Discontinuous bifurcation and coexistence of attractors in a piecewise linear map with a gap

Coexistence of attractors with striking characteristics is observed in this work, where a stable period-5 attractor coexists successively with chaotic band-ll, period-6, chaotic band-12 and band-6 attractors. They are induced by dif- ferent mechanisms due to the interaction between the discontinuity and the non-invertibility. A characteristic boundary collision bifurcation, is observed. The critical conditions are obtained both analytically and numerically.

Noise destroys the coexistence of periodic orbits of a piecewise linear map

The effects of Gaussian white noise and Gaussian colored noise on the periodic orbits of period-5（P-5） and period-6（P-6） in their coexisting domain of a piecewise linear map are investigated numerically.The probability densities of some orbits are calculated.When the noise intensity is D = 0.0001,only the orbits of P-5 exist,and the coexisting phenomenon is destroyed.On the other hand,the self-correlation time τ of the colored noise also affects the coexisting phenomenon.When τc〈τ〈τc,only the orbits of P-5 appear,and the stability of the orbits of P-5 is enhanced.However,when τ〉τc（τc and τc are critical values）,only the orbits of P-6 exist,and the stability of the P-6 orbits is enhanced greatly.When τ〈τc,the orbits of P-5 and P-6 coexist,but the stability of the P-5 orbits is enhanced and that of P-6 is weakened with τ increasing.

Linear Maps Preserving Idempotency of Products of Matrices on Upper Triangular Matrix Algebras

Let Tn be the algebra of all n × n complex upper triangular matrices. We give the concrete forms of linear injective maps on Tn which preserve the nonzero idempotency of either products of two matrices or triple Jordan products of two matrices.

Galois linear maps and their construction

The condition of an algebra to be a Hopf algebra or a Hopf(co)quasigroup can be determined by the properties of Galois linear maps.For a bialgebra H,if it is unital and associative as an algebra and counital coassociative as a coalgebra,then the Galois linear maps T1 and T2 can be defined.For such a bialgebra H,it is a Hopf algebra if and only if T1 is bijective.Moreover,T1^-1 is a right H-module map and a left H-comodule map(similar to T2).On the other hand,for a unital algebra(no need to be associative),and a counital coassociative coalgebra A,if the coproduct and counit are both algebra morphisms,then the sufficient and necessary condition of A to be a Hopf quasigroup is that T1 is bijective,and T1^-1 is left compatible with ΔT1-11^r and right compatible with mT1-1^l at the same time(The properties are similar to T2).Furthermore,as a corollary,the quasigroups case is also considered.

An Adaptive Non-Linear Map and Its Application

A novel adaptive non-linear mapping （ANLM）, integrating an adaptive mapping error （AME） with a chaosgenetic algorithm （CGA） including chaotic variable, was proposed to overcome the deficiencies of non-linear mapping （NLM）. The value of AME weight factor is determined according to the relative deviation square of distance between the two mapping points and the corresponding original objects distance. The larger the relative deviation square between two distances is, the larger the value of the corresponding weight factor is. Due to chaotic mapping operator, the evolutional process of CGA makes the individuals of subgenerations distributed ergodieally in the defined space and circumvents the premature of the individuals of subgenerations. The comparison results demonstrated that the whole performance of CGA is better than that of traditional genetic algorithm. Furthermore, a typical example of mapping eight-dimenslonal olive oil samples onto two-dimensional plane was employed to verify the effectiveness of ANLM. The results showed that the topology-preserving map obtained by ANLM can well represent the classification of original objects and is much better than that obtained by NLM.

Non-linear Maps on Borel Subalgebras of Simple Lie Algebras Preserving Abelin Ideals

Let g be a complex simple Lie algebra of rank ι, b the standard Borel subalgebra. An invertible map on Ь is said to preserve abelian ideals if it maps each abelian ideal to some such ideal of the same dimension. In this article, by using some results of Chevalley groups, the theory of root systems and root space decomposition, the author gives an explicit description on such maps of Ь.

On Representations Associated with Completely n-Positive Linear Maps on Pro-C^＊-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C^＊-algebra A to L（H）, which verifies the condition of complete positivity, is of the form [V^＊TijФ（·）V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^＊-algebra of all n×n matrices over the commutant of Ф（A） in L（K）. This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112（3）, 1991, 709-712. Also, a covariant version of this construction is given.

Similarity-Invariant Subspaces and Similarity-Preserving Linear Maps

In this paper, the similarity-invariant subspaces of B(H), which is tile Banach algebra of all bounded linear operators on a separable infinite-dimensional Hilbert space H, are completely characterized and the representations of bounded linear maps on B(H) which preserve similarity in both directions are given.

Inverse-preserving Linear Maps Between Spaces of Matrices over Fields

Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.

LMM:A Fixed-Point Linear Mapping Based Approximate Multiplier for IoT

The development of IoT(Internet of Things)calls for circuit designs with energy and area efficiency for edge devices.Approximate computing which trades unnecessary computation precision for hardware cost savings is a promising direction for error-tolerant applications.Multipliers,as frequently invoked basic modules which consume non-trivial hardware costs,have been introduced approximation to achieve distinct energy and area savings for data-intensive applications.In this paper,we propose a fixed-point approximate multiplier that employs a linear mapping technique,which enables the configurability of approximation levels and the unbiasedness of computation errors.We then introduce a dynamic truncation method into the proposed multiplier design to cover a wider and more fine-grained configuration range of approximation for more flexible hardware cost savings.In addition,a novel normalization module is proposed for the required shifting operations,which balances the occupied area and the critical path delay compared with normal shifters.The introduced errors of our proposed design are analyzed and expressed by formulas which are validated by experimental results.Experimental evaluations show that compared with accurate multipliers,our proposed approximate multiplier design provides maximum area and power savings up to 49.70%and 66.39%respectively with acceptable computation errors.

New Developments on Operator Reverse AM-GM Inequality for Positive Linear Maps

In this research article,we shall give some reverse Arithmetic-Geometric mean inequalities for unital positive linear maps on Hilbert space operators under some different conditions.Our results are sharper and more precise as compared to some recent published results.Moreover,we shall present refinements of the Lin conjecture.

A NEW SIMPLE 2-D PIECEWISE LINEAR MAP

A new simple piecewise linear map of the plane is presented and analyzed, then a detailed study of its dynamical behaviour is described, along with some other dynamical phenomena, especially fixed points and their stability, observation of a new chaotic attractors obtained via border collision bifurcation. An important resuk about coexisting chaotic attractors is also numerically studied and discussed.

Non-linear Invertible Maps that Preserve Staircase Subalgebras

Let g be the general linear Lie algebra consisting of all n x n matrices over a field F and with the usual bracket operation {x, y} =xy - yx. An invertible map φ ： g →g is said to preserve staircase subalgebras if it maps every staircase subalgebra to some staircase subalgebra of the same dimension. In this paper, we devote to giving an explicit description on the invertible maps on g that preserve staircase subalgebras.

LINEAR DILATATION OF QUASIREGULAR MAPPINGS IN SPACE

This paper presents an explicit upper bound for the linear dilatation of K- quasiregular (K-qr) mappings, which improves S. Rickman's [6, P.37] corresponding re- sult for K-qr mappings and generalizes P. Seittenranta's [7, Theorem 1.5] result for K- quasiconformal (K-qc) maps.

Solution of Nonlinear Advection-Diffusion Equations via Linear Fractional Map Type Nonlinear QCA

Linear fractional map type (LFMT) nonlinear QCA (NLQCA), one of the simplest reversible NLQCA is studied analytically as well as numerically. Linear advection equation or Time Dependent Schrödinger Equation (TDSE) is obtained from the continuum limit of linear QCA. Similarly it is found that some nonlinear advection-diffusion equations including inviscid Burgers equation and porous-medium equation are obtained from LFMT NLQCA.

Linear-fitting-based similarity coefficient map for tissue dissimilarity analysis in T2^＊-w magnetic resonance imaging

Similarity coefficient mapping（SCM） aims to improve the morphological evaluation of T＊2weighted magnetic resonance imaging（T＊2-w MRI）. However, how to interpret the generated SCM map is still pending. Moreover, is it probable to extract tissue dissimilarity messages based on the theory behind SCM？ The primary purpose of this paper is to address these two questions. First, the theory of SCM was interpreted from the perspective of linear fitting. Then, a term was embedded for tissue dissimilarity information. Finally, our method was validated with sixteen human brain image series from multiecho T＊2-w MRI. Generated maps were investigated from signal-to-noise ratio（SNR） and perceived visual quality, and then interpreted from intra- and inter-tissue intensity. Experimental results show that both perceptibility of anatomical structures and tissue contrast are improved. More importantly, tissue similarity or dissimilarity can be quantified and cross-validated from pixel intensity analysis. This method benefits image enhancement, tissue classification, malformation detection and morphological evaluation.

Dissipative Properties of ω-Order Preserving Partial Contraction Mapping in Semigroup of Linear Operator

This paper consists of dissipative properties and results of dissipation on infinitesimal generator of a C0-semigroup of ω-order preserving partial contraction mapping (ω-OCPn) in semigroup of linear operator. The purpose of this paper is to establish some dissipative properties on ω-OCPn which have been obtained in the various theorems (research results) and were proved.

Non-Linear Effects in Optical Systems by Lie Algebra and Symplectic Mapping

The use of signals of different frequencies determines the geometrical deviation with respect to the optical axes of a given beam. This angle can be determined by Sympletic Map (SM), a powerful and simple mathematical tool for the characterization and construction of images in Geometrical Optics. The Sympletic Map constitutes a Lie Group, with an algebra associated: the Lie Algebra. In general, the SM can be expressed as an infinite series, where each term corresponds to different contributions produced by the optical devices that constitute the optical system (lenses, apertures, bandwidth cutoff, etc.). The level of correction to be performed on the image to recover the original object is clear and controllable by SM. This formalism can be extended easily to physical optics to describe diffraction and interference phenomena.

An Automatic Approach for Satisfying Dose-Volume Constraints in Linear Fluence Map Optimization for IMPT

Prescriptions for radiation therapy are given in terms of dose-volume constraints (DVCs). Solving the fluence map optimization (FMO) problem while satisfying DVCs often requires a tedious trial-and-error for selecting appropriate dose control parameters on various organs. In this paper, we propose an iterative approach to satisfy DVCs using a multi-objective linear programming (LP) model for solving beamlet intensities. This algorithm, starting from arbitrary initial parameter values, gradually updates the values through an iterative solution process toward optimal solution. This method finds appropriate parameter values through the trade-off between OAR sparing and target coverage to improve the solution. We compared the plan quality and the satisfaction of the DVCs by the proposed algorithm with two nonlinear approaches: a nonlinear FMO model solved by using the L-BFGS algorithm and another approach solved by a commercial treatment planning system (Eclipse 8.9). We retrospectively selected from our institutional database five patients with lung cancer and one patient with prostate cancer for this study. Numerical results show that our approach successfully improved target coverage to meet the DVCs, while trying to keep corresponding OAR DVCs satisfied. The LBFGS algorithm for solving the nonlinear FMO model successfully satisfied the DVCs in three out of five test cases. However, there is no recourse in the nonlinear FMO model for correcting unsatisfied DVCs other than manually changing some parameter values through trial and error to derive a solution that more closely meets the DVC requirements. The LP-based heuristic algorithm outperformed the current treatment planning system in terms of DVC satisfaction. A major strength of the LP-based heuristic approach is that it is not sensitive to the starting condition.

UMTS Turbo码的滑动窗-线性-Log-MAP算法及VLSI设计

讨论UMTS中turbo码的设计,提出采用前向状态度量作为初始化精确值的滑动窗方法和改进的Log-MAP(线性Log-MAP)相结合的算法(称为滑动窗-线性-Log-MAP算法),设计出turbo码的VLSI结构,且在归一化处理时,介绍了一种可节约12.5%存储量的方法。FPGA设计结果显示在BER=10-5时,与最优算法Log-MAP相比,性能损失0.2dB,硬件资源节约30%,译码时延小,吞吐量大。