Factorisation Of Normalised Sequences Generated By z^i/(z^i+/-1) - an interim report

by

Huen Y.K.

CAHRC, P.O.Box 1003, Singapore 911101
http://web.singnet.com.sg/~huens/
email: huens@mbox3.singnet.com.sg

(A short communication - 1st released: 11/7/97. Revised: 12/7,16/7)


Abstract

Normalised sequences can be generated by the generalised even generating function z^i/(z^i-1). However, in factorisation, it is necessary to bring in the odd generating function z^i/(z^i+1) as well. By ordering finite sequences from these two functions in ascending number of terms starting from unity, these form an algebraic sequence number set. Sequence numbers have properties which are analogous to those in the natural number system especially concerning primeness and compositeness. There is no connection between this and algebraic numbers since the former is defined in the integral domain whereas the latter is defined in the complex domain. A finite sequence number is either reducible or irreducible. If reducible, it is a sequence primes, otherwise it is a sequence composite. There are three types of sequence primes. Sequence numbers with uniform unity intervals which are irreducible are called intrinsic sequence primes. Sequence primes having intervals greater than unity are called secondary sequence primes. Those which have irregular intervals and may also be unnormalised are called irregular sequence primes. This paper reports on interim findings on these three types of sequence primes for the first three sequence number systems only. Global prediction of sequence primes faces the same difficulties met in the deterministic prediction of primes.


======================================== How To Introduce HotSpots In .avi Video Clips Using VB5's Animated Control by Huen Y.K. CAHRC, P.O.Box 1003, Singapore 911101 http://web.singnet.com.sg/~huens/ email: huens@singnet.com.sg email: huens@mbox3.singnet.com.sg (A short communication released by the author on: 11/7/97) Abstract The paper describes how to get more out of the animation control in VB5 without doing more work. Animation controls are used to run .avi video clips. The development of avi files can be done by a well known third party softwares such as Corel 7. Although .avi files can run both sound and video, animation control can only run silent videos. This is not a great loss since sound files are still hogging too much bandwidth in the Internet. Unless really needed, the author desists from including .wav files in webpages. Multiple animation controls can be mounted in layers so that some can be assigned as video hotspots so that whenever the mouse is dragged in the down position over an assigned area, another video clip will pop and play. You can cause it to stop by lifting your finger from the mouse button whilst within the rectangle. Videos within videos can thus be introduced with great simplicity. Potential applications are found in the design of visually attractive instructional manuals. Since VB5 includes actiVEX technology, distributing these over the Internet is a real possibility. 1. Introduction Nowadays most academics might look askance at me if I tell them that I used solely VB5 for academic computing and publications. They wrongly associate modern VB with the primitive GWBasic packaged into DOS-1 in those pioneering days of Bill Gates. All computing languages are evolving. Even Fortran has transformed into Fortran 90 which hosted many new features and is still the language of choice in Supercomputer programming. The learning curve of C++ is just too steep and for busy scientists or mathematicians who have other preoccupations, I recommend VB. I switch from C++ to VB for that very reason and I never regretted on my choice. With each upgrade of VB, it is hosting almost as many OOP features as C++ so that the need to learn the latter has receded. Most of us seldom need to drive a compiler to its very limit. So why not use VB instead for programs with short shelf lives? 2. Animation Control In VB5 An .avi clip is a series of bitmap frames like a movie. One example is the piece of paper that "flies" between folders when copying files in the Windows 95 system. Although AVI clips can have sound, such clips cannot be used with the Animation control as only silent AVI clips can be used. To display avi files with sound one should use the Multimedia (MCI) control. The basic operations for animation control are few being confined to Open, Play, Stop, and Close methods. You open an .avi file using the Open method, play it using the Play method, and stop it with the Stop method. All gif files can run multithreaded and it is not necessary to close one file after another like what we do with file read/write operations. Assuming that you have an avi file called test.avi which you have designed yourself using either a shareware gif animator or Corel 7. In the first example, we setup two command buttons controls alled cmdPlay and cmdStop. We set up an animation control called anmAvi. You can place these control interactively in design mode and test it immediately in run mode. We assume that the VB5 project called video.vbj will be placed in the same directory as the video clip called test.avi. The three lines of code below will do it although it is the author's habit to introduce two menuitem called Cls and Exit, the purpose of which is self explanatory. Example 1: Project: video1.vbj Form1: video1.frm Module: video1.bas Option Explicit '====Start video clip by clicking Play command button===== Private Sub cmdPlay_Click anmAvi.Open "c:\vb5\test.avi" anmAvi.Play End Sub '====Stop video clip by clicking on Stop command button== Private Sub cmdStop_Click anmAvi.Stop End Sub '====Click menuitem to clear screen================= Private Sub memCLS_Click() Cls End Sub '====Click menutitem to end project================= Private Sub memEND_Click() End End Sub '=========================================== Here are the screen layouts: Menuitem Bar: --------------------------------------------------------------------------------------------------- Cls Exit ---------------------------------------------------------------------------------------------------- cmdPlay ----------------------------- | Open and Play | Visible at run time ----------------------------- cmdStop ----------------------------- | Stop | Visible at run time ----------------------------- Animation Control ------------------------------------------------------------ | Animation Control (Invisible at run time) | ------------------------------------------------------------ Expected results: The test.avi video clip will start playing when cmdPlay is clicked. It will stop play when cmdStop is clicked. It will be cleared from the screen when cmdClose is clicked. You can repeat by clicking these command buttons again. Exit menuitem will exit the project. Cls menuitem should clear the screen but it is doing nothing since there is nothing to clear. Example 2: This example is more ambitious. Athough you can stop the video in example 1, you cannot clear it from the screen. The Cls menuitem only works on Form1 itself. test1.avi is a larger rectangle and is assigned as the container of test2.avi. Both are invisible at run time until the cmdPlay button is clicked. Then test1.avi will play but tes2.avi remains static but its small thumnail is visible. If you clikc within it, it will also start running. Both can be stopped by clicking cmdStop but can only be clear from the screen by clicking on cmdClose. Close method is not enough to clear the videos from the screen. You have got to make them invisible. Option Explicit '================================= Private Sub anmAvi2_Click() anmAvi2.Visible = True anmAvi2.Open "c:\vb5\yyy.avi" anmAvi2.Play End Sub '================================== Private Sub cmdPlay_Click() 'dlgOpen.Filter = "laser.wav" 'dlgOpen.ShowOpen anmAvi.Visible = True anmAvi2.Visible = True anmAvi.Open "c:\vb5\xxx.avi" anmAvi.Play End Sub '================================== Private Sub cmdStop_Click() anmAvi.Stop anmAvi2.Stop End Sub '================================== Private Sub cmdClose_Click() anmAvi.Visible = False anmAvi2.Visible = False anmAvi.Close anmAvi2.Close End Sub '=================================== Private Sub memCLS_Click() Cls End Sub '==================================== Private Sub memEND_Click() End End Sub '===================================== Menuitem Bar: --------------------------------------------------------------------------------------------------- Cls Exit ---------------------------------------------------------------------------------------------------- cmdPlay ----------------------------- | Open and Play | Visible at run time ----------------------------- cmdStop ----------------------------- | Stop | Visible at run time ----------------------------- cmdClose ----------------------------- | Close | Visible at run time ----------------------------- Animation Control ------------------------------------------------------------ | AnmAvi (Invisible at run time) | | | | -------------------------------------------- | | |AnmAvi2 (Invisible at run time)| | | -------------------------------------------- | | | | | ------------------------------------------------------------ 3. Brief Description Of Making .avi Video Clips I found that Corel 5's Corelmove can only generate animated gif files, not avi files. On the other hand Corel 7's Corel Paint can generate both gif and avi files. This is very handy since it means that we can convert from one format to the other without leaving the software environment. If one is only interested in presenting Cartoon type of animations, animated gifs are most suitable but if one wants to present a video clip of some homemade movies or photographic stills snapped by digital cameras, then video clips are more suitable. I am sure readers have got their own preference from previous experience. So I choose to be brief here. 4. Summary Basically, one could improvise video hotspots using VB5's animation controls by mounting them in layers. The author has experiemented briefly with it and consider that there are more potentials than what has been discovered in an afternoon on the PC. I leave it to the imagination of readers to do more with it than what I have done. Since it is possible to roll one's own control in VB5, it will make it easier to improvise simple hotspot systems or video within video systems in a webpage provided one is experience in activeVEX programming. 5 References 1. Microsoft Reference Manual: Component Tools Guide, Microsoft Visual Basic 5 - Programming System for Windows. Document No. DD93014-1296 pp 14-16. (Printed in Singapore). 2.Danesh A. and Tatters W.(1996) : ActiveX Controls, chapter 34, pp426 to 439, JavaScript 1.1 Developer's Guide, Sams net (U.S.A.) 3. Spencer, K.L., Miller, K.C. & Lassesen L.(1996 ):ActiveX animation, chapter 12, pp231 to 258, Introducing VBScript and Activex, IDG Books (U.S.A.) =========================END OF PAPER ============================== ======================================= 1. Introduction

From textbooks, an algebraic number u is defined as a complex number which satisfies a polynomial equation with rational coefficients not all zero. An algebraic number is any complex number which is algebraic over the field Q of rationals [1,2]. In sequence algebra, a sequence number is a normalised sequence with unity values of coefficients not all zero and is algebraic over the integer domaim. Thus there is no connection between algebraic numbers and sequence numbers. The name sequence number is chosen because these have properties analogous to those of the natural number system inclusive of primeness and compositeness. This paper is confined to the investigations of sequence numbers in the integer domain.

In this paper, the general sequence number system is described by the generalised generating function Sni(z) given by equation (1) for i ranging from unity to infinity. Thus the sequence number system generated by putting i = 1 is given by Sn1(z)=z/(z+/-1) and those by putting i =2 is given by Sn2(z)=z^2/(z^2+/-1) and so on.

...................................................................i
.................................................................z
............................................Sni(z):=..---------------.............(1).
..............................................................i
...........................................................z...+/-...1

2. Factorisation Of Sequence Numbers

For each fixed index value of i, we can expand Sni(z) into a normalised sequence with uniform intervals of i units. Sequence numbers are formed by arranging finite sequences from expansion of Sni(z) with ascending number of terms starting from unity as shown in Tables 1a to 3b for system-1 to system-3. All series expansions are based on the Laurent series form although the same properties could also be elicited using Taylor's power series form.

(i) System-1; Sn1(z):=z/(z+/-1)

An even generating function generates sequences with positive signed terms. An odd generating function generates sequences with alternative positive and negative signed terms. Subsets of sequences generated by the even and odd Sn1(z) are listed in Tables 1a and 1b. Sequence primes are tagged with asterisks * in these tables. It is observed that for both even and odd Sn1(z), primes sequences only occur when the number of terms in a sequence equal to a prime greater than 1.

.....Table 1a - Even sequence number system-1: Sn1(z) = z/(z-1), i.e. i = 1.
.....(Maple program line: Sn1(z) := series(sum(z^i/(z^i-1),i=1..1),z=infinity,ub).
.....Intrinsic sequence primes are those tagged by asterisks with ub = a prime.
..........Actual program line used: sort(factor(sum(1/z^i,i=0..n)));
------------------------------------------------------------------------------------------------------------------
ub=....................................Even algebraic sequence numbers
number of terms..................(Sequence primes are tagged by *).
-------------------------------------------------------------------------------------------------------------------
1............................1
2*..........................1+1/z
3*..........................1+1/z+1/z^2
4............................1+1/z+1/z^2+1/z^3
5*...........................1+1/z+1/z^2+1/z^3+1/z^4
6.............................1+1/z+1/z^2+1/z^3+1/z^4+1/z^5
7*...........................1+1/z+1/z^2+1/z^3+1/z^4+1/z^5+1/z^6
8.............................(not displayed for page economy)
9........................... ..
10......................... ..
11* ...................... ..
12........................ ..
13* ...................... ..
14........................ ..
16........................ ..
17* ...................... ..
19* ..................... ..
23* ...................... ..
................................................................
----------------------------------------------------------------------------------------------------------------------

Table 1b - Odd sequence number system-1: Sn1(z) = z/(z+1), i.e. i = 1.
(Maple program line: Sn1(z) := series(sum(z^i/(z^i+1),i=1..1),z=infinity,ub).
Intrinsic sequence primes are those tagged by asterisks with ub = a prime.
Actual program line used: sort(factor(sum((-1)^i/z^i,i=0..n)));
----------------------------------------------------------------------------------------------------------------------------
ub=.....................................Odd algebraic sequence numbers
number of terms...................(Sequence primes are tagged by asterisks *).
----------------------------------------------------------------------------------------------------------------------------
1.............................1
2*...........................1-1/z
3*...........................1-1/z+1/z^2
4.............................1-1/z+1/z^2-1/z^3
5*...........................1-1/z+1/z^2-1/z^3+1/z^4
6.............................1-1/z+1/z^2-1/z^3+1/z^4-1/z^5
7*...........................1-1/z+1/z^2-1/z^3+1/z^4-1/z^5+1/z^6
8.............................(not displayed for page economy)
9............................. ..
10........................... ..
11* ......................... ..
12........................... ..
13* ......................... ..
14........................... ..
16........................... ..
17* ......................... ..
19* ........................ ..
23* ......................... ..
...................................................................................
---------------------------------------------------------------------------------------------------------------------------

Factorisation of larger sequence numbers may generate type 3 sequence primes which have irregular intervals and unnormalised sequences. These have not been exhaustively investigated but here are some examples:

....................Table 1c - A list of irregular sequence primes
...........................selected from sequences by factorising Sn1(z).
--------------------------------------------------------------------------------------------------------------------------
Irregular Intervals:

z^8-z^7+z^5-z^4+z^3-z+1
z^12-z^11+z^9-z^8+z^6-z^4-z+1
z^20-z^19+z^17-z^16+z^14-z^13+z^11-z^10+z^9-z^7+z^6-z^4+z^3-z+1
z^24-z^23+z^19-z^18+z^17-z^16+z^14-z^13+z^12-z^11+z^10-z^8+z^7-z^6+z^5-z+1
z^24-z^23+z^21-z^20+-z^18-z^17+z^15-+z^14+z^12-z^10+z^9-z^7+z^6-z^4-z+z^3-z+1

Irregular Intervals and Unnormalised:

Sn1(z):=sort(factor(z^165/(z^165-1)));

....................165 /..............10......9.....8......7.....6......5......4......3......2
Sn1(z) := z........./..((z - 1) (z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ z ..+ 1)
......................./

........4.....3......2
.....(z ..+ z ..+ z ..+ z ..+ 1)

........40...39....35...34...30...28....25...23....20...17...15...12....10....6.....5
.....(z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ 1)

........2 .................20...19....17....16...14...13...11...10......9.....7......6.....4.....3
.....(z ..+ z ..+ 1) (z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ z ..- z ..+ 1)

........8.....7......5.....4......3.................80....79....78...75...74...73...69...68...67....65
.....(z ..- z ..+ z ..- z ..+ z ..- z ..+ 1) (z ..+ z ..+ z ..- z ..- z ..- z ..- z ..- z ..- z ..+ z

..............64.......63....62...60...59...58...54...53...52....50 .....49......48......47.....46
......+ 2 z ..+ 2 z ..+ z ..- z ..- z ..- z ..- z ..- z ..- z ..+ z ..+ 2 z ..+ 2 z ..+ 2 z ..+ z

...........44...43...42...41...40...39...38...37...36....34......33......32.......31.....30....28
...... - z ..- z ..- z ..- z ..- z ..- z ..- z ..- z ..- z ..+ z ..+ 2 z ..+ 2 z ..+ 2 z ..+ z ..- z

..........27....26...22...21...20...18.....17.......16....15....13...12...11...7.....6.....5......2
......- z ..- z ..- z ..- z ..- z ..+ z ..+ 2 z ..+ 2 z ..+ z ..- z ..- z ..- z ..- z ..- z ..- z ..+ z

......+ z ..+ 1))

-------------------------------------------------------------------------------------------------------------------

(ii) System-2

Tables 2a and 2b show sequence numbers for system-2. The pattern of sequence primes is less obvious since these are not confined to those with number of terms equal to a prime. Furthermore, the distribution of sequence primes are not symmetrical, there being more sequence primes in the odd series than the even series.

Table 2a - Even sequence number system-2: Sn2(z) = z^2/(z^2-1), i.e. i = 2.
.......(Maple program line: Sn2(z) := series(sum(z^i/(z^i-1),i=2..2),z=infinity,ub).
.......Actual program line used: sort(factor(sum(1/z^(2*i),i=0..n)));
---------------------------------------------------------------------------------------------------------------------------
ub.................................................Even algebraic sequence numbers
=number of terms.........................(All sequence primes are tagged with asterisks *)
(Term counts include zeroes).........All without exceptions are sequence primes.
--------------------------------------------------------------------------------------------------------------------------
1........................................1
3*......................................1+1/z^2
5........................................1+1/z^2+1/z^4
7........................................1+1/z^2+1/z^4 +1/z^6
9........................................1+1/z^2+1/z^4 +1/z^6+1/z^8
11......................................1+1/z^2+1/z^4 +1/z^6+1/z^8+1/z^10
13......................................1+1/z^2+1/z^4 +1/z^6+1/z^8+1/z^10+1/z^12
15......................................1+1/z^2+1/z^4 +1/z^6+1/z^8+1/z^10+1/z^12+1/z^14
...................................................................................
-------------------------------------------------------------------------------------------------------------------------

Table 2b - Odd sequence number system-2: Sn2(z) = z^2/(z^2+1), i.e. i = 2.
.........(Maple program line: Sn2(z) := series(sum(z^i/(z^i-1),i=2..2),z=infinity,ub).
.........Actual program line used: sort(factor(sum((-1)*i/z^(2*i),i=0..n)));
---------------------------------------------------------------------------------------------------------------------------
ub................................................Odd algebraic sequence numbers
=number of terms.........................(Odd algebraic sequence numbers are generated by
(Term counts include zeroes).........Sn2(z)=z^2/(z^2-1). These are not shown
--------------------------------------------------------------------------------------------------------------------------
1.........................................1
3.........................................1-1/z^2
5*.......................................1-1/z^2+1/z^4
7.........................................1-1/z^2+1/z^4 -1/z^6
9*.......................................1-1/z^2+1/z^4 -1/z^6+1/z^8
11 ......................................1-1/z^2+1/z^4 -1/z^6+1/z^8-1/z^10
13*.....................................1-1/z^2+1/z^4 -1/z^6+1/z^8-1/z^10+1/z^12
15.......................................1-1/z^2+1/z^4 -1/z^6+1/z^8-1/z^10+1/z^12-1/z^14
21* ....................................(not displayed for page economy)
25* ................................... ..
33* ................................... ..
37* ................................... ..
57* ................................... ..
61* ................................... ..
73* ................................... ..
81* ................................... ..
85* ................................... ..
89 ................................... ..
93* ................................... ..
97 ................................... ..
101.................................. ..
105*.................................. ..
113.................................. ..
117*.................................. ..
121*.................................. ..
-------------------------------------------------------------------------------------------------------------------------

(iii) System-3

The invesitgation was not exhaustive but exploration up to 117 terms (i=39) did not uncover anymore sequence primes.

Table 3a - Even sequence number system-2: Sn3(z) = z^3/(z^3-1), i.e. i = 3.
.........(Maple program line: Sn2(z) := series(sum(z^i/(z^i-1),i=3..3),z=infinity,ub).
..........Acutal program line used: sort(factor(sum(1/z^(3*i),i=0..n)));
---------------------------------------------------------------------------------------------------------------------------
ub.................................................Even algebraic sequence numbers
=number of terms.........................(All sequence primes are tagged with asterisks *)
(Term counts include zeroes)..........All without exceptions are sequence primes.
--------------------------------------------------------------------------------------------------------------------------
1........................................1
3........................................1+1/z^3
7*......................................1+1/z^3+1/z^6
10......................................1+1/z^3+1/z^6+1/z^9
13......................................1+1/z^3+1/z^6+1/z^9+1/z^12
16......................................1+1/z^3+1/z^6+1/z^9+1/z^12+1/z^15
............Up to i = 39, no more sequence primes have been uncovered. i=34...................................irregular and unnormalised sequence composite
i=39...................................irregular sequence composite
.....................................................................
-------------------------------------------------------------------------------------------------------------------------

Table 3b - Odd sequence number system-2: Sn3(z) = z^3/(z^3+1), i.e. i = 3.
..........(Maple program line: Sn2(z) := series(sum(z^i/(z^i-1),i=3..3),z=infinity,ub).
..........Actual program line used: sort(factor(sum((-1)^i/z^(3*i),i=0..n)));
---------------------------------------------------------------------------------------------------------------------------
ub.................................................Odd algebraic sequence numbers
=number of terms.........................(All sequence primes are tagged with asterisks *)
(Term counts include zeroes)..........All without exceptions are sequence primes.
--------------------------------------------------------------------------------------------------------------------------
1........................................1
3........................................1-1/z^3
7*......................................1-1/z^3+1/z^6
10......................................1-1/z^3+1/z^6-1/z^9
13......................................1-1/z^3+1/z^6-1/z^9+1/z^12
16......................................1-1/z^3+1/z^6-1/z^9+1/z^12-z^15
............Up to i = 39, no more sequence primes have been uncovered. i=34...................................irregular and unnormalised sequence composite
i=39...................................irregular sequence composite
.....................................................................
-------------------------------------------------------------------------------------------------------------------------

3. Postulates and Theorems

(i) Postulates

Postulate 1a: A normalised sequence in system-1 which contains p terms where p is a prime is irreducible. This can be checked using the Maple program line sort(factor(sum(1/z^i,i=0..p-1))); and sort(factor(sum((-1)^i/z^i,i=0..p-1))) where p is a prime.

Postulate 1b: A normalised sequence which contain p terms with uniform intervals of p unit where p is a prime is irreducible. This can be checked using the Maple program line sort(factorsum(1/z^(p*i),i=0..p-1))) and sort(factor(sum((-1)^i/z^(p*i),i=0..p-1))) where p is a prime.

Remarks: In both Theorems 1a and 1b above, the distributions of primes are symmetrical.

Postulate 2: All number sequences can be reduced to a product of sequence primes.

Postulate 3: There is no end to the number of sequence primes.

(ii) Theorems

Theorem 1: The generating function z^i/(z^i-1) is factorisable for i > 1 and for all sequence numbers with number of terms greater than two.

Proof: Using a well known algebraic identity given by

(z^i-1) = (z-1)*(z^(i-1)+z^(i-2)+......+z+1)

we show that this generating function for i > 1 is factorisable as follows:

Factor(z^i/(z^i-1)) = z^i/((z-1)*(z^(i-1)+z^(i-2)+......+z+1)).

Only the first two terms of 1 and 1+1/z^i are irreducbile, i.e., sequence primes. Q.E.D.

Theorem 2: Sni(z):=z^i/(z^i-1) can be derived from the equation of divisibles.

Proof:

................................................(2 i)
...............................................z
..........................Sni(z) := -------------- = z^(2*i)* (equation of divisibles) .....(2). .............................................i....i
...........................................z (z - 1)
................................................................................................................Q.E.D.

Theorem 3: The product of an even generating function and an odd generating function will generate a normalised sequence with double the intervals.

Proof: An even generating function is given by z^i/(z^i-1) and that for an odd generating function is given by z^i(z^i+1). The product of the two is given by z^(2*i)/(z^(2*i)-1) which of course will generate a normalised sequence with doubled the intervals compared to the orginal generating functions. Q.E.D.

Comment: The well known identity (z^2-1) = (z-1)*(z+1) is the simplest case of this theorem.

4. Conclusions

This paper reports interim findings on sequence primes on system-1, system-2, and system-3 sequence numbers. It is found that all sequence numbers in system-1 which contain the number of terms equal to a prime are irreducible. This remains an unsolved problem as there is no proof of this. Just like in the prediction of primes, it is very unlikely that a deterministic formulation could be found for the global prediction of sequence primes. This is because it is found that factorisation of larger sequence numbers result in the appearance of irregular sequence primes which are difficult to charaterise. Nevertheless the findings should fuel interest amongst number theorists interested in finding explanations for the findings reported in this paper.


5. References

1. Pinter C.C.: A Book Of Abstract Algebra (2nd Edition) Chapter 25, pp 251 to 257.

2. Garret Birkhoff and Sauders MacLano: A survey of modern algebra, (4th Edition), pp435 to 439.


The references below are not necessarily related to the present paper but are provided as a convenient source of information on sequence algebra.


3. Huen Y.K.: A Simple Introduction To Sequence Algebra, URL site: http://web.singnet.com.sg/~huens/.

4. Huen Y.K.: The Canonical Generating Function or CGF(z) - a Swiss-knife function. URL site: http://web.singnet.com.sg/~huens/ .

5. Huen Y.K.: Information Contents Of Number Theoretic Functions. URL site: http://web.singnet.com.sg/~huens/ .

6. Huen Y.K.: In Search Of Exotic Arithmetic Operators, URL site: http://web.singnet.com.sg /~huens/ .

7. Huen Y.K.: Visual Solutions Of Number Theoretic Functions in Multidimensional Sequence Space, URL site: http://web.singnet.com.sg /~huens/ .

8. Huen Y.K.: Final Value Theorems Applied To Number Sequences -- its strengths and weaknesses, URL site: http://web.singnet.com.sg /~huens/ .

9. Huen Y.K.: Unsolved Problems In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .

10. Huen Y.K.: Explicit Formulation For Modular Arithmetic In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .

11. Huen Y.K.: Cyclic Generating Functions In Sequence Algebra, URL site: http://web.singnet.com.sg /~huens/ .

12. Huen Y.K. : Methods Of Developing Sequence Algebraic Formulations For Comp(z) and Prime(z). URL site: http://web.singnet.com.sg /~huens/.

13. Huen Y.K. : Information Contents Of Hypothetical DNA Sequences. URL site: http://web.singnet.com.sg/~huens/ .

14. Huen Y.K. : Composite Number Sequence Challenge 1/97. URL site: http://web.singnet.com.sg/~huens/ .

15. Huen Y.K. : Lemmata, Corollaries, And Theorems In Sequence Order Analysis. URL site: http://web.singnet.com.sg/~huens/ .

16. Huen Y.K.: A Matrix Map for Prime and Non-prime Numbers, INT. J. Math. Educ. Sci. Technol., 1994, VOL. 25, NO.6, pp 913-920.

17. Huen Y.K.: Some Interesing Properties Of The Natural Number System, Int. J. Math. Educ. Sci. Technol., 1996, VOL.27, NO. 5, 685-691.

18. Huen Y.K.: Visual algebra and its applications, INT. J. Math. Educ. Sci. Technol.,1996, VOL.??, NO.?, ???-??? (In the press as proof paper mes 100421).

19. Huen Y.K.: Twin primes revisited: INT. J. Math. Educ. Sci. Technol., 1997, VOL.??,NO.?, ???-???. (In the press as proof paper mes 100488).

20. Huen Y.K.: Is Pie Periodic? : INT. J. Math. Educ. Sci. Technol., 1997, VOL??,NO.?,???- ???,(In the press as proof paper mes100495).

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