\r\n\tSome studies should be linked to the late-stage tumorigenesis promoting metastasis in cancer. In addition, deregulated cellular processes such as cell proliferation, apoptosis, and differentiation as related to different tumor types should be investigated in this book. Besides tumorigenesis, spontaneous tumor regression and its potential formation mechanisms should be reviewed or researched. In addition, the role of the deregulated immunity in tumorigenesis should be explored. The drug targets and treatment alternatives in various cancer types should be described or investigated in some studies. The studies relating to the laboratory tests used as diagnostic and prognostic in cancer patients should also be presented. Consequently, this book may include but is not limited to these topics.
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She is interested in cancer, molecular biology, human genetics, cytogenetics, molecular cytogenetics, genomics, and bioinformatics. She has participated in many research projects on neuroblastoma, human gross gene deletions, non-B DNA-forming sequences, solid tumors, HCV, and leukemia, resulted in six articles, one book chapter, and numerous reports. She performed many molecular biological methods: PCR, real-time PCR, bacterial transformation, plasmid vector transfection, RNA interference, fluorescence in situ hybridization (FISH), cytogenetic, DNA sequencing, and cell culture. She also performed genomics and biostatistics analyses using some bioinformatics tools and SPSS program. She reviewed several manuscripts for some medical, genetics, and genomics journals. 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\n
1. Introduction
\n
Optimum design of heat exchanger using nanotechnology is a burning field to reduce the energy consumption. Recently, application of nanosolids, nanofluids, and nanogases is the promising nanolevel research area of interest for energy savings in heat exchanger. Investigation of the nanolevel heat transfer using molecular dynamics (MD) simulation is only a new pioneer concept for the last few years [1, 2, 3, 4, 5]. Here investigations are based on atomic movement within a nanosystem during MD simulation. Experimental study on the heat transfer of nano-size devices or equipments are very time-consuming and expensive for the existing testing capabilities [6, 7, 8, 9]. Many studies have been done for enhancing the energy consumption of heat exchanger by improving thermal properties. This thermal behavior of nanocomposite can bring a huge transformation and innovation in the heat transfer. The ultrathin thermal polyurethane heat transfer material can be applied to a number of different fabrics in heat exchanger. The use of graphene (Gr) in thermoplastic polyurethane (TPU), that is, Gr/TPU nanocomposite in place of traditional material in heat exchanger, increases the heat transfer rate in a significant manner. Hussein studied to calculate thermal properties of metals and nonmetals at room temperature for applications in heat exchanger and found that metallic materials are preferably suitable for heat transfer application [10]. But there is some limitation for application of metals due to corrosion. To overcome the situation, implementation of polymer heat exchanger technology for the past decades is a pioneering innovation for heat exchanger materials [11]. The major limitation of polymer for application in heat exchanger is very low thermal conductivity. To improve that property graphene-reinforced polymer nanocomposite is a suitable candidate material in evaporation and condensation applications within heat exchanger [12]. Thermal performance of polymer nanocomposite heat exchangers mainly deals with on shell and tube heat exchangers, plate heat exchangers, finned tube heat exchangers, immersed tube heat exchangers, and hollow fiber heat exchangers [13]. Currently, thermoplastic elastomer is used in heat exchanger applications. Thermoplastics elastomer can be repeatedly softened by heating and solidified by cooling as long as the material is not thermally damaged by overheating [14]. The thermal expansion of thermoplastic polymer can be beneficial with regard to fouling because repeated expansion and contraction of the polymer channels can lead to scale detachment [15]. It is seen from previous studies that there are less number of simulation-based studies like MD simulation on enhanced heat transfer in heat exchanger materials. Detail results of MD simulation are obtained by solving Newton’s equation of motion of every atom within nanoscopic system. The basic dynamics parameters of all atoms, that is, position, velocity, and interaction force, play a vital rule during MD simulation. Nanoheat transfer problems of nanocomposites are related to thermo-mechanical properties of nanomaterials. For the design of nanodevices like nanoheat exchanger (NHE), the concepts of the nanothermal properties with temperature variation and dimension of the nanodevice is very much promising ideas. Determination of the thermal properties of nanocomposites by MD simulations is very time-consuming and a challenging task. The objective of the chapter is to characterize thermal properties like thermal conductivity and thermal expansion coefficient of graphene-reinforced polyurethane nanocomposite using molecular dynamics (MD) modeling for nanoheat exchanger material application.
\n
MD simulation consists of many parts which are mainly (a) molecular interactions, (b) molecular minimization, (c) algorithms, (d) ensemble, (e) boundary conditions, (f) atomistic stress calculation, etc. MD simulation helps to determine the position (ri) and velocity (vi) vectors of atom i with the time integration method (e.g., the velocity Verlet algorithm):
where (\na\ni) is the acceleration of atom i, t is the current time, and Δt is the time step.
\n
\n\n\na\ni\n\n=\n\n\nF\ni\n\n\nm\ni\n\n\n\nE3
\n
where m\n\ni\n is mass of atom ith and \nF\ni is the force vector of atom ith obtain from the gradient of the total potential energy (E) on atom i\n
\n
\n\n\nF\ni\n\n=\n\ndE\n\ndr\ni\n\n\n\nE4
\n
In MD simulation, a model system is built at the atomistic level with prescribed potentials (also known as the force field) acting between the atoms. The potential energy is dependent on the force field that is applied to the system. The potential energies of the system are determined from both bonded and nonbonded energies. The total potential energy combines all energetic contributions shown in the following equation:
where q1 and q2 are the charges on the interacting atoms, ε is the dielectric constant, and rij is the interatomic distance.
\n
The Condensed-phase Optimized Molecular Potentials for Atomistic Simulation Studies (COMPASS) [16] which is incorporated in both amorphous and forcite plus atomistic simulation modules in the Material Studio is used for this present study. COMPASS functional form has 11 valence terms (including diagonal and off-diagonal cross coupling terms) and 2 nonbond interaction terms (the Coulombic and Lennard-Jones functions for electrostatic and van der Waals (vdW) interactions, respectively). During all the simulations, the temperature and pressure are maintained by Andersen and Berendsen method. The calculation of nonbonded interactions is simulated by applying a cutoff distance of 12.5 Å. The spline and buffer widths are 1 and 0.5 Å, respectively.
\n
Experimental methods for prediction of the enhanced thermal properties of graphene-reinforced nanocomposites are limited because nanometer scale measurements are difficult and costly. Thus, MD simulation techniques are only an economical path to characterize nanomaterial and nanocomposites for heat exchanger material within small length and small time.
\n
\n
\n
2. MD simulation models and methods for thermal property calculation
\n
Heat can transfer through electrons and phonons, by excitations and by scattering in the nanocomposites [17]. This different ways heat transfer methods help to understand the mechanism of thermal properties enhancement in graphene-based nanocomposites. In order to study the enhanced thermal properties of graphene-reinforced thermoplastic polyurethane nanocomposites, the wide range thermal parameters of the nanocomposite material (like thermal conductivity, coefficient of the thermal expansion, glass transition temperature, etc.) have to be calculated. To calculate these parameters, different simulation models are constructed using molecular modeling software package by Materials Studio 2017. In MD simulation study, there are three types of models which are developed, namely concentrated model (CM), layer model (LM), and interfacial model (IM). Different models have been used to study different properties of the nanocomposite with respect to different parameters. In this study, we have focused mainly on layer models to characterize enhanced thermal properties. Both graphene and polyurethane models are simulated separately before constructing the graphene-reinforced thermoplastic polyurethane (Gr/TPU) nanocomposites. The condensed-phase optimized molecular potential for atomistic simulation studies (COMPASS) force field has been selected to describe the atomistic behavior for the simulation models. In MD simulation, each atom is modeled as a point mass and interacts with other atoms through force field. The position and momentum of atoms are updated based on Newton’s equation of motion.
\n
After the construction all of the graphene and TPU model using amorphous cell module within the Material Studio, build layer option is used to construct the layer models of the Gr/TPU nanocomposites. It is seen that with high weight fraction condition of graphene, nanocomposites behave likely more brittle than polymer matrix due to growth of void and chain disentanglement. So, 1% weight fraction of graphene-reinforced nanocomposites is considered in this study. The constructed nanocomposite lattice parameters with dimensions a = 22 Å, b = 22 Å, and c = 95 Å are constrained in such a way that after dynamic equilibrium process density of Gr/TPU, nanocomposite is 1.38 g/cc (nearly experimental value). MD simulation run is divided into two parts, namely equilibration run and production run. Equilibration run helps to develop the molecular structure under the condition of the desired thermodynamic state, while the production run helps to calculate different thermal parameters, namely specific heat, thermal expansion, and thermal conductivity. In equilibration run, two important conditions have to be fulfilled. One is the minimum energy stabilized condition at a prescribed temperature, and another is initial stress-free structure within periodic boundary condition. So, first steps nanocomposite models are moved through energy minimization, canonical ensemble (constant number of atoms, volume, and temperature) (NVT)) dynamic simulations, and temperature annealing cycle, respectively. The duration of the dynamic run is considered 200 ps with an integration time step of 1 fs (femto-second). This process is followed by graphene as a rigid structure so that the lattice dimension (c) in the z-direction will be changed. Temperature annealing cycle involves temperature up and down from 300 to 600 K to get the minimization of energy in the structure. The annealing time is set for 500 ps, during which the temperature is raised from 300 to 600 K with a rate of 6 K/ps and cool down to 300 K with the same rate. In the second step, the non-constrained parts (TPU) within lattices are compressed in such a way so that the final density of nanocomposite will be 1.38 g/cc (nearly experimental value) after using isothermal-isobaric (NPT) ensembles. The lowest energy structure models are fully relaxed under an isothermal-isobaric NPT ensemble (i.e., constant numbers of atoms, pressure, and temperature) at 300 K and 1 atm for 500 ps. The isothermal-isobaric (NPT) ensembles help to relax the lattices parameters and angles in order to obtain a final reasonable equilibrated structure. These steps generate various curves of various parameters such as energy, pressure, volume, and temperature versus simulation run time. These curves are very important to study thermal properties of the nanocomposite. Figure 1 shows the developed 1% Gr/TPU nanocomposite model by MD simulation.
\n
Figure 1.
Developed 1% Gr/TPU nanocomposite model.
\n
After dynamic equilibrium process density of 1% Gr/TPU, the nanocomposite is 1.38 g/cc which is shown in Figure 2.
\n
Figure 2.
Density (g/cc) versus time (ps) in dynamic equilibrium run.
\n
Heat capacity is one of the important thermal properties for the nanocomposite system. In this work, MD simulation is applied to calculate the isobaric heat capacity (Cp), and the value of Cp can be determined according to the following equation:
where KE is the kinetic energy, PE is the potential energy, P is the pressure, V is the volume, KB is the Boltzmann constant, and T is the temperature. The specific heat at constant volume (Cv) is obtained from the following equation:
where δE is the fluctuation of the energy, kB and T are Boltzmann constant, and absolute temperature, respectively.
\n
In order to study the glass transition temperature and coefficient of thermal expansion (CTE), a high-temperature annealing protocol is followed. At each temperature, the system is equilibrated by isothermal-isobaric (NPT) ensemble in MD simulation at atmospheric pressure for 500 ps. The temperature is raised up to 600 K and equilibrated for 500 ps using an NPT ensemble under atmospheric pressure and then dropped by 20 K each time until it reached 300 K. The cooling down method is applied after the heating up method by decreasing the temperature with the same settings and simulation time. Since each temperature drop is only 20 K, the structure is re-equilibrated very quickly every time its temperature is decreased. For each temperature, the volume of the simulation box V is examined over the duration time of the MD simulation, and the average value is calculated. From the volume versus temperature relationship curve, it is seen that there is a discontinuity in the volume versus temperature slope, which gives the glass transition temperature (Tg) of nanocomposite. The volume versus temperature (V-T) results are important to know two factors; first, this provides a means of determining the quality of the force field used in the simulations, and second, prediction of the volumetric glass transition temperature (Tg) [18]. The simulation result is in good agreement with experiment and demonstrates the accuracy of COMPASS force field. The volumetric coefficient of thermal expansion (VCTE) is defined by (α) [19]:
\n
\n\nα\n=\n\n1\n\nV\n0\n\n\n\nΔV\nΔT\n\n\nE16
\n
where V0 is the equilibrated system volume before the cooling simulation starts. The fractional change of volume with respect to temperature (ΔV/ΔT) is obtainable from volume versus temperature relationship curve. It is seen that change of volume with respect to temperature (ΔV/ΔT) is a different value above glass transition temperature (Tg) for graphene-reinforced thermoplastic polyurethane nanocomposite. So, volumetric coefficient of thermal expansion (VCTE) has two values due to glass transition temperature (Tg) [20]. The glass transition temperature and volumetric coefficient of thermal expansion (VCTE) of nanocomposite can also be obtained by calculating the free volume as a function of temperature, since the free volume undergoes an abrupt change when the material goes through glass transition. By probing the lattice cell with a spherical probe, using the “atom volume and surfaces” tool of the Materials Studio (MS), the free volume in the nanocomposite is calculated as a function of temperature during the annealing process as shown in Figure 3.
\n
Figure 3.
Free volume calculation using atom volume and surfaces tool in MS.
\n
Free volume is the volume that is not occupied by either the graphene or the TPU chains. The free volume fraction (FVF) can be obtained by the following equation [21]:
where Vf is the free volume and V0 is the occupied volume of the polymer chains.
\n
Thermal conductivity is the sum of the phonon contribution and the electronic contribution. Therefore, total thermal conductivity (K)
\n
\n\nK\n=\n\nK\ne\n\n+\n\nK\np\n\n\nE18
\n
where Kp and Ke are the phonon contribution and the electronic contribution to the thermal conductivity, respectively. Though, electrons contributed thermal conductivity is neglected in most graphene-reinforced nanocomposite. Thermal conductivity is an important thermal property relevant to thermal management applications. Thermal conductivity is generally calculated using equilibrium or nonequilibrium MD approaches. Equilibrium molecular dynamic (EMD) facilitates thermal conductivity prediction in all directions using one simulation, whereas nonequilibrium molecular dynamic (NEMD) requires the use of a thermal gradient and therefore only enables the calculation of thermal conductivity in one direction. The indirect method is an equilibrium molecular dynamic (EMD) method which is derived from Green-Kubo approach [22, 23], where current fluctuations are used to compute the thermal conductivity via the fluctuation-dissipation theorem [24]:
where kB is the Boltzmann constant, V and T denote the volume and temperature of the system, Jα is the heat flux in the α direction, and the angular brackets denote the ensemble average. The heat flux vector can be written as
where ri and Ei are the position and total energy of the ith atom, respectively. EMD is particularly useful for geometries where periodic boundary conditions can be applied. EMD is often computationally more expensive, and the results are more sensitive to the simulation parameters. In EMD, the system is set to the desired temperature, and then a constant energy scheme is used with the well-known Green-Kubo relations to calculate the thermal conductivity tensor. The direct method is a NEMD method in which a temperature difference is introduced into the simulation domain and the thermal conductivity is computed according to Fourier’s law as K = −J/ΔT, where J and ΔT are heat flux and temperature gradient across the system, respectively. For nonequilibrium MD (NEMD) methods, a long slab of polymer nanocomposite is constructed, and a difference in temperature is established between a heat source and a sink at the ends of the slab, and the flux is calculated. Equilibrium systems are simulated by nonequilibrium MD (NEMD) based on our homemade PERL script to compute their thermal conductivities. The nonequilibrium state can be established either by applying two thermostats at different temperatures to maintain a constant temperature at the two ends of the system or by artificially swapping atom velocities in different regions to impose a constant heat flux also known as the reverse nonequilibrium MD (RNEMD) method based on Muller-Plathe’s approach [25]. In the reverse nonequilibrium MD (RNEMD) method, the energy exchange is carried out by exchanging the kinetic energy of two particles: the hottest particle in the cold layer and the coldest particle in the hot layer. The energy E is therefore variable and needs averaging over many exchanges. In the RNEMD method, the simulation box was divided into a number of slabs in the heat flux (z) direction with the same thickness. The heat flux was generated by exchanging the kinetic energy between the highest kinetic (the hottest) atom in the heat sink and the lowest kinetic energy (the coldest) atom in the heat source. The larger momentum exchange rate in RNEMD method suggests higher energy exchange frequency between heat source and heat sink. The thermal conductivity was calculated using Fourier’s law:
\n
\n\nK\n=\n−\n\n\nJ\nq\n\n\n∇\nT\n\n\n\nE21
\n
where ∇T denoted the time-integrated temperature gradient from least squares approximation of the discrete temperatures according to the heat flow direction. The temperature of each slab was calculated using the virial theorem, and Jq is the heat flux given as
where E is the subtracted energy from the heat sink. Ac is cross-sectional area and Δt are the time step.
\n
Interface thermal conductance (ITC) is developed between graphene and polymer matrix within nanocomposite due to their weak interactions. However, the thermal conductivity (TC) of graphene-reinforced TPU nanocomposites are far below the thermal conductivity (TC) of graphene. Such a low filler efficiency is most likely attributed to the low interface thermal conductance (ITC) between graphene and polymer. The simulating method (pump-probe transient thermo-reflectance method) is used to calculate the interface thermal conductance (ITC) between the filler (graphene) and matrix (TPU). The graphene-TPU interfacial thermal resistance (R) is then calculated by following equation:
\n
\n\nR\n=\n\nΔT\n\nJ\nq\n\n\n\nE23
\n
\n
\n
3. Results and discussion
\n
In the equilibration stage within MD simulation, NPT dynamic run is carried out for 500 ps at room temperature and pressure to generate curves of energy, density, pressure, and temperature versus time. These curves are used to decide the cutoff between equilibration and production runs. It is observed that all the models are equilibrated at 50 ps, that is, no fluctuations after 50 ps. At the end of the equilibrations, the density of the nanocomposites stabilized at an average density of 1.3 g/cc with a standard deviation of 0.02 g/cc. The reason behind for the density differs from experimental value because MD simulation deals with material defect-free and impurities. It is seen in the volume versus temperature (V-T) curve during NPT dynamic run that free volume change affects the thermal properties of the nanocomposites. Further it is also observed that free volume within lattice increases according to the strain application. The glass transition mainly depends on two factors: (a) free volume and (b) the mobility of chain segments. Initially there is no graphene in TPU chains for that free volume is zero in the simulated cell and polymers are free to move within this cell volume. As a result, high values for the entropy and low values for the bulk, shear, and Young’s moduli. Further, the incorporation of the graphene into the simulated cells increases free volume and decrease the entropy and increase the values of the bulk, shear, and Young’s moduli. Since the entropy of the system is related to the free volume and the Connolly surface of the TPU nanocomposites, the prediction of these parameters may give a concept on the enhancement of the nanocomposite thermal properties. The simulation results show that the TPU matrix reinforced with graphene have a tendency to increase glass transition temperature (Tg) for stronger interlocking between the graphene and TPU molecules. The Connolly surface to volume ratio is small values for the neat TPU and increase with the increase in the graphene loading. The free volume is defined as the volume on the side of the Connolly surface without atoms. This simulation study reveals that Tg of TPU is in the range of 285–305°C and for the Gr/TPU nanocomposite is 350 K (experimental 223–323 K). From the V-T curve, volumetric coefficient of thermal expansion (α) is evaluated from the slope above and below Tg, which is between 2.6 × 10−5 and 2.4 × 10−4/K−1 (experimental 3.15 × 10−4/K−1). The free volume change rapidly when the material goes through glass transition, according to Fox and Flory’s theory of glass transition [20]. Further simulation study reveals that graphene/TPU nanocomposite thermal conductivity is 1.5 W/mK, whereas TPU thermal conductivity is 0.2 W/mK. NEMD simulations are used to calculate the thermal conductivity either by imposing a thermal gradient into the system of particles or by introducing a heat flux flow in the reverse nonequilibrium MD (RNEMD) method. In the present study, heat flux flow method is used to calculate thermal conductivity in the longitudinal direction. The total number of layers is 40. Two types of exchange method are used in the present study, namely VARIABLE and FIXED. About 1 kcal/mol energy exchange is taken in FIXED method. The number of exchanges is taken as 500 for equilibrium stage under NVT and 1000 for production stage under NVE. The time steps in between two exchanges are fixed at 100. Due to the presence of the graphene-TPU interface, there exists a temperature jump ΔT at the interface. The present study obtained values are in good agreement with previous values obtained from simulations and experimental measurements in the literatures [26, 27, 28, 29, 30, 31]. The present study contributes some novel procedures during MD simulation work which will not be done in previous researchers.
\n
\n
\n
4. Conclusions
\n
After all earlier studies, it can be concluded that the development of new technologies are giving a new attention on to investigate nanoscale phenomena (including nanoscale heat transfer). Therefore, MD simulation is the only nanoscale tool to investigate the enhancement of thermal properties of graphene-reinforced nanocomposites for heat exchanger material. Based on the current simulation results, it is found that graphene-reinforced TPU nanocomposites demonstrate higher moduli, higher glass transition temperature, and lower values of CTE than pure TPU, that is, without reinforcements. This provides useful information to understand the nanoheat transport behaviors within TPU nanocomposites for the future development of thermal nanodevice. By taking advantage of low-cost simulations to establish material designs, overall materials development costs can be dramatically reduced, and development times can be expedited.
\n
\n
Acknowledgments
\n
The authors would like to thank the organizer committee members of TEQIP which funded short-term course on mechanics of composite using Material Studio in NIT, Durgapur, and further license software support to conduct MD simulation. Thanks to nanoHUB Pro for instructions and online helps. Also thanks to Accelrys (recently BIOVIA) Materials Studio community members help for simulation study.
\n
\n
Nomenclature
\n\n\nE\n\n
total potential energy
\n\n\n\nkb\n\n\n
the stretching force constant
\n\n\n\nkθ\n\n\n
the angle-bending force constant
\n\n\n\nkφ\n\n\n
the torsional barrier
\n\n\n\nKω\n\n\n
the force constant
\n\n\n\nT\n\n
thermodynamic temperature
\n\n\n\nV\n\n
volume of nanocomposites
\n\n\n\nCp\n\n\n
specific heat of nanoparticle bulk material
\n\n\n\nk\n\n
thermal conductivity
\n\n\n\nkB\n\n\n
Boltzmann constant
\n\n\n\nKp\n\n\n
the phonon contribution
\n\n\n\nKe\n\n\n
the electronic contribution
\n\n\n\nJq\n\n\n
the heat flux
\n\n\n\nR\n\n
interfacial thermal resistance
\n\n\n\nα\n\n
the volumetric coefficient of thermal expansion
\n\n\n\nρ\n\n
density of nanomaterial
\n\n\n\nri\n\n\n
the position of the ith atom
\n\n\n\nEi\n\n\n
total energy of the ith atom
\n\n\n\n
\n',keywords:"molecular dynamics (MD) simulations, thermo-mechanical properties, glass transition temperature, coefficient of thermal expansion, thermal conductivity",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/67407.pdf",chapterXML:"https://mts.intechopen.com/source/xml/67407.xml",downloadPdfUrl:"/chapter/pdf-download/67407",previewPdfUrl:"/chapter/pdf-preview/67407",totalDownloads:193,totalViews:0,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,dateSubmitted:"January 16th 2019",dateReviewed:"April 25th 2019",datePrePublished:"October 23rd 2019",datePublished:null,dateFinished:null,readingETA:"0",abstract:"Molecular dynamics (MD) simulation-based development of heat resistance nanocomposite materials for nanoheat transfer devices (like nanoheat exchanger) and applications have been studied. In this study, MD software (Materials Studio) has been used to know the heat transport behaviors of the graphene-reinforced thermoplastic polyurethane (Gr/TPU) nanocomposite. The effect of graphene weight percentage (wt%) on thermal properties (e.g., glass transition temperature, coefficient of thermal expansion, heat capacity, thermal conductivity, and interface thermal conductance) of Gr/TPU nanocomposites has been studied. Condensed-phase optimized molecular potentials for atomistic simulation studies (COMPASS) force field which is incorporated in both amorphous and forcite plus atomistic simulation modules within the software are used for this present study. Layer models have been developed to characterize thermal properties of the Gr/TPU nanocomposites. It is seen from the simulation results that glass transition temperature (Tg) of the Gr/TPU nanocomposites is higher than that of pure TPU. MD simulation results indicate that addition of graphene into TPU matrix enhances thermal conductivity. The present study provides effective guidance and understanding of the thermal mechanism of graphene/TPU nanocomposites for improving their thermal properties. Finally, the revealed enhanced thermal properties of nanocomposites, the interfacial interaction energy, and the free volume of polymer nanocomposites are examined and discussed.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/67407",risUrl:"/chapter/ris/67407",book:{slug:"inverse-heat-conduction-and-heat-exchangers"},signatures:"Animesh Talapatra and Debasis Datta",authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. MD simulation models and methods for thermal property calculation",level:"1"},{id:"sec_3",title:"3. Results and discussion",level:"1"},{id:"sec_4",title:"4. Conclusions",level:"1"},{id:"sec_5",title:"Acknowledgments",level:"1"},{id:"sec_5",title:"Nomenclature",level:"1"}],chapterReferences:[{id:"B1",body:'\nKomarneni S. Feature article. Nanocomposites. Journal of Materials Chemistry. 1992;2:1219-1230\n'},{id:"B2",body:'\nJordan J, Jacob KI, Tannenbaum R, Sharaf MA, Jasiuk IM. Experimental trends in polymer nanocomposites—A review. Materials Science and Engineering A. 2005;393(1–2):1-11. DOI: 10.1016/j.msea.2004.09.044\n'},{id:"B3",body:'\nPodsiadlo P, Tang Z, Shim BS, Kotov NA. Counterintuitive effect of molecular strength and role of molecular rigidity on mechanical properties of layer-by-layer assembled nanocomposites. Nano Letters. 2007;7(5):1224-1231\n'},{id:"B4",body:'\nChan L, Zuo X, Wang L, Wang E, Song S, Wang J, et al. Flexible carbon nanotube-polymer composite films with high conductivity and super hydrophobicity made by solution process. Nano Letters. 2008;8(12):4454-4458\n'},{id:"B5",body:'\nLehn JM, Fendler JH, Meldrum F. The colloid chemical approach to nanostructured materials. Advanced Materials. 1995;7(7):607-632\n'},{id:"B6",body:'\nSeol JH, Jo I, Moore AL, Lindsay L, Aitken ZH, Pettes MT, et al. Two-dimensional phonon transport in supported graphene. Science. 2010;328(5975):213-216\n'},{id:"B7",body:'\nKlemens PG, Pedraza DF. Thermal conductivity of graphite in the basal plane. Carbon. 1994;32(4):735-741\n'},{id:"B8",body:'\nAllen PB, Feldman JL. Thermal conductivity of glasses: Theory and application to amorphous Si. Physical Review Letters. 1989;62(6):645\n'},{id:"B9",body:'\nAllen PB, Feldman JL, Fabian J, Wooten F. Diffusons, locons and propagons: Character of atomic vibrations’ in amorphous Si. Philosophical Magazine B. 1999;79(11–12):1715-1731\n'},{id:"B10",body:'\nHussein AM, Bakar RA, Kadirgama K, Sharma KV. Experimental measurement of nanofluids thermal properties. International Journal of Automotive and Mechanical Engineering. 2013;7:850-863\n'},{id:"B11",body:'\nCevallos JG, Bergles AE, Bar-Cohen A, Rodgers P, Gupta SK. Polymer heat exchangers-history, opportunities, and challenges. Heat Transfer Engineering. 2012;33(13):1075-1093\n'},{id:"B12",body:'\nChen H, Ginzburg VV, Yang J, Yang Y, Liu W, Huang Y, et al. Thermal conductivity of polymer-based composites: Fundamentals and applications. Progress of Polymer Science. 2016;59:41-85\n'},{id:"B13",body:'\nChen X, Su Y, Reay D, Riffat S. Recent research in polymer heat exchangers—A review. Renewable and Sustainable Energy Reviews. 2016;60:1367-1386\n'},{id:"B14",body:'\nT’Joen C, Park Y, Wang Q, Sommers A, Han X, Jacobi A. A review on polymer heat exchangers for HVAC&R applications. International Journal of Refrigeration. 2009;32:763-779\n'},{id:"B15",body:'\nZarkadas DM, Sirkar KK. Polymeric hollow fiber heat exchangers: An alternative for lower temperature applications. Industrial and Engineering Chemistry Research. 2004;43:8093-8106\n'},{id:"B16",body:'\nSun H. COMPASS: An ab initio force-field optimized for condensed-phase applications overview with details on alkane and benzene compounds. The Journal of Physical Chemistry B. 1998;102(38):7338-7364\n'},{id:"B17",body:'\nXu Z. Heat transport in low-dimensional materials: A review and perspective. Theoretical and Applied Mechanics Letters. 2016;6(3):113-121\n'},{id:"B18",body:'\nChoi J, Yu S, Yang S, Cho M. The glass transition and thermo elastic behavior of epoxy-based nanocomposites: A molecular dynamics study. Polymer. 2011;52(22):5197-5203\n'},{id:"B19",body:'\nShiu SC, Tsai JL. Characterizing thermal and mechanical properties of graphene/epoxy nanocomposites. Composites Part B: Engineering. 2014;56:691-697\n'},{id:"B20",body:'\nYang S, Qu J. Computing thermo mechanical properties of cross-linked epoxy by molecular dynamic simulations. Polymer. 2012;53(21):4806-4817\n'},{id:"B21",body:'\nSchmidtke E, Günther-Schade K, Hofmann D, Faupel F. The distribution of the unoccupied volume in glassy polymers. Journal of Molecular Graphics & Modelling. 2004;22(4):309-316\n'},{id:"B22",body:'\nGreen MS. Mark off random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids. The Journal of Chemical Physics. 1954;22(3):398-413\n'},{id:"B23",body:'\nKubo R. Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems. Journal of the Physical Society of Japan. 1957;12(6):570-586\n'},{id:"B24",body:'\nSellan DP, Landry ES, Turney JE, Mc Gaughey AJH, Amon CH. Size effects in molecular dynamics thermal conductivity predictions. Physical Review B. 2010;81:214-305\n'},{id:"B25",body:'\nMüller-Plathe F. A simple nonequilibrium molecular dynamics method for calculating the thermal conductivity. The Journal of chemical physics. 1997;106(14):6082-6085\n'},{id:"B26",body:'\nFox TG, Loshaek S. Influence of molecular weight and degree of crosslinking on the specific volume and glass temperature of polymers. Journal of Polymer Science. 1955;15(80):371-390\n'},{id:"B27",body:'\nWu SL, Shi TJ, Zhang LY. Latex co-coagulation approach to fabrication of polyurethane/graphene nanocomposites with improved electrical conductivity, thermal conductivity, and barrier property. Journal of Applied Polymer Science. 2016;133(11):13\n'},{id:"B28",body:'\nLee SH, Jung JH, Oh IK. 3D networked graphene-ferromagnetic hybrids for fast shape memory polymers with enhanced mechanical stiffness and thermal conductivity. Small. 2014;10(19):3880-3886\n'},{id:"B29",body:'\nLi A, Zhang C, Zhang YF. Thermal conductivities of PU composites with graphene aerogels reduced by different methods. Composites Part A: Applied Science and Manufacturing. 2017;103:161-167\n'},{id:"B30",body:'\nYadav SK, Cho JW. Functionalized graphene nanoplatelets for enhanced mechanical and thermal properties of polyurethane nanocomposites. Applied Surface Science. 2013;266:360-367\n'},{id:"B31",body:'\nMaterials Studio. User’s Manual, Version 1.2. San Diego, CA: Accelrys, Inc.; 2001\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Animesh Talapatra",address:"animesh_talapatra@yahoo.co.in",affiliation:'
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Marais",authors:[{id:"112229",title:"Prof.",name:"Chris",middleName:null,surname:"Aldrich",fullName:"Chris Aldrich",slug:"chris-aldrich"},{id:"112232",title:"Prof.",name:"Hansie",middleName:null,surname:"Knoetze",fullName:"Hansie Knoetze",slug:"hansie-knoetze"},{id:"135327",title:"Ms.",name:"Corne",middleName:null,surname:"Marais",fullName:"Corne Marais",slug:"corne-marais"}]},{id:"36189",title:"Optical Technologies for Determination of Pesticide Residue",slug:"optical-technology-for-determination-of-pesticide-residue",signatures:"Yankun Peng, Yongyu Li and Jingjing Chen",authors:[{id:"113343",title:"Prof.",name:"Yankun",middleName:null,surname:"Peng",fullName:"Yankun Peng",slug:"yankun-peng"},{id:"116636",title:"Dr.",name:"Yongyu",middleName:null,surname:"Li",fullName:"Yongyu Li",slug:"yongyu-li"},{id:"116637",title:"Dr.",name:"Jingjing",middleName:null,surname:"Chen",fullName:"Jingjing Chen",slug:"jingjing-chen"}]},{id:"36190",title:"High Resolution Far Infrared Spectra of the Semiconductor Alloys Obtained Using the Synchrotron Radiation as Source",slug:"high-resolution-spectra-of-semiconductor-s-alloys-obtained-using-the-far-infrared-synchrotron-radi",signatures:"E.M. 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\n
1. Introduction
\n
The notion of statistical convergence was introduced independently by Fast and Schoenberg in [1, 2], and the notion of \n\nI\n\n–convergence introduced by Kostyrko et al. in the paper [3] coresponds to the natural generalization of statistical convergence (see also [4] where \n\nI\n\n–convergence is defined by means of filter – the dual notion to ideal). These notions have been developed in several directions in [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] and have been used in various parts of mathematics, in particular in Number Theory and Ergodic Theory, for example [15, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] also in Economic Theory [29, 30] and Political Science [31]. Many authors deal with average and normal order of the well-known arithmetical functions (see [20, 21, 23, 24, 26, 28, 32, 33] and the monograph [34] for basic properties of the well-known arithmetical functions). In what follows, we shall strengthen these results from the standpoint of \n\nI\n\n–convergence of sequences, mainly by \n\n\nI\nc\n\nq\n\n\n\n–convergence and \n\n\nI\nu\n\n\n–convergence. On connection with that we can obtain a good information about behaviour and properties of the well-known arithmetical functions by investigating \n\nI\n\n–convergence of these functions or some sequences connected with these functions. Specifically in [28] by means of \n\n\nI\nd\n\n\n–convergence, there is recalled the result that normal order of \n\nΩ\n\nn\n\n\n or \n\nω\n\nn\n\n\n respectively is \n\nlog\nlog\nn\n\n. We managed to completely determine for which \n\nq\n∈\n\n0\n1\n\n\n the sequences \n\n\n\nΩ\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n and \n\n\n\nω\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n are \n\n\nI\nc\n\nq\n\n\n\n–convergent. As consequence of our results, we have that the above sequences are \n\n\nI\nd\n\n\n–convergent to \n\n1\n\n, what is equivalent that normal order of \n\nΩ\n\nn\n\n\n or \n\nω\n\nn\n\n\n respectively is \n\nlog\nlog\nn\n\n. Further in [26], there is proved that the sequence \n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\n is \n\n\nI\nd\n\n\n–convergent to \n\n0\n\n (see also [21]). We shall extend this result by means of \n\n\nI\nu\n\n\n–convergence of the sequence \n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\n. So we can get a better view of the structure of the set \n\nB\n\nε\n\n=\n\n\nn\n∈\nN\n:\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n<\nε\n\n\n\n, \n\nε\n>\n0\n\n. We also want to investigate the \n\n\nI\nc\n\nq\n\n\n\n–convergence of further arithmetical functions.
\n
\n
\n
2. Basic notions
\n
Let \n\nN\n\n be the set of positive integers. Let \n\nA\n⊆\nN\n\n. If \n\nm\n,\nn\n∈\nN\n\n, \n\nm\n≤\nn\n\n, we denote by \n\nA\n\nm\nn\n\n\n the cardinality of the set \n\nA\n∩\n\nm\nn\n\n\n. \n\nA\n\n1\nn\n\n\n is abbreviated by \n\nA\n\nn\n\n\n. We recall the concept of asymthotic, logarithmic and uniform density of the set \n\nA\n⊆\nN\n\n (see [35, 36, 37, 38]).
\n
\nDefinition 1.1.\nLet\n\n\nA\n⊆\nN\n\n.
Put \n\n\nd\nn\n\n\nA\n\n=\n\n\nA\n\nn\n\n\nn\n\n=\n\n1\nn\n\n\n∑\n\nk\n=\n1\n\nn\n\n\nχ\nA\n\n\nk\n\n\n, where \n\n\nχ\nA\n\n\n is the characteristic function of the set \n\nA\n\n. Then the numbers \n\n\nd\n¯\n\n\nA\n\n=\nlim\n\n\ninf\n\nn\n→\n∞\n\n\n\nd\nn\n\n\nA\n\n\n and \n\n\nd\n¯\n\n\nA\n\n=\n\nlimsup\n\nn\n→\n∞\n\n\n\nd\nn\n\n\nA\n\n\n are called the lower and upper asymptotic density of the set \n\nA\n\n, respectively. If there exists \n\n\nlim\n\nn\n→\n∞\n\n\n\n\nd\nn\n\n\nA\n\n\n, then \n\nd\n\nA\n\n=\n\nd\n¯\n\n\nA\n\n=\n\nd\n¯\n\n\nA\n\n\n is said to be the asymptotic density of \n\nA\n\n.
Put \n\n\nδ\nn\n\n\nA\n\n=\n\n1\n\ns\nn\n\n\n\n∑\n\nk\n=\n1\n\nn\n\n\n\n\nχ\nA\n\n\nk\n\n\nk\n\n\n, where \n\n\ns\nn\n\n=\n\n∑\n\nk\n=\n1\n\nn\n\n\n1\nk\n\n\n. Then the numbers \n\n\nδ\n¯\n\n\nA\n\n=\nlim\n\n\ninf\n\nn\n→\n∞\n\n\n\n\nδ\nn\n\n\nA\n\n\n and \n\n\nδ\n¯\n\n\nA\n\n=\nlim\n\n\nsup\n\nn\n→\n∞\n\n\n\n\nδ\nn\n\n\nA\n\n\n are called the lower and upper logarithmic density of \n\nA\n\n, respectively. Similarly, if there exists \n\n\nlim\n\nn\n→\n∞\n\n\n\n\nδ\nn\n\n\nA\n\n\n, then \n\nδ\n\nA\n\n=\n\nδ\n¯\n\n\nA\n\n=\n\nδ\n¯\n\n\nA\n\n\n is said to be the logarithmic density of \n\nA\n\n. Since \n\n\ns\nn\n\n=\nlog\nn\n+\nγ\n+\nO\n\n\n1\nk\n\n\n\n for \n\nn\n→\n∞\n\n and \n\nγ\n\n is the Euler constant, \n\n\ns\nn\n\n\n can be replaced by \n\nlog\nn\n\n in the definition of \n\n\nδ\nn\n\n\nA\n\n\n.
Put \n\n\nα\ns\n\n=\n\nmin\n\nn\n≥\n0\n\n\n\nA\n\n\nn\n+\n1\n\n\nn\n+\ns\n\n\n\n and \n\n\nα\ns\n\n=\n\nmax\n\nn\n≥\n0\n\n\n\nA\n\n\nn\n+\n1\n\n\nn\n+\ns\n\n\n\n. The following limits \n\n\nu\n¯\n\n\nA\n\n=\n\nlim\n\ns\n→\n∞\n\n\n\n\nα\ns\n\ns\n\n\n, \n\n\nu\n¯\n\n\nA\n\n=\n\nlim\n\ns\n→\n∞\n\n\n\n\nα\ns\n\ns\n\n\n exist (see [17, 37, 39, 40]) and they are called lower and upper uniform density of the set \n\nA\n\n, respectively. If \n\n\nu\n¯\n\n\nA\n\n=\n\nu\n¯\n\n\nA\n\n\n, then we denote it by \n\nu\n\nA\n\n\n and it is called the uniform density of \n\nA\n\n. It is clear that for each \n\nA\n⊆\nN\n\n we have
Further densities can be found in papers [11, 12].
\n
Let \n\nn\n=\n\np\n1\n\nα\n1\n\n\n\np\n2\n\nα\n2\n\n\n⋯\n\np\nk\n\nα\nk\n\n\n\n be the canonical representation of the integer \n\nn\n∈\nN\n\n. Recall some arithmetical functions, which belong to our interest.
\n\n\nω\n\nn\n\n\n – the number of distinct prime factors of \n\nn\n\n\n\n\n\n\nω\n\nn\n\n=\nk\n\n\n\n,
\n\n\nΩ\n\nn\n\n\n – the number of prime factors of \n\nn\n\n counted with multiplicities \n\n\n\nΩ\n\nn\n\n=\n\nα\n1\n\n+\n⋯\n+\n\nα\nk\n\n\n\n\n,
\n\n\nd\n\nn\n\n\n – the number of divisors of \n\nn\n\n\n\n\n\n\nd\n\nn\n\n=\n\n∑\n\nd\n∣\nn\n\n\n1\n\n\n\n,
define h(n) and H(n), put \n\nh\n\n1\n\n=\n1\n\n, \n\nH\n\n1\n\n=\n1\n\n and for \n\nn\n>\n1\n\n denote
\n\n\nf\n\nn\n\n=\n\n∏\n\nd\n∣\nn\n\n\nd\n\n, \n\n\nf\n∗\n\n\nn\n\n=\n\n\nf\n\nn\n\n\nn\n\n\n, where \n\nn\n=\n1\n,\n2\n,\n…\n\n,
let p be a prime number, \n\n\na\np\n\n\nn\n\n\n is defined as follows: \n\n\na\np\n\n\n1\n\n=\n0\n\n and if \n\nn\n>\n0\n\n, then \n\n\na\np\n\n\nn\n\n\n is a unique integer \n\nj\n≥\n0\n\n satisfying \n\n\np\nj\n\n∣\nn\n\n, but \n\n\np\n\nj\n+\n1\n\n\n∤\nn\n\n i.e., \n\n\np\n\n\na\np\n\n\nn\n\n\n\n∥\nn\n\n,
\n\n\nγ\n\nn\n\n\n and \n\nτ\n\nn\n\n\n – were introduced in connection with representation of natural numbers of the form \n\nn\n=\n\na\nb\n\n\n, where \n\na\n,\nb\n\n are positive integers. Let
It is clear that \n\nγ\n\nn\n\n≥\n1\n\n, because for any \n\nn\n>\n1\n\n there exist representation in the form \n\n\nn\n1\n\n\n.
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3. Ideals
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A lot of mathematical disciplines use the term small (large) set from different point of view. For instance a final set, a set having the measure zero and nowhere dense set is a small set from point of view of cardinality, measure (probability) and topology, respectively. The notion of ideal \n\nI\n⊆\n\n2\nX\n\n\n is the unifying principle how to express that a subset of \n\nX\n≠\nØ\n\n is small. We say a set \n\nA\n⊆\nX\n\n is a small set if \n\nA\n∈\nI\n\n. Recall the notion of an ideal \n\nI\n\n of subsets of \n\nN\n\n.
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Let \n\nI\n⊆\n\n2\nN\n\n\n. \n\nI\n\n is said to be an ideal in \n\nN\n\n, if \n\nI\n\n is additive (if \n\nA\n,\nB\n∈\nI\n\n then \n\nA\n∪\nB\n∈\nI\n\n) and hereditary (if \n\nA\n∈\nI\n\n and \n\nB\n⊂\nA\n\n then \n\nB\n∈\nI\n\n). An ideal \n\nI\n\n is said to be non-trivial ideal if \n\nI\n≠\nØ\n\n and \n\nN\n∉\nI\n\n. A non-trivial ideal \n\nI\n\n is said to be admissible ideal if it contains all finite subsets of \n\nN\n\n. The dual notion to the ideal is the notion filter. A non-empty family of sets \n\nF\n⊂\n\n2\nN\n\n\n is a filter if and only if \n\nØ\n∉\nF\n\n, for each \n\nA\n,\nB\n∈\nF\n\n we have \n\nA\n∩\nB\n∈\nF\n\n and for each \n\nA\n∈\nF\n\n and each \n\nB\n⊃\nA\n\n we have \n\nB\n∈\nF\n\n (for definitions see e.g. [4, 41, 42]). Let \n\nI\n\n be a proper ideal in \n\nN\n\n (i.e. \n\nN\n∉\nI\n\n). Then a family of sets \n\nF\n\nI\n\n=\n\n\nB\n⊆\nN\n:\nthere\n\nexists\n\nA\n∈\nI\n\nsuch\n\nthat\n\nB\n=\nN\n\\\nA\n\n\n\n is a filter in \n\nN\n\n, so called the associated filter with the ideal \n\nI\n\n.
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The following example shows the most commonly used admissible ideals in different areas of mathematics.
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\nExample 1.2.\n
The class of all finite subsets of \n\nN\n\n forms an admissible ideal usually denoted by \n\n\nI\nf\n\n\n.
Let \n\nϱ\n\n be a density function on \n\nN\n\n, the set \n\n\nI\nϱ\n\n=\n\n\nA\n⊂\nN\n:\nϱ\n\nA\n\n=\n0\n\n\n\n is an admissible ideal. We will use namely the ideals \n\n\nI\nd\n\n\n, \n\n\nI\nδ\n\n\n and \n\n\nI\nu\n\n\n related to asymptotic, logarithmic and uniform density, respectively.
A wide class of ideals \n\nI\n\n can be obtained by means of regular non negative matrixes \n\nT\n=\n\n\n\nt\n\nn\n,\nk\n\n\n\n\nn\n,\nk\n∈\nN\n\n\n\n (see [43]). For \n\nA\n⊂\nN\n\n, we put \n\n\nd\nT\n\nn\n\n\n\nA\n\n=\n\n∑\n\nk\n=\n1\n\n∞\n\n\nt\n\nn\n,\nk\n\n\n\nχ\nA\n\n\nk\n\n\n for \n\nn\n∈\nN\n\n. If \n\n\nlim\n\nn\n→\n∞\n\n\n\n\nd\nT\n\nn\n\n\n\nA\n\n=\n\nd\nT\n\n\nA\n\n\n exists, then \n\n\nd\nT\n\n\nA\n\n\n is called \n\nT\n\n\n–density of \n\nA\n\n (see [3, 44]). Put \n\n\nI\n\nd\nT\n\n\n=\n\n\nA\n⊂\nN\n:\n\nd\nT\n\n\nA\n\n=\n0\n\n\n\n. Then \n\n\nI\n\nd\nT\n\n\n\n is a non-trivial ideal and \n\n\nI\n\nd\nT\n\n\n\n contains both \n\n\nI\nd\n\n\n and \n\n\nI\nδ\n\n\n ideals as a special case. Indeed \n\n\nI\nd\n\n\n can be obtained by choosing \n\n\nt\n\nn\n,\nk\n\n\n=\n\n1\nn\n\n\n for \n\nk\n≤\nn\n\n, \n\n\nt\n\nn\n,\nk\n\n\n=\n0\n\n for \n\nk\n>\nn\n\n and \n\n\nI\nδ\n\n\n by choosing \n\n\nt\n\nn\n,\nk\n\n\n=\n\n\n1\nk\n\n\ns\nn\n\n\n\n for \n\nk\n≤\nn\n\n, \n\n\nt\n\nn\n,\nk\n\n\n=\n0\n\n for \n\nk\n>\nn\n\n where \n\n\ns\nn\n\n=\n\n∑\n\nk\n=\n1\n\nn\n\n\n1\nk\n\n\n for \n\nn\n∈\nN\n\n.
For the matrix \n\nT\n=\n\n\n\nt\n\nn\n,\nk\n\n\n\n\nn\n,\nk\n∈\nN\n\n\n\n, where \n\n\nt\n\nn\n,\nk\n\n\n=\n\n\nφ\n\nk\n\n\nn\n\n\n for \n\nk\n≤\nn\n\n, \n\nk\n∣\nn\n\n and \n\n\nt\n\nn\n,\nk\n\n\n=\n0\n\n otherwise we obtain \n\n\nI\nφ\n\n\n ideal of Schoenberg (see [2]), where \n\nφ\n\n is Euler function.
Another special case of \n\n\nI\n\nd\nT\n\n\n\n is the following. Take an arbitrary divergent series \n\n\n∑\n\nn\n=\n1\n\n∞\n\n\nc\nn\n\n\n, where \n\n\nc\nn\n\n>\n0\n\n for \n\nn\n∈\nN\n\n and put \n\n\nt\n\nn\n,\nk\n\n\n=\n\n\nc\nk\n\n\nS\nn\n\n\n\n for \n\nk\n≤\nn\n\n, where \n\n\nS\nn\n\n=\n\n∑\n\ni\n=\n1\n\nn\n\n\nc\ni\n\n\n, and \n\n\nt\n\nn\n,\nk\n\n\n=\n0\n\n for \n\nk\n>\nn\n\n.
Let \n\nμ\n\n be a finitely additive normed measure on a field \n\nS\n⊆\n\n2\nN\n\n\n. Suppose that \n\nS\n\n contains all singletons \n\n\nn\n\n\n, \n\nn\n∈\nN\n\n. Then the family \n\n\nI\nμ\n\n=\n\n\nA\n⊂\nN\n:\nμ\n\nA\n\n=\n0\n\n\n\n is an admissible ideal. In the case if \n\nμ\n\n is the Buck measure density (see [13, 45]), \n\n\nI\nμ\n\n\n is an admissible ideal and \n\n\nI\nμ\n\n⊊\n\nI\nd\n\n\n.
Suppose that \n\n\nμ\nn\n\n:\n\n2\nN\n\n→\n\n0\n1\n\n\n is a finitely additive normed measure for \n\nn\n∈\nN\n\n. If for \n\nA\n⊆\nN\n\n there exists \n\nμ\n\nA\n\n=\n\nlim\n\nn\n→\n∞\n\n\n\n\nμ\nn\n\n\nA\n\n\n, then the set \n\nA\n\n is said to be measurable and \n\nμ\n\nA\n\n\n is called the measure of \n\nA\n\n. Obviously \n\nμ\n\n is a finitely additive measure on some field \n\nS\n⊆\n\n2\nN\n\n\n. The family \n\n\nI\nμ\n\n=\n\n\nA\n⊂\nN\n:\nμ\n\nA\n\n=\n0\n\n\n\n is a non-trivial ideal. For \n\n\nμ\nn\n\n\n we can take for instance \n\n\nd\nn\n\n\n, \n\n\nδ\nn\n\n\n or \n\n\nd\nT\n\nn\n\n\n\n.
Let \n\nN\n=\n\n⋃\n\nj\n=\n1\n\n∞\n\n\nD\nj\n\n\n be a decomposition on \n\nN\n\n (i.e. \n\n\nD\nk\n\n∩\n\nD\nl\n\n=\nØ\n\n for \n\nk\n≠\nl\n\n). Assume that \n\n\nD\nj\n\n\n\n\n\n\n\nj\n=\n1\n\n2\n…\n\n\n are infinite sets (e.g. we can choose \n\n\nD\nj\n\n=\n\n\n\n2\n\nj\n−\n1\n\n\n⋅\n\n\n2\ns\n−\n1\n\n\n:\ns\n∈\nN\n\n\n\n for \n\nj\n=\n1\n,\n2\n,\n…\n\n). Denote \n\n\nI\nN\n\n\n the class of all \n\nA\n⊂\nN\n\n such that \n\nA\n\n intersects only a finite number of \n\n\nD\nj\n\n\n. Then \n\n\nI\nN\n\n\n is an admissible ideal.
For an \n\nq\n∈\n\n0\n1\n\n\n the set \n\n\nI\nc\n\nq\n\n\n=\n\n\nA\n⊂\nN\n:\n\n∑\n\na\n∈\nA\n\n\n\na\n\n−\nq\n\n\n<\n+\n∞\n\n\n\n is an admissible ideal (see [23]). The ideal \n\n\nI\nc\n\n1\n\n\n=\n\n\nA\n⊂\nN\n:\n\n∑\n\na\n∈\nA\n\n\n\na\n\n−\n1\n\n\n<\n+\n∞\n\n\n\n is usually denoted by \n\n\nI\nc\n\n\n. It is easy to see, that for any \n\n\nq\n1\n\n,\n\nq\n2\n\n∈\n\n0\n1\n\n\n, \n\n\nq\n1\n\n<\n\nq\n2\n\n\n we have
The fact \n\n\nI\nc\n\n⊊\n\nI\nd\n\n\n in Eq. (2) follows from the following result. Let \n\nA\n⊆\nN\n\n and \n\n\n∑\n\na\n∈\nA\n\n\n\n1\na\n\n<\n∞\n\n then \n\nd\n\nA\n\n=\n0\n\n (see [46]) thus if \n\nA\n∈\n\nI\nc\n\n\n then \n\nA\n∈\n\nI\nd\n\n\n. The opposite is not true, consider the set of primes \n\nP\n\n, for which we have \n\nd\n\nP\n\n=\n0\n\n but \n\n\n∑\n\np\n∈\nP\n\n\n\n1\np\n\n=\n∞\n\n thus \n\nP\n∈\n\nI\nd\n\n\n but \n\nP\n∉\n\nI\nc\n\n\n\n\n\n\n\n\nI\nc\n\n≠\n\nI\nd\n\n\n\n\n.
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The fact that for any \n\n\nq\n1\n\n,\n\nq\n2\n\n∈\n\n0\n1\n\n\n, \n\n\nq\n1\n\n<\n\nq\n2\n\n\n we have \n\n\nI\nc\n\n\nq\n1\n\n\n\n⊊\n\nI\nc\n\n\nq\n2\n\n\n\n\n in Eq. (2) is clear. For showing that \n\n\nI\nc\n\n\nq\n1\n\n\n\n≠\n\nI\nc\n\n\nq\n2\n\n\n\n\n it suffices to find a set \n\nH\n=\n\n\n\nh\n1\n\n<\n\nh\n2\n\n<\n⋯\n<\n\nh\nk\n\n<\n⋯\n\n\n⊂\nN\n\n such that \n\n\n∑\n\nk\n=\n1\n\n∞\n\n\nh\nk\n\n−\n\nq\n1\n\n\n\n=\n+\n∞\n\n and \n\n\n∑\n\nk\n=\n1\n\n∞\n\n\nh\nk\n\n−\n\nq\n2\n\n\n\n<\n+\n∞\n\n. Put \n\n\nh\nk\n\n=\n\n\nk\n\n1\n\nq\n1\n\n\n\n\n\n. Since \n\n\nh\n1\n\n<\n\nh\n2\n\n<\n⋯\n<\n\nh\nk\n\n<\n⋯\n\n and \n\n\nh\nk\n\nq\n1\n\n\n≤\nk\n\n we have \n\n\n∑\n\nk\n=\n1\n\n∞\n\n\nh\nk\n\n−\n\nq\n1\n\n\n\n≥\n\n∑\n\nk\n=\n1\n\n∞\n\n\nk\n\n−\n1\n\n\n=\n+\n∞\n\n. On the other side \n\n\nh\nk\n\n>\n\nk\n\n1\n\nq\n1\n\n\n\n−\n1\n≥\n\n1\n2\n\n\nk\n\n1\n\nq\n1\n\n\n\n\n for \n\nk\n≥\n2\n\n, so we obtain \n\n\n∑\n\nk\n=\n1\n\n∞\n\n\nh\nk\n\n−\n\nq\n2\n\n\n\n≤\n\n2\n\nq\n2\n\n\n\n∑\n\nk\n=\n1\n\n∞\n\n\nk\n\n−\n\n\nq\n2\n\n\nq\n1\n\n\n\n\n<\n+\n∞\n\n since \n\n\n\nq\n2\n\n\nq\n1\n\n\n>\n1\n\n.
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4. \n\nI\n\n– and \n\n\nI\n∗\n\n\n–convergence
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The notion of statistical convergence was introduced in [1, 2] and the notion of \n\nI\n\n–convergence introduced in [3] corresponds to the natural generalization of the notion of statistical convergence.
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Let us recall notions of statistical convergence, \n\nI\n\n– and \n\n\nI\n∗\n\n\n–convergence of sequence of real numbers (see [3]).
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\nDefinition 1.3. We say that a sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n is statistically convergent to a number \n\nL\n∈\nR\n\n and we write \n\nlim\n\nstat\n\n\nx\nn\n\n=\nL\n\n, provided that for each \n\nε\n>\n0\n\n we have \n\nd\n\n\nA\n\nε\n\n\n\n=\n0\n\n, where \n\nA\n\nε\n\n=\n\n\nn\n∈\nN\n:\n\n\n\nx\nn\n\n−\nL\n\n\n≥\nε\n\n\n\n.
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\nDefinition 1.4.\n
We say that a sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n is \n\nI\n\n\n–convergent to a number \n\nL\n∈\nR\n\n and we write \n\nI\n−\nlim\n\nx\nn\n\n=\nL\n\n, if for each \n\nε\n>\n0\n\n the set \n\nA\n\nε\n\n=\n\n\nn\n∈\nN\n:\n\n\n\nx\nn\n\n−\nL\n\n\n≥\nε\n\n\n\n belongs to the ideal \n\nI\n\n.
Let \n\nI\n\n be an admissible ideal on \n\nN\n\n. A sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n of real numbers is said to be \n\n\nI\n∗\n\n\n\n–convergent to \n\nL\n∈\nR\n\n, if there is a set \n\nH\n∈\nI\n\n, such that for \n\nM\n=\nN\n\\\nH\n=\n\n\n\nm\n1\n\n<\n\nm\n2\n\n<\n⋯\n<\n\nm\nk\n\n<\n⋯\n\n\n∈\nF\n\nI\n\n\n we have \n\n\nlim\n\nk\n→\n∞\n\n\n\n\nx\n\nm\nk\n\n\n=\nL\n\n, where the limit is in the usual sense.
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In the definition of usual convergence the set \n\nA\n\nε\n\n\n is finite, it means that it is small from point of view of cardinality, \n\nA\n\nε\n\n∈\n\nI\nf\n\n\n. Similarly in the definition of statistical convergence the set \n\nA\n\nε\n\n\n has asymptotic density zero, it is small from point of view of density, \n\nA\n\nε\n\n∈\n\nI\nd\n\n\n. The natural generalization of these notions is the following, let \n\nI\n\n be an admissible ideal (e.g. anyone from Example 1.2) then for each \n\nε\n>\n0\n\n we ask whether the set \n\nA\n\nε\n\n\n belongs in the ideal \n\nI\n\n. In this way we obtain the notion of the \n\nI\n\n–convergence. For the following use, we note that the concept of \n\nI\n\n–convergence can be extended for such sequences that are not defined for all \n\nn\n∈\nN\n\n, but only for “almost” all \n\nn\n∈\nN\n\n. This means that instead of a sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n we have \n\n\n\n\nx\ns\n\n\n\ns\n∈\nS\n\n\n\n, where \n\ns\n\n runs over all positive integers belonging to \n\nS\n⊆\nN\n\n and \n\nS\n∈\nF\n\nI\n\n\n.
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Remember that \n\nI\n\n–convergence in \n\nR\n\n has many properties similar to properties of the usual convergence. All notions which are used next we considered in real numbers \n\nR\n\n. The following theorem can be easily proved.
If \n\nI\n−\nlim\n\nx\nn\n\n=\nL\n\n and \n\nI\n−\nlim\n\ny\nn\n\n=\nK\n\n, then \n\nI\n−\nlim\n\n\n\nx\nn\n\n±\n\ny\nn\n\n\n\n=\nL\n±\nK\n\n.
If \n\nI\n−\nlim\n\nx\nn\n\n=\nL\n\n and \n\nI\n−\nlim\n\ny\nn\n\n=\nK\n\n, then \n\nI\n−\nlim\n\n\n\nx\nn\n\n⋅\n\ny\nn\n\n\n\n=\nL\n⋅\nK\n\n.
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The following properties are the most familiar axioms of convergence (see [47]).
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(S) Every constant sequence \n\n\nx\nx\n…\nx\n…\n\n\n converges to \n\nx\n\n.
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(H) The limit of any convergent sequence is uniquely determined.
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(F) If a sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n has the limit \n\nL\n\n, then each of its subsequences has the same limit.
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(U) If each subsequence of the sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n has a subsequence which converges to \n\nL\n\n, then \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n converges to \n\nL\n\n.
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A natural question arises which above axioms are satisfied for the concept of \n\nI\n\n–convergence.
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\nTheorem 1.6 (see [14] and Proposition 3.1 from [3], where the concept of \n\nI\n\n–convergence has been investigated in a metric space) Let\n\n\nI\n⊂\n\n2\nN\n\n\n\nbe an admissible ideal.\n
\n\n\nI\n\n–convergence satisfies (S), (H) and (U).
If \n\nI\n\n contains an infinite set, then \n\nI\n\n–convergence does not satisfy (F).
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\nTheorem 1.7 (see [3]) Let\n\n\nI\n\n\nbe an admissible ideal in\n\n\nN\n\n\n. If\n\n\n\nI\n∗\n\n−\nlim\n\nx\nn\n\n=\nL\n\n\nthen\n\n\nI\n−\nlim\n\nx\nn\n\n=\nL\n\n.
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The following example shows that the converse of Theorem 1.7 is not true.
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\nExample 1.8. Let \n\nI\n=\n\nI\nN\n\n\n be an ideal from Example 1.2 f). Define \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n as follows: For \n\nn\n∈\n\nD\nj\n\n\n we put \n\n\nx\nn\n\n=\n\n1\nj\n\n\n for \n\nj\n=\n1\n,\n2\n,\n…\n\n. Then obviously \n\nI\n−\nlim\n\nx\nn\n\n=\n0\n\n. But we show that \n\n\nI\n∗\n\n−\nlim\n\nx\nn\n\n=\n0\n\n does not hold.
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If \n\nH\n∈\nI\n\n then directly from the definition of \n\nI\n\n there exists \n\np\n∈\nN\n\n such that \n\nH\n⊆\n\nD\n1\n\n∪\n\nD\n2\n\n∪\n⋯\n∪\n\nD\np\n\n\n. But then \n\n\nD\n\np\n+\n1\n\n\n⊆\nN\n\\\nH\n=\n\n\n\nm\n1\n\n<\n\nm\n2\n\n<\n⋯\n<\n\nm\nk\n\n<\n⋯\n\n\n∈\nF\n\nI\n\n\n and so we have \n\n\nx\n\nm\nk\n\n\n=\n\n1\n\np\n+\n1\n\n\n\n for infinitely many indices \n\nk\n∈\nN\n\n. Therefore \n\n\nlim\n\nk\n→\n∞\n\n\n\nx\n\nm\nk\n\n\n=\n0\n\n cannot be true.
\n
In [3] was formulated a necessary and sufficient condition for an admissible ideal \n\nI\n\n under which \n\nI\n\n– and \n\n\nI\n∗\n\n\n–convergence to be equivalent. Recall this condition (AP) that is similar to the condition (APO) in [7, 35].
\n
\nDefinition 1.9 (see also [40]) An admissible ideal \n\nI\n⊂\n\n2\nN\n\n\n is said to satisfy the condition (AP) if for every countable family of mutually disjoint sets \n\n\n\nA\n1\n\n\nA\n2\n\n…\n\n\n belonging to \n\nI\n\n there exists a countable family of sets \n\n\n\nB\n1\n\n\nB\n2\n\n…\n\n\n such that symmetric difference \n\n\nA\nj\n\nΔ\n\nB\nj\n\n\n is finite for \n\nj\n∈\nN\n\n and \n\nB\n=\n\n∪\n\nj\n=\n1\n\n∞\n\n\nB\nj\n\n∈\nI\n\n.
\n
\nRemark. Observe that each \n\n\nB\nj\n\n\n from the previous Definition belong to \n\nI\n\n.
\n
\nTheorem 1.10 (see [14]) From\n\n\nI\n−\nlim\n\nx\nn\n\n=\nL\n\n\nthe statement\n\n\n\nI\n∗\n\n−\nlim\n\nx\nn\n\n=\nL\n\n\nfollows if and only if\n\n\nI\n\n\nsatisfies the condition (AP).\n
\n
In [44] it is proved that \n\n\nI\n\nd\nT\n\n\n\n– and \n\n\nI\n\nd\nT\n\n∗\n\n\n–convergence are equivalent in \n\nR\n\n provided that \n\nT\n=\n\n\n\nt\n\nn\n,\nk\n\n\n\n\nn\n,\nk\n∈\nN\n\n\n\n from Example 1.2 c) is a non-negative triangular matrix with \n\n\n∑\n\nk\n=\n1\n\nn\n\n\nt\n\nn\n,\nk\n\n\n=\n1\n\n for \n\nn\n∈\nN\n\n. From this we get that \n\n\nI\nd\n\n\n, \n\n\nI\nδ\n\n\n, \n\n\nI\nφ\n\n\n–convergence coinside with \n\n\nI\nd\n∗\n\n\n, \n\n\nI\nδ\n∗\n\n\n, \n\n\nI\nφ\n∗\n\n\n–convergence, respectively. On the other hand for further ideals from Example 1.2 e.g. \n\n\nI\nu\n\n\n, \n\n\nI\nN\n\n\n and \n\n\nI\nμ\n\n\n, respectively, we have that they do not fulfill the assertion that their \n\nI\n\n–convergence coincides with \n\n\nI\n∗\n\n\n–convergence. Since these ideals do not fulfill condition (AP) (see [13, 38, 40]).
\n
The following Theorem shows that also for all ideals \n\n\nI\nc\n\nq\n\n\n\n for \n\nq\n∈\n\n0\n1\n\n\n the concepts \n\nI\n\n– and \n\n\nI\n∗\n\n\n–convergence coincide.
\n
\nTheorem 1.11 (see, [20, 23]) For any\n\n\nq\n∈\n\n0\n1\n\n\n\nthe notions\n\n\n\nI\nc\n\nq\n\n\n\n\n– and\n\n\n\nI\nc\n\n\nq\n\n∗\n\n\n\n\n–convergence are equivalent.\n
\n
\nProof. It suffices to prove that for any \n\n\nI\nc\n\nq\n\n\n\n, \n\nq\n∈\n\n0\n1\n\n\n and any sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n of real numbers such that \n\n\nI\nc\n\nq\n\n\n−\nlim\n\nx\nn\n\n=\nL\n\n for \n\nq\n∈\n\n0\n1\n\n\n there exists a set \n\nM\n=\n\n\n\nm\n1\n\n<\n\nm\n2\n\n<\n⋯\n<\n\nm\nk\n\n<\n⋯\n\n\n⊆\nN\n\n such that \n\nN\n\\\nM\n∈\n\nI\nc\n\nq\n\n\n\n and \n\n\nlim\n\nk\n→\n∞\n\n\n\nx\n\nm\nk\n\n\n=\nL\n\n.
\n
For any positive integer \n\nk\n\n let \n\n\nε\nk\n\n=\n\n1\n\n2\nk\n\n\n\n and \n\n\nA\nk\n\n=\n\n\nn\n∈\nN\n:\n\n\n\nx\nn\n\n−\nL\n\n\n≥\n\n1\n\n2\nk\n\n\n\n\n\n. As \n\n\nI\nc\n\nq\n\n\n−\nlim\n\nx\nn\n\n=\nL\n\n, we have \n\n\nA\nk\n\n∈\n\nI\nc\n\nq\n\n\n\n, i.e.
Therefore there exists an infinite sequence \n\n\nn\n1\n\n<\n\nn\n2\n\n<\n⋯\n<\n\nn\nk\n\n<\n⋯\n\n of integers such that for every \n\nk\n=\n1\n,\n2\n,\n…\n\n\n
Thus \n\nH\n∈\n\nI\nc\n\nq\n\n\n\n. Put \n\nM\n=\nN\n\\\nH\n=\n\n\n\nm\n1\n\n<\n\nm\n2\n\n<\n⋯\n<\n\nm\nk\n\n<\n⋯\n\n\n\n. Now it suffices to prove that \n\n\nlim\n\nk\n→\n∞\n\n\n\nx\n\nm\nk\n\n\n=\nL\n\n. Let \n\nε\n>\n0\n\n. Choose \n\n\nk\n0\n\n∈\nN\n\n such that \n\n\n1\n\n2\n\nk\n0\n\n\n\n<\nε\n\n. Let \n\n\nm\nk\n\n>\n\nn\n\nk\n0\n\n\n\n. Then \n\n\nm\nk\n\n\n belongs to some interval \n\n\n\nn\nj\n\n\nn\n\nj\n+\n1\n\n\n\n\n where \n\nj\n≥\n\nk\n0\n\n\n and does not belong to \n\n\nA\nj\n\n\n\n\n\n\n\nj\n≥\n\nk\n0\n\n\n\n\n. Hence \n\n\nm\nk\n\n\n belongs to \n\nN\n\\\n\nA\nj\n\n\n, and then \n\n∣\n\nx\n\nm\nk\n\n\n−\nL\n∣\n<\nε\n\n for every \n\n\nm\nk\n\n>\n\nn\n\nk\n0\n\n\n\n, thus \n\n\nlim\n\nk\n→\n∞\n\n\n\nx\n\nm\nk\n\n\n=\nL\n\n.
\n
\nCorollary 1.12\nIdeals\n\n\n\nI\nc\n\nq\n\n\n\n\nfor\n\n\nq\n∈\n\n0\n1\n\n\n\nhave the property (AP).\n
\n
It is easy to prove the following lemma.
\n
\nLemma 1.13 (see [3]). If\n\n\n\nI\n1\n\n⊆\n\nI\n2\n\n\n\nthen the statement\n\n\n\nI\n1\n\n−\nlim\n\nx\nn\n\n=\nL\n\n\nimplies\n\n\n\nI\n2\n\n−\nlim\n\nx\nn\n\n=\nL\n\n.
\n
\n
\n
5. \n\nI\n\n–convergence of arithmetical functions
\n
We can obtain a good information about behaviour and properties of the well-known arithmetical functions by investigating \n\nI\n\n–convergence of these functions or some sequences connected with these functions. Recall the concept of normal order.
\n
\nDefinition 1.14. The sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n has the normal order\n\n\n\n\n\ny\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n if for every \n\nε\n>\n0\n\n and almost all (almost all in the sense of asymptotic density) values \n\nn\n\n we have \n\n\n\n1\n−\nε\n\n\n\ny\nn\n\n<\n\nx\nn\n\n<\n\n\n1\n+\nε\n\n\n\ny\nn\n\n\n.
\n
Schinzel and Šalát in [28] pointed out that one of equivalent definitions to have the normal order is as follows. The sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n has the normal order \n\n\n\n\ny\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n if and only if \n\n\nI\nd\n\n−\nlim\n\n\nx\nn\n\n\ny\nn\n\n\n=\n1\n\n. The results concerning the normal order will be formulated using the concept of statistical convergence, which coincides with \n\n\nI\nd\n\n\n–convergence. For equivalent definitions of the normal order and more examples concerning this notion see [34, 38, 48].
\n
In the papers [21, 27, 28] and in the monograph [38] there are studied various kinds of convergence of arithmetical functions which were mentioned at the beginning. The following equalities were proved in the paper [28] by using the concept of the normal order.
Let us recall one more result from [26], let p be a prime number, there was proved that the sequence \n\n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is \n\n\nI\nd\n\n\n–convergent to \n\n0\n\n. Moreover the sequence \n\n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is \n\n\nI\nc\n\nq\n\n\n\n–convergent to \n\n0\n\n for \n\nq\n=\n1\n\n and it is not \n\n\nI\nc\n\nq\n\n\n\n–convergent for all \n\nq\n∈\n\n0\n1\n\n\n, as it was shown in [21]. In [19] it was proved that this sequence is also \n\n\nI\nu\n\n\n–convergent.
\n
The following theorem shows that the assertions using the notion \n\n\nI\nu\n\n\n instead of \n\n\nI\nc\n\nq\n\n\n\n, \n\nq\n∈\n\n0\n1\n\n\n need to use a different technique for their proofs. First of all we recall a new kind of convergence so called the uniformly strong \n\nℓ\n\n–Cesàro convergence. This convergence is an analog of the notion of strong almost convergence (see [6]).
\n
\nDefinition 1.15. A sequence \n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n is said to be uniformly strong\n\n\nℓ\n\n–Cesàro convergent\n\n\n\n\n0\n<\nℓ\n<\n∞\n\n\n\n to a number \n\nL\n\n if \n\n\nlim\n\nN\n→\n∞\n\n\n\n1\nN\n\n\n∑\n\nn\n=\nk\n+\n1\n\n\nk\n+\nN\n\n\n\n\n\n\nx\ni\n\n−\nL\n\n\nℓ\n\n=\n0\n\n uniformly in \n\nk\n\n.
\n
The following Theorem shows a connection between uniformly strong \n\nℓ\n\n–Cesàro convergence and \n\n\nI\nu\n\n\n–convergence.
\n
\nTheorem 1.16 (see [6]). If\n\n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n\nis a bounded sequence, then\n\n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n\nis\n\n\n\nI\nu\n\n\n\n–convergent to\n\n\nL\n\n\nif and only if\n\n\n\n\n\nx\nn\n\n\n\nn\n=\n1\n\n∞\n\n\n\nis uniformly strong\n\n\nℓ\n\n\n–Cesàro convergent to\n\n\nL\n\n\nfor some\n\n\nℓ\n\n,\n\n0\n<\nℓ\n<\n∞\n\n.
\n
The sequence \n\n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is \n\n\nI\nu\n\n\n–convergent to zero i.e. for arbitrary \n\nε\n>\n0\n\n the set \n\nA\n\nε\n\n=\n\n\nn\n∈\nN\n:\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n≥\nε\n>\n0\n\n\n\n has uniform density equal to zero.
\n
\nTheorem 1.17 (see [19]). We have\n\n\n\nI\nu\n\n−\nlim\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n=\n0\n\n.
\n
\nProof. The sequence \n\n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is bounded. Using Theorem 1.16, it is sufficient to show that the sequence \n\n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is uniformly strong \n\nℓ\n\n–Cesàro convergent to \n\n0\n\n for \n\nℓ\n=\n1\n\n. For the reason that all members of \n\n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\nn\n=\n2\n\n∞\n\n\n are positive, we shall prove that \n\n\nlim\n\nN\n→\n∞\n\n\n\n1\nN\n\n\n∑\n\nn\n=\nk\n+\n1\n\n\nk\n+\nN\n\n\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n=\n0\n\n, uniformly in \n\nk\n\n. \n\n\na\np\n\n\nn\n\n=\nα\n\n if \n\n\np\nα\n\n∥\nn\n\n. Let \n\n\nα\n0\n\n=\n\n\n\nlog\nN\n\n\nlog\np\n\n\n\n\n. This immediately implies that \n\n\np\n\nα\n0\n\n\n≤\nN\n<\n\np\n\n\nα\n0\n\n+\n1\n\n\n\n. Then for all \n\nn\n∈\n\nk\n\nk\n+\nN\n\n\n\n we have \n\n\na\np\n\n\nn\n\n=\nα\n<\n\nα\n0\n\n\n with the possible exception of one \n\n\nn\n1\n\n∈\n\nk\n\nk\n+\nN\n\n\n\n for which we could have \n\n\na\np\n\n\n\nn\n1\n\n\n=\n\nα\n1\n\n>\n\nα\n0\n\n\n. Assume that there exist two such numbers \n\n\nn\n1\n\n,\n\nn\n2\n\n∈\n\nk\n\nk\n+\nN\n\n\n\n for which \n\n\na\np\n\n\n\nn\n1\n\n\n=\n\nα\n1\n\n>\n\nα\n0\n\n\n and \n\n\na\np\n\n\n\nn\n2\n\n\n=\n\nα\n2\n\n>\n\nα\n0\n\n\n, then \n\n\nn\n1\n\n=\n\nm\n1\n\n\np\n\nα\n1\n\n\n\n, \n\n\nn\n2\n\n=\n\nm\n2\n\n\np\n\nα\n2\n\n\n\n hence \n\n\np\n\n\nα\n0\n\n+\n1\n\n\n∣\n\nn\n1\n\n−\n\nn\n2\n\n\n. We have \n\n\np\n\n\nα\n0\n\n+\n1\n\n\n\n\n\n\n<\n∣\n\nn\n1\n\n−\n\nn\n2\n\n∣\n≤\nN\n\n, what is a contradiction with \n\n\np\n\n\nα\n0\n\n+\n1\n\n\n>\nN\n\n. When we omit such an \n\n\nn\n1\n\n\n from the sum, the error is less than \n\n\n1\nN\n\n\n\n\na\np\n\n\n\nn\n1\n\n\n\n\nlog\n\nn\n1\n\n\n\n≤\n\n1\nN\n\n\n\nα\n1\n\n\nα\n1\n\n\nlog\np\n\n. Using the Hölder’s inequality we get
Formula \n\n\n∑\n\nα\n=\n0\n\n\nα\n0\n\n\n\nα\n2\n\n=\nP\n\n\nα\n0\n\n\n\n, where \n\nP\n\nx\n\n=\n\n\nx\n\n\nx\n+\n1\n\n\n\n\n2\nx\n+\n1\n\n\n\n6\n\n\n and simple estimations give \n\n\n∑\n\nα\n=\n0\n\n\nα\n0\n\n\n\nα\n2\n\n\n1\n\np\nα\n\n\n≤\n\n∑\n\nα\n=\n0\n\n∞\n\n\n\nα\n2\n\n\np\nα\n\n\n<\n∞\n\n.
\nRemark. It is known that \n\n\nI\nu\n\n⊊\n\nI\nd\n\n\n (see e.g. [5, 6]) but the ideals \n\n\nI\nc\n\n\n and \n\n\nI\nu\n\n\n are not disjoint, and moreover \n\n\nI\nu\n\n⊈\n\nI\nc\n\n\n and \n\n\nI\nc\n\n⊈\n\nI\nu\n\n\n. For example the set of all prime numbers belongs to \n\n\nI\nu\n\n\n but not belongs to \n\n\nI\nc\n\n\n. On the other hand there exists the set \n\nB\n=\n\n∪\n\nk\n=\n1\n\n∞\n\n\nB\nk\n\n\n, where \n\n\nB\nk\n\n=\n\n\n\nk\n3\n\n+\n1\n\n\n\nk\n3\n\n+\n2\n\n…\n\n\nk\n3\n\n+\nk\n\n\n\n which not belongs to \n\n\nI\nu\n\n\n but it belongs to \n\n\nI\nc\n\n\n.
\n
Under the fact that \n\n\nI\nc\n\nq\n\n\n⊊\n\nI\nd\n\n\n for all \n\nq\n∈\n\n0\n1\n\n\n and Lemma 1.13 it is useful to investigate \n\n\nI\nc\n\nq\n\n\n\n–convergence of special sequences described in the introduction. Under the Lemma 1.13 it is clear that if there exists the \n\n\nI\nc\n\nq\n\n\n\n–limit of some sequence for any \n\nq\n∈\n\n0\n1\n\n\n, then it is equal to the \n\n\nI\nd\n\n\n–limit of the same sequence. There are no other options.
\n
Consider the sequences \n\n\n\n\n\nh\n\nn\n\n\n\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n and \n\n\n\n\n\nH\n\nn\n\n\n\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n. In [28] it was proved that these sequences are dense on \n\n\n0\n\n1\n\nlog\n2\n\n\n\n\n and moreover they both are statistically convergent to zero. The same result we have for \n\n\nI\nc\n\nq\n\n\n\n–convergence, but only for the sequence \n\n\n\n\n\nh\n\nn\n\n\n\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n for all \n\nq\n∈\n\n0\n1\n\n\n.
Then for \n\nq\n>\n\n1\nk\n\n\n, the product on the right-hand side of the previous equality converges. Thus, the series on the left-hand side of Eq. (6) converges.
\n
Let \n\nε\n>\n0\n\n. Put \n\nA\n\nε\n\n=\n\n\nn\n∈\nN\n:\n\n\nh\n\nn\n\n\n\nlog\nn\n\n\n≥\nε\n>\n0\n\n\n\n. There exists an \n\n\nn\n0\n\nk\n\n\n∈\nN\n\n for all \n\nk\n≥\n2\n\n such that for all \n\nn\n>\n\nn\n0\n\nk\n\n\n\n and \n\nn\n∈\nA\n\nε\n\n\n we have \n\nh\n\nn\n\n≥\nε\n⋅\nlog\nn\n>\nk\n\n (it is sufficient to put \n\n\nn\n0\n\nk\n\n\n=\n\n\ne\n\nk\nε\n\n\n\n\n).
\n
From this \n\nA\n\nε\n\n∩\n\n\n\nn\n0\n\nk\n\n\n+\n1\n\n\n\nn\n0\n\nk\n\n\n+\n2\n\n…\n\n⊆\n\n\nn\n∈\nN\n:\nh\n\nn\n\n≥\nk\n\n\n\n for all \n\nk\n≥\n2\n\n, \n\nk\n∈\nN\n\n.
\n
Therefore \n\n\n∑\n\nn\n∈\nA\n\nε\n\n\n\n\nn\n\n−\nq\n\n\n<\n+\n∞\n\n for all \n\nk\n≥\n2\n\n and \n\n\nI\nc\n\nq\n\n\n−\nlim\n\n\nh\n\nn\n\n\n\nlog\nn\n\n\n=\n0\n\n since the series Eq. (6) converges for all \n\nq\n>\n\n1\nk\n\n\n. If \n\nk\n→\n∞\n\n for sufficient large then \n\n\nI\nc\n\nq\n\n\n−\nlim\n\n\nh\n\nn\n\n\n\nlog\nn\n\n\n=\n0\n\n for all \n\nq\n∈\n\n0\n1\n\n\n.
For the sequence \n\n\n\n\n\nH\n\nn\n\n\n\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n we get the result of different character.
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\nTheorem 1.20 (see [20]). The sequence\n\n\n\n\n\n\nH\n\nn\n\n\n\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n\nis not\n\n\n\nI\nc\n\nq\n\n\n\n\n–convergent for every\n\n\nq\n∈\n\n0\n1\n\n\n.
\n
\nProof. In the paper [21] is proved, that the sequence \n\n\n\n\nlog\np\n⋅\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is not \n\n\nI\nc\n\nq\n\n\n\n–convergent for any \n\nq\n∈\n\n0\n1\n\n\n. The sequence \n\n\n\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is also not \n\n\nI\nc\n\nq\n\n\n\n–convergent to zero. The inequality \n\nH\n\nn\n\n≥\n\na\np\n\n\nn\n\n\n holds for all \n\nn\n=\n1\n,\n2\n,\n…\n\n and for any prime number \n\np\n\n. Then we have \n\n\n\nH\n\nn\n\n\n\nlog\nn\n\n\n≥\n\n\n\na\np\n\n\nn\n\n\n\nlog\nn\n\n\n\n for all \n\nn\n=\n2\n,\n3\n,\n…\n\n. This implies that the sequence \n\n\n\n\n\nH\n\nn\n\n\n\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is also not \n\n\nI\nc\n\nq\n\n\n\n–convergent to zero for every \n\nq\n∈\n\n0\n1\n\n\n.
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\nTheorem 1.21 (see [20]). For\n\n\nq\n=\n1\n\n\n, we obtain\n
Every non-negative integer \n\nn\n\n can be represented as \n\nn\n=\n\nab\n2\n\n\n, where \n\na\n\n is a square-free number. Hence \n\nH\n\na\n\n=\n1\n\n and
We use the inequality \n\n\nS\nk\n\n=\n\n∑\n\nj\n=\n1\n\nk\n\n\n1\nj\n\n≤\n1\n+\nlog\nk\n\n for the harmonic series. Then we have the following inequality
For any \n\nn\n∈\nN\n\n we have \n\nn\n=\n\np\n1\n\na\n1\n\n\n⋯\n\np\nk\n\na\nk\n\n\n≥\n\n2\n\nH\n\nn\n\n\n\n\n and from this \n\nH\n\nn\n\n≤\n\n\nlog\nn\n\n\nlog\n2\n\n\n\n. Therefore
We have shown that the sum in Eq. (8) is finite and therefore the sum in Eq. (7) is also finite.
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Moreover \n\nB\n∈\n\nI\nc\n\n\n and because \n\nA\n\nε\n\n⊆\nB\n\n we have \n\nA\n\nε\n\n∈\n\nI\nc\n\n\n.
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The situation for sequences \n\n\n\n\n\nω\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n, \n\n\n\n\n\nΩ\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is following.
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\nTheorem 1.22 (see [20]). The sequences\n\n\n\n\n\n\nω\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n\nand\n\n\n\n\n\n\nΩ\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n\nare not\n\n\n\nI\nc\n\nq\n\n\n\n\n–convergent for all\n\n\nq\n∈\n\n0\n1\n\n\n.
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\nProof. We prove this assertion only for \n\n\n\n\n\nω\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n. The proof for the sequence \n\n\n\n\n\nΩ\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n is analogous. Let \n\nq\n=\n1\n\n. On the basis of the Theorem 2.2 of [28] and Lemma 1.13 we can assume that \n\n\nI\nc\n\n−\nlim\n\n\nω\n\nn\n\n\n\nlog\nlog\nn\n\n\n=\n1\n\n. Take \n\nε\n∈\n\n0\n\n1\n2\n\n\n\n and consider the set
Put \n\nn\n=\np\n\n, where \n\np\n\n is a prime number, then \n\nω\n\np\n\n=\n1\n\n and \n\n\n\n\n1\n\nlog\nlog\np\n\n\n−\n1\n\n\n≥\nε\n\n holds for all prime numbers \n\np\n>\n\np\n0\n\n\n. Therefore the set \n\n\nA\nε\n\n\n contains all prime numbers greater than \n\n\np\n0\n\n\n. For these \n\np\n\n we have: \n\n\n∑\n\np\n>\n\np\n0\n\n\n\n\n1\np\n\n=\n+\n∞\n\n and so \n\nA\n\nε\n\n∉\n\nI\nc\n\n\n. From this \n\n\nI\nc\n\n−\nlim\n\n\nω\n\nn\n\n\n\nlog\nlog\nn\n\n\n≠\n1\n\n. Under the inclusion \n\n\nI\nc\n\nq\n\n\n⊊\n\nI\nc\n\n1\n\n\n≡\n\nI\nc\n\n\n and according to Lemma 1.13 we have \n\n\nI\nc\n\nq\n\n\n−\nlim\n\n\nω\n\nn\n\n\n\nlog\nlog\nn\n\n\n≠\n1\n\n for \n\nq\n∈\n\n0\n1\n\n\n. This complete the proof.
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Further possibility where the results can be strengthened by the way that the statistical convergence in them is replaced by \n\n\nI\nc\n\nq\n\n\n\n–convergence is the concept of the famous Pascal’s triangle. The \n\nn\n\n-th row of the Pascal’s triangle consists of the numbers \n\n\n\n\n\nn\n\n\n\n\n0\n\n\n\n\n,\n\n\n\n\nn\n\n\n\n\n1\n\n\n\n\n,\n…\n,\n\n\n\n\nn\n\n\n\n\nn\n−\n1\n\n\n\n\n,\n\n\n\n\nn\n\n\n\n\nn\n\n\n\n\n\n. Their sum equals to \n\n\n2\nn\n\n=\n\n\n\n1\n+\n1\n\n\nn\n\n=\n\n∑\n\nk\n=\n0\n\nn\n\n\n\n\n\nn\n\n\n\n\nk\n\n\n\n\n\n. Let \n\nΓ\n\nt\n\n\n denote the number of times the positive integer \n\nt\n\n, \n\nt\n>\n2\n\n occurs in the Pascal’s triangle. That is, \n\nΓ\n\nt\n\n\n is the number of binomial coefficients \n\n\n\n\n\nn\n\n\n\n\nk\n\n\n\n\n\n satisfying \n\n\n\n\n\nn\n\n\n\n\nk\n\n\n\n\n=\nt\n\n. From this point of view \n\nΓ\n\n is the function which maps the set \n\nN\n\n in the set \n\nN\n∪\n\n\nℵ\n0\n\n\n\n\n\n\n\n\nΓ\n\n1\n\n=\n\nℵ\n0\n\n\n\n\n. Let us observe that for every \n\nt\n∈\nN\n\n, \n\nΓ\n\nt\n\n≥\n1\n\n.
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In [32] it is proved that the average and normal order of the function \n\nΓ\n\n is \n\n2\n\n. Since the normal order is \n\n2\n\n, we have
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\n\n\nI\nd\n\n−\nlim\nΓ\n\nt\n\n=\n2\n\n
\n
(see [28]). We are going to show two results which strengthen the result of [32] and their proofs are outlined in [24].
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\nTheorem 1.23 (see [24]). \n\n\nI\nc\n\n−\nlim\nΓ\n\nt\n\n=\n2\n.\n\n\n
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\nProof. The values of the function \n\nΓ\n\n are positive integers for \n\nt\n≠\n1\n\n. Thus for \n\nε\n>\n0\n\n the set \n\n\nA\nε\n\n=\n\n\nt\n∈\nN\n:\n\n\nΓ\n\nt\n\n−\n2\n\n\n≥\nε\n\n\n\n is a subset of the set \n\nH\n=\n\n1\n\n∪\n\n2\n\n∪\nM\n\n, where \n\nM\n=\n\n\nt\n∈\nN\n:\nΓ\n\nt\n\n>\n2\n\n\n\n. Note that \n\nΓ\n\n2\n\n=\n1\n\n. Therefore is suffices to show that \n\n\n∑\n\nn\n∈\nH\n\n\n\n1\nn\n\n<\n+\n∞\n\n. Evidently this is equivalent with
In [32] it is shown that \n\nM\n\nx\n\n=\nO\n\n\nx\n\n\n\n. Therefore there exists such \n\n\nc\n1\n\n>\n0\n\n that for every \n\nk\n∈\nN\n\n, \n\nM\n\nk\n\n≤\n\nc\n1\n\n\nk\n\n\n holds. But then
According these inequalities by comparison test of the convergence of the series in Eq. (9) follows.
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Now we shall use the concept of \n\n\nI\nc\n\nq\n\n\n\n–convergence.
\n
\nTheorem 1.24 (see [24]). For every\n\n\nq\n>\n\n1\n2\n\n\n,\n\n\nI\nc\n\nq\n\n\n−\nlim\nΓ\n\nt\n\n=\n2\n\n\nholds and\n\n\n\nI\nc\n\nq\n\n\n−\nlim\nΓ\n\nt\n\n=\n2\n\n\ndoes not hold for any\n\n\nq\n\n,\n\n0\n<\nq\n≤\n\n1\n2\n\n\n.
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\nProof. Let \n\n0\n<\nq\n<\n1\n\n and let \n\nM\n\n have the same meaning as in the proof of Theorem 1.23. Let us examine the series \n\n\n∑\n\nn\n∈\nM\n\n\n\n1\n\nn\nq\n\n\n\n. We write it in the form
We have already seen, that \n\nM\n\nk\n\n≤\n\nc\n1\n\n\nk\n\n\n, \n\n\n\nk\n=\n1\n\n2\n…\n\n\n (in the proof of Theorem 1.23). Consider that every binomial coefficient \n\nt\n=\n\n\n\n\nn\n\n\n\n\n2\n\n\n\n\n\n, \n\nn\n≥\n4\n\n occurs in Pascal’s triangle at least four times \n\n\n\n\nas\n\n\n\n\n\nn\n\n\n\n\n2\n\n\n\n\n\n\n\n\n\nn\n\n\n\n\nn\n−\n2\n\n\n\n\n\n\n\n\nt\n\n\n\n\n1\n\n\n\n\n\n\n\n\nt\n\n\n\n\nt\n−\n1\n\n\n\n\n\n\n. Therefore every number of this form belongs to \n\nM\n\n. Consequently for \n\nx\n>\n4\n\n, \n\nx\n∈\nN\n\n the number \n\nM\n\nx\n\n\n is greater then or equal to the number \n\nV\n\nx\n\n\n of all numbers of the form \n\n\n\n\n\nn\n\n\n\n\n2\n\n\n\n\n\n, \n\nn\n≥\n4\n\n not exceeding \n\nx\n\n. But \n\nV\n\nx\n\n≥\ns\n−\n3\n\n, where \n\ns\n\n is the integer satisfying
Thus \n\n\n∑\n\nn\n∈\nM\n\n\n\n1\n\nn\nq\n\n\n=\n+\n∞\n\n, and so \n\n\n∑\n\nn\n∈\n\nA\nε\n\n\n\n\n1\n\nn\nq\n\n\n=\n+\n∞\n\n for every \n\nε\n>\n0\n\n.
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Similar results we can prove for functions \n\nf\n\nn\n\n\n and \n\n\nf\n∗\n\n\nn\n\n\n.
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\nTheorem 1.25 (see [20, 27]). The sequence\n\n\n\n\n\n\nlog\nlog\nf\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n\nis not\n\n\n\nI\nc\n\nq\n\n\n\n\n–convergent for all\n\n\nq\n∈\n\n0\n1\n\n\n.
\n
\nProof. According to Theorem 2.1 of [27] suppose that the
Put \n\nn\n=\np\n\n, where \n\np\n\n is a prime number, then \n\nf\n\np\n\n=\np\n\n and \n\n\n\nlog\nlog\np\n\n\nlog\nlog\np\n\n\n=\n1\n\n. Therefore the set \n\nA\n\nε\n\n\n contains all prime numbers. Next we have:
Hence \n\nA\n\nε\n\n∉\n\nI\nc\n\nq\n\n\n\n and \n\n\nI\nc\n\nq\n\n\n−\nlim\n\n\nlog\nlog\nf\n\nn\n\n\n\nlog\nlog\nn\n\n\n≠\n1\n+\nlog\n2\n\n for all \n\nq\n∈\n\n0\n1\n\n\n.
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\nTheorem 1.26 (see [20, 27]). The sequence\n\n\n\n\n\n\nlog\nlog\n\nf\n∗\n\n\nn\n\n\n\nlog\nlog\nn\n\n\n\n\nn\n=\n2\n\n∞\n\n\n\nis not\n\n\n\nI\nc\n\nq\n\n\n\n\n–convergent for all\n\n\nq\n∈\n\n0\n1\n\n\n.
\n
\nProof. According to Theorem 2.2 of [27] again suppose that the
where \n\nq\n∈\n\n0\n1\n\n\n. The proof is going similar as in the previous Theorem. Put \n\nn\n=\n\np\ni\n\n\np\nj\n\n\n, \n\ni\n≠\nj\n\n, where \n\n\np\ni\n\n\n, \n\n\np\nj\n\n\n are distinct prime numbers. Then \n\n\nf\n∗\n\n\nn\n\n=\n\nf\n∗\n\n\n\n\np\ni\n\n\np\nj\n\n\n\n=\n\n\nf\n\n\n\np\ni\n\n\np\nj\n\n\n\n\n\n\np\ni\n\n\np\nj\n\n\n\n=\n\n\n\np\ni\n\n\np\nj\n\n\n\n\np\ni\n\n\np\nj\n\n\n\n\n\n\np\ni\n\n\np\nj\n\n\n\n=\n\np\ni\n\n\np\nj\n\n\n, \n\ni\n≠\nj\n\n. Hence \n\n\n\nlog\nlog\n\nf\n∗\n\n\n\n\np\ni\n\n\np\nj\n\n\n\n\n\nlog\nlog\n\np\ni\n\n\np\nj\n\n\n\n=\n1\n\n. Let \n\nε\n∈\n\n0\n\nlog\n2\n\n\n\n and define the set
Since the series \n\n\n∑\n\nj\n=\n1\n\n∞\n\n\n1\n\n2\n\np\nj\n\n\n\n\n diverges, we have \n\nA\n\nε\n\n∉\n\nI\nc\n\nq\n\n\n\n for all \n\nq\n∈\n\n0\n1\n\n\n. Therefore \n\n\nI\nc\n\nq\n\n\n−\nlim\n\n\nlog\nlog\n\nf\n∗\n\n\nn\n\n\n\nlog\nlog\nn\n\n\n≠\n1\n+\nlog\n2\n\n and the proof is complete.
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There exists a relationship between functions \n\nf\n\nn\n\n\n and \n\nd\n\nn\n\n\n (where \n\nd\n\nn\n\n\n is the number of divisors of \n\nn\n\n). The following equality holds: \n\nlog\nf\n\nn\n\n=\n\n\nd\n\nn\n\n\n2\n\n⋅\nlog\nn\n\n, \n\n\n\nn\n>\ne\n\n\n\n (see [34]). From this we have
For a sufficiently large number \n\nk\n\n\n\n\n\n\nk\n>\n\nk\n0\n\n\n\n\n we have \n\n\n\nk\nα\n\n\n\nk\nα\n\n−\n1\n\n\n<\n2\n\n. We can estimate the series on the right-hand side of Eq. (14) with
ii. \n\n\nI\nc\n\nq\n\n\n\n–divergent for \n\nq\n∈\n\n0\n\n1\n2\n\n\n\n and \n\n\nI\nc\n\nq\n\n\n\n–convergent to \n\n1\n\n for \n\nq\n∈\n\n\n1\n2\n\n1\n\n\n.
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\nProof.\n
Let \n\nε\n>\n0\n\n. The set of numbers \n\n\n\nn\n∈\nN\n\n\nn\n>\n1\n:\n\n\nγ\n\nn\n\n−\n1\n\n\n≥\nε\n\n\n\n is a subset of \n\nH\n=\n\n\nn\n=\n\nt\ns\n\n:\nn\n∈\nN\n\n\nt\n>\n1\n\n\ns\n>\n1\n\n\n\n and \n\n\n∑\n\na\n∈\nH\n\n\n\n1\na\n\n<\n+\n∞\n\n. From the definition of \n\n\nI\nc\n\n\n–convergence Cor. 1.29 i. (Cor. is the abbreviation for Corollary) follows.
Let \n\nε\n>\n0\n\n and denote \n\n\nA\nε\n\n=\n\n\nn\n∈\nN\n:\n\n\nγ\n\nn\n\n−\n1\n\n\n≥\nε\n\n\n\n. When \n\n0\n<\nq\n≤\n\n1\n2\n\n\n then for the numbers \n\nn\n∈\nK\n\n, \n\nK\n=\n\n\n\nk\n2\n\n:\nk\n∈\nN\n\n\nk\n>\n1\n\n\n\n considering Eq. (13) for \n\nq\n=\nα\n\n holds
The convergence of the series on the right-hand side we proved previously in Theorem 1.28. Therefore \n\nγ\n\nn\n\n\n is \n\n\nI\nc\n\nq\n\n\n\n–convergent to \n\n1\n\n if \n\nq\n∈\n\n\n1\n2\n\n1\n\n\n.
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\nRemark. We have \n\nlim\n\nstat γ\n\nn\n\n=\n1\n\n.
We shall try to use a similar method to Mycielski’s proof of the convergence of \n\n\n∑\n\nn\n=\n2\n\n∞\n\n\n\nτ\n\nn\n\n−\n1\n\n\nn\nα\n\n\n\n to explain the equality Eq. (15). Since \n\n\ns\n\nk\nαs\n\n\n=\n−\n\nk\nα\n\n\nd\ndt\n\n\n\n\n1\n\nt\nαs\n\n\n\n\nt\n=\nk\n\n\n\n and \n\n\n∑\n\ns\n=\n2\n\n∞\n\n\n1\n\nt\nαs\n\n\n=\n\n1\n\n\nt\nα\n\n\n\n\nt\nα\n\n−\n1\n\n\n\n\n\n the right-hand side of Eq. (15) is equal to
Denote by \n\n\nb\nk\n\n=\n\n1\n\nk\n\n2\nα\n\n\n\n\n and consider that \n\n\nlim\n\nk\n→\n∞\n\n\n\n\na\nk\n\n\nb\nk\n\n\n=\n2\n\n. Hence the series \n\n\n∑\n\ns\n=\n2\n\n∞\n\n\na\nk\n\n\n converges (diverges) if and only if the series \n\n\n∑\n\ns\n=\n2\n\n∞\n\n\nb\nk\n\n\n converges (diverges). Since \n\n∑\n\nb\nk\n\n\n is convergent (divergent) for any \n\nα\n>\n\n1\n2\n\n\n\n\n\n\n\n0\n<\nα\n≤\n\n1\n2\n\n\n\n\n so does the series \n\n∑\n\na\nk\n\n\n and therefore the series \n\n∑\n\n\nτ\n\nn\n\n−\n1\n\n\nn\nα\n\n\n\n.
\nRemark. We have \n\nlim\n\nstat\n\nτ\n\nn\n\n=\n1\n\n.
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6. Conclusions
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It turns out that the study of \n\nI\n\n–convergence of arithmetical functions or some sequences related to these arithmetical functions for different kinds of ideals \n\nI\n\n (see [18]) gives a deeper insight into the behaviour and properties of these arithmetical functions.
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On the other hand Algebraic number theory has many deep applications in cryptology. Many basic algorithms, which are widely used, have its security due to ANT. The theory of arithmetic functions has many connections to the classical ciphers, and to the general theory as well.
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Acknowledgments
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This part was partially supported by The Slovak Research and Development Agency under the grant VEGA No. 2/0109/18.
\n
\n',keywords:"sequences, I–convergence, arithmetical functions, normal order, binomial coefficients",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/71745.pdf",chapterXML:"https://mts.intechopen.com/source/xml/71745.xml",downloadPdfUrl:"/chapter/pdf-download/71745",previewPdfUrl:"/chapter/pdf-preview/71745",totalDownloads:131,totalViews:0,totalCrossrefCites:0,dateSubmitted:"November 25th 2019",dateReviewed:"February 28th 2020",datePrePublished:"May 7th 2020",datePublished:null,dateFinished:null,readingETA:"0",abstract:"Let \n\nn\n>\n1\n\n be an integer with its canonical representation, \n\nn\n=\n\np\n1\n\nα\n1\n\n\n\np\n2\n\nα\n2\n\n\n⋯\n\np\nk\n\nα\nk\n\n\n\n. Put \n\nH\n\nn\n\n=\nmax\n\n\nα\n1\n\n…\n\nα\nk\n\n\n\n, \n\nh\n\nn\n\n=\nmin\n\n\nα\n1\n\n…\n\nα\nk\n\n\n\n, \n\nω\n\nn\n\n=\nk\n\n, \n\nΩ\n\nn\n\n=\n\nα\n1\n\n+\n⋯\n+\n\nα\nk\n\n\n, \n\nf\n\nn\n\n=\n\n∏\n\nd\n∣\nn\n\n\nd\n\n and \n\n\nf\n∗\n\n\nn\n\n=\n\n\nf\n\nn\n\n\nn\n\n\n. Many authors deal with the statistical convergence of these arithmetical functions. For instance, the notion of normal order is defined by means of statistical convergence. The statistical convergence is equivalent with \n\n\nI\nd\n\n\n–convergence, where \n\n\nI\nd\n\n\n is the ideal of all subsets of positive integers having the asymptotic density zero. In this part, we will study \n\nI\n\n–convergence of the well-known arithmetical functions, where \n\nI\n=\n\nI\nc\n\nq\n\n\n=\n\n\nA\n⊂\nN\n:\n\n∑\n\na\n∈\nA\n\n\n\na\n\n−\nq\n\n\n<\n+\n∞\n\n\n\n is an admissible ideal on \n\nN\n\n such that for \n\nq\n∈\n\n0\n1\n\n\n we have \n\n\nI\nc\n\nq\n\n\n⊊\n\nI\nd\n\n\n, thus \n\n\nI\nc\n\nq\n\n\n\n–convergence is stronger than the statistical convergence (\n\n\nI\nd\n\n\n–convergence).",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/71745",risUrl:"/chapter/ris/71745",signatures:"Vladimír Baláž and Tomáš Visnyai",book:{id:"8142",title:"Number Theory and Its Applications",subtitle:null,fullTitle:"Number Theory and Its Applications",slug:"number-theory-and-its-applications",publishedDate:"November 4th 2020",bookSignature:"Cheon Seoung Ryoo",coverURL:"https://cdn.intechopen.com/books/images_new/8142.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"230100",title:"Prof.",name:"Cheon Seoung",middleName:null,surname:"Ryoo",slug:"cheon-seoung-ryoo",fullName:"Cheon Seoung Ryoo"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Basic notions",level:"1"},{id:"sec_3",title:"3. Ideals",level:"1"},{id:"sec_4",title:"4. \n\nI\n\n– and \n\n\nI\n∗\n\n\n–convergence",level:"1"},{id:"sec_5",title:"5. \n\nI\n\n–convergence of arithmetical functions",level:"1"},{id:"sec_6",title:"6. Conclusions",level:"1"},{id:"sec_7",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'\nFast H. Sur la convergence statistique. Colloquia Mathematica. 1951;2(3–4):241-244\n'},{id:"B2",body:'\nSchoenberg IJ. The Integrability of certain functions and related Summability methods. American Mathematical Monthly. 1959;66(5):361-375\n'},{id:"B3",body:'\nKostyrko P, Wilczyński W, Šalát T. \n\nI\n\n–convergence. Real Analysis Exchange. Bratislava; 2000;26(2):669-686\n'},{id:"B4",body:'\nBourbaki N. Éléments de mathématique: Topologie générale. In: Livre III, (Russian translation) Obščaja topologija. Moskow, Nauka: Osnovnye struktury; 1968\n'},{id:"B5",body:'\nBaláž V, Strauch O, Šalát T. Remarks on several types of convergence of bounded sequences. Acta Mathematica Universitatis Ostraviensis. 2006;14(1):3-12\n'},{id:"B6",body:'\nBaláž V, Šalát T. Uniform density u and corresponding \n\n\nI\nu\n\n\n–convergence. Mathematical Communications. 2006;11(1):1-7\n'},{id:"B7",body:'\nConnor J. The statistical and strong p-Cesaro convergence of sequences. Analysis. 1988;8(1–2):47-64\n'},{id:"B8",body:'\nFridy JA. On statistical convergence. Analysis. 1985;5(4):301-314\n'},{id:"B9",body:'\nKostyrko P, Mačaj M, Šalát T, Sleziak M. \n\nI\n\n–convergence and extremal \n\nI\n\n–limit points. Mathematica Slovaca. 2005;55(4):443-464\n'},{id:"B10",body:'\nFridy J, Miller H. A matrix characterization of statistical convergence. Analysis. 1991;11(1):59-66\n'},{id:"B11",body:'\nFilipów R, Tryba J. Representation of ideal convergence as a union and intersection of matrix summability methods. Journal of Mathematical Analysis and Applications. 2020;484(2). [Accessed: 15 April 2020]\n'},{id:"B12",body:'\nGiuliano R, Grekos G. On the upper and lower exponential density functions. Mathematica Slovaca. 2017;67(5):1105-1128\n'},{id:"B13",body:'\nPaštéka M, Šalát T, Visnyai T. Remarks on Buck’s measure density and a generalization of asymptotic density. Tatra Mountains Mathematical Publications. 2005;31(2):87-101\n'},{id:"B14",body:'\nKostyrko P, Mačaj M, Šalát T. Statistical convergence and \n\nI\n\n–convergence; 2000. Available from: http://thales.doa.fmph.uniba.sk/macaj/ICON.pdf [Accessed: 20 January 2020]\n'},{id:"B15",body:'\nKostyrko P, Mačaj M, Šalát T, Strauch O. On statistical limit points. Proceedings of American Mathematical Society. 2001;129(9):2647-2654\n'},{id:"B16",body:'\nŠalát T. On statistically convergent sequences of real numbers. Mathematica Slovaca. 1980;30(2):139-150\n'},{id:"B17",body:'\nŠalát T, Visnyai T. Subadditive measures on N and the convergence of series with positive terms. Acta Math. 2003;6:43-52\n'},{id:"B18",body:'\nTóth JT, Filip F, Bukor J, Zsilinszky L. On \n\n\nI\n\n<\nq\n\n\n\n and \n\n\nI\n\n≤\nq\n\n\n\n convergence of arithmetical functions. Periodica Mathematica Hungarica. 2020 [Accessed: 24 January 2020]\n'},{id:"B19",body:'\nBaláž V. Remarks on uniform density u. In: Proceedings IAM Workshop on Institute of Information Engineering, Automation and Mathematics. Bratislava: Slovak University of Technology; 2007. pp. 43-48\n'},{id:"B20",body:'\nBaláž V, Gogola J, Visnyai T. \n\n\nI\nc\n\nq\n\n\n\n–convergence of arithmetical functions. Journal of Number Theory. 2018;183:74-83\n'},{id:"B21",body:'\nFehér Z, László B, Mačaj M, Šalát T. Remarks on arithmetical functions \n\n\na\np\n\n\nn\n\n,\nγ\n\nn\n\n,\nτ\n\nn\n\n\n. Annals of Mathematics and Informaticae. 2006;33:35-43\n'},{id:"B22",body:'\nFurstenberg H. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton: Princeton University Press; 1981\n'},{id:"B23",body:'\nGogola J, Mačaj M, Visnyai T. On \n\n\nI\nc\n\nq\n\n\n\n–convergence. Annals of Mathematics and Informaticae. 2011;38:27-36\n'},{id:"B24",body:'\nGubo Š, Mačaj M, Šalát T, Tomanová J. On binomial coefficients. Acta Math. 2003;6:33-42\n'},{id:"B25",body:'\nRenling J. Applications of nonstandard analysis in additive number theory. The Bulletin of Symbolic Logic. 2000;6(3):331-341\n'},{id:"B26",body:'\nŠalát T. On the function \n\n\np\n\n\na\np\n\n\nn\n\n\n\n‖\nn\n\n\n\nn\n>\n1\n\n\n\n. Mathematica Slovaca. 1994;44(2):143-151\n'},{id:"B27",body:'\nŠalát T, Tomanová J. On the product of divisors of a positive integer. Mathematica Slovaca. 2002;52(3):271-287\n'},{id:"B28",body:'\nSchinzel A, Šalát T. Remarks on maximum and minimum exponents in factoring. Mathematica Slovaca. 1994;44(5):505-514\n'},{id:"B29",body:'\nPetri H. Asymptotic properties of welfare relations. Economic Theory. 2019;67(4):853-874\n'},{id:"B30",body:'\nRamsey FP. A mathematical theory of saving. The Economic Journal. 1928;38(152):543-559\n'},{id:"B31",body:'\nFey M. May’s theorem with an infinite population. Social Choice and Welfare. 2004;23:275-293\n'},{id:"B32",body:'\nAbbott H, Erdös P, Hanson D. On the number of times an integer occurs as a binomial coefficient. American Mathematical Monthly. 1974;81(3):256-261\n'},{id:"B33",body:'\nMycielski J. Sur les représentations des nombres naturels par des puissances à base et exposant naturels. Colloquium Mathematicum. 1951;2(3–4):254-260\n'},{id:"B34",body:'\nHardy GH, Wright EM. An Introduction to the Theory of Numbers. 5th ed. Oxford: Clarendon Press; 1979\n'},{id:"B35",body:'\nOstmann HH. Additive Zahlentheorie I. Berlin: Springer; 1956\n'},{id:"B36",body:'\nPaštéka M. On Four Approaches to Density. Bratislava: Peter Lang AG; 2014\n'},{id:"B37",body:'\nPaštéka M. Density and Related Topics. Nakladatelství Academia: Praha; 2017\n'},{id:"B38",body:'\nStrauch O, Porubský Š. Distribution of Sequences: A Sampler. Bern, Switzerland: Peter Lang; 2005\n'},{id:"B39",body:'\nBrown T, Freedman A. Arithmetic progressions in lacunary sets. Rocky Mountain Journal of Mathematics. 1987;17(3):587-596\n'},{id:"B40",body:'\nFreedman AR, Sember JJ. Densities and summability. Pacific Journal of Mathematics. 1981;95(2):293-305\n'},{id:"B41",body:'\nKuratowski C. Topologie I. Warszawa: Panstwowe Wydawnictwa Naukowe; 1958\n'},{id:"B42",body:'\nNagata JI. Modern General Topology. 2nd ed. Amsterdam: Elsevier; 1985\n'},{id:"B43",body:'\nPetersen GM. Regular Matrix Transformations. London: McGraw-Hill; 1966\n'},{id:"B44",body:'\nMiller HI. A measure theoretical subsequence characterization of statistical convergence. Transactions of the American Mathematical Society. 1995;347(5):1811-1819\n'},{id:"B45",body:'\nBuck RC. The measure theoretic approach to density. American Journal of Mathematics. 1946;68(4):560-580\n'},{id:"B46",body:'\nPowell BJ, Šalát T. Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers. Publications de l’Institut Mathématique Nouvelle Série. 1991;50:60-70\n'},{id:"B47",body:'\nMikusiński P. Axiomatic theory of convergence. Pr Nauk Uniw ŚI Katow. 1982;12:13-21\n'},{id:"B48",body:'\nMitrinoviċ DS, Sándor J, Crstici B. Handbook of Number Theory (Mathematics and its Applications). Vol. 351. Dordrecht: Kluwer Academic Publishers; 1995\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Vladimír Baláž",address:"vladimir.balaz@stuba.sk",affiliation:'
Institute of Information Engineering, Automation, and Mathematics, Faculty of Chemical and Food Technology STU in Bratislava, Bratislava, Slovakia
Institute of Information Engineering, Automation, and Mathematics, Faculty of Chemical and Food Technology STU in Bratislava, Bratislava, Slovakia
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