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The mature and scale utilization of renewable energy is restricted by the deficient efficiency of latent heat storage (LHS) technology. To address this, an innovative Vicsek fractal fin is first proposed herein for the heat charging/discharging rate enhancement. The phase transition behavior and thermal characteristics of the innovative LHS unit are solved numerically and compared to the traditional one, employing the enthalpy–porosity algorithm and a two-dimensional transient model. The results indicate that Vicsek fractal fins remarkably promote the thermal efficiency of LHS units. Compared to the traditional LHS units, innovative LHS units with Vicsek fractal fins, respectively, reduce the melting/solidification duration by 16.1% and 28.7%. The melting process is divided into early thermal conduction, subsequent convective thermal transport, and later thermal conduction, while solidification processes are dominated by the initial natural convection and the subsequent continuous heat conduction. Moreover, the impact of free convection is higher on melting processes than on solidification processes. Importantly, the efficient heat transfer path of Vicsek fractal fins allows the inherent natural convection suppression to be much weaker than the improvement level of heat transfer efficiency, facilitating the interstitial thermal transport between the phase change material (PCM) and fins and improving the overall temperature uniformity of LHS units.

A substrate integrated waveguide based leaky wave antenna with Vicsek-cross fractal shaped slot is proposed. Vicsek cross fractal is used here to obtain wide band and compact structure for wideband applications in X-band. The proposed design achieves a wide impedance bandwidth of 1.98 GHz ranging from 8.96 to 10.94 GHz. A near identical pattern bandwidth with peak realized gain of 8.41dBi is achieved over the resonance frequency band of the proposed antenna. A prototype in X-band of the proposed design is examined and measured for verification.

We investigate a class of self-similar networks modeled on the three-dimensional Vicsek fractal, and obtain the asymptotic formula of node-weighted average geodesic distances, which is an integral of geodesic distance in terms of the self-similar measure.

This paper presents a fast, accurate, and multifunctional electromagnetic optimization bioinspired technique that uses the social spider optimization algorithm (SSO). The proposed technique is applied to the design of a frequency selective surface (FSS), with Vicsek fractal patch elements, and an ultra wideband antenna (UWB), consisting of a ring monopole. In the proposed (synthesis) technique, the use of a database or the construction of a neural network is not required. The SSO algorithm communicates via scripts directly with the Ansoft HFSS software tool, enabling a good flexibility in the number of parameters to be optimized and reducing the required computation time. The accuracy and efficiency of the proposed technique is confirmed by comparison with experimental results and with other simulation techniques. © 2016 Wiley Periodicals, Inc. Microwave Opt Technol Lett 58:1912–1917, 2016

Fractal generally has self-similarity. Using the self-similarity of fractal, we can obtain some important theories about complex networks. In this paper, we concern the Vicsek fractal in three-dimensional space, which provides a natural generalization of Vicsek fractal. Concretely, the Vicsek fractal in three-dimensional space is obtained by repeatedly removing equilateral cubes from an initial equilateral cube of unit side length, at each stage each remaining cube is divided into [Formula: see text] smaller cubes of which [Formula: see text] are kept and the rest discarded, where [Formula: see text] is odd. In addition, we obtain the skeleton network of the Vicsek fractal in three-dimensional space. Then we focus on weighted average geodesic distance of the Vicsek fractal in three-dimensional space. Take [Formula: see text] as an example, we define a similar measure on the Vicsek fractal in three-dimensional space by weight vector and calculate the weighted average geodesic distance. At the same time, asymptotic formula of weighted average geodesic distance on the skeleton network is also obtained. Finally, the general formula of weighted average geodesic distance should be applicable to the models when [Formula: see text], the base of a power, is odd.

Fractal phenomena may be widely observed in a great number of complex systems. In this paper, motivated by the well-known Vicsek fractal, we propose the generalized Vicsek fractal whose seed is not necessarily a single edge but an arbitrary tree, and study random walks on it for the purpose of understanding how the underlying topology influences dynamic behaviors. Meanwhile, we determine the exact solution to the mean first-passage time for random walks on the generalized version in a more convenient mapping-based manner, while other methods suitable for the typical Vicsek fractal will become prohibitively complicated and even fail in this kind of situation. The analytic results suggest that the scaling relation between mean first-passage time and vertex number in generalized versions of Vicsek fractal keeps unchanged in the large graph size limit. Lastly, we conduct extensive computer simulations, and observe that experimental results are well consistent with the theoretical analysis.

We consider the role of local orientational mobility for the NMR relaxation of hyperbranched macromolecules modeled in form of Vicsek fractals. The spectral density J ( ω ) , that is relevant for the NMR relaxation, is studied for labeled segments of flexible and semiflexible Vicsek fractals. We show that the semiflexibility leads to strong effects. For flexible Vicsek fractals the function ω J ( ω ) has a single maximum, whose position is determined by the local-scale modes and therefore independent of the location of the labeled segments. For the labeled segments of a semiflexible Vicsek fractal, we observe a broad spectral density. It is characterized by the overall relaxation of the branch that originates from the labeled segment and by the overall relaxation of the subbranches that build this branch. In this way, going from the periphery to the inner part of the macromolecule, the labeled segments display a broader spectral density. However, this effect saturates when the labeled segments get closer to the core of the hyperbranched macromolecule, so that they do not feel their remoteness from the periphery anymore.

This paper presents the design and simulation of a passive chipless RFID Tag using fractal shapes. To design the fractal antenna the vicsek fractal was choosen on its different iterations. In order to obtain data regarding its detection, simulations were done covering the frequency range of 5GHz to 15GHz. The simulation will have two stages: In the first one, the tag will be illuminated by two horn antennas to obtain the return loss and forward transmission parameters, which shows the medium disturbance generated by the TAG induction. The second stage shows the monostatic Radar Cross Section (RCS) using a probe, located in a certain distance away from the chipless fractal TAG. Finally, the results shows that fractal antenna design after multiple iterations, improves their detection. The software used in this resarch is CST Microwave Studio.

This paper concerns weighted average distances of Vicsek fractal and Vicsek networks by using the integral of geodesic distance in terms of self-similar measure with respect to the weight vector.

In this work, we study the eccentric distance sum of Vicsek networks. To obtain the eccentric distance sum of networks, we investigate the corresponding integral on self-similar measure for Vicsek fractals. We use the self-similarity of distance and measure to solve the integral.

We suggest a flexible way to study the self-similar interface of two different fractals. In contrast to previous methods the participating energies are modified in the neighborhood of the intersection of the fractals. In the example of the Vicsek snowflake and the 3-gasket, a variant of the Sierpinski gasket, we calculate the admissible transition constants via the “Short-cut Test”. The resulting range of values is reinterpreted in terms of traces of Lipschitz spaces on the intersection. This allows us to describe the interface effects of different transition constants and indicates the techniques necessary to generalize the present interface results. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson's groups F, T, and V . Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite CAT(0) cubical complex, and we use this action to prove that certain rearrangement groups are of type F infinity., 48 pages, 37 figures

It has been shown analytically that a proper tuning of the boundary condition can ensure an infinity of extended electronic eigenstates on a loopless semi-infinite Vicsek fractal. First we map this particular loopless fractal exactly onto a one-dimensional hierarchical chain. Within the framework of real space renormalization group and the transfer matrix methods we then demonstrate how a suitable choice of the on-site potential of an edge atom can generate a whole hierarchy of extended Bloch-like eigenfunctions.

In this work, we study the relaxation dynamics of a multihierarchical polymer network built by replicating the Vicsek fractal in dendrimer shape. The relaxation dynamics is investigated in the framework of the generalized Gaussian structure model by employing both Rouse and Zimm approaches. In the Rouse-type approach, we show the iterative procedure whereby the whole eigenvalue spectrum of the connectivity matrix of the multihierarchical structure can be obtained. Remarkably, the general picture that emerges from both approaches, even though we have a mixed growth algorithm, is that the obtained multihierarchical structure preserves the individual relaxation behaviors of its components. The theoretical findings with respect to the splitting of the intermediate domain of the relaxation quantities are well supported by experimental results.

We present an exact analytical method of engineering the localization of electromagnetic waves in a fractal waveguide network. It is shown that, a countable infinity of localized electromagnetic modes with a multitude of localization lengths can exist in a Vicsek fractal geometry built with diamond shaped monomode waveguides as the 'unit cells'. The family of localized modes form clusters of increasing size. The length scale at which the onset of localization for each mode takes place can be engineered at will, following a well defined prescription developed within the framework of a real space renormalization group. The scheme leads to an exact evaluation of the wave vector for every such localized state, a task that is non-trivial, if not impossible for any random or deterministically disordered waveguide network., 8 pages, 4 figures. arXiv admin note: text overlap with arXiv:1204.4986

We present exact analytical results revealing the existence of a countable infinity of unusual single particle states, which are localized with a multitude of localization lengths in a Vicsek fractal network with diamond shaped loops as the 'unit cells'. The family of localized states form clusters of increasing size, much in the sense of Aharonov-Bohm cages [J. Vidal et al., Phys. Rev. Lett. 81, 5888 (1998)], but now without a magnetic field. The length scale at which the localization effect for each of these states sets in can be uniquely predicted following a well defined prescription developed within the framework of real space renormalization group. The scheme allows an exact evaluation of the energy eigenvalue for every such state which is ensured to remain in the spectrum of the system even in the thermodynamic limit. In addition, we discuss the existence of a perfectly conducting state at the band center of this geometry and the influence of a uniform magnetic field threading each elementary plaquette of the lattice on its spectral properties. Of particular interest is the case of extreme localization of single particle states when the magnetic flux equals half the fundamental flux quantum., 9 pages, 8 figures

We present an exact real space renormalization group approach to study the critical properties of the Vicsek fractal. This fractal lattice has a finite order of ramification. The physical model defined on it is exactly solvable. We argue that the short-range spin model on the Vicsek fractal shows no finite-temperature phase transition.

The nature of transverse vibrational eigenmodes of a regular Vicsek fractal with hierarchical nearest-neighbor interactions has been analyzed using a real-space renormalization-group approach. Different types of spatial scaling behavior for the nondegenerate transverse vibrational modes are obtained depending on a positive hierarchical parameter R: for R>Rc, the nondegenerate modes are all extended in nature, while for R

Fractal systems exhibit anomalous diffusion where the mean square distance covered in a random walk (RW) is related to time t as 〈r2〉 ∼ t2/dw with dw ≠ 2. The presence of a bias along any direction in fractal systems gives rise to competing effects - enhanced diffusion as well as trapping at dead ends. A random walk on a Vicsek fractal (VF) is studied. The same exponent dw is obtained from a similar vibrating mass-spring system. Introducing a bias along the positive Y-direction in RWs, the competing effects are also observed. Results are compared with another well-studied deterministic fractal, the Sierpinski gasket (SG).

The small-angle scattering curves of deterministic mass fractals are studied and analyzed in the momentum space. In the fractal region, the curve I(q)q^D is found to be log-periodic with a good accuracy, and the period is equal to the scaling factor of the fractal. Here D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q^D in the momentum space is related to the log-periodicity of the quantity g(r)r^{3-D} in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in the real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed., 12 pages, 4 figures

In this paper, we investigate the effects of free-end boundary condition on the transverse vibrations of a Vicsek fractal. The unique feature of the eigenvalue spectrum corresponding to this case is that both non-degenerate and degenerate modes of a given generation are present in all subsequent generations, whereas in the case of fixed-end boundary condition only the degenerate modes persist. In the first case, the nature of the persistent degenerate modes is found to be either edge-localized or localized in the region between two adjacent clusters or adjacent sub-clusters, whereas in the second case the persistent modes are found to be edge-confined superlocalized modes. In both cases one finds that: (i) the frequency spectrum consists of atomic-like levels superimposed on a point-dense, cantor-set-like background; and (ii) there is side-by-side coexistence of the extended non-degenerate modes and superlocalized degenerate persistent modes.

A new method to determine eigenvectors based on a real space Green's function is reported. This method is particularly useful for determining the eigenvectors corresponding to degenerate states. Using this approach, the lowest persistent degenerate mode exhibited by the third-stage Vicsek fractal has been examined and has been shown to be an edge-confined, superlocalized vibrational mode

The Vicsek fractal patch antenna is introduced to reduce the antenna size in this paper. Compared to the normal square patch antenna, this antenna reveals lower resonant frequency. Some results of experiments and simulations using the method of momentum are given here to show the miniaturizing benefits of using Vicsek fractal shape.

We propose a simple model of a waveguide network designed following the growth rule of a Vicsek fractal. We show, within the framework of real space renormalization group (RSRG) method, that such a design may lead to the appearance of unusual electromagnetic modes. Such modes exhibit an extended character in RSRG sense. However, they lead to a power law decay in the end-to-end transmission of light across such a network model as the size of the network increases. This, to our mind, may lead to an observation of power law localization of light in a fractal waveguide network. The general occurence of photonic band gaps and their change as a function of the parameters of the system are also discussed., Comment: 13 pages, 6 figures

In this paper, we study consensus problems in a family of tree networks and investigate first and second order consensus denoted as network coherence characterized by Laplacian spectrum. According to the tree structures, we obtain the recursive relationships of Laplacian matrix and its Laplacian eigenvalues at two successive generations. We then obtain the analytical expressions for the sum of the reciprocals and the square reciprocals of all nonzero Laplacian eigenvalues. Finally, we calculate first and second order network coherence and see that the scalings of first and second order coherence with network size N are lnN and N, which are smaller than some studied tree graphs, such as Peano basin fractal, T-graph, and generalized Vicsek fractal.

We present a class of extended electronic wave functions on a Vicsek fractal. The transmittivity of arbitrarily large fractal lattices corresponding to these particular extended-state eigenvalues exhibits a power-law decay with increasing system size. The eigenvalues corresponding to the above extended states as well as the scaling law for the transmittivity have been exactly calculated using a real-space renormalization-group method. {copyright} {ital 1996 The American Physical Society.}

An exact renormalization-group approach is used to study the spectral properties of the Vicsek fractal (VF). Particular attention is focused on the spectral clustering. We highlight the effect of the complete interactions within each cluster on the spectral structure of the VF Results are given on a general characterization of the spectra.

In this paper, we investigate the effects of free-end boundary condition on the transverse vibrations of a Vicsek fractal. The unique feature of the eigenvalue spectrum corresponding to this case is that both non-degenerate and degenerate modes of a given generation are present in all subsequent generations, whereas in the case of fixed-end boundary condition only the degenerate modes persist. In the first case, the nature of the persistent degenerate modes is found to be either edge-localized or localized in the region between two adjacent clusters or adjacent sub-clusters, whereas in the second case the persistent modes are found to be edge-confined superlocalized modes. In both cases one finds that: (i) the frequency spectrum consists of atomic-like levels superimposed on a point-dense, cantor-set-like background; and (ii) there is side-by-side coexistence of the extended non-degenerate modes and superlocalized degenerate persistent modes.

The nature of the frequency spectrum of a Vicsek fractal of any stage has been predicted using an analysis based on local symmetry and self-similarity. The analysis was carried out for transverse vibrations of the fractal with nearest-neighbor interactions. The complete frequency spectrum was found to contain both nondegenerate and degenerate components. The degenerate component of the frequency spectrum of the nth-stage fractal consists of threefold degenerate, D 2 -fold degenerate,..., and D n -fold degenerate modes with D 2 =8 and D n =5D n-1 -8 for n≥3. In particular, D 2 -fold degenerate, D 3 -fold degenerate modes,..., etc., are all persistent modes

The vibrational dynamics of a general Vicsek fractal with nearest-neighbor interactions has been solved exactly. The most surprising finding of this paper is that the nondegenerate modes are all extended in nature. Furthermore, the nondegenerate modes occur side by side with the persistent degenerate modes, which are superlocalized. There is no single demarcation frequency separating the extended states from the localized states as one would expect in a three-dimensional disordered system.

By an exact real-space renormalization-group approach, we investigate ultradiffusion with a hierarchical pattern of hopping rates defined on the regular Vicsek fractal. Anomalous long-time behavior of the autocorrelation function of ultradiffusion is observed. It is found that the transition from anomalous to normal diffusion occurs in the general n-dimensional case, and the transition point is independent of dimension. In the presence of bias, we find another unusual transition that leaves hopping particles trapped in some areas in contrast to the diffusive state for which the average square distance 〈${\mathit{R}}^{2}$(t)〉\ensuremath{\sim}\ensuremath{\infty} for t\ensuremath{\rightarrow}\ensuremath{\infty}. This transition is essentially different from the one for biased ultradiffusion of the one-dimensional multiple-furcation hierarchical system.

The electronic spectra for the Vicsek fractal at any stage have been discussed using the tight-binding method with the nearest-neighbor interaction approximation. The calculated results have shown the relationship between the persistent modes from one stage to the next. The particularity of the spectra for the fifth stage of Vicsek fractal obtained by numerical calculation was analyzed.