The “high school algebra” laws of exponentiation fail in the ordinal arithmetic of sets that generalizes the arithmetic of the von Neumann ordinals. The situation can be remedied by using an alternative arithmetic of sets, based on the Zermelo ordinals, where the high school laws hold. In fact the Zermelo arithmetic of sets is uniquely characterized by its satisfying the high school laws together with basic properties of addition and multiplication. We also show how in both arithmetics the behavior of (...) exponentiation depends on whether the empty set is an element of the base. (shrink)

We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.

In this paper, the stability of switched neural networks with interval parameter uncertainties and time delays is investigated. First, the conditions for the existence and uniqueness of the equilibrium point of the system are discussed. Second, the average dwell time approach and M-matrix property are employed to obtain conditions to ensure the globally exponential stability of the delayed SNNs under constrained switching. Third, by resorting to inequality technique and the idea of vector Lyapunov function, sufficient condition to ensure the robust (...) exponential stability of the delayed SNNs under arbitrary switching is derived. The form of the constructed Lyapunov functions is simple, which has certain commonality in studying delayed SNNs, and the proposed results not only are explicit but also reveal the relationship between the constrained switching and the arbitrary switching of the SNNs. Finally, two numerical examples are presented to illustrate the effectiveness and less conservativeness of the main results compared with the existing literature. (shrink)

We study the deviations from the exponential decay law, both in quantum field theory (QFT) and quantum mechanics (QM), for an unstable particle which can decay in (at least) two decay channels. After a review of general properties of non-exponential decay in QFT and QM, we evaluate in both cases the decay probability that the unstable particle decays in a given channel in the time interval between t and t+dt. An important quantity is the ratio of the probability of decay (...) into the first and the second channel: this ratio is constant in the Breit-Wigner limit (in which the decay law is exponential) and equals the quantity Γ1/Γ2, where Γ1 and Γ2 are the respective tree-level decay widths. However, in the full treatment (both for QFT and QM) it is an oscillating function around the mean value Γ1/Γ2 and the deviations from this mean value can be sizable. Technically, we study the decay properties in QFT in the context of a superrenormalizable Lagrangian with scalar particles and in QM in the context of Lee Hamiltonians, which deliver formally analogous expressions to the QFT case. (shrink)

We present theories of gravitation and electric potentials with exponential dependence on the reciprocal distance. In the context of this kind of electric potential we investigate the dynamics of a relativistic electron interacting with a proton.

In this paper, dynamic behavior analysis has been discussed for a class of switched complex-valued neural networks with interval parameter uncertainties and impulse disturbance. Sufficient conditions for guaranteeing the existence, uniqueness, and global robust exponential stability of the equilibrium point have been obtained by using the homomorphism mapping theorem, the scalar Lyapunov function method, the average dwell time method, and M-matrix theory. Since there is no result concerning the stability problem of switched neural networks defined in complex number domain, the (...) stability results we describe in this paper generalize the existing ones. The effectiveness of the proposed results is illustrated by a numerical example. (shrink)

We present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, (...) using techniques from tropical geometry.We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.Abstract prepared by Francesco Paolo GallinaroE-mail: [email protected]: https://etheses.whiterose.ac.uk/31077/. (shrink)

In this paper, we prove that a pseudo-exponential field has continuum many nonisomorphic countable real closed exponential subfields, each with an order-preserving exponential map which is surjective onto the nonnegative elements. Indeed, this is true of any algebraically closed exponential field satisfying Schanuel’s conjecture.

Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers and Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper, we will present a natural refinement of their definition that allows for stability results to hold within the wider class of [Formula: see text]-minimal structures. One of the main technical improvements is that we remove the requirement of definable Skolem functions from the proofs. As (...) a result, we obtain stability in particular for all intermediate structures between the semi-algebraic and the sub-analytic languages. (shrink)

We show that the set of natural numbers has an exponential diophantine definition in the rationals. It follows that the corresponding decision problem is undecidable.

We extend the field of Laurent series over the reals in a canonical way to an ordered differential field of “logarithmic-exponential series” , which is equipped with a well behaved exponentiation. We show that the LE-series with derivative 0 are exactly the real constants, and we invert operators to show that each LE-series has a formal integral. We give evidence for the conjecture that the field of LE-series is a universal domain for ordered differential algebra in Hardy fields. We define (...) composition of LE-series and establish its basic properties, including the existence of compositional inverses. Various interesting subfields of the field of LE-series are also considered. (shrink)

The combinatorial parity principle states that there is no perfect matching on an odd number of vertices. This principle generalizes the pigeonhole principle, which states that for a fixed bipartition of the vertices, there is no perfect matching between them. Therefore, it follows from recent lower bounds for the pigeonhole principle that the parity principle requires exponential-size bounded-depth Frege proofs. Ajtai previously showed that the parity principle does not have polynomial-size bounded-depth Frege proofs even with the pigeonhole principle as an (...) axiom schema. His proof utilizes nonstandard model theory and is nonconstructive. We improve Ajtai's lower bound from barely superpolynomial to exponential and eliminate the nonstandard model theory. Our lower bound is also related to the inherent complexity of particular search classes . In particular, oracle separations between the complexity classes PPA and PPAD, and between PPA and PPP also follow from our techniques. (shrink)

We prove an exponential lower bound on the size of proofs in the proof system operating with ordered binary decision diagrams introduced by Atserias, Kolaitis and Vardi [2]. In fact, the lower bound applies to semantic derivations operating with sets defined by OBDDs. We do not assume any particular format of proofs or ordering of variables, the hard formulas are in CNF. We utilize (somewhat indirectly) feasible interpolation. We define a proof system combining resolution and the OBDD proof system.

Simpson has claimed that “ATR0 is the weakest set of axioms which permits the development of a decent theory of countable ordinals” [8]. This paper provides empirical support for Simpson's claim. In particular, Cantor's Normal Form Theorem and Sherman's Inequality for countable well-orderings are both equivalent to ATR0. The proofs of these results require a substantial development of ordinal exponentiation and a strengthening of the comparability result in [3].

V i 2 ⊢A iff for some term t :S i 2 ⊢ “2 i exists→ A”, a bounded first-order formula, i ≥1. V i 2 is not Π b 1 -conservative over S i 2 . Any model of V 2 not satisfying Exp satisfies the collection scheme BΣ 0 1 . V 1 3 is not Π b 1 -conservative over S 2.

This paper deals with the exponential stabilization problem for a class of nonlinear switched systems with mixed delays under asynchronous switching. The switching signal of the switched controller involves delay, which results in the asynchronous switching between the candidate controllers and subsystems. By constructing the parameter-dependent Lyapunov-Krasovskii functional and the average dwell time approach, some sufficient conditions in forms of linear matrix inequalities are presented to ensure the exponential stability of the switched nonlinear system under arbitrary switching signals. In addition, (...) through the special deformation of the matrix and Schur complement, the controllers with asynchronous switching are designed. Finally, a numerical example and a practical example of river pollution control are provided to show the validity and potential of the developed results. (shrink)

This paper investigates the exponential synchronization of complex dynamical networks based on the sampled-data control method. The sampled-data control means that the control input remains unchanged for a long time after each sampling, which can reduce the sampling number. By using the stability theory of the dynamical systems, this paper provides a novel sampling controller and estimates the bound of the sampling interval. Finally, a numerical example is given to demonstrate the effectiveness of the proposed design technique.

In the present paper some tools are given to state the exact number of roots for some simple classes of exponential terms . The result were obtained by generalizing Sturm's technique for real closed fields. Moreover for arbitrary non-zero terms t certain estimations concerning the location of roots of t are given. MSC: 03C65, 03C60, 12L12.

The lexicographic power ΔΓ of chains Δ and Γ is, roughly, the Cartesian power Πγ∈Γ Δ, totally ordered lexicographically from the left. Here the focus is on certain powers in which either Δ = R or Γ = R, with emphasis on when two such powers are isomorphic and on when ΔΓ is 2-homogeneous. The main results are: (1) For a countably infinite ordinal α, Rα* +α ≃ Rα. (2) RR ≄ RQ. (3) For Δ a countable ordinal ≥ 2. (...) ΔR, with its smallest element deleted, is 2-homogeneous. (shrink)

In [F.-V. Kuhlmann, S. Kuhlmann, S. Shelah, Exponentiation in power series fields, Proc. Amer. Math. Soc. 125 3177–3183] it was shown that fields of generalized power series cannot admit an exponential function. In this paper, we construct fields of generalized power series with bounded support which admit an exponential. We give a natural definition of an exponential, which makes these fields into models of real exponentiation. The method allows us to construct for every κ regular uncountable cardinal, 2κ pairwise non-isomorphic (...) models of real exponentiation , but all isomorphic as ordered fields. Indeed, the 2κ exponentials constructed have pairwise distinct growth rates. This method relies on constructing lexicographic chains with many automorphisms. (shrink)

We provide a first order axiomatization of the expansion of the complex field by the exponential function restricted to the subring of integers modulo the first order theory of (Z, +, ·).

My aim is to evaluate a new realist strategy for addressing the pessimistic induction, Ludwig Fahrbach’s (Synthese 180:139–155, 2011) appeal to the exponential growth of science. Fahrbach aims to show that, given the exponential growth of science, the history of science supports realism. I argue that Fahrbach is mistaken. I aim to show that earlier generations of scientists could construct a similar argument, but one that aims to show that the theories that they accepted are likely true. The problem with (...) this is that from our perspective on the history of science we know their argument is flawed. Consequently, we should not be impressed or persuaded by Fahrbach’s argument. Fahrbach has failed to identify a difference that matters between today’s theories and past theories. But realists need to find such a difference if they are to undermine the pessimistic induction. (shrink)

In this work a double exponential time inseparability result is proven for a finitely axiomatizable first order theory Q₊. The theory, subset of Presburger theory of addition S₊, is the additive fragment of Robinson system Q. We prove that every set that separates Q₊` from the logically false sentences of addition is not recognizable by any Turing machine working in double exponential time. The lower bound is given both in the non-deterministic and in the linear alternating time models. The result (...) implies also that any theory of addition that is consistent with Q₊—in particular any theory contained in S₊—is at least double exponential time difficult. Our inseparability result is an improvement on the known lower bounds for arithmetic theories. Our proof uses a refinement and adaptation of the technique that Fischer and Rabin used to prove the difficulty of S₊. Our version of the technique can be applied to any incomplete finitely axiomatizable system in which all of the necessary properties of addition are provable. (shrink)

We explain how the field of logarithmic-exponential series constructed in 20 and 21 embeds as an exponential field in any field of exponential-logarithmic series constructed in 9, 6, and 13. On the other hand, we explain why no field of exponential-logarithmic series embeds in the field of logarithmic-exponential series. This clarifies why the two constructions are intrinsically different, in the sense that they produce non-isomorphic models of Thequation image; the elementary theory of the ordered field of real numbers, with the (...) exponential function and restricted analytic functions. (shrink)

According to Craig Callender [2022], the ‘received view’ across the social sciences is that, when it comes to time and preference, only exponential time discounting is rational. Callender argues that this view is false, even pernicious. Here I endorse what I take to be Callender’s main argument, but only in so far as the received view is understood in a particular way. I go on to propose a different way of understanding the received view that makes it true. In short: (...) When time discounting is suitably conceived, the exponential form of the discounting function is indeed uniquely rational. (shrink)

Education is increasingly vocational and structured to serve the ongoing exponential increase in economic growth. Climate change is an outcome of these same economic values and praxes. Attempts to shift these values and our approach to technology are continually absorbed and overcome by the pressing motif of economic growth. In this article, Ruth Irwin uses Martin Heidegger's concept of the technological enframing of modernity to view economic growth. John Maynard Keynes's notion of economic growth has impacted the pace of consumerism (...) that now permeates every aspect of knowing about the world we live in. Irwin asks us to think through technological enframing anew by looking to an early mechanical Greek artifact, the Antikythera mechanism, which depicts a cyclical notion of time used by ancient cultures to define the rhythm of economic productivity. The earth-centric cosmos embodied by this mechanism helped the Mesopotamian economy stay within the parameters of the local ecology and demonstrates that cyclical economic growth may enable a civilization to maintain a steady state over time and survive for millennia. An earth-centric cosmology creates a different set of values, one that emphasizes the need to regulate the pace of consumerism rather than allowing it free rein “as by an Invisible Hand.” The role of education in this exploration is twofold: first, it is a pivotal site for cultural exploration and transformation; and second, the expectations of the state strictly limit the forms education can take, so that as long as promoting economic growth defines state expectations, education will remain subservient to these values. If we aim to overcome climate change, Irwin concludes, we need to transform the expectations for education from society as well as from teachers. (shrink)

The exponential Papapetrou metric induced by scalar field conforms to observational data not worse than the vacuum Schwarzschild solution. Here, we analyze the origin of this metric as a peculiar space-time within a wide class of scalar and antiscalar solutions of the Einstein equations parameterized by scalar charge. Generalizing the three families of static solutions obtained by Fisher, Janis et al. :878. https://doi.org/10.1103/PhysRevLett.20.878, 1968), and Xanthopoulos and Zannias :2564, 1989), we prove that all three reduce to the same exponential metric (...) provided that scalar charge is equal to central mass, thereby suggesting the universal character of such background scalar field. (shrink)

This paper deals with stochastically globally exponential stability for stochastic impulsive differential systems with discrete delays and infinite distributed delays. By using vector Lyapunov function and average dwell-time condition, we investigate the unstable impulsive dynamics and stable impulsive dynamics of the suggested system, and some novel stability criteria are obtained for SIDSs with DDs and IDDs. Moreover, our results allow the discrete delay term to be coupled with the nondelay term, and the infinite distributed delay term to be coupled with (...) the nondelay term. Finally, two examples are given to verify the effectiveness of our theories. (shrink)

We study the relative strength of the two axioms Every Pell equation has a nontrivial solution Exponentiation is total over weak fragments, and we show they are equivalent over IE1. We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.

Four decades have passed since the EU and China established diplomatic relations in 1975, and now became mutually indispensable economic partners, presenting both an opportunity and challenge. During that time, after the first market reforms were introduced in 1978, China has transitioned from a predominantly agricultural to industrial and service-oriented economy. On 11 December 2001, China also became the 143rd member of the WTO. The aim of this research is to quantitatively compare the US, EU and Chinese GDP from 1995 (...) to 2014, the US and Chinese outward FDI from 1995 to 2013, and the EU 28 Total Imports from and Exports to the US and China between 2002 to 2014, and analyse the impact of the exponentially rising Chinese investments in the European Union (EU). We have found that while the US economy experienced a regression in general, and exhibited outward FDI exponential decrease in particular, the Chinese outward FDI sustained an exponential growth. Our analysis has shown that the now weakened US dollar and diminished US economic and industrial power, along with the US designed and archaic post World War II European institutions, vitiated by the recent US Economic Recession, which has swept along the EU and all other US dependent foreign economies globally, are and will continue to experience the primary impact of the exponentially rising economic investment of China in the EU. Furthermore, we have found that the Chinese economic growth has been affecting both the EU identity and policies. Yet, its full effect is waiting to be realized. The Chinese investment in the EU is estimated to peak during the next decade, when China will become by far the foremost economic partner of the EU. This crossroad may be reached as soon as in few years’ time and contribute towards the EU – Asian regional stabilization. In the changing context of the Sino – EU - US relations, due to the exponentially growing economic prosperity and rising global influence of China, the EU international institutions and socio-political structures will re-emerge with the once forgotten access to the 21st Century Silk Road Economic Belt, “The Silk Road II”, between China and the EU in the foreseeable future. (shrink)

How should one discount utility across time? The conventional wisdom in social science is that one should use an exponential discount function. Such a function is a representation of the axioms that provide a well-defined utility function plus a condition known as stationarity. Yet stationarity doesnt really have much intuitive normative pull on its own. Here I try to cast it in a normative glow by deriving stationarity from two explicitly normative premises, both suggested by the philosophical thesis of temporal (...) neutralism. Putting the argument in this form helps us better understand exponential discounting and challenges to it. (shrink)

This paper is about time preferences, the phenomenon that the very same things are usually considered the less valuable the farther in the future they are obtained. The utilities of those things are discounted at a certain rate. The paper presents a novel normative argument for exponential discount rates, whatever their empirical adequacy. It proposes to take indexical utility seriously, i.e. utilities referring to indexical propositions (that speak of ‘I’, ‘now’, etc.) as opposed to non-indexical propositions. Economic focus is only (...) on the latter, while the former are neglected. The potential ignorance of when is now generates a difference between indexical and non-indexical utility that can be exploited for a novel argument in favour of exponential discount rates. (shrink)

We show that Zilber’s conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary (...) class, answering a question of Kesälä and Baldwin. (shrink)

Working in the weakening of constructive Zermelo-Fraenkel set theory in which the subset collection scheme is omitted, we show that the binary refinement principle implies all the instances of the exponentiation axiom in which the basis is a discrete set. In particular binary refinement implies that the class of detachable subsets of a set form a set. Binary refinement was originally extracted from the fullness axiom, an equivalent of subset collection, as a principle that was sufficient to prove that the (...) Dedekind reals form a set. Here we show that the Cauchy reals also form a set. More generally, binary refinement ensures that one remains in the realm of sets when one starts from discrete sets and one applies the operations of exponentiation and binary product a finite number of times. (shrink)

Developmental states are criticized for rapid “industrialization without enlightenment.” In the last 30 years, China’s breathtaking growth has been achieved at a high environmental and food safety cost. This article, utilizing a recent survey of China’s livestock industry, illustrates the initiating role of China’s developmental state in the exponential expansion of the country’s livestock production. The enthusiastic response of the livestock industry to the many state policy incentives has made China the world’s biggest animal farming nation. Shortage of meat and (...) dairy supply is history. Yet, the Chinese government is facing new challenges of no less a threat to political stability. Production intensification has created a welfare crisis impacting the world’s biggest number of farm animals. The resulting food safety incidents are affecting consumer confidence and health. Untreated waste contributes to the nation’s environmental degradation. Developmental states may have a proud record of growth in the initial stage of industrialization. Their prospects for sustained development have long been questioned. China has come to an important juncture to march towards a sustained development. (shrink)