# 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.

20 January 2009 | formal relationship

Formal Relationships: Introduction and Orientation

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

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

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

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

Volume: 20 Article ID: 29

Pages: 721–730

DOI: 10.4054/DemRes.2009.20.29

18 November 2009 | formal relationship

The metastable birth trajectory

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

Volume: 21 Article ID: 29

Pages: 879–884

DOI: 10.4054/DemRes.2009.21.29

22 January 2010 | formal relationship

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

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

Volume: 23 Article ID: 3

Pages: 63–72

DOI: 10.4054/DemRes.2010.23.3

07 September 2010 | formal relationship

Volume: 23 Article ID: 19

Pages: 531–548

DOI: 10.4054/DemRes.2010.23.19

12 October 2010 | formal relationship

Attrition in heterogeneous cohorts

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

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

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

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

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

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"

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

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

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

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

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

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?

Volume: 30 Article ID: 56

Pages: 1561–1570

DOI: 10.4054/DemRes.2014.30.56

29 April 2015 | formal relationship

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

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

Volume: 35 Article ID: 14

Pages: 381–398

DOI: 10.4054/DemRes.2016.35.14

27 September 2018 | formal relationship

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

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

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

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

Volume: 44 Article ID: 2

Pages: 49–66

DOI: 10.4054/DemRes.2021.44.2

16 February 2021 | formal relationship

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

Volume: 44 Article ID: 35

Pages: 853–864

DOI: 10.4054/DemRes.2021.44.35

11 June 2021 | formal relationship

Volume: 44 Article ID: 49

Pages: 1165–1184

DOI: 10.4054/DemRes.2021.44.49

17 May 2023 | formal relationship

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

Volume: 50 Article ID: 6

Pages: 171–184

DOI: 10.4054/DemRes.2024.50.6