Partitions sentence example

partitions
  • The Daniell type consists of a teak trough divided into five cells by slate partitions coated with marine glue.
    6
    2
  • If the coenosteum of Millepora be broken across, each pore-canal (perhaps better termed a polypcanal) is seen to be interrupted by a series of transverse partitions, representing successive periods of growth with separation from the underlying dead portions.
    6
    2
  • The warehouse was lit, but metal-framed partitions covered in sheeting formed a maze.
    4
    1
  • Recalling the formulae above which connect s P4 and a m, we see that dP4 and Dp q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10 P 01 4) respectively.
    4
    1
  • Jackson began ripping down the sheets and knocking over the partitions as he yelled for Elisabeth.
    1
    0
    Advertisement
  • The living tissues at the surface are cut off from the underlying dead portions by horizontal partitions termed tabulae, which are formed successively as the coenosteum increases in age and size.
    1
    4
  • pores may be confluent with the cyclosystem presents itself as a radiating partitions, thus having a superficial resemblance to a madreporarian coral with its radiating septa (figs.
    0
    0
  • This is the typical arrangement, which is exhibited in the majority of the Polychaeta and Oligochaeta; in these the successive chambers of the coelom are separated by the intersegmental septa, sheets of muscle fibres extending from the body wall to the gut and thus forming partitions across the body.
    0
    0
  • Even so, Prussia was bereft of half of her territories; those west of the river Elbe went to swell the domains of Napoleon's vassals or to form the new kingdom of Westphalia for Jerome Bonaparte; while the spoils which the House of Hohenzollern had won from Poland in the second and third partitions were now to form the duchy of Warsaw, ruled over by Napoleon's ally, the elector (now king) of Saxony.
    0
    0
  • Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.
    0
    0
    Advertisement
  • Those in the middle are thin, having only the pavement of the cella to support, and are provided with doors and partitions that make a sort of subterranean labyrinth.
    0
    0
  • 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.
    0
    0
  • A partition is separated into separates so as to produce a separation of the partition by writing down a set of partitions, each separate partition in its own brackets, so that when all the parts of these partitions are reassembled in a single bracket the partition which is separated is reproduced.
    0
    0
  • It is convenient to write the distinct partitions or separates in descending order as regards weight.
    0
    0
  • be any partitions of X, respectively, the function isexpressible by means of functions symbolized by separation of X1A2X 3.
    0
    0
    Advertisement
  • The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula.
    0
    0
  • Every symmetric function denoted by partitions, not involving the figure unity (say a non-unitary symmetric function), which remains unchanged by any increase of n, is also a seminvariant, and we may take if we please another fundamental system, viz.
    0
    0
  • The general term in a solution involves the product ao°ai 1 a2 2 ...an" wherein Tr =0, Zs7r s =w; the number of such products that may appear depends upon the number of partitions of w into B or fewer parts limited not to exceed n in magnitude.
    0
    0
  • The extraordinary advantage of the transformation of S2 to association with non-unitary symmetric functions is now apparent; for we may take, as representative forms, the symmetric functions which are symbolically denoted by the partitions referred to.
    0
    0
  • If we wish merely to enumerate those whose partitions contain the figure 0, and do not therefore contain any power of a as a factor, we have the generator ze 1-z2.1-z3.1-z4....1-z0.
    0
    0
    Advertisement
  • In general the coefficient, of any product A n A m A 7, 3 ..., will have, as coefficient, a seminvariant which, when expressed by partitions, will have as leading partition (preceding in dictionary order all others) the partition (Tr1lr2lr3�..).
    0
    0
  • Remark, too, that we are in association with non-unitary symmetric functions of two systems of quantities which will be denoted by partitions in brackets ()a, ()b respectively.
    0
    0
  • 1111 The Number Of Such Terms Is The Number Of Partitions Of W Into 0 0 Parts, The Part Magnitudes, In The Two Portions, Being Limited Not To Exceed P And Q Respectively.
    0
    0
  • The circulation of air in any given division of the mine is further controlled and its course determined by temporary or permanent partitions, known as brattices, by the erection of stoppings, or by the insertion of doors in the mine passages and by the use of special airways.
    0
    0
  • Thus the main features of the Japanese dwelling-house were evolved, and little change took place subsequently, except that the brush of the painter was freely used for decorating partitions, and in aristocratic mansions unlimited care was exercised in the choice of rare woods.
    0
    0
    Advertisement
  • Further divisions followed, and the weakness caused by these partitions was accentuated by a rivalry between the two main branches of the family.
    0
    0
  • In his process a current was passed through a tank divided into two or three cells by porous partitions, hoods and tubes were arranged to carry off chlorine and hydrogen respectively, and the whole was heated to r20° F.
    0
    0
  • Accordingly the Dual State was involved in a common downfall, and in the three partitions of 1772, 1792 and 1795 to which it was subjected at the hands of Russia, Prussia and Austria, Lithuania fell a prey to Russia and Prussia.
    0
    0
  • The precipitating boxes, generally made of wood but sometimes of steel, and set on an incline, are divided by partitions into alternately wide and narrow compartments, so that the liquor travels upwards in its passage through the wide divisions and downwards through the narrow divisions.
    0
    0
  • The ventilation of ends is effected by means of brattices or temporary partitions of thin boards placed midway in the drift, and extending to within a few feet of the face.
    0
    0
    Advertisement
  • The influence of the Swabian branch of the Hohenzollerns was weakened by several partitions of its lands; but early in the 16th century it rose to some eminence through Count Eitel Frederick II.
    0
    0
  • It not unfrequently happens that some of the larger houses have one or more partitions, but these are only posts of the main building left by the builders to support the roof, for the apartments have usually no communication with each other except by water.
    0
    0
  • When Louis was dying in 840, he sent the imperial insignia to Lothair, who, disregarding the various partitions, claimed the whole of the Empire.
    0
    0
  • The three carpels forming the pepo are separated by partitions (cl).
    0
    0
  • The Poles were, together with the Ruthenians, the youngest Austrian nation; the repeated partitions of Poland since the 18th century brought them unwillingly under Austrian rule.
    0
    0
    Advertisement
  • Lithuania and the Ruthenian Palatinates, the spoil of former partitions, continued to be incorporated with Russia.
    0
    0
  • 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.
    0
    0
  • Arbogast's rule of the last and the last but one; in fact, taking the value of a to be unity, and, understanding this letter in each term, the rule gives b; c, b2; d, bc, b; e, bd, c, b c, b, &c., which, if b, c, d, e, &c., denote I, 2, 3, 4, &c., respectively, are the partitions of 1, 2, 3, 4, &c., respectively.
    0
    0
  • An important notion is that of conjugate partitions.
    0
    0
  • We have for the number of partitions an analytical theory depending on generating functions; thus for the partitions of a number n with the parts I, 2, 3, 4, 5, &c., without repetitions, writing down the product I +x.
    0
    0
    Advertisement
  • F.) for the number of such partitions.
    0
    0
  • observing that any factor 1/I-x l is=l+x l +x 2l +..., we see that in the term Nx the coefficient is equal to the number of partitions of n, with the parts I, 2,, ..,withh repetitions.
    0
    0
  • +Nxnzk+.., we see that in the term Nx n z k of the development the coefficient N is equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4,, without repetitions.
    0
    0
  • The formula shows that except for numbers of the form (3n 2 n) the number of partitions without repetitions into an odd number of parts is equal to the number of partitions without repetitions into an even number of parts, whereas for the excepted numbers these numbers differ by unity.
    0
    0
  • Thus for the number II, which is not an excepted number, the two sets of partitions are in each set 6.
    0
    0
    Advertisement
  • we see that in the term Nx z of the development the coefficient Nis equal to the number of partitions of n into k parts, with the parts I, 2, 3, 4, ..., with repetitions.
    0
    0
  • The cathammal areas may remain very small, mere wedge-shaped partitions dividing up the coelenteron into a four-lobed stomach, the lobes of which communicate at the periphery of the body by a spacious ring-canal.
    0
    0
  • This geographical division was not reproduced by Rome in any administrative partitions of the province.
    0
    0
  • There is little evidence for partitions inside, and in wealthy establishments the place of rooms seems to have been supplied by separate buildings within the same enclosure.
    0
    0
  • When this failed, as happened during the wars and partitions which followed the death of Louis, the fate of this Church, with no effective head and under no regular direction, was very uncertain.
    0
    0
    Advertisement
  • In European Russia, as a result of the partitions of Poland under Catherine II.
    0
    0
  • This prosperity, however, was checked by a growing tendency among the Silesian dynasties to make partitions of their territories at each new succession.
    0
    0
  • There were 18 cells or cubicles on each side, divided by partitions, the bases of which remain.
    0
    0
  • from east to west) into three partitions by a double row of columns.
    0
    0
  • The cella - divided, like that of Zeus, into three partitions by a double row of columns - had four " tongue-walls," or small screens, projecting at right angles from its north wall, and as many from the south wall.
    0
    0
  • The efforts of the dukes to increase their power and to give unity to the duchy had met with a fair measure of success; but they were soon vitiated by partitions among different members of the family which for 250 years made the history of Bavaria little more than a dejune chronicle of territorial divisions bringing war and weakness in Division their train.
    0
    0
  • of parenchyma, like the palm of the hand, as in the sycamore, castoroil plant, &c. The divisions of leaves with radiating venation may extend to near the base of the leaf, and the names bipartite, tripartite, quinquepartite, &c., are given according as the partitions are two, three, five or more.
    0
    0
  • This was the first of the many partitions which effectually divided the kingdom of the Franks into an eastern and a western portion, that is to say, into divisions which eventually became Germany and France respectively, and the district ruled by Theuderich was almost identical with that which afterwards bore the name of Austrasia.
    0
    0
  • (1700-1746) Italianised the rooms, and completed the degradation by running up partitions which blocked up whole apartments, gems of taste and patient ingenuity.
    0
    0
  • 2) is a capsule opening between the partitions and containing generally a large number of roundish or angular seeds.
    0
    0
  • Both first and second compartments are remarkable for the presence of a number of pouches or cells in their walls, with muscular partitions, and a sphincter-like arrangement of their orifices, by which they can be shut off from the rest of the cavity, and into which the fluid portion only of the contents of the stomach is allowed to enter.
    0
    0
  • The kernel consists mainly of the abundant endosperm, which is firm, whitish in colour and marbled with numerous reddish-brown vein-like partitions, into which the inner seedcoat penetrates, forming what is known botanically as ruminated endosperm.
    0
    0
  • At Baltimore he gave an enormous impetus to the study of the higher mathematics in America, and during the time he was there he contributed to the American Journal of Mathematics, of which he was the first editor, no less than thirty papers, some of great length, dealing mainly with modern algebra, the theory of numbers, theory of partitions and universal algebra.
    0
    0
  • Both these reactions are carried out in tall cylindrical columns or " stills," consisting of a number of superposed cylinders, having perforated horizontal partitions, and provided with a steam-heating arrangement in the enlarged bottom portion.
    0
    0
  • In the galleries are situated the cells, separated from one another by transverse partitions, which are formed of chips of wood, cemented by the saliva of the bee.
    0
    0
  • The Church was broken up by the successive partitions of Poland, and those parts of it which fell to Russia were, notably under Catherine II.
    0
    0
  • The Hydromedusae are distinguished from the Scyphozoa chiefly by negative characters; they have no stomodaeum, that is, no ingrowth of ectoderm at the mouth to form an oesophagus; they have no mesenteries (radiating partitions) which incompletely subdivide the coelenteron; and they have no concentration of digestive cells into special organs.
    0
    0
  • The Scyphozoa have the following features in common: - They typically exhibit an ectodermal stomodaeum; partitions or mesenteries project into their coelenteron from the body-wall, and on these are generally concentrated digestive cells (to form mesenterial filaments, phacellae or gastric filaments, &c.); the external musculature of the body-wall is circular (except in Cerianthus); the internal, longitudinal; and the sexual cells probably always arise in the endoderm.
    0
    0
  • All partitions must be of incombustible material."
    0
    0
  • Some architects depend solely upon partitions, and a building with a well-constructed iron frame should be safe if provided with brick partitions or if the exterior of the iron framework is covered with well-built masonry of sufficient thickness.
    0
    0
  • In all structures exposed to wind, if the resisting moments of the ordinary materials of construction, such as masonry, partitions, floors and connexions, are not sufficient to resist the moment of distortion due to wind pressure, taken in any direction on any part of the structure, additional bracing shall be introduced sufficient to make up the difference in the moments.
    0
    0
  • With very rapid building, no method of construction of floors furrings, or partitions should be adopted which will not dry out with great speed.
    0
    0
  • Such fire-proof coverings, and also interior partitions, are composed of hollow, hard-burned terra-cotta blocks, of porous (sawdust) terra cotta, or various plastic compositions applied to metallic lath, many of which are patented both as to material and method of application.
    0
    0
  • The most simple test for the value of a system of fire-proof coverings, and of partitions and furrings, is to erect a large sample of the work and to subject it alternately to the continued action of an intensely hot flame which is allowed to impinge upon it, and to a stream of cold water directed upon it from the ordinary service nozzle of a steam fire engine.
    0
    0
  • It is important in all fire-proofing of columns and girders, and in all floor construction, furring and partitions, that there shall be no continuous voids, either vertical or horizontal, which may possibly serve as flues for the spread of heat or flame in case of fire.
    0
    0
  • All furrings and partitions must be started on the solid masonry of the floors to prevent the possible passage of fire from the room in which it may occur.
    0
    0
  • The cavities both of the calices and coenenchymal tubes of Heliopora are closed below by horizontal partitions or tabulae, hence the genus was formerly included in the group Tabulata, and was supposed to belong to the madreporarian corals, both because of its lamellar skeleton, which resembles that of a Ma.drepore, and because each calicle has from twelve to fifteen radial partitions or septa projecting into its cavity.
    0
    0
  • In addition to these parts the following structures may exist in corals:- Dissepiments are oblique calcareous partitions, stretching from septum to septum, and closing the interseptal chambers below.
    0
    0
  • Tabulae are stout horizontal partitions traversing the centre of the calicle and dividing it into as many superimposed chambers.
    0
    0
  • The barriers between these groups may be regarded as horizontal planes cutting across the branches of the ascending tree of life at levels determined chiefly by our ignorance; as knowledge increases, and as the conception of a genealogical classification gains acceptance, they are being replaced by vertical partitions which separate branch from branch.
    0
    0
  • (After Duchartre.) the calyx being trifid (three-cleft), quinquefid (five-cleft), &c., according to their number; or they reach to near the base in the form of partitions, the calyx being tripartite, quadripartite, quinquepartite, &c. The union of the parts may be complete, and the calyx may be quite entire or truncate, as in some Correas, the venation being the chief indication of the different parts.
    0
    0
  • The number of parts forming such a corolla can be determined by the divisions, whether existing as teeth, crenations, fissures or partitions, or if, as rarely happens, the corolla is entire, by the venation.
    0
    0
  • The three carpels forming the pepo are separated by partitions.
    0
    0
  • - Diagrammatic section of a compound unilocular ovary, in which there are no indications of partitions.
    0
    0
  • Anhalt-Bernburg had been weakened by partitions, but its princes had added several districts to their lands; and in 1812, on the extinction of a cadet branch, it was again united under a single ruler.
    0
    0
  • The interiors are cut up into numerous small rooms by transverse partitions, while numerous beamholes and dumb-sheaves indicate other divisions.
    0
    0
  • acetate film or panels to be hung over existing glass partitions.
    0
    0
  • disk partitions can be saved to image files that can be further saved to cd/dvd's.
    0
    0
  • Composition and manufacture Partitions and doors are manufactured using an 18mm, moisture-resistant chipboard, faced both sides with high-pressure laminate.
    0
    0
  • The integration is done in frequency partitions that use the midpoints between adjacent spectral peak frequencies.
    0
    0
  • Just install the 2 NT os to separate partitions, the dual boot menu will be automatically created.
    0
    0
  • Menu driven, it is a breeze to create and delete partitions, multiple primary partitions, configure the MBR area.
    0
    0
  • The other hard disk, which has two DOS partitions, was fine.
    0
    0
  • XXioctl swap device There are swap partitions in the XX type device.
    0
    0
  • So I mounted both / and /home partitions from Solaris in Linux by performing the following.
    0
    0
  • plasterboard partitions are more susceptible to damage than solid walls.
    0
    0
  • The evolutionary idea has revolutionized and unified geography as it did biology, breaking down the old hard-and-fast partitions between the various departments, and substituting the study of the nature and influence of actual terrestrial environments for the earlier motive, the discovery and exploration of new lands.
    0
    0
  • Shell with very low spire, without umbilicus, internal partitions frequently absorbed; a single ctenidium; a cephalic penis present.
    0
    0
  • Terrestrial and usually littoral; genital duct monaulic, the penis being connected with the aperture by an open or closed groove; shell with a prominent spire, the internal partitions often absorbed and the aperture denticulated.
    0
    0
  • It can be shown that the number 0 enumerates distributions of a certain nature defined by the partitions (,�i,�2...), (sT1s°2...), 1212 an = a 1 a 2 a 3 ...
    0
    0
  • x3 (ll'1T2...) and it is seen intuitively that the number 0 remains unaltered when the first two of these partitions are interchanged (see Combinatorial Analysis).
    0
    0
  • and now expanding the (P1 right-hand side _ I ql)(P 2 +1721..Q1 /2172�..), h pg - pi p2 / J L' the summation being for all partitions of the biweight.
    0
    0
  • d p4sp4 +Dp4(pg)+1; d P4 causes every other signle part function to vanish, and must cause any monomial function to vanish which does not comprise, one of the partitions of the biweight pq amongst its parts.
    0
    0
  • The general term in a solution involves the product ao°ai 1 a2 2 ...an" wherein Tr =0, Zs7r s =w; the number of such products that may appear depends upon the number of partitions of w into B or fewer parts limited not to exceed n in magnitude.
    0
    0
  • In general the coefficient, of any product A n A m A 7, 3 ..., will have, as coefficient, a seminvariant which, when expressed by partitions, will have as leading partition (preceding in dictionary order all others) the partition (Tr1lr2lr3�..).
    0
    0
  • (iv.) Other developments of the theory deal with distributions, partitions, &c. (see Combinatorial Analysis).
    0
    0
  • The circulation of air in any given division of the mine is further controlled and its course determined by temporary or permanent partitions (" brattices "), by the erection of stoppings, or by the insertion of doors in the mine passages and by the use of special air-ways (see Coal).
    0
    0
  • From these and other considerations it is obvious that (I) the electrolyte must be such as will freely dissolve the metal to be refined; (2) the electrolyte must be able to dissolve the major portion of the anode, otherwise the mass of insoluble matter on the outer layer will prevent access of electrolyte to the core, which will thus escape refining; (3) the electrolyte should, if possible, be incapable of dissolving metals more electro-negative than that to be refined; (4) the proportion of soluble electro-positive impurities must not be excessive, or these substances will accumulate too rapidly in the solution and necessitate its frequent purification; (5) the current density must be so adjusted to the strength of the solution and to other conditions that no relatively electro-positive metal is deposited, and that the cathode deposit is physically suitable for subsequent treatment; (6) the current density should be as high as is consistent with the production of a pure and sound deposit, without undue expense of voltage, so that the operation may be rapid and the "turnover" large; (7) the electrolyte should be as good a conductor of electricity as possible, and should not, ordinarily, be altered chemically by exposure to air; and (8) the use of porous partitions should be avoided, as they increase the resistance and usually require frequent renewal.
    0
    0
  • In his process a current was passed through a tank divided into two or three cells by porous partitions, hoods and tubes were arranged to carry off chlorine and hydrogen respectively, and the whole was heated to r20° F.
    0
    0
  • The political history of the parts of Saxony left by the capitulation of Wittenberg to the Ernestine line, which occupy the region now generally styled Thuringia (Thuringen), is mainly a recital of partitions, reunions, redivisions and fresh combinations of territory among the various sons of the successive dukes.
    0
    0
  • We consider in it a number as made up by the addition of other numbers: thus the partitions of the successive numbers 1, 2, 3, 4, 5, 6, &c., are as follows: I; 2, I I, 3, 21, III; 4, 31, 22, 211, 1111; 5, 41, 32, 311, 221, 2111, I I II I; 6, 51, 42, 411, 33, 321, 311I, 222, 221I, 21III, IIIIII.
    0
    0
  • These are formed each from the preceding ones; thus, to form the partitions of 6 we take first 6; secondly, 5 prefixed to each of the partitions of 1 (that is, 51); thirdly, 4 prefixed to each of the partitions of 2 (that is, 42, 411); fourthly, 3 prefixed to each of the partitions of 3 (that is, 321, 3111); fifthly, 2 prefixed, not to each of the partitions of 4, but only to those partitions which begin with a number not exceeding 2 (that is, 222, 2211, 21111); and lastly, 1 prefixed to all the partitions of 5 which begin with a number not exceeding 1 (that is, 11111 I); and so in other cases.
    0
    0
  • If we form all the partitions of 6 into not more than three parts, these are 6, 51, 4 2, 33, 411, 321, 222, and the conjugates are Iiiiii, 21iii, 221i, 222, 311i, 321, 33, where no part is greater than 3; and so in general we have the theorem, the number of partitions of n into not more than k parts is equal to the number of partitions of n with no part greater than k.
    0
    0
  • =n, then we have in the development of the product a term x n, and hence that in the term Nx of the product the coefficient N is equal to the number of partitions of n with the parts I, 2, 3, ..., without repetitions; or say that the product is the generating function (G.
    0
    0
  • When in a radiating leaf there are three primary partitions, and the two lateral lobes are again cleft, as in hellebore (fig.
    0
    0
  • The scope of his researches was described by Arthur Cayley, his friend and fellow worker, in the following words: "They relate chiefly to finite analysis, and cover by their subjects a large part of it - algebra, determinants, elimination, the theory of equations, partitions, tactic, the theory of forms, matrices, reciprocants, the Hamiltonian numbers, &c.; analytical and pure geometry occupy a less prominent position; and mechanics, optics and astronomy are not absent."
    0
    0
  • In forms such as Lucernaria and Charybdaea, in which the umbrella is of deep form and the stomachcavity consequently of great extent in the vertical direction, the concrescence-areas or septal nodes are drawn out into vertical partitions or taeniolae (fig.
    0
    0
  • He subsequently examined the passage of gases through septa or partitions of indiarubber, unglazed earthenware and plates of metals such as palladium, and proved that gases pass through these septa neither by diffusion nor effusion nor by transpiration, but in virtue of a selective absorption which the septa appear to exert on the gases in contact with them.
    0
    0
  • The inside of the case has a mirror on the lid and partitions to keep items in place.
    0
    0
  • In general, most physicians believe that IUGR is the consequence of a disease process within one or more of the three partitions that maintain and regulate fetal growth, i.e., the maternal compartment, the placenta, or the fetus.
    0
    0
  • Enjoy the restaurant's modern accents as well as the old fashioned elegance, complete with century-old glass panel partitions.
    0
    0
  • Standing partitions broke up areas of the expanse.
    0
    1
  • Multiplying out the right-hand side and comparing coefficients X1 = (1)x1, X 2 = (2) x2+(12)x1, X3 = (3)x3+(21)x2x1+ (13)x1, X4 = (4) x 4+(31) x 3 x 1+(22) x 2+(212) x2x 1 +(14)x1, Pt P2 P3 P1 P2 P3 Xm=?i(m l m 2 m 3 ...)xmlxm2xm3..., the summation being for all partitions of m.
    0
    1
  • certain partitions of those numbers (vide the definition of the specification of a separation).
    0
    1
  • The number of partitions of a biweight pq into exactly i biparts is given (after Euler) by the coefficient of a, z xPy Q in the expansion of the generating function 1 - ax.
    0
    1
  • The partitions with one bipart correspond to the sums of powers in the single system or unipartite theory; they are readily expressed in terms of the elementary functions.
    0
    1
  • We can verify the relations s 30 -a310 -3a 20 a 10 + a30, S 21 - 02100 01 -a 2C a 01 -0 11 0 10 021 The formula actually gives the expression of q) by means of separations of (10P01'), which is one of the partitions of (pq).
    0
    1
  • The keeper opens the two center partitions to examine the brood cells.
    0
    1