Concept version: 050919
Ingo Piepers, GLOBAL4CAST, Amsterdam, The Netherlands
Preliminary findings based on research presented and discussed in this paper, show that during the period 1982-2019, American society produced a series of mass shooting cycles (MS-cycles), that evolved from cycles with periodic characteristics, with T = 8 years, to T= 4 years, to aperiodic dynamics with chaotic characteristics, starting around 2016/2017. Mass shootings qualify as system-behavior. During the period 1982 until 2016/2017, MS-cycles had a typical configuration; until 2012, mass shooting cycles and four-year presidential (election) cycles were “exactly” aligned/synchronized; after 2012 – and certainly after the 2016-election – this was no longer the case. Despite the alignment of MS- and presidential (cyclic) cycles, this analysis also shows that the political signature of a president (Democrat or Republican) does not impact on mass shooting dynamics, at least not during the period 1985-2012. The “mass shooting system” (MS-system), is an integral part of American society and is powered and controlled by tensions in American society; tension-production is the parameter that drives the MS-system. The changes in the behavior of MS-dynamics, during the period 1982-2019, from cycles with periodic characteristics (with T = 8 years, to T= 4 years), to aperiodic dynamics with chaotic characteristics, starting around 2016/2017, involved two bifurcations (respectively in 2008 and in 2016/2017). This process – characterized by a typical sequence of bifurcations and dynamics that can also be observed in other systems – is sometimes also referred to as the “route to chaos. A question is how American society can (and will) respond to an increasing frequency of mass shootings, with increasing numbers of fatalities, now that mass shooting dynamics have become chaotic: Will the route to chaos, a phrase/term in dynamical system sciences, also cause chaos in American society, in a social and political sense? The patterns that can be identified in mass shootings provide valuable insights in the mechanisms and systems that produce them, and furthermore provide clues to develop more effective strategies and policies to prevent and control these events. These preliminary conclusions are based on a limited data set, and further research and validation are required.
Following the latest mass shooting (MS) in the United States (Odessa, Texas, United States, 31/8/19), I have made a (preliminary) analysis of mass shootings in the United States (US), based on data of the Mother Jones’ investigation, concerning the period 1982-2019 (1), and based on insights provided by my research related to war-dynamics in the international System (period 1480 – present) (4), (5).
Although there are some fundamental differences between war (dynamics) and mass shootings and the respective systems that produce them, there are also some (possible) similarities, that are worthwhile exploring.
Concerning this analysis, relevant questions include: What causes these mass shootings? And do insights in the war dynamics of the (international) System (4), (5), provide useful clues to better understand the phenomenon of mass shootings in the United States?
Initial results provide indications that it is possible to identify cycles in the number of fatalities of mass shootings, during certain periods. I refer to these cycles as mass shooting cycles (MS-cycles).
In this preliminary paper, I discuss these “cycles” and “patterns”, and propose a mechanism that could produce them. These are still speculative findings; more research is required.
I discuss the following subjects in this preliminary paper:
- The perspective – framework – I use to study the mass shooting data and MS-cycles;
- Some preliminary results;
- A preliminary interpretation of the results.
Before analyzing mass shooting data, I discuss some insights research into war dynamics has produced. This short introduction provides the reader with some basic understanding of these types of dynamics, and my approach to study them.
Based on my research (4), (5), I argue that the (international) System produced four accelerating war-cycles, during the period 1480-1945. From a thermodynamic perspective, these war-cycles can be understood as cycles of kinetic activity (KA-cycles).
I argue that the four accelerating KA/war-cycles accompanied a so-called dissipative structure, the System produced through a so-called bifurcation in 1480. This dissipative structure (1480-1945) was instrumental in a phase transition (accomplished in 1945), that resulted in thermodynamic equilibrium in the core of the System (Europe, albeit with a delay until 1989), and the globalization of the (State-)System.
As Kondepudi (3) and Prigogine (6) explain, systems far-from-equilibrium can produce dissipative structures, when certain conditions are met. Dissipative structures and their (typical) dynamics are manifestations of the laws of thermodynamics and related principles that apply far-from-equilibrium.
The research shows that the entropy the System produced through interactions and exchanges between its components (e.g. societies and state-structures), manifests itself as social tensions (threats and opportunities), the System and its components have to deal with (must regulate), to maintain a certain coherence. In case of the System, entropy (tensions) is periodically transferred into kinetic activity (war), to regulate these tensions, and to ensure a certain “balance”; In fact, entropy – tensions – serve as a source of order in social systems.
In case of the System, tensions are regulated through an emergent structure; a dissipative structure. This is a so-called self-organized – emergent – structure, “produced by” the laws of thermodynamics and related principles that apply.
As I mentioned, during the period 1480-1945, the dissipative structure was instrumental in a phase transition. The four KA/war-cycles that accompanied the dissipative structure (1480-1945) had a typical configuration: In all four cases a relatively long stable period, was always followed by a critical period, when through system-wide KA/war, the order of the System was “upgraded”. Through the upgraded order, a new relatively long stable period was enabled.
In fact, the dissipative structure harnessed and utilized tensions (entropy): Tensions periodically triggered KA/war, that was used to upgrade – reorganize – the System. This was a natural – emergent – process.
Societies also produce entropy (tensions), which must be regulated. State-structures – which include governments, government institutions, rules and regulations, etc. – play an important role in tension-regulation, not only in the international system, but also “within” societies.
To optimize their control and regulation, states have centralized certain control functions and monopolized the use of violence.
The system of government also impacts on its ability and “method” of (tension-) regulation.
Democracy is a system of government where citizens exercise power by periodic voting. Through this system of government, societies/states also (try to) regulate tensions (threats and opportunities). Citizens are actively involved in “choices”, and through their participation have a stake in the development and the stability of the society they are part of, is the assumption.
Through periodic voting, citizens are asked for their opinion (in practice to chose for alignment with a political party that represents a (more or less consistent) set of ideas). Voting results in the selection of a president (in the US), who represents certain ideas, and the forming of a new – “upgraded” – government. In the US, presidential (election) cycles have a duration of four years and follow a preset timetable (…. 2000 – 2004 – 2008- etc.)
I argue that a presidential election is the societal/state equivalent of a systemic KA/war in the international System: While elections are deliberate (“organized’), systemic KA/wars in the System are self-organized. In both cases an “upgraded” order is implemented, respectively a new government and a new international order, to ensure a certain stability and (renewed) viability for the system itself, and its constituents.
In case of the System, kinetic activity (KA, war) – organized violence – is a manifestation of – a reaction to – tensions in the System that require regulation; the emergent dissipative structure ensures adequate regulation, as I explained.
In case of US-society, mass shootings can be considered self-organized “kinetic” responses to (societal) tensions; tensions the government/state cannot (or will not) adequately regulate.
From this system- and thermodynamic perspective – KA/war in the System, and mass shootings in US society (in this case) constitute system-behavior; self-organized responses of these systems to tensions, that obviously have reached a certain threshold and consequently trigger kinetic responses.
From such a “top down” perspective, motivations of states/societies to start a war and for individuals to start a mass-shooting, are not of direct interest.
KA/war-cycles in the System have a consistent configuration, my research shows. The question is, if this is also the case for mass shootings during presidential (election) cycles – the four-year government periods – in the United States. This research shows that this is indeed the case for MS-cycles during the period (1985-2008); I will elaborate on the typical configuration of these MS-cycles later.
3. PRELIMINARY RESULTS.
These are some preliminary results of data-analysis, based on the Mother Jones’ dataset (1):
(a) Six peaks. Based on the analysis of mass shooting data, it is possible to identify six peak-years in mass shooting fatalities, respectively the years: 1984, 1991, 1999, 2007, 2012 and 2017. Each peak is preceded by a relatively long period with “medium-sized” mass-shootings.
(b) Regular development of peaks. The development of the number of fatalities during successive peak-years developed very regularly:
(c) Accelerating growth in the number of fatalities during successive peak-years. Initially (peaks 1984, 1991 and 1999), the number of fatalities during peak years, increased linearly, then exponential, and subsequently (during the year 2017); the increase of the number fatalities during peak-years accelerated, on the longer term.
(d) Increasing number/frequency of mass shootings per year. Not only the number of fatalities during peak-years increased over time, but also the frequency of mass-shootings during successive years, on average. It seems that around 2011/2012, the frequency of mass shootings started increasing more structurally.
(e) Cycles in mass shootings. Based on the available data, it is possible to distinguish five cycles of mass shootings (MS-cycles), it seems, during the period 1985-2018:
(f) Alignment/synchronization of MS- and presidential (election) cycles. The analysis shows that mass shooting cycles and presidential (election) cycles are highly synchronized; see also above table (peak timings). This alignment suggests that both dynamics interact, and “feed” on each other.
This assumption does make sense, typically during (pre-)election years, polarization in American society (further) intensifies, as the contesting parties magnify their differences, at the expense of shared values. Over time, the past decades, elections in the US have become more “extreme” and intense, and increasingly focus implicitly and explicitly on basic ideas and values, and disagreements about that.
(g) The number of mass shootings during MS-cycles was “stable”. The number of mass shootings was more or less stable during the four MS-cycles (1985-2012). The fifth period (2013-2018) produced a significant increase in the number/frequency of mass shootings, that can be especially attributed to the years 2017 and 2018.
(h) The number of fatalities of mass shootings during successive cycles. The number of fatalities of mass shootings during successive cycles developed remarkably regular; however, the last cycle (2013-2018) causes a significant distortion, caused by mass shootings during the period 2017-2018.
(i) The number of fatalities during successive four-year presidential (election) cycles. The number of fatalities during successive four-year presidential (election) cycles also increased, but not as regular as of MS-cycles; it is possible to identify three oscillations in presidential (election) cycles, which are aligned with the three eight-year MS-cycles.
- Six peak(years) in mass shooting fatalities can be identified, during the period 1982-2019.
- The number of fatalities during these peak-years develops very regular.
- There is accelerating growth in the number of fatalities during successive peak-years.
- There is an increase in the number of mass shootings per year; the increase is somewhat irregular, but around 2011/2012 a significant increase can be observed.
- It is possible to identify four cycles in mass shootings: three eight-year and one four-year cycles. These MS-cycles have a similar “configuration” (profile): In all case the peak in fatalities is during a year preceding a presidential election, or in the election year itself, and in all four cases preceded by a period with a “moderate” number of fatalities.
- The four (periodic) cycles (they concern the period 1985-2012) were aligned/synchronized with presidential (election) cycles, during this period.
- The number of mass shootings during MS-cycles was stable during the period 1985-2012; a very significant increase in the number of mass shootings can be observed starting in 2017 and 2018 (complete 2019-data is not yet available).
- The number of fatalities of mass shootings during successive MS-cycles developed remarkably regular; however, the last period/cycle (2013-2018) causes a significant distortion, caused by mass shootings during the period 2017-2018.
- The number of fatalities during successive four-year presidential (election) cycles also increased, but not as regular as of MS-cycles; it is possible to identify three oscillations, which are aligned with the three eight-year MS-cycles.
In this paragraph, I make a tentative interpretation; the findings (“conclusions”) are preliminary, and still explorative. Further research is required.
(a) Three types of MS-dynamics; two bifurcations.
It seems that three distinctly different types of mass shooting dynamics can be distinguished, during the period 1982-2019:
I assume that tensions and (the rate of) tension-production “drive” MS-dynamics. If these observations are correct, this means that the system concerned, experienced two bifurcations. Bifurcations typically result in a change of behavior. These bifurcations are triggered by slight changes in parameters (related to the production and buildup of tensions, in this case, I argue). This system experienced period-doubling bifurcation in/around 2008 (T = 8 years becomes T = 4 years), and a chaotic bifurcation (a switch to chaotic properties of mass shootings) around 2016/2017.
(b) Alignment of MS-cycles and four-year presidential (election) cycles, during the period 1985-2012.
During the period 1985-2012, the MS-system produced four MS-cycles (three 8-year MS-cycles, and one 4-year MS-cycle), with more or less identical “configurations”. The MS-cycles, and the timing of the peaks in fatalities, were aligned with four-year presidential (election) cycles. The first three (8-year) MS-cycles (1985-1992, 1993-2000 and 2001-2008), “contained” six presidential (election) cycles in total, two each, respectively: (1) 1985-1988 and 1989-1992, (2) 1993-1996 and 1997-2000, and (3) 2001-2004 and 2005-2008); the fourth MS-cycle (this is a four-year cycle, 2009-2012), contained only one presidential (election) cycle (2009-2012). I propose that during the period 1985-2012, a relationship existed between MS-cycles and presidential (election) cycles, that is (also) related to tensions created by presidential elections.
(c) Oscillations point to corrective action of the system.
The oscillations during the three eight-year cycles, each consisting of two four-year presidential (election) cycles, is interesting.
Oscillation is one of three modes of dynamic behavior (9): Exponential growth, goal seeking, and oscillation. Exponential growth arises from positive (self-reinforcing) feedback; in case of goal seeking, negative feedback loops dominate dynamics and act to bring the state of the system concerned in line with a goal or desired state (9). Oscillation is caused by negative feedback loops, with time delays in their corrective actions (9).
The three oscillations during the period 1985-2008, with a significantly increased amplitude in case of the third oscillation, during the third eight-year cycle (2001-2008), not only point to corrective behavior of the system, but also that there is growth (of fatalities and underlying tensions), that could at a certain point no longer be contained in an eight-year cycle, and consequently triggered a switch (bifurcation) to a four-year-cycle.
In case of the MS-system there is a combination of two modes of dynamic behavior at play, during the period 1985-2008: (exponential) growth and oscillation.
(d) The political signature of a US president does not impact on MS-dynamics.
The oscillations the system produced during the period 1985-2008 are a manifestation of corrective behavior (with delays) of the system, as I explained. This system behavior (oscillation) is not related to the political signature of the respective presidents, during the presidential (election) cycles. The very regular corrections (oscillations) took place independent of the political signature of respective presidents.
Furthermore, as above figure shows, from the presidential (election) cycle 2005-2008 onwards, the number of fatalities during the next three next presidential (election) cycles, steadily increased, also irrespective of the political signature of the respective presidents.
(e) A closer look at the oscillations during the period 1985-2008.
A closer look at the three oscillations the MS-system produced during the period 1985-2008, not only reveals remarkable regularities (see above), but also that the bifurcation in 2008 (causing a switch to a MS-cycle with T = 4 years), was preceded by an oscillation with a much larger amplitude than its two predecessors. The increased amplitude points to the increasing instability of the MS-system, and the need for a switch (bifurcation) of the system, to maintain a certain “regulation”. “More” corrective action was required, in a shorter (now four-year) period of time. However, it was only a matter of time, before the “corrective action” was (again) insufficient, and mass shootings and increased tensions “enforced” another bifurcation, that resulted in aperiodic (I assume chaotic) MS-dynamics. From that point in time, mass shootings (corrective actions) were no longer aligned with presidential (election) cycles and were “uncoupled”, so to speak.
Above discussed process and dynamics can also be viewed as follows: The oscillations in fatalities during the first three (8-year) MS-cycles (1985-1992 and 1993-2000), “contain” four presidential (election) cycles, two each, respectively: 1985-1988 and 1989-1992, and 1993-1996 and 1997-2000; the two oscillations that are produced seem to be regular. However, during the third (8-year) MS-cycle (2001-2008), which contains two presidential (election) cycles (2001-2004 and 2005-2008), the amplitude of the MS-cycle significantly increases, while the period is still eight years.
It seems that the increasing size of the number of fatalities in mass shootings – the violence (kinetic activity) generated by tensions in American society – causes a change in period to four years, through a bifurcation in 2008. The configuration of the MS-cycle (but with a peak now in the actual election year, and not in the preceding year as was the case during the three preceding MS-cycles), is more or less identical.
The number of fatalities during the fourth MS-cycle (2009-2012), the number of 2012 peak-year fatalities, and the total number of fatalities during the fourth MS-cycle, follow trends in these characteristics of the preceding three eight-year MS-cycles: There is remarkable consistency.
It can be argued that the three eight-year MS-cycles were produced – and “controlled” – by two presidential (election) cycles each, while the fourth MS-cycle was produced during a single presidential (election) cycle. However – as I mentioned – it was also only a matter of time before a four-year cycle could not be sustained and increased tensions in American society consequently forced the MS-system to a stability domain with a chaotic (aperiodic) attractor and accompanying dynamics(2016/2017): The mass shootings were no longer (primarily) induced by tensions related to elections, but to an overall increase in societal tensions, that could no longer be “regulated” by four-year presidential (election) cycles.
(f) On the “route to chaos”.
The changes – bifurcations – that can be observed in the MS-dynamics, from periodic dynamics with T = 8 years, to periodic dynamics with T = 4 years, to aperiodic/chaotic dynamics, are not arbitrary; this is a typical sequence of developments that can also be observed in other systems, and is sometimes referred to as the “route to chaos”, that is a route – path – of dynamics of a system to dynamics with chaotic properties. The route to chaos refers to the process by which a simple attracting set for a dynamical system (like a fixed point or periodic orbit) becomes chaotic as an external parameter is varied (8); it is a “technical” term.
This analysis suggests that an increase in societal tensions was (and still is) the driver – the control (bifurcation) parameter – of the MS-system, that triggered several bifurcations, and put and sustained the MS-system on the route to chaos.
The question is, if this technical “chaos” (deterministic chaos from a dynamical system perspective) concerning MS-dynamics, can (eventually) also result in political and social chaos in US-society (a high degree of disruptive social disorder).
To answer this question, it is necessary to define (operationalize) the concept (term) “political and social chaos”. Also, the question “How much tensions, mass shootings, etc. can US-society and its political system absorb and sustain, without causing more damage to its social fabric and political institutions?” must be answered. How resilient are US-society and its political system and institutions?
The fact that US-society and its political system are (now) further polarized – also because of disagreement over the cause of mass shootings and how to contain them – and that this polarization will be further boosted and politically exploited during the upcoming 2020 presidential elections, are reasons for serious concern: Can US society collapse if MS-dynamics further increase and become more organized and large scale?
(g) Mass shootings are boosted by several – interacting – sources of tensions.
I have suggested that tensions and tension-production – which typically flare up in the year preceding presidential elections and in the election year itself – are the (main) driver of MS-dynamics. I assume that there are two important “sources” – that are probably also related – for tensions in US-society: (1) Internal sources: including (increasing) dissatisfaction and uncertainty in general, and a fundamental contradiction at the core of American identity/society related to the Second Amendment, and (2) external sources, e.g. related to increasing tensions and uncertainty in the international System. The “complex” variables and factors involved in these sources, reinforce each other; presently positive (self-reinforcing) feedback structures control (MS-)dynamics.
Increasing dysfunctionality and ineffectiveness of the US political system (e.g. deadlocks in political institutions) also go hand in hand – interact with – increasing dysfunctionality and fragmentation of the international System, and its institutions and arrangements.
Obviously, exchanges of tensions between US society and the international System take place continuously: In the international System, the US is the most dominant great power; the US can be considered the node with the greatest centrality in the network of international relations: The US is inseparably linked to – and a part of – the international System, and vice versa.
It can be argued that a very significant increase in terrorist activity (worldwide), starting in 2011 (2), is related to a significant increase in mass shootings per year in the US , starting in 2011/2012 (see above).
(h) Mass shootings – and their dynamics – are system-behavior.
In this paragraph, I make some remarks about the phenomenon of mass shootings in US society. I also address the following questions: Why are mass shootings a particular American phenomenon? (which points to the presence of a set of unique conditions)? And: Why are the US-government, US society and political system not able – powerless – to take decisive measures to stop mass shootings, and to break this deadly – potentially self-destructive – dynamic?
The idea about the right to self-defense in the US is intrinsically related to the idea of freedom in the US, anchored in the US constitution, and central to American identity. I argue that this set of ideas and convictions is an important contributing factor to mass shootings, and to the inability of the US (government) to stop them.
The United States allows its citizens – with just a view conditions and only limited supervision – to keep and bear arms, for self-defense (the Second Amendment to the United States Constitution). This right to self-defense and to keep and bear arms, are central to American identity (at least for a significant group). The Second Amendment “enables” the proliferation – and easy availability – of a range of arms.
It can be argued, that the right to self-defense and to keep and bear arms, “competes” with and undermines the monopoly of violence in American society; this monopoly is not as strict/absolute as in for example European societies/states. The American state is – “by design” – more restricted in the “services” it provides to its citizens, not only concerning social welfare and health care, but also concerning safety/security. A certain do-it-yourself attitude – considered (an aspect of) freedom – is nourished in the US.
This deeply ingrained and institutionalized contradiction American society harbors at its “core”, concerning the American idea of individual freedom (which includes the right to self-defense and the right to keep and bear arms) on the one hand, and the requirement for collective security (implying a more restricted/constrained version of the “basic” right to self-defense, and an effective monopoly on violence by the state) on the other hand, generates tensions, that at a certain point trigger individuals to “perform” their right to self-defense (at least their individual interpretation of this right; offenders always argue that the mass shootings they committed must be understood as a form of “self-defense” of rightful retribution). This basic contradiction in American society can be understood as a conflict between forces that produces tensions; these tensions power the mass shooting system and its dynamics. This competition between forces has produced a faultline, that (potentially) threatens the cohesion and stability of American society.
Mass shootings have (for above mentioned reasons) a contradictory effect on American society: On the one hand, it is a release of tensions, but on the other hand, these mass shootings contribute to additional tension-production. It is simultaneous a cause and an effect.
The fact that mass shooting dynamics intensify – as this analysis shows – means that related political issues, will become more central to the political process. The self-reinforcing nature of mass shooting dynamics drives the evolution – determines (at least to a degree) – the evolutionary path – of US-society. This self-reinforcing and potentially self-destructive behavior can – at least in some respects – be compared with an auto-immune response of American society against itself.
The American response to mass shootings is also always ambivalent: A mass outcry of indignation follows, but there is never any meaningful legal follow-up to prevent and contain mass shootings. The response is highly ritualistic, but due to the absence of any (effective) action – the right to self-defense is in fact (re)confirmed. This deeply ingrained contradiction in American society and its political system will doubtlessly be further politically exploited.
(i) Containment of violence.
In the introduction of this paper, I explained the need for tension regulation in social systems to maintain a certain coherence, including in the international System and societies. From a system-perspective, state-structures have the function of regulators.
The international System lacks an overarching regulator (it is anarchistic), and consequently regulation is accomplished through emergent (self-organized) processes and dynamics. The international System and states that make up this system are – so to speak – at the mercy of the “uncensored” laws of thermodynamics and related principles that apply. As I explained, a dissipative structure regulates the international System. In the international System, “orders” (its organization, like the United Nations order, and its predecessors) are periodically “upgraded” through systemic KA/war. In (most) societies upgrades are accomplished through a political process; in case of democracies through voting. In (most) societies, the laws of thermodynamics are tamed (a better word is accommodated), at least to a certain degree.
This analysis shows that in American society, violence (mass shootings) are still an integral part of its societal dynamics; not all violence is effectively contained and harnessed in American society.
As is the case in the international System, tensions in American society drive kinetic activity (war in the international system, mass shootings in American society). Both dynamics are self-organized and produce cyclic-behavior and other patterns. These patterns provide us with valuable insights in the underlying mechanisms that produce these destructive dynamics, and with practical clues to control and prevent them.
The initial results of this research are hopeful. But I remind the reader: This is (still) exploratory research, and the findings discussed in this paper are only preliminary conclusions; more research is required.
(1) Follman M, Aronsen G, Pan D (2019) Mother Jones’ Investigation, US Mass Shootings, 1982-2019, Mother Jones, https://www.motherjones.com/politics/2012/12/mass-shootings-mother-jones-full-data/ Dataset extracted: 01/09/2019.
(2) Global terrorism Index, produced by the Institute for Economics & Peace, http://globalterrorismindex.org/.
(3) Kondepudi D, Prigogine I (2015) Modern Thermodynamics. From Heat Engines to Dissipative Structures, Wiley.
(4) Piepers I (2016) 2020 Warning, Patterns in war dynamics reveal disturbing developments, IP-Publishing, Amsterdam.
(5) Piepers I (2019) On the Thermodynamics of war and Social Evolution, IP-Publishing, Amsterdam.
(6) Prigogine I, Nicolis G (1971) Biological order, structure and instabilities, Quarterly Reviews of Biophysics 4, 2 & 3, pp. 107-148.
(7) Solé RV, Bascompte J (2006) Self-Organization in Complex Ecosystems, Princeton University Press, Princeton and Oxford.
(8) Springer, Encyclopedia of Mathematics, https://www.encyclopediaofmath.org/index.php/Routes_to_chaos, extracted 03/09/19.
(9) Sterman J (2000) Business Dynamics. Systems Thinking and Modeling for a complex World, Irwin McGraw-Hill.
 The amount of presidential campaign funding that is spent over time, is probably a good measure for the (development of the) intensity of campaign battles.
 With System (system with capital “S”), I refer to what is normally understood as the international System; for further explanation see (5).