平成 19 年度 ( 第 29 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 19 ~8 年月 72 月日開催 30 日 ) 1 PCF (Programming language for Computable Functions) PCF adequacy adequacy
|
|
|
- ゆうりゅう こしの
- 5 years ago
- Views:
Transcription
1 1 PCF (Programming language for Computable Functions) PCF adequacy adequacy 2 N X Y X Y f (x) f x f x y z (( f x) y) z = (( f (x))(y))(z) X Y x e X Y λx. e x x 2 + x + 1 λx. x 2 + x PCF 3.1 PCF PCF PCF (call-by-value) PCF (call-by-name) PCF M, N, L PCF
2 PCF ( ) PCF PCF 1. PCF 0, 1, 2, ( PCF ) PCF succ M, pred M M 1 / pred 0 succ (succ 5) 7 PCF succ pred 2 ( ) 2. x, y, z, x 5 pred x 4 3. PCF 0 ifzero L then M else N L 0 M N ifzero (pred x) then M else N x 0 1 M N 4. PCF fun(x : τ). M x M 1 x M 1 M
3 PCF y 7 fun(x : τ). succ y y 7 x succ y ( 8 ) y 8 τ x τ nat 5. (M N) M 1 M 2 M n (( (M 1 M 2 ) ) M n ) N M M N (fun(x : nat). pred x) (succ 3) ( L ) (a) succ 3 4 (b) fun(x : nat). pred x pred V (c) V 4 4 x pred x 3 L 6. PCF rec f (x : τ) : τ. M fun(x : τ). M x M x M M f Dbl rec dbl(x : nat) : nat. ifzero x then 0 else succ (succ (dbl (pred x))) n n 0 0 n 0 (dbl ) Dbl n 1 2 n Dbl n 2n
4 M N M N (= ) (call-by-value) (call-by-name) (Algol, Haskell ) PCF PCF 0, 1, 2, 0, 1, 2, succ M, pred M x ifzero L then M else N fun(x : τ). M M N rec f (x : τ) : τ. M M 1 / x L 0 M N x M M N M f 1: PCF PCF F = fun(x : nat). fun(y : nat). M m, n F m n 1 PCF add = fun(x : nat). rec l(y : nat) : nat. ifzero y then x else succ (l (pred y)) sub = fun(x : nat). rec l(y : nat) : nat. ifzero y then x else pred (l (pred y)) mul = fun(x : nat). rec l(y : nat) : nat. ifzero y then 0 else add x (l (pred y)) PCF ( ) 3.2 PCF L x ( L ) M x L fun(x : τ). M rec f (x : τ) : τ. M rec x(y : τ) : τ. M
5 L x x L fun(x : τ). M x rec f (x : τ) : τ. M f x M FV(M) M FV(M) = M FV(M) M fun(x : nat). succ (succ x) fun(y : nat). succ (succ y) α M x x z M[z/x] ((fun(x : nat). x) x)[z/x] = (fun(x : nat). x) z fun(x : τ). M α fun(z : τ). M[z/x] (z FV(M)) rec f (x : τ) : τ. M α rec f (z : τ) : τ. M[z/x] (z FV(M) { f }) rec x(y : τ) : τ. M α rec z(y : τ) : τ. M[z/x] (z FV(M) {y}) α = α α 2 M = α N M, N Barendregt variable convention (BVC) M 1,, M n M i (1 i n) x x M 1,, M n M 1,, M n ( ) M 1,, M n BVC M 1,, M n M i = α M i (1 i n) BVC 3.3 PCF PCF succ (fun(x : nat). x) fun(x : nat). x 1 1. nat 2. τ τ (τ τ ) τ 1 τ n 1 τ n (τ 1 ( (τ n 1 τ n ) )) 2 R (M, N) R C[ ] (C[M], C[N]) R C[M], C[N] C[ ] M, N
6 PCF nat (nat nat) (nat nat) nat x : τ Γ x : τ Γ x : τ Γ Γ M : τ ( ) Γ M τ : Y 1 Y n X Y 1 Y n X Γ n : nat x : τ Γ Γ x : τ Γ M : nat Γ succ M : nat Γ M : nat Γ pred M : nat Γ, x : τ M : τ Γ M : τ τ Γ N : τ Γ fun(x : τ). M : τ τ Γ M N : τ Γ L : nat Γ M : τ Γ N : τ Γ ifzero L then M else N : τ Γ, f : τ τ, x : τ M : τ Γ rec f (x : τ) : τ. M : τ τ Γ M : τ Γ M : τ (fun(x : nat). x) (succ 3) : nat x : nat x : nat x : nat x : nat fun(x : nat). x : nat nat 3 : nat succ 3 : nat (fun(x : nat). x) (succ 3) : nat PCF 3.1 Γ M τ, τ Γ M : τ Γ M : τ τ = τ 3.2 Γ M : τ τ Γ x Γ, x : τ M : τ
7 x 1 : τ 1,, x n : τ n M : τ N i : τ i (1 i n) M x i N i M[N 1 /x 1,, N n /x n ] M[N 1 /x 1,, N n /x n ] : τ 4 PCF 3.1 PCF 0, 1, 2, fun(x : τ). M τ C τ C τ = {V V : τ V } M V M V n n M n succ M n + 1 L 0 M V ifzero L then M else N V M 0 pred M 0 fun(x : τ). M fun(x : τ). M M n n 1 pred M n 1 L k N V k 0 ifzero L then M else N V M fun(x : τ). M N W M [W/x] V M N V rec f (x : τ) : τ. M fun(x : τ). M[rec f (x : τ) : τ. M/ f ] M V V M V M big-step semantics 1 small-step semantics Dbl 2 Dbl fun(x : nat). ifzero x then 0 else succ (succ (Dbl (pred x))) W n N B n B n = ifzero n then 0 else succ (succ (Dbl (pred n))) 3 BVC BVC α
8 B 0 0 Dbl 2 4 Dbl W Dbl 2 4 Dbl W 2 2 pred Dbl (pred 2) 2 succ (Dbl (pred 2)) 3 Dbl W succ (succ (Dbl (pred 2))) 4 B pred 1 0 Dbl (pred 1) 0 succ (Dbl (pred 1)) 1.. B 0 0 succ (succ (Dbl (pred 1))) 2 B V V V 2. M M V V 3. M V, V M V M V V = V 4. M : τ V M V V : τ M 1. M V V M succ (fun(x : τ). M) 2. M V M Ω = (rec f (x : nat) : nat. f x) Ω 0 : nat Ω error succ M fun(x : τ). M succ M error M error succ M error M M error M M error ( ) M : τ M V V error
9 4 M M ( 2) 5 PCF PCF ( ) 1. τ [[τ]] [[x 1 : τ 1,, x n : τ n ]] = [[τ 1 ]] [[τ n ]] 1 { } 2. Γ M : τ [[M]] [[Γ]] [[τ]] Γ M [[τ]] PCF 5 [[nat]] = N, [[τ τ ]] = [[τ]] [[τ ]] 1. PCF M : nat M 2. rec f (x : nat) : M. : nat nat f M N N f M x N f M (x) = [[M]]( f M, x) (1) P M (N N) (N N) (1) P M = λh. λx. [[M]](h, x) f M = P M ( f M ) 4 ;? 5 PCF rec PCF
10 rec f (x : τ) : M. f M P M N N ( / ) f M ( 2 ) Scott PCF D D D (D, D ) ( ) d D. d D d ( ) d, d, d D. d D d d D d = d D d ( ) d, d D. d D d d D d = d = d 2. ω- (ω-complete partial order ω-cpo) (D, D ) d 0 D d 1 D n N d n 3. ω- (ω-complete pointed partial order ω-cppo) ω-cpo (D, D ) D 6 D ω-cpo (ω-cppo) D D ( D ) ω-cpo 1. X = X (X, = X ) ω-cpo 2. X P(X) ω-cppo d 0 d 1 i N d i 3. { } ω-cppo Sierpinski space O 4. N N { } ω-cppo ω ω-cpo 5.2 D, D ω-cpo 6 ω-cpo predomain ω-cppo domain
11 1. f D D f D x, y D x D y = f (x) D f (y) 2. f D D f D d 0 D d 1 D D f d n = f (d n ) n N D D [D D ] k N ω O x k δ k (x) = x < k n N δ 5.2 ω-cpo D, D ω-cpo D D (d, d ) D D (e, e ) d D e d D e (D D, D D ) ω-cpo D D 5.3 D, D, D ω-cpo 1. f [D D ], g [D D ] λx. ( f (x), g(x)) [D D D ] 2. π D D D, π D D D π(x, y) = x, π (x, y) = y π [D D D], π [D D D ]
12 D, D ω-cpo [D D ] f [D D ] g x D. f (x) D g(x) ([D D ], [D D ]) ω-cpo [D D ] 5.4 D, D, D ω-cpo 1. f [D D D ] λ( f ) D (D D ) f λ( f )(x) = λy. f (x, y) λ( f ) [D [D D ]] 2. ev ((D D ) D) D ev( f, x) = f (x) ev [[D D ] D D ] D ω-cpo D ω-cppo [D D ] ω-cppo D ω-cpo D D { } D x D y x = x D y (D { }, D ) ω-cppo D D D ( ) 5.5 D, D ω-cpo seq [D [D D ] D ] seq(, f ) =, seq( x, f ) = f (x) let e D e x D D let x be e in e seq(e, λx. e ) ω-cppo 5.6 D ω-cppo f [D D]
13 1. D f ( D ) f ( f ( D )) f (n) ( D ) f (n) ( D ) f n N 2. fix : f n N f (n) ( D ) [D D] D ( fix [[D D] D]) 5.4 PCF PCF τ ω-cpo [[τ]] [[nat]] = (N, = N ), [[τ τ ]] = [[[τ]] [[τ ]] ] [[x 1 : τ 1,, x n : τ n ]] = [[τ 1 ]] [[τ n ]] { } Γ M : τ [[M]] [[[Γ]] [[τ]] ] [[τ]] M Γ [[M]] [[τ]] Γ i x i ρ (v 1,, v n ) [[x i ]] = λ(v 1,, v n ). v i (n Γ ) [[n]] = λρ. n [[fun(x : τ). M]] = λρ. λx. [[M]](ρ, x) [[M N]] = λρ. let n be [[N]]ρ in let m be [[M]]ρ in ev(m, n) [[succ M]] = λρ. let m be [[M]]ρ in m m = 0 [[pred M]] = λρ. let m be [[M]]ρ in m 1 m > 0 [[M]]ρ (l = 0) [[ifzero L then M else N]] = λρ. let l be [[L]]ρ in [[N]]ρ (l 0) [[rec f (x : τ) : τ. M]] = λρ. fix(λh. λv. [[M]](ρ, h, v)) 3.1 add : nat nat nat [[add]] [N [N N ] ] [[add]] = λx. fix(h x)
14 h [N [[N N ] [N N ]]] x (y = 0) h = λx. λ f. λy. let z be f (y 1) in z + 1 (y > 0) add 5.7 n, m N [[add n m]] = [[n + m]] = n + m [[add n m]] = fix(h n)(m) = h n (fix(h n)) m fix(h n) h n h n (fix(h n)) m = n + m m m = 0 h n (fix(h n)) 0 = n m > 0 h n (fix(h n)) m = let z be fix(h n) (m 1) in z + 1 fix(h n) (m 1) = n + m 1 h n (fix(h n)) m = n + m = n + m [[add n m]] = [[n + m]] = n + m 6 Adequacy adequacy PCF adequacy 6.1 M : τ M V [[M]] = [[V]] M V [[fun(x : nat). 0]] = [[fun(x : nat). pred 1]] = λx. 0 fun(x : nat). 0 fun(x : nat). pred 1 (= ) nat nat nat 6.1
15 6.2 M : nat n N M n [[M]] = n 6.2 M : nat n N [[M]] = n M n M : τ [[M]] V C τ M V nat V, W [[V]] = [[W]] V = W Γ M : τ M fun(x : τ). M N W M [W/x] V M N V M N M [W/x] Plotkin [6] adequacy relation 7 Adequacy τ R τ [[τ]] C τ (n, m) R nat n = m ( f, fun(x : τ). M) R τ τ (v, V) R τ. ( f (v), M[V/x]) T(R τ ) T(R τ ) [[τ]] {M M : τ } (x, M) T(R τ ) y [[τ]]. x = y = ( V. M V (y, V) R τ ) 6.3 τ M : τ 1. (, M) T(R τ ) 2. d 0 d 1 [[τ]] (d i, M) T(R τ ) ( i N d i, M) T(R τ ) 6.4 (Basic Lemma) x 1 : τ 1,, x n : τ n M : τ ( v i, M i ) T(R τ i ) (1 i n) ([[M]](v 1,, v n ), M[M 1 /x 1,, M n /x n ]) T(R τ ) x 1 : τ 1,, x n : τ n M : τ 6.5 M : nat n N [[M]] = n M n 6.6 (Adequacy) M : nat n N [[M]] = n M n Adequacy M : nat [[M]] = 7 adequacy relation PCF
16 7 Full Abstraction full abstraction M, N : τ τ nat M, N M, N M C[M] : nat M C[ ] N C[N] : nat C[M] n = C[N] n M C[ ] N C[M] M N C[ ] : nat M PCF N M N M N ( 5.1 ) M N M N N M adequacy 7.1 τ M, N : τ [[M]] [[N]] M N 7.2 τ M, N : τ [[M]] = [[N]] M N τ [[τ]] M : τ [[M]] x x [[τ]] parallel or 7.3 por τ [[[[τ]] N ] [[[τ]] N ] [[τ]] N ] 0 f (x) = 0 0 g(x) = 0 por τ ( f, g, x) = 1 n, m > 0. f (x) = n g(x) = m, otherwise Por τ = λ f. λg. λx. por τ ( f, g, x) [[(τ nat) (τ nat) τ nat]] f (x) ( g(x)) g(x) ( f (x)) 0 0 f (x), g(x) 0 1 por f (x) g(x) 0 0 PCF PCF 7.4 τ PCF M : (τ nat) (τ nat) τ nat [[M]] = Por τ
17 7.2 M 1, M 2 : ((nat nat) (nat nat) nat nat) nat M 1 M 2 [[M 1 x]](por nat ) [[M 2 x]](por nat ) 7.1 fully abstract fully abstract fully abstract full abstraction PCF full abstraction PCF por PCF PCF full abstraction [6, 2] PCF por [9, 5, 7] por [4, 1] 1990 PCF fully abstract Milner PCF fully abstract fully abstract PCF full abstraction [8, 3] proof reading Mikhov Rossen [1] S. Abramsky, R. Jagadeesan, and P. Malacaria. Full abstraction for PCF. Inf. Comput., 163(2): , [2] R. Cartwright, P.-L. Curien, and M. Felleisen. Fully abstract semantics for observably sequential languages. Inf. Comput., 111(2): , [3] K. Honda and N. Yoshida. Game-theoretic analysis of call-by-value computation. Theor. Comput. Sci., 221(1-2): , [4] J. Hyland and C.-H. Ong. On full abstraction for PCF: I, II, and III. Inf. Comput., 163(2): , 2000.
18 [5] P. O Hearn and J. Riecke. Kripke logical relations and PCF. Inf. Comput., 120(1): , [6] G. Plotkin. LCF considered as a programming language. Theor. Comput. Sci., 5: , [7] J. Riecke and A. Sandholm. A relational account of call-by-value sequentiality. Inf. Comput., 179(2): , [8] K. Sieber. Relating full abstraction results for different programming languages. In FST and TC 10: Proceedings of the tenth conference on Foundations of software technology and theoretical computer science, pages , New York, NY, USA, Springer-Verlag. [9] K. Sieber. Reasoning about sequential functions via logical relations. In Proc. LMS Symposium on Applications of Categories in Computer Science, LMS Lecture Note Series 177, pages Cambridge University Press, 1992.
I: 2 : 3 +
I: 1 I: 2008 I: 2 : 3 + I: 3, 3700. (ISBN4-00-010352-0) H.P.Barendregt, The lambda calculus: its syntax and semantics, Studies in logic and the foundations of mathematics, v.103, North-Holland, 1984. (ISBN
平成 28 年度 ( 第 38 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 ~8 28 月年 48 日開催月 1 日 semantics FB 1 x, y, z,... FB 1. FB (Boolean) Functional
1 1.1 semantics F 1 x, y, z,... F 1. F 38 2016 9 1 (oolean) Functional 2. T F F 3. P F (not P ) F 4. P 1 P 2 F (P 1 and P 2 ) F 5. x P 1 P 2 F (let x be P 1 in P 2 ) F 6. F syntax F (let x be (T and y)
shift/reset [13] 2 shift / reset shift reset k call/cc reset shift k shift (...) k 1 + shift(fun k -> 2 * (k 3)) k 2 * (1 + 3) 8 reset shift reset (..
![shift/reset [13] 2 shift / reset shift reset k call/cc reset shift k shift (...) k 1 + shift(fun k -> 2 * (k 3)) k 2 * (1 + 3) 8 reset shift reset (..](/thumbs/93/111616220.jpg "shift/reset [13] 2 shift / reset shift reset k call/cc reset shift k shift (...) k 1 + shift(fun k -> 2 * (k 3)) k 2 * (1 + 3) 8 reset shift reset (..")
arisa@pllab.is.ocha.ac.jp asai@is.ocha.ac.jp shift / reset CPS shift / reset CPS CPS 1 [3, 5] goto try/catch raise call/cc [17] control/prompt [8], shift/reset [5] control/prompt, shift/reset call/cc (continuationpassing
Emacs ML let start ::= exp (1) exp ::= (2) fn id exp (3) ::= (4) (5) ::= id (6) const (7) (exp) (8) let val id = exp in
Emacs, {l06050,sasano}@sic.shibaura-it.ac.jp Eclipse Visual Studio Standard ML Haskell Emacs 1 Eclipse Visual Studio variable not found LR(1) let Emacs Emacs Emacs Java Emacs JDEE [3] JDEE Emacs Java 2
Jacques Garrigue
Jacques Garrigue Garrigue 1 Garrigue 2 $ print_lines () > for i in $1; do > echo $i > done $ print_lines "a b c" a b c Garrigue 3 Emacs Lisp (defun print-lines (lines) (dolist (str lines) (insert str)
ML λ λ 1 λ 1.1 λ λ λ e (λ ) ::= x ( ) λx.e (λ ) e 1 e 2 ( ) ML λx.e Objective Caml fun x -> e x e let 1
2005 sumii@ecei.tohoku.ac.jp 2005 6 24 ML λ λ 1 λ 1.1 λ λ λ e (λ ) ::= x ( ) λx.e (λ ) e 1 e 2 ( ) ML λx.e Objective Caml fun x -> e x e let 1 let λ 1 let x = e1 in e2 (λx.e 2 )e 1 e 1 x e 2 λ 3 λx.(λy.e)
64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () m/s : : a) b) kg/m kg/m k
63 3 Section 3.1 g 3.1 3.1: : 64 3 g=9.85 m/s 2 g=9.791 m/s 2 36, km ( ) 1 () 2 () 3 9.8 m/s 2 3.2 3.2: : a) b) 5 15 4 1 1. 1 3 14. 1 3 kg/m 3 2 3.3 1 3 5.8 1 3 kg/m 3 3 2.65 1 3 kg/m 3 4 6 m 3.1. 65 5
( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................
医系の統計入門第 2 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, 第 2 版 1 刷発行時のものです.
医系の統計入門第 2 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/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
28 Horizontal angle correction using straight line detection in an equirectangular image
28 Horizontal angle correction using straight line detection in an equirectangular image 1170283 2017 3 1 2 i Abstract Horizontal angle correction using straight line detection in an equirectangular image
n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m
![n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m](/thumbs/94/118912662.jpg "n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m")
1 1 1 + 1 4 + + 1 n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m a n < ε 1 1. ε = 10 1 N m, n N a m a n < ε = 10 1 N
II (Percolation) ( 3-4 ) 1. [ ],,,,,,,. 2. [ ],.. 3. [ ],. 4. [ ] [ ] G. Grimmett Percolation Springer-Verlag New-York [ ] 3
![II (Percolation) ( 3-4 ) 1. [ ],,,,,,,. 2. [ ],.. 3. [ ],. 4. [ ] [ ] G. Grimmett Percolation Springer-Verlag New-York [ ] 3](/thumbs/94/121840826.jpg "II (Percolation) ( 3-4 ) 1. [ ],,,,,,,. 2. [ ],.. 3. [ ],. 4. [ ] [ ] G. Grimmett Percolation Springer-Verlag New-York [ ] 3")
II (Percolation) 12 9 27 ( 3-4 ) 1 [ ] 2 [ ] 3 [ ] 4 [ ] 1992 5 [ ] G Grimmett Percolation Springer-Verlag New-York 1989 6 [ ] 3 1 3 p H 2 3 2 FKG BK Russo 2 p H = p T (=: p c ) 3 2 Kesten p c =1/2 ( )
‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í
Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I
Optical Flow t t + δt 1 Motion Field 3 3 1) 2) 3) Lucas-Kanade 4) 1 t (x, y) I(x, y, t)
http://wwwieice-hbkborg/ 2 2 4 2 -- 2 4 2010 9 3 3 4-1 Lucas-Kanade 4-2 Mean Shift 3 4-3 2 c 2013 1/(18) http://wwwieice-hbkborg/ 2 2 4 2 -- 2 -- 4 4--1 2010 9 4--1--1 Optical Flow t t + δt 1 Motion Field
untitled
c 645 2 1. GM 1959 Lindsey [1] 1960 Howard [2] Howard 1 25 (Markov Decision Process) 3 3 2 3 +1=25 9 Bellman [3] 1 Bellman 1 k 980 8576 27 1 015 0055 84 4 1977 D Esopo and Lefkowitz [4] 1 (SI) Cover and
LLG-R8.Nisus.pdf
d M d t = γ M H + α M d M d t M γ [ 1/ ( Oe sec) ] α γ γ = gµ B h g g µ B h / π γ g = γ = 1.76 10 [ 7 1/ ( Oe sec) ] α α = λ γ λ λ λ α γ α α H α = γ H ω ω H α α H K K H K / M 1 1 > 0 α 1 M > 0 γ α γ =
cm H.11.3 P.13 2 3-106-
H11.3 H.11.3 P.4-105- cm H.11.3 P.13 2 3-106- 2 H.11.3 P.47 H.11.3 P.27 i vl1 vl2-107- 3 h vl l1 l2 1 2 0 ii H.11.3 P.49 2 iii i 2 vl1 vl2-108- H.11.3 P.50 ii 2 H.11.3 P.52 cm -109- H.11.3 P.44 S S H.11.3
2000年度『数学展望 I』講義録
2000 I I IV I II 2000 I I IV I-IV. i ii 3.10 (http://www.math.nagoya-u.ac.jp/ kanai/) 2000 A....1 B....4 C....10 D....13 E....17 Brouwer A....21 B....26 C....33 D....39 E. Sperner...45 F....48 A....53
微分積分 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, 初版 1 刷発行時のものです.
微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
³ÎΨÏÀ
2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x
![I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x](/thumbs/103/158868201.jpg "I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ) modular symbol., notation. H = { z = x")
I. (CREMONA ) : Cremona [C],., modular form f E f. 1., modular X H 1 (X, Q). modular symbol M-symbol, ( ). 1.1. modular symbol., notation. H = z = x iy C y > 0, cusp H = H Q., Γ = PSL 2 (Z), G Γ [Γ : G]
n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x
三石貴志.indd
流通科学大学論集 - 経済 情報 政策編 - 第 21 巻第 1 号,23-33(2012) SIRMs SIRMs Fuzzy fuzzyapproximate approximatereasoning reasoningusing using Lukasiewicz Łukasiewicz logical Logical operations Operations Takashi Mitsuishi
1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π
![1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π](/thumbs/101/149329485.jpg "1. 4cm 16 cm 4cm 20cm 18 cm L λ(x)=ax [kg/m] A x 4cm A 4cm 12 cm h h Y 0 a G 0.38h a b x r(x) x y = 1 h 0.38h G b h X x r(x) 1 S(x) = πr(x) 2 a,b, h,π")
. 4cm 6 cm 4cm cm 8 cm λ()=a [kg/m] A 4cm A 4cm cm h h Y a G.38h a b () y = h.38h G b h X () S() = π() a,b, h,π V = ρ M = ρv G = M h S() 3 d a,b, h 4 G = 5 h a b a b = 6 ω() s v m θ() m v () θ() ω() dθ()
1. ( ) 1.1 t + t [m]{ü(t + t)} + [c]{ u(t + t)} + [k]{u(t + t)} = {f(t + t)} (1) m ü f c u k u 1.2 Newmark β (1) (2) ( [m] + t ) 2 [c] + β( t)2
![1. ( ) 1.1 t + t [m]{ü(t + t)} + [c]{ u(t + t)} + [k]{u(t + t)} = {f(t + t)} (1) m ü f c u k u 1.2 Newmark β (1) (2) ( [m] + t ) 2 [c] + β( t)2](/thumbs/80/80934335.jpg "1. ( ) 1.1 t + t [m]{ü(t + t)} + [c]{ u(t + t)} + [k]{u(t + t)} = {f(t + t)} (1) m ü f c u k u 1.2 Newmark β (1) (2) ( [m] + t ) 2 [c] + β( t)2")
212 1 6 1. (212.8.14) 1 1.1............................................. 1 1.2 Newmark β....................... 1 1.3.................................... 2 1.4 (212.8.19)..................................
1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1.2 R A 1.3 X : (1)X (2)X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f
1 X X A, B X = A B A B A B X 1.1 R R I I a, b(a < b) I a x b = x I 1. R A 1.3 X : (1)X ()X X (3)X A, B X = A B A B = 1.4 f : X Y X Y ( ) A Y A Y A f 1 (A) f X X f 1 (A) = X f 1 (A) = A a A f f(x) = a x
放射線化学, 92, 39 (2011)
V. M. S. V. 1 Contents of the lecture note by Prof. V. M. Byakov and Dr. S. V. Stepanov (Institute of Theoretical and Experimental Physics, Russia) are described in a series of articles. The first article
T rank A max{rank Q[R Q, J] t-rank T [R T, C \ J] J C} 2 ([1, p.138, Theorem 4.2.5]) A = ( ) Q rank A = min{ρ(j) γ(j) J J C} C, (5) ρ(j) = rank Q[R Q,
![T rank A max{rank Q[R Q, J] t-rank T [R T, C \ J] J C} 2 ([1, p.138, Theorem 4.2.5]) A = ( ) Q rank A = min{ρ(j) γ(j) J J C} C, (5) ρ(j) = rank Q[R Q,](/thumbs/93/114160334.jpg "T rank A max{rank Q[R Q, J] t-rank T [R T, C \ J] J C} 2 ([1, p.138, Theorem 4.2.5]) A = ( ) Q rank A = min{ρ(j) γ(j) J J C} C, (5) ρ(j) = rank Q[R Q,")
(ver. 4:. 2005-07-27) 1 1.1 (mixed matrix) (layered mixed matrix, LM-matrix) m n A = Q T (2m) (m n) ( ) ( ) Q I m Q à = = (1) T diag [t 1,, t m ] T rank à = m rank A (2) 1.2 [ ] B rank [B C] rank B rank
# let rec sigma (f, n) = # if n = 0 then 0 else f n + sigma (f, n-1);; val sigma : (int -> int) * int -> int = <fun> sigma f n ( : * -> * ) sqsum cbsu
II 4 : 2001 11 7 keywords: 1 OCaml OCaml (first-class value) (higher-order function) 1.1 1 2 + 2 2 + + n 2 sqsum 1 3 + 2 3 + + n 3 cbsum # let rec sqsum n = # if n = 0 then 0 else n * n + sqsum (n - 1)
() n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (5) (6 ) n C + nc + 3 nc n nc n (7 ) n C + nc + 3 nc n nc n (
3 n nc k+ k + 3 () n C r n C n r nc r C r + C r ( r n ) () n C + n C + n C + + n C n n (3) n C + n C + n C 4 + n C + n C 3 + n C 5 + (4) n C n n C + n C + n C + + n C n (5) k k n C k n C k (6) n C + nc
Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.
![Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.](/thumbs/93/113887245.jpg "Basic Math. 1 0 [ N Z Q Q c R C] 1, 2, 3,... natural numbers, N Def.(Definition) N (1) 1 N, (2) n N = n +1 N, (3) N (1), (2), n N n N (element). n/ N.")
Basic Mathematics 16 4 16 3-4 (10:40-12:10) 0 1 1 2 2 2 3 (mapping) 5 4 ε-δ (ε-δ Logic) 6 5 (Potency) 9 6 (Equivalence Relation and Order) 13 7 Zorn (Axiom of Choice, Zorn s Lemma) 14 8 (Set and Topology)
1 1 1 1-1 1 1-9 1-3 1-1 13-17 -3 6-4 6 3 3-1 35 3-37 3-3 38 4 4-1 39 4- Fe C TEM 41 4-3 C TEM 44 4-4 Fe TEM 46 4-5 5 4-6 5 5 51 6 5 1 1-1 1991 1,1 multiwall nanotube 1993 singlewall nanotube ( 1,) sp 7.4eV
Int Int 29 print Int fmt tostring 2 2 [19] ML ML [19] ML Emacs Standard ML M M ::= x c λx.m M M let x = M in M end (M) x c λx.
![Int Int 29 print Int fmt tostring 2 2 [19] ML ML [19] ML Emacs Standard ML M M ::= x c λx.m M M let x = M in M end (M) x c λx.](/thumbs/93/111616248.jpg "Int Int 29 print Int fmt tostring 2 2 [19] ML ML [19] ML Emacs Standard ML M M ::= x c λx.m M M let x = M in M end (M) x c λx.")
1, 2 1 m110057@shibaura-it.ac.jp 2 sasano@sic.shibaura-it.ac.jp Eclipse Visual Studio ML Standard ML Emacs 1 ( IDE ) IDE C C++ Java IDE IDE IDE IDE Eclipse Java IDE Java Standard ML 1 print (Int. 1 Int
K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X
![K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X](/thumbs/91/106734137.jpg "K 2 X = 4 MWG(f), X P 2 F, υ 0 : X P 2 2,, {f λ : X λ P 1 } λ Λ NS(X λ ), (υ 0 ) λ : X λ P 2 ( 1) X 6, f λ K X + F, f ( 1), n, n 1 (cf [10]) X, f : X")
2 E 8 1, E 8, [6], II II, E 8, 2, E 8,,, 2 [14],, X/C, f : X P 1 2 3, f, (O), f X NS(X), (O) T ( 1), NS(X), T [15] : MWG(f) NS(X)/T, MWL(f) 0 (T ) NS(X), MWL(f) MWL(f) 0, : {f λ : X λ P 1 } λ Λ NS(X λ
x V x x V x, x V x = x + = x +(x+x )=(x +x)+x = +x = x x = x x = x =x =(+)x =x +x = x +x x = x ( )x = x =x =(+( ))x =x +( )x = x +( )x ( )x = x x x R
V (I) () (4) (II) () (4) V K vector space V vector K scalor K C K R (I) x, y V x + y V () (x + y)+z = x +(y + z) (2) x + y = y + x (3) V x V x + = x (4) x V x + x = x V x x (II) x V, α K αx V () (α + β)x
a n a n ( ) (1) a m a n = a m+n (2) (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 552
3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55 3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n
…J…−†[†E…n…‘†[…hfi¯„^‚ΛžfiüŒå
takuro.onishi@gmail.com II 2009 6 11 [A] D B A B A B A B DVD y = 2x + 5 x = 3 y = 11 x = 5 y = 15. Google Web (2 + 3) 5 25 2 3 5 25 Windows Media Player Media Player (typed lambda calculus) (computer
koba/class/soft-kiso/ 1 λ if λx.λy.y false 0 λ typed λ-calculus λ λ 1.1 λ λ, syntax τ (types) ::= b τ 1 τ 2 τ 1
http://www.kb.ecei.tohoku.ac.jp/ koba/class/soft-kiso/ 1 λ if λx.λy.y false 0 λ typed λ-calculus λ λ 1.1 λ 1.1.1 λ, syntax τ (types) ::= b τ 1 τ 2 τ 1 τ 2 M (terms) ::= c τ x M 1 M 2 λx : τ.m (M 1,M 2
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
「計算と論理」 Software Foundations その3
Software Foundations 3 cal17@fos.kuis.kyoto-u.ac.jp October 24, 2017 ( ) ( 3) October 24, 2017 1 / 47 Lists.v ( ) ( ) ( ) ( 3) October 24, 2017 2 / 47 ( ) Inductive natprod : Type := pair : nat nat natprod.
TOP URL 1
TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy,
変 位 変位とは 物体中のある点が変形後に 別の点に異動したときの位置の変化で あり ベクトル量である 変位には 物体の変形の他に剛体運動 剛体変位 が含まれている 剛体変位 P(x, y, z) 平行移動と回転 P! (x + u, y + v, z + w) Q(x + d x, y + dy, z + dz) Q! (x + d x + u + du, y + dy + v + dv, z +
2 1 1 α = a + bi(a, b R) α (conjugate) α = a bi α (absolute value) α = a 2 + b 2 α (norm) N(α) = a 2 + b 2 = αα = α 2 α (spure) (trace) 1 1. a R aα =
1 1 α = a + bi(a, b R) α (conjugate) α = a bi α (absolute value) α = a + b α (norm) N(α) = a + b = αα = α α (spure) (trace) 1 1. a R aα = aα. α = α 3. α + β = α + β 4. αβ = αβ 5. β 0 6. α = α ( ) α = α
. T ::= x f n t 1 t n F n,m (x(t 1 t n )t 1 t m) x, f n n, F n,m n, m-., F n,m (x(t 1 t n )t 1 t m), x, t 1,..., t n, t 1,..., t m. F n,m (x(t 1 t n )
Kazuki Nakamura Department of Mathematical and Computing Science, Tokyo Institute of Technology * 1 Kashima Ryo Department of Mathematical and Computing Science, Tokyo Institute of Technology 1,,., Σ,..,.
(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
第5章 偏微分方程式の境界値問題
October 5, 2018 1 / 113 4 ( ) 2 / 113 Poisson 5.1 Poisson ( A.7.1) Poisson Poisson 1 (A.6 ) Γ p p N u D Γ D b 5.1.1: = Γ D Γ N 3 / 113 Poisson 5.1.1 d {2, 3} Lipschitz (A.5 ) Γ D Γ N = \ Γ D Γ p Γ N Γ
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P
![Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P](/thumbs/94/118231293.jpg "Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, P")
Milnor 1 ( ), IX,. [KN].,. 2 : (1),. (2). 1 ; 1950, Milnor[M1, M2]. Milnor,,. ([Hil, HM, IO, St] ).,.,,, ( 2 5 )., Milnor ( 4.1)..,,., [CEGS],. Ω m, PC ( 4 5 )., 5, Milnor Milnor., ( 6 )., (I) Z modulo
,. 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 3.,. 4. F. (), ,,. 8.,. 1. (75%) (25%) =7 20, =7 21 (. ). 1.,, (). 3.,. 1. ().,.,.,.,.,. () (12 )., (), 0. 2., 1., 0,.
24(2012) (1 C106) 4 11 (2 C206) 4 12 http://www.math.is.tohoku.ac.jp/~obata,.,,,.. 1. 2. 3. 4. 5. 6. 7.,,. 1., 2007 (). 2. P. G. Hoel, 1995. 3... 1... 2.,,. ii 3.,. 4. F. (),.. 5... 6.. 7.,,. 8.,. 1. (75%)
II
II 2009 1 II Euclid Euclid i 1 1 2 7 3 11 4 18 5 20 6 22 7 26 8 Hausdorff 29 36 1 1 1.1 E n n Euclid E n n R n = {(x 1,..., x n ) x i R} x, y = x 1 y 1 + + x n y n (x, y E n ), x = x, x 1/2 = { (x 1 )
ML Edinburgh LCF ML Curry-Howard ML ( ) ( ) ( ) ( ) 1
More Logic More Types ML/OCaml GADT Jacques Garrigue ( ) Jacques Le Normand (Google) Didier Rémy (INRIA) @garriguejej ocamlgadt ML Edinburgh LCF ML Curry-Howard ML ( ) ( ) ( ) ( ) 1 ( ) ML type nebou and
http://www.ns.kogakuin.ac.jp/~ft13389/lecture/physics1a2b/ pdf I 1 1 1.1 ( ) 1. 30 m µm 2. 20 cm km 3. 10 m 2 cm 2 4. 5 cm 3 km 3 5. 1 6. 1 7. 1 1.2 ( ) 1. 1 m + 10 cm 2. 1 hr + 6400 sec 3. 3.0 10 5 kg
Input image Initialize variables Loop for period of oscillation Update height map Make shade image Change property of image Output image Change time L
1,a) 1,b) 1/f β Generation Method of Animation from Pictures with Natural Flicker Abstract: Some methods to create animation automatically from one picture have been proposed. There is a method that gives
Design of highly accurate formulas for numerical integration in weighted Hardy spaces with the aid of potential theory 1 Ken ichiro Tanaka 1 Ω R m F I = F (t) dt (1.1) Ω m m 1 m = 1 1 Newton-Cotes Gauss
基礎数学I
I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............
Powered by TCPDF ( Title 第 11 講 : フィッシャー統計学 II Sub Title Author 石川, 史郎 (Ishikawa, Shiro) Publisher Publication year 2018 Jtitle コペンハーゲン解
Powered by TCPDF (www.tcpdf.org) Title 第 11 講 : フィッシャー統計学 II Sub Title Author 石川, 史郎 (Ishikawa, Shiro) Publisher Publication year 018 Jtitle コペンハーゲン解釈 ; 量子哲学 (018. 3),p.381-390 Abstract Notes 慶應義塾大学理工学部大学院講義ノート
xia2.dvi
Journal of Differential Equations 96 (992), 70-84 Melnikov method and transversal homoclinic points in the restricted three-body problem Zhihong Xia Department of Mathematics, Harvard University Cambridge,
1. A0 A B A0 A : A1,...,A5 B : B1,...,B
1. A0 A B A0 A : A1,...,A5 B : B1,...,B12 2. 3. 4. 5. A0 A, B Z Z m, n Z m n m, n A m, n B m=n (1) A, B (2) A B = A B = Z/ π : Z Z/ (3) A B Z/ (4) Z/ A, B (5) f : Z Z f(n) = n f = g π g : Z/ Z A, B (6)
IPSJ SIG Technical Report 1, Instrument Separation in Reverberant Environments Using Crystal Microphone Arrays Nobutaka ITO, 1, 2 Yu KITANO, 1
1, 2 1 1 1 Instrument Separation in Reverberant Environments Using Crystal Microphone Arrays Nobutaka ITO, 1, 2 Yu KITANO, 1 Nobutaka ONO 1 and Shigeki SAGAYAMA 1 This paper deals with instrument separation
Akito Tsuboi June 22, T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1
Akito Tsuboi June 22, 2006 1 T ϕ T M M ϕ M M ϕ T ϕ 2 Definition 1 X, Y, Z,... 1 1. X, Y, Z,... 2. A, B (A), (A) (B), (A) (B), (A) (B) Exercise 2 1. (X) (Y ) 2. ((X) (Y )) (Z) 3. (((X) (Y )) (Z)) Exercise
21 2 26 i 1 1 1.1............................ 1 1.2............................ 3 2 9 2.1................... 9 2.2.......... 9 2.3................... 11 2.4....................... 12 3 15 3.1..........
006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................
kokyuroku.dvi
On Applications of Rigorous Computing to Dynamical Systems (Zin ARAI) Department of Mathematics, Kyoto University email: arai@math.kyoto-u.ac.jp 1 [12, 13] Lorenz 2 Lorenz 3 4 2 Lorenz 2.1 Lorenz E. Lorenz
ｎｓｇ０４－２８／ｋｙ２０８６８４３５６１０００４３０７７
δ!!! μ μ μ γ UBE3A Ube3a Ube3a δ !!!! α α α α α α α α α α μ μ α β α β β !!!!!!!! μ! Suncus murinus μ Ω! π μ Ω in vivo! μ μ μ!!! ! in situ! in vivo δ δ !!!!!!!!!! ! in vivo Orexin-Arch Orexin-Arch !!
keisoku01.dvi
2.,, Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 4 Mon, 2006, 401, SAGA, JAPAN Dept. of Mechanical Engineering, Saga Univ., JAPAN 5 Mon, 2006, 401, SAGA, JAPAN Dept.
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
m dv = mg + kv2 dt m dv dt = mg k v v m dv dt = mg + kv2 α = mg k v = α 1 e rt 1 + e rt m dv dt = mg + kv2 dv mg + kv 2 = dt m dv α 2 + v 2 = k m dt d
m v = mg + kv m v = mg k v v m v = mg + kv α = mg k v = α e rt + e rt m v = mg + kv v mg + kv = m v α + v = k m v (v α (v + α = k m ˆ ( v α ˆ αk v = m v + α ln v α v + α = αk m t + C v α v + α = e αk m
2011de.dvi
211 ( 4 2 1. 3 1.1............................... 3 1.2 1- -......................... 13 1.3 2-1 -................... 19 1.4 3- -......................... 29 2. 37 2.1................................ 37
V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V
I (..2) (0 < d < + r < u) X 0, X X = 0 S + ( + r)(x 0 0 S 0 ) () X 0 = 0, P (X 0) =, P (X > 0) > 0 0 H, T () X 0 = 0, X (H) = 0 us 0 ( + r) 0 S 0 = 0 S 0 (u r) X (T ) = 0 ds 0 ( + r) 0 S 0 = 0 S 0 (d r)
main.dvi
SGC - 70 2, 3 23 ɛ-δ 2.12.8 3 2.92.13 4 2 3 1 2.1 2.102.12 [8][14] [1],[2] [4][7] 2 [4] 1 2009 8 1 1 1.1... 1 1.2... 4 1.3 1... 8 1.4 2... 9 1.5... 12 1.6 1... 16 1.7... 18 1.8... 21 1.9... 23 2 27 2.1
inkiso.dvi
Ken Urai May 19, 2004 5 27 date-event uncertainty risk 51 ordering preordering X X X (preordering) reflexivity x X x x transitivity x, y, z X x y y z x z asymmetric x y y x x = y X (ordering) completeness
..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A
![..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A](/thumbs/93/112175219.jpg "..3. Ω, Ω F, P Ω, F, P ). ) F a) A, A,..., A i,... F A i F. b) A F A c F c) Ω F. ) A F A P A),. a) 0 P A) b) P Ω) c) [ ] A, A,..., A i,... F i j A i A")
.. Laplace ). A... i),. ω i i ). {ω,..., ω } Ω,. ii) Ω. Ω. A ) r, A P A) P A) r... ).. Ω {,, 3, 4, 5, 6}. i i 6). A {, 4, 6} P A) P A) 3 6. ).. i, j i, j) ) Ω {i, j) i 6, j 6}., 36. A. A {i, j) i j }.
[I486S] 暗号プロトコル理論
![[I486S] 暗号プロトコル理論](/thumbs/91/105754538.jpg "[I486S] 暗号プロトコル理論")
[I486S] 2018 5 1 (JAIST) 2018 5 1 1 / 22 : I486S I URL:https://wwwjaistacjp/~fujisaki/i486S (Tuesdays) 5 17:10 18:50 4/17, 4/24, 5/1, 5/15, 5/22, 5/29, 6/5, 6/19, 6/26, 7/3, 7/10, 7/17, 7/24, 7/31 (JAIST)
20 15 14.6 15.3 14.9 15.7 16.0 15.7 13.4 14.5 13.7 14.2 10 10 13 16 19 22 1 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 2,500 59,862 56,384 2,000 42,662 44,211 40,639 37,323 1,500 33,408 34,472
- 2 -
- 2 - - 3 - (1) (2) (3) (1) - 4 - ~ - 5 - (2) - 6 - (1) (1) - 7 - - 8 - (i) (ii) (iii) (ii) (iii) (ii) 10 - 9 - (3) - 10 - (3) - 11 - - 12 - (1) - 13 - - 14 - (2) - 15 - - 16 - (3) - 17 - - 18 - (4) -
2 1980 8 4 4 4 4 4 3 4 2 4 4 2 4 6 0 0 6 4 2 4 1 2 2 1 4 4 4 2 3 3 3 4 3 4 4 4 4 2 5 5 2 4 4 4 0 3 3 0 9 10 10 9 1 1
1 1979 6 24 3 4 4 4 4 3 4 4 2 3 4 4 6 0 0 6 2 4 4 4 3 0 0 3 3 3 4 3 2 4 3? 4 3 4 3 4 4 4 4 3 3 4 4 4 4 2 1 1 2 15 4 4 15 0 1 2 1980 8 4 4 4 4 4 3 4 2 4 4 2 4 6 0 0 6 4 2 4 1 2 2 1 4 4 4 2 3 3 3 4 3 4 4
I? 3 1 3 1.1?................................. 3 1.2?............................... 3 1.3!................................... 3 2 4 2.1........................................ 4 2.2.......................................
1 (1) (2)
1 2 (1) (2) (3) 3-78 - 1 (1) (2) - 79 - i) ii) iii) (3) (4) (5) (6) - 80 - (7) (8) (9) (10) 2 (1) (2) (3) (4) i) - 81 - ii) (a) (b) 3 (1) (2) - 82 - - 83 - - 84 - - 85 - - 86 - (1) (2) (3) (4) (5) (6)
[Ver. 0.2] 1 2 3 4 5 6 7 1 1.1 1.2 1.3 1.4 1.5 1 1.1 1 1.2 1. (elasticity) 2. (plasticity) 3. (strength) 4. 5. (toughness) 6. 1 1.2 1. (elasticity) } 1 1.2 2. (plasticity), 1 1.2 3. (strength) a < b F
0.45m1.00m 1.00m 1.00m 0.33m 0.33m 0.33m 0.45m 1.00m 2
24 11 10 24 12 10 30 1 0.45m1.00m 1.00m 1.00m 0.33m 0.33m 0.33m 0.45m 1.00m 2 23% 29% 71% 67% 6% 4% n=1525 n=1137 6% +6% -4% -2% 21% 30% 5% 35% 6% 6% 11% 40% 37% 36 172 166 371 213 226 177 54 382 704 216
10 117 5 1 121841 4 15 12 7 27 12 6 31856 8 21 1983-2 - 321899 12 21656 2 45 9 2 131816 4 91812 11 20 1887 461971 11 3 2 161703 11 13 98 3 16201700-3 - 2 35 6 7 8 9 12 13 12 481973 12 2 571982 161703 11
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1
QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann
1/2 ( ) 1 * 1 2/3 *2 up charm top -1/3 down strange bottom 6 (ν e, ν µ, ν τ ) -1 (e) (µ) (τ) 6 ( 2 ) 6 6 I II III u d ν e e c s ν µ µ t b ν τ τ (2a) (
August 26, 2005 1 1 1.1...................................... 1 1.2......................... 4 1.3....................... 5 1.4.............. 7 1.5.................... 8 1.6 GIM..........................
1 1.1 / Fik Γ= D n x / Newton Γ= µ vx y / Fouie Q = κ T x 1. fx, tdx t x x + dx f t = D f x 1 fx, t = 1 exp x 4πDt 4Dt lim fx, t =δx 3 t + dxfx, t = 1
1 1.1......... 1............. 1.3... 1.4......... 1.5.............. 1.6................ Bownian Motion.1.......... Einstein.............. 3.3 Einstein........ 3.4..... 3.5 Langevin Eq.... 3.6................
外国語学部 紀要３０号（横書）／０３＿菊地俊一
Information Technology: IT e-learning e-japan e-japan e-japan IT IT IT IT IT IT e-japan e-japan e-japan e-japan e-japan e-japan IT e-japan e-japan e-japan e-japan IIe-Japan e-japan II e-japan IT e-japan
gr09.dvi
.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 {
JKR Point loading of an elastic half-space 2 3 Pressure applied to a circular region Boussinesq, n =
JKR 17 9 15 1 Point loading of an elastic half-space Pressure applied to a circular region 4.1 Boussinesq, n = 1.............................. 4. Hertz, n = 1.................................. 6 4 Hertz
2 4 202 9 202 9 6 3................................................... 3.2................................................ 4.3......................................... 6.4.......................................
I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )
I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17
平成 30 年度 ( 第 40 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 30 ~8 年月 72 月日開催 30 日 [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1,
![平成 30 年度 ( 第 40 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 30 ~8 年月 72 月日開催 30 日 [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1,](/thumbs/94/121497492.jpg "平成 30 年度 ( 第 40 回 ) 数学入門公開講座テキスト ( 京都大学数理解析研究所, 平成 30 ~8 年月 72 月日開催 30 日 [6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1,")
[6] 1 4 A 1 A 2 A 3 l P 3 P 2 P 1 B 1 B 2 B 3 m 1 l 3 A 1, A 2, A 3 m 3 B 1, B 2, B 3 A i 1 B i+1 A i+1 B i 1 P i i = 1, 2, 3 3 3 P 1, P 2, P 3 1 *1 19 3 27 B 2 P m l (*) l P P l m m 1 P l m + m *1 A N
1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji
8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi