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The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution operators, where unboundedness of evolution operator is an essential ingredient of nonlinear analysis. In conclusion a general framework for the identification between the infinitesimal generators with evolution operators is established. A mathematical framework for such an identification is indispensable to the rigorous treatment of nonlinear transforms: e.g., transforms appearing in the theory of integrable systems., To appear in Math. Meth. Appl. Sci

Novel Branching on Integrals takes a single integration example derived through the standardized dingbat expression permutation formation regulations and expands deprograms the zero using the deprogramming function.What remains is the missing component - but where did it go? Read on to find out. What we find is that there are new branches of mathematics hanging by the integrals.

The uniqueness of each viewpoint, each point of effect, can be "overcome" only by changing the viewpoint to other viewpoints and returning. Such an alternation, which can also appear as constant change, makes up the unity of the world. The wholeness of an alternation, however, is a consciousness structure because of the special relationship between the circumscribing periphery and the infinitesimal center. This process structure unites determinacy and indeterminacy at every point also totally. We are dealing, therefore, with forms of consciousness everywhere, with more or less freedom of choice and an increasingly unknown depth. We live in a world of choosing consciousness or better: awareness. In this respect, our environment expresses a deep truth about ourselves.Published in: Journal of Consciousness Exploration & Research (2020), 11(1), p.1-16.

The kinetic description of relativistic plasmas in the presence of time-varying and spatially non-uniform electromagnetic fields is a fundamental theoretical issue both in astrophysics and plasma physics. This refers, in particular, to the treatment of collisionless and strongly-magnetized plasmas in the presence of intense radiation sources. In this paper the problem is investigated in the framework of a covariant gyrokinetic treatment for Vlasov-Maxwell equilibria. The existence of a new class of kinetic equilibria is pointed out, which occur for spatially-symmetric systems. These equilibria are shown to exist in the presence of non-uniform background EM fields and curved space-time. In the non-relativistic limit this feature permits the determination of kinetic equilibria even for plasmas in which particle energy is not conserved due to the occurrence of explicitly time-dependent EM fields. Finally, absolute stability criteria are established which apply in the case of infinitesimal symmetric perturbations that can be either externally or internally produced., Comment: 8 pages

Over a period of 350 years or more, calculus has evolved conceptually and in notation. Up until recently, calculus was described using infinitesimals, which are numbers so small, they can be ignored in certain products. This led to arguments about “ratios of infinitesimally small quantities” and “ratios of evanescent quantities”. Eventually, it was the French mathematician Augustin-Louis Cauchy (1789–1857), and the German mathematician Karl Weierstrass (1815–1897), who showed how limits can replace infinitesimals. However, in recent years, infinitesimals have bounced back onto the scene in the field of “non-standard analysis”, pioneered by the German mathematician Abraham Robinson (1918–1974). Robinson showed how infinitesimal and infinite quantities can be incorporated into mathematics using simple arithmetic rules: $$\begin{aligned} \text {infinitesimal} \times \text {bounded} &= \text {infinitesimal} \\ \text {infinitesimal}\times \text {infinitesimal}&=\text {infinitesimal} \end{aligned}$$ where a bounded number could be a real or integer quantity. So, even though limits have been adopted by modern mathematicians to describe calculus, there is still room for believing in infinitesimal quantities.

Up to equivalence, this paper classifies all the irreducible unitary representations with non-zero Dirac cohomology for the simple Lie group $E_{6(-14)}$, which is of Hermitian symmetric type. Each FS-scattered Dirac series of $E_{6(-14)}$ is realized as a composition factor of certain $A_{\mathfrak{q}}(\lambda)$ module. Along the way, we have also obtained all the fully supported irreducible unitary representations of $E_{6(-14)}$ with integral infinitesimal characters., Comment: 32 pages, some strings are folded

Leibniz described imaginary roots, negatives, and infinitesimals as useful fictions. But did he view such 'impossible' numbers as mathematical entities? Alice and Bob take on the labyrinth of the current Leibniz scholarship., 12 pages, to appear in The Mathematical Intelligencer

In this paper, we study $$\eta$$ η -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , μ ) -manifolds. As an application, it is proved that if an almost cosymplectic $$(k,\mu )$$ ( k , μ ) -metric with $$k k < 0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by $$k k < 0 . In addition, a concrete example is constructed to illustrate the above result.

Based on the classical Lie group method, a generalized two-component Hunter-Saxton system is studied in this paper. All of the its geometric vector fields, infinitesimal generators and the commutation relations of Lie algebra are derived. Furthermore, the similarity variables and symmetry reductions of this new generalized two-component Hunter-Saxton system are derived. Under these Lie symmetry reductions, some exact solutions are obtained by using the symbolic computation. Moreover, a conservation law of this system is presented by using the multiplier approach.

In this paper, we introduce a concept called the finite-time expected deviation exponent (FTEDE), which measures the expected separation rate of a particle with another initially infinitesimally close but randomly sampled particle over a finite time period. The proposed FTEDE can be viewed as a stochastic version of the traditional finite-time Lyapunov exponent (FTLE) and is also a useful tool to measure the chaotic behaviors of continuous dynamical systems.

A universal method of grinding force modeling in arbitrary 2D freeform grinding is proposed, which is based on the infinitesimal approach. The main idea of this method is to discretize the freeform surface into a group of surfaces with very small widths along the profile of the surface cross section, which can be considered as flat surfaces inclined at different angles. The form grinding process can be considered as a series of inclined flat surface grinding, where the normal grinding force of each flat inclined surface is divided into portions along specific directions. The process of inclined flat surface grinding at certain angle is equivalently converted to flat surface grinding in terms of the same abrasive wheel peripheral velocity and workpiece velocity but different cutting depth and contacting curvature in inclined view plane. A mathematical model of grinding force in flat surface grinding (FSG) is established, where the sliding stage, plowing stage and chip formation stage are considered respectively with 9 coefficients which are determined with linear regression from the grinding experiments. The forces in form grinding are then obtained by summing the individual inclined surfaces grinding forces, which are verified via rail form grinding (RFG). The algorithm was validated by means of the grinding experiments with average relative percentage errors of 6.08% and 7.01% respectively on tangential grinding force and normal grinding force in FSG and average relative percentage errors of 8.47% and 10.37% respectively on longitudinal grinding force and vertical grinding force in RFG.

We introduce a new approach to bargaining, with strategic and axiomatic foundations, into models of decentralized asset markets. According to this approach, whichencompasses the Nash (1950) solution as a special case, bilateral negotiations follow an agenda that partitions assets into bundles to be sold sequentially. We construct twoalternating-offer games consistent with this approach and characterize their subgame perfect equilibria. We show the revenue of the asset owner is maximized when assetsare sold one infinitesimal unit at a time. In a general equilibrium model with endogenous asset holdings, gradual bargaining reduces asset misallocation and preventsmarket breakdowns.

George Berkeley is known as “The Good Bishop,” a name celebrating his faith, pastoral ministry and earnest commitment to his philosophical views. To mathematicians, he is known for his agitated per...

In this research, we discussed the equilibruim points in the restricted three body problem (in brief RTBP) such as computing infinitesimal orbits. We investigated the dynamics of the orbits around the equilibrium points in the oblate restricted of three bodies problem. We costructed the problem formulation of infinitesimal orbit by applying the perturbation forces, the Hamiltonian of the dynamical problem is formulated. The transformed Hamiltonian near any one of the equilibrium points is obtained. A Lie operator method is applied to find the solution of the dynamical problem. The explicit first and second order solutions for the coordinates and their conjugate momenta of a particle in the infinitesimal orbit around equilibrium points are obtained. Numerical simulation of infinitesimal orbit around equilibrium points in the Earth-Moon (EM) system are deduced.

UDC 519.218.2 We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Markov branching processes. We treat the critical case so that the infinitesimal generating function of the process has the infinite second moment, but it regularly varies with the remainder. We improve the basic lemma of the theory of critical Markov branching processes and refine known limit results.

In the present paper, some new exact solutions of the (3+1)-dimensional sinh-Gorden equation are displayed. First, based on the Lie point symmetry, basic infinitesimal generators given; also, and the Lie algebra and commutative relations are shown. Second, some exact solutions are obtained using the F-expansion method, and a great many Jacobian-elliptic function solutions are presented. Finally, other forms of solutions are derived using the traveling wave transform.

Our aim in this study is to establishing nonstandard foundations, definitions and theorems for completion a noncomplete metric spaces. We have a lot of space or sets X which agree with all usual properties of complete, except at a small size subset of it. In this paper, by using nonstandard analysis tools founded by A. Robinson and axiomatized by E. Nelson, we try to introduce a new notion of completion corresponding to nonstandard modified metric , and to use it to reprove the classical (standard) completion theorem and to overcome the incompetence of those spaces which has deprivation at a small size subset.

On the present work it is presented a new category of continued fraction functions (CFF) of a real variable $$\lambda$$ , named “the $$\epsilon$$ class”, such that $$\lambda$$ enters in the CFF through small or infinitesimal contributions in the partial quotients. These CFF emerge on the secular equation of the Sturm–Liouville equation (SLE) by a finite difference treatment. Also, it was obtained the eigenvalues of the one-dimensional Schrodinger equation (a particular case of SLE) from the CFF secular equation. The case of the two-dimensional Schrodinger equation was partially studied.

The tensor product of $${\mathbb {A}}^1$$ -invariant sheaves with transfers introduced by Voevodsky is generalized to reciprocity sheaves via the theory of modulus presheaves with transfers. We prove several general properties of this construction and compute it in some cases. In particular we obtain new (motivic) presentations of the absolute Kahler differentials and the first infinitesimal neighborhood of the diagonal.

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2 + 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots.

It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers$$\widetilde{\mathbb {R}}$$R~does not generalize classical results. E.g. the sequence$$\frac{1}{n}\not \rightarrow 0$$1n↛0and a sequence$$(x_{n})_{n\in \mathbb {N}}$$(xn)n∈Nconvergesifand only if$$x_{n+1}-x_{n}\rightarrow 0$$xn+1-xn→0. This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that$$\widetilde{\mathbb {R}}$$R~is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.

The information dynamics in finance and insurance applications is usually modelled by a filtration. This paper looks at situations where information restrictions apply so that the information dynamics may become non-monotone. A fundamental tool for calculating and managing risks in finance and insurance are martingale representations. We present a general theory that extends classical martingale representations to non-monotone information generated by marked point processes. The central idea is to focus only on those properties that martingales and compensators show on infinitesimally short intervals. While classical martingale representations describe innovations only, our representations have an additional symmetric counterpart that quantifies the effect of information loss. We exemplify the results with examples from life insurance and credit risk.

This article investigates the localization problem of a network in 2-D and 3-D spaces given the positions of anchor nodes in a global frame and internode relative measurements in local coordinate frames. It is assumed that the local frames of different nodes have different unknown orientations. First, an angle-displacement rigidity theory is developed, which can be used to localize all the free nodes by the known positions of the anchor nodes and local relative measurements (local relative position, distance, local relative bearing, angle, or ratio-of-distance measurements). Then, necessary and sufficient conditions for network localizability are given. Finally, a distributed network localization protocol is proposed, which can globally estimate the locations of all the free nodes of a network if the network is infinitesimally angle-displacement rigid. The proposed method unifies local-relative-position-based, distance-based, local-relative-bearing-based, angle-based, and ratio-of-distance-based distributed network localization approaches. The novelty of this article is that the proposed method can be applied in both generic and nongeneric configurations with an unknown global coordinate frame in both 2-D and 3-D spaces. National Research Foundation (NRF) Accepted version

This paper presents an analysis of the sorites paradox for collective nouns and gradable adjectives within the framework of classical logic. The paradox is explained by distinguishing between qualitative and quantitative representations. This distinction is formally represented by the use of a different mathematical model for each type of representation. Quantitative representations induce Archimedean models, but qualitative representations induce non-Archimedean models. By using a non-standard model of $$ {\mathbb{R}} $$ called $$ the\,hyperreal\,numbers $$ , which contains infinite and infinitesimal numbers, the two paradoxes are shown to have distinct structures. The sorites paradox for collective nouns arises from the use of infinite numbers, whereas the sorites paradox for gradable adjectives arises from the use of infinitesimal numbers. Each paradox can be traced to a different source of vagueness. The sorites paradox for collective nouns is caused by $$ semantic \,\,indeterminacy $$ , and the sorites paradox for gradable adjectives is caused by $$ epistemic $$ $$ indiscriminability $$ . If correct, this analysis implies that infinite and infinitesimal numbers are cognitively real, and that they play a role in the semantic interpretation of natural language.

Suppose the 300+ published asset pricing factors are all spurious. How much p-hacking is required to produce these factors? If 10,000 researchers generate 8 factors every day, it takes hundreds of years. This is because dozens of published t-statistics exceed 6.0, while the corresponding p-value is infinitesimal, implying an astronomical amount of p-hacking in a general model. More structure implies p-hacking cannot address ≈100 published t-statistics that exceed 4.0, as they require an implausibly non-linear preference for t-statistics or even more p-hacking. These results imply mispricing, risk, and/or frictions have a key role in stock returns.

Fil: Esquisabel, Oscar M. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Esquisabel, Oscar M. Universidad Nacional de Quilmes. Centro de Estudios de Filosofía e Historia de la Ciencia; Argentina Fil: Esquisabel, Oscar M. Pontificia Universidad Católica Argentina; Argentina Fil: Raffo Quintana, Federico. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Raffo Quintana, Federico. Pontificia Universidad Católica Argentina. Facultad de Filosofía y Letras; Argentina This paper is concerned with the status of mathematical fictions in Leibniz’s work and especially with infinitary quantities as fictions. Thus, it is maintained that mathematical fictions constitute a kind of symbolic notion that implies various degrees of impossibility. With this framework, different kinds of notions of possibility and impossibility are proposed, reviewing the usual interpretation of both modal concepts, which appeals to the consistency property. Thus, three concepts of the possibility/impossibility pair are distinguished; they give rise, in turn, to three concepts of mathematical fictions. Moreover, such a distinction is the base for the claim that infinitesimal quantities, as mathematical fictions, do not imply an absolute impossibility, resulting from self-contradiction, but a relative impossibility, founded on irrepresentability and on the fact that it does not conform to architectonic principles. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz view, a presumptive or “conjectural” status.