The theory of multivariate numbers.
Multivariate numbers of a sum from unit.
- Multivariate numbers.
1.1 The common concept about multivariate numbers.
Multivariate number of kind Аnaction f(t){xa ; ya ; za ; … ; Ran } represents object (a point, a piece, a rectangular …), formed of sequence of numbers f (t) by principle of an action (a sum, a multiplication), describing the given multivariate number. Value of multivariate number is a real number.
The n-dimensional number of a sum is designated Аnsumf(t){ xa ; ya ; za ; … ; Ran }[1] or sumnf(t){ xa ; ya ; za ; … ; Ran }; the n-dimensional number of multiplication is designated Bnmulf(t){ xb; yb; zb; … ; Rbn}[2] or mulnf(t){ xb; yb; zb; … ; Rbn}, where
-
A, B is a capital letter of the Latin alphabet for a designation in general multivariate number. -
n is the number of coordinates of multivariate number, is measurement of n-dimensional number, for example, Аnsum - n-dimensional number of a sum. -
sum, mul is an actions, principles of formation of multivariate number. -
f (t) is the function of formation of one-dimensional number from which the further multivariate numbers are formed, where t - a variable (if ranges of definition t and x coincide it is possible to write f (x)) .
f(t)
t=
0
1
2
3
4
5
N(t), tN - function of natural numbers
N(t)=
0
1
2
3
4
5
2t, tN – indicative function
2t=
1
1
2
4
8
16
Table 1.
- {xa; ya; za; …; Ran}, {xb; yb; zb; …; Rbn } are coordinates of multivariate number (the whole consecutive numbers).
Number Аnmulf(t){xa; ya; za; …; Ran} is read: A - n-dimensional number of multiplication from function f (t), with coordinates x, y, z, …, Rn.
1.2 Formation of multivariate numbers.
Formation of multivariate numbers occurs according to formation of n-dimensional axes of Descartes i.e. distribution occurs in two opposite directions to perpendicularly previous distribution. In conditionally positive direction of distribution of number t in function f (t) undertake with "plus", in opposite - conditionally negative direction of distribution, number t undertake with "minus".
It is possible to make distribution and only in a positive direction.
Definition 1 (dimension of space on which there is a given multivariate number). The n-dimensional number exists on space on which first of last coordinates is more than unit, i.e.
An{xa;ya;za;…;Ran-n1>1;1;1…;1; Ran =1}=Bn-n1{xa;ya;za;…; Ran-n1}.
Consequence of definition 1. Value of number of n-dimensional space with last coordinate to equal unit coincides with value of number (n−1)-dimensional space with the same first coordinates as in the given number, only without last coordinate, i.e.
An{xa; ya; za; …; Ran =1}=Bn-1{xa; ya; za; …; Ran-1}.
Thus, number An{ xa ; ya ; za ; … ; Ran =1} is simultaneously and n-dimensional number and (n−1) … (n−n1)-dimensional number, but exists on space (n−n1), last coordinate on the given space is more than unit.
Definition 2 (n-dimensional number of a special case). We shall name n-dimensional number with last coordinate to equal unit n-dimensional number of a special case to designate, that its existence occurs on space below n-dimensional (as it is already told, such number exists on (n−n1)-dimensional space on which last coordinate of the given number is more than unit).
1.2.1 One-dimensional number.
One-dimensional number А1 represents a number - point with one coordinate {x}. The point is formed functionally А1{x}=f(t) where a variable, at concurrence of ranges of definition of variables x and t, is x (coordinate of one-dimensional number).
Example:
t=
0
1
2
3
4
Straight line of one-dimensional numbers from constant number C=1
1
1
1
1
1
Straight line of one-dimensional numbers from function t! (tN).
1
1
2
6
24
Straight line of one-dimensional numbers from function 3t (tN).
1
3
9
27
81
Table 2.
One-dimensional number А1 {x} is a special case of bidimentional number А2 {x; 1} and their values coincide (as bidimentional number А2 {x; 1} exists on one-dimensional space А1 {x} (look definition 1)). The sequence of one-dimensional numbers (points) is a straight line - sequence of bidimentional numbers, with the second coordinate to equal unit.
By definition 1:
A1f(x){x} = f(x) = A2f(x){x;1} , (at t=x) the formula 1.
Definition 3 (Formation of n-dimensional number) for number of the n-dimensional space which is not being n-dimensional number of a special case, i.e. for number which last coordinate is more than unit (for n-dimensional number of a special case it is necessary to find dimension of space in which it is not number of a special case and then to apply this definition).N-dimensional number An with coordinates {xa; ya; za; …; Ran} represents action (a sum, a multiplication) of n-dimensional numbers on (n-1) the-dimensional space[3] limited to the beginning of coordinates and coordinates xa; ya; za; …; Ran-1 at the constant last coordinate equal Rn=Ran−1=const. I.e., number An represents action of exhaustive search of all n-dimensional numbers with coordinates {x[0;xa] ; y[]0;ya]; z[0;za]; …; Rn-1[0;Ran-1]; Rn = Ran−1}, notice - last coordinate is constant!
For example:
a) sum2f(t){x;y}=.
b) mul2 f(t){x;y}=(mul2 f(t){i;y-1})[4].
1.2.2 Bidimentional number.
Bidimentional number A2 {x; y} represents number - straight line with two coordinates {x; y}. At y=1 number A2 is a straight line of one-dimensional numbers, at y [2; ) number A2 is a straight line of the bidimentional numbers formed from a straight line of bidimentional numbers with the second coordinate equal (y-1). Formation occurs by a principle of the action describing the given multivariate number. Value of this number is the action (a sum, a multiplication) values of numbers of this straight line.
1.2.3 Three-dimensional number.
Three-dimensional number A3{xa; ya; za} represents a number - plane with three coordinates, made at za=1 from bidimentional numbers, and at za[2; ) from three-dimensional numbers with the third coordinate equal (za−1). Value of this number is the action (a sum, a multiplication) values of numbers of this plane.
1.2.4 N-dimensional number.
The N-dimensional number is analogy of the numbers described above.
For presentation, multivariate numbers should be presented as values of the given n-dimensional numbers on n-dimensional Cartesian space, as points with the coordinates corresponding to coordinates of the given multivariate number.
1.3 Function of multivariate number.
Function of multivariate number is the regularity, allowing expressing values of n-dimensional number through coordinates of this number. This function enters the name f (x; y; z; …; Rn) actionnf (t), where
-
x, y, z..., Rn - constant or independent variables which are coordinates of n-dimensional number. -
n - the common number of coordinates, the order of measurement of multivariate number. -
f (t) - function of formation of one-dimensional number.
Function f (x; y; z; …; Rn) sumnf (t) it is read: function of a sum of n-dimensional numbers from function f (t) with variables x, y …; with constants Ra=C1, Rb=C2, ….
So, function f(xa; ya; za; …; Ran)действиеn f(t)= Rn+1,
where Rn+1 the dependent variable being value of n-dimensional number An. For example:
а) See the formula 5.1 - function of bidimentional number of a sum from unit.
б) f (x; 2) mul2N(t) =x! - function of multiplication of bidimentional numbers from sequence of natural numbers (the function known in theory of probabilities as a n-factorial).
- Multivariate numbers of a sum from constant number C=1.
2.1 Formation of multivariate numbers of a sum from unit.
According to definition 3, multivariate number of a sum from unit Ansum1{xa; ya; za; …; Ran} represents the sum of n-dimensional numbers on the (n−1)-dimensional space limited to the beginning of coordinates and coordinates xa , ya , za, …, Ran-1, at Rn= Ran−1. We shall agree to count the sum of n-dimensional numbers at an exhaustive search of the coordinates accepting values from 1 up to , instead of from 0 up to - i.e. Rn[1;)[5]. Mathematically definition 3 will be written down:
sumn1{x;y;z;…;Rn}= formula 2[6].
2.2 Values of Multivariate numbers of a sum from unit.
2.2.1 sum11 {x}.
Let's find values of one-dimensional numbers of a sum from unit.
By definition, the one-dimensional number is value of function f (t), but f (t) in this case this constant number C=1, therefore A1sum1{x} =1 at anyone x.
2.2.2 sum21 {x; y}.
Values of bidimentional numbers of the sum from unit of a special case coincide with values of one-dimensional numbers. Therefore
A2sum1{x;1}= A1sum1{x}=1 formula 3.1
Straight line (x; 1) in the appendix 1 there are values of bidimentional numbers of a special case.
Condition on the order of n-dimensional number. There are n-dimensional numbers with coordinates {x;y;…;Rn-1;Rn=const} where last coordinate is constant, we shall agree to name such numbers n-dimensional numbers Rn=const of the order, for example:
a) sum21 {x; 2} - bidimentional number of a sum from unit of the second order.
b) mul2N (t) {x; 2} - bidimentional number of multiplication from sequence of natural numbers of the second order.
Condition of brief record. We shall agree for simplicity to designate n-dimensional number of a sum from unit only in its coordinates in braces, i.e. Ansum1{xa;ya;za;…;Ran}={xa;ya;za;…;Ran}.
Values of bidimentional numbers of a sum from unit at y> 1 we shall find under the formula 2:
{x;y}={1;y-1} + {2;y-1} + … + {x-1;y-1} + {x;y-1}.
So for y=2
{x;2}=, but sum21{x;1}=1, then
{x;2}==x formula 3.2
For y=3
{x;3}=, but sum21{x;2}=x, then
{x;3}== 1+2+3+…+x formula 3.3
For y=4
{x;4}= = {1;3} + {2;3} + … + {x-1;3} + {x;3}, but sum21{x;3}=, then
{x;4}= formula 3.4
And for anyone y
{x;y}= formula 3.y
Values for {x; y} there are straight lines {x; y=const} in the appendix 1.
2.2.3 sum31 {x; y; z}.
Values of numbers {x; y; 1} a special case coincide with values {x; y}, then
{x;y;1}= formula 4.1
Under the formula 2 we find values of other three-dimensional numbers of the sum from unit
formula 4.z
2.2.4 sumn1{x;y;z;…;Rn}.
Values of more complex n-dimensional numbers it is calculated similarly[7].
2.3 Functions of Multivariate numbers of a sum from unit.
Calculation of values of n-dimensional numbers of a sum from unit under the formula 2 is too bulky. But there are the regularities, allowing to calculate values of some multivariate numbers of a sum from unit through their coordinates.
2.3.1 f(x;y)sum21.
The function, allowing to express values of bidimentional numbers of a sum from unit through its coordinates, looks like:
f{x;y}sum2(C=1)= ,
function is valid for ;
at f{x;y}sum21=0 , at f{x;y}sum21=1,
I.e., if a point with coordinates of bidimentional number {x; y} is on one of numerical axes value of this number is equal to zero (exception {0;1}=1[8]):
{0;y≠1}=0,
{x;0}=0.
So, any bidimentional number {x; y} with coordinates x> 0, y> 0, y≠1 it is possible to find under the formula:
A2sum1{xa;ya}= , formula 5.1
Where xa, ya - coordinates of bidimentional number A2sum1.
The formula 5.1 for can be presented as
A2sum1{xa;ya}= formula 5.2
The schedule of function f {x; y} sum2 (C=1). Values of function z= are integers - separate points on space of the Cartesian coordinate axes if to connect these points among themselves we shall receive a surface which is a spiral twisting on an axis 0Y on 90º from plane X0Y in infinity (see schedule 1).
Schedule 1. A spiral z = {x; y}
2.3.2 f (x; y; z) sum31.
The way of decomposition of sequence on summands[9] we shall find regularity between values of three-dimensional numbers of a sum from unit from its coordinates (expressed through bidimentional numbers).
{x;y;z}=
formula 6.[10]
The schedule of function R4 = {x; y; z} is rotating around of an axis 0Y a spiral z = {x; y} with simultaneous, respective to rotation, extension of a petal of this spiral (extension of forming z={x;i[1;y]}).
2.4 Symmetry of values of n-dimensional numbers of a sum from unit.
Values of n-dimensional numbers of a sum from unit, located on the Cartesian space as points (coordinates of points coincide with coordinates of n-dimensional number) are symmetric concerning a straight line, a plane, a cube … with coordinates (x=y; y=x; z; …; Rn), i.e. value of number Ansum1 with coordinates { xa; ya; za; …; Ran } is equal to value of number Bnsum1 with coordinates {ya; xa; za; …; Ran }.
sumn1{ xa; ya; za; …; Ran }=sumn1{ya; xa; za; …; Ran } formula 7[11].
- Application of Multivariate numbers of a sum from unit.
3.1 Life[12].
{x; 1} - there is a point, unit of something, the base of mathematics[13].
{x; 2} - there is a sum of units of something, represents calculation of quantity. It is initial, fundamental action in mathematics[14].
3.2 Theory of probabilities.
a) {x; y} - there is a combination (x+y-2) elements on (x-1) the elements differing though one element, i.e.
{x;y}= или ={n-m+1;m+1}
b) Regularities, actions with the multivariate numbers, facilitating calculations:
-
By definition of multivariate number
-
Decomposition of bidimentional number on other bidimentional numbers
-
Decomposition of three-dimensional number on bidimentional numbers
-
The sum of multiplications of bidimentional numbers at consistently varying first coordinates
,
sense
- Disclosing complex function
3.3 Mathematics.
The way of decomposition of sequence on summands.
The sum of bidimentional numbers {x; y}, everyone increased on constant number, is function of a kind a1xn+ a2xn-1+…+ an+1x0=f(x) [1][15], and as bidimentional number {x; y} there is a sum of the sum, the sum, …, the sum of units it is possible to find regularities (function) for some only decreasing or only growing sequences described by function of a kind [1], representing a sums of the sum, of the sum, …, of the sum of units. We shall name such sequences - sequences of the bidimentional sum.
Let's consider an example of a finding of regularity for sequence of values of factor of the fifth member of function x(x+1)(x+2)…(x+(a-1))(x+a)=f(x) [2]. The fifth member after disclosing brackets looks like kpxp, where p=a−3. Let's make the table of values of factor kp depending on a degree p a variable x.
a
(variable
number of function[2])
3
4
5
6
7
8
9
10
11
12
13
14
15
p
(a degree of a variable x)
0
1
2
3
4
5
6
7
8
9
10
11
12
kp
(value of factor of the fifth
member of function [2])
0
24
274
1624
6769
22449
63273
157773
357423
749463
1474473
2749747
4899622
Table 3.
We have growing sequence. We shall find dependence between value of factor kp and a degree p of a variable x.
Let's make the table of members of sequence and differences of two next members of this sequence, and a difference we shall write down under the bigger number. We shall name a line of differences the level, and a line of sequence the maximum level. We shall designate each difference in coordinates (p; Ln): the first coordinate p - a degree of a variable x and number of the column, the second coordinate Ln - number of a level. We shall find differences of the next numbers for all levels (in the table the difference is allocated by color). At low enough level we shall reach a line consisting of zero. Let this level will be zero level, the next higher line will be the first level, and so on up to a maximum level.
level Ln
p
-1
0
1
2
3
4
5
6
7
8
9
10
L9max
kp
0
0
24
274
1624
6769
22449
63273
157773
357423
749463
1474473
L8
0
0
24
250
1350
5145
15680
40824
94500
199650
392040
725010
L7
0
0
24
226
1100
3795
10535
25144
53676
105150
192390
332970
L6
0
0
24
202
874
2695
6740
14609
28532
51474
87240
140580
L5
1
0
24
178
672
1821
4045
7869
13923
22942
35766
53340
L4
-26
-1
24
154
494
1149
2224
3824
6054
9019
12824
17574
L3
130
25
25
130
340
655
1075
1600
2230
2965
3805
4750
L2
-210
-105
0
105
210
315
420
525
630
735
840
945
L1
105
105
105
105
105
105
105
105
105
105
105
105
L0
0
0
0
0
0
0
0
0
0
0
0
0
Table 4.
The formula for a finding of any number of the table
, but formula 8.
,
,
… ,
i.e. the number of a n-level is the sum of numbers (n−1) - a level, since one of columns (we shall name it column of the beginnings, it can be chosen any way), and, finishing with column in which there is a required number (we shall name its column-coordinate), plus number of a n-level of a column before column of the beginnings (we shall name it column of a minimum):
, formula 9.1
or
.
Let's express the sum of numbers Ln-1-уровня of the formula 9.1 through the sums of numbers Ln-2-уровня
Opening the sums of the given equation, number (p; Ln-2) we shall meet once, number (p−1; Ln−2) we shall meet two times, …, number (pmin+1; Ln-2) we shall meet (p−pmin) times, i.e.
formula 9.2[16]
Let's express numbers Ln-2-уровня through numbers Ln-3-уровня
Let's open the sums and we shall group identical numbers
formula 9.3
Let's express numbers Ln-3-уровня through numbers Ln-4-уровня
Let's open the sums and let's group identical numbers
formula 9.4
Let's notice, that the sums of numbers of formulas 9.1-9.4 are bidimentional numbers of a sum from unit, having it in a kind, a method of deduction we shall express number (p; Ln) through the sums of numbers Ln-n-уровня and the sums of numbers of a column of the minimum, increased on bidimentional numbers.
The formula 10.1
But number Ln-n=0-уровня is equal to zero, therefore the sum in a square brackets and composed (in the formula 10.1) are equal to zero, then
Or
The formula 10.2
For number of a maximum level (sequence) the formula 10.2 will be written down
The formula 10.3[17]
So, we have received the formula for a finding of any member of the growing sequence expressed through numbers of the column of a minimum[18].
For simplification of calculations the column of a minimum should be chosen with the minimal numbers, with a plenty of zero or with identical numbers. It is visible, that the minimal numbers (table 4) are in columns with numbers close to zero. To fill such columns, it is necessary to continue a zero level to the left and under the formula 8 to find numbers of other levels (in table 4 such numbers on a transparent background).
Now we shall return to a finding of regularity for sequence of table 4.
We have growing sequence, its function we shall find under the formula 10.3, but all over again we shall choose a column of a minimum. Most of all approach a column p=0 (a) (a plenty of zero) and p=1 (b) (many identical numbers). We shall substitute numbers of the give columns and value of maximum level Lnmax=9 in the formula 10.3:
a)
b)
Resulting in standard record, we shall receive the equation of a kind [1] eighth order[19]
Let's consider necessary and sufficient conditions that the sequence was sequence of the bidimentional sum.
Sufficient conditions:
a) If function of sequence looks like a1xn+ a2xn-1+…+ an+1x0=f(x) the given sequence is sequence of the bidimentional sum[20].
b) If the sequence is decomposed up to zero[21] the given sequence is sequence of the bidimentional sum.
Necessary conditions:
a) If the sequence is characterized by the formula
[22],
that this sequence can be sequence of the bidimentional sum.
b) ???
3.4 Multivariate figures.
Calculation of quantity of connections in multivariate figures.
For example, quantity of connections in a n-dimensional parallelepiped equally
,
And the quantity of connections in a n-dimensional pyramid is equal
,
where n - dimension of a figure,
p - a kind of connections:
p=0 - connection - a point,
p=1 - connection - a straight line,
p=2 - connection − a plane,
and so on.
- Appendices.
The appendix 1.
The table of values f(x;y)sum21.
x
1
2
3
4
5
6
7
8
9
10
11
12
y
1
1
1
1
1
1
1
1
1
1
1
1
1
2
1
2
3
4
5
6
7
8
9
10
11
12
3
1
3
6
10
15
21
28
36
45
55
66
78
4
1
4
10
20
35
56
84
120
165
220
286
364
5
1
5
15
35
70
126
210
330
495
715
1001
1365
6
1
6
21
56
126
252
462
792
1287
2002
3003
4368
7
1
7
28
84
210
462
924
1716
3003
5005
8008
12376
8
1
8
36
120
330
792
1716
3432
6435
11440
19448
31824
9
1
9
45
165
495
1287
3003
6435
12870
24310
43758
75582
10
1
10
55
220
715
2002
5005
11440
24310
48620
92378
167960
11
1
11
66
286
1001
3003
8008
19448
43758
92378
184756
352716
12
1
12
78
364
1365
4368
12376
31824
75582
167960
352716
705432
13
1
13
91
455
1820
6188
18564
50388
125970
293930
646646
1352078
14
1
14
105
560
2380
8568
27132
77520
203490
497420
1144066
2496144
15
1
15
120
680
3060
11628
38760
116280
319770
817190
1961256
4457400
16
1
16
136
816
3876
15504
54264
170544
490314
1307504
3268760
7726160
17
1
17
153
969
4845
20349
74613
245157
735471
2042975
5311735
13037895
18
1
18
171
1140
5985
26334
100947
346104
1081575
3124550
8436285
21474180
19
1
19
190
1330
7315
33649
134596
480700
1562275
4686825
13123110
34597290
20
1
20
210
1540
8855
42504
177100
657800
2220075
6906900
20030010
54627300
21
1
21
231
1771
10626
53130
230230
888030
3108105
10015005
30045015
84672315
22
1
22
253
2024
12650
65780
296010
1184040
4292145
14307150
44352165
1,29E+08
23
1
23
276
2300
14950
80730
376740
1560780
5852925
20160075
64512240
1,94E+08
24
1
24
300
2600
17550
98280
475020
2035800
7888725
28048800
92561040
2,86E+08
25
1
25
325
2925
20475
118755
593775
2629575
10518300
38567100
1,31E+08
4,17E+08
26
1
26
351
3276
23751
142506
736281
3365856
13884156
52451256
1,84E+08
6,01E+08
27
1
27
378
3654
27405
169911
906192
4272048
18156204
70607460
2,54E+08
8,55E+08
28
1
28
406
4060
31465
201376
1107568
5379616
23535820
94143280
3,48E+08
1,2E+09
29
1
29
435
4495
35960
237336
1344904
6724520
30260340
1,24E+08
4,73E+08
1,68E+09
30
1
30
465
4960
40920
278256
1623160
8347680
38608020
1,63E+08
6,36E+08
2,31E+09
31
1
31
496
5456
46376
324632
1947792
10295472
48903492
2,12E+08
8,48E+08
3,16E+09
32
1
32
528
5984
52360
376992
2324784
12620256
61523748
2,73E+08
1,12E+09
4,28E+09
33
1
33
561
6545
58905
435897
2760681
15380937
76904685
3,5E+08
1,47E+09
5,75E+09
34
1
34
595
7140
66045
501942
3262623
18643560
95548245
4,46E+08
1,92E+09
7,67E+09
35
1
35
630
7770
73815
575757
3838380
22481940
1,18E+08
5,64E+08
2,48E+09
1,02E+10
36
1
36
666
8436
82251
658008
4496388
26978328
1,45E+08
7,09E+08
3,19E+09
1,33E+10
37
1
37
703
9139
91390
749398
5245786
32224114
1,77E+08
8,86E+08
4,08E+09
1,74E+10
38
1
38
741
9880
101270
850668
6096454
38320568
2,16E+08
1,1E+09
5,18E+09
2,26E+10
39
1
39
780
10660
111930
962598
7059052
45379620
2,61E+08
1,36E+09
6,54E+09
2,91E+10
40
1
40
820
11480
123410
1086008
8145060
53524680
3,14E+08
1,68E+09
8,22E+09
3,74E+10
The appendix 2.
The table of values f (x; y) f(x;y)mul2N(t).
x
1
2
3
4
5
6
7
8
9
10
y
1
1
2
3
4
5
6
7
8
9
10
2
1
2
6
24
120
720
5040
40320
362880
3628800
3
1
2
12
288
34560
24883200
1,25411E+11
5,0566E+15
1,8349E+21
6,6586E+27
4
1
2
24
6912
238878720
5,94407E+15
7,45453E+26
3,7694E+42
6,9167E+63
4,6055E+91
5
1
2
48
331776
7,92542E+13
4,71092E+29
3,51177E+56
1,324E+99
9,156E+162
4,217E+254
6
1
2
96
31850496
2,52429E+21
1,18917E+51
4,1761E+107
5,528E+206
#ЧИСЛО!
#ЧИСЛО!
7
1
2
192
6115295232
1,54368E+31
1,8357E+82
7,6661E+189
#ЧИСЛО!
#ЧИСЛО!
#ЧИСЛО!
8
1
2
384
2,34827E+12
3,62497E+43
6,6543E+125
#ЧИСЛО!
#ЧИСЛО!
#ЧИСЛО!
#ЧИСЛО!
9
1
2
768
1,80347E+15
6,53754E+58
4,3503E+184
#ЧИСЛО!
#ЧИСЛО!
#ЧИСЛО!
#ЧИСЛО!
10
1
2
1536
2,77014E+18
1,81099E+77
7,8784E+261
#ЧИСЛО!
#ЧИСЛО!
#ЧИСЛО!
#ЧИСЛО!
The appendix 3.
Value of n-dimensional number is convenient for presenting as the Latin letter with coordinates of the given number:
For one-dimensional number Ax=f (x),
For bidimentional number Bxy,
For three-dimensional number Cxyz,
and so on.
Then value of one-dimensional number of a sum from function f (t) is Ax. Value of bidimentional number of a sum from function f (t) is Bxy. We shall make the table of formulas for bidimentional number Bxy.
Bidimentional number of a sum of the first order
Bidimentional number of a sum of the second order
Bidimentional number of a sum of the third order
…
Bidimentional number of a sum of the y-order
Bx1=Ax
Bx2=
Bx3=→
Bx3=
…
Bxy=Ai→
Bxy=
Value of three-dimensional number of a sum from function f (t) is Cxyz.
Three-dimensional number of a sum of the first order
Three-dimensional number of a sum of the second order
Three-dimensional number of a sum of the third order
…
Three-dimensional number of a sum of the z-order
Cxy1=Bxy
Cxy2=
Cxy3=→
Cxy3=
…
Cxyz=Bij→
Cxyz=
Value of n-dimensional number of a sum from function f (t) is Nxyz … Rn.
n-dimensional number of a sum of the Rn-order
Nxyz…Rn=Oijk…(n-2)(n-1) →
Nxyz…Rn=
The appendix 4.
The formula 6 it is evident:
{i;x} i=
{i;z-1} i =
{x+z-1;y+z-1}
The appendix 5.
Let's prove, that the sum of bidimentional numbers {x; y}, everyone increased on constant number, is function of a kind a1xn+ a2xn-1+…+ an+1x0=f(x) [1], i.e.
[3]
So,
At disclosing brackets we shall receive a multinomial of a kind
Let's notice, that the maximal order of bidimentional number in the formula [3] is more on unit of a degree of function [1].
-
It is necessary to finish.
To add comments.
Item 1.1. a) Multivariate numbers from various functions f (t).
b) Existence of multivariate numbers from function f (t), tR, i.e. at nonintegral t.
Item 3.3. a) The finding of regularity for sequence of bidimentional multiplication.
b) Necessary conditions that the sequence has been decomposed on composed (was sequence of the bidimentional sum).
c) A provisional finding of regularity for sequence is similar to sequence of the bidimentional sum (the first level of the given sequence is approximately equal).
d) A finding of regularity for sequence, growing and decreasing.
e) An opportunity of decomposition of nonintegral sequence.
f) An application of multivariate numbers of a sum from unit.
Table of contents.
- Multivariate numbers.
1.1 The common concept about multivariate numbers.
1.2 Formation of multivariate numbers.
1.2.1 One-dimensional number.
1.2.2 Bidimentional number.
1.2.3 Three-dimensional number.
1.2.4 N-dimensional number.
1.3 Function of multivariate number.
- Multivariate numbers of a sum from constant number C=1.
2.1 Formation of multivariate numbers of a sum from unit.
2.2 Values of Multivariate numbers of a sum from unit.
2.2.1 sum11{x}.
2.2.2 sum21{x;y}.
2.2.3 sum31{x;y;z}.
2.2.4 sumn1{x;y;z;…;Rn}.
2.3 Functions of Multivariate numbers of a sum from unit.
2.3.1 f(x;y)sum21.
2.3.2 f(x;y;z)sum31.
2.4 Symmetry of values of n-dimensional numbers of a sum from unit.
- Application of Multivariate numbers of a sum from unit.
3.1 Life
3.2 Theory of probabilities.
3.3 Mathematics.
3.4 Multivariate figures.
-
Appendices.
-
It is necessary to finish.
[1] - Sum.
[2] - Multiplication.
[3] For example, at n =2 action is made on one-dimensional space - a piece, at n=3 - in a rectangular, at n=4 - in a parallelepiped and so on
[4] By analogy to the formula n! - multiplication of sequence of natural numbers.
[5] As at any of coordinates of multivariate number of a sum from unit equal to zero, value of this number is equal to zero (exception {0; 1; …; 1} =1, look a footnote 8).
[6] Incorrect record of the sum.
[7] Probably it is more convenient to present calculation of multivariate numbers how it is shown in the appendix 3.
[8] As {0; 1} there is a bidimentional number of a special case or one-dimensional number, and value of one-dimensional number is function f (t), in this case f (t) =1.
[9] About the way of decomposition of sequence on summands read in section 3 “Appendix of Multivariate numbers of a sum from unit”.
[10] The formula 6 is evidently submitted in the appendix 4.
[11] Exception number {0; 1; 1; …; 1} =1 (look a footnote 8).
[12] Pathos.
[13] Certainly, not number {x;1} is a prototype of unit, and on the contrary.
[14] Certainly, not number {x; y} is a prototype of the sum, and the sum is action due to which the given number is formed.
[15] Look the appendix 5.
[16] Numbers 1, 2, 3 … it was necessary to present as the sum of units 1, 1+1, 1+1+1 …
[17] This formula is for growing sequence. For decreasing sequence the first coordinate in bidimentional numbers is not (p-pmin), and this one (pmin-p), that is the formula for calculation of numbers of decreasing sequence will be written down
formula 10.4
[18] If we have found the formula not through a degree p a variable x, and through a variable a the sense would not be lost since value of a variable p or a, and a difference about a column of required number with the order of a column of a minimum is important not, and this difference for any variables is equal
r=p-pmin=a-amin,
that is numbering has no value
[19] The maximal order of bidimentional number is more on unit of a degree of required function (look the appendix 5).
[20] Look the appendix 5.
[21] Look table 4.
[22] If xn=1, sequence a constant, if xn> 1 the sequence can be sequence of bidimentional multiplication.
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