Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons...

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Heat equa(on:

Transcript of Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons...

Page 1: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Heatequa(on:

Page 2: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Heatequa(on:

Numericalsolu(onrequiresadiscre(sa(onInbothspaceand(me

Page 3: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 4: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

?

Page 5: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 6: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 7: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 8: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 9: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 10: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 11: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 12: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 13: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Example:Fourierseries

Page 14: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Arbitraryfunc(on

Orthogonalbasisfunc(ons

Spectralcoefficients

1.4 0.8 0.3

Page 15: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

Page 16: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

Page 17: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

Page 18: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

Page 19: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

T1

T2

Page 20: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

T1

T2

Page 21: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

N1(x) N2(x)

T1

T2

Page 22: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Finiteelements?

N1(x) N2(x)

T1

T2

T1.N1(x)T2.N2(x)

Page 23: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 24: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 25: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 26: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 27: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Insufficientequa(on->Risunknown!

Page 28: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Welookforthesolu(onsuchthattheprojec(onoftheresidualoverthebasisfunc(onsiszero

Page 29: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Welookforthesolu(onsuchthattheprojec(onoftheresidualoverthebasisfunc(onsiszero

Page 30: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Welookforthesolu(onsuchthattheprojec(onoftheresidualoverthebasisfunc(onsiszero

Page 31: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

FEM:

Fourier:

Projec(ons

Commonspace:

~V = Vx

~ex

+ Vy

~ey

Vx

= ~V · ~ex

Vy

= ~V · ~ey

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~ex

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~ey<latexit sha1_base64="PjoTtd5EXoKyNv+X0cBFQCrtmso=">AAAB8HicbVBNS8NAEJ34WetX1aOXYBE8laQIeix48VjBfkgbymY7aZfubsLuphBCf4UXD4p49ed489+4bXPQ1gcDj/dmmJkXJpxp43nfzsbm1vbObmmvvH9weHRcOTlt6zhVFFs05rHqhkQjZxJbhhmO3UQhESHHTji5m/udKSrNYvlosgQDQUaSRYwSY6Wn/hRpjoNsNqhUvZq3gLtO/IJUoUBzUPnqD2OaCpSGcqJ1z/cSE+REGUY5zsr9VGNC6ISMsGepJAJ1kC8OnrmXVhm6UaxsSeMu1N8TORFaZyK0nYKYsV715uJ/Xi810W2QM5mkBiVdLopS7prYnX/vDplCanhmCaGK2VtdOiaKUGMzKtsQ/NWX10m7XvO9mv9wXW3UizhKcA4XcAU+3EAD7qEJLaAg4Ble4c1Rzovz7nwsWzecYuYM/sD5/AEnv5CT</latexit><latexit sha1_base64="PjoTtd5EXoKyNv+X0cBFQCrtmso=">AAAB8HicbVBNS8NAEJ34WetX1aOXYBE8laQIeix48VjBfkgbymY7aZfubsLuphBCf4UXD4p49ed489+4bXPQ1gcDj/dmmJkXJpxp43nfzsbm1vbObmmvvH9weHRcOTlt6zhVFFs05rHqhkQjZxJbhhmO3UQhESHHTji5m/udKSrNYvlosgQDQUaSRYwSY6Wn/hRpjoNsNqhUvZq3gLtO/IJUoUBzUPnqD2OaCpSGcqJ1z/cSE+REGUY5zsr9VGNC6ISMsGepJAJ1kC8OnrmXVhm6UaxsSeMu1N8TORFaZyK0nYKYsV715uJ/Xi810W2QM5mkBiVdLopS7prYnX/vDplCanhmCaGK2VtdOiaKUGMzKtsQ/NWX10m7XvO9mv9wXW3UizhKcA4XcAU+3EAD7qEJLaAg4Ble4c1Rzovz7nwsWzecYuYM/sD5/AEnv5CT</latexit><latexit sha1_base64="PjoTtd5EXoKyNv+X0cBFQCrtmso=">AAAB8HicbVBNS8NAEJ34WetX1aOXYBE8laQIeix48VjBfkgbymY7aZfubsLuphBCf4UXD4p49ed489+4bXPQ1gcDj/dmmJkXJpxp43nfzsbm1vbObmmvvH9weHRcOTlt6zhVFFs05rHqhkQjZxJbhhmO3UQhESHHTji5m/udKSrNYvlosgQDQUaSRYwSY6Wn/hRpjoNsNqhUvZq3gLtO/IJUoUBzUPnqD2OaCpSGcqJ1z/cSE+REGUY5zsr9VGNC6ISMsGepJAJ1kC8OnrmXVhm6UaxsSeMu1N8TORFaZyK0nYKYsV715uJ/Xi810W2QM5mkBiVdLopS7prYnX/vDplCanhmCaGK2VtdOiaKUGMzKtsQ/NWX10m7XvO9mv9wXW3UizhKcA4XcAU+3EAD7qEJLaAg4Ble4c1Rzovz7nwsWzecYuYM/sD5/AEnv5CT</latexit><latexit sha1_base64="PjoTtd5EXoKyNv+X0cBFQCrtmso=">AAAB8HicbVBNS8NAEJ34WetX1aOXYBE8laQIeix48VjBfkgbymY7aZfubsLuphBCf4UXD4p49ed489+4bXPQ1gcDj/dmmJkXJpxp43nfzsbm1vbObmmvvH9weHRcOTlt6zhVFFs05rHqhkQjZxJbhhmO3UQhESHHTji5m/udKSrNYvlosgQDQUaSRYwSY6Wn/hRpjoNsNqhUvZq3gLtO/IJUoUBzUPnqD2OaCpSGcqJ1z/cSE+REGUY5zsr9VGNC6ISMsGepJAJ1kC8OnrmXVhm6UaxsSeMu1N8TORFaZyK0nYKYsV715uJ/Xi810W2QM5mkBiVdLopS7prYnX/vDplCanhmCaGK2VtdOiaKUGMzKtsQ/NWX10m7XvO9mv9wXW3UizhKcA4XcAU+3EAD7qEJLaAg4Ble4c1Rzovz7nwsWzecYuYM/sD5/AEnv5CT</latexit>

~V<latexit sha1_base64="LxDJGm7UlXzLn/2r/NkQO7fnYF0=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWy2k3bpZhN2N4US+iO8eFDEq7/Hm//GTZuDtj4YeLw3w8y8IBFcG9f9dkpb2zu7e+X9ysHh0fFJ9fSso+NUMWyzWMSqF1CNgktsG24E9hKFNAoEdoPpfe53Z6g0j+WTmSfoR3QsecgZNVbqDmbIss5iWK25dXcJskm8gtSgQGtY/RqMYpZGKA0TVOu+5ybGz6gynAlcVAapxoSyKR1j31JJI9R+tjx3Qa6sMiJhrGxJQ5bq74mMRlrPo8B2RtRM9LqXi/95/dSEd37GZZIalGy1KEwFMTHJfycjrpAZMbeEMsXtrYRNqKLM2IQqNgRv/eVN0mnUPbfuPd7Umo0ijjJcwCVcgwe30IQHaEEbGEzhGV7hzUmcF+fd+Vi1lpxi5hz+wPn8AXlpj5g=</latexit><latexit sha1_base64="LxDJGm7UlXzLn/2r/NkQO7fnYF0=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWy2k3bpZhN2N4US+iO8eFDEq7/Hm//GTZuDtj4YeLw3w8y8IBFcG9f9dkpb2zu7e+X9ysHh0fFJ9fSso+NUMWyzWMSqF1CNgktsG24E9hKFNAoEdoPpfe53Z6g0j+WTmSfoR3QsecgZNVbqDmbIss5iWK25dXcJskm8gtSgQGtY/RqMYpZGKA0TVOu+5ybGz6gynAlcVAapxoSyKR1j31JJI9R+tjx3Qa6sMiJhrGxJQ5bq74mMRlrPo8B2RtRM9LqXi/95/dSEd37GZZIalGy1KEwFMTHJfycjrpAZMbeEMsXtrYRNqKLM2IQqNgRv/eVN0mnUPbfuPd7Umo0ijjJcwCVcgwe30IQHaEEbGEzhGV7hzUmcF+fd+Vi1lpxi5hz+wPn8AXlpj5g=</latexit><latexit sha1_base64="LxDJGm7UlXzLn/2r/NkQO7fnYF0=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWy2k3bpZhN2N4US+iO8eFDEq7/Hm//GTZuDtj4YeLw3w8y8IBFcG9f9dkpb2zu7e+X9ysHh0fFJ9fSso+NUMWyzWMSqF1CNgktsG24E9hKFNAoEdoPpfe53Z6g0j+WTmSfoR3QsecgZNVbqDmbIss5iWK25dXcJskm8gtSgQGtY/RqMYpZGKA0TVOu+5ybGz6gynAlcVAapxoSyKR1j31JJI9R+tjx3Qa6sMiJhrGxJQ5bq74mMRlrPo8B2RtRM9LqXi/95/dSEd37GZZIalGy1KEwFMTHJfycjrpAZMbeEMsXtrYRNqKLM2IQqNgRv/eVN0mnUPbfuPd7Umo0ijjJcwCVcgwe30IQHaEEbGEzhGV7hzUmcF+fd+Vi1lpxi5hz+wPn8AXlpj5g=</latexit><latexit sha1_base64="LxDJGm7UlXzLn/2r/NkQO7fnYF0=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mKoMeCF48V7Ae0oWy2k3bpZhN2N4US+iO8eFDEq7/Hm//GTZuDtj4YeLw3w8y8IBFcG9f9dkpb2zu7e+X9ysHh0fFJ9fSso+NUMWyzWMSqF1CNgktsG24E9hKFNAoEdoPpfe53Z6g0j+WTmSfoR3QsecgZNVbqDmbIss5iWK25dXcJskm8gtSgQGtY/RqMYpZGKA0TVOu+5ybGz6gynAlcVAapxoSyKR1j31JJI9R+tjx3Qa6sMiJhrGxJQ5bq74mMRlrPo8B2RtRM9LqXi/95/dSEd37GZZIalGy1KEwFMTHJfycjrpAZMbeEMsXtrYRNqKLM2IQqNgRv/eVN0mnUPbfuPd7Umo0ijjJcwCVcgwe30IQHaEEbGEzhGV7hzUmcF+fd+Vi1lpxi5hz+wPn8AXlpj5g=</latexit>

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Page 33: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 34: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 35: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 36: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 37: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 38: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Solvingforthe(me-dependence

Page 39: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Solvingforthe(me-dependence

Page 40: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Solvingforthe(me-dependence

Page 41: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Solvingforthe(me-dependence

Page 42: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Solvingforthe(me-dependence

Page 43: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Morethanoneelement?

Page 44: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y
Page 45: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

R =

R11 R12

R21 R22

�

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Globalmatrices

Page 48: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Globalmatrices

Page 49: Heat equaon - macOS Serverjupiter.ethz.ch/.../2019/docs/Presentation1.pdfFEM: Fourier: Projecons Common space: V~ = V xe~x + Vy e~y Vx = V~ · e~x Vy = V~ · e~y

Globalmatrices

Tn+1 = LG�1b

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Hitachi VY-25E

Hitachi VY-25E

N vy componen Var etlands, Bangalore, India...Vol.:(0123456789) SN Applied Sciences (2020) 2:1449 | Research Article N vy componen Var etlands, Bangalore, India T. V ...

N vy componen Var etlands, Bangalore, India...Vol.:(0123456789) SN Applied Sciences (2020) 2:1449 | Research Article N vy componen Var etlands, Bangalore, India T. V ...

(Microsoft PowerPoint - vy\232etrovacie matody v nefrl .ppt [iba na ...

(Microsoft PowerPoint - vy\232etrovacie matody v nefrl .ppt [iba na ...

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**Elicit expert knowledge to select relevant climate variablesfloridawca.org/sites/.../Terando_secsc_downscaling... · 1) Projecons that reflect reality given constraints of GCMs

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Egd Z`i d`¨VY`^ - gandalf.com.pl fileEgd_Z`i d`¨VY`^

Egd Z`i d`¨VY`^ - gandalf.com.pl fileEgd_Z`i d`¨VY`^

VY ŠNÉ RUŽBACHY

VY ŠNÉ RUŽBACHY

mabdalla/papers/dhes.pdf · 2000-09-08 · ... vy vy ¢

mabdalla/papers/dhes.pdf · 2000-09-08 · ... vy vy ¢

TYlimat - fiu.azzaman1, mYsul _Yxs i_dYn azad olunduqda vY ya i_ yerini dYyi_dikdY vY s. hallar ba_ verdikdY sertifikat1 lY˜v edib yeni i_çinin ad1na sertifikat yaratmaq tYlYb olunur.

TYlimat - fiu.azzaman1, mYsul _Yxs i_dYn azad olunduqda vY ya i_ yerini dYyi_dikdY vY s. hallar ba_ verdikdY sertifikat1 lY˜v edib yeni i_çinin ad1na sertifikat yaratmaq tYlYb olunur.

Novi vky v blokádach dol vej ko včati vy Grgac Blokady dolnej...Novi vky v blokádach dol vej ko včati vy MUDr. Ivan GrgačAbteilug für Aästhesie, Notfall - und Allgemeine Intensivmedizin

Novi vky v blokádach dol vej ko včati vy Grgac Blokady dolnej...Novi vky v blokádach dol vej ko včati vy MUDr. Ivan GrgačAbteilug für Aästhesie, Notfall - und Allgemeine Intensivmedizin

GRADUATION PROJECT VY

GRADUATION PROJECT VY

121-F · 121F YxVyVz(x = yvx = z vy = z) A Bx£y(x = y) YxVyVz(x = y v x = z vy = z) VxVy(x = y) (Y 3xBy3z(x = y v x —z v y —z) A 3x3y(x — y) (t

121-F · 121F YxVyVz(x = yvx = z vy = z) A Bx£y(x = y) YxVyVz(x = y v x = z vy = z) VxVy(x = y) (Y 3xBy3z(x = y v x —z v y —z) A 3x3y(x — y) (t

v ktorom aj vy počujúc slovo

v ktorom aj vy počujúc slovo

vy final report

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Presentación de PowerPoint - Facultad de Ingeniería...Sean Vy W dos K-espacios vectoriales, V de dimensión finita. Sea B - {VI, v } una base de Vy sean WI, w2, . vectores arbitrarios.

Presentación de PowerPoint - Facultad de Ingeniería...Sean Vy W dos K-espacios vectoriales, V de dimensión finita. Sea B - {VI, v } una base de Vy sean WI, w2, . vectores arbitrarios.

leasingnews.orgleasingnews.org/PDF/CITGROUP10Q_8_17_09.pdf · vy))) ))) )) [vy

leasingnews.orgleasingnews.org/PDF/CITGROUP10Q_8_17_09.pdf · vy))) ))) )) [vy

1 A³Vy~a · 1 A³Vy~a 2019 ^m

1 A³Vy~a · 1 A³Vy~a 2019 ^m

V¡$n Vy$X¡$ RNI No.GUJGUJ/2016/74805 · V¡$n Vy$X¡$ Tax Today (Monthly) (V¡$n k„gÁ_ s\p AÞe dlÐh_p kdpQpf ârkÙ L$fsy„ dprkL$) u hj®: 4, u A„L$: 02, u km„N A„L$:

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VY Canis Majoris

VY Canis Majoris