We prove that any C transformation, possibly with a (non-flat) critical or singular region, admits an invariant probability measure absolutely continuous with respect to any expanding measure whose Jacobian satisfies a mild distortion condition. This is an extension to arbitrary dimension of a famous theorem of Keller [37] for maps of the interval with negative Schwarzian derivative. We also sh...

In [Hz], Heinz proved that there is no harmonic diffeomorphism from the unit disk D onto the complex plane C. The result was generalized by Schoen [S] and he proved that there is no harmonic diffeomorphism from the unit disk onto a complete surface of nonnegative curvature. Unlike conformal or quasi-conformal maps between Riemann surfaces, the inverse of a harmonic map is not harmonic in genera...

H. Brezis and L. Nirenberg proved that if (gk) ⊂ C(S , S ) and g ∈ C(S , S ) (N ≥ 1) are such that gk → g in BMO(S ), then deg gk → deg g. On the other hand, if g ∈ C(S , S ), then Kronecker’s formula asserts that deg g = 1 |SN | ∫ SN det(∇g) dσ. Consequently, ∫ SN det(∇gk) dσ converges to ∫ SN det(∇g) dσ provided gk → g in BMO(S N ). In the same spirit, we consider the quantity J(g, ψ) := ∫ SN...

In this paper, kinematic modeling and singularity analysis of a three DOF redundant parallel manipulator has been elaborated in detail. It is known, that on the contrary to series manipulators, the forward kinematic map of parallel manipulators involves highly coupled nonlinear equations, whose closed-form solution derivation is a real challenge. This issue is of great importance noting that th...

A kinematic singularity occurs when the Jacobian matrix associated with the forward kinematic map loses rank. At a singular connguration, the end-eeector velocities are restricted to a subspace of the task space velocities. Also, rate control schemes that require the inversion of the Jacobian matrix fail at such singulari-ties. However, even at a singular connguration, if the kinematic map is g...

In this paper, we show that the Jacobian conjecture holds for gradient maps in dimension n ≤ 3 over a field K of characteristic zero. We do this by extending the following result for n ≤ 2 by F. Dillen to n ≤ 3: if f is a polynomial of degree larger than two in n ≤ 3 variables such that the Hessian determinant of f is constant, then after a suitable linear transformation (replacing f by f(Tx) f...

We show that the iterated images of a Jacobian pair f : C 2 → C 2 stabilize; that is, all the sets f k (C 2) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of C N , and f : X → X an open polynomial map with X − f (X) a finite set. We show that the sets f k (X) stabilize, and for any cofinite subset Ω ⊆ X with f (Ω) ⊆ Ω, the sets f k (Ω) stabilize. We appl...

In this paper we give a detailed proof of the Jacobian Conjecture (posed in 1939) which says that a polynomial map Cn → Cn is invertible if and only if its Jacobian is a non-zero constant. To prove it we equip both the set of all polynomial automorphisms of Cn and the set of all polynomial endomorphisms of Cn whose Jacobians are non-zero constants with structures of ind-varieties. Then using al...

Given a curve C of genus 2 defined over a field k of characteristic different from 2, with Jacobian variety J , we show that the two-coverings corresponding to elements of a large subgroup of H ` Gal(k/k), J [2](k) ́ (containing the Selmer group when k is a global field) can be embedded as intersection of 72 quadrics in P k , just as the Jacobian J itself. Moreover, we actually give explicit equ...

Sufficient conditions have been recently given for a classs of ergodic maps of an interval onto itself: I = [0, 1] ⊂ R1 → I and its associated binary function to generate a sequence of independent and idetically distributed (i.i.d.) random variables. Jacobian elliptic Chebyshev map, its derivative and second derivative induce Jacobian elliptic space curve. A mapping of the space curve onto itse...