Numerical Examples

In this section, we show availability of the proposed method by solving the some numerical examples. The proposed method is implemented in C++ with Boost1.52.0, CRlibm1.0beta4, MPFR, and gcc4.6.3. In this method, if the num-ber of ε exceeds 300, we reduce 50 εs by (3.2.7). We use AWA-Package[25]

implemented in PascalXSC by Lohner as comparison. In the following, we de-notes:

Method AAWA

Method BProposed Method

The experimental environment consists is below: CPU: Intel Core i7 2.7GHz, memory: 4GB, OS: Fedora16 64bit. We choose step size and degree of the Taylor expansion such that they seem to be the best for each example and each method.

Example 1

We consider the following Van der Pol equations:

dx₀

dt = x₁ (5.29)

dx1

dt = e(1−x²₀)x₁−x₀ (5.30)

e=0.01, x₀(0) = [1,1.01], x₁(0) = [1,1.01]. (5.31)

Let step size and degree of the Taylor expansion be Table 5.1. Figure 5.2 is a single logarithmic chart and it shows increasing widths of (x0). Table 5.2 shows execution time.

Table 5.1: Parameters

Method A Method B

Degree of Taylor Expansion 18 10

Step Size auto (average0.272) 0.125

Figure 5.2: Comparison of error bounds

Table 5.2: Computation Time (sec) Method A Method B t_e=250 1.41 0.18 t_e=500 2.77 0.39 t_e=750 4.29 0.56 t_e=1000 5.89 0.75

Example 2

We consider the following Lorenz equations:

dx₀

dt = −px0+px1 (5.32)

dx₁

dt = −x₀x₂+rx₀−x₁ (5.33) dx₂

dt = x₀x₁−bx₂ (5.34)

p=10, r=28, b=8

3 (5.35)

x0(0) =15, x1(0) =15, x2(0) =36 (5.36)

Let step size and degree of the Taylor expansion be Table 5.3. Figure 5.3 is a single logarithmic chart and it shows increasing widths of (x0). Table 5.4 shows execution time.

Table 5.3: Parameters

Method A Method B

Degree of Taylor Expansion 16 22

Step Size auto (average0.01411) 0.03125

Figure 5.3: Comparison of error bounds

Table 5.4: Computation Time (sec) t Method A Method B

t_e=5 0.9 0.07

t_e=10 1.79 0.13

t_e=15 2.71 0.21

t_e=20 3.67 0.28

Chapter 6

Method to Find All Solutions

In this chapter, we introduce a method to find all solutions using Krawczyk’s method [4, 5].

6.1 Krawczyk’s Method

For f :Rⁿ→Rⁿ, I∈IRⁿ,c(center ofI),R f(c)⁻¹(non-singular matrix), E (unit matrix), and Krawczyk operatorK(I)defined as follows:

K(I) =c−R f(c) + (E−RF(I))(I−c) (6.1) F(I)⊃ {f(x)|x∈I}, (6.2)

If the conditions:

K(I) ⊂ I (6.3)

E−RF(I) < 1 (6.4)

hold, we have follows:

(1) a unique solution of the equation f(x) =0 exists inI.

(2) For arbitrary pointx₀inI, point sequence{x_k}defined as follows:

x_k+1=x_k−R f(x_k),

x_k convergesx^∗and rate of convergence is as follows:

x_k−x^∗ ≤ αⁿ

1−αR f(x₀). (6.5)

Here,α=E−RF(I).

Thus, we can examine the existence of solutions by Krawczyk operator.

6.2 Method to Find All Solutions using Krawczyk’s Method

In this section, we introduce a method to find all solutions of f(x) =0 using 6.1. To begin with, we introduce the existence test of solutions and the non-existence test of solutions.

Non-Existence Test For f(I), if

0∈/ f(I) (6.6)

holds, there is no solution inI.

Existence Test

For Krawczyk operatorK(I), if both

K(I) ⊂ I (6.7)

||E−RF(I)|| < 1 (6.8)

hold, there is a unique solution inI. Besides, ifK(I)∩I= /0holds, there is no solution inI.

Then, we introduce a method to find all solutions. It finds all solutions by the repetition of the non-existence test, the existence test, and division of the search area. The algorithm of this method is as follows:

Algorithm 6.1 This algorithm gives the all solutions of f(x) =0. Let interval vectorI0be search area.

Step-1 Initialize interval vector list

L

^as

L

={I0}.

Step-2 If

L

is empty, terminate. Otherwise letI be the last element of

L

^{, then}

remove the last element.

Step-3 Calculate f(I). If 0∈ f(I)holds, there is no solution; go toStep-2. Oth-erwise gotoStep-4.

Step-4 Calculate

K(I) =c−R f(c) + (E−RF(I))(I−c). (6.9)

If both

K(I) ⊂I (6.10)

E−RF(I) < 1 (6.11)

hold, there is a unique solution in I; go to Step-2for searching remaining areas. Otherwise, gotoStep-5.

Step-5 IfK(I)∩I= /0holds, there is no solutions; go toStep-2. Otherwise, goto Step-6.

Step-6 DivideI into two areasI₁, I₂ by the side which has the maximum width.

AddI1,I2to the end of

L

. Then go toStep-2.

Chapter 7

Method to Find All Periodic

Solutions of ODEs

In this chapter, we explain a method to find all periodic solutions of ODEs.

This method is combination of our proposed method to solve the initial value prob-lems mentioned in Chapter 5 and the method to find all solutions using Krawczyk’s method mentioned in Chapter 6.

7.1 Find All Periodic Solutions

We consider all periodic solutions of the following ODEs:

dx(t)

dt = f(x(t),t), f :Rⁿ×R→Rⁿ.

Here, f(x,t) = f(x,t+2π). For initial valuex(0) =v, functiong2π which returns the value at 2π, and functionh_2π:

h2π(v) =g2π(v)−v, (7.1)

we can obtain all periodic solutions by finding all solutions of the following equa-tions:

h_2π(v) =0. (7.2)

Here,g_2π(v)is calculated by our proposed method mentioned in Chapter 5. In the Krawczyk’s method, we need differential values ofh_2π below:

h_2π(v) =g_2π(v)−E, whereEis unit matrix.

For 0=t₀<t₁<t₂< ... <t_m=t_2π, we can obtaing_2π(v)as follows:

g_2π(v) =^m−1

∏

i=0

φ_ti,ti+1(x(t_i)).

7.2 Numerical Examples

In this section, we find all periodic solutions of some numerical examples. The experimental environment is the same as Section 5.6.

Example 1

We consider to find all periodic solutions of the following ODEs:

dx₀

dt = x₁ dx1

dt = −0.1x₁−x⁵₀+0.2 cos(t) +0.05

Let search area, step size, and degree of Taylor be Table 7.1. We find all periodic solutions of a period of 2π, 4π, and 6π. Table 7.2 shows all periodic solutions of 2π and Table 7.3 shows number of trials of non-existence test, existence test and calculation time. Table 7.4 shows all periodic solutions of 4π and Table 7.5 shows number of trials of non-existence test, existence test and calculation time.

Table 7.6 shows all periodic solutions of 6πand Table 7.7 shows number of trials of non-existence test, existence test and calculation time.

Table 7.1: Parameters

Search Area x0=[-1,1],x1=[-1,1]

Degree of Taylor Expansion 20

Step size 0.125

Table 7.2: 2πPeriodic Solutions Solution 1 x0 [0.85912, 0.85914]

x₁ [0.94242, 0.94244]

Solution 2 x0 [0.159, 0.160]

x₁ [0.033, 0.034]

Solution 3 x0 [-0.8491, -0.8490]

x₁ [0.67683, 0.67686]

Table 7.3: Results (2π)

Non-Existence Test Existence Test Calculation Time (sec)

6953 1286 319.75

Table 7.4: 4πPeriodic Solutions Solution 1 x0 [0.8591, 0.8592]

x₁ [0.9424, 0.9425]

Solution 2 x₀ [0.7315, 0.7316]

x₁ [0.1332, 0.1333]

Solution 3 x₀ [0.0749, 0.0750]

x₁ [0.2548, 0.2550]

Solution 4 x₀ [0.1600, 0.1601]

x₁ [0.0337, 0.0338]

Solution 5 x₀ [0.0137, 0.0146]

x1 [0.1899, 0.1903]

Solution 6 x₀ [-0.850, -0.849]

x1 [0.6768, 0.6769]

Solution 7 x₀ [-0.761, -0.760]

x1 [0.0873, 0.0874]

Solution 8 x₀ [-0.244, -0.243]

x1 [-0.100, -0.099]

Solution 9 x₀ [-0.644, -0.643]

x1 [-0.140, -0.139]

Table 7.5: Results (4π)

Non-Existence Test Existence Test Calculation Time (sec)

101833 6245 8875.25

Table 7.6: 6πPeriodic Solutions Solution 1 x0 [0.8591, 0.8592]

x₁ [0.9424, 0.9425]

Solution 2 x0 [0.5922, 0.5923]

x₁ [0.3161, 0.3162]

Solution 3 x0 [0.5133, 0.5134]

x₁ [0.3212, 0.3213]

Solution 4 x₀ [0.1600, 0.1601]

x₁ [0.0337, 0.0338]

Solution 5 x₀ [-0.736, -0.735]

x₁ [0.5338, 0.5339]

Solution 6 x₀ [-0.850, -0.849]

x1 [0.6768, 0.6769]

Solution 7 x₀ [-0.755, -0.754]

x1 [0.5289, 0.5290]

Solution 8 x₀ [-0.792, -0.791]

x1 [0.5289, 0.5290]

Solution 9 x₀ [-0.775, -0.774]

x1 [-0.201, -0.200]

Table 7.7: Results (6π)

Non-Existence Test Existence Test Calculation Time (sec)

703364 37651 91636.5

Example 2

We consider to find all periodic solutions of the following Duffing equations:

dx₀

dt = x1

dx₁

dt = −0.1x₁−x³₀+Bcos(t)

Let search area, step size, and degree of Taylor be Table 7.8. We find all periodic solutions of a period of 2π. Table 7.9 shows all periodic solutions of B=0.1 and Table 7.10 shows number of trials of non-existence test, existence test and calculation time. Table 7.11 shows all periodic solutions of B=0.3 and Table 7.12 shows number of trials of non-existence test, existence test and calculation time. Table 7.13 shows all periodic solutions of B=1.5 and Table 7.14 shows number of trials of non-existence test, existence test and calculation time. Table 7.15 shows all periodic solutions of B =2.5 and Table 7.16 shows number of trials of non-existence test, existence test and calculation time. Table 7.17 shows all periodic solutions of B=3.5 and Table 7.18 shows number of trials of non-existence test, non-existence test and calculation time.

Table 7.8: Parameters

Search Area x0=[-5,5],x1=[-5,5]

Degree of Taylor Expansion 20

Step size 0.0625

Table 7.9: 2πPeriodic Solution (B=0.1) Solution 1 x₀ [-0.1006, -0.0976]

x1 [0.00985, 0.01031]

Table 7.10: Results (B=0.1)

Non-Existence Test Existence Test Calculation Time (sec)

370053 23103 19877.1

Table 7.11: 2πPeriodic Solutions (B=0.3) Solution 1 x0 [1.1381, 1.1382]

x₁ [0.7445, 0.7447]

Solution 2 x0 [-0.3253, -0.3202]

x₁ [0.03394, 0.03797]

Solution 3 x₀ [-0.9171, -0.9168]

x₁ [0.38111, 0.38124]

Table 7.12: Results (B=0.3)

Non-Existence Test Existence Test Calculation Time (sec)

369901 22942 18553.7

Table 7.13: 2πPeriodic Solution (B=1.5) Solution 1 x₀ [1.6889, 1.6890]

x1 [0.35295, 0.35300]

Table 7.14: Results (B=1.5)

Non-Existence Test Existence Test Calculation Time (sec)

364513 21467 18830.8

Table 7.15: 2πPeriodic Solutions (B=2.5) Solution 1 x₀ [1.5364, 1.5365]

x₁ [1.8866, 1.8867]

Solution 2 x₀ [1.9396, 1.9397]

x₁ [0.3263, 0.3264]

Solution 3 x₀ [1.7727, 1.7728]

x1 [-1.4127, -1.4126]

Table 7.16: Results (B=2.5)

Non-Existence Test Existence Test Calculation Time (sec)

360704 22184 17788.9

Table 7.17: 2πPeriodic Solutions (B=3.5) Solution 1 x₀ [0.6217, 0.6218]

x₁ [3.4790, 3.4791]

Solution 2 x₀ [0.5017, 0.5018]

x1 [2.8491, 2.8492]

Solution 3 x₀ [2.3812, 2.3813]

x1 [2.0884, 2.0885]

Solution 4 x₀ [2.1564, 2.1565]

x1 [0.3341, 0.3342]

Solution 5 x₀ [1.1295, 1.1296]

x1 [-0.546, -0.545]

Solution 6 x0 [0.5599, 0.5600]

x1 [-2.094, -2.093]

Solution 7 x0 [0.9175, 0.9176]

x1 [-2.546, -2.545]

Table 7.18: Results (B=3.5)

Non-Existence Test Existence Test Calculation Time (sec)

353874 21390 17738.67

Chapter 8

Conclusion

We proposed a high-speed and high-accuracy method to solve IVPs of ODEs and implemented this method in this paper. Moreover, we found all periodic so-lutions by using above method.

In Chapter 5, we introduced and implemented our proposed method to solve IVPs of ODEs by using mean value form and affine arithmetic. In addition, we showed availability of our proposed method by solving some numerical examples and comparing with AWA-Package implemented by Lohner. Automated adjust-ment of step size, more speeding up, and higher accuracy are challenges for the future.

In Chapter 7, we explain a method that is combination of our proposed method to solve the IVPs of ODEs and the method to find all solutions using Krawczyk’s method. Furthermore, we found all periodic solutions of some numerical exam-ples. Applying to ODEs which does not have forced oscillation terms is a chal-lenge for the future.

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ݚڀۀ੷

(Research Achievements)

म࢜࿦จ(Master’s thesis)

1. ദ໦ܒҰ࿠: “ΞϑΟϯϕΩڃ਺ԋࢉΛར༻ͨ͠ৗඍ෼ํఔࣜͷਫ਼

౓อূ”,ૣҴాେֶ, 2008.

࿦จ(Papers)

1. ദ໦ܒҰ࿠, ٶౡ৴໵, ദ໦խӳ,ʠGPUΛར༻ͨ͠ඇઢܗํఔࣜ

ͷฒྻશղ୳ࡧ๏ ʡ, ೔ຊԠ༻਺ཧֶձ࿦จࢽ, vol. 18, no. 3, pp.

347-362, 2008.

2. ദ໦ܒҰ࿠,ദ໦խӳ,ʠ ฏۉ஋ܗࣜͱΞϑΟϯԋࢉΛ༻͍ͨৗඍ

෼ํఔࣜͷਫ਼౓อূ๏ ʡ,೔ຊԠ༻਺ཧֶձ࿦จࢽ, vol. 21, no. 1, pp. 37-58, 2011.

3. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "ϒϩάۭؒͷ৘ใ఻೻ωοτϫʔ ΫಛੑͷఆྔԽ",ਓ޻஌ೳֶձ࿦จࢽ, vol. 25, no. 3, pp. 404-409, 2010.

4. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "ϒϩάۭؒͷ৘ใ఻೻ಛੑΛ༻

͍ͨ৘ใݯͷଟ໘తϥϯΩϯά",৘ใॲཧֶձ࿦จࢽ, vol. 3, no.

2, pp. 102-110, 2010.

5. Kazuhiro Kazama, Miyuki Imada, and Keiichiro Kashiwagi : “Char-acteristics Estimation of Information Sources by Information Diffu-sion Analysis”, WI-IAT, vol. 1, pp.484-491, 2010.

6. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "৘ใ఻೻ಛੑͷఆྔԽख๏",೔ ຊιϑτ΢ΣΞՊֶձ࿦จࢽ, vol. 28, no. 1, pp. 162-172, 2011.

7. தଜོ޾,ߥ઒๛,ࢁຊ३,ദ໦ܒҰ࿠,౦ౡ༝Ղ,ೆ༟໵,தଜݩل, দඌਅਓ: "ϢϏΩλεσʔλڞ༗ػߏuTupleSpaceͷఏҊͱϑΟʔ ϧυ࣮ݧ΁ͷద༻ධՁ",ిࢠ৘ใ௨৴ֶձ࿦จࢽB, vol. J95-B, no.

11, pp. 1414-1426, 2012.

ߨԋ(Technical reports and others)

1. ദ໦ܒҰ࿠,ٶౡ৴໵,ദ໦խӳʠ, GPUʹΑΔߴ଎ͳඇઢܗํఔࣜ

ͷશղ୳ࡧ๏ ʡ,೔ຊԠ༻਺ཧֶձ2007೥౓೥ձ,๺ւಓେֶ(๺

ւಓࡳຈࢢ), pp. 222-223, 2007.

2. ദ໦ܒҰ࿠,ٶౡ৴໵,ദ໦խӳ,಺ଜ૑ʠ, GPGPUʹΑΔඇઢܗํఔ

ࣜͷશղ୳ࡧ๏ ʡ,ిࢠ৘ใ௨৴ֶձඇઢܗ໰୊ݚڀձ(NLP),ٕज़ ݚڀใࠂ,๺ւಓେֶ(๺ւಓࡳຈࢢ), vol. 107, no. 477, NLP2007-129, pp. 1-6, 2008.

3. ಺ଜ૑,ദ໦խӳ,ദ໦ܒҰ࿠: "ΞϑΟϯԋࢉΛ༻͍ͨࡾ֯ܗ-ޫઢ

ަࠩͷਫ਼౓อূ",ిࢠ৘ใ௨৴ֶձඇઢܗ໰୊ݚڀձ(NLP),ٕज़ ݚڀใࠂ,๺ւಓେֶ(๺ւಓࡳຈࢢ), vol. 107, no. 477, NLP2007-130, pp. 7-11, 2008.

4. ෩ؒҰ༸, ࠓాඒ޾, ദ໦ܒҰ࿠: "ϒϩάۭؒʹ͓͚Δ৘ใ఻೻

ωοτϫʔΫͷநग़ͱ෼ੳ", WebDB Forum 2008,Ұൠߨԋ,ֶश Ӄ૑ཱඦप೥ه೦ձؗ(౦ژ౎๛ౡ۠), 2A-4, 2008.

5. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "ϒϩάۭؒͷ৘ใ఻೻ͷࣄྫ෼

ੳ",ୈ13ճిࢠ৘ใ௨৴ֶձWebΠϯςϦδΣϯεͱΠϯλϥ

Ϋγϣϯݚڀձ(WI2),ϩϯάൃද,ਆಸ઒ۙ୅จֶؗ(ਆಸ઒ݝԣ

඿ࢢ), WI2-2008-62, pp. 62, 2008.

6. ߥ઒๛,ദ໦ܒҰ࿠,தଜོ޾,தଜݩل,দඌਅਓ: "޿ҬϢϏΩλ εϓϥοτϑΥʔϜͷಈతεέʔϧԽ",ిࢠ৘ใ௨৴ֶձ2009૯

߹େձ,Ұൠߨԋ,Ѫඤେֶ(Ѫඤݝদࢁࢢ), B-7-7, 2009.

7. ߥ઒๛, ദ໦ܒҰ࿠, தଜོ޾, தଜݩل, দඌਅਓ: "࣮ੈքσʔ λڞ༗ػߏuTupleSpaceʹ͓͚ΔಈతεέʔϧԽํࣜͷධՁ",ి ࢠ৘ใ௨৴ֶձ৘ใωοτϫʔΫݚڀձ(IN2009),Ұൠߨԋ,෱Ҫ

େֶ (෱Ҫݝ෱Ҫࢢ), ৴ֶٕใ, vol. 109, no. 79, IN2009-21, pp.

49-54, 2009.

8. ദ໦ܒҰ࿠, ദ໦խӳ: "΂͖ڃ਺ԋࢉΛར༻ͨ͠ॳظ஋໰୊ͷਫ਼

౓อূ๏-ฏۉ஋ܗࣜʹΑΔਪਐΦϖϨʔλͷΞϑΟϯԽ",ୈ38

ճ਺஋ղੳγϯϙδ΢Ϝ(NAS2009),Ұൠߨԋ,೤઒ϋΠπ(੩Ԭ ݝլໜ܊), ୈ38ճ਺஋ղੳγϯϙδ΢Ϝߨԋ༧ߘू, pp. 87-90, 2009.

9. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "ϒϩάۭؒͷ৘ใ఻೻ͷಛੑͷ ఆྔԽ",ୈ23ճ࣮ߦ஌ೳֶձશࠃେձ(JSAI2009),ΦʔΨφΠζ υηογϣϯ,αϯϙʔτϗʔϧߴদ(߳઒ݝߴদࢢ), 2E1-OS5-4, 2009.

10. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "ϒϩά͔Β஫໨͞Ε͍ͯΔ৘ใ ݯͷಛੑͷఆྔԽ",೔ຊιϑτ΢ΣΞՊֶձωοτϫʔΫ͕૑ൃ

͢Δ஌ೳݚڀձ(JWEIN09),ޱ಄ൃද,ஜ೾େֶ౦ژΩϟϯύε(౦ ژ౎จژ۠),ηογϣϯ7-22, 2009.

11. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "ϒϩάۭؒͷ৘ใ఻೻ಛੑΛ༻

͍ͨ৘ใݯͷଟ໘తϥϯΩϯά", WebDB Forum 2009, Ұൠߨԋ, ܚԠٛक़େֶ೔٢Ωϟϯύε, 3A-2, 2009.

12. Takayuki Nakamura, Motonori Nakamura, Atsushi Yamamoto, Kei-ichiro Kashiwagi, Yutaka Arakawa, Masato Matsuo, and Hiroya

Mi-nami: "uTupleSpace: A Bi-Directional Shared Data Space for Wide-Area Sensor Network", pdcat, pp. 396-401, 2009 International Con-ference on Parallel and Distributed Computing, Applications and Tech-nologies, 2009.

13. ෩ؒҰ༸, ࠓాඒ޾, ദ໦ܒҰ࿠: "Twitterʹ͓͚Δ৘ใ఻೻ܦ࿏

ͷநग़๏", ୈ17ճిࢠ৘ใ௨৴ֶձWebΠϯςϦδΣϯεͱΠ

ϯλϥΫγϣϯݚڀձ(WI2),Ұൠߨԋ,େࡕେֶத೭ౡηϯλʔ, WI2-2010-9, 2010.

14. தଜོ޾, ദ໦ܒҰ࿠, ߥ઒๛, தଜݩل: "͔ͨ·Γੜ੒ॲཧʹΑ Γޮ཰Խͨ͠ηϯα৘ใ஝ੵγεςϜͷఏҊ",ిࢠ৘ใ௨৴ֶձ 2010૯߹େձ,Ұൠߨԋ,౦๺େֶ઒಺Ωϟϯύε, B-20-3, 2010.

15. Yutaka Arakawa, Keiichiro Kashiwagi, Takayuki Nakamura, Motonori Nakamura, Masato Matsuo: "Dynamic Scaling Method of uTupleSpace Data Sharing Mechanism for Wide Area Ubiquitous Network", AP-SITT, A-5-1, 2010.

16. ෩ؒҰ༸,ࠓాඒ޾,ദ໦ܒҰ࿠: "Twitterͷ৘ใ఻೻ωοτϫʔΫ

ͷ෼ੳ",ୈ24ճਓ޻஌ೳֶձશࠃେձʮωοτϫʔΫ͕૑ൃ͢Δ஌

ೳʯ,ΦʔΨφΠζυηογϣϯ,௕࡚ϒϦοΫϗʔϧ, 1F2-OS8-4, 2010.

17. ദ໦ܒҰ࿠: "ඍ෼ํఔࣜͷॳظ஋໰୊ͷਫ਼౓อূ๏",ϋΠςΫϦ αʔνηϯλʔϓϩδΣΫτ̍एखަྲྀձ,ૣҴాେֶཧ޻ֶ෦, 2010.

18. ദ໦ܒҰ࿠,ߥ઒๛,தଜོ޾,தଜݩل,দඌਅਓ: "େྔεΩʔϚ Ϩεσʔλͷ஝ੵɾݕࡧΛ࣮ݱ͢Δ৽͍͠uTupleSpaceͷઃܭͱ

࣮૷", DICOMO2010γϯϙδ΢Ϝ, Ұൠߨԋ, ذෞݝԼ࿊ࢢ, pp.

76-82, 2010.

19. ദ໦ܒҰ࿠, ߥ઒๛,தଜོ޾, தଜݩل: "ϢϏΩλεσʔλڞ༗

ػߏuTupleSpaceʹ͓͚Δޮ཰తͳΞΫηε੍ޚํࣜͷఏҊ",ి

ࢠ৘ใ௨৴ֶձ৘ใωοτϫʔΫݚڀձ(IN2010),Ұൠߨԋ,ԭೄ

ίϯϕϯγϣϯηϯλʔ,৴ֶٕใ, vol. 110, no. 449, pp. 241-245, 2011.

20. ৿ᚠฏ, ദ໦ܒҰ࿠, தଜོ޾, ߥ઒๛, ౦ౡ༝Ղ, தଜݩل:

"uTu-pleSpaceΛ༻͍ͨηϯαར༻αʔϏεͷ։ൃΛࢧԉ͢Δʮ࣮ੈք

։ൃελδΦʯͷఏҊ",ిࢠ৘ใ௨৴ֶձ2011૯߹େձ,Ұൠߨ ԋ,ट౎େֶ౦ژ, B-20-51, 2011.

21. Takayuki Nakamura, Keiichiro Kashiwagi, Yutaka Arakawa, Motonori Nakamura: "Design and Implementation of New uTupleSpace En-abling Storage and Retrieval of Large Amount of Schema-less Sensor Data", SAINT2011, pp. 414-420, 2011.

22. தଜݩل, தଜོ޾, ߥ઒๛, ౦ౡ༝Ղ, ദ໦ܒҰ࿠, ৿ᚠฏ, দଜ Ұ, ੴాൟາ, Ԑ౉ढ़հ,ԧ௕ٱ, ৿઒ത೭: "uTupleSpaceΛར༻͠

ͨCO2ഉग़ྔՄࢹԽͷ࣮ূ࣮ݧ",ిࢠ৘ใ௨৴ֶձ2011ιαΠ ΤςΟେձ,Ұൠߨԋ,๺ւಓେֶ, B-19-21, 2011.

23. ৿ᚠฏ, தଜོ޾, ౦ౡ༝Ղ, ദ໦ܒҰ࿠, தଜݩل: "uTupleSpace Λ༻͍ͨηϯαར༻αʔϏεͷ։ൃΛࢧԉ͢Δʮ࣮ੈք։ൃελ δΦʯͷ༗ޮੑݕূ", ిࢠ৘ใ௨৴ֶձ2011ιαΠΤςΟେձ, Ұൠߨԋ,๺ւಓେֶ, B-19-22, 2011.

24. தଜོ޾,ߥ઒๛,ദ໦ܒҰ࿠,৿ᚠฏ,தଜݩل: "ϢϏΩλεσʔ λڞ༗ػߏuTupleSpaceʹ͓͚Δ৽͍͠νϟϯΫܗࣜͱߴ଎୳ࡧ",

ిࢠ৘ใ௨৴ֶձ2011ιαΠΤςΟେձ,Ұൠߨԋ, ๺ւಓେֶ, B-19-23, 2011.

25. ദ໦ܒҰ࿠, ߥ઒๛,தଜོ޾, தଜݩل: "ϢϏΩλεσʔλڞ༗

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