Infinitesimal Length Research Articles

Comment on “Groundwater Affects the Geomorphic and Hydrologic Properties of Coevolved Landscapes” by Litwin et al.

AbstractThe objective of this comment is to correct two sets of statements in Litwin et al. (2022, https://doi.org/10.1029/2021JF006239), which consider our research work (Bonetti et al., 2018, https://doi.org/10.1098/rspa.2017.0693; Bonetti et al., 2020, https://doi.org/10.1073/pnas.1911817117). We clarify here that (a) the specific contributing area is defined in the limit of an infinitesimal contour length instead of the product of a reference contour width (Bonetti et al., 2018, https://doi.org/10.1098/rspa.2017.0693), and (b) not all solutions obtained from the minimalist landscape evolution model of Bonetti et al. (2020, https://doi.org/10.1073/pnas.1911817117) are rescaled copies of each other. We take this opportunity to demonstrate that the boundary conditions impact the obtained solutions, which has not been considered in the dimensional analysis of Litwin et al. (2022, https://doi.org/10.1029/2021JF006239). We clarify this point by using dimensional analysis and numerical simulations for a square domain, where only one horizontal length scale (the side length l) enters the physical law.

Bohm Criterion of Plasma Sheaths away from Asymptotic Limits.

The plasma exit flow speed at the sheath entrance is constrained by the Bohm criterion. The so-called Bohm speed regulates the plasma particle and power exhaust fluxes to the wall, and it is commonly deployed as a boundary condition to exclude the sheath region in quasineutral plasma modeling. Here the Bohm criterion analysis is performed in the intermediate plasma regime away from the previously known limiting cases of adiabatic laws and the asymptotic limit of infinitesimal Debye length in a finite-size system, using the transport equations of an anisotropic plasma. The resulting Bohm speed has explicit dependence on local plasma heat flux, temperature isotropization, and thermal force. Comparison with kinetic simulations demonstrates its accuracy over the plasma-sheath transition region in which quasineutrality is weakly perturbed and the Bohm criterion applies.

A finite Zener-Stroh crack interacting with a blunt crack under in-plane deformations

A Zener-Stroh crack is the name given to a microcrack resulting from the coalescence of edge dislocations along a slip plane. The process by which the crack is initiated was proposed in the first instance by Zener and later examined in detail by Stroh. The Zener-Stroh crack can be thought of as being equivalent, in some sense, to the Griffith crack in classical linear elastic fracture mechanics. We study a finite mode I and II Zener-Stroh crack interacting with a nearby mode I and II blunt crack represented by a parabolic cavity. In this respect, the Zener-Stroh crack can be considered as a coalescence of edge dislocations emitted from the blunt crack. A conformal mapping function is introduced which maps the parabolic boundary onto a straight line in the image plane. By treating the Zener-Stroh crack as a pileup of edge dislocations, two Cauchy singular integral equations are constructed in the image plane. The Gauss-Chebyshev integration formula is applied to numerically solve the resulting singular integral equations. The local mode I and II stress intensity factors at the two tips of the Zener-Stroh crack are determined. We establish conditions on the propagation direction of the Zener-Stroh crack away from or toward the blunt crack. Also established are the conditions for the emission of an edge dislocation from the blunt crack by treating the edge dislocation as a Zener-Stroh crack with infinitesimal length. The dislocation emission criteria from the blunt crack are established here in elementary closed form and will thus be particularly convenient for application in engineering practice.

Derivation of Einstein–Cartan theory from general relativity

This paper derives the elements of classical Einstein–Cartan theory (EC) from classical general relativity (GR) in two ways. (I) Derive discrete versions of torsion (translational holonomy) and the spin-torsion field equation of EC from one Kerr solution in GR. (II) Derive the field equations of EC as the continuum limit of a distribution of many Kerr masses in classical GR. The convergence computations employ “epsilon-delta” arguments, and are not as rigorous as convergence in Sobolev norm. Inequality constraints needed for convergence restrict the limits from continuing to an infinitesimal length scale. EC enables modeling exchange of intrinsic and orbital angular momentum, which GR cannot do. Derivation of EC from GR strengthens the case for EC and for new physics derived from EC.

Multiple Crack Detection in Thick-walled Pipes Using Artificial Bee Colony Algorithm

An analytical method for diagnosis of cracks in thick-walled pipes with a circular hollow section is investigated in this study. In the proposed method, the defect is assumed to be a non-leaking crack, which is modeled by a massless linear spring with infinitesimal length at the crack location. In order to find the cracks in the pipe, the vibration-based method related to the modal properties of the pipe is utilized. In the modal analysis, the mass and stiffness matrices influence the dynamic properties of the pipe. It is assumed that the mass matrix remains unchanged after the crack initiation, while the corresponding stiffness matrix changes. The stiffness matrix of a cracked element can be formulated by the finite element method with two unknown parameters: location and depth of the crack. Using the eigensolution for an undamped dynamic system to formulate the objective function yields to a complicated optimization problem, which can be solved by an iterative numerical optimization method. Among the optimization approaches, the Artificial Bee Colony (ABC) algorithm is a simple and flexible technique for minimizing the objective function. In this paper, the analytical model is utilized to find the size and position of cracks in a pipe using the ABC algorithm and subsequently some numerical examples are examined in order to assess the accuracy of the method. The results show that the proposed method is able to acceptably estimate the location and depth of multiple cracks in the straight pipes as well as curved ones.

A financial market with singular drift and no arbitrage

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803–2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay theta > 0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as theta > 0. This implies that there is no arbitrage in the market in that case. However, when theta goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223–262, 2016) and the references therein.

Order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing

Journal bearing has been widely used in the wide range of rotating machineries to support large loads and to add significant damping to the system. Conventionally, its fluid film force is represented by the linear model of spring and damper around its equilibrium position in the vibration analysis of the rotor system. However, the fluid film force of the journal bearing is essentially nonlinear, and it is necessary to consider its nonlinearity to expect the characteristics of the limit cycle at around the instability point. This paper investigated the order reduction and bifurcation analysis of a flexible rotor system supported by a full circular journal bearing. The fluid film force is derived under the conditions of both infinitesimal length approximation and half-Zommerfeld boundary condition, and its polynomial function approximation expression is used. Order reduction in the FEM rotor model retaining the nonlinearity of the journal bearing was performed by utilizing the substructure synthesis method. Then, its bifurcation phenomena at around the instability point are investigated by applying the center manifold theory and using the normal form theory. The influences of various parameters, such as kinematic viscosity, bearing length, and the disk position, on the bifurcation phenomena at around the instability point were investigated and explained. Furthermore, the validity of the derived analytical observation was confirmed by numerical simulation and experiment. By invoking these analytical techniques and obtained results, the bifurcation characteristics can be expected theoretically at the design stage of the journal bearing and rotor system.

ON THE SCATTERING OF SURFACE WAVES BY UNDERWATER OBSTACLES

The paper considers a 2D linear problem on scattering of the low-amplitude surface wave in an incompressible fluid that propagates over the bottom obstacle in form of a vertical rectangular wall of the finite of infinitesimal length. The bottoms of the channel and obstacle walls are absolutely rigid. The solution is found by the method of partial domains implying the subdivision of the complex geometry of the overall domain of wave field existence into the subdomains of canonical shapes. This approach allows the formal representing of the corresponding general solutions as the series with respect to eigenfunctions satisfying part of the boundary conditions in each subdomain. The unknown coefficients present in these general solutions may be found from matching conditions that postulate continuity of pressure and normal velocity fields at the boundaries of adjacent subdomains. Applying algebraization procedure to matching conditions for the corresponding general solutions leads to generation of infinite systems of linear equations with respect to the unknown coefficients. The obtained systems possess slow convergence that is determined by the presence of velocity singularities in angular points of the obstacle. To accelerate the convergence, the method of improved reduction is applied in analyzing the infinite systems. It is based on the a priori hypotheses about the asymptotic behavior of the unknown coefficients in general solutions that is controlled by the order of field singularity in the vicinity of the corresponding angular point. The dependence of calculated energy discrepancy on the amount of considered terms in the reduced system is assessed. The method of improved reduction is shown to provide the better convergence quality at moderate number of kept terms (50 to 70), in comparison with the ordinary reduction. The use of the improved reduction also leads to some widening of the region for which the numerical error remains within acceptable limits.

Development of Finite Element Models for Predicting the Mechanical Behavior of the Cracked Pipe

In this paper, two theoretical models, micro and macro model, are presented to investigate the effect of circumferential cracks on the structural characteristics of a pipe. In the micro model, the equivalent bending stiffness is obtained based on the energy method and fracture mechanics. Because of the high change of strain energy near the crack location, the variation is represented by reduction in the bending stiffness along the length of the pipe. The development of the macro model is based on a simplified procedure, i.e., the pipe is separated into two segments without any change in the cross section property. Two separated pieces are connected by a rotational massless spring of the infinitesimal length inserted at the crack location. Moreover, it is shown that in the macro model so smaller number of elements are required than the micro model to provide a sufficient accuracy. Finally, the experimental measurements are utilized to validate the proposed models. The deflections and frequencies obtained for several cases through the displacement and vibration method, using the proposed models agree reasonably well with the experimental measurement data.

An analytical method for free vibration of multi cracked and stepped nonlocal nanobeams based on wave approach

Various discontinuities and boundary conditions affect natural frequencies of nanobeams. In this paper, free lateral vibration of Euler-Bernoulli nanobeam with multiple discontinuities is studied. The governing equations are developed by using Eringen's nonlocal elasticity theory. Cracks and steps are considered as discontinuities. Based on wave approach, vibrations take into account as moving waves along the structure. Waves propagate through waveguide and are reflected and transmitted at the discontinuities and boundaries. The propagation, reflection and transmission matrices are derived for discontinuities and boundaries. Cracks are modelled by a massless torsional spring with infinitesimal length and steps are considered as two connected nanobeams with different cross-sectional areas and mechanical properties. Boundary is formed by combining a torsional and a translational spring. Also buckyball (as a lumped mass) at the tip of the beam is considered as a boundary condition. Using propagation and appropriated reflection and transmission matrices for each case, leads to an analytical comprehensive succinct method so-called wave approach to analyse the free lateral vibration of nanobeams. The flexural natural frequencies are derived analytically for cracked or/and stepped nanobeams with ordinary boundary conditions or buckyball at the tip. The effects of crack severity, changing ratio of cross-sectional area as step, cracks and steps location, mass of buckyball and small-scale parameter on natural frequencies are deliberated. This approach is demonstrated with a number of examples that can be used as benchmarks for other works. The results are compared with other methods.

Measurement-Based Thermal Modeling Using Laser Thermography

There have been several studies in using infrared thermography for nondestructive evaluation and testing of materials. Some studies have explored the potential of using thermography for classification of materials based on their thermal response. However, the modeling behind these efforts is rather simplistic, and in some ways not corresponding well with practical applications. This paper presents a realistic model-based approach for characterizing unknown materials using laser thermography. The process being modeled is laser excitation step thermography with finite but not infinitesimal excitation length and uniform distribution over a finite radius. A mathematical model, based on the heat equation that governs the flow of heat in materials, is introduced to characterize objects based on their response to thermal excitation. Experimental results consistently demonstrated the ability of the proposed approach to classify different materials based on their thermal properties. This contactless characterization method might prove to be suitable for a range of applications including teleoperation, haptic mapping and multimodal human computer interaction.

Longitudinal vibration analysis of nanorods with multiple discontinuities based on nonlocal elasticity theory using wave approach

Discontinuities such as cracks and steps on the length, and the mass of attached buckyball on the tip, of nanoresonators have large effects on nanoresonators natural frequencies. In this paper, free longitudinal vibration of multiple cracked, stepped nanorods with a buckyball at tip is studied using the Eringen nonlocal elasticity theory. From wave viewpoint, vibrations can be considered as travelling waves along structures. Waves propagate in a waveguide and transmit and reflect at discontinuities. The propagation, transmission and reflection functions are derived for nanorods. Cracks and steps as discontinuities, and buckyball at the end, clamped end and free end for boundary conditions, are considered. Cracks are modelled by an infinitesimal length, massless linear spring. Steps are formed by connecting two nanorods with different cross section areas and mechanical properties and buckyballs are considered as lumped masses. These propagation, transmission and reflection functions are combined to provide a short comprehensive systematic analytical approach to analyze the free longitudinal vibration of nanorods. Also, explicit expressions for natural frequencies are derived. The effects of crack intensity, sectional change as step, mass of buckyball, location of crack and step and also small-scale effect on natural frequencies are discussed. The approach is explained using several examples and the closed-form solutions are derived for some cases. The results are compared with the existing methods and can be used as benchmark for other future works.

Analysis of the dependence of spinodal decomposition in nanoparticles on boundary reaction rate and free energy of mixing

The mathematical model for intercalation dynamics in phase-separating materials (Singh et al., 2008) is a powerful tool for the investigation of the spinodal decomposition in nanoparticles. By means of this model, we conduct a careful mathematical analysis of the intercalation dynamics in nanoparticles to study the dependence of spinodal gap on the boundary reaction rate and the particle size, which can be used for LiFePO4 battery material application. Consistent with previous investigations, we found that for some range of the boundary reaction rate and the particle size the concentration spinodal gap is not continuous, but it has stable “islands” where no spinodal decomposition is expected. The new important observation is that the presence of an infinitesimally small boundary reaction rate will destabilize nanoparticles even for infinitesimal length. In particular for nanoparticles having the size of order or less than interphase width λ, the spontaneous charge or discharge will occur at the reaction rate of order 0.1 D/λ. The further raise of the intercalation rate will stabilize the system until some size limit of order two diffusion length. The intercalation effects are proven by means of numerical simulations. We also show that the increasing enthalpy of the spinodal mixture as well as increasing elastic energy due to the lattice misfit can destabilize the particles and increase the spinodal gap.

A ${Z}$ -Transform Method for Synthesis of Unequal-Length Multisection Transmission Lines for Multiband Applications

A method based on the $Z$ -transform is demonstrated to cascade transmission line sections of unequal lengths and achieve multiband and/or wideband properties. Unlike previous methods based on the $Z$ -transform, the proposed method associates the $z^{-1}$ term with the infinitesimal length of the unit segment. Consequently, any electrical length can be expressed in the $z^{-1}$ form. Therefore, while the previous methods are restricted to cascading lines of a commensurate length only, the proposed method can be applied to those of incommensurate lengths. Furthermore, the $z$ -polynomials for triple or more bands can be constructed by multiplying a combination of single- and dual-band polynomials, which remarkably simplifies the entire procedure to develop multiband and/or wideband circuits with virtually no limitation in terms of the number and/or combination of frequencies. The validity and practicality of the proposed method is verified with experimental results for various impedance transformers for real loads and Wilkinson power dividers; the results show the independence in the number and combination of operating frequencies as well as a dual-band operation with a center frequency ratio as large as 7.75.

A hybrid analytical-numerical method for solving water flow equations in root hydraulic architectures

Abstract In this manuscript, we propose a new method to calculate water flow and xylem water potential distribution in hydraulic architectures (such as root systems) of any complexity. It is based on the extension of the water flow equation analytical resolution of Landsberg and Fowkes for single roots. It consists in splitting the root systems in zones of homogeneous or homogeneously changing properties and deriving the xylem potential and water flow under any given boundary conditions (plant transpiration or collar potential, and potential at soil-root interfaces) without assuming a uniform xylem potential within each zone. The method combines analytical solutions of water flow within the segmented zones with the numerical solution of flow connectivity for the whole root system. We demonstrate that the proposed solution is the asymptote of the exclusively numerical solution for infinitesimal root segment lengths (and infinite segment number). As water uptake locations and magnitudes predicted by the latter solution for finite segmentation lengths deviate from the exact solution, and are computationally more intensive, we conclude that the new methodology should always be privileged for future applications. The proposed solution can be easily coupled to soil modules (as already done with existing solutions) and further implemented in functional-structural plant models to predict water flow in the soil-plant atmosphere continuum with a better accuracy than current models. Finally the new solution may be used to calculate more accurately plant scale macroscopic parameters for crop models.

A proposal for unification of fatigue crack growth law

In the present paper, the new fractional-differential dependences of cycles to failure for a given initial crack length upon the stress amplitude in the linear fracture approach are proposed. The anticipated unified propagation function describes the infinitesimal crack length growths per increasing number of load cycles, supposing that the load ratio remains constant over the load history. Two unification fractional-differential functions with different number of fitting parameters are proposed. An alternative, threshold formulations for the fractional-differential propagation functions are suggested. The mean stress dependence is the immediate consequence from the considered laws. The corresponding formulas for crack length over the number of cycles are derived in closed form.

Electric charge in hyperbolic motion: the early history

We revisit the early work of Minkowski and Sommerfeld concerning hyperbolic motion, and we describe some geometrical aspects of the electrodynamic interaction. We discuss the advantages of a time symmetric formulation in which the material points are replaced by infinitesimal length elements.

Three-Dimensional Parabolic Cable Element for Static Analysis of Cable Structures

AbstractThis paper presents a three-dimensional inclined parabolic cable element for static analysis of cable structures. After a catenary cable element is briefly reviewed, the three dimensional parabolic cable considered in this paper is presented. The derivation of the stiffness matrix for the element is described by equating the total differentiations of the cable length and its elastic elongation, and defining the stiffness as the ratio of the infinitesimal force increment to the infinitesimal length increment. The static equilibrium equation for a two-node cable element and the procedure for nonlinear structural analysis using the Newton-Raphson iterative method are presented as well. The proposed element can consider an inclined cable without the need for a transformation, and has explicit and simple stiffness formulas compared with those of a catenary element. To validate the proposed parabolic cable element, four example models subjected to static loadings are analyzed using the proposed element,...

The exact (up to infinitesimals) infinite perimeter of the Koch snowflake and its finite area

The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the infinity Computer being a new supercomputer patented in USA and EU. It is revealed in the paper that at infinity the snowflake is not unique, i.e., different snowflakes can be distinguished for different infinite numbers of steps executed during the process of their generation. It is then shown that for any given infinite number n of steps it becomes possible to calculate the exact infinite number, Nn, of sides of the snowflake, the exact infinitesimal length, Ln, of each side and the exact infinite perimeter, Pn, of the Koch snowflake as the result of multiplication of the infinite Nn by the infinitesimal Ln. It is established that for different infinite n and k the infinite perimeters Pn and Pk are also different and the difference can be infinite. It is shown that the finite areas An and Ak of the snowflakes can be also calculated exactly (up to infinitesimals) for different infinite n and k and the difference An−Ak results to be infinitesimal. Finally, snowflakes constructed starting from different initial conditions are also studied and their quantitative characteristics at infinity are computed.

Zeno’s arrow and the infinitesimal calculus

I offer a novel solution to Zeno’s paradox of The Arrow by introducing nilpotent infinitesimal lengths of time. Nilpotents are nonzero numbers that yield zero when multiplied by themselves a certain number of times. Zeno’s Arrow goes like this: during the present, a flying arrow is moving in virtue of its being in flight. However, if the present is a single point in time, then the arrow is frozen in place during that time. Therefore, the arrow is both moving and at rest. In “Zeno’s Arrow, Divisible Infinitesimals, and Chrysippus,” White suggests using an infinitesimal value as the length of the present. Contra Zeno, this allows the arrow to be moving in the present, rather than frozen in place. In this paper, I follow the basic outline of White’s solution but argue that his solution suffers from arbitrariness and a related theoretical artificiality in relation to the system of infinitesimals he invokes, viz. in relation to the hyperreal infinitesimals of nonstandard analysis. After arguing that any solution to the paradox must satisfy certain theoretical requirements, I examine White’s solution alongside two nilpotent solutions. One of these solutions is inspired by F.W. Lawvere’s Smooth Infinitesimal Analysis and the other is inspired by Paolo Giordano’s ring of Fermat Reals. I argue that Giordano’s nilpotents supply the best answer to Zeno’s paradox.