Binormal Research Articles

In silico method for identifying the key residues in a knotted protein: with MJ0366 as an example.

The knotted proteins are a class of rare but biologically important proteins, due to the special topology of their native structure. Here we present a simple in silico method to identify the key residues for knotting and unknotting in a knotted protein, using the trefoil protein MJ0366 as an example. We first simulate the folding process via the annealing molecular dynamics (AMD) simulations in the coarse-grained "Go"-like model. From the folding trajectories, we monitor the knotting process using the quantity "length of knot tails". In the meantime, we analyze the evolution of the local geometry of the Cα trace with the help of the Discrete Frenet Frame (DFF). We identify the key residues by correlating the local geometry at each residue with the variable "length of knot tails" in the folding process, where a higher correlation coefficient indicates that the residue is more important for knotting. We validate our method by comparing with the experimental results in the literature. With the same method, we further predict the key residues for unknotting MJ0366 using the AMD simulations in both the coarse-grained "Go"-like model and all-atom (AA) force field model, respectively. We find that the key residues for unknotting are partially overlapped with those for knotting, indicating that the pathways for unknotting and knotting are generally similar except for the existence of some non-native contact interactions in the unknotting process. This in silico method can provide a new insight for understanding the knotting and unknotting processes of a knotted protein.

On Involute-Evolute Curve Pair in Semi-Euclidean Space

In this study, a kind of generalized involute and evolute curve pair is considered in 4 dimensional semi Euclidean space with 2 index. The curvatures and the associated Frenet Frame of this kind of generalized
 involute-evolute curve pair are presented.

(k,m)-type Slant Helices According to Parallel Transport Frame in Euclidean 4-Space

In this work, we describe a Frenet frame in 4-dimensional Euclidean space and call this frame as parallel transport frame (PTF). PTF is an alternative approach to defining a moving frame. This frame is obtained by rotating the tangent vector and the first binormal vector of a unit speed curve by an euler angle and then we give curvature functions according to PTF of the curve. Also, we introduce $(k,m)$-type slant helices according to PTF in Euclidean 4-Space. Additionally, we obtain the characterization of slant helices according to this frame in $\mathbb{E}^{4}$ and give an example of our main result.

THE RELATION BETWEEN FRENET FRAME OF THE NATURAL LIFT CURVE AND BISHOP FRAME OF THE CURVE

In this study, the relation between Frenet Frame of the natural lift curve γ- of the curve γ and Bishop Frame vectors of γ is given in IR^3 and IR1^3 .

Autofocusing drift tube linac envelopes

To date, beam dynamics studies and design of combined zero degree drift tube linac (DTL) structures (Kombinierte Null Grad Struktur; KONUS) have only been carried out in multiparticle codes. Quantities such as the beam envelopes are obtained by averaging over particles, whose tracking is computationally intensive for large bunch populations. Tune computations, which depend on this average, are burdensome to obtain. This has motivated the implementation of a simulation of KONUS DTLs in the code transoptr, whose Hamiltonian treatment of beam dynamics enables the integration of energy gain from the longitudinal electric field on axis while simultaneously elaborating the field in transverse directions to obtain the linear optics. The code also features an in-built space charge capability. The evolution of the beam matrix including longitudinal optics is computed in a reference Frenet-Serret frame through the time-dependent DTL cavity fields. This enables fast envelope simulations for DTLs, resulting in a variable energy sequential tune optimization capability. The implementation methodology and optimization techniques, applicable for any combination of DTL tanks and bunchers, is outlined. Comparisons with the code lorasr, in addition to beam-based $E/A$ measurements of a DTL are presented.

Occlusion-Aware Path Planning to Promote Infrared Positioning Accuracy for Autonomous Driving in a Warehouse

Infrared positioning is a critical module in an indoor autonomous vehicle platform. In an infrared positioning system, the ego vehicle is equipped with an infrared emitter while the infrared receivers are fixed onto the ceiling. The infrared positioning result is accurate only when the number of valid infrared receivers is more than three. An infrared receiver easily becomes invalid if it does not receive light from the infrared emitter due to indoor occlusions. This study proposes an occlusion-aware path planner that enables an autonomous vehicle to navigate toward the occlusion-free part of the drivable area. The planner consists of four layers. In layer one, a homotopic A* path is searched for in the 2D grid map to roughly connect the initial and goal points. In layer two, a curvature-continuous reference line is planned close to the A* path using numerical optimal control. In layer three, a Frenet frame is constructed along the reference line, followed by a search for an occlusion-aware path within that frame via dynamic programming. In layer four, a curvature-continuous path is optimized via quadratic programming within the Frenet frame. A path planned within the Frenet frame may violate the curvature bounds in a real-world Cartesian frame; thus, layer four is implemented through trial and error. Simulation results in CarSim software show that the derived paths reduce the poor positioning risk and are easily tracked by a controller.

Intersection curve of two parametric surfaces in Euclidean n-space

The aim of this paper is to study the differential geometric properties of the intersection curve of two parametric surfaces in Euclidean n-space. For this aim, we first present the mth order derivative formula of a curve lying on a parametric surface. Then, we obtain curvatures and Frenet vectors of the transversal intersection curve of two parametric surfaces in Euclidean n-space. We also provide computer code produced in MATLAB to simplify determining the coefficients relative to Frenet frame of higher order derivatives of a curve.

Slant Helices of (k,m)-Type According to the ED-Frame in Minkowski 4-Space

In differential geometry, relations between curves are a large and important area of study for many researchers. Frame areas are an important tool when studying curves, specially the Frenet–Serret frame along a space curve and the Darboux frame along a surface curve in differential geometry. In this paper, we obtain slant helices of k-type according to the extended Darboux frame (or, for brevity, ED-frame) field by using the ED-frame field of the first kind (or, for brevity, EDFFK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} and the ED-frame field of the second kind (or, for brevity, EDFSK), which is formed with an anti-symmetric matrix for ε1=ε2=ε3=ε4∈{−1,1} in four-dimensional Minkowski space E14. In addition, we present some characterizations of slant helices and determine (k,m)-type slant helices for the EDFFK and EDFSK in Minkowski 4-space.

Control Design and Assessment for a Reversing Tractor–Trailer System Using a Cascade Controller

In recent years, control design for unmanned systems, especially a tractor–trailer system, has gained popularity among researchers. The emergence of such interest is caused by the potential reduction in cost and shortage of number of workers and labors. Two industries will benefit from the advancements of these types of systems: agriculture and cargo. By using the unmanned tractor–trailer system, harvesting and cultivating plants will become a safe and easy task. It will also cause a reduction in cost which in turn reduces the price on the end consumers. On the other hand, by using the unmanned tractor–trailer system in the cargo industry, shipping cost and time for the item delivery will be reduced. The work presented in this paper focuses on the development of a path tracking and a cascaded controller to control a tractor–trailer in reverse motion. The path tracking controller utilizes the Frenet–Serret frame to control the kinematics of the tractor–trailer system on a desired path, while the cascade controller’s main objective is to stabilize the system and to perform commands issued by the path tracker. The controlled parameters in this proposed design are the lateral distance to a path, trailer’s heading angel, articulated angel, and articulated angle’s rate. The main goal of such controller is to follow a path while the tractor–trailer system is moving in reverse and controlling the stability of the articulated vehicle to prevent the occurrence of a jackknife incident (uncontrolled state). The proposed controller has been tested in a different scenario where a successful implementation has been shown.

Coupled vortex-induced modeling for spatially large-curved beam with elastic support

The existing studies on mathematical modeling for the forced vibration response of naturally curved beams under elastic boundary conditions mainly focus on slightly curved structures. This work proposes the mathematical model for naturally curved beam with a large curvature under the elastic boundary. Under the assumption of small deformations, the six degrees of freedom are reduced to four degrees of freedom through spatial bending beam theory and differential geometry. The dynamics governing equations of the naturally curved beam are derived using Hamilton’s principle and Frenet–Serret formula. An improved Van der Pol model is introduced to investigate the vortex induced vibration (VIV) or vortex induced divergence (VID) response. Hammer test, wind tunnel experiment and finite element method validate the proposed model for spatially curved helical beam. It is found that under the fixed boundary condition, transposition occurs in the first two coupled modes in the normal direction of the helical beam. The errors of the first two nature frequencies are 0.016% and 0.088%, respectively. The modal analysis under the elastic boundary shows that the first-order decoupled modes in each direction will “disappear” with the variation of stiffness. Under the boundary condition of small stiffness, the first two modes of the normal direction change abruptly at the subtended angle of 76∘∼90∘. This abrupt change produces negative damping and leads to structural unstability. When the first two frequencies are close to each other, the water velocity region of VIV or VID response will intersect and the orthogonal modes in normal direction should be calculated coupled. In this case, the amplitude of the first order mode in the normal direction will always be greater than that of the second order mode. The maximum amplitude in u direction can reach 0.014 m. The obtained results provide useful information for further research on the fluid-induced vibration response of natural large-curved beam .

The pseudo-null geometric phase along optical fiber

In this study, a pseudo-null space curve in Minkowski 3-space is used to describe an optical fiber that is injected into monochromatic linear polarized light. The direction of the electric field vector with respect to the Frenet frame of a pseudo-null curve determines the state polarization of a monochromatic linearly polarized light wave traveling along an optical fiber. For the Frenet frame of a pseudo-null curve in Minkowski 3-space, the polarization vector [Formula: see text] is assumed to be perpendicular to the tangent vector [Formula: see text] with respect to anholonomic coordinates. Anholonomic coordinates for the Frenet frame of a pseudo-null curve are used to describe pseudo-null electromagnetic curves in the normal and binormal directions along an optical fiber. For the Frenet frame of the pseudo-null curve, Lorentz force equations in the normal and binormal directions along the optical fiber are presented. Pseudo-normal and binormal Rytov parallel transport laws for electric fields in the normal and binormal directions along with the optical fiber for the Frenet frame of the pseudo-null curve via anholonomic coordinates are presented. For anholonomic coordinates in Minkowski 3-space, rotations of the polarization planes of a light wave traveling in the normal and binormal directions along with the optical fiber with respect to the Frenet frame of the pseudo-null curve are obtained. Finally, a pseudo-null curve’s Maxwellian evolution is determined.

Structured time-delay models for dynamical systems with connections to Frenet-Serret frame.

Time-delay embedding and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal; for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet–Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet–Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modelling procedure on data from several nonlinear synthetic and real-world examples.

A Study on $f$-Rectifying Curves in Euclidean $n$-Space

A rectifying curve in the Euclidean $n$-space $\mathbb{E}^n$ is defined as an arc-length parametrized curve $\gamma$ in $\mathbb{E}^n$ such that its position vector always lies in its rectifying space (i.e., the orthogonal complement of its principal normal vector field) in $\mathbb{E}^n$. In this paper, in analogy to this, we introduce the notion of an $f$-rectifying curve in $\mathbb{E}^n$ as a curve $\gamma$ in $\mathbb{E}^n$ parametrized by its arc-length $s$ such that its $f$-position vector field $\gamma_f$, defined by $\gamma_f(s) = \int f(s) d\gamma$, always lies in its rectifying space in $\mathbb{E}^n$, where $f$ is a nowhere vanishing real-valued integrable function in parameter $s$. The main purpose is to characterize and classify such curves in $\mathbb{E}^n$.

Surfaces and Curves Induced by Nonlinear Schrödinger-Type Equations and Their Spin Systems

In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.

1-TYPE AND HARMONIC 1-TYPE TIMELIKE CURVES IN SEMI-EUCLIDEAN SPACE 42

In the present study we consider 1-type, biharmonic, weak biharmonic and harmonic 1-type curves in semi-Euclidean space 42R according to the timelike Frenet frame. We give some characterizations and classifications of these type curves.

The evolution of the electric field with Frenet frame in Lorentzian Lie groups

In this paper, we introduce three classes of curve evolution associated with the visco repulsive type generalized nonlinear nonlinear Schrödinger equation and the visco generalized nonlinear heat system with respect to the Frenet frame in 3 dimensional Lorentzian Lie groups. We express an electric field via the Frenet frame apparatus for three classes of curve evolution in three dimensional Lorentzian Lie groups. Later, we obtain the variations of electric fields and Lorentz force equations for three classes of curve evolution for the Frenet frame in 3D Lorentzian Lie groups.

The visco modified Heisenberg ferromagnet equation and physical applications

In this paper, we introduce three classes of a spacelike curve evolution in terms of Frenet frame for the visco modified Heisenberg ferromagnet (visco MHF) equation in Minkowski 3-space. By using those frames and the visco MHF equation, we obtain the visco modified nonlinear heat system and the visco modified Schrödinger equation. Also, we study the Lorentz force equations of the electric fields for the t,n,b-magnetic curves of the magnetic vector fields with respect to Frenet frame of the spacelike curve. From this, we discuss the geometric phases, the Frenet geometric electric flow phases and the Frenet magnetic flow densities for Lorentz forces connected with the visco MHF equation.

On a surface pencil with a common new type of special surface curve in Galilean space G 3

In this study, we investigate a new type of surface curve called a D-type special curve. We also show that this special curve is more general than a geodesic curve or an asymptotic curve. Then, we give the necessary and sufficient conditions for a curve to be the new D-type special curve using Frenet frame in Galilean space. Moreover, we investigate some results by taking account of a D-type special curve as a helix, a Salkowski curve and an anti-Salkowski curve. After then, for the sake of visualizing this study, we plot some examples for this surface pencil (i.e. surface family).

Fitting Splines to Axonal Arbors Quantifies Relationship Between Branch Order and Geometry.

Neuromorphology is crucial to identifying neuronal subtypes and understanding learning. It is also implicated in neurological disease. However, standard morphological analysis focuses on macroscopic features such as branching frequency and connectivity between regions, and often neglects the internal geometry of neurons. In this work, we treat neuron trace points as a sampling of differentiable curves and fit them with a set of branching B-splines. We designed our representation with the Frenet-Serret formulas from differential geometry in mind. The Frenet-Serret formulas completely characterize smooth curves, and involve two parameters, curvature and torsion. Our representation makes it possible to compute these parameters from neuron traces in closed form. These parameters are defined continuously along the curve, in contrast to other parameters like tortuosity which depend on start and end points. We applied our method to a dataset of cortical projection neurons traced in two mouse brains, and found that the parameters are distributed differently between primary, collateral, and terminal axon branches, thus quantifying geometric differences between different components of an axonal arbor. The results agreed in both brains, further validating our representation. The code used in this work can be readily applied to neuron traces in SWC format and is available in our open-source Python package brainlit: http://brainlit.neurodata.io/.

Geometrical effect of graded index on the transport of polarized lights.

We investigate the geometrical effect of graded index on the transport of polarized lights in the scope of geometrical optics, and present three effective methods based on the Runge-Kutta ray tracing technique in the Frenet-Serret frame and a fixed global Cartesian frame, respectively, for obtaining the polarization state of a polarized light that propagates in graded-index media. The three methods have their own advantages in accuracy, computational efficiency, and difficulty of implementation, respectively. Simulations on a three-dimensional graded-index model show that the results obtained by the three methods are in perfect agreement with each other.