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Notes: Necessary and sufficient conditions for unitary ray representations of connected Lie groups are reexamined. Thus a systematic constructive method is obtained for calculating the admissible exponent factors (two-cocycles). The gauge freedom of the unitary ray representation formalism is also considered. This introduces the distinction between trivial and genuine ray representations. A special gauge is then adopted, within which the two-cocycle is almost unique. The only prerequisite of the exponent factor calculus is the knowledge of the binary combination rules for the essential parameters of the group. The attained method affords a simple, general, and explicit (i.e., coordinate-dependent) two-cocycle calculus. The aim of this paper is merely instrumental.

Notes: The converse problem of similarity analysis is discussed for the infinitesimal symmetry transformations of ordinary second-order differential equations which are nonlinear in x(overdot) (and may be linear or nonlinear in x). A natural classification of the problem arises, according to the highest order N of nonlinearity in x(overdot). The completely general maximal Lie algebra is obtained for the case N≤3. In the case N≥4 one has, besides the system of differential equations for the infinitesimal generators, an extra set of anholonomic constraints, which operates as a symmetry-breaking mechanism producing a strong reduction in the number of surviving parameters. Miscellaneous examples are given, which illustrate some features of similarity analysis of nonlinear systems. The infinitesimal point transformation symmetries of the Van der Pol oscillator are also briefly discussed.

Notes: The two-dimensional space-time realizations of the Lie algebra of SL(3,R) are obtained, when the group acts as the maximal point symmetry group of any given one-dimensional Newtonian linear system. It is shown that these realizations are isomorphic with the realization of the Lie algebra of the projective group in the plane. Next an active point of view is introduced, and SL(3,R) is interpreted as a group of mappings that transform one admissible world line of the system into another. Thus a new mechanical realization of the sl(3,R) algebra comes to the fore. Some miscellaneous examples are included.

Notes: The converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x¨+f2(t)x(overdot)+f1(t)x =f0(t). The eight-parameter realizations of the symmetry group are obtained in the form F−1P2 F, where F stands for transformations of (t,x) that depend exclusively on the fundamental solutions of the equation, and where P2 is an arbitrary projective transformation in the plane. Thus it is shown that the full point symmetry group corresponds to SL(3,R) indeed, without recourse to the Lie algebra. Also, a technique is obtained for calculating the finite point symmetry realization of SL(3,R) for any given one-dimensional linear system. Some miscellaneous examples are given.

Notes: The importance of species-specific swim speeds for the formation of multi-species fish shoals in a Canadian freshwater system was investigated. Swim speed was positively correlated with total length and differed between species but was unrelated to shoal size.

Notes: Abstract The theory of plate tectonics provides a basic framework for evaluating the potential for future great earthquakes to occur along major plate boundaries. Along most of the transform and convergent plate boundaries considered in this paper, the majority of seismic slip occurs during large earthquakes, i.e., those of magnitude 7 or greater. The concepts that rupture zones, as delineated by aftershocks, tend to abut rather than overlap, and large events occur in regions with histories of both long- and short-term seismic quiescence are used in this paper to delineate major seismic gaps. In detail, however, the distribution of large shallow earthquakes along convergent plate margins is not always consistent with a simple model derived from plate tectonics. Certain plate boundaries, for example, appear in the long term to be nearly aseismic with respect to large earthquakes. The identification of specific tectonic regimes, as defined by dip of the inclined seismic zone, the presence or absence of aseismic ridges and seamounts on the downgoing lithospheric plate, the age contrast between the overthrust and underthrust plates, and the presence or absence of back-arc spreading, have led to a refinement in the application of plate tectonic theory to the evaluation of seismic potential. The term seismic gap is taken to refer to any region along an active plate boundary that has not experienced a large thrust or strike-slip earthquake for more than 30 years. A region of high seismic potential is a seismic gap that, for historic or tectonic reasons, is considered likely to produce a large shock during the next few decades. The seismic gap technique provides estimates of the location, size of future events and origin time to within a few tens of years at best. The accompanying map summarizes six categories of seismic potential for major plate boundaries in and around the margins of the Pacific Ocean and the Caribbean, South Sandwich and Sunda (Indonesia) regions for the next few decades. These categories range from what we consider high to low potential for being the site of large earthquakes during that period of time. Categories 1, 2 and 6 define a time-dependent potential based on the amount of time elapsed since the last large earthquake. The remaining categories, 3, 4, and 5, are used for areas that have ambiguous histories for large earthquakes; their seismic potential is inferred from various tectonic criteria. These six categories are meant to be interpreted as forecasts of the location and size of future large shocks and should not be considered to be predictions in which a precise estimate of the time of occurrence is specified. Several of the segments of major plate boundaries that are assigned the highest potential, i.e., category 1, are located along continental margins, adjacent to centers of population. Some of them are hundreds of kilometers long. High priority should be given to instrumenting and studying several of these major seismic gaps since many are now poorly instrumented. The categories of potential assigned here provide a rationale for assigning prorities for instrumentation, for future studies aimed at predicting large earthquakes and for making estimates of tsunami potential.