On existence of the sigma(600) - Its physical implications and related problems

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Existence of Homoclinic Solutions

Advances in Dynamical Systems and ApplicationsISSN0973-5321,V olume5,Number1,pp.75–85(2010)/adsaExistence of Homoclinic Solutions for a Class of Second-Order Differential Equations with Multiple LagsChengjun GuoGuangdong University of TechnologySchool of Applied Mathematics,510006,P.R.Chinaguochj817@Donal O’ReganNational University of IrelandDepartment of Mathematics,Galway,Irelanddonal.oregan@nuigalway.ieRavi P.AgarwalFlorida Institute of TechnologyDepartment of Mathematical SciencesMelbourne,Florida32901,U.S.A.agarwal@AbstractThis paper is concerned with the existence of homoclinic orbits for second-order differential equations with multiple lags.By using Mawhin’s continuationtheorem,a nontrivial homoclinic orbit is obtained as a limit of a certain sequenceof periodic solutions of the equation.AMS Subject Classiﬁcations:34K15,34C25.Keywords:Homoclinic orbit,multiple lags,Mawhin’s continuation theorem.1IntroductionIn recent years several authors studied homoclinic orbits for Hamiltonian systems via critical point theory.In particular second-order systems were considered in[1,2,4–6, Received December4,2009;Accepted December11,2009Communicated by Martin Bohner76Chengjun Guo,Donal O’Regan and Ravi P.Agarwal 12–16,19]andﬁrst-order systems in[3,7–9,11,17,18].In this paper we consider the existence of homoclinic orbits for FDE by using Mawhin’s continuation theorem.In particular we discuss the existence of homoclinic orbits for the equationx (t)+a1(t)x (t)−a2(t)x(t)=g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f(t),(1.1) whereτi(i=1,2,···,n)are constants,a1(t)and a2(t)are real continuous functions deﬁned on R with positive period T,f:R→R is a continuous and bounded function, g(t,x1,x2,···,x n)∈C(R×R×R×···×R,R),g(t,0,0,···,0)=0,and is T-periodic in t.A solution x of(1.1)is said to be homoclinic(to0)if x(t)→0as t→±∞.In addition,if x≡0then x is called a nontrivial homoclinic solution.This paper is largely motivated by the work of Rabinowitz[15]in which the exis-tence of nontrivial homoclinic solutions for the second-order Hamiltonian system¨q+V q(t,q)=0was proved.For the sake of completeness,weﬁrst state Mawhin’s continuation theorem [10].Assume X and Y are two Banach spaces,L:Dom L⊂X→Y is a linear mapping and N:X→Y is a continuous mapping.The mapping L will be called a Fredholm mapping of index zero if dim Ker L=codim Im L<+∞and Im L is closed in Y.If L is a Fredholm mapping of index zero,then there exist continuous projections P:X→X and Q:Y→Y such that Im P=Ker L and Im L=Ker Q=Im(I−Q). It follows that L|Dom L∩Ker P:(I−P)X→Im L has an inverse which will be denoted by K P.IfΩis an open and bounded subset of X,the mapping N will be called L-compact onΩif QN(Ω)is bounded and K P(I−Q)N(Ω)is compact.Since Im Q is isomorphic to Ker L,there exists an isomorphism J:Im Q→Ker L.Theorem1.1(Mawhin’s continuation theorem[10]).Let L be a Fredholm mapping of index zero,and let N be L-compact onΩ.Suppose(1)for eachλ∈(0,1)and x∈∂Ω,Lx=λNx;(2)for each x∈∂Ω∩Ker(L),QNx=0and deg(QN,Ω∩Ker(L),0)=0.Then the equation Lx=Nx has at least one solution inΩ∩D(L).2Main ResultNow we make the following assumptions on a1(t),a2(t)and f(t):(H1)0≤m1≤|a1(t)|≤M1;(H2)M2=maxt∈[0,T]a2(t)≥a2(t)≥m2=mint∈[0,T]a2(t)>0;Existence of Homoclinic Solutions77(H3)f:R→R is continuous and bounded,f≡0andR |f(t)|2dt12≤η,whereη>0is a positive constant.Our main result is the following theorem. Theorem2.1.Suppose(H1)–(H3)and assume(H4)|g(t,x1,x2,···,x n)|≤rni=1|x i|and m2−M214−2rn>0.Then system(1.1)possesses a nontrivial homoclinic solution x∈C2(R,R)such that x (t)→0as t→±∞.In order to prove the main theorem we need some preliminaries.For each k∈N, setX k:={x|x∈C1(R,R),x(t+2kT)=x(t),∀t∈R}and x(0)(t)=x(t),deﬁne the norm on X k byx =maxmaxt∈[−kT,kT]|x(t)|,maxt∈[−kT,kT]|x (t)|,and setY k:={y|y∈C(R,R),y(t+2kT)=y(t),∀t∈R}.We deﬁne the norm on Y k as y 0=maxt∈[−kT,kT]|y(t)|.Thus both(X k, · )and(Y k, · 0) are Banach spaces.Remark2.2.If x∈X k,then it follows that x(i)(0)=x(i)(2kT)(i=0,1).In the works of Izydorek and Janczewska[12]and Tanaka[19],a homoclinic so-lution of(1.1)is obtained as a limit,as k→±∞,of a certain sequence of functions x k∈X k.So here we will consider a sequence of systems of functional differential equationsx (t)+a1(t)x (t)−a2(t)x(t)=g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t),(2.1) where for each k∈N,f k:R→R is a2kT-periodic extension of the restriction of f to the interval[−kT,kT]and x k is a2kT-periodic solution of(2.1)obtained via Mawhin’s continuation theorem.Deﬁne the operators L k:X k→Y k and N k:X k→Y k byL k x(t)=x (t),t∈R,(2.2) andN k x(t)=−a1(t)x (t)+a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t),t∈R.(2.3)78Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Clearly,Ker L k={x∈X k:x(t)=c∈R}(2.4)andIm L k=y∈Y k:kT−kTy(t)dt=0(2.5)is closed in Y k.Thus L k is a Fredholm mapping of index zero.Let us deﬁne P k:X k→X k and Q k:Y k→Y k/Im(L k)byP k x(t)=12kTkT−kTx(t)dt=x(0),t∈R,(2.6)for x=x(t)∈X andQ k y(t)=12kTkT−kTy(t)dt,t∈R(2.7)for y=y(t)∈Y k.It is easy to see that Im P k=Ker L k and Im L k=Ker Q k=Im(I k−Q k).It follows that L k|Dom Lk∩Ker P k :(I k−P k)X k→Im L k has an inversewhich will be denoted by K Pk.LetΩk be an open and bounded subset of X k.We can easily see that Q k N k(Ωk)isbounded and K Pk (I k−Q k)N k(Ωk)is compact.Thus the mapping N k is L-compact onΩk.That is,we have the following result.Lemma2.3.Let L k,N k,P k and Q k be deﬁned by(2.2),(2.3),(2.6)and(2.7)respec-tively.Then L k is a Fredholm mapping of index zero and N k is L-compact onΩk,where Ωk is any open and bounded subset of X k.In order to prove our main result,we need the following lemma[15].Lemma2.4(See Remark2.2and[15]).There is a positive constant such that for each k∈N and x∈X k the following inequality holds:max t∈[−kT,kT]|x(t)|≤kT−kT(|x(t)|2+|x (t)|2)dt12.Now,we consider the auxiliary equationx (t)+λ[a1(t)x (t)−a2(t)x(t)]=λ[g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)],(2.8) where0<λ<1.Existence of Homoclinic Solutions79 Lemma2.5.Suppose that the conditions of Theorem2.1are satisﬁed.If x k(t)is a2kT-periodic solution of Eq.(2.8),then there are positive constants D i,i=0,1,which are independent ofλ,such thatx(i)k0≤D i,t∈[−kT,kT],i=0,1.(2.9) Proof.Suppose that x k is a2kT-periodic solution of Eq.(2.8).We have from(2.8)thatkT −kT [xk(t)+λa1(t)xk(t)−λa2(t)x k(t)]x k(t)dt=λkT−kT[g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))+f k(t)]x k(t)dt.(2.10)From(2.10),we havekT −kT {|xk(t)|2+λa2(t)|x k(t)|2}dt=−λkT−kT[g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))+f k(t)−a1(t)xk(t)]x k(t)dt≤λkT−kT |g(t,x k(t−τ1),x k(t−τ2),···,x k(t−τn))|2dt12×kT−kT |x k(t)|2dt12+λkT−kT|f k(t)|2dt12+M1kT−kT|xk(t)|2dt12×kT−kT |x k(t)|2dt12≤2rnλkT−kT |x k(t)|2dt+λη+M1kT−kT|xk(t)|2dt12 kT−kT|x k(t)|2dt12,soλm2−M214kT−kT|x k(t)|2dt≤λkT−kT|xk(t)|2dt12−M12kT−kT|x k(t)|2dt12 2+λa2(t)−M214kT−kT|x k(t)|2dt+1λ−λkT−kT|xk(t)|2dt ≤2rnλkT−kT|x k(t)|2dt+ηλkT−kT|x k(t)|2dt12,80Chengjun Guo,Donal O’Regan and Ravi P .Agarwalwhich givesm 2−M 214−2rnkT−kT |x k (t )|2dt ≤ηkT−kT|x k (t )|2dt 12.(2.11)From (H 4)and (2.11),there exists a positive constant C 1such thatkT−kT |x k (t )|2dt ≤η2(m 2−M 214−2rn )2=C 1.(2.12)From (2.8),we havekT−kT[x k (t )+λa 1(t )xk (t )−λa 2(t )x k (t )]x k (t )dt=λkT−kT[g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))+f k (t )]x k (t )dt,(2.13)som 1kT−kT|x k (t )|2dt≤kT−kT |a 1(t )||x k (t )|2dt ≤(M 2+2rn )kT−kT |x k (t )|2dt 12+ηkT−kT|x k (t )|2dt12≤[(M 2+2rn )C 1+η]kT−kT|x k (t )|2dt 12,and as a result there exists a positive constant C 2such thatkT−kT|x k (t )|2dt ≤C 2.(2.14)Moreover,for x ∈X k and t,τ∈[−kT,kT ],we have|x (t )|≤x (τ)+ t τx (s )ds .(2.15)Integration of (2.15)over t −12,t +12shows|x (t )|≤ t +12t −12|x (τ)|dτ+t +12t −12t τx (s )dsdτ≤2t +12t −12(|x (τ)|2+|x (t )|2)dτ12.(2.16)Existence of Homoclinic Solutions 81Hence (2.15)and (2.16)implymax t ∈[−kT,kT ]|x (t )|≤kT−kT(|x (t )|2+|x (t )|2)dt 12,x ∈X k ,(2.17)where is given in Lemma 2.4.From Lemma 2.4,(2.12)and (2.14),we havemax t ∈[−kT,kT ]|x k (t )|≤kT−kT(|x k (t )|2+|x k (t ))|2dt 12≤ (C 1+C 2)12=D 0.(2.18)On the other hand,we have from (2.8)thatkT−kT[x k (t )+λa 1(t )x k (t )−λa 2(t )x k (t )]xk (t )dt=λkT−kT[g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))+f k (t )]x k (t )dt,(2.19)so we have kT−kT|x k (t )|2dt≤kT−kT|x k (t )|2dt12M 1 kT−kT|x k (t )|2dt 12+M 2kT−kT|x k (t )|2dt12+ kT−kT |g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))|2dt 12×kT−kT |x k (t )|2dt12+ kT−kT|f k (t )dt12kT−kT|x k (t )|2dt12≤[(2rn +M 2)C 1+M 1 C 2+η]kT−kT|x k (t )|2dt12,and as a result there exists a positive constant C 4such thatkT−kT|x k (t )|2dt ≤[(2rn +M 2) C 1+M 1C 2+η]2=C 4.(2.20)From Lemma 2.4and (2.20),we havemaxt ∈[−kT,kT ]|x k (t )|≤kT−kT(|x k (t )|2+|x k (t )|2)dt 12≤ (C 2+C 4)12=D 1.(2.21)The proof iscomplete.82Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Lemma2.6.Let k∈N.If(H1)–(H4)hold,then the system(2.1)possesses a2kT-periodic solution.Proof.Suppose that x is a2kT-periodic solution of Eq.(2.8).By Lemma2.5,there exist positive constants D i(i=0,1)which are independent ofλsuch that(2.9)is true.Consider any positive constantαk>max0≤i≤1{D i}+ξ,whereξ=maxt∈R|f(t)|.SetΩk:={x∈X k: x <αk}.We know that L k is a Fredholm mapping of index zero and N k is L-compact onΩk (see[2]).RecallKer(L k)={x∈X k:x(t)=c∈R}and the norm on X k isx =maxmaxt∈[−kT,kT]|x(t)|,maxt∈[−kT,kT]|x (t)|.Then we havex=αk or x=−αk for x∈∂Ωk∩Ker(L k).(2.22) From(H4),we have(ifαk is chosen large enough)a2(t)αk+g(t,αk,αk,···,αk)− f k 0>0t∈[−kT,kT](2.23) andx (t)=0,∀x∈∂Ωk∩Ker(L k).(2.24) Finally from(2.3),(2.7)and(2.22)–(2.24),we have(Q k N k x)=12kTkT−kT[−a1(t)x (t)+a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)]dt=12kTkT−kT[a2(t)x(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))+f k(t)]dt=0,∀x∈∂Ωk∩Ker(L k).Then,for any x∈Ker L k∩∂Ωk andη∈[0,1],we havexH(x,η)=−ηx2−x2kT(1−η)kT−kT[−a1(t)x (t)+a2(t)x(t)+f k(t)+g(t,x(t−τ1),x(t−τ2),···,x(t−τn))]dt =0.Existence of Homoclinic Solutions 83Thusdeg {Q k N k ,Ωk ∩Ker(L k ),0}=deg −12kT kT−kT[−a 1(t )x (t )+a 2(t )x (t )+f k (t )+g (t,x (t −τ1),x (t −τ2),···,x (t −τn ))]dt,Ωk ∩Ker(L k ),0}=deg {−x,Ωk ∩Ker(L k ),0}=0.From Lemma 2.5,for any x ∈∂Ωk ∩Dom(L k )and λ∈(0,1)we have L k x =λN k x .By Theorem 1.1,the equation L k x =N k x has at least one solution in Dom(L )∩Ωk .So there exists a 2kT -periodic solution x k of the system (2.1).The proof iscomplete.Lemma 2.7.Let {x k }k ∈N be the sequence given by Lemma 2.6.Then there exists x 0anda subsequence of {x n }n ∈N (again we call it {x n }n ∈N )such that x k →x 0in C 1loc (R ,R )as k →+∞.Proof.By (2.18),(2.21)and the Arzel`a –Ascoli theorem,we obtain that a subsequenceof {x k }k ∈N converges in C 1loc (R ,R )to a solution x 0of (1.1)satisfying∞−∞(|x 0(t )|2+|x 0(t )|2)dt <∞.(2.25)To see this note from (2.1)thatlim k →∞[x k (t )+a 1(t )xk (t )−a 2(t )x k (t )−g (t,x k (t −τ1),x k (t −τ2),···,x k (t −τn ))]=x 0(t )+a 1(t )x0(t )−a 2(t )x 0(t )−g (t,x 0(t −τ1),x 0(t −τ2),···,x 0(t −τn ))=lim k →∞f k (t )=f (t ),so x 0is a solution of (1.1).Also,we have∞−∞[|x 0(t )|2+|x 0(t )|2]dt =limk →∞kT−kT[|x k (t )|2+|x k (t )|2]dt <∞.This shows that (2.25)holds.Lemma 2.8.The function x 0determined by Lemma 2.7is the desired homoclinic solu-tion of (1.1).Proof.The proof will be divided into two steps.Step 1:We prove that x 0(t )→0,as t →±∞.By (2.25),we havelim j →∞|t |≥j[|x 0(t )|2+|x 0(t )|2]dt =0.(2.26)84Chengjun Guo,Donal O’Regan and Ravi P.Agarwal Hence(2.18)and(2.26)shows that our claim holds.Step2:We now show that x 0(t)→0as t→±∞.By(2.16),(2.18)and(2.26),it sufﬁces to prove thatj+1 j |x(t)|2dt→0,as j→+∞.(2.27)On the other hand,we obtain from(1.1)thatj+1 j |x(t)|2dt=j+1j|−a1(t)x(t)+a2(t)x0(t)+f(t)+g(t,x0(t−τ1),x0(t−τ2),···,x0(t−τn))|2dt.Since g(t,0,0,···,0)=0for all t∈R,x0(t)→0as t→±∞,j+1j |x(t)|2dt→0and j+1j|f(t)|2dt→0as j→±∞,so(2.27)follows.Proof of Theorem2.1.The result follows now from Lemma2.8.AcknowledgementSupport by grant10871213from NNSF of China and by grant093051from Guangdong University of Technology of China is acknowledged.References[1]A.Ambrosetti,V.Coti Zelati,Multiple homoclinic orbits for a class of conserva-tive system,Rend.Sem.Mat.Univ.Padova.89(1993),177–194.[2]P.C.Carri˜a o,O.H.Miyagaki,Existence of homoclinic solutions for a class oftime-dependent Hamiltonian systems,J.Math.Anal.Appl.230(1999),157–172.[3]V.Coti Zelati,I.Ekeland,E.S´e r´e,A variational approach to homoclinic orbits inHamiltonian 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应用地球化学元素丰度数据手册-原版

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Cumulative dominance and probabilistic sophistication*Rakesh Sarin a, Peter P. Wakker ba The Anderson Graduate School of Management, UCLA, Los Angeles, CAb CentER, Tilburg University, Tilburg, The NetherlandsDecember 1998AbstractMachina and Schmeidler gave preference conditions for probabilistic sophistication, i.e., decision making where uncertainty can be expressed in terms of (subjective) probabilities without commitment to expected utility maximization. This note shows that simpler and more general results can be obtained by combining results from qualitative probability theory with a "cumulative dominance" axiom.Keywords: Probabilistic sophistication; Subjective probability; Qualitative probability Corresponding author:Peter WakkerCentER for Economic ResearchTilburg UniversityP.O. Box 90153Tilburg, 5000 LE, The Netherlands--31-13-466.24.57 (O)--31-13-466.32.80 (F)--31-13-466.23.40 (S)*****************.LeidenUniv.NL* The support for this research was provided in part by the Decision, Risk, and Management Science branch of the National Science Foundation.Machina & Schmeidler (1992) pose an interesting question: What conditions on an agent's preferences justify probabilistic sophistication; that is, when can beliefs be represented by probabilities? A probabilistically sophisticated agent may deviate from expected utility in his preferences over lotteries but must express uncertainty in terms of additive probabilities. In this note, we show that a cumulative dominance condition can be used to simplify and extend the results of Machina & Schmeidler (1992). In a previous paper (Sarin & Wakker 1992), we showed that cumulative dominance is a necessary condition for Choquet expected utility, and it can be used to extend expected utility under risk to Schmeidler's (1989) Choquet expected utility under uncertainty. It turns out that cumulative dominance is necessary for probabilistic sophistication as well, and here we show that this axiom can be used to extend qualitative probability for two-consequence acts to probabilistically sophisticated preferences for many-consequence acts.Let S denote the state space; is the algebra of subsets of S, the elements of which are called events; is the set of consequences; is the set of acts, i.e., maps from S to that are finite-valued and measurable (f-1(x)∈ for all consequences x); x denotes both a consequence and the related constant act. By u we denote the preference relation over acts, that also denotes the induced ordering of consequences. The notation s (strict preference), ~ (equivalence), e, and c is as usual. In this note, probability measures are finitely additive and need not be countably additive unless stated otherwise.Following Machina & Schmeidler (1992), an agent is probabilistically sophisticated if there exists a probability measure P over S such that:(i)The agent chooses between acts based on the probability distributions generatedover the consequences.(ii)First-order stochastic dominance is satisfied.Stochastic dominance in (ii) is taken in the strict sense (see Machina & Schmeidler 1992, Section 3.1). It relates to general, possibly nonmonetary, outcomes, and is defined as follows: f s g whenever P(f u x) ≥ P(g u x) for all consequences x with strict inequality for at least one x. We have modified the definition of probabilistic sophistication of Machina & Schmeidler by omitting their "mixture continuity" condition, in order to allow for general models with atoms (e.g., in Theorem 5 below). Machina & Schmeidler show that mixture continuity is implied by Savage's P6, hence it can be seen to be implied in our Theorem 2.We next derive a more likely than relation on events from preferences between two-consequence acts. We write A uλB if there exist consequences x s y such that the act [x if A; y if not A] is preferred to the act [x if B; y if not B]. Obviously, probabilistic sophistication requires that A uλB if and only if P(A)≥P(B), i.e., P is an agreeing probability measure for uλ. For the case where contains only two consequences, one immediately observes that probabilistic sophistication holds if and only if an agreeing probability measure exists. Machina & Schmeidler (1992, Section 3.4) demonstrate that the existence of an agreeing probability measure need not imply probabilistic sophistication when there are more than two consequences. To obtain probabilistic sophistication for general acts, Machina & Schmeidler assume the restrictions of Savage's axioms that guarantee the existence of a probability measure agreeing with uλ, and then add an additional axiom (P4* in their paper) for acts with many consequences.We will follow the first part of their analysis but use cumulative dominance instead of their P4* to ensure probabilistic sophistication. In line with the strict stochastic dominance condition of Machina & Schmeidler, we strengthen the definition of cumulative dominance of Sarin & Wakker (1992) somewhat and define a strict version thereof. Cumulative dominance holds iff ug whenever {s∈S: f(s)u x} uλ {s∈S: g(s)u x} for all consequences x,where the preference between f and g is strict whenever one of the antecedentuλ orderings is strict (sλ).This condition says that if one considers the cumulative event [x or more] more likely under f than under g for every consequence x, then f should be preferred to g, where the more likely than relation uλ is derived from choices. The following elementary lemma shows the way in which cumulative dominance ensures probabilistic sophistication for many-consequence acts in situations where an agreeing probability measure exists for uλ.Lemma 1. The following two statements are equivalent.(i)Probabilistic sophistication holds.(ii)There exists an agreeing probability measure P for uλ and cumulative dominance holds.Proof. If a probability measure P agrees with uλ and if preferences depend only on probability distributions generated over consequences through P, then one immediately sees that cumulative dominance and stochastic dominance are equivalent. So we only have to show:•Under (i), P agrees with uλ: This is elementarily verified.•Under (ii), preferences depend only on probability distributions generated over consequences: If f and g generate the same probability distribution overconsequences, then {s∈S: f(s)u x} ~λ {s∈S: g(s)u x} for all consequences x, and hence f u g and f u g, by twofold application of cumulative dominance. That is, f~ g.∼Preference characterizations of probabilistic sophistication can be derived from the above lemma by substituting preference conditions, obtained from qualitative probability theory, to guarantee the existence of an agreeing probability measure P foruλ. Several such conditions have been presented in qualitative probability theory. Hence, our approach shows a way to derive probabilistic sophistication from results of qualitative probability theory. As a first illustration, we invoke the qualitative probability conditions of Savage (1954); for definitions the reader is referred to Savage (1954) or Machina & Schmeidler (1992). We assume their P1 (weak ordering), P3 (tastes are independent of beliefs), P4 (beliefs are independent of tastes), P5 (nontriviality), and P6 (fineness of the state space); P7 is not needed because we only consider finite-valued acts. We note that P4 need not be stated because it is implied by cumulative dominance (see the proof below), and restrict P2 to two-consequence acts:Postulate P2* (sure-thing principle for two-consequence acts). For all consequences x s y and events A,B,H with A∩H = B∩H = ∅:[x if A; y if not A]u[x if B; y if not B]if and only if[x if A∪H; y if not A∪H] u[x if B∪H; y if not B∪H]Theorem 2. Assume Savage's (1954) P1, P3, P5, and P6. Then the following two statements are equivalent.(i)Probabilistic sophistication holds.(ii)P2* and cumulative dominance hold.Proof. It is obvious that (i) implies (ii), therefore we assume (ii) and derive (i). P4 follows from the restriction of cumulative dominance to two-consequence acts. P1,P2*, P3, P4, P5, and P6 are all that Savage needs to derive an agreeing probability measure. His analysis, as well as that of Machina & Schmeidler (1992), assumes that is the collection of all subsets of S. It is well-known that these proofs and results immediately extend to any sigma-algebra of events (Savage 1954, Section III.4; Machina & Schmeidler 1992, Section 6.2). Our theorem extends the result further to any algebra of events: Savage's axioms P1, P2*, P3, P4, P5, and P6 also imply theexistence of an agreeing probability measure for that case (Wakker 1981). Now the theorem follows from Lemma 1. ∼Machina & Schmeidler (1992) eloquently pose the question what additional conditions are needed to characterize probabilistic sophistication over many-consequence acts once an agreeing probability measure exists for uλ. Once one assumes qualitative probability theory, then our characterization seems to be the simplest way to obtain probabilistic sophistication in the Savage framework. Note here that qualitative probability theory is the special case of additive conjoint measurement with two-element component sets (Fishburn & Roberts 1988).For many results of probability theory, countable additivity of a probability measure is essential. Hence Machina & Schmeidler (1992, Section 6.2) address the characterization of countable additivity in probabilistic sophistication. They conjecture that conditions of Villegas (1964) and Arrow (1965) may be used to imply countable additivity. We prove that that is indeed the case. Following Villegas (1964), monotone continuity holds if for each sequence of events A j that increases to an event A (A j+1⊃A j and ∪A j = A), A j eλB for all j implies A eλB. In the presence of this condition, Savage's structural condition P6 can be weakened to the requirement that no atoms exist. An event A is an atom if it is nonnull but it cannot be partitioned into two nonnull events.Theorem 3. Assume Savage's (1954) P1, P3, and P5, assume that is a sigma algebra, and that there are no atoms. Then the following two statements are equivalent.(i)Probabilistic sophistication holds with respect to a countably additive probabilitymeasure.(ii)P2*, monotone continuity, and cumulative dominance hold.Proof. The implication (i) ⇒ (ii) is straightforward, hence we assume (ii) and derive (i). Cumulative dominance implies P4. P4 in turn implies weak ordering of uλ(i.e.,Villegas 1964, Conditions Q1(a), Q1(b), and Q1(c)). Now, mainly because of P3 and P5, it follows that ∅cλS and, for any event A, ∅eλA eλS (Villegas' ConditionQ1(d)). P2* implies Villegas' condition Q2 for uλ. Now, by Theorem 4.3 of Villegas, there exists a unique countably additive probability measure agreeing with uλ. By our Lemma 1, Statement (i) holds. ∼By means of the above result, it can be demonstrated that the conjecture in Section 6.2 of Machina & Schmeidler (1992) is correct.Corollary 4. In Theorem 2 of Machina & Schmeidler (1992), the probability measure in (ii) is countably additive if in Statement (i) their monotone continuity axiom is added, and the domain of events is assumed a sigma algebra.Proof. Monotone continuity as defined here and in Villegas (1964) follows from the first part of the monotone continuity axiom of Machina & Schmeidler (1992, Section6.2) by defining, for outcomes x c y, f = [y if A, x if A c], E j = A\A j, and g = [y if B, x ifB c]. Their preference conditions imply probabilistic sophistication and hence, by Lemma 1, cumulative dominance. Also there exist no atoms in their model, mainly by P6. Now the result follows from Lemma 1 and Theorem 3. ∼Finally, we give an example of an alternative derivation of probabilistic sophistication using a qualitative probability result different than that of Savage. We invoke the remarkable result provided by Chateauneuf (1985). Chateauneuf provided necessary and sufficient conditions for the existence of an agreeing probability measure in full generality, i.e., without assuming any restrictive condition such as Savage's P6 or nonatomicity. His result extends the earlier results by Kraft, Pratt, & Seidenberg (1959) and Scott (1964), who gave necessary and sufficient conditions for finite state spaces. For brevity, we do not describe the details of these conditions, but refer the reader to the respective papers. Extensions were provided by Lehrer (1991).Theorem 5 (general probabilistic sophistication). The following two statements are equivalent.(i)Probabilistic sophistication holds.(ii)Cumulative dominance holds, and Chateauneuf's (1985) conditions hold for uλ. Proof. See Chateauneuf (1985) and Lemma 1. ∼By means of Chateauneuf's conditions and our Lemma 1, we thus obtain a characterization of probabilistic sophistication in full generality. The above result can be extended to the case where in (i) the probability measure is countably additive. Obviously, many alternative derivations of probabilistic sophistication can be obtained by applying Lemma 1 to other results of qualitative probability theory. For instance, Zhang (1998) shows how to extend results to domains more general than algebras. Fishburn (1986) gives an extensive survey of such results. An alternative approach, using the model of Anscombe and Aumann (1963), is presented by Machina & Schmeidler (1995).A possible objection to our result may be that cumulative dominance is too close to the functional form it seeks to characterize and therefore the proof is trivial. We note, however, that the purpose of characterizations is to provide preference conditions that are simple to understand and that can be directly tested through observable choices. The focus should be on the insights that preference conditions generate and not on the complexity of proofs they entail. The condition of Machina & Schmeidler (1992) is a generalization of independence of beliefs from tastes and in that manner it provides new insights. Our cumulative dominance condition is an extension of stochastic dominance to the case of uncertainty and thus it reveals the nature of probabilistic sophistication in a transparent and empirically meaningful way.ReferencesAnscombe, F.J. & Robert J. Aumann (1963), "A Definition of Subjective Probability,"Annals of Mathematical Statistics 34, 199−205.Arrow, Kenneth J. (1965), "Aspects of the Theory of Risk-Bearing." Academic Bookstore, Helsinki. Elaborated as Kenneth J. Arrow (1971), "Essays in the Theory of Risk-Bearing." North-Holland, Amsterdam.Chateauneuf, Alain (1985), "On the Existence of a Probability Measure Compatible with a Total Preorder on a Boolean Algebra," Journal of Mathematical Economics 14, 43−52.Fishburn, Peter C. (1986), "The Axioms of Subjective Probability," Statistical Science 1, 335−358.Fishburn, Peter C. & Fred S. Roberts (1988), "Unique Finite Conjoint Measurement,"Mathematical Social Sciences 16, 107−143.Kraft, Charles H., John W. Pratt, & A. Seidenberg (1959), "Intuitive Probability on Finite Sets," Annals of Mathematical Statistics 30, 408−419.Lehrer, Ehud (1991), "On a Representation of a Relation by a Measure," Journal of Mathematical Economics 20, 107−118.Machina, Mark J. & David Schmeidler (1992), "A More Robust Definition of Subjective Probability," Econometrica 60, 745−780.Machina, Mark J. & David Schmeidler (1995), "Bayes without Bernoulli: Simple Conditions for Probabilistically Sophisticated Choice," Journal of Economic Theory 67, 106−128.Sarin, Rakesh K. & Peter P. Wakker (1992), "A Simple Axiomatization of Nonadditive Expected Utility," Econometrica 60, 1255-1272.Savage, Leonard J. (1954), "The Foundations of Statistics." Wiley, New York.(Second edition 1972, Dover, New York.)Schmeidler, David (1989), "Subjective Probability and Expected Utility without Additivity," Econometrica 57, 571-587.Scott, Dana (1964), "Measurement Structures and Linear Inequalities," Journal of Mathematical Psychology 1, 233-247.Villegas, C. (1964), "On Quantitative Probability σ-Algebras," Annals of Mathematical Statistics 35, 1787-1796.Wakker, Peter P. (1981), "Agreeing Probability Measures for Comparative ProbabilityStructures," The Annals of Statistics 9, 658-662.Zhang, Jiangkang (1998), "Qualitative Probabilities on Lambda-Systems,"Mathematical Social Sciences, forthcoming.。

Energy-Momentum Tensor and Particle Creation in the de Sitter Universe

a rXiv:g r-qc/97188v 118Oct1997Energy-Momentum Tensor and Particle Creation in the de Sitter Universe Carmen Molina-Par´ıs ∗Los Alamos National Laboratory,Theoretical Division,Los Alamos,NM,87544(February 7,2008)Particle creation in a conformally ﬂat spacetime (e.g.,FRW universe)requires a non-conformal ﬁeld.The choice of state is crucial,as one may misunderstand the physics of particle creation by choosing a too restrictive vacuum for the quantum ﬁeld.We exhibit a vacuum state in which the expectation values of the energy and pressure allow an intuitive physical interpretation.We apply this general result to the de Sitter universe.I.INTRODUCTION We ﬁrst consider a charged scalar ﬁeld Φin Minskowski spacetime.Suppose that the electric ﬁeld is E =E z ,and the vector potential is A (t )=−Et z .The wave equation for a charged ﬁeld in a Minkowski universe is given by:[(∂µ−ieA µ)2+m 2]Φ(t,x )=0.(1.1)We can ﬁnd solutions of the form (Fourier mode decomposition)Φ(t,x )=1V k (k z +eEt )|f k (t )|2.(1.4)Therefore in the vacuum state deﬁned by a k |0 =0=b k |0 ,we have j z =0.On the other hand,we know that there are solutions with adiabatic asymptotic behaviour,such that lim t →±∞f k (±)(t )=˜f k (t )with ˜f k (t )=[2ωk (t )]−1/2exp[−i tdt ′ωk (t ′)].(1.5)These two families of solutions {f k (−)}and {f k (+)}are related by a Bogoliubov transformation and they represent two diﬀerent vacua.If our initial vacuum state is |0(−) (adiabatic vacuum at early times),it is easy to show that in the remote future,when the natural choice for a set of adiabatic observers is {fk (+)},these inertial observers woulddetect particle productiongiven by N k (+)|0(−) =[a †k (+)a k (+)+b †−k (+)b −k (+)]|0(−) =0.(1.6)These observers will measure a nonvanishing j z .This is the Schwinger eﬀect which would be completely missed in the T −invariant vacuum state [1].II.SCALAR FIELD IN DE SITTER SPACETIMEWe next consider a scalarﬁeldΦin a de Sitter gravitational background.We use a coordinate system[2]in which the spatial sections have curvatureκ=+1,and the scale factor is a(t)=Z−1cosh(Zt)with u=Zt.The wave equation is[−2+m2+ξR]Φ(t,x)=0.(2.1) In the de Sitter universe this equation is separable and theﬁeldΦcan be written in terms of creation and annihilation operators.The equation of motion for the mode functions is¨f(t)+Ω2k(t)f k(t)=0,(2.2)kwithΩ2k(t)=Z2 γ2+ k+12 sech2(Zt) andγ2=m26 −1III.ENERGY-MOMENTUM TENSORThe most important physical quantity to compute is the expectation value of the energy-momentum tensor in the state deﬁned by the modes f k.We consider the caseξ=1/6and m=0.It can be shown that there is a particular choice of adiabatic observers(those with zeroth adiabatic frequency)for which the energy takes the very simple formǫξ=1/6=12(1+2N k).(3.1)The pressure can be obtained from the conservation equation˙ ǫξ=1/6+3˙a(t)[1]E.Mottola and Y.Kluger,Private communication.[2]N.D.Birrell and P.C.W.Davies,Quantumﬁelds in curved space,(Cambridge University Press,1982).[3]E.Mottola,Phys.Rev.D31,754(1985).[4]M.Gutzwiller,Helv.Phys.Acta29,313(1956).[5]T.S.Bunch,J.Phys.A:Math.Gen.13,1297(1980).[6]Paul R.Anderson and Leonard Parker,Phys.Rev.D36,2963(1987).3。

SIGMA DP3 Merrill摄影机说明书

The lens mounted on the Sigma DP3 Merrill is a medium-telephoto macro lens (equivalent to a 75mm lens on a 35mm SLR) specifically developed for this camera and exemplifyingSigma’s optical engineering expertise. Whether you’re shooting at infinity or the minimum focusing distance, imaging performance is impeccable, and the cropping effect of medium-telephoto macro shooting enables you to capture your vision with frame-filling impact. From macro images to portraiture, this lens offers an artistic palette capable of exquisite delicacy, profundity and narrative power.Each camera in the DP series features a state of the art lens matched to a state-of the art sensor. Now the DP3 Merrill expands your expressive potential.Y ou are the conductor and the Sigma DP series is your orchestra. The DP3 Merrill is a new instrument with which to enrich your photographic repertoire.CONCEPTThe Sigma DP3 traces its lineage back to March 2008 when the very first model in the DP series was born. 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Repeatedly.True high-end performancethat’s always at the ready.Y our hands will recognize the feel ofquality — the finely crafted precision ofa high-end digital SLR housed ina compact body.A fixed-lens compact camera equippedwith a large imaging sensor.It is a category of camera created bySigma and one that Sigma continues torefine with the spirit of a true artisan.From tiny components to the completedproduct, Sigma’s Aizu factory delivers“made in Japan” quality usingstate-of-the-art manufacturing technology.Fuelled by a passion for what really mattersin photography, Sigma’s craftsmanship livesin every detail of the Sigma DP3 Merrill.like using an SLR.Easy-to-grip body designThe lens position is offset to enhance your grip ofthe camera body. This also helps you operate theshutter and command dial with a steady hand.Command dial for intuitive controlQS (quick set) button for favorite functionsInstantly recall ISO sensitivity, white balance and otherfrequently used functions.Handy mode button for ‘scene’ switchingLocated next to the command dial for easy accessand quick selection.3" LCD for high viewabilityHigh-definition 920,000 pixel wide-viewing-angleLCD monitor assists framing and focusing.ACCESSORIESSIGMA DP3 Merrill AccessoriesELECTRONIC FLASH: EF-140 DGA compact flashgun designed exclusivelyfor the DP series and featuringa Guide Number of 14. This flashgunextends the camera’s photographicpossibilities with such features as fill-in flashand full-flash for night photography.UPC Code : 0085126-925703FIL TER DG UV 52mmThe UV filter, does not only reduce ultravioletlight but are also ideal for protecting the lensduring general photography.UPC Code : 0085126-923693FIL TER DG WIDE CIRCULAR PL 52mmCircular Polarizing filters eliminate reflectionsfrom subject surfaces such as glass and waterand deepen the intensity of blue skies.UPC Code : 0085126-923808LENS HOOD: LH3-01The dedicated bayonet lens hood can beattached to block out extraneous light.UPC Code : 0085126-929565LENS HOOD : LH3-01FIL TER DG WIDE CIRCULAR PL 52mmFIL TER DG UV 52mmELECTRONIC FLASH : EF-140 DG2322IMAGE SENSORFormatImage Sensor Size Number of Pixels Aspect RatioFoveon X3® direct image sensor CMOS 23.5 x 15.7mmT otal Pixels: 48 MPEffective Pixels: 46 MP (4,800 x 3,200 x 3)3 : 2Focal Length35mm Equivalent Focal Length Lens F numberNumber of Diaphragm Blades Lens Construction Shooting RangeMaximum Magnification ShootingLENS50mmApprox. 75mm F2.8 ~ F167 Blades10 Elements in 8 Groups22.6cm ~ o o, LIMIT Mode (It is possible to choose Macro, Portrait and Scenery.)1 : 3Approx.Approx.Approx.Approx.Approx.Approx.Approx.Approx.Approx.Approx.Approx.Approx.MB MB MB MBMBMBMBMBMBMBMB MB452412 105.64.2 5 2.72 2.5 1.4 1SD Card, SDHC Card, SDXC Card, Multi Media CardLossless compression RAW data(12-bit, High, Medium, Low), JPEG(High, Medium, Low), RAW+JPEG, Movie(AVI)RAWHigh Medium Low 4,7043,2642,336x x x JPEGHigh4,7044,7044,704x x x Fine Normal Basic :::VGA :Medium 3,2643,2643,264x x x Fine Normal Basic :::Low2,3362,3362,336x x x Fine Normal Basic :::3,1362,1761,5683,1363,1363,1362,1762,1762,1761,5681,5681,568333x x x :::640 x 480 (Image area 640 x 426)RECORDING SYSTEMStorage Media File FormatFile SizeMovie ISO 100 ~ ISO 6400 (1/3 steps for appropriate sensitivity), AUTO: High limit, low limit setting is possible between ISO 100 ~ ISO 6400. When using with flash, it changes depending on the low limit setting.ISO SensitivityISO SENSITIVITY8 types (Auto, Daylight, Shade, Overcast, Incandescent, Fluorescent, Flash, Custom)WHITE BALANCESettings8 types (Standard, Vivid, Neutral, Portrait, Landscape, B&W , Sepia, FOV Classic Blue)COLOR MODESettingsContrast Detection Type9 points select mode, Free move mode (It is possible tochange the size of Focus Frame to Spot, Regular and Large.)Face Detection AF modeShutter release halfway-down position(AF lock can be done by AE lock button from menu setting)Focus Ring TypeAUTOFOCUSAuto Focus Type AF PointFocus Lock Manual FocusEvaluative Metering, Center-Weighted Average Metering, Spot Metering[ P ] Program AE(Program Shift is possible), [ S ] Shutter Speed Priority AE,[ A ] Aperture Priority AE, [ M ] Manual +- 3EV (1/3 stop increments)AE lock buttonAppropriate, under, over; 1/3EV steps up to +- 3EV for appropriate exposureEXPOSURE CONTROLMetering SystemExposure Control SystemExposure Compensation AE LockAuto Bracketing1/2000* - 30sec.(*Depending on the aperture value, shutter speed changes)Shutter SpeedSHUTTERSingle, Continuous, Self Timer (2sec. /10sec.), Interval timer, Unlimited ShootingDrive ModeDRIVE MODETFT color LCD monitor 3.0 inchesApprox. 920,000 PixelsLCD MONITORTypeMonitor Size LCD PixelsPC/IFAUDIO/VIDEOUSB (USB2.0)Video Out (NTSC/PAL), Audio Out (Monaural)INTERFACEEnglish / Japanese / German / French / Spanish / Italian / Chinese (Simplified) / Korean / Russian /Chinese (Traditional) / Nederlands / Polski / Português / Dansk / Svenska / Norsk / SuomiLCD Monitor LanguageMENUPower Battery LifeLi-ion Battery BP-41, Battery Charger BC-41,AC Adapter SAC-5 (with DC Connector CN-11) (Optional)Approx. 97 (+25°c)POWER SOURCE121.5mm/4.8"(W), 66.7mm/2.6"(H), 80.6mm/3.2"(D)400g/14.1oz (without battery and memory card)Dimensions WeightDIMENSIONS AND WEIGHTNote : T o help with the correct and safe of this product, please read the manual carefully first.Copyright© 2013 Sigma Corporation All Rights Reserved.The appearance, specifications and other aspects of this product are subject to change without notice for improvement purposes.SIGMA DP3 Merrill | PRINCIPAL SPECIFICA TIONSFor more information,visit the SIGMA DP3 Merrill website atSIGMA CORPORATION 2-4-16, Kurigi, Asao-ku, Kawasaki-shi, Kanagawa, 215-8530 Japan T el: +81-44-989-7437 Fax: +81-44-989-7448 www.sigma-photo.co.jp。

letter m 的小文章

Letter MIntroductionThe letter “M” is the thirteenth letter in the modern English alphabet. It is a consonant and is frequently used in various words acrossdifferent languages. In this article, we will explore the significance and usage of the letter “M” in various aspects of language, science, and culture.Linguistic Significance of “M”Phonetics and PronunciationThe letter “M” is a bilabial nasal consonant, which means it is produced by closing both lips and releasing air through the nasal passage. It is one of the most common sounds made by humans. In English, “M” is mostly pronounced with a voiced sound at the beginning orwithin words, such as in “moon,” “man,” and “hamster.”Morphology and Word Formation“M” is also frequently used as a morpheme, the smallest grammaticalunit that has a meaning. It can serve as a prefix or a suffix in many words, altering their meaning. For example, the prefix “multi-” denotes multiple or many, such as in “multinational” or “multipurpose.” On the other hand, the suffix “-ism” indicates a belief or practice, as in “capitalism” or “communism.”Mathematical Symbolism of “M”Roman NumeralThe letter “M” represents the value 1,000 in Roman numerals. It is derived from the Latin word “mille,” which means a thousand. Inancient times, this numeral system was widely used for counting and enumeration.Mathematical OperationsIn mathematics, “M” is often used to represent variables or unknown quantities. It can also denote different operations, such as the uppercase sigma symb ol (∑) used in summation, which adds up a series of numbers. Additionally, “M” is often used to represent slopes in linear equations, where it signifies the change in y-values for a given change in x-values.Significance of “M” in ScienceChemistryIn the periodic table of elements, “M” represents the element “Molybdenum.” Molybdenum is a transition metal with various applications in industry, including as an alloying agent in steel production and as a catalyst in chemical reactions.BiologyThe letter “M” has significance in biology as it represents various biological processes and molecules. For instance, “M” is used to denote “mitochondria,” which are the powerhouse of the cell responsible for generating energy. It also signifies “mRNA” (messenger RNA), a vital molecule in the process of protein synthesis.PhysicsIn physics, “M” is often used to denote mass. Mass is a fundamental property of matter that determines an object’s resistance to acceleration. It plays a crucial role in various physical equations, such as Newton’s second law of motion (F = ma), where “m” represents mass.Cultural References and SymbolsRoman Numeral TattooThe letter “M” in Roman numerals, representing 1,000, has become a popular choice for tattoos. People often have their birth year, a significant date, or a meaningful number represented in Roman numerals, adding a personal and aesthetic touch.Logo and BrandingMultiple companies and organizations incorporate the letter “M” into their logos and branding. Notable examples include McDonald’s, with its iconic golden arches forming the letter “M,” and BMW, where the “M” symbolizes their performance division—BMW M.Music and ArtIn the realm of music, the letter “M” holds significance in various ways. For instance, “M” is often associ ated with the musical term “majesty,” indicating a grand and majestic style of performance or composition. Furthermore, “M” is frequently used in notations to indicate a specific metronome marking, representing the tempo or speed of the music.ConclusionThe letter “M” possesses linguistic, mathematical, scientific, and cultural significance. From its phonetic usage in language to its representation of variables in mathematics, “M” plays a multifaceted role in our daily lives. Whether as a symbol of power in a logo or as a meaningful tattoo, “M” holds its place in various aspects of human existence, making it an integral part of our world’s communication, knowledge, and creativity.。

六西格玛指导手册说明书

@RISK and Six-Sigma GuideThis short guide is designed to give you a very brief introduction to Six Sigma, and an overview of the features that @RISK provides to aid your Six Sigma analyses.In today’s competitive business environment, quality is more important than ever. @RISK is the perfect companion for any Six Sigma or quality professional.ContentsWhat is Six Sigma? (2)The Importance of Variation (2)Six Sigma Methodologies (3)Six Sigma / DMAIC (3)Design for Six Sigma (DFSS) (4)Lean or Lean Six Sigma (4)@RISK and Six Sigma (5)@RISK and DMAIC (5)@RISK and Design for Six Sigma (DFSS) (6)@RISK and Lean Six Sigma (6)Using @RISK for Six Sigma (7)RiskSixSigma Property Function (7)Six Sigma Statistics Functions (9)Six Sigma and the Results Summary Window (10)Six Sigma Markers on Graphs (11)Six Sigma Example Models (13)Version 1 - Last Updated 4/17/2020@RISK | Six Sigma Guide What is Six Sigma?Six Sigma is a set of practices to systematically improve processes by reducing process variation and thereby eliminating defects. A defect is defined as nonconformity of a product or service to its specifications. While the particulars of the methodology were originally formulated by Motorola in the mid-1980s, Six Sigma was heavily inspired by six preceding decades of quality improvement methodologies such as quality control, TQM, and Zero Defects. Like its predecessors, Six Sigma asserts the following:•Continuous efforts to reduce variation in process outputs is key to business success•Manufacturing and business processes can be measured, analyzed, improved and controlled •Succeeding at achieving sustained quality improvement requires commitment from the entire organization, particularly from top-level managementSix Sigma is driven by data, and frequently refers to “X” and “Y” variables. X variables are independent input variables that affect the dependent output variables, Y. Six Sigma focuses on identifying and controlling variation in X variables to maximize quality and minimize variation in Y variables.The term Six Sigma (or in symbols, 6σ) is very descriptive.The Greek letter sigma (σ) signifies standard deviation, an important measure of variation. The variation of a process refers to how tightly all outcomes are clustered around the mean. The probability of creating a defect can be estimated and translated into a “Sigma level.” The higher the Sigma level, the better the performance. Six Sigma refers to having six standard deviations between the average of the process center and the closest specification limit or service level. That translates to fewer than 3.4 defects per one million opportunities (DPMO).The cost savings and quality improvements that have resulted from Six Sigma corporate implementations are significant. Motorola has reported billions in savings since implementation in the mid-1980s. Lockheed Martin, GE, Honeywell, and many others have also experienced tremendous benefits from Six Sigma.The Importance of VariationM any Six Sigma practitioners rely on static models that don’t account for inherent uncertainty and variability in their processes or designs. In the quest to maximize quality, it’s vital to consider as many scenarios as possible.That’s where @RISK comes in. @RISK uses Monte Carlo simulation to analyze thousands of different possible outcomes, showing you the likelihood of each occurring. Uncertain factors are defined with probability distribution functions that describe the possible range of values your inputs couldtake. @RISK allows you to define Upper and Lower Specification Limits and Target values for each output, and it includes a wide range of Six Sigma statistics and capability metrics on the outputs.@RISK | Six Sigma Guide@RISK Industrial edition also includes RISKOptimizer, which combines the power of Monte Carlo simulation with genetic algorithm-based optimization. This gives you the ability to tackle optimization problems that have inherent uncertainty, such as:•Resource allocation to minimize cost•Project selection to maximize profit•Optimize process settings to maximize yield or minimize cost•Optimize tolerance allocation to maximize quality•Optimize staffing schedules to maximize serviceSix Sigma Methodologies@RISK can be used in a variety of Six Sigma and related analyses. The three principal areas of analysis are:•Six Sigma / DMAIC•Design for Six Sigma (DFSS)•Lean or Lean Six SigmaEach of these is described in a little more detail below.Six Sigma / DMAICWhen most people refer to Six Sigma, they are in fact referring to the DMAIC methodology. The DMAIC methodology should be used when a product or process is in existence but is not meeting customer specification or is not performing adequately.DMAIC focuses on evolutionary and continuous improvement in manufacturing and services processes, and is almost universally defined as being comprised of five phases - Define, Measure, Analyze, Improve and Control:1. Define the project goals and customer (internal and external Voice of Customer or VOC)requirements2. Measure the process to determine current performance3. Analyze and determine the root cause(s) of the defects4. Improve the process by eliminating defect root causes5. Control future process performance@RISK | Six Sigma Guide Design for Six Sigma (DFSS)DFSS is used to design or re-design a product or service from the ground up. The expected process Sigma level for a DFSS product or service is at least 4.5 (no more than approximately 1 defect per thousand opportunities), but can be 6 Sigma or higher depending on the product. Producing such a low defect level from a product or service launch means that customer expectations and needs (Critical-To-Qualities or CTQs) must be completely understood before a design can be completed and implemented. Successful DFSS programs can reduce unnecessary waste at the planning stage and bring products to market more quickly.Unlike the DMAIC methodology, the steps of DFSS are not universally recognized or defined; almost every company or training organization will define DFSS differently. One popular Design for Six Sigma methodology is called DMADV, and retains the same number of letters, number of phases, and general feel as the DMAIC acronym. The five phases of DMADV are defined as: Define, Measure, Analyze, Design and Verify:1. Define the project goals and customer (internal and external VOC) requirements2. Measure and determine customer needs and specifications; benchmark competitors andindustry3. Analyze the process options to meet the customer needs4. Design (detailed) the process to meet the customer needs5. Verify the design performance and ability to meet customer needsLean or Lean Six Sigma“Lean Six Sigma” is the combination of Lean manufacturing (originally developed by Toyota) and Six Sigma statistical methodologies in a synergistic tool. Lean deals with improving the speed of a process by reducing waste and eliminating non-value added steps. Lean focuses on a customer “pull” strategy, producing only those products demanded with “just in time” delivery. Six Sigma improves performance by focusing on those aspects of a process that are critical to quality from the customer perspective and eliminating variation in that process. Many service organizations, for example, have already begun to blend the higher quality of Six Sigma with the efficiency of Lean into Lean Six Sigma.Lean utilizes “Kaizen events” -- intensive, typically week-long improvement sessions -- to quickly identify improvement opportunities and goes one step further than a traditional process map in its use of value stream mapping. Six Sigma uses the formal DMAIC methodology to bring measurable and repeatable results.Both Lean and Six Sigma are built around the view that businesses are composed of processes that start with customer needs and should end with satisfied customers using your product or service.@RISK | Six Sigma Guide@RISK and Six SigmaWhether it’s in DMIAC, DFSS, or Lean Six Sigma, uncertainty and variability lie at the core of any Six Sigma analysis. @RISK uses Monte Carlo simulation to identify, measure, and root out the causes of variability in your production and service processes. Each of the Six Sigma methodologies can benefit from @RISK throughout the stages of analysis.@RISK and DMAIC@RISK is useful at each stage of the DMAIC process to account for variation and hone in on problem areas in existing products.1. Define. Define your process improvement goals, incorporating customer demand and business strategy. Value-stream mapping, cost estimation, and identification of CTQs (Critical-To-Qualities) are ************************************************************************@RISKzoomsin on CTQs that affect your bottom-line profitability.2. Measure. Measure current performance levels and their variations. Distribution fitting and over 35 probability distributions make defining performance variation accurate. Statistics from @RISK simulations can provide data for comparison against requirements in the Analyze phase.3. Analyze. Analyze to verify relationship and cause of defects, and attempt to ensure that all factors have been considered. Through @RISK simulation, you can be sure all input factors have been considered and all outcomes presented. You can pinpoint the causes of variability and risk with sensitivity and scenario analysis, and analyze tolerance. Use @RISK’s Six Sigma statistics functions to calculate capability metrics which identify gaps between measurements and requirements. Here we see how often products or processes fail and get a sense of reliability.4. Improve. Improve or optimize the process based upon the analysis using techniques like Design of Experiments. Design of Experiments includes the design of all information-gathering exercises where variation is present, whether under the full control of the experimenter or not. Using @RISK simulation, you can test different alternative designs and process changes. @RISK is also used for reliability analysis and – using RISKOptimizer - resource optimization at this stage.5. Control. Control to ensure that any variances are corrected before they result in defects. In the Control stage, you can set up pilot runs to establish process capability, transition to production and thereafter continuously measure the process and institute control mechanisms. @RISK automatically calculates process capability and validates models to make sure that quality standards and customer demands are met.@RISK | Six Sigma Guide @RISK and Design for Six Sigma (DFSS)One of @RISK’s main us es in Six Sigma is with DFSS at the planning stage of a new project. Testing different processes on physical manufacturing or service models or prototypes can be cost prohibitive. @RISK allows engineers to simulate thousands of different outcomes on models without the cost and time associated with physical simulation. @RISK is helpful at each stage of a DFSS implementation in the same way as the DMAIC steps. Using @RISK for DFSS gives engineers the following benefits: •Experiment with different designs / Design of Experiments•Identify CTQs•Predict process capability•Reveal product design constraints•Cost estimation•Project selection – using RISKOptimizer to find the optimal portfolio•Statistical tolerance analysis•Resource allocation – using RISKOptimizer to maximize efficiency@RISK and Lean Six Sigma@RISK is the perfect companion for the synergy of Lean manufacturing and Six Sigma. “Quality only” Six Sigma models may fail when applied to reducing variation in a single process step, or to processes which do not add value to the customer. For example, an extra inspection during the manufacturing process to catch defective units may be recommended by a Six Sigma analysis. The waste of processing defective units is eliminated, but at the expense of adding inspection which is itself waste. In a Lean Six Sigma analysis, @RISK identifies the causes of these failures. Furthermore, @RISK can account for uncertainty in both quality (ppm) and speed (cycle time) metrics.@RISK provides the following benefits in Lean Six Sigma analysis:•Project selection – using RISKOptimizer to find the optimal portfolio•Value stream mapping•Identification of CTQs that drive variation•Process optimization•Uncover and reduce wasteful process steps•Inventory optimization – using RISKOptimizer to minimize costs•Resource allocation – using RISKOptimizer to maximize efficiency@RISK | Six Sigma GuideUsing @RISK for Six Sigma@RISK’s standard simulation capabilities have been enhanced for use in Six Sigma modeling through the addition of four key features. These are:1. The RiskSixSigma property function for entering specification limits and target values forsimulation outputs.2. Six Sigma statistics functions, including process capability indices such as RiskCpk, RiskCpmand others which return Six Sigma statistics on simulation results directly in spreadsheet cells.3. Columns in the Results Summary window that provide Six Sigma statistics on simulationresults in table form.4. Markers on graphs of simulation results that display specification limits and the target value. The standard features of @RISK, such as entering distribution functions, fitting distributions to data, running simulations and performing sensitivity analyses, are also applicable to Six Sigma models.RiskSixSigma Property FunctionIn an @RISK simulation the RiskOutput function identifies a cell in a spreadsheet as a simulation output. A distribution of possible outcomes is generated for every output cell selected. These probability distributions are created by collecting the values calculated for a cell for each iteration of a simulation.When Six Sigma statistics are to be calculated for an output, the RiskSixSigma property function should be entered as an argument to the RiskOutput function. This property function specifies the lower specification limit, upper specification limit, target value, long term shift, and the number of standard deviations for the Six Sigma calculations for an output. These values are used in calculating Six Sigma statistics displayed in the Results window and on graphs for the output. For example:=RiskOutput(“Part Height”,,RiskSixSigma(0.88,0.95,0.915,1.5,6))specifies an LSL of 0.88, a USL of 0.95, target value of 0.915, long term shift of 1.5, and a number of standard deviations of 6 for the output Part Height. You can also use cell referencing in the RiskSixSigma property function.These values are used in calculating Six Sigma statistics displayed in the Results window and as markers on graphs for the output.When @RISK detects a RiskSixSigma property function in an output, it automatically displays the available Six Sigma statistics on the simulation results for the output in the Results Summary window and adds markers for the entered LSL, USL and Target values to graphs of simulation results for the output.You can type the RiskOutput function, together with the RiskSixSigma function, directly into the cell’s formula, or you can have @RISK help you do this using the user interface.@RISK | Six Sigma Guide From the Add Output dialog, click the Settings/Actions button at the bottom of the window and select ‘Sho w Advanced Properties’:The Six Sigma tab of the dialog contains fields for configuring all the options:Clicking the OK button will add the RiskOutput function, together with the RiskSixSigma function, to the ce ll’s formula.The options available in the Six Sigma tab are:Calculate Capability Metrics for This Output - Specifies that capability metrics will be calculated and displayed in reports and graphs for the output. These metrics will use the entered LSL, USL and Target values.LSL, USL and Target - Sets the LSL (Lower Specification Limit), USL (Upper Specification Limit) and Target values for the output.Use Long Term Shift and Shift -Specifies an optional shift for calculation of long-term capability metrics.@RISK | Six Sigma GuideUpper/Lower X Bound - The number of standard deviations to the right or the left of the mean for calculating the upper or lower X-axis values.Six Sigma Statistics Functions@RISK includes a set of Six Sigma statistics functions which can be entered directly into a spreadsheet model to perform Six Sigma calculations. For example, consider the simple model shown below:Cell C15 contains a RiskOutput function with a RiskSixSigma property function:=RiskOutput(C14,,,RiskSixSigma(C4,C5,C6,0,6)) +RiskNormal(C10,C11)The green cells in column C contain the following Six Sigma statistics functions:=RiskCpk(C15)=RiskPNC(C15)=RiskDPM(C15)These statistics functions, like other @RISK statistics functions, show relevant results only after a simulation has been run. They rely on the parameter values (LSL, USL, and so on) in the RiskSixSigma property function in cell C15 for their values.Note also in this screenshot how the graph of the output in C15 shows the LSL, Target, and USL as markers. These markers also rely on information provided by the RiskSixSigma property function in cell C15.@RISK | Six Sigma Guide The complete list of Six Sigma statistic function can be found on @RISK’s Function menu:Six Sigma and the Results Summary Window@RISK’s Results Summary window summarizes the results of your model and displays thumbnail graphs and summary statistics for your simulated output cells and input distributions. When @RISK detects a RiskSixSigma property function in an output, it also will automatically display the available Six Sigma statistics for the simulation results for any output that utilizes Six Sigma.@RISK | Six Sigma GuideThis table can be exported to Excel, the printer, or a PDF file by clicking the Export button in the bottom right corner of the window.Clicking the Table Settings item from the Settings/Actions menu displays a dialog from which you can customize which statistics to display in the window:Six Sigma Markers on GraphsWhen @RISK detects a RiskSixSigma property function in an output, it adds markers for the LSL, USL and Target values to graphs of simulation results for the output. It also adds Six Sigma statistics to the statistics grid to the right of the graph.@RISK | Six Sigma GuideYou can configure the display of both the markers and the grid from by choosing the Graph Formatting Options item on the Settings/Action menu.@RISK | Six Sigma GuideSix Sigma Example ModelsA number of examples models that demonstrate the use o f Six Sigma can be found on Palisade’s website. Please visit https:///models/ and search for Six Sigma (results pictured below).。

苦学英文之英文标点符号翻译大全

英文标点符号翻译大全+plus加号；正号-minus减号；负号±plus or minus正负号×is multiplied by乘号÷is divided by除号＝is equal to等于号≠is not equal to不等于号≡is equivalent to全等于号≌is equal to or approximately equal to等于或约等于号≈is approximately equal to约等于号＜is less than小于号＞is more than大于号≮is not less than不小于号≯is not more than不大于号≤is less than or equal to小于或等于号≥is more than or equal to大于或等于号%per cent百分之…‰per mill千分之…∞infinity无限大号∝varies as与…成比例√(square) root平方根∵since; because因为∴hence所以∷equals, as (proportion)等于，成比例∠angle角≲semicircle半圆≰circle圆○circumference圆周πpi 圆周率△triangle三角形≱perpendicular to垂直于∪union of并，合集∩intersection of 交，通集∫the integral of …的积分∑(sigma) summation of总和°degree度′minute分″second秒℃Celsius system摄氏度{open brace, open curly左花括号}close brace, close curly右花括号(open parenthesis, open paren左圆括号)close parenthesis, close paren右圆括号() brackets/ parentheses括号[open bracket 左方括号]close bracket 右方括号[] square brackets方括号.period, dot句号，点|vertical bar, vertical virgule竖线&ampersand, and, reference, ref和，引用*asterisk, multiply, star, pointer星号，乘号，星，指针/slash, divide, oblique 斜线，斜杠，除号//slash-slash, comment 双斜线，注释符#pound井号\backslash, sometimes escape反斜线转义符，有时表示转义符或续行符~tilde波浪符full stop句号,comma逗号:colon冒号;semicolon分号?question mark问号!exclamation mark (英式英语) exclamation point (美式英语)'apostrophe撇号-hyphen连字号-- dash 破折号...dots/ ellipsis省略号"single quotation marks 单引号""double quotation marks 双引号‖parallel 双线号&ampersand = and～swung dash 代字号§section; division 分节号→arrow 箭号；参见号词汇询价make an inquiry报价quotation报/发盘offer底盘floor offer实/虚盘firm/non-firm offer开/收盘opening/closing price现/期货价spot/forward price还盘counter-offer回佣return commission到岸价C.I.F.（即Cost, Insurance and Freight)到岸加佣金价C.I.F.C.（即Cost, Insurance, Freight and Commission)现货spot goods库存有限limited stock批发价wholesale price零售价retail price净利润net profit定金down payment分期付款payment by installment现金结算cash settlement 信用证结算payment by letter of credit(L/C)股东shareholder; stockholder我方on our part双赢战略win-win strategy中止合同terminate the contract提出索赔lodge a claim要求赔偿损失claim for a compensation of the loss/damage贸易索赔business claim补偿贸易compenstion trade商品交易会Commodities Fair经营范围line/scope of business独家经销代理exclusive selling agency市场准入market access机床machine tools汽车零部件auto parts电子商务e-commerce; e-business请给我一个有效期为90天的C.I.F.报价，目的港为洛杉矶，报价含5%的佣金。

北京工业大学《六西格玛管理》考试参考答案

北京工业大学2015级硕士研究生“六西格玛管理” 课程考试试题一、叙述题(40分，每小题5分)1. 六西格玛的定义；答：西格玛是一套系统的业务改进方法体系，是旨在持续改进企业业务流程，实现客户满意的管理方法。

它通过系统地、集成地采用业务改进流程，实现无缺陷的过程设计，并对现有过程进行过程定义(Define)、测量(Measure)、分析(Analyze)、改进(Improve)、控制(Control)，简称DMAIC流程，消除过程缺陷和无价值作业，从而提高质量和服务、降低成本、缩短运转周期，达到客户完全满意，增强企业竞争力。

2. 试验设计的三个基本原理；答：试验设计的三个基本原理是重复化，随机化以及区组化（1）重复化：是基本试验的重复进行。

重复有两条重要的性质。

第一，允许试验者得到试验误差的一个估计量。

这个误差的估计量成为确定数据的观察差是否是统计上的试验差的基本度量单位。

第二，如果样本均值用作为试验中一个因素的效应的估计量，则重复允许试验者求得这一效应的更为精确的估计量。

（2）随机化：是指试验材料的分配和试验的各个试验进行的次序，都是随机地确定的。

（3）区组化：是用来提高试验的精确度的一种方法。

3. 流程图的作用；答：流程图是将过程（如工艺过程、检验过程、质量改进过程等）的步骤用图的形式表示出来，通过过程中各步骤间关系的研究，能够发现故障或问题存在的潜在原因。

4. 因果图的作用；答：(1)分析因果关系(2)表达因果关系(3)通过识别症状、分析原因，寻找措施、促进问题解决5. 系统图法的主要用途答：系统图法是把要实现的目的、需要采取的措施或手段，系统的展开分析，并绘制成图，以明确问题的重点，并寻找最佳手段或措施。

在质量管理活动中，下面几个方面经常用到系统分析图法：①在开发新产品中，将满足用户要求的设计质量进行系统地展开；②在质量目标管理中，将目标层层分解和系统地展开，使之落实到各个单位；③在建立质量保证体系中，可将各部门的质量职能展开，进一步开展质量保证活动；④在处理量、本、利之间的关系及制订相应措施时，可用系统图法分析并找出重点措施；⑤在减少不良品方面，有利于找出主要原因，采取有效措施。

anhydrous for analysis emsure -回复

anhydrous for analysis emsure -回复Anhydrous for Analysis EMSURE: Understanding Its Importance and ApplicationsIntroduction:Anhydrous for Analysis EMSURE is a high-quality reagent widely used in various scientific disciplines and industries. It plays a crucial role in ensuring accurate and reliable analytical results. In this article, we will explore in detail the significance, properties, and applications of Anhydrous for Analysis EMSURE, thereby providing a comprehensive understanding of this essential reagent.1. What is Anhydrous for Analysis EMSURE?Anhydrous for Analysis EMSURE is a term used to describe a broad range of reagents that are completely free from water molecules. These reagents are produced using advanced techniques to remove any moisture content, ensuring maximum stability and purity. Anhydrous for Analysis EMSURE is typically available in ultra-pure forms, meeting the highest quality standards demanded by analytical laboratories.2. Importance of Anhydrous for Analysis EMSURE:2.1. Eliminating Water Interference:Water is a common impurity in many chemicals used in analytical processes. However, the presence of water can interfere with various reactions and measurements, leading to inaccurate results. Anhydrous for Analysis EMSURE eliminates this interference, allowing for precise and reliable analysis.2.2. Enhanced Stability:Water can initiate degradation processes in certain substances, affecting their stability over time. Anhydrous for Analysis EMSURE, being entirely free from water, exhibits superior stability and prolonged shelf life. This property is especially critical for long-term storage of reagents and standards.2.3. Prevention of Hydrate Formation:Certain compounds readily react with water, forming hydrates—a chemically combined form where water molecules are incorporated into the substance's crystal lattice. Anhydrous for Analysis EMSURE prevents hydrate formation, maintaining the integrity of thecompound and ensuring accurate analysis.3. Properties of Anhydrous for Analysis EMSURE:3.1. Low Water Content:Anhydrous for Analysis EMSURE reagents typically have an extremely low moisture content, often in the range of parts per million (ppm) or below. This ensures minimal water-related interference during analytical procedures.3.2. High Purity:To meet the stringent requirements of analytical applications, Anhydrous for Analysis EMSURE reagents are manufactured to possess high purity levels. They undergo rigorous quality control measures, including multiple purification steps, to eliminate impurities that could affect the accuracy of analytical results.3.3. Traceable Certification:Anhydrous for Analysis EMSURE reagents are accompanied by comprehensive certificates of analysis, detailing the quality, purity, and conformity of the product. These certificates provide traceability and help maintain consistency in analytical procedures.4. Applications of Anhydrous for Analysis EMSURE:4.1. Chemical Analysis:Anhydrous for Analysis EMSURE reagents are widely used in various chemical analyses, including titrations, spectrophotometry, chromatography, and atomic absorption spectroscopy. Their water-free nature ensures accurate measurements and consistent results.4.2. Pharmaceutical Industry:In the pharmaceutical industry, Anhydrous for Analysis EMSURE is invaluable for conducting quality control tests, formulation development, and stability studies. It helps ensure the purity and stability of drug substances and excipients, thus contributing to the production of safe and effective medications.4.3. Food and Beverage Industry:Anhydrous for Analysis EMSURE reagents find extensive utility in the food and beverage industry. They are employed for the analysis of food components, additives, and contaminants, ensuring compliance with regulatory standards and ensuring consumersafety.4.4. Environmental Analysis:In environmental analysis, Anhydrous for Analysis EMSURE reagents aid in monitoring pollution levels, assessing the quality of water and air, and investigating the impact of pollutants on the environment. The absence of water interference allows for precise measurements and reliable data.5. Conclusion:Anhydrous for Analysis EMSURE is an indispensable reagent that plays a vital role in ensuring accurate and reliable analysis across various scientific disciplines and industries. Its ability to eliminate water interference, enhance stability, and prevent hydrate formation makes it a preferred choice for a wide range of applications. By understanding the significance and properties of Anhydrous for Analysis EMSURE, researchers and analysts can confidently employ this high-quality reagent to obtain precise and consistent results.。

六西格玛管理概述英文版

© 1994 Six Sigma Academy
• Seven articles lost per hour
• One unsafe minute every seven months
• 1.7 incorrect operations per week • One short or long landing every
five years
• 68 wrong prescriptions per year
S SIX IGMA 2
What is Six Sigma?
S SIX IGMA
• Bottom line: Six Sigma
– Defines the goals of the business – Defines performance metrics that tie to the
business goals – Identifies, using performance metrics, projects that
– Customer focus – Breakthrough improvement – Continuous improvement – People Involvement
© 1999 Sigma Consultants, L.L.C. Revision 2.0 – October 1, 2000
Six Sigma Focus
S SIX IGMA
• Delighting the customer through flawless execution
• Rapid breakthrough improvement • Advanced breakthrough tools that work • Positive and deep culture change • Real financial results that impact the bottom

小学下册第5次英语第三单元真题试卷(含答案)

小学下册英语第三单元真题试卷(含答案)英语试题一、综合题(本题有50小题，每小题1分，共100分.每小题不选、错误，均不给分)1 The flowers are ___. (beautiful)2 I drink _____ (water/coffee) with lunch.3 My mom makes _____ for dinner. (pasta)4 We have a ______ (精彩的) plan for the weekend.5 The sun is very _____ (热).6 My dad _____ a new car last week. (bought)7 The boy likes to play ________.8 What do you drink in the morning?A. JuiceB. SandC. GrassD. Stone答案: A9 What is the capital of Iceland?A. ReykjavikB. OsloC. HelsinkiD. Stockholm答案: A10 What do you call a small, furry animal that hops?A. CatB. RabbitC. DogD. Mouse11 The __________ (树皮) protects the tree from harm.12 I enjoy playing ______ in the playground.13 The cat chases after a _____ fluttering moth.14 What do we call a scientist who studies the climate?A. ClimatologistB. MeteorologistC. GeologistD. Ecologist答案: A15 Gravity helps keep the planets in ______.16 The main gas present in the Earth's atmosphere is _______.17 He __________ breakfast every morning.18 What is the name of the story about a boy who never grows up?A. Alice in WonderlandB. Peter PanC. The Little MermaidD. Pinocchio19 My mom enjoys __________ (锻炼) and keeping fit.20 The ________ likes to swim in the pool.21 My favorite holiday is ______ (春节) because we celebrate with family and have delicious ______ (食物).22 A manatee is gentle and loves ______ (水).23 I think it’s important to ________ (保持好奇心).24 The teacher is very ___ (kind).25 What is the name of the famous river in China?A. YangtzeB. MekongC. GangesD. Indus答案:A26 What do you call an animal that hunts for food?A. PredatorB. PreyC. ScavengerD. Herbivore答案: A. Predator27 Which of these is a dessert?A. SoupB. SaladC. Ice CreamD. Rice答案: C28 The __________ is a large expanse of open water.29 The process of ______ contributes to soil formation.30 The __________ (历史的传递方式) affect comprehension.31 What is the name of the phenomenon where the moon blocks the sun?A. Solar eclipseB. Lunar eclipseC. SupermoonD. Blood moon答案: A32 Which insect makes honey?A. AntB. ButterflyC. BeeD. Fly答案: C33 The __________ (历史的交汇) creates new possibilities.34 What do we call the process of taking in food and digesting it?A. IngestionB. AbsorptionC. DigestionD. Metabolism答案:A35 My ________ (玩具) is full of potential.36 The ______ (小鹿) watches cautiously from the edge of the ______ (树林).37 What is the name of the rabbit in "Alice in Wonderland"?A. White RabbitB. Peter RabbitC. ThumperD. Roger Rabbit答案:A38 His favorite book is about a ________.39 The snow is _______ (soft) and white.40 What do we call the act of jumping from an airplane and free-falling before opening a parachute?A. SkydivingB. Bungee JumpingC. ParaglidingD. Base Jumping答案：A41 Community gardens promote ______ (邻里关系).42 Mount Everest is located in the __________ mountains.43 I love to _______ (进行) science experiments.44 The __________ is a famous national park.45 I like to draw ______ (漫画) in my free time. It allows me to express my creativity and tell funny ______ (故事).46 The sun rises in the ______ (east).47 It is ___ outside today. (sunny)48 The capital of Jordan is ________ (安曼).49 The sun is ___ (setting/rising) in the evening.50 Caves are formed by the erosion of ______ rock.51 What do we call the spiral-shaped galaxies?A. Elliptical GalaxiesB. Irregular GalaxiesC. Spiral GalaxiesD. Lenticular Galaxies52 I like to draw ______ in my spare time.53 I love _______ (春天) because of the flowers.54 What do we call a person who helps sick people?A. TeacherB. DoctorC. EngineerD. Lawyer55 The _____ (山) is beautiful.56 A ______ is a measurement of how much a substance can dissolve.57 What is the name of the popular social media platform for sharing photos?A. FacebookB. InstagramC. TwitterD. TikTok答案: B58 The teacher gives _____ (作业) every week.59 How many days are there in a week?A. 5B. 6C. 7D. 8答案: C60 A chemical that can donate protons is called an ______.61 The capital of Venezuela is __________.62 A plant’s ______ (高度) can vary greatly from species to species.63 How do you say "hello" in Spanish?A. BonjourB. HalloC. HolaD. Ciao答案:C64 What is the color of a typical watermelon?A. YellowB. GreenC. RedD. Blue答案:C65 We have a ______ (精彩的) event planned for next month.66 What is the name of the famous tower in Paris, France?A. Big BenB. Leaning Tower of PisaC. Eiffel TowerD. CN Tower答案:C67 The Earth's crust is divided into ______ sections.68 A ________ (海鸥) flies over the ocean looking for food.69 A ______ (温室) helps protect plants from harsh weather.70 What is the capital of the United Arab Emirates?b. Abu Dhabic. Sharjahd. Ajman答案:b71 What do you call a story passed down from generation to generation?A. MythB. NovelC. BiographyD. Fairytale答案:A72 The _______ is the amount of space occupied by an object.73 I want to _____ (become/learn) an artist.74 The rock cycle illustrates how rocks can change from one form to another over ______.75 We can _______ a treasure hunt.76 The ________ (sculpture) is made of stone.77 Which tool is used for measuring temperature?A. BarometerB. ThermometerC. RulerD. Stopwatch答案: B78 What is the color of a ripe banana?A. GreenC. BrownD. Red答案:B79 I _____ (love) chocolate.80 The _____ (花蕾) opens to reveal blossoms.81 The ________ (discussion) was productive.82 My sister is a ______. She studies very hard.83 I like to watch ___ (sports) on TV.84 What is the opposite of "happy"?A. ExcitedB. JoyfulC. SadD. Angry85 The ______ is known for her artistic flair.86 My grandmother loves to __________. (织毛衣)87 The ______ writes poems and songs.88 My cat loves to sleep in a _______ (阳光) spot.89 I can ________ my name.90 The bird watches me from the _______ (鸟从_______上看着我).91 The _____ (field) is green.92 The chemical formula for potassium sulfite is _______.93 The _____ (plantation) produces coffee beans.94 What is 18 ÷ 2?A. 6B. 8C. 9D. 1095 The chemical formula for isopropyl alcohol is ______.96 The city of Muscat is the capital of _______.97 What is the name of the chemical element with the symbol O?A. OxygenB. GoldC. SilverD. Iron答案: A. Oxygen98 a Desert is mainly located in __________. (非洲) The Saha99 What is the chemical symbol for potassium?A. KB. KrC. PD. Pt100 A bumblebee is important for _______ (授粉).。

绿带培训(1)

第二天(Variance Reduction):- Concept of precision 和accuracy How does it link to the mean 和标准偏差 Precision 和accuracy example (i.e. Selection of fund manager)Statapult exercise I-----------------------------------Break----------------------------------------------- Introducing concept of variance reduction (i.e. PF/CE/CNX/FMEA/SOP)- Variance reduction 脑力风暴exercise for Statapult Statapult exercise II Computation of Statapult exercise result after variance reduction Discussion of variance contributors- 第二天wrap upDAY 3 (Measure Phase): Recap of statistical terminology Histogram 和a normal data Construction of histogram Transformation of data Calculate Cp, Cpk from non-normal data-----------------------------------Break----------------------------------------------来自2020/8/9 Sunday

A Lefschetz formula for flows

A LEFSCHETZ FORMULA FOR FLOWS
2
ﬁrst glance to be replaced by the Lefschetz index indL (c) of the Poincar´ e map around the orbit, but it turns out that one has to take into account the multiplicity µ(c) so we deﬁne the Fuller index to be indF (c) := indL (c)/µ(c). Our local side would then be the sum over all closed orbits of indF (c)tr ϕ(c). Unfortunately this sum does not converge. So we have to replace it by a zeta-regularized version ˆ c indF (c)tr (Φl(c) | E ). D. Fried [6] has shown that in case of geodesic ﬂows on hyperbolic spaces we have an identity − ˆ
c
indF (c)tr (Φl(c) | E ) = log τ (E )
mod 2πiZ,
where τ (E ) is the analytic torsion of the bundle E . This formula is an analogue of the Hopf formula but not of the Atiyah-Bott-Lefschetz formula since on the global side the presence of the ﬂow is not visible. In this paper we present an analogue of (1). Let H be the generator of the ﬂow. We deﬁne a twisted version H . (Cϕ ) of the tangential cohomology of the stable foliation. Then our Lefschetz formula is

Six Sigma Quality 六西格玛理论

Six Sigma QualityThe Definition of The Theory6Sigma(6σ)is the Greek letter used to designate the estimated standard deviation or variation in a process. This theory is a kind of quality management aimed at pursuing zero of defect production and improving customer satisfaction. 6σ management theory not only focuses on the quality of production and service, but also concerned about the process of improvement. 6σ is a target, this level of quality means that in all of the processes and results,99.99966% is not defective. That is, less than 3.4 defects or mistakes per million. It is evolved from Total Quality Management( TQM) and becomes the statistical tools to analyze the causes of product defects. The higher the “sigma level”, the l ower the amount of variation.) 6σ is the highly standard process of assisting companies focus on developing and providing the nearly perfect products and services.The History of 6σThe 6σ concept- as the quality management concept, originally came up and developed by Bill Smith( Motorola. Inc, 1986.) It was formed and put into practice. Three years later, the Motorola got great success: the product failure rate decreased from 6210 per million(about 4σ)to 32 per million.(about 5.5σ). General Electric Company make the highly effective strategy into management philosophy and practice, and eventually form the corporate culture. At present, 6σ management theory is spreading fast among industries and many companies.The characters of 6σAs a continuous method of quality improvement, 6 sigma theory has its special characters:1, Highly focus on the customer requirement6σ theory centers on customer, it pays much attention on the need of customers.2, Based on statistic dataStatistic data is an important tool to measure 6σ. The numbers, data are useful for making decisions and supervising the error rate of product.3, Highly concern about business processesThe traditional quantity management just focus on the results, not seriously deal with the Processes are as important as results. After sales of service and repair cause more cost of companies and waster of the available resources.4, No border cooperationNo border cooperation is a secret of GE to be success. 6σ management widens the Opportunities of cooperate, eliminates barriers between high and lower levels at the Same or different departments.The advantages of 6σ1, Improve the ability of corporate management2, Save the costs of business operations.3, increase customer value and satisfaction4, enhance service levels.5, form a positive corporate cultureThe levels of 6σ6σ = 3.4 defects per million, equals almost perfect management, very strong competitive and loyal and regular customer.5σ = 230 defects per million, equals excellent management, strong competitive and regular customer.4σ = 6,210 defects per million, equals good management and operational ability and have some satisfying customer.3σ = 66,800 defects per million, equals ordinary management and lack of competitive.2σ = 308,000 defects per million, equals one third resources of corporate waste everyday.1σ = 690,000 defects per million, equals two third of business go wrong, the corporate can not survive.The Organizational structure of 6σ Personnel Management6σ management need s a style of rational and efficient structure to make sure the business can be done successfully. In the past ,almost 80% implementer of TQM got failed. The biggest reason of failures is that the management lacks of a personnel organization.(1) 6σ Management CommitteeThe committee is the highest leadership agency. Its main duty is setting kinds of positions at the beginning stage, distributing resources and instruct the projects. When the teams meet some barriers, the members assist them to work out.(2) ExecutivesThis position is very essential for the whole processes. The detailed responsibilities are: set goals, directions and scope for projects; coordinate resources for each terms needs and requirement; strengthen communication between each project team.（3）Black BeltBlack belt are the backbone of reform. They are voted out from internal selection with Six Sigma Black Belt Certification(CSSBB). They take charge of their programs and training the green belt. They could demonstrate teamwork ability and take their responsibilities seriously.(4 ) Green BeltGreen Belt take charge of some little difficult teamwork. Their training includes Program Management, Quality Management tools and date analysis, which are involved with 6σmanagement theory.The Application of 6σ: D-M-A-I-CDMAIC means the five stages when people implement a program. The five letters are the abbreviation of Define – Measure – Analyze – Improve – Control. DefineDefine the definite problems, know the target of the project and hear the feedback from customer.MeasureMeasure the process of core business and collect operating data. AnalysisAnalysis and understand which methods and tools can use to reduce the current sales to target sales. Take advantage of statistic tools to guide analysis. Make sure the causes of defect.ImprovementImprove the current processes based on the analysis data. Improve the operation faster and more efficient.ControlControl the further stages of process, reduce the defect rate. Keep the productivity at a new level.Citation/certification/six-sigma//certification/six-sigma-green-belt//wiki/Six_Sigma#DMAIC。

数学符号英文

数学符号英文Mathematical Symbols in EnglishMathematics is a universal language, and just like any language, it has its own set of symbols to represent various mathematical concepts and operations. These symbols are essential for communication and understanding within the field of mathematics. In this article, we will explore some commonly used mathematical symbols in English.1. Basic Arithmetic Symbols1.1 Addition (+): This symbol represents the operation of addition, where two or more quantities are combined to form a sum. For example, 2 + 3 = 5.1.2 Subtraction (-): The subtraction symbol is used to subtract one quantity from another. For instance, 7 - 4 = 3.1.3 Multiplication (×): Multiplication is denoted by the symbol '×.' It represents repeated addition of the same quantity. For example, 3 × 4 = 12.1.4 Division (÷): Division is indicated by the symbol '÷.' It represents splitting a quantity into equal parts. For instance, 10 ÷ 2 = 5.2. Equality and Inequality Symbols2.1 Equality (=): The equality symbol is used to indicate that two quantities are equal. For example, 2 + 3 = 5.2.2 Greater Than (>): The greater than symbol compares two quantities and indicates that the left quantity is greater than the right one. For instance, 7 > 4.2.3 Less Than (<): The less than symbol compares two quantities and indicates that the left quantity is smaller than the right one. For example, 3 < 5.3. Mathematical Operations Symbols3.1 Square Root (√): The square root symbol is used to represent finding th e square root of a number. For example, √25 = 5.3.2 Exponentiation (^): The exponentiation symbol denotes raising a number to a power. For instance, 2^3 = 8.3.3 Summation (∑): The summation symbol represents the sum of a series of terms. For example, ∑(1 to 5) = 1+2+3+4+5 = 15.3.4 Integral (∫): The integral symbol is used to represent the process of finding the area under a curve. For example, ∫(0 to 1) f(x) dx.4. Sets and Logic Symbols4.1 Union (∪): The union symbol denotes combining elements from two or more sets. For instance, A ∪ B represents the union of sets A and B.4.2 Intersection (∩): The intersection symbol is used to indicate the common elements between two or more sets. For example, A ∩ B represents the intersection of sets A and B.4.3 Negation (¬): The negation symbol represents the opposite or non-existence of a condition. For instance, ¬P denotes the negation of statement P.4.4 Implication (→): The implication symbol is used to show the logical consequence between two statements. For ex ample, P → Q represents if statement P is true, then statement Q is also true.5. Greek LettersIn addition to the English alphabet, Greek letters are commonly used as symbols in mathematics. Some frequently used Greek letters include:- Alpha (α)- Beta (β)- Gamma (γ)- Delta (δ)- Sigma (σ)- Pi (π)- Omega (ω)These Greek letters often represent variables, constants, or special mathematical functions.In conclusion, mathematical symbols in English play a crucial role in expressing mathematical ideas and operations. By understanding and correctly utilizing these symbols, mathematicians and students can effectively communicate and solve complex mathematical problems.。