We consider the nonlocal boundary value problem for difference
equations <mml:math alttext="$(u_{k}-u_{k-1})/\tau+Au_k=\varphi_k$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>&#8722;</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mi>k</mml:mi><mml:mo>&#8722;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:mi>&#964;</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:msub><mml:mi>u</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>&#966;</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:math>, <mml:math alttext="$1\leq k \leq N$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn><mml:mo>&#8804;</mml:mo><mml:mi>k</mml:mi><mml:mo>&#8804;</mml:mo><mml:mi>N</mml:mi></mml:math>, <mml:math alttext="$N\tau = 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mi>&#964;</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math>, and <mml:math alttext="$u_0 = u_{[\lambda/\tau]}+\varphi $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mrow><mml:mi>&#955;</mml:mi><mml:mo>/</mml:mo><mml:mi>&#964;</mml:mi></mml:mrow></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mi>&#966;</mml:mi></mml:math>, <mml:math alttext="$ 0&#60;\lambda\leq 1$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>0</mml:mn><mml:mo>&#60;</mml:mo><mml:mi>&#955;</mml:mi><mml:mo>&#8804;</mml:mo><mml:mn>1</mml:mn></mml:math>, in an
arbitrary Banach space <mml:math alttext="$E$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>E</mml:mi></mml:math> with the strongly positive operator
<mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>. The well-posedness of this nonlocal boundary value problem
for difference equations in various Banach spaces is studied. In
applications, the stability and coercive stability estimates in
H&#246;lder norms for the solutions of the difference
scheme of the mixed-type boundary value problems for the
parabolic equations are obtained. Some results of numerical
experiments are given.

A.  Ashyralyev

I.  Karatay

P. E. Sobolevskii

On well-posedness of the nonlocal boundary value problem for parabolic difference equations

The baker map is investigated by
two different theories of irreversibility by Prigogine and his
colleagues, namely, the <mml:math alttext="$\Lambda$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Λ</mml:mi></mml:math>-transformation and complex
spectral theories, and their structures are compared. In both
theories, the evolution operator <mml:math alttext="$U^\dag$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>U</mml:mi><mml:mo>†</mml:mo></mml:msup></mml:math> of observables (the
Koopman operator) is found to acquire dissipativity by
restricting observables to an appropriate subspace <mml:math alttext="$\Phi$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Φ</mml:mi></mml:math> of the
Hilbert space <mml:math alttext="$L^2$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> of square integrable functions. Consequently,
its spectral set contains an annulus in the unit disc. However,
the two theories are not equivalent. In the
<mml:math alttext="$\Lambda$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Λ</mml:mi></mml:math>-transformation theory, a bijective map <mml:math alttext="$\Lambda^{\dag -1}:\Phi\rightarrow L^2$" id="E6" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>Λ</mml:mi><mml:mrow><mml:mo>†</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo>:</mml:mo><mml:mi>Φ</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math> is looked for and the evolution operator
<mml:math alttext="$U$" id="E7" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>U</mml:mi></mml:math> of densities (the Frobenius-Perron operator) is transformed
to a dissipative operator <mml:math alttext="$W=\Lambda U\Lambda^{-1}$" id="E8" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>W</mml:mi><mml:mo>=</mml:mo><mml:mi>Λ</mml:mi><mml:mi>U</mml:mi><mml:msup><mml:mi>Λ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>. In the
complex spectral theory, the class of densities is restricted
further so that most values in the interior of the annulus are
removed from the spectrum, and the relaxation of expectation
values is described in terms of a few point spectra in the
annulus (Pollicott-Ruelle resonances) and faster decaying terms.

S. Tasaki

On Prigogine's approaches to irreversibility: a case study by the
baker map

We present state of the art, the new results, and discuss open
problems in the field of spectral analysis for a class of
integral-difference operators appearing in some nonequilibrium
statistical physics models as collision operators. The author
dedicates this work to the memory of Professor Ilya Prigogine,
who initiated this activity in 1997 and whose
interesting and most enlightening advices had gudided the author during all these years.

Yuri B. Melnikov

Spectral analysis for a class of integral-difference operators:
known facts, new results, and open problems

We provide here a qualitative summary of the main ideas from
econophysics and finance theory, starting with a thorough
criticism of the standard ideas taught in typical economics
textbooks. The emphasis is on the Galilean or physicists'
approach to market dynamics, as opposed to the standard
nonempirical postulatory one.

Joseph L. McCauley

Cornelia M. K&#252;ffner

Economic system dynamics

We describe the role of symmetries in formation of quantum dynamics.
A quantum version of d'Alembert's principle is
proposed to take into account the symmetry constrains more exact.
It is argued that the time reversibility of quantum process, as
the quantum analogy of d'Alembert's principle, makes the
measure of the corresponding path integral <mml:math alttext="$\delta$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#948;</mml:mi></mml:math>-like. The
argument of this <mml:math alttext="$\delta$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#948;</mml:mi></mml:math>-function is the sum of all classical
forces of the problem under consideration plus the random force
of quantum excitations. Such measure establishes the one-to-one
correspondence with classical mechanics and, for this reason,
allows a free choice of the useful dynamical variables. The
analysis shows that choosing the action-angle variables, one may
get to the free-from-divergences quantum field theory. Moreover,
one can try to get an independence from necessity to extract the
degrees of freedom constrained by the symmetry. These properties
of new quantization scheme are vitally essential for such
theories as the non-Abelian Yang-Mills gauge theory and quantum
gravity.

J.  Manjavidze

A.  Sissakian

Symmetries, variational principles, and quantum dynamics

We review the formulation of the problem of the electromagnetic
self-interaction of a relativistic charged particle in the
framework of the manifestly covariant classical mechanics of
Stueckelberg, Horwitz, and Piron. The gauge fields of this theory,
in general, cause the mass of the particle to change. We show
that the nonlinear Lorentz force equation for the
self-interaction resulting from the expansion of Green's function
has chaotic solutions. We study the autonomous equation for the
off-shell particle mass here, for which the effective charged
particle mass achieves a macroscopic average value determined by
what appears to be a strange attractor.

L. P. Horwitz

N. Katz

O. Oron

Could the classical relativistic electron be a strange attractor?

Elementary excitations (electrons, holes, polaritons, excitons,
plasmons, spin waves, etc.) on discrete substrates (e.g.,
 polymer chains, surfaces, and lattices) may move
coherently as quantum waves (e.g., Bloch waves), but also
incoherently (&#147;hopping&#148;) and may lose their phases due to their
interaction with their substrate, for example, lattice
vibrations. In the frame of Heisenberg equations for projection
operators, these latter effects are often phenomenologically
taken into account, which violates quantum mechanical consistency,
however. To restore it, quantum mechanical fluctuating forces
(noise sources) must be introduced, whose properties can be
determined by a general theorem. With increasing miniaturization,
in the nanotechnology of logical devices (including quantum
computers) that use interacting elementary excitations, such
fluctuations become important. This requires the determination of
quantum noise sources in composite quantum systems. This is the
main objective of my paper, dedicated to the memory of Ilya
Prigogine.

Hermann  Haken

Quantum fluctuations of elementary excitations in discrete media

First, I will recount the substance of several important
conversations I had with Ilya Prigogine over the years. There is
no doubt in my mind that Professor Prigogine firmly believed in
the underlying stochasticity of the universe. Second, I will
summarize my curiosity about the principle of detailed balance.
In my opinion, so far it has always been put in by hand. Third, I
will advance my own theory of microscopic irreversibility, that
is, irreversibility at the quantum level. This involves new
concepts of probability preference and probability violation. A
manifestation of this irreversibility may be seen during emission
of bound-free photons during field-matter interactions. Entropy
increase is tied to radiation increase, and the notion of heat
bath becomes light bath.

Karl  Gustafson

Microscopic irreversibility

We describe bi-Hamiltonian structure and Lax-pair formulation with the
spectral parameter of the generalized fermionic Toda lattice
hierarchy as well as its bosonic and fermionic symmetries for
different (including periodic) boundary conditions.
Its two reductions<mml:math alttext="$N=4$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> and <mml:math alttext="$N=2$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>2</mml:mn></mml:math> supersymmetric Toda lattice
hierarchiesin different (including canonical) bases are
investigated. Its <mml:math alttext="$r$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>r</mml:mi></mml:math>-matrix description, monodromy matrix, and
spectral curves are discussed.

V. V. Gribanov

V. G. Kadyshevsky

A. S. Sorin

Generalized fermionic discrete Toda hierarchy

Ioannis   Antoniou

Vladimir  Gontar

In memory of Ilya Prigogine

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Discrete Dynamics in Nature and Society publishes research that links basic and applied research relating to discrete dynamics of complex systems encountered in the natural and social sciences.

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Discrete Dynamics in Nature and Society