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.

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.

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.

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.

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.

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.

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

它被看那从 pro-C 的连续线性地图的一个 n × n 矩阵[*]到 L (H)的代数学 A ，验证完全的确实的条件，具有形式[ V [*] T ij Φ(· ) V ]我， j=1 [ n ]，在Φ 是 Hilbert 空间 K 上的 A 的一个代表的地方， V 是从 H 的一个围住的线性操作员到 K ，；[T ij ] n i， j=1 [n ] 是在 C 的一个积极元素[*] 在Φ(A 的 commutant 上的所有 n × n 矩阵的代数学) 在 L (K) 。这概括 C. Y 的结果。在 Proc 的 Suen。Amer。数学。Soc， 112 (3 ) ， 1991， 709 712。另外，这构造的一个 covariant 版本被给。

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.

A NEW SIMPLE 2-D PIECEWISE LINEAR MAP

A new simple piecewise linear map of the plane is presented and analyzed,then a detailedstudy of its dynamical behaviour is described,along with some other dynamical phenomena,especiallyfixed points and their stability,observation of a new chaotic attractors obtained via border collisionbifurcation.An important result about coexisting chaotic attractors is also numerically studied anddiscussed.

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.

Combinations of Right Half-Plane Mappings and Vertical Strip Mappings Convex in the Vertical Direction

In this paper, we explore the linear combinations of right half-plane mappings and vertical strip mappings. We demonstrate that the combinations of these harmonic mappings are convex in the vertical direction provided they are locally univalent and sense-preserving. Furthermore, we extend this analysis to a more general case by setting specific conditions. Additionally, we take some common parameters such as as the dilatation of these harmonic mappings, and prove the sufficient conditions that their combinations are locally univalent and convex in the vertical direction. Several examples are constructed by the Mathematica software to demonstrate our main results.

基于LLM的金融市场波动率高频数据异常检测方法

金融市场高频数据包括时间序列数据和其他宏观经济指标,通常具有高维特征.其处理需要更复杂的算法,易产生较高的模型过拟合风险.基于此,提出基于局部线性映射(Local Linear Mapping,LLM)的金融市场波动率高频数据异常检测方法,对各个高频数据目标的日平均序列数据进行标准化处理,在数据筛选时,使用标准化处理设定相关阈值,将不同维度的数据转化为相同的尺度,并利用连通图算法,将具有边连接的金融市场波动率高频数据划分至一个群组内,计算待检测高频数据阈值,采用局部线性映射,完成金融市场波动率高频数据异常检测.实验结果表明:所提方法在TPR为0.98时,ROC曲线稳定运行,贡献因子为1.287,重构误差为1.6%,能够以最快速度使训练集异常检测的损失值达到稳定.

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.

2种低秩矩阵恢复优化模型的误差估计定理

近年来低秩矩阵恢复问题逐渐引起人们的关注,类似于向量稀疏恢复的充分条件是需要测量矩阵满足限制等距性质,低秩矩阵恢复的充分条件是需要一个线性映射满足限制等距性质.低秩矩阵恢复时所需的模型大致分为有噪和无噪两种恢复模型,恢复出来的结果需要不同的限制等距常数界去保证.文章证明了这2种优化模型的误差界估计定理,并得出了2种不同的限制等距常数界.