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### List of publications of Chandan Dasgupta

(a) Research papers published in refereed journals:

1. A. B. Harris, T. C. Lubensky, W. K. Holcomb and C. Dasgupta (1975), Renormalization group approach to percolation problems, Phys. Rev. Lett. 35, 327.
2. C. Dasgupta (1976), Renormalization group calculation of the critical exponents for percolation, Phys. Rev. B 14, 1221.
3. C. Dasgupta (1977), Renormalization-group study of the Ashkin-Teller-Potts model in two dimensions, Phys. Rev. B 15, 3460.
4. C. Dasgupta, A. B. Harris and T. C. Lubensky (1978), Renormalization-group treatment of the random resistor network in $6-\epsilon$ dimensions}, Phys. Rev. B 17}, 1375.
5. T. C. Lubensky, C. Dasgupta and C. M. Chaves (1978), Statistics of trees and branched polymers from a generalized Hilhorst model, J. Phys. A 11, 2219.
6. C. Dasgupta, S. K. Ma and C. K. Hu (1979), Dynamic properties of a spin-glass model at low temperatures, Phys. Rev. B 20, 3837.
7. S. K. Ma, C. Dasgupta and C. K. Hu (1979), The random antiferromagnetic chain, Phys. Rev. Lett. 43, 1434.
8. C. Dasgupta and S. K. Ma (1980), Low-temperature properties of the random Heisenberg antiferromagnetic chain, Phys. Rev.B 22, 1305.
9. S. Chakravarty and C. Dasgupta (1980), Absence of crystalline order in two dimensions, Phys. Rev. B 22, 369.
10. C. Dasgupta and P. Pfeuty (1981), Real-space renormalization-group study of the one-dimensional Hubbard model, J. Phys. C 14, 717.
11. C. Dasgupta and B. I. Halperin (1981), Phase transition in a lattice model of superconductivity, Phys. Rev. Lett. 47, 1556.
12. C. Dasgupta and H. Sompolinsky (1983), Equivalence of statistical-mechanical and dynamic descriptions of the infinite-range Ising spin-glass, Phys. Rev. B (Rapid Communications) 27, 4511.
13. C. Dasgupta (1983), Duality maps for a lattice model of the smectic-A-nematic transition, Phys. Rev. A (Rapid Communications) 27, 1262.
14. C. Dasgupta and K. L. Yao (1984), Macroscopic anisotropy in transition-metal spin-glass alloys, Phys. Rev. B 29, 4071.
15. C. Dasgupta (1985), Monte Carlo study of the nematic-to-smectic-A transition, Phys. Rev. Lett. 55, 1771.
16. C. Dasgupta and R. Pandit (1986), Kinetics of domain growth: The relevance of two-step quenches, Phys. Rev.B 33, 4752.
17. A. Chakrabarti and C. Dasgupta (1985), Phase transition in the Ruderman-Kittel-Kasuya-Yosida model of spin-glass , Phys. Rev. Lett. 56 , 1404.
18. M. H. Lau and C. Dasgupta (1987), Critical behaviour of the $n$-vector model for 1 < n < 2 , Phys. Rev. B 35 , 329.
19. A. Chakrabarti and C. Dasgupta (1987), Anisotropy-induced phase transition in metallic spin-glass alloys , Phys. Rev. B 36 , 793.
20. C. Dasgupta (1987), Critical behaviour of a lattice model of the nematic to smectic-A transition , J. Physique (Paris) 48 , 957.
21. C. Dasgupta and R. Pandit (1987), Testing approximate theories of first-order phase-transitions on the two-dimensional Potts model , J. Stat. Phys. 47, 375.
22. M. H. Lau and C. Dasgupta (1988), Role of topological defects in the phase transition of the three-dimensional Heisenberg model, J. Phys. A 21, L 51.
23. A. Chakrabarti and C. Dasgupta (1988), A Monte Carlo study of realistic models of metallic spin glass, J. Phys. C \ 21, 1613.
24. A. Chakrabarti and C. Dasgupta (1988), Phase transition in positionally disordered Josephson-junction arrays in a transverse magnetic field, Phys. Rev. B 37, 7557.
25. M. H. Lau, C. Dasgupta and O. T. Valls (1988), Domain growth in the field-theoretic version of the Potts model, Phys. Rev. B. 38, 9024.
26. M. H. Lau and C. Dasgupta (1989), Numerical investigation of the role of topological defects in the three-dimensional Heisenberg transition , Phys. Rev. B 39 , 7212.
27. C. Dasgupta (1989), Variational calculation for the spin-1/2 Heisenberg antiferromagnet on a square lattice, Phys. Rev. B 39, 386.
28. V. Deshpande and C. Dasgupta (1991), Heirarchical neural networks for the storage of correlated memories , J. Stat. Phys. 64 , 755.
29. C. Dasgupta and T. V. Ramakrishnan (1991), Flux pinning and creep in high-$T_c$ superconductors, Physica C 183, 62.
30. C. Dasgupta, A. V. Indrani, S. Ramaswamy, and M. K. Phani (1991), Is there a growing correlation length near the glass transition ?, Europhys. Lett. 15, 307.
31. V. Deshpande and C. Dasgupta (1991), A neural network for storing individual memories in limit cycles, J. Phys. A. 24, 5105.
32. S. Sengupta, C. Dasgupta, H. R. Krishnamurthy, G. I. Menon and T. V. Ramakrishnan (1991), Freezing of the flux liquid in high-$T_c$ superconductors: a density functional approach , Phys. Rev. Lett. 67, 3444.
33. C. Dasgupta (1992), Glass transition in the density functional theory of freezing, Europhys. Lett. 20, 131.
34. C. Dasgupta (1992), Neural network modeling of associative memory: beyond the Hopfield model, Physica A 186 , 49.
35. C. Dasgupta and S. Ramaswamy (1992), Search for a thermodynamic basis of the glass transition , Physica A 186 , 314.
36. C. Dasgupta and J. W. Halley (1993), Phase diagram of the two-dimensional disordered Hubbard model in the Hartree-Fock approximation , Phys. Rev. B (Rapid Communications) 47, 1126.
37. M. R. Mehta, C. Dasgupta and G. R. Ullal (1993), A neural network model for kindling of focal epilepsy: basic mechanism, Biological Cybernetics 68, 335.
38. C. C. Huang, I. M. Jiang, A. J. Lin, T. R. Geer and C. Dasgupta (1993), Heat capacity anomalies from four-layer liquid crystal films: experimental results and simulation results, Phys. Rev. E 47, 2938.
39. A. Chakrabarti and C. Dasgupta (1993), Comment on Finite temperature phase transition in metallic spin glass alloys'', Phys. Rev. Lett. 70, 3178.
40. L. M. Lust, O. T. Valls and C. Dasgupta (1993), Nonlinear hydrodynamics of a hard-sphere fluid near the glass transition , Phys. Rev. E 48, 1787.
41. L. M. Lust, O. T. Valls and C. Dasgupta (1994), Langevin simulation of the dynamics of a dense hard-sphere liquid , Phase Transitions 50 , 47.
42. C. Dasgupta (1994), Statisical mechanics of flux lines in oxide superconductors, Physica A 205, 140.
43. G. I. Menon and C. Dasgupta (1994), Effects of pinning disorder on the correlations and freezing of the flux liquid in layered superconductors . Phys. Rev. Lett. 73, 1023.
44. C. Dasgupta and O. T. Valls (1994), Two distinct time scales in the dynamics of a dense hard sphere liquid , Phys. Rev. E 50 , 3916.
45. M. P. Singh, C. X. Zhang and C. Dasgupta (1995), Fixed points of a Hopfield model with random asymmetric interactions , Phys. Rev. E 52 , 5261.
46. M. R. Mehta, C. Dasgupta and G. R. Ullal (1995), Kindling of focal epilipsy occurs due to learning: a neural network model, Int. J. Neur. Sys. 6 (Supp. 1995), 107.
47. C. Dasgupta and O. T. Valls (1996), Time scales for transitions between free energy minima of a dense hard sphere system , Phys. Rev. E 53 , 2603.
48. C. Dasgupta, S. Das Sarma and J. M. Kim (1996), Controlled instability and multiscaling in models of epitaxial growth, Phys. Rev. E 54, R4552.
49. S. S. Ghosh and C. Dasgupta (1996), Domain-wall scaling study of the structural glass transition, Phys. Rev. Lett. 77, 1310.
50. G. I. Menon, C. Dasgupta, H. R. Krishnamurthy, T. V. Ramakrishnan and S. Sengupta (1996), Density-functional theory of flux-lattice melting in high-T$_c$ superconductors, Phys. Rev. B 54, 16192.
51. C. Dasgupta, J. M. Kim, M. Dutta and S. Das Sarma (1997), Instability, intermittency and multiscaling in discrete growth models of kinetic roughening , Phys. Rev. E 55, 2235.
52. G. Athithan and C. Dasgupta (1997), On the problem of spurious patterns in neural associative memory models, IEEE Transactions on Neural Networks 8, 1483.
53. A. Kundagrami, C. Dasgupta, P. Punyindu and S. Das Sarma (1998), Extended self-similarity in kinetic surface roughening, Phys. Rev. E 57, R3703.
54. C. Dasgupta and D. Feinberg (1998), Structure and magnetization of a two dimensional vortex liquid in the presence of strong pinning, Phys. Rev. B 57, 11,730.
55. C. Dasgupta and O. T. Valls (1998), Entropic origin of the growth of relaxation times in simple glassy liquids , Phys. Rev. E. 58, 801.
56. C. Dasgupta and O. T. Valls (1999), Free energy landscape of a dense hard sphere system, Physical Review E 59, 3123.
57. G. I. Menon, C. Dasgupta and T. V. Ramakrishnan (1999), Muon-spin rotation spectra in the mixed phase of high-T$_c$ superconductors: thermal fluctuation and disorder effects, Physical Review B 60, 7607.
58. A. Kundagrami and C. Dasgupta (1999), Multiscaling in discrete models of epitaxial growth, Physica A 270 , 135.
59. Zhang Chengxiang, C. Dasgupta and M. P. Singh (2000), Retrieval properties of a Hopfield model with random asymmetric interactions, Neural Computation 12, 865.
60. A. Dhar, P. Choudhury and C. Dasgupta (2000), The triangular Ising antiferromagnet in a staggered field, Phys. Rev. B 61 , 6227.
61. F. Thalmann, C. Dasgupta and D. Feinberg (2000), Phase diagram of a classical fluid in a quenched random potential, Europhys. Lett. 50, 54.
62. C. Dasgupta and O. T. Valls (2000), Free energy landscape of simple liquids near the glass transition, J. Phys. Cond. Mat. 12, 6553.
63. A. Dhar, B. S. Shastry and C. Dasgupta (2000), Equilibrium and dynamic properties of the ANNNI chain at the multiphase point, Phys. Rev E 62, 1592.
64. C. Dasgupta and O. T. Valls (2000), Phase diagram of a hard-sphere system in a quenched random potential: a numerical study , Phys. Rev. E 62 , 3648.
65. C. Dasgupta and O. T. Valls (2001), Vortex Lattice Melting in Layered Superconductors with Periodic Columnar Pins, Phys. Rev. Lett. 87, 257002; (E) Phys. Rev. Lett. 89, 049901.
66. C. Dasgupta (2002), Freezing of classical fluids in a quenched random potential , Phase transitions 75 , 441.
67. B. Biswal and C. Dasgupta (2002), Neural Network Model for Apparent Deterministic Chaos in Spontaneously Bursting Hippocampal Slices, Phys. Rev. Lett. 88 , 088102.
68. T. Joseph and C. Dasgupta (2002), Continuous melting of a partially pinned two-dimensional vortex lattice in a square array of pinning centers, Phys. Rev. B 66 , 212506.
69. B. Biswal and C. Dasgupta (2002), Stochastic neural network model for spontaneous bursting in hippocampal slices , Phys. Rev. E 66, 051908.
70. D. B. Dougherty, I. Lyubinetsky, E. D. Williams, M. Constantin, C. Dasgupta, and S. Das Sarma (2002), Experimental Persistence Probability for Fluctuating Steps , Phys. Rev. Lett. 89, 136102.
71. C. Dasgupta and O. T. Valls (2002), Vortices in layered superconductors with columnar pins: A density-functional study, Phys. Rev. B 66, 064518.
72. C. Dasgupta (2002), Glassy behavior in neural network models of associative memory, Physica A 315, 137.
73. B. Chakrabarti and C. Dasgupta (2003), Nonequilibrium phase transition in surface growth , Europhys. Lett. 61 547.
74. T. Joseph and C. Dasgupta (2003), Structure and magnetization of two-dimensional vortex arrays in the presence of periodic pinning , Phys. Rev. B 67, 214514.
75. M. Constantin, S. Das Sarma, C. Dasgupta, O. Bondarchuk, D. B. Dougherty, and E. W. Williams (2003), Infinite family of persistence exponents for interface fluctuations, Phys. Rev. Lett. 91, 086103 (cond-mat/0303214).
76. C. Dasgupta and O. T. Valls (2003), Two-step melting of the vortex solid in layered superconductors with random columnar pins, Phys. Rev. Lett. 91 127002 (cond-mat/0308084).
77. M. P. Singh and C. Dasgupta (2003), Mean-field Monte Carlo approach to the dynamics of a one-pattern model of associative memory, submitted for publication (cond-mat/0303061).
78. M. Constantin, S. Das Sarma and C. Dasgupta (2004), Spatial  persistence and survival probabilities for fluctuating interfaces,
Phys. Rev. E 69, 051603  (cond-mat/0312461).
79. C. Dasgupta, M. Constantin, S. Das Sarma and S. N. Majumdar (2004), Survival in equilibrium step fluctuations,
Phys. Rev. E  69, 022102  (cond-mat/0307086).
80. B. Chakrabarti and C. Dasgupta (2004),  Mound formation and coarsening from a nonlinear instability in surface growth
Phys. Rev. E  69, 011601   (cond-mat/0212646).
81. B. Chakrabarti, M. Das, C. Dasgupta, S. Ramaswamy and A. K. Sood (2004),  Spatiotemporal rheochaos in nematic hydrodynamics, Phys. Rev. Lett.  92, 055501 (cond-mat/0311101).
82. M. Constantin, C. Dasgupta, P. Punyindu Chatraphorn, S. N. Majumdar, and S. Das Sarma (2004), Persistence in nonequilibrium surface growth, Phys. Rev. E 69, 061608  (cond-mat/0401438).
83. C. Dasgupta and O.T. Valls (2004), Melting and structure of the vortex solid in highly anisotropic layered superconductors with random columnar pins, Phys. Rev. B 69, 214520 (cond-mat/0401589).
84. O. Bondarchuk, D. B. Dougherty, M. Degawa, E. D. Williams, M.Constantin, C. Dasgupta and S. Das Sarma (2005),  Correlation time for step structural fluctuations, Phys. Rev. B  71, 045426  (cond-mat/0408181).
85. M. Das, B. Chakrabarti, C. Dasgupta, S. Ramaswamy and A. K. Sood (2005),  Routes to spatiotemporal chaos in the rheology of nematogenic fluids, Phys. Rev. E 71, 021707  (cond-mat/0408282).
86. B. Chakrabarti and C. Dasgupta (2005),  Interplay of instabilities in mounded surface growth, Phys. Rev. E  71, 020601 (cond-mat/0408501).
87. D. B. Dougherty, C. Tao, O. Bondarchuk, W. G. Cullen, E. D. Williams, M. Constantin, C. Dasgupta and S. Das Sarma (2005),  Sampling time effects for persistence and survival in step structural fluctuations, Phys. Rev. E  71, 021602. (cond-mat/0410094).
88. C. Dasgupta and O. T. Valls (2005), Phase diagram of  the vortex system in layered superconductors with strong columnar pinning, to be published Phys. Rev. B. (cond-mat/0505167).
89. P. Chaudhuri, S. Karmakar, C. Dasgupta, H. R. Krishnamurthy, and A. K. Sood (2005),  Equilibrium glassy phase in a polydisperse hard sphere system,  submitted to Phys. Rev. Lett. (cond-mat/0503685).

(b) Review papers:

1. J. W. Halley and C. Dasgupta (1983), Percolation and related systems in equilibrium statistical mechanics , in The Mathematics and Physics of Disordered Media , edited by B. D. Hughes and B. W. Ninham (Springer-Verlag, Berlin).
2. C. Dasgupta (1991), Neural networks: a review , J. Indian Inst. Sci. 71 , 491.
3. C. Dasgupta (1992), Renormalization group studies of the statics and dynamics of liquid crystal phase transitions , in Phase Transitions in Liquid Crystals , edited by S. Martellucci (Plenum Press, New York).
4. O. T. Valls and C. Dasgupta (1995), Numerical studies of Langevin equations for the dynamics of a dense hard sphere fluid, Transport Theory and Statistical Physics 24, 1199.
5. C. Dasgupta (1995), Mathematical modeling of neurons and neural networks , Current Science , 68 , 162.
6. C. Dasgupta (1995), Numerical studies of phase transitions and critical phenomena in liquid crystals, Int. J. Mod. Phys. B 9 , 2219. Reprinted in the book Liquid Crystals in the Nineties and Beyond, edited by S. Kumar (World Scientific, 1995).
7. C. Dasgupta and G. I. Menon (2000), Equilibrium properties of the mixed phase of high-T$_c$ superconductors in the presence of pinning , in Studies of High Temperature Superconductors , Vol. 31, edited by A. V. Narlikar (Nova Science Publishers, New York).
8. C. Dasgupta and S. Ramaswamy (2001), Smectic A liquid crystals: Continuum theory, in Encyclopedia of Materials: Science and Technology (Elsevier, Amsterdam).
9. C. Dasgupta (2002), Nonequilibrium kinetics of surface growth , to be published in a special issue of Indian Journal of Physics.
(c) Papers published in conference proceedings:

1. C. Dasgupta (1987), Relaxation in spin glasses at low temperatures , in Non-Debye Relaxation in Condensed Matter}, edited by T. V. Ramakrishnan and M. Raj Lakshmi (World Scientific, Singapore .
2. A. Chakrabarti and C. Dasgupta (1987), Phase transition in metallic spin-glass, in Condensed Matter Theory, Vol.II, edited by P. Vashishta, R. K. Kalia and R. F. Bishop (Plenum Press, New York).
3. C. Dasgupta and T. V. Ramakrishnan (1990), Intrinsic flux pinning and flux creep in oxide superconductors, in Proceedings of the International Conference on Superconductivity - ICSC , edited by S. K. Joshi, C. N. R. Rao and S. V. Subramanyam, (World Scientific, Singapore).
4. C. Dasgupta and V. Deshpande (1991), Neural network models of associative memory, in Proceedings (Invited Papers) of the DAE Solid State Physics Symposium, 1990, Volume 33 A, 228.
5. C. Dasgupta (1992), Statistical mechanics of flux lines in high-temperature superconductors, in Superconductivity: New Insights and Current Trends in High-$T_c$ Superconductors, edited by S. Gregoli, B. Raveau and C.N.R. Rao (Office of the Official Publications of the European Community, Luxembourg).
6. J. W. Halley, C. Dasgupta, S. Davis and X. F. Wang (1992), Point-defect disorder in models of high-temperature superconductivity , in Recent Progress in Many Body Theories, Vol. 3 , edited by T. Ainsworth et al (Plenum Press, New York).
7. M. R. Mehta, C. Dasgupta and G. R. Ullal (1996), A neural network model for kindling of focal epilepsy , in Neural Modeling of Brain and Cognitive Disorders , edited by J. A. Reggia, E. Rupin and R. S. Berndt (World Scientific, Singapore).
8. C. Dasgupta and O. T. Valls (1997), Glass transition in the hard sphere system , in Complex Behaviour of Glassy Systems (Proceedings of the XIV Sitges Conference), edited by M. Rubi and C. Perez-Vicente (Springer-Verlag, Berlin).
9. B. Biswal, C. Dasgupta and G. R. Ullal (1999), Controlling chaos'' in a stochastic neural network model for epileptic brain activity, in Nonlinear Dynamics and Brain Functions, edited by Paul E. Rapp, N. Pradhan and R. Sreenivasan (Nova Science Publishers, New york).
10. C. Dasgupta, Zhang Chengxiang, M. P. Singh and G. Athithan (2000), Role of synaptic asymmetry in neural network models of associative memory, in Advances in Pattern Recognition and Digital Techniques, edited by Nikhil R. Pal, Arun K. De and J. Das (Narosa Publishing House, New Delhi, 2000).


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# What is the general definition of exponentiation? [duplicate]

The definition of exponentiation is commonly defined as:

$$x^n:=\underbrace{x\cdot x\cdot x\cdot ...\cdot x}_\text{n times}$$

when $$n \in \mathbb N$$.

Given that, what is the definition when $$n \notin \mathbb N$$ ? for example when $$n$$ is a real number, what about complex numbers, irrational numbers, matrices? does the definition extends or is it a completely different one, if so, what is it for every case? I also would like to know who defined exponentiation for these cases and in which paper.

I read for a non-reliable source that the actual definition of a number exponentiated itself is $$x^x := \exp(x \ln(x))$$ is this an extension of the real exponent definition, or am I missing something and this is different than an exponent?

In 702414 Gyu Eun Lee explains what exponentiation really means, I am not looking for that, I am looking for the strict definition.

• There are many possible definitions. There is no single "right" one. A very common very general definition is to use the power series definition, $e^x = 1 + x + x^2/2! + x^3/3! + \cdots$. This also works for complex numbers, matrices, and elsewhere. A slightly different question is to show that the various definitions are all equivalent, but this is well-trodden on this site. Jun 21, 2021 at 18:00
• The general way to extend exponentiation to other fields (although I'm not sure whether it relates to the actual definitions) is to say $a^b = \exp(b \ln(a)),$ where you can define the natural exponential and logarithm by their limit definitions or power series Jun 21, 2021 at 18:00
• Welcome to MSE! Exponentiation $a^b$ is defined to be $e^{b \ln(a)}$. Then, depending on which book you read, there are a handful of definitions for what $e^x$ and $\ln(x)$ should be (though the power series definitions are quite good, since you can plug any old complex number, square matrix, etc into the power series and get something that might be meaningful). You can read more here Jun 21, 2021 at 18:00
• Note that different definitions work for different domains (in both base and exponent) - but the different definitions fortunately tend to agree when several are applicable . For example $a^b=\exp(b\ln a))$ fails when $a=0$ which even makes many believe that $0^0$ is undefined when it clearly is $=1$ (because the number of maps from the empty set to the empty set is one) Jun 21, 2021 at 18:03
• I also would like to know who defined exponentiation for these cases and in which paper. --- FYI, elementary things like this from well before the last hundred years or so often have such complicated histories (involving parallel and independent paths, rediscoveries, different interpretations from how we presently think about things, issues over whether to count limited readership locally published books/papers or private letters or lectures given but not formally recorded or . . .) that the question is not "well posed", let alone having a known answer. Jun 21, 2021 at 18:04

• $$x^x = \exp(x\cdot\ln(x))$$ is not the extension of exponentiation,
it's the consequence of this : $$f(f^{-1}(x)) = x$$ Where the input is $$x^x$$ instead of just $$x$$, and $$f(x) = \exp(x) = e^x, f^{-1}(x) = \ln(x)$$.
• If you want to compute the value of an exponential function $$a^{\textstyle x}$$ at some $$x = x_0$$ where $$x_0 \notin \mathbb{Z}$$ you would have to compute it manually via what's called "Taylor Series" or "Maclaurin Series" (which is Taylor series centered at x = 0), search for this series on Google on your own.
• For some $$x < 0, x\in \mathbb{Z}$$, $$a^{x} = \dfrac{1}{a^{x}}$$ by consequence of the basic definition of exponention $$\displaystyle a^x = \prod_{k=1}^x a$$
• For $$x = 0$$, the definition of $$a^x$$ is that $$a^x = 1$$ for any real number $$a$$ (including $$a = 0$$)

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# Thread: maclaurin series

1. ## maclaurin series

How would you find the maclaurin series for this
sin(x^5)
thanks.

2. Originally Posted by davecs77
How would you find the maclaurin series for this
sin(x^5)
thanks.
recall that the Maclaurin series for sine is:

$\sin x = \sum_{n = 0}^{\infty} \frac {(-1)^n}{(2n + 1)!} x^{2n + 1} = x - \frac {x^3}{3!} + \frac {x^5}{5!} - \frac {x^7}{7!} +...$ for ALL $x$

so for $\sin \left( x^5 \right)$, just replace $x$ with $x^5$ in the series above and simplify

3. Originally Posted by davecs77
How would you find the maclaurin series for this
sin(x^5)
thanks.
A McLaurin series is a Taylor series for x = 0. So we have:
$f(x) \approx f(0) + \frac{1}{1!}f^{\prime}(0)x +
\frac{1}{2!}f^{\prime \prime}(0)x^2 + ~ ...$

$f(x) = sin(x^5)$

So
$f(0) = 0$

$f^{\prime}(x) = cos(x^5) \cdot 4x^5$ ==> $f^{\prime}(0) = 0$

$f^{\prime \prime}(x) = 20x^3 cos(x^5) - 25x^8sin(x^5)$ ==> $f^{\prime \prime}(0) = 0$

$f^{\prime \prime \prime}(x) = -125x^{12}cos(x^5) + 60x^2 cos(x^5) - 300x^7sin(x^5)$ ==> $f^{\prime \prime \prime}(0) = 0$

etc.

I'll give you a hint. After a lot of hard work on your part, you will find that $f^{(6)}(0)$ is the first non-zero term in the series.

(This one's just nasty to do by hand!)

-Dan

4. Originally Posted by Jhevon
recall that the Maclaurin series for sine is:

$\sin x = \sum_{n = 0}^{\infty} \frac {(-1)^n}{(2n + 1)!} x^{2n + 1} = x - \frac {x^3}{3!} + \frac {x^5}{5!} - \frac {x^7}{7!} +...$ for ALL $x$

so for $\sin \left( x^5 \right)$, just replace $x$ with $x^5$ in the series above and simplify
So you are just memorizing that for sin(x) or Maclaurn series?
Would that be the same for Taylor series?

5. Originally Posted by davecs77
So you are just memorizing that for sin(x) or Maclaurn series?
Would that be the same for Taylor series?
I don't like telling people to just memorize stuff, but yes, you should have the Maclaurin series for functions like sine, cosine, e^x, 1/(1 - x) ... memorized.

the Maclaurin series is just the Taylor series centered at zero. topsquark showed you how to derive it, but in this case, i think the memorization route is easiest. unless your professor requires you to derive it

6. Originally Posted by Jhevon
recall that the Maclaurin series for sine is:

$\sin x = \sum_{n = 0}^{\infty} \frac {(-1)^n}{(2n + 1)!} x^{2n + 1} = x - \frac {x^3}{3!} + \frac {x^5}{5!} - \frac {x^7}{7!} +...$ for ALL $x$

so for $\sin \left( x^5 \right)$, just replace $x$ with $x^5$ in the series above and simplify
Or you can take Jhevon's pansy-butted approach and skip all the work. But why do that when you could do all that work?

-Dan

7. how would you find the Maclaurin series for this then:
1/(1+x^5)

8. Originally Posted by topsquark
But why do that when you could do all that work?

-Dan
because mathematicians don't like hard work.

there are some things you want to try once just to see if you can do them, and then afterwards you take the shortcut. finding Taylor/Maclaurin series expansions are such things.

"pansy-butted"?

9. Originally Posted by Jhevon
"pansy-butted"?
It was the best I could come up with on short notice. Hey, I'm a Mormon. I'm not supposed to swear so that limits my insulting remarks.

-Dan

10. Originally Posted by davecs77
how would you find the Maclaurin series for this then:
1/(1+x^5)

Does anyone know? I don't think you can memorize this one. Thanks.

11. Originally Posted by davecs77
how would you find the Maclaurin series for this then:
1/(1+x^5)
$\frac{1}{1+x}=1-x+x^2-x^3+...$

You do the rest.

Note: You need to remember the most important power series as well as you name.

12. Originally Posted by davecs77
Does anyone know? I don't think you can memorize this one. Thanks.
actually you can. this happens to be one of the easiest to memorize in fact. the appropriate one to remember is

$\frac {1}{1 - x} = \sum_{n = 0}^{\infty} x^n = 1 + x + x^2 + x^3 + ...$ for $|x|<1$

13. Now me, I tend to forget that $\sum_{i = 1}^n 1 = n$. (I try not to memorize too much! )

-Dan

14. Originally Posted by ThePerfectHacker
$\frac{1}{1+x}=1-x+x^2-x^3+...$

You do the rest.

Note: You need to remember the most important power series as well as you name.
The answer in the book is summation from n = 1 to infinity of (-1)^n times x^(5n)

I see the (-1)^n since the values are changing from positive to negative every time, but where does the x^(5n) come from? Also how did you find your results... 1 - x + x^2 - x^3 ... I don't quite understand. Thank you.

15. Originally Posted by davecs77
The answer in the book is summation from n = 1 to infinity of (-1)^n times x^(5n)

I see the (-1)^n since the values are changing from positive to negative every time, but where does the x^(5n) come from? Also how did you find your results... 1 - x + x^2 - x^3 ... I don't quite understand. Thank you.
he treated $\frac {1}{1 + x}$ as $\frac {1}{1 - (-x)}$ and then used the formula i gave you.

where did the $x^{5n}$ come from? did you actually look at the series? $\sum_{n = 0}^{\infty}(-1)^n\left( x^5 \right)^n = \sum_{n = 0}^{\infty}(-1)^nx^{5n}$


text:
## Abstract

As a consequence of the reversal of the gender gap in education, the female partner in a couple now typically has as much as or more education compared with the male partner in most Western countries. This study addresses the implications for the earnings of women relative to their male partners in 16 European countries. Using the 2007 and 2011 rounds of the European Union Statistics on Income and Living Conditions (N = 58,292), we investigate the extent to which international differences in women’s relative earnings can be explained by educational pairings and their interaction with the motherhood penalty on women’s earnings, by international differences in male unemployment, or by cultural gender norms. We find that the newly emerged pattern of hypogamy is associated with higher relative earnings for women in all countries and that the motherhood penalty on relative earnings is considerably lower in hypogamous couples, but neither of these findings can explain away international country differences. Similarly, male unemployment is associated with higher relative earnings for women but cannot explain away the country differences. Against expectations, we find that the hypogamy bonus on women’s relative earnings, if anything, tends to be stronger rather than weaker in countries that exhibit more conservative gender norms.

## Introduction

Universities largely remained a male domain until well into the second half of the twentieth century. Male enrollment and completion rates in advanced education were higher than female rates virtually everywhere. Since the 1990s, however, more women than men are enrolled in college-level education, and women also graduate more successfully (Buchmann and DiPrete 2006; Esteve et al. 2016).

The reversal of the gender gap in education implies that for the first time in history, more highly educated women than highly educated men reach the ages of partnering and parenthood. Recent studies have shown that the gender-specific trends in educational enrollment have indeed undermined the traditional pattern of educational hypergamy (women marrying up). Hypogamy (women marrying down) has become more prevalent than hypergamy in most countries where the reversal of the gender gap in education has occurred (De Hauw et al. 2017; Esteve et al. 2012, 2016; Grow and Van Bavel 2015).

Here we investigate how the new pattern of educational assortative mating shapes the relative contribution of partnered women to joint couple income across European countries. Drawing on the European Union Statistics on Income and Living Conditions (EU-SILC), we attempt to explain country differences in women’s relative earnings, compared with their male partners, by accounting for micro-level compositional differences in terms of educational pairings and parenthood as well as the interaction between these two. After accounting for these micro-level factors, we investigate the extent to which the remaining country differences can be explained by male unemployment and country-level gender norms.

The interplay between the relative education and relative earnings of female and male partners is an important topic for family demography because of its potential implications for fertility and divorce (Esteve et al. 2016; Schwartz and Han 2014; Van Bavel 2012). A switch from hypergamy to hypogamy is expected to affect women’s contribution to the joint couple income. If the female partner has more education than her male partner, she will tend to have a higher earning potential in the paid labor market (Wang et al. 2013; Winkler 1998; Winkler et al. 2005). Theoretically, this shift should affect the decision-making processes concerning both partners’ labor market participation and thus the timing and quantum of fertility (Van Bavel 2012).

Studies have consistently reported that mothers tend to scale down their paid labor market activity and earn less income after childbirth (Budig and England 2001; Budig et al. 2012). Yet, a recent study by Klesment and Van Bavel (2017) showed that this motherhood penalty may be modified by the relative education of the partners. The authors found that college-educated mothers of school-aged children with less-educated partners are about as likely to be the main breadwinners as college-educated childless women who are with a similarly educated partner. The Klesment and Van Bavel (2017) study had a number of limitations, however. First, it focused on a rather crude measure of relative earnings: namely, whether the wife is earning more than one-half of the joint couple income. We offer a more nuanced approach here, addressing the full continuum of the female partner’s relative income rather than grouping it into broad categories. Second, the earlier study is based on gross income measures, which may yield misleading results because couples in Europe may be taxed differently from country to country depending on the relative earnings of each partner. The current article therefore uses net income data, after taxation (which varies widely across Europe). Third, the earlier study did not include unemployment and sick leave benefits. Because such benefits are linked to employment in European welfare states, leaving them out may yield a distorted picture of the importance of his and her employment career for the family budget. We therefore include these benefits here. Finally, the earlier study looked only at the overall picture in the pooled sample of all countries. The current study aims to explore country differences. The primary focus here is the extent to which international differences in women’s relative earnings can be explained by compositional differences in women’s relative education.

## Background

In the male breadwinnerfemale homemaker family model he earns (almost) all the income, and she allocates most of her time to childcare and household work. In contrast, the dual-earner model refers to families in which both partners earn a significant if not equal share of the family income (Nock 2001; Raley et al. 2006). In this section, we formulate a set of hypotheses about how the female contribution to the couple’s joint earnings depends on educational assortative mating and its interaction with the motherhood status, and about macro-level factors that may account for cross-country variation in women’s relative earnings.

### Educational Assortative Mating

Differences between the earnings of spouses are partly driven by their relative human capital. The traditional male breadwinner model was linked with educational hypergamy. Yet, with the expansion of female participation in college-level education, educational homogamy became the modal marriage pattern in most Western countries during the twentieth century (Blossfeld 2009; Kalmijn 1991; Mare 1991; Schwartz 2013; Schwartz and Mare 2005). Correspondingly, dual-income households became the norm, but with the share of the male partner in the joint income typically largely exceeding the share of the female partner (Blossfeld 2009; Blossfeld and Timm 2003; Buss et al. 2001; Raley et al. 2006; Schwartz 2013; Winkler et al. 2005).

The reversal of the gender gap in education implies that more highly educated women than men enter the marriage market. Some of the highly educated women will then either remain single or select a partner with less education. Recent evidence indeed showed that hypogamy has become more prevalent than hypergamy in many countries, including the United States and most European countries (De Hauw et al. 2017; Esteve et al. 2012; Grow and Van Bavel 2015). Our baseline hypothesis is that a women’s relative education is positively associated with her relative earnings in the couple (Hypothesis 1). That is, women who are in hypogamous partnerships will have higher relative earnings than those who are in other pairing types. If so, compositional differences in educational pairings might at least partially explain country differences in women’s relative earnings.

### Motherhood and Relative Income

Women with children earn less relative to both their male partners and women without children. The presence of dependent children tends to lead to lower involvement in the paid labor market and, hence, to lower relative earnings (Budig and England 2001; Budig et al. 2012; Dotti Sani 2015; Gangl and Ziefle 2009). Several explanations for this motherhood penalty have been advanced, including selection effects as well as causal mechanisms (Anderson et al. 2002; Budig and England 2001; Petersen et al. 2010; Waldfogel 1998). Selection effects involve cases in which women who are more family-oriented opt out of better-paying but competitive and time-consuming careers in order to spend more time with their children (Chevalier 2007; Hakim 2003; Lück and Hofäcker 2008). However, the lower earnings of mothers may also be caused directly by the fact that most women at least temporarily retreat from paid work following childbirth (Budig et al. 2012; OECD 2011; Stier et al. 2001). Thus, motherhood implies an opportunity cost via reduced income simultaneous with the household need for increased income to account for the significant costs that children entail.

The extent of retreat from the labor market after entry into parenthood is expected to depend on his and her income potential, which is determined partly by the partners’ education. Both the level and the field of education affect income potential (Van Bavel 2010), but because our data lack information on field of study, we consider only educational attainment. For both men and women with lower levels of attainment, the gains from labor market activity and the opportunity costs of staying at home are relatively low. Although the need for additional income might produce the incentive to work more hours in paid labor, the opportunity costs of staying at home to provide childcare are low. For those with higher levels of education, the gains from paid labor and the opportunity costs of staying at home are higher. Parents with greater earning potential would need less work hours to achieve a given income level, but they will also lose more financially by staying at home (Becker 1993; Mincer 1963). Research in wealthy countries has typically found the opportunity costs predominating for women and the income effect for men (England 2010).

Women more often stay at home when they have children, given that the costs of outsourcing childcare undermine the incentives for paid labor, particularly among women with a low level of education. But when a woman has more education than her partner, the gender balance of income effects and opportunity costs may shift because she is then more likely to have a higher earning potential than her partner. In this case, her attachment to the labor market will be stronger because the income she generates is relatively more important to the family’s standard of living. In micro-economic terms, if she has higher potential income than he, the opportunity costs of staying at home to take care of children will tend to be relatively higher for her than for him. Among childless couples, the influence of opportunity costs is less critical given the lack of imperative to divide partners’ time between paid work and childcare. We therefore expect a smaller motherhood penalty on relative earnings in case of educational hypogamy and a larger motherhood penalty in case of hypergamy compared with homogamy (Hypothesis 2). If that is the case, compositional differences between countries in terms of the combined distribution of educational pairings and motherhood may explain at least part of the country gradient.

### Male Unemployment

Both economic and cultural factors may affect women’s relative earnings. Male unemployment is a major economic factor because the share of the female partner in the joint couple income will obviously be higher when her male partner is unemployed (Bettio et al. 2013; Chesley 2011; Vitali and Arpino 2016). Therefore, the economic crisis in the wake of the 2008 financial crisis may have affected women’s relative earnings because of high male unemployment rates independent of their relative education or any prevailing motherhood penalty. The hardest-hit economic sectors tended to employ more men than women (such as heavy industry), while sectors that were less affected tended to employ more women (such as education). Recent evidence indeed suggests that women’s relative income share increased in the wake of the 2008 financial crisis (Klesment and Van Bavel 2015; Vitali and Mendola 2014). In Europe, Bettio et al. (2013) found that the proportion of dual-earner couples declined during the downturn, while the share of female breadwinner couples increased to almost 10 %. Similarly, in the United States, Chesley (2011) saw the decline of men’s employment as a major factor explaining the rise of breadwinner women. Vitali and Arpino (2016) found that regional male unemployment rates were positively associated with women’s self-reported share in the household income in Europe. Based on these prior studies, we expect that part of the country gradient in women’s relative income is due to male unemployment. Hypothesis 3 therefore states that the country gradient in women’s relative earnings will be reduced after we account for male unemployment.

### Gender Norms

International differences in women’s relative earnings may also be due to cultural factors related to gender roles. Female labor market participation and earnings of mothers vary substantially across countries in accordance with prevailing gender role expectations and cultural family ideals (Budig et al. 2012; Craig and Mullan 2010; Harkness and Waldfogel 2003; Stier et al. 2001). The effect of gender role attitudes on relative earnings may be indirect as well as direct. In some countries, widespread reluctance to enroll young children in formal childcare may prolong career interruptions after childbearing (Davies and Pierre 2005; Lundberg and Rose 2000; Molina and Montuenga 2009; Sigle-Rushton and Waldfogel 2007), especially lowering the earnings of mothers. Beliefs, values, and attitudes about gender roles may also sustain gender inequality more directly (Lewis 1992; Pascall and Lewis 2004). Institutional arrangements hindering gender equality in the labor market (Lewis 1992; Neyer and Andersson 2008; Thévenon 2011) typically reflect cultural beliefs and attitudes that may underpin the male breadwinner–female homemaker family model (Creighton 1999; Esping-Andersen 2009; Janssens 1997; Lewis 2001; Pfau-Effinger 1998). Gender roles embedded in this family model are explicitly at odds with women earning more than their partners, especially if they have children. To the extent that such traditional views prevail, women may modify their labor supply in order to avoid a gender role reversal in earnings. Hence, traditional gender attitudes may undermine the hypogamy bonus on relative earnings by preventing women to harvest the returns to education in the labor market. We therefore expect the hypogamy bonus on relative earnings to be smaller in countries that have more conservative gender role attitudes (Hypothesis 4).

## Data and Methods

### Sample Selection

We use the EU-SILC, an annual household survey collecting both cross-sectional and longitudinal data. In most countries, the survey applies a rotating panel design with a length of four waves. Each subsequent wave replaces part of the sample, and the entire sample is renewed across a four-year period (Atkinson and Marlier 2010). We analyze cross-sectional data from the 2007 and 2011 waves, ensuring that samples do not overlap. The income reference period is the year before the year of interview, so our income data cover earnings in 2006 and 2010, respectively. Our focus on woman’s contribution to the joint net earnings of the couple requires limiting the study sample to the 16 countries that provide this measure (see Table 1).

We select women who are living with a partner at the time of the survey, either married or unmarried, and who are aged 25 to 45. To calculate the contribution of the female partner to the couple’s combined income, only couples in which at least one partner had earned income are included. We include income earned through both employment and self-employment. Our measure incorporates unemployment benefits and sick leave benefits, which compensate for the temporary absence from the labor market (Dotti Sani 2015). Excluding individuals with missing information, couples without any earnings, and observations with extreme values of the relative earnings measure (as explained in the next section) yields a study sample of 58,292 couples (30,702 in 2007 and 27,590 in 2011, respectively). Table 1 displays sample sizes per country included, along with basic descriptive statistics of the main variables featuring in the analysis.

### Measures

Our main dependent variable is the women’s relative income, defined as follows:
$wicf=yicf−yicmycm.$
1

In this definition, $yicf$ represents the earnings of the female partner of couple i in country c, $yicm$ is the earnings of the male partner in the same couple and country, and the denominator $ycm$ represents the average earnings of men in that country. The relative income is standardized by the country’s average male earnings to prevent scores from reflecting country differences in income level rather than gender difference because the potential raw differences between his and her income are greater in high-earning countries than in low-income countries. Women’s relative earnings, as defined in Eq. (1), therefore represent her relative earnings proportional to the average male earnings level in a given country. If he and she earn exactly the same amount, this variable equals 0. Positive values signify that she makes more money than he, and negative values signify that she earns less than he. For example, a value of 0.10 means that she outearns him by 10 % of the national male earnings; a value of –0.50 means that she earns 50 % of the national male average earnings less than her partner. We exclude 118 observations with very extreme values in which her relative earnings were less than –5.00 (i.e., earnings difference between the spouses in favor of the man is five times the average male earnings); we exclude 24 more cases in which relative earnings are above +4.00 (i.e., earnings difference in favor of the woman is four times the average male earnings). Figure S1 in Online Resource 1 plots the country-specific distributions of the dependent variable.

Our baseline Hypothesis 1 states that educational hypogamy is associated with higher relative earnings for women. To measure the educational attainment of the male and female partners, EU-SILC used a measure to facilitate international comparison, the ISCED-97 scale (UNESCO 2003). We collapse this scale into three categories: low (ISCED levels 0–2, up to the second stage of basic education, equivalent to 7th to 9th grades in the United States); medium (ISCED 3–4, secondary education or postsecondary but not tertiary; up to 12th grades, vocational education, or junior and community colleges); and higher education (ISCED 5–6, university level bachelors, masters, and PhD). Using these broad categories implies a loss of information and obviously amplifies the level of educational homogamy, but it ensures that the categories represent substantive differences in educational attainment in Europe.

Hypothesis 2 holds that the motherhood penalty on relative earnings will be smaller in case of educational hypogamy. To measure motherhood status, we discern whether any children are living in the couple’s household. EU-SILC does not collect direct fertility information on the number of children ever born. Hence, what we will capture in this article is really the effect of having a child living at home rather than the effect of parenthood per se.

To address Hypothesis 3 concerning male unemployment, we use individual-level data from the EU-SILC on the number of months men spent unemployed during the income reference period. We enter the months of unemployment as a continuous variable in regression models. Average values of the unemployment variable are shown in Table 1.

The EU-SILC does not contain data that can be used to test Hypothesis 4, which relates to attitudes about gender roles. To obtain relevant country-level indicators, we incorporate data from the European Social Survey (ESS), which samples people aged 15 and older for almost all European countries (Jowell et al. 2007). Two questions about conservative male breadwinner–female homemaker attitudes were asked in Rounds 2 and 5 (fieldwork around 2004 and 2010, respectively). Respondents were asked to indicate on a five-point scale whether they strongly agreed, agreed, neither agreed nor disagreed, disagreed, or strongly disagreed with the following statements: “Women should be prepared to cut down on paid work for sake of family,” and “Men should have more right to job than women when jobs are scarce.” We calculated the percentages agreeing or strongly agreeing with these items and took the average of the two ESS rounds. Bulgaria and Lithuania participated in Round 5 but not in Round 2, whereas Italy and Luxembourg participated in Round 2 but not Round 5; thus, we can use data for only one survey year for these four countries. Table 1 gives country-specific means of both ESS variables.

As control variables, we include the woman’s age and age squared in years, a dummy variable for the year of income reference period (2006 vs. 2010), and the absolute level of the couple’s joint earnings (because women in poorer families tend to be more often the main earners; see Winslow-Bowe 2006) in country-specific quartiles.

### Analytic Strategy

To test our hypotheses, we fit linear models with country fixed effects and with standard errors adjusted to account for the clustering at the country level. We also fit binary logistic versions of our fixed-effects models with the dependent variable redefined as the probability that the female earns more than her male partner. Details about the binary approach are reported in Online Resource 1, but the picture that emerges from it is the same as that reported here.

We first test whether the woman’s relative earnings are associated with being in a hypergamous, hypogamous, or homogamous union, controlling for her own level of educational attainment (Hypothesis 1). Next, we test whether the motherhood penalty on relative earnings is smaller in hypogamous unions than in other union types (Hypothesis 2) by including an interaction term between motherhood and the educational pairing variable. For Hypothesis 3, we test whether the variance of the country fixed effects becomes smaller after we add a measure of male unemployment to the model. Finally, we evaluate the correlation between (1) the country-level indicators for conservative gender role attitudes and (2) the country-specific effects of hypogamy on women’s relative earnings (Hypothesis 4). Given the limited number of countries that can be included from the EU-SILC data, fitting a multilevel model with both individual- and country-level variables is not feasible. Instead, we interact country effects with the effect of educational pairings to obtain country-specific estimates of the effect of educational hypogamy and hypergamy compared with homogamy.

## Results

Table 1 shows basic descriptive statistics about the distribution of couples by type of educational pairing in each country. Equally educated couples are the majority of the sample in every country. Among heterogamous unions, hypogamy is more common than hypergamy in all countries except Austria and Romania. The observation that hypogamy has become more prevalent than hypergamy confirms findings from other recent studies (De Hauw et al. 2017; Esteve et al. 2012; Grow and Van Bavel 2015).

Figure 1 shows women’s relative earnings by educational pairing and country. Overall, across pairing types, women’s relative earnings are lowest in Austria, Italy, Greece, and Estonia (where women’s earnings are lower than their partners’ by approximately 50 % of the national male average) and highest in Portugal, Sweden, and Slovenia (where the earnings gaps are less than 35 %). In all countries, women in hypogamous couples have higher relative earnings than women in homogamous or hypergamous couples, although the differences between pairing types are very small in Sweden. Figure 2 shows that in all countries, mothers consistently have lower relative earnings than childless women.

Table 2 reports the estimates from the regression models. The baseline Model 1 contains only control variables: her age (centered on age 35) and age squared, her educational level, the joint couple earnings, the income reference year, and country fixed effects. Unsurprisingly, higher educational attainment is associated with women’s higher relative earnings. The couple’s combined earnings are negatively correlated with relative income: the higher the joint earnings, the lower the woman’s share tends to be. As expected from the economic climate of the period, women’s relative earnings were somewhat higher in 2010 than in 2006.

The country fixed effects for Model 1, plotted in Fig. 3, are very much in line with the country differences described with the black solid points in Figs. 1 and 2. The major difference is that after controls, Belgium is found in the middle of the country distribution, whereas it was near the top for women’s relative earnings in the unadjusted distribution. This shift reflects the fact that the female population of Belgium is very highly educated. After we control for education, we find that Belgium lands in the middle of the distribution.

In Model 2, we focus on the key substantive variables of the study: educational pairing and motherhood status. Mothers have lower relative earnings than childless women. On average across countries, they earn approximately 23 % less than their male partners (–0.232, standardized to the country-specific male average). The parameters for the educational pairings are consistent with Hypothesis 1: net of the control variables and motherhood status, women in hypogamous unions have almost 14 % higher relative earnings than those in homogamous ones and approximately 28 % higher than the women in hypergamous couples.

The results shown in Model 3 for the interaction between motherhood and educational pairings are consistent with Hypothesis 2. The motherhood effect on relative earnings is significantly smaller for women in hypogamous unions than for their homogamously paired peers; the difference is approximately 7 percentage points (as implied by parameter estimate 0.074 in Model 3) and is statistically significant at the p < .01 level. The same interaction can also be considered the other way around: the hypogamy bonus on relative earnings is approximately 7 percentage points larger for mothers than for childless women. Adding the effects of educational pairings and the interactions with motherhood, the differences in predicted values are quite substantial. For mothers in hypogamous unions, the predicted relative earnings are approximately –10 % (–0.254 + 0.080 + 0.740 = –0.100; all other variables at their reference levels), compared with approximately –25 % for mothers in homogamous unions and –39 % for mothers in hypergamous unions (–0.254 – 1.154 + 0.015 = –0.393).

We have no clear evidence in support of the second part of Hypothesis 2, regarding the difference between hypergamous and homogamous women. The difference in motherhood effect between these two groups is not statistically significant. In additional tests not reported here, we also distinguished between couples with children younger than 5 and those with children older than 5, but this hardly made a difference for the estimates and does not affect our conclusions. The negative motherhood effect is stronger when the youngest child is younger than 5, but the interaction with hypogamy stays the same.

So far, the findings suggest that women’s relative earnings are a function of their absolute and relative education, motherhood status, and the interaction between the latter two variables. International differences in relative earnings may therefore be due to the fact that these variables are differentially distributed in different countries. The country effects plotted in Fig. 3 address this issue. These reflect the residual country differences in women’s relative earnings after accounting for the other variables in the model. The country effects from Model 3 are considerably closer to the zero line than the ones from baseline Model 1, confirming that motherhood and (relative) education are important reasons why women’s relative earnings are generally lower than men’s in all countries. However, accounting for these factors does not reduce the country gradient. The variability of unexplained country differences is not smaller for Model 3 compared with Model 1; the standard deviation of the country fixed effects in Model 3 is 0.0843, which is actually a bit larger than the one for Model 1 (0.0831).

Model 4 includes male unemployment with the expectation that this factor would account for some of the remaining cross-country differences (Hypothesis 3). Male unemployment is clearly associated with higher relative earnings on the micro level: a month of unemployment for the male partner is associated with an increase of 5 % of the female partner’s yearly relative earnings. However, this factor cannot explain the country differences given that the standard deviation for the residual country effects in Model 4 is 0.0854, which is not smaller than the one for the country effects in Model 3 (0.0843). Interestingly, all the residual country effects are more strongly negative in Model 4 than in Model 3, indicating that the share of women’s earnings in the joint couple income is elevated by male unemployment in all countries and that women’s expected relative earnings are lower again after this factor is accounted for. We illustrate this point by comparing Spain and Sweden. These two countries are similar in the baseline rank ordering of women’s relative income (Model 1). However, after we account for male unemployment, the expected relative earnings of women are much lower in Spain than in Sweden. The rank order based on Model 4 places Spain in a much lower position—somewhere between Italy and Poland—because the male unemployment rate in Spain is three times as high as in Sweden (see Table 1), which results in the comparably high relative earnings in Spain in the baseline ranking. In sum, male unemployment explains an important part of the gender earnings gap in all countries, and it explains part of the relative position between specific countries, but it cannot explain the overall country gradient in women’s relative earnings.

Figure 4 plots our two country-level measures of conservative gender role attitudes against country-specific estimates of the effect of hypogamy compared with homogamy (upper panels) and hypergamy (lower panels). Hypothesis 4 is that the positive effect of hypogamy on women’s relative earnings is lower in countries that exhibit more traditional gender norms. Yet, if anything, we find the opposite to be true, as indicated by the positive slopes of the lines in Fig. 4: the more people tend to agree with these conservative statements about gender roles, the higher the positive effect of hypogamy tends to be. The correlations are weak, however, and most of them are not statistically significant at the 5 % or even 10 % level. The only exception is the contrast between hypogamy and hypergamy in the lower-left panel, which shows that the positive effect of hypogamy on women’s relative earnings compared with women in hypergamous unions is statistically significantly correlated with the percentage strongly agreeing/agreeing that women should be prepared to reduce paid work for the sake of family (r = .56, p < .04). Again, against Hypothesis 4, the correlation is positive, not negative.

Additional descriptive data explorations, depicted in Fig. 5, suggest that the reason for the significant correlation in the case of the contrast between hypogamy and hypergamy is the fact that hypergamous women tend to have lower relative earnings in countries that exhibit more conservative gender norms, while the country-level correlation between gender norms and relative earnings is absent or much weaker for hypogamous women.

## Conclusion

The reversal of the gender gap in education represents a major social development, not just in the West but also in many other regions of the world (Esteve et al. 2016). It has potentially wide-ranging implications for gender roles, including female and male economic roles in the family. Our aim was to investigate the extent to which international differences in women’s relative earnings could be explained by couple-level gender differences in educational attainment and their interaction with parenthood status. We also investigated male unemployment and attitudes about gender roles as additional candidates to explain country differences. Based on EU-SILC data on net earnings of partnered men and women in 16 European countries, we found considerable heterogeneity in women’s relative income, even though women, on average, earned less than their male partners in all countries.

As a baseline, we examined educational pairings and found that in all countries included, women who are more educated than their male partners have higher relative earnings. Next, considering the reversed gender gap in education led us to hypothesize that the negative association between motherhood and relative earnings would be weaker for women who are more educated than their male partners, compared with women who have the same or a lower attainment level than their male partners. Indeed, our results indicate that the motherhood penalty on relative earnings is weaker for hypogamous women than for women in other pairing types and further implies that the hypogamy bonus on women’s relative earnings is stronger for mothers than for childless women.

At the macro level, we found that neither the country distributions of women’s absolute and relative education nor differences in motherhood status and their interaction with relative education can explain the country heterogeneity in women’s relative earnings. Additional analyses showed that male unemployment has a positive effect on women’s relative earnings in all countries, but it cannot explain the remaining country heterogeneity of those earnings. Male unemployment helps to account for why women’s relative earnings tend to be higher in some particular countries than in others (much in line with recent findings by Vitali and Arpino 2016), but the unexplained variance in country differences remains just as high as before unemployment is accounted for.

Turning to cultural factors, we examined the association between the country-specific effects of hypogamy on women relative earnings and two country-level measures of prevailing attitudes about gender roles regarding women’s position in the labor market. We expected that the positive effect of hypogamy on relative earnings would be more limited in countries where conservative attitudes toward gender roles prevail. Yet, if anything, we found the opposite to be the case. The rationale for our hypothesis was that conservative gender norms may hinder well-educated women to cash the returns to their degrees in the labor market, but this is apparently not the case. Instead, we found that conservative norms hinder the earnings of less-educated, hypergamous women. The more-educated, hypogamous women are not really bothered by traditional gender norms that may prevail in their country: for them, we found at best a weak association between their relative earnings and country-level gender role attitudes, whereas less-educated hypergamous women clearly earn less (compared with their partners) in countries with traditional attitudes. Yet, our macro-level correlations should not be considered as conclusive evidence because our tests were based on a limited number of countries and because our sample-based measures of gender role attitudes obviously are subject to error.

Apart from that, our study has other important limitations. First, the number of countries in the study sample was limited to 16 because the data for the other European countries did not include information on net income. Given the important differences between European countries in fiscal arrangements, it was important to use a net income measure to evaluate international differences in women’s relative contribution to the joint disposable income. This requirement reduced the number of countries to a level that precluded robust estimation of multilevel models. For the analysis of gender role attitudes, we were further limited to 14 countries, thus undermining the statistical power of our macro-level analysis.

A second limitation is that we examined only the educational attainment level of male and female partners. Information on their respective fields of education was not available in our data set. However, the field of study clearly has strong implications for the earning potential in the labor market for women and men and is strongly related to the transition to motherhood as well as to attitudes about gender roles (Van Bavel 2010). Choice of study disciplines along gender-stereotypical lines might explain why women often earn less than their partners even in hypogamous couples. Third, the reported associations between educational pairings and women’s share in the family income cannot be interpreted as pure causal effects. The selection into union, union type, and motherhood status are likely to be correlated with women’s earnings and educational level (Dribe and Nystedt 2013). Our results therefore reflect both selection and causal effects.

Nevertheless, the micro-level findings from this study are very robust. They have important implications for family demography, regardless of whether they reflect causal or selection effects. The rising prevalence of hypogamy, the decline of hypergamy, and high male unemployment mean that women’s earnings represent a growing share of joint couple income. This is bound to affect the couple’s negotiations about the division of paid and unpaid work and, hence, their decision-making about the timing and quantum of fertility (Esping-Andersen and Billari 2015; Goldscheider et al. 2015).

## Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Union Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement no. 312290 for the GENDERBALL project. The authors thank Gray Swicegood as well as the anonymous reviewers for many useful comments and suggestions.

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text:
Codeforces celebrates 10 years! We are pleased to announce the crowdfunding-campaign. Congratulate us by the link https://codeforces.com/10years. ×

witua's blog

By witua, 6 years ago, translation, ,

CodeChef invites you to participate in the August Cook-off 2014 at http://www.codechef.com/COOK49.

Time: 24th August 2014 (2130 hrs) to 25th August 2014 (0000 hrs). (Indian Standard Time — +5:30 GMT) — Check your timezone.

Details: http://www.codechef.com/COOK49/

Registration: Just need to have a CodeChef user id to participate.

- Problem Setter : Vitaliy Herasymiv
- Problem Tester and Russian Translator : Sergey Kulik
- Editorialist: Constantin Sokol
- Mandarin Translator: Gedi Zheng & Minako Kojima

It promises to deliver on an interesting set of algorithmic problems with something for all.
The contest is open for all and those, who are interested, are requested to have a CodeChef userid, in order to participate.

• +67

 » 6 years ago, # |   +9 Does anyone have problems with codechef.com?
•  » » 6 years ago, # ^ |   0 yes, no page that starts with http://www.codechef.com seems to be loading (not even old contests).
•  » » 6 years ago, # ^ |   0 Yes :(
•  » » » 6 years ago, # ^ |   -8 yes ：<
•  » » 6 years ago, # ^ | ← Rev. 2 →   +7 Yes. But then again, you didn't expect it? See more on Know Your Meme.
•  » » » 6 years ago, # ^ |   -8 you are so cute :)
•  » » » » 6 years ago, # ^ |   0 but you, with these type of spammy comments, are not!
 » 6 years ago, # |   +1 Can someone provide statements at least? Between, i hate cookoffs due to system issues.
•  » » 6 years ago, # ^ | ← Rev. 2 →   0 first ~10 seconds worked for me, so i can provide the problem names and codes. i haven't managed to open any of the other pages, so nothing else. :(
•  » » » 6 years ago, # ^ |   +12 So lets think solutions from statement titles:)
 » 6 years ago, # |   +22 Here's one problem that I can see : Alice and Bob are playing the game. Initially, there is a single rooted tree. Players take turns alternately, Alice starts.In a single turn, Alice should choose any non-empty subset of roots of tree (or trees) and remove them. On the other hand, Bob should choose any non-empty subset of leafs and remove them from the tree (or trees). It's not allowed to skip the turn. The game ends when a player can't take a turn.The Alice's goal is to minimize the number of turns made by the both players, while Bob wants this number to be the maximal. Considering they both play optimally, what is the number of turns eventually made? InputThe first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.The first line of each test case contains a single integer N denoting the number of nodes of the tree.The next N-1 lines contain pairs of nodes' numbers Ui and Vi (one pair per line), denoting the edges of the tree.The nodes are numbered from 1 to N, inclusive, and the node with the index 1 is the root of the tree. OutputFor each test case, output a single line containing the answer to the corresponding test case. Constraints 1 ≤ T ≤ 40 1 ≤ N ≤ 10000 1 ≤ Ui, Vi ≤ NExampleInput: 3 2 1 2 7 1 3 1 4 4 2 4 5 4 6 6 7 1Output: 2 5 1
•  » » 6 years ago, # ^ |   0 You're a hero.
 » 6 years ago, # |   0 Same here don't know what's wrong with codechef.com.
•  » » 6 years ago, # ^ |   -8 it's a bad thing!
 » 6 years ago, # |   +24 Permutation ShuffleYou are given a permutation of natural integers from 1 to N, inclusive. Initially, the permutation is 1, 2, 3, ..., N. You are also given M pairs of integers, where the i-th is (Li Ri). In a single turn you can choose any of these pairs (let's say with the index j) and arbitrarily shuffle the elements of our permutation on the positions from Lj to Rj, inclusive (the positions are 1-based). You are not limited in the number of turns and you can pick any pair more than once. The goal is to obtain the permutation P, that is given to you. If it's possible, output "Possible", otherwise output "Impossible" (without quotes). Input The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows. The first line of each test case contains two space separated integers N and M denoting the size of the permutation P and the number of pairs described above. The next line contains N integers — the permutation P. Each of the following M lines contain pair of integers Li and Ri. Output For each test case, output a single line containing the answer to the corresponding test case. Constraints 1 ≤ T ≤ 35 1 ≤ N, M ≤ 100000 1 ≤ Li ≤ Ri ≤ NExample Input: 2 7 4 3 1 2 4 5 7 6 1 2 4 4 6 7 2 3 4 2 2 1 3 4 2 4 2 3Output: Possible ImpossibleShootingYou are given a rectangular grid with N rows and M columns. Each its cell is either empty, contains an enemy, or contains a laser. The lasers are capable of shooting. Each one can shoot in one of three directions: either left, right or up.When a laser shoots at some direction, it kills all the enemies on its way. Find out whether it's possible to kill all the enemies on the grid. If it's possible, output "Possible", otherwise output "Impossible". Input The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows. The first line of each test case contains the pair of integers N and M denoting the size of the grid. The next N lines contain M characters each. The characters can be one of the following ones: '.' denoting an empty cell. 'E' denoting an enemy. 'L' denoting a laser. Output For each test case, output a single line containing the answer to the corresponing test case. Constraints 1 ≤ T ≤ 10 1 ≤ N, M ≤ 50 The number of lasers will be between 1 and 16, inclusive.Example Input: 3 2 2 E. L. 2 4 E.EL LL.. 3 3 EE. L.L L..Output: Possible Possible ImpossiblePant The TreeYou are given a rooted tree with N nodes. You would like to paint each node of the tree in one of three possible colors: red, green, and blue. The following conditions must be fulfilled: For each node painted red, there must be no more than R red nodes in its subtree (including this node). For each node painted green, there must be no more than G green nodes in its subtree (including this node). For each node painted blue, there must be no more than B blue nodes in it's subtree (including this node). Find the number of ways to paint the tree and output it modulo 1000000007. Input The first line contains four integers N, R, G and B. The following N-1 lines contain pairs of the nodes' numbers Ui and Vi (one pair per line), describing the edges of the tree. The nodes are numbered from 1 to N, inclusive, and the node with the index 1 is the root of the tree. Output Output the answer to the problem on the first line. Constraints 1 ≤ N ≤ 300 0 ≤ R, G, B ≤ 300Example Input #1: 3 1 1 1 1 2 3 1Output #1: 12 Input #2: 8 2 1 3 1 2 1 4 5 4 7 4 3 2 4 6 8 6Output #2: 613Count DigitsYou are given an integer N. For each pair of integers (L, R), where 1 ≤ L ≤ R ≤ N you can find the number of distinct digits that appear in the decimal representation of at least one of the numbers L L+1 ... R. Find the sum of all that numbers. Since the answer can be large, output it modulo 1000000007. Input The only line of the input contains a single integer N without leading zeros. Output Output the answer to the problem on the first line. Constraints 1 ≤ N ≤ 10100Example Input: 7Output: 84
•  » » 6 years ago, # ^ |   +1 Time limits: 2sec, 1sec, 1sec, 3sec
•  » » » 6 years ago, # ^ |   0 Are they real statements? Does it not provide an advantage for codeforces users?
•  » » » » 6 years ago, # ^ |   +10 Better than just providing an advantage to people who were lucky enough to open a problem. And a huge ethical plus for the poster.
•  » » 6 years ago, # ^ |   0 At counting digits limit is 10^100 or 10,100?
•  » » » 6 years ago, # ^ |   0 10100
•  » » » » 6 years ago, # ^ |   -8 right
 » 6 years ago, # |   +22 This is ridiculous, why I have to wait half an hour for a reply on my question (which is crucial for solving the problem)...
•  » » 6 years ago, # ^ | ← Rev. 2 →   -8 but i don't see any question posted by u? EDIT: oops, i got trolled! :D
•  » » » 6 years ago, # ^ |   0 Exactly! It hasn't even been approved by the moderator.
•  » » » 6 years ago, # ^ |   -8 The funniest thing is that I have to debug two programs because both give WA. :D
•  » » » » 6 years ago, # ^ |   0 @gen what is your id on codechef?
•  » » » » » 6 years ago, # ^ |   0 jevi
•  » » » » » » 6 years ago, # ^ |   0 I am surprised that your comment was not published, I had myself asked problem setter to reply to this comment. Probably it was not answered due to mis-communication on our part. Really sorry for it :(
 » 6 years ago, # |   +16 Ow. The second hardest problem is of the type I hate: taking care of special cases arising from zeroes and the number of digits having to satisfy some constraint. That in itself would be manageable, but if I make a mistake somewhere, it takes a long time to find it because there are so many independent parts of the code where it could be.
•  » » 6 years ago, # ^ | ← Rev. 2 →   0 Pretty much...Time to code: ~40 min Time to debug: ~1h 30min(These are unfortunately real statistics :( ).Edit: tourist's code looks pretty special case free due to his excellent state choice. I'm not that good, though.
 » 6 years ago, # |   -8 502 Bad Gateway！！
 » 6 years ago, # |   0 Hi. I asked a question on codechef.com, but there are still no answers.http://discuss.codechef.com/questions/50210/problems-with-javaI've tried to send several solutions on Java, but there always was NZEC verdict. I wasn't able to make any Java solution get AC. But the C++ and C# ones are okay. So.. I hope that somebody could explain me, what I do wrong using Java on codechef. Thanks.


text:
# Math Help - Question in MATLAB

[IMG] [/IMG]

2. ## Re: Question in MATLAB

Originally Posted by nezar
[IMG] [/IMG]
What ideas have you had so far?

actually no

4. ## Re: Question in MATLAB

I would probably start by defining a vector that contains the $R_{L}$ values. How could you do that?

5. ## Re: Question in MATLAB

Originally Posted by nezar
actually no
Why not this is not the first Matlab plot question you have asked?

In fact I see no reason why you should have a problem with this since you have code to plot the other one.

CB

6. ## Re: Question in MATLAB

i am sorry i am really try to get the answer ,but i don't get any thing
rl=(1,10,1000);
p=12^2.*rl/(rl+2.5)^2;
plot(p);

7. ## Re: Question in MATLAB

Originally Posted by nezar
i am sorry i am really try to get the answer ,but i don't get any thing
rl=(1,10,1000);
p=12^2.*rl/(rl+2.5)^2;
plot(p);
I would modify your last line. What your current last line will do is plot p as a function of the index. But you really want to plot as a function of rl. Hence, I would do this:

plot(rl,p)

See what that does.

8. ## Re: Question in MATLAB

Originally Posted by nezar
i am sorry i am really try to get the answer ,but i don't get any thing
rl=(1,10,1000);
p=12^2.*rl/(rl+2.5)^2;
plot(p);
Have you tried typing any of this at the command line?

CB

9. ## Re: Question in MATLAB

Originally Posted by Ackbeet
I would modify your last line. What your current last line will do is plot p as a function of the index. But you really want to plot as a function of rl. Hence, I would do this:

plot(rl,p)

See what that does.
i try but he gave me
??? Error: File: nezarn.m Line: 1 Column: 6
Expression or statement is incorrect--possibly unbalanced (, {, or [.

10. ## Re: Question in MATLAB

Originally Posted by nezar
i try but he gave me
??? Error: File: nezarn.m Line: 1 Column: 6
Expression or statement is incorrect--possibly unbalanced (, {, or [.
I suspect your first line is not the correct syntax for defining a vector in MATLAB. You're after, apparently, a vector containing 1000 numbers ranging from 1 to 10. Try this expression instead:

rl=1:0.009:10;

11. ## Re: Question in MATLAB

Originally Posted by CaptainBlack
Have you tried typing any of this at the command line?

CB
no

12. ## Re: Question in MATLAB

Originally Posted by Ackbeet
I suspect your first line is not the correct syntax for defining a vector in MATLAB. You're after, apparently, a vector containing 1000 numbers ranging from 1 to 10. Try this expression instead:

rl=1:0.009:10;
error
??? Error using ==> mpower
Matrix must be square.

Error in ==> nezarn at 2
p=12^2.*rl/(rl+2.5)^2;

13. ## Re: Question in MATLAB

Originally Posted by nezar
error
??? Error using ==> mpower
Matrix must be square.

Error in ==> nezarn at 2
p=12^2.*rl/(rl+2.5)^2;
Try using dots everywhere, since you're wanting component-wise arithmetic. That is, try

p=12^2.*rl./(rl+2.5).^2;

Note: you actually don't need a dot for the first one, because that's a scalar. So you could probably do

p=12^2*rl./(rl+2.5).^2;

14. ## Re: Question in MATLAB

Originally Posted by Ackbeet
Try using dots everywhere, since you're wanting component-wise arithmetic. That is, try

p=12^2.*rl./(rl+2.5).^2;

Note: you actually don't need a dot for the first one, because that's a scalar. So you could probably do

p=12^2*rl./(rl+2.5).^2;
thank you for help me

15. ## Re: Question in MATLAB

Originally Posted by nezar
thank you for help me
So, in paying attention to CB's posts, let me ask you these questions:

1. Do you know why you got the errors you got?
2. Do you know how to fix them once you see them again?
3. Do you see why MATLAB didn't like your previous syntax?
4. Does your graph make sense?

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text:
After some time stepper motor is hot

• I have three stepper motors.
One Nema 17 - 2.4 ohm, the second smaller noname from color printer - 9.5 ohm and third the smallest noname from cdrom - 10.5 ohm.

I have connected them to arduino mega 2560 with ramps 1.4(set to 1/32 micro stepping) and drivers drv8825. See my previous question.

After some time (less than one minute) the first is cold. The second motor is hot. And the third is very hot. I can not even touch it.

What can I do to fix it.

there should be labels what they are rated for on the smaller motors.

Note that you should unlock your steppers when not in use. When an "idle" stepper gets hot, it's using energy to keep rigidly locked at a precise rotation (often for no good reason).

@Davo, they work constantly.

• The second motor is hot. And the third is very hot. I can not even touch it.

This is to some degree, completely normal and expected. From the datasheet for a typical NEMA 17 stepper, the rated temperature rise is 80 °C above ambient and the maximum operating temperature is 130 °C (implying an ambient temperature of 50 °C). It is normal that stepper motors (in general) get a bit hot.

"Too hot to touch" is still relatively cold. 60 °C is already too hot to touch, and that's only a 40 °C rise above a 20 °C ambient temperature.

You can reduce the temperature rise of the motors by reducing the current they receive. The stepper driver has a small potentiometer that can be turned to adjust the current, but keep in mind that doing so will also reduce the torque of the motors and thus they might skip steps if you reduce the current too much.

Technical details: Note that stepper motor drivers used in 3D printers are constant current drivers, and the little potentiometer controls the current. If you had not paid much attention to this potentiometer, the drivers might all have been set for the same constant current of $$1.0\ \text A$$. The stepper driver would (to achieve the same constant current) send a higher voltage to the higher resistance motors. This would imply a power dissipation of $$2.4\ \text W$$ in the Nema 17, and a power dissipation of $$10.5\ \text W$$ in the small stepper. $$2.4\ \text W$$ in the Nema 17 would only heat it up by about $$20\ °\text C$$ above ambient. A dissipation of $$10\ \text W$$ in the small stepper, which also has much less surface area to dissipate the power, would heat it up by a lot (and probably, given that you didn't fry it, the current was set lower -- or a technical peculiarity limited the current given that the motor likely also has very low inductance).

2 of his motors are *not* NEMA 17 but salvaged from other machines.

Stepper motors, in general, may be rated to operate at higher temperatures. "Too hot to touch" could still be completely normal. I am just using that datasheet (for a NEMA17) as one example of how hot steppers can normally get.

true. note though that a typical CD-drive Stepper with 5V/10 Ohm is like *20* (or 24) steps per turn...

Thank you! I need to check it.

I have reduced current by reduced voltage from 0.65 V to 0.15(cold)-0.2(warm) V for cd motor and 0.2(cold)-0.4(warm) V for color printer motor.

@burtsevyg can you measure the current instead? but reducing the current to about a third means that about that is for the current too, which means about 1/9th of the power dissipated.

@Trish A reference voltage of 0.65V translates to a current of 1.2A, so the current has been reduced to around 300-500mA for the small motor and 500-700mA for the large one.

@Trish I will try. Should I measure the current on single сoil?