Unit Vector Research Articles

Streaming algorithms for monotone non-submodular function maximization under a knapsack constraint on the integer lattice

The study of non-submodular maximization on the integer lattice is an important extension of submodular optimization. In this paper, streaming algorithms for maximizing non-negative monotone non-submodular functions with knapsack constraint on integer lattice are considered. We first design a two-pass StreamingKnapsack algorithm combining with BinarySearch as a subroutine for this problem. By introducing the DR ratio γ and the weak DR ratio γw of the non-submodular objective function, we obtain that the approximation ratio is min⁡{γ2(1−ε)/2γ+1,1−1/γw2γ−ε}, the total memory complexity is O(Klog⁡K/ε), and the total query complexity for each element is O(log⁡Klog⁡(K/ε2)/ε). Then, we design a one-pass streaming algorithm by dynamically updating the maximal function value among unit vectors along with the currently arriving element. Finally, in order to decrease the memory complexity, we design an improved StreamingKnapsack algorithm and reduce the memory complexity to O(K/ε2).

Learning how a tree branches out: A statistical modeling approach.

The increasingly large size of the graphical and numerical data sets collected with modern technologies requires constant update and upgrade of the statistical models, methods and procedures to be used for their analysis in order to optimize learning and maximize knowledge and understanding. This is the case for plant CT scanning (CT: computed tomography), including applications aimed at studying leaf canopies and the structural complexity of the branching patterns that support them in trees. Therefore, we first show after a brief review, how the CT scanning data can be leveraged by constructing an analytical representation of a tree branching structure where each branch is represented by a line segment in 3D and classified in a level of a hierarchy, starting with the trunk (level 1). Each segment, or branch, is characterized by four variables: (i) the position on its parent, (ii) its orientation, a unit vector in 3D, (iii) its length, and (iv) the number of offspring that it bears. The branching structure of a tree can then be investigated by calculating descriptive statistics on these four variables. A deeper analysis, based on statistical models aiming to explain how the characteristics of a branch are associated with those of its parents, is also presented. The branching patterns of three miniature trees that were CT scanned are used to showcase the statistical modeling framework, and the differences in their structural complexity are reflected in the results. Overall, the most important determinant of a tree structure appears to be the length of the branches attached to the trunk. This variable impacts the characteristics of all the other branches of the tree.

The explicit elastic field for two perfectly bonded half-spaces with an ellipsoidal thermal inclusion

The analytical solution for a full-space or half-space with an ellipsoidal inclusion is available in literature, but the explicit analytical solution for two bonded half-spaces with a three-dimensional inclusion was more complex and less reported. Deriving a complete explicit analytical solution for two bonded half-spaces with an ellipsoidal inclusion involves complicated domain integrations/derivatives. This work first derived the explicit analytic solution of two perfectly bonded half-spaces due to an ellipsoidal thermal inclusion and presented it in a more concise and compact tensorial structure. These explicit expressions became easier to program and geometrically meaningful after introducing the normal unit vector of the imaginary confocal ellipsoid. Further, the current solution can be simplified and matches exactly with the published analytical solutions in the case of either semi-infinite space or full space, by varying Young's modulus ratio of the two half-spaces. The closed-form solutions of two perfectly bonded half-spaces embedded with a spherical or spheroidal inclusion were derived and validated as well. The strain/stress jump conditions across the bonded interface were comprehensively discussed for a mechanistic understanding of the delamination around the bonded interface. Moreover, the agreement between the present analytical solution and the photo-elastic experimental results justifies the engineering practicability of this work.

An Optimized SIFT-OCT Algorithm for Stitching Aerial Images of a Loblolly Pine Plantation

When producing orthomosaic from aerial images of a forested area, challenges arise when the forest canopy is closed, and tie points are hard to find between images. The recent development in deep leaning has shed some light in tackling this problem with an algorithm that examines each image pixel-by-pixel. The scale-invariant feature transform (SIFT) algorithm and its many variants are widely used in feature-based image stitching, which is ideal for orthomosaic production. However, although feature-based image registration can find many feature points in forest image stitching, the similarity between images is too high, resulting in a low correct matching rate and long splicing time. To counter this problem by considering the characteristics of forest images, the inverse cosine function ratio of the unit vector dot product (arccos) is introduced into the SIFT-OCT (SIFT skipping the first scale-space octave) algorithm to overcome the shortfalls of too long a matching time caused by too many feature points for matching. Then, the fast sample consensus (FSC) algorithm was introduced to realize the deletion of mismatched point pairs and improve the matching accuracy. This optimized method was tested on three sets of forest images, representing the forest core, edge, and road areas of a loblolly pine plantation. The same process was repeated by using the regular SIFT and SIFT-OCT algorithms for comparison. The results showed the optimized SIFT-OCT algorithm not only greatly reduced the splicing time, but also increased the correct matching rate.

Involute evolute curve family induced by the coupled dispersionless equations

In this paper, we derive the coupled dispersionless (CD) equation from the motion of the involute evolute curve pairs, which provides a favourable interpretation for the integrability conditions. This nonlinear differential equation allows us to introduce the Lax pair providing the integrability of this equation. The gauge equivalence between the CD equation and the Zhaidary-III equation is established. Integrable generalized Heisenberg ferromagnet type equations for the tangent, principal normal, and binormal unit vectors T,N,B are presented. Each of these three integrable nonlinear equations for the orthogonal frame is, separately, geometrical equivalent counterparts of the CD equation. Finally, we study the link between the involute evolute curves and the CD equation.

Freeform Optical Design of Smooth Hermite Radial Basis Function Implicit Surface under Point-Normal Constraints

The freeform surface has become a primary optical design method due to its higher degree of freedom, and the quadratic supporting method is one of the mainstream methods of freeform optical design because of its inherent integrability. Nevertheless, the quadratic supporting surface is usually not smooth, and its smooth enveloping surface needs to be further solved to satisfy the optical manufacturing requirements. The sampling value points and corresponding unit normal vectors on the quadratic supporting surface are used as constraints, and the implicit surface method based on the Hermite radial basis is adopted to successfully generate a smooth freeform optical surface that can project a uniform square spot on the target screen. The point-normal constraints are optimized in terms of value point interpolation and increasing the supporting sub-surface array. And the optical simulation results show that proposed algorithm has more minor errors of value point and normal vector, and better shapes the light beam.

Matrix coupling and generalized frustration in Kuramoto oscillators.

The Kuramoto model describes the synchronization of coupled oscillators that have different natural frequencies. Among the many generalizations of the original model, Kuramoto and Sakaguchi (KS) proposed a frustrated version that resulted in dynamic behavior of the order parameter, even when the average natural frequency of the oscillators is zero. Here, we consider a generalization of the frustrated KS model that exhibits new transitions to synchronization. The model is identical in form to the original Kuramoto model but written in terms of unit vectors and with the coupling constant replaced by a coupling matrix. The matrix breaks the rotational symmetry and forces the order parameter to point in the direction of the eigenvector with the highest eigenvalue, when the eigenvalues are real. For complex eigenvalues, the module of order parameter oscillates while it rotates around the unit circle, creating active states. We derive the complete phase diagram for the Lorentzian distribution of frequencies using the Ott-Antonsen ansatz. We also show that changing the average value of the natural frequencies leads to further phase transitions where the module of the order parameter goes from oscillatory to static.

Optimization of Power Quality Using the Unified Power Quality Conditioner (UPQC) with Unbalanced Loads

The power quality (PQ) is a major issue for both electrical utilities and their customers. The nonlinear loads cause PQ problems like current harmonics, voltage harmonics, frequency deviation, voltage sag and voltage swell. A unified power quality conditioner (UPQC) is utilized in this research to minimize these PQ problems. The UPQC is made up of two active power filters (APFs), one of which is connected to the line in series and the other in parallel. The Unit Vector Template Generation (UVTG) approach is used to control the series APF, whereas the Synchronous Reference Frame (SRF) technique is used to control the shunt APF. The compensating properties of a series-shunt APFs when the loads become imbalanced have been explored. The system performance has been tested under conditions current harmonics, voltage harmonics, voltage sag, and voltage swell. The voltage harmonics are identified and reparation in a series APF using the UVTG technique, whereas the harmonics and unbalanced currents are identified and reparation in a shunt APF using the SRF method. An IEEE-519-compliant THD (Total Harmonic Distortion) of less than 5% is achieved by UPQC in simulations with unbalanced loads. The findings indicate that harmonic currents and supply voltage fluctuations were lessened by UPQC.

Financial Sector Development and Capital Formation: A Comparative Analysis of Nigeria and South Africa

This study examined the effect of financial sector development on capital formation of Nigeria and South Africa using time series data from 1987-2019.Time series data was sourced from Central Bank of Nigeria Statistical bulletin and World Bank data base. percentage of capital formation to gross domestic products was used as the function of credit to private sector, broad money supply, interest rate spread and market capitalization ratio. Ordinary least square methods of cointegration, granger causality test, unit root test and Vector error correction model. The study found that the financial sector development explained 64.1 percent variation in Nigeria capital formation as against 46.4 percent variation from South Africa; this implies that the variables have more explanatory powers in Nigeria than South Africa. From Nigeria, the study conclude that credit to private sector have positive and significant effect, interest rate spread have negative and no significant effect while money supply and market capitalization ratio have positive and significant effect on Nigeria capital formation. However, from south Africa, credit to private sector have negative and no significant effect, interest rate spread have positive and significant effect, money supply have positive and no significant effect while market capitalization ratio have positive and significant effect on south African capital formation. It recommends that the need to increase the size of the markets in both countries by increasing the number of financial instruments available to investors so as to increase trading as well as improve liquidity in the markets and government effort to increase the operational efficiency of the financial sector; the banking habit shall be increase and banking density reduced through effective branch banking policies to enhance effective savings mobilization and credit allocation that will bridge the wide savings-investment gap in the economy

Control strategy without phase‐locked loop based on power estimation and natural frame for single‐phase cascaded H‐bridge converter

AbstractBased on a single‐phase cascaded H‐bridge (CHB) multilevel converter, this paper investigates a control strategy combining power estimation and natural frame control (NFC) with the omission of grid voltage sensors. Besides, this control method, without phase‐locked loop (PLL), can meet the continuous operation of the system and also eliminates grid voltage feedforward. First, natural frame control is introduced to derive the implementation process of control strategy without PLL, and the calculation formula of AC reference currents is given. Furthermore, the separate control of active and reactive currents is achieved. Second, the principle of proposed control algorithm is presented, and the sinusoidal voltage unit vector for synchronization can be accurately estimated. Next, considering the errors, the article corrects the method through parameter compensation. In addition, the balanced control method stabilized the DC side voltage of the converter. Finally, the effectiveness of this method is verified by computer simulation and experimental prototype.

On the construction of polynomial minimal surfaces with Pythagorean normals

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples of complex polynomials is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construction generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean–hodograph (PH) preserving property — a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau problem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non–isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.

A Robust Convolutional Neural Network for 6D Object Pose Estimation from RGB Image with Distance Regularization Voting Loss

Six-degree (6D) pose estimation of objects is important for robot manipulation but at the same time challenging when dealing with occluded and textureless objects. To overcome this challenge, the proposed method presents an end-to-end robust network for real-time 6D pose estimation of rigid objects using the RGB image. In this proposed method, a fully convolutional network with a features pyramid is developed that effectively boosts the accuracy of pixelwise labeling and direction unit vector field that take part in the voting process for object keypoints estimation. The network further takes into account measuring the distance between pixel and keypoint, which aims to help select accurate hypotheses in the RANSAC process. This avoids hypothesis deviations caused by the errors due to direction unit vectors in cases of distant pixels from keypoints. A vectorial distance regularization loss function is used to help Perspective-n-Point find 2D-3D correspondences between 3D object keypoints and their estimated corresponding 2D counterparts. Experiments are performed on widely used LINEMOD and occlusion LINEMOD datasets with ADD (-S) and 2D projection evaluation metrics. The results show that our method improves pose estimation performance compared to the state-of-the-art while still achieving real-time efficiency.

Proposing a Developed Gram-Schmidt Algorithm to Construct Orthogonal Unit Vectors

Proposing a Developed Gram-Schmidt Algorithm to Construct Orthogonal Unit Vectors

Grid-connected solar photovoltaic-fed brushless DC motor drive for water pumping system using colliding body optimization technique

PurposeThis paper aims to present an improvement to the power quality of the grid by using a colliding body optimization (CBO) based proportional-integral (PI) compensated design for a grid-connected solar photovoltaic-fed brushless DC motor (BLDC)-driven water pumping system with a bidirectional power flow control. The system with bidirectional power flow allows driving the pump at full proportions uninterruptedly irrespective of the weather conditions and feeding a grid when water pumping is not required.Design/methodology/approachHere, power quality issue is taken care of by the optimal generation of the duty cycle of the voltage source converter. The duty cycle is optimally generated by optimal selection of the gains of the current controller (i.e. PI), with the CBO technique resulting in a nearly unity power factor as well as lower total harmonic distortion (THD) of input current. In the CBO technique, the gains of the PI controller are considered as agents and collide with each other to obtain the best value. The system is simulated using MATLAB/Simulink and validated in real time with OPAL RT simulator, OP5700.FindingsIt was found that the power quality of grid using the CBO technique has improved much better than the particle swarm optimization and Zeigler–Nichols approach. The bidirectional flow of control of VSC allowed for optimum resource utilization and full capacity of water pumping whatever may be weather conditions.Originality/valueImproved power quality of grid by optimally generation of the duty cycle for the proposed system. A unit vector tamplate generation technique is used for bidirectional power transfer.

An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form

The first extended greatest common right divisor (GCRD) algorithm for parametric univariate polynomial matrices is presented. The starting point of this GCRD algorithm is the free property of submodules over univariate polynomial rings. We convert the computation of GCRDs to that of free basis for modules and prove that a free basis of the submodule generated by row vectors of input matrices forms just a GCRD of these matrices. The GCRD algorithm is obtained by computing a minimal Gröbner basis for the corresponding submodule since a minimal Gröbner basis of submodules is a free basis for univariate cases. While the key idea of extended algorithm is to construct a special module by adding the unit vectors which can record the representation coefficients. This method based on modules can be naturally generalized to the parametric case because of the comprehensive Gröbner systems for modules. As a consequence, we obtain an extended GCRD algorithm for parametric univariate polynomial matrices. More importantly, we apply the proposed extended GCD algorithm for univariate polynomials (as a special case of matrices) to the computation of Smith normal form, and give the first algorithm for reducing a univariate polynomial matrix with parameters to its Smith normal form.

A Convolutional Neural-Network-Based Training Model to Estimate Actual Distance of Persons in Continuous Images.

Distance and depth detection plays a crucial role in intelligent robotics. It enables drones to understand their working environment to avoid collisions and accidents immediately and is very important in various AI applications. Image-based distance detection usually relies on the correctness of geometric information. However, the geometric features will be lost when the object is rotated or the camera lens image is distorted. This study proposes a training model based on a convolutional neural network, which uses a single-lens camera to estimate humans’ distance in continuous images. We can partially restore depth information loss using built-in camera parameters that do not require additional correction. The normalized skeleton feature unit vector has the same characteristics as time series data and can be classified very well using a 1D convolutional neural network. According to our results, the accuracy for the occluded leg image is over 90% at 2 to 3 m, 80% to 90% at 4 m, and 70% at 5 to 6 m.

A Machine Learning Approach to a Multidetector Array Response Function for Nuclear Search

During nuclear search operations, the localization of radioactive sources can be a time-consuming process that requires mapping relative radiation intensity in a large area to determine the position of a source. This article introduces the use of machine learning, specifically a temporal convolutional network (TCN), to estimate the direction between a detector array and a static <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">137</sup> Cs source. This application of machine learning provides a directional vector in <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4\pi $ </tex-math></inline-formula> with a 90% confidence of 5.6° and a 99% confidence within 11.2°. With the use of low-cost NaI(Tl) detectors, the effects of self-shielding within the array creates gamma-ray shadows depending on the orientation to the source. Using the convolved detector array response function, we apply supervised machine learning with a neural network to predict a unit vector that points toward the observed source. The directional vector is expected to reduce search times once implemented in future work.

Vectorial-Opinion Dynamics With Familiarity Neighborhoods in Virtual Social Groups

In this work, we mimic interactions among agents on social networks and examine their impact on the evolution of agents’ opinions. Each agent is entitled to an opinion represented by unit vectors of varying orientation. Agents are also characterized based on their inclination to alter their opinion and their susceptibility to the influence of peers with contrasting opinions. The agents constitute the nodes of a directed and time-varying influence network whose edges form their respective familiarity neighborhoods and govern their interactions. We also devise an interpersonal influence score to quantify the strength of the influence and the probability of acquisition, retention, or loss of a tie. This characterization of agents’ behavior and their interpersonal relationships differs from traditional bounded-confidence models based on the agreement in opinions alone. The agents with different traits form groups—liberal and conservative—distinguished by the distribution in opinions of their constituents. We analyze their interactions with different initial opinions, group compositions, network structures, and network evolution rates through simulations. The results reveal that flexible agents facilitate consensus among group members, while stubborn agents are instrumental in forming opinion clusters. This is substantiated by observing the average connectedness of the influence network and the average number of opinion clusters; groups with mostly flexible agents have better connectedness and fewer clusters than groups with stubborn agents in a majority. The simulations with stubborn agents reveal that new ties among agent pairs do not facilitate group consensus.

Fully distributed containment of multiple‐input‐multiple‐output linear multi‐agent systems with multiple nonautonomous leaders

SummaryIn this article, we study the fully distributed containment problem for multiple‐input‐multiple‐output linear multi‐agent systems (MASs) with multiple nonautonomous leaders (i.e., leaders with nonzero inputs). First, based on an unknown input observer (UIO) with full order and by using unit vector method, we design an adaptive discontinuous controller, where the adaptive law is used to adjust network coupling strengths. And by using Barbǎlat lemma, we show that fully distributed containment can be achieved if the graph is connected and the subgraph among multiple followers is detail balanced. Second, we study fully distributed containment for MASs under an adaptive discontinuous controller based on a UIO with reduced order. Moreover, we study the case with directed switching topology. Finally, we perform simulations on a team of aircrafts to validate the theoretical results.

The General Dispersion Relation for the Vibration Modes of Helical Springs

A system of mathematical equations was developed for the calculation of the natural frequencies of helical springs, its predictions being compared with finite element simulation with ANSYS®. Authors derive the general equations governing the helical spring vibration relative to the Frenet trihedral representing the normal, binormal and tangent unit vectors to the spring medium line. The dispersion relation ω=f(k) has been obtained to model a wave traveling along the axis of the wire.