Hadamard's Theorem Research Articles

Classical Global Solutions for a Class of Hamilton–Jacobi Equations

Let V be a real-valued function of class C2 on \({\mathbb{R}^n}\), n ≥ 2, which vanishes if |x| ≤ R and, for some \({\epsilon >0 }\), satisfies \({\partial_x^\alpha V(x)=O(|x|^{-\epsilon-|\alpha|}) }\), as |x| → ∞, for |α| ≤ 2. Using a global inverse function theorem of Hadamard, we show that if R is sufficiently large, then the Hamilton–Jacobi equation of eikonal type |∇u|2 + V(x) = k2, with k > 0, has a C1 solution on \({\mathbb{R}^n\setminus\{0\}}\).

A BOUNDARY VERSION OF CARTAN–HADAMARD AND APPLICATIONS TO RIGIDITY

The classical Cartan–Hadamard theorem asserts that a closed Riemannian manifold Mn with non-positive sectional curvature has universal cover [Formula: see text] diffeomorphic to ℝn, and a by-product of the proof is that [Formula: see text] is homeomorphic to Sn-1. We prove analogues of these two results in the case where Mn has a non-empty totally geodesic boundary. More precisely, if [Formula: see text], [Formula: see text] are two negatively curved Riemannian manifolds with non-empty totally geodesic boundary, of dimension n ≠ 5, we show that [Formula: see text] is homeomorphic to [Formula: see text]. We show that if [Formula: see text] and [Formula: see text] are a pair of non-positively curved Riemannian manifolds with totally geodesic boundary (possibly empty), then the universal covers [Formula: see text] and [Formula: see text] are diffeomorphic if and only if the universal covers have the same number of boundary components. We also show that the number of boundary components of the universal cover is either 0, 2 or ∞. As a sample application, we show that simple, thick, negatively curved P-manifolds of dimension ≥ 6 are topologically rigid. We include some straightforward consequences of topological rigidity (diagram rigidity, weak co-Hopf property, and the Nielson problem).

Some remarks on holomorphic functions and taylor series in C n

Some previous results on convergence of Taylor series in C n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy—Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C n are constructed and the Taylor series expansion is deduced.

Bounds for Zeros of Entire Functions

Let f be an entire function. Denote by z1(f),z2(f),… the zeros of f with their multiplicities. In the paper, estimates for the sums $$\sum_{k=1}^{j}\frac{1}{|z_{k}(f)|}\quad (j=1,2,\ldots )$$ and for the counting function of the zeros of f are established. If f is of finite order ρ(f), we derive bounds for the series $$S_{p}(f):=\sum_{k=1}^{\infty}\frac{1}{|z_{k}(f)|^{p}}\quad (p\geq \rho(f))\quad \mbox{and}\quad \sum_{k=1}^{\infty}\biggl(\mathop{Im}\frac{1}{z_{k}(f)}\biggr)^{2}\quad (\rho(f)<2),$$ as well as relations between the series $$\sum_{k=1}^{\infty}\frac{1}{z_{k}^{m}(f)}\quad (m\geq \rho(f))$$ and the traces of certain matrices. The contents of the paper is closely connected with the following well-known results: the Hadamard theorem on the convergence exponent of the zeros of an entire function, the Jensen inequality for the counting function, the Cauchy theorem on the comparison of the zeros of polynomials, Ostrowski’s inequalities for the real and imaginary parts of the zeros of polynomials and the Cartwright–Levinson theorem. The suggested approach is based on the recent development of the spectral theory of linear operators.

A parametrization method for solving nonlinear two-point boundary value problems

A sharper version of the local Hadamard theorem on the solvability of nonlinear equations is proved. Additional parameters are introduced, and a two-parameter family of algorithms for solving nonlinear two-point boundary value problems is proposed. Conditions for the convergence of these algorithms are given in terms of the initial data. Using the right-hand side of the system of differential equations and the boundary conditions, equations are constructed from which initial approximations to the unknown parameters can be found. A criterion is established for the existence of an isolated solution to a nonlinear two-point boundary value problem. This solution is shown to be a continuous function of the data specifying the problem.

The Hadamard Theorem and Borel–Carathéodory Theorem on Riemann Surfaces

In this paper, we first define a kind of pseudo–distance function and annulus domain on Riemann surfaces, then prove the Hadamard Theorem and the Borel–Caratheodory Theorem on any Riemann surfaces.

Absolutely exponential stability of Cohen–Grossberg neural networks with unbounded delays

In this paper, the Cohen–Grossberg neural networks (CGNNs) with variable and unbounded delays are studied. By using the Lipschitzian Hadamard Theorem and a property of homeomorphism mapping, some new sufficient conditions are obtained to ensure the existence, uniqueness and stability of the equilibrium point. The activation functions need only to be partially Lipschitz continuous and monotone nondecreasing, but do not require to be bounded or differentiable.

LOCAL INEQUALITIES FOR SIDON SUMS AND THEIR APPLICATIONS

The authors consider Sidon sets of first kind. By comparing them with the Steinhaus sequence, they prove a local Khintchine-Kahane inequality on compact sets. As consequences, they prove the following results for Sidon series taking values in a Banach space: the summability on a set of positive measure implies the almost everywhere convergence; the contraction principle of Billard-Kahane remains true for Sidon series. As applications, they extend a uniqueness theorem of Zygmund concerning lacunary Fourier series and an analytic continuation theorem of Hadamard concerning lacunary Taylor series. Some of their results still hold for Sidon sets of second kind.

On sums of roots of infinite order entire functions

A class of infinite order entire functions is considered. Estimates for sums of the roots are derived. These estimates supplement the Hadamard theorem. Moreover, we establish a new estimate for the counting function of the roots, which in appropriate situations can be more useful than the Jensen inequality.

Relations for Imaginary Parts of Zeros of Entire Functions

Finite order entire functions are considered. New relations for the imaginary parts of the zeros are derived. They particularly generalize the Cartwright-Levinson theorem. By virtue of these relations, under some restriction, the Hadamard theorem on the convergence exponent of the zeros is improved. 1. The main result. Consider the finite order entire function (1.1) f(λ) = ∞ ∑ k=0 akλ k (k!)γ , λ ∈ C, a0 = 1, γ > 0, with complex, in general, coefficients. Assume that (1.2) w(f) ≡ ∞ ∑ k=1 |ak| m. AMS Mathematics Subject Classification. 30D20, 30C15.

Matrix inequalities and the complex Monge–Ampère operator

We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge&#8211;Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of f

Determining radii of meromorphy via orthogonal polynomials on the unit circle

Using a convergence theorem for Fourier–Padé approximants constructed from orthogonal polynomials on the unit circle, we prove an analogue of Hadamard's theorem for determining the radius of m-meromorphy of a function analytic on the unit disk and apply this to the location of poles of the reciprocal of Szegő functions.

On numerical solutions of periodically perturbed conservative systems

A nonlinear perturbed conservative system is discussed. By means of Hadamard's theorem, the existence and uniqueness of the solution of the continuous problem are proved. When the equation is discreted on the uniform meshes, it is proved that the corresponding discrete problem has a unique solution. Finally, the accuracy of the numerical solution is considered and a simple algorithm is provided for solving the nonlinear difference equation.

Determinant Forms for Extended Heat Functions and Expansions of Solutions of Associated Cauchy Problems

We briefly review series solutions of differential equations problems of the second order that lead to coefficients expressed in terms of determinants. Derivative type formulas involving a generating function with several parameters are developed for these determinant coefficients in first order problems. These permit constructing determinant forms for the heat polynomials and their Appell transforms. Hadamard's theorem for bounding determinants and conical regions are used to deduce simplified versions of expansion theorems involving these polynomials and associated Appell transforms. Extended versions of the heat equation are also considered.

Hele–Shaw flow on hyperbolic surfaces

Consider a complete simply connected hyperbolic surface. The classical Hadamard theorem asserts that at each point of the surface, the exponential mapping from the tangent plane to the surface defines a global diffeomorphism. This can be interpreted as a statement relating the metric flow on the tangent plane with that of the surface. We find an analogue of Hadamard's theorem with metric flow replaced by Hele–Shaw flow, which models the injection of (two-dimensional) fluid into the surface. The Hele–Shaw flow domains are characterized implicitly by a mean value property on harmonic functions.

A Cartan–Hadamard Theorem for Banach–Finsler Manifolds

In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.

Global inversion via the Palais-Smale condition

Fixing a complete Riemannian metric g on $\mathbb R^n$, we show that a local diffeomorphism $f : \mathbb R^n\to \mathbb R^n$ is bijective if the height function $f\cdot v$ (standard inner product) satisfies the Palais-Smale condition relative to $g$ for each for each nonzero $v\in \mathbb R^n$. Our method substantially improves a global inverse function theorem of Hadamard. In the context of polynomial maps, we obtain new criteria for invertibility in terms of Lojasiewicz exponents and tameness of polynomials.

A generalization of Hadamard's Theorem

lim sup n→∞ | n(0)| = ‘(61) then S−1(z) can be extended to an analytic function in the disk D0 = [|z|iR0]; R0 = 1=‘; where it coincides with the sum of the series above (see [4,3]). This formula reminds us of Cauchy’s formula for the radius of convergence of a power series. Let Dm = [|z|iRm] be the largest disk centered at z = 0 inside of which S−1 can be extended to a meromorphic function with no more than m poles (counting multiplicities). For each +xed m= 0; 1; : : : ; set

Strictly Convex Submanifolds and Hypersurfaces of Positive Curvature

We construct smooth closed hypersurfaces of positive curvature with prescribed submanifolds and tangent planes. Further, we develop some applications to boundary value problems via Monge-Ampére equations, smoothing of convex polytopes, and an extension of Hadamard's ovaloid theorem to hypersurfaces with boundary.

Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function f(z) from the Hardy space Hp, p ≥ 1, in disks of radii ρ, ρ1, and ρ2, 0 < ρ1 < ρ < ρ2 < 1.