Position vector of a time-like slant helix in Minkowski 3-space

In this paper, position vector of a time-like slant helix with respect to standard frame of Minkowski space E³₁ is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine position vector of an arbitrary time-like slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the representation of a time-like Salkowski and time-like anti-Salkowski curves as examples of a slant helices, by means of intrinsic equations. Moreover, we present some new characterizations of slant helices and illustrate some examples of our main results.     

Position vectors of slant helices in Euclidean 3-space

In this paper, position vector of a slant helix with respect to standard frame in Euclidean space E³ is studied in terms of Frenet equations. First, a vector differential equation of third order is constructed to determine a position vector of an arbitrary slant helix. In terms of solution, we determine the parametric representation of the slant helices from the intrinsic equations. Thereafter, we apply this method to find the parametric representation of a Salkowski curve, anti-Salkowski curve and a curve of constant precession, as examples of a slant helices, by means of intrinsic equations.     

Spacelike helices in Minkowski 4-space $${E_1^4 }$$

In this paper, we give some characterizations for spacelike helices in Minkowski space–time E₁⁴{E_1^4}. We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space–time E₁⁴{E_1^4}.     

Position vectors of spacelike general helices in Minkowski 3-space

In this paper, the position vectors of a spacelike general helix with respect to the standard frame in Minkowski space are studied in terms of the Frenet equations. First, a vector differential equation of third order is constructed to determine the position vectors of an arbitrary spacelike general helix. In terms of solution, we determine the parametric representation of the general helices from the intrinsic equations. Moreover, we give some examples to illustrate how to find the position vectors of spacelike general helices with a spacelike and timelike principal normal vector.     

On contact equivalence of Abel cubic differential equations of second order

We study the geometry of cubic Abel differential equations on one-dimensional real curve. We prove that such an equation is the kernel of some nonlinear differential operator. This operator is defined by a cubic on the Cartan distribution on the space of 1-jets. Based on this observation, we construct a contact-invariant {e}-structure associated with a non-degenerate Abel equation and obtain contact classification of such equations.     

On variational approach to differential invariants of rank two distributions

We construct differential invariants for generic rank 2 vector distributions on n-dimensional manifolds, where n?5. Our method for the construction of invariants is completely different from the Cartan reduction-prolongation procedure. It is based on the dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the theory of unparameterized curves in the Lagrange Grassmannian, developed in [A. Agrachev, I. Zelenko, Geometry of Jacobi curves I, J. Dynam. Control Syst. 8 (1) (2002) 93-140; II, 8 (2) (2002) 167-215]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n?5. In the next paper [I. Zelenko, Fundamental form and Cartan's tensor of (2,5)-distributions coincide, J. Dynam. Control. Syst., in press, SISSA preprint, Ref. 13/2004/M, February 2004, math.DG/0402195] we show that in the case n=5 our fundamental form coincides with the Cartan covariant biquadratic binary form, constructed in 1910 in [E. Cartan, Les systemes de Pfaff a cinque variables et les equations aux derivees partielles du second ordre, Ann. Sci. Ecole Normale 27 (3) (1910) 109-192; reprinted in: Oeuvres completes, Partie II, vol. 2, Gautier-Villars, Paris, 1953, pp. 927-1010]. Therefore first our approach gives a new geometric explanation for the existence of the Cartan form in terms of an invariant degree four differential on an unparameterized curve in Lagrange Grassmannians. Secondly, our fundamental form provides a natural generalization of the Cartan form to the cases n>5. Somewhat surprisingly, this generalization yields a rational function on the fibers of the appropriate vector bundle, as opposed to the polynomial function occurring when n=5. For n=5 we give an explicit method for computing our invariants and demonstrate the method on several examples.     

Lyapunov regularity of impulsive differential equations

For linear impulsive differential equations, we give a simple criterion for the existence of a nonuniform exponential dichotomy, which includes uniform exponential dichotomies as a very special case. For this we introduce the notion of Lyapunov regularity for a linear impulsive differential equation, in terms of the so-called regularity coefficient. The theory is then used to show that if the Lyapunov exponents are nonzero, then there is a nonuniform exponential behavior, which can be expressed in terms of the Lyapunov exponents of the differential equation and of the regularity coefficient. We also consider the particular case of nonuniform exponential contractions when there are only negative Lyapunov exponents. Having this relation in mind, it is also of interest to provide alternative characterizations of Lyapunov regularity, and particularly to obtain sharp lower and upper bound for the regularity coefficient. In particular, we obtain bounds expressed in terms of the matrices defining the impulsive linear system, and we obtain characterizations in terms of the exponential growth rate of volumes. In addition we establish the persistence of the stability of a linear impulsive differential equation under sufficiently small nonlinear perturbations.     

Some curvature properties of Cartan spaces with mth root metrics

In this paper, we define some non-Riemannian curvature properties for Cartan spaces. We consider a Cartan space with the mth root metric. We prove that every mth root Cartan space of isotropic Landsberg curvature, or isotropic mean Landsberg curvature, or isotropic mean Berwald curvature reduces to a Landsberg, weakly Landsberg, and weakly Berwald spaces, respectively. Then we show that the mth root Cartan space of almost vanishing H-curvature satisfies H?=?0.     

Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities

In 1956, R. Penrose studied best-approximate solutions of the matrix equation AX = B. He proved that A⁺B (where A⁺ is the Moore-Penrose inverse) is the unique matrix of minimal Frobenius norm among all matrices which minimize the Frobenius norm of AX ? B. In particular, A⁺ is the unique best-approximate solution of AX = I. The vector version of Penrose's result (that is, the fact that the vector A⁺b is the best-approximate solution in the Euclidean norm of the vector equation Ax = b) has long been generalized to infinite dimensional Hilbert spaces.In this paper, an infinite dimensional version of Penrose's full result is given. We show that a straightforward generalization is not possible and provide new extremal characterizations (in terms of the Hermitian order) of A⁺ and of the classes of generalized inverses associated with minimal norm solutions of consistent operator equations or with least-squares solutions. For a certain class of operators, we can phrase our characterizations in terms of a whole class of norms (including the Hilbert-Schmidt and the trace norms), thus providing new extremal characterizations even in the matrix case. We treat both operators with closed range and with not necessarily closed range. Finally, we characterize A⁺ as the unique inner inverse of minimal Hilbert-Schmidt norm if ∥A⁺∥₂ < ∞. We give an application of the new extremal characterization to the compensation problem in systems analysis in infinite-dimensional Hilbert spaces.     

The differential equation satisfied by a plane curve of degree n

Eliminating the arbitrary coefficients in the equation of a generic plane curve of order n by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to conics, has been obtained by Monge. Sylvester, Halphen, Cartan used invariants of higher order. The expression of these invariants is rather complicated, but becomes much simpler when interpreted in terms of symmetric functions.     

On Darboux helices in Euclidean 4‐space

The notion of Darboux helix in Euclidean 3‐space was introduced and studied by Yayl? et al. 2012. They show that the class of Darboux helices coincide with the class of slant helices. In a special case, if the curvature functions satisfy the equality κ² + τ² = constant, then these curves are curve of the constant precession. In this paper, we study Darboux helices in Euclidean 4‐space, and we give a characterization for a curve to be a Darboux helix. We also prove that Darboux helices coincide with the general helices. In a special case, if the first and third curvatures of the curve are equal, then Darboux helix, general helix, and V₄‐slant helix are the same concepts.     

On projective symmetries on Finsler spaces

There are two definitions of Einstein-Finsler spaces introduced by Akbar-Zadeh, which we will show is equal along the integral curves of I-invariant projective vector fields. The sub-algebra of the C-projective vector fields, leaving the H-curvature invariant, has been studied extensively. Here we show on a closed Finsler space with negative definite Ricci curvature reduces to that of Killing vector fields. Moreover, if an Einstein-Finsler space admits such a projective vector field then the flag curvature is constant. Finally, a classification of compact isotropic mean Landsberg manifolds admitting certain projective vector fields is obtained with respect to the sign of Ricci curvature.     

The prescribed p-mean curvature equation of low regularity in the Heisenberg group

This work reports on the author’s recent study about regularity and the singular set of a C ¹ smooth surface with prescribed p (or H)-mean curvature in the 3-dimensional Heisenberg group. As a differential equation, this is a degenerate hyperbolic and elliptic PDE of second order, arising from the study of CR geometry. Assuming only the p-mean curvature H ∈ C ⁰, it is shown that any characteristic curve is C ² smooth and its (line) curvature equals ?H. By introducing special coordinates and invoking the jump formulas along characteristic curves, it is proved that the Legendrian (horizontal) normal gains one more derivative. Therefore the seed curves are C ² smooth. This work also obtains the uniqueness of characteristic and seed curves passing through a common point under some mild conditions, respectively. In an on-going project, it is shown that the p-area element is in fact C ² smooth along any characteristic curve and satisfies a certain ordinary differential equation of second order. Moreover, this ODE is analyzed to study the singular set.     

Minimal harmonic graphs and their Lorentzian cousins

Motivated by the observation that the only surface which is locally a graph of a harmonic function and is also a minimal surface in E³ is either a plane or a helicoid, we provide similar characterizations of the elliptic, hyperbolic and parabolic helicoids in L³ as the nontrivial zero mean curvature surfaces which also satisfy the harmonic equation, the wave equation, and a degenerate equation which is derived from the harmonic equation or the wave equation. This elementary and analytic result shows that the change of the roles of dependent and independent variables may be useful in solving differential equations.     

Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

The Cauchy problem for critical and subcritical semilinear parabolic equations in Lr (II), initial data in critical Sobolev spaces Ḣ−s,rs

By presenting some time–space L^p−L^r estimates, we will establish the local and global existence and uniqueness of solutions for semilinear parabolic equations with the Cauchy data in critical Sobolev spaces of negative indices. Our results contain the complex (derivative) Ginzburg–Landau equation and the Cahn–Hilliard equation as special cases.     

A direct method for solving an anisotropic mean curvature flow of plane curves with an external force

A new method for solution of the evolution of plane curves satisfying the geometric equation v=β(x,k,ν), where v is the normal velocity, k and ν are the curvature and tangential angle of a plane curve Γ ? ?² at the point x∈Γ, is proposed. We derive a governing system of partial differential equations for the curvature, tangential angle, local length and position vector of an evolving family of plane curves and prove local in time existence of a classical solution. These equations include a non‐trivial tangential velocity functional governing a uniform redistribution of grid points and thus preventing numerically computed solutions from forming various instabilities. We discretize the governing system of equations in order to find a numerical solution for 2D anisotropic interface motions and image segmentation problems. Copyright © 2004 John Wiley & Sons, Ltd.     

两种特殊的螺旋线

通过对螺旋线主法线面上一些特殊曲线的研究,得到了两种螺旋线的特征.     

Hyperelliptic functions solutions of some nonlinear partial differential equations using the direct method

In this paper, we will use a simple and direct method to obtain particular solutions of some (2+1) dimensional nonlinear partial differential equations expressed in terms of the Kleinian hyperelliptic functions for a given curve y²=f(x) whose genus is two. We observe that this method generalizes the auxiliary method.