Problem Of Small Denominators Research Articles

The snapshot problem for the wave equation

By definition, a wave is a C∞ solution u(x,t) of the wave equation on Rn, and a snapshot of the wave u at time t is the function ut on Rn given by ut(x)=u(x,t). We show that there are infinitely many waves with given C∞ snapshots f0 and f1 at times t=0 and t=1 respectively, but that all such waves have the same snapshots at integer times. We present a necessary condition for the uniqueness, and a compatibility condition for the existence, of a wave u to have three given snapshots at three different times, and we show how this compatibility condition leads to the problem of small denominators and Liouville numbers. We extend our results to wave equations on noncompact symmetric spaces. Finally, we consider the two-snapshot problem and corresponding small denominator results for the wave equation on the n-sphere.

Travelling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media

Travelling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrödinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearised equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localised on a finite spatial domain with small tails in far fields.

Inverse Problems for the Equation of Vibrations of a Canister Beam to Find the Source

For the beam vibration equation, inverse problems are studied to find the right side, i.e. vibration source. Solutions of the problems by methods of spectral analysis and Volterra integral equations are constructed explicitly as sums of series, and the corresponding uniqueness and existence theorems are proved. When substantiating the existence of a solution to the inverse problem by determining the factor of the right-hand side, which depends on the spatial coordinate, the problem of small denominators arises. In this regard, estimates of the denominators are established that guarantee their separation from zero, with an indication of the corresponding asymptotics. On the basis of these estimates, the convergence of the series in the class of regular solutions of the beam oscillation equation is substantiated.

Forward and Inverse Source Reconstruction Problems for the Equations of Vibrations of a Rectangular Plate

For the equation of vibrations of a rectangular plate, the initial-boundary value and inverse problems of finding the right-hand side (the source of vibrations) are studied. Solutions of the problems are constructed explicitly as sums of series, and the corresponding uniqueness and existence theorems are proved. When substantiating the existence of a solution to the inverse problem of determining the factor on the right-hand side, which depends on spatial coordinates, the problem of small denominators of two natural variables arises, for which estimates of the separation from zero with the corresponding asymptotics are established. These estimates made it possible to prove the existence theorem for this problem in the class of regular solutions by imposing certain smoothness conditions on the given boundary functions.

Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

The so-called ``small denominator problem'' was a fundamental problem of dynamics, as pointed out by Poincar\'{e}. Small denominators appear most commonly in perturbative theory. The Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small denominators. In this paper, using the forced Duffing equation as an example, we illustrate that the famous ``small denominator problems'' never appear if a non-perturbative approach based on the homotopy analysis method (HAM), namely ``the method of directly defining inverse mapping'' (MDDiM), is used. The HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully obtained. So, from the viewpoint of the HAM, the famous ``small denominator problems'' are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods but using the HAM-based MDDiM, one would be never troubled by ``small denominators''. The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called ``small denominators''.

NONHOMOGENEOUS BOUNDARY VALUE PROBLEM WITH NONLOCAL CONDITIONS FOR A PARTIAL DIFFERENTIAL EQUATION WITH THE OPERATOR OF THE GENERALIZED DIFFERENTIATION

The article is devoted to investigation of nonlocal boundary value problem for nonhomogeneous partial differential equation with the operator of the generalized differentiation $B=z\frac{\partial}{\partial z}$, which operate on function of scalar complex variable $z$. Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Particularly, this is due with the fact that these problems are models of the propagation of heat, process of moisture transfer in capillary-porous environments, diffusion of particles in the plasma, inverse problems, and also problems of mathematical biology. One of the most important question of the general theory of partial differential equations is the establishment of conditions for the correctness of boundary value problems. However, the investigation of problems with nonlocal conditions for partial differential equations in bounded domains connected with the problem of small denominators. This problem connected with the fact, that the denominators of coefficients of the series, which represented the solutions of nonlocal problems may be arbitrary small. Specific of the present work is the investigation of a nonlocal boundary-value problem for nonhomogeneous partial differential equation with the operator of the generalized differentiation $B=z\frac{\partial}{\partial z}$, which operate on functions of one scalar complex variable $z$. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the case of one scalar complex variable we showed, that the problem is Hadamard correct. The conditions of correct solvability of the nonlocal boundary value problem in Sobolev spaces are established. The uniqueness theorem and existence theorem of the solution of problem in these spaces are proved.

Overlap weight and propensity score residual for heterogeneous effects: A review with extensions

Individual responses to a treatment D=0,1 differ, depending on covariates X. Averaging such a heterogeneous effect is usually done with the density of X, but averaging with ‘overlap weight (OW)’ is also often done, where OW is the normalized version of PS×(1-PS) with PS denoting the propensity score. OW attains its maximum at PS=0.5, i.e., when subjects in one group have the best overlap with the other group, and OW accords several advantages to treatment effect estimators as reviewed in this paper. First, matching with OW addresses the non-overlapping support problem in a built-in way, without an arbitrary user intervention. Second, inverse probability weighting with OW overcomes the “too small denominator problem”, and can be efficient as well. Third, regression adjustment with OW is robust to misspecified outcome regression models. Fourth, covariate balance holds exactly, if OW is estimated by the generalized method of moment. In these advantages, the PS residual ‘D−PS’ plays a central role. We also discuss some shortcomings of OW, and show how seemingly unrelated estimators are in fact closely related through OW. Finally, we provide an empirical illustration.

Characterization of Low-Energy Quasiperiodic Orbits in the Elliptic Restricted 4-Body Problem with Orbital Resonance

In this work, we investigate the behavior of low-energy trajectories in the dynamical framework of the spatial elliptic restricted 4-body problem, developed using the Hamiltonian formalism. Introducing canonical transformations, the Hamiltonian function in the neighborhood of the collinear libration point L1 (or L2), can be expressed as a sum of three second order local integrals of motion, which provide a compact topological description of low-energy transits, captures and quasiperiodic libration point orbits, plus higher order terms that represent perturbations. The problem of small denominators is then applied to the order three of the transformed Hamiltonian function, to identify the effects of orbital resonance of the primaries onto quasiperiodic orbits. Stationary solutions for these resonant terms are determined, corresponding to quasiperiodic orbits existing in the presence of orbital resonance. The proposed model is applied to the Jupiter-Europa-Io system, determining quasiperiodic orbits in the surrounding of Jupiter-Europa L1 considering the 2:1 orbital resonance between Europa and Io.

Аналог задачи Дезина для уравнения параболо-гиперболического типа с условиями периодичности

В прямоугольной области рассматривается неоднородное уравнение смешанного параболо-гиперболического типа второго порядка. Для данного уравнения исследуется аналог задачи А. А. Дезина, который заключается в отыскании решения уравнения, удовлетворяющего внутренне-краевому условию, связывающему значение искомой функции на линии изменения типа уравнения со значением нормальной производной на границе в области гиперболичности, и неоднородным нелокальным краевым условиям периодичности. Приводится подстановка, позволяющая свести задачу к эквивалентной и, не теряя общности, ограничиться исследованием задачи с однородными условиями для неоднородного уравнения. Доказаны теоремы единственности и существования решения задачи, решение выписано в явном виде. Решение поставленной задачи ищется в виде суммы ряда Фурье по ортонормированной системе собственных функций соответствующей одномерной спектральной задачи. Установлен критерий единственности решения задачи. Для случая, когда нарушен критерий единственности, приведен пример нетривиального решения однородной задачи и получено необходимое и достаточное условие существования решения неоднородной задачи. При обосновании существования решения возникает проблема малых знаменателей в сумме ряда относительно соотношения сторон прямоугольника в гиперболической части области. Получена оценка отделенности знаменателя от нуля при некоторых условиях относительно параметров задачи, которая при определенных условиях на заданные функции позволяет доказать абсолютную и равномерную сходимость как формально построенного решения, так и соответствующих производных, входящих в уравнение.

NONLOCAL BOUNDARY VALUE PROBLEM IN SPACES OF EXPONENTIAL TYPE OF DIRICHLET-TAYLOR SERIES FOR THE EQUATION WITH COMPLEX DIFFERENTIATION OPERATOR

Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Such problems are mainly ill possed in the Hadamard sence, and their solvability is connected with the problem of small denominators. A specific feature of the present work is the study of a nonlocal boundary-value problem for partial differential equations with the operator of the generalized differentiation $B=zd/dz$, which operate on functions of scalar complex variable $z$. A criterion for the unique solvability of these problems and a sufficient conditions for the existence of its solutions are established in the spaces of functions, which are Dirichlet-Taylor series. The unity theorem and existence theorems of the solution of problem in these spaces are proved. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the article shown that in the case of one variable the corresponding denominators are not small and are estimated from below by some constants. Correctness after Hadamard of the problem is shown. It distinguishes it from an illconditioned after Hadamard problem with many spatial variables.

On the solvability of a Dirichlet‐type problem for one equation with variable coefficients

The paper investigates a boundary value problem in a rectangular domain for a high even order equation with discontinuous coefficients. A criterion for the uniqueness of a solution is obtained using the spectral method. The solution is built in the form of a Fourier series. When justifying the convergence of the series, the problem of small denominators arises. We obtain sufficient conditions for the convergence of the series.

Conditions of solvability of the nonlocal boundary-value problem for a differential-operator equation with weak nonlinearity

We study the solvability of a nonlocal boundary-value problem for a differential equation with nonlinearity. The linear part of the equation has complex coefficients which together with the coefficient in nonlocal conditions are considered to be the parameters of the problem. The area of change of each parameter is limited by a complex circle with its center at the origin. The nonlinear part of the equation is given by a smooth function that satisfies, together with its derivatives, some conditions of growth in the Dirichlet--Fourier space scale. The proofs are based on the differentiable Nash--Moser iteration scheme, where the main difficulty is to get estimates of the interpolation type for the inverse linearized operators obtained at each step of the iteration. The estimation is connected with the problem of small denominators which is solved, by using a metric approach on the set of parameters of the problem.

Ubiquitous Overlap Weight and Propensity Score Residual for Heterogeneous Treatment Effect and Its Estimation

Individual responses to a treatment D=0,1 differ, depending on covariates X. Averaging such a heterogeneous effect is usually done with the density of X, but the average with `overlap weight (OW)' is done surreptitiously with X controlled, where OW is the normalized version of PS×(1-PS) with PS denoting the propensity score. This paper brings OW out of shadow to advocate more use of OW. OW attains its maximum at PS=0.5, i.e., when subjects in one group have the best overlap with the other group, and OW accords several advantages to treatment effect estimators. First, matching with OW addresses the well-known non-overlapping support problem in a built-in way, without an arbitrary user intervention. Second, inverse probability weighting with OW overcomes the too small denominator problem, and can become efficient as well. Third, regression adjustment with OW is robust to misspecified outcome regression models. Fourth, covariate balance holds exactly if OW estimated by the method of moment is used. In these advantages, PS residual `D-PS' plays a central role. We also show how seemingly unrelated estimators are in fact closely related through OW, and an empirical analysis illustrates how similar the estimates can be.

The First Boundary Problem for a Mixed Type Equation with Characteristic Degeneracy with Discontinuous Frankl Condition

For an equation of mixed elliptic-hyperbolic type of the second kind, the boundary problem in a rectangular region is studied. The criterion of uniqueness is established. The solution is built as a sum row. When substantiating convergence, the problem of small denominators. In this connection, estimates of separation from zero small denominators with the corresponding asymptotics, which and allowed us to justify the existence of a solution in the class of regular solutions and solution stability from boundary data.

Initial-Boundary Problem for a Three-Dimensional Inhomogeneous Equation of Parabolic-Hyperbolic Type

For an inhomogeneous three-dimensional equation of mixed parabolic-hyperbolic type in a rectangular parallelepiped, the initial-boundary problem is studied. A criterion for the uniqueness of a solution is established. The solution is constructed as the sum of an orthogonal series. In substantiating the convergence of the series, the problem of small denominators of two natural arguments arose. Estimates are established for the separation from zero of the small denominators with the corresponding asymptotics. These estimates made it possible to justify the convergence of the constructed series in the class of regular solutions of this equation.

The First Boundary Problem with an Integral Condition for a Mixed-Type Equation with a Characteristic Degeneration

In this paper, we study the first boundary problem for a mixed-type elliptic-hyperbolic equation of the second kind in a rectangular domain. We construct the problem solution as the sum of a biorthogonal series and prove the uniqueness criterion for it. When proving the existence of the problem solution, we encounter the problem of small denominators. In this connection, we establish estimates for the separation of denominators from zero with the corresponding asymptotics; this allows us to justify the existence of a solution in the class of regular solutions.

The Dirichlet Problem for an Equation of Mixed Type with Two Internal Lines of Type Change

In a rectangular domain, for the equation of mixed elliptic-hyperbolic type with the Lavrent’ev–Bitsadze operator and two perpendicular lines of type change, we investigate the first boundary value problem. A criterion for the uniqueness of its solution is established. The solution to the problem is constructed as the sum of a series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. On the basis of the completeness of the biorthogonal system in the space of square-summable functions, we prove the uniqueness of a solution to the problem. When proving the existence of a solution, a problem of small denominators arises. Therefore, we obtain estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.

Two-point boundary value problem for a partial differential equation in spaces of periodic functions

We investigate the two-point in time boundary value problem for the partial differential equations of the second-order with one spatial variable and constant coefficients. The problem is considered in in the spaces of functions which Fourier coefficients are characterized by exponential behavior on the Cartesian product of the time interval and spatial domain $\mathbb{R}/2\pi\mathbb{Z}$. The correct solvability of the problem is established, the formulas for solutions are presented, the kernel is described and the smoothness of the solution is established in the spaces of functions that are periodic in one spatial variable. We have established the conditions which are close to the necessary conditions of solvability of the problem in scale of spaces of functions with exponentially increasing (or decreasing) Fourier coefficients.We also found the asymptotic estimates demonstrating the absence of the problem of small denominators, which arises of many spatial variables and makes the boundary value problem incorrect. We have established sufficient conditions of the finite-dimensionality of the kernel of the problem and found upper bounds for its dimension. The results are obtained under the condition of minimum smoothness on the right-hand sides of two-point conditions, which is close to the necessary condition.

NONLOCAL PROBLEM FOR A MIXED TYPE FOURTH-ORDER DIFFERENTIAL EQUATION WITH HILFER FRACTIONAL OPERATOR

In this paper, we consider a non-self-adjoint boundary value problem for a fourth-order differential equation of mixed type with Hilfer operator of fractional integro-differentiation in a positive rectangular domain and with spectral parameter in a negative rectangular domain. The mixed type differential equation under consideration is a fourth order differential equation with respect to the second variable. Regarding the first variable, this equation is a fractional differential equation in the positive part of the segment, and is a second-order differential equation with spectral parameter in the negative part of this segment. A rational method of solving a nonlocal problem with respect to the Hilfer operator is proposed. Using the spectral method of separation of variables, the solution of the problem is constructed in the form of Fourier series. Theorems on the existence and uniqueness of the problem are proved for regular values of the spectral parameter. For sufficiently large positive integers in unique determination of the integration constants in solving countable systems of differential equations, the problem of small denominators arises. Therefore, to justify the unique solvability of this problem, it is necessary to show the existence of values of the spectral parameter such that the quantity we need is separated from zero for sufficiently large \(n\). For irregular values of the spectral parameter, an infinite number of solutions in the form of Fourier series are constructed. Illustrative examples are provided.

Inverse Problems of Determining the Right-Hand Side and the Initial Conditions for the Beam Vibration Equation

For the beam vibration equation, we study the inverse problems of finding the right-hand side (the source of vibrations) and the initial conditions. The solutions of the problems are constructed in closed form as sums of series, and the corresponding existence and uniqueness theorems are proved. The problem of small denominators arises when justifying the convergence of the series. In this connection, we establish estimates for the denominators that guarantee their boundedness away from zero and indicate the corresponding asymptotics. Based on these estimates, we prove the convergence of the series in the class of regular solutions of the beam vibration equation.