Partitioning

“ Estimated detection ranges still vary much less than emitted intensities, but some bats achieve almost twice as long maximum detection distances as other, mainly due to lower emitted frequency. Thus, if call frequency was to play an important role in resource partitioning, we would expect dietary differences in the amount of large insects taken by the bats and less so in the amount of small insects. ”

“ However, the intensity of bat calls has been largely neglected although intensity is a key factor determining echolocation range and interactions with other bats and prey. Differences in detection range, in turn, are thought to constitute a mechanism promoting resource partitioning among bats, which might be particularly important for the species-rich bat assemblages in the tropics. Here we present data on emitted intensities for 11 species from 5 families of insectivorous bats from Panamá hunting in open or background cluttered space or over water. ”

“ Three particular cases are of great importance. Case I. is the “one-to-one distribution,” in which the number of parcels is equal to the number of objects, and one object is distributed in each parcel. Case II. is that in which the parcels are all different, being defined by the partition (1111. . .) , conveniently written (1m) ; this is the theory of the compositions of unipartite and multipartite numbers. Case III. is that in which the parcels are all similar, being defined by the partition (m) ; this is the theory of the partitions of unipartite and multipartite numbers. ”

“ If the successive weights of the separates be enclosed in a bracket we obtain a partition of the weight which appertains to the separated partition. This partition is termed the specification of the separation. The degree of the separation is the sum of the degrees of the component separates. A separation is the symbolic representation of a product of monomial symmetric functions. A partition, , can be separated in the manner , and we may take the general form of a partition to be and that of a separation when denote the distinct separates involved. ”

“ Another method of partition will be found in the fields "checky" (or "chequy") and lozengy; but these divisions, as also the foregoing, will be treated more specifically under the different ordinaries. A field which is party need not necessarily have all its lines of partition the same. This peculiarity, however, seldom occurs except in the case of a field quarterly, the object in coats of this character being to prevent different quarters of one coat of arms being ranked as or taken to be quarterings representing different families. ”

“ The method gives all the partitions of a number, but we may consider different classes of partitions: the partitions into a given number of parts, or into not more than a given number of parts; or the partitions into given parts, either with repetitions or without repetitions, &c. It is possible, for any particular class of partitions, to obtain methods more or less easy for the formation of the partitions either of a given number or of the successive numbers 1, 2, 3, &c. And of course in any case, having obtained the partitions, we can count them and so obtain the number of partitions. ”

“ When an accurate indexing or dividing wheel is required on a machine, the method of securing accurate divisions of the circle illustrated at C, Fig. 22, is sometimes employed. There is a series of circular disks or bushings equal in number to the divisions required, and these disks are all in contact with each other and with a circular boss or shoulder on the plate to which they are attached. The space between adjacent disks is used to accurately locate the dividing wheel, engagement being made with a suitable latch or indexing device. ”

“ Many different schemes of partition upon this principle may be found, each including forms subordinated one to the other, descending from the more comprehensive to the less comprehensive. It is only when you can find no more parts which are forms, that you must be content to divide into parts which are not forms. ”

“ According to this then, a partition, which under ordinary circumstances should facilitate the observance of laws, would prove a detriment; for were there no partition between the larger and smaller court and vines were planted anywhere within the two courts, a man would simply be obliged to measure off four ells whence the vines grew and could then plant whatever he chose. ”

“ Here is the partition at my ear down in the deep on the lee side. Are we ever coming up again together? I think not; the partition and I are so long about it that I really do believe we have overdone it this time. Heavens, what a scoop! What a deep scoop, what a hollow scoop, what a long scoop! Will it ever end, and can we bear the heavy mass of water we have taken on board, and which has let loose all the table furniture in the officers’ mess, and has beaten open the door of the little passage between the purser and me, and is swashing about, even there and even here? ”

“ If the cluster be slightly touched, all the branches instantly fold up, and form a small round mass. The stem which supports the cluster contracts also at the same time, folding up like a workman’s measuring rule, that consists of three or four joints. This extraordinary assemblage constitutes one organized whole, formed of a multitude of similar and particular ones. A new species of society, in which all the individuals are members of each other in the strictest sense, and all participate of the same life. ”

“ If we recur to what we have called displacements, the aggregate of which forms, as we have seen, a group, we shall be brought to distinguish those in which a finger does not move; and by what has preceded, those are the displacements which characterise a point in space, and their aggregate will form a sub-group of our group. To each sub-group of this kind, then, will correspond a point in space. We might be tempted to conclude that experiment has taught us the number of dimensions of space ”

“ A radial partition (Fig. 145, A) , starting from the apex of the quadrant, is at once excluded, for a reason similar to that just referred to; our choice must lie therefore between two modes of division such as are illustrated in Fig. 145, where the partition is either (as in B) {360} concentric with the outer border of the cell, or else (as in C) cuts that outer border; in other words, our partition may (B) cut both radial walls, or (C) may cut one radial wall and the periphery. These are the two methods of division which Sachs called, respectively, (B) periclinal, and (C) anticlinal391. ”

“ Whatever the number of pieces in a cluster in the closest possible packing of solids may be for any particular shape, we may consider each cluster as itself a given single body, and thus reduce the problem to the packing closely together of assemblages of individuals all sameways oriented ”

“ The totality of entities which lend themselves to a given type of order, form a collection, or an aggregate; and if the order of a given aggregate is to be unambiguous, the members of this aggregate must be uniquely determined, and must be mutually exclusive: they must be what we shall later denote by the term elements of the aggregate. Externality and ‘elementarity’ are only different consequences of the same relation, which we denote by the term extension, and which thus together with the relation of order, will constitute the basis of our definition of space and time. ”

“ Thus, if the heap of straw was ten spans high, it still serves the purpose of a partition, even though it does not reach the ceiling. We may adduce therefrom, that any partition if it be only ten spans high, though it should not reach the ceiling, is valid. ”

“ Spencer holds that this can be done. For any finite aggregate, being unequally exposed to surrounding forces, will become more diverse in structure, every differentiated part will become the parent of further differences; at the same time, dissimilar units in the aggregate tend to separate, and those which are similar, to cluster together ("segregation") ; and this subdivision and dissipation of forces, so long as there are any forces unbalanced by opposite forces, must end at last in rest ”

“ A division of property had now become inevitable, and the principle of partition seems to have been that primitive one in virtue of which the stronger gets the larger share. There being only one frying-pan remaining, each of the stalwart leaders simultaneously caught hold of this handy domestic article, and the poor pan went to pieces in the struggle, the result being such as would have followed the adoption of Solomon's advice to halve the living child. ”

“ If at any time, through this process of concurrent elongation and subdivision, the apical cell become equivalent to, or less than, a hemisphere, it will next divide by means of a longitudinal, or vertical partition; and similar longitudinal partitions will arise in the other segments of the cylinder, as soon as it comes about that their length (in the direction of the axis) is less than their breadth. ”

“ Simple time is that in which the normal subdivision of its beats is by two, whether the number of the beats themselves is duple or triple. Compound time is that in which the beats are regularly divided by three, which three subdivisions are reckoned as subordinate beats. ”

“ The tendency to separate is the tendency not to act together, probably arising from inability to act together any longer; and the process of separation is the process of ceasing to act together. How truly non-co-ordination is the essence of the matter will be seen on observing that fission takes place more or less rapidly, according as the co-ordinating apparatus is less or more developed. Thus, "the capability of spontaneous division is one of the most distinctive attributes of the acrite type of structure;" ”

“ The usual way to gather, is by laying piles of sheets upon a long table, and for the gatherer to take from each pile a sheet in succession. By the new method a round table is made to revolve by machinery, and upon it are placed the piles of sheets. As the table revolves the gatherer takes a sheet from each pile as it passes him. It will at once be seen that not only is space saved, but that a number of gatherers may be placed at the table; and that there is no possibility of the gatherers shirking their work, as the machine is made to register the revolutions. ”

“ Where it extends over upon the next page or plate, it is always to be understood that the one which precedes is to have its left margin placed to the right margin of that which follows, and that the day column at the left of a division refers to all that stand to the right of it, when thus placed, until another day column is reached. In order to make this clear, let me illustrate by an example. On the middle division of Plate V we find a day column with two figures to the right of it. Turning to Plate IV, we find other figures of a similar character, but no day column. ”

“ The partition method of treating symmetrical algebra is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies. ”

“ Hence the set of permutations constitutes a group which is simply isomorphic with the given group.
Every group of finite order N can therefore be represented in concrete form as a transitive group of permutations on N symbols.
The order of any subgroup or operation of G is necessarily finite. If T1 (= S1) , T2, . . ., Tn are the operations of a subgroup H of G, and if Σ is any operation of G which is not contained in H, the set of operations ΣT1, ΣT2, . . ., ΣTn, or ΣH, are all distinct from each other and from the operations of H. Properties of a group which depend on the order. ”

“ Fig. 186. Development of antheridium of Riccia. (After Campbell.)
The method of division by means of oblique partitions is a common one in the case of ‘growing points’; for it evidently {404} includes all cases in which the act of cell-division does not lag far behind that elongation which is determined by the specific rate of growth. And it is also obvious that, under a common type, there must here be included a variety of cases which will, at first sight, present a very different appearance one from another. ”

“ We can have matters so arranged that we may have to make, if we like, ten turns of the screw in order to move the cross wires from one graduation to the next, and we may have the milled head of the screw itself divided into 100 divisions, so that we shall be able to divide each of the ten turns into 100, or the whole division into 1,000 parts. ”

“ Suppose for instance we want to subdivide the divisions marked on the arc of a circle, Fig. 102 a b, and say we wish to divide them into tenths, what we have to do is this—First, take a length equal to nine of these divisions on a piece of metal, c, called the vernier, carried on an arm from the centre of the circle, and then, on a separate scale altogether, divide that distance not into nine, as it is divided on the circle, but into ten portions. Now mark what happens as the vernier sweeps along the circle, instead of having Tycho’s pointer sweeping across the diagonal scale. ”

“ It leads us to conclusions that are of the first importance. For instance, when once the mass is formed, concealment and flexibility are at an end. The further, therefore, from the formation of the ultimate mass we can stop the process of concentration the better designed it will be. The less we are committed to any particular mass, and the less we indicate what and where our mass is to be, the more formidable our concentration. To concentration, therefore, the idea of division is as essential as the idea of connection. ”