Fakultt für Mathematik und Informatik

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基督圣体节德语简介

基督圣体节⼜称“基督圣体瞻礼”。

天主教规定恭敬“耶稣圣体”的节⽇。

弥撒时，供“耶稣圣体”于祭台上称为圣体发光的器⽫中。

教徒⼿持烛⽕或彩旗花束，唱赞美诗或者朗诵经⽂，在教堂附近巡游⼀周。

始于13世纪⽐利时的⼀些教区，后渐渐推⾏⾄各地的天主教会。

1311年教皇克雷芒五世(Clemens V.)定于三⼀主⽇后的星期四举⾏。

2007年的基督圣体节在6⽉7⽇。

Das Fronleichnamsfest, kurz: Fronleichnam (von althochdeutsch: fron Herr, liknam Leib; kirchenlat.: Corpus Domini), offiziell …Hochfest des Leibes und Blutes Christi“, regional auch Prangertag oder Blutstag genannt, ist ein Fest im römisch-katholischen Festjahr, in dem die leibliche Gegenwart Jesu Christi im Sakrament(圣礼) der Eucharistie(圣餐礼) gefeiert wird. Fronleichnam wird am Donnerstag nach dem Dreifaltigkeitsfest(圣三⼀主⽇), dem zweiten Donnerstag nach Pfingsten(圣灵降临⽇), begangen und fällt somit frühestens auf den 21. Mai und spätestens auf den 24. Juni. Der Donnerstag als Festtermin knüpft an den eigentlichen Gedenktag des letzten Abendmahls an, den Gründonnerstag(圣周四,濯⾜节), der aber wegen des stillen Charakters der Karwoche(圣周) keine größere Festlichkeit erlaubt. Aus diesem Grund wurde Fronleichnam auf den ersten Donnerstag nach der österlichen Zeit und der ehemaligen Pfingstoktav gelegt.FeierlichkeitenDer wichtigste Teil der Fronleichnamsliturgie ist die Heilige Messe(弥撒) des Tages, die in Lesungen und Gebeten das Geheimnis der Eucharistie umkreist. Vielerorts wird diese Messe heute im Freien, etwa auf öffentlichen Plätzen gefeiert.An die hl. Messe schließt sich in der Regel die Prozession an, bei der die Gläubigen die vom Priester getragene Monstranz(圣体光座) mit dem Allerheiligsten (nach römisch-katholischem Glauben durch den Heiligen Geist in den Leib Jesu Christi verwandeltes Brot, die Hostie) in einem Festzug unter Gesang zu mehreren geschmückten Außenaltären begleiten. Dort wird eine Statio mit Evangelienlesung, Fürbitten und abschließender Eucharistischer Segnung derGläubigen und des Ortes gehalten. Die Prozession schließt meist in der Pfarrkirche mit dem "Te deum" (Großer Gott wir loben dich!)GeschichteDas Fest der leibhaften Gegenwart Christi im Altarsakrament wurde erstmals 1246 im Bistum(教区) Lüttich gefeiert und 1264 von Papst Urban IV. zum Fest der Gesamtkirche erhoben. Es feiert den eucharistisch anwesenden Christus als Opfer, Kommunion (Opferspeise) und als Gegenstand der Anbetung.Die wichtigste Voraussetzung des Festes war das 4. Laterankonzil(第四届拉特朗⼤公会议), das die Verwandlung der eucharistischen Gestalten mit der Transsubstantiationslehre(化体说) präzisiert und zum Dogma erhoben hatte. Danach glauben Katholiken, dass in der Messe Brot und Wein in den Leib und das Blut Christi verwandelt werden und Gott darin gegenwärtig ist.Parallel zu dieser Lehrentscheidung wurde von zahlreichen Hostienwundern berichtet, wie etwa dem, das einemungläubigen böhmischen Mönch (Peter von Prag) auf seiner Pilgerreise nach Rom 1263 in dem mittelitalienischenStädtchen Bolsena widerfahren sein soll: Als er während der Eucharistie die Hostie brach, tropfte daraus Blut auf den Altartisch; die Hostie war also (über die Transsubstantiation hinaus) auch der Form nach zum Leib Christi geworden.Der entscheidende Anstoß zur Schaffung des Festes war jedoch eine Vision der später heilig gesprochenen Augustinernonne(奥斯定会修⼥) Juliana von Lüttich im Jahre 1209. Sie habe, so wird berichtet, beim Beten den Mond gesehen, der an einer Stelle verdunkelt gewesen sei. Christus habe ihr erklärt, dass der Mond das Kirchenjahr(礼仪年) bedeute, der dunkle Fleck das Fehlen eines Festes des Eucharistie-Sakraments.Mit einer Sakramentsprozession wurde das Fest erstmals 1279 in Köln begangen. Die erste Fronleichnamsprozession in Bayern fand 1273 in Benediktbeuern statt.Vor dem Hintergrund der mittelalterlichen Verzweckung des Festes (wie parallel des Reliquienkults) für die Gewinnung von Ablassgeldern(免罪券) ist es nicht verwunderlich, dass der Reformator Martin Luther ein ausdrücklicher Gegner des Fronleichnamsfestes war; er bezeichnete es 1527 als das …schädlichste aller Feste“ und betrachtete die Prozessionen als unbiblisch und als Gotteslästerung. Heute dagegen wirken nicht selten evangelische Pastoren in Amtstracht bei der Fronleichnamsfeier mit.In der Orthodoxen Kirche(东正教) ist die Verehrung des zur Anbetung ausgesetzten eucharistischen Brotes unbekannt. Hier lautet ein Grundsatz: Wir verehren die heiligen Gaben, weil wir sie - etwa zur Krankenkommunion - aufbewahren, aber wir bewahren sie nicht auf, um sie zu verehren. Ebenso wie die scholastisch-rationalistische Transsubstantiationslehre wird das Fronleichnamsfest als Kult des Kultes abgelehnt.Das Konzil von Trient(天特会议) (1545–1563) bestätigte das Fronleichnamsfest und wertete es gleichsam zu einer gegenreformatorischen Machtdemonstration auf. Als Reaktion darauf wurde es in manchen gemischt-konfessionellen Gebieten (z.B. der Schweiz) üblich, dass die protestantischen Bauern als Provokation den Mist gerade an Fronleichnam aufdie Felder ausbrachten; die katholischen Bauern antworteten dann am Karfreitag(基督受难⽇) mit gleicher Münze. Andererseits verzichten die Katholiken in den Vereinigten Staaten und anderen Diasporagebieten oft auf die Prozession als demonstratives Zeichen des Katholizismus, um keine Kontroversen mit der nichtkatholischen Bevölkerung auszulösen. Wo Fronleichnam kein arbeitsfreier Feiertag ist (s.u.), wird das Fest meist erst am Abend oder am darauffolgenden Sonntag gefeiert.Deutung und BrauchtumNur wenigen ist bekannt, was die römische Ritenkongregation bereits 1959 erklärt hat: Die Fronleichnamsprozession ist keine Liturgie römischen Rechtes, sondern sie fällt als "pium exercitium" (fromme Übung) in die Zuständigkeit der Bischöfe.Die heutige Sinngebung der Prozession geht in der Regel vom Bild des …wandernden Gottesvolks“ aus, dessen Mitte Christus, das …Brot des Lebens“, ist.Das Schmücken ganzer Straßenzüge für die Prozession ist bis heute weit verbreitet. Vielerorts werden traditionell auch Bilder, Ornamente und Schriften aus vielen einzelnen Blütenteilen als sog. Blumenteppich vor Altäre oder/und auf Prozessionswegen gelegt. Der eucharistische Christus wird außer mit Weihrauch und Schellen oft auch mit gestreuten Blumen, mancherorts mit Salutschüssen gegrüßt.Regional gibt es die Fronleichnamprozession als …Prozession zu Pferd“ (Antlassritt) oder als Schiffsprozession.Im nordhessischen Fritzlar wird nach alter Tradition das Fronleichnamsfest am Vorabend mit dem …Kazenkopfschießen“eröffnet. Bevor diese Kanonen abgeschossen werden, erklingen alle acht Glocken der Fritzlarer Basilika. Diese Prozedur wird drei Mal wiederholt. Am Festtag selbst findet nach dem feierlichen Hochamt mit den Chorherren derPrämonstratenser(⽩⾐修⼠团) die traditionelle Fronleichnamsprozession durch die Fritzlarer Altstadt statt. Nach vier Altären zieht die Prozession wieder unter vollem Domgeläut in die Basilika ein, wo dann der sakramentale Segen gespendet wird. FeiertagsregelungenDeutschlandIn Deutschland ist Fronleichnam ein gesetzlicher Feiertag in den Bundesländern Baden-Württemberg, Bayern, Hessen, Nordrhein-Westfalen, Rheinland-Pfalz und Saarland sowie in einigen Gemeinden mit überwiegend katholischerBevölkerung in den Bundesländern Sachsen (bestimmt durch die Fronleichnamsverordnung) und Thüringen (im Landkreis Eichsfeld). In den übrigen Bundesländern gibt es Sonderregelungen wie etwa Anspruch auf unbezahlte Freistellung und Unterrichtsbefreiung für katholische Schulkinder.SchweizIn der Schweiz ist Fronleichnam ein gesetzlicher Feiertag in den überwiegend katholisch bevölkerten Kantonen Appenzell Innerrhoden, Jura, Luzern, Nidwalden, Obwalden, Schwyz, Tessin, Uri, Wallis und Zug sowie in bestimmten Gemeinden der Kantone Aargau, Basel-Landschaft, Freiburg, Graubünden, Neuenburg und Solothurn.Österreich und LiechtensteinFronleichnam ist ein gesetzlicher Feiertag in ganz Österreich sowie in Liechtenstein.ItalienIn Italien, wo Corpus Domini vor einigen Jahren als gesetzlicher Feiertag abgeschafft wurde, kann das Fest gemäßKirchenrecht am darauffolgenden Sonntag gefeiert werden (vgl. Christi Himmelfahrt基督升天⽇). Die Bürger Roms lassen sich jedoch nicht davon abhalten, weiterhin in den Abendstunden des Fronleichnamstages gemeinsam mit dem Papst vom Vorplatz der Lateransbasilika zum Vorplatz der Basilika Santa Maria Maggiore zu ziehen, wo der Eucharistische Segen erteilt wird.UngarnIn Ungarn wird Fronleichnam - ähnlich wie in Italien - nicht an einem Donnerstag, sondern an dem darauffolgenden Sonntag gefeiert。

HtnklaF法语交际口语渐进初级上

秋风清，秋月明,落叶聚还散,寒鸦栖复惊。

应力应变张量分解

Comparison of the multiplicative decompositions F = FΘ FM and F = FM FΘ in ﬁnite strain thermo-elasticity
Stefan Hartmann
Institute of Applied Mechanics, Clausthal University of Technology, Adolph-Roemer-Str. 2a, 38678 Clausthal-Zellerfeld, Germany
2 Strain and stress measures of decomposition F = FΘ FM
According to the sketch in Fig. 1 the multiplicative decomposition (2) is applied, ˆ M FM , F = FΘ FM = FΘ F where the mechanical part ˆ M FM FM = F ˆ M, is decomposed into a volume-preserving part FM and a volume-changing part F ˆ M = (det FM )1/3 I, F FM = (det FM )
Impressum
Publisher: Fakultät für Mathematik/Informatik und Maschinenbau, Technische Universität Clausthal Leibnizstraße 32, 38678 Clausthal-Zellerfeld, Germany Editor-in-chief: Alfons Esderts Technical editor: Martina Wächter Contact: martina.waechter@tu-clausthal.de URL: http://www.fakultaet3.tu-clausthal.de/forschung/technical-reports/ ISSN: 1869-8018

Invited Lecture

8.4 ER-schemes
9 Conclusions and further research
We have introduced the use of translation schemes into database design theory. We have shown how they capture disparate notions such as information preservation and dependency preservation in a uniform way. We have shown how they relate to normal form theory and have stated what we think to be the Fundamental Problem of Database Design. Several resulting research problems have been explicitly stated in the paper. We have shown that the Embedded Implicational Dependencies are all needed, when we deal with stepwise re nements of database schemes speci ed by Functional and Inclusion Dependencies. As the material presented grew slowly while teaching database theory, its foundational and didactic merits should not be underestimated. Over the years our students of the advanced database theory course con rmed our view that traditional database design lacks coherence and that this approach makes many issues accessible to deeper understanding. Our approach via dependency preserving translation{re nements can be extended to a full edged design theory for Entity{Relationship design or, equivalently, for database schemes in ER-normal form, cf. MR92]. It is also the appropriate framework to compare transformations of ER-schemes and to address the Fundamental Problem of ER{Database Design. Translation schemes can also be used to deal with views and view updates, as views are special cases of translation schemes. The theory of complementary views from BS81] can be rephrased elegantly in this framework. It is connected with the notion of translation schemes invariant under a relation and implicit denability, Kol90]. Order invariant translation schemes play an important role in descriptive complexity theory, Daw93] and Mak94]. The theory of independent complementary views of KU84] exhibits some severe limitations on the applicability of BS81]. In spite of these limitations it seems worthwhile to explore the connection between independent views and transformation invariant for certain relations further. The latter two applications are currently being developed by the authors and their students and will be included in the full paper.

Free-Knot Spline Approximation of Stochastic Processes

a rX iv:mat h /612313v1[mat h.PR]12Dec26Free-Knot Spline Approximation of Stochastic Processes Jakob Creutzig ∗Thomas M¨u ller-Gronbach †Klaus Ritter ∗Dedicated to Henryk Wo ´z niakowski on the occasion of his 60th birthday December 6,2006Abstract We study optimal approximation of stochastic processes by polynomial splines with free knots.The number of free knots is either a priori ﬁxed or may depend on the particular trajectory.For the s -fold integrated Wiener process as well as for scalar diﬀusion processes we determine the asymptotic behavior of the average L p -distance to the splines spaces,as the (expected)number k of free knots tends to inﬁnity.Keywords:Integrated Wiener process,diﬀusion process,stochastic dif-ferential equation,optimal spline approximation,free knots 1Introduction Consider a stochastic process X =(X (t ))t ≥0with continuous paths on a proba-bility space (Ω,A ,P ).We study optimal approximation of X on the unit interval by polynomial splines with free knots,which has ﬁrst been treated in [10].For k ∈N and r ∈N 0we let Πr denote the set of polynomials of degree at most r ,and we consider the space Φk,r of polynomial splinesϕ=kj =11]t j −1,t j ]·πj ,where 0=t 0<...<t k =1and π1,...,πk ∈Πr .Furthermore,for 1≤p ≤∞and 1≤q <∞we let N k,r denote the class of measurable mappingsX :Ω→Φk,rwithΦk,r being considered as a subset of L p[0,1],and we deﬁnee k,r(X,L p,q)=inf E X− X q L p[0,1] 1/q: X∈N k,r .This quantity is an average L p-distance from X to the spaceΦk,r.A natural extension of this methodology is not to work with an a priori chosen number of free knots,but only to control the average number of knots needed.This leads to the deﬁnitionΦr= ∞k=1Φk,r and to the study of the class N r of measurable mappingsX:Ω→ΦrwithΦr being considered as a subset of L p[0,1].For a spline approximation method X∈N r we deﬁneζ( X)=E(min{k∈N: X(·)∈Φk,r}),i.e.,ζ( X)−1is the expected number of free knots used by X.Subject to the boundζ( X)≤k,the minimal achievable error for approximation of X in the class N r is given bye av k,r(X,L p,q)=inf E X− X q L p[0,1] 1/q: X∈N r,ζ( X)≤k . We shall study the asymptotics of the quantities e k,r and e av k,r as k tends to inﬁnity.The spline spacesΦk,r form nonlinear manifolds that consist of k-term linear combinations of functions of the form1]t,1]·πwith0≤t<1andπ∈Πr. Hence we are addressing a so-called nonlinear approximation problem.While nonlinear approximation is extensively studied for deterministic functions,see [7]for a survey,much less is known for stochastic processes,i.e.,for random functions.Here we refer to[2,3],where wavelet methods are analyzed,and to [10].In the latter paper nonlinear approximation is related to approximation based on partial information,as studied in information-based complexity,and spline approximation with free knots is analyzed as a particular instance.2Main ResultsFor two sequences(a k)k∈N and(b k)k∈N of positive real numbers we write a k≈b k if lim k→∞a k/b k=1,and a k b k if lim inf k→∞a k/b k≥1.Additionally, a k≍b k means c1≤a k/b k≤c2for all k∈N and some positive constants c i.Fix s∈N0and let W(s)denote an s-fold integrated Wiener process.In[10], the following result was proved.Theorem1.For r∈N0with r≥s,e k,r(W(s),L∞,1)≍e av k,r(W(s),L∞,1)≍k−(s+1/2).2Ourﬁrst result reﬁnes and extends this theorem.Consider the stopping time τr,s,p=inf t>0:infπ∈Πr W(s)−π L p[0,t]>1 ,which yields the length of the maximal subinterval[0,τr,s,p]that permits best approximation of W(s)fromΠr with error at most one.We have0<Eτr,s,p<∞,and we putβ=s+1/2+1/pas well asc r,s,p=(Eτr,s,p)−βandb s,p=(s+1/2)s+1/2·p−1/p·β−β.Theorem2.Let r∈N0with r≥s and1≤q<∞.Then,for p=∞,e av k,r(W(s),L∞,q)≈e k,r(W(s),L∞,q)≈c r,s,∞·k−(s+1/2).(1) Furthermore,for1≤p<∞,b s,p·c r,s,p·k−(s+1/2) e k,r(W(s),L p,q) c r,s,p·k−(s+1/2)(2) ande av k,r(W(s),L p,q)≍k−(s+1/2).(3)Note that the bounds provided by(1)and(2)do not depend on the averaging parameter q.In particular,asymptotic constants cannot explode for q tending to inﬁnity.Furthermore,b s,p=1limp→∞for every s∈N,butlimb s,p=0s→∞for every1≤p<∞.We conjecture that the upper bound in(i)is sharp. We have an explicit construction of methods X∗k∈N k,r that achieve the upper bounds in(1)and(2),i.e.,E W(s)− X∗k q L p[0,1] 1/q≈c r,s,p·k−(s+1/2),(4) see(10).Moreover,these methods a.s.satisfyW(s)− X∗k L p[0,1]≈c r,s,p·k−(s+1/2)(5) as well,whileW(s)− X k L p[0,1] b s,p·c r,s,p·k−(s+1/2)(6) holds a.s.for every sequence of approximations X k∈N k,r.Here,b s,∞=1.3Our second result deals with approximation of a scalar diﬀusion process given by the stochastic diﬀerential equationdX(t)=a(X(t))dt+b(X(t))dW(t),t≥0,X(0)=x0.(7)Here x0∈R,and W denotes a one-dimensional Wiener process.Moreover,we assume that the functions a,b:R→R satisfy(A1)a is Lipschitz continuous,(A2)b is diﬀerentiable with a bounded and Lipschitz continuous derivative,(A3)b(x0)=0.Theorem3.Let r∈N0,1≤q<∞,and1≤p≤∞.Thene k,r(X,L p,q)≍e av k,r(X,L p,q)≍k−1/2holds for the strong solution X of equation(7).For a diﬀusion process X piecewise linear interpolation with free knots is fre-quently used in connection with adaptive step-size control.Theorem3provides a lower bound for the L p-error of any such numerical algorithm,no matter whether just Wiener increments or,e.g.,arbitrary multiple Itˆo-integrals are used.Error estimates in[8,16]lead to reﬁned upper bounds in Theorem3for the case1≤p<∞,as follows.Putκ(p1,p2)= E b◦X p2L p1[0,1] 1/p2for1≤p1,p2<∞.Furthermore,let B denote a Brownian bridge on[0,1]and deﬁneη(p)= E B p L p[0,1] 1/p.Thene k,1(X,L p,p) η(p)·κ(2p/(p+2),p)·k−1/2ande av k,1(X,L p,p) η(p)·κ(2p/(p+2),2p/(p+2))·k−1/2.We add that these upper bounds are achieved by numerical algorithms with adaptive step-size control for the Wiener increments.In the case p=∞it is interesting to compare the results on free-knot spline approximation with average k-widths of X.The latter quantities are deﬁned byd k(X,L p,q)=infΦ E infϕ∈Φ X−ϕ q L p[0,1] 1/q,where the inﬁmum is taken over all linear subspacesΦ⊆L p[0,1]of dimension at most k.For X=W(s)as well as in the diﬀusion case we haved k(X,L∞,q)≍k−(s+1/2),4see[4,13,14,15]and[6].Almost optimal linear subspaces are not known ex-plicitly,since the proof of the upper bound for d k(X,L∞,q)is non-constructive. We add that in the case of an s-fold integrated Wiener process piecewise poly-nomial interpolation of W(s)at equidistant knots i/k only yields errors of order (ln k)1/2·k−(s+1/2),see[19]for results and references.Similarly,in the diﬀusion case,methods X k∈N r that are only based on pointwise evaluation of W and satisfyζ( X k)≤k can at most achieve errors of order(ln k)1/2·k−1/2,see[17]. 3Approximation of Deterministic FunctionsLet r∈N0and1≤p≤∞beﬁxed.We introduce error measures,which allow to determine suitable free knots for spline approximation.For f∈C[0,∞[and 0≤u<v we putδ[u,v](f)=infπ∈Πr f−π Lp[u,v].Furthermore,forε>0,we putτ0,ε(f)=0,and we deﬁneτj,ε(f)=inf{t>τj−1,ε(f):δ[τj−1,ε(f),t](f)>ε}for j≥1.Here inf∅=∞,as usual.Put I j(f)={ε>0:τj,ε(f)<∞}. Lemma4.Let j∈N.(i)Ifε∈I j(f)thenδ[τj−1,ε(f),τj,ε(f)](f)=ε.(ii)The set I j(f)is an interval,and the mappingε→τj,ε(f)is strictly in-creasing and right-continuous on I j(f).Furthermore,τj,ε(f)>τj−1,ε(f) ifε∈I j−1(f),and limε→∞τj,ε(f)=∞.(iii)If v→δ[u,v](f)is strictly increasing for every u≥0,thenε→τj,ε(f)is continuous on I j(f).Proof.First we show that the mapping(u,v)→δ[u,v](f)is continuous.Put J1=[u/2,u+(v−u)/3]as well as J2=[v−(v−u)/3,2v].Moreover,let πα(t)= r i=0αi·t i forα∈R r+1,and deﬁne a norm on R r+1byα = πα Lp[u+(v−u)/3,v−(v−u)/3].If(x,y)∈J1×J2andf−πα Lp[x,y]=δ[x,y](f)thenα ≤ πα Lp[x,y]≤δ[u/2,2v](f)+ f Lp[u/2,2v].Hence there exists a compact set K⊆R r+1such thatδ[x,y](f)=infα∈K f−πα Lp[x,y]5for every(x,y)∈J1×J2.Since(x,y,α)→ f−πα Lp[x,y]deﬁnes a continuousmapping on J1×J2×K,we conclude that(x,y)→infα∈K f−πα Lp[x,y]iscontinuous,too,on J1×J2.Continuity and monotonicity of v→δ[u,v](f)immediately imply(i).The monotonicity stated in(ii)will be veriﬁed inductively.Let0<ε1<ε2withε2∈I j(f),and suppose thatτj−1,ε1(f)≤τj−1,ε2(f).Note that the latterholds true by deﬁnition for j=1.From(i)we getδ[τj−1,ε1(f),τj,ε2(f)](f)≥δ[τj−1,ε2(f),τj,ε2(f)](f)=ε2.This impliesτj,ε1(f)≤τj,ε2(f),and(i)excludes equality to hold here.Sinceδ[u,v](f)≤ f Lp[u,v],the mappingsε→τj,ε(f)are unbounded andτj,ε(f)>τj−1,ε(f)ifε∈I j−1(f).For the proof of the continuity properties stated in(ii)and(iii)we also proceed inductively,and we use(i)and the monotonicity from(ii).Consider a sequence(εn)n∈N in I j(f),which converges monotonically toε∈I j(f),andput t=lim n→∞τj,εn (f).Assume that lim n→∞τj−1,εn(f)=τj−1,ε(f),whichobviously holds true for j=1.Continuity of(u,v)→δ[u,v](f)and(i)implyδ[τj−1,ε(f),t](f)=ε,so that t≤τj,ε(f).For a decreasing sequence(εn)n∈N wealso haveτj,ε(f)≤t.For an increasing sequence(εn)n∈N we use the strict monotonicity of v→δ[u,v](f)to derive t=τj,ε(f).Let F denote the class of functions f∈C[0,∞[that satisfyτj,ε(f)<∞(8) for every j∈N andε>0as well aslimε→0τj,ε(f)=0(9) for every j∈N.Let k∈N.We now present an almost optimal spline approximation method of degree r with k−1free knots for functions f∈F.Putγk(f)=inf{ε>0:τk,ε(f)≥1}and note that(9)together with Lemma4.(ii)impliesγk(f)∈]0,∞[.Letτj=τj,γk (f)(f)for j=0,...,k and deﬁneϕ∗k(f)=kj=11]τj−1,τj]·argminπ∈Πr f−π L p[τj−1,τj].(10)Note that Lemma4guaranteesf−ϕ∗k(f) Lp[τj−1,τj]=γk(f)(11) for j=1,...,k andτk≥1.(12) The splineϕ∗k(f)|[0,1]∈Φk,r enjoys the following optimality properties.6Proposition5.Let k∈N and f∈F.(i)For1≤p≤∞,f−ϕ∗k(f) Lp[0,1]≤k1/p·γk(f).(ii)For p=∞and everyϕ∈Φk,r,f−ϕ L∞[0,1]≥γk(f).(iii)For1≤p<∞,everyϕ∈Φk,r,and every m∈N with m>k,f−ϕ Lp[0,1]≥(m−k)1/p·γm(f). Proof.For p<∞,f−ϕ∗k(f) pL p[0,1]≤kj=1 f−ϕ∗k(f) p L p[τj−1,τj]=k·(γk(f))pfollows from(11)and(12).For p=∞,(i)is veriﬁed analogously.Consider a polynomial splineϕ∈Φk,r and let0=t0<...<t k=1denote the corresponding knots.Furthermore,letρ∈]0,1[.For the proof of(ii)we putσj=τj,ρ·γk (f)(f).for j=0,...,k.Thenσk<1,which implies[σj−1,σj]⊆[t j−1,t j] for some j∈{1,...,k}.Consequently,by Lemma4,f−ϕ L∞[0,1]≥ f−ϕ L∞[σj−1,σj]≥infπ∈Πrf−π L∞[σj−1,σj]=ρ·γk(f).For the proof of(iii)we deﬁneσℓ=τℓ,ρ·γm(f)(f) forℓ=0,...,m.Thenσm<1,which implies[σℓi−1,σℓi]⊆[t ji−1,t ji]for some indices1≤j1≤...≤j m−k≤k and1≤ℓ1<...<ℓm−k≤m. Hence,by Lemma4,f−ϕ pL p[0,1]≥m−ki=1infπ∈Πr f−π p L p[σℓi−1,σℓi]=(m−k)·ρp·(γm(f))p.for1≤p<∞.Lettingρtend to one completes the proof.74Approximation of Integrated Wiener Processes Let W denote a Wiener process and consider the s-fold integrated Wiener pro-cesses W(s)deﬁned by W(0)=W andW(s)(t)= t0W(s−1)(u)dufor t≥0and s∈N.We brieﬂy discuss some properties of W(s),that will be important in the sequel.The scaling property of the Wiener process implies that for everyρ>0the process(ρ−(s+1/2)·W(s)(ρ·t))t≥0is an s-fold integrated Wiener process,too. This fact will be called the scaling property of W(s).While W(s)has no longer independent increments for s≥1,the inﬂuence of the past is very explicit.For z>0we deﬁne z W(s)inductively byzW(0)(t)=W(t+z)−W(z)andzW(s)(t)= t0z W(s−1)(u)du.Then it is easy to check thatW(s)(t+z)=s i=0t iLemma6.The s-fold integrated Wiener process W(s)satisﬁesP(W(s)∈F)=1.For everyε>0the random variablesξj,εform an i.i.d.sequence withξ1,εd=ε1/β·ξ1,1and E(ξ1,1)<∞.Proof.We claim thatE(τj,ε(W(s)))<∞(14) for every j∈N.For the case j=1let Z=δ[0,1](W(s))and note thatδ[0,t](W(s))d=tβ·Zfollows for t>0from the scaling property of W(s).Hence we haveP(τ1,ε(W(s))<t)=P(δ[0,t](W(s))>ε)=P(Z>ε·t−β),(15) which,in particular,yieldsτ1,ε(W(s))d=ε1/β·τ1,1(W(s)).(16) According to Corollary17,there exists a constant c>0such thatP(Z≤η)≤exp(−c·η−1/(s+1/2))holds for everyη∈]0,1].We conclude thatP(τ1,1(W s))>t)≤exp(−c·t)if t≥1,which implies E(τ1,1(W(s)))<∞.Next,let j≥2,putτ=τj−1,ε(W(s))andτ′=τj,ε(W(s)),and assume that E(τ)<∞.From the representation(13)and the fact that r≥s we deriveδ[τ,t](W(s))=δ[0,t−τ](τW(s)),and hence it follows thatτ′=τ+τ1,ε(τW(s)).(17) We have E(τ1,ε(τW(s)))<∞,sinceτW(s)is an s-fold integrated Wiener process again,and consequently E(τ′)<∞.We turn to the properties of the sequenceξj,ε.Due to(16)and(17)we have ξj,ε=τ1,ε(τW(s))d=τ1,ε(W(s))d=ε1/β·ξ1,1. Furthermore,ξj,εand(1[0,τ](t)·W(s)(t))t≥0are independent because of the strong Markov property of W(s),and thereforeξj,εand(ξ1,ε,...,ξj−1,ε)are independent as well.9It remains to show that the trajectories of W(s)a.s.satisfy(9).By the properties of the sequenceξj,εwe haveτj,ε(W(s))d=ε1/β·τj,1(W(s)).(18) Observing(14)we conclude thatP(limε→0τj,ε(W(s))≥t)=limε→0P(τj,ε(W(s))≥t)=limε→0P(τj,1(W(s))≥t/ε1/β)=0for every t>0,which completes the proof.Because of Lemma6,Proposition5yields sharp upper and lower bounds for the error of spline approximation of W(s)in terms of the random variableV k=γk(W(s)).Remark7.Note that W(s)a.s.satisﬁes W(s)|[u,v]∈Πr for all0≤u<v. Assume that p<∞.Then v→δ[u,v](W(s))is a.s.strictly increasing for all u≥0.We use Lemma4.(iii)and Lemma6to conclude that,with probability one,V k is the unique solution ofτk,Vk(W(s))=1.Consequently,we a.s.have equality in Proposition5.(i)for1≤p<∞,too. Note that with positive probability solutionsεof the equationτk,ε(W(s))=1 fail to exist in the case p=∞.To complete the analysis of spline approximation methods we study the asymptotic behavior of the sequence V k.Lemma8.For every1≤q<∞,(E V qk)1/q≈(k·E(ξ1,1))−β.Furthermore,with probability one,V k≈(k·E(ξ1,1))−β.Proof.PutS k=1/k·k j=1ξj,1and use(18)to obtainP(V k≤ε)=P(τk,ε(W(s))≥1)=P(k−β·S−βk≤ε).(19) ThereforeE(V qk )=k−βq·E(S−βqk),10and for theﬁrst statement it remains to show thatE(S−βqk)≈(E(ξ1,1))−βq.The latter fact follows from Proposition15,if we can verify thatξ1,1has a proper lower tail behavior(28).To this end we use(15)and the large deviation estimate(32)to obtainP(ξ1,1≤η)=P(δ[0,1](W(s))≥η−β)≤P( W(s) Lp[0,1]≥η−β)≤exp(−c·η−2β)with some constant c>0for allη≤1.In order to prove the second statement,putS∗k=(k·σ2)−1/2·kj=1(ξj,1−µ),whereµ=E(ξ1,1)andσ2denotes the variance ofξ1,1.Letρ>1.Then P(V k>ρ·(k·µ)−β)=P(S k<ρ−1/β·µ)=P(S∗k<k1/2· ρ)withρ=(ρ−1/β−1)/σ·µ<0,due to(19).We apply a local version of the central limit theorem,which holds for i.i.d.sequences with aﬁnite third moment,see[18,Thm.V.14],to obtain P(V k>ρ·(k·µ)−β)≤c1·k−1/2·(1+k1/2·| ρ|)−3+(2π)−1/2· k1/2·eρ−∞exp(−u2/2)du≤c2·k−2with constants c i>0.For everyρ<1we getP(V k<ρ·(k·µ)−β)≤c2·k−2(20) in the same way.It remains to apply the Borel-Cantelli Lemma.Observe Remark7and use Lemma8to obtain(4)and(5)for the methods X∗k=ϕ∗k(W(s)).Clearly,(4)implies the upper bounds in(1),(2),and(3).For the proof of the lower bound in(2)we apply Proposition5.(iii)withm=⌊β/(s+1/2)·k⌋.Then we have(m−k)1/p·(E V q m)1/q≈k−(s+1/2)·p−1/p·β−β·(s+1/2)s+1/2·(E(ξ1,1))−β,11as ing the second statement in Lemma 8,the lower bound in (6)is shown in the same way.It remains to prove the lower bounds for e av k,r (W (s ),L p ,q )in Theorem 2.Proof of the lower bound in (1).Let k ∈N and consider X k ∈N r such that ζ( Xk )≤k ,i.e.,∞ℓ=1ℓ·P (B ℓ)≤k (21)for B ℓ= X(·)∈Φℓ,r \Φℓ−1,r,where Φ0,r =∅.By Proposition 5.(ii),E W (s )− X k q L ∞[0,1]≥∞ ℓ=1E (1B ℓ·V q ℓ).For ̺∈]0,1[,µ=E (ξ1,1),and L ∈N we deﬁneA ℓ=V ℓ>ρ·(ℓ·µ)−β ,andC L =Lℓ=1B ℓ.Since γℓ(f )≥γℓ+1(f )for f ∈F ,we obtainEW (s )− X k q L ∞[0,1]≥L ℓ=1E (1B ℓ·V qL )+∞ℓ=L +1E (1B ℓ·V q ℓ)≥L ℓ=1E (1B ℓ∩A L ·V qL )+∞ℓ=L +1E (1B ℓ∩A ℓ·V q ℓ)≥ρq µ−βq· L −βq ·P (C L ∩A L )+∞ l =L +1ℓ−βq ·P (B ℓ∩A ℓ) .From (20)we infer that P (A c ℓ)≤c 1·ℓ−2with a constant c 1>0.Hence there exists a constant c 2>0such thatΓ(L )=L−βq·P (C L )+∞ℓ=L +1ℓ−βq ·P (B ℓ)−c 2·L −βq −1satisﬁes ρ−q µβq ·EW (s )− X k q L ∞[0,1]≥Γ(L )(22)for every L ∈N .12Putα=(1+2βq)/(2+2βq),and take L(k)∈[kα−1,kα].We claim that there exists a constant c3>0such thatkβq·Γ(L(k))≥ 1−k−(1−α)βq 1+βq−c3·k−1/2.(23) First,assume that P(C L)≥k−(1−α)βq.Thenkβq·Γ(L(k))≥kβq· k−αβq·P(C L)−c2·(kα−1)−βq−1≥1−c3·k−1/2with a constant c3>0.Next,assume P(C L)<k−(1−α)βq and use(21)to derive1−k−(1−α)βq<∞ℓ=L+1P(Bℓ)=∞ℓ=L+1(ℓ·P(Bℓ))βq/(1+βq)·(ℓ−βq·P(Bℓ))1/(1+βq)≤ ∞ ℓ=L+1ℓ·P(Bℓ) βq/(1+βq)· ∞ ℓ=L+1ℓ−βq·P(Bℓ) 1/(1+βq)≤kβq/(1+βq)· ∞ ℓ=L+1ℓ−βq·P(Bℓ) 1/(1+βq). Consequently,kβq·Γ(L(k))≥kβq· ∞ ℓ=L+1ℓ−βq·P(Bℓ)−c2·(kα−1)−βq−1≥ 1−k−(1−α)βq 1+βq−c3·k−1/2,which completes the proof of(23).By(22)and(23),E W(s)− X k q L∞[0,1] ρqµ−βq·k−βqfor everyρ∈]0,1[.Finally,for the proof of the lower bound in(3)it suﬀuces to establish the lower bound claimed for e av k,r(W(s),L1,1).For further use,we shall prove a more general result.Lemma9.For every s∈N there exists a constant c>0with the following property.For every X∈N r,every A∈A with P(A)≥4/5,and every t∈]0,1] we haveE 1A· W(s)− X L1[0,t] ≥c·t s+3/2·k−(s+1/2).13Proof.Because of the scaling property of W(s)it suﬃces to study the particular case t=1.PutB={ X∈Φ2k,r},and observe that P(B)≥1/2follows fromξ( X)≤k.Due to Lemma6and Proposition5.(iii),1B· W(s)− X L1[0,1]≥1B·2k·V4k.Putµ=E(ξ1,1),choose0<c<(2µ)−β,and deﬁneD k={V k≥c·k−β}.By(19)we obtainP(D k)=P(S k<c−1/β)≥P(S k<2µ).HenceP(D k)=1limk→∞due to the law of large numbers,and consequently P(B∩D k)≥2/5if k is suﬃciently large,say k≥k0.We conclude that1A∩B∩D4k· W(s)− X L1[0,1]≥1A∩B∩D4k·c·21−2β·k−(s+1/2)and P(A∩B∩D4k)≥1/5if4k≥k0.Take expectations to complete the proof.Lemma9with A=Ωand t=1yields the lower bound in(3)5Approximation of Diﬀusion ProcessesLet X denote the solution of the stochastic diﬀerential equation(7)with initial value x0,and recall that the drift coeﬃcient a and the diﬀusion coeﬃcient b are supposed to satisfy conditions(A1)–(A3).In the following we use c to denote unspeciﬁed positive constants,which may only depend on x0,a,b and the averaging parameter1≤q<∞.Note that<∞(24)E X qL∞[0,1]andE sup t∈[s1,s2]|X(t)−X(s1)|q ≤c·(s2−s1)q/2(25) for all1≤q<∞and0≤s1≤s2≤1,see[9,p.138].145.1Upper BoundsIn order to establish upper bounds,it suﬃces to consider the case of p=∞and r=0,i.e.,nonlinear approximation in supremum norm with piecewise constant splines.We dissect X into its martingale partM(t)= t0b(X(s))dW(s)andY(t)=x0+ t0a(X(s))ds.Lemma10.For all1≤q<∞and k∈N,there exists an approximation Y∈N k,0such thatE Y− Y q L∞[0,1] 1/q≤c·k−1.Proof.Put g Lip=sup0≤s<t≤1|g(t)−g(s)|/|t−s|for g:[0,1]→R,and deﬁneY=k j=11](j−1)/k,j/k]·Y((j−1)/k).By(A1)and(24),E Y− Y q L∞[0,1]≤E Y q Lip·k−q≤c· 1+E X q L∞[0,1] ·k−q≤c·k−q.Lemma11.For all1≤q<∞and k∈N,there exists an approximation M∈N k,0such thatE M− M q L∞[0,1] 1/q≤c·k−1/2.Proof.LetX=k j=11](j−1)/k,j/k]·X((j−1)/k).Clearly,by(25),E X− X q L2[0,1] 1/q≤c·k−1/2.DeﬁneR(t)= t0b( X(s))dW s.15By the Burkholder-Davis-Gundy inequality and(A2),E M−R q L∞[0,1] 1/q≤c· E 10(b(X(s))−b( X(s)))2ds q/2 1/q≤c· E X− X q L2[0,1] 1/q≤c·k−1/2.(26) Note thatR= R+V,whereR=k j=11](j−1)/k,j/k]·R((j−1)/k)andV=kj=11](j−1)/k,j/k]·b(X((j−1)/k))·(W−W((j−1)/k)).According to Theorem2,there exists an approximation W∈N k,0such thatE W− W 2q L∞[0,1] 1/(2q)≤c·k−1/2. Using W we deﬁne V∈N2k,0byV=k j=11](j−1)/k,j/k]·b(X((j−1)/k))·( W−W((j−1)/k)).Clearly,V− V L∞[0,1]≤ b(X) L∞[0,1]· W− W L∞[0,1].Observing(24)and(A2),we conclude thatE V− V q L∞[0,1] 1/q≤ E b(X) 2q L∞[0,1] 1/(2q)· E W− W 2q L∞[0,1] 1/(2q)≤c·k−1/2.(27) Weﬁnally deﬁne M∈N2k,0by M= R+ V.SinceM− M=(M−R)+(V− V),it remains to apply the estimates(26)and(27)to complete the proof.The preceding two lemma imply e k,0(X,L∞,q)≤c·k−1/2.165.2Lower BoundsFor establishing lower bounds it suﬃces to study the case p=q=1.Moreover, we assume without loss of generality that b(x0)>0.Chooseη>0as well as a function b0:R→R such that(a)b0is diﬀerentiable with a bounded and Lipschitz continuous derivative,(b)inf x∈R b0(x)≥b(x0)/2,(c)b0=b on the interval[x0−η,x0+η].We will use a Lamperti transform based on the space-transformationg(x)= x x01dW(s).b0(X(s))The idea of the proof is as follows.We show that any good spline approxima-tion of X leads to a good spline approximation of H(X).However,since with a high probability,X stays within[x0−η,x0+η]for some short(but nonrandom) period of time,approximation of H(X)is not easier than approximation of W, modulo constants.First,we consider approximation of H1(X).Lemma12.For every k∈N there exists an approximation X1∈N k,0such thatE H1(X)− X1 L1[0,1]≤c·k−1. Proof.Observe that g′a+g′′/2·b2 (x)≤c·(1+x2),and proceed as in the Proof of Lemma10.Next,we relate approximation of X to approximation of H2(X).Lemma13.For every approximation X∈N r there exists an approximation X2∈N r such thatζ( X2)≤2·ζ( X)andE H2(X)− X2 L1[0,1]≤c· E X− X L1[0,1]+1/ζ( X) .17Proof.For aﬁxedω∈Ωlet X(ω)be given byX(ω)=k j=11]t j−1,t j]·πj.We reﬁne the corresponding partition to a partition0= t0<...< t e k=1 that contains all the points i/ℓ,whereℓ=ζ( X).Furthermore,we deﬁne the polynomials πj∈Πr byX(ω)=e k j=11]e t j−1,e t j]· πj.Put f=X(ω)and deﬁneX2(ω)=e k j=11]e t j−1,e t j]·q jwith polynomialsq j=g(f( t j−1))+g′(f( t j−1))·( πj−f( t j−1))∈Πr.Let f2= X2(ω).If t∈ t j−1, t j ⊆](i−1)/ℓ,i/ℓ],then|H2(f)(t)− f2(t)|= g(f(t))−g(f( t j−1))−g′(f( t j−1))·( πj(t)−f( t j−1))≤ g(f(t))−g(f( t j−1))−g′(f( t j−1))·(f(t)−f( t j−1))+ g′(f( t j−1)) ·|f(t)− πj(t)|≤c· |f(t)−f( t j−1)|2+|f(t)− πj(t)|≤c· sup|f(t)−f((i−1)/ℓ)|2+|f(t)− πj(t)| .t∈](i−1)/ℓ,i/ℓ]Consequently,we may invoke(25)to deriveE H2(X)− X2 L1[0,1]≤c· 1/ζ( X)+E X− X L1[0,1] . Moreover,ζ( X2)≤2·ζ( X).Finally,we establish a lower bound for approximation of H(X). Lemma14.For every approximation X∈N r,E H(X)− X L1[0,1]≥c·(ζ( X))−1/2.18Proof.Choose t0∈]0,1]such thatA= sup t∈[0,t0]|X(t)−x0|≤ηsatisﬁes P(A)≥4/5.Observe that1A· H(X)− X L1[0,1]≥1A· W− X L1[0,t0],and apply Lemma9for s=0.Proof of the lower bound in Theorem3.Consider any approximation X∈N r with k−1<ζ( X)≤k,and choose X1and X2according to Lemma12and Lemma13,respectively.ThenE H(X)−( X2− X1) L1[0,1]≤E H2(X)− X2 L1[0,1]+E H1(X)− X1 L1[0,1]≤c· E X− X L1[0,1]+(ζ( X))−1+k−1≤c· E X− X L1[0,1]+k−1 .On the other hand,ζ( X2− X1)≤ζ( X2)+k≤3·k,so thatE H(X)−( X2− X1) L1[0,1]≥c·k−1/2follows from Lemma14.We conclude thatE X− X L1[0,1]≥c·k−1/2,as claimed.A Convergence of Negative Moments of MeansLet(ξi)i∈N be an i.i.d.sequence of random variables such thatξ1>0a.s.and E(ξ1)<∞.PutS k=1/k·k i=1ξi.Proposition15.For everyα>0,lim inf k→∞E(S−αk)≥(E(ξ1))−α.IfP(ξ1<v)≤c·vρ,v∈]0,v0],(28) for some constants c,ρ,v0>0,thenlim k→∞E(S−αk)=(E(ξ1))−α.19Proof.Put µ=E (ξ1)and deﬁneg k (v )=α·v −(α+1)·P (S k <v ).Thanks to the weak law of large numbers,P (S k <v )tends to 1]µ,∞[(v )for every every v =µ.Hence,by Lebesgue’s theorem,lim k →∞∞µ/2g k (v )dv =µ−α.(29)SinceE (S −αk )=∞P (S −αk>u )du =∞g k (v )dvthe asymptotic lower bound for E (S −αk )follows from (29).Given (28),we may assume without loss of generality that c ·v ρ0<1.We ﬁrst consider the case ξ1≤1a.s.,and we putA k = µ/2x 0/kg k (v )dv and B k = x 0/kg k (v )dv.For v 0/k ≤v ≤µ/2we use Hoeﬀding’s inequality to obtaing k (v )≤α·v −(α+1)·P (|S k −µ|>µ/2)≤α·(k/v 0)α+1·2exp(−k/2·µ2),which implieslim k →∞A k =0.On the other hand,if ρk >α,thenB k =kα+1·α·v 0v−(α+1)·Pk i =1ξi <vdv≤k α+1·α·v 0v −(α+1)·(P (ξ1<v ))k dv≤k α+1·α·c k ·v 0v ρk −(α+1)dv =kα+1·α·(ρk −α)−1·c k ·v ρk −α0,and thereforelim k →∞B k =0.In view of (29)we have thus proved the proposition in the case of boundedvariables ξi .In the general case put ξi,N =min {N,ξi }as well as S k,N =1/k · k i =1ξi,N ,and apply the result for bounded variables to obtainlim sup k →∞E (S −αk )≤inf N ∈N lim sup k →∞E (S −αk,N )=inf N ∈N(E ξ1,N )−α=(E ξ1)−αby the monotone convergence theorem.20B Small Deviations of W(s)fromΠrLet X denote a centered Gaussian random variable with values in a normed space(E, · ),and consider aﬁnite-dimensional linear subspaceΠ⊂E.We are interested in the small deviation behavior ofd(X,Π)=infπ∈ΠX−π .Obviously,P( X ≤ε)≤P(d(X,Π)≤ε)(30) for everyε>0.We establish an upper bound for P(d(X,Π)≤ε)that involves large deviations of X,too.Proposition16.If dim(Π)=r thenP(d(X,Π)≤ε)≤(4λ/ε)r·P( X ≤2ε)+P( X ≥λ−ε)for allλ≥ε>0.Proof.Put Bδ(x)={y∈E: y−x ≤δ}for x∈E andδ>0,and consider the sets A=Π∩Bλ(0)and B=Bε(0).Then{d(X,Π)≤ε}⊂{X∈A+B}∩{ X ≥λ−ε},and therefore it suﬃces to proveP(X∈A+B)≤(4λ/ε)r·P( X ≤2ε).(31) Since1/λ·A⊂Π∩B1(0),theε-covering number of A is not larger than (4λ/ε)r,see[1,Eqn.(1.1.10)].HenceA⊂n i=1Bε(x i)for some x1,...,x n∈E with n≤(4λ/ε)r,and consequently,A+B⊂ni=1B2ε(x i).Due to Anderson’s inequality we haveP(X∈B2ε(x i))≤P(X∈B2ε(0)),which implies(31).Now,we turn to the speciﬁc case of X=(W(s)(t))t∈[0,1]and E=L p[0,1], and we consider the subspaceΠ=Πr of polynomials of degree at most r.21。

★留学院校★ 【德国】-曼海姆大学

曼海姆大学（Universität Mannheim）是德国最年轻的大学之一，成立于1967年。

大学座落于有200多年历史的巴罗克式的曼海姆宫(Mannheimer Barockschloss)中，宫殿是1763年选帝侯卡尔-西奥多（Karl Theodor）曼海姆大学的前身是成立于1907年的曼海姆市立贸易专科高校（Handelshochschule Mannheim），1933年曾一度关闭。

1946年以国立经济专科高校（Staatliche Wirtschaftshochschule）之名重新开放，当时有学生545人。

1963年学校增设了法学系和哲学系，此时学生已增加到1715人。

1967年大学正式更名为曼海姆大学，学生人数增加到3150人。

如今大学有学生11500名（2007年），共有5大院系29个专业。

这5个院系为：社会科学系（Fakultät für Sozialwissenschaften）、数学信息学系（Fakultät für Mat hematik und Informatik）、法学和国民经济学系（Fakultät für Rechtswissenschaft und Volkswirtschaftslehre）、哲学系（Philosophische Fakultät）和企业管理学系（Betriebswirtschaftslehre）。

其经济学和社会科学专业历史悠久，教育水平久享盛名。

选修企业管理、国民经济学、政治经济学、社会学专业的学生多达6500人。

曼海姆大学的企业管理专业在德国声望极高，在教学内容上偏重数理分析。

2003年《经济周刊》（Wirtschaftswoche）对工商管理专业的排名中，曼海姆大学位居第一；在国民经济学排名中其位居第二（第一是科隆大学）。

曼海姆大学也是一所国际化的大学，大约每十名学生中就有一名外国学生。

慕尼黑工业大学世界排名情况怎么样

慕尼黑工业大学作为德国的精英大学之一、TU-9院校之一，凭借着历年来优秀的学术成绩，成为了很多德国留学生的梦想院校。

那今年该大学的排名情况如何呢？一起跟小编来看看吧！一、慕尼黑工业大学世界性排名基于其专业的学科和高质量的教学，除了21年QS世界大学排名里名列前茅，在其他知名的世界排名中实力也是相当强劲：2021年泰晤士高等教育世界大学排名（THE）中，慕尼黑工业大学名列德国第2，全球第41位。

2020年软科世界大学学术排名（ARWU）中，慕尼黑工业大学名列德国第2，全球第54位。

在2020年《美国新闻与世界报导》的全球大学排名中，慕尼黑工业大学名列德国第3，全球第77位二、慕尼黑工业大学院系设置数学与自然科学部（Mathematics and Natural Sciences）：数学学院-Fakult?t für Mathematik物理学院-Fakult?t für Physik化学学院-Fakult?t für Chemie工程与建筑学部（Engineering and Architecture）：航天航空与测绘学院-Fakult?t für Luftfahrt， Raumfahrt und Geod?sie土木与环境工程学院-Ingenieurfakult?t Bau Geo Umwelt建筑学院-Fakult?t für Architektur机械工程学院-Fakult?t für Maschinenwesen计算机科学学院-Fakult?t für Informatik电子与信息工程学院-Fakult?t für Elektrotechnik und Informationstechnik生命与健康科学部（Life Sciences and Health Sciences）：生命学院-Wissenschaftszentrum Weihenstephan für Ern?hrung， Landnutzung und Umwelt医学院-TUM Fakult?t für Medizin体育与健康科学学院-Fakult?t für Sport- und Gesundheitswissenschaften社会科学部（Social Sciences）：教育学院-TUM School of Education政治、公共关系学院-TUM School of Governance商业管理学（企业/经济）-TUM School of Management三、慕尼黑工业大学强势学科慕尼黑工业大学的系科设置主要基于工程科学和部分自然科学，另外还设有部分医学学科（易萨河畔的大学医院，德文：Klinikum rechts der Isar der TU München，以及慕尼黑德意志心脏研究中心），体育科学，经济学，营养科学中心，土地规划和设立于Weihenstephan（弗莱辛，德文：Freising）的环境科学。

大学生活(听力词汇)德语

Bewerbungs unterlagen:文字材料Originale 原件Beglaubigung 公正Übersetzung 翻译Verfassen:撰写Motivation, formlos,Antragsformular, Anmeldeformular ausfüllen,tabellarischer Lebenslauf1 Zeugnis Abschlusszeugnis Sprachzeugnis2 Verleihungsurkunde 学位证书Geburtsurkunde 出生证Heiratsurkunde 结婚证3 Zertifikat APS i (veraltend) (amtliche) Bescheinigung, Beglaubigung ii Zeugnis über ein abgelegte Prüfung z.B. ein Z erwerben, ausstellen4 bestätigen vt.i für richtig, zutreffend erklären z.B. e-e Nachricht amtlich, offiziell bii als richtig erweisen z.B. Das Ereignis bestätigt meine Vermutung.iii sich b sich als richtig, zutreffend erweisenz.B. Die Nachricht, der Verdacht hat sich bestätigt.iv mitteilen, dass man e-e Sendung o.ä. empfangen hat:z.B. das Eintreffen e-r Ware, einen Brief, e-e Bestellung bBestätigung f mit der etw. 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Abitur 中学毕业考试Abiturient anwerben 招募Fächer:Jura ---- Rechtwissenschaften juristisch Jurist 法学学生Germanistik Anglistik英美文学Linguistik Philosophie----- Geistes wissenschaftenBWL / VWL Betriebswirtschaft V olkswirtschaftNatur wissenschaften ---- Physik Mathematik ChemieWirtschaftsmathematik Mathe 学数学专业的人Pharmazie 药学Medizin ----- StaatsexamenIngenieur wesen --- Elektrotechnik Informatik Maschinenbau Bauingenieur 土木工程Vorgang 大学学习过程-Gymnasium Gymnasiast 文理中学学生Abitur ablegen 考试Abiturient--- Hochschul reife erwerben 获得上大学资格- nicht anerkenntFeststellungsprüfung 鉴定考试--- Studienkolleg 预科---Hochschul reife erwerben 1, sich bewerben um + A 申请Studienplatz Bewerbungsunterlagen2, aufnehmen 录取Zulassung(jemandenA zur Prüfung/zum Studium zu/lassen) 3, Immatrikulation4, Sprachkurse besuchen / belegen Sprachniveau erreichenSprachzeugnis n. Zertifikat5, 选课belegen Lehrveranstaltungen 教学活动--- Belegbogen 选课单Teilnahmeschein Leistungsnachweis 成绩证明benoteter ScheinNote Zensur分数Woher:1, V orlesungs verzeichnis课程目录Gesamt- 总目录------ BuchhandlungKommentiertes 分目录（有注释的）------ im Sekretariat n. 秘书处（Fachschaft 系办）2, am Schwarzbrett 布告栏Aushang 告示Prüfung1, Einstufungstest 分级考试2, mündliche Prüfung- Auswahl gespräch在录取之前- Eignungstest 技能考试（艺术类）- Feststellungsprüfung3, Klausur f. 正式的笔试Note Zensur4, Abschlussprüfung 毕业考试5, Staatsexamenbestehen/ bei D gut abschneiden/abgeschnitten haben 通过ablegen考---- durchfallen 不过动词：1, 举行：stattfinden2, ausüben 做什么事Studenten üben vielen Prüfung aus.Studien phasen大学学习阶段Grundstudium Zwischenprüfung Hauptstudium / Aufbaustudium 专业课学习Praktika (pl.) 实习Praktikum machen Praktikant 实习生----LohnHonorar 酬谢Honorarstelle 临时报酬工作--- FeststelleAusbeuten 剥削Firmenliste von Fachschaft ( Fachbereich )erreichen 联系上某人jemand ist zu erreichen.Abteilung, f Fakultät wofür zuständig sind ------ Referenz推荐，介绍，推荐人，介绍单位Austauschprogramm Erasmusprojekt Erasmus studenten 交换生Stipendium Lebensunterhalt Zulage 补助Finanzhilfekürzen停了奖学金停了（缩短了）Mein Stipendium wird gekürzt. (被动)找房子WohnungssucheGeld: Miete 房租kalt oder warmNebenkosten inklusive/einschließlich +G: Heizung 暖气Gas-Anschluss 煤气接口Zentralheizung, Strom, Wasser, Gas, Elektroherd 电炉Kochplatte 做饭的地方Müllabfuhr 垃圾清运费Gerage 车库Internet- Zugang, Internet-Anschluss, Kabelanschluss（电视接口）Die Miete ist am Anfang des Monats fällig应支付的（少用sollen）Miete: Betrag 金额Empfänger 收款人Zimmervermittlung / Makler 经纪人，中介Provision 手续费，回扣einen Mietvertrag ab/schließen/ unterschreiben 签合同vermieten mietenVermittlungsgebühren 中介费hinterlegen留下来Belohnung 酬劳Kaution 押金: 3 Monatsmiete (etwas schädigen)m Hausverwalter-, der Hausmeister-Form1, zur Untermiete （住在德国人家里）转租Balkon 阳台, Terrasse露台bei Dachzimmer, oder in Erdgeschoss，im Keller nach Süden2, WG WohngemeinschaftFlur Wohnzimmer Toilette Bad Küche sind gemeinsam3, Studenten(wohn)heim: Wartelisteeintragen 登记beantragen 申请– einen Antrag stellenSuche: die Anzeigeam Schwarzbrett Wochenblätter 报纸（免费）Immobilien 不动产(正式报纸) Forum 论坛Foren (pl.)auskommen miteinander 和别人相处得好图书馆一，BücherStandardwerk 标准手册Lehrbuch, Lehrwerk 教材n. Lexikon 专业词典--- Lexika (pl.) Nachschlagewerk 参考书Roman 长篇小说/ Novelle 中篇小说/ Erzählung 短篇小说Geschichte 故事Fachliteratur 专业文献Primärliteratur, Sekudärliteratur einsprachige Wörterbuch f. Lektüre (总称) 阅读材料Zeichen: roter Punkt / roter Streifen 条码，die Signatur 书目号，书架号fachliche Begriffe 专业概念nachschlagen 查找recherchieren 检索（电脑）二，地点Lesesaal 阅览室Lesesäle (pl.) Präsenzbibliothek 不许外借Magazin（书库）Computerraum 机房Formalitäten 手续erledigen 办理Personalausweis Passfoto照片Studentenausweis vorlegen 出示jemandem einen Bibliotheksausweis ausstellen（lassen）签发Matrikelnummer注册号immatrikulieren 注册Verwaltung 管理部门Ausweis ausstellen lassen 证件签发Abstempeln 盖章gelten gilt als.... sein gültig 有效的läuft ab 过期Leihfrist 借期Rückgabetermin 还书日期versäumen 错过，verlängern vt.延长V orwort 前言( geschichtlich ) Register 索引( Zitat 引言Quelle)Entstehung Hintergrund 背景Urheberrecht 版权Lizenzgebühren 版权费Patent 专利Buchhandlung 书店Taschenbuch gebundene Bücher 装订bestellen 订书Abkürzung 缩写Terminologie 专业术语Pädagogik 教育学/ Studiengang für Lehramteinige Formalitäten⏹sich (an einer Uni/für das Wintersemester 2006/07)anmelden/einschreiben/immatrikulieren/ist eingeschrieben(die Immatrikulation)（注册）⏹sich für… zurückmelden⏹sich abmelden/sich exmatrikulieren (大学生离校或毕业时的）注销⏹das Stidium abbrechen/unterbrechen（中断）Beiträge bezahlen⏹Sozialbeitrag/gebühren（杂费）⏹Verwaltungskostenbeitrag/gebühren（管理费）⏹Grundstudienbeitrag/gebühren（学费）⏹wie bezahlen? (an/zahlen vt.首付（现金）und überweisen转账) Lehrveranstaltungsarten an deutschen Universitäten⏹die Einführungsveranstaltung（引导课）wann? in der Orientierungwoche, kurz vorm Vorlesungsbeginn, z.B. das Erstsemesterfrühstück von wem? von der Fachschaft eines Fachbereichs/einer Fakultät(系), älteren Studentenwas? ein gutes Forum für alle Fragen zum Studiumwie? organisieren oder veranstaltenwozu? die neuen Studienanfänger orientieren（引导）und schnell eingewöhnenWo? auf dem Universitätscampus(m)（校园）, sich anmelden（报名）⏹die Vorlesungwie? über ein Thema sprechen, zuhören und mitschreiben, oft Skripte（讲稿，教案）und Literaturhinweise ausgehändigt(把…交给）werdenwichtige Verben: sich für eine V orlesung/ ein Seminar/ eine Prüfung/ einen Studiengang einschreiben（报名）oder eine V orlesung belegen vt.（选课）/ eine V orlesung abwählen vt.（不再选）⏹das Seminarvon wem? zwischen dem Dozenten（讲师）und den Studierendenwozu? etwas erarbeiten und diskutierenwodurch? Referat（报告）vorbereiten und halten⏹Übungder Tutor und die Tutorin（辅导老师）(Tutorium辅导课)Fachrichtungen⏹Naturwissenschaft自然科学⏹Geisteswissenschaft人文科学⏹Ingenieurwissenschaft理工科⏹MedizinWo kann man Informationen bekommen?⏹das Studentensekretariat (für Anmeldung, Zurückmeldung, Abmeldung)⏹die Anschlagtafel für Aushänge=auf dem schwarzen Brett/Info-Brett（信息板）(Zettelaushängen)⏹das Studentenwerk (für Jobsuchen und Wohnungen)Erstellen eines Studienplans⏹Veranstaltungen frei zu wählen und das Studium selbst zu gestalten⏹das Gesamt-V orlesungsveryeichtins（总课程目录）,das kommentiertesV orlesungsverzeichtnis(有注释的课程总目录）was? Wie man einen Leistungsnachweis für die jeweilige Veranstaltung bekommt, z.B. durch Klausuren(schreiben,machen), Referat(halten,vorbereiten), Hausarbeit(bearbeiten, abgeben).⏹Wie viele Semesterwochenstunden(SWS)?für jede SWS eine Stunde V or- und Nachbereitungszeit⏹Prüfungsordnungwelche Scheine（证书）=Leistungsnachweiseder Schein=LeistungsnachweisArten der Scheine☆unbenotete Scheine（打分的）: nur teilgenommen hat☆benotete Scheine: eine Lehrveranstaltung belegen, eine Prüfung belegenPrüfungsarten⏹Einstufungstest（分级考试）⏹mündliche Prüfung⏹Auswahlsgespräch⏹Eignungsprüfung（资格技能考试，艺术类院校的考试）⏹Feststellungsprüfung （下结论的考试）⏹Zwischenprüfung⏹Klausur （eine schriftliche Prüfung, besonders ander Uni) f Note=f Zensur-en⏹Abschlussprüfung⏹Staatsexamenablegen,machen, schreiben, bestehen=bei D gut abgeschnitten haben（结果好）, durchfallen vi.（没通过）Er ist durchgefalleneine vernünfige Zeitplanung⏹Wozu? Zeit sparen und Stress ersparen⏹Was zu planen? Alle wichtige Daten wie Klausuren- und Referatstermin,Lerngruppentermine eintragenTipps an Erstsemester⏹Kontakt mit anderen Kommilitonen zu knüpfenWie? z.B. eine Lerngruppe zu gründen(gemeinsam die V orlesung nachzuarbeiten, sich auf die Prüfung vorzubereiten), der Uni-SportLernen, mit der vielen Freizeit/dem Freiraum, umzugehen⏹Sofort nachfragenwo und wie? einen Termin bei der Studienberatung zu machen, beim AAA(das Akademische Auslandsamt), sich ans Studentensekretariat kim wendenWichtige Lernfähigkeiten⏹starke Lern- und Leistungsbereitschaft⏹hohe Selbstständigkeit und Motivation⏹besondere kommunikative Fähigkeiten（交流能力）⏹hohe Ausdauer und Belastbarkeiten（毅力）⏹intellektuelle Neugier （求知心）⏹X-kompetenz能力(Sprachkompetenz)ein Austauschprogramm⏹mit Stipendium oder ohne Stipendium (Zuschüsse für die Krankenversicherung und fürdie Reisekosten, einen Aufenthalt finazieren vt. m Stipendiat-en)⏹AnerkennungDie Studienleistungen im Ausland werden anerkannt/sich die Studienleistung anrechnen（估价，评价）lassen.=Man braucht die versäumten Semester nicht nachzuholen.⏹die Bewerbungsfrist und die Informationsbroschüre (um Informationen zu holen)⏹die Studienzeit verlängernAbschlüsse⏹Diplom-, Magister-, Bachlor- und Master(studiengänge/Titel) erwerben⏹promomieren(Promotion)⏹das Studium abschließen/absolvieren（毕业）/abbrechen⏹m Absolvent-en, m Doktorand-en（攻读博士学位的人）, m Studienabbrecher⏹ein Zeugnis/einen Titel erlangen（得到）, erwerben（得到）, anstreben, ausstellen（发）,verleihen（颁发）andere wichtige Wörter und Formulierungen⏹sich qualifizieren(die Qualifikation)⏹das Hauptfach und das Nebenfach⏹obligatorisch(必修的）/fakultativ(选秀的）Fächer⏹die Studienzeit⏹das Akademische Auslandsamt (AAA)⏹das Stipendium beantragen/sich bewerben um⏹das Studentenwerk⏹die Aufbaustudiengänge (nach dem Studium, für weitere Qualifikation)。

从文本类型理论角度浅析字幕翻译

Modern Linguistics 现代语言学, 2023, 11(9), 4026-4032 Published Online September 2023 in Hans. https:///journal/ml https:///10.12677/ml.2023.119541从文本类型理论角度浅析字幕翻译——以电影《绿皮书》为例仰林凤安徽大学外语学院，安徽合肥收稿日期：2023年7月31日；录用日期：2023年9月8日；发布日期：2023年9月22日摘要 20世纪70年代，凯瑟琳娜·赖斯以对等理念为基础提出了文本类型理论，并列举了一些操作准则，用于评估译文的充分性。

在文本类型理论下，电影属于视听类文本，但是又具有其它类型文本的特色。

本文以文本类型理论为基本框架，借助赖斯的操作准则，对电影《绿皮书》中的字幕翻译实例进行简要评析，主要阐述文本类型理论对电影字幕翻译实践的指导作用，提供借鉴意义。

关键词文本类型理论，字幕翻译，《绿皮书》Subtitle Translation Study from the Perspective of Text Typology—A Case Study of Green BookLinfeng YangSchool of Foreign Studies, Anhui University, Hefei Anhui Received: Jul. 31st , 2023; accepted: Sep. 8th , 2023; published: Sep. 22nd , 2023AbstractIn the 1970s, Katharina Reiss put forward the text typology based on the concept of equivalence and listed a series of instruction criteria for evaluating the adequacy of translation. Under the theory of text typology, film belongs to audio-visual text, but it has the characteristics of other types of text. With the text typology theory as the basic framework and with the help of Reiss’s in-struction criteria, this paper conducts a brief evaluation and analysis of the subtitle translation仰林凤examples in the film Green Book, aiming to expound the guiding role of the text typology theory in film subtitle translation practice and provide reference significance.KeywordsText Typology, Subtitle Translation, Green BookThis work is licensed under the Creative Commons Attribution International License (CC BY 4.0)./licenses/by/4.0/1. 引言随着全球影视业的蓬勃发展，大量国际影视作品不断涌入中国市场，并迎来了蓬勃发展的大好时期。

T眉RK_ES_ZL眉KLER璇嶆眹

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Reihe Informatik2/1998The Evaluation of Content-Based Web Queries .Thorsten Fiebig and Guido MoerkotteTheEvaluation of Content-Based Web QueriesThorsten Piebig Guido M oerkotteLehrstuhl für Informatik IIIFakultät für Mathematik und InformatikUniversität MannheimSeminargebäude A568131MannheimGermanyemail:thorstenjmoer@rmatik.uni-mannheim.deweb:rmatik.uni-mannheim.defon:+496212925403fax:+496212923394AbstractWe introduce the notions of syntactically and semantically structured data to refine the notion of semi-structured data.As we will see,most data found on the Web is syntactically structured.In order to evaluate content-based Web queries,semantically structured data is needed.The problem occurs to transform syntactically structured data into semantically struc-tured data.Syntactically and semantically structured data can be represented by trees.Our main con-tribution is a powerful restructuring mechanism that allows to express the transformation of trees representing syntactically structured data to trees that represent semantically structured data.We embed our restructuring mechanism into RAW(Relational Algebra for the Web)and demonstrate its expressiveness by several example queries.11IntroductionAdvanced applications demand queries to be evaluated against a variety of data sources.These data sources can either be traditional database systems or non-traditional data sources containing semi-or unstructured data.Furthermore,both kinds data sources are often distributed world-wide.Several approaches exist to integrate different kinds of data sources for querying[1,2,20,16, 21,16,25,12,11,8].Among the predominant approaches are wrappers that wrap data sources in order to provide a uniform interface for the integrating query engine[10,29,16].One necessary requirement is a common data model into which wrappers transform the data of their wrapped data sources.Many existing approaches build upon tree-based data models for this purpose[7,31, 5,14,15,31,3]'others build on object-oriented data models[1,2,8].Another major problem is to describe the query capabilities of wrappers or data sour ces and exploit these for query processing [25,26,32].Howevet,we are not going to discuss this issue here.Instead,we dig deeper into the problems involved in processing semi-structured data representable as trees.The actual differences between tree data models are minor.Some allow for labels only at the leaf nodes[31,5]'others allow labels at edges only[14,15];some support only unordered trees,others ordered trees[3].The main advantage common to all tree models is that data in all standard data models(e.g.hierarchical,relational or object-oriented data model)as weIl as semi-structured data can be represented using trees[6,15].In fact,it seems that the term semi-structured is often used in exactly this case,that is,if a tree-based data model is used,the data represented with it is called semi-structured.Hence,the not ion of semi-structured data is not very useful if data sources have to be classified.To see the point,consider two examples.A HTML-page can be represented by its parse tree.The nodes(or edges)might be labeled by the syntactic categories(e.g.list item).These labels differ from the semantic labels like"restaurant"found in work on tree-based data models [6,15].Obviously,the former carries less semantics than the latter although there mostly exists a correspondence between the syntactic representation and the semantic interpretation.However,this2correspondence is-without any furt her information-not E)asilyaccessible to computers.Hence, we distinguish between syntactically and semantically structured data(trees).Figures1and2 represent a syntactically and a semantically structured data tree.The example is taken from the Ley server[27J.Since most data on the Web is syntactically structured,content-based Web queries-i.e.queries stated using semantic categories such as author name or publication date-demand the transformation of syntactically structured data to semantically structured data.However,note that todays query languages far the Web do not provide such a mechanism[28,24,23J.Instead,they rely on predicates based on string matching and syntactic structure.Other approaches directly work on semantically structured data only[1,2J.The problem focussed on in this paper is to introduce a means to express the missing information needed to transform syntactically structured data into semantically structured data-as required for content-based Web queries.As a first step we introduce RAW data trees which are able to represent both kinds of data.Our working hypothesis is that tree restructuring mechanisms are capable of transforming syntactically structured trees into semantically structured data trees.Hence,we introduce a powerful restructuring mechanism.Prom RAW data trees representing semantically structured data,subtrees relevant to process a query must often be extracted.We present the extraction mechanism used within RAW.It supports the10different kinds of tree pattern matching discussed in[plex example queries against the Ley server illustrate how the restructuring and extraction mechanisms are embedded into RAW expressions that express content-based queries.The rest of the paper is organized as follows.Section2reviews RAW.Its definition differs slightly from the original proposal[17J.The main contribution of the paper can be found in Sec-tion 3.The new domain RAW data tree-our variant of a tree-based data model-is introduced. Besides different constructors it provides for a powerful restructuring mechanism allowing to trans-form syntactically structured data trees into semantically structured data trees.Relevant parts of a data tree can be extracted using different tree matching algorithms producing variable(at-tribute)bindings.The restructuring mechanism and theextraction mechanism are both described3in Section 3.In Section4,we give several examples of complex queries against Ley's server.They demonstrate how the restructuring and extraction mechanismsare embedded into the algebra. Section5concludes the paper.2RAWOne major design goal of RAW I is to bridge the gap between web query languages and traditional query optimization since we believe that there is much unexploited optimization potential in the currently prevailing interpreter approach.As a consequence,RAW must be able to cope with syntactically structured data found on the biggest data source there is-the Web.Instead of designing a totally new approach,we decided to enhance the traditional relational algebra by as few new algebraic operators as possible and mainly by new domains.Each new domain type comes along with certain functions and,hence,can be thought of as an ADT.This will open a road between web queries and standard database query evaluation and optimization techniques.2.1RAW's Relational Model and DomainsA relational schema n is a finite set of attribute names{Al,'",An}.With each attribute Ai a)domain Di is associated.A Relation R for a schema n={Al,'",An}is a subset R~D I x...xD n. This is standard[4].The only new algebraicoperator in RAW is the MAP operator.This operator is rat her standard in object-oriented systems where it occurs under a variety of names[22,13].Apart from the standard domains(int,float,string,bool)RAW features the following domains:1.URL2.Web-Path3.HTML-Document4.RAW Data Tree(RDT)Isee[17]for a full account of the design goals4A URL denotes an universal resource identifier(URI)as defined in[18].A Web-Path is used to represent a path within the Web.It can be thought of as an alternating sequence of URLs and documents-except that instead of the actual documents special identifiers are used.For the purpose of the paper,the following two functions defined for Web-Paths suflice:•length:Web-Path-+intreturns the number of URLs contained in a Web-Path.•get_last:Web-Path-+URLreturns the last URL contained in the Web-PathWeb-Paths can be generated by applying the functiongeLpaths:URL,pred,patt-+set-of(Web-Path)Starting at the argument URL,it traverses the whole net to generate all possible paths starting at this URL.Since often not all paths are required,there exist two possibilities to rest riet the number of actually generated paths.The first is a unary predicate that is evaluated on every generated path,only if the predicate is fulfilled,the path is extended and put into the result set.The second possibility to restriet the number of results is a pattern.This pattern describes those parts of the starting document where geLpaths is allowed to look for URL's.Patterns are described in more detail in the next section.2.2The Algebraic OperatorsAs the common Relational Algebra RAW consists of a setof operations.The operator set contains the standard set operators union,intersection,set-difference.Further,it contains the standard relational algebra operators.Let us denote the type of relation with schema{Al,""An}by {[Al:Tl,".,An:T n]}.Each Attribute of the relation is represented by its name Ai and its domain type Ti.5•BELEGT p:{[Al:Ti,...,An:T n]}-7{[Al:Tl,'..,An:Tn]},where P:[Al:Tl,'",An:TnJ-7bool is the selection predicate•PROJEGT{Ai:Ti,,Apj r{[A l:Tl,...,An:Tn]}-7{[Ai:Ti,...,Aj:Tj]},where{Ai:Ti,,Aj:Tj}~{Al:Tl,'",An:Tn}.where P:[Al:TAl'".,An:TAnl,[BI:TBl""Bn:TBnJ-7bool is the join predicateAs selection and join predicates,RAW accepts arbitrary boolean expressions possibly contain func-tions for the new data types introduced above.Additionally theMAP.operator belongs to the operator set of RAW.MAP f,B:{[Al:TAl"'"An:TA n]}-7{[Al:TAl""~An:TA n,B:T]}f:[Al:TAl"",An:TAnJ-7TFor each input tuple the map operator extends the input relation by the attribute B,that is the result of the evaluation of the function f found in the subscript.Possibly the result of function f is a set,in thiscase the operator generates one output tuple for each element in the set or in case of multiple set valued expressions,one output tuple for each combination.In the latter case the MAP operator can be thought~f as a standard MAP operator follow:edby an unnest.3From Syntaetically to Semantically Struetured DataIn order to facilitate content-based Web queries,more domains are needed to represent syntactically and semantically structured data.This sections introduces the RAW data tree(or RDT for short). The RDT is the new core domain used to represent syntactically and semantically structured data. RAW data trees will be presented in the next subsection.A typical RAW query works in several phases.First,the information is fetched from the net and presented as a RAW data tree.This6RAW data tree is typically syntactically structured.The conversion from syntactically structured data to semantically structured data is achieved by a powerful restructuring mechanism represented in subsection 3.2.Then the relevant parts of the semantically structured data tree are extracted and mostly represented as atomic attribute values.This part is introduced in subsection st, additional selection predicates are evaluated and the result is projected.Note that this is only a: common pattern found in RAW expressions.Queries do not have to follow this pattern.Typical queries following this pattern are presented in Section 4.3.1RAW Data Trees and their ConstructionA RAW data tree is a labeled ordered tree.Every node consists of a label and a unique identifier. Additionally,inner nodes have an ordered list of successor nodes.Anode is termed leaf node if this list is empty.Example RAW data trees can be found in Figures1and 2.(We do not give the note identifiers since they are not relevant for our discussion here.)For a given URL,the referenced document is transformed into a RAW data tree in two steps. First,the URL is dereferenced and the HTML-Document is parsed.As in traditional compilers, thisresults in an abstract syntax tree(AST)[9].The domain HTML-Doc consists of the set of all possibleabstract syntax trees possible for HTML-Documents.Second,the abstract syntax tree is ttansformed into a RAW data tree.Why this indirection?If in the near future HTML will be replaced by XML then we only have toexchange the parser for HTML in order to enable RAW to deal with XML documents.Hence, the necessary changes to the RAW system remain small.The functions supporting the transformation are:•letch:URL--+HTML-Doc•build_tree:HTML-Doc--+RDTThe letch function retrieves a HTML-Document referenced by a given URL from the Web and parses7it.The result is its abstract syntax tree in the domain HTML-Doc.Within the AST representation, two kinds of no des exist.Those representing HTML-Tags together with their attributes and plain text nodes representing the displayed text or contents of the tags.Consider for example the A-Tag.It consists of q,ttributes(e.g.the href attribute)and its contents(the anchor text).The implementation of fetch is rat her standard.After fetching the document via the hypertext transfer protocol[18]'standard parsing takes place[9].The function build_tree transforms the abstract syntax tree into the RAW data tree represen-tation.For every AST node representing a HTML-Tag,a small subtree consisting of three nodes is generated.The root of the subtree is anode labeled by the HTML-Tag.Its successor nodes are the attribute node and the content node.Every AST node representing plain text is mapped to a single RAW data tree node representing this text.All nodes representing plain text are collected under a newly constructed nodes labeled by"Text".Again,the implementation is rat her straight forward.For both functions,the Cocktail Tools[19]were used for the implement at ion.3.2RestructuringRAW Data TreesThe application of the build_tree function typically results in a syntactically structured data tree. The crucial and most difficult step is to transform this syntactically structured information into semantically structured information.Let us illuminate the occurring problems by an example situ-ation.Ley's Logic Programming and Database server provides plenty of HTML-pages representing bibliographical data[27].Among the different views given upon this data is the author page.For a given author,all publications of this author are presented on a single page.Let our task be to add this information for a given set of authors to a relational database containing bibliographical data.Obviously,doing so manually is a tedious ing RAW this task becomes less laborious.Using the mechanisms explained so far,it is not difficult to retrieve the page for a given author and produce the according syntactically structured data tree.For every publication of the given author it contains a subtree representing this publication.Unfortunately,the syntactic structure8Figure1:A Syntactically Structured Data Theedoes not match the semantic structure we would expect for a bibliographie reference.Figure1 shows one subtree for a publication by the authors Kemper and Moerkotte as found on the page for"Moerkotte".As adefault,while parsing a HTML-page,longer strings are cut into pieces whenever a blank or newline character occurs.For every coauthor,except the one to whom the page belongs,there is a reference to the coauthor's publication page.This(justified)irregularity is directly reflected within the data tree resulting from the abstract syntax tree for the entries:the tree in Figure1has a plain text node for the author and the title since this part is no furt her structured by HTML~Tags.The resulting problem is that a single node represents information on two semantic categories:coauthor and paper title.Hence,this node needs to be refined by splitting.The opposite can also occur:one semantic concept is represented by different nodes in the syntactically structured data tree.Take the whole list of authors in Figure1as an example.A step towards a semantically structured data tree for the same entry can be found in Figure 2. The two main differences are that(1)there exists a single subtree for every semantic category and(2)the according root nodes are labeled by the category names.It should be clear that this tree could be gained from the one in Figure1by restructuring.It is our working hypothesis that9Figure2:A Semantically Structured Data Treethe transformation of syntactically structured data trees to semantically structured data trees can be achieved by a sequence of restructuring steps.Each step thereby enrichesjrestructures a given RAW data tree.In RAW,a single restructuring step corresponds to an application of the func-tion tree_restructure which takes a RAW data tree and a restructuring specification as arguments. Restructuring specifications bear some resemblance to traditional context free grammars.A restructuring specification consists of a set of non-terminal symbols NT and a set of rules. As building blocks rules contain non-terminal symbols and regular expressions to denote tokens. The regular expressions have to be enclosed by quotation marks.Further,two types of rules exist. The first type consists of rules with no non-terminal symbol on the left-hand side(Type-I).These rules are merely used to describe patterns to be found in the data tree.The second type of rules exhibits a non-terminal symbol on the left-hand side(Type-2).These rules additionally introduce new nodes labeled by their left-hand side's non-terminals into the data tree.Here is part of the restructuring specification needed for author pages of the Ley server:1.NT={AuthorList,Title,BookTitle,Pages,Year}102.;=AuthorList"."Title"."BookTitle Year";"Pages;3.AuthorList;="f-,;}+";4.Title;="f-,.}+";5.BookTitle;="f-,.}+";6.Pages;="fO-9}+-fO-9}+";7.Year;="fO-9}+";The right-hand sides of the rules consist of a sequence of non-terminal symbols and regular ex-pressions.Within the regular expressions,we preceded a character c by-,in order to denote any character but c.As usual,"+"denotes non-empty repetition.The complete right-hand sides of a rule constitute a pattern to be matched against the given data tree.During the matching process, the actual boundaries of the plain text fragments in leaf nodes are crossed if necessary.For exam-pIe,the first rule specifies that the author list and the title must be separated by a colon.This colon is not present as a singular node in the data tree,instead it is contained in the node labeled by"Moerkotte:".Further,the complete author list is contained in four data tree nodes.Though the rules are easy to specify,the matching process is rat her complex.After a succesful match,new nodes are created for those rules whose left-hand side is non-empty.The label of the node directly corresponds to the left-hand side's non-terminal.More specifically,restnicturing takes place in three steps.Step1expands Type-l rules by replacing the non-terminals by the according rules.This requires the rule set to be non-recursive. If there exist several rules with the same left-hand side,all combinations are generated.Step2 tries to match the expanded rules against the data tree.Thereby,the total sequence of leaf nodes is considered as a single string.The goal of the matching step is to find fragments of the data tree that correspond to the regular express ions of the expanded rules.A fragment consists of a sequence of leaf nodes corresponding to exactly one regular expression in the expanded rule.During this11Figure3:Intermediate State Istep leaf nodes are splitted if several regular express ions share the label of anode.The shared label is then distributed among the nodes resulting from the split.Inner nodes are splitted,if they point to more than one fragment.For splitted inner nodes,the label is duplicated.This step takes place iteratively until no more inner node points to more than one fragment.Step3creates new nodes for the matching rules with non-empty left-hand side.Their label is the non-terminal on the left-hand side of these rules.Let us illustrate the three steps using the example.During the first step,the first nile is expanded to;="[.;]+"";""[•.]+"".""[•.]+""[0-9]+"";""[0-9]+-[0-9]+";The second step matches this expanded rule against the given data tree.The first two regular express ions("[.:J+"and":")match the four nodes labeled"Alfons","Kemper,","Guido",and Moerkotte:".Hence,since"Moerkotte:"contains parts for two regular expressions,it must be split into anode labeled"Moerkotte"and anode labeled":".Likewise,the node labeled"1995:" must be split.Figure3shows the tree after splitting the leaf nodes.Fragments corresponding to a regular expression in the expanded rule are framed by dotted lines.Note that the first node and12------------------------------.---------------------~Figure4:Intermediate8tate IIthe second node labeled"Text"point two more than one fragment.Hence,they must be split.The result of splitting/duplicating the inner nodes is shown in Figure 4.In thethird step,newnodes with labels"AuthorList","Title","BookTitle","Year",and"Pages"have to be introduced(see the last5rules of the restructuring specification).The least upper node whose subtree pointsto all fragments comprising the body of the rule is labeled"Content".Hence,the new nodes areintroduced directly below this node.Figure2shows the result after introducing the new nodes.3.3Extraction of Raw Data SubtreesOften the interesting information is placed only in a small part of the data tree.In order to extractthe relevant parts of a data tree,the function•tree_match:RDT,RPT-+VariableBinding(=Tuple)is introduced.It takes a RAW data tree and a RAW pattern tree as arguments.A RAW patterntree(RPT)is an adorned RAW data tree.Nodes in RAW pattern trees can be labeled by variables.Additional adornments allow to express different kinds of tree matchings(see below).13RPT:~00matches with OTI:RDTl,RDT2,RDT3matches with OPI:RDTl,RDT3RDT3:RDTl:~80'0Figure5:Different Kinds of Tree MatchingThe function tree_match matches the RPT against the given data tree.A match consists of a correspondence relation between the nodes of the data tree and the nodes of the pattern tree.For every successful match a variable binding is produced for every variable found in the pattern tree. More specifically,if a pattern node n p is labeled by a variable v and for a successful match nd is the corresponding node in the data tree,the variable v is bound to the subtree rooted at nd.For every variable binding a tuple is constructed where the attribute names equal the variable names and the attribute's value are the bindings of the corresponding variables.With adornments,RAW pattern trees are able to express different kinds of matching.These different kinds of matching express variations in the degree of correspondence that has to be achieved during the matching process.For example,if the order of the nodes in the pattern tree might be relevant or irrelevant for the matching process.Another distinction is whether the subtree of the data tree must be isomorphie to the pattern tree or whether it must contain the pattern tree.Let us explain the difference between ordered tree inclusion and ordered path inclusion.For ordered tree inclusion,every node of the pattern tree must match withsome node in the data tree but a child node of the pattern tree does not have to match with a child of the matching node of its parent pattern node.Instead the matching successor node can be several levels down.Consider14for example the pattern RPT and the three RAW data trees RDT1to RDT3in Figure 5.Under ordered tree inclusion,the pattern tree RPT matches with all three data trees even with the second one.Obviously,the root node of the pattern tree matches with the root node of the data tree RDT2.The left child of the pattern tree(labeled by"c")matches the middle child of RDT2's root. The right child(labeled"d"),however,does not match any direct child of RDT2's root.Instead,it matches the grandchild labeled"d".For tree inclusion matching this is a regular situation.Hence, tree inclusion can be used to traverse unknown paths down the tree.In situations where this implicit traversal of paths is not allowed,path inclusion matching is applied.Here,the children of a pattern tree must match the children of the corresponding node in the data tree.Other possible restrictions exist.For example,one might want the match to be complete,that is,every node in the data tree must also be present in the pattern tree.In total,ten variants of tree matching are distinguished[30]:•Unordered Tree Inclusion(UTI)•Ordered Tree Inclusion(OTI),•Unordered Path Inclusion(UPI),•Ordered Path Inclusion(OPI),•Unordered Regiön Inclusion(URI),•Orderd Region Inclusion(ORI),•Unordered Child Inclusion(UCI),•Ordered Child Inclusion(OCI),•Unordered Subtree Problem(USP),and•Ordered Subtree Problem(OSP).15With according adorn:r:nents,RAW pattern trees are ahle to express which kind of matching is demanded at a certain node.The necessary matching algorithms have been implemented according the lines of[30].However,some modifications in order to retrieve the variable bindings and to enhance performance had to be made.But these algorithms as weIl as a detailed description of the different matching variants is weIl beyond the scope of the paper.For the purpose of the paper,it suffices that relevant parts of a data tree can be extracted using RAW pattern trees together with the function tree_match.4Example QueriesThis section presents three queries against the Ley server[27].Every query will first be stated in natural language.Then we give their algebraic equivalent.We are aware of the fact that these algebraic expressions exhibit optimization potential but optimizing them is beyond the scope of this paper.Query1Query1retrieves the names of all authors that have published at least a SIGMOD or a VLDB paper.Figure6shows the RAW expression for this query.We explain the expression from bottom to top.The lowest line-representing the input relation-contains a single tuple with a single attribute.This attribute points to the author index of the Ley server.Here,authors are put into small alphabetical groups.For example,the second group contains authors from Abr to Ade.Every group is represented as a hyperlink referencing the list of according authors.These are hyperlinks pointing to the actual author pages.On this page,all the publications of the author are collected.The second line of the RAW expression retrieves all the links on the start page and dereferences them two times.The step is implemented by applying the geLpaths function via the MAP operator. This function returns a set of Web-paths.Since it is set-valued,the MAP operator unnests this set.For every element in the set,an output tuple is constructed.These output tuples differ from16HrefPattem:VLDBPattem:SigmodPattem:PROJECT[NameStr]IMAP[geUeaCstr(Name ),NameStr]I MAP[tree_match(Tree,NamePattem),Name]ISELECT[tree_match(Tree,SigmodPattem)not emptyll tree_match(Tree,VLDBPattem)not empty]IMAP[tree_restructure(Tree,RestrucSpec)]IMAP[build_tree(Content),Tree]IMAP[fetch[geUast(Path)],Content]I MAP[gecpaths(Address,lengthO=2,HrefPattem),Path]IAddress =rmatik.uni-trier.de/-ley /db/indicesl a-treelindex.htmlNamePattern:Figure 6:The RAW expression for Query 1the input tuple in that a single attributePath is added that holds one of the paths contained in the result set of geLpaths.The next two MAP operators fetch the documents and build the data trees.Then the re-structuring specification is applied to transform the syntactically structured author pages intosemantically structured author pages as discussed in the previous section.The BELEGToperator uses the tree matching algorithms to verify that the paper is a SIGMOD or a VLDB paper.The according patterns are given on the right-hand side of the figure.Then,the name pattern is applied to retrieve the subtree of the data tree that represents the author's name.The function geUeaf_str then collects the tree labels and builds astring by concatenating st,the name is projected.Query 2This query retrieves the years when "Carey"published a SIGMOD paper.The RAW expression can be found in Figure 7.The bottom four lines are identical with those for Query 1.17。

Fakultt für Mathematik und Informatik

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Fakultt für Mathematik und Informatik

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★留学院校★ 【德国】-曼海姆大学

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