Circle Method Research Articles

Polynomial sequences in discrete nilpotent groups of step 2

Abstract We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

Infinitely many zeros of additively twisted L-functions on the critical line

For f a cuspidal modular form for the group Γ0(N) of integral or half-integral weight, N a multiple of 4 in case the weight is half-integral, we study the zeros of the L-function attached to f twisted by an additive character e2πinpq with pq∈Q. We prove that for certain f and pq∈Q, the additively twisted L-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses automorphic distributions to prove the result.

Sums of singular series with large sets and the tail of the distribution of primes

Abstract In 1976, Gallagher showed that the Hardy–Littlewood conjectures on prime k-tuples imply that the distribution of primes in log-size intervals is Poissonian. He did so by computing average values of the singular series constants over different sets of a fixed size k contained in an interval $[1,h]$ as $h \to \infty$, and then using this average to compute moments of the distribution of primes. In this paper, we study averages where k is relatively large with respect to h. We then apply these averages to the tail of the distribution. For example, we show, assuming appropriate Hardy–Littlewood conjectures and in certain ranges of the parameters, the number of intervals $[n,n +\lambda \log x]$ with $n\le x$ containing at least k primes is $\ll x\exp(-k/(\lambda {\rm e}))$.

Magnetic Resonance Imaging Analysis Demonstrates Improved Reliability in Measuring Shoulder Glenoid Bone Loss Using a Two-Thirds Glenoid Height Technique Compared to the “Best-fit Circle”

To evaluate the superior to inferior glenoid height as a reliable reference in best-fit circle creation for glenoid anatomy. The morphology of the native glenoid was evaluated using magnetic resonance imaging (MRI) in patients without shoulder instability. Using T1 sagittal MRI images, two reviewers independently estimated glenoid size using the two-thirds technique and the "best-fit circle" technique at two different times. A student's t-test was used to determine significant difference between the two methodologies. Inter- and intra-rater reliability were calculated using interclass and intraclass coefficients. This study included 112 patients. Using the results of glenoid height and "best-fit circle" diameter, the diameter of the "best-fit circle" was found to intersect the glenoid line at 67.8% of the glenoid height on average. We found no significant difference between the two measures of glenoid diameter (27.6 vs 27.9, p =.456). The interclass and intraclass coefficients for the two-third method were 0.85 and 0.88, respectively. The inter-class and intra-class coefficients for the perfect circle methods were 0.84 and 0.73, respectively. We determined that the diameter of a circle placed on the inferior glenoid using the 'best-fit circle' technique corresponds to 67.8% of the glenoid height. Additionally, we found that constructing a perfect circle using a diameter equal to two-thirds the height of the glenoid may improve intra-class reliability.

Motion detection in diffraction tomography by common circle methods

The method of common lines is a well-established reconstruction technique in cryogenic electron microscopy (cryo-EM), which can be used to extract the relative orientations of an object given tomographic projection images from different directions. In this paper, we deal with an analogous problem in optical diffraction tomography. Based on the Fourier diffraction theorem, we show that rigid motions of the object, i.e., rotations and translations, can be determined by detecting common circles in the Fourier-transformed data. We introduce two methods to identify common circles. The first one is motivated by the common line approach for projection images and detects the relative orientation by parameterizing the common circles in the two images. The second one assumes a smooth motion over time and calculates the angular velocity of the rotational motion via an infinitesimal version of the common circle method. Interestingly, using the stereographic projection, both methods can be reformulated as common line methods, but these lines are, in contrast to those used in cryo-EM, not confined to pass through the origin and allow for a full reconstruction of the relative orientations. Numerical proof-of-the-concept examples demonstrate the performance of our reconstruction methods.

Unexpected biases between congruence classes for parts in k-indivisible partitions

For integers k,t≥2, and 1≤r≤t let Dk×(r,t;n) be the number of parts among all k-indivisible partitions of n (i.e., partitions where all parts are not divisible by k) of n that are congruent to r modulo t. Using Wright's circle method, we derive an asymptotic for Dk×(r,t;n) as n→∞ when k,t are coprime. The main term of this asymptotic does not depend on r, and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo t. This induces an ordering on the congruence classes modulo t, which we call the k-indivisible ordering. We prove that for k≥6(t2−1)π2 the k-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when k<6(t2−1)π2.

Quantitative Characteristics of Micro Bedding Fractures in the Wufeng–Longmaxi Formation Based on High-Resolution Map Imaging Technology

The study of microfractures in shale is mainly based on qualitative description. Conversely, quantitative description of the parameters of shale microfractures can provide a quantitative basis for shale fracture characterization and shale physical properties. Nine shale reservoir samples of the Wufeng–Longmaxi Formation in the Jiaoshiba area were studied, using the backscattered two-dimensional multiscale resolution imaging technology, combined with high-resolution map imaging technology (MAPS), and thousands of images were obtained using scanning electron microscopy. Gray image analysis was used to extract microfracture information from images (2 × 2 cm multiresolution). The “maximum circle method” was used to calculate the length and aperture characteristics of the fractures. Parameters such as the area of the bedding fractures, the surface rate of the fractures, and the linear density of the fractures were obtained by the integration of apertures. The fracture length was between 2~7 mm, the aperture was between 1~6 μm, the linear density was between 1~6/m and the surface rate was 1%. The bedding fractures do not contribute much to the porosity of the shale reservoir; however, shale reservoirs with high porosity have a high development of bedding fractures and good permeability. The development of a bedding fracture is controlled by the lithology within shale reservoirs. Different types of lithology contain different bedding fractures, but they have a certain regularity. Moreover, the content of organic matter and TOC (total organic content) in the shale reservoir control the development of a bedding fracture, where a high organic and TOC content are accompanied by a high number of fractures.

Seaweed algebras and the index statistic for partitions

In 2018 Coll, Mayers, and Mayers conjectured that the q-series (q,−q3;q4)∞−1 is the generating function for a certain parity statistic related to the index of seaweed algebras. We prove this conjecture. Thanks to earlier work by Seo and Yee, the conjecture would follow from the non-negativity of the coefficients of this infinite product. Using a variant of the circle method along with Euler–Maclaurin summation, we establish this non-negativity, thereby confirming the Coll–Mayers–Mayers Conjecture.

RECONSTRUCTION OF TOPOLOGY OPTIMIZED GEOMETRY WITH CASTING CONSTRAINTS IN A FEATURE-BASED APPROACH

AbstractTopology Optimization (TO) is an established method for the development of high strength and lightweight structural components. However, its results have to be geometrically revised to obtain a computational model that meets product development requirements. Design has to be in accordance with manufacturing constraints. Geometry reconstruction therefore still is a typically manual and tedious task, but increasingly supported by computational and automated approaches. In this paper, the inclusion of a casting constraint in an automated Medial Axis based reconstruction method is presented. Since the Medial Axis provides cross-section values by the computation of maximally inscribed spheres, this information is used for geometry reconstruction and even further for the purposeful adaption of the cross-section to match Heuver's circle method. Thereby, the directed solidification of molten material is considered. With a predefined feeder position, the demonstrator of a suspension control arm is used to show the application of the method. Resulting CAD-models are also structurally evaluated for their stiffness characteristics.

Bivariate asymptotics for eta-theta quotients with simple poles

We employ a variant of Wright’s Circle Method to determine the bivariate asymptotic behavior of Fourier coefficients for a wide class of eta-theta quotients with simple poles in mathbb {H}.

Design of a Web-Based Learning Management System for Physics Education FKIP University of Riau

21st-century technology requires everyone to see and continue to advance in learning in various forms. Technology is a reference in the success of the learning process supported by the quality of teaching, teaching staff, and curriculum. The Physics Education Study Program at the University of Riau in its learning still uses conventional methods. Learning is only carried out by students and lecturers in class in a short time so students often do not understand the material provided. Based on these problems, we need a learning management system. The goal is to add a limited amount and short meeting time. The system was built using the PHP and MySQL programming languages with the CodeIgniter framework. This study uses the Research and Development (R&amp;D) method and software development uses the System Development Life Circle (SDLC) method with a variation of the v-model. Based on software testing using a black box, all features run 100% well and the User Experience Questionnaire (UEQ) test found that the Learning Management System has good appeal, clarity, and certainty. While efficiency is categorized as very good, stimulation and novelty are included as above average.

An asymptotic formula for the number of $n$-dimensional representations of SU(3)

We prove an asymptotic formula for the number of n -dimensional representations of the group SU(3) Main tools for the proof are Wright’s circle method and the saddle point method.

Transversals in quasirandom latin squares

AbstractA transversal in an latin square is a collection of entries not repeating any row, column, or symbol. Kwan showed that almost every latin square has transversals as . Using a loose variant of the circle method we sharpen this to . Our method works for all latin squares satisfying a certain quasirandomness condition, which includes both random latin squares with high probability as well as multiplication tables of quasirandom groups.

Research and Design of RF Ion Source Impedance Matching Network

The radio frequency (RF) ion source is the key component of the neutral beam injector to achieve high power and long pulse operation. The key technology is feeding RF power from the power source to the ion source and producing stable plasma. It is impossible to calculate the exact equivalent impedance of plasma during discharge, and the impedance will change with the change in discharge pressure and feed power. In order to transfer the maximum output power of the RF power source to the coil antenna of the ion source, an impedance-matching network must be added between the power source and the ion source. Four possible matching network structures are designed by the Smith diagram circle method, and two kinds of capacitor structures are calculated and analyzed. The influence of capacitance change on impedance matching state is analyzed, and the electrical parameters of capacitance under the two structures are compared. After comparison and analysis, the structure of the parallel capacitor is first, and then the series capacitor is selected. Finally, the plasma discharge can be stable through experiments. The research on the matching network is of great significance for the development and experimental operation of the RF feeding system of the high-power RF ion source in the future

Solving Geometry Problems by Alternative Methods in Mathematics Education

Solving geometry problems is both difficult and interesting. Difficult because there is no general algorithm to solve more or less non-trivial problems as every single problem requires individual and creative approach. At the same time, this is a very interesting activity, because for almost every problem there are plenty of ways to solve it. In this work, we present the method of auxiliary circle divided into equal parts. This method allows finding solution algorithm for some geometry problems which are hard to solve by the method of additional constructions.

A Wafer Pre-Alignment Algorithm Based on Weighted Fourier Series Fitting of Circles and Least Squares Fitting of Circles.

The wafer pre-aligner is a crucial component in the lithography process to correct the wafer center and notch orientation. To improve the precision and the efficiency of pre-alignment, a new method to calibrate the center and the orientation of a wafer based on the weighted Fourier series fitting of circles (WFC) method and the least squares fitting of circles (LSC) method, respectively, is proposed. The WFC method effectively suppressed the influence of the outliers and had high stability compared with the LSC method when fitted to the center of the circle. While the weight matrix degenerated to the identity matrix, the WFC method degenerated into the Fourier series fitting of circles (FC) method. The fitting efficiency of the FC method is 28% higher than that of the LSC method, and the fitting accuracy of the center of the FC method is the same as that of the LSC method. In addition, the WFC method and the FC method perform better than the LSC method in radius fitting. The pre-alignment simulation results showed that the absolute position accuracy of the wafer was ±2 µm, the absolute direction accuracy was 0.01°, and the total calculation time was less than 3.3 s in our platform.

Nut Geometry Inspection Using Improved Hough Line and Circle Methods.

Nuts are the cornerstone of human industrial construction, especially A-grade nuts that can only be used in power plants, precision instruments, aircraft, and rockets. However, the traditional nuts inspection method is to manually operate the measuring instrument for conducting an inspection, so the quality of the A-grade nut cannot be guaranteed. In this work, a machine vision-based inspection system was proposed, which performs a real-time geometric inspection of the nuts before and after tapping on the production line. In order to automatically screen out A-Grade nuts on the production line, there are 7 inspections within this proposed nut inspection system. The measurements of parallel, opposite side length, straightness, radius, roundness, concentricity, and eccentricity were proposed. To shorten the overall detection time regarding nut production, the program needed to be accurate and uncomplicated. By modifying the Hough line and Hough circle, the algorithm became faster and more suitable for nut detection. The optimized Hough line and Hough circle can be used for all measures in the testing process.

Free rational curves on low degree hypersurfaces and the circle method

We use a function field version of the Hardy-Littlewood circle method to study the locus of free rational curves on an arbitrary smooth projective hypersurface of sufficiently low degree. On the one hand this allows us to bound the dimension of the singular locus of the moduli space of rational curves on such hypersurfaces and, on the other hand, it sheds light on Peyre's reformulation of the Batyrev-Manin conjecture in terms of slopes with respect to the tangent bundle.

Identification and comparative analysis of vibration modal parameters of rice planter frame

Modal parameter analysis is a common method for structural characteristic analysis. To accurately identify the modal parameters that describe the vibration characteristics of the structure, the chassis frame of the rice transplanter is taken as the research object. The least squares complex frequency domain method, the admittance circle method, and the modal peak picking method are used to process the measured vibration signal. The modal parameters, such as the natural frequency, damping ratio, and modal shape of the chassis frame, are obtained. Except for the eight-order frequency error greater than 10%, the errors of all other orders are all less than 10%. In the modal test of the frame structure of the rice transplanter, the identification results of the modal parameters are generally reliable. The recognition accuracy of the PolyLeast-Squares Complex Frequency (PolyLSCF) algorithm is lower than the recognition accuracy of the admittance circle method and modal peak picking. The values of some off-diagonal elements in the modal mass matrix calculated by the PolyLSCF algorithm and the admittance circle method are greater than 0.2. The diagonal elements of the modal mass matrix calculated by the modal peak picking method are all 1.

Acromion morphology is associated with glenoid bone loss in posterior glenohumeral instability

The acromion morphology in a shoulder with posterior instability differs from that of a shoulder without glenohumeral instability. Specifically, the acromion with a flatter sagittal tilt, greater posterior acromial height, and less posterior coverage is associated with posterior instability. However, the association between acromion morphology and glenoid bone loss (GBL) in the setting of posterior glenohumeral instability has not previously been investigated. The purpose of this study was to determine whether acromial morphology influences the extent or pattern of posterior GBL in a cohort of patients with posterior glenohumeral instability. This multicenter retrospective study identified 89 shoulders with unidirectional posterior glenohumeral instability. Total area GBL was measured using the best-fit circle method on magnetic resonance imaging (MRI). Shoulders were divided into 3 groups: (1) no GBL (n=30), (2) GBL 0%-13.5% (n=45), or (3) GBL ≥13.5% (n=14). Acromion measurements were performed on MRI and included acromial tilt, posterior acromial height, anterior acromial coverage, and posterior acromial coverage. Patients without GBL had a steeper acromial tilt (58.5°±1.4°) compared with those with 0%-13.5% GBL (64.3°±1.5°) or GBL ≥13.5% (67.7°±1.8°) (P=.004). Patients without GBL also had greater posterior coverage (65.4°±1.7°) compared with those with GBL (60.3°±1.4°) (P=.015). Posterior acromion height was not significantly different among groups. The results demonstrate that an acromion with a flatter sagittal tilt and less posterior coverage is associated with GBL in the setting of posterior glenohumeral instability. This is important to consider as posterior GBL has been identified as a risk factor for failure of posterior soft tissue-stabilizing procedures.