Force distribution for double-walled carbon nanotubes and gigahertz oscillators

Advances in nanotechnology have led to the creation of many nano-scale devices and carbon nanotubes are representative materials to construct these devices. Double-walled carbon nanotubes with the inner tube oscillating can be used as gigahertz oscillators and form the basis of possible nano-electronic devices that might be instrumental in the micro-computer industry which are predominantly based on electron transport phenomena. There are many experiments and molecular dynamical simulations which show that a wave is generated on the outer cylinder as a result of the oscillation of the inner carbon nanotube and that the frequency of this wave is also in the gigahertz range. As a preliminary to analyze and model such devices, it is necessary to estimate accurately the resultant force distribution due to the inter-atomic interactions. Here we determine some new analytical expressions for the van der Waals force using the Lennard–Jones potential for general lengths of the inner and outer tubes. These expressions are utilized together with Newton’s second law to determine the motion of an oscillating inner tube, assuming that any frictional effects may be neglected. An idealized and much simplified representation of the Lennard–Jones force is used to determine a simple formula for the oscillation frequency resulting from an initial extrusion of the inner tube. This simple formula is entirely consistent with the existing known behavior of the frequency and predicts a maximum oscillation frequency occurring when the extrusion length is (L ₂ – L ₁)/2 where L ₁ and L ₂ are the respective half-lengths of the inner and outer tubes (L ₁ < L ₂).     

Suction energy and offset configuration for double-walled carbon nanotubes

Nanostructures such as carbon nanotubes and fullerenes offer the means to create new mechanical devices operating at the nanoscale. Such devices include oscillators constructed from an inner carbon nanotube sliding inside another carbon nanotube. The resultant oscillatory frequency is found to be in the gigahertz range and they have applications in the computing industry for signalling devices, such as an ultra-fast optical filter. While most research in the area is dominated by molecular dynamics simulations, our approach here is to use elementary mechanical principles and classical applied mathematical modelling techniques to formulate explicit analytical criteria and ideal model behaviour. In this paper, we first investigate the suction force experienced by a single-walled carbon nanotube located near an open end of a semi-infinite single-walled carbon nanotube, using the Lennard–Jones potential and the continuum approximation. Second the equilibrium position of an offset inner tube with reference to the cross-section of the outer tube is determined.     

Mechanics of spheroidal fullerenes and carbon nanotubes for drug and gene delivery

There is considerable interest in the mechanics of carbon nanostructures,such as carbon nanotubes and fullerenes, and the manner of theirinteractions at the intermolecular level. Medical applicationsinclude the use of carbon nanotubes for targeted drug and genedelivery, for which issues relating to the acceptance and containmentof drugs or genes are not properly understood. A spheroid isan ellipsoid with two equal axes and the general spheroidalshape includes a wide variety of possible molecular configurationssuch as spheres, capped cylindrical tubes and ellipsoids ofrevolution, and therefore the determination of the interactionforces for this general shape may have many applications. Phenomenasuch as the suction of fullerenes into carbon nanotubes dueto the van der Waals interatomic interactions and ultra-lowfriction of a molecule moving inside a carbon nanotube giverise to the possibility of constructing nanoscaled oscillatorswith frequencies in the gigahertz range. This paper models themechanics of such a system by employing a six-twelve Lennard–Jonespotential taken over two surfaces assumed to be composed ofmean distributions of atoms over the two idealized surfacesof an open-ended semi-infinite circular cylinder and a spheroid.Following the methodology of previous work with spherical surfaces,the acceptance energy and suction energy for spheroidal moleculesare given and the special case of spherical molecules is alsoreproduced to validate the method. The results for ellipticalmolecules are novel and cannot be validated experimentally atthis stage, but the results for the special case of sphericalmolecules are given and shown to be in good agreement with publishedmolecular dynamical simulations. Finally, a general numerical-analyticalprocedure is proposed to calculate the Lennard–Jones potentialfor any axially symmetric surface, and the prior results obtainedfor the spheroid are used to validate the procedure.     

A comprehensive study on the oscillation frequency of spherical fullerenes in carbon nanotubes under different system parameters

Amongst possible new nanomechanical devices created based on carbon nanostructures, high-frequency nanoscale oscillators, or the so-called gigahertz oscillators have attracted much attention. In this paper, the oscillatory behavior of spherical fullerenes inside carbon nanotubes is thoroughly investigated. To this end, the continuum approximation together with Lennard-Jones potential is used to evaluate the van der Waals potential energy and interaction force. The equation of motion is directly solved based on the actual force distribution between the two nanostructures, without any simplifying assumption. A semi-analytical expression is obtained for the oscillation frequency into which the effect of initial conditions is incorporated. Thereafter, this newly derived expression is utilized in order to present a comprehensive study on the effects of different system variables such as geometrical parameters and initial conditions on the oscillation frequency. Based upon these studies, some new features of such oscillations have been revealed.     

The effects of geometrical parameters on force distributions and mechanics of carbon nanotubes: A critical study

In this paper, using the continuum approximation together with Lennard–Jones potential, a new semi-analytical expression is given to evaluate the van der Waals interaction between two single-walled carbon nanotubes. Based on this expression, two new formulations are also proposed to model multi-walled carbon nanotubes. In the first one, the interactions between each pair of shells from the inner and outer tubes are summed up over all of the pairs, whereas in the second formulation, a set of correction factors are applied to convert the results of double-walled carbon nanotubes to the correlated multi-walled ones. With respect to the present formulations, extensive studies on the variations of force distributions are performed by varying nanotube geometries so that the important features of the geometrical parameters are explored. Moreover, an acceptance condition for a nanotube at rest which is to be sucked into a semi-infinite nanotube is obtained. The influence of different geometrical parameters on the acceptance condition and suction energy, two main characteristics of nanotube-based systems for applications such as drug delivery and so on, is fully demonstrated. Lastly, an interesting relation for the maximum value of suction energy in terms of geometrical parameters is also extracted in this study.     

The Farrell–Hsiang method revisited

We present a sufficient condition for groups to satisfy the Farrell–Jones Conjecture in algebraic K-theory and L-theory. The condition is formulated in terms of finite quotients of the group in question and is motivated by work of Farrell–Hsiang.     

Homogenization with corrector for a periodic elliptic operator near an edge of inner gap

For an ellipticoperator with rapidly oscillating coefficients we consider a homogenization procedure near the edge of an interior gap in the spectrum of this operator. At a point close to the edge, we obtain an approximation of the resolvent in the operator L ₂(ℝ)-norm. The first order corrector is taken into account in the approximation. Bibliography: 11 titles. Dedicated to dear Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 127–142.     

Lp-Lq DECAY ESTIMATES FOR HYPERBOLIC EQUATIONS WITH OSCILLATIONS IN COEFFICIENTS

This work is concerned with the proof of Lp -Lq decay estimates for solutions of the Cauchy problem for utt -λ2(t)b2(t)/Δu =0. The coefficient consists of an increasing smooth function λ and an oscillating smooth and bounded function b which are uniformly separated from zero. The authors‘ main interest is devoted to the critical case where one has an interesting interplay between the growing and the oscillating part.     

Hilbert Transforms on the Sphere with the Clifford Algebra Setting

Through a double-layer potential argument the inner and outer Poisson kernels, the Cauchy-type conjugate inner and outer Poisson kernels, and the kernels of the Cauchy-type inner and outer Hilbert transformations on the sphere are deduced. We also obtain Abel sum expansions of the kernels and prove the L ^p -boundedness of the inner and outer Hilbert transformations for 1<p<∞.     

The Frattini Subalgebra of Restricted Lie Superalgebras

In the present paper, we study the Frattini subalgebra of a restricted Lie superalgebra （L, [p]）. We show first that if L = A1 ＋ A2 ＋… ＋An, then Фp（L） = Фp（A1） ＋Фp（A2） ＋…＋Фp（An）, where each Ai is a p-ideal of L. We then obtain two results： F（L） = Ф（L） = J（L） = L if and only if L is nilpotent; Fp（L） and F（L） are nilpotent ideals of L if L is solvable. In addition, necessary and sufficient conditions are found for Фp-free restricted Lie superalgebras. Finally, we discuss the relationships of E-p-restricted Lie superalgebras and E-restricted Lie superalgebras.     

Gigahertz nanomechanical oscillators based on ions inside cyclic peptide nanotubes: a continuum study

The present work aims to investigate the mechanical oscillatory behavior of ions, and in particular \({{\rm Li}^{+}, {\rm Na}^{+}, {\rm Rb}^{+}}\) and \({{\rm Cl}^{-}}\) ions, inside a cyclo[(–d-Ala–l-Ala)₄–] peptide nanotube using the continuum approximation along with the 6–12 Lennard–Jones (LJ) potential function. Assuming that each peptide unit is comprised of an inner and an outer tube, the van der Waals (vdW) potential energy and interaction force between an ion and a cyclic peptide nanotube (CPN) are determined analytically. With respect to the present formulations, a detailed parametric study is conducted on the vdW potential energy and interaction force distributions by varying the number of peptide units. Employing the conservation of mechanical energy principle, a novel expression for precise evaluation of oscillation frequency is introduced. To verify the accuracy of the proposed frequency expression, the results obtained from energy equation are compared with the ones predicted through solving the equation of motion numerically. The effects of number of peptide units and initial conditions including initial separation distance and velocity on the oscillatory behavior of various ions inside CPNs are explored. Among the considered ions, \({{\rm Cl}^{-}}\) ion is found to generate the highest frequency. According to the potential energy profile, one oscillatory zone for one peptide unit and different oscillatory zones for more than one peptide unit are observed. Numerical results indicate that optimal frequency decreases with increasing the number of peptide units and almost remains unchanged when the number of peptide units exceeds four.     

Contribution to the bandwidth choice for kernel density estimates

In the present paper we focus on the problem of the bandwidth choice for the kernel density estimates. The problem of finding the optimal bandwidth belongs to the crucial problems of the kernel estimates. As a criterion of quality of the estimates the L ₂ type measure is used. A special iterative method based on a relevant estimation of mean integrated square error given in papers Müller and Wang (Prob Theor Relat Fields 85:523–538, 1990), Jones et al. (Ann Stat 19:1919–1932, 1991) is suggested. Moreover the idea of maximal smoothing principle (Terrell in J Am Stat Assoc 85:470–477, 1990) is extended to the higher order kernels. A simulation study brings a comparison of the proposed method and the cross-validation method. Research supported by the GACR:402/04/1308.     

On the boundedness of some potential‐type operators with oscillating kernels

We consider a class of multidimensional potential‐type operators with kernels that have singularities at the origin and on the unit sphere and that are oscillating at infinity. We describe some convex sets in the (1/p, 1/q)‐plane for which these operators are bounded from L_p into L_q and indicate domains where they are not bounded. We also reveal some effects which show that oscillation and singularities of the kernels may strongly influence on the picture of boundedness of the operators under consideration. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Pseudodifferential Operators Approach to Singular Integral Operators in Weighted Variable Exponent Lebesgue Spaces on Carleson Curves

The main results of the paper are: (1) The boundedness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) on a class of composed Carleson curves Γ where the weights w have a finite set of oscillating singularities. The proof of this result is based on the boundedness of Mellin pseudodifferential operators on the spaces L^p(·)(\mathbbR ₊,dm){L^{p(\cdot )}(\mathbb{R} _{+},d\mu)} where dμ is an invariant measure on multiplicative group ${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}${\mathbb{R}_{+}=\left\{r\in \mathbb{R}:r >0 \right\}}. (2) Criterion of local invertibility of singular integral operators with piecewise slowly oscillating coefficients acting on L ^p(·)(Γ, w) spaces. We obtain this criterion from the corresponding criteria of local invertibility at the point 0 of Mellin pseudodifferential operators on \mathbbR₊{\mathbb{R}_{+}} and local invertibility of singular integral operators on \mathbbR{\mathbb{R}}. (3) Criterion of Fredholmness of singular integral operators in the variable exponent Lebesgue spaces L ^p(·)(Γ, w) where Γ belongs to a class of composed Carleson curves slowly oscillating at the nodes, and the weight w has a finite set of slowly oscillating singularities.     

A Generalization of the Mean Size Formula of Wavelet Packets in Lp

The purpose of this paper is to investigate the mean size formula of wavelet packets in Lp for 0 〈 p ≤ ∞. We generalize a mean size formula of wavelet packets given in terms of the p-norm joint spectral radius and we also give some asymptotic formulas for the Lp-norm or quasi-norm on the subdivision trees. All results will be given in the general setting,     

Linear stability of radially-heated circular Couette flow with simulated radial gravity

The stability of circular Couette flow between vertical concentric cylinders in the presence of a radial temperature gradient is considered with an effective “radial gravity.” In addition to terrestrial buoyancy − ρg e _z we include the term − ρg _m f(r)e _r where g _m f(r) is the effective gravitational acceleration directed radially inward across the gap. Physically, this body force arises in experiments using ferrofluid in the annular gap of a Taylor–Couette cell whose inner cylinder surrounds a vertical stack of equally spaced disk magnets. The radial dependence f(r) of this force is proportional to the modified Bessel function K ₁(κr), where 2π/κ is the spatial period of the magnetic stack and r is the radial coordinate. Linear stability calculations made to compare with conditions reported by Ali and Weidman (J. Fluid Mech., 220, 1990) show strong destabilization effects, measured by the onset Rayleigh number R, when the inner wall is warmer, and strong stabilization effects when the outer wall is warmer, with increasing values of the dimensionless radial gravity γ = g _m /g. Further calculations presented for the geometry and fluid properties of a terrestrial laboratory experiment reveal a hitherto unappreciated structure of the stability problem for differentially-heated cylinders: multiple wavenumber minima exist in the marginal stability curves. Transitions in global minima among these curves give rise to a competition between differing instabilities of the same spiral mode number, but widely separated axial wavenumbers.     

Sublattices of a lattice of clones of functions on a 3-element set. I

We deal with a latticeL ₃ of clones on a 3-element set. The lattice of suborders of the chain of natural numbers is proved to be dually isomorphic to some sublattice inL ₃. Translated fromAlgebra i Logika, Vol. 38, No. 1, pp. 3–23, January–February, 1999.     

An example concerning the nonlinear pointwise ergodic theorem

It is shown that the pointwise ergodic theorem for Markov operators inL ₁, having a finite invariant measure, fails to extend to the case of nonlinear operators. Recall thatT is called nonexpansive inL _p if ‖Tf – Tg ‖ _p ≦‖f – g‖ _p holds for allf andg.     

Linear stability of radially-heated circular Couette flow with simulated radial gravity

The stability of circular Couette flow between vertical concentric cylinders in the presence of a radial temperature gradient is considered with an effective “radial gravity.” In addition to terrestrial buoyancy − ρg e _z we include the term − ρg _m f(r)e _r where g _m f(r) is the effective gravitational acceleration directed radially inward across the gap. Physically, this body force arises in experiments using ferrofluid in the annular gap of a Taylor–Couette cell whose inner cylinder surrounds a vertical stack of equally spaced disk magnets. The radial dependence f(r) of this force is proportional to the modified Bessel function K ₁(κr), where 2π/κ is the spatial period of the magnetic stack and r is the radial coordinate. Linear stability calculations made to compare with conditions reported by Ali and Weidman (J. Fluid Mech., 220, 1990) show strong destabilization effects, measured by the onset Rayleigh number R, when the inner wall is warmer, and strong stabilization effects when the outer wall is warmer, with increasing values of the dimensionless radial gravity γ = g _m /g. Further calculations presented for the geometry and fluid properties of a terrestrial laboratory experiment reveal a hitherto unappreciated structure of the stability problem for differentially-heated cylinders: multiple wavenumber minima exist in the marginal stability curves. Transitions in global minima among these curves give rise to a competition between differing instabilities of the same spiral mode number, but widely separated axial wavenumbers.     

Quantifying curvelike structures of measures by using L2 Jones quantities

We study the curvelike structure of special measures on ?ⁿ in a multiscale fashion. More precisely, we consider the existence and construction of a sufficiently short curve with a sufficiently large measure. Our main tool is an L₂ variant of Jones' β numbers, which measure the scaled deviations of the given measure from a best approximating line at different scales and locations. The Jones function is formed by adding the squares of the L₂ Jones numbers at different scales and the same location. Using a special L₂ Jones function, we construct a sufficiently short curve with a sufficiently large measure. The length and measure estimates of the underlying curve are expressed in terms of the size of this Jones function. © 2003 Wiley Periodicals, Inc.