Self-similar Energy Research Articles

Box dimension of harmonic functions on higher dimensional Sierpinski gasket and Sierpinski gasket with bilateral energy

This paper concentrates on the bounds of the box-counting dimension of the graph of various fractal functions on the (N-1) dimensional Sierpinski gasket for N≥3. The bounds of the box-counting dimension of the graph of harmonic functions and functions having finite energy associated with the standard self-similar energy are found. Besides that, the bounds for the graph of harmonic functions and functions having finite energy associated with the bilateral energy on the Sierpinski gasket are also examined.

A Sierpinski carpet like fractal without standard self-similar energy

We construct a Sierpinski carpet like fractal, on which a diffusion with sub-Gaussian heat kernel estimate does not exist, in contrast to previous researches on the existence of such diffusions, on the generalized Sierpinski carpets and recently introduced unconstrained Sierpinski carpets.

An asymptotic expansion method vs a self-similar solution for convective heat transfer in rotating cone-disk systems

This paper compares an asymptotic expansion method and a self-similar solution for modeling Couette flow and convective heat transfer in a conical gap at small conicity angles up to 4°. The cases of rotation of a cone with a stationary disk and rotation of a disk with a stationary cone are considered. The self-similar system of equations provides the best agreement with experiments compared to the asymptotic expansion method. In any case, both methods are applicable only to conicity taper angles up to 4°, while at large conicity angles, the calculation results become significantly inaccurate. Calculations also showed that, at small conicity angles, convective heat transfer can be modeled using the self-similar energy equation in the boundary-layer approximation without considering radial heat conduction. In this study, analytical solutions were also obtained for limiting cases of a stationary fluid in a gap at small conicity angles without and with allowance for radial heat conduction.

Concerning the effect of radial thermal conductivity in a self-similar solution for rotating cone-disk systems

PurposeThus, the purposes of this study are to study the limits of applicability of the self-similar solution to the problem of fluid flow, heat and mass transfer in conical gaps with small conicity angles, to substantiate the impossibility of using a self-similar formulation of the problem in the case of large conicity angles and to substantiate the absence of the need to take into account the radial thermal conductivity in the energy equation in its self-similar formulation for the conicity angles up to 4°.Design/methodology/approachIn the present work, an in-depth and extended analysis of the features of fluid flow and heat transfer in a conical gap at small angles of conicity up to 4° is performed. The Couette-type flow arising, in this case, was modeled using a self-similar formulation of the problem. A detailed analysis of fluid flow calculations using a self-similar system of equations showed that they provide the best agreement with experiments than other known approaches. It is confirmed that the self-similar system of flow and heat transfer equations is applicable only to small angles of conicity up to 4°, whereas, at large angles of conicity, this approach becomes unreasonable and leads to significantly inaccurate results. The heat transfer process in a conical gap with small angles of conicity can be modeled using the self-similar energy equation in the boundary layer approximation. It was shown that taking into account the radial thermal conductivity in the self-similar energy equation at small conicity angles up to 4° leads to maximum deviations of the Nusselt number up to 1.5% compared with the energy equation in the boundary layer approximation without taking into account the radial thermal conductivity.FindingsIt is confirmed that the self-similar system of fluid flow equations is applicable only for small conicity angles up to 4°. The inclusion of radial thermal conductivity in the model unnecessarily complicates the mathematical formulation of the problem and at small conicity angles up to 4° leads to insignificant deviations of the Nusselt number (maximum 1.5%). Heat transfer in a conical gap with small conicity angles up to 4° can be modeled using the self-similar energy equation in the boundary layer approximation.Originality/valueThis paper investigates the question of the validity of taking into account the radial heat conduction in the energy equation.

Band conductivity oscillations in a gate-tunable graphene superlattice

Electrons exposed to a two-dimensional (2D) periodic potential and a uniform, perpendicular magnetic field exhibit a fractal, self-similar energy spectrum known as the Hofstadter butterfly. Recently, related high-temperature quantum oscillations (Brown-Zak oscillations) were discovered in graphene moiré systems, whose origin lies in the repetitive occurrence of extended minibands/magnetic Bloch states at rational fractions of magnetic flux per unit cell giving rise to an increase in band conductivity. In this work, we report on the experimental observation of band conductivity oscillations in an electrostatically defined and gate-tunable graphene superlattice, which are governed both by the internal structure of the Hofstadter butterfly (Brown-Zak oscillations) and by a commensurability relation between the cyclotron radius of electrons and the superlattice period (Weiss oscillations). We obtain a complete, unified description of band conductivity oscillations in two-dimensional superlattices, yielding a detailed match between theory and experiment.

Gromov hyperbolic graphs arising from iterations

For a contractive iterated function system (IFS), it is known that there is a natural hyperbolic graph structure (augmented tree) on the symbolic space of the IFS that reflects the relationship among neighboring cells, and its hyperbolic boundary with the Gromov metric is Hölder equivalent to the attractor K[14,26,28]. This setup was taken up to study the probabilistic potential theory on K[21,20], and the bi-Lipschitz equivalence on K[29]. In this paper, we formulate a broad class of hyperbolic graphs, called expansive hyperbolic graphs, to capture the most essential properties from the augmented trees and the hyperbolic boundaries (e.g., the special geodesics, bounded degree property, metric doubling property, and Hölder equivalence). We also study a new setup of “weighted” IFS and investigate its connection with the self-similar energy form in the analysis of fractals.

Electronic properties and quasi-zero-energy states of graphene quantum dots

In this paper, research has been carried out on the electronic properties of nanostructured graphene. We focus our attention on trapped states of the proposed systems such as spherical and toroidal graphene quantum dots (GQDs). Using a continuum model, by solving the Dirac-Weyl equation, and applying periodic boundary conditions of two types, i.e., either with zigzag edges only or with both armchair and zigzag edges, we obtain analytical results for energy levels yielding self-similar energy bands located subsequently one after another on the energy scale. Only for the toroidal quantum dot (owing to the lack of curvature) the distribution of electron density is like Bohr atomic orbitals. However, although the quasi-zero-energy band exists for both spherical and toroidal quantum dots, no electron density is present on this band for the toroidal quantum dot. This causes the formation of a pseudogap between the hole and electron bands because of the absence of the electron density at the quantum dot center, like in the case of an ordinary atom. Conversely, the confinement of the charge-carrier density is observed for both geometries of GQDs.

The mean logarithm emerges with self-similar energy balance

Abstract

A p.c.f. self-similar set with no self-similar energy

An example of a p.c.f. ( post-critically finite ) self-similar set without eigenform for any set of weights, is provided. The existence of an eigenform on such sets was an important, long-standing open problem in analysis on fractals. This problem is related to that of existence of self-similar energies on fractals, because self-similar energies on p.c.f. self-similar sets are obtained by using eigenforms (and only in this way). A general existence result of self-similar energies was previously established by the present author with respect to a weaker version of self-similarity.

Spectral triples for nested fractals

It is shown that, for nested fractals [31], the main structural data, such as the Hausdorff dimension and measure, the geodesic distance (when it exists) induced by the immersion in \mathbb R^n , and the self-similar energy can all be recovered by the description of the fractals in terms of the spectral triples considered in [18].

Hofstadter butterfly evolution in the space of two-dimensional Bravais lattices

The self-similar energy spectrum of a particle in a periodic potential under a magnetic field, known as the Hofstadter butterfly, is determined by the lattice geometry as well as the external field. Recent realizations of artificial gauge fields and adjustable optical lattices in cold atom experiments necessitate the consideration of these self-similar spectra for the most general two-dimensional lattice. In a previous work, we investigated the evolution of the spectrum for an experimentally realized lattice which was tuned by changing the unit cell structure but keeping the square Bravais lattice fixed. We now consider all possible Bravais lattices in two dimensions and investigate the structure of the Hofstadter butterfly as the lattice is deformed between lattices with different point symmetry groups. We model the optical lattice by a sinusoidal real space potential and obtain the tight binding model for any lattice geometry by calculating the Wannier functions. We introduce the magnetic field via Peierls substitution and numerically calculate the energy spectrum. The transition between the two most symmetric lattices, i.e. the triangular and the square lattice displays the importance of bipartite symmetry featuring deformation as well as closing of some of the major energy gaps. The transition from the square to rectangular and from the triangular to centered rectangular lattices are analyzed in terms of coupling of one-dimensional chains. We calculate the Chern numbers of the major gaps and Chern number transfer between bands during the transitions. We use gap Chern numbers to identify distinct topological regions in the space of Bravais lattices.

Impact-induced damage accumulation at micro- and nanostructural scale levels in sintered powders SiO2, SiC, and Al2O3 and in their single crystal counterparts

The acoustic emission (AE) and fractoluminescence (FL) techniques were applied to study the impact damage in sintered powders of SiO2, SiC and Al2O3 and their single crystal counterparts. The measured AE intensities and FL amplitudes are proportional to the energy release in events of microcrack nucleation and chemical bond breakage, respectively. In crystals, the AE method showed the random energy release in crystals, and the scaling energy distributions in powders. The FL method evidenced the self-similar energy distributions in α-SiO2 and α-SiC due to nanostructural heterogeneity inherent even to homogeneous solids, while luminescence from α-Al2O3 was specific for the random process.

Turbulence Spectra for Boundary-Layer Winds in Tropical Cyclones: A Conceptual Framework and Field Measurements at Coastlines

A conceptual model is proposed for the characteristic sub-ranges in the velocity and temperature spectra in the boundary layer of tropical cyclones (hurricanes or typhoons). The model is based on observations and computation of radial and vertical profiles of the mean flow and turbulence, and on the interpretation of eddy mechanisms determined by shear (namely roll and streak structures near the surface), convection, rotation, blocking and sheltering effects at the ground/sea surface and in internal shear layers. The significant sub-ranges, as the frequency increases, are associated with larger energy containing eddies, shear and blocking, inertial transfer between large and small scales, and intense small-scale eddies generated near the surface caused by waves, coastal roughness change, and the buoyancy force associated with the evaporation of spray droplets. These sub-ranges vary with the locations at which the spectra are measured, i.e. the level \(z\) in relation to the height \(z_{max}\) of the peak mean velocity and the depth \(h\) of the boundary layer, and the radius \(r\) in relation to the eyewall radius \(R_{ew}\) and the outer-vortex radius \(R_{ov}\). For two tropical cyclones (Nuri and Hagupit), experimental data were analyzed. Spectra were measured where \(r\) is near to \(R_{ew}\) and \(R_{ov}\) using four 1-h long datasets at coastal towers, at 10- and 60-m heights for tropical cyclone Nuri, and at 60-m height for tropical cyclone Hagupit at the south China coast. The field measurements of spectra within the boundary layer show significant sub-ranges of self-similar energy spectra (lying between the length scale 1,000 m and the smallest scales less than 40 m) that are consistent with the above conceptual model of the surface layer. However, with very high wind speeds near the eyewall, the energy of the independently generated intense surface eddy motions, associated with surface waves and water droplets in the airflow, greatly exceeds the energies of the small scales in the inertial sub-range of the boundary layer, over scales less than about 3–40 m depending on the height \(z\) and the radius \(r\). This rise in the small-scale frequency weighted spectra (\(nS_u (n)\), where \(n\) is natural frequency, and \(S_u (n)\) is the energy spectrum of the longitudinal wind component) is consistent with the hypothesis that these processes are only weakly correlated with the main boundary-layer turbulence.

Existence of self-similar energies on finitely ramified fractals

A self-similar energy on finitely ramified fractals can be constructed starting from an eigenform, i.e., an eigenvector of a special operator defined on the fractal. In this paper, we prove two existence results for regular eigenforms that consequently are existence results for self-similar energies on finitely ramified fractals. The first result proves the existence of a regular eigenform for suitable weights on fractals, assuming only that the boundary cells are separated and the union of the interior cells is connected. This result improves previous results and applies to many finitely ramified fractals usually considered. The second result proves the existence of a regular eigenform in the general case of finitely ramified fractals in a setting similar to that of P.C.F. self-similar sets considered, for example, by R. Strichartz in [11]. In this general case, however, the eigenform is not necessarily on the given structure, but is rather on only a suitable power of it. Nevertheless, as the fractal generated is the same as the original fractal, the result provides a regular self-similar energy on the given fractal.

Dirac and magnetic Schrödinger operators on fractals

In this paper we define (local) Dirac operators and magnetic Schrödinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with Röckner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet.

Energy efficiency of consecutive fragmentation processes

Motivated by a problem arising in the mining industry, we present a first study of the energy required to reduce a unit mass fragment by consecutively using several devices. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced in Bertoin and Martínez (2005), we compute the average energy required to attain a size η0 with this two-device procedure. We then asymptotically compare, as η0 goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that, for a certain range of parameters of the fragmentation processes and of their energy cost functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or vice versa.

Energy efficiency of consecutive fragmentation processes

Motivated by a problem arising in the mining industry, we present a first study of the energy required to reduce a unit mass fragment by consecutively using several devices. Two devices are considered, which we represent as different stochastic fragmentation processes. Following the self-similar energy model introduced in Bertoin and Martínez (2005), we compute the average energy required to attain a size η0 with this two-device procedure. We then asymptotically compare, as η0 goes to 0 or 1, its energy requirement with that of individual fragmentation processes. In particular, we show that, for a certain range of parameters of the fragmentation processes and of their energy cost functions, the consecutive use of two devices can be asymptotically more efficient than using each of them separately, or vice versa.

Self-Similar Analysis on Vortex Shedding Process

AbstractChaos theory has been seen as an efficient tool for studying the turbulent flow, the findings of attractor were also important and made in the study to investigate the wake flow behind the bluff body. Here, the fractal dimension value would then be found by Hurst analysis. According to the results found, the Hurst empirical formula derived by the self-similar laceration of vortex plane would be applied by self-similar property to decide the band of the frequency variations in the vortex shedding process. The three kinds of flow mode with their individual attractors and characteristics could be decomposed and shown as following: self-similar laceration, energy input and white noise band. Finally, the energy ratio for the three kinds of flow mode had been confirmed. Hence, these findings would be helpful to further study the wake flow in the vortex shedding process.

Surface plasmon–polaritons' contribution to the specific heat of quasiperiodic GaAs/SiO2 structures

It is quite interesting that the energy spectra of a variety of collective modes on quasiperiodic structures, exhibit a complex fractal profile due to the underlying long range order induced by the specific deterministic sequences used to build them up. In order to have a deep insight on the relevant thermodynamic implications of the above mentioned long range order, we have performed analytical and numerical calculations of the surface plasmon–polariton's contribution to the specific heat associated with successive hierarchical sequences of quasiperiodic Fibonacci structures. They can be traced back to the surface plasmon–polariton's self-similar energy spectra. We consider the building blocks, used to set up these structures, as the semiconductor GaAs and the insulator SiO2. At the interfaces there is a two-dimensional electron gas, modeled due to the presence of a surface density of current. We have also calculated the specific heat at the two temperature limit cases, namely when T goes to zero and infinity.

Spin wave specific heat in quasiperiodic Fibonacci structures

The energy spectra of a variety of collective modes on quasiperiodic structures exhibit a complex fractal profile. Among the modes that have attracted particular attention in this context are the spin wave spectra of quasiperiodic magnetic multilayers that obey a substitutional sequence of the Fibonacci type. They are described within the framework of the Heisenberg theory. In order to have a deep insight on the relevant thermodynamical implications of the above mentioned energy spectra's fractal profile, we have performed analytical and numerical calculations of the spin wave specific heat associated with successive hierarchical sequences of the Fibonacci quasiperiodic structures. The spectra show interesting oscillatory behavior in the low-temperature region, which can be traced back to the spin wave's self-similar energy spectrum.