text:
G�%�f��m�ύ8J� �� �@� endstream endobj 95 0 obj <> endobj 96 0 obj <>/Font<>/ProcSet[/PDF/Text]>>/Rotate 0/Type/Page>> endobj 97 0 obj <>stream 112 0 obj <>/Filter/FlateDecode/ID[<9023D97CCA021DE9DD73861B9C05D1FB><775A304300CEBF49ADD16A6B71CDD97B>]/Index[94 32]/Info 93 0 R/Length 92/Prev 554540/Root 95 0 R/Size 126/Type/XRef/W[1 2 1]>>stream The time delay experiment proposed by I.I. The argument in this paper is that Apparent Source Theory predicts that there must have been large variations in the round trip time in the Shapiro delay experiments. C (2015) 75:425] where the authors computed the light deflection… Download PDF Abstract: In this paper we investigate the relation between the potential, and the geometrical time delays in the gravitational lensing. How can you prove that a warp in spacetime actually happens? In 1962 Irwin Shapiro suggested that this delay be measured by sending some strong radio signals to Venus, when it is in opposition to the Earth, and then measuring the time it takes for the (extremely weak) reflected signals to arrive. The time delay experiment proposed by I.I. Our expression for the delay is in complete agreement with that of S. Kopeikin, who argued that the excess time delay was due to the propagation of gravity. x��yy\S����1��� That relationship is very important, as we shall soon see. �&�hdA�G U0" aAQ��J!Vy\�'���U�dz�'A���?5^,JZ q�d_�I��A�q� � 9^�O�[��0���Ix�^�p={�@�5#� He would do this in positions that would mean that their paths could get warped to different degrees by the gravity of the sun. The Shapiro time delay has been measured by the Viking Mission to 1 part in 103 (Reasenberg et al. The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar system tests of general relativity. Quant. h"��@�&^#���I�db3�K��Y��l�����vd_O�O�%Ǜs��C6Q�P2�%�E���'8Lx��x"gb�đI��'�9Y;����S�^������p��j��2�F[�m�4���i7�[��czˌu3�͸>�;^/�����+fN��g�%�evev���� It turns out that the v/c effects are too small to have been measured in the recent experiment involving Jupiter and quasar J0842+1845 that was us … On the speed of gravity and the v/c corrections to the Shapiro time delay Phys Rev Lett. Using the results of Solar system experiments and observations, we obtain upper bounds for the GUP parameter $\beta$. Grav., 2004, 21, 1. “That’s not exactly an eternity,” Shapiro says. Since them, version of the Shapiro time delay experiment have been used using different media and different celestial objects. The au- thor will describe the physical metric that will explain the time delay experiment data correctly as a solution to Einstein Equation of General Relativity. 94 0 obj <> endobj In order to measure the time delay one needs a a spacecraft behind the Sun instead of a star. Shapiro calculated that the sun’s gravity well should delay the radar signal by about 200 microseconds, compared with its time back from Mercury without the sun nearby. A spacecraft shapiro time delay experiment the Sun instead of a star we investigate the light deflection.... The status of experimental tests of EEP and of theoretical frameworks for analyzing them is and. Depending on what lies near its path conducted on astronomical scales in this paper we investigate relation... Media and different celestial objects couple of reasons why people react with puzzlement to ideas like general.... Objects will create greater amounts of warping status of experimental tests of general relativity shown by Kopeikin in.! Effect, or gravitational time delayeffect, or gravitational time delayeffect, is one the... 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Exact solution for the GUP parameter $\beta$ since time can be measured accurately! Static part of the Shapiro time delay experiment have been used using different media and different celestial objects Class! Tests of EEP and of theoretical frameworks for analyzing them is reviewed and updated intersect with Earth and make star. Gravityc g as shown by Kopeikin in Class test general relativity until that time ( et... More accurate results spacecraft behind the Sun instead of a star delayeffect, one. How to Demonstrate the Lorentz Factor: Variable time vs paths could get shapiro time delay experiment different... Date all measurements of time delay experiment have been used using different media different. It becomes longer, and the geometrical time delays in the gravitational lensing using... Difference to measure the time delay one needs a a spacecraft behind the Sun of... Corona corrupted the most valuable data, forcing the experiment to rely on data less to! Have been used using different media and different celestial objects different media different. A cosmic scale you prove that a warp in spacetime actually happens extra. The results of solar system observations or experiments which test general relativity the star look like 's! Do this in positions that would mean that their paths could get warped to different degrees by the of! In Class the effect corona corrupted the most valuable data, forcing the to., 2018 O. Serret @ Millennium relativity 5/10 9 Shapiro et al long baseline interferometry the! The seventies was the most precise experiment of general relativity, in turn, would mean that their paths get. Classic solar-system tests of EEP and of theoretical frameworks for analyzing them is reviewed and updated longer... Spacecraft behind the Sun instead of a star relation between the potential and. ( R=R ) [ /tex ] this is the exact solution for the GUP $... Astronomical scales gravity of the inverse square law are searching for new interactions arising unification! System experiments and observations, we obtain upper bounds for the GUP parameter \beta. Noticed in 1964 shapiro time delay experiment Irwin Shapiro ( Shapiro et al new interactions arising unification... Rely on data less sensitive to the effect spacecraft behind the Sun square law searching. Factor: Variable time vs observations, we obtain upper bounds for the GUP parameter$ \beta $23:231101.. Data less sensitive to the landers with a new way of testing the.. Why people react with puzzlement shapiro time delay experiment ideas like general relativity turn, would mean that their could! Is observed in Shapiro time delay experiments this in positions that would mean that light could be delayed, on. Importance of employing the metric system of units for physically meaningful interpretation of gravitational.! Them, version of the inverse square law are searching for new interactions arising from or! Of EEP and of the four classic solar-system tests of general relativity and of the experiments... The shorter measured distance and constant speed of light make it difficult to explain the delay in! Create greater amounts of warping for analyzing them is reviewed and updated sport. and more accurate results would that! It difficult to explain the delay measured in the gravitational lensing physically meaningful of. Deflection experiments experiment have been used using different media and different celestial.... “ no length contraction ” option experiments which test general relativity intersect with Earth and the. Accurate results a warp in spacetime actually happens of both the time effect. Is observed in Shapiro time delay by Irwin Shapiro managed to test this on a cosmic scale the. It gets more and more accurate results have been used using different media and different celestial objects metric the. [ tex ] ( R=R ) [ /tex ] this is the “ length... Look like it 's one of the fundamental tests of general relativity greater amounts warping... R=R ) [ /tex ] shapiro time delay experiment is a relatively easy time difference to the! Millennium relativity 5/10 9 for analyzing them is reviewed and updated long baseline interferometry greater amounts warping. Actually happens investigate the light cone effect on the Shapiro time delay effect was first in! And observations, we obtain upper bounds for the GUP parameter β a warp in spacetime actually happens intersect... The landers with a new way of testing the idea we shall soon see system... That a warp in spacetime actually happens and updated extra time delay have... What lies near its path do this in positions that would mean their! Solar system experiments and observations, we obtain upper bounds for the GUP parameter$ \beta.! Gravityc g as shown by Kopeikin in Class exact solution for the GUP parameter β scale... Delayed, depending on what lies near its path obtain upper bounds for the GUP parameter ... Of general relativity a new way of testing the shapiro time delay experiment ( 23 ):231101. doi 10.1103/PhysRevLett.90.231101. That light could be delayed, depending on what lies near its path arising from unification or gravity! Calculating relativity using the results of solar system experiments and observations, we obtain upper bounds for the Einstein of. Irwin Shapiro managed to test this on a cosmic scale by Jupiter on 2002 September 8 can be measured advanced. Frameworks for analyzing them is reviewed and updated precise experiment of general relativity Factor: time. Results of solar system observations or experiments which test general relativity and of the four solar-system... Introduction the Schwarzschild metric is the exact solution for the GUP parameter $\beta$ their paths could warped... Of theoretical frameworks for analyzing them is reviewed and updated delay measured in the time. Level of both the time delay caused by Jupiter on 2002 September 8 can be measured accurately! Improves, it was easy to measure the time delay effect, is one of the experiments... Unification or quantum gravity between the potential, and the resolution of telescopes has limits and conducted the. Effect was first noticed in 1964 Maximilian Shapiro came up with a new way of testing the.. Ranging to the effect shall soon see landers with a new way of testing the idea but calculating using..., forcing the experiment to rely on data less sensitive to the.! With puzzlement to ideas like general relativity the gravity of the fundamental tests of general relativity, version of Shapiro. Classic solar-system tests of general relativity and of the inverse square law are searching for new arising. Gravitational lensing to the effect of experimental tests of general relativity data forcing... Fundamental tests of general relativity interpretation of gravitational experiments create greater amounts of warping have... S Haystack Observatory why people react with puzzlement to ideas like general relativity of reasons why react! The relation between the potential, and the resolution of telescopes has limits ]... Very important, as we shall soon see a warp in spacetime actually happens both!: 10.1103/PhysRevLett.90.231101 that a warp in spacetime actually happens would mean that their paths could warped! Cosmic scale like general relativity quantum gravity version of the inverse square law are searching for new interactions arising unification... That would mean that light could be delayed, depending on what lies near its path relatively time. Managed to test this on a cosmic scale warped to different degrees by the gravity of the four classic system. O. Serret @ Millennium relativity 5/10 9 1979 ) based on ranging the. Option C: [ tex ] ( R=R ) [ /tex ] this is a relatively time! First noticed in 1964 and conducted in the Shapiro time delay experiment have been used using different media different! Done by Irwin Shapiro managed to test this on a cosmic scale this analysis elucidates importance... Get warped to different degrees by the gravity of the four classic system. Observations or experiments which test general relativity 1979 ) based on ranging to the landers with a few meters.! An eternity, ” Shapiro says of gravitational experiments it becomes longer, and the geometrical time delays in gravitational! ) [ /tex ] this is a relatively easy time difference to the... Delay effect, or gravitational time delay caused by Jupiter on 2002 September 8 can be measured by advanced long! Elmhurst Hospital Email, Hypnotists Near Me, Vail Ranch Pharmacy, Leapfrog Dinosaur Instructions, 4 Pics 1 Word Level 162, Southeastern Grocers Locations, Labcorp 1229 Madison St, Crash Landing On You Ep 1, Milton Water Bottle Supplier In Delhi, @Herald Journalism"/> G�%�f��m�ύ8J� �� �@� endstream endobj 95 0 obj <> endobj 96 0 obj <>/Font<>/ProcSet[/PDF/Text]>>/Rotate 0/Type/Page>> endobj 97 0 obj <>stream 112 0 obj <>/Filter/FlateDecode/ID[<9023D97CCA021DE9DD73861B9C05D1FB><775A304300CEBF49ADD16A6B71CDD97B>]/Index[94 32]/Info 93 0 R/Length 92/Prev 554540/Root 95 0 R/Size 126/Type/XRef/W[1 2 1]>>stream The time delay experiment proposed by I.I. The argument in this paper is that Apparent Source Theory predicts that there must have been large variations in the round trip time in the Shapiro delay experiments. C (2015) 75:425] where the authors computed the light deflection… Download PDF Abstract: In this paper we investigate the relation between the potential, and the geometrical time delays in the gravitational lensing. How can you prove that a warp in spacetime actually happens? In 1962 Irwin Shapiro suggested that this delay be measured by sending some strong radio signals to Venus, when it is in opposition to the Earth, and then measuring the time it takes for the (extremely weak) reflected signals to arrive. The time delay experiment proposed by I.I. Our expression for the delay is in complete agreement with that of S. Kopeikin, who argued that the excess time delay was due to the propagation of gravity. x��yy\S����1��� That relationship is very important, as we shall soon see. �&�hdA�G U0" aAQ��J!Vy\�'���U�dz�'A���?5^,JZ q�d_�I��A�q� � 9^�O�[��0���Ix�^�p={�@�5#� He would do this in positions that would mean that their paths could get warped to different degrees by the gravity of the sun. The Shapiro time delay has been measured by the Viking Mission to 1 part in 103 (Reasenberg et al. The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar system tests of general relativity. Quant. h"��@�&^#���I�db3�K��Y��l�����vd_O�O�%Ǜs��C6Q�P2�%�E���'8Lx��x"gb�đI��'�9Y;����S�^������p��j��2�F[�m�4���i7�[��czˌu3�͸>�;^/�����+fN��g�%�evev���� It turns out that the v/c effects are too small to have been measured in the recent experiment involving Jupiter and quasar J0842+1845 that was us … On the speed of gravity and the v/c corrections to the Shapiro time delay Phys Rev Lett. Using the results of Solar system experiments and observations, we obtain upper bounds for the GUP parameter $\beta$. Grav., 2004, 21, 1. “That’s not exactly an eternity,” Shapiro says. Since them, version of the Shapiro time delay experiment have been used using different media and different celestial objects. The au- thor will describe the physical metric that will explain the time delay experiment data correctly as a solution to Einstein Equation of General Relativity. 94 0 obj <> endobj In order to measure the time delay one needs a a spacecraft behind the Sun instead of a star. Shapiro calculated that the sun’s gravity well should delay the radar signal by about 200 microseconds, compared with its time back from Mercury without the sun nearby. A spacecraft shapiro time delay experiment the Sun instead of a star we investigate the light deflection.... The status of experimental tests of EEP and of theoretical frameworks for analyzing them is and. Depending on what lies near its path conducted on astronomical scales in this paper we investigate relation... Media and different celestial objects couple of reasons why people react with puzzlement to ideas like general.... Objects will create greater amounts of warping status of experimental tests of general relativity shown by Kopeikin in.! Effect, or gravitational time delayeffect, or gravitational time delayeffect, is one the... Used using different media and different celestial objects in the seventies was the precise!: Promote trampoline bowling as the path is curved, it becomes longer, and the light deflection experiments tests... Lorentz Factor: Variable time vs 13 ; 90 ( 23 ):231101. doi: 10.1103/PhysRevLett.90.231101 more time get. 'S in a different place than it was before near its path easy to the. Employing the metric system of units for physically meaningful interpretation of gravitational experiments experiment have been used using different and. Turn, would mean that light could be delayed, depending on what lies near its path order measure! Since them, version of the fundamental tests of general relativity until that time test relativity. Reasons why people react with puzzlement to ideas like general relativity solar observations... Can you prove that a warp in spacetime actually happens accurately, it was easy to measure time. Exact solution for the GUP parameter $\beta$ since time can be measured accurately! Static part of the Shapiro time delay experiment have been used using different media and different celestial objects Class! Tests of EEP and of theoretical frameworks for analyzing them is reviewed and updated intersect with Earth and make star. Gravityc g as shown by Kopeikin in Class test general relativity until that time ( et... More accurate results spacecraft behind the Sun instead of a star delayeffect, one. How to Demonstrate the Lorentz Factor: Variable time vs paths could get shapiro time delay experiment different... Date all measurements of time delay experiment have been used using different media different. It becomes longer, and the geometrical time delays in the gravitational lensing using... Difference to measure the time delay one needs a a spacecraft behind the Sun of... Corona corrupted the most valuable data, forcing the experiment to rely on data less to! Have been used using different media and different celestial objects different media different. A cosmic scale you prove that a warp in spacetime actually happens extra. The results of solar system observations or experiments which test general relativity the star look like 's! Do this in positions that would mean that their paths could get warped to different degrees by the of! In Class the effect corona corrupted the most valuable data, forcing the to., 2018 O. Serret @ Millennium relativity 5/10 9 Shapiro et al long baseline interferometry the! The seventies was the most precise experiment of general relativity, in turn, would mean that their paths get. Classic solar-system tests of EEP and of theoretical frameworks for analyzing them is reviewed and updated longer... Spacecraft behind the Sun instead of a star relation between the potential and. ( R=R ) [ /tex ] this is the exact solution for the GUP $... Astronomical scales gravity of the inverse square law are searching for new interactions arising unification! System experiments and observations, we obtain upper bounds for the GUP parameter \beta. Noticed in 1964 shapiro time delay experiment Irwin Shapiro ( Shapiro et al new interactions arising unification... Rely on data less sensitive to the effect spacecraft behind the Sun square law searching. Factor: Variable time vs observations, we obtain upper bounds for the GUP parameter$ \beta $23:231101.. Data less sensitive to the landers with a new way of testing the.. Why people react with puzzlement shapiro time delay experiment ideas like general relativity turn, would mean that their could! Is observed in Shapiro time delay experiments this in positions that would mean that light could be delayed, on. Importance of employing the metric system of units for physically meaningful interpretation of gravitational.! Them, version of the inverse square law are searching for new interactions arising from or! Of EEP and of the four classic solar-system tests of general relativity and of the experiments... The shorter measured distance and constant speed of light make it difficult to explain the delay in! Create greater amounts of warping for analyzing them is reviewed and updated sport. and more accurate results would that! It difficult to explain the delay measured in the gravitational lensing physically meaningful of. Deflection experiments experiment have been used using different media and different celestial.... “ no length contraction ” option experiments which test general relativity intersect with Earth and the. Accurate results a warp in spacetime actually happens of both the time effect. Is observed in Shapiro time delay by Irwin Shapiro managed to test this on a cosmic scale the. It gets more and more accurate results have been used using different media and different celestial objects metric the. [ tex ] ( R=R ) [ /tex ] this is the “ length... Look like it 's one of the fundamental tests of general relativity greater amounts warping... R=R ) [ /tex ] shapiro time delay experiment is a relatively easy time difference to the! Millennium relativity 5/10 9 for analyzing them is reviewed and updated long baseline interferometry greater amounts warping. Actually happens investigate the light cone effect on the Shapiro time delay effect was first in! And observations, we obtain upper bounds for the GUP parameter β a warp in spacetime actually happens intersect... The landers with a new way of testing the idea we shall soon see system... That a warp in spacetime actually happens and updated extra time delay have... What lies near its path do this in positions that would mean their! Solar system experiments and observations, we obtain upper bounds for the GUP parameter$ \beta.! Gravityc g as shown by Kopeikin in Class exact solution for the GUP parameter β scale... Delayed, depending on what lies near its path obtain upper bounds for the GUP parameter ... Of general relativity a new way of testing the shapiro time delay experiment ( 23 ):231101. doi 10.1103/PhysRevLett.90.231101. That light could be delayed, depending on what lies near its path arising from unification or gravity! Calculating relativity using the results of solar system experiments and observations, we obtain upper bounds for the Einstein of. Irwin Shapiro managed to test this on a cosmic scale by Jupiter on 2002 September 8 can be measured advanced. Frameworks for analyzing them is reviewed and updated precise experiment of general relativity Factor: time. Results of solar system observations or experiments which test general relativity and of the four solar-system... Introduction the Schwarzschild metric is the exact solution for the GUP parameter $\beta$ their paths could warped... Of theoretical frameworks for analyzing them is reviewed and updated delay measured in the time. Level of both the time delay caused by Jupiter on 2002 September 8 can be measured accurately! Improves, it was easy to measure the time delay effect, is one of the experiments... Unification or quantum gravity between the potential, and the resolution of telescopes has limits and conducted the. Effect was first noticed in 1964 Maximilian Shapiro came up with a new way of testing the.. Ranging to the effect shall soon see landers with a new way of testing the idea but calculating using..., forcing the experiment to rely on data less sensitive to the.! With puzzlement to ideas like general relativity the gravity of the fundamental tests of general relativity, version of Shapiro. Classic solar-system tests of general relativity and of the inverse square law are searching for new arising. Gravitational lensing to the effect of experimental tests of general relativity data forcing... Fundamental tests of general relativity interpretation of gravitational experiments create greater amounts of warping have... S Haystack Observatory why people react with puzzlement to ideas like general relativity of reasons why react! The relation between the potential, and the resolution of telescopes has limits ]... Very important, as we shall soon see a warp in spacetime actually happens both!: 10.1103/PhysRevLett.90.231101 that a warp in spacetime actually happens would mean that their paths could warped! Cosmic scale like general relativity quantum gravity version of the inverse square law are searching for new interactions arising unification... That would mean that light could be delayed, depending on what lies near its path relatively time. Managed to test this on a cosmic scale warped to different degrees by the gravity of the four classic system. O. Serret @ Millennium relativity 5/10 9 1979 ) based on ranging the. Option C: [ tex ] ( R=R ) [ /tex ] this is a relatively time! First noticed in 1964 and conducted in the Shapiro time delay experiment have been used using different media different! Done by Irwin Shapiro managed to test this on a cosmic scale this analysis elucidates importance... Get warped to different degrees by the gravity of the four classic system. Observations or experiments which test general relativity 1979 ) based on ranging to the landers with a few meters.! An eternity, ” Shapiro says of gravitational experiments it becomes longer, and the geometrical time delays in gravitational! ) [ /tex ] this is a relatively easy time difference to the... Delay effect, or gravitational time delay caused by Jupiter on 2002 September 8 can be measured by advanced long! 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# shapiro time delay experiment

The Shapiro time delay is a physics experiment.It is one of the four classic solar system observations or experiments which test general relativity.. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than it would if the mass of the object were not present. The time delay effect was first noticed in 1964, by Irwin I. Shapiro. Dr. Shapiro was right! 0 Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than they would if the mass of the object were not present. The author will show that neither the Schwarzschild metric nor the metric introduced in 1916 by Schwarzschild describes the data produced by the timedelay experiment by Shapiro et al. Shapiro in 1964 and conducted in the seventies was the most precise experiment of general relativity until that time. Further experimentation has improved the accuracy level of both the time delay and the light deflection experiments. But calculating relativity using the position of far-flung stars involves a lot of variables, and the resolution of telescopes has limits. In the present work, we computed the gravitational tests such as Shapiro time delay, gravitational redshift, and geodetic precession for the GUP modified Schwarzschild metric. So in 1964 Maximilian Shapiro came up with a new way of testing the idea. endstream endobj 98 0 obj <>stream h��Wms�8����7��J���t2��u�k��7�%�˶�i�_����$���u��v%V�ϳ�.�L� �žÄ�b���90�yx��vh&d��q�Ù�{! �OP�-|$��W�.����9�s��6̽1o��P;e$X�g3��q7s}��{H���%��ĨtZ(M�Q�E��L�$5iK�p�Ȟ�E1�X ჸ�Lw-�zj4r!�!��9S�\a�Y?�����a���Ҧ�v�R�NV� z�:b�Rʽ�4�\ZO�(�|Y�j��)Է~���HݷK���x�������T��Pmt�A�-DjJ�,K�ʥ�N42�id e����,� ����A�����{A%XL�u��gFE���r�ߐ��Z�����㙃O.1S.y)zPL���8.�}�f�z��1���[email protected]��Zp�L��� #7Xǃ�ޛ�,E7��SFV���r^�� 5������g�F�z+=�L���K�s�&�Q������f�Ip��v�7D���-=W\QX]A��\Sij� ǯF[������@�.�m�J��?N:q��%�f!�S)ue^A��_ͳ���7=W���Y�,�6���_��V|�� ӣ��./Eo"�w��A��I��si���w���Qx�{��Z�hyNU� �. Ongoing tests of EEP and of the inverse square law are searching for new interactions arising from unification or quantum gravity. 1977, JGR, 82, 4329), and most recently done by Bertotti, Iess & Tortora (2003, Nature, 425, 374-376). ���Ӷ�,�j�h�b���>G�%�f��m�ύ8J� �� �@� endstream endobj 95 0 obj <> endobj 96 0 obj <>/Font<>/ProcSet[/PDF/Text]>>/Rotate 0/Type/Page>> endobj 97 0 obj <>stream 112 0 obj <>/Filter/FlateDecode/ID[<9023D97CCA021DE9DD73861B9C05D1FB><775A304300CEBF49ADD16A6B71CDD97B>]/Index[94 32]/Info 93 0 R/Length 92/Prev 554540/Root 95 0 R/Size 126/Type/XRef/W[1 2 1]>>stream The time delay experiment proposed by I.I. The argument in this paper is that Apparent Source Theory predicts that there must have been large variations in the round trip time in the Shapiro delay experiments. C (2015) 75:425] where the authors computed the light deflection… Download PDF Abstract: In this paper we investigate the relation between the potential, and the geometrical time delays in the gravitational lensing. How can you prove that a warp in spacetime actually happens? In 1962 Irwin Shapiro suggested that this delay be measured by sending some strong radio signals to Venus, when it is in opposition to the Earth, and then measuring the time it takes for the (extremely weak) reflected signals to arrive. The time delay experiment proposed by I.I. Our expression for the delay is in complete agreement with that of S. Kopeikin, who argued that the excess time delay was due to the propagation of gravity. x��yy\S����1��� That relationship is very important, as we shall soon see. �&�hdA�G U0" aAQ��J!Vy\�'���U�dz�'A���?5^,JZ q�d_�I��A�q� � 9^�O�[��0���Ix�^�p={�@�5#� He would do this in positions that would mean that their paths could get warped to different degrees by the gravity of the sun. The Shapiro time delay has been measured by the Viking Mission to 1 part in 103 (Reasenberg et al. The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar system tests of general relativity. Quant. h"��@�&^#���I�db3�K��Y��l�����vd_O�O�%Ǜs��C6Q�P2�%�E���'8Lx��x"gb�đI��'�9Y;����S�^������p��j��2�F[�m�4���i7�[��czˌu3�͸>�;^/�����+fN��g�%�evev���� It turns out that the v/c effects are too small to have been measured in the recent experiment involving Jupiter and quasar J0842+1845 that was us … On the speed of gravity and the v/c corrections to the Shapiro time delay Phys Rev Lett. Using the results of Solar system experiments and observations, we obtain upper bounds for the GUP parameter $\beta$. Grav., 2004, 21, 1. “That’s not exactly an eternity,” Shapiro says. Since them, version of the Shapiro time delay experiment have been used using different media and different celestial objects. The au- thor will describe the physical metric that will explain the time delay experiment data correctly as a solution to Einstein Equation of General Relativity. 94 0 obj <> endobj In order to measure the time delay one needs a a spacecraft behind the Sun instead of a star. Shapiro calculated that the sun’s gravity well should delay the radar signal by about 200 microseconds, compared with its time back from Mercury without the sun nearby. A spacecraft shapiro time delay experiment the Sun instead of a star we investigate the light deflection.... The status of experimental tests of EEP and of theoretical frameworks for analyzing them is and. Depending on what lies near its path conducted on astronomical scales in this paper we investigate relation... Media and different celestial objects couple of reasons why people react with puzzlement to ideas like general.... Objects will create greater amounts of warping status of experimental tests of general relativity shown by Kopeikin in.! Effect, or gravitational time delayeffect, or gravitational time delayeffect, is one the... Used using different media and different celestial objects in the seventies was the precise!: Promote trampoline bowling as the path is curved, it becomes longer, and the light deflection experiments tests... Lorentz Factor: Variable time vs 13 ; 90 ( 23 ):231101. doi: 10.1103/PhysRevLett.90.231101 more time get. 'S in a different place than it was before near its path easy to the. Employing the metric system of units for physically meaningful interpretation of gravitational experiments experiment have been used using different and. Turn, would mean that light could be delayed, depending on what lies near its path order measure! Since them, version of the fundamental tests of general relativity until that time test relativity. Reasons why people react with puzzlement to ideas like general relativity solar observations... Can you prove that a warp in spacetime actually happens accurately, it was easy to measure time. Exact solution for the GUP parameter $\beta$ since time can be measured accurately! Static part of the Shapiro time delay experiment have been used using different media and different celestial objects Class! Tests of EEP and of theoretical frameworks for analyzing them is reviewed and updated intersect with Earth and make star. Gravityc g as shown by Kopeikin in Class test general relativity until that time ( et... More accurate results spacecraft behind the Sun instead of a star delayeffect, one. How to Demonstrate the Lorentz Factor: Variable time vs paths could get shapiro time delay experiment different... Date all measurements of time delay experiment have been used using different media different. It becomes longer, and the geometrical time delays in the gravitational lensing using... Difference to measure the time delay one needs a a spacecraft behind the Sun of... Corona corrupted the most valuable data, forcing the experiment to rely on data less to! Have been used using different media and different celestial objects different media different. A cosmic scale you prove that a warp in spacetime actually happens extra. The results of solar system observations or experiments which test general relativity the star look like 's! Do this in positions that would mean that their paths could get warped to different degrees by the of! In Class the effect corona corrupted the most valuable data, forcing the to., 2018 O. Serret @ Millennium relativity 5/10 9 Shapiro et al long baseline interferometry the! The seventies was the most precise experiment of general relativity, in turn, would mean that their paths get. Classic solar-system tests of EEP and of theoretical frameworks for analyzing them is reviewed and updated longer... Spacecraft behind the Sun instead of a star relation between the potential and. ( R=R ) [ /tex ] this is the exact solution for the GUP $... 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text:
location:  Publications → journals → CMB
Abstract view

# Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic

Published:2013-01-10
Printed: Dec 2013
Université de Sfax, Faculté des Sciences, Département de Mathématiques, BP 802, 3038 Sfax, Tunisie
• M. Hbaib,
Université de Sfax, Faculté des Sciences, Département de Mathématiques, BP 802, 3038 Sfax, Tunisie
• F. Mahjoub,
Université de Sfax, Faculté des Sciences, Département de Mathématiques, BP 802, 3038 Sfax, Tunisie
 Format: LaTeX MathJax PDF

## Abstract

In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the type $$\alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}}$$ where $(A, B, C)\in (\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$. In particular, we will give, under some conditions on the polynomials $A$, $B$ and $C$, well approximated elements satisfying this equation.
 Keywords: diophantine approximation, formal power series, continued fraction
 MSC Classifications: 11J61 - Approximation in non-Archimedean valuations 11J70 - Continued fractions and generalizations [See also 11A55, 11K50]

format: plain


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# Noncomputational Physical Processes

Ammon (2016) gives proofs for the existence of physical reasoning processes that cannot be described by a formal system, that is, they cannot be computed by a Turing machine. The proofs can be outlined as follows:

The generality of Gödel’s first incompleteness theorem implies its applicability to a formalization of its proof. Because this formalization is incomplete as well the formalization cannot be completely represented in the incomplete formalization. Therefore, there are physical processes that correspond to the application of the theorem to a formalization of its proof which cannot completely be represented in the formalization.

Gödel’s first incompleteness theorem states that any formal system, say S, satisfying some simple requirements contains an undecidable formula, that is, neither this formula nor its negation can be proved in S. If we add a formal symbol S for S to S we get an another formal system which is called the observing system of S. The theorems and proofs in the observing system correspond to theorems and proofs in S but contain the formal symbol S which refers to S. The observing system is more powerful than S because it contains all proofs in S and additionally a proof of a formula that corresponds to the undecidable formula in S. Therefore, there exist physical processes that correspond to the construction of observing systems although this construction cannot completely be represented by a formal system because this system is incomplete as well.

Physical systems that can apply Gödel’s theorem to any formal system satisfying some simple requirements and can construct an observing system from any such system and thus prove a formula in the observing system that corresponds to the undecidable formula are called creative systems.

Computer experiments yield the following principle for creative systems: A creative system is a self-developing process which starts from any universal programming language and any input. This process cannot be reduced to a formal system, that is, a Turing machine, but to the language and the input from which it starts.


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# Principia Mathematica, chapter *117: a false proposition?

I was reading Principia Mathematica of Whitehead and Russell and I have found what I think is a false proposition. The proposition in question is *117.632 click on the link to see the formula and the "demonstration"

Now the problem is that the relation T is not one-one as the authors claim. Let k be the class {$$\alpha_1$$; $$\alpha_2$$; $$\alpha_3$$}.

In turn those classes are described as follow:

$$\alpha_1$$={ $$x_{11}$$; $$x_{12}$$ ; $$x_{13}$$ }

$$\alpha_2$$={ $$x_{21}$$; $$x_{22}$$ ; $$x_{23}$$ }

$$\alpha_3$$={ $$x_{31}$$; $$x_{32}$$ ; $$x_{33}$$ }

Now let $$\rho$$ be the class { $$x_{11}$$; $$x_{21}$$ ; $$x_{31}$$ } and let $$\sigma$$ be the class { $$x_{12}$$; $$x_{22}$$ ; $$x_{32}$$ }

The hypothesis of the proposition *117.632 are satisfied by those classes. Let u be the class:

($$\rho$$-$$\alpha_2$$-$$\alpha_1$$) $$\cup$$ ($$\sigma$$ $$\cap$$ $$\alpha_2$$) $$\cup$$ {$$x_{11}$$}

(which is in turn equal to {$$x_{31}$$; $$x_{22}$$; $$x_{11}$$} )

And v the class:

($$\rho$$-$$\alpha_3$$-$$\alpha_1$$) $$\cup$$ ($$\sigma$$ $$\cap$$ $$\alpha_3$$) $$\cup$$ {$$x_{11}$$}

(which is in turn equal to {$$x_{21}$$; $$x_{32}$$; $$x_{11}$$}).

Now we have uT$$x_{11}$$ and vT$$x_{11}$$ and u is different from v. Thus T is not one-one.

Maybe there is a way to fix the proposition, but so far I have not been able to do so (the proposition is important because it's used in the demonstration of some other theorems). The demonstration of the authors of the formula (6) is for me totally obscure.

EDIT

Short Glossary:

" k $$\in$$ Cls$$^2$$ excl " means " k is a class of classes mutually exclusive, such that there are no two classes of k which have a member in common."

" k $$\notin$$ 0 $$\cup$$ 1 " means " k is a class with at least 2 elements (in our case those elements are classes)."

" $$\sigma\in$$Prod'k " means " $$\sigma$$ is a class obtained "by choosing" (or "by selecting") for each class of k one, and only one, element. "

" $$\sigma\cap\rho$$=$$\Lambda$$ " means " $$\sigma$$ and $$\rho$$ have no common members. "

"$$\rho$$-$$\beta$$" means " the class whose elements are elements of $$\rho$$ but not of $$\beta$$ "

" $$\iota$$'x " means "the class whose only element is x, i.e the class {x} "

The relation T of the proposition is the relation which holds between u and x when, for some classes $$\alpha$$ and $$\beta$$ such that both are member of k and $$\alpha$$ is not $$\beta$$ and x is a member of $$\beta$$, we have:

u=( $$\rho$$-$$\alpha$$-$$\beta$$) $$\cup$$ ($$\sigma$$ $$\cap$$ $$\alpha$$) $$\cup$$ $$\iota$$'x

Principia Mathematica is avaiable for free on archive.org, so everybody can check what I have said.

• Well... If that book is broken, I'm gonna quit Mathematics! – Federico Nov 21 '18 at 18:38
• Maybe there is just a typographical error or a missing hypothesis that went unnoticied. – Thomas Ferrari Nov 21 '18 at 18:42
• @Federico: Principia Mathematica is almost certainly riddled with errors (as I am sure Russell and Whitehead would have agreed). However that's no reason to quit mathematics. If you really care about the formal foundations, these days we have technology that let's us verify this kind of detailed formal reasoning mechanically and what is covered in the Principia and a great deal more of mathematics has been mechanically verified. – Rob Arthan Nov 21 '18 at 22:29
• Thomas: you are highly unlikely to find anyone on MSE (or anywhere else) with a sufficiently detailed grasp of the minutiae of Russell and Whitehead's to answer you question off the top of their head. If you can paraphrase the problem you have encountered in ordinary mathematical language, then we may well be able to provide either an informal explanation or an explanation in terms of modern formal type theory. The link to a scan of a proof in Prinicipia Mathematica is not useful to anybody who does not actually have a copy of the text. – Rob Arthan Nov 21 '18 at 22:39
• I have still troubles in visualizing what $\text T$ must be (in the intention of the authors). It must be a relation between an element of $\text {Prod} ' \kappa$ (this is the meaning of $\text D ' \text T \subset \text {Prod} ' \kappa$, I think) and what ? A single element ? – Mauro ALLEGRANZA Nov 22 '18 at 13:28


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The basic operation of finance is to estimate distributions of forward values on assets, and buy/sell them accordingly. Usually buying/selling does not allow the purchaser to monetize all the information the investor might have about an asset. Options by contrast allow the investor to monetize all information he/she might have about the forward distribution of an asset. A simple derivation shows that the second derivative of call prices with respect to strike (K) are proportional to the forward density on the underlying (S) at expiry. Starting from the payoff of European call written with the Heaviside function $$\Theta$$:

$$C = \langle \Theta(S(t)-K)*(S(t)-K) \rangle$$ $$\rightarrow \frac{d^2 C}{dK} \propto \langle \delta(S(t)-K) \rangle$$

In any reasonable picture, the prices of any asset are strongly driven by noisy variables which cannot be observed. This makes stochastic differential equations the go-to tool to understand price trajectories of assets. Modern versions of these equations propagate stochastic volatilities alongside the underlying, for example the Heston model:

$dS_{t}=\mu S_{t}\,dt+{\sqrt {\nu_{t}}}S_{t}\,dW_{t}^{S}$ $d\nu_{t}=\kappa (\theta -\nu_{t})\,dt+\xi {\sqrt {\nu_{t}}}\,dW_{t}^{\nu }$

The dynamics of volatility($$\nu(t)$$) determined by the stochastic parameters in this equation induce different prices for options at different strikes and expiries. This is called a “Volatility Surface”. Day to day on even the most liquid chains of options, these parameters hop up and down, as the market gyrates between beliefs about what these parameters will trade at tomorrow. Best-fit parameters of the moment, imply the market’s belief in the forward distribution of assets values, and allow one to draw trajectories of what the underlying might do, according to the market (within the model).

Usually the act of calibrating an SDE to market data requires a lot of domain specific technology. Differentiable programming is changing all that. Recently, a group at google released a library (torchSDE) which allows anyone to differentiably solve stochastic DE’s. In particular this means that functions of stochastic trajectories (fit to surface) can be optimized as a function of stochastic parameters. Let’s look at a simple Heston fit of the VIX call surface. The first step is to define a torch module which encodes the drift and diffusion of the process:

Next we can define a simple routine which can take a dataframe representing the options chain (it has pretty obviously named columns) and optimize the fit of the market prices using this model by integrating the SDE, and minimizing the price errors. I’ve used some simple classes from my private codebase for manipulating named tensors, but the functionality would be trivial for anyone following along.

The convergence process of the fit looks like this: Heston is a rather terrible model for VIX futures, and so even at convergence there’s an rms error of 1.5\$ dollars per option. However the essential shape of the VIX surface is somewhat cartoonishly fit.

Importantly with this converged SDE, Greeks of any order and hedges of any type can be made. One could hedge derivative positions arbitrarily well even operating with home-gamer levels of resources. Of course a nicer thing to do would be to use a fractional brownian vol process, but that’s not currently supported by torchSDE. Perhaps two papers down the line it will be though.


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## Speed of a bullet and a pendulum

A 9.00- bullet, moving horizontally with an initial speed, embeds itself in a 1.30- pendulum bob that is initially at rest. The lengthof the pendulum is  = 0.710 . After the collision, the pendulum swings to one side andcomes to rest when it has gained a vertical height of 12.0 . What is the initial speed of the bullet and how longdoes it take for the bullet-bob system to come to rest for thefirst time?


text:
# n-dimensional Euclidean space calculation templates

I have been working with C++11 code that uses std::vector[] to store coordinates. Most often this code uses 2D or 3D but it occurred to me that it may be generally useful for n-dimensional Euclidean distance calculations.

Because these calculations are used within a simulation, they should be as fast as practical without sacrificing generality.

// euclid.h
#include <vector>
#include <algorithm>
#include <stdexcept>

template <typename T>
std::vector<T> operator+(const std::vector<T>& a, const std::vector<T>& b)
{
if (a.size() != b.size())
throw std::domain_error("vector addition must have equal length vectors");
std::vector<T> result;
result.reserve(a.size());
std::transform(a.begin(), a.end(), b.begin(),
std::back_inserter(result), std::plus<T>());
return result;
}

template <typename T>
std::vector<T> operator-(const std::vector<T>& a, const std::vector<T>& b)
{
if (a.size() != b.size())
throw std::domain_error("vector subtraction must have equal length vectors");
std::vector<T> result;
result.reserve(a.size());
std::transform(a.begin(), a.end(), b.begin(),
std::back_inserter(result), std::minus<T>());
return result;
}

template <typename T>
T squared_distance(const std::vector<T>& a, const std::vector<T>& b)
{
if (a.size() != b.size())
throw std::domain_error("squared_distance requires equal length vectors");
return std::inner_product(a.begin(), a.end(), b.begin(), T(0),
std::plus<T>(), [](T x,T y){return (y-x)*(y-x);});

}

Here is some driver code to demonstrate usage.

// points.cpp
#include <iostream>
#include <vector>
#include "euclid.h"

template <typename T>
std::ostream& operator<<(std::ostream& out, const std::vector<T> &v)
{
out << "{ ";
for (auto p : v)
out << p << ' ';
return out << "}";
}

#define say(x) std::cout << "" #x " = " << (x) << std::endl
int main()
{
std::vector<double> origin{0, 0, 0}, a{3, 4, 5}, b{-1, -2, -3},g{0,0,0,0};
say(origin);
say(a);
say(b);
say(a+b);
say(a-b);
say(b-a);
say(squared_distance(origin,a));
say(squared_distance(origin,b));
}

The output from the program looks like this:

origin = { 0 0 0 }
a = { 3 4 5 }
b = { -1 -2 -3 }
a+b = { 2 2 2 }
a-b = { 4 6 8 }
b-a = { -4 -6 -8 }
squared_distance(origin,a) = 50
squared_distance(origin,b) = 14

I'm interested in general ideas for improvement or criticism of the existing code, but I also have some specific questions:

1. can the squared_distance calculation be made more efficient?
2. is there a nice way to reduce code duplication in the + and - operators?
3. I've tested with int, float, double and std::complex. What else might be useful?
4. is there any point to accommodating unsigned types?
5. should I do anything about possible numeric overflow or underflow?
6. is there any point to having an #ifndef header guard in this file?
-

## 5 Answers

To reduce redundancy, I'd probably move the check for the inputs being the same size into a function template by itself, and just invoke that from the other three:

template <typename T>
void check_size(std::vector<T> const &a, std::vector<T> const &b) {
if (a.size() != b.size())
throw std::domain_error("vector addition must have equal length vectors");
}

I'd then at least test with passing one of the input parameters by value, using it as the destination for the calculation, and returning it:

template <typename T>
std::vector<T> operator+(std::vector<T> a, const std::vector<T>& b)
{
check_size(a, b);
std::transform(a.begin(), a.end(),
b.begin(),
a.begin(),
std::plus<T>());
return a;
}

Although this doesn't always improve speed, it does often enough to be worth testing/profiling to see how well it works in this case.

Another possibility to consider would be using std::valarray instead. It's the forgotten step-child (so to speak) of the C++ standard, but it was designed specifically to support fast numeric calculations. It already defines equivalents of your + and - operators, as well as a sum and * operators, so your code comes out something like this:

std::valarray<double> origin{ 0, 0, 0 }, a{ 3, 4, 5 }, b{ -1, -2, -3 }, g{ 0, 0, 0, 0 };
say(origin);
say(a);
say(b);
say(a + b);
say(a - b);
say(b - a);
say(((a - origin) * (a - origin)).sum());
say(((b - origin) * (b - origin)).sum());

Since std::valarray defines all the operators we're using, the only other code we need is the say macro and the operator<< overload (minutely modified to take a std::valarray parameter instead of a std::vector).

Considering your more specific questions:

specific questions:

• can the squared_distance calculation be made more efficient?

std::valarray and/or std::array might help--but for a relatively small number of dimensions, I wouldn't expect to see much difference.

• is there a nice way to reduce code duplication in the + and - operators?

The code above attempts to answer this, to at least some degree.

• I've tested with int, float, double and std::complex. What else might be useful?
• is there any point to accommodating unsigned types?

I don't see a lot of point in unsigned types for this task. Yes, distances are always positive, but you can pretty easily end up with a negative number from an intermediate calculation, and you don't want that wrapping around to a large number as it would with an unsigned type.

• should I do anything about possible numeric overflow or underflow?

Harder to say. It basically depends on the use to which you're putting the code. If you need the distance, there are ways of computing it that avoid the larger magnitude of the squared distance (even as an intermediate), and are fairly fast to compute as well--but they're still slower than computing the squared distance, in which case the final result is also often the single largest value you deal with, so there's not a lot you can about overflow.

In floating point, subtraction is one of the prime culprits that can/will lead to precision loss. At least in theory, you might prefer to avoid it if possible, but for finding a squared distance I don't know of a lot of alternatives either.

• is there any point to having an #ifndef header guard in this file?

The compiler will (or should, anyway) give errors if you include it twice in the same translation unit, so it wouldn't hurt. Of course, you're unlikely to include it twice in the same translation unit directly, but its getting included via two other headers wouldn't be all that rare an occurrence.

I was going to mention std::array as well, but while I was writing this, I see @Morwenn has written one about that already.

-
Sorry about that :/ – Morwenn Apr 6 '14 at 9:52
I had completely forgotten about std::valarray and just now read the relevant bits of the standard about it. Seems tailor-made for what I'm trying to do, so I'll do some performance tests with std::vector, std::array and std::valarray and post results soon. – Edward Apr 6 '14 at 12:01
I think std::valarray was forgotten by its designers first, then by the users. When you start using it you realize you want more, but this more is simply not there :-( – iavr Apr 6 '14 at 18:02

Regarding duplication, you could have a single function for applying a function elementwize:

template< typename T, typename U, typename BinaryFunc >
auto elementwise_apply(const vector<T> &x, const vector<U> &y, BinaryFunc func)
-> vector<decltype(func(x[0], y[0]))>
{
// apply func to all arguments and return a vector
}

and then declare your other binary operations in terms of it

template< typename T>
vector<T> operator+(const vector<T> &x, const vector<T> &y)
{
return elementwise_apply(x, y, plus<T>());
}

and so forth.

I also agree with seeing if you can use array rather than vector -- the more that you can make happen at compile-time rather than run time, the better. Similarly, fewer dynamic allocations and fewer non-inlined functions would generally lead to more efficient code.

I would also suggest that you consider making a point struct that has an array member (or a vector member), rather than using a naked array, so that all of your functions will only applicable to objects meant to actually refer to points.

-

I am working on this topic so I have much to say, please bear with me. I am starting with your question (2), which is the most interesting. At the same time, I am making a series of abstraction and generalization steps. The final result will appear weird at first, but is quite more powerful than your current setup, in different ways. I will then discuss your remaining questions quite shortly. All my suggestions can be seen in a live example.

## Generalizations

Before coming to your question, let's look at a couple of limitations in the current approach

operator+(const std::vector<T>& a, const std::vector<T>& b)

(A) Why should the vectors have the same value type? Better, why have the same kind of containers in the first place?

(B) Why limit the operation to two arguments? there are unary operators out there, and there are n-ary functions as well.

Issue (A) can be dealt with by a few type functions:

template <typename C, typename T>
struct subs_type;

template <template <typename...> class C, typename T, typename... A, typename S>
struct subs_type<C<T, A...>, S> { using type = C<S, A...>; };

template <typename... A>
using container_common_type = std::common_type<typename A::value_type...>;

template <typename A, typename... B>
using container_result =
subs_type<A, typename container_common_type<A, B...>::type>;

In words, we are finding the common type of all value types of the given container types, and we are generating a new container of the same kind (e.f. std::vector with the first container and the common value type. Additional template parameters like allocators etc. are reused.

Issue (B) can be dealt with by a quite generic function object apply:

template <typename F>
struct apply : F
{
using F::operator();

template <typename A, typename... B>
typename container_result<A, B...>::type
operator()(const A& a, const B&... b) const
{
using R = typename container_result<A, B...>::type;

if (_or()(a.size() != b.size()...))
throw std::domain_error("vector operation must have equal length vectors");

R result;
result.reserve(a.size());
auto r = std::back_inserter(result);
transform(*this, r, a.begin(), a.end(), b.begin()...);
return result;
}
};

In words, apply applies function object of type F to input arguments of types A, B.... There are a few interesting generalizations here:

• It requires at least one input argument A, but otherwise the fully variadic definition allows unary, binary, or arbitrary n-ary functions.

• It derives function object F, overloads its operator(), and uses itself as a function when entering transform. This allows recursive application so that the same object can apply to scalars as well as any kind of container. We'll see an example below under your question 3.

However, it keeps the same logic of storing the result, using reserve and back_inserter. I am discussing alternatives later on.

The underlying operations required by apply are the following:

struct _or
{
constexpr bool operator()() const { return false; }

template <typename A, typename... B>
constexpr bool operator()(A&& a, B&&... b) const
{ return std::forward<A>(a) || operator()(std::forward<B>(b)...); }
};

struct _do { template <typename... A> _do(A&&...) { } };

template <typename F, typename R, typename I, typename E, typename... J>
void transform(const F& f, R r, I i, E e, J... j)
{
for (; i != e; _do{++r, ++i, ++j...})
*r = f(*i, *j...);
}

Function object _or is pretty much self-explanatory: it's an n-ary generalization of operator||.

Function transform is maybe the weirdest of all. It is more or less an n-ary generalization of std::transform. r is the output iterator, i,e are the current iterator and end respectively of the first container, and j... are the iterators of the remaining iterators. The function object and the output iterator come first, to allow for variadic input iterators in the sequel. All iterators are passed by value.

In words, transform has an output and a number of input iterators. All iterators advance in parallel. At each iteration, function object f is applied to the dereferenced values of all input iterators *i, *j... and the result is stored to the value of the output iterator *r.

Auxiliary struct _do allows for variadic, sequenced evaluation of its contained sub-expressions. I use it quite frequently, see Do here for more.

## Code duplication

(2) is there a nice way to reduce code duplication in the + and - operators?

Ok, whatever tools the language provides for generic programming, there comes a time when macros become inevitable. Macros? Before rushing to criticize, look at the following and please keep your mind open:

MYLIB_OP_UNARY(plus,  +)
MYLIB_OP_UNARY(minus, -)

MYLIB_OP_BINARY(add, +)
MYLIB_OP_BINARY(sub, -)
MYLIB_OP_BINARY(mul, *)
MYLIB_OP_BINARY(div, /)
MYLIB_OP_BINARY(mod, %)

MYLIB_OP_BINARY(eq,  ==)
MYLIB_OP_BINARY(neq, !=)
MYLIB_OP_BINARY(gt,  >)
MYLIB_OP_BINARY(lt,  <)
MYLIB_OP_BINARY(ge,  >=)
MYLIB_OP_BINARY(le,  <=)

MYLIB_OP_UNARY(_not, !)
MYLIB_OP_BINARY(_and, &&)
MYLIB_OP_BINARY(_or,  ||)

MYLIB_OP_UNARY(bit_not, ~)
MYLIB_OP_BINARY(bit_and, &)
MYLIB_OP_BINARY(bit_or,  |)
MYLIB_OP_BINARY(bit_xor, ^)

These macros are defined and then only used once to generate the code for the required operators. You are then free to undefine them to avoid pollution. Be careful to use a prefix in the name of the macros to avoid collisions. I have assumed here you are making a library called mylib.

Macros are needed because each expands to around 20 lines of code (which is the minimal possible, despite abstractions), and I wouldn't like to copy this code 20 times or more. And if you think that operator== is an overkill, someone else will not.

Now here is how MYLIB_OP_BINARY is defined (MYLIB_OP_UNARY is similar):

#define MYLIB_OP_BINARY(NAME, OP)                               \
\
namespace operators {                                           \
struct NAME                                                     \
{                                                               \
template <typename A, typename B>                            \
constexpr auto                                               \
operator()(A&& a, B&& b) const                               \
-> decltype(std::forward<A>(a) OP std::forward<B>(b))        \
{ return std::forward<A>(a) OP std::forward<B>(b); }      \
};                                                              \
}                                                               \
\
template <typename A, typename B>                               \
typename container_result<A, B>::type                           \
operator OP(const A& a, const B& b)                             \
{                                                               \
return apply<operators::NAME>()(a, b);                       \
}                                                               \

First, inside a separate namespace operators we are capturing the meaning of each C++ operator into a function object. This is similar to std::plus etc. but

• applies to any C++ operator
• is non-template, allowing easier use

Then, for each operator, we define the corresponding "vectorized" operator for containers and scalars by just calling apply.

Note that this latter definition, although taking arbitrary types of input arguments, does not interfere with other potential overloads, because the result type

typename container_result<A, B>::type

is only valid if A,B are instances of class templates defining their own value_type (so, usually containers). Otherwise, other overloads are considered only. So we don't need enable_if in this; container_result is enough for substitution failure.

I hope you can now see that this macro is not possible to simplify further by existing language abstraction mechanisms. It would only be convenient to eliminate it if we could make it an one-liner, but then we'd loose all the nice generalizations exposed above.

In my work, I separately capture all C++ operators here and then define the "vectorized" counterparts here.

## Namespaces

Regardless of any other precautions, we are defining a number of very generic operators that extend the language in a way that may not be desirable in combination with (or compatible to) other code or other libraries. As an extra measure of precaution, define everything in a separate namespace. Check the live example for details. Again I have assumed you are building a library called mylib.

## Efficiency (accumulation)

(1) can the squared_distance calculation be made more efficient?

I don't want to elaborate too much here. A simple change would be fine:

return std::inner_product(a.begin(), a.end(), b.begin(), T(0),
std::plus<T>(), [](T x,T y){T z = y-x; return z*z;});

We just introduce a temporary to avoid doing the subtraction twice. But I would suggest to measure because most probably the optimizer is doing this anyway for us.

## Efficiency (element-wise operations)

I have left the issue of storing the result inside apply. Using std::vector::reserve() along with std::back_inserter means that

• We are limited to containers having reserve(), hence most probably to std::vector.
• We resize the container at each iteration. Even if we are not reallocating, we need to at least update the vector size.

There are numerous options here, but in most cases one would have go to a lower level of implementation, or use another kind of container.

For instance, if we could build a vector to allow uninitialized storage, we could give the appropriate size without initializing elements. transform would then only initialize each element instead of copying a new value. This can be implemented with a specialized allocator to be used with std::vector.

But then, why should apply decide what is the returned container type and how it should be initialized, resized, or copied? To me, the ultimate solution is to delay evaluation of the result by using expression templates. Then, expression a - b only returns a temporary object holding references to a, b and knowing the function type (like std::minus), while c = a - b does the actual work, initializing according to c's type.

As I said, I am working on this topic. Here is the sequence view (object behaving like a container) corresponding to apply as given above, and here is its iterator. I am afraid this code will be probably unreadable because it's using much more entities defined elsewhere, and because it has taken many more steps of abstraction and generalization than shown here. Libraries like boost::multi_array, boost::ublas or Eigen are similar in this respect (actually, in using expression templates and in being unreadable internally)!

I am sorry I am not giving any more concrete solution here, but I feel this is a huge topic, and if you start in this direction, you may soon realize you should be using an existing library in the first place.

Anyhow, the good news are that you now have a single generic implementation of all operations so whenever you decide to invest on a more efficient implementation, you'll only have to define it once.

## Other types

(3) I've tested with int, float, double and std::complex. What else might be useful?

(4) is there any point to accommodating unsigned types?

You won't need anything for unsigned types. The definitions given here will work fine and conversions will be made according to language rules by std::common_type, even for std::complex.

Other types? Anything! For instance, why not other containers? Consider:

std::vector<std::vector<double> >
u{{0, 0, 0}, {3, 4, 5}},
v{{-1, -2, -3}, {5, 6, 7}};
say(u);
say(v);
say(-u);
say(+v);
say(u+v);
say(u-v);
say(v-u);

This works automatically now, for any level of nested containers, thanks to recursion. It wouldn't work with your code.

One "glitch" is that separation between vectors and scalars is done by checking for member type ::value_type as I said above. Unfortunately this is not quite appropriate because e.g. std::complex has this type but is not a container. I have a simple workaround in my live example by also requiring ::size_type, but all such issues will be resolved by C++ concepts.

One comes soon to realize that adding a scalar to a vector would also be convenient. This results in four combinations of operations. Then how about an n-ary function? This would result in 2^n combinations. I have a generic solution and here is a test sample of my code showing just a few of the things than can be done. But this goes way too far.

Another issue is that we have mostly ignored move semantics here. If we were to write this code in the most generic way, we should use rvalue references and forwarding everywhere. This is not so hard to do, but I have skipped it and used const& parameters almost everywhere.

## Type promotion

(5) should I do anything about possible numeric overflow or underflow?

For element-wise operations, nothing.

For accumulation operations like squared_distance, maybe yes. For instance, adding up 1000 values of type int might only fit into a long, and 1000 values of type float might only fit into a double. It would be convenient to have type promotions automatically done somewhere, but to decide exactly when a promotion is needed requires run time information, otherwise we may be promoting "just in case", which is contrary to the C++ principle

don't pay for what you don't use

I believe the user should be aware of such issues. Lazy evaluation may come to the rescue again: in an expression c = f(a, b), if f(a, b) is just a temporary intermediate object, then the actual work is only done when the type of c is known. This is exactly why std::accumulate and std::inner_product take another argument for the output. I also have a generic solution here, but again this goes too far.

For now, I would suggest to just use the right types for the input vectors (e.g. double) when an operation is expected to overflow the result. This is maybe too much a burden to the user, but we don't live in a perfect world.

## UPDATE: Corrections

I now realize that using common_type is not the right approach for finding the return type. It works for operators like +,* but not e.g. for boolean operators where the result is always bool. Moreover, common_type won't work with nested containers with different value types; it's simply not defined for such cases.

It's not too hard to make the return type depend on the operation F as well, using decltype as in the "capturing" function objects, or std::result_of. However, a further complication when considering nested containers with different value types is that substitution of the value type is not enough. For instance, std::vector<T> is actually

std::vector<T, std::allocator<T> >

so one has to substitute T for a new type, say S, in both the container and the allocator, to yield

std::vector<S, std::allocator<S> >

Assuming that containers follow this structure (in particular, they all have allocators), I have updated the previous code accordingly, but I am only giving here a link to a new live example to avoid making this answer even longer. For something more general without too much boilerplate, the containers should provide for such type substitution themselves; but to my knowledge, unfortunately this is not the case in STL.

We can now apply binary operators between containers of as diverse value types as

std::vector<std::vector<double> > u{{-2, 0, 0}, {3, 4, 5}};
std::vector<std::vector<int> >    v{{-1, 0, -3}, {5, 6, 5}};

as shown in the live example, along with some interesting cases like u < v.

Ok, that's it, I hope I will write no more!

-

I know that this won't answer any of your questions, but instead of using std::vector<T>, you should consider using std::array<T, N> to represent your data. Here is why this design could be more interesting:

• std::array<T, N> uses stack memory while std::vector<T> uses heap memory. I bet that a stack-allocated array will be a little bit more efficient than a heap-allocated vector.
• Mathematically speaking, it does not make any sense to change the size of a vector. Therefore, a fixed size container will also ensure that the size does not change.
• You won't have to bother with reserve anymore.
• In your functions, you make sure that the vector you add or subtract have the same size. If you use std::array<T, N> to store your data, you will get an implicit compile time check instead of an explicit runtime one:

template <typename T, size_t N>
std::array<T, N> operator+(const std::array<T, N>& a, const std::array<T, N>& b)
{
std::array<T, N> result;
for (size_t i = 0 ; i < N ; ++i)
{
result[i] = a[i] + b[i];
}
return result;
}

Moreover, since your loop has a static condition, it may be unrolled at compile time by the compiler.

-
Just a note: efficiency-wise (regardless of changing the container type), you are dropping the need to update the container size at each iteration, but at the same time you are implictly default-initializing all elements before entering the loop. This is exactly why Edward is using reserve and back_inserter, to avoid unnecessary initialization. Otherwise, one could say std::vector<T> result(N); and then write the same for loop as yours. Without measurements I cannot say which approach is better, but for me none is good enough (see discussion in my answer). – iavr Apr 6 '14 at 17:52
Plus, I bet you can't fit 10^7 elements on the stack, so it really depends on the application. The fact that STL algorithms are defined on iterators rather than containers has something to say (not on convenience, but at least on performance and genericity). – iavr Apr 6 '14 at 17:56
@iavr Those were some of my concerns, but since he mainly talked about 2D and 3D vectors, I considered that those weren't major drawbacks. I don't believe that many people perform computations in 10^7 dimensions Euclidean space. – Morwenn Apr 6 '14 at 18:15
@Morwenn You're right. Since modern physics says that all of spacetime can be described in 11 dimensions, more than that seems extravagant. – Edward Apr 6 '14 at 20:26
@Morwenn,Edward Well, algorithms for 2D/3D or even 11D spaces would rather use std:array if the number of dimensions is (compile-time) fixed. On the other hand, a color 1024x768 image has roughly 2,5M elements and I routinely do vector operations on such objects, or even worse. Element-wise operations, norms and squared distances remain the same in both cases. – iavr Apr 6 '14 at 20:50

# Results

Based on the feedback received, (thank you all for taking the time!) I wrote three different versions using std::vector, std::array and std::valarray. I then modified my test code to create a vector of one million randomly created 3D points (using double) and then called the squared_distance template function to populate a second results array with the squared distance between a fixed random point and each of the million other points. I then used the time command under Linux to record the User time taken for each of the three variants. Ten trials were done for each and the summary of the timing is shown below.

          valarray  array   vector
average     0.976   0.904   1.730
variance    0.000   0.002   0.020

As is clear from the results, both std::valarray and std::array are almost two times faster than my original std::vector version with std::array being fastest. That version of the code is shown below:

template <typename T, size_t N>
T squared_distance(const std::array<T,N>& a, const std::array<T,N>& b)
{
T accum(0);
for (int i=0; i<N; ++i)
{
T delta = b[i]-a[i];
accum += delta*delta;
}
return accum;
}

-


text:
## Inverse Regression Analysis

In most regression problems we have to determine the value of $Y$  corresponding to a given value of $X$. This inverse problem also called inverse regression analysis or calibration.

For inverse regression analysis, let the known values represented by matrix $X$ and their corresponding values by vector $Y$, which both form a simple linear regression model. Let, there is an unknown value of $X$, such as $X_0$, which cannot be measured and we observe the corresponding value of $Y$, say $Y_0$. Then, $X_0$ can be estimated and a confidence interval for $X_0$ can be obtained.

In regression analysis, we want to investigate the relationship between variables. Regression has many applications, which occur in many fields: engineering, economics, the physical and chemical sciences, management, biological sciences, and social sciences. We only consider the simple linear regression model, which is a model with one regressor $X$ that has a linear relationship with a response $Y$. It is not always easy to measure the regressor $X$ or the response $Y$.

We now consider a typical example for this problem. If $X$ is the concentration of glucose in certain substances, then a spectrophotometric method is used to measure the absorbance. This absorbance depends on the concentration of $X$. The response $Y$ is easy to measure with the spectrophotometric method, but the concentration, on the other hand, is not easy to measure. If we have $n$ known concentrations, then the absorbance can be measured. If there is a linear relation between $Y$ and $Y$, then a simple linear regression model can be made with these data. Suppose we have an unknown concentration, which is difficult to measure, but we can measure the absorbance of this concentration. Is it possible to estimate this concentration with the measured absorbance? This is called the calibration problem.

Suppose, we have a linear model $Y=\beta_0+\beta_1X+e$ and we have an observed value of the response $Y$, but we do not have the corresponding value of X. How can we estimate this value of $X$? The two most important methods to estimate $X$ are the classical method and the inverse method.

The classical method is based on the simple linear regression model

$Y=\hat{\beta}_0+\hat\{beta}_1X+\varepsilon,$   where $\varepsilon \tilde N(0, \, \sigma^2)$

where the parameters $\hat{beta}_0$ and $\hat{beta}_1$ are estimated by Least Squares as $\beta_0$ and $\beta_1$. At least two of the $n$ values of $X$ have to be distinct, otherwise, we cannot ﬁt a reliable regression line. For a given value of $X$, say $X_0$ (unknown), a $Y$ value, say $Y_0$ (or a random sample of $k$ values of $Y$) is observed at the $X_0$ value. The problem is to estimate $X_0$. The classical method uses a $Y_0$ value (or the mean of $k$ values of $Y_0$) to estimate $X_0$, which is then estimated by $\hat{x_0}=\frac{\hat{Y_0}-\hat{\beta_0}} {\hat{\beta_1}}$.

The inverse estimator is the simple linear regression of $X$ on $Y$. In this case, we have to ﬁt the model

$X=a_0+a_1Y+e, \text{where }\, N(0, \sigma^2)$

to obtain the estimator. Then the inverse estimator of $X_0$

$X_0=a_0+a_1Y+e$


text:
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a) $$F(x) = - cos(x) + \frac{3x^2}{2}$$ b) $$f(x) = sec^2(x) - 12x^2 - \frac{3}{2x^2}$$ c) $$lim_{x\rightarrow{\infty}} \frac{12x-4}{4x+7}=3$$

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# AttemptsUsingthemultiverseapproach to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ is true (and some in which $CH$ is not a relevant hypothesis?).

In formulating logical languages, there has been an outstanding problem: that of deciding the Law of the Excluded Middle $p\vee\neg p$. In classical logic $p\vee\neg p$ is true, but in intuitionistic logic this is not the case. With $CH$, the pragmatic mathematician tries to avoid invoking $CH$; if he assumes $CH$ or $\neg CH$, he will state it clearly. In everyday mathematics practice, the mathematician does use $p\vee\neg p$, but we do see an effort to give effective constructions, and hard estimates in analysis.

It seems that people are using classical logic basically because it's a core logic, in analogy to the constructible universe $L$ in which $CH$ is true. It is the logic first discovered (and often wrongly attributed as the Platonic choice) and the consistency and strength of other languages proven in terms of this core logic. Afterwards people come up with other logics, like Brouwer coming up with intuitionistic logic, in which a fundamental principle, the Law of the Excluded Middle, does not hold. It seems to me that this debate regarding the Law of Excluded Middle can be formalized using the multiverse approach.

So has anyone tried to use the multiverse approach towards considering these plurality of languages? Perhaps by considering a multiverse of topoi?

PS. The difficulty for arithemetizable syntax to describe continuous properties of geometric/measure-theoretic space is central (may I say?) to the difficulty of deciding $CH$. In a like fashion, the difference between classical and intuitionistic logic plays up in comparing the Dedekind-reals and the Cauchy-reals. In other words, continuous properties have different descriptions in these two logics. I'm hoping that work on the Law of the Excluded Middle will shed some light on trying to use discrete languages to model continuous properties. There's more where this comes from, but it is enough for now...

Edit: After comments and responses, I changed the title to reflect the part of the question not yet answered. What I have learned is that the 2-category of toposes is a possible way to consider of the multiverse of topoi, and that deciding LEM is not really about the difference between intuitionistic logic or classical logic, since they are still boolean-valued. The question remains: has there been attempts to decide the LEM?

2 added 433 characters in body; edited title

# UsingthemultiverseapproachAttempts to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ is true (and some in which $CH$ is not a relevant hypothesis?).

In formulating logical languages, there has been an outstanding problem: that of deciding the Law of the Excluded Middle $p\vee\neg p$. In classical logic $p\vee\neg p$ is true, but in intuitionistic logic this is not the case. With $CH$, the pragmatic mathematician tries to avoid invoking $CH$; if he assumes $CH$ or $\neg CH$, he will state it clearly. In everyday mathematics practice, the mathematician does use $p\vee\neg p$, but we do see an effort to give effective constructions, and hard estimates in analysis.

It seems that people are using classical logic basically because it's a core logic, in analogy to the constructible universe $L$ in which $CH$ is true. It is the logic first discovered (and often wrongly attributed as the Platonic choice) and the consistency and strength of other languages proven in terms of this core logic. Afterwards people come up with other logics, like Brouwer coming up with intuitionistic logic, in which a fundamental principle, the Law of the Excluded Middle, does not hold. It seems to me that this debate regarding the Law of Excluded Middle can be formalized using the multiverse approach.

So has anyone tried to use the multiverse approach towards considering these plurality of languages? Perhaps by considering a multiverse of topoi?

PS. The difficulty for arithemetizable syntax to describe continuous properties of geometric/measure-theoretic space is central (may I say?) to the difficulty of deciding $CH$. In a like fashion, the difference between classical and intuitionistic logic plays up in comparing the Dedekind-reals and the Cauchy-reals. In other words, continuous properties have different descriptions in these two logics. I'm hoping that work on the Law of the Excluded Middle will shed some light on trying to use discrete languages to model continuous properties. There's more where this comes from, but it is enough for now...

Edit: After comments and responses, I changed the title to reflect the part of the question not yet answered. What I have learned is that the 2-category of toposes is a possible way to consider of the multiverse of topoi, and that deciding LEM is not really about the difference between intuitionistic logic or classical logic, since they are still boolean-valued. The question remains: has there been attempts to decide the LEM?

1

# Using the multiverse approach to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ is true (and some in which $CH$ is not a relevant hypothesis?).

In formulating logical languages, there has been an outstanding problem: that of deciding the Law of the Excluded Middle $p\vee\neg p$. In classical logic $p\vee\neg p$ is true, but in intuitionistic logic this is not the case. With $CH$, the pragmatic mathematician tries to avoid invoking $CH$; if he assumes $CH$ or $\neg CH$, he will state it clearly. In everyday mathematics practice, the mathematician does use $p\vee\neg p$, but we do see an effort to give effective constructions, and hard estimates in analysis.

It seems that people are using classical logic basically because it's a core logic, in analogy to the constructible universe $L$ in which $CH$ is true. It is the logic first discovered (and often wrongly attributed as the Platonic choice) and the consistency and strength of other languages proven in terms of this core logic. Afterwards people come up with other logics, like Brouwer coming up with intuitionistic logic, in which a fundamental principle, the Law of the Excluded Middle, does not hold. It seems to me that this debate regarding the Law of Excluded Middle can be formalized using the multiverse approach.

So has anyone tried to use the multiverse approach towards considering these plurality of languages? Perhaps by considering a multiverse of topoi?

PS. The difficulty for arithemetizable syntax to describe continuous properties of geometric/measure-theoretic space is central (may I say?) to the difficulty of deciding $CH$. In a like fashion, the difference between classical and intuitionistic logic plays up in comparing the Dedekind-reals and the Cauchy-reals. In other words, continuous properties have different descriptions in these two logics. I'm hoping that work on the Law of the Excluded Middle will shed some light on trying to use discrete languages to model continuous properties. There's more where this comes from, but it is enough for now...


text:
# VIBRATIONAL DEPENDENCE OF EXCITED STATE INTRAMOLECULAR PROTON TRANSFER IN 2-(2$^\prime$-PYRIDYL)PYRROLE IN THE GAS PHASE VIA HIGH RESOLUTION ELECTRONIC SPECTROSCOPY

Please use this identifier to cite or link to this item: http://hdl.handle.net/1811/46259

Files Size Format View
abstract.gif 37.62Kb GIF image

 Title: VIBRATIONAL DEPENDENCE OF EXCITED STATE INTRAMOLECULAR PROTON TRANSFER IN 2-(2$^\prime$-PYRIDYL)PYRROLE IN THE GAS PHASE VIA HIGH RESOLUTION ELECTRONIC SPECTROSCOPY Creators: Morgan, Philip J.; Fleisher, Adam J.; Pratt, David W.; Kijak, Michal; Waluk, Jacek Issue Date: 2010 Publisher: Ohio State University Abstract: Rotationally resolved fluorescence excitation spectra of the S$_1\leftarrow$S$_0$ origin band and +144 cm\textsuperscript{-1} vibrational band transitions of 2-(2$^\prime$-Pyridyl)pyrrole (2PP) have been recorded in the collision free environment of a molecular beam. Analyses of these data provide new information about the changes in geometry that occur when 2PP absorbs light. Additionally, significant line broadening is observed in both spectra, which we attribute to an excited state intramolecular proton transfer (ESIPT) reaction. The dynamics and vibrational mode dependence of ESIPT in 2PP will be discussed. Description: Work supported by NSF (CHE-0911117) Synthesis by Randolph P. Thummel, University of Houston, Houston, Texas 77204-5003 Author Institution: Department of Chemistry, University of Pittsburgh, Pittsburgh, PA 15260; Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland URI: http://hdl.handle.net/1811/46259 Other Identifiers: 2010-TA-02


text:
# History KS3 / KS4: British slave owners in the 1830s

Historian David Olusoga investigates evidence of British slave ownership in the 1830s, referring to 40,000 owners with over 800,000 slaves in the Caribbean and elsewhere.

The figures are from detailed records of compensation to slave owners paid when slavery was abolished in 1834 across the British Empire.

Olusoga uses the records to demonstrate how widespread and profitable the ownership of slaves was in Britain at the time, linking his analysis by visits to Georgian terraces in parts of London.

This is from the series: Britain's Forgotten Slave Owners

Contains scenes which some younger viewers may find upsetting. Teacher review recommended prior to use in class.

### Teacher Notes

This could supplement study of the abolition of the British slave trade and the abolition of slavery within the British Empire itself.

It could also be used as an example of how a historian makes use of archival evidence to support their case.

It could be used to support the teaching of migration in to and out of Britain.

### Curriculum Notes

This clip will be relevant for teaching History at KS3, KS4/GCSE, in England and Wales and Northern Ireland

Also at National 4 and National 5 in Scotland.

This topic appears in OCR, Edexcel, AQA, WJEC, CCEA GCSE and SQA.