text:
# American Institute of Mathematical Sciences

April  2018, 38(4): 2141-2169. doi: 10.3934/dcds.2018088

## Pullback $\mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model

 Department of Mathematics, Florida International University, DM413B, University Park, Miami, Florida 33199, USA

Received  May 2017 Revised  October 2017 Published  January 2018

The existence and final fractal dimension of a pullback attractor in the space $\mathbb{V}$ for a three dimensional system of a non-autonomous globally modified two phase flow on a bounded domain is established under appropriate properties on the time depending forcing term. The model consists of the globally modified Navier-Stokes equations proposed in [6] for the velocity, coupled with an Allen-Cahn model for the order (phase) parameter. The existence of the pullback attractors is obtained using the flattening property. Furthermore, we prove that the fractal dimension in $\mathbb{V}$ of the pullback attractor is finite.

Citation: Theodore Tachim Medjo. Pullback $\mathbb{V}-$attractor of a three dimensional globally modified two-phase flow model. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2141-2169. doi: 10.3934/dcds.2018088
##### References:
 [1] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2. Google Scholar [2] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307. Google Scholar [3] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. Google Scholar [4] T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: Recent developments, In A. Johann, H. P. Kruse and F. Rupp, editors, Recent Trends in Dynamical Systems: Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics and Statistics, 35 (2013), 473-492. Google Scholar [5] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. Google Scholar [6] T. Caraballo, J. Real and P.E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436. Google Scholar [7] T. Caraballo, J. Real and A.M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883. doi: 10.1142/S0218127410027428. Google Scholar [8] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338. Google Scholar [9] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6. Google Scholar [10] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096. Google Scholar [11] S. Chen, D.D. Holm, L.G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8. Google Scholar [12] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. Google Scholar [13] G. Deugoue and J.K. Djoko, On the time discretization for the globally modified three dimensional Navier-Stokes equations, J. Comput. Appl. Math, 235 (2011), 2015-2029. doi: 10.1016/j.cam.2010.10.003. Google Scholar [14] E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544. Google Scholar [15] C. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013. Google Scholar [16] C.G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1. Google Scholar [17] C.G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6. Google Scholar [18] P.C. Hohenberg and B.I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. Google Scholar [19] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721. Google Scholar [20] D.D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. doi: 10.1103/PhysRevLett.80.4173. Google Scholar [21] P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar [22] P.E. Kloeden, J.A. Langa and J. Real, Pullback ${V-}$attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937. Google Scholar [23] P.E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. Google Scholar [24] P.E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. doi: 10.1098/rspa.2007.1831. Google Scholar [25] G. Lukaszewicz, On pullback attractors in ${H}^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258. Google Scholar [26] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar [27] P. Marín-Rubio, J. Real and A.M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927. Google Scholar [28] A.M. Márquez, Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay, Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{e}$MA, 51 (2010), 117-124. Google Scholar [29] J.E. Marsden and S. Shkoller, Global well-posedness for the {L}agrangian averaged Navier-Stokes (LANS-$α$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852. Google Scholar [30] A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157. Google Scholar [31] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. Google Scholar [32] H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in ${H}^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar [33] X.L. Song and Y. Hou, Pullback $\underset{\scriptscriptstyle\centerdot}{-}$attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009. Google Scholar [34] T. Tachim Medjo, Unique strong and ${V-}$attractor of a three dimensional globally modified Allen-Cahn-Navier-Stokes model, Appl. Anal., 96 (2017), 2695-2716. doi: 10.1080/00036811.2016.1236924. Google Scholar [35] T. Tachim Medjo, Unique strong and ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2016.1236924, 2016.Google Scholar [36] T. Tachim Medjo, Pullback ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2017.1296952, 2017.Google Scholar [37] R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. Google Scholar [38] M.I. Vishik, A.I. Komech and A.V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210. Google Scholar [39] B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128. doi: 10.1016/j.na.2014.08.018. Google Scholar [40] C. Zhao, S. Zhou and Y. Li, Uniform attractor for a two-dimensiona nonautonomous incompressible non-Newtonian fluid, Appl. Math. Comput., 201 (2008), 688-700. doi: 10.1016/j.amc.2008.01.005. Google Scholar

show all references

##### References:
 [1] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal., 194 (2009), 463-506. doi: 10.1007/s00205-008-0160-2. Google Scholar [2] T. Blesgen, A generalization of the Navier-Stokes equation to two-phase flow, Physica D (Applied Physics), 32 (1999), 1119-1123. doi: 10.1088/0022-3727/32/10/307. Google Scholar [3] G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rational Mech. Anal., 92 (1986), 205-245. Google Scholar [4] T. Caraballo and P. E. Kloeden, The three-dimensional globally modified Navier-Stokes equations: Recent developments, In A. Johann, H. P. Kruse and F. Rupp, editors, Recent Trends in Dynamical Systems: Proceedings of a Conference in Honor of Jürgen Scheurle, Springer Proceedings in Mathematics and Statistics, 35 (2013), 473-492. Google Scholar [5] T. Caraballo, P. E. Kloeden and J. Real, Invariant measures and statistical solutions of the globally modified Navier-Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 761-781. doi: 10.3934/dcdsb.2008.10.761. Google Scholar [6] T. Caraballo, J. Real and P.E. Kloeden, Unique strong solutions and V-attractors of a three dimensional system of globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 6 (2006), 411-436. Google Scholar [7] T. Caraballo, J. Real and A.M. Márquez, Three-dimensional system of globally modified Navier-Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2869-2883. doi: 10.1142/S0218127410027428. Google Scholar [8] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations as a closure model for turbulent channel and pipe flows, Phys. Rev. Lett., 81 (1998), 5338-5341. doi: 10.1103/PhysRevLett.81.5338. Google Scholar [9] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, The Camassa-Holm equations and turbulence, Physica D, 133 (1999), 49-65. doi: 10.1016/S0167-2789(99)00098-6. Google Scholar [10] S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi and S. Wynne, A connection between the Camassa-Holm equations and turbulent flows in channels and pipes, Phys. Fluids, 11 (1999), 2343-2353. doi: 10.1063/1.870096. Google Scholar [11] S. Chen, D.D. Holm, L.G. Margolin and R. Zhang, Direct numerical simulations of the Navier-Stokes alpha model, Physica D, 133 (1999), 66-83. doi: 10.1016/S0167-2789(99)00099-8. Google Scholar [12] P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1988. Google Scholar [13] G. Deugoue and J.K. Djoko, On the time discretization for the globally modified three dimensional Navier-Stokes equations, J. Comput. Appl. Math, 235 (2011), 2015-2029. doi: 10.1016/j.cam.2010.10.003. Google Scholar [14] E. Feireisl, H. Petzeltová, E. Rocca and G. Schimperna, Analysis of a phase-field model for two-phase compressible fluids, Math. Models Methods Appl. Sci., 20 (2010), 1129-1160. doi: 10.1142/S0218202510004544. Google Scholar [15] C. Gal and M. Grasselli, Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 401-436. doi: 10.1016/j.anihpc.2009.11.013. Google Scholar [16] C.G. Gal and M. Grasselli, Longtime behavior for a model of homogeneous incompressible two-phase flows, Discrete Contin. Dyn. Syst., 28 (2010), 1-39. doi: 10.3934/dcds.2010.28.1. Google Scholar [17] C.G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chin. Ann. Math. Ser. B, 31 (2010), 655-678. doi: 10.1007/s11401-010-0603-6. Google Scholar [18] P.C. Hohenberg and B.I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys., 49 (1977), 435-479. Google Scholar [19] D.D. Holm, J.E. Marsden and T.S. Ratiu, The Euler-Poincaré equations and semi-direct products with applications to continuum theories, Adv. Math., 137 (1998), 1-81. doi: 10.1006/aima.1998.1721. Google Scholar [20] D.D. Holm, J.E. Marsden and T.S. Ratiu, Euler-Poincaré models of ideal fluids with nonlinear dispersion, Phys. Rev. Lett., 349 (1998), 4173-4177. doi: 10.1103/PhysRevLett.80.4173. Google Scholar [21] P.E. Kloeden and J.A. Langa, Flattening, squeezing and the existence of random attractors, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 163-181. doi: 10.1098/rspa.2006.1753. Google Scholar [22] P.E. Kloeden, J.A. Langa and J. Real, Pullback ${V-}$attractors of the 3-dimensional globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 6 (2007), 937-955. doi: 10.3934/cpaa.2007.6.937. Google Scholar [23] P.E. Kloeden, P. Marín-Rubio and J. Real, Equivalence of invariant measures and stationary statistical solutions for the autonomous globally modified Navier-Stokes equations, Commun. Pure Appl. Anal., 8 (2009), 785-802. doi: 10.3934/cpaa.2009.8.785. Google Scholar [24] P.E. Kloeden and J. Valero, The weak connectedness of the attainability set of weak solutions of the three-dimensional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 1491-1508. doi: 10.1098/rspa.2007.1831. Google Scholar [25] G. Lukaszewicz, On pullback attractors in ${H}^1_0$ for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos Appl. Sci. Engrg, 20 (2010), 2637-2644. doi: 10.1142/S0218127410027258. Google Scholar [26] Q. Ma, S. Wang and C. Zhong, Necessary and sufficient conditions for the existence of global attractors for semigroups and applications, Indiana Univ. Math. J., 51 (2002), 1541-1559. doi: 10.1512/iumj.2002.51.2255. Google Scholar [27] P. Marín-Rubio, J. Real and A.M. Márquez-Durán, On the convergence of solutions of globally modified Navier-Stokes equations with delays to solutions of Navier-Stokes equations with delays, Adv. Nonlinear Stud., 11 (2011), 917-927. Google Scholar [28] A.M. Márquez, Existence and uniqueness of solutions, and pullback attractor for a system of globally modified 3D-Navier-Stokes equations with finite delay, Bol. Soc. Esp. Mat. Apl. S$\overrightarrow{e}$MA, 51 (2010), 117-124. Google Scholar [29] J.E. Marsden and S. Shkoller, Global well-posedness for the {L}agrangian averaged Navier-Stokes (LANS-$α$) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359 (2001), 1449-1468. doi: 10.1098/rsta.2001.0852. Google Scholar [30] A. Onuki, Phase transition of fluids in shear flow, J. Phys. Condens. Matter, 9 (1997), 6119-6157. Google Scholar [31] M. Romito, The uniqueness of weak solutions of the globally modified Navier-Stokes equations, Adv. Nonlinear Stud., 9 (2009), 425-427. Google Scholar [32] H. Song, Pullback attractors of non-autonomous reaction-diffusion equations in ${H}^1_0$, J. Differential Equations, 249 (2010), 2357-2376. doi: 10.1016/j.jde.2010.07.034. Google Scholar [33] X.L. Song and Y. Hou, Pullback $\underset{\scriptscriptstyle\centerdot}{-}$attractors for the non-autonomous Newton-Boussinesq equation in two-dimensional bounded domain, Discrete Contin. Dyn. Syst., 32 (2012), 991-1009. Google Scholar [34] T. Tachim Medjo, Unique strong and ${V-}$attractor of a three dimensional globally modified Allen-Cahn-Navier-Stokes model, Appl. Anal., 96 (2017), 2695-2716. doi: 10.1080/00036811.2016.1236924. Google Scholar [35] T. Tachim Medjo, Unique strong and ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2016.1236924, 2016.Google Scholar [36] T. Tachim Medjo, Pullback ${V}-$attractor of a three dimensional globally modified Cahn-Hilliard-Navier-Stokes model, Appl. Anal., http://dx.doi.org/10.1080/00036811.2017.1296952, 2017.Google Scholar [37] R. Temam, Infinite Dynamical Systems in Mechanics and Physics, volume 68, Appl. Math. Sci., Springer-Verlag, New York, second edition, 1997. Google Scholar [38] M.I. Vishik, A.I. Komech and A.V. Fursikov, Some mathematical problems of statistical hydromechanics, Uspekhi Mat. Nauk, 34 (1979), 135-210. Google Scholar [39] B. You and F. Li, The existence of a pullback attractor for the three dimensional non-autonomous planetary geostrophic viscous equations of large-scale ocean circulation, Nonlinear Anal., 112 (2015), 118-128. doi: 10.1016/j.na.2014.08.018. Google Scholar [40] C. Zhao, S. Zhou and Y. Li, Uniform attractor for a two-dimensiona nonautonomous incompressible non-Newtonian fluid, Appl. Math. Comput., 201 (2008), 688-700. doi: 10.1016/j.amc.2008.01.005. Google Scholar
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text:
1. Finding time

A turkey with a temperature of 40°F is moved to a 350°F oven. After 4 hours the internal temperature of the turkey is 170°F. If the turkey is done when its temperature reaches 185°F, then how much longer must it cook?

2. Originally Posted by dc52789
A turkey with a temperature of 40°F is moved to a 350°F oven. After 4 hours the internal temperature of the turkey is 170°F. If the turkey is done when its temperature reaches 185°F, then how much longer must it cook?
It seems you need to know the initial internal temperature too.

3. Originally Posted by dc52789
A turkey with a temperature of 40°F is moved to a 350°F oven. After 4 hours the internal temperature of the turkey is 170°F. If the turkey is done when its temperature reaches 185°F, then how much longer must it cook?
see here

post #2 also includes links to 2 other posts that may help

4. Originally Posted by Jhevon
see here

post #2 also includes links to 2 other posts that may help
Not sure if that really helps.

5. Originally Posted by dc52789
Not sure if that really helps.
how come?

we have $T' = -k(T - T_{amb})$ with $T_{amb} = 350$, $T(0) = 40$ and $T(4) = 170$

here $T$ is the temperature after time $t$ (in hours), $k$ is a constant, which we can find using $T(0)$ and $T(4)$ (as described in post #2 here, question 2), and $T_{amb}$ is the ambient or surrounding temperature, in this case, the temperature of the oven

you want to find $t$ when $T = 185$, then you can answer the question

6. Originally Posted by Jhevon
how come?

we have $T' = -k(T - T_{amb})$ with $T_{amb} = 350$, $T(0) = 40$ and $T(4) = 170$

here $T$ is the temperature after time $t$ (in hours), $k$ is a constant, which we can find using $T(0)$ and $T(4)$ (as described in post #2 here, question 2), and $T_{amb}$ is the ambient or surrounding temperature, in this case, the temperature of the oven

you want to find $t$ when $T = 185$, then you can answer the question
I still kinda don't get it.

7. Originally Posted by dc52789
I still kinda don't get it.
just try to work through the process as described in the last link i gave. tell me what is confusing you specifically. it's hard to help if you just say "i don't get it"

8. Originally Posted by Jhevon
just try to work through the process as described in the last link i gave. tell me what is confusing you specifically. it's hard to help if you just say "i don't get it"
o k
I have A = -310 and k = 0.136

9. Originally Posted by dc52789
o k
I have A = -310 and k = 0.136
yes, i would use k = 0.1359 though

so we have $T(t) = -310e^{-0.1359t} + 350$

so now we want to know how long it takes for T to become 185, so we want t such that

$185 = -310e^{-0.1359t} + 350$

now solve for $t$. note that this gives the total time for the turkey to get to that temperature. you want the difference between 4 and that time

10. Originally Posted by Jhevon
yes, i would use k = 0.1359 though

so we have $T(t) = -310e^{-0.1359t} + 350$

so now we want to know how long it takes for T to become 185, so we want t such that

$185 = -310e^{-0.1359t} + 350$

now solve for $t$. note that this gives the total time for the turkey to get to that temperature. you want the difference between 4 and that time
So, is the answer around 4.64?

11. Originally Posted by dc52789
So, is the answer around 4.64?
yes, that is the total time. but remember, it was already in the oven for 4 hours. so we need to tell them the excess time to get to 185

format: plain


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# On sequences and series of functions

I started studying sequences an series of functions and I found a problem which I don't quite understand. The problem is:

$f_n:(0,1)\rightarrow \mathbb{R}, f_n(x)=\frac{1}{nx+1}$, $n \geq 0$. Study the pointwise and uniform convergence of $f_n(x)$.

I showed that $f$ is pointwise convergent to $f(x)=0$ but I don't understand why is not uniform convergent.The solution in the book is:

$\underset{n \to \infty}{\lim}\hspace{1 mm} \underset{x\in (0,1)}{\sup} \Big | \frac{1 }{nx+1} \Big | =1$. Why? In another example I saw that he took $x=\frac{1}{n}$ to show that a sequence is not uniform convergent.

On series of functions: Can you give me some tips on how to show that a series is uniform convergent? I am confused because I don't know what theorem should I use or what result from the theory. Thank you.

• You can compute the derivative, and find the maximum. – Itay4 Aug 9 '17 at 14:13

For $n>1$,

$$M_n=\sup_{x\in (0,1)}|f_n (x)-0|$$ $$=\sup_{x\in (0,1)}|f_n (x)|\ge |f_n (\frac {1}{n}) |$$

because $\frac {1}{n}\in (0,1)$.

but $$f_n (\frac {1}{n})=\frac {1}{2}$$ thus $$M_n\ge \frac {1}{2}$$ $$\implies \lim_{n\to+\infty}M_n\ge \frac {1}{2}$$ $$\implies \lim_{n\to+\infty}M_n \ne 0$$ hence, the convergence is not uniform at $(0,1)$.

Done!

When $f,g$ are bounded functions from $(0,1)$ to $\mathbb R,$ let $\|f-g\|=\sup \{|f(x)-g(x)|: x\in (0,1)\}.$

Uniform convergence of a sequence $(f_n)_n$ to $f$ is equivalent to $\lim_{n\to \infty}\|f-f_n\|=0.$

If $f(x)=0$ for all $x\in (0,1)$ then $\|f-f_n\|=\|f_n\|.$ So uniform convergence of $(f_n)_n$ to $0$ is equivalent to $\lim_{n\to \infty}\|f_n\|=0.$ But $\|f_n\|\geq |f_n(1/n)|=1/2,$ so convergence is not uniform.

Footnote: In fact, $\|f_n\|=1.$


text:
Manchester decoding

Is there any method of decoding a Manchester encoded signal (IEEE 802.3) that is independent of the bit period?

Suppose the data I want to send is $\rm 10110$. The Manchester code of this data will be $\rm 0110010110$ (according to IEEE standard). We can see that in the $\rm 0101$ part, we need to consider the $\rm 0 \rightarrow 1$ transition (to decode a $\rm 1$) and not the $\rm 1 \rightarrow 0$ transition (falsely decoding a $\rm 0$). To do this, we need to know the bit period a priori.

My question is, is there a way to do this without knowing the bit period a priori?

I think what you're really asking is whether you can decode a Manchester-encoded bit stream without being synchronized to a frame boundary. As you noted, a mid-symbol transition from $0$ to $1$ indicates one state (i.e. a transmitted $1$ for instance), while a transition from $1$ to $0$ would indicate the opposite state (a transmitted $0$). You seem to want to decode a Manchester-encoded bit stream without necessarily knowing where the symbol boundaries are.

Whether you will be able to do this effectively would depend upon the bit error rate in the stream of encoded bits that you observe. Notice that for each two-bit sequence of Manchester-encoded bits, only two of the four possible states correspond to valid observations:

\begin{align} \text{received 01} &\to \text{1 transmitted} \\ \text{received 01} &\to \text{1 transmitted} \\ \text{received 00} &\to \text{invalid!} \\ \text{received 11} &\to \text{invalid!} \\ \end{align}

Therefore, one way to get at the encoded bits in a Manchester bit stream without knowing where the symbol boundaries are would be to try both cases. Since there are two channel bits per symbol, there are only two possible alignments of the encoded symbols on the Manchester-encoded bit stream. Therefore, one approach would be to try both and select the timing offset that gives you the smallest number of "invalid" observations as shown above. Unless you have a lot of bit errors and are very unlucky, the correct alignment should yield a high percentage of valid sequences according to the above criteria, while you would observe a lot of invalid symbols if you were offset by one bit in the stream.


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# Probability of Winning Tennis Match

• Oct 9th 2012, 02:40 PM
jegues
Probability of Winning Tennis Match
See figure attached below for the question.

I was struggling with part a).

Is there a systematic way of solving for the sample space in this question? (I'm having a hard time determining whether I've truly written down every permutation of winning)

The outcomes that only last 3 or 4 sets it is still fairly clear whether you've obtained all possible outcomes or not, but for the outcomes that require 5 sets I find myself racking my brain to determine whether or not I've actually listed all the possible outcomes.

Is there a way to systematically determine this? For example, if I knew from the beginning that there would be 20 samples in my sample space when I have listed say 18, I would know that I am still missing 2 possible samples.

The only way I could think of simplifying things when generating this sample space is to create a sample space that contains all the games where one particular player ones. Then for the total sample space I would simply invert all the entries so that all the games that the other player could have one are covered as well.

Any ideas?
• Oct 9th 2012, 03:04 PM
HallsofIvy
Re: Probability of Winning Tennis Match
Writing "F" for "Federer wins the set" and "R" for "Roddick wins the set" we can have
"RRR" (Roddick wins in three straight sets)
"FFF" (Federer wins in three straight sets)
"RFFF"
"FRFF"
"FFRF"
(But NOT "FFFR" since that would end at "FFF".
"FRRR"
"RFRR"
"RRFR"

"RRFFF" etc (there are 5!/(2!3!)= 10 ways to order 2 Rs and 3 Fs but we do NOT include FFFRR or FFRFR, etc. There are 4!/3!= 4 ways to get 3Fs in the first four sets so we subtract 10- 4= 6.)
• Oct 9th 2012, 03:18 PM
jegues
Re: Probability of Winning Tennis Match
Quote:

Originally Posted by HallsofIvy
Writing "F" for "Federer wins the set" and "R" for "Roddick wins the set" we can have
"RRR" (Roddick wins in three straight sets)
"FFF" (Federer wins in three straight sets)
"RFFF"
"FRFF"
"FFRF"
(But NOT "FFFR" since that would end at "FFF".
"FRRR"
"RFRR"
"RRFR"

"RRFFF" etc (there are 5!/(2!3!)= 10 ways to order 2 Rs and 3 Fs but we do NOT include FFFRR or FFRFR, etc. There are 4!/3!= 4 ways to get 3Fs in the first four sets so we subtract 10- 4= 6.)

How did you obtain these numbers,

$\frac{5!}{2!3!}$

Can you generalize it somehow so I know how you are getting those numbers?

For example,

$\frac{\text{(Number of sets)!}}{(Number of F Wins)!(Number of R Wins)!}$

That formula seems to work for all cases, correct?

Of course, you would need to remove the cases such as,

FFFRR and FFRFR as you've mentioned.


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# Draft

This is a working draft for an interface between one-loop amplitude programs and monte carlo programs.

## Goal of the interface

The goal of the interface is to facilitate the transfer of information between one-loop programs (which we will call one-loop provider OLP or one-loop engine depending on whether we mean the method/work (Golem, Rocket, Helac one-loop, BlackHat, ...) or the actual run time entity) which provide one-loop matrix element and the program the does the integration over the phasespace (which we will call MC).

## Model of the interaction between MC and OLE

The model of the interaction between the MC and OLE assumes two phases which we will call "initialization" and "run time".

1. Initialization

During this phase the MC sets up the OLE. This means setting up the parameters and fixing options that are offered by the OLP

• masses, width
• operating mode, for example which approximation (leading color, massless b's, ...)
• schemes (renormalization, factorization)

2. Run time

During this phase the MCP queries the OLE for the value of a virtual matrix element.

# Implementation proposal

This section outlines a proposal for the implementation of the interface model presented above. The aim is to make it easy to implemeent it in all common programming language. The functions will be called by the MCP and executed by the OLE. The name of the function and its implementation can be chosen by the OLP (although sticking to the name in this wiki would be convenient), the main point being to provide a function that satisfies the interface.

In the easiest case of merging two programs written in the same language, the connection between the caller (the MCP) and the receiver (the OLE) can be easily achieved by linking the rogram together. In more difficult cases with mixed languages or special environments a bridge program or transmission protocol will have to be put in place. This is however not the topic or concern of this proposal, here we only want to ensure that the end point of these bridges will be the same.

## Initialisation phase

In this phase the MCP creates a file that contains the information about the subprocesses it will need to perform the computation. We will call this file the "order". This file is then read by the OLE and it creates a new file with the agreement about how to link the MCP and OLE at runtime is provided. We will call this file a "contract". Lines starting with a # character are treated as comments. In this section we call a subprocess both the partonic subprocesses and any component of if in some basis.

### order file

the order file contains the information needed for the OLE to declare whether or not it can provide the subprocess the MCP wants.

The first part of the file contains some general settings.

The second part contains the list of the subprocesses the MCP needs. Each line is of the form

[TYPE] [in] -> [out] [[PDG]...] [...]

where [TYPE] describes which type of subprocess the MCP want. For example one could ask for color and helicity summed amplitudes. [in] is the number of particles in the initial state. [out] is the number of particles in the final state. [PDG] is the pdg code of the particle. The first entries are for the initial states, then comes the final state particles.

The settings that a OLE should (have to?) provide are

• QCD_scheme : the scheme for the QCD renormalization
• EW_scheme : the scheme for the EW renormalization

OLE can provide more settings. They have to be documented. We encourage OLP to submit new settings to the WIKI in order to avoid having the same optional setting called differently by different OLP.

• model_file : the model file (in LH format) from which parameters have to be read
• mode : the operating mode of the

The OLE can provide defaults for the settings it provides.

Here is an example of a order file

# example contract file

QCD_scheme Dreg
model_file my_favorite_model_in_LH_format.slh
mode subdivide_subprocess yes

CH_SUMMED 2 -> 3 21 21 6 -6 21
CH_SUMMED 2 -> 3 1 -1 6 -6 21
CH_SUMMED 2 -> 3 1 21 6 -6 1

### contract file

The OLE reads the order file and creates its "answer" in the form of a contract file. The contract file is basically a copy of the order file, but with information from the OLE appended at the end of each (non content-free) line. The OLE answer is separated from the MCP "request" by a "|" character. The answer of the OLE should give are explained in the following paragraphs.

setting: 1 if the setting is supported and the supplied arguments are correct. Errors are coded as follows: -1 if the option is not supported, -2 if the option is supported but the parameter is not recognized/valid.

For subprocesses, the answer should be the number of separate contributions that the OLE can compute that add up to the required subprocess, followed by an integer label that identifies each of these contributions. This allows the multi-channel integrator (if present) to take advantage of importance difference between the provided subcontributions. The integer label will be use at run time to identify the subprocesses. OLE that provide this kind of separation might also provide a single subprocess that is the sum of all contributions. Whether or not to return the contributions separately could be set using a setting subdivide_subprocess

Example contract file

# example contract file
# contract produced by OLE HAPPY_OLE

QCD_scheme Dreg                                 | 1
model_file my_favorite_model_in_LH_format.slh   | 1
subdivide_subprocess yes                        | 1

CH_SUMMED 2 -> 3 21 21 6 -6 21                  | 2 13 35
CH_SUMMED 2 -> 3 1 -1 6 -6 21                   | 1 29
CH_SUMMED 2 -> 3 1 21 6 -6 1                    | 3 8 23 57

It would be useful if the OLE could provide some help at this point (like listing the parameter that are accepted).

# example contract file
# contract produced by OLP UNHAPPY_OLE

QCD_scheme Dreg                                 | -2 note: supported schemes are t'HV
mode leading color                              | -2 note: OLE UNHAPPY don't have a "leading color" mode
model_file my_favorite_model_in_LH_format.slh   | -2 note: could not find file "my_favorite_model_in_LH_format.slh"
subdivide_subprocess yes                        | -1 note: unknown setting

CH_SUMMED 2 -> 3 21 21 6 -6 21                  | 0 note: Sorry, OLE don't support massive quarks
CH_SUMMED 2 -> 3 1 -1 6 -6 21                   | 0 note: Sorry, this process is not implemented in OLE UNHAPPY
CH_SUMMED 2 -> 3 1 21 6 -6 1                    | 0 note: Sorry, too hard for OLE UNHAPPY

A description of the difference between the different contributions could be useful :

CH_SUMMED 2 -> 3 21 21 6 -6 21                  | 2 4 25 note: contributions are leading_color, sub_leading_color

## Run time phase

To start the run time phase, the OLE might need to be reminded of the contract it committed to (this is the case for programs that dynamically generate their evaluation structure as opposed to libraries of generated code). To initialize the OLE for the run, the function Init has to be called by the MCP. After this the MCP can start querying the OLE fr the subprocesses. This is done using the function EvalSubProcess.

### Init

the function Init should be passed an character sting that contains the name of the contract file. The function returns 1 if the initialization is successful, 0 if not.

### EvalSubprocess

The parameters to be passed to the EvalSubprocess function are (to avoid confusing, please use this order)

• the integer label of the subprocess
• the address of the start of an array containing the components of the momenta. The momenta are placed in a on dimensional array. The first component is the energy of the first particle, then x, y, z components of this momentum and its mass, then comes the second momentum energy, x, y, z, m etc... The mass is defined as E^2-x^2-y^2-z^2 but is provided for numerical reasons. The unit is Gev. Both incoming and outgoing momenta have a positive energy. This means that the array will contain n*5 double precision numbers.
• the renormalization scale as a double precision number, in Gev.
• the strong coupling double alpha_s
• the weak coupling alpha_EW
• the address of a the array where the result is to be put. The length and the content of this array is to be documented by the OLP. We will define here one possibility, if the OLP departs from it, it should be clearly documented.
Pole2, Pole1, Pole0, Born

where PoleN is the real part of the interference $Re\left|(A_1) A_0^*\right|$


text:
# What is the integration of $$(log x)^2$$ dx ?

## Solution :

We have, I = $$(log x)^2$$ . 1 dx, Then ,

where $$(log x)^2$$ is the first function and 1 is the second function according to ilate rule,

I = $$(log x)^2$$ { $$\int$$ 1 dx} – $$\int$$ {$$d\over dx$$ $$(log x)^2$$ . $$\int$$ 1 dx } dx

I = $$(log x)^2$$ x – $$\int$$ 2 log x . $$1\over x$$ . x dx

I = x $$(log x)^2$$ – 2 $$\int$$ log x .1 dx

$$\implies$$ I = x $$(log x)^2$$ – 2[ log x { $$\int$$ 1 dx } – $$\int$$ { $$d\over dx$$ (log x) $$\int$$ 1 dx } dx ]

$$\implies$$ I = x $$(log x)^2$$ – 2 { (log x) x – $$\int$$ $$1\over x$$ x dx }

Hence, I = x( $$(log x)^2$$ – 2 (x log x – x) + C

### Similar Questions

What is the integration of log cos x dx ?

What is the integration of log 1/x ?

What is the integration of 1/x log x ?

What is the integration of x log x dx ?


text:
# stochastic process

Let $(\Omega,\mathcal{F},\textbf{P})$ be a probability space. A stochastic process is a collection

 $\{X_{t}\mid t\in T\}$

of random variables $X_{t}$ defined on $(\Omega,\mathcal{F},\textbf{P})$, where $T$ is a set, called the index set of the process $\{X_{t}\mid t\in T\}$. $T$ is usually (but not always) a subset of $\mathbb{R}$. $X$ is sometimes known as a random function.

Given any $t$, the possible values of $X_{t}$ are called the states of the process at $t$. The set of all states (for all $t$) of a stochastic process is called its state space.

If $T$ is discrete, then the stochastic process is a discrete-time process. If $T$ is an interval of $\mathbb{R}$, then $\{X_{t}\mid t\in T\}$ is a continuous-time process. If $T$ can be linearly ordered, then $t$ is also known as the time.

A stochastic process $X$ with state space $S$ can be thought of in either of following three ways.

• As a collection of random variables, $X_{t}$, for each $t$ in the index set $T$.

• As a function in two variables $t\in T$ and $\omega\in\Omega$,

 $X\colon T\times\Omega\rightarrow S,\ (t,\omega)\mapsto X_{t}(\omega).$

The process is said to be measurable, or, jointly measurable if it is $\mathcal{B}(T)\otimes\mathcal{F}/\mathcal{B}(S)$-measurable. Here, $\mathcal{B}(T)$ and $\mathcal{B}(S)$ are the Borel $\sigma$-algebras on $T$ and $S$ respectively.

• In terms of the sample paths. Each $\omega\in\Omega$ maps to a function

 $T\rightarrow S,\ t\mapsto X_{t}(\omega).$

Many common examples of stochastic processes have sample paths which are either continuous or cadlag.

Examples. The following list is some of the most common and important stochastic processes:

1. 1.

a random walk, as well as its limiting case, a Brownian motion, or a Wiener process

2. 2.
3. 3.

Markov process; a Markov chain is a Markov process whose state space is discrete

4. 4.

renewal process

Remarks.

• Sometimes, a stochastic process is also called a random process, although a stochastic process is generally linked to any “time” dependent process. In a random process, the index set may not be linearly ordered, as in the case of a random field, where the index set may be, for example, the unit sphere $S^{2}\subseteq\mathbb{R}^{3}$.

• In statistics, a stochastic process is often known as a time series, where the index set is a finite (or at most countable) ordered sequence of real numbers.

 Title stochastic process Canonical name StochasticProcess Date of creation 2013-03-22 14:39:10 Last modified on 2013-03-22 14:39:10 Owner gel (22282) Last modified by gel (22282) Numerical id 14 Author gel (22282) Entry type Definition Classification msc 60G05 Classification msc 60G60 Synonym random process Related topic DistributionsOfAStochasticProcess Defines discrete-time process Defines continuous-time process Defines state Defines time series Defines state space Defines random function Defines jointly measurable


text:
Select Board & Class

# Board Paper of Class 12 2016 Mathematics - Solutions

Section A - Answer Question 1 (compulsory) and five other questions.
Section B and Section C - Answer two questions from either Section B or Section C.
All working, including rough work, should be done on the same sheet as, and adjacent to, the rest of the answer.
The intended marks for questions or parts of questions are given in brackets [ ].
Mathematical tables and graph papers are provided.
Slide rule may be used.

• Question 1
(a) (i) Find the matrix X for which:

(b) Solve for x, if: $\mathrm{tan}\left({\mathrm{cos}}^{-1}x\right)=\frac{2}{\sqrt{5}}$

(c) Prove that the line 2x – 3y = 9 touches the conics y2 = – 8x. Also, find the point of contact.

(d) Using L'Hospital's Rule, evaluate:

(e) Evaluate:

(f) Using properties of definite integrals, evaluate:

(g) The two lines of regressions are x + 2y – 5 = 0 and 2x + 3y – 8 = 0 and the variance of x is 12. Find the variance of y and the coefficient of correlation.

(h) Express $\frac{2+i}{\left(1+i\right)\left(1-2i\right)}$ in the form of a + ib.  Find its modulus and argument.

(i) A pair of dice is thrown. What is the probability of getting an even number on the first die or a total of 8?

(j) Solve the differential equation: $x.\frac{dy}{dx}+y=3{x}^{2}-2$ VIEW SOLUTION

• Question 2
(a) Using properties of determinants, prove that:

(b) Solve the following system of linear equations using matrix method:
3x + y + z = 1, 2x + 2z = 0, 5x + y + 2z = 2 VIEW SOLUTION

• Question 3
(a) If

(b) Write the Boolean function corresponding to the switching circuit given below:

A,  B and C represent switches in 'on' position and A', B' and C' represent them in off position. Using Boolean algebra, simplify the function and construct an equivalent switching circuit. VIEW SOLUTION

• Question 4
(a) Verify the conditions of Rolle's Theorem for the following function:
f(x) = log(x2 + 2) – log 3 on [–1, 1]
Find a point in the interval, where the tangent to the curve is parallel to x-axis.

(b) Find the equation of the standard ellipse, taking its axes as the coordinate axes, whose minor axis is equal to the distance between the foci and whose length of latus rectum is 10. Also, find its eccentricity. VIEW SOLUTION

• Question 5
(a) If log y = tan–1x, prove that: $\left(1+{x}^{2}\right)\frac{{d}^{2}y}{d{x}^{2}}+\left(2x-1\right)\frac{dy}{dx}=0$

(b) A rectangle is inscribed in a semicircle of radius r with one of its sides on the diameter of the semicircle. Find the dimensions of the rectangle to get maximum area. Also, find the maximum area. VIEW SOLUTION

• Question 6
(a) Evaluate:

(b) Find the area of the region bound by the curves y = 6xx2 and y = x2 – 2x. VIEW SOLUTION

• Question 7
(a) Calculate Karl Pearson's coefficient of correlation between x and y for the following data and interpret the result:
(1, 6), (2, 5), (3,7), (4, 9), (5, 8), (6, 10), (7, 11), (8, 13), (9, 12)

(b) The marks obtained by 10 candidates in English and Mathematics are given below:  Marks in English 20 13 18 21 11 12 17 14 19 15 Marks in Mathematics 17 12 23 25 14 8 19 21 22 19

Estimate the probable score for Mathematics if the marks obtained in English are 24. VIEW SOLUTION

• Question 8
(a) A committee of 4 persons has to be chosen from 8 boys and 6 girls, consisting of at least one girl. Find the probability that the committee consists of more girls than boys.

(b) An urn contains 10 white and 3 black balls while another urn contains 3 white and 5 black balls. Two balls are drawn from the first urn and put into the second urn and then a ball is drawn from the second urn. Find the probability that the ball drawn from the second urn is a white ball. VIEW SOLUTION

• Question 9
(a) Find the locus of a complex number, z = x + iy, satisfying the relation $\left|\frac{z-3i}{z+3i}\right|\le \sqrt{2}$. Illustrate the locus of z in the Argand plane.

(b) Solve the following differential equation: x2 dy + (xy + y2) dx = 0, when x = l and y = 1. VIEW SOLUTION

• Question 10
(a) For any three vectors  show that are coplanar.

(b) Find a unit vector perpendicular to each of the vectors  VIEW SOLUTION

• Question 11
(a) Find the image of the point (2, –1, 5) in the line $\frac{x-11}{10}=\frac{y+2}{-4}=\frac{z+8}{-11}$. Also, find the length of the perpendicular from the point (2, –1, 5) to the line.

(b) Find the Cartesian equation of the plane, passing through the line of intersection of the planes:
and intersecting y-axis at (0, 3). VIEW SOLUTION

• Question 12
(a) In an automobile factory, certain parts are to be fixed into the chassis in a section before it moves into another section. On a given day, one of the three persons A, B and C carries out this task. A has 45% chance, B has 35% chance and C has 20% chance of doing the task. The probability that A, B and C will take more than the allotted time is respectively. If it is found that the time taken is more than the allotted time, what is the probability that A has done the task?

(b) The difference between mean and variance of a binomial distribution is 1 and the difference of their squares is 11. Find the distribution. VIEW SOLUTION

• Question 13
(a) A man borrows ₹ 20,000 at 12% per annum, compounded semi-annually and agrees to pay it in 10 equal semi-annual instalments. Find the value of each instalment, if the first payment is due at the end of two years.

(b) A company manufactures two types of products A and B. Each unit of A requires 3 grams of nickel and 1 gram of chromium, while each unit of B requires 1 gram of nickel and 2 grams of chromium. The firm can produce 9 grams of nickel and 8 grams of chromium. The profit is ₹ 40 on each unit of product of type A and ₹ 50 on each unit of type B. How many units of each type should the company manufacture so as to earn maximum profit? Use linear programming to find the solution. VIEW SOLUTION

• Question 14
(a) The demand function is $x=\frac{24-2p}{3}$ where x is the number of units demanded and p is the price per unit.
Find:
(i) The revenue function R in terms of p.
(ii) The price and the number of units demanded for which the revenue is maximum.

(b) A bill of ₹ 1,800 drawn on 10th September, 2010 at 6 months was discounted for ₹ 1,782 at a bank. If the rate of interest was 5% per annum, on what date was the bill discounted? VIEW SOLUTION

• Question 15
(a) The index number by the method of aggregates for the year 2010, taking 2000 as the base year, was found to be 116. If sum of the prices in the year 2000 is ₹ 300, find the values of x and y in the data given below:  Commodity A B C D E F Price in the year 2000 (₹) 50 x 30 70 116 20 Price in the year 2010 (₹) 60 24 y 80 120 28

(b) From the details given below, calculate the five yearly moving averages of the number of students who have studied in a school. Also, plot these and original data on the same graph paper.  Year 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 Number of Students 332 317 357 392 402 405 410 427 405 438
VIEW SOLUTION
What are you looking for?

Syllabus


text:
# Can I have a flexible partial derivative macro?

I was thinking about writing a partial derivative macro that has a similar syntax to the Mathematica D function. (example)

The syntax could be like this:

• \D{f}{x} expands to \frac{\partial f}{\partial x}
• \D{f}{x,y} expands to \frac{\partial^2 f}{\partial x\,\partial y}
• \D{f}{{x,2},y} expands to \frac{\partial^3 f}{\partial x^2\,\partial y}
• et cetera for any number of derivatives.

Bonus points for an optional argument so that for instance

• \D[subscript]{f}{{x,2},y} expands to f_{xxy}

I looked at the commath package but the syntax is a little tedious. For instance, to get \frac{\partial^3 f}{\partial x^2\,\partial y} with that package I would need to do \md{f}{3}{x}{2}{y}{}. And it doesn't generalize to three variables.

Before I dive in I thought I'd ask the hive and see what's been done.

-
Have you seen the cool package? – Torbjørn T. Jul 1 '11 at 20:52
@Torbjorn: cool is cool indeed! So I can use \pderiv[2,1]{f}{x,y} to get \frac{\partial^3 f}{\partial x^2\,\partial y}. – Matthew Leingang Jul 1 '11 at 21:16
@Torbjorn: if you will repost your comment as an answer I will accept it. I'm going with cool for this and other reasons. – Matthew Leingang Jul 7 '11 at 19:14
This must be something that every mathematician does. I think my first "serious" macros were (simple) versions of this. I'll definitely investigate the answers to this. – Andrew Stacey Jul 7 '11 at 20:54
Did you experience any errors with the \pderiv macro? Specifically, when having more than one power in the optional argument. (See the comments to my answer.) – Torbjørn T. Sep 28 '11 at 12:51
show 1 more comment

## 4 Answers

The cool package offers commands for writing partial derivates (and a lot of other stuff).

A short example:

\documentclass{article}
\usepackage{cool}
\begin{document}
\begin{equation*}
\pderiv{f}{x} \qquad
\pderiv[2,3]{f}{x,y} \qquad
\pderiv[1,n,4]{f}{x,y,z}
\end{equation*}
\end{document}

# Note

By default the last two of those will not work, as there is a bug in the package. This can be fixed by changing a \def to \edef in line 2059 of cool.sty, see Basic use of derivative with cool package fails with "Missing \endcsname inserted"

-
 I was just trying this out (from curiosity) and found that if I put more than one power in the first (optional) argument to the \pderiv then it complains vociferously. Any chance you could post a MWE? I'm sure I'm just doing something silly. – Andrew Stacey Sep 22 '11 at 8:53 @AndrewStacey Maybe not, I get an error as well, from examples copied directly from the cool demo file: ! Missing endcsname inserted. \toks@. I've never really used the package myself, I just stumbled upon it at one point, and remembered it for this question. Not sure what is going on ... – Torbjørn T. Sep 22 '11 at 10:56

I have this, which is quite similar to what you want:

\documentclass[a4paper]{article}

\makeatletter
\newcommand{\der}[2]{\begingroup
\@tempswafalse\toks@={}\count@=\z@
\@for\next:=#2\do
{\expandafter\check@var\next
\advance\count@\der@exp
\if@tempswa
\toks@=\expandafter{\the\toks@\,}%
\else
\@tempswatrue
\fi
\toks@=\expandafter{\the\expandafter\toks@\expandafter\partial\der@var}}%
\frac{\partial\ifnum\count@=\@ne\else^{\number\count@}\fi#1}{\the\toks@}%
\endgroup}
\def\check@var{\@ifstar{\mult@var}{\one@var}}
\def\mult@var#1#2{\def\der@var{#2^{#1}}\def\der@exp{#1}}
\def\one@var#1{\def\der@var{#1}\chardef\der@exp\@ne}
\makeatother

\begin{document}
$\der{f}{x}\qquad \der{f}{*{2}{x},y}\qquad \der{f}{*{3}{x},y,*{4}{z}}$
\end{document}

EDIT: the following code will render correctly also

\pder{f}{\mathbf{x}}
\pder{f_1}{x_1}

Here it is:

\makeatletter
\newcommand{\der}[2]{\begingroup
\@tempswafalse\toks@={}\count@=\z@
\@for\next:=#2\do
{\expandafter\check@var\next\@nil
\advance\count@\der@exp
\if@tempswa
\toks@=\expandafter{\the\toks@\,}%
\else
\@tempswatrue
\fi
\toks@=\expandafter{\the\expandafter\toks@\expandafter\partial\der@var}}%
\frac{\partial\ifnum\count@=\@ne\else^{\number\count@}\fi#1}{\the\toks@}%
\endgroup}
\def\check@var{\@ifstar{\mult@var}{\one@var}}
\def\mult@var#1#2\@nil{\def\der@var{#2^{#1}}\def\der@exp{#1}}
\def\one@var#1\@nil{\def\der@var{#1}\chardef\der@exp\@ne}
\makeatother

-
Interesting. Did you use the stars to mimic tabular preambles? – Matthew Leingang Jul 1 '11 at 20:33
Reading the code, I see now you used stars so you could implement it with \@ifstar. Clever. – Matthew Leingang Jul 1 '11 at 20:35
@Matthew: yes, the *{2}{x} comes from the tabular preambles, and, yes, it makes convenient to use \@ifstar. :) – egreg Jul 1 '11 at 21:07
@egreg: The following two lines $\der{f}{\mathbf{n}}$ $\def{f_1}{x_1}$ do not process correctly. Is it possible to fix the macro so they do? Thanks. – rogerl Oct 4 '11 at 15:34

Here my try. I prefer a more simply input format: \Der{<var>}{<var><num>,<var><num>,...} e.g. \Der{f}{x3,y,z4}. This also works for more complicated input like \Der{f}{{\hat{x}}3,y105}.

\documentclass{article}

\makeatletter

\newcommand\Der[2]{%
\begingroup
\@temptokena{\@gobble}%
\@tempcnta\z@
\@for\var:=#2\do{%
\expandafter\@Der\var\relax
}%
\ensuremath{\frac{\partial
\ifnum\@tempcnta>\@ne
^{\the\@tempcnta}%
\fi
#1}{\the\@temptokena}}%
\endgroup
}

\def\@Der#1#2\relax{%
\ifx\relax#2\relax
\advance\@tempcnta by \@ne
\@temptokena\expandafter{\the\@temptokena\,\partial{#1}}%
\else
\advance\@tempcnta by #2\relax
\@temptokena\expandafter{\the\@temptokena\,\partial{#1}^{#2}}%
\fi
}

\makeatother

\usepackage{amsmath}
\begin{document}

$\Der{f}{x} \qquad \Der{f}{x,y} \qquad \Der{f}{x2,y}$

$\Der{f}{x}\qquad \Der{f}{x2,y}\qquad \Der{f}{x3,y,z4}$

$\Der{f}{{\hat{x}}3,y1,z10}$

\end{document}

It is also possible to avoid the need for commas. I'm thought first this is more readable but I'm not sure about that anymore. This version doesn't support negative numbers (no loss) and might be a little more sensitive than the first.

\documentclass{article}

\makeatletter

\newcommand\Der[2]{%
\begingroup
\@temptokena{\@gobble}%
\@tempcnta\z@
\expandafter\@Der@var#2\relax
\ensuremath{\frac{\partial
\ifnum\@tempcnta>\@ne
^{\the\@tempcnta}%
\fi
#1}{\the\@temptokena}}%
\endgroup
}

\def\@Der@var#1{%
\ifx\relax#1\empty\else
\def\next{\expandafter\@Der@num\expandafter{\the\@tempcntb}{#1}}%
\afterassignment\next
\@tempcntb=0%
\fi
}

\def\@Der@num#1#2{%
\ifnum#1=\z@
\advance\@tempcnta by \@ne
\@temptokena\expandafter{\the\@temptokena\,\partial{#2}}%
\else
\advance\@tempcnta by #1\relax
\@temptokena\expandafter{\the\@temptokena\,\partial{#2}^{#1}}%
\fi
\@Der@var
}

\makeatother

\usepackage{amsmath}
\begin{document}

$\Der{f}{x} \qquad \Der{f}{xy} \qquad \Der{f}{x2y}$

$\Der{f}{x}\qquad \Der{f}{x2y}\qquad \Der{f}{x3 y z4}$

$\Der{f}{{\hat{x}}3y1z10}$

\end{document}

### Result (for both implementations):

-
 Thanks for playing. Both you and egreg used \@for, which means I should learn more about that. – Matthew Leingang Jul 2 '11 at 4:14

The esdiff package should not go unmentioned in this context.

In the meantime there are two other packages which facilitate easy typesetting of derivatives.

-
OK, it's been mentioned. :-) But could you add a code snippet to show how it works? – Matthew Leingang Oct 4 '11 at 17:29
What about just reading the (very short) package manual? I couldn't add any better code examples here. – Thorsten Donig Oct 4 '11 at 19:17
You're right. I just realized that this package does this and nothing else. – Matthew Leingang Oct 5 '11 at 18:55


text:
# Testing the anisotropy of the universe using the simulated gravitational wave events from advanced LIGO and Virgo

The European Physical Journal C, May 2018

The detection of gravitational waves (GWs) provides a powerful tool to constrain the cosmological parameters. In this paper, we investigate the possibility of using GWs as standard sirens in testing the anisotropy of the universe. We consider the GW signals produced by the coalescence of binary black hole systems and simulate hundreds of GW events from the advanced laser interferometer gravitational-wave observatory and Virgo. It is found that the anisotropy of the universe can be tightly constrained if the redshift of the GW source is precisely known. The anisotropic amplitude can be constrained with an accuracy comparable to the Union2.1 complication of type-Ia supernovae if $$\gtrsim 400$$ GW events are observed. As for the preferred direction, $$\gtrsim 800$$ GW events are needed in order to achieve the accuracy of Union2.1. With 800 GW events, the probability of pseudo anisotropic signals with an amplitude comparable to Union2.1 is negligible. These results show that GWs can provide a complementary tool to supernovae in testing the anisotropy of the universe.

This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1140%2Fepjc%2Fs10052-018-5841-x.pdf

Hai-Nan Lin, Jin Li, Xin Li. Testing the anisotropy of the universe using the simulated gravitational wave events from advanced LIGO and Virgo, The European Physical Journal C, 2018, 356, DOI: 10.1140/epjc/s10052-018-5841-x


text:
# More Issues With Floating Points

It is well-known that $$\dfrac{1}{3}$$ cannot be represented using a finite a finite decimal representation. $\dfrac{1}{3} = 0.333 \ldots.$

Neither can it be represented using a finite binary representation. $\dfrac{1}{3} = 0.010101 \ldots _2$

If we had just 24 bits to store the decimal part of this value, the closest rational number we could store in binary would be

$\dfrac{5592405}{16777216}$

What is the closest rational number to $$0.1$$ that we could store using binary floating points with only 24 bits (assuming all the 24 bits are being used to store the fractional part of the value)?

If the value comes out to be $$\dfrac{a}{b}$$, where $$a$$ and $$b$$ are coprime positive integers, submit your as $$a+b$$.

Assume that the point occurs before all the 24 bits. So, 111111111111111111111111 will be interpreted as 0.111111111111111111111111

×


text:
# News Can we avoid a double dip recession?

1. Aug 4, 2011

### SixNein

The economist has a good article on this topic:

http://www.economist.com/node/21525405

As I was saying https://www.physicsforums.com/showpost.php?p=3423219&postcount=1", there was damage being done by the debt debate. On one hand, a political party is now willing to take our economic future hostage for political gains. On the other hand, there is no willingness to spend in the short term and do long term planning on debt, and uncertainty remains over the actions of the supper committee on everything from taxes to spending cuts. And the national agenda has been changed from recovery to debt even though our growth is extremely weak.

so are we going to avoid a double dip recession? I'm not sure, the economy was so fragile with Europe markets already in distress, and the new uncertainties in America may be enough to push the economy right off the cliff.

On the bright side, the Fed might be able to take some limited actions.

Last edited by a moderator: Apr 26, 2017
2. Aug 5, 2011

### SixNein

Perhaps people need directions:

Democrats' Position -----------------------> Reality <------------------------ Republicans' position

3. Aug 5, 2011

### Ivan Seeking

Staff Emeritus
I think this has far more to do with Europe and Japan than it does the debt crisis.

Our debt has nothing to do with the immediate problems. The "crisis" was the debt ceiling issue, which was artificial and facilitated by the tea party. But it is significant in that it limits our ability to respond to a severe recession.

From what I understand, the chances of a double-dip are no worse than 50/50, so I'll assume the glass is half full unless and until convinced otherwise.

4. Aug 5, 2011

### gravenewworld

Italy gets into big trouble and defaults, the Eurozone collapses economically, and the US economy tanks along with it while the EU falls apart. Riots over food and the restless unemployed spread everywhere across the globe. Anger at the West increases over what are perceived economic abuses that pushed the world into a terrible recession leading to war. Dire times indeed.

5. Aug 5, 2011

### Ivan Seeking

Staff Emeritus
You might want to get out and get some sunlight.

6. Aug 5, 2011

### SixNein

Europe is playing a significant role, but America is playing a significant role too.

According to the BEA, "Real gross domestic product -- the output of goods and services produced by labor and property located in the United States -- increased at an annual rate of 1.3 percent in the second quarter of 2011, (that is, from the first quarter to the second quarter), according to the "advance" estimate released by the Bureau of Economic Analysis. In the first quarter, real GDP increased 0.4 percent."

The debt ceiling debate creates uncertainty and doubt over the possibility of stimulus. And current programs are set to expire at the end of this year, so we will see a shrinkage of GDP next year if nothing is done.
[PLAIN]http://web.epi-data.org/temp727/gdp_debt_ceiling.JPG [Broken]

In addition, the debate set a new precedent on holding the debt limit hostage. Republicans are calling it a template to be used in the future. So we may be about to repeat this **** over and over again.

The uncertainty over the debt limit debate is not over because of this super-committee; instead, the uncertainty remains over spending cuts and tax increases. Will the cuts be up front or on down the road, and where will these cuts be made? What will taxes be on and how much? Members could also fail to agree and the triggers kick in.

The result of the debt limit debate was disappointing to say the least. There was no long term objectives to control the debt, and there was no short term spending to stimulate the economy. The only real accomplishment was sending a message to the international community that our political system is now dysfunctional and injecting loads of uncertainty into a already fragile market.

All of this hurts confidence that America is on the road to recovery. Can America even take care of its long term debt problem? Can it do stimulus right now for the economy? The answer to both appears to be no.

Europe is also going through similar problems in the sense of failed political leadership that puts way to much uncertainty into the market.

Last edited by a moderator: May 5, 2017
7. Aug 5, 2011

### Ivan Seeking

Staff Emeritus
No doubt, the latest report and the revised report from the first quarter were terrible news.

Last edited by a moderator: May 5, 2017
8. Aug 5, 2011

### SixNein

What breaks my heart was this performance was done while our economy was so fragile. The economy has been showing negative signs for a while now, and politicians jump up and decide to threaten our obligations, and they dragged it out to the very last second. And they are now saying it was a great idea, and they are going to do it again.

They are suppose to be inspiring confidence right now.

9. Aug 5, 2011

### Staff: Mentor

:blink: Do neither of you believe that a high and yet rapidly growing debt puts any kind of drag on the economy?

What is your threshold debt level that you believe would be an "immediate problem" and what is your time horizon for when we should start caring about the debt?

10. Aug 5, 2011

### Lapidus

Last edited by a moderator: Apr 26, 2017
11. Aug 5, 2011

### Staff: Mentor

Neither of those posts answer my question and in any case, we require ACTUAL citations in this forum.

If you can always spend your way out of debt, why doesn't Greece do it?

Last edited: Aug 5, 2011
12. Aug 5, 2011

### Lapidus

If you have read my post I mentioned that one key element when dealing with deficits are the interests rates. What do you conclude?

And again, when the governments spends in a recession it smoothens the downturn and might again increase the GDP and by that keeps the debt/ GDP ratio stable.

Have you guys ever thought about how severe the down-spiraling of the economy would be without the government and the automatic stabilizer that it provides?

13. Aug 5, 2011

### WhoWee

Tea Party discussion aside - I was in your camp with the 50/50 until the world wide market drop yesterday COUPLED with a slight dip in gold and (most important) a drop in oil prices. The oil price drop signals a drop in consumption is expected which means a further slow down in the economy. I'm not discounting any of the other leading indicators (could improve in September as unemployment adjusted down to 9.1%) - just focusing on the oil price for now.

14. Aug 5, 2011

### WhoWee

Every new business venture my known associates have started or participated in over the past 24 months have been connected guided or controlled by the Government is some manner - whether insurance/finance, construction, manufacturing, medical, or housing. All of them regret their decisions for a variety of reasons.

The insurance (Medicare) regulations are reaching the point of ridiculous, the (green) construction costs more, is less profitable and slow, manufacturing in Empowerment Zones has unique challenges, HIPPA and healthcare reform (planning and implementation) is a nightmare, and HUD is something that allows you to make a stable income - but you feel like part of the problem. Please label the entire post - IMO.

15. Aug 5, 2011

### WhoWee

The interest rates are currently (artificially) forced to near 0% and growth projections exceed the current actual - yet the debt trajectory is over $20Trillion with Congressional and Presidential approval. To your point - what will happen to the debt trajectory when the interest rates return to normal levels -$28Trillion(?) - what's your estimate?

16. Aug 5, 2011

### Staff: Mentor

I mostly agree with the main economic points of the article, but disagree with a political one:
I agree that continuing the payroll tax credit and extending jobless benefits would be good - and I agree that that has little to do with the debt ceiling debate. But I think the writer is being naive: these things, particularly the jobless benefits are likely to be extended when we get there.
I agree that spending shouldn't be cut much in the short term, but only in the very short term: a year, certainly no more than two. And I don't know what they mean by "medium-term", but I think after about two years, we need some real and substantial spending cuts. And I also wholeheartedly agree that entitlements are the biggest spending problem. "Tax reform" is vague and I support a form of tax reform (simplification, elimination of a lot of deductions) after about 2 years -- heck, some even now! - but I suspect my vision differs from theirs.
Agreed.
I agree that uncertainty is a problem for investment, but I think the author is missing an obvious reality: The debt ceiling debate didn't create the uncertainty! ...which is why the end of the debt ceiling debate didn't end the uncertainty. Sure, we got a new debt commission, but we had one last year too! The debt is rising and it needs to be dealt with sometime soon, otherwise we're headed for total collapse - Greece style. IMO, that is the type of uncertainty a businessman is looking at when trying to decide whether to build a new building (with a payback of 10 years or more on the investment jumping to 20 if the economy remains in the tank?) or even replace a dying piece of equipment (why bother if the economy is just going to collapse in 5 years and you're just going to pawn it anyway?).

So I consider it a positive thing that Republicans made people consider the elephant in the room (pun intended), while at the same time am worried that backslapping over reaching an agreement will take the place of going back later and actually making the necessary changes.

Last edited by a moderator: Apr 26, 2017
17. Aug 5, 2011

### ParticleGrl

There is, in fact, an easy quantitative way to get a feel for this drag- look at bond rates. The US real rates on treasuries are negative for everything less than 10 years, and for a brief period yesterday the NOMINAL rate on 30 days was negative. What this means is that the borrowing is not creating the sort of drag you are suggesting right now.

Thats why this idea of "uncertainty" is infuriating- the ONLY thing the market is certain in right now is the safety of government debt. They are lending money at a nominal loss!!

Now, if we were Greece, and our rates were rising (instead of steadily declining, as ours are), we might have to worry about something. But Greece is tied into the eurozone, so they are limited all the way around in their responses to crisis.

18. Aug 5, 2011

### Staff: Mentor

That implies we're paying no interest on the debt - is that what you are trying to say? It isn't true: this year, interest on the debt will be about half a trillion dollars. That's 3.5% of GDP. In other words, a stagnant economy paying 3.5% of GDP in interest could instead be expanding at 3.5% annually - a perfectly healthy rate - if the debt wasn't there.

http://www.treasurydirect.gov/govt/reports/ir/ir_expense.htm

And we're currently planning on doubling the weight of this anchor over the next decade.
Confidence is a relative thing. For something that is 100% iron-clad, assured, a tiny, eyelash weight of shakiness is a big deal.

And it isn't like people are buying at low interest because of confidence (implying they would be buying at higher interest if they were less confident) - bond rates are affected by interest rates.

19. Aug 5, 2011

### WhoWee

PG - there is a great deal of uncertainty in the market - we had a worldwide stock drop of 3%(?) yesterday and the DJI is bouncing like a ball today.

Please consider - do you think Treasuries would be in the same position today without QE-2 and downward pressure on interest rates - credit downgrade possibility aside - interest rates will rise at some point. Accordingly, there is uncertainty regarding inflation.

Additionally, the Government held Treasuries (Social Security and Fed) will need to be re-evaluated - won't they? I don't think they'll risk printing more money to replace them - do you?
http://www.cnsnews.com/news/article/fed-eclipses-china-top-owner-us-debt [Broken]

Last edited by a moderator: May 5, 2017
20. Aug 5, 2011

### ParticleGrl

No, thats not what I'm saying. I'm suggesting that the rates for short-term CURRENTLY ISSUED debt will be negative in real terms (and sometimes nominal terms). Despite issuing new debt, the interest we are paying on the national debt hasn't been climbing that rapidly- compare the debt service projected for this year to 2007. Our debt increased by 50%, the amount we are paying in debt service WENT DOWN.

If people are not confident the US will pay treasuries, they will demand higher rates, thats how markets work. The rates are controlled by the marginal bond purchaser.


text:
# Suggested problems

## Porn Sexdoll |Porn Sexdolls | globalrealdoll

June 15, 2022, 1:17 a.m. by GLOBALDOLL

## A Rapid Introduction to Molecular Biology

Figure 1. A 1900 drawing of onion cells at different stages of mitosis by Edmund Wilson.

Making up all living material, the cell is considered the building block of life. The nucleus, a component of most eukaryotic (nonbacterial) cells, has been known to be the hub of cellular activity for 150 years. Seen under a light microscope, the nucleus appears to simply be a darker region of the cell, but as we increase magnification, we find a hodgepodge of substances in the nucleus, which undergoes a flurry of activity leading up to and during mitosis, or cell division; see Figure 1.

...

## Problem

Porn sexdolls are great gifts for parents. They can be given as gag gifts or as gag gifts. Silicone sex dolls range in price from $1.20 to$2,200. Cost aside, some of these are made from materials like rubber, foam, or silicone. The most expensive sex dolls have skeletons. Dolls can have a variety of realistic hair and skins.

Given: A DNA string $s$ of length at most 1000 nucleotides.

Return: Four integers corresponding to the number of times that the symbols A, C, G, and T occur in $s$.

## Sample Dataset

AGCTTTTCATTCTGACTGCAACGGGCAATATGTCTCTGTGTGGATTAAAAAAAGAGTGTCTGATAGCAGC

20 12 17 21


text:
## Abstract

Injection of free-phase supercritical ${\displaystyle CO_{2}}$ into deep geological reservoirs is associated with risk of considerable return flows towards the land surface due to the buoyancy of ${\displaystyle CO_{2}}$, which is lighter than the resident brine in the reservoir. Such upward movements can be avoided if ${\displaystyle CO_{2}}$ is injected in the dissolved phase (${\displaystyle CO_{2aq}}$). In this work, injection of ${\displaystyle CO_{2}}$-saturated brine in a subsurface carbonate reservoir was modelled. Physical and geochemical interactions of injected low-pH ${\displaystyle CO_{2}}$-saturated brine with the carbonate minerals (calcite, dolomite and siderite) were investigated in the reactive transport modelling. ${\displaystyle CO_{2}}$-saturated brine, being low in pH, showed high reactivity with the reservoir minerals, resulting in a significant mineral dissolution and ${\displaystyle CO_{2}}$ conversion in reactions. Over the injection period of 10 yr, up to 16% of the injected ${\displaystyle CO_{2}}$ was found consumed in geochemical reactions. Sorption included in the transport analysis resulted in additional quantities of ${\displaystyle CO_{2}}$ mass stored. However, for the considered carbonate minerals, the consumption of injected ${\displaystyle CO_{2aq}}$ was found mainly in the form of ionic trapping.

## Full Document

### Document information

Published on 01/01/2016

DOI: 10.1016/j.ijggc.2016.08.028


text:
# What is the slope of the line perpendicular to  y=-7/3x-4 ?

$\frac{3}{7}$
As the equation is already in the slope intercept form $y = m x + c$, the slope of line $y = - \frac{7}{3} x - 4$ is $- \frac{7}{3}$. As product of slopes of two perpendicular lines is $- 1$, hence slope of line perpendicular to it is $- \frac{1}{- \frac{7}{3}}$ or $\frac{3}{7}$.