On Empathy Theory

On the Thesis:

 

 

 

The Publishers Abctales don't usually publish books in mathematics research. My essay was written with the idea of explaining in layman's terms what the work is about and provides general background. It is prose and the equations now and then are not really meant to be understood. The essay will definitely help any person with a basic training in the natural sciences to get some grasp of the theory and its significance as well as of possible real-world uses.

 

 

It is necessary to attempt to clarify the title of my thesis. Mathematical terms are used and the words therefore necessarily each have a clearly defined and distinct meaning; however as a metaphor in an intuitive manner it is pleasingly descriptive and appeals very strongly to the imagination, and in an appropriate manner, not only in a purely human sense but also in a religious context or in philosophy.

 

The metaphor in a typical physical model is quite accurate and it clearly conveys the idea of the evolution in time of the state of a material body and its boundary, in agreement and in harmony. A layman's understanding of the phrase might be descriptive also of the history, the way in which ideas & concepts evolved driven by applications and the mathematical maturing of the ideas since 1982.

 

Persons not familiar with academic works in mathematics always find the use of the royal “we” very unusual. In works in mathematics it is used freely and it has historically always been the custom. There are sound practical reasons for this.

 

 

The study of Partial Differential Equations has always been closely linked and bound to applications. It brings together many different disciplines of mathematics and of science. Two very simple real-world situations are sketched in the thesis the one is described here and the other just briefly.

 

For the heat equation our problem involves the conduction of heat in an isolated rod with a container at one endpoint containing gas, vigorously stirred, so that the temperature can be assumed to be uniform throughout the gas. The other endpoint is kept at a fixed temperature, and the initial distribution (spreading) of temperature is prescribed for Time = 0 . The question is to find the temperature distribution along the rod at any given time t. The boundary value at the one endpoint is known, it is fixed. At other endpoint the boundary value ( the temperature of the gas at Time = t ) is not known beforehand, it is free, it depends on what is happening inside the rod itself and is called a Dynamic Boundary Condition.

 

Another example but involving the wave equation now, is a lengthwise vibrating stiff elastic rod with one endpoint fixed i.e. held still, while a small (free) mass is attached to the other. The end with the small mass on it has a DBC.

 

The standard form for the heat equation is u_xx ( x, t ) = - K u_t ( x, t ) and in turn

v_tt ( x, t ) + c^2 v_xx ( x, t ) = 0 is the basic form of the wave equation.

 

Many realistic applications applications in science and engineering are modeled in terms of PDEs and amongst many others, problems such as in these examples are appropriately modeled with DBCs. Mathematical equations involving quantities that are in flux and are changing continuously as a function of more than one variable, without prior prescribed boundary values.

 

Such a time-evolution problem where the solution is to be found in terms of the variable time normally is modeled as a system of equations with a PDE and boundary conditions. This often turns out as an Abstract Cauchy Problem.

 

 

The academic work is primarily a study in advanced mathematics and the methods and technicalities of proofs are not within reach of most people. However as is often the case in modern mathematics the results should still be able to be understood and applied meaningfully by non-specialist scientists. We hope that one will be able to achieve this.

 

A simple operator equation giving the empathy relation, a dual time-evolution in two distinct vector spaces, was formally introduced by Niko Sauer (1997). It is given by S(t + s) = S(t ) E(s ), where the S(t ) is the family of “solutions”. For each given time value t , S (t ) maps (at that exact moment) from the space X into itself, X ; whereas the family E(t ) maps the space Y into X. Empathy is an extension of the concept of a semi-group which is the simpler one, a semi-group involves only one vector space so in effect X and Y coincide. In the same way a semi-group describes how a system evolves, flows as a function of time.

 

Semi-groups have been exhaustively studied in mathematical analysis since the middle 1900s. As a possible analogy in the pair < S, E >; the family of mappings { S( t ) } can be thought of as a “video film” where each individual value, each instant in time produces one frame, a frozen picture on the space X, or the “screen”. Exactly the same idea applies to { E( t ) } but here giving a different movie, which though plays in precise synchronization. The S family could then be thought of as the visible one, the physical observations.

 

The operator S(t ) would then for a given t, determine the complete state of the system (in an appropriate mathematical framework), and effects the observations, i.e. the “solution” in X at the moment t, and as t increases our solution (our “picture”) is in flux and smoothly flowing with time. Such a situation would in practice usually be determined (described) mathematically by a partial differential equation and an initial state ( a starting “frame”, i.e. at t = 0 ) in the form of an Abstract Cauchy Problem.

 

The mathematical equation describing the integrated empathy relation (2004) is more sophisticated and similarly is an extension of Wolfgang Arendt's notion of an integrated semi-group (1987). The fact that the Laplace transform can be extended to a vector-valued setting (H&P) simplifies the calculations immensely, and so facilitates investigations.

 

Each empathy has its unique corresponding integrated empathy, but it is not true going the other way round. The empathy condition is mathematically the stricter, the stronger more constrained of the two relations, so that technically our integrated empathy equation is a relaxation of that of the empathy, and this integrated form in turn applies to a weaker, “integrated” form of the given Cauchy problem. However on the other hand the setting (vector spaces) for integrated empathies is much more general.

 

It is the price to be paid in order to extend the characterization results of Sauer (1997), to a general setting, i.e. all Banach spaces. The need to extend the theory and introduce an “integrated” form of empathy is also exactly this – many real-world models are not properly modeled in those original “more restricted” spaces.

 

Closer now to philosophy, we have a new explicit causality where we may have “cause” and “effect” living in different worlds, where only effect is in the one experienced or observed, whereas the real, actual, deeper causes are unknown and exists in another, a higher world. The space X would be the the world of effect, and Y would be the world of cause, our observed reality.

 

It could be interesting in theology to speculate along these lines too.

 

 

The reports from the internal examiner Prof Sauer and the externals, Prof Banasiak of the University of KwaZulu Natal (SA) and especially that of Prof Neubrander (USA) raised deeper questions regarding some difficult aspects as far as the conceptual mathematics goes. To publish a monograph would be an ideal opportunity to clarify these and make necessary corrections where there might still be inaccuracies.

 

One specific mathematical technicality, the question of a separate (seemingly redundant) boundedness assumption on the Laplace Transforms (resolvents then) is controversial and of some importance, (ultimately the reason is the “unusual” definition of the Laplace-transform as a Lebesgue-integral in our study but the ideas turn out to be subtle).

 

 

The following are suggestions for possible (new) additional material especially in mind to make the work more widely accessible to engineers and scientists. It is probably also a good idea to make the book of some value to laymen.

 

One would like to give more, and more diverse examples for model applications worked in more detail and provide clear explanations. These must be made much more generally accessible. Extra (colour) illustrations, diagrams and graphs can be added easily it will help and it would compliment the appearance of the manuscript. More discussion on possibilities for further research also should be of value.

 

In fairness other researchers' ideas and theories in abstract Cauchy problems deserve more attention. I.e. a discussion on some alternative approaches giving comparisons, and weaknesses and strengths of some individual theories. Indeed some practical remarks of Prof Banasiak must be addressed, he insisted that this research should be put in a proper context in the literature. There have been many such investigations and he has also made significant original research contributions to the study of ACPs. Also, he recommended a more formal style of presentation for the mathematics.

 

We believe this work is of considerable originality for to start with an approach with two distinct spaces, and operators (mappings) involving both, and then correspondingly a double family of evolution operators, definitely was new. Some new prospects for research have opened there is a lot of scope. A purely mathematical study by itself is already a worthwhile but challenging pursuit.

 

It is hoped that the study of empathy theory will benefit the physical and other sciences in future and in a very concrete manner. There is potential in such real-world problems our research has in fact always been inherently motivated by applications. It is encouraging also to find that the word “empathy” is used these days in the popular science press e.g. Scientific American magazine and in an appropriate context.

 

And yes, wouldn't it be a great idea to make the book more entertaining!!

 

 

“These principles have given me a way of explaining naturally the union or rather mutual

 

 

The Monadology and other Philosophical Writings (1714), trans. Robert Latta (1898), p.262

 

 

A few references to the print are mentioned now, of present interest in particular are two doctoral theses that were submitted at the University of Pretoria. Many examples especially in classical mechanics involving applications and their modeling with dynamic boundary values were studied by WJ Rossouw “Behoudwet formulerings vir Randvoorwaardes” (1983), (“Conservation law formulations of Boundary conditions”); and of course “The Theory of Integrated Empathies”, TJ Brown (2005).

 

Many relevant textbooks include a standard reference, (1958) Hille & Phillips “Functional Analysis and Semi-groups”; and a more recent monograph of Arendt, Batty, Hieber & Neubrander, “Vector valued Laplace Transforms and Cauchy Problems” (2001). For a general mathematics background a standard university textbook for example “A first Course in Partial Differential Equations” (1965) by HF Weinberger may be consulted.

 

Research articles by Niko Sauer & collaborators (UP) follow the historical development of the concepts of empathy theory. Notably are “Linear evolution equations in two Banach spaces” (1982); “Empathy Theory and the Laplace Transform” (1997); and “Evolution Operators in Empathy with a Semi-group” (1985), the last in collaboration with Joy Singleton; and finally “Double families of Integrated Evolution Operators” co-authored by Thomas Brown (2004).

 

For an idea of the diversity of approaches to the abstract Cauchy problem see the monograph Favini & Yagi “Degenerate Differential equations” (1999); and Banasiak & Singh, “B-bounded Semi-groups and C-existence families (2002).

 

Prof Sauer also wrote three excellent textbooks on the partial differential equations of mathematical physics, ranging from an undergraduate textbook to a book in advanced abstract analysis, all are available from the University of Pretoria electronically in pdf free of cost. He has also been preparing a research monograph “Dynamic Boundary Conditions” which is directly related to this study.

 

Of more general interest in the field of the economic sciences is the seminal article of Black & Scholes “The pricing of Options and Corporate Liabilities” (1973). The modeling introduced a famous partial differential equation that is still a cornerstone in the mathematical analysis of financial markets.

 

Finally and maybe as a curiosity the works in philosophy by the great German scholar Wolfgang Leibniz (1664 – 1716) are available in Penguin Paperback in an English translation “Philosophical Writings” from the original French.