Hadamard's Theorem Research Articles

On some generalizations of Hadamard's inversion theorem beyond differentiability

A recognized trend of research investigates generalizations of the Hadamard's inversion theorem to functions that may fail to be differentiable. In this vein, the present paper explores some consequences of a recent result about the existence of global Lipschitz continuous inverse by translating its assumptions of metric nature in terms of nonsmooth analysis constructions. This exploration focuses on continuous, but possibly not locally Lipschitz mappings, acting in finite-dimensional Euclidean spaces. As a result, sufficient conditions for global invertibility are formulated by means of strict estimators, ∗ -difference of convex compacta and regular/basic coderivatives. These conditions qualify the global inverse as a Lipschitz continuous mapping and provide quantitative estimates of its Lipschitz constant in terms of the above constructions.

A note on an analogue of Hadamard's theorem for determining the radii of m-meromorphy

A note on an analogue of Hadamard's theorem for determining the radii of m-meromorphy

Paradigm for the creation of scales and phases in nonlinear evolution equations

The transition from regular to apparently chaotic motions is often observed in nonlinear flows. The purpose of this article is to describe a deterministic mechanism by which several smaller scales (or higher frequencies) and new phases can arise suddenly under the impact of a forcing term. This phenomenon is derived from a multiscale and multiphase analysis of nonlinear differential equations involving stiff oscillating source terms. Under integrability conditions, we show that the blow-up procedure (a type of normal form method) and the Wentzel-Kramers-Brillouin approximation (of supercritical type) introduced in [7,8] still apply. This allows to obtain the existence of solutions during long times, as well as asymptotic descriptions and reduced models. Then, by exploiting transparency conditions (coming from the integrability conditions), by implementing the Hadamard's global inverse function theorem and by involving some specific WKB analysis, we can justify in the context of Hamilton-Jacobi equations the onset of smaller scales and new phases.

Parabolic subgroups of two‐dimensional Artin groups and systolic‐by‐function complexes

We extend previous results by Cumplido, Martin and Vaskou on parabolic subgroups of large-type Artin groups to a broader family of two-dimensional Artin groups. In particular, we prove that an arbitrary intersection of parabolic subgroups of a (2,2)-free two-dimensional Artin group is itself a parabolic subgroup. An Artin group is (2,2)-free if its defining graph does not have two consecutive edges labeled by 2. As a consequence of this result, we solve the conjugacy stability problem for this family by applying an algorithm introduced by Cumplido. All of this is accomplished by considering systolic-by-function complexes, which generalize systolic complexes. Systolic-by-function complexes have a more flexible structure than systolic complexes since we allow the edges to have different lengths. At the same time, their geometry is rigid enough to satisfy an analogue of the Cartan–Hadamard theorem and other geometric properties similar to those of systolic complexes.

On the Stability of Incommensurate h-Nabla Fractional-Order Difference Systems

This work aims to present a study on the stability analysis of linear and nonlinear incommensurate h-nabla fractional-order difference systems. Several theoretical results are inferred with the help of using some theoretical schemes, such as the Z-transform method, Cauchy–Hadamard theorem, Taylor development approach, final-value theorem and Banach fixed point theorem. These results are verified numerically via two illustrative numerical examples that show the stabilities of the solutions of systems at hand.

Analog of the Hadamard Theorem and Related Extremal Problems on the Class of Analytic Functions

We study several related extremal problems for analytic functions in a finitely connected domain \(G\) with rectifiable Jordan boundary \(\Gamma\). A sharp inequality is established between values of a function analytic in \(G\) and weighted means of its boundary values on two measurable subsets \(\gamma_{1}\) and \(\gamma_{0}=\Gamma\setminus\gamma_{1}\) of the boundary: \(|f(z_{0})|\leq\mathcal{C}\,\|f\|^{\alpha}_{L^{q}_{\varphi_{1}}(\gamma_{1})}\, \|f\|^{\beta}_{L^{p}_{\varphi_{0}}(\gamma_{0})},z_{0}\in G,0<q,p\leq\infty.\) The inequality is an analog of Hadamard’s three-circle theorem and the Nevanlinna brothers’ two-constant theorem. In the case of a doubly connected domain \(G\) and \(1\leq q,p\leq\infty\), we study the cases where the inequality provides the value of the modulus of continuity for a functional of analytic extension of a function from the part \(\gamma_{1}\) of the boundary to a given point of the domain. In these cases, the corresponding problem of optimal recovery of a function from its approximate boundary values on \(\gamma_{1}\) and the problem of the best approximation of a functional by bounded linear functionals are solved. The case of a simply connected domain \(G\) has been completely investigated previously.

Landis' conjecture for general second order elliptic equations with singular lower order terms in the plane

In this article, we study the order of vanishing and a quantitative form of Landis' conjecture in the plane for solutions to second-order elliptic equations with variable coefficients and singular lower order terms. Precisely, we let A be real-valued, bounded and elliptic, but not necessary symmetric or continuous, and we assume that V and Wi are real-valued and belong to Lp and Lqi, respectively. We prove that if u is a real-valued, bounded and normalized solution to an equation of the form −div(A∇u+W1u)+W2⋅∇u+Vu=0 in Bd, then under suitable conditions on the lower order terms, for any r sufficiently small, the following order of vanishing estimate holds||u||L∞(Br)≥rCM, where M depends on the Lebesgue norms of the lower order terms. In a number of settings, a scaling argument gives rise to a quantitative form of Landis' conjecture,inf|z0|=R⁡||u||L∞(B1(z0))≥exp⁡(−CRβlog⁡R), where β depends on p, q1, and q2. The integrability assumptions that we impose on V and Wi are nearly optimal in view of a scaling argument. We use the theory of elliptic boundary value problems to establish the existence of positive multipliers associated to the elliptic equation. Then the proofs rely on transforming the equations to Beltrami systems and applying a generalization of Hadamard's three-circle theorem.

Hadamard’s theorem for mappings with relaxed smoothness conditions

The paper puts forward sufficient conditions for a mapping from to to be a global homeomorphism. As an application, the Hadamard theorem for differentiable mappings and conditions for the existence and uniqueness of a coincidence point of a covering mapping and a Lipschitz mapping on are derived. Covering mappings of metric spaces and mappings covering at a point are studied. Bibliography: 23 titles.

High-order cluster integrals for the lattice-gas model

In the paper, for the known statistical lattice-gas model, a new method is proposed to define the high-order reducible cluster integrals on the basis of the Cauchy--Hadamard theorem and recently established strict information about the convergence radius of the virial expansions for pressure and density in powers of activity. Compared with previous studies (where the whole set of reducible cluster integrals were evaluated on the basis of the extremely limited number of low-order irreducible integrals), the proposed complex method to approximate the reducible cluster integrals yields the theoretical values of pressure at the saturation and boiling points considerably close to each other as well as to the Lee--Yang solution for the two-dimensional lattice gas.

A coarse Cartan–Hadamard theorem with application to the coarse Baum–Connes conjecture

We establish a coarse version of the Cartan–Hadamard theorem, which states that proper coarsely convex spaces are coarsely homotopy equivalent to the open cones of their ideal boundaries. As an application, we show that such spaces satisfy the coarse Baum–Connes conjecture. Combined with the result of Osajda–Przytycki, it implies that systolic groups and locally finite systolic complexes satisfy the coarse Baum–Connes conjecture.

The Cartan–Hadamard Theorem for metric spaces with local geodesic bicombings

We prove the Cartan–Hadamard Theorem for spaces which are not necessarily uniquely geodesic but locally possess a suitable selection of geodesics, a so-called convex geodesic bicombing. Furthermore, we deduce a local-to-global theorem for injective (or hyperconvex) metric spaces, saying that under certain conditions a complete, simply-connected, locally injective metric space is injective. A related result for absolute 1-Lipschitz retracts follows.

On global inverse theorems

Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inversion and implicit theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of $f$ at every $x$, namely: $f$ is a local homeomorphism or $f$ is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of $f$, ideally, expecting to conclude that $f$ is a global diffeomorphism or equivalent to a global projection. A review of such these results and some relationships between different criteria are shown. Also, in this context, a global version of Graves Theorem is obtained.

Locally convex hypersurfaces immersed in $$\mathbb {H}^n\times \mathbb {R}$$ H n × R

We prove a theorem of Hadamard–Stoker type: a connected locally convex complete hypersurface immersed in \(\mathbb {H}^n\times \mathbb {R}\) (\(n\ge 2\)), where \(\mathbb {H}^n\) is n-dimensional hyperbolic space, is embedded and homeomorphic either to the n-sphere or to \(\mathbb {R}^n\). In the latter case it is either a vertical graph over a convex domain in \(\mathbb {H}^n\) or has what we call a simple end.

Distributed shape derivativeviaaveraged adjoint method and applications

The structure theorem of Hadamard–Zolésio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. Actually the domain representation, also known as distributed shape derivative, is more general than the boundary expression as it is well-defined for shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithms. In this paper we describe several advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We identify a tensor representation of the distributed shape derivative, study its properties and show how it allows to recover the boundary expression directly. We use a novel Lagrangian approach, which is applicable to a large class of shape optimization problems, to compute the distributed shape derivative. We also apply the technique to retrieve the distributed shape derivative for electrical impedance tomography. Finally we explain how to adapt the level set method to the distributed shape derivative framework and present numerical results.

On Kakeya–Nikodym averages, $L^p$-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions

We extend a result of the second author [27, Theorem 1.1] to dimensions d \geq 3 which relates the size of L^p -norms of eigenfunctions for 2 &lt; p &lt; 2(d+1) / d-1 to the amount of L^2 -mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee [22] and a variable coefficient variant of an " \epsilon removal lemma" of Tao and Vargas [35]. We also use Hörmander's [20] L^2 oscillatory integral theorem and the Cartan–Hadamard theorem to show that, under the assumption of nonpositive curvature, the L^2 -norm of eigenfunctions e_{\lambda} over unit-length tubes of width \lambda^{-1/2} goes to zero. Using our main estimate, we deduce that, in this case, the L^p -norms of eigenfunctions for the above range of exponents is relatively small. As a result, we can slightly improve the known lower bounds for nodal sets in dimensions d \ge 3 of Colding and Minicozzi [10] in the special case of (variable) nonpositive curvature.

Recursion rules for the hypergeometric zeta function

The hypergeometric zeta function is defined in terms of the zeros of the Kummer function M(a, a+b; z). It is established that this function is an entire function of order 1. The classical factorization theorem of Hadamard gives an expression as an infinite product. This provides linear and quadratic recurrences for the hypergeometric zeta function. A family of associated polynomials is characterized as Appell polynomials and the underlying distribution is given explicitly in terms of the zeros of the associated hypergeometric function. These properties are also given a probabilistic interpretation in the framework of beta distributions.

Non-Positive Curvature and Global Invertibility of Maps

Let F:M→N be a C 1 map between Riemannian manifolds of the same dimension, M complete, N Cartan–Hadamard. We show that F is a C 1 diffeomorphism if inf x∈M |d(B ζ ∘F)(x)|>0 for all ζ∈N(∞) and Busemann functions B ζ . This generalizes the Cartan–Hadamard theorem and the Hadamard invertibility criterion, which requires inf x∈M ∥DF(x)−1∥−1=inf ζ∈N(∞)inf x∈M |d(B ζ ∘F)(x)|>0. Our proofs use a version of the shooting method for two-point boundary value problems. These ideas lead to new results about the size of the critical set of a function f∈C 2(ℝ n ,ℝ): a) If \(\inf_{x\in \mathbb{R}^{n}}|\operatorname{Hess} f(x)v|>0\) for all v≠0 then the function f has precisely one critical point. (b) If g∈C 2(ℝ n ,ℝ) is the C 1 local uniform limit of functions as in a), and \(\operatorname{Hess} g(x)\) is nowhere singular, then g has at most one critical point. The totality of functions described in (b) properly contains the class consisting of all C 2 strictly convex functions defined on ℝ n .

On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree

The ill-posedness degree for the controllability of the one-dimensional heat equation by a Dirichlet boundary control is the purpose of this work. This problem is severely (or exponentially) ill-posed. We intend to shed more light on this assertion and the underlying mathematics. We start by discussing the framework liable to fit an efficient numerical implementation without introducing further complications into the theoretical analysis. Afterward we expose the Fourier calculations that transform the ill-posedness issue to the one related to linear algebra. This consists of investigating the singular values of some infinite structured matrices that are obtained as sums of Cauchy matrices. Utilising the Gershgorin–Hadamard theorem and the Collatz–Wielandt formula, we are able to provide the lower and upper bounds for the largest singular value of these matrices. After checking whether they are also solutions of some symmetric Lyapunov (or Sylvester) equations with a very low displacement rank, we use an estimate that improves Penzl's former result to bound the ratio's smaller/largest singular values of these matrices. Accordingly, the controllability problem is confirmed to be severely ill-posed. The bounds proved here will be supported by computations made using Matlab procedures. At last, the well-known explicit inverse of Cauchy-type matrices allows us to provide a formal exponential series representation of the Dirichlet control in a long horizon controllability. That series has to be checked afterward to discover whether it is convergent or not and to find out if the desired state can be reached. Here again, some examples ran within Matlab will be discussed and commented upon.

The growth and zeros of Bargmann functions

The Bargmann formalism is presented with emphasis on the growth and zeros of Bargmann functions. Hadamard's theorem is expressed in a physical language and it is used to construct quantum states, with given zeros in their Bargmann function. Various examples are discussed. The time evolution of Bargmann functions and the motion of their zeros, for various Hamiltonians, is studied.

Foliations and global inversion

We consider topological conditions under which a locally invertible map admits a global inverse. Our main theorem states that a local diffeomorphism f: M → ℝ^n is bijective if and only if H_{n − 1}(M) = 0 and the pre-image of every affine hyperplane is non-empty and acyclic. The proof is based on some geometric constructions involving foliations and tools from intersection theory. This topological result generalizes in finite dimensions the classical analytic theorem of Hadamard–Plastock, including its recent improvement by Nollet–Xavier. The main theorem also relates to a conjecture of the aforementioned authors, involving the well-known Jacobian conjecture in algebraic geometry.