MRS. Jesse Sakari Hyttinen

Transcription

1 MRS Jesse Sakari Hyttinen

2 MRS HIERARCHY AND BASIC STRUCTURE MRS als knwn as mnrutesystemsis means a grup which includes all mnrutesystemsis als knwn as Mrsystemsis (MrS) As there can be infinitely many Mr-systemsis, is the size f MrS-grup (MRS) infinite Yet a bigger entity is a MRS-grup whse MrS-grups are all different cmpared t each ther based upn MRS-appinted restrictins als knwn as cnditins As there can be infinitely many restrictins and cmbinatins f restrictins is als the size f MRS-grup infinite MrS aka Mnrutesystemsi cnsists f mnrutesystems aka Mr-systems (Mrs) The quantity f Mr-systems is variable, and thus the size f MrS aka Mrs-grup is between zer and infinite Mrs aka mnrutesystem cnsists f pints and pint cnnecting trails Mr-systems f an arbitrary Mr-systemsi are all different cmpared t each ther, but cnsist f always the same amunt a certain pint type r a cmbinatin f pint types based upn mutual cnditins A pint is ne f Mr-system s cmpnents A pint can be divided int three pint types, what is useful especially in PT-arithmetics and when the starting pint in a structure des matter The three types f a pint are a prime pint, a nrm pint and a side pint Prime pint aka p-pint (p p ) is the starting pint f examining a structure and besides nrm and side pint ne f PTarithmetics s crnerstnes Every ther pints are seen t riginate frm prime pint and that s why p p is always marked visible p p can be cnnected t many different pints (nrm and side pints) at the same time Whether the prime pint aka the bservatin pint f Mr-system s structure is remarkable, affects cnsiderably the quantity f Mrsystems and thus the whle Mr-systemsi That s why it is always marked apart, whether the MrS is p p [+] (p p is remarkable aka the bservatin pint des matter) r p p [ ] (p p is nt remarkable aka the bservatin pint des nt matter) Nrm pint aka the n-pint (n p ) creates with side pints and ther nrm pints arbitrary structures sectrs n p can be cnnected t all the three classes f a pint and like the prime pint t many at the same time Side pint aka s-pint (s p ) ends the structures begun by prime pint, and the n-pints between the prime pint and side pints can be thught as gap pints aka the middle sectin f the structures s p is the nly class f pint which can be cnnected t nly ne pint, either p- r n-pint A trail is the secnd cmpnent f an Mr-system (there are tw cmpnents) and creates the cnnectin between tw pints When the quantities f the tw cmpnents pints and trails are cmpared t each ther, is there always ne trail less A sectr cnsists f an arbitrary cmbinatin f the trails and pints (n- and s-pints) and leaves f a pint which is cnnected t at least tw ther trails in additin t the trail between the sectr and the pint A sectr (symbl m ) can be divided in t tw classes: prime sectrs (symbl m p ) leaving f the prime pint and nrm sectrs (symbl m n ) leaving f the nrm pint, if and nly if neither f the nrm sectrs cnsist f the prime pint A sectr has always a beginning (p- r n-pint) and an end (s-pint r s-pints)

3 A rute cnsists f ne r mre trails The system cnstructed by pints and trails is Mrs if and nly if there is nly ne rute between each pint pair (There are ( m ) = m!!(m )! pint pairs, when the ttal quantity f pints m If m <, there are n pint pairs) Frm this ne can cnstruct the basis f PT-arithmetics and the mst imprtant rule in the MRS-thery: The MRS-therem aka L MRS L MRS : Frm ne arbitrary pint t anther arbitrary pint can ne g thrugh nly ne rute The system cnsisting f pints and trails is Mrs if and nly if the system qualifies the L MRS PRS PRS-thery aka plyrutesystemsis-thery is t be expected a much mre cmplicated thery than the MRS-thery, as there can be an arbitrary number f trails between pints (hwever there has t be at least ne trail) It s difficult t prduce a wrking PT-arithmetic analysis methd, s in this wrk nly the MRS-thery is being cncentrated n EXAMPLES E Let frm the MRS-grup a particular MRS be chsen, in which the p-pint and every n-pint has t be cnnected t at least three ther pints Let frm the MrS-grup a particular MrS be chsen, in which there are seven pints Let it be decided that the MrS is p p [ ], and an MrS which includes tw Mr-systems is gtten When the MrS is p p [ ], the p-pint is highlighted as a white pint amng ther pints, which are all black pints Trails are black lines E The same MrS as in example E, nly that MrS is nw p p [+] Let the terminlgy f the MRS-thery be demnstrated

4 The MrS nw cnsists f three Mr-systems The marking habit f the upper Mr-systems is used when the pints are numbered; The marking habit f the lwer Mr-systems (same systems as upper nes) is used, when the pints are nt numbered The numbering is based n a mre specific analysis, which is later returned t Lines are trails, arrws r arrw cmbinatins are rutes The parts circled with dashed line are prime sectrs and the parts circled with cntinuus line are nrm sectrs PRESENTATION FORMS Mrs and due t it the bigger structures (MrS, MRS, MRS-grup) can be presented in three different ways The presentatin frms are tag, sum and map frm Tag frm aka [TF] means a symblic presentatin The general tag frm fr Mr-system is m a Σ φ (a is the rdinal number f tag frms crrespnding sum frm, Σ φ is a cnditin matrix, φ is the pint number assigned by the cnditin matrix, is the number f cnditin matrice s cnditins and the tag m a Σ φ is read m a, when the cnditin matrice s frm is Σ φ ), fr Mr-systemsi M φ Σ φ, fr MrS-grup (M φ Σ φ ) φ=0,,, = M 0 Σ 0, M Σ, M Σ, (grup presentatin) and fr MRS-grup {(M φ Σ φ ) φ=0,,, φ ) φ=0,,, φ ) φ=0,,, φ ) φ=0,,, x } =x0, x, x, = (M φ Σ x0, (M φ Σ x, (M φ Σ x, (the grup presentatin f grups) It s bserved that in particular tag frms m a and M φ can be alternatively used fr Mr-system and -systemsis, if the cnditin matrix has been presented and clearly cnnected t Mr-system and -systemsis Let als the fllwing facts be bserved: Σ φ desn t mean a sum In tag Σ φ the φ isn t an expnent but a tag clearing identifier and as a pint number is a part f the cnditin matrix There isn t a system that identifies precisely cnditin matrices, as the system is hard t be executed and benefits a little r nne at all PT-arithmetic analyses In general presentatins exceptinally Σ φ xm means cnditin matrice s type x m and (Σ φ x ) =x0, x, x, means all the pssible cnditin matrice s types Althugh the pint numbers and quantities f cnditins fr a tw r mre cnditin matrices are the same, the cnditins fr the cnditin matrices and thus the cnditin matrices itself can be arbitrarily different a Sum frm aka [SF] means PT-arithmetic presentatin The general sum frm fr Mr-system is (φ) φ Σ φ (a is the rdinal number f sum frm, φ is the pint number assigned by the cnditin matrix, φ is the pint ntatin assigned

5 by the cnditin matrix, Σ φ is the cnditin matrix, is the number f cnditin matrice s cnditins and a (φ) φ Σ φ a is read (φ) φ Σ ({φ} φ Σ φ Σ ) φ=0,,, = {0} φ Σ 0 Σ, {} φ Σ Σ, {} φ Σ {({φ} φ Σ φ x ) φ=0,,, } =x0, x, x, = ( Σ {φ} φ, when the cnditin matrice s frm is Σ φ Σ ), fr Mr-systemsi {φ} φ Σ, (grup presentatin) and fr MRS-grup Σ φ x0 ) φ=0,,, Σ, ({φ} φ Σ φ x ) φ=0,,, a grup presentatin f grups) Let it be bserved that in particular tag frms (φ) φ Σ, ({φ} φ Σ and {φ} φ Σ φ, fr MrS-grup Σ φ x ) φ=0,,,, (the can be alternatively used fr Mr-system and -systemsis, if the cnditin matrix has been presented and clearly cnnected t Mr-system and -systemsis Map frm aka [MF] means graphic presentatin As map frm is an arbitrary figure drawn n an underlay (paper, blackbard, drawing applicatin s drawing state fr a cmputer and et cetera), there s n general presentatins fr map frm Often map and tag frms are inferred directly frm the pened sum frms Opening sum frms is the subject f discuss in the part f PT-arithmetics Let it yet be mentined that ne can use abbreviatins fr tag, sum and map frms: T-frm fr tag frm, S-frm fr sum frm and M-frm fr map frm EXAMPLES E Let frm an MRS-grup an MRS be chsen which has n restrictins Let frm an MrS-grup an MrS be chsen which has fur pints (φ = 4, φ = snp) Let it be decided that MrS is p p [ ], and an MrS which includes tw Mr-systems is gtten Let presentatin frms be demnstrated, when the cnditin matrice s frm is Σ 4 MrS: Tag frm: M 4 Σ 4 Sum frm: Σ {4} snp Map frm: Picture ) Mrs: [TF]: m [SF]: (4) snp Mrs: [TF]: m [SF]: (4) snp [MF]: Picture ) [MF]: Picture ) E

6 The same MrS as in example E with the exceptin f Mrs being nw p p [+] Let presentatin frms be demnstrated MrS: [TF]: M 4 Σ 4 [SF]: Σ {4} snp [MF]: Picture 4) Mrs: [TF]: m [SF]: (4) snp Mrs: [TF]: m [SF]: (4) snp Mrs: [TF]: m [SF]: (4) snp 4 4 Mrs: [TF]: m 4 [SF]: (4) snp [MF]: Picture 5) [MF]: Picture 6) [MF]: Picture 7) [MF]: Picture 8) NUMBERS AND PARAMETERS Numbers and parameters, which define certain prperties and affect the Mr-system and the larger structures (MrS, MRS, MRS-grup), are an essential part f the MRS-thery and are thus a part f the PT-arithmetics In the MRSthery (and thus in the PT-arithmetics) all numbers and values belng t the grup = (n) n=0,,, = 0,,,, excluding the identity numbers l x and l which are presented in the furth chapter Map frm s structures Pint number φ is an MrS-parameter and describes the quantity f an arbitrary pint s class r the cmbinatin f classes The quantity must always be fulfilled An essential part f the pint number is the pint ntatin φ, which defines pint s class r the cmbinatin f pint s classes which bth have t be cunted The pint ntatin als wrks separately as an Mrs-parameter All the pssible values f the pint ntatin are the fllwing: s, n, p, sn, sp, np and snp (value s matches pint s class s p, value n matches pint s class n p and value p matches pint s class p p ) The mst practical values f the pint ntatin are sp, np and snp and when the value f the pint ntatin isn t snp, pints ttal quantity s frntier s presentatin φ snp [a, b] can be defined t guarantee that the MrS-number M(φ) is limited The a is the frntier s minimum and the b is the frntier s maximum (the frntier s presentatin can have ther defining areas as well, fr example φ snp < 8 r φ snp = 0) The MrS-number M(φ) is the functin f the pint number φ M(φ) describes the quantity f Mr-systems fr a certain pint number φ Mrs-number aka pint quantity m a (φ ) is the functin f the pint ntatin φ and describes the quantity f Mrsystem s m a certain pint s class r the cmbinatin f pint s classes Althugh the pint number φ has a certain pint ntatin s φ value, m a (φ ) can still get values fr all the values f pint ntatin φ Let it be bserved, that m a (s) + m a (n) + m a (p) = m a (s) + m a (np) = m a (n) + m a (sp) = m a (p) + m a (sn) = m a (snp) Als the fllwing things are valid: m a : m a (p) =, m a : m a (n) = 0 and m a : m a (s) > 0

7 The general trail number τ x is amng the pint number φ and the pint ntatin φ ne f the mst imprtant numbers in PT-arithmetics The value r grup f values f the general trail number defines directly the allwed quantities f trails cnnected t the pint x Presentatin habits f the value r grup f values f the general trail number, when A is a part f a sentence, which is read The quantity f the trails cnnected t the pint x has t be : τ x > a: A bigger than a τ x a: A bigger than r as big as a τ x < a: A smaller than a τ x a: A smaller than r as small as a τ x ]a, b[, a < b: A smething between numbers a and b τ x ]a, c[ \{b}, a < b < c: A smething between numbers a and c, excluding the value b τ x ]a, d[ \{b, c}, a < b < c < d: A smething between numbers a and d, excluding the values b and c τ x [a, b], a < b: A a r b r smething between numbers a and b τ x [a, c] \{b}, a < b < c: A a r c r smething between numbers a and c, excluding the value b τ x [a, d] \{b, c}, a < b < c < d: A a r d r smething between numbers a and d, excluding the values b and c τ x = a: A is equal t a τ x a: A isn t equal t a τ x = a, b, c, a < b < c: A is equal t a, b r c τ x a, b, c, a < b < c: A isn t equal t a, b and c Let it be bserved that the general trail number s lgical maximum value desn t crss the ttal quantity f trails and the alternative values f the general trail number usually rise and nt s ften keep the MrS-number s value at bay In ther wrds M(φ) τ x = a, b, c; Σ φ M(φ) τ x = a, b; Σ φ, when the cnditin matrices are ttally identical t each ther The presentatin frms f the general trail number τ x are τ p, τ n and τ s τ p (x = p) is the trail number f the p-pint The thery f general trail number τ x applies directly t the trail number f the p-pint τ n (x = n) is the trail number f the n-pint The thery f general trail number τ x applies directly t the trail number f the p-pint, nly if τ n > τ s (x = s) is the trail number f the s-pint The thery f general trail number τ x desn t apply t the trail number f the p-pint, as the value f the trail number f the s-pint is always ne The trail numbers τ p, τ n and τ s are always presented tgether in the cnditin matrix In additin t the trail numbers the cnditin matrix includes the infrmatin whether the bservatin pint is remarkable, the pint number and ntatin and pssibly the presentatin φ snp [a, b], a < b and mre specific restrictins The cnditin matrice s Σ φ frm is Σ φ = 0,,,, 4, 5, 6,, in which the cnditins 0,,,, 4, 5 and 6 are determined and the cnditins 7, 8, 9, are free t be determined The determined cnditins are the fllwing, when the whle cnditin itself cmes after the mark : 0 p p [+] r 0 p p [ ] φ = q (q is sme value) φ = q (q is sme value f the fllwing values: s, n, p, sn, sp, np, snp)

8 τ p X (X is sme arbitrary grup f numbers, whse size is at least ne) 4 τ n X (X is sme arbitrary grup f numbers, whse size is at least ne) 5 τ s = 6 φ snp [a, b], a < b The cnditin matrice s frm is thus Σ φ = ( j ) j=0,,, = 0,,,, 4, 5, 6, = p p [+/ ], φ = q, φ = q, τ p X, τ n X, τ s =, φ snp [a, b], The cnditin matrix defines with its cnditins aka restrictins the whle bservable Mr-systemsi, s Σ φ is created due t its significance befre PT-arithmetic analyses The cnditin matrix is, simply put, a grup f determined cnditins (the size f the grup is ) in a matrix frm The parenthesis f the matrix frm vanish, if Σ φ is put inside the parenthesis The fllwing numbers and presentatins are nly practical when the bserved MrS is p p [+] and has been applied with the MPN MPN MPN aka the mnrutesystem s pints numbering is a methd, which is applied when the bserved MrS is p p [+] With the numbering the arbitrary Mr-system s map frm can be bserved in detail Let MPN be presented as a catalgue: The bserved map frm f the Mr-system is drawn in a way that sectrs with different pint quantities (snp) are in rder frm the smallest t the biggest (the bigger the pint quantity, the bigger the sectr), sectrs with same pint quantities (snp) and different M-frms are in accrdance t sum frm s rder number in rder frm the smallest t the biggest and ttally identical sectrs (the same pint quantity (snp) and M-frm) are in arbitrary rder The rder s directin is clckwise (rtatin s directin ), when the rder is frm the smallest t the biggest Apprximatin: Sectrs with same pint quantities and different M-frms are in length rder frm the shrtest t the lngest, when the length f the sectr is determined by its lngest rute r rutes (the lnger the rute r rutes the lnger the sectr) The numbering is started The prime pint is always the pint number ne The pints are numbered after the prime pint in the same rder as the M-frm itself is drawn (in accrdance t catalgued instructins) Lcal trail numbers and rute presentatins can be defined fr MPN-adapted Mr-system The lcal trail number t ma (b) is the functin f a pint number b t ma (b) expresses the quantity f trails cnnected t the Mr-system s m a pint number b The rute presentatin r ma (b, c) is the functin f pint numbers b and c r ma (b, c) expresses the rute frm pint b t c in the Mr-system m a Let it be bserved, that b c (b > c r b < c) aka the case b = c isn t determined (r ma (b, c) = [Ø] isn t a rute presentatin, as b = c)

9 Uniqueness: Let there be an Mrs m a in which there are numbered pints c, d and e If bth pint c and pint e are cnnected with a trail directly t the pint d, then uniquely r ma (c, e) = [c d e], r ma (e, c) = [e d c], r ma (c, d) = [c d], r ma (d, c) = [d c], r ma (d, e) = [d e] and r ma (e, d) = [e d] Quantity: Let there be an Mrs m a fr which m a (snp) = q Thus there are ( q ) = q! arbitrary pint pairs which!(q )! are different t themselves In additin there are ( q ) = q! = q! = q! = q! (tw rute!(q )!!(q )! (q )! (q )! presentatins per ne pint pair) arbitrary rute presentatins which t are different t themselves The quantity f rute presentatins is even, as in Mr-system m a and in vectr-based thinking (pints are deleted, every trail in the rute is a vectr with the same directin as the rute) there are tw vectrs with equal sizes and ppsite directins aka rute presentatins r ma (a, b) and r ma (b, a) fr every arbitrary pint pair a and b (pints numbers) S are there tw rute presentatins per ne pint pair As any given number ( q ) N = (n) n=,,, =,,, (als q N\{}) multiplied by tw is an even number thus is the quantity f rute presentatins, ( q ) when q = m a (snp) > is an arbitrary pint number always even The demnstratin f vectr-based thinking: ab = ba, r ma (a, b) ab (in a vectr-based thinking) and r ma (b, a) = ba (in a vectr-based thinking) EXAMPLES E Let there be the fllwing MrS: M 9 Σ 9 7, when Σ = ( j ) j=0,,, = 0,,,, 4, 5, 6 = p p [ ], φ = 9, φ = snp, τ p >, τ n >, τ s =, φ snp = 9 The fllwing [MF] f the Mr-systemsi is gtten: m (s) = 8, m (n) = 0, m (p) =, m (s) = 7, m (n) =, m (p) =, m (s) = 6, m (n) =, m (p) =, m 4 (s) = 7, m 4 (n) =, m 4 (p) =, m 5 (s) = 6, m 5 (n) =, m 5 (p) = M(9) = 5

10 E Let there be the fllwing MrS: M 9 Σ 9 7, when Σ = ( j ) j=0,,, = 0,,,, 4, 5, 6 = p p [+], φ = 9, φ = snp, τ p >, τ n >, τ s =, φ snp = 9 The fllwing [MF] f the Mr-systemsi is gtten: E Let there be an Mrs m a, which is p p [+], fr which m a (snp) = 6 and whse [MF] is the fllwing: Let Mrs m a be applied with the methd MPN and let t ma (i), i =, 9, 4, 0, and r ma (p, q), (p, q) = (,), (,), (8,6) be determined Frm [MF] is gtten the fllwing: t ma () = 7, t ma (9) = 4, t ma (4) =, t ma (0) =, t ma () = r ma (,) = [ 4 8 ], r ma (,) = [ 9 ], r ma (8,6) = [ ]

11 E4 Let there be the fllwing MrS: M Σ 7, when Σ 7 = p p [+], φ =, φ = np, τ p =, τ n =, τ s =, φ snp < 0 Let all rute presentatins be determined As φφ = np, is m a (np) = and thus there has t be a p-pint and an n-pint cnnected t the p-pint with a trail As τ p =, τ n = and (naturally) τ s =, are there tw s-pints cnnected t the p-pint, in additin t ne n-pint There are n the ther hand the p-pint s n-pint and als an s-pint cnnected t the n-pint (let it be bserved that the infrmatin φ snp < 0 des nt help at all) Thus is the [MF] f the Mr-system the fllwing: There are ( m a(snp) ) pint pairs in an arbitrary Mr-system m a, s in the Mr-system m there are ( m (snp) ) = ( 5 ) = 5! = 5! = 5 4 = 5 4 = 0 = 0 pint pairs and thus there are 0 = 0 rute presentatins!(5 )!!! Rute presentatins: r m (,) = [ ], r m (,) = [ ], r m (,) = [ ], r m (,) = [ ], r m (,4) = [ 4], r m (4,) = [4 ], r m (,5) = [ 4 5], r m (5,) = [5 4 ], r m (,) = [ ], r m (,) = [ ] r m (,4) = [ 4], r m (4,) = [4 ], r m (,5) = [ 4 5], r m (5,) = [5 4 ], r m (,4) = [ 4], r m (4,) = [4 ], r m (,5) = [ 4 5], r m (5,) = [5 4 ], r m (4,5) = [4 5], r m (5,4) = [5 4] E5 Let there be an arbitrary Mrs m a, fr which the fllwing things are applied: φ = 6, φ = sp and the quantity f rute presentatins is 600 Let an arbitrary prime sectr m px be bserved m px includes m a(n) n-pints and m a(s) s-pints Let prime sectr s parameterized rute presentatins r ma (x y, x z ) be determined, when x z > x y and x y and x z are the numbers f tw n-pints

12 600 = ( m a(snp) m a (snp)! )! (m a (snp) )! = 600 m a(snp)!! (m a (snp) )! = 600 m a(snp)! (m a (snp) )! = 600 m a(snp)! (m a (snp) )! = 600 5! (5 )! = 5! 5 4! = = 5 4 = 5 (0 + 4) = = = 600!!,s m a (snp) = 5 m a (n) + m a (sp) = m a (n) + φ = m a (snp) m a (n) = m a (snp) φ = 5 6 = 9 m a (p) + m a (s) = m a (sp) m a (s) = m a (sp) m a (p) = φ m a (p) = 6 = 5 S are there m a(n) = 9 = n-pints and m a(s) = 5 = 5 s-pints in the prime sectr m p x The fllwing basic hulls frmed by n-pints fr the prime sectr m px are gtten The s-pints can be set in the hull ne in fur and in the hull tw in six different ways (pints a, b and c; a < b < c), when numbers are the quantities f the s-pints Tw basic hulls, fur cmbinatins fr the first and six cmbinatins fr the secnd hull and ( ) =! =!( )!! = = rutes fr ne cmbinatin (the cnditin x!! z > x y halves the quantity f rutes, as nrmally x z x y x z > x y r x z < x y ) = 8 + = 0 parameterized rute presentatins (x n x + n) Let there be a cmbinatin α which is Kα and a hull β which is Rβ Let all the hull s cmbinatins and rute presentatins be presented, when the first pint f the prime sectr is x (Mrs is arbitrary)

13 Rute presentatins: HC: r ma (x 0, x 4 ) = [x 0 x 4 ], r ma (x 0, x 6 ) = [x 0 x 6 ], r ma (x 4, x 6 ) = [x 4 x 0 x 6 ] HC: r ma (x 0, x 4 ) = [x 0 x 4 ], r ma (x 0, x 6 ) = [x 0 x 4 x 6 ], r ma (x 4, x 6 ) = [x 4 x 6 ] Rute presentatins: HC:

14 r ma (x 0, x ) = [x 0 x ], r ma (x 0, x 4 ) = [x 0 x 4 ], r ma (x, x 4 ) = [x x 0 x 4 ] HC: r ma (x 0, x 6 ) = [x 0 x x 6 ], r ma (x 0, x ) = [x 0 x ], r ma (x, x 6 ) = [x x 6 ] Rute presentatins: HC: HC: r ma (x 0, x ) = [x 0 x ], r ma (x 0, x 4 ) = [x 0 x x 4 ], r ma (x, x 4 ) = [x x 4 ] Rute presentatins: HC4:

15 r ma (x 0, x ) = [x 0 x ], r ma (x 0, x 5 ) = [x 0 x 5 ], r ma (x, x 5 ) = [x x 0 x 5 ] HC4: r ma (x 0, x ) = [x 0 x ], r ma (x 0, x 6 ) = [x 0 x x 6 ], r ma (x, x 6 ) = [x x 6 ] Rute presentatins: HC5: HC5: r ma (x 0, x ) = [x 0 x ], r ma (x 0, x 5 ) = [x 0 x x 5 ], r ma (x, x 5 ) = [x x 5 ] Rute presentatins: HC6:

16 r ma (x 0, x ) = [x 0 x ], r ma (x 0, x 5 ) = [x 0 x 5 ], r ma (x, x 5 ) = [x x 0 x 5 ] HC6: r ma (x 0, x ) = [x 0 x ], r ma (x 0, x 5 ) = [x 0 x x 5 ], r ma (x, x 5 ) = [x x 5 ] 4 MAP FORM S STRUCTURE There are different numbers and methds t examine and administer the Mr-system and the bigger entities t it The numbers are the lngitudinalness number l, the furcateness number l x and the degree number f freedm v q (q = +, if the MrS is p p [+] and q =, if the MrS is p p [ ]) The methds are the PFS and the already presented MPN The lngitudinalness number l and the furcateness number l x are applied with the fllwing: l + l x = l and l x can be defined, when the Mrs is either p p [ ] r p p [+] The lngitudinalness number l depicts the lngitudinalness nature f the Mr-system: l = l x Fr the lngitudinalness number l the fllwing is applied: l [0,], aka the lngitudinalness number is either 0 r r sme fractin between the numbers 0 and The fractin quality cmes frm the calculatin habit f the lngitudinalness number: Let a pint b in the Mr-system m a be chsen, fr which t ma (b) (if the pint isn t fund, then directly l = ) The leaving sectrs lngitudinal parts (tw trails r mre in the line) ttal quantities f trails are each subtracted ne trail and the sum f brn quantities is divided by the ttal trail quantity f the Mr-system t get the lngitudinalness number s value If l = (l x = 0), is the Mrs nt furcate at all and is thus bserved as a clean rute The sectr, fr which l =, is marked by a symbl m (m is the ttal quantity f sectr s pints, when φ = snp) aka the symbl f the clean rute is m A deeper view is granted in the PT-arithmetics The furcateness number l x depicts the furcateness nature f the Mr-system: l x = l Fr the furcateness number l x the fllwing is applied: l x [0,], aka the furcateness number is either 0 r r sme fractin between numbers 0 and The fractin quality cmes frm the calculatin habit f the furcateness number: Let a pint b in the Mr-system m a be chsen, fr which t ma (b) (if the pint isn t fund, then directly l x = 0) The quantity f the leaving sectrs is divided by the ttal trail quantity f the Mr-system t get the furcateness number s value If l x = (l = 0), is the Mrs nt lngitudinal at all and is thus bserved as an intersectin The sectr, fr which l x =, is marked by a symbl m x (m is the ttal quantity f sectr s pints, when φ = snp) aka the symbl f the intersectin is m x A deeper view is granted in the PT-arithmetics The degree number f freedm v q is determined separately fr cases The Mr-system is p p [ ] (q = ) and The Mrsystem is p p [+] (q = +) The degree number f freedm v q depicts the freedm degree f the Mr-systemsi aka hw many different mdes aka Mr-systems can the Mr-systemsi have v q can get the value,,, 4 r 5, depending n the cnditin matrix Σ φ The dmain φ snp r the parameters φ and φ aren t taken int accunt when the degree number f freedm is being determined v q = 5: The MrS is free The example frms f the cnditin matrix Σ φ : Σ φ = p p [+], τ n >, τ s = fr the case q = + Σ 4 φ = p p [ ], τ p >, τ n >, τ s = fr the case q =

17 v q = 4: The MrS is quite free The example frms f the cnditin matrix Σ φ : Σ 4 φ = p p [+], τ p >, τ n >, τ s = fr the case q = + Σ 4 φ = p p [ ], τ p > (r τ p > ), τ n >, τ s = fr the case q = v q = : The MrS is restricted The example frms f the cnditin matrix Σ φ : Σ 4 φ = p p [+], τ p [,4], τ n [,4], τ s = fr the case q = + Σ 4 φ = p p [ ], τ p [,4], τ n [,4], τ s = fr the case q = v q = : The MrS is quite determined The example frms f the cnditin matrix Σ φ : Σ 4 φ = p p [+], τ p =, τ n =, τ s = fr the case q = + Σ 4 φ = p p [ ], τ p =, τ n =, τ s = fr the case q = v q = : The MrS is determined The example frms f the cnditin matrix Σ φ : Σ φ x x 4 = p p [+], τ p =, m p =, m m p = m fr the case q = + Σ φ x 7 = p p [ ], τ p =, τ n =, m p =, m m p = m, m n =, m m 4n = m 4 fr the case q = Σ φ 4 = p p [ ], τ p =, τ n =, m p = m fr the case q = CONVERGENT STRUCTURES Because f cnvergent structures is the bservatin f the Mr-systemsi mre difficult in case 0 = p p [ ] (MrS is p p [ ]) than in case 0 = p p [+] (MrS is p p [+])Cnvergent structures are nly namely taken int accunt when the MrS is p p [ ] and frm this it can be cncluded that M(φ) 0 = p p [+], Σ φ \{ 0 } M(φ) 0 = p p [ ], Σ φ \{ 0 }, when the cnditin matrices are ttally identical t each ther (after the deductins) The structures are cnvergent t each ther if and nly if their M-frms are exactly the same (in cnvergence bservatin all pints belng t the same class) The arbitrary psitinings f the trails and the pints d nt affect the cnvergences Let the cnvergent structures be demnstrated: Let it be that φφ = 4snp and the Mr-system m a has the fllwing [MF]:

18 Withut restrictins there wuld be at least ten cnvergent structures fr the Mr-system s m a map frm, s let it be put that τ p > (the s-pints as p-pints are abandned) and the fllwing result is gtten: a snp is the [SF] fr the Mr-system m a and snp is the [SF] fr the Mr-system (4) a n (4) m nx a nx m a is the Mr-system s m a n cnvergent frm aka M-frmally equivalent presentatin Mark = x depicts the sum frms map-frmal equivalence aka the mark = x is the equivalence mark f map frms Accrdingly the mark x means that the map frms f sum frms are nt cnvergent Let the equivalence marks be demnstrated: Let it be that Σ 6 8 = p p [ ], φ = 8, φ = snp, τ p >, τ n >, τ s = The fllwing M 8 Σ 6 8 (Mr-systems frms) is gtten: The Mrs m 5 in parenthesis isn t actually a part f the Mr-systemsi M 8, but is nly nted in this example fr demnstratin purpses Clearly (8) snp (8) snp x (8) snp x (8) snp x 4 (8) snp x (8) snp (8) snp x 4 (8) snp (8) snp x 4 (8) snp If m 5 is taken int accunt, then (8) snp = x 5 (8) snp There is a frmatin law fr the Mr-system s m a cnvergent frms m nx a The precise frmatin law: The cnvergent frms are traversed in the PT-arithmetic rder

19 The apprximated frmatin law: After the Mr-system s m a p-pint the pint is searched fr, whse biggest sectr is the smallest ne amng the rest f pints biggest sectrs (The size f the sectr is defined by the ttal quantity f its pints; the bigger the sectr the mre the pints) If there are tw r mre pints which have an identical biggest sectr in a way that sum frms aren t equivalent (map frms aren t the same when examined thrugh the p-pints perspective), the equivalent frms are traversed frm the shrtest t the lngest (the main fcus f the length is the fact that hw early the lngitudinal structure is impersnated) If the sum frms are equivalent, nly ne representing structure f sum frm in questin is accepted If there are tw r mre pints with equal pint quantities and biggest sectrs, the cnvergent frms are traversed frm the shrtest t the lngest SYMMETRY Symmetry in the MRS-thery can be divided in t tw classes: perfect symmetry and trail symmetry Perfect symmetry: A line and infinitely many ther lines are dragged thrugh an arbitrary and PFS-applied Mr-system s M-frm s p-pint s center The first line n 0 frms with infinitely many ther lines n, n, n, singly by pairs a genuinely cntinuus and grwing angle functin α(m) = (n 0, n m ); m = 0,,, ; α [0, 80 ] If the brn halves are in each line s case the identical mirrr images f each ther, is the Mrs perfectly symmetric In ther cases Mrs is nt perfectly symmetric The perfect symmetry is very impractical due t its mathematically demanding cnditin There are thus sme special exceptins, fr which the perfect symmetry is applied: A prime pint (Mrs f ne single pint, n trails) and a PFSapplied Mrs, in which there are infinitely many s-pints cnnected t the p-pint In a certain manner nly the latter specialty is accepted, as the pint nly as the secnd f Mr-system s cmpnents isn t classified as an Mr-system (the vacuum isn t either classified as an Mr-system) The trail symmetry in its cnditins is much mre mdest than the perfect symmetry and is thus practical in the bservatin f the Mr-system s ( 0 = p p [+]) M-frm The Mr-system has t be p p [+] t avid incherence: When 0 = p p [ ], can the Mr-system s prime pint be anywhere in which case the Mr-system in example can be trail symmetric frm ne pint s view and in example cannt be trail symmetric frm the ther pint s view In the trail symmetry and in a PFS-applied Mr-system a line is dragged as a trail s parallel and bilateral extensin In this case the trail cnnects the p-pint and a chsen prime sectr If the brn halves are identical t each ther, is the Mr-system trail symmetric in relatin t the chsen sectr In an arbitrary sectr s case the trail symmetry divides in t fur classes: p p t-, p p t SS -, p p t LS - and p p t np -symmetry p p ts aka prime pint trail symmetry is in its cnditins the mst demanding The Mrs is p p t-symmetric, if the Mrs is trail symmetric in every prime sectr s case p p t-symmetric structures are called as snwflakes, when in the structures τ p [5,6] p p t SS s aka prime pint trail symmetry regarding the smallest prime sectr is nt in its cnditins as demanding as the p p t-symmetry The Mrs is p p t SS -symmetric, if the Mrs is trail symmetric regarding every smallest prime sectr (there can be many smallest prime sectrs) p p t LS s aka prime pint trail symmetry regarding the largest prime sectr is in its cnditins as demanding as the p p t SS -symmetry The Mrs is p p t LS -symmetric, if the Mrs is trail symmetric regarding every largest prime sectr (there can be many largest prime sectrs)

20 p p t np s aka prime pint trail symmetry regarding an arbitrary prime sectr n p is in its cnditins variably demanding p p t np -symmetry fills every size class f the sectrs, and is thus mre practical cmpared t the fact that there wuld be prime pint trail symmetry regarding the third largest r the seventh smallest prime sectr The Mrs is p p t np - symmetric, if the Mrs is trail symmetric regarding the arbitrary prime sectr n p (there can be many f the same arbitrary sectr) Let it be bserved, that in particular in the Mr-system, in which there are nly identical prime sectrs cmpared t each ther, are p p ts, p p t SS s, p p t LS s and p p t np s in their cnditins cnvergent In additin if sme Mrs is p p tsymmetric, is it abslutely p p t SS -, p p t LS - ja p p t np -symmetric The PFS aka the principle f finding symmetry tries t adapt the psitining f the arbitrary Mr-system s M-frm in a way that the M-frm fulfils a cnditin (in pririty rder is the p p ts first) f a class f the trail symmetry If the PFS fails is the bserved Mrs nt trail symmetric PFS: Phase : Let there be m prime sectrs cnnected t the p-pint The trails cnnecting the prime sectrs t the p-pint are set cmpared t each ther in a way that the angle between tw arbitrary and cntiguus trails is 60 m Phase : A line is dragged thrugh the prime sectr s starting pint The line is perpendicular t a trail s imaginary extensin The trail cnnects the prime sectr s starting pint and the p-pint Pssible nrm sectrs and trails cnnecting the nrm sectrs and the prime sectr s starting pint are in the line s side where the p-pint is nt Let it be imagined that the line s tw halves (the starting pint is between the halves) are tw trails and let the restricted half by the line be bserved where the p-pint is nt If the lcal trail number f the prime sectr s starting pint is n, let the angle between tw cntiguus trails be set t 80 (the trail between the starting pint and the p-pint is (n ) cunted in) The phase is repeated (in need by thinking the bserved nrm sectr as a prime sectr) s lng, that the [MF] ends The rder f different prime and nrm sectrs when watching clckwise are cncluded purely with lgic EXAMPLES E4 Let there be the fllwing map frms (the p p is nt marked): Let the lngitudinalness number l and with the lngitudinalness number the furcateness number l x be determined fr each map frm and the fllwing result is gtten (pints a i, fr which t mx (a i ), are marked):

21 m x :l = 4, l x = l = 4 = 4 m x :l = =, l x = l = = 0 m x :l = 4 =, l x = l = = m x4 :l = 0 9 = 0, l x = l = 0 = m x5 :l = 4 8 =, l x = l = = E4 Let there be the fllwing map frms (p p is nt marked): Let the the furcateness number l x and with the furcateness number the lngitudinalness number l be determined fr each map frm and the fllwing result is gtten (pints a i, fr which t mx (a i ), are marked): m y :l x = 5 6, l = l x = 5 6 = 6 m y :l x = 5 5 =, l = l x = = 0 m y :l x = 5 0 =, l = l x = = m y4 :l x = 5 5 =, l = l x = = 0 m y5 :l x = 0 4 = 0, l = l x = 0 =

22 E4 Let structural examples be demnstrated fr the degree numbers f freedm v = 5, v + = 4, v =, v + = and v + = v = 5: Asterism v + = 4: Data structure v = : Cmpund v + = : Chart f radiactive decay v + = : Genealgy

23 E44 Let the graph f the perfect symmetry s angle functin α(m) = (n 0, n m ), m = 0,,,, α [0, 80 ] be drawn and let it be indicated why the pint is perfectly symmetric Let the line be limited as finite, let the ne head be jinted and let it be circled with the free, drawing head ne cycle arund the jinted head Let the limited line be deleted and a circle is gtten Thus by the analgy t the cnditin f the perfect symmetry is a circle (and a ball) perfectly symmetric As the infinitesimally small pint can be magnified a circle is thus als the pint perfectly symmetric E45 Let there be the fllwing map frms:

24 Let the map frms be applied with PFS and let every map frm s class f trail symmetry be defined m a is p p t-symmetric and τ p [5,6], s m a is a snwflake m a is p p t-symmetric and τ p [5,6], s m a is a snwflake m a is p p t LS - and p p t x -symmetric m a4 is p p t x -symmetric

25 m a5 is nt trail symmetric m a6 is p p t SS - and p p t LS -symmetric 5 PT-ARITHMETICS PT-arithmetics aka pint trail arithmetics and the insight f PT-arithmetics the MFMF aka the methd f finding map frms are the mst imprtant field in the MRS-thery, but d need the ther fields as a backup The PT-arithmetics is an idea t express an arbitrary Mr-system s pened sum frm and the MFMF is the executin f the idea Usually in PT-arithmetics the map frms are directly read frm the pened sum frms, but als vice versa the sum frm can be read frm the map frm (the significance f a marked p-pint is highlighted) In the pened Mr-systemsi s sum frm the sum frms f the Mr-systems are equivalent in pint sums (φ and φ determine directly the pint sum) The expressed and used number values in the PT-arithmetics belng t a grup = (n) n=0,,, = 0,,, The Mrsystems sum frms expressed in the pened Mr-systemsi s sum frm are divided in t tw classes: basic and sub frms The cncept f the PT-arithmetics cmes frm the fact that any Mrs can be expressed as the sum f pints and in the sum the sum marks summing the pints are trails 5 BASIC FORM The basic frms are in the Mr-systemsi s pened sum frm the PT-arithmetic presentatins, in which there are n arbitrary sectrs m i bracket frms A singular basic frm is expressed usually always befre pssible sub frms The bjects that belng t the basic frm are the fllwing: Σ Σ,, +,, =,, =,, =,, n x x,, n and [ ] is the prime pint and always the first term in the basic frm The quantity f the prime pint in the basic frm is always ne starts the reading f a map frm (n)

26 is the side pint and always the last term in the basic frm The quantity f the side pints in the basic frm is arbitrary (within the given limits), whereupn in the case f many side pints the grup f the side pints is the last bject in the basic frm ends the reading f a map frm (if the bject is expressed in the basic frm) + is the trail and is always between tw terms (the terms are,, n x, (n) and n ) The quantity f the visible trails is at least ne in maximum the quantity f pints minus ne (a part f the trails can be hidden in a prduct r an arbitrary sectr m ) is the multiplicatin sign and a way t reduce the quantity f the visible trails and thus the length f the PTarithmetic presentatin If an arbitrary sectr m is multiplied by a number n aka there exists a prduct m n, then there is n plus-marks aka trails in the prduct In the prduct marking the arbitrary sectr is expressed always befre the number expressing its quantity fr clarity s sake = is the equality sign is the inequality sign Σ = is the Mr-systemsi s sum frm s equivalence sign which is used between the Mr-systemsi s sum frm and the sum frm s pened frm Σ Σ Σ is the ppsite f the sign = If the sign is between the Mr-systemsi s sum frm and the sum frm s pened frm are nt the frms cncerned cnvergent in sum frm equivalence = is the pint sum equivalence s sign aka the Mr-system s sum frm equality s sign and is always between tw different sum frms The sign = can nly be used when the sum frm cming after the sign = is a new frm aka there are n cnvergent frms If the cnvergent sum frm has t be nevertheless expressed, is it afterwards discarded with square brackets [ ] is the ppsite f the sign = The sign between tw different Mr-system s sum frms means that the sum frms cncerned are nt cnvergent in sum frm equivalence Let it be bserved that arbitrary sectrs m i are expressed in basic frm in rder frm the biggest t the smallest (the case where there are identical sectrs in pint quantities is becme clear later) in the fllwing habit: + m x + m x + + m n x n,m m m n, x i 0 (can als be applied cnversely when the [SF] is read frm the map frm), m i 0, i: m i x i > 0 m x is the intersectin and ne f the arbitrary sectr s m nrm frms m x includes m pints and m trails in a way that m side pints cnnected t the nrm pint with a trail intersect frm the ne nrm pint The smallest pssible intersectin is x x (m) means that the arbitrary sectr m has bracket frms in additin t the nrm frms m x and m x (m) intersectin x als means the m is the clean rute and the secnd ne f the arbitrary sectr s m nrm frms, when there are tw nrm frms includes m pints and m trails in a way that m nrm pints are cnnected t each ther with trails in a line and the last nrm pint is yet cnnected in additin t the nrm pint the ne side pint (in ther wrds the fllwing is applied in the clean rute: τ n = ) The smallest pssible clean rute is (the side pint isn t cunted as a clean rute) [ ] arund the basic frm means that the basic frm cncerned is discarded The sign [ ] is usually used nly in the Mrsystemsi, fr which 0 = p p [ ] (cnvergent frms) The sign [ ] is called the square brackets m

27 EXAMPLES E5 Let the fllwing map frms f the sum frms be read, when 0 = p p [+]: a () a 5 () a (4) a (4) snp = +, snp = +, snp = +, snp = + x snp = + +, snp = a 6 () The fllwing map frms are gtten: x (5) x a 4 () E5 Let there be the fllwing map frms, when 0 = p p [+]: The fllwing perceptins are gtten frm the map frms: a (6) snp a (6) a (6) = snp snp a (7) snp a (6) snp a (6) sp a () np = a (6) sp = a () np a (6) sp a () np a (5) sp a () np a (6) sp a () np Let it be bserved that althugh a b c it culd still be that a = c

28 E5 Let there be the fllwing map frms ( 0 = p p [+]): The fllwing sum frms can be read frm the map frms: a (5) a (6) a () snp = + 4 x snp = + + x x snp = SUB FORM Sub frms are in Mr-systemsi s pened sum frm PT-arithmetic presentatins, in which there are arbitrary sectrs m i bracket frms Sub frms are usually nly expressed after a singular basic frm The bjects that belng t the sub frm are the fllwing: Σ Σ,, +,, =,, =,, =,, m x x,, m, [ ],, a a a m, ( ) and ) is the prime pint and always the first term in the sub frm The quantity f the prime pint in the sub frm is always ne starts the reading f the map frm and is never expressed in the parenthesis (bracket frm) is the side pint and always the last term in bth the bracket frms and the sub frm itself The quantity f the side pint in the sub frm is arbitrary (within the given limits), when in many side pints case the grup f the side pints is the last bject in the bracket frms and the sub frm itself ends the reading f the map frm (if the bject is expressed in the bracket frms r the sub frm itself) + is the trail and is always between tw terms (the terms are,, m x, (m), m, and a m ) The quantity f the visible trails is at least tw (if the nrm frms aren t seen as bracket frms, therwise the quantity has t be at least ne) and in maximum the quantity f pints minus ne (a part f the trails can be hidden in the prduct r in the arbitrary sectr m ) (m) is the multiplicatin sign and a way t reduce the quantity f the visible trails and the length f the PT-arithmetic presentatins If the arbitrary sectr m is multiplied by number n aka there exists a prduct m n, then in the prduct there is n plus-marks aka trails In the prduct marking the arbitrary sectr is always expressed befre the number expressing its quantity fr clarity s sake = is the equality sign is the inequality sign = Σ is the Mr-systemsi s sum frm s equivalence sign which is used between the Mr-systemsi s sum frm and the sum frm s pened frm x (

29 Σ Σ Σ is the ppsite f the sign = If the sign is between the Mr-systemsi s sum frm and the sum frm s pened frm are nt the frms cncerned cnvergent in sum frm equivalence = is the pint sum equivalence s sign aka the Mr-system s sum frm equality s sign and is always between tw different sum frms The sign = can nly be used when the sum frm cming after the sign = is a new frm aka there are n cnvergent frms If the cnvergent sum frm has t be nevertheless expressed, is it afterwards discarded with square brackets [ ] is the ppsite f the sign = The sign between tw different Mr-system s sum frms means that the sum frms cncerned are nt cnvergent in sum frm equivalence Let it be bserved that arbitrary sectrs m i are expressed in sub frm in rder frm the biggest t the smallest (the case where there are identical sectrs in pint quantities is becme clear in the cmbinatin system) in the fllwing habit: + m x + m x + + m n x n,m m m n, x i 0 (can als be applied cnversely when the [SF] is read frm the map frm), m i 0, i: m i x i > 0 m x is the intersectin and ne f the arbitrary sectr s m nrm frms m x includes m pints and m trails in a way that m side pints cnnected t the nrm pint with a trail intersect frm the ne nrm pint The smallest pssible intersectin is x x (m) means that the arbitrary sectr m has bracket frms in additin t the nrm frms m x and m x (m) intersectin als means the m is the clean rute and the secnd ne f the arbitrary sectr s m nrm frms, when there are tw nrm frms includes m pints and m trails in a way that m nrm pints are cnnected t each ther with trails in a line and the last nrm pint is yet cnnected in additin t the nrm pint the ne side pint (in ther wrds the fllwing is applied in the clean rute: τ n = ) The smallest pssible clean rute is (the side pint isn t cunted as a clean rute) [ ] arund the sub frm means that the sub frm cncerned is discarded The sign [ ] is usually used nly in the Mrsystemsi, fr which 0 = p p [ ] (cnvergent frms) The sign [ ] is called the square brackets is the nrm pint and always the first term in the bracket frm (als the inner bracket frms f the bracket frm) The quantity f the nrm pint in the bracket frm is always ne, but in the subfrm itself the quantity f the nrm pint is arbitrary (within given limits) Usually is never expressed utside the bracket frm a m is the abbreviatin fr the arbitrary sectr s m bracket frm, which includes parenthesis and whse running a a number is a The marking m is especially handy when m as pened frm is a lng expressin and when there are many terms (m x a i) and the cmbinatin system is used ( m fr clarity s sake is thugh pened nce within Mr-systemsi s x reach) If there is an arbitrary sectr m, which can be expressed in the frm (m) and which has n bracket frms, it can be thught that m x 0 = m ( m x = ( + (m ))) and m = m ( m = ( + ( + + ( + ) )), m bracket pairs and n-pints) The sectrs m x and m are thught as bracket frms in the cmbinatin system In general if there x are sectrs (m) and (n) x x, m > n, sectr s (m) bracket frms quantity is S m and nrm frms quantity is tw, sectr s bracket frms quantity is S n and nrm frms quantity is tw, then S m + S n + S m S n x (n) ( ) aka the parenthesis defines the starting pint (the first bracket ( ) and the ending pint (the secnd bracket ) ) f the bracket frm (n+) a ( a ) aka the bracket frm s numbered parenthesis is a tl t bring clarity especially when there are many inner a bracket frms The numbering rder f the parenthesis is the fllwing: a ( b ( b ( b ( b ) c ) + c ( c ( c ) a ) a ) ), c > c > b > b > a > a aka the number grws when ne mves frm inner t mre inner parenthesis and when ne mves frm ne bracket elabratin t anther m

30 INNER BRACKET FORMS If there is a bracket frm r there are bracket frms in the bracket frm and if there is a bracket frm r there are bracket frms in the bracket frm r in the bracket frms that were inside the riginal bracket frm and et cetera then inner bracket frms are talked abut Althugh ne wuld find inner bracket frms in a sum frm, is the sum frm cncerned seen nrmally still as a sub frm The arbitrary sectr s m bracket frms can be assembled by many different cmbinatins f inner bracket frms THE COMBINATIONS OF THE SECTORS x n x x x x If there is a grup f sectrs ( m i x i ) i=0,,, = m 0 x 0, m x, m x,, m n x n, m 0 m m m n, n 0, m i n, x i 0 in the basic frm in a way that i: m i x i > 0 aka the basic frm s frm wuld be + x i=0,,, ( m i x i ) + y + y, n 0, y 0, y 0, then the frmatin rder f the sectrs nrm and bracket frms cmbinatins is defined in the cmbinatin system THE COMBINATION SYSTEM x n x x x x Let there be a grup f sectrs ( m i x i ) i=0,,, = m 0 x 0, m x, m x,, m n x n, m 0 m m m n, n 0, m i, x i 0 in the basic frm in a way that i: m i x i > 0 and thus the basic frm s frm wuld be + n x i=0,,, ( x i ) + y + y, n 0, y 0, y 0 Nw the fllwing cmbinatin system C S is frmed: m i As the sectrs f the Mr-systemsi aren t lcked dwn but they can be mved as ne wuld like, the fllwing laws are gtten:

31 n i) Let in the cmbinatin system there be a part where there is a grup (a i ) i=0,,, = a 0, a, a,, a n in a way that the bject a i has b i bracket frms and tw nrm frms (the bjects a i are different t each n n ther) Nw the cmbinatin cefficient Z = i=0,,, ( + b i ), when nly the grup (a i ) i=0,,, is examined There are n cnvergent cmbinatins (fr example y = XY = YX = y, y = 0 X + Y, y = 0 Y + X) n ii) Let in the cmbinatin system there be a part where there is a grup (a i ) i=0,,, = a 0, a, a,, a n = a, a, a,, a in a way that the bject a i = a has b i = b bracket frms and tw nrm frms (the bjects a i are similar t each ther) Nw the cmbinatin cefficient Z < ( + b) n+,when n > 0 and Z = ( + b) n+ n,when n = 0, when nly the grup (a i ) i=0,,, = a 0, a, a,, a n = a, a, a,, a is examined There are cnvergent cmbinatins (fr example y = XY = YX = y, y = 0 X + Y, y = 0 Y + X) Example cases (the cmbinatins circled are discarded): x x 00 = m 0 m 0 = 0 cmbinatin = c 0 x 0 = m 0 m 0 = cmbinatin = c = m 0 m 0 = cmbinatin = c 00 = m 0 0 = m 0 0 = m 0 = m 0 x m x m m x = c 0 = c m x = c = c

32 x x 00 = = c 0 m 0 m 0 0 = m 0 0 = m 0 = m 0 = m 0 x m 0 = c x m 0 m 0 m 0 = m 0 = c = c = c 4 m 0 = c 5 00 = m 0 0 = m 0 0 = m 0 0 = m 0 = m 0 = m 0 0 = m 0 = m 0 = m 0 x m x m x = c 0 = c x m m m m = c x = c = c 4 m m = c 5 x = c 6 = c 7 m = c 8

33 x x x 000 = = c 0 m 0 m m x x 00 = m 0 m m = c x x 00 = m 0 m m = c x 0 = m 0 m m = c x x 00 = m 0 m m = c 4 x 0 = m 0 m m = c 5 x x 00 = m 0 m m = c 6 x 0 = m 0 m m = c 7 x 0 = m 0 m m = c 8 = m 0 m m = c 9 x 0 = m 0 m m = c 0 = m 0 m m = c x x 00 = m 0 m m = c x 0 = m 0 m m = c x 0 = m 0 m m = c 4 = m 0 m m = c 5 x 0 = m 0 m m = c 6 = m 0 m m = c 7 x x 00 = m 0 m m = c 8 x 0 = m 0 m m = c 9 x 0 = m 0 m m = c 0

34 = = c m 0 m m x 0 = m 0 m m = c = m 0 m m = c 0000 = m = m = m 0 00 = m 0 00 = m 0 0 = m = m 0 00 = m 0 00 = m 0 0 = m 0 0 = m 0 = m = m 0 00 = m 0 00 = m 0 0 = m 0 0 = m 0 = m 0 x x x m m m x x x m m m x x m m m x x m m m x m m m x m m m x x m m m x x m m m x m m m x m m m m m m m m m x x m m m x x m m m x m m m x m m m m m m m m m x = c 0 = c x = c = c x = c 4 = c 5 x = c 6 = c 7 x = c 8 = c 9 x = c 0 = c x = c = c x = c 4 = c 5 x = c 6 = c 7

35 EXAMPLES E5 Let the map frms f the fllwing sum frms be drawn, when the Mrs is p p [+] a () snp a (5) snp a (6) snp a 4 (8) snp a 5 (0) snp a 6 (4) snp = + ( = + ( = + ( = ) + ) + + ) + + x + ( + (5) = + ( = + ( ) x (4) x + + ) + x ) + + The fllwing map frms are gtten: Let it be bserved that fr example + ( + ) = + and + ( + ) = + x E5 Let the map frms f the fllwing sum frms be drawn, when the Mrs is p p [ ] a (8) snp a (5) snp a (0) snp a 4 () snp = + ( = + ( = + + ) + ( + ) + + ) + ( + ) + ( + ) + + x (4) + + ( + ) + ( + ) + = + ( + ( + )) + +