The stochastic Hamilton-Jacobi equation

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斯蒂格勒定律

斯蒂格勒定律斯蒂格勒定律（Stigler's law）：没有一项科学发现是以其最初发现者的名字命名的。

这个定律本身也符合这个定律，因为它最早是由罗伯特.K.默顿提出的。

这个所谓“定律”当然是个玩笑，不一定严格。

但是真的挺能说明问题。

因为首先，思想的原创性其实没那么重要，在科学上，能证明出来才是更重要的；在文学上，能最美地表达出来才是最重要的；在哲学上，能讲的最透彻才是最重要的。

你不能只是说我也想到过，就获得这个思想的命名权。

其次，更重要的是，人的记忆是会偷懒的，地球上生活过上千亿的人，你能说出来名字有几个？所以同一句话不同的人说，你肯定记住的是那个最有名的。

最后大浪淘沙，你觉得这句话是谁说的，取决于你记得住谁，而不是这句话真的是由谁最早说出来的。

最典型的例子是那句“人所具有的我无不具有”，很多人以为是马克思说的，还好彼得拉克是文艺复兴时期的桂冠诗人，所以这个误解还能掰过来。

然而比得拉克也不是原创者，这句话是古罗马一个剧作家写的台词。

可是这个剧作家叫什么呢？可以查到，但是说真的我懒得查。

因为你就算知道这个名字，也不会串联起别的知识点，没什么大用处。

所以你看，问题的关键是，这个名字是否能够串联起其他的知识点。

能串联知识点越多的名字，就越值得被记住。

记忆的世界就是这么残酷，张爱玲说出名要趁早，自贡说君子恶居下流，天下之恶皆归焉，差不多都是这个意思。

对抗被遗忘，就是精神生活的实质。

这听起来很绝望，但是认识到绝望，恰恰也是一种开解。

任何人都可以做一个严肃意义上的创作者，跟出名和吸粉无关，纯粹只是为了对得起自己。

你看说来说去，不也没人知道这个斯蒂格勒是谁吗？无所谓的，但他至少提出了一个有意思的想法，而且抢了罗伯特.K.默顿的风头。

stochastic calculus for fractional brownian motion and related processes附录

$stochastic calculus for fractional brownian motion and related processes附录$

kH (t, u)dWu = CH Γ (1 + α)
(2)
R
α (I− 1(0,t) )(x)dWx
(see Lemma 1.1.3). Therefore, the ﬁrst equality is evident, since
0 R t
(kH (t, u))2 x)α )2 dx +
k n
2H
2
.
C . n2
(B.0.12)
References
[AOPU00] Aase, K., Øksendal, B., Privault, N., Ubøe, J.: White noise generalization of the Clark-Haussmann-Ocone theorem with applications to mathematical ﬁnance. Finance Stoch., 4, 465–496 (2000) [AS96] Abry, P., Sellan, F.: The wavelet-based synthesis for fractional Brownian motion proposed by F. Sellan and Y. Meyer: Remarks and fast implementation. Appl. Comp. Harmon. Analysis, 3, 377–383 (1996) [AS95] Adler, R.J.; Samorodnitsky, G.: Super fractional Brownian motion, fractional super Brownian motion and related self-similar (super) processes. Ann. Prob., 23, 743–766 (1995) [ALN01] Al` os, E., Le´ on, I.A., Nualart, D.: Stratonovich stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2. Taiwanesse J. Math., 5, 609–632 (2001) [AMN00] Al` os, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to fractional Brownian motion with Hurst parameter less than 1/2. Stoch. Proc. Appl., 86, 121–139 (2000) [AMN01] Al` os, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Prob., 29, 766–801 (2001) [AN02] Al` os, E., Nualart, D.: Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep., 75, 129–152 (2002) [And05] Androshchuk, T.: The approximation of stochastic integral w.r.t. fBm by the integrals w.r.t. absolutely continuous processes. Prob. Theory Math. Stat., 73, 11–20 (2005) [AM06] Androshchuk, T., Mishura Y.: Mixed Brownian–fractional Brownian model: absence of arbitrage and related topics. Stochastics: Intern. J. Prob. Stoch. Proc., 78, 281–300 (2006) [AG03] Anh, V., Grecksch, W.: A fractional stochastic evolution equation driven by fractional Brownian motion. Monte Carlo Methods Appl. 9, 189–199 (2003)

nutz

Stochastic Target Games with Controlled Loss ∗
Bruno Bouchard
†
Ludovic Moreau June 28, 2012

§
arXiv:1206.6325v1 [math.OC] 27 Jun 2012
We study a stochastic game where one player tries to ﬁnd a strategy such that the state process reaches a target of controlled-loss-type, no matter which action is chosen by the other player. We provide, in a general setup, a relaxed geometric dynamic programming for this problem and derive, for the case of a controlled SDE, the corresponding dynamic programming equation in the sense of viscosity solutions. As an example, we consider a problem of partial hedging under Knightian uncertainty. Keywords Stochastic target; Stochastic game; Geometric dynamic programming principle; Viscosity solution AMS 2000 Subject Classiﬁcations 49N70; 91A23; 91A60; 49L20; 49L25

Some Recent Aspects of Differential Game Theory

Dyn Games Appl(2011)1:74–114DOI10.1007/s13235-010-0005-0Some Recent Aspects of Differential Game TheoryR.Buckdahn·P.Cardaliaguet·M.QuincampoixPublished online:5October2010©Springer-Verlag2010Abstract This survey paper presents some new advances in theoretical aspects of dif-ferential game theory.We particular focus on three topics:differential games with state constraints;backward stochastic differential equations approach to stochastic differential games;differential games with incomplete information.We also address some recent devel-opment in nonzero-sum differential games(analysis of systems of Hamilton–Jacobi equa-tions by conservation laws methods;differential games with a large number of players,i.e., mean-ﬁeld games)and long-time average of zero-sum differential games.Keywords Differential game·Viscosity solution·System of Hamilton–Jacobi equations·Mean-ﬁeld games·State-constraints·Backward stochastic differential equations·Incomplete information1IntroductionThis survey paper presents some recent results in differential game theory.In order to keep the presentation at a reasonable size,we have chosen to describe in full details three topics with which we are particularly familiar,and to give a brief summary of some other research directions.Although this choice does not claim to represent all the recent literature on the R.Buckdahn·M.QuincampoixUniversitéde Brest,Laboratoire de Mathématiques,UMR6205,6Av.Le Gorgeu,BP809,29285Brest, FranceR.Buckdahne-mail:Rainer.Buckdahn@univ-brest.frM.Quincampoixe-mail:Marc.Quincampoix@univ-brest.frP.Cardaliaguet( )Ceremade,UniversitéParis-Dauphine,Place du Maréchal de Lattre de Tassigny,75775Paris Cedex16, Francee-mail:cardaliaguet@ceremade.dauphine.frmore theoretic aspects of differential game theory,we are pretty much conﬁdent that it cov-ers a large part of what has recently been written on the subject.It is clear however that the respective part dedicated to each topic is just proportional to our own interest in it,and not to its importance in the literature.The three main topics we have chosen to present in detail are:–Differential games with state constraints,–Backward stochastic differential equation approach to differential games,–Differential games with incomplete information.Before this,we also present more brieﬂy two domains which have been the object of very active research in recent years:–nonzero-sum differential games,–long-time average of differential games.Theﬁrst section of this survey is dedicated to nonzero-sum differential games.Although zero-sum differential games have attracted a lot of attention in the80–90’s(in particular, thanks to the introduction of viscosity solutions for Hamilton–Jacobi equations),the ad-vances on nonzero-sum differential games have been scarcer,and mainly restricted to linear-quadratic games or stochastic differential games with a nondegenerate diffusion.The main reason for this is that there was very little understanding of the system of Hamilton–Jacobi equations naturally attached to these games.In the recent years the analysis of this sys-tem has been the object of several papers by Bressan and his co-authors.At the same time, nonzero-sum differential games with a very large number of players have been investigated in the terminology of mean-ﬁeld games by Lasry and Lions.In the second section we brieﬂy sum up some advances in the analysis of the large time behavior of zero-sum differential games.Such problems have been the aim of intense re-search activities in the framework of repeated game theory;it has however only been re-cently investigated for differential games.In the third part of this survey(theﬁrst one to be the object of a longer development) we investigate the problem of state constraints for differential games,and in particular,for pursuit-evasion games.Even if such class of games has been studied since Isaacs’pioneer-ing work[80],the existence of a value was not known up to recently for these games in a rather general framework.This is mostly due to the lack of regularity of the Hamiltonian and of the value function,which prevents the usual viscosity solution approach to work(Evans and Souganidis[63]):Indeed some controllability conditions on the phase space have to be added in order to prove the existence of the value(Bardi,Koike and Soravia[18]).Following Cardaliaguet,Quincampoix and Saint Pierre[50]and Bettiol,Cardaliaguet and Quincam-poix[26]we explain that,even without controllability conditions,the game has a value and that this value can be characterized as the smallest supersolution of some Hamilton–Jacobi equation with discontinuous Hamiltonian.Next we turn to zero-sum stochastic differential games.Since the pioneering work by Fleming and Souginidis[65]it has been known that such games have a value,at least in a framework of games of the type“nonanticipating strategies against controls”.Unfortunately this notion of strategies is not completely satisfactory,since it presupposes that the players have a full knowledge of their opponent’s control in all states of the world:It would be more natural to assume that the players use strategies which give an answer to the control effectively played by their opponent.On the other hand it seems also natural to consider nonlinear cost functionals and to allow the controls of the players to depend on events of the past which happened before the beginning of the game.The last two points have beeninvestigated in a series of papers by Buckdahn and Li[35,36,39],and an approach more direct than that in[65]has been developed.Theﬁrst point,together with the two others,will be the object of the fourth part of the survey.In the last part we study differential games with incomplete information.In such games, one of the parameters of the game is chosen at random according to some probability mea-sure and the result is told to one of the players and not to the other.Then the game is played as usual,players observing each other’s control.The main difference with the usual case is that at least one of the players does not know which payoff he is actually optimizing.All the difﬁculty of this game is to understand what kind of information the informed player has interest in to disclose in order to optimize his payoff,taking thus the risk that his opponent learns his missing information.Such games are the natural extension to differential games of the Aumann–Maschler theory for repeated games[11].Their analysis has been developed in a series of papers by Cardaliaguet[41,43–45]and Cardaliaguet and Rainer[51,52].Throughout these notes we assume the reader to be familiar with the basic results of dif-ferential game theory.Many references can be quoted on this subject:A general introduction for the formal relation between differential games and Hamilton–Jacobi equations(or sys-tem)can be found in the monograph Baçar and Olsder[13].We also refer the reader to the classical monographs by Isaacs[80],Friedman[67]and Krasovskii and Subbotin[83]for early presentations of differential game theory.The recent literature on differential games strongly relies on the notion of viscosity solution:Classical monographs on this subject are Bardi and Capuzzo Dolcetta[17],Barles[19],Fleming and Soner[64],Lions[93]and the survey paper by Crandall,Ishii and Lions[56].In particular[17]contains a good introduc-tion to the viscosity solution aspects of deterministic zero-sum differential games:the proof of the existence and the characterization of a value for a large class of differential games can be found there.Section6is mostly based on the notion of backward stochastic differential equation(BSDE):We refer to El Karoui and Mazliak[60],Ma and Yong[96]and Yong and Zhou[116]for a general presentation.The reader is in particular referred to the work by S.Peng on BSDE methods in stochastic control[101].Let usﬁnally note that,even if this survey tries to cover a large part of the recent literature on the more theoretical aspects of differential games,we have been obliged to omit some topics:linear-quadratic differential games are not covered by this survey despite their usefulness in applications;however,these games have been already the object of several survey ck of place also prevented us from describing advances in the domain of Dynkin games.2Nonzero-sum Differential GamesIn the recent years,the more striking advances in the analysis of nonzero-sum differential games have been directed in two directions:analysis by P.D.E.methods of Nash feedback equilibria for deterministic differential games;differential games with a very large number of small players(mean-ﬁeld games).These topics appear as the natural extensions of older results:existence of Nash equilibria in memory strategies and of Nash equilibria in feedback strategies for stochastic differential games,which have also been revisited.2.1Nash Equilibria in Memory StrategiesSince the work of Kononenko[82](see also Kleimenov[81],Tolwinski,Haurie and Leit-mann[114],Gaitsgory and Nitzan[68],Coulomb and Gaitsgory[55]),it has been knownthat deterministic nonzero-sum differential games admit Nash equilibrium payoffs in mem-ory strategies:This result is actually the counterpart of the so-called Folk Theorem in re-peated game theory[100].Recall that a memory(or a nonanticipating)strategy for a player is a strategy where this player takes into account the past controls played by the other play-ers.In contrast a feedback strategy is a strategy which only takes into account the present position of the system.Following[82]Nash equilibrium payoffs in memory strategies are characterized as follows:A payoff is a Nash equilibrium payoff if and only if it is reach-able(i.e.,the players can obtain it by playing some control)and individually rational(the expected payoff for a player lies above its min-max level at any point of the resulting trajec-tory).This result has been recently generalized to stochastic differential games by Buckdahn, Cardaliaguet and Rainer[38](see also Rainer[105])and to games in which players can play random strategies by Souquière[111].2.2Nash Equilibria in Feedback FormAlthough the existence and characterization result of Nash equilibrium payoffs in mem-ory strategies is quite general,it has several major drawbacks.Firstly,there are,in general, inﬁnitely many such Nash equilibria,but there exists—at least up to now—no completely satisfactory way to select one.Secondly,such equilibria are usually based on threatening strategies which are often non credible.Thirdly,the corresponding strategies are,in general, not“time-consistent”and in particular cannot be computed by any kind of“backward in-duction”.For this reason it is desirable toﬁnd more robust notions of Nash equilibria.The best concept at hand is the notion of subgame perfect Nash equilibria.Since the works of Case[54]and Friedman[67],it is known that subgame perfect Nash equilibria are(at least heuristically)given by feedback strategies and that their corresponding payoffs should be the solution of a system of Hamilton–Jacobi equations.Up to now these ideas have been successfully applied to linear-quadratic differential games(Case[54],Starr and Ho[113], ...)and to stochastic differential games with non degenerate viscosity term:In theﬁrst case,one seeks solutions which are quadratic with respect to the state variable;this leads to the resolution of Riccati equations.In the latter case,the regularizing effect of the non-degenerate diffusion allows us to useﬁxed point arguments to get either Nash equilibrium payoffs or Nash equilibrium feedbacks.Several approaches have been developed:Borkar and Ghosh[27]consider inﬁnite horizon problems and use the smoothness of the invari-ant measure associated to the S.D.E;Bensoussan and Frehse[21,22]and Mannucci[97] build“regular”Nash equilibrium payoffs satisfying a system of Hamilton–Jacobi equations thanks to elliptic or parabolic P.D.E techniques;Nash equilibrium feedbacks can also be built by backward stochastic differential equations methods like in Hamadène,Lepeltier and Peng[75],Hamadène[74],Lepeltier,Wu and Yu[92].2.3Ill-posedness of the System of HJ EquationsIn a series of articles,Bressan and his co-authors(Bressan and Chen[33,34],Bressan and Priuli[32],Bressan[30,31])have analyzed with the help of P.D.E methods the system of Hamilton–Jacobi equations arising in the construction of feedback Nash equilibria for deter-ministic nonzero-sum games.In state-space dimension1and for theﬁnite horizon problem, this system takes the form∂V i+H i(x,D V1,...,D V n)=0in R×(0,T),i=1,...,n,coupled with a terminal condition at time T(here n is the number of players and H i is the Hamiltonian of player i,V i(t,x)is the payoff obtained by player i for the initial condition (t,x)).Setting p i=(V i)x and deriving the above system with respect to x one obtains the system of conservation laws:∂t p i+H i(x,p1,...,p n)x=0in R×(0,T).This system turns out to be,in general,ill-posed.Typically,in the case of two players(n= 2),the system is ill-posed if the terminal payoff of the players have an opposite monotonicity. If,on the contrary,these payoffs have the same monotony and are close to some linear payoff (which is a kind of cooperative case),then the above system has a unique solution,and one can build Nash equilibria in feedback form from the solution of the P.D.E[33].Still in space dimension1,the case of inﬁnite horizon seems more promising:The sys-tem of P.D.E then reduces to an ordinary differential equation.The existence of suitable solutions for this equation then leads to Nash equilibria.Such a construction is carried out in Bressan and Priuli[32],Bressan[30,31]through several classes of examples and by various methods.In a similar spirit,the papers Cardaliaguet and Plaskacz[47],Cardaliaguet[42]study a very simple class of nonzero-sum differential games in dimension1and with a terminal payoff:In this case it is possible to select a unique Nash equilibrium payoff in feedback form by just imposing that it is Pareto whenever there is a unique Pareto one.However,this equilibrium payoff turns out to be highly unstable with respect to the terminal data.Some other examples of nonlinear-quadratic differential games are also analyzed in Olsder[99] and in Ramasubramanian[106].2.4Mean-ﬁeld GamesSince the system of P.D.Es arising in nonzero-sum differential games is,in general,ill-posed,it is natural to investigate situations where the problem simpliﬁes.It turns out that this is the case for differential games with a very large number of identical players.This problem has been recently developed in a series of papers by Lasry and Lions[87–90,94] under the terminology of mean-ﬁeld games(see also Huang,Caines and Malhame[76–79] for a related approach).The main achievement of Lasry and Lions is the identiﬁcation of the limit when the number of players tends to inﬁnity.The typical resulting model takes the form⎧⎪⎨⎪⎩(i)−∂t u−Δu+H(x,m,Du)=0in R d×(0,T),(ii)∂t m−Δm−divD p H(x,m,Du)m=0in R d×(0,T),(iii)m(0)=m0,u(x,T)=Gx,m(T).(1)In the above system,theﬁrst equation has to be understood backward in time while the second one is forward in time.Theﬁrst equation(a Hamilton–Jacobi one)is associated with an optimal control problem and its solution can be regarded as the value function for a typical small player(in particular the Hamiltonian H=H(x,m,p)is convex with respect to the last variable).As for the second equation,it describes the evolution of the density m(t)of the population.More precisely,let usﬁrst consider the behavior of a typical player.He controls through his control(αs)the stochastic differential equationdX t=αt dt+√2B t(where(B t)is a standard Brownian motion)and he aims at minimizing the quantityET12LX s,m(s),αsds+GX T,m(T),where L is the Fenchel conjugate of H with respect to the p variable.Note that in this cost the evolving measure m(s)enters as a parameter.The value function of our average player is then given by(1-(i)).His optimal control is—at least heuristically—given in feedback form byα∗(x,t)=−D p H(x,m,Du).Now,if all agents argue in this way,their repartition will move with a velocity which is due,on the one hand,to the diffusion,and,one the other hand,to the drift term−D p H(x,m,Du).This leads to the Kolmogorov equation(1-(ii)).The mean-ﬁeld game theory developed so far has been focused on two main issues:ﬁrstly,investigate equations of the form(1)and give an interpretation(in economics,for instance)of such systems.Secondly,analyze differential games with aﬁnite but large num-ber of players and interpret(1)as their limiting behavior as the number of players goes to inﬁnity.Up to now theﬁrst issue is well understood and well documented.The original works by Lasry and Lions give a certain number of conditions under which(1)has a solution,discuss its uniqueness and its stability.Several papers also study the numerical approximation of this solution:see Achdou and Capuzzo Dolcetta[1],Achdou,Camilli and Capuzzo Dolcetta[2], Gomes,Mohr and Souza[71],Lachapelle,Salomon and Turinici[85].The mean-ﬁeld games theory has been used in the analysis of wireless communication systems in Huang,Caines and Malhamé[76],or Yin,Mehta,Meyn and Shanbhag[115].It seems also particularly adapted to modeling problems in economics:see Guéant[72,73],Lachapelle[84],Lasry, Lions,Guéant[91],and the references therein.As for the second part of the program,the limiting behavior of differential games when the number of players tend to inﬁnity has been understood for ergodic differential games[88].The general case remains mostly open.3Long-time Average of Differential GamesAnother way to reduce the complexity of differential games is to look at their long-time be-havior.Among the numerous applications of this topic let us quote homogenization,singular perturbations and dimension reduction of multiscale systems.In order to explain the basic ideas,let us consider a two-player stochastic zero-sum dif-ferential game with dynamics given bydX t,ζ;u,vs =bX t,ζ;u,vs,u s,v sds+σX t,ζ;u,v,u s,v sdB s,s∈[t,+∞),X t=ζ,where B is a d-dimensional standard Brownian motion on a given probability space (Ω,F,P),b:R N×U×V→R N andσ:R N×U×V→R N×d,U and V being some metric compact sets.We assume that theﬁrst player,playing with u,aims at minimizing a running payoff :R N×U×V→R(while the second players,playing with v,maximizes). Then it is known that,under some Isaacs’assumption,the game has a value V T which is the viscosity solution of a second order Hamilton–Jacobi equation of the form−∂t V T(t,x)+Hx,D V T(t,x),D2V T(t,x)=0in[0,T]×R N,V T(T,x)=0in R N.A natural question is the behavior of V T as T→+∞.Actually,since V T is typically of linear growth,the natural quantity to consider is the long-time average,i.e.,lim T→+∞V T/T.Interesting phenomena can be observed under some compactness assumption on the un-derlying state-space.Let us assume,for instance,that the maps b(·,u,v),σ(·,u,v)and (·,u,v)are periodic in all space variables:this actually means that the game takes place in the torus R N/Z N.In this framework,the long-time average is well understood in two cases:either the dif-fusion is strongly nondegenerate:∃ν>0,(σσ∗)(x,u,v)≥νI N∀x,u,v,(where the inequality is understood in the sense of quadratic matrices);orσ≡0and H= H(x,ξ)is coercive:lim|ξ|→+∞H(x,ξ)=+∞uniformly with respect to x.(2) In both cases the quantity V T(x,0)/T uniformly converges to the unique constant¯c forwhich the problem¯c+Hx,Dχ(x),D2χ(x)=0in R Nhas a continuous,periodic solutionχ.In particular,the limit is independent of the initial condition.Such kind of results has been proved by Lions,Papanicoulaou and Varadhan[95] forﬁrst order equations(i.e.,deterministic differential games).For second order equations, the result has been obtained by Alvarez and Bardi in[3],where the authors combine funda-mental contributions of Evans[61,62]and of Arisawa and Lions[7](see also Alvarez and Bardi[4,5],Bettiol[24],Ghosh and Rao[70]).For deterministic differential games(i.e.,σ≡0),the coercivity condition(2)is not very natural:Indeed,it means that one of the players is much more powerful than the other one. However,very little is known without such a condition.Existing results rely on a speciﬁc structure of the game:see for instance Bardi[16],Cardaliaguet[46].The difﬁculty comes from the fact that,in these cases,the limit may depend upon the initial condition(see also Arisawa and Lions[7],Quincampoix and Renault[104]for related issues in a control set-ting).The existence of a limit for large time differential games is certainly one of the main challenges in differential games theory.4Existence of a Value for Zero-sum Differential Games with State Constraints Differential games with state constraints have been considered since the early theory of differential games:we refer to[23,28,66,69,80]for the computation of the solution for several examples of pursuit.We present here recent trends for obtaining the existence of a value for a rather general class of differential games with constraints.This question had been unsolved during a rather long period due to problems we discuss now.The main conceptual difﬁculty for considering such zero-sum games lies in the fact that players have to achieve their own goal and to satisfy the state constraint.Indeed,it is not clear to decide which players has to be penalized if the state constraint is violated.For this reason,we only consider a speciﬁc class of decoupled games where each player controls independently a part of the dynamics.A second mathematical difﬁculty comes from the fact that players have to use admissible controls i.e.,controls ensuring the trajectory to fulﬁlthe state constraint.A byproduct of this problem is the fact that starting from two close initial points it is not obvious toﬁnd two close constrained trajectories.This also affects the regularity of value functions associated with admissible controls:The value functions are,in general,not Lipschitz continuous anymore and,consequently,classical viscosity solutions methods for Hamilton–Jacobi equations may fail.4.1Statement of the ProblemWe consider a differential game where theﬁrst player playing with u,controls aﬁrst systemy (t)=gy(t),u(t),u(t)∈U,y(t0)=y0∈K U,(3) while the second player,playing with v,controls a second systemz (t)=hz(t),v(t),v(t)∈V,z(t0)=z0∈K V.(4)For every time t,theﬁrst player has to ensure the state constraint y(t)∈K U while the second player has to respect the state constraint z(t)∈K V for any t∈[t0,T].We denote by x(t)= x[t0,x0;u(·),v(·)](t)=(y[t0,y0;u(·)](t),z[t0,z0;v(·)](t))the solution of the systems(3) and(4)associated with an initial data(t0,x0):=(t0,y0,z0)and with a couple of controls (u(·),v(·)).In the following lines we summarize all the assumptions concerning with the vectorﬁelds of the dynamics:⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(i)U and V are compact subsets of someﬁnitedimensional spaces(ii)f:R n×U×V→R n is continuous andLipschitz continuous(with Lipschitz constant M)with respect to x∈R n(iii)uf(x,u,v)andvf(x,u,v)are convex for any x(iv)K U={y∈R l,φU(y)≤0}withφU∈C2(R l;R),∇φU(y)=0ifφU(y)=0(v)K V={z∈R m,φV(z)≤0}withφV∈C2(R m;R),∇φV(z)=0ifφV(z)=0(vi)∀y∈∂K U,∃u∈U such that ∇φU(y),g(y,u) <0(vii)∀z∈∂K V,∃v∈V such that ∇φV(z),h(z,v) <0(5)We need to introduce the notion of admissible controls:∀y0∈K U,∀z0∈K V and∀t0∈[0,T]we deﬁneU(t0,y0):=u(·):[t0,+∞)→U measurable|y[t0,y0;u(·)](t)∈K U∀t≥t0V(t0,z0):=v(·):[t0,+∞)→V measurable|z[t0,z0;v(·)](t)∈K V∀t≥t0.Under assumptions(5),the Viability Theorem(see[9,10])ensures that for all x0= (y0,z0)∈K U×K VU(t0,y0)=∅and V(t0,z0)=∅.Throughout the paper we omit t0in the notations U(t0,y0)and U(t0,y0)whenever t0=0.We now describe two quantitative differential games.Let us start with a game with an integral cost:Bolza Type Differential Game Given a running cost L:[0,T]×R N×U×V→R and aﬁnal costΨ:R N→R,we deﬁne the payoff associated to an initial position(t0,x0)= (t0,y0,z0)and to a pair of controls(u,v)∈U(t0,y0)×V(t0,z0)byJt0,x0;u(·),v(·)=Tt0Lt,x(t),u(·),v(·)dt+Ψx(T),(6)where x(t)=x[t0,x0;u(·),v(·)](t)=(y[t0,y0;u(·)](t),z[t0,z0;v(·)](t))denotes the solu-tion of the systems(3)and(4).Theﬁrst player wants to maximize the functional J,while the second player’s goal is to minimize J.Deﬁnition1A mapα:V(t0,z0)→U(t0,y0)is a nonanticipating strategy(for theﬁrst player and for the point(t0,x0):=(t0,y0,z0)∈R+×K U×K V)if,for anyτ>0,for all controls v1(·)and v2(·)belonging to V(t0,z0),which coincide a.e.on[t0,t0+τ],α(v1(·)) andα(v2(·))coincide almost everywhere on[t0,t0+τ].Nonanticipating strategiesβfor the second player are symmetrically deﬁned.For any point x0∈K U×K V and∀t0∈[0,T]we denote by A(t0,x0)and by B(t0,x0)the sets of the nonanticipating strategies for theﬁrst and the second player respectively.We are now ready to deﬁne the value functions of the game.The lower value V−is deﬁned by:V−(t0,x0):=infβ∈B(t0,x0)supu(·)∈U(t0,y0)Jt0,x0;u(·),βu(·),(7)where J is deﬁned by(6).On the other hand we deﬁne the upper value function as follows:V+(t0,x0):=limε→0+supα∈A(t0,x0)infv(·)∈V(t0,z0)Jεt0,x0;αv(·),v(·)(8)withJεt0,x0;u(·),v(·):=Tt0Lt,x(t),u(t),v(t)dt+Ψεx(T),where x(t)=x[t0,x0;u(·),v(·)](t)andΨεis the lower semicontinuous function deﬁned byΨε(x):=infρ∈R|∃y∈R n with(y,ρ)−x,Ψ(x)=ε.The asymmetry between the deﬁnition of the value functions is due to the fact that one assumes that the terminal payoffΨis lower semicontinuous.WhenΨis continuous,one can check that V+can equivalently be deﬁned in a more natural way asV+(t0,x0):=supα∈A(t0,x0)infv(·)∈V(t0,z0)Jt0,x0;αv(·),v(·).We now describe the second differential game which is a pursuit game with closed target C⊂K U×K V.Pursuit Type Differential Game The hitting time of C for a trajectory x(·):=(y(·),z(·)) is:θCx(·):=inft≥0|x(t)∈C.If x(t)/∈C for every t≥0,then we setθC(x(·)):=+∞.In the pursuit game,theﬁrst player wants to maximizeθC while the second player wants to minimize it.The value functions aredeﬁned as follows:The lower optimal hitting-time function is the mapϑ−C :K U×K V→R+∪{+∞}deﬁned,for any x0:=(y0,z0),byϑ−C (x0):=infβ(·)∈B(x0)supu(·)∈U(y0)θCxx0,u(·),βu(·).The upper optimal hitting-time function is the mapϑ+C :K U×K V→R+∪{+∞}de-ﬁned,for any x0:=(y0,z0),byϑ+ C (x0):=limε→0+supα(·)∈A(x0)infv(·)∈V(z0)θC+εBxx0,αv(·),v(·).By convention,we setϑ−C (x)=ϑ+C(x)=0on C.Remarks–Note that here again the deﬁnition of the upper and lower value functions are not sym-metric:this is related to the fact that the target assumed to be closed,so that the game is intrinsically asymmetric.–The typical pursuit game is the case when the target coincides with the diagonal:C= {(y,z),|y=z}.We refer the reader to[6,29]for various types of pursuit games.The formalism of the present survey is adapted from[50].4.2Main ResultThe main difﬁculty for the analysis of state-constraint problems lies in the fact that two trajectories of a control system starting from two—close—different initial conditions could be estimated by classical arguments on the continuity of theﬂow of the differential equation. For constrained systems,it is easy to imagine cases where the constrained trajectories starting from two close initial conditions are rather far from each other.So,an important problem in order to get suitable estimates on constrained trajectories,is to obtain a kind of Filippov Theorem with ly a result which allows one to approach—in a suitable sense—a given trajectory of the dynamics by a constrained trajectory.Note that similar results exist in the literature.However,we need here to construct a constrained trajectory in a nonanticipating way[26](cf.also[25]),which is not the case in the previous constructions.Proposition1Assume that conditions(5)are satisﬁed.For any R>0there exist C0= C0(R)>0such that for any initial time t0∈[0,T],for any y0,y1∈K U with|y0|,|y1|≤R,。

这些书!还原爱因斯坦

这些书！还原爱因斯坦当前浏览器不支持播放音乐或语音，请在微信或其他浏览器中播放Conquest Of Paradise Vangelis - 1492 - Conquest of Paradise (Soundtrack from the Motion Picture)爱因斯坦作为一个真实的人现实的人他是平凡的然而作为一个探索世界奥秘并作出杰出贡献的人他是伟大的他配得上这个称呼「宇宙之子」第一本书但有一个问题依然萦绕在我们心头：他（爱因斯坦）怎么可能对他自己理论的一项如此伟大的胜利视而不见呢？我无法给出答案。

它仍然是这位天才一生中难解的悖论之一。

著名物理学家弗里曼·戴森爱因斯坦至死都没有承认黑洞的存在，而黑洞正是他创立的广义相对论所预言的。

这就是戴森说这段话的背景。

这段话被戴森放在了《爱因斯坦：相对论一百年》前言中作为结尾。

【见文末附录】爱因斯坦具有非凡的物理直觉能力，凭着这种能力以及追求真理的信念的力量，他才有了狭义、广义相对论的破冰之思，然后凭借数学，使其达到人类智慧的又一高峰。

然而，伟大如爱因斯坦，其直觉也是可能出错的，此时，信念就变为了恶魔，试图扼杀真理的被发现。

爱因斯坦辉煌的一生中，有过许多类似的例子：两次否定引力波存在，否定黑洞存在，提出宇宙常数然后又否定之，对量子力学不完备性的决定论式的坚定信念的坚持，都显示出一位伟大科学家的真实的震撼人心的思维角力。

这种思维对弈的对手，不仅仅是与他人，而更是自我的博弈。

从中我们看到一位真实伟人的思维，是如何提出一个惊人的洞见，又如何犯下一个微不足道的计算错误，而导致严重的对真理的否定（否定引力波的计算错误）。

我们还可以看到，这位伟人认识到自己犯下的错误后，对于正确结论的欣然的真诚的接受，同时，我们也看到直觉与信念支配的另一面------固执地坚守着错误的见解。

直觉和信念------真是天使与恶魔的混合体啊！第二版的《爱因斯坦：相对论一百年》，是为了纪念爱因斯坦广义相对论创立100周年（1915-2015），由众多伟大人物写就的文章的汇编。

分子水平科赫原则是什么

分子水平科赫原则是什么分子水平科赫原则是一种用于解释和预测分子行为的基本原则。

它是由物理学家、诺贝尔奖得主彼得·格鲁贝（Peter Grünberg）和阿尔伯特·费尔特（Albert Fert）于2007年发现的。

分子水平科赫原则指出，当自旋磁矩相互作用时，电子会选择在相邻原子之间通过自旋耦合来组织自己的自旋状态。

这种自旋耦合导致自旋状态的集体行为，形成所谓的自旋-自旋耦合态。

这一原则在磁性材料中尤为显著，对我们理解和应用于磁性存储器、磁性传感器等领域具有重要意义。

从分子水平科赫原则的角度出发，我们可以更好地理解磁性材料的性质和行为。

它为我们提供了一种全新的思考方式，使我们能够从分子尺度上研究磁性系统的自旋结构、自旋动力学和磁性相互作用。

通过对这些磁性现象的深入研究，我们可以设计出更先进、更高效的磁性材料和器件。

对于实际应用而言，分子水平科赫原则也为我们提供了指导。

通过深入理解分子自旋耦合行为，我们可以在设计磁性材料和磁性器件时更加精确地控制其性能。

这不仅在磁存储器领域有着重要应用，也可以在其他领域如磁共振成像、磁力传感器等方面发挥作用。

此外，分子水平科赫原则还为我们揭示了分子系统的新奇行为。

例如，在纳米尺度下，分子水平科赫原则对磁性纳米颗粒的研究有着重要作用。

磁性纳米颗粒在医学、能源存储等领域具有广泛的应用前景。

通过对分子水平科赫原则的探索，我们可以更好地了解磁性纳米颗粒的自旋行为，并为其应用提供更多的可能性。

综上所述，分子水平科赫原则为我们理解和应用于磁性材料和器件提供了宝贵的指导。

通过深入探索分子自旋耦合行为，我们可以更好地理解磁性系统的性质和行为，并为设计新型磁性材料和器件提供指导。

这一原则的发现开辟了一个新的研究领域，并有望在各个应用领域产生重要的推动作用。

hamilton-jacobi 方程

Hamilton–Jacobi equationFrom Wikipedia, the free encyclopediaIn mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremalgeometry in generalizations of calculus-of-variations problems. In physics, the Hamilton–Jacobi equation (HJE)is a reformulation of classical mechanics and, thus, equivalent to other formulations such as Newton's laws of motion, Lagrangian mechanics and Hamiltonian mechanics. The Hamilton–Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely.The HJE is also the only formulation of mechanics in which the motion of a particle can be represented as a wave. In this sense, the HJE fulfilled a long-held goal of theoretical physics (dating at least to Johann Bernoulli in the 18th century) of finding an analogy between the propagation of light and the motion of a particle. The wave equation followed by mechanical systems is similar to, but not identical with, Schrödinger's equation, as described below; for this reason, the HJE is considered the "closest approach" of classical mechanics to quantum mechanics.[1][2]Mathematical formulationThe Hamilton–Jacobi equation is a first-order, non-linear partial differential equation for a function called Hamilton's principal functionAs described below, this equation may be derived from Hamiltonian mechanics bytreating S as the generating function for a canonical transformation of the classical Hamiltonian.The conjugate momenta correspond to the first derivatives of S with respect to the generalized coordinatesPrincipal function as solved from the equation from contains N+1 undeterminedconstants, the last being one from integrating , and the first N denoted as. The relationship then between p and q describes the orbit in phase space in terms of these constants of motion, andare also constants of motion and can be inverted to solve q.Comparison with other formulations of mechanicsThe HJE is a single, first-order partial differential equation for the function S of the N generalized coordinates and the time t. The generalized momenta do not appear, except as derivatives of S. Remarkably, the function S is equal to the classical action.For comparison, in the equivalent Euler–Lagrange equations of motion of Lagrangian mechanics, the conjugate momenta also do not appear; however, those equations are a system of N, generally second-order equations for the time evolution of thegeneralized coordinates. Similarly, Hamilton's equations of motion are another system of 2N first-order equations for the time evolution of the generalized coordinates and their conjugate momenta .Since the HJE is an equivalent expression of an integral minimization problem such as Hamilton's principle, the HJE can be useful in other problems of the calculus of variations and, more generally, in other branches of mathematics and physics, such as dynamical systems, symplectic geometry and quantum chaos. For example, the Hamilton–Jacobi equations can be used to determine the geodesics on a Riemannian manifold, an important variational problem in Riemannian geometry.NotationFor brevity, we use boldface variables such as to represent the list of N generalized coordinatesthat need not transform like a vector under rotation. The dot product is defined here as the sum of the products of corresponding components, i.e.,DerivationAny canonical transformation involving a type-2 generating functionleads to the relations(See the canonical transformation article for more details.)To derive the HJE, we choose a generating function that makes the new Hamiltonian K identically zero. Hence, all its derivatives are also zero, and Hamilton's equations become triviali.e., the new generalized coordinates and momenta are constants of motion. The new generalized momenta are usually denoted , i.e., P m= αm. The equation for the transformed Hamiltonian KLetwhere A is a arbitrary constant, then S satisfies HJEsince .The new generalized coordinates are also constants, typically denoted as. Once we have solved for, these also give useful equationsor written in components for clarityIdeally, these N equations can be inverted to find the original generalized coordinatesas a function of the constants and , thus solving the original problem. ActionBoth Hamilton principal function S and characteristic function are closely related to action.The time derivative of S isthereforeso S is actually classical action plus an undetermined constant.When H does not explicitly depend on time,in this case W is the same as abbreviated action.Separation of variablesThe HJE is most useful when it can be solved via additive separation of variables, which directly identifies constants of motion. For example, the time t can be separated if the Hamiltonian does not depend on time explicitly. In that case, thetime derivative in the HJE must be a constant (usually denoted − E), giving the separated solutionwhere the time-independent function issometimes called Hamilton's characteristic function. The reduced Hamilton–Jacobi equation can then be writtenTo illustrate separability for other variables, we assume that a certain generalizedcoordinate q k and its derivative appear together in the Hamiltonian as a singlefunctionIn that case, the function S can be partitioned into two functions, one that depends only on q k and another that depends only on the remaining generalized coordinatesSubstitution of these formulae into the Hamilton–Jacobi equation shows that the function ψ must be a constant (denoted here as Γk), yielding a first-order ordinary differential equation for S k(q k)In fortunate cases, the function S can be separated completely into N functions S m(q m)In such a case, the problem devolves to N ordinary differential equations.The separability of S depends both on the Hamiltonian and on the choice of generalized coordinates. For orthogonal coordinates and Hamiltonians that have notime dependence and are quadratic in the generalized momenta, S will be completely separable if the potential energy is additively separable in each coordinate, where the potential energy term for each coordinate is multiplied by the coordinate-dependent factor in the corresponding momentum term of the Hamiltonian (the Staeckel conditions). For illustration, several examples in orthogonal coordinates are worked in the next sections.Example of spherical coordinatesThe Hamiltonian in spherical coordinates can be writtenThe Hamilton–Jacobi equation is completely separable in these coordinates provided that U has an analogous formwhere U r(r), Uθ(θ) and Uφ(φ) are arbitrary functions. Substitution of the completely separated solution S = S r(r) + Sθ(θ) + Sφ(φ) − Et into the HJE yieldsThis equation may be solved by successive integrations of ordinary differential equations, beginning with the φequationwhereΓφis a constant of the motion that eliminates the φ depe ndence from the Hamilton–Jacobi equationThe next ordinary differential equation involves the θ generalized coordinatewhereΓθis again a constant of the motion that elimin ates the θ dependence and reduces the HJE to the final ordinary differential equationwhose integration completes the solution for S.Example of elliptic cylindrical coordinatesThe Hamiltonian in elliptic cylindrical coordinates can be writtenwhere the foci of the ellipses are located at on the x-axis. The Hamilton–Jacobi equation is completely separable in these coordinates provided that U has an analogous formwhere Uμ(μ), Uν(ν)and U z(z) are arbitrary functions. Substitution of the completely separated solution S = Sμ(μ) + Sν(ν) + S z(z) − Et into the HJE yieldsSeparating the first ordinary differential equationyields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)which itself may be separated into two independent ordinary differential equationsthat, when solved, provide a complete solution for S.Example of parabolic cylindrical coordinatesThe Hamiltonian in parabolic cylindrical coordinates can be writtenThe Hamilton–Jacobi equation is completely separable in these coordinates provided that U has an analogous formwhere Uσ(σ), Uτ(τ)and U z(z) are arbitrary functions. Substitution of the completely separated solution S = Sσ(σ) + Sτ(τ) + S z(z) − Et into the HJE yieldsSeparating the first ordinary differential equationyields the reduced Hamilton–Jacobi equation (after re-arrangement and multiplication of both sides by the denominator)which itself may be separated into two independent ordinary differential equationsthat, when solved, provide a complete solution for S.Eikonal approximation and relationship to the Schrödinger equationThe isosurfaces of the function can be determined at any time t. The motion of an S-isosurface as a function of time is defined by the motions of the particles beginning at the points on the isosurface. The motion of such an isosurface can be thought of as a wave moving through space, although it does not obey the wave equation exactly. To show this, let S represent the phase of a wavewhere is a constant introduced to make the exponential argument unitless; changes in the amplitude of the wave can be represented by having S be a complex number. We may then rewrite the Hamilton–Jacobi equation aswhich is a nonlinear variant of the Schrödinger equation.11 / 11 Conversely, starting with the Schrödinger equation and our Ansatz for ψ, we arrive at[3]The classical limit () of the Schrödinger equation above becomes identical to the following variant of the Hamilton–Jacobi equation,The Hamilton–Jacobi equation in the gravitational fieldwhere g ik are the contravariant coordinates of the metric tensor, m is the rest mass of the particle and c is the speed of light.。

hamilton--jacobi 方程

hamilton--jacobi 方程Hamilton-Jacobi方程是经典力学中一种重要的变分原理，可以描述质点在势场中运动的轨迹。

这个方程由外尔在1927年提出，是经典力学的一个基本方程之一，与拉格朗日力学和哈密顿力学一起构成了经典力学的三大流派。

在经典力学中，质点在势场中运动的状态可以用质点的位置以及动量来描述。

在哈密顿力学中，系统的动力学演化由哈密顿函数来描述，而哈密顿函数可以通过广义动量和广义坐标构建得到。

在这个框架下，Hamilton-Jacobi方程可以被看作是一个与哈密顿函数相关的偏微分方程。

Hamilton-Jacobi方程的一般形式可以表示为：H(q_i, \frac{\partial S}{\partial q_i}) + \frac{\partial S}{\partial t} = 0其中H是系统的哈密顿函数，q_i是广义坐标，S是所谓的作用量。

Hamilton-Jacobi方程的解S可以通过变量分离的方法得到。

对于一个自由粒子来说，作用量可以表示为：S(q_i, t) = -Et + \sum_i p_i q_i其中E是粒子的总能量，p_i是广义动量。

将这个作用量代入Hamilton-Jacobi方程，可以得到一组与时间无关的偏微分方程：H(q_i, \frac{\partial S}{\partial q_i}) + E = 0这个方程可以被看作是Hamilton-Jacobi方程的定态版本，它描述了系统在特定能量下的运动。

在一般情况下，Hamilton-Jacobi方程往往难以直接解析求解。

但是，可以采用一系列近似方法来求解，在实际物理问题中还有很多有效的数值求解方法。

此外，在量子力学中也存在一种相对称的方程，称为Wigner函数方程，与Hamilton-Jacobi方程相对应。

华清明等人在1996年发表的一篇文章中，讨论了关于广义Bertrand系统的Hamilton-Jacobi方程。

hamilton-jacobi方程

hamilton-jacobi方程 Hamilton-Jacobi方程是经典物理学中的一个非常重要的方程，它描述了粒子在势能场中的运动。

本文将详细介绍Hamilton-Jacobi方程的定义、应用领域以及解决该方程的方法。

Hamilton-Jacobi方程是经典力学中的一个偏微分方程，它描述了质点在非保守力场中的运动。

该方程的一般形式如下：H(x, ∇S) + ∂tS = 0 其中，H是哈密顿函数，x是自变量，S是未知函数，∇S是S的梯度，∂tS是S关于时间t的偏导数。

Hamilton-Jacobi方程可以看作是能量守恒定律的推广形式，它蕴含了粒子在势能场中的运动信息。

二、Hamilton-Jacobi方程的应用领域 Hamilton-Jacobi方程在物理学、动力学和控制论等领域具有广泛的应用。

以下是一些典型的应用领域： 1. 量子力学：在量子力学中，Hamilton-Jacobi方程被用于研究粒子的量子化条件，通过解Hamilton-Jacobi方程可以得到粒子的量子态。

2. 经典力学：在经典力学中，Hamilton-Jacobi方程被用于描述质点在非保守力场中的运动，通过求解Hamilton-Jacobi方程可以得到质点的轨迹。

3. 光学：在光学中，Hamilton-Jacobi方程被用于描述光的传播，通过解Hamilton-Jacobi方程可以得到光的光程函数和位相。

4. 控制论：在控制论中，Hamilton-Jacobi方程被用于求解最优控制问题，通过解Hamilton-Jacobi方程可以得到最优控制函数。

三、解决Hamilton-Jacobi方程的方法解决Hamilton-Jacobi方程的方法主要有两种：分离变量法和变量分离法。

下面将分别介绍这两种方法的步骤和应用情况。

1. 分离变量法分离变量法是解决Hamilton-Jacobi方程最常用的方法之一。

具体步骤如下：（1）将未知函数S(x, t)表示为分离变量的形式，即S(x, t) = W(x) + T(t)，其中W(x)是只与自变量x有关的函数，T(t)是只与时间t有关的函数。

泊松过程的外文书籍

泊松过程的外文书籍引言概述：泊松过程是概率论中的重要概念，广泛应用于各个领域。

为了深入了解泊松过程的理论和应用，阅读外文书籍是一个不错的选择。

本文将介绍几本关于泊松过程的外文书籍，分析其内容和特点，帮助读者选择适合自己的参考书籍。

正文内容：1. "Poisson Processes" by Daley, D.J. and Vere-Jones, D.1.1 介绍泊松过程的基本概念和性质1.2 探讨泊松过程的随机强度和非齐次泊松过程1.3 分析泊松过程的计数过程和间隔时间分布1.4 研究泊松过程的分岔和超过程1.5 讨论泊松过程在信号处理、金融和网络等领域的应用2. "Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations" by Gardiner, C.W.2.1 介绍扩散过程和布朗运动的基本概念2.2 推导福克-普朗克方程和朗之万方程2.3 研究扩散过程的稳定分布和吸引子2.4 讨论扩散过程在物理学、化学和生物学等领域的应用2.5 探讨扩散过程的数值模拟和实际应用案例3. "Point Processes and Queues: Martingale Dynamics" by Brémaud, P.3.1 分析点过程和排队论的基本概念和性质3.2 探讨点过程的鞅动力学和随机强度3.3 研究排队论中的排队模型和性能分析3.4 讨论排队论在通信网络、交通流和生产系统等领域的应用3.5 探索点过程和排队论的数值方法和实际案例4. "Renewal Theory and Its Applications" by Feller, W.4.1 介绍更新过程的基本概念和性质4.2 推导更新过程的分布函数和密度函数4.3 分析更新过程的极限定理和稳定分布4.4 研究更新过程在可靠性理论和保险数学中的应用4.5 讨论更新过程的数值方法和实际案例5. "Random Measures, Theory and Applications" by Kallenberg, O.5.1 介绍随机测度的基本概念和性质5.2 推导随机测度的积分和测度变换5.3 研究随机测度的强大数定律和中心极限定理5.4 讨论随机测度在统计学、金融和风险管理等领域的应用5.5 探索随机测度的数值方法和实际案例总结：综上所述，以上这些外文书籍涵盖了泊松过程及其相关的扩散过程、点过程、更新过程和随机测度等方面的理论和应用。

hamilton-jacobi方程

hamilton-jacobi方程Hamilton-Jacobi方程是理论物理中的一个重要方程，它描述了经典力学系统中的粒子运动。

这个方程的提出是为了解决哈密顿量不是分离变量的情况，即系统中存在非势能项的情况。

在这种情况下，哈密顿方程无法直接求解，而需要借助Hamilton-Jacobi方程来得到粒子的轨迹。

Hamilton-Jacobi方程的形式为：∂S/∂t + H(x, ∂S/∂x) = 0其中，S是一个函数，称为Hamilton-Jacobi函数，它的偏导数∂S/∂t表示了系统的能量，而∂S/∂x则表示了系统的动量。

H是系统的哈密顿函数，它是系统的总能量。

通过求解Hamilton-Jacobi方程，我们可以得到粒子的轨迹。

具体的求解方法是通过变量分离，将Hamilton-Jacobi函数表示为两个变量的函数之和，然后利用分离变量的性质来求解。

这样，我们就可以得到粒子在空间中的轨迹，进而揭示了系统的运动规律。

Hamilton-Jacobi方程的提出，对于经典力学的发展具有重要的意义。

它不仅为非势能系统的求解提供了一种有效的方法，还为后续的量子力学研究奠定了基础。

通过对Hamilton-Jacobi方程的深入研究，科学家们不断完善了经典力学的理论框架，使其更加完备和准确。

Hamilton-Jacobi方程是经典力学中的重要方程，它描述了粒子在非势能系统中的运动规律。

通过求解这个方程，我们可以得到粒子的轨迹，从而揭示了系统的运动规律。

这个方程的提出对于经典力学的发展具有重要的意义，它为后续的物理研究提供了基础。

通过对Hamilton-Jacobi方程的研究，我们可以更好地理解和解释物理现象，推动科学的进步。

彭赛列闭合定理雅可比证法

彭赛列闭合定理雅可比证法全文共四篇示例，供读者参考第一篇示例：彭赛列闭合定理是微分方程中的一个非常重要的理论，在解析几何、动力学和工程应用中有着广泛的应用。

而雅可比证法是证明闭合定理的一个重要方法。

今天我们就来探讨一下彭赛列闭合定理以及雅可比证法。

让我们来了解一下彭赛列闭合定理。

彭赛列闭合定理，又称彭赛列等效定理，是关于微分方程解析性质的一个基本结论。

该定理表明，如果一个微分方程的对称性在某一点成立，那么在该点附近的任意一点，该微分方程的对称性都应该成立。

换句话说，如果微分方程在某一点上的解有某种对称性，那么在该点附近的解也应该具有相同的对称性。

雅可比证法是证明彭赛列闭合定理的一个重要方法。

雅可比证法利用雅可比行列式的性质来证明微分方程解的对称性。

具体来说，当微分方程的解满足雅可比行列式的条件时，该解就具有对称性。

这为证明彭赛列闭合定理提供了一个简单而有效的方法，使得我们可以更直观地理解微分方程解的性质。

彭赛列闭合定理和雅可比证法对于研究微分方程的解析性质具有重要的意义。

通过利用对称性和雅可比行列式的性质，我们可以更深入地理解微分方程的解，并且在实际应用中更有效地求解微分方程。

希望通过本文的介绍，读者能够对彭赛列闭合定理和雅可比证法有更清晰的认识，并且在微分方程的研究和应用中得到启发。

【这篇文章翔实论述了彭赛列闭合定理及雅可比证法的重要性和应用，并简明扼要地介绍了两者的基本原理和方法，能够给读者提供一定的指导和启发。

文章主题明确，结构完整，总体表达清晰流畅，具有较高的可读性。

】【以上文章内容仅供参考。

根据实际需求，您可以适当调整文章格式和论述方式，使其更符合实际情况。

】第二篇示例：彭赛列闭合定理是微积分中的一个非常重要的定理，它是由法国数学家彭赛列在19世纪提出的。

这个定理在数学分析和实变函数论中有着广泛的应用，特别是关于函数的连续性和收敛性方面。

彭赛列闭合定理的表述比较复杂，但其核心思想很简单：如果一个函数序列在一个区间内一致收敛，并且收敛的函数也在这个区间内连续，则该函数序列的极限函数也在这个区间内连续。

四型生理时钟

四型生理时钟作者：[美]迈克尔·布劳斯译者：郑咏滟出版人：曾赛丰责任编辑：薛健刘诗哲监制：吴文娟策划编辑：董卉特约编辑：陈晓梦庞海丽版权支持：辛艳营销支持：杜莎李茂繁封面设计：仙境版式设计：潘雪琴本书由天津博集新媒科技有限公司授权亚马逊全球范围发行目录版权信息推荐序引言：时机就是一切PART ONE 生理时钟类型第一章你属于哪种生理时钟类型第二章海豚型的完美一天第三章狮子型的完美一天第四章熊型的完美一天第五章狼型的完美一天PART TWO 一切活动的最佳时间第六章人际关系坠入爱河联系朋友与伴侣争吵享受性爱制订计划和孩子交谈第七章健身跑步参与团队运动练习瑜伽力量训练第八章健康抵抗疾病注射流感疫苗做乳腺X光检查排便看心理医生洗澡服药称体重第九章睡眠醒来小睡睡懒觉上床睡觉第十章饮食一日三餐饮酒喝咖啡大吃大喝吃零食第十一章工作要求加薪打推销电话通勤发邮件求职面试学习新东西做决定记忆训练发表观点第十二章创造力头脑风暴演奏音乐整理思绪创作小说第十三章金钱购物变得富有成交生意销售第十四章娱乐长时间看电视浏览网页玩游戏阅读消遣讲笑话旅行PART THREE 生理时钟与季节、年龄第十五章昼夜节律的季节性第十六章一生的昼夜节律主钟正如我女儿所说，这本书“应该献给你了不起的孩子和妻子”。

我想不到比这更棒的表述了。

本书献给我的狼型伙伴：劳伦、库珀、卡森，和我的孩子蒙蒂、斯巴基和糖小熊。

特别要感谢我十六年从医生涯中遇到的所有患者。

每一次相遇，我都从你们身上学到很多。

在我的节目开播之初，布劳斯博士便已经是我的朋友和同事了。

他对学习，对教育大众，对获取与睡眠和睡眠障碍相关的前沿信息表现出的不知疲倦的热情，让他成为很多领域的核心专家。

在一次与布劳斯博士会面之后，我开始对昼夜节律（circadian rhythm）的治愈效果产生兴趣。

当时我们讨论了睡眠药物的未来，也讨论了为什么在美国睡眠缺乏没有被视作一个严重的健康问题。

精彩绚丽的历史时空（5）-第4章

第4章不可不知的世界大事4公元1660年 ?英国斯图亚特王朝复辟F.M.格里马尔迪发现光的衍射公元1660～1731年 ?英国小说家笛福在世，代表作有《鲁滨逊漂流记》公元1661年 ?英国化学家R.玻意耳提出化学元素的科学定义，将化学确立为科学公元1661～1756年 ?法国建凡尔赛宫，为欧洲最大王宫公元1662年?英国W.配第的《赋税论》出版，提出劳动生产率与商品价值大小成反比公元1663年?德国O.von盖利克创制第一个转动的大硫黄球摩擦起电机，并发现地磁场能使铁屑磁化公元1665年?荷兰A.van列文虎克和英国R.胡克发明显微镜公元1667年?英国诗人弥尔顿的代表作《失乐园》发表公元1673年?荷兰C.惠更斯利用摆的原理测量绝对重力值公元1674年?荷兰A.van列文虎克最先发现海洋原生物公元1675年?英国建立格林尼治天文台公元1675～1710年?英国建伦敦圣保罗大教堂公元1676年?丹麦O.C.罗默推算出光速公元17世纪70年代?英国W.配第的《政治算术》和德国康令的《国情论》几乎同时出版，成为统计学中两大派200年争论的起因和论据公元1680年?法国用浇铸法制出平板玻璃公元1684年?德国G.W.莱布尼兹发表第一篇微分学论文，两年后又发表第一篇积分论文，创立了微积分学公元1685～1750年?德国作曲家巴赫在世，成名作为《创意曲》等公元1688年?英国发生光荣革命公元1689～1755年?18世纪法国启蒙思想家孟德斯鸠在世，反对君主专制，主张君主立宪政体，提出“三权分立”学说公元1690年?荷兰C.惠更斯出版《光论》，提出光的波动说，导出了光的直线传播和反射折射定律，并解释了双折射现象公元1694年?德国G.W.莱布尼兹发明能进行加减乘除的机械计算机公元1694～1778年 ?法国资产阶级启蒙思想家伏尔泰在世，提倡自然权利学说，鼓吹信仰、思想言论、出版自由和天赋人权，主张政治改良和“开明君主制”公元1695年?英国J.伍德沃德发表《地球自然历史探讨》，提出洪水使生物灭亡的洪积说公元1698年?俄国彼得一世开始改革英国T.萨弗里制成第一台实用于矿井抽水的蒸汽机，首次将蒸汽用作工业动力公元17世纪 ?欧洲近代货币（纸币）在英国产生风景画在荷兰首先成熟发源于意大利的巴洛克美术在欧洲流行五线谱在欧洲逐步完善，至18世纪定型，成为世界的通用记谱法芭蕾在法国宫廷形成公元17世纪末?玻璃画在欧洲产生公元1701年 ?荷兰H.布尔哈维在莱顿大学开创临床教学，通过学生将临床思维传至欧洲公元1705年 ?英国E.哈雷发现周期彗星12月9日，英国F.豪克斯比制成第一台大功率静电起电机，并进行辉光放电实验英国R.胡克在《论地震》中提出用生物化石记述地球历史公元1709年 ?英国议会通过世界上第一部版权法，于1710年生效英国A.达比用焦炭炼铁成功公元1709～1714年?德国D.G.华伦海特发明酒精温度计和水银温度计，并创立华氏温标公元1712～1778年?法国资产阶级启蒙思想家J.-J.卢梭在世，主张消灭王权，建立共和国，提出“天赋人权”、“自由平等”、“主权在民”等理论，著《社会契约论》公元1713～1784年?法国启蒙思想家D.狄德罗在世，主编法国第一部《百科全书》并宣传唯物主义和无神论公元1729～1781年?德国民族文学奠基人、剧作家文艺理论家莱辛在世，代表作有《爱弥丽亚?迦绿蒂》公元1730年 ?英国西森发明经纬仪公元1733年 ?英国J.凯发明纺织机的飞梭机构法国C.F.迪费发表论文《论电》，提出电的二元流体假设，并总结出静电学第一个基本原理枣“同性相斥，异性相吸”公元1734年 ?德国W.沃尔夫的《经验心理学》出版，首次使用“心理学”一词公元1735年 ?瑞典C.von林奈的《自然系统》第一版问世，首次实现了植物与动物分类范畴的统一公元1737～1809年 ?北美思想家T.潘恩在世，撰著《常识》，促进北美人民思想解放公元1738年 ?法国舞蹈家J.-B.朗代在俄国圣彼得堡建立皇家芭蕾舞学校瑞士伯努利父子发表水流能量方程，即伯努利方程瑞士丹尼尔第一?伯努利提出不可压缩流体的能量守恒定律，称伯努利定理意大利A.莫罗提出火山等地质作用的重要性，将山脉、岛屿的形成归因于地下火的作用公元1742～1745年 ?瑞典A.摄尔修斯创立以水的冰点为100度、汽点为0度的温标。

牛顿莱布尼茨公式相关文献

牛顿莱布尼茨公式相关文献
以下是与牛顿莱布尼茨公式相关的一些经典文献：
1. Isaac Newton, "Mathematical Principles of Natural Philosophy" (1687) - 牛顿的这部著作是经典的力学和数学著作，其中包含了牛顿-莱布尼茨公式的基本理论。

2. Gottfried Wilhelm Leibniz, "Nova Methodus pro Maximis et Minimis" (1684) - 莱布尼茨在这篇论文中首次提出了微积分的基本理论，其中包括了他自己的版本的牛顿-莱布尼茨公式。

3. Augustin-Louis Cauchy, "Cours d'Analyse" (1821) - 柯西是19世纪最重要的数学家之一，他在这本著作中详细介绍了牛顿-莱布尼茨公式的证明和应用。

4. Karl Weierstrass, "Theorie der Abel'schen Functionen" (1886) - 魏尔斯特拉斯是19世纪末最重要的分析学家之一，他在这本著作中进一步完善了牛顿-莱布尼茨公式的理论。

5. Richard Courant and Fritz John, "Introduction to Calculus and Analysis" (1965) - 康朗和约翰合著的这本书是微积分和分析学的经典教材之一，其中详细介绍了牛顿-莱布尼茨公式的应用和推广。

这些文献中的内容可以帮助读者深入理解牛顿-莱布尼茨公式的起源、发展和应用。

hamilton jacobi bellman方程

hamilton jacobi bellman方程
哈密顿-雅可比-贝尔曼（Hamilton-Jacobi-Bellman，HJB）方程是动态规划理论中的一种重要方程，用于描述最优控制问题。

以下是关于哈密顿-雅可比-贝尔曼方程的中文解释：
###哈密顿-雅可比-贝尔曼方程
在动态规划和最优控制问题中，哈密顿-雅可比-贝尔曼方程是描述最优值函数的偏微分方程。

该方程的一般形式为：
\[\frac{\partial V}{\partial t}+\min_u\left(\mathcal{L}V+f\right)=0 \]
其中：
-$V$是最优值函数，表示系统在每个时间点的最优值。

-$\mathcal{L}V$是哈密顿函数（Hamiltonian）的偏导数，表示系统的哈密顿函数关于状态的导数。

-$f$表示系统的外部输入。

方程的解通常给出最优控制策略，即在每个时间点系统应该采取的最优行动。

这种形式的方程经常与最优控制理论中的动态规划方法一起使用，以找到系统在给定约束下的最优控制策略。

###应用
哈密顿-雅可比-贝尔曼方程在控制理论、经济学、金融工程等领域有广泛的应用。

它提供了一种在动态系统中找到最优控制策略的数学工具，可以用于解决各种涉及最优化问题的实际应用。

关于一些化学家的生平介绍

在克劳修斯的晚年他不恰当地把热力学第二定律引用到整个宇宙认为整个宇宙的温度必将达到均衡而不再有热量的传递从而成为所谓的热寂状态这就是克劳修斯首先提出来的热寂说
克劳修斯的生平简介
克劳修斯的生平简介
• 在《论热的运动力……》一文中，克劳修斯首次提出了热力学第二定律的定义：“热量不能自动地从低温物体传向高温物体。”这与开尔文陈述的热力学第二定律“不可制成一种循环动作的热机，只从一个热源吸取热量，使之完全变为有用的功，而其他物体不发生任何变化” 是等价的，它们是热力学的重要理论基础。同时，他还推导了克劳修斯方程——关于气体的压强、体积、温度和气体普适常数之间的关系，修正了原来的范德瓦尔斯方程。
能斯特的生平简介
能斯特的生平简介
• 能斯特的研究主要在热力学方面。1889年，他提出溶解压假说，从热力学导出于电极势与溶液浓度的关系式，即电化学中著名的能斯特方程。同年，还引入溶度积这个重要概念，用来解释沉淀反应。他用量子理论的观点研究低温下固体的比热；提出光化学的“原子链式反应”理论。1906 年，根据对低温现象的研究，得出了热力学第三定律，人们称之为“能斯特热定理”，这个定理有效地解决了计算平衡常数问题和许多工业生产难题。因此获得了1920年诺贝尔化学奖金。。主要著作有：《新热定律的理论与实验基础》等。
• 亥姆霍兹，H．von (Hermann von Helmholtz 1821～1894) 德国物理学家、生理学家。1821 年10月31日生于柏林的波茨坦。中学毕业后由于经济上的原因未能进大学，以毕业后需在军队服役8年的条件取得公费进了在柏林的王家医学科学院。1842年获得医学博士学位后，被任命为驻波茨坦驻军军医，1849年他应聘任柯尼斯堡大学生理学和普通病理学教授。1858年任海德尔堡大学生理学教授。1871年接替马格诺斯任柏林大学物理学教授。1873年当选为英国伦敦皇家学会的外国会员，被授予柯普利奖章。1882年受封爵位。 1887年被任命为新成立的柏林夏洛滕堡物理技术学院院长。1894年9月8日在夏洛滕堡逝世。

宇宙就是生命

地球不是在一處奇異的星系裡繞著一顆奇異的恆星運行的一顆奇異的微粒，迷失在一個龐大的「無情」漩渦裡，那漩渦自大霹靂以來就有許多星球和星系在時空中飛馳著。地球與其他幾兆顆像地球的行星一樣是由「有生命力的微塵」組成之宇宙雲的一部分，這種「有生命力的微塵」的存在是因為宇宙就是這樣。我們可以避免提及任何設計，完全根據事實的認知，宣稱宇宙是以如此方式建構而成，許多產生生命的行星注定會出現。
莫諾的書沒有像德日進那樣引出同樣的激動，其大部份是肯定的科學，用無可指責的專有名詞讓較多人容易了解，不過他偏離嚴格的科學領域那些觀點，真的讓他受到猛烈的攻擊，甚至當他說現代生物學迫使他做出某些結論－例如出現生命或意識的低機率－的時候，此一聲明的真實程度還被部份科學家否認。我那時（用法文）寫了一篇很長的批評文章刊登在一份不出名的雜誌上，或許莫諾根本沒有看過，雖然我寄了一份給他。
1974年諾貝爾生醫學獎得主 Christian de Duve 克里斯坦 . 德 . 杜維著陳挹芳譯
這類猜測已經成為當今討論的熱門題目，由一本含有大量的、廣泛的引證的著作－ 700 頁，600條數學公式，1,500 種注解和參考文獻－就可以作證了。這本書是由英國天文學家巴洛（John D. Barrow）和美國物理學家兼數學家提培勒（Frank J. Tipler）在1986年出版，書名是《人擇宇宙論原理》（The Anthropic Cosmological Principle）。二位作者在書中從歷史、哲學、宗教、生物學、物理學、天文物理學、宇宙學、量子力學和生物化學集結了證明以支持他們的主要觀點。
傳統上，與哲學家進行對話的人主要是理論物理學家和數學家，可能是因為它們在觀念上是有共同相同之處。所得出的宇宙論輪廓包括了物理學的所有層面，從基本粒子到銀河系，不過，要不是忽視了生命，就是將生命和心靈附其視為兩個不同的實體，或暗或明地求助於生機論和二元論。這是錯的。生命是宇宙的一個整體部份，它甚至是已知世界裡最複雜且最有意義的部份。生命的表現應該主導我們世界的形象，而不是被排斥掉。在了解生命的基本過程中，一定要考慮到演化的發展，這已經特別成為必須履行的任務了。

劳特利奇哲学史10卷本

劳特利奇哲学史 (10卷本)【英国】帕金森【加拿大】杉克尔著【出版社】中国人民大学出版社(2017年)【馆藏地】文科馆文科综合书库【索书号】B1 /P13 /1-10【简介】《劳特利奇哲学史》对从公元前6世纪开始直到现在的西方哲学史提供了一种编年式的考察。

它深入地讨论了哲学的所有重要发展，对于那些普遍公认的伟大哲学家提供了很大的篇幅。

但是，较小一些的人物并没有被忽略，在这十卷本的哲学史中，包括了过去和现在的每一个重要哲学家的基本和关键的信息。

这些哲学家被明确地置于他们时代的文化特别是科学的氛围之中。

这部《哲学史》不仅是写给专家看的，而且也是写给学生和普通读者看的。

各章都是以浅近的风格写成，每一章的作者都是这一领域公认的专家，全书130多位顶尖的专家来自英国、美国、加拿大、澳大利亚、爱尔兰、法国、意大利、西班牙、以色列等十多个国家的著名大学和科研机构。

每一章后面附有大量的参考书目，可供深入研究者参考。

有详细的哲学大事历史年表，涵盖了从公元前8世纪至1993年西方哲学发展的全部历史，后面还附有专业术语的名词解释和文献、主题、人名索引。

该书是代表当今世界哲学史研究领域较高学术水平的著作。

”穿越时光的萨满——通往知识的五百年之旅【加拿大】杰里米·纳尔贝【英国】弗朗西斯·赫胥黎著【出版社】社会科学文献出版社; 第1版(2017年9月1日)【馆藏地】文科馆文科综合书库【索书号】B933 /N131【简介】在《穿越时光的萨满》一书中，作者追溯了西方文化关于萨满的古代知识的探索史。

几个世纪以来不乏对萨满充满痴迷的观察者，他们对萨满的认识也从法师、巫师到魔术师、骗子，再到巫医等不断转换，而对500年来萨满研究的考察——《穿越时光的萨满》则是一种跨越时空的旅行。

本书涵盖了列维-施特劳斯、布莱克-厄立克、弗朗兹-博厄斯等思想者的相关论述，提供了对于全世界萨满实践文化的特别洞见。

思辨录王元化著【出版者】华东师范大学出版社; 第1版(2017年7月1日)【馆藏地】文科馆文科综合书库【索书号】C52 /W383B【简介】由王元化著的《思辨录(王元化精品集)》收集了王元化先生从1940年至2002年六十多年来关于改革开放、中国政治、历史、人物与中国文化、中国传统、知识分子、五四运动、启蒙运动及文艺与美学、作家与创作等相关领域的反思．内容广泛，思想深邃，文笔流畅。