Special Collection 8

Formal Relationships

Published 20 January 2009

Special Collection 8 is an ongoing project containing short reports on mathematical relationships in formal demography. The collection was initiated by the journal’s founding publishers James W. Vaupel and Joshua R. Goldstein. It is currently being edited by Monica Alexander and Joshua R. Goldstein.

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20 January 2009 | formal relationship

Formal Relationships: Introduction and Orientation

Joshua R. Goldstein, James W. Vaupel

Volume: 20 Article ID: 1
Pages: 1–2
DOI: 10.4054/DemRes.2009.20.1

20 January 2009 | formal relationship

Life lived equals life left in stationary populations

Joshua R. Goldstein

Volume: 20 Article ID: 2
Pages: 3–6
DOI: 10.4054/DemRes.2009.20.2

20 January 2009 | formal relationship

Life lived and left: Carey’s equality

James W. Vaupel

Volume: 20 Article ID: 3
Pages: 7–10
DOI: 10.4054/DemRes.2009.20.3

29 May 2009 | formal relationship

The effect of changes in fertility on the age distribution of stable populations

Michel Guillot

Volume: 20 Article ID: 24
Pages: 595–598
DOI: 10.4054/DemRes.2009.20.24

18 June 2009 | formal relationship

The age separating early deaths from late deaths

Zhen Zhang, James W. Vaupel

Volume: 20 Article ID: 29
Pages: 721–730
DOI: 10.4054/DemRes.2009.20.29

18 November 2009 | formal relationship

The metastable birth trajectory

Robert Schoen

Volume: 21 Article ID: 25
Pages: 759–764
DOI: 10.4054/DemRes.2009.21.25

11 December 2009 | formal relationship

Survival as a Function of Life Expectancy

Maxim Finkelstein, James W. Vaupel

Volume: 21 Article ID: 29
Pages: 879–884
DOI: 10.4054/DemRes.2009.21.29

22 January 2010 | formal relationship

Life expectancy is the death-weighted average of the reciprocal of the survival-specific force of mortality

Joel E. Cohen

Volume: 22 Article ID: 5
Pages: 115–128
DOI: 10.4054/DemRes.2010.22.5

30 June 2010 | formal relationship

Total daily change with age equals average lifetime change

James W. Vaupel

Volume: 22 Article ID: 36
Pages: 1143–1148
DOI: 10.4054/DemRes.2010.22.36

09 July 2010 | formal relationship

Sensitivity of life disparity with respect to changes in mortality rates

Peter Wagner

Volume: 23 Article ID: 3
Pages: 63–72
DOI: 10.4054/DemRes.2010.23.3

07 September 2010 | formal relationship

Reproductive value, the stable stage distribution, and the sensitivity of the population growth rate to changes in vital rates

Hal Caswell

Volume: 23 Article ID: 19
Pages: 531–548
DOI: 10.4054/DemRes.2010.23.19

12 October 2010 | formal relationship

Attrition in heterogeneous cohorts

James W. Vaupel, Zhen Zhang

Volume: 23 Article ID: 26
Pages: 737–748
DOI: 10.4054/DemRes.2010.23.26

08 February 2011 | formal relationship

Life expectancy: Lower and upper bounds from surviving fractions and remaining life expectancy

Joel E. Cohen

Volume: 24 Article ID: 11
Pages: 251–256
DOI: 10.4054/DemRes.2011.24.11

15 March 2011 | formal relationship

Linking period and cohort life-expectancy linear increases in Gompertz proportional hazards models

Trifon Missov, Adam Lenart

Volume: 24 Article ID: 19
Pages: 455–468
DOI: 10.4054/DemRes.2011.24.19

10 May 2011 | formal relationship

The reproductive value as part of the shadow price of population

Gustav Feichtinger, Michael Kuhn, Alexia Prskawetz, Stefan Wrzaczek

Volume: 24 Article ID: 28
Pages: 709–718
DOI: 10.4054/DemRes.2011.24.28

10 June 2011 | formal relationship

Age-specific growth, reproductive values, and intrinsic r

Robert Schoen

Volume: 24 Article ID: 33
Pages: 825–830
DOI: 10.4054/DemRes.2011.24.33

18 April 2012 | formal relationship

Stationary populations with below-replacement fertility

Carl Schmertmann

Volume: 26 Article ID: 14
Pages: 319–330
DOI: 10.4054/DemRes.2012.26.14

06 March 2012 | formal relationship

Historical Addendum to "Life lived equals life left in stationary populations"

Joshua R. Goldstein

Volume: 26 Article ID: 7
Pages: 167–172
DOI: 10.4054/DemRes.2012.26.7

04 September 2012 | formal relationship

How life expectancy varies with perturbations in age-specific mortality

Tomasz Wrycza, Annette Baudisch

Volume: 27 Article ID: 13
Pages: 365–376
DOI: 10.4054/DemRes.2012.27.13

12 September 2012 | formal relationship

The difference between alternative averages

James W. Vaupel, Zhen Zhang

Volume: 27 Article ID: 15
Pages: 419–428
DOI: 10.4054/DemRes.2012.27.15

12 February 2013 | formal relationship

Gamma-Gompertz life expectancy at birth

Trifon Missov

Volume: 28 Article ID: 9
Pages: 259–270
DOI: 10.4054/DemRes.2013.28.9

06 May 2014 | formal relationship

Entropy of the Gompertz-Makeham mortality model

Tomasz Wrycza

Volume: 30 Article ID: 49
Pages: 1397–1404
DOI: 10.4054/DemRes.2014.30.49

07 May 2014 | formal relationship

Variance in age at death equals average squared remaining life expectancy at death

Tomasz Wrycza

Volume: 30 Article ID: 50
Pages: 1405–1412
DOI: 10.4054/DemRes.2014.30.50

16 May 2014 | formal relationship

Is the fraction of people ever born who are currently alive rising or falling?

Joel E. Cohen

Volume: 30 Article ID: 56
Pages: 1561–1570
DOI: 10.4054/DemRes.2014.30.56

29 April 2015 | formal relationship

The force of mortality by life lived is the force of increment by life left in stationary populations

Tim Riffe

Volume: 32 Article ID: 29
Pages: 827–834
DOI: 10.4054/DemRes.2015.32.29

20 May 2015 | formal relationship

The Gompertz force of mortality in terms of the modal age at death

Trifon Missov, Adam Lenart, Laszlo Nemeth, Vladimir Canudas-Romo, James W. Vaupel

Volume: 32 Article ID: 36
Pages: 1031–1048
DOI: 10.4054/DemRes.2015.32.36

19 August 2016 | formal relationship

Symmetries between life lived and left in finite stationary populations

Francisco Villavicencio, Tim Riffe

Volume: 35 Article ID: 14
Pages: 381–398
DOI: 10.4054/DemRes.2016.35.14

27 September 2018 | formal relationship

The impact of proportional changes in age-specific mortality on life expectancy when the mortality rate is a log-linear function of age

Michael Væth, Mette Vinther Skriver, Henrik Støvring

Volume: 39 Article ID: 23
Pages: 671–684
DOI: 10.4054/DemRes.2018.39.23

26 October 2018 | formal relationship

Life lived and left: Estimating age-specific survival in stable populations with unknown ages

James W. Vaupel, Francisco Villavicencio

Volume: 39 Article ID: 37
Pages: 991–1008
DOI: 10.4054/DemRes.2018.39.37

09 July 2019 | formal relationship

The threshold age of the lifetable entropy

Jose Manuel Aburto, Jesús-Adrián Alvarez, Francisco Villavicencio, James W. Vaupel

Volume: 41 Article ID: 4
Pages: 83–102
DOI: 10.4054/DemRes.2019.41.4

12 March 2020 | formal relationship

Revivorship and life lost to mortality

Carl Schmertmann

Volume: 42 Article ID: 17
Pages: 497–512
DOI: 10.4054/DemRes.2020.42.17

06 January 2021 | formal relationship

Born once, die once: Life table relationships for fertility

Annette Baudisch, Jesús-Adrián Alvarez

Volume: 44 Article ID: 2
Pages: 49–66
DOI: 10.4054/DemRes.2021.44.2

16 February 2021 | formal relationship

How do populations aggregate?

Dennis Feehan, Elizabeth Wrigley-Field

Volume: 44 Article ID: 15
Pages: 363–378
DOI: 10.4054/DemRes.2021.44.15

15 April 2021 | formal relationship

Outsurvival as a measure of the inequality of lifespans between two populations

James W. Vaupel, Marie-Pier Bergeron-Boucher, Ilya Kashnitsky

Volume: 44 Article ID: 35
Pages: 853–864
DOI: 10.4054/DemRes.2021.44.35

11 June 2021 | formal relationship

Measuring the concentration of urban population in the negative exponential model using the Lorenz curve, Gini coefficient, Hoover dissimilarity index, and relative entropy

Joel E. Cohen

Volume: 44 Article ID: 49
Pages: 1165–1184
DOI: 10.4054/DemRes.2021.44.49

17 May 2023 | formal relationship

Improved bounds and high-accuracy estimates for remaining life expectancy via quadrature rule-based methods

Oscar Fernandez

Volume: 48 Article ID: 27
Pages: 809–828
DOI: 10.4054/DemRes.2023.48.27

19 January 2024 | formal relationship

The optimal transition to a stationary population for concentrated vitality rates

Gustav Feichtinger, Stefan Wrzaczek

Volume: 50 Article ID: 6
Pages: 171–184
DOI: 10.4054/DemRes.2024.50.6

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