Equality of Schur and skew Schur functions

㊃综述㊃基金项目:湖北省卫健委面上项目金蝉花对糖尿病肾病足细胞保护及机制的研究(W J 2019H 140)通信作者:刘伦志,E m a i l :l i u l u n z h i @163.c o m维生素K 依赖性基质G l a 蛋白与慢性肾脏病患者的血管钙化刘 炀,向海燕,刘伦志(湖北民族大学医学部附属民大医院肾病内科,湖北恩施445000) 摘 要:基质G l a 蛋白(m a t r i xG l a p r o t e i n ,MG P )是由软骨和动脉壁细胞分泌的维生素K 依赖性蛋白㊂MG P 为了具有生物活性,必须经过进行羧化和磷酸化,这个过程高度依赖于维生素K 的状态㊂MG P 是目前被发现的最强大的天然的钙化抑制因子,在活化过程中由于经历了不同的转化方式产生了4种亚型㊂血管钙化(v a s c u l a r c a l c i f i c a t i o n,V C )在慢性肾脏病(c h r o n i ck i d n e y di s e a s e ,C K D )患者中非常普遍,并且与心血管疾病的发病率和死亡率密切相关㊂本文旨在介绍C K D 患者中MG P 激活和发挥功能的病理生理机制㊂关键词:肾功能不全,慢性;基质G l a 蛋白;血管钙化;维生素K ;维生素K 依赖蛋白中图分类号:R 692.5 文献标志码:A 文章编号:1004-583X (2020)09-0857-04d o i :10.3969/j.i s s n .1004-583X.2020.09.018 在慢性肾脏病(c h r o n i c k i d n e y di s e a s e ,C K D )患者中,心血管疾病的患病率逐渐上升,并且与其高死亡率直接相关㊂心血管疾病的风险随着C K D 的进展而逐渐增加㊂肾小球滤过率每降低10m l /m i n,C K D 患者发生心血管疾病猝死的风险就增加5%,而且心血管疾病是导致大约50%的终末期肾脏病患者死亡的直接因素[1]㊂C K D 患者所伴随的这种沉重的心血管疾病负担不能仅用传统的危险因素来解释㊂例如氧化应激㊁炎症和钙/磷代谢异常等这些新发现的与尿毒症和透析有关的危险因素,已被证明可促进终末期肾脏病患者发生血管钙化(v a s c u l a rc a l c i f i c a t i o n ,V C )和心血管疾病[2-4]㊂V C 是心血管疾病和尿毒症患者死亡的独立危险因素,在C K D 早期阶段就可出现,并随着疾病的发展逐渐加重㊂钙在人体动脉血管壁内的沉积会导致发生心血管死亡和心血管事件的风险分别增加近4倍和3.4倍[5]㊂尽管V C 在一个多世纪以来都被认为是一种被动性疾病,但近几年来,V C 逐渐被认为是一个涉及调节蛋白和分子的活跃㊁持续的过程㊂这些蛋白和分子促进或抑制钙和羟基磷灰石在血管壁内的沉积㊂因此,V C 应该是其抑制因子和促进因子之间失衡所导致的㊂在C K D 患者中,V C 的特征是钙化抑制因子水平的下降,常表现为血管内膜㊁中膜和心脏瓣膜的钙化[6]㊂肾脏基质G l a 蛋白(m a t r i x G l a p r o t e i n ,MG P )的表达随C K D 的进展而逐渐增加,并可以预测肾脏病的预后[7]㊂1 肾脏基质G l a 蛋白(M G P )1.1 MG P 功能的发现 MG P 是由软骨和动脉壁细胞分泌的一种分子量为12k D a 的小蛋白质,包含了84个氨基酸㊁5个谷氨酸残基和3个丝氨酸残基㊂MG P 是第一种在体内外被公认为V C 抑制剂的蛋白,也是人体中最有效的天然钙化抑制剂[8]㊂此外,这种小蛋白质是唯一已知的不仅可以抑制,而且可以逆转钙化的因素㊂MG P 的功能最早是由L u o等[9]发现的,他们使用了MG P 基因被敲除的小鼠(MG P -/-),发现它们出生6~8周后全部死于主动脉钙化㊂另有研究发现K e u t e l 综合征与基因突变所导致的无功能的MG P 表达有关㊂K e u t e l 综合征是一种常染色体隐性遗传的罕见病,其特征是软组织和软骨过度异位钙化[10]㊂这一事实也提示MG P 在V C 预防中起着关键作用㊂在一项随访了7年的118名已确诊的糖尿病肾病患者的队列研究中发现,MG PT -138C (r s 1800802)多态性是V C 和心血管疾病死亡率的强而独立的预测因子[11]㊂S h e n g 等[12]对23项涉及5773名对照和5280例病例的研究进行了荟萃分析,发现MG P G -7A (r s 1800801)多态性是血管内膜和中层钙化的独立预测因子㊂这些发现提示V C 发病机制和MG P 分子活化可能存在遗传基础㊂MG P 基因的表达可以通过各种机制进行调节,这些机制有可能成为预测V C 进展的基因组生物标志物[13]㊂1.2 MG P 的活化过程和4种亚型 MG P 是维生素K 依赖蛋白家族的成员之一,该家族是一组涉及凝血㊁V C 和骨代谢的17种人类蛋白[14]㊂所有维生素K 依赖蛋白均具有无活性的谷氨酸残基,需要维㊃758㊃‘临床荟萃“ 2020年9月20日第35卷第9期 C l i n i c a l F o c u s ,S e pt e m b e r 20,2020,V o l 35,N o .9Copyright ©博看网. All Rights Reserved.生素K 才能使谷氨酸的γ-羧基转化为γ-羧基谷氨酸(γ-c a r b o x y gl u t a m a t e ,G l a )㊂维生素K 是MG P 的羧化过程中的辅助因子,被进一步循环再用于另一次羧化㊂MG P 因为谷氨酸转化为G l a 导致分子结构和形态发生变化而被激活㊂此外,MG P 在羧化之后,还需要进行丝氨酸残基的磷酸化后才具有生物活性㊂羧化是维生素K 的高度依赖性反应,被认为是MG P 激活中最关键的步骤㊂MG P 只有在羧化和磷酸化之后,才能获得与钙㊁羟基磷灰石和骨形成蛋白2(B o n e M o r p h o g e n e t i cP r o t e i n -2,B M P -2)结合的能力,从而抑制V C [15]㊂因此,MG P 由于其经历不同的转化方式而以各种形式存在于循环中:完全活化的羧化和磷酸化的MG P ,完全未活化的未磷酸化㊁未羧化的MG P (d p -u c MG P ),部分活化的未磷酸化但羧化的MG P (d p -c MG P )以及部分活化的未羧化但磷酸化的MG P (u c MG P 或p u c MG P ),如图1㊂有学者研究建立了一个健康人群中总MG P 的参考区间:6~108μg/L ,认为较高的总MG P 可以确定患有血管疾病的患者㊂M a a s t r i c h t 的科学团队是第一个开发出可定量u c MG P 总量的特异性抗体的团队㊂R o i je r s 等[16]在尸体解剖过程中收集了12个人体冠状动脉组织,并利用质子显微镜评估了样本的微钙化组成状态㊂将所有动脉粥样硬化病变根据微钙化的严重程度分成了4个亚型:Ⅰ型为动脉粥样硬化前病变,Ⅱ~Ⅳ型为内膜层钙化范围逐渐扩大阶段㊂尽管在Ⅰ型病变中未检测到u c MG P ,但在Ⅱ~Ⅳ型中u c MG P 染色呈增强趋势㊂相反,完全羧化的MG P 和B M P -2的染色在Ⅰ型中较弱,而在Ⅳ型病变中则明显增强㊂由此可见,羧化的MG P 和B M P -2均与微钙化程度相关,MG P 可能通过直接结合B M P -2来充当局部钙化的抑制因子㊂此外,MG P的羧化不足伴随着冠状动脉的早期亚临床微钙化[17]㊂图1 M G P 的活化过程1.3 d p -u c MG ㊁维生素K 和V C S c h u r g e r s 等[18]通过动物模型发现,维生素K 拮抗剂华法林治疗6周导致实验动物体内的MG P 羧化不足,从而加快V C ㊂由此可见,MG P 需要维生素K 才能具有生物活性㊂为了探讨是否可以通过摄入维生素K 来逆转现有的V C ,他们将所有大鼠分为低或高剂量的维生素K 或华法林组后继续接受8周的实验㊂与补充维生素K 的大鼠相比,华法林组的大鼠表现出V C 加速,动脉粥样硬化状态增强和循环u c MG P 水平明显升高㊂补充高剂量维生素K 导致V C 下降了37%㊂所以,维生素K 介导的MG P 羧化过程是激活这种强大的V C 天然抑制因子的必不可少的步骤㊂这项研究首次表明补充维生素K 可能停止㊁甚至逆转V C ㊂尽管在体内㊁外研究中均取得了初步的结论,但基于人群的研究未能显示u c MG P 与V C 和心血管疾病有明确的关联[19-20]㊂这可能是由于测量u c MG P 的方法对MG P 磷酸化状态不敏感,因为它同时评估了d p -u c MG P 和p u c MG P ㊂与非磷酸化相比,磷酸化形式的MG P (无论羧化状态)对结合游离钙㊁羟基磷灰石晶体和B M P -2的亲和力更高,因此对V C 的发展具有不同的影响㊂另有研究发现丝氨酸残基的磷酸化是MG P 活化的关键步骤[21]㊂特异性夹心抗体的发展使得d p -u c MG P 可以与其他MG P 形式分开定量,表明循环中的d p -u c MG 相比p u c MG P 是更可靠的维生素K 状态指标㊁更强的V C 标志物和更佳的心血管疾病的预测指标[8,21-22]㊂㊃858㊃‘临床荟萃“ 2020年9月20日第35卷第9期 C l i n i c a l F o c u s ,S e pt e m b e r 20,2020,V o l 35,N o .9Copyright ©博看网. All Rights Reserved.在一般人群中,d p-u c MG P与V C[23-24]㊁动脉张力[25]和心血管疾病[26]的各种标志物密切相关㊂类似的结果在以高动脉粥样硬化状态为特征的队列中(如心力衰竭和心血管疾病患者)也有报道[20,27]㊂由于C K D会使血管内膜和中膜以及软组织处于加速钙化的状态,因此一些研究者探讨了这些患者中d p-u c MG P与V C和心血管疾病之间的关系㊂M a a s t r i c h t研究小组对107名不同C K D分期的肾病患者进行了前瞻性研究,发现d p-u c MG P与主动脉钙化评分㊁肾功能恶化和全因死亡率密切相关[28]㊂在67例合并糖尿病的C K D2-5期患者队列中,d p-u c MG P随着疾病进展为终末期肾病而逐渐增高,并强烈预测了全因和心血管疾病死亡率[11]㊂同样,一些研究者在C K D患者中发现d p-u c MG P与各种V C 标记物和肾功能之间关系密切[19,29-30]㊂在透析患者中,越来越多的证据表明d p-u c MG P水平与心血管疾病死亡率和发病率之间有很强的独立联系[31]㊂L e e s等[32]对27项研究进行荟萃分析后表明,补充维生素K虽然不能降低血管硬度,但是可以显著改善V C㊂2M G P抑制V C的主要机制目前MG P抑制V C的机制不完全清楚,主要有以下几条作用途径:(1)结合游离钙离子㊂由于MG P 带负电,所以它不仅对游离钙表现出高亲和力,还直接与循环中的钙分子和堆积在血管壁内的羟基磷灰石晶体结合,形成非活性复合物㊂MG P进一步通过吸引吞噬细胞和巨噬细胞来激活这些复合物的自噬清除[33]㊂MG P可以将血液循环中的游离钙清除并引导其进入骨骼内㊂(2)拮抗骨形态发生蛋白的作用㊂MG P除了作为钙螯合剂外,改善V C的另一种分子途径是下调B M P-2,这是一种众所周知的V C 促进因子㊂B M P-2可使血管平滑肌细胞转化为成骨细胞表型㊂MG P通过抑制B M P-2与其受体的结合来消除B M P-2的表达㊂S w e a t t等[34]在衰老动物模型的动脉粥样硬化血管中发现了未活化的MG P和游离活化的B M P-2(与其受体结合后),而在无病变的动脉壁中发现了MG P与B M P-2紧密结合而成的无活性的复合物㊂Z h a n g等[35]的研究首次确定了MG P在破骨细胞的分化和功能中起着至关重要的作用,MG P可以抑制破骨细胞的分化和骨吸收㊂(3)结合平滑肌细胞分泌的基质囊泡㊂平滑肌细胞具有表型可塑性,其释放的基质囊泡可以抑制钙超载导致的平滑肌细胞分泌凋亡小体㊂MG P可以与平滑肌细胞分泌的基质囊泡结合抑制V C的形成㊂(4)抗凋亡作用㊂血管平滑肌细胞中的凋亡小体有利于钙结晶盐的沉积,在V C的形成过程中发挥重要重用,MG P的表达会随着凋亡小体的增长而增高,MG P在体内发挥一定的抗凋亡作用[11]㊂(5)与细胞外基质成分结合[36]㊂MG P可以与玻连蛋白和弹力蛋白这两种细胞外基质成分结合,发挥抑制V C的作用,具体机制目前尚不明确㊂(6)形成胎球蛋白-MG P-矿化复合物㊂胎球蛋白和MG P与矿物质核形成稳定的复合物,可以抑制矿物质的增长㊁聚集和沉积㊂3研究展望MG P是V C强效抑制剂之一,需要维生素K才能被活化㊂在C K D和透析患者中,维生素K含量较低的情况已被反复证实㊂目前尚无关于在C K D和透析患者中补充维生素K的建议㊂但是,越来越多的证据表明补充维生素K可能会改善V C,以及通过激活MG P延缓心血管疾病的进展㊂至今还没有关于维生素K摄入的毒性或严重不良反应的报道,因此它被认为是一种安全的疗法,具有潜在的重要临床意义㊂所以,C K D和透析患者是否需要监测维生素K缺乏症?如果是,用什么方法?他们应该补充维生素K吗?目前,一些研究正在调查摄入维生素K是否有抗动脉粥样硬化作用㊂这些研究的结果可能会在这一领域有所启发,可能会建议C K D和透析患者每日适当摄入维生素K㊂参考文献:[1] M a n j u n a t hG,T i g h i o u a r tH,C o r e s h J,e t a l.L e v e l o f k i d n e yf u n c t i o na sar i s kf a c t o rf o rc a r d i o v a s c u l a ro u t c o m e si nt h ee l d e r l y[J].K i d n e y I n t,2003,63(3):1121-1129.[2] R o u m e l i o t i sS,R o u m e l i o t i s A,D o u n o u s iE,e ta l.D i e t a r ya n t i o x i d a n t s u p p l e m e n t s a n d u r i c a c i d i n c h r o n i c k i d n e yd i se a s e:a r e v i e w[J].N u t r i e n t s,2019,11(8):1911.[3] L i a k o p o u l o sV,R o u m e l i o t i s S,Z a r o g i a n n i s S,e t a l.O x i d a t i v es t r e s s i n h e m o d i a l y s i s:c a u s a t i v e m e c h a n i s m s,c l i n i c a li m p l i c a t i o n s,a n d p o s s i b l e t h e r a p e u t i c i n t e r v e n t i o n s[J].S e m i nD i a l,2019,32(1):58-71.[4] R o u m e l i o t i sS,E l e f t h e r i a d i s T,L i a k o p o u l o s V.I so x i d a t i v es t r e s s a n i s s u e i n p e r i t o n e a l d i a l y s i s?[J].S e m i nD i a l,2019,32(5):463-466.[5] R e n n e n b e r g R J,K e s s e l s A G,S c h u r g e r sL J,e ta l.V a s c u l a rc a l c i f i c a t i o na sa m a r k e ro fi n c r e a s e dc a rd i o v a s c u l a rr i s k:am e t a-a n a l y s i s[J].V a s cH e a l t hR i s k M a n a g,2009,5(1):185-197.[6]S c h l i e p e rG,W e s t e n f e l dR,B r a n d e n b u r g V,e t a l.I n h i b i t o r so f c a l c i f i c a t i o n i nb l o o da n du r i n e[J].S e m i n D i a l,2007,20(2):113-121.[7] M i y a t aK N,N a s tC C,D a iT,e t a l.R e n a lm a t r i xG l a p r o t e i ne x p r e s s i o n i n c r e a s e s p r o g r e s s i v e l y w i t hC K Da n d p r e d i c t s r e n a lo u t c o m e[J].E x p M o l P a t h o l,2018,150(1):120-129. [8] R o u m e l i o t i s S,D o u n o u s i E,E l e f t h e r i a d i sT,e t a l.A s s o c i a t i o no f t h e i n a c t i v ec i r c u l a t i n g M a t r i x G l a p r o t e i n w i t hv i t a m i n Ki n t a k e,c a l c i f i c a t i o n,m o r t a l i t y,a n dc a r d i o v a s c u l a rd i s e a s e:a㊃958㊃‘临床荟萃“2020年9月20日第35卷第9期 C l i n i c a l F o c u s,S e p t e m b e r20,2020,V o l35,N o.9Copyright©博看网. 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五邑大学学报（自然科学版）JOURNAL OF WUYI UNIVERSITY （Natural Science Edition ）第35卷 第2期 2021年 5月V ol.35 No.2 May 2021文章编号：1006-7302（2021）02-0015-05三个矩阵乘积的Moore -Penrose 逆的正序律周婉娜，熊志平（五邑大学 数学与计算科学学院，广东 江门 529020）摘要：Moore-Penrose 逆（简称M-P 逆）是矩阵理论中的一个重要分支，它在线性控制理论、投影算法、统计学等领域的广泛应用使其成为一个热点研究问题. 本文利用秩等式和广义Schur 补，研究了3个矩阵乘积的M-P 逆的正序律，得出了正序律()123123++++=A A A A A A 成立的充要条件.关键词：Moore-Penrose 逆；秩等式；广义Schur 补；正序律中图分类号：O151.21 文献标志码：AA Note on the Forward Order Law for Moore-Penrose Inverse of Three Matrix ProductsZHOUWan-na, XIONGZhi-ping(School of Mathematics and Computational Science, Wuyi University, Jiangmen 529020, China)Abstract: Moore -Penrose inverse (M -P inverse) is an important branch of matrix theory due to its extensive applications in linear control theory, projection algorithm, statistics and other fields. In this paper, we study the forward order law for M -P inverse of product of three matrices by using the rank equality and the generalized Schur complement, and the necessary and sufficient conditions for the forward order law()123123++++=A A A A A A are obtained.Key words: Moore -Penrose inverse; Rank equality; Generalized Schur complement; Forward order law1 引言及预备知识矩阵乘积广义逆的反序律和正序律在统计学、微分方程、电网络分析等领域都有着不可或缺的重要作用[1-2]. 20世纪60年代以来，很多学者研究了矩阵乘积广义逆的反序律，例如矩阵乘积的{}1-逆、{}1,3-逆、{}1,2,3-逆、M-P 逆的反序律成立的充要条件，得到了很多有趣的结果和一些重要的应用算法[2-3]. 关于矩阵乘积广义逆的正序律的理论与应用研究相对较少，很多相关问题还需要进一步的解决，因此矩阵乘积广义逆的正序律成为了一个热点研究课题.在本文中，m n ⨯C 表示复数域中所有m n ⨯矩阵，m I 为m 阶单位矩阵，m n ⨯0为m n ⨯零矩阵（常用0代表合适的零矩阵）. 对任意的m n ⨯∈A C ，*A 为A 的共轭转置，()r A 为A 的秩，()R A 为A 的值域，收稿日期：2020-11-03基金项目：广东省普通高校特色创新类项目（2018KTSCX234）；广东省本科高校教学质量与教学改革工程项目（GDJX2018004；GDJX2018014）；江门市基础与理论科学研究类科技计划项目（2020JC01010）；五邑大学港澳联合研发基金资助项目（2019WGALH20）作者简介：周婉娜（1996—），女，广东江门人，在读硕士生，研究方向为矩阵与算子广义逆；熊志平，教授，博士，硕士生导师，通信作者，主要从事矩阵与算子广义逆的研究.五邑大学学报（自然科学版） 2021年16 ()N A 为A 的零空间，相关概念参见文献[1,3].定义1[4] 设m n ⨯∈A C ，满足下列4个Penrose 条件：1）=AXA A ，2）=XAX X ，3）()*=AX AX ，()*=XA XA 的矩阵n m ⨯∈X C 称为A 的M-P 逆，记+=X A .引理1[1] 矩阵的M-P 逆满足以下性质：********()()()++++===A AA A A A A A AA A A .引理2[5] 设矩阵A 、B 、C 和D 满足以下条件：()()R R ⊆B A ，**()()R R ⊆C A 或()()R R ⊆C D ，**()()R R ⊆B D ，则()()r r r +⎛⎫=+- ⎪⎝⎭A B A D CA B C D 或()()r r r +⎛⎫=+- ⎪⎝⎭A B D A BD C C D . 引理3[6] 设i i s t i ⨯∈A C ，1,2,,i n = ；1i i s t i +⨯∈B C ，1,2,,1i n =- ，再设1i i i i +=B A X A ，1,2,,1i n =- ， （1）则对于某些i X 有：()()i i R R ⊆B A 且**1()()i i R R +⊆B A ，1,2,,1i n =- ， （2）而且n n ⨯分块矩阵112211n n n n --⎛⎫ ⎪ ⎪ ⎪= ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭0000000000000 A A B J A B A B 的M-P 逆可以表示为：(1,)(1,1)(1,2)(1,1)(2,)(2,1)(2,2)(1,)(1,1)(,)n n n n n n n n n n n +-⎛⎫ ⎪- ⎪ ⎪= ⎪--- ⎪ ⎪⎝⎭00000000 F F F F F F F J F F F ，其中，(,)i i i +=F A ，1,2,,i n = ， （3）111(,)(1)j i i i i i j j i j i j -++++++--=- F A B A B A B A ，1i j n ≤≤≤. （4）2 主要结果本节，我们将给出3个矩阵乘积的M-P 逆的正序律()123123++++=A A A A A A 成立的充要条件，相关结论会在下面的3个定理中给出.定理1 设123,,m m ⨯∈A A A C ，则55⨯分块矩阵 ***3333****22223****11112*1m m⎛⎫ ⎪ ⎪⎪= ⎪ ⎪ ⎪⎝⎭000000000000000I A A A A M A A A A A A A A A A I A 的M-P 逆可以表示为： (1,5)(1,4)(1,3)(1,2)(1,1)(2,5)(2,4)(2,3)(2,2)(3,5)(3,4)(3,3)(4,5)(4,4)(5,5)+⎛⎫⎪⎪ ⎪=⎪⎪ ⎪⎝⎭0000000000M M M M M M M M M M M M M M M M ，第35卷 第2期 17周婉娜等：三个矩阵乘积的Moore -Penrose 逆的正序律其中，(),i j M 可根据引理3中的式（3）和（4）给出，特别地，()1,5+=M PM Q 51************111112222233333(1)()()()m m -+++++=-=I A A A A A A A A A A A A A A A I123+++A A A ， （5）其中，()()*,,,,,,,,,m m ==00000000P I Q I .证明 因为*****111111111==()m m ++A I A A A I A A A A A ，所以*1()()m R R ⊆A I ，*1111()()R R ⊆A A A A . （6）因为**********121112221111122222==()()++++A A A A A A A A A A A A A A A A A A ，所以****12111()()R R ⊆A A A A A ，*21222()()R R ⊆A A A A A . （7）因为**********232223332222233333==()()++++A A A A A A A A A A A A A A A A A A ，所以****23222()()R R ⊆A A A A A ，*32333()()R R ⊆A A A A A . （8）因为*****333333333==()m m ++A A A A I A A A A A I ，所以***3333()()R R ⊆A A A A ，3()()m R R ⊆A I . （9）结合式（6-9）以及引理3中的式（1-2），可以得出定理1的结论. 特别地，根据引理1，我们知道****()++=A A AA A A . 因此，可得：123(1,5)++++==M PM Q A A A .为了得到定理2，我们首先证明以下秩等式：对于任意的矩阵i A 来说，****()()()()i i i i i i i r r r r ===A A A A A A A . （10）证明 因为**********()()()()[()]()i i i i i i i i i i i i i i i i i r r r r r r ++≤≤==≤A A A A A A A A A A A A A A A A A 且*()()i i r r =A A ，所以****()()()()i i i i i i i r r r r ===A A A A A A A .定理2 设123,,m m⨯∈A A A C且123=A A A A ，M ,P 和Q 由定理1给出，则： 123()2()()()r m r r r =+++M A A A ， ()()R R ⊆Q M ，**()()R R ⊆P M ， ()()R R ⊆QA M ，***()()R R ⊆P A M .证明 构造可逆矩阵1234,,,D D D D 和列矩阵5D 如下：*11mm m m m ⎛⎫- ⎪ ⎪ ⎪= ⎪ ⎪ ⎪⎝⎭00000000000000000I A I D I I I ，**1122()m m m m m +⎛⎫ ⎪- ⎪ ⎪= ⎪ ⎪ ⎪⎝⎭000000000000000I I A A A D I I I ，**3223()mm m mm I +⎛⎫ ⎪ ⎪ ⎪=- ⎪ ⎪ ⎪⎝⎭00000000000000000I I D I A A A I ，4*33()m m m m m +⎛⎫ ⎪ ⎪ ⎪= ⎪- ⎪ ⎪⎝⎭0000000000000000I I D I I A A I ，5m ⎛⎫ ⎪ ⎪ ⎪= ⎪⎪⎪⎝⎭0000D I 则五邑大学学报（自然科学版） 2021年18 **333**1234222**111m m⎛⎫ ⎪ ⎪ ⎪= ⎪ ⎪ ⎪⎝⎭000000000000000000I A A A MD D D D A A A A A A I ，且12345m ⎛⎫ ⎪ ⎪⎪== ⎪ ⎪⎪⎝⎭0000I MD D D D D Q . 因此1234123()()2()()()r r m r r r ==+++M MD D D D A A A ，且12345()()()()R R R R ⊆=⊆QA Q MD D D D D M .另一方面，构造可逆矩阵1234,,,T T T T 和行矩阵5T 如下：*31mm m m m ⎛⎫ ⎪- ⎪ ⎪=⎪ ⎪ ⎪⎝⎭0000000000000000000I A I T I I I ，**2233()mm m m m +⎛⎫ ⎪ ⎪ ⎪=- ⎪ ⎪ ⎪⎝⎭0000000000000000000I I T A A A I I I ， 3**122()m mmm m +⎛⎫⎪ ⎪ ⎪= ⎪- ⎪ ⎪⎝⎭0000000000000000000I I T I A A A I I ，4*11()m m m m m +⎛⎫ ⎪ ⎪ ⎪= ⎪⎪⎪-⎝⎭0000000000000000000I I T I I A A I ，5(,,,,)m =0000T I . 则 **333**4321222**111m m⎛⎫⎪ ⎪⎪= ⎪ ⎪ ⎪⎝⎭00000000000000000I A A A T T T T M A A A A A A I ，且54321(,,,,)m ==0000T T T T T M I P ，因此，**********12345()()()()R R R R ⊆=⊆P A P M T T T T T M .利用定理1和定理2可以得到定理3.定理3 设123,,m m ⨯∈A A A C ，123=A A A A ，123+++=X A A A 且M ,P 和Q 由定理1给出，则以下等式等价：1）()123123+++++===A A A A A A A X ； 2）()()()()11232r r r r r ****⎛⎫-=+++ ⎪⎝⎭A AA A E A A A A E AN . 其中，()1,,,m =000E I ，()2,,,m *=000E I ，以及***3333****22223****11112*1⎛⎫ ⎪⎪= ⎪ ⎪ ⎪⎝⎭000000000A A A A A A A A A N A A A A A A . 证明 由定理1可得123++++==X A A A PM Q 且12⎛⎫= ⎪⎝⎭0E M E N . 显然，()123123+++++===A A A A A A A X成立的充要条件为()()0r r +++-=-=A X A PM Q . （11）由上式构造一个33⨯分块矩阵****⎛⎫ ⎪=- ⎪ ⎪⎝⎭000MQ G A AA A P A ，根据引理2，第35卷 第2期 19周婉娜等：三个矩阵乘积的Moore -Penrose 逆的正序律()()*******,,R R R R ⎛⎫⎛⎫⎛⎫⊆⊆⎪ ⎪ ⎪--⎝⎭⎝⎭⎝⎭000Q M M P A A A AA AA A ，则()()******,r r r +⎡⎤⎛⎫⎛⎫⎛⎫=+=⎢⎥⎪ ⎪ ⎪--⎢⎥⎝⎭⎝⎭⎝⎭⎣⎦0000MM Q G P A A AA A AA A ()()******,r r ++⎡⎤⎛⎫⎛⎫⎛⎫⎢⎥ ⎪+= ⎪ ⎪ ⎪-⎢⎥-⎝⎭⎝⎭⎝⎭⎣⎦0M M Q P A A AA A A AA ()()()******r r r ++⎡⎤++-⎢⎥⎣⎦M A AA PM Q A A AA A . 根据引理2、定理2以及式（10），可得：()()()()()()()()()1232r r r r m r r r r r ++++=++-=+++++-G M A PM Q A A A A A PM Q A . （12）另一方面，()1*22**********1***m m m mr r r r ⎛⎫⎛⎫⎛⎫ ⎪ ⎪-⎪ ⎪ ⎪=-=== ⎪ ⎪ ⎪--- ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭0000000000000000000E I I MQ E N N E A G A AA A A AA A A E A AA A P A I A I A *****221*******11222m mr m r m r ⎛⎫⎪-⎛⎫⎛⎫-- ⎪=+=+ ⎪ ⎪ ⎪----⎝⎭⎝⎭ ⎪⎝⎭0000000000I N E A N E A A AA A E A E A AA A E A AA E A N I （13） 结合式（11-13）可知，定理3成立.3 结论对任意的矩阵,1,2,3m m i i ⨯∈=A C ，本文利用秩等式和广义Schur 补的性质，得出了3个矩阵乘积的M-P 逆正序律()123123++++=A A A A A A 成立的充要条件.参考文献[1] 王国荣. 矩阵与算子广义逆[M]. 北京：科学出版社，1994.[2] BEN-ISRAEL A, GREVILLE T N E. Generalized inverse: theory and application [M]. 2nd Edition. New York:Springer-Verlag, 2003: 35-54.[3] WANG Guorong, WEI Yimin, QIAO Sanzheng. Generalized inverse: theory and computations [M]. Beijing:Science Press, 2004: 9-26.[4] PENROSE R. A generalized inverse for matrix [J]. Proc Cambridge Philos Soc, 1955, 51: 406-413.[5] MATSAGLIA G, STYAN G P H. Equalities and inequalities for ranks of matrices [J]. Linear and MultilinearAlgebra, 1974, 2: 269-292.[6] TIAN Yongge. Reverse order laws for the generalized inverses of multiple matrix products [J]. Linear Algebraand its Applications, 1994, 211: 85-100.[责任编辑：熊玉涛]。

Multi-agent actions under uncertainty: situation calculus,discrete time,plans andpoliciesDavid PooleDepartment of Computer ScienceUniversity of British ColumbiaV ancouver,B.C.,Canada V6T1Z4poole@cs.ubc.cahttp://www.cs.ubc.ca/spider/pooleApril23,1997AbstractWe are working on a logic to combine the advantages offirst-order logic, but using Bayesian decision theory(or more generally game theory)as a ba-sis for handing uncertainty.This forms a logic for multiple agents under un-certainty.These agents act asynchronously,can have their own goals,have noisy sensors,and imperfect effectors.Recently we have developed the in-dependent choice logic that incorporates all of these features.In this paper we discuss two different representations of time within this framework:the situation calculus and what is essentially the event calculus.We show how they both can be used,and compare the different ontological commitments made by each.Uncertainty is handled in terms of a logic which allows for independent choices and a logic program that gives the consequences of the choices.There are probabilities over the choices by nature.As part of theconsequences are a specification of the utility of(final)states.In the situa-tion calculus,agents adopt programs and programs lead to situations in pos-sible worlds(which correspond to the possible outcomes of complete histo-ries);given a probability distribution over possible worlds,we can get theexpected utility of a program.In the event calculus view,actions are propo-sitions and agents adopt policies which are logic programs to imply what theagent will do based on what it observes.Again the expected value of a policycan be computed.The aim is to choose the plan or policy that maximizes theexpected utility.This paper overviews both approaches,and explains why Ithink the event calculus is the most promising approach.1Introduction1.1Logic and UncertaintyThere are many normative arguments for the use of logic in AI(see e.g.,Nilsson 1991,Poole,Mackworth&Goebel1997).These arguments are usually based on reasoning with symbols with an explicit denotation,allowing relations amongst individuals,and quantification over individuals.This is often translated as need-ing(at least)thefirst-order predicate calculus.However,thefirst-order predicate calculus has very primitive mechanisms for handling uncertainty.There are also normative arguments for Bayesian decision theory and game theory(see e.g.,V on Neumann&Morgenstern1953,Savage1972)for handling uncertainty.These are based on theorems that show that,under certain reasonable assumptions,rational decision makers will act as though they are using probabili-ties and utilities and maximizing their expected utilities.Game theory(Von Neu-mann&Morgenstern1953,Myerson1991,Fudenberg&Tirole1992)is the ex-tension of Bayesian decision theory to include multiple intelligent agents.We would like to combine the advantages of logic with those of Bayesian de-cision/game theory.The independent choice logic(ICL)(Poole1997)is designed with the goal of including the advantages of logic,but handling all uncertainty us-ing Bayesian decision or game theory.The idea is to not use disjunction,but rather to allow agents,including nature, to make choices from a choice space,and use a restricted underlying logic to spec-ify the consequences of the choices.To start off without disjunction,we can adopt acyclic logic programs(Apt&Bezem1991)as the underlying logical formalism. What is interesting is that simple logic programming solutions to the frame prob-lem(see e.g.,Shanahan1997,Chapter12)seem to be directly transferrable to the2ICL which has more sophisticated mechanisms for handling uncertainty than the predicate calculus.I would even dare to venture that the main problems with for-malizing action within the predicate calculus arise because of the inadequacies of disjunction to represent the sort of uncertainty we need.When mixing logic and probability,one can extend a rich logic with probabil-ity,and have two sorts of uncertainty:that uncertainty from the probabilities and that from disjunction in the logic(Bacchus1990,Halpern&Tuttle1993).An al-ternative that is pursued in the independent choice logic is to have all of the uncer-tainty in terms of probabilities.The underlying logic is restricted so that there is no uncertainty in the logic1;every set of sentences has a unique model.In particular we choose the logic of acyclic logic programs under the stable model semantics; this is a practical language with the unique model property.All uncertainty is han-dled by what can be seen as independent stochastic mechanisms.A deterministic logic program that gives the consequences of the agent’s choices and the random outcomes.In this manner we can get a simple mix of logic and Bayesian decision theory(Poole1997).1.2Actions and UncertaintyThe combination of decision or game theory and planning(Feldman&Sproull 1975)is very appealing.The general idea of planning is to construct a sequence of steps,perhaps conditional on observations that solves a goal.In decision-theoretic planning,this is generalised to the case where there is uncertainty about the envi-ronment and we are concerned,not only with solving a goal,but what happens under any of the contingencies.Goal solving is extended to the problem of max-imizing the agent’s expected utility,where the utility is an arbitrary function of thefinal state(or the accumulation of rewards received earlier).Moreover in the multi-agent case,each agent may have a different utility,they carry out actions asynchronously,and the actions of all of the agents affect the utility for each agent. Moreover there is uncertainty about the environment,and it is often optimal for an agent to act stochastically.It is our goal to write a logic that allows for prac-tical reasoning for domains that include multiple agents,utilities,uncertainty and stochastic actions.Within the independent choice logic,we have considered both the situation cal-culus(Poole1996),discrete time(Poole1997),and continuous time(Poole1995). In this paper we compare the situation calculus(McCarthy&Hayes1969)and the discrete time frameworks in this framework.Recently there have been claims made that Markov decision processes(MDPs) (Puterman1990)are the appropriate framework for developing decision theoretic planners(e.g.,Boutilier,Dearden&Goldszmidt1995).MDPs,and dynamical sys-tem in general(Luenberger1979)are based on the notion of a state:what is true at a time such that the past at that time can only affect the future from that time by affecting the state.In terms of probability:the future is independent of the past given the state.This is called the Markov property.In the most general frame-work the agents individually have states(these are called belief states)and idea is to construct a state transition function that specify how the agent’s belief state is updated from its previous belief state and its observations,and a command func-tion(policy)that specifies what the agent should do based on its observations and belief state(Poole et al.1997,Chapter12).In fully observable MDPs,the agent can observe the actual state and so doesn’t need belief states.In partially observ-able MDPs,the belief state is a probability distribution over the actual states of the system,and the state transition function is given by Bayes’rule.In between these are agents that have limited memory or limited reasoning capabilities.For a representation for time,actions and states we can adopt a number of dif-ferent representations.One is to consider robot programs as policies(Poole1995) where actions are propositions,and agents adopt logic programs that specify what they will do based on their observations and belief state.An alternative is to rep-resent actions in the situation calculus,where the agents have conditional plans (Poole1996).Both are discussed here.All of the uncertainty in our rules is relegated to independent choices as in the independent choice logic(Poole1997)(an extension of probabilistic Horn abduc-tion(Poole1993)to include multiple agents and negation as failure).2The Independent Choice LogicAn independent choice space theory is make of two principal components:a choice space that specifies what choice can be made,and a set of facts that specifies what follows from the choices.Definition2.1A choice space is a set of sets of ground atomic formulae,such that if,and are in the choice space,and then.An element4of a choice space is called a choice alternative(or sometimes just an alternative). An element of a choice alternative is called an atomic choice.Definition2.2Given choice space,a selector function is a mapping such that for all.The range of selector function,written is the set.The range of a selector function is called a to-tal choice.In other words,a total choice is a selection of one member from each element of.Definition2.3The Facts,,are sentences in some base logic with the following two properties:For every selector function,the set of sentences is definitive on every proposition;that is,for every proposition either or,where is logical consequence in the underlying logic.For all atomic choices,iff.The restriction really means that the base logic contains no uncertainty.All uncer-tainty is handled in the choice space.The semantics of an ICL is defined in terms of possible worlds.There is a pos-sible world for each selection of one element from each alternative.The atoms which follow using the consequence relation from these atoms together with are true in this possible world.Definition2.4Suppose we are given a base logic and ICL theory.For each selector function there is a possible world.We write,read“is true in world based on”,iff.When understood from context,the is omitted as a subscript of.The fact that every proposition is either true or false in a possible world follows from the definitiveness of the base logic.Note that,for each alternative and for each world,there is exactly one element of that’s true in.In particular,,and for all .The base logic we use is that of acyclic logic programs(Apt&Bezem1991), such that no atomic choice is in the head of any rule.means is true in the (unique)stable model of.This logic is definitive in the above sense and is rich enough to axiomatise many of the domains we are interested in.Note that acyclic logic programs allow recursion,but all recursion must be well founded.This semantic construction is the core of the ICL.Other components we require for different applications include:5is afinite set of agents.There is a distinguished agent called“nature”.is a function from.If then agent is said to control alternative.If is an agent,the set of alternatives controlled by is.Note that.is a function such that,.That is,for each alternative controlled by nature,is a probability measure over the atomic choices in the alternative.When each agent(other than nature)makes a choice(possibly stochastic)from each alternative it controls,we can determine the probability of any proposition. The probability of a proposition is defined in the standard way.For afinite choice space2,the probability of any proposition is the sum of the probabilities of the worlds in which it is true.The probability of a possible world is the product of the probabilities of the atomic choices that are true in the world.That is,the atomic choices are(unconditionally)probabilistically independent.Poole(1993)proves that such independent choices together with an acyclic logic program can represent anyfinite probability distribution.Moreover the structure of the rule-base mirrors the structure of Bayesian networks(Pearl1988)3.Similarly we can define the ex-pectation of a function that has a value in each world,as the value averaged over all possible worlds,weighted by their probability.See Poole(1997)for more details on the ICL.3The Situation Calculus and the ICLIn this section we sketch how the situation calculus can be embedded in the ICL. What we must remember is that we only need to axiomatise the deterministic as-pects in the logic programs;the uncertainty is handled separately.What gives us confidence that we can use simple solutions to the frame problem,for example,is that every statement that is a consequence of the facts that doesn’t depend on the atomic choices is true in every possible world.Thus,if we have a property thatdepends only on the facts and is robust to the addition of atomic choices,then it will follow in the ICL;we would hope than any logic programming solution to the frame problem would have this property.Before we show how to add the situation calculus to the ICL,there are some design choices that need to be made even to consider just single agents.In the deterministic case,the trajectory of actions by the(single)agent up to some time point determines what is true at that point.Thus,the trajectory of actions,as encapsulated by the‘situation’term of the situation calculus (McCarthy&Hayes1969,Reiter1991)can be used to denote the state,as is done in the traditional situation calculus.However,when dealing with un-certainty,the trajectory of an agent’s actions up to a point,does not uniquely determine what is true at that point.What random occurrences or exoge-nous events occurred also determines what is true.We have a choice:we can keep the semantic conception of a situation(as a state)and make the syntactic characterization more complicated by perhaps interleaving exoge-nous actions,or we can keep the simple syntactic form of the situation calcu-lus,and use a different notion that prescribes truth values.We have chosen the latter,and distinguish the‘situation’denoted by the trajectory of actions, from the‘state’that specifies what is true in the situation.In general there will be a probability distribution over states resulting from a set of actions by the agent.It is this distribution over states,and their corresponding utility, that we seek to model.This division means that agent’s actions are treated very differently from ex-ogenous actions that can also change what is true.The situation terms define only the agent’s actions in reaching that point in time.The situation calculus terms indicate only the trajectory,in terms of steps,of the agent and essen-tially just serve to delimit time points at which we want to be able to say what holds.This is discussed further in Section5.None of our representations assume that actions have preconditions;all ac-tions can be attempted at any time.The effect of the actions can depend on what else is true in the world.This is important because the agent may not know whether the preconditions of an action hold,but,for example,may be sure enough to want to try the action.When building conditional plans,we have to consider what we can condi-tion these plans on.We assume that the agent has passive sensors,and that7it can condition its actions on the output of these sensors.We only have one sort of action,and these actions only affect‘the world’(which includes both the robot and the environment).All we need to do is to specify how the agent’s sensors depend on the world.This does not mean that we can-not model information-producing actions(e.g.,looking in a particular place)—these information producing actions produce effects that make the sensor values correlate with what is true in the world.The sensors can be noisy;the value they return does not necessarily correspond with what is true in the world(of course if there was no correlation with what is true in the world, they would not be very useful sensors).Before we introduce the probabilistic framework we present the situation cal-culus(McCarthy&Hayes1969).The general idea is that robot actions take the world from one‘situation’to another situation.We assume there is a situation that is the initial situation,and a function4that given action and a situa-tion returns the resulting situation.An agent that knows what it has done,knowswhat situation it is in.It however does not necessarily know what is true in that sit-uation.The robot may be uncertain about what is true in the initial situation,what the effects of its actions are and what exogenous events occurred.3.1The ICL SCWithin the ICL we can use the situation calculus as a representation for change. Within the logic,there is only one agent,nature,who controls all of the alterna-tives.These alternatives thus have probability distributions over them.The prob-abilities are used to represent our ignorance of the initial state and the outcomes of actions.We can then use the situations to reflect the“time”at which somefluents are true or not.We model all randomness as independent stochastic mechanisms,such that an external viewer that knew the initial state(i.e.,what is true in the situation),and knew how the stochastic mechanisms resolved themselves would be able to pre-dict what was true in any situation.Given a probability distribution over the initial state and the stochastic mechanisms,we have a probability distribution over the effects of actions.This is modelled by having the mechanisms as atomic choices controlled by nature(and so with a probability distribution).We use logic to specify the transitions specified by actions and thus what is true in a situation.What is true in a situation depends on the action attempted,what was true before and what stochastic mechanism occurred.Afluent is a predicate (or function)whose value in a world depends on the situation;we use the situation as the last argument to the predicate(function).We assume that for eachfluent we can axiomatise in what situations it is true based on the action that was performed, what was true in the previous state and the outcome of the stochastic mechanism.Note that a possible world in this framework corresponds to a complete history.A possible world specifies what is true in each situation.In other words,given a possible world and a situation,we can determine what is true in that situation. Example3.1We can write rules such as,the robot is carrying the key after it has (successfully)picked it up:is true if the agent would succeed if it picks up the key and is false if the agent would fail to pick up the key.The agent typically does not know the value ofwould be an atomic choice.That is5This is now a reasonably standard logic programming solution to the frame problem(Shanahan 1997,Chapter12),(Apt&Bezem1991).It is essentially the same as Reiter’s(1991)solution to the frame problem.It is closely related to Kowalski’s(1979)axiomatization of action,but for each proposition,we specify which actions are exceptional,whereas Kowalski specifies for every every action which propositions are exceptional.Kowalski’s representation could also be used here.9Example3.2For example,an agent is carrying the key as long as the action was not to put down the key or pick up the key,and the agent did not accidentally drop the key while carrying out another action:may be something that the agent does not know whether it is true—there may be a probability that the agent will drop the key.If dropping the key is independent at each situation,we can model this as:The prize depends on whether the robot reached its destination or it crashed.No matter what the definition of any other predicates is,the following definition of will ensure there is a unique prize for each world and situation:The resources used depends not only on thefinal state but on the route taken.To model this we make afluent,and like any otherfluent we axiomatise it: Here we have assumed that non-goto actions cost,and that paths have costs. Paths and their costs can be axiomatised usingthat is true if the path from to via has cost.3.3Axiomatising SensorsWe also need to axiomatise how sensors work.We assume that sensors are pas-sive;this means that they receive information from the environment,rather than “doing”anything;there are no sensing actions.This seems to be a better model of actual sensors,such as eyes or ears,and makes modelling simpler than when sensing is an action.So called“information producing actions”(such as opening the eyes,or performing a biopsy on a patient,or exploding a parcel to see if it is (was)a bomb)are normal actions that are designed to change the world so that the sensors correlate with the value of interest.Note that under this view,there are no information producing actions,or even informational effects of actions;rather var-ious conditions in the world,some of which are under the robot’s control and some of which are not,work together to give varying values for the output of sensors.11A robot cannot condition its action on what is true in the world;it can only condition its actions on what it senses and what it remembers.The only use for sensors is that the output of a sensor depends,perhaps stochastically,on what istrue in the world,and thus can be used as evidence for what is true in the world.Within our situation calculus framework,can write axioms to specify how sensed values depend on what is true in the world.What is sensed depends on the situa-tion and the possible world.We assume that there is a predicate that is true if is sensed in situation.Here is a term in our language,that representsone value for the output of a sensor.is said to be observable.Example3.4A sensor may be to be able to detect whether the robot is at the sameposition as the key.It is not reliable;sometimes it says the robot is at the same po-sition as the key when it is not(a false positive),and sometimes it says that the robot is not at the same position when it is(a false negative).Suppose that noisy sensoris true(in a world)if the robot senses that it is at the key in situation.It can be axiomatised as:Thefluent is true if the sensor is giving a false-positive value in situation,and is true if the sensor is not giving a false negative in situation.Each of these could be part of an atomic choice,which would let us model sensors whose errors at different times are independent.If we had a theory about how sensors break,we could write rules that imply these fluents.123.4Conditional PlansThe idea behind the ICL SC is that agents get to choose situations(they get to choose what they do,and when they stop),and‘nature’gets to choose worlds(there is a probability distribution over the worlds that specifies the distribution of effects ofthe actions).Agents do not directly adopt situations,they adopt‘plans’or‘programs’.In general these programs can involve atomic actions,conditioning on observations, loops,nondeterministic choice and procedural abstraction(Levesque,Reiter,Lesp´e rance, Lin&Scherl1996).In this paper we only consider simple conditional plans whichare programs consisting only of sequential composition and conditioning on obser-vations(Levesque1996,Poole1996)).An example conditional plan is:if then else endIfAn agent executing this plan will start in situation,then do action,then it will sense whether is true in the resulting situation.If is true,it will do then,and if is false it will do then then.Thus this plan either selects the situa-tion or the situation.It selectsthe former in all worlds where is true,and selects the latter inall worlds where is false.Note that each world is definitive on eachfluent for each situation.The expected utility of this plan is the weighted av-erage of the utility for each of the worlds and the situation chosen for that world. The only property we need of is that its value in situation will be able tobe observed.The agent does not need to be able to determine its value beforehand.Definition3.5A conditional plan,or just a plan,is of the formwhere is a primitive actionwhere and are plansif then else endIfwhere is observable;and are plansNote that“”is not an action;the plan means that the agent does not do anything—time does not pass.This is introduced so that the agent can stop with-out doing anything(this may be a reasonable plan),and so we do not need an“if then endIf”form as well;this would be an abbreviation for“if then elseendIf”.13Plans select situations in worlds.We can define a relation:that is true if doing plan in world from situation results in situation. This is similar to the macro of Levesque et al.(1996)and the of Levesque (1996),but here what the agent does depends on what it observes,and what the agent observes depends on which world it happens to be in.We can define the relation in pseudo Prolog as:if then else endIfif then else endIfNow we are at the stage where we can define the expected utility of a plan.The expected utility of a plan is the weighted average,over the set of possible worlds, of the utility the agent receives in the situation it ends up in for that possible world: Definition3.6If our theory is utility complete,the expected utility of plan is6:(summing over all selector functions on)whereifwhere(this is well defined as the theory is utility complete),andis the utility of plan P in world.is the probability of world. The probability is the product of the independent choices of nature.It is easy to show that this induces a probability measure(the sum of the probabilities of the worlds is).4Independent Choice Logic and Reactive PoliciesThere is a completely different way to use the ICL to model time and action.Here we can only sketch the idea;see Poole(1997)for details.We only consider discrete time here.The idea is to made agents and nature in the same way.For the situation cal-culus axiomatization above,the single agent was treated in a completely different way to nature.Symmetry is important when we consider multiple agents.We represent time in terms of the integers.The fact that the agent attempted an an action is represented a propositions indexed by time.We can use a predicate that is true if the agent attempted action at time.What is true at a time depends on what was true at the previous times and what actions have occurred, and the outcome of stochastic mechanisms.This places actions by the agent at the same level as actions by nature.There are two parts to axiomatise.Thefirst is to axiomatise the effect of ac-tions,and the second is to specify what an agent will do based on what it observes (i.e.,its policy).To axiomatise the effect of actions,for the discrete time case we can simply write how what is true at one time depends on what was true at the previous time (including what actions occurred).We would write similar axioms to the situation calculus,but indexed by time,and using as a predicate.Example4.1The axiom for carrying of Example3.1can be stated as:The frame axiom for in Example3.2would look like:Shanahan(1997)in that it is reasoning about a particular course of events.This is true for each possible world,but we can have a probability distribution over possible worlds.We have a mechanism for allowing multiple agents to choose what events that they can control occur,and to allow a probability distribution over events that nature controls.For each world,we only need to worry about what events are true in that world.5DiscussionWe have given a(too)brief sketch of two different representations of change for a single agent under uncertainty in the ICL.See Poole(1997)and Poole(1996)for more details.In some sense the axioms look similar;there is not really much difference be-tween the situation calculus and event calculus axiomatization given here.The ma-jor difference is that the robot programs that are natural for the situation calculus are very different from the reactive policies that are natural for the event calculus.When we extend this to multiple agents,and stochastic actions(as is often needed for agents with limited sensing and communication),the event calculus framework can easily be adapted.Nothing needs to be changed to allow for concurrent actions (the actions by each agent).Each agent adopts its own(private)policy based on what it can sense and what it can do.Extending the situation calculus version to multiple agents isn’t so straightfor-ward.The way we have treated the situation calculus(and we have tried hard to keep it as close to the original as possible)really gives an agent-oriented view of time—the‘situations’in some sense mark particular time points that correspond to the agent completing its actions.Everything else(e.g.,actions by nature or other agents)then has to meld in with this division of time.This is even trickier when we realize that when agents have sloppy actuators and noisy sensors,the actions defining the situations correspond to action attempts;the agent doesn’t really know what it did it only knows what it attempted and what its sensors now tell it.When there are multiple agents,either there has to be a common clock,we need some master agent which the other agents define their state transition,or complex ac-tions(Reiter1996,Lin&Shoham1995).These all mean that the actions need to be carried out lock-step,removing the intuitive appeal of the situation calculus,and making it much closer to the event calculus.The work of Reiter(1996)and Lin& Shoham(1995)assumes a very deterministic world.Not only must the world un-fold deterministically,but you must know how it unfolds.This is very different17。

我的大学老师石焕南教授的专著。

原文地址：拙著新版《Schur-凸函数与不等式》前言和目录作者：延川老猫Schur-凸函数与不等式(Schur convex functions and Inequalities)前言拙著《受控理论与解析不等式》自2012年4月经哈尔滨工业大学出版社出版后，受到国内同行的关注。

5年间，书中所涉及的几乎所有问题都有了后续的研究成果. 本书《Schur凸函数与不等式》是《受控理论与解析不等式》的再版，之所以更名为《Schur凸函数与不等式》，是因为“受控理论”易与浑然不同的“控制理论”混淆，而Schur凸函数是受控理论的核心概念，故以它替代“受控理论”. 与《受控理论与解析不等式》相比较，本书的参考文献新增了近160余篇，基本上是近5年发表的，其中95篇是国内作者发表的(包括笔者及合作者的28篇), 本书收录了这些新成果, 并修补、纠正了《受控理论与解析不等式》一书中的诸多疏漏和错误.本书共分九章, 第一、二章介绍Schur凸函数理论的基本概念和主要定理. 为减少篇幅, 本书略去了一些基础定理的详细证明(这些内容可查阅专著[1]和[26]), 而较为详细地介绍了国内学者对Schur凸函数的新的推广. 第三、四章介绍Schur凸函数在对称函数不等式上的丰富应用.第五、六章分别介绍Schur凸函数在序列不等式和积分不等式上的应用, 第七、八章介绍Schur凸函数在均值不等式上的应用, 新增的第九章是介绍Schur凸函数在几何不等式上的应用.这些年, 国内受控理论的研究方兴未艾, 硕果累累, 愈加受到国际同行的关注. 令人欣慰的是涌现了一些受控理论研究的新人, 例如张静、何灯、许谦、王文、龙波涌、王东生等等.感谢哈尔滨工业大学出版社刘培杰副社长建议我撰写此书，并得到哈尔滨工业大学出版社的出版资助，感谢刘培杰数学工作室这个优秀团队的精心编辑.感谢李明老师等国内同行指出了《受控理论与解析不等式》中的多处疏漏.感谢我的母校北京师范大学的王伯英教授和刘绍学教授对我科研工作的关心和鼓励. 本书保留了《追念胡克教授》一文，并补充了我与胡克教授的两封通信及原文影印件.衷心感谢我的家人对我始终不渝的呵护与照料，使我得以有足够的体力、精力和时间从事我钟爱的科研与写作. 深深地怀念和感恩不久前去世的父亲石承忠，他含辛茹苦地养育了我及五位弟妹，教我一辈子老老实实做人，踏踏实实做事.本人努力使本书不出疏漏, 不留遗憾, 但学识水平所限必有不妥之处, 敬请读者指教.石焕南2016-07-20序前言一般记号目录引言第一章控制不等式1.1 增函数与凸函数1.2 凸函数的推广1.2.1．对数凸函数1.2.2．弱对数凸函数1.2.3. 几何凸函数1.2.4．调和凸函数1.2.5. MN凸函数1.2.6. Wright-凸函数1.3 控制不等式的定义及基本性质1.4 一些常用控制不等式1.5 凸函数与控制不等式1.6 Karamata不等式的推广第二章 Schur凸函数的定义和性质2.1 Schur凸函数的定义和性质2.2 凸函数与Schur凸函数2.3 Karamata不等式的若干应用2.3.1 整幂函数不等式的控制证明2.3.2 一个有理分式不等式的加细2.3.3 一类含有幂平均,算术平均和几何平均的不等式 2.3.4 钟开来不等式的加强2.3.5 凸函数的两个性质的控制证明2.4 Schur凸函数的推广2.4.1 Schur几何凸函数2.4.2 Schur调和凸函数2.4.3 Schur幂凸函数2.4.4 一类条件不等式的控制证明2.5 凸函数和Schur凸函数的对称化2.6 抽象受控不等式2.6.1 抽象受控不等式2.6.2 抽象受控不等式的同构映射第三章 Schur凸函数与初等对称函数不等式3.1 初等对称函数及其对偶式的性质3.2 初等对称函数的商或差的Schur凸性3.2.1 初等对称函数商的Schur凸性3.2.２初等对称函数差的Schur凸性3.2.3 初等对称函数差或商的复合函数3.3 初等对称函数的某些复合函数的Schur凸性 3.3.1 复合函数E_k(x/(1-x))的Schur凸性3.3.2复合函数E_k((1-x)/(x)的Schur凸性3.3.3 复合函数E_k((1+x)/(1-x))的Schur凸性 3.3.4 复合函数E_k(1/x-x)的Schur凸性3.3.5 复合函数E_k(1/x-μ)的Schur调和凸性 3.3.6 复合函数 E_k(f(x)）的Schur凸性3.4 几个著名不等式的推广3.4.1 Weierstrass不等式3.4.２ Adamovic不等式3.4.3 Chrystal不等式3.4.4 Bernoulli不等式3.4.5 Rado-Popoviciu不等式3.4.6 幂平均不等式3.4.7 算术-几何-调和平均值不等式第四章 Schur凸函数与其它对称函数不等式4.1 完全对称函数的Schur凸性4.1.1 完全对称函数的Schur凸性4.1.2 完全对称函数的推广4.1.3 一个完全对称函数复合函数的Schur凸性 4.2 Hamy对称函数的Schur凸性4.2.1 Hamy对称函数及其推广4.2.2 Hamy对称函数的对偶式4.2.3 Hamy对称函数对偶式的复合函数4.3 Muirhead对称函数的Schur凸性及其应用 4.3.1 Muirhead对称函数的Schur凸性4.3.2 涉及Muirhead对称函数的不等式4.3.3 Jensen-Pečarić-Svrtan-Fan型不等式4.3.4 含剩余对称平均的不等式4.4 Kantorovich不等式的推广4.5 一对互补对称函数的Schur凸性第五章 Schur凸函数与序列不等式5.1 凸数列的定义及性质5.2 各种凸数列5.3 关于凸序列一个不等式5.4 凸数列的几个加权和性质的控制证明5.5 离散Steffensen不等式的加细5.6 凸函数单调平均不等式的改进5.7 一类跳阶乘不等式5.8 等差数列和等比数列的凸性和对数凸性5.8.1 等差数列的凸性和对数凸性5.8.2 等比数列的凸性和对数凸性第六章 Schur凸函数与积分不等式6.1 涉及Hadamard积分不等式的Schur凸函数 6.2 涉及Hadamard型积分不等式的Schur凸函数 6.2.1. 涉及Dragomir积分不等式的Schur凸函数 6.2.2 涉及Lan He积分不等式的Schur凸函数6.2.3 涉及广义积分拟算术平均的Schur凸函数 6.3 涉及Schwarz积分不等式的Schur凸函数6.4 涉及Chebyshev积分不等式的Schur凸函数 6.5 受控型积分不等式6.6 Schur凸函数与其他积分不等式6.7 Schur凸函数与伽马函数第七章 Schur凸函数与二元平均值不等式7.1 Stolarsky 平均的Schur凸性7.2 Gini平均的Schur凸性7.3 Gini平均与Stolarsky平均的比较7.4 广义Heron平均的Schur凸性7.5 其他二元平均的Schur凸性7.5.1 广义Muirhead平均7.5.2 Seiffert型平均7.5.3 指数型平均7.5.4 三角平均7.5.5 Lehme平均7.5.6 “奇特”平均7.5.7 Toader型积分平均7.5.8 椭圆纽曼平均7.6 某些均值差的Schur凸性7.6.1 某些均值差的凸性和Schur凸性7.6.2 某些均值差的Schur几何凸性7.6.3 某些均值差的Schur几何凸性和调和凸性 7.6.4 某些均值商的Schur凸性7.7 双参数齐次函数第八章 Schur凸函数与多元平均值不等式8.1 第三类次对称平均的Schur凸性8.1.1 第三类次对称平均8.1.2 第三类次对称平均的函数推广8.1.3 第三类次对称平均的变形8.2 n元加权广义对数平均的Schur凸性8.3 关于幂平均不等式的最优值8.4 n元平均商的p阶Schur-幂凸性8.5 Bonferroni平均的Schur凸性第九章 Schur凸函数与几何不等式9.1 Schur凸函数与三角形不等式9.1.1 三角形中的控制关系9.1.2 某些三角形内角不等式的控制证明9.1.3 其他三角形不等式的控制证明9.1.4 多边形不等式的控制证明9.2 Schur凸函数与单形不等式9.2.1 单形中的记号与等式9.2.2 单形的伍德几何不等式9.2.3 单形的Berker不等式9.2.4 单形的Milosević不等式9.2.5 对称函数与单形不等式附录1 参考文献附录2 追念胡克教授附录2 我与胡克教授的两封通信及原文影印件。

矩阵迹的若干个性质与应用指导老师：某某摘要：根据矩阵迹的定义，首先给出了矩阵迹的性质，然后依据方阵的F —范数定义Cauchy —Schwarz 不等式，给岀了零矩阵，不相似矩阵，数幂矩阵，列矩阵，幂等矩阵及矩阵不等式的证法。

矩阵的迹在解题中的应用给出了实例。

关键词：迹矩阵范数特征值1引言矩阵的迹及其应用是高等数学的重要内容，也是工程理论研究中的重要工具。

本文在前人研究的基础上，首先介绍了矩阵迹的相关性质，然后给出了零矩阵，不相似矩阵，数幕矩阵，列矩阵，幕等矩阵及矩阵不等式的证法，最后对矩阵的应用给出实例。

2预备知识n定义1 设人二⑻）C nn，则trA a H称为A的迹。

i£定义2 设人=@耳）・C n n，记与向量范数AX 2相容的A的F —范数为：n n21 》aj1 )2i =1 j i(1) A^O二A 尸A O⑵|KA|F =K ||A|F,\7K E C⑶|A +B|F WI A L +||B|F,$A B E C n⑷|AB F乞A F|B F, -A,B C n n(5) |AX〔2 勻A F UI2引理：矩阵迹的性质：1 tr （A 二B）二trA - trB证明：设in i h i hA =(引)佃，B = (b j )代则tr(A)=》an，tr (B)=为0，tr (A ±B)=为佝二0)姓名：某某i=1 i=±i=1i -n i -n i -n又tr(A) 土tr(B)=迟a H±S b H=S 佝+6)7 i 4 i —所以tr(A _B) =tr(A) _tr(B)得证2 tr(kA)二k trA ( k为任意常数)证明：设人=佝人n则tr(A)八a H.k tr(A)二k' a ii；tr(kA)=為(k aj =k' a.tr(kA) = k tr (A)由( 1)与(2) 知tr(mA _nB) = m tr (A) _n tr (B),m, n C3 tr(AB) =tr(BA)证明：设A = (a j )n n, B = (b j )n nk z=n则AB =(c ij)n n，其中c ij - 'a ik b kj 所以有t「(AB) = ' ' a j b jik 二k =nBA=(d j)nn其中d j \ b k Qkj ,所以有tr(AB)八a0口k=1.tr(AB) =tr(BA)得证4 trA = trA证明：矩阵取转置运算主对角线上的元素不变，所以等式很显然成立。

• 138 •Modern Practical Medicine, January 2021, Vol.33, No. 1功能性消化不良病理生理机制及治疗进展沈少英，许丰doi:10.3969/j.issn,1671-0800.2021.01.070【中图分类号】R57 【文献标志码】C【文章编号】1671-0800(2021)01-0138-03功能性消化不良（FD)是指起源于 胃十二指肠区域的一组症状，包括上腹 部疼痛、上腹部烧灼感、餐后饱胀和早 饱，并排除其他引起这些症状的器质性、系统性或代谢性疾病。

根据罗马IV的 FD诊断标准，FD分为餐后不适综合征 (PDS)和上腹痛综合征(EPS)。

PDS主 要表现为主要表现为餐后饱胀和早饱, 每周至少发作3 d;EPS主要表现为上腹 痛和烧灼感,每周至少发作1d;两者的 症状均持续至少3个月w。

本文拟FD 病理生理机制及治疗进展综述如下。

1流行病学总体来说，西方国家FD发病率为 10%〜40%,亚洲地区为5%〜30%,我 国发病率普遍高于其他亚洲国家|21。

在 未成年人中,FD发病率约为7.6%'— 项基于美国、英国及加拿大人口的调查发 现，在FD患者中，PD S占61%, EPS占18%,相互重叠占21%w。

根据一项荟萃 分析，未经调查的FD占21%,女性、吸烟 及幽门螺杆菌(HP)阳性等为高危因素|51。

2病理生理机制2.1胃运动功能障碍胃运动功能障碍主要包括胃排空延迟和胃容受性功能受 损，两者均是FD发病的重要机制。

研究 发现f c l,FD患者中有20%〜50%#在胃 排空障碍，约40%存在胃容受性舒张功 能紊乱。

除上述两个主要原因外，胃移行 性复合运动(MMC)异常也是重要因素之 一。

它是指消化间期的胃运动呈现以间歇 性强力收縮,伴有较长的静息期为特征的 周期性活动，最具特征的是第III期，调查发 现约有30%的症状与MMC异常有关|71。

利特伍德理查德森规则英文English:The Littwood-Richardson rule is a combinatorial algorithm for calculating the Littlewood-Richardson coefficients, which arise in representation theory and algebraic geometry. These coefficients are a fundamental and important part of algebraic combinatorics and have applications in various fields such as quantum physics, quantum computation, and computer science. The rule provides a method for decomposing the tensor product of irreducible representations of a symmetric group into a direct sum of irreducible representations. It is an essential tool for understanding the structure of tensor products and has important implications in the study of symmetric functions and Schur functions.中文翻译:利特伍德理查德森规则是一种组合算法，用于计算利特伍德理查德森系数，这些系数出现在表示论和代数几何中。

这些系数是代数组合学的基本和重要部分，并在量子物理学、量子计算和计算机科学等各个领域得到应用。

国家重点基础研究发展计划(973)项目“数学机械化方法及其在信息技术中的应用”学术交流与汇报会第二届全国计算机数学学术会议(CM 2008)2008年10月24-27日青岛目录●973项目学术交流与汇报会日程●第二届全国计算机数学学术会议日程●报告摘要●会议须知第二届全国计算机数学学术会议组织主办：中国数学学会计算机数学专业委员会承办：中国石油大学中国科学院系统科学研究所中国科学院数学机械化重点实验室会议主席：高小山程序委员会：李洪波(主席)、曾振柄、陈永川、李子明、杨路、刘木兰、查红彬、陈发来、李华组织委员会：李树荣(主席)、周代珍、黄雷国家重点基础研究发展计划(973)项目“数学机械化方法及其在信息技术中的应用”学术交流与汇报会地点：青岛金港大酒店时间：2008年10月24日09:00-09:30 项目介绍、领导讲话09:30-10:10 数学机械化理论与核心算法10:10-10:30 休息10:30-11:10 差分与微分方程的机械化算法11:10-11:50 实几何与实代数的高效能算法12:00-14:00 午餐14:00-14:40 数学机械化与信息安全和编码基础理论研究14:40-15:20 数学机械化在生物特征识别中的应用15:20-15:40 休息15:40-16:20 数学机械化在几何建模中的应用16:20-17:00 基于网络的数学机械化软件开发17:00 总结18:00- 晚餐第二届全国计算机数学大会日程(CSCM 2008)2008年10月25-27日青岛金港大酒店10月25日地点： ***08:30-09:00 开幕式主会场1（主席：高小山）09:00-09:45 邀请报告: 徐宗本, 西安交通大学基于视觉认知的数据建模09:45-10:30 邀请报告: 齐东旭, 澳门科技大学关于非连续的正交函数10:30-10:50 休息10月25日10:50-12:05 分组报告：**会议室**会议室**会议室分组1：微分代数（主席：张鸿庆）分组2：应用研究（主席：王定康）分组3：代数方法（主席：符红光）10:50-11:15李子明, 吴敏Computing dimension of solution spaces forlinear functionalsystems10:50-11:15李邦和酶动力学中的拟稳态假设10:50-11:15张树功多元有理插值的Groebner基方法11:15-11:40王怀富A criterion for thesimilarity of length-two elements ina PID11:15-11:40LEI YANG, 李树荣Optimization ofinjection strategiesfor polymer floodingbased on a real-codedgenetic algorithm11:15-11:40Erich Kaltofen, 李斌,杨争锋, 支丽红Exact Certification ofGlobal Optimality ofApproximateFactorizations ViaRationalizingSums-Of-Squares withFloating Point Scalars11:40-12:05 郑大彬, 吴敏Testing algebraic dependence of hyperexponentialelements11:40-12:05侯春望因子优化法在控制系统根轨迹绘制中的应用11:40-12:05王明生Prime factorization ofmultivariatepolynomial matrices12:00-2:00 午餐10月25日2:00-3:40 分组报告：**会议室**会议室**会议室分组4：微分代数（主席：李志斌）分组5：编码与密码（主席：邢朝平）分组6：应用与算法（主席：齐东旭）2:00-2:25朝鲁Differential Characteristic SetAlgorithm for theComplete Symmetry Classification of PDEs2:00-2:25林东岱, 邓炎炎密码学理论中的挑战2:00-2:25黄雷, 李洪波基于共形几何和复数法的几何计算新方法2:25-2:50刘姜, 李洪波, 曹源昊涉及坐标变换的微分多项式在求和约定下的化简和标准型2:25-2:50吴文玲Improved ImpossibleDifferentialCryptanalysis ofReduced-Round Camellia2:25-2:50廖啟征四元数的复数形式及其在机构求解中的应用2:50-3:15李子明, Martin Ondera,王怀富Simplifying skewfractions modulodifferential and difference relations2:50-3:15刘峰, 武传坤, 林喜军Color VisualCryptography Schemes2:50-3:15李忠, 王爱玲一种基于D-S证据推理的信息融合改进算法3:15-3:40袁春明差分素理想的一个判定准则3:15-3:40邓映蒲攻破Cai-Cusick基于格的公钥密码系统3:15-3:403:40-4:00 休息10月25日4:00-5:40 分组报告：**会议室**会议室**会议室分组7：组合与图论（主席：王明生）分组8：有限域（主席：刘卓军）分组9：计算机视觉与模式识别（主席：查红彬）4:00-4:25陈永川, 唐凌, 王星炜,杨立波Schur positivity and q-log-convexity4:00-4:25高小山, 黄震宇有限域上求解多项式方程的特征列方法4:00-4:25阮秋琦基于偏微分方程的最具可分性人脸特征融合的预处理算法4:25-4:50Burcin Erocal, 侯庆虎, Peter PauleAn implement of MacMahon's partitionanalysis4:25-4:50赵尚威有限域上二次方程组求解的近似算法4:25-4:50罗定生汉语词汇的一体化联合分析方法研究4:50-5:15冯荣权Enumerating typical abelian prime-fold coverings of a circulant graph4:50-5:15孙瑶, 王定康有限域F2上Groebner基的计算4:50-5:15张超Multivariate LaplaceFilter: a Heavy-TailedModel for TargetTracking5:15-5:40谢应泰A polynomial time algorithm for judgingH-graph5:15-5:40张艳硕基于身份的短代理签名方案及其扩展5:15-5:40许超多媒体检索中的转移学习10月26日主会场2（主席：李洪波）08:30-09:15 邀请报告: 邢朝平, 南洋理工大学Space-time codes--introduction and constructions09:15-10:00 邀请报告: 张健, 中科院软件所有限模型和反例的搜索10:00-10:20 休息10月26日10:20-12:00 分组报告：**会议室**会议室**会议室分组10：实代数方法（主席：冯勇）分组11：计算机图形学与辅助设计（主席：陈发来）分组12：优化算法（主席：支丽红）10:20-10:45张景中直观几何代数基础问题10:20-10:45陈冲, 徐国良几何设计中的水平集方法10:20-10:45黄文奇, 叶涛等圆Packing问题完全拟物算法的进一步研究10:45-11:10 邵俊伟, 侯晓荣基于区间分析的不等式自动证明系统10:45-11:10汪国昭混合B样条的统一表示10:45-11:10谢福鼎时序波动周期关联规则挖掘的一个算法11:10-11:35曾振柄基于区域剖分的不等式证明11:10-11:35李华基于几何不变量的三维形状分析和检索11:10-11:35纪哲基于层次分析法的购房策略模型11:35-12:00 张志海, 马蕾, 夏壁灿判定一类线性程序终止性的加速算法11:35-12:00宋瑞霞数字图象自适应非均匀分割及其应用11:35-12:00刘新平, 刘颖基于最大最小距离的改进遗传算法12:00-2:00 午餐10月26日2:00-3:40 分组报告：**会议室**会议室**会议室分组13：逻辑与网络（主席：张健）分组14：模式识别（主席：李华）分组15：微分方程（主席：李子明）2:00-2:25吴尽昭基于代数符号计算的形式化验证方法及其若干关键问题研究2:00-2:25杨国为, 王守觉判定一点是否属于高维复杂形体的算法2:00-2:25张鸿庆一类非线性偏微分方程组的解析解2:25-2:50Guang Zheng, 李廉, 吴尽昭, Wenbo Chen Weaker bisimulation: how to make a+b and tau.a+b equivalent?2:25-2:50査红彬, 裴玉茹The CraniofacialReconstruction fromthe Local StructuralDiversity of Skulls2:25-2:50李志斌Darboux变换与多孤子解算法研究2:50-3:15杜玉越逻辑工作流网及其应用2:50-3:15林通流形学习理论与应用2:50-3:15陆征一Computer aided anal-ysis for differentialpolynomial systems3:15-3:40刘家保, 潘向峰Estrada Index of Hypercubes Networks3:15-3:40邓九英, 王钦若,毛宗源, 杜启亮基于粗糙集的支持向量回归机混合算法3:15-3:40闫振亚The MKdV eqs withvariable coefficients:Exact uni/bi-variabletraveling wave-likesolutions3:40-4:00 休息10月26日4:00-5:40 分组报告：**会议室**会议室**会议室分组16：实代数方法（主席：曾振柄）分组17：计算机辅助设计与数控（主席：徐国良）分组18：控制方法（主席：李树荣）4:00-4:25符红光Dixon结式的三类多余因子4:00-4:25杨周旺点云曲线/曲面的微分信息计算4:00-4:25王峰, 杨永青多目标随机规划在区域水资源优化调度中的应用4:25-4:50冯勇, 张景中Obtaining Exact Inter- polation Multivariate Polynomial byApproximations4:25-4:50韩丽基于复杂截面点云的三角网格模型重建和特征检测方法研究4:25-4:50张玉斌基于MPI的迭代动态规划并行化4:50-5:15 Zhen-Yi Ji, 李永彬Some Improvements upon Unmixed Decomposition of An Algebraic Variety4:50-5:15张梅, 曹源昊数控系统中的数据压缩4:50-5:15田华阁, 车荣杰, 王平,田学民基于FP-EFCM的聚丙烯熔融指数软测量5:15-5:40王云诚, 方伟武,吴天骄A New Bisection-Newton Method for Finding Real Roots of UnivariatePolynomials5:15-5:40李家代数曲线与曲面拓扑的确定与逼近5:15-5:40张晓东聚合物驱最优控制问题的必要条件及数值求解第二届全国计算机数学大会报告摘要10月25日主会场1（主席：高小山）09:00-09:45 邀请报告: 徐宗本, 西安交通大学题目：基于视觉认知的数据建模摘要：数据建模是信息技术的共有基础,是当今信息化社会数学应用的主要形式之一,其目的是揭示数据中所隐含的信息(结构、模式与规律等)。