Inhomogeneous Approximation Research Articles

A Note on Well Distributed Sequences

Abstract We prove an easy statement about inhomogeneous approximation for non-singular vectors in metric theory of Diophantine approximation.

Inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems

In this paper, we investigate inhomogeneous and simultaneous Diophantine approximation in beta dynamical systems. For β>1 let Tβ be the β-transformation on [0,1]. We determine the Lebesgue measure and Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβnx−f(x,y)|<φ(n) for infinitely many n∈N}, where f:[0,1]2→[0,1] is a Lipschitz function and φ is a positive function on N. Let β2≥β1>1, f1,f2:[0,1]→[0,1] be two Lipschitz functions, τ1,τ2 be two positive continuous functions on [0,1]. We also determine the Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβ1nx−f1(x)|<β1−nτ1(x)|Tβ2ny−f2(y)|<β2−nτ2(y) for infinitely many n∈N}. Under certain additional assumptions, the Hausdorff dimension of the set{(x,y)∈[0,1]2:|Tβ1nx−g1(x,y)|<β1−nτ1(x)|Tβ2ny−g2(x,y)|<β2−nτ2(y) for infinitely many n∈N} is also determined, where g1,g2:[0,1]2→[0,1] are two Lipschitz functions.

Splitting methods for solution decomposition in nonstationary problems

In approximating solutions of nonstationary problems, various approaches are used to compute the solution at a new time level from a number of simpler (sub-)problems. Among these approaches are splitting methods. Standard splitting schemes are based on one or another additive splitting of the operator into “simpler” operators that are more convenient/easier for the computer implementation and use inhomogeneous (explicitly-implicit) time approximations. In this paper, a new class of splitting schemes is proposed that is characterized by an additive representation of the solution instead of the operator corresponding to the problem (called problem operator). A specific feature of the proposed splitting is that the resulting coupled equations for individual solution components consist of the time derivatives of the solution components. The proposed approaches are motivated by various applications, including multiscale methods, domain decomposition, and so on, where spatially local problems are solved and used to compute the solution. Unconditionally stable splitting schemes are constructed for a first-order evolution equation, which is considered in a finite-dimensional Hilbert space. In our splitting algorithms, we consider the decomposition of both the main operator of the system and the operator at the time derivative. Our goal is to provide a general framework that combines temporal splitting algorithms and spatial decomposition and its analysis. Applications of the framework will be studied separately.

Mechanical Tuning of the Terahertz Photonic Bandgap of 3D-Printed One-Dimensional Photonic Crystals

Mechanical tuning of a 3D-printed, polymer-based one-dimensional photonic crystal was demonstrated in the terahertz spectral range. The investigated photonic crystal consists of 13 alternating compact and low-density layers and was fabricated through single-step stereolithography. While the compact layers are entirely polymethacrylate without any intentional internal structures, the low-density layers contain sub-wavelength sized slanted columnar inclusions to allow the mechanical compression in a direction normal to the layer interfaces of the photonic crystal. Terahertz transmission spectroscopy of the photonic crystal was performed in a spectral range from 83 to 124 GHz as a function of the compressive strain. The as-fabricated photonic crystal showed a distinct photonic bandgap centered at 109 GHz, which blue shifted under compressive stress. A maximum shift of 12 GHz in the bandgap center frequency was experimentally demonstrated. Stratified optical models incorporating simple homogeneous and inhomogeneous compression approximations were used to analyze the transmission data. A good agreement between the experimental and model-calculated transmission spectra was found.

Inhomogeneous functionals and approximations of invariant distributions of ergodic diffusions: Central limit theorem and moderate deviation asymptotics

The paper studies asymptotics of inhomogeneous integral functionals of an ergodic diffusion process under the effect of discretization. Convergence to the corresponding functionals of the invariant distribution is shown for suitably chosen discretization steps, and the fluctuations are analyzed through central limit theorem and moderate deviation principle. The results will be particularly useful for understanding accuracy of an Euler discretization based numerical scheme for approximating functionals of invariant distribution of an ergodic diffusion. This is an infinite-time horizon problem, and the accuracy of numerical schemes in this context are comparatively much less studied than the ones used for generating approximate trajectories of diffusions over finite time intervals. The potential applications of these results also extend to other areas including mathematical physics, parameter inference of ergodic diffusions and analysis of multiscale dynamical systems with averaging.

Inhomogeneous Diophantine approximation over fields of formal power series

We prove a sharp analogue of Minkowski's inhomogeneous approximation theorem over fields of power series $\mathbb {F}_q((T^{-1}))$. Furthermore, we study the approximation to a given point $\underline {y}$ in $\mathbb {F}_q((T^{-1}))^2$ by the $\mathrm {SL}_2(\mathbb {F}_q[T])$-orbit of a given point $\underline {x}$ in $\mathbb {F}_q((T^{-1}))^2$.

Explicit–Implicit Schemes for First-Order Evolution Equations

Computational practice widely employs inhomogeneous time approximations in which one part of the problem operator is taken from the lower time level and the other, from the upper one. We discuss questions of constructing explicit–implicit schemes for the approximate solution of the Cauchy problem for a first-order evolution equation based on additive two-component splitting of the main problem operator. Novel explicit–implicit schemes are proposed when splitting the operator multiplying the time derivative. The stability of the proposed explicit–implicit two- and three-level operator-difference schemes is investigated.

New technique for calculating the ionospheric phase advance and dual frequency mode of measuring ionospheric TEC

The earlier developed alternative technique for calculating the ionospheric contribution into the full phase of the signal on the transionospheric paths of propagation, based on the perturbation theory in spherical co-ordinates with the spherically symmetric inhomogeneous zero-order approximation for the ionosphere, is further extended in order to apply it to the two-frequency mode of operation of GNSS. When doing this, the method itself is additionally validated and verified in the numerical experiment, where the reconstruction of the slant TEC calculated by the alternative technique for the two-frequency mode of operation is compared to that directly generated by the NeQuick model of the ionosphere.

Discrete eigenmodes of filamentation instability in the presence of a q-nonextensive distribution.

Discrete eigenmodes of the filamentation instability in a weakly ionized current-driven plasma in the presence of a q-nonextensive electron velocity distribution is investigated. Considering the kinetic theory, Bhatnagar-Gross-Krook collision model, and Lorentz transformation relations, the generalized longitudinal and transverse dielectric permittivities are obtained. Taking into account the long-wavelength limit and diffusion frequency limit, the dispersion relations are obtained. Using the approximation of geometrical optics and linear inhomogeneity of the plasma, the real and imaginary parts of the frequency are discussed in these limits. It is shown that in the long-wavelength limit, when the normalized electron velocity is increased the growth rate of the instability increases. However, when the collision frequency is increased the growth rate of the filamentation instability decreases. In the diffusion frequency limit, results indicate that the effects of the electron velocity and q-nonextensive parameter on the growth rate of the instability are similar. Finally, it is found that when the collision frequency is increased the growth rate of the instability increases in the presence of a q-nonextensive distribution.

The Generalized Baker–Schmidt Problem on Hypersurfaces

Abstract The generalized Baker–Schmidt problem (1970) concerns the $f$-dimensional Hausdorff measure of the set of $\psi $-approximable points on a nondegenerate manifold. There are two variants of this problem concerning simultaneous and dual approximation. Beresnevich–Dickinson–Velani (in 2006, for the homogeneous setting) and Badziahin–Beresnevich–Velani (in 2013, for the inhomogeneous setting) proved the divergence part of this problem for dual approximation on arbitrary nondegenerate manifolds. The corresponding convergence counterpart represents a major challenging open question and the progress thus far has only been attained over planar curves. In this paper, we settle this problem for hypersurfaces in a more general setting, that is, for inhomogeneous approximations and with a non-monotonic multivariable approximating function.

On homogeneous and inhomogeneous Diophantine approximation over the fields of formal power series

We prove over fields of power series the analogues of several Diophantine approximation results obtained over the field of real numbers. In particular we establish the power series analogue of Kronecker's theorem for matrices, together with a quantitative form of it, which can also be seen as a transference inequality between uniform approximation and inhomogeneous approximation. Special attention is devoted to the one dimensional case. Namely, we give a necessary and sufficient condition on an irrational power series $\alpha$ which ensures that, for some positive $\eps$, the set $$ \liminf_{Q \in \mathbb{F}_q[z], \,\, \deg Q \to \infty} \| Q \| \cdot |\langle Q \alpha - \theta \rangle| \geq \eps $$ has full Hausdorff dimension.

Numerical Solution of Time-Dependent Problems with Different Time Scales

Problems for time-dependent equations in which processes are characterized by different time scales are studied. Parts of the equations describing fast and slow processes are distinguished. The basic features of such problems related to their approximation are taken into account using finer time grids for fast processes. The construction and analysis of inhomogeneous time approximations is based on the theory of additive operator–difference schemes (splitting schemes). To solve time-dependent problems with different time scales, componentwise splitting schemes and vector additive schemes are used. The capabilities of the proposed schemes are illustrated by numerical examples for the time-dependent convection–diffusion problem. If convection is dominant, the convective transfer is computed on a finer time grid.

Inhomogeneous random graphs, isolated vertices, and Poisson approximation

AbstractConsider a graph on randomly scattered points in an arbitrary space, with any two pointsx,yconnected with probability ϕ(x,y). Suppose the number of points is large but the mean number of isolated points isO(1). We give general criteria for the latter to be approximately Poisson distributed. More generally, we consider the number of vertices of fixed degree, the number of components of fixed order, and the number of edges. We use a general result on Poisson approximation by Stein's method for a set of points selected from a Poisson point process. This method also gives a good Poisson approximation for U-statistics of a Poisson process.

Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin–Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Szüsz’s inhomogeneous (1958) version of Khintchine’s Theorem (1924). For example, given any real sequence [Formula: see text], we build a divergent series of non-negative reals [Formula: see text] such that for any [Formula: see text], almost no real number is inhomogeneously [Formula: see text]-approximable with inhomogeneous parameter [Formula: see text]. Furthermore, given any second sequence [Formula: see text] not intersecting the rational span of [Formula: see text], and assuming a dynamical version of Erdős’ Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously [Formula: see text]-approximable with any inhomogeneous parameter [Formula: see text]. Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin–Schaeffer Conjecture and Duffin–Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher’s Zero-One Law (1961) for inhomogeneous approximation by reduced fractions.

Solution of Cassels’ Problem on a Diophantine Constant over Function Fields

This paper deals with the analogue of Inhomogeneous Diophantine Approximation in function fields. The inhomogeneous approximation constant of a Laurent series $\theta\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ with respect to $\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)$ is defined to be $c(\theta,\gamma)=\inf_{0\neq N\in\mathbb F_q\left[t\right]}|N|\cdot|\langle N\theta - \gamma \rangle|$. We show that for every $\theta$ there exists $\gamma$ such that $c(\theta,\gamma)\geq q^{-2}$, and find a sufficient condition on $\theta$ which forces $c(\theta,\gamma) \leq q^{-2}$ for every $\gamma$. Given $\theta$, we prove that the set $BA_{\theta}=\left\{\gamma\in\mathbb F_q\left(\left(\frac{1}{t}\right)\right)\;:\; c(\theta,\gamma)>0\right\}$ has full Hausdorff dimension. Our methods allow us to solve the case of vectors in $\mathbb F_q\left(\left(\frac{1}{t}\right)\right)^d$ as well. Our results offer a strengthening to analogues of results for real inhomogeneous approximation.

Describability via ubiquity and eutaxy in Diophantine approximation

We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of Levy processes.

DRBEM solution of mixed convection flow of nanofluids in enclosures with moving walls

This paper presents the results of a numerical study on unsteady mixed convection flow of nanofluids in lid-driven enclosures filled with aluminum oxide and copper–water based nanofluids. The governing equations are solved by the Dual Reciprocity Boundary Element Method (DRBEM), and the time derivatives are discretized using the implicit central difference scheme. All the convective terms and the vorticity boundary conditions are evaluated in terms of the DRBEM coordinate matrix. Linear boundary elements and quadratic radial basis functions are used for the discretization of the boundary and approximation of inhomogeneity, respectively. Solutions are obtained for several values of volume fraction (φ), the Richardson number (Ri), heat source length (B), and the Reynolds number (Re). It is disclosed that the average Nusselt number increases with the increase in volume fraction, and decreases with an increase in both the Richardson number and heat source length.

Grids with dense values

Given a continuous function from Euclidean space to the real line, we analyze (under some natural assumption on the function), the set of values it takes on translates of lattices. Our results are of the flavor: For almost any translate the set of values is dense in the set of possible values. The results are then applied to a variety of concrete examples obtaining new information in classical discussions in different areas in mathematics; in particular, Minkowski’s conjecture regarding products of inhomogeneous forms and inhomogeneous Diophantine approximations.

Field effect on surface states in a doped Mott-insulator thin film

Surface effects of a doped thin film made of a strongly correlated material are investigated both in the absence and presence of a perpendicular electric field. We use an inhomogeneous Gutzwiller approximation for a single band Hubbard model in order to describe correlation effects. For low doping, the bulk value of the quasiparticle weight is recovered exponentially deep into the slab, but with increasing doping, additional Friedel oscillations appear near the surface. We show that the inverse correlation length has a power-law dependence on the doping level. In the presence of an electrical field, considerable changes in the quasiparticle weight can be realized throughout the system. We observe a large difference (as large as five orders of magnitude) in the quasiparticle weight near the opposite sides of the slab. This effect can be significant in switching devices that use the surface states for transport.

Inhomogeneous diophantine approximations and distribution of fractional parts

A new estimate of the remainder term is obtained in the problem of the distribution of fractional parts of nα over an arbitrary interval.