Representation of Algebraic Integers as Sum of Units over the Real Quadratic Fields

في هذا البحث تم تعميم نتائج الباحث جاكوبسن بأستخدام الوحدات الأساسية للحقل التربيعي الحقيقي ممثلة في الشرطين التاليين: عندما ,  والشرط الثاني هو , وبأستخدام هذه الشروط أستطعنا أثبات أن الحقل التربيعي   عندما الحقيقي  في الجموعة  حيث أن المجموعة  تمثل مجموعة كل الحقول التربيعية التي تمثل اعدادها كتجمع لوحدات بتكرار عدده . وبذلك يتم تصنيف الحقول  وفق للقيم t في المجاميع  حيث أن جاكوبسن برهن أن الحقلين التربيعيين ينتميان للمجموعة  كذلك برهن الباحث (J.Silwa) بأن  هما الحقلان الوحيدان في المجموعة .  In this paper we generalize Jacobsons results by proving that any integer  in   is a square-free integer), belong to . All units of  are generated by the fundamental unit  having the forms our generalization build on using the conditions This leads us to classify the real quadratic fields  into the sets  Jacobsons results shows that  and Sliwa confirm that  and  are the only real quadratic fields in

Sums of Divisors, Units and Algebraic Integers

Abstract not provided

Explicit upper bound for the function of sum of divisors

we developed the result proved by A.Eviwhere he proved the following  theore

A Combined EMD-ICA Analysis of Simultaneously Registered EEG-fMRI Data

Within a combined EEG-fMRI study of contour integration, we analyze responses to Gabor stimuli with an Empirical Mode Decomposition combined with an Independent Component Analysis. Generally, responses to different stimuli are very similar thus hard to differentiate. EMD and ICA are used intermingled and not simply in a sequential way. This novel combination helps to suppress redundant modes resulting from an application of ensemble EMD alone. The simulation results show an improved mode separation quality. Hence, the proposed method is an efficient data analysis tool to clearly reveal differences between similar response signals and activity distributions

Combining EMD with ICA to analyze combined EEG-fMRI data.

Within a combined EEG-fMRI study of contour integration, we analyze responses to Gabor stimuli with an Empirical Mode Decomposition combined with an Independent Component Analysis. Generally, responses to different stimuli are very similar thus hard to differentiate. EMD and ICA are used intermingled and not simply in a sequential way. This novel combination helps to suppress redundant modes resulting from an application of ensemble EMD alone. The simulation results show an improved mode separation quality. Hence, the proposed method is an efficient data analysis tool to clearly reveal differences between similar response signals and activity distributions

EMDLAB: A toolbox for analysis of single-trial {EEG} dynamics using empirical mode decomposition

Background Empirical mode decomposition (EMD) is an empirical data decomposition technique. Recently there is growing interest in applying EMD in the biomedical field. New method EMDLAB is an extensible plug-in for the EEGLAB toolbox, which is an open software environment for electrophysiological data analysis. Results EMDLAB can be used to perform, easily and effectively, four common types of EMD: plain EMD, ensemble EMD (EEMD), weighted sliding EMD (wSEMD) and multivariate EMD (MEMD) on EEG data. In addition, EMDLAB is a user-friendly toolbox and closely implemented in the EEGLAB toolbox. Comparison with existing methods EMDLAB gains an advantage over other open-source toolboxes by exploiting the advantageous visualization capabilities of EEGLAB for extracted intrinsic mode functions (IMFs) and Event-Related Modes (ERMs) of the signal. Conclusions EMDLAB is a reliable, efficient, and automated solution for extracting and visualizing the extracted IMFs and ERMs by EMD algorithms in EEG study

Functional Biomedical Images of Alzheimer's Disease a Green's Function based Empirical Mode Decomposition Study

Positron emission tomography (PET) provides a functional imaging modality to detect signs of dementias in human brains. Two-dimensional empirical mode decomposition (2D-EMD) provides means to analyze such images. It decomposes the latter into characteristic modes which represent textures on different spatial scales. These textures provide informative features for subsequent classification purposes. The study proposes a new EMD variant which relies on a Green's function based estimation method including a tension parameter to fast and reliably estimate the envelope hypersurfaces interpolating extremal points of the two-dimensional intensity distrubution of the images. The new method represents a fast and stable bi-dimensional EMD which speeds up computations roughly 100-fold. In combination with proper classifiers these exploratory feature extraction techniques can form a computer aided diagnosis (CAD) system to assist clinicians in identifying various diseases from functional images alone. PET images of subjects suffering from Alzheimer's disease are taken to illustrate this ability

Functional biomedical images of Alzheimer’s disease. A green’s function-based empirical mode decomposition study

International audiencePositron emission tomography (PET) provides a functional imaging modality to detect signs of dementias in human brains. Two-dimensional empirical mode decomposition (2D-EMD) provides means to analyze such images. It decomposes the latter into characteristic modes which represent textures on different spatial scales. These textures provide informative features for subsequent classification purposes. The study proposes a new EMD variant which relies on a Green’s function based estimation method including a tension parameter to fast and reliably estimate the envelope hypersurfaces interpolating extremal points of the twodimensional intensity distrubution of the images. The new method represents a fast and stable bi-dimensional EMD which speeds up computations roughly 100-fold. In combination with proper classifiers these exploratory feature extraction techniques can form a computer aided diagnosis (CAD) system to assist clinicians in identifying various diseases from functional images alone. PET images of subjects suffering from Alzheimer’s disease are taken to illustrate this ability