Veröffentlichungen in wissenschaftlichen Zeitschriften

First we discuss the hierarchy of of possibilities for defining a Gibbs process either by a grandcanonial specification constructed from a point process with independent increments and a potential, a local energy or conditional energy or by way of the Papangelou kernel. Secondly we discuss the question of whether a given point process can be described as a Gibbs process and if so in which sense. As an answer to this question we give necessary and sufficient conditions for this process to be described by a Papangelou kernel, a conditional energy, a local energy or potential.

Point processes P are probability measures on the set N of all integer valued Radon measures ~ on a certain phase space T which for most of the applications is ad (lattice models) or mD (con tinuous models). The name "point process" comes from the fact., that an integer valued Radon measure can be identified with a corresponding configuration of pOints in the phase space. The concept of Gibbsian processes as a special class of point processes arose in statistical mechanics to describe systems of interacting particles. The interaction is usually defined 1n terms of specifications, conditional energies, local energy, or po tentials. A survey of the various definitions of Gibbsian processes one can find in Glötzl [4J. It turns out that the conditions under which a general point pro cess can be described as Gibbsian pOint process are very weak.

In [4] WEISS showed that POISSON Processes P,, and PA* with a-finite mean measures I, and I, are singular if and only if P,,/dt, and P,,l@lo are singular.

In Theorem 1 we give a new characterisation of point processes with property S which makes it possible to show that the result of WEISS is true not only for

POISSON processes but also for the much larger class of point processes with property Z, which include as the most important cam the class of GIBBS processes [i], [3]. (For the property L' see also [2], [3].)

The concept of Gibbsian point processes is mainly stimulated by applications in statistical physics (for a survey see PRESTON [9]). Recently MATTHES, WARMUTH and MECKE [5] showed the connections between the work of PAPANGELOU [8] and KALLENBERG [3] on the conditional intensity of a point process and the work of GEORGII [l], NGUYEN and ZESSIN [6] and PRESTON [9] on the local characterisation of GIBBS processes.

In dieser Arbeit behandeln wir das Problem, unter welchen Bedingungen ein Punktprozess mit Hilfe einer Energiefunktion bzw. eines Wechselwirkungspotentials als Gibbsprozess beschrieben werden kann. Wie bei vielen Fragen der statistischen Mechanik wurde dieses Problem zuerst fur Gitterprozesse (Punktprozesse auf Zd) behandelt (vgl. SULLIVAN [ lo], KOZLOV [ci]). An diese Arbeiten anschlieBend hat KOZLOV [7] dieses Problem auch fur Punktprozesse auf einem beliebigen Phasenraum behandelt und hinreichende Bedingungen dafür angegeben.

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For lattice models it is well known (LOGAN [3], GEORGII [l]) that canonical

GIBBSjan point processes with respect to some local energy E(z, p) are (under

suitable conditions) just the probability measures which are time reversible for

the particle jump rate c(x, y, p) satisfying the condition

c(x, y, p) . ~-~@,{3=c(g, 2, p) * e-B(~~~)

We generalize this result to general phase spaces including, in pwticular, models

with phase space Rd and 2". The method we use is similar to that of [r]. To

make this paper selfcontained we shall introduce once again our basic notations.

Für einen Punktprozess P wird gezeigt, dass die Bedingung SIGMA ("nicht Vakuum") zusammen mit der Bedingung, dass P absolut stetig in Bezug zu einem Prozess Q notwendig und hinreichend dafür ist, dass P als Gibbs`scher Prozess interpretiert werden kann, dessen Wechselwirkungen durch eine bedingte Energie definiert sind.

xxx

For lattice models it is well known (LOGAN [3], GEORGII [l]) that canonical

GIBBSjan point processes with respect to some local energy E(z, p) are (under

suitable conditions) just the probability measures which are time reversible for

the particle jump rate c(x, y, p) satisfying the condition

c(x, y, p) . ~-~@,{3=c(g, 2, p) * e-B(~~~)

We generalize this result to general phase spaces including, in particular, models

with phase space Rd and Z". The method we use is similar to that of [2].

We deal with the problem of predicting some hours ahead the concentration of a certain pollutant(either dust or SO2), measured at one particular station of the urban air quality monitoring system of Linz.

Im Laufe der Geschichte ist es immer wieder vorgekommen, daß gesellschaftliche Katastrophen unterschiedlichen Ausmaßes durch ein Versagen der Wirtschaftsmechanismen hervorgerufen worden sind. Erinnert sei dabei beispielsweise an die römische Proskription 43 v. Chr., die Französische Revolution 1789 und die Weltwirtschaftskrise 1929. Die Vorgänge der letzten Vergangenheit in Japan, Ostasien und zuletzt in Rußland weisen auf die Aktualität dieses Themas hin. Dieses weitgehend unabhängig von der Art der Wirtschaftssysteme periodisch immer wieder auftretende Phänomen gleicht einem immer wiederkehrenden Wechselfieber.

Die herkömmliche Ökonomie erklärt dieses Phänomen in der Regel mit menschlichem Versagen bzw. Politikversagen oder auch mit einem zyklischen Auftreten von Innovationen. Wesentlichste These dieser Arbeit ist, daß es sich dabei nicht immer um Politikversagen oder um menschliches Versagen sondern sehr oft um ein Systemversagen handelt, das von der herkömmlichen Ökonomie tabuisiert wird. Von Politikversagen kann nur insofern gesprochen werden, als die Politik (noch?) nicht bereit ist, ernsthaft über Systemänderungen zu diskutieren oder diese Diskussion von der Wissenschaft einzufordern.

Geschäftsbanken können kein selbstgeschöpftes Geld kaufkraftwirksam verleihen. Sie können nur Geld, das sie von der Notenbank oder von Einzahlern bekommen haben, kaufkraftwirksam weiter verleihen. Sie können Geld zum kaufkraftwirksamen Verleihen also nur bekommen 

• durch Verkauf von Gold, Devisen oder Wertpapieren an die Notenbank (dabei wird Notenbankgeld geschöpft); 

• auf Kredit von der Notenbank gegen Hinterlegung von Wertpapieren und Bezahlung von Zinsen an die Notenbank (dabei wird Notenbankgeld geschöpft);

• durch Geldeinzahlungen von Kunden, denen sie in der Regel dafür Zinsen zahlen müssen (wenig wenn der Kunde sein Geld auf ein Girokonto einzahlt, mehr wenn ein Kunde sein Geld auf ein Sparbuch einzahlt). Dabei wird kein Notenbankgeld geschöpft. Es entstehen dabei Guthaben, die aus verschiedenen Gründen in der Literatur oft zur Geldmenge gezählt werden;

• durch Einsatz von Eigenkapital. 

Warum diese im Bankwesen unbestrittene Tatsache in verschiedenen Kreisen immer wieder in Zweifel gezogen und damit unnötig Verwirrung gestiftet wird, hat folgende Ursachen:

• Die Darstellungen, die sich in Büchern im Zusammenhang mit der Ablösung von Gold als Geld durch Papiergeld finden;

• die unpräzise Verwendung der verschiedenen Begriffe von Geld;

• durch die Verwendung des Begriffes „der aktiven Giralgeldschöpfung“ (im Gegensatz zum Begriff der passiven Giralgeldschöpfung) und der damit im Zusammenhang stehenden englischen Buchungsmethode, die heute bei der Vergabe von Krediten verwendet wird;

• Mißverständnisse beim Vorgang der „multiplen Geldschöpfung.

For more than 100 years economists have tried to describe economics in analogy to physics, more precisely to classical Newtonian mechanics. The development of the Neoclassical General Equilibrium Theory has to be understood as the result of these efforts. But there are many reasons why General Equilibrium Theory is inadequate: 1. No true dynamics. 2. The assumption of the existence of utility functions and the possibility to aggregate them to one “Master” utility function. 3. The impossibility to describe situations as in “Prisoners Dilemma”, where individual optimization does not lead to a collective optimum. This paper aims at overcoming these problems. It illustrates how not only equilibria of economic systems, but also the general dynamics of these systems can be described in close analogy to classical mechanics. 

To this end, this paper makes the case for an approach based on the concept of constrained dynamics, analyzing the economy from the perspective of “economic forces” and “economic power” based on the concept of physical forces and the reciprocal value of mass. Realizing that accounting identities constitute constraints in the economy, the concept of constrained dynamics which is part of the standard models of classical mechanics, can be applied to economics. Therefore it is reasonable to denote such models as Newtonian Constraint Dynamic Models (NCD-Models) 

Such a framework allows understanding both Keynesian and neoclassical models as special cases of NCD-Models in which the power relationships with respect to certain variables are one-sided. As mixed power relationships occur more frequently in reality than purely one-sided power constellations, NCD-models are better suited to describe the economy than standard Keynesian or Neoclassic models. 

A NCD-model can be understood as “Continuous Time”, “Stock Flow Consistent”, “Agent Based Model”, where the behavior of the agents is described with a general differential equation for every agent. In the special case where the differential equations can be described with utility functions, the behavior of every agent can be understood as an individual optimization strategy. He thus seeks to maximize his utility. However, while the core assumption of neoclassical models is that due to the “invisible hand” such egoistic individual behavior leads to an optimal result for all agents, reality is often defined by “Prisoners Dilemma” situations, in which individual optimization leads to the worst outcome for all. One advantage of NCD-models over standard models is that they are able to describe also such situations, where an individual optimization strategy does not lead to an optimum result for all agents. This will be illustrated in a simple example. 

In conclusion, the big merit and effort of Newton was, to formalize the right terms (physical force, inertial mass, change of velocity) and to set them into the right relation. Analogously the appropriate terms of economics are force, economic power and change of flow variables. NCD-Models allow formalizing them and setting them into the right relation to each other.

General Constrained Dynamic models (GCD –models) in economics are inspired by classical mechanics with constraints. Most macroeconomic models can be understood as special cases of GCD – models. Moreover, in this paper it will be shown that not only macroeconomic models but also game theoretic models are strongly related to GCD – models. 

GCD models are characterized by a system of differential equations in continuous time while most game theoretical models are set up in discrete time. Therefore it is necessary to build a bridge from game theoretical models denominated in discrete time to game theoretical models using continuous time. This bridge is illustrated in the following using the example of a continuous time, continuous decision space prisoner’s dilemma. Furthermore, it is shown that the differential equations which determine other continuous game theoretic models can be understood to a certain extent as special cases of the GCD – differential equations. Well known types of continuous game theoretic models include for instance “Evolutionary Game Theory” with the replicator equation, “Adaptive Dynamics” with the canonical equation, which is nothing else than a replicator – mutator equation, and the so called “Differential Games”, which are strongly related to optimal control theory with two controls and two different objectives (goals).

Most of the GCD – models are characterised by 3 key feature:

• mutual influence,

• Power-factors

• Constraints

Nowak (2006b) and Taylor & Nowak (2007) show that there are five mechanisms which, under certain conditions, can lead to the evolution of cooperation in an iterated prisoner’s dilemma. Inspired by this, we apply the 3 key features of GCD –models to the standard prisoner’s dilemma in discrete time which yields 3 additional mechanisms which enable the evolution of cooperation. 

The assumption or axiom of the free market economy is that an individual optimisation strategy will lead to an overall optimum by virtue of Adam Smith’s invisible hand. Without additional conditions this assumption alone is fundamentally wrong. As in prisoner’s dilemma also in economics cooperation is essential to get an overall optimum. The big question of political economy is to analyse which additional measures could guarantee that the individual optimisation strategy characterising a free market economy leads to cooperation as precondition to get an overall optimum. 

From this point of view the different economic theories could be characterised in terms of which measures they assume to be sufficient to guarantee an overall optimum without abandoning the principle of individual optimisation.

Economic equilibrium models have been inspired by analogies to stationary states in classical mechanics. To extend these mathematical analogies from constrained optimization to constrained dynamics, we formalize economic (constraint) forces and economic power in analogy to physical (constraint) forces and the reciprocal value of mass. Agents employ forces to change economic variables according to their desire and their power to assert their interest. These ex-ante forces are completed by constraint forces from unanticipated system constraints to yield the ex-post dynamics. The differential-algebraic equation framework seeks to overcome some restrictions inherent to the optimization approach and to provide an out-of-equilibrium foundation for general equilibrium models. We transform a static Edgeworth box exchange model into a dynamic model with procedural rationality (gradient climbing) and slow price adaptation, and discuss advantages, caveats, and possible extensions of the modeling framework.

In monetary Stock-Flow Consistent (SFC) models, accounting identities reduce the number of behavioral functions to avoid an overdetermined system of equations. We relax this restriction using a differential-algebraic equation framework of constrained dynamics. Agents exert forces on the variables according to their desire, for instance to gradually improve their utility. The parameter ‘economic power’ corresponds to their ability to assert their interest. In analogy to Lagrangian mechanics, system constraints generate additional constraint forces that lead to unintended dynamics. We exemplify the procedure using a simple SFC model and reveal its implicit assumptions about power relations and agents’ preferences.

Abstract

In economics balance identities as e.g. C+K'-Y(L,K) = 0 must always apply. Therefore, they are called constraints. This means that variables C,K,L cannot change independently of each other. In the general equilibrium theory (GE) the solution for the equilibrium is obtained as an optimisation under the above or similar constraints. The standard method for modelling dynamics in macroeconomics is DSGE. Dynamics in DSGE models result from the maximisation of an intertemporal utility function that results in the Euler-Lagrange equations. The Euler-Lagrange equations are differential equations that determine the dynamics of the system. In Glötzl, Glötzl und Richters (2019) we have introduced an alternative method to model dynamics, which is a natural extension of GE theory. It is based on the standard method in physics for modelling dynamics under constraints. We therefore call models of this type "General Constrained Dynamic (GCD)" models. In this paper we apply this method to macroeconomic models of increasing complexity. The target of this labour is primarily to show the methodology of GCD models in principle and why and how it can be useful to analyse the macroeconomy with this method. Concrete economic statements play only a subordinate role. All calculations, even for GCD models of any complexity, can be easily performed with the open-source program GCDconfigurator. 

Abstract

In economics balance identities as e.g. C+K'-Y(L,K) = 0 must always apply. Therefore, they are called constraints. This means that variables C,K,L cannot change independently of each other. In General Equilibrium Theory (GE), the solution for equilibrium is obtained as optimisation under the above or similar constraints. The standard method for modelling dynamics in macroeconomics are Dynamic Stochastic General Equilibrium (DSGE) models. Dynamics in DSGE models result from the maximisation of an intertemporal utility function that results in the Euler-Lagrange equations. The Euler-Lagrange equations are differential equations that determine the dynamics of the system. In Glötzl, Glötzl und Richters (2019) we have introduced an alternative method to model dynamics, which is constitutes a natural extension of GE theory. It is based on the standard method for modelling dynamics under constraints in physics. We therefore call models of this type "General Constrained Dynamic (GCD)" models. GCD models can be seen as an alternative to DSGE models to model the dynamics of economic processes. DSGE models are used in particular to analyse economic shocks. For this reason, the aim of this article is to show how GCD models are formulated and how they can be used to model economic shocks such as demand, supply, and price shocks. Since the goal of this paper is to lay out the fundamental principles to the formulation of such GCD models, very simple macroeconomic models are used for illustrative purposes. All calculations can easily be carried out with the open-source program GCDconfigurator, which also allows for the integration of shocks.

Abstract

In economics balance identities as e.g. C+K'-Y(L,K)=0 must always apply. Therefore, they are called constraints. This means that variables C,K,L cannot change independently of each other. In general equilibrium theory (GE) the solution for the equilibrium is obtained as an optimisation under the above or similar constraints. The standard method for modelling dynamics in macroeconomics are Dynamic Stochastic General Equilibrium (DSGE) models. Dynamics in DSGE models result from the maximisation of an intertemporal utility function that results in the Euler-Lagrange equations. The Euler-Lagrange equations are differential equations that determine the dynamics of the system. In Glötzl, Glötzl und Richters (2019) we have introduced an alternative method to model dynamics, which constitutes a natural extension of GE theory. This approach is based on the standard method for modelling dynamics under constraints in physics. We therefore call models of this type "General Constrained Dynamic (GCD)" models. In Glötzl (2022b) this modelling method is described for non-intertemporal utility functions in macroeconomics. Since intertemporal utility functions are, however, essential for many economic models, this paper sets out to extend the GCD modelling framework to intertemporal GCD models, referred to as IGCD models in the following. This paper sets out to define the principles of formulating IGCD models and show how IGCD can be understood as a generalisation and alternative to DSGE models. 

Abstract: 

This paper introduces a novel method to extend the Helmholtz Decomposition to

n-dimensional sufficiently smooth and fast decaying vector fields. The rotation is described

by a superposition of n(n+1)/2 rotations within the coordinate planes. The source potential

and the rotation potential are obtained by convolving the source and rotation densities

with the fundamental solutions of the Laplace equation. The rotation-free gradient of

the source potential and the divergence-free rotation of the rotation potential sum to the

original vector field. The approach relies on partial derivatives and Newton integrals and

allows for a simple application of this standard method to high-dimensional vector fields,

without using concepts from differential geometry and tensor calculus.

Keywords: 

Helmholtz Decomposition, Fundamental Theorem of Calculus, Curl Operator

Abstract:

We present a Helmholtz Decomposition for many n-dimensional, continuously

differentiable vector fields on unbounded domains that do not decay at infinity. Existing

methods are restricted to fields not growing faster than polynomially and require solving

n-dimensional volume integrals over unbounded domains. With our method only one-

dimensional integrals have to be solved to derive gradient and rotation potentials.

Analytical solutions are obtained for smooth vector fields f(x) whose components are

separable into a product of two functions: fk(x) = uk(xk) · vk(x,k), where uk(xk) depends

only on xk and vk(x,k) depends not on xk. Additionally, an integer λk must exist such that

the 2λk-th integral of one of the functions times the λk-th power of the Laplacian applied

to the other function yields a product that is a multiple of the original product. A similar

condition is well-known from repeated partial integration: If the shifting of derivatives

yields a multiple of the original integrand, the calculation can be terminated.

Also linear combinations of such vector fields can be decomposed. These conditions

include periodic and exponential functions, combinations of polynomials with arbitrary

integrable functions, their products and linear combinations, and examples such as Lorenz

or Rössler attractor.

Keywords: 

Helmholtz Decomposition, Fundamental Theorem of Vector Calculus, Gradi-

ent and Rotation Potentials, Unbounded Domains, Poisson Equation.

Abstract: 

The Helmholtz decomposition splits a sufficiently smooth vector field into a gradient field and a divergence-free rotation field. Existing decomposition methods impose constraints on the behavior of vector fields at infinity and require solving convolution integrals over the entire coordinate space. To allow a Helmholtz decomposition in Rn, we replace the vector potential in R3 by the rotation potential, an n-dimensional, antisymmetric matrix-valued map describing n(n-1)=2 rotations within the coordinate planes. We provide three methods to derive the Helmholtz decomposition: (1) a numerical method for fields decaying at infinity by using an n-dimensional convolution integral, (2) closed-form solutions using line-integrals for several unboundedly growing fields including periodic and exponential functions, multivariate polynomials and their linear combinations, (3) an existence proof for all analytic vector fields. Examples include the Lorenz and Rössler attractor and the competitive Lotka–Volterra equations with n species.

Keywords: 

Partial Differential Equations, Helmholtz Decomposition, Fundamental Theorem of Calculus, Gradient Potential, Rotation Potential, Analytic Functions

Abstract: 

The general evolutionary theory can be seen as a comprehensive generalization and extension of Darwin's theory. The basic idea is to consider not only the evolution of genetic information - as Darwin did - but also the evolution of very general information. It shows that evolution is characterized by the fact that new types of information have developed in leaps and bounds, each with new storage technologies, new duplication technologies and new processing technologies. This unified concept of evolution makes it possible, among other things, to 1) achieve a unified view of biological and cultural evolution; 2) find a natural periodization of the evolution from the formation of the earth until today; and 3) understand the exponential acceleration of evolution through the emergence of targeted variation mechanisms.

1. So

Abstract:

 The general evolutionary theory can be seen as a comprehensive generalization and extension of Darwin's theory. The basic idea is to consider not only the evolution of genetic information - as Darwin did - but also the evolution of very general information. It shows that evolution is characterized by the fact that new types of information have developed in leaps and bounds, each with new storage technologies, new duplication technologies and new processing technologies. This unified concept of evolution makes it possible, among other things, to 1) achieve a unified view of biological and cultural evolution; 2) find a natural periodization of the evolution from the formation of the earth until today; and 3) understand the exponential acceleration of evolution through the emergence of targeted variation mechanisms.