The tangential k-Cauchy-Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group

The tangential k-Cauchy-Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables,respectively.In this paper,we introduce a Lie group that the Heisenberg group can be imbedded into and call it generalized complex Heisenberg.We investigate quaternionic analysis on the generalized complex Heisenberg.We also give the Penrose integral formula for k-CF functions and construct the tangential k-Cauchy-Fueter complex.

A novel fractional-order hyperchaotic complex system and its synchronization

A novel fractional-order hyperchaotic complex system is proposed by introducing the Caputo fractional-order derivative operator and a constant term to the complex simplified Lorenz system. The proposed system has different numbers of equilibria for different ranges of parameters. The dynamics of the proposed system is investigated by means of phase portraits, Lyapunov exponents, bifurcation diagrams, and basins of attraction. The results show abundant dynamical characteristics. Particularly, the phenomena of extreme multistability as well as hidden attractors are discovered. In addition, the complex generalized projective synchronization is implemented between two fractional-order hyperchaotic complex systems with different fractional orders. Based on the fractional Lyapunov stability theorem, the synchronization controllers are designed, and the theoretical results are verified and demonstrated by numerical simulations. It lays the foundation for practical applications of the proposed system.

Two-Soliton Solutions and Interactions for the Generalized Complex Coupled Kortweg-de Vries Equations

Kortweg-de Vries （KdV）-typed equations have been used to describe certain nonlinear phenomena in fluids and plasmas. Generalized complex coupled KdV （GCCKdV） equations are investigated in this paper. Through the dependent variable transformations and symbolic computation, GCCKdV equations are transformed into their bilinear forms, based on which the one- and two-soliton solutions are obtained. Through the interactions of two solitons, the regular elastic collision are shown. When the wave numbers are complex, three kinds of solitonie collisions are presented： （i） two solitons merge and separate from each other periodically; （ii） two solitons exhibit the attraction and repulsion nearly twice, and finally separate from each other after such type of interaction; （iii） two solitons are ftuctuant in the central region of the collision. Propagation features of solitons are investigated with the effects of the coefficients in the GCCKdV equations considered. Velocity of soliton increase with the a increasing. Amplitude of v increase with the a increasing and decrease with the β increasing.

A New Generalized Complex Orthogonal Space-Time Block Codes

Recent research challenges in the wireless communication include the usage of diversity and efficient coding to improve data transmission quality and spectral efficiency. Space diversity uses multiple transmitting and/or receiving antennas to create independent fading channels without penalty in bandwidth efficiency. Space-time block coding is an encoding scheme for communication over Rayleigh fading channels using multiple transmitting antennas. Space-time block codes from complex orthogonal designs exist only for two transmitting antennas. This paper generalizes a new complex orthogonal space-time block code for four transmitting antennas, whose decoding complexity is very low. Simulations show that the generalized complex orthogonal space-time block code has low bit error rate, full rate and possibly large diversity.

Dual Characteristic Pairs in Generalized Complex Structures

This paper presents the relations between spinors and dual characteristic pairs, and gives a way to get the dual characteristic pairs of Dirac structure associated to a generalized complex structure.

On Integrable Conditions of Generalized Almost Complex Structures

Generalized complex geometry is a new kind of geometrical structure which contains complex and symplectic geometry as its special cases. This paper gives the equivalence between the integrable conditions of a generalized almost complex structure in big bracket formalism and those in the general framework.

A Generalized Schwarz Lemma

In this paper, we establish a boundary Schwarz Lemma for holomorphic mapping on the generalized complex ellipsoid in C^n.

On precise time integration method for non-classically damped MDOF systems

In the complex mode superposition method, the equations of motion for non-classically damped multiple-degree-of-freedom （MDOF） discrete systems can be transferred into a combination of some generalized SDOF complex oscillators. Based on the state space theory, a precise recurrence relationship for these complex oscillators is set up; then a delicate general solution of non-classically damped MDOF systems, completely in real value form, is presented in this paper. In the proposed method, no calculation of the matrix exponential function is needed and the algorithm is unconditionally stable. A numerical example is given to demonstrate the validity and efficiency of the proposed method.

Dissipative Discrete System with Nearest-Neighbor Interaction for the Nonlinear Electrical Lattice

A generalized dissipative discrete complex Ginzburg-Landau equation that governs the wave propagation in dissipative discrete nonlinear electrical transmission line with negative nonlinear resistance is derived. This equation presents arbitrarily nearest-neighbor nonlinearities. We analyze the properties of such model both in connection to their modulational stability, as well as in regard to the generation of intrinsic localized modes. We present a generalized discrete Lange-Newell criterion. Numerical simulations are performed and we show that discrete breathers are generated through modulational instability.

PREDICTIVE MODELS AND GENERATIVE COMPLEXITY

The causal states of computational mechanics define the minimal sufficient memory for a given discrete stationary stochastic process. Their entropy is an important complexity measure called statistical complexity （or true measure complexity）. They induce the s-machine, which is a hidden Markov model （HMM） generating the process. But it is not the minimal one, although generative HMMs also have a natural predictive interpretation. This paper gives a mathematical proof of the idea that the s-machine is the minimal HMM with an additional （partial） determinism condition. Minimal internal state entropy of a generative HMM is in analogy to statistical complexity called generative complexity. This paper also shows that generative complexity depends on the process in a nice way. It is, as a function of the process, lower semi-continuous （w.r.t. weak-, topology）, concave, and behaves nice under ergodic decomposition of the process.

THE MULTIPLICATIVE COMPLEXITY AND ALGORITHM OF THE GENERALIZED DISCRETE FOURIER TRANSFORM(GFT)

In this paper, we have proved that the lower bound of the number of real multiplications for computing a length 2(t) real GFT(a,b) (a = +/-1/2, b = 0 or b = +/-1/2, a = 0) is 2(t+1) - 2t - 2 and that for computing a length 2t real GFT(a,b)(a = +/-1/2, b = +/-1/2) is 2(t+1) - 2. Practical algorithms which meet the lower bounds of multiplications are given.

Solution to the Generalized Champagne Problem on simultaneous stabilization of linear systems

The well-known Generalized Champagne Problem on simultaneous stabilization of linear systems is solved by using complex analysis and Blonders technique. We give a complete answer to the open problem proposed by Patel et al., which automatically includes the solution to the original Champagne Problem. Based on the recent development in automated inequality-type theorem proving, a new stabilizing controller design method is established. Our numerical examples significantly improve the relevant results in the literature.

The new AI is general and mathematically rigorous

Most traditional artificial intelligence(AI)systems of the past decades are either very limited,or based on heuristics,or both.The new millennium,however,has brought substantial progress in the field of theoretically optimal and practically feasible algorithms for prediction,search,inductive inference based on Occam’s razor,problem solving,decision making,and reinforcement learning in environments of a very general type.Since inductive inference is at the heart of all inductive sciences,some of the results are relevant not only for AI and computer science but also for physics,provoking nontraditional predictions based on Zuse’s thesis of the computer-generated universe.We first briefly review the history of AI since Godel’s 1931 paper,then discuss recent post-2000 approaches that are currently transforming general AI research into a formal science.