Real Analytic Functions Research Articles

On the boundedness of solutions of a forced discontinuous oscillator

We study the global boundedness of the solutions of a non-smooth forced oscillator with a periodic and real analytic forcing. We show that the impact map associated with this discontinuous equation becomes a real analytic and exact symplectic map when written in suitable canonical coordinates. By an accurate study of the behaviour of the map for large amplitudes and by employing a parametrization KAM theorem, we show that the periodic solutions of the unperturbed oscillator persist as two-dimensional tori under conditions that depend on the Diophantine conditions of the frequency, but are independent on both the amplitude of the orbit and of the specific value of the frequency. This allows the construction of a sequence of nested invariant tori of increasing amplitude that confine the solutions within them, ensuring their boundedness. The same construction may be useful to address such persistence problem for a larger class of non-smooth forced oscillators.

Fast computation of function composition derivatives for flatness-based control of diffusion problems

The chain rule is a standard tool in differential calculus to find derivatives of composite functions. Faà di Bruno’s formula is a generalization of the chain rule and states a method to find high-order derivatives. In this contribution, we propose an algorithm based on Faà di Bruno’s formula and Bell polynomials (Bell in Ann Math 29:38–46, 1927; Parks and Krantz in A primer of real analytic functions, 2012) to compute the structure of derivatives of function compositions. The application of our method is showcased using trajectory planning for the heat equation (Laroche et al. in Int J Robust Nonlinear Control 10(8):629–643, 2000).

A 2-dimensional real Banach space with constant of analyticity less than one

We show that on the real 2-dimensional Banach space ℓ12 there is an analytic function f:Bℓ12→R such that its power series at origin has radius of uniform convergence one, but for some a∈Bℓ12 the power series centred at that point has radius of uniform convergence strictly less than 1−‖a‖. This result highlights a fundamental distinction in real analytic functions (compared to complex analytic functions), where the radius of analyticity can differ from the radius of uniform convergence. Moreover, this example provides the first non-trivial upper bound for the constant of analyticity.

Unstability problem of real analytic maps

AbstractAs well known, the stability of proper maps is characterized by the infinitesimal stability. In the present paper, we study the counterpart in real analytic context. In particular, we show that the infinitesimal stability does not imply stability; for instance, a Whitney umbrella is not stable. A main tool for the proof is a relative version of Whitney's analytic approximation theorem that is shown by using H. Cartan's Theorems A and B.

Radial behavior of Mahler functions

Many papers have been recently devoted to the study of the radial behavior as [Formula: see text] of transcendental r-Mahler functions holomorphic in the open unit disk. In particular, Bell and Coons showed in 2017 that, in a generic sense, r-Mahler functions behave like [Formula: see text] for some [Formula: see text] and [Formula: see text] is a real analytic function of [Formula: see text] such that [Formula: see text]. They did not provide a formula for [Formula: see text] which was made explicit only in a few examples of r-Mahler functions of orders 1 and 2, and for specific values of r. In this paper, we first provide an explicit expression of [Formula: see text] as an exponential of a Fourier series in the variable [Formula: see text] for every r-Mahler function of order 1. Then, extending to a large setting a method introduced by Brent–Coons–Zudilin in 2016 to compute [Formula: see text] associated to the Dilcher–Stolarsky function (a 4-Mahler function of order 2 in [Formula: see text]), we provide an explicit expression of [Formula: see text] for every r-Mahler function of order 2 under mild assumptions on the coefficients in [Formula: see text] of the underlying r-Mahler equations. This applies in particular to the generating function of the Baum–Sweet sequence. We do the same for r-Mahler functions solutions of inhomogeneous Mahler equations of order 1.

Some remarks about $$ \rho $$-regularity for real analytic maps

Some remarks about $$ \rho $$-regularity for real analytic maps

On real analytic functions on closed subanalytic domains

We show that a function f:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: X \\rightarrow {\\mathbb {R}}$$\\end{document} defined on a closed uniformly polynomially cuspidal set X in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document} is real analytic if and only if f is smooth and all its composites with germs of polynomial curves in X are real analytic. The degree of the polynomial curves needed for this is effectively related to the regularity of the boundary of X. For instance, if the boundary of X is locally Lipschitz, then polynomial curves of degree 2 suffice. In this Lipschitz case, we also prove that a function f:X→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: X \\rightarrow {\\mathbb {R}}$$\\end{document} is real analytic if and only if all its composites with germs of quadratic polynomial maps in two variables with images in X are real analytic; here it is not necessary to assume that f is smooth.

Improved algebraic lower bound for the radius of spatial analyticity for the generalized KdV equation

We consider the initial value problem (IVP) for the generalized Korteweg–de Vries (gKdV) equation ∂tu+∂x3u+μuk∂xu=0,x∈R,t∈R,u(x,0)=u0(x),where u(x,t) is a real valued function, u0(x) is a real analytic function, μ=±1 and k≥4. We prove that if the initial data u0 has radius of analyticity σ0, then there exists T0>0 such that the radius of spatial analyticity of the solution remains the same in the time interval [−T0,T0]. In the defocusing case, for k∈2N, k≥4, we prove that when the local solution extends globally in time, then for any T≥T0, the radius of analyticity cannot decay faster than cT−2kk+4+ϵ, ϵ>0 arbitrarily small and c>0 a constant. The result of this work improves the one by Bona et al. (2005) where the authors proved the decay rate is no faster than cT−(k2+3k+2).

Zadacha Koshi dlya nagruzhennogo uravneniya Kortevega–de Friza v klasse periodicheskikh funktsiy

The inverse spectral problem method is applied to finding a solution of the Cauchyproblem for the loaded Korteweg–de Vries equation in the class of periodic infinite-gap functions.A simple algorithm for constructing a high-order Korteweg–de Vries equation with loaded termsand a derivation of an analog of Dubrovin’s system of differential equations are proposed. Itis shown that the sum of a uniformly convergent function series constructed by solving theDubrovin system of equations and the first trace formula actually satisfies the loaded nonlinearKorteweg–de Vries equation. In addition, we prove that if the initial function is a real π-periodicanalytic function, then the solution of the Cauchy problem is a real analytic function in thevariable x as well, and also that if the number π/n, n ∈ N, n ≥ 2, is the period of the initialfunction, then the number π/n is the period for solving the Cauchy problem with respect to thevariable x.

Cauchy problem for the nonlinear Hirota equation in the class of periodic infinite-zone functions

In this paper, the method of inverse spectral problem is used to integrate the nonlinear Hirota equation in the class of periodic infinite-zone functions. An evolution of the spectral data of the periodic Dirac operator is introduced, where the coefficient of this operator is the solution of the nonlinear Hirota equation. The solvability of the Cauchy problem for an infinite system of Dubrovin differential equations in the class of five times continuously differentiable periodic infinite-zone functions is shown. In addition, it is proved that if the initial function is a π \pi -periodic real-analytic function, then the solution of the Cauchy problem for the Hirota equation is also a real-analytic function in the variable x x ; and if the number π / 2 \pi /2 is a period (antiperiod) of the initial function, then the number π / 2 \pi /2 is a period (antiperiod) in the variable x x of the solution of the Cauchy problems for the Hirota equation.

Resolution of singularities for C∞ functions and meromorphy of local zeta functions

In this paper, we attempt to resolve the singularities of the zero variety of a C∞ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the “almost” normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with C∞ functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) C∞ function has a singularity different from the pole. From this observation, the following questions are naturally raised in the C∞ case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each C∞ function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region.

Invariant curves of quasi-periodic reversible mappings and its application

We consider the existence of invariant curves of real analytic reversible mappings which are quasi-periodic in the angle variables. By the normal form theorem, we prove that under some assumptions, the original mapping is changed into its linear part via an analytic convergent transformation, so that invariants curves are obtained. In the iterative process, by solving the modified homological equations, we ensure that the transformed mapping is still reversible. As an application, we investigate the invariant curves of a class of nonlinear resonant oscillators, with the Birkhoff constants of the corresponding Poincaré mapping all zeros or not.

Fibration theorems à la Milnor for analytic maps with non-isolated singularities

AbstractWe study the topology of real analytic maps in a neighborhood of a (possibly non-isolated) critical point. We prove fibration theorems à la Milnor for real analytic maps with non-isolated critical values. Here we study the situation for maps with arbitrary critical set. We use the concept of d-regularity introduced in an earlier paper for maps with an isolated critical value. We prove that this is the key point for the existence of a Milnor fibration on the sphere in the general setting. Plenty of examples are discussed along the text, particularly the interesting family of functions $$(f,g):{\mathbb {R}}^n \rightarrow {\mathbb {R}}^2$$ ( f , g ) : R n → R 2 of the type $$\begin{aligned} (f,g) = \left( \sum _{i=1}^n a_i x_i^p, \sum _{i=1}^n b_i x_i^q \right) , \end{aligned}$$ ( f , g ) = ∑ i = 1 n a i x i p , ∑ i = 1 n b i x i q , where $$a_i, b_i \in {\mathbb {R}}$$ a i , b i ∈ R are constants in generic position and $$p,q \ge 2$$ p , q ≥ 2 are integers.

A unified approach to nonlinear Perron-Frobenius theory

Let f:R>0n→R>0n be an order-preserving and homogeneous function. We show that the set of eigenvectors of f in R>0n is nonempty and bounded in Hilbert's projective metric if and only if f satisfies a condition involving upper and lower Collatz-Wielandt numbers of readily computed auxiliary functions. This condition generalizes a test for the existence of eigenvectors using hypergraphs that was proved by Akian, Gaubert, and Hochart. We include several examples to show how the new condition can be combined with the hypergraph test to give a systematic approach to determine when homogeneous and order-preserving functions have eigenvectors in R>0n. We also observe that if the entries of f are real analytic functions on R>0n, then the set of eigenvectors of f in R>0n is nonempty and bounded in Hilbert's projective metric if and only if the eigenvector is unique, up to scaling.

A global in time existence and uniqueness result of a multidimensional inverse problem

ABSTRACT This study is the multidimensional inverse problem for a viscoelasticity equation. This inverse problem aims to reconstruct the displacement and convolutional kernel of the integral term simultaneously, from the measurement described as the overdetermination condition. The main result is the theorem of global unique solvability of the inverse problem in the class of functions continuously in the time variable t and analytic in the space variable. The methods of scales of Banach spaces of real analytic functions of variable and weight norms in the class of continuous functions are applied.

A KAM algorithm for two-dimensional nonlinear Schrödinger equations with spatial variable

We consider the two-dimensional nonlinear Schrödinger equationiut−△u+|u|2pu+H(x,u,u¯)=0,t∈R,x∈T2 with periodic boundary conditions, where the nonlinearity H(x,u,u¯)=∑m=1∞αm(x)|u|2p+2mu is a real analytic function in a neighborhood of the origin. We obtain, through a KAM algorithm, a Whitney smooth family of small–amplitude quasi–periodic solutions.

Hyper-power series and generalized real analytic functions

This article is a natural continuation of the paper Tiwari, D., Giordano, P., Hyperseries in the non-Archimedean ring of Colombeau generalized numbers in this journal. We study one variable hyper-power series by analyzing the notion of radius of convergence and proving classical results such as algebraic operations, composition and reciprocal of hyper-power series. We then define and study one variable generalized real analytic functions, considering their derivation, integration, a suitable formulation of the identity theorem and the characterization by local uniform upper bounds of derivatives. On the contrary with respect to the classical use of series in the theory of Colombeau real analytic functions, we can recover several classical examples in a non-infinitesimal set of convergence. The notion of generalized real analytic function reveals to be less rigid both with respect to the classical one and to Colombeau theory, e.g. including classical non-analytic smooth functions with flat points and several distributions such as the Dirac delta. On the other hand, each Colombeau real analytic function is also a generalized real analytic function.

Finding the Two-Dimensional Relaxation Kernel of an Integro-Differential Wave Equation

We consider the multidimensional inverse problem of determining the kernel of the integral term in an integro-differential wave equation. In the direct problem, it is required to find the displacement function from an initial–boundary value problem, and in the inverse one, to determine the kernel of the integral term depending on both time and one of the spatial variables. The local unique solvability of the problem in the class of functions continuous in one of the variables and analytic in the other one is proved on the basis of the method of scales of Banach spaces of real analytic functions.

Real analytic functions and monomial curves

Real analytic functions and monomial curves

Sparse analytic systems

Abstract Erdős [7] proved that the Continuum Hypothesis (CH) is equivalent to the existence of an uncountable family $\mathcal {F}$ of (real or complex) analytic functions, such that $\big \{ f(x) \ : \ f \in \mathcal {F} \big \}$ is countable for every x. We strengthen Erdős’ result by proving that CH is equivalent to the existence of what we call sparse analytic systems of functions. We use such systems to construct, assuming CH, an equivalence relation $\sim $ on $\mathbb {R}$ such that any ‘analytic-anonymous’ attempt to predict the map $x \mapsto [x]_\sim $ must fail almost everywhere. This provides a consistently negative answer to a question of Bajpai-Velleman [2].