Use of Complex Number in the Analysis of Increase in Dry Matter Indigestibility with Growth of Forages

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1 九州大学学術情報リポジトリ Kyushu University Institutional Repository Use of Complex Number in the Analysis of Increase in Dry Matter Indigestibility with Growth of Forages Shimojo, Masataka Bungo, Takashi Imura, Yoshimi Tobisa, Manabu 他出版情報 : 九州大学大学院農学研究院紀要. 43 (1/2), pp , Kyushu University バージョン :published 権利関係 :

2 .I. Fac....-'\gr., Kyus]m Univ., 43 (1 ' 2), l (1998) Use of Complex Number in the Analysis of Increase in Dry Matter Indigestibility with Growth of Forages Masataka Shlmojo, Takashi BUllgO, Yoshimi Imura, Manabu Tobisa, Naoki Koga, Shao Tao, Muhammad Yunus, Yutaka Nakano, Ichiro Goto, Mitsuhiro Furuse and Yasuhisa Masuda L..1.bOralOry of Animal Feed Science, Faculty of Agric.ulture, KY11Shu University, Fukuoka , Japan (R (~ceived July 31, 1998 an d u.ccepteda.ugust 7, 1998) This study v..'as conducted to give a trigonometric representation, using complex numbers and plane, of th ~ innease in dry mat ter indigcst.ibil ily [1O:\f11 with growt.h of forages. The ro llowing results were obtained. Di"isiollaJ t.:alculations between two coordinales of dry matter indigestibility [DMTI ~x pr e!ise( l llsing complex numbers were related 1.0 t.he subtraction theorem or t.angent. This relation might give Sllrne ill fo rmation on DMI changes, namely TOMI, t.he geometric mean of DMI [CMDMI] and relat.ive IDMt (=IDMI/G1lDMl). Changes in dr~y weight and indigest.ible dry weight with forage growth were visualized by the geometric representation. Therefore, there were four infonnation 011 IDMI given by the trigonometric representation of 1)\11 on the coltllllex plane. It was suggested t.hat I.he complex representation 01" D~r of forages d erived, from til e subf.racl. ion theorem of tangent, some information on IOMI with fo rage growth. INTRODUCTION The data on the Jlut.ritive value of forages are normally described using real numbers, There are probahly no or few cases where imaginary nwnbers have been liseo in thi<; field, It seems to be strange, t he refore, that the p resent study will deal with the complex representation of indigestibility increases with growth of forages. The present motive of the use of complex munbers comes from an attempt at tjle geometric representation of forage indigestibility using trigonometric fun ctions. It is suggested that trigonometric functions, when combined with complex nllmbprs, give some useful information, for example, the description of electric power on the electric circuit, In our previous two brief reports, some aspects on the digestibility decrease "rjth forage growth were sh o ~'1l by its t.rigonometric represent.ation on the complex plane (Shimojo et at., 1998c) and on the orthogonal toordinates (Shimojo et ai., 1998e). The analysis of indigestibility increases seems to be more attractive than that of digestibility decreases (Masuda, 1985), because the former describes the forage maturation process using the increase in anti- digestion component.s (Shirnojo et al., 1995, 1997a, b, 1998a, b, d,o. The present study was designed to suggest a trigonometric representation, using eomplex numbers and plane, of the increase in dry matter indigestibijity with growth of forages. '" Kyushu Lniver.::ity l'mm, Fukuoka :

3 138 ih. Shinwjo et al. DRY MATTER INDIGESTIBILITY INCREASES EXPRESSED USING COMPLEX NLMIlERS The increase in dry matter indigestibility with grmvth of forages [IDMI] expressed using real numbers is as follows O\.'fasuda, 1985; Shimojo el ai., 1995): IDMI = I, _ I,. ~V2 WI' where 1V=[orage dry weight (IV, < lve), J=dry weight of indigestible materials in forage (I, <[,), Tc/1V,=dry matter indigestibility [DMI] at time t,. Then, the follov;ing two coordinates are t.aken up and plotted on the complex plane (Fig. 1), (I) Z2=VV 2 +jj Z ' (2) wherej=the imaginary unit, J,lW,=tanO, [llmi at t,l, ]JW,=tanO, [DMI at ld, another allgle (ij,-o,) is also shown, 0< 0, < 0,< 1</4, 0< tanij, < tanij,,< 1. (ij '"."... '" '" E ,z2 W, Forage dry weight (real axis) W2 Fig. 1. Plotting of forage dry vi'eight [J.f l and dry weight of indigestible materials [I] on the complex plane (8 shows the angle).

4 ~-.. Complex Representation of IndigestibUitl/lnrrease with Forage GruuHI 139 Thus, equation (1) is re-written as follows: IDMI ~ tano, - tano,. (3) The two coordinates (equation (2)) rewritten llsing trigoilolnetric functions are as follows: Z 2 ~ J W,' +1,'. (coso, +.isin0 2 ) (4) How equation (3) will be related to equation (4) is the present subject. For this purpose we take up the following equation that is kno\vi1 as the subtraction theorem of tangent, 1 + (VtanO,. tano, )'. (5) Equation (5) shows that the numerator of the right-hand side gives IDMI. 1t is also shown that the denominator includes the geometric mean of t"vo DMIs at tj and t~ [GMDMI], namely (tano', tano',)". GMDMI is also described using tano',., tano'" and tan(o', - 0',) as follows: _-_._- mmmi ~.j"tan-' -e--.-ta-n-e- j _! l1me2- tane, - tan (e, - e,) 2 -V tan(ez-e j ) (6) GMD1.fI seems to show, as it were, the ihdigestlon level associated with the two DMIs. Then, t.he IDMI related to its indigestion level [relative IDMI] might be estimated as follows: Relative IDMI (7) Thus, some infonnation on DMI changes might be derived from the right-hand side of'the subtraction theorem of tangent (equation (5)). namely IDMI, GMDMI and relative IDML This leads us to the investigation into how equation (4) YVill be related to equation (5). The two coordinates, ZJ and Z2l will be related with the subtraction theorem of tangent according to the follmving calculation:

5 140 M. Shimojo et al W 2' W L {( ) ( )), 1 + tano,,tano, + j tan0 2 - tano,, J'V i 1 '.!. therefore, _ W_J 2 +, I J:!. Zz ( 0 8) (0 0), - ~ l+tan " tan, +j tan,- tan, ' W,' W, Z, (8) In the right- hand side of equation (8), the real part [Re(ZJZJ J, 1 + tan6'"tan6'" gives the denominator of the right-hand side of equation (5), and the imaginary part [1m (Z';Z,)], tan 6', - tan 6'" gives the numerator ohhe right.-hand side of equation (5), ZJZ, is also related "ith the left-hand side of equation (5) as follows: J Z, = W., +J, { (0 0) "(0 0)) 0 " COS.) -, +Jsm?-, Z J I W, +1 2 therefore, (9) In the right- hand side of equation (9), Re(Z,IZ,), I, gives the denominator of the left-hand side of equation (5), and lm(zjz,), tan(8,.-8j, gives the numerator of the

6 Curnple:(; Nt!jJll:senl (l(iml oj"lndig e:;tibility Increase with F U'f'(!(JIf (J rou.h h 141 loft- hand side of equation (f)). Therefore, equations (8) and (9) suggest a sort of rclatlonship between ZjZ, and the suht.ractiun theorem of tangent (equation (5)), when some modifi cations are made on the calculation of ZjZl. This relationship might give some information on dry matter indigeslibility changes with gro"th of forages, namely IDM1, G:l-I!)M I and relative IDMI. This seems to be due probabjy to the c haracle ri~ lj cs of complex representation combining things which are different and do not. speir! CO he related. Another benefit obtained from the geometric representation of DlvlI seems to be the visualization of changes in dry weight and those in indigestible dry weight with growth of forages (Fig. 1). Thus, there are four infotillation on IDMl derived from the trigonometric represent.ation of DM1 on the complex pla.ne. The present results are comparable to those for the digestibility decrease with forage growth (Shimojo el " l., 1998c). There '''ill he a strong objection to the complication, llsing complcx representation, of simple subjects such as changes in dry mattcr digestibilit.y and indigestibility "ith growth of forages. It seems to be, however, of interest to make anal~yt ic approaches to t.his sort of subj ects using various methods we can try. This will lead to an understanding of interrelationships bet\,,"een some aspects of forage utilization. Conclusions It is suggested from this study that the complex representation of D~lI of forages de rives, from the subtraction theorem of tangent, sume information on IOMI occurring with forage grm\'th. ACKNOWLE DGE MENTS The present work is based on our prp,\ious works on forage indigestibilit.y \vhich \vere slii>ported by skilled technical assistance by ~:fr. Yasukatsu Yano. REFERENC ES \ fa!:h1da, Y. 1!}85 Analysis of dry matter digestibility of }(reen panic as affected by the change in temperat.ure. PrO G. X V Int. Grassl. Congr., loon- l Oll Shimojo, M., Y. l\lasuda, T. Rungo, T. Kawamura and I. (tot.o UJ95 A.nalytical C'xpression for formation of indigestible mat.erials and increase ill dry matter indigestibilit.y \\;l.h growt.h of some tropical gra..<.;se!:i. 1. Fac. Agr., Kyu~<;hl.l. ljniv., 40: Shimojo, M., T. BUllgo, Y. irnllra. M. Tobisa, N. Ko~. Y. Nakano, I. Coto and Y. M(l s ud ~ 1997a An anatysis of d igeslibility change \\ith 8rO\,,1.h of forage. In "Present and futurp of rumen research", Proc. Firs/. Joint Symp. Japan o,ru/. Koma on Rumen M etal>. PhYS10l., P-ll Shimojn, M., Y.lmura, M. Tobisa, N. Koga. T. tlungo, Y. Nakano, T. Nishihira. I. (joto and Y. Masuda 1997b ~' o rma ti ()n of indigestible materials and increase ill dry matter indiges tibilit.y in the growth of n... o tropical forage s. J. Pac. Agr., Kyu..<;hu [Init'., 42: Shimojo, M., Y. Irnura, T. Bungo, M. Tohisa, N. Koga, S. Tao, rvl. Yunus. Z. Yin. Y. Nakano, 1. Goto and Y. Masuda 1998a Formation of indigestihle materials from digestible mate ri Hls ami photosynthatcs in lhe gro\o\-th of Rhodes grass (ChkJ'n"::i gayanl1 Kunth). 1. Fac. Agr., K:l)u.';hll, Univ., 42: Shimojo, M., T. RUlIgo, M. Tobisa;. Imura. :-f. Koga, S. Tao, M. unus, Y. Nak'lno, I. Got.o, lo.f. Furw;e and Y. Masuda. 190Rb Analyses u f ro rmation of indigestible malerials and inc rease in dry ma( Ler ilu ligcslibility \\; th growt.h of lwo t.ropical forages. Proc. 8th World ConI. An-hn. Prod., Seoul, Korea, Vol. D,

7 14 2 1\-1. /'ihi mojo et al. Shi rf1(~io. M., T. l:sungo, Y. Imura, M. Tohisa, N. Koga, $. Tao, M. YUIlUS, Y. Nakano, l. Gota, M. rllnlst" and Y. Masuda. 19!}fk D i~es l.i b ili ty decrease willi forage growth as intcrpretecll L<:;illg complex plane. JAroc. 8th World Cmif. Anim. P"rod., Seoul, Korea, Vol. n: Shirnojo, M., T. Bungo, y, Imura, M. TobiS<l, N. Koga, S. Tao, M. 'iunljs, Y. Nakano, l. Golo, M. F'Un1se and Y. Ma~ llda. 1998<1 Some aspects on the analysis of the formation of indigestihle materials with growl II of foragp,f;. Proc. 811, TVor(a Con.1 Anim. Prod., Seoul, Kc)rea, Vol. ll: H Shimojo, M., T.llungo, Y. hnura, M. Tobis.:'l, N. Koga, S. Tao, M. YlInUS, Y. Nakano, I. Goto, M. FUnis!:' and Y. Masuda. 1998e Digestibility decrease with forage growth as interpreted using orthogonal coordinates. Proc. 8th rvorld Canf Aru:m. Prod., Seuul, Korea, Vol. H: fi20-i)21 Shimojo, M., T.13ungo, Y. lmura, M. Tobisa, 1\. Koga, S. Tao, M. YUHllS, Y. Nakano, T. Gata, M. Furuse and Y. Masuda 19B5f Accumulation rate of digestihle materials Clnd formation rate of indigestible materials in the description of relative grov.th rate of forages. J. Fo,o. Aqr., Kyushu Un'iu., in press

Visualization of the Hollowness in Unit Particles of Allophane and Imogolite

九州大学学術情報リポジトリ Kyushu University Institutional Repository Visualization of the Hollowness in Unit Particles of Allophane and Imogolite Wada, Shin-Ichiro Department of Agro-environmental Sciences, Faculty