「すざく」 が探る銀河・銀河団の重元素汚染史
9
0
0
全文
(2) . . . .
(3) . 920ῌ1192 . e-mail: [email protected]. . . . .
(4)
(5). 162ῌ8601 1ῌ3. e-mail: [email protected]. . . . !
(6). 192ῌ0397 "#$% 1ῌ1. e-mail: [email protected].
(7)
(8) "$%&. !"#" . ' ()*+,- ./0" 123 456712. 8 9:.;'- 12 <=>.*12> ?@ABC0"D< EF.G"./0" HIJ
(9) KLM 'INO6" G PQR S,O.TU'VWXYI5+,- Z"Z" X [\ ],^_` a'12b1 2>c,defgh" iIjklm- ! no pqA /r 1 2. 12 8s%t'_uvwl0"- 12> wl. ?' x y z{ pq H |' D}~xyI,z{ + 3 _xG' GI5
(10) Z+-. 1. !"#$% &%. ῍13B LU- 0lm
(11) U- -B * Zd. &'()*+,-./01234567389. 3KaW|<tpqm Ia 0Se qD. :;<=>?@AB CDEFGH$IJ-KL. cwx-yz{-ZU[\01SmM7a. M7 NOM73PQRST 1C-,-UV0-. ¡¢£L|lFRSe. WXYZU[\0]^_DB ./`abc. _tB CD01¤?5-E¥¢M7|5-uc. R=DR<de fDfD-gGhiGjk-lm. L./a>?<tLm-0<¦¥=e O a<. noapqrdUYZU[\0bcR=Dd. t67| 70§B ¨L089:;| 28§>?. M70s^_DtLRSe ZU[\a1B K. <B ©F|OM73¢FRSe C-OM7-¥ªB. uvtwx-yz{|lFRSe }- 10. «dª]a1¬=@¢L7| ®¯°R. ~OLU|O pq<t[\Sm II. Se ErtB 7-X^±|./-OM7X^±. ῍1 II -2=aG Ib, Ic GO 0S|B ²01³<t II 3<t´<RSe I/II -£L1µ¶¡a 67-·¸¹º= NII TB ¹R¢L= NI T 0 <tLRSe 34. »¼. 2010 ½ 1 ¼.
(12)
(13) . . !". II #$ %&'()*! +* ,-./01
(14) 2 3 45 6 7 +892 II # %&'()*! :" Ia #;!%
(15)
(16) 2 7 <=>()*! %&'" ?@2 7< 2 >)A@ *8B8 C!(". 1. B! Ia/II #<2 %
(17) D EB& *<F* G ; 82!(" 2 %
(18) <2!HIJKH. f (Digitized Sky Survey) 8 X q UROSAT rX µ}¶0w·RS` Z5 QO 6 25 ¸ NASA/skyview *) ¹7. !'" ;'L. 9& *8M J N. <F* RS` ) . B8<O " II # . 1014ῌ1015 * " ; #+ RS8x. J P Q EO 7RST (). vw8,< 1 : 5 : 30 ,% D 8. UB!RS8VWX 8Y)!Z".
(19) &' X q>xvwRS . Ia #$[\ ?@ EB&'. % 5 *,% >" ; Q-RS
(20) !
(21). 8Y)!' %&'" 7 ]. *&vwo8Y)!' *". *)G2 %
(22) 7<J . B xvw "
(23) 8& 7. 1). !" ^<&'. & *8_<)!. <!'B8<F*&' " RS. 2. X
(24) RS`abc*)adc RS Qe`. *)". !vw&'B8'. '" xvw ! 7 ) ῍2 a 1 ,%
(25) <". " B!)<a*)adf g e. xvwRS a*o " RS. ' RS` h . !Mi. UX <)8.,%H88" RS. Q jM klm*I'n&> 8. ! 78xvw ! 7. ''" ?@ o8Y. +.>)8
(26) 5). )!'2) 1970 p X q. !r %&. ;!" B ad%8xvw . !'B8<. ! 7 X q2 %'. !3), 4) B!RS`#$v. B8<O * B %
(27) x. w x%<RS` yz{/|}&~'. ¡" ¢£¤018&7¥. '" ad%*) 1 % x
(28) &'. ¦ /0'"
(29)
(30) 2 7 1 c
(31) . X q'*) 1. ac /5§'/0'¨©A@. % f8 X qRS`'. '6) ;'ª{/5«1W. u8" fc RS e`8'u. ¬&' ª{ h . RS`" X qu8c RSRS`. 258 ®3 % " ¯.}°±². $xvw '('u
(32) >
(33) &' . 4E. 'RS`*)s X q<"t u. !'/5. ! ; ³" ´W ¯.}°±². ῍2 ) !¢ :'º7 45a 1 8»± ¼ 103 ½. ¼16. 35.
(34) x%0 78) 7 MN& y X1z'yX { 1 ()(F d Ou-7 3 |#$ IFOL GH !u- b8-}~(AKQ? * "[ +AE79) O Q? Ia/II t,H -.(X0 7 8 Ia/II t \D? /1}0% ( 7 2000 1 2 3 u - j k XMM 2. Newton GH!E %{ MN. IJKL GH(./0 7 ABELL 1060 M N X #$%&'
(35) >?OP( )#¬- dO.
(36) <E >? ./ +AE 710)ῌ12) ./
(37) < >? yX 1z'yX 4 ()(F. Q?.
(38) . yX1z'5()(F dO.
(39) 2 . E7 Ia/II t! /"[P 7. !" X #$%&'
(40) ()# *+ X. 13) } II tX6OP 7 8K1u. #, -./0
(41) 122. E71dOEm(7 - MN. !345 2 6 78῍ 0.1ῌ10 9.
(42) <1 cD MN῍4(fu7 Ia t,. : ;
(43) ' X #
(44) <1= 7>?@. H9:!u-X0 7D?H p . AB CD? ./AAE. !u-0 7D?(0 -2E ;. 2 . -F 1 X #GH IJ. MN 71(m. KL (./0 7 ABELL 1060 MNOP./0. !u- MN <781O. 7 X #$%&'
(45) ( MN./OPP. P 7
(46) <(@Q?}~(AK II t,. X #$%&'
(47) 1C ῌQRST!. H(mB=d >?@ . UV!WX YZ[7!"A. u7D?./> FE ¡. \D? )#(]X0 -. KP%MNBOP . 3. ^_`a HEAO-1 GH!E b8-MN ./ cd 7 1 X #
(48) 7). < )# (e f K
(49) g D? )#OP -78 h. .. 7
(50) <(1)) ?KAu-!i1./ (m(7. 3. ῌῐ῏῎῍ ῒῑΐ. / !i(D? (f!K"[P -. ¢7£ <E7¤u-7MN¥¦ >. 7 jk EXOSAT GH@#$ Ilm L GH. ?@AB \D? * #$ X #GH I. ./OP =no$ p MNq. JKL !E%POAu-7 IJKL. rstMN p dO. @§0 7 X # CCD a_¨ 1 >?). E MN!u-=no$ vw0 7. #©Sª =«)?¬AB ©®&. ῍3 +¯? ¨QS a )#1 10.2 ;
(51) '(FE 7 ° 7>? 25 ° 7 * ± 7²C, ¨QS a )#13 ³ 0.65, 6.97 9: ;
(52) 'AE ῍4 MN D´C µ9$ D( D t D0 MN "£MNFE¶%·¸MN p 1z' p 1012ῌ1013 F¶ MN 10ῌ100 GFE {¹H1º»¼:(<P 36. ½7. 2010 1 1 ½.
(53) "3. ¯n AWM 7 X $%&' + MNOPU.-./.
(54) AWM 7 X . !" 3 #. "4. $%&'()*+ ,-./01!2& 0'. AWM 7 $% &'I; °±H ¡px X 1 1F1,;¡px. 3456 /.$%&'. 34X&F-6 $%+&. +789: ;<=#6 >;9?@AB. ' ῍5 0.3 + X. C 7!DE1# F 006 ?@. }. F 0+ 86 : ;I. ABC7DE#G01HI. ;! 5G-DE1# F. #J#'6 22K7(LMN. /.'$%+x-+. OP!QRF S&' T/>;U . /I;H.6 SH. VLHW#G+X /.. I#. H Y,Z6 [\-]^_ `5G-S. 4.1.
(55) . 5G-6 2. >;6 ?@ABC6 : ;6 6 8. 2K
(56) >;1
(57) >;. $%&' 0.3 /. TXc /01!dF#G-, S0. X.b&L5 F . >;1ef5gSh1ij! II klmn. 1F-" 4 AWM 7 ;. op/?@ABC!2&a6 5. $%&'XF S. qr Ia/II ksXtu /01!DE1. 6 : ;6 6 8H$%H ¡p. #/. ¢6 `&G-{£F-., ¡p. VL7a'b,!. 14), 15). 4..
(58) . 0.3 ¤L1# F ¥ 6 >;H$ % ¡ph¦`&G-7 {£F §iF!6 ?@ABCH$. vwH X.-xyz{N# 1 |. %H ¡p¤L<G76 {£. L&' 3 |LX}. F. ¨©H8#j1x/199ª9&#G-./. ~H>;F-./6 05. «`!¬' F II klmnop/>. 7NL!M#G-F g1>; T!. ;9?@ABC!6 Ia k#op /8. {#G-!dF#G-F g. #jx-99®!G-./5g#«`!. . 16)ῌ22). ῍5 ² L!³´ µ L 180 ¶· ¸¹F-./º 1#/ » 103 ¼. »1½. 37.
(59)
(60) " II #$%&. !". '( )*+ ,-.. /01232"45 Ia #678 5!";5. 9:. <=5( )*.
(61) 2 <=>?;5 ;@ ABC*; DEDF GHICJK. " L. MNO 5PQ)* R S
(62) T5U V2 W)*23), ῌ6
(63) T. .
(64) 2X:*5Y)*;Z [52 LMNO. "\]Y)*. ^_'U<=`<; ]. abPQ
(65) T5 Ia #5c )*. Vd'e*; ;fQ5g
(66) T 100 hUi.
(67) 2X:] @C. 2jN5k)*. . "
(68) . }5. AWM 7 5PQ)*+ (O) -. /01 (Mg) (Si) (S) V (Fe) Ia/II #$%&5( )*& } <'PQ@C"
(69) ¸ ¹ #8 º»(¼)ZU. ' *. lgm D II #$%&5c ). " ^_ 2. *
(70) 2 no. II/Ia #b 5U 3.5 5d. 32";pq. 2r@' 9: <5 II #. U5 4.2. d'!";Z[2 . Ia/II ῎ῑ῏ῐῌ῍ῒ. LMNO 5\]+ ,-./01;" II #$%&5( )*
(71)
(72) 2. :'U ;&2NC* [. 9. ;.
(73) 2;ABC*. r@"5 Ia/II #5( )*
(74) . ab\] X PQ@C II #$%&2. ; PQ)* 0.3 v. ]C*5 5PQ)*& @. stu. 24), 25). wux
(75) T
(76) y Ia/II #z. Cf)* II #$%& ;. {|26), ῌ7 } 5 AWM 7 5. r*r* X PQS2 5. Ia/II #zp; + ,-./. !" II #5d. 01~[ II #5( )*. . @Cf5PQ)*¡!. V 8 2 Ia #5( )*U5. &( "27)@C¢£5g *C¤¥. 4 , Ia/II #z2R. ¦;#§]. 5d; S2 C* . ¡!5&( "¬@C9:. 9:
(77) @C2"@ C*5. |b Ia/II #$%&2!". @U . {| *5. !. & . $¨%5©ª«. 5®¯°u±5²U5 ³S *5. !" II #$%&. $¨U 5 } 4 5p. ῌ6 DEDF GHI ´ http://vega.ess.sci.osaka-u.ac.jp/¥µ7Efujita/article/suzaku/a399a401.htm ῌ7 ¶N http://astro.s.kanazawa-u.ac.jp/·ksato/gingadan.html 38. )*½+. 2010 1 ½.
(78) kA6 "-.&A[\"XY lmn*op 0.1 , q& 0.3 , q r=s&$ 0.1 ,I1t q a kA"OI1bbu"vwx "yY z{|-.Wv.1 /&}~"$ ^ " .0Y$ a&O 3 " 1 T20F~ KIv+&<! =" -.W0 zIWFv~ U6. D2VWG !&a$ cdefgh$ O :&<! 4MN <6 = a$ cdefgh :<!= " D2VWG Iv`#i#Y l mn*op 0.1 ,& 0.3 , ´ r =s&$ bba kA"OI1 u"vwxy . <! =s 2FY 0Y !0vI0 2"$ -.0 kA"
(79) IW+v F$ WJ90v+f* "J9#tv F ~ XMM-Newton H5 FY W.
(80) . $ + kA"I1u"v. . !". wx"E112) D2VWG XY+. #$ %#& '()* +. :ῌ<!= FY op. % 2 ,. $ J9W^_9 0.3 , lm. 4.3. !-./&012 . n*op tWk-.W01 . ῌ
(81) . %%+3 456 789. +"89 1gg. $ 3 + :&;82. z&Y/ IA0opT90. <! =$ +>?@. $ +"tu"vvF$ W. A +0BC&012 DE2FG H5. " ¡vF$ ¢Y. I1$ JK 4L 6 <! 4MN. F&012 ¢Y$ £¤0+. <6 E1 O :=P- 4 6 I. 8¥2¦FY§¨
(82) & ©. 1 1 QRSJ9/&"T9 U 6. ª«¬.& "® 2 $. T2IA$ D2VWG XYO:&. P-¯°῍8 "E1$ ± ;8"k. <!=FY Z[\ ]$ ^_`#. &0῍9 XMM-Newton FY$. abcdefgh :& <!=i. / IA0 +&<!= op. #/&" . abcdefgh <. "t ²]wxE&A[\XY. !=j -.0 kA 4 !" . 30) / &$ +78³. 28). 29). ῍8 µ;82¶ ¯° 2 3 ¯° ·¸W$ §¨ 58k& E1~ ῍9 / IA0 Fossil group &¹º2 » 103 ¼. »1½. 39.
(83) / L'&"#&R
(84) o R pm ,
(85) Our galaxy.
(86) qr. ,sKL. ;. Q( + ( &Ro R, D 1). &t. *. Ia uvOa,w7'L. ;. Q(+ ,
(87) cD ! "#01234. 567' K. xy' . (= + .& / &z{Y\| 32). }G~ I4 ~ :& m7. <= > :? , B C 7 ' NGC 72031), NGC 1399, NGC 140416) ´µM NGC 425837), 33 NGC 4631 & 3¡ £ 36) M 82 & “cap” 35)& =K¶"#&@· Gn &+ ¸ ¹!ºD'»' Ia/II uvOa ,w7'K"#&"#¼}24), 25)+ ¸$ º;';'&¢ po ¢r &!"#R 33)Gn&+.
(88)
(89) . . !"#$% &'()*+. 5. . ,-'!"#.& / 0123. 3
(90) :; ()*+ 8 %&&&R_(K Ia u vOaw G~ s K& Q 1 7: ( (=+ L&` Ia uvOaw
(91) 0 a& Ia uvOaw 0 5.2. xy' (=5)+. . 33
(92) &
(93) t. LK,!"#&e d7'K xy' (=34)+ <=>:? , M 82 NGC 4631. /33&BC. kl7' (=+ m 7 n= / M 82 , “cap” . *+ <=>:? ,&@A&&BC. 'K &!"#RG. D&EFGHIJKL. a&"#R L. ;Q(35)+ (. '&¢ po ¢r &"#R & £
(94) II uvOa L. ;Q(. + L'
(95) ¤¥ II uvOa&w .
(96) . 36). ,V# WXY Z[W\]^&RG_`K ,M(. L II uvO. NGC 4631 ,
(97) & 3¡
(98) ;';. <=>:? & L. xy. O: <=>:? ,. S TU M(=+ 5.1. /& Q. D&PQ&!"#R. '(,; KN8 @A&. *.. J'()* & II uvOa . 4567'&8
(99) (9 :; () ,M(=+ LL,
(100) L. /. 12), 31). + &aI23
(101) . e4~ KL. Gn¦ K&. '()*+ 33. y.£.
(102) §: (V#¨Z[W\]^G© :. b aca K da&ef. ª`'K&
(103) <=>:? «` &, L'. &23a&g,h 7'&,=+ . &¬
(104) (9®(¯Q. <=>:? Q_`'aI23&V#. 5.3. WXY Z[W\]^&R
(105) a&ef&R Ei=K
(106) j,=+ BCkl m 7 n= 40. °y(=+. . ±e. ²;'K'()* ;';'&. ¢ ³ /´µM &01aI23 %&½. 2010 1 ½.
(107)
(108) . TO!38)'%&J ῍10 %&J. . !"#$. ) X T1!wx$$.3. %&' ( )(*+, - . ' r*0'jk~H. ./01'%&23./$0. 'H+ 3/
(109) . NGC 4258 /45267. '- +. - 8 7 ./$
(110)
(111) 67. }~%&(3'2h- 'y. 9$.+:3'
(112) ;;. ^67J $03C. - 45267
(113)
(114) 67<./$. /
(115)
(116) ¡,^-¢@_(3'- H. . ./$'@.jk$H /
(117) 3. 37). !'=$03./0>?. @ABC/ D. - 45267. TO!@zN{|O. $- ;/c+0 . 67 E F' G3 H/ IJ '- KL67M NO1%& $0 +C/.
(118) %£¤- ISAS/JAXA ¥0. [¦§- ¨©. 1ª«¦§- ¬2[3¥©®¯«¦§ \#( $°±@4 (. 6. ῌ ῌ KPQR- S$ X T%&UV@
(119) W$XY2 '- 67 Z[0\Z @H3#$- ]^67 @_(3 `a67&0Hb+%& ' - c67 d&^67 ^67 e>f0Hg$0Gh0. X T1%&$ijk
(120)
(121) 67l mnop$ q'r ( st$ 20uvOw!xO@(y J 3'zN{|O}~( zN{|O}~ '9$ +- jkS. CCD. ~ 10 !' ( 2014 "$ '#J3$% X T ASTRO-H $zN{|O}~&'#J i - +'@(y J ( d- ;'' " y ^67@ 3$0. ( $- '' ( 6 "L. . 1) ²³´ª- 2007, 56µ7 100- 384 2) Sarazin C. L., 1988, Cambridge Astrophysics Series, Cambridge, Cambridge University Press 3) Mitchell R. J., Culhane J. L., Davison P. J. N., Ives J. C., 1976, MNRAS 175, 29P 4) Serlemitsos P. J., Smith B. W., Boldt E. A., Holt S. S., Swank J. H., 1977, ApJ 211, L63 5) Renzini A., Ciotti L., D’Ercole A., Pellegrini S., 1993, ApJ 419, 52 6) 8 h )¶{O·j956[ 4 67 I, ¸$%:; ¹- 2007º 7) Mushotzky R. F., 1984, Physica Scr. T7, 157 8) Arnaud M., Rothenflug R., Boulade O., Vigroux L., Vangioni-Flam E., 1992, A&A 254, 49 9) Fukazawa Y., Makishima K., Tamura T., Ezawa H., Xu H., Ikebe Y., Kikuchi K., Ohashi T., 1998, PASJ 50, 187 10) Matsushita K., Finoguenov A., Bo ¨hringer H., 2003, A&A 401, 443 11) Tamura T., Kaastra J. S., Makishima K., Takahashi I., 2003, A&A 399, 497 12) Matsushita K., Bo ¨hringer H., Takahashi I., Ikebe Y., 2007, A&A 462, 953 13) de Plaa J., Werner N., Bleeker J. A. M., Vink J., Kaastra J. S., Me ´ndez M., 2007, A&A 465, 345. ῍10 NASA $.3»¼QR http://exist.gsfc.nasa.gov/papers/presentations/apsῌmay09.html < 103 5. <1½. 41.
(122) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23). 24). 25) 26) 27) 28) 29) 30). 31) 32) 33) 34) 35) 36) 37) 38). 42. Yoshino T., et al., 2009, PASJ, in press 2009, 102, 370 Matsushita K., et al., 2007, PASJ 59, 327 Sato K., et al., 2007, PASJ 59, 299 Sato, K., Matsushita K., Ishisaki Y., Yamasaki N. Y., Ishida M., Sasaki S., Ohashi T., 2008, PASJ 60, 333 Tokoi K., et al., 2008, PASJ 60, 317 Komiyama M., Sato K., Nagino R., Ohashi T., Matsushita K., 2009, PASJ 61, 337 Sato K., Matsushita, K., Ishisaki Y., Yamasaki N.-Y., Ishida M., Ohashi T., 2009, PASJ 61, 353 Sato K., Matsushita K., Gastaldello F., 2009, PASJ 61, 365 Fujita Y., Tawa N., Hayashida K., Takizawa M., Matsumoto H., Okabe N., Reiprich T. H., 2008, PASJ 60, 343 Iwamoto K., Brachwitz F., Nomoto K., Kishimoto N., Umeda H., Hix W.-R., Thielemann F.-K., 1999, ApJS 125, 439 Nomoto K., Tominaga N., Umeda H., Kobayashi C., Maeda K., 2006, Nuclear Physics A 777, 424 Sato, K., Tokoi K., Matsushita K., Ishisaki Y., Yamasaki N. Y., Ishida M., Ohashi T., 2007, ApJ 667, L41 Madau P., Pozzetti L., Dickinson M., 1998, ApJ 498, 106 Ciotti L., D’Ercole A., Pellegrini S., Renzini A., 1991, ApJ 376, 380 Makishima K., et al., 2001, PASJ 53, 401 Kawaharada M., Makishima K., Kitaguchi T., Okuyama S., Nakazawa K., Matsushita K., Fukazawa Y., 2009, ApJ 691, 971 Tawara Y., Matsumoto C., Tozuka M., Fukazawa Y., Matsushita K., Anabuki N., 2008, PASJ 60, 307 Bo ¨hringer H., Matsushita K., Churazov E., Finoguenov A., Ikebe Y., 2004, A&A 416, L21 Lodders K., 2003, ApJ 591, 1220
(123) 2007, 100, 378 Tsuru T. G., et al., 2007, PASJ 59, 269 Yamasaki N. Y., Sato K., Mitsuishi, I., Ohashi T., 2009, PASJ 61, 291 Konami S., et al., 2009, PASJ, 61, 941 2009, 102, 248. Suzaku Observations of Metal Enrichment History in Galaxies and Clusters of Galaxies Kosuke SATO Graduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa, Ishikawa 920ῌ1192, Japan Kyoko MATSUSHITA Department of Physics, Tokyo University of Science, 1ῌ3 Kagurazaka, Shinjuku-ku, Tokyo 162 ῌ8601, Japan Takaya OHASHI Department of Physics, Tokyo Metropolitan University, 1ῌ1 Minami-Ohsawa, Hachioji, Tokyo 192ῌ0397, Japan Abstract : In order to investigate how stars have evolved and synthesized elements from the beginning of the universe, we need to know the amount of various elements in the present-day universe. Elements were synthesized in stars, and then distributed not only inside galaxies but also in intergalactic space. In clusters of galaxies, the strong gravity prevents heavy elements synthesized in the member galaxies from escaping. We measured the amount and spatial distribution of metals in the thermal plasma in galaxies and clusters of galaxies observed with the X-ray astronomy satellite “Suzaku”. The distribution we observed suggests that the metals have escaped to intergalactic space from galaxies probably in the starburst phase. Abundances of di#erent elements from oxygen to iron indicated that roughly three times more Type II supernovae, caused by massive stars, have occurred than Type Ia ones, by white dwarfs.. . 2010 1 .
(124)
関連したドキュメント
, Graduate School of Medicine, Kanazawa University of Pathology , Graduate School of Medicine, Kanazawa University Ishikawa Department of Radiology, Graduate School of
Department of Cardiovascular and Internal Medicine, Kanazawa University Graduate School of Medicine, Kanazawa (N.F., T.Y., M. Kawashiri, K.H., M.Y.); Department of Pediatrics,
3 Department of Respiratory Medicine, Cellular Transplantation Biology, Graduate School of Medicine, Kanazawa University, Japan. Reprints : Asao Sakai, Respiratory Medicine,
＊2 Kanazawa University, Institute of Science and Engineering, Faculty of Geosciences and civil Engineering, Associate Professor. ＊3 Kanazawa University, Graduate School of
of Internal Medicine II, School dicine, University of Kanazawa.. Takaramachi 13-1,
熱力学計算によれば、この地下水中において安定なのは FeSe 2 (cr)で、Se 濃度はこの固相の 溶解度である 10 -9 ～10 -8 mol dm
For staggered entry, the Cox frailty model, and in Markov renewal process/semi-Markov models (see e.g. Andersen et al., 1993, Chapters IX and X, for references on this work),
A lemma of considerable generality is proved from which one can obtain inequali- ties of Popoviciu’s type involving norms in a Banach space and Gram determinants.. Key words
