Voronezh State University

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures.

Aging is a general degenerative process related to deterioration of cell functions in the entire organism. Mitochondria, which play a key role in energy homeostasis and metabolism of reactive oxygen species (ROS), require lifetime control and constant renewal. This explains recently peaked interest in the processes of mitochondrial biogenesis and mitophagy. The principal event of mitochondrial metabolism is regulation of mitochondrial DNA (mtDNA) transcription and translation, which is a complex coordinated process that involves at least two systems of transcription factors. It is commonly believed that its major regulatory proteins are PGC-1 and PGC-1, which act as key factors connecting several regulator cascades involved in the control of mitochondrial metabolism. In recent years, the number of publications on the essential role of Nrf2/ARE signaling in the regulation of mitochondrial biogenesis has grown exponentially. Nrf2 is induced by various xenobiotics and oxidants that oxidize some Nrf2 negative regulators. Thus, ROS, in particular H 2 O 2 , were found to be strong Nrf2 activators. At present, there are two major concepts of mitochondrial biogenesis. Some authors suggest direct involvement of Nrf2 in the regulation of this process. Others believe that Nrf2 regulates expression of the antioxidant genes, while the major and only regulator of mitochondrial biogenesis is PGC-1. Several studies have demonstrated the existence of the regulatory loop involving both PGC-1 and Nrf2. In this review, we summarized recent data on the Nrf2 role in mitochondrial biogenesis and its interaction with PGC-1 in the context of extending longevity.

Measures of Noncompactness.- The Linear Theory.- The Fixed-Point Index of Condensing Operators.- Applications.

Urkowitz (1967) has discussed the detection of a deterministic signal of unknown structure in the presence of band-limited Gaussian noise. That analysis is developed to the case of a signal with random (Rayleigh, Rice, Nakagami, and other) amplitude. For such amplitude the distribution of a decision statistic of an energy detector is retrieved and expressions for detection probability are obtained.

Cardiovascular disease (CVD) is the Nation's leading killer for both men and women among all racial and ethnic groups. Development and progression of CVD is linked to the presence of risk factors such as hyperlipidemia, hypertension, obesity, and diabetes mellitus. It is known that cholesterol is an indicator of increased risk of heart attack and stroke. Low-density cholesterol (LDL) above 130 mg/dl high-density cholesterol (HDL) cholesterol below 35 mg/dl and total blood cholesterol above 200 mg/dl are indicators of problematic cholesterol. Proper ranges of cholesterol are important in the prevention of CVD. It has been suggested that a reduction in the consumption of saturated and an increase in unsaturated fatty acids is beneficial and prevents CVD. Amaranth grain contains tocotrienols and squalene compounds, which are known to affect cholesterol biosynthesis. The cholesterol precursors squalene, lanosterol and other methyl sterols, reflect cholesterol synthesis 123, whereas plant sterols and cholestanol, a metabolite of cholesterol, reflect the efficiency of cholesterol absorption in normal and hyperlipidemic populations 456. Qureshi with co-authors 7 showed that feeding of chickens with amaranth oil decreases blood cholesterol levels, which are supported by the work of others 8. Previously, we have shown that Amaranth oil modulates the cell membrane fluidity 9 and stabilized membranes that could be one reason as to why it is beneficial to those who consume it. It is known that in hypertension, the cell membrane is defective and hence, the movement of the Na and K ions across the cell membranes could defective that could contribute to the development of increase in blood pressure. Based on these properties of amaranth oil we hypothesize that it could be of significant benefit for patients with CVD.

Organic acids are synthesized in plants as a result of the incomplete oxidation of photosynthetic products and represent the stored pools of fixed carbon accumulated due to different transient times of conversion of carbon compounds in metabolic pathways. When redox level in the cell increases, e.g., in conditions of active photosynthesis, the tricarboxylic acid (TCA) cycle in mitochondria is transformed to a partial cycle supplying citrate for the synthesis of 2-oxoglutarate and glutamate (citrate valve), while malate is accumulated and participates in the redox balance in different cell compartments (via malate valve). This results in malate and citrate frequently being the most accumulated acids in plants. However, the intensity of reactions linked to the conversion of these compounds can cause preferential accumulation of other organic acids, e.g., fumarate or isocitrate, in higher concentrations than malate and citrate. The secondary reactions, associated with the central metabolic pathways, in particularly with the TCA cycle, result in accumulation of other organic acids that are derived from the intermediates of the cycle. They form the additional pools of fixed carbon and stabilize the TCA cycle. Trans-aconitate is formed from citrate or cis-aconitate, accumulation of hydroxycitrate can be linked to metabolism of 2-oxoglutarate, while 4-hydroxy-2-oxoglutarate can be formed from pyruvate and glyoxylate. Glyoxylate, a product of either glycolate oxidase or isocitrate lyase, can be converted to oxalate. Malonate is accumulated at high concentrations in legume plants. Organic acids play a role in plants in providing redox equilibrium, supporting ionic gradients on membranes, and acidification of the extracellular medium.

The scattering-based approach for calculating the ballistic conductance of open quantum systems is generalized to deal with magnetic transition metals as described by ultrasoft pseudopotentials. As an application we present quantum-mechanical conductance calculations for monatomic Co and Ni nanowires with a magnetization reversal. We find that in both Co and Ni nanowires, at the Fermi energy, the conductance of $d$ electrons is blocked by a magnetization reversal, while the $s$ states (one per spin) are perfectly transmitted. $d$ electrons have a nonvanishing transmission in a small energy window below the Fermi level. Here, transmission is larger in Ni than in Co.

INTRODUCTION . . . . . . xiii § 1. LINEAR EQUATIONS. BASIC NOTIONS . 3 § 2. EQUATIONS WITH A CLOSED OPERATOR 6 § 3. THE ADJOINT EQUATION . . . . . . 10 § 4. THE EQUATION ADJOINT TO THE FACTORED EQUATI

We present a single lane car- following model of traffic flow which is inertial and free of collisions. It demonstrates observed features of traffic flow such as existence of three regimes: free, nonhomogeneous congested (NHC) or synchronized, and homogeneous congested (HC) or jammed flow; bistability of free and NHC flow states in a range of densities, hysteresis in transitions between these states; jumps in the density-flux plane in the NHC regime; gradual spatial transition from synchronized to free flow; long survival time of jams in the HC regime. The model predicts that in the NHC regime there exist many stable states with different wavelengths, and noise can cause transitions between them.

Single ionization of He by two oppositely circularly polarized, time-delayed attosecond pulses is shown to produce photoelectron momentum distributions in the polarization plane having helical vortex structures sensitive to the time delay between the pulses, their relative phase, and their handedness. Results are obtained by both ab initio numerical solution of the two-electron time-dependent Schrödinger equation and by a lowest-order perturbation theory analysis. The energy, bandwidth, and temporal duration of attosecond pulses are ideal for observing these vortex patterns.

9R57. Non-Smooth Thermomechanics. - M Fremond (Laboratoire Lagrange, LCPC, 58 Blvd Lefebve, Paris Cedex 15, 75732, France). Springer-Verlag, Berlin. 2002. 480 pp. ISBN 3-540-66500-5. $89.95.Reviewed by MV Shitikova (Dept of Struct Mech, Voronezh State Univ of Architec and Civil Eng, ul Kirova 3-75, Voronezh, 394018, Russia).Non-smooth thermomechanical processes occurring in different solids are considered in this book consisting of 14 chapters, an appendix, list of 256 references, and a subject index. A finite number of the quantities that characterize the state of a material or of a structure, ie, the state quantities, are introduced in Chapter 1. The author deals not only with the discontinuous state quantities and their time-derivatives, but also includes into consideration different constraints and limitations imposed on the state quantities due to physical and/or geometrical properties of each particular problem. The account for these two aspects results in non-smooth equations with the corresponding non-smooth solutions. The basic mechanical relationships and constitutive laws involving the properties induced by the constraints and limitations are derived in Chapters 3–5 using the principle of virtue power, which is formulated in Chapter 2, either from the free energy or from the pseudopotential of dissipation that define a material. Deformable solids with and without interaction at a distance are considered in Chapters 6 and 7, respectively. Collision of rigid bodies is analyzed using the proposed technique in Chapter 8 and is illustrated by the problem of the collisions of three aligned balls. Evolution of two colliding deformable solids is described in Chapter 9 starting from the principle of virtue work. Chapter 10 presents the evaluation of the evolution of fibre reinforced materials under traction. Chapter 11, one of the most interesting in the opinion of this reviewer, is devoted to the analysis of solid-liquid phase change using classical ice-water phase change, the supercooling phenomena, and phase changes in porous media during soil freezing as the examples. Such problems can be of interest to engineers dealing with fall and winter maintenance and control of highways in countries with rather a long, cold season and frequent changes in temperatures, resulting in generation of so called black ice on a road surface. The damage theory founded on the principle of virtual power is formulated in Chapter 12, in so doing numerical examples related to simple concrete beams are given, which are useful for evaluating the damage state of civil engineering structures. The evolution of structures made of shape memory alloys is considered in Chapter 13 at the macroscopic level involving internal quantities describing the mixture of martensites and austenite, which can transform into one another. Such problems are also very important due to wide application of such materials in modern civil engineering structures. The last chapter, 14 presents contact problem with and without adhesion. But a reader cannot find well-defined boundaries for applicability of the approach proposed in this book. The circle of problems, wherein it works effectively, is not sharply outlined. Having read the book, this reviewer is under impression that the procedure developed is well suited to static and quasi-static problems, but needs some correction for application in boundary-value dynamic problems. By the way, dynamic problems are the least presented. One cannot find the boundary-value problems, resulting in the shock wave propagation in solids, but it is precisely these problems of non-smooth thermomechanics that are of prime interest, since they lead to non-equilibrium thermomechanics. Therefore, some doubts are cast upon the applicability of the author’s approach for solving such problems. The question arises as to whether it can be useful for determining the location of the shock wave front, its velocity, polarization, phase transitions which may occur on the wave front, the smearing of the wave front, and so on. It is not evident how to choose the optimal number of internal variables, which can help to solve successfully the problems that challenge engineers today. Since there are no general-purpose recipes for choosing the internal values, then it seems likely that the optimal choosing of the internal quantities is sort of a special skill, but one that an ordinary engineer without special training will probably not possess. This reviewer thinks that Non-Smooth Thermomechanics can be useful for individuals and students of mechanical engineering or civil engineering departments who want to improve mathematical knowledge in non-smooth thermomechanics, as well as for those who are interested in problems of heat and mass transfer.

We report vapor-cell magneto-optical trapping of Hg isotopes on the (1)S(0)-(3)P(1) intercombination transition. Six abundant isotopes, including four bosons and two fermions, were trapped. Hg is the heaviest nonradioactive atom trapped so far, which enables sensitive atomic searches for "new physics" beyond the standard model. We propose an accurate optical lattice clock based on Hg and evaluate its systematic accuracy to be better than 10;{-18}. Highly accurate and stable Hg-based clocks will provide a new avenue for the research of optical lattice clocks and the time variation of the fine-structure constant.

The hypothesis that a non-coupled alternative oxidase of plant mitochondria operates as an antioxygen defence mechanism [Purvis, A.C. and Shewfelt, R.L., Physiol. Plant. 88 (1993) 712-718; Skulachev, V.P., Biochemistry (Moscow) 59 (1994) 1433-1434] has been confirmed in experiments on isolated soybean and pea cotyledon mitochondria. It is shown that inhibitors of the alternative oxidase, salicyl hydroxamate and propyl gallate strongly stimulate H2O2 production by these mitochondria oxidizing succinate. Effective concentrations of the inhibitors proved to be the same as those decreasing the cyanide-resistant respiration. The inhibitors proved to be ineffective in stimulating H2O2 formation in rat liver mitochondria lacking the alternative oxidase.

A closed-form analytic formula for high-order harmonic generation (HHG) rates for atoms (that generalizes an HHG formula for negative ions [M. V. Frolov, J. Phys. B 42, 035601 (2009)10.1088/0953-4075/42/3/035601]) is used to study laser wavelength scaling of the HHG yield for harmonic energies in the cutoff region of the HHG plateau. We predict increases of the harmonic power for HHG by Ar, Kr, and Xe with increasing wavelength lambda over atom-specific intervals of lambda in the infrared region, lambda approximately (0.8-2.0) microm.

Describing harmonic generation (HG) in terms of a system's complex quasienergy, the harmonic power ${P}_{\ensuremath{\Delta}E}(\ensuremath{\lambda})$ (over a fixed interval, $\ensuremath{\Delta}E$, of harmonic energies) is shown to reproduce the wavelength scaling predicted recently by two groups of authors based on solutions of the time-dependent Schr\"odinger equation: ${P}_{\ensuremath{\Delta}E}(\ensuremath{\lambda})\ensuremath{\sim}{\ensuremath{\lambda}}^{\ensuremath{-}x}$, where $x\ensuremath{\approx}5--6$. Oscillations of ${P}_{\ensuremath{\Delta}E}(\ensuremath{\lambda})$ on a fine $\ensuremath{\lambda}$ scale are then shown to have a quantum origin, involving threshold phenomena within a system of interacting ionization and HG channels, and to be sensitive to the bound state wave function's symmetry.

DERIVATION OF THE EQUATIONS WELL-POSEDNESS OF THE LINEARIZED SYSTEM AND GENERAL ASYMPTOTICS Linear Well-Posedness First Results on the Time-Asymptotic Behavior ASYMPTOTIC BEHAVIOR FOR LINEARIZED ONE-DIMENSIONAL MODELS Large Time Behavior Bounded Domains The Cauchy Problem The Semi-Axis Propagation of Singularities ASYMPTOTIC BEHAVIOR FOR LINEARIZED MULTI-DIMENSIONAL MODELS Large-Time Behavior Bounded Domains The Cauchy Problem Isotropic Media Cubic media Propagation of singularities LOCAL EXISTENCE Initial Boundary Value Problems The Cauchy Problem NONLINEAR ONE-DIMENSIONAL THERMOELASTICITY Bounded Domains The Cauchy Problem The Semi-Axis Stationary Forces Blow-up of Smooth Solutions for Large Data Weak Solutions NONLINEAR MULTI-DIMENSIONAL THERMOELASTICITY Bounded Domains The Cauchy Problem Blow-Up CONTACT PROBLEMS Fully Dynamical Contact Problems Quasi-Static Contact Problems Linear Quasi-Static Problems Quasi-Static Contact Smoothing Property RELATED RESULTS Exponential Decay in the Case of Damping Asymptotic Behavior of Solutions as 1/2x1/2 (R)* Numerical Analysis APPENDIX Existence Theory for Linear Equations Existence for Linear Evolution Systems Linear Hyperbolic Systems Linear Parabolic Equations Regularity for Linear Elliptic Systems and Inequalities

Membranes are widely used in modern technology. The demand for different types of membranes and membrane processes is increasing every year. This review summarizes the current state of the art and prospects of membrane science developments including membrane materials for gas separation, pervaporation, and pressure-driven membrane processes; ion-exchange, hybrid, and track-etched membranes; membranes for electrochemical sensors; and mathematical modeling of membrane separation processes and ion and water transport in membrane systems. Studies aimed at improving the selectivity and performance of membranes and their stability are surveyed. New approaches to the synthesis and modification of membranes as well as their advanced applications are discussed.

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.

Biofilms, the communities of surface-attached bacteria embedded into extracellular matrix, are ubiquitous microbial consortia securing the effective resistance of constituent cells to environmental impacts and host immune responses. Biofilm-embedded bacteria are generally inaccessible for antimicrobials, therefore the disruption of biofilm matrix is the potent approach to eradicate microbial biofilms. We demonstrate here the destruction of Staphylococcus aureus and Staphylococcus epidermidis biofilms with Ficin, a nonspecific plant protease. The biofilm thickness decreased two-fold after 24 hours treatment with Ficin at 10 μg/ml and six-fold at 1000 μg/ml concentration. We confirmed the successful destruction of biofilm structures and the significant decrease of non-specific bacterial adhesion to the surfaces after Ficin treatment using confocal laser scanning and atomic force microscopy. Importantly, Ficin treatment enhanced the effects of antibiotics on biofilms-embedded cells via disruption of biofilm matrices. Pre-treatment with Ficin (1000 μg/ml) considerably reduced the concentrations of ciprofloxacin and bezalkonium chloride required to suppress the viable Staphylococci by 3 orders of magnitude. We also demonstrated that Ficin is not cytotoxic towards human breast adenocarcinoma cells (MCF7) and dog adipose derived stem cells. Overall, Ficin is a potent tool for staphylococcal biofilm treatment and fabrication of novel antimicrobial therapeutics for medical and veterinary applications.

Dedicated to Professor Stanislav Meshkov on the occasion of his 75th birthdayInterest in fractional calculus has quickened profoundly in the past few decades, resulting in a large body of articles devoted to this challenge, and sometimes researchers, especially young ones, who have tried to or will attempt the fractional calculus in problems of mechanics, may be hard pressed to orient themselves in such information explosion. Thus, as a result, certain findings are rediscovered, references are cited improperly or even incorrectly, priorities are placed erroneously, and some results remain concealed. In this connection, a story about two papers dealing with fractional calculus application in mechanics is rather instructive.The first paper to be under consideration is “A New Dissipation Model Based on Memory Mechanism” by the Italian researchers Caputo and Mainardi (1), which was submitted in March 5, 1971 to Pure and Applied Geophysics and was published in its December 1971 issue, while the second article “Integral Representation of ∍γ-functions and Their Application to Problems in Linear Viscoelasticity” by the Russian scholars Meshkov et al. (2) was submitted to International Journal of Engineering Science in June 5, 1970 and was published in its April 1971 issue.The story began from a letter (3) I received by e-mail in the summer of 1997 from Prof. Mainardi, wherein he complained that his paper (1), which he considered as a classical contribution to linear viscoelasticity, had been unfairly forgotten, and that is why he called Ref. 1 as a “phantom-paper.” I offered to join our forces in order to improve a historical unfairness in priorities ordering, since I was also one of the co-authors of the other phantom-paper, i.e., Ref. 2. The years of creating both papers, i.e., 1969–1970, belong to the period of so-called “Cold War,” so in those times there was no opportunity for us to know about each other. But Prof. Mainardi declined my offer.A decade has since gone by. Meanwhile, many publishing houses such as Elsevier, Springer, and Wiley have launched tremendous projects for publishing online technical articles using their hard copies published in peer-reviewed journals, including Russian academic journals, beginning from their first issues, in so doing covering the 1950s through the 1970s, and even earlier. The digital object identifier (DOI), as an identification system for intellectual property in the digital environment, has been developed by the International DOI Foundation on behalf of the publishing industry with the goals to provide a framework for managing intellectual content, link customers with publishers, facilitate electronic commerce, and enable automated copyright management.Thus, Ref. 2 has been available online since February 27, 2003 via ScienceDirect website, while Ref. 1 was published online on December 29, 2004 by SpringerLink website. Nevertheless, the situation with the citation of Refs. 12 was not dramatically changed till 2007. In January 2007 I received a new e-mail message from Prof. Mainardi (4) notifying me and all the research community dealing with fractional calculus applications that Ref. 1 has been published online, and from now on it is available via SpringerLink website for anyone who is interested. At the same time I was aware that Prof. Virginia Kiryakova, Managing Editor of Fractional Calculus and Applied Analysis (FCAA), had a project to reproduce Ref. 1 in the 10th Jubilee volume of FCAA.Since I saw little reason for reprinting the paper (1), which is available for the research audience both in hard and digital versions, I sent to Prof. Kiryakova a copy of Ref. 2 and, as an alternative, made a suggestion to write a tutorial survey (5) with a comparative analysis of Refs. 12 and their contribution to the linear viscoelasticity theory. But it went unnoticed.By the end of 2007, Prof. Kiryakova has formalized her plans. First, Ref. 1 was reprinted (6) with the Editor’s comments (7) involving the following:“Some other authors worked also on similar problems either simultaneously but independently, or years afterwards. Although they recognized the Caputo–Mainardi achievements and priorities in private discussions, correspondence and public lectures, the Caputo-Mainardi PAGEOPH-1971 paper (our Ref. 1) was often forgotten to be (or not properly) referred to…”It remains a surprising thing that the editor of the journal “closely specialized on Applied Fractional Calculus and Special Functions” (7) published in Bulgaria, has been unaware of, or has forgotten, a major contribution of Russian researchers to this field of knowledge.The next FCAA issue devoted to the 80th anniversary of Prof. Caputo and the 40th anniversary of the so-called “Caputo derivative” involves the reprint of Caputo’s paper (8) originally published in 1967 in Geophysical Journal of the Royal Astronomical Society (9) and recently online (on April 2, 2007 in Wiley InterScience website). As this takes place, the historical facts concerning the of this fractional order have been In her Prof. Kiryakova about the Caputo the of the for of fractional as applications in to linear of viscoelasticity, first in Caputo’s paper of i.e., in Ref. and the same in his paper he an for of order of the referred to as the Caputo 5, the research community in fractional calculus its since this in his first paper on the in As for the first application of this in mechanics, it was by in his “A of Linear of and Application to the Problems of made in the of Russian of Science on 29, which was published as a technical paper in In Ref. 12 the of the order for the the linear the and the for the of and the and the for the 2 was in Ref. 12 for the of a the of its two one of which is but the other to the first one to the and (2) by the this paper by was not since the Russian academic journal Applied and was since the that some authors cited Ref. 12 that the of has and this in one with the papers by as the first who the fractional 2 in the is that the fractional 1 in viscoelasticity problems years Caputo fractional was also by in while the for the to which is years Ref. was was with the by few years the same fractional order was also by with to in the published in the Caputo’s paper (9) which was in and has available for since as the volume of the in Applied and was by that 1 is the of the fractional order to and the was in mechanics of by But this researchers and was of no and this for problems of of fractional of the be to Prof. Caputo for his in this via the and its in mechanics and but not for the of Caputo of is by other researchers as be also that this of the fractional has a large this I have on to the and of all co-authors of the both papers since Ref. 2 remains in the of is a major contribution of Prof. Meshkov to the of applications of fractional calculus in problems of mechanics and the papers that are cited who was a of my and I worked with from 1967 to my a in the by was the first to write the fractional linear in 1967 in i.e., years Caputo and Mainardi which is a little for our as the article with Prof. which applications of fractional calculus to problems of linear and mechanics of it has been that of the by Russian and researchers in the field are not by their since they published in Russian journals, and But now this situation has been since of by both and are available is references DOI of papers will be as as the of their i.e., of online of of and technical research have i.e., and Engineering covering all of and from of and the that paper in was by reason of its be Nevertheless, a tremendous contribution of Russian researchers in fractional calculus viscoelasticity is the papers which in the are cited and all of have been unfairly in the historical of the recently published and technical papers but not a that has me to write this was Professor as a to the FCAA issue to the years of Caputo and to the 80th anniversary of Prof. wherein he his of on of those But I that the research community in viscoelasticity aware of the and now it is my to some both papers in 1971 the same time and simultaneously had gone in the i.e., they have been unfairly The the findings of articles have been many times by researchers, and few know or have been in the Refs. 12 are devoted to the same it is rather to the comparative analysis of their results and applications it be that one and the same in both papers in the two in Ref. in the of the of the linear by the with of order one those Caputo and Mainardi unaware that Meshkov had this years but from Refs. it is that this has not been the while in Ref. 2, in the of the with the fractional as a of The was by in and in who had the as a and its which was by as the fractional the same as the of the is the fractional and is the or the was the first to in about the of two of the for the fractional linear he considered fractional as some and and to the mechanics of the of fractional in the his the via fractional in the fractional Caputo’s paper (9) and the fractional and in Meshkov the fractional and fractional linear be that was the first to in a of fractional the of a by its on with the and Meshkov the of the fractional while Caputo and the Caputo it is that the two of a linear and as Mainardi as from the past a of authors have or the fractional calculus as an of the of is to the of mechanics developed by that fractional my the of the two or application of fractional calculus in viscoelasticity is not and the of the is that they are and the same for all of the under consideration will be in by the results of Refs. Thus, Meshkov the of the of a linear with fractional and the with the fractional as a of The in the two is in in the article of the first researchers who recognized that of mechanics is to that the be to the fractional of the was in he with his via of second in and from the fractional 2 in Refs. and about Refs. the fractional in In so authors the the a fractional and the on the of linear viscoelasticity via the was in Refs. for the fractional of two as the or of fractional in viscoelasticity, it is to the of the via with of and in of fractional as the first and second the the was developed and in problems of mechanics in while the was in and the of in those and of one and the two papers under consideration i.e., Refs. are the Caputo and Mainardi (1), of the second from the Meshkov et al. a of the first and is the is the and are the and and are the of or the and of or the of the and are the and of the is the time of the fractional order and is the fractional which is to the for in Refs. 12 the authors the for the same and for the of the are for of from both in Ref. the and the in of the which not in Ref. in Ref. 2, the and the of the and the are as the the the and to and and the of the of the of which have the and not in Ref. but the which is to the the of to the fractional was in Ref. 2. that the for the fractional linear was by Meshkov in of the and time the was in both their for of the fractional 2 in Ref. 1 is with 1 in Ref. 2, it is that they little from each the of of the which has no has been in Ref. 2 the of the findings of the two papers the the two developed in Refs. and (2) that the in the both articles is to that in his had the in the involving the fractional as is the of the fractional order is to that the be also in the are in and have the the which is by the involving and the of the fractional linear by in in the and are the in Refs. and Ref. the some applications in both Thus, in Ref. the of the was for the and the with the available for and the and a was In Ref. 2, of an and in a using the will not on their in a was and by Meshkov and in with the of the of the the of the fractional for the is to the is the of the and is the has been that the of the as the to and to a some our paper on this published in is cited sometimes but not Ref. which is available in the of our the fractional calculus linear has been in the of in Ref. and in Ref. that and in considered the for the is the fractional it is to that from the of not belong to the of the it is the linear the But rather of the fractional order in the by in Caputo (9) in and are some using the fractional and of and be in Refs. us the research on the the of which are by the fractional linear In it be that the on the fractional was for the first time in Ref. in wherein the fractional was but this remains in the of the of the from the that the results in Ref. 2 years by and and are referred to authors for in Ref. as as Ref. has been the in Ref. of all the of of a was in the is the is the and is the was in the the via the in was As a result, the was in the in Ref. the be all for the the of the has been that two and the in Ref. has been for their The of the in the as of the for of the fractional are in in Ref. the of as the was in the with and in Ref. and are by a of of the the of the and of the involving the of the referred the of the of the system as is one to the that not in by which are to the fractional calculus linear it was a to in in Ref. that results in by and the authors of Ref. our on the that they considered a is from and the of two for this have to since the similar is the about the of on the fractional which was in Ref. in As this takes place, the was in the of the the two in the as of the is in for of 2 in Ref. may in the authors of Ref. know about Refs. but a few years the of Ref. in wherein Refs. and in et al. their for the of a as it was in Ref. that the authors of Refs. that the of the which in Ref. years the first in Ref. the historical of the about 12 it may be that the paper by Caputo and Mainardi has received recently its as Ref. with the that it is in Ref. contribution of this paper is of one of the first a of the those times to of and this was and Prof. Mainardi is of our paper (2) has not been reprinted our it is since it is for an to it via the ScienceDirect but there are to be of the that it was by Prof. one of the of the the the my co-authors of Ref. 2 in those I had the of the with fractional and the with by its in I not to a is or of is rather a of But one is and the two in of the each other and in the historical be considered as classical papers in that field of which is with the application of fractional calculus in linear viscoelasticity Refs. and the of the I to research in this field two and the of the of a a was in my research to in linear and as as and to of major results have been in a of papers in the in there was an of in fractional calculus viscoelasticity and it the of in of on the dealing with fractional received a by a and Russian academic such papers as while it was in to a paper to journal to an article was those is one In Prof. Mainardi has in Ref. about the same and he had in those times in which to to other of research as and in fractional calculus was a the in Ref. was in that I went to the applications of Fractional Calculus I aware of the or on the application of fractional order by the of similar has to a of years my in with fractional was to rather In January for the first the on Applied in to the from the International Science The period of the had but it was to all the for such a from to from a Russian a But all problems I had to as with the of an of this I a of from the and I had by their by one of the by Prof. who the of Applied two years I had the by the scholars from concerning of a fractional with so-called i.e., fractional of the order of its it that and about the of Russian in the field they even had no of the the of fractional and of in the and and with they me that they had made a for and aware of all papers in the field their I that a had for a to be with of Russian in the applications of fractional calculus in problems of the mechanics of and, in with the contribution of the of by the which was published in his two as as in technical papers of his not the articles published in Russian as they be in Ref. was a for the of our first article including references devoted to applications of fractional calculus to problems of linear and mechanics of the of references in Ref. is from and some papers in the field published by have been be consideration in Ref. was under had no to journal via which is a and in Russian the in that there was a for journal to I to the opportunity to the of Ref. who sent us some papers and some I will this some through for papers on the application of fractional order in mechanics in order to has been in this field in the past I with for the of the in this in there was the of in the research of fractional of viscoelasticity the second and especially in their application for problems of In the in the of and et al. and et al. the of has been using and but from the references cited in Refs. it was that their authors had about the contribution of Russian researchers in the The papers of and and in the have the of the of the of fractional the a fractional calculus application in mechanics began to in and my the Prof. in and Prof. in As for the on the application of fractional in mechanics, and other of and it that there is not a a research dealing with this in one or the of Ref. in which was our second paper in this journal the first one was published in on Prof. me to as an Editor of which was my and that my with this journal and it has been to the June of one of my International for a on and Fractional Calculus in which was by and Mainardi, he made a of our results it was Prof. Mainardi aware of Russian in the field through our paper first with the contribution of Russian authors and in he was one of the of the on In Ref. Prof. Mainardi on his this a I had the by the of in the of I aware that the by the Russian in from our of with the Fractional Linear the Prof. Mainardi a correspondence with me by e-mail that there was a of in our each of us has two of research in fractional in viscoelasticity, as years of and little resulting in the of Refs. by Mainardi with and Refs. by and other and (2) the period in and for Mainardi and the first from the comparative analysis of Refs. 12 made in it is that worked on for the our have Mainardi has been the fractional by the second in the Refs. as as the references on his while the major of our is devoted to and both a both of us in of But even in that period there was one of the application of the to the the paper our results in this and also the of and his co-authors is reason to for a this is a or from my research has been in of linear and and and other fractional and other fractional the of linear our was with the analysis of the of the of the with fractional their with to the of the fractional their and so As for the the of received the of the of the fractional and the on the as as the of the and so that the two in the analysis of the problems in the mechanics of i.e., the on the with of and (2) the on the fractional and each other. Their for of some and to using the correspondence to the of a the of which is by the linear fractional it is to the by the with the fractional as the there is a to the in they be by of the of the fractional in the and of the the with the In this it is not to the with an the one the of the involving fractional or fractional The of is that the is its the of while for the there is a each time to their as a results in some for the and fractional that Thus, for and the fractional linear with fractional of the for the and in of its with In on the of the to that it a of and a it has been that the of the fractional on the and be to each other. The for involving fractional of have been in the article our papers, it has been that the application of the involving fractional to the problems of of the for the of the the one the as the in the to the fractional from both the fractional and fractional the on such a the under some of its and fractional In other the fractional of other of the fractional or may to new years have since the of our paper body of has in this period of which the fractional calculus in of are many and involves fractional calculus its of theory. fractional in the of published in that it to such by with for and applications in many to of with their be in Ref. recently the of research in the fractional calculus application to problems to the audience of researchers and in the field of and has a article papers, has been recently with Prof. and submitted for in Applied in December of and results in the field by the of both which are in the the of the and of fractional calculus in problems and of this I to the of the made to by and Russian researchers in the field of fractional calculus applications in linear viscoelasticity, which are in the of was for the of the of fractional calculus The fractional calculus the of using the two i.e., via the with and via fractional or fractional as as other of fractional are in the first while the Russian and authors who for the first time our in problems of viscoelasticity are cited in the second and The the papers wherein for the first time for problems of mechanics of and of the is from the fractional by and using the second and first fractional calculus was by as the with the it has the property that its has from the to the was by Caputo to of an fractional calculus linear by and was with in the by Russian scholars for a of problems Thus, in and the first to the of an the order of the fractional as the and are which results in the of the is now by the authors dealing with in with fractional The in the of the in 1970 by et al. for the fractional linear and in Refs. for the in in by Meshkov and and the in a made of the fractional linear was by and this paper was originally published in a Russian the results of Ref. in of Ref. The of a with the of a was by et al. in The of the in a by the application to its end the time was in by and years the in the of the in a similar for a made of the fractional was by and Mainardi using the second for was a in the of the fractional calculus in in Ref. the which the with times on the fractional such in Ref. for an of the and for The of a large of one to the of the the and involving and in of as a which was in the of a fractional from the fractional or fractional of was by in In Meshkov the for the while it for in with a of years Meshkov and to the as the in the and this to the of a and the in Ref. for the of a in a and its fractional and linear first by and Meshkov in and and but a time by Caputo (9) and Caputo and Mainardi (1), The applications of made by Caputo (9) in 1967 and Caputo and Mainardi in 1971 for the problems dealing with In the fractional was by Caputo to of an the 1970s, an on fractional and their application to the problems of was in the by and Their first paper in the field to was devoted to an as a fractional the next authors to the fractional linear the for from the papers cited in the of 1 it is a tremendous was in the and by Russian and Italian researchers in the application of fractional calculus or in the same period using the two as in the first of 1) for problems in the mechanics of and to first of all Professor Stanislav the of my who first me in the of the fractional calculus viscoelasticity, years all my to I have the to on his 75th I Professor Meshkov and mechanics research community Professor who was the of Applied for for his to a new in the volume of from has been publishing by authors on their and of my there is no other research journal to have such a I to for this and opportunity to my on fractional calculus viscoelasticity application in of and the contribution of all my co-authors in the papers in the I to Professor for her in of this paper and the of this