On regularization of plurisubharmonic functions on manifolds

We study the question of when a -plurisubharmonic function on a complex manifold, where is a fixed -form, can be approximated by a decreasing sequence of smooth -plurisubharmonic functions. We show in particular that it is always possible in the compact Kähler case.

     

Adaptive Arnoldi-Tikhonov regularization for image restoration

In the framework of the numerical solution of linear systems arising from image restoration, in this paper we present an adaptive approach based on the reordering of the image approximations obtained with the Arnoldi-Tikhonov method. The reordering results in a modified regularization operator, so that the corresponding regularization can be interpreted as problem dependent. Numerical experiments are presented.     

Geometric Hermite interpolation with Tschirnhausen cubics

Explicit formulae are found that give the unique Tschirnhausen cubic that solves a geometric Hermite interpolation problem. That solution is used to create a planar G¹ spline by joining segments of Tschirnhausen cubics. If the geometric Hermite data is from a smooth function, the Tschirnhausen cubic approximates the smooth function. The error in the approximation of a short segment of length h can be expressed as a power series in h. The error is O(h⁴) and the coefficient of the leading term is found.     

Solving cubics by polynomial fitting

A top-performance algorithm for solving cubic equations is introduced. This algorithm uses polynomial fitting for a decomposition of the given cubic into a product of a quadratic and a linear factor. This factorization can be computed extremely accurately and efficiently using a fixed-point iteration of the linearized fitting error. The polynomial fitting concept performs orders of magnitude better in terms of numerical accuracy and precision than any of the currently known and available algorithms for solving cubic equations. A special exception handler is presented for a reliable operation in the event of double, triple and tightly clustered roots.     

A new regularization of equilibrium problems on Hadamard manifolds: applications to theories of desires

In this paper, we introduce a new proximal algorithm for equilibrium problems on a genuine Hadamard manifold, using a new regularization term. We first extend recent existence results by considering pseudomonotone bifunctions and a weaker sufficient condition than the coercivity assumption. Then, we consider the convergence of this proximal-like algorithm which can be applied to genuinely Hadamard manifolds and not only to specific ones, as in the recent literature. A striking point is that our new regularization term have a clear interpretation in a recent “variational rationality” approach of human behavior. It represents the resistance to change aspects of such human dynamics driven by motivation to change aspects. This allows us to give an application to the theories of desires, showing how an agent must escape to a succession of temporary traps to be able to reach, at the end, his desires.

     

All cubics are self-isogonal

We develop techniques to prove that a cubic curve is invariant under the isogonal transformation of the projective plane determined by some triangle.     

Duality and Riemannian cubics

Riemannian cubics are curves used for interpolation in Riemannian manifolds. Applications in trajectory planning for rigid bodiy motion emphasise the group SO(3) of rotations of Euclidean 3-space. It is known that a Riemannian cubic in a Lie group G with bi-invariant Riemannian metric defines a Lie quadratic V in the Lie algebra, and satisfies a linking equation. Results of the present paper include explicit solutions of the linking equation by quadrature in terms of the Lie quadratic, when G is SO(3) or SO(1,2). In some cases we are able to give examples where the Lie quadratic is also given in closed form. A basic tool for constructing solutions is a new duality theorem. Duality is also used to study asymptotics of differential equations of the form , where β₀,β₁ are skew-symmetric 3×3 matrices, and x :ℝ→ SO(3). This is done by showing that the dual of β₀+tβ₁ is a null Lie quadratic. Then results on asymptotics of x follow from known properties of null Lie quadratics. To Charles Micchelli, with warm greetings and deep respect, on his 60th birthday Mathematics subject classifications (2000) 53A17, 53B20, 65D18, 68U05, 70E60.     

Primitive free cubics with specified norm and trace

The existence of a primitive free (normal) cubic over a finite field with arbitrary specified values of () and (primitive) is guaranteed. This is the most delicate case of a general existence theorem whose proof is thereby completed.

     

Twisted cubics, bis

We consider the smooth compactification constructed in [12] for a space of varieties like twisted cubics. We show this compactification embeds naturally in a product of flag varieties.Partially supported by CNPq, Pronex (ALGA)     

Adaptive total variation regularization based scheme for Poisson noise removal

To better preserve the edge features, this paper investigates an adaptive total variation regularization based variational model for removing Poisson noise. This edge‐preserving scheme comprises a spatially adaptive diffusivity coefficient, which adjusts the diffusion strength automatically. Compared with the classical total variation based one, numerical simulations distinctly indicate the superiority of our proposed strategy in maintaining the small details while denoising Poissonian image. Copyright © 2012 John Wiley & Sons, Ltd.     

Frobenius nonclassicality of Fermat curves with respect to cubics

For Fermat curves F: aX ⁿ + bY ⁿ = Z ⁿ defined over F _q , we establish necessary and sufficient conditions for F to be F _q -Frobenius nonclassical with respect to the linear system of plane cubics. In the new F _q -Frobenius nonclassical cases, we determine explicit formulas for the number N _q (F) of F _q -rational points on F. For the remaining Fermat curves, nice upper bounds for N _q (F) are immediately given by the Stöhr–Voloch Theory.     

Double cubics and double quartics

We study a double cover branched over a smooth divisor such that R is cut on V by a hypersurface of degree 2(n−deg(V)), where n ≥ 8 and V is a smooth hypersurface of degree 3 or 4. We prove that X is nonrational and birationally superrigid.     

On the Jacobians of plane cubics

Let S be a scheme and f a ternary cubic form whose ten coefficients are sections of O_S without common zero. The equation f=0 defines a family of plane cubic curves parametrized by S. We prove that the family of generalized Jacobians of those cubic curves is a group scheme J/S which is the locus of smoothness of a scheme f*=0, where f* is a Weierstrass cubic formf*=f*(x,y,z)=y²z+a₁xyz+a₂yz²-x³-a₂x²z-a₄xz²-a₆z³, in which the coefficient a_i is a homogeneous polynomial with integral coefficients, of degree i in the ten coefficients of f, which we give explicitly. A key ingredint of the proof is a characterization, over sufficiently nice bases, of group algebraic spaces which can be described by such a Weierstrass equation.