Localization in Stationary Non-equilibrium Solutions for Multicomponent Coagulation Systems

We consider the multicomponent Smoluchowski coagulation equation under non-equilibrium conditions induced either by a source term or via a constant flux constraint. We prove that the corresponding stationary non-equilibrium solutions have a universal localization property. More precisely, we show that these solutions asymptotically localize into a direction determined by the source or by a flux constraint: the ratio between monomers of a given type to the total number of monomers in the cluster becomes ever closer to a predetermined ratio as the cluster size is increased. The assumptions on the coagulation kernel are quite general, with isotropic power law bounds. The proof relies on a particular measure concentration estimate and on the control of asymptotic scaling of the solutions which is allowed by previously derived estimates on the mass current observable of the system.

An introduction to analysis of R\’enyi complexity ratio of quantum states for central potential

R\’enyi complexity ratio of two density functions is introduced for three and multidimensional quantum systems. Localization property of several density functions are defined and five theorems about near continuous property of R\’enyi complexity ratio are proved by Lebesgue measure. Some properties of R\’enyi complexity ratio are demonstrated and investigated for different quantum systems. Exact analytical forms of R\’enyi entropy, R\’enyi complexity ratio, statistical complexities based on R\’enyi entropy for integral order have been presented for solutions of pseudoharmonic and a family of isospectral potentials. Some properties of R\’enyi complexity ratio are verified for six diatomic molecules (CO, NO, N$_2$, CH, H$_2$, and ScH) and for other quantum systems.

Some results concerning localization property of generalized Herz, Herz-type Besov spaces and Herz-type Triebel-Lizorkin spaces

In this paper, based on generalized Herz-type function spaces $\dot{K}_{q}^{p}(\theta)$ were introduced by Y. Komori and K. Matsuoka in 2009, we define Herz-type Besov spaces $\dot{K}_{q}^{p}B_{\beta }^{s}(\theta)$ and Herz-type Triebel-Lizorkin spaces $\dot{K}_{q}^{p}F_{\beta }^{s}(\theta)$, which cover the Besov spaces and the Triebel-Lizorkin spaces in the homogeneous case, where $\theta=\left\{\theta(k)\right\} _{k\in\mathbb{Z}}$ is a sequence of non-negative numbers $\theta(k)$ such that \begin{equation*} C^{-1}2^{\delta (k-j)}\leq \frac{\theta(k)}{\theta(j)} \leq C2^{\alpha (k-j)},\quad k>j, \end{equation*} for some $C\geq 1$ ($\alpha$ and $\delta $ are numbers in $\mathbb{R}$). Further, under the condition mentioned above on ${\theta }$, we prove that $\dot{K}_{q}^{p}\left({\theta }\right)$ and $\dot{K}_{q}^{p}B_{\beta }^{s}\left({\theta }\right)$ are localizable in the $\ell _{q}$-norm for $p=q$, and $\dot{K}_{q}^{p}F_{\beta }^{s}\left({\theta }\right)$ is localizable in the $\ell _{q}$-norm, i.e. there exists $\varphi \in \mathcal{D}({\mathbb{R}}^{n})$ satisfying $\sum_{k\in \mathbb{Z}^{n}}\varphi \left( x-k\right) =1$, for any $x\in \mathbb{R}^{n}$, such that \begin{equation*} \left\Vert f|E\right\Vert \approx \Big(\underset{k\in \mathbb{Z}^{n}}{\sum }\left\Vert \varphi (\cdot-k)\cdot f|E\right\Vert ^{q}\Big)^{1/q}. \end{equation*} Results presented in this paper improve and generalize some known corresponding results in some function spaces.

On Computational Aspects of Krawtchouk Polynomials for High Orders

Discrete Krawtchouk polynomials are widely utilized in different fields for their remarkable characteristics, specifically, the localization property. Discrete orthogonal moments are utilized as a feature descriptor for images and video frames in computer vision applications. In this paper, we present a new method for computing discrete Krawtchouk polynomial coefficients swiftly and efficiently. The presented method proposes a new initial value that does not tend to be zero as the polynomial size increases. In addition, a combination of the existing recurrence relations is presented which are in the n- and x-directions. The utilized recurrence relations are developed to reduce the computational cost. The proposed method computes approximately 12.5% of the polynomial coefficients, and then symmetry relations are employed to compute the rest of the polynomial coefficients. The proposed method is evaluated against existing methods in terms of computational cost and maximum size can be generated. In addition, a reconstruction error analysis for image is performed using the proposed method for large signal sizes. The evaluation shows that the proposed method outperforms other existing methods.

Signature Recognition using 2D Discrete Wavelet Transforms

Signature recognition is one of the most secure techniques used for person identification. Wavelet transforms possess an extra time localization property over Fourier transforms, despite both are frequency localized. That is why; it is more useful to analyze the one dimensional and two dimensional signals both. First of all the features of a signature are extracted and then matching is performed. We have proposed feature extraction technique using discrete wavelet transforms level-1 and matching of signatures are performed using some statistical parameters of discrete wavelet coefficients using matching percentage, sum of absolute difference, mean square error and city block distance.

Face Identification Based on Discrete Wavelet Transform and Neural Networks

The subject paper presents implementation of a new automatic face recognition system. To formulate an automated framework for the recognition of human faces is a highly challenging endeavor. The face identification problem is particularly very crucial in the context of today’s rapid emergence of technological advancements with ever expansive requirements. It has also significant relevance in the related engineering disciplines of computer graphics, pattern recognition, psychology, image processing and artificial neural networks. This paper proposes a side-view face authentication approach based on discrete wavelet transform and artificial neural networks for the solution of the problem. A subset determination strategy that expands on the number of training samples and permits protection of the global information is discussed. The authentication technique involves image profile extraction, decomposition of the wavelets, splitting of the subsets and finally neural network verification. The procedure exploits the localization property of the wavelets in both the frequency and spatial domains, while maintaining the generalized properties of the neural networks. The realization strategy of the methodology was executed using MATLAB, demonstrating that the performance of the technique is quite satisfactory.