Below are abstracts and downloadable preprints of my published (and some unpublished) papers in physics and chemistry.

(online versions of my mathematical publications - complete list of my publications)

For manuscripts with an e-print number, you can also get the latex source (of some version of the paper) from an e-print archive such as http://xxx.uni-augsburg.de

For the published version see the references given.

I do **not** send out paper copies of my manuscripts.
If you have problems downloading, decoding, or printing a file, look at
*downloading/printing problems?*

P000.

Contains over 240 sections with explanations answering questions from theoretical physics, collected from my answers to postings to various physics discussion groups. Most topics are related to quantum mechanics, quantum field theory, renormalization, the measurement problem, randomness, and philosophical issues in physics.

P000.

This paper gives a thorough critique of the foundations of quantum physics in its mainstream interpretation (i.e., treating pure states as primitives, without reference to hidden variables, and without modifications of the quantum laws).

This is achieved by cleanly separating a concise version of the (universally accepted) formal core of quantum physics from the (controversial) interpretation issues. The latter are primarily related to measurement, but also to questions of existence and of the meaning of basic concepts like 'state' and 'particle'. The requirements for good foundations of quantum physics are discussed.

Main results:

P000.

This paper presents the thermal interpretation of quantum physics. The insight from Part I of this series that Born's rule has its limitations -- hence cannot be the foundation of quantum physics -- opens the way for an alternative interpretation -- the thermal interpretation of quantum physics. It gives new foundations that connect quantum physics (including quantum mechanics, statistical mechanics, quantum field theory and their applications) to experiment.

The thermal interpretation resolves the problems of the foundations of quantum physics revealed in the critique from Part I. It improves the traditional foundations in several respects:

P000.

This paper presents the measurement problem from the point of view of the thermal interpretation of quantum physics introduced in Part II. Unlike most work on the foundations of quantum mechanics, the present paper involves a multitude of connections to the actual practice of quantum theory and quantum measurement.

The measurement of a Hermitian quantity A is regarded as giving an uncertain value approximating the q-expectation of A rather than (as tradition wanted to have it) as an exact revelation of an eigenvalue of A. Single observations of microscopic systems are (except under special circumstances) very uncertain measurements only.

The thermal interpretation

The thermal interpretation is an interpretation of quantum physics that is in principle refutable by theoretical arguments leading to a negative answer to a number of open issues collected at the end of the paper, since there is plenty of experimental evidence for each of the points mentioned there.

P000.

This paper is the second in a series of papers on coherent spaces and their applications. It begins the study of coherent quantization - the way operators in a quantum space can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization to the non-unitary case.

Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties.

The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps.

The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization operator that associates homomorphically with every coherent map a linear operator from the quantum space into itself. This operator generalizes to general symmetry groups of coherent spaces the second quantization procedure for free classical fields. The latter is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. A coordinate-free derivation is given of the basic properties of creation and annihilation operators in Fock spaces.

LP000.

The notion of a coherent space is a nonlinear version of the notion of a complex Euclidean space: The vector space axioms are dropped while the notion of inner product is kept.

Coherent spaces provide a setting for the study of geometry in a different direction than traditional metric, topological, and differential geometry. Just as it pays to study the properties of manifolds independently of their embedding into a Euclidean space, so it appears fruitful to study the properties of coherent spaces independent of their embedding into a Hilbert space.

Coherent spaces have close relations to reproducing kernel Hilbert spaces, Fock spaces, and unitary group representations, and to many other fields of mathematics, statistics, and physics.

This paper is the first of a series of papers and defines concepts and basic theorems about coherent spaces, associated vector spaces, and their topology. Later papers in the series discuss symmetries of coherent spaces, relations to homogeneous spaces, the theory of group representations, C

P970.

Insight Article

This article is a sequel of several earlier ones that make precise what a virtual particle is, what being real means, document some of the liberties taken in physics textbooks in the use of this concept, mention the most prominent misuses, and document the origin of some of the associated myths. In short, the concept of virtual particles is well-defined and useful when restricted to its use in Feynman diagrams and associated technical discussions. But it is highly misleading when used to argue about vacuum fluctuations, as if these were processes happening in space and time. The latter is a frequent misunderstanding, a myth that has not the slightest basis in particle physics.

However, one meets occasionally mythical claims even in the scientific literature. Therefore we look at a representative recent paper in which vacuum fluctuations play a seemingly prominent role, and answer the question: How do vacuum fluctuations look like in practice?

P971.

Insight Article

This article explains how the widespread but misleading informal practice of treating - when explaining the subject of subatomic particles to the general public - virtual particles as real objects popping in and out of existence for a tiny time could arise. This is done at the example of Steve Carlip's page on Hawkings radiation, where Steve Carlip, a well-known theorical physicist working on quantum gravity, gave a lucid but completely mythical narrative about how vacuum fluctuations create Hawking radiation.

P000.

In this lecture, a timeline is traced from Huygens' wave optics to the modern concept of light according to quantum electrodynamics. The lecture highlights the closeness of classical concepts and quantum concepts to a surprising extent. For example, it is shown that the modern quantum concept of a qubit was already known in 1852 in fully classical terms.

P000.

In this lecture the results of the historical review given in my lecture ''Classical models for quantum light'' are utilized to reassess the meaning of observables and stochastic processes for the classical and quantum description of light.

In particular we discuss the description of partially coherent, fluctuating light through classical stochastic Maxwell equations (with uncertainty in the initial conditions only), and look at a generalization that works for all quantum aspects of arbitrary quantum systems.

P972.

Insight Article

This article goes though a number of wikipedia pages and comments on their misleading statements about virtual particles and Feynman diagrams. In addition, the article discusses some textbooks on quantum field theory and the extent to which they contain problematic formulations about virtual particles.

P175.

In 1960, E.H. Brown defined a set of characteristic curves (also known as ideal curves) of pure fluids, along which some thermodynamic properties match those of an ideal gas. These curves are used for testing the extrapolation behaviour of equations of state. This work is revisited, and an elegant representation of the first-order characteristic curves as level curves of a master function is proposed. It is shown that Brown's postulate - that these curves are unique and dome-shaped in a double-logarithmic p,T representation - may fail for fluids exhibiting a density anomaly. A careful study of the Amagat curve (Joule inversion curve) generated from the IAPWS-95 reference equation of state for water reveals the existence of an additional branch.

P973.

Insight Article

In discussions on the internet (including a number of wikipedia pages) and in books and articles for non-experts in particle physics, there is considerable confusion about various notions around the concept of particles of subatomic size, and in particular about the notion of a virtual particle. This is partly due to misunderstandings in the terminology used, and partly due to the fact that subatomic particles manifest themselves only indirectly, thus leaving a lot of leeway for the imagination to equip these invisible particles with properties, some of which sound very magical.

The aim of this Insight article is a definition of physical terms essential for an informed discussion of which of these properties have a solid basis in physics, and which of these are gross misconceptions or exaggerations that shouldn't be taken seriously.

P174.

Brown's characteristic curves (also known as ideal curves) describe states at which one thermodynamic property of a real pure fluid matches that of an ideal gas; these curves can be used for testing the extrapolation behaviour of equations of state. In this work, some characteristic curves are computed directly - without recourse to an equation of state - for some pair potentials by Monte Carlo computer simulation. The potentials used are an ab-initio potential for argon, the 1-center Lennard-Jones potential, and a softer pair potential whose short-range part is in accordance with quantum mechanical predictions. The influence of the short-distance repulsion on the characteristic curves is found to be significant even in the 10-100 MPa pressure range.

P974.

Insight Article

Relativistic quantum field theory is notorious for the occurrence of divergent expressions that must be renormalized by recipes that on first sight sound very arbitrary and counterintuitive. This Insight article shows that it doesn't have to be this way!

PN165.

pdf file (460K)

We propose a new form for equations of state (EOS) of thermodynamic systems in the Ising universality class. The new EOS guarantees the correct universality and scaling behavior close to critical points and is formulated in terms of the scaling fields only -- unlike the traditional Schofield representation, which uses a parametric form.

Close to a critical point, the new EOS expresses the square of the strong scaling field as an explicit function of the thermal scaling field and the dependent scaling field. A numerical expression is derived, valid close to critical points.

As a consequence of the construction it is shown that the dependent scaling field can be written as an explicit function of the relevant scaling fields without causing strongly singular behavior of the thermodynamic potential in the one-phase region.

Augmented by additional scaling correction fields, the new EOS also describes the state space further away from critical points. It is indicated how to use the new EOS to model multiphase fluid mixtures, in particular for vapor-liquid-liquid equilibrium (VLLE) where the traditional revised scaling approach fails.

P000.

phenTherm.pdf (207K)

This paper gives a concise, mathematically rigorous description of phenomenological equilibrium thermodynamics for single-phase systems in the absence of chemical reactions and external forces. From the formulas provided, it is an easy step to go to various examples and applications discussed in standard textbooks (such as those by Callen or Reichl). A full discussion of global equilibrium would also involve the equilibrium treatment of multiple phases and chemical reactions. Since their discussion offers no new aspects compared with traditional textbook treatments, they are not treated here.

The present phenomenological approach is similar to that of Callen, who introduces in his well-known thermodynamics book the basic concepts by means of a few postulates from which everything else follows. The present setting is a modified version designed to match the more fundamental approach based on statistical mechanics. By specifying the kinematical properties of states outside equilibrium, his informal thermodynamic stability arguments (which depend on a dynamical assumption close to equilibrium) can be replaced by rigorous mathematical arguments.

AN000.
**
A. Neumaier,
A multi-phase, multi-component critical equation of state,**
Manuscript (2013).
arXiv:1307.8391

pdf file (193K)

Realistic equations of state valid in the whole state space of a
multi-component mixture should satisfy at least three important
constraints:

(i) The Gibbs phase rule holds.

(ii) At low densities, one can deduce a virial equation of state with
the correct multicomponent structure.

(iii) Close to critical points, plait points, and consolute points,
the correct universality and scaling behavior is guaranteed.

This paper discusses semiempirical equations of state for mixtures that
express the pressure as an explicit function of temperature and the
chemical potentials. In the first part, expressions are derived for
the most important thermodynamic quantities. The main result of the
second part is the construction of a large family of equations of
state with the properties (i)--(iii).

P000.

arXiv:0810.1019

pdf file (3165K)

The goal of this book is to present classical mechanics, quantum mechanics, and statistical mechanics in an almost completely algebraic setting, thereby introducing mathematicians, physicists, and engineers to the ideas relating classical and quantum mechanics with Lie algebras and Lie groups. The book emphasizes the closeness of classical and quantum mechanics, and the material is selected in a way to make this closeness as apparent as possible.

Much of the material covered here is not part of standard textbook treatments of classical or quantum mechanics (or is only superficially treated there). For physics students who want to get a broader view of the subject, this book may therefore serve as a useful complement to standard treatments of quantum mechanics.

Almost without exception, this book is about precise concepts and exact results in classical mechanics, quantum mechanics, and statistical mechanics. The structural properties of mechanics are discussed independent of computational techniques for obtaining quantitatively correct numbers from the assumptions made. The standard approximation machinery for calculating from first principles explicit thermodynamic properties of materials, or explicit cross sections for high energy experiments can be found in many textbooks and is not repeated here.

P000.
**
A. Neumaier,
Renormalization without infinities - an elementary tutorial**,
Manuscript (2011).

pdf file (362K)

Renormalization is an indispensable tool for modern theoretical
physics. At the same time, it is one of the least appealing techniques,
especially in cases where naive formulations result in divergences
that must be cured - a step that is often done in a mathematically
dubious way.

In this paper, it is shown how the renormalization procedure works
both in singular cases where it removes naive divergences and in
regular cases where a naive approach is possible but renormalization
improves the quality of perturbation theory. In fact, one can see
immediately that the singular situation is simply a limiting case of the
regular situation.

After discussing generalities, the paper introduces a large family of
toy examples, defined by special perturbations of an arbitrary
Hamiltonian with a discrete spectrum.
The examples show explicitly many of the renormalization effects
arising in realistic quantum field theories such as quantum
chromodynamics: logarithmic divergences, running couplings,
asymptotic freedom, dimensional transmutation, the renormalization
group, and renormalization scheme dependent results at any order of
perturbation theory.

Unlike in more realistic theories, everything is derived rigorously
and nonperturbatively in terms of simple explicit formulas. Thus one
can understand in detail how the infinities arise (if they arise) -
namely as an unphysical infinitely sensitive dependence on the bare
coupling constants. One also sees that all spurious infinities are
cured automatically by the same renormalization process that gives
robust physical results in the case where no infinities occur.

P000.

This lecture (the second of three) discusses work towards a new, classical view of quantum mechanics. It is based on an analysis of polarized light, of the meaning of quantum ensembles in a field theory, of classical simulations of quantum computing algorithms, and resulting optical models for the simulation of quantum mechanics.

In particular, it is shown that classical second-order stochastic optics is precisely the quantum mechanics of a single photon, with all its phenomenological bells and whistles.

P000.

This lecture (the first of three) discusses foundational problems on the nature of light revealed by 1. attempts to define a probability concept for photons, 2. quantum models for photons on demands (and their realization through laser-induced emission by a calcium ion in a cavity), 3. models explaining the photo effect, and 4. Bell-type experiments for single photon nonlocality.

P000.

An experiment is described which proves, using single photons only, that the standard hidden variables assumptions (commonly used to derive Bell inequalities) are inconsistent with quantum mechanics. The analysis is very simple and transparent. In particular, it demonstrates that a classical wave model for quantum mechanics is not ruled out by experiments demonstrating the violation of the traditional hidden variable assumptions.

P000.

On the basis of new, concise foundations, this paper establishes the four laws of thermodynamics, the Maxwell relations, and the stability requirements for response functions, in a form applicable to global (homogeneous), local (hydrodynamic) and microlocal (kinetic) equilibrium.

The present, self-contained treatment needs very little formal machinery and stays very close to the formulas as they are applied by the practicing physicist, chemist, or engineer. From a few basic assumptions, the full structure of phenomenological thermodynamics and of classical and quantum statistical mechanics is recovered.

Care has been taken to keep the foundations free of subjective aspects (which traditionally creep in through information or probability). One might describe the paper as a uniform treatment of the nondynamical part of classical and quantum statistical mechanics ``without statistics'' (i.e., suitable for the definite descriptions of single objects) and ``without mechanics'' (i.e., independent of microscopic assumptions). When enriched by the traditional examples and applications, this paper may serve as the basis for a course on thermal physics.

P000.

The collapse challenge for interpretations of quantum mechanics is to build from first principles and your preferred interpretation a complete, observer-free quantum model of the described experiment (involving a photon and two screens), together with a formal analysis that completely explains the experimental result. The challenge is explained in detail, and discussed in the light of the Copenhagen interpretation and the decoherence setting.

P208.

pdf file (116K)

P117.

Linear optical networks are devices that turn classical incident modes by a linear transformation into outgoing ones. In general, the quantum version of such transformations may mix annihilation and creation operators. We derive a simple formula for the effective Hamiltonian of a general linear quantum network, if such a Hamiltonian exists. Otherwise we show how the scattering matrix of the network is decomposed into a product of three matrices that can be generated by Hamiltonians.

P000.

A mathematically well-defined, manifestly covariant theory of classical and quantum field is given, based on Euclidean Poisson algebras and a generalization of the Ehrenfest equation, which implies the stationary action principle. The theory opens a constructive spectral approach to finding physical states both in relativistic quantum field theories and for flexible phenomenological few-particle approximations.

In particular, we obtain a Lorentz-covariant phenomenological multiparticle quantum dynamics for electromagnetic and gravitational interaction which provides a representation of the Poincaré group without negative energy states. The dynamics reduces in the nonrelativistic limit to the traditional Hamiltonian multiparticle description with standard Newton and Coulomb forces.

The key that allows us to overcome the traditional problems in canonical quantization is the fact that we use the algebra of linear operators on a space of wave functions slightly bigger than traditional Fock spaces.

PN113.

A Gaussian resolution method for the computation of equilibrium density matrices rho(T) for a general multidimensional quantum problem is presented. The variational principle applied to the ``imaginary time'' Schroedinger equation provides the equations of motion for Gaussians in a resolution of rho(T) described by their width matrix, center and scale factor, all treated as dynamical variables.

The method is computationally very inexpensive, has favorable scaling with the system size and is surprisingly accurate in a wide temperature range, even for cases involving quantum tunneling. Incorporation of symmetry constraints, such as reflection or particle statistics, is also discussed.

PS112.

A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization.

Extending the `probability via expectation' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotations of unlimited repeatability; hence it can be applied to unique systems such as the universe.

Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

P000.

The projection formalism for calculating effective Hamiltonians and resonances is generalized to the nonlocal and/or nonhermitian case, so that it is applicable to the reduction of relativistic systems (Bethe-Salpeter equations), and to dissipative systems modeled by an optical potential.

It is also shown how to recover

For practical calculations, it is important that the resulting formulas can be used without computing any projection operators. This leads to a modified coupled reaction channel/resonating group method framework for the calculation of multichannel scattering information.

P103.

We review and further develop the recently introduced numerical approach for scattering calculations based on a so called pseudo-time Schrödinger equation, which is in turn a modification of the damped Chebyshev polynomial expansion scheme.

The method utilizes a special energy-dependent form for the absorbing potential in the time-independent Schrödinger equation, in which the complex energy spectrum E_k is mapped to u_k inside the unit disk, where u_k are the eigenvalues of some explicitly known sparse matrix U.

Most importantly for the numerical implementation, all the physical eigenvalues u_k are extreme eigenvalues of U, which allows one to extract these eigenvalues very efficiently by harmonic inversion of a pseudo-time autocorrelation function using the filter diagonalization method. The computation of 2T steps of the autocorrelation function requires only T sparse real matrix-vector multiplications.

We describe and compare different schemes, effectively corresponding to different choices of the energy-dependent absorbing potential, and test them numerically by calculating resonances of the HCO molecule. Our numerical tests suggest an optimal scheme that provide accurate estimates for most resonance states.

PN97.

The Schrödinger equation with an energy-dependent complex absorbing potential, associated with a scattering system, can be reduced for a special choice of the energy-dependence to a harmonic inversion problem of a discrete pseudo-time correlation function. An efficient formula for Green's function matrix elements is also derived. Since the exact propagation up to time 2t can be done with only t real matrix-vector products, this gives an unprecedently efficient scheme for accurate calculations of quantum spectra for possibly very large systems.

P000.

It is shown that, for a harmonic oscillator in the ground state, Bohmian mechanics and quantum mechanics predict values of opposite sign for certain time correlations.

The discrepancy can be explained by the fact that Bohmian mechanics has no natural way to accomodate the Heisenberg picture, since the local expectation values that define the beables of the theory depend on the Heisenberg time being used to define the operators.

Relations to measurement are discussed, too, and are shown to leave no loophole for claiming that Bohmian mechanics reproduces

CP000.

(in German; English title: Quantum designs - foundations of a non-commutative theory of designs)

Quantum designs are sets of subspaces, or equivalent sets of orthogonal projection matrices, in complex finite dimensional vector spaces with certain properties. These structures are generalizations of classical t-designs (the special case of pairwise commuting matrices), spherical designs, complex t-designs and equi-isoclinic subspaces. All elements of quantum design theory have a natural interpretation in terms of quantum theory.

Apart from general theory (e.g., absolute and special bounds), constructions are given for two classes of quantum designs which generalize the classical balanced incomplete block designs and affine designs. One of them gives rise to the first known class of infinitely many complex 2-designs. Also new tight complex 2-designs are constructed. The constructions have a close analogy to formalisms of quantum theory in infinite-dimensional vector spaces.

P000.

The best mathematical arguments against a realistic interpretation of quantum mechanics - that gives definite but partially unknown values to all observables - are analysed and shown to be based on reasoning that is not compelling.

This opens the door for an interpretation that, while respecting the indeterministic nature of quantum mechanics, allows to speak of definite values for all observables at any time that are, however, only partially measurable.

The analysis also suggests new areas where the foundations of quantum theory need to be tested.

P000.

These comments intend to show that quantum paradoxes are not resolved by the "many-worlds" interpretation or metatheory of quantum mechanics; instead, the latter is full of home-made puzzles and ambiguities.

PS000.

Based on a principal component analysis of 47 published attempts to quantify hydrophobicity in terms of a single scale, we define a representation of the 20 amino acids as points in a 3-dimensional hydrophobicity space and display it by means of a minimal spanning tree. The dominant scale is found to be close to two scales derived from contact potentials.

OPS90.

A smooth empirical potential is constructed for use in off-lattice protein folding studies. Our potential is a function of the amino acid labels and of the distances between the C(alpha) atoms of a protein. The potential is a sum of smooth surface potential terms that model solvent interactions and of pair potentials that are functions of a distance, with a smooth cutoff at 12 Å. Techniques include the use of a fully automatic and reliable estimator for smooth densities, of cluster analysis to group together amino acid pairs with similar distance distributions, and of quadratic programming to find appropriate weights with which the various terms enter the total potential. For nine small test proteins, the new potential has minima within 1.3-4.7Å of the PDB geometry, with one exception that has an error of 8.5Å.

Moreover, a nonuniqueness theorem is given that shows that no set of equilibrium geometries can determine the true effective potential energy function.

NOP85.

This paper discusses the mathematical formulation of and solution attempts for the so-called protein folding problem. The static aspect is concerned with how to predict the folded (native, tertiary) structure of a protein, given its sequence of amino acids. The dynamic aspect asks about the possible pathways to folding and unfolding, including the stability of the folded protein.

From a mathematical point of view, there are several main sides to the static problem:

- the selection of an appropriate potential energy function;

- the parameter identification by fitting to experimental data; and

- the global optimization of the potential.

The dynamic problem entails, in addition, the solution of (because of multiple time scales very stiff) ordinary or stochastic differential equations (molecular dynamics simulation), or (in case of constrained molecular dynamics) of differential-algebraic equations. A theme connecting the static and dynamic aspect is the determination and formation of secondary structure motifs.

The present paper gives a self-contained introduction to the necessary background from physics and chemistry and surveys some of the literature. It also discusses the various mathematical problems arising, some deficiencies of the current models and algorithms, and possible (past and future) attacks to arrive at solutions to the protein folding problem.

P000.

Measurements can be adequately described without reference to ``the collapse of the wave function'' (or to wave functions at all). The collapse, as far as it occurs (i.e., the convergence of the density matrix to one that commutes with the Hamiltonian of the system), is a natural consequence of the reduced description of macroscopic systems in the thermodynamic limit since that leads to a dissipative dynamics. However, in the presence of spin, there is no complete collapse: macroscopic polarization phenomena remain that need 2-state quantum physics, a fact that seems to have escaped notice before. Since polarization is well-understood as a macroscopic phenomenon (no one ever talked about philosophical problems related to macroscopic polarization!), there is no reason to consider the microscopic world as essentially different from the macroscopic world.

P000.

In this paper, an elementary and self-contained axiomatic treatment is given of equilibrium thermodynamics including fluctuations. Among other things, this leads to a natural explanation of the Hilbert space underlying quantum physics, using only a simple quantization condition related to the third law of thermodynamics.

NP73.

The Thiele-Wilson system, a simple model of a linear, triatomic molecule, has been studied extensively in the past. The system exhibits complex molecular dynamics including dissociation, periodic trajectories and bifurcations. In addition, it has for a long time been suspected to be chaotic, but this has never been proved with mathematical rigor.

In this paper, we present numerical results that, using interval methods, rigorously verify the existence of transversal homoclinic points in a Poincarè map of this system. By a theorem of Smale, the existence of transversal homoclinic points in a map rigorously proves its mixing property, i.e., the chaoticity of the system.

NP71.

scanned text

**
Mathematik - Sprache und Werkzeug der Naturwissenschaften
**

I do **not** send out paper copies of my manuscripts;
but here are some hints of how to make your own copy.

To uncompress the gz files obtained you need the GNU program gunzip.

Some browsers seem to save *.dvi.gz files as *.dvi, so that one thinks one has a dvi-file while one actually may still have to do a gunzip. And some other browsers appear to do an automatic gunzip, while leaving the file name with the suffix .gz! In both cases, the file needs to be renamed appropriately for further processing.

See The gzip homepage or Compression for gunzip, dvi.exe or TeX Facilities or LaTeX Home Page for a dvi-viewer, Ghostscript for the popular free postscript viewer, and

Help for printing PostScript Files.

If you still have difficulties obtaining or printing one of the papers above, please tell me details about the difficulties you encountered.

Global Optimization

Protein Folding

Interval Methods

Regularization

Mathematical Software

Computational Mathematics Links

Mathematics Links

Statistics Links

my home page (http://www.mat.univie.ac.at/~neum)

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)