Very Original Proofs of Two Famous Problems: “Are There Any Odd Perfect Numbers?” (Unsolved until to Date) and “Fermat’s Last Theorem: A New Proof of Theorem (Less than One and a Half Pages) and Its Generalization”

This article presents very original and relatively brief or very brief proofs about of two famous problems: 1) Are there any odd perfect numbers? and 2) “Fermat’s last theorem: A new proof of theorem and its generalization”. They are achieved with elementary mathematics. This is why these proofs can be easily understood by any mathematician or anyone who knows basic mathematics. Note that, in both problems, proof by contradiction was used as a method of proof. The first of the two problems to date has not been resolved. Its proof is completely original and was not based on the work of other researchers. On the contrary, it was based on a simple observation that all natural divisors of a positive integer appear in pairs. The aim of the first work is to solve one of the unsolved, for many years, problems of the mathematics which belong to the field of number theory. I believe that if the present proof is recognized by the mathematical community, it may signal a different way of solving unsolved problems. For the second problem, it is very important the fact that it is generalized to an arbitrarily large number of variables. This generalization is essentially a new theorem in the field of the number theory. To the classical problem, two solutions are given, which are presented in the chronological order in which they were achieved. Note that the second solution is very short and does not exceed one and a half pages. This leads me to believe that Fermat, as a great mathematician was not lying and that he had probably solved the problem, as he stated in his historic its letter, with a correspondingly brief solution. To win the bet on the question of whether Fermat was telling truth or lying, go immediately to the end of this article before the General Conclusions.

A three-dimensional CFD study of the hydrodynamic behavior of equal and unequal-sized in-line methane bubbles at high pressure

The hydrodynamic behavior of multiple bubbles rising upward is a field of ongoing research since various aspects of their interaction require further analysis. Shape deformation, rise velocity, and drag coefficient are some of the uncertainties to be determined in a bubble upward flow. For this study the predictions of the three-dimensional numerical simulations of the volume of fluid(VOF) CFD model were first compared with experimental results available in the literature, serving as benchmark cases. Next, 28 cases of pairs of equal and unequal-sized in-line pairs of bubbles moving upwards were simulated. The bubble size varied between 2.0–10 mm. Breakthrough of the present study is the small initial distance of 2.5 R between the center of the bubbles. To provide a more practical nature in this study material properties were selected to match methane gas and seawater properties at deepsea conditions of 15 MPa and 4 °C, thus yielding a fluid-to-bubble density ratio λ = 7.45 and viscosity ratio n = 100.46. This is one of the few studies to report results of the coalescence procedure in this context. The hydrodynamic behavior of the leading and trailing bubbles was thoroughly studied. Simulation results of the evolution of the rise velocity and the shape deformation with time indicate that the assumption that the leading bubble is rising as a free rising single one is not valid for bubbles between 2.0–7.0 mm. Finally, results of the volume of the daughter bubble exhibited an oscillating nature.

Modeling and optimization of the multichannel spark discharge

This paper reports a novel analytic model of this multichannel spark discharge, considering the delay time in the breakdown process, the electric transforming of the discharge channel from a capacitor to a resistor induced by the air breakdown, and the varying plasma resistance in the discharge process. The good agreement between the experimental and the simulated results validated the accuracy of this model. Based on this model, the influence of the circuit parameters on the maximum discharge channel number（MDCN） is investigated. Both the input voltage amplitude and the breakdown voltage threshold of each discharge channel play a critical role. With the increase of the input voltage and the decrease of the breakdown voltage, the MCDN increases almost linearly. With the increase of the discharge capacitance, the MDCN first rises and then remains almost constant. With the increase of the circuit inductance, the MDCN increases slowly but decreases quickly when the inductance increases over a certain value. There is an optimal value of the capacitor connected to the discharge channel corresponding to the MDCN. Finally, based on these results, to shorten the discharge time, a modified multichannel discharge circuit is developed and validated by the experiment. With only 6-kV input voltage, 31-channels discharge is achieved. The breakdown voltage of each electrode gap is larger than 3 kV. The modified discharge circuit is certain to be widely used in the PSJA flow control field.

Ionization wave propagation and cathode sheath formation due to surface dielectric-barrier discharge sustained in pulsed mode

This work deals with the experimental study of a surface dielectric-barrier discharge,as a part of the ongoing interest in the control of plasma induced electro-fluid dynamic effects(e.g.plasma actuators).The discharge is generated using a plasma reactor consisting of a fused silica plate which is sandwiched between two printed circuit boards where the electrodes are developed.The reactor is driven by narrow high voltage square pulses of asymmetric rising(25 ns)and falling(2.5μs)parts,while the discharge evolution is considered in a temporarily and spatially resolved manner over these pulses.That is,conventional electrical and optical emission analyzes are combined with high resolution optical emission spectroscopy and ns-resolved imaging,unveiling main characteristics of the discharge with a special focus on its propagation along the dielectric-barrier surface.The voltage rising part leads to cathode-directed ionization waves,which propagate with a speed up to 105m s~(-1).The voltage falling part leads to cathode sheath formation on the driven electrode.Τhe polarization of the dielectric barrier appears critical for the discharge dynamics.

Effect of Cilia Orientation in Metachronal Transport of Microparticles

A biomimetic approach is used to generate a directed transversal transportation of micron-sized particles in liquids based on the principle of cilia-type arrays in coordinated motion. Rows of flaps mimicking planar cilia are positioned off-centre along an array of cavities covered with membranes that support the flaps. These membranes are deflected from a concave to a convex shape and vice versa by pneumatic actuation applying positive and negative pressures (relative to the ambient) inside the cavities. As a result, the flap on top of the membrane tilts to the left or right within such a pressure cycle, performing a beat stroke. Since each cavity can be addressed in the device individually and in rapid succession, waves of coordinated flap motion can be run along the wall. Such metachronal waves are generated and transport of particles along the cilia surface is achieved in both symplectic and antiplectic direction. It is shown that the initial tilt of the flaps relative to the wall-normal determines the direction of transport.

In vitro calcification studies on bioprosthetic and decellularized heart valves under quasi-physiological flow conditions

The lifespan of biological heart valve prostheses available in the market is limited due to structural alterations caused by calcium phosphate deposits formed from blood plasma in contact with the tissues.The objective of this work is to present a comparative methodology for the investigation of the formation of calcium phosphate deposits on bioprosthetic and tissue-engineered scaffolds in vitro and the influence of mechanical forces on tissue mineralization.Based on earlier investigations on biological mineralization at constant supersaturation,a circulatory loop simulating dynamic blood flow and physiological pressure conditions was developed.The system was appropriately adapted to evaluate the calcification potential of decellularized(DCV)and glutaraldehyde-fixed(GAV)porcine aortic valves.Results indicated that DCV calcified at higher,statistically nonsignificant,rates in comparison with GAV.This difference was attributed to the tissue surface modifications and cell debris leftovers from the decellularization process.Morphological analysis of the solids deposited after 20 h by scanning electron microscopy in combination with chemical microanalysis electron-dispersive spectroscopy identified the solid formed as octacalcium phosphate(Ca8(PO4)6H2·5H2O,OCP).OCP crystallites were preferentially deposited in high mechanical stress areas of the test tissues.Moreover,GAV tissues developed a significant transvalvular pressure gradient increase past 36 h with a calcium deposition distribution similar to the one found in explanted prostheses.In conclusion,the presented in vitro circulatory model serves as a valuable prescreening methodology for the investigation of the calcification process of bioprosthetic and tissue-engineered valves under physiological mechanical load.

Improvement of Interlaminar Fracture Properties of Out of Autoclave Manufactured Carbon Fiber Reinforced Polymers Using Multi-Walled Carbon Nanotubes

The present study aims to the development of Out of Autoclave (OoA) Carbon Fiber Reinforced Polymers (CFRPs) with increased interlaminar fracture toughness by using MWCNTs. The introduction of MWCNTs into the structure of CFRPs has been succeeded by using carbon nanotube-enriched sizing agent for the pretreatment of the fiber preform using an in-house developed methodology that can be easily scaled up. The positive effect of the proposed methodology on the interlaminar fracture toughness of the CFRP laminate was assessed by the increase of Mode I and Mode II interlaminar fracture toughness of the composites. Different wt% MWCNTs concentrations were used (namely 0.5%, 1%, 1.5% and 2.5%). It was found that the nanomodified composites exhibit a significant increase of the interlaminar critical strain energy release rate GIC and GIIC of the order of 103% and 62% respectively, in the case of 1.5 wt% MWCNTs weight content. Scanning Electron Microscopy (SEM) of the fracture surfaces of CFRP samples revealed the contribution and the associated synergistic mechanisms of MWCNTs to the increase of the crack propagation resistance in the case of nano-modified CFRPs compared to the reference material.

A Meshless LBIE/LRBF Method for Solving the Nonlinear Fisher Equation: Application to Bone Healing

A simple Local Boundary Integral Equation(LBIE)method for solving the Fisher nonlinear transient diffusion equation in two dimensions(2D)is reported.The method utilizes,for its meshless implementation,randomly distributed nodal points in the interior domain and nodal points corresponding to a Boundary Element Method(BEM)mesh,at the global boundary.The interpolation of the interior and boundary potentials is accomplished using a Local Radial Basis Functions(LRBF)scheme.At the nodes of global boundary the potentials and their fluxes are treated as independent variables.On the local boundaries,potential fluxes are avoided by using the Laplacian companion solution.Potential gradients are accurately evaluated without RBFs via a LBIE,valid for gradient of potentials.Nonlinearity is treated using the Newton-Raphson scheme.The accuracy of the proposed methodology is demonstrated through representative numerical examples.Fisher equation is solved here via the LBIE/LRBF method in order to predict cell proliferation during bone healing.Cell concentrations and their gradients are numerically evaluated in a 2D model of fractured bone.The results are demonstrated and discussed.

Review of the dynamic stiffness method for free-vibration analysis of beams

The application of the dynamic stiffness method(DSM)for free-vibration analysis of beams is surveyed in this paper.The historical development of the DSM,which has taken place in several stages,is discussed in detail with reference to the free-vibration problems of beams.In particular,the suitability of the DSM in solving the free-vibration problems of beams through the application of the well-known Wittrick–Williams algorithm as a solution technique is highlighted.The literature concerning homogeneous isotropic metallic beams,for which the DSM is well established,is reviewed first,after which,with the rapid and ongoing emergence of advanced composite materials,the development of the DSM in solving the free-vibration problems of anisotropic beams is discussed.The free-vibration analysis of functionally graded beams using the DSM is also highlighted.The survey covers the DSM application for free-vibration analysis of a wide range of beams,including sandwich beams,rotating beams,twisted beams,moving beams and bending-torsion coupled beams,amongst others.Some aspects of the contributions made by the author and his research team are also highlighted.Finally,the future potential of the DSM in solving complex engineering problems is projected.