Schulz Method Research Articles

Improving Newton–Schulz Method for Approximating Matrix Generalized Inverse by Using Schemes with Memory

Some iterative schemes with memory were designed for approximating the inverse of a nonsingular square complex matrix and the Moore–Penrose inverse of a singular square matrix or an arbitrary m×n complex matrix. A Kurchatov-type scheme and Steffensen’s method with memory were developed for estimating these types of inverses, improving, in the second case, the order of convergence of the Newton–Schulz scheme. The convergence and its order were studied in the four cases, and their stability was checked as discrete dynamical systems. With large matrices, some numerical examples are presented to confirm the theoretical results and to compare the results obtained with the proposed methods with those provided by other known ones.

On the Estimation and Comparison of Lifetime Morbid Risks

Lifetime morbid risks are usually determined either by the Kaplan- Meier product limit estimator or by simpler estimators such as the lifetime prevalence, the Weinberg method or the Schulz method, which can be con- sidered an elaboration of the Weinberg method. We show that the Kaplan-Meier product limit estimator of lifetime mor- bid risk may yield unreliable estimates. Although the simplicity of the Schulz method and the Weinberg method is appealing, we suggest that under a proper model, those methods can be replaced by the original Stromgren estimator which is almost equally simple, and more accurate. Increased ac- curacy is achieved when the investigators have prior indication regarding the distribution of the ages at onset for those affected by the disorder, and even when that indication is vague and only limited knowledge of the distribution is available.

Development of a multi-agent system for solving domain dictionary construction problem

The object of research is the use of multi-agent systems for text data mining. The need for this study arose with a tendency to increase the amount of textual information generated in the world. Accordingly, it is necessary to develop and research methods of its processing, as well as ways to use the results of this processing, because the methods can’t exist in isolation from practice. At the same time, there is a development of multi-agent systems (MAS), where agents are endowed with some kind of intelligence, these systems can be easily scaled. The use of MAS for text analysis is a promising area.The following methods of text data analysis were used in this study: TF-IDF and RAKE methods, Word2Vec neural network models, and TextRank. The algorithms were compared for their work and the results were compared. The corpus of documents (10–12 texts, 5732–12331 words) from the subject areas of physics and biology were used as a test set. According to the results of the study, one method was chosen, on the basis of which the MAS was built to solve the problem. Additionally, Schulze methods (with one and several winners) were used for voting. With the received system additional researches concerning accuracy and speed of work, and also – influence are carried out system parameters for its operation.It has been found that TF-IDF-based analysis is useful for finding terms in documents with a weak context. The resulting system shows an accuracy of 75 % (3 of the 4 words proposed by the system are terms). The maximum operating time on test cases is 2–3 seconds, which is achieved through the use of parallel calculations and modification of the Schulze method. The results obtained in this paper are heuristic (ontology is a rather vague concept) and require additional elaboration by experts in the relevant fields. However, the results are positive within this experiment

Dedicated Poster Abstracts

Background/Purpose: Virtual dissection is performed with CT scans on near life-size touchscreens, which function like hospital radiology workstations. Students work together to manipulate the data and perform their dissection. The purpose of this study was to develop and qualitatively assess the educational value of virtual dissection laboratories for first year medical students as well as to understand students' preferred pedagogical approaches for learning from this new technology.Methods: 105 first-year medical students participated in a case-based virtual dissection curriculum and were invited to complete a theory-based post-experience survey. Eight unique cases were selected based on the first-year curricular objectives and in groups of 6-8, students reviewed the cases with a radiologist. First, students' reactions to virtual dissection were measured by three constructs using a 5-point Likert scale: quality of curriculum design (11 questions), impact on learning (7 questions), and comfort with technology (3 questions). Second, students ranked the usefulness of six pedagogical approaches for this technology. Responses were tabulated and rank order item lists were generated statistically using the Schulze method where appropriate.Results: The survey response rate was 83% (87/105). Overall, students' reactions to virtual dissection were positive across all three measured constructs. Most students indicated that the cases were of an appropriate level of difficulty (90%) and that virtual dissection improved their understanding of disease and pathology (89%), the clinical relevance of anatomy (77%), and visuospatial relationships (64%). Almost all students (94%) reported that the curriculum improved understanding of the role of the radiologist in patient care. Students felt that the “very useful” pedagogical approaches were small group demonstration (68%) and problem-based learning (51%).Conclusion: First-year medical students perceive virtual dissection as a valuable tool for learning anatomy and radiology. This technology allows real-life clinical cases to be introduced into pre-clinical learning, thereby providing opportunities to strengthen the vertical integration in medical curricula.

Private computation of the Schulze voting method over the cloud

In this work, we propose an algorithm that computes the Schulze voting method privately and finds the winner of the candidate without revealing the preferences of the voter. Users often outsource data to the cloud to get the result of an intended computation. Many a time, the user would like to keep the data and the result of the delegated computation private due to its sensitive nature. It is possible using data analytics to extract private information from a person. Hence, there is a need to perform computation on encrypted data, which can protect from a leak of private information. Homomorphic encryption (HE), allows computation on encrypted data. HE scheme takes input data in encrypted form and produces output in encrypted form. This encrypted output can not be decrypted without the private key. The Schulze method involves computation of a more complex function known as strength of strongest path. This is challenging to implement privately because it requires the evaluation of several functions over the ciphertexts. We use the Levelled-Brakerski–Gentry–Vaikuntanathan (BGV) fully homomorphic encryption (FHE) scheme to privately compute the strongest paths in a weighted graph using a modified image result of the Floyd–Warshall algorithm. We evaluated our proposed algorithm using an FHE library HElib. Besides, we implemented it for various parameters like time and number of levels in the modulus chain and also evaluated the size of the public key and secret key used for encryption and decryption. From the implementation results, we found that if we increase the number of levels, then the computation and communication complexity will also increase. Therefore, for efficient computation, we need to choose the optimal level.

Virtual Dissection with Clinical Radiology Cases Provides Educational Value to First Year Medical Students

In virtual dissection, three-dimensional computed tomography scans are viewed on a near-life size virtual dissection table and through touchscreen technology, students work together to manipulate the data to perform their dissection. The purpose of this study was to develop a Virtual Dissection Curriculum for first year medical students and to assess its educational value as well as students' preferred pedagogy for learning with this new technology. One hundred and five first-year medical students participated in a case-based virtual dissection curriculum and were invited to complete a theory-based post experience survey. Eight unique clinical cases were selected based on the first-year curricular objectives and divided into four 30-minute sessions. In groups of 6-8, students reviewed the cases with a radiologist. First, students' reactions to virtual dissection were measured by three constructs using a 5-point Likert scale: quality of curriculum design (11 questions), impact on learning (7 questions), and comfort with technology (3 questions). Second, students ranked the usefulness of six pedagogical approaches for this technology. Responses were tabulated and rank order item lists were generated statistically using the Schulze method where appropriate. The survey response rate was 83% (87/105). Overall, students' reactions to virtual dissection were positive across all three measured constructs. Most students indicated that the cases were of an appropriate level of difficulty (90%) and that virtual dissection improved their understanding of disease and pathology (89%), the clinical relevance of anatomy (77%), and visuospatial relationships (64%). Almost all students (94%) reported that the curriculum improved understanding of the role of the radiologist in patient care. Students felt that the "very useful" pedagogical approaches were small group demonstration (68%) and problem-based learning (51%). First-year medical students perceive the use of virtual dissection as a valuable tool for learning anatomy and radiology. This technology enables the integration of clinical cases and radiology content into preclinical learning.

Virtual Dissection Adds Educational Value to a Traditional Medical Undergraduate Cadaveric Anatomy Course

BackgroundVirtual dissection is performed on near life‐size anatomy visualization tables (AVTs), which are like hospital radiology workstations. Patient CT scans are loaded into these tables and through powerful software interactions students work together to manipulate the data and perform their dissection. The purpose of this study was to develop virtual dissection laboratories for first year medical students and to qualitatively assess the educational value of these sessions as well as students' preferred pedagogical approaches for this new technology.MethodsAll students in the first‐year medical undergraduate program were included in this study (n = 292). A Basic Virtual Dissection Curriculum focused on normal anatomy was offered to all students concurrently with the Cadaveric Laboratory Sessions and an Advanced Virtual Dissection Curriculum focused on pathology was offered as 4 extra‐curricular sessions. 36.6% of students participated in the Advanced Curriculum. Following the course, both groups of students were surveyed to determine their attitude toward virtual dissection and the pedagogical approaches they perceived to be the most useful for this technology. Results were statistically analyzed using the Schulze method.ResultsThe response rate for the Basic Curriculum was 69.2% and the response rate for the Advanced Curriculum was 82.9%. 93% indicated that virtual dissection was “definitely” a valuable addition to the anatomy lab. 89% of respondents “agreed” or “strongly agreed” that AVT virtual dissection improved their understanding of disease and pathology. They reported that the aortic aneurysm case was the most memorable case because the imaging made it easier to understand the pathogenesis of the disease. Students felt that small group demonstration and problem‐based learning would be the best teaching approaches for this technology.ConclusionsVirtual dissection adds educational value to undergraduate anatomy teaching, primarily because it provides students with a clinical context for the anatomy they are learning.This abstract is from the Experimental Biology 2018 Meeting. There is no full text article associated with this abstract published in The FASEB Journal.

Rethinking the PGY-1 Basic Clinical Year: A Canadian National Survey of Its Educational Value for Diagnostic Radiology Residents

Recently, the relevance of the postgraduate year 1 (PGY-1) Basic Clinical Year for radiology residents has been questioned. The purpose of this study was to determine the attitude of radiologists and trainees toward this year and which clinical rotations they perceived as most valuable to clinical practice. Following institutional review board approval, an anonymous online survey was administered to Canadian radiologists and radiology trainees. In addition to reporting demographic information, respondents were asked to rank the usefulness of individual rotations on a five-point Likert scale. To assess whether there are differences in the ratings and therefore rankings of the rotations by gender, position, and level of training, the Kruskal-Wallis one-way analysis of variance test was used with significance defined as P < .05. The Schulze method was used to rank the perceived usefulness of clinical rotations considered. Of the 275 respondents, 73.1% were male and 47.3% were trainees. A total of 71.3% of respondents were in favor of the basic clinical year, whereas 16.4% opposed. There was a statistically significant difference between the responses of staff radiologists and trainees, with the staff more strongly favoring the Basic Clinical Year (84.1%) than the trainees (56.9%) (P < .0001). As a whole, the respondents favored general surgery rotations as most relevant to their clinical practice (agreement rate of 48.3%). Interventional radiologists found general and subspecialty surgical rotations to be equally relevant. The rotations deemed to be "essential" were emergency medicine (48.7%) and general surgery (46.6%), and the rotations deemed to be "very useful" were orthopedics (45.8%), trauma (44.4%), neurosurgery (43.3%), neurology (42.2%), and hepatobiliary surgery (38.9%). There was no statistical difference between the respondents' choices based on their level of experience and scope of practice. Most radiologists and radiology trainees were in favor of completing the PGY-1 Basic Clinical Year. However, programs should maximize the education value of this year by including more of the top-ranked rotations. As the practice of radiology evolves, it is important to ensure that training paradigms continue to prepare residents for independent practice.

Физиологические критерии динамики овладения методикой аутогенной тренировки

The purpose of study was to investigate the short-term and long-term effects of the autogenous training (AT) and to search the evaluation criteria of successfulness of AT mastering. It was carried out a comparative analysis of dynamics of changes of the galvanic skin response (GSR), ECG and infrared irradiation of hands, as during the session, as well as at the initial and final stage of the AT course. 10 patients have been studied: 5 women and 5 men aged 19-21 years. AT training was implemented basing on the classic Schulz method for 20 sessions. On the basis of the dynamics of physiological characteristics it was established that the GSR reduction is the most representative indicator of the short-term adaptation to AT. The ability to voluntarily prolong the R-R interval was determined as a criterion of successfulness of mastering of AT. The ability to voluntarily increase temperature of the hands during the AT session was determined as the most significant parameter of the long-term adaptation to AT. The revealed physiological criteria of dynamics of acquisition of AT can be used in practice; availability of instrumental diagnostics means, including the portable infrared thermometer CEM TermoDiagnostics and the automatic tonometer, can contribute to a more widespread application of AT tech-niques.

On the extension of Householder’s method for weighted Moore–Penrose inverse

This note discusses how to extend and use the well-known method of Householder (1964) [6] for finding weighted Moore–Penrose inverse. The discussions cover both theoretical and numerical aspects. An approach for accelerating the initial phase of convergence will be contributed. Some tests will also be employed to check the validity and superiority of the results over the Schulz method.

Some Major Features in Texture and Anisotropy Field

Some Major Features in Texture and Anisotropy Field

A new inversion-free method for a rational matrix equation

Motivated by the classical Newton–Schulz method for finding the inverse of a nonsingular matrix, we develop a new inversion-free method for obtaining the minimal Hermitian positive definite solution of the matrix rational equation X + A ∗ X - 1 A = I , where I is the identity matrix and A is a given nonsingular matrix. We present convergence results and discuss stability properties when the method starts from the available matrix AA ∗ . We also present numerical results to compare our proposal with some previously developed inversion-free techniques for solving the same rational matrix equation.

Geophysical exploration for gold and associated minerals, case study: Wadi El Beida area, South Eastern Desert, Egypt

The occurrences of gold and disseminated sulfides lie as a part of the shearing fault zone that extends from the north to the south of the study area for a length of about 25 km. The gold and disseminated sulfides are located on the alteration shear zone which is composed of quartz–feldspathic highly ferruginated rock (gossans) occupying the eastern and central parts of the area. Mineralogical analyses that were done on bedrock samples of the oxidized and alteration zones indicated that there are two anomalous spots of gold contents; the first one has values ranging from 5 to 49 g ton−1 and the second anomaly has values ranging from 150 to 502.5 g ton−1. Magnetic, self-potential, resistivity and induced polarization surveys were applied at Wadi El Beida area to delineate the mineral ore deposits in terms of depths and extensions through the structural shearing zone. The quantitative interpretation of magnetic data was carried out by using two techniques; the first is 3D magnetic inversion using Euler deconvolution and the second is magnetic models using the MAGMOD program. The results of the magnetic interpretation indicated that the depths of such ore deposits range from 35.9 to 52.7 m and the half width ranged from 27.2 to 87.8 m. The SP contour maps show negative anomalies with ranges from −70 to 20 mV. Most of these anomalies occupy the shear, silicified zones, alterations and rock contacts. The SP anomalies are correlated with other geophysical ones and also with the geological sources. Quantitative interpretation was done on the selected anomalies along the coded lines on the normal SP contour map. The quantitative interpretation of self-potential anomalies (SP) was carried out using two techniques; the first is a new algorithm constructed by Monteiro Santos (2009) using particle swarm optimization (PSO) and the second is the code constructed by Caglar (2000). The depths range from 20 to 60 m. The gradient resistivity survey was carried out simultaneously with IP measurements. The low-resistivity zones coincide with the altered and sheared acidic meta-volcanics. The quantitative interpretation technique determined the conductive bodies' parameters using the Schulz method (1985) where the depth to the top of the ore body ranged from 21 to 62 m while the maximum width ranged from 52 to 165 m. The induced polarization-chargeability data were measured in the time domain. The positive anomalies on the IP-chargeability map coincide with the sites of alterations, shears and contact zones. Four dipole–dipole sections were carried out along the anomalous sites selected from the constructed maps in the study area and were inverted using the RES2DINV program. The results of resistivity and IP inversions indicated that there are conductive and chargeable bodies at depths ranging from 15 to 65 m. According to the integrated geophysical results, two suggested borehole sites were selected before carrying out any future mining projects.

Crystallization Kinetics and Texture Evolution in Iron-Based Amorphous Alloys under a Magnetic Field

Crystallization kinetics and texture evolution in iron-based amorphous alloys (Fe78Si9B13, Fe73.5Si13.5B9Nb3Cu1) under a magnetic field have been studied. We have found that the application of a magnetic field during crystallization at a temperature ranging the Curie point of the amorphous phase and that of the crystalline bcc-Fe(Si) phase can produce a {110} texture in the alloys. X-ray diffraction with the Schulz method revealed that {110} oriented grains were preferentially nucleated due to the magnetic field. The studies of crystallization kinetics found that the nucleation rate of the α-Fe(Si) phase in amorphous Fe73.5Si13.5B9Nb3Cu1 was enhanced approximately three times as high with the 6T magnetic field than without field. The volume fraction of crystalline α-Fe(Si) phase produced was increased by the magnetic field.

Incidence Angle Dependences of Columnar Grain Structure and Texture in Obliquely Deposited Iron Films

From the viewpoint of the crystallographic contribution to the formation of columnar grain structure we investigated the incidence angle dependences of the columnar grain structure and the texture for iron films prepared at various deposition rates R. The films were vapor-deposited at incidence angles of 15°, 30°, 45° and 65° on glass substrates heated at 473 K. The X-ray analysis using the Schulz method revealed that there exist three kinds of texture: [100], type I [111] and type II [111] textures. From the observation using replica electron microscopy and the measurement of electrical resistivity, it was found that the two characteristics of the columnar grain structure (inclination and bundling of columnar grains) depend on the kind of texture irrespective of the incidence angle. For the films with type I [111] texture, the inclination angle and the degree of bundling of columnar grains take constant values with R at a given incidence angle. For the films with [100] or type II [111] texture, on the other hand, the inclination angle and the degree of bundling change markedly with R even if the incidence angle is constant. These changes in columnar grain structure with R depending on the kind of texture can be understood by considering the appearance of the {100} crystal habit.

Determining the mass density along magnetic field lines from toroidal eigenfrequencies

Toroidal eigenfrequencies can be used to remotely sense the equatorial mass density ρeq and the density dependence along a magnetic field line. Here we present improvements to the method of Schulz [1996], which allows ρeq and the power law index α (for mass density along a field line ∝ R−α, where R is the radial distance from the center of the Earth) to be determined from the y intercept and slope of a plot of toroidal frequency versus toroidal harmonic number n. Our modifications include a model form for eigenfrequencies with a fractional precision of 0.0005 for −6 ≤ α ≤ 6 and 2 ≤ L ≤ 8 (accuracy is doubtful beyond L = 5) and an iterative procedure for getting more accurate results than those found using Schulz's method. In addition, we do an analysis of the effect of random measurement errors. Observed frequencies need to be accurate to ∼6% (3%) of the fundamental frequency in order to determine ρeq (α) to a precision of 30% (unity). We then apply our method to data generated using the Global Core Plasma Model for plasmaspheric mass density; our analysis demonstrates clearly how the α index represents the mass density dependence on the outer part of the field line (R/(LRE) ≳ 2/3).

Methods of computing weighted pseudoinverse matrices with singular weights

Three iterative processes are constructed and investigated for computing weighted pseudoinverse matrices with singular weights and ML-weighted pseudoinverse matrices. Two of them are based on the decompositions of the weighted pseudoinverse matrix with singular weights into matrix power series, and the third is a generalization of the Schulz method to nonsingular square matrices.

A class of iteration methods based on the moser formula for nonlinear equations in Markov chains

Many stochastic models in queueing, inventory, communications, and dam theories, etc., result in the problem of numerically determining the minimal nonnegative solutions for a class of nonlinear matrix equations. Various iterative methods have been proposed to determine the matrices of interest. We propose a new, efficient successive-substitution Moser method and a Newton-Moser method which use the Moser formula (which, originally, is just the Schulz method). These new methods avoid the inverses of the matrices, and thus considerable savings on the computational workloads may be achieved. Moreover, they are much more suitable for implementing on parallel multiprocessor systems. Under certain conditions, we establish monotone convergence of these new methods, and prove local linear convergence for the substitution Moser method and superlinear convergence for the Newton-Moser method.

Effect of oblique sputtering on microstructural modification of ZnO thin films

The effects of oblique sputtering on microstructural modification of ZnO films were investigated. Surface morphologies of the films were profoundly modified with an incidence angle. By using pure Ar gas, etch pits, pyramids, ridge, and ledgelike structures were developed on the film surface. By using Ar/O2 mixture gas, large elongated grains, which aligned perpendicular to the direction of incidence vapor at incidence angle above 80°, were formed on the fine columnar matrix. The x-ray Schulz method showed that the (002) pole located away from the direction of incoming vapor at low incidence angles, while the (101) pole uniformly distributed as a concentric ring. For films deposited at high angles, the (002) pole inclined toward an incoming vapor, while the (101) pole moved to the central region. In this case, (110) poles had a tendency to align biaxially. The topological modification and crystallographic orientation change could be explained with an adatom mobility and crystallographic geometry contribution.

Hysteresis loops of cobalt films deposited obliquely by sputtering

The geometric and crystallographic alignments of crystallites in cobalt films deposited obliquely by sputtering were investigated by analyzing hysteresis loops. The incidence angle was 45° and the substrate temperature was 332 K. The film thickness ranged from 0.4 to 3.8 μm. The temperature dependence of the anisotropy of the remanence ( M r⊥ − M r∥)/ M S revealed that irrespective of film thickness crystallites in the film geometrically align in the direction parallel to the incidence plane and have a tendency to direct their c-axes in the perpendicular direction. These alignments have been recognized by replica electron microscopy and the X-ray Schulz method.