The latter is therefore called the generating function of the former.
An effect of the greater tide-generating force will also be instability of the liquid magmas underlying volcanic areas, leading to violent eruptions and earthquakes.
Such a constellation can be shown to occur at intervals of about 1,800 years and about those times the tide-generating force will be at an absolute maximum.
There are therefore maxima and minima in the value of the tide-generating force, depending on the relative positions of the sun, earth and moon.
Given, then, that the variations in tide-generating force are big enough, the periods when the maxima occur will be critical with regard to oceanographical and meteorological phenomena.
There will be great outbursts of polar ice, but this will melt at higher latitudes than in the periods when the tide-generating force is minimal.
Many economic changes probably occurred in consequence of the variations in tide-generating force, as, for instance, the decline in the mediaeval Baltic herring fisheries controlled by the Hanseatic League.
In addition he added certain spark-generating coils across the contacts of the relay and tapper.
Pettersson's papers on tide-generating force are published in Publications de Circonstance, Conseil Internat.
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.
Enumerating Generating Functions Professor Michael Roberts (Quart.
The enumerating generating function of asyzygetic seminvariants.
- zn +9 1 -z2.1 -z3....1-z8; and since this expression is unaltered by the interchange of n and B we prove Hermite's Law of Reciprocity, which states that the asyzygetic forms of degree 0 for the /t ie are equinumerous with those of degree n for the The degree of the covariant in the variables is e=nO-2w; consequently we are only concerned with positive terms in the developments and (w, 0, n) - (w - r; 0, n) will be negative unless nO It is convenient to enumerate the seminvariants of degree 0 and order e=n0-2w by a generating function; so, in the first written generating function for seminvariants, write z2 for z and az n for a;.
The complete function may be written ll A2(z) i 2A2 (z/ ' A 2 z 1az2 1.1-a2; and this is the reduced generating function which tells us, by its.
A little further progress has been made by Cayley, who established the two generating functions for the quintic 1 -a3s 11 -a8.1 a12.
Putting n equal to co, in a generating function obtained above, we find that the function, which enumerates the asyzvgetic seminvariants of degree 0, is 1 1-z2.1-z3.1-z4....1-z0 that is to say, of the weight w, we have one form corresponding to each non-unitary partition of w into the parts 2, 3, 4,...0.
The generating function is I - z2' 52 For 0 =3, (alai +a2a2+a3a3) 10; the condition is clearly a1a2a3 = A3 = 0, and since every seminvariant, of proper degree 3, is associated, as coefficient, with a product containing A3, all such are perpetuants.
The general form is (3'2 A and the generating function 3 3.
The general form of perpetuant is (4 K 3 A 2"`) and the generating function 1-z2.1-z3.1-z4 In general when 0 is even and =20, the condition is a l a 2 ...U 24 II(v 1 +a 2)II(a l +a 2 +cr 3)...II(Q 1 +a 2 -}-...
The generating function is thus z2e-1 - 1 (1 -z 2) (1 -z 3) (1 -z 4)...
Taking The First Generating Function, And Writing Az P, Bz4, 2 For A, B And Z Respectively, We Obtain The Coefficient Of Aobe'Zpo 0' 2W That Is Of A E B E 'Z ï¿½, In 1 Z 2 1 Azp. 1 Azp 2....1 A2 P 2.1 Az P .
1 And The Actual Forms For The First Three Weights Are 1 Aobzo, (Ao B 1 A 1 B O) Bo, (A O B 2 A 1 2 0 Bo, Ao(B2, 3 A1B2 A2B1 A O (B L B 2 3B O B 3) A I (B 2 1 2B 0 B 2); Amongst These Forms Are Included All The Asyzygetic Forms Of Degrees 1, 1, Multiplied By Bo, And Also All The Perpetuants Of The Second Binary Form Multiplied By Ao; Hence We Have To Subtract From The 2 Generating Function 1Z And 1 Z Z2, And Obtain The Generating Function Of Perpetuants Of Degrees I, 2.
Their Number For Any Weight W Is The Number Of Ways Of Composing W 3 With The Parts I, 2, And Thus The Generating Function Is Verified.
Proceeding as we did in the case of the single binary form we find that for a given total degree 0+0', the condition which expresses reducibility is of total degree in the coefficients a and T; combining this with the knowledge of the generating function of asyzygetic forms of degrees 0, 0', we find that the perpetuants, of these degrees are enumerated by z26"'-11 -z.
We will choose from the forms in such manner that the product of letters A is either a power of A i, or does not contain A i; this rule leaves us with A2B 1 B 2 and A 2 B,Bs; of these forms we will choose that one which in letters B is earliest in ascending dictionary order; this is A2B 1 B 21 and our earliest perpetuant is (22)a(21)b - (221)a(2)b, and thence the general form enumerated by the generating function Z7 is (1-z)(1 - z2)2 (2 A2+2) a (2ï¿½2 +1 1ï¿½1 +1) b - (2 A2+2 1)a (2 M2+1 1, ai)b ...
By the rules adopted we take A?B 2 B 3, which gives (12)a(32)b - (1)a(321)b+ao(3212)b, the simplest perpetuant of weight 7; and thence the general form enumerated by the generating function 1 -z.1-z2.1 - z3 ?
(2A2+ 41ï¿½i +2)a(3ï¿½3+12ï¿½2+ 1) b, due to the generating function 2 15 (1 -z)(1 -z 2) 2 (1 -z3) For the case 0=r, 0'=4., the condition is a 1 r 1 T 2 r 3 7 4 (a i + T 1)(a + T 2)(a 1 T (a i + T 4) II (T s T t) 0; the calculation gives Selecting the product A;B 4 B 3 B2, we find the simplest perpetuant A1B4(A 1 B2+A1B3+B4)(-B3-A1B2B3-ATB4) =0.
(14) a (4322) b - (13) a (432 2 1) b + (12)a (432212) b - (1) a (432213)b +ao(432214)b, and thence the general form (1 A i + 4) a (4ï¿½4 + 1.3 ï¿½ 3 + 121 1, 2 + 2) b - ... ?ao(4ï¿½4+13ï¿½a+l2ï¿½2+21 Al+4)b, due to the generating function 2 15 1 -z.
If a longitudinally magnetized wire is twisted, circular magnetization is developed; this is evidenced by the transient electromotive force induced in the iron, generating a current which will deflect a galvanometer connected with the two ends of the wire.
Laplace published in 1779 the method of generating functions, the foundation of his theory of probabilities, and the first part of his Theorie analytique is devoted to the exposition of its principles, which in their simplest form consist in treating the successive values of any function as the coefficients in the expansion of another function with reference to a different variable.
A direct and an inverse calculus is thus created, the object of the former being to determine the coefficients from the generating function, of the latter to discover the generating function from the coefficients.
Thirdly, on the grounds that logical thinking adds the notion of substance, as substrate, to experience of the physical, but not of the psychical, and that the most proper being of mind is will, he concludes that wills are not active substances, but substance-generating activities (" nicht thatige Substanzen sondern substanzerzeugende Thdtigkeiten," System, 429) What kind of metaphysics, then, follows from this compound of psychology and epistemology?
Thus, according to him, in the first place reason forms a cosmological " ideal " of a multitude of simple units related; secondly, it forms a psychological " ideal " of a multitude of wills, or substance-generating activities, which communicate with one another by ideas so that will causes ideas in will, while together they constitute a collective will, and it goes on to form the moral ideal of humanity (das sittliche Menschheitsideal); and, thirdly, it forms an ontological " ideal " of God as ground of this moral " ideal," and therewith of all being as means to this end, and an " ideal " of God as world-will, of which the world is development, and in which individual wills participate each in its sphere.
We can only explain it by supposing that Wundt wishes to believe that, beyond the " ideal," there really is proof of a transcendent, ideating, substance-generating will of God; and that he is approaching the noumenal voluntarism of his younger contemporary Paulsen.
The great extension during the same period of the use of water-power has been of immense importance to Canada, most of the provinces possessing numerous swift-flowing streams or waterfalls, capable of generating a practically unlimited supply of power.
The city has a large jobbing trade, a coal supply from rich deposits in Pierce county, and abundant water-power from swift mountain streams, which is used for generating electricity for municipal and industrial use.
This last point is important in connexion with voltmeters used on the switchboards of electric generating stations, where relatively strong electric or magnetic fields may be present, due to strong currents passing through conductors near or on the board.
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.
=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.
And so in other cases we obtain a generating function.
It sprang into importance through the utilization of the falls in the river Glommen for driving saw-mills and generating electric power.
The subject received little attention in the United Kingdom, owing to the relatively high cost of home-produced alcohol as compared with that of imported petrol; and the use of alcohol in England for generating mechanical power was neither contemplated nor provided for by the Legislature before 1920, when, as the result of the consideration of the position by the Government, following on a report by a Departmental Committee appointed towards the end of 1918, clauses were inserted in the Finance Act of 1920 legalizing the use of alcohol for power purposes.