橡CompSimmAllcpct.PDF
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- こうた あいしま
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1 3 1
2
3 M
4
5
6
7 dx 1 dx/ 1/
8 x P 0 (x) x dx x P 0 (x) x+dx P 0 (x+dx) x dx P 0 (x+dx)=p 0 (x)(1-dx/ dx/)
9 x P 0 (x) P 0 (x+dx)=p 0 (x)(1-dx/ dx/) Tayler (dx) P 0 (x)+(dp 0 /dx)dx (dp 0 (x)/dx)dx= -P 0 (x) dx/ dx/ 1/P 0 (x) (dp 0 (x)/dx)= -1/ 1/ 1/P 0 (x) dp 0 (x)= -1/ 1/ dx logp 0 (x)=-x/ x/+const. P 0 (x)=e -x/ x/ x=
10 x x+dx f (x)dx x dx f (x)dx=p 0 (x) dx/ dx/=e -x/ x/ / dx f (x)=e -x/ x/ / [ ] x / / λ x λ e / λdx = e 0 = 0 ( 1) = 1 0 λ x / x e dx λ = λ 0
11 = y=e -x =1/ y=e -x/ x/ y=e / 1/ =1 y=e -x =1 =1/ y=e -x/ y=e / x/ / = 1
12 [ ] λ λ λ λ λ / / 0 / 0 / 1 1) ( / x x x x x x e e e dx e = = = = =1 =1/ 1/ y x y x x y e y x 1 log log / log / λ λ λ λ = = = = /λ 1 x e y =
13 y = 1 e x /λ
14 y = 1 e x /λ y = e x/ λ log y = x / λ x = λlog y 1 x = λlog y
15 rnd() 0 1 t ( t ) double exp_rnd(int t){ return t*(log(1/rnd()));} ta tb exp_rnd(ta) exp_rnd(tb)
16
17 exp_rnd(ta). ( exp_rnd(tb)) a ( exp_rnd(ta)). 3b.
18 queue.c 1 #include <stdio.h> #include <math.h> 3 #include <stdlib.h> 4 #define NofClerk 3 5 enum event_type {arival,finish}; 6 struct event_node { 7 long time; 8 enum event_type type; 9 int clerk_id; 10 struct event_node *next; 11 }; 1 long mean_arival_interval; 13 long mean_operation_interval; 14 long quit_time; 15 int clerk_busy[nofclerk]; 16 int nof_person_in_queue; 17 long current_time; 18 struct event_node *event_root; 19 void event_add(long interval,enum event_type type,int clerk_id); 0 long exp_rnd(long mean_interval); BSD
19 queue.c main 1 int main(int argc,char **argv){ int i; 3 struct event_node *rm_event; 4 if(argc!=4){ 5 printf("usage: queue arival_interval operation_interval quit_time n"); 6 return 1; 7 } 8 sscanf(argv[1],"%d",&mean_arival_interval); 9 sscanf(argv[],"%d",&mean_operation_interval); 30 sscanf(argv[3],"%d",&quit_time); 31 for(i=0;i<nofclerk;i++) clerk_busy[i]=0; 3 nof_person_in_queue=1; 33 current_time=0; 34 event_root=null; 35 event_add(exp_rnd(mean_arival_interval),arival,0);. 66 }
20 36 for(;;){ 37 if(current_time>quit_time)return 0; 44 for(;;){ 45 if(!nof_person_in_queue) break; 46 for(i=0;i<nofclerk;i++){ 47 if(!clerk_busy[i]) break; 48 } 49 if(i==nofclerk) break; 50 clerk_busy[i]++; 51 nof_person_in_queue--; 5 event_add(exp_rnd(mean_operation_interval),finish,i); 53 } 54 rm_event=event_root; 55 event_root=rm_event->next; 56 current_time=rm_event->time; 57 if(rm_event->type==arival){ 58 nof_person_in_queue++; 59 event_add(exp_rnd(mean_arival_interval),arival,0); 60 } 61 else if(rm_event->type==finish) 6 clerk_busy[rm_event->clerk_id]=0; 63 else ; 64 free(rm_event); 65 } queue.c main 38 /* */ 39 printf("time=%3d nof_p=%3d", current_time,nof_person_in_queue); 40 for(i=0;i<nofclerk;i++) 41 printf(" busy[%d]=%d",i,clerk_busy[i]); 4 printf(" n"); 43 /* */
21 67 void event_add(long interval,enum event_type type,int clerk_id){ 68 struct event_node *new_event; 69 struct event_node *event; 70 new_event=(struct event_node *)malloc(sizeof(struct event_node)); 71 new_event->time=current_time+interval; 7 new_event->type=type; 73 new_event->clerk_id=clerk_id; 74 if(event_root==null){ 75 new_event->next=null; 76 event_root=new_event; 77 return; 78 } 79 if(event_root->time>new_event->time){ 80 new_event->next=event_root; 81 event_root=new_event; 8 return; 83 } 84 for(event=event_root;;event=event->next){ 85 if(event->next==null){ 86 new_event->next=null; 87 event->next=new_event; 88 return; 89 } 90 if(event->next->time>new_event->time){ 91 new_event->next=event->next; 9 event->next=new_event; 93 return; 94 } 95 } 96 } queue.c event_add
22 queue.c exp_rnd 97 long exp_rnd(long mean_interval){ 98 double r; 99 r=(double)mean_interval*(log((double)rand_max/(double)random())); 100 return (long)r; 101 } random()0 RAND_MAX
23 v1 e v e3 e1 e5 e4 e6 e7 v3 e8 v4
24 G V E E V G=( =(V, E,), G=( =(V, E) E 1 V
25 e v3 v1 e1 e5 e4 e8 v e6 v4 e7 e3
26 (1/6) e=(v i,v j ) e v i v j v i,v j e e=(v i,v j ) v i e v j v
27 (/6)
28 (3/6) (n-1) (v 1,v ),(v,v 3 ),...,(v n-1,v n ) (v 1,v ),(v,v 3 ),...,(v n-1,v 1 ) (v 1,v ),(v,v 3 ),...,(v n-1,v n )
29 (4/6)
30 (5/6) 0
31 (6/6) G G 1 A G=( =(V, E)V A G
32 G=( =(V, E) V V 1,V V 1 V v1 e v e3 V 1 ={v 1,v 3 } V ={v,v 4 } e1 v3 e5 e4 e8 e6 v4 e7 E 1 ={e,e 4,e 5,e 8 } C(E 1 )=(V 1,V )
33 G 1 =(V 1,E 1 )G =(V,E ) E 1 E 1 G 1 G G 1 G G 1 G 11
34
35 G=( =(V,E) G G
36
37 G G G G G
38
39 4
40 I 1 I I 3 1 I 4 4 3
41 (1/) (m) (NP ) (?) (NP ) (NP ) 4 (NP )
42 (/) (m ) (NP ) (m ) G (NP ) (NP ) (NP )
43 P NP P NP
44 NP NP NP NP NP NP NP
45 ( ) ( )
46
47 1 1
48
49 OS
50 A B readera writer readerb
51
52 1 M 1 M M M 1 M i,m i+1,m i+,...,m j M j M i M R(M)
53 k- k 1-
54 M,...,M i,...,m j
55
56
57 n< +n= +n= -n= n=
58
59 p 1 t (1,0,0,0) t 1 (0,1,1,0) t 3 t 1 t 3 (0,0,0,1) (1,1,0,0) t 4 p t p 3 t 4 (1,0,0,0) t (1,,0,0) t 1 (0,,1,0) p 4 (0,,0,1) t 3 (1,,0,0) t 4 (1,,0,0)
60 »»
61 »»
62 state machine
63 marked graph
64 NG OK OK NG
65 (1) 1 1 free choice net
66 () fork join select
67 (1) 1 simple Petri net
68 ()
69 Petri Petri
70
71 1
72
73 M M' M' R(M)
74 (1) M M' M''
75 () M' M'' M M'
76 (3) 1 1 M M M
77 (1) M' M M' MM' M'
78 (1) M M 3
79 M M' M' M
80 1 1
81 1
82
83 1
84 Dijkstra 0,1,,... V +1 P P -1 V 0 V
85
86 not
87
88 » What How T. DeMarco
89 DFD: Data Flow Diagram CASE: Computer Assisted Software Engineering : :
90 DFD DFD DFD 4
91 TOP DFD
92 DFD
93
94 DFD
95 TOP MiniSpec
96 DFD MiniSpec DD(Data Dictionary) (a b): aa b a+b: aa b {a}: aa 0 [a]: aa
97 D. J. Hatley DFDCFD(Control flow Diagram DFD CFD MiniSpec ControlSpec DD TD (Timing Dictionary) Petri net
98 (State Transition Diagram) DFD DFD DFD STD
99 Mealey
100 Moore
101 Moore
102
103
104
105 (symbol) symbol) a,b,c,...,... (alphabet) 0,1 {0,1}
106 01101 {0,1} {0,1} a ai a i
107 0 (concatenation) =abc abc=ca =abcca abcca =
108 (reverse) =
109 =0 abc =3
110 * {} * ={ ={a,b} *={ *={,a,b,aa,bb,ab,ba,aaa,...} * + * +
111 L 1 1 L { L 1 L } L 1 L L 1 L
112 (1) L * (closure) L L 0 ={ } n1 n L n =LL n-1 L L * (n0 ) L n
113 () L * L L L * L * L L +
114
115
116 (rewriting rule) UWUW UW»» ( )OK
117 UWUW ( ) ()
118 Chomsky
119 Chomsky V N : V T : V N V T a,b,...,... A,B,...
120 Chomsky (V N V T ) * V N (V N V T ) *» (V N V T ) *» (V N V T ) * V N (V N V T ) * (V N V T ) * ( )
121 Chomsky G 4 G = (V( N, V T, P, S ) V N V T P» V = (V( N V T ) P V*V N V* S V N
122 V N ={S} V T ={0,1} P={ ={S01S, S1} S S01S3 S
123 G = (V( N, V T, P, S ) S uw P uw uw V = (V( N V T ) V* G L(G) G
124 G uw G uw a G* a m a G a 1, a 1 G a,..., a m-1 G a m a G* a k Gk
125 L(G)={ )={ V T* S} G* G L(G) 1... n 1... n L(G)={(01) n 1 n 0}
126 3 (1) 3 1 1
127 0 () 0 Chomsky 4
128
129 3 A,B ABA ABA 3
130 V N ={S} V T ={0,1} P={ ={S01S, S1} S S01S3 S L(G)={(01) n 1 n 0}
131 A ( ) A A 3
132 V N ={S} (1) V T ={+,*,a} P={ ={S+SS * *SS a} S*SS*S+SS*a+aa a + *
133 () V N ={A,S} V T ={0,1} P={ ={S0A1, A0A1, A } S0A100A11000A L(G)={0 n 1 n n 1}
134 ,,1
135 V N ={A, B, S } V T ={a, b, c } P={ ={SaSAB abb, BA AB, ba bb, bb bc, cb cc} SaSABaaSABABaaabBABAB aaabbaabb aaabbaabbaaabababb aaabaabbb aaabaabbbaaabbabbb aaabbbbbb aaabbbbbbaaabbbcbb aaabbbccb aaabbbccbaaabbbccc L(G)={a n b n c n n 1}
136 0 OK 0 3,,1 0
137 (3)
138
139 ... a a a b b b c c c d d d... a 1 b 3 1 1
140 1 Last-In In-First-Out(LIFO) Turing
141 1»
142 ... h i a d c k... a b b b c c c
143 Turing ( ) Turing Turing
144 Fork
145 Turing
146 ( ) {} a a {a} s,t S,T s+t ST st ST (( ) s* S* (( )
147 01 {01} (0+1) {0,1} (0+1)(0+1)=(0+1) {00,01,10,11} (0+1)* {0,1}*={ {0,1}*={,0,1,00,01,10,11,...} 1(0+1)*1 {11,101,111,1001,1011,1101,..} ={0,1} ={0,1} (0+1)
148 ,, = (0+1)*= (0*1*)* (0+1)* {0,1}* (0*1*)* {{0}*{1}*}*={ {{0}*{1}*}*={,0,1,00,01,...}* L L* L
149 s S S R ={x R x S} S} s R R = a a R =a u,v» s=u+v s R R +v R» s=uv s R =v R u R» s=u* s R =(u R )*
150 sed hello hello [.?!].?! [A-Za-z] [^0-9] [hh]ello hello Hello. ^hello hello (bye )* bye 0 bye$ bye ^$ bye *bye bye.*.*bye bye..
151 Entiti-Relation Model: ERM
152 Entiti-Action Model Michael A. Jackson
153 A B C D (A=BCD) (E=F+G+H) (J=K*) E F o G o H o J K*
154 A B C D E F o G o H o J K*
155 * o o
156 BDD BDD: Binary Decision Diagram SROBDD: Shared Reduced Ordered BDD ZBDD: Zero-suppressed BDD ( )
157 (decision tree) 5g 90 50g160 5g110 50g190 5g130 50g30
158 5g 50g 50g 100g 50g 500g 1kg kg 5g 50g 50g 100g 50g 500g 1kg kg 5g 50g 50g 100g 50g 500g 1kg kg
159
160
161 (decision table) decision table) 1 3
162
163 1 3 don't care
164
165 Binary Decision Diagram ( ) Binary Decision Tree Binary Decision Diagram
166 BDD (1) 5g 90 50g 90 5g 50g 5g 50g
167 BDD () 5g yes 90 no yes no yes no yes no
168 BDD (3) 90 no yes 5g no yes no no yes yes
169 Shared Reduced Ordered BDD BDD
170 n y n y n y n y
171 SROBDD ( BDD ) ~ ~ ~ 1 3 n y
172 ZBDD n y no
173 M
174 GF(q) q m p ( ) p { 0,1,, L, p 1} p mod p
175 ( ) x ( )1 1 i ( ) + F ( F F ) GF( p) GF( p) m GF ( p 1 ( ) m )
176 m F(x) GF ( m p ) F(x) α ( ) m GF ( p ) α m 0 1 p α ( = 1), α, α, L, α m ( p 1) α p m 1 = 1 m GF ( p )
177 α m F(x) F( α) = 0 α m G(α ) m 1 α m = G(α) m m GF( p ) i α α m 1 i m 1 α = a0 + a1α + L + am 1α, a,, ) ( 0 1 a m 1 a L ( 0 1 a m 1 a, a, L, ) m GF( p )
178 GF() x --x 0 1 x 1/x GF ( m ) α + α = ( 1+ 1) α = 0 α = α x + x 4
179 (1) b b 1 F ( x) = f 0 + f1x + L + fb 1x + x b α GF( b ) α α
180 () ) ( a0, a1, L, a b 1 GF( b ) a = a 0 + aα + L + 1 a b 1 α b 1 α α
181 (3) (3) b a b a a a α α α α = L b α 0 ) ( = = b b f b f f F α α α α L = b b b f f f α α α L ) ( ) ( = b b b b b b f a a f a a f a a α α α L a α ),,, ( b b b b b f a a f a a f a L
182 a α ),,, ( b b b b b f a a f a a f a L ),,, ( b b f f f f a a a L L L L L L L L L L a ),,, ( a b a a L
183 f f a 0 a 1 a b-1 f 0 f f 1 f b-1 M
184 1+x+x 4 M 1+x+x 4 =1+1x+0 +0x +0x 3 +1x 4 a 0 a 1 a a 3 a 0 a 1 a a 3
185 x 1,x,x 3,... x k x k < c c
186 XX< x F (x)x F (x) f (x)=f F ' (x)
187 1 f(x)=f'(x) F(x) 0 1
188 rand() U 0 U<1 U=(1.0 / (M + 1.0) (rand() + 0.5) 0 rand() M=99 99 M U=(1/100) (rand()+0.5) U01
189 F(x) X U0 U<1 X=F - 1 (U) 1 0 1
190 (pseudo random number)
191 (pseudo random number) Knuth Wichmann-Hill Hill M Mersenne Twister
192 x i = (a( a x i-1 + c) ) mod M, i=1,,3... x 0 0 x i < M M a a mod 8 = 5 c M
193 Knuth T E X
194 M=8, a=5, c=1, x 0 =0 x i = (5x i-1 +1) mod 8 0,1,6,7,4,5,,3,0,...
195 Mersenne Twister 1996~1997 ^ C rand() 64 OK
196 MT GF() (=1), 1,, 3,..., ^ ^ ^ ,,3,...
197 M -1 M,3,5,7,13,17,19,31,61,89, 107,17,51,607,179,03,81,317,45 3,443, 9689,9941,1113, ,1701,
198 MT» ^
199 Wichmann-Hill Hill 16 x 0 = x i = x i-1 mod x i /
200 rand() ANSI CC x i =( x i ) mod 31 31
201 Knuth x 55 +x 4 +1 M
202 0 d p(x,t) t x x = md (m ) t = nt (n ) p(0,0)=1 p(x,0)=0 (x( 0) p(x,t+t)=1/p(x-d,t)+1/p(x+d,t)
203 d d t d x p t x p d t x p t d x p T d T t x p T t x p ), ( ), ( ), ( ), ( ), ( ), ( + = + )), ( ), ( ( )), ( ), ( ( ), ( ), ( t d x p t x p t x p t d x p t x p T t x p + = +
204 d d t d x p t x p d t x p t d x p T d T t x p T t x p ), ( ), ( ), ( ), ( ), ( ), ( + = + T d, 0 d T ), ( t x p ), ( ), ( x t x p T d t t x p = ) (,0) ( x x p δ =
205 d d t d x p t x p d t x p t d x p T d T t x p T t x p ), ( ), ( ), ( ), ( ), ( ), ( + = + ), ( t x p md x = nt t = T d ), ( ), ( x t x p T d t t x p = ) (,0) ( x x p δ =
206 ), ( ), ( x t x p T d t t x p = ), ( ), ( x t x p D t t x p = D > 0 t ) (,0) ( x x p δ = Dt x e Dt t x p ), ( = π
207 µ σ N( µ, σ ) ( x µ ) ( σ 1 f x) = e πσ x 1 p( x, t) = e 4Dt 4πDt Dt = σ
208 (1) T d 1/ n + n - nt md p(md, nt) + n n = + n n + n = m p( md, nt ) n n + 1 n n! 1 p ( md, nt ) = nc + n ( ) = ( ) + + n!( n n )! n
209 () 1 n! 1 p ( md, nt ) ) n!( n n )! n n = nc + n ( ) = ( + + n + + n = n n + n = m p( md, nt ) = n + n!! n 1 ( )! n = n ( n! + m n )!( 1 ( ) m )! n
210 (3) (3) n n n ne n n π! ) log( )log (!) log( 1 1 π + + n n n n n m e n nt md p ), ( = π
211 (1) (1) Dt x e Dt t x p ), ( = π n m e n nt md p ), ( = π md x = nt t = T d D = n m nt T d d m e nd e nt T d nt md p ), ( = = π π
212 () m p( md, nt) = e n πn d m 1 p( md, nt ) = e n πnd
213 (3) m 1 p( md, nt ) = e n πnd p n m 1 md d md d
214 (1) -l n l x 1 n (x,t) v th l 1/ l l τc = v th
215 () -l A B l A 0 l n( x) dx n τc x B n( x) dx l 0
216 (3) -l A B l n x τ c 0 n( x) dx l 0 l n( x) dx
217 (4) n(x) -l A n B l x 0 n( x) dx l 0 l n( x) dx 1/ 1 dn dx l
218 (5) A B 1 dn dx l τ c -l l l dn τ c dx l τ c D
219 (6) x x +x n( x, t) D n td( ) x t x+ x n td ( ) x x n n n x t = td ( ) x+ x ( ) x t x x
220 (7) (7) x x +x = + x x x x n x n td t t n x ) ( ) ( ) ( ) ( x n D x x n x n D t n x x x = = +
221 Langevin (1) Langevin m 1 (x) m d dt x = X f dx dt 0 f
222 Langevin Langevin () () dt dx f X dt x d m = ) ( ) ( dt x d x dt dx dt dx dt dx x dt d dt dx dt d dt x d + = = = ) ( ) ) (( dt x d mx dt dx m dt x d x dt dx m dt x m d + = + = x dt dx f X dt dx m dt x m d ) ( ) ( + =
223 Langevin Langevin (3) (3) x dt dx f X dt dx m dt x m d ) ( ) ( + = dt dx f Xx dt dx fx Xx dt dx m dt x m d 1 ) ( = = dt dx f Xx dt dx m dt x m d ) ( =
224 Langevin Langevin (4) (4) dt dx f Xx dt dx m dt x m d ) ( = dt x d f Xx dt dx m dt x d m > < > >=< < > < ) ( ( ) x X = >= >< >=< < x X Xx dt x d f dt dx m dt x d m > < >= < > < ) (
225 Langevin (5) m d < x > dx f d < x > m < ( ) >= dt dt dt d < x > = u dt m du dx f m < ( ) >= dt dt u du dt dx f < ( ) >= u dt m du dt + f m u = < ( dx dt ) >
226 1 kt pv = nrt p: V: n: T: R: k k: 1
227 Langevin (6) du dt + f m u = < ( dx dt ) > dx 1 1 m < ( ) kt dt >= du f + u = kt dt m m
228 Langevin (7) du f + u = kt du f = u + kt dt m m dt m m 1 du f u + kt m m = dt 1 dx = log ax + ax+ b a 1 b m log u + kt = t + f f m m C f m log u + m kt = f m t + C
229 Langevin (8) f m log u + m kt = f m t + C C e f t m = f m u + m kt m f u = C e f t m + f kt Langevin
230 Einstein u = C e f t m + f kt u d < x > dt Langevin f t m e < x >= ktt f Dt = σ D = kt f Dt =σ =< x >= ktt f
231 f (1) (1) c l D τ = f kt D = > < = ) ( dt dx m kt kt dt dx m 1 ) ( 1 >= < f c c c c c c c f ml f v mv f mv l D τ τ τ τ τ τ τ = = = = c c c f ml l τ τ τ = c f m τ = 1 c m f τ =
232 f () d x Langevin m dt = X f dx dt f = m τ c d x m dx m = X dt dt τ c d dt x 1 dx dv 1 = X = X v m dt dt m τ c τ c
233 f (3) X dv dt = 1 m X τ c v dv 0 dt 1 0 = X m τc v 1 m X = τc v Xτ c = vm
橡試験直前配布.PDF
d d/ / P 0 d P 0 +d P 0 +d d P 0 +d=p 0 -d/ d/ P 0 P 0 +d=p 0 -d/ d/ Tayler d P 0 +dp 0 /dd dp 0 /dd= -P 0 d/ d/ /P 0 dp 0 /d= -/ / /P 0 dp 0 = -/ / d logp 0 =-/ /+cons. P 0 =e -/ / =0 0 0 +d f d d f d=p
(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
入試の軌跡
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C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 新装版 1 刷発行時のものです.
C による数値計算法入門 ( 第 2 版 ) 新装版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009383 このサンプルページの内容は, 新装版 1 刷発行時のものです. i 2 22 2 13 ( ) 2 (1) ANSI (2) 2 (3) Web http://www.morikita.co.jp/books/mid/009383
A B P (A B) = P (A)P (B) (3) A B A B P (B A) A B A B P (A B) = P (B A)P (A) (4) P (B A) = P (A B) P (A) (5) P (A B) P (B A) P (A B) A B P
1 1.1 (population) (sample) (event) (trial) Ω () 1 1 Ω 1.2 P 1. A A P (A) 0 1 0 P (A) 1 (1) 2. P 1 P 0 1 6 1 1 6 0 3. A B P (A B) = P (A) + P (B) (2) A B A B A 1 B 2 A B 1 2 1 2 1 1 2 2 3 1.3 A B P (A
Part () () Γ Part ,
Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
x h = (b a)/n [x i, x i+1 ] = [a+i h, a+ (i + 1) h] A(x i ) A(x i ) = h 2 {f(x i) + f(x i+1 ) = h {f(a + i h) + f(a + (i + 1) h), (2) 2 a b n A(x i )
![x h = (b a)/n [x i, x i+1 ] = [a+i h, a+ (i + 1) h] A(x i ) A(x i ) = h 2 {f(x i) + f(x i+1 ) = h {f(a + i h) + f(a + (i + 1) h), (2) 2 a b n A(x i )](/thumbs/92/107866678.jpg "x h = (b a)/n [x i, x i+1 ] = [a+i h, a+ (i + 1) h] A(x i ) A(x i ) = h 2 {f(x i) + f(x i+1 ) = h {f(a + i h) + f(a + (i + 1) h), (2) 2 a b n A(x i )")
1 f(x) a b f(x)dx = n A(x i ) (1) ix [a, b] n i A(x i ) x i 1 f(x) [a, b] n h = (b a)/n y h = (b-a)/n y = f (x) h h a a+h a+2h a+(n-1)h b x 1: 1 x h = (b a)/n [x i, x i+1 ] = [a+i h, a+ (i + 1) h] A(x
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
, 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 =
x n 1 1.,,.,. 2..... 4 = 2 2 12 = 2 2 3 6 = 2 3 14 = 2 7 8 = 2 2 2 15 = 3 5 9 = 3 3 16 = 2 2 2 2 10 = 2 5 18 = 2 3 3 2, 3, 5, 7, 11, 13, 17, 19.,, 2,.,.,.,?.,,. 1 , 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x
x () g(x) = f(t) dt f(x), F (x) 3x () g(x) g (x) f(x), F (x) (3) h(x) = x 3x tf(t) dt.9 = {(x, y) ; x, y, x + y } f(x, y) = xy( x y). h (x) f(x), F (x
[ ] IC. f(x) = e x () f(x) f (x) () lim f(x) lim f(x) x + x (3) lim f(x) lim f(x) x + x (4) y = f(x) ( ) ( s46). < a < () a () lim a log xdx a log xdx ( ) n (3) lim log k log n n n k=.3 z = log(x + y ),
koji07-01.dvi
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35-8585 7 8 1 I I 1 1.1 6kg 1m P σ σ P 1 l l λ λ l 1.m 1 6kg 1.1m 1.m.1m.1 l λ ϵ λ l + λ l l l dl dl + dλ ϵ dλ dl dl + dλ dl dl 3 1. JIS 1 6kg 1% 66kg 1 13 σ a1 σ m σ a1 σ m σ m σ a1 f f σ a1 σ a1 σ m
,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.
9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,
II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx
i B5 7.8. p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x
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.1, θ, ϕ d = A, t dt + B, t dtd + C, t d + D, t dθ +in θdϕ.1.1 t { = f1,t t = f,t { D, t = B, t =.1. t A, tdt e φ,t dt, C, td e λ,t d.1.3,t, t d = e φ,t dt + e λ,t d + dθ +in θdϕ.1.4 { = f1,t t = f,t {
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TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
r 1 m A r/m i) t ii) m i) t B(t; m) ( B(t; m) = A 1 + r ) mt m ii) B(t; m) ( B(t; m) = A 1 + r ) mt m { ( = A 1 + r ) m } rt r m n = m r m n B
1 1.1 1 r 1 m A r/m i) t ii) m i) t Bt; m) Bt; m) = A 1 + r ) mt m ii) Bt; m) Bt; m) = A 1 + r ) mt m { = A 1 + r ) m } rt r m n = m r m n Bt; m) Aert e lim 1 + 1 n 1.1) n!1 n) e a 1, a 2, a 3,... {a n
1. (8) (1) (x + y) + (x + y) = 0 () (x + y ) 5xy = 0 (3) (x y + 3y 3 ) (x 3 + xy ) = 0 (4) x tan y x y + x = 0 (5) x = y + x + y (6) = x + y 1 x y 3 (
1 1.1 (1) (1 + x) + (1 + y) = 0 () x + y = 0 (3) xy = x (4) x(y + 3) + y(y + 3) = 0 (5) (a + y ) = x ax a (6) x y 1 + y x 1 = 0 (7) cos x + sin x cos y = 0 (8) = tan y tan x (9) = (y 1) tan x (10) (1 +
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ad bc A A A = ad bc ( d ) b c a n A n A n A A det A A ( ) a b A = c d det A = ad bc σ {,,,, n} {,,, } {,,, } {,,, } ( ) σ = σ() = σ() = n sign σ sign(
I n n A AX = I, YA = I () n XY A () X = IX = (YA)X = Y(AX) = YI = Y X Y () XY A A AB AB BA (AB)(B A ) = A(BB )A = AA = I (BA)(A B ) = B(AA )B = BB = I (AB) = B A (BA) = A B A B A = B = 5 5 A B AB BA A
熊本県数学問題正解
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() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y
![() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y](/thumbs/85/92979039.jpg "() (, y) E(, y) () E(, y) (3) q ( ) () E(, y) = k q q (, y) () E(, y) = k r r (3).3 [.7 ] f y = f y () f(, y) = y () f(, y) = tan y y ( ) () f y = f y")
5. [. ] z = f(, y) () z = 3 4 y + y + 3y () z = y (3) z = sin( y) (4) z = cos y (5) z = 4y (6) z = tan y (7) z = log( + y ) (8) z = tan y + + y ( ) () z = 3 8y + y z y = 4 + + 6y () z = y z y = (3) z =
1 return main() { main main C 1 戻り値の型 関数名 引数 関数ブロックをあらわす中括弧 main() 関数の定義 int main(void){ printf("hello World!!\n"); return 0; 戻り値 1: main() 2.2 C main
C 2007 5 29 C 1 11 2 2.1 main() 1 FORTRAN C main() main main() main() 1 return 1 1 return main() { main main C 1 戻り値の型 関数名 引数 関数ブロックをあらわす中括弧 main() 関数の定義 int main(void){ printf("hello World!!\n"); return
1. 2 P 2 (x, y) 2 x y (0, 0) R 2 = {(x, y) x, y R} x, y R P = (x, y) O = (0, 0) OP ( ) OP x x, y y ( ) x v = y ( ) x 2 1 v = P = (x, y) y ( x y ) 2 (x
. P (, (0, 0 R {(,, R}, R P (, O (0, 0 OP OP, v v P (, ( (, (, { R, R} v (, (, (,, z 3 w z R 3,, z R z n R n.,..., n R n n w, t w ( z z Ke Words:. A P 3 0 B P 0 a. A P b B P 3. A π/90 B a + b c π/ 3. +
II Time-stamp: <05/09/30 17:14:06 waki> ii
II waki@cc.hirosaki-u.ac.jp 18 1 30 II Time-stamp: ii 1 1 1.1.................................................. 1 1.2................................................... 3 1.3..................................................
No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
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240-8501 79-2 Email: nakamoto@ynu.ac.jp 1 3 1.1...................................... 3 1.2?................................. 6 1.3..................................... 8 1.4.......................................
4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
1 c Koichi Suga, ISBN
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A(6, 13) B(1, 1) 65 y C 2 A(2, 1) B( 3, 2) C 66 x + 2y 1 = 0 2 A(1, 1) B(3, 0) P 67 3 A(3, 3) B(1, 2) C(4, 0) (1) ABC G (2) 3 A B C P 6
1 1 1.1 64 A6, 1) B1, 1) 65 C A, 1) B, ) C 66 + 1 = 0 A1, 1) B, 0) P 67 A, ) B1, ) C4, 0) 1) ABC G ) A B C P 64 A 1, 1) B, ) AB AB = 1) + 1) A 1, 1) 1 B, ) 1 65 66 65 C0, k) 66 1 p, p) 1 1 A B AB A 67
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(3) (2),,. ( 20) ( s200103) 0.7 x C,, x 2 + y 2 + ax = 0 a.. D,. D, y C, C (x, y) (y 0) C m. (2) D y = y(x) (x ± y 0), (x, y) D, m, m = 1., D. (x 2 y
[ ] 7 0.1 2 2 + y = t sin t IC ( 9) ( s090101) 0.2 y = d2 y 2, y = x 3 y + y 2 = 0 (2) y + 2y 3y = e 2x 0.3 1 ( y ) = f x C u = y x ( 15) ( s150102) [ ] y/x du x = Cexp f(u) u (2) x y = xey/x ( 16) ( s160101)
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7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±
![7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±](/thumbs/91/106462075.jpg "7 π L int = gψ(x)ψ(x)φ(x) + (7.4) [ ] p ψ N = n (7.5) π (π +,π 0,π ) ψ (σ, σ, σ )ψ ( A) σ τ ( L int = gψψφ g N τ ) N π * ) (7.6) π π = (π, π, π ) π ±")
7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2
![6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2](/thumbs/92/109118076.jpg "6 2 T γ T B (6.4) (6.1) [( d nm + 3 ] 2 nt B )a 3 + nt B da 3 = 0 (6.9) na 3 = T B V 3/2 = T B V γ 1 = const. or T B a 2 = const. (6.10) H 2 = 8π kc2")
1 6 6.1 (??) (P = ρ rad /3) ρ rad T 4 d(ρv ) + PdV = 0 (6.1) dρ rad ρ rad + 4 da a = 0 (6.2) dt T + da a = 0 T 1 a (6.3) ( ) n ρ m = n (m + 12 ) m v2 = n (m + 32 ) T, P = nt (6.4) (6.1) d [(nm + 32 ] )a
) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4
![) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4](/thumbs/92/107866707.jpg ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4")
1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
2011de.dvi
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