Sampling expansions associated with quaternion difference equations

Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete transform, we form a quaternion function space whose elements have sampling expansions. Moreover, through formulating boundary-value problems, we make a connection between a class of tridiagonal quaternion matrices and polynomials with quaternion coefficients. We show that for a tridiagonal symmetric quaternion matrix, one can always associate a quaternion characteristic polynomial whose roots are eigenvalues of the matrix. Several examples are given to illustrate the results

Floquet multipliers and the stability of periodic linear differential equations: a unified algorithm and its computer realization

Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers) and determine the stability of the periodic linear differential equations on time scales unifying discrete, continuous, and hybrid dynamics. Our approach is based on calculating the value of A and B (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. We obtain an explicit expression of A (see Theorem 4.1) by the method of variation and approximation theory and an explicit expression of B by Liouville's formula. Furthermore, a computer program is designed to realize our algorithm. Specifically, you can determine the stability of a second order periodic linear system, whether they are discrete, continuous or hybrid, as long as you enter the program codes associated with the parameters of the equation. In fact, few literatures have dealt with the algorithm to compute the Floquet multipliers, not mention to design the program for its computer realization. Our algorithm gives the explicit expressions of all Floquet multipliers and our computer program is based on the approximations of these explicit expressions. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of Floquet multipliers (see Theorem 4.2). Therefore, for any discrete periodic system, we can accurately determine the stability of the system even without computer! Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm

Flame Boundary Measurement Using an Electrostatic Sensor Array

Flame boundary is an important geometrical characteristic for the evaluation of flame properties such as heat release rate and radiation. Reliable and accurate measurement of flame boundary is desirable for the prediction of flame structure and the optimization of combustion systems. Such measurement will inform the designers and operators of the combustion systems. This paper presents for the first time a study of using an electrostatic sensor array for flame boundary measurement. The electrostatic sensor is placed in the vicinity of the flame to sense its movement through charge transfer. The principle, design, implementation and assessment of a measurement system based on this methodology are introduced. Comparative experimental investigations with a digital camera conducted on a laboratory-scale combustion test rig show that the electrostatic sensor can respond to the variation of the distance between the electrode and the flame boundary. Reconstruction of the flame boundary is achieved using a set of distance measurements obtained from a sensor array. For diffusion flames over the range of fuel flow rate 0.60-0.80 L/min and premixed flames over the range of equivalence ratio 1.27-3.81, experimental results show that the measurement system is capable of providing reliable measurement of the flame boundary. The correlation coefficients under all test conditions are mostly larger than 0.96, the mean relative errors within 7.4% and the relative root mean square errors within 0.09. More accurate flame boundary measurements are achieved for diffusion flames. In addition, the overall polarity of charges in a flame can be determined from the polarity of the sensor signal

Training Deeper Neural Machine Translation Models with Transparent Attention

While current state-of-the-art NMT models, such as RNN seq2seq and Transformers, possess a large number of parameters, they are still shallow in comparison to convolutional models used for both text and vision applications. In this work we attempt to train significantly (2-3x) deeper Transformer and Bi-RNN encoders for machine translation. We propose a simple modification to the attention mechanism that eases the optimization of deeper models, and results in consistent gains of 0.7-1.1 BLEU on the benchmark WMT'14 English-German and WMT'15 Czech-English tasks for both architectures.Comment: To appear in EMNLP 201