Lebesgue可測性に関するSoloayの定理と実数の集合の正則性=1This slide is available on ` `%%%`#`&12_`__~~~ౡ氀猀e
|
|
|
- さみ うばら
- 5 years ago
- Views:
Transcription
1 Khomskii Lebesgue Soloay 1 Friday 27 th November This slide is available on 1 / 34
2 Khomskii Khomskii 2 / 34
3 Khomskii Solovay 3 / 34
4 Khomskii Lebesgue Theorem 1 (Vitali) R/Q Lebesgue R/Q R/Q R µ(r) = 0 4 / 34
5 Khomskii DC R [ x y (x R y) = x n n < ω (x n R x n+1 )] R/Q? Solovay-Shelah 5 / 34
6 Khomskii Solovay-Shelah Theorem 2 (Solovay 1970 [5]) ZF + DC + (LM) Theorem 3 (Shelah 1984 [4]) ZF + DC + LM ZFC ZFC Solovay 6 / 34
7 Khomskii Khomskii 7 / 34
8 Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 8 / 34
9 Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 8 / 34
10 Khomskii Definition 3 V On ZFC ω N ω 1 ω 9 / 34
11 Khomskii (cont.) Definition 4 κ 2 κ := P(κ). κ = def α < κ f : α κ γ < κ [sup f < γ]. i.e. κ κ = def λ < κ 2 λ < κ. i.e. κ κ = def κ > ω κ κ κ V κ ZFC ZFC 10 / 34
12 Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 11 / 34
13 Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 11 / 34
14 Khomskii (x y M = x M) ZF Gödel L On M L V M Γ = def φ φ M ω Γ! 0 12 / 34
15 Khomskii HOD ω Solovay HOD ω A A OD ω = def φ(x, y) σ ω On A = { x φ(x, σ) } V Levy φ V = φ V α = φ α On V α A A HOD ω def = trcl({ A }) OD ω, i.e. A OD ω HOD ω 13 / 34
16 Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 14 / 34
17 Khomskii Solovay Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 14 / 34
18 Khomskii V G V[G] On V G V[G] ω 0 15 / 34
19 Khomskii V G V[G] G P On P V G V[G] ω 0 15 / 34
20 Khomskii V G V[G] G P p P V G V[G] On P ω V G V[G] 0 15 / 34
21 Khomskii V G V[G] G P p P V G V[G] p P φ p P G V[G] φ On 0 ω V G V[G] x P-name ẋ V P- P V[G] 15 / 34
22 Khomskii V G V[G] G P p P V G V[G] p P φ p P G V[G] φ On 0 ω V G V[G] x P-name ẋ V P- V[G] V P V[G] 15 / 34
23 Khomskii Levy Col(ω, < κ) Col(ω, < κ) : Levy κ ω 1 On V V ω κ κ ω V 2 ω V 1 ω 0 16 / 34
24 Khomskii Levy Col(ω, < κ) Col(ω, < κ) : Levy κ ω 1 On κ V V[G] V ω κ ω < λ < κ ω λ ω V 2 ω V 1 ω 0 16 / 34
25 Khomskii Levy Col(ω, < κ) Col(ω, < κ) : Levy κ ω 1 On 0 κ = ω V[G] 1 ω V[G] V ω κ ω < λ < κ ω λ V[G] ω κ V[G] κ ω 1 16 / 34
26 Khomskii Theorem 2 (Solovay 1970) κ Col(ω, < κ) HOD ω ZF + DC + LM 17 / 34
27 Khomskii Theorem 2 (Solovay 1970) κ κ ω 1 HOD ω ZF + DC + LM 17 / 34
28 Khomskii Theorem 2 (Solovay 1970) κ κ ω 1 HOD ω Lebesgue Shelah Solovay 17 / 34
29 Khomskii Theorem 2 (Solovay 1970) κ κ ω 1 HOD ω Lebesgue Shelah Solovay V[G] HOD ω Solovay 17 / 34
30 Khomskii Khomskii 18 / 34
31 Khomskii Solovay 1 κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A 19 / 34
32 Khomskii Solovay 1 κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A 19 / 34
33 Khomskii Solovay 1 κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A Col(ω, < κ) 19 / 34
34 Khomskii Solovay κ ω 1 2 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A HOD( ω On) 19 / 34
35 Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } 3 σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A κ σ 19 / 34
36 Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ 4 V[G λ] 5 φ, σ A M Borel B 6 Borel A 19 / 34
37 Khomskii 4. I Definition 4 M x M def = B M [B : M Borel = x / B ]. B M B V[G] Borel k U k c(0) = 0 B c = U c(1), c(0) = 1 B c = ω ω \ B c c(0) = 2 B c = n<ω B (c)n c ω ω c (k) = c(k + 1), (c) n (k) = c( n, k ). Borel 20 / 34
38 Khomskii 4. II N M := V[G λ] Borel N N N V[G] M Borel (2 ℵ 0 ) M λ M κ (2 ℵ 0 ) M < κ V[G] M Borel V[G] N 21 / 34
39 Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] 5 φ, σ A M Borel B 6 Borel A 22 / 34
40 Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] 5 φ, σ A M Borel B 6 Borel A 22 / 34
41 Khomskii Borel Borel MB 23 / 34
42 Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ 23 / 34
43 Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p 23 / 34
44 Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p MB MB M - ṙ 23 / 34
45 Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p MB MB M - ṙ x A M[x] Col(ω,<κ) φ(ṙ, ˇσ) [ ] D M D M MB Col(ω,<κ) φ(ṙ, ˇσ) 23 / 34
46 Khomskii Borel Borel MB MB M MB M Col(ω, < κ) V[G] Col(ω, < κ) <ω µ Col(ω, < κ) V[G] p MB MB M - ṙ x A M[x] Col(ω,<κ) φ(ṙ, ˇσ) [ ] D M D M MB Col(ω,<κ) φ(ṙ, ˇσ) B := { } D D MB M, D MB Col(ω,<κ) φ(ṙ, š) Borel Borel 23 / 34
47 Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] φ, σ A M Borel B 6 Borel A 24 / 34
48 Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] φ, σ A M Borel B 6 Borel A 24 / 34
49 Khomskii Solovay κ ω 1 A HOD( ω On) V[G] φ σ ω On A = { x φ(x, σ) } σ M := V[G λ] λ < κ V[G λ] λ V[G λ] φ, σ A M Borel B Borel A 24 / 34
50 Khomskii Khomskii 25 / 34
51 Khomskii Baire Baire? Khomskii [2] Definition 5 I P(X) σ- 1 A I, B A = B I, 2 { A n I n < ω } n<ω A n I I + := { B B B / I }. B Borel 26 / 34
52 Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A 27 / 34
53 Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
54 Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
55 Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
56 Khomskii Definition 6 (Khomskii 2012) A R I- def = B B \ I C B \ I [C \ B I (C A C A = )]. I- A A C C B 27 / 34
57 Khomskii σ- null := { A R A : Lebesgue }, meager := { A R A : } A A: null- A Baire A: meager- Ramsey 28 / 34
58 Khomskii Khomskii Theorem 7 (Khomskii 2012) I σ-i + (proper) Solovay I- I- 29 / 34
59 Khomskii Future Work I Solovay Wright [6] ZF + DC + BP + LM 30 / 34
60 Khomskii Future Work II Solovay Solovay Shelah Solovay Shelah Khomskii 31 / 34
61 Khomskii I Thomas Jech. Set Theory: The Third Millennium Edition, revised and expanded. Springer Monographs in Mathematics. Springer, Yurii Khomskii. Regularity Properties and Definability in the Real Number Continuum. Idealized forcing, polarized partitions, Hausdorff gaps and mad families in the projective hierarchy. English. PhD thesis. Institute for Logic, Language and Computation, Universiteit van Amsterdam, Ralf Schindler. Set Theory: Exploring Independence and Truth. Universitext. Springer International Publishing Switzerland, isbn: / 34
62 Khomskii II Saharon Shelah. Can You Take Solovay s Inaccessible Away? In: Israel Journal of Mathematics 48.1 (1984), pp issn: doi: /BF url: Robert M. Solovay. A model of set theory in which every set of reals is Lebesgue measurable. In: The Annals of Mathematics. 2nd ser (July 1970), pp issn: X. doi: / url: 33 / 34
63 Khomskii III J. D. Maitland Wright. Functional Analysis for the Practical Man. English. In: Functional analysis: surveys and recent results. Proceedings of the Conference on Functional Analysis. (Paderborn, Germany). Ed. by Klaus-Dieter Bierstedt and Benno Fuchssteiner. Mathematics Studies 27. North-Holland, 1977, pp isbn: / 34
( ) 7 29 ( ) meager (forcing) [12] Sabine Koppelberg 1995 [10] [15], [2], [3] [15] [2] [3] [11]
![( ) 7 29 ( ) meager (forcing) [12] Sabine Koppelberg 1995 [10] [15], [2], [3] [15] [2] [3] [11]](/thumbs/91/106835374.jpg "( ) 7 29 ( ) meager (forcing) [12] Sabine Koppelberg 1995 [10] [15], [2], [3] [15] [2] [3] [11]")
(Sakaé Fuchino) fuchino@isc.chubu.ac.jp 2002 8 24 2002 11 11 2002 11 23 2002 11 29 2003 10 30 footnote 0 2 1 3 1.1...................... 3 1.2....................... 5 1.3.......................... 6 1.4..........................
1980年代半ば，米国中西部のモデル 理論，そして未来－モデル理論賛歌
2016 9 27 RIMS 1 2 3 1983 9-1989 6 University of Illinois at Chicago (UIC) John T Baldwin 1983 9-1989 6 University of Illinois at Chicago (UIC) John T Baldwin Y N Moschovakis, Descriptive Set Theory North
1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) / 25
.. IV 2012 10 4 ( ) 2012 10 4 1 / 25 1. R n Ω ε G ε 0 Ω ε B n 2 Ωε = with Bu = 0 on Ω ε i=1 x 2 i ε +0 B Bu = u (Dirichlet, D Ω ε ), Bu = u ν (Neumann, N Ω ε ), Ω ε G ( ) 2012 10 4 2 / 25 1. Ω ε B ε t
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
第86回日本感染症学会総会学術集会後抄録（I）
κ κ κ κ κ κ μ μ β β β γ α α β β γ α β α α α γ α β β γ μ β β μ μ α ββ β β β β β β β β β β β β β β β β β β γ β μ μ μ μμ μ μ μ μ β β μ μ μ μ μ μ μ μ μ μ μ μ μ μ β
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
III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). T
III III 2010 PART I 1 Definition 1.1 (, σ-),,,, Borel( ),, (σ-) (M, F, µ), (R, B(R)), (C, B(C)) Borel Definition 1.2 (µ-a.e.), (in µ), (in L 1 (µ)). Theorem 1.3 (Lebesgue ) lim n f n = f µ-a.e. g L 1 (µ)
untitled
1 kaiseki1.lec(tex) 19951228 19960131;0204 14;16 26;0329; 0410;0506;22;0603-05;08;20;0707;09;11-22;24-28;30;0807;12-24;27;28; 19970104(σ,F = µ);0212( ); 0429(σ- A n ); 1221( ); 20000529;30(L p ); 20050323(
( 9 1 ) 1 2 1.1................................... 2 1.2................................................. 3 1.3............................................... 4 1.4...........................................
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 ( )
1
1 2 3 4 5 6 7 A.C. Littleton, The Purpose of Accounting Education Proceedings of International Conference on Accounting Education, 1962.10, pp.12~20. 87 2 40 1 144 Robert T. Trueboold, The Future of Accounting
Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2
Exercise in Mathematics IIB IIB (Seiji HIRABA) 0.1, =,,,. n R n, B(a; δ) = B δ (a) or U δ (a) = U(a;, δ) δ-. R n,,,, ;,,, ;,,. (S, O),,,,,,,, 1 C I 2 C II,,,,,,,,,,, 0.2. 1 (Connectivity) 3 2 (Compactness)
I , : ~/math/functional-analysis/functional-analysis-1.tex
I 1 2004 8 16, 2017 4 30 1 : ~/math/functional-analysis/functional-analysis-1.tex 1 3 1.1................................... 3 1.2................................... 3 1.3.....................................
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
(Basic of Proability Theory). (Probability Spacees ad Radom Variables , (Expectatios, Meas) (Weak Law
I (Radom Walks ad Percolatios) 3 4 7 ( -2 ) (Preface),.,,,...,,.,,,,.,.,,.,,. (,.) (Basic of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................
2 (March 13, 2010) N Λ a = i,j=1 x i ( d (a) i,j x j ), Λ h = N i,j=1 x i ( d (h) i,j x j ) B a B h B a = N i,j=1 ν i d (a) i,j, B h = x j N i,j=1 ν i
1. A. M. Turing [18] 60 Turing A. Gierer H. Meinhardt [1] : (GM) ) a t = D a a xx µa + ρ (c a2 h + ρ 0 (0 < x < l, t > 0) h t = D h h xx νh + c ρ a 2 (0 < x < l, t > 0) a x = h x = 0 (x = 0, l) a = a(x,
(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi
I (Basics of Probability Theory ad Radom Walks) 25 4 5 ( 4 ) (Preface),.,,,.,,,...,,.,.,,.,,. (,.) (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios,
Lebesgue Fubini L p Banach, Hilbert Höld
II (Analysis II) Lebesgue (Applications of Lebesgue Integral Theory) 1 (Seiji HIABA) 1 ( ),,, ( ) 1 1 1.1 1 Lebesgue........................ 1 1.2 2 Fubini...................... 2 2 L p 5 2.1 Banach, Hilbert..............................
ε ε x x + ε ε cos(ε) = 1, sin(ε) = ε [6] [5] nonstandard analysis 1974 [4] We shoud add that, to logical positivist, a discussion o
![ε ε x x + ε ε cos(ε) = 1, sin(ε) = ε [6] [5] nonstandard analysis 1974 [4] We shoud add that, to logical positivist, a discussion o](/thumbs/87/95875098.jpg "ε ε x x + ε ε cos(ε) = 1, sin(ε) = ε [6] [5] nonstandard analysis 1974 [4] We shoud add that, to logical positivist, a discussion o")
dif engine 2017/12/08 Math Advent Calendar 2017(https://adventar.org/calendars/2380) 12/8 IST(Internal Set Theory; ) 1 1.1 (nonstandard analysis, NSA) ε ε (a) ε 0. (b) r > 0 ε < r. (a)(b) ε sin(x) d sin(x)
Mathematical Logic I 12 Contents I Zorn
Mathematical Logic I 12 Contents I 2 1 3 1.1............................. 3 1.2.......................... 5 1.3 Zorn.................. 5 2 6 2.1.............................. 6 2.2..............................
page: 1 1 Club guessing sequence , 1). M. Foreman ([ 7 ], [ 8 ] ). 1960,,,, ZFC,,.,,., ZFC,, 20.. G. Cantor.., Zermelo ZFC 2).,., ZFC,., ZFC
![page: 1 1 Club guessing sequence , 1). M. Foreman ([ 7 ], [ 8 ] ). 1960,,,, ZFC,,.,,., ZFC,, 20.. G. Cantor.., Zermelo ZFC 2).,., ZFC,., ZFC](/thumbs/94/118646775.jpg "page: 1 1 Club guessing sequence , 1). M. Foreman ([ 7 ], [ 8 ] ). 1960,,,, ZFC,,.,,., ZFC,, 20.. G. Cantor.., Zermelo ZFC 2).,., ZFC,., ZFC")
page: 1 1 Club guessing sequence 1 1.1 1980, 1). M. Foreman ([ 7 ], [ 8 ] ). 1960,,,, ZFC,,.,,., ZFC,, 20.. G. Cantor.., Zermelo ZFC 2).,., ZFC,., ZFC ZFC, ZFC., K. Gödel [19] L P. Cohen [ 4 ], [ 5 ].,
Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S
![Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S](/thumbs/94/118569863.jpg "Riemann-Stieltjes Poland S. Lojasiewicz [1] An introduction to the theory of real functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,,. Riemann-S")
Riemnn-Stieltjes Polnd S. Lojsiewicz [1] An introduction to the theory of rel functions, John Wiley & Sons, Ltd., Chichester, 1988.,,,, Riemnn-Stieltjes 1 2 2 5 3 6 4 Jordn 13 5 Riemnn-Stieltjes 15 6 Riemnn-Stieltjes
sequentially Cohen Macaulay Herzog Cohen Macaulay 5 unmixed semi-unmixed 2 Semi-unmixed Semi-unmixed G V V (G) V G V G e (G) G F(G) (G) F(G) G dim G G
Semi-unmixed 1 K S K n K[X 1,..., X n ] G G G 2 G V (G) E(G) S G V (G) = {1,..., n} I(G) G S square-free I(G) = (X i X j {i, j} E(G)) I(G) G (edge ideal) 1990 Villarreal [11] S/I(G) Cohen Macaulay G 2005
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 =
COE-RES Discussion Paper Series Center of Excellence Project The Normative Evaluation and Social Choice of Contemporary Economic Systems Graduate Scho
COE-RES Discussion Paper Series Center of Excellence Project The Normative Evaluation and Social Choice of Contemporary Economic Systems Graduate School of Economics and Institute of Economic Research
ルベーグ積分 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.
ルベーグ積分 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. http://www.morikita.co.jp/books/mid/005431 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. Lebesgue 1 2 4 4 1 2 5 6 λ a
(Basics of Proability Theory). (Probability Spacees ad Radom Variables,, (Ω, F, P ),, X,. (Ω, F, P ) (probability space) Ω ( ω Ω ) F ( 2 Ω ) Ω σ (σ-fi
II (Basics of Probability Theory ad Radom Walks) (Preface),.,,,.,,,...,,.,.,,.,,. (Basics of Proability Theory). (Probability Spacees ad Radom Variables...............2, (Expectatios, Meas).............................
.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
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)
- 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
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
I? 3 1 3 1.1?................................. 3 1.2?............................... 3 1.3!................................... 3 2 4 2.1........................................ 4 2.2.......................................
1 (1997) (1997) 1974:Q3 1994:Q3 (i) (ii) ( ) ( ) 1 (iii) ( ( 1999 ) ( ) ( ) 1 ( ) ( 1995,pp ) 1
1 (1997) (1997) 1974:Q3 1994:Q3 (i) (ii) ( ) ( ) 1 (iii) ( ( 1999 ) ( ) ( ) 1 ( ) ( 1995,pp.218 223 ) 1 2 ) (i) (ii) / (iii) ( ) (i ii) 1 2 1 ( ) 3 ( ) 2, 3 Dunning(1979) ( ) 1 2 ( ) ( ) ( ) (,p.218) (
2016 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 16 2 1 () X O 3 (O1) X O, O (O2) O O (O3) O O O X (X, O) O X X (O1), (O2), (O3) (O2) (O3) n (O2) U 1,..., U n O U k O k=1 (O3) U λ O( λ Λ) λ Λ U λ O 0 X 0 (O2) n =
1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b
![1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b](/thumbs/92/108011004.jpg "1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)? (2) () f(x)? b lim a f n (x)dx = b")
1 Introduction 2 2.1 2.2 2.3 3 3.1 3.2 σ- 4 4.1 4.2 5 5.1 5.2 5.3 6 7 8. Fubini,,. 1 1 Introduction 1 (1) (2) (3) () {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a, b] lim f n (x) f(x) (1) f(x)?
1 a b = max{a, b}, a b = mi{a, b} a 1 a 2 a a 1 a = max{a 1,... a }, a 1 a = mi{a 1,... a }. A sup A, if A A A A A sup A sup A = + A if A = ± y = arct
27 6 2 1 2 2 5 3 8 4 13 5 16 6 19 7 23 8 27 N Z = {, ±1, ±2,... }, R =, R + = [, + ), R = [, ], C =. a b = max{a, b}, a b = mi{a, b}, a a, a a. f : X R [a < f < b] = {x X; a < f(x) < b}. X [f] = [f ],
8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a
![8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a](/thumbs/92/108011076.jpg "8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) Carathéodory 10.3 Fubini 1 Introduction 1 (1) (2) {f n (x)} n=1 [a, b] K > 0 n, x f n (x) K < ( ) x [a")
% 100% 1 Introduction 2 (100%) 2.1 2.2 2.3 3 (100%) 3.1 3.2 σ- 4 (100%) 4.1 4.2 5 (100%) 5.1 5.2 5.3 6 (100%) 7 (40%) 8 Fubini (90%) 2007.11.5 1 8.1 Fubini 8.2 Fubini 9 (0%) 10 (50%) 10.1 10.2 Carathéodory
_0212_68<5A66><4EBA><79D1>_<6821><4E86><FF08><30C8><30F3><30DC><306A><3057><FF09>.pdf
Shunsuke Kobayashi 1 [6] [11] [7] u t = D 2 u 1 x 2 + f(u, v) + s L u(t, x)dx, L x (0.L), t > 0, Neumann 0 v t = D 2 v 2 + g(u, v), x (0, L), t > 0. x
![Shunsuke Kobayashi 1 [6] [11] [7] u t = D 2 u 1 x 2 + f(u, v) + s L u(t, x)dx, L x (0.L), t > 0, Neumann 0 v t = D 2 v 2 + g(u, v), x (0, L), t > 0. x](/thumbs/97/133795870.jpg "Shunsuke Kobayashi 1 [6] [11] [7] u t = D 2 u 1 x 2 + f(u, v) + s L u(t, x)dx, L x (0.L), t > 0, Neumann 0 v t = D 2 v 2 + g(u, v), x (0, L), t > 0. x")
Shunsuke Kobayashi [6] [] [7] u t = D 2 u x 2 + fu, v + s L ut, xdx, L x 0.L, t > 0, Neumann 0 v t = D 2 v 2 + gu, v, x 0, L, t > 0. x2 u u v t, 0 = t, L = 0, x x. v t, 0 = t, L = 0.2 x x ut, x R vt, x
2001 年度 『数学基礎 IV』 講義録
4 A 95 96 4 1 n {1, 2,,n} n n σ ( ) 1 2 n σ(1) σ(2) σ(n) σ σ 2 1 n 1 2 {1, 2,,n} n n! n S n σ, τ S n {1, 2,,n} τ σ {1, 2,,n} n τ σ σ, τ τσ σ n σ 1 n σ 1 ( σ σ ) 1 σ = σσ 1 = ι 1 2 n ι 1 2 n 4.1. 4 σ =
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......................
ver F = i f i m r = F r = 0 F = 0 X = Y = Z = 0 (1) δr = (δx, δy, δz) F δw δw = F δr = Xδx + Y δy + Zδz = 0 (2) δr (2) 1 (1) (2 n (X i δx
ver. 1.0 18 6 20 F = f m r = F r = 0 F = 0 X = Y = Z = 0 (1 δr = (δx, δy, δz F δw δw = F δr = Xδx + Y δy + Zδz = 0 (2 δr (2 1 (1 (2 n (X δx + Y δy + Z δz = 0 (3 1 F F = (X, Y, Z δr = (δx, δy, δz S δr δw
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,
3 1 5 1.1........................... 5 1.1.1...................... 5 1.1.2........................ 6 1.1.3........................ 6 1.1.4....................... 6 1.1.5.......................... 7 1.1.6..........................
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)
‚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
I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( )
I (Analysis I) Lebesgue (Lebesgue Integral Theory) 1 (Seiji HIRABA) 1 ( ),,, ( ) 1 (Introduction) 1 1.1... 1 1.2 Riemann Lebesgue... 2 2 (Measurable sets and Measures) 4 2.1 σ-... 4 2.2 Borel... 5 2.3...
Kullback-Leibler
Kullback-Leibler 206 6 6 http://www.math.tohoku.ac.jp/~kuroki/latex/206066kullbackleibler.pdf 0 2 Kullback-Leibler 3. q i.......................... 3.2........... 3.3 Kullback-Leibler.............. 4.4
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
(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t
![(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t](/thumbs/67/56612994.jpg "(2 X Poisso P (λ ϕ X (t = E[e itx ] = k= itk λk e k! e λ = (e it λ k e λ = e eitλ e λ = e λ(eit 1. k! k= 6.7 X N(, 1 ϕ X (t = e 1 2 t2 : Cauchy ϕ X (t")
6 6.1 6.1 (1 Z ( X = e Z, Y = Im Z ( Z = X + iy, i = 1 (2 Z E[ e Z ] < E[ Im Z ] < Z E[Z] = E[e Z] + ie[im Z] 6.2 Z E[Z] E[ Z ] : E[ Z ] < e Z Z, Im Z Z E[Z] α = E[Z], Z = Z Z 1 {Z } E[Z] = α = α [ α ]
SAMA- SUKU-RU Contents p-adic families of Eisenstein series (modular form) Hecke Eisenstein Eisenstein p T
SAMA- SUKU-RU Contents 1. 1 2. 7.1. p-adic families of Eisenstein series 3 2.1. modular form Hecke 3 2.2. Eisenstein 5 2.3. Eisenstein p 7 3. 7.2. The projection to the ordinary part 9 3.1. The ordinary
7... 1... 34... 55... 86... 108 15 2003 1 2 3 1 7 pdf 2 8 1996 2-1 9010 5 247 362 4 7 5 4 11 7 22 4 29 45 5 59 6 61 4 63 10 2 820 3 1646 19 1944 1 2 4 11 3 11 22 4 1 340 2 20 3 13 4 450 1 2 3 4 2-2 29
Tricorn
Triorn 016 3 1 Mandelbrot Triorn Mandelbrot Robert L DevaneyAn introdution to haoti dynamial Systems Addison-Wesley, 1989 Triorn 1 W.D.Crowe, R.Hasson, P.J.Rippon, P.E.D.Strain- Clark, On the struture
1).1-5) - 9 -
- 8 - 1).1-5) - 9 - ε = ε xx 0 0 0 ε xx 0 0 0 ε xx (.1 ) z z 1 z ε = ε xx ε x y 0 - ε x y ε xx 0 0 0 ε zz (. ) 3 xy ) ε xx, ε zz» ε x y (.3 ) ε ij = ε ij ^ (.4 ) 6) xx, xy ε xx = ε xx + i ε xx ε xy = ε
provider_020524_2.PDF
1 1 1 2 2 3 (1) 3 (2) 4 (3) 6 7 7 (1) 8 (2) 21 26 27 27 27 28 31 32 32 36 1 1 2 2 (1) 3 3 4 45 (2) 6 7 5 (3) 6 7 8 (1) ii iii iv 8 * 9 10 11 9 12 10 13 14 15 11 16 17 12 13 18 19 20 (2) 14 21 22 23 24
untitled
0 ( L CONTENTS 0 . sin(-x-sinx, (-x(x, sin(90-xx,(90-xsinx sin(80-xsinx,(80-x-x ( sin{90-(ωφ}(ωφ. :n :m.0 m.0 n tn. 0 n.0 tn ω m :n.0n tn n.0 tn.0 m c ω sinω c ω c tnω ecω sin ω ω sin c ω c ω tn c tn ω
: 2005 ( ρ t +dv j =0 r m m r = e E( r +e r B( r T 208 T = d E j 207 ρ t = = = e t δ( r r (t e r r δ( r r (t e r ( r δ( r r (t dv j =
72 Maxwell. Maxwell e r ( =,,N Maxwell rot E + B t = 0 rot H D t = j dv D = ρ dv B = 0 D = ɛ 0 E H = μ 0 B ρ( r = j( r = N e δ( r r = N e r δ( r r = : 2005 ( 2006.8.22 73 207 ρ t +dv j =0 r m m r = e E(
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
Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull
![Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull](/thumbs/94/118365979.jpg "Feynman Encounter with Mathematics 52, [1] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull")
Feynman Encounter with Mathematics 52, 200 9 [] N. Kumano-go, Feynman path integrals as analysis on path space by time slicing approximation. Bull. Sci. Math. vol. 28 (2004) 97 25. [2] D. Fujiwara and
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
y = x x R = 0. 9, R = σ $ = y x w = x y x x w = x y α ε = + β + x x x y α ε = + β + γ x + x x x x' = / x y' = y/ x y' =
y x = α + β + ε =,, ε V( ε) = E( ε ) = σ α $ $ β w ( 0) σ = w σ σ y α x ε = + β + w w w w ε / w ( w y x α β ) = α$ $ W = yw βwxw $β = W ( W) ( W)( W) w x x w x x y y = = x W y W x y x y xw = y W = w w
x 3 a (mod p) ( ). a, b, m Z a b m a b (mod m) a b m 2.2 (Z/mZ). a = {x x a (mod m)} a Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} a + b = a +
1 1 22 1 x 3 (mod ) 2 2.1 ( )., b, m Z b m b (mod m) b m 2.2 (Z/mZ). = {x x (mod m)} Z m 0, 1... m 1 Z/mZ = {0, 1... m 1} + b = + b, b = b Z/mZ 1 1 Z Q R Z/Z 2.3 ( ). m {x 0, x 1,..., x m 1 } modm 2.4
Anderson ( ) Anderson / 14
Anderson 2008 12 ( ) Anderson 2008 12 1 / 14 Anderson ( ) Anderson 2008 12 2 / 14 Anderson P.W.Anderson 1958 ( ) Anderson 2008 12 3 / 14 Anderson tight binding Anderson tight binding Z d u (x) = V i u
,, Poisson 3 3. t t y,, y n Nµ, σ 2 y i µ + ɛ i ɛ i N0, σ 2 E[y i ] µ * i y i x i y i α + βx i + ɛ i ɛ i N0, σ 2, α, β *3 y i E[y i ] α + βx i
![,, Poisson 3 3. t t y,, y n Nµ, σ 2 y i µ + ɛ i ɛ i N0, σ 2 E[y i ] µ * i y i x i y i α + βx i + ɛ i ɛ i N0, σ 2, α, β *3 y i E[y i ] α + βx i](/thumbs/87/95421301.jpg ",, Poisson 3 3. t t y,, y n Nµ, σ 2 y i µ + ɛ i ɛ i N0, σ 2 E[y i ] µ * i y i x i y i α + βx i + ɛ i ɛ i N0, σ 2, α, β *3 y i E[y i ] α + βx i")
Armitage.? SAS.2 µ, µ 2, µ 3 a, a 2, a 3 a µ + a 2 µ 2 + a 3 µ 3 µ, µ 2, µ 3 µ, µ 2, µ 3 log a, a 2, a 3 a µ + a 2 µ 2 + a 3 µ 3 µ, µ 2, µ 3 * 2 2. y t y y y Poisson y * ,, Poisson 3 3. t t y,, y n Nµ,
JFE.dvi
,, Department of Civil Engineering, Chuo University Kasuga 1-13-27, Bunkyo-ku, Tokyo 112 8551, JAPAN E-mail : atsu1005@kc.chuo-u.ac.jp E-mail : kawa@civil.chuo-u.ac.jp SATO KOGYO CO., LTD. 12-20, Nihonbashi-Honcho
,.,. NP,., ,.,,.,.,,, (PCA)...,,. Tipping and Bishop (1999) PCA. (PPCA)., (Ilin and Raiko, 2010). PPCA EM., , tatsukaw
,.,. NP,.,. 1 1.1.,.,,.,.,,,. 2. 1.1.1 (PCA)...,,. Tipping and Bishop (1999) PCA. (PPCA)., (Ilin and Raiko, 2010). PPCA EM., 152-8552 2-12-1, tatsukawa.m.aa@m.titech.ac.jp, 190-8562 10-3, mirai@ism.ac.jp
特別寄稿 1931 Kurt Gödel, inexhaustibility Jean Cavaillès,
Title < 特別寄稿 > 数学の無尽蔵性 と二つの哲学 -- カヴァイエスとゲーデル -- Author(s) 中村, 大介 Citation 哲学論叢 (2016), 43: 27-39 Issue Date 2016 URL http://hdl.handle.net/2433/219150 Right Type Departmental Bulletin Paper Textversion
BayesfiI‡É“ÅfiK‡È−w‘K‡Ì‡½‡ß‡ÌChow-Liu…A…‰…S…−…Y…•
1 / 21 Kruscal V : w i,j R: w i,j = w j,i i j Kruscal (w i,j 0 ) 1 E {{i, j} i, j V, i i} 2 E {} 3 while(e = ϕ) for w i,j {i, j} E 1 E E\{i, j} 2 G = (V, E {i, j}) = E E {i, j} G {i,j} E w i,j 2 / 21 w
浜松医科大学紀要
On the Statistical Bias Found in the Horse Racing Data (1) Akio NODA Mathematics Abstract: The purpose of the present paper is to report what type of statistical bias the author has found in the horse
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
2 A id A : A A A A id A def = {(a, a) A A a A} 1 { } 1 1 id 1 = α: A B β : B C α β αβ : A C αβ def = {(a, c) A C b B.((a, b) α (b, c) β)} 2.3 α
20 6 18 1 2 2.1 A B α A B α: A B A B Rel(A, B) A B (A B) A B 0 AB A B AB α, β : A B α β α β def (a, b) A B.((a, b) α (a, b) β) 0 AB AB Rel(A, B) 1 2 A id A : A A A A id A def = {(a, a) A A a A} 1 { } 1
Gelfand 3 L 2 () ix M : ϕ(x) ixϕ(x) M : σ(m) = i (λ M) λ (L 2 () ) ( 0 ) L 2 () ϕ, ψ L 2 () ((λ M) ϕ, ψ) ((λ M) ϕ, ψ) = λ ix ϕ(x)ψ(x)dx. λ /(λ ix) ϕ,
A spectral theory of linear operators on Gelfand triplets MI (Institute of Mathematics for Industry, Kyushu University) (Hayato CHIBA) chiba@imi.kyushu-u.ac.jp Dec 2, 20 du dt = Tu. (.) u X T X X T 0 X
<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C602E646F63>
スピントロニクスの基礎 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/077461 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 1 2 ii 3 5 4 AMR (anisotropic magnetoresistance effect) GMR (giant magnetoresistance
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
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)
newmain.dvi
数論 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/008142 このサンプルページの内容は, 第 2 版 1 刷発行当時のものです. Daniel DUVERNEY: THÉORIE DES NOMBRES c Dunod, Paris, 1998, This book is published
2,., ,. 8.,,,..,.,, ,....,..,... 4.,..
Contents 1. 1 2. 2 3. 2 4. 2 5. 3 6. 3 7. 3 8. 4 9. 5 10. 6 11. 8 12. 9 13. - 10 14. 12 15. 13 16. 14 17. 14 18. 15 19. 15 20. 16 21. 16 References 16 1......, 1 2,.,. 4. 2. 2.,. 8.,,,..,.,,... 3....,....,..,...
<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C60202E646F63>
単純適応制御 SAC 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/091961 このサンプルページの内容は, 初版 1 刷発行当時のものです. 1 2 3 4 5 9 10 12 14 15 A B F 6 8 11 13 E 7 C D URL http://www.morikita.co.jp/support
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
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
研修コーナー
l l l l l l l l l l l α α β l µ l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l
i iii 1 ZFC Skolem
i iii 1 ZFC 1 1.1................... 2 1.2........................... 4 1.3 Skolem..................... 6 1.4...................... 8 11 2 13 2.1..................... 13 2.2........ 15 2.3.........................
On the Limited Sample Effect of the Optimum Classifier by Bayesian Approach he Case of Independent Sample Size for Each Class Xuexian HA, etsushi WAKA
Journal Article / 学術雑誌論文 ベイズアプローチによる最適識別系の有限 標本効果に関する考察 : 学習標本の大きさ がクラス間で異なる場合 (< 論文小特集 > パ ターン認識のための学習 : 基礎と応用 On the limited sample effect of bayesian approach : the case of each class 韓, 雪仙 ; 若林, 哲史
,. 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,
Jacobi, Stieltjes, Gauss : :
Jacobi, Stieltjes, Gauss : : 28 2 0 894 T. J. Stieltjes [St94a] Recherches sur les fractions continues Stieltjes 0 f(u)du, z + u f(u) > 0, z C z + + a a 2 z + a 3 +..., a p > 0 (a) Vitali (a) Stieltjes