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This paper shows that a potential system with one multiple eigenvalue of multiplicity two or more can assuredly be made unstable by infinitesimal positional perturbations. The explicit nature of th...

In this paper, a variable-coefficient cubic–quintic nonlinear Schrodinger equation involving five arbitrary real functions of space and time is analyzed from the point of view of symmetry analysis by using Lie’s invariance infinitesimal criterion. The infinitesimal generators of corresponding equivalence transformations are presented. The first-order differential invariants are constructed to identify when the equation can be mapped to a constant-coefficient cubic–quintic nonlinear Schrodinger equation. The constrained conditions on the variable coefficients we arrived here extend the cases discussed before and present more general results. Some brightlike and darklike solitary wave solutions for special potentials and cubic–quintic nonlinearities are obtained.

Utility-indifference approach is a useful approach to be adopted for pricing financial derivatives in an incomplete market and is an ongoing hot research topic in quantitative finance. One interesting question associated with this approach is whether or not it renders to the same option prices, degenerately, when the market becomes infinitesimally close to a complete market. The answer for such a question has been provided for European-style options as there is a well-documented theoretical proof in Davis et al. (1993). However, a theoretical proof for the case of pricing American-style options is unavailable at this stage and the answer for this question must be at least numerically confirmed before it can be comfortably used to price American-style options in incomplete markets. The contribution of this paper is to provide such a numerical verification.

The paper presents a review of studies on the optimal control of the boundary layer on permeable surfaces under various flow modes. The main attention is paid to the studies carried out in 1964–2019 at Kazan National Research Technical University–KAI at the departments of Aerohydrodynamics, Flight and Control Dynamics, and Special Mathematics. The essential role of the first integrals for conjugate systems obtained using the theory of the Noether–Ibragimov invariant variational problems and the Lie–Ovsyannikov infinitesimal apparatus was noted in a number of papers.

In Bertrand Russell’s 1903 The Principles of Mathematics, he offers an apparently devastating criticism of The Principle of the Infinitesimal Method and Its History (PIM) by the neo-Kantian Hermann...

We develop a martingale approach for a class of singular stochastic PDEs of Burgers type (including fractional and multi-component Burgers equations) by constructing a domain for their infinitesimal generators. It was known that the domain must have trivial intersection with the usual cylinder test functions, and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions. Using the new domain, we are able to prove existence and uniqueness for the Kolmogorov backward equation and the martingale problem. We also extend the uniqueness result for "energy solutions" of the stochastic Burgers equation of [GP18a] to a wider class of equations., 47 pages

Since Loday introduction of Leibniz algebras as a generalisation of Lie algebras, many results of the theory of Lie algebras have been extended to Leibniz algebras. Cyclic Leibniz algebras, which are generated by one element, have no equivalent into Lie algebras, though. This fact provides cyclic Leibniz algebras with important properties. Throughout the present paper we extend the concept of cyclic to Leibniz superalgebras, obtaining then the definition, as long as the description and classification of finite-dimensional complex cyclic Leibniz superalgebras. Furthermore, we prove that any cyclic Leibniz superalgebra can be obtained by means of infinitesimal deformations of the null-filiform Leibniz superalgebra. We also obtain a description of irreducible components in the variety of Leibniz algebras and superalgebras.

Hyperbolic towers are towers in the shape of a single-sheet hyperboloid, and they are interesting in architecture. In this paper, we deal with the infinitesimal bending of a curve on a hyperboloid of one sheet; that is, we study the flexibility of the net-like structures used to make a hyperbolic tower. Visualization of infinitesimal bending has been carried out using Mathematica, and some examples are presented and discussed.

Whenever a litigant needs to prove that a certain result was caused in a specific way, what could be more compelling than citing the infinitesimal probability of that result emanating from an alternative natural cause? Contrary to this intuitive position, in the present article, I argue that the contention that a result was due to a certain cause should remain unaffected by statistical evidence of the extremely low probability of an alternative cause. The only scenario in which the low probability of a natural cause would be relevant to the case at hand is if it were contrasted with another piece of statistical evidence: the frequency of the criminal activity among people who are similar to the accused. However, by connecting the use of probabilistic generalisations in legal fact-finding to the issue of free will, I hold that, in Criminal Law, contrasting frequencies in this manner is objectionable—as a matter of principle—regardless of how reliable the statistical analysis is. Consequently, if the low probability of a natural cause is probative only if contrasted with another piece of statistical evidence that is objectionable, then neither piece of evidence should be admitted in criminal trials.

The reducibility problem of optimal processes is posed, and a group-theoretic approach to its solution is proposed. This approach is based on the Lie–Ovsiannikov infinitesimal apparatus [1].

Reseña de: Timothy Morton: El pensamiento ecológico (trad. Fernando Borrajo). Barcelona: Paidós, 2018. 205 pp. “La ecología equivale a menos Naturaleza y a más conciencia”. Tal sería la propuesta del filósofo Timothy Morton (Londres, 1968) en esta obra. El pensamiento ecológico es un ensayo ecléctico donde se nos presenta una forma ecológica de pensar, que no tiene tanto que ver con la ciencia como con las humanidades. La clave del pensamiento ecológico según Morton es precisamente ésta: su ramificación y diversidad. Parte de la consideración de que todos los seres están conectados entre sí en una “malla” (the mesh); como nada existe por sí mismo, nada es plenamente “sí mismo”. Lo que contemplamos es una inmensidad de diferencias infinitesimales, que nos requerirá “pensar a lo grande” (así se titula el capítulo 1 de la obra). De la mano del arte, la filosofía, la literatura, la música y la cultura popular, el autor intenta trazar en tres capítulos una nueva estética ecológica: la ecología oscura (dark ecology, noción que da título a una obra anterior de Morton en 2018), un pensamiento capaz de expresar la ironía y la incertidumbre en la que nos sitúa la ecología mortoniana.

We explore new analytical solutions for the two-dimensional nonlinear elliptic Bratu equation. Through the point transformation, the integrable form of Bratu equation was investigated then we obtain the Lie infinitesimals for the new equation. These vectors reduce the integrable equation to solvable ODEs then we use the boundary conditions (BCs) to spin two new exact solutions for Bratu equation in a unit square domain. A three-dimensional plot illustrates some resulting solutions. Comparison with other work has been presented.

Lagrangian techniques, such as the finite-time Lyapunov exponent (FTLE) and hyperbolic Lagrangian coherent structures (LCS), have become popular tools for analyzing unsteady fluid flows. These techniques identify regions where particles transported by a flow will converge to and diverge from over a finite-time interval, even in a divergence-free flow. Lagrangian analyses, however, are time consuming and computationally expensive, hence unsuitable for quickly assessing short-term material transport. A recently developed method called OECSs [Serra, M. and Haller, G., `Objective Eulerian Coherent Structures', Chaos 26(5), 2016] rigorously connected Eulerian quantities to short-term Lagrangian transport. This Eulerian method is faster and less expensive to compute than its Lagrangian counterparts, and needs only a single snapshot of a velocity field. Along the same line, here we define the instantaneous Lyapunov Exponent (iLE), the instantaneous counterpart of the FTLE, and connect the Taylor series expansion of the right Cauchy-Green deformation tensor to the infinitesimal integration time limit of the FTLE. We illustrate our results on geophysical fluid flows from numerical models as well as analytical flows, and demonstrate the efficacy of attracting and repelling instantaneous Lyapunov exponent structures in predicting short-term material transport., 44 pages, 16 figures

In this article, the formation control problem of multiagent systems with nonholonomic constraints by bearing-only measurements is studied. Many previous works used the notation of infinitesimal rigidity to define the communication topology in formation control, but the global stability is difficult to achieve without leaders. To overcome this shortage, the maximal clique graph is used to define the topological relationship between agents. By minimizing the designed cost function, a novel distributed bearing-only formation controller is designed. Then, the close-loop system can achieve the global stability due to the existence of common agents between the maximal cliques under the designed controller. Besides, the global collision avoidance control strategy is also investigated with assistant distance measurement, which can prevent collisions between agents during the whole formation process. In addition, the embedded rotation angles between the final formation and the target formation are presented to optimize the motion process in real time. Finally, the simulation results verify the effectiveness of the designed controllers.

In Quantum ElectroDynamics (QED) the propagator is a function that describes the probability amplitude of a particle going from point A to B. It summarizes the many paths of Feynman’s path integral approach. We propose a reverse propagator (R-propagator) that, prior to the particle’s emission, summarizes every possible path from B to A. Wave function collapse occurs at point A when the particle randomly chooses one and only one of many incident paths to follow backwards with a probability of one, so it inevitably strikes detector B. The propagator and R-propagator both calculate the same probability amplitude. The R-propagator has an advantage over the propagator because it solves a contradiction inside QED, namely QED says a particle must take EVERY path from A to B. With our model the particle only takes one path. The R-propagator had already taken every path into account. We propose that this tiny, infinitesimal change from propagator to R-propagator would vastly simplify the mathematics of Nature. Many experiments that currently describe the quantum world as weird, change their meaning and no longer say that. The quantum world looks and acts like the classical world of everyday experience.

The objective of this document is to contribute to the modelling of engineering changes and their propagation. Usable in preliminary redesign activities, a new approach is suggested that allows greater efficiency. Given a product model, the idea is to use the experiments to calculate in advance the consequences of potential changes in continuous variables. These consequences are collected, analysed and structured in a dependency model, noted $$\langle \varGamma , \varPhi \rangle$$ , composed of a dependency graph $$\varGamma$$ and its associated set of influence functions $$\varPhi$$ . Bilateral influence functions, associated with arcs, quantify the dependencies between node pairs. Multilateral influence functions, identified by the application of the total differential theorem, define the dependencies between a node and all influential nodes on which it depends. Finally, the relative error is calculated by following the infinitesimal assumption of the total differential for each variables. Our findings show that such a dependency model is informative and allows effective prediction and evaluation of changes. The approach is illustrated by a geometric bicycle model. The results are discussed and future areas of research are finally presented.

In this paper we consider the effect of symmetry on the rigidity of bar-joint frameworks, spherical frameworks and point-hyperplane frameworks in $\mathbb{R}^d$. In particular we show that, under forced or incidental symmetry, infinitesimal rigidity for spherical frameworks with vertices in $X$ on the equator and point-hyperplane frameworks with the vertices in $X$ representing hyperplanes are equivalent. We then show, again under forced or incidental symmetry, that infinitesimal rigidity properties under certain symmetry groups can be paired, or clustered, under inversion on the sphere so that infinitesimal rigidity with a given group is equivalent to infinitesimal rigidity under a paired group. The fundamental basic example is that mirror symmetric rigidity is equivalent to half-turn symmetric rigidity on the 2-sphere. With these results in hand we also deduce some combinatorial consequences for the rigidity of symmetric bar-joint and point-line frameworks., Comment: 30 pages, 5 figures

The net force is enough to predict the motion of an object, but to predict its deformation, the distribution of the force inside the object should be known. In the continuum electromechanics, various expressions are used to calculate the electric forces acting on a dielectric object. The Kelvin and the Korteweg–Helmholtz forces are the most common electric body forces. The net force resulting from both of these “forces” is equal, but the distribution of these two forces inside a dielectric material is different because the expressions of both these forces lose their physical manifestation in the derivation process. This article is primarily aimed at revealing the common mistakes that are made when deriving volumetric force density. Then, the concept of an infinitesimal particle and external electric field is introduced. By using this concept, the electric force density inside a dielectric material is uniquely determined irrespective of the force expression, which is shown analytically and numerically.

The elliptic case of restricted four-body problem with variable mass of infinitesimal body is studied here. The three primary bodies which are placed at the vertices of an equilateral triangle and moving in the elliptical orbits around their common center of mass. Out of these primaries we have considered that one massive body is having radiating effect and other two bodies are oblate in shapes. The fourth body which have infinitesimal mass, are varying its mass according to Jeans law. We derive the equations of motion of the infinitesimal body under the generalized sense in the elliptic restricted four-body problem by using the Meshcherskii-space time transformations. Further we numerically study about the equilibrium points, Poincare surfaces of section, regions of possible motion and basins of the attracting domain by considering the variation of parameters used. Further more we examine the stability of these equilibrium points and found them unstable.

Ceramic balls represent a new type of damaging element, and studies on their damaging power of composite armor are required for a comprehensive evaluation of the effectiveness of various types of weapons. The goal of this study was to determine the impact of ϕ7 mm toughened Al2O3 ceramic balls on a composite ceramic/metal armor. The influences of the ceramic panel and the thickness of the metal backing material on the destroying power of the ceramic balls were first determined. Based on the agreement between numerical simulation, experimental results, and calculation models of the target plate resistance, the response mechanism of the ceramic balls was further analyzed. The results indicate that for a back plate of Q235 steel, with an increasing thickness of the ceramic panel, the piercing speed limit of the ceramic balls gradually increases and the diameter of the out-going hole on the metal back decreases. Different conditions were tested to assess the effects on the piercing speed, the diameter of the out-going hole, the micro-element stress, and the integrity of the recovered ceramic bowl.

We analyze the deformation theory of equivariant vector bundles. In particular, we provide an effective criterion for verifying whether all infinitesimal deformations preserve the equivariant structure. As an application, using rigidity of the Frobenius homomorphism of general linear groups, we prove that projectivizations of Frobenius pullbacks of tautological vector bundles on Grassmanians are arithmetically rigid, that is, do not lift over rings where $$p \ne 0$$ . This gives the same conclusion for Totaro’s examples of Fano varieties violating Kodaira vanishing. We also provide an alternative purely geometric proof of non-liftability mod $$p^2$$ and to characteristic zero of the Frobenius homomorphism of a reductive group of non-exceptional type. In the appendix, written jointly with Piotr Achinger, we provide examples of non-liftable Calabi–Yau varieties in every characteristic $$p \geqslant 5$$ .

A new formal model of belief dynamics is proposed, in which the epistemic agent has both probabilistic beliefs and full beliefs. The agent has full belief in a proposition if and only if she considers the probability that it is false to be so close to zero that she chooses to disregard that probability. She treats such a proposition as having the probability 1, but, importantly, she is still willing and able to revise that probability assignment if she receives information that gives her sufficient reasons to do so. Such a proposition is (presently) undoubted, but not undoubtable (incorrigible). In the formal model it is assigned a probability 1 − δ, where δ is an infinitesimal number. The proposed model employs probabilistic belief states that contain several underlying probability functions representing alternative probabilistic states of the world. Furthermore, a distinction is made between update and revision, in the same way as in the literature on (dichotomous) belief change. The formal properties of the model are investigated, including properties relevant for learning from experience. The set of propositions whose probabilities are infinitesimally close to 1 forms a (logically closed) belief set. Operations that change the probabilistic belief state give rise to changes in this belief set, which have much in common with traditional operations of belief change.

Let $M$ be a real-analytic connected CR-hypersurface of CR-dimension $n>0$ having a point of Levi-nondegeneracy. The following alternative is demonstrated for both the symmetry algebra $s$ and the automorphism group $G$ of $M$. Denote by $d$ the dimension of $s$ or $G$. Then (i) either $d=n^2+4n+3$ and $M$ is spherical everywhere; (ii) or $d\le n^2+2n+2+\delta_{2,n}$ and in the case of equality $M$ is spherical of fixed Levi signature in the open dense subset of Levi-nondegenerate points. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry., Comment: A bit more corrections and improvements, essentially supported in Section 2

We consider the optimal stopping problem with non-linear $f$-expectation (induced by a BSDE) without making any regularity assumptions on the reward process $\xi$. and with general filtration. We show that the value family can be aggregated by an optional process $Y$. We characterize the process $Y$ as the $\mathcal{E}^f$-Snell envelope of $\xi$. We also establish an infinitesimal characterization of the value process $Y$ in terms of a Reflected BSDE with $\xi$ as the obstacle. To do this, we first establish a comparison theorem for irregular RBSDEs. We give an application to the pricing of American options with irregular pay-off in an imperfect market model.

We study homeomorphisms and mappings with branching in domains of the Euclidean space. We establish pointwise HOlder and Lipschitz properties of mappings whose characteristics satisfy a Dini-type condition or whose mean values over infinitesimal balls are finite at the corresponding points. Moreover, we find conditions on the complex coefficients of the Beltrami equations in the unit disk under which their homeomorphic solutions are HOlder-continuous on the boundary.

Amid the debate over infinitesimals, Bolzano introduced the alternative notion of variable quantities ω in his 1816 work on the binomial theorem. It has often been assumed that his 1817 definition of continuity using them is practically the modern one. This paper explores Bolzano's early mathematical works and diaries to gain insight into the subtleties in his definition of ω and some of his mathematical procedures. We show that those quantities are not clearly ‘proto-Weierstrassian’ and argue that Bolzano was in the process of refinement and increasing abstraction of the idea of quantity that eventually led to the development of a theory of real numbers.

Let ‎$‎ (M,g) $ be a Riemannian manifold and ‎$‎ TM $ be its tangent bundle. ‎In the present paper‎, ‎we study infinitesimal projective transformations on $ TM $ with respect to the Levi-Civita connection of a class of (pseudo-)Riemannian metrics ‎$ ‎‎\tilde{g}‎ $ which is a generalization of the three classical lifts of the metric ‎$‎g$. We characterized this type of transformations and then ‎we prove that if $ (TM,‎\tilde{g}‎) $ admits a non-affine infinitesimal projective transformation‎, ‎then $ M $ and $ TM‎‎ $ are locally flat.

A theoretical model of an anisotropic material, Tensylon®, under large strains is proposed. This model is capable to describe the material’s response in in-plane tension at different angles to the fibrils. At 0° and at 90°, i.e., along the fibrils in either “criss” or “cross” plies, it quantitatively predicts the experimentally observed elastic behaviour until failure. At 45° to the fibrils, it quantitatively describes the experi- mental data in the elastic and plastic domains. The description remains accurate up to strains of 35%, that corresponds to 30÷40% of deforma- tion gradient components. The infinitesimal strains model would give at least 25% of error under such circumstances.

Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable $${{\mathcal {I}}}$$. Homeostasis plays an important role in the regulation of biological systems, cf. Ferrell (Cell Syst 2:62–67, 2016), Tang and McMillen (J Theor Biol 408:274–289, 2016), Nijhout et al. (BMC Biol 13:79, 2015), and Nijhout et al. (Wiley Interdiscip Rev Syst Biol Med 11:e1440, 2018). A method for finding homeostasis in mathematical models is given in the control theory literature as points where the derivative of the output variable with respect to $${{\mathcal {I}}}$$ is identically zero. Such points are called perfect homeostasis or perfect adaptation. Alternatively, Golubitsky and Stewart (J Math Biol 74:387–407, 2017) use an infinitesimal notion of homeostasis (namely, the derivative of the input–output function is zero at an isolated point) to introduce singularity theory into the study of homeostasis. Reed et al. (Bull Math Biol 79(9):1–24, 2017) give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to 78 three-node input–output network configurations, under the assumption that there is one input node, one output node, and they are distinct. The different configurations are based on which node is the input node and which node is the output node. We show nonetheless that there are only three basic mechanisms for three-node input–output networks that lead to infinitesimal homeostasis and we call them structural homeostasis, Haldane homeostasis, and null-degradation homeostasis. Substantial parts of this classification are given in Ma et al. (Cell 138:760–773, 2009) and Ferrell (2016) among others. Our contributions include giving a complete classification using general admissible systems (Golubitsky and Stewart in Bull Am Math Soc 43:305–364, 2006) rather than specific biochemical models, relating the types of infinitesimal homeostasis to the graph theoretic existence of simple paths, and providing the basis to use singularity theory to study higher codimension homeostasis singularities such as the chair singularities introduced in Nijhout and Reed (Integr Comp Biol 54(2):264–275, 2014. https://doi.org/10.1093/icb/icu010) and Nijhout et al. (Math Biosci 257:104–110, 2014). See Golubitsky and Stewart (2017). The first two of these mechanisms are illustrated by feedforward excitation and substrate inhibition. Structural homeostasis occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, when the network is just the feedforward loop motif itself, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Haldane homeostasis occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path has strength 0. Null-degradation homeostasis is illustrated by a biochemical example from Ma et al. (2009); this kind of homeostasis can occur only when the degradation constant of the third node is 0. The paper ends with an analysis of Haldane homeostasis infinitesimal chair singularities.

Both conservative and nonconservative systems can be studied simultaneously using the differential variational principle of Herglotz type. For a perturbed system, in which parameters change with time, it is useful to find adiabatic invariants. Based on the Herglotz differential variational principle, we study the perturbation of infinitesimal transformations and adiabatic invariants for perturbed nonconservative Lagrangian systems. From the generalized Euler-Lagrange equation and the invariance condition for the Hamilton-Herglotz action under the group of infinitesimal transformations, we obtain an exact invariant of Herglotz type for a holonomic nonconservative system. We propose a definition of higher-order adiabatic invariants of Herglotz type and obtain such adiabatic invariants for nonconservative Lagrangian systems with small perturbations. We prove the corresponding inverse theorem of adiabatic invariants. As examples of using the obtained results, we consider an oscillator with square damping and a system with two degrees of freedom.

Nonstationary environments are ubiquitous in communications and acoustic systems. The ability to track their dynamics is one of the most desirable features of adaptive processing algorithms. The designers of these algorithms employ guidelines derived from stochastic analyses to adjust user-defined parameters to maximize performance or avoid stability issues. It is therefore important that analyzes of adaptive processing algorithms take into account the built-in sophisticated learning capabilities. This work presents a comprehensive model of the performance of the least mean square algorithm, operating under Markovian time-varying channels. Our advanced analysis considers both transient and steady-state regimes. Furthermore, in our analysis, the popular independence assumption is not adopted, resulting in a stochastic model which is accurate even when: (i) the step size is not infinitesimally small; or (ii) when the unknown system presents a high nonstationary degree. In addition, our evaluation is also able to provide a deterministic theoretical step-size sequence that optimizes algorithmic performance, as well as an accurate step size upper bound that guarantees algorithm stability. Computer simulations performed are in accordance with our theoretical predictions.

General problems of optimal trajectory generation and of optimal space-time rendezvous for autonomous underwater vehicles affected by time-varying fluid flows are formulated and solved in the framework of dynamic programming. The optimal solutions include optimal trajectories, as well as departure times and positions. The approach consists in using the principle of optimality (PO) to embed, for example, an optimal time to reach a target problem from some fixed position and time into a more general problem of finding the optimal time to reach a target from any point and time. The solution of this general problem is given by the value function, the solution of a Hamilton-Jacobi-Bellman equation (HJBE) which expresses the PO in an infinitesimal form. The HJBE is solved using an efficient parallel numerical solver. The problems of interest are solved either by minimizing the value function over one or more variables (e.g., time) or by using level sets of the value function to coordinate departure times for multiple vehicles to rendezvous at a given target. The paper presents a description and an illustration of the approach and briefly discusses how value-function-based calculations provide a very effective way to solve complex motion planning and coordination problems. The discussion is aided by examples modeling real operational scenarios using current velocity forecasts from a state-of-the-art model of the Sado river estuary in Portugal.

This paper presents the first general framework to explain the magnetic force as a result of electric force interactions between current charges moving at any constant speed and combination. The explanation depends on analyzing the spreading electric field in the space and the movement of charges inside current elements. Previous work used special relativity to describe the magnetic force as an electric one, but this description contradicts the fact that electrically neutral wires stay neutral with or without current flowing through them, as well as, it does not facilitate the derivation of the infinitesimal laws of magnetism. In this paper, the provided explanation is proved by deriving the infinitesimal magnetic force law and Biot-Savart law using the basis of electric forces. This work lies at the intersection between Electrical Engineering and Physics, and it is important to understand what is magnetism and its origin. Such understanding may help engineers and scientists in making new advancements in magnetic materials and applied magnetic technologies.

During the Enlightenment period the concept of the infinitesimal was developed as a means to solve the mathematical problem of the incommensurability between human reason and the movements of physical beings. In this essay, the author analyzes the metaphysical prejudices subtending Enlightenment Humanism through the lens of the infinitesimal calculus. One of the consequences of this analysis is the perception of a two-fold possibility occasioned by the infinitesimal. On the one hand, it occasions an extreme form of humanism, “transhumanism,” which exhibits limitless confidence in the possibility of human science. On the other hand, the concept of the infinitesimal also contains within itself a source for a critical “posthumanism,” that is to say, a source which initiates the dissolution of the presuppositions of humanism while simultaneously announcing a different ontological organization. In , Tostoy’s novel takes up the problem of the relation between reason and motion and makes the two-fold possibility visible by presenting a contrast between its theoretical presentations and the lived experiences of the characters in the novel. Thus, is the setting in which the author has chosen to conduct this analysis.

The limit idea is the basic idea of calculus. Almost all concepts in mathematical analysis are inseparable from the limit, but the limit is a concept that is difficult to understand accurately. By assuming that there is a certain point closest to the limit value, this paper provides a reasonable explanation for the infinitesimal paradox and a new answer to the question why the limit value is accurate in teaching. At the same time, this method is applied to the derivative and used to understand a common practical problem in mathematics. The analysis shows that this method is effective for the accurate understanding of limit problems.

In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.

We study non-flat planar 3-webs with infinitesimal symmetries. Using multi-dimensional Schwarzian derivative we give a criterion for linearization of such webs and present a projective classification thereof. Using this classification we show that the Gronwall conjecture is true for 3-webs admitting infinitesimal symmetries., 20 pages

The calculation of rail potential and stray current in the DC traction power system can guide the protection against the rail overvoltage problem and mitigate the electrolytic corrosion due to stray current. To obtain the accurate stray current distribution, the infinitesimal method based calculation of metro stray current in multiple power supply sections (PPSs) is studied in this paper. The through-type structure characteristics of power supply system and that of return current system which is constituted of rails - stray current collection mats - tunnel reinforcements - ground (RSTG) are considered in the proposed calculation method. In the traction power supply system, the currents injected into the rail by train, and absorbed by traction power substation (TPS) or regeneratively braking train, are solved by the power flow calculation which considers stray current. In the return current system, these currents are used as the boundary conditions for stray current calculation. Then the rail potential and stray current in multiple PPSs are solved by the infinitesimal method. The verification result shows that the proposed calculation method can effectively calculate the rail potential and stray current in multiple metro PPSs. At last, the novel calculation method is used to gain the dynamic stray current along the metro line and evaluate the influence factors of stray current.

The equation for fluid flow in porous media is analyzed in this paper with the aid of Lie symmetry method (LSM) and invariant subspace method (ISM). Infinitesimal generators, the entire geometric fields of the vectors and the symmetry groups of the equation being considered are given. One-dimensional optimal systems of sub-algebra are reported with corresponding reduced nonlinear ordinary differential equations. By means of ISM, we determine the exact solutions and invariant subspaces (ISs) for the equation under consideration. Moreover, with the aid of the new theorem of conservation, we establish the conservation laws (CLs) for the governing equation. The construction of the conserved vectors reveals the integrability and existence of soliton solutions of the equation for fluid flow in porous media.

The H2 norm of an exponentially stable system described by Delay Differential Algebraic Equations (DDAEs) might be infinite due to the existence of hidden feedthrough terms and, as shown in this paper, it might become infinite as a result of infinitesimal changes to the delay parameters. We first introduce the notion of strong H2 norm of semi-explicit DDAEs, a robustified measure that takes into account delay perturbations, and we analyze its properties. Next, we derive necessary and sufficient finiteness criteria for the strong H2 norm, in terms of a frequency sweeping test over a hypercube, and in terms of a finite number of equalities involving multi-dimensional powers of a finite set of matrices. As the main contribution, we present a strengthened, sufficient, condition for finiteness of the strong H2 norm, along with an algorithm for checking it, which has significantly better scalability properties in terms of both the dimension of the system and the number of delays. We show that the satisfaction of the novel condition is equivalent to the existence of a simultaneous block triangularization of the matrices of the delay difference equation associated to the DDAE. The latter is instrumental to a novel regularization procedure that allows to transform the DDAE to a neutral type system with the same transfer matrix, without any need for differentiation of inputs or outputs. As we illustrate, this transformation enables for instance to compute the strong H2 norm using an established approach grounded in Lyapunov matrices. Finally, we investigate the conservatism of the sufficient finiteness condition.We show by a counterexample that the condition is in general not necessary, inducing open problems, but we also list several classes of DDAEs for which it is necessary and sufficient.

For a long time, physicists always directly use mathematical tools to deal with physical problems, and few people pay attention to the difference between mathematical theory and physical theory. Just like the dilemma that physicists once faced when dealing with the problem of blackbody radiation function. By analyzing the difference between the theoretical basis of mathematics and the theoretical basis of physics, this paper draws the following conclusions: (1) The theoretical basis of mathematics and the theoretical basis of physics are different, so when we use mathematical tools for physics research, we need to be very careful. (2) Finiteness and discreteness should be the basis of the whole physical theory; This paper points out that it is not advisable to use infinite " " and infinitesimal "0" without restriction and demonstration in physics, as well as the continuity of functions, which will bring a lot of trouble to physical theory. At the same time, through the analysis of Banach-Tarski paradox and Bertrand paradox, this paper proposes that if we revise the basic assumptions of probability theory: assuming that "points" have quantized sizes, and "lines" also have quantized widths. After the correction, we can not only avoid the troubles caused by Bertrand paradox, but also make probability theory better for practical application.

Let F be a non-archimedean o-field, $$C=F(i)$$ the imaginary quadratic extension field of F with $$i^2=-1$$ . In this paper, all directed partial orders on C are classified via the new concept of doubly convex set consisting of some infinitesimals. This unifies the previous work Ma et al. (Order 34(1):37–44, 2017; Order 35(3):461–466, 2018; Positivity 24(3):1001–1007, 2019). It is surprising that this new theory applies well to the quaternions $$H=F+Fi+Fj+Fk$$ over F and all directed partial orders on H are classified. As an application, the Fuchs’ problem is answered negatively for H.

A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint frameworks in a variety of such contexts. The main results are combinatorial characterisations for (i) frameworks restricted to surfaces with both Euclidean and geodesic distance constraints, (ii) frameworks in the plane with Euclidean and non-Euclidean distance constraints, and (iii) direction-length frameworks in the non-Euclidean plane., Comment: Revised version, 23 pages, 6 figures. The analysis of the projective plane frameworks (whose proofs were incomplete) has been removed for discussion elsewhere. This double-distance context is defined in the final section of the revision together with other settings with underlying (2,1)-sparsity

This article provides a computational formula for the the stability of the probability of ruin of the compound Poisson risk process. This stability is the infinitesimal standardized variation of th...

In this paper, finite time rigidity-based formation maneuvering control of single integrator multiagent systems is considered. The target formation graph is assumed to be minimally and infinitesimally rigid, and the desired group velocity is considered to be available only to a subset of the agents. A distributed nonsmooth velocity estimator is used for each agent to estimate the desired group velocity in finite time. Using Lyapunov and input to state stability notions, a finite time distance-based formation maneuvering controller is presented and it is proved that by using the controller, agents converge to the target formation and track the desired group velocity in finite time. Furthermore, it is demonstrated that the designed controller is implementable in local coordinate frames of the agents. Simulation results are provided to show the effectiveness of the proposed control scheme.

This paper investigates conformal invariance and conserved quantities of nonmaterial volumes. An infinitesimal transformation group and infinitesimal transformations vectors of generators are proposed. The definition of conformal invariance and the determining equation for the systems are presented. The necessary and sufficient conditions of conformal invariance being Noether symmetrical simultaneously by the action of infinitesimal transformations are given. The conformal factor is deduced from the conformal invariance and Noether symmetry. The corresponding Noether conserved quantities are derived with the aid of a structure equation. Three examples are given to illustrate application of the method, the corresponding conserved quantities are obtained under the conformal invariance and Noether symmetry.