In [S, pp. 77–88], Smullyan introduced the class of rudimentary relations, and showed that they form a basis for the recursively enumerable sets. He also asked [S, p. 81] if the addition and multiplication relations were rudimentary. In this note we answer one of these questions by showing that the addition relation is rudimentary. This result was communicated to Smullyan orally in 1960 and is announced in [S, p. 81, footnote 1]. However, the proof has not yet appeared in print. (...) Let us begin by reviewing Smullyan's definition [S, p. 10] of dyadic notation for the positive integers. Each positive integerais identified with the unique stringanan−1…a1a0of 1's and 2's such thata= Σin=0ai2iBecause of this identification, we are able to speak of typographical properties of numbers. (shrink)

A photoaffinity analog of cGMP has been used to biochemically identify a new ligand-binding subunit of the retinal rod cGMP-activated ion channel, as well as amino acids in contact with cGMP in the original subunit. Covalent tethering of this probe to channels in excised menbrane patches has revealed a functional heteogeneity in the ligand-binding sites that may arise from the two biochemically identified subunits.

PREFACE. THE Author of this very practical treatise on Scotch Loch - Fishing desires clearly that it may be of use to all who had it. He does not pretend to have written anything new, but to have attempted to put what he has to say in as readable a form as possible. Everything in the way of the history and habits of fish has been studiously avoided, and technicalities have been used as sparingly as possible. The writing of this (...) book has afforded him pleasure in his leisure moments, and that pleasure would be much increased if he knew that the perusal of it would create any bond of sympathy between himself and the angling community in general. This section is interleaved with blank shects for the readers notes. The Author need hardly say that any suggestions addressed to the case of the publishers, will meet with consideration in a future edition. We do not pretend to write or enlarge upon a new subject. Much has been said and written-and well said and written too on the art of fishing but loch-fishing has been rather looked upon as a second-rate performance, and to dispel this idea is one of the objects for which this present treatise has been written. Far be it from us to say anything against fishing, lawfully practised in any form but many pent up in our large towns will bear us out when me say that, on the whole, a days loch-fishing is the most convenient. One great matter is, that the loch-fisher is depend- ent on nothing but enough wind to curl the water, -and on a large loch it is very seldom that a dead calm prevails all day, -and can make his arrangements for a day, weeks beforehand whereas the stream- fisher is dependent for a good take on the state of the water and however pleasant and easy it may be for one living near the banks of a good trout stream or river, it is quite another matter to arrange for a days river-fishing, if one is looking forward to a holiday at a date some weeks ahead. Providence may favour the expectant angler with a good day, and the water in order but experience has taught most of us that the good days are in the minority, and that, as is the case with our rapid running streams, -such as many of our northern streams are, -the water is either too large or too small, unless, as previously remarked, you live near at hand, and can catch it at its best. A common belief in regard to loch-fishing is, that the tyro and the experienced angler have nearly the same chance in fishing, -the one from the stern and the other from the bow of the same boat. Of all the absurd beliefs as to loch-fishing, this is one of the most absurd. Try it. Give the tyro either end of the boat he likes give him a cast of ally flies he may fancy, or even a cast similar to those which a crack may be using and if he catches one for every three the other has, he may consider himself very lucky. Of course there are lochs where the fish are not abundant, and a beginner may come across as many as an older fisher but we speak of lochs where there are fish to be caught, and where each has a fair chance. Again, it is said that the boatman has as much to do with catching trout in a loch as the angler. Well, we dont deny that. In an untried loch it is necessary to have the guidance of a good boatman but the same argument holds good as to stream-fishing... (shrink)

Mr. Last's very interesting note is so ingenious and the Egyptian evidence falls so pat that it deserves to be right, but I very much doubt if it is. In fact the Aithiopians do not stand alone, and the context of their longevity deserves consideration. If that context is recalled, it may appear that ‘to say that the legend was attached by the Greeks to the Aithiopians through their remoteness from the Mediterranean world is no explanation’ is itself a hard (...) saying. (shrink)