Scale, Landscape and Indigenous Bedouin Land Use: Spatial Order and Agricultural Sedentarisation in the Negev Highland

By taking a small-scale perspective, Bedouin pastoral space in the Israeli Negev in the modern period has been misinterpreted as chaotic by various Israeli institutions. In critiquing this ontology we suggest that a knowledge gap with regard to an appropriate scale of understanding Bedouin settlement patterns and mechanisms of sedentarisation is at its root, and that a larger-scale analysis indicates that their space is in fact highly ordered. Field surveys and interviews with the local Bedouin showed that household cultivation plots in the Negev Highland during the period of the British Mandate were organised at a large scale through natural and man-made landscape features reflecting their structure, development and deployment in a highly ordered space. This analysis carries significant implications for understanding pastoral spaces at the local scale, particularly offering better comprehension of various sedentary forms and suggesting new approaches to sustainable planning and development for the Bedouin.

Binary solutions to large systems of linear equations

The concept of generic computational complexity has been extended to generalized register machines over an ordered field. In this case, the machine halts at every input and gives a meaningful answer at almost every input, but it can abandon the calculation using explicit notification, that is, there exists the vague halting state. Note that the machine does not make any error. A generic polynomial time algorithm is proposed to recognize systems of linear equations without any binary solution, when the number of equations m is close to the number of unknowns n. More precisely, two conditions are required. Firstly, the inequality 2n ≥(n-m+1)(n-m+2) holds. Such systems are called large because the number of equations is close to the number of unknowns. Secondly, some assumptions of generality of the system of equations are fulfilled. Our approach is based on finding a positive definite quadratic form among the set of forms that depend on parameters. On the other hand, a counterexample has been found, whicht shows the inapplicability of this method for checking the absence of any binary solution to one equation.

A Representation Theorem for Generic Line Arrangements in the Plane

In this article, we prove a representation theorem that any generic line arrangement in the plane over an ordered field can be represented isomorphically by a very generic line arrangement in the sense of C. A. Athanasiadis [2] with a given set of distinct slopes of the same cardinality.

Defining integer-valued functions in rings of continuous definable functions over a topological field

Let [Formula: see text] be an expansion of either an ordered field [Formula: see text], or a valued field [Formula: see text]. Given a definable set [Formula: see text] let [Formula: see text] be the ring of continuous definable functions from [Formula: see text] to [Formula: see text]. Under very mild assumptions on the geometry of [Formula: see text] and on the structure [Formula: see text], in particular when [Formula: see text] is [Formula: see text]-minimal or [Formula: see text]-minimal, or an expansion of a local field, we prove that the ring of integers [Formula: see text] is interpretable in [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] is definably connected of pure dimension [Formula: see text], then [Formula: see text] defines the subring [Formula: see text]. If [Formula: see text] is [Formula: see text]-minimal and [Formula: see text] has no isolated points, then there is a discrete ring [Formula: see text] contained in [Formula: see text] and naturally isomorphic to [Formula: see text], such that the ring of functions [Formula: see text] which take values in [Formula: see text] is definable in [Formula: see text].

Magnetic fields and cosmic rays in M 31

Context. Magnetic fields play an important role in the dynamics and evolution of galaxies; however, the amplification and ordering of the initial seed fields are not fully understood. The nearby spiral galaxy M 31 is an ideal laboratory for extensive studies of magnetic fields. Aims. Our aim was to measure the intrinsic structure of the magnetic fields in M 31 and compare them with dynamo models of field amplification. Methods. The intensity of polarized synchrotron emission and its orientation are used to measure the orientations of the magnetic field components in the plane of the sky. The Faraday rotation measure gives information about the field components along the line of sight. With the Effelsberg 100-m telescope three deep radio continuum surveys of the Andromeda galaxy, M 31, were performed at 2.645, 4.85, and 8.35 GHz (wavelengths of 11.3, 6.2, and 3.6 cm). The λ3.6 cm survey is the first radio survey of M 31 at such small wavelengths. Maps of the Faraday rotation measures (RMs) are calculated from the distributions of the polarization angle. Results. At all wavelengths the total and polarized emission is concentrated in a ring-like structure of about 7–13 kpc in radius from the centre. Propagation of cosmic rays away from the star-forming regions is evident. The ring of synchrotron emission is wider than the ring of the thermal radio emission, and the radial scale length of synchrotron emission is larger than that of thermal emission. The polarized intensity from the ring in the plane of the sky varies double-periodically with azimuthal angle, indicating that the ordered magnetic field is oriented almost along the ring, with a pitch angle of −14 ° ±2° at λ6.2 cm. The RM varies systematically along the ring. The analysis shows a large-scale sinusoidal variation with azimuthal angle, signature of an axisymmetric spiral (ASS) regular magnetic field, plus a superimposed double-periodic variation of a bisymmetric spiral (BSS) regular field with about six times smaller amplitude. The RM amplitude of (118 ± 3) rad m−2 between λ6.2 cm and λ3.6 cm is about 50% larger than between λ11.3 cm and λ6.2 cm, indicating that Faraday depolarization at λ11.3 cm is stronger (i.e. with a larger Faraday thickness) than at λ6.2 cm and λ3.6 cm. The phase of the sinusoidal RM variation of −7 ° ±1° is interpreted as the average spiral pitch angle of the regular field. The average pitch angle of the ordered field, as derived from the intrinsic orientation of the polarized emission (corrected for Faraday rotation), is significantly smaller: −26 ° ±3°. Conclusions. The dominating ASS plus the weaker BSS field of M 31 is the most compelling case so far of a field generated by the action of a mean-field dynamo. The difference in pitch angle of the regular and the ordered fields indicates that the ordered field contains a significant fraction of an anisotropic turbulent field that has a different pattern than the regular (ASS + BSS) magnetic field.

ABSTRACT REAL NUMBERS

In this paper we represent a result of our study in field of analysis , that is, an abstraction of the system of real numbers. We start by defining a set of positive elements in a countable infinite (denumerable) field and, hence, we obtain a linearly ordered field which we call a field of rational elements or a rational field. After that we may introduce irrationals elements in our rational field. And, at last we have a system of real abstract numbers.

NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: A GENERALIZATION OF CONWAY’S THEORY OF SURREAL NUMBERS II

In [16], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\bf{No}}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of ${\bf{No}}$, i.e., a subfield of ${\bf{No}}$ that is an initial subtree of ${\bf{No}}$. In this sequel to [16], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of ${\bf{No}}$ that are themselves initial. It is further shown that an initial subdomain of ${\bf{No}}$ is discrete if and only if it is a subdomain of ${\bf{No}}$’s canonical integer part ${\bf{Oz}}$ of omnific integers. Finally, making use of class models the results of [16] are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\bf{No}}$.