q-Shock Soliton Evolution

By generating function based on the Jackson's q-exponential function and standard exponential function, we introduce a new q-analogue of Hermite and Kampe-de Feriet polynomials. In contrast to standard Hermite polynomials, with triple recurrence relation, our polynomials satisfy multiple term recurrence relation, derived by the q-logarithmic function. It allow us to introduce the q-Heat equation with standard time evolution and the q-deformed space derivative. We found solution of this equation in terms of q-Kampe-de Feriet polynomials with arbitrary number of moving zeros, and solved the initial value problem in operator form. By q-analog of the Cole-Hopf transformation we find a new q-deformed Burgers type nonlinear equation with cubic nonlinearity. Regular everywhere single and multiple q-Shock soliton solutions and their time evolution are studied. A novel, self-similarity property of these q-shock solitons is found. The results are extended to the time dependent q-Schr\"{o}dinger equation and the q-Madelung fluid type representation is derived.Comment: 15 pages, 6 figure

Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem

In this paper, we develop a Bayesian evidence maximization framework to solve the sparse non-negative least squares (S-NNLS) problem. We introduce a family of probability densities referred to as the Rectified Gaussian Scale Mixture (R- GSM) to model the sparsity enforcing prior distribution for the solution. The R-GSM prior encompasses a variety of heavy-tailed densities such as the rectified Laplacian and rectified Student- t distributions with a proper choice of the mixing density. We utilize the hierarchical representation induced by the R-GSM prior and develop an evidence maximization framework based on the Expectation-Maximization (EM) algorithm. Using the EM based method, we estimate the hyper-parameters and obtain a point estimate for the solution. We refer to the proposed method as rectified sparse Bayesian learning (R-SBL). We provide four R- SBL variants that offer a range of options for computational complexity and the quality of the E-step computation. These methods include the Markov chain Monte Carlo EM, linear minimum mean-square-error estimation, approximate message passing and a diagonal approximation. Using numerical experiments, we show that the proposed R-SBL method outperforms existing S-NNLS solvers in terms of both signal and support recovery performance, and is also very robust against the structure of the design matrix.Comment: Under Review by IEEE Transactions on Signal Processin

A Unified Framework for Sparse Non-Negative Least Squares using Multiplicative Updates and the Non-Negative Matrix Factorization Problem

We study the sparse non-negative least squares (S-NNLS) problem. S-NNLS occurs naturally in a wide variety of applications where an unknown, non-negative quantity must be recovered from linear measurements. We present a unified framework for S-NNLS based on a rectified power exponential scale mixture prior on the sparse codes. We show that the proposed framework encompasses a large class of S-NNLS algorithms and provide a computationally efficient inference procedure based on multiplicative update rules. Such update rules are convenient for solving large sets of S-NNLS problems simultaneously, which is required in contexts like sparse non-negative matrix factorization (S-NMF). We provide theoretical justification for the proposed approach by showing that the local minima of the objective function being optimized are sparse and the S-NNLS algorithms presented are guaranteed to converge to a set of stationary points of the objective function. We then extend our framework to S-NMF, showing that our framework leads to many well known S-NMF algorithms under specific choices of prior and providing a guarantee that a popular subclass of the proposed algorithms converges to a set of stationary points of the objective function. Finally, we study the performance of the proposed approaches on synthetic and real-world data.Comment: To appear in Signal Processin

q-Analogue of Shock Soliton Solution

By using Jackson's q-exponential function we introduce the generating function, the recursive formulas and the second order q-differential equation for the q-Hermite polynomials. This allows us to solve the q-heat equation in terms of q-Kampe de Feriet polynomials with arbitrary N moving zeroes, and to find operator solution for the Initial Value Problem for the q-heat equation. By the q-analog of the Cole-Hopf transformation we construct the q-Burgers type nonlinear heat equation with quadratic dispersion and the cubic nonlinearity. In q -> 1 limit it reduces to the standard Burgers equation. Exact solutions for the q-Burgers equation in the form of moving poles, singular and regular q-shock soliton solutions are found.Comment: 13 pages, 5 figure

Radiographic Evaluation of Current Status of Permanent Lower First and Second Molars in Geriatric Patients in Turkish Population

Objective:Our study aimed to examine the effects of age and gender for both tooth groups by determining the current status of the permanent mandibular first and second molars, presence/absence status of them and disease/health status of them in geriatric patients.Methods:Panoramic radiographs of 1,500 patients, 815 women and 685 men, aged 65 and over, who were admitted to Bezmialem Vakıf University Faculty of Dentistry between 2019-2021, were examined. A single investigator reviewed each patient’s X-ray. The patients were classified according to their age groups and genders. Age classification was made as 65-74 years, 75-84 years and ≥85 years. Conditions of permanent lower first and second molars were recorded as present or absent. If present, it was reported whether healthy, canal treated, filled orroot canal treated + prosthetic restoration. Root residue and the presence of implants were also noted.Results:Among 1,500 geriatric patients who were admitted to Bezmialem Vakıf University Faculty of Dentistry between 2019-2021, 1,127 (75.1%) were in the 65-74 years of age group, 321 (21.4) in the 75-84 years of age group, and 52 (3.5%) in the ≥85 years of age group. Of the permanent left mandibular first molars, 6.7% were healthy and 72.1% were absent. Of the permanent left mandibular second molars, 10.3% were healthy and 67% were absent. While 6.1% of the permanent right mandibular first molars were healthy, 73.3% were absent. On the other hand, 10% of the permanent right mandibular second molars were healthy, while 66.8% were absent. While the number of geriatric patients with no missing teeth was 97, the number of patients with four missing teeth was found to be 785.Conclusion:The survival percentage of permanent mandibular second molars is higher than permanent mandibular first molars. Despite the prolongation of life expectancy, there is no increase in the frequency of permanent molars

New description of vagal nerve commanted intrapancreatic taste buds and blood glucose level: An experimental analysis

Introduction: There have been thousands of neurochemical mechanism about blood glucose level regulation, but intrapancreatic taste buds and their roles in blood glucose level has not been described. We aimed to investigate if there are taste buds cored neural networks in the pancreas, and there is any relationship between blood glucose levels. Methods: This examination was done on 32 chosen rats with their glucose levels. Animals are divided into owned blood glucose levels. If mean glucose levels were equal to 105 ± 10 mg/dL accepted as euglycemic (G-I; n = 14), 142 ± 18 mg/dL values accepted as hyperglycemic (G-II; n = 9) and 89 ± 9 mg/dL accepted as hypoglycemic (G-III; n = 9). After the experiment, animals were sacrificed under general anesthesia. Their pancreatic tissues were examined histological methods and numbers of newly described taste bud networks analyzed by Stereological methods. Results compared with Mann-Whitney U test P < 0.005 considered as significant. Results: The mean normal blood glucose level (mg/dL) and taste bud network densities of per cm3 were: 105 ± 10 mg/dL; 156±21 in G-I; 142 ± 18 mg/dL and 95 ± 14 in G-II and 89 ± 9 mg/dL and 232 ± 34 in G-III. P values as follows: P < 0.001 of G-II/G-I; P < 0.005 of G-III/G-I and P < 0.0001 of G-III/G-II. We detected periarterial located taste buds like cell clusters and peripherally located ganglia connected with Langerhans cells via thin nerve fibers. There was an inverse relationship between the number of taste buds networks and blood glucose level. Conclusion: Newly described intrapancreatic taste buds may have an important role in the regulation of blood glucose level

Rectified Sparse Bayesian Learning and Effects and Limitations of Nuisance Regression in Functional MRI

This dissertation considers the problems of sparse signal recovery (SSR) and nuisance regression in functional MRI (fMRI). The first part of the dissertation introduces a Bayesian framework to recover sparse non-negative solutions in under-determined systems of linear equations. A novel class of probability density functions named Rectified Gaussian Scale Mixtures (R-GSM) is proposed to model the sparse non-negative solution of interest. A Bayesian inference algorithm called Rectified Sparse Bayesian Learning (R-SBL) is developed, which robustly recovers the solution in numerous experimental settings and outperforms the state-of-the-art SSR approaches by a large margin.The rest of the dissertation investigates the effects of nuisance regression in fMRI. Chapter 3 proposes a mathematical framework to investigate the effects of global signal regression (GSR). GSR is a widely used nuisance removal approach in resting-state fMRI, however its use has been controversial since it introduces artifactual anti-correlations between pairs of fMRI signals. The proposed framework shows that the main effects of GSR can be well-approximated as a temporal down-weighting or temporal censoring process, in which the data from time points with relatively large GS magnitudes are greatly attenuated (or censored) while data from time points with relatively small GS magnitudes are largely retained. The censoring approximation reveals that the anti-correlated networks are intrinsic to the brain's functional organization and are not simply an artifact of GSR.In Chapters 4 and 5, the effects of nuisance terms on the relationship between pairs of fMRI signals both before and after nuisance regression are investigated. It is shown that geometric norms of various nuisance regressors can significantly influence the correlation-based functional connectivity (FC) estimates in both static FC and dynamic FC studies. It is demonstrated that nuisance regression is largely ineffective in removing the significant correlations observed between FC estimates and nuisance norms. Consequently, a mathematical bound is derived on the difference between correlation coefficients before and after nuisance regression. This bound restricts the removal of nuisance norm effects from FC estimates