WPCH 2BPZCourier 10cpi#|x`ax6X@8 X@Panasonic KX-P1180PAKXP118.PRSx  @XqX@221B Zs#|xCourier 10cpiCourier 7cpiCourier 10cpi ItalicPanasonic KX-P1180PAKXP118.PRSx  @XqX@2XvWpk=?xxxx`ax6X@8 X@?xl\6X@9 X@?xxxxUx6NhM XH^rHF_^ZY[XùP( 8 ްPvPPSRVW"a8DocumentgDocument Style StyleXX` `  ` a4DocumentgDocument Style Style . a6DocumentgDocument Style Style GX  2kEvt`a5DocumentgDocument Style Style }X(# a2DocumentgDocument Style Style<o   ?  A.  a7DocumentgDocument Style StyleyXX` ` (#` BibliogrphyBibliography:X (# 2   C a1Right ParRight-Aligned Paragraph Numbers:`S@ I.  X(# a2Right ParRight-Aligned Paragraph Numbers C @` A. ` ` (#` a3DocumentgDocument Style Style B b  ?  1.  a3Right ParRight-Aligned Paragraph Numbers L! ` ` @P 1. ` `  (# 2  } <  a4Right ParRight-Aligned Paragraph Numbers Uj` `  @ a. ` (# a5Right ParRight-Aligned Paragraph Numbers _o` `  @h(1)  hh#(#h a6Right ParRight-Aligned Paragraph Numbersh` `  hh#@$(a) hh#((# a7Right ParRight-Aligned Paragraph NumberspfJ` `  hh#(@*i) (h-(# 2  +a8Right ParRight-Aligned Paragraph NumbersyW"3!` `  hh#(-@p/a) -pp2(#p a1DocumentgDocument Style StyleXqq   l ^) I. ׃  Doc InitInitialize Document Style  0*0*  I. A. 1. a.(1)(a) i) a) I. 1. A. a.(1)(a) i) a)DocumentgTech InitInitialize Technical Style. k I. A. 1. a.(1)(a) i) a) 1 .1 .1 .1 .1 .1 .1 .1 Technical2ka5TechnicalTechnical Document Style)WD (1) . a6TechnicalTechnical Document Style)D (a) . a2TechnicalTechnical Document Style<6  ?  A.   a3TechnicalTechnical Document Style9Wg  2  1.   2Ca4TechnicalTechnical Document Style8bv{ 2  a.   a1TechnicalTechnical Document StyleF!<  ?  I.   a7TechnicalTechnical Document Style(@D i) . a8TechnicalTechnical Document Style(D a) . 2CF`FPleadingHeader for numbered pleading paperP@n   $] X X` hp x (#%'0*,.8135@8:1 have a special name given to them since they require the juggler to be able to catch and throw more than one ball using the same hand. They are called multiplex patterns.  ?H& Patterns are valid if they satisfy the equation of continuity. It is interesting to note, however, that a valid pattern is not necessarily physically realizable. A valid pattern can have throws which move the ball backwards in time! We will not deal with these rather abstract patterns until later. Therefore we add the jugglability condition a pattern is@h)0*0*0*!X( %! !X %$A@  ? jugglable if it is valid and has no throws backwards in time and if there is a throw with zero time component, it must be a selfthrow. The latter statement takes care of the "rest" which occurs in a frame when a hand has no ball to catch. Periodicity The general equation of continuity is of little practical help because it  ? requires an infinite summation. In practice, we encounter periodic patterns, which repeat after a certain number of frames. Therefore, we confine our interest to this class. Suppose a pattern repeats after L frames. We say its period is L and since it is periodic we have x! r m(h,t) = r m(h,t+L) where r m(h,t) is the displacement vector representing the M throw(s) from position (h,t) and is given by  r m(h,t) = p m(h,t) (h,t). Given that the pattern is periodic, we can restrict (h,t) to one period of pattern space. It is sufficient to know only one period of the pattern in order to know the entire pattern. We define the period space to be ca#(#X8ddddd ddaax0i ~==~ LBRACE (h,t):~1<=h<=H,~1<=t<=L RBRACE x6X@8 X@x6X@8 X@x6X@8 X@8i8888 88 88(8,8)M8:81u8,E81m8h]8t 8h8H5 8t% 8Lc$XX%%XX%%X!aX%$ The equivalent definition of periodicity for the positional form of the pattern can now be found. If a pattern P is periodic with period L, then #(#XdddddddaaSUMNxhDbold p sub m (h,t)~+~k(0,L)~=~bold p sub m (h,t+kL)~~~forall k in Z x6X@8 X@x6X@8 X@x6X@8 X@_p+mZ_p+m_(_,_)_(R_0_,_)* _( _,r _)Z_hJ_tb_kB_L _h _t _kLj_kZ_Z__ _ _z_h$XX%%XX%%X!X%$ The pattern can be represented by an HxLxM matrix of vectors, p m(h,t). This brings us to multihand notation (MHN), which is devised to represent periodic patterns. @h)0*0*0*!X8%1aX%@Ԍ MultiHand Notation (MHN) A throw takes an object (a ball) to a new location in (h,t) space. This destination vector is placed at the location in (h,t) space from which the throw initiates. A matrix of vectors describes the throws making up the  ? pattern. This is the positional form of the pattern. The subscript m is used for multiplex patterns. For M>1, a different 2d matrix would be needed for each m=1,2,...M. Thus, the multiplex pattern matrix becomes three dimensional. #(#>X ddddd]vddaaxP~ = ~left [matrix{bold p sub m (1,1) & bold p sub m (1,2) & ... & bold p sub m (1,L) # bold p sub m (2,1) # . # . # bold p sub m (H,1) & bold p sub m (H,2) & ... & bold p sub m (H,L)} right ] x6X@8 X@x6X@8 X@x6X@8 X@P L%_H_H0 _H _Ldvxxwx(xxx;xxxNxxwd}w(;N~p]mvpm ph mp]jm_p]+mv_p+m _ph +m(%1,1)F(16,2&) . . . (0 1 , )(%2,1)5o.5g._(_,_1_)F_(6_,_2&_) _. _. _. _( _, _)$XX%%XX%%!X%$ The zero permutation matrix, P0, takes each (h,t) location to itself. #(#Xddddd= 'ddaaPATMATx>gP_0~=~left [matrix{(1,1) & (1,2) & ... & (1,L) # (2,1) # . # . # (H,1) & (H,2) & ... & (H,L)} right ] x6X@8 X@x6X@8 X@x6X@8 X@DP WL7Hn7H 7H 7L0-W(W1W,W1 W)W(nW1W,^W2W)W.7W.W. W( W1 W,x W)-O(O2O,O1 O)G.?.-7(7,71 7)7(7,^727)7.77.7. 7( 7,x 7)2Dvxwx(xxx;xxxNxxw }  w (   ;   N  ~>߻$XX%%0XX%%!X%$ The relative form of a pattern is found by subtracting P0 from P. %R  P P0ă As we shall see later, P0, is the identity for composition of permutations. Equation of Continuity for Periodic Patterns In this section we develop a method of verifying continuity for a periodic pattern which involves a summation over just one period. Again the concept is to count the number of balls being thrown to a particular hand in a frame and check to see if this count equals M, the number of balls being thrown from the hand. If we restrict our count to just one period, we will miss those balls whose throws originate outside this period. How do we account for these? Consider a throw that originates in the previous period. The concept of periodicity tells us that the same throw takes place exactly L frametimes later, which is in the period of interest. Therefore, we count this as one of the balls being thrown to the hand of interest. @h)0*0*0*!X %!X%2@ԌTheorem 1 #(#XxdddddW #ddaaxyosum from {bold x in i } sum from {m=1} to M delta(bold zbold p_m'(bold x))~=~M ~~~~~forall bold z in l x6X@8 X@x6X@8 X@x6X@8 X@"IOI+x_zOpmx zb++i+WW zG  lME+m'M+1(())l :y$XX%%XX%%!X%$ where we define p m'( x ) as follows: Let (h p,tp) = p m( x ) and let (hz,tz) = z .   #(#XH ddddd*ddaaxRbold p_m'(bold x)~==~bold p_m(bold x)~~(0, LDBRACK (t_p~~t_z)/L~RDBRACK~CDOT~L) x6X@8 X@x6X@8 X@x6X@8 X@_p:mZ_xr_p+m_x_(_)B_(2_)_(J_0_,x_( _)0 _/_)__ __:_x __tp +p _th +z _L_L$XX%%XX%%X!X%$ Proof We start with the general equation of continuity and sum it period by period. !#(#Xdddddv#ddaa(#sum from {tau=inf} to inf~ sum from {t=1tau L} to {Ltau L}~ sum from {h=1} to H~ sum from {m=1} to M~ delta(bold z bold p_m(h,t))~=~Mx6X@8 X@x6X@8 X@x6X@8 X@@IXILII[+++z*++++ +))+)@ *LL+tc+LHB+hM+mk h[ t Mz+1+1a+1( ( , )K )3z# p m$XX%%XX%%!!X%$ Applying the definition of periodicity to the equation of continuity, we obtain: A#(#Xddddd#ddaaxDsum from {tau= INF } to INF~sum from {t=1tau L} to {Ltau L} ~sum from {h=1} to H~sum from {m=1} to M~ delta(bold z~~bold p_m(h,t+tau L)~+~(0,tau L))~~=~M x6X@8 X@x6X@8 X@x6X@8 X@@IXILII[+++z*++++  m+))+)@  ))*LL+tc+LHB+hM+m h th LUL=Mz+1+1a+1( ( , )(0p,)E)3z pS mD$XX%%PXX%%!AX%$ Now we change the order of the sum and let t' = t + )L. 1dddddddd (1) A1dddddddd (1) 7a#(#Xxddddd?ddaaxsum from {t'=1} to L~sum from {h=1} to H~sum from {m=1} to M~sum from {tau=INF} to INF~ delta(bold z~~bold p_m(h,t')~+~(0,tau L))~~=~M x6X@8 X@x6X@8 X@x6X@8 X@4IIKIIzL+tHGhMAGmc hS tLmMddrY+GG4 GYGGK = +1bG1G1( ( , ) ((0,)u)G) ){z p m7$XX%%XX%%!aX%$ There is but one integer ) which can possibly satisfy: $#(#X$ddddd ddaax%bold z~=~bold p_m(h,t')~~(0,tau L). x6X@8 X@x6X@8 X@x6X@8 X@_z_p2+m_{w__(r_,_)G_(_07_,_) _._h_t_L_)$$XX%% XX%%X!X%$ It is #(#X(dddddbddaaxh$tau'~=~LDBRACK~(t_p~t_z)/L~RDBRACK x6X@8 X@x6X@8 X@x6X@8 X@_)_ ___i_(Q_)_/_ta+p_t+zA_Lh$XX%%#XX%%X!X%$ Thus, we can drop the summation over ) by replacing p m by p m' as defined in theorem 1. When we do this, we get equation of Theorem 1. h)0*0*0*qXx%XH %X%!X%NAXx%!aX$%&X(%*Ԍ Multiplex Space So far, we have been denoting throwvectors by p m(h,t). The subscript m can take on integer values from 1 to M, the multiplex limit. The multiplex limit is the number of balls which can be caught and thrown from one hand at one time. We have been considering these M different throws occuring at (h,t) as being interchangeable. From now on, however, we will identify each throw  ?x with a number from 1 to M. We begin by defining multiplex space. #(#X dddddddaa(#;MU ~==~ LBRACE (m,h,t): 1<=m<=M,~1<=h<=H,~INFV z  =+ f I(~(n))&14 .R=/߬$XX%%XX%% !X%$where !#(#X(#dddddddaa(#8MU '~==~LBRACE~(m,h,t):~1<=m<=M,~1<=h<=H,~1<=t<=L~RBRACEx6X@8 X@x6X@8 X@x6X@8 X@8z88 88H 88 8p 8`8888(8,p8,`8)8:81 8, 81( 8, 818m8h8t8m8M 8h 8H 8t8L$XX%%xXX%%X!!X%$and where both the domain and range of p ' is restricted to one period of multiplex space in such a way that KA#(#X'dddddddaa(#^bold p'(bold x)~=~bold p(bold x)~~(0,0,tau(bold x) cdot L)~~~where~tau(bold x)~is~an~integer.x6X@8 X@x6X@8 X@x6X@8 X@8pG8x_8pO8x 8xS8xz8(8)8(8)g8(80W8,80G8,, 8( 8)> 8)8(8)s8.88 88)n8) 8L 8where8is8an+8integerK$XX%%`"XX%%X!AX%$ The equation of continuity shows that p ' is a bijection and therefore it is a permutation on restricted multiplex space. We have proven a valuable theorem about periodic patterns. It allows one to quickly generate hosts of valid patterns by simple permutations and additions. ph)0*0*0*QX % X%^X@%*"X(#%=%!X'%%)ApԌTheorem 2. 2a#(#Xxddddd *ddaa(# P~=~perm(P_0')~+~(0,0,L) cdot E.x6X@8 X@x6X@8 X@x6X@8 X@_P_perm _P_L _E__ __(:0_)r_(_0b_,_0R_,B _)d _.2$XX%%XX%%X!aX%$ where E is the excitation matrix of integers #(#X( ddddd|'ddaa(#E~=~left [ matrix{tau (m,1,1) & tau (m,1,2) & ... & tau (m,1,L) # tau (m,2,1) & tau (m,2,2) & ... & tau (m,2,L) # . # . # tau (m,H,1) & tau (m,H,2) & ... & tau (m,H,L)} right ].x6X@8 X@x6X@8 X@x6X@8 X@DEWmWm WmWLOmOm OmOL7m7H7m7H 7m 7H7LDvxwx(xxx;xxxNxxw}w(;N~W)W) W)O)O) O)7)7) 7)JW(:W,W1*W,W1W)pW(`W,W1PW,W2@ W)) W. W. W.o W(_ W, W1OW,?W)JO(:O,O2*O,O1O)pO(`O,O2PO,O2@ O)) O. O. O.o O(_ O, O2OO,?O){G.{?.J7(:7,*7,717)p7(`7,P7,72@ 7)) 7. 7. 7.o 7(_ 7,O7,?7)D.ߎ$XX%%xXX%%!X%$ E is really a threedimensional matrix whose elements, )( x ), are the integers used in defining p '( x ). Theorem 2 defines p only in restricted multiplex space. The domain of p can be extended to multiplex space by using the fact that p is periodic. Any element of multiplex space can be written as ( x + k(0,0,L)) where x is an element of restricted multiplex space and k is an integer.  p ( x + k(0,0,L)) = p ( x ) + k(0,0,L) = p '( x ) + (0,0,L)()( x )+k). Theorem 2 can be stated in words: Any valid periodic pattern is representable by the sum of some permutation matrix and (0,0,L) times some excitation matrix. Pattern Operations which Preserve Validity There are three useful operations which can be easily proven to preserve validity. They are permutation, local translation, and global translation. Consider an arbitrary permutation on restricted multiplex space, q ( x ). Suppose that p is a valid pattern. q ( x ) is in restricted multiplex space so Theorem 2 says: |  p ( q ( x )) = p '( q ( x )) + (0,0,L))( q ( x )) p '( q ) is a permutation on restricted multiplex space and )( q ( x )) is defined and is an integer. As done before, we extend the domain of p ( q ) to multiplex space by using the fact that it is periodic and this gives a valid pattern. A local translation is defined as an addition of any integer number times L to any one of the time components of the throwvectors in the matrix. It is also easily proven from Theorem 2. S#(#X*dddddA*ddaa(#FP~+~(0,0,L) cdot E_{local}~=~perm(P_0')~+~(0,0,L) cdot (E_{local}~+~E)x6X@8 X@x6X@8 X@x6X@8 X@_P _L,_E+local\_perm _Pt_L_E+local>_E___E L _d_n__(*_0_,_0_,_)< _(, :0| _) _( _0 _,_0_,_)_(_)S$XX%%H&XX%%X!X%$ Ph)0*0*0*1Xx%aX( %zX*%M-PԌThe new excitation matrix for (P + (0,0,L)Elocal) is (E + Elocal). Hence, the local translation of P is also valid. A global translation is defined as the addition of some integer, g, to the time components of all of the vectors in the matrix. It is proven using the general equation of continuity. ##(#kX( dddddddaa(#^sum from {bold x in MU}~delta(bold p(bold x)~~[bold z(0,0,g)]~)~=~1~~~~~forall bold z in MU.x6X@8 X@x6X@8 X@x6X@8 X@*I+xpx zpzb+ z+ `((()[(0,x0, )X ]( ) 1.h g#$XX%%xXX%% !X%$  #(#kXH dddddddaa(#[sum from {bold x in MU}~delta([bold p(bold x)+(0,0,g)]bold z)~=~1~~~~~forall bold z in MU.x6X@8 X@x6X@8 X@x6X@8 X@*I+xpx zhzb+0   z+ X(([()p(0`,0P,@)] ) 1.g $XX%%XX%% !X%$ The elements of the globally translated matrix are p ( x )+(0,0,g). Therefore, global translation preserves validity. States and Transitions So far we have only looked at patterns by themselves. That is, we have not thought about what a juggler might do to go from pattern X to pattern Y. Our present notation does not tell us where all the balls are at every frame of the pattern. It only tells us what throws to make at each frame. To denote  ?P the configuration of balls in (h,t) space we define a state matrix. It should be noted that the state matrix as defined here is only for those valid patterns whose throws are all forward into time. #(#Xddddd ddaax kS_0~=~left [ matrix{s(1,1)&s(1,2)&...&s(1,W)# s(2,1)&s(2,2)&...&s(2,W)#.#s(H,1)&s(H,2)&...&s(H,W)} right ] x6X@8 X@x6X@8 X@x6X@8 X@S-OsnOs Osh OW-GsnGs Gsh GW-7s7Hn7s^7H 7sx 7Hh 7W0O(O1O, O1O)O(^O1O,NO2O)O.' O. O. O(x O1 O, O)G(G2G, G1G)G(^G2G,NG2G)G.' G. G. G(x G2 G, G)Y?.7(7, 717)7(7,N727)7.' 7. 7. 7( 7, 7)2vx;xxxNxxwX }X X ;X X X NX X ~ ߇$XX%%8XX%%x!X%$ !#(#X(#dddddddaaxMBW~ ==~ max t_r~~where~ we~ let~ (h_r,t_r)~=~r_m(x)~~~~(x in i )x6X@8 X@x6X@8 X@x6X@8 X@_W_t+r_whereJ_we_let _hJ +r _t +r_r+m_x_x_ _ __i_maxR _( _, _)R_(B_)_( _)M$XX%%xXX%%X!!X%$  ?`" This matrix is HxW where W, the width of the matrix, is the maximum throw height. Each element tells how many balls there are at that location in (h,t) space. Elements beyond this width, W, are all taken to be zero. This  ?$ state space is a subset of pattern space and is defined eA#(#X+ddddd ddaax2sigma ~=~ lbrace (h,t):~~1<=h<=H,~~0<=t=Wx6X@8 X@x6X@8 X@x6X@8 X@sa*tijJat+l+aiH+hM+mR hB lijatfortW*J++i++:*Jz(j,Z):+1+18 +1 ( ( , )Z(J,))TI#II   p m<߹$XX%%XX%%!X%$ From the general equation of continuity we have _!#(#Xddddd#ddaa xtM~=~sum from {l=INF} to INF~sum from {h=1} to H~sum from {m=1} to M~delta(bold p_m(h,l)(i,j+a+t))~~>=~s_{a+t}(i,j)x6X@8 X@x6X@8 X@x6X@8 X@M+lH+hPM+mh l iz jj aZtBsaZt"ij*+R+++P+   r II I!+1+1(B(2 ," ) ( ,)J)(,) rpm_$XX%%XX%%!!X%$ Intuitively, we know that a component of S must not exceed the multiplex limit, M. M represents the total number of balls one hand can handle at one time. If s(i,j) were to exceed M, then when it came time to catch the s(i,j) balls, there would be too many to handle. The following theorem is an extension of Shannon's Theorem. It relates the number of balls, N, to the period, L, and the relative pattern matrix, R. Theorem 6 &A#(#XH&dddddY ddaa x5sum from {bold x in MU '}~bold r(bold x)~~=~~(0,0,NL)x6X@8 X@x6X@8 X@x6X@8 X@:I+xrxb+ddY^+F(6)(0v,0f,)NL&$XX%%!XX%% !AX%$ where a#(#kX+dddddHddaa xz7N~=~sum from {bold x in sigma}~s_a(bold x)~~~~forall a.x6X@8 X@x6X@8 X@x6X@8 X@N5saEa+zI+x}xR+%().z$XX%%'XX%%!aX%$ 0*0*0*aX@%  X( %v  XP% X%! XH&%2)A X+%.a Ԓ Proof From Theorem 2 we have, 2#(#Xddddd *ddaa (# P~=~perm(P_0')~+~(0,0,L) cdot E.x6X@8 X@x6X@8 X@x6X@8 X@_P_perm _P_L _E__ __(:0_)r_(_0b_,_0R_,B _)d _.2$XX%% XX%%X!X%$ where P is the pattern matrix of vectors (m,h,t). Summing over restricted multiplex space and applying the commutative property of addition to perm(P'0), #(#Xddddd`ddaa (#hsum from {bold x in MU'}~bold p(bold x)~=~sum from {bold x in MU'}~bold p_0'(bold x)~+~sum from {bold x in MU'}~(0,0,L) cdot ~tau(bold x)x6X@8 X@x6X@8 X@x6X@8 X@:II I+xpx+xpx +x]xb+ddY&+ddY : +dd Y+v+ +x)F(6)Z(J) ( 0 , 0 ,v)()#V 0 Lh$XX%%` XX%% !X%$ Since p '0( x ) = x , #(#kXdddddddaa (#hsum from {bold x in MU}~(bold p(bold x)bold x)~=~sum from {bold x in MU}~bold r(bold x)~=~(0,0,L) cdot sum from {bold x in MU}~tau(bold x).x6X@8 X@x6X@8 X@x6X@8 X@*III+x%px}x+x0r x+x2xb++h ++5+R+M)(())( )8 ( 0( , 0 ,)()". Lh$XX%%XX%% !X%$ This proves the theorem for the first two components. I have not found a simple proof for the last component. We start with the transition theorem for a=0 and t=L. ( p m maps pattern space into pattern space.) #(#Xddddd-#ddaa x rs_L(i,j)~=~s_0(i,j+L)~+~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~delta(bold p_m(h,l)~~(i,j+L))x6X@8 X@x6X@8 X@x6X@8 X@sLZiJjbsijL L +li H +h M +mRhBlijL(,)0*(,): +1 +18 +1((,)Z(J,*))  +i + +: I# I I  pm ߉$XX%%XX%%!X%$ Now we multiply both sides by j and form a sum over state space. #(#X ddddd#ddaa xtusum from {i=1} to H~sum from {j=1} to INF~j s_L(i,j)~=~sum from {i=1} to H~sum from {j=1} to INF~j s_0(i,j+L)~+~term2x6X@8 X@x6X@8 X@x6X@8 X@IIIa IbH+i+jjsLij(H+iW +j jN s i~ jnLterm2b++(+  + +11+1X(H,8)x+1 +1 0 ( ,)t$XX%% XX%%!X%$ !#(#X%ddddd#ddaa xTterm2~=~sum from {i=1} to H~sum from {j=1} to INF~j~sum from {bold x in i }~sum from {m=1} to M~delta(bold p_m(bold x)~~(i,j+L))x6X@8 X@x6X@8 X@x6X@8 X@term2H+i+jj; M+m5i%jL+aa+++i; + IIpII2+1+1 +1 (- ( )(,))`+x] p m xj  T$XX%% XX%%!!X%$ ~A#(#X0*ddddd #ddaa xterm1~=~sum from {h=1} to H~sum from {j=1} to INF~j~s_0(i,j+L)~=~sum from {h=1} to H~sum from {j=1} to INF~(j+L~~L)~s_0(i,j+L)x6X@8 X@x6X@8 X@x6X@8 X@term1H+h+jj`si j L H +h+jjLLs&ijL+aa+   +gg+II I!I2+1+10(( , )8 +1+1()^0(,~)~$XX%%%XX%%!AX%$h) 0*0*0*qX%%  X% X%V X% X %$ X%%(! X0*%~-A Ԍa#(#Xddddd9#ddaa x}q=~sum from {i=1} to H~sum from {j=1+L} to INF~j~s_0(i,j)~~sum from {i=1} to H~sum from {j=1+L} to INF~L~s_0(i,j)x6X@8 X@x6X@8 X@x6X@8 X@2++Q+ + g + +II Iq I2H+ia+j+LjPsij H +i +jW +L6 LsFi6j+1+10(,)8 +1 +1~0(,)}$XX%%XX%%!aX%$'#(#(Xddddd^`ddaa x{=~left [ sum from {i=1} to H~sum from {j=1} to INF~j~s_0(i,j)~~sum from {i=1} to H~sum from {j=1} to L~j~s_0(i,j) right ] x6X@8 X@x6X@8 X@x6X@8 X@]++Ss + +NvxwN}~II- I I]H +i+j jsi js H# +i L +j! j s1i!j+1,+1S0(,) +1B +1i 0 (,)'$XX%%XX%%!X%$#(#(X ddddd`ddaa x~~~~~~~~L~left [ sum from {i=1} to H~sum from {j=1} to INF~s_0(i,j)~~sum from {i=1} to H~sum from {j=1} to L~s_0(i,j) right ]x6X@8 X@x6X@8 X@x6X@8 X@=++c  + +L=H+il+js+i j H3 +i L +j1sqiajNvxwQN}QQ~IvI= I I+1 +1c0(, ) +1R +10(,)$XX%%XX%%!X%$#(#(Xhddddd`ddaa xK=~left [sum from {h=1} to H~sum from {j=1} to INF~j~s_0(i,j) right ]~~L cdot N~+~sum from {i=1} to H~sum from {j=1} to L~(Lj)~s_0(i,j).x6X@8 X@x6X@8 X@x6X@8 X@]++   +v +NvxwN}~II I0 I]H +h+j jsi j] L N H +iv L& +jL jUsij+1,+1S0(,)G +1 +1()0( ,)u.K$XX%% XX%%!X%$  ?h We now define the excitation value, XV, of a state, S. #(#Xddddd #ddaa x<XV(S)~==~sum from {i=1} to H~sum from {j=1} to INF~j~s(i,j).x6X@8 X@x6X@8 X@x6X@8 X@XVzSH+i+jj`sPi@ j()2+1+1(, )0 .+aa+IIߎ$XX%%XX%%!X%$ Our total equation now becomes #(#Xddddd#ddaa x$VXV(S_L)~=~XV(S_0)~~LN~+~sum from {i=1} to H~sum from {j=1} to L~(Lj)s_0(i,j)~+~term2x6X@8 X@x6X@8 X@x6X@8 X@XVzSLXVRSLN" H +i LQ +jH L8j(shiXjpterm2(J)(0)r +1 +1 ()0(,) " + +  I[ I$ߡ$XX%%8XX%%!X%$ We know that since S0=SL, XV(S0)=XV(SL). Thus, the equation becomes !#(#X(#ddddd#ddaa xNL~=~sum from {j=1} to L~sum from {i=1} to H~(Lj)s_0(i,j)~+~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~sum from {i=1} to H~sum from {j=1} to INF~j~delta(p_m(h,l)~~(i,j+L)).x6X@8 X@x6X@8 X@x6X@8 X@NLzL*+jH+iLjsi j L +l HG +hM+mHE+i+jCjpmNh>lijLZz++  + ++++64II IQ IIOII+1I+1(()0H(8 ,( )h +1 +1f+1+1d+1((,)V(F,&)). $XX%%xXX%%!!X%$ \A#(#X'ddddd#ddaa xterm2~=~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~sum from {i=1} to H~sum from {j=1} to INF~j delta(bold p_m(h,l)~~(i,j+L))x6X@8 X@x6X@8 X@x6X@8 X@term2L+laH+hM+m_H+i +j j hl@i0j L+a++_+  +IIII I2+1+10+1+1. +1 (H (8,()(,))  x p m\$XX%%(#XX%%!AX%$a #(#(Xddddd`ddaax0=~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~sum from {i=1} to H~left [sum from {j=INF} to INF~j delta(bold p_m(h,l)~~(i,j+L))~~sum from {j=INF} to 0~j delta(bold p_m(h,l)~~(i,j+L)) right ]x6X@8 X@x6X@8 X@x6X@8 X@2++0++Y+++ s++k+5IkIIiINvxwN}~;II2L+lHa+h0M+mH_+i +jj h{ l ijL{+jJjhl}imj]L+1+1+1+1 ( ( , )(,c))0=((u,e)(,)M)P   C p mp5m0߭' #(#X;ddddd#ddaa(#=~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~(t_p~~L)~~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~sum from {i=1} to H~sum from {j=INF} to L~(j~~L) delta(bold p_m(h,l)~(i,j))x6X@8 X@x6X@8 X@x6X@8 X@2++0+ 7 + +5 ++3+++-IkII Ip I InII2L+lHa+h0M+mtWpL7 L +l Hf +h5 M +mHd+i[L+j*jLhli j+1+1+1_(G) +1 +1 +1+1(B)5(}(m,])(,)) p-m' 8 #(#X ddddd0#ddaa(#=~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~(t_R~+~l~~L)~+~sum from {j=1} to L~sum from {i=1} to H~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~(L~~j) delta(bold p_m(h,l)~~(i,j))x6X@8 X@x6X@8 X@x6X@8 X@2++0+  +V ++T++J}IkII I IIII2L+lHa+h0M+mtWRlo L L +jV H +iL+lTH+hM+mzLjEh5lij+1+1+1_( )' +1 +1%+1+1#+1()((,)M(=,-))  p}m8 #(#Xddddd#ddaa(#~~~~~~~~~~~~+~sum from {j=INF} to 0~sum from {i=1} to H~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~(L~~j) delta(bold p_m(h,l)~~(i,j))x6X@8 X@x6X@8 X@x6X@8 X@2R+++!++ + +H4IIZI IX Iz0q+1+1o +1 +1 (])P((,x)(,)p)+j!H+iLP+l H +h MN +mE Ljhlij pHm߄ #(#X%ddddd#ddaa(#Vsum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~(t_R(h,l)~+~l~~L)~=~sum from {bold x in i }~sum from {m=1} to M~t_R(bold x)~+~sum from {j=1} to L~sum from {i=1} to H~sum from {m=1} to M~(j~~L).x6X@8 X@x6X@8 X@x6X@8 X@III IIIIIbL+lH+h`M+mtROh?lW l LM+mtRLy+jHH+iMw+mnjLb++`+' ? _ + +i++H++>+11+1+1((,)o ):+1()+1+1+1(). +xaxV Collecting terms we have,  #(#(Xsddddd `ddaa(# NL~=~sum from {j=1} to L~sum from {i=1} to H~(L~~j) left [s_0(i,j)~~M~+~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~delta(bold p_m(h,l)~~(i,j)) right ] x6X@8 X@x6X@8 X@x6X@8 X@NLzL*+jH+iL@j[s i j MLC+lH+hMA+mhl{ikjZz++p s +++34II0Nv0x0wN}~MIIKI+1I+1(()0# ( , )+1b+1+1;((s,c)(,)[) p3m ߝ! #(#XdddddE#ddaa(#/+~sum from {bold x in i }~sum from {m=1} to M~t_R(bold x)~+~sum from {j=INF} to 0~sum from {i=1} to H~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~(L~~j) delta(bold p_m(h,l)~~(i,j)) x6X@8 X@x6X@8 X@x6X@8 X@2++i+|++<+k + +i + +_IwI~I% I I# I I+x4xpmMm+mtlRL+jk H +i L +li H +h M +mL/jZhJlij +1()0 +1: +1 +18+1()((,)b(R,B)) /߬ڌ$XX%%XX%%8!aX%$ԙ For the first term, we make the following substitutions for s0(i,j) and M: 7A#(#X@ddddd#ddaa(#ys_0(i,j)~=~sum from {l=inf} to 0~sum from {h=1} to H~sum from {m=1} to M~delta(bold p_m(h,l)~~(i,j))~~~(Corollary~ 2A) x6X@8 X@x6X@8 X@x6X@8 X@sRiBjZ+lyH)+hM+mb hR lij CorollaryA0(,)0+1H+1 ( ( , )j(Z,J))B(J2:)++J+y++ I3II'   p m7$XX%%XX%%!AX%$a#(#X(#ddddd#ddaa(#6sM~=~sum from {l=inf} to inf~sum from {h=1} to H~sum from {m=1} to M~delta(bold p_m(h,l)~~(i,j))~~~~~(continuity) x6X@8 X@x6X@8 X@x6X@8 X@M+lH+hPM+mh l: i* j continuity*+R+++P+ II I!+1+1(B(2 ," ) ( , ))J(r) rpm6߳$XX%%xXX%%!aX%$ The equation reduces to #(#Xh)ddddd5#ddaa(#SNL~=~sum from {bold x in i }~sum from {m=1} to M~t_R(bold x)~~sum from {j=1} to L~sum from {i=1} to H~(L~~j)~sum from {l=L+1} to INF~sum from {h=1} to H~sum from {m=1} to M~delta(bold p_m(h,l)~~(i,j))x6X@8 X@x6X@8 X@x6X@8 X@NLM+m4tRL+jc H +i L j+l+LaH+hM+mJh:lijZz++i++c + B++a++:III ILIII*+x|xpmU+1()4 +1 +1 (")2+1+10+1((,)R(B,2)) S$XX%%$XX%%!X%$ #(#XdddddC#ddaa(#~~~~~~~+~sum from {j=inf} to 0~sum from {i=1} to H~sum from {l=1} to L~sum from {h=1} to H~sum from {m=1} to M~(L~~j) delta(bold p_m(h,l)~~(i,j))x6X@8 X@x6X@8 X@x6X@8 X@z++:+i++g+ +] |I#II!I I0+18+1+16 +1 ( )((,)`(P,@))J+jiH+iL+lgH+h M +m L- jXhHlij pm $XX%%XX%%!X%$ԙ The last two terms are both identically zero. The reason is that throws cannot be made backwards in time. This means #(#X ddddd ddaa(#1t_R(bold x)~>=~0~~~~implies~~~~t_p(bold x)~>=~t_xx6X@8 X@x6X@8 X@x6X@8 X@_t+RC_t+p _t# +x_(_)r_0_( _)Z_x_x_J_8 _$XX%%XX%%X!X%$ where p m( x ) = (hp( x ),tp( x )),    r m( x ) = (hR( x ),tR( x )), and x = (hx,tx). In the second to last sum, the condition of jugglability implies tp(h,l)l. Clearly, j0, there is a positive integer T such that when jT, the time component of p j( x )  ? is greater than B. Similarly, a path diverges to negative infinity if given any A<0, there is a negative integer T such that when jT, the time component of p j( x ) is less than A. Consider p ' to be the permutation on restricted multiplex space (M') as defined in theorem 2. From abstract algebra, we know that p ' can be partitioned into disjoint cycles. This means that for any member x of M', x is moved by a cycle or is a fixed point. In other words, there exists an integer, l, such that p 'l( x ) = x .  ?$ We define the orbit to be 3a #(#X*dddddddaa(#\O_x~==~LBRACE~{bold p'}^j(bold x):~~0<=j0 then choose positive j such that j > (Bt)/KL. This implies tp > B. p jl( x ) is in the positive p path of x . Hence, the positive p path of x diverges to positive infinity. Now choose negative j such that j < (At)/KL. This implies tp < A. Hence, the negative p path of x diverges to negative infinity. If K<0 then choose positive j such that j > (At)/KL. This implies jKL < (At) since K is negative. Hence tp < A and the positive path diverges to negative infinity. For negative j such that j < (Bt)/KL, we get jKL > (Bt), so tp > B and the negative path diverges to positive infinity. A path with zero gain is simply an orbit with l elements.  ?x  Definitions: We define the p path of x to be the union of positive and  ?@ negative p paths of x together with x itself. The p path is forward if the  ? positive p path diverges to positive infinity. It is backward if the  ? positive p path diverges to negative infinity. The gain of the p path equals the gain of its positive p path. Given these definitions, it is clear from the proof above that paths with positive gains are forward and paths with negative gains are backward. Theorem 9 The following statements about a p path are equivalent: (i) A p path is an orbit. (ii) The positive and negative p paths do not diverge. (iii) Its gain is zero. Proof We have noted above that p 's path forms an orbit if and only if it has zero gain so (i) is equivalent to (iii). We now proceed to show (i) is equivalent to (ii). Let x be some point in multiplex space. Suppose the p path of x is an orbit. Then the set of elements (m,h,t) of this path is finite. Let B be the maximum value of t for all such elements. Then there is no t>B. This implies the positive and negative p paths do not diverge to positive infinity. Now let B be the minimum value of t for all such elements. Then there is no t0 then it belongs to an equivalence class with K members. If K<0 then the equivalence class has K members. For both cases, these members are Pathp( x '), Pathp( x '+(0,0,L)), Pathp( x '+(0,0,2L)), .. Pathp( x '+(0,0, K LL)). We must show these members are spatially equivalent and that these represent all paths which are spatially equivalent. By theorem 11, Pathp( x ')  Pathp( x ' + j(0,0,L)) for any integer j. So all members are spatially equivalent. & #(#Xddddd)ddaa(#zPath_p(bold x)~=~LBRACE~{bold p}^j(bold x):~~ j in Z~RBRACE~=~Path_p({bold p}^l(bold x))~= ~Path_p(bold x+K cdot (0,0,L)).x6X@8 X@x6X@8 X@x6X@8 X@_Path+pj_j _Z _Path +p<l_Path|+p4_K6_LJ_(:_)K_(;_)_:+_(_(|_)_)_(_(V_0_,F_0_,_)&_)_._xb_p_x_p_xD_x _S _# _____ _&$XX%%H XX%%X! X%$ Now consider a spatially equivalent path, Pathp( x '+j(0,0,L)). Let j=qK+r where 0r0~RBRACEx6X@8 X@x6X@8 X@x6X@8 X@_Let_OJ_O+xB_Path*+p _c _ orresponds_tw_oG_O+x_a?_nd_hasG_KJ+____:z_( _)_>7_0_x v$XX%%$XX%%X! X%$! #(#X+dddddJddaa(#uQLet~O_~==~LBRACE~O_x:~~Path_p(bold x')~c orresponds~t o~O_x~~a nd~has~K<0~RBRACEx6X@8 X@x6X@8 X@x6X@8 X@_Let_OJ_O+xB_Path*+p _c _ orresponds_tw_oG_O+x_a?_nd_hasG_KJ+____:z_( _)_<7_0_x u$XX%%'XX%%X!! X%$A #(#Xddddd ddaa(#X]N_+~=~sum from {bold x in O_+} tau(bold x) ~~~~~~ N_~=~sum from {bold x in O_} tau(bold x)x6X@8 X@x6X@8 X@x6X@8 X@NGOJNR GO2RGdd%j:  Gdd %*I IGxJx Gx xe) )() (r )Xڌ$XX%%XX%% !A X%$The union of O+ and Oé is M'. The summation of )( x ) for x in M' is therefore equal to the number of forward paths minus the number of backward paths. a #(#X( dddddddaa(#1N_+~~N_~=~sum from {bold x in MU'} tau(bold x).x6X@8 X@x6X@8 X@x6X@8 X@NN2z"B+ddYI+x;x+V)()+.$XX%%xXX%% !a X%$ Theorem 13  #(#Xddddd+ ddaa(#JCsum from {bold x in MU'} bold r(bold x)~=~(N_+~~N_) cdot (0,0,L).x6X@8 X@x6X@8 X@x6X@8 X@:I+xvrfxb+ddYn^&+()~()X(0H , 08 ,( ) .NN LJ$XX%%( XX%% ! X%$ Proof  #(#Xdddddddaa(#Osum from {bold x in MU'} bold r(bold x)~=~sum from {bold x in MU'} (bold p(bold x)~~bold x)~=~sum from {bold x in MU'} (bold p'(bold x)~+~tau(bold x) cdot (0,0,L)~~bold x)x6X@8 X@x6X@8 X@x6X@8 X@:II I+xvrfx~+xZpJxb xz +xV pxxxb+ddY+ddY  +dd YxJ++ +)()(() ) (()()("0,0,z)) =LO$XX%%XX%% ! X%$ #(#Xddddd'ddaa(#d=~sum from {bold x in MU'} bold p'(bold x)~~~sum from {bold x in MU'} bold x~~+~sum from {bold x in MU'} tau(bold x) cdot (0,0,L)~=~~(N_+~~N_) cdot (0,0,L).x6X@8 X@x6X@8 X@x6X@8 X@2+ddY;+ddYw +ddS Y RjZ" IIo I+xFp{x+xOxG +x x++ + )=() ( ) (* 0 ,0,)z()T(0D,04,$). LNNLd$XX%%XX%% ! X%$ Notice that if there are no throws backwards in time, there can be no backward paths and the equations becomes V #(#Xddddd ddaa(#5sum from {bold x in MU'} bold r(bold x)~=~(0,0,N_+L).x6X@8 X@x6X@8 X@x6X@8 X@:I+xvrfxb+ddYN+()~(0n,0^,).NLV$XX%%8XX%% ! X%$ Looking back to theorem 6 we see that N+ = N. Recall that when the concept of a state was introduced, no throws backward in time were allowed. We know that this means there cannot be any backward paths so our definition for N gave the number of forward paths. Theorem 13 is more general. Although mathematically interesting, it is not useful to ordinary jugglers, few of whom can actually throw a ball into the past!! This theorem brings up some interesting questions. Suppose we allow throwing balls into the past as well as into the future. What do N+ and Né really represent in terms of numbers of balls being juggled? If a person naturally progressed backward in time, he would throw all balls backward in time. In his view, Né would represent the number of real balls while N+ would always be zero. According to Allen Knutson, the constant (N+ Né) represents the number of real balls minus the number of antiballs seen in the pattern at any given moment. This is a very plausible idea. It is no more absurd than the concept of "antimatter", which can be thought of as matter moving backward in time. Let us entertain the question further. Suppose at frame 1 we throw a ball into the future so it lands at frame 2. At this time, we throw the ball back into the past exactly one frame so it lands at frame 1 where it started. Now suppose we observeh)0*0*0*aX%A X( % a X% X%r X% X%  the pattern. What do we see? At frame 0 nothing has happened so we see nothing. At frame 1, we see two balls suddenly appear. One is a real ball but the other is an antiball. At frame 2 they vanish! This violates our intuitive sense of continuity. We started with no ball; from frame 1 until frame 2 we had two balls; and thereafter we had none again. It is not just the number of balls that must remain constant. It is the difference between real and antiballs which does not change. We see that a ball can be created from nothing if at the same time and place, an antiball is created. Definitions: Let t be some real number representing a point in time. Let a ball be thrown from (m1,h1,t1) to (m2,h2,t2), which are points in multiplex space. At time  ?` t, the ball is real if t1t< t2. The ball is called an antiball if t2tt and there exists some integer, B, such that for jB, tjt then the ball thrown from p C( x ) to p C+1( x ) is real. Since tC+1 and tD are on the same side of t, NrealĩNanti is zero considering only these throws. When all throws are considered, Nreal increases by 1 so NrealĩNanti is 1. If tC+1t then the ball thrown from p C( x ) to p C+1( x ) is neither a real nor an antiball. Since tC+1tt then the ball thrown from p C( x ) to p C+1( x ) is neither a real nor an antiball. Since tDtt we find the ball thrown from p C( x ) to p C+1( x ) is real. NrealĩNanti is 1 considering only the other throws. Therefore when all throws are considered, NrealĩNanti is zero. On the other hand, suppose tC and tD are both greater than t. If tC+1>t then no real balls or antiballs are found so NrealĩNanti remains zero. If tC+1t we find the ball thrown from p C( x ) to p C+1( x ) is an antiball. NrealĩNanti is 1 considering only the other throws. NrealĩNanti is zero when all throws are considered. The case for a backward p path is proved in a similar fashion. Theorem 15 For any periodic pattern, p , and any time t, the number of real balls minus the number of antiballs is equal to (N+ Né). Proof The contribution of each forward path to NrealĩNanti as seen at some time t is 1. The contribution of each backward path to it is 1. Therefore the collection of forward paths (N+) and backward paths (Né) contributes (N+ĩNé) to NrealĩNanti. Since these represent a partition of multiplex space, there can be no other contributions to it so N+ĩNé must equal NrealĩNanti. Extension of the Mathematics to Include Any Pattern Although the mathematics presented this far is able to describe a large class of juggling patterns, it does not describe all of them. For example, patterns which require the juggler to throw in unusual rythms are not described by the mathematics if the time intervals between throws cannot be represented by integers multiples of some frametime. At the beginning, we stipulated that a pattern consisted of catches and throws which had to occur at discrete points in pattern space. Pattern space was then extended to multiplex space so that the pattern would be a permutation. In reality, time is continuous not discretized. All throws and catches in a pattern take place in continuous multiplex space. It is the set of times at which these throws and catches occur which is discrete. We already have developed many theorems that deal with multiplex space. As we shall see, it is not necessary to redevelop these for our new discretized space. Definition of a Pattern A pattern, p a, is a selfmapping of an unbounded countable subset, Md, of continuous multiplex space (Mc). Theorem 16 ( p a:MdMd, ) forms a group which is isomorphic to ( p :MM, ). Proof Choose some element of Md and call it t0. Label all elements by tj for some integer j so that their order is reflected by their subscripts. (i.e. tj>tk implies j>k). Let f map M to Md by f (m,h,j) = (m,h,tj). Note that f is a bijection because it is both onetoone and onto. Now let , map from p a to p : p = ,( p a) = f é1( p a( f )). , is welldefined because f is a bijection. h)0*0*0* Furthermore, , is a bijection and we show ,( p a q a) = ,( p a) ,( q a). ,( p a q a) = f é1 p a q a f = f é1 p a f f é1 q a f = ,( p a) ,( q a). What does this mean? Every definition and theorem before was based on p and the operation of composition. We have proved this new group, which encompasses all patterns, is isomorphic to our old group. Thus, to analyze any pattern, p a, we look at p = ,( p a). To notate a periodic p a, it suffices to use MHN in conjunction with the set, {t1,t2,.. tL}. As a side note, it should be mentioned that aperiodic patterns can be viewed as a concatenation of periodic patterns. To link pattern X to pattern Y requires that they have one state in common and the link is made at this state. Thus, the mathematics is capable of describing aperiodic patterns as well as periodic patterns. A Brief Summary with Conclusions The axiom of continuity provided a test for pattern validity. It was found that a pattern was represented by a permutation, p , on multiplex space (M), an infinite space. Periodicity and restricted multiplex space (M') were defined and it was determined that for x in M', p ( x ) was the sum of p '( x ) and an integer multiple, )( x ) of (0,0,L) where p ' was a permutation on M' (see theorem 2). A way of notating p by means of a matrix of vectors was introduced called "MultiHand Notation" or "MHN". The operations of permutation, local translation, and global translation were found to preserve validity. The concept of states and transitions was presented and a useful theorem was proved which related the pattern, its period, and the number of balls being juggled (see theorem 6). It was shown that the entire class of patterns together with the operation of composition formed a group. Multiplex space (M) was partitioned into ball paths. It was shown that each path in M corresponded to an orbit, Ox in M', and also to a gain, K. K was found to equal the summation of )( x ) for x in Ox. Paths which corresponded to the same orbit in M' were termed spatially equivalent and M was partitioned into spatial equivalence classes. It was proved that each of these spatial equivalence classes had K forward paths if K was positive or K backward paths if K was negative. The difference between the numbers of forward and backward paths, (N+ Né), was found to be the summation of K for all equivalence classes which was equal to the summation of )( x ) for x in M' (see theorem 12). A theorem more general than theorem 6 was proved which related a pattern, its period, and the constant, (N+ Né). Antiballs and real balls were defined in terms of being visible at a specific time t. It was determined that the number of real balls minus the number of antiballs was equal to (N+ Né). Finally, a broader definition for a pattern was given which included arhythmic throws. It was determined that any pattern whatsoever could be identified with a rhythmic pattern through an isomorphism, ,. Bibliography  ? Buchthal, David C., and Douglas E. Cameron. Modern Abstract Algebra. Boston Mass.: PWS Publishers, 1987.  ? Magnusson, Bengt, and Bruce Tiemann. The Physics of Juggling. The Physics Teacher November 1989, p. 586. Author's Address Ed Carstens RR2 Box 645 Rolla, MO 65401 (314) 3647536