Weinberg's 1972 work, in his description, had two purposes. The first was practical to bring together and assess the wealth of data provided over the previous decade while realizing that newer data would come in even as the book was being printed. He hoped the comprehensive picture would prepare the reader and himself to that new data as it emerged. The second was to produce a textbook about general relativity in which geometric ideas were not given a starring role for (...) (in his words) too great an emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics. (shrink)

Penrose's conformal cyclic cosmology describes the cosmos as a collection of successive universes, the so-called aeons. The beginning and ending of our universe are directly connected to two other, anterior and posterior, universes. Penrose considers but rules out a different interpretation of conformal cyclic cosmology: that the beginning of our universe is connected to its own end in a cosmic loop. The paper argues that the view, aeon monism, should be regarded as a natural interpretation of conformal cyclic (...) cosmology and discusses its implications for the concept of eternal return in light of the most popular metaphysics of time. (shrink)

In earlier writings, I argued that neither of the two major physical cosmologies of the 20th century support divine creation, so that atheism has nothing to fear from the explanations required by these cosmologies. Yet theists ranging from Augustine, Aquinas, Descartes, and Leibniz to Richard Swinburne and Philip Quinn have maintained that, at every instant anew, the existence of the world requires divine creation ex nihilo as its cause. Indeed, according to some such theists, for any given moment t. God's (...) volition that the-world-should-exist-at-t supposedly brings about its actual existence at t. In an effort to reestablish the current viability of this doctrine of perpetual divine conservation. Philip Quinn argued (1993) that it is entirely compatible with physical energy conservation in the Big Bang cosmology, as well as with the physics of the steady-state theories. But I now contend that instead, there is a logical incompatibility on both counts. Besides, the stated tenet of divine conservation has an additional defect: It speciously purchases plausibility by trading on the multiply disanalogous volitional explanations of human actions. (shrink)

This book is a sequel to Heaven and Earth in Ancient Greek Cosmology. With the help of many pictures, the reader is introduced into the way of thinking of ancient believers in a flat earth. The first part offers new interpretations of several Presocratic cosmologists and a critical discussion of Aristotle’s proofs that the earth is spherical. The second part explains and discusses the ancient Chinese system called gai tian. The last chapter shows that, inadvertently, ancient arguments and ideas (...) return in the curious modern flat earth cosmologies. (shrink)

Isaac Newton's Scientific Method examines Newton's argument for universal gravity and his application of it to resolve the problem of deciding between geocentric and heliocentric world systems by measuring masses of the sun and planets. William L. Harper suggests that Newton's inferences from phenomena realize an ideal of empirical success that is richer than prediction. Any theory that can achieve this rich sort of empirical success must not only be able to predict the phenomena it purports to explain, but also (...) have those phenomena accurately measure the parameters which explain them. Harper explores the ways in which Newton's method aims to turn theoretical questions into ones which can be answered empirically by measurement from phenomena, and to establish that propositions inferred from phenomena are provisionally accepted as guides to further research. This methodology, guided by its rich ideal of empirical success, supports a conception of scientific progress that does not require construing it as progress toward Laplace's ideal limit of a final theory of everything, and is not threatened by the classic argument against convergent realism. Newton's method endorses the radical theoretical transformation from his theory to Einstein's. Harper argues that it is strikingly realized in the development and application of testing frameworks for relativistic theories of gravity, and very much at work in cosmology today. (shrink)

This book examines Aristotle's critical reaction to the mathematical cosmology of Plato's Academy, and traces the aporetic method by which he developed his own ...

Through criticism and analysis of ancient traditions, Kahn reconstructs the pattern of Anaximander’s thought using historical methods akin to the reconstructive techniques of comparative linguists.

Process and Reality, Whitehead’s magnum opus, is one of the major philosophical works of the modern world, and an extensive body of secondary literature has developed around it. Yet surely no significant philosophical book has appeared in the last two centuries in nearly so deplorable a condition as has this one, with its many hundreds of errors and with over three hundred discrepancies between the American and the English editions, which appeared in different formats with divergent paginations. The work itself (...) is highly technical and far from easy to understand, and in many passages the errors in those editions were such as to compound the difficulties. The need for a corrected edition has been keenly felt for many decades. (shrink)

Of all the traditional sciences it is alchemy based as it is in metallurgy that is directly concerned with the coming of the industrial order. In alchemical terms modern man lives in the Ferric Age and his state is best analogized to the properties of the metal iron, hard, cold, unbending but quick to succumb to corrosion and rust. The great ancient wisdom traditions of the world all anticipated this present age for it was already implicit in the technological and (...) other changes that brought on the dawn of history. These ancient traditions - dismissed as childish superstitions by the scientist contain ideas essential to the self-understanding that contemporary man so desperately needs. This is the central contention of the writings brought together in Primordial Alchemy & Modern Religion. Drawing upon many unexpected sources especially Plato and the ancient Greeks these wide-ranging, generous essays provide timely reiterations of ancient alchemical traditions and trace some pivotal themes that have continued in the theology and symbolism of the modern religions. This work restores some important keys for the recovery of a lost heritage of primordial wisdom, offering fresh perspectives on aspects of the Western and Islamic alchemical traditions in particular, and at the same time exploring the most distant roots of the modern impasse. (shrink)

This volume is the first systematic presentation of the work of Albert Einstein, comprising fourteen essays by leading historians and philosophers of science that introduce readers to his work. Following an introduction that places Einstein's work in the context of his life and times, the book opens with essays on the papers of Einstein's 'miracle year', 1905, covering Brownian motion, light quanta, and special relativity, as well as his contributions to early quantum theory and the opposition to his light quantum (...) hypothesis. Further essays relate Einstein's path to the general theory of relativity (1915) and the beginnings of two fields it spawned, relativistic cosmology and gravitational waves. Essays on Einstein's later years examine his unified field theory program and his critique of quantum mechanics. The closing essays explore the relation between Einstein's work and twentieth-century philosophy, as well as his political writings. (shrink)

I review some theoretical ideas in cosmology different from the standard “Big Bang”: the quasi-steady state model, the plasma cosmology model, non-cosmological redshifts, alternatives to non-baryonic dark matter and/or dark energy, and others. Cosmologists do not usually work within the framework of alternative cosmologies because they feel that these are not at present as competitive as the standard model. Certainly, they are not so developed, and they are not so developed because cosmologists do not work on them. It (...) is a vicious circle. The fact that most cosmologists do not pay them any attention and only dedicate their research time to the standard model is to a great extent due to a sociological phenomenon . We might well wonder whether cosmology, our knowledge of the Universe as a whole, is a science like other fields of physics or a predominant ideology. (shrink)

This study is based on the many cosmological problems in Islam as aspects of thought that receive serious attention. In fact, there are also many polemics of thought that occur amongst Muslim scholars, which can be divided into two main groups: traditionalists and rationalists. The traditionalists, represented by Al-Ghazali and the Ash'ariyah theologians, put forward their cosmological thinking on the principle of God's absolute will, while the rationalists, especially those represented by Avicenna, proposed their cosmological thinking based on the theory (...) of emanation from Plotinus in terms of its creation and the concept of a geocentric Ptolameus in terms of its structure. In this conflict of thought between the two groups, Averroes proposed a cosmological thought different from the two. This study seeks to elaborate on the thought of Ibn Rushd's cosmology which is different from that of Ibn Sina and Al-Ghazali.CONTRIBUTION: This research provides a clear understanding of the cosmological thoughts put forward by earlier Muslim thinkers. In particular, it wants to bridge the differences regarding the concept of cosmology as put forward by Ibn Sina and Al-Ghazali and how Ibn Rushd bridges the two. (shrink)

The concept of the vacuum in quantum field theory is a subtle one. Vacuum states have a rich and complex set of properties that produce distinctive, though usually exceedingly small, physical effects. Quantum vacuum noise is familiar in optical and electronic devices, but in this paper I wish to consider extending the discussion to systems in which gravitation, or large accelerations, are important. This leads to the prediction of vacuum friction: The quantum vacuum can act in a manner reminiscent of (...) a viscous fluid. One result is that rapidly changing gravitational fields can create particles from the vacuum, and in turn the backreaction on the gravitational dynamics operates like a damping force. I consider such effects in early universe cosmology and the theory of quantum black holes, including the possibility that the large-scale structure of the universe might be produced by quantum vacuum noise in an early inflationary phase. I also discuss the curious phenomenon that an observer who accelerates through a quantum vacuum perceives a bath of thermal radiation closely analogous to Hawking radiation from black holes, even though an inertial observer registers no particles. The effects predicted raise very deep and unresolved issues about the nature of quantum particles, the role of the observer, and the relationship between the quantum vacuum and the concepts of information and entropy. © 2001 American Institute of Physics. (shrink)

A number of years ago we argued that Hartshorne’s psychicalism and his doctrine of divine memory are incompatible with contemporary big bang cosmology. Theodore Walker has responded to our objection by arguing that our understanding of psychicalism is flawed and that Hartshorne’s metaphysics has the resources for accommodating what the big bang theory says about the origin and fate of the universe. In the present article we attempt to show that Walker’s defense of Hartshorne fails.

I explain the difficulty of making various concepts of and relating to probability precise, rigorous and physically significant when attempting to apply them in reasoning about objects living in infinite-dimensional spaces, working through many examples from cosmology. I focus on the relation of topological to measure-theoretic notions of and relating to probability, how they diverge in unpleasant ways in the infinite-dimensional case, and are even difficult to work with on their own. Even in cases where an appropriate family of (...) spacetimes is finite-dimensional, and so admits a measure of the relevant sort, however, it is always the case that the family is not a compact topological space, and so does not admit a physically significant, well behaved probability measure. Problems of a different but still deeply troubling sort plague arguments about likelihood in that context, which I also discuss. I conclude that most standard forms of argument used in cosmology to estimate the likelihood of the occurrence of various properties or behaviors of spacetimes have serious mathematical, physical and conceptual problems. (shrink)

This article seeks to interpret the phrase Πάτερ, ημων ο εν τοῖς οὐρανοῖς in the invocation of the Lord’s Prayer in the light of Ewe-Ghanaian cosmology. The article employs a combination of the historical-critical and indigenous mother tongue biblical hermeneutical approaches to explore the implication of the invocation for Ewe-Ghanaian Christian spirituality today. The article firstly discusses the various theological and hermeneutical positions of the invocation in dialogue with Ewe-Ghanaian concept of God and the plurality of his dwelling place. (...) The article argues that Matthew’s use of οὐρανοῖς to suggest a plurality in God’s dwelling place resonates with Ewe cosmology, where it is believed that there are seven states of existence and that Mawugã, the Supreme Being, dwells in the first – the absolute state. Thus God exists in ‘seven heavens’ in Ewe cosmology, with the highest heaven being the eternal state of abode. On the other hand, the inclusive interpretation of the fatherhood of God in Ewe cosmology is discontinuous with the exclusive interpretation in the works of the church fathers such as Cyprian, Origen, Gregory of Nissa, Augustine and Clement of Alexandria. This hermeneutical position, the article observes, was responsible for the theological tensions that characterised the encounter between missionary Christianity and Ewe indigenous religion in the middle of the 19th century. However, the introduction of social services as evangelisation strategy, the legacy of the Ewe Bible and liturgy and the handing over of the Ewe church to the indigenous coworkers may have contributed to a large extent in ensuring religious tolerance among followers of the two religions. Today, Ewe-Ghanaian popular Christianity has shifted from its apologetic stance to a more liberal stance and employs indigenous religious and cultural categories in theologising.Contribution: Matthew’s rendition of the invocation of the Lord’s Prayer in the context of Ewe-Ghanaian cosmology is the focus of this article. The article forms part of the researcher’s contribution to the academic knowledge on the Lord’s Prayer and inspires the use of mother tongue biblical hermeneutics in the development of theological materials for the Ewe-Ghanaian Christian communities in Ghana and Togo. (shrink)

This contribution explores Gilles Personne de Roberval’s 1644 Aristarchi Samii de mundi systemate, partibus, & motibus eiusdem, libellus. I focus on the complex circumstances of publication, the intellectual context of the polemics of Copernicanism within the scientific community, as well as the natural philosophy of the treatise. Roberval’s strategy of publication provides a very sophisticated example of authorship in early modern natural philosophy. The strategy lies at the conflux of certain specific motivations. I contextualize these motivations by accounting for the (...) delicate debates around the motion of the Earth in mid-seventeenth century Europe, the institutional communities of the Collège Royal and Mersenne’s circle, and the disciplinary bounds of Roberval’s professional authority. Weighing in all the elements, I argue that Roberval’s publication is an interesting intellectual game, playing with the notion of ancient authority and humanistic recovery. By this somewhat libertine attitude, Roberval takes a stance in a complicated debate around heliocentrism and the status of hypothetical cosmology. Roberval’s position is supported by his natural philosophical speculations, which I situate in the contemporary debates of the community. Roberval borrows from many philosophers, but he aims at achieving a highly systematic speculative cosmology. One of the functions of this system is to confirm Copernicanism, while maintaining a very pessimistic attitude on the prospects of precise astronomical knowledge and, among others, prognostication issues. (shrink)

Precision Medicine has become a common label for data-intensive and patient-driven biomedical research. Its intended future is reflected in endeavours such as the Precision Medicine Initiative in the USA. This article addresses the question whether it is possible to discern a new ‘medical cosmology’ in Precision Medicine, a concept that was developed by Nicholas Jewson to describe comprehensive transformations involving various dimensions of biomedical knowledge and practice, such as vocabularies, the roles of patients and physicians and the conceptualisation of (...) disease. Subsequently, I will elaborate my assessment of the features of Precision Medicine with the help of Michel Foucault, by exploring how precision medicine involves a transformation along three axes: the axis of biomedical knowledge, of biomedical power and of the patient as a self. Patients are encouraged to become the managers of their own health status, while the medical domain is reframed as a data-sharing community, characterised by changing power relationships between providers and patients, producers and consumers. While the emerging Precision Medicine cosmology may surpass existing knowledge frameworks; it obscures previous traditions and reduces research-subjects to mere data. This in turn, means that the individual is both subjected to the neoliberal demand to share personal information, and at the same time has acquired the positive ‘right’ to become a member of the data-sharing community. The subject has to constantly negotiate the meaning of his or her data, which can either enable self-expression, or function as a commanding Superego. (shrink)

Informed by Gloria Anzaldúa's and José Carlos Mariátegui's work, as well as by Andean cosmology, Omar Rivera turns to Inka stonework and architecture as an example of a "Cosmological Aesthetics." He articulates ways of sensing, feeling and remembering that are attuned to an aesthetic of water, earth and light. On this basis, Rivera brings forth a corporeal orientation that can be inhabited by the oppressed, one that withdraws from predominant modern/Western conceptions of the human. By providing an aesthetic analysis (...) of cosmological sensing, Rivera sets the stage for exploring physical dimensions of anti-colonial resistance, and furthers the Latinx and Latin American tradition of anti-colonial and liberatory philosophy. Seeingaesthetic involvements with the cosmos as a source for embodied modes of resistance, Rivera turns to the work of María Lugones and Enrique Dussel in order to make explicit the aesthetic dimensions of their work. Andean Aesthetics and Anticolonial Resistance creates a new dialogue between art historians, artists, and philosophers working on Latin American thought, phenomenology, and hermeneutics. It weaves together a Latin American philosophy that connects pre-Columbian cosmologies with contemporary thinkers. Rivera's original approach introduces us to the living, evolving and aesthetic alternatives to coloniality of power and of knowledge, overhauling current understandings of decolonial theory and opening the tradition in transformative ways. (shrink)

During the last decade new developments in theoretical and speculative cosmology have reopened the old discussion of cosmology’s scientific status and the more general question of the demarcation between science and non-science. The multiverse hypothesis, in particular, is central to this discussion and controversial because it seems to disagree with methodological and epistemic standards traditionally accepted in the physical sciences. But what are these standards and how sacrosanct are they? Does anthropic multiverse cosmology rest on evaluation criteria (...) that conflict with and go beyond those ordinarily accepted, so that it constitutes an “epistemic shift” in fundamental physics? The paper offers a brief characterization of the modern multiverse and also refers to a few earlier attempts to introduce epistemic shifts in the science of the universe. It further discusses the several meanings of testability, addresses the question of falsifiability as a sine qua non for a theory being scientific, and briefly compares the situation in cosmology with the one in systematic biology. Multiverse theory is not generally falsifiable, which has led to proposals from some physicists to overrule not only Popperian standards but also other evaluation criteria of a philosophical nature. However, this is hardly possible and nor is it possible to get rid of explicit philosophical considerations in some other aspects of cosmological research, however advanced it becomes. (shrink)

Philosopher Alan Watts describes his experiences with consciousness-changing drugs and the levels of insight that they can facilitate, illuminating questions about the nature of existence and the existence of the sacred. Originally published in 1962; new edition includes article on psychedelics written for the California Law Review"--Provided by publisher.

Barbour, Hawking, Misner and others have argued that time cannot play an essential role in the formulation of a quantum theory of cosmology. Here we present three challenges to their arguments, taken from works and remarks by Kauffman, Markopoulou and Newman. These can be seen to be based on two principles: that every observable in a theory of cosmology should be measurable by some observer inside the universe, and all mathematical constructions necessary to the formulation of the theory (...) should be realizable in a finite time by a computer that fits inside the universe. We also briefly discuss how a cosmological theory could be formulated so it is in agreement with these principles. (shrink)

Introduction THE FACES OF GOD How many faces has God? Egyptologists have wrestled with the problem over many years ...

This book presents the history of how the universe at large became the object of scientific understanding. Starting with the ancient creation myths, it offers an integrated and comprehensive account of cosmology that covers all major events from Aristotle's Earth-centred cosmos to the recent discovery of the accelearting universe.

In the contemporary philosophy of mind and consciousness studies, views such as panpsychism or theories of universal consciousness, have enjoyed a recent renaissance of metaphysical speculations in Western philosophy. Its similarities with Eastern philosophical traditions went not unnoticed. However, the potential contribution that the evolutionary cosmology of the Indian poet, mystic and philosopher Sri Aurobindo can offer to these ontologies, remains largely unknown or unexplored. Here, consciousness, mind, life, matter and evolution are interpreted in an extended metaphysical framework, uniting (...) Western and Eastern traditions inside a modern philosophical system where materialism, panpsychism, cosmopsychism and philosophical idealism are not seen as irreconcilable exclusives, but different points of view that can be reconciled in a synthesis of knowledge, as envisioned from higher states of consciousness. This integral view of reality offers a conceptual platform that can complement and incorporate current theoretical approaches in modern philosophy, where their present internal inconsistencies do not emerge, and can be suggestive for scientists as well, especially in the fields of biology and physical sciences. (shrink)

Anne-Lise Rey’s account of Maupertuis’s physical theology challenges the efforts on the part of commentators to align his work with Newton, Leibniz, or critics of Leibniz by examining the innovative epistemological principles that inform his Essay on Cosmology (1750). Through an analysis of his proofs of the existence of God in this work, she investigates Maupertuis’s view on how we can obtain certain knowledge of nature and the place he affords to experimentation and contingency in this regard. Rey claims (...) that Maupertuis, despite his hostility toward Descartes, Leibniz, and Wolff, did not abolish the idea of a metaphysics of nature per se, but tried to strike a new balance between, on the one hand, the deduction of laws from the nature of bodies and, on the other hand, experimentation. More specifically, she argues that Maupertuis in the Essay on Cosmology affirms our epistemic incapacity to grasp the whole system of nature and, hence, argues that the probable truths we can obtain in physics by means of experiments are superior to the geometrical truths that Descartes and his followers aimed to establish in both physics and metaphysics. (shrink)

Quantum theory plays an increasingly significant role in contemporary early-universe cosmology, most notably in the inflationary origins of the fluctuation spectrum of the microwave background radiation. I consider the two main strategies for interpreting standard quantum mechanics in the light of cosmology. I argue that the conceptual difficulties of the approaches based around an irreducible role for measurement - already very severe - become intolerable in a cosmological context, whereas the approach based around Everett's original idea of treating (...) quantum systems as closed systems handles cosmological quantum theory satisfactorily. Contemporary cosmology, which indeed applies standard quantum theory without supplementation or modification, is thus committed - tacitly or explictly - to the Everett interpretation. (shrink)

A reprint of the Routledge edition of 1935.

This book deals with a topic that has been largely neglected by philosophers of science to date: the ability to refer and analyze in tandem. On the basis of a set of philosophical case studies involving both problems in number theory and issues concerning time and cosmology from the era of Galileo, Newton and Leibniz up through the present day, the author argues that scientific knowledge is a combination of accurate reference and analytical interpretation. In order to think well, (...) we must be able to refer successfully, so that we can show publicly and clearly what we are talking about. And we must be able to analyze well, that is, to discover productive and explanatory conditions of intelligibility for the things we are thinking about. The book’s central claim is that the kinds of representations that make successful reference possible and those that make successful analysis possible are not the same, so that significant scientific and mathematical work typically proceeds by means of a heterogeneous discourse that juxtaposes and often superimposes a variety of kinds of representation, including formal and natural languages as well as more iconic modes. It demonstrates the virtues and necessity of heterogeneity in historically central reasoning, thus filling an important gap in the literature and fostering a new, timely discussion on the epistemology of science and mathematics. (shrink)

I argue that quantum decoherence—understood as a dynamical process entailed by the standard formalism alone—carries us beyond conceptual aspects of non-relativistic quantum mechanics deemed insurmountable by many contributors to the recent quantum gravity and cosmology literature. These aspects include various incarnations of the measurement problem and of the quantum -to-classical puzzle. Not only can such problems be largely bypassed or dissolved without default to a particular interpretation, but theoretical work in relativistic arenas stands to gain substantial physical and philosophical (...) insight by incorporating decoherence phenomena. (shrink)

This wide-ranging book reconstructs this debate and places within their contemporary contexts the rival claims concerning the nature of the cosmos and the ...

Beginning with a reflection on ‘conceptual schemes’ and ‘very’ ideas and proceeding to examine different approaches to thinking philosophy of science not only with Kant but also between traditional analytic and hermeneutico-phenomenological approaches, this essay features a review of Kant’s 1755 solar nebular hypothesis and a reading of Nietzsche and Kant on cosmology along with a reflection on chemistry and the properties of cinnabar. Overall it is argued that a philosophy of science must be critical rather than normative/prescriptive. Seeking (...) to raise the question of science along with the question of truth, the essay also raises the question of esoteric historiography in science, including alchemy. (shrink)

Two different versions of the ending of the first additament to C. S. Peirce's 1908 article, "A Neglected Argument for the Reality of God," appear in the Collected Papers but were omitted from The Essential Peirce. In one, he linked the hypothesis of God's Reality to his entire theory of logic as semeiotic, claiming that proving the latter would also prove the former. In the other, he offered a final outline of his cosmology, in which the Reality of God (...) as Ens necessarium is indispensable to both the origin and order of our existing universe of Signs. Exploring these passages, as well as the unpublished manuscript drafts of the article, provides important insights into the key concepts of instinct and continuity within Peirce's comprehensive system of thought. (shrink)

Inspired by “One Note Samba,” a standard jazz repertoire, we present an outline of Bose-Einstein Condensate Cosmology (BECC). Although this approach seems awkward and a bit off the wall at first glance, it is not impossible to connect altogether BEC, Scalar Field Cosmology and Feshbach Resonance with Ermakov-Pinney equation. We also discuss shortly possible link with our previous paper, where we describe Newtonian Universe with Vortex in terms of Ermakov equation.

No pre-Socratic philosopher, perhaps, has caused more disagreement, or been more variously interpreted, than Anaxagoras of Clazomenae. Among recent attempts to reconstruct his system some of the more notable are those of Tannery, Bailey, Cornford, Peck, and Vlastos. Each of these reconstructions, and especially that of Tannery, has its adherents; and since none of them has much in common with any other, a universally acceptable solution to the fundamental problems involved may well by now seem unattainable. It is my belief, (...) however, and it is the object of this paper to try to prove, that all these modern reconstructions have at least one quality in common, namely an undue complication. The actual system of Anaxagoras was, I believe, considerably simpler than any reconstruction yet forthcoming. Unfortunately what I take to be the basis of the whole system, Anaxagoras' reaction to his Eleatic predecessors, is by.no means easy to convey in brief and simple terms. But if I should succeed in conveying it, then it will, I hope, be seen in itself to possess such a simplicity and neatness that my contention will carry with it a fair warrant of its own truth. I shall, for convenience, divide my argument into ten sections, to which I shall append a brief summary of my conclusions. (shrink)

It has been known for long time that most of the existing cosmology models have singularity problem. Cosmological singularity has been a consequence of excessive symmetry of flow, such as “Hubble’s law”. More realistic one is suggested, based on Newtonian cosmology model but here we include the vertical-rotational effect of the whole Universe. We review a Riccati-type equation obtained by Nurgaliev, and solve the equation numerically with Mathematica. It is our hope that the new proposed method can be (...) verified with observation data. (shrink)