Manifolds Of Points Research Articles

The benefits of peer mentoring in higher education: findings from a systematic review

The integration of peer mentoring into academic programmes as a mechanism for supporting new students has received considerable attention in higher education worldwide, especially in developed countries. Despite a growing body of research on peer mentoring in higher education, an extensive review of the benefits of this approach is needed to establish a nuanced understanding of the impacts of peer mentoring on students across various contexts. This study employed a systematic review method to examine the benefits of peer mentoring in higher education. The review covered research articles about peer mentoring published within the last decade (2013−2023). The findings showed that the benefits of peer mentoring in higher education could be categorised into four fundamental aspects, including academic performance, retention rates, emotional and psychological wellbeing, and social integration. These categories of benefits could be developed into a conceptual framework for studying the impacts or benefits of peer mentoring in higher education. The study highlights the significance of adopting peer mentoring as a supportive scheme for students encountering critical challenges due to its manifold benefits and points to the importance of identifying the varying factors contributing to the success of peer-mentoring schemes. The study concludes with a discussion and the implications of the findings, as well as suggestions for future research to examine how peer mentoring could be effectively implemented in different educational contexts.

Bifurcation Mechanism of Quasi-Halo Orbit from Lissajous Orbit

This paper presents a general analytical method to describe the center manifolds of collinear libration points in the restricted three-body problem (RTBP). It is well known that these center manifolds include Lissajous orbits, halo orbits, and quasi-halo orbits. Previous studies have traditionally treated these orbits separately by iteratively constructing high-order series solutions using the Lindstedt–Poincaré method. Instead of relying on resonance between their frequencies, this study identifies that halo and quasi-halo orbits arise due to intricate coupling interactions between in-plane and out-of-plane motions. To characterize this coupling effect, a novel concept, coupling coefficient η, is introduced in the RTBP, incorporating the coupling term ηΔx in the z-direction dynamics equation, where Δ represents a formal power series concerning the amplitudes. Subsequently, a uniform series solution for these orbits is constructed up to a specified order using the Lindstedt–Poincaré method. For any given paired in-plane and out-of-plane amplitudes, the coupling coefficient η is determined by the bifurcation equation Δ=0. When η=0, the proposed solution describes Lissajous orbits around libration points. As η transitions from zero to nonzero values, the solution describes quasi-halo orbits, which bifurcate from Lissajous orbits. Particularly, halo orbits bifurcate from planar Lyapunov orbits if the out-of-plane amplitude is zero. The proposed method provides a unified framework for understanding these intricate orbital behaviors in the RTBP.

Efficient estimation of the modified Gromov–Hausdorff distance between unweighted graphs

Gromov–Hausdorff distances measure shape difference between the objects representable as compact metric spaces, e.g. point clouds, manifolds, or graphs. Computing any Gromov–Hausdorff distance is equivalent to solving an NP-hard optimization problem, deeming the notion impractical for applications. In this paper we propose a polynomial algorithm for estimating the so-called modified Gromov–Hausdorff (mGH) distance, a relaxation of the standard Gromov–Hausdorff (GH) distance with similar topological properties. We implement the algorithm for the case of compact metric spaces induced by unweighted graphs as part of Python library scikit-tda, and demonstrate its performance on real-world and synthetic networks. The algorithm finds the mGH distances exactly on most graphs with the scale-free property. We use the computed mGH distances to successfully detect outliers in real-world social and computer networks.

Integration of Manifold Learning and Density Estimation for Fine-Tuned Face Recognition

With the rapid advancements in data analysis and the increasing complexity of high-dimensional datasets, traditional dimensionality reduction techniques like Local Linear Embedding (LLE) often face challenges in maintaining accuracy and efficiency. This research aims to overcome the limitations of LLE, specifically its reliance on the nearest neighbor concept, its inability to distinguish differences among manifold points, and its underutilization of data discrimination information. To address these issues, we propose an advanced LLE algorithm that integrates decision tree-based neighbor recognition with Gaussian kernel density estimation. Decision trees accurately determine neighboring relationships, which are then optimized using Gaussian kernel density estimation to better reflect the distribution of sample points on the manifold. The algorithm also incorporates data discrimination information to enhance classification accuracy and efficiency. Evaluations in facial recognition tasks using SVM classifiers demonstrate significant improvements. Integrating decision trees (LLE-DT) yielded accuracy gains, with LFW at 98.75%, CFP 96.10%, and Olivetti 92.18%. Gaussian density estimation (LLE-GDE) achieved further enhancements, especially in LFW (99.13%), with CFP at 96.85%, and Olivetti at 91.82%. Combining both methods (LLE-DT-GDE) led to substantial improvements: LFW 99.61%, CFP 97.23%, and Olivetti 93.56%, highlighting the synergy between decision trees and Gaussian estimation. This advanced LLE algorithm effectively addresses the limitations of traditional approaches, showing promising results in complex data processing tasks such as facial recognition. These findings suggest its potential for broader applications in fields requiring robust data analysis and classification.

Distributed Learning for Principal Eigenspaces without Moment Constraints

Distributed Principal Component Analysis (PCA) has been studied to deal with the case when data are stored across multiple machines and communication cost or privacy concerns prohibit the computation of PCA in a central location. However, the sub-Gaussian assumption in the related literature is restrictive in real application where outliers or heavy-tailed data are common in areas such as finance and macroeconomics. In this article, we propose a distributed algorithm for estimating the principal eigenspaces without any moment constraints on the underlying distribution. We study the problem under the elliptical family framework and adopt the sample multivariate Kendall’s tau matrix to extract eigenspace estimators from all submachines, which can be viewed as points in the Grassmann manifold. We then find the “center” of these points as the final distributed estimator of the principal eigenspace. We investigate the bias and variance for the distributed estimator and derive its convergence rate which depends on the effective rank, eigengap of the scatter matrix and the number of submachines. We show that the distributed estimator performs as if we have full access to the whole data. Simulation studies show that the distributed algorithm performs comparably with the existing one for light-tailed data, while showing great advantages for heavy-tailed data. We also extend the distributed algorithm to cases with limited communication constraints and with elliptical factor structure. Thorough simulation studies and a real application to a macroeconomic dataset verify the advantages of the proposed distributed algorithms. Supplementary materials for this article are available online.

Computing parametrised large intersection sets of 1D invariant manifolds: a tool for blender detection

A dynamical system given by a diffeomorphism with a three-dimensional phase space may have a blender, which is a hyperbolic set Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} with, say, a one-dimensional stable invariant manifold that behaves like a surface. This means that the stable manifold of any fixed or periodic point in Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} weaves back and forth as a curve in phase space such that it is dense in some projection; we refer to this as the carpet property. We present a method for computing very long pieces of such a one-dimensional manifold so efficiently and accurately that a very large number of intersection points with a specified section can reliably be identified. We demonstrate this with the example of a family of Hénon-like maps H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} on R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document}, which is the only known, explicit example of a diffeomorphism with proven existence of a blender. The code for this example is available as a Matlab script as supplemental material. In contrast to earlier work, our method allows us to determine a very large number of intersection points of the respective one-dimensional stable manifold with a chosen planar section and render each as individual curves when a parameter is changed. With suitable accuracy settings, we not only compute these parametrised curves for the fixed points of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document} over the relevant parameter interval, but we also compute the corresponding parametrised curves of the stable manifolds of a period-two orbit (with negative eigenvalues) and of a period-three orbit (with positive eigenvalues). In this way, we demonstrate that our algorithm can handle large expansion rates generated by (up to) the fourth iterate of H\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {H}$$\\end{document}.

Similarity between a many-body quantum avalanche model and the ultrametric random matrix model

In the field of ergodicity-breaking phases, it has been recognized that quantum avalanches can destabilize many-body localization at a wide range of disorder strengths. This has in particular been demonstrated by the numerical study of a toy model, sometimes simply called the “avalanche model” or the “quantum sun model” [], which consists of an ergodic seed coupled to a perfectly localized material. In this paper, we connect this toy model to a well-studied model in random matrix theory, the ultrametric ensemble. We conjecture that the models share the following features. (1) The location of the critical point may be predicted sharply by analytics. (2) On the localized site, both models exhibit Fock space localization. (3) There is a manifold of critical points. On the critical manifold, the eigenvectors exhibit nontrivial multifractal behavior that can be tuned by moving on the manifold. (4) The spectral statistics at criticality is intermediate between Poisson statistics and random matrix statistics, also tunable on the critical manifold. We confirm numerically these properties. Published by the American Physical Society 2024

When the moduli space is an orbifold: spontaneous breaking of continuous non-invertible symmetries

We investigate theories of Nambu-Goldstone bosons where the spontaneously broken continuous symmetry is non-invertible. In such theories, the vacua generically parameterize an orbifold. We study in detail the simplest example of a single free scalar with shift symmetry, modded by reflection symmetry. At singular points of the vacuum manifold, we show that the spectrum of NG excitations is reduced, in particular there are no single-particle states. At the smooth points, on the other hand, single NG modes are present. We show that this is a consequence of the fact that at those points one can construct invertible operators implementing the continuous symmetry on the Hilbert space.

Learning Provably Stable Local Volt/Var Controllers for Efficient Network Operation

This paper develops a data-driven framework to synthesize local Volt/Var control strategies for distributed energy resources (DERs) in power distribution grids (DGs). Aiming to improve DG operational efficiency, as quantified by a generic optimal reactive power flow (ORPF) problem, we propose a two-stage approach. The <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">first</i> stage involves learning the manifold of optimal operating points determined by an ORPF instance. To synthesize local Volt/Var controllers, the learning task is partitioned into learning local surrogates (one per DER) of the optimal manifold with voltage input and reactive power output. Since these surrogates characterize efficient DG operating points, in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">second</i> stage, we develop local control schemes that steer the DG to these operating points. We identify the conditions on the surrogates and control parameters to ensure that the locally acting controllers collectively converge, in a global asymptotic sense, to a DG operating point agreeing with the local surrogates. We use neural networks to model the surrogates and enforce the identified conditions in the training phase. AC power flow simulations on the IEEE 37-bus network empirically bolster the theoretical stability guarantees obtained under linearized power flow assumptions. The tests further highlight the optimality improvement compared to prevalent benchmark methods.

Scenario of stable transition from diffeomorphism of torus isotopic to identity one to skew product of rough transformations of circle

In this paper, we consider gradient-like diffeomorphisms of a two-dimensional torus isotopic to the identical one. The isotopicity of diffeomorphisms $f_0$, $f_1$ on an $n$-manifold $M^n$ means the existence of some arc $\{f_t:M^n\to M^n,t\in[0,1]\}$ connecting them in the space of diffeomorphisms. If isotopic diffeomorphisms are structurally stable (qualitatively not changing their properties with small perturbations), then it is natural to expect the existence of a stable arc (qualitatively not changing its properties under small perturbations) connecting them. In this case, one says that the isotopic diffeomorphisms $f_0$, $f_1$ are stably isotopic or belong to the same class of stable isotopic connectivity. The simplest structurally stable diffeomorphisms on surfaces are gradient-like transformations having a finite hyperbolic non-wandering set, stable and unstable manifolds of various saddle points of which do not intersect. However, even on a two-dimensional sphere, where all orientation-preserving diffeomorphisms are isotopic, gradient-like diffeomorphisms are generally not stably isotopic. The countable number of pairwise different classes of stable isotopic connectivity is constructed on the base of a rough transformation of the circle $\phi_{\frac{k}{m}}$ with exactly two periodic orbits of the period $m$ and the rotation number $\frac{k}{m}$, which can be continued to a diffeomorphism $F_{\frac k m}:\mathbb S^2\to\mathbb S^2$ with two fixed sources at the North and South poles. On the torus $\mathbb T^2$, the model representative in the considered class is the skew products of rough transformations of a circle. We show that any isotopic gradient-like diffeomorphism of a torus is connected by a stable arc with some model transformation.

Common population codes produce extremely nonlinear neural manifolds

Populations of neurons represent sensory, motor, and cognitive variables via patterns of activity distributed across the population. The size of the population used to encode a variable is typically much greater than the dimension of the variable itself, and thus, the corresponding neural population activity occupies lower-dimensional subsets of the full set of possible activity states. Given population activity data with such lower-dimensional structure, a fundamental question asks how close the low-dimensional data lie to a linear subspace. The linearity or nonlinearity of the low-dimensional structure reflects important computational features of the encoding, such as robustness and generalizability. Moreover, identifying such linear structure underlies common data analysis methods such as Principal Component Analysis (PCA). Here, we show that for data drawn from many common population codes the resulting point clouds and manifolds are exceedingly nonlinear, with the dimension of the best-fitting linear subspace growing at least exponentially with the true dimension of the data. Consequently, linear methods like PCA fail dramatically at identifying the true underlying structure, even in the limit of arbitrarily many data points and no noise.

Link as a complete invariant of Morse-Smale 3-diffeomorphisms

In this paper we consider gradient-like Morse-Smale diffeomorphisms defined on the three-dimensional sphere S3 . For such diffeomorphisms, a complete invariant of topological conjugacy was obtained in the works of C. Bonatti, V. Grines, V. Medvedev, E. Pecu. It is an equivalence class of a set of homotopically non-trivially embedded tori and Klein bottles embedded in some closed 3-manifold whose fundamental group admits an epimorphism to the group Z . Such an invariant is called the scheme of the gradient-like diffeomorphism f:S3→S3 . We single out a class G of diffeomorphisms whose complete invariant is a topologically simpler object, namely, the link of essential knots in the manifold S2×S1 . The diffeomorphisms under consideration are determined by the fact that their non-wandering set contains a unique source, and the closures of stable saddle point manifolds bound three-dimensional balls with pairwise disjoint interiors. We prove that, in addition to the closure of these balls, a diffeomorphism of the class G contains exactly one nonwandering point, which is a fixed sink. It is established that the total invariant of topological conjugacy of class G diffeomorphisms is the space of orbits of unstable saddle separatrices in the basin of this sink. It is shown that the space of orbits is a link of non-contractible knots in the manifold S2×S1 and that the equivalence of links is tantamount to the equivalence of schemes. We also provide a realization of diffeomorphisms of the considered class along an arbitrary link consisting of essential nodes in the manifold S2×S1 .

Analysis of SVEIL Model of Tuberculosis Disease Spread with Imperfect Vaccination

This study proposes a SVEIL model of tuberculosis disease spread with imperfect vaccination. Susceptible individuals can receive imperfect vaccination, but over the time the vaccine efficacy will decrease. Vaccinated individuals are in vulnerable class since they still have probability to get reinfected. The proposed model includes treatment for both high-risk latent and active TB patients. In fact, after getting appropriate treatment (get recovered) the individuals still have bacteria in their body and it is classified to low-risk laten class. Dynamical behaviour of the model is analyzed to understand the local stability equilibrium. The Routh-Hurwitz criterion is used to analyze the local stability equilibrium in disease free equilibrium (DFE) point and Center Manifold theorem is used to prove the local stability of the endemic equilibrium (EE) point. The local stability equilibrium state totally depends on the effective reproduction number R_v. If R_v&lt;1 , then the DFE point is locally asymtotically stable, while if R_v&gt;1 the EE point is locally asymptotically stable . The parameter used in this paper is based on the previous researches related to TB and the initial subpopulations are assumed. Numerical simulations show that the disease transmission rate affect the effective reproduction number, therefore it influences the stability of equilibrium points..

Event-triggered global sliding mode controller design for anti-sway control of offshore container cranes

This paper proposes an event-triggered global sliding mode based anti-sway control for offshore container cranes. Its main objective is to guarantee the robust tracking performance of containers transferred by cranes installed on cargo ships, while saving the transmission resources. To this end, a suitable sliding manifold is formulated with the desired motion properties. A global sliding mode control is then synthesized to maintain stability of the system and ensure robustness against disturbances such as sea waves and wind induced movements. Finally, the event-triggered strategy is considered in the implementation phase to generate the sampling instant at which the control signal is updated, thereby minimizing communication resources and execution time. Using the Lyapunov's stability theory, it is proven that the controlled system reaches the manifold and equilibrium point from any initial state. We also prove that the proposed approach is free from the Zeno solution, i.e., a phenomenon when the infinite triggering instants exist in a limited time range. Eliminating the reaching phase of traditional sliding mode control, overcoming the effect of the disturbances during the entire dynamics, optimizing resources at the implementation phase, and removing the Zeno phenomenon are among the main advantages of the proposed control approach.

Properties of Orbits Under Lie Group Action

Bu çalışmada, Lie grubunun manifold yapısının geometrik formu ile Lie dönüşüm grubu olarak diferensiyellenebilir bir manifoldun noktalarına etki ettirildiğinde, Lie grubunun geometrik yapısı ile Lie dönüşüm grubu etkisi altındaki noktaların yörüngelerinin geometrik yapıları arasındaki ilişkiler incelendi. Matlab uygulamaları yapıldı.

Secure and Reliable Indoor Localization Based on Multitask Collaborative Learning for Large-Scale Buildings

Accurate and reliable indoor location estimate is crucial for many Internet-of-Things (IoT) applications in the era of smart buildings. However, the positioning accuracy and security of the existing positioning works cannot meet the demands in the large-scale smart buildings scenarios covering multiple multifloor buildings. Therefore, in this article, we focus on the reliable and accurate localization under multibuilding and multifloor environments. We propose two novel designs, including a two-step reliable feature selector and a multitask collaborative positioning model. First, we design a two-step reliable feature selector based on an access point (AP) confidence model and manifold learning, to help select the most representative and reliable fingerprint features. Second, we propose a multitask cooperative positioning model, which consists of a multiscale feature fusion module to adaptively fuse multiscale features and a multitask joint learning module to effectively constrain the cumulative error of multiscale position. Finally, based on the above two, we propose a reliable multibuilding and multifloor localization method (RMBMFL), which can achieve accurate and reliable location estimates with low computational complexity in a smart building complex. We did real-world experiments in a 20 000 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m^{2}}$ </tex-math></inline-formula> site that covers three multistory buildings to evaluate the performance of the proposed RMBMFL. The experimental results show that RMBMFL achieves a building identification accuracy and a floor identification accuracy of 99%, and a room-level indoor localization with an average positioning error within 2 m, and outperforms state-of-the-art solutions.

Controlling the configuration space topology of mechanical structures.

Linkages are mechanical devices constructed from rigid bars and freely rotating joints studied both for their utility in engineering and as mathematical idealizations in a number of physical systems. Recently, there has been a resurgence of interest in designing linkages in the physics community due to the concurrent developments of mechanical metamaterials, topological mechanics, and the discovery of anomalous rigidity in fiber networks and vertex models. These developments raise a natural question: to what extent can the motion of a linkage or mechanical structure be designed? Here, we describe a method to design the topology of the configuration space of a linkage by first identifying the manifold of critical points, then perturbing around such critical configurations. Unlike other methods, our methods are tractable and provide a simple visual toolkit for mechanism design. We demonstrate our procedure by designing a mechanism to gate the propagation of a soliton in a Kane-Lubensky chain of interconnected rotors.

Development of A Prototype Extractor of CO₂ from Exhaust Gases

A Prototype Extractor of CO₂ from Exhaust Gases adopting Post-combustion capture process is developed in this study and its performance was evaluated. The system consists of a small petrol engine of 1kw power, an absorber with volume of 0.00054 m³, a balloon that can contain sizeable amount of exhaust gas, a washer of 0.0181m³, a compressor of 1hp, a dryer which consist of two compartment, with internal volume of 0.00093 m³insulated with fibre glass to prevent heat loss. All of these component parts works in a circle, to convert stockpiled exhaust gases from exhaust manifold point source of an electrical power generating set into carbon (iv) oxide (CO2). This system was carefully developed to ensure extraction of pure CO₂ from the waste emission gases. Study was conducted using the developed system to extract suspected CO₂ gaseous substance from exhaust gases. The obtained gaseous substance was assay for chemical analysis test using sodalime solution (CaHNaO2). The result of the test confirmed the extracted gaseous substance as CO₂ gas by changing a colorless sodalime solution to a milk color. It is also found that the quantity of CO₂ gas generated increases with the increment of the quantity of the fuel consumed by the Petrol engine. The produced CO₂ gas can be utilized as commodity in the industrial production of materials and fuels. This work has elucidated one of the means of converting waste emissions (exhaust gases) to wealth and could help to reduce greenhouse effect thus providing a friendly environment for humans’ inhabitants.

On Embedding of the Morse–Smale Diffeomorphisms in a Topological Flow

This review presents the results of recent years on solving of the Palis problem on finding necessary and sufficient conditions for the embedding of Morse–Smale cascades in topological flows. To date, the problem has been solved by Palis for Morse–Smale diffeomorphisms given on manifolds of dimension two. The result for the circle is a trivial exercise. In dimensions three and higher new effects arise related to the possibility of wild embeddings of closures of invariant manifolds of saddle periodic points that leads to additional obstacles for Morse–Smale diffeomorphisms to be embedded in topological flows. The progress achieved in solving of Palis’s problem in dimension three is associated with the recently obtained complete topological classification of Morse–Smale diffeomorphisms on threedimensional manifolds and the introduction of new invariants describing the embedding of separatrices of saddle periodic points in a supporting manifold. The transition to a higher dimension requires the latest results from the topology of manifolds. The necessary topological information, which plays key roles in the proofs, is also presented in the survey.

Finding geodesics joining given points

Finding a geodesic joining two given points in a complete path-connected Riemannian manifold requires much more effort than determining a geodesic from initial data. This is because it is much harder to solve boundary value problems than initial value problems. Shooting methods attempt to solve boundary value problems by solving a sequence of initial value problems, and usually need a good initial guess to succeed. The present paper finds a geodesic gamma :[0,1]rightarrow M on the Riemannian manifold M with γ(0) = x0 and γ(1) = x1 by dividing the interval [0,1] into several sub-intervals, preferably just enough to enable a good initial guess for the boundary value problem on each subinterval. Then a geodesic joining consecutive endpoints (local junctions) is found by single shooting. Our algorithm then adjusts the junctions, either (1) by minimizing the total squared norm of the differences between associated geodesic velocities using Riemannian gradient descent, or (2) by solving a nonlinear system of equations using Newton’s method. Our algorithm is compared with the known leapfrog algorithm by numerical experiments on a 2-dimensional ellipsoid Ell(2) and on a left-invariant 3-dimensional special orthogonal group SO(3). We find Newton’s method (2) converges much faster than leapfrog when more junctions are needed, and that a good initial guess can be found for (2) by starting with Riemannian gradient descent method (1).