Real Analytic Functions Research Articles

Boundedness in asymmetric oscillations under the non-resonant case

In this paper, we are concerned with the boundedness of all solutions of the asymmetric oscillationx″+ax+−bx−=p(t), where x+=max⁡{x,0}, x−=max⁡{−x,0}, p(t) is a real analytic 2π periodic function, a and b are two different positive constants satisfying ω0:=12(1a+1b)∈R\\Q and the condition|kω0−l|≥c0Ω(|k|),k∈Z\\{0},l∈Z, where Ω is an approximation function and c0 is a small positive constant. In particular, when a=5,b=1,p(t)=cos⁡4t, the boundedness of all solutions will be proved.

Polynomial configurations in sets of positive upper density over local fields

Let F(x) = (f1(x), …, fm(x)) be such that 1, f1, …, fm are linearly independent polynomials with real coefficients. Based on ideas of Bachoc, DeCorte, de Oliveira and Vallentin in combination with estimating certain oscillatory integrals with polynomial phase we will show that the independence ratio of the Cayley graph of ℝm with respect to the portion of the graph of F defined by a ≤ log åså ≤ T is at most O(1/(T — a)). We conclude that if I ⊆ ℝm has positive upper density, then the difference set I — I contains vectors of the form F(s)for an unbounded set of values s ∈ ℝ. It follows that the Borel chromatic number of the Cayley graph of ℝm with respect to the set {±F(s): s ∈ ℝ} is infinite. Analogous results are also proven when ℝ is replaced by the field of p-adic numbers ℚp. At the end, we will also show the existence of real analytic functions f1, …, fm, for which the analogous statements no longer hold.

Topological Classification and Finite Determinacy of Knotted Maps

We show that the knot type of the link of a real analytic map germ with isolated singularity f : ( R 2 , 0 ) → ( R 4 , 0 ) is a complete invariant for C 0 - A -equivalence. Moreover, we also prove that isolated singularity implies finite C 0 -determinacy, giving an explicit estimate for its degree. For the general case of real analytic map germs, f : ( R n , 0 ) → ( R p , 0 ) ( n ≤ p ), we use the Lojasiewicz exponent associated with Mond’s double point ideal I 2 ( f ) to obtain some criteria of Lipschitz and analytic regularity.

Quasi-Periodic Solution of Nonlinear Beam Equation on $${\mathbb T}^d$$ with Forced Frequencies

In this paper, we study the higher dimensional nonlinear beam equation under periodic boundary condition: utt+Δx2u+Mξu+f(ω¯t,u)=0,u=u(t,x),t∈R,x∈Td,d≥2where Mξ is a real Fourier multiplier, f=f(θ¯,u) is a real analytic function with respect to (θ¯,u), and f(θ¯,u)=O(|u|2). This equation can be viewed as an infinite dimensional nearly integrable Hamiltonian system. We establish an infinite dimensional KAM theorem, and apply it to this equation to prove that there exist a class of small amplitude quasi-periodic solutions corresponding to finite dimensional invariant tori.

Global graph of metric entropy on expanding Blaschke products

We study the global picture of the metric entropy on the space of expanding Blaschke products. We first construct a smooth path in the space tending to a parabolic Blaschke product. We prove that the metric entropy on this path tends to 0 as the path tends to this parabolic Blaschke product. It turns out that the limiting parabolic Blaschke product on the unit circle is conjugate to the famous Boole map on the real line. Thus we can give a new explanation of Boole's formula discovered more than one hundred and fifty years ago. We modify the first smooth path to get a second smooth path in the space of expanding Blaschke products. The second smooth path tends to a totally degenerate map. We see that the first and second smooth paths have completely different asymptotic behaviors near the boundary of the space of expanding Blaschke products. However, they represent the same smooth path in the space of all smooth conjugacy classes of expanding Blaschke products. We use this to give a complete description of the global graph of the metric entropy on the space of expanding Blaschke products. We prove that the global graph looks like a bell. It is the first result to show a global picture of the metric entropy on a space of hyperbolic dynamical systems. We apply our results to the measure-theoretic entropy of a quadratic polynomial with respect to its Gibbs measure on its Julia set. We prove that the measure-theoretic entropy on the main cardioid of the Mandelbrot set is a real analytic function and asymptotically zero near the boundary.

Steklov spectral problems in a set with a thin toroidal hole

The paper concerns the Steklov spectral problem for the Laplace operator, and some variants in a 3-dimensional bounded domain, with a cavity Γ ε having the shape of a thin toroidal set, with a constant cross-section of diameter ε ≪ 1 . We construct the main terms of the asymptotic expansion of the eigenvalues in terms of real-analytic functions of the variable | ln ε | − 1 , and we prove that the relative asymptotic error is of much smaller order O ( ε | ln ε | ) as ε → 0 + . The asymptotic analysis involves eigenvalues and eigenfunctions of a certain integral operator on the smooth curve Γ , the axis of the cavity Γ ε .

Kolmogorov–Arnold–Moser theorem for nonlinear beam equations with almost-periodic forcing

In this study, we construct a Kolmogorov–Arnold–Moser theorem regarding the existence of almost-periodic solutions for some infinitely dimensional Hamiltonian systems with almost-periodic forcing. This theorem is applied to an almost-periodically forced nonlinear beam equation with periodic boundary conditions to obtain the almost-periodic solutionsutt+(−∂xx+μ)2u+ψ(ωt)f(u)=0,μ>0,t∈R,x∈R, where ψ(ωt) is real analytic and almost periodic on t and the nonlinearity f is a real-analytic function near u=0 with f(0)=f′(0)=0.

Finite $C^0$-determinacy of real analytic map germs with isolated instability

Let f : (Rn; 0) (Rp; 0) be a real analytic map germ with isolated instability. We prove that if n = 2 and p = 2; 3, then f is finitely C0-determined. This result can be seen as a weaker real counterpart of Mather-Gaffney finite determinacy criterion.

Generating series of all modular graph forms from iterated Eisenstein integrals

We study generating series of torus integrals that contain all so-called modular graph forms relevant for massless one-loop closed-string amplitudes. By analysing the differential equation of the generating series we construct a solution for their low-energy expansion to all orders in the inverse string tension α′. Our solution is expressed through initial data involving multiple zeta values and certain real-analytic functions of the modular parameter of the torus. These functions are built from real and imaginary parts of holomorphic iterated Eisenstein integrals and should be closely related to Brown’s recent construction of real-analytic modular forms. We study the properties of our real-analytic objects in detail and give explicit examples to a fixed order in the α′-expansion. In particular, our solution allows for a counting of linearly independent modular graph forms at a given weight, confirming previous partial results and giving predictions for higher, hitherto unexplored weights. It also sheds new light on the topic of uniform transcendentality of the α′-expansion.

Identities for Correlation Functions in Classical Statistical Mechanics and the Problem of Crystal States

Let $z$ be the activity of point particles described by classical equilibrium statistical mechanics in ${\bf R}^\nu$. The correlation functions $\rho^z(x_1,\dots,x_k)$ denote the probability densities of finding $k$ particles at $x_1,\dots,x_k$. Letting $\phi^z(x_1,\dots,x_k)$ be the cluster functions corresponding to the $\rho^z(x_1,\dots,x_k)/z^k$ we prove identities of the type $$ \phi^{z_0+z'}(x_1,\dots,x_k) $$ $$ =\sum_{n=0}^\infty{z'^n\over n!}\int dx_{k+1}\dots\int dx_{k+n}\,\phi^{z_0}(x_1,\dots,x_{k+n}) $$ It is then non-rigorously argued that, assuming a suitable cluster property (decay of correlations) for a crystal state, the pressure and the translation invariant correlation functions \- $\rho^z(x_1,\dots,x_k)$ are real analytic functions of $z$.

The Betti map associated to a section of an abelian scheme

Given a point $$\xi $$ on a complex abelian variety A, its abelian logarithm can be expressed as a linear combination of the periods of A with real coefficients, the Betti coordinates of $$\xi $$ . When $$(A, \xi )$$ varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often $$\xi $$ takes a torsion value (for instance, Manin’s theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when $$\xi $$ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira–Spencer map associated to $$(A, \xi )$$ (assuming A without fixed part, and $${\mathbb {Z}}\xi $$ Zariski-dense in A), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension $$\le 3$$ , and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if $$A\rightarrow S$$ is a principally polarized abelian scheme of relative dimension g which has no non-trivial endomorphism (on any finite covering), and if the image of S in the moduli space $${{\mathcal {A}}}_g$$ has dimension at least g, then the Betti map of any non-torsion section $$\xi $$ is generically a submersion, so that $$\,\xi ^{-1}A_{\mathrm{tors}}$$ is dense in $$S({\mathbb {C}})$$ .

The Zero Set of a Real Analytic Function

A brief proof of the statement that the zero-set of a nontrivial real-analytic function in $d$-dimensional space has zero measure is provided.

Domains with a continuous exhaustion in weakly complete surfaces

In previous works, Tomassini and the authors studied and classified complex surfaces admitting a real-analytic plurisubharmonic exhaustion function; let X be such a surface and $$D\subseteq X$$ a domain admitting a continuous plurisubharmonic exhaustion function: what can be said about the geometry of D? If the exhaustion of D is assumed to be smooth, the second author already answered this question; however, the continuous case is more difficult and requires different methods. In the present paper, we address such question by studying the local maximum sets contained in D and their interplay with the complex geometric structure of X; we conclude that, if D is not a modification of a Stein space, then it shares the same geometric features of X.

Double points in families of map germs from ℝ2 to ℝ3

We show that a 1-parameter family of real analytic map germs [Formula: see text] with isolated instability is topologically trivial if it is excellent and the family of double point curves [Formula: see text] in [Formula: see text] is topologically trivial. In particular, we deduce that [Formula: see text] is topologically trivial when the Milnor number [Formula: see text] is constant.

One-Sided Invertibility of Toeplitz Operators on the Space of Real Analytic Functions on the Real Line

We show that a Toeplitz operator on the space of real analytic functions on the real line is left invertible if and only if it is an injective Fredholm operator, it is right invertible if and only if it is a surjective Fredholm operator. The characterizations are given in terms of the winding number of the symbol of the operator. Our results imply that the range of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite codimension. Similarly, the kernel of a Toeplitz operator (and also its adjoint) is complemented if and only if it is of finite dimension.

The Universal Lie $\infty$-Algebroid of a Singular Foliation

We consider singular foliations \mathcal{F} as locally finitely generated \mathscr{O} -submodules of \mathscr{O} -derivations closed under the Lie bracket, where \mathscr{O} is the ring of smooth, holomorphic, or real analytic functions on a correspondingly chosen manifold. We first collect and/or prove several results about the existence of resolutions of such an \mathcal{F} in terms of sections of vector bundles. For example, these exist always on a compact smooth manifold M if \mathcal{F} admits real analytic generators. We show that every complex of vector bundles (E_\bullet,\mathrm{d}) over M providing a resolution of a given singular foliation \mathcal{F} in the above sense admits the definition of brackets on its sections such that it extends these data into a Lie \infty -algebroid. This Lie \infty -algebroid, including the chosen underlying resolution, is unique up to homotopy and, moreover, every other Lie \infty -algebroid inducing the given \mathcal{F} or any of its sub-foliations factors through it in an up-to-homotopy unique manner. We therefore call it the universal Lie \infty -algebroid of \mathcal{F} . It encodes several aspects of the geometry of the leaves of \mathcal{F} . In particular, it permits us to recover the holonomy groupoid of Androulidakis and Skandalis. Moreover, each leaf carries an isotropy Lie \infty -algebra structure that is unique up to isomorphism. It extends a minimal isotropy Lie algebra, that can be associated to each leaf, by higher brackets, which give rise to additional invariants of the foliation. As a byproduct, we construct an example of a foliation \mathcal{F} generated by r vector fields for which we show by these techniques that, even locally, it cannot result from a Lie algebroid of the minimal rank r .

Semi-Fredholm Toeplitz operators on the space of real analytic functions

Semi-Fredholm Toeplitz operators on the space of real analytic functions

Error-functions in double-sided Taylor's approximations

In this paper we introduce the error-functions for one-sided and double-sided Taylor's approximations of real analytic functions. We illustrate the application of error-functions in the process of generalization of one trigonometric inequality.

Interpolation Problems for Functions with Zero Integrals over Balls of Fixed Radius

Let $${{V}_{r}}({{\mathbb{R}}^{n}})$$, n ≥ 2, be the set of functions $$f \in {{L}_{{{\text{loc}}}}}({{\mathbb{R}}^{n}})$$ with zero integrals over all balls in $${{\mathbb{R}}^{n}}$$ of radius r. Various interpolation problems for the class $${{V}_{r}}({{\mathbb{R}}^{n}})$$ are studied. In the case when the set of interpolation nodes is finite, the multiple interpolation problem is solved under general assumptions. For problems with an infinite set of nodes, sufficient solvability conditions are founded. Additionally, we construct a new example of a subset in $${{\mathbb{R}}^{n}}$$ for which some nontrivial real analytic function of the class $${{V}_{r}}({{\mathbb{R}}^{n}})$$ vanishes.

Systems with vanishing time-variant delay: Structural properties based on the Pantograph (Scale-Delay) equation.

This paper studies the scale-delay (SD) system as a paradigm for systems with vanishing time-variant delay. After a brief review of the scale-delay equation and its associated solution, we derive new identities for the zeros of the deformed exponential. A solution method for the higher order scale-delay equation is given in terms of these deformed exponentials. We then consider the inverse problem: “Given a function x, which differential equation does it solve?” but with a twist: We are interested in time-variant realizations of the solution, as described in a companion paper. This extends the well known solution for Bohl functions, where an LTI-ODE solves the problem. If the function belongs to a subclass of real-analytic functions with only single real zeros, a regular smooth second-order linear time-variant annihilator exists. Finally this is applied to obtain a second order time-variant approximation of the SD-equation.