Existing bounds on the minimum weight d of the dual 7-ary code of a projective plane of order 49 show that this must be in the range of 75 < d < 99. We use combinatorial arguments to improve this range to 87 < d < 99, noting that the upper bound can be taken to be 91 if the plane has a Baer subplane, as in the desarguesian case. A brief survey of known results for the minimum weight of the dual...

Proof. First note that the map φ is uniquely determined by its components φ11, φ12, φ21, φ22. Indeed, by the universal property of the coproduct M1 ⊕ M2 we see that φ is uniquely determined by the maps φ∗1 := φ◦ι1 and φ∗2 := φ◦ι2. Then from the universal property of the product N1⊕N2 we see that the map φ∗1 is uniquely determined by the maps φ11 = π1 ◦φ∗1 and φ21 = π2 ◦ φ∗1. Similarly, φ∗2 is u...

for −∞ < z < ∞. The properties of this distribution and its generalizations have been studied by several authors. Of particular eminence are the numerous papers on this topic by Professor N. Balakrishnan and his colleagues; see, for example, Balakrishnan [1, 2, 3], Balakrishnan and Aggarwala [4], Balakrishnan et al. [5, 7, 12], Balakrishnan and Chan [6], Balakrishnan and Joshi [8], Balakrishnan...

We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M , generalizing a result of Dolich on o-minimal theories in [4].

In ring theory, the notion of annihilator is an important tool for studying the structures. Many characterizations and structure theorems can be derived by using this notion. On the other hand, certain classes of rings (e.g., Baer rings and Rickart rings) are defined by considering annihilators ideals. In the present work, we introduce a class of rings which is close to the class of Rickart rin...

While environmental pollution is becoming more and more serious, many countries are adopting policies to control pollution. At the same time, the environmental regulation will inevitably affect economic and social development, especially employment growth. The environmental regulation will not only affect the scale of employment directly, but it will also have indirect effects by stimulating up...

Background in music performance research For the past few decades there has been considerable scientific interest in expression in music performance (Gabrielsson, 2003). A particularly relevant aspect of music performance is expressive timing, that is, the fluctuations of tempo during a performance. Accordingly, expressive timing has been one of the major topics in music performance research. A...

Browne, Lockhart Clarke, Kesteven, Moxon, &c., the journal cannot fail to secure the support of the profession. With the exception of one or two articles, those which have already appeared are of an essentially practical and valuable character. We are sorry to find that Dr. Bucknill, in his review of Dr. Bateman's " Darwinism tested by Language," in the first number of Brain, has allowed his od...

Let T be an almost strongly minimal theory. Work inside a monster model M of T . Let M be a model (a small elementary substructure of M). If G is an M -definable group, and H is an M -definable subgroup, we can consider the homogeneous space G/H with the left-action by G. We will define “etale cohomology groups” H M(G/H,A) for all n ≥ 0 and all abelian groups A. We would like the following thin...