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In this paper, by the complexification of the wave number of the soliton given by the Hirota bilinear method, we get the breather solutions. Lumps of the BKP equations are constructed by full degeneration of the breathers, i.e., the limit of infinitely large period of the breathers. The localization characters of the 1-order lump by contour line method are studied analytically. The partial degeneration of the breathers yields hybrid solutions including soliton, lump and breathers.

In 1935 Ervin Bauer formulated a basic principle of organization of living matter defined as the stable non-equilibrium state. The homeostatic stable non-equilibrium is realized internally by selecting the trajectories maintaining stable configurations from those disturbing it and supported by the dynamical structure of metabolism consisting of metabolic cycles, feedback loops and feedforward constraints. In the developing systems, according to Bauer, the principle of stable non-equilibrium is transformed into the principle of increasing external work which is grounded in the hyper-restorative non-equilibrium. The basis of this principle, as formulated by Lev Beloussov, is a hyper-restorative process during conformational relaxation of biomacromolecules which adds extra energy in the system and governs its complexification in which a new optimal coordinate pattern is searched. At the quantum level, this complexification is determined by the parametric refinement process in the field of possibilities. The complexification process takes place at the level of the cytoskeleton which represents a macroscopic enzymatic system and can generate differentiation waves that spread between cells. This may be associated with the function of cytoskeleton of transmitting signals and generating impulses. Different types of waves during morphogenesis correspond to different ranges of wavelengths of emission, from biophoton radiation to the hypersound waves in transmission of neural impulses. It is concluded that biological morphogenesis is based on the hyper-restorative non-equilibrium supported by the functional structure of the cytoskeleton.

We construct an explicit family of irreducible representations of the canonical commutation relations parametrized by the space of pairs of transverse Lagrangian subspaces in the complexification of a symplectic vector space. We identify different restrictions of this family with previously studied families of Lion-Vergne, Satake, Grossmann, and Mumford. The classical Schrodinger and Fock-Segal-Bargmann representations are also derived.

We report the discovery of wide nested sequences of period-adding complexification routes spawning the whole control parameter space of the Henon map, a discrete-time proxy of single-mode loss-modulated CO 2 lasers. For a realistic map, the new adding routes found reproduce analogous phenomena previously observed in continuous-time differential equations describing, for instance, oscillatory phases in an enzyme reaction model. In contrast to differential equations, the map is analytically tractable to a large extent, and provides a prototypic framework to investigate intricate features of generic dissipative flows.

Classical equations that govern the information in a Boltzmann–Gibbs Hamiltonian dynamical system are shown to entail the evolution of a phenomenological fraction of the information. The evolution of this fraction of information can be predicted along a preferential time direction provided a structural condition on the dynamics time scale is fulfilled. This preferential direction points towards the future or the past, starting from an initial time, provided an appropriate initial condition is fulfilled by the Boltzmann–Gibbs distribution at this time. Choosing one direction defines an arrow of time. That fraction of information is thus submitted to a phenomenological determinism. It permanently generates non-phenomenological components of the information, which accumulate, and experience an exponentially fast complexification. This complexification makes the accumulated non-phenomenological information unable to influence the evolution of the phenomenological information. That evolution is thus autonomous and deterministic.

We carry out multivariate time series analysis on price indices of individual goods and services collected over the last 35 years in Japan. Adoption of the complex principal component analysis (CPCA) enables us to have a new insight into dynamic correlation structure involved in the price data. The CPCA is based on complexification of real data using the Hilbert transformation; lead-lag relations between individual prices manifest in a form of instantaneous phases of the complex time series. The correlation matrix in the CPCA is purified by adopting the random matrix theory as a null hypothesis for removal of statistical noises. We identify four significant eigenmodes for price movement which are free from seasonal variations. Each of them has different characteristics of dynamical correlations and is shown to be responsive to different economic events.

We introduce tensor products in the category of lattice seminormed spaces. We show that the reasonable cross vector seminorms on the complexification of a lattice seminormed space are the same as the admissible vector seminorms. We then specialize these results to complexifications of Archimedean Riesz spaces.

A σ-operator on a complexification V C of an R -vector space V R is an operator A ∈ End C ( V C ) such that σ ( A ) = 0 , where σ ( z ) denotes the Weierstrass σ-function. In this paper, we define the notion of strongly pseudo-real σ-operator and prove that there is a one-to-one correspondence between real mixed Hodge structures and strongly pseudo-real σ-operators.

Generalized equipped posets and their representations over a Galois field extension K ⊂ L are introduced and studied. An equipment of a poset P is a pair ( Δ , S ) consisting of a finite group Δ and a collection S = { Δ x y | x ⩽ y } of its non-empty subsets such that Δ x y Δ y z ⊂ Δ x z for any chain x ⩽ y ⩽ z in P (equipped posets in the old sense of our former papers correspond to the case | Δ | = 2 with some additional restriction). Since the category of representations of a (strictly) equipped poset ( P , Δ , S ) is isomorphic to the category of representations of its evolvent ( Q , Δ ) (which is a poset with automorphisms), we study in fact representations of posets with automorphisms over the pair ( K , L ) . In the case Δ = Γ , where Γ = Gal ( L / K ) , we define the complexification functor C : rep ( K , L ) ( Q , Γ ) → rep L Q and the realification functor R : rep L Q → rep ( K , L ) ( Q , Γ ) and show that they induce reciprocal bijections between the isoclasses of indecomposables of the category rep ( K , L ) ( Q , Γ ) and the Γ-orbits of isoclasses of indecomposables of the category rep L Q . In this way, the problem on classification of indecomposables of equipped posets and posets with automorphisms over ( K , L ) actually is reduced to that one for representations of ordinary posets over L.

A functional is a function from the space of functions to a number field. We constructed an integral theory for a functional, using double extensions of real, complex number field in nonstandard arguments. In this paper, we consider a good system of infinite harmonic oscillators. Our functional integral implies that the propagator of the system is represented as the Riemann zeta function. The result is a direct generalization of our previous work [T. Nitta, Complexification of the propagator for the harmonic oscillator, in: Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics, World Scientific, 2006, pp. 261–268]. We remark that the variables are not only real numbers but also complex numbers. We will assume that the reader is familiar with Nonstandard Analysis.

Multidimensional fractional actionlike variational problem with time-dependent dynamical fractional exponents is constructed. Fractional Euler–Lagrange equations are derived and discussed in some details. The results obtained are used to explore some novel aspects of fractional quantum field theory where many interesting consequences are revealed, in particular the complexification of quantum field theory, in particular Dirac operators and the novel notion of “mass without mass”.

We investigate iterative algorithms for solving complex symmetric constrained singular systems arising in magnetized multicomponent transport. The matrices of the corresponding linear systems are symmetric with a positive semi-definite real part and an imaginary part with a compatible nullspace. We discuss well posedness, the symmetry of generalized inverses and Cholesky methods. We investigate projected stationary iterative methods as well as projected orthogonal residuals algorithms generalizing previous results on real systems. As an application, we consider the linear systems arising from the kinetic theory of gases and providing transport coefficients of partially ionized gas mixtures subjected to a magnetic field.

To continue the investigations of ecosemiotics, the topos of complexification is proposed as a filter through which humans perceive and interact with nature. The topos is characterized by the qualities of self-referentiality, stochasticity or heterogeneity, discontinuity, simultaneous multiple perspectives, radical subjectivity, and nonlocality. The topos is formally constructed by employing an iterated function system as the morphism that transitions between succeeding world states, eventually resulting in fractal world states. The topos is presented in a general form which is universally applicable to all worldly processes. Descriptive examples are provided from ecosystem ecology, law and Cubism. The relations between these examples, and the insights they provide, are discussed. A new direction for ecosystem studies is suggested.

We consider the Segal-Bargmann transform for a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities., Comment: Final version. To appear in Journal of Functional Analysis. Minor revisions

We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different classes of real and autonomous Newtonian dynamics in the plane. The rectilinear motion has been studied in a number of papers, while the cyclic motion is much less understood. For small data, the cyclic time trajectories lead to isochronous dynamics. For bigger data the situation is quite complicated; computer experiments show that, for sufficiently small degree of the monomial, the motion is generically isochronous with integer period, which depends in a quite sensitive way on the initial data. If the degree of the monomial is sufficiently high, computer experiments show essentially chaotic behavior. We suggest a possible theoretical explanation of these different behaviors. We also introduce a two-parameter family of two-dimensional mappings, describing the motion of the center of the circle, as a convenient representation of the cyclic dynamics; we call such a mapping the center map. Computer experiments for the center map show a typical multifractal behavior with periodicity islands. Therefore the above complexification procedure generates dynamics amenable to analytic treatment and possessing a high degree of complexity.

It is well-known that the Schrodinger equation reduces to a classical diffusion equation by means of Wick rotation (t → it), suggesting a correspondence between quantum and classical mechanics. Nonetheless, this result does not admit a clear conceptual interpretation. In the framework of his fractal E ( ∞ ) space-time theory, El Naschie showed that great conceptual advantage could be achieved by extending the imaginary time, it, to a perfectly symmetric, complex conjugate time 0 ± it. In this note we show through a simple analysis, involving formal analytic continuation (t → 0 ± it), that El Naschie’s time complexification provides the basis for a physical interpretation of the correspondence between quantum and classical mechanics in terms of quantum decoherence. We find that decoherent states inevitably arise due to time symmetry breaking as we go from the micro Cantorian space-time, where the two symmetric times, 0 + it and 0 − it, coexist to our 4-dimensional smooth space-time, where t is the only time.

In this article the fabrication of advanced masks will be briefly outlined and some specific aspects will be discussed. It will be shown that the increasing design complexity and the complexification of the optical lithographic process generate major issues on all aspects of the fabrication and control of such masks. These issues have not only technical but also economical impacts that will also be outlined. To cite this article: C. Reita, C. R. Physique 7 (2006).

We derive explicit inversion formulae for the attenuated geodesic and horocyclic ray transforms of functions and vector fields on two-dimensional manifolds equipped with the hyperbolic metric. The inversion formulae are based on a suitable complexification of the associated vector fields so as to recast the reconstruction as a Riemann–Hilbert problem. The inversion formulae have a very similar structure to their counterparts in Euclidean geometry and may therefore be amenable to efficient discretizations and numerical inversions. An important field of application is geophysical imaging when absorption effects are accounted for.

We prove that a symmetric space is Osserman if its complexification is a complex hyper-Kahler symmetric space. This includes all pseudo-hyper-Kahler as well as para-hyper-Kahler symmetric spaces. We extend the classification of pseudo-hyper-Kahler symmetric spaces obtained by the first and the third author to the class of para-hyper-Kahler symmetric spaces. These manifolds are possible targets for the scalars of rigid N=2 supersymmetric field theories with hypermultiplets on four-dimensional space-times with Euclidean signature.

Though the Kerr–Newman metric which represents the field of a charged spinning black hole was deduced by Newman decades ago by introducing complex coordinates, the physical motivation has remained a mystery, as also the fact that this metric infact represents also the field of the electron including the purely Quantum Mechanical gyromagnetic ratio g=2. These two circumstances were characterized by Newman as an accident. In this note, we argue on the other hand that Newman's complexification is infact symptomatic of the fuzzy nature of the space time at the micro scale which can be described by a non-commutative geometry, which would also explain why the Kerr–Newman metric describes the field of the Quantum Mechanical electron. From these considerations, we argue that a time varying gravitational constant is implied, which apart from other things, explains the otherwise inexplicable anomalous accelerations of the Pioneer spacecrafts.

We consider singular perturbation problems depending on a parameter e ⩾0 such that for e >0 the solution u e belongs to a Sobolev space on a domain Ω , but the limit u 0 is not a distribution on Ω . A very simple model problem, solvable by Fourier transform allows us to study the complexification process of u e as e↘0 . The limit holds in the topology of a space of analytical functionals. To cite this article: C.A. De Souza, E. Sanchez-Palencia, C. R. Mecanique 332 (2004).

There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151 , 111–133]. We also make comments on several problems raised by others.

Viro's construction of real smooth hypersurfaces uses regular (also called convex or coherent) subdivisions of Newton polytopes. Nevertheless, Viro's construction, sometimes calledpatchworking, can be applied as well to arbitrary subdivisions as a purely combinatorial procedure. Are the schemes coming from nonregular subdivisions, still topological types of some real smooth hypersurfaces? In the first part of this paper we prove a combinatorial version of Hilbert's Lemma (a consequence of Bezout's Theorem) that bounds the depth of nests in aT-curve, and we use this result and a previous work by I. Itenberg to answer the question affirmatively forT-curves of degree less than 6.According to V. A. Rokhlin, a real algebraic scheme has complex orientation of type I (alternatively of type II) if any curve with this real scheme divides (does not divide) its complexification. A real algebraic scheme hasindefinite typeif there are type I and type II curves with that particular scheme. In the second part of this paper, we describe a combinatorial algorithm, due to I. Itenberg and O. Viro, that allows one to determine the type of aT-curve. We then present a partial list of indefinite schemes forT-curves of degrees 7 and 8.

We previously proposed to use complex numbers in neural networks to take into consideration the phase differences among impulse trains. In the present paper, we propose a complex associative memory in which input and output patterns are composed of the values 1 and − 1 and weights, and hence the membrane potentials, take complex values. The learning rule of the complex weights is a direct extension of that of the complex perceptron that we previously proposed. We also propose the “sigmoidal learning” for the real-valued associative memory, which replaces the signature function in the orthogonal learning rule with the sigmoid function that yields weight matrices different from those obtained by the generalized inverse method. The complex associative memory yielded better performance than would be predicted by the increase of the degree of freedom due to complexification. It also showed some interesting characteristics in its dynamics and in the distribution of its weights, which are quite distinguished from those of the real-valued counterpart.

The purpose of this paper is to introduce a "sheaf" of hyperfunctions with Cω-parameters at the boundary, to study its behavior under the trace morphism, and to state some criteria on extension of solutions of systems of P.D.E. across singular sets of codimension ≥ 1. Let M be a Cω-manifold, X a complexification of M, Ω an open set with Cω-boundary N = ∂Ω, Y a complexilication of N, V a closed conic regular involutive submanifold of T*MX\ T*YX transversal to N × MT*MX with regular involutjve intersection. Let M be a DX-module for which Y is non-characteristic. We introduce a complex BaM | X = Ba, VM | X whose 0th cohomology consists of those hyperfunctions on M which depend real analytically on the variables transversal to the leaves of V. We also introduce a new complex BaΩ | X = Ba, VΩ | X whose main feature is that traces on N of H0(Ba, VΩ | X)-so1utions of M belong to H0(Ba, V′N | Y)). (Here V′ = ρω−1(V) with T*Y ← ρY × XT*X → ωT*X.) We are then able to state in Section 3 several principles on continuation of hyperfunction solutions with Cω-parameters (resp. real analytic solutions) to systems M across a subset S contained in a non-characteristic boundary N. The method (inspired by [Kan2]) consists in proving that under some hypotheses on non-microcharacteristicity of N, ΓΩ(H0(BaM | X))-solution f of M belongs automatically to H0(BaΩ | X) (Ω = Ω± denoting the components of M \ N), and therefore their traces γ ± (f) on N satisfy SSγ ± (f) ∩ V′ = Ø. The extendability of f across S is then an immediate consequence of the propagation 0 for γ+ (f) − γ−(f) from N \ S up to N.

The recent suggestion by Umeki [Phys. Lett. A 236 (1997) 69] of generating new solutions of nonlinear evolution equations by complexification of wave numbers is commented on. It is shown that this procedure leads in many cases to the breather solutions of the respective equation. An extension of the strategy to obtain genuine new solutions even in the case of equations with breather solutions is sketched.

Advances in hydrologic modeling techniques typically involves the incorporation of higher complexity into the hydrology model by use of hydraulic submodels. With over 150 models reported in the open literature, it is appropriate to review the progress achieved by the complexification of hydrologic models. That is, it is still not clear whether the general level of success afforded by the many types of complex models provide a marked improvement over that achieved by the more commonly used and simpler models such as the unit hydrograph method. Such a review indicates that it is still not clear, in general, whether as modeling complexity increases, modeling accuracy increases.

We survey the realization of quantum mechanics in quaternionic Hilbert spaces following the methods of Mackey, who examined the complex and real cases exploiting the imprimitivity theorem. We show that there exists a unique unitary skew-adjoint operator which commutes with all the observables. This operator not only plays the role of the imaginary unit in the complex case, but allows a complexification of the Hilbert space by the choice of any quaternionic imaginary unit. Difficulties in the definition of time reversal, however, arise because of the properties of the quaternionic field. The introduction of an extra imaginary unit, commuting with the others, is suggested in order to implement time reversal properly. In the Appendix we give the proof of the imprimitivity theorem, in the quaternionic case, that we use in the paper.

We show that N=1, D=4 euclidean supersymmetry leads to the complexification of gauge potentials, and that to have a supersymmetric euclidean D=4 theory which allows for the selfduality of real gauge potentials, the supersymmetry has to be extended to even N, i.e. at least N=2. We also discuss the supersymmetrizatin of Manton's procedure of using the dimensional reduction D=4→D=3 to derive self-dual monopoles from instantons.