Initial and Stopping Condition in Possibility Principal Factor Rotation

Uemura [1] discovered the mapping formula for Type 1 Vague events and presented an alternative problem as an example of its application. Since it is well known that the alternative problem leads to sequential Bayesian inference, the flow of subsequent research was to make the mapping formula multidimensional, to introduce the concept of time, and to derive a Markov (decision) process. Furthermore, we formulated stochastic differential equations to derive them [2]. This paper refers to type 2 vague events based on a second-order mapping equation. This quadratic mapping formula gives a certain rotation named as possibility principal factor rotation by transforming a non-mapping function by a relation between two mapping functions. In addition, the derivation of the Type 2 Complex Markov process and the initial and stopping conditions in this rotation are mentioned. .

Construction of Multivariate Tight Framelet Packets Associated with Dilation Matrix

In this paper, we present a method for constructing multivariate tight framelet packets associated with an arbitrary dilation matrix using unitary extension principles. We also prove how to construct various tight frames for L2 （JRa） by replacing some mother framelets.

A Characterization of MRA Based Wavelet Frames Generated by the Walsh Polynomials

Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly- nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11（6） （2013）, 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.

Characterizations of L_p(R) Using Tight Wavelet Frames

In this paper,a Littlewood-Paley function characterization of the spaces L p（R）,1〈p〈∞,is first established by means of the equivalent conditions of tight wavelet frames,wherein the Littlewood-Paley function is associated with a tight wavelet frame generated by the so-called extension principles.With the above characterization,another characterization of L p（R）,1〈p〈∞,is also established in terms of the weighted l 2-norm of the wavelet frame coefficients,which can be a useful tool in harmonic analysis,approximation theory,and image processing.

Linguistic dynamic systems based on computing with words and their stabilities

Linguistic dynamic systems （LDS） are the systems based on computing with words （CW） instead of computing with numbers or symbols. In this paper, LDS are divided into two types： type-I LDS being converted from conventional dynamical systems （CDS） by using extension principle and type-II LDS by using fuzzy logic rules. For type-I LDS, the method of endograph is provided to discuss the stabilities of type-I LDS and two cases of stabilities of logistic mappings： one is the states being abstracted and the other is parameters also being abstracted. For type-Ⅱ LDS, the method of degree of match is used to discuss the dynamical behavior of arbitrary initial words under fuzzy rule.

Construction of periodic wavelet frames with dilation matrix

An important tool for the construction of periodic wavelet frame with the help of extension principles was presented in the Fourier domain by Zhang and Saito [Appl. Comput. Harmon. Anal., 2008, 125： 68-186]. We extend their results to the dilation matrix cases in two aspects. We first show that the periodization of any wavelet frame constructed by the unitary extension principle formulated by Ron and Shen is still a periodic wavelet frame under weaker conditions than that given by Zhang and Saito, and then prove that the periodization of those generated by the mixed extension principle is also a periodic wavelet frame if the scaling functions have compact supports.

General Schemes of Iterative Optimization with Applications to Optimal Control Problems

A general(abstract)scheme of iterative improvement and optimization on the base of extension,localization principles which would help to generate new concrete methods and algorithms for new problems is proposed.Application to optimal control problems for continuous systems is considered.Visual example is given.

Multivariate quasi-tight framelets with high balancing orders derived from any compactly supported refinable vector functions

Generalizing wavelets by adding desired redundancy and flexibility,framelets(i.e.,wavelet frames)are of interest and importance in many applications such as image processing and numerical algorithms.Several key properties of framelets are high vanishing moments for sparse multiscale representation,fast framelet transforms for numerical efficiency,and redundancy for robustness.However,it is a challenging problem to study and construct multivariate nonseparable framelets,mainly due to their intrinsic connections to factorization and syzygy modules of multivariate polynomial matrices.Moreover,all the known multivariate tight framelets derived from spline refinable scalar functions have only one vanishing moment,and framelets derived from refinable vector functions are barely studied yet in the literature.In this paper,we circumvent the above difficulties through the approach of quasi-tight framelets,which behave almost identically to tight framelets.Employing the popular oblique extension principle(OEP),from an arbitrary compactly supported M-refinable vector functionφwith multiplicity greater than one,we prove that we can always derive fromφa compactly supported multivariate quasi-tight framelet such that:(i)all the framelet generators have the highest possible order of vanishing moments;(ii)its associated fast framelet transform has the highest balancing order and is compact.For a refinable scalar functionφ(i.e.,its multiplicity is one),the above item(ii)often cannot be achieved intrinsically but we show that we can always construct a compactly supported OEP-based multivariate quasi-tight framelet derived fromφsatisfying item(i).We point out that constructing OEP-based quasi-tight framelets is closely related to the generalized spectral factorization of Hermitian trigonometric polynomial matrices.Our proof is critically built on a newly developed result on the normal form of a matrix-valued filter,which is of interest and importance in itself for greatly facilitating the study of refinable vector functions and multiwavelets/multiframelets.This paper provides a comprehensive investigation on OEP-based multivariate quasi-tight multiframelets and their associated framelet transforms with high balancing orders.This deepens our theoretical understanding of multivariate quasi-tight multiframelets and their associated fast multiframelet transforms.

On a Class of Weak Nonhomogeneous Affine Bi-frames for Reducing Subspaces of L^2(R^d)

For refinable functiombased affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame （WNABF）. Under the setting of reducing subspaces of L2（Rd）, we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.