in the output for independent samples is the two-tailed p-value for the null hypothesis that the two groups have the same variances. A small p-value indicates a difference in
variances. If you have a significant result here, your data violates the assumption for equal variances.
What does SIG mean in SPSS?
significance level
The p-value is labeled as “Sig.” in the SPSS output (“Sig.” stands for significance level).
Is p 0.02 statistically significant?
The smaller the p-value the greater the discrepancy: “If p is between 0.1 and 0.9, there is certainly no reason to suspect the hypothesis tested, but if it is below 0.02, it strongly indicates that
the hypothesis fails to account for the entire facts.
Is P 0.01 statistically significant?
The degree of statistical significance generally varies depending on the level of significance. For example, a p-value that is more than 0.05 is considered statistically significant while a figure
that is less than 0.01 is viewed as highly statistically significant.
How do you tell if a difference is statistically significant?
You may be able to detect a statistically significant difference by increasing your sample size. If you have a very small sample size, only large differences between two groups will be significant.
If you have a very large sample size, both small and large differences will be detected as significant.
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How to work out stock turnover ratio
Apply the formula to calculate the inventory turnover ratio. Once you know the COGS and the average inventory, you can calculate the inventory turnover ratio. Using the information from the above
examples, in this 12 month period, the company had a COGS of $26,000 and an average inventory of $6,000. The calculus for figuring out inventory turnover ratio is fairly straightforward. Basically,
here's the formula: Inventory Turnover Ratio = cost of products or goods sold / average inventory Here's
Inventory turnover ratio calculator measures company's efficiency in turning its inventory into sales, the number of times the inventory is sold and replaced. Inventory turnover ratio is often linked
with the measurement of profitability. Though this ratio does not in itself measure profitability, but an increase in the rate of 6 Dec 2019 But first, you need to know how to calculate your
inventory turnover ratio. What is the inventory turnover formula? There are a few steps to Like a typical turnover ratio, inventory turnover details how much inventory is sold over a period. To
calculate the inventory turnover ratio, cost of goods sold is divided by the average inventory Related Articles 1. Estimate the average inventory during the period for which you want to calculate 2.
Calculate the cost of goods sold for the period. 3. Divide the value of your average inventory into the cost of goods sold to calculate 4. Calculate your stock turnover ratio for Similarly, the
inventory turnover ratio of General Motors Company is calculated by entering the formula =C4/C3 into cell C5. The resulting inventory turnover ratio is 11.43. Ford's higher inventory The inventory
turnover ratio is calculated by dividing the cost of goods sold for a period by the average inventory for that period. Average inventory is used instead of ending inventory because many companies’
merchandise fluctuates greatly throughout the year.
Inventory turnover, or the inventory turnover ratio, is the number of times a business sells and replaces its stock of goods during a given period. It considers the cost
Calculating your inventory turnover ratio is fairly simple. To get the ratio for a given time period, you need to find how many times the inventory was sold or used 1 May 2019 Stock / inventory
turnover ratio is an important financial ratio to evaluate the efficiency and effectiveness of inventory management of the firm. This tool will calculate your business' inventory turnover ratio and
compare the results to your industry's benchmark. The formula for the inventory turnover ratio measures how well a company is turning their inventory into sales. The costs associated with retaining
excess 19 Feb 2019 How do you calculate stock turn? The formula for calculating inventory turnover ratio is: Cost of Goods Sold (COGS) divided by the Average
The calculus for figuring out inventory turnover ratio is fairly straightforward. Basically, here's the formula: Inventory Turnover Ratio = cost of products or goods sold / average inventory Here's
For example business operations with low stock turnover tend to require higher working capital. The calculation used to obtain the ratio is: Stock Turnover Ratio = . 31 Oct 2018 Good inventory
management depends on knowing a company's inventory turnover ratio. Learn how to calculate it and what it means. Inventory/stock turnover ratio (in days). Method of calculation. Formula for inventory
(stock) turnover ratio in days (inventories cycle): inventory. Ratio's Inventory turnover is an efficiency ratio that shows how many times a company sells and replaces inventory in a given 13 May
2019 Inventory turnover is an efficiency/activity ratio which estimates the number of times per period a business sells and replaces its entire batch of 28 Jan 2018 Inventory turnover ratio (ITR) is
an activity ratio and is a tool to evaluate the liquidity of company's inventory. It measures how many times a 11 Jul 2018 One of the most critical tools you can use to achieve this is the inventory
turnover ratio. Find out what this figure is, why it's important, and how to
You can also use your stock turnover rate to calculate the minimum levels of stock you need. Your minimum stock levels, and the types of incidents that could affect them, should be addressed in your
business continuity plan. Benchmarking your business. You can compare your stock turnover rate to other similar businesses when benchmarking your business. This can help you work out how well you are
performing and what areas you might need to improve on. Valuing your business
24 Jul 2013 Inventory Turnover Ratio Calculation. Inventory turnover ratio calculations may appear intimidating at first but are fairly easy once a person To calculate your inventory turnover:
Inventory Turnover = COGS / Average Inventories. The result you come up with will give you the inventory turnover ratio. 27 Feb 2020 Calculating The Financial Ratio. 1. Deciding the Inventory
Turnover Period. Inventory turnover is calculated over a certain time period. The time Calculating your inventory turnover ratio is fairly simple. To get the ratio for a given time period, you need
to find how many times the inventory was sold or used 1 May 2019 Stock / inventory turnover ratio is an important financial ratio to evaluate the efficiency and effectiveness of inventory management
of the firm. This tool will calculate your business' inventory turnover ratio and compare the results to your industry's benchmark.
For example business operations with low stock turnover tend to require higher working capital. The calculation used to obtain the ratio is: Stock Turnover Ratio = .
This tool will calculate your business' inventory turnover ratio and compare the results to your industry's benchmark. The formula for the inventory turnover ratio measures how well a company is
turning their inventory into sales. The costs associated with retaining excess 19 Feb 2019 How do you calculate stock turn? The formula for calculating inventory turnover ratio is: Cost of Goods
Sold (COGS) divided by the Average By calculating your stock turnover ratio, you can determine whether it's a factor of your newfound popularity and increased demand, or if you're simply not buying
Estimate the average inventory during the period for which you want to calculate the stock turnover ration. Add the cost of your inventory at the beginning of the Inventory turnover, or the
inventory turnover ratio, is the number of times a business sells and replaces its stock of goods during a given period. It considers the cost
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lgebra equations
algebra equations reviews and downloads
Search results for «algebra equations»:
Showing 1 to 10 from 10 matches
Basic Algebra Shape-Up 4.0
Hands-on algebra tutorials covering integers, expressions, equations, and more.
Linear Algebra 2
Performs computations associated with matrices.
Graphmatica 2.4
powerful, easy-to-use, equation plotter with numerical and calculus features
DeadLine 2.36
Analyze, solve, and plot various mathematical equations.
.NET Matrix Library 2.5.5000.8
Provides classes for object-oriented linear algebra in the .NET platform.
Algebra - One On One 1.1
Makes learning algebra fun. 21 skill levels. Learn step by step.
AlgeWorksheets 2.0
AlgeWorksheets - Generate and print pre-algebra and algebra worksheets or test papers in minutes.
.NET Matrix Library 32-bit Developer 5.0
Provides classes for object-oriented linear algebra in the .NET platform.
Kalkulator 2.41
Advanced Sci/Eng calculator, from simple expressions to differential equations
UltimaCalc 4.1.941
Algebraic calculator. User functions. Charts. Regression. Std Deviation. Plots.
Related: algebra equations equations differential equations equations for windows algebra
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Performance Considerations
Managing Memory
There are two general approaches for managing memory when solving most problems supported by the SDE engine:
Managing Memory with Outputs
Perform a traditional simulation to simulate the underlying variables of interest, specifically requesting and then manipulating the output arrays.
This approach is straightforward and the best choice for small or medium-sized problems. Since its outputs are arrays, it is convenient to manipulate simulated results in the MATLAB^® matrix-based
language. However, as the scale of the problem increases, the benefit of this approach decreases, because the output arrays must store large quantities of possibly extraneous information.
For example, consider pricing a European option in which the terminal price of the underlying asset is the only value of interest. To ease the memory burden of the traditional approach, reduce the
number of simulated periods specified by the required input NPeriods and specify the optional input NSteps. This enables you to manage memory without sacrificing accuracy (see Optimizing Accuracy:
About Solution Precision and Error).
In addition, simulation methods can determine the number of output arguments and allocate memory accordingly. Specifically, all simulation methods support the same output argument list:
where Paths and Z can be large, three-dimensional time series arrays. However, the underlying noise array is typically unnecessary, and is only stored if requested as an output. In other words, Z is
stored only at your request; do not request it if you do not need it.
If you need the output noise array Z, but do not need the Paths time series array, then you can avoid storing Paths two ways:
• It is best practice to use the ~ output argument placeholder. For example, use the following output argument list to store Z and Times, but not Paths:
• Use the optional input flag StorePaths, which all simulation methods support. By default, Paths is stored (StorePaths = true). However, setting StorePaths to false returns Paths as an empty
Managing Memory Using End-of-Period Processing Functions
Specify one or more end-of-period processing functions to manage and store only the information of interest, avoiding simulation outputs altogether.
This approach requires you to specify one or more end-of-period processing functions, and is often the preferred approach for large-scale problems. This approach allows you to avoid simulation
outputs altogether. Since no outputs are requested, the three-dimensional time series arrays Paths and Z are not stored.
This approach often requires more effort, but is far more elegant and allows you to customize tasks and dramatically reduce memory usage. See Price European Stock Options Using Monte Carlo Simulation
Enhancing Performance
The following approaches improve performance when solving SDE problems:
• Specifying model parameters as traditional MATLAB arrays and functions, in various combinations. This provides a flexible interface that can support virtually any general nonlinear relationship.
However, while functions offer a convenient and elegant solution for many problems, simulations typically run faster when you specify parameters as double-precision vectors or matrices. Thus, it
is a good practice to specify model parameters as arrays when possible.
• Use models that have overloaded Euler simulation methods, when possible. Using Brownian motion (BM) and geometric Brownian motion (GBM) models that provide overloaded Euler simulation methods
take advantage of separable, constant-parameter models. These specialized methods are exceptionally fast, but are only available to models with constant parameters that are simulated without
specifying end-of-period processing and noise generation functions.
• Replace the simulation of a constant-parameter, univariate model derived from the SDEDDO class with that of a diagonal multivariate model. Treat the multivariate model as a portfolio of
univariate models. This increases the dimensionality of the model and enhances performance by decreasing the effective number of simulation trials.
This technique is applicable only to constant-parameter univariate models without specifying end-of-period processing and noise generation functions.
• Take advantage of the fact that simulation methods are designed to detect the presence of NaN (not a number) conditions returned from end-of-period processing functions. A NaN represents the
result of an undefined numerical calculation, and any subsequent calculation based on a NaN produces another NaN. This helps improve performance in certain situations. For example, consider
simulating paths of the underlier of a knock-out barrier option (that is, an option that becomes worthless when the price of the underlying asset crosses some prescribed barrier). Your
end-of-period function could detect a barrier crossing and return a NaN to signal early termination of the current trial.
Optimizing Accuracy: About Solution Precision and Error
The simulation architecture does not, in general, simulate exact solutions to any SDE. Instead, the simulation architecture provides a discrete-time approximation of the underlying continuous-time
process, a simulation technique often known as a Euler approximation.
In the most general case, a given simulation derives directly from an SDE. Therefore, the simulated discrete-time process approaches the underlying continuous-time process only in the limit as the
time increment dt approaches zero. In other words, the simulation architecture places more importance on ensuring that the probability distributions of the discrete-time and continuous-time processes
are close, than on the pathwise proximity of the processes.
Before illustrating techniques to improve the approximation of solutions, it is helpful to understand the source of error. Throughout this architecture, all simulation methods assume that model
parameters are piecewise constant over any time interval of length dt. In fact, the methods even evaluate dynamic parameters at the beginning of each time interval and hold them fixed for the
duration of the interval. This sampling approach introduces discretization error.
However, there are certain models for which the piecewise constant approach provides exact solutions:
More generally, you can simulate the exact solutions for these models even if the parameters vary with time, if they vary in a piecewise constant way such that parameter changes coincide with the
specified sampling times. However, such exact coincidence is unlikely; therefore, the previously discussed constant parameter condition is commonly used in practice.
One obvious way to improve accuracy involves sampling the discrete-time process more frequently. This decreases the time increment (dt), causing the sampled process to more closely approximate the
underlying continuous-time process. Although decreasing the time increment is universally applicable, however, there is a tradeoff among accuracy, run-time performance, and memory usage.
To manage this tradeoff, specify an optional input argument, NSteps, for all simulation methods. NSteps indicates the number of intermediate time steps within each time increment dt, at which the
process is sampled but not reported.
It is important and convenient at this point to emphasize the relationship of the inputs NSteps, NPeriods, and DeltaTime to the output vector Times, which represents the actual observation times at
which the simulated paths are reported.
• NPeriods, a required input, indicates the number of simulation periods of length DeltaTime, and determines the number of rows in the simulated three-dimensional Paths time series array (if an
output is requested).
• DeltaTime is optional, and indicates the corresponding NPeriods-length vector of positive time increments between successive samples. It represents the familiar dt found in stochastic
differential equations. If DeltaTime is unspecified, the default value of 1 is used.
• NSteps is also optional, and is only loosely related to NPeriods and DeltaTime. NSteps specifies the number of intermediate time steps within each time increment DeltaTime.
Specifically, each time increment DeltaTime is partitioned into NSteps subintervals of length DeltaTime/NSteps each, and refines the simulation by evaluating the simulated state vector at (NSteps
- 1) intermediate times. Although the output state vector (if requested) is not reported at these intermediate times, this refinement improves accuracy by causing the simulation to more closely
approximate the underlying continuous-time process. If NSteps is unspecified, the default is 1 (to indicate no intermediate evaluation).
• The output Times is an NPeriods + 1-length column vector of observation times associated with the simulated paths. Each element of Times is associated with a corresponding row of Paths.
The following example illustrates this intermediate sampling by comparing the difference between a closed-form solution and a sequence of Euler approximations derived from various values of NSteps.
Improve Solution Accuracy
Consider a univariate geometric Brownian motion (GBM) model using gbm with constant parameters:
Assume that the expected rate of return and volatility parameters are annualized, and that a calendar year comprises 250 trading days.
Use simBySolution to simulate approximately four years of univariate prices for both the exact solution and the Euler approximation (using simByEuler) for various values of NSteps.
nPeriods = 1000;
dt = 1/250;
obj = gbm(0.1,0.4,'StartState',100);
[X1,T1] = simBySolution(obj,nPeriods,'DeltaTime',dt);
[Y1,T1] = simByEuler(obj,nPeriods,'DeltaTime',dt);
[X2,T2] = simBySolution(obj,nPeriods,'DeltaTime', ...
[Y2,T2] = simByEuler(obj,nPeriods,'DeltaTime', ...
[X3,T3] = simBySolution(obj,nPeriods, 'DeltaTime', ...
[Y3,T3] = simByEuler(obj,nPeriods,'DeltaTime', ...
[X4,T4] = simBySolution(obj,nPeriods,'DeltaTime', ...
[Y4,T4] = simByEuler(obj,nPeriods,'DeltaTime', ...
Compare the error (the difference between the exact solution and the Euler approximation) graphically.
plot(T1,X1 - Y1,'red')
hold on;
plot(T2,X2 - Y2,'blue')
plot(T3,X3 - Y3,'green')
plot(T4,X4 - Y4,'black')
hold off
xlabel('Time (Years)')
ylabel('Price Difference')
title('Exact Solution Minus Euler Approximation')
legend({'# of Steps = 1' '# of Steps = 2' ...
'# of Steps = 10' '# of Steps = 100'}, ...
'Location', 'Best')
hold off
As expected, the simulation error decreases as the number of intermediate time steps increases. Because the intermediate states are not reported, all simulated time series have the same number of
observations regardless of the actual value of NSteps.
Furthermore, since the previously simulated exact solutions are correct for any number of intermediate time steps, additional computations are not needed for this example. In fact, this assessment is
correct. The exact solutions are sampled at intermediate times to ensure that the simulation uses the same sequence of Gaussian random variates in the same order. Without this assurance, there is no
way to compare simulated prices on a pathwise basis. However, there might be valid reasons for sampling exact solutions at closely spaced intervals, such as pricing path-dependent options.
See Also
sde | bm | gbm | merton | bates | drift | diffusion | sdeddo | sdeld | cev | cir | heston | hwv | sdemrd | rvm | roughbergomi | roughheston | ts2func | simulate | simByEuler | interpolate |
simByQuadExp | simBySolution | simBySolution
Related Examples
More About
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Bit Shift calculator - hex, octal, binary, decimal
Bit Shift Calculator
The bitwise calculator is a tool to perform Bit Shift operation on numbers. The input can combine binary, decimal, hexadecimal, or octal numbers.
How it Works?
To use this calculator, follow the below steps:
• Enter the number in the first input box.
• Select the operator in the dropdown list. ( << or >>)
• Enter the number in the second input box.
• You will see the calculated number in the output boxes for each base system
• The results is displayed as binary, decimal, hexadecimal, or octal numbers.
Left Shift
The left shift operator is a binary operator which shifts some number of bits, in the given bit pattern, to the left and appends 0 at the end. The left shift is equivalent to multiplying the bit
pattern with 2^k ( if we are shifting k bits ).
Right Shift
The right shift operator is a binary operator which shifts some number of bits, in the given bit pattern, to the right and appends 1 at the end. The right shift is equivalent to dividing the bit
pattern with 2^k ( if we are shifting k bits ).
Bitwise Operations
• A wide variety of programming languages supports bitwise operations on two integer integers.
• The evaluation and manipulation of particular bits within an integer are possible with bitwise operators.
• Each subsequent pair of bits in the operands is subjected to the operation.
• Bitwise AND, OR, and XOR are the three most popular operations.
What is a Base System?
A base system is a mechanism of representing numbers. When we talk about base-n, the system can show a number with n characters (including 0). Numbers are represented by digits that are less than or
equal to n. As a result, 3 in base-3 equals 10: because that system lacks a "3," it starts anew (1, 2, 10, 11, 12, 20, 21, 22, 100, etc.).
We commonly utilize base-10 since we have 10 (including 0) digits until we start anew (8,9,10). We only have two characters in base-2 (binary), 0 and 1, until we begin anew. In our (base-10) system,
the binary number 10 is 2 in this example.
JavaScript Bitwise Operators
Operator Name Description
& AND Sets each bit to 1 if both bits are 1
| OR Sets each bit to 1 if one of two bits is 1
^ XOR Sets each bit to 1 if only one of two bits is 1
~ NOT Inverts all the bits
<< Zero fill left shift Shifts left by pushing zeros in from the right and let the leftmost bits fall off
>> Signed right shift Shifts right by pushing copies of the leftmost bit in from the left, and let the rightmost bits fall off
>>> Zero fill right shift Shifts right by pushing zeros in from the left, and let the rightmost bits fall off
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Practical implementation of a scalable discrete Fourier transform using logical phi-bits: nonlinear acoustic qubit analogues (Journal Article) | NSF PAGES
Abstract We present a model of an externally driven acoustic metamaterial constituted of a nonlinear parallel array of coupled acoustic waveguides that supports logical phi-bits, classical analogues
of quantum bits (qubit). Descriptions of correlated multiple phi-bit systems emphasize the importance of representations of phi-bit and multiple phi-bit vector states within the context of their
corresponding Hilbert space. Experimental data are used to demonstrate the realization of the single phi-bit Hadamard gate and the phase shift gate. A three phi-bit system is also used to illustrate
the development of multiple phi-bit gates as well as a simple quantum-like algorithm. These demonstrations set the stage for the implementation of a digital quantum analogue computing platform based
on acoustic metamaterial that can implement quantum-like gates and may offer promise as an efficient platform for the simulation of materials.
more » « less
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Numbers to Words Converter (Number to text)
Numbers to Words Converter
Enter your number in the given input box of numbers to words converter and hit the calculate button to convert your number into the words (number to text).
Enter a number in the input box and hit the calculate button to converts from numbers to words
The numbers to words converter gives you the text form of the number you enter. You can enter up to 9 digit values to convert numbers to words. Use this numbers to words calculator to know your
amount in words in Indian rupees
How to use our numbers to text calculator?
1. Enter the number in the input box.
2. Hit Calculate.
3. Reset for another conversion
How to convert the number in words?
It is a simple enough method once you memorize the place values of the numbers. But converting larger numbers to words can take time and hence the above online numbers in words in English converter
is recommended.
To convert numbers or fractions to words, letters and text, see the places below where you start placing your numbers from the right.
│Ten Crore│Crore│Ten Lakh│Lakh│Ten Thousand │Thousand│Hundred│Ten│Unit│
│10^8 │10^7 │10^6 │10^5│10^4 │1000 │100 │10 │1 │
If you have a number then start placing its rightmost digit under the unit’s place. After that, the second right number should be placed under ten’s place, and so on.
Rules to convert numbers to word
To write the numbers in words.
1. Always start from right.
2. Write the numbers in English e.g 1 will be written as one, 2 as two e.t.c
3. Places with the same suffixes like hundred thousand, ten thousand, and one thousand will be written jointly.
For instance, consider you have the number 111000. It will be read as One hundred eleven thousand. It will not be read as one hundred thousand one ten thousand and one thousand.
Also, it doesn’t make sense.
4. Write ‘and’ before the last word or compound (e.g twenty-three)
Convert 12300 into words.
Step 1: Make the place value table.
│Ten Crore│Crore│Ten Lakh│Lakh│Ten Thousand │Thousand│Hundred│Ten│Unit│
Step 2: Place the rightmost digit under the unit.
The rightmost digit is 0.
│Ten Crore│Crore│Ten Lakh│Lakh│Ten Thousand │Thousand│Hundred│Ten│Unit│
│ │ │ │ │ │ │ │ │0 │
Step 3: keep placing numbers from the right side.
│Ten Crore│Crore│Ten Lakh│Lakh│Ten Thousand │Thousand│Hundred│Ten│Unit│
│ │ │ │ │1 │2 │3 │0 │0 │
Step 4: Write in words.
Twelve Thousand and three hundred.
Example 2:
Convert following numbers to words.
1. 230
2. 10001
3. 6789
4. 1000278
1. Two hundred and thirty
2. Ten thousand and one
3. Six thousand seven hundred and eighty-nine
4. Ten lakh two hundred and seventy-eight
International numeric words for numbers
The words lakh and Crore are Hindi numeric units but they are equally famous all around the world similarly million and billions are the units that are used internationally. The alternative words are
1. Lakh = Hundred thousand
2. Ten lakh = Million
3. Crore = Ten Million
4. Ten Crore = Hundred Million
In the international number system, a new word is introduced after three places from thousand onwards. Like
• 1000 is one thousand (1st place)
• 10,000 is ten thousand (2nd place)
• 100,1000 is a hundred thousand. (3rd place)
Now, a new word will be used for the digit places after a hundred thousand.
• 1,000,000 is one million (1st place)
• 10,000,000 is ten million (2nd place)
• 100,000,000 is a hundred million (3rd place)
The new word is billion (10^9) and after that trillion (10^12), quadrillion (10^15), and quintillion (10^15) respectively.
1. International number system - thousands, million, billion, trillion | Teachoo.com
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[Solved] Consider a spherical fluid particle in an | SolutionInn
Consider a spherical fluid particle in an inviscid fluid (no shear stresses). If pressure and gravitational forces
Consider a spherical fluid particle in an inviscid fluid (no shear stresses). If pressure and gravitational forces are the only forces acting on the particle, can they cause the particle to rotate?
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How do you calculate volume flow rate?
How do you calculate volume flow rate?
How to calculate flow rate? Flow rate formulas
1. Volumetric flow rate formula: Volumetric flow rate = A * v. where A – cross-sectional area, v – flow velocity.
2. Mass flow rate formula: Mass flow rate = ρ * Volumetric flow rate = ρ * A * v. where ρ – fluid density.
How do you calculate flow rate in a pipe?
The equation for pipe diameter is the square root of 4 times the flow rate divided by pi times velocity. For example, given a flow rate of 1,000 inches per second and a velocity of 40 cubic inches
per second, the diameter would be the square root of 1000 times 4 divided by 3.14 times 40 or 5.64 inches.
How do you calculate volume and rate?
These three calculations can be represented by the following formulas:
1. Rate Var = (Actual Rate – Budgeted Rate) * Actual Average Balance * Basis.
2. Volume Var = (Actual Avg Bal – Budgeted Avg Bal) * Budgeted Rate * Basis.
3. Mix Var = (Actual Rate – Budgeted Rate) * (Actual Avg Bal – Budgeted Avg Bal) * Basis.
What is the unit of volume flow rate?
meters cubed per second
In S.I. units (International System of Units), volume flow rate has units of meters cubed per second, m 3 s \dfrac{\text m^3}{\text s} sm3start fraction, start text, m, end text, cubed, divided by,
start text, s, end text, end fraction, since it tells you the number of cubic meters of fluid that flow per second.
Is volume flow rate constant?
Volume flow rate at standard conditions remains constant under changing thermodynamic conditions because it is calculated at a constant (imaginary) set of conditions.
How do you calculate pressure from flow rate?
The first one defines fluid flow at one point where pressure is P1, velocity is v1, and height is h1. The second equation defines the fluid flow at another point where pressure is P2. Velocity
and height at that point are v2 and h2. Define the pressure and flow rate.
Is volume a rate?
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, rate of fluid flow, or volume velocity) is the volume of fluid which passes per
unit time; usually it is represented by the symbol Q (sometimes V̇)….Volumetric flow rate.
Volume flow rate
SI unit m3/s
Does volume flow rate change?
No. The flow rate / discharge increases when diameter increases. For given pressure drop, mass density, and effective length, the square of the volumetric flow-rate, given by Bernoullis principle, is
proportional to the diameter to the fifth divided by the friction factor.
What is LPM in flow rate?
LPM is an abbreviation of litres per minute (l/min). When used in the context of a particle counter’s flow rate, it is a measurement of the velocity at which air flows into the sample probe. For
example, a flow rate of 2.83 LPM means the particle counter will sample 2.83 litres of air per minute.
Volume flow rate is calculated from the cross sectional area and the velocity or by dividing the mass flow rate by the fluid density.
What are the units for volume flow rate?
In physics and engineering, in particular fluid dynamics and hydrometry, the volumetric flow rate (also known as volume flow rate, rate of fluid flow or volume velocity) is the volume of fluid which
passes per unit time; usually represented by the symbol Q (sometimes V̇). The SI unit is m 3/s (cubic metres per second).
What is the formula to calculate flow rate?
The flow rate formula, in general, is Q = A × v, where Q is the flow rate, A is the cross-sectional area at a point in the path of the flow and v is the velocity of the liquid at that point.
How do you calculate average flow rate?
Select the shape of the cross-section of the channel
Input all the measurements required to compute the cross-sectional area
Input the average velocity of the flow
Choose the unit of the flow rate
Click on the “Calculate” button to compute the flow rate.
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Is Your 401k Too Big - Part 2 - Go Curry Cracker!
Is Your 401k Too Big – Part 2
by Go Curry Cracker | Nov 19, 2018 | Taxes | 52 comments
(This post is the 2nd in a series. Subsequent posts forthcoming… soon. See the first post: Is Your 401k Too Big?)
A Traditional 401k / IRA allows us to invest for the future in a tax advantaged way. However, in some cases these accounts can become tax disadvantageous due to sheer size.
When the IRS forces withdrawals after our 70 1/2 birthday (the RMD), large accounts may get hit with higher tax bills. Those taxes could even be greater than what we saved on contribution.
We already saw this in the first post in this series. Even account values at age 70 1/2 of $350k or more (Married Filing Jointly) would most likely fail to Never Pay Taxes Again. But this isn’t
necessarily tax disadvantageous.
Due to tax savings on contributions, are there higher account values that can be reached before our 401k becomes too big? And if so, how do we evaluate additional contributions?
Is Your 401k Too Big?
Higher Account Thresholds
When making contributions to Traditional 401k/IRAs we get an immediate tax deduction at our current marginal tax rate. We come out ahead If withdrawals are taxed at a lower rate.
Using the same methodology that we used to determine account value thresholds that allow zero tax burden, we can find glide paths that coast under the RMD without crossing over the 10%, 12%, and 22%
tax brackets.
Methodology reminder: we make regular annual withdrawals to the top of a tax bracket, assuming a range of forward CAGRs (cumulative annual growth rates.) State taxes and ACA subsidy reductions not
The following charts are for the 10%, 12%, and 22% tax brackets, with CAGRs from 4% – 9%. All 2018 numbers for Married Filing Jointly (Single filers divide by 2.) Scales are the same for easy
comparison. For years after 2018, all numbers can be scaled; X = Current Year Standard Deduction / 2018 Standard Deduction.
From this chart we can see that account values of ~$650k at traditional retirement age could likely never pay more than 10% tax. This is about 2x the account values that allow paying no tax. With
higher acceptable marginal tax rates the account values can be larger… as can be seen in the 12% and 22% charts that follow.
Using these charts we can find target 401k values to achieve our tax minimization goals. All raw data is here, in both chart and table format.
Example Scenario 1:
At age 60, if our 401k was heavy on stocks (assumed 7%+ real CAGR), and we had an arbitrary goal of never paying tax at a rate higher than 10%, we might struggle to meet that goal if our 401k value
was greater than $600k. (Or $1400k at 12%, or $2600k at 22%.)
Example Scenario 2:
If throughout our accumulation phase we were earning at the 12% marginal rate, and now Traditional 401k/IRA values exceed $2 million at age 70, then will likely pay some tax at 22%+. Could be
considered slightly tax disadvantageous.
Other Income and Credits
What about other income? Many “early retirees” have some rental properties, a pension, non-qualified dividends, interest, or accidental blogging income (basically anything that isn’t qualified
dividends or long term capital gains.) And most people will have some amount of Social Security income. How does that factor into this?
The math is straightforward… albeit a bit messy the first time through.
Ordinary income: $43,050. This completely fills up the 0% and 10% tax brackets ($24,000 standard deduction, $19,050 10% bracket – 2018 numbers, MFJ)
Goal: max 12% tax rate
Target retirement age: 50
Allocation: 50/50 stock/bond (assume 6% CAGR)
Target 401k value at age 50 = Tax 12% 401k value @ 50 – Tax 10% 401k value @ 50 = 1630k – 692k = $938k
(Data here.)
For credits, we can choose to increase the size of IRA conversion / withdrawal. For example, if we have a Foreign Tax Credit of $600 from ownership of International Stock funds, we can convert/
withdraw an additional $6,000 at 10% tax rate, $5,000 at 12% rate, etc…. Our target IRA size can be increased by ratio of increased withdrawal/tax bracket, e.g. 6,000/19,050 = 30%, 5,000/58,350 =
8.5%, etc…
Reevaluating Additional Contributions
This is all well and good, but how do we know if we are on trend to exceed certain disadvantageous levels?
It helps to look at a specific target return of return. I’ve chosen 7% real since that is the actual long term CAGR of the stock market since the beginning of time.
We saw in the first post in this series that any 401k/IRA values exceeding the Tax 0% Blue Line in the above chart will result in tax being paid.
Similarly, if values exceed the Tax 10% Orange Line, then taxes in excess of 10% will be paid at some point. Taxes in excess of 12% (Taxes 12% Gray Line) and 22% (Tax 22% Gold Line) will be required
as values grow further.
We can use these thresholds to determine if additional contributions are warranted.
Age 50, 401k/IRA value = $1 million.
If Marginal rate on contribution is 22%, then there is definite value in more Traditional contributions.
If Marginal rate is 12%, then contributions to Roth or brokerage account would have better long term value.
Summary and Next Steps
In this, the second post in this series, we evaluated higher thresholds where 401ks can become tax disadvantageous, and how to evaluated if there is benefit to additional contributions. We also
explored how to make adjustments for other income and tax credits.
Whereas account values of more than $250k will potentially struggle to achieve a lifetime 0% tax rate, our 401k can reach into the millions while still being advantageous to higher income earners.
All of the raw data in chart and table format is located here.
So far we’ve looked at things purely from a marginal rate perspective, and the analysis has been orderly and predictable. It even seems like we have some semblance of choice or control in the matter.
In the next post… I’ll dissuade that delusion.
(This post is the 2nd in a series. Subsequent posts forthcoming… soon.)
(See the 1st post here.)
Are you tracking your 401k / IRA values?
We monitor our 401k/IRA values using free tools from Personal Capital (affiliate link).
52 Comments
steveark on November 19, 2018 at 5:20 am
I always maxed out both my 401k and a Roth or post tax regular IRA depending on income limits and put another 15% of income in a brokerage account. I was in the 33% tax bracket most of the time
and in retirement I’m in the 24% bracket with side gig income. I think maxing the 401k and not doing Roth conversions saved me money.
Go Curry Cracker on November 19, 2018 at 5:55 am
Most likely it did. It might not when you hit 70.5.
David Ann Arbor on November 19, 2018 at 5:53 am
Won’t the standard deduction and the income tax brackets change over time? For example won’t the standard deduction become 20% greater in about 7 years? Wouldn’t this affect the charts you
Go Curry Cracker on November 19, 2018 at 5:55 am
Kinda. Using real returns means everything is inflation adjusted. Just like the standard deduction.
Joe on November 19, 2018 at 6:00 am
This one was a bit confusing, but I think I got it by the end. Our combined traditional 401k/IRA are worth over a million and we’re 45. I’ll double check our tax marginal rate, but I’m pretty
sure it’s more than 22%. Anyway, we’ll keep contributing for now.
Once Mrs. RB40 retires, then we’ll do some conversion at a lower tax rate.
Go Curry Cracker on November 19, 2018 at 7:11 am
Leave OR first.
snowcanyon on November 19, 2018 at 7:53 am
Oregon is truly tax hell for early retirees, well most people, I guess. Funny how twenty years ago Portland was considered a place for younger people to escape California and retire
early. I guess someone has to pay the piper for their huge social programs despite half the state having no economy.
Will Herbert on November 19, 2018 at 6:21 am
While I can’t disagree with the maths, I think there’s still a considerable amount of control you have over this. There are extremes (a strategic year abroad) but also more quotidian methods –
doing a Roth conversion during a market slump for example. What’s your opinion on timing conversions for market slumps? Am I ignoring something critical here or is that a reasonable idea?
Go Curry Cracker on November 19, 2018 at 7:11 am
It’s not unreasonable.
If you do a conversion at a “bottom” and you pay the tax 100% from existing cash then it could help, but that is assuming your 401k was big enough to warrant taking those steps.
Ryan Stubbs on November 19, 2018 at 6:34 am
Completely Greek to me! You are writing above our heads. Please do a “Baby” version of your exquisite article so that we ordinary folks can benefit also. Thanks for sharing your amazing insights
(this one just did not translate for me)
Go Curry Cracker on November 19, 2018 at 7:07 am
Think of it like a text book. You only need to really read the intro to each chapter.
Rounding the Bend on November 19, 2018 at 6:55 am
What do you think about taking our more than the RMD requires early in the RMD timeline (but not spend it all) so that RMDs will be smaller later? Could that have the effect of smoothing the
marginal tax rate?
Go Curry Cracker on November 19, 2018 at 9:27 pm
The answer is the same as should you make additional contributions. Saving 12% now is better than paying 10% later but worse than paying 22% later.
Mark S on November 19, 2018 at 7:29 am
It might also be interesting to add potentially higher taxes on social security or Medicare IRMAA that could occur with higher income later.
Go Curry Cracker on November 19, 2018 at 9:35 pm
See the examples above where income fills the 0% or 10% bracket. SS is just a form of income. If you get taxed on 24k worth of SS every year, then that occupies your standard deduction / 0%
tax bracket. This reduces target 401k values by ~$300k
East Coast on November 19, 2018 at 6:30 pm
I second the comment about including the impact of RMDs on the taxation of social security and the tax bracket one ends up in… Thanks. (I also share the experience of this post being over my
Little Miss Fire on November 21, 2018 at 1:59 am
Really interesting take on this, as from what I understand (I’m a UKer) its pretty much a given that everyone needs a 401K! I guess our equivalent is an ISA? please someone correct me if I’#m
Go Curry Cracker on November 21, 2018 at 8:58 pm
No idea about comparison to UK. But not everyone needs a 401k. A married couple making median income could save 20% of income in IRAs alone
Chris on November 21, 2018 at 4:31 am
I love this series can’t wait for part 3. I think I’ve read each part 5 times, slower each time to digest it.
I have long had the fear that there is a point where you have too much in a tax deferred. I don’t think the charts do it justice since SS isn’t calculated in. I think as higher income earners
(especially if you have 2 high income earners), when they start taking SSN and then have RMD’s they could potentially be forced into higher brackets. I’m not too worried about being 100+ and
RMD’s accelerating, heck if I make it over 100 I will say that is a good problem to have. I am personally trying to decide this year if I want to max out pre-tax since if I don’t I could
potentially use more of a 199A deduction now. So that 22% may only be 17.4 (22% x .20 = 4.4, 22-4.4=17.4).
Go Curry Cracker on November 21, 2018 at 9:01 pm
SS income (and all other income) is different for everybody, but you can calculate it yourself for your specific situation.
Jim on December 4, 2018 at 4:02 pm
Thanks for an excellent, clear presentation of some complex math. I would like to add one point to the conversation.
I would prefer to optimize around the 4% or so assumed return. If a retiree achieves 7% real CAGR for 20-30 years, they are not going to have any problems. In that case paying a little extra tax
is not a big deal.
On the other hand, if they experience 4% CAGR, then they may struggle with their 4% withdrawal rate and will regret paying some taxes upfront, having optimized around the 7% return case.
If you optimize for 4 and get 7, no big deal. If you optimize for 7 and get 4, you have cost yourself a difference-making amount as you struggle with your 4% withdrawal rate.
Also note Bogle predicts 4% stock returns over the next decade or so: https://goo.gl/qj5N5D
Go Curry Cracker on December 5, 2018 at 7:30 pm
We do have to make a choice/prediction, and your guess is as good as mine (or Bogle’s.)
One thing I’ll critique about Mr. Bogle’s method is you have to look at combined buybacks and dividend yield, not dividend yield alone.
Nooj on December 7, 2018 at 3:39 pm
Am I reading this correctly? I am single with a little more than $500K in Traditional IRA value as of today (market correction included). Looks like I really need to max my ROTH conversion if my
goal is to minimize RMDs. I have $100K in 457b and $1M in after-tax account. Am currently retired. Thanks
Go Curry Cracker on December 8, 2018 at 12:02 am
I dunno, it depends on what you mean by minimize RMDs and how old you are.
An RMD by itself is no big deal unless it forces you to pay more taxes than you wanted, and is mostly a consequence of being financially successful. (Also, about RMDs – they ramp up slowly
and half of all tax paid is after age 90 or so. Most people are dead by then, so isn’t necessarily something to worry about.)
But, if you never want to pay tax at a rate greater than 10% then that might be difficult, but if 12% is ok then you are probably fine. ymmv
Steve (NWOutlier) on December 15, 2018 at 12:31 pm
Ugh – help me out here:
I’m 50 will retire in 5-10 years
my tax deferred total is 660k
my tax free total is 98k
I am 22% married filing jointly
I could stop contributing to tax deferred 401k and make all 401k contribs as ROTH 401k 19k + 6k (25k/yr with catch up contrib)
plus I still contribute max to roth ira for both my wife and myself (additional 12k/yr)
total roth contributions: 37k per year for the next 10 years to beef up my tax free total
simple math tells me my tax deferred total will grow to about 1M give or take a hundred k or so… when I retire I can convert the 24k annually into roth and continue that growth so I don’t have to
deal with complex taxes as I age….
I can’t tell, should I? or should I NOT continue tax deferred contributions?
Thanks in advance!
Go Curry Cracker on December 18, 2018 at 7:35 am
Hey Steve,
The following is a lengthy walk through of the data in this post with your numbers, to show how I would explore the decision:
Looking at the source data, it says if your Traditional accounts at age 60 are greater in value than $1.8 million w/ 5% forward growth or $1.4 million with 7% forward growth, then you will
probably pay some tax at a rate higher than 12%.
If we assume that you have $24k in taxable social security income, then this reduces the target Traditional account values by $300k – $400k since it consumes the standard deduction. These are
the values from the 0% table in the source data / never pay taxes again example from part 1. Subtracting this, we get target values of $1.4 million @ 5% or $1.1 million at 7%. If you have
other regular income, you’ll need to make additional adjustments.
By my math, $660k in tax deferred accounts today would grow to $1.1 million at age 60 with 5% real growth or $1.3 million with 7%, even without additional contributions. With an additional
$25k/year for 10 more years these numbers become $1.3 million and $1.6 million respectively. In other words, your past contributions are doing most of the heavy lifting at this point (ain’t
compound interest a beautiful thing?)
So, if you have 5% growth you are all good and if you get 7% growth you might pay some tax at a rate greater than 12%, someday.
Higher than 12% is just the next tax bracket up (22%), and paying 22% now is exactly the same as paying 22% on future withdrawals. Same same.
Someday is whenever larger withdrawals happen, probably from the RMD. RMD withdrawals don’t really get that big until age 85, and half of all taxes paid is after age 90. So you might never
see the 22% rates, and if you do it is 40 or more years from now, after you have had some wonderful growth.
So personally, I would continue to make deductible tax deferred contributions, and invest the current year tax savings in a taxable brokerage account. (If you contribute $25k, that saves
$5500 in taxes today.)
In the years between retirement and RMD, you’ll want to make large withdrawals up to the top of the 12% tax bracket. That is about $100k/year minus SS. Post-retirement / pre-Medicare, this
will impact ACA subsidies.
Hopefully this helps. Let me know if you have any Qs.
NWOutlier (Steve) on December 18, 2018 at 8:00 am
This is great! Thank you. I have read this once over, i will do so again so i am confident i inderstand. I was going to switch to all roth to beef it up, but another 5-10 years
contributions to tax deferred seem safe for now and i agree i can keep pumping funds into my taxable accout..
Sure do wish i had the roth significantly higher, but – i will continue those contributions as well.
Btw – i worked in building 42 back in the late 90’s on the operating system teams – fun times. Maybe we can share a beer sometime when you are around the NW again
Best Regards,
Steve (NWOutlier)
Steve (NWOutlier) on December 18, 2018 at 8:05 am
Just realized – if I feel uncomfortable with 25k tax deferred, I can do my “catch up contributions” as ROTH and normal contributions (19k) as tax deferred. hhmmm
Go Curry Cracker on December 19, 2018 at 5:36 am
Yessir, you can dial it anywhere from 0% to 100%.
re: I wish I had the Roth significantly higher – we are all really just playing a tax now vs tax later game. It can be nice to have the simplicity of the Roth, but too often people
pay way more tax today than they would pay later if they chose Traditional. I’d rather have a little complexity and be a whole lot richer.
Steve (NWOutlier) on December 28, 2018 at 12:53 pm
Have I mentioned how much you ROCK!?!?! :) — I’ve read this many times over and gone back to my spreadsheets, dividends, savings rates… etc.. the suggestions you make are laser focused and will
benefit me huge! Thank you! Sure wish I had this clarity when I was younger, I fall into the “if you save early you only need a few hundred a month, but if you wait, it will cost thousands a
month”… past 3 years I’ve saved between 50-70k/yr each year not because I know what’s right – but because I have to to catch up…. I hope others see the benefits of your work… and save early, save
I wish I did not have to save so much now, because I’m in the stage of life for (unfortunately) family passing, kids going to college… and I would like some time to enjoy some of the money…. but
it’s not possible yet…
anyway – thank you for all your work, while retired. :)
Go Curry Cracker on May 21, 2019 at 11:14 pm
Don’t forget to enjoy the ride!
Shane on May 16, 2019 at 11:03 am
Love your posts, they are always super insightful. This one in particular is my favorite, and I noticed you left us hanging indicating that there will be a follow up post dissuading the delusion
that we have some semblance of choice or control in the matter. Hope to see that one soon! I appreciate all that you do and love all your content.
Go Curry Cracker on May 21, 2019 at 11:13 pm
Hi Shane. I hope to see that one soon too, sorry. I see I posted this one over 6 months ago… I’ll try to bump it up my to do list.
JW from NJ on November 16, 2019 at 3:16 pm
Hello, thank you for this post. I think I understand it to a degree, but not enough to definitively apply it to my situation and get an answer. With that, can I ask you: with minimal 401k
contribution to trigger employer match leading to approx 1.7 mil at my likely retire age of 63 (6% CAGR), and assuming social security annual income at 26K and a pension of about 80K, is it safe
to say that increasing my 401k contribution at this point won’t help me in terms of minimizing taxes?
Go Curry Cracker on November 16, 2019 at 5:09 pm
Hard to say with just what you shared here, but with only pension and SS you’ll be in the 22% tax bracket (MFJ) or 24% tax bracket (single.)
(See tax bracket info here.)
If you aggressively Roth convert post-retirement you’ll probably pay 22%+ on <=100% of 401k withdrawals. Maybe some at 24-32%. Compare that to your current top marginal rate.
JW from NJ on November 16, 2019 at 5:25 pm
Thank you, I’m currently MFJ, 39 years old, and paying 22% marginal. I think that means I don’t have to increase my contributions beyond employer match.
Go Curry Cracker on November 16, 2019 at 7:45 pm
A lot can happen in 25-30 years… but yeah, no sense in saving 22% now to pay 24% later.
The question then becomes what you do with your savings. If you can do Roth 401k contributions and/or backdoor Roth (mega or normal) then you can at least get the growth to be tax
JW from NJ on November 17, 2019 at 8:29 am
I’m on the fence about it at the moment…should I pound the brokerage account and try to retire earlier, say around 55, and fill in the gap between 55 and 63 (or 65) with tax free
gains and Roth conversion, or focus more on just putting it in a Roth for tax free growth, or just split it 50/50 between the two? Then again, I can always take the principal out
of the Roth without penalty during those years, so just focus on the Roth and when I hit 55 , start taking out as much as possible from the 401K up to 12% marginal and do the Roth
conversion…yeah this is just too damn hard.
Go Curry Cracker on November 17, 2019 at 8:35 am
Seems like you have a pretty good handle on it though.
But here is something to keep in perspective. Even if you make the absolutely worst possible choices in every case, what happens… you are super rich and you pay 24% tax
instead of 22%.
JW from NJ on November 17, 2019 at 1:40 pm
Yes, that’s true. Thanks again.
Pete on December 12, 2019 at 12:29 pm
Thanks for your hard work! Loving this series and patiently waiting for the next one to drop.
KM on December 18, 2019 at 8:02 am
So, looking at the charts: I am 33, MFJ, 22% marginal with 325k in 401ks. That means that continuing to contribute to 401k will likely put me over 22% upon withdrawal, correct?
Go Curry Cracker on December 18, 2019 at 6:48 pm
Potentially. 30 years of 7% real cagr on $325k would put you at ~$2.5 million, in which case the marginal dollar might be taxed at 22%+.
chris on September 15, 2021 at 12:37 pm
This is one of my favorite articles I have probably read it 10 times both part 1 and part 2. I also read the comments and I think what people fail to think of is, you don’t have to wait until
RMD’s to start withdrawing at lower tax rates. You can fill up the 10 and 12% brackets before RMD’s which have been changed from 70.5 to 72 since this article was originally published. Maybe CA
will inspire you to write the part 3 cliff hanger to the series. I keep leaning towards traditional since, if I’m completely wrong I will likely be in the 22% bracket, otherwise I win with the 10
or 12. Kids are getting older (1 or 3 out on her own, 2nd close behind) and its actually quite shocking how quickly expenses are dropping. Focus on expenses and fill up those lower 2 brackets
each year and it seems very controllable to me.
John P. on February 23, 2022 at 4:50 pm
The recent Secure act affects these calculations in 2 significant ways. The first and beneficial change is the delay of RMD’s to age 72. The other and detrimental change is the requirement that
inherited tax deferred accounts be drained in 10 years. This can force you into a higher tax bracket and interfere with Roth laddering plans. It also adds a lot of uncertainty as is not something
you can precisely plan for in advance. For those that do not retire early it can mean taking IRA withdrawals at the same time as your peak earning and tax rate years. You also cannot do Roth
conversions from inherited IRA withdrawals. I have not found any clever strategies to mitigate this.
Go Curry Cracker on February 24, 2022 at 9:00 pm
Multigenerational tax planning is more complex (if you or your elders care about tax rates after death.)
Multi-Generational Tax Minimization
The SECURE Act – Compression of the Stretch IRA
andrew on March 22, 2023 at 7:54 am
very useful ideas – was there a part 3 in this series?
Go Curry Cracker on March 28, 2023 at 1:59 pm
I never did get around to writing part 3. fwiw it says it is entirely out of your control anyway… just continue contributing
Marcy on August 24, 2023 at 4:30 am
Interesting indeed and thank you for sharing. I think I’ve absorbed this at a surface level and I will continue to study it. We have nearly $4M in tax-advantaged accounts. If we contribute to
taxable instead of tax-advantaged now, our marginal rate rises to 45% (inc state tax). We will have some rental income in retirement and will be retiring after 59 1/2. Does this make us not early
retirees. We havent’ been putting much in taxable so we aren’t well-positioned to do the 401k-ROTH conversions when our income drops unless we reduce tax advantaged savings for the next 5 years
when we may retire. Looking at the RMD schedule, I don’t see us having such high marginal tax rates until one of us inherits, if then. Unless taxes go up significantly (possible), then all bets
are off. Does this sound like I’m following? Thank you
Go Curry Cracker on August 24, 2023 at 3:09 pm
Given that the highest (federal) tax bracket is 37% the likelihood of paying more than 45% in retirement is low unless you remain in a high-tax state. Then it might just be a wash.
With $4KK in tax-advantaged accounts, it may not matter what you do at this point. Adding another $60k/year is peanuts compared to normal market activity.
At age 59.5, you have unrestricted access to all retirement accounts -> You don’t need funds in taxable accounts to fund daily expenses. You could do Roth conversions, locking in tax rates of
24-35% or so, depending on your target annual budget. This could result in some small tax savings vs just letting it all ride.
Howie on January 8, 2024 at 2:10 pm
It seems if one had a significant amount in TIRAs, even with Roth conversion activity for a number of years (even longer given the recent shift to 75 years of age for RMD commencement) that there
could be huge RMD hit unless the TIRAs were drawn down (sacrificing ongoing appreciation) during one’s decumlation phase.
Trying to get a feel if I’m thinking about this wrong- that is, while during retirement, spending TIRA money at any point before exhausting one’s taxable investments?
Go Curry Cracker on January 8, 2024 at 4:01 pm
Larger Roth conversion(s) will have the same impact (make the TIRA smaller.)
If you foresee that TIRAs are growing too fast, you can fill the next higher tax bracket for a few/many years (e.g. Roth convert to top of 24% or 32% bracket instead of 22%.)
Submit a Comment Cancel reply
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List-Decodable Sparse Mean Estimation via Difference-of-Pairs Filtering
Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track
Ilias Diakonikolas, Daniel Kane, Sushrut Karmalkar, Ankit Pensia, Thanasis Pittas
We study the problem of list-decodable sparse mean estimation. Specifically, for a parameter $\alpha \in (0, 1/2)$, we are given $m$ points in $\mathbb{R}^n$, $\lfloor \alpha m \rfloor$ of which are
i.i.d. samples from a distribution $D$ with unknown $k$-sparse mean $\mu$. No assumptions are made on the remaining points, which form the majority of the dataset. The goal is to return a small list
of candidates containing a vector $\hat \mu$ such that $\|\hat \mu - \mu\|_2$ is small. Prior work had studied the problem of list-decodable mean estimation in the dense setting. In this work, we
develop a novel, conceptually simpler technique for list-decodable mean estimation. As the main application of our approach, we provide the first sample and computationally efficient algorithm for
list-decodable sparse mean estimation. In particular, for distributions with ``certifiably bounded'' $t$-th moments in $k$-sparse directions and sufficiently light tails, our algorithm achieves error
of $(1/\alpha)^{O(1/t)}$ with sample complexity $m = (k\log(n))^{O(t)}/\alpha$ and running time $\mathrm{poly}(mn^t)$. For the special case of Gaussian inliers, our algorithm achieves the optimal
error guarantee $\Theta (\sqrt{\log(1/\alpha)})$ with quasi-polynomial complexity. We complement our upper bounds with nearly-matching statistical query and low-degree polynomial testing lower
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Get Strong at the Endgame
Get Strong at the Endgame, by Richard Bozulich. Kiseido K57; 1997.
This book starts with 42 endgame problems on an 11x11 board, has 120 endgame tesuji problems, 101 endgame calculation problems, and then 28 more 11x11 problems. There's also an appendix showing the
same endgame being played by a professional against an amateur and by a professional against a professional.
They want you to do the first 42 problems but not look at the answers; then, come back to them after you've done the tesuji and calculation problems and see how much your game has improved. I was a
little bit dubious of that when I read it, but I tried it anyways and kept copies of my results both times, and the results were impressive: I got eight of the problems correct both times, I got 15
of them wrong the first time and right the second time, I got 10 of them equally wrong both times, I got eight of them wrong both times but less wrong the second time, and I got one of them wrong
both times but less wrong the first time. So on 23 of the 42 problems I did better the second time, I did equally well both times on 18 of them, and on one of them I did better the first time. I did
spend somewhat more time thinking the second time through, but not enough more time to make that kind of difference. (By the way, I recommend that you xerox the problem pages twice when doing this so
you'll have something to write down your answers on both times.)
On the whole, this book is really quite nice. The theory of endgame play seems straightforward, but you really need to go through a book like this to understand how it works in practice. The final
set of 11x11 problems is much more realistic than the earlier set, and I didn't actually manage to make it through all of them; it will take a while practicing these techniques in real games and
going through this book again before I'm really up to speed with the endgame. But I think that this book is already started paying off in my real games, getting me to spend more time counting various
situations on the board and occasionally spotting a tesuji that I might not have spotted otherwise. So I wholeheartedly recommend it.
This book doesn't explain the theory to you (though some of that does appear in the solutions to some of the 11x11 problems); for that, they refer you to The Endgame, by Ogawa and Davies, which is
the only other book on the endgame that has been published in English. I would definitely recommend reading it before reading this one, though you certainly don't have to have mastered its contents
to be able to get a lot out of this book. I wish that another publisher would put out a more advanced theoretical endgame book, but in the mean time you can get a long way by going through those two
The 11x11 problems were taken from Kano Yoshinori's Yose Jiten (with a few exceptions).
Georg Snatzke (IGS 4k*, European 1d) says:
I have only worked through about half of the book, but that really intensively. There is no doubt that one gets stronger by working through Get strong at the Endgame. You learn better reading and
a better spotting of tesujis as well. But in my eyes there is a really big flaw in this book, bigger than in any other go-book I know: A lot of (yes, not just some) the solutions in the
Bozulich-part of the book seem to be simply wrong. Either that or they are far too difficult for me to grasp. And the bad feeling that I can never trust the given answers slows me down and
diminishes many of the positive aspects of the book.
david carlton <carlton@bactrian.org>
Last modified: Sun Aug 10 20:54:54 PDT 2003
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Kilometers to Meters Conversion (km to m) - Inch Calculator
Kilometers to Meters Converter
Enter the length in kilometers below to convert it to meters.
Do you want to convert meters to kilometers?
How to Convert Kilometers to Meters
To convert a measurement in kilometers to a measurement in meters, multiply the length by the following conversion ratio: 1,000 meters/kilometer.
Since one kilometer is equal to 1,000 meters, you can use this simple formula to convert:
meters = kilometers × 1,000
The length in meters is equal to the length in kilometers multiplied by 1,000.
For example,
here's how to convert 5 kilometers to meters using the formula above.
meters = (5 km × 1,000) = 5,000 m
How Many Meters Are in a Kilometer?
There are 1,000 meters in a kilometer, which is why we use this value in the formula above.
1 km = 1,000 m
Kilometers and meters are both units used to measure length. Keep reading to learn more about each unit of measure.
What Is a Kilometer?
One kilometer is equal to 1,000 meters, which are defined as the distance light travels in a vacuum in ^1/[299,792,458] of a second.^[1] One kilometer is equal to 0.621371 miles.
The kilometer, or kilometre, is a multiple of the meter, which is the SI base unit for length. In the metric system, "kilo" is the prefix for thousands, or 10^3. Kilometers can be abbreviated as km;
for example, 1 kilometer can be written as 1 km.
Learn more about kilometers.
What Is a Meter?
According to the most recent 2019 definition, the meter is defined as the distance traveled by light in vacuum during a time interval with a duration of ^1/[299,792,458] of a second.^[2] One meter is
equal to 100 centimeters, 3.28084 feet, or 39.37 inches.
The meter, or metre, is the SI base unit for length in the metric system. Meters can be abbreviated as m; for example, 1 meter can be written as 1 m.
Learn more about meters.
Kilometer to Meter Conversion Table
Table showing
various kilometer
converted to meters.
Kilometers Meters
0.001 km 1 m
0.002 km 2 m
0.003 km 3 m
0.004 km 4 m
0.005 km 5 m
0.006 km 6 m
0.007 km 7 m
0.008 km 8 m
0.009 km 9 m
0.01 km 10 m
0.02 km 20 m
0.03 km 30 m
0.04 km 40 m
0.05 km 50 m
0.06 km 60 m
0.07 km 70 m
0.08 km 80 m
0.09 km 90 m
0.1 km 100 m
0.2 km 200 m
0.3 km 300 m
0.4 km 400 m
0.5 km 500 m
0.6 km 600 m
0.7 km 700 m
0.8 km 800 m
0.9 km 900 m
1 km 1,000 m
1. Ambler Thompson and Barry N. Taylor, Guide for the Use of the International System of Units (SI), National Institute of Standards and Technology, https://physics.nist.gov/cuu/pdf/sp811.pdf
2. International Bureau of Weights and Measures, The International System of Units, 9th Edition, 2019, https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf
More Kilometer & Meter Conversions
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Age Calculator Online | Calculate Your Age Online
Age Calculator Online
• Age Calculator Online: Determine the time between nowadays and your next birthday mistreatment the date calculator. This is free online Age Calculator computes age in terms of years, months,
weeks, days, hours, minutes, and seconds, given a date of birth.
Calculated Age Results
Age in Years:
Age in Months:
Age in Weeks:
Age in Days:
Age in Hours:
Age in Minutes:
Age in Seconds:
Upcoming Birthday: left for your next birthday
How can I calculate my birthday?
So now, the instructions bellow:
• 1. Enter the number 7.
• 2. Multiply by the month of your birth (Eg, if your birthday was 29 February, you'd multiply by 2)
• 3. Subtract 1.
• 4. Multiply by 13.
• 5. Add the day of your birth
• 6. Add 3
• 7. Multiply by 11
• 8. Subtract the month of your birth
• 9. Subtract the day of your birth
• 10. Divide by 10
• 11. Add 11
• 12. Divide by 100.
How do you calculate age in months and days?
If you would like to understand the precise age, i.e. how many years, months and days there are between someone's birth date and the current date, write 3 different DATEDIF functions: To get the
number of years: =DATEDIF(B2, TODAY(), "Y") to induce the amount of months: =DATEDIF(B2, TODAY(), "YM").
What's the most common birthday UK?
More babies ar born in late Sept and early October than at the other purpose of the year. According to new analysis by the Office for National Statistics (ONS) the most common birthday in England and
Wales was 26 September - nearly nine months after Christmas Day.
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Long decimal results instead of scientific notation?
06-23-2021, 09:08 PM
Post: #1
matalog Posts: 356
Senior Member Joined: May 2021
Long decimal results instead of scientific notation?
Is there a way to get the prime to show all possible leading zeroes in CAS (or normal) mode, for example, to show 0.00000000000000000000981 instead of 9.81E-21?
I have it in standard notation mode.
06-23-2021, 09:17 PM
Post: #2
rprosperi Posts: 6,632
Super Moderator Joined: Dec 2013
RE: Long decimal results instead of scientific notation?
I'm no Prime guru, but it seems unlikely, 'cuz what should it show if the number was 1.0E-250? 250 leading zeros?
--Bob Prosperi
06-23-2021, 09:55 PM
Post: #3
Liamtoh Resu Posts: 142
Member Joined: May 2021
RE: Long decimal results instead of scientific notation?
I would try using the logarithm of the number to create a string of zeroes and padding
the significant numbers to the end.
The log of 9.81e-21 is about -20.00833988260. Use the absolute value of the
integer. Create a loop to make the zero padded string. Pad the string with 981.
There may be a simpler way.
It is like programming in c where you have to create your own string functions.
I hope this will help get you started.
06-24-2021, 03:33 AM
Post: #4
Joe Horn Posts: 2,018
Senior Member Joined: Dec 2013
RE: Long decimal results instead of scientific notation?
(06-23-2021 09:08 PM)matalog Wrote: Is there a way to get the prime to show all possible leading zeroes in CAS (or normal) mode, for example, to show 0.00000000000000000000981 instead of
I have it in standard notation mode.
Sort of. In CAS, the command:
returns the string "0.00000000000000000000981"
A program could be written, I suppose, which would calculate the necessary second argument of format() to obtain the desired output for any given input.
If you're looking for a display mode that automatically displays all leading zeros, sorry, there is no such mode.
06-24-2021, 01:16 PM
Post: #5
Liamtoh Resu Posts: 142
Member Joined: May 2021
RE: Long decimal results instead of scientific notation?
I came up with a program to print 9.81e-21 in the format you wanted.
It is free form, off the cuff, coding at the keyboard with no optimization.
I hope to recode it later.
EXPORT numstr01()
local i;
local n,nl,na,nal;
local s,z;
local zi,zl;
local t1,t2,t3;
local t4;
local newline;
newline := "\n";
z :="0";
n := 9.81e-21;
nl := log(n);
// print(n + ", " + nl + ", " + z );
// print(newline);
na := abs(nl);
zl := trunc(na);
// print (na + ", " + zl);
t1 := nl + zl;
t2 := 10^t1;
// print(newline);
// print(t1 + ", " + t2);
t3 := trunc(1000*t2);
// print(t3);
t4 := "0.";
for i from 1 to zl
t4 := t4 + z;
print(n+" = " + t4+t3);
Not one for the software engineers but it gets the job done.
06-25-2021, 04:09 AM
(This post was last modified: 06-25-2021 05:09 AM by Liamtoh Resu.)
Post: #6
Liamtoh Resu Posts: 142
Member Joined: May 2021
RE: Long decimal results instead of scientific notation?
Take a look at this program.
Type a number, eg. 9.81e-21 on the command line in HOME.
Type PLZ to launch. It will print and return the desire format on to the stack.
You can change the print() statement to msgbox if you want.
There is a variable named sf if you want to change the number of siginificant
Thanks for the tip about getting input via Ans(1).
edit: This is very alpha software. It seems to work with numbers less
than one. It will not work in CAS mode. Perhaps the second trunc
function should be a Round function. Perhaps this program can
serve as a base for another one. There are limitations when the
exponent is less that -100. Perhap a version that did not depend
on logarithms and exponents would work better.
EXPORT plz()
local a,b,c;
local d1,f,g;
local h,i,k;
local sf; // significant figures
local z;
sf := 3;
z := "0";
a := gans();
b := log(a);
c := abs(b);
d1 := trunc(c);
f := b + d1;
g := 10 ^ f;
h := trunc((10^sf) * g);
k := "0.";
for i from 1 to d1
k := k + z;
print (k+h);
return Ans(1);
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Assignment 3: Introduction to Data Science and AI
import pandas as pd import numpy as np import matplotlib.pyplot as plt import matplotlib.cm as cm import math from scipy.ndimage.filters import gaussian_filter import seaborn as sns from matplotlib
import mlab as ml from sklearn.datasets import make_blobs from sklearn import metrics from sklearn.cluster import KMeans from sklearn.preprocessing import StandardScaler
# Importing the data set to a data frame file_name = "data_all.csv" df = pd.read_csv(file_name) # Checking for any NaN values in the data set df.isnull().values.any()
# Extracting the variables X_phi = df['phi'] X_psi = df['psi'] # Making a scatterplot plt.figure(figsize=(14,9)) plt.scatter(X_phi, X_psi, s = 10, c = 'darkcyan') plt.grid(True) plt.title(
'Distribution of Phi and Psi combinations for protein molecules') plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.show()
# Generating some test data plt.figure(figsize=(14,9)) heatmap, xedges, yedges = np.histogram2d(X_phi, X_psi, bins = 220) heatmap = gaussian_filter(heatmap, sigma = 32) extent = [xedges[0], xedges[-1
], yedges[0], yedges[-1]] plt.clf() plt.axis([X_phi.min(), X_phi.max(), X_psi.min(), X_psi.max()]) plt.imshow(heatmap.T, extent=extent, origin='lower', cmap=cm.jet) cb = plt.colorbar() cb.set_label(
'Number of samples per bin') plt.title("Distribution of Phi and Psi combinations for protein molecules") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.grid(True) plt.show()
# Function for conducting K-means Clustering and plotting the results def kmeans_clustering(X, n_clusters): plt.figure(figsize=(6,4)) # Perform K-means clustering kmeans = KMeans(n_clusters =
n_clusters, random_state = 42) y_pred = kmeans.fit_predict(X) plt.scatter(X[:,0], X[:,1], c = y_pred, cmap='gist_rainbow', edgecolor='black', s = 20) plt.scatter(kmeans.cluster_centers_[:, 0], kmeans
.cluster_centers_[:, 1], marker='x', c='black') plt.title(f'Data points with K-means algorithm when k = {n_clusters}') plt.show()
# Scaling the data as the euclidean distance is used for K-means algorithm X = df[['phi', 'psi']] scaler = StandardScaler() X = scaler.fit_transform(X) # Setting the range of k that we want to test
between 2 - 8 k_values = range(2, 8) # For each value in our lenght, do the K-means clustering algorithm and display the results for i in k_values: kmeans_clustering(X, i)
distortions = [] n_clusters = range(1,10) # We try out different k:s and get the inertia for cluster in n_clusters: kmean_model = KMeans(n_clusters = cluster) kmean_model.fit(X) distortions.append(
plt.figure(figsize=(6,4)) plt.plot(n_clusters, distortions, 'bx-') plt.xlabel('Number of Clusters') plt.ylabel('Sum of Squared Distance') plt.title('Elbow Method Showing The Optimal # Clusters') plt.
grid(True) plt.show()
# Selecting k as 4 k = 4 percentages = [1,0.75,0.5,0.25,0.10] plots = len(percentages) fig, axs = plt.subplots(1, plots, figsize=(20,4)) for i in range(0, plots): # choosing different sample lengths
for the plots to show difference in centroids being selected n_sample_size = math.floor(len(X) * percentages[i]) # Creating blobs of the data in order to change the sample sizes X, y = make_blobs(
n_samples = n_sample_size, centers = df[['phi', 'psi']]) kmeans = KMeans(n_clusters = k, random_state = 0) y_pred = kmeans.fit_predict(X) axs[i].scatter(X[:,0], X[:,1], c = y_pred, cmap =
'gist_rainbow', edgecolor='black', s = 20) # Plot kmeans cluster centers axs[i].scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1], marker='x', c= 'black') axs[i].set_title(
f'Displaying {percentages[i]*10}% of the total data points') plt.tight_layout() plt.show()
This plot illustrates how the clustering is affected by the removal of random data points. The results show a strong consistency in the clustering, using k=4. The groups remain consistent through out
the iterations and keep the overarching structure, where you can clearly see that points which were together in a previous iteration remain clustered together.
# Selecting k as 4 X = df[['phi', 'psi']] scaler = StandardScaler() X = scaler.fit_transform(X) random_init = [0, 1, 2, 3, 4, 5] plots = len(random_init) fig, axs = plt.subplots(1, plots, figsize=(20
,4)) for i in range(0, plots): kmeans = KMeans(n_clusters = k, random_state = i) y_pred = kmeans.fit_predict(X) axs[i].scatter(X[:,0], X[:,1], c = y_pred, cmap= 'gist_rainbow', edgecolor='black', s =
20) axs[i].scatter(kmeans.cluster_centers_[:, 0], kmeans.cluster_centers_[:, 1], marker = 'x',c = 'black') axs[i].set_title(f'Random state {i}') plt.tight_layout() plt.show()
The plot displays an incredible consistency in the clustering, when using 4 groups, regardless of initial position for the centroids in K-means algorithm. This speaks very highly of the consistency
of k=4 and the validity of the clusters.
X = df[['phi', 'psi']] k_values = range(2, 8) k_opt = 0 high_score = 0 for k in k_values: kmeans = KMeans(n_clusters = k, random_state = 0).fit(X) labels = kmeans.labels_ score = metrics.
silhouette_score(X, labels, metric = 'euclidean') print(f"Silhoutte score for k = {k} is: {score}") if (score > high_score): k_opt = k high_score = score print(f"The optimal silhoutte score is for k
= {k_opt} and is: {high_score}")
The silhoutte score ranges from -1 to 1. Values closer to 1 indicate a good clustering method. Looking for the different suggestions for k (ranging from 2 to 8) we find that k=3 and k=4 provides the
best sihoutte scores, as they are so similar it is hard to make a judgement. Using the findings from the 'elbow method' and visual inspection of plots, we choose k=4 as the best fit.
df_mod = df.copy() df_mod['phi']=(df['phi']+360)%360 df_mod['psi']=(df['psi']+360)%360 # Extracting the variables X_phi_mod = df_mod['phi'] X_psi_mod = df_mod['psi'] # Making a scatterplot plt.figure
(figsize=(14,9)) plt.scatter(X_phi_mod, X_psi_mod, s = 10, c = 'darkcyan', label = 'Distribution of phi and psi') plt.grid(True) plt.title('Distribution of Phi and Psi combinations for protein
molecules (shifted by 360 degrees)') plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left') plt.show()
X = df_mod[['phi', 'psi']] X = StandardScaler().fit_transform(X) k_values = range(2, 8) for value in k_values: kmeans_clustering(X, value)
It appears as though the most intuitive fit is now instead 3 clusters. Let's see how this looks for the silhoutte score. Let's compare k=3 and k=4 (which was the most effective before shifting the
data set).
X = df_mod[['phi', 'psi']] k_values = [3, 4] k_opt=0 high_score = 0 for k in k_values: kmeans = KMeans(n_clusters=k, random_state=0).fit(X) labels = kmeans.labels_ score = metrics.silhouette_score(X,
labels, metric='euclidean') print(f"Silhoutte score for k = {k} is: {score}") if (score > high_score): k_opt = k high_score = score print(f"The optimal silhoutte score is for k = {k_opt} and is: {
The silhoutte score is now clearly optimal for k=3 instead. Which is consistent with the graphical displays above.
from sklearn.cluster import DBSCAN from sklearn import metrics from sklearn.preprocessing import StandardScaler from sklearn.neighbors import NearestNeighbors import collections
X = df[['phi', 'psi']] scaler = StandardScaler() X = scaler.fit_transform(X)
# Function for creating and plotting a DBSCAN for different values of eps and min_samples def createDBSCAN(X, eps = 0.5, min_samples = 100, add_bar_plot = False): # Fitting and predicting given
values provided for eps and min_samples dbscan = DBSCAN(eps = eps, min_samples = min_samples) y = dbscan.fit_predict(X) labels = dbscan.labels_ # Number of clusters in labels, ignoring noise if
present. n_clusters_ = len(set(labels)) - (1 if -1 in labels else 0) n_noise_ = list(labels).count(-1) print('Estimated number of clusters: %d' % n_clusters_) print('Estimated number of noise points:
%d' % n_noise_) outliers_df = df[labels == -1] clusters_df = df[labels != -1] color_clusters = labels[labels != -1] color_outliers = 'black' plt.figure(figsize=(7,4)) plt.scatter(clusters_df['phi'],
clusters_df['psi'], c = color_clusters, edgecolors = 'black', cmap='gist_rainbow', s = 30) plt.scatter(outliers_df['phi'], outliers_df['psi'],c = color_outliers, edgecolors = 'black', label =
'Outliers', s = 30) plt.title(f"Datapoints with DBSCAN, minimum samples:{min_samples}, eps: {eps}") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left') plt.show(
) if add_bar_plot == True: bar = outliers_df['residue name'].value_counts(sort=True).plot.bar() bar.set_title('Amino acid residue types that are most frequently outliers') # Tryng the function with
eps = 0.5 and different values for min_samples createDBSCAN(X, min_samples = 10) createDBSCAN(X, min_samples = 100) createDBSCAN(X, min_samples = 500)
# Minimum samples to test min_samples = [200, 250, 300]
color_list = ['orchid', 'darkcyan', 'darkviolet'] i = 0 for value in min_samples: neigh = NearestNeighbors(n_neighbors = value) # Fitting NearestNeighbors to the data nbrs = neigh.fit(X) # Retrieving
the distances and indices from Kneigbors distances, indices = nbrs.kneighbors(X) distances = np.sort(distances, axis=0) distances = distances[:,1] plt.plot(distances, c = color_list[i], linewidth = 3
) plt.xlabel('Number of points') plt.ylabel('Average Distance') plt.title(f'Finding optimal eps, # of nearest neighbours:{value}') plt.grid(True) plt.show() i = i + 1
eps = [0.3, 0.4, 0.5] for i in eps: print(f'DBSCAN with eps = {i} and various values for min_samples') for j in min_samples: createDBSCAN(X, eps = i, min_samples = j)
# Plotting the cluster found with DBSCAN with epsilon = 0.4 and min_samples = 150 createDBSCAN(X, 0.4, 200, True)
print('For non-translated data, k = 4 is optimal') kmeans_clustering(X, 4) createDBSCAN(X, 0.4, 200)
pro_df = df[(df['residue name'] == 'PRO')].copy() X = pro_df[['phi', 'psi']] X = StandardScaler().fit_transform(X) min_samples = 200 eps = 0.5 # Fitting and predicting given values provided for eps
and min_samples dbscan = DBSCAN(eps = eps, min_samples = min_samples) y = dbscan.fit_predict(X) labels = dbscan.labels_ # Number of clusters in labels, ignoring noise if present. n_clusters_ = len(
set(labels)) - (1 if -1 in labels else 0) n_noise_ = list(labels).count(-1) print('Estimated number of clusters: %d' % n_clusters_) print('Estimated number of noise points: %d' % n_noise_)
outliers_df = pro_df[labels == -1] clusters_df = pro_df[labels != -1] color_clusters = labels[labels != -1] color_outliers = 'black' plt.figure(figsize=(7,4)) plt.scatter(clusters_df['phi'],
clusters_df['psi'], c = color_clusters, edgecolors = 'black', cmap='gist_rainbow', s = 30) plt.scatter(outliers_df['phi'], outliers_df['psi'],c = color_outliers, edgecolors = 'black', label =
'Outliers', s = 30) plt.title(f"Datapoints with DBSCAN for PRO, minimum samples:{min_samples}, eps: {eps}") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left')
The initial parameters seem to produce consistent results, even varying them slightly does not impact the solution.
The clustering using only the residue type PRO differs from the general DBSCAN clustering by not having any clusters with positive Phi values. Furthermore, it produces two well defined clusters, and
does not find any values in the top left corner, which was very prevalent in previous DBSCAN clusters. This is interesting, as DBSCAN never seems to cluster these exact spots, however, for large k,
the k means algorithm seems to find these clusters (found in residue type PRO) more accurately.
pro_df = df[(df['residue name'] == 'GLY')].copy() X = pro_df[['phi', 'psi']] X = StandardScaler().fit_transform(X) min_samples = 200 eps = 0.5 # Fitting and predicting given values provided for eps
and min_samples dbscan = DBSCAN(eps = eps, min_samples = min_samples) y = dbscan.fit_predict(X) labels = dbscan.labels_ # Number of clusters in labels, ignoring noise if present. n_clusters_ = len(
set(labels)) - (1 if -1 in labels else 0) n_noise_ = list(labels).count(-1) print('Estimated number of clusters: %d' % n_clusters_) print('Estimated number of noise points: %d' % n_noise_)
outliers_df = pro_df[labels == -1] clusters_df = pro_df[labels != -1] color_clusters = labels[labels != -1] color_outliers = 'black' plt.figure(figsize=(7,4)) plt.scatter(clusters_df['phi'],
clusters_df['psi'], c = color_clusters, edgecolors = 'black', cmap='gist_rainbow', s = 30) plt.scatter(outliers_df['phi'], outliers_df['psi'],c = color_outliers, edgecolors = 'black', label =
'Outliers', s = 30) plt.title(f"Datapoints with DBSCAN for GLY, minimum samples:{min_samples}, eps: {eps}") plt.xlabel('Phi, in degrees') plt.ylabel('Psi, in degrees') plt.legend(loc ='upper left')
The initial parameters seem to produce consistent results, even varying them slightly does not impact the solution.
The residue type GLY seems to represent somewhat more of the clusters found in the general case. We see one cluster with phi>0, we find clusters both in the upper left and the middle left.
However, some data points fall in the remaining clusters found in the general case, however, these are deemed outliers by the DBSCAN method.
It is important to consider that in previous tasks we found that the GLY residue had the highest number of outliers, by multiple factors. This can be displayed in the clustering of only GLY residues
also, as we can see there are no clear clusters, there seems to be data points in each quadrant of the graph, and some almost randomly scattered.
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Fuzzy Concepts. Hedges
Another important feature of fuzzy systems is the ability to define "hedges," or
modifier of fuzzy values. These operations are provided in an effort to maintain
close ties to natural language, and to allow for the generation of fuzzy statements
through mathematical calculations. As such, the initial definition of hedges and
operations upon them will be quite a subjective process and may vary from one
project to another. Nonetheless, the system ultimately derived operates with the
same formality as classic logic. The simplest example is in which one transforms
the statement "Jane is old" to "Jane is very old." The hedge "very" is usually
defined as follows:
m"very"A(x) = mA(x)^2 Thus, if mOLD(Jane) = 0.8, then mVERYOLD(Jane) = 0.64.
Other common hedges are "more or less" [typically SQRT(mA(x))], "somewhat,"
"rather," "sort of," and so on. Again, their definition is entirely subjective, but
their operation is consistent: they serve to transform membership/truth values in a
systematic manner according to standard mathematical functions.
A more involved approach to hedges is best shown through the work of Wenstop
in his attempt to model organizational behavior. For his study, he constructed
arrays of values for various terms, either as vectors or matrices. Each term and
hedge was represented as a 7-element vector or 7x7 matrix. He ten intuitively
assigned each element of every vector and matrix a value between 0.0 and 1.0,
inclusive, in what he hoped was intuitively a consistent manner. For example, the
term "high" was assigned the vector 0.0 0.0 0.1 0.3 0.7 1.0 1.0
and "low" was set equal to the reverse of "high," or 1.0 1.0 0.7 0.3 0.1 0.0 0.0
Wenstop was then able to combine groupings of fuzzy statements to create new
fuzzy statements, using the APL function of Max-Min matrix multiplication.
These values were then translated back into natural language statements, so as to
allow fuzzy statements as both input to and output from his simulator. For
example, when the program was asked to generate a label "lower than sortof
low," it returned "very low;" "(slightly higher) than low" yielded "rather low,"
etc. The point of this example is to note that algorithmic procedures can be
devised which translate "fuzzy" terminology into numeric values, perform
reliable operations upon those values, and then return natural language statements
in a reliable manner.
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Research projects
Tackling the worlds most pressing challenges
In the last decade, machine learning has positively impacted numerous areas of society, particularly in chemistry and innovative materials. Applications include discovering eco-friendly chemical
processes, creating efficient materials for energy, and advancing drug discovery. However, these challenges are computationally demanding due to the complexity of simulating quantum systems on
classical computers. The integration of machine learning with quantum simulation and quantum computing now offers unprecedented opportunities to address these complex problems, surpassing the
capabilities of even the largest supercomputers.
Open positions
Project 1. Machine learning and quantum simulations for quantum state generation
We are looking for candidates to join our research project focused on realizing versatile multiqubit Hamiltonians in Rydberg arrays. The objectives include using multifrequency coupling fields for
machine-assisted generation and optimization of strongly-correlated quantum states, with applications in quantum sensing and chemistry simulations. This position offers an exciting opportunity to
contribute to cutting-edge research at the intersection of experimental quantum physics, machine learning, and numerical methods.
1. Upgrade the Strasbourg quantum simulator to a modular Python-based control environment with a remote interface and integrate machine learning algorithms.
2. Experimentally implement a recent theoretical proposal using programmable multifrequency coupling fields for continuous- and discrete-time quantum dynamics.
3. Implement hybrid quantum-classical variational methods for steering quantum dynamics to strongly-correlated states, capitalizing on parallel quantum operations in the Rydberg platform.
Scientific Training Goals:
1. Acquire expert knowledge in the principles and operation of Rydberg atom-based quantum simulators.
2. Develop a deep understanding of machine learning techniques, programming, and data management best practices.
3. Gain proficiency in state-of-the-art numerical methods and hybrid quantum-classical techniques for strongly-correlated quantum systems and dynamics.
Contact: Shannon Whitlock (whitlock at unistra.fr)
Project 2. Machine learning for numerical simulations and quantum state tomography for near-term quantum devices
We are seeking candidates for a research project focusing on the machine-learning assisted acceleration of Markov Chain-based Quantum Monte Carlo (MCQMC) methods for strongly correlated systems.
Additionally, we aim to develop approximate tomography methods using machine learning techniques to characterize quantum states. This position offers an exciting opportunity to contribute to the
advancement of quantum Monte Carlo methods and the application of machine learning techniques in quantum state characterization.
1. Develop strategies employing neural networks to accelerate Markov chain Monte Carlo methods, facilitating efficient global updates in path-integral and perturbation-series representations.
2. Train artificial neural networks using measurement data from numerical simulations and quantum computers, enabling the determination of properties of underlying quantum states.
Scientific Training Goals:
1. Acquire expert knowledge in Quantum Monte Carlo approaches for strongly interacting bosonic and fermionic systems.
2. Gain a deep understanding of machine learning, neural network techniques, and programming using existing open-source frameworks and libraries.
3. Develop expertise in benchmarking and characterization techniques for quantum machines.
Contact: Guido Pupillo (pupillo at unistra.fr)
Project 3. Qubit decoherence simulation on a PARSEC machine
We are looking for candidates to contribute to a research project with the following objectives: Understanding how to simulate the instantaneous exploration of probability space by entangled quantum
particles as they decohere following an observation, and Implementing the algorithm on a PARSEC massively parallel emergent machine and evaluating its performance compared to parallelization on a
standard supercomputer. This position offers an exciting opportunity to contribute to advancing quantum computing simulation methodologies and exploring their performance on emergent parallel
computing architectures.
1. Upgrade the EASEA software platform, utilizing PARSEC machines, to efficiently perform massively parallel quantum qubits decoherence simulations.
2. Use the developed method to simulate and experiment with quantum computing on a massively parallel synchronous/asynchronous system.
3. Compare the results with theoretical and actual outcomes obtained on a real Quantum Computer, considering both quality and quantity.
Scientific Training Goals:
1. Acquire knowledge on efficiently discretizing a quantum search space based on the energy received by entangled qubits.
2. Understand how to perform global or local searches in the discretized hypervolumes.
3. Develop expertise in massively parallel synchronous and asynchronous stochastic optimization within the defined hypervolumes to approach the actual results of the observation of entangled qubits.
Contact: Aline Deruyver / Pierre Collet (aline.deruyver at unistra.fr)
Project 4. Machine learning interface for the MIMIQ quantum emulator
This research project will explore quantum machine learning on state-of-the-art emulators for quantum circuits. A fundamental goal is to analyze the applicability of quantum neural network algorithms
and to determine their usefulness for practical tasks. The project will analyze resource requirements and fundamentally explore new ways for obtaining a quantum advantage in machine learning. The
quantum emulation will simulate execution of perfect and noisy quantum circuits using matrix product and tensor network concepts. For this we are seeking a highly motivated candidate skilled in both
analytical and numerical quantum many-body theory concepts.
1. Collaborate with other fellows to develop quantum machine learning methods on quantum circuits, utilizing exact state-vector simulation and Matrix Product State (MPS) methods.
2. Deploy the Quantum Machine Learning (QML) ideas on the MIMIQ quantum emulator.
Scientific Training Goals:
1. Acquire expert knowledge in matrix product state/tensor network algorithms, QML, entanglement as resource for quantum computing and QML.
2. Develop expert skills in quantum software design and development.
Contact: Johannes Schachenmayer (schachenmayer at unistra.fr)
Project 5. Molecular qudits for quantum machine learning
We are seeking candidates to contribute to our research project focused on the synthesis, implementation, and utilization of qudits in quantum computing. The primary objectives are threefold: (i)
obtain a significant reduction of the number of quantum operations. (ii) Improvement of the circuit depth in the realization of quantum gates, such as the Toffoli gate. (iii) Application of
qudit-based gates for optimization tasks in Quantum Machine Learning. This position offers an exciting opportunity to contribute to the advancement of quantum computing by exploring the potential
advantages of qudits and their application in quantum algorithms, particularly in the field of Quantum Machine Learning.
1. Produce enlarged Hilbert space qudits.
2. Implement, read-out, and steer the enlarged Hilbert space of qudits within quantum devices.
3. Perform qubit-to-qudit mapping and compare it to a standard realization of quantum algorithms to highlight the potential advantages of qudits.
Scientific Training Goals:
1. Design appropriate molecular qudits.
2. Implement qudits in Quantum Machine Learning algorithms.
3. Program qudit-based quantum algorithms.
Contact: Mario Ruben (mario.ruben at kit.edu)
Project 6. Tensor network machine learning
We are looking for candidates to contribute to our research project with the following two objectives: Development of a tensor network machine learning suite. Application of the developed suite to
relevant problems in Rydberg atoms quantum technology physics.
1. Integrate tensor network methods into machine learning codes.
2. Investigate the application of tensor network machine learning to Rydberg quantum simulators and quantum computers.
Scientific Training Goals:
1. Acquire knowledge of tensor networks and their application to machine learning tasks.
2. Develop an optimizer for Rydberg quantum computers and compare its performance with standard machine learning methods.
Contact: Simone Montangero (simone.montangero at unipd.it and matilde.cassin at unipd.it)
Project 7. Machine learning for developing better error mitigation schemes and gate decompositions for quantum algorithms
We are seeking candidates to contribute to our research project with the following objectives: Scaling up currently available error mitigation schemes through the application of machine learning
techniques and developing machine-learning assisted gate decomposition methods. This position offers an exciting opportunity to contribute to the advancement of error mitigation schemes and gate
decomposition methods in quantum computing, leveraging machine learning techniques for practical applications in variational quantum circuits and near-term quantum devices.
1. Develop neural network-based techniques for learning readout errors, testing them on prototype quantum processors.
2. Mitigate noise detectors using a neural network description of readout errors, applying it for variational quantum algorithms.
3. Optimize gate decompositions using reinforcement learning (RL).
4. Tailor RL-based gate decompositions for near-term devices, considering native gates and device specifications.
5. Apply developed gate-decomposition methods for near-term applications, such as variational quantum circuits.
Scientific Training Goals:
1. Gain an overview of current quantum hardware, noise models, and error mitigation schemes.
2. Acquire a deep understanding of both neural network methods and reinforcement learning techniques.
3. Acquire expert knowledge on gate decompositions with different native gate sets and on variational quantum algorithms.
Contact: Zoltan Zimboras (zimboras.zoltan at wigner.hu)
Project 8. Machine learning for simulation in quantum materials discovery
We are seeking candidates to contribute to our research project with the following objectives. Firstly, to develop Machine Learning universal models capable of predicting specific quantum mechanical
properties of molecules, solids, and quantum materials with maximum data efficiency. Secondly, to apply these new Machine Learning models to quantum materials discovery. This position offers an
exciting opportunity to contribute to the development of efficient Machine Learning models for quantum materials discovery and gain expertise in cutting-edge computational methods and technologies.
1. Construct databases and descriptors for supervised learning for use in material discovery with specified sets of electronic structure methods.
2. Validate and test Machine Learning potentials.
3. Enable the fast and precise prediction of new materials with optimized required electronic properties.
Scientific Training Goals:
1. Gain knowledge and expertise in high-throughput ab-initio methods and technologies.
2. Gain knowledge and expertise in the development of Machine Learning interatomic potentials and density functionals.
3. Gain knowledge and expertise on a variety of electronic structure computational methods, both wave function-based and density-based, complemented by Machine Learning, to predict molecular
4. Use and develop cloud-based material databases.
Contact: Guido Goldoni (guido.goldoni at unimore.it)
Project 9. Quantum algorithms, including quantum machine learning, for the prediction of materials phase equilibria and molecular energies
We are seeking candidates to contribute to our research project with the following objectives: improving the prediction of quantum properties of molecules using quantum algorithms and developing and
applying quantum machine learning algorithms for phase equilibria prediction. This position offers an exciting opportunity to contribute to the advancement of quantum algorithms for predicting
molecular properties and the development of quantum machine learning approaches for phase equilibria prediction.
1. Predicting materials phases and phase transitions through computations based on quantum machine learning.
2. Improving the precision of predicting quantum properties (electronic structure, optical response, transport, etc.) of large molecules by complementing electronic structure calculations with
machine learning approaches.
Scientific Training Goals:
1. Gain knowledge and expertise in formulating quadratic unconstrained binary optimization problems (QUBO) to represent different metastable configurations of selected materials and using quantum
annealers to solve these problems.
2. Gain knowledge and expertise in translating classical machine learning algorithms for materials optimization to QUBO formulations and using quantum optimization algorithms for materials
optimization for specific applications.
3. Gain knowledge and expertise in a variety of electronic structure computational methods, both wave function-based and density-based, complemented by machine learning, to predict molecular
Contact: Rosa Di Felice (difelice at usc.edu)
Project 10. Machine Learning for Quantum Information Processing in Hybrid Quantum Systems
The research project aims to optimize control of hybrid quantum systems using machine learning. Objectives include quantum state preparation, optimized measurements, and feedback protocols. Tasks
involve developing reinforcement learning strategies for state control, finding optimized measurement strategies, and exploring feedback protocols. Training goals include understanding open quantum
systems theory, mastering numerical simulations, and integrating technical skills with scientific knowledge. Expected results include ML-based strategies for state control, optimized measurements,
and adaptive control techniques.
1. Develop continuous reinforcement learning strategies for state preparation and stabilization in multipartite systems of linear and nonlinear oscillators (cavities, mechanical modes, qubits,
2. Find optimized measurement strategies beyond the standard quantum limit for amplitude and phase estimation.
3. Explore how feedback protocols can improve and/or speed up the protocols developed above.
Scientific Training Goals:
1. Acquire knowledge about the theory of open quantum systems with a focus on noise and decoherence in hybrid quantum systems.
2. Achieve competence in numerical simulations of quantum trajectories using classical stochastic algorithms to generate training data for neural networks and learn machine learning techniques.
3. Combine the technical skills acquired under goal 2 with the scientific knowledge gained in goal 1 to develop protocols for quantum information processing.
Contact: Anja Metelmann (anja.metelmann at kit.edu)
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vertical movement physics
There are three planes of movement: Sagittal plane - a vertical plane that divides the body into left and right sides. This article contains an example physics engine which can be used in a Scratch
project. A pitch of a body with initial velocity in the vertical direction is called the Vertical Throw. Vertical motion is referred as the movement of the object against the gravitational pull. 1.
Yet in actuality, gravity causes the cannonball to accelerate downwards at a rate of 9.8 m/s/s. We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the
Basic Physics Concepts. Unit 04 "Vertical Motion" Special Place. If there were no gravity, the cannonball would continue in motion at 20 m/s in the horizontal direction. Suppose that the cannonball
is launched horizontally with no upward angle whatsoever and with an initial speed of 20 m/s. By continuing to use the site, you agree to the use of cookies. The horizontal motion of the projectile
is the result of the tendency of any object in motion to remain in motion at constant velocity. This is to say that the vertical velocity changes by 9.8 m/s each second and the horizontal velocity
never changes. Projectiles travel with a parabolic trajectory due to the influence of gravity. You can think of a rotor as a fan, because they work pretty much the same. In this live Grade 12
Physical Sciences show we discuss Projectile Motion (Equations & Graphs). There is no acceleration … Lab 02: Vertical Motion Webquest. movement physically results from the gradient force of
per-turbation pressures. These same two concepts could be depicted by a table illustrating how the x- and y-component of the velocity vary with time. These same two concepts could be depicted by a
table illustrating how the x- and y-component of the velocity vary with time. For example, a passenger sitting on a moving bus is at rest because he is not changing his position with respect to other
passengers or objects on the bus. We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts. To do 6 min read. The cable attached to
the elevator exerts an upward force of 18 000N. If a vector diagram (showing the velocity of the cannonball at 1-second intervals of time) is used to represent how the x- and y-components of the
velocity of the cannonball is changing with time, then x- and y- velocity vectors could be drawn and their magnitudes labeled. Kinematic Equations for a Projectile Horizontal motion Vertical motion a
x = 0 a y = g = - 10 m/s 2 t x v x v y = v oy + g t x = v x t 2 2 1 gt t v y oy Notice the minus sign in the equations in the right column. Spinning blades push air down. Not enough information to
draw a con- clusion. Vertical movement of the chart display depends on the selected storage cycle, and can be adjusted for mm/h, time/grid or storage cycle during parameter adjustment. E- None of the
above choices are correct. © 1996-2021 The Physics Classroom, All rights reserved. Two objects m1 and m2 each with a mass of 6 kg and 9 kg separated by a distance of 5... 1. Graphs of Vertical
Projectile Motion. Introduction to Free Fall. There is a vertical acceleration caused by gravity; its value is 9.8 m/s/s, down. As it moves upwards vertically its velocity reduces gradually under the
influence of earth’s gravity working towards the opposite direction of the ball’s motion. Vertical motion is any type of upwards or downwards motion that is constant. Along y-axis: uniform
acceleration, responsible for the vertical (downwards) motion of the particle. 1 second later, from the same position, Ball B is thrown vertically upward at … Some examples of circular motion are a
ball tied to a string and swung in a circle, a car taking a curve on a track etc. Linear motion at an angle downward. The horizontal and vertical motion are independent of each other and are only
linked by time. Bestimmen Sie die Lagrange-Gleichungen. 1. Why, because the equations of rectilinear motion can be applied to any motion in a straight line with constant acceleration. Related
Resources. In most of the cases of projectile motion, the vertical component is due to the action of gravity. (d) Use Equation 4.20 to write an expression for R in terms of vi, E, the charge and mass
of the proton, and the angle u. By using this website, you agree to our use of cookies. Vertical Motion Problems 1 Just prior to satellite recovery in the ocean, a parachute attached to a 250 kg
satellite exerts a retarding force of 1600. Vertical Motion. The horizontal motion of a projectile is independent of its vertical motion. Vibratory motion “The motion of a body about its mean
position is known as vibratory motion.” Consider a baby in a swing as shown in the above figure. Since the acceleration g and the initial vertical velocity v oy are in opposite directions, we must
give one of them a negative sign, and here we’ve chosen to make g negative. Lab 01: Galielo's Theory . The diagram below reveals the answers to these questions. Example: Moving water in a river. The
state of rest or motion of a body is relative. Motion of a Particle in a Vertical Circle (i) Minimum value of velocity at the highest point (i.e. The object moves from equilibrium point to the... 1
second later, from the same position, Ball B is thrown vertically upward at the same path, with the speed of. (d) Use Equation 4.20 to write an expression for R in terms of vi, E, the charge and mass
of the proton, and the angle u. A sailor drops a wrench from the top of a sailboat's vertical mast while the boat is moving rapidly and steadily straight forward. The motion of the wheel about its
axis and that of the steering wheel are examples of rotatory motion. (e) Find the two possible values of the angle u. ), Wanted : The height of ball B when it encounters ball A (h), 10x + ½ (-10) x2
= 25 (x-1) + ½ (-10) (x-1)2, – 5x2 + 5x2 + 10x – 25x – 10x + 25 + 5 = 0, Time interval ball A in air before it encounters ball B = 1.2 seconds. Some examples of circular motion are a ball tied to a
string and swung in a circle, a car taking a curve on a track etc. 0.5 HP 3. Drones use rotors for propulsion and control. Since the acceleration g and the initial vertical velocity v oy are in
opposite directions, we must give one of them a negative sign, and here we’ve chosen to make g negative. An understanding of the physics of the vertical jump can help you maximize your jump height.
4. The equations of rectilinear motion that you learnt about in Grade 10 can be used for vertical projectile motion, with acceleration from gravity: \(\vec{a}=\vec{g}\). As it is pushed, the swing
moves back and … 3. Vectors - Motion and Forces in Two Dimensions - Lesson 2 - Projectile Motion. In the developing stage, unsaturated air above cloud top moves upwards with small vertical
velocities. A projectile is any object upon which the only force is gravity. University Physics Book: Calculus-Based Physics (Schnick) Volume A: Kinetics, Statics, and Thermodynamics ... One of the
interesting things about projectile motion is that the horizontal motion is independent of the vertical motion. University Physics Book: Calculus-Based Physics (Schnick) Volume A: Kinetics, Statics,
and Thermodynamics ... One of the interesting things about projectile motion is that the horizontal motion is independent of the vertical motion. The vertical jump test is a common test of leg power.
Upward motion is not limited to regions of saturation. 1. Advertisement Finally, the symmetrical nature of the projectile's motion can be seen in the diagram above: the vertical speed one second
before reaching its peak is the same as the vertical speed one second after falling from its peak. 5. Force of gravity and gravitational field – problems and solutions, Parabolic motion, work and
kinetic energy, linear momentum, linear and angular motion – problems and solutions, Transverse waves – problems and solutions, Speed of the mechanical waves – problems and solutions, Simple harmonic
motion – problems and solutions. 53:54. You will learn how the numerical values of the x- and y-components of the velocity and displacement change with time (or remain constant). Projectiles That are
Not Launched. In this lesson we do calculations from graphs of motion representing vertical projectile motion. A vertical force causes a vertical acceleration - in this case, an acceleration of 9.8 m
/s/s. The important concept depicted in the above vector diagram is that the horizontal velocity remains constant during the course of the trajectory and the vertical velocity changes by 9.8 m/s
every second. 1. In solving the problem of vertical motion, the vector quantity which direction upward is given a positive sign, the vector quantity that direction downward is given a negative sign.
It is a type of motion in which the distance of the body remains constant from a fixed plane. The above diagrams, tables, and discussion pertain to how the horizontal and vertical components of the
velocity vector change with time during the course of projectile's trajectory. Kinematic Equations for a Projectile Horizontal motion Vertical motion a x = 0 a y = g = - 10 m/s 2 t x v x v y = v oy +
g t x = v x t 2 2 1 gt t v y oy Notice the minus sign in the equations in the right column. Curved motion downward with initial horizontal velocity. Physics Page. The numerical information in both
the diagram and the table above further illustrate the two key principles of projectile motion - there is a horizontal velocity that is constant and a vertical velocity that changes by 9.8 m/s each
second. Vertical motion – problems and solutions. The numerical description of the displacement of a projectile is discussed in the next section of Lesson 2. Such a diagram is shown below. It is nice
to be able to get advice on how to reduce your body's aches and pains as well as how to do simple and quick exercise at the comfort of your own home to improve the health of your body. But what if
the projectile is launched upward at an angle to the horizontal? Ball A threw vertically upward with the speed of 10 m/s. Its trajectory is rectilinear and vertical, and due to gravity, the movement
is classified as Uniformly Varied. July 23, 2015. by Nipun. Now in this unit we will apply both kinematic principles and Newton's laws of motion to understand and explain … (c) Argue that Equation
4.20 would be applicable to the protons in this situation. In physics, engineering and construction, the direction designated as vertical is usually that along which a plumb-bob hangs. Physics
engines are commonly designed once and used for many projects or borrowed by other users to save the time of recreating one. s $^{-2}$}\)directed downwards towards the centre of the earth. To find If
we drop a feather and a stone of the same height, we observe that the stone will reach the ground first. To reach a maximum jump height an athlete tries to vertically accelerate his body as fast as
possible. In Unit 1 of the Physics Classroom Tutorial, we learned a variety of means to describe the 1-dimensional motion of objects. Linear and Vertical Motion review Sheet Honors Physics Oct 28/29,
2020 Terms/Concepts to know Define and give an example of each : Kinematics Definition: The branch of mechanics that studies the motion of a body or a system of bodies without consideration given to
its mass or the forces acting on it. Calculate the satellite’s acceleration. Linear and Vertical Motion review Sheet Honors Physics Oct 28/29, 2020 Terms/Concepts to know Define and give an example
of each : Kinematics Definition: The branch of mechanics that studies the motion of a body or a system of bodies without consideration given to its mass or the forces acting on it. We will learn that
all projectiles fall freely under gravity and accelerate at 'g' whether they are moving up or down. Physics, 10.01.2021 17:40 tiannacorreastamas (b) What analysis model describes the vertical motion
of the protons above the plane? Physical Sciences / Grade 12 / Vertical Projectile Motion in 1D. It is also useful knowing the physics to understand the formula used to calculate jump height from the
time of the jump, one of the several methods you can use to measure vertical jump height. Physics. The vertical velocity of a projectile changes by 9.8 m/s each second. 1. The distance between the
two troughs of the water surface waves is 20 m. An object floats on the surface of... 1. For every object, the gravitational force causes a constant acceleration of 32.2 ft/s2 or 9.8 m/s2 towards the
Earth. This article contains an example physics engine which can be used in a Scratch project. 2. pma-online.de Der Vorschub der Diagramm-Darstellungen is t vom e ingestellten Speicherzyklus
abhängig, die Vorschubanzeige kann in mm/h, Zeit/ Raster oder Speicherzyklus in der Parametrierung eingestellt werden. Here we will be discussing a special type of motion known as vertical circular
motion. (adsbygoogle = window.adsbygoogle || []).push({}); There are no horizontal forces acting upon projectiles and thus no horizontal acceleration. A projectile that is not launched is one that
has no initial vertical velocity. Notes 02 & 03: Introduction to Vertical Motion. For every object, the gravitational force causes a constant acceleration of 32.2 ft/s2 or 9.8 m/s2 towards the Earth.
Motion in Vertical Circle | Physics – Motion in a Plane. The diagram depicts an object launched upward with a velocity of 75.7 m/s at an angle of 15 degrees above the horizontal. In other words,
during upward movement, the ball is moving with retardation. Revision Video . The horizontal velocity of a projectile is constant (a never changing in value), There is a vertical acceleration caused
by gravity; its value is 9.8 m/s/s, down, The vertical velocity of a projectile changes by 9.8 m/s each second, The horizontal motion of a projectile is independent of its vertical motion. It is
further classified as uniform and non-uniform circular motion. To solve for instantaneous velocity we will need to take the derivative of our position function. Vertical Projectile Motion. (i)
Minimum value of velocity at the highest point (i.e. COVID update: Vertical Movement has updated their hours and services. Vertical motion straight down. at point C) is. Physics engines are commonly
designed once and used for many projects or borrowed by other users to save the time of recreating one. Vertical Projectile Motion. The human body is no different to any other object, and must follow
the laws of physics, and when jumping, Newton's laws of motion apply. On a smaller scale, a mountain to one side may deflect the plumb bob away from the true zenith. Physics » One-Dimensional Motion
» Vertical Projectile Motion. Notes 04: Dollar Bill Trick Video. A pitch of a body with initial velocity in the vertical direction is called the Vertical Throw. at point C) is \(\sqrt{g r}\) (ii) The
minimum velocity at the bottom required to complete the circle, i.e. Please give a complete description of your method of solution since partial credit will be given. So we think the heavier the
body, the faster it will fall. Home » Science » Physics » How to Solve Vertical Circular Motion Problems. This acceleration acts vertically downward. Vertical Motion – numericals (for AP physics,
JEE, NEET, WBJEE) 3) A parachutist descending at a constant rate of 2.0 m/s drops a smoke canister at a height of 300 m. Find the time for the smoke canister to reach the ground and its velocity when
it strikes the ground. When a ball is thrown vertically upward it starts its vertical motion with an initial velocity. Projectiles That are Not Launched. Vertical movement. View Lab Report - 4.12
Physics Lab from PHYSICS 1 at Marathon High School. Revision Video . 6 min read. Ignore frictional losses and assume that the 0.760 m mass of the ball is distributed uniformly. How to Solve Vertical
Circular Motion Problems. Which forms of motion do they describe? It is a non homogeneous, non spherical, knobby planet in motion, and the vertical not only need not lie along a radial, it may even
be curved and be varying with time. Consider again the cannonball launched by a cannon from the top of a very high cliff. home / study / engineering / mechanical engineering / mechanical engineering
definitions / vertical motion. 10154 | 32 | 4. In Unit 2 of the Physics Classroom Tutorial, we learned how Newton's laws help to explain the motion (and specifically, the changes in the state of
motion) of objects that are either at rest or moving in 1-dimension. Vertical Motion : Vertical motion is the movement of an object in a straight line. Curved motion downward with initial ver- tical
velocity. The horizontal and vertical motion are independent of each other and are only linked by time. A physics engine is a scripting method used for creating the physics, or movement, in a
project.This movement can include jumping, side-motion, wall-jumping, and more. (Don’t miss the 5th problem on the final page.) he tangential acceleration can be found with this equation (I'll skip
how I got it): Vertical motion – problems and solutions. An object vibrates with a frequency of 5 Hz to rightward and leftward. Accelerations in the horizontal projectile motion and vertical
projectile motion of a particle: When a particle is projected in the air with some speed, the only force acting on it during its time in the air is the acceleration due to gravity (g). A ball is
thrown from the top of a building with an initial speed of 8 m/s at an angle of... 1. Projectile Motion. Vertical Motion PT looks forward to the opportunity to guide you to lasting Physical Therapy
results! A- at the base of the mast. The path of a projectile is parabolic and is referred to as its trajectory. Another vector quantity that can be discussed is the displacement. In this live Grade
12 Physical Sciences show we take a look at Graphs of Motion. Horizontal Motion Definition In physics, the projectile motion is broken into two components: horizontal and vertical. vertical mobility
synonyms, vertical mobility pronunciation, vertical mobility translation, English dictionary definition of vertical mobility. In this chapter we are going to look at the motion of objects that are
either projected, thrown, or shot directly into the air, be it vertically upwards, downwards or when objects are dropped. For such an initial velocity, the object would initially be moving 19.6 m/s,
upward and 73.1 m/s, rightward. Ball A threw vertically upward with the speed of 10 m/s. At a Point P , $ \\displaystyle T – mg cos\\theta = \\frac{m v^2}{r} $ At highest Point C , θ = 180° $ \\
displaystyle T_c – mg cos180 = \\frac{m v^2}{r} $ $ … Continue reading "Vertical Circular Motion" D- on the windward side of the base of the mast. We will use simple equations of motion to describe
their motion. As you proceed through this part of Lesson 2, pay careful attention to how a conceptual understanding of projectiles translates into a numerical understanding. The horizontal velocity
of a projectile is constant (a never changing in value). 6. The path of a projectile is parabolic and is referred to as its trajectory. How to Solve Vertical Circular Motion Problems. And finally,
the velocity of the ball becomes zero at a height. The faster the rotors spin, the greater the lift, and vice-versa. Gravity acts to influence the vertical motion of the projectile, thus causing a
vertical acceleration. In a vertical motion problem, you may be asked about instantaneous velocity, and/or average velocity. Projectile motion is the motion of an object thrown or projected into the
air, subject to only the acceleration of gravity. Trajectory - Horizontally Launched Projectiles Questions, Vectors - Motion and Forces in Two Dimensions, Circular, Satellite, and Rotational Motion,
Horizontally Launched Projectile Problems, Non-Horizontally Launched Projectile Problems, there is a vertical force acting upon a projectile but no horizontal force, Continue for a Discussion of
Displacement. Wood lath: Cross section about 5 mm x 5 mm, length about 1 m. Airfoil about 15 cm x 60 cm. and vertical motion consistent with the new state is then generated by an explicit adjustment,
or ‘spin-up’, of the model. Vertical Motion – numericals (for AP physics, JEE, NEET, WBJEE) 3) A parachutist descending at a constant rate of 2.0 m/s drops a smoke canister at a height of 300 m. Find
the time for the smoke canister to reach the ground and its velocity when it strikes the ground. Vertical Jump Physics . As the projectile rises towards its peak, it is slowing down (19.6 m/s to 9.8
m/s to 0 m/s); and as it falls from its peak, it is speeding up (0 m/s to 9.8 m/s to 19.6 m/s to ...). Physical Sciences / Grade 12 / Vertical Projectile Motion in 1D. A physics engine is a scripting
method used for creating the physics, or movement, in a project.This movement can include jumping, side-motion, wall-jumping, and more. In this lesson we do calculations from graphs of motion
representing vertical projectile motion. This is indeed consistent with the fact that there is a vertical force acting upon a projectile but no horizontal force. Primary qualification factors in
promotions are the employee’s record of performance in his or her present job and projected performance in the advanced position. The tension force of the rope is... 1. B- ahead of the base of the
mast . 10154 | 32 | 4. 1 second later, from the same position, Ball B is thrown vertically upward … Time interval ball B in air before it encounters ball A = 1.2 seconds – 1 seconds = 0.2 seconds.
Viewed from the aircraft the airfoil deflects the horizontal flow of the air down-wards. Physics Chemistry Statistics Economics Accounting Computer Science. Chapter 3: Vertical projectile motion in
one dimension 3.1 Introduction (ESCJV). What is the height of ball B when it encounters ball A. 7 reviews of Vertical Movement "What a great website for our body and soul. Notes. Grade 12 | Learn
Xtra Live 2014. n sociol the movement of individuals or groups to positions in society that involve a change in class, status, and power. Problem Set. Again, the important concept depicted in the
above diagram is that the horizontal velocity remains constant during the course of the trajectory and the vertical velocity changes by 9.8 m/s every second. Where will the wrench hit the deck?
Related Resources. A projectile is an object upon which the only force is gravity. Graphs of Vertical Projectile Motion. In this portion of Lesson 2 you will learn how to describe the motion of
projectiles numerically. We can describe vertical projectile motion through a series of graphs: position, velocity and acceleration versus time graphs. In physics, the motion of a projectile can be
broken down into two components: horizontal and vertical.In most cases, the vertical component is the motion caused by gravity acting on the object. Notes 01: Dropping a Penny Reading. The motion of
the earth about its geographic axis that causes day and night is rotatory motion. Physics, 10.01.2021 17:40 tiannacorreastamas (b) What analysis model describes the vertical motion of the protons
above the plane? A boat crosses a river of width 158 m in which the current has a uniform speed of 2.29 m/s. work to move the object when recognizing the amount of gravity and force on the object.
The numerical information in both the diagram and the table above illustrate identical points - a projectile has a vertical acceleration of 9.8 m/s/s, downward and no horizontal acceleration. 2 An
elevator, including passengers, has a mass of 1500 kg. The translational speed of the ball is 3.50 m/s at the bottom of the rise. July 23, 2015. by Nipun. This is called as vertical velocity. Physics
Multiple Choice Question!!? It is a type of motion in which the distance of the body remains constant from a fixed plane. This means that the vertical velocity is changing by 9.8 m/s every second.
Motion of a Particle in a Vertical Circle. Due to the absence of horizontal forces, a projectile remains in motion with a constant horizontal velocity. Vertical circular motion with variable speed:
Let mass of the particle is m and radius of circular path is r . Both A and D 4. Air Resistance and Gravity Review. Home. So far in Lesson 2 you have learned the following conceptual notions about
projectiles. Revision Video . In this article, we will look at how to solve vertical circular motion problems. Because vertical movement in the organization is an emotionally charged event,
management’s objective in promotion is always to identify and select the best qualified candidate. Define vertical mobility. The speed of the transverse wave on a 25 meters rope is 50 m/s. Physics
UN2801 Professor Christ Final Examination (Practice) 12/15/2020 Answer three (3) of the following five (5) questions. 53:54. Its trajectory is rectilinear and vertical, and due to gravity, the
movement is classified as Uniformly Varied. (c) Argue that Equation 4.20 would be applicable to the protons in this situation. Here we will be discussing a special type of motion known as vertical
circular motion. Right: Vertical movement of the cardboard indicating vertical movement of the air induced by the airfoil. The lengths of the vector arrows are representative of the magnitudes of
that quantity. marble) moving on a frictionless, vertical guide of equation y = f(x) under the gravity force. Example: Moving water in a river. Planes and axes of movement Planes of movement.
Vertical circular motion with variable speed: Let mass of the particle is m and radius of circular path is r . Acceleration due to gravity (g) = -10 m/s2 (given minus sign because the direction of
gravity is downward Surroundings are the places in their neighborhood where various objects are present. (Wir betrachten ein Doppelpendel wie in Abbildung 2 dargestellt. Then again it starts falling
downwards vertically and this tim… The height of ball A when it encounters ball B : h = vo t + ½ g t2 = (10)(1.2) + 1/2 (-10)(1.2)2 = 12 – 5(1.44) = 12 – 7.2 = 4.8 meters. 6 min read. We use cookies
to provide you with a great experience and to help our website run effectively. Nehmen Sie dazu an, dass ‘ 1 ’ ‘ 2, m 1 ’ m 2. b) L osen Sie die Lagrangegleichungen f ur kleine Auslenkungen, d.h. f
ur das linearisierte Pro-blem. These values are x- and y-components of the initial velocity and will be discussed in more detail in the next part of this lesson. 1 second later, from the same
position, Ball B is thrown vertically upward at the same path, with the speed of 25 m/s. In this live Grade 12 Physical Sciences show we take a look at Graphs of Motion. However, if we place the
stone and quill in an airless tube (vacuum), we will notice that both objects take the same time to fall. C- behind the base of the mast. 2 Physical School of Peking University, Beijing 100871
(Received May 4, 2015; in final form November 11, 2015) ABSTRACT The complete form of the vertical vorticity tendency equation (the complete-form vorticity equation) is derived from the Ertel
potential vorticity equation to contain thermodynamic factors. These concepts are further illustrated by the diagram below for a non-horizontally launched projectile that lands at the same height as
which it is launched. How would the horizontal and vertical velocity values change with time? Recall that in freefall, an object continually experiences a downward acceleration of \(9.80\dfrac{m}{s^
2}\) but has no horizontal acceleration. Horizontal Motion Definition In physics, the projectile motion is broken into two components: horizontal and vertical. Revision Video . In most of the cases
of projectile motion, the vertical component is due to the action of gravity. Ball A threw vertically upward with the speed of 10 m/s. Summary:: To find the equation of motion of a point like
particle (eg. It is the motion that is perpendicular to the straight or flat surface. At the peak itself, the vertical velocity is 0 m/s; the velocity vector is entirely horizontal at this point in
the trajectory. A projectile that is not launched is one that has no initial vertical velocity. When an object is thrown up, it always comes down, as it is affected by the acceleration due to
gravity. (e) Find the two possible values of the angle u. At a Point P , $ \\displaystyle T – mg cos\\theta = \\frac{m v^2}{r} $ At highest Point C , θ = 180° $ \\displaystyle T_c – mg cos180 = \\
frac{m v^2}{r} $ $ … Continue reading "Vertical Circular Motion" In this article, we will look at how to solve vertical circular motion problems. The height of ball B when it encounters ball A : h =
vo t + ½ g t2 = (25)(0.2) + 1/2 (-10)(0.2)2 = 5 – 5(0.04) = 5 – 0.2 = 4.8 meters. No horizontal force is not launched is one that has no initial vertical velocity, acceleration!: Physics page. m and
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Problems in the Geometry of the Siegel-Jacobi Space
Problems in the Geometry of the Siegel-Jacobi Space
Jae-Hyun Yang
Department of Mathematics, Inha University, Incheon 22212, Republic of Korea
e-mail : jhyang@inha.ac.kr; jhyang8357@gmail.com
Received: September 5, 2023; Revised: October 25, 2023; Accepted: October 26, 2023
The Siegel-Jacobi space is a non-symmetric homogeneous space which is very important geometrically and arithmetically. In this short paper, we propose the basic problems in the geometry of the
Siegel-Jacobi space.
Keywords: Siegel-Jacobi space, Invariant metrics, Laplace operator, Invariant differential operators, Compactification
For a given fixed positive integer n, we let
be the Siegel upper half plane of degree n and let
be the symplectic group of degree n, where F(k,l) denotes the set of all k×l matrices with entries in a commutative ring F for two positive integers k and l, ^tM denotes the transposed matrix of a
matrix M and
Then Sp(n,R) acts on Hn transitively by
where M=ABCD∈Sp(n,R) and Ω∈Hn. Let
be the Siegel modular group of degree n. This group acts on Hn properly discontinuously. C. L. Siegel investigated the geometry of Hn and automorphic forms on Hn systematically. Siegel[16] found a
fundamental domain Fn for Γn\Hn and described it explicitly. Moreover he calculated the volume of Fn. We also refer to [13, 16] for some details on Fn.
For two positive integers m and n, we consider the Heisenberg group
endowed with the following multiplication law
with (λ,μκ),(λ',μ'κ')∈HR(n,m). We define the Jacobi group GJ of degree n and index m that is the semidirect product of Sp(n,R) and HR(n,m)
endowed with the following multiplication law
with M,M'∈Sp(n,R),(λ,μκ),(λ',μ'κ')∈HR(n,m) and (λ˜,μ˜)=(λ,μ)M'. Then GJ acts on Hn×C(m,n) transitively by
where M=ABCD∈Sp(n,R),(λ,μκ)∈HR(n,m) and (Ω,Z)∈Hn×C(m,n). We note that the Jacobi group GJ is not a reductive Lie group and the homogeneous space Hn×C(m,n) is not a symmetric space. From now on, for
brevity we write Hn,m=Hn×C(m,n). The homogeneous space Hn,m is called the Siegel-Jacobi space of degree n and index m.
In this short article, we propose the basic and natural problems in the geometry of the Siegel-Jacobi space.
Notations: We denote by
Q,R and C the field of rational numbers, the field of real numbers and the field of complex numbers respectively. We denote by Z the ring of integers. The symbol “:=” means that the expression on the
right is the definition of that on the left. For two positive integers k and l, F(k,l) denotes the set of all k×l matrices with entries in a commutative ring F. For a square matrix A∈F(k,k) of degree
k, σ(A) denotes the trace of A. For any M∈F(k,l)tM denotes the transpose of a matrix M. In denotes the identity matrix of degree n. For a complex matrix A, A¯ denotes the complex conjugate of A. For
a number field F, we denote by AF the ring of adeles of F. If F=Q, the subscript will be omitted.
2. Brief Review on the Geometry of the Siegel Space
We let G:=Sp(n,R) and K=U(n). The stabilizer of the action (1.1) at iIn is
Thus we get the biholomorphic map
Hn is a Hermitian symmetric manifold.
For Ω=(ωij)∈Hn, we write Ω=X+iY with X=(xij),Y=(yij) real. We put dΩ=(dωij) and dΩ¯=(dω¯ij). We also put
∂∂Ω=1+δij2∂∂ωijand∂∂Ω¯=1+δij2∂∂ ω ¯ ij.
C. L. Siegel [16] introduced the symplectic metric dsnA2 on Hn invariant under the action (1.1) of Sp(n,R) that is given by
It is known that the metric dsnA2 is a Kähler-Einstein metric. H. Maass [12] proved that its Laplace operator ΔnA is given by
is a Sp(n,R)-invariant volume element on Hn(cf.[17, p.130]).
Siegel proved the following theorem for the Siegel space (Hn,dsn12).
Theorem 2.1. (Siegel[16]). (1) There exists exactly one geodesic joining two arbitrary points Ω0,Ω1 in Hn. Let R(Ω0,Ω1) be the cross-ratio defined by
For brevity, we put R*=R(Ω0,Ω1). Then the symplectic length ρ(Ω0,Ω1) of the geodesic joining Ω0 and Ω1 is given by
(2) For M∈Sp(n,R), we set
Then R(Ω1,Ω0) and R(Ω˜1,Ω˜0) have the same eigenvalues.
(3) All geodesics are symplectic images of the special geodesics
where a1,a2,⋯,an are arbitrary positive real numbers satisfying the condition
The proof of the above theorem can be found in [16, pp.289-293].
Let D(Hn) be the algebra of all differential operators on Hn invariant under the action (1.1). Then according to Harish-Chandra [5, 6],
where D1,⋯,Dn are algebraically independent invariant differential operators on Hn. That is, D(Hn) is a commutative algebra that is finitely generated by n algebraically independent invariant
differential operators on Hn. Maass [13] found the explicit D1,⋯,Dn. Let gC be the complexification of the Lie algebra of G. It is known that D(Hn) is isomorphic to the center of the universal
enveloping algebra of gC(cf.[7]).
Example 2.2. We consider the simplest case n=1 and A=1. Let H be the Poincaré upper half plane. Let ω=x+iy∈H with x,y∈R and y>0. Then the Poincaré metric
is a SL(2,R)-invariant Kähler-Einstein metric on H. The geodesics of (H,ds2) are either straight vertical lines perpendicular to the x-axis or circular arcs perpendicular to the x-axis (half-circles
whose origin is on the x-axis). The Laplace operator Δ of (H,ds2) is given by
is a SL(2,R)-invariant volume element. The scalar curvature, i.e., the Gaussian curvature is -1. The algebra D(H) of all SL(2,R)-invariant differential operators on H is given by
The distance between two points ω1=x1+iy1 and ω2=x2+iy2 in (H,ds2) is given by
3. Basic Problems in the Geometry of the Siegel-Jacobi Space
For a coordinate (Ω,Z)∈Hn,m with Ω=(ωμν) and Z=(zkl), we put dΩ,dΩ¯,∂∂Ω,∂∂Ω¯ as before and set
Z=U+iV,U=(ukl),V=(vkl)real,dZ=(dzkl),dZ¯=(dz¯kl), ∂∂Z=∂∂z11 ⋯∂∂zm1 ⋮⋱⋮∂∂z1n ⋯∂∂zmn ,∂∂Z¯=∂∂ z ¯ 11 ⋯∂∂ z ¯ m1 ⋮⋱⋮∂∂ z ¯ 1n ⋯∂∂ z ¯ mn .
The author proved the following theorems in [18].
Theorem 3.1. For any two positive real numbers A and B,
dsn,mA,B2=Aσ(Y-1dΩY-1dΩ¯) +B{σ(Y-1tVVY-1dΩY-1dΩ¯)+σ(Y-1t(dZ)dZ¯) -σ(VY-1dΩY-1t(dZ¯))-σ(VY-1dΩ¯Y-1t(dZ))}
is a Riemannian metric on Hn,m which is invariant under the action (1.2) of GJ.
Proof. See [18, Theorem 1.1].
Theorem 3.2. The Laplace operator Δm,mA,B of the GJ-invariant metric dsn,mA,B2 is given by
Furthermore M1 and M2 are differential operators on Hn,m invariant under the action (1.2) of GJ.
Proof. See [18, Theorem 1.2].
Remark 3.3. Erik Balslev [2] developed the spectral theory of Δ1,11,1 on H1,1 for certain arithmetic subgroups of the Jacobi modular group to prove that the set of all eigenvalues of Δ1,11,1
satisfies the Weyl law.
Remark 3.4. The scalar curvature of (H1,1,ds1,1A,B2) is - and hence is independent of the parameter B. We refer to [21] for more detail.
Remark 3.5. Yang and Yin [22] showed that dsn,mA,B2 is a Kähler metric. For some applications of the invariant metric dsn,mA,B2 we refer to [22].
Now we propose the basic and natural problems.
Problem 1. Find all the geodesics of (Hn,m,dsn,mA,B2) explicitly.
Problem 2. Compute the distance between two points (Ω1,Z1) and (Ω2,Z2) of Hn,m explicitly.
Problem 3. Compute the Ricci curvature tensor and the scalar curvature of (Hn,dsn,mA,B2).
Problem 4. Find all the eigenfunctions of the Laplace operator Δn,mA,B.
Problem 5. Develop the spectral theory of Δn,mA,B.
Problem 6. Describe the algebra of all GJ-invariant differential operators on Hn,m explicitly. We refer to [19, 20, 22] for some details.
Problem 7. Find the trace formula for the Jacobi group GJ(A).
Problem 8. Discuss the behaviour of the analytic torsion of the Siegel-Jacobi space Hn,m or the arithmetic quotients of Hn,m.
We make some remarks on the above problems.
Remark 3.6. Problem 1 reduces to trying to solve a system of ordinary differential equations explicitly. If Problem 2 is solved, the distance formula would be a very beautiful one that generalizes
the distance formula ρ(Ω0,Ω1) given by Theorem 2.1 (the Siegel space case).
Remark 3.7. Problem 3 was recently solved in the case that n=1 and m is arbitrary. Precisely the scalar and Ricci curvatures of the Siegel-Jacobi space (H1,m,ds1,mA,B2)(m≥1) were completely computed
by G. Khan and J. Zhang [8, Proposition 8, pp.825–826]. Furthermore Khan and Zhang proved that (H1,m,ds1,mA,B2)(m≥1) has non-negative orthogonal anti-bisectional curvature (cf.[8, Proposition 9,
Remark 3.8. Concerning Problem 4 and Problem 5, computing eigenfunctions explicitly is a tall order, but if this can be done it will shed a lot of light onto the geometry of this space. And
understanding the spectral geometry seems to be a central question which will likely have applications in number theory and other areas.
Remark 3.9. The algebra D(Hn,m) of all GJ-invariant differential operators on Hn,m is not commutative. Concerning Problem 6, the case n=m=1 was completely solved by M. Itoh, H. Ochiai and J.-H. Yang
in 2013. They proved that the noncommutative algebra D(H1,1) is generated by four explicit generators D1,D2,D3,D4, and found the relations among those Di(1≤i≤4). For more precise statements, we refer
to [19, pp.56–58] and [20, pp.285–290]. We note that the above four generators Di(1≤i≤4) are not algebraically independent.
Remark 3.10 The solution of Problem 7 will provide lots of arithmetic properties of the Siegel-Jacobi space.
Let Γn(N) be the principal congruence subgroup of the Siegel modular group Γn. Let Xn(N):=Γn(N)\Hn be the moduli of n-dimensional principally polarized abelian varieties with level N-structure. The
Mumford school [1] found toroidal compactifications of Xn(N) which are usefully applied in the study of the geometry and arithmetic of Xn(N). D. Mumford [14] proved the Hirzebruch's Proportionality
Theorem in the non-compact case introducing a good singular Hermitian metric on an automorphic vector bundle on a smooth toroidal compactification of Xn(N) with N≥3.
We set
be the universal abelian variety. An arithmetic toroidal compactification of Xn,m(N) was intensively investigated by R. Pink [15]. D. Mumford described very nicely a toroidal compactification of the
universal elliptic curve X1,1(N) (cf.[1, pp.14–25]). The geometry of Xn,m(N) is closely related to the theory of Jacobi forms (cf. [3, 9-11]). Jacobi forms play an important role in the study of the
geometric and arithmetic of Xn,m(N). We refer to [4, 23] for the theory of Jacobi forms.
1. A. Ash, D. Mumford, M. Rapoport and Y.-S. Tai, Smooth Compactifications of Locally Symmetric Varieties, Cambridge Math. Lib., Cambridge University Press, Cambridge, 2010, x+230 pp.
2. E. Balslev, Spectral theory of the Laplacian on the modular Jacobi group manifold, preprint, Aarhus University (2012).
3. J.I. Burgos Gil, J. Kramer and U. Kühn. The Singularities of the invariant metric on the Jacobi line bundle, London Math. Soc. Lecture Note Ser., Cambridge University Press, Cambridge, 427(2016),
4. M. Eichler and D. Zagier. The Theory of Jacobi Forms, Progr. Math., 55, Birkhäuser Boston, Inc., Boston, MA, 1985, v+148 pp.
5. Harish-Chandra. Representations of a semisimple Lie group on a Banach space. I, Trans. Amer. Math. Soc., 75(1953), 185-243.
6. Harish-Chandra. The characters of semisimple Lie groups, Trans. Amer. Math. Soc., 83(1956), 98-163.
7. S. Helgason. Groups and geometric analysis, Pure Appl. Math., 113, Academic Press, Inc., Orlando, FL, 1984, xix+654 pp.
8. G. Khan and J. Zhang. A hall of statistical mirrors, Asian J. Math., 26(6)(2022), 809-846.
9. J. Kramer. A geometrical approach to the theory of Jacobi forms, Compositio Math., 79(1)(1991), 1-19.
10. J. Kramer. A geometrical approach to Jacobi forms, revisited, 2014, Oberwolfach reports from the workshop on modular forms held April 22-May 2, 2014, organized by J. H. Bruinier, A. Ichino, T.
Ikeda and Ö. Imamoglu, Report No. 22/2014. DOI: 10.4171/OWR/2014/22.
11. J. Kramer and A.-M. von Pippich. Snapshots of modern mathematics from Oberwolfach: Special Values of Zeta Functions and Areas of Triangles, Notices Amer. Math. Soc., 63(8)(2016), 917-922.
12. H. Maass. Die Differentialgleichungen in der Theorie der Siegelschen Modulfunktionen, Math. Ann., 126(1953), 44-68.
13. H. Maass. Siegel's modular forms and Dirichlet series, Lecture Notes in Math., 216, Springer-Verlag, Berlin-New York, 1971, v+328 pp.
14. D. B. Mumford. Hirzebruch's Proportionality Theorem in the Non-Compact Case, Invent. Math., 42(1977), 239-272.
15. R. Pink. Arithmetical compactification of mixed Shimura varieties, Bonner Math. Schriften, 209, [Bonn Mathematical Publications] Universität Bonn, Mathematisches Institut, Bonn, 1990, xviii+340
16. C. L. Siegel. Symplectic Geometry, Amer. J. Math., 65(1943), 1-86. Academic Press, New York and London (1964); Gesammelte Abhandlungen, no. 41, vol. II, Springer- Verlag (1966), 274-359.
17. C. L. Siegel, Topics in complex function theory. Vol. III: Abelian Functions and Modular Functions of Several Variables, Wiley Classics Lib. Wiley-Intersci. Publ., John Wiley & Sons, Inc., New
York, 1989, x+244 pp.
18. J.-H. Yang. Invariant metrics and Laplacians on Siegel-Jacobi space, J. Number Theory, 127(1)(2007), 83-102.
19. J.-H. Yang. Sectional Curvatures of the Siegel-Jacobi Space, Bull. Korean Math. Soc., 50(3)(2013), 787-799.
20. J.-H. Yang. Differential Operators for Siegel-Jacobi forms, Sci. China Math., 59(6)(2016), 1029-1050.
21. J.-H. Yang, Y.-H. Yong, S.-N. Huh, J.-H. Shin and G.-H. Min. Sectional Curvatures of the Siegel-Jacobi Space, Bull. Korean Math. Soc., 50(3)(2013), 787-799.
22. J. Yang and L. Yin. Differential Operators for Siegel-Jacobi forms, Sci. China Math., 59(6)(2016), 1029-1050.
23. C. Ziegler. Jacobi Forms of Higher Degree, Abh. Math. Sem. Univ. Hamburg, 59(1989), 191-224.
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s - Petr Khomyakov
« on: May 17, 2016, 19:14 »
In 'Iteration control parameters', you may increase 'Maximum steps' from 100 to whatever you think is reasonable, e.g., 200.
However, it might also be that your computational settings are not appropriate, e.g., the number of k-points, electron temperature or some other setting. This may cause an instability of the
self-consistent calculation. In this case, you should identify which of the computational settings (parameters) are responsible for the slow convergence.
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The earth is said to be a sphere with a radius of 6370 km.
Inside the...
The earth is said to be a sphere with a radius of 6370 km. Inside the...
The earth is said to be a sphere with a radius of 6370 km. Inside the earth there is a nucleus and its density is 11.0g/cm^3 and radius is 3000Km, the rest of earth is a mantle (density is 5.00g/cm^
The core of the nucleus and mantle coincides with the mass center of the earth. In this case, calculate the gravity(received from the earth) acting on particles with a mass of 1.00g as a function of
distance from the center of the earth.
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Order of Operations Lesson Plan - 6th Grade Math
Bell ringer
Start the lesson with a warm-up activity on slide one of the slideshow where students must choose which student solved the problem accurately. Give students a few minutes to decide who solved it
correctly. They should write their reasoning in their notebook.
Turn and talk
Students turn to their partner and share their thoughts on who is correct and why. Once each student is done sharing with their partner, come together as a class to discuss the answer.
Call on students to share what their partner said, whether they agree with them, and why. We want students to notice that Maia solved the problem by working from left to right. Dakota solved the
problem by first using multiplication and division operations, then moving to addition. Share this with the class if it was not brought up.
Why should we follow an order?
Ask students whether it is important for everybody to follow an order. Imagine what could happen if we did not follow the same order. Tell students that both Maia and Dakota should have gotten the
same answer. We want them to follow the order of operations. Tell them that it is important to follow the order of operations so that everyone gets the same answer.
What is the order of operations?
Move onto slide `2` where you will teach the order of operations using an anchor chart. You can also print this anchor chart slide to post in your class for students to reference.
What comes first? Addition or subtraction?
This is an important rule that students are likely to misinterpret. Present a problem where they have to decide whether addition or subtraction comes first.
Students tend to add before subtracting since A comes before S in PEMDAS. Talk about how when choosing between addition and subtraction, you have to go from left to right. You might want to give a
similar problem where they have to choose between multiplication and division.
Class practice
Move onto slides `3` and `4` to work through a few problems with students. A strategy for simplifying using the order of operations is to identify and underline what needs to be done at each step.
Follow the examples below as a way to teach your students these habits!
Example 1: Simplify `3 \times (6 + 8) – 10 \times 4`
Example 2: Simplify `8\div 4+7(6-2)+3^2`
Partner activity
Once you go over the two examples as a class, pass out the partner activity where students get to work through some practice problems. Let students know that they will work together on this activity
to decide which students solved the problem using the correct order of operations and that they will need to explain where the error occurred and accurately solve the problem.
Order of Operations Practice
After you’ve completed the examples with the whole class, it’s time for some independent practice! ByteLearn gives you access to tons of order of operations activities. Check out their online
practice and assign them to your students for classwork and/or homework!
Order of Operations Practice
Problem 1 of 4
<p>Simplify:</p><p>`3 + 9 × 5 + 6^2`</p>
View this practice
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