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Mathematical models of heat and moisture transfer for anisotropic materials, based on the use of the fractional calculus of integro-differentiation, are considered because such two-factor fractal models have not been proposed in the literature so far. The numerical implementation of mathematical models for determining changes in heat exchange and moisture exchange is based on the adaptation of the fractal neural network method, grounded in the physics of processes. A fractal physics-informed neural network architecture with a decoupled structure is proposed, based on loss functions informed by the physical process under study. Fractional differential formulas are applied to the expressions of non-integer operators, and finite difference schemes are developed for all components of the loss functions. A step-by-step method for network training is proposed. An algorithm for the implementation of the fractal physics-informed neural network is developed. The efficiency of the new method is substantiated by comparing the obtained numerical results with numerical approximation by finite differences and experimental data for particular cases. [ABSTRACT FROM AUTHOR]

Compression behavior of nanoparticle powder is still unclear, while it could largely affect its applications in additive manufacturing. Considering the fractal structure of aggregates in nanoparticle powder, from the van der Waal's state equation, the relation between porosity and compression pressure of nanoparticle powder was modeled theoretically, i.e. (P + c)(φ -a)=(1-a)c, here P is pressure, φ is porosity, a and c are parameters defined in main text. The model could capture the compression behavior of nanoparticle powders well. Under the same compression pressure, smaller nanoparticles will lead to larger porosity and thus lower thermal conductivity of nanoparticle packed beds, due to the easier aggregate of smaller nanoparticles. The compression behavior and the thermal conductivity of nanoparticle powders or packed beds are expected to supply information for powder spreading and sintering treatment respectively in additive manufacturing. [ABSTRACT FROM AUTHOR]

This study examines the structure of the universe during its high-energy era. In this context, we examine Barrow’s proposal of entropy, which forecasts a fractal configuration for the geometric properties of the black hole’s horizon. By applying the principles of thermodynamics, we get the equation of motion using the Barrow entropic consideration. A quasi-de Sitter phase, in which fractal and non-fractal states may coexist, is determined. This is an uncertainty state (coexistence of fractal and non-fractal states), and so a deformation for de Sitter spacetime, which contains a high-energy scale in the framework of the COBE normalization. With the kinetic energy term of the inflation field, we remove this ambiguity owing to the field oscillations. However, the oscillatory behavior of the inflation field continues and causes the universe to enter a new phase state. At this point, in which the inflation field oscillations are over, we obtain a solution that provides the universe transition to the radiation and dust phases. [ABSTRACT FROM AUTHOR]

Living cells have spontaneous ultraweak photon emission derived from metabolic reactions associated with physiological conditions. The ORCA-Quest CMOS camera (Hamamatsu Photonics, Japan) is a highly sensitive and essential tool for photon detection; its use with a microscope incubator (Olympus) enables the detection of photons emitted by embryos with the exclusion of harmful visible light. With the application of the second law of thermodynamics, the low-entropy energy absorbed and used by embryos can be distinguished from the higher-entropy energy released and detectable in their environment. To evaluate higher-entropy energy data from embryos, we developed a unique algorithm for the calculation of the entropy-weighted spectral fractal dimension, which demonstrates the self-similar structure of the energy (photons) released by embryos. Analyses based on this structure enabled the distinction of living and degenerated mouse embryos, and of frozen and fresh embryos and the background. This novel detection of ultra-weak photon emission from mouse embryos can provide the basis for the development of a photon emission embryo control system. The ultraweak photon emission fingerprints of embryos may be used for the selection of viable specimens in an ideal dark environment. [ABSTRACT FROM AUTHOR]

We offer a detailed description of our previously published new concept of the evolution of the biosphere as an integral system of its states over time, united by development trends. The structure of the biosphere is considered as a hierarchical fractal system, and the elementary unit of the biosphere is an ecosystem. The fractal structure of the biosphere corresponds to the emergent principle: each lower level is an element of a more complex system and has its own properties. The proposed concept of biosphere evolution is based on the general categories of dialectics: states and interstates, continuity and discreteness, reproducibility and uniqueness. The evolutionary history of the biosphere is a metameric picture of changing states and interstates. The most important feature of the biosphere organization in space–time is a complex system of continua. The development of an integral biospheric system occurs in a time continuum: in the biosphere, the differences between the early and subsequent states are quite significant and obvious. Moreover, these differences are associated with fundamental complications, development, which is, in fact, evolution. The states of the biosphere in certain periods are linked by trends that form an evolutionary system. Continuing states, when the system remains self-identical for a long period, are replaced by new states through interstates. Based on the principle of actualism, the problems of the biosphere's future and evolutionary trends of the biosphere under anthropogenic impact are considered. [ABSTRACT FROM AUTHOR]

In this study, a two-port network-based microwave sensor for liquid characterization is presented. The suggested sensor is built as a miniature microwave resonator using the third iteration of Hilbert's fractal architecture. The suggested structure is used with the T-resonator to raise the sensor quality factor. The suggested sensor is printed on a FR4 substrate and has a footprint of 40 × 60 × 1.6 mm 3 . Analytically, a theoretical investigation is made to clarify how the suggested sensor might function. The suggested sensor is created and put to the test in an experiment. Later, two pans to contain the urine Sample Under Test (SUT) are printed on the sensor. Before loading the SUT, it is discovered that the suggested structure's frequency resonance is 0.46 GHz. An 18 MHz frequency shift is added to the initial resonance after the pans are printed. They monitor the S-parameters in terms of S12 regarding the change in water content in the urine samples, allowing for the sensing component to be completed. As a result, 10 different samples with varying urine percentages are added to the suggested sensor to evaluate its ability to detect the presence of urine. Finally, it is discovered that the suggested process' measurements and corresponding simulated outcomes agreed quite well. [ABSTRACT FROM AUTHOR]

We argue the merits of having a simplified method to simulate equatorial plasma bubble (EPB) plume structures for practical usage. The capability to realistically model EPB plume structures in simulations would be advantageous when assessing the severity of ionospheric threats. Such advantages would arise because a realistic model of EPB plume structures could allow nonstationary scintillation signals to be simulated. Although EPB plume structures can be modeled via first-principle physics-based models, these models tend to be computationally demanding. High-performance computing facilities might be able to offer some remedy, but serious handicaps would remain for those without access to such advanced facilities. We investigated multiple options that utilize the diffusion-limited aggregation (DLA) fractal process to generate bifurcating structures that resemble typical EPB plume structures. We combined the DLA algorithm with the International Reference Ionosphere model to simulate EPBs in three dimensions. Initial tests of this modeling approach indicate promising results. [ABSTRACT FROM AUTHOR]

[Objectives]In order to solve the problem of poor ventilation areas of grain pile, the effect of the fractal structure on the ventilation uniformity in the ventilation process was explored. [Methods]A porous medium(sand)packing wrapped non-uniform section ventilation pipeline ventilation system was proposed. The relevant parameters of fractal structure were obtained by the minimum flow resistance optimization method of rectangular unit configuration. Combining the fractal branch pipelines designed according to the optimization parameters with the non-uniform section ventilation pipelines, a porous medium packing wrapped non-uniform section pipelines and fractal structure ventilation system was proposed. The grain type was selected as rice, and the porous medium material of the packing zone was selected as sand. The ventilation and cooling simulation of the two ventilation systems was carried out. The study simulated the cooling process of the granary with an initial temperature of 300 K and cold inlet air with a constant temperature of 293 K. The ventilation time lasted for 7.5 days. The ventilation velocity field, temperature field of the two ventilation systems were compared and verified. [Results]In the non-fractal ventilation system, most of the poor ventilation areas were in the packing zone, and the ventilation intensity of the grain pile was guaranteed. However, the strong ventilation and strong cooling areas were concentrated near and above the ventilation pipelines, and the cooling uniformity was not good. In the fractal ventilation system, the poor ventilation areas were suppressed under the ventilation pipelines and the fractal branch pipelines. From the velocity field, the ventilation uniformity and the cooling uniformity in the grain pile were better, and the maximum temperature difference was reduced from 2.8 K to 1.3 K. From the ventilation uniformity index, the ventilation uniformity of the fractal ventilation system was good, and exponent of uniformity was 0.919. [Conclusions]The sand packing zone could help the grain pile avoid most of the poor ventilation areas. The fractal structure could effectively enhance the ventilation intensity in the weak ventilation areas of the velocity field, and the temperature field had excellent effect in improving the uniformity of cooling. [ABSTRACT FROM AUTHOR]

Polymer‐based magnetic particles have been widely used for the separation of biological samples including nucleic acids, proteins, virus, and cells. Existing magnetic particles are almost prepared by coating polymers on magnetic nanoparticles (NPs). However, this strategy usually encounters the problem of poor magnetic NPs loading capacity. Here, a series of nanofractal magnetic particles (nanoFMPs) synthesized by a strategy of mediator monomer regulated emulsion interfacial polymerization is presented, which allows effective magnetic NPs loading and show efficient nucleic acid separation performance. The mediator monomers facilitate the dispersion of magnetic NPs in internal phase to achieve higher loading, and the hydrophilic monomers use electrostatic interactions to form surface nanofractal structures with functional groups. Compared with magnetic particles without nanofractal structure, nanoFMPs exhibit a higher nucleic acid extraction capability. This strategy offers an effective and versatile way for the synthesis of nanoFMPs toward efficient separation in various fields from clinical diagnosis to food safety and environmental monitoring. [ABSTRACT FROM AUTHOR]

The distribution of velocity and temperature field inside the granary can be used to evaluate ventilation quality. A well-ventilated system with uniform air-flow and cooling can effectively prevent moisture accumulation. To address the issue of poor uniformity in ventilation and cooling, this research introduces two new ventilation systems: the non-fractal ventilation system and the fractal ventilation system. The fractal structure was designed based on optimized parameters obtained from rectangular elements with minimum flow resistance. Numerical simulation methods were employed to verify the effectiveness of these ventilation systems. A comparative analysis was conducted, examining the velocity field, temperature field, and velocity uniformity of the two systems. The results demonstrated that the presence of a sand packing zone helped the grain pile zone avoid most of the areas with poor ventilation. The discontinuous pores at the interface between the sand packing zone and the grain pile zone significantly increased flow resistance, effectively reducing the velocity of high speed ventilation into the grain pile zone and decreasing the permeability of the grain pile zone towards the poorly ventilated areas of the sand pile zone. Furthermore, the fractal structure greatly enhanced ventilation intensity in areas with poor ventilation in the velocity field, while also improving the uniformity of cooling in the temperature field. This research presents an innovative solution for ventilation in room-type warehouses, offering a new option for effective and efficient ventilation. [ABSTRACT FROM AUTHOR]

In this paper, we propose an optically transparent dual-band metamaterial absorber (MMA) that uses Ag nanowire screen-printed fractal structures. The proposed MMA exhibits near-perfect absorption in the C- and K-bands. This dual-band absorption property is achieved through two inductive–capacitive (L-C) resonances located at 6.45 and 21.14 GHz, which are generated by the second-order fractal structures. We analyzed the microwave absorbing mechanisms through the distributions of the surface current and electromagnetic field on the top and bottom layers. The MMA demonstrates an optical transmittance of 63.1% at a wavelength of 550 nm. This high optical transmittance is attained by screen printing transparent Ag nanowire ink onto a transparent PET substrate. Since screen printing is a simple and low-cost fabrication method, the proposed MMA offers the advantages of being low cost while having the properties of optical transparency and effective dual-band absorption. Consequently, it holds great potential for the radar stealth application of C- and K-bands in that it can be attached to the windows of stealth aircraft due to its optical transparency and dual-band near-perfect absorption property. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to obtain some variable separation solitary solutions of (2 + 1)-dimensional nonlinear long water wave (LWW) equation through symbolic computation. By applying Riccati equation mapping method, the localized dromion excitations are constructed and different number of M × N multi-dromions are described by selecting appropriate functions for the LWW equations, which provide a model of the interaction between two waves propagation consequently, the solition solutions are obtained in different forms of dynamical structures. Moreover, the collision phenomena between two multidromions are explored he fractal structure was also investigated. The results have rich physical structures that are helpful to explain the nonlinear solitary phenomena in nonlinear physics. [ABSTRACT FROM AUTHOR]

Spotted hyaenas live in unusually large social groups for a carnivore. Since, all else equal, the ‘infertility trap’ (a negative relationship between fertility and the number of females in a group) limits social group sizes to ~5 reproductive females in mammals, hyaena must, like other very social species, have found a way to mitigate the stresses involved in order to do so. From a comparative analysis of data from 19 well‐studied Crocuta crocuta populations, I show (1) that the distribution of hyaena clan sizes is multimodal, with a fractal scaling ratio close to 3 and a base unit of 12–15 individuals (3–5 reproductive females), (2) that fertility is a negative function of number of females in the group and (3) that there is a trade‐off between the benefits of having more males in the group and the costs incurred by having more females. Although females do buffer themselves against the infertility trap by forming matrilineal alliances (thereby creating a primate‐like multilevel structure), males seem to play an important role, such that, in areas with a low density of males, clan sizes are much smaller. [ABSTRACT FROM AUTHOR]

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel σ -algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each μ -measurable function is Riemann-integrable with respect to μ. Moreover, if μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of R n meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets. [ABSTRACT FROM AUTHOR]

In the 1970s, it was proved that a bounded linearly convex domain with smooth boundary in is homeomorphic to an open ball. If the boundary of a bounded linearly convex domain in is allowed not to be smooth, then the domain may be of a different topological type. The projection of the complex plane onto the Hartogs diagram in with symmetry plane has a simple geometric shape only for : in that case, this is a circular cone with vertex in the plane . This fact allows one to construct linearly convex Hartogs domains in with symmetry plane whose projections onto the Hartogs diagram have a fractal structure. [ABSTRACT FROM AUTHOR]

Nonintegrable dynamical systems have complex structures in their phase space. Due to the axisymmetry of the system, motion of a test charged particle in a dipole magnetic field can be reduced to a two-degree-of-freedom (2-DoF) nonintegrable Hamiltonian system (Störmer, Radium (Paris) 4(1):2–5, 1907; Dragt, Rev Geophys, https://doi.org/10.1029/RG003i002p00255, 1965). We carried out a systematic study of orbits of charged particles with an azimuthal initial velocity in a dipole field via calculation of their Lyapunov characteristic exponents (LCEs) and escape times for a dimensionless energy, respectively, less and greater than 1/32, above which most particles will escape from the magnetic dipole to infinity. Meridian plane periodic orbits symmetric with respect to the equatorial plane of the magnetic dipole are then identified. We found that (1) symmetric periodic orbits can be classified into several classes based on their number of crossing points on the equatorial plane; (2) the initial conditions of these classes locate on closed loops or closed curves going through the origin; (3) most isolated regions of stable quasi-periodic orbits are associated asymmetric stable periodic orbits; (4) classes of asymmetric periodic orbits either go through the origin or terminate at flat equatorial plane orbits with the other end approaching centers of spiral structures; (5) there are apparent self-similarities in the above features with the decrease of energy. These results can be used to guide the search for stable orbits that may have applications in broad physical contexts. [ABSTRACT FROM AUTHOR]

A sinuous antenna is a frequency-independent antenna known for its wide bandwidth and consistent gain, which makes it valuable in broadband applications such as ultrawideband (UWB) radar and ground-penetrating radar (GPR). However, sinuous antennas tend to be rather large. Consequently, numerous studies have explored miniaturization methods, with the gap-loading method emerging as a prominent approach. Unfortunately, it is still difficult to achieve broad bandwidths for conventional miniaturized sinuous antennas. In this paper, we use a novel approach incorporating a meander shape into the sinuous curve and employing gap loading with meandering. This innovative technique results in the development of a fractal-structured two-arm sinuous antenna characterized by an ultra-compact size and significantly expanded bandwidth. Adding a meander line in the outermost part maximizes the capacitance, thereby enhancing the gap-loading effect and minimizing the overall size of the sinuous antenna. In addition, the introduction of an inner meander line increases the inductance, contributing to a further expansion of the antenna's bandwidth. For example, the electrical length of the antenna without the meander line is 0.552 × 0.552 × 0.052 λ g 3 , while the electrical length of the antenna with the meander line is only 0.445 × 0.445 × 0.036   λ g 3 , i.e., 19.4% smaller. The antenna lacking the outermost meander line exhibits a 10 dB impedance bandwidth, spanning from 0.74 to 10.53 GHz. In contrast, the antenna featuring the outermost meander line has a 10 dB impedance bandwidth, extending from 0.51 to 10.72 GHz, which results in a remarkable enhancement in the fractional bandwidth (by 8.1%). Hence, the proposed antenna design is a good candidate for broadband applications that require miniaturization. [ABSTRACT FROM AUTHOR]

This paper systematically classifies and summarizes the current state of research on the enhancement of heat transfer in microchannels with bionic fractal structure, including fractal structures such as leaf veins, tree branches, lung trachea, spider webs and other biomimetic microchannels. The fractal topology microchannels are also discussed with theoretical models and experimental numerical verification, which shows their advantages by comparing with conventional microchannels. Finally, the existing challenges and future directions of fractal structured microchannel are analyzed in depth and prospected, respectively. This study can provide new research ideas for the development of heat transfer process strengthening and provide the corresponding theoretical basis and scientific guidance for industrial application. [ABSTRACT FROM AUTHOR]

In order to obtain the local structural solutions of nonlinear integrable systems, a (3 + 1)-dimensional nonlinear integrable equation is studied by using the multi-linear variable separation method, and the soliton, dromion, breather and instanton solutions containing arbitrary functions are obtained. Then, the abundant local excitations for the proposed equations are constructed by appropriately setting arbitrary function forms, and the evolution characteristics of system's dromion solutions with time are investigated. In addition, the fractal structure of the separable solution of the system was described. The results show that the proposed method can obtain some special solutions and this method has been extended in different ways so as to enroll more low-dimensional functions in the solution. [ABSTRACT FROM AUTHOR]

The fractal structure was applied to the fin optimization of latent heat storage (LHS) technology based on the fractal principle. By numerical simulation, the performances of LHS units with fractal tree-shaped fins were compared with those with traditional plate fins, and the effects of fin structure parameters based on the fractal principle on heat storage and release performances were studied. The results show that the heat storage performance and heat release performance of LHS units with fractal tree-shaped fins are clearly improved, and the optimal fractal structure parameters allow the LHS units to achieve optimal performance. A nonlinear surface fitting function method was used to obtain the functional relation between the branching angle and length index and the heat storage time of the LHS unit. In fractal tree-shaped fins, the optimal length index is 1.141, the optimal width ratio is 1, and the branching angle is 73.55°. In comparison to the LHS unit with traditional plate fins, the LHS unit with the optimum fractal structure parameters has a 35.85% faster melting rate and a 63.39% faster heat release rate. The least entransy dissipation, least thermal resistance and best reversibility are all characteristics of the LHS unit with the optimal fractal structure parameters. [ABSTRACT FROM AUTHOR]

The model of (2 + 1)-dimensional breaking soliton equation is proposed, and the novel soliton solutions are derived by using the variable separation and extended mapping approach of Riccati equation and then, the propagation dynamics of the solutions which include multi-dromions, ring breather, fusion solitons and other multi-solitons are also constructed. Furthermore, the rich local excitations and fractal structures are investigated. The simulation results verify the proposed scheme. It can get more abundant local excitation structures than the (1 + 1)-dimensional nonlinear systems and it has potential values for certain applications in optical field. [ABSTRACT FROM AUTHOR]

Fractal structure investigations of materials have a recent tradition in physics with applications to the natural sciences and technology. A new approach of fractal geometry in solids is analyzed, which allows the accurate evaluation of thermodynamic quantities for various values of fractional dimensions. Based on a fractal concept, the modified Einstein–Debye model and its application of thermodynamics are presented, which contribute to the evaluation of thermal properties as well as many materials. The reliable modified analytical expressions of the energy, heat capacities, entropy, enthalpy, and Gibbs free energy are established on the basis of modified Einstein–Debye approximation. This shows the perspective of the utilizing the fractal, describing quantities and parameters that analyze properties of nature. [ABSTRACT FROM AUTHOR]

In conventional ceramic–metal matrix composites, with the addition of the ceramic phase, although it can significantly improve the performance of the material in one aspect, it tends to weaken some of the excellent properties of the metal matrix as well. In order to meet the high toughness and high strength requirements of composites for practical production applications, researchers have searched for possible reinforcing structures from nature. They found that fractal structures, which are widely found in nature, have the potential to improve the mechanical properties of materials. However, it is often not feasible to manufacture these geometric structures using conventional processes. In this study, alumina ceramic fractal structures were prepared by 3D printing technology, and aluminum composites containing fractal ceramic structures were fabricated by spark plasma sintering technology. We have studied the effect of the fractal structure of alumina ceramics on the mechanical properties of composites. The compression strength of samples was measured by a universal testing machine and the torsional properties of samples were measured by a torsional modulus meter. The results show that a fractal structure improves the compressive strength of aluminum/alumina ceramic composites by 10.97% and the torsional properties by 17.45%. The results of the study will provide a new method for improving the mechanical properties of materials. [ABSTRACT FROM AUTHOR]

Studies on the erosion products of tungsten plasma-facing components (films, surfaces, and dust) for thermonuclear fusion reactors by spectroscopic methods are considered and compared with those of carbon deposits. The latter includes: carbon–deuterium CDx (x ~ 0.5) smooth films deposited at the vacuum chamber during the erosion of the graphite limiters in the T-10 tokamak and mixed CHx-Me films (Me = W, Fe, etc.) formed by irradiating a tungsten target with an intense H-plasma flux in a QSPA-T plasma accelerator. It is shown that the formerly developed technique for studying CDx films with 15 methods, including spectroscopic methods, such as XPS, TDS, EPR, Raman, and FT-IR, is universal and can be supplemented by a number of new methods for tungsten materials, including in situ analysis of the MAPP type using XPS, SEM, TEM, and probe methods, and nuclear reaction method. In addition, the analysis of the fractality of the CDx films using SAXS + WAXS is compared with the analysis of the fractal structures formed on tungsten and carbon surfaces under the action of high-intensity plasma fluxes. A comparative analysis of spectroscopic studies on carbon and tungsten deposits makes it possible to identify the problems of the safe operation of thermonuclear fusion reactors. [ABSTRACT FROM AUTHOR]

• This study proposes a group tree modeling approach to examine the modal properties of fractal sympodial trees analytically. • Modal properties of sympodial trees are characterized by the emerging and the self-similar modes. • The self-similar modes prevent the trunk and lower order branches from vibration. • Recursive closed-form formulas are presented to estimate the modal properties of trees. • A sympodial tree with infinite tree order and infinite branching number resembles to a branchless tree. This study examines the intricate relationship between the hierarchical architecture of fractal trees and their dynamic properties, namely modal frequencies and mode shapes. It focuses on idealized sympodial trees under small, linear elastic lateral vibrations. This study proposes a group tree modeling approach that condenses the branches into simple structures, reducing the complexity of the model as appropriate. With the aid of a renormalization technique, the analysis proves that the sympodial tree is characterized by two families of vibration modes, i.e., the "emerging" and the "self-similar" modes. It further shows that the sympodial tree inherits all modes from the previous tree order (i.e., fractal level). Crucially, these inherited modes are self-similar, and beneficial in mitigating the vibration of the trunk by localizing the vibration of the tree at higher order branches. The study derives recursive closed-form solutions for both self-similar modal frequencies and mode shapes. Further, the analysis of sympodial trees with infinite tree order shows that their modal frequencies converge to stable values as the tree order increases. Accordingly, the study proposes analytical formulas to estimate these frequency limits from the fractal parameters of the sympodial trees. Results reveal that the natural frequency of a sympodial tree with infinite tree order and infinite branching number approaches asymptotically to that of a branchless tree. The proposed approach is then validated with numerical simulations and compared with independent studies. These findings are promising towards new bio-mimicking design concepts for civil engineering structures with improved dynamic behaviour. [ABSTRACT FROM AUTHOR]

This article presents the design of a compact quasi-elliptic dual-mode microstrip band-pass filter deploying a novel Sierpinski–Knopp fractal geometry on a patch resonator. In this work, a rectangular perturbation is inserted at the diagonal edge of the resonator with fractal slots to realize dual-mode response with two transmission zeros near each skirt. The fractal BPF is designed, simulated, and measured at the resonant frequency of 2.38 GHz for wireless local area network applications. The fabricated filter exhibits insertion loss less than 1.43 dB, return loss greater than 14.33 dB with a 3-dB narrow fractional bandwidth of 4.62%. Moreover, it has a compact size of 24.6 × 24.6 mm2, suitable for the requirements of wireless communication systems. A good agreement is obtained between the simulated and measured results for the proposed filter design. [ABSTRACT FROM AUTHOR]

With wind energy utilization shifting to the deep sea, floating offshore wind turbines (FOWTs) have huge potential to improve the competitiveness of offshore wind energy generation. However, traditional FOWT systems are often in force imbalance due to the unique swaying characteristics caused by the unfixed foundation and high center of mass of the platform. Therefore, a novel bionic semi-submersible floating platform based on Victoria Amazonia (with random fractal structure) is proposed to increase the hydrodynamic stability of FOWT. A regular fractal structure with similar number of perforations and perforated area is also implemented to verify the effectiveness of the bionic FOWT. In this study, the unsteady computational fluid dynamic approach is adopted to simulate the aerodynamic and hydrodynamic response. More specifically, the dynamic fluid-body interaction module integrated in STAR-CCM+ is employed to establish a FOWT numerical model with rotating blades, tower, and mooring line system. Furthermore, the volume of fluid model in conjunction with the 6-DOF solver was used to efficiently solve the fluid-induced dynamic motion of FOWT in a multi-phase flow composed of air and water. Finally, the fully coupled calculation of the fractal structure FOWT was performed using the established reliable numerical model. The maximum average thrust and power of 736.43 kN and 5294.04 kW are obtained by random fractal FOWT in aerodynamic responses, while in hydrodynamic amplitude responses, the maximum decreases of 19.16% are obtained by both random fractal and regular fractal structures. As for amplitude responses in heave and surge, the random fractal FOWT of 12.91% and 5.05% decreases performs relatively better than the regular one (11.48% and 4.09% decreases). In addition, the visualization of unsteady flow fields and vortices in fractal structures is investigated in detail. Compared with the regular fractal structure, vortices within random fractal structure interact adequately with the pontoon walls, resulting in higher energy absorption effect and stability. • The present study firstly establishes a reliable full-coupled system for FOWT through the CFD method. • The hydrodynamic performance of FOWT is improved via a novel bionic fractal (random fractal structure) floating platform. • A regular fractal structure is tested to verify the effectiveness of random fractal structure. • Three-dimensional visualization is presented to further analyze the influence of fractal structures on FOWT. [ABSTRACT FROM AUTHOR]

A spectral problem is considered in a domain Ωε$$ {\Omega}_{\varepsilon } $$ that is the union of a domain Ω0$$ {\Omega}_0 $$ and a lot of thin trees situated ε$$ \varepsilon $$‐periodically along some manifold on the boundary of Ω0.$$ {\Omega}_0. $$ The trees have finite number of branching levels. The perturbed Robin boundary condition ∂νuε+εαiki,muε=0$$ {\partial}_{\nu }{u}^{\varepsilon }+{\varepsilon}^{\alpha_i}{k}_{i,m}{u}^{\varepsilon }=0 $$ is given on the i$$ i $$th branching layer; {αi}$$ \left\{{\alpha}_i\right\} $$ are real parameters. The asymptotic analysis of this problem is made as ε→0$$ \varepsilon \to 0 $$, that is, when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the Hausdorff convergence of the spectrum to the spectrum of the corresponding nonstandard homogenized spectral problem is proved, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions. [ABSTRACT FROM AUTHOR]

A frequency-selective surface (FSS) optimization method combining a curve-fitting technique and an improved bacterial foraging optimization (IBFO) algorithm is proposed. In the method, novel Koch curve-like FSS and Minkowski fractal islands FSS were designed with a desired resonance center frequency and bandwidth. The bacteria foraging optimization (BFO) algorithm is improved to enhance the performance of the FSS. A curve-fitting technique is provided to allow an intuitive and numerical analysis of the correspondence between the FSS structural parameters and the frequency response. The curve-fitting results are used to evaluate the fitness function of the IBFO algorithm, replacing multiple repeated calls to the electromagnetic simulation software with the curve-fitting equation and thus speeding up the design process. IBFO is compared with the classical BFO algorithm, the hybrid BFO-particle swarm optimization algorithm (BSO), and the artificial bee colony algorithm (ABC) to demonstrate its superior performance. The designed fractal FSS is fabricated and tested to verify the experimental results. The simulation and measurement results show that the proposed FSS has a fractional bandwidth of 91.7 % in the frequency range of 3.41–9.19 GHz (S, C, and X-bands). In addition, the structure is very thin, with only 0.025 λ and 0.067 λ at the lowest and highest frequencies, respectively. The proposed fractal FSS has shown stable performance for both TE and TM polarizations at oblique incidence angles up to 45°. according to simulations and measurements. [ABSTRACT FROM AUTHOR]

Recent studies have shown that it is possible to construct a probability measure from a fractal structure defined on a space. On the other hand, a theory on cumulative distribution functions from an order on a separable linearly ordered topological space has been developed. In this paper, we show how to define a linear order on a space with a fractal structure, so that these two theories can be used interchangeably in both topological contexts. [ABSTRACT FROM AUTHOR]

[Display omitted] • Hydrophilic/hydrophobic dual metal–organic frameworks Janus network is prepared on bio-inspired leaf vein (LV) structure for fog harvesting. • The integrated strategy by mimicking cactus needles, Janus wettability and bionic LV network in fog collection can be regulated by the multi-cooperation theory. • The Janus LV devices yields a high fog collection efficiency and intelligent interfacial materials. The leaf vein (LV) structure is an evolutionarily optimized quasi-fractal network responsible for efficient capture and transport of water with minimum energy consumption. For the first time, a bio-inspired dual-metal organic framework (MOF) functionalized LV network was proposed for water collection. Two MOFs with distinct wettabilities and microstructures are in-situ grown on opposite side of a metallic LV network, yielding a Janus LV (J-LV) network. The integrated strategy by mimicking cactus needles, Janus structures and bionic LV network in water harvesting can be regulated by the multi-cooperation theory in the processes of droplet caption, transport and collection. As such, the fog collection efficiency of J-LV devices can reach 1.23 ± 0.07 g cm-2h−1, which is a 292.9 % increase compared to that of the bare LV. Moreover, the J-LV has excellent interfacial floatability, showing capability of not only carrying objects weighing 149.5 times its own weight while keeping afloat, but also attaching firmly to the air–water interface under harsh airflow. We demonstrate that such Janus quasi-fractal LVs networks by in-situ dual-MOF growth will inspire new insight for designing highly efficient fog water collectors and intelligent Janus interface materials. [ABSTRACT FROM AUTHOR]

In this paper, a filtering antenna based on a metasurface is designed using slot-coupled feeding, and the metasurface unit is a fractal pattern. Replacing the rectangular patch on the metasurface with a fractal patch can introduce a radiation zero point on the upper sideband of the antenna, thereby enhancing the sideband selectivity. In addition, the current is reversed by loading shorting pin, and a radiation zero point is also introduced in the lower sideband. After measurement, the - 10 dB impedance bandwidth of the filtering antenna is 23.6% (3.22-4.08 GHz), and the average antenna gain in the passband is 8.1 dBi. The filtering antenna does not involve an additional filter circuit, has a simple structure, and a small size, and ideally eliminates insertion loss. To verify, the real object was made and tested and found that the reflection coefficient, pattern, gain, etc. were in good agreement with the simulation results, it can be applied to the 5G communication frequency band. [ABSTRACT FROM AUTHOR]

Hierarchical networks as fundamental models to describe the complex networks, have many applications in networks science, engineering technology and so on. In this paper, we first propose a new class of hierarchical networks with fractal structure, which are the networks with triangles compared to traditional hierarchical networks. Second, we study the precise results of some structural properties to derive small-world effect and scale-free feature. Third, it is found that the constructed network is sparse through the average degree and density. Fourth, it is also demonstrated that the degree distributions of hub nodes and the bottom nodes are the power law and exponential, respectively. Finally, we prove that clustering coefficient with a definite value z tends to stabilize at a lower bound as t iterates to a certain number, and the average distance of G t z has an increasing relationship along with the value of ln N t . [ABSTRACT FROM AUTHOR]

Hydrogen production by methanol steam reforming is one of the best choices for portable proton exchange membrane fuel cells (PEMFC). The channel is a dominant role in hydrogen production performance as the main reaction site. In this study, a fractal network channel architecture was designed and a three-dimensional (3D) numerical simulation was conducted in accordance with the Uniform Latin Hypercube design simulation scheme to explore the influence mechanism of branch grade (m), branch angle (θ), branch length ratio (α) and branch width ratio (β) on hydrogen production performance. Furthermore, four factors (m , θ , α , β) for the hydrogen production performance of fractal network channel were optimized by combining response surface method and non dominated sorting genetic algorithm II. The results showed that the hydrogen production performance is most sensitive to changes in β with the growth of the fractal network, followed by m , α , and θ. And the optimum parameter combination in terms of yield hydrogen is m of 3, θ of 134°, α of 0.88, and β of 0.85. The mechanism of the influence of fractal network structure parameters on the methanol reforming reaction as revealed in this study may provide a workable guideline for effective heat and mass transfer in new fractal reaction channel. [Display omitted] • A fractal network was developed with constant reaction area and initial width. • The influence mechanism is a trade-off between fluid velocity and channel length. • Combining RSM and NSGA-Ⅱ algorithms to optimize the structural parameters. • Hydrogen production performance is most sensitive to variations in β [ABSTRACT FROM AUTHOR]

A low‐profile dual‐band metasurface antenna is demonstrated for 4G LTE and 5G Sub‐6GHz applications. Based on the characteristic mode analysis, a wideband metasurface with stable radiation pattern is proposed by introducing Minkowski fractal structure in the edge patches and scaling the corner patches. The metasurface is operating in Sub‐6GHz band and excited by a Vivaldi bowtie slot antenna, which also serves as the radiator in LTE band. To validate the concept, a prototype with an overall size of 1.05λ0 × 1.05λ0 × 0.045λ0 (λ0 is the free‐space wavelength at 2.3 GHz) is fabricated. The experiment results show that the −10 dB impedance bandwidths in two bands are 1.7–2.78 (48.2%) and 3.2–5.01 GHz (44.1%), respectively. Moreover, broadside radiation performances in two bands are obtained. [ABSTRACT FROM AUTHOR]

We propose a toy model to reproduce the fractal structure of kink and antikink collisions on one topological sector of ϕ 6 theory. Using the toy model, we investigate the missing bounce windows observed in the fractal structure, and show that the coupling between two oscillation modes modulate the emergence of the missing bounce windows. By numerical calculation, we present that the two bounce resonance corresponds to the lower eigenfrequency of the toy model. [ABSTRACT FROM AUTHOR]

Using natural clinoptilolite (NCP) as a carrier and alginate (Alg)-calcium as an active species, the porous silicon calcium alginate nanocomposite (Alg-Ca-NCP) was successfully fabricated via adsorption-covalence-hydrogen bond. Its structural features and physicochemical properties were detailed investigated by various characterizations. The results indicated that Alg-Ca-NCP presented the disordered lamellar structures with approximately uniform particles in size of 300–500 nm. Specially, their surface fractal evolutions between the irregular roughness and dense structures were demonstrated via the SAXS patterns. The results elucidated that the abundant micropores of NCP were beneficial for unrestricted diffusing of Alg-Ca, which was conducive to facilitate a higher loading and sustainable releasing. The Ca content of leaf mustard treated with Alg-Ca-NCP-0.5 was 484.5 mg/100g on the 21st day, higher than that by water (CK) and CaCl 2 solution treatments, respectively. Meanwhile, the prepared Alg-Ca-NCPs presented the obvious anti-aging effects on peroxidase drought stress of mustard leaves. These demonstrations provided a simple and effective method to synthesize Alg-Ca-NCPs as delivery nanocomposites, which is useful to improve the weak absorption and low utilization of calcium alginate by plants. [Display omitted] • The alginate-calcium loaded natural clinoptilolites (Alg-Ca-NCPs) were fabricated. • The SAXS patterns demonstrated the surface fractal evolution of the Alg-Ca-NCPs. • Micropore in Alg-Ca-NCP favored to improve higher loading and sustainable releasing. • Alg-Ca-NCPs presented a better release behavior on malondialdehyde drought stress. • Alg-Ca-NCPs had an anti-aging effect on peroxidase drought stress of mustard leaves. [ABSTRACT FROM AUTHOR]

Fractal structures appear very often in open Hamiltonian systems, and can be identified in the deposition of chaotic magnetic field line on the plates of a tokamak divertor. Indeed, tokamaks with divertors are used to control the magnetic confinement of plasmas, such that the field lines created by electric currents have escape channels, through which plasma particles can be diverted out of the tokamak wall and redirected to divertor plates. In this work, we use a symplectic map to investigate the deposition patterns on the plates of a divertor. We show that the pattern of magnetic footprints on divertor plates (deposition patterns) underlying the chaotic orbits involves a number of fractal structures related to the existence of a nonattractive invariant chaotic set. In order to investigate the fractal characteristic of magnetic footprints, we analyze quantitatively the degree of stability of the fractal structure by calculating the entropy of the basin. These numerical analyses indicate that the fractal pattern on the divertor plates depends sensibly on the magnetic field structure. We show qualitative evidences of the Wada property. [ABSTRACT FROM AUTHOR]

Let η (s) = ∑ n = 1 ∞ (− 1) n + 1 n − s be the alternating zeta function. For a real number τ we define certain complex numbers b M , m (τ) and consider finite Dirichlet series υ M (τ , s) = ∑ m = 1 M b M , m (τ) m − s and η N (τ , s) = ∑ M = 1 N υ M (τ , s). Computations demonstrate some remarkable properties of these finite Dirichlet series, but nothing was supported by a proof so far. First, numerical data show that η N (τ , s) approximates η (s) with high accuracy for s in the vicinity of 1 / 2 + i τ ; this allows one to surmise that (*) η (s) = ∑ M = 1 ∞ υ M (τ , s). Moreover, it looks that lim M → ∞ ⁡ m υ M (τ , 1 − σ + i t) ‾ m υ M (τ , σ + i t) = η (σ + i t) m η (1 − σ + i t) ‾ m ; in other words, the individual summands in expected expansion (⁎) satisfy with an increasing accuracy a counterpart of the classical functional equation. Let ϒ M (τ , σ + i t) = υ M (τ , σ + i t) / η (σ + i t). When M , τ , and either σ or t are fixed, and the fourth parameter varies, the plot of ϒ M (τ , σ + i t) on the complex plane contains numerous almost ideally circular arcs with geometrical parameters closely related to the non-trivial zeros of the zeta function. [ABSTRACT FROM AUTHOR]

In a feedback control system like the manufacturing equipment or physiological system, its observed data usually manifests a specific autocorrelation structure and mixes with two external signals: disturbance and noise. The external disturbance and noise need to be estimated and compensated by the feedback control mechanism. However, the autocorrelation structure could mask problem signals and affect the monitoring of the system's inner health. Without separating the mixed signals, neither the evaluation of inner health nor the compensation of external disturbance can be achieved. A new signal separation approach, fractal component analysis (FCA), is developed to identify the autocorrelation structure for improving the health of the feedback control system. As a result, our proposed method outperformed FastICA and principal component analysis (PCA). [ABSTRACT FROM AUTHOR]

In this study, we report the emergence of two-dimensional (2D) branching fractal structures (BFS) in the nanoconfinement between the active and the support layer of a thin-film-composite polyamide (TFC-PA) nanofiltration membrane. These BFS are crystal dendrites of NaCl formed when salts are either added to the piperazine solution during the interfacial polymerization process or introduced to the nascently formed TFC-PA membrane before drying. The NaCl dosing concentration and the curing temperature have an impact on the size of the BFS but not on the fractal dimension (~1.76). The BFS can be removed from the TFC-PA membranes by simply dissolving the crystal dendrites in deionized water, and the resulting TFC-PA membranes have substantially higher water fluxes (three-to fourfold) without compromised solute rejection. The flux enhancement is believed to be attributable to the distributed reduction in physical binding between the PA active layer and the support layer, caused by the exertion of crystallization pressure when the BFS formed. This reduced physical binding leads to an increase in the effective area for water transport, which, in turn, results in higher water flux. The BFS-templating method, which includes the interesting characteristics of 2D crystal dendrites, represents a facile, low-cost, and highly practical method of enhancing the performance of the TFC-PA nanofiltration membrane without having to alter the existing infrastructure of membrane fabrication. [ABSTRACT FROM AUTHOR]

A new sound insulation metamaterial (SIM) is designed by combining a Helmholtz resonator and a fractal structure for sound insulation in the frequency range of 600 to 1700 Hz. Using a fractal-based internal structure, the designed hybrid SIM shows a relatively constant value of sound transmission loss (STL) over the target frequency range. The basic concept of this work is to develop two functional parts working in each effective frequency range. One is a Helmholtz resonance structure that operates with a high value of STL from 600 to 1000 Hz, and the other is an internal fractal structure that shows a high STL at higher frequencies in the 1000 to 1700 Hz range. In this work, the proposed metamaterial has a mean of 16.34 dB with a standard deviation of 1.73 dB within the wide target range of 600 to 1700 Hz. Additionally, the proposed SIM is approximately 40 % smaller in size than the previous resonator and has a higher mean STL with a small ΔSTL of 1.7 dB between highest and lowest values. [ABSTRACT FROM AUTHOR]

In this letter, a novel wideband circularly polarized (CP) crossed dipole antenna with frequency reconfiguration is proposed. By introducing two quasi‐Sierpinski fractal orthogonal dipoles printed on both sides of the substrate, compact, broadband and low cross‐polarization characteristics are achieved. The proposed antenna can be switched flexibly over two frequency bands via controlling the states of the PIN diodes inserted in each arm of the crossed dipole. The measured −10 dB impedance bandwidths of the antenna are 63% in 2.23–4.27 GHz and 12.5% in 4.04–4.58 GHz, respectively. Meanwhile, the 3 dB axial ratio bandwidths are 34.7% in 2.5–3.55 GHz and 4.16% in 4.23–4.41 GHz, respectively. The proposed antenna is desired to be applied in the future wireless communication systems. [ABSTRACT FROM AUTHOR]

• A novel cold-set rice starch (RS)/heat-set curdlan composite gel systems prepared. • The RS-curdlan interaction affects the composite gel's structure and properties. • The 5:5 composite gel starts to show rheological properties dominated by curdlan. • Curdlan inclusion in RS enhances gel structure density and mechanical properties. • A schematic proposed for temperature-induced conformational change. In pursuit of advancing the understanding of composite gel systems, this study delves into the intricate realm of rheology, structural elucidation, and mechanical attributes. Specifically, it scrutinizes the symbiotic interplay between rice starch, a cooling-set gel, and curdlan, a thermo-irreversible heating-set gel. A higher curdlan content enhances the inter-chain hydrogen bonding between rice starch and curdlan, resulting in a denser gel structure and thus increased moduli, solid-like behavior, and mechanical properties, and reduced frequency-dependence, especially at high temperatures (>65 °C). For example, with 50 % curdlan incorporation, G ′ (90 °C) improved by 252 %. Notably, thermal treatment can compromise the structural integrity of the rice starch gel, reducing strength and softening texture. However, this textural degradation can be effectively mitigated with, for example, 30 % curdlan incorporation, resulting in a 55-fold hardness increase at 85 °C. The knowledge gained from this work offers valuable guidance for tailoring starch-based gel products to specific properties. [ABSTRACT FROM AUTHOR]

• The Thermal conductivity of nano-grained graphite was investigated with MD method. • Both the equilibrium and non-equilibrium molecular dynamics methods were adopted. • Nano-grained graphite models with various grain sizes and arrangements were designed. • The atomic fractal structures of different fractal dimensions and stages were built. • Obtained results of MD simulations were verified with theoretical-model predictions. The thermal conductivity of isotropic graphite is a crucial parameter in mechanical and nuclear engineering. However, obtaining its value accurately is still difficult, especially under irradiation and high ambient temperature conditions. In this study, the nano-grained graphite models with different grain sizes and arrangements are designed, and corresponding thermal conductivity is calculated with equilibrium and non-equilibrium molecular dynamics methods. The results show that the thermal conductivity of nano-grained graphite varies between 3 to 5 W/(m·K) and decreases exponentially with the increase of the number of nanograins, while the arrangement of nanograins has almost no effect on the thermal conductivity. In addition, the effect of porosity of nano-polycrystalline graphite is analyzed by constructing fractal porous structures with different fractal dimensions and stages, of which the pore characteristics are determined with the initial zero-stage structure. The thermal conductivity of porous graphite increases with increasing fractal dimension and decreasing fractal porosity and the outcomes of molecular dynamics simulations are verified with the theoretical model results. [ABSTRACT FROM AUTHOR]

Fabrication of ring-like nanostructures of noble metal, particularly silver (Ag) nanostructures are challenging but long expected and suitable for various applications. Here, in this work, a strategy has been reported to fabricate Ag nanorings from ultrathin layer of the same. The edges of as-fabricated Ag nanorings were found continuous in nature rather than linear aggregation of adjacent nanoclusters as confirmed by high-resolution scanning electron microscope (SEM) micrographs. Interestingly, none of the Ag nanorings was found broken or cracked. Such clues led ones to a plausible hypothesis in demonstrating the fabrication mechanism. In this context, an in-depth morphological investigation and quantitative analysis have been carried out to understand the evolution of nanorings. SEM micrographs captured at the intermediate stage of Ag nanorings formation revealed that ultrathin layer of Ag turned into spherical nanoparticles before ending at the formation of such nanorings. High-resolution SEM micrographs confirmed that Ag nanoparticles were mostly spherical along with dark sheds at the center. A detailed morphological investigation of such Ag nanoparticles confirmed average estimated outer diameter of 376.74 ± 37.24 nm and average wall thickness of 102.89 ± 18.10 nm. Further estimation indicated Ag nanorings of 213.58 nm edge height which was verified in high-resolution SEM investigations. An attempt was taken to slice an individual Ag nanoparticle with SEM-aided focus ion beam (FIB) facility. A small deviation in heights was observed by high-resolution SEM micrographs of such SEM-aided FIB treated nanoparticles. Silver nanorings of different sizes and shapes (background) along with fractal features at the rim (inset) and plausible formation mechanism thereof (right). [ABSTRACT FROM AUTHOR]

We conducted a longitudinal study involving 2 4 0 patients grouped according to the classification of periodontal diseases agreed in the World Workshop by the different groups of specialists gathered there. We proceed to select images of Cone Beam Computed Tomography (CBCT) that were used to perform a study of bone density through a precise algorithm allowing an accurate calculation of the fractal dimensions of such images. A detailed anthropometric analysis was also carried out. Our objective was to demonstrate that there exists a direct relationship between either the loss of bone or the changes related to its height and diameter and the variations in bone density. Our results highlight significant differences among the initial and moderate periodontal groups with respect to both the control and the periodontal groups, where patients experience a severe and controlled periodontal disease. We conclude that there is a variation in the architecture of patients with periodontal disease that have an acute component and have not been treated or their treatment is not effective and their bone loss does not slow down. [ABSTRACT FROM AUTHOR]

Summary: The design of non–noble metal binderless electrode for enhanced electrocatalytic oxygen evolution is an unremitting challenge. Herein, we first report the construction of fractal structural alloy electrodes using controllable electrodeposition. Meanwhile, we have explored the regulation and morphology transition of metal dendrites in the electrolytic process. The metal wire‐type "one‐body" electrode prepared in the full‐immersion reactor demonstrates the unique three‐dimensional fractal structure with high branching degree and surface area. Moreover, the compositional turning of metal crystal via element doping is beneficial to improving the conductivity and crystal structure. The Ni and Zn elements are introduced to form Ni‐Fe and Zn‐Fe alloy dendrites, showing a preferable electrochemical catalytic oxygen evolution performance. The as‐prepared Ni‐Fe dendrite demonstrates a lower overpotential about 248 mV (vs RHE) at 10 mA cm−2 and Tafel slope of 55 mV dec−1 in the alkaline condition. [ABSTRACT FROM AUTHOR]

City-size distributions are known to be well approximated by power laws across a wide range of countries. But such distributions are also meaningful at other spatial scales, such as within certain regions of a country. Using data from China, France, Germany, India, Japan, and the United States, we first document that large cities are significantly more spaced out than would be expected by chance alone. We next construct spatial hierarchies for countries by first partitioning geographic space using a given number of their largest cities as cell centers and then continuing this partitioning procedure within each cell recursively. We find that city-size distributions in different parts of these spatial hierarchies exhibit power laws that are, again, far more similar than would be expected by chance alone--suggesting the existence of a spatial fractal structure. [ABSTRACT FROM AUTHOR]

A novel extremely miniaturized dual‐mode defected ground structure (DGS) duplexer based on fractal structure is presented. To further reduce the circuit size, the novel DGS resonator with Hilbert fractal structure is used. The equivalent circuits of the presented duplexer are given to analyze and design the duplexer. An extremely miniaturized dual‐mode DGS‐resonator duplexer with the center frequencies of 1.9 GHz and 2.14 GHz, respectively is designed and assembled. The total size of the assembled duplexer is only 0.23λg × 0.1λg. The measured results are in good agreement with the simulated results. [ABSTRACT FROM AUTHOR]