Arbitrary Curve Research Articles

On the Properties of Algebraic Geometric Codes as Copy Protection Codes

Traceability schemes which are applied to the broadcast encryption can prevent unauthorized parties from accessing the distributed data. In a traceability scheme a distributor broadcasts the encrypted data and gives each authorized user unique key and identifying word from selected error-correcting code for decrypting. The following attack is possible in these schemes: groups of c malicious users are joining into coalitions and gaining illegal access to the data by combining their keys and identifying codewords to obtain pirate key and codeword. To prevent this attacks, classes of error-correcting codes with special c-FP and c-TA properties are used. In particular, c -FP codes are codes that make direct compromise of scrupulous users impossible and c -TA codes are codes that make it possible to identify one of the a‹ackers. We are considering the problem of evaluating the lower and the upper boundaries on c, within which the L-construction algebraic geometric codes have the corresponding properties. In the case of codes on an arbitrary curve the lower bound for the c-TA property was obtained earlier; in this paper, the lower bound for the c-FP property was constructed. In the case of curves with one infinite point, the upper bounds for the value of c are obtained for both c-FP and c-TA properties. During our work, we have proved an auxiliary lemma and the proof contains an explicit way to build a coalition and a pirate identifying vector. Methods and principles presented in the lemma can be important for analyzing broadcast encryption schemes robustness. Also, the c-FP and c-TA boundaries monotonicity by subcodes are proved.

A General Exact Closed-Form Solution for Nonlinear Differential Equation of Pendulum

In the present paper, the nonlinear differential equation of pendulum is investigated to find an exact closed form solution, satisfying governing equation as well as initial conditions. The new concepts used in the suggested method are introduced. Regarding the fact that the governing equation for any arbitrary system represents its inherent properties, it is shown that the nonlinear term causes that the system to have a variable identity. Hence, the original function is included as a variable in the solution to can take into account the variation of governing equation. To find the exact closed form solution, the variation of the nonlinear differential equation tends to zero, where in this case the system with a local linear differential equation has a definite identity with a definite local answer. It is shown that the general answer is an arbitrary curve on a surface, a newly developed concept known as super function, and different initial conditions give different curves as particular solutions. The comparison of the results with those of finite difference shows an exact agreement for any arbitrary amplitude and initial conditions.

Synthesizing Mechanical Chains for Morphing Between Spatial Curves

Abstract This paper extends the kinematic synthesis methodology for designing a chain of bodies to match a set of arbitrary curves to the spatial case. The methodology initiates with an arbitrary set of spatial curves, and concludes with a set of bodies defined by their spatial features. The bodies synthesized can be one of three types: a rigid segment, a helical segment with constant curvature and torsion but varying length, and a growth segment that maintains its geometry but may be scaled to become larger or smaller. To realize mechanical chains for mechanisms that achieve spatial shape change, only rigid and helical segments are used. After designing the segments, they may be aligned with the original spatial curves with their ends connected via an optimization. For two curves, these connections may be made with revolute joints to obtain high accuracy. For three or more curves, spherical joint connections allow for best accuracy. To compare curves as is useful in morphometry, all three segment types may be employed. In this case, an accurate description of the changes between curves is important, and optimizing to connect the segments is not needed. The procedure for redefining the curves in a way that the techniques in this paper may be applied, as well as the methodologies for synthesizing the three segment types are presented. Examples include a continuum robot problem and the morphometric analyses of cochlear curves and the lambdoidal suture located on a human skull. This work extends the established planar techniques for synthesizing mechanisms and addressing morphometric issues that are motivated with curves in two-dimensions.

Multi-band Acousto-Optical Filters for Correlation Spectroscopy

The problem of creating optical filters with an arbitrary transmission curve for correlation spectroscopy problems is considered. Experiments showed that acousto-optical filters with a digital synthesis of a high-frequency control signal make it possible to create transmission functions of various types: multi-band, with variable positions and widths of windows, including asymmetric shapes, which makes it possible to synthesize transmission functions of complex form.

Stiffness Simulation of Joints of Large Manipulators based on Cam Torsion Spring Mechanisms

There is elasticity in the joint of large manipulators, and the transfer torque is related to the relative rotation angle between input and output terminal. In order to simulate this stiffness characteristic in the test experiment, a cam torsion spring mechanism and its design method are proposed in this paper. The mechanism can simulate arbitrary angle-torque curve by directional design. After the necessary analysis and design, the actual characteristics of the mechanism are tested by experiments, which proves that the mechanism can simulate the stiffness characteristics of joints of large manipulator arm very well.

Generalization of the least uncomfortable journey problem

Abstract The variational problem of the least uncomfortable journey between two locations on a straight line is simplified by a choice of the dependent variable. It is shown that taking the position, instead of the velocity, as the optimal function of time to be determined does away with the isoperimetric constraint. The same results as those found with the velocity as the dependent variable are obtained in a simpler and more concise way. Next the problem is generalized for motion on an arbitrary curve. In the case of acceleration-induced discomfort, it is shown that, as expected, motion on a curved path is always more uncomfortable than motion on a straight line. It is not clear that this is necessarily the case for jerk-induced discomfort, which appears to indicate that the acceleration provides a more reasonable measure of the discomfort than the jerk. The example of motion on a circular path is studied. Although we have been unable to solve the problem analytically, approximate solutions have been constructed by means of trial functions and the exact solution has been found numerically for some choices of the relevant parameters.

Self-Reproducing Hidden Attractors in Fractional-Order Chaotic Systems Using Affine Transformations

This article proposes a unified approach for hidden attractors control in fractional-order chaotic systems. Hidden attractors have small basins of attractions and are very sensitive to initial conditions and parameters. That is, they can be easily drifted from chaotic behavior into another type of dynamics, which is not suitable for encryption applications that require quite wide initial conditions and parameters ranges for encryption key design. Hence, a systematic coordinate affine transformation framework is utilized to construct transformed systems with self-reproducing attractors. Simulation results of two three-dimensional fractional-order chaotic systems with hidden attractors validate that the proposed framework supports attractors geometric structure design and multi-wing generation. Hidden attractor size, polarity, phase, shape and position control while preserving the chaotic dynamics is indicated by strange attractors, spectral entropy, maximum Lyapunov exponent and bifurcation diagrams. Simulations demonstrate the capability of multi-wing generation from fractional-order hidden attractors with no equilibria using non-autonomous parameters as opposed to the classical equilibria extension techniques suitable only for self-excited attractors. The self-reproduced multiple wings can share the same center point or be distributed along an arbitrary line, curve or surface thanks to the non-autonomous translation parameters. Multi-wing attractors widen the basin of attraction and enlarge the state space volume. For practical applications, the proposed technique makes fractional-order systems with hidden attractors suitable for circuit implementations that require specific signal level and polarity conditions. In addition, for digital encryption applications, the relatively wide range of the extra parameters enhances the key space and hence the robustness against brute force attacks.

A Vehicle Trajectory Tracking Method With a Time-Varying Model Based on the Model Predictive Control

The vehicle trajectory tracking algorithm is one of the key and difficult issues of intelligent driving technologies. In current control algorithms for the vehicle trajectory tracking, there are three main assumptions: the standard working condition for the driving path, the same control model used for the entire control process, and a fixed value for the longitudinal vehicle speed. However, the above determinations in current control algorithms are inconsistent with the actual vehicle driving conditions. To overcome those problems, a vehicle trajectory tracking method with a time-varying model is proposed. The time-varying model is developed by using a two-dimensional vehicle kinematics model. This method considers the influences of the longitudinal speed and road curvature on the vehicle trajectory tracking stability under the low-speed complex driving condition. Thus, the proposed method can improve the trajectory tracking accuracy when the unmanned vehicle is located at the road with a sharp curvature under the low-speed complex driving condition. Moreover, the proposed model can achieve the real time calculation. Meanwhile, the prediction accuracy of the vehicle kinematics model is ensured. The proposed approach with the above characteristics can complete the trajectory tracking for the route composed of arbitrary curves. The results show that the proposed method can effectively improve the trajectory tracking stability of the unmanned vehicle on the roads with different curvatures under complex driving conditions.

Metal removal from aqueous solutions: insights from modeling precipitation titration curves

To provide a good reference for many metal ion removal processes, we developed an analytical model (AM) and used the PHREEQC computer program to numerically model (NM) arbitrary metal hydroxide precipitation titration curves (PTCs). For validation, these models were compared to slow titration experiments of ZnCl2 with NaOH to precipitate Zn(OH)2. The AM effectively demonstrates how initial pH, metal ion concentration and - valence and metal hydroxide equilibrium constant affect the shape of the PTCs. Comparison of AM and NM exposed AM limitations that include the effect of NaCl concentration, activity coefficients, and multi-species ions, on the shape of the PTCs. The experimental results agree well with either model for different concentrations of ZnCl2 at slow titration rates. However, the models and experiments deviated at higher titration rates. These deviations were attributed to inhomogeneities in the solution concentration, which caused precipitation at a bulk pH below the theoretical precipitation pH (i.e., pre-precipitation).

A modified 'Hawkyard's method to predict the deformation of projectiles in Taylor impact tests

The shapes of projectiles after Taylor impact tests can be predicted by using simple equations that were proposed by Hawkyard, in which elastic-perfectly plastic materials and instant strain-hardening materials are assumed to represent the projectiles. In order to improve the accuracy, the force on the undeformed volume of the projectile to depend on the strain of the deformed volume. Based on the equation of motion, equations that represent the shape of the deformed projectile by assuming an instant strain-hardening model are developed. In the same way, those assuming an arbitrary continuous stress-strain curve are developed, and the solution of using the power-law hardening rule to represent the stress strain relation is derived. To validate the formulations, Taylor impact tests were conducted, and the deformed shapes of the projectiles were compared with the calculations made by the developed formulations. The lengths of the deformed projectiles are reproduced within several percent by the proposed models using the stress-strain relation obtained from split-Hopkinson bar tests. The proposed models tend to underestimate the diameters of the deformed projectiles around the contact face and to overestimate the diameters of the other part of the deformed projectiles.

Avocado: Photometric Classification of Astronomical Transients with Gaussian Process Augmentation

Abstract Upcoming astronomical surveys such as the Large Synoptic Survey Telescope (LSST) will rely on photometric classification to identify the majority of the transients and variables that they discover. We present a set of techniques for photometric classification that can be applied even when the training set of spectroscopically confirmed objects is heavily biased toward bright, low-redshift objects. Using Gaussian process regression to model arbitrary light curves in all bands simultaneously, we “augment” the training set by generating new versions of the original light curves covering a range of redshifts and observing conditions. We train a boosted decision tree classifier on features extracted from the augmented light curves, and we show how such a classifier can be designed to produce classifications that are independent of the redshift distributions of objects in the training sample. Our classification algorithm was the best-performing among the 1094 models considered in the blinded phase of the Photometric LSST Astronomical Time-Series Classification Challenge, scoring 0.468 on the organizers’ logarithmic-loss metric with flat weights for all object classes in the training set, and achieving an AUC of 0.957 for classification of SNe Ia. Our results suggest that spectroscopic campaigns used for training photometric classifiers should focus on typing large numbers of well-observed, intermediate-redshift transients, instead of attempting to type a sample of transients that is directly representative of the full data set being classified. All of the algorithms described in this paper are implemented in the avocado software package (https://www.github.com/kboone/avocado).

Particle sliding down an arbitrary concave curve in the Lagrangian formalism

We apply the method of Lagrange multipliers to the problem of a particle sliding on an arbitrary concave downward surface under the action of gravity to obtain the point where it leaves the surface.

The correlated linking numbers of a Brownian loop with two arbitrary curves

The standard kinetic path integral for all spatially closed Brownian paths (loops) of duration t weighted by the product mn is evaluated, where m and n are the linking numbers of the Brownian loop with two arbitrary curves in 3D space. The path integral thus indicates the extent to which these two linking numbers are correlated, ranging from the value zero for far apart curves when it is unlikely that the Brownian loop links with both, to ±infinity for nearly coincident curves. The result takes a form that loosely resembles that for the mutual inductance of two current carrying circuits in magnetostatics, a double line integral, but is also dependent on a single extra parameter, the duration t of the path. The result for the equivalent 2D problem was given previously (Hannay ).

NUFFT for the Efficient Spectral Domain MoM Analysis of a Wide Variety of Multilayered Periodic Structures

In this paper the Method of Moments (MoM) in the spectral domain is used for the analysis of multilayered structures containing periodic arrays of either patches or apertures. The patches and apertures may have many different geometries including complex surfaces limited by two parallel lines and two arbitrary curves, circular and elliptic rings, circular and elliptic arcs, and circular and elliptic sectors. Basis functions accounting for edge singularities are used in the approximation of the electric/magnetic current density on the patches/apertures, which enables a fast convergence of MoM with respect to the number of basis functions. Since the 2-D Fourier transforms of the basis functions cannot be obtained in closed-form, these Fourier transforms are efficiently computed by means of the nonuniform fast Fourier transform (NUFFT) algorithm. Results have been obtained for frequency-selective surfaces (FSSs), and for the elements used in the design of both reflectarray and metasurface antennas. The results obtained indicate that the software based on the NUFFT is only 15% slower than the standard spectral domain MoM software used for structures in which the 2-D Fourier transform of the basis functions is analytical, and between 50 and 80 times faster than CST.

Conservation Laws and Other Formulas for Families of Rays and Wavefronts and for the Eikonal Equation

In the previous studies, the author has obtained conservation laws for the 2D eikonal equation in an inhomogeneous isotropic medium. These laws are divergent identities of the form div F = 0. The vector field F is expressed through a solution to the eikonal equation (the time field), the refractive index (the equation parameter), and their partial derivatives. Besides that, equivalent conservation laws (divergent identities) were found for families of rays and families of wavefronts in terms of their geometric characteristics. Thus, the geometric essence (interpretation) of the conservation laws obtained for the 2D eikonal equation was found. In this paper, 3D analogs of the results obtained are presented: differential conservation laws for the 3D eikonal equation and conservation laws (divergent identities of the form div F = 0) for families of rays and families of wavefronts, the vector field F expressed through classical geometric characteristics of the ray curves: their Frenet basis (the unit tangent vector, principal normal, and binormal), the first curvature, and the second curvature, or through the classical geometric characteristics of the wavefront surfaces: their normal, principal directions, principal curvatures, Gaussian curvature, and mean curvature. All the results have been obtained on the basis of the general vector and geometric formulas (differential conservation laws and some formulas) obtained for families of arbitrary smooth curves, families of arbitrary smooth surfaces, and arbitrary smooth vector fields.

ARCHED TRANSMISSIONS TEETH GEOMETRY WITHIN OFFSET OF INITIAL PROFILE

There has been researched teeth geometry of cylindrical arched teeth gears, cut with rod instrument, teeth of which profiled by arbitrary curve in normal cross-section and in linear direction within initial profile offset.Toothing of instrumental rod with arched teeth with gears within initial profile offset is an analogue of the process of cutting teeth with rod instrument by means of enveloping. Surfaces of gear teeth within toothing with rod are enveloping of teeth surfaces of the last one.If teeth surfaces of instrumental rods for cutting pinion teeth and gear are non-congruous, then we have case of dotty toothing of pinion and gear. If initial profiles are non-congruous, then contact dot moves from one side of the tooth to another.It was obtained equations of teeth surfaces, while writing down coordinates of toothing surfaces in coordinate systems X 1 Y 1 Z 1 andX 2 Y 2 Z 2 , connected with pinion and gear. While making such a transition, we have equations of surfaces of convex side of pinion teeth and concave side of gear teeth. In common case those equations are equations of helical lines of variable pitch. These equations can be used within the determination of toothing field borders, appropriate to the tops of the pinion and the gear teeth. Within this case two types of toothed gears with arched teeth will differ: gears with symmetric arched teeth (teeth are symmetric relatively to the plane and asymmetric arched teeth (working area of teeth surface is situated from one side of plane.Received results can be used within the determination of indices of loading ability and other characteristics of cylindrical arched transmissions with generalized teeth geometry within initial profile offset.

Finite generation of the algebra of type A conformal blocks via birational geometry II: higher genus

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed curves, whose fiber over a smooth curve is a moduli space of semistable parabolic bundles. This generalizes a construction of a degeneration of the moduli space of vector bundles presented in a recent work of Belkale and Gibney.

Modeling curved interfaces without element‐partitioning in the extended finite element method

SummaryIn this paper, we model holes and material interfaces (weak discontinuities) in two‐dimensional linear elastic continua using the extended finite element method on higher‐order (spectral) finite element meshes. Arbitrary parametric curves such as rational Bézier curves and cubic Hermite curves are adopted in conjunction with the level set method to represent curved interfaces. Efficient computation of weak form integrals with polynomial integrands is realized via the homogeneous numerical integration scheme—a method that uses Euler's homogeneous function theorem and Stokes' theorem to reduce integration to the boundary of the domain. Numerical integration on cut elements requires the evaluation of a one‐dimensional integral over a parametric curve, and hence, the need to partition curved elements is eliminated. To improve stiffness matrix conditioning, ghost penalty stabilization and the Jacobi preconditioner are used. For material interface problems, we develop an enrichment function that captures weak discontinuities on spectral meshes. Taken together, we show through numerical experiments that these advances deliver optimal algebraic rates of convergence with h‐refinement (p=1,2,…,5) and exponential rates of convergence with p‐refinement (p=1,2,…,7) for elastostatic problems with holes and material inclusions on Cartesian pth‐order spectral finite element meshes.

Holograms to Focus Arbitrary Ultrasonic Fields through the Skull

We report 3D-printed acoustic holographic lenses for the formation of ultrasonic fields of complex spatial distribution inside the skull. Using holographic lenses, we experimentally, numerically and theoretically produce acoustic beams whose spatial distribution matches target structures of the central nervous system. In particular, we produce three types of targets of increasing complexity. First, a set of points are selected at the center of both right and left human hippocampi. Experiments using a skull phantom and 3D printed acoustic holographic lenses show that the corresponding bifocal lens simultaneously focuses acoustic energy at the target foci, with good agreement between theory and simulations. Second, an arbitrary curve is set as the target inside the skull phantom. Using time-reversal methods the holographic beam bends following the target path, in a similar way as self-bending beams do in free space. Finally, the right human hippocampus is selected as a target volume. The focus of the corresponding holographic lens overlaps with the target volume in excellent agreement between theory in free-media, and experiments and simulations including the skull phantom. The precise control of focused ultrasound into the central nervous system is mainly limited due to the strong phase aberrations produced by refraction and attenuation of the skull. Using the present method, the ultrasonic beam can be focused not only at a single point but overlapping one or various target structures simultaneously using low-cost 3D-printed acoustic holographic lens. The results open new paths to spread incoming biomedical ultrasound applications including blood-brain barrier opening or neuromodulation.

Collapse mode of rock mass induced by a concealed karst cave above a deep cavity

Reliable prediction of the potential collapse region of rock mass is a key challenge for deep underground cavity excavation, especially if a concealed karst cave is located above the excavated cavity. Because of the effect of the blast vibration, a possible collapse would occur at a certain region between the concealed karst cave and the excavated cavity. This paper aims to study the roof collapse of a deep buried cavity induced by a concealed karst cave base on a two-dimensional failure mechanism by using upper bound theorem. The failure mechanism is constituted by arbitrary curves which is similar to the collapse observed in an actual cavity excavation. The shape and range of the collapse block is determined by virtual work equation in conjunction with variational approach. The results obtained by the presented approach are approximate with the numerical results provided by finite difference code, which indicates that the proposed method in this work is valid.