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In this chapter we introduce three very important components of variational data assimilation: tangent linear modeling, perturbation forecast modeling, and the adjoint operator. We shall present a technique to linearize the semi-Lagrangian advection as well as the tangent linear model and the adjoint of parts of spectral modeling. Given these three different techniques, it is possible to determine different properties of the numerical model as well as observation impacts. We shall introduce the singular vectors which are used to identify properties of the model.

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the Levy-type behaviour of the spatial solution, a fractional derivative in time to depict the intermittency of its temporal solution, and is driven by vector-valued fractional Brownian motion on the unit sphere to characterize its temporal long-range dependence. The solution to the SPDE is presented in the form of the Karhunen-Loeve expansion in terms of vector spherical harmonics. Its covariance matrix function is established as a tensor field on the unit sphere that is an expansion of Legendre tensor kernels. The variance of the increments and approximations to the solutions are studied and convergence rates of the approximation errors are given. It is demonstrated how these convergence rates depend on the decay of the power spectrum and variances of the fractional Brownian motion.

To achieve $$G^3$$ Hermite interpolation with a lower degree curve, this paper studies planar $$G^3$$ Hermite interpolation using a quintic Bezier curve. First, the first and second derivatives of the quintic Bezier curve satisfying $$G^2$$ condition are constructed according to the interpolation conditions. Four parameters are introduced into the construction. Two of them are set as free design parameters, which represent the tangent vector module length of the quintic Bezier curve at the two endpoints, and the other two parameters are derived from $$G^3$$ condition. Then, to match $$G^3$$ condition, it is necessary to ensure that the first derivative of curvature with respect to arc length is equal. Nevertheless, the direct calculation of the derivative of curvature involves the calculation of square root. Alternatively, an equivalent condition is derived by investigating the first derivative of curvature square. Based on this condition, the two parameters can be computed as the solutions of linear systems. Finally, the control points of the quintic Bezier curve are obtained. Several comparative examples are provided to demonstrate the effectiveness of the proposed method. A variety of complex shape curves can be obtained by adjusting the two free design parameters. Applications to shape design are also shown.

Some researches have investigated that a Bézier curve can be treated as circular arcs. This work is to proposea new scheme for approximating an arbitrary degree Bézier curve by a sequence of circular arcs. The sequenceof circular arcs represents the shape of the given Bézier curve which cannot be expressed using any other algebraicapproximation schemes. The technique used for segmentation is to simply investigate the inner anglesand the tangent vectors along the corresponding circles. It is obvious that a Bézier curve can be subdivided intothe form of subcurves. Hence, a given Bézier curve can be expressed by a sequence of calculated points on thecurve corresponding to a parametric variable t. Although the resulting points can be used in the circular arcconstruction, some duplicate and irrelevant vertices should be removed. Then, the sequence of inner angles arecalculated and clustered from a sequence of consecutive pixels. As a result, the output dots are now appropriateto determine the optimal circular path. Finally, a sequence of circular segments of a Bézier curve can be approximatedwith the pre-defined resolution satisfaction. Furthermore, the result of the circular arc representationis not exceeding a user-specified tolerance. Examples of approximated nth-degree Bézier curves by circular arcsare shown to illustrate efficiency of the new method.

In this article, we study $$C^1$$ regular curves in the 2-sphere that start and end at given points with given directions, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in an open interval. Especially, we show that a $$C^1$$ regular curve is such a curve if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in the same interval.

The main motive of the paper is to look on rectifying and osculating curves on a smooth surface. In this paper we find the normal and geodesic curvature for a rectifying curve on a smooth surface and we also prove that geodesic curvature is invariant under the isometry of surfaces such that rectifying curves remain. We find a sufficient condition for which an osculating curve on a smooth surface remains invariant under isometry of surfaces and also we prove that the component of the position vector of an osculating curve α(s) on a smooth surface along any tangent vector to the surface at α(s) is invariant under such isometry.

This paper proposes a lightweight bidirectional scattering distribution function (BSDF) model for layered materials with anisotropic reflection and refraction properties. In our method, each layer of the materials can be described by a microfacet BSDF using an anisotropic normal distribution function (NDF). Furthermore, the NDFs of layers can be defined on tangent vector fields, which differ from layer to layer. Our method is based on a previous study in which isotropic BSDFs are approximated by projecting them onto base planes. However, the adequateness of this previous work has not been well investigated for anisotropic BSDFs. In this paper, we demonstrate that the projection is also applicable to anisotropic BSDFs and that the BSDFs are approximated by elliptical distributions using covariance matrices.

The present paper is devoted to the problem of boundary conditions formulation in the growing micropolar solid mechanics. The static equations of the micropolar continuum in terms of relative tensors (pseudotensors) are derived due to virtual work principle for a solid of constant staff. The constitutive quadratic form of the elastic potential (treated as an absolute scalar) for a linear hemitropic micropolar solid is presented and discussed. The constitutive equations for symmetric and antisymmetric parts of force and couple stress tensors are given. The final forms of the static equations for the hemitropic micropolar continuum in terms of displacements and microrotations rates are obtained including the case of growing processes. A transformation of the equilibrium equations is proposed to obtain boundary conditions on the propagating growing surface in terms of relative tensors in the form of differential constraints. Those are valid for a wide range of materials and metamaterials. The algebra of rational relative invariants is intensively used for deriving the constitutive relations on the growing surface. Systems of joint algebraic rational relative invariants for force, couple stress tensors and also unit normal and tangent vectors to propagating growing surface are obtained, including systems of invariants sensitive to mirror reflections and 3D-space inversions.

In this article, several examples of mechanical systems which configuration spaces are smooth manifolds with a unique singular point are considered. Configuration spaces are the following: two smooth curves with a common point (or tangent) on the two-dimensional torus, four smooth curves on the four-dimensional torus with a common point, twodimensional cone (cusp) in the space R6. The main problem in the article is the calculation of (co)tangent space at a singular point by using different theoretical approaches. Outside of the singular point, the motion could be described in the frames of classical mechanics. But in the neighborhood of the singular points the terms like “tangent vector” and “cotangent vector” must have new conceptual definitions. In this article, the approach of differential spaces is used. Two differential structures for the modeling conical singular point are studied in order to construct (co)tangent space at singular points: locally-constants functions near to the cone vertex and the algebra of the restrictions of smooth functions in the comprehensive Euclidean space on the cone. In the first case, tangent and cotangent spaces at the singular points are zero. In the second case, the value of the functions on the cotangent bundle is constant on the cotangent layer under the singular point.

The aim of this study is to investigate the set of critical points of solutions to quasilinear elliptic boundary value problems with Dirichlet null condition on the border of a region convex in one direction, satisfying a symmetry condition. The numberof critical points is estimated by counting the number of connected components of the border having a prescribed unitary tangent vector.

Pythagorean-hodograph (PH) curves have nice properties and have found important applications in geometric modeling and CNC machining. While the unit normals of PH curves of degree n are generally rational curves of degree n − 1 , this paper investigates PH curves of arbitrary degrees but with only quadratic rational unit normals when the curves are convex or with quartic rational unit normals when the curves have single inflection points. PH curves with quadratic or quartic rational normals have simple Gauss maps and hodographs of the curves are given by low degree tangent vector fields together with simple real scaling functions. Practical algorithms for interpolation of point-normal pairs or point-normal-curvature pairs together with unit normals at selected parameter coordinates or at inflection points by the investigated PH curves without or with the constraint of arc lengths have been given. The parameters for defining the interpolating PH curves are either obtained directly from the input data or by solving simple linear systems. This method of PH curve interpolation has unique solutions and the shapes of the interpolating PH curves are controlled well by the interpolated data. Even though the interpolating curves may have cusps, the regularity of the PH curves can be checked easily based on the signs of the real scaling functions within the hodographs.

This paper describes a method for efficiently computing parallel transport of tangent vectors on curved surfaces, or more generally, any vector-valued data on a curved manifold. More precisely, it extends a vector field defined over any region to the rest of the domain via parallel transport along shortest geodesics. This basic operation enables fast, robust algorithms for extrapolating level set velocities, inverting the exponential map, computing geometric medians and Karcher/Fr\'{e}chet means of arbitrary distributions, constructing centroidal Voronoi diagrams, and finding consistently ordered landmarks. Rather than evaluate parallel transport by explicitly tracing geodesics, we show that it can be computed via a short-time heat flow involving the connection Laplacian. As a result, transport can be achieved by solving three prefactored linear systems, each akin to a standard Poisson problem. To implement the method we need only a discrete connection Laplacian, which we describe for a variety of geometric data structures (point clouds, polygon meshes, etc). We also study the numerical behavior of our method, showing empirically that it converges under refinement, and augment the construction of intrinsic Delaunay triangulations (iDT) so that they can be used in the context of tangent vector field processing., Comment: 19 pages

In this paper, monotonicity-preserving interpolation is generalized to direction-consistent interpolation. The conditions for constructing direction-consistent tangent vectors are given. The conditions on the tangent vectors are also obtained such that the piecewise cubic Hermite interpolation curves are tangent direction-consistent with the direction of data polygon. Based on geometric insights, the balanced tangent vectors are presented and proved to be direction-consistent tangent vectors. With the balanced tangent vectors, the generated cubic Hermite interpolation curves are tangent direction-consistent, and the generated quintic Hermite interpolation curves are also tangent direction-consistent provided that we use the accumulative chord length parametrization. Some graphic examples are given to show that the generated interpolation curves preserve satisfactorily the shape of the given data control polygon.

A method for performing a spherical harmonic analysis, using observed horizontal components of a tangent vector on a sphere, is presented. The vector data samples are assumed to be provided in an equiangular grid, which essentially simplifies the least-squares analysis by making use of (1) the block diagonal structure of the normal equations of least squares, (2) the even–odd symmetry of the associated Legendre functions , and (3) the fast Fourier transform of mix-radix. The correct function of the program and its numerical precision is verified by applying it to a data set, derived by evaluating a given set of vector spherical harmonic coefficients. That the program works correctly is demonstrated by the excellent agreement between the input and output spherical harmonic coefficients. Program summary Program Title: SHAVEL Program Files doi: http://dx.doi.org/10.17632/nppz4y7wg7.1 Licensing provisions: GPLv3 Programming language: Fortran 2003, Linux External routines: FFTPACK5.1, CPC program library SPHAN Classification: 4.9, 4.10, 4.11 Nature of problem: The least-squares analysis of horizontal vector field sampled in an equiangular grid on a sphere in terms of horizontal vector spherical harmonics. Solution method: The vector spherical harmonic coefficients of a horizontal vector field are estimated by the method of least-squares adjustment of data samples distributed in an equiangular grid on a sphere. For such a regular grid the normal matrix is sparse and allows the system of the normal equations to be decomposed into a series of subsystems according to azimuthal order m . The solution of each subsystem is sought by the Gauss elimination . The fast Fourier transform of mix-radix is implemented in (i) setting up the right-hand sides of the normal equations, and (ii) performing the spherical harmonic synthesis where the series of spherical harmonics are summed.

Dynamical systems, such as the second-order ODEs that govern the motion of finite-sized objects in fluids, describe the evolution of a state by a trajectory living in a high-dimensional phase space. The high dimensionality leads to visualization challenges and, for the case of inertial particles, multiple models exist that pose different assumptions. In this paper, we thoroughly address the extraction of a specific feature, namely the vortex corelines of inertial particles. Based on a general template model that comprises two of the most commonly used inertial particle ODEs, we first transform their high-dimensional tangent vector field into a Galilean reference frame in which the observed inertial particle flow becomes as steady as possible. In the optimal frame, we derive first-order and second-order vortex coreline criteria, allowing us to extract straight and bent inertial vortex corelines using 3D and 6D parallel vectors operators, respectively. With this, we generalize existing work in multiple ways: not only do we handle two inertial particle models at once, we extend the concept of second-order vortex corelines to the inertial case and make them Galilean-invariant by deriving the criteria from a steady reference frame, rather than from a geometric characterization.

In this paper, we revisit the notion of length measures associated to planar closed curves. These are a special case of area measures of hypersurfaces which were introduced early on in the field of convex geometry. The length measure of a curve is a measure on the circle $\mathbb{S}^1$ that intuitively represents the length of the portion of curve which tangent vector points in a certain direction. While a planar closed curve is not characterized by its length measure, the fundamental Minkowski-Fenchel-Jessen theorem states that length measures fully characterize convex curves modulo translations, making it a particularly useful tool in the study of geometric properties of convex objects. The present work, that was initially motivated by problems in shape analysis, introduces length measures for the general class of Lipschitz immersed and oriented planar closed curves, and derives some of the basic properties of the length measure map on this class of curves. We then focus specifically on the case of convex shapes and present several new results. First, we prove an isoperimetric characterization of the unique convex curve associated to some length measure given by the Minkowski-Fenchel-Jessen theorem, namely that it maximizes the signed area among all the curves sharing the same length measure. Second, we address the problem of constructing a distance with associated geodesic paths between convex planar curves. For that purpose, we introduce and study a new distance on the space of length measures that corresponds to a constrained variant of the Wasserstein metric of optimal transport, from which we can induce a distance between convex curves. We also propose a primal-dual algorithm to numerically compute those distances and geodesics, and show a few simple simulations to illustrate the approach., 30 pages, 10 figures

In this work, a kind of normal partner curves of a pseudo null curve on dual space forms is defined and studied. The Frenet frames and curvatures of a pseudo null curve and its associate normal curve on de-Sitter space, its associate normal curve on hyperbolic space, are related by some particular function and the angles between their tangent vector fields, respectively. Meanwhile, the relationships between the normal partner curves of a pseudo null curve are revealed. Last but not least, some examples are given and their graphs are plotted by the aid of a software programme.

WOS:000537530300013 In this paper, we have explained how to define the basic concepts of differential geometry on Dual space. To support this, dual tangent vectors that have (p) over bar as dual point of application have been defined. Then, the dual analytic functions defined by Dimentberg have been examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps have been introduced. TUBITAK (The Scientific and Technological Research Council of Turkey)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) The first and the second authors would like to thank TUBITAK (The Scientific and Technological Research Council of Turkey) for their financial supports during their Ph.D. studies.

Riemann's non-differentiable function is a classic example of a continuous function which is almost nowhere differentiable, and many results concerning its analytic regularity have been shown so far. However, it can also be given a geometric interpretation, so questions on its geometric regularity arise. This point of view is developed in the context of the evolution of vortex filaments, modelled by the Vortex Filament Equation or the binormal flow, in which a generalisation of Riemann's function to the complex plane can be regarded as the trajectory of a particle. The objective of this document is to show that the trajectory represented by its image does not have a tangent anywhere. For that, we discuss several concepts of tangent vectors in view of the set's irregularity., 27 pages, 6 figures. v2: Typos corrected, references added. v3: Accepted manuscript

Given a compact convex set F in R n , with the origin in its interior, and a point on its boundary, near which it is given by an implicit equation, we present a formula to compute the curvature in the direction of any tangent vector. For this, we consider the intersection curve between the boundary of F and a suitable plane, but without using the plane equations or the curve expression. Furthermore, we see that, when we use the equations of the plane and the equation that define the boundary of F near the fixed point, the formula that we obtain is equivalent to the existing ones, but it is easier to apply.

This paper presents a parametric level set method for the optimization of composite structures with curvilinear fibers. We use iso-contours of a scalar function , also called the level set function , to represent the fibers, and the level set function is constructed by a set of radial basis functions with compact support. The orientation of fiber at any arbitrary point is defined by the orientation of the tangent vector of iso-contour. In order to prevent gaps and overlaps between neighboring fiber tows from arising during the manufacturing of composite structures, a constraint is formulated that the norm of gradient vector of the level set function is equal to 1 almost everywhere. Furthermore, these gradient constraints are aggregated into a single one by using the p -norm method to improve the efficiency of optimization. The compliance minimization problem is considered, and the coefficients of radial basis functions are taken as the design variables. The results of optimizations proved that the proposed method can provide an optimized composite structure with equally spaced curvilinear fibers.

The present paper deals with some characterizations of rectifying and osculating curves on a smooth surface with respect to the reference frame { T → , N → , T → × N → } . We have computed the components of position vectors of rectifying and osculating curves along T → , N → , T → × N → and then investigated their invariancy under isometry of surfaces, and it is shown that they are invariant iff either the normal curvature of the curve is invariant or the position vector of the curve is in the direction of the tangent vector to the curve.

We present a systematic method for producing double-sided minimal surfaces by wire-cut machines. A link between minimal surfaces and ruled surfaces is pursued through wire cutting. Weierstrass parameterization is employed to define minimal surfaces ( $$\mathbb {R}^3$$ ) over a complex plane ( $$\mathbb {C}$$ ). Our method consists of three components. First, the orthogonal double-sided cuts match a pair of orthonormal tangent vectors on the surface. Second, A closed-form expression for the principal directions facilitates the global quadrangulation of minimal surfaces. Third, the CNC machine’s toolpath results from the surface’s analytic characterization. Asymptotic cutting and principal cutting are compared in terms of collisions and cutting error. We employed a general-purpose language (Java) to create machine instructions from the Weierstrass representation of minimal surfaces. Thus, the entire workflow from mathematical modeling to production involves no 3D models or CAD/CAM software. Both a 5-axis wire cutter and a customized robotic system were tested.

The analysis of members that can be modeled as extensible elastic slender rods is investigated. A meshfree formulation using a Local Radial Point Interpolation Method (LRPIM) is developed that utilizes radial basis functions in curvilinear coordinates. This approach bypasses the need to utilize more conventional element meshes and significantly reduces the number of equations needed for the numerical solution. The slender rod formulation presented allows for tension variation, axial stretch, incremental loading and distributed load variation along the rod. It is well suited for nonlinear problems that involve large deflections and rigid rotations. The position and tangent vectors are expressed using Hermite-type approximations, and radial basis functions, while the interpolation of tension variation and distributed loads are described using polynomials. The solution procedure of weighted residuals Galerkin weak formulation combined with an incremental iterative numerical scheme is introduced to address the incremental loading and large deflection issues for static and quasi-static problems. The implementation of the analytical formulation and the numerical procedure are illustrated using three nonlinear problems. The first two examples provide insight into the validity, accuracy, and efficiency of the methodology. The third example presents the case of a moving boundary condition problem which models a cable entangled by fishing boat-trawling equipment.

Tangent categories were introduced by Rosicky as a categorical setting for differential structures in algebra and geometry; in recent work of Cockett, Crutwell and others, they have also been applied to the study of differential structure in computer science. In this paper, we prove that every tangent category admits an embedding into a representable tangent category---one whose tangent structure is given by exponentiating by a free-standing tangent vector, as in, for example, any model of Kock and Lawvere's synthetic differential geometry. The key step in our proof uses a coherence theorem for tangent categories due to Leung to exhibit tangent categories as a certain kind of enriched category., 17 pages. v3: final journal version

We present a nonlinear discrete Kirchhoff–Love four-node shell finite element that is based on the cubic Hermite edge curves and the bilinear Coons surface patch spanning the surface between them. The cubic Hermite edge curves are constructed by minimizing the bending curvature of a spatial curve connecting two adjacent nodes of the element. The G 1 -continuity is obtained at each node of the finite element mesh. Namely, the tangent vectors of the set of the edge curves attached to a given node of the mesh share the same tangent plane to the shell mid-surface for any configuration. To avoid the membrane locking, common in shell elements with higher-order interpolations, the assumed natural strains are adopted, solving the plate compatibility equation. The derived element has 5 degrees of freedom per node, 3 mid-surface displacements and 2 rotations of the mid-surface normal vector, which also rotate the corresponding mid-surface tangent plane. Several numerical examples illustrate its performance in linear and nonlinear tests, for both regular and distorted meshes.

A current ideal control issue in the region of financial aspects has numerical properties that do not fall into the standard optimal control problem detailing. In our concern the state an incentive at the final time, y(T ) = z, is free and obscure, and furthermore the integrand is a piecewise consistent capacity of the obscure esteem y(T ). This is not a standard optimal control problem and cannot be settled utilizing Pontryagin’s Minimum Principle with the standard limit conditions at the final time. In the standard issue a free final state y(T ) yields an important limit condition p(T ) = 0, where p(t) is the costate. Since the integrand is a component of y(T ), the new fundamental condition is that y(T ) yield to be equivalent to a specific necessary that is a consistent capacity of z. We present a continuous estimation of the piecewise consistent integrand function through hyperbolic tangent approach and tackle a case utilizing a C++ shooting method with Newton emphasis for tackling the two point boundary value problem (TPBVP). The limiting free y(T ) value is computed in an external circle emphasis through the Golden Section method.

The shape operator is one of the most important research tools in differential geometry of surfaces. It uses the tangent vectors on the surface, extensively the tangent vectors of the parameter curves on the surface, and the normal vectors of the surfaces. Quaternions have the practical method in the rotation in the Euclidean 3-space. The main goal of our paper is to use quaternions in the research of the surfaces. Firstly, we have given the shape operator with Darboux frame along the curve in the surface. Then, the quaternionic shape operator is defined by the quaternion. Also, we have obtained some results related to the quaternionic shape operator and its matrix representation.

The integrability of certain classes of dynamic systems is shown on the tangent bundles to twodimensional manifolds. In this case, the force fields have the so-called variable dissipation and generalize the previously considered fields.

Path following is an indispensable function for autonomous vehicles. Desired paths may be of arbitrary shape, not just the mostly investigated straight lines and circles. This paper addresses the path following problem of arbitrary twice differentiable curves using vector-field-based approach. A tangent vector field is constructed through coordinate transformation, and a sufficient condition for its feasibility concerning the input saturation is given out. A saturated turning velocity controller is designed and its Lyapunov stability is discussed in detail. Numerical simulation results show us that the path following performance of the proposed approach is comparable with that of the literature while involving 5 less parameters to be set.

The integrability of certain classes of dynamic systems arising in the dynamics of a multidimensional solid and in the dynamics of a point on a tangent bundle of a multidimensional sphere is shown. In this case, the force fields under consideration have the so-called variable dissipation with zero average and generalize previously considered fields.

The Holder convexity of the composition of inverse hyperbolic tangent function and Jacobian sine function is investigated in this paper.

We propose a novel closed curve blending technique that describes continuous and gradual transitions between a pair of closed curves in a plane. The main contribution of this study is to formulate the closed curve blending problem in terms of curvature flow. In this formulation, closed curve blending can be represented as a simple linear flow under linear constraints. This approach enables rapid blending, which is preferred in the production of 2D animation. In the case of curves with different turning numbers, our technique also offers smooth blending by suppressing sharp changes in the tangent vector by incorporating the curvature smoothing flow. This has not been realized with previous methods. We conducted experiments and confirmed that our technique achieved aesthetic blending in under a tenth of a second, two orders of magnitude reduction in computation time compared to state-of-the-art methods.

The paper contains a sufficient condition for an intersection of regular tangent cones to be a tangent cone. Regular tangent cones and tents for sets given by locally Lipschitz functions are constructed. The cones are described in terms of generalized K-derivatives.

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of K\"ahler-Einstein metrics with cone singularities., Comment: Reference to Panov's Polyhedral Kahler Manifolds added

Methods to estimate surface areas of geometric objects in 3D are well known. A number of these methods are of Monte Carlo type, and some are based on the Cauchy–Crofton formula from integral geometry. Employing this formula requires the generation of sets of random lines that are uniformly distributed in 3D. One model to generate sets of random lines that are uniformly distributed in 3D is called the tangent model (see [4]). In this paper, we present an extension of this model to higher dimensions, and we examine its performance by estimating hypersurface areas of n-ellipsoids. Then we apply this method to estimate surface areas of hypersurfaces defined by Fermat-type varieties of even degree.

Integrability in elementary functions is demonstrated for some classes of dynamic systems on tangent bundles of two- and three-dimensional spheres. The force fields possess the so-called variable dissipation with a zero mean and generalize those considered earlier.

Consider a problem consisting of conservation laws coupled with ordinary differential equations through boundary conditions. We provide a characterization of the solutions by means of metric tangent vectors, thus guaranteeing the uniqueness of the solutions.

We deal with irregular curves contained in smooth, closed, and compact surfaces. For curves with finite total intrinsic curvature, a weak notion of parallel transport of tangent vector fields is well-defined in the Sobolev setting. Also, the angle of the parallel transport is a function with bounded variation, and its total variation is equal to an energy functional that depends on the "tangential" component of the derivative of the tantrix of the curve. We show that the total intrinsic curvature of irregular curves agrees with such an energy functional. By exploiting isometric embeddings, the previous results are then extended to irregular curves contained in Riemannian surfaces. Finally, the relationship with the notion of displacement of a smooth curve is analyzed., 30 pages, 1 figure; v2: published on Rendiconti del Circolo Matematico di Palermo; v3: with Erratum

We investigate the spectrum of the Dirichlet Laplacian in a unbounded strip subject to a new deformation of "shearing": the strip is built by translating a segment oriented in a constant direction along an unbounded curve in the plane. We locate the essential spectrum under the hypothesis that the projection of the tangent vector of the curve to the direction of the segment admits a (possibly unbounded) limit at infinity and state sufficient conditions which guarantee the existence of discrete eigenvalues. We justify the optimality of these conditions by establishing a spectral stability in opposite regimes. In particular, Hardy-type inequalities are derived in the regime of repulsive shearing., Comment: 21 pages, 3 figures

For a large class of dynamical systems, the optimally time-dependent (OTD) modes, a set of deformable orthonormal tangent vectors that track directions of instabilities along any trajectory, are known to depend "pointwise" on the state of the system on the attractor but not on the history of the trajectory. We leverage the power of neural networks to learn this "pointwise" mapping from the phase space to OTD space directly from data. The result of the learning process is a cartography of directions associated with strongest instabilities in the phase space. Implications for data-driven prediction and control of dynamical instabilities are discussed.

According to the wave power rule, the second derivative of a function with respect to the variable t is equal to negative n times the function raised to the power of 2n-1. Solving the ordinary differential equations numerically results in waves appearing in the figures. The ordinary differential equation is very simple; however, waves, including the regular amplitude and period, are drawn in the figure. In this study, the function for obtaining the wave is called the leaf function. Based on the leaf function, the exact solutions for the undamped and unforced Duffing equations are presented. In the ordinary differential equation, in the positive region of the variable, the second derivative becomes negative. Therefore, in the case that the curves vary with the time under the condition x(t)>0, the gradient constantly decreases as time increases. That is, the tangential vector on the curve of the graph changes from the upper right direction to the lower right direction as time increases. On the other hand, in the negative region of the variable, the second derivative becomes positive. The gradient constantly increases as time decreases. That is, the tangent vector on the curve changes from the lower right direction to the upper right direction as time increases. Since the behavior occurring in the positive region of the variable and the behavior occurring in the negative region of the variable alternately occur in regular intervals, waves appear by these interactions. In this paper, I present seven types of damped and divergence exact solutions by combining trigonometric functions, hyperbolic functions, hyperbolic leaf functions, leaf functions, and exponential functions. In each type, I show the derivation method and numerical examples, as well as describe the features of the waveform.

International audience; There is a class of planar 1D-continua which can be described exclusively by their placement functions which in turn are curves in a two-dimensional space. In contrast to the Elastica for which the deformation energy depends on the projection of the second gradient to the normal vector of the placement function, i.e. the material curvature, the proposed continuum does also depend on the projection onto the tangent vector, introduced as the stretch gradient. Thus, the deformation energy takes into account the complete second gradient of the placement function. In such a model, non-standard boundary conditions and more generalized forces such as double forces do appear. The deformation energy of the continuum is obtained by applying a heuristic homogenization procedure to a family of slender discrete pantographic structures constituted by extensional and rotational springs. Within the homogenization process, the overall length of the system is kept fixed, the number of the periodically appearing sub-systems, called cells, is increased, and the stiffnesses are appropriately scaled. For two examples, we numerically compare the family of discrete systems with the continuum. The analysis shows that the continuum represents the behaviour of the discrete system already for a relatively moderate number of cells. In particular, the behaviour of the deformation energy error between the discrete and the continuum models when the number of cells tends to infinity is determined by the homogenization process.