Alternative numerical systems

The article is devoted to the construction of numerical systems, alternative to the system of real numbers and applicable in curvilinear space-time. Examples of such systems are given. Within the framework of a stationary numerical system, it is admissible to sum the diverging series like the Dirichlet series for the Riemann zeta function without resorting to its analytic continuation in the plane of the complex argument. In the framework of a non-stationary numerical system, a description of the Hubble effect is obtained, taking into account the corrections that correspond to the apparently accelerated recession of galaxies without invoking the hypothesis of dark energy.

Paul Ricoeur on Evil

Paul Ricoeur represents an important source in Western culture who refuses to adopt a sharp separation between humanity and the rest of nature, while recognizing the importance of human distinctiveness. This chapter will engage Ricoeur’s works, beginning with Freedom and Nature, where he emphasizes the preconditions for human sin and the distinctions between scientific explanations and philosophical understanding. Another work, Fallible Man, distinguishes between the finite and infinite and describes the preconditions for human sin. Here, Ricoeur takes steps to fill in the gap between what he terms the pathétique of misery and the transcendental. He resists the idea that the source of evil arises directly from animal passions, but presents a more complex argument related to the force of what he terms ‘the fault’. In The Symbolism of Evil, Ricoeur further describes his recognition that the Fall of humanity admits a voluntary quality to specifically human sin; therefore, guilt is distinct from suffering. Ricoeur’s interpretation of the significance and problematic nature of Augustine’s account of the Fall is instructive in this respect. How far is the explicit human propensity for sin also dependent on prior language and symbolic thought? Ricoeur’s thought also frames the discussion that follows as a dialectical relationship between the natural propensity for evil and its voluntary, symbolic/semiotic character.

ON LOCAL TWO-POINT PROBLEM FOR THE EQUATION WITH THE OPERATOR OF GENERALIZED DIFFERENTIATION

A local two-point problem with the oprator of Gelfond-Leontiev generalized differentiation with a complex argument is investigated. The conditions of unity and existence of the solution of the problem are obtained. It is proved that such conditions are satisfied for almost all (relative to Lebesgue measure) node of interpolation.

Early Māori photography as commodified object: Mementoes, miniatures and material culture

During the late nineteenth and early twentieth centuries, there was a boom in the different forms of material culture of the photographic image with the emergence of cheap methods for its mass (re)production. The material culture extended into postcards, illustrated books, magic lantern slides and stereoviews, but also into the much-less discussed area of souvenir china. These commodified objects of illustrated porcelain were popular mementoes of places visited, physical reminders of spaces encountered, made possible through newly developing modes of leisure culture and organized travel. Edwardian New Zealand was no exception, where images of the Māori were a striking presence within its visual culture. This was a country that was beginning to promote its cultural uniqueness partly through its Indigenous population, with early tourism literature referring to the country as Maoriland. New Zealand souvenirs depicted images of the Māori and Māoritanga (Māori culture) on decorative china essentially for consumption by local tourists and travellers. This article considers these commodified objects in the context of photography as material culture, exploring their social biography and the manner in which the images were reproduced and altered. It contends that in addressing keepsake china as objects bearing photographic images, and in positioning these souvenirs as popular artefacts within a scopic culture, a more complex argument of variant readings emerges.

Integrity in Law’s Empire

In this paper I focus on Dworkin's arguments for the distinctive political virtue of integrity, arguing that we have serious reasons to doubt that the case for integrity has been made. I approach Dworkin’s complex argument in two main steps, following his two main arguments for the distinct value of integrity. The first, and more direct argument, is based on what Dworkin takes to be the grounds for rejecting “checkerboard laws”. The second argument is the one that ties the value of integrity to political legitimacy by way of articulating the value of integrity in light of its affinity with Fraternity, the idea of a “true community”, and the associative obligations such communities engender. I try to show in this paper that both of these lines of argument are not convincing.

Conservation of the numbers of characteristic values of polynomial and entire matrix pencils inside and outside a circle under perturbations

Let [Formula: see text] and [Formula: see text] be entire matrix-valued functions of a complex argument [Formula: see text] (entire matrix pencils) and [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the numbers of the characteristic values of [Formula: see text] taken with their multiplicities located inside and outside [Formula: see text], respectively. Besides [Formula: see text] can be infinite. We consider the following problem: how “close” should be [Formula: see text] and [Formula: see text] in order to provide the equalities [Formula: see text] and [Formula: see text]? We restrict ourselves by the entire pencils of order not more than two. Our results are new even for polynomial pencils.

On a Fire Fighter’s Problem

Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed [Formula: see text]. How large must [Formula: see text] be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results. First, we analyze the natural curve [Formula: see text] that develops when the fighter keeps building, at speed [Formula: see text], a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function [Formula: see text], where [Formula: see text] and [Formula: see text] are real functions of [Formula: see text]. For [Formula: see text] all zeroes are complex conjugate pairs. If [Formula: see text] denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs [Formula: see text] rounds before the fire is contained. As [Formula: see text] decreases towards [Formula: see text] these two zeroes merge into a real one, so that argument [Formula: see text] goes to 0. Thus, curve [Formula: see text] does not contain the fire if the fighter moves at speed [Formula: see text]. (That speed [Formula: see text] is sufficient for containing the fire has been proposed before by Bressan et al. [6], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that for any curve that visits the four coordinate half-axes in cyclic order, and in increasing distances from the origin the fire can not be contained if the speed [Formula: see text] is less than 1.618…, the golden ratio.