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.

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 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.

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)

The works of Plato have been a cornerstone of Western philosophy for centuries, inspiring countless readers and thinkers over the course of millennia. But not all of the writings attributed to Plato are genuine. In this scholarly investigation, W.A. Heidel explores the origins and authenticity of some of these so-called 'pseudo-Platonic' texts, providing insights into the ways in which ancient cultures valued and appropriated the ideas of its greatest thinkers. This work has been selected by scholars as being culturally (...) important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (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.

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 develop and test a model of pseudo-transformational leadership. Pseudo-transformational leadership is manifested by a particular combination of transformational leadership behaviors, and is differentiated from both transformational leadership and laissez-faire -leadership. Survey data from senior managers show differential outcomes of transformational, pseudo-transformational, and laissez-faire leadership. Possible extensions of the theoretical model and directions for future research are offered.

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.

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)

According to some cosmologists, the big bang cosmogony and even the (now largely defunct) steady-state theory pose a scientifically insoluble problem of matter-energy creation. But I argue that the genuine problem of the origin of matter-energy or of the universe has been fallaciously transmuted into the pseudo-problem of creation by an external cause. A fortiori, it emerges that the initial "true" and "false" vacuum states of quantum cosmology do not vindicate biblical divine creation ex nihilo at all.

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 explores the influence of the fifth-century Christian Neoplatonist Pseudo-Dionysius the Areopagite (Denys) on the twentieth-century philosopher of science Paul Feyerabend. I argue that the later Feyerabend took from Denys a metaphysical claim—the ‘doctrine of ineffability’—intended to support epistemic pluralism. The paper has five parts. Part one introduces Denys and Feyerabend’s common epistemological concern to deny the possibility of human knowledge of ultimate reality. Part two examines Denys’ arguments for the ‘ineffability’ of God as presented in On the (...) Divine Names . Part three then explores how Feyerabend imported Denys’ account of divine ineffability into his own metaphysics to provide a novel argument for epistemic pluralism. Part four explains the significance of an appreciation of Dionyius’ influence for our understanding of Feyerabend. I conclude that Denys was a significant and neglected influence upon the later Feyerabend. (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)

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)

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)

This work is the second part of a critical textual edition that is devoted to an extraordinarily rich collection of manuscripts with equally remarkable historical significance dating from late Christian antiquity.

This volume is a new edition of a collection of problems in natural science and medicine which has been variously attributed to Aristotle and to Alexander of Aphrodisias; the collection includes material from both the Hellenistic and Roman periods.

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)

We present a game mechanic called pseudo-visibility for games inhabited by non-player characters (NPCs) driven by reinforcement learning (RL). NPCs are incentivized to pretend they cannot see pseudo-visible players: the training environment simulates an NPC to determine how the NPC would act if the pseudo-visible player were invisible, and penalizes the NPC for acting differently. NPCs are thereby trained to selectively ignore pseudo-visible players, except when they judge that the reaction penalty is an acceptable tradeoff (e.g., (...) a guard might accept the penalty in order to protect a treasure because losing the treasure would hurt even more). We describe an RL agent transformation which allows RL agents that would not otherwise do so to perform some limited self-reflection to learn the training environments in question. (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.

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)

De mundo is a protreptic to philosophy in the form of a letter to Alexander the Great and is traditionally ascribed to Aristotle. It offers a unique view of the cosmos, God and their relationship, which was inspired by Aristotle but written by a later author. The author provides an outline of cosmology, geography and meteorology, only to argue that a full understanding of the cosmos cannot be achieved without a proper grasp of God as its ultimate cause. To ensure (...) such a grasp, the author provides a series of twelve carefully chosen interlocking analogies, building a complex picture in the reader's mind. The work develops a distinctly Aristotelian picture of God and the cosmos while paying tribute to pre-Aristotelian philosophers and avoiding open criticism of rival schools of philosophy. De mundo exercised considerable influence in late antiquity and then in the Renaissance and Early Modern times. (shrink)

De mundo is a protreptic to philosophy in the form of a letter to Alexander the Great and is traditionally ascribed to Aristotle. It offers a unique view of the cosmos, God and their relationship, which was inspired by Aristotle but written by a later author. The author provides an outline of cosmology, geography and meteorology, only to argue that a full understanding of the cosmos cannot be achieved without a proper grasp of God as its ultimate cause. To ensure (...) such a grasp, the author provides a series of twelve carefully chosen interlocking analogies, building a complex picture in the reader's mind. The work develops a distinctly Aristotelian picture of God and the cosmos while paying tribute to pre-Aristotelian philosophers and avoiding open criticism of rival schools of philosophy. De mundo exercised considerable influence in late antiquity and then in the Renaissance and Early Modern times. (shrink)

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)

According to the comparative Bayesian concept of confirmation, rationalized versions of creationism come out as empirically confirmed. From a scientific viewpoint, however, they are pseudo-explanations because with their help all kinds of experiences are explainable in an ex-post fashion, by way of ad-hoc fitting of an empirically empty theoretical framework to the given evidence. An alternative concept of confirmation that attempts to capture this intuition is the use novelty criterion of confirmation. Serious objections have been raised against this criterion. (...) In this paper I suggest solutions to these objections. Based on them, I develop an account of genuine confirmation that unifies the UN-criterion with a refined probabilistic confirmation concept that is explicated in terms of the confirmation of evidence-transcending content parts of the hypothesis. (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.

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.

One of the few studies to date which considers in a comprehensive way the relation between these remarkable thinkers. By concrete example and continual reference it illustrates both the pervasive influence of Pseudo-Dionysius and the profound originality of Aquinas.

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.

This paper generalizes rationalizability of a choice function by a single acyclic binary relation to rationalizability by a set of such relations. Rather than selecting those options in a menu that are maximal with respect to a single binary relation, a weakly pseudo-rationalizable choice function selects those options that are maximal with respect to at least one binary relation in a given set. I characterize the class of weakly pseudo-rationalizable choice functions in terms of simple functional properties. This (...) result also generalizes Aizerman and Malishevski's characterization of pseudo-rationalizable choice functions, that is, choice functions rationalizable by a set of total orders. (shrink)

Pseudo-Dionysius’ first principle is hyperousios. By definition, that concept is not theistic. In his oeuvre, however, Pseudo-Dionysius promotes Trinitarianism. A majority of Pseudo-Dionysius’ interpreters have maintained that these concepts are compatible. This article makes a case for the incoherence of that position.