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, 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.

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)

We use some notions from computability in an uncountable setting to describe a difference between the “Zilber field” of size ℵ1ℵ1 and the “Zilber cover” of size ℵ1ℵ1.

In this paper we study predimension inequalities in differential fields and define what it means for such an inequality to be adequate. Adequacy was informally introduced by Zilber, and here we give a precise definition in a quite general context. We also discuss the connection of this problem to definability of derivations in the reducts of differentially closed fields. The Ax-Schanuel inequality for the exponential differential equation (proved by Ax) and its analogue for the differential equation of the j-function (...) (established by Pila and Tsimerman) are our main examples of predimensions. We carry out a Hrushovski construction with the latter predimension and obtain a natural candidate for the first-order theory of the differential equation of the j-function. It is analogous to Kirby's axiomatisation of the theory of the exponential differential equation (which in turn is based on the axioms of Zilber's pseudo-exponentiation), although there are many significant differences. In joint work with Sebastian Eterović and Jonathan Kirby we have recently proven that the axiomatisation obtained in this paper is indeed an axiomatisation of the theory of the differential equation of the j-function, that is, the Ax-Schanuel inequality for the j-function is adequate. (shrink)

Can quantum communication be more efficient than its classical counterpart? Holevo's theorem rules out the possibility of communicating more than n bits of classical information by the transmission of n quantum bits—unless the two parties are entangled, in which case twice as many classical bits can be communicated but no more. In apparent contradiction, there are distributed computational tasks for which quantum communication cannot be simulated efficiently by classical means. In some cases, the effect of transmitting quantum bits cannot be (...) achieved classically short of transmitting an exponentially larger number of bits. In a similar vein, can entanglement be used to save on classical communication? It is well known that entanglement on its own is useless for the transmission of information. Yet, there are distributed tasks that cannot be accomplished at all in a classical world when communication is not allowed, but that become possible if the non-communicating parties share prior entanglement. This leads to the question of how expensive it is, in terms of classical communication, to provide an exact simulation of the spooky power of entanglement. (shrink)

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)

A pseudotree is a partially ordered set for which is a linear ordering for each . Define , the pseudo treealgebra over T, as the subalgebra of the power set of T generated by where . It is shown that every pseudo treealgebra is embeddable into an interval algebra; thus it is a retractive Boolean algebra. Moreover, superatomicity of is described using conditions on.

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.

The sunspot data seems to indicate that the Sun is likely to enter Maunder Minimum, then it will mean that low Sun activity may cause low temperature in Earth. If this happens then it will cause a phenomenon which is called by some climatology experts as “The Little Ice Age” for the next 20-30 years, starting from the next few years. Therefore, the Earth climate in the coming years tend to be cooler than before. This phenomenon then causes us to (...) ask: what can we do as human being in Earth to postpone or avoid the worsening situation in terms of Earth cooling temperature in the coming years? We think this is a more pressing problem for the real and present danger that we are facing in the Earth. What we are suggesting in this paper is that perhaps it is possible to model Sun-Earth interaction in terms of Shannon entropy. Since Shannon entropy can be expressed as bits of information, then it would mean that perhaps we can do something with Earth temperature by controlling the amount of information transfer and storage in the Earth. This proposal is somewhat in resemblance with message of a 2012 movie “A Thousand words” where we shall strive to love our neighbours and nature, instead of being absorbed in a culture of less-meaningful fast-talk (starred by Eddie Murphy). (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)

Pseudo‐mechanistic explanations in psychology and cognitive neuroscienceThis paper focuses on the level of systems/cognitive neuroscience. It argues that the great majority of explanations in psychology and cognitive neuroscience is “pseudo‐mechanistic.” On the basis of various case studies, Hommel argues that cognitive neuroscience should move beyond what he calls an “Aristotelian phase” to become a mature “Galilean” science seeking to discover actual mechanisms of cognitive phenomena.

Este artículo se concentra en la discusión crítica que Pseudo Justino dirigió hacia ciertos principios básicos de la filosofía de la naturaleza de Aristóteles. Pseudo Justino dirige su crítica a la doctrina aristotélica de la eternidad del movimiento, del tiempo y del mundo. También argumenta contra Aristóteles para mostrar la no-eternidad del éter, implicando una unificación de los reinos terrestre y celestial.

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.

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)

This paper argues that Hindi incorporation is, in fact, pseudoincorporation, involving noun phrases rather than nouns. Furthermore, it shows that there is no requirement that the incorporated nominal form a morphological or even a syntactic unit with the verb. Such loosely aligned nominals can nevertheless be identified as incorporation on the basis of semantic intuitions having to do with number interpretation, anaphora, and certain properties typically associated with lexical processes. Contrary to standard assumptions, it is argued that the targets of (...) pseduo-incorporation are specified for number. Singular incorporated nominals in Hindi are shown to be semantically singular, with number neutrality arising as a consequence of interaction with aspectual specification. Taking aspectual information into account is also shown to have interesting implications for current approaches to the semantics of incorporation, one in which the incorporated nominal introduces a discourse referent, and one in which it functions as a predicate modifier. A closer look at the effect of aspect on anaphora, for example, does not unequivocally support the predicate modification view of pseudoincorporation. The paper also explores the cross-linguistic applicability of the claims made on the basis of Hindi. Most notably, a distinction between Hungarian verbs with respect to incorporation of bare singulars provides striking confirmation of the claims that number morphology is semantically visible in pseudo-incorporation. It also discusses the restrictions on the productivity of pseudo-incorporation in light of the findings for Hindi and Hungarian. (shrink)

This study employs a newly developed methodology called the variational homotopy perturbation transformation method to study fractional acoustic wave equations. The motivation for this study is to extend the variational homotopy perturbation technique to the variational homotopy perturbation transformation technique in the sense of the Yang–Caputo–Fabrizio operator. The suggested method demonstrated a straightforward and accurate technique for investigating fractional-order partial differential equations. The technique’s validity is demonstrated through the use of several illustrative instances. The obtained answers were found to be (...) extremely near to the precise solutions. Additionally, the proposed strategy achieves the best degree of accuracy. Indeed, the current technique can be seen as one of the analytic strategies for solving nonlinear fractional partial differential equations compared to other analytical techniques. (shrink)

Irregularity may occur on the earth’s surface in the form of mountains due to the imperfection of the earth’s crust. To explore the influence of horizontally polarized shear waves on mountains, we considered the fluid-saturated porous medium over an orthotropic semi-infinite medium with rigid and soft mountain surfaces for wave propagation. The mountain surface is defined mathematically as a periodic function of the time domain. The physical interpretation of materials’ structure has been explained in rectangular Cartesian coordinate system originated at (...) the contact interface of layer and half-space. The displacement of the mountains has been derived by solving energy equations analytically. The influence of rigid and soft mountain surfaces on the phase velocity of shear waves has been demonstrated graphically. (shrink)

The article presents the preliminary results of the author’s study of the unedited homily De transfiguratione et eleemosyna, ascribed to John Chrysostom. The question of the manuscript tradition is first discussed. The article shows that Maurice Sachot is right when he indicates only the manuscript Romanus Angelicus gr. 125 as a manuscript witness of this homily and that the other two witnesses indicated by Pinakes are erroneous. Then, the descriptions of the folios that preserve the homily are analysed in the (...) light of a new examination of them. This re-examination shows that hitherto several sources that inspired the compiler have gone unnoticed. This is followed by a brief summary of the contents of the homily and an analysis of the passages in which almsgiving is mentioned.Contribution: The article offers a re-examination of the folios 353v–361 from the manuscript Romanus Angelicus gr. 125, the first presentation of the content of the homily De transfiguratione et eleemosyna, and tries to shed light on the association of the theme of the Transfiguration with almsgiving. (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.

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)

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)

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)

In Lucian's highly competitive and exhibitionist world, hyper-Atticism, the use of recondite, archaic words for the sake of impression, has become a sort of plague. In this article, I discuss how Lexiphanes focuses precisely on the literal and metaphorical associations of hyper-Atticism as a disease, by paying particular attention on the medical verdict - articulated in the text by Lucian's authorial double, Lycinus - that the dialogue's eponymous character suffers from melancholia. Rather than constitute a passing reference to the colloquial (...) vocabulary of insanity, melancholia, I argue, helps Lucian forcefully assimilate the accumulation of pretentious words to the raving of the insane, a non-sensical blabbering that is void of meaning. At the same time, Lucian aims also to expose the presence of melancholy in the cultural and medical idiom of his time as a disease that typically affects 'great spirits', people of exceptional intelligence. By calling Lexiphanes 'melancholic', Lucian scolds Lexiphanes' pretentiousness both as a hyper-Atticist and, no less importantly, as a pseudo-intellectual who is shaping his public image, temperamentally and physiognomically, as a genius whose atrabilious constitution entitles him to act and speak in strange ways. (shrink)

Those rights are human rights which, in Professor Gewirth's phrase, “all persons equally have simply insofar as they are human.” His task is to demonstrate that there are human rights, and to demonstrate that such demonstration is necessary to the very existence of these rights. “That human rights exist…is a proposition whose truth depends upon the possibility, in principle, of constructing a body of moral justificatory argument from which that proposition follows as a logical consequence.” As philosophers we should no (...) doubt like to be able to prove the existence of human rights – prove that there are such rights in the event that the fool shall have said in his heart that there are none, even using his folly against him by showing his denial to entail its denial – but it is a bold claim that rights are things whose esse est demonstrari. (shrink)

A new structure, called pseudo equality algebras, will be introduced. It has a constant and three connectives: a meet operation and two equivalences. A closure operator will be introduced in the class of pseudo equality algebras; we call the closed algebras equivalential. We show that equivalential pseudo equality algebras are term equivalent with pseudo BCK-meet-semilattices. As a by-product we obtain a general result, which is analogous to a result of Kabziński and Wroński: we provide an equational (...) characterization for the equivalence operations of pseudo BCK-meet-semilattices. Our result treats a much more general algebraic structure, namely, pseudo BCK-meet-semilattice instead of Heyting algebras, on the other hand, we also need to use the meet operation. Finally, we prove that the variety of pseudo equality algebras is a subtractive, 1-regular, arithmetical variety. (shrink)

Includes Greek text of Epinomis (p. 182-202).

This chapter presents a critical edition and translation of so-called Pseudo-Joscelin’s ‘Tractatus de generibus et speciebus’, an anonymous work from the early twelfth century, which offers a sustained treatment of mereological metaphysics unlike any other work we know, as well as providing information about philosophical views of universals held at the time. The presentation includes discussion of the authorship, unity, date, and philosophical significance of the treatise, with special attention given to its defense of atomism and ‘collective realism’.

Pseudo-patient studies are studies in which a medical sociologist or anthropologist masquerades as a patient. Medical treatment is sought without revealing that the 'patient' is really a covert research worker. When access has thus been gained to a medical setting--typically a hospital ward--social interaction between medical staff and patients is then observed over a period of days or weeks. Important studies have been carried out in this way of psychiatric treatment and of the care of the terminally-ill. Is the (...) use of the method justified? What ethical problems does its use raise? How do the undoubted advantages of the approach compare with the clear drawbacks and objections which can be made to its use? (shrink)