Azimuthal anisotropy of K^0_S and Lambda production at mid-rapidity from Au+Au collisions a

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泊松融合原理和python代码

泊松融合原理和python代码【原创版】目录1.泊松融合原理概述2.Python 代码实现泊松融合原理3.泊松融合原理的应用正文1.泊松融合原理概述泊松融合原理是一种概率论中的经典理论，由法国数学家泊松提出。

泊松融合原理描述了一个事件在特定时间间隔内发生的概率与另一个事件在相同时间间隔内发生的次数之间的关系。

具体而言，泊松融合原理表明，一个事件在时间间隔Δt 内发生的次数服从泊松分布，即P(X=k)=e^(-λΔt) * (λΔt)^k / k!，其中λ是事件的平均发生率，X 是事件在时间间隔Δt 内发生的次数，k 是事件发生的次数。

2.Python 代码实现泊松融合原理为了验证泊松融合原理，我们可以使用 Python 编写代码模拟事件的发生过程。

以下是一个简单的 Python 代码示例：```pythonimport randomimport mathdef poisson_fusion(lambda_value, dt, num_trials):"""泊松融合原理模拟:param lambda_value: 事件的平均发生率:param dt: 时间间隔:param num_trials: 模拟次数:return: 事件在时间间隔内发生的次数"""count = 0for _ in range(num_trials):# 随机生成一个 0 到 dt 之间的时间间隔t = random.uniform(0, dt)# 计算在时间间隔内事件发生的次数k = int(lambda_value * t)# 计算泊松分布的概率poisson_prob = math.exp(-lambda_value * t) * (lambda_value * t) ** k / k# 根据泊松分布概率随机生成事件发生的次数count += random.choice([0, 1, 2, 3, 4],p=poisson_prob)return count# 示例lambda_value = 1 # 事件的平均发生率为 1dt = 1 # 时间间隔为 1um_trials = 1000 # 模拟次数为 1000# 模拟事件在时间间隔内发生的次数counts = [poisson_fusion(lambda_value, dt, num_trials) for _in range(num_trials)]# 计算平均值和方差mean = sum(counts) / num_trialsvariance = sum((count - mean) ** 2 for count in counts) / num_trialsprint("平均值:", mean)print("方差:", variance)```3.泊松融合原理的应用泊松融合原理在实际应用中有很多场景，例如统计学、保险、生物学等领域。

Constraints on Inflation in Einstein-Brans-Dicke Frame

a r X i v :g r -q c /9808057v 2 19 M a r 1999Constraints on inﬂation in the Einstein-Brans-Dicke frameYungui Gong∗Physics Department,University of Texas at Austin,Austin,Texas 78712The density perturbation during inﬂation seeds the large scale structure of the Universe.We consider both new inﬂation-type and chaotic inﬂation-type potentials in the framework of Einstein-Brans-Dicke gravity.The density perturbation gives strong constraints on the parameters in these potentials.For both potentials,the constraints are not much diﬀerent from those obtained in the original inﬂationary models by using Einstein gravity.98.80.Cq,04.50.+hI.INTRODUCTIONThe successful explanation of cosmological puzzles,such as the horizon,ﬂatness,and monopole problems,is achieved by the inﬂationary scenarios.The inﬂationary scenarios also predict the spectrum of the density perturbation which seeds the formation of the large scale structure of the Universe.The basic assumption is that,at the early times,the Universe experienced an accelerated expansion,while the Hubble radius changed very slowly.Consequently the wavelength of a quantum ﬂuctuation soon exceeds the Hubble radius.The amplitude of the ﬂuctuation is frozen after the horizon crossing.After the end of inﬂation,the Hubble radius increases faster than the scale factor;so the ﬂuctuations eventually reenter the Hubble radius during the radiation-dominated (RD)or matter-dominated (MD)eras.The original model,the so-called “old inﬂation”model [1],is based on the ﬁrst-order phase transition.It failed because of the “graceful exit”problem.Soon after the “new”and “chaotic”inﬂationary models were proposed to solve this problem [2].These models use a simple scalar ﬁeld as the matter source.The scalar ﬁeld (inﬂaton ﬁeld)slow-rolls down the potential during the inﬂationary phase.All the above models are based on Einstein gravity.However,inﬂation may be driven by non-Einstein gravity.“Extended inﬂation”employs Jordan-Brans-Dicke (JBD)gravity [3].The introduction of the Brans-Dicke (BD)ﬁeld slows down the inﬂation and solves the graceful exit problem.But it was soon found that there was a “big bubble”problem for the original extended inﬂation [4].The interaction between the BD ﬁeld and the inﬂaton ﬁeld can change the spectrum of the density perturbation.The spectra of density perturbations were analyzed in the extended new and chaotic inﬂationary models by several authors [5].But the density perturbations given by those papers are not correct [6].The correct density perturbation in BD inﬂation was given in [6]1.In this paper,I use the slow-roll approximation and work with BD gravity in the Einstein frame (let us call it Einstein-Brans-Dicke gravity).There are strong arguments identifying the Jordan frame 2as the physical one.The possibility of identifying the Einstein frame as the physical one was ﬁrst raised by Cho [8].Cho [8]and Damour and Nordtvedt [9]pointed out that only in the Einstein frame does the Pauli metric represent the massless spin-2graviton and the scalar ﬁeld represent the massless spin-0ﬁeld.In the Jordan frame the graviton is described by both the Pauli metric tensor and the BD scalar ﬁeld.Cho also pointed out that in the compactiﬁcation of Kaluza-Klein theory,the physical metric must be identiﬁed as the Pauli metric because of the wrong sign of the kinetic term of the scalar ﬁeld in the Jordan frame.In string theory,the dilaton ﬁeld appears naturally.The Einstein frame is greatly favored over the string frame although the string frame is chose for the pre-big-bang cosmology.For further discussions about the two frames,see [10]and references therein.Because the dilaton ﬁeld evolves very slowly during the RD and MD eras,we then assume that the dilaton ﬁeld at the end of inﬂation takes the same value as that at present.By this assumption,we can ﬁx the value of the inﬂaton ﬁeld at the beginning of inﬂation.We ﬁnd that the results are diﬀerent from those in [6].Inﬂationary models based on general scalar-tensor gravity in the Einstein frame were also discussed in [11]3and [12].The BD Lagrangian in Einstein frameisL=√2κ2ℜ−1−g 1√2e aσ˙ψ2+V(ψ),p=1R2=κ22˙σ2+e−2aσρ ,(4a)¨σ+3H˙σ=12e2aσ˙σ2≪V(ψ),13e−2aσV(ψ),(6a)3H˙σ≈2ae−2aσV(ψ),(6b)3H˙ψ≈−e−aσV′(ψ).(6c) The consistency conditions for the above approximations areβ2≪36κ2V2(ψ)≪1, e aσV′′(ψ)R(0) 2β2=2κ2˜m2P l1−β2H(ψ)= 3e−aσ(ψ)R(0)=1˜m2P l.(9)It is easy to get N tot≥65.The conditions(7)tell us that inﬂation ends whenm2P l V′2(ψ(t e))≈3V2(ψ(t e)),or|2m2P l V′′(ψ(t e))|≈3V(ψ(t e)).(10) The densityﬂuctuation due to primordially adiabaticﬂuctuation is given by[6]δρ2π e3aσ/2˙σ t k=f H16πV′(ψ)+κ2+Bψ4 ln(ψ2/η2)−12−λ4ψ4,(13)where V0=Bη4/2andλ=|4B ln(ψ2/η2)|is approximately a constant.From the potential(13),we know that inﬂation will end whenm2P l V′2(ψ(t e))≈3V2(ψ(t e)),if 15V0,|2m2P l V′′(ψ(t e))|≈3V(ψ(t e)),if15V0. The inﬂatonﬁeld at the end of inﬂation takes the valueψ2f=ψ2(t e)≈−4m2P l+2λV0+4λ2m4P l≈V0In the last step,we use the result V0≪4λm4P l.Becauseλψ4f/4≪V0,it is well justiﬁed to approximate V(ψ)≈V0 during inﬂation.By using this approximation,weﬁnd the time evolutions of the scale factor,dilatonﬁeld,and inﬂaton ﬁeld asexp[aσ(t)]=2κ2˜m2P l(1+2β2H v t),R(t)=R(0)(1+2β2H v t)1/2β2,1ψ2i−λψ2i−2λ3V0κt,where H v=√3κ˜m2P lis the Hubble parameter at the beginning of inﬂation.The function g(ψ)isg(ψ)= dψV(ψ)λψ3=V02λ[(1+β2)m2P l−˜m2P l]≈β2ψf,(16a)ψ2(t k)=β2V02√V0ρ=13πβ3eβ2N k 1+β2−e−2β2N k 3/2√3V0ρ=47.6√V0λin Eq.(17b)is about38.4whenβ≈0.13.Ifβ2≥0.04,thecoeﬃcient increases quickly and it requires smallerλto get the right density perturbation.In[16],the author did not get the constraint onλbecause they considered theﬂuctuation from the dilatonﬁeld only.However,it is clear form Eq.(17b)that theﬂuctuation from the inﬂatonﬁeld is larger than that from the dilatonﬁeld.In fact,our result is easily understood.Note that theﬂuctuation due to the inﬂatonﬁeld isδψ(t)=e aσ(t)/2H(t)2π√2√V02π 6˜m2P l.(19)Here we useψ(t)>ψi andψi is given in Eq.(16a).To make our semiclassical discussions valid,we must require that the classical value of the inﬂatonﬁeld be larger than its quantumﬂuctuation.So we haveλ<12π2β2˜m2P lIII.CHAOTIC TYPE POTENTIALFor chaotic inﬂation,the potential takes the power law typeV (ψ)=λnV ′(ψ)=dψψ2n.(22)Because n ≥2,the end of inﬂation will happen when|2m 2P l V ′′(ψ(t e ))|≈3V (ψ(t e )),(23a)ψ2f =2n (n −1)β2+2n (n −1)β2˜m 2P l ≈2n3m 2P l ,(24a)ψ2(t k )=2n3−exp(−2β2N k )m 2P l .(24b)In terms of N k ,we ﬁndH (t k )=124πe 2β2N k2n3−exp(−2β2N k ) m 2P l n/4 nm 2P l,(25a)δρ5πnm n/2−2P l [h (N k )]n/4−16e β2N k4√β21+(n −1)β2ρ=1.7√m P l.So the bounds on the anisotropy of the microwave background giveλ2<4×10−11m 2P l .(26)For n =4,we haveδρλ4.In order to get the small density perturbation,we requireλ4<1×10−13.(27)5At last,let us look at the exponential potential V(ψ)=V0exp(−ψ/ψ0)[18].In this case,both V′(ψ)/V(ψ)and V′′(ψ)/V(ψ)are constants.Then the slow-roll condition cannot determine when the inﬂation will end under the assumption that exp(aσ)=16πat the end of inﬂation.Therefore,this kind of potential is not workable in our concern.In this model,we must consider the interaction between the inﬂatonﬁeld and otherﬁelds to let the Universe exit from the inﬂationary epoch.In[7],the authors considered the exponential potential in extended inﬂation.They used the consistency conditions(7)to give the value of the BDﬁeld at the end of inﬂation.Since we know the evolution of the BDﬁeld during the RD and MD eras,it may be a problem to match the value of the BDﬁeld from the end of inﬂation to the present.We would like to say a few more words about the diﬀerence between our results and those in[6].As a result of the assumption that the dilatonﬁeld takes the same value at the end of inﬂation and the present,we canﬁx the value of the inﬂatonﬁeld at both the beginning and the end of inﬂation.If we take the approximation exp(−2β2N k)≈1−2β2N k,then our results(17b)and(25b)are similar to those in[6].For the choices ofβ2and N k in this paper,we can take this approximation.That is why our numerical values do not diﬀer much from those in[6].Instead of thinking the physical frame to be the Jordan frame,we work in the Einstein frame.ACKNOWLEDGMENTSThe author would like to thank Professor Yuval Ne’eman for his helpful comments.。

The primal-dual method for approximation algorithms and its application to network design p

1
Introduction
Many problems of interest in combinatorial optimization are considered unlikely to have eﬃcient algorithms; most of these problems are N P -hard, and unless P = N P they do not have polynomialtime algorithms to ﬁnd an optimal solution. Researchers in combinatorial optimization have considered several approaches to deal with N P -hard problems. These approaches fall into one of two classes. The ﬁrst class contains algorithms that ﬁnd the optimal solution but do not run in polynomial time. Integer programming is an example of such an approach. Integer programmers attempt to develop branch-and-bound (or branch-and-cut, etc.) algorithms for dealing with particular problems such that the algorithm runs quickly enough in practice for instances of interest, although the algorithm is not guaranteed to be eﬃcient for all instances. The second class contains algorithms that run in polynomial time but do not ﬁnd the optimal solution for all instances. Heuristics and metaheuristics (such as simulated annealing or genetic algorithms) are one approach in this class. Typically researchers develop a heuristic for a problem and empirically demonstrate its eﬀectiveness on instances of interest. In this survey, we will consider another approach in this second class called approximation algorithms. Approximation algorithms are polynomial-time heuristics for N P -hard problems whose solution values are provably close to optimum for all instances of the problem. More formally, an α-approximation algorithm for an optimization problem is an algorithm that runs in polynomial time and produces a solution whose value is within a factor of α of the value of an optimal solution. The parameter α is called the performance guarantee or the approximation ratio of the algorithm. We assume that the value of any feasible solution is nonnegative for the problems we consider; extensions of the notion of performance guarantee have been developed in other cases, but we will not discuss them here. This survey will follow the convention that α ≥ 1 for minimization problems and α ≤ 1 for maximization problems, so that a 2-approximation algorithm for a minimization problem produces a solution of value no more than twice the optimal value, 1 and a 2 -approximation algorithm for a maximization problem produces a solution of value at least

Quasi-Exactly Solvable Models with Spin-Orbital Interaction

a r X i v :h e p -t h /9810045v 1 7 O c t 1998TPI-MINN -98/04Quasi-Exactly Solvable Models with Spin-Orbital Interaction 1Alexander UshveridzeTheoretical Physics Institute,University of MinnesotaandDepartment of Theoretical Physics,University of Lodz,Pomorska 149/153,90-236Lodz,Poland 2Abstract First examples of quasi-exactly solvable models describing spin-orbital interaction are constructed.In contrast with other examples of matrix quasi-exactly solvable models discussed in the literature up to now,our models admit inﬁnite (but still incomplete)sets of exact (algebraic)solutions.The hamiltonians of these models are hermitian operators of the form H =−∆2+V 1(r )+(s ·l )V 2(r )+(s +l )·h V 3(r )where V 1(r ),V 2(r )and V 3(r )are scalar functions,l is a vector of the angular momentum operator,s is a matrix-valued vector spin-operator and h is an external (constant)vector magnetic ﬁeld.1Introduction Quasi-exactly solvable (QES)problems are distinguished by the fact that only some of their energy levels and corresponding wavefunctions admit explicit construction.In the last decade a big progress has been acieved in elaborating the concepts of the phenomenon of quasi-exact solvability and formulating methods of constructing and solving QES models of a variety oftypes (for more detail see e.g.the reviews [[4],[7],[3],[5]]and book [[8]]).In this paper we undertake a new step in this direction and construct new classes of QES models which (up to now)have never been discussed in the QES-literature.These are QES models with a spin-orbital interaction.The potentials of such models have the following general formV (x,y,z )=V 1(r )+(s ·l )V 2(r )+(j ·h )V 3(r )where r =1This work was supported by the grant No DOE/DE-FG02-94ER408232E-mail address:alexush@mvii.uni.lodz.pl and alexush@krysia.uni.lodz.plenergy levels and corresponding eigenvalues.They are however quasi-exactly solvable because the set of their exact solutions is still incomplete and does notﬁll all the spectrum of a model. First examples of such models were presented in our recent work[[1]]where they have been called”inﬁnite QES models”.Another unusual feature of these models is that they(in contrast with models discussed in paper[[1]])are matrix models with physically realistic hermitian hamiltonians.Everybody who has some experience with quasi-exact solvability in the matrix(multi-channel)case knows how diﬃcult is to satisfy the condition of hermiticity when constructing such models.It is hardly neccessary to remind the reader that up to now only a couple of hermitian matrix QES models have been constructed(see e.g.[[4],[2]]).2Starting pointTo demonstrate how does our construction procedure work we start with the simplest one-dimensional QES model with hamiltonianH=−∂2r2+[b2−2a(2m+c+1)]r2+2abr4+a2r6(1)acting in Hilbert space of functions deﬁned on the positive half axis r∈[0,∞]and vanishing suﬃciently fast at its ends r=0and r=∞.Here a,b,c are real parameters satisfying the conditions(a>0,c>0)and m is a non-negative integer.As it was demonstrated in[[8]], for anyﬁxed m the Schroedinger equationHψ(r)=Eψ(r)(2) for model1admits algebraic solutions whose general form is given by the formulasψ(r)=r c−1/2mi=1 r2/2−ξi exp −ar42 (3)E=2b(2m+c)+8ami=1ξi(4)The m complex numbersξi in expressions3and4satisfy the system of m algebraic equationsmk=1,k=i12ξi−b−2aξi=0,i=1,...,m.(5)It turns out that system5has only m+1permutationally invariant solutions for any given m which are represented by the sets of real pointsξi.Each solution is completely characterized by a(quantum)number k=0,1,...,m which indicates the number of positiveξi-points. According to formula3,the number of positiveξi-points determines the number of(real) wavefunction zeros,which,in turn,determines the ordinal number of an excitation(oscillation theorem).This means that model1has m+1exactly constructable solutions describing the ground state and mﬁrst excited states.A more detailed exposition of properties of model1 and its algebraic solutions can be found in the book[[8]].3The modiﬁed equationIt is not diﬃcult to see that the transformationψ(r)=rϕ(r)(6)reduces the equation2to the form−∂2r∂r2+V(r,l,m) ϕ(r)=Eϕ(r)(7) in which we used the notationV(r,l,m)=[b2−2a(2m+5/2+l)]r2+2abr4+a2r6(8) andl=c−3/2(9)Hereafter we shall consider l as a new independent parameter taking(by agreement)only non-negative integer values.The form of theﬁrst three terms in the equation7coincides with the form of the radial part of a tree-dimensional Laplace operator.For this reason it seems quite natural to interpret l as the3-dimensional angular momentum and try to relate the equation7to a certain3-dimensional quantum problem.In the following three sections we show that there are three such possibilities leading to three diﬀerent kinds of quasi-exactly solvable problems in the3-dimensional space.4Theﬁrst possibilityOne of the simplest possibilities of interpreting equation7is based on the assumption that function8entering into7is l independent:V(r,l,m)=V0(r,N)=[b2−2a(N+5/2)]r2+2abr4+a2r6(10) For this the numberN=l+2m(11)must beﬁxed.In this case,equation7takes the form of a typical radial Schroedinger equation for a spherically symmetric3-dimensional equation(−∆+V0(r,N))Ψ(x,y,z)=EΨ(x,y,z).(12) Since both m and l are assumed to be positive,the condition11leads to aﬁnite number of possibilities with m=0,1,...,[N/2],and l=N,N−2,...,N−2[N/2],respectively.For this reason,for any given N,the model12is quasi-exactly solvable and has(as usually)only a ﬁnite([N/2]([N/2]+1)/2)number of explicit solutions.The model of such a form and even its more complicated spherically non-symmetric versions were considered many years ago in papers[[6],[7]].5The second possibilityAnother possibility of interpreting equation7is to consider m as aﬁxed number not restricting the value of l.In this case the function8becomes linearly dependent on l and can be represented in the formV(r,l,m)=V1(r,m)−l·V2(r)={[b2−2a(2m+5/2)]r2+2abr4+a2r6}−l·{2ar2}(13)It is quite obvious that,in order to associate the equation13with a certain3-dimensional Schroedinger equation,we mustﬁnd a proper3-dimensional source for the term which is linear in the momentum l.Theﬁrst think which comes in ones head is to look for the3-dimensional scalar operators O which wuld commute with both the Laplace operator and r and would have the eigenvalues linear in l.In this case we could consider7as a reduction of a3-dimensional problem(−∆+V1(r,m)−O·V2(r))Ψ(x,y,z)=EΨ(x,y,z)(14)The linearity in l means that the operator must be proportional to the operator of the angular momentuml=(l x,l y,l z)=(i(y∂z−z∂y),i(z∂x−x∂z),i(x∂y−y∂x)).(15) But this is a3-dimensional vector while the operator we are looking for must be a scalar. The only possibility to construct a scalar from15is to take a scalar product of l with another vector operator.It is quite obvious that there is no such operator if we restrict ourselves to the single-channel problems.However,if we admit the consideration of multi-channel problems, then a good candidate for the second operator can immediately be found.This is obviously the spin operator s!Rerstricting ourselves(for the sake of simplicity)to the1/2-spin case(2 by2matrices),we can easily check that the spectrum of the operatorO=2·s·l(16)(which,obviously commutes with both∆and r)is linear in l.Indeed,representing operator 16in the formO=j2−l2−s2(17) (where j=l+s is a total momentum)and taking for concreteness a particular case with j=l+1/2we easilyﬁnd the corresponding branch of the spectrumo=j(j+1)−l(l+1)−s(s+1)=(l+1/2)(l+3/2)−l(l+1)−3/4=l.(18)Thisﬁnally leads us to a3-dimensional matrix QES models−∆+{[b2−2a(2m+5/2)]r2+2abr4+a2r6}−2(s·l)·{2ar2} Ψ(x,y,z)=EΨ(x,y,z)(19) describing spin-orbital interaction.It is a time to ask ourselves of what kind of models did we obtain?First of all,one should stress again that these models are actually quasi-exactly solvable.This follows from the fact that for any given m and l they have an inﬁnite number of normalizable solutions,but only m+1of them are exactly(algebraically)constructable.Second,and this is may be the most important thing,despite the fact that the set of exactly constructable solutions is incomplete,this set is inﬁnitely large.This is so because the number l is notﬁxed by the3-dimensional model19.It appears as a solution of the eigenvalue problem for operators O and may take arbitrary non-negative integer values.In conclusion of this section note that the hamiltonians of models we obtained are hermi-tian by construction.6The third possibilityThe last interesting possibility of reducing the equation7to a3-dimensional form appears when the function8depends on both parameters l and m.In this case the function8becomes linearly dependent on both l and m and can be represented in the formV(r,l,m)=V1(r)−l·V2(r)−m·V3(r)={(b2−5a)r2+2abr4+a2r6}−l·{2ar2}−m·{4ar2}(20) By analogy with the previous section we can consider the numbers l as the eigenvalues of the operator of spin-orbital interaction,and the only thing which remains to do is to interpret m as an independent quantum number appearing in equation7as an eigenvalue of a certain operator M commuting with the variable r,Laplasian∆and the spin-orbital operator s·l.A good candidate for such an operator is the z-projection of the total momentum s+l.In fact,it should not neccessarily be a z-projection.Because of the spherical symmetry,it could be equally weel a x-or y-projection,or any other projection.We can therefore represent this operator in a covariant formM=2(s+l)·h(21) where h is a unit magneticﬁeld.We introduced an additional factor2to make the eigenvalues of operator M integer rather than half integer.Of course,the negative integers are not interesting for us,because,as we remember,only for non-negative integer values of m the system admits algebraic solution.Summarizing,we can consider7as a reduction of a3-dimensional problem(−∆+V1(r)−(s·l)·V2(r)−2(s+l)·h·V3(r))Ψ(x,y,z)=EΨ(x,y,z)(22) which can be treated as a spectral problem for a matrix quantum model describing spin-orbital interaction together with the interaction of a total momentum with an external magneticﬁeld. It is remarkable,that the model22does not contain any integer parameters anylonger.All these parameters appear dynamically as solutions of the eigenvalue problems for additionally introduced symmetry operators.At the same time,the model22remains quasi-exactly solvable,because for any particular values of these eigenvalues the equation7has only a certain incomplete set of solutions.7ConclusionThe method of construction inﬁnite(matrix)QES models exposed in this paper is,obviously, quite general and can easily be used for building other spin-orbital models with more com-plicated potentials and higher matrix dimensions.For this it is suﬃcient to start with other known one-dimensional QES modelsﬁrst rewritting them in the form of a radial Schroedinger equation and then interpreting the l-dependent terms appearing in their potential as the eigenvalues of a spin-orbital operators.8AcknowledgementsI would like to express my sincere gratitude to my colleagues from the Theoretical Physics Institute of the University of Minnesota(where this work has been written)for their kind hos-pitality.I am especially grateful to Professor M.Shifman for very intersting and fruitful dis-cussions during my visit.This work was supported by the grant DOE/DE-FG02-94ER40823. References[1]H.Doebner,K. L azarow and hveridze Inﬁnite Quasi-Exactly Solvable models,preprint hep-th No9707254,1997[2]Y.Brihaye and P.Kosi´n ski,Quasi-exactly solvable2×2matrix equations,J.Math.Phys.35,3089-3098(1994)[3]A.Gonz´a lez-L´o pez,N.Kamran and P.J.Olver,Quasi-exact solvability,Contemp.Math.160113-140(1994)[4]M.A.Shifman,Newﬁndings in quantum mechanics:partial algebraization of the spectralproblem,Int J.Mod.Phys.A4,2897-2952(1989)[5]M.A.Shifman,Quasi-exactly solvable spectral problems and conformalﬁeld theory,Con-temporary Mathematics160,237-262(1994)[6]hveridze,Exact solutions of one-and multi-dimensional Schr¨o dinger equations,Sov.Phys.-Lebedev Inst.Rep.254-58(1988)[7]hveridze,Quasi-exactly solvable models in quantum mechanics,Sov.J.Part.Nucl.20,504-528(1989)[8]hveridze,Quasi-exactly solvable problems in quantum mechanics,IOP Publishing:Bristol1994[9]hveridze,Quasi-exact solvability in localﬁeld theory,Preprint hep-th No9607080(1996)。

数学家--中英文对照

数学家--中英文对照[zz]Weierstrass 魏尔斯特拉斯（古典分析学集大成者，德国人）Cantor 康托尔（Weiestrass的学生，集合论的鼻祖）Bernoulli 伯努力（这是一个17世纪的家族，专门产数学家物理学家）Fatou 法都（实变函数中有一个Fatou引理，为北大实变必考的要点）Green 格林(有很多姓绿的人，反正都很牛）S.Lie 李 (创造了著名的Lie群，是近代数学物理中最重要的一个概念)Euler 欧拉（后来双目失明了，但是其伟大很少有人能与之相比）Gauss 高斯（有些人不需要说明，Gauss就是一个）Sturm 斯图谟（那个Liouvel-Sturm定理的人，项武义先生很推崇他)Riemann 黎曼(不知道这个名字，就是说不知道世界上存在着数学家）Neumann 诺伊曼（造了第一台电脑，人类历史上最后一个数学物理的全才）Caratheodory 卡拉西奥多礼（外测度的创立者，曾经是贵族）Newton 牛顿（名字带牛，实在是牛）Jordan 约当（Jordan标准型，Poincare前的法国数学界精神领袖）Laplace 拉普拉斯（这人的东西太多了，到处都有）Wiener 维纳（集天才变态于一身的大家，后来在MIT做教授）Thales 泰勒斯（古希腊著名哲学家，有一个他囤积居奇发财的轶事）Maxwell 麦克斯韦（电磁学中的Maxwell方程组）Riesz 黎茨（泛函里的Riesz表示定理，当年匈牙利数学竞赛第一）Fourier 傅立叶(巨烦无比的Fourier变换，他当年黑过Galois)Noether 诺特（最最伟大的女数学家，抽象代数之母）Kepler 开普勒（研究行星怎么绕着太阳转的人)Kolmogorov 柯尔莫戈洛夫(苏联的超级牛人烂人，一生桀骜不驯）Borel 波莱尔（学过数学分析和实分析都知道此人）Sobolev 所伯列夫（著名的Sobolev空间，改变了现代PDE的写法）Dirchlet 狄利克雷（Riemann的老师，伟大如他者廖若星辰）Lebesgue 勒贝格（实分析的开山之人，他的名字经常用来修饰测度这个名词） Leibniz 莱不尼兹（和Newton争谁发明微积分，他的记号使微积分容易掌握） Abel 阿贝尔（天才，有形容词形式的名字不多，Abelian就是一个）Lagrange 拉格朗日（法国姓L的伟人有三个，他，Laplace，Legendre) Ramanujan 拉曼奴阳（天资异禀，死于思乡病）Ljapunov 李雅普诺夫（爱微分方程和动力系统，但更爱他的妻子）Hold er 赫尔得（Holder不等式，L-p空间里的那个）Poisson 泊松（概率中的Poisson过程，也是纯数学家）Nikodym 发音很难的说（有著名的Ladon-Nikodym定理）H.Hopf 霍普夫（微分几何大师，陈省身先生的好朋友）Pythagoras 毕达哥拉斯(就是勾股定理在西方的发现者）Baire 贝尔(著名的Baire纲)Haar 哈尔（有个Haar测度,一度哥廷根的大红人）Fermat 费马（Fermat大定理，最牛的业余数学家，吹牛很牛的）Kronecker 克罗内克（牛人，迫害Cantor至疯人院）udau 朗道（巨富的数学家，解析数论超牛）Markov 马尔可夫(Markov过程)Wronski 朗斯基（微分方程中有个Wronski行列式，用来解线性方程组的）Zermelo 策梅罗（集合论的专家，有以他的名字命名的公理体系）Rouche 儒契（在复变中有Rouche定理Rouche函数）Taylor 泰勒（Taylor有很多，最熟的一个恐怕是Taylor展开的那个）Urysohn 乌里松(在拓扑中有著名的Urysohn定理)Frechet 发音巨难的说，泛函中的Frechet空间Picard 皮卡（大小Picard定理，心高气敖，很没有人缘）Schauder 肖德尔（泛函中有Schauder基Schauder不动点定理）Lipschiz 李普西茨（Lipshciz条件，研究函数光滑性的）Liouville 刘维尔（用Liouville定理证明代数基本定理应该是最快的方法）Lindelof 林德洛夫（证明了圆周率是超越数，讲课奇差）de Moivre 棣莫佛（复数的乘法又一个他的定理，很简单的那个）Klein 克莱因（著名的爱尔兰根纲领，哥廷根的精神领袖）Bessel 贝塞尔（Hilbert空间一个东西的范数用基表示有一个Bessel定理）Euclid 欧几里德（我们的平面几何学的都是2000前他的书）Kummer 库默尔（数论中最有影响的几个人之一）Ascoli 阿斯克里（有Ascoli－Arzela定理，要一致有界等度连续的那个）Chebyschev 切比雪夫(他证明了n和2n之间有一个素数）Banach 巴拿赫（波兰的牛人，泛函分析之父）Hilbert 希尔伯特（这个也没有介绍的必要）Minkowski 闵可夫斯基（Hilbert的挚友，Einstein的“恩师”）Hamilton 哈密尔顿（第一个发现了４元数，在一座桥上）Poincare 彭加莱(数学界的莎士比亚）Peano 皮亚诺（有Peano公理，和数学归纳法有关系）Zorn 佐恩(Zorn引理，看起来显然的东西都用这个证明)一、伯努利家族Bernoulli（伯努利）家族(1)Euler（欧拉）停止了生命，也就停止了计算。

数据科学家用R语言揭秘特朗普推特幕后经过

数据科学家用R语言揭秘特朗普推特幕后经过美国数据科学家David Robinson在2016年8月9日用了12个小时，对特朗普推特文本数据进行分析挖掘，得出系列政治结论。

这次数据分析使用的工具主要是R语言，涉及到社交媒体数据的采集、清洗、加工和可视化等流程。

David Robinson在本次特朗普推特数据分析中运用了声量分析、时间序列分析、文本挖掘、十种情感分析、多维度剖析、英文分词、样本统计、数据可视化技术，并选用了R语言的twitteR包、tidytext包、dplyr包、purr包、tidyr 包、lubridate包、scales包、broom包，还调用了NRC Word-Emotion Association 词库、停用词库等。

数据科学家 David Robinson擅长 R语言和Python语言I don’t normally post about politics (I’m not particularly savvy about polling, which is where data science has had the largest impact onp olitics). But this weekend I saw a hypothesis about Donald Trump’s twitter account that simply begged to be investigated with data:我很少有发表过关于政治的文章（我一般不特别关注总统选举投票，这其实是数据科学对政治影响最大的地方）。

但是这个周末，我看到一个希望通过数据研究得出唐纳德·特朗普推特账号结论的推测性论断：Every non-hyperbolic tweet is from iPhone (his staff).Every hyperbolic tweet is from Android (from him)./GWr6D8h5ed— Todd Vaziri (@tvaziri) August 6, 2016所有非夸张性的推特来自于iPhone（他的员工）。

Clones and Genoids in Lambda Calculus and First Order Logic

a r X i v :0712.3088v 2 [c s .L O ] 19 D e c 2007Clones and GenoidsinLambda Calculus and First Order LogicZhaohua Luo Content :0.Introduction.1.Genoids.2.Clones.3.Binding Algebras mbda Calculus.5.First Order Logic 6.Clones Over A Subcategory.7.Relate Work.IntroductionA genoid (A,G )consists of a monoid G with an element +,a right act A of G with an element x ,such that for any a ∈A and u ∈G there is a unique element [a,u ]∈G such that x [a,u ]=a and +[a,u ]=u .A genoid represents a category with two objects such that one is the dense product of itself with the other.Denote by Act G the category of right acts of G.The inﬁnite sequence ofﬁnite powers of A in Act G determines a Lawvere theory T h(A,G):A0,A,A2,A3,...A genoid may be viewed as a Lawvere theory with extra capacity provided by G.For any right act P of G,we deﬁne a new right act P A=(P,◦),which has the same universe as P,but the action for any u∈G is deﬁned by a◦u=a[x,u+]. Let ev:P A×A→P be the map deﬁned by ev(p,a)=p[a,e]for any p∈P and a∈A.ThenΛ:hom(T×A,P)→hom(T,P A)deﬁned by(Λf)t=f(t+,x) for any t∈T is bijective,with the inverseΛ′:hom(T,P A)→hom(T×A,P)deﬁned by(Λ′g)(t,a)=ev(g(t),a).Thus(P A,ev)is the exponent in the cartesian closed category Act G.In particular if T=P=A we obtain a canonical bijectionΛ:hom(A×A,A)→hom(A,A A).This is the starting point of lambda calculus.We deﬁne an extensive lambda genoid to be a genoid(A,G)together with two homomorphismsλ:A A→A and•:A×A→A such that(Λ•)λ=id A (β-conversion)andλ(Λ•)=id A(η-conversion).This means that A and A A are isomorphic as right acts of G.Conversely,any genoid(A,G)such that A and A A are isomorphic determines an extensive lambda genoid.A quantiﬁer algebra of a genoid(A,G)is a Boolean algebra P which is also a right act of G with Boolean algebra endomorphisms as actions,together with a homomorphism∃:P A→P such that∃(p∨q)=(∃p)∨(∃q)and p<(∃p)+ for any p,q∈P.The study of aﬁrst order theory can also be reduced to the study of a quantiﬁer algebra for a genoid(A,G).We say a genoid(A,G)is a clone if G is the countable power Aωof A.Alge-braically the class of clones forms a(non-ﬁnitary)variety.A general theory of clones over any full subcategory of a category is presented at the end of this paper.The theory of clones considered in this paper originated from the theory of monads.Two equivalent deﬁnitions of monads,namely monads in clone form and monads in extension form given by E.Mane[7],can be interpreted as only deﬁned over a given subcategory of a category.These are clones in al-gebraic form and clones in extension form over a subcategory respectively.It turns out that these two forms of clones are no longer equivalent unless the subcategory is dense.But morphisms of clones,algebras of clones,and mor-phisms of algebras can all be deﬁned for these two types of clones.Since manyfamiliar algebraic structures,such as monoids,unitary Menger algebras,Law-vere theories,countable Lawvere theories,classical and abstract clones are all special cases of clones over various dense subcategories of Set,the syntax and semantics of these algebraic structures can be developed in a uniﬁed way,so that it is much easier to extend these results to many-sorted sets.1GenoidsA genoid theory is a category(A,G)of two objects together with two mor-phisms x:G→A and+:G→A such that(G,x,+)is the product of A and G,i.e.G=A×G.We also assume that G is a dense object,although this is not essential.A left algebra of(A,G)is a functor from(A,G)to the category Set of sets preserving the product.A morphism of left algebras is a natural transformation.Recursively we haveG=A n×G=A×...×A×Gfor any positive integer n.If G is the countable power of A,i.e.G=Aω=A×A×...then we say that(A,G)is a clone theory.Let A=hom(G,A)and G=hom(G,G).Then G is a monoid and A is a right act of G.For any pair(a,u)∈A×G let[a,u]:G→G be the unique morphism such that x[a,u]=a and+[a,u]=u.Then u=[xu,+u]for any u∈G.Deﬁnition1A genoid(A,G,x,+,[])consisting of a monoid(G,e),a right act A of G,x∈A,+∈G,and a map[]:A×G→G such that for any a∈A and u∈G we have(G1)x[a,u]=a.(G2)+[a,u]=u.(G3)u=[xu,+u].A genoid is simply denoted by(A,G).Clearly any genoid theory(A,G)deter-mines a genoid(A,G).Since we assume G is a dense object,(A,G)is uniquely determines by(A,G).Conversely if(A,G)is a genoid then the subcategory of right acts of G generated by A and G is a genoid theory.Hence the notions of genoid theory and genoid are equivalent.Remark2Genoids form a variety of2-sorted heterogeneousﬁnitary algebras with universes A and G.A genoid(A,G)is called standard if it is generated by A as a2-sorted algebra.Suppose(A,G)is a genoid.We have e=[x,+]and[a,u]v=[au,vu]for any a∈A and u∈G.We shall write[a1,a2,..a n,u]for[a1,[a2,[...[a n,u]...]]].Let x1=x,x i+1=x i+for any i>0.Then axiom(G3)extends tou=[x1u,x2u,...,x n u,+n u]for any n>0.It is easy to deﬁne a many-sorted genoid for a nonempty set S of sorts:Deﬁnition3An S-genoid theory is a category({A s}s∈S,G)of objects{A s}s∈S and G,together with morphisms{x s:G→A s}s∈S and{+s:G→G}s∈S such that(G,x s,+s)is the product of A s and G for all s∈S,and x s+t=x s, +s+t=+t+s for any distinct s,t∈S.We also assume that G is a dense object.A left algebra of({A s}s∈S,G)is a functor from this category to Set preserving the products.Deﬁnition4An S-genoid is a pair(A,G)consisting of a monoid G and a set A={(A s,G,x s,+s,[]s)}s∈S of genoids such that x s+t=x s and+s+t=+t+s for any two distinct elements s,t∈S.Suppose(A,G)is an S-genoid.For any s∈S letκs n:G→(A s)n be the map sending each u∈G to[x s1u,x s2u,...,x n u]s∈(A s)n.Let(T,n)be a pair consisting of aﬁnite subset T of S and an integer n>0.We say an element p of P has aﬁnite support(T,n)(or p has aﬁnite rank n)if pu=pv for anyu,v∈G withκsn (u)=κs n(v)for any s∈T.We say p has aﬁnite rank0(orp is closed)if pu=pv for any u,v∈G.An element of P is calledﬁnitary if it has aﬁnite support.We say P is locallyﬁnitary if any of its element is ﬁnitary.We say(A,G)is a locallyﬁnitary genoid if A is locallyﬁnitary as a right act of G.Example1.1An algebraic genoid is a monoid G together with two elements x,+∈G such that xx=+x=x,and(xG,G,x,+)is a genoid.Algebraic genoids form aﬁnitary variety.Example1.2An algebraic S-genoid with a zero element0is a monoid G with a zero element0together with a set{x s,+s}s∈S of pairs of elements ofG such that1.x s x t=0for any distinct s,t∈S.2.+s x t=x t x t=x t for any s,t∈S.3.(A,G)is an S-genoid with A={(x s G,G,x s,+s)}s∈S.Algebraic S-genoids form aﬁnitary variety.2ClonesLet N be the set of positive integers.Deﬁnition5A clone theory over N is a category(A,G)of two objects to-gether with an inﬁnite sequence of morphisms x1,x2,..from G to A such that (G,{x1,x2,..})is a countable power of A.A left algebra of(A,G)is a functor from(A,G)to Set preserving the countable power.Let A=hom(G,A)and G=hom(G,G).Then G is a monoid and A is a right act of G.For any inﬁnite sequence a1,a2,....of elements of A let[a1,a2...]∈G be the unique morphism such that x i[a1,a2...]=a i for any integer i>0.Then u=[x1u,x2u,...]for any u∈G.Any clone theory determines a genoid theory with x=x1,and+=[x2,x3,...].Deﬁnition6A clone in extension form over N is a nonempty set A such that(i)The set A∗of all the inﬁnite sequences[a1,a2,...]of elements of A is a monoid with a unit[x1,x2,...].(ii)A is a right act of A∗.(iii)x i[a1,a2,...]=a i for any[a1,a2,...]and i>0.Any clone A in extension form determines a genoid(A,A∗,x1,+,[])with +=[x2,x3,...]and[a1,[b1,b2,..]]=[a1,b1,b2,..].Thus we may speak of locally ﬁnitary clones.Conversely,assume(A,G)is a any genoid.Denote by F(A) the set ofﬁnitary elements of A.For any a∈F(A)with aﬁnite rank n>0 and[a1,a2,...]∈F(A)∗deﬁnea[a1,a2,...]=a[a1,a2,...,a n,e],which is independent of the choice of n.Let[a1,a2,...][b1,b2,...]=[a1[b1,b2,...],a2[b1,b2,...],...].Then F(A)∗is a monoid with the unit[x1,x2,...],F(A)is a right act of A∗,and x i[a1,a2,...]=a i.Thus F(A)is a locallyﬁnitary clone.If A=F(A)is locally ﬁnitary then we have a canonical homomorphism of genoids(A,G)→(A,A∗) sending each u∈G to[x1u,x2u,...x n u,...].Remark7Clones form a variety of inﬁnitary algebras with universe A. Remark8The notion of a locallyﬁnitary clone is equivalent to that of a Lawvere theory(without the0-ary object).Deﬁnition9Let A be a clone.A left algebra of A(or a left A-algebra)is a set D together with a multiplication A×D N→D such that for any a∈A, [a1,a2,...]∈A N and[d1,d2,...]∈D N1.(a[a1,a2,...])[d1,d2,...]=a([a1[d1,d2,...],a2[d1,d2,...],...].2.x i[d1,d2,...]=d i.Remark10Left algebras of clones are main objects of study in universal algebra(cf.[8]).Deﬁnition11A clone in algebraic form over N is a nonempty set A such that the set A N of maps from N to A is a monoid and(ru)v=r(uv)for any maps r:N→N and u,v:N→A.Remark12Since N is dense in Set,one can show easily that the two forms of clones over N are equivalent.Therefore in the following we shall not distin-guish these two types of clones.3Binding AlgebrasLet(A,G)be a genoid.The mapδ:G→G sending u to[x,u+]is an endomorphism of monoid G.Let−=[x,e].One can show that(δ,+,−)is a monad on the one-object category determined by the monoid G,as we have +−=(δ+)−=e and−−=(δ−)−.The Kleisli category of this monad is the monoid(G,∗)with u∗v=u[x1,v].If P is any right act of G denote by P A the new right act(P,◦)of G deﬁned by p◦u=p(δu)=p[x,u+]for any p∈P and u∈G.Let ev:P A×A→P be the map deﬁned by ev(p,a)=p[a,e].DeﬁneΛ:hom(T×A,P)→hom(T,P A) by(Λf)t=f(t+,x)for any t∈T,andΛ′:hom(T,P A)→hom(T×A,P)by (Λ′g)(t,a)=ev(g(t),a).Then bothΛ′ΛandΛΛ′are identities(which implies thatΛis bijective).Thus(P A,ev)is the exponent in the the cartesian closed category Act G of right acts of G.Let∆:Act G→Act G be the functor sending each act P to P A,and each morphism f:P→Q to f:P A→Q A.The actions of+and−induces two natural transformations+:Id→∆and−:∆2→∆.It is easy to see that (∆,+,−)is a monad on Act G.Deﬁnition13A binding operation is a homomorphism P A→P.A cobind-ing operation is a homomorphism P→P A.Remark14We assume y,z,w,...,y1,y2,...,z1,z2,...∈{x1,x2,x3...},which are called syntactical variables.Supposeσis a binding operation.The tradi-tional operationσx i:P→P(for each variable x i)is deﬁned as the derived operation:σx i.p=σ(p[x2,x3,...,x i,x1,+i+1]).If y=x i thenσy.p meansσx i.p.Example3.1For any p∈P we have1.σx1.p=σ(p[x1,++])=(σp)+.2.σp=(σx1.p)−.3.If p has aﬁnite rank n>0thenσp has aﬁnite rank n−1.Thusσn p is closed.4.If p is closed thenσp is closed.Deﬁnition15Let S be any set of sorts.Let k be a non-negative integer.An S-arity of rank k is aﬁnite sequenceα=<(s1,n1),...,(s k,n k),(s k+1,n k+1)> with s i∈S and n i≥0.Anα-binding operation on a right act P of an S-genoid(A,G)is a homomorphism of right acts(∆s1)n1P×...×(∆s k)n k P→(∆s k+1)n k+1P(assume(∆s i)0P=P),where∆s i is the functor sending each right act Q of G to Q A s i.Deﬁnition16An S-signature is a setΣof operation symbols such that for each symbol f∈Σan S-arity ar(f)is attached.AΣ-binding algebra for an S-genoid(G,A)is a right act P of G such that for each symbol f∈Σan ar(f)-binding operation f P on P is assigned.Remark17We shall drop all the references to the elements of S if S is a sin-gleton.Thus an arity of rank k is simply aﬁnite sequenceα=<n1,...,n k,n k+1> of non-negative integers.Example3.21.A binding operation is a<1,0>-operation.2.A cobinding is a<0,1>-operation.3.A homomorphism P2→P is a<0,0,0>-operation.4.A homomorphism P0→P is a<0>-operation,which reduces to a closed element of P.Lambda genoids and predicate algebras deﬁned below are examples of binding algebras.4Lambda CalculusA genoid(A,G)is reﬂexive if A A is a retract of A(as right acts of G).It is extensive if A A is isomorphic to A.A lambda genoid is a genoid(A,G)together with two homomorphismsλ: A A→A and A2→A of right acts of G.If((λa)+)x=a for any a∈A we say A is a reﬂexive lambda genoid(or aλβ-genoid).If furthermoreλ((a+)x)=a for any a∈A then we say A is an extensive lambda genoid(or aλβη-genoid). Thus a genoid is reﬂexive(resp.extensive)iﬀit is the underlying genoid of a reﬂexive(resp.extensive)lambda genoid.Remark18Lambda clones(resp.reﬂective lambda clones,resp.extensive lambda clones)form a variety of(inﬁntary)algebras.The initial lambda clone is precisely the clone determined by terms inλσ-calculus(cf[2]).Remark19Lambda algebraic genoids(resp.reﬂective lambda algebraic genoids, resp.extensive lambda algebraic genoids)form a variety ofﬁnitary algebras.Remark20The classical operationλx i:A→A(for each variable x i)is deﬁned as the derived operation:λx i.a=λ(a[x2,x3,...,x i,x1,+i+1]).If y=x i thenλy.a meansλx i.a.Suppose(A,G)is an extensive lambda genoid.Assume a,b,c∈A and u∈G. Here are some useful formulas:(1)(λa)b=a[b,e].(2)((λa)u)b=a[b,u].(3)(λa+)b=a.(4)(λn a)+n x n...x1=a for any integer n>0.(5)If a has aﬁnite rank n>0thenλn a is closed and(λn a)x n...x1=a.Thus(λn a)a n...a1=(λn a)x n...x1[a1,...,a n,e]=a[a1,...,a n](6)An element a has aﬁnite rank n>0if and only if there is a closed elementc such thata=cx n...x1.The following closed terms play important roles in lambda calculus(notation:λy1...y n.a=λy1.(λy2.(..(λy n.a)...)).)I=λy.y=λx1.K=λyz.y=λλx2.S=λyzw.yw(zw)=λλλx3x1(x2x1).It follows from(5)we haveI a=x1[a,e]=a.K ab=x2[b,a,e]=a.S abc=(x3x1(x2x1))[c,b,a,e]=ac(bc).Deﬁnition21Let S be a nonempty set carrying a binary operation→.An S-simply typed lambda genoid is an S-genoid(A,G)together with homomor-phisms{λs:(A t)A s→A s→t}and{A s→t×A s→A t}such that for any a∈A t and c∈A s→t we have(λs a)+s x s=a andλs(c+s x s)=c.5First Order LogicA predicate algebra of an S-genoid(A,G)is a right act P of G together with homomorphisms of right acts{∃s:P A s→P}s∈S,F:P0→P,and ⇒:P×P→P.Deﬁne the following derived operations on P:¬p=(p⇒F),T=¬F,p∨q=(¬p)⇒q,p∧q=¬(p⇒¬q).We say P is a reduced predicate algebra if for any p,q,∈P and s∈S(i)(∨,∧,¬,F,T)deﬁnes a Boolean algebra P.(ii)∃s(p∨q)=(∃s p)∨(∃s q).(iii)p<(∃s p)+sA reduced predicate algebra is also called a quantiﬁer algebra.Remark22The class of predicate algebras(resp.quantiﬁer algebras)of a genoid is a variety ofﬁnitary algebras.An interpretation of a predicate algebra P is a pair(Q,µ)consisting of a reduced predicate algebra Q and a homomorphismµ:P→Q of predicate algebras.We say p∈P is logical valid(written|=p)if for any interpretation (Q,µ)we haveµ(p)=T.If p,q∈P then we say that p and q are logically equivalent(written p≡q)if(p⇒q)∧(q⇒p)is logically valid.Then≡is a congruence on P.The set of congruence classes of P with respect to the congruence≡is a reduced predicate algebra called the Lindenbaum-Tarski algebra of P.(see[8]for a further development of the theory of predicate algebras).Theorem23Suppose A is a locallyﬁnitary clone.Any left algebra D of A determines a locallyﬁnitary reduced predicate algebra P(D N),where P(D N) is the power set of D N.A locallyﬁnitary predicate algebra of A is reduced iﬀit belongs to the variety generated by predicate algebras P(D N)for all left algebras D of A.6Clones Over A SubcategoryDeﬁnition24Let N be a full subcategory of a category X.A clone(in ex-tension form,or Kleisli triple)over N is a system T=(T,η,∗−)consisting of functions(a)T:Ob N→Ob X.(b)ηassigns to each object A∈N a morphismηA:A→T A.(c)∗−maps each morphism f:B→T C with B,C∈N to a morphism ∗f:T B→T C,such that for any g:C→T D with D∈N(i)∗f∗g=∗(f∗g).(ii)ηB∗f=f.(iii)∗ηC=id T C.Remark25If N=X we obtain the original deﬁnition for a Kleisli triple over a category,which is an alternative description of a monad.Deﬁnition26Let N be a full subcategory of a category X.A clone theory in extension form(resp.in algebraic form)over N is a pair(K,T)where K is a category and T is a functor T:K→X(resp.T is a function T:Ob K→Ob X)such that for any A,B,C,D∈N(i)Ob N=Ob K.(ii)K(A,B)=X(A,T B).(iii)f(T g)=fg(resp.r(fg)=(rf)g)for any f∈K(A,B),g∈K(B,C) and r∈N(D,A).Remark27If N is dense then these two forms of clone theory are equivalent. In particular,a clone theory over N=X in both forms corresponds to a monad on X.Remark28Any clone over N determines a clone theory in extension form over N,called its Kleisli category.Conversely any clone theory in extension form over N induces a clone over N(see[8]for details).Example6.1Let X=Set be the category of sets.1.A clone over a singleton is equivalent to a monoid.2.A clone over aﬁnite set is equivalent to a unitary Menger algebra.3.A clone over a countable set is equivalent to a clone over N deﬁned above.4.A clone over the subcategory{0,1,2,...}ofﬁnite sets is equivalent to a clone in the classical sense(i.e.a Lawvere theory).Remark291.A clone theory over N=X is equivalent to a monad on X.2.A clone(or a monad)over a one-object category is called a Kleisli algebra.3.Any genoid(A,G)determines a Kleisli algebra since(δ,+,−)is a monad on the one-object category G.7Relate WorkIn classical universal algebra one studies left algebras of a clone over N.Such a clone can be deﬁned in many diﬀerent ways(see[3][9][10][15]).Our approach to binding algebras and untyped lambda calculus was greatly inspired by[1].For other algebraic approaches to untyped lambda calculus see[2][5][6][12][14][16].Our deﬁnition of a quantiﬁer algebra of a genoid was based on Pinter[13](see also[4][11]).References[1]M.Fiore,G.Plotkin,D.Turi,Abstract syntax and variable binding Proceedingsof the14th Annual IEEE Symposium on Logic in Computer Science,LICS’99, (1999)193–202[2] C.Hankin,An Introduction to Lambda Calculus for computer Scientists CollegePublications(February2,2004)[3]P.M.Cohn,Universal Algebra,Kluwer Academic Publishers(1981)[4]J.Cirulis,An algebraization ofﬁrst order logic with terms Colloq.Math.Soc.J.Bolyai54,Algebraic logic,1991,125146[5] B.Jacobs,Simply typed and untyped lambda calculus revisited In:M.P.Fourman,P.T.Johnstone and A.M.Pitts(eds.)Applications of Category Theory in Computer Science(LMS177,Camb.Univ.Press,1992),119-142.[6]M.Maggesi,A.Hirschowitz,The algebraicity of the lambda-calculus preprint2007arXiv:0704.2900.[7] E.Manes,Algebraic Theories,Springer-Verlag,1976.[8]Z.Luo,Clone theory,its syntax and semantics,and applications to lambdacalculus and algebraic logic(2007).[9]W.D.Neumann,Representing varieties of algebras by algebras,J.Austral.Math.Soc.11(1970),1–8.[10]B.Pareigis and H.Rohrl,Left linear theories–A generalization of moduletheory,Applied Categorical Structures.Vol.11,No.2,(1994),145-171. [11]B.Plotkin,Algebraic logic,varieties of algebras and algebraic varietiesarXiv:math/0312420[12]A.Obtulowicz,A.Wiweger,Categorical,functorial,and algebraic aspects ofthe.type-free lambda calculus Banach Center Publications9,Warsaw(1982), 399-422[13]C,C.Pinter,A simple algebra ofﬁrst order logic,Notre Dame Journal of FormalLogic,Vol.XIV,No.3,(1973),361-366.[14]A.Salibra,R.Goldblatt Aﬁnite equational axiomatization of the functionalalgebras for the lambda calculus Information and Computation,vol.148,n.1, pp.71-130(1999).[15]B.M.Schein,V.S.Trohimenko,Algebras of multiplace functions,Semi-groupForum17,(1979),1-64[16]P.Selinger,The lambda calculus is algebraic Journal of Functional Programming(2002),12:549-566。

Continued Fractions and Dynamics

Continued Fractions and DynamicsStefano Isola【期刊名称】《应用数学（英文）》【年(卷),期】2014(5)7【摘要】Several links between continued fractions and classical and less classical constructions in dynamical systems theory are presented and discussed.【总页数】24页(P1067-1090)【关键词】Continued;Fractions;Fast;and;Slow;Convergents;Irrational;Rotations;Farey;a nd;Gauss;Maps;Transfer;Operator;Thermodynamic;Formalism【作者】Stefano Isola【作者单位】Dipartimento di Matematica e Informatica, Università degli Studi di Camerino, Camerino Macerata, Italy【正文语种】中文【中图分类】O1【相关文献】1.Quantitative Poincare recurrence in continued fraction dynamical system [J], PENG Li;TAN Bo;WANG BaoWei2.MULTIFRACTAL ANALYSIS OF THE CONVERGENCE EXPONENT INCONTINUED FRACTIONS [J], 房路路;马际华;宋昆昆;吴敏3.Continued Fraction Method for Approximation of Heat Conduction Dynamics in a Semi-Infinite Slab [J], Jietae Lee;Dong Hyun Kim4.Gravity Field Imaging by Continued Fraction Downward Continuation: A Case Study of the Nechako Basin(Canada) [J], ZHANG Chong;ZHOU Wenna;LV Qingtian;YAN Jiayong5.On Continued Fractions and Their Applications [J], Zakiya M. Ibran;EfafA. Aljatlawi;Ali M. Awin因版权原因，仅展示原文概要，查看原文内容请购买。

基于叠前AVAZ反演的火山岩裂缝预测方法研究

（a）叠前AVAZ反演裂缝
（b）断裂相干属性图4 裂缝与断层分布
将本文方法与椭圆拟合法的裂缝预测结果进行对比，椭圆拟合法预测结果显示裂缝在研究区中部较发育，和目的层均方根振幅属性的强振幅规律基本一致，而在北部断裂带附近无明显异常，说明椭圆拟合法较易受振幅影响。相比于椭圆拟合方法，本文叠前AVAZ反演方法裂缝预测结果更加突出了强各向异性的裂缝发育区，减小了强振幅对预测结果的影响，对断裂带附近的裂缝
图2 不同入射角下振幅随方位角的变化
2.2 叠前 AVAZ 反演及应用效果分析
研究区C5井成像测井显示，目的层火石岭组井周裂缝较为发育，主要包括高导缝、诱导缝和高阻缝。裂缝走向玫瑰图见图3，三种类型裂缝的走向都为近南-北向，近似垂直于最大水平主应力方向。
基于校正后的OVT道集开展叠前AVAZ反演，得到各向同性梯度、各向异性梯度（裂缝发育密度）和方位角（裂缝发育方向）数据。图4a为沿目的层裂缝发育矢量图，颜色代表裂缝发育密度，矢量棒方向代表裂缝方向，图4b为沿层提取的断裂相干属性。图4显示裂缝与断层分布相关
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1 叠前 AVAZ 反演方法原理
根据等效介质理论，可将定向发育裂缝的地层可看作HTI介质模型，在实际中地震波反射系数与反射角和传播方向均有关。当来自各个方向的地震波穿过裂缝时，其振幅、速度、旅行时差都

Supersymmetric Nonlinear Sigma Models on Ricci-flat Kahler Manifolds with O(N) Symmetry

Kiyoshi Higashijima∗ , Tetsuji Kimura† and Muneto Nitta‡
Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
N −1
AN =
a2 −
Aa Aa .
a=1
(4)
To obtain the Ricci-ﬂat K¨ ahler manifold with O(N ) symmetry, we calculate the Ricci form and solve the Ricci-ﬂat condition. To do this, we ﬁrst calculate the K¨ ahler metric gab∗ = ∂a ∂b∗ K of the 1
where AN is given by (4). Therefore, we only need the solution of K′ but not K itself, to calculate the Ricci-ﬂat K¨ ahler metric.
2
To solve the nonlinear diﬀerential equation (10), we transform it to a linear diﬀerential equation: dy (x) x c(N − 1) + (N − 1) 2 y (x) = , 4 dx x −a x2 − a4 The general solution of y (x) is immediately obtained as y (x) = 1 2 x − a4

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a r X i v :n u c l-ex /208013v 1 22 A u g 2002Azimuthal anisotropy of K 0Sand Λproduction at mid-rapidity from Au+Au collisions at √s NN =130GeV at RHIC.The value of v 2asa function of transverse momentum p t and collision centrality is presented for both particles and compared with model calculations.A strong p t dependence in v 2is observed up to p t ∼2.0GeV/c where v 2begins to saturate.INTRODUCTIONMeasurements of azimuthal anisotropies in the transverse momentum distributions of particles probe early stages of ultra-relativistic heavy-ion collisions [1,2,3].After an ini-tial geometric anisotropy is established in an off-center or non-central collision,rescat-tering in the overlapping region of the colliding nuclei amongst collision participants transfers the spatial anisotropy into an anisotropy in momentum space.The extent of the transformation depends on the initial conditions and the dynamical evolution of the collision.As a result,anisotropy measurements for nucleus-nucleus collisions at RHIC energies may increase our understanding of the processes governing the evolution of the collision system and in particular,may provide information about an early partonic stage in the evolution of the system [1,4,5,6,7,8].For the purpose of studying azimuthal anisotropies it is advantageous to write the triple differential distribution of particles in the form of a Fourier seriesE d 3N2πd 2N1The reaction plane is the plane deﬁned by the beam line and the line connecting the centers of the colliding nuclei.particles up to a p t of about 2.0GeV/c,and for identiﬁed π±,K ±and p (s NN =130GeV .Our measurements of v 2for identiﬁedparticles using the Solenoidal Tracker At RHIC (STAR)are the ﬁrst to extend beyond the p t range where particles are identiﬁed by their speciﬁc energy loss (dE/dx)in the gas of the Time Projection Chamber (TPC),and up to p t ∼3.0GeV/c.Previously v 2in this higherp t range had only been measured for unidentiﬁed charged particles [15].ANALYSISThe STAR detector [16],due to its azimuthal symmetry and large acceptance,is ideally suited for measuring elliptic ﬂow.For collisions in its center,the STAR TPC measures the tracks of charged particles in the pseudo-rapidity range |η|<1.8with 2πazimuthal coverage.A scintillator barrel,the Central Trigger Barrel (CTB),surrounding the TPC that measures the charged particle multiplicity within |η|<1was used for a central trigger.Two Zero-Degree Calorimeters [17]at both ends of the TPC in coincidence provided a minimum bias trigger.020004000600080000.40.50.6mass = 496 ± 4 MeV/c2width = 15 ± 1 MeV/c2K S0M inv (π+π-) (GeV/c 2)C o u n t s1.11.1251.15mass = 1116 ± 7 MeV/c2width = 5 ± 1 MeV/c2ΛM inv (p π-) (GeV/c 2)FIGURE 1.Invariant mass distributions for π+π−showing an enhancement at the K 0Smass (left panel)and p π−(Λ)mass (right panel).Fitting results are shown as dashed lines in the ﬁgure.The masses and kinematic properties of both K 0S→π++π−and Λ(p +π+),are reconstructed via their decay topologies in the TPC [18,19,20].Figure 1shows the invariant mass distributions for a π+π−mass hypothesis and a p π−(Λare summed together.To reduce the combinatoricbackground,for K 0S,pion-like tracks are required to have a distance of closest approach dca >1.0cm,while for Λ,the pion-like tracks have a dca >1.5cm and the proton-liketracks have a dca>0.8cm.Tracks are determined to be either proton-like or pion-like based on their energy loss(dE/dx)in the TPC gas.The yield from the enhancement in the invariant mass peaks in eachφ,p t bin is used to evaluate v2= cos(2φ) as a function of p t.This method enables us to measure identiﬁed particleﬂow beyond the p t range where dE/dx particle identiﬁcation fails.The event plane,an experimental estimator of the true reaction plane[10],is cal-culated from the azimuthal distribution of tracks using cuts similar to those used in reference[11].To avoid auto-correlations,only tracks excluded from the neutral vertex reconstruction are used in the event plane calculation.The observed v2is corrected to account for the imperfect event plane resolution estimated using the method of subevents described previously[10].The maximum resolution correction factor for the K0S andΛanalysis is found to be0.681±0.004and0.582±0.007respectively and is reached in the centrality corresponding to25–35%of the measured cross section,where the relative multiplicity distribution is used to estimate the event centrality as in reference[11]. For this analysis,three sources contribute to systematic errors in the measured anisotropy parameters:(1)particle identiﬁcation;(2)background subtraction;(3)non-reaction plane related correlations contributing to v2such as resonance decays or Coulomb and Bose-Einstein correlations[22,23].Theﬁrst two sources are estimated by examining the variation in v2after changing several track,event and neutral vertex cuts and are found to contribute an error of less than±0.005to v2.A previous study used the correlation of event plane angles from subevents to estimate the magnitude of non-reaction plane related correlations[12].That analysis showed that these effects which always act to increase the measured value of v2above its true value typically contribute a systematic error to v2of-0.005,but that the magnitude is larger in the more peripheral events where the error increases to about-0.035for the centrality corresponding to58–85%of the measured cross section.RESULTSThe centrality dependence of v2as a function of transverse momentum calculated from 201thousand minimum bias and180thousand central events is shown inﬁgure2.The two particles show a similar p t dependence in the respective centralities with moreﬂow in the more peripheral collisions.This is similar to observations made previously at the same energy[12]where the agreement with hydrodynamic calculations in the lower p t region was interpreted as evidence for early local thermal equilibrium in all but the most peripheral events(45–85%of the measured cross section).Inﬁgure3(left)we plot v2(p t)for K0S andΛfrom minimum bias collisions with results from hydrodynamic model calculations[5]and v2(p t)for negatively charged particles[15].The K0S results are in agreement with the v2for K±in the p t range they share(300≤p t≤700MeV/c)[12].We observe that v2for both strange particles increases as a function of p t similar to the hydrodynamic model prediction,up to about 1.5GeV/c.In the higher p t region however(p t≥2GeV/c),the values of v2seem to be saturated.It has been suggested[6]that the shape and height of v2above2–3GeV/c is related to energy loss in an early,high parton-density stage of the evolution.0.050.10.150.20.25Transverse momentum p t (GeV/c)A n i s o t r o p y p a r a m e t e r v 2FIGURE 2.Elliptic ﬂow,v 2(p t )for K 0Sand Λfor central (0–11%)and mid-central (11–48%)collisions.The p t integrated v 2from minimum bias collisions for K 0S,Λand negatively charged particles are shown in ﬁgure3(right).The integrated values of v 2are calculated byparameterizing the yield with the inverse slope parameter of exponential ﬁts to the K 0Sor Λtransverse mass distributions and are dominated by the region near the particles meanp t .The relatively larger v 2of Λreﬂects the higher mean p t of the Λcompared to the K 0S.Hydrodynamic model calculations [5],shown as a gray-band and central line,are,within errors,in agreement with this result.The width of the gray-band indicates the uncertainty of the model calculation,mostly due to the choice of the freeze-out conditions.The increase of v 2with particle mass in ﬁgure 3points to a signiﬁcant commonality in velocities between particles of different masses that is perhaps,established early in the collision.The nature of the particles during this process however,whether parton or hadron,and the degree of thermalization remains unclear.0.050.10.150.20.25Transverse momentum p t (GeV/c)A n i s o t r o p y p a r a m e t e r v 2Particle Mass (GeV/c 2)FIGURE 3.Elliptic ﬂow,v 2for K 0S and Λas a function of p tfrom minimum bias Au+Au collisions compared to results from hydrodynamic model calculations and v 2of negatively charged particles [15](left).Integrated azimuthal anisotropy parameters v 2as a function of particle mass with a gray-band and central line indicating hydrodynamic model results [5](right).SUMMARYWe have reported theﬁrst measurement of v2for K0S andΛfrom Au+Au collisions at √。