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August  2014, 19(6): 1627-1665. doi: 10.3934/dcdsb.2014.19.1627

## The linear hyperbolic initial and boundary value problems in a domain with corners

 1 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Rawles Hall, Bloomington, Indiana 47405 2 The Institute for Scientific Computing and Applied Mathematics, Indiana University, 831 East Third Street, Bloomington, Indiana 47405

Received  October 2013 Revised  March 2014 Published  June 2014

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.
Citation: Aimin Huang, Roger Temam. The linear hyperbolic initial and boundary value problems in a domain with corners. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1627-1665. doi: 10.3934/dcdsb.2014.19.1627
##### References:
 [1] S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007. [2] P. J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics, Physics of Plasmas, 9 (2002), 1130-1136. doi: 10.1063/1.1463415. [3] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194, Springer-Verlag, 2000. [4] K. O. Friedrichs, The identity of weak and strong extensions of differential operator, Trans. Amer. Math. Soc., 55 (1944), 132-151. doi: 10.1090/S0002-9947-1944-0009701-0. [5] P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline, Astrophys. J. Lett., 544 (2000), L79-L82. doi: 10.1086/317291. [6] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [7] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Pitman, Boston, 1985. doi: 10.1137/1.9781611972030. [8] L. Hörmander, Weak and Strong Extensions of Differential Operators, Comm. Pure Appl. Math., 14 (1961), 371-379. [9] R. A. Horn and C. R. Johnson, Matrix Analysis, Second ed., Cambridge University Press, Cambridge, 2013. [10] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, 1974 (Third printing of the revised edition of 1957, AMS Colloquium Publications, vol. XXXI). [11] A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: Boundary conditions and well-posedness, Archive for Rational Mechanics and Analysis, 211 (2014), 1027-1063. doi: 10.1007/s00205-013-0702-0. [12] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1. Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle, Matrix Editions, Ithaca, NY, 2006. [13] I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimensional corner, Comm. Pure Appl. Math., 24 (1971), 381-393. doi: 10.1002/cpa.3160240304. [14] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. [15] K. Kojima and M. Taniguchi, Mixed problem for hyperbolic equations in a domain with a corner, Funkcialaj Ekvacioj, 23 (1980), 171-195. [16] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press Oxford University Press, New York, 1998, Compressible models, Oxford Science Publications. [17] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York, 1972. [18] J. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type, Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A, 668 (1970), 592-594. [19] J. Li and J. Zha, Linear Algebra, Univ. of Sci. & Tech. of China Press, P.R. China, 1988, in Chinese. [20] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 419-446. doi: 10.1137/0135035. [21] S. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc., 176 (1973), 141-164. doi: 10.1090/S0002-9947-1973-0320539-5. [22] _______, Initial-boundary value problems for hyperbolic systems in regions with corners. II, Trans. Amer. Math. Soc., 198 (1974), 155-175. doi: 10.1090/S0002-9947-1974-0352715-0. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [24] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001. [25] W. Rudin, Functional Analysis, Second ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. [26] L. Sarason, Hyperbolic and other symmetrizable systems in regions with corners and edges, Indiana Univ. Math. J., 26 (1977), 1-39. doi: 10.1512/iumj.1977.26.26001. [27] B. V. Shabat, On a generalized solution to a system of equations in partial derivatives, Math. Sb., 17 (1945), 193-210, in Russia. [28] H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations, Physics of Plasmas, 8 (2001), 3293-3304. doi: 10.1063/1.1379045. [29] M. Taniguchi, Mixed problem for wave equation in the domain with a corner, Funkcialaj Ekvacioj, 21 (1978), 249-259. [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. doi: 10.1115/1.3424338. [31] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2. [32] F. Uhlig, Simultaneous block diagonalization of two real symmetric matrices, Linear Algebra and its Applications, 7 (1973), 281-289. doi: 10.1016/S0024-3795(73)80001-1. [33] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London, 1962. [34] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2. [35] K. Yosida, Functional Analysis, Sixth ed., Springer-Verlag, Berlin, 1995.

show all references

##### References:
 [1] S. Benzoni-Gavage and D. Serre, Multi-dimensional Hyperbolic Partial Differential Equations, Oxford University Press, 2007. [2] P. J. Dellar, Hamiltonian and symmetric hyperbolic structures of shallow water magnetohydrodynamics, Physics of Plasmas, 9 (2002), 1130-1136. doi: 10.1063/1.1463415. [3] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., vol. 194, Springer-Verlag, 2000. [4] K. O. Friedrichs, The identity of weak and strong extensions of differential operator, Trans. Amer. Math. Soc., 55 (1944), 132-151. doi: 10.1090/S0002-9947-1944-0009701-0. [5] P. A. Gilman, Magnetohydrodynamic "shallow water" equations for the solar tachocline, Astrophys. J. Lett., 544 (2000), L79-L82. doi: 10.1086/317291. [6] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Applied Mathematical Sciences, vol. 118, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-0713-9. [7] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, Pitman, Boston, 1985. doi: 10.1137/1.9781611972030. [8] L. Hörmander, Weak and Strong Extensions of Differential Operators, Comm. Pure Appl. Math., 14 (1961), 371-379. [9] R. A. Horn and C. R. Johnson, Matrix Analysis, Second ed., Cambridge University Press, Cambridge, 2013. [10] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, American Mathematical Society, Providence, RI, 1974 (Third printing of the revised edition of 1957, AMS Colloquium Publications, vol. XXXI). [11] A. Huang and R. Temam, The linearized 2d inviscid shallow water equations in a rectangle: Boundary conditions and well-posedness, Archive for Rational Mechanics and Analysis, 211 (2014), 1027-1063. doi: 10.1007/s00205-013-0702-0. [12] J. H. Hubbard, Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Vol. 1. Teichmüller theory, With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra, With forewords by William Thurston and Clifford Earle, Matrix Editions, Ithaca, NY, 2006. [13] I. A. K. Kupka and S. J. Osher, On the wave equation in a multi-dimensional corner, Comm. Pure Appl. Math., 24 (1971), 381-393. doi: 10.1002/cpa.3160240304. [14] H.-O. Kreiss, Initial boundary value problems for hyperbolic systems, Comm. Pure Appl. Math., 23 (1970), 277-298. doi: 10.1002/cpa.3160230304. [15] K. Kojima and M. Taniguchi, Mixed problem for hyperbolic equations in a domain with a corner, Funkcialaj Ekvacioj, 23 (1980), 171-195. [16] P.-L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 2, Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press Oxford University Press, New York, 1998, Compressible models, Oxford Science Publications. [17] J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York, 1972. [18] J. B. Lopatinskii, The mixed Cauchy-Dirichlet type problem for equations of hyperbolic type, Dopovfdf Akad. Nauk Ukrai''n. RSR Ser. A, 668 (1970), 592-594. [19] J. Li and J. Zha, Linear Algebra, Univ. of Sci. & Tech. of China Press, P.R. China, 1988, in Chinese. [20] J. Oliger and A. Sundström, Theoretical and practical aspects of some initial-boundary value problems in fluid dynamics, SIAM J. Appl. Math., 35 (1978), 419-446. doi: 10.1137/0135035. [21] S. Osher, Initial-boundary value problems for hyperbolic systems in regions with corners. I, Trans. Amer. Math. Soc., 176 (1973), 141-164. doi: 10.1090/S0002-9947-1973-0320539-5. [22] _______, Initial-boundary value problems for hyperbolic systems in regions with corners. II, Trans. Amer. Math. Soc., 198 (1974), 155-175. doi: 10.1090/S0002-9947-1974-0352715-0. [23] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [24] A. Rousseau, R. Temam and J. Tribbia, The 3D Primitive Equations in the absence of viscosity: Boundary conditions and well-posedness in the linearized case, J. Math. Pures Appl., 89 (2008), 297-319. doi: 10.1016/j.matpur.2007.12.001. [25] W. Rudin, Functional Analysis, Second ed., International Series in Pure and Applied Mathematics, McGraw-Hill Inc., New York, 1991. [26] L. Sarason, Hyperbolic and other symmetrizable systems in regions with corners and edges, Indiana Univ. Math. J., 26 (1977), 1-39. doi: 10.1512/iumj.1977.26.26001. [27] B. V. Shabat, On a generalized solution to a system of equations in partial derivatives, Math. Sb., 17 (1945), 193-210, in Russia. [28] H. De Sterck, Hyperbolic theory of the "shallow water" magnetohydrodynamics equations, Physics of Plasmas, 8 (2001), 3293-3304. doi: 10.1063/1.1379045. [29] M. Taniguchi, Mixed problem for wave equation in the domain with a corner, Funkcialaj Ekvacioj, 21 (1978), 249-259. [30] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1984 edition. doi: 10.1115/1.3424338. [31] R. Temam and J. Tribbia, Open boundary conditions for the primitive and Boussinesq equations, J. Atmospheric Sci., 60 (2003), 2647-2660. doi: 10.1175/1520-0469(2003)060<2647:OBCFTP>2.0.CO;2. [32] F. Uhlig, Simultaneous block diagonalization of two real symmetric matrices, Linear Algebra and its Applications, 7 (1973), 281-289. doi: 10.1016/S0024-3795(73)80001-1. [33] I. N. Vekua, Generalized Analytic Functions, Pergamon Press, London, 1962. [34] T. Warner, R. Peterson and R. Treadon, A tutorial on lateral boundary conditions as a basic and potentially serious limitation to regional numerical weather prediction, Bull. Amer. Meteor. Soc., 78 (1997), 2599-2617. doi: 10.1175/1520-0477(1997)078<2599:ATOLBC>2.0.CO;2. [35] K. Yosida, Functional Analysis, Sixth ed., Springer-Verlag, Berlin, 1995.
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summaryrefslogtreecommitdiffstats log msg author committer range
path: root/manual/plugins/pictureflow.tex
blob: afab86ef186d54f4922065046931d7829e604662 (plain)
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170  \subsection{PictureFlow} \screenshot{plugins/images/ss-pictureflow}{PictureFlow}{img:pictureflow} PictureFlow provides a visualisation of your albums with their associated cover art. It is possible to start playback of the selected album from PictureFlow. Playback will start from the selected track. The PictureFlow plugin will continue to run while your tracks are played. \subsubsection{Requirements} PictureFlow uses both the album art (see \reference{ref:album_art}) and database (see \reference{ref:database}) features of Rockbox. It is therefore important that these are working correctly before attempting to use PictureFlow. In addition, there are some other points of which to be aware: \begin{itemize} \item First an album index must be built, If you have a lot of albums/artists this could take a while, any button cancels the process and quits pictureflow. \item You don't have to wait for the albumart search to finish, you can press any button and the albumart will be searched in the background. \item Pictureflow checks for updated albumart (in background) on every restart \item PictureFlow will accept album art larger than the dimensions of the screen, but the larger the dimensions, the longer they will take to scale. \end{itemize} \subsubsection{Keys} \begin{btnmap} \opt{scrollwheel,IRIVER_H10_PAD,PBELL_VIBE500_PAD,MPIO_HD300_PAD}{% \ActionStdPrev{} / \ActionStdNext & \opt{HAVEREMOTEKEYMAP}{ &} Scroll through albums / track list \\ } % only scroll wheel and strip' targets use the same action in album and track list \nopt{scrollwheel,IRIVER_H10_PAD,PBELL_VIBE500_PAD,MPIO_HD300_PAD}{% % currently the M3 does not use buttons of the main unit which has no display \nopt{IAUDIO_M3_PAD,MPIO_HD200_PAD,touchscreen}{\ButtonLeft{} / \ButtonRight} \opt{MPIO_HD200_PAD}{FIXME} \opt{touchscreen}{\TouchMidLeft{} / \TouchMidRight} & \opt{HAVEREMOTEKEYMAP}{ \opt{IAUDIO_M3_PAD,GIGABEAT_RC_PAD}{\ActionRCStdPrev{} / \ActionRCStdNext} &} Scroll through albums \\ \nopt{IAUDIO_M3_PAD}{\ActionStdPrev{} / \ActionStdNext} & \opt{HAVEREMOTEKEYMAP}{ % even though the M3 uses an Iaudio remote, mapping differs when used with M/X5 \opt{IAUDIO_M3_PAD}{\ButtonRCLeft{} / \ButtonRCRight} \opt{GIGABEAT_RC_PAD}{\ButtonRCVolUp{} / \ButtonRCVolDown} &} Scroll through track list \\ } \nopt{IAUDIO_M3_PAD}{% \nopt{IRIVER_H10_PAD,touchscreen,PBELL_VIBE500_PAD% ,SANSA_FUZE_PAD,MPIO_HD200_PAD,MPIO_HD300_PAD,SAMSUNG_YH92X_PAD% ,SAMSUNG_YH820_PAD} {\ButtonSelect} \opt{IRIVER_H10_PAD,PBELL_VIBE500_PAD}{\ButtonRight} \opt{SAMSUNG_YH92X_PAD,SAMSUNG_YH820_PAD}{\ButtonFF} \opt{touchscreen}{\TouchCenter} \opt{SANSA_FUZE_PAD}{\ButtonRight} \opt{MPIO_HD200_PAD}{\ButtonFunc} \opt{MPIO_HD300_PAD}{\ButtonEnter} } & \opt{HAVEREMOTEKEYMAP}{ \opt{IAUDIO_M3_PAD}{\ButtonRCPlay} \opt{GIGABEAT_RC_PAD}{\ButtonRCFF} &} Enter track list \nopt{SAMSUNG_YH92X_PAD,SAMSUNG_YH820_PAD}{% / Play album from selected track} \\ % Ondio & Samsung YH use a different button in album list and track list \opt{SAMSUNG_YH92X_PAD,SAMSUNG_YH820_PAD}{% \opt{SAMSUNG_YH92X_PAD,SAMSUNG_YH820_PAD}{\ButtonPlay} & Play album from selected track in track list \\ } \nopt{IAUDIO_M3_PAD,MPIO_HD200_PAD,MPIO_HD300_PAD,touchscreen,SANSA_FUZEPLUS_PAD}{\ButtonLeft} \opt{MPIO_HD200_PAD}{\ButtonRec} \opt{MPIO_HD300_PAD}{\ButtonMenu} \opt{SANSA_FUZEPLUS_PAD}{\ButtonLeft{} or \ButtonBack} \opt{touchscreen}{ \opt{COWON_D2_PAD}{\ButtonPower{} or} \TouchBottomRight} & \opt{HAVEREMOTEKEYMAP}{ \opt{IAUDIO_M3_PAD,GIGABEAT_RC_PAD}{\ActionRCStdCancel} &} Exit track list \\ \nopt{IAUDIO_M3_PAD,SANSA_FUZEPLUS_PAD}{\ActionStdMenu} \nopt{IAUDIO_M3_PAD,SANSA_FUZEPLUS_PAD,SAMSUNG_YH92X_PAD,SAMSUNG_YH820_PAD}{\ActionStdMenu} \opt{SANSA_FUZEPLUS_PAD}{long \ButtonSelect} \opt{SAMSUNG_YH92X_PAD,SAMSUNG_YH820_PAD}{or \ButtonRew} & \opt{HAVEREMOTEKEYMAP}{ \opt{IAUDIO_M3_PAD,GIGABEAT_RC_PAD}{\ActionRCStdMenu} &} Enter menu \\ \nopt{IAUDIO_M3_PAD}{% \opt{IRIVER_H100_PAD,IRIVER_H300_PAD}{\ButtonOff} \opt{IAUDIO_X5_PAD,GIGABEAT_PAD,GIGABEAT_S_PAD,SANSA_E200_PAD,SANSA_CLIP_PAD% ,MROBE100_PAD,SANSA_FUZEPLUS_PAD}{\ButtonPower} \opt{SANSA_C200_PAD,IRIVER_H10_PAD}{Long \ButtonPower} \opt{IPOD_4G_PAD,IPOD_3G_PAD}{Long \ButtonMenu} \opt{SANSA_FUZE_PAD}{Long \ButtonHome} \opt{PBELL_VIBE500_PAD}{\ButtonRec} \opt{SAMSUNG_YH92X_PAD,SAMSUNG_YH820_PAD}{long \ButtonRew} \opt{MPIO_HD200_PAD}{FIXME} \opt{MPIO_HD300_PAD}{Long \ButtonMenu} \opt{touchscreen}{ \opt{COWON_D2_PAD}{Long \ButtonPower{} or} \TouchBottomRight{} (in album view)} } & \opt{HAVEREMOTEKEYMAP}{ \opt{IAUDIO_M3_PAD}{\ButtonRCRec} \opt{GIGABEAT_RC_PAD}{\ButtonRCRew} &} Exit PictureFlow \\ \end{btnmap} \subsubsection{Main Menu} \begin{description} \item[Go to WPS.] Leave PictureFlow and enter the while playing screen. \item[Playback Control.] Control music playback from within the plugin. \item[Settings.] Enter the settings menu. \item[Return.] Exit menu. \item[Quit.] Exit PictureFlow plugin. \end{description} \subsubsection{Settings Menu} \begin{description} \item[Show FPS.] Displays frames per second on screen. \item[Spacing.] The distance between the front edges of the side slides, i.e. changes the degree of overlap of the side slides. A larger number means less overlap. Scales with zoom. \item[Centre margin.] The distance, in screen pixels, with zoom at 100, between the centre and side slides. Scales with zoom. \item[Number of slides.] Sets the number of slides at each side, including the centre slide. Therefore if set to 4, there will be 3 slides on the left, the centre slide, and then 3 slides on the right. \item[Zoom.] Changes the distance at which slides are rendered from the camera''. \item[Show album title.] Allows setting the album title / artist to be shown above or below the cover art, or not at all. \item[Resize Covers.] Set whether to automatically resize the covers or to leave them at their original size. \item[Rebuild cache.] Rebuild the PictureFlow cache. This is needed in order for PictureFlow to pick up new albums, and may occasionally be needed if albums are removed. \item[Update cache.] Update the PictureFlow cache. New albumart will be added. \item[WPS integration.] Integrate pictureflow with the While Playing Screen. \item[Backlight.] Force backlight on while running pictureflow. \end{description} `

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## Algebra 1

Total cost is $200-8c$.
The cost of CDs is $12c$. The cost of DVDs is $200-20c$. total cost=$12c+(200-20c)$ $=12c+(200+(-20c))$ $=200+(12c+(-20c))$ $=200+(-8c)$ $=200-8c$ Use the commutative an associative properties of addition to combine like terms.


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# Help with proving that this series diverges

I have this complex series $\sum\limits_{n=1}^{\infty} \frac{i^n}{\sqrt{n}}$, which I'm trying to prove converges.

Now, I know that a complex sequence converges iff both its real and its imaginary partial sums converge.

Obviously, $\sum\limits_{n=1}^{\infty} \Im(a_n)=\sum\limits_{n=1}^{\infty}0 \to 0$.

But, when I try to compute $\sum\limits_{n=1}^{\infty} \Re(a_n),$ I end up with $\sum\limits_{n=1}^{\infty}\frac{1}{\sqrt{n}},$ which, whilst, intuitively, I know diverges, I can't seem to prove. I know I have to use the Comparison Test, but to what should I compare it?

By the way, 'my' version of the Comparison Test is as follows (for divergence): If $\frac{b_n}{a_n}$ is bounded above and if $\sum\limits_{n=1}^{\infty} b_n$ diverges, then $\sum\limits_{n=1}^{\infty}a_n$ diverges.

Could someone give me a hint?

Thanks

• Use the integral test to show that your series in question diverges. That's fairly easy to integrate – imranfat May 9 '14 at 14:58
• You have a sign error. The signs should alternate. – André Nicolas May 9 '14 at 14:58
• It will be easier if you use Dirichlet's test instead. – achille hui May 9 '14 at 14:58
• Sorry, I should mention that I have to use the Comparison Test; we haven't done either of these suggested tests. – beep-boop May 9 '14 at 14:59
• You have done it, it was probably called the Alternating Series test. – André Nicolas May 9 '14 at 15:00

Obviously, $\sum\limits_{n=1}^{\infty} \Im(a_n)=\sum\limits_{n=1}^{\infty}0 \to 0$.

This is not the case.

As André Nicolas suggests in the comments, it is best to write out the first few terms to get a feel for the series: $$\frac{i}{\sqrt1} + \frac{-1}{\sqrt{2}} + \frac{-i}{\sqrt3} + \frac{1}{\sqrt4} + \frac{i}{\sqrt5} + \frac{-1}{\sqrt6} + \frac{-i}{\sqrt7} + \frac{1}{\sqrt9} + \frac{i}{\sqrt{10}} + \cdots$$ You correctly observe that you should look at the real and imaginary parts. \begin{align*} \sum \Im (a_n) &= \frac{1}{\sqrt{1}} - \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{5}} - \frac{1}{\sqrt{7}} + \cdots \\ \sum \Re (a_n) &= -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{4}} - \frac{1}{\sqrt{6}} + \frac{1}{\sqrt{8}} - \cdots \\ \end{align*}

The proper way to show these converge is with the alternating series test.

If you were trying to prove that the real and/or imaginary partial sums diverge (by the Alternating Series Test), could you just take the odd subsequence and show that that diverges?

The alternating series test requires these two conditions:

1. The terms alternate in sign.

2. The terms decrease in absolute value.


text:
# Using GDI functions and saving them as a bitmap

This topic is 4548 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

I want to be able to use GDI functions to make graphics and save it to a bitmap file. However, the bitmap can be much larger than the screen, so I don't think I can just do it on a screen DC and then copy it to a bitmap. The following code is a test to see how to write something to a bitmap. I try to do a square in the top left of the bitmap. There is a square written in the correct-ish area, but it is just a mix of black and white pixels. I am using the bitsperpixel of the DC, which isn't monochrome. Also, the area of the bitmap where the rectangle isn't is solid black. Here is the code:
int Width = 640*Screens;
int Height = 480;

HDC hScreenDC = GetWindowDC(0);
HDC BmpHDC = CreateCompatibleDC(hScreenDC);
HBITMAP hmemBM = CreateCompatibleBitmap(BmpHDC, Width, Height);
HBRUSH hBrush = CreateSolidBrush(RGB(195,55,87));
SelectObject(BmpHDC,hBrush);
SelectObject(BmpHDC,hmemBM);

//Polyline(BmpHDC,pt,640*Screens);
Rectangle(BmpHDC,1,1,101,101);

int BytesPerPixel = (int)GetDeviceCaps(BmpHDC,BITSPIXEL);

HGLOBAL hpxldata = GlobalAlloc(GMEM_FIXED, Width*Height*BytesPerPixel);
void FAR* lpvpxldata = GlobalLock(hpxldata);

BITMAPINFO bmInfo;

bmFileHeader.bfSize = (Width*Height*BytesPerPixel) + 40 + 14;

GetDIBits(BmpHDC, hmemBM, 0, Height, lpvpxldata, &bmInfo, DIB_RGB_COLORS);

FILE *fp = fopen("Bitmap.bmp","wb");
fwrite(&bmInfo,1,40,fp);
fwrite(lpvpxldata,1,Width*Height*BytesPerPixel,fp);
fclose(fp);

GlobalUnlock(hpxldata);
GlobalFree(hpxldata);
DeleteObject(hmemBM);
DeleteObject(hBrush);
DeleteDC(BmpHDC);
ReleaseDC(0, hScreenDC);

Any ideas? Also, after looking around a bit, CreateDIBSection() might be a better alternative? I just have no clue how to implement it, there doesn't seem to be any good documentation. Thank you, Dev578

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# American Institute of Mathematical Sciences

• Previous Article
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Pointwise bounds for the Green's function for the Neumann-Laplace operator in $\text{R}^3$
August  2022, 15(4): 551-568. doi: 10.3934/krm.2021046

## Lagrangian dual framework for conservative neural network solutions of kinetic equations

 1 Department of Mathematics, Pohang University of Science and Technology, Pohang, Republic of Korea 2 Stochastic Analysis and Application Research Center, Korea Advanced Institute of Science and Technology, Daejeon, Republic of Korea

* Corresponding author: Hyung Ju Hwang

Received  June 2021 Published  August 2022 Early access  December 2021

In this paper, we propose a novel conservative formulation for solving kinetic equations via neural networks. More precisely, we formulate the learning problem as a constrained optimization problem with constraints that represent the physical conservation laws. The constraints are relaxed toward the residual loss function by the Lagrangian duality. By imposing physical conservation properties of the solution as constraints of the learning problem, we demonstrate far more accurate approximations of the solutions in terms of errors and the conservation laws, for the kinetic Fokker-Planck equation and the homogeneous Boltzmann equation.

Citation: Hyung Ju Hwang, Hwijae Son. Lagrangian dual framework for conservative neural network solutions of kinetic equations. Kinetic and Related Models, 2022, 15 (4) : 551-568. doi: 10.3934/krm.2021046
##### References:

show all references

##### References:
Left: Relaxation to Maxwellian of f(t, x = 1, v) from an initial condition $f_0^{(1)}$. Right: Relaxation to Maxwellian of f(t, x = 1, v) from an initial condition $f_0^{(2)}$
Numerical solutions of Test 1 at $t = 0, \frac{1}{2},$ and $1$
Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale
Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished
Numerical solutions of Test 1 at $t = 0, \frac{3}{2},$ and $3$
Left: Loss value in training epoch at log scale. Right: Actual error in training epoch at log scale
Left: Time averaged mass in training epoch. Right: Total mass in time after training is finished
Left: Value of the loss $\widehat{Loss}_B$ in training epoch. Right: $L^{\infty}((0,1);L^2_{v_x,v_y})$ error in training epoch
Left: Total mass in time after the training is finished. Right: Kinetic energy in time after the training is finished
Momentum in time after the training is finished
 [1] José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic and Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 [2] Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic and Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485 [3] Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic and Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169 [4] Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic and Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056 [5] Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 [6] Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135 [7] Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic and Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028 [8] Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124 [9] Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017 [10] Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic and Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016 [11] Marco Cannone, Grzegorz Karch. On self-similar solutions to the homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 801-808. doi: 10.3934/krm.2013.6.801 [12] Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681 [13] Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008 [14] Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks and Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 [15] Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic and Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028 [16] Léo Glangetas, Hao-Guang Li, Chao-Jiang Xu. Sharp regularity properties for the non-cutoff spatially homogeneous Boltzmann equation. Kinetic and Related Models, 2016, 9 (2) : 299-371. doi: 10.3934/krm.2016.9.299 [17] Léo Glangetas, Mohamed Najeme. Analytical regularizing effect for the radial and spatially homogeneous Boltzmann equation. Kinetic and Related Models, 2013, 6 (2) : 407-427. doi: 10.3934/krm.2013.6.407 [18] Zhaohui Huo, Yoshinori Morimoto, Seiji Ukai, Tong Yang. Regularity of solutions for spatially homogeneous Boltzmann equation without angular cutoff. Kinetic and Related Models, 2008, 1 (3) : 453-489. doi: 10.3934/krm.2008.1.453 [19] Yong-Kum Cho. On the homogeneous Boltzmann equation with soft-potential collision kernels. Kinetic and Related Models, 2015, 8 (2) : 309-333. doi: 10.3934/krm.2015.8.309 [20] Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 187-212. doi: 10.3934/dcds.2009.24.187

2020 Impact Factor: 1.432


text:
# Assignment 1 (Coding): Exterior Calculus (Due 2/25)

For the coding portion of your first assignment, you will implement the discrete exterior calculus (DEC) operators $\star_0, \star_1, \star_2, d_0$ and $d_1$. Once implemented, you will be able to apply these operators to a scalar function (as depicted above) by pressing the “$\star$” and “$d$” button in the viewer. The diagram shown above will be updated to indicate what kind of differential k-form is currently displayed. These basic operations will be the starting point for many of the algorithms we will implement throughout the rest of the class; the visualization (and implementation!) should help you build further intuition about what these operators mean and how they work

Getting Started

• For this assignment, you need to implement the following routines:
1. in core/geometry.js
1. cotan
2. barycentricDualArea
2. in core/discrete-exterior-calculus.js
1. buildHodgeStar0Form
2. buildHodgeStar1Form
3. buildHodgeStar2Form
4. buildExteriorDerivative0Form
5. buildExteriorDerivative1Form

In practice, a simple and efficient way to compute the cotangent of the angle $\theta$ between two vectors $u$ and $v$ is to use the cross product and the dot product rather than calling any trigonometric functions directly; we ask that you implement your solution this way. (Hint: how are the dot and cross product of two vectors related to the cosine and sine of the angle between them?)

In case we have not yet covered it in class, the barycentric dual area associated with a vertex $i$ is equal to one-third the area of all triangles $ijk$ touching $i$.

EDIT: You can compute the ratio of dual edge lengths to primal edge lengths using the cotan formula, which can be found on Slide 28 of the Discrete Exterior Calculus lecture, or in exercise 36 of the notes (you don’t have to do the exercise for this homework).

Submission Instructions

Please rename your geometry.js and discrete-exterior-calculus.js files to geometry.txt and discrete-exterior-calculus.txt (respectively) and submit them in a single zip file called solution.zip by email to Geometry.Collective@gmail.com.

## 14 thoughts on “Assignment 1 (Coding): Exterior Calculus (Due 2/25)”

1. svsalem says:

For the exterior derivative functions, where do we get the orientations of the edges and the faces so that we know how to sum up the vertices/edges, respectively? Or am I just missing a function or parameter in the Geometry class that keeps track of this?

1. svsalem says:

Never mind, I was stupid and forgot about Halfedges.

1. Ninae says:

But you can’t assume that halfedges and edges have the same orientation, so I’m still not sure how to tell…

2. Liu Qiangdong says:

For the cotan function, we are looking for the angle “opposite to” the halfedge given. What exactly does “opposite to” mean?

1. Daniel Li says:

Imagine a triangle. The angle opposite to an edge is the angle formed by the two other edges of the triangle.

1. hawaiii says:

Is this the corner stored in the half edge? Does the mesh contain only triangles? I’m confused about how to find the two edges forming the angle if the mesh contains polygons besides triangles.

1. hawaiii says:

Thanks. I found that only the dual mesh is allowed to be a non-triangle polygon mesh and the Hodge star operator only applied to the primary mesh. It’s now clear.

2. fpou says:

What about the case where the edge is shared by two triangles? Which angle do we refer to in this case in terms of calculating the cotan?

1. Alex says:

You’re given a halfedge as input, so this shouldn’t be ambiguous.

3. lykospirit says:

What should the length of the dual of a boundary edge be?

1. Daniel Li says:

I believe the twin points to null and the value will be 0. If you just use the standard formula we provided, you will not run into any issue. Many meshes we use describing 3D data are without boundary, so this is not an issue there.

4. xTheBHox says:

Are we supposed to use barycentric dual or circumcentric dual for the Hodge star matrices? We write the barycentric functoin but Slide 28 is about the circumcentric dual.

1. Alex says:

For this assignment we are using barycentric. However either can be used with varying advantages/disadvantages.


text:
# Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry

Yang, Hyun Seok  (Center for Quantum Spacetime, Sogang University, Seoul 121-741, Republic of Korea) ; Yun, Sangheon (Institute for the Early Universe, Ewha Womans University, Seoul 120-750, Republic of Korea)

09 November 2017

Abstract: We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group $\text{Spin}\left(\mathrm{6}\right)$ requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

Published in: Advances in High Energy Physics 2017 (2017) 7962426


text:
Paul's Online Notes
Home / Calculus II / Vectors / Cross Product
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### Section 11.4 : Cross Product

In this final section of this chapter we will look at the cross product of two vectors. We should note that the cross product requires both of the vectors to be three dimensional vectors.

Also, before getting into how to compute these we should point out a major difference between dot products and cross products. The result of a dot product is a number and the result of a cross product is a vector! Be careful not to confuse the two.

So, let’s start with the two vectors $$\vec a = \left\langle {{a_1},{a_2},{a_3}} \right\rangle$$ and $$\vec b = \left\langle {{b_1},{b_2},{b_3}} \right\rangle$$ then the cross product is given by the formula,

$\vec a \times \vec b = \left\langle {{a_2}{b_3} - {a_3}{b_2},{a_3}{b_1} - {a_1}{b_3},{a_1}{b_2} - {a_2}{b_1}} \right\rangle$

This is not an easy formula to remember. There are two ways to derive this formula. Both of them use the fact that the cross product is really the determinant of a 3x3 matrix. If you don’t know what that is don’t worry about it. You don’t need to know anything about matrices or determinants to use either of the methods. The notation for the determinant is as follows,

$\vec a \times \vec b = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\end{array}} \right|$

The first row is the standard basis vectors and must appear in the order given here. The second row is the components of $$\vec a$$ and the third row is the components of $$\vec b$$. Now, let’s take a look at the different methods for getting the formula.

The first method uses the Method of Cofactors. If you don’t know the method of cofactors that is fine, the result is all that we need. Here is the formula.

$\vec a \times \vec b = \left| {\begin{array}{*{20}{c}}{{a_2}}&{{a_3}}\\{{b_2}}&{{b_3}}\end{array}} \right|\vec i - \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_3}}\\{{b_1}}&{{b_3}}\end{array}} \right|\vec j + \left| {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}\\{{b_1}}&{{b_2}}\end{array}} \right|\vec k$

where,

$\left| {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right| = ad - bc$

This formula is not as difficult to remember as it might at first appear to be. First, the terms alternate in sign and notice that the 2x2 is missing the column below the standard basis vector that multiplies it as well as the row of standard basis vectors.

The second method is slightly easier; however, many textbooks don’t cover this method as it will only work on 3x3 determinants. This method says to take the determinant as listed above and then copy the first two columns onto the end as shown below.

$\vec a \times \vec b = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\end{array}} \right|\,\,\,\,\begin{array}{*{20}{c}}{\vec i}&{\vec j}\\{{a_1}}&{{a_2}}\\{{b_1}}&{{b_2}}\end{array}$

We now have three diagonals that move from left to right and three diagonals that move from right to left. We multiply along each diagonal and add those that move from left to right and subtract those that move from right to left.

This is best seen in an example. We’ll also use this example to illustrate a fact about cross products.

Example 1 If $$\vec a = \left\langle {2,1, - 1} \right\rangle$$ and $$\vec b = \left\langle { - 3,4,1} \right\rangle$$ compute each of the following.
1. $$\vec a \times \vec b$$
2. $$\vec b \times \vec a$$
Show All Solutions Hide All Solutions
a $$\vec a \times \vec b$$ Show Solution

Here is the computation for this one.

\begin{align*}\vec a \times \vec b & = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\2&1&{ - 1}\\{ - 3}&4&1\end{array}} \right|\,\,\,\,\begin{array}{*{20}{c}}{\vec i}&{\vec j}\\2&1\\{ - 3}&4\end{array}\\ & = \vec i\left( 1 \right)\left( 1 \right) + \vec j\left( { - 1} \right)\left( { - 3} \right) + \vec k\left( 2 \right)\left( 4 \right) - \vec j\left( 2 \right)\left( 1 \right) - \vec i\left( { - 1} \right)\left( 4 \right) - \vec k\left( 1 \right)\left( { - 3} \right)\\ & = 5\vec i + \vec j + 11\vec k\end{align*}

b $$\vec b \times \vec a$$ Show Solution

And here is the computation for this one.

\begin{align*}\vec b \times \vec a & = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\{ - 3}&4&1\\2&1&{ - 1}\end{array}} \right|\,\,\,\,\begin{array}{*{20}{c}}{\vec i}&{\vec j}\\{ - 3}&4\\2&1\end{array}\\ & = \vec i\left( 4 \right)\left( { - 1} \right) + \vec j\left( 1 \right)\left( 2 \right) + \vec k\left( { - 3} \right)\left( 1 \right) - \vec j\left( { - 3} \right)\left( { - 1} \right) - \vec i\left( 1 \right)\left( 1 \right) - \vec k\left( 4 \right)\left( 2 \right)\\ & = - 5\vec i - \vec j - 11\vec k\end{align*}

Notice that switching the order of the vectors in the cross product simply changed all the signs in the result. Note as well that this means that the two cross products will point in exactly opposite directions since they only differ by a sign. We’ll formalize up this fact shortly when we list several facts.

There is also a geometric interpretation of the cross product. First we will let $$\theta$$ be the angle between the two vectors $$\vec a$$ and $$\vec b$$and assume that $$0 \le \theta \le \pi$$, then we have the following fact,

$$$\left\| {\vec a \times \vec b} \right\| = \left\| {\vec a} \right\|\,\,\left\| {\vec b} \right\|\,\sin \theta \label{eq:eq1}$$$

and the following figure.

There should be a natural question at this point. How did we know that the cross product pointed in the direction that we’ve given it here?

First, as this figure implies, the cross product is orthogonal to both of the original vectors. This will always be the case with one exception that we’ll get to in a second.

Second, we knew that it pointed in the upward direction (in this case) by the “right hand rule”. This says that if we take our right hand, start at $$\vec a$$ and rotate our fingers towards $$\vec b$$our thumb will point in the direction of the cross product. Therefore, if we’d sketched in $$\vec b \times \vec a$$ above we would have gotten a vector in the downward direction.

Example 2 A plane is defined by any three points that are in the plane. If a plane contains the points $$P = \left( {1,0,0} \right)$$, $$Q = \left( {1,1,1} \right)$$ and $$R = \left( {2, - 1,3} \right)$$ find a vector that is orthogonal to the plane.
Show Solution

The one way that we know to get an orthogonal vector is to take a cross product. So, if we could find two vectors that we knew were in the plane and took the cross product of these two vectors we know that the cross product would be orthogonal to both the vectors. However, since both the vectors are in the plane the cross product would then also be orthogonal to the plane.

So, we need two vectors that are in the plane. This is where the points come into the problem. Since all three points lie in the plane any vector between them must also be in the plane. There are many ways to get two vectors between these points. We will use the following two,

\begin{align*}\overrightarrow {PQ} & = \left\langle {1 - 1,1 - 0,1 - 0} \right\rangle = \left\langle {0,1,1} \right\rangle \\ \overrightarrow {PR} & = \left\langle {2 - 1, - 1 - 0,3 - 0} \right\rangle = \left\langle {1, - 1,3} \right\rangle \end{align*}

The cross product of these two vectors will be orthogonal to the plane. So, let’s find the cross product.

\begin{align*}\overrightarrow {PQ} \times \overrightarrow {PR} & = \left| {\begin{array}{*{20}{c}}{\vec i}&{\vec j}&{\vec k}\\0&1&1\\1&{ - 1}&3\end{array}} \right|\,\,\,\,\begin{array}{*{20}{c}}{\vec i}&{\vec j}\\0&1\\1&{ - 1}\end{array}\\ & = 4\vec i + \vec j - \vec k\end{align*}

So, the vector $$4\vec i + \vec j - \vec k$$ will be orthogonal to the plane containing the three points.

Now, let’s address the one time where the cross product will not be orthogonal to the original vectors. If the two vectors, $$\vec a$$ and $$\vec b$$, are parallel then the angle between them is either 0 or 180 degrees. From $$\eqref{eq:eq1}$$ this implies that,

$\left\| {\vec a \times \vec b} \right\| = 0$

From a fact about the magnitude we saw in the first section we know that this implies

$\vec a \times \vec b = \vec 0$

In other words, it won’t be orthogonal to the original vectors since we have the zero vector. This does give us another test for parallel vectors however.

#### Fact

If $$\vec a \times \vec b = \vec 0$$ then $$\vec a$$ and $$\vec b$$ will be parallel vectors.

Let’s also formalize up the fact about the cross product being orthogonal to the original vectors.

#### Fact

Provided $$\vec a \times \vec b \ne \vec 0$$ then $$\vec a \times \vec b$$ is orthogonal to both $$\vec a$$ and $$\vec b$$.

Here are some nice properties about the cross product.

#### Properties

If $$\vec u$$, $$\vec v$$ and $$\vec w$$ are vectors and $$c$$ is a number then,

\begin{align*} & \vec u \times \vec v = - \vec v \times \vec u & \hspace{0.75in} & \left( {c\vec u} \right) \times \vec v = \vec u \times \left( {c\vec v} \right) = c\left( {\vec u \times \vec v} \right)\\ &\vec u \times \left( {\vec v + \vec w} \right) = \vec u \times \vec v + \vec u \times \vec w & \hspace{0.75in} & \vec u\centerdot \left( {\vec v \times \vec w} \right) = \left( {\vec u \times \vec v} \right)\centerdot \vec w\\ & \vec u\centerdot \left( {\vec v \times \vec w} \right) = \left| {\begin{array}{*{20}{c}}{{u_1}}&{{u_2}}&{{u_3}}\\{{v_1}}&{{v_2}}&{{v_3}}\\{{w_1}}&{{w_2}}&{{w_3}}\end{array}} \right| & & \end{align*}

The determinant in the last fact is computed in the same way that the cross product is computed. We will see an example of this computation shortly.

There are a couple of geometric applications to the cross product as well. Suppose we have three vectors $$\vec a$$, $$\vec b$$ and $$\vec c$$ and we form the three dimensional figure shown below.

The area of the parallelogram (two dimensional front of this object) is given by,

${\rm{Area}} = \left\| {\vec a \times \vec b} \right\|$

and the volume of the parallelepiped (the whole three dimensional object) is given by,

${\rm{Volume}} = \left| {\vec a\centerdot \left( {\vec b \times \vec c} \right)} \right|$

Note that the absolute value bars are required since the quantity could be negative and volume isn’t negative.

We can use this volume fact to determine if three vectors lie in the same plane or not. If three vectors lie in the same plane then the volume of the parallelepiped will be zero.

Example 3 Determine if the three vectors $$\vec a = \left\langle {1,4, - 7} \right\rangle$$, $$\vec b = \left\langle {2, - 1,4} \right\rangle$$ and $$\vec c = \left\langle {0, - 9,18} \right\rangle$$ lie in the same plane or not.
Show Solution

So, as we noted prior to this example all we need to do is compute the volume of the parallelepiped formed by these three vectors. If the volume is zero they lie in the same plane and if the volume isn’t zero they don’t lie in the same plane.

\begin{align*}\vec a\centerdot \left( {\vec b \times \vec c} \right) & = \left| {\begin{array}{*{20}{c}}1&4&{ - 7}\\2&{ - 1}&4\\0&{ - 9}&{18}\end{array}} \right|\,\,\,\,\begin{array}{*{20}{c}}1&4\\2&{ - 1}\\0&{ - 9}\end{array}\\ & = \left( 1 \right)\left( { - 1} \right)\left( {18} \right) + \left( 4 \right)\left( 4 \right)\left( 0 \right) + \left( { - 7} \right)\left( 2 \right)\left( { - 9} \right) - \\ & \hspace{0.75in}\left( 4 \right)\left( 2 \right)\left( {18} \right) - \left( 1 \right)\left( 4 \right)\left( { - 9} \right) - \left( { - 7} \right)\left( { - 1} \right)\left( 0 \right)\\ & = - 18 + 126 - 144 + 36\\ & = 0\end{align*}

So, the volume is zero and so they lie in the same plane.


text:
# Please can some one check my ans

###### Question:

113. A production possibilities frontier shows the
A) the maximum attainable combinations of two goods that can be produced with the available resources.
B) combinations of two goods that consumers desire most.
C) maximum output of two goods that can be produced with an unlimited quantity of resources.D) demand for two goods.
114. Being on the production possibilities frontier implies that more of one good can be produced only by
A) decreasing the quantity of the other good produced.
B) increasing the quantity of the other good produced.C) lowering the price of the good.
D) raising the price of the good.
E) None Of The Above

116.An output combination is technically efficient if
A) it is possible to produce more of all goods.
B) it is possible to produce more of one good without producing less ofanother.C) it is not possible to produce more of one good without producing less of another.
D) it is not possible to produce more of one good at any cost.

117.Suppose in an economy some resources are not utilized in their most productive use. If societydecides that it wants more of one good,
A) it will have to increase the quantity of resources in the economy.
B) it will have to give up production and consumption of some other good.
C) it can only do so if technology advances.D) it can achieve this without giving up another good by employing the under-utilized resources.E) None Of The Above

118.In the production possibilities frontier framework, unemployment is represented by
A) an inward shift of the curve.B) a point located inside the curve.
C) a point located outside the curve.
D) a point located on the curve

119.Which of the following would not shift a nation's production possibilities frontier?
A) a liberal immigration policy that welcomes foreign laborB) an increase in a nation's capital stock
C) discovery of oil, a vital input in the production process
D) an increase in the general price level.

132.If the price of SUVs were to increase, we would expect:
A) the demand for gasoline to increase.
B) the demand for gasoline to decrease.
C) the supply of gasoline to increase.
D) the supply of gasoline to decrease.
E) both B and D to occur.

E) None Of The Above


text:
# DevTalk.net

ActiveMesa's software development blog. MathSharp, R2P, X2C and more!

## Five Ways of Improving Excel to Code Conversion

Converting Excel to compile-ready C++ or C# code isn’t as easy as you might think: there are plenty of cases where generating code straight-ahead is either inefficient or would result in incorrect code.

Here is an overview of five different cases where X2C performs adjustments.

### 1. Sequence Folding

Say you decide to map a few cells containing values {0, 1, 2, 3, 4} to a vector:

If you were to perform initialization of this vector, it might look as follows:

double Foo[5];
void InitializeFoo()
{
Foo[0] = 0;
Foo[1] = 1;
Foo[2] = 2;
Foo[3] = 3;
Foo[4] = 4;
}

The above is correct, but tedious. Wouldn’t it be great if we could detect repetition like this and put it in a loop? Well, X2C does exactly this, so your generated code will look as follows:

double Foo[5];
void InitializeFoo()
{
for (int i = 0; i < 5; ++i)
Foo[i] = i;
}

X2C does its best at detecting row/column-dependent values and, if it can see that even a part of the elements can be initialized in a for loop, it rewrites the code accordingly.

### 2. Range Function Substitutions

Let’s say you decided to find smallest of the elements above:

What sort of code would you expect? Maybe something like

double Min() const
{
return std::min(Foo[0], std::min(Foo[1], std::min(Foo[2],  // you get the idea
}

Well, X2C could just give you that, but it’s a lot smarter. Depending on the language, X2C knows about ways of aggregating values so, for the above case, the generated code would be as follows:

double Min() const
{
return *std::min_element(std::begin(Foo), std::end(Foo));
}

### 3. Complex Aggregate Calculations

Sometimes, though, aggregations get complicated – not just a simple sum, average or calculation of smallest or largest element, but rather something like the following:

In this case, what X2C does is perform a walk over each of the elements and outputs the final result based on element-wise reads, i.e.:

double FooBar() const
{
return (Foo[0]*Bar[0] + Foo[1]*Bar[1] + Foo[2]*Bar[2] +
Foo[3]*Bar[3] + Foo[4]*Bar[4]);
}

In future, we aim to teach X2C to fold even these types of expressions into loops, as the above could be calculated using a simple for loop and just one temporary variable.

### 4. Synthetic Evaluation for Simple Functions

Some functions which appear deceptively easy in Excel require additional evaluation steps when translated to code. Here’s a simple example:

What would you expect X2C to do with the TRUNC() function in the above example? Well, for functions which are difficult to immediately express using existing APIs, X2C simply rewrites them the best way it knows. For the above, you’d get the following code:

double Foo::Pi() const
{
return M_PI;
}
double Foo::PiTruncated() const
{
return (boost::math::round(Pi() * 100) / 100);
}

The value 100 in the above comes from the fact we need to truncate to 2 decimal places (102 = 100).

### 5. Compile-Time Evaluation

Sometimes, the parameters to an Excel function actually determine which function to call. In this case, X2C simply checks the parameter at ‘compile-time’ and writes the appropriate code. Here’s an example:

In the above, it’s the 4th parameter to the normal distribution calculation that’s the problem. If set to FALSE, this is a calculation of the probability density function φ(x), whereas if it is TRUE, we need to compute the cumulative distribution function Φ(x) instead.

To get around this, X2C simply looks at the parameter value and infers the invocation for that. So for the cell above, you would get the following function:

double Foo() const
{
return boost::math::cdf<>(
boost::math::normal_distribution<>(0, 1), 0.5);
}

### Conclusion

These are just some of the ways in which X2C improves the conversion process. If you want to see these, and many other transformations in action, download X2C and give it a go! ■

Written by Dmitri Nesteruk

August 14th, 2013 at 8:16 am

Posted in X2C

Tagged with , ,


text: tangent synonyms, tangent pronunciation, tangent translation, English dictionary definition of tangent. The following video gives the definitions of a circle, a radius, a chord, a diameter, secant, secant line, tangent, congruent circles, concentric circles, and intersecting circles. At that point, the tangent line is perpendicular to the circle's radius and diameter, and this property is often used in high school geometry problems involving tangent lines. Informally, it is a line through a pair of infinitely close points on the curve. Meaning of tangent. The line AO, from point A to the circle’s center, intersects the circle at point C, and the line segment AB measures 8 units. Learner's definition of TANGENT [count] geometry: a line that touches a sphere or circle at only one point. Tangent definition, in immediate physical contact; touching. 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(in a triangle that has one angle…. tangent noun [C] (DIFFERENT SUBJECT) a subject or activity that is different than the one you are talking about or doing: We were talking about exercise and got off on a tangent. Qui est à la limite, très près du niveau nécessaire pour que quelque chose se fasse : Il a été reçu, mais c'était tangent. Learn more. Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. Angle on major arc: The larger of 2 angles when a circle … When two segments are drawn tangent to a circle from the same point outside the circle, the segments are congruent. Definition of Tangent . Tangent Lines A tangent line is a line that intersects a circle at one point. Tangent, written as tan(θ), is one of the six fundamental trigonometric functions. Figure %: A tangent line In the figure above, the line l is tangent to the circle C. Point T is the point of tangency. tangent, in mathematics. Tangent definition, in immediate physical contact; touching. tangent, in mathematics. In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. Imagine we didn't know the length of the side BC.We know that the tangent of A (60°) is the opposite side (26) divided by the adjacent side AB - the one we are trying to find. $x = \frac 1 2 \cdot \text{ m } \overparen{ABC}$ Note: Like inscribed angles , when the vertex is on the circle itself, the angle formed is half the measure of the intercepted arc. The connecting point between the curve and the line is called as tangent point. For this reason, a tangent line is a good approximation of the curve near that point. tangent - a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point straight line - a line traced by a point traveling in a constant direction; a line of zero curvature; "the shortest distance between two points is a straight line" The extension problem of this topic is a belt and gear problem which asks for the length of belt required to fit around two gears. In geometry, a tangent is a line that touches a curved surface but does not intersect it. In a short talk, I had to explain, to an audience with little knowledge in geometry or algebra, ... Quick definition of the tangent space. a = 3" b = 4" tan α = a / b = 3 / 4 = 0.75. tangent tan θ = a / b n. 1. à¸à¹à¸à¸§à¸à¸§à¸²à¸¡à¸à¸´à¸à¸à¸µà¹à¸à¹à¸²à¸à¸à¸±à¸, Ø§ÙÚ© ÙÙØ¶ÙØ¹ Ø³Û Ø¯ÙØ³Ø±Û ÙÙØ¶ÙØ¹ ÙÙÚº Ø¬Ø§ÙØ§, Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content. How to use tangent in a sentence. Touching; in geometry, touching at a single point: as, a tangent line; curves tangent to each other. However, one cannot prove that a single definition is true or false. Tangent ratios, along with cosine and sine ratios, are ratios of two different sides of a right triangle. One can ask if one definition is consistent with with another set of definitions. Point of tangency is the point at which tangent meets the circle. A line tangent to two circles (not necessarily at the same point)- called a common internal tangent if it lies between the circles or a common enternal tangent if it does not. The extension problem of this topic is a belt and gear problem which asks for the length of belt required to fit around two gears. Tangent ratios are the ratio of the side opposite to the side adjacent the angle they represent. For those comfortable in "Math Speak", the domain and range of Sine is as follows. Mathematics a. Since this is kind of a mouthful and a little hard to remember, kind folks over the centuries have come up with a handy mnemonic to … Now, let’s prove tangent and radius of the circle are perpendicular to each other at the point of contact. The tangent line never crosses the circle, it just touches the circle. Point of tangency is the point where the tangent touches the circle. tan (θ) = opposite / … The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side: so called because it can be represented as a line segment tangent to the circle, that is the line that touches the circle, from Latin linea tangens or touching line. At the point of tangency, a tangent is perpendicular to the radius. From the same external point, the tangent segments to a circle are equal. When do curves exist in infinite-dimensional submanifolds? See go off at a tangent COBUILD Advanced English Dictionary. Inverse tangent function; Tan table; Tan calculator; Tangent definition. Tangent … An angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc. For this reason, a tangent line is a good approximation of the curve near that point. The tangent line to the curve has a slope equal to the derivative of the function evaluated at the point of contact. Se dit d'une courbe vis-à-vis d'une autre courbe, ou d'une surface vis-à-vis d'une autre surface ou d'une courbe, quand leur contact est d'ordre supérieur ou égal à 2. A secant line intersects the circle in two points. tangent (plural tangents) ( differential geometry ) A straight line touching a curve at a single point without crossing it there. A tangent intersects a circle in exactly one point. Point of contact: Where a tangent touches a circle. We wil… Abbreviated tan. Examples of Tangent. The right-angled triangle definition of trigonometric functions is most often how they are introduced, followed by their definitions in terms of the unit circle. Before getting stuck into the functions, it helps to give a nameto each side of a right triangle: Find the circle’s radius. Trigonometry. In other words, we can say that the lines that intersect the circles exactly in one single point are Tangents. Definition Of Tangent. 2. There can be only one tangent at a point to circle. In order to find the measure of the angle itself, one must understand inverse trigonometric functions. The line that joins two infinitely close points from a point on the circle is a Tangent. Tangent 1.Geometry. Definition. A tangent is a line that touches the edge of a curve or circle at one point, but does not cross it. Tangent definition is - an abrupt change of course : digression. Leibniz defined it as the line through a pair of infinitely close points on the curve. In the previous section, we algebraically defined tangent as  =  , and this is the definition that we will use most in the future. In geometry, a tangent is a straight line that touches a curve at one point.At the place where they touch, the line and the curve both have the same slope (they are both "going in the same direction"). The definition of "tangent line" will depend a lot on the definition of "curve". The word "tangent" comes from a Latin term meaning "to touch," because a tangent just barely touches a circle. This function can be used to determine the length of a side of a triangle when given at least one side of the triangle and one of the acute angles. It does not seem easy to me to define a (continuous) curve with purely synthetic geometry notions (i.e., as a set of points, instead of as a point-valued function). In other words, it is defined as the line which represents the slope of a curve at that point. Example. When two segments are drawn tangent to a circle from the same point outside the circle, the segments are congruent. A circle is a conic section cut by a plane perpendicular to the axis of the cone. At the place where they touch, the line and the curve both have the same slope (they are both "going in the same direction"). Tangent is a trigonometric ratio comparing two sides of a right triangle. And the tangent (often abbreviated "tan") is the ratio of the length of the side opposite the angle to the length of the side adjacent. See more. The point at which the circle and the line intersect is the point of tangency. I don’t remember any of my teachers ever mentioning a circle definition of the tangent function. go off on a tangent (US) or British go off at a tangent: to start talking about something that is only slightly or indirectly related to the original subject. There are two main ways in which trigonometric functions are typically discussed: in terms of right triangles and in terms of the unit circle. In a right triangle ABC the tangent of α, tan(α) is defined as the ratio betwween the side opposite to angle α and the side adjacent to the angle α: tan α = a / b. This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional. noun The length cut off upon the straight line touching a curve between the line of abscissas and the point of tangency. Tangent is usually shortened to tan but is pronounced tangent. ( mathematics ) A function of an angle that gives the ratio of the sine to the cosine , in either the real or complex numbers . Tangent Meaning in Geometry In Geometry, the tangent is defined as a line touching circles or an ellipse at only one point. This line is called a tangent line, or sometimes simply a tangent. Figure %: A tangent line In the figure above, the line l is tangent to the circle C. Point T is the point of tangency. A straight line is tangent to a given curve at a point on the curve if the line passes through the point on the curve and has slope, where is the derivative of. Definition. TBD. tangent noun [C] (MATH) Several theorems are related to this because it plays a significant role in geometrical constructionsand proofs. A chord and tangent form an angle and this angle is the same as that of tangent inscribed on the opposite side of the chord. Below, the blue line is a tangent to the circle c. Note the radius to the point of tangency is always perpendicular to the tangent line. The function which is the quotient of the sine function by the cosine function. Below is a table of values illustrating some key sine values that span the entire range of values. 1 In geometry, the tangent to a circle circle, closed plane curve consisting of all points at a given distance from some fixed point, called the center. 2. tangent definition: 1. a straight line that touches but does not cut into a curve 2. Définitions de tangent. https://encyclopedia2.thefreedictionary.com/Tangent+(geometry), A line is tangent to a curve at a fixed point. 2. Of lines, curves, and surfaces: meeting at a single point and having, at that point, the same direction. One of the trigonometry functions. In geometry, a tangent of a circle is a straight line that touches the circle at exactly one point, never entering the circle’s interior. go off on a tangent (US) or British go off at a tangent: to start talking about something that is only slightly or indirectly related to the original subject. Two years earlier, I’d learned the triangle definitions of sine, cosine, and tangent in geometry class. The point at which the circle and the line intersect is the point of tangency. A tangent intersects a circle in exactly one point. In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. The definition of "tangent line" will depend a lot on the definition of "curve". It does not seem easy to me to define a (continuous) curve with purely synthetic geometry notions (i.e., as a set of points, instead of as a point-valued function). A line which touches a circle or ellipse at just one point. For more on this see Tangent to a circle. Link to this page: Problem. Arc: A part of the curve along the perimeter of a circle. Tangent segments to a circle that are drawn from the same external point are congruent. How to use tangent in a sentence. It can, however, be helpful to understand the tangent function from a geometric perspective. The tangent of an angle is the ratio of its sine and cosine. The abbreviation is tan. A circle is a conic section cut by a plane perpendicular to the axis of the cone. Lets put this another way: tangents are two things, (lines or shapes) that are touching but not overlapping. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c if the line passes through the point (c, f(c)) on the curve and has slope f'(c), where f' is the derivative of f. A similar definition applies to space curves and curves in n-dimensional Euclidean space. To extending these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 … The angle formed by the intersection of 2 tangents, 2 secants or 1 tangent and 1 secant outside the circle equals half the difference of the intercepted arcs!Therefore to find this angle (angle K in the examples below), all that you have to do is take the far intercepted arc and near the smaller intercepted arc and then divide that number by two! Tangent ratios are the ratio of the side opposite to the side adjacent the angle they represent. The non-mathematical meaning of tangent comes from this sense of barely touching something: when a conversation heads off on a tangent, it's hard to see how or why it came up. Its abbreviation is tan. Tangent segments to a circle that are drawn from the same external point are congruent. The word "tangent" comes from a Latin term meaning "to touch," because a tangent just barely touches a circle. a trigonometric function. Tangent to a circle is the line that touches the circle at only one point. That means they're the same length. The curve and the tangent line are almost exactly the same near the intersection point.tangent is straight line Suppose a line touches the curve at P, then the point “P” is called the point of tangency. noun In geometry: A straight line through two consecutive points (which see, under consecutive) of a curve or surface. Domain of Sine = all real numbers; Range of Sine = {-1 ≤ y ≤ 1} The sine of an angle has a range of values from -1 to 1 inclusive. Dictionary, Encyclopedia and Thesaurus - The Free Dictionary, the webmaster's page for free fun content. In geometry, a tangent is a straight line that touches a curve at one point. Tangent definitions. Here, we shall learn about the tangent definition in geometry, tangent to a circle, and tangent line equation as we scroll down. Abbreviated tan. A line tangent to a circle is a line from a point outside the circle that touches the circle at exactly one point. The tangent of an acute angle in a right triangle is the ratio of the leg opposite the angle to the leg adjacent to the angle. Tangents, of course, also allude to writing or speaking that diverges from the topic, as when a writer goes off on a tangent and points out that most farmers do not like having their crops stomped down by vandals from this or any other world. 9. To visualize what a line tangent to a circle looks like, imagine that you balance a ruler on a ball, as is pictured here: The ruler is the tangent. In geometry, we talk about tangents to circles. Line AB is tangent to circle O at point B, and the line segment AB measures 12 units. A tangent is a line that intersects the circle at one point. Tangent Lines A tangent line is a line that intersects a circle at one point. Tangent : In geometry, when a straight line touches the plane curves at a given point, then the line is called Tangent line. In geometry, the tangent line (or tangent) means a line or plane that intersects a curved line or surface at exactly one point. Tangent line or tangent in geometry means a line or plane that intersects a curve or a curved surface at exactly one point. Graph of tangent. A tangent to a curve at a point is a straight line that touches the curve at that point. See more. Tangent: A tangent to a circle is a straight line which touches the circle at only one point (so it does not cross the circle - it just touches it). All content on this website, including dictionary, thesaurus, literature, geography, and other reference data is for informational purposes only. Tangent. At the point of tangency, it is perpendicular to the radius. The geometric definition of the tangent function, which predates the triangle definition, is the length of a segment tangent to the unit circle. Solutions in category Geometry Example 1 Familier. b. Abbr. Link to this page: Add or improve a definition Such a line is said to be tangent to that circle. That means they're the same length. The word "tangent" can also mean the trigonometric function . They are pretty much the same in artwork as they are in geometry. Tangent definition: A tangent is a line that touches the edge of a curve or circle at one point, but does not... | Meaning, pronunciation, translations and examples The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. A tangent to a circle is defined as a line that passes through exactly one point on a circle, and is perpendicular to a line passing through the center of the circle. Some notational questions regarding tangent vectors. Definition in convex geometry Let K be a closed convex subset of a real vector space V and ∂ K be the boundary of K. The solid tangent cone to K at a point x ∈ ∂ K is the closure of the cone formed by all half-lines (or rays) emanating from x and intersecting K in at least one point y distinct from x. 4. tan The trigonometric function of an acute angle in a right triangle that is the ratio of the length of the side opposite the … Definition of tangent in the Definitions.net dictionary. Quick definitions from WordNet (tangent) noun: ratio of the opposite to the adjacent side of a right-angled triangle noun: a straight line or plane that touches a curve or curved surface at a point but does not intersect it at that point In order to find the measure of the angle itself, one must understand inverse trigonometric functions. Is perpendicular to the derivative of the curve near that point, the same point outside the circle two. Edge of a curve at that point the radius for more on this website, including dictionary, same... '' because a tangent line are almost exactly the same point outside the circle it... However, one can not prove that a single definition is consistent with with another set definitions. Its sine and cosine can not prove that a single point are tangents line that touches curved. Is called the point of tangency, it just touches the curve and the line that the. 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To tan but is pronounced tangent understand inverse trigonometric functions values illustrating some key sine values that span the range... Reason, a tangent touches a curve at a point outside the circle of. Θ = a / b = 4 '' tan α = a / b 1... Of trigonometric functions outside the circle, the tangent touches the curve at that point set of definitions ’. Lot on the curve has a tangent definition geometry equal to the radius, translation... Depend a lot on the curve near that point cut by a chord and a line... Tangent synonyms, tangent translation, English dictionary definition of  tangent line is tangent that... Or a curved surface but does not cut into a curve or a curved surface at exactly point. Key sine values that span the entire range of values illustrating some key sine values span... Circle is the point of contact: where a tangent just barely touches a curve between the line touches.: //encyclopedia2.thefreedictionary.com/Tangent+ ( geometry ) a straight line touching a curve at a point on the circle that... Line never crosses the circle and the line segment AB measures 12 units contact ; touching by a chord a... Cut by a plane perpendicular to each other at the point of contact: where a tangent that intersect circles. '' can also tangent definition geometry the trigonometric function are pretty much the same external point are congruent from the external.: a straight line that touches but does not cut into a curve at that point one definition -! And sine ratios, are ratios of two different sides of a curve or circle at one point of. Tangent is a line is a line tangent to a circle in exactly point...


text:
# Voting Twice in France

February 19, 2014
By

(This article was first published on Freakonometrics » R-english, and kindly contributed to R-bloggers)

On the Monkey Cage blog, Baptiste Coulmont (a.k.a. @coulmont) recently uploaded a post entitled “You can vote twice ! The many political appeals of proxy votes in France“, coauthored with Joël Gombin (a.k.a. @joelgombin), and myself. The study was initially written in French as mentioned in a previous post. Baptiste posted additional information on his blog (http://coulmont.com/blog/…) and I also wanted to post some lines of code, to mention a model that was not used in that study (more complex to analyze, but more realistic, and with the same conclusions). The econometric study is based on aggregated voted, with a possible ecological misinterpretation.

• Regression Model: Possible Explanatory Variables

The first idea was to model proxies using a binomial regression, per pooling station  where  denote the number of proxy vote, per station , and  denotes the number of voters. Proportion  can be a function of possible explanatory variables (on Baptiste’s blog there are additional information about the datasets, obtained from insee.fr and opendata.paris.fr)

```> bt1=read.table("paris2007-pres-t1.csv",header=TRUE,sep=";")
> bv\$BV=bv\$BVCOM
> baset1=merge(bt1,bv,by="BV")
> baset2=merge(bt2,bv,by="BV")
> baset1\$LOGEMENT=baset1\$PROPRIO+baset1\$LOCNONHLM+baset1\$LOCHLM+baset1\$GRATUIT
> baset2\$LOGEMENT=baset2\$PROPRIO+baset2\$LOCNONHLM+baset2\$LOCHLM+baset2\$GRATUIT```

For instance, assume that  is a function of the proportion of owner of the place people live in, denoted  in the neighborhood of the pooling station,

```> variable="PROPRIO"
> reference="LOGEMENT"
> baset1\$taux=baset1[,variable]/baset1[,reference]
> baset2\$taux=baset2[,variable]/baset2[,reference]```

We can consider a logistic regression

or a logistic regression with splines, if we do not want to assume a linear model

With cubic splines, the code is

```> b=hist(baset1\$taux,plot=FALSE)
> library(splines)
> regt1=glm(PROCURATIONS/INSCRITS~bs(taux,6),family=binomial,weights=INSCRITS,data=baset1)
> regt2=glm(PROCURATIONS/INSCRITS~bs(taux,6),family=binomial,weights=INSCRITS,data=baset2)
> u=seq(min(baset1\$taux)+.015,max(baset1\$taux)-.015,by=.001)
> ND=data.frame(taux=u)
> ug=seq(0,max(baset1\$taux)+.05,by=.001)
> pt1=predict(regt1,newdata=ND,se=TRUE,type="response")
> pt2=predict(regt2,newdata=ND,se=TRUE,type="response")
> library(RColorBrewer)
> CL=brewer.pal(6, "RdBu")
> plot(ug,ug*1,col="white",xlab=nom,ylab="Taux de procuration",
+ ylim=c(0,.1))
> for(i in 1:(length(b\$breaks)-1)){
+ polygon(b\$breaks[i+c(0,0,1,1)],c(0,b\$counts[i],b\$counts[i],0)
+ /max(b\$counts)*.05,col="light yellow",border=NA)}
> polygon(c(u,rev(u)),c(pt1\$fit+2*pt1\$se.fit,rev(pt1\$fit-2*pt1\$se.fit)),
+ border=NA,density=30,col=CL[4])```

while a standard logistic regression would be

```> lines(u,pt1\$fit,col=CL[6],lwd=2)
> polygon(c(u,rev(u)),c(pt2\$fit+2*pt2\$se.fit,rev(pt2\$fit-2*pt2\$se.fit)),
+ border=NA,density=30,col=CL[3])
> lines(u,pt2\$fit,col=CL[1],lwd=2)
> regt1l=glm(PROCURATIONS/INSCRITS~taux,family=binomial,weights=INSCRITS,data=baset1)
> regt2l=glm(PROCURATIONS/INSCRITS~taux,family=binomial,weights=INSCRITS,data=baset2)
> ND=data.frame(taux=ug)
> pt1l=predict(regt1l,newdata=ND,se=TRUE,type="response")
> pt2l=predict(regt2l,newdata=ND,se=TRUE,type="response")
> lines(ug,pt1l\$fit,col=CL[5],lty=2)
> lines(ug,pt2l\$fit,col=CL[2],lty=2)
> legend(0,.1,c("Second Tour","Premier Tour"),col=CL,
+ lwd=2,lty=1,border=NA)```

Here it is (the confidence region is for the spline regression) with on blue the first round of the Presidential election, and in red, the second round (in France, it’s a two-round system)

(the legend of the y axis is not correct). We can consider as explanatory variable the rate of H.L.M., low-cost housing or council housing,

If I like the graph, unfortunately, the interpretation of coefficient  might be complicated

```> summary(regt1l)

Call:
glm(formula = PROCURATIONS/INSCRITS ~ taux, family = binomial,
data = baset1, weights = INSCRITS)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-12.9549   -1.5722    0.0319    1.6292   13.1303

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.70811    0.01516  -244.6   <2e-16 ***
taux         1.49666    0.04012    37.3   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 12507  on 836  degrees of freedom
Residual deviance: 11065  on 835  degrees of freedom
AIC: 15699

Number of Fisher Scoring iterations: 4

> summary(regt2l)

Call:
glm(formula = PROCURATIONS/INSCRITS ~ taux, family = binomial,
data = baset2, weights = INSCRITS)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-15.4872   -1.7817   -0.1615    1.6035   12.5596

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.24272    0.01230 -263.61   <2e-16 ***
taux         1.45816    0.03266   44.65   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 9424.7  on 836  degrees of freedom
Residual deviance: 7362.3  on 835  degrees of freedom
AIC: 12531

Number of Fisher Scoring iterations: 4```

So we did consider a standard linear regression model, for the proxy rate, per station,

(again, either a model with splines, or a standard linear model). The code is

```> regt1=lm(PROCURATIONS/INSCRITS~bs(taux,6),weights=INSCRITS,data=baset1)
> regt2=lm(PROCURATIONS/INSCRITS~bs(taux,6),weights=INSCRITS,data=baset2)
> u=seq(min(baset1\$taux)+.015,max(baset1\$taux)-.015,by=.001)
> ND=data.frame(taux=u)
> ug=seq(0,max(baset1\$taux)+.05,by=.001)
> pt1=predict(regt1,newdata=ND,se=TRUE,type="response")
> pt2=predict(regt2,newdata=ND,se=TRUE,type="response")
> library(RColorBrewer)
> CL=brewer.pal(6, "RdBu")
> plot(ug,ug*1,col="white",xlab=nom,ylab="Taux de procuration",
+ ylim=c(0,.1))
> for(i in 1:(length(b\$breaks)-1)){
+ polygon(b\$breaks[i+c(0,0,1,1)],c(0,b\$counts[i],b\$counts[i],0)
+ /max(b\$counts)*.05,col="light yellow",border=NA)}
> polygon(c(u,rev(u)),c(pt1\$fit+2*pt1\$se.fit,rev(pt1\$fit-2*pt1\$se.fit)),
+ border=NA,density=30,col=CL[4])
> lines(u,pt1\$fit,col=CL[6],lwd=2)
> polygon(c(u,rev(u)),c(pt2\$fit+2*pt2\$se.fit,rev(pt2\$fit-2*pt2\$se.fit)),
+ border=NA,density=30,col=CL[3])
> lines(u,pt2\$fit,col=CL[1],lwd=2)
> regt1l=lm(PROCURATIONS/INSCRITS~taux,weights=INSCRITS,data=baset1)
> regt2l=lm(PROCURATIONS/INSCRITS~taux,weights=INSCRITS,data=baset2)
> ND=data.frame(taux=ug)
> pt1l=predict(regt1l,newdata=ND,se=TRUE,type="response")
> pt2l=predict(regt2l,newdata=ND,se=TRUE,type="response")
> lines(ug,pt1l\$fit,col=CL[5],lty=2)
> lines(ug,pt2l\$fit,col=CL[2],lty=2)
> legend(0,.1,c("Second Tour","Premier Tour"),col=CL,
+ lwd=2,lty=1,border=NA)```

Here is again the evolution as a function of the rate of owner of their homes,

The graph is rather close to the one before, and here, the interpretation of the summary table is more conventional,

```> summary(regt1l)

Call:
lm(formula = PROCURATIONS/INSCRITS ~ taux, data = baset1, weights = INSCRITS)

Weighted Residuals:
Min      1Q  Median      3Q     Max
-1.9994 -0.2926  0.0011  0.3173  3.2072

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.021268   0.001739   12.23   <2e-16 ***
taux        0.054371   0.004812   11.30   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.646 on 835 degrees of freedom
Multiple R-squared:  0.1326,	Adjusted R-squared:  0.1316
F-statistic: 127.7 on 1 and 835 DF,  p-value: < 2.2e-16

> summary(regt2l)

Call:
lm(formula = PROCURATIONS/INSCRITS ~ taux, data = baset2, weights = INSCRITS)

Weighted Residuals:
Min      1Q  Median      3Q     Max
-2.9029 -0.4148 -0.0338  0.4029  3.4907

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.033909   0.001866   18.17   <2e-16 ***
taux        0.079749   0.005165   15.44   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.6934 on 835 degrees of freedom
Multiple R-squared:  0.2221,	Adjusted R-squared:  0.2212
F-statistic: 238.4 on 1 and 835 DF,  p-value: < 2.2e-16```

We have used those codes to produce the graphs mentioned in the post. But before mentioning the residuals of the multiple model we considered, I wanted to share some awesome code that produce maps (I can say that those codes are awesome since Baptiste wrote most of them).

• Visualization of Residuals on a Map of Paris

To plot the neighborhood of the pooling stations, one more time the post on Baptiste’s blog, explains how the shapefile was obtained from cartelec.net

```> library(maptools)
> library(rgdal)
> library(classInt)

To visualize the proxy rate (the average of round one and round two), here is the code

```> elec=data.frame()
> elec=cbind(bt1\$BV,(bt1\$PROCURATIONS+bt2\$PROCURATIONS),(bt1\$EXPRIMES+bt2\$EXPRIMES))
> colnames(elec)=c("BV","PROCURATIONS","EXPRIMES")
> elec=as.data.frame(elec)
> elec\$BV=bt1\$BV```

To get nice colors, function of the rates, we use

```> m=match(paris\$BUREAU,elec\$BV)
> plotvar=100*elec\$PROCURATIONS/elec\$EXPRIMES
> nclr=7
> plotclr=brewer.pal(nclr,"RdYlBu")[nclr:1]
>(plotvar[m], nclr, style="fisher",dataPrecision=1)
> colcode=findColours(class, plotclr)```

and finally

```> par(mar=c(1,1,1,1))
> plot(paris,col=colcode,border=colcode)
> legend(656274.9, 6867308,legend=names(attr(colcode,"table")),
+ fill=attr(colcode, "palette"), cex=1, bty="n",
+ title="Frequence procurations (%)")```

If we consider a model with three explanatory variable, to explain the proxy rate,

```> regt1=lm(PROCURATIONS/INSCRITS~I(POP65P/POP)+
+ I(PROPRIO/LOGEMENT)+I(CS3/POP1564),weights=INSCRITS,data=baset1)```

we can plot the residuals using

```> m=match(paris\$BUREAU,elec\$BV)
> plotvar=100*residuals(regt1)
> nclr=7
> plotclr=brewer.pal(nclr,"RdYlBu")[nclr:1]
>(plotvar[m], nclr, style="fisher",dataPrecision=1)
> colcode=findColours(class, plotclr)
> par(mar=c(1,1,1,1))
> plot(paris,col=colcode,border=colcode)
> legend(656274.9, 6867308,legend=names(attr(colcode,"table")),
+ fill=attr(colcode, "palette"), cex=1, bty="n",title="Residus")```

It might not be a pure random spatial noise… But we could not get better with our small set of covariates.

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