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Marissa is drawing coins from a bag that contains
5
pennies (yellow),
4
nickels (green),
5
dimes (red), and
2
quarters (blue). Test your ideas by creating a bag of coins. Use the various colors to represent pennies, nickels, dimes, and quarters.
What is the probability that she will draw a nickel? Write your answer as a fraction, as a decimal, and as a percent.
Remember, probability is the number of successful outcomes out of the total number of possible outcomes.
\frac{\text{number of nickels}}{\text{total number of coins}}
How many nickels are in the bag?
\frac{4}{16}=\frac{1}{4}
If one penny, two dimes, and one quarter are added to the bag, what is the new probability that Marissa will draw a nickel? Write your answer as a fraction, as a decimal, and as a percent.
What is the new number of nickels?
What is the new total number of coins?
In which situation is it more likely that Marissa will draw a nickel?
Compare the answer in (a) to the answer in (b).
Which answer is greater?
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Naidu, V. , Whalley, J. and Narayanan, A. (2018) Generating Rule-Based Signatures for Detecting Polymorphic Variants Using Data Mining and Sequence Alignment Approaches. Journal of Information Security, 9, 265-298. doi: 10.4236/jis.2018.94019.
D\left(I,G\right)=\sqrt{\underset{k=1}{\overset{n}{\sum }}{\left({w}_{k}\frac{d\left({I}_{k},{G}_{k}\right)}{{I}_{k}^{\mathrm{max}}-{I}_{k}^{\mathrm{min}}}\right)}^{2}}
{I}_{k}^{\mathrm{min}}
{I}_{k}^{\mathrm{max}}
{G}_{k}^{\mathrm{min}}
{G}_{k}^{\mathrm{max}}
\begin{array}{l}{d}_{num}\left({I}_{k},{G}_{k}\right)=\left\{\begin{array}{l}0,\text{}{I}_{k}\text{belongsto}{G}_{k}\hfill \\ 1,\text{otherwise}\hfill \end{array},\text{}\\ {d}_{num}\left({I}_{k},{G}_{k}\right)=\left\{\begin{array}{l}0,\text{}{G}_{k}^{\mathrm{min}}\le {I}_{k}\le {G}_{k}^{\mathrm{max}}\hfill \\ {G}_{k}^{\mathrm{min}}-{I}_{k},\text{}{I}_{k}<{G}_{k}^{\mathrm{min}}\hfill \\ {I}_{k}-{G}_{k}^{\mathrm{max}},\text{}{I}_{k}>{G}_{k}^{\mathrm{max}}\hfill \end{array}\end{array}
[1] Thompson, G.R. and Flynn, L.A. (2007) Polymorphic Malware Detection and Identification via Context-Free Grammar Homomorphism. Bell Labs Technical Journal, 12, 139-147.
[2] Harley, D. and Lee, A. (2007) The Root of All Evil?โRootkits Revealed.
http://eset.version-2.sg/softdown/manual/Whitepaper-Rootkit_Root_Of_All_Evil.pdf
[3] Mishra, U. (2010) An Introduction to Virus Scanners.
[4] Rad, B.B., Masrom, M. and Ibrahim, S. (2011) Evolution of Computer Virus Concealment and Anti-Virus Techniques: A Short Survey. International Journal of Computer Science Issues, 8, 113-121.
[5] Vinod, P., Laxmi, V. and Gaur, M.S. (2009) Survey on Malware Detection Methods. Proceedings of the 3rd Hackersโ Workshop on Computer and Internet Security, Kanpur, 17-19 March 2009, 74-79.
[6] Cesare, S. and Xiang, Y. (2010) A Fast Flowgraph Based Classification System for Packed and Polymorphic Malware on the Endhost. Proceedings of the 24th IEEE International Conference on Advanced Information Networking and Applications, Perth, 20-23 April 2010, 721-728.
[7] Cesare, S., Xiang, Y. and Zhou, W. (2013) MalwiseโAn Effective and Efficient Classification System for Packed and Polymorphic Malware. IEEE Transactions on Computers, 62, 1193-1206.
[8] Rad, B.B., Masrom, M. and Ibrahim, S. (2012) Camouflage in Malware: From Encryption to Metamorphism. International Journal of Computer Science and Network Security, 12, 74-83.
[9] Cabrera, A. and Calix, R.A. (2016) On the Anatomy of the Dynamic Behavior of Polymorphic Viruses. Proceedings of the 2016 International Conference on Collaboration Technologies and Systems, Orlando, 31 October-4 November 2016, 424-429.
https://doi.org/10.1109/CTS.2016.0081
[10] Naidu, V. and Narayanan, A. (2016) A Syntactic Approach for Detecting Viral Polymorphic Malware Variants. In: Chau, M., et al., Eds., Pacific Asia Workshop on Intelligence and Security Informatics (PAISI), LNCS 9650, Springer, Berlin, 146-165.
[11] Naidu, V. and Narayanan, A. (2016) The Effects of Using Different Substitution Matrices in a String-Matching Technique for Identifying Viral Polymorphic Malware Variants. Proceedings of the IEEE Congress on Evolutionary Computation, Vancouver, 24-29 July 2016, 2903-2910.
[12] Naidu, V. and Narayanan, A. (2016) Needleman-Wunsch and Smith-Waterman Algorithms for Identifying Viral Polymorphic Malware Variants. Proceedings of the 14th IEEE International Conference on Dependable, Autonomic and Secure Computing, Auckland, 8-12 August 2016, 326-333.
[13] Naidu, V., Whalley, J. and Narayanan, A. (2017) Exploring the Effects of Gap-Penalties in Sequence-Alignment Approach to Polymorphic Virus Detection. Journal of Information Security, 8, 296-327. https://doi.org/10.4236/jis.2017.84020
[14] Gold, E. (1967) Language Identification in the Limit. Information and Control, 5, 447-474.
[15] Xinguang, T., Miyi, D., Chunlai, S. and Xin, L. (2009) Detecting Network Intrusions by Data Mining and Variable-Length Sequence Pattern Matching. Journal of Systems Engineering and Electronics, 20, 405-411.
[16] Chen, Y., Narayanan, A., Pang, S. and Tao, B. (2012) Malicious Software Detection Using Multiple Alignment and Data Mining. Proceedings of 26th International Conference on Advanced Information Networking and Applications, Fukuoka, 26-29 March 2012, 8-14.
[17] Narayanan, A. and Chen, Y. (2013) Bio-Inspired Data Mining: Treating Malware Signatures as Biosequences. Computing Research Repository (CoRR), 1-33.
[18] Srakaew, S., Piyanuntcharatsr, W. and Adulkasem, S. (2015) On the Comparison of Malware Detection Methods Using Data Mining with Two Feature Sets. International Journal of Security and Its Applications, 9, 293-318. https://doi.org/10.14257/ijsia.2015.9.3.23
[19] Rieck, K., Holz, T., Willems, C., Dรผssel, P. and Laskov, P. (2008) Learning and Classification of Malware Behavior. In: Zamboni, D., Ed., International Conference on Detection of Intrusions and Malware, and Vulnerability Assessment, LNCS 5137, Springer, Berlin, 108-125.
[20] Singhal, P. and Raul, N. (2012) Malware Detection Module Using Machine Learning Algorithms to Assist with Centralized Security in Enterprise Networks. International Journal of Network Security & Its Applications, 4, 61-67.
[21] Naidu, V. and Narayanan, A. (2014) Further Experiments in Biocomputational Structural Analysis of Malware. 10th International Conference on Natural Computation, Xiamen, 19-21 August 2014, 605-610.
[22] Cendrowska, J. (1988) PRISM: An Algorithm for Inducing Modular Rules. International Journal of Man-Machine Studies, 27, 349-370. https://doi.org/10.1016/S0020-7373(87)80003-2
[23] Witten, I.H. More Data Mining with Weka. Class 3, Lesson 1, Decision Trees and Rules. University of Waikato, Hillcrest. http://www.cs.waikato.ac.nz/
[24] Koklu, M., Kahramanli, H. and Allahverdi, N. (2015) Applications of Rule Based Classification Techniques for Thoracic Surgery. Management, Knowledge and LearningโJoint International Conference 2015โTechnology, Innovation and Industrial Management, Bari, 27-29 May 2015, 1991-1998.
[25] Wespi, A., Dacier, M. and Debar, H. (1999) An Intrusion-Detection System Based on the Teiresias Pattern-Discovery Algorithm. IBM Thomas J. Watson Research Division.
[27] Kim, H.-A. and Karp, B. (2004) Autograph: Toward Automated, Distributed Worm Signature Detection. USENIX Security Symposium, San Diego, 286.
[28] Singh, S., Estan, C., Varghese, G. and Savage, S. (2004) Automated Worm Fingerprinting. OSDI, 4.
[29] Newsome, J., Karp, B. and Song, D. (2005) Polygraph: Automatically Generating Signatures for Polymorphic Worms. IEEE Symposium on Security and Privacy, Oakland, 8-11 May 2005, 226-241.
[30] Yegneswaran, V., Giffin, J.T., Barford, P. and Jha, S. (2005) An Architecture for Generating Semantic-Aware Signatures. 14th USENIX Security Symposium, Baltimore, 1-5 August 2005, 97-112.
[31] Wang, K., Cretu, G. and Stolfo, S.J. (2005) Anomalous Payload-Based Worm Detection and Signature Generation. In: International Workshop on Recent Advances in Intrusion Detection, Springer, Berlin, 227-246.
[32] Li, Z., Sanghi, M., Chen, Y., Kao, M.-Y. and Chavez, B. (2006) Hamsa: Fast Signature Generation for Zero-Day Polymorphic Worms with Provable Attack Resilience. IEEE Symposium on Security and Privacy, Berkeley, 21-24 May 2006, 15.
[33] Cui, W., Peinado, M., Wang, H.J. and Locasto, M.E. (2007) Shieldgen: Automatic Data Patch Generation for Unknown Vulnerabilities with Informed Probing. IEEE Symposium on Security and Privacy, Oakland, 20-23 May 2007, 252-266.
[34] Xie, Y., Yu, F., Achan, K., Panigrahy, R., Hulten, G. and Osipkov, I. (2008) Spamming Botnets: Signatures and Characteristics. ACM SIGCOMM Computer Communication Review, 38, 171-182.
[37] Wurzinger, P., Bilge, L., Holz, T., Goebel, J., Kruegel, C. and Kirda, E. (2009) Automatically Generating Models for Botnet Detection. In: European Symposium on Research in Computer Security, Springer, Berlin, 232-249.
[38] Rieck, K., Schwenk, G., Limmer, T., Holz, T. and Laskov, P. (2010) Botzilla: Detecting the Phoning Home of Malicious Software. Proceedings of the ACM Symposium on Applied Computing, Sierre, 22-26 March 2010, 1978-1984.
[39] Zhao, Y., Tang, Y., Wang, Y. and Chen, S. (2013) Generating Malware Signature Using Transcoding from Sequential Data to Amino Acid Sequence. International Conference on High Performance Computing and Simulation, Helsinki, 1-5 July 2013, 266-272.
[40] Rossow, C. and Dietrich, C.J. (2013) Provex: Detecting Botnets with Encrypted Command and Control Channels. In: International Conference on Detection of Intrusions and Malware, and Vulnerability Assessment, Springer, Berlin, 21-40.
[41] Rafique, M.Z. and Caballero, J. (2013) Firma: Malware Clustering and Network Signature Generation with Mixed Network Behaviors. In: International Workshop on Recent Advances in Intrusion Detection, Springer, Berlin, 144-163.
[43] Kirat, D. and Vigna, G. (2015) Malgene: Automatic Extraction of Malware Analysis Evasion Signature. Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security, Denver, 12-16 October 2015, 769-780.
[44] Kumar, V. and Mishra, S.K. (2013) Detection of Malware by Using Sequence Alignment Strategy and Data Mining Techniques. International Journal of Computer Applications, 61, 16-19.
[45] Prabha, A.P.M. and Kavitha, P. (2012) Malware Classification through HEX Conversion and Mining. Proceedings of International Conference on E-Governance & Cloud Computing Services, December 2012, 6-12.
[46] Salzberg, S. (1991) A Nearest Hyperrectangle Learning Method. Machine Learning, 6, 277-309.
[47] Panda, M. and Patra, M.R. (2009) Semi-Naรฏve Bayesian Method for Network Intrusion Detection System. In: Leung, C.S., et al., Eds., 16th International Conference on Neural Information Processing, Part I, LNCS 5863, Springer, Berlin, 614-621.
[48] Zaharie, D., Perian, L. and Negru, V. (2011) A View inside the Classification with Non-Nested Generalized Exemplars. IADIS European Conference Data Mining.
[49] Martin, B. (1995) Instance-Based Learning: Nearest Neighbour with Generalisation. Working Paper Series 95/18 Computer Science, Hamilton, University of Waikato, Hillcrest, 90.
[50] Wettschereck, D. and Dietterich, T.G. (1995) An Experimental Comparison of the Nearest-Neighbor and Nearest-Hyperrectangle Algorithms. Machine Learning, 19, 1-25.
[51] Garyโs Hood (2016) Online Virus Scanner. http://www.garyshood.com/virus/
[52] Linux Mint (2016) From Freedom Came Elegance. https://www.linuxmint.com/
[53] Contagio (2013) 16,800 Clean and 11,960 Malicious Files for Signature Testing and Research.
http://contagiodump.blogspot.co.nz/2013/03/16800-clean-and-11960-malicious-files.html
[54] Oracle VM VirtualBox (2016) VirtualBox. https://www.virtualbox.org/
[55] JS. Cassandra by Second Part to Hell (2015) rRlF#4 (Redemption).
[56] Viruses (2004) Second Part to Hellโs Artworks VIRUSES.
http://spth.virii.lu/Cassandra-testset.rar
[57] ClamavNet (2015) ClamAV Is an Open Source Antivirus Engine for Detecting Trojans, Viruses, Malware & Other Malicious Threats. https://www.clamav.net/
[58] Weka 3: Data Mining Software in Java (2016) Weka 3โData Mining with Open Source Machine Learning Software in Java. http://www.cs.waikato.ac.nz/ml/weka/
[59] JAligner (2010) JAligner: Java Implementation of the Smith-Waterman Algorithm for Biological Sequence AlignmentโSourceForge. http://jaligner.sourceforge.net/
[61] Katoh, K., Misawa, K., Kuma, K. and Miyata, T. (2002) MAFFT: A Novel Method for Rapid Multiple Sequence Alignment Based on Fast Fourier Transform. Nucleic Acids Research, 30, 3059-3066. https://doi.org/10.1093/nar/gkf436
[62] Katoh, K. and Standley, D.M. (2013) MAFFT Multiple Sequence Alignment Software Version 7: Improvements in Performance and Usability. Molecular Biology and Evolution, 30, 772-780.
[63] MAFFT Alignment and NJ/UPGMA Phylogeny (2016) MAFFT Version 7.
http://mafft.cbrc.jp/alignment/server/index.html
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Define Custom Deep Learning Layer with Learnable Parameters - MATLAB & Simulink - MathWorks Benelux
This example shows how to create a PReLU layer, which is a layer with a learnable parameter and use it in a convolutional neural network. A PReLU layer performs a threshold operation, where for each channel, any input value less than zero is multiplied by a scalar learned at training time.[1] For values less than zero, a PReLU layer applies scaling coefficients
{\alpha }_{i}
f\left({x}_{i}\right)=\left\{\begin{array}{cc}{x}_{i}& \text{if }{x}_{i}>0\\ {\alpha }_{i}{x}_{i}& \text{if }{x}_{i}\le 0\end{array}
{x}_{i}
{\alpha }_{i}
{\alpha }_{i}
f\left({x}_{i}\right)
. The PReLU layer does not require memory or a different forward function for training, so you can remove the forward function from the class file. Add a comment to the top of the function that explains the syntaxes of the function.
|
\left(m+1\right)
-dimensional spacelike parallel
{p}_{i}
-equidistant ruled surfaces in the Minkowski space
{R}_{1}^{n}
Masal, Melek (2006)
\left(n-1\right)
-dimensional generalized null scrolls in
{R}_{1}^{n}
Balgetir, Handan, Ergรผt, Mahmut (2003)
A characterization for isometries and conformal mappings of pseudo-Riemannian manifolds.
Jan Peleska (1984)
A class of metrics on tangent bundles of pseudo-Riemannian manifolds
H. M. Dida, A. Ikemakhen (2011)
We provide the tangent bundle
TM
of pseudo-Riemannian manifold
\left(M,g\right)
with the Sasaki metric
{g}^{s}
and the neutral metric
{g}^{n}
. First we show that the holonomy group
{H}^{s}
\left(TM,{g}^{s}\right)
contains the one of
\left(M,g\right)
. What allows us to show that if
\left(TM,{g}^{s}\right)
is indecomposable reducible, then the basis manifold
\left(M,g\right)
is also indecomposable-reducible. We determine completely the holonomy group of
\left(TM,{g}^{n}\right)
according to the one of
\left(M,g\right)
. Secondly we found conditions on the base manifold under which
\left(TM,{g}^{s}\right)
( respectively
\left(TM,{g}^{n}\right)
) is Kรคhlerian, locally symmetric or Einstein...
A generalization of Frenet's frame for non-degenerate quadratic forms with any index
Lionel Bรฉrard Bergery, Xavier Charuel (2001/2002)
r
r
r
r
r
A particular conformally symmetric space
M. C. Chaki, K. K. Sharma (1976)
A study on the one parameter Lorentzian spherical motions.
Tosun, M., Gungor, M.A., Hacisalihoglu, H.H., Okur, I. (2006)
About twistor spinors with zero in Lorentzian geometry.
Leitner, Felipe (2009)
Affine surfaces with planar affine normals in 3-dimensional Minkowski space
{\Re }_{1}^{3}
Manhart, Friedrich (2006)
An alternative moving frame for tubular surfaces around time-like curves in the Minkowski 3-space.
Karacan, Murat Kemal, Bukcu, Bahaddin (2007)
An example of an almost hyperbolic Hermitian manifold.
Bejan, Cornelia-Livia, Ornea, Liviu (1998)
An Osserman-type condition on g.f.f-manifolds with Lorentz metric
Letizia Brunetti (2014)
A condition of Osserman type, called the ฯ-null Osserman condition, is introduced and studied in the context of Lorentz globally framed f-manifolds. An explicit example shows the naturality of this condition in the setting of Lorentz ๐ข-manifolds. We prove that a Lorentz ๐ข-manifold with constant ฯ-sectional curvature is ฯ-null Osserman, extending a well-known result in the case of Lorentz Sasaki space forms. Then we state a characterization of a particular class of ฯ-null Osserman ๐ข-manifolds....
Area-stationary surfaces in neutral Kรคhler 4-manifolds.
Guilfoyle, Brendan, Klingenberg, Wilhelm (2008)
Bernstein type properties of two-sided hypersurfaces immersed in a Killing warped product
Antonio W. Cunha, Eudes L. de Lima, Henrique F. de Lima, Eraldo A. Lima Jr., Adriano A. Medeiros (2016)
Our purpose is to apply suitable maximum principles in order to obtain Bernstein type properties for two-sided hypersurfaces immersed with constant mean curvature in a Killing warped product
Mโฟ{ร}_{\rho }โ
, whose curvature of the base Mโฟ satisfies certain constraints and whose warping function ฯ is concave on Mโฟ. For this, we study situations in which these hypersurfaces are supposed to be either parabolic, stochastically complete or, in a more general setting, Lยน-Liouville. Rigidity results related to entire...
Bianchi identities, Yang-Mills and Higgs field produced on
{\stackrel{ห}{S}}^{\left(2\right)}M
-deformed bundle.
Stavrinos, P.C. (1996)
Biharmonic curves in Minkowski 3-space. II.
Inoguchi, Jun-Ichi (2006)
Canonical forms for
{๐ฎ}_{n-3}
-structures in pseudo-riemannian manifolds
|
The Application of Gormanโs Eigen Values to The Industrial Sewing Machineโs Needle Vibration - Difference between revisions of "Gholmy Hawary 2016a" - Scipedia
(Created page with "==Abstract== The free vibration of the sewing needle is divided into lateral free vibrations and an axial free vibration. In this work a theoretical study that concerns the f...")
m (Scipediacontent moved page Draft Content 699436700 to Gholmy Hawary 2016a)
{\textstyle I_{i}\quad {\mbox{where}}\quad i=1{\mbox{,}}2{\mbox{,}}3\quad {\mbox{and}}\quad 4}
{\displaystyle I=I_{1}\times {\left({\frac {15}{45}}\right)}^{2}+I_{2}\times {\left({\frac {25}{45}}\right)}^{2}+}
{\displaystyle I_{3}\times {\left({\frac {2}{45}}\right)}^{2}+I_{4}\times {\left({\frac {3}{45}}\right)}^{2}=}
{\displaystyle 2.4850\times {10}^{-13}\times {\left({\frac {15}{45}}\right)}^{2}+}
{\displaystyle 1.0417\times {10}^{-14}\times {\left({\frac {25}{45}}\right)}^{2}+}
{\displaystyle 7.8125\times {10}^{-16}\times {\left({\frac {2}{45}}\right)}^{2}+}
{\displaystyle 1.9175\times {10}^{-16}\times {\left({\frac {3}{45}}\right)}^{2}=}
{\displaystyle 3.0982\quad {\mbox{m}}^{4}}
{\displaystyle {\frac {\pi {\overline {\varphi }}^{4}}{64}}=3.0982\times {10}^{-14}}
{\displaystyle {\overline {\varphi }}=0.89\quad {\mbox{mm}}}
{\displaystyle f={\frac {{\beta }^{2}}{2\pi L^{2}}}{\sqrt {\frac {EI}{\rho A}}}}
{\displaystyle {\beta }^{4}={\frac {\rho A{\acute {\omega }}^{2}L^{4}}{EI}}\rightarrow }
{\textstyle \rho }
{\textstyle =7850\quad {\mbox{kg}}/{\mbox{m}}^{3}}
{\textstyle A}
{\textstyle \omega }
{\textstyle L}
{\textstyle E}
{\textstyle =206\quad {\mbox{MPa}}}
{\textstyle I}
{\textstyle \beta }
{\textstyle \xi ={\frac {X}{L}}isr(\xi )}
{\displaystyle r(\xi )=sin\beta \xi -sinh\beta \xi -\gamma (cos\beta \xi -}
{\displaystyle cosh\beta \xi )\quad {\mbox{and}}\quad (0\leqslant \xi \leqslant 1)}
{\displaystyle \gamma ={\frac {sin\beta +sin{\mbox{h}}\beta }{cos\beta +cosh\beta }}}
{\displaystyle r(\xi )=sinh\beta \xi -sin\beta \xi +\gamma (cosh\beta \xi -}
{\displaystyle cos\beta \xi )\quad {\mbox{and}}\quad (0\leqslant \xi \leqslant 1)}
{\displaystyle \gamma ={\frac {sinh\beta -sin\beta }{cos\beta -cosh\beta }}}
{\displaystyle r(\xi )=sin\beta \xi -sinh\beta \xi +\gamma (cos\beta \xi -}
{\displaystyle cosh\beta \xi )\quad {\mbox{and}}\quad (0\leqslant \xi \leqslant 1)}
{\displaystyle \gamma ={\frac {sin\beta -sinh\beta }{cos\beta -cos\beta }}}
{\textstyle L=45\quad {\mbox{mm}}=0.045\quad {\mbox{m}}}
{\textstyle {\overline {\varphi }}=0.89\quad {\mbox{mm}}=8.9\times {10}^{-4}\quad {\mbox{m}}}
{\textstyle I}
{\displaystyle I=3.0783\times {10}^{-14}\quad {\mbox{m}}^{4}}
{\displaystyle E=206\times {10}^{9}\quad {\mbox{pascal}}({\mbox{pa}})}
{\displaystyle EI=6.3413\times {10}^{-3}\quad {\mbox{m}}^{2}}
{\displaystyle \rho \times A=7850\times {\frac {\pi \times {0.89}^{2}}{4\times {1000}^{2}}}=}
{\displaystyle 4.811\times {10}^{-3}\quad {\mbox{kg}}/{\mbox{m.}}}
{\displaystyle {\begin{array}{rl}&n=1\\&f_{cs}={\frac {{3.927}^{2}}{2\pi {\left(0.045\right)}^{2}}}\cdot {\sqrt {\frac {6.3413\times {10}^{-3}}{4.811\times {10}^{-3}}}}\\&\quad =1575.35\quad {\mbox{cps}}\\&\quad =94{\mbox{,}}522\quad {\mbox{cpm}}\end{array}}}
{\displaystyle {\begin{array}{rl}&n=1\\&f_{Cf}={\frac {{1.875}^{2}}{2\pi {\left(0.041\right)}^{2}}}\cdot {\sqrt {\frac {206\times {10}^{9}\times 3.6783\times {10}^{-14}}{4.811\times {10}^{-3}}}}\\&\quad =21{\mbox{,}}548\quad {\mbox{cpm}}\end{array}}}
{\displaystyle {\begin{array}{rl}&n=1\\&\beta =0\\&F=0\\&n=2\\&f_{FF}={\frac {{4.73}^{2}}{2\pi {\left(0.045\right)}^{2}}}.{\sqrt {\frac {6.3413\times {10}^{-3}}{4.811\times {10}^{-3}}}}\\&=137{\mbox{,}}130\quad {\mbox{cpm}}\end{array}}}
{\displaystyle K^{{_{\ast }}{_{\ast }}}={\frac {{\beta }^{2}(1+cos\beta cosh\beta )}{sinh\beta \cdot cos\beta -sin\beta cosh\beta }}}
{\textstyle K^{_{\ast }}}
{\displaystyle ={\frac {{KL}^{3}}{EI}}}
{\textstyle K}
{\textstyle L}
{\textstyle E}
{\displaystyle r(\xi )=sinh\beta \xi -sin\beta \xi +\gamma (cosh\quad \beta \xi -}
{\displaystyle cos\beta \xi )\quad (0\leqslant \xi \leqslant 1)}
{\displaystyle \gamma ={\frac {-(sin\beta +sinh\beta )}{cosh\beta +cos\beta }}}
{\displaystyle \left(f_{CE})=f_{n}=f_{CF}+Q(f_{CF}-f_{CF}\right)}
{\textstyle r(\xi )}
{\textstyle \xi ={\frac {x}{L}}(dimensionless)}
{\textstyle x}
{\textstyle Q}
{\textstyle \beta }
{\textstyle F_{CS}}
{\textstyle CF}
{\textstyle CS}
{\textstyle \left(Q=1.0\right)}
{\textstyle Q\quad {\mbox{and}}\quad K^{{_{\ast }}{_{\ast }}}}
{\textstyle Q}
{\displaystyle Q=1-0.2\times {\frac {K_{a=0.8}^{{_{\ast }}{_{\ast }}}}{K^{_{\ast }}}}}
{\textstyle K_{a=0.8}^{{_{\ast }}{_{\ast }}}}
{\textstyle K^{{_{\ast }}{_{\ast }}}}
{\textstyle Q=0.8}
{\textstyle K_{\varphi =0.8}^{{_{\ast }}{_{\ast }}}=89.96}
{\displaystyle {\begin{array}{rl}&K^{_{\ast }}={\frac {2000\times {\left(0.045\right)}^{3}}{6.3413\times {10}^{-3}}}=28.74\\&\therefore Q=1-0.2\left({\frac {89.96}{28.74}}\right)=0.3740\end{array}}}
{\textstyle A_{s}}
{\displaystyle f_{CF}=94{\mbox{,}}522\quad {\mbox{SPM}}}
{\displaystyle F_{CS}=21.548\quad {\mbox{SPM}}}
{\displaystyle f_{n}(f_{CE})=94.522+0.3740(21.548-94.522)}
{\displaystyle r_{1}(\xi )=sin{\beta }_{1}\xi -sinh{\beta }_{1}\xi +B_{1}(cos{\beta }_{1}\xi -}
{\displaystyle cosh{\beta }_{1}\xi )\quad (0\leqslant \xi \leqslant 1)}
{\displaystyle r_{2}(\xi )=A_{2}sin{\beta }_{1}\theta \xi -sinh{\beta }_{1}\theta \xi +}
{\displaystyle B_{2}(cos{\beta }_{1}\theta \xi -cosh{\beta }_{1}\theta \xi )}
{\displaystyle B_{1}(cos{\beta }_{1}\mu -cosh{\beta }_{1}\mu )+A_{2}(-sin{\beta }_{1}\theta \gamma -}
{\displaystyle sinh{\beta }_{1}\theta \gamma )+B_{2}(-cos{\beta }_{1}\theta \gamma -}
{\displaystyle cosh{\beta }_{1}\theta \gamma )=sinh{\beta }_{1}\mu -sin{\beta }_{1}\mu }
{\displaystyle B_{1}(-sin{\beta }_{1}\mu -sinh{\beta }_{1}\mu )+A_{2}\theta (cos{\beta }_{1}\theta \gamma +}
{\displaystyle B_{2}\theta (-sin{\mbox{B}}_{1}\theta \gamma +sinhcosh{\beta }_{1}\theta \gamma )=}
{\displaystyle cosh{\beta }_{1}\mu -cos{\beta }_{1}\mu }
{\displaystyle B_{1}(-cos{\beta }_{1}\mu -cosh{\beta }_{1}\mu )+A_{2}{\alpha }^{4}(sin{\beta }_{1}\theta \gamma -}
{\displaystyle sinh{\beta }_{1}\theta \gamma )+B_{2}{\alpha }^{4}{\theta }^{2}(cos{\beta }_{1}\theta \delta -}
{\displaystyle cosh{\beta }_{1}\theta \delta )=sin{\beta }_{1}\mu +sinh{\beta }_{1}\mu }
{\displaystyle {\begin{array}{l}\propto ={\left({\frac {E_{2}I_{2}}{E_{1}I_{1}}}\right)}^{\frac {1}{4}}\\\varnothing ={\left({\frac {{\rho }_{2}A_{2}}{{\rho }_{1}A_{1}}}\right)}^{\frac {1}{4}}\\\theta ={\frac {\varnothing }{\propto }}\end{array}}}
{\displaystyle I_{1}=2.4850\times {10}^{-13}\quad {\mbox{m}}^{4}/I_{2}=1.0417\times {10}^{-14}\quad {\mbox{m}}^{4}}
{\displaystyle \mu ={\frac {15}{40}}=0.375/\gamma =1-\mu =0.625}
{\textstyle E_{1}I_{1}=206\times {10}^{9}\times 2.4850\times {10}^{-13}=}
{\displaystyle 0.0512}
{\displaystyle {\begin{array}{rl}&{\rho }_{1}A_{1}=7850\times {\frac {{\left(1.5\times {10}^{3}\right)}^{2}\times \pi }{4}}=0.01387\quad {\mbox{kg}}/{\mbox{m}}\\&\propto ={\left({\frac {I_{2}}{I_{1}}}\right)}^{\frac {1}{4}}={\left({\frac {1.0417\times {10}^{-14}}{2.4850\times {10}^{-13}}}\right)}^{\frac {1}{4}}=0.4524\\&\varphi ={\left({\frac {A_{2}}{A}}\right)}^{\frac {1}{4}}={\left({\frac {0.5204}{1.7663}}\right)}^{\frac {1}{4}}=0.7303\\&\theta ={\frac {0.7303}{0.4524}}=0.6196\end{array}}}
{\textstyle {\beta }_{1}=3.966}
{\displaystyle =6.244\quad {\mbox{for}}\quad n=2}
{\textstyle n=1}
{\displaystyle f={\frac {{3.966}^{2}}{2\pi \cdot {\left(0.04\right)}^{2}}}{\sqrt {\frac {0.0512}{0.01387}}}=}
{\displaystyle 3.007\quad {\mbox{cps}}=180{\mbox{,}}400\quad {\mbox{cpm}}({\mbox{SPM}})}
{\textstyle b_{0}/b_{1}=0}
{\textstyle {\beta }^{_{\ast }}(Eigenvalue)=2.2}
{\displaystyle {\mbox{the frequency equation is}}}
{\displaystyle f={\frac {{\beta }^{{_{\ast }}2}}{2\pi L^{2}}}{\sqrt {\frac {pA1}{EI1}}}=}
{\displaystyle {\frac {{2.2}^{2}}{6.78\times {\left(0.045\right)}^{2}}}{\sqrt {\frac {7850\times \pi \times {\left(1.5\times {10}^{-3}\right)}^{2}}{4\times 206\times {10}^{9}\times 2.4850\times {10}^{-13}}}}=}
{\displaystyle 1981\quad {\mbox{Sps}}=118.832\quad {\mbox{cpm}}}
{\displaystyle {\frac {{\varphi }_{2}}{{\varphi }_{1}}}={\frac {40}{45}}=0.89\quad {\mbox{and}}\quad {\varphi }_{2}=}
{\displaystyle {\varphi }_{1}\times 0.89=1.5\times 0.89=1.33\quad {\mbox{mm}}}
{\textstyle {\frac {b_{0}}{b_{1}}}={\frac {1.33}{1.5}}=0.89{\mbox{,}}\quad \therefore \quad {\beta }^{_{\ast }}}
{\textstyle f={\frac {{1.8}^{2}}{2\pi \times {0.04}^{2}}}\times {\sqrt {\frac {\rho A_{1}}{{EI}_{1}}}}=}
{\displaystyle 1678\quad n=100{\mbox{,}}678\quad {\mbox{SPM}}}
{\textstyle {\frac {x}{l}}}
|
Trigonometric Functions, Popular Questions: CBSE Class 11-science ENGLISH, English Grammar - Meritnation
Saniyya P.m asked a question
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draw the graph of sin x , 0 less than or equal to x less than or equal to 2 pi
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Aastha Sharma asked a question
rajnsicsherry... asked a question
Prove that sin(a) + sin(a+h) + sin(a+2h) +.........+ sin(a+(n-1)h) = (sin(nh/2) sin(a+(n-1)h/2)/sin(h/2)
Tessa Paul asked a question
How to find the value of sec 480 degrees???
Mohammed Salmaan asked a question
Find the value of tan 18
Shagun Garg asked a question
cos4A=1-8cos2a+8cos4a
Shivanshu Gupta asked a question
Soumyadeep Banerjee asked a question
if a sec alpha -c tan alpha=d and b sec alpha+d tan alpha=c then show that a2+b2=c2+d2
prove that cos20 cos 40 cos 60 cos 80=1/16.
The exact value of cosec10 - 4sin70
prove that tan 50 = 2 tan 10 + tan40
Snigdha Pankaj asked a question
Solve cosec (- 1410 degree)
Walid & 1 other asked a question
1.(1+cos pi/ 8) (1+cos 3pi / 8) (1+cos 5pi / 8) (cos 7pi / 8)=1/8
If A+B+C = pai, prove that
sinA + sinB + sinC = 4 cosA/2 cosB/2 cosC/2
Vinit Mishra asked a question
Prove that sin3x + sin 2x โ sinx = 4 sinx cos x/2 cos3x/2
Prove that ( cos alpha + 2 cos 3 alpha + cos 5 alpha) / cos 3 alpha + 2 cos 5 alpha + cos 7 alpha) = cos 3 alpha sec 5 alpha?
How to download the solutions of Elements book class 11th Maths ??
sinA+sinnB-sinC=4sinA/2 sinB/2 cosC/2
Abhishek Mathur asked a question
suzanasharma21... asked a question
why the function changes on 90 degree and 270 degree. for eg. cos becomes sine and tan becomes cot.Reply fast. thnx.......
Shailee Kampani asked a question
Prove the following: cosA / 1 - sinA = tan (45 + A/2)
Shruthi Keerthi asked a question
Prove:Cos18+Sin18/Cos18-Sin18 = Tan63Please provide me the answer for this question as soon as possible...
Sri Janani & 1 other asked a question
cos4 pi /8+ cos4 3pi /8+ cos4 5pi /8 +cos4 7pi/ 8=3/2 prove it fast rely plssss
If tanx= -4/3 and x is in the second quadrant; find the values of sin x/2, cos x/2 and tan x/2.
Aishwarya Banerjee asked a question
cos4 pi / 8+cos4 3pi /8 + cos4 5pi / 8 +cos4 7pi /8 =3/2
Prove that-
sin2A = cos2(A-B) + cos2B - 2cos(A-B)cosAcosB
Sir I recieved an answer from you (meritnation) but it is cos(A-B) in both the places not cos(A+B).
Siddhesh Sarage & 2 others asked a question
if A+B=45 degree.show that {1+tanA}{1+tanB}=2
Reet Chaudhary asked a question
Prove that Sin (4pi/9 + 7). Cos (pi/9 + 7) - Cos (4pi/9 + 7). Sin (pi/9 + 7) = Root3/2
in any triangleABC P.T
(b2-c2)cotA + (c2-a2)cotB + (a2-b2)cotC = 0
Srilakshmi Mahesh asked a question
sin5A= 5sinA - 20sin3A + 16sin5A
Goku Raj asked a question
findx and y cos(2x+3y)=1/2 and cos(3x+2y)=squareroot3/2
Augustinerobson... asked a question
prove that cos11+sin11/cos11-sin11=tan56
If sin ( alpha + beta) = 4/5, sin ( alpha - beta) =5/13, alpha + beta, alpha - beta, being acute angles prove that tan 2 alpha = 63/16.
prove that sin (A+B)+sin (A-B)/ cos (A+B)+cos (A-B) = tan A
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FIND THE VALUE OS COS(-1710)degree
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dhilllu asked a question
Consider a triangle with sides 3,6,8 cm respectivelty.Now if a man runs around the triangle in such a way that he is always at a distance of 1cm from the sides of triangle then how much distance will he travel.
Nimisha Gupta asked a question
What is the value of cos 1500 degrees ?
Harsh Gupta & 1 other asked a question
Jobin Reji Abraham & 3 others asked a question
Q.47. Find the value of
4 \mathrm{sin}50ยฐ-\sqrt{3} \mathrm{tan}50ยฐ
geometrically prove that :
1.sin(X-Y) = sinx.cosy-cosx-siny
2. sin(x+y)=sinx.cosy + cosx.siny
3, cos(x+y)= cosx.cosy-sinx.siny
4.tan(x+y)= tanx+tany/1-tanxtany
5. tan(x'y) =tanx-tany/1+tanxtany
If 3sinx +4cosx =5 then find sin2x
find the derivative of cotx by the first principle?
1) Sin (45ยฐ + a) * Sin (45ยฐ - a) = 1/2 Cos 2a
2) (2Cos2a + 1) / (2Cos2a - 1) = tan (60ยฐ + a) * tan (60* - a)
3) 4SinA*Sin(A +ฯ/3)*Sin(A + 2ฯ/3) = Sin3A
4) Sin10ยฐ.Sin50ยฐ.Sin60ยฐ.Sin70ยฐ = (โ3) / 16
5) Cos20ยฐ.Cos30ยฐ.Cos40ยฐ.Cos80ยฐ =(โ3) / 16
The angles of a triangle are in AP and the ratio of the number of degrees in the least to the number of radians in the greatest is 60:ฯ. Find the angles in degrees and radians.
sin^2 A/2 +sin^2 B/2 + sin^2 C/2 = 1 - 2sinA/2 sinB/2 sinC/2
prove sin2A - sin2B/sinA .cosA- sinB. cosB = tan(A+B)
Shayesta Habeeb asked a question
if sina=1/2 (x+1/x) show that sin 3a +1/2 (x^3+1/x^3)=0
Aneeta Ajith asked a question
Prove that cos10x+cos8x+3cos4x+3cos2x = 8cosxcoscube3x
cosec2o - cot2o = 1
EVALUATE cos(-1125)
sinA.cosB.cosc + cosA.sinB.cosC + sinC.cosC.cosA = sinA.sinB.sinC
Aneeta Ajith & 1 other asked a question
The set of values of x, satisfying the equation tan2(sinโ1 x) > 1 is
(A) [โ1, 1] (B)
\left[-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right]
\left(-1,1\right)-\left[-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right]
\left[-1,1\right]-\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)
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System of resistors โ lesson. Science State Board, Class 10.
A system of resistors is a combination or a group of resistors in a circuit.
The resistors can be connected in different combinations to form an electric circuit. But, there are only two basic methods of joining resistors. These are:
a) Resistors in series
b) Resistors in parallel
We can easily compute the effective resistance of resistors with different resistance values when it is connected in series and parallel.
Resistors connected in series:
In a series circuit, the electrical components are connected in a single loop, one after the other. The electric charge in a series circuit can only flow in one direction.
If the circuit is broken or disturbed at any point in the loop, the current cannot flow through the circuit, and hence electric appliances linked to it will not function.
Let the resistances of three resistors be \(R_1\), \(R_2\) and \(R_3\), connected in series, and 'I' be the current flowing through the circuit. According to Ohm's law, the potential differences \(V_1\), \(V_2\) and \(V_3\) across \(R_1\), \(R_2\) and \(R_3\), respectively, are given by
\begin{array}{l}{V}_{1}=I{R}_{1}\\ \\ {V}_{2}=I{R}_{2}\\ \\ {V}_{3}=I{R}_{3}\end{array}
Substituting the values of \(V_1\), \(V_2\) and \(V_3\) in the above equation, we get
A single resistor that can effectively replace all the resistors in the electric circuit to maintain the same current is known as an effective resistor.
Let \(R_S\) be the effective resistance of the series-combination of the resistors. Then, the (eq. 1) becomes
\(V\) \(=\) \(I\ R_S\)
On substituting the values of effective resistance \(V\) in (eq. 1), we get
In a series circuit, the effective or equivalent resistance is equal to the sum of the individual resistances of the resistors.
When '\(n\)' number of resistors with an equal resistance '\(R\)' are connected in series, their equivalent resistance is given as '\(n R\)'. Hence,
\(R_S\) \(=\) \(n R\)
From the diagram, it is deduced that the current '\(I\)' remains constant throughout the series circuit. In other words, the current in every point or junction of the circuit is the same.
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Application Stats | APWine Finance
The APWine application provides useful statistics for all users willing to interact with the protocol.
Displayed statistics#
Global Statistics#
The global statistics section provides information about the state of the APWine protocol and AMM.
Tokenized TVL#
The Tokenized TVL displays the amount of IBT (Interest Bearing Tokens) deposited in the protocol.
โน๏ธ The data displayed corresponds to the network you are currently connected to (e.g Mainnet, Polygon).
Exchange liquidity#
The Exchange Liquidity displays the amount of liquidity deposited in each pair of the AMM (including the underlying).
Current Period#
The current period displays the status of the different futures. It includes the start date, the end date (term of the future period) and shows the time remaining before the future reach maturity.
Future statistics#
The Future statistics displays an aggregation of data for each future.
The Tokenized TVL corresponds to the amount of IBT (Interest Bearing Tokens) deposited in the corresponding Future Vault.
Pools TVL#
The Pools TVL displays the amount of liquidity deposited in each pair of the corresponding AMM.
Spot APR#
The Spot APR displays the APR of the IBT given by the corresponding platform contracts.
Market APR#
The Market APR represents the APRs implied by the spot price of the derivatives on the amm.
The PT APR is derived from the PT price in the first pair of the AMM. At expiry of the future the PT token will be worth one underlying token. The spot PT price in the AMM represent therefore the discounted underlying token. From this price the discount rate is computed for the time remaining in the period. This gives the APR of the period implied by the first pair of the AMM.
P^t_{PT} = PV(P^T_{PT}) = \frac{P^T_{PT}}{1+r_{remaining}}
P^t_{PT}
the price of the PT in underlying at time
t
T
correspond to the term of the future. Hence
P^T_{PT}=1
r_{remaining}
represents the discount rate for the time remaining in the period.
\implies P^t_{PT} = \frac{1}{1+r_{remaining}} \implies r_{remaining} = \frac{1}{P^t_{PT}}-1
The APR is then derived as
APR_{PT} = r_{remaining} \times \frac{t_{1year}}{t_{remaining}}
The FYT APR is derived from the ratio of FYT and PT in the second pair of the AMM. The ratio of FYT/PT in the second pair of the AMM represent directly the APR of the period predicted by the market.
APR_{FYT} = P^t_{FYT} \times \frac{t_{1year}}{t_{remaining}}
P^t_{FYT}
the FYT price in PT at time
t
To understand how prices are computed from the AMM state follow this link.
Period Switch ยป
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(2-1)-beta-D-fructan:(2-1)-beta-D-fructan 1-beta-D-fructosyltransferase Wikipedia
(2-1)-beta-D-fructan:(2-1)-beta-D-fructan 1-beta-D-fructosyltransferase
In enzymology, a 2,1-fructan:2,1-fructan 1-fructosyltransferase (EC 2.4.1.100) is an enzyme that catalyzes the chemical reaction
[beta-D-fructosyl-(2->1)-]m + [beta-D-fructosyl-(2->1)-]n
{\displaystyle \rightleftharpoons }
[beta-D-fructosyl-(2->1)-]m-1 + [beta-D-fructosyl-(2->1)-]n+1
Thus, the two substrates of this enzyme are [[[beta-D-fructosyl-(2->1)-]]m]] and [[beta-D-fructosyl-(2->1)-]]n]], whereas its two products are [[beta-D-fructosyl-(2->1)-]]m-1]] and [[beta-D-fructosyl-(2->1)-]]n+1]].
This enzyme belongs to the family of glycosyltransferases, specifically the hexosyltransferases. The systematic name of this enzyme class is 2,1-beta-D-fructan:2,1-beta-D-fructan 1-beta-D-fructosyltransferase. Other names in common use include 1,2-beta-D-fructan 1F-fructosyltransferase, fructan:fructan fructosyl transferase, FFT, 1,2-beta-fructan 1F-fructosyltransferase, 1,2-beta-D-fructan:1,2-beta-D-fructan, 1F-beta-D-fructosyltransferase, and fructan:fructan 1-fructosyl transferase.
Henry RJ, Darbyshire B (1980). "Sucrose:sucrose fructosyltransferase and fructan:fructan fructosyltransferase from Allium cepa". Phytochemistry. 19 (6): 1017โ1020. doi:10.1016/0031-9422(80)83050-0.
Vergauwen R, Van Laere A, Van den Ende W (2003). "Properties of fructan:fructan 1-fructosyltransferases from chicory and globe thistle, two Asteracean plants storing greatly different types of inulin". Plant Physiol. 133 (1): 391โ401. doi:10.1104/pp.103.026807. PMC 196615. PMID 12970504.
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G
{}^{c}
G
3-K-contact Wolf spaces
Wลodzimierz Jelonek (2003)
The aim of this paper is to give an easy explicit description of 3-K-contact structures on SO(3)-principal fibre bundles over Wolf quaternionic Kรคhler manifolds.
A canonical connection associated with certain
G
Josรฉ M. Sierra, Antonio Valdรฉs (1997)
A Characterization of Affine Cylinders.
Barbara Opozda (1996)
A characterization of isometries between Riemannian manifolds by using development along geodesic triangles
Petri Kokkonen (2012)
In this paper we characterize the existence of Riemannian covering maps from a complete simply connected Riemannian manifold
\left(M,g\right)
onto a complete Riemannian manifold
\left(\stackrel{^}{M},\stackrel{^}{g}\right)
in terms of developing geodesic triangles of
M
\stackrel{^}{M}
. More precisely, we show that if
{A}_{0}:T{{|}_{{x}_{0}}M\to T|}_{{\stackrel{^}{x}}_{0}}\stackrel{^}{M}
is some isometric map between the tangent spaces and if for any two geodesic triangles
\gamma
\omega
M
{x}_{0}
the development through
{A}_{0}
of the composite path
\gamma ยท\omega
\stackrel{^}{M}
results in a closed path based at
{\stackrel{^}{x}}_{0}
, then there exists a Riemannian covering map...
A classification of the torsion tensors on almost contact manifolds with B-metric
Mancho Manev, Miroslava Ivanova (2014)
The space of the torsion (0,3)-tensors of the linear connections on almost contact manifolds with B-metric is decomposed in 15 orthogonal and invariant subspaces with respect to the action of the structure group. Three known connections, preserving the structure, are characterized regarding this classification.
A classification theorem for connections.
Teleman, Kostake (2000)
A cohomological description of connections and curvature over posets.
Roberts, John E., Ruzzi, Giuseppe (2006)
A complement to the paper โOn the Kolรกล connectionโ [Arch. Math. (Brno) 49 (2013), 223โ240]
W.M. Mikulski (2015)
A few remarks on the geometry of the space of leaf closures of a Riemannian foliation
Maลgorzata Jรณzefowicz, R. Wolak (2007)
The space of the closures of leaves of a Riemannian foliation is a nice topological space, a stratified singular space which can be topologically embedded in
{โ}^{k}
for k sufficiently large. In the case of Orbit Like Foliations (OLF) the smooth structure induced by the embedding and the smooth structure defined by basic functions is the same. We study geometric structures adapted to the foliation and present conditions which assure that the given structure descends to the leaf closure space. In Section...
A functorial approach to differential characters.
Turner, Paul (2004)
A generalization of the exterior product of differential forms combining Hom-valued forms
Christian Gross (1997)
This article deals with vector valued differential forms on
{C}^{\infty }
-manifolds. As a generalization of the exterior product, we introduce an operator that combines
Hom\left({โจ}^{s}\left(W\right),Z\right)
-valued forms with
Hom\left({โจ}^{s}\left(V\right),W\right)
-valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.
A generalization of the torsion form
Ivan Kolรกล (1975)
A geometrical analysis of the field equations in field theory.
Echeverrรญa-Enrรญquez, A., Muรฑoz-Lecanda, M.C., Romรกn-Roy, N. (2002)
โณ=\left(M,{๐ช}_{โณ}\right)
\nabla
{๐ช}_{โณ}\cong {\Gamma }_{\Lambda {E}^{*}}
\left(โณ,\nabla \right)
E\to M
\nabla
{๐ช}_{โณ}\cong {\Gamma }_{\Lambda {E}^{*}}
โณ
{โ}^{1|1}
โณ
E
A new cohomology on special kinds of complex manifolds
Kostadin Trenฤevski (2003)
K. Guruprasad, Shrawan Kumar (1990)
Szilasi, Jรณzsef, Vincze, Csaba (2000)
A note on characteristic classes
Jianwei Zhou (2006)
This paper studies the relationship between the sections and the Chern or Pontrjagin classes of a vector bundle by the theory of connection. Our results are natural generalizations of the Gauss-Bonnet Theorem.
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โก Angles in a triangle (each side of which is an arc of a great circle) add up to more than
180
The minimal distance between two points is known as a geodesic and is the spherical analog of a line segment: an arc of a great circle. The distance between two points is therefore
R \varphi
R
\varphi
is the measure (in radians) of the central angle subtended by the radii to the two points.
What is the minimal distance on the sphere, centered at the origin and of radius
2
, between points
\big(1, \, 1, \, \sqrt{2}\big)
\big(-1, \, 1, \, \sqrt{2}\big)?
The dot product of the vectors to the two points is
R^2 \cos \varphi = 1 \cdot (-1) + 1 \cdot 1 + \sqrt{2} \cdot \sqrt{2} = 2
\cos \varphi = \tfrac{1}{2}
\varphi = \tfrac{\pi}{3}
, and the distance along the sphere is
R \varphi = \frac{2\pi}{3}.\ _\square
The equator has a latitude of
0^\text{o}.
Find its azimuthal position as measured from the North Pole.
In order to get from the North Pole to the equator, one must travel
\frac14
of the way around the circumference of the earth, so the azimuthal position of the equator is
\frac14
360^\text{o}
90^\text{o}.
So, since latitude is measured off the equator, the convention is actually
90^\text{o}
off of a true azimuthal measurement.
_\square
The angle between two geodesics is taken to be the angle between the planes of their great circles. The measure of that angle is obtained normally, yielding a value between
0^\circ
180^\circ
If a spherical triangle has angles of measure (in radians)
\alpha
\beta
\gamma
, then it has area
(\alpha + \beta + \gamma - \pi) \cdot R^2
. All polygons may be created from triangles sharing edges.
A circle on a sphere has the same boundary as a sphere in three-dimensional space: namely, the intersection of a plane with the sphere. The interior of the circle is the portion of the sphere on one side of the boundary. A circle with radius
r
in three-dimensional space has area on the sphere
2\pi R \cdot \big(R \pm \sqrt{R^2 - r^2}\big),
where the sign is determined by whether or not a great circle is contained within the interior of the circle.
8000000\pi
16000000\pi
\frac{16000000\pi^3}{9}
2000000\pi^3
\frac{8000000\pi^3}{3}
What is the area of the region within
\frac{4000\pi}{3}
miles from the North Pole?
l
, there exist two antipodal points through which any line perpendicular to
l
must pass.
(x, \, y, \, z) \mapsto \left(\frac{x}{R - z}, \, \frac{y}{R - z}\right)
is a bijective, smooth map from points on the sphere to the real plane and the point at infinity. This projection preserves angle measures.
Three-dimensional Cartesian space could be interpreted with spherical geometry by selecting various values of the sphere radius
R = \rho
(x, \, y, \, z)
may be converted into spherical coordinates with the following formulas:
\begin{aligned} x &= \rho \sin \varphi \cos \theta \\ y &= \rho \sin \varphi \sin \theta \\ z &= \rho \cos \varphi. \end{aligned}
x^2 + y^2 + z^2 = \rho^2
\rho
represents the radius of the sphere,
\varphi
represents the angle off of the
z
\theta
x
-axis. Note that
\theta
serves the same purpose as in polar coordinates. By convention, the variables take values satisfying
\rho \ge 0
0 \le \theta < 2\pi
0 \le \varphi \le \pi
. With the exception of the origin, there is a one-to-one correspondence between
(x, \, y, \, z)
(\rho, \, \theta, \, \varphi)
Spherical coordinates are useful because they allow triple integrals to be rewritten. For instance, given a multivariable function
F
\int \int \int_E F(x, \, y, \, z) \, dV = \int \int \int_E \rho^2 \sin \varphi F(\rho \sin \varphi \cos \theta, \, \rho \sin \varphi \sin \theta, \, \rho \cos \varphi) \, d\rho \, d\theta \, d\varphi.
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Recursion is a powerful tool for solving problems that require the execution of an action multiple times until a condition is met. For many problems, a recursive solution will result in fewer lines of code and will be easier to comprehend than a solution that uses a for or while loop.
You may find that recursion is a difficult concept to wrap your head around at first. Thatโs fine! This lesson is meant as an introduction. As you see more examples and get more practice, you will start to feel comfortable with the concept.
In this lesson, you will learn to implement a recursive method that returns the factorial of a number. An integerโs factorial is the product of an integer and all positive numbers less than it.
Letโs consider 4 factorial:
4! = 4 \times 3 \times 2 \times 1 = 24
Four factorial is equal to the product of 4 x 3 x 2 x 1, which is 24. The exclamation mark denotes that the number 4 is being factorialized.
1! and 0! are both valid base cases of factorial. The factorial product of both numbers is 1.
Before we dive into recursion, you will consider how factorial is implemented with an iterative approach.
Factorial.java includes a method called .iterativeFactorial(). The method accepts an integer as an argument, and returns the factorial of it.
Take a look at how we implemented the method. Run the code when youโre ready to move to the next checkpoint.
Set an int named fourFactorial equal to the value returned from iterativeFactorial(), with 4 as the argument.
Print the value saved to fourFactorial.
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Analytic Continuation | Brilliant Math & Science Wiki
Yao Liu, Arturo Presa, Pham Khanh, and
The principle of analytic continuation is one of the most essential properties of holomorphic functions. Even though it could be stated simply and precisely as a theorem, doing so would obscure many of the subtleties and how remarkable it is. It is perhaps more instructive to take a step back to real (analytic) functions and Taylor series, and to see why complex numbers is the natural setting. Along the way, we shall encounter other fundamental concepts in complex analysis, such as branch cuts, isolated singularities (including poles), meromorphic functions, monodromy, and even Riemann surfaces.
It may serve as a prologue to a formal study of complex analysis, only assuming basic acquaintance with Taylor series and complex numbers. This largely is the perspective of Weierstrass; for a more complete view, there are Cauchy's theory based on contour integration, Riemann's geometric theory, as well as the perspective of PDE (partial differential equations).
The video Visualizing the Riemann zeta function and analytic continuation by 3Blue1Brown is excellent in giving the geometric intuition, and this article was largely written to complement it.
Natural Domains
Definitions of Holomorphic and Meromorphic Functions
Methods of Analytic Continuation
Most functions that come up "in nature" โ either in describing the (ideal) physical world or in pure mathematics โ particularly those that are given a special symbol or a name, are in fact analytic: if we take its Taylor series at any point, which only uses the data arbitrarily close to that point, we could recover the function completely. For example, knowing the function
\sin x
x\in \big[0, \frac{\pi}{2} \big]
, or even for a tiny interval, is enough to determine the entire function: Simply take its Taylor series at
x=0:
x-\frac{x^3}{3!} + \frac{x^5}{5!} - \cdots,
which converges for all
x\in\mathbb R
, and agrees with the standard, periodic definition of
\sin x
over the reals. Taking the Taylor series at any other point will result in the exact same function (see Taylor's theorem). This is the simplest and best-case scenario of analytic continuation โ from a small interval to the whole real line โ for the radius of convergence is always infinite. Such functions are called entire functions, which include all polynomials, the exponential function, certain "special functions" (e.g., Bessel functions), and their sums, products, and compositions.
The problem becomes more subtle, and hence more interesting, if the radius of convergence is finite. Suppose we only know
f(x)=\frac{1}{x}
on a small neighborhood around
x=1
. The Taylor series takes the form
\sum_{n=0}^\infty (-1)^{n} (x-1)^n, \qquad (*)
which, being a geometric series, converges only for
|x-1|<1
0<x<2
, where it indeed converges to
\frac{1}{x}
. Now, knowing the values of the function near
x=1.5
, for instance, we may "Taylor expand" at
x=1.5
, and the new Taylor series in fact converges for
x\in (0, 3)
while agreeing with the previous values on
(0, 2)
. We can say that we have analytically continued the function to
(0,3)
. Thus, by successive Taylor series expansions, we could "recover" the function
\frac{1}{x}
x>0
, but no way whatsoever could we extend it to
x<0
x=0
poses as an insurmountable barrier, called a singularity of the function. It seems that we are free to define
f(x)
to be any (analytic) function on
x<0
, and no criterion on the function could favor one over the countless others.
This is where complex numbers come into play so that we might be able to "circumvent" the barrier by going into the complex plane. In fact, the Taylor series in general (or power series) makes perfect sense, as a series, when
x
is any complex number, so long as we know how to add and multiply complex numbers. Examining the derivation of the geometric series, we see that
1 + r + r^2 + \cdots = \lim_{n\to\infty}\frac{1-r^{n+1}}{1-r}=\frac{1}{1-r}
holds for all complex numbers
r
of modulus strictly less than 1. Thus, the series
(*)
converges for all complex
|x-1|<1
, which is a disk of radius
1
x=1
. The radius of convergence is literally a radius, and this phenomenon holds true for all (convergent) power series. In particular, entire functions are naturally defined on the whole complex plane.
Now, by taking the Taylor series at a point off from the real axis, we may get around the singularity at
x=0
. There are two ways to get to the negative real axis: through the upper half of the complex plane, or through the lower half. It turns out that we'd end up with the exact same result, which as one might expect is simply
\frac{1}{x}
x<0
. In fact, as illustrated above, each Taylor series in the process of analytic continuation converges in an (open) disk just short of the singularity at
x=0
a\in\mathbb C\setminus\{0\}
, we may expand
\frac{1}{x}
as a Taylor series centered at
a
\frac{1}{x} = \frac{1}{a+(x-a)} = \frac{1}{a}\cdot \frac{1}{1+\frac{x-a}a}=\frac{1}{a}\sum_{n=0}^\infty (-1)^n\left(\frac{x-a}{a}\right)^n=\sum_{n=0}^\infty \frac{(-1)^n}{a^{n+1}}(x-a)^n
|\frac{x-a}{a}|<1
|x-a|<|a|
. This also illustrates that the precise procedure of analytic continuation (choices of the centers of Taylor series expansions) does not matter, and the end result is the same, namely
f(x)=\frac{1}{x}
on the punctured plane
\mathbb C\setminus\{0\}
It should be noted right away that not all functions, when analytically continued around a singularity from above and below, have the same result. The two prototypical examples are
\log x
\sqrt x
; they are not typically defined for negative
x
for this reason.
The complex numbers also provide more insight even in the case when we could analytically continue over the reals. For example,
f(x)=\frac{1}{1+x^2}
is defined and infinitely differentiable for all
x\in\mathbb R
. The Taylor series at
x=0
, however, has a radius of convergence of 1 (again by geometric series). If we take the complex perspective, we see that
f(x)
does have singularities at
x=\pm i
, which are at a distance 1 from the origin, so it couldn't have a larger radius of convergence. In fact, it is true in general that the Taylor series of any analytic function converges to the function itself within a disk as large as possible (before hitting a "singularity"), when viewed as a complex function.
It may already be enough evidence that analytic functions, which include all the familiar functions, really should be regarded as living on the complex plane or subsets or extensions thereof. They are not confined to a particular domain (per the modern concept of a function) but have the ability to extend or continue in all directions as far as possible, to what can be called its natural domain. Our modern definition of a function, an arbitrary assignment of a value
y
x
in a prescribed domain, has a very different flavor: no way whatsoever to extend its domain, or rather infinitely many choices of extension that do not single out any particular one. If the original function happens to be continuous, one may require the extension to be continuous too, which would narrow down the choices but still leave infinitely many possibilities (unless the extension is just for one extra point); if the original function was differentiable, one may ask the same for the extension, which would further narrow down the choices. Analyticity is the strongest criterion of all, and it turns out it is enough to single out a unique choice of extension if one exists. That is the principle of analytic continuation.
To phrase the principle of analytic continuation differently: the identity of an analytic function is "encoded" in each and every point of its natural domain, in the sequence of Taylor series coefficients (or the derivatives) at that point, traditionally known as the germ of the function at the point (in the sense of the seed of a crop). One could easily write down the rules for the basic operations โ addition, multiplication, division, inversion, differentiation, etc. โ on the set of germs at the same point. To carry on the agrarian analogy, a collection
(
often a
\mathbb C
-vector subspace
)
of germs at a point is called a stalk, and putting all the stalks (of the same sort) over various points of a domain together, endowed with some topology, we get a sheaf, which is semantically the same as a bundle. This is the beginning of sheaf theory.
From now on, we shall use
z
\big(
\zeta
s
\big)
x
for the variable of our functions.
Despite the fact that an analytic function, by its very nature, is fully determined by a sequence of (complex) numbers, the general theory of functions in the complex domain is a vast subject that goes under many names: complex analysis, (complex) function theory, theory of functions of a (single) complex variable, etc. From the point of view of analytic continuation, the most natural question
Given a convergent power series
f(z)=\sum_{n=0}^\infty a_n (z-z_0)^n,
determine the largest domain in the complex plane to which
f(z)
can be analytically continued.
is hopelessly difficult. Nevertheless, it offers a panorama of a wide variety of functions, with connections to different areas of mathematics, if we wish to look past some of the detailed justifications. In increasing level of "complexity" (by some measure), we have the following:
Entire functions: those that can be analytically continued to the whole complex plane. It generalizes polynomials. For example, the Fourier (and Laplace) transform
f(\zeta)=\int_{\mathbb R} e^{-ix\zeta}\phi(x)\,dx \qquad \zeta=\xi+i\eta\in\mathbb C
of a compactly supported continuous function
\phi\in C_0(\mathbb R)
\big(
or more generally a distribution
\phi\in\mathcal E'(\mathbb R)
of compact support
\big)
is entire, and furthermore the support of
\phi
is governed by the growth of
f(\zeta)
in the imaginary direction, i.e. as
\eta\to\pm\infty
(Paley-Wiener theorem).
Meromorphic functions on
\mathbb C
: the barriers are all isolated points (called singularities), but analytic continuation is possible around each singularity, and the result does not depend on which way to go around them. (One extra technical condition is often imposed so that all the singularities are "poles" instead of "essential singularities.") It generalizes rational functions. For any non-constant polynomial
P
n
P(x)\geq 0
x\in\mathbb R^n
, and any compactly supported smooth
\phi\in C^\infty_0(\mathbb R^n),
f(s) = \int_{\mathbb R^n} P(x)^s \phi(x)\,dx \qquad \operatorname{Re} s>0
can be analytically continued to the whole complex plane except for isolated, albeit infinitely many, points on the negative real axis (Bernstein's theorem). For another important class of examples, the so-called
L
-functions, such as the Dirichlet
L
L(s)=\sum_{n=1}^\infty \frac{\chi(n)}{n^s} \qquad \operatorname{Re} s>1
associated to a Dirichlet character
\chi:\mathbb Z\to\mathbb C
, can be analytically continued to all of
\mathbb C
except possibly for a few points such as
s=1
Functions such as
\log z
\sqrt z
could be analytically continued around the singularity at
z=0,
but the result depends on the path taken. To remove this ambiguity, one would need to agree on a continuous "borderline" or "cut" extending from
z=0
to infinity (e.g. the negative real axis), across which no analytic continuation is permitted. Due to the presence of the cut,
z=0
shall not be considered an isolated singularity even though it is the only "barrier" of analytic continuation.
(
\sqrt z
does not go to infinity when
z\to 0.)
Alternatively, we could analytically continue across the cut by "jumping" to another copy of the complex plane: Thus the natural domain of
\log z
\sqrt z
is not a subset of the complex plane but consists of multiple copies of the complex plane properly glued together; this is an example of a Riemann surface. In a sense that could be made precise, the point
z=0
is no longer a singularity of the function.
The barriers of analytic continuation may not even be isolated points but form a "wall." In fact, for any open, connected subset
U\subsetneq\mathbb C
, there exists an analytic function on
U
that cannot be extended past any point of the boundary. In other words,
U
is the natural domain of that function. This class may seem exotic, but in fact it is as rich as non-analytic functions of a real variable. To illustrate it, consider
f(z)=\int_{\mathbb R} \frac{\phi(x)}{x-z}dx \qquad z\in\mathbb C\setminus\operatorname{supp}\phi,
\phi
only needs to be integrable
\big(\phi\in L^1(\mathbb R)\big).
z
x_0
on the real axis from above and below, the limits
f(x_0\pm i\epsilon)
2\pi i\phi(x_0)
; moreover, analytic continuation across the real axis is possible in a neighborhood of
x_0
\phi
is (real) analytic at
x_0
. Thus, by choosing appropriate
\phi
, we can construct many functions on the upper (or lower) half plane that cannot be analytically continued across part or all of the real line. Another way for a function to fail to analytically continue past a boundary is when the value of the function approaches infinity along the boundary. An important class of functions of this sort is modular forms, which are defined on the upper half plane, and have very stringent transformation properties; they have deep connections with
L
-functions, and are likewise important in many areas of mathematics, most notably in number theory.
Cite as: Analytic Continuation. Brilliant.org. Retrieved from https://brilliant.org/wiki/analytical-continuation/
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if the numerator of 25/35 is 65 then its denominator is equal to____________________ plz plz fast i have a FA - Maths - Rational Numbers - 7527805 | Meritnation.com
if the numerator of 25/35 is 65 then its denominator is equal to____________________.plz plz fast i have a FA 1 TOMMORROW PLZ
\frac{25}{35}
And the simplest form of
\frac{25}{35}
\frac{5}{7}
Now multiplying both numerator and denominator of
\frac{5}{7}
by 13 , we get;
\frac{5ร13}{7ร13} = \frac{65}{91}
Therefore the denominator will be 91 when the numerator is 65.
Tanya Mondal answered this
please put a thums up for me
yes it is correct answer TANYA
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14D24 Geometric Langlands program: algebro-geometric aspects
3-Instantons et rรฉseaux de quadriques.
L. Gruson, M. Skiti (1994)
Pavel Belorousski, Rahul Pandharipande (2000)
Mario Maican (2010)
A Global Moduli Space of Ample Subvarieties on Compact Kรคhler Manifolds with Very Strongly Negative Curvature.
B. Wong (1987)
A New Compactification of the Siegel Space and Degeneration of Abelian Varieties. I.
Yukihiko Namikawa (1976)
A New Compactification of the Siegel Space and Degeneration of Abelian Varieties. II.
A note on a Brill-Noether locus over a non-hyperelliptic curve of genus 4
Sukmoon Huh (2009)
We prove that a certain Brill-Noether locus over a non-hyperelliptic curve C of genus 4, is isomorphic to the Donagi-Izadi cubic threefold in the case when the pencils of the two trigonal line bundles of C coincide.
A propos de l' espace des modules de fibrรฉs de rang 2 sur une courbe.
Yves Laszlo (1994)
A remark on moduli spaces of complete intersections.
P. Brรผckmann (1996)
A remark on the Picard group of spin moduli space
Maurizio Cornalba (1991)
We describe a number of classes in the Picard group of spin moduli space and determine the relations they satisfy; as an application we show that the Picard group in question contains 4-torsion elements.
A sharp slope inequality for general stable fibrations of curves.
Atsushi Moriwaki (1996)
A Theorem of Local-Torelli Type.
D. Lieberman, R. Wilsker, G. Peters (1977)
Indranil Biswas, Norbert Hoffmann (2012)
X
{X}^{\prime }
be compact Riemann surfaces of genus
\ge 3
G
{G}^{\prime }
be nonabelian reductive complex groups. If one component
{โณ}_{G}^{d}\left(X\right)
of the coarse moduli space for semistable principal
G
โbundles over
X
is isomorphic to another component
{โณ}_{{G}^{\prime }}^{{d}^{\prime }}\left({X}^{\prime }\right)
X
{X}^{\prime }
Algebraic Surfaces of General Type with Small ... . II.
Eiji Horikawa (1976)
An effective Shafarevich theorem for elliptic curves
Clemens Fuchs, Rafael von Kรคnel, Gisbert Wรผstholz (2011)
Apolarity and its applications.
N.I. Shepherd-Barron (1989)
Appendix : An example of thick wall
Nicolas Ressayre (1998)
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Gamma Function | Brilliant Math & Science Wiki
Aareyan Manzoor, Ronak Agarwal, Kishlaya Jaiswal, and
Sir Hรกjek
The gamma function, denoted by
\Gamma(s)
, is de๏ฌned by the formula
\Gamma (s)=\int_0^{\infty} t^{s-1} e^{-t}\, dt,
which is defined for all complex numbers except the nonpositive integers. It is frequently used in identities and proofs in analytic contexts.
The above integral is also known as Euler's integral of second kind. It serves as an extension of the factorial function which is defined only for the positive integers. In fact, it is the analytic continuation of the factorial and is defined as
\Gamma (n)=(n-1)! .
However, the gamma function is but one in a class of multiple functions which are also meromorphic with poles at the nonpositive integers.
Weierstrass Representation
Connection to Beta Function
Connection to Digamma Function
Connection to Zeta Function
Connection to Polylogarithms
\Gamma(s+1)=s\Gamma(s)
is true for all values of
s \in \mathbb{C}
; this can be derived from an application of integration by parts.
Recalling the definition of the gamma function above, we can see that by applying integration by parts,
\Gamma(s+1) = \int_0^{\infty} t^{s} e^{-t}\, dt = -t^{s}e^{-t}\Big|_0^{\infty} + s \int_0^{\infty} t^{s-1} e^{-t}\, dt = s\Gamma(s).
Hence, the functional equation holds.
_\square
\Gamma(1) = 1
, we can easily see
\Gamma(s) = (s-1)!
for all positive integers
by simple induction.
\frac{\Gamma\left(\frac{5}{2}\right)}{\Gamma\left(\frac{1}{2}\right)}.
Using the functional equation for the gamma function, we obtain that
\frac{\Gamma\left(\frac{5}{2}\right)}{\Gamma\left(\frac{1}{2}\right)} = \frac{\dfrac{3}{2}\Gamma\left(\frac{3}{2}\right)}{\Gamma\left(\frac{1}{2}\right)} = \frac{\dfrac{3}{2}\frac{1}{2}\Gamma\left(\frac{1}{2}\right)}{\Gamma\left(\frac{1}{2}\right)} =\frac{3}{4}. \ _\square
\dfrac{\Gamma\left(\frac {16}{3}\right)}{\Gamma\left(\frac{10}{3}\right)},
\Gamma (z)
denotes the gamma function.
If the value of the above expression can be expressed in the form of
\frac{a}{b}
and
b
a+b
\Gamma(s)=\lim_{n\to \infty}\left(\dfrac{n^s}{s} \prod_{k=1}^n \dfrac{k}{s+k}\right)
S=\int_0^n t^{s-1} \left(1-\dfrac{t}{n}\right)^n\, dt.
S=\left.\dfrac{t^s}{s} \left(1-\dfrac{t}{n}\right)^n\right|_0^n+\dfrac{n}{ns} \int_0^n t^{s} \left(1-\dfrac{t}{n}\right)^{n-1}dt=\dfrac{n}{ns} \int_0^n t^{s} \left(1-\dfrac{t}{n}\right)^{n-1}dt.
We can deduce that by integrating by parts
n-1
times, we will get
S=\dfrac{n}{ns}\dfrac{n-1}{n(s+1)}\dfrac{n-2}{n(s+2)}\cdots \dfrac{1}{n(s+n-1)} \int_0^n t^{s+n-1} dt.
Evaluating the integral, we have
S=\dfrac{n^s}{s} \prod_{k=1}^n \dfrac{k}{s+k}.
n
toward infintiy,
\lim_{n\to \infty}\int_0^n t^{s-1} \left(1-\dfrac{t}{n}\right)^ndt=\lim_{n\to \infty}\left(\dfrac{n^s}{s} \prod_{k=1}^n \dfrac{k}{s+k}\right).
\lim_{n\to \infty}\int_0^n t^{s-1} \left(1-\dfrac{t}{n}\right)^ndt=\int_0^\infty t^{s-1} e^{-t}dt=\Gamma(s).
\Gamma(s)=\lim_{n\to \infty}\left(\dfrac{n^s}{s} \prod_{k=1}^n \dfrac{k}{s+k}\right).\ _\square
\Gamma(s)=\dfrac{e^{-\gamma s}}{s} \prod_{k=1}^\infty e^{s/k} \left(1+\dfrac{s}{k}\right)^{-1}
\Gamma(s)=\lim_{n\to \infty}\left(\dfrac{n^s}{s} \prod_{k=1}^n \dfrac{k}{s+k}\right).
Then rewrite it as
\begin{aligned} \Gamma(s)&=\lim_{n\to \infty}\left(\dfrac{1}{s}\exp\big(sH_k-sH_k+s\ln(n)\big) \prod_{k=1}^n \dfrac{k}{s+k}\right)\\\\ &=\lim_{n\to \infty}\left(\dfrac{1}{s}\exp\left(\sum_{k=1}^n \dfrac{s}{k}-s\gamma\right)\prod_{k=1}^n \left(1+\dfrac{s}{k}\right)^{-1}\right) \\\\ &=\lim_{n\to \infty}\left(\dfrac{e^{-\gamma s}}{s} \prod_{k=1}^n e^{s/k} \left(1+\dfrac{s}{k}\right)^{-1}\right)\\\\ &=\dfrac{e^{-\gamma s}}{s} \prod_{k=1}^\infty e^{s/k} \left(1+\dfrac{s}{k}\right)^{-1}.\ _\square \end{aligned}
The gamma function has a very nice duplication formula. It can be stated as
\sqrt { \pi } \ \Gamma (2z)={ 2 }^{ 2z-1 }\Gamma (z)\Gamma \left (z+\frac { 1 }{ 2 } \right ).
This is notoriously simple to prove.
We start with the representation of beta function and a relation between beta function and gamma function:
\displaystyle \int _{ 0 }^{ \pi /2 }{ \sin ^{ 2m-1 }{ x } \cos ^{ 2n-1 }{ x }\, dx } = \dfrac{B(m,n)}{2} = \frac { \Gamma (m)\Gamma (n) }{2\Gamma (m+n) }.
m=z
n=z
\begin{aligned} \displaystyle I = \frac { \Gamma (z)\Gamma (z) }{ 2\Gamma (2z) } &= \int _{ 0 }^{ \pi /2 }{ \sin ^{ 2z-1 }{ x } \cos ^{ 2z-1 }{ x }\, dx } \\ &= \frac { 1 }{ { 2 }^{ 2z-1 } } \int _{ 0 }^{ \pi /2 }{ \sin ^{ 2z-1 }{ 2x }\, dx }. \end{aligned}
2x=t
2\,dx=dt
\displaystyle I = \frac { 1 }{ { 2 }^{ 2z } } \int _{ 0 }^{ \pi }{ \sin ^{ 2z-1 }{ t }\, dt }.
Now, using the property
\displaystyle \int _{ 0 }^{ 2a }{ f(x)\, dx } =2\int _{ 0 }^{ a }{ f(x)\, dx }
f(x)=f(2a-x),
\displaystyle I = \frac { 1 }{ { 2 }^{ 2z-1 } } \int _{ 0 }^{ \pi /2 }{ \sin ^{ 2z-1 }{ t }\, dt }.
Again using beta function, we have
I = \dfrac { 1 }{ { 2 }^{ 2z } } \dfrac { \Gamma (z)\Gamma \left(\dfrac { 1 }{ 2 } \right) }{ \Gamma \left(z+\dfrac { 1 }{ 2 } \right) }.
I = \dfrac { \Gamma (z)\Gamma (z) }{ 2\Gamma (2z) },
\dfrac { \Gamma (z)\Gamma (z) }{ \Gamma (2z) } =\dfrac { 1 }{ { 2 }^{ 2z-1 } } \dfrac { \Gamma (z)\Gamma \left(\dfrac { 1 }{ 2 } \right) }{ \Gamma \left(z+\dfrac { 1 }{ 2 } \right) }.
\Gamma\left(\dfrac{1}{2}\right) = \sqrt{\pi}
{ 2 }^{ 2z-1 }\Gamma (z)\Gamma \left(z+\dfrac { 1 }{ 2 } \right)=\sqrt { \pi }\, \Gamma (2z).
_\square
\Gamma(z)\Gamma(1-z) = \frac{\pi}{\sin \pi z},
z \in \mathbb{C}.
_\square
\Gamma(s)=\lim_{n\to \infty}\left(\dfrac{n^s}{s} \prod_{k=1}^n \dfrac{k}{s+k}\right).
s
-s
\Gamma(-s)=\lim_{n\to \infty}\left(\dfrac{n^{-s}}{-s} \prod_{k=1}^n \dfrac{k}{-s+k}\right).
Multiply both of these to get
\Gamma(s)\Gamma(-s)=\dfrac{-1}{s^2} \prod_{k=1}^\infty \left(1-\dfrac{s^2}{k^2}\right)^{-1}.
The product on the RHS is the famous Weierstrass product, so we have
\begin{aligned} \Gamma(s)\Gamma(-s)&=\dfrac{-1}{s^2} \dfrac{\pi s}{\sin(\pi s)}\\\\ \Gamma(s)\big(-s\Gamma(-s)\big)&=\dfrac{\pi}{\sin(\pi s)}\\\\ \Gamma(s)\Gamma(1-s)&=\dfrac{\pi}{\sin(\pi s)}.\ _\square \end{aligned}
\Gamma\left(\frac{1}{2}\right)^{4}
Using Euler's reflection formula,
\Gamma\left(\frac{1}{2}\right)^{4} = \left[\Gamma\left(\frac{1}{2}\right)\Gamma\left(1-\frac{1}{2}\right)\right]^{2} = \left(\frac{\pi}{\sin \frac{\pi}{2}}\right)^{2} = \pi^{2}. \ _\square
There is also an Euler reflection formula for the digamma function
Main article: Beta Function
The gamma and beta functions satisfy the identity
B(x, y) = \dfrac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)} = \int_0^1 t^{x-1} (1-t)^{y-1}\, dt= \int_0^{\pi/2}2 \sin^{2x-1} (t)\cos^{2y-1}(t)\, dt.
The above integral is indeed known as Euler's integral of first kind.
Main article: Digamma Function
The digamma function is defined as
\psi(s)=\dfrac{\Gamma'(s)}{\Gamma(s)}.
Using this on the Euler reflection formula and Legendre duplication formula, we have
\begin{aligned} \psi(1-s)-\psi(s)&=\pi\cot\big(\pi(s)\big)\\ 2\psi(2s)&=2\ln(2)+\psi(s)+\psi\left(s+\dfrac{1}{2}\right). \end{aligned}
The zeta function and gamma functions are very closely related.
The gamma function turns up in the zeta functional equations:
\begin{aligned} \pi^{-\frac{s}{2}} \Gamma\left(\frac{s}{2}\right)\zeta(s)&=\pi^{-\frac{1-s}{2}} \Gamma\left(\frac{1-s}{2}\right)\zeta(1-s)\\\\ \zeta(s)&=2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s)\zeta(1-s). \end{aligned}
It also has an integral closely related to it:
\Gamma(s) \zeta(s) =\int_0^\infty\dfrac{x^{s-1}}{e^x-1} dx.
Main article: Polylogarithm
An integral representation of gamma function is
\Gamma(s)\text{Li}_s(z)= \int_0^\infty\frac{x^{s-1}}{\frac{e^x}{z}-1} dx.
The Bohr-Mollerup theorem states that the gamma function
\Gamma
is the unique function on the interval
x > 0
\Gamma(1) = 1, \Gamma(s+1) = s\Gamma(s),
\ln \Gamma(s)
Cite as: Gamma Function. Brilliant.org. Retrieved from https://brilliant.org/wiki/gamma-function/
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GetIsomorphism - Maple Help
Home : Support : Online Help : Mathematics : Algebra : Magma : GetIsomorphism
return an isomorphism between isomorphic magmas
GetIsomorphism( m1, m2 )
The GetIsomorphism( 'm1', 'm2' ) command returns an isomorphism, in the form of a permutation of 1..n, where m1 and m2 are of order n, which effects an isomorphism from m1 to m2. In case m1 and m2 are not isomorphic, then the value false is returned.
\mathrm{with}โก\left(\mathrm{Magma}\right):
\mathrm{m1}โโฉโฉโฉ1|2|3โช,โฉ2|3|1โช,โฉ3|1|2โชโชโช
\textcolor[rgb]{0,0,1}{\mathrm{m1}}\textcolor[rgb]{0,0,1}{โ}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\end{array}]
\mathrm{m2}โโฉโฉโฉ2|3|1โช,โฉ3|1|2โช,โฉ1|2|3โชโชโช
\textcolor[rgb]{0,0,1}{\mathrm{m2}}\textcolor[rgb]{0,0,1}{โ}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{GetIsomorphism}โก\left(\mathrm{m1},\mathrm{m2}\right)
[\textcolor[rgb]{0,0,1}{3}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{1}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{2}]
\mathrm{m3}โโฉโฉโฉ1|2|1โช,โฉ2|3|2โช,โฉ3|1|3โชโชโช
\textcolor[rgb]{0,0,1}{\mathrm{m3}}\textcolor[rgb]{0,0,1}{โ}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{1}\\ \textcolor[rgb]{0,0,1}{2}& \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{2}\\ \textcolor[rgb]{0,0,1}{3}& \textcolor[rgb]{0,0,1}{1}& \textcolor[rgb]{0,0,1}{3}\end{array}]
\mathrm{GetIsomorphism}โก\left(\mathrm{m1},\mathrm{m3}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The Magma[GetIsomorphism] command was introduced in Maple 15.
|
Some Remarks on the Mathieu Series
Robert Frontczak, "Some Remarks on the Mathieu Series", International Scholarly Research Notices, vol. 2014, Article ID 985782, 8 pages, 2014. https://doi.org/10.1155/2014/985782
Robert Frontczak 1
1Landesbank Baden-Wรผrttemberg (LBBW), Am Hauptbahnhof 2, 70173 Stuttgart, Germany
Academic Editor: G. Fikioris
The object of this note is to present new expressions for the classical Mathieu series in terms of hyperbolic functions. The derivation is based on elementary arguments concerning the integral representation of the series. The results are used afterwards to prove, among others, a new relationship between the Mathieu series and its alternating companion. A recursion formula for the Mathieu series is also presented. As a byproduct, some closed-form evaluations of integrals involving hyperbolic functions are inferred.
The infinite series is called a Mathieu series. It was introduced in 1890 by ร. L. Mathieu (1835โ1890) who studied various problems in mathematical physics. Since its introduction the series has been studied intensively. Mathieu himself conjectured that . The conjecture was proved in 1952 by Berg in [1]. Nowadays, the mathematical literature provides a range of papers dealing with inequalities for the series . In 1957 Makai [2] derived the double inequality More recently, Alzer et al. proved in [3] that Here, as usual, denotes the Riemann zeta function defined by , . The constants and are the best possible. Other lower and upper bound estimates for the Mathieu series can be found in the articles of Qi et al. [4] and Hoorfar and Qi [5].
An integral representation for the Mathieu series (1) is given by The integral representation was used by Elbert in [6] to derive the asymptotic expansion of : where denote the even indexed Bernoulli numbers defined by the generating function See also [7] for a derivation.
The Mathieu series admits various generalizations that have been introduced and investigated intensively in recent years. The generalizations include the alternating Mathieu series, the -fold generalized Mathieu series, Mathieu -series, and Mathieu -series [8โ11]. The generalizations recapture the classical Mathieu series as a special case. On the other hand, the alternating Mathieu series, although connected to its classical companion, is a variant that allows a separate study. It was introduced by Pogรกny et al. in [11] by the equation It possesses the integral representation Recently derived bounding inequalities for alternating Mathieu-type series can be found in the paper of Pogรกny and Tomovski [12]. The latest research article on integral forms for Mathieu-type series is the paper of Milovanoviฤ and Pogรกny [13]. The authors present a systematic treatment of the subject based on contour integration. Among others, the following new integral representation for is derived [13, Corollary 2.2]: In this note we restrict the attention to the classical Mathieu series. Starting with the integral form (4) new representations for are derived. The derivation is based on elementary arguments concerning the integrand combined with related integral identities. The results are used afterwards to establish interesting properties of the series. Among others, a new relationship between the Mathieu series and its alternating variant is derived. A recursion formula for the Mathieu series is also presented. As a byproduct, some closed-form evaluations of integrals involving hyperbolic functions are inferred. Finally, a new proof is given for an exact evaluation of an infinite series related to .
In what follows we will use the following integral identities. The identities are well known and are stated without proof (see [14] for a reference).
Lemma 1. For and it holds that where denotes the hyperbolic cosecant of defined by Similarly, for and an integer where denotes the hyperbolic cotangent of defined by Finally, for , , and an integer
The main theorem of this note is stated next. It gives three expressions for the Mathieu series in a semi-integral form.
Theorem 2. The Mathieu series has the following representations where and denote the hyperbolic sine and cosine functions, respectively, , and denotes its first derivative.
Proof. Let . The main argument in the proof is the observation that satisfies the nonlinear differential equation which may be also written in the form Inserting the relation in (4) and using (12) with give Since we see that Integration by parts gives which is easily evaluated using (10) with . Finally, the elementary relation establishes (15). Integrating (4) by parts results in Equations (18) and (10) give Let denote the last component of (26); that is, Then, can be evaluated in two different ways. First, the relation from (21) in combination with (10) shows that This proves (16). Finally, by the rule of lโHospital, Splitting the integrand of into two parts and integrating by parts in each case, we obtain In the last equation we have used the fact that Since is equal to For the first integral we apply once more the result from (12) with . Direct calculation shows that the expression in brackets under the second integral sign is equal to . This completes the proof.
The elementary relation in (18) seems to have been overseen in the literature. It allows to express in terms of the hyperbolic sine and cosine functions, respectively. The representations may turn out to be useful to study the properties of . For the remainder of this note, however, we will mainly work with (15). The equation will be used to infer interesting consequences, which we are going to present immediately. Additionally, in Section 3 it will be outlined that (18) is also useful to study some topics that are related to the evaluation of .
Concerning the integrands in Theorem 2 we make the following observations. Firstly, standard arguments show that for and Secondly, for and Finally, since for , it is clear that and Figure 1 illustrates the results from Theorem 2 numerically. is plotted together with (15)โ(17), where the new representations are decomposed into nonintegral and integral parts (termed first and second parts, resp.). More precisely, this means that (15) is decomposed in the following manner: Equations (16) and (17) are decomposed analogously.
Numerical (graphical) illustrations of the results from Theorem 2.
A first interesting consequence of the theorem is the following closed-form evaluation of an integral.
Corollary 3. Let . Then, it holds that
Proof. From (1) it is obvious that . Further, from the identity for Bernoulli numbers (see [14]) and the fact that , we easily deduce that Hence, The assertion follows from (15).
It is clear that a successive repetition of integration by parts in Theorem 2 will produce a range of other expressions for . Among others, from (15) and the fact that it is fairly easy to produce the following characterization: In the above characterization the sine and cosine functions appear simultaneously as integrands. Now, in view of the previous proof and (18) it is straightforward to get where we have used the relation . Furthermore, since direct computation verifies that the last integral may be written in the form A further interesting consequence of the previous theorem is the following new integral-type representation for the alternating Mathieu series .
Proposition 4. The alternating Mathieu series may be represented as with
Proof. We start with the identity In view of (15) and applying the relation we obtain where For the second integral, a change of variable () is used in combination with the multiple-angle formulas for the hyperbolic sine and cosine functions, respectively, to get and hence Next, since can be simplified to The relation shows that . The final formula follows from and the half-angle formula
Proof. We have . Additionally, using similar arguments as in Corollary 3, it can be shown that This proves the first expression. The second follows from combining (38) and (60).
The next assertion provides a recursion formula for the Mathieu series. Surprisingly, it is also a consequence of Proposition 4.
Corollary 6. For define the function as where is given in (48). Then, for one has the recursion formula
Proof. Combine (47) and (49) to get Successive repetition of the identity establishes the stated formula.
3. Evaluation of a Related Series
It is interesting to mention that (18) may be used to prove a result for a closed-form evaluation of an infinite series that is related to . More precisely, if we define the function for by the infinite series then and are connected via the following relationship. Consider the (complex) function with being the Gamma function. Note first that Comparing the two equations we see that The function admits an exact evaluation (see [7]) for which we can provide a new elementary proof applying the analytical structure of . However, in contrary to the pervious section where we have focused on integral-type expressions, we change the point of view and work with summation-type representations of .
Proposition 7. The function can be evaluated exactly as
Proof. Once more, let . We start with (18) and (26). Since we have Thus, after interchanging summation and integration and applying (14) where This gives From the identity we arrive at Next, we simplify the sums in the above equation for further. Using the fact that it is straightforward to show that The last identity is true since the first sum on the right-hand side of the equality telescopes. This gives which may be written as Finally, notice that This leads to canceling out in (80) and the proof is complete.
Remark 8. Using the further observation that may be expressed as it is possible to derive the following summation-type form of :
In this paper new expressions for the Mathieu series were derived in terms of hyperbolic functions. To derive the new identities, a basic property of the function was utilized. Using a particular identity it was possible to prove a new relationship between the Mathieu series and the alternating variant. Also, a new recursion formula and some interesting closed-form evaluations of definite integrals involving hyperbolic functions were established. Finally, a new elementary proof for an evaluation of an infinite series related to the Mathieu series was presented.
It would be interesting to know whether the new identities can be used to derive new (double) inequalities for the series.
The statements and conclusions made in this paper are entirely those of the author. They do not necessarily reflect the views of LBBW.
The author thanks two anonymous referees for a careful reading of the paper and valuable comments that helped to improve the content of the paper.
L. Berg, โรber eine Abschรคtzung von Mathieu,โ Mathematische Nachrichten, vol. 7, pp. 257โ259, 1952 (German). View at: Google Scholar | MathSciNet
E. Makai, โOn the inequality of Mathieu,โ Publicationes Mathematicae Debrecen, vol. 5, pp. 204โ205, 1957. View at: Google Scholar | MathSciNet
H. Alzer, J. L. Brenner, and O. G. Ruehr, โOn Mathieu's inequality,โ Journal of Mathematical Analysis and Applications, vol. 218, no. 2, pp. 607โ610, 1998. View at: Publisher Site | Google Scholar | MathSciNet
F. Qi, C.-P. Chen, and B.-N. Guo, โNotes on double inequalities of Mathieu's series,โ International Journal of Mathematics and Mathematical Sciences, no. 16, pp. 2547โ2554, 2005. View at: Publisher Site | Google Scholar | MathSciNet
A. Hoorfar and F. Qi, โSome new bounds for Mathieu's series,โ Abstract and Applied Analysis, vol. 2007, Article ID 94854, 10 pages, 2007. View at: Publisher Site | Google Scholar | MathSciNet
ร. Elbert, โAsymptotic expansion and continued fraction for Mathieu's series,โ Periodica Mathematica Hungarica, vol. 13, no. 1, pp. 1โ8, 1982. View at: Publisher Site | Google Scholar | MathSciNet
D. C. Russell, โA note on Mathieu's inequality,โ Aequationes Mathematicae, vol. 36, no. 2-3, pp. 294โ302, 1988. View at: Publisher Site | Google Scholar | MathSciNet
P. Cerone and C. T. Lenard, โOn integral forms of generalised Mathieu series,โ Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 5, 2003. View at: Google Scholar | MathSciNet
H. M. Srivastava and ลฝ. Tomovski, โSome problems and solutions involving Mathieu's series and its generalizations,โ Journal of Inequalities in Pure and Applied Mathematics, vol. 5, no. 2, 2004. View at: Google Scholar | MathSciNet
T. K. Pogรกny and ลฝ. Tomovski, โOn multiple generalized Mathieu series,โ Integral Transforms and Special Functions, vol. 17, no. 4, pp. 285โ293, 2006. View at: Publisher Site | Google Scholar | MathSciNet
T. K. Pogรกny, H. M. Srivastava, and ลฝ. Tomovski, โSome families of Mathieu
๐
-series and alternating Mathieu
๐
-series,โ Applied Mathematics and Computation, vol. 173, no. 1, pp. 69โ108, 2006. View at: Publisher Site | Google Scholar | MathSciNet
T. K. Pogรกny and ลฝ. Tomovski, โBounds improvement for alternating Mathieu type series,โ Journal of Mathematical Inequalities, vol. 4, no. 3, pp. 315โ324, 2010. View at: Publisher Site | Google Scholar | MathSciNet
G. V. Milovanoviฤ and T. K. Pogรกny, โNew integral forms of generalized Mathieu series and related applications,โ Applicable Analysis and Discrete Mathematics, vol. 7, no. 1, pp. 180โ192, 2013. View at: Publisher Site | Google Scholar | MathSciNet
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Elsevier Academic Press, Amsterdam, The Netherlands, 7th edition, 2007. View at: MathSciNet
Copyright ยฉ 2014 Robert Frontczak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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{S}^{H}=\left\{{S}_{t}^{H},t\ge 0\right\}
be a sub-fractional Brownian motion with
H\in \left(0,1\right)
. We establish the existence, the joint continuity and the Hรถlder regularity of the local time
{L}^{H}
{S}^{H}
. We will also give Chungโs form of the law of iterated logarithm for
{S}^{H}
. This results are obtained with the decomposition of the sub-fractional Brownian motion into the sum of fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. This decomposition is given by Ruiz de Chavez and Tudor [10].
Mots clรฉs : Sub-fractional Brownian motion, local time, local nondeterminism, Chungโs type law of iterated logarithm
author = {Mendy, Ibrahima},
title = {On the local time of sub-fractional {Brownian} motion},
AU - Mendy, Ibrahima
TI - On the local time of sub-fractional Brownian motion
Mendy, Ibrahima. On the local time of sub-fractional Brownian motion. Annales Mathรฉmatiques Blaise Pascal, Tome 17 (2010) no. 2, pp. 357-374. doi : 10.5802/ambp.288. http://www.numdam.org/articles/10.5802/ambp.288/
[1] Adler, Robert J. The geometry of random fields, John Wiley & Sons Ltd., Chichester, 1981 (Wiley Series in Probability and Mathematical Statistics) | MR 611857 | Zbl 0478.60059
[2] Adler, Robert J. An introduction to continuity, extrema, and related topics for general Gaussian processes, Institute of Mathematical Statistics Lecture NotesโMonograph Series, 12, Institute of Mathematical Statistics, Hayward, CA, 1990 | MR 1088478 | Zbl 0747.60039
[3] Baraka, D.; Mountford, T.; Xiao, Y. Hรถlder properties of local times for fractional Brownian motions, Metrika, Volume 69 (2009) no. 2-3, pp. 125-152 | Article | MR 2481918
[4] Berman, S. M. Local times and sample function properties of stationary Gaussian processes, Trans. Amer. Math. Soc., Volume 137 (1969), pp. 277-299 | Article | MR 239652 | Zbl 0184.40801
[5] Berman, S. M. Gaussian processes with stationary increments: Local times and sample function properties, Ann. Math. Statist., Volume 41 (1970), pp. 1260-1272 | Article | MR 272035 | Zbl 0204.50501
[6] Berman, S. M. Local nondeterminism and local times of Gaussian processes, Indiana University Mathematical Journal, Volume 23 (1973), pp. 69-94 | Article | MR 317397 | Zbl 0264.60024
[7] Berman, Simeon M. Gaussian sample functions: Uniform dimension and Hรถlder conditions nowhere, Nagoya Math. J., Volume 46 (1972), pp. 63-86 | MR 307320 | Zbl 0246.60038
[8] Bojdecki, T. L. G.; Gorostiza, L. G.; Talarczyk, A. Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particule systems, Electron. Comm. Probab., Volume 32 (2007), pp. 161-172 | MR 2318163 | Zbl 1128.60025
[9] Boufoussi, B.; Dozzi, M.; Guerbaz, R. On the local time of the multifractional brownian motion, Stochastics and stochastic repports, Volume 78 (2006), pp. 33-49 | MR 2219711 | Zbl 1124.60061
[10] Ruiz de Chรกvez, J.; Tudor, C. A decomposition of sub-fractional Brownian motion, Math. Rep. (Bucur.), Volume 11(61) (2009) no. 1, pp. 67-74 | MR 2506510
[11] Csรถrgล, Miklรณs; Lin, Zheng Yan; Shao, Qi Man On moduli of continuity for local times of Gaussian processes, Stochastic Process. Appl., Volume 58 (1995) no. 1, pp. 1-21 | Article | MR 1341551 | Zbl 0834.60088
[12] Ehm, W. Sample function properties of multi-parameter stable processes, Z. Wahrsch. Verw. Gebiete, Volume 56 (1981), pp. 195-228 | Article | MR 618272 | Zbl 0471.60046
[13] Geman, D.; Horowitz, J. Occupation densities, Annales of probability, Volume 8 (1980), pp. 1-67 | Article | MR 556414 | Zbl 0499.60081
[14] Guerbaz, R. Local time and related sample paths of filtered white noises, Annales Mathematiques Blaise Pascal, Volume 14 (2007), pp. 77-91 | Article | Numdam | MR 2298805 | Zbl 1144.60029
[15] Kรดno, N. Hรถlder conditions for the local times of certain gaussian processes with stationary increments, Proceeding of the Japan Academy, Volume 53 (1977), pp. 84-87 | Article | MR 494453 | Zbl 0437.60057
[16] Kรดno, N.; Shieh, N. R. Local times and related sample path proprieties of certain selfsimilar processes, J. Math. Kyoto Univ., Volume 33 (1993), pp. 51-64 | MR 1203890 | Zbl 0776.60054
[17] Lei, P.; Nualart, D. A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett, Volume 779 (2009), pp. 619-624 | Article | MR 2499385 | Zbl 1157.60313
[18] Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and applications, Stochastic processes: theory and methods (Handbook of Statist.), Volume 19, North-Holland, Amsterdam, 2001, pp. 533-597 | MR 1861734 | Zbl 0987.60053
[19] Monrad, D.; Rootzรฉn, H. Small values of Gaussian processes and functional laws of the iterated logarithm, Probab. Th. Rel. Fields, Volume 101 (1995), pp. 173-192 | Article | MR 1318191 | Zbl 0821.60043
[20] Pitt, L. Local times for gaussian vector fields, Indiana Univ. Math. J., Volume 27 (1978), pp. 204-237 | Article | MR 471055 | Zbl 0382.60055
[21] Tudor, C. Some propreties of sub-fractional Brownian motion, Stochastics., Volume 79 (2007), pp. 431-448 | MR 2356519 | Zbl 1124.60038
[22] Tudor, C. Inner product spaces of integrands associated to sub-fractional Brownian motion, Statist. Probab. Lett., Volume 78 (2008), p. 2201-2209. | Article | MR 2458028
[23] Xiao, Y. Hรถlder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields, Probab. Th. Rel. fields, Volume 109 (1997), pp. 129-157 | Article | MR 1469923 | Zbl 0882.60035
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Leonhard Euler | Brilliant Math & Science Wiki
Patrick Corn, Sravanth C., Sandeep Bhardwaj, and
Leonhard Euler. [1]Leonhard Euler (1707-1783) was a Swiss mathematician and physicist who made fundamental contributions to countless areas of mathematics. He studied and inspired fundamental concepts in calculus, complex numbers, number theory, graph theory, and geometry, many of which bear his name. (A common joke about Euler is that to avoid having too many mathematical concepts named after him, the convention is to give them the name of the first person after Euler to have discovered them.)
Euler also made immense contributions to the language of modern mathematics. He was the first writer to define a function and write it as
f(x)
; he was the first to use summation notation using the Greek letter
\Sigma
; he introduced the standard notation for trigonometric functions; and many more.
What follows is a list of some of the highlights of Euler's work. It will necessarily be incomplete, as Euler led a long and extremely prolific mathematical life.
Euler's criterion for quadratic residues
Euler's modular form; partitions
Euclid-Euler theorem on perfect numbers
Euler was not the first to identify
e
as an interesting mathematical constant (depending on what one takes "discover" to mean, the constant was discovered by either Napier in the early 1600s or Bernoulli in the late 1600s), but he did make several important contributions to its study. In one of his most famous books, Introductio in Analysin infinitorum (1748), he showed that the infinite sum
\frac1{0!} + \frac1{1!} + \frac1{2!} + \cdots = \sum_{k=0}^{\infty} \frac1{k!}
and the value of the limit
\lim_{n\to\infty} \left( 1+\frac1{n}\right)^n
a
\int_1^a \frac1{x} \, dx = 1
were all the same number, and called that number
e
. He also gave an approximation to
23
decimal places. He was also the first to prove that
e
is irrational, by proving that its continued fraction is
e = [2;1,2,1,1,4,1,1,6,1,1,8,1,\cdots] = 2+\frac1{1+\frac1{2+\frac1{1+\frac1{1+\frac1{4+\frac1{\cdots}}}}}}.
Since rational numbers have finite continued fractions, this gives a proof that
e
For more on the history of
e
, see the discovery of the number e.
In his analysis of the Riemann zeta function (see below), Euler analyzed the harmonic numbers
H_n = 1 + \frac12 + \cdots + \frac1{n}.
In a 1731 paper, he showed that
\lim_{n\to\infty} (H_n - \ln(n))
exists and equals the sum of the conditionally convergent series
\sum_{n=2}^{\infty} (-1)^n \frac{\zeta(n)}{n}.
The constant is now known as
\gamma
, due to its connection with the gamma function, and is usually called the Euler-Mascheroni constant. It arises in sums involving the zeta function and integrals involving logarithms and exponentials, e.g.
\int_0^{\infty} \frac{\ln(x)}{e^x} \, dx = -\gamma.
\large \int_{-\infty}^{\infty} {xe^{2x}e^{-{e}^{2x}} \, dx}
The above Integral can be expressed as
-\dfrac{\gamma}{a},
\gamma
denotes the Euler-Mascheroni constant
\displaystyle \gamma = \lim_{n\to\infty} \left( - \ln n + \sum_{k=1}^n \dfrac1k \right) \approx 0.5772 .
a
Euler often manipulated infinite series in ways that were not rigorous by modern standards, but his intuition generally steered him to correct results. These manipulations could often be justified after the fact by modern mathematicians.
As a pioneering user of Taylor series, Euler noticed that plugging in
z=i\theta
e^z = 1+z+\frac{z^2}{2!}+\cdots= \sum_{k=0}^{\infty} \frac{z^k}{k!}
\begin{aligned} e^{i\theta} &= \left( 1-\frac{\theta^2}{2!}+\frac{\theta^4}{4!}-\cdots\right) +i\left(\frac{\theta}{1!}-\frac{\theta^3}{3!}+\frac{\theta^5}{5!}-\cdots\right) \\ &= \cos \theta+i\sin \theta. \end{aligned}
This result is fundamental in modern complex analysis, and has many applications for trigonometry as well. The special case
\theta = \pi
e^{i\pi} +1 = 0,
which is often cited as the most beautiful formula in all of mathematics.
For more, see Euler's formula.
Euler developed the Euler product formula for the Riemann zeta function,
\zeta(s) = \sum_{n=1}^{\infty} \frac1{n^s} = \prod_{p \text{ prime}} \left( 1-\frac1{p^s} \right)^{-1}.
He used this to show that the sum of the reciprocals of the primes diverges, and used a similar (and easier) argument to show that there were infinitely many primes.
Euler defined the totient function
\phi(n)
in 1736, and proved several facts about it, including the product formula
\frac{\phi(n)}{n} = \prod_{\stackrel{p|n}{p \text{ prime}}} \left( 1- \frac1{p} \right)
and Euler's theorem
a^{\phi(n)} \equiv 1 \pmod n
a
n
. This function is of paramount importance in number theory and cryptography (as in the RSA cryptosystem).
Euler studied the problem of determining whether a number is a quadratic residue modulo a prime
p
, and proved in 1748 the following simple criterion:
\left( \frac{a}{p} \right) = 1 \Leftrightarrow a^{\frac{p-1}{2}} \equiv 1 \pmod p .
\left( \frac{a}{p} \right)
This result is used to prove the theorem of quadratic reciprocity.
Euler showed that the infinite product
\prod_{k=1}^{\infty} (1-x^k)
\sum_{m=-\infty}^{\infty} (-1)^m x^{m(3m-1)/2}
and used this to derive a recursive formula for the partition function
p(n)
p(n) = p(n-1)+p(n-2)-p(n-5)-p(n-7)+p(n-12)\cdots,
where the numbers that appear are precisely the pentagonal numbers. This is Euler's pentagonal number theorem.
Among Euler's contributions to graph theory is the notion of an Eulerian path. This is a path that goes through each edge of the graph exactly once. If it starts and ends at the same vertex, it is called an Eulerian circuit.
Euler proved in 1736 that if an Eulerian circuit exists, every vertex has even degree, and stated without proof the converse that a connected graph with all vertices of even degree contains an Eulerian circuit. (This was proved in the late
19^\text{th}
century.)
The motivation for his exploration was the famous Seven Bridges of Konigsberg problem.
Euler showed that for convex polyhedra,
V-E+F = 2,
V,E,F
are the number of vertices, edges, and faces, respectively. Using similar definitions, he showed that the same was true for a connected planar graph (planar means that the edges can be drawn on a piece of paper without crossing each other). In modern language, we say that the right side of the equation is the Euler characteristic of the object. Generalizations of this idea (including, most notably, the genus) were among the earliest foundational problems in topology.
Euler proved in 1765 that the orthocenter, circumcenter, and centroid of a triangle are collinear (they all lie on one line). This line, called the Euler line, has many other beautiful properties, detailed in the wiki: Euler line.
Euler proved a theorem characterizing all even perfect numbers. He showed that they were of the form
\frac{q(q+1)}2
q
is a Mersenne prime. The fact that all such numbers were perfect was known as far back as Euclid's Elements, but Euler's result was new. The proof is quite elementary and is in the wiki on perfect numbers.
[1] Image from https://upload.wikimedia.org/wikipedia/commons/d/d7/Leonhard_Euler.jpg under the creative Commons licensing for reuse and modification.
Cite as: Leonhard Euler. Brilliant.org. Retrieved from https://brilliant.org/wiki/leonhard-euler/
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Infinite Arrays - MATLAB & Simulink - MathWorks รฆโยฅรฆลยฌ
What Are Infinite Arrays?
Infinite Array Solver
Create Infinite Array Using Antenna Toolbox
Choose a Unit Cell
Scan Infinite Arrays
Scan Impedance and Scan Element Pattern
Scan Element Pattern
Compare Scan Element Pattern of Finite and Infinite Arrays
Case 1: Compare finite array and infinite array with unit cell of dimensions 0.5รยป รโ 0.5รยป
Impact of Infinite Double Summation
Infinite arrays are rectangular arrays of infinite extent. In an infinite array, a single element called a unit cell, is repeated uniformly an infinite number of times along a plane.
All arrays used in real-world scenarios are finite. But antenna arrays used in radio astronomy, air defense, or surveillance radar can have more than 1000 antenna elements. In such large arrays, the electromagnetic analysis of each element is tedious and time consuming.
Infinite array analysis ignores the effect of truncation (edge effect) at array edges. The method analyzes the behavior of the active antenna element as a function of frequency and scan. The goal of infinite array analysis is to extract the behavior of the active antenna element embedded in the array.
For infinite array analysis, array size must be greater than 10x10. The technique makes other assumptions:
Each element is identical.
Each element is uniformly excited in amplitude.
All elements are spaced uniformly in two dimensions.
To model an infinite array, the method of moments (MoM) formulation is changed to account for the infinite behavior by replacing Green's functions with periodic Green's functions. The periodic Green's function is an infinite double summation.
Periodic Greenรขโฌโขs Function
\begin{array}{l}g=\frac{{e}^{รขยยjkR}}{R}\\ R=|\stackrel{รขยย}{r}รขยย{\stackrel{รขยย}{r}}^{รขยยฒ}|\end{array}
\begin{array}{l}{g}_{\text{periodic}}=\underset{m=รขยย\infty }{\overset{\infty }{รขยย}}\underset{n=รขยย\infty }{\overset{\infty }{รขยย}}{e}^{j{\mathrm{รย}}_{mn}}\frac{{e}^{รขยยjk{R}_{mn}}}{{R}_{mn}}\\ {R}_{mn}=\sqrt{{\left(xรขยย{x}^{รขยยฒ}รขยย{x}_{m}\right)}^{2}+{\left(yรขยย{y}^{รขยยฒ}รขยย{y}_{n}\right)}^{2}+{\left(zรขยย{z}^{รขยยฒ}\right)}^{2}}\\ {\mathrm{รย}}_{mn}\text{รขยย}\text{รขยย}\text{รขยย}\text{รขยย}=รขยยk\left({x}_{m}\mathrm{sin}\mathrm{รยธ}\mathrm{cos}\mathrm{รย}+{y}_{n}\mathrm{sin}\mathrm{รยธ}\mathrm{sin}\mathrm{รย}\right)\\ {x}_{m}\text{รขยย}\text{รขยย}\text{รขยย}\text{รขยย}=mรขยย
{d}_{x},\text{}{y}_{n}=nรขยย
{d}_{y}\end{array}
The periodic Green's function has an additional exponential term added to the infinite sum. The รยฆmn term accounts for the scanning of the infinite array. The periodic Green's function also accounts for the effect of mutual coupling.
To create an infinite array, use the infiniteArray object to repeat a single antenna element (unit cell), infinitely along the X-Y plane. The layout function displays a typical unit cell.
layout(infarray)
You can use any antenna from the Antenna Toolboxโข as the unit cell. The unit cell requires a ground plane to specify the boundaries. You can use a reflector to back antennas that do not have a ground plane.
The default reflector properties are:
The default unit cell in an infinite array is a reflector that has a dipole as an exciter. The Spacing property gives the distance between the reflector and the exciter. The default infinite array properties are:
infarray = infiniteArray
show (infarray)
The dotted blue box bounds the unit cell. Ground plane length and ground plane width of the unit cell are the dimensions of the antenna element of the infinite array.
An antenna with a ground plane, such as a microstrip patch antenna, is specified directly as an Element of an infinite array.
infarray = infiniteArray('Element', patchMicrostrip)
infarray =
Element: [1x1 patchMicrostrip]
The Antenna Toolbox infinite array is located in the X-Y plane. Unit cells consisting of antennas with ground planes are also located in the X-Y plane. For antennas used as unit cells, such as the one in this example, you ignore the value of the Tilt property.
You scan a finite array by specifying the appropriate phase shift for each antenna element. In Antenna Toolbox, you specify the scan angle (in azimuth and elevation) and frequency for infinite array analysis. By default, an array always scans at boresight (azimuth = 0 degrees and elevation = 90 degrees).
To change the scan angles, change the values of ScanAzimuth and ScanElevation.
To calculate the scan impedance for an infinite array, use the impedance function as a function of scan angle. Fixing the scan angle pair and sweeping the frequency variable reveals the frequency dependency in the scan impedance. Because ScanAzimuth and ScanElevation are scalar values, you must use a for-loop to calculate the complete scan impedance of the array. For more information on calculating the scan impedance and the scan element pattern see, Infinite Array Analysis.
To calculate the scan element pattern using scan impedance, use these expressions:
{g}_{s}\left(\mathrm{รยธ}\right)=\frac{4{R}_{g}{R}_{\text{iso}}{g}_{\text{iso}}\left(\mathrm{รยธ}\right)}{{|{Z}_{s}\left(\mathrm{รยธ}\right)+{Z}_{g}|}^{2}}
Rg โ Resistance of generator
Zg โ Impedance of generator
Zs โ Scan impedance
giso(รยธ) โ Pattern of isolated element
Risoโ Resistance of isolated element
The scan element pattern can also be expressed in terms of the reflection coefficient, รโ(รยธ):
{g}_{s}\left(\mathrm{รยธ}\right)=\frac{4{R}_{\text{iso}}{g}_{\text{iso}}\left(\mathrm{รยธ}\right)}{{R}_{s}\left(\mathrm{รยธ}\right)}\text{รขยย}\left(1รขยย{|\mathrm{รย}\left(\mathrm{รยธ}\right)|}^{2}\right)
The Antenna Toolbox software calculates the scan element pattern of a finite array by driving just a single element. You terminate all the other elements using a suitable impedance. The resulting element pattern includes mutual coupling and is valid for all scan angles.
To calculate the scan element pattern of the finite arrays, first, create a reflector-backed dipole. Set the dipole dimensions to
\text{Length}\left(L\right)=0.495\mathrm{รยป}
\text{Width}\left(W\right)=\mathrm{รยป}/160
and the ground plane dimensions to 0.5รยป รโ 0.5รยป. Place the dipole at a distance of h = รยป/4 from the reflector. The ground plane dimensions set the boundaries of the unit cell. Create finite arrays of sizes 11x11, 15x15, and 17x17 using this unit cell.
For finite arrays, calculate the scan element pattern by driving a single element in the array. Terminate all other finite array elements using the broadside resistance of the infinite array. For an infinite array with the unit cell of dimensions 0.5รยป รโ 0.5รยป, the broadside resistance is 176 รขโยฆ. Calculate the scan element pattern for E-, D-, and H-planes of all three finite arrays.
To calculate the scan element pattern of an infinite array, create an infinite array using the same unit cell and the infiniteArray class. Calculate the scan impedance for three scan planes: E, D, and H. Compute the pattern of the isolated element (dipole backed by reflector). Finally, use the equations from the previous section to generate the scan element pattern for the infinite array.
Perform all analysis at 10 GHz. To compare the patterns of finite and infinite array, overlay them on the same plot.
To compare the scan element pattern of these array types and infinite arrays, repeat the process in case 1. Using these unit cell dimensions creates grating lobes. Terminate the finite arrays using 86-รขโยฆ resistance. For an infinite array with unit cell of dimensions 0.7รยป รโ 0.7รยป, the broadside resistance is 86 รขโยฆ.
For finite arrays of size greater than 10 x 10, the scan element patterns in the E-, D-, and H-planes match the patterns of the infinite array scan element.
As show in the Greenรขโฌโขs equations, the periodic Green's function has an infinite double summation in (m, n). When performing infinite array analysis, the number of terms in the double summation affects the accuracy of the final solution. Higher number of terms results in better accuracy but increases computation time.
By default, Antenna Toolbox uses 10 terms for each summation term (m, n) to perform infinite array analysis. The total summation term length is 2*10+1 (-10 to +10). To modify the number of terms, use the method numSummationTerms.
Higher number of terms are required if:
You observe negative values for scan resistance for certain scan angles at certain frequencies.
You must investigate for convergence when scan impedance shows slow variations.
[1] Mailloux, R. J. Phased Array Antenna Handbook. Norwood, MA: Artech House. 2nd Edition. 2005.
[2] Hansen, R. C. Phased Array Antennas. Hoboken, NJ: John Wiley & Sons Inc. 2nd Edition. 1998, pp. 221โ313.
[3] Allen, J. "Gain and impedance variation in scanned dipole arrays." IRE Transactions on Antennas and Propagation. Vol. 10, Number 5, September 1962, pp. 566โ572.
[4] Wasylkiwskyj, W., and W. Kahn. "Efficiency as a measure of size of a phased-array antenna." IEEE Transactions on Antennas and Propagation. Vol. 21, Number 6, November 1973, pp. 879โ884.
[5] Holter, H., and H. Steyskal. "On the size requirement for finite phased-array models." IEEE Transactions on Antennas and Propagation. Vol. 50, Number 6, June 2002, pp. 836โ840.
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This article is about the letter. For the programming language, see C (programming language). For other uses, see C (disambiguation).
For technical reasons, "C#" redirects here. For uses of C#, see C-sharp.
For technical reasons, "C# (programming language)" redirects here. For the programming language by Microsoft, see C Sharp (programming language).
7 Use as a number
See also: Hard and soft C
{\displaystyle \mathbb {C} }
Retrieved from "https://en.wikipedia.org/w/index.php?title=C&oldid=1086171480"
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Binomial Distribution - MATLAB & Simulink - MathWorks รญโขลรชยตยญ
0รขยยคpรขยยค1
f\left(x|N,p\right)=\left(\begin{array}{c}N\\ x\end{array}\right){p}^{x}{\left(1รขยยp\right)}^{Nรขยยx}\text{รขยย};\text{รขยย}x=0,1,2,...,N\text{รขยย},
F\left(x|N,p\right)=\underset{i=0}{\overset{x}{รขยย}}\left(\begin{array}{c}N\\ i\end{array}\right){p}^{i}{\left(1รขยยp\right)}^{Nรขยยi}\text{รขยย};\text{รขยย}x=0,1,2,...,N\text{รขยย},
pci2 = 2รโ1
Normal Distribution โ The normal distribution is a two-parameter continuous distribution that has parameters รยผ (mean) and รฦ (standard deviation). As N increases, the binomial distribution can be approximated by a normal distribution with รยต = Np and รฦ2 = Np(1 โ p). See Compare Binomial and Normal Distribution pdfs.
Poisson Distribution โ The Poisson distribution is a one-parameter discrete distribution that takes nonnegative integer values. The parameter รยป is both the mean and the variance of the distribution. The Poisson distribution is the limiting case of a binomial distribution where N approaches infinity and p goes to zero while Np = รยป. See Compare Binomial and Poisson Distribution pdfs.
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Predicting Stock Prices using classical machine Learning (Time Series #2) | Londogard Blog
[CA]: Time Series #2 - Predicting Stock Prices (Time Series) using classical Machine Learning
Today we will move from learning how to analyze Time Series to actually predicting them using simple models and data.
We'll be predicting Stocks from the top tech companies like Apple & Google.
In part #3 we'll move back to the crypto world!
Feel free to ignore the cells and simply run them, the lazy style ๐ฅฑ
Installing the important libraries...
And importing them...
Minor Analysisโ
df = pdr.get_data_yahoo(['AAPL', 'GOOGL', 'AMZN', 'MSFT', 'GE'])
32.807190 865.909973 852.530029 60.125767 212.658173 34.747501 865.909973 852.530029 64.410004 227.230774 ... 34.825001 863.750000 853.549988 64.529999 228.923080 61236400.0 1061700.0 2130600.0 14280200.0 2964208.0
33.154175 868.390015 852.969971 60.443142 214.241959 35.115002 868.390015 852.969971 64.750000 228.923080 ... 34.852501 867.940002 854.330017 64.550003 227.307693 102767200.0 1332900.0 2562200.0 24833800.0 3268564.0
Looks fine, but how much data did we download?
We can view the .index which is a DateTimeIndex and figure out how it stretches.
df.index[0],df.index[-1]
Hmm, 5 years, that should be enough to find some kind of patterns.
Now let us analyze this data further by looking at if the stocks correlate somehow! ๐ค
N.B. this analysis was first done by Heidi Mach, it's something I would never have done myself. Really cool results incoming!
df['Adj Close'].corr().style.background_gradient(cmap="Blues")
Holy macaron, that's a lot more correlated data than I expected! ๐
The seaborn library has a function called pairplot which plots this correlation, but using the points which is visually interesting in comparison to simply seeing the table above.
df = df.drop(columns="GE")
sns.pairplot(df.drop_duplicates())
<seaborn.axisgrid.PairGrid at 0x7f1fbb4ef650>
Does this in fact mean what that we can predict prices of a stock based on their competition? The correlation does suggest so.
First we'll try using a LinearRegression which simply said fits a line to be as close to all points as possible.
First we import LinearRegression through scikit-learn and then we add train_test_split which allows us to split our data into a training and testing dataset.
Whenever you test your Machine Learning or Deep Learning Models you never want to test it on data that it has trained on, as you might've overfitted the data and have a really good result until you see new data points.
The end-goal of a model is to generalize a problem and find the local minima which optimizes the funtion for the data points. By only looking at the same data we can't be sure we generalized correctly.
And the code ๐ฉโ๐ป
non_google_df = df.drop(columns="GOOGL")
X_train, X_valid, y_train, y_valid = train_test_split(non_google_df, df['GOOGL'], test_size=0.2)
We got our data divided into valid and train, we got a regression model in our clf.
Let us predict the data and view our r2_score and mean_absolute_error.
r2_score: (coefficient of determination) regression score function.
Best possible score is 1.0 and it can be negative (because the model can be arbitrarily worse). A constant model that always predicts the expected value of y, disregarding the input features, would get a score of 0.0.
mean_absolute_error: Mean absolute error regression loss.
preds = clf.predict(X_valid)
r2_score(y_valid, preds), mean_absolute_error(y_valid, preds)
(0.9431732611282428, 130.75344061010207)
R^2 = 93 \%
That's actually not bad at all, the mean_absolute_error being 129.7 is not very telling. Either we have to view the data to understand the magnituide, or we can apply MAPE which is the Mean Absolute Percentage Error.
Not sure if I'm lazy or simply want to show you the other function ๐ค, but I'll use MAPE!
mean_absolute_percentage_error(y_valid, preds)
< 9\%
Pretty acceptable considering we have not done anything except deliver data to one of the simplest models that exists!
Let's show this visually!
# px.line(y=[y_valid, preds])
Looks pretty good, but it is very messy... Something is off right?
The index is not a DateTimeIndex anymore because we shuffled the data in train_test_split -- a big difference is thereby applied.
y_valid.plot()
<matplotlib.axes._subplots.AxesSubplot at 0x7f1fb5e5f310>
y_valid.plot(legend="Valid")
pd.Series(preds, index=y_valid.index).plot(legend="Pred")
<matplotlib.axes._subplots.AxesSubplot at 0x7f1fb5dd5190>
Looks pretty fly, but can we take it further?
...yes we can! ๐
I see a few options, the two first being:
Scaling the data as errors at the end are larger than in the beggining based on stocks rising.
LinearRegression is a very simple yet efficient model that we can try to replace.
Let's start with the second point, scikit-learn has a multitude of regression-models, one being RandomForestRegressor that's pretty strong.
r2_score(y_valid, preds), mean_absolute_percentage_error(y_valid, preds)
R^2 >99\%
That's actually crazy. And MAPE is not even 2%.
<matplotlib.axes._subplots.AxesSubplot at 0x7f1fcaeacad0>
That's an incredibly fitted curve.
We most likely overfit the data.
We are looking at AMZN, AAPL and more data that is highly correlated during the same day as the one we wish to predict.
In the end this is a useless task, if we know the prices of today we'd also know GOOGL's prices!
We're using shuffled data, meaning that in a way we've seen the future and past values surrounding the predicted one. This is a regression problem and not really a forecasting problem, which is simpler than forecasting.
Impressive nontheless
Even as I'm aware of all the drawbacks I'm thouroughly impresed by the results we're seeing.
We should make use of the previous days data to make sure we are not "cheating".
Let's get on it! ๐ฏ
We'll be able to move, or shift, the data using หpd.DataFrame.shiftห which shifts the data either forwad (
+X
) or backwards (
-X
And while we're at it, let's group this up into a function.
pd.DataFrame.shift: Shift index by desired number of periods with an optional time freq.
def fit_validate_plot(X_train, X_valid, y_train, y_valid):
pd.DataFrame({'Valid': y_valid, 'Preds': preds}, index=y_valid.index).plot()
$R^2$: {r2_score(y_valid, preds)}
MAPE: {mean_absolute_percentage_error(y_valid, preds)}
MAE: {mean_absolute_error(y_valid, preds)}
And making use of it will now be easy! ๐
Refactoring and abstractions are incredibly important.
X_train, X_valid, y_train, y_valid = train_test_split(df.drop(columns="GOOGL").shift(1).iloc[1:], df['GOOGL'].iloc[1:], test_size=0.2)
fit_validate_plot(X_train, X_valid, y_train, y_valid)
$R^2$: 0.9948464033958241
MAPE: 0.019439064157954267
๐คฏ this is crazy impressive!
We made the task at hands legit by only using historical data of GOOGL's competitors. The
R^2
and MAPE is incredible.
It'd be interesting to investigate how badly we overfit the data, but that's for another day.
And how about if we don't shuffle the data? E.g. we do an actual forecast and not regression!
X_train, X_valid, y_train, y_valid = train_test_split(df.drop(columns="GOOGL").shift(1).iloc[1:], df['GOOGL'].iloc[1:], test_size=0.2, shuffle=False)
$R^2$: -7.02034763602467
MAPE: 0.24152517366886156
MAE: 660.6506098187159
๐คฏ๐ญ
What are we seeing and why?
Regression algorithms/models try to fit a line to multiple points and it should be able to guess what point the data has depending on its features. In our case the regression algorithm has never seen data as high as above y_train.max(), which means it can't guess the data.
Don't trust me? Simply validate by looking at the chart ๐.
What's one way to fix this? Scaling
How will we try to achieve this practically? LogReturn
๐ก You can also take the %-difference, which according to Taylors Theorem will approximate the LogReturn.
def log_return(x: pd.DataFrame) -> pd.DataFrame:
return x.apply(lambda x: np.log(x/x.shift(1))).dropna()
log_return(df).head()
df_lr = log_return(df)
X_train, X_valid, y_train, y_valid = train_test_split(df_lr.drop(columns="GOOGL").shift(1).iloc[1:], df_lr['GOOGL'].iloc[1:], test_size=0.2, shuffle=False)
$R^2$: -0.15979886803424925
MAPE: 33272784735.11796
Most certainly not perfect... Forecasting seems harder than expected based on our initial results...
And that's really because we weren't forecasting before, we were solving a regression-problem
Perhaps we need to use more data than simply the previous day?
Predicting Based on historical performanceโ
We might predict based on historical performance.
32.807190 865.909973 852.530029 60.125767
df = df[['GOOGL']]
โ
Only Google Data
โ Historical Data
So what should we do? One way to solve this is to use shift multiple times.
def build_history(df: pd.DataFrame, num_back: int) -> pd.DataFrame:
for i in range(num_back):
df.loc[:, f"t_{i}"] = df['GOOGL'].shift(i + 1)
build_history(df, 3).head()
865.909973 NaN NaN NaN
868.390015 865.909973 NaN NaN
870.000000 868.390015 865.909973 NaN
t_0
is the previous value,
t_1
two steps back, and so on.
This is actually very memory intense as our data grows X times, one time per time step we build. In part #3 we'll go through how one can solve this issue.
No we need to drop all places where we don't have any history. That is easily achieved by dropping NaN.
pd.DataFrame.dropna: Remove missing values.
axis attribute tells if you wish to drop rows or columns based on NaN, default is row.
df = build_history(df, 7)
Let's scale our data and then make predictions.
As previously,
X_train, X_valid, y_train, y_valid = train_test_split(df_lr.iloc[:, 1:], df_lr['GOOGL'], test_size=0.2, shuffle=False)
MAPE: 10166738051.820312
Not great, not awful. Some self-exercises:
How would we do without scaling?
How would we do without shuffling?
Any other ideas? Try 'em out!
# Test your own ideas
If you didn't try previously, try appling a rolling mean and rerun fit_validate_plot as this should reduce the "swings" and thereby be a little bit more predictable.
๐ก pd.DataFrame.Rolling: Provide rolling window calculations.
In other words: We slide a window on our data and do calculations, in our case mean. This window includes window, min_periods, center & more attributes which impacts size of window, how large minimal window can be, and more.
Validating what rolling.mean() does to our data:
df['GOOGL_ROLLING'] = df['GOOGL'].rolling(3).mean() # Rolling over 3 days mean
df[-100:].plot(y=['GOOGL', 'GOOGL_ROLLING'])
<matplotlib.axes._subplots.AxesSubplot at 0x7f1fb531bc90>
Zooming ๐
df_last_months = df[df.index > datetime(2021, 6, 6)]
# df_last_months.plot(y=['GOOGL', 'GOOGL_ROLLING'], backend='plotly')
The curve is very similar, but different.
Self-exercise: Test applying different functions like min, max and expanding window size into more days.
And validating what this does to our prediction.
df_lr = df.pct_change().dropna().rolling(3).mean().dropna()
MAPE: 0.8209516085248725
We're back! ๐ฅณ
It's not perfect, but we got something. And we can work with something. We can work with something... :)
Self-exercise: Validat how rolling would affect our non-history-based forecasting
Let's reverse our transformation to see what we'd actually predict in the end.
y_rolling = df['GOOGL'].rolling(3).mean().dropna()
y_train_non_scaled, y_valid_non_scaled = train_test_split(y_rolling, test_size=0.2, shuffle=False)
preds = (preds + 1).cumprod() # Cummulative multiplication, first day + 1%, but then we got -1%, that's 1.01 * 0.99
preds = preds * y_train_non_scaled.iloc[-1] # Scaling it up based on the last training value
# pd.DataFrame({'Preds': preds, 'Valid Rolling': y_valid_non_scaled[1:], 'Valid': df['GOOGL'].iloc[-len(preds):]}).plot(backend='plotly')
Seems as we're a little low in our predictions, but the curve is followed after all.
What issues are left?
We are not using an AutoRegressive model which might be interesting.
More about this in the next session
We are not using the "better" models, e.g. Neural Networks or statistic-model for Time Series like ARIMA.
Personally I'm very pleased with the results and can't wait to get started on part #3!
Extra Self Exercisesโ
Try different window-sizes with rolling
Try different length of history to predict new result on
Test new architectures
Find your own way to improve the results
Predicting Based on historical performance
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Dynamics of FPSO in the Horizontal Plane | J. Offshore Mech. Arct. Eng. | ASME Digital Collection
Antonio C. Fernandes and,
Antonio C. Fernandes and
J. Offshore Mech. Arct. Eng. Nov 2002, 124(4): 173 (1 pages)
Fernandes and , A. C., and Chakrabarti, S. K. (October 22, 2002). "Dynamics of FPSO in the Horizontal Plane ." ASME. J. Offshore Mech. Arct. Eng. November 2002; 124(4): 173. https://doi.org/10.1115/1.1514497
ships, stability, dynamics
Dynamics (Mechanics), FPSO, Ships, Stability
The Floating Production Storage and Offloading (FPSO) system is a novel application for hydrocarbon production, which is usually designed to stay permanently moored in an offshore, open sea location. The traditional FPSO has been designed for quick intervention and for temporary applications in a well-defined environment in benign waters. The prevailing FPSO hulls are ship-shaped and their permanent applications are, in a way, a โback-to-the-pastโ scenario. The ship form often has inadequate station-keeping and seakeeping behavior. This led to the invention and development of alternative hull shapes, such as Semi-Submersibles, TLPs and recently, Spars. In fact, the list now includes FPSOs adapted from Very Large Crude Carriersโ (VLCC) hulls, which, due to their size (of more than 320 m), may be considered the forerunners of Very Large Floating Structures (VLFS). However, with these latter developments came several traditional pitfalls that were earlier considered solved.
The station-keeping of FPSOs with regard to dynamic stability and consequences to mooring line sizes and arrangements as well as offloading operations are being re-stated and solved nowadays in a different context. This new effort became apparent during the organization of OMAE2001. For this reason, a Special Session, which is reflected in this Special Issue, was organized including experts presently dedicated to the dynamics of FPSO in the horizontal plane.
Basically, there are several complementary but competing ways that this problem may be understood. The so-called Cross Flow Principle was well developed for FPSO, and several time-domain analysis software programs were based on it. On the other hand, due to their continued support to the Simulators for ship steering training, the maneuvering models have also been implemented both for stability analysis and for time-domain numerical analysis. The maneuvering models may be either cubic and usually used for fast motions or quadratic (meaning
|v|v
to maintain its odd nature) and used for slow motions. Sometimes, even fifth order models have been suggested.
Besides these techniques, the intrinsic dynamic quality of the three degrees of freedom system must also be realized. This was pioneered into directional stability for moving ships, and what gave the initial elements for the Single Point Mooring (SPM) system, and subsequently, for the turret mooring system. This requires evaluations in a multi-component environment with wind, waves and current acting simultaneously, each one imposing its own dynamic condition, in a vessel with ever-varying draft due to the offloading requirement. Finally, it is also important to mention another aspect of this technology development that is the small-scale model testing, since besides the traditional ones, new ones are being proposed.
These combinations of traditional and novel approaches for an FPSO as a permanent platform in open sea are difficult to find in one collection. Hence, by organizing this special JOMAE issue, we hope to be contributing to further developments.
Extensions and Improvements to the Solutions for Linear Tank Dynamics
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NDF_CGET
The routine obtains the value of the specified character component of an NDF (i.e. the value of the LABEL, TITLE or UNITS component).
CALL NDF_CGET( INDF, COMP, VALUE, STATUS )
\ast
\ast
Name of the character component whose value is required: โLABELโ, โTITLEโ or โUNITSโ.
\ast
\ast
If the requested component is in an undefined state, then the VALUE argument will be returned unchanged. A suitable default should therefore be established before calling this routine.
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Global Constraint Catalog: Cdiscrepancy
<< 5.120. diffn_include5.122. disj >>
[Focacci01] and [vanHoeve05]
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๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐๐๐}\right)
๐บ\ge 0
๐บ\le |\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
๐บ
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
that take their value in their respective sets of bad values.
\left(\begin{array}{c}โฉ\begin{array}{cc}\mathrm{๐๐๐}-4\hfill & \mathrm{๐๐๐}-\left\{1,4,6\right\},\hfill \\ \mathrm{๐๐๐}-5\hfill & \mathrm{๐๐๐}-\left\{0,1\right\},\hfill \\ \mathrm{๐๐๐}-5\hfill & \mathrm{๐๐๐}-\left\{1,6,9\right\},\hfill \\ \mathrm{๐๐๐}-4\hfill & \mathrm{๐๐๐}-\left\{1,4\right\},\hfill \\ \mathrm{๐๐๐}-1\hfill & \mathrm{๐๐๐}-\varnothing \hfill \end{array}โช,2\hfill \end{array}\right)
\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
constraint holds since exactly
๐บ=2
variables (i.e., the first and fourth variables) of the
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection take their value within their respective sets of bad values.
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|>1
๐บ<|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
๐บ
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}\left(\mathrm{๐๐๐๐๐}\right)
๐บ\left(+\right)
Limited discrepancy search was first introduced by M. L. Ginsberg and W. D. Harvey as a search technique in [GinsbergHarvey95]. Later on, discrepancy based filtering was presented in the PhD thesis of F. Focacci [Focacci01]. Finally the
\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
constraint was explicitly defined in the PhD thesis of W.-J. van Hoeve [vanHoeve05].
\mathrm{๐๐๐๐๐}
\mathrm{๐๐}_\mathrm{๐๐๐}
heuristics: heuristics, limited discrepancy search.
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐ธ๐ฟ๐น}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐๐๐๐๐}\right)
\mathrm{๐๐}_\mathrm{๐๐๐}
\left(\mathrm{๐๐๐๐๐๐๐๐๐}.\mathrm{๐๐๐},\mathrm{๐๐๐๐๐๐๐๐๐}.\mathrm{๐๐๐}\right)
\mathrm{๐๐๐๐}
=๐บ
The arc constraint corresponds to the constraint
\mathrm{๐๐}_\mathrm{๐๐๐}\left(\mathrm{๐๐๐๐๐๐๐๐๐}.\mathrm{๐๐๐},\mathrm{๐๐๐๐๐๐๐๐๐}.\mathrm{๐๐๐}\right)
defined in this catalogue. We employ the
\mathrm{๐๐ธ๐ฟ๐น}
\mathrm{๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
|
Your science teacher needs to prepare a solution that is
60\%
acid for your upcoming class experiment. The problem is that the company your teacher orders supplies from messed up the last order. They sent
45\%
65\%
acid solutions instead of
60\%
. Your science teacher needs your math help! If you start with
350
mL of the
45\%
solution, how much of
65\%
solution must be added to end up with the
60\%
solution needed for the experiment?
350\left(0.45\right)+x\left(0.65\right)=\left(350+x\right)\left(0.60\right)
|
Both experimental and theoretical investigations on the heat transfer and flow friction characteristics of compact cold plates have been performed. From the results, the local and average temperature rises on the cold plate surface increase with increasing chip heat flux or decreasing air mass flow rate. Besides, the effect of chip heat flux on the thermal resistance of cold plate is insignificant; while the thermal resistance of cold plate decreases with increasing air mass flow rate. Three empirical correlations of thermal resistance in terms of air mass flow rate with a power of
โ0.228
are presented. As for average Nusselt number, the effect of chip heat flux on the average Nusselt number is insignificant; while the average Nusselt number of the cold plate increases with increasing Reynolds number. An empirical relationship between
Nuยฏcp
and Re can be correlated. In the flow frictional aspect, the overall pressure drop of the cold plate increases with increasing air mass flow rate; while it is insignificantly affected by chip heat flux. An empirical correlation of the overall pressure drop in terms of air mass flow rate with a power of 1.265 is presented. Finally, both heat transfer performance factor
โjโ
and pumping power factor
โfโ
decrease with increasing Reynolds number in a power of 0.805; while they are independent of chip heat flux. The Colburn analogy can be adequately employed in the study.
|
Poynting's theorem | Less Than Epsilon
If we have some charge and current configuration which, produces some fields
\mathbf{E}
\mathbf{B}
. After a while, the charges move around.
The question is, how much work
dW
is done by the electromagnetic forces in the interval
dt
To do this, we simply compute the work, which is
dW = \mathbf{F} \cdot d \mathbf{l}=q(\mathbf{E}+\mathbf{v} \times \mathbf{B}) \cdot \mathbf{v} d t=q \mathbf{E} \cdot \mathbf{v} d t
We can rewrite this in terms of charge and current densities. Swap out
q \rightarrow \rho d\tau
\rho \mathbf{v} \rightarrow \mathbf{J}
\frac{d W}{d t}=\int_{\mathcal{V}}(\mathbf{E} \cdot \mathbf{J}) d \tau
\mathbf{E} \cdot \mathbf{J}
is the work done per time, per unit volume, or the power per volume. We would like to know what this quantity is.
Begin with the Ampere-Maxwellโs Law:
\nabla \times \mathbf{B}=\mu_{0} \mathbf{J}+\mu_{0} \varepsilon_{0} \frac{\partial \mathbf{E}}{\partial t}
and so we can dot both sides with
\mathbf{E}
, using this equation to get rid of
\mathbf{J}
\mathbf{E} \cdot \mathbf{J}=\frac{1}{\mu_{0}} \mathbf{E} \cdot(\nabla \times \mathbf{B})-\epsilon_{0} \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t}
We now have to deal with two terms. The first we can use the following vector calculus identity:
\nabla \cdot(\mathbf{A} \times \mathbf{B})=\mathbf{B} \cdot(\nabla \times \mathbf{A})-\mathbf{A} \cdot(\nabla \times \mathbf{B})
and now plugging in the fields, we have
\nabla \cdot(\mathbf{E} \times \mathbf{B})=\mathbf{B} \cdot(\nabla \times \mathbf{E})-\mathbf{E} \cdot(\nabla \times \mathbf{B})
using Faradayโs Law (
(\nabla \times \mathbf{E}=-\partial \mathbf{B} / \partial t
\mathbf{E} \cdot(\nabla \times \mathbf{B})=-\mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t}-\nabla \cdot(\mathbf{E} \times \mathbf{B})
Using another calculus identity, we have:
\mathbf{B} \cdot \frac{\partial \mathbf{B}}{\partial t}=\frac{1}{2} \frac{\partial}{\partial t}\left(B^{2}\right), \quad \text { and } \quad \mathbf{E} \cdot \frac{\partial \mathbf{E}}{\partial t}=\frac{1}{2} \frac{\partial}{\partial t}\left(E^{2}\right)
\mathbf{E} \cdot \mathbf{J}=-\frac{1}{2} \frac{\partial}{\partial t}\left(\epsilon_{0} E^{2}+\frac{1}{\mu_{0}} B^{2}\right)-\frac{1}{\mu_{0}} \nabla \cdot(\mathbf{E} \times \mathbf{B})
And plugging it into our original expression for work, then calling the divergence theorem
\int_{\mathcal{V}}(\nabla \cdot \mathbf{v}) d \tau=\oint_{S} \mathbf{v} \cdot d \mathbf{a}
on the second term allows to convert a volume integral into a surface integral. Finally, we have
\frac{d W}{d t}=-\frac{d}{d t} \int_{\mathcal{V}} \frac{1}{2}\left(\epsilon_{0} E^{2}+\frac{1}{\mu_{0}} B^{2}\right) d \tau-\frac{1}{\mu_{0}} \oint_{\mathcal{S}}(\mathbf{E} \times \mathbf{B}) \cdot d \mathbf{a}
Which is the work energy theorem of electrodynamics: The first term is the total energy stored in electromagnetic fields:
u = \frac{1}{2} \left( \epsilon_0 E^2 + \frac{1}{\mu_0} B^2 \right)
The second term is the rate at which energy is transported out of the surface. The mysterious second term is defined as the Poynting vector. It is interpreted as the energy per unit time, per unit area.
\boxed{\mathbf{S} \equiv \frac{1}{\mu_{0}}(\mathbf{E} \times \mathbf{B})}
\mathbf{S} \cdot d \mathbf{a}
is the energy leaving surface
d \mathbf{a}
Finally, we can write the above equation into a more compact form:
\frac{d W}{d t}=-\frac{d}{d t} \int_{\mathcal{V}} u d \tau-\oint_{\mathcal{S}} \mathbf{S} \cdot d \mathbf{a}
Now what is the meaning of this equation? Imagine we do work on some charge configuration. Either the energy stored in the fields had to have decreased, or the energy must have went outside the surface.
The second interpretation could use a little more work. What does it mean for energy to leave a surface? After all, we said that the volume
\mathcal{V}
is arbitrary, and
\mathcal{S}
is only required to be the boundary of such a volume.
To be concrete, letโs say our system is a battery, and pick
\mathcal{V}
to be the volume of the battery. In a circuit, the battery is clearly doing work to drive say a lightbulb. (increasing
dW/dt
So where does the energy come from? If we say there arenโt any fields in the battery, then energy really is leaving the battery to drive the circuit, in order word the second term must decrease.
dW/dt = 0
, then using the divergence theorem again gives
\int \frac{\partial u}{\partial t} d \tau=-\oint \mathbf{S} \cdot d \mathbf{a}=-\int(\mathbf{\nabla} \cdot \mathbf{S}) d \tau
and removing the integrals gives us:
\frac{\partial u}{\partial t}=-\nabla \cdot \mathbf{S}
which is the continuity equation for energy! This says that energy is locally conserved!
If we compare this to the continuity equation for fluids, we see that the Poynting vector
\mathbf{S}
really is the energy flux.
Image: Wikipedia. Dipole radiation of a dipole vertically in the page showing electric field strength (colour) and Poynting vector (arrows) in the plane of the page.
|
Setting Up Equations | Brilliant Math & Science Wiki
Jordan Calmes, Janae Pritchett, A Former Brilliant Member, and
Setting up equations, or writing equations, involves translating a problem from words into a mathematical statement.
Basic Steps to Setting Up Equations
"Translating" Words Into Numbers
Determine what the question is asking.
Write down the relevant information in simple statements.
Assign symbols to unknown values that need to be found.
Determine how the statements relate to each other mathematically.
An exhibit at the zoo contains four times as many giraffes as it has elephants. If the zoo has 12 giraffes, how many animals are there?
1:4 ratio of elephants to giraffes.
Find the total number of giraffes and elephants.
Write down relevant information:
Total animals is giraffes and elephants. Four times as many giraffes (More giraffes than elephants).
Assign symbols:
Here, we're interested in knowing the total number of animals, and in order to find that, we'll need to know the values for elephants and giraffes, so assign each variable a symbol: A=Animals, E=Elephants, G=Giraffes.
While the sentences from step 2 may not win any literary prizes, words like is, and, or times are easy to translate into mathematical symbols (
=, +, \text{ and }\times
There are two equations, one for each sentence. A = G + E, and E * 4 = G.
English and mathematics can be thought of as two separate languages, each with its own symbols, grammar, and style rules. Setting up an equation is similar to translating a paragraph between two languages. The result should contain the same information as the original piece. A direct translation is not always possible, because words that exist in one language may not exist in another, or a word-for-word translation may not make sense in the final language.
Write two equations from the following information:
A bat and ball cost $100, but the bat costs $90 more than the ball.
Bat - Ball = 90.
_\square
In the above equations, we have two unknowns: the price of the bat and the price of the ball. The unknown parts of the equation, or variables, may have one or more answers, depending on the problem. Variables are usually symbolized with a letter in algebraic equations.
Rewrite the two equations using
x
y
to represent the variables.
Bat - Ball = 90
x=
y=
x+y=100
x-y=90
_\square
3J+J+0.5J=M
E+S=J
J+2.5J=M
3S+J+0.5E=M
Sam (
S
) owns three times as many marbles as Jenny (
J
). Ed (
E
) owns half as many marbles as Jenny. How many marbles (
M
) do they own together?
Story problems can contain lots of information, including details that are not necessary to solve the problem. Additionally, the problem may assume general knowledge on the reader's part, and not explain all the information explicitly. Writing that information down in a streamlined form is an efficient way to solve problems.
At the beginning of January, Johnny and Mary decide they want to buy a new car together. The car costs $8,000 and they want to buy it in July. Each of them will pay half the total amount. Mary has $2,000 in savings and her income is $1,200 per month. Johnny has no savings and his income is $2,500 per month. How much money should Johnny save per month in order to reach this goal?
In reading this question, the person solving it may start streamlining information in her head. January to July is six months. They need a total of $8,000 and have $2,000, so they need another $6,000, or $1,000 per month. While this answer is true, it's not correct, because the solver tackled the wrong question. The problem asks how much money Johnny needs to save per month.
Another reader might be overwhelmed by the number of details in the problem, and not start with an equation in their head at all. This reader might try to guess how much money Johnny would need per month. Let's see, if Johnny saves $1,000 per month for six months, he would have $6,000 by the time they go to buy the car. That's more than enough. While that solution may work in real life, it may also leave Johnny without enough money left to pay his rent.
Neither of these students is going to get points on an exam, or give Johnny the answer he's really looking for.
While no single detail in this problem is complicated, the number of facts the student is being asked to remember may keep her from finding the answer quickly and efficiently without writing out an equation. Additionally, the details presented about Mary are distracting.
Translating words into symbols is a great way to get started solving word problems. The above example asks us to find the money Johnny needs to save. The problem states that he needs to have half of the $8,000 by July. Those two sentences can be rewritten as follows:
Johnny = J = (\$8,000/2) = \$4,000 \implies J = \$4,000.
Johnny needs $4,000 by July. He has six months to save, and needs to save an unknown number of dollars per month. Assigning the unknown amount the symbol
x
allows those pieces of information to be added to the equation. The result is a single equation with a single variable, which can be solved with division.
\begin{aligned} \$4,000 &= 6x\\ \$4,000/6 &= 6x/6\\ \Rightarrow \$667 &= x. \end{aligned}
As the problems become more complicated, multiple steps or multiple variables may be involved. A problem may also require writing and solving several equations, using information found in one step to solve the next.
F=2A+B; A=2B
A+B=F; 2A=B
F=A+B; A=2B
F=A+2B; F=24
There are 24 pieces of fruit (
F
) in a basket. Each piece of fruit is either an apple or a banana. There are twice as many apples (
A
) as there are bananas (
B
Phil had 3 times as much money as Anne. After Phil gave $285 to Anne, he had twice as much money as she did. How much money did Phil have at first (in $)?
Now try solving some equations once you get them set up!
When I was 14 years old, my father was three times my age.
Now, he is twice my age. How old am I?
In your dining room, there are
P
people,
C
armless chairs, and one table with four legs. There are 10 arms in the room and 18 legs. How many more chairs do you need to get so that everyone can sit down?
Cite as: Setting Up Equations. Brilliant.org. Retrieved from https://brilliant.org/wiki/setting-up-equations/
|
Probability Problem on Variance - Definition: Sunny Shoot-out - Worranat Pakornrat | Brilliant
Sunny Shoot-out
You are planting 5 sunflowers in each of the 2 gardens, where these sets of plants shoot out in varying heights.
Shown above is the graph depicting the height of each sunflower, where the red line indicates the mean height of sunflower population
\mu
For example, the shortest sunflower in Garden A is 5 cm shorter than average while the highest one in Garden B is 7 cm higher than average.
Which set of sunflowers has higher population variance?
Garden A Garden B They have the same variance Not enough information
|
Global Constraint Catalog: Csymmetric_gcc
<< 5.398. symmetric_cardinality5.400. temporal_path >>
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐}\left(\mathrm{๐
๐ฐ๐๐},\mathrm{๐
๐ฐ๐ป๐}\right)
\mathrm{๐๐๐๐}
\mathrm{๐
๐ฐ๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐}-\mathrm{๐๐๐},\mathrm{๐๐๐}-\mathrm{๐๐๐๐},\mathrm{๐๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐
๐ฐ๐ป๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐}-\mathrm{๐๐๐},\mathrm{๐๐๐}-\mathrm{๐๐๐๐},\mathrm{๐๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐},\left[\mathrm{๐๐๐๐๐},\mathrm{๐๐๐},\mathrm{๐๐๐๐}\right]\right)
|\mathrm{๐
๐ฐ๐๐}|\ge 1
\mathrm{๐
๐ฐ๐๐}.\mathrm{๐๐๐๐๐}\ge 1
\mathrm{๐
๐ฐ๐๐}.\mathrm{๐๐๐๐๐}\le |\mathrm{๐
๐ฐ๐๐}|
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐},\mathrm{๐๐๐๐๐}\right)
\mathrm{๐
๐ฐ๐๐}.\mathrm{๐๐๐๐}\ge 0
\mathrm{๐
๐ฐ๐๐}.\mathrm{๐๐๐๐}\le |\mathrm{๐
๐ฐ๐ป๐}|
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐},\left[\mathrm{๐๐๐๐๐},\mathrm{๐๐๐},\mathrm{๐๐๐๐}\right]\right)
|\mathrm{๐
๐ฐ๐ป๐}|\ge 1
\mathrm{๐
๐ฐ๐ป๐}.\mathrm{๐๐๐๐๐}\ge 1
\mathrm{๐
๐ฐ๐ป๐}.\mathrm{๐๐๐๐๐}\le |\mathrm{๐
๐ฐ๐ป๐}|
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐},\mathrm{๐๐๐๐๐}\right)
\mathrm{๐
๐ฐ๐ป๐}.\mathrm{๐๐๐๐}\ge 0
\mathrm{๐
๐ฐ๐ป๐}.\mathrm{๐๐๐๐}\le |\mathrm{๐
๐ฐ๐๐}|
Put in relation two sets: for each element of one set gives the corresponding elements of the other set to which it is associated. In addition, enforce a cardinality constraint on the number of occurrences of each value.
\left(\begin{array}{c}โฉ\begin{array}{ccc}\mathrm{๐๐๐๐๐}-1\hfill & \mathrm{๐๐๐}-\left\{3\right\}\hfill & \mathrm{๐๐๐๐}-1,\hfill \\ \mathrm{๐๐๐๐๐}-2\hfill & \mathrm{๐๐๐}-\left\{1\right\}\hfill & \mathrm{๐๐๐๐}-1,\hfill \\ \mathrm{๐๐๐๐๐}-3\hfill & \mathrm{๐๐๐}-\left\{1,2\right\}\hfill & \mathrm{๐๐๐๐}-2,\hfill \\ \mathrm{๐๐๐๐๐}-4\hfill & \mathrm{๐๐๐}-\left\{1,3\right\}\hfill & \mathrm{๐๐๐๐}-2\hfill \end{array}โช,\hfill \\ โฉ\begin{array}{ccc}\mathrm{๐๐๐๐๐}-1\hfill & \mathrm{๐๐๐}-\left\{2,3,4\right\}\hfill & \mathrm{๐๐๐๐}-3,\hfill \\ \mathrm{๐๐๐๐๐}-2\hfill & \mathrm{๐๐๐}-\left\{3\right\}\hfill & \mathrm{๐๐๐๐}-1,\hfill \\ \mathrm{๐๐๐๐๐}-3\hfill & \mathrm{๐๐๐}-\left\{1,4\right\}\hfill & \mathrm{๐๐๐๐}-2,\hfill \\ \mathrm{๐๐๐๐๐}-4\hfill & \mathrm{๐๐๐}-\varnothing \hfill & \mathrm{๐๐๐๐}-0\hfill \end{array}โช\hfill \end{array}\right)
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
3\in \mathrm{๐
๐ฐ๐๐}\left[1\right].\mathrm{๐๐๐}โ1\in \mathrm{๐
๐ฐ๐ป๐}\left[3\right].\mathrm{๐๐๐}
1\in \mathrm{๐
๐ฐ๐๐}\left[2\right].\mathrm{๐๐๐}โ2\in \mathrm{๐
๐ฐ๐ป๐}\left[1\right].\mathrm{๐๐๐}
1\in \mathrm{๐
๐ฐ๐๐}\left[3\right].\mathrm{๐๐๐}โ3\in \mathrm{๐
๐ฐ๐ป๐}\left[1\right].\mathrm{๐๐๐}
2\in \mathrm{๐
๐ฐ๐๐}\left[3\right].\mathrm{๐๐๐}โ3\in \mathrm{๐
๐ฐ๐ป๐}\left[2\right].\mathrm{๐๐๐}
1\in \mathrm{๐
๐ฐ๐๐}\left[4\right].\mathrm{๐๐๐}โ4\in \mathrm{๐
๐ฐ๐ป๐}\left[1\right].\mathrm{๐๐๐}
3\in \mathrm{๐
๐ฐ๐๐}\left[4\right].\mathrm{๐๐๐}โ4\in \mathrm{๐
๐ฐ๐ป๐}\left[3\right].\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐}\left[1\right].\mathrm{๐๐๐}=\left\{3\right\}
\mathrm{๐
๐ฐ๐๐}\left[2\right].\mathrm{๐๐๐}=\left\{1\right\}
\mathrm{๐
๐ฐ๐๐}\left[3\right].\mathrm{๐๐๐}=\left\{1,2\right\}
\mathrm{๐
๐ฐ๐๐}\left[4\right].\mathrm{๐๐๐}=\left\{1,3\right\}
\mathrm{๐
๐ฐ๐ป๐}\left[1\right].\mathrm{๐๐๐}=\left\{2,3,4\right\}
\mathrm{๐
๐ฐ๐ป๐}\left[2\right].\mathrm{๐๐๐}=\left\{3\right\}
\mathrm{๐
๐ฐ๐ป๐}\left[3\right].\mathrm{๐๐๐}=\left\{1,4\right\}
\mathrm{๐
๐ฐ๐ป๐}\left[4\right].\mathrm{๐๐๐}=\varnothing
|\mathrm{๐
๐ฐ๐๐}|>1
|\mathrm{๐
๐ฐ๐ป๐}|>1
\mathrm{๐
๐ฐ๐๐}
\mathrm{๐
๐ฐ๐ป๐}
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
is a variant of personnel assignment problem, where one person can be assigned to perform between
m
\left(n\le m\right)
p
q
\left(p\le q\right)
\mathrm{๐
๐ฐ๐๐}
\mathrm{๐
๐ฐ๐ป๐}
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
constraint by allowing a variable to take more than one value. It corresponds to a variant of the
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
constraint described in [KocjanKreuger04] where the occurrence variables of the
\mathrm{๐
๐ฐ๐๐}
\mathrm{๐
๐ฐ๐ป๐}
collections are replaced by fixed intervals.
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
\mathrm{๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐ก๐๐}
\mathrm{๐๐๐๐๐๐๐๐}
\mathrm{๐๐}_\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐}
\mathrm{๐
๐ฐ๐ป๐}
\mathrm{๐๐
๐๐ท๐๐ถ๐}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐},\mathrm{๐๐๐๐}\right)
โข
\mathrm{๐๐}_\mathrm{๐๐๐}
\left(\mathrm{๐๐๐๐}.\mathrm{๐๐๐๐๐},\mathrm{๐๐๐๐}.\mathrm{๐๐๐}\right)โ
\mathrm{๐๐}_\mathrm{๐๐๐}
\left(\mathrm{๐๐๐๐}.\mathrm{๐๐๐๐๐},\mathrm{๐๐๐๐}.\mathrm{๐๐๐}\right)
โข\mathrm{๐๐๐๐}.\mathrm{๐๐๐๐}=\mathrm{๐๐๐๐}_\mathrm{๐๐๐}\left(\mathrm{๐๐๐๐}.\mathrm{๐๐๐}\right)
โข\mathrm{๐๐๐๐}.\mathrm{๐๐๐๐}=\mathrm{๐๐๐๐}_\mathrm{๐๐๐}\left(\mathrm{๐๐๐๐}.\mathrm{๐๐๐}\right)
\mathrm{๐๐๐๐}
=|\mathrm{๐
๐ฐ๐๐}|*|\mathrm{๐
๐ฐ๐ป๐}|
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐๐๐๐๐}
Parts (A) and (B) of Figure 5.399.1 respectively show the initial and final graph. Since we use the
\mathrm{๐๐๐๐}
\mathrm{๐๐ข๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐
๐๐ท๐๐ถ๐}
\mathrm{๐
๐ฐ๐๐}
\mathrm{๐
๐ฐ๐ป๐}
|\mathrm{๐
๐ฐ๐๐}|ยท|\mathrm{๐
๐ฐ๐ป๐}|
|\mathrm{๐
๐ฐ๐๐}|ยท|\mathrm{๐
๐ฐ๐ป๐}|
\mathrm{๐๐๐๐}=|\mathrm{๐
๐ฐ๐๐}|ยท|\mathrm{๐
๐ฐ๐ป๐}|
\mathrm{๐๐๐๐}\ge |\mathrm{๐
๐ฐ๐๐}|ยท|\mathrm{๐
๐ฐ๐ป๐}|
\underline{\overline{\mathrm{๐๐๐๐}}}
\overline{\mathrm{๐๐๐๐}}
|
Home : Support : Online Help : Science and Engineering : Units : Environments : Simple : frem
floating-point remainder function in the Simple Units environment
frem(x1, x2)
In the Simple Units environment, the frem procedure overrides the top-level frem procedures. The units for the arguments need to have the same dimension, and the result is given a unit corresponding to that dimension.
More precisely, if x1 and x2 have the same unit, then the result has that unit. If they have different units of the same dimension, then the result has the default unit for that dimension (as set by the UseUnit or UseSystem commands).
\mathrm{unit}
\mathrm{with}โก\left(\mathrm{Units}[\mathrm{Simple}]\right):
\mathrm{frem}โก\left(\frac{3}{2}โข\mathrm{Unit}โก\left('m'\right),1.49โข\mathrm{Unit}โก\left('m'\right)\right)
\textcolor[rgb]{0,0,1}{0.010000000}\textcolor[rgb]{0,0,1}{โข}โฆ\textcolor[rgb]{0,0,1}{m}โง
\mathrm{frem}โก\left(\frac{3}{2}โข\mathrm{Unit}โก\left('\mathrm{ft}'\right),1.49โข\mathrm{Unit}โก\left('\mathrm{ft}'\right)\right)
\textcolor[rgb]{0,0,1}{0.010000000}\textcolor[rgb]{0,0,1}{โข}โฆ\textcolor[rgb]{0,0,1}{\mathrm{ft}}โง
\mathrm{frem}โก\left(\frac{3}{2}โข\mathrm{Unit}โก\left('m'\right),1.49โข\mathrm{Unit}โก\left('\mathrm{ft}'\right)\right)
\textcolor[rgb]{0,0,1}{0.137544000}\textcolor[rgb]{0,0,1}{โข}โฆ\textcolor[rgb]{0,0,1}{m}โง
The Units[Simple][frem] command was introduced in Maple 2021.
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Entering Commands in 2-D Math - Maple Help
Home : Support : Online Help : Create Maple Worksheets : Enter Expressions : Enter 2-D Math : Entering Commands in 2-D Math
Maple has an extensive library of mathematical commands. These can be entered using 1-D Math or 2-D Math. With 2-D Math, you can use the command form or the notation form, both described below. When calling commands with options, the command form must be used in most cases.
Command and Notation Forms
Commands may be entered in 2-D math as they are entered in 1-D math. This is called command form, as the command name is typed in full. Here is an example using the int command, using both 1-D and 2-D math input.
int(x^2+x/5, x=0..4);
\frac{\textcolor[rgb]{0,0,1}{344}}{\textcolor[rgb]{0,0,1}{15}}
\mathrm{int}\left({x}^{2}+\frac{x}{5}, x=0..4\right);
\frac{\textcolor[rgb]{0,0,1}{344}}{\textcolor[rgb]{0,0,1}{15}}
Some commands allow alternative forms of entry that use mathematical notation. Many of these forms are available in Maple's palettes. For example, the integral computed above can be entered using the definite integration template from the Expression palette:
{โซ}_{0}^{4}{x}^{2}+\frac{x}{5} โ
x
\frac{\textcolor[rgb]{0,0,1}{344}}{\textcolor[rgb]{0,0,1}{15}}
Additional forms are available through command completion. For example, if you type int and invoke command completion, you will be offered, among other choices, the option to compute the Cauchy principal value:
\mathrm{PV}โข{\int }_{-1}^{2}\frac{1}{{x}^{3}} โ
x
\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{8}}
Many Maple commands offer options that let you control various aspects of the command's behavior or the form of the output. For example, the int/details help page shows several options for the int command. In general, when using an option, it is necessary to use the command form rather than mathematical notation. You can still use 2-D math notation for the command's arguments. However, the command name must be given explicitly. In the example below, the option 'numeric' causes the command to return a floating-point result.
\mathrm{int}\left({x}^{2}+\frac{x}{5}, x=0..4, \mathrm{numeric}\right)
\textcolor[rgb]{0,0,1}{22.93333333}
2-D Math, Entering Expressions
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Musiker, Gregg ; Roby, Tom
Algebraic Combinatorics, Tome 2 (2019) no. 2, pp. 275-304.
Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on order ideals (equivalently on antichains) of a poset
P
, which when iterated on special posets, has unexpectedly nice properties in terms of periodicity, cyclic sieving, and homomesy (statistics whose averages over each orbit are constant). In this context, rowmotion appears to be related to AuslanderโReiten translation on certain quivers, and birational rowmotion to
Y
-systems of type
{A}_{m}ร{A}_{n}
described in Zamolodchikov periodicity.
We give a formula in terms of families of non-intersecting lattice paths for iterated actions of the birational rowmotion map on a product of two chains. This allows us to give a much simpler direct proof of the key fact that the period of this map on a product of chains of lengths
r
s
r+s+2
(first proved by D. Grinberg and the second author), as well as the first proof of the birational analogue of homomesy along files for such posets.
DOI : https://doi.org/10.5802/alco.43
Mots clรฉs : birational rowmotion, dynamical algebraic combinatorics, homomesy, periodicity, toggling.
@article{ALCO_2019__2_2_275_0,
author = {Musiker, Gregg and Roby, Tom},
title = {Paths to {Understanding} {Birational} {Rowmotion} on {Products} of {Two} {Chains}},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
doi = {10.5802/alco.43},
url = {http://www.numdam.org/articles/10.5802/alco.43/}
AU - Musiker, Gregg
AU - Roby, Tom
TI - Paths to Understanding Birational Rowmotion on Products of Two Chains
JO - Algebraic Combinatorics
PB - MathOA foundation
UR - http://www.numdam.org/articles/10.5802/alco.43/
UR - https://doi.org/10.5802/alco.43
DO - 10.5802/alco.43
ID - ALCO_2019__2_2_275_0
Musiker, Gregg; Roby, Tom. Paths to Understanding Birational Rowmotion on Products of Two Chains. Algebraic Combinatorics, Tome 2 (2019) no. 2, pp. 275-304. doi : 10.5802/alco.43. http://www.numdam.org/articles/10.5802/alco.43/
[1] Armstrong, Drew; Stump, Christian; Thomas, Hugh A uniform bijection between nonnesting and noncrossing partitions, Trans. Am. Math. Soc., Volume 365 (2013) no. 8, pp. 4121-4151 | Article | MR 3055691 | Zbl 1271.05011
[2] Brouwer, Andries; Schrijver, Lex On the period of an operator, defined on antichains, Stichting Mathematisch Centrum. Zuivere Wiskunde, Volume ZW 24/74 (1974), pp. 1-13 | Zbl 0282.06003
[3] Cameron, Peter; Fon-Der-Flaass, Dmitrii Orbits of antichains revisited, Eur. J. Comb., Volume 16 (1995) no. 6, pp. 545-554 | Article | MR 1356845 | Zbl 0831.06001
[4] Di Francesco, Philippe; Kedem, Rinat
T
-systems with boundaries from network solutions, Electron. J. Comb., Volume 20 (2013) no. 1, p. Paper 3, 62 | MR 3015686 | Zbl 1266.05176
[5] Einstein, David; Propp, James Combinatorial, piecewise-linear, and birational homomesy for products of two chains (2013) (https://arxiv.org/abs/1310.5294v1)
[6] Einstein, David; Propp, James Piecewise-linear and birational toggling, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) (Discrete Mathematics and Theoretical Computer Science), Discrete Mathematics & Theoretical Computer Science (DMTCS), 2014, pp. 513-524 | MR 3466399 | Zbl 1394.06005
[7] Farber, Miriam; Hopkins, Sam; Trongsiriwat, Wuttisak Interlacing networks: birational RSK, the octahedron recurrence, and Schur function identities, J. Comb. Theory, Ser. A, Volume 133 (2015), pp. 339-371 | Article | MR 3325638 | Zbl 1315.05144
[8] Fon-Der-Flaass, Dmitrii Orbits of antichains in ranked posets, Eur. J. Comb., Volume 14 (1993) no. 1, pp. 17-22 | Article | MR 1197471 | Zbl 0777.06002
[9] Frieden, Gabriel The geometric
R
-matrix for affine crystals of type
A
[10] Fulmek, Markus Bijective proofs for Schur function identities (2009) (https://arxiv.org/abs/0909.5334)
[11] Fulmek, Markus; Kleber, Michael Bijective proofs for Schur function identities which imply Dodgsonโs condensation formula and Plรผcker relations, Electron. J. Comb., Volume 8 (2001) no. 1, 16, 22 pages | MR 1855857 | Zbl 0978.05005
[12] Galashin, Pavel; Pylyavskyy, Pavel
R
-systems (2017) (https://arxiv.org/abs/1709.00578) | Zbl 07036397
[13] Goncharov, Alexander; Shen, Linhui Donaldson-Thomas transformations of moduli spaces of G-local systems, Adv. Math., Volume 327 (2018), pp. 225-348 | Article | MR 3761995 | Zbl 06842126
[14] Goulden, Ian P. Quadratic forms of skew Schur functions, Eur. J. Comb., Volume 9 (1988) no. 2, pp. 161-168 | Article | MR 939866 | Zbl 0651.05011
[15] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion (2014) (https://arxiv.org/abs/1402.6178)
[16] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion II: rectangles and triangles, Electron. J. Comb., Volume 22 (2015) no. 3, 3.40, 49 pages | MR 3414186 | Zbl 1339.06001
[17] Grinberg, Darij; Roby, Tom Iterative properties of birational rowmotion I: generalities and skeletal posets, Electron. J. Comb., Volume 23 (2016) no. 1, 1.33, 40 pages | MR 3484738 | Zbl 1338.06003
[18] Henriques, Andrรฉ A periodicity theorem for the octahedron recurrence, J. Algebr. Comb., Volume 26 (2007) no. 1, pp. 1-26 | Article | MR 2335700 | Zbl 1125.05106
[19] Kirillov, Anatol N.; Berenstein, Arkady D. Groups generated by involutions, Gelfand-Tsetlin patterns, and combinatorics of Young tableaux, Algebra Anal., Volume 7 (1995) no. 1, pp. 92-152 | MR 1334154 | Zbl 0848.20007
[20] Panyushev, Dmitri I. On orbits of antichains of positive roots, Eur. J. Comb., Volume 30 (2009) no. 2, pp. 586-594 | Article | MR 2489252 | Zbl 1165.06001
[21] Propp, James; Roby, Tom Homomesy in products of two chains, Electron. J. Comb., Volume 22 (2015) no. 3, 3.4, 29 pages | MR 3367853 | Zbl 1319.05151
[22] Reiner, Victor; Stanton, Dennis; White, Dennis The cyclic sieving phenomenon, J. Comb. Theory, Ser. A, Volume 108 (2004) no. 1, pp. 17-50 | Article | MR 2087303
[23] Reiner, Victor; Stanton, Dennis; White, Dennis What is
...
cyclic sieving?, Notices Am. Math. Soc., Volume 61 (2014) no. 2, pp. 169-171 | Article | MR 3156682 | Zbl 1338.05012
[24] Roby, Tom Dynamical algebraic combinatorics and the homomesy phenomenon, Recent trends in combinatorics (The IMA Volumes in Mathematics and its Applications), Volume 159, Springer, 2016, pp. 619-652 (Also available at http://www.math.uconn.edu/~troby/homomesyIMA2015Revised.pdf) | Article | MR 3526426 | Zbl 1354.05146
[25] Rush, David B.; Shi, XiaoLin On orbits of order ideals of minuscule posets, J. Algebr. Comb., Volume 37 (2013) no. 3, pp. 545-569 | Article | MR 3035516 | Zbl 1284.06008
[26] Rush, David B.; Wang, Kelvin On Orbits of Order Ideals of Minuscule Posets II: Homomesy (2015) (https://arxiv.org/abs/1509.08047)
[27] Speyer, David E. Perfect matchings and the octahedron recurrence, J. Algebr. Comb., Volume 25 (2007) no. 3, pp. 309-348 | Article | MR 2317336 | Zbl 1119.05092
[28] Stanley, Richard P. Two poset polytopes, Discrete Comput. Geom., Volume 1 (1986) no. 1, pp. 9-23 | Article | MR 824105 | Zbl 0595.52008
[29] Stanley, Richard P. Enumerative combinatorics. Vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, 2012, xiii+626 pages (Also available at http://math.mit.edu/~rstan/ec/ec1/.) | Zbl 1247.05003
[30] Striker, Jessica Rowmotion and generalized toggle groups, Discrete Math. Theor. Comput. Sci., Volume 20 (2018) no. 1, 17, 26 pages | MR 3811480 | Zbl 06991639
[31] Striker, Jessica; Williams, Nathan Promotion and rowmotion, Eur. J. Comb., Volume 33 (2012) no. 8, pp. 1919-1942 | Article | MR 2950491 | Zbl 1417.06003
[33] Thomas, Hugh; Williams, Nathan Rowmotion in slow motion (2017) (https://arxiv.org/abs/1712.10123) | Zbl 07143125
[34] Volkov, Alexandre Yu. On the periodicity conjecture for
Y
-systems, Commun. Math. Phys., Volume 276 (2007) no. 2, pp. 509-517 | Article | MR 2346398
[35] Yฤฑldฤฑrฤฑm, Emine The Coxeter transformation on Cominuscule Posets (2017) (https://arxiv.org/abs/1710.10632) | Zbl 07075207
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Attention Models: What They Are and Why They Matter - Blog | AI Exchange
With attention models you can design ML algorithms that learn which parts of the input are essential to solving a problem and which are irrelevant.
The concept of attention in AI and machine learning was inspired by psychology, which describes it as the cognitive processes involved in focusing or โattending toโ specialized pieces of information while ignoring the rest.
When reading a book, people tend to concentrate on a few selected words to understand the full meaning of a sentence. Similarly, when presented with an image of a scene, people need only focus on a few objects to comprehend the theme of the image. People tend to ignore the minute details in the picture.
In machine learning, โattentionโ refers to assigning relevance and importance to specific parts of the input data while ignoring the rest to mimic the cognitive attention process. It lets you design algorithms that can learn which parts of the input are essential to solving a problem and which are irrelevant.
Why Is Attention Important, and What Are the Real-World Use Cases?
The attention mechanism has revolutionized the world of deep learning and helped to solve many challenging real-world problems. Research has shown that adding an attention layer to different types of deep learning neural architectures, such as encoder-decoder networks and recurrent neural networks, improves their performance.
Attention is now also an inherent part of many deep learning networks. These include transformers, memory networks, and graph attention networks.
In practice, the attention model has been shown to successfully solve many real-life problems in the domains of natural language processing and computer vision. Applications include translating languages, classifying documents, paraphrasing and summarizing text, recognizing images, answering questions visually, generating images, synthesizing text and images, and more.
Apart from NLP and computer vision, healthcare applications use attention-based networks to analyze medical data, classify diseases in MRI images, develop chatbots for conversational AI, among other applications.
How Does the Attention Model Work?
The example dataset below illustrates how the attention mechanism works. This provides a feature or predictor x to determine the value of the target variable y. If you use a least squares linear regression model, you can predict the value of the target variable by using a fixed weight w regardless of the input values xtest. An example expression is given below, with b as a constant.
y_{predict} = wx_{test} + b
The attention mechanism takes into account the relevance of the input values xtest to the predictors in the dataset. Instead of using a static value w, the weights are generated according to how similar xtest is to a data point in the training instances. The generalized attention model predicts the target y as:
y_{predict} = \sum_i f\big(a(K_i, Q)\big) V_i
From the above equation, the main ingredients of the generalized attention model are given below. The figure below provides a worked-out example.
Keys K: These correspond to the values of the predictors in the training dataset. In the figure, keys are the values in column x.
Values V: These correspond to the target values in the training dataset. Column y of the figure corresponds to values.
Query Q: This is the value of the test data point xtest.
Alignment function a: This function determines the similarity of keys with the query.
Distribution function f: This function normalizes the weights.
Figure 1: An example of the generalized attention model. The alignment function has arbitrarily been chosen as a product of key and query. The distribution function simply normalizes the weights to sum to one. Source: Mehreen Saeed
The alignment function computes the similarity between the query Q and keys K. You can choose the function that works best for your use case. For example, you can use dot product, cosine similarity, or scaled dot product as an alignment function.
The main guideline is to return a high value if the query is close to the key, and a low value if they are very different.
There are also other alignment functions such as concat alignment, where keys and queries are both concatenated. You can also use area attention, which does not consider the keys and depends on the query only.
Use a distribution function to ensure that the attention weights lie between 0 and 1 and are normalized to sum to 1. The logistic sigmoid or softmax function suffices for this purpose. The output values from the distribution function can be seen as probabilistic relevance scores.
Different Types of Attention Models
There are several variations of the generalized attention model. โAn Attentive Survey of Attention Modelsโ describes the following taxonomy of attention models:
There are different attention models based on the type of input sequences. This mainly differentiates how you define query Q and keys K.
Distinctive attention: This is used in tasks such as language translation, speech recognition, and sentence paraphrasing when the key and query states correspond to distinct input and output sequences. The distinctive attention mechanism was used by Bahdanau et al. in โNeural Machine Translation by Jointly Learning to Align and Translate.โ
Co-attention: This model learns attention weights by using multiple input sequences at the same time. For example, co-attention was used for visual question answering, where both image sequences and words in questions were used to model attention.
Self attention: In self attention, the key and query states belong to the same sequence. For example, in a document categorization task, the input is a sequence of words, and the output is the category, which is not a sequence. In this scenario, the attention mechanism learns the relevance of tokens in the input sequence for every token of the same input sequence.
Abstraction LevelโBased
You can also define attention models based on the hierarchical levels at which the attention weights are computed.
Single-level: In single-level attention, the weights are calculated only for the original input sequence.
Multilevel: For multilevel attention, the attention weights are repeatedly calculated at different levels of abstraction. The lower abstraction level serves to generate the query state for the higher level. A good example is the word-level and document-level abstraction used in document categorization.
Position-based attention models determine the actual positions in the input sequence on which attention weights are calculated.
Soft/global attention: Global attention (also called โsoft attentionโ) uses all the tokens in the input sequence to compute the attention weights and decides the relevance of each token. For example, in text translation models, all of the words in a sentence are considered to calculate the attention weights in order to decide the relevance of each word.
Hard attention: Hard attention computes the attention weights from some tokens of the input sequence while leaving out the others. Deciding which tokens to use is also a part of hard attention. So for the same text translation task, some words in the input sentence are left out when computing the relevance of different words.
Local attention: Local attention is the middle ground between hard and soft attention. It picks a position in the input sequence and places a window around that position to compute an attention model.
There are different types of representation based attention models:
Multi-representational: In a multi-representational representation of attention, the inputs are represented using different features; hence, the attention weights are used to assign relevance to all the representations.
Multidimensional: Similar to multi-representational models, multidimensional attention models can be used to determine the relevance of various dimensions if the input data has a multidimensional representation.
What Are the Challenges and Future Directions of Attention Models?
Attention models implemented in conjunction with deep learning networks have their challenges. Training and deploying these large models is costly in terms of time and computational power. This includes the self-attention model implemented in transformers that involves a large number of computations. Research is now geared toward reducing these costs.
Online attention is another area of future research to implement real-time applications such as machine translation while a person is speaking. Generating correct, partially translated text before a person finishes speaking the entire sentence is a challenge in itself.
Determining the structure and network architecture of a deep learning model that employs attention is another aspect that needs further research. This has given rise to auto-learning attention, where the goal is to use attention itself to determine the optimal structure and design of an attention model.
The attention mechanism has served to motivate future research into alternatives to attention models. Active memory is one such alternative that does not focus on a specialized area of memory as implemented by attention models but operates uniformly in parallel on all of it.
An attention layer integrated within a mathematical model assigns importance weights to different tokens in an input sequence. This determines which parts of the input are more relevant to solving a given problem and which input tokens can be ignored. The concept of attention works in conjunction with other machine learning models, which are typically based on deep learning architectures.
To get started implementing your own attention model, the first step is to understand how it is incorporated into different deep learning networks. The deep learning model you choose, along with the type of attention model you incorporate within it, depends mostly on the nature of the problem you are solving and the application you are developing.
Show, Attend, and Tell: Neural Image Caption Generation with Visual Attention
5 Robotics Machine Learning Techniques: How to Choose
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Logical constant โ Wikipedia Republished // WIKI 2
{\displaystyle {\mathcal {L}}}
is a symbol that has the same semanticโ
value under every interpretation of
{\displaystyle {\mathcal {L}}}
. Two important types of logical constants are logicalโ
connectives and quantifiers. The equality predicate (usually written '=') is also treated as a logical constant in many systemsโ
ofโ
logic.
One of the fundamental questions in the philosophyโ
ofโ
logic is "What is a logical constant?";[1] that is, what special feature of certain constants makes them logical in nature?[2]
โ "forโ
all"
โ "thereโ
exists", "for some"
{\displaystyle \Box }
{\displaystyle \Diamond }
Many of these logical constants are sometimes denoted by alternate symbols (e.g., the use of the symbol "&" rather than "โง" to denote the logicalโ
and).
Defining logical constants is a major part of the work of Gottlobโ
Frege and Bertrandโ
Russell. Russell returned to the subject of logical constants in the preface to the second edition (1937) of Theโ
Principlesโ
ofโ
Mathematics noting that logic becomes linguistic: "If we are to say anything definite about them, [they] must be treated as part of the language, not as part of what the language speaks about."[3] The text of this book uses relations R, their converses and complements as primitiveโ
notions, also taken as logical constants in the form aRb.
Joseph Abrahamson on "On the Meanings of the Logical Constants"
+2 LOGIC, SYMBOLIC LOGIC, TRUTH FUNCION, logical constants, propositional variable,E-Teaching ODISHA
^ Peacocke, Christopher (May 6, 1976). "Whatโ
isโ
aโ
Logicalโ
Constant?". The Journal of Philosophy. 73 (9): 221โ240. doi:10.2307/2025420. Retrieved Jan 12, 2022.
^ Bertrandโ
Russell (1937) Preface to The Principles of Mathematics, pages ix to xi
Stanfordโ
Encyclopediaโ
ofโ
Philosophyโ
entryโ
onโ
logicalโ
constants
|
mdecamp
From mdecamp
Welcome to Dr. DeCamp's Lab Wiki
Our group is a member of the Ultrafast division of the University of Delaware
The group focuses on the study of sub-picosecond (
{\displaystyle 10^{-12}}
seconds) dynamics of complex systems. Specifically, we utilize time-domain THz, optical pump-probe techniques, and time-resolved x-ray spectroscopies to study the dynamics of thin films and biomaterials.
Matthew DeCamp Department of Physics and Astronomy Colloquium September 12, 2012
Matthew DeCamp and Yuan Gao Optical Society of America Annual Meeting, Rochester NY (2012)
Congratulations to Zhiyuan Chen for being a finalist for the Daicar-Bata prize for best student paper in the Physics Department (2011)
Congratulations to Yuan Gao for defending his PhD dissertation PROBING AND RECONSTRUCTING TRANSIENT ONE-DIMENSIONAL ACOUSTIC STRAINS USING TIME-RESOLVED X-RAY DIFFRACTION (4/4/13)
Z. Chen, Y. Gao, and M.F. DeCamp, "Retrieval of terahertz spectra through ultrafast electro-optic modulation," Applied Physics Letters 99 011106 (2011)
Y. Gao and M.F. DeCamp "Generation of acoustic pulses from a photo-acoustic transducer measured by time-resolved x-ray diffraction" Applied Physics Letters 100 191903 (2012)
Z. Chen and M.F. DeCamp "Coherent optical phonons in bismuth detected through a surface plasmon resonance" J. Applied Physics 112 191903 (2012)
J. Shen et al. "Damping modulated terahertz emission for ferromagnetic films excited by ultrafast laser pulses" Applied Physics Letters 101 072401 (2012)
Retrieved from "https://wiki.physics.udel.edu/wiki_mdecamp/index.php?title=Main_Page&oldid=90"
About mdecamp
|
Prove the following: cosA / 1 - sinA = tan (45 + A/2) - Maths - Trigonometric Functions - 7523351 | Meritnation.com
L.H.S=\frac{\mathrm{cos} A}{1-\mathrm{sin} A}
=\frac{\mathrm{cos} 2\left(\frac{A}{2}\right)}{1-\mathrm{sin} 2\left(\frac{A}{2}\right)}
=\frac{{\mathrm{cos}}^{2}\left(\frac{A}{2}\right)-{\mathrm{sin}}^{2}\left(\frac{A}{2}\right)}{{\mathrm{sin}}^{2}\left(\frac{A}{2}\right)+{\mathrm{cos}}^{2}\left(\frac{A}{2}\right)+2\mathrm{sin}\frac{A}{2}\mathrm{cos}\frac{A}{2}}
=\frac{\left(\mathrm{cos}\frac{A}{2}+\mathrm{sin}\frac{A}{2}\right)\left(\mathrm{cos}\frac{A}{2}-\mathrm{sin}\frac{A}{2}\right)}{{\left(\mathrm{cos}\frac{A}{2}+\mathrm{sin}\frac{A}{2}\right)}^{2}}
=\frac{\mathrm{cos}\frac{A}{2}-\mathrm{sin}\frac{A}{2}}{\mathrm{cos}\frac{A}{2}+\mathrm{sin}\frac{A}{2}}
=\frac{1-\mathrm{tan}\frac{A}{2}}{1+\mathrm{tan}\frac{A}{2}}
(Dividing both numerator and denominator by
\mathrm{cos}\frac{A}{2}
=\mathrm{tan}\left(45ยฐ+A\right)
=R.H.S
|
Options Jargon - Zeta
Glossary of options terminology - don't worry you don't have to know it all, feel free to use this just as a reference!
A list of orders on a particular asset reflecting all the orders from the different buyers and sellers open in a market. It effectively shows the price and volume that participants are willing to buy/sell the asset for.
Is an Automated Market Maker (AMM) which uses an algorithm to define the price that it will buy or sell an asset at.
Contracts, which let the buyer of a 1) call (put) choose whether or not they want to 2) buy (sell) the underlying asset at the 3) strike price once the contract hits its 4) expiry date
Options that are based on a binary success or failure outcome. The parties that chose the correct outcome (success of failure) win the entire pot.
Gives the buyer the right (but not obligation) to purchase an asset at the particular price and date specified in the contract.
Gives the buyer the right (but not obligation) to sell an asset at the particular price and date specified in the contract.
Is the market price of purchasing an option. By paying the premium you purchase the right to exercising an option. The seller receives this premium as their payoff for selling the option.
Profit and Loss. It represents the change in the value of a traderโs position. Whilst a trade is still open PnL is considered "unrealised" and when the trade is close it becomes "realised" PnL.
The event in which the buyer of an option choses to execute the option contract they purchased in order to buy or sell the underlying at the strike price.
Occurs when an option is exercised. In the case of physical settlement, there is a transfer of assets between the seller and buyer of an option to reflect the contract that was exercised. In the case of cash settlement, the trader exercising the option is paid out in cash (no exchange of assets) based on their PnL.
Shorting an option without holding any (or enough) of the underlying asset to protect from adverse price movements, exposing the trader to high amounts of unhedged risk.
An asset accepted as security for a loan or credit risk. In the case of options collateral is required to make sure that the trader can cover their position if they get margin called.
\delta
- The price sensitivity of the option relative to the underlying asset i.e. how much the option price changes when the underlying assets price increases by $1. When buying a call option the delta is positive, when buying a put option the delta is negative.
\theta
- Reflects the options price sensitivity with respect to time i.e. the $ change in the option price given time moved 1 day closer to the expiry.
\gamma
- Reflects the rate of change between the Delta and the underlying asset price i.e. given a $1 change in price the Delta will change by the Gamma.
\nu
- The price sensitivity of an option with respect to a change in the underlying asset's implied volatility i.e. the $ change in the option given a 1% change in the underlying assets implied volatility.
The act of purchasing complementary (usually inverse) assets to reduce the trader's risk exposure to sudden adverse price movements. Options are a popular method to hedge risk as they allow traders to limit their losses to a fixed amount, almost acting as insurance.
S
Is the current market price of the underlying asset.
Strike Price,
K
The price defined in an option contract specifying the price that the underlying asset will be bought/sold at.
Expiry Time,
T
The date specified in the option contract at which the option can be exercised (European options) or that time before which options must be exercised (American options).
Risk-free Rate of Return,
r
The theoretical rate of return on an investment that carries no risk.
\sigma
Reflects the extent to which the underlying asset is expected to fluctuate between now and the asset expiry.
V_{mark}
The last price at which the option was purchased/sold on the market.
Index Price,
V_{index}
The price of the underlying asset, where this price is derived from more than 1 source.
Long coin + short call.
Short coin + short put.
Long lower strike call + short higher strike call, long higher strike put + short lower strike put.
Short lower strike call + long higher strike call, short higher strike put + long lower strike put.
Short call + short put on the same strike.
Short call on higher strike + short put on lower strike.
How To Trade On Zeta - Previous
|
coordinates of vertices of triangle ABC are A(X1,Y1),B(X2,Y2),C(X3,Y3) if BC=a CA=b and AB=c, then find the coordinates of the - Maths - Coordinate Geometry - 8799063 | Meritnation.com
coordinates of vertices of triangle ABC are A(X1,Y1),B(X2,Y2),C(X3,Y3). if BC=a.CA=b and AB=c, then find the coordinates of the incentre of triangle ABC.
Given coordinates of vertices of triangle = A( X1 , Y1 ) , B ( X2 , Y2 ) and C ( X3 , Y3 ) , And BC = a , CA = b and AB = c
And we know formula for coordinates of incenter , As :
Let incenter at " O " , So
O ( X , Y ) =
\left(\frac{BCร{A}_{x} + CA ร {B}_{x} + AB ร {C}_{x}}{P} , \frac{BCร{A}_{y} + CA ร {B}_{y} + AB ร {C}_{y}}{P}\right)
Here Ax , Bx and Cx are x coordinates of vertices A , B and C respectively
Ay , By and Cy are y coordinates of vertices A , B and C respectively .
P = Perimeter of triangle = AB + BC + CA
Coordinates of Incenter O ( X , Y ) =
\left(\frac{a{X}_{1} + b{X}_{2} + c{X}_{3}}{a + b + c} , \frac{a{Y}_{1} + b{Y}_{2} + c{Y}_{3}}{a + b + c}\right)
|
C
be a curve in the plane. The area of the surface obtained when
C
is revolved around an external axis is equal to the product of the arc length of
C
and the distance traveled by the centroid of
C.
R
be a region in the plane. The volume of the solid obtained when
R
is revolved around an external axis is equal to the product of the area of
R
R.
Consider the cylinder obtained by revolving a rectangle with horizontal side
r
and vertical side
h
around one of its vertical sides (say its left side). The surface area of the cylinder, not including the top and bottom, can be computed from Pappus's theorem since the surface is obtained by revolving its right side around its left side. The arc length of its right side is
h
and the distance traveled by its centroid is simply
2\pi r,
so its area is
2 \pi r h.
The volume of the cylinder is the area
rh
of the rectangle multiplied by the distance traveled by its centroid. The centroid of the rectangle is its center, which is a distance of
\frac r2
from the axis of revolution. So it travels a distance of
2\pi\big(\frac r2\big) = \pi r
as it revolves. The volume of the cylinder is
(rh)(\pi r) = \pi r^2 h.
To compute the volume of a solid formed by rotating a region
R
around an external axis (a similar argument applies for surface area), one can break the region up into small regions of area
\Delta A
that are located a distance
r
from the axis. Since these regions travel a distance
2\pi r
when revolved around the axis, their contribution to the volume of the solid is roughly
2\pi r \Delta A.
Adding up all the contributions gives the volume (as usual, passing to the limit where
\Delta A
dA,
and the sum becomes an integral, makes this an exact computation).
The difficulty is that all the different small regions are different distances away from the axis. To make this method computationally feasible, one would need to know the average distance
r_{\text{avg}}
of all the small pieces, so that the volume of the region is
2\pi r_{\text{avg}}
times the area
A.
This is precisely what Pappus' centroid theorem gives: it identifies
r_{\text{avg}}
as the distance from the centroid of the region to the axis of revolution.
The surface area and volume of a torus are quite easy to compute using Pappus' theorem. A torus is a circle of radius
r< R,
(R,0)
and rotated around the
y
-axis. The centroid of both the surface of the circle and the region enclosed by the circle is just the center of the circle. This travels a distance of
2\pi R
when it revolves, so the surface area is
2\pi R
times the arc length
2\pi r
of the circle, and the volume is
2\pi R
\pi r^2
of the circle. That is,
\begin{aligned} \text{surface area of the torus} &= 4\pi^2 Rr \\ \text{volume of the torus} &= 2\pi^2 Rr^2. \end{aligned}
Consider the single rectangle in
\mathbb{R}^2
A = (1,2), B = (2,1), C = (4,3), D = (3,4)
rotating around
x
\mathbb{R}^3
The volume of the surface of revolution obtained can be written as
A\pi \text{ unit}^3
A
Revolving a right triangle with legs of length
r
h
around the leg of length
produces a cone. The surface of the cone (not including the circular base) is obtained by revolving the hypotenuse around that leg. The centroid of the hypotenuse is just the midpoint, located halfway up the side of the cone, which travels a distance
\frac{2\pi r}2
as it rotates. So the surface area is
\pi r \sqrt{r^2+h^2}
by Pappus' theorem.
The centroid of the triangle is the center of mass of the three vertices (see the Triangle Centroid wiki), which is located at a distance of
\frac r3
from the axis of revolution
\big(
\frac h3
above the base
\big).
So the volume is
2\pi \big(\frac r3\big)
times the area of the triangle, which is
\frac{rh}2.
The product is
\frac13 \pi r^2 h,
the familiar formula for the volume of a cone.
An equilateral triangle of side length
r
in the first quadrant, one of whose sides lies on the
x
-axis, is revolved around the line
y= -r.
The volume of the resulting solid is
c\pi r^3
c.
c?
C
be the curve
y = \sqrt{r^2-x^2}.
The sphere of radius
r
is obtained by revolving
C
x
-axis. The arc length
L
C
\pi r
since it is half a circle.
C
is on the
y
-axis by symmetry. Its
y
-coordinate is given by the formula
y = \frac1{L} \int_{-r}^r y\sqrt{1+\left( \frac{dy}{dx}\right)^2} \, dx.
The reasoning here is that the centroid is a center of mass and the mass of a small piece of the arc over a small length
dx
x
-axis is proportional to the arc length, which is
\sqrt{1+\left( \frac{dy}{dx}\right)^2} \, dx.
(See the Arc Length wiki for a thorough explanation.) So the location of the centroid is
\begin{aligned} \frac1{\pi r} \int_{-r}^r \sqrt{r^2-x^2} \sqrt{1+\frac{x^2}{r^2-x^2}} dx &= \frac1{\pi r} \int_{-r}^r \sqrt{r^2-x^2} \sqrt{\frac{r^2}{r^2-x^2}} dx \\ &= \frac1{\pi r} \int_{-r}^r r \, dx\\t &= \frac{2r^2}{\pi r} = \frac{2r}{\pi}. \end{aligned}
So the surface area is
2\pi
y
-coordinate of the centroid, times the arc length, which is
2\pi \left( \frac{2r}{\pi} \right) (\pi r) = 4\pi r^2.
y
-coordinate of the centroid of the disk is the (double) integral of
y
over the region, divided by the area of the region:
\begin{aligned} \frac1{\frac12 \pi r^2} \iint_R y \, dy \, dx &= \frac2{\pi r^2} \left. \int_{-r}^r \frac{y^2}2\right|_0^{\sqrt{r^2-x^2}} dx \\ &= \frac1{\pi r^2} \int_{-r}^r \big(r^2-x^2\big) dx \\ &= \frac1{\pi r^2}\left. \left(r^2 x - \frac{x^3}3\right)\right|_{-r}^r \\ &= \frac{4r}{3\pi}. \end{aligned}
2\pi
y
-coordinate of the centroid, times the area, which is
2\pi \left( \frac{4r}{3\pi} \right) \frac{\pi r^2}2 = \frac43 \pi r^3,
|
TL;DR visit the able.md shortcut domain to try our updated markdown editor.
Able is an independent developer community for people to read and write about building things with technology. Weโre building a clean place for high-quality technical knowledge sharing without the ad networks, dark patterns or data lock-in. You can read more about this in our manifesto.
As a website for people to write, we naturally want to have a great editor that makes writing feel enjoyable. We announced the initial version of our new markdown editor in October last year and since then weโve been working on features to improve data portability and technical writing. We also wanted to do this in a way that allows most of the editor to be used by anyone who wants to write markdown, without requiring an account.
Here are some of the new features weโve added. You can now:
visit the able.md shortcut domain to start a new markdown doc or post
use the editor to write and download markdown without an Able account
drag and drop a markdown file into the editor
import posts into your Able account from Medium, Markdown or your own RSS feed. You can preview imports for the latter two without signing up.
set up a webhook in your Able account to call any URL with your postโs data whenever you create, update or delete a post
use our Push to GitHub extension for relaying webhook calls to create, update or delete markdown files in a GitHub repo whenever you do the same on Able
toggle between rich text and Markdown mode in the editor, using the keyboard shortcut Ctrl/โ + M
write math equations, inline or as full-width content blocks, which are rendered using KaTeX
create and manage markdown compatible tables in the editor
share drafts saved in your account for others to review
toggle a table of contents for your post
export all of your posts from your Able account as a single JSON file, with content formatted as markdown and HTML, and images included in an accompanying folder
properly delete all of your data from Able when deleting your account
use a lot more keyboard shortcuts to write productively
All of the above features are responsive and work with mobile browsers, where applicable, and itโs about 920kB of data transfer when loading up for the first time.
Weโve also replaced Google Analytics with a more privacy-oriented, non-cookie based analytics service.
Read on if youโre interested in more details about some of the new features.
All imports can be started here.
If you write on your own blog, you can now syndicate your posts into Able by setting up imports from your RSS feed. Weโll check it for updates once a day and import any new posts as drafts into your Able account.
Weโll automatically set your original blog post as the canonical URL and weโve added visual cues to posts so that you can display your own domain when posting on Able.
For example, if you look at Ansonโs post or the screenshot below youโll notice that the website domain he has set on his Able profile appears under his name at the top of the page and there is a link to the postโs canonical URL displayed under the post title.
You can now import a zip file that contains one or many .md files. We also parse basic frontmatter like title and date. If you have a single markdown file, then you can just drag it directly into the markdown editor and start editing.
You can also import your Medium posts. Just export your data from Medium and upload the zip file. Weโll import your posts into drafts for you to review before publishing. You can also specify if you want embedded gists in your Medium posts to be converted to Markdown code blocks when theyโre being imported.
From the options sidebar (click โขโขโข or press Ctrl/โ + O), you can click on Webhook to go to your accountโs webhook settings page. You can set this up to call a URL with your postโs data any time you create, update or delete a post. You can use this for real-time back-ups to GitHub, pushing yours posts to other APIs or anything else you might want to point the editor at. You can also set whether the body of your content should be in Markdown or HTML format. Hereโs what a typical request body looks like:
"body_type": "markdown"
Webhook requests include your webhook token in the header that you can use to verify requests when handling them and webhook calls are limited to 1 call per domain every second. If you want to try this out you can read our webhook documentation and use something like Webhook.site to set up a temporary handler to get familiar with requests.
When the webhook is enabled and you save an article, youโll see a toast notification pop out to show the HTTP response code. This can be expanded to show the response body too, to help you with debugging integrations.
You can now quickly toggle between rich text and Markdown mode with the keybord shortcut Ctrl/โ + M. Itโs handy for toggling markdown mode when you want more control and back to rich text mode to preview your writing. The editor remembers your preferred writing format so if you just prefer writing in markdown mode then you can stick to that.
You can now write inline math equations or math equation blocks in your posts. We use KaTeX to render them. To write an inline equation just enclose your text within single dollar signs followed by a space. So typing something like $-b \pm \sqrt{b^2 - 4ac} \over 2a$ into the editor followed by a space will render an inline equation that looks like this
-b \pm \sqrt{b^2 - 4ac} \over 2a
in your post.
For block equations, just enclose your text in double dollar signs, so typing $$-b \pm \sqrt{b^2 - 4ac} \over 2a$$ will give you a block-width equation like the one below.
-b \pm \sqrt{b^2 - 4ac} \over 2a
Check out Patrickโs post or Shonโs post to see some of these in action.
You can now easily create and manage markdown compatible tables in the editor. Hereโs an example of a table that shows all of the shortcuts that can be used when writing in the Able editor.
Enclose text in double $$
Creating and managing tables is also possible on mobile devices.
You can now export all of your post data from Able in a single JSON file under Settings > Account > Download your data. This will email you a zip file that contains a JSON file with all of your post objects. The zip file also includes images for all of your posts. This is what a typical extracted zip file looks like:
โposts
โ โโโ 33cacf41
โ โ โโโ image_name_b.jpg
โ โ โโโ image_name_a.jpg
โ โโโ 21c76577
โ โ โโโ image_name_d.png
โ โโโ 19b7ce11
โ โ โโโ image_name_k.jpg
โ โ โโโ image_name_s.jpg
โ โ โโโ image_name_w.jpg
โ โโโ 421c5845
โ โ โโโ image_name_i.gif
Images for your posts are saved in a folder named with the postโs ID. Image references inside the markdown and HTML content also have these relative references which should make it easy for you to find and replace the paths if you want to drop the images into another storage service somewhere.
This is what posts.json looks like inside:
โ[
"title": "My amazing post",
"markdown": "Some **body** text goes here",
"html": "<p>Some <b>body</b> text goes here</p>",
"topics": ["Able"],
"id": "33cacf41",
"slug": "my-amazing-post",
"pub_date": "2020-05-30 11:47:23.267728+00:00",
Additionally, when deleting your account weโve gone through a lot of care to make sure that all of your data actually gets deleted. This includes:
removing your profile, posts, comments from our production database. If some of your comments have replies then the comment will be dis-associated from your account and the comment text is replaced with the text โDeletedโ, to preserve the comment thread.
all images youโve ever uploaded to your account are deleted
all posts and images are purged from our edge cache
your data will roll out of our database backups within a few days
Aside from database backups, all of your data is typically deleted within 10 - 15 minutes.
Thereโs a lot more that we still want to do with Able, this includes support for post series, creating publications, custom domains and improving the Able job board. This is what weโll be working on in the future.
Finally, a big thank you to Soumyajit once again for all of the outstanding JavaScript work and other bits along the way. Your contributions have been invaluable.
Weโve spent a fair amount of our spare time, effort and resources building Able because we want something like this to exist on the web. Just another place for knowledge to thrive without clutter. And by doing that we hope to build enough of a community to create and monetise a non-intrusive job board where companies can present their vacancies to great people who might be looking for new opportunities.
If you have any feedback or something isnโt working as expected then please create an issue in our suggestion box or contact us at hello@able.bio.
Next post in the Building Able series
Kriss June 30, 2020, 4:10 a.m.
thank you loving your editor give me feel comfortable then DEV
|
Applying Differentiation Rules to Trigonometric Functions | Brilliant Math & Science Wiki
Hemang Agarwal, Aditya Virani, Gene Keun Chung, and
By applying the differentiation rules we have learned so far, we can find the derivatives of trigonometric functions. The differentiation of the six basic trigonometric functions (which are
\sin, \cos, \tan, \csc, \sec,
\cot
) can be done as shown below:
y=\sin x ,
\sin a - \sin b = 2\cos \left ( \frac{a+b}{2} \right ) \sin \left ( \frac{a-b}{2} \right)
y' :
\begin{aligned} y' &= \lim_{h \rightarrow 0} \frac{\sin(x+h)-\sin x}{h} \\ &= \lim_{h \rightarrow 0} \frac{2\cos \left ( x+ \frac{h}{2} \right) \sin \frac{h}{2}}{h} \\ &= \lim_{h \rightarrow 0} \cos \left(x + \frac{h}{2} \right) \cdot \frac{\sin \frac{h}{2}}{\frac{h}{2}}\\ &= \cos x. & \left(\text{since }\lim_{x\rightarrow0}\frac{\sin x}{x}=1\right) \end{aligned}
y=\cos x ,
\cos a - \cos b = -2\sin \left ( \frac{a+b}{2} \right ) \sin \left ( \frac{a-b}{2} \right)
y' :
\begin{aligned} y' &= \lim_{h \rightarrow 0} \frac{\cos(x+h)-\cos x}{h} \\ &= \lim_{h \rightarrow 0} \frac{-2\sin \left ( x+ \frac{h}{2} \right) \sin \frac{h}{2}}{h} \\ &= -\lim_{h \rightarrow 0} \sin \left(x + \frac{h}{2} \right) \cdot \frac{\sin \frac{h}{2}}{\frac{h}{2}} \\ &= -\sin x. &\left(\text{since }\lim_{x\rightarrow0}\frac{\sin x}{x}=1\right) \end{aligned}
y= \tan x,
we convert it to
\frac{\sin x}{\cos x}
and use the quotient rule, which gives
\begin{aligned} y' &= (\tan x)' \\ &= \left ( \frac{\sin x}{\cos x} \right )' \\ &= \frac{\cos x \cdot \cos x- \sin x \cdot (-\sin x)}{\cos^2 x} \\ &= \frac{1}{\cos^2 x} &\left(\text{since }\sin^2 x+\cos^2 x=1\right)\\ &= \sec^2 x. \end{aligned}
y= \cot x,
we use the same method as for
y=\tan x,
\begin{aligned} y' &= (\cot x)' \\ &= \left ( \frac{\cos x}{\sin x} \right )' \\ &= \frac{-\sin x \cdot \sin x- \cos x \cdot \cos x}{\sin^2 x} \\ &= \frac{-1}{\sin^2 x} \\ &= -\csc^2 x. \end{aligned}
y= \sec x ,
\begin{aligned} y' &= (\sec x)' \\ &= \left (\frac{1}{\cos x} \right )' \\ &= \frac{0-1 \cdot (-\sin x)}{\cos^2 x} \\ &= \frac{\sin x}{\cos^2 x} \\ &= \sec x \cdot \tan x. \end{aligned}
y= \csc x ,
\begin{aligned} y' &= (\csc x)' \\ &= \left (\frac{1}{\sin x} \right )' \\ &= \frac{0-1 \cdot (\cos x)}{\sin^2 x} \\ &= \frac{-\cos x}{\sin^2 x} \\ &= -\csc x \cdot \cot x. \end{aligned}
The box below summarizes the derivative of the six trigonometric functions, which of course should be memorized:
\ y = \sin x \implies y' = \cos x
\ y = \cos x \implies y' = -\sin x
\ y= \tan x \implies y' = \sec^2 x
\ y= \cot x \implies y' = -\csc^2 x
\ y = \sec x \implies y' = \sec x \cdot \tan x
\ y= \csc x \implies y' = -\csc x \cdot \cot x
y= \tan x - 3 \cot x .
\begin{aligned} y' &= \sec^2 x -3\big(-\csc^2 x\big) \\ &= \sec ^2 x + 3\csc^2 x.\ _ \square \end{aligned}
f(x)=\sec^2 x .
By applying the chain rule, we have
\begin{aligned} f'(x) &= 2\sec x\cdot(\sec x)' \\ &= 2\sec x \cdot \sec x \cdot \tan x \\ &= 2\sec^2 x \cdot \tan x.\ _ \square \end{aligned}
f(x)=(2x^2+1) \sin 2x .
Applying the product rule gives
\begin{aligned} f'(x) &= (2x^2+1)'\sin 2x + (2x^2+1)(\sin 2x)' \\ &= 4x \sin 2x + (2x^2 +1) \cdot 2\cos 2x \\ &= 4x \sin 2x + 2(2x^2+1) \cos 2x.\ _ \square \end{aligned}
\frac{dy}{dx}
\sin x + \sin y = 3?
This equation should be differentiated implicitly. Differentiating both sides with respect to
x
\begin{aligned} \cos x + \cos y \frac{dy}{dx} &= 0 \\ \Rightarrow \frac{dy}{dx} &= -\frac{\cos x}{\cos y}.\ _ \square \end{aligned}
g(x)
f(x) = \cos x\ \ \left ( 0 < x < \frac{\pi}{2} \right) ,
g'\left( \frac{1}{2} \right) ?
g(x)
f(x) ,
f\big(g(x)\big) = x \implies \cos g(x) = x.
Differentiating both sides of
\cos g(x) = x
\begin{aligned} \big( - \sin g(x) \big) \cdot g'(x) &= 1 \\ \Rightarrow g'(x) &= -\frac{1}{\sin g(x)}. \qquad (1) \end{aligned}
\cos \frac{\pi}{3}=\frac{1}{2},
g \left( \frac{1}{2} \right) = \frac{\pi}{3}.
Plugging this into (1) gives
\begin{aligned} g'\left( \frac{1}{2} \right) &= -\frac{1}{\sin g \left( \frac{1}{2} \right)} \\ &= -\frac{1}{\sin \frac{\pi}{3}} \\ &= -\frac{1}{\hspace{3mm} \frac{\sqrt{3}}{2}\hspace{3mm} } \\ &= - \frac{2\sqrt{3}}{3}.\ _ \square \end{aligned}
f(x) = \sin^2 \frac{x}{2} ,
{\displaystyle\lim_{x \rightarrow \pi}} \frac{f'(x)}{x-\pi}?
\begin{aligned} f'(x) &= 2\sin \frac{x}{2} \cdot \cos \frac{x}{2} \cdot \frac{1}{2} \\ &= \frac{1}{2} \sin x &\qquad (\text{since } \sin 2A = 2 \sin A \cdot \cos A )\\ \Rightarrow f'(\pi) &= 0. \end{aligned}
\begin{aligned} \lim_{x \rightarrow \pi} \frac{f'(x)}{x-\pi} &= \lim_{x \rightarrow \pi} \frac{f'(x) - f'(\pi)}{x-\pi} \\ &= f''(\pi). &\qquad (1) \end{aligned}
f''(x) = \frac{1}{2} \cos x ,
(1)
\begin{aligned} \lim_{x \rightarrow \pi}\frac{f'(x)}{x-\pi} &= f''(\pi) \\ &= \frac{1}{2} \cos \pi \\ &= - \frac{1}{2}.\ _ \square \end{aligned}
Cite as: Applying Differentiation Rules to Trigonometric Functions. Brilliant.org. Retrieved from https://brilliant.org/wiki/applying-differentiation-rules-to-trigonometric/
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Torque - Equilibrium Practice Problems Online | Brilliant
Three objects are hanging on a scale system, as shown in the figure above. The distances between the objects and the pivots satisfy
r_1=4\text{ m}, r_2=8\text{ m}, r_3=20\text{ m}, r_4=8\text{ m}.
If the weight of object
A
w_A = 12 \text{ N}
and the whole scale system is in balance, how much does object
C
weigh?
60 \text{ N}
72 \text{ N}
90 \text{ N}
108 \text{ N}
\begin{array}{c}&F_1, &F_2 = 6 \text{ N}, &F_3 = 3 \text{ N}, &F_4 = 6 \text{ N}, &F_5 \end{array}
are exerted perpendicularly on a rod of length
l,
as shown in figure above. If the rod is in static equilibrium, what is the magnitude of
F_5?
6 \text{ N}
7 \text{ N}
8 \text{ N}
9 \text{ N}
A fastener is a system of 2 objects - a bolt and a nut. You come across such a bolt/nut system tightened all the way, so that the nut and the top of the bolt are pressing against each other with a force of 5 N. If the nut is held fixed, how much torque in mJ does one need to put on the bolt to begin to unscrew it?
The radius of the bolt is
R = 2
The threads on the bolt are such that the bolt needs to rotate N = 20 cycles to move
d = 1~cm
vertically away from the nut.
The friction coefficient between the bolt and nut is
k = 0.5
Between doing physics problems on Brilliant, some people like to unicycle. A unicyclist is cycling up a hill angled
15^\circ
with respect to the horizontal. The center of mass of the cyclist is directly over the axle of the wheel and the cyclist/unicycle system have a combined mass of
\SI{100}{\kilo\gram}.
The radius of the wheel is
\SI{0.5}{\meter}
and the coefficient of static friction between the wheel and the asphalt is
1.
What is the magnitude of the torque (in
\si{\newton\meter}
) that the cyclist needs to exert on the pedals in order to cycle up the hill at a constant speed?
The unicycle does not slip against the hill.
You may take the acceleration of gravity to be
-\SI[per-mode=symbol]{9.8}{\meter\per\second\squared}.
Cheerleaders at sporting events will often use a bullhorn to make themselves louder. A bullhorn is simply a hollowed out cone - you speak into the narrow end and your voice comes out the wide end, channeled by the cone so it is louder than it would be otherwise. At a sporting event, a cheerleader holds a bullhorn up to their mouth as in the figure below. The very tip of the bullhorn rests on their mouth and they use their hand on the other end to hold the bullhorn horizontal and still. We model the bullhorn as the rigid 2-d hollow cone of horizontal length
0.75~\mbox{m}
, opening angle
30^\circ
, and (surface) mass density of
0.4~\mbox{kg/m}^2
(see figure). How much force in Newtons does the cheerleader exert at point A?
You may take
g
9.8~\mbox{m/s}^2
Assume the force at point A is directed straight up.
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Global Constraint Catalog: Csliding_sum
<< 5.350. sliding_distribution5.352. sliding_time_window >>
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}\left(\mathrm{๐ป๐พ๐},\mathrm{๐๐ฟ},\mathrm{๐๐ด๐},\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}\right)
\mathrm{๐๐๐๐๐๐๐๐}
\mathrm{๐ป๐พ๐}
\mathrm{๐๐๐}
\mathrm{๐๐ฟ}
\mathrm{๐๐๐}
\mathrm{๐๐ด๐}
\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐๐ฟ}\ge \mathrm{๐ป๐พ๐}
\mathrm{๐๐ด๐}>0
\mathrm{๐๐ด๐}\le |\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐๐๐}\right)
\mathrm{๐๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
so that the sum of the variables belongs to interval
\left[\mathrm{๐ป๐พ๐},\mathrm{๐๐ฟ}\right]
\left(3,7,4,โฉ1,4,2,0,0,3,4โช\right)
The example considers all sliding sequences of
\mathrm{๐๐ด๐}=4
consecutive values of
โฉ1,4,2,0,0,3,4โช
collection and constraints the sum to be in
\left[\mathrm{๐ป๐พ๐},\mathrm{๐๐ฟ}\right]=\left[3,7\right]
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
constraint holds since the sum associated with the corresponding subsequences
1420
4200
2003
0034
are respectively 7, 6, 5 and 7.
\mathrm{๐ป๐พ๐}\ge 0
\mathrm{๐๐ฟ}>0
\mathrm{๐๐ด๐}>1
\mathrm{๐๐ด๐}<|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}\ge 0
\mathrm{๐๐ฟ}<
\mathrm{๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}\right)
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐ด๐}=1
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
constraint. In 2008, Maher et al. showed in [MaherNarodytskaQuimperWalsh08] that the
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
constraint has a solution โif and only there are no negative cycles in the flow graph associated with the dual linear programโ that encodes the conjunction of inequalities. They derive a bound-consistency filtering algorithm from this fact.
sliding_sum in MiniZinc.
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐ก๐๐}_\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐}_\mathrm{๐๐๐}
characteristic of a constraint: hypergraph, sum.
filtering: linear programming, flow, bound-consistency.
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐ด๐๐ป}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐ด๐}
โข
\mathrm{๐๐๐}_\mathrm{๐๐๐}
\left(\mathrm{๐๐๐๐๐๐๐๐๐๐},\ge ,\mathrm{๐ป๐พ๐}\right)
โข
\mathrm{๐๐๐}_\mathrm{๐๐๐}
\left(\mathrm{๐๐๐๐๐๐๐๐๐๐},\le ,\mathrm{๐๐ฟ}\right)
\mathrm{๐๐๐๐}
=|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\mathrm{๐๐ด๐}+1
\mathrm{๐๐๐}_\mathrm{๐๐๐}
as an arc constraint.
\mathrm{๐๐๐}_\mathrm{๐๐๐}
takes a collection of domain variables as its first argument.
\mathrm{๐๐๐๐}=|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\mathrm{๐๐ด๐}+1
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\left(3,7,\mathbf{4},โฉ1,\mathbf{4},\mathbf{2},\mathbf{0},\mathbf{0},\mathbf{3},\mathbf{4}โช\right)
\mathrm{๐๐ด๐}=\mathbf{4}
vertices (e.g., the first ellipse represents the constraint
1+4+2+0\in \left[3,7\right]
\mathrm{๐๐ด๐๐ป}
\mathrm{๐๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\mathrm{๐๐ด๐}+1
\mathrm{๐๐๐๐}=|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\mathrm{๐๐ด๐}+1
\mathrm{๐๐๐๐}\ge |\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\mathrm{๐๐ด๐}+1
\underline{\overline{\mathrm{๐๐๐๐}}}
\overline{\mathrm{๐๐๐๐}}
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Global Constraint Catalog: Csoft_alldifferent_ctr
<< 5.358. soft_all_equal_min_var5.360. soft_alldifferent_var >>
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}\left(๐ฒ,\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}\right)
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐ก}_\mathrm{๐๐๐}
๐ฒ
\mathrm{๐๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐๐}\right)
๐ฒ\ge 0
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐๐๐}\right)
Consider the disequality constraints involving two distinct variables
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}\left[i\right].\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}\left[j\right].\mathrm{๐๐๐}
\left(i<j\right)
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
๐ฒ
is greater than or equal to the number of disequality constraints that do not hold.
\left(4,โฉ5,1,9,1,5,5โช\right)
\left(1,โฉ5,1,9,1,2,6โช\right)
\left(0,โฉ5,1,9,0,2,6โช\right)
โฉ5,1,9,1,5,5โช
the first and fifth values, the first and sixth values, the second and fourth values, and the fifth and sixth values are identical. Consequently, the argument
๐ฒ=4
is greater than or equal to the number of disequality constraints that do not hold (i.e, 4) and the
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
๐ฒ>0
๐ฒ\le |\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|*\left(|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-1\right)/2
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|>1
๐ฒ
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐ก}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
constraint, the original article [PetitReginBessiere01] that introduces this constraint describes how to evaluate the minimum value of
๐ฒ
๐ฒ
. The corresponding filtering algorithm does not achieve arc-consistency. W.-J. van Hoeve [Hoeve04] presents a new filtering algorithm that achieves arc-consistency. This algorithm is based on a reformulation into a minimum-cost flow problem.
n
Solutions 15 208 3625 72576 1630279 40632320 1114431777
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
0..n
n
Total 15 208 3625 72576 1630279 40632320 1114431777
2 - 60 540 6120 80640 1169280 18144000
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
0..n
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐ก}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐ข}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}
filtering: minimum cost flow.
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐ถ๐ฟ๐ผ๐๐๐ธ}
\left(<\right)โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{1},\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{2}\right)
\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{1}.\mathrm{๐๐๐}=\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{2}.\mathrm{๐๐๐}
\mathrm{๐๐๐๐}
\le ๐ฒ
\mathrm{๐ถ๐ฟ๐ผ๐๐๐ธ}\left(<\right)
in order to avoid counting twice the same equality constraint. The graph property states that
๐ฒ
is greater than or equal to the number of equalities that hold in the final graph.
\mathrm{๐๐๐๐}
graph property, the arcs of the final graph are stressed in bold. Since four equality constraints remain in the final graph the cost variable
๐ฒ
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
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K
A
M. Furi, M. Martelli (1974)
Josรฉ Antonio Ezquerro, Daniel Gonzรกlez, Miguel รngel Hernรกndez (2013)
From Kantorovichโs theory we present a semilocal convergence result for Newtonโs method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newtonโs method. We illustrate this study with applications to nonlinear integral equations of mixed Hammerstein type.
From Kantorovichโs theory we present a semilocal convergence result for Newtonโs method which is based mainly on a modification of the condition required to the second derivative of the operator involved. In particular, instead of requiring that the second derivative is bounded, we demand that it is centered. As a consequence, we obtain a modification of the starting points for Newtonโs method. We illustrate this study with applications to nonlinear...
A generalization of Yano's extrapolation theorem
Capri, O.N., Fava, N.A. (1987)
A new method for the obtaining of eigenvalues of variational inequalities of the special type (Preliminary communication)
A nonlinear Banach-Steinhaus theorem and some meager sets in Banach spaces
Jacek Jachymski (2005)
We establish a Banach-Steinhaus type theorem for nonlinear functionals of several variables. As an application, we obtain extensions of the recent results of Balcerzak and Wachowicz on some meager subsets of Lยน(ฮผ) ร Lยน(ฮผ) and cโ ร cโ. As another consequence, we get a Banach-Mazurkiewicz type theorem on some residual subset of C[0,1] involving Kharazishvili's notion of ฮฆ-derivative.
A note on fixed point theorem of Schauder type with applications
A note on geometric characterization of Frรฉchet differentiability
Josef Daneลก, Jiลรญ Durdil (1976)
Guillermo Restrepo (1971)
A strong convergence in
{L}^{p}
q
-continuous operators
Alexander Haลกฤรกk (1988)
An application of the Hahn-Banach theorem in convex analysis
Ludฤk Jokl (1981)
An Existence Theorem for Class of Non-Coercive Optimization and Variational Problems.
An Iteration Scheme for Boundary Value-Alternative Problems.
D.A. Sรขnchez (1974/1975)
Application of Rothe's method to perturbed linear hyperbolic equations and variational inequalities
Approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces.
Igbokwe, D.I. (2002)
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Error, invalid input: f uses a 1st argument, x, which is missing - Maple Help
Home : Support : Online Help : Error, invalid input: f uses a 1st argument, x, which is missing
i
\mathrm{sum}\left(i\right)
i.
\mathrm{sum}\left(i,i\right)
\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{โข}{\textcolor[rgb]{0,0,1}{i}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\frac{\textcolor[rgb]{0,0,1}{1}}{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{โข}\textcolor[rgb]{0,0,1}{i}
\mathrm{factor}\left(\right)
\mathrm{factor}\left(6 {x}^{2}+18\cdot x-24\right)
\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{โข}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\right)\textcolor[rgb]{0,0,1}{โข}\left(\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{1}\right)
\mathrm{Adder}โ\mathbf{proc}\left(aโท\mathrm{integer} ,b\right) a+b \mathbf{end} \mathbf{proc};
\textcolor[rgb]{0,0,1}{\mathrm{Adder}}\textcolor[rgb]{0,0,1}{:=}\textcolor[rgb]{0,0,1}{\mathbf{proc}}\left(\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{::}\textcolor[rgb]{0,0,1}{\mathrm{integer}}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{b}\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{a}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{b}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}\textcolor[rgb]{0,0,1}{\mathbf{end proc}}
\mathrm{Adder}\left(3\right)
\mathrm{Adder}\left(3,2.5\right)
\textcolor[rgb]{0,0,1}{5.5}
\mathrm{eliminate}โก\left(\left\{{x}^{2}+{y}^{2}-1,{x}^{3}-{y}^{2}โขx+xโขy-3\right\}\right)
\mathrm{eliminate}โก\left(\left\{{x}^{2}+{y}^{2}-1,{x}^{3}-{y}^{2}โขx+xโขy-3\right\},x\right)
\left[\left\{\textcolor[rgb]{0,0,1}{x}\textcolor[rgb]{0,0,1}{=}\frac{\textcolor[rgb]{0,0,1}{3}}{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{โข}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{1}}\right\}\textcolor[rgb]{0,0,1}{,}\left\{\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{7}\textcolor[rgb]{0,0,1}{โข}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{4}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{โข}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{6}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{6}\textcolor[rgb]{0,0,1}{โข}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{3}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{โข}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{5}}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{4}\textcolor[rgb]{0,0,1}{โข}{\textcolor[rgb]{0,0,1}{y}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{-}\textcolor[rgb]{0,0,1}{2}\textcolor[rgb]{0,0,1}{โข}\textcolor[rgb]{0,0,1}{y}\textcolor[rgb]{0,0,1}{+}\textcolor[rgb]{0,0,1}{8}\right\}\right]
|
\displaystyle{\int_a^b f(x)\, dx}:
S(x)
f(x)
S(x)=\int _{ a }^{ x }{ f(t)\, dt },
f
[a,b]
x
f(t)
a
x+\Delta x
f(t)
a
x
\begin{aligned} \Delta S&=A(x+\Delta x)-A(x)\\\\ \frac{\Delta S}{\Delta x}&=\frac{A(x+\Delta x)-A(x)}{\Delta x}. \end{aligned}
S'(x)=\frac{dS}{dx}=\lim_ {\Delta x\rightarrow 0 } \frac { S(x+\Delta x)-S(x) }{ \Delta x } .
\overline{x}
x
x+\Delta x
f(\overline{x})\Delta x
\begin{aligned} S'(x) &=\lim_{\Delta x\rightarrow 0 }\frac { S(x+\Delta x)-S(x) }{ \Delta x } \\ &=\lim_{\Delta x\rightarrow 0 }\frac { f(\overline { x } )\Delta x }{ \Delta x } \\ &=\lim_{\Delta x\rightarrow 0 } f(\overline { x } )\\ &=f(x). \end{aligned}
\Delta x\rightarrow 0
x
x+\Delta x
x
If
[a,b]
S(x)=\int _{ a }^{ x }{ f(t)\, dt }
[a,b]
(a,b)
S'(x)=f(x)
\frac { d }{ dx } \int _{ a }^{ x }{ f(t)\, dt=f(x) }.
\displaystyle k(x)=\int _{ 2 }^{ x }{ ({ 4}^{ t }+t)\, dt }
f
k'(x)={ 4 }^{ x }+x. \ _\square
\displaystyle h(x)=\int_{ 2 }^{ { x }^{ 2 } }{ \frac {1}{ 1+{ t }^{ 2 } }\, dt }?
u=x^{2}
\begin{aligned} \frac{d}{dx} \int_{2}^{x^2}{\frac{1}{1+t^2}\, dt} &= \frac { d }{ du } \left[ \int _{ 1 }^{ u }{ \frac { 1 }{ 1+{ t }^{ 2 } } dt} \right] \cdot \frac { du }{ dx } \\ &=\frac { 1 }{ 1+{ u }^{ 2 } } \cdot 2{ x }\\ &=\frac { 2x }{ 1+{ x }^{ 4 } }. \ _\square \end{aligned}
\displaystyle\int_{1}^{x^2}\cos t \, dt.
u=x^2
\begin{aligned} \displaystyle\int_{1}^{x^2}\cos t\, dt&=\dfrac{d}{du}\left[\displaystyle\int_{1}^{u}\cos t\, dt\right]\cdot\dfrac{du}{dx}\\ &=\cos{u}\cdot\dfrac{d}{dx}\left(x^2\right)\\ &=\cos u\cdot 2x\\ &=2x\cos x^2. \ _\square \end{aligned}
\displaystyle h(x)=\int _{ x }^{ 3x }{ \sin\theta \, d\theta }
\int _{ b }^{ a }{ f(x)\, dx } =\int _{ b }^{ 0 }{ f(x)\, dx+\int _{ 0 }^{ a }{ f(x)\, dx } } .
\begin{aligned} h'(x) &=\frac { d }{ dx } \int _{ x }^{ 3x }{ \sin\theta \, d\theta } \\ &=\frac { d }{ dx } \int _{ x }^{ 0 }{ \sin\theta \, d\theta +\frac { d }{ dx } \int _{ 0 }^{ 3x }{ \sin\theta \, d\theta } } \\ &=\frac { d }{ dx } -\int _{ 0 }^{ x }{ \sin\theta \, d\theta } +\frac { d }{ dx } \int _{ 0 }^{ 3x }{ \sin\theta \, d\theta } \\ &=-\sin x+\frac { d }{ du } \int _{ 0 }^{ u }{ \sin\theta \, d\theta } \cdot \frac { du }{ dx } \\ &=-\sin x+3\sin3x. \ _\square \end{aligned}
If
[a,b]
\int _{ a }^{ b }{ f(x)\, dx=F(b)-F(a) } ,
F
is an anti-derivative of
F'=f
_\square
\displaystyle S(x)=\int _{ a }^{ x }{ f(t)\, dt }
S
is an anti-derivative of
S'(x)=f(x)
F'(x)=f(x),
S'(x)=F'(x).
x
\displaystyle S(x)=\int _{ a }^{ x }{ f(t)\, dt }=F(x)+C, \qquad \qquad(1)
C
x=a
S(a)=\int _{ a }^{ a }{ f(t)\, dt }=F(a)+C.
\displaystyle S(a)=\int _{ a }^{ a }{ f(t)\, dt }=0
\begin{aligned} F(a)+C&=0\\ \Rightarrow C&=-F(a). \end{aligned}
C=-F(a)
(1)
\displaystyle S(x)=\int _{ a }^{ x }{ f(t)\, dt }=F(x)-F(a).
x=b
\begin{aligned} S(b)=\int _{ a }^{ b }{ f(t)\, dt }&=F(b)-F(a)\\ \Rightarrow \int _{ a }^{ b }{ f(x)\, dx }&=F(b)-F(a).\ _\square \end{aligned}
\displaystyle S(x)=\int _{ a }^{ x }{ f(t)\, dt }
S
is an anti-derivative of
S'(x)=f(x)
\displaystyle f(x) = F'(x)
P = \{ a= x_0 < x_1 < x_2 < \dots < x_n = b \}
F(b) - F(a) = \sum_{i=0}^{n-1} \big[F(x_{i+1}) - F(x_i)\big].
\displaystyle \forall i = 0,1,2, \dots, n -1 \text{ } \exists t_i \in \text{ } ( x_i , x_{i+1} )
\displaystyle F(x_{i+1}) - F(x_i) = F'(t_i) \left( x_{i+1} - x_{i} \right) = f(t_i) \left( x_{i+1} - x_{i} \right)
F(b) - F(a) = \sum_{i=0}^{n-1} f(t_i) \left( x_{i+1} - x_{i} \right);~ t_i \in \text{ } ( x_i , x_{i+1} ).
P
F(b)-F(a)=\int _{ a }^{ b }{ f(t)\, dt } . \ _\square
\displaystyle \int _{ a }^{ b }{ f(x)\, dx }
f(x)
\displaystyle{\int_0^1}x^2\, dx.
\displaystyle{\int_0^1}x^2\, dx=F(1)-F(0),
F(x)
x^2.
x^2
\int x^2dx=\frac{1}{3}x^3+C,
C
F(1)-F(0)=\left(\frac{1}{3}\times1^3+C\right)-\left(\frac{1}{3}\times0+C\right)=\frac{1}{3}.\ _\square
C
y=x^3+1
x=-1
x=1.
x^3+1\geq0
[-1,1],
y=x^3+1
x=-1
x=1
\displaystyle{\int_{-1}^1\left(x^3+1\right)\, dx}.
\frac{1}{4}x^4+x
x^3+1,
\begin{aligned} \int_{-1}^1\left(x^3+1\right)\, dx&=\left[\frac{1}{4}x^4+x\right]_{-1}^1\\ &=\left(\frac{1}{4}\cdot1^4+1\right)-\left(\frac{1}{4}\cdot(-1)^4+(-1)\right)\\ &=\frac{5}{4}-\left(-\frac{3}{4}\right)\\ &=2.\ _\square \end{aligned}
\displaystyle{\int_{-3}^{2}\left(2x^2-3x+4\right)\, dx.}
\displaystyle{\int\left(2x^2-3x+4\right)\, dx=\frac{2}{3}x^3-\frac{3}{2}x^2+4x+C,}
\begin{aligned} \displaystyle{\int_{-3}^{2}\left(2x^2-3x+4\right)\, dx}&=\left[\frac{2}{3}x^3-\frac{3}{2}x^2+4x\right]_{-3}^{2}\\ &=\left(\frac{2}{3}\cdot2^3-\frac{3}{2}\cdot2^2+4\cdot2\right)-\left(\frac{2}{3}\cdot(-3)^3-\frac{3}{2}\cdot(-3)^2+4\cdot(-3)\right)\\ &=\frac{305}{6}.\ _\square \end{aligned}
\displaystyle{\int_{4}^{9}\sqrt{x}\, dx.}
\begin{aligned} \int_{4}^{9}\sqrt{x}\, dx&=\int_4^9 x^{\frac{1}{2}}\, dx\\ &=\left[\frac{2}{3}x^{\frac{3}{2}}\right]_4^9\\ &=\frac{2}{3}\cdot9^{\frac{3}{2}}-\frac{2}{3}\cdot4^{\frac{3}{2}}\\ &=\frac{38}{3}.\ _\square \end{aligned}
y=\frac{1}{x}
x=\frac{1}{2}
x=\frac{5}{2}.
\displaystyle{\int\frac{1}{x}\, dx=\ln x+C,}
\begin{aligned} \int_{\frac{1}{2}}^{\frac{5}{2}}\frac{1}{x}\, dx&=\Big[\ln x\Big]_{\frac{1}{2}}^{\frac{5}{2}}\\ &=\ln\frac{5}{2}-\ln\frac{1}{2}\\ &=\ln5.\ _\square \end{aligned}
\displaystyle\int_{-\frac{\pi}{4}}^{0} \sec x\tan x \ dx.
\dfrac{d}{dx}\sec x = \sec x\tan x.
\begin{aligned} \displaystyle\int_{-\frac{\pi}{4}}^{0} \sec x\tan x \ dx &= \Big[\sec x\Big]_{-\frac{\pi}{4}}^{0} \\ &=\sec 0 - \sec\left(\dfrac{-\pi}4\right)\\ &=1-\sqrt 2. \ _\square \end{aligned}
\displaystyle{\int_{-\frac{\pi}{2}}^{\frac{5}{6}\pi}\sin x\, dx.}
\begin{aligned} \int_{-\frac{\pi}{2}}^{\frac{5}{6}\pi}\sin x\, dx&=\Big[-\cos x\Big]_{-\frac{\pi}{2}}^{\frac{5}{6}\pi}\\ &=\frac{\sqrt{3}}{2}.\ _\square \end{aligned}
y=2x-x^2
x=1
x=2.
2x-x^2\geq0
[1,2],
y=2x-x^2
x=1
x=2
\displaystyle{\int_1^2\left(2x-x^2\right)\, dx}.
x^2-\frac{1}{3}x^3
2x-x^2,
\begin{aligned} \int_1^2\left(2x-x^2\right)\, dx&=\left[x^2-\frac{1}{3}x^3\right]_1^2\\ &=\left(2^2-\frac{1}{3}\cdot2^3\right)-\left(1^2-\frac{1}{3}\cdot1^3\right)\\ &=\frac{2}{3}.\ _\square \end{aligned}
|
Find minimum of unconstrained multivariable function using derivative-free method - MATLAB fminsearch
\underset{x}{\mathrm{min}}f\left(x\right)
f\left(x\right)=100\left({x}_{2}-{x}_{1}^{2}{\right)}^{2}+\left(1-{x}_{1}{\right)}^{2}.
f\left(x\right)=100\left({x}_{2}-{x}_{1}^{2}{\right)}^{2}+\left(1-{x}_{1}{\right)}^{2}.
f\left(x,a\right)=100\left({x}_{2}-{x}_{1}^{2}{\right)}^{2}+\left(a-{x}_{1}{\right)}^{2}.
{x}_{1}=a
{x}_{2}={a}^{2}
a=3
\underset{x}{\mathrm{min}}{โf\left(x\right)โ}_{2}^{2}=\underset{x}{\mathrm{min}}\left({f}_{1}{\left(x\right)}^{2}+{f}_{2}{\left(x\right)}^{2}+...+{f}_{n}{\left(x\right)}^{2}\right)
|
Liquidation Mechanism - Zeta
An important component of a robust margining system is liquidation to ensure that one trader does not adversely impact other traders or the security of the platform.
When accounts do not have sufficient capital they will need to be liquidated to ensure that the system at large does not become bankrupt and to ensure that risks are isolated to individual user accounts where possible. The element underpinning this process is the margin check:
AB + UP - IM - MM > 0
AB = Account Balance
UP = Unrealised PnL
IM = Initial Margin
MM = Maintenance Margin
If a user fails the margin check, the liquidation process will begin:
3rd party liquidators can cancel all open orders on a personโs book. The user will not be able to send orders during this time until all orders are cancelled.
If the user is still below maintenance margin after the above cancels, they are able to be liquidated by sending a liquidation instruction. This instruction specifies a particular margin account, the marketโs position and the size to be liquidated.
Upon a successful liquidation, the liquidator will trade with the liquidatee at the current mark price stored by Zeta, as well as gain a fixed % of the liquidateeโs collateral put up for maintenance margin as a reward. This reward is currently set at 35% of the maintenance margin for both options and futures positions, with 25% going to the liquidator and 10% going to the insurance fund.
Liquidation is a permissionless instruction, open to anyone. Check out our naive liquidator example here that shows you how to check for positions at risk and liquidate their positions.
Note: Liquidation is a risky endeavour and not guaranteed to be profitable.
The Zeta DEX - Previous
Next - The Zeta DEX
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Maximum likelihood - Wikipรฉdia Sunda, รฉnsiklopรฉdi bรฉbas
Dina statistik, mรฉtodeu maximum likelihood, dimimitian ku ahli genetika/ahli statistik Sir Ronald A. Fisher, mangrupa mรฉtodeu titik estimasi, nu digunakeun keur nga-estimasi anggota populasi nu teu ka-observasi tina paramรฉter ruang nu di-maksimal-keun ku fungsi likelihood. Tingali p ngalambangkeun paramรฉter populasi teu ka-observasi nu bakal di-estimasi. Tingali X ngalambangkeun random variable nu di-observasi (which in general will not be scalar-valued, but often will be a vector of probabilistically independent scalar-valued random variables. The probability of an observed outcome X=x (this is case-sensitive notation!), or the value at (lower-case) x of the probability density function of the random variable (Capital) X, as a function of p with x held fixed is the likelihood function
{\displaystyle L(p)=P(X=x\mid p).}
{\displaystyle L(p)={n \choose x}p^{x}(1-p)^{n-x}.}
The value of p that maximizes L(p) is the maximum-likelihood estimate of p. By finding the root of the first derivative one will obtain x/n as the maximum-likelihood estimate. In this case, as in many other cases, it is much รฉasier to take the logarithm of the likelihood function before finding the root of the derivative:
{\displaystyle {\frac {x}{p}}-{\frac {n-x}{1-p}}=0}
If we replace the lower-case x with capital X then we have, not the observed value in a particular case, but rather a variabel acak, which, like all random variables, has a probability distribution. The value (lower-case) x/n observed in a particular case is an estimate; the variabel acak (Capital) X/n is an estimator. The statistician may take the nature of the probability distribution of the รฉstimator to indicate how good the รฉstimator is; in particular it is desirable that the probability that the รฉstimator is far from the paramรฉter p be small. Maximum-likelihood รฉstimators are sometimes better than unbiased estimators. They also have a property called "functional invariance" that unbiased รฉstimators lack: for any function f, the maximum-likelihood รฉstimator of f(p) is f(T), where T is the maximum-likelihood รฉstimator of p.
However, the bias of maximum-likelihood รฉstimators can be substantial. Consider a case where n tickets numbรฉrรฉd from 1 through to n are placed in a box and one is selected at random, giving a value X. If n is unknown, then the maximum-likelihood รฉstimator of n is X, even though the expectation of X is only n/2; we can only be certain that n is at lรฉast X and is probably more.
Dicomot ti "https://su.wikipedia.org/w/index.php?title=Maximum_likelihood&oldid=490911"
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Global Constraint Catalog: Csoft_all_equal_min_var
<< 5.357. soft_all_equal_min_ctr5.359. soft_alldifferent_ctr >>
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}\left(๐ฝ,\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}\right)
๐ฝ
\mathrm{๐๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐๐}\right)
๐ฝ\ge 0
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐๐๐}\right)
M
be the number of occurrences of the most often assigned value to the variables of the
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
๐ฝ
is greater than or equal to the total number of variables of the
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection minus
M
๐ฝ
is greater than or equal to the minimum number of variables that need to be reassigned in order to obtain a solution where all variables are assigned a same value).
\left(1,โฉ5,1,5,5โช\right)
โฉ5,1,5,5โช
, 3 is the number of occurrences of the most assigned value. Consequently, the
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
๐ฝ=1
is greater than or equal to the total number of variables 4 minus 3.
๐ฝ>0
๐ฝ<|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
๐ฝ<|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|/10+2
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|>1
๐ฝ
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
m
denote the total number of potential values that can be assigned to the variables of the
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection. In [HebrardMarxSullivanRazgon09], E. Hebrard et al. provides an
O\left(m\right)
filtering algorithm achieving arc-consistency on the
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
constraint. The same paper also provides an algorithm with a lower complexity for achieving range consistency. Both algorithms are based on the following ideas:
In a first phase, they both compute an envelope of the union
๐
of the domains of the variables of the
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection, i.e., an array
A
that indicates for each potential value
v
๐
, the maximum number of variables that could possibly be assigned value
v
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}
denote the maximum value over the entries of array
A
{๐ฑ}_{\mathrm{๐๐๐ฅ}}_\mathrm{๐๐๐}
denote the set of values which all occur in
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection. The quantity
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}
is a lower bound of
๐ฝ
In a second phase, depending on the relative ordering between
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-๐ฝ
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\overline{๐ฝ}
, we have the three following cases:
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}<|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\overline{๐ฝ}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
simply fails since not enough variables of the
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection can be assigned the same value.
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}=|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\overline{๐ฝ}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
can be satisfied. In this context, a value
v
can be removed from the domain of a variable
V
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection if and only if:
v
{๐ฑ}_{\mathrm{๐๐๐ฅ}}_\mathrm{๐๐๐}
the domain of variable
V
contains all values of
{๐ฑ}_{\mathrm{๐๐๐ฅ}}_\mathrm{๐๐๐}
On the one hand, the first condition can be understand as the fact that value
v
is not a value that allows to have at least
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\overline{๐ฝ}
variables assigned the same value. On the other hand, the second condition can be interpreted as the fact that variable
V
is absolutely required in order to have at least
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\overline{๐ฝ}
variables assigned the same value.
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}>|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\overline{๐ฝ}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
can be satisfied, but no value can be pruned.
Note that, in the context of range consistency, the first phase of the filtering algorithm can be interpreted as a sweep algorithm were:
On the one hand, the sweep status corresponds to the maximum number of occurrence of variables that can be assigned a given value.
On the other hand, the event point series correspond to the minimum values of the variables of the
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection as well as to the maximum values (
+1
) of the same variables.
Figure 5.358.1. Illustration of the two phases filtering algorithm of the
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\left(1,โฉ{V}_{1},{V}_{2},{V}_{3},{V}_{4}โช\right)
constraint with
{V}_{1}\in \left[1,3\right]
{V}_{2}\in \left[3,7\right]
{V}_{3}\in \left[0,8\right]
{V}_{4}\in \left[5,6\right]
Figure 5.358.1 illustrates the previous filtering algorithm on an example where
๐ฝ
is equal to 1, and where we have four variables
{V}_{1}
{V}_{2}
{V}_{3}
{V}_{4}
respectively taking their values within intervals
\left[1,3\right]
\left[3,7\right]
\left[0,8\right]
\left[5,6\right]
(see Part (A) of Figure 5.358.1, where the values of each variable are assigned a same colour that we retrieve in the other parts of Figure 5.358.1).
Part (B) of Figure 5.358.1 illustrates the first phase of the filtering algorithm, namely the computation of the envelope of the domains of variables
{V}_{1}
{V}_{2}
{V}_{3}
{V}_{4}
. The start events
{s}_{1}
{s}_{2}
{s}_{3}
{s}_{4}
(i.e., the events respectively associated with the minimum value of variables
{V}_{1}
{V}_{2}
{V}_{3}
{V}_{4}
) where the envelope is increased by 1 are represented by the character
โ
. Similarly, the end events (i.e., the events
{e}_{1}
{e}_{2}
{e}_{3}
{e}_{4}
respectively associated with the maximum value (
+1
{V}_{1}
{V}_{2}
{V}_{3}
{V}_{4}
are represented by the character
โ
). Since the highest peak of the envelope is equal to 3 we have that
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}
is equal to 3. The values that allow to reach this highest peak are equal to
{๐ฑ}_{\mathrm{๐๐๐ฅ}}_\mathrm{๐๐๐}=\left\{3,5,6\right\}
(i.e., shown in red in Part (B) of Figure 5.358.1).
Finally, Part (C) of Figure 5.358.1 illustrates the second phase of the filtering algorithm. Since
\mathrm{๐๐๐ฅ}_\mathrm{๐๐๐}=3
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-\overline{๐ฝ}=4-1
we remove from the variables whose domains contain
{๐ฑ}_{\mathrm{๐๐๐ฅ}}_\mathrm{๐๐๐}=\left\{3,5,6\right\}
(i.e., variables
{V}_{2}
{V}_{3}
) all values not in
{๐ฑ}_{\mathrm{๐๐๐ฅ}}_\mathrm{๐๐๐}=\left\{3,5,6\right\}
(i.e., values 4, 7 for variable
{V}_{2}
and values 0, 1, 2, 4, 7, 8 for variable
{V}_{3}
n
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
0..n
n
2 9 64 505 1656 4039 8632 16713
4 - - 625 7776 112609 751472 2852721
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
0..n
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐ก}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}
\mathrm{๐ก๐๐}
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}
filtering: arc-consistency, sweep.
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐ถ๐ฟ๐ผ๐๐๐ธ}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{1},\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{2}\right)
\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{1}.\mathrm{๐๐๐}=\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{2}.\mathrm{๐๐๐}
\mathrm{๐๐๐}_\mathrm{๐๐๐๐}
\ge |\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|-๐ฝ
We generate an initial graph with binary equalities constraints between each vertex and its successors. The graph property states that
๐ฝ
is greater than or equal to the difference between the total number of vertices of the initial graph and the number of vertices of the largest strongly connected component of the final graph.
\mathrm{๐๐๐}_\mathrm{๐๐๐๐}
graph property we show one of the largest strongly connected components of the final graph.
\mathrm{๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐}_\mathrm{๐๐๐}
|
Global Constraint Catalog: Csum_of_weights_of_distinct_values
<< 5.387. sum_of_increments5.389. sum_powers4_ctr >>
[BeldiceanuCarlssonThiel02]
\mathrm{๐๐๐}_\mathrm{๐๐}_\mathrm{๐ ๐๐๐๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐
๐ฐ๐ป๐๐ด๐},\mathrm{๐ฒ๐พ๐๐}\right)
\mathrm{๐๐ ๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐
๐ฐ๐ป๐๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐},\mathrm{๐ ๐๐๐๐๐}-\mathrm{๐๐๐}\right)
\mathrm{๐ฒ๐พ๐๐}
\mathrm{๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐๐๐}\right)
|\mathrm{๐
๐ฐ๐ป๐๐ด๐}|>0
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐๐ด๐},\left[\mathrm{๐๐๐},\mathrm{๐ ๐๐๐๐๐}\right]\right)
\mathrm{๐
๐ฐ๐ป๐๐ด๐}.\mathrm{๐ ๐๐๐๐๐}\ge 0
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐๐ด๐},\mathrm{๐๐๐}\right)
\mathrm{๐๐}_\mathrm{๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐๐๐},\mathrm{๐
๐ฐ๐ป๐๐ด๐},\mathrm{๐๐๐}\right)
\mathrm{๐ฒ๐พ๐๐}\ge 0
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
collection take a value in the
\mathrm{๐
๐ฐ๐ป๐๐ด๐}
collection. In addition
\mathrm{๐ฒ๐พ๐๐}
\mathrm{๐ ๐๐๐๐๐}
attributes associated with the distinct values taken by the variables of
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\left(\begin{array}{c}โฉ1,6,1โช,\hfill \\ โฉ\begin{array}{cc}\mathrm{๐๐๐}-1\hfill & \mathrm{๐ ๐๐๐๐๐}-5,\hfill \\ \mathrm{๐๐๐}-2\hfill & \mathrm{๐ ๐๐๐๐๐}-3,\hfill \\ \mathrm{๐๐๐}-6\hfill & \mathrm{๐ ๐๐๐๐๐}-7\hfill \end{array}โช,12\hfill \end{array}\right)
\mathrm{๐๐๐}_\mathrm{๐๐}_\mathrm{๐ ๐๐๐๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}
constraint holds since its last argument
\mathrm{๐ฒ๐พ๐๐}=12
is equal to the sum
5+7
of the weights of the values 1 and 6 that occur within the
โฉ1,6,1โช
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|>1
\mathrm{๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}\right)>1
|\mathrm{๐
๐ฐ๐ป๐๐ด๐}|>1
\mathrm{๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐๐ด๐}.\mathrm{๐ ๐๐๐๐๐}\right)>1
\mathrm{๐๐๐ก๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐๐ด๐}.\mathrm{๐ ๐๐๐๐๐}\right)>0
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐ป๐๐ด๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐ป๐๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐
๐ฐ๐ป๐๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐ฒ๐พ๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐ป๐๐ด๐}
\mathrm{๐๐๐๐๐๐}
(all values have a weight of 1).
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}_\mathrm{๐ ๐๐๐}_\mathrm{๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐ ๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐ ๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}
(weighted assignment).
constraint type: relaxation.
filtering: cost filtering constraint.
problems: domination, weighted assignment, facilities location problem.
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐
๐ฐ๐ป๐๐ด๐}
\mathrm{๐๐
๐๐ท๐๐ถ๐}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐๐๐๐๐},\mathrm{๐๐๐๐๐๐}\right)
\mathrm{๐๐๐๐๐๐๐๐๐}.\mathrm{๐๐๐}=\mathrm{๐๐๐๐๐๐}.\mathrm{๐๐๐}
โข
\mathrm{๐๐๐๐๐๐๐}
=|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
โข
\mathrm{๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐๐ด๐},\mathrm{๐ ๐๐๐๐๐}\right)=\mathrm{๐ฒ๐พ๐๐}
\mathrm{๐๐
๐๐ท๐๐ถ๐}
arc generator, the number of sources of the final graph cannot exceed the number of sources of the initial graph. Since the initial graph contains
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
sources, this number is an upper bound of the number of sources of the final graph. Therefore we can rewrite
\mathrm{๐๐๐๐๐๐๐}=|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
\mathrm{๐๐๐๐๐๐๐}\ge |\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
\underline{\overline{\mathrm{๐๐๐๐๐๐๐}}}
\overline{\mathrm{๐๐๐๐๐๐๐}}
\mathrm{๐๐๐๐๐๐๐}
graph property, the source vertices of the final graph are shown in a double circle. Since we also use the
\mathrm{๐๐๐}
graph property we show the vertices from which we compute the total cost in a box.
\mathrm{๐๐๐}_\mathrm{๐๐}_\mathrm{๐ ๐๐๐๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}
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Basic Differential Equations Tutorial - AudioPlastic
Basic Differential Equations Tutorial
This is a brief tutorial showing how to use differential equations to do make predictions about the state of a simple dynamic analogue electronics circuit. I have ported this post over from my discontinued Wordpress blog, as some readers may find the information useful.
Put simply, a differential equation is any equation that contains derivatives. A derivative is basically a gradient, and so it is an equation that describes a system containing something that changes. In audio, this will likely be something that changes over time (t), so an example might be
I looks kinda like a polynomial, but with various order derivatives. Weโd call this example a second order differential equation, as the highest order derivative is 2.
I quickly want to move onto a real world example, showing some useful stuff we can do with differential equations. For this, Iโm going to use simple electronic components in a circuit and then try and predict the state of the circuit at a given time. Iโm going to use basic analogue electronic components for my first real-world example, as digital filter theory is rooted in basic analogue electronics theory.
The complex impedance (
Z
) of a circuit is made up of real resistive terms (
R
) and imaginary reactive terms (
X
The reactance term comes from the combination of inductance and capacitance. Like resistance, capacitance and inductance are inherent properties of electronic components. The capacitative effect comes from built up electric field that resists voltage changes. The inductive effect comes from the build up of magnetic field that resists changes in current. Reactance effects are only exhibited when circuit conditions change. Therefore, we can use differential equations to predict the state of a circuit after a change. For the following example, the change is the circuit being switched on.
Consider a simple circuit containing a prefect voltage source, a resistor and an inductor โฆ
When the circuit is switched on, the inductor chokes the initial burst of current, giving rise to a gradual increase in current through the inductor until a maximum value is reached.
One way to visualise the current flow through this circuit is to build it and measure it, but this is a bit of a faff. Another way is to use electronic simulation software. For simple circuits (and even for relatively complex ones too) I am a huge fan of the excellent and free web app, CircuitLab. CircuitLab allows the user to just drag and drop various components, set their values, then produce simulation plots like that shown below.
The plot shows that the rise in current through the inductor is not linear with time. This is because the relationship between the inductance (L), voltage (V), and current (I) is differential - it changes over time โฆ
We will now see if we can predict the current for any given time analytically using differential equations. We know from Kirchoffโs Voltage Law that the sum total voltage drop across series components (VR and VL) equals the supply voltage (VS) โฆ
We want to predict the current, so using Ohmโs law (V=IR), we can express this in terms of current (I) โฆ
The next thing we want to do is get this into a suitable form for integration so we can remove the differential terms (dI and dt) and find a solution. THe first step is to split dI and dt โฆ
โฆ then separate the rest of the variables โฆ
This is now in a suitable form for integration.
This is a great time to introduce a web app that I find particularly useful, Wolfram Alpha. This tool can be used to perform calculus. It will show the intermediate steps towards a solution, so there is no black-box trickery to get in the way of our understanding. I have posted a screenshot of the output of the web app when asked to integrate the left-hand-side (LHS) of the above equation. It saves all the
LaTeX
typing anyway!
Integration of the right-hand-side (RHS) is trivial, so we end up with the following โฆ
We are still trying to solve this for current, so the next step is to isolate the logarithmic term so that both sides of the equation can be easily exponentiated โฆ
Now simply exp() both sides โฆ
Now save a little writing by using the following alias for K, where K is some constant term โฆ
This allows us to finally get the current term by itself โฆ
Great! We can almost state the current at a given time after the circuit is switched on, but there are still some pesky unknown constant terms in K that would cause some uncertainty in the result. However, this is not a problem because we know the initial conditions at switch on. At time t=0, the current through the circuit I=0. Substituting this in โฆ
And so it follows โฆ
Now we can determine the current at any given time after switch on!
It is then easy to check if our analytical solution matches our hardware equivalent by making a quick Matlab script โฆ
tAxis = 0:1e-4:5e-3; % Time Axis
% We can use a Lambda to neatly represent our solution
current = @(V,R,L,t)(V/R)*(1-exp(-t*R/L));
plot(1000*tAxis,1000*current(9,20,10e-3,tAxis),'Linewidth',2);
xlabel('Time [ms]'); ylabel('Current [mA]'); ylim([0 500])
So there we have it. I differential equation solved that allows us to predict the behaviour of a real physical system.
Posted by Nick Clark Jan 7th, 2013
ยซ Matlab-like profiling in C++ using RAII Cobalt theme for Xcode ยป
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Home : Support : Online Help : Education : Student Packages : Linear Algebra : Computation : Overview
Overview of Computation in Student[LinearAlgebra]
For a general introduction to the LinearAlgebra subpackage of the Student package and a list of the linear algebra computation routines, see Student[LinearAlgebra].
The computation routines in the Student[LinearAlgebra] subpackage are interfaces to the corresponding routines in the top-level LinearAlgebra package. There are two principal differences that these interfaces implement, however.
First, the top-level LinearAlgebra routines use hardware floating-point computations whenever possible. While this is important for large scale problems, it is potentially confusing, so in the Student[LinearAlgebra] subpackage this feature is turned off by default.
Second, the top-level LinearAlgebra routines generally treat symbols as complex-valued rather than real-valued. For example, a calculation such as
โฉa,bโชยทโฉc,dโช
results in complex conjugates being applied to some of the symbols. Again, this working environment, while important in the context of the full Maple program, is less essential in the Student[LinearAlgebra] context, and symbols are generally treated as real-valued in this package.
To use hardware floating-point computations and treat symbols as complex-valued, use the SetDefault command in the (main) Student subpackage. Local control is available for the complex-versus-real assumption by appropriate use of the conjugate option on relevant Student[LinearAlgebra] commands. This local control is not available for the hardware-versus-software floating-point context. These variations are illustrated in the following examples.
Norm(<a,b>, 2);
\sqrt{{\textcolor[rgb]{0,0,1}{a}}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{\textcolor[rgb]{0,0,1}{b}}^{\textcolor[rgb]{0,0,1}{2}}}
To assume that the symbols are complex for a particular computation:
Norm(<a,b>, 2, conjugate);
\sqrt{{|\textcolor[rgb]{0,0,1}{a}|}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{|\textcolor[rgb]{0,0,1}{b}|}^{\textcolor[rgb]{0,0,1}{2}}}
To assume that symbols are complex in any computation:
SetDefault(conjugate = true);
\textcolor[rgb]{0,0,1}{\mathrm{conjugate}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}
\sqrt{{|\textcolor[rgb]{0,0,1}{a}|}^{\textcolor[rgb]{0,0,1}{2}}\textcolor[rgb]{0,0,1}{+}{|\textcolor[rgb]{0,0,1}{b}|}^{\textcolor[rgb]{0,0,1}{2}}}
Normal floating-point computation:
<1.2,3.4> . <1.3,4.2>;
\textcolor[rgb]{0,0,1}{15.8399999999999999}
For floating-point computations to take place in hardware whenever possible:
SetDefault(hardwarefloats=true);
\textcolor[rgb]{0,0,1}{\mathrm{hardwarefloats}}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{\mathrm{false}}
\textcolor[rgb]{0,0,1}{15.8399999999999999}
There is special notation for the transpose and Hermitian transpose operations for Matrices and Vectors:
{M}^{+}
computes the transpose of
M
M
is a Matrix or Vector (or scalar); and
{M}^{*}
computes the Hermitian (conjugate) transpose of
M
A := <<a,b>|<c,d>|<e,f>>;
\textcolor[rgb]{0,0,1}{A}\textcolor[rgb]{0,0,1}{โ}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{e}\\ \textcolor[rgb]{0,0,1}{b}& \textcolor[rgb]{0,0,1}{d}& \textcolor[rgb]{0,0,1}{f}\end{array}]
v := <a | b | c>;
\textcolor[rgb]{0,0,1}{v}\textcolor[rgb]{0,0,1}{โ}[\begin{array}{ccc}\textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{b}& \textcolor[rgb]{0,0,1}{c}\end{array}]
A^+, v^+;
[\begin{array}{cc}\textcolor[rgb]{0,0,1}{a}& \textcolor[rgb]{0,0,1}{b}\\ \textcolor[rgb]{0,0,1}{c}& \textcolor[rgb]{0,0,1}{d}\\ \textcolor[rgb]{0,0,1}{e}& \textcolor[rgb]{0,0,1}{f}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\textcolor[rgb]{0,0,1}{a}\\ \textcolor[rgb]{0,0,1}{b}\\ \textcolor[rgb]{0,0,1}{c}\end{array}]
A^*, v^*;
[\begin{array}{cc}\stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{a}}& \stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{b}}\\ \stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{c}}& \stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{d}}\\ \stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{e}}& \stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{f}}\end{array}]\textcolor[rgb]{0,0,1}{,}[\begin{array}{c}\stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{a}}\\ \stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{b}}\\ \stackrel{\textcolor[rgb]{0,0,1}{&conjugate0;}}{\textcolor[rgb]{0,0,1}{c}}\end{array}]
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find 10 rational numbers between 3/4 and 4/6 using mean method - Maths - Rational Numbers - 7507127 | Meritnation.com
We have two numbers
\frac{3}{4}
\frac{4}{6}
Taking L.C.M. of denominator we write our numbers As :
\frac{9}{12}
\frac{8}{12}
For finding 10 rational number between then we multiply and divide both numbers by 11 , And get
\frac{99}{132}
\frac{88}{132}
Now we can easily find out 10 rational number between them As :
\frac{99}{132}
\frac{98}{132}>\frac{97}{132}>\frac{96}{132}>\frac{95}{132}>\frac{94}{132}>\frac{93}{132}>\frac{92}{132}>\frac{91}{132}>\frac{90}{132}>\frac{89}{132}
\frac{88}{132}
There fore we can write it As:
\frac{3}{4}
\frac{98}{132}>\frac{97}{132}>\frac{96}{132}>\frac{95}{132}>\frac{94}{132}>\frac{93}{132}>\frac{92}{132}>\frac{91}{132}>\frac{90}{132}>\frac{89}{132}
\frac{4}{6}
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Global Constraint Catalog: Klogigraphe
<< 3.7.138. Logic3.7.140. Magic hexagon >>
\mathrm{๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐}
modelling: logigraphe A constraint which can be used for modelling the logigraphe problem. The logigraphe problem, see Figure 3.7.38 for an instance taken from [Pitrat08], consists of colouring a board of squares in black or white, so that each row and each column contains a specific number of sequences of black squares of given size. A sequence of integers
{s}_{1},{s}_{2},\cdots ,{s}_{m}
\left(p\ge 1\right)
enforces:
a first block of
{s}_{1}
consecutive black squares,
a second block of
{s}_{2}
a last block of
{s}_{p}
consecutive black squares.
Each block of consecutive black squares must be separated by at least one white square. Finally, white squares may possibly precede (respectively follow) the first (respectively the last) block of black squares. The logigraphe problem is NP-complete [UedaNagao96].
Figure 3.7.38. Part (A): an instance of a logigraphe and the initial deductions achieved after posting the constraints, Part (B): the corresponding unique solution.
Part (A) of Figure 3.7.38 shows an instance of a logigraphe and the corresponding initial deductions achieved after posting the
\mathrm{๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐}
constraints associated with each row and each column. We assume that each constraint achieves arc-consistency, which is actually the case when the
\mathrm{๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐}
constraint is represented as a counter free automaton. A white or black square indicates an initial deduction (i.e., setting a variable to 0 or to 1). Part (B) of Figure 3.7.38 provides the unique solution found after developing three choices,Each time we try to assign a value to a not yet fixed variable, the number of choices is incremented by 1 just before making the assignment. assuming that variables are assigned from the uppermost to the lowermost row. Within a given row, variables are assigned from the leftmost to the rightmost column. Value 0 is tried first before value 1. Seven additional choices are required for proving that this solution is unique. Figure 3.7.39 displays the corresponding search tree. Within this figure, a variable
{V}_{i,j}
\left(1\le i,j\le 10\right)
denotes the 0-1 variable associated with the
{i}^{th}
{j}^{th}
column of the board.
Figure 3.7.39. Search tree developed for the logigraphe instance of Figure 3.7.38 (variables that are fixed by propagation were removed from the search tree)
|
<< 3.6.13. Modelling3.6.15. Problems >>
Assigning and scheduling tasks that run in parallel: inspired by a modelling question on the Choco mailing list about an assignment and scheduling problem involving nurses and surgeons, use one
\mathrm{๐๐๐๐๐}
constraint as well as inequalities for breaking symmetries with respect to groups of identical persons. The keyword relaxation dimension shows how to extend the previous model in order to take into account over-constrained assignment and scheduling problems.
Assignment to the same set of values: inspired by a presentation of F. Hermenier about a task assignment problem where subtasks have to be assigned a same group of machines, use several
\mathrm{๐๐๐๐๐๐๐}
constraints and a single resource constraint that has an assignment dimension (e.g.,
\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐}
\mathrm{๐๐๐๐๐}
Degree of diversity of a set of solutions: inspired by a discussion with E. Hebrard, how to find out 9 completely different solutions for the 10-queens problem, use the
\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐ก}_\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐๐}
Logigraphe: inspired by an instance from [Pitrat08], use a conjunction of
\mathrm{๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐}_\mathrm{๐๐๐๐}
Magic series: a special case of Autoref, use a single
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐ข}
Metro: a model from H. Simonis, use only
\mathrm{๐๐๐}_\mathrm{๐๐๐}
constraints and propagation (i.e., no enumeration) for modelling the shortest path problem in a network.
Multi-site employee scheduling with calendar constraints: a timetabling problem, inspired by H. Simonis, where tasks have to be assigned groups of employes located in different countries subject to different calendars, use resource constraints as well as the
\mathrm{๐๐๐๐๐๐๐๐}
n-Amazons: an extension of the n-queens problem, use one
\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐}
constraints and three
\mathrm{๐๐๐๐๐๐}
relaxation dimension: illustrate how to model over-constrained placement problems by introducing an extra dimension in the context of the
\mathrm{๐๐๐๐๐}
\mathrm{๐๐๐๐๐}
Scheduling with machine choice, calendars and preemption: a scheduling problem with crossable and non-crossable unavailability periods as well as resumable and non-resumable tasks, illustrate the use of two time coordinates systems within the same model, use precedence and resource constraints as well as the
\mathrm{๐๐๐๐๐๐๐๐}
Sequence dependent set-up: a classical scheduling problem, use the
\mathrm{๐๐๐}_\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐}
Zebra puzzle: illustrate the duality of choice of what is a variable and what is a value in a constraint model as well as the difficulty of stating the constraints in one of the two models, use the
\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}
โ with variables in the table โ and the
\mathrm{๐๐๐๐๐๐๐}
Denotes that a keyword describes a constraint modelling exercise.
|
A bijective proof of Macdonaldโs reduced word formula
Billey, Sara C. ; Holroyd, Alexander E. ; Young, Benjamin J.
We give a bijective proof of Macdonaldโs reduced word identity using pipe dreams and Littleโs bumping algorithm. This proof extends to a principal specialization due to Fomin and Stanley. Such a proof has been sought for over 20 years. Our bijective tools also allow us to solve a problem posed by Fomin and Kirillov from 1997 using work of Wachs, Lenart, Serrano and Stump. These results extend earlier work by the third author on a Markov process for reduced words of the longest permutation.
author = {Billey, Sara C. and Holroyd, Alexander E. and Young, Benjamin J.},
title = {A bijective proof of {Macdonald{\textquoteright}s} reduced word formula},
AU - Billey, Sara C.
AU - Holroyd, Alexander E.
AU - Young, Benjamin J.
TI - A bijective proof of Macdonaldโs reduced word formula
Billey, Sara C.; Holroyd, Alexander E.; Young, Benjamin J. A bijective proof of Macdonaldโs reduced word formula. Algebraic Combinatorics, Tome 2 (2019) no. 2, pp. 217-248. doi : 10.5802/alco.23. http://www.numdam.org/articles/10.5802/alco.23/
[1] Bergeron, Nantel; Billey, Sara C. RC-graphs and Schubert polynomials, Exp. Math., Volume 2 (1993) no. 4, pp. 257-269 | Zbl 0803.05054
[2] Billey, Sara C.; Hamaker, Zachary; Roberts, Austin; Young, Benjamin Coxeter-Knuth graphs and a signed Little map for type B reduced words, Electron. J. Comb., Volume 21 (2014) no. 4, P4.6, 39 pages | MR 3284055 | Zbl 1298.05006
[3] Billey, Sara C.; Jockusch, William; Stanley, Richard P. Some Combinatorial Properties of Schubert Polynomials, J. Algebr. Comb., Volume 2 (1993) no. 4, pp. 345-374 | Article | MR 1241505 | Zbl 0790.05093
[4] Bressoud, David M. Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture, Spectrum Series, Cambridge University Press, 1999, xv+274 pages | Zbl 0944.05001
[5] Carlitz, Leonard A combinatorial property of
q
-Eulerian numbers, Am. Math. Mon., Volume 82 (1975), pp. 51-54 | Article | MR 0366683 | Zbl 0296.05007
[6] Chevalley, Claude Sur les dรฉcompositions cellulaires des espaces
G/B
, Algebraic Groups and their generalizations: Classical methods (University Park, PA, 1991) (Proceedings of Symposia in Pure Mathematics), Volume 56, Part 1, American Mathematical Society, 1994, pp. 1-23 | Zbl 0824.14042
[7] Edelman, Paul; Greene, Curtis Balanced tableaux, Adv. Math., Volume 63 (1987) no. 1, pp. 42-99 | Article
[8] Foata, Dominique On the Netto inversion number of a sequence, Proc. Am. Math. Soc., Volume 19 (1968), pp. 236-240 | Article | Zbl 0157.03403
[9] Fomin, Sergey; Kirillov, Anatol N. Grothendieck polynomials and the Yang-Baxter equation, Proceedings of the Sixth Conference in Formal Power Series and Algebraic Combinatorics (Series in Discrete Mathematics and Theoretical Computer Science) (1994), pp. 183-190
[10] Fomin, Sergey; Kirillov, Anatol N. Yang-Baxter Equation, Symmetric Functions, and Schubert Polynomials, Discrete Math., Volume 153 (1996) no. 1-3, pp. 123-143 | Article | Zbl 0852.05078
[11] Fomin, Sergey; Kirillov, Anatol N. Reduced words and plane partitions, J. Algebr. Comb., Volume 6 (1997) no. 4, pp. 311-319 | MR 1471891 | Zbl 0882.05010
[12] Fomin, Sergey; Stanley, Richard P. Schubert Polynomials and the NilCoxeter Algebra, Adv. Math., Volume 103 (1994) no. 2, pp. 196-207 | Article | Zbl 0809.05091
[13] Garsia, Adriano The Saga of Reduced Factorizations of Elements of the Symmetric Group, Publications du Laboratoire de Combinatoire et dโInformatique Mathรฉmatique, 29, Laboratoire de combinatoire et dโinformatique mathรฉmatique, 2002
[14] Gessel, Ira M.; Viennot, Xavier Determinants, paths, and plane partitions (1989) (manuscript)
[15] Gupta, Hansraj A new look at the permutations of the first
natural numbers, Indian J. Pure Appl. Math., Volume 9 (1978) no. 6, pp. 600-631 | MR 495467 | Zbl 0386.05005
[16] Haglund, James; Loehr, Nicholas; Remmel, Jeffrey B. Statistics on wreath products, perfect matchings, and signed words, Eur. J. Comb., Volume 26 (2005) no. 6, pp. 835-868 | Article | MR 2143200 | Zbl 1063.05009
[17] Hamaker, Zachary; Marberg, Eric; Pawlowski, Brendan Transition formulas for involution Schubert polynomials (2016) (https://arxiv.org/abs/1609.09625) | Zbl 06941774
[18] Hamaker, Zachary; Young, Benjamin Relating Edelman-Greene insertion to the Little map, J. Algebr. Comb., Volume 40 (2014) no. 3, pp. 693-710 | Article
[19] Kasteleyn, Pieter W. The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice, Physica, Volume 27 (1961) no. 12, pp. 1209-1225 | Article | Zbl 1244.82014
[20] Knutson, Allen Schubert polynomials and symmetric functions; notes for the Lisbon combinatorics summer school 2012, 2012 (http://www.math.cornell.edu/~allenk/schubnotes.pdf)
[21] Knutson, Allen; Miller, Ezra Grรถbner geometry of Schubert polynomials, Ann. Math., Volume 161 (2005) no. 3, pp. 1245-1318 | Article | Zbl 1089.14007
[22] Kogan, Mikhail Schubert geometry of Flag Varieties and Gelfand-Cetlin theory (2000) (Ph. D. Thesis) | MR 2716977
[23] Koike, Kazuhiko; Terada, Itaru Young-diagrammatic methods for the representation theory of the classical groups of type
{B}_{n},{C}_{n},{D}_{n}
, J. Algebra, Volume 107 (1987) no. 2, pp. 466-511 | Article
[24] Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike k-Schur Functions and Affine Schubert Calculus, Fields Institute Monographs, 33, The Fields Institute for Research in the Mathematical Sciences, 2014 | MR 3379711 | Zbl 1360.14004
[25] Lam, Thomas; Shimozono, Mark A Little bijection for affine Stanley symmetric functions, Sรฉmin. Lothar. Comb., Volume 54A (2005), B54Ai, 12 pages | MR 2264936 | Zbl 1178.05009
[26] Lascoux, Alain Transition on Grothendieck polynomials, Proceedings of the Nagoya 2000 International Workshop (Physics and Combinatorics), World Scientific, 2000, pp. 164-179
[27] Lascoux, Alain; Schรผtzenberger, Marcel-Paul Polynรดmes de Schubert, C. R. Acad. Sci., Paris, Sรฉr. I, Volume 294 (1982), pp. 447-450 | Zbl 0495.14031
[28] Lascoux, Alain; Schรผtzenberger, Marcel-Paul Symmetry and flag manifolds, Invariant theory (Montecatini, 1982) (Lecture Notes in Mathematics), Volume 996, Springer, 1983, pp. 118-144 | Article | MR 718129 | Zbl 0542.14031
[29] Lascoux, Alain; Schรผtzenberger, Marcel-Paul Schubert Polynomials and the Littlewood-Richardson Rule, Lett. Math. Phys., Volume 10 (1985), pp. 111-124 | Article | MR 815233 | Zbl 0586.20007
[30] Lenart, Cristian A unified approach to combinatorial formulas for Schubert polynomials, J. Algebr. Comb., Volume 20 (2004) no. 3, pp. 263-299 | Article | Zbl 1056.05146
[31] Little, David P. Combinatorial aspects of the Lascoux-Schรผtzenberger tree, Adv. Math., Volume 174 (2003) no. 2, pp. 236-253 | MR 1963694 | Zbl 1018.05102
[32] Little, David P. Factorization of the RobinsonโSchenstedโKnuth correspondence, J. Comb. Theory, Ser. A, Volume 110 (2005) no. 1, pp. 147-168 | Article | MR 2128971 | Zbl 1059.05106
[33] Macdonald, Ian G. Notes on Schubert Polynomials, Publications du Laboratoire de combinatoire et dโinformatique mathรฉmatique, 6, Universitรฉ du Quรฉbec, 1991 | MR 1161461
[34] Manivel, Laurent Symmetric Functions, Schubert Polynomials and Degeneracy Loci, SMF/AMS Texts and Monographs, 6, American Mathematical Society, 2001 | Zbl 0998.14023
[35] Monks Gillespie, Maria A combinatorial approach to the
q,t
-symmetry relation in Macdonald polynomials, Electron. J. Comb., Volume 23 (2016) no. 2, P2.38, 64 pages | MR 3512660 | Zbl 1337.05109
[36] Novick, Mordechai A bijective proof of a major index theorem of Garsia and Gessel, Electron. J. Comb., Volume 17 (2010) no. 1, 64, 12 pages | MR 2644850 | Zbl 1189.05010
[37] Postnikov, Alexander; Stanley, Richard P. Chains in the Bruhat order, J. Algebr. Comb., Volume 29 (2009) no. 2, pp. 133-174 | Article | Zbl 1238.14036
[38] Proctor, Robert A. Odd symplectic groups, Invent. Math., Volume 92 (1988) no. 2, pp. 307-332 | Article | MR 936084 | Zbl 0621.22009
[39] Proctor, Robert A. New symmetric plane partition identities from invariant theory work of De Concini and Procesi, Eur. J. Comb., Volume 11 (1990) no. 3, pp. 289-300 | Article
[40] Reiner, Victor; Tenner, Bridget Eileen; Yong, Alexander Poset edge densities, nearly reduced words, and barely set-valued tableaux (2016) (https://arxiv.org/abs/1603.09589) | Zbl 1391.05269
[41] Serrano, Luis; Stump, Christian Generalized triangulations, pipe dreams, and simplicial spheres, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (Discrete Mathematics and Theoretical Computer Science), The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS), 2011, pp. 885-896 | Zbl 1355.05078
[42] Serrano, Luis; Stump, Christian Maximal fillings of moon polyominoes, simplicial complexes, and Schubert polynomials, Electron. J. Comb., Volume 19 (2012) no. 1, P16, 18 pages | Zbl 1244.05239
[43] Skandera, Mark An Eulerian partner for inversions, Sรฉmin. Lothar. Comb., Volume 46 (2001), B46d, 19 pages | MR 1848722 | Zbl 0982.05006
[44] Stanley, Richard P. On the Number of Reduced Decompositions of Elements of Coxeter Groups, Eur. J. Comb., Volume 5 (1984), pp. 359-372 | Article | MR 782057 | Zbl 0587.20002
[45] Stanley, Richard P. Permutations, 2009 (http://www-math.mit.edu/~rstan/papers/perms.pdf)
[46] Stembridge, John R. A weighted enumeration of maximal chains in the Bruhat order, J. Algebr. Comb., Volume 15 (2002) no. 3, pp. 291-301
[47] Wachs, Michelle L. Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Comb. Theory, Ser. A, Volume 40 (1985) no. 2, pp. 276-289 | Article | MR 814415 | Zbl 0579.05001
[48] Weigandt, Anna; Yong, Alexander The prism tableau model for Schubert polynomials, J. Comb. Theory, Ser. A, Volume 154 (2018), pp. 551-582 | Article | MR 3718077 | Zbl 1373.05219
[49] Young, Benjamin A Markov growth process for Macdonaldโs distribution on reduced words (2014) (https://arxiv.org/abs/1409.7714)
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Asymptotic computational complexity - Wikipedia
In computational complexity theory, asymptotic computational complexity is the usage of asymptotic analysis for the estimation of computational complexity of algorithms and computational problems, commonly associated with the usage of the big O notation.
2 Types of algorithms considered
With respect to computational resources, asymptotic time complexity and asymptotic space complexity are commonly estimated. Other asymptotically estimated behavior include circuit complexity and various measures of parallel computation, such as the number of (parallel) processors.
Since the ground-breaking 1965 paper by Juris Hartmanis and Richard E. Stearns[1] and the 1979 book by Michael Garey and David S. Johnson on NP-completeness,[2] the term "computational complexity" (of algorithms) has become commonly referred to as asymptotic computational complexity.
Further, unless specified otherwise, the term "computational complexity" usually refers to the upper bound for the asymptotic computational complexity of an algorithm or a problem, which is usually written in terms of the big O notation, e.g..
{\displaystyle O(n^{3}).}
Other types of (asymptotic) computational complexity estimates are lower bounds ("Big Omega" notation; e.g., ฮฉ(n)) and asymptotically tight estimates, when the asymptotic upper and lower bounds coincide (written using the "big Theta"; e.g., ฮ(n log n)).
A further tacit assumption is that the worst case analysis of computational complexity is in question unless stated otherwise. An alternative approach is probabilistic analysis of algorithms.
Types of algorithms considered[edit]
In most practical cases deterministic algorithms or randomized algorithms are discussed, although theoretical computer science also considers nondeterministic algorithms and other advanced models of computation.
^ Hartmanis, J.; Stearns, R. E. (1965). "On the computational complexity of algorithms". Transactions of the American Mathematical Society. 117: 285โ306. doi:10.1090/S0002-9947-1965-0170805-7.
^ Michael Garey, and David S. Johnson: Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: W. H. Freeman & Co., 1979.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Asymptotic_computational_complexity&oldid=993888102"
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Graph fundamentals Practice Problems Online | Brilliant
For which of the following scenarios would a DAG (Directed acyclic graph) data structure be most appropriate?
A model showing the dependencies between the steps needed to assemble a car Representing a Sudoku grid. The map of New york city None of the above
Match the following graph with its corresponding adjacency matrix:
Consider traversing the graph below using a depth-first search, beginning with vertex
1
. Which of the choices below lists vertices in the order they are visited in such a depth-first search traversal?
Consider traversing the graph below using breadth-first search, beginning with vertex
1
. Which of the choices below lists vertices in the order they are visited in a breadth-first search traversal?
Which of the following choices describes the graph below?
Directed graph Self-loop graph Incomplete graph Complete graph
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five thousand forty
(five thousand fortieth)
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 48, 56, 60, 63, 70, 72, 80, 84, 90, 105, 112, 120, 126, 140, 144, 168, 180, 210, 240, 252, 280, 315, 336, 360, 420, 504, 560, 630, 720, 840, 1008, 1260, 1680, 2520, 5040
,ฮฮยด
5040 is a factorial (7!), a superior highly composite number, abundant number, colossally abundant number and the number of permutations of 4 items out of 10 choices (10 ร 9 ร 8 ร 7 = 5040). It is also one less than a square, making (7, 71) a Brown number pair.
2 Number theoretical
Plato mentions in his Laws that 5040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state or polis) into lesser parts, making it an ideal number for the number of citizens (heads of families) making up a polis. He remarks that this number can be divided by all the (natural) numbers from 1 to 12 with the single exception of 11 (however, it is not the smallest number to have this property; 2520 is). He rectifies this "defect" by suggesting that two families could be subtracted from the citizen body to produce the number 5038, which is divisible by 11. Plato also took notice of the fact that 5040 can be divided by 12 twice over. Indeed, Plato's repeated insistence on the use of 5040 for various state purposes is so evident that Benjamin Jowett, in the introduction to his translation of Laws, wrote, "Plato, writing under Pythagorean influences, seems really to have supposed that the well-being of the city depended almost as much on the number 5040 as on justice and moderation."[1]
Jean-Pierre Kahane has suggested that Plato's use of the number 5040 marks the first appearance of the concept of a highly composite number, a number with more divisors than any smaller number.[2]
Number theoretical[edit]
{\displaystyle \sigma (n)}
is the divisor function and
{\displaystyle \gamma }
is the EulerโMascheroni constant, then 5040 is the largest of the known numbers (sequence A067698 in the OEIS) for which this inequality holds:
{\displaystyle \sigma (n)\geq e^{\gamma }n\log \log n}
This is somewhat unusual, since in the limit we have:
{\displaystyle \limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\ \log \log n}}=e^{\gamma }.}
Guy Robin showed in 1984 that the inequality fails for all larger numbers if and only if the Riemann hypothesis is true.
5040 has exactly 60 divisors, counting itself and 1.
5040 is the largest factorial (7! = 5040) that is also a highly composite number. All factorials smaller than 8! = 40320 are highly composite.
5040 is the sum of 42 consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 +163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211 + 223 + 227 + 229).
^ Laws, by Plato, translated By Benjamin Jowett, at Project Gutenberg; retrieved 7 July 2009.
^ Kahane, Jean-Pierre (February 2015), "Bernoulli convolutions and self-similar measures after Erdลs: A personal hors d'oeuvre" (PDF), Notices of the American Mathematical Society, 62 (2): 136โ140 .
Mathworld article on Plato's numbers
Retrieved from "https://en.wikipedia.org/w/index.php?title=5040_(number)&oldid=1086013350"
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Complete elliptic integral of the first kind - MATLAB ellipticK - MathWorks Deutschland
Find Complete Elliptic Integrals of First Kind
Differentiate Complete Elliptic Integral of First Kind
Plot Complete Elliptic Integral of First Kind
ellipticK(m)
ellipticK(m) returns the complete elliptic integral of the first kind.
Compute the complete elliptic integrals of the first kind for these numbers. Because these numbers are not symbolic objects, you get floating-point results.
s = [ellipticK(1/2), ellipticK(pi/4), ellipticK(1), ellipticK(-5.5)]
1.8541 2.2253 Inf 0.9325
Compute the complete elliptic integrals of the first kind for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, ellipticK returns unresolved symbolic calls.
s = [ellipticK(sym(1/2)), ellipticK(sym(pi/4)),...
ellipticK(sym(1)), ellipticK(sym(-5.5))]
[ ellipticK(1/2), ellipticK(pi/4), Inf, ellipticK(-11/2)]
[ 1.854074677, 2.225253684, Inf, 0.9324665884]
Differentiate these expressions involving the complete elliptic integral of the first kind. ellipticE represents the complete elliptic integral of the second kind.
diff(ellipticK(m))
diff(ellipticK(m^2), m, 2)
- ellipticK(m)/(2*m) - ellipticE(m)/(2*m*(m - 1))
(2*ellipticE(m^2))/(m^2 - 1)^2 - (2*(ellipticE(m^2)/(2*m^2) -...
ellipticK(m^2)/(2*m^2)))/(m^2 - 1) + ellipticK(m^2)/m^2 +...
(ellipticK(m^2)/m + ellipticE(m^2)/(m*(m^2 - 1)))/m +...
ellipticE(m^2)/(m^2*(m^2 - 1))
Call ellipticK for this symbolic matrix. When the input argument is a matrix, ellipticK computes the complete elliptic integral of the first kind for each element.
ellipticK(sym([-2*pi -4; 0 1]))
[ ellipticK(-2*pi), ellipticK(-4)]
[ pi/2, Inf]
Plot the complete elliptic integral of the first kind.
fplot(ellipticK(m))
title('Complete elliptic integral of the first kind')
ylabel('ellipticK(m)')
K\left(m\right)=F\left(\frac{\pi }{2}|m\right)=\underset{0}{\overset{\pi /2}{\int }}\frac{1}{\sqrt{1-m{\mathrm{sin}}^{2}\theta }}d\theta
For most symbolic (exact) numbers, ellipticK returns unresolved symbolic calls. You can approximate such results with floating-point numbers using vpa.
If m is a vector or a matrix, then ellipticK(m) returns the complete elliptic integral of the first kind, evaluated for each element of m.
ellipke | ellipticCE | ellipticCK | ellipticCPi | ellipticE | ellipticF | ellipticPi | vpa
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Hold-And-Modify โ Wikipedia Republished // WIKI 2
Hold-And-Modify, usually abbreviated as HAM, is a displayโ
mode of the Commodore Amiga computer. It uses a highly unusual technique to express the color of pixels, allowing many more colors to appear on screen than would otherwise be possible. HAM mode was commonly used to display digitized photographs or video frames, bitmap art and occasionally animation. At the time of the Amiga's launch in 1985, this near-photorealistic display was unprecedented for a home computer and it was widely used to demonstrate the Amiga's graphical capability. However, HAM has significant technical limitations which prevent it from being used as a general purpose display mode.
Forensic Pixology Episode 4: Classic Amiga Computers Part 1
2โ
Hold-And-Modifyโ
mode
3.1โ
Originalโ
Chipโ
Setโ
HAMโ
modeโ
(HAM6)
3.2โ
Slicedโ
HAMโ
modeโ
(SHAM)
3.3โ
Advancedโ
Graphicsโ
Architectureโ
HAMโ
modeโ
(HAM8)
3.4โ
HAMโ
emulation
3.5โ
Third-partyโ
HAMโ
implementations
Main article: Originalโ
Chipโ
Set
The original Amigaโ
chipset uses a planar display with a 12-bit RGBโ
colorโ
space that produces 4096 possible colors.
The bitmap of the playfield was held in a section of mainโ
memory known as chipโ
RAM, which was shared between the display system and the main CPU. The display system usually used an indexedโ
color system with a colorโ
palette.
The hardware contained 32 registers that could be set to any of the 4096 possible colors, and the image could access up to 32 values using 5 bits per pixel. The sixth available bit could be used by a display mode known as Extraโ
Half-Brite which reduced the luminosity of that pixel by half, providing an easy way to produce shadowing effects.[1]
Hold-And-Modify mode
The Amiga chipset was designed using a HSV (hue, saturation and luminance) color model, as was common for early homeโ
computers and gamesโ
consoles which relied on televisionโ
sets for display. HSV maps more directly to the YUV colorspace used by NTSC and PAL color TVs, requiring simpler conversion electronics compared to RGBโ
encoding.
Color television, when transmitted over an RF or composite video link, uses a much reduced chroma bandwidth (encoded as two color-difference components, rather than hue + saturation) compared to the third component, luma. This substantially reduces the memory and bandwidth needed for a given perceived fidelity of display, by storing and transmitting the luminance at full resolution, but chrominance at a relatively lower resolution - a technique shared with image compression techniques like JPEG and MPEG, as well as in other HSV/YUV based video modes such as the YJK encoding of the V9958โ
MSX-Video chip (first used in the MSX2+).
{\displaystyle Y}
{\displaystyle U}
{\displaystyle V}
As the Amiga design migrated from a gamesโ
console to a more general purpose homeโ
computer, the video chipset was itself changed from HSV to the modern RGBโ
colorโ
model, seemingly negating much of the benefit of HAM mode. Amiga project lead Jayโ
Miner relates:
Hold and Modify came from a trip to see flightโ
simulators in action and I had a kind of idea about a primitive type of virtualโ
reality. NTSC on the chip meant you could hold the hue and change the luminance by only altering four bits. When we changed to RGB I said that wasn't needed any more as it wasn't useful and I asked the chip layout guy to take it off. He came back and said that this would either leave a big hole in the middle of the chip or take a three-month redesign and we couldn't do that. I didn't think anyone would use it. I was wrong again as that has really given the Amiga its edge in terms of the color palette.
The final form of Hold-And-Modify was, hardware-wise, functionally the same as the original HSV concept, but instead of operating on those three descriptive components (mostly prioritising the V component), it modifies one of the three RGB color channels. HAM can be considered a lossyโ
compression technique, similar in operation and efficiency to JPEG minus the DCT stage; in HAM6 mode, an effective 4096-color (12-bit) playfield is encoded in half the memory that would normally be required - and HAM8 reduces this still further, to roughly 40%. There is a however a payoff for this simplistic compression: a greater overall color fidelity is achieved at the expense of horizontal artifacts, caused by the inability to set any single pixel to an arbitrary 12- (or 18, 24) bit value. At the extreme, it can take three pixels to change from one color to another, reducing the effective resolution at that point from a "320-pixel" to approximately "106-pixel" mode, and causing smears and shadows to spread along a scanline to the right of a high contrast feature if the 16 available palette registers prove insufficient.
When the Amiga was launched in 1985, HAM mode offered a significant advantage over competing systems. HAM allows display of all 4096 colors simultaneously, though with the aforementioned limitations. This pseudo-photorealistic display was unprecedented for a home computer of the time and allowed display of digitized photographs and rendered 3D images. In comparison, the then IBM-PC standard EGA allowed 16 on-screen colors from a palette of 64. EGA's successor VGA released in 1987 with its flagship games mode, Modeโ
13h, allowed 256 on-screen colors from 262,144. HAM mode was frequently used to demonstrate the Amiga's ability in store displays and trade presentations, since competing hardware could not match the color depth. Due to the limitations described above HAM was mainly used for display of static images and developers largely avoided its use with games or applications requiring animation.
HAM mode places restrictions on the value of adjacent pixels on each horizontal line of the playfield. In order to render two arbitrary colors adjacently, it may take up to two intermediary pixels to change to the intended color (if the red, green and blue components must all be modified). In the worst case this reduces the horizontal usable chroma resolution in half, from 320~360 pixels to 106~120. Even so, it compares favorably to contemporary video technologies like VHS, that has a chroma resolution of around 40 Televisionโ
lines, roughly equivalent to 80 pixels.
Displaying such images over a compositeโ
video connection provides some horizontal smoothing that minimizes color artifacts. But if an RGBโ
monitor is used, artifacts becomes particularly noticeable in areas of sharp contrast (strong horizontal imageโ
gradients), where an undesirable multi-hued artifact or "fringe" may appear. Various rendering techniques were used to minimize the impact of "fringing" and HAM displays were often designed to incorporate subtle horizontal color gradients, avoiding vertical edges and contrasts.
Displaying a full color image in HAM mode requires some careful preprocessing. Because HAM can only modify one of the RGB components at a time, rapid color transitions along a scan line may be best achieved by using one of the preset color registers for these transitions. To render an arbitrary image, a programmer may choose to first examine the original image for the most noticeable of these transitions and then assign those colors to one of the registers, a technique known as adaptiveโ
palettes. However, with only 16 available registers in the original HAM mode, some loss in color fidelity is common.
Additionally, HAM mode does not easily permit arbitrary animation of the display. For example, if an arbitrary portion of the playfield is to be moved to another on-screen position, the Hold-and-Modify values may have to be recomputed on all source and target lines in order to display the image correctly (an operation not well-suited to animation). Specifically, if the left-most edge of the animated object contains any 'modify' pixels, or if the image immediately to the right of the object contains any 'modify' pixels, then those Hold-and-Modify values must be recomputed. An attempt to move an object around the screen (such as with the use of the blitter) will create noticeable fringing at the left and right borders of that image, unless the graphics are specially designed to avoid this. In order to avoid recomputing Hold-and-Modify values and circumvent fringing, the programmer would have to ensure the left-most pixel of every blitter object and the left-most pixel of every line of a scrolling playfield is a 'set' pixel. The palette would have to be designed so that it incorporates every such left-most pixel. Alternatively, a HAM display can be animated by generating pixel values through proceduralโ
generation, though this is generally useful for synthetic images only, for example, the 'rainbow' effects used in demos.
Original Chip Set HAM mode (HAM6)
HAM6 mode, named for the 6 bits of data per pixel, was introduced with the Originalโ
Chipโ
Set and was retained in the later Enhancedโ
Chipโ
Set and Advancedโ
Graphicsโ
Architecture. HAM6 allows up to 4096 colors to be displayed simultaneously at resolutions from 320ร200 to 360ร576.
Sliced HAM mode (SHAM)
The SHAM idea was deprecated when HAM8 was introduced with the AGAโ
chipset,[2] since even an unsliced HAM8 image has far more color resolution than a sliced HAM6 image. However, SHAM remains the best available HAM mode on those Amigas with the original or ECS chipsets.
Advanced Graphics Architecture HAM mode (HAM8)
With the release of the Advancedโ
Graphicsโ
Architecture (AGA) in 1992, the original HAM mode was renamed "HAM6", and a new "HAM8" mode was introduced (the numbered suffix represents the bitplanes used by the respective HAM mode). With AGA, instead of 4 bits per color component, the Amiga now had up to 8 bits per color component, resulting in 16,777,216 possible colors (24-bit colorโ
space).
HAM8 operates in the same way as HAM6, using two "control" bits per pixel, but with six bits of data per pixel instead of four. The set operation selects from a palette of 64 colors instead of 16. The modify operation modifies the six mostโ
significantโ
bits of either the red, green or blue color component - the two leastโ
significantโ
bits of the color cannot be altered by this operation and remain as set by the most recent set operation. Compared to HAM6, HAM8 can display many more on-screen colors. The maximum number of on-screen colors using HAM8 was widely reported to be 262,144 colors (18-bit RGB color space). In fact, the maximum number of unique on-screen colors can be greater than 262,144, depending on the two least significant bits of each color component in the 64 color palette. In theory, all 16.7 million colors could be displayed with a large enough screen and an appropriate base palette, but in practice the limitations in achieving full precision mean that the two least significant bits are typically ignored. In general, the perceived HAM8 color depth is roughly equivalent to a highโ
color display.
HAM emulation
HAM is unique to the Amiga and its distinct chipsets. To allow direct rendering of legacy images encoded in HAM format software-based HAM emulators have been developed which do not require the original display hardware. Pre-4.0 versions of AmigaOS can use HAM mode in the presence of the native Amiga chipset. AmigaOS 4.0 and up, designed for radically different hardware, provides HAM emulation for use on modern chunky graphics hardware. Dedicated Amigaโ
emulators running on non-native hardware are able to display HAM mode by emulation of the display hardware. However, since no other computer architecture used the HAM technique, viewing a HAM image on any other architecture requires programmatic interpretation of the image file. Faithful software-based decoding will produce identical results, setting aside variations in color fidelity between display setups.
Third-party HAM implementations
Amigaโ
Halfbriteโ
mode
Sonyโ
ARWโ
2.0 (ARW 2.0+ raw image files use a similar technique for their lossy delta-compression[4])
^ "Blackโ
Beltโ
HAM-E". 2004-12-22. Retrieved 2017-11-06.
^ Paul, Matthias R. (2014-03-18) [2013-01-07]. "SLT-A99V:โ
14-Bit-Aufnahmenโ
nurโ
beiโ
Einzelaufnahmen,โ
12โ
Bitโ
oderโ
14โ
Bitโ
inโ
RAWs?". Minolta-Forum (in German). Archived from theโ
original on 2016-08-08. Retrieved 2016-08-08.
|
Batting average (cricket) - WikiMili, The Best Wikipedia Reader
{\displaystyle \mathrm {Batting~average} ={\frac {\mathrm {Runs~scored} }{\mathrm {Number~of~times~out} }}}
In cricket, a player's batting average is the total number of runs they have scored divided by the number of times they have been out, usually given to two decimal places. Since the number of runs a player scores and how often they get out are primarily measures of their own playing ability, and largely independent of their teammates, batting average is a good metric for an individual player's skill as a batter (although the practice of drawing comparisons between players on this basis is not without criticism [1] ). The number is also simple to interpret intuitively. If all the batter's innings were completed (i.e. they were out every innings), this is the average number of runs they score per innings. If they did not complete all their innings (i.e. some innings they finished not out), this number is an estimate of the unknown average number of runs they score per innings.
Leading male batting averages
Leading female batting averages
One Day Internationals 2
T20 Internationals 2
Most players have career batting averages in the range of 20 to 40. This is also the desirable range for wicket-keepers, though some fall short and make up for it with keeping skill. Until a substantial increase in scores in the 21st century due to improved bats and smaller grounds among other factors, players who sustained an average above 50 through a career were considered exceptional, and before the development of the heavy roller in the 1870s (which allowed for a flatter, safer cricket pitch) an average of 25 was considered very good. [2]
Career records for batting average are usually subject to a minimum qualification of 20 innings played or completed, in order to exclude batsmen who have not played enough games for their skill to be reliably assessed. Under this qualification, the highest Test batting average belongs to Australia's Sir Donald Bradman, with 99.94. Given that a career batting average over 50 is exceptional, and that only 4 other players have averages over 60, this is an outstanding statistic. The fact that Bradman's average is so far above that of any other cricketer has led several statisticians to argue that, statistically at least, he was the greatest athlete in any sport. [3]
Disregarding this 20 innings qualification, the highest career test batting average is 112, by Andy Ganteaume, a Trinidadian Keeper-batsman, who was dismissed for 112 in his only test innings. [4] Amongst active players, Kurtis Patterson has the highest average, having scored 144 runs for the loss of one wicket in his two test innings, giving him a batting average of 144. He then fell out of the Australian squad due to a loss of form and injury.
However, for a batter with one or more innings which finished not out, the true mean or average number of runs they score per innings is unknown as it is not known how many runs they would have scored if they could have completed all their not out innings. In this case, this statistic is an estimate of the average number of runs they score per innings. If their scores have a geometric distribution, then this statistic is the maximum likelihood estimate of their true unknown average. [5]
Batting averages can be strongly affected by the number of not outs. For example, Phil Tufnell, who was noted for his poor batting, [6] has an apparently respectable ODI average of 15 (from 20 games), despite a highest score of only 5 not out, as he scored an overall total of 15 runs from 10 innings, but was out only once. [7]
A batting average of above 50 is considered by many as a benchmark to distinguish between a good and a great batsman. [8] Highest male career batting averages in Test matches as follows:
Jacques Henry Kallis is a South African cricket coach and former cricketer. Widely regarded as one of the greatest cricketers of all time and as South Africa's greatest batsman ever, he is a right-handed batsman and right-arm fast-medium swing bowler. As of 2021 he is the only cricketer in the history of the game to score more than 10,000 runs and take over 250 wickets in both ODI and Test match cricket; he also took 131 ODI catches. He scored 13,289 runs in his Test match career and took 292 wickets and 200 catches.
In cricket, the batting order is the sequence in which batters play through their team's innings, there always being two batters taking part at any one time. All eleven players in a team are required to bat if the innings is completed.
โ Date, Kartikeya (29 May 2014). "The calculus of the batting average". ESPN Cricinfo. Retrieved 10 March 2020.
โ Rae, Simon (1998). W.G. Grace: A Life. London: Faber and Faber. p. 26. ISBN 0571178553.
โ "Sir Donald Bradman". Players and Officials. Cricinfo.com. Retrieved 27 April 2006.
โ "Andy Ganteaume | West Indies Cricket | Cricket Players and Officials | ESPNcricinfo". statsguru. Cricinfo.com. Retrieved 25 July 2018.
โ Das, S. (2011). "On Generalized Geometric Distributions: Application to Modeling Scores in Cricket and Improved Estimation of Batting Average in Light of Notout Innings". Social Science Research Network. SSRN 2117199.
โ "The Jack of all rabbits". 23 July 2007.
โ "Phil Tufnell". Cricinfo.
โ "A genuine matchwinner".
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Help:Wiki markup examples - Wikiquote
This help page may require cleanup to meet Wikipedia's quality standards. No cleanup reason has been specified. Please help improve this help page if you can; the talk page may contain suggestions.
1 Wikitext markup -- making your page look the way you want
1.1 Organizing your writing -- sections, paragraphs, lists and lines
1.3 Images, video, and sounds
1.4 Text formatting -- controlling how it looks
1.5 Spacing things out -- spaces and tables
1.7 Complicated mathematical formulae
1.9 Including another page -- transclusion and templates
Wikitext markup -- making your page look the way you want[edit]
This is page Help:Wiki markup examples, transcluded in Help:Editing.
If you want to try out things without danger of doing any harm, you can do so in the Wikiquote:Sandbox.
More information on HTML tags in wikitext
Organizing your writing -- sections, paragraphs, lists and lines[edit]
;Sections and subsections
If appropriate, place subsections in an appropriate order. If listing countries, for example, place them in alphabetical order rather than, say, relative to population of OECD countries, or some random order.
{{center|Centered text.}}
Enclose the target name in double square brackets -- "[[" and "]]"
First letter of target name is automatically capitalized
Spaces are represented as underscores (but don't do underscores yourself)
Links to nonexistent pages are shown in red -- Help:Starting a new page tells about creating the page.
When the mouse cursor "hovers" over the link, you see a "hover box" containing...
For more info see m:Help:Interwiki linking.
*For more info see [[m:Help:Interwiki linking]].
*[[:fr:Wikipรฉdia:Aide]].
List of cities by country#Sealand.
If the section doesn't exist, the link goes to the top of the page. If there are multiple sections by the same name, link to specific ones by adding how many times that header has alreay appeared (e.g. if there are 3 sections entitled "Example header," and you wish to link to the third one, then use [[#Example section 3]]. For more info, see Help:Editing FAQ.
*[[List of cities by country#Sealand]].
*[[List of cities by country#Morocco|
Cities in Morocco]]
Words in parentheses: kingdom.
Namespace: Requests for adminship.
*In parentheses: [[kingdom (biology)|]].
*Namespace: [[Meta:Requests for adminship|]].
Your user name: Karl Wick
Or your user name plus date/time: Karl Wick 08:10 Oct 5, 2002 (UTC)
Five tildes gives the date/time alone: 08:10 Oct 5, 2002 (UTC)
: Or your user name plus date/time: ~~~~
To include links to a Category page.
will all appear as 20 July-1969 if you set your date display preference to 1 January 2001.
"What links here" and "Recent changes" can be linked as:
Special:Whatlinkshere/Help:Wiki markup examples and Special:Recentchangeslinked/Help:Wiki markup examples
Help:Editing]] and
Help:Editing]]
Images, video, and sounds[edit]
See also: Help:Images and other uploaded files
In-line picture
A picture: [[Image:Wikiquote-logo-en.png]]
or, with alternative text (strongly encouraged) vbgf
[[Image:Wikiquote-logo-en.png|
Wikipedia - The Free Encyclopedia]]
Other ways of linking to pictures
The Image page: Image:Wikiquote-logo-en.png
A link to just the picture: Wikipedia
[[:Image:Wikiquote-logo-en.png]]
[[media:Wikiquote-logo-en.png|Wikipedia]]
Other Media Links -- Video and Sounds
Text formatting -- controlling how it looks[edit]
You can also write in small caps. If the wiki has the templates, this can be even easier to write.
{{smallcaps|be even easier to write}}.
1 [[hectare]] = [[1 E4 m²]]
Spacing things out -- spaces and tables[edit]
<i>x</i><sup>2</sup>
≥
0 true.
Using Wikitext piped tables
x2 โฅ0 true.
|<i>x</i><sup>2</sup>
| width=20px | || width=20px | โฅ0 || true.
| a || || b
reformat text (removing newlines and multiple spaces)
don't reformat text
<pre>arrow &rarr;
The text between here and here won't be displayed
Complicated mathematical formulae[edit]
{\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}
See Formula or TeX markup
A useful resource for understanding TeX can be found at the TeX User's Group.
Problem symbols:
∉ ℵ
Including another page -- transclusion and templates[edit]
This transclusion demo is a little bit of text to be included into any file.
Retrieved from "https://en.wikiquote.org/w/index.php?title=Help:Wiki_markup_examples&oldid=2860132"
|
Prediction & Confidence intervals โ Web Education in Chemistry
How good are the titrations?
In order to get an idea of the agreement between the results of fifteen students, each having done three replicate measurements, one could count how many titration results are not further away than 0.005 M from the mean value of 0.1005 M. You can verify this by clicking on the "Submit" box below. Next, find how many results differ less than 0.0017 M, the standard deviation, from the true value. How many are closer to the mean value than 2 times and 3 times the standard deviation?
The reverse procedure is also possible. Using the next "Submit" button, you can calculate what interval around the mean value contains 95% of all titration results.
The principal idea in statistics is the notion that more or less the same results would be expected if the same group of students would perform the same titrations again. Although the individual results would be different, the distribution of the results would be similar. The distribution of the measurements is often approximated by a normal distribution. This is completely defined by only two values: the mean and the standard deviation of the distribution. The spread around the mean value, as measured by the standard deviation, is directly related to the width of a prediction interval. Now, what is a prediction interval anyway?
A prediction interval of 95 percent simply means that we expect 95% of all future measurements to fall within this interval. This also means that 5% of all measurements are expected to fall outside! Likewise, prediction intervals of 90% and 99% are often used. The exact calculation of a prediction interval requires a bit of background which is beyond this course; however, approximate values for prediction intervals can easily be explained. We already hinted that the width of a prediction interval is related to the standard deviation of the data. Now, as a rule of thumb, a prediction interval of 95% is obtained by taking the mean plus or minus twice the standard deviation. A prediction interval of 99% (approximately) is given by the mean plus or minus three times the standard deviation:
Question: why are 99% prediction intervals wider than 95% prediction intervals?
We now see that prediction intervals routinely are constructed from previous data. This implies that the intervals are only valid if we expect the future data to behave in the same way!
A direct application of prediction intervals is the determination of the limit of detection (LOD) of quantitative analytical methods. A definition of the LOD is: the LOD is the smallest signal value that is significantly (e.g. with 99% confidence) different from the signal of a true blank. To assess the LOD, a sufficient number of true blank values should be measured (preferably more than 20). The LOD is then equal to the mean signal of these measurements plus three times the standard deviation.
Using this procedure (because the LOD is the upper bound of a 99% prediction interval), you are 99% sure that a sample yielding a larger signal value than the LOD is not a blank, so the signal is actually due to analyte.
A 95-percent prediction interval implies that there is a 95 percent chance that another titration experiment would find a value in that range (provided it is executed in exactly the same way as all the other volume determinations, and by the same people).
However, each student performed 10 volume determinations, and took the mean value of these as the final result. Obviously, the histogram of all these mean values shows considerably less variation than the individual volume determinations (remember, errors cancel out!). This means that the standard deviation of a mean value is smaller than the standard deviation for individual values.
The histograms of the individual measurements and the mean values are depicted below.
Clearly, there is one student with quite a low mean value. The means of the other students are very close indeed. The relation between the standard deviation of the individual titration results and the standard deviation for mean values is given by
n
is the number of measurements used to calculate the mean.
\sigma
, the Greek lowercase letter sigma, is often used as the symbol for the standard deviation and
\mu
, the Greek lowercase letter mu, as the symbol for the mean (but the latter one does not occur in the equation).
Confidence intervals are calculated in exactly the same way as prediction intervals for individual measurements, only the standard deviation for the mean is used instead of the standard deviation of the individual measurements. This formula also explains why the mean is more precise when we use more data: its prediction interval becomes narrower. Again, note that this does not mean that the standard deviation of the individual measurements gets smaller!
Now, continue with the questions on this subject.
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Queues - Basic Practice Problems Online | Brilliant
The java code below is a simple implementation of an array based data structure that is either a stack or a queue. What type of structure is this, and how should the three methods one(), two(), and three() be renamed so that they describe their functions?
public class data_structure {
private int limit = 1000000;
private int[] arr = new int[limit];
private int rear = limit - 1;
public void one(int element){
arr[ rear ] = element;
rear --;
public int two(){
rear ++;
int temp = arr[rear];
arr[rear] = 0;
public int three(){
return arr[rear + 1];
Queue, 1โถenqueue, 2โถpeek, 3โถdequeue Stack, 1โถpush, 2โถpeek, 3โถpop Stack, 1โถpush, 2โถpop, 3โถpeek Queue, 1โถenqueue, 2โถdequeue, 3โถpeek
Suppose the following operations are made on a circular queue structure.
Insert entry
A
B
C
D
E
F
After completing these operations, what are the remaining items in the queue?
E,F E,C,D A,B,D E,D
The code below makes use of a deque object. A deque object is similar to a queue except that it allows the insertion/deletion of an element to both the back and front of the queue. What does the pseudocode below output?
function isTrue(string) //Defines a new function 'isTrue' that returns boolean
deq = new Deque(); //Creates a new deque object
for char in string{
deq.addRear(char)
boolean equal = true //Creates a boolean variable called true
while (deq.size() > 1 and equal){
first = deq.removeFront()
last = deq.removeRear()
if (first != last){
boolean value_a = isTrue("radar")
boolean value_b = isTrue("plus")
boolean value_c = isTrue("madam")
if (value_a and value_b and !value_c)
else if (value_a and !value_b and value_c)
else if (!value_a and value_b and value_c)
else if (!value_a and !value_b and value_c)
else if (!value_a and !value_b and !value_c)
The addRear(e) method adds an item to the rear of a deque,the addFront(e) to the front and removeFront() and removeRear() remove the front and rear items of a deque respectively.
Which of the following best represents a queue data structure?
A) A pile of dirty dishes requiring washing
B) Requests on a single shared resource, like a printer
C) A line of people waiting to go to an amusement park ride
A pile of dirty dishes requiring washing Requests on a single shared resource, like a printer A line of people waiting to go to an amusement park ride B and C
The below is incomplete python code to implement a queue that uses only two int variables(self.size and self.val) to save its contents.
class Integral_queue:
def enqueue(self,value):
self.val += value * pow(10,self.size)
#def deque(self...) INCOMPLETE
return self.val % 10
However, this queue only supports single digit integers as elements. Specifically, the only elements that can be pushed to the queue are numbers in the range
1
9
. The dequeue() method of the queue class is incomplete. Which of the following methods correctly implements the dequeue() method?
power = pow(10,self.size-1)
ret = self.val / power
self.val = self.val % power
ret = self.val % 10
self.val = self.val // 10
ret = floor(log(val,10))
self.val /= 10
ret = int(str(self.val)[0])
self.val = int(str(self.val)[1::])
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๏ปฟ Risk Assessment at the Design Phase of Construction Projects in Ghana
The study was carried out exclusively in Ghana to explore the approaches employed by consultants in risk assessment at the design phase of projects in Ghana. One hundred and fourteen (114) consultants were selected out of a population of one hundred and eighty six (186) from three main professional associations in Ghana made up of the Ghana Institute of Architects, Ghana Institution of Engineers and the Ghana Institution of Surveyors (Quantity Surveying Division) practicing in Ghana for the study. Both primary and secondary data were collected. A descriptive survey was also used to observe and describe the presence, frequency or absence of characteristics of a phenomenon as it naturally occurred, in order to gain additional information. A questionnaire was also designed to collect data from the architects, engineers and quantity surveyors. The data was analyzed using Statistical Package for the Social Scientists (SPSS) 17.0. Descriptive and inferential statistics, such as frequency tables, percentages and cross tabulations were used in the data analysis and summaries. Simple tests of associations were undertaken by using Chi square and Cramerโs V statistics to compare relationships between variables. Again, relative importance index was also used to analyze some of the data by computing to deduce their rankings. The relative importance index was used to analyze some of the data by computing to deduce their rankings. The research revealed that majority of consultants had an average knowledge of risk management. Based on the findings it was recommended that consultants undergo advanced training in risk assessment. It was therefore suggested that consultancy firms should develop a set of laid down procedures for consultants to use in risk assessment in order that the use of intuition employed by majority is lessened. The challenges observed in risk assessment and the remedial steps suggested curtailing the detrimental effects of risks would be of wide importance to many developing economies.
Construction, Industry, Project, Risk, Management, Assessment, Design Phase
n=\frac{{n}^{1}}{\left(1+{n}^{1}/N\right)}
{n}^{1}=\frac{{S}^{2}}{{V}^{2}}
{S}^{2}=P\left(1-P\right)=0.5\left(1-0.5\right)=0.25\text{=}\frac{{n}^{1}}{\left(1+{n}^{1}/N\right)}
Awuni, M.A. (2019) Risk Assessment at the Design Phase of Construction Projects in Ghana. Journal of Building Construction and Planning Research, 7, 39-58. https://doi.org/10.4236/jbcpr.2019.72004
1. Anvuur, A. and Kumaraswamy, M.S. (2006) Taking Forward Public Project Reforms in Ghana. CIB W107 Construction in Developing Countries International Symposium โConstruction in Developing Economies: New Issues and Challengesโ, Santiago, 18-20 January 2006.
2. Agyakwa-Baah, A. (2007) Stakeholdersโ Perceptions of the Causes of Delay on Construction Projects. BSc Dissertation, Kwame Nkrumah University of Science and Technology, Kumasi.
3. Tuuli, M.M., Baiden, B.K. and Badu, E. (2007) Assessment and Enforcement of Liquidated Damages in Construction Contracts in Ghana. Structural Survey, 25, 204-219. https://doi.org/10.1108/02630800710772809
4. Agyakwa-Baah, A., Chileshe, N. and Stephenson, P. (2010) A Risk Assessment and Management Framework to Support Project Delivery. Proceedings: The 5th Scientific Conference on Project Management, Advancing Project Management for the 21st Century โConcepts, Tools and Techniques for Managing Successful Projectsโ, Heraklion, 29-31 May 2010, 52-59.
5. Latham, M. (1994) Constructing the Team. HMSO, London.
6. Egan, S.J. (1998) Rethinking Construction. Department of the Environment, London.
7. Ericsson, L.E. (2002) Skรคrpning Gubbar. SOU 2002:15, Byggkommissionen, Stockholm.
8. Chan, A., Chan, D. and Ho, K. (2003) An Empirical Study of the Benefits of Construction Partnering in Hong Kong. Construction Management and Economics, 5, 523-533. https://doi.org/10.1080/0144619032000056162
9. Mahamid, I. (2013) Common Risks Affecting Time Overrun in Road Construction Projects in Palestine: Contractorsโ Perspective. Australian Journal of Construction Economics and Building, 12, 45-53. https://doi.org/10.5130/AJCEB.v13i2.3194
10. Flanagan, R. and Norman, G. (1993) Risk Management and Construction. Blackwell Scientific, Oxford.
11. Thompson, P.A. and Perry, J.G. (1992) Engineering Construction Risks: A Guide to Project Risk Analysis and Risk Management. Thomas Telford, London.
12. Tadayon, M., Jaafar, M. and Nasri, E. (2012) An Assessment of Risk Identification in Large. Construction Projects in Iran. Journal of Construction in Developing Countries, 1, 57-69.
13. Andi, S. and Minato, T. (2003) Design Documents Quality in the Japanese Construction Industry: Factors Influencing and Impacts on Construction Process. International Journal of Project Management, 21, 537-546. https://doi.org/10.1016/S0263-7863(02)00083-2
14. Faridi, A. and El-Sayegh, S. (2006) Significant Factors Causing Delay in the UAE Construction Industry. Construction Management and Economics, 24, 1167-1176. https://doi.org/10.1080/01446190600827033
15. Uher, E. and Toakley, R.A. (1999) Risk Management in the Conceptual Phase of a Project. International Journal of Project Management, 17, 161-169. https://doi.org/10.1016/S0263-7863(98)00024-6
16. Aniekwu, A.N. and Okpala, D.C. (1988) The Effect of Systemic Factors in Contract Services in Nigeria. Construction Management and Economics, 6, 171-182. https://doi.org/10.1080/01446198800000015
17. Kumaraswamy, M.M. (1994) Industry Development through Creative Project Packaging and Integrated Management. Engineering, Construction and Architectural Management, 5, 228-237. https://doi.org/10.1046/j.1365-232X.1998.00041.x
18. Rwelamila, P.D., Talukhaba, A.A. and Ngowi, A.B. (1999) Tracing the African Project Failure Syndrome: The Significance of โUbuntuโ. Engineering, Construction and Architectural Management, 6, 335-346. https://doi.org/10.1046/j.1365-232x.1999.00120.x
19. Hillebrandt, P.M. (2000) Economic Theory and the Construction Industry. 2nd Edition, Macmillan, Basingstoke. https://doi.org/10.1057/9780230372481
20. Ofori, G. (2012) Developing the Construction Industry in Ghana: The Case for a Central Agency. MSc Project Dissertation, National University of Singapore, Singapore.
21. Lopes, J. (2012) Construction in the Economy and Its Role in Socio-Economic Development. Journal of Management in Engineering, 13, 41-49.
22. World Bank (2003) Public Procurement Act of GhanaโCountry Procurement Assessment Report. The World Bank, Washington DC, Report No. 29055.
23. Chileshe, N. and Yirenkyi-Fianko, A.B. (2011) Perception of Threat Risk Frequency and Impact on Construction Projects in Ghana. Journal of Construction in Developing Countries, 16, 115-149.
24. Smith, N.J., Tony, M. and Jobling, P. (2006) Managing Risk in Construction Projects. 2nd Edition, Blackwell Publishing, Oxford.
25. Osipova, E. (2008) The Impact of Procurement Options on Risk Management in Swedish Construction Projects. Research Report, Lulea University of Technology, Luleรฅ.
26. Ayirebi-Dansoh, K. (2005) Strategic Planning Practice of Construction Firms in Ghana. Construction Management and Economics, 23, 163-168. https://doi.org/10.1080/0144619042000241435
27. Eyiah, A.K. and Cook, P. (2003) Financing Small and Medium-Scale Contractors in Developing Countries: A Ghana Case Study. Construction Management and Economics, 21, 357-367. https://doi.org/10.1080/0144619032000111241
28. World Bank (1996) Ghana 1996 Country Project Assessment Report. The World Bank, Washington DC.
29. Bajaj, D., Oluwoye, J. and Lenard, D. (1997) An Analysis of Contractorโs Approaches to Risk Identification in New South Wales, Australia. Construction Management and Economics, 15, 363-369. https://doi.org/10.1080/014461997372917
30. Baker, S., Ponniah, D. and Smith, S. (1999) Risk Response Techniques Employed Currently for Major Projects. Construction Management and Economics, 4, 205-213. https://doi.org/10.1080/014461999371709
31. Carter, E.E. (2002) What Are the Risk Analyses? Harvard Business Review, 5, 72-82.
32. Wang, S., Dulaimi, M. and Aguira, M. (2004) Risk Management Framework for Construction Projects in Developing Countries. Construction Management and Economics, 22, 237-252. https://doi.org/10.1080/0144619032000124689
33. Hull, J.C. (1990) The Evaluation of Risk in Business Investment. Pergamon Press, Oxford, 177-182.
34. Hertz, D.B. and Thomas, H. (1995) Risk Analysis and Its Applications. John Wiley and Sons, Inc., New York.
35. Tah, J.H.M. and Carr, V. (2001) Knowledge-Based Approach to Construction Project Risk Management. Journal of Computing in Civil Engineering, 13, 170-177. https://doi.org/10.1061/(ASCE)0887-3801(2001)15:3(170)
36. Dehdasht, G., Zin, R. and Keyvanfar, A. (2015) Risk Classification and Barrier of Implementing Risk Management in Oil and Gas Construction Companies. Journal Technology of Sciences & Engineering, 77, 161-169. https://doi.org/10.11113/jt.v77.6413
37. Paulson, U. (2007) On Managing Disruption Risks in the Supply ChainโThe DRISC Model. KFS I Lund Ab. Lund.
38. Deviprasadh, A. (2007) Risk Assessment and Management in Construction Projects. MSc Dissertation, Anna University, Chennai.
39. Baloi, D. and Price, A.D.F. (2003) Modelling Global Risk Factors Affecting Construction Cost Performance. International Journal of Project Management, 11, 33-40.
40. Barber, R.B. (2005) Understanding Internally Generated Risks in Projects. International Journal of Project Management, 10, 252-260.
41. Chapman, C. and Ward, S. (2002) Managing Project Risk and Uncertainty: A Constructively Simple Approach to Decision Making. Wiley, Chichester.
42. IEC (2001) Project Risk ManagementโApplication Guidelines, International Standard. IEC, Genรจve.
43. Jaafari, A. (2001) Management of Risks, Uncertainties and Opportunities on Projects: Time for a Fundamental Shift. International Journal of Project Management, 19, 89-101. https://doi.org/10.1016/S0263-7863(99)00047-2
44. Project Management Institute S.C. (2008) A Guide to the Project Management Body of Knowledge. 7th Edition, PMI Communications Publishing, Pennsylvania.
45. Songer, A.D., Diekmann, J. and Pecsok, R.S. (1997) Risk Analysis for Revenue Dependent Infrastructure Projects. Construction Management and Economics, 15, 377-382. https://doi.org/10.1080/014461997372935
46. Mubarak, S. (2010) Construction Project Scheduling and Control. 2nd Edition, John Wiley and Sons Inc., Hoboken. https://doi.org/10.1002/9780470912171
47. Standards Australia/Standards New Zealand (1999) Risk Management. Australian/New Zealand Standard. AS/NZS4360.
48. BSI (2000) British Standards Institution.
49. Canadian Standards Association (1997) Risk Management: Guideline for Decision-Makers (CAN/CSA-Q850-97). Canadian Standards Association, Rexdale.
50. Akintoye, A.S. and MacLeod, M.J. (1997) Risk Analysis and Management in Construction. International Journal of Project Management, 15, 31-38. https://doi.org/10.1016/S0263-7863(96)00035-X
51. Lyons, T. and Skitmore, M. (2004) Project Risk Management in the Queensland Engineering Construction Industry: A Survey. International Journal of Project Management, 22, 51-61. https://doi.org/10.1016/S0263-7863(03)00005-X
52. Burns, N. and Grove, S.K. (2001) The Practice of Nursing Research: Conduct, Critique and Utilization. 4th Edition, WB Saunders, Philadelphia.
53. Kish, L. (1965) Survey Sampling. John Wiley and Sons Inc., New York.
54. Ploit, D. and Beck, C. (2006) The Content Validity Index: Are You Sure You Know Whatโs Being Reported? Critique and Recommendations. Research in Nursing and Health, 29, 489-497. https://doi.org/10.1002/nur.20147
55. Mouton, J. (2001) How to Succeed in Your Masterโs and Doctoral Studies: A South African Guide and Resource Book. Van Schaik Publishers, Hatfield.
56. Robson, C. (2002) Real World Research: Recourse for Social Scientists and PractitionerโResearcher. Oxford Blackwell Publishing, Oxford.
57. Naoum, S.G. (2007) Dissertation Research and Writing for Construction Students. 2nd Edition, Butterworth-Heinemann, Cambridge.
58. Tang, W., Qiang, M., Duffield, C., Young, D.M. and Lu, Y. (2007) Risk Management in the Chinese Construction Industry. Journal of Construction Engineering and Management, 133, 944-955. https://doi.org/10.1061/(ASCE)0733-9364(2007)133:12(944)
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Search results for: Huihui Pang
Positive solutions to n-dimensional ฮฑ1+ฮฑ2
\alpha _{1}+\alpha _{2}
order fractional differential system with p-Laplace operator
Tian Wang, Guo Chen, Huihui Pang
Advances in Difference Equations > 2019 > 2019 > 1 > 1-21
In this paper, we study an n-dimensional fractional differential system with p-Laplace operator, which involves multi-strip integral boundary conditions. By using the LeggettโWilliams fixed point theorem, the existence results of at least three positive solutions are established. Besides, we also get the nonexistence results of positive solutions. Finally, two examples are presented to validate the...
Iterative positive solutions to a coupled fractional differential system with the multistrip and multipoint mixed boundary conditions
Xiaodi Zhao, Yuehan Liu, Huihui Pang
Using the monotone iterative technique, we investigate the existence of iterative positive solutions to a coupled system of fractional differential equations supplemented with multistrip and multipoint mixed boundary conditions. It is worth mentioning that the nonlinear terms of the system depend on the lower fractional-order derivatives of the unknown functions and the boundary conditions involve...
Existence criteria of solutions for a fractional nonlocal boundary value problem and degeneration to corresponding integer-order case
Chaoqun Gao, Zihan Gao, Huihui Pang
In this paper, we mainly discuss the existence and uniqueness results of solutions to fractional differential equations with multi-strip boundary conditions. When the fractional order ฮฑ becomes integer, the existence theorem of positive solutions can be established by a monotone iterative technique. Also, some examples are presented to illustrate the main results.
Existence results for the fractional differential equations with multi-strip integral boundary conditions
Bin Di, Huihui Pang
Journal of Applied Mathematics and Computing > 2019 > 59 > 1-2 > 1-19
In this paper, we consider the existence of solutions for the fractional differential equations with multi-point and multi-strip boundary conditions. The existence results are obtained by applying LerayโSchauderโs alternative, while the uniqueness of solution is established via Banachโs contraction principle. We also consider the existence of positive solutions for the fractional differential equations...
The shooting method and positive solutions of fourth-order impulsive differential equations with multi-strip integral boundary conditions
Yuke Zhu, Huihui Pang
In this paper, we investigate the existence results of a fourth-order differential equation with multi-strip integral boundary conditions. Our analysis relies on the shooting method and the Sturm comparison theorem. Finally, an example is discussed for the illustration of the main work.
Existence and uniqueness results for a coupled fractional order systems with the multi-strip and multi-point mixed boundary conditions
Mengyan Cui, Yuke Zhu, Huihui Pang
This paper is concerned with the existence and uniqueness of solutions for a coupled system of fractional differential equations supplemented with the multi-strip and multi-point mixed boundary conditions. The existence of solutions is derived by applying Leray-Schauderโs alternative, while the uniqueness of the solution is established via Banachโs contraction principle. We also show the existence...
The shooting method and integral boundary value problems of third-order differential equation
Wenyu Xie, Huihui Pang
In this paper, the existence of at least one positive solution for third-order differential equation boundary value problems with Riemann-Stieltjes integral boundary conditions is discussed. By applying the shooting method and the comparison principle, we obtain some new results which extend the known ones. Meanwhile, an example is worked out to demonstrate the main results.
Existence and multiplicity of positive solutions to a fourth-order impulsive integral boundary value problem with deviating argument
Jian Dou, Dongyuan Zhou, Huihui Pang
In this paper, we study the existence of multiple positive solutions for fourth-order impulsive differential equation with integral boundary conditions and deviating argument. The main tool is based on the Avery and Peterson fixed point theorem. Meanwhile, an example to demonstrate the main results is given.
Successive iteration and positive solutions for a third-order boundary value problem involving integral conditions
Huihui Pang, Wenyu Xie, Limei Cao
This paper investigates the existence of concave positive solutions and establishes corresponding iterative schemes for a third-order boundary value problem with Riemann-Stieltjes integral boundary conditions. The main tool is a monotone iterative technique. Meanwhile, an example is worked out to demonstrate the main results.
The method of upper and lower solutions to impulsive differential equations with integral boundary conditions
Huihui Pang, Meng Lu, Chen Cai
This paper considers a second-order impulsive differential equation with integral boundary conditions. Some sufficient conditions for the existence of solutions are proposed by using the method of upper and lower solutions and Leray-Schauder degree theory.
Symmetric positive solutions to a second-order boundary value problem with integral boundary conditions
Huihui Pang, Yulong Tong
Boundary Value Problems > 2013 > 2013 > 1 > 1-9
This paper investigates the existence of concave symmetric positive solutions and establishes corresponding iterative schemes for a second-order boundary value problem with integral boundary conditions. The main tool is a monotone iterative technique. Meanwhile, an example is worked out to demonstrate the main results.
Yulong Tong, Huihui Pang
2011 International Conference on Multimedia Technology > 2698 - 2700
This paper investigates a fourth-order boundary value problem with integral boundary conditions. The existence of iterative solutions is obtained via the lower and upper solutions method.
Necessary and sufficient conditions for the existence of quasi-symmetric positive solutions of singular boundary value problem
Huihui Pang, Chunmei Miao, Weigao Ge
Nonlinear Analysis > 2009 > 71 > 1-2 > 654-665
This paper investigates the existence of quasi-symmetric positive solutions for fourth-order singular three-point boundary value problem. We first give definitions of quasi-symmetric lower and upper solutions. By constructing quasi-symmetric lower and upper solutions and using the Schauder fixed point theorem, necessary and sufficient conditions for existence results are obtained.
Existence results for some fourth-order multi-point boundary value problem
Huihui Pang, Weigao Ge
Mathematical and Computer Modelling > 2009 > 49 > 7-8 > 1319-1325
In this paper, we consider the following fourth-order multi-point boundary value problem {u(iv)(t)=f(t,u(t),uโฒ(t),uโณ(t),uโด(t)),0<t<1,u(0)=0,uโฒ(1)=โi=1mโ2aiuโฒ(ฮพi),uโณ(0)=0,uโด(1)=โi=1mโ2biuโด(ฮพi),ฮพiโ(0,1), where nonlinear f is depending on all lower-order derivatives of u. By using the method of upper and lower solutions and Schauderโs fixed point theorem, existence result is obtained.
A class of three-point boundary-value problems for second-order impulsive integro-differential equations in Banach spaces
Meiqiang Feng, Huihui Pang
Nonlinear Analysis > 2009 > 70 > 1 > 64-82
In this paper, we consider the following boundary-value problems for second-order three-point nonlinear impulsive integro-differential equation of mixed type in a real Banach space E: xโณ(t)+f(t,x(t),xโฒ(t),(Ax)(t),(Bx)(t))=ฮธ,tโJ,tโ tk,ฮx|t=tk=Ik(x(tk)),ฮxโฒ|t=tk=Iฬk(x(tk),xโฒ(tk)),k=1,2,โฆ,m,x(0)=ฮธ,x(1)=ฯx(ฮท), where ฮธ is the zero element of E, (Ax)(t)=โซ0tg(t,s)x(s)ds,(Bx)(t)=โซ01h(t,s)x(s)ds,gโC[D,R+],D={(t,s)โJรJ:tโฅs},hโC[JรJ,R],...
Multiplicity of symmetric positive solutions for a multipoint boundary value problem with a one-dimensional p-Laplacian
Hanying Feng, Huihui Pang, Weigao Ge
Nonlinear Analysis > 2008 > 69 > 9 > 3050-3059
In this paper we consider the multipoint boundary value problem for the one-dimensional p-Laplacian (ฯp(uโฒ(t)))โฒ+q(t)f(t,u(t),uโฒ(t))=0,tโ(0,1), subject to the boundary conditions u(0)=โi=1nฮผiu(ฮพi),u(1)=โi=1nฮผiu(ฮทi), where ฯp(s)=|s|pโ2s,p>1,ฮผiโฅ0,0โคโi=1nฮผi<1,0<ฮพ1<ฮพ2<โฏ<ฮพn<1/2,ฮพi+ฮทi=1,i=1,2,โฆ,n. Applying a fixed point theorem of functional type in a cone, we study the existence of...
Existence and monotone iteration of positive solutions for a three-point boundary value problem
Huihui Pang, Meiqiang Feng, Weigao Ge
Applied Mathematics Letters > 2008 > 21 > 7 > 656-661
In this work, we obtain the existence of quasi-symmetric monotone positive solutions and establish a corresponding iterative scheme for the following three-point boundary value problem: uโณ(t)+f(t,u(t),uโฒ(t))=0,0<t<1,ฮฑu(0)โฮฒuโฒ(0)=0,uโฒ(ฮท)+uโฒ(1)=0. The main tool is the monotone iterative technique. The interesting point is that the nonlinear term involves the first-order derivative explicitly.
INTEGRAL BOUNDARY CONDITION (3)
LOWER AND UPPER SOLUTIONS (3)
THREE-POINT BOUNDARY VALUE PROBLEM (3)
FOURTH-ORDER (2)
FRACTIONAL DIFFERENTIAL EQUATIONS (2)
IMPULSIVE DIFFERENTIAL EQUATION (2)
MULTI-STRIP BOUNDARY CONDITIONS (2)
MULTI-STRIP INTEGRAL BOUNDARY CONDITIONS (2)
MULTIPLE POSITIVE SOLUTIONS (2)
QUASI-SYMMETRIC (2)
SHOOTING METHOD (2)
BOUNDARY CONDITIONS INCLUDING STIELTJES INTEGRALS (1)
COINCIDENCE DEGREE THEORY (1)
EIGENVALUE INTERVAL (1)
FINITE DIFFERENCE EQUATION (1)
FIXED-POINT INDEX THEORY (1)
FOURTH-ORDER BOUNDARY VALUE PROBLEM (1)
FRACTIONAL DIFFERENTIAL EQUATION (1)
FRACTIONAL DIFFERENTIAL SYSTEM (1)
FRACTIONAL DIFFERENTIAL SYSTEMS (1)
HALF LINE (1)
HIGHER ORDER DIFFERENTIAL EQUATION (1)
INFINITE INTERVALS (1)
INTEGRAL BOUNDARY CONDITIONS (1)
ITERATIVE SOLUTION (1)
LERAY-SCHAUDERโS ALTERNATIVE (1)
LERAYโSCHAUDER ALTERNATIVE PRINCIPLE (1)
LERAYโSCHAUDERโS ALTERNATIVE (1)
LOWER (UPPER) SOLUTION (1)
MEASURE OF NONCOMPACTNESS (1)
MULTI-POINT BOUNDARY VALUE CONDITIONS (1)
MULTI-STRIP AND MULTI-POINT MIXED BOUNDARY CONDITIONS (1)
MULTIPOINT BOUNDARY VALUE PROBLEM (1)
MULTISTRIP AND MULTIPOINT MIXED BOUNDARY CONDITIONS (1)
N-DIMENSIONAL (1)
NAGUMO CONDITION (1)
NECESSARY AND SUFFICIENT CONDITION (1)
NONLOCAL BOUNDARY VALUE PROBLEM (1)
ONE-DIMENSIONAL P-LAPLACIAN (1)
P-LAPLACE OPERATOR (1)
RIEMANN-STIELTJES INTEGRAL BOUNDARY CONDITIONS (1)
SCHAUDER FIXED POINT THEOREM (1)
SYMMETRIC POSITIVE SOLUTION (1)
THE COINCIDENCE THEORY (1)
THE FIXED POINT THEOREM (1)
THIRD-ORDER (1)
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Optimum Nusselt Number for Simultaneously Developing Internal Flow Under Conjugate Conditions in a Square Microchannel | J. Heat Transfer | ASME Digital Collection
Manoj Kumar Moharana,
, Kanpur, UP 208016,
Piyush Kumar Singh,
e-mail: samkhan@iitk.ac.in
Moharana, M. K., Singh, P. K., and Khandekar, S. (May 22, 2012). "Optimum Nusselt Number for Simultaneously Developing Internal Flow Under Conjugate Conditions in a Square Microchannel." ASME. J. Heat Transfer. July 2012; 134(7): 071703. https://doi.org/10.1115/1.4006110
A numerical study has been carried out to understand and highlight the effects of axial wall conduction in a conjugate heat transfer situation involving simultaneously developing laminar flow and heat transfer in a square microchannel with constant flux boundary condition imposed on bottom of the substrate wall. All the remaining walls of the substrate exposed to the surroundings are kept adiabatic. Simulations have been carried out for a wide range of substrate wall to fluid conductivity ratio (ksfโโผโ0.17โ703), substrate thickness to channel depth (ฮดsfโโผโ1โ24), and flow rate (Reโโผโ100โ1000). These parametric variations cover the typical range of applications encountered in microfluids/microscale heat transfer domains. The results show that the conductivity ratio, ksf is the key factor in affecting the extent of axial conduction on the heat transport characteristics at the fluidโsolid interface. Higher ksf leads to severe axial back conduction, thus decreasing the average Nusselt number (โ
Nuยฏ
โ ). Very low ksf leads to a situation which is qualitatively similar to the case of zero-thickness substrate with constant heat flux applied to only one side, all the three remaining sides being kept adiabatic; this again leads to lower the average Nusselt number (โ
Nuยฏ
โ ). Between these two asymptotic limits of ksf, it is shown that, all other parameters remaining the same (ฮดsf and Re), there exists an optimum value of ksf which maximizes the average Nusselt number (โ
Nuยฏ
โ ). Such a phenomenon also exists for the case of circular microtubes.
microchannel, axial heat conduction, conjugate heat transfer, thermally developing flow, optimum Nusselt number
Boundary-value problems, Flow (Dynamics), Fluids, Heat conduction, Heat flux, Heat transfer, Microchannels, Thermal conductivity, Heat, Temperature, Wall thickness, Simulation, Internal flow
Pogrebhyak
Effect of Axial Conduction on the Heat Transfer in Micro-Channels
Conduction and Entrance Effects on Laminar Liquid Flow and Heat Transfer in Rectangular Microchannels
Heat Transfer in Micro-Channels: Comparison of Experiments With Theory and Numerical Results
Sobierska
Experimental Results of Flow Boiling of Water in a Vertical Microchannel
M. K,
Axial Conduction in Single-Phase Simultaneously Developing Flow in a Rectangular Mini-Channel Array
The Effect of Longitudinal Heat Conduction on Periodic-Flow Heat Exchanger Performance
Size Limits for Regenerative Heat Engines
Numerical Modeling of Conduction Effects in Microscale Counter Flow Heat Exchangers
Experimental and Numerical Studies of Liquid Flow and Heat Transfer in Microtubes
Numerical Studies of Simultaneously Developing Laminar Flow and Heat Transfer in Microtubes With Thick Wall and Constant Outside Wall Temperature
Heat Transfer and Drag of Laminar Flow of Liquid in Pipes
,โ Energiya, Moscow.
The Conjugate Heat Transfer of the Partially Heated Microchannels
Simultaneous Wall and Fluid Axial Conduction in Laminar Pipe-Flow Heat Transfer
The Effect of Heat Conduction in a Tube Wall Upon Forced Convection Heat Transfer in the Thermal Entry Region
The Advancement of Compact Heat Exchanger Theory Considering the Effects of Longitudinal Heat Conduction and Flow Nonuniformity
ASME-HTD Symposium on Compact Heat Exchangers
eds., 2009,
Micro Process Engineering: A Comprehensive Handbook, Volume 1: Fundamentals, Operations and Catalysis
Weinheim, Germnay
Size Effect on Single-Phase Channel Flow and Heat Transfer at Microscale
The Heat Transfer and Fluid Flow of a Partially Heated Microchannel Heat Sink
Conjugate Forced Convection and Heat Conduction in Circular Microchannels
Effect of Substrate Thickness and Material on Heat Transfer in Microchannel Heat Sinks
Experimental Study on Axial Wall Heat Conduction for Conductive Heat Transfer in Stainless Steel Microtube
Distributed Hydrogen Production From Ethanol in a Microfuel Processor: Issues and Challenges
Micro-Structured Reactors for Gas Phase Reactions
Evaluation of Viscous Dissipation in Liquid Flow in Microchannels
Scaling Effects for Liquid Flows in Microchannels
ANSYS-FLUENTยฎ 12.0 Users Guide, Ansys Inc., USA.
Modeling Nusselt Number for Thermally Developing Laminar Flow in Non-Circular Ducts
Proceedings of the 7th AIAA/ASME Joint Thermophysics and Heat Transfer Conference
Thermally Developing Flow and Heat Transfer in Rectangular Microchannels of Different Aspect Ratios
Laminar Flow Forced Convection in Ducts (Advances in Heat Transfer)
Analysis of Conjugate Heat Transfer in Microchannels: Solution via Hybrid Analytical-Numerical Method
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Cryptography | Free Full-Text | A Searchable Encryption Scheme with Biometric Authentication and Authorization for Cloud Environments
Mihailescu, M. Iulian
Nita, S. Loredana
Scientific Research Center in Mathematics and Computer Science, Spiru Haret University of Bucharest, 030045 Bucharest, Romania
Department of Computers and Cyber Security, Ferdinand I Military Technical Academy, 050141 Bucharest, Romania
Academic Editors: Cheng-Chi Lee, Mehdi Gheisari, Mohammad Javad Shayegan, Milad Taleby Ahvanooey and Yang Liu
{C}_{ANNL}
\left({C}_{ANLU}\right)
Keywords: applied cryptography; theoretical cryptography; information security; cybersecurity; searchable encryption applied cryptography; theoretical cryptography; information security; cybersecurity; searchable encryption
Mihailescu, M.I.; Nita, S.L. A Searchable Encryption Scheme with Biometric Authentication and Authorization for Cloud Environments. Cryptography 2022, 6, 8. https://doi.org/10.3390/cryptography6010008
Mihailescu MI, Nita SL. A Searchable Encryption Scheme with Biometric Authentication and Authorization for Cloud Environments. Cryptography. 2022; 6(1):8. https://doi.org/10.3390/cryptography6010008
Mihailescu, Marius I., and Stefania L. Nita. 2022. "A Searchable Encryption Scheme with Biometric Authentication and Authorization for Cloud Environments" Cryptography 6, no. 1: 8. https://doi.org/10.3390/cryptography6010008
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Specifies what 2D math transition should appear when working in document mode and performing context menu entries in-line. The default value for this option is Typesetting:-mo("→") which is displayed as a right arrow
\to
. When customizing this it is recommended to use the template Typesetting:-mover(Typesetting:-mo("oper"),Typesetting:-mtext("text")) Where oper should be an operator symbol, like → or =, and text should be a text description like transpose. Most MathML operator symbols are supported as a choice for oper. For example, setting operator to Typesetting:-mover(Typesetting:-mo("⇒"),Typesetting:-mtext("implies")) displays as
\stackrel{\text{implies}}{\mathbf{โ}}
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Template-free Synthesis of One-dimensional Cobalt Nanostructures by Hydrazine Reduction Route | Nanoscale Research Letters | Full Text
Tianmin Lan2,
One-dimensional cobalt nanostructures with large aspect ratio up to 450 have been prepared via a template-free hydrazine reduction route under external magnetic field assistance. The morphology and properties of cobalt nanostructures were characterized by scanning electron microscopy, X-ray diffractometer, and vibrating sample magnetometer. The roles of the reaction conditions such as temperature, concentration, and pH value on morphology and magnetic properties of fabricated Co nanostructures were investigated. This work presents a simple, low-cost, environment-friendly, and large-scale production approach to fabricate one-dimensional magnetic Co materials. The resulting materials may have potential applications in nanodevice, catalytic agent, and magnetic recording.
In recent years, nanostructure materials have been actively studied due to their novel properties and potential applications. Among them, much attention has been focused on one-dimensional (1D) magnetic materials such as Fe, Co, and Ni due to their potential applications in nanodevice, biosensor, and magnetic recording [1โ3]. 1D cobalt nanostructures with uniform shape and high purity have become increasingly required for specific uses in many areas, such as in high-density information storage, magnetic sensors, commercial batteries, and catalysts.
Various approaches have been developed to synthesize cobalt nanostructures [4โ6]. For example, Puntes et al. prepared Co nanodisks by rapid decomposition of carbonyl cobalt in the presence of trioctylphosphane (TOP) and oleic acid (OA) [7]. Legrand et al. applied a physical method to synthesize 3D supra-organization of Co nanocrystals [8]. The most common method to fabricate 1D Co nanostructures is based on porous anodic aluminum oxide (AAO) templates. Li et al. prepared Co nanowire arrays in alumina arrays by using a chemical electrodeposition method [9]. Other templates such as polyaniline or polycarbonate membranes, diblock copolymer, and mesoporous silica have also been applied to fabricate magnetic Co nanowires [10โ12]. In view of the complexity of multi-step template preparation and low production, it is imperative to develop simpler template-free methods for the fabrication of magnetic Co nanowires.
In recent years, our research group has reported a chemical solution reduction approach for fabricating 1D Ni nanostructures assisted by magnetic fields [13โ15]. This may be a more promising method to prepare cobalt nanostructures in terms of its low cost and potential for large-scale production. In this study, we fabricated Co nanowires with large aspect ratio under normal pressure in absence of any templates or surfactants. The influence of reaction temperature, Co ion concentration and magnetic field intensity on the formation and morphology of Co nanowires were investigated. The magnetic properties of Co nanostructures with different morphology were also evaluated.
All chemicals were of analytical grade without further purification. In a typical synthesis, an appropriate amount of CoCl2ยท6H2O was dispersed in 50 ml of ethylene glycol in a 250-ml beaker, where the concentration of CoCl2ยท6H2O varied from 0.01 to 0.1 M. An appropriate amount of 85 wt% N2H4ยทH2O solution and 5 M NaOH solution was then added to the mixed solution with constant stirring. An NdFeB permanent magnet was placed beneath the beaker to apply an external magnetic field to the reaction system. The magnetic field intensity on the inner surface of the beaker was controlled from 0.005 to 0.40 T at room temperature by adjusting the distance between the beaker and the magnet, and the intensity of the magnetic field was measured by using a Tesla meter. The final mixture was then allowed to react at temperature of 40, 60, and 80ยฐC for 30 min. The chemical reaction for the synthesis of the Co nanowires can be expressed as below:
2{\text{Co}}^{2+}+{\text{N}}_{2}{\text{H}}_{4}+4{\text{OH}}^{-}=2\text{Co}โ+\phantom{\rule{0.5em}{0ex}}{\text{N}}_{2}โ+\phantom{\rule{0.5em}{0ex}}4{\text{H}}_{2}\text{O}
After the beaker cooled down to room temperature, the gray solid product floating on the beaker was collected by using a magnet. Then it was washed with distilled water and absolute ethanol several times and finally dried in a vacuum oven at 60ยฐC for 24 h.
The size and morphology analyses were performed using a field emission scanning electron microscope (SEM, Ultra55, Zeiss). The crystal structure was characterized by an X-ray polycrystalline diffractometer (XRD, D8 Advance, Bruker) using Cu Ka radiation (ฮป = 1.54056 nm) with graphite monochromator. The hysteresis properties were measured on a vibration sample magnetometer (VSM, Lake Shore 7400).
Figure 1 shows the SEM images of resulting products prepared under different reaction temperatures of 40, 60, and 80ยฐC, in which Co ion concentration was 0.01 M and the magnetic field of 0.4 T. It is apparent that the mean diameter of the wires prepared at higher temperature is much less than at lower temperature, whereas the aspect ratio of the wires increased with the temperature firstly and reached its highest value of about 450 at 60ยฐC and then decreased with increased temperature. The diameter is about 4 ฮผm at 40ยฐC (see Figure 1a) and 800 nm at 60 and 80ยฐC (see Figure 1b, c), respectively. The wires fabricated at 60ยฐC are much smoother, longer, and more uniform. This may be derived from the thermodynamic influence on the nucleation velocity and the subsequent fast growth of the crystals, which results in smaller size of the Co particles [16, 17]. When the reaction temperature is sufficiently high up to 80ยฐC, the average length of the corresponding Co nanowires appears to decrease from 350 to 200 ฮผm as a result of the influence of thermal kinetics interaction. Consequently, we choose temperature of 60ยฐC as the optimal reaction temperature for the preparation of Co wires.
SEM images with the same magnification times of Co nanostructures prepared at different reaction temperatures: a 40ยฐC; b 60ยฐC; c 80ยฐC.
The influence of Co ion concentration on morphology of Co wires has also been investigated. Figure 2 shows the SEM images of products with different Co ion concentration under the same external magnetic field of 0.4 T. It can be seen that the Co ion concentration has a strong influence on the morphology of products. The average diameter of the wires increases with increased Co ions concentration. It is about 800 nm for concentration of 0.01 M (see Figure 2a) and about 1.5 ฮผm for 0.1 M (see Figure 2b), whereas the average length decreases from 350 to 50 ฮผm. When the concentration reached 0.5 M, the corresponding products tend to further agglomerate and grow thicker, which results in the potato-like structure with the average diameter of 2 ฮผm (see Figure 2c). The possible reason is that the primary Co crystallites tend to aggregate together to form spherical particles to decrease the surface energy when the concentration increased.
SEM images of products prepared in different Co ion concentrations under a 0.40 T magnetic field: a 0.01 mol/l; b 0.1 mol/l; c 0.5 mol/l.
In order to investigate the effect of the external magnetic field on the morphology of the resulting products, magnetic fields with different intensity were applied, where the Co ion concentration and the reaction temperature were 0.01 M and 60ยฐC, respectively. As shown in Figure 3a, only some bulky particles were observed in the absence of the external magnetic field. Under a low magnetic field of 0.15 T, some short, unsmooth, and thick wires were obtained. The average length and diameter are respectively about 200 and 1.5 ฮผm (see Figure 3b). The surface of the wires was irregular, at which lots of Co particles are not aligned along its magnetic anisotropy direction in order to reduce the surface energy. When the intensity of the external magnetic field was increased to 0.40 T, the corresponding Co nanowires appeared to be significantly elongated. The orientation of the Co nanoparticles is further promoted, which results in some parallel self-assembled arrays of Co nanowires. The wires have aspect ratio of about 450 with average diameter of 800 nm and length up to 350 ฮผm (see Figure 3c). It can be concluded that the magnetic field had played a very important role in forming 1D nanostructure, and therefore the length of the wires can be easily controlled by adjusting the intensity of the external magnetic field.
SEM images of samples prepared under different magnetic fields: a 0 T; b 0.15 T; c 0.40 T.
The XRD pattern of the Co nanowires prepared at 0.01 M concentration under an external magnetic field of 0.40 T is shown in Figure 4. All the diffraction peaks can be well indexed to hexagonal-phase cobalt, with lattice constants of a = 2.492 ร
and c = 4.025 ร
, which is well consistent with the standard card (JCPDS 89-4308, P63/mmc, a = 2.505 ร
, c = 4.089 ร
). Compared with the neighboring peak (101), the relative intensity of peak (002) in patterns increases significantly, which indicates the oriented growth of cobalt crystallites. There is no other impurity observed, suggesting the prepared Co nanowires have high purity.
XRD pattern of Co nanowires prepared at 60ยฐC under a 0.40 T magnetic field.
Figure 5 displays the hysteresis loops of the samples prepared without an external magnetic field (Zero-field), with a low external magnetic field of 0.15 T (0.15 T-field) and with an external magnetic field of 0.40 T (0.40 T-field), respectively. The saturation magnetization (M S ) of the samples is respective 127, 138, and 160 emu/g, which are all smaller than that of bulk Co materials (168 emu/g). The reduced magnetizations result from the surface effect of Co nanostructures, in which lots of Co atoms of the surface are not aligned along its magnetic anisotropy direction in order to reduce the surface energy. Therefore, the magnetic moments of these Co atoms cannot be aligned along the magnetic field due to the strong exchange interaction. On the other hand, the Zero-field sample had coercivity (H C ) value of 103 Oe, whereas the 0.15 T-field sample was 84 Oe and 65 Oe for 0.40 T-field one. These values are much lower than that of bulk cobalt material of 1500 Oe. These differences may be attributed to the different magnetic anisotropy manner [18]. The magnetocrystalline anisotropy and shape anisotropy are two main anisotropy energies existed in magnetic materials, which can induce different coercivity. These two anisotropies cause the nanowires to exhibit a vortex-like magnetization distribution so that the moment directions can easily turn parallel to the external magnetic field, which leads to the reduction in the coercivity of the products.
Hysteresis loops of Co nanostructures measured at different intensity of applied field: Zero-field, 0.15 T-field, and 0.40 T-field.
The possible mechanism for the formation of Co nanowires under applied magnetic field may be expressed as following: At first, Co ions were reduced by strong reduction agent of hydrazine hydrate and turned to tiny spherical particles. Then the magnetic Co particles aligned along the magnetic field direction to form one-dimensional nanostructures under the magnetic driving force. The cobalt nanowires retained their linear structure after kept in ultrasonic bath for 10 min, which proved that the nanowires displayed a good mechanical strength.
A simple, low-cost, environment-friendly approach of preparation magnetic Co nanowires was developed. In this method, the nanowires were fabricated in an ethylene glycol solution at normal pressure by assistant of magnetic field without any templates or surfactants. The prompt wires have average length of up to 350 ฮผm and aspect ratio of up to 450. It was found that the nanowires are elongated with the increasing intensity of magnetic field, and there is no wires formed in absence of magnetic field. The reaction temperature and Co ion concentration have also strong influences on formation and morphology of nanowires. This method provides a new approach to fabricate magnetic nanowires under normal pressure and may be the most promising candidate to produce large-scale magnetic nanowires, which broads their practical applications.
Puntes VF, Krishnan KM, Alivisatos AP: Science. 2001, 291: 2115. 10.1126/science.1057553
Zeng H, Skomski R, Menon L, Liu Y, Bandyopadhyay S, Sellmyer D: Phys Rev B. 2002, 65: 134426. 10.1103/PhysRevB.65.134426
Maaz K, Karim S, Usman M, Mumtaz A, Liu J, Duan JL, Maqbool M: Nanoscale Res Lett. 2010, 5: 1111. 10.1007/s11671-010-9610-5
Liu XG, Wu NQ, Wunsch BH, Barsotti RJ, Stellacci F: Small. 2006, 2: 1046. 10.1002/smll.200600219
Narayanan TN, Shaijumon MM, Ajayan PM, Anantharaman MR: Nanoscale Res Lett. 2010, 5: 164. 10.1007/s11671-009-9459-7
Nielsch K, Castano FJ, Ross CA, Krishnan R: J Appl Phys. 2005, 98: 034318. 10.1063/1.2005384
Puntes VF, Zanchet D, Erdonmez CK, Alivisatos AP: J Am Chem Soc. 2002, 124: 12874. 10.1021/ja027262g
Legrand JL, Ngo AT, Petit C, Pileni MP: Adv Mater. 2001, 13: 58. 10.1002/1521-4095(200101)13:1<58::AID-ADMA58>3.0.CO;2-A
Li DD, Thompson RS, Bergmann G, Lu JG: Adv Mater. 2008, 20: 4575. 10.1002/adma.200801455
Qin J, Nogues J, Mikhaylova M, Roig A, Munoz JS, Muhammed M: Chem Mater. 2005, 17: 1829. 10.1021/cm047870q
Cao HQ, Xu Z, Sang H, Sheng D, Tie CY: Adv Mater. 2001, 13: 121. 10.1002/1521-4095(200101)13:2<121::AID-ADMA121>3.0.CO;2-L
Ge SH, Li C, Ma X, Li W, Xi L, Li CX: J Appl Phys. 2001, 90: 509. 10.1063/1.1327599
Liu P, Li ZJ, Yadian BL, Zhang YF: Mater Lett. 2009, 63: 1650. 10.1016/j.matlet.2009.04.031
Zhang LY, Wang J, Wei LM, Liu P, Wei H, Zhang YF: Nano-Micro Lett. 2009, 1: 49. 10.5101/nml.v1i1.p49-52
Wang J, Zhang LY, Liu P, Lan TM, Zhang J, Wei LM, Kong ES, Jiang CH, Zhang YF: Nano-Micro Lett. 2010, 2: 134. 10.5101/nml.v2i2.p134-138
Peng XG, Manna L, Yang WD, Wiekham J, Scher E, Kadavanieh A, Alivisatos AP: Nature. 2000, 404: 5. 10.1038/35003535
Sun SH, Murray CB: J Appl Phys. 1999, 85: 4325. 10.1063/1.370357
Kisielewski M, Maziewsk A, Zablotskii VJ: Magn Magn Mater. 2005, 290โ291: 776. 10.1016/j.jmmm.2004.11.403
This research was supported by the Hi-Tech Research and Development Program of China No. 2007AA03Z300, Shanghai-Applied Materials Research and Development fund No. 07SA10, National Natural Science Foundation of China (No. 50730008, 50902092), Shanghai Science and Technology Grant (No. 0752 nm015, 1052 nm06800), National Basic Research Program of China No. 2006CB300406, and the fund of Defence Key Laboratory of Nano/Micro Fabrication Technology.
National Key Laboratory of Nano/Micro Fabrication Technology, Key Laboratory for Thin Film and Microfabrication of the Ministry of Education, Institute of Micro and Nano Science and Technology, Shanghai Jiao Tong University, 200240, Shanghai, China
Liying Zhang, Jian Wang, Liangmin Wei, Zhi Yang & Yafei Zhang
School of Materials Science and Engineering, Shanghai Jiao Tong University, 200240, Shanghai, China
Tianmin Lan & Jian Wang
Tianmin Lan
Correspondence to Yafei Zhang.
Zhang, L., Lan, T., Wang, J. et al. Template-free Synthesis of One-dimensional Cobalt Nanostructures by Hydrazine Reduction Route. Nanoscale Res Lett 6, 58 (2011). https://doi.org/10.1007/s11671-010-9807-7
Magnetic field assistance
Hydration hydrazine reduction
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Home : Support : Online Help : Programming : Document Tools : Components : Microphone Component
generate XML for a Microphone Component
Microphone( opts )
enabled : truefalse:=true; Indicates whether the component is enabled. The default is true. If enabled is false then the inserted component is grayed out and interaction with it cannot be initiated.
record : truefalse:=false; Indicates whether the component is initially recording. The default value is false.
The Microphone command in the Component Constructors package returns an XML function call which represents a Microphone Component.
\mathrm{with}โก\left(\mathrm{DocumentTools}\right):
\mathrm{with}โก\left(\mathrm{DocumentTools}:-\mathrm{Components}\right):
\mathrm{with}โก\left(\mathrm{DocumentTools}:-\mathrm{Layout}\right):
Executing the Microphone command produces a function call.
Sโ\mathrm{Microphone}โก\left(\mathrm{identity}="Microphone0"\right)
\textcolor[rgb]{0,0,1}{S}\textcolor[rgb]{0,0,1}{โ}\textcolor[rgb]{0,0,1}{\mathrm{_XML_EC-Microphone}}\textcolor[rgb]{0,0,1}{โก}\left(\textcolor[rgb]{0,0,1}{"id"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Microphone0"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"enabled"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"true"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"visible"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"true"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"samplerate"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"16000"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"stereo"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"false"}\textcolor[rgb]{0,0,1}{,}\textcolor[rgb]{0,0,1}{"record"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"false"}\right)
\mathrm{xml}โ\mathrm{Worksheet}โก\left(\mathrm{Group}โก\left(\mathrm{Input}โก\left(\mathrm{Textfield}โก\left(S\right)\right)\right)\right):
\mathrm{InsertContent}โก\left(\mathrm{xml}\right):
Sโ\mathrm{Microphone}โก\left(\mathrm{identity}="Microphone0",\mathrm{tooltip}="My microphone",\mathrm{stereo}=\mathrm{false}\right):
\mathrm{xml}โ\mathrm{Worksheet}โก\left(\mathrm{Group}โก\left(\mathrm{Input}โก\left(\mathrm{Textfield}โก\left(S\right)\right)\right)\right):
The previous example's call to the InsertContent command inserted a component with identity "Microphone0", which still exists in this worksheet. Inserting additional content whose input contains another component with that same identity "Microphone0" incurs a substitution of the input identity in order to avoid a conflict with the identity of the existing component.
\mathrm{lookup}โ\mathrm{InsertContent}โก\left(\mathrm{xml},\mathrm{output}=\mathrm{table}\right)
\textcolor[rgb]{0,0,1}{\mathrm{lookup}}\textcolor[rgb]{0,0,1}{โ}\textcolor[rgb]{0,0,1}{table}\textcolor[rgb]{0,0,1}{โก}\left([\textcolor[rgb]{0,0,1}{"Microphone0"}\textcolor[rgb]{0,0,1}{=}\textcolor[rgb]{0,0,1}{"Microphone1"}]\right)
\mathrm{lookup}["Microphone0"]
\textcolor[rgb]{0,0,1}{"Microphone1"}
\mathrm{GetProperty}โก\left(\mathrm{lookup}["Microphone0"],\mathrm{stereo}\right)
\textcolor[rgb]{0,0,1}{\mathrm{false}}
The DocumentTools:-Components:-Microphone command was introduced in Maple 2015.
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Contents: (Phys. Status Solidi B 1/2010)
physica status solidi (b) > 247 > 1 > 3 - 6
physica status solidi (b) > 247 > 1 > 226 - 227
Optical and magnetooptical properties of Ho3+:YGG
Uygun V. Valiev, Sharof A. Rakhimov, Nafisa I. Juraeva, Romano A. Rupp, more
The luminescence and excitation spectra in the paramagnetic holmiumโyttrium gallium garnet Ho0.2Y2.8Ga5O12 (Ho3+:YGG) have been studied experimentally at temperatures Tโ=โ80 and 293โK. Identification of the radiative 4fโโโ4f transitions observed between 5S2, 5F4, and 5I8 multiplets of the ground 4f(10)โconfiguration of Ho3+ in sites of D2 symmetry was performed on the base of the results of numerical...
Front Cover (Phys. Status Solidi B 1/2010)
physica status solidi (b) > 247 > 1 > n/a - n/a
With the cover illustration the Editors would like to call your attention to two Feature Articles published in this issue. Both articles are among the most accessed Early View papers in Wiley InterScienceยฎ. Becquart and Domain present a versatile technique called kinetic Monte Carlo which can be used to simulate the evolution of a complex microstructure at the atomic scale. Starting on page 9, they...
The limits of the total crystalโfield splittings
The crystalโfields (CFs) causing
|J\rangle
electron states splittings of the same second moment ฯ2 can produce different total splitting ฮE magnitudes. Based on the numerical data on CF splittings for the representative sets of CF Hamiltonians ${\cal H}_{\rm CF} = \sum\nolimits_k \sum\nolimits_q B_{kq} C_q^{(k)} $ with fixed indexes either k or q, the potentials leading to the extreme ฮE have been...
Firstโprinciples study of W, WN, WN2, and WN3
Lu Song, YuanโXu Wang
physica status solidi (b) > 247 > 1 > 54 - 58
Firstโprinciples calculations were performed to investigate the structural, elastic, and electronic properties of W and its nitrides. The results show that the bulk modulus decreases with the increase in the nitrogen content for the tungsten nitrides. A comparison of the calculated ratio of the shear modulus to the bulk modulus for W, WN, WN2, and WN3 suggests the more pronounced directional bonding...
Mรถssbauer study of nanosized CoโNi ferrite Cox Ni1โx Fe2O4 (0โโคโxโโคโ1) particles
Z. P. Niu, Y. Wang, F. S. Li
Nanocrystalline CoโNi ferrite CoxNi1โx Fe2O4 (0โโคโxโโคโ1) have been synthesized by using the PVA solโgel method. All the samples were examined by powder Xโray diffraction (XRD) measurements and found to be single phase spinel. The Mรถssbauer spectra at room temperature were recorded by using a Mรถssbauer spectrometer and were fitted by two magnetic sextets A and B. An electron configuration of 3d54sk...
Coupled magnetic plasmons in metamaterials [Phys. Status Solidi B 246, No. 7, 1397โ1406 (2009)]
H. Liu, Y. M. Liu, T. Li, S. M. Wang, more
Copyright lines are added for reproduction of Figs. 4, 5 and 8โ10 in the Feature Article by Liu et al. [Phys. Status Solidi B 246, 1397 (2009)].
Investigation of the localization effect in InGaNAs/GaAs SQWs using the LSE model
Esmaeil Abdoli, Hamid Haratizadeh
In this paper, the temperature behaviors of photoluminescence (PL) spectra of asโgrown and annealed InGaASN/GaAs single quantumโwell (SQW) samples with different nitrogen levels have been investigated by means of the localizedโstate ensemble (LSE) model. The variations of PL peak position and linewidth versus temperature are attributed to the creation of a fluctuation potential in the band edge of...
A. Audzijonis, R. Sereika, R. ลฝaltauskas, L. ลฝigas
We present the results of the ab initio theoretical study of the optical properties for paraelectric BiSCl crystal using the full potential linearized augmented plane wave (FPโLAPW) method as implanted in the Wien 2k code. For theoretical calculations of optical constants and functions we used the generalized gradient approximation (PBEโGGA), an improvement of the local spinโdensity approximation...
Electron localization and emission mechanism in wurtzite (Al, In, Ga)N alloys
Qihang Liu, Jing Lu, Zhengxiang Gao, Lin Lai, more
The electronic structures of wurtzite InGaN and AlGaN alloys are investigated using the firstโprinciple density functional theory calculation. The results indicate that some short InโNโIn atomic chains and small InโN atomic condensates composed of a few In and N atoms can be randomly formed in InGaN alloys. The electrons at the top of valence bands can be effectively localized in the vicinity of the...
Flexoelectric charge separation and the associated size dependent piezoelectricity are investigated in centrosymmetric dielectric solids. Direct piezoelectricity can exist as external mechanical stress is applied to nonโpiezoelectric dielectrics with shapes such as truncated pyramids, due to elastic strain gradient induced flexoelectric polarization. Effective piezoelectric coefficient is analyzed...
Effect of pressure on the secondโorder Raman scattering intensities of zincblende semiconductors
C. TralleroโGiner, K. Syassen
A microscopic description of the twoโphonon scattering intensities in directโgap zincblendeโtype semiconductors as a function of hydrostatic pressure and for nonโresonant excitation is presented. The calculations were performed according to the electronโtwoโphonon deformation potential interaction for the ฮ1 and ฮ15 components of the Raman tensor. It is shown that the effect of pressure on the Raman...
Multishell structure and size effect of barium titanate nanoceramics induced by grain surface effects
Chao Fang, Dongxiang Zhou, Shuping Gong, Wei Luo
A quantitative multishell structure model of the grain size effect of nanoโBaTiO3 ceramics is proposed on the basis of GinsburgโLandauโDevonshire thermodynamic theory. The existence of surface defects is considered by assuming that surface energy varies inversely with the distance to the grain surface. The surface effects lead to the multishell structure in BaTiO3 nanoparticles consisting of a surface...
Quantum interference through parallelโcoupled double quantum dots with Rashba spinโorbit interaction
HaiโTao Yin, TianโQuan Lรผ, XiaoโJie Liu, HuiโJie Xue
The electronic transport through parallelโcoupled double quantum dots with Rashba spinโorbit interaction is studied theoretically in the framework of the equation of motion of Green's function. Based on molecular state representation, the Fano interference in the conductance spectrum is readily explained. The possibility of manipulation of the Fano line shape of each spin component is explored by...
Electrically controlled Fano lines in double quantum dot system with intraโdot Coulomb interaction
M. Pylak, R. ลwirkowicz
Theoretical study of quantum interference processes which take place during electron transport through artificial molecule formed by double quantum dot system is presented. The nonโequilibrium Green function formalism based on the equation of motion method within HartreeโFock approximation is used to calculate the conductance spectrum. Interplay between intraโdot Coulomb correlations described by...
Elastic and electronic properties of the new perovskiteโlike superconductor ZnNNi3 in comparison with MgCNi3
I. R. Shein, V. V. Bannikov, A. L. Ivanovskii
Fullโpotential linearized augmented plane wave (FLAPW) method with the generalized gradient approximation (GGA) for the exchangeโcorrelation potential has been applied for the study of structural, elastic, and electronic properties of the newly synthesized nitrogenโcontaining perovskiteโlike 3K superconductor ZnNNi3. The optimized lattice parameter, independent elastic constants (C11, C12, and C44...
Influence of surface roughness scattering on electron lowโfield mobility in thin undoped GaAsโinโAl2O3 nanowires with rectangular crossโsection
The influence of surface roughness scattering on electron lowโfield mobility in thin undoped GaAsโinโAl2O3 nanowires with rectangular crossโsection is studied by means of the direct numerical solution of the Boltzmann transport equation at the electric quantum limit for different values of the nanowire crossโsection dimensions, the nanowire temperature and the Fermi level in a oneโdimensional electron...
Density functional calculations of the electronic structure and optical properties of magnesium oxide
ZiโJiang Liu, YingโXue Du, XiuโLu Zhang, JianโHong Qi, more
The electronic structure and optical properties of magnesium oxide (MgO) are investigated at the structural phase transition pressure using the planeโwave pseudoโpotential density functional method within the generalized gradient approximation (GGA). Good agreement between the calculated lattice parameters and experimental results is obtained, and a direct energy gap of 3.72โeV is estimated in the...
C. S. Becquart, C. Domain
physica status solidi (b) > 247 > 1 > 9 - 22
The evolution of alloy microstructures under nonโequilibrium conditions such as irradiation is an important academic as wellโindustrial issue. Atomistic kinetic Monte Carlo is one of the most versatile method which can be used to simulate the evolution of a complex microstructure at the atomic scale, dealing with elementary atomic mechanisms. It was developed more than 40 years ago to investigate...
AUXETICS (60)
NEGATIVE POISSON'S RATIO (59)
AUXETIC (56)
IIโVI SEMICONDUCTORS (56)
IIIโV SEMICONDUCTORS (46)
SINGLEโWALLED CARBON NANOTUBES (38)
71.27.+A (36)
DENSITYโFUNCTIONAL THEORY (33)
ELECTRONIC TRANSPORT (33)
RAMAN SPECTRA (31)
POISSON'S RATIO (28)
NANORIBBONS (27)
MAGNETIC SEMICONDUCTORS (24)
LATTICE DYNAMICS (23)
MULTIFERROICS (23)
FIRST-PRINCIPLES CALCULATIONS (22)
MOLECULAR DYNAMICS SIMULATIONS (21)
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Straddle / Strangle | Brilliant Math & Science Wiki
A straddle is an option strategy in which a call and put with the same strike price and expiration date is bought. A strangle is an option strategy in which a call and put with the same expiration date but different strikes is bought.
These strategies are useful to pursue if you believe that the underlying price would move significantly, but you are uncertain of the direction of the movement. However, if the direction of the movement is not significant enough, then a loss will still be incurred. Hence, these are extremely risky strategies if the realized volatility isn't higher than the implied volatility of the options.
A straddle refers to both a call and a put option on the same strike, with the same expiration. Usually these options are near ATM.
The straddle at strike
X
Y _ X
An investor bought the straddle on the $50 strike for $6. What price must the stock expire at in order for the investor to make money?
In order to make money, the value of the options must be more than $6. On expiation, the put and call options cannot be both in the money.
If the call was worth $6 or more, that means that it has an intrinsic value of $6 or more, or that the stock price was at least
\$50 + \$6 = \$56
If the put was worth $6 or more, that means that it has an intrinsic value of $6 or more, or that the stock price was at most
\$50 - \$6 = \$44
Hence, in order for the investor to make money, the stock must be either above $56, or under $44.
A strangle refers to a call and a put option on distinct strikes, with the same expiration. Usually these options are OTM. If both of these options are ITM, then it is known as a gut strangle.
\$0 \leq P < \$45 \text{ or } P > \$55
\$40 < P < \$60
\$45 < P < \$55
\$0 \leq P < \$40 \text{ or } P > \$60
When the underlying price is close to the strikes, long straddles and strangles are
1. Option value is mostly extrinsic
2. High Gamma
3. High Vega
4. Paying a lot of theta
When the underlying price has moved through the strike, long straddles and strangles are
1. Option value is mostly intrinsic
2. Lower Gamma
3. Lower Vega
4. Paying less theta
5. High skew risk
Short straddle on the 30 strike Long the 28-32 strangle Long straddle on the 30 strike Short the 28-32 strangle
The stock is currently trading at 30. Which option strategy has the most (positive) vega?
The options all have the same expiry date.
Consider buying straddles and strangles in the following situations:
1. You believe that the underlying will move more than the implied volatility
2. You believe that the volatility will increase soon
3. You believe that the underlying will move significantly in one direction, and do not know which direction or for how far.
Note: Despite having high Gamma, owing straddles/strangles are useless if the underlying is highly volatile within a tight range, that does not allow you to do any delta hedging. In particular, if you own the 95-105 strangle, and the underlying trades back and forth between 99-101 a lot, it will be very hard to make money. In such a situation, it is better to be short the strangle / straddle and not hedge too aggressively.
Cite as: Straddle / Strangle. Brilliant.org. Retrieved from https://brilliant.org/wiki/straddle-strangle/
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Effect of Side Wind on a Simplified Car Model: Experimental and Numerical Analysis | J. Fluids Eng. | ASME Digital Collection
Emmanuel Guilmineau,
Laboratoire de Mรฉcanique des Fluides, CNRS UMR 6598, Equipe Modรฉlisation Numรฉrique,
, 1 Rue de la Noรซ, BP 921001, 44321 Nantes Cedex 3, France
e-mail: emmanuel.guilmineau@ec-nantes.fr
Francis Chometon
Laboratoire dโAรฉrodynamique,
, 15 Rue Marat, 78210 Saint Cyr lโEcole, France
Guilmineau, E., and Chometon, F. (January 15, 2009). "Effect of Side Wind on a Simplified Car Model: Experimental and Numerical Analysis." ASME. J. Fluids Eng. February 2009; 131(2): 021104. https://doi.org/10.1115/1.3063648
A prior analysis of the effect of steady cross wind on full size cars or models must be conducted when dealing with transient cross wind gust effects on automobiles. The experimental and numerical tests presented in this paper are performed on the Willy square-back test model. This model is realistic compared with a van-type vehicle; its plane underbody surface is parallel to the ground, and separations are limited to the base for moderated yaw angles. Experiments were carried out in the semi-open test section at the Conservatoire National des Arts et Mรฉtiers, and computations were performed at the Ecole Centrale de Nantes (ECN). The ISIS-CFD flow solver, developed by the CFD Department of the Fluid Mechanics Laboratory of ECN, used the incompressible unsteady Reynolds-averaged NavierโStokes equations. In this paper, the results of experiments obtained at a Reynolds number of
0.9ร106
are compared with numerical data at the same Reynolds number for steady flows. In both the experiments and numerical results, the yaw angle varies from 0 deg to 30 deg. The comparison between experimental and numerical results obtained for aerodynamic forces, wall pressures, and total pressure maps shows that the unsteady ISIS-CFD solver correctly reflects the physics of steady three-dimensional separated flows around bluff bodies. This encouraging result allows us to move to a second step dealing with the analysis of unsteady separated flows around the Willy model.
generic car body, numerical simulation, experimental data, cross wind effect, vehicle aerodynamics, aerodynamics, automobiles, computational fluid dynamics, flow instability, flow separation, Navier-Stokes equations, turbulence
Aerodynamics, Automobiles, Computational fluid dynamics, Flow (Dynamics), Pressure, Turbulence, Vehicles, Vortices, Wind, Yaw, Cylinders, Reynolds-averaged NavierโStokes equations, Computer simulation, Computation, Numerical analysis, Wakes, Wind tunnels, Physics
Assement of the Adequacy of Various Wind Tunnel Techniques to Obtain Aerodynamic Data for Ground Vehicles in Cross Winds
The Aerodynamic Forces Induced on a Passenger Vehicle in Response to a Transient Cross-Wind Gust at a Relative Incidence of 30ยฐ
An Experimental Study of Unsteady Vehicle Aerodynamics
Ground Vehicles in High Cross Winds, Part 1: Steady Aerodynamic Forces
Wake Surveys Behind a Passenger Car Subjected to a Transient Cross-Wind Gust
Some Effects of Ground Clearance and Ground Plane Boundary Layer Thickness on the Mean Base Pressure of a Bluff Vehicle Type Body
The Side Load Distribution on a Rover 800 Saloon Car Under Crosswind Conditions
Turbulent Structure of Three-Dimensional Flow Behind a Model Car: 2. Exposed to Crosswind
Comparison of Quasi-Static and Dynamic Wind Tunnel Measurements on Simplified Tractor-Trailer Models
Experimental Study of Unsteady Wakes Behind an Oscillating Car Model
Some Salient Features of the Time-Averaged Ground Vehicle Wake
Flow and Turbulence in the Wake of a Simplified Car Model
Computational Analysis of Three-Dimensional Turbulent Flow Around a Bluff Body in Ground Proximity
Experimental and Computation Investigation of Ahmed Body for Ground Vehicle Aerodynamics
The Ahmed Model Unsteady Wake: Experimental and Computational Analyses
Flow Around a Simplified CarโPart 1: Large Eddy Simulation
Unsteady Flow Simulation of the Ahmed Reference Body Using a Lattice Boltzmann Approach
Detached-Eddy Simulation of the Flow Around the Ground Transportation System
Aerodynamic Drag of Heavy Vehicles (Calls 7-8): Simulation and Benchmarking
Flow Structure Around Trains Under Side Wind Conditions: A Numerical Study
Large-Eddy Simulation of the Flow Around Simplified High-Speed Trains Under Side Wind Conditions
,โ Ph.D. thesis, Chalmers University of Technology, Gรถteborg, Sweden.
Resolution NVD Differencing Scheme for Arbitrarily Unstructured Meshes
A Numerical Study of the Turbulent Flow Past an Isolated Aerofoil With Trailing Edge Separation
An Interface Capturing Method for Free-Surface Hydrodynamic Flows
On the Role Played by Turbulence Closures in Hull Ship Optimization at Model and Full Scale
Comparison of Explicit Algebraic Stress Models and Second-Order Turbulence Closures for Steady Flow Around Ships
Seventh Symposium on Numerical Ship Hydrodynamics
, Nantes, France, pp.
Three-Dimensional Flow Computation With Reynolds Stress and Algebraic Stress Models
Proceedings of the ERCOFTAC International Symposium on Engineering Turbulence Modelling and MeasurementsโETMM6
A Unified Analysis of Planar Homogeneous Turbulence Using Single-Point Closure Equations
Zonal Two-Equation k-ฯ
Turbulence Models for Aerodynamic Flows,โ
Turbulence et Couche Limite
Turbulent Flow Properties of Large-Scale Vortex Systems
Separated Flows Around the Rear Window of a Simplified Car Geometry
Numerical and Experimental Investigations of Rotating Wheel Aerodynamics on the DrivAer Model With Engine Bay Flow
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Global Constraint Catalog: Cchange_partition
<< 5.63. change_pair5.65. change_vectors >>
\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด},\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}\right)
\mathrm{๐
๐ฐ๐ป๐๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐}\right)
\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}
\mathrm{๐๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(๐-\mathrm{๐
๐ฐ๐ป๐๐ด๐}\right)
|\mathrm{๐
๐ฐ๐ป๐๐ด๐}|\ge 1
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐๐ด๐},\mathrm{๐๐๐}\right)
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐ป๐๐ด๐},\mathrm{๐๐๐}\right)
\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}\ge 0
\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}<|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐},\mathrm{๐๐๐}\right)
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐},๐\right)
|\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}|\ge 2
\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}
is the number of times that the following constraint holds:
X
Y
do not belong to the same partition of the collection
\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}
X
Y
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\left(\begin{array}{c}2,โฉ6,6,2,1,3,3,1,6,2,2,2โช,\hfill \\ โฉ๐-โฉ1,3โช,๐-โฉ4โช,๐-โฉ2,6โชโช\hfill \end{array}\right)
In the example we have the following two changes:
One change between values 2 and 1 (since 2 and 1 respectively belong to the third and the first partition),
One change between values 1 and 6 (since 1 and 6 respectively belong to the first and the third partition).
Consequently the
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐}
\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}
is assigned to 2.
\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}>0
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|>1
\mathrm{๐๐๐๐๐}
\left(\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}\right)>1
|\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}|>|\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}|
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}
\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}.๐
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}.\mathrm{๐๐๐}
\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}
\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}
This constraint is useful for the following problem: Assume you have to produce a set of orders, each order belonging to a given family. In the context of the Example slot we have three families that respectively correspond to values
1,3
, to value 4 and to values
2,6
. We would like to sequence the orders in such a way that we minimise the number of times two consecutive orders do not belong to the same family.
\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐ข}\mathrm{๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐}_\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐}
\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}
\mathrm{๐๐ด๐๐ป}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{1},\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{2}\right)
\mathrm{๐๐}_\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐}
\left(\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{1}.\mathrm{๐๐๐},\mathrm{๐๐๐๐๐๐๐๐๐}\mathtt{2}.\mathrm{๐๐๐},\mathrm{๐ฟ๐ฐ๐๐๐ธ๐๐ธ๐พ๐ฝ๐}\right)
\mathrm{๐๐๐๐}
=\mathrm{๐ฝ๐ฒ๐ท๐ฐ๐ฝ๐ถ๐ด}
โข
\mathrm{๐ฐ๐ฒ๐๐ฒ๐ป๐ธ๐ฒ}
โข
\mathrm{๐ฑ๐ธ๐ฟ๐ฐ๐๐๐ธ๐๐ด}
โข
\mathrm{๐ฝ๐พ}_\mathrm{๐ป๐พ๐พ๐ฟ}
\mathrm{๐๐๐๐}
\mathrm{๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐}
|
Integration of Exponential Functions | Brilliant Math & Science Wiki
Satyajit Mohanty, Mahindra Jain, Samir Khan, and
Exponential functions are those of the form
f(x)=Ce^{x}
C
, and the linear shifts, inverses, and quotients of such functions. Exponential functions occur frequently in physical sciences, so it can be very helpful to be able to integrate them.
Nearly all of these integrals come down to two basic formulas:
\int e^x\, dx = e^x + C, \quad \int a^x\, dx = \frac{a^x}{\ln(a)} +C.
\int (3e^x+2^x)\, dx,
C
\begin{aligned} \int (3e^x+2^x)\, dx &=3\int e^x dx+\int 2^x\, dx \\ &=3e^x+\frac{2^x}{\ln 2}+C, \end{aligned}
C
_\square
\int e^{x+2}\, dx,
C
\begin{aligned} \int e^{x+2}\, dx &=\int e^x e^2\, dx \\ &=e^2\int e^x\, dx \\ &=e^2 e^x +C \\ &=e^{x+2}+C, \end{aligned}
C
_\square
Case 1: Suppose we have an exponential function clubbed as
\displaystyle \int e^x\big(f(x) + f'(x)\big)\, dx
. In this case, the integral is
e^xf(x) + C.
\int e^x\big(\sin(x) + \cos(x)\big)\, dx,
C
We have the integral in the form of
\displaystyle \int e^x\big(f(x) + f'(x)\big)\, dx,
f(x) = \sin(x)
. So our integral is
e^x\sin(x) + C.\ _\square
Case 2: Suppose we have an integration of the form
\displaystyle I = \int e^{ax}\cos(bx+c).
Its integral is
I=\dfrac{e^{ax}\big(a\cos(bx+c)+b\sin(bx+c)\big)}{a^2 + b^2}.
We'll integrate the above using integration by parts as follows:
\begin{aligned} I &= \int e^{ax}\cos(bx+c)\ dx\\ &= \cos(bx+c)\frac{e^{ax}}{a} + \frac{b}{a} \int e^{ax}\sin(bx+c)\ dx\\\\ &= \cos(bx+c)\frac{e^{ax}}{a} + \frac{b}{a}\left(\frac{e^{ax}}{a}\sin(bx+c) - \frac{b}{a} \int e^{ax}\cos(bx+c)\right)\ dx\\\\ &=\frac{e^{ax}\big(a\cos(bx+c)+b\sin(bx+c)\big)}{a^2} - \frac{b^2}{a^2}I. \end{aligned}
\begin{aligned} I\left(1+\frac{b^2}{a^2}\right)& = \frac{e^{ax}\big(a\cos(bx+c)+b\sin(bx+c)\big)}{a^2}\\ \Rightarrow I&=\frac{e^{ax}\big(a\cos(bx+c)+b\sin(bx+c)\big)}{a^2 + b^2}.\ _\square \end{aligned}
Note: The above example is also applicable for the form
\displaystyle I = \int e^{ax}\sin(bx+c).
\int e^{2x}\cos(5x+3)\, dx,
C
By the above result, we obtain our answer as
\frac{e^{2x}\big(2\cos(5x+3)+5\sin(5x+3)\big)}{29} + C.\ _\square
Case 3: If the integration is of the form
\displaystyle \int \frac{ae^x + be^{-x}}{pe^x + qe^{-x}} dx,
\text{(NUM)} =\alpha \text{(DEN)} + \beta \frac{d}{dx} \text{(DEN)},
where NUM=(the numerator of the integrand) and DEN=(the denominator of the integrand), and then integrate as usual.
\int \frac{2e^x + 3e^{-x}}{e^x - 5e^{-x}}\, dx,
C
2e^x + 3e^{-x} = \alpha(e^x - 5e^{-x}) + \beta(e^x + 5e^{-x}).
e^x \text{ and } e^{-x}
\alpha + \beta = 2
\alpha - \beta = -\frac{3}{5},
\alpha = \frac7{10}, \beta=\frac{13}{10}
\int \frac{2e^x + 3e^{-x}}{e^x - 5e^{-x}} dx=\alpha \int dx + \beta \int \frac{e^x + 5e^{-x}}{e^x - 5e^{-x}} dx. \qquad (1)
e^x - 5e^{-x} = t,
(e^x + 5e^{-x})dx = dt,
\begin{aligned} (1) &=\frac7{10} \int dx +\frac{13}{10}\int \frac{dt}{t}\\ &=\frac{7x}{10} +\frac{13}{10}\ln |t| + C \\ &=\frac{7x}{10} +\frac{13}{10}\ln \big|e^x - 5e^{-x}\big| + C, \end{aligned}
C
_\square
Cite as: Integration of Exponential Functions. Brilliant.org. Retrieved from https://brilliant.org/wiki/integration-of-exponential-functions/
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NDF_HINFO
The routine returns character information about an NDFโs history component or about one of the history records it contains.
CALL NDF_HINFO( INDF, ITEM, IREC, VALUE, STATUS )
ITEM = CHARACTER
\ast
\ast
Name of the information item required: โAPPLICATIONโ, โCREATEDโ, โDATEโ, โDEFAULTโ, โHOSTโ, โMODEโ, โNLINESโ, โNRECORDSโ, โREFERENCEโ, โUSERโ, โWIDTHโ or โWRITTENโ (see the "General Items" and "Specific Items" sections for details). This value may be abbreviated, to no less than three characters.
History record number for which information is required. This argument is ignored if information is requested about the history component as a whole. See the "Specific Items" section for details of which items require this argument.
\ast
\ast
The history information requested (see the "Returned String Lengths" section for details of the length of character variable required to receive this value).
The following ITEM values request general information about the history component and do not use the IREC argument:
โCREATEDโ: return a string giving the date and time of creation of the history component as a whole in the format โYYYY-MMM-DD HH:MM:SS.SSSโ (e.g. โ1993-JUN-16 11:30:58.001โ).
โDEFAULTโ: return a logical value indicating whether default history information has yet to be written for the current application. A value of โFโ is returned if it has already been written or has been suppressed by a previous call to NDF_HPUT, otherwise the value โTโ is returned.
โMODEโ: return the current update mode of the history component (one of the strings โDISABLEDโ, โQUIETโ, โNORMALโ or โVERBOSEโ).
โNRECORDSโ: return the number of history records present (an integer formatted as a character string). Note that for convenience this value may also be obtained directly as an integer via the routine NDF_HNREC.
โWRITTENโ: return a logical value indicating whether the current application has written a new history record to the NDFโs history component. A value of โTโ is returned if a new record has been written, otherwise โFโ is returned.
The following ITEM values request information about specific history records and should be accompanied by a valid value for the IREC argument specifying the record for which information is required:
โAPPLICATIONโ: return the name of the application which created the history record.
โDATEโ: return a string giving the date and time of creation of the specified history record in the format โYYYY-MMM-DD HH:MM:SS.SSSโ (e.g. โ1993-JUN-16 11:36:09.021โ).
โHOSTโ: return the name of the machine on which the application which wrote the history record was running (if this has not been recorded, then a blank value is returned).
โNLINESโ: return the number of lines of text contained in the history record (an integer formatted as a character string).
โREFERENCEโ: return a name identifying the NDF dataset in which the history component resided at the time the record was written (if this has not been recorded, then a blank value is returned). This value is primarily of use in identifying the ancestors of a given dataset when history information has been repeatedly propagated through a sequence of processing steps.
โUSERโ: return the user name for the process which wrote the history record (if this has not been recorded, then a blank value is returned).
โWIDTHโ: return the width in characters of the text contained in the history record (an integer formatted as a character string).
Returned String Lengths
If ITEM is set to โCREATEDโ, โDATEโ, โMODEโ, โNLINESโ, โNRECORDSโ or โWIDTHโ, then an error will result if the length of the VALUE argument is too short to accommodate the returned result without losing significant (non-blank) trailing characters.
If ITEM is set to โAPPLICATIONโ, โHOSTโ, โREFERENCEโ or โUSERโ, then the returned value will be truncated with an ellipsis โ...โ if the length of the VALUE argument is too short to accommodate the returned result without losing significant (non-blank) trailing characters. No error will result.
When declaring the length of character variables to hold the returned result, the constant NDF__SZHDT may be used for the length of returned date/time strings for the โCREATEDโ and โDATEโ items, the constant NDF__SZHUM may be used for the length of returned update mode strings for the โMODEโ item, and the constant VAL__SZI may be used for the length of returned integer values formatted as character strings.
Use of the constant NDF__SZAPP is recommended when declaring the length of a character variable to hold the returned application name for the โAPPLICATIONโ item. Similarly, use of the constant NDF__SZHST is recommended when requesting the โHOSTโ item, NDF__SZREF when requesting the โREFERENCEโ item and NDF__SZUSR when requesting the โUSERโ item. Truncation of the returned values may still occur, however, if longer strings were specified when the history record was created.
The NDF__SZAPP, NDF__SZHDT, NDF__SZHST, NDF__SZHUM, NDF__SZREF and NDF__SZUSR constants are defined in the include file NDF_PAR. The VAL__SZI constant is defined in the include file PRM_PAR (see SUN/39).
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From the graph of
\Large \lim_{x\to6}f(f(x)).
5 6 7 Does not exist
0 ^ 0 = 1
\Large \lim_{ x \rightarrow 0^+ } x ^{^ { \frac{ 1}{\ln x} }}?
\pi
A man stuck in a small sailboat on a perfectly calm lake throws a stone overboard. It sinks to the bottom of the lake.
When the water again settles to a perfect calm, is the water level in the lake higher, lower, or in the same place compared to where it was before the stone was cast in?
Hint: You can use limits to solve this problem!
The water level stays the same. The water level is lower. It depends on the size of the stone. The water level rises.
\lim_{x \rightarrow 0^+ } \sqrt{ x + \sqrt{ x + \sqrt{ x + \ldots } } }.
\frac{1}{2}
1 Does not exist 0
\large \lim_{x \to 1} \left( \frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right) = \, ?
by Kazem Sepehrinia
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How to Use a Scientific Calculator For Algebra: 13 Steps
How to Use a Scientific Calculator For Algebra
1 Basic Operational Functions
2 Common Functions for Algebra
A scientific calculator provides functions that make common calculations easy. While each calculator is slightly different, every model has the basic functions needed for middle and high school math courses. Once you understand how to use its features, it will help you on your way towards mathematical success.
This article will use the TI-30X IIS calculator.[1] X Research source Be sure to check the instruction manual for your own calculator. Instructions for pressing buttons will be in
{\displaystyle \mathrm {math} }
code.
Basic Operational Functions Download Article
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/9\/94\/Use-a-Scientific-Calculator-For-Algebra-Step-1-Version-2.jpg\/v4-460px-Use-a-Scientific-Calculator-For-Algebra-Step-1-Version-2.jpg","bigUrl":"\/images\/thumb\/9\/94\/Use-a-Scientific-Calculator-For-Algebra-Step-1-Version-2.jpg\/aid2850740-v4-728px-Use-a-Scientific-Calculator-For-Algebra-Step-1-Version-2.jpg","smallWidth":460,"smallHeight":348,"bigWidth":728,"bigHeight":551,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/en.wikipedia.org\/wiki\/Fair_use\">Fair Use<\/a> (screenshot)<br>\n<\/p><\/div>"}
Use the basic operation symbols to perform basic operations. These operations include addition (+), subtraction (-), multiplication (ร), and division (รท).
You have to hit the equals (=) sign to complete a calculation using these symbols.
Use these functions the same way you would on a basic calculator. For example, to find 12/4, enter
{\displaystyle 12\div 4=}
{"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/1\/1b\/Use-a-Scientific-Calculator-For-Algebra-Step-2.jpg\/v4-460px-Use-a-Scientific-Calculator-For-Algebra-Step-2.jpg","bigUrl":"\/images\/thumb\/1\/1b\/Use-a-Scientific-Calculator-For-Algebra-Step-2.jpg\/aid2850740-v4-728px-Use-a-Scientific-Calculator-For-Algebra-Step-2.jpg","smallWidth":460,"smallHeight":348,"bigWidth":728,"bigHeight":551,"licensing":"<div class=\"mw-parser-output\"><p>License: <a target=\"_blank\" rel=\"nofollow noreferrer noopener\" class=\"external text\" href=\"https:\/\/en.wikipedia.org\/wiki\/Fair_use\">Fair Use<\/a> (screenshot)<br>\n<\/p><\/div>"}
Donโt worry about order of operations. A scientific calculator will automatically calculate using the correct order.
For example, if your problem is 2 - 4 ร -3 รท 2, you can type
{\displaystyle 2-4\times -3\div 2=}
, and the calculator will automatically do the order of operations for you.
Change the order of operations using the parentheses keys. This will override the calculatorโs order of operations.
Hit the beginning parentheses before you hit your first number, and hit the ending parentheses after you hit your last number. The calculator will complete that calculation before you enter your next functions.
If you need, you can also nest parentheses, though be sure that you can keep track of them.
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Fix your mistakes. If you accidentally hit a wrong key, hit the
{\displaystyle \mathrm {DEL} }
button. This will clear the last button you pressed, but will not clear any calculations youโve made.
Clear your work. To clear the line, hit the
{\displaystyle \mathrm {CLEAR} }
button. Chances are, you can scroll up to see past calculations. You can hit delete with each line in order to get rid of these.
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Explore all the functions on the calculator. To use some functions, you might have to use the
{\displaystyle \mathrm {2nd} }
key. These functions are listed above the button, similar to how symbols are listed above the number keys on a keyboard. To use these functions, hit the 2nd key first. Getting the syntax right for some operations may involve switching the order between the operation and the number.
For the TI-30X IIS, one of the 2nd functions is to turn off the calculator. Simply press
{\displaystyle \mathrm {2nd} \ \mathrm {ON} }
Common Functions for Algebra Download Article
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Make a fraction. While there are buttons to do this, fractions are much easier to type in manually. The reason is that most scientific calculators only output in decimal format.
For example, to evaluate 4/5 + 6/7, type in
{\displaystyle 4\div 5+6\div 7=}
to get approximately 1.657. Use parentheses for more complex operations.
To convert back to fractions, use
{\displaystyle \mathrm {2nd} \ \mathrm {PRB} }
to get the answer in mixed fractions.
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Square a number. Do this using the
{\displaystyle x^{2}}
button. Type in the number you want to square, then hit the that button.
For example, to square the number 12, type
{\displaystyle 12\ x^{2}=}
To find powers of any number, use the
{\displaystyle \wedge }
key. For example, type
{\displaystyle 3\wedge 5=}
to get 243.
Find a square root. Type
{\displaystyle \mathrm {2nd} \ x^{2}}
to display the square root.
To find the square root of 9, type
{\displaystyle \mathrm {2nd} \ x^{2}\ 9=}
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Find logarithms. There are two buttons for logarithms. The
{\displaystyle \mathrm {LOG} }
is the logarithm base 10. Use whichever one is appropriate.
For example, to find the logarithm base 10 of 100, simply type
{\displaystyle \mathrm {LOG} \ 100=}
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Use the exponential function. The exponential function
{\displaystyle e^{x}}
is found by typing
{\displaystyle \mathrm {2nd} \ \mathrm {LN} }
. If you need to use Euler's number only, use e to the first power.
For example, to find the natural log of e, type in
{\displaystyle \mathrm {LN} \ \mathrm {2nd} \ \mathrm {LN} \ 1=}
Find trigonometric functions. Sine, cosine, and tangent come standard with any scientific calculator. To use these buttons, be sure that you know whether to use degrees or radians. To convert, use the
{\displaystyle \mathrm {DRG} }
button to highlight DEG or RAD. Other calculators will generally have a button that converts between these two systems.
For example, if you wanted to find the sine of 60 degrees, make sure that you are in degree mode by checking the lower right of the display. Then, type
{\displaystyle \mathrm {SIN} \ 60=}
Find the reciprocal of a number. Do this using the
{\displaystyle x^{-1}}
button. On some calculators, it may be labeled as
{\displaystyle 1/x}
instead. Type in the number you want to find the reciprocal of, then hit the reciprocal button.
For example, to find the reciprocal of 3, type
{\displaystyle 3\ x^{-1}=}
How do I make 7/3 on a calculator?
Fractions mean division. Type in 7รท3.
How do I calculate 2 to the 64th power?
Press the "2" button, then the "xสธ" (the y is above the x on a calculator) button. Then type in "64".
How do I do brackets on the calculator?
You can't, but I'm sure parentheses will work fine in place of square/curly brackets. Brackets are used by some people because the expression has groups in groups (e.g. 1+[4(2+1)+3]).
How can I make the parenthesis with my scientific calculator?
If your scientific calculator has them (and most of them do), press each parenthetical in the order you want it to be used. The calculator will base it's answer on the order of operations where the stuff inside the parenthesis would be evaluated first, if placed inside the parenthesis correctly. If you try to place the ending parenthesis before the starting parenthesis, the calculator should give you an error - making sure that you start again with the correct sign. Each scientific calculator has the parentheticals in a different spot, but your owner's manual should tell you where to find them.
How do I do x on a scientific calculator when doing an algebra problem?
When doing an equation that specifies an unknown as X, usually when you go to enter the equation as a posed question the calculator knows that you are going to use the X as the designation for an unknown amount, and reads it like all the other letters. See your instruction book where the topic is equations.
How can I use a calculator to solve a math question?
You probably have to figure out the formula first, if it's not already written down for you. Then, you just replace the formula with numbers and type it into the calculator.
Remember that you are responsible for converting the answer after the calculator does its job. It is much more useful to have those skills than to be skilled at using a particular model.
The TI-30X IIS is pretty loose with closing parentheses for operations. For example, you do not need to close the parentheses when evaluating logarithms or trig functions.
When choosing a scientific calculator, consider what math classes besides algebra you'll be using it for. You should choose a calculator that will take you through several years of the math classes you anticipate taking. In recent years, it is more beneficial to use a graphing calculator for more advanced courses, like calculus.
Take a look at several calculators before you buy to familiarize yourself with their layouts. You can work with the calculator application available with your operating system, if it has a scientific calculator setting, or an online calculator, to familiarize yourself with some of the functions of a standalone scientific calculator before you buy.
โ https://education.ti.com/en/us/guidebook/details/en/706B2B75F7D3464EBE6A8F8BE9F00EAC/30xii?id=6134
Categories: Algebra | Calculators
ะ ัััะบะธะน:ะธัะฟะพะปัะทะพะฒะฐัั ะฝะฐััะฝัะน ะบะฐะปัะบัะปััะพั ะดะปั ะฐะปะณะตะฑัั
Espaรฑol:usar una calculadora cientรญfica para el รกlgebra
Italiano:Usare una Calcolatrice Scientifica per l'Algebra
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Global Constraint Catalog: Ctrack
<< 5.401. tour5.403. tree >>
[Marte01]
\mathrm{๐๐๐๐๐}\left(\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป},\mathrm{๐๐ฐ๐๐บ๐}\right)
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}
\mathrm{๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐}-\mathrm{๐๐๐},\mathrm{๐๐๐๐๐๐}-\mathrm{๐๐๐๐},\mathrm{๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}>0
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}\le |\mathrm{๐๐ฐ๐๐บ๐}|
|\mathrm{๐๐ฐ๐๐บ๐}|>0
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐๐ฐ๐๐บ๐},\left[\mathrm{๐๐๐๐๐},\mathrm{๐๐๐๐๐๐},\mathrm{๐๐๐}\right]\right)
\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐๐}\le \mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐}
constraint forces that, at each point in time overlapped by at least one task, the number of distinct values of the
\mathrm{๐๐๐๐๐}
attribute of the set of tasks that overlap that point, is equal to
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}
\left(\begin{array}{c}2,โฉ\begin{array}{ccc}\mathrm{๐๐๐๐๐}-1\hfill & \mathrm{๐๐๐๐๐๐}-1\hfill & \mathrm{๐๐๐}-2,\hfill \\ \mathrm{๐๐๐๐๐}-2\hfill & \mathrm{๐๐๐๐๐๐}-1\hfill & \mathrm{๐๐๐}-2,\hfill \\ \mathrm{๐๐๐๐๐}-1\hfill & \mathrm{๐๐๐๐๐๐}-2\hfill & \mathrm{๐๐๐}-4,\hfill \\ \mathrm{๐๐๐๐๐}-2\hfill & \mathrm{๐๐๐๐๐๐}-2\hfill & \mathrm{๐๐๐}-3,\hfill \\ \mathrm{๐๐๐๐๐}-2\hfill & \mathrm{๐๐๐๐๐๐}-3\hfill & \mathrm{๐๐๐}-4\hfill \end{array}โช\hfill \end{array}\right)
Figure 5.402.1 represents the tasks of the example: to the
{i}^{th}
task of the
\mathrm{๐๐ฐ๐๐บ๐}
collection corresponds a rectangle labelled by
i
\mathrm{๐๐๐๐๐}
The first and second tasks both overlap instant 1 and have a respective trail of 1 and 2. This makes two distinct values for the trail attribute at instant 1.
The third and fourth tasks both overlap instant 2 and have a respective trail of 1 and 2. This makes two distinct values for the trail attribute at instant 2.
The third and fifth tasks both overlap instant 3 and have a respective trail of 1 and 2. This makes two distinct values for the trail attribute at instant 3.
Figure 5.402.1. The tasks associated with the example of the Example slot, at each instant we have two distinct values for the
\mathrm{๐๐๐๐๐}
attribute (
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}=2
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}<|\mathrm{๐๐ฐ๐๐บ๐}|
|\mathrm{๐๐ฐ๐๐บ๐}|>1
\mathrm{๐๐๐๐๐}
\left(\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐}\right)>1
\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐๐}<\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}
\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}
\mathrm{๐๐๐๐๐}
2ยท|\mathrm{๐๐ฐ๐๐บ๐}|
\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right],\mathrm{๐๐ฐ๐๐บ๐}\left[j\right]
\left(i,j\in \left[1,|\mathrm{๐๐ฐ๐๐บ๐}|\right]\right)
\mathrm{๐๐ฐ๐๐บ๐}
{T}_{ij}^{\mathrm{๐๐๐๐๐๐}}
which is set to the
\mathrm{๐๐๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right]
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right]
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
, and to the
\mathrm{๐๐๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
i=j
{T}_{ij}^{\mathrm{๐๐๐๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐}
i\ne j
{T}_{ij}^{\mathrm{๐๐๐๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐}\vee {T}_{ij}^{\mathrm{๐๐๐๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐}
\left(\left(\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐๐}\le \mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐๐}\wedge
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐}>\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐๐}\right)\wedge \left({T}_{ij}^{\mathrm{๐๐๐๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐}\right)\right)\vee
\left(\left(\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐๐}>\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐๐}\vee
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐}\le \mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐๐}\right)\wedge \left({T}_{ij}^{\mathrm{๐๐๐๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐}\right)\right)
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
\left(i\in \left[1,|\mathrm{๐๐ฐ๐๐บ๐}|\right]\right)
we impose the number of distinct trails associated with the tasks that overlap the origin of task
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
overlaps its own origin) to be equal to
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}
\mathrm{๐๐๐๐๐๐}
\left(\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป},โฉ{T}_{i1}^{\mathrm{๐๐๐๐๐๐}},{T}_{i2}^{\mathrm{๐๐๐๐๐๐}},\cdots ,{T}_{i|\mathrm{๐๐ฐ๐๐บ๐}|}^{\mathrm{๐๐๐๐๐๐}}โช\right)
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right],\mathrm{๐๐ฐ๐๐บ๐}\left[j\right]
\left(i,j\in \left[1,|\mathrm{๐๐ฐ๐๐บ๐}|\right]\right)
\mathrm{๐๐ฐ๐๐บ๐}
{T}_{ij}^{\mathrm{๐๐๐}}
\mathrm{๐๐๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right]
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right]
overlaps the end attribute of task
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
\mathrm{๐๐๐๐๐}
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
i=j
{T}_{ij}^{\mathrm{๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐}
i\ne j
{T}_{ij}^{\mathrm{๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐}\vee {T}_{ij}^{\mathrm{๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐}
\left(\left(\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐๐}\le \mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐}-1\wedge
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐}>\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐}-1\right)\wedge \left({T}_{ij}^{\mathrm{๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐}\right)\right)\vee
\left(\left(\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐๐๐๐}>\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐}-1\vee
\mathrm{๐๐ฐ๐๐บ๐}\left[j\right].\mathrm{๐๐๐}\le \mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐}-1\right)\wedge \left({T}_{ij}^{\mathrm{๐๐๐}}=\mathrm{๐๐ฐ๐๐บ๐}\left[i\right].\mathrm{๐๐๐๐๐}\right)\right)
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
\left(i\in \left[1,|\mathrm{๐๐ฐ๐๐บ๐}|\right]\right)
we impose the number of distinct trails associated with the tasks that overlap the end of task
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
\mathrm{๐๐ฐ๐๐บ๐}\left[i\right]
overlaps its own end) to be equal to
\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป}
\mathrm{๐๐๐๐๐๐}
\left(\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป},โฉ{T}_{i1}^{\mathrm{๐๐๐}},{T}_{i2}^{\mathrm{๐๐๐}},\cdots ,{T}_{i|\mathrm{๐๐ฐ๐๐บ๐}|}^{\mathrm{๐๐๐}}โช\right)
With respect to the Example slot we get the following conjunction of
\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ1,2,1,1,1โช\right)
constraint corresponding to the
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the origin of the first task (i.e., instant 1) that has a trail of 1.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ1,2,2,2,2โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the origin of the second task (i.e., instant 1) that has a trail of 2.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ1,1,1,2,1โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the origin of the third task (i.e., instant 2) that has a trail of 1.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ2,2,1,2,2โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the origin of the fourth task (i.e., instant 2) that has a trail of 2.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ2,2,1,2,2โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the origin of the fifth task (i.e., instant 3) that has a trail of 2.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ1,2,1,1,1โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the last instant of the first task (i.e., instant 1) that has a trail of 1.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ1,2,2,2,2โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the last instant of the second task (i.e., instant 1) that has a trail of 2.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ1,1,1,1,2โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the last instant of the third task (i.e., instant 3) that has a trail of 1.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ2,2,1,2,2โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the last instant of the fourth task (i.e., instant 2) that has a trail of 2.
\mathrm{๐๐๐๐๐๐}
\left(2,โฉ2,2,1,2,2โช\right)
\mathrm{๐๐๐๐๐}
attributes of the tasks that overlap the last instant of the fifth task (i.e., instant 3) that has a trail of 2.
\mathrm{๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐}\left(\begin{array}{c}\mathrm{๐๐ธ๐ผ๐ด}_\mathrm{๐ฟ๐พ๐ธ๐ฝ๐๐}-\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\begin{array}{c}\mathrm{๐๐๐๐๐๐}-\mathrm{๐๐๐๐},\hfill \\ \mathrm{๐๐๐}-\mathrm{๐๐๐๐},\hfill \\ \mathrm{๐๐๐๐๐}-\mathrm{๐๐๐๐}\hfill \end{array}\right),\hfill \\ \left[\begin{array}{c}\mathrm{๐๐๐๐}\left(\begin{array}{c}\mathrm{๐๐๐๐๐๐}-\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐๐},\hfill \\ \mathrm{๐๐๐}-\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐},\hfill \\ \mathrm{๐๐๐๐๐}-\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐๐}\hfill \end{array}\right),\hfill \\ \mathrm{๐๐๐๐}\left(\begin{array}{c}\mathrm{๐๐๐๐๐๐}-\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐๐},\hfill \\ \mathrm{๐๐๐}-\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐},\hfill \\ \mathrm{๐๐๐๐๐}-\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐}-1\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}\right)
\mathrm{๐๐ฐ๐๐บ๐}
\mathrm{๐๐ธ๐ฟ๐น}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐}\right)
\mathrm{๐๐๐๐๐}.\mathrm{๐๐๐๐๐๐}\le \mathrm{๐๐๐๐๐}.\mathrm{๐๐๐}
\mathrm{๐๐๐๐}
=|\mathrm{๐๐ฐ๐๐บ๐}|
\mathrm{๐๐ธ๐ผ๐ด}_\mathrm{๐ฟ๐พ๐ธ๐ฝ๐๐}
\mathrm{๐๐ฐ๐๐บ๐}
\mathrm{๐๐
๐๐ท๐๐ถ๐}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐},\mathrm{๐๐๐๐๐}\right)
โข\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐}.\mathrm{๐๐๐}>\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐}.\mathrm{๐๐๐๐๐๐}
โข\mathrm{๐๐๐๐๐}.\mathrm{๐๐๐๐๐๐}\le \mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐}.\mathrm{๐๐๐๐๐}
โข\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐๐๐}.\mathrm{๐๐๐๐๐}<\mathrm{๐๐๐๐๐}.\mathrm{๐๐๐}
\begin{array}{c}\mathrm{๐ฒ๐ด๐ข๐ข}โฆ\hfill \\ \left[\begin{array}{c}\mathrm{๐๐๐๐๐๐},\hfill \\ \mathrm{๐๐๐๐๐๐๐๐๐}-\mathrm{๐๐๐}\left(\begin{array}{c}\mathrm{๐
๐ฐ๐๐ธ๐ฐ๐ฑ๐ป๐ด๐}-\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐๐๐}\right),\hfill \\ \mathrm{๐๐๐๐}\left(\mathrm{๐๐๐}-\mathrm{๐๐ฐ๐๐บ๐}.\mathrm{๐๐๐๐๐}\right)\right]\hfill \end{array}\right)\hfill \end{array}\right]\hfill \end{array}
\mathrm{๐๐๐๐๐๐}
\left(\mathrm{๐ฝ๐๐๐ฐ๐ธ๐ป},\mathrm{๐๐๐๐๐๐๐๐๐}\right)
Parts (A) and (B) of Figure 5.402.2 respectively show the initial and final graph of the second graph constraint of the Example slot.
\mathrm{๐๐๐๐๐}
\mathrm{๐๐ธ๐ฟ๐น}
\mathrm{๐๐ฐ๐๐บ๐}
|\mathrm{๐๐ฐ๐๐บ๐}|
\mathrm{๐๐๐๐}=|\mathrm{๐๐ฐ๐๐บ๐}|
\mathrm{๐๐๐๐}\ge |\mathrm{๐๐ฐ๐๐บ๐}|
\underline{\overline{\mathrm{๐๐๐๐}}}
\overline{\mathrm{๐๐๐๐}}
|
Linear regression โ Web Education in Chemistry
The best line through a set of points
Lambert-Beer's law states that there is a linear relationship between the concentration of a compound and the absorbance at a certain wavelength. This property is exploited in the use of calibration curves, which enable us to estimate the concentration in an unknown sample.
A straight line is defined by the equation
b
denotes the intercept of the line with the
y
a
denotes the slope. The usual method to calculate the best line is called "Least Squares (LS)" regression. This method finds values for
and
b
in such a way that the sum of the squared differences of the datapoints with the fitted line is minimal. This is done by setting partial derivatives for
and
b
Two examples of data sets are given below. The first is a calibration line for the determination of Cd with Atomic Absorption Spectroscopy (AAS); the second a comparison of two analytical methods, a reference method and a new one. Select one of the examples and click the "Submit" button. It is also possible to fill in your own data for
x
y
LS regression: assumptions
1. Errors (in this case: deviations from the ideal straight line) are only present in the
y
-variable (i.e. the absorption) and not in
x
(concentration). Does it make a difference which variable we put on the
x
-axis and which on the
y
-axis? To check this, select one of the two prefab datasets or enter your own data, and press the "Submit" button.
2. The errors in y are independent and normally distributed with a constant variance over the whole range of the calibration line. Several violations of this assumption can be seen in practice (make a choice and be sure that you check all three options!):
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Whoops! Maybe you were looking for Awesomeness, or Michael Bay? How about Asplode, or Cosby and Hitler? Possibly even Game?
An explosion is the rapid combustion of megahurtz, and can be recognised by such things as smoke, fire and the rather unpleasant smell of CGI. Explosions are mostly caused by trying to flush a glock down the toilet while dropping a lit match. Despite their regular appearances in such countries as Iraq and Movieland, explosions can technically be classified as "rare". Zoologist Alan Shepard explains this: "Explosions are penis because, uumm, you know your kitchen? You see explosions less than your kitchen. Yeah." Explosions are the mortal enemies of lasers, and some people think that years ago, long before the birth of mankind and the sticky placenta that followed, lasers and explosions duked it out in a vicious battle for supremacy to see who could be totally awesomer. The life cycle of explosions is peculiar, as it starts off with a sort of larvae called "bombs" or "explosives", or "everything" if you are in a high-budget action flick. Likewise, the cycle ends with remenants of the explosion, named "dead henchmen", strewn across the location of detonation.
{\displaystyle x(c+d+e+f)-(qa/qb)/cz.}
Retrieved from "http://en.uncyclopedia.co/w/index.php?title=Explosion&oldid=6057427"
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User talk:Pretenderrs - Meta
User talk:Pretenderrs
IโLL PLEASED YOUR MESSAGES!Edit
{\displaystyle Insertformulahere}
Tell me, please, how edit the number of articles on www.wikipedia.org in Russian section. Very often incompatible with Russian Main page. I ll can help you in this question. Thank you. From Wikipedia ---- Pretenderrs =TALK=
If you know HTML, please edit this page. Otherwise, please tell us what exactly is wrong, so we can correct the problem. (Please note that the front page only reports numbers by the thousands.) Thanks. โ Minh Nguyแป
n (talk, contribs) 03:07, 8 July 2009 (UTC)
I ll try it. And I will look after this numbers. Thank you for the help! ---- Pretenderrs =TALK=
I was correct this page, but alteration is absend on www.wikipedia.org. Don t anderstand. ---- Pretenderrs =TALK=
To answer your question, the top 10 ring is for the ten wikis with the most visits per hour according to Wikipedia Statistics. We used to have the largest 10 wikis around the logo, but the community voted to change the criterion in 2008. The Chinese Wikipedia's visitor statistics can change drastically depending on whether the site is blocked in the PRC, so I wouldn't be surprised if Dutch reclaims that spot at some point in the future. For more information on how the portal is structured, see the documentation. โ Minh Nguyแป
n (talk, contribs) 10:01, 22 September 2011 (UTC)
OK! Thank you for the information! ---- PretenderrsTalk16:57, 22 September 2011 (UTC)
Retrieved from "https://meta.wikimedia.org/w/index.php?title=User_talk:Pretenderrs&oldid=2921613"
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Socialized Loss - Zeta
If both the liquidation process and insurance fund fail to cover over-bankruptcy, losses will be shared across depositors in the Zeta platform.
This will be implemented by applying an over-bankruptcy ratio to withdrawals ensuring that the platform has sufficient capital to continue operating. Withdrawals will be subject to the below ratio:
AB * ((DP)/(DP+BA))
DP = Deposits To Platform
BA = Bankruptcy Amount
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Global Constraint Catalog: Celem
<< 5.136. domain_constraint5.138. elem_from_to >>
\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐}\left(\mathrm{๐ธ๐๐ด๐ผ},\mathrm{๐๐ฐ๐ฑ๐ป๐ด}\right)
\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐}
\mathrm{๐๐๐๐๐ข}
\mathrm{๐ธ๐๐ด๐ผ}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐ก}-\mathrm{๐๐๐๐},\mathrm{๐๐๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐๐ก}-\mathrm{๐๐๐},\mathrm{๐๐๐๐๐}-\mathrm{๐๐๐๐}\right)
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐ธ๐๐ด๐ผ},\left[\mathrm{๐๐๐๐๐ก},\mathrm{๐๐๐๐๐}\right]\right)
\mathrm{๐ธ๐๐ด๐ผ}.\mathrm{๐๐๐๐๐ก}\ge 1
\mathrm{๐ธ๐๐ด๐ผ}.\mathrm{๐๐๐๐๐ก}\le |\mathrm{๐๐ฐ๐ฑ๐ป๐ด}|
|\mathrm{๐ธ๐๐ด๐ผ}|=1
|\mathrm{๐๐ฐ๐ฑ๐ป๐ด}|>0
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐๐ฐ๐ฑ๐ป๐ด},\left[\mathrm{๐๐๐๐๐ก},\mathrm{๐๐๐๐๐}\right]\right)
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}.\mathrm{๐๐๐๐๐ก}\ge 1
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}.\mathrm{๐๐๐๐๐ก}\le |\mathrm{๐๐ฐ๐ฑ๐ป๐ด}|
\mathrm{๐๐๐๐๐๐๐๐}
\left(\mathrm{๐๐ฐ๐ฑ๐ป๐ด},\mathrm{๐๐๐๐๐ก}\right)
\mathrm{๐ธ๐๐ด๐ผ}
is equal to one of the entries of the table
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
\left(\begin{array}{c}โฉ\mathrm{๐๐๐๐๐ก}-3\mathrm{๐๐๐๐๐}-2โช,\hfill \\ โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-1\hfill & \mathrm{๐๐๐๐๐}-6,\hfill \\ \mathrm{๐๐๐๐๐ก}-2\hfill & \mathrm{๐๐๐๐๐}-9,\hfill \\ \mathrm{๐๐๐๐๐ก}-3\hfill & \mathrm{๐๐๐๐๐}-2,\hfill \\ \mathrm{๐๐๐๐๐ก}-4\hfill & \mathrm{๐๐๐๐๐}-9\hfill \end{array}โช\hfill \end{array}\right)
\mathrm{๐๐๐๐}
\mathrm{๐ธ๐๐ด๐ผ}
corresponds to the third item of the
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
|\mathrm{๐๐ฐ๐ฑ๐ป๐ด}|>1
\mathrm{๐๐๐๐๐}
\left(\mathrm{๐๐ฐ๐ฑ๐ป๐ด}.\mathrm{๐๐๐๐๐}\right)>1
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
\mathrm{๐ธ๐๐ด๐ผ}.\mathrm{๐๐๐๐๐}
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}.\mathrm{๐๐๐๐๐}
\mathrm{๐ธ๐๐ด๐ผ}.\mathrm{๐๐๐๐๐}
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}.\mathrm{๐๐๐๐๐}
\mathrm{๐ธ๐๐ด๐ผ}.\mathrm{๐๐๐๐๐}
\mathrm{๐ธ๐๐ด๐ผ}.\mathrm{๐๐๐๐๐ก}
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
Makes the link between the discrete decision variable
\mathrm{๐ธ๐ฝ๐ณ๐ด๐}
\mathrm{๐
๐ฐ๐ป๐๐ด}
according to a given table of values
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
. We now give five typical uses of the
\mathrm{๐๐๐๐}
In some problems we may have to represent a function
y=f\left(x\right)
x\in \left[1,m\right]
). In this context we generate the following
\mathrm{๐๐๐๐}
\mathrm{๐ธ๐ฝ๐ณ๐ด๐}
is a domain variable taking its values in
\left\{1,2,\cdots ,m\right\}
\mathrm{๐๐๐๐}\left(\begin{array}{c}โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-x\hfill & \mathrm{๐๐๐๐๐}-y\hfill \end{array}โช,\hfill \\ โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-1\hfill & \mathrm{๐๐๐๐๐}-f\left(1\right),\hfill \\ \mathrm{๐๐๐๐๐ก}-2\hfill & \mathrm{๐๐๐๐๐}-f\left(2\right),\hfill \\ & โฎ\hfill \\ \mathrm{๐๐๐๐๐ก}-m\hfill & \mathrm{๐๐๐๐๐}-f\left(m\right)\hfill \end{array}โช\hfill \end{array}\right)
Figure 5.137.1.
y={x}^{3}
1\le x\le 3
As an example, consider the problem of finding the smallest integer that can be decomposed in two different ways in the sum of two cubes [HardyWright75]. The
\mathrm{๐๐๐๐}
constraint can be used for representing the function
y={x}^{3}
(Figure 5.137.1). The unique solution
1729={12}^{3}+{1}^{3}={10}^{3}+{9}^{3}
can be obtained by the following set of constraints:
\left\{\begin{array}{c}\mathrm{๐๐๐๐}\left(โฉ\mathrm{๐๐๐๐๐ก}-{x}_{1}\mathrm{๐๐๐๐๐}-{y}_{1}โช,\hfill \\ โฉ\mathrm{๐๐๐๐๐ก}-1\mathrm{๐๐๐๐๐}-1,\mathrm{๐๐๐๐๐ก}-2\mathrm{๐๐๐๐๐}-8,\cdots ,\mathrm{๐๐๐๐๐ก}-20\mathrm{๐๐๐๐๐}-8000โช\right)\hfill \\ \mathrm{๐๐๐๐}\left(โฉ\mathrm{๐๐๐๐๐ก}-{x}_{2}\mathrm{๐๐๐๐๐}-{y}_{2}โช,\hfill \\ โฉ\mathrm{๐๐๐๐๐ก}-1\mathrm{๐๐๐๐๐}-1,\mathrm{๐๐๐๐๐ก}-2\mathrm{๐๐๐๐๐}-8,\cdots ,\mathrm{๐๐๐๐๐ก}-20\mathrm{๐๐๐๐๐}-8000โช\right)\hfill \\ \mathrm{๐๐๐๐}\left(โฉ\mathrm{๐๐๐๐๐ก}-{x}_{3}\mathrm{๐๐๐๐๐}-{y}_{3}โช,\hfill \\ โฉ\mathrm{๐๐๐๐๐ก}-1\mathrm{๐๐๐๐๐}-1,\mathrm{๐๐๐๐๐ก}-2\mathrm{๐๐๐๐๐}-8,\cdots ,\mathrm{๐๐๐๐๐ก}-20\mathrm{๐๐๐๐๐}-8000โช\right)\hfill \\ \mathrm{๐๐๐๐}\left(โฉ\mathrm{๐๐๐๐๐ก}-{x}_{4}\mathrm{๐๐๐๐๐}-{y}_{4}โช,\hfill \\ โฉ\mathrm{๐๐๐๐๐ก}-1\mathrm{๐๐๐๐๐}-1,\mathrm{๐๐๐๐๐ก}-2\mathrm{๐๐๐๐๐}-8,\cdots ,\mathrm{๐๐๐๐๐ก}-20\mathrm{๐๐๐๐๐}-8000โช\right)\hfill \\ {y}_{1}+{y}_{2}={y}_{3}+{y}_{4}\hfill \\ {x}_{1}<{x}_{2}\hfill \\ {x}_{3}<{x}_{4}\hfill \\ {x}_{1}<{x}_{3}\hfill \end{array}\right\
The last three inequalities constraints in the conjunction are used for breaking symmetries. The constraints
{x}_{1}<{x}_{2}
{x}_{3}<{x}_{4}
respectively order the pairs of variables
\left({x}_{1},{x}_{2}\right)
\left({x}_{3},{x}_{4}\right)
from which the sums
{x}_{1}^{3}+{x}_{2}^{3}
{x}_{3}^{3}+{x}_{4}^{3}
are generated. Finally the inequality
{x}_{1}<{x}_{3}
enforces a lexicographic ordering between the two pairs of variables
\left({x}_{1},{x}_{2}\right)
\left({x}_{3},{x}_{4}\right)
In some optimisation problems a classical use of the
\mathrm{๐๐๐๐}
constraint consists expressing the link between a discrete choice and its corresponding cost. For each discrete choice we create an
\mathrm{๐๐๐๐}
constraint of the form:
\mathrm{๐๐๐๐}\left(\begin{array}{c}โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-\mathrm{๐ฒ๐๐๐๐๐}\hfill & \mathrm{๐๐๐๐๐}-\mathrm{๐ฒ๐๐๐}\hfill \end{array}โช,\hfill \\ โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-1\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ฒ๐๐๐}}_{1},\hfill \\ \mathrm{๐๐๐๐๐ก}-2\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ฒ๐๐๐}}_{2},\hfill \\ & โฎ\hfill \\ \mathrm{๐๐๐๐๐ก}-m\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ฒ๐๐๐}}_{m}\hfill \end{array}โช\hfill \end{array}\right)
\mathrm{๐ฒ๐๐๐๐๐}
is a domain variable that indicates which alternative will be finally selected,
\mathrm{๐ฒ๐๐๐}
is a domain variable that corresponds to the cost of the decision associated with the value of the
\mathrm{๐ฒ๐๐๐๐๐}
variable,
{\mathrm{๐ฒ๐๐๐}}_{1},{\mathrm{๐ฒ๐๐๐}}_{2},\cdots ,{\mathrm{๐ฒ๐๐๐}}_{m}
are the respective costs associated with the alternatives
1,2,\cdots ,m
In some problems we need to express a disjunction of the form
\mathrm{๐
๐ฐ๐}={\mathrm{๐
๐ฐ๐}}_{1}\vee \mathrm{๐
๐ฐ๐}={\mathrm{๐
๐ฐ๐}}_{2}\vee \cdots \vee \mathrm{๐
๐ฐ๐}={\mathrm{๐
๐ฐ๐}}_{n}
. This can be directly reformulated as the following
\mathrm{๐๐๐๐}
\mathrm{๐ธ๐ฝ๐ณ๐ด๐}
is a domain variable taking its value in the finite set
\left\{1,2,\cdots ,n\right\}
and where the
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
argument corresponds to the domain variables
{\mathrm{๐
๐ฐ๐}}_{1},{\mathrm{๐
๐ฐ๐}}_{2},\cdots ,{\mathrm{๐
๐ฐ๐}}_{n}
\mathrm{๐๐๐๐}\left(\begin{array}{c}โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-\mathrm{๐ธ๐ฝ๐ณ๐ด๐}\hfill & \mathrm{๐๐๐๐๐}-\mathrm{๐
๐ฐ๐}\hfill \end{array}โช,\hfill \\ โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-1\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐
๐ฐ๐}}_{1},\hfill \\ \mathrm{๐๐๐๐๐ก}-2\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐
๐ฐ๐}}_{2},\hfill \\ & โฎ\hfill \\ \mathrm{๐๐๐๐๐ก}-n\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐
๐ฐ๐}}_{n}\hfill \end{array}โช\hfill \end{array}\right)
In some scheduling problems the duration of a task depends on the machine where the task will be assigned in final schedule. In this case we generate for each task an
\mathrm{๐๐๐๐}
constraint of the following form:
\mathrm{๐๐๐๐}\left(\begin{array}{c}โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-\mathrm{๐ผ๐๐๐๐๐๐}\hfill & \mathrm{๐๐๐๐๐}-\mathrm{๐ณ๐๐๐๐๐๐๐}\hfill \end{array}โช,\hfill \\ โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-1\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ณ๐๐}}_{1},\hfill \\ \mathrm{๐๐๐๐๐ก}-2\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ณ๐๐}}_{2},\hfill \\ & โฎ\hfill \\ \mathrm{๐๐๐๐๐ก}-m\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ณ๐๐}}_{m}\hfill \end{array}โช\hfill \end{array}\right)
\mathrm{๐ผ๐๐๐๐๐๐}
is a domain variable that indicates the resource to which the task will be assigned,
\mathrm{๐ณ๐๐๐๐๐๐๐}
is a domain variable that corresponds to the duration of the task,
{\mathrm{๐ณ๐๐}}_{1},{\mathrm{๐ณ๐๐}}_{2},\cdots ,{\mathrm{๐ณ๐๐}}_{m}
are the respective duration of the task according to the hypothesis that it runs on machine
1,2
m
Figure 5.137.2. A task
t
for which the duration depends on the machine 1, 2 or 3 to which it is assigned
Figure 5.137.2 illustrates this particular use of the
\mathrm{๐๐๐๐}
constraint for modelling that a task has a duration of 4, 6 and 4 when we respectively assign it on machines 1, 2 and 3.
In some vehicle routing problems we typically use the
\mathrm{๐๐๐๐}
constraint to express the distance between location
i
and the next location visited by a vehicle. For this purpose we generate for each location
i
\mathrm{๐๐๐๐}
\mathrm{๐๐๐๐}\left(\begin{array}{c}โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-{\mathrm{๐ฝ๐๐ก๐}}_{i}\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐๐๐๐๐๐๐๐}}_{i}\hfill \end{array}โช,\hfill \\ โฉ\begin{array}{cc}\mathrm{๐๐๐๐๐ก}-1\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ณ๐๐๐}}_{{i}_{1}},\hfill \\ \mathrm{๐๐๐๐๐ก}-2\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ณ๐๐๐}}_{{i}_{2}},\hfill \\ & โฎ\hfill \\ \mathrm{๐๐๐๐๐ก}-m\hfill & \mathrm{๐๐๐๐๐}-{\mathrm{๐ณ๐๐๐}}_{{i}_{m}}\hfill \end{array}โช\hfill \end{array}\right)
{\mathrm{๐ฝ๐๐ก๐}}_{i}
is a domain variable that gives the index of the location the vehicle will visit just after location
i
{\mathrm{๐๐๐๐๐๐๐๐}}_{i}
is a domain variable that corresponds to the distance between location
i
and the location the vehicle will visit just after,
{\mathrm{๐ณ๐๐๐}}_{{i}_{1}},{\mathrm{๐ณ๐๐๐}}_{{i}_{2}},\cdots ,{\mathrm{๐ณ๐๐๐}}_{{i}_{m}}
are the respective distances between location
i
and locations
1,2,\cdots ,m
An other example where the table argument corresponds to domain variables is described in the keyword entry assignment to the same set of values.
Originally, the parameters of the
\mathrm{๐๐๐๐}
constraint had the form
\mathrm{๐๐๐๐๐๐๐}\left(\mathrm{๐ธ๐ฝ๐ณ๐ด๐},\mathrm{๐๐ฐ๐ฑ๐ป๐ด},\mathrm{๐
๐ฐ๐ป๐๐ด}\right)
\mathrm{๐ธ๐ฝ๐ณ๐ด๐}
\mathrm{๐
๐ฐ๐ป๐๐ด}
were two domain variables and
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
was a list of non-negative integers.
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
p
\mathrm{๐๐๐๐}
I=J-p+1
\wedge
\mathrm{๐๐๐๐}\left(โฉ\mathrm{๐๐๐๐๐ก}-I\mathrm{๐๐๐๐๐}-Vโช,\mathrm{๐๐ฐ๐ฑ๐ป๐ด}\right)
I
J
|\mathrm{๐๐ฐ๐ฑ๐ป๐ด}|
p
p+|\mathrm{๐๐ฐ๐ฑ๐ป๐ด}|-1
nth in Choco, element in Gecode, element in JaCoP, element in SICStus.
\mathrm{๐๐๐๐}_\mathrm{๐๐๐๐}_\mathrm{๐๐}
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐ก}
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}
(single item replaced by two variables),
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐๐๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐}\left(\mathrm{๐ธ๐๐ด๐ผ},\mathrm{๐๐ฐ๐ฑ๐ป๐ด}\right)
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}.\mathrm{๐๐๐๐๐}\ge 0
\mathrm{๐๐๐}_\mathrm{๐๐๐๐๐๐๐}_\mathrm{๐๐๐๐}
\left(\mathrm{๐๐ฐ๐ฑ๐ป๐ด},\mathrm{๐ธ๐๐ด๐ผ}\right)
\mathrm{๐ธ๐๐ด๐ผ}
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
\mathrm{๐๐
๐๐ท๐๐ถ๐}
โฆ\mathrm{๐๐๐๐๐๐๐๐๐๐}\left(\mathrm{๐๐๐๐},\mathrm{๐๐๐๐๐}\right)
โข\mathrm{๐๐๐๐}.\mathrm{๐๐๐๐๐ก}=\mathrm{๐๐๐๐๐}.\mathrm{๐๐๐๐๐ก}
โข\mathrm{๐๐๐๐}.\mathrm{๐๐๐๐๐}=\mathrm{๐๐๐๐๐}.\mathrm{๐๐๐๐๐}
\mathrm{๐๐๐๐}
=1
We regroup the
\mathrm{๐ธ๐ฝ๐ณ๐ด๐}
\mathrm{๐
๐ฐ๐ป๐๐ด}
parameters of the original
\mathrm{๐๐๐๐๐๐๐}
\mathrm{๐๐๐๐๐๐๐}\left(\mathrm{๐ธ๐ฝ๐ณ๐ด๐},\mathrm{๐๐ฐ๐ฑ๐ป๐ด},\mathrm{๐
๐ฐ๐ป๐๐ด}\right)
into the parameter
\mathrm{๐ธ๐๐ด๐ผ}
. We also make explicit the different indices of the table
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
\mathrm{๐๐๐๐}
\mathrm{๐๐๐๐}
\mathrm{๐๐๐๐๐ก}
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
are distinct and because of the first condition of the arc constraint the final graph cannot have more than one arc. Therefore we can rewrite
\mathrm{๐๐๐๐}=1
\mathrm{๐๐๐๐}\ge 1
\underline{\overline{\mathrm{๐๐๐๐}}}
\overline{\mathrm{๐๐๐๐}}
\mathrm{๐๐๐๐}
\mathrm{๐ธ๐ฝ๐ณ๐ด๐}
\mathrm{๐
๐ฐ๐ป๐๐ด}
\mathrm{๐๐๐๐๐ก}
\mathrm{๐๐๐๐๐}
\mathrm{๐ธ๐๐ด๐ผ}
{\mathrm{๐ธ๐ฝ๐ณ๐ด๐}}_{i}
{\mathrm{๐
๐ฐ๐ป๐๐ด}}_{i}
\mathrm{๐๐๐๐๐ก}
\mathrm{๐๐๐๐๐}
attributes of item
i
\mathrm{๐๐ฐ๐ฑ๐ป๐ด}
\left(\mathrm{๐ธ๐ฝ๐ณ๐ด๐},\mathrm{๐
๐ฐ๐ป๐๐ด},{\mathrm{๐ธ๐ฝ๐ณ๐ด๐}}_{i},{\mathrm{๐
๐ฐ๐ป๐๐ด}}_{i}\right)
{S}_{i}
\left(\left(\mathrm{๐ธ๐ฝ๐ณ๐ด๐}={\mathrm{๐ธ๐ฝ๐ณ๐ด๐}}_{i}\right)\wedge \left(\mathrm{๐
๐ฐ๐ป๐๐ด}={\mathrm{๐
๐ฐ๐ป๐๐ด}}_{i}\right)\right)โ{S}_{i}
\mathrm{๐๐๐๐}
\left(\mathrm{๐ธ๐๐ด๐ผ},\mathrm{๐๐ฐ๐ฑ๐ป๐ด}\right)
constraint (once one finds the right item โ index and value โ in the table, one switches from the initial state
s
t
\mathrm{๐๐๐๐}
|
Find domain and range from graphs | College Algebra | Course Hero
We can observe that the graph extends horizontally from
-5
to the right without bound, so the domain is
\left[-5,\infty \right)
. The vertical extent of the graph is all range values
5
and below, so the range is
\left(\mathrm{-\infty },5\right]
. Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.
Example 6: Finding Domain and Range from a Graph
f
whose graph is shown in Figure 7.
We can observe that the horizontal extent of the graph is โ3 to 1, so the domain of
\left(-3,1\right]
The vertical extent of the graph is 0 to โ4, so the range is
\left[-4,0\right)
Example 7: Finding Domain and Range from a Graph of Oil Production
f
Figure 9. (credit: modification of work by the U.S. Energy Information Administration)
The input quantity along the horizontal axis is "years," which we represent with the variable
t
for time. The output quantity is "thousands of barrels of oil per day," which we represent with the variable
b
for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as
1973\le t\le 2008
and the range as approximately
180\le b\le 2010
Given the graph in Figure 10, identify the domain and range using interval notation.
Precalculus. Authored by: Jay Abramson, et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/[email protected]. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]
Ex 1: Determine the Domain and Range of the Graph of a Function. Authored by: Mathispower4u. Located at: https://www.youtube.com/watch?v=QAxZEelInJc. License: All Rights Reserved. License terms: Standard YouTube LIcense
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Finding the intercepts, asymptotes, domain, and range from the graph of a rational function.pdf
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Domain and range from the graph of a continuous function.pdf
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|
The arc length formula uses the language of calculus to generalize and solve a classical problem in geometry: finding the length of any specific curve. Given a function
f
that is defined and differentiable on the interval
[a, \, b]
L
y = f(x)
in that interval is
L = \int_{a}^{b} \sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx.
The crux of the issue lies in the fact that a line segment of
y = mx + b
x
x + w
\sqrt{w^2 + (wm)^2} = w \sqrt{1 + m^2}
The following sections aim to suggest that
(1)
it is possible to approximate a function with piecewise-linear functions, and
(2)
no matter how a function is modeled by piecewise-linear functions, the arc length remains the same.
(1)
The first approach is to choose many points distributed across the function and draw tangent lines to the function at those points. By choosing enough points, it is always possible to make subsequent tangent lines intersect (i.e., not be parallel). The line segments between tangent line intersections then provides a rough estimate of the function itself and therefore of the function's length (and one that can be easily calculated!).
If the tangent line segments have slopes
m_1, \, m_2, \, \dots, \, m_n
and widths
w_1, \, w_2, \, \dots, \, w_n
, then the length of the tangent line construction is
\sum_{i=1}^n \sqrt{1 + (m_i)^2} \cdot w_i.
Recalling each
m_i
\tfrac{dy}{dx}
for the function and each
w_i
x
's, this formula looks remarkably similar to the above formula (just with a summation instead of an integral). In fact, as
n
grows very large, they become equal.
Intuitively, this makes sense. Tangent lines serve to model a function well near their point of tangency, so their length should model a function's arc length well too. The main difficulty in formalizing this method is that, in order to stay "near" the function, subsequent tangent lines should intersect in between their points of tangency (this also allows for a more natural translation between
w_i
x_i
's). This issue is resolved by choosing more points along the curve wherever the tangent lines are not "near" the function.
(2)
The second approach is to choose points regularly distributed across the function and connect them with line segments. If the interval has length
W = b - a
n+1
points can be chosen according to the rule
x_k = a + k \cdot \left( \frac{b-a}{n} \right)
y_k = f(x_k)
. Then, the line connecting
(x_k, \, y_k)
(x_{k+1}, \, y_{k+1})
is both an approximation of the function itself and, as
n
grows large, an approximation of a tangent line to the function at
x = x_k
k^\text{th}
line segment has length
\sqrt{1 + \left( \frac{y_{k+1} - y_k}{x_{k+1} - x_k} \right)^2} \cdot (x_{k+1} - x_k)
. Summing over all segments,
\sum_{k = 1}^n \sqrt{1 + \left( \frac{y_{k+1} - y_k}{x_{k+1} - x_k} \right)^2} \cdot (x_{k+1} - x_k),
yields an expression that looks very much like a Riemann sum (another way of expressing an integral). Recalling that
\frac{y_{k+1} - y_k}{x_{k+1} - x_k}
is the approximation of the function's derivative (the slope of its tangent line), the result looks remarkably like the above formula. As
n
grows large, the values are, in fact, equal.
Both these methods
(1)
(2)
take a function and attempt to decompose it into a connected series of line segments (a polygonal path). This presupposes that the length of a curve is equal to the limit of a sequence of lengths of polygonal paths. Happily, this is the leading definition of curve length, and curves whose lengths are not measurable by such a method are known as unrectifiable.
Find the length of the curve
\displaystyle y=1+6x^{\frac{3}{2}},
0 \leq x \leq 1.
\begin{array}{c}\displaystyle{\frac{dy}{dx}}=9x^{\frac{1}{2}} & \Rightarrow & 1+\left(\frac{dy}{dx}\right)^2=1+81x, \end{array}
and so the arc length formular gives
L=\int_{0}^{1} \sqrt{1+81x}\, dx.
u=1+81x
du=81\,dx
, note that when
x=0,
u=1;
x=1,
u=82.
L=\int_{1}^{82} u^{\frac{1}{2}} \left(\frac{1}{81}\, du\right)=\frac{2}{243}\left[u^{\frac{3}{2}}\right]_{1}^{82}=\frac{2}{243}(82\sqrt{82}-1). \ _\square
L
\displaystyle{y = \frac{1}{4}{x}^2 - \frac{1}{2} \ln x }
4 \leq x \leq 8.
L = a+\frac{1}{2}\ln{b} ,
and
b
a+b?
4
5
6
7
8
f(n)
y=\ln\left(x\right)
x\in\left [1,n\right ]
\lim_{n\to\infty}\big(n-f(n)\big).
\ln\left(\sqrt{a}-b\right)+\sqrt{c}-d
a
b
c
, and
and
c
a+b+c+d
Suppose that a curve
C
y=f(x),
f
a \leq x \leq b,
and divide the interval
[a,b]
n
subintervals with endpoints
x_0, x_1, \ldots , x_n
and equal width
\Delta x.
y_i=f(x_i),
P_i=(x_i,y_i)
C
and the line segments connecting adjacent two points of
P_0,P_1, \ldots, P_n,
shown above, are an approximation to
C.
L
C
is approximately the sum of the lengths of these line segments. Therefore, it is possible to define the length
L
C
y=f(x),
a \leq x \leq b,
as the limit of the sum of the lengths of these line segments:
L=\lim_{n \to \infty} \sum_{i=1}^{n} \lvert{P_{i-1}P_i}\rvert, \qquad (1)
\lvert{P_{i-1}P_i}\rvert
denotes the distance between the two points
P_{i-1}
P_i.
\Delta y_i=y_i-y_{i-1},
\lvert{P_{i-1}P_i}\rvert=\sqrt{(x_i-x_{i-1})^2+(y_i-y_{i-1})^2}=\sqrt{(\Delta x)^2+(\Delta y)^2} .
By applying the mean value theorem to
f
[x_{i-1},x_i],
note that there is a number
{x_i}^{*}
x_{i-1}
x_i
f(x_i)-f(x_{i-1})=f'({x_i}^{*})(x_i-x_{i-1}),
\Delta y_i=f'({x_i}^{*}) \Delta x.
\begin{aligned} \lvert{P_{i-1}P_i}\rvert &= \sqrt{(\Delta x)^2+(\Delta y)^2} \\ &= \sqrt{(\Delta x)^2+(f'({x_i}^{*}) \Delta x)^2} \\ &= \sqrt{1+(f'({x_i}^{*}))^2} ~\Delta x. \end{aligned}
Therefore, by definition
(1),
L=\lim_{n \to \infty} \sum_{i=1}^{n} \lvert{P_{i-1}P_i}\rvert = L=\lim_{n \to \infty} \sum_{i=1}^{n} \sqrt{1+(f'({x_i}^{*}))^2} ~ \Delta x.
\int_{a}^{b} \sqrt{1+(f'(x))^2}\, dx.
L=\int_{a}^{b} \sqrt{1+\left(\frac{dy}{dx}\right)^2}\, dx.
y=\frac{a}{2}\left(e^{\frac{x}{a}}+e^{\frac{-x}{a}}\right)
x=0
x=a
y=a\cosh\left(\dfrac{x}{a}\right) \Rightarrow \dfrac{dy}{dx} = \sinh\left(\dfrac{x}{a}\right)
(using chain rule) . Now,
\sqrt{1+\left(\dfrac{dy}{dx}\right)^2} = \sqrt{1+\sinh^2\left(\dfrac{x}{a}\right)} = \left| \cosh\left(\dfrac{x}{a}\right) \right|.
Now length of the arc from
x=0
x=a
is given by the integral:
\displaystyle\int_{0}^{a} \cosh\left(\dfrac{x}{a}\right) \ dx.
y=\dfrac{x}{a} \Rightarrow dx=a \ dy
and changing limits, the integral becomes
a\big[\sinh (y)\big]_{0}^{1} = a\sinh(1) = \dfrac{a\big(e^2-1\big)}{2e}.\ _\square
{f}'(x) = \sqrt{ {x}^4-1 },
y = f(x)
x = 2
x = 8?
The arc length
L
of the graph of a parametric function
(x,y) = \left(x(t),y(t)\right)
t=a
t=b
L = \int_a^b \sqrt{\left( \frac{dx}{dt} \right)^2+\left( \frac{dy}{dt} \right )^2 }\, dt.
This follows readily from the earlier discussion by making the substitution
dx = \frac{dx}{dt} \, dt
. Formally, the same method of proof applies.
L
of the graph of a polar function
r(\theta)
t = a
t = b
L = \int_a^b \sqrt{r^2 + \left( \frac{dr}{d\theta} \right)^2} \, d\theta.
This formula follows immediately from the parametric form, upon noting that the parameter is
\theta
x'(\theta) = \frac{d}{d\theta} \left( r(\theta) \cos \theta \right) = r'(\theta) \cos \theta - r(\theta) \sin \theta, \\ y'(\theta) = \frac{d}{d\theta} \left( r(\theta) \sin \theta \right) = r'(\theta) \sin \theta + r(\theta) \cos \theta.
|
Model the dynamics of simplified three-phase synchronous machine - Simulink
\begin{array}{c}\Delta \omega \left(t\right)=\frac{1}{2H}\underset{0}{\overset{t}{\int }}\left(Tm-Te\right)\text{\hspace{0.17em}}-Kd\Delta \omega \left(t\right)dt\\ \omega \left(t\right)=\Delta \omega \left(t\right)+{\omega }_{0},\end{array}
\frac{\delta }{{P}_{m}}=\frac{{\omega }_{s}/\left(2H\right)}{{s}^{2}+2\zeta {\omega }_{n}s+{\omega }_{n}^{2}},
\sqrt{{\omega }_{s}{P}_{\text{max}}/\left(2H\right)}
\left({K}_{d}/4\right)\sqrt{2/\left({\omega }_{s}H{P}_{\text{max}}\right)}
{K}_{d}=4\zeta \sqrt{{\omega }_{s}H{P}_{\text{max}}/2}.
{K}_{d}=4\zeta \sqrt{{\omega }_{s}H{P}_{\text{max}}/2}=64.3
Pe=\frac{{V}_{t}E\mathrm{sin}\delta }{X}=\frac{1.0\cdot 1.0149\cdot \mathrm{sin}\left({5.65}^{\circ }\right)}{0.2}=0.5\text{ p}\text{.u}\text{.}
{f}_{n}=\frac{1}{2\pi }\sqrt{\frac{{\omega }_{s}{P}_{\text{max}}}{2H}}=2.84\text{ Hz}\text{.}
|
Coordinate Bonds | Brilliant Math & Science Wiki
Jordan Calmes, Aditya Virani, Sรฉbastien Bernt, and
A coordinate bond is a type of covalent bond where both of the electrons that form the bond originate from the same atom (more generally, a "dative" covalent bond).
Coordinate bonds form between a central electrophile (low electron density, such as metal cations) and one or more nucleophiles ( high electron density, such as the hydroxide anion) oriented around the former. Nucleophiles acts as "ligands" by supplying two electrons per coordinate bond to the electrophile to satisfy the octet rule. A "polydentate" (from Latin "dฤns": tooth) ligand molecule may form multiple coordinate bonds.
As with any covalent bond, the atoms in a coordinate bond redistribute electrons in order to satisfy the octet rule, which states that each atom will lose, gain, or share electrons in order to have a full valence of eight electrons in its outer shell. Hydrogen is a common exception to the octet rule, and instead follows the duet rule because it only has one 1s orbital and its outer valence is full when it has two electrons.
It is important to note that coordinate bonding between oppositely charged ionic species is of a covalent character, where the bonding electrons are shared, as opposed to ionic compounds.
In the reaction below, ammonia and hydrochloric acid react to form an ammonium ion and a chloride ion. Chloride is more electronegative than hydrogen, so it is able to strip the hydrogen atom of its single electron. The hydrogen ion shifts to the ammonia molecule. As a result, the hydrogen ion carries no electrons. As the Lewis structure shows, nitrogen has two unpaired electrons (also called a lone pair, or nonbonding electrons) that attract the positively charged hydrogen ion. Imgur
At this point, nitrogen has an octet. However, the lone pair creates a polarity with a partial negative charge on nitrogen. The molecular geometry of ammonia is affected as well. Rather than having the hydrogens arranged in a single plane at 120-degree angles, the hydrogens are squeezed closer to each other to form the base of a pyramid, with the negatively-charged nitrogen atom forming the apex.
Meanwhile, hydrogen has no electrons at all, leaving it with a net positive charge that is attracted to the partial negative charge of nitrogen.
The orbital for that lone pair of electrons now includes the hydrogen atom, holding the two species together in a single, positively charged ammonium ion. The nucleus of the hydrogen atom is participating in the bond, but hydrogen did not donate any electrons to the process. All octets and duets are satisfied, and the ammonium has four evenly-spaced hydrogen atoms forming a tetrahedral structure around the nitrogen atom.
Is the coordinate bond a different length or strength than the other covalent bonds in ammonium?
No. Though the coordinate bond is often drawn differently on paper, in a molecule, it is indistinguishable from the other N-H bonds in terms of both bond length and strength.
C-O C-H O-Cl H-Cl
In a reaction between chloromethane
\ce{CH_3Cl}
and hydroxide
\ce{OH^-},
where would you expect a new coordinate bond to form?
In this second example, ammonia reacts with boron trifluoride. Before the reaction, nitrogen has eight valence electrons, including a lone pair of electrons. Boron has only six valence electrons, so it is two electrons short of an octet. The two unpaired electrons form a bond between nitrogen and boron, resulting in complete octets for both atoms. A coordinate bond is sometimes represented by an arrow, as shown in the figure below. The direction of the arrow indicates that the electrons are moving from nitrogen to boron. Imgur
There is another little concept called solubility with vast applications hidden in the above example. Notice that ammonia is a polar molecule due to the presence of a lone pair of electrons, and boron trifluoride a non polar compound. Conventionally like dissolves like, concluding that
NH_3
BF_3
are insoluble in each other.
But, as seen in the figure above, there is an acid-base reaction which happens.
NH_3
is a Lewis base and
BF_3
is a Lewis acid. So
NH_3
forms a coordinate bond with
BF_3
. Therefore, they are considered as soluble in each other because of the reaction between them.
Understanding how coordinate bonds form can be instrumental in designing complex organic molecules. Rotaxanes, shown in the righthand column, are one of many mechanically interlocked molecular shapes. [1] Rotaxanes are an example of a mechanically interlocked molecule, a complex organic molecule that is physically joined. These molecules are interesting to synthetic chemists as "potential building blocks for future molecular-scale devicesโmotors, sensors, and machines on the nanometer scale" [1].
The image below shows an organic molecule that can self-assemble to form a rotaxane because the molecule was synthesized with coordinate bonds in targeted places. Those bonds are then broken and re-formed to give the molecule its desired shape [2]. Precursor molecules to a rotaxane shape [2] Schematic showing how coordinate bond dissolution allows the rotaxane to self-assemble [2]
[1] Lawrence Livermore National Lab. "Machines from Interlocking Molecules." Science and Technology Review, December 2002. Accessed from https://str.llnl.gov/str/December02/Vance.html on March 1, 2016.
[2] Ballester, Pablo et al. "Self-Assembly Of Rotaxane Exploiting Reversible Pt(II)- Pyridine Coordinate Bonds" Molecules 9.5 (2004): 278-286.
Cite as: Coordinate Bonds. Brilliant.org. Retrieved from https://brilliant.org/wiki/coordinate-bonds/
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Convert prices to returns - MATLAB price2ret - MathWorks United Kingdom
price2ret
Compute Return Series from Price Series in Vector of Data
Compute Simple Periodic Return Series from Table of Price Series
Specify Observation Times and Units
price2ret supports name-value argument syntax for all optional inputs
[Returns,intervals] = price2ret(Prices)
ReturnTbl = price2ret(PriceTbl)
[___] = price2ret(___,Name=Value)
[Returns,intervals] = price2ret(Prices) returns the matrix of numVars continuously compounded return series Returns, and corresponding time intervals intervals, from the matrix of numVars price series Prices.
ReturnTbl = price2ret(PriceTbl) returns the table or timetable of continuously compounded return series ReturnTbl of each variable in the table or timetable of price series PriceTbl. To select different variables in Tbl from which to compute returns, use the DataVariables name-value argument.
[___] = price2ret(___,Name=Value) uses additional options specified by one or more name-value arguments, using any input-argument combination in the previous syntaxes. price2ret returns the output-argument combination for the corresponding input arguments. For example, price2ret(Tbl,Method="periodic",DataVariables=1:5) computes the simple periodic returns of the first five variables in the input table Tbl.
Load the Schwert Stock data st Data_SchwertStock.mat, which contains daily prices of the S&P index from 1930 through 2008, among other variables (enter Description for more details).
numObs = height(DataTableDly)
numObs = 20838
dates = datetime(datesDly,ConvertFrom="datenum");
Convert the S&P price series to returns.
prices = DataTableDly.SP;
returns = price2ret(prices);
returns is a 20837-by-1 vector of daily S&P returns compounded continuously.
r9 = returns(9)
returns(9) = 0.0033 is the daily return of the prices in the interval [21.45, 21.52].
plot(dates,DataTableDly.SP)
plot(dates(1:end-1),returns)
title("S&P Index Prices and Returns")
Convert the price series in a table to simple periodic return series.
numObs = height(TT);
Convert the NASDAQ and NYSE prices to simple periodic and continuously compounded returns.
varnames = ["NASDAQ" "NYSE"];
TTRetC = price2ret(TT,DataVariables=varnames);
TTRetP = price2ret(TT,DataVariables=varnames,Method="periodic");
Because TT is a timetable, TTRetC and TTRetP are timetables.
Plot the return series with the corresponding prices for the last 50 observations.
idx = ((numObs - 1) - 51):(numObs - 1);
plot(dates(idx + 1),TT.NYSE(idx + 1))
h = plot(dates(idx),[TTRetC.NYSE(idx) TTRetP.NYSE(idx)]);
h(2).Color = 'k';
legend(["Price" "Continuous" "Periodic"],Location="northwest")
plot(dates(idx + 1),TT.NASDAQ(idx + 1))
title("NASDAQ Index Prices and Returns")
h = plot(dates(idx),[TTRetC.NASDAQ(idx) TTRetP.NASDAQ(idx)]);
In this case, the simple periodic and continuously compounded returns of each price series are similar.
price2ret returns rates matching the rates from the simulated series. price2ret assumes prices are recorded in a regular time base. Therefore, all durations between prices are 1.
Prices โ Time series of prices
Time series of prices, specified as a numObs-by-numVars numeric matrix. Each row of Prices corresponds to an observation time specified by the optional Ticks name-value argument. Each column of Prices corresponds to an individual price series.
PriceTbl โ Time series of prices
Time series of prices, specified as a table or timetable with numObs rows. Each row of Tbl is an observation time. For a table, the optional Ticks name-value argument specifies observation times. For a timetable, PriceTbl.Time specifies observation times and it must be a datetime vector.
Specify numVars variables, from which to compute returns, by using the DataVariables argument. The selected variables must be numeric.
Example: price2ret(Tbl,Method="periodic",DataVariables=1:5) computes the simple periodic returns of the first five variables in the input table Tbl.
numeric vector | datetime vector
Observation times ฯ, specified as a length numObs numeric or datetime vector of increasing values.
When the input price series are in a matrix or table, the default is 1:numObs.
When the input price series are in a timetable, price2ret uses the row times in PriceTbl.Time and ignores Ticks. PriceTbl.Time must be a datetime vector.
Example: Ticks=datetime(1950:2020,12,31) specifies the end of each year from 1950 through 2020.
Example: Ticks=datetime(1950,03,31):calquarters(1):datetime(2020,12,31) specifies the end of each quarter during the years 1950 through 2020.
Time units to use when observation times Ticks are datetimes, specified as a value in this table.
price2ret requires time units to convert duration intervals to numeric values for normalizing returns.
When the value of the Ticks name-value argument is a numeric vector, price2ret ignores the value of Units.
"continuous" Compute continuously compounded returns
"periodic" Compute simple periodic returns
DataVariables โ Variables in PriceTbl
Variables in PriceTbl, from which price2ret computes returns, specified as a string vector or cell vector of character vectors containing variable names in PriceTbl.Properties.VariableNames, or an integer or logical vector representing the indices of names. The selected variables must be numeric.
Returns โ Return series
Return series, returned as a (numObs โ 1)-by-numVars numeric matrix. price2ret returns Returns when you supply the input Prices.
Returns in row i ri are associated with price interval [pi,pi+1], i = 1:(numObs - 1), according to the compounding method Method:
{r}_{i}=\frac{\mathrm{log}\left({p}_{i+1}/{p}_{i}\right)}{{\tau }_{i+1}-{\tau }_{i}}.
{r}_{i}=\frac{\left({p}_{i+1}/{p}_{i}\right)-1}{{\tau }_{i+1}-{\tau }_{i}}.
When observation times ฯ (see Ticks) are datetimes, the magnitude of the normalizing interval ฯi+1 โ ฯi depends on the specified time units (see Units).
intervals โ Time intervals between observations
Time intervals between observations ฯi+1 โ ฯi, returned as a length numObs โ 1 numeric vector. price2ret returns intervals when you supply the input Prices.
When observation times (see Ticks) are datetimes, interval magnitudes depend on the specified time units (see Units).
ReturnTbl โ Return series and time intervals
Return series and time intervals, returned as a table or timetable, the same data type as PriceTbl, with numObs โ 1 rows. price2ret returns ReturnTbl when you supply the input PriceTbl.
ReturnTbl contains the outputs Returns and intervals.
ReturnTbl associates observation time ฯi+1 with the end of the interval for the returns in row i ri.
p is a price series (Prices).
ฯ is a vector of the observation times (Ticks).
R2022a: price2ret supports name-value argument syntax for all optional inputs
price2ret accepts the observation times ticktimes and compounding method method as the name-value arguments Ticks and Method, respectively. However, the function will continue to accept the previous syntax.
price2ret(Prices,ticktimes,method)
price2ret(Prices,Ticks=ticktimes,Method=method)
ret2price | tick2ret
|
The relative strength of two systems of formal logic can be defined via model theory. Specifically, a logic
{\displaystyle \alpha }
is said to be as strong as a logic
{\displaystyle \beta }
if every elementary class in
{\displaystyle \beta }
is an elementary class in
{\displaystyle \alpha }
^ Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors, Model-theoretic logics, 1985 ISBN 0-387-90936-2 page 43
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Exact Solutions and Conservation Laws of the Drinfelโd-Sokolov-Wilson System
Catherine Matjila, Ben Muatjetjeja, Chaudry Masood Khalique, "Exact Solutions and Conservation Laws of the Drinfelโd-Sokolov-Wilson System", Abstract and Applied Analysis, vol. 2014, Article ID 271960, 6 pages, 2014. https://doi.org/10.1155/2014/271960
Catherine Matjila,1 Ben Muatjetjeja ,1 and Chaudry Masood Khalique 1
We study the Drinfel'd-Sokolov-Wilson system, which was introduced as a model of water waves. Firstly we obtain exact solutions of this system using the -expansion method. In addition to exact solutions we also construct conservation laws for the underlying system using Noether's approach.
The classical Drinfelโd-Sokolov-Wilson (DSW) system given by where , , , and are nonzero constants, has been studied by [1]. The authors obtained various types of explicit solutions for (1) by using the bifurcation method and qualitative theory of dynamical systems. Also, Yao and Li [2] and C. Liu and X. Liu [3] obtained some exact solutions for the DSW system (1) by using a direct algebra method. A special case of the classical DSW system, namely, was studied by several authors [4โ8]. Hirota et al. [4] investigated the soliton structure of (2) and by employing an algebraic method, Fan [5] constructed some exact solutions. By using the improved generalized Jacobi elliptic function method, Yao [6] obtained some traveling wave solutions of (2), whereas by applying the Adomian decomposition method, Inc [7] obtained approximate doubly periodic wave solutions of (2). Zhao and Zhi [8] constructed exact doubly periodic solutions of (2) by using an improved -expansion method.
In this paper, we study the Drinfelโd-Sokolov-Wilson (DSW) system given bywhich can be derived from (1) by taking and . We obtain exact solutions and construct conservation laws of the DSW systems (3a) and (3b).
Nonlinear partial differential equations (PDEs) model diverse nonlinear phenomena in natural and applied sciences such as mechanics, fluid dynamics, biology, plasma physics, and mathematical finance. Therefore finding exact solutions of nonlinear PDEs is very important. Unfortunately, this is a very difficult task and there are no systematic methods that can be used to find exact solutions of the nonlinear PDEs. However, in the past few decades a number of new methods have been developed to obtain exact solutions to nonlinear PDEs. Some of these methods include the exp-function method, the homogeneous balance method, the sine-cosine method, the hyperbolic tangent function expansion method, and the -expansion function method [9โ15].
We recall that conservation laws are mathematical expressions of the physical laws, such as conservation of energy, mass, and momentum. They are of great significance in the solution process and reduction of PDEs. In the literature, one finds that the conservation laws have been widely used in studying the existence, uniqueness, and stability of solutions of nonlinear partial differential equations [16โ18], as well as in the development and use of numerical methods [19, 20]. Also, conserved vectors associated with Lie point symmetries have been employed to find exact solutions of partial differential equations [21โ23]. There are various methods of constructing conservation laws. One of the methods for variational problems is by means of Noetherโs theorem [24]. In order to apply Noetherโs theorem, the knowledge of a suitable Lagrangian is necessary. For nonlinear differential equations that do not have a Lagrangian, several methods have been developed (see, e.g., [25โ30]).
This paper is structured as follows. In Section 2, exact solutions of (3a) and (3b) are obtained using the -expansion function method. In Section 3, we construct Noetherโs symmetries and the conserved vectors for the DSW system (3a) and (3b). Concluding remarks are presented in Section 4.
2. Exact Solutions of the DSW System (3a) and (3b)
In this section, we obtain exact solutions of the DSW system (3a) and (3b). We first transform the system (3a) and (3b) into a system of ordinary differential equations by using the substitutions where . Substituting (4) into the system (3a) and (3b) and integrating with respect to , we obtain the following ordinary differential equations (ODEs): where integration constants are taken to be zero. From the first equation, we obtain . Substituting this value of in the second equation of the system, we obtain Now multiplying the above equation by and integrating while taking the constant of integration to be zero, we arrive at a first-order variables separable equation. Integrating this equation and reverting back to our original variables, we obtain where is an arbitrary constant of integration. Since , we have Thus, we have obtained one exact solution of the DSW system (3a) and (3b).
To obtain more exact solutions of the DSW system (3a) and (3b), we employ the -expansion function method [15]. We assume that the solutions of the ODE (6) can be expressed as a polynomial in by where is the balancing number to be determined and the function satisfies the second-order linear ODE given by with and being arbitrary constants. In our case, the balancing procedure gives . Thus Substituting (11) into (6) and making use of (10) and then equating all the terms with the same powers of to zero yield the following system of algebraic equations: Solving the above equations, with the aid of Mathematica, we obtain Consequently, we obtain the following two types of travelling wave solutions of the DSW system.
For , we obtain the hyperbolic functions travelling wave solutions For , we obtain the trigonometric function travelling solutions
3. Conservation Laws of the DSW Equations (3a) and (3b)
In this section, we construct the conservation laws of the DSW system (3a) and (3b). Since the third-order DSW system (3a) and (3b) does not have a Lagrangian, we cannot apply the Noether theorem. However, if we transform the third-order DSW system (3a) and (3b) to a fourth-order with the aid of the transformation , [31], we obtainThis system has a Lagrangian given by and it satisfies the Euler-Lagrange equations where and are defined by respectively.
Let us now consider the vector field
The second prolongation operator, of , is given by where We recall that , given by (20), is a Noether symmetry of (16a) and (16b), if it satisfies where and are the gauge functions. Expanding the above equation gives This leads to an overdetermined system of PDEs for the functions , and . Solving the system of PDEs gives We may choose and as they contribute to the trivial part of the conserved vector. The conserved vector for the second-order Lagrangian is given by [24, 31] where and are the characteristic functions. Now using (26) in conjunction with (25), , and , we obtain the following independent conserved vectors for system (3a) and (3b): We note that (27) is a nonlocal conserved vector, whereas (28) is a local conserved vectors. Also, for the arbitrary functions and , we obtain the following conserved vectors: which gives us infinitely many nonlocal conservation laws.
The third-order DSW system (3a) and (3b) was studied. Exact solutions of the DSW system were obtained using direct integration and the -expansion function method. The solutions obtained were hyperbolic and trigonometric solutions. In addition conservation laws were also derived. This system does not have a Lagrangian. In order to invoke Noetherโs theorem we used the transformations and to convert the DSW system to a fourth-order system, which has a Lagrangian. The conservation laws were then obtained and consisted of a local and infinite number of nonlocal conserved vectors.
All authors would like to thank NRF and North-West University, Mafikeng Campus, for financial support.
Z. Wen, Z. Liu, and M. Song, โNew exact solutions for the classical Drinfel'd-Sokolov-Wilson equation,โ Applied Mathematics and Computation, vol. 215, no. 6, pp. 2349โ2358, 2009. View at: Publisher Site | Google Scholar | MathSciNet
C. Liu and X. Liu, โExact solutions of the classical Drinfel'd-Sokolov-Wilson equations and the relations among the solutions,โ Physics Letters A, vol. 303, no. 2-3, pp. 197โ203, 2002. View at: Publisher Site | Google Scholar | MathSciNet
R. Hirota, B. Grammaticos, and A. Ramani, โSoliton structure of the Drinfel'd-Sokolov-Wilson equation,โ Journal of Mathematical Physics, vol. 27, no. 6, pp. 1499โ1505, 1986. View at: Publisher Site | Google Scholar | MathSciNet
E. Fan, โAn algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations,โ Journal of Physics A: Mathematical and General, vol. 36, no. 25, pp. 7009โ7026, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Y. Yao, โAbundant families of new traveling wave solutions for the coupled Drinfel'd-Sokolov-Wilson equation,โ Chaos, Solitons & Fractals, vol. 24, no. 1, pp. 301โ307, 2005. View at: Publisher Site | Google Scholar
M. Inc, โOn numerical doubly periodic wave solutions of the coupled Drinfel'd-Sokolov-Wilson equation by the decomposition method,โ Applied Mathematics and Computation, vol. 172, no. 1, pp. 421โ430, 2006. View at: Publisher Site | Google Scholar | MathSciNet
X.-Q. Zhao and H.-Y. Zhi, โAn improved
F
-expansion method and its application to coupled Drinfel'd-Sokolov-Wilson equation,โ Communications in Theoretical Physics, vol. 50, no. 2, pp. 309โ314, 2008. View at: Publisher Site | Google Scholar | MathSciNet
J.-H. He and X.-H. Wu, โExp-function method for nonlinear wave equations,โ Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 700โ708, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Z. Y. Yan and H. Q. Zhang, โOn a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics,โ Applied Mathematics and Mechanics, vol. 21, no. 4, pp. 383โ388, 2000. View at: Publisher Site | Google Scholar | MathSciNet
A. M. Wazwaz, โCompactons and solitary patterns structures for variants of the KdV and the KP equations,โ Applied Mathematics and Computation, vol. 139, no. 1, pp. 37โ54, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. M. Wazwaz, โThe tanh method for compact and noncompact solutions for variants of the KdV-Burger and the
K\left(n,n\right)
-Burger equations,โ Physica D, vol. 213, no. 2, pp. 147โ151, 2006. View at: Publisher Site | Google Scholar | MathSciNet
\left({G}^{\text{'}}/G\right)
T. B. Benjamin, โThe stability of solitary waves,โ Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 328, pp. 153โ183, 1972. View at: Publisher Site | Google Scholar | MathSciNet
E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, Berlin, Germany, 1996. View at: MathSciNet
\left(2+1\right)
E. Noether, โInvariante variationsprobleme,โ Nachrichten von der Kรถniglichen Gesellschaft der Wissenschaften zu G รถttingen, Mathematisch-Physikalische Klasse, Heft, vol. 2, pp. 235โ257, 1918. View at: Google Scholar
P. S. Laplace, Traite de Mecanique Celeste, vol. 1, Paris, France, 1978, (English translation Celestial Mechanics, New York, NY, USA, 1966).
H. Steudel, โรber die Zuordnung zwischen Invarianzeigenschaften und Erhaltungssรคtzen,โ Zeitschrift fรผr Naturforschung, vol. 17a, pp. 129โ132, 1962. View at: Google Scholar | MathSciNet
S. C. Anco and G. Bluman, โDirect construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications,โ European Journal of Applied Mathematics, vol. 13, no. 5, pp. 545โ566, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
A. H. Kara and F. M. Mahomed, โNoether-type symmetries and conservation laws via partial Lagrangians,โ Nonlinear Dynamics, vol. 45, no. 3-4, pp. 367โ383, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
R. Naz, F. M. Mahomed, and T. Hayat, โConservation laws for third-order variant Boussinesq system,โ Applied Mathematics Letters, vol. 23, no. 8, pp. 883โ886, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
Copyright ยฉ 2014 Catherine Matjila et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Cross Product - Properties Practice Problems Online | Brilliant
Consider a vector
\vec{v}
\sqrt{26}.
If this vector is perpendicular to both
\vec{a}=(2,3,1)
\vec{b}=(1,3,5),
what is the square of its
x
Given two vectors such that
\lvert\vec{a}\rvert=3,\lvert\vec{b}\rvert=4,
\vec{a}\cdot\vec{b}=8,
\vec{b}\times\vec{a}?
6\sqrt{2}
5\sqrt{3}
4\sqrt{5}
9
Torque is vector that measures the tendency of a force to rotate an object about an axis. It is given by the formula
\vec{\tau}=\vec{r}\times\vec{F},
\vec{r}
is the displacement vector (pointing from the axis to the point at which the force is applied) and
\vec{F}
is the force vector. If a disc rotates counter-clockwise on a record player, as shown in the figure above, what is the direction of the torque?
It will point into the plane. It will point out of the plane. Its direction keeps changing. Its direction cannot be determined using the given information.
Find the volume of the parallelepiped spanned by the vectors
\vec{a}=(1,1,2), \vec{b}=(2,1,3),
\vec{c}=(3,4,1).
What is the volume of the tetrahedron whose vertices are
O=(0,0,0), A=(4,0,1), B=(-2,-4,3),
C=(3,-1,-1)?
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