V r,k j F k m N k+1 N k N k+1 H j

n = 7 n = 16 Ṽ r ñ,ñ j Ṽ Ṽ j x / Ṽ

W

2r V r D N T T

2r 2r

N k F k N 2r

Ω R 2 n Ω I n = { N: n} n N

R 2 x R 2, I n Ω R 2 u R 2, I n x k+1 = x k + u k, u, x R 2, I n. k = 0,1,2... x r C C = { x R 2 : x x r }, I n. S x S = { x R 2 } : x x R, In, R k = + p n, p N

Ω Ω P R 2 P P P p j, j I N(P) N(P) P A( ) A(P) = 1 2 (p j p j+1 ) j I, N(P) p j p N(P)+1 p 1 Ω Ω I n C x Ω, I n A ( Ω ) I n C X = ( x1 T ) T,xT 2,...xT n X R 2n x Ω, I n A X B A, B ω j Ω j I N(Ω) ( ) X : A Ω, A X B I n C

Ω Ω n n V, I n

V = { x Ω: x x x x j, j In }, In. V Ω R 2 I n V = Ω Int V Int V j = /0,, j I n, j Int V v, j, j I N(V ) N N = { j I n : V V j /0, j }, I n. V V j N Ω j = V V j, I n, j N. Ω

C r V r = V C, I n. Ω,V,C V r Ω V U = C \V r, I n. V,V r Ω, I n U I n U Ω U U = U Ω j N U j, I n, U Ω U j = C \Ω Ω = U V j Ω H = A ( Ω I n C ) = A(V r ). I n H Int V r Int V r j = /0,, j I n, j r

Blnd regons x r Unexploted regons r-lmted Vorono cell Vorono cell Sensng regon H Ω k

N k N k+1 k k + 1 (b) k k + 1 N k N k+1 x k j N k

N j N k \ Nk+1 j / N k j N k+1 \N k V j N k Nk+1 {} N k Nk+1 {} k V k W k Fk x k,xk+1 F k = N k {} W k x k+1 W k V k xk N k+1, k W k F k x k+1 W k k F k W k V k

F k k W k x k+1 W k q l I q l I I n k l = k + 1 x k+1 k r j F k k Vj r F k x k+1 H k+1 H k x k+1 F k H k+1 F k H k F k k r

x x (a) (b) V r,k j F k V r j F k r V r,k+1 j F k j F k l {k,k + 1} ( A V r,l j ) ( ) ( = A V r,l j F k A m N l j \Fk l = k j F k ( A(C j ) = A ( = A V r,k j V r,k j ) + ) m N k j ( A + A m N k j Fk U m,k j ( ) ( + A U m,k j U Ω,k j ) = ) ( + A m N k j \Fk U m,l j U m,k j ), ) ( + A U Ω,k j ).

A(C j ) ( ) A(C j ) = A V r,k j F k + m N k j Fk ( ) = A V r,k j F k + A m N k j Fk ( ) ( ) A U m,k j F k + A U Ω,k j F k = ( U m,k j ) ( + A U Ω,k j F k ), m N k j Fk N k j Fk F k A(C j ) ( ) ( ) ( ) ( ( ) ( )) A V r,k j = A V r,k j F k A U m,k j + A U Ω,k j F k A U Ω,k j. m N k j \Fk U Ω,k j = C j \Ω ( ) ( A U Ω,k j F k = A U Ω,k j l = k + 1 F k ), H k+1 F k H k F k = H k+1 H k. [ ( ) ( H k+1 H k = j I n A V r,k+1 j A V r,k j = j F k [ ( A V r,k+1 j ) ( A V r,k j )] = )] [ ( + A j I n \F k V r,k+1 j ) ( A ( ) ( ) A,A [ A H k+1 H k = j F k ( V r,k+1 j F k ) A V r,k j ( V r,k j F k )] j F k V r,k+1 j [ ( A m N k j \Fk U m,k+1 j ) V r,k j )]. ( A U m,k j )].

F k N k j \ Fk jm ( m / F k ) [ ( ) ( H k+1 H k = A V r,k+1 j F k A V r,k j F k )]. j F k I n \ F k H k+1 H k = j F k j I n \F k [ A ( ) ( V r,k+1 j F k A V r,k j F k )]+ [ ( ) ( )] A V r,k+1 j F k A V r,k j F k = H k+1 F k H k F k, W k { W k = x R 2 : x x k αd (x k,v k )}, 0 < α 1 d (x,m) x M d (x,m) := nf{ x y : y M}. W k x k xk+1 W k x k+1 W k x k+1 k x k+1 W k : } {H k+1 F H k k F, k : H k+1 F k> H k F k

W k u k = x k+1 x k, u α W k x k W k F k k F k W k V k N k R k (I) Rk (I) wcs k I I n

N k k ( ) { ( ) } R k N k = 2max d x k, k j : j N k = max{ x k,x k j : j N k }, j d (x,m) S k Ω ( ) { x R k N k = 2max k v k }, j : j IN(V k wcs ), N( ) N k V k N k R k ( ) N k wcs R k ( ) N k wcs N k R k ( ) N k wcs

N k R k 0 Sk /0 ˆV k Ω ˆN k /0 { } R k x 2max k ˆv k, j : j I N( ˆV k V k R k S k j ˆNk ˆN k j ˆV k ˆV k N k ˆN k ) S k Ω x k+1 x k+1 W k V k W k N k+1 W k k k x k ( ) Rk N k N k+1 wcs N k x k+1 N k V k+1 N k j N k x k+1 V k+1 N k N k+1

m I n N k+1 x k+1 m h j h j = { x R 2 : x x = x x j },, j In, j. m N k+1 h k+1 m V k+1 N k v k+1, j N k V k+1 N k j j = argmax { x k+1 v, k+1 j N k : j IN ) (V k+1 N k }. a,b R 2 L b l, l L a d (a,l) a b m x k+1 v k+1, j N k xk+1 x k+1 m N k+1 V k+1 N k

m m ( xm k+1 = x k+1 + 2 v k+1, j N k ) x k+1. x k m R k ( N k+1 ) wcs = x k ( ( x k+1 + 2 v k+1, j N k )) ( x k+1 = x k + x k+1 ) 2v k+1, j N k, j x k x k+1 k m N k+1 N k N k Nk+1

k N k N k k+1 V N k k ˆV k+1 N k m x k+1 xm k+1 v k+1, j N k m R k ( ) N k+1 wcs x k+1 V k+1 N k R k ( ) N k+1 wcs k x k+1 R k ( ) N k+1 wcs x k+1 W k N k+1 wcs { ( = sup R k N k+1 ) : wcs xk+1 W k N k+1 } W k, R k R k ( ) F k wcs N k R k ( ) F k wcs

F k ( ) R k x k+1 W k N k+1 wcs N k V k W k x k+1 W k V k+1 N k v k+1, j N k ) R k ( N k+1 wcs { )wcs max ( R k x k+1 R k W k N k+1 ( F k )wcs max { R k S k F k ( N k ) R k ( x k+1 wcs,rk ( x k+1 W k W k ) N k+1 wcs,rk } ) N k+1 wcs ( ) } N k+1 wcs k R k ( ) F k wcs = max Rk ( ) N k, wcs Rk x k+1 W k N k+1 wcs, Ω R 2 Ω

Ω n = 18 n = 10 Ω r = 1.5 r = 3 supa ( I n C ) = nπr 2 127.23 2 282.74 2 A(Ω) = 226.37 2 nπr2 A(Ω) = 56.2% 100% Ω 100% α α = 0.1 H = nπr 2

100 100 90 90 80 80 70 70 A cov (%) 60 50 40 A cov (%) 60 50 40 30 30 20 20 10 10 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 k 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 k Ω

n r C = { x R 2 : x x r }, In. r

V V = Ω j I n H j, I n, H j R 2 j H j = { x R 2 : x x x x j },, j In. H = R 2, I n Ω Ω I n : x U j I n, j : x C j, x Ω. x Ω U x Ω (C \V ) x C x x r x V x k H k j, j : x H j x x j x x x x j r x Cj C = Ω C = I n V r, I n A(C) = A(V r ), I n A( )

C =V r U, I n C = Ω I n C = Ω I n (V r U ) = I n (Ω (V r U )) = I n ((Ω V r) (Ω U )) = I n (V r (Ω U )) = ( I n V r ) ( I n (Ω U ) ) I n (Ω U ) I n V r x ( ) I n (Ω U ) j : x C j x Vj r U j, U j V x Vj r x I n V r x U j x V U j V x U x U V = /0 I n V r I n (Ω U ) I n V r A(C) = A ( I n V r ) V r Vj r = /0,, j I n, j A(C) = In A(V r) Ω Ω = A( Ω I n C ) A(Ω). C r V r V r ACP(V ˆ r ) (V ˆ r ) = A( A(Ω) I n V r ) = I n A(V r). A(Ω) = (V ˆ r )

x r r r

r R 2 Ω r

Ṽ = Ω j I n H j, I n, H j H j H j n 1 ~ H j x j d j x d ~ H j H j Ω Ω d d j j H j H j j w = x x j d j x

H j H j x x 2 = d 2 + ( x x j 2 d 2 j ) d j d x x 2 = d 2 + x x j 2 (w d ) 2 = x x j 2 w 2 +2d w H j H j = { x R 2 : x x x x j + w(2d w) },, j I n, j H j H j w > 0 w = 0 Ω = R 2 H j H j d d = d j = w 2 d j j j r r j H j j H j j r j > r H j H j d > 0 d < 0 d j w r +r j C C j = /0

d d x x j x x j (a) (b) d x x j x x j (c) (d) d = r r + r j w, d j = r j r + r j w. ( r + r j w ) j d + d j = w d,d j

j C C j /0 C k C k, k I n r r j w r +r j d d j C, C j x c C, C j x c x 2 = r 2 xc x j 2 = r 2 j x c x 2 xc x j 2 = r 2 r2 j (d 2 + ) ( xc x j 2 d 2 j + ) xc x j 2 = r 2 r2 j d2 d2 j = r2 r2 j d w d +d j d < 0 w d + d j d + d j = w d j d 2 (w d ) 2 = r 2 r2 j w2 + 2d w = r 2 r2 j d d j d + d j = w d = r2 r2 j + w2 2w d + d j = w, d j = r2 j r2 + w2, ( r Rr j w r + r j ) 2w C C j C j C 0 < w r r j H j = R 2 H j = /0 Ṽ = /0

Ω I n : x Ũ j I n, : x C j, x Ω. m l Ṽ = /0 n = m l x Ω Ũ x ( C \Ṽ ) x C x x r x Ṽ x k H k j, j : x H j x x j 2 x x 2 + w(2d j w) r 2 + w(2d j w) w = x x j r r j w r + r j d j ( ) x x j 2 r 2 +w(2d j w) = r 2 +w 2 r2 j r2 +w2 2w w = r 2 +r2 j r2 +w2 w 2 = r 2 j x C j

w > r +r k k I n Ũ = /0 Ũ /0 d < r d = r r +r j w > r r +r j (r + r j ) = r Ũ = /0 Ṽ r C = Ω I n C = Ṽ r, I n Ṽ r = Ṽ C, I n. A(C) = In A ( Ṽ r ) Ṽ r ACP ˆ ( Ṽ r) r ˆ ( Ṽ r) = A( A(Ω) I n Ṽ r ) = I n A ( Ṽ r ). A(Ω) r = r j r ACP = ACP ˆ ( Ṽ r)

Ω Ω H j H j R 2 Ṽ, I n d,d j w d,d j

H j ~ H 12 ~ ~ H H 23 21 ~ 2 H 32 R 2 sole cell 3 1 R 1 R 3 ~ ~ H 13 H 31 O

Ṽ, I n x O j : x C j, x Ω x Ω O j : x C j Ω I n C = I n Ṽ r Ω C j I n Ṽ r Ω C j O I n Ṽ r x Ω C j O I n Ṽ r I n Ṽ x I n Ṽ x O x I n Ṽ j : x C j r O I n Ṽ Ω

O no cell assgned r r r Ṽ r = Ṽ C = j I n H j C Ω. H j C H j C j H j Ṽ r Ṽ r j /0 C j

Ṽ r Ω R 2 A(Ω) = 5.08 2 Ω n = 7 r = 0.08, I n Ω r ACP(V ˆ r ) = 38.20% ACP = 50.01% Ṽ

(a) (b) n = 7 ˆ ACP ( Ṽ r) = 50.01 = ACP% r Ω n = 16 r Ω ˆ ACP(V r ) = 68.32% =

(a) (b) n = 16 ˆ ACP ( Ṽ r) = 84.66% Ṽ C /0 ϕ : Ω R 2 R + x Ω x

Ω u R 2 ẋ = u, u R 2, x Ω, I n, f : Ω R 2 R + x 1 C Ω 1 D (x) = 1 x D 0 x / D D R 2 Ω H ϕ Ω Ω ϕ Ω I n C H = max f (x)ϕ (x)dx. Ω I n Ω

C f 1 C Ω H = f (x)ϕ (x)dx = I n V I n V r ϕ (x)dx = I n H. ϕ (H) (H )

n x V r n (x) = 1 n (x) V r x x S V r V r S S r = r > 0, I n u = n ϕ dx, I n, V r C H V r C C V V r V r n (x), x V r C V r C ( B ) H H = V r ϕdx, I n ñ x Ṽ r u = ñ ϕ dx, I n Ṽ r C

H = max f ϕ dx = max Ω I n O I n f ϕ dx + max f ϕ dx. I n Ṽ I n C O = /0, I n f (x) = 0, x O, I n max j I n f j (x) = f (x), x Ṽ. x Ṽ r max j In f j (x) = 1 = f (x) x B f (x) = 0 f j (x) = 0, j I n \ {} f j (x) = 1 j I n x Ũ j x C x B Ṽ r Ṽ H H = f ϕ dx = I n Ṽ I n Ṽ r ϕ dx, B f = 1 C H x H = ( ϕ dx x x Ṽ r ) ( + x ϕ x x Ṽ r j Ṽ r Ṽ r j /0 H = x Ṽ r j υ ñ ϕ dx + j Ṽ r Ṽ j r Ṽ r j ϕ dx ). υ j ñ j ϕ dx, ñ ñ j Ṽ r Ṽj r υ, υ j x x Ṽ r x Ṽj r υ j (x) x x, x Ṽ r j,, j I n.

Ṽ r Ω C O r j Ṽ r Ṽj r Ṽ r { } Ṽ r = { Ṽ r Ω } { Ṽ r C } { Ṽ r O } Ṽ r Ṽ r Ṽj r j. H x sole cell x Ṽ r H = x Ṽ r υ Ω ñ ϕ dx + Ṽ r j Ṽ r Ṽ r j Ṽ r C υ ñ ϕ dx + j υ ñ ϕ dx + Ṽ r Ṽ r j Ṽ r O υ ñ ϕ dx+ υ j ñ j ϕ dx. Ṽ r O /0 C O Ṽ r O υ = 0 x Ṽ r Ω x Ω x Ṽ r j Ṽ r Ṽj r = j Ṽ r Ṽj r H = x Ṽ r C υ ( ñ ϕ dx + υ ñ + υ ) j Ṽ r Ṽ j r j ñ j ϕ dx. Ṽ Ṽ r C x x

υ (x) = I N I N N N υ (x)ñ (x) = υ j (x)ñ j (x) x Ṽ r Ṽj r υ (x) = υ j (x) Ṽ r Ṽj r υ ñ = ñ j x Ṽ r Ṽj r Ṽ,Ṽ j Ṽ r Ṽj r ~ n ~ n j x x j x x j ~ V ~ V j ~ V ~ V j ñ,ñ j Ṽ Ṽ j x / Ṽ H x H r

r r R Ṽ r r r = max{r, I n }. r r R Ṽ r r + r

r Ṽ r H j Ṽ r H j j x x j = r + r j j r r j = r k r + r x x k r + r r d H k H k d x x k r + r k d = r x x k r (r + r) r. r + r k r + r k H k C Ṽ r R Ṽ r R r r = { } r j, j I n, I n r

r = r j j r ( ) (k) k r (k) rem k R m r k Ṽ r(k) = Ṽ r(k 1) H m m r (k) rem = r (k 1) rem \ {r m } r k R wcs(k) = ( ) 1 + maxr(k) { } rem sup x x, x Ṽ r(k). r m r m = maxr rem m H m, H m Ṽ r

halfplane margnally crossng r the boundary of V (worst case scenaro) x r wcs R maxr rem nodes already dentfed artfcal node m (worst case scenaro) d = sup { x x, x Ṽ r }. d = r r + r m x x m, x x m ( x x m = 1 + r ) m sup { x x, x Ṽ r }. r r m = maxr rem R wcs x x m R { } sup x x, x Ṽ r(k) r Ṽ r C maxr (k) rem r

r rem r \ {r } R wcs r + maxr rem & r rem /0 R < R wcs R m Ṽ r r rem R wcs Ω R 2 ϕ n = 10 {r, I n } σ = 0.1 Ω

In πr 2 = 4.01 2 Ω dx = 5.081 2 Ω In πr2 Ω dx = 78.9% Ω (a) (b) (c) (d) (e) (f)

n = 7 σ 0.2

(a) (b) (c) (d) (e) (f) H Ω

n Ω R 2 ϕ : Ω R 2 R + x Ω x I a a I a = {k N k a} x, I n ẋ = u, x Ω, u R 2, I n. C (x ) C C C := Ω I n C (x ) Ω 1 C(x ) (x) = 1(0) x (/ )C (x ) I n C (x ) H = max1 C(x ) (x)ϕ (x)dx = ϕ (x)dx. I n Ω C ϕ

C (x ) maxmal nscrbed node-centred dsk maxmal nscrbed convex set maxmal nscrbed node-centred dsk (approxmated) maxmal nscrbed convex set orgnal footprnts

n {V, I n } Ω r r 1 C(x )

C D R 2 C: = D D R 2 λd + υ, λ > 0, υ R 2 D λ = 1 D C C W = O {W, I n } C W C O C W W = j I n W j, W j C, j C (x ), C (x j ) W j C (x ) C (x j ) 1 W j := C (x ) Ω

C (x ) C (x j ) (1) = 2 x j,x (2) j C (x ) C (x j ) C (1) j,c (2) j W j := C (l) j Ω, l {1,2} : C (l) j C (x ). W j W j := Ω { C (x ) C (x j ) } \W j 3 2 4 5 2 3 6 4 6 1 7 1 5 j C (x ) C (x j ) 1 W {W, I n } C O O = C \ {W, I n } W = O {W, I n } C

u H u = α W C(x ) n ϕ dx, α > 0, I n, n W W, I n H x H = ϕ + ϕ x x O x W + j x W j ϕ, H = υ0 n 0 ϕ + x W O W υ n ϕ + j W W j υ j n j ϕ,

υ,υ j x x W x W j υ0 υ x 0 (x) := x, x O, I n n 0 O W Ω C (x ) O H x H = x W O W O υ0 n 0 ϕ + υ n ϕ + j W Ω υ n ϕ + W C(x ) W W j υ n ϕ + j υ n ϕ + W W j υ j n j ϕ. Ω W,O W,W j H = n ϕ. x W C(x ) H H dh dt H = ẋ = I n x α I n W C(x ) n ϕ 2 0. W

u W sup{ x x j C(x ) C(x j ) = 1}. W C(x ) C(x j ) {1,2} W C(x ) C(x j ) = 1

worst case scenaro topology geometrc locus of nodes centres mnmum communcaton radus to ensure dstrbutedness, j, j ϕ

v j, j I 5 v j 1 v j : (0,0), (1,0), (1, 3 /8), ( 1 /2, 7 /8), (0, 11 /16). m j, j I 5 v j,v j+1 v 6 v 1 m 6 m 1 C = j I 5 B j B j = (1 τ) 3 P j,0 + (1 τ) 2 τp j,1 + (1 τ)τ 2 P j,2 + τ 3 P j,3, τ [0,1] P j,k, k = 0,...3 P j,0 = m j, P j,1 = P j,2 = v j+1, P j,3 = m j+1, j I 5. ( 3 /7, 3 /10) v j+2 B j m = P j+ 1 j,3 x v j mj= Pj,0 vj+ 1 = Pj,1=P j,2

ũ = α B r V r n ϕ, α > 0, I n, α B r := {x x x r} r x V r := V B r r C(x ) B r W V n = 12 Ω λ = 0.6 Ω Ω ϕ = 5.0809 unts 2 C ϕ = 0.2443 unts 2 57.6952% Ω r = 0.18 unts α α = α = 3, I n

Ω C H / Ω ϕ 14.0857% 36.3818% Ω 57.6952%

57.6952% n = 6 λ = 1.1 C ϕ = 0.8211 unts 2 n C ϕ = 96.9600% Ω ϕ Ω 26.4291% Ω α = α = 2 r = 0.33 unts C 69.7661% 88.6032%

n = 10 Ω 100% C(x )

H/ Ω ϕ

(a) (b) (c) maxmal nscrbed node-centred dscs rotaton-nvarant (approx) maxmal nscrbed convex set common orentaton demanded no approxmaton requred no demand for common orentaton C C C := x + R(θ ) λ C, x θ λ R( ) [ cos( ) sn( ) R( ) = sn( ) cos( ) ]. ẋ = u, x D R 2, u R 2, θ = ω, θ,ω R,

f x f (x;x ) f (x;x,θ,λ ) (a) (b) (c )

C, I n n x θ W Ω W : = Ω C \ j C j, I n. {W, I n } W c W c : = C j Ω, jc W c W W c = In C \ In W {W 1,W 2,...W n,w c } C j C j W = /0

5 3 4 6 2 1 7 (a) (b) W c H H = ϕ + ϕ, I n W W c W,W c Ω

S n(x) S x S S S W W c n(x), x W n(x), x W c n n c u = α,u n ϕ, W C ω = α,ω R(90 W o )(x x ) n ϕ, C α,u,α,ω > 0 H dh dt { H = ẋ + H } θ. I n x θ u = α,u H x, ω = α,ω H θ,

u x ω H x x = x n j ϕ + n c ϕ x x j I n W j = j W j x x n j ϕ + W c W x x n ϕ + W c x x n c ϕ. W W Ω W c W W c W C regon boundary x W c x j neghborng node n c x n n j W c W j x j W W x Ω x x = 0 x x n j = x x n c, j I n

H = x W C x x n ϕ. H θ H = θ W C x θ n ϕ. x x, x θ x W C x = x + ρ (x) [ cos(ξ (x) + θ ) sn(ξ (x) + θ ) ], x x x =0 x ρ,ξ x x,θ x x = I 2 x θ = ρ (x) [ sn(ξ (x) + θ ) cos(ξ (x) + θ ) [ ][ ] x 0 1 cos(ξ (x) + θ ) = ρ (x) θ 1 0 sn(ξ (x) + θ ) = R(90 o )(x x ), ].

{ dh H 2 = dt α,u H 2} I n x + α,ω 0, θ α,u,α,ω > 0 W sup { x x j : C C j = 1 },

a = 0.5 unts, b = 0.3 unts Ω 0 o 30 o 0 o 30 o 330 o 330 o 300 o 60 o 300 o 60 o 270 o 90 o 270 o 90 o 240 o 120 o 240 o 120 o 210 o 180 o 150 o 210 o 180 o 150 o

Ω

Ω

r

R R r G c x x j R, j G c l, j l + 1,k 1,k 2,...k l 1, j k 1 k 2... k l 1 j

2r r N 2r = { j I n : V r V r j /0 (non sngleton), j }, I n. G = {V,E} V E G d 2r G 2r d G 2r d G d r 2 1 2r G 2r d 2r r H r 2r

R 2r r 2r R 2r, j G c N N j N N = 1 j

D N N { } D N l = j I n : j, l N, j, I n. k = 0 N 2r N I n 1 j N 2r j D N, I n. N N 2r r N 2r V r N R = 1.5r r 3 2 1 V r

2 1 2 2 2r N N 2r m = 1: N......

N N N D N N = 3 N 1-hop 2-hops 3-hops T-nterval D N T T

k T N 2r 1 2 1 2 2r

2r R = r 1 2 1 2 2r 2r

Ω N N r

V r N 2r { V r G 2r d 4 3 1 2 2r 2r H

r r N E r { } E: = j I n : Vj r C j /0 I: = I n \ E r 2r r

2r

±90 r-lmted Vorono cell area coverage gradent drecton drectons ncreasng coverage δθ m v m = R(( 1) m m /2 δθ) H, m /2δθ π /2, m R( )

2r N G 2r D V r N 2r m: = 0 2r { V r m m + 1 G 2r d N 2r

N 2r N G 2r d 2r D N N D N N D N N 2r N 2r N D N d d 2r d N j,k d N 2r

j k d j q 1 q 2...q m d q m+1 q m+2...q m+p k j q 1 q 2...q m d q m+1 q m+2...q m+p k (a) (b) m, p N d j q 1 q 2 q m d q m+1 q m+2 q m+ p k j,k j j D N k 2r (m + 1) + 1 + (p + 1) N j j / D N j N + 1 m + 2 N + 1 m + p + 3 N m + 2 N + 1 m + p + 3 > N + p + 2 N N + p + 2 p 2 p N m+ p+3 N j m+1

m + 2 m + 1 N + 1 m + 1 N + 1 m + p + 3 > N + p + 3 N N + p + 3 p 3 p N j k d j,k D N N j,k 2r Ω dx = 6.2 unts2 n = 10 r = 0.2unts (R = r) R = 2r N N

2r H 2r N

Ω

B r B: { = j I n : Vj } r Ω /0 Ī r { Ī: = j I n \ B: } Vj r C = /0 Ē: = I n \ { Ī B } r r I n

Ω Ω V r N 2r

N 2r r

20 th 19 th

14 th

47 th 19 th

n 20 th