Stabilized Kuramoto-Sivashinsky equation A useful model for secondary instabilities and rel

KitAlysis™ Suzuki-Miyaura Cross-CouplingReaction Screening KitCatalog Number KITALYSIS-SMStore at Room TemperatureTECHNICAL BULLETINProduct DescriptionThe KitAlysis™ Suzuki-Miyaura Cross-Coupling Reaction Screening Kit enables chemists to quickly and efficiently screen conditions for the coupling of aryl halides with organoboronic acids.Quick identification of optimal conditions allows faster scale-up of the desired synthetic transformation. All required chemicals come pre-weighed for ease of use.Each Suzuki-Miyaura Cross-Coupling Reaction Screening Kit contains 4 individual screening sets. Each screening set has components to run 24 unique Suzuki-Miyaura reactions:∙10 μmol (het)aryl halide (user supplied)∙15–20 μmol (het)aryl boronate (user supplied)∙ 1 μmol catalyst per vial (1:1 Pd:Ligand); 6 in this system∙30 μmol base (1.5 M K3PO4)∙0.1 M solvent concentration (DMAc, Toluene, n-butanol, and THF) Access /kitalysis for detailed step-by-step Protocols and Scale-up Guide. The Protocols include an introductory video, instructions for the Labware, and work-up/internal standard addition procedures.A downloadable spreadsheet for stock solution recipes for each screening set is available from the step-by-step Protocol.The spreadsheet provides:∙Calculations for substrate recipes based on the molecular weight of user supplied substrates∙Quick directions for experienced users∙Print button allows you to take experimental recipes to the lab∙Contains all experimental information, which can easily be saved as a pdf file and appended toelectronic laboratory notebooksScreening Kit Components∙Four screening sets individually sealed under nitrogen in mylar bags. Each bag includes 24 glass vials loaded with stir bars and pre-weighedcatalysts (1 μmol), see Figures 1 and 2∙Ampule boxes containing degassed, anhydrous liquids:4 ⨯ 2 mL Toluene4 ⨯ 2 mL THF4 ⨯ 2 mL n-Butanol4 ⨯ 2 mL DMAc4 ⨯ 2 mL aq. 1.5 M K3PO4∙Biphenyl Internal Standard (15.4 mg)∙ 4 New KitAlysis 24-Well Reaction BlockReplacement Films∙ 4 substrate stock solution vials with stir bars2 Figure 2. CatalystsFigure 3.Positions of CatalystsSuggested Equipment (Not Provided)∙ 96-well plate for automated HPLC analysis(Catalog Number Z688711)∙ 96-well plate cap mat for automated HPLC analysis(Catalog Number C1111)∙ 10–100 μL pipette (Catalog Number Z683809) ∙ 2–200 μL pipette tip refill (Catalog NumberZ640220)∙ 100–1,000 μL pipette (Catalog Number Z683825) ∙ 50–1,000 μL pipette tip refill (Catalog NumberZ640247)∙ 2-inch needles for ease of ampule solventextraction (Catalog Number Z192570)∙ 1 mL syringe for accurate solvent volume extractionfrom ampule (Catalog Number Z230723)Specially designed KitAlysis Labware allows screening set up in a hood without the use of a glove box.∙ KitAlysis Benchtop Inertion Box (Catalog NumberZ742064)∙ KitAlysis 24-well Reaction Block and Screwdriver(Catalog Number Z742107)∙ KitAlysis 24-well Reaction Block and Intertion BoxStarter Set (Catalog Number Z742108)Instructions for the Labware can be accessed in the Articles section of the website product display page for the kit.Precautions and DisclaimerThis product is for R&D use only, not for drug,household, or other uses. Please consult the Safety Data Sheet for information regarding hazards and safe handling practices.Storage/StabilityStore the kit at room temperature.KitAlysis is a trademark of Sigma-Aldrich Co. LLC.DR,MAM 08/16-2©2016 Sigma-Aldrich Co. LLC. All rights reserved. SIGMA-ALDRICH is a trademark of Sigma-Aldrich Co. LLC, registered in the US and other countries. Aldrich brand products are sold through Sigma-Aldrich, Inc. Purchaser must determine the suitability of the product(s) for theirparticular use. Additional terms and conditions may apply. Please see product information on the Sigma-Aldrich website at and/or on the reverse side of the invoice or packing slip.。

Integrating Factor Methods asExponential IntegratorsBorislav V.MinchevDepartment of Mathematical Science,NTNU,7491Trondheim,NorwayBorko.Minchev@ii.uib.noAbstract.Recently a lot of effort has been placed in the constructionand implementation of a class of methods called exponential integrators.These methods are preferable when one has to deal with stiffand highlyoscillatory semilinear problems,which often arise after spatial discretiza-tion of Partial Differential Equations(PDEs).The main idea behind themethods is to use the exponential and some closely related functionsinside the numerical scheme.In this note we show that the integratingfactor methods,introduced by Lawson in1967,are also examples of expo-nential integrators with very special structure for the related exponentialfunctions.In order to prove this relation,we use the approach based onbi-coloured rooted trees and B-series.We also show under what condi-tions every bi-coloured rooted tree can be express as a linear combinationof standard non-coloured rooted trees.1IntroductionRealistic models of many physical processes require effective numer-ical solvers for a special class of partial differential equations,which after semidiscretization in space can be written in the following formu =Lu+N(u(t)),u(t0)=u0,(1.1)where u:R→R d,L∈R d×d,N:R d→R d and d is a discretiza-tion parameter equal to the number of spatial grid points.Several interesting problems can be brought to this form.Examples are Allen-Cahn,Burgers,Cahn-Hilliard,Kuramoto-Sivashinsky,Navier-Stokes,Swift-Hohenberg,nonlinear Scr¨o dinger equations.Typically the linear part of the problem will be stiffand the nonlinear part will be nonstiff.Many numerical integrators have been developed to overcome the phenomenon of stiffness.Exponential integrators was introduced in the early sixties as an alternative approach for solving2stiffsystems.The main idea behind these methods is to integrate exactly the linear part of the problem and then use an appropriate approximation of the nonlinear part.Thus the exponential function, and functions which are closely related to the exponential function, appear in the format of the method.This was the reason why,until recently,these methods have not been regarded as practical.The latest achievements in thefield of computing approximations to the matrix exponential,have provided a new interest in the construction and implementation of exponential integrators[2,3,6,7,9].The main requirements imposed on the functions,which appear in the format of an exponential integrator are,to be analytic,map the spectrum of L into a bounded region in C and can be computed exactly or up to arbitrary high order cheaply.Suppose that,for all l∈N andλ∈R,the operatorsφ[l](λ):R d×d→R d×d satisfy the above conditions and can be expanded in the formφ[l](λ)(hL)= j≥0φ[l]j(λ)(hL)j.Theφ[l]functions,which are used in practice are associated with the so called Exponential Time Differencing methods[3,4,10,11], and can be written explicitly asφ[l](λ)(hL)=(λhL)−l eλhL−l−1 k=0(λhL)k k! .(1.2)If h represents the stepsize and U i denotes the internal stage ap-proximation of the exact solution for i=1,2,...,s then the compu-tations performed in step number n of an exponential Runge–Kutta (RK)method are related by the equationsU i=sj=1s l=1α[l]ijφ[l](c i)(hL)hN(U j)+e c i hL u n−1,u n=sj=1s l=1β[l]jφ[l](1)(hL)hN(U j)+e hL u n−1,(1.3)whereα[l]ij andβ[l]j are the parameters of the method and the vector c=(c1,c2...,c s)T is the abscissae vector.Ifα[l]ij=0for all j≥i3 the method is explicit and implicit otherwise.Alternatively the com-putations performed in step number n can be represented in a more Runge–Kutta type formulation as followscα[1]α[2]···α[s],β[1]Tβ[2]T···β[s]Twhere each element in row number i of the matrixα[l]is multiplied byφ[l](c i)(hL)and each element in the vectorβ[l]T is multiplied by φ[l](1)(hL).The resulting matrices are then added in a component by component sense.Other important class of methods which are also used for solving the semilinear problem(1.1)are the Integrating Factor(IF)methods. The idea behind these methods goes back to the work of Lawson [8].He proposes to ameliorate the effect of the stifflinear part of equation(1.1)by using change of variables(also known as Lawson transformation),v(t)=e−tL u(t).The initial value problem(1.1)written in the new variable isv (t)=e−tL N(e tL v(t))v(t0)=v0,(1.4) where v0=e−t0L u0.The approach now is to apply an arbitrary s-stage Runge–Kutta method to the transformed equation(1.4)and then to transform the result back into the original variable.Thus,a method which satisfy just the nonstifforder conditions will not suffer from sever order reduction,when it is applied to stiffproblems.The aim of this paper is to show that the IF methods are ex-amples of exponential RK methods with special choices for theφ[l] functions and the parametersα[l]andβ[l]T.We also prove that in this special case the nonstifforder theory for the exponential RK methods reduces to the classical Runge–Kutta order theory,which explains why the IF methods exhibit the expected order.The paper is organized as follows:we briefly survey the nonstifforder theory for the exponential RK methods in Section2.Next,in Section3,we define the structure the matricesα[l]and the vectors β[l]T as well as the form of the functionsφ[l],which correspond to the IF methods.Finally,in Section4,we conclude with several remarks and questions of future interest.42Nonstifforder conditionsThe nonstifforder theory for the exponential RK methods was first constructed in [4]and later developed in [11].Here we follow the approach suggested in [9].It is based on bi-coloured rooted trees and B-series.For those not familiar with these concepts we suggest the monographs [1,5]for a complete treatment.Let 2T ∗denote the set of all bi-coloured (black and white)rooted trees with the requirement that the valency of the white nodes is always one.This correspods to the fact that the first term on the right hand side of (1.1)is linear.Let ∅represents the empty set which remains if the root of the one node tree or is removed.The order of the tree τ∈2T ∗is defined as the number of vertices in the tree,and it is denoted by |τ|.The density γof the tree is defined as the product over all vertices of the order of the subtree rooted at that vertex.An exponential Runge–Kutta method with elementary weight function a :2T ∗→R ,has nonstifforder p ,if for all τ∈2T ∗,such that |τ|≤p ,a (τ)=1/γ(τ).In order to give a practical representation of the elementary weight function a of the numerical solution,it is convenient to in-troduce some notations.Let for l =0,1,...and k =1,...,m the s ×smatrix φ[k ]l (c )=diag φ[k ]l (c 1),...,φ[k ]l (c s ) and C =diag(c 1,...,c s ).DefineA [l ]=s k =1φ[k ]l (c )α[k ],b [l ]T =s k =1φ[k ]l (1)β[k ]T ,C [l ]=1(l +1)!C l +1.(2.1)The elementary weight function a of the numerical solution can be computed using the following non-recursive rule:–Attach b [j ]Tto the root black node.–Attach A [j ]to all remaining nonterminal black nodes.–Attach A [j ]e to all terminal black nodes.–Attach C [j ]e to all terminal white nodes.–Attach I to all remaining white nodes.5 The value j is the number of white nodes directly below the corre-sponding node,I is the s×s identity matrix and e=(1,...,1)T. Now for each tree multiply from the root to the leaf as in the case for Runge–Kutta methods,then multiply these expressions in a compo-nent by component sense.3IF methods as a special caseApplying a standard s-stage Runge–Kutta method to the trans-formed equation(1.4)and then transforming back the result into the original variable,leads us to the followingφ[l]functionsφ[l](λ)(hL)=e(λ−c l)hL.(3.1) Every IF method can be represented in the form(1.3)withφ[l]func-tions given by(3.1)and with a special choice of the coefficient ma-tricesα[l]and the coefficient vectorsβ[l]T.This choice reduces the set of all order conditions to a set,which consists only of the order conditions corresponding to the black trees.To proof of this fact we need the following lemma.Lemma1.Let t∈R\{0,−1,−2,...},then for j=0,1,2,...jk=0(−1)k k!(j−k)!1(k+t)=1.t(t+1)···(t+j)Proof.The proof of this statement is by induction on j.The following theorem defines the structure of the matricesα[l] and the vectorsβ[l]T for the IF methods.With this structure of the coefficients,to achieve certain nonstifforder,it is sufficient to satisfy only the black trees.This implies that the transformed differential equation(1.4)is solved using a Runge–Kutta method.Theorem1.Let all the non-zero coefficients of an exponential Runge–Kutta method(1.3),withφ[l]functions given by(3.1)be located in column number l of the matrixα[l]and in position number l of the vectorβ[l]T for l=1,2,...,s.The method has nonstifforder p iffall order conditions corresponding to the black trees are satisfied.6Proof.It follows directly that if the exponential Runge–Kutta method has nonstifforder p then all order conditions corresponding to the black trees are satisfied.Let us assume that all the order conditions corresponding to the black trees are satisfied.We need to prove that all the remaining order conditions are also satisfied.From the defi-nition of theφ[l]functions(3.1),it follows that for j=0,1,2,...,φ[l]j(1)=(1−c l)jj!,φ[l]j(c)=1j!diag((c1−c l)j,...,(c s−c l)j).(3.2)Since all order conditions corresponding to the black trees involve only the coefficients A[0],b[0]T and c,we need to express every other order condition in terms of these coefficients.Having in mind the special structure of the matricesα[l]and the vectorsβ[l]T,after sub-stituting(3.2)into(2.1),we obtain for j=1,2,...,A[j]=jk=0(−1)k k!(j−k)!C[0]j−k A[0]C[0]k,b[j]T=jk=0(−1)k k!(j−k)!b[0]T C[0]k.(3.3)From the fact that all order conditions corresponding to the black trees are satisfied,it follows that A[0],b[0]T and c form a Runge–Kutta method.Therefore,C[0]e=A[0]e,C[0]ζ=(A[0]e)ζ,C[0]kζ=(A[0]e)...(A[0]e)ζ,(3.4)whereζis an arbitrary vector and the multiplications between the elements in the brackets are in a component by component sense.Now,we are in a position to define a procedure which transforms every coloured treeτinto a linear combination of black trees of order at most|τ|.Each treeτcan be decomposed asτ=(τb,τj,τt), whereτb is a coloured tree on the bottom with less number of white nodes thanτ;τj is tall white tree with j≥1white nodes andτt is black tree on the top.First applying formula(3.3)and then(3.4), for the order condition corresponding to a treeτ,we obtain the following three representations in terms of black trees or trees with7less white vertices.If τt =∅then τreduces toτ==If τb =∅then τreduces toτ==δ0+···+δ+···+δ,where δk =(−1)k k !(j −k )!for k =0,1,2,...,j .In the general case when τ{t,b }=∅,then τcan be representedas τ==δ+···+δk+···+δj (3.5)For each of the trees in the linear combination we apply the same procedure.Thus,after a finite number of steps all the trees in the combination will be black.From (2.1)it is clear that the order of every single black tree cannot exceed the order of coloured tree.To complete the proof we need to show that a (τ)=1/γ(τ)for all coloured trees τ,where |τ|≤p .We prove this by induction on the number of steps θin the transformation process.Let θ=1.Every coloured tree τhas representation τ= jk =0δk τk ,where all τk areblack trees.If γ(τt )=x 1|τt |x 2then a (τk )=1γ(τk )=1x 1(|τt |+k )x 2and by Lemma 1for t =|τt |it follows thata (τ)=j k =0(−1)k k !(j −k )!a (τk )=jk =0(−1)k k !(j −k )!1x 1(|τt |+k )x 2=1x 1|τt |(|τt |+1)···(|τt |+j )x 2=1γ(τ).Assume that a (τ)=1/γ(τ)for all coloured trees τwith θsteps in the transformation process.Let τbe a tree with θ+1steps in the transformation process.From (3.5)it follows that τ= j k =0δk τk ,8whereτk are coloured trees withθsteps in the transformation pro-cess and hence a(τk)=1/γ(τk).Ifγ(τt)=x1|τt|x2then a(τk)=1γ(τk)=1x1(|τt|+k)x2and by Lemma1for t=|τt|it again follows thata(τ)=1/γ(τ).4ConclusionsWe have shown that the IF methods are examples of exponential Runge–Kutta methods with special structure of the coefficients ma-trices and the relatedφ[l]functions.We have also proven that,in this special case,the nonstifforder theory for the exponential RK meth-ods reduces to the classical Runge–Kutta order theory.This explains why the IF methods exhibit the expected order.Other examples of theφ[l]functions,rather than(1.2)and(3.1),arise from the frame-work of Lie group methods see[9].The question how tofind the best set ofφ[l]functions is open and needs further investigation. References1.J.C.Butcher,Numerical methods for ordinary differential equations,John Wi-ley&Sons,2003.2.M.Calvo and C.Palencia,A class of explicit multistep exponential integrators forsemilinear problems,preprint,2005.3.P.M.Cox and P.C.Matthews,Exponential time differencing for stiffsystems,J.Comput.Phys.176(2002),430–455.4. A.Friedli,Verallgemeinerte Runge–Kutta verfahren zur l¨o esung steifer differen-tialgleichungssysteme,Lect.Notes Math.631(1978).5. E.Hairer,C.Lubich,and G.Wanner,Geometric numerical integration,Springer,2003,Number31in Springer Series in Computational Mathematics.6.M.Hochbruck and A.Osterman,Explicit exponential Runge–Kutta methods forsemilinear parabolic problems,Submitted to SIAM J.Numer.Anal.,2004.7.Exponential Runge–Kutta methods for parabolic problems,To appear in Appl.Nu-mer.Math.,2004.wson,Generalized Runge–Kutta processes for stable systems with large Lips-chitz constants,SIAM J.Numer.Anal.4(1969),372–390.9. B.Minchev,Exponential integrators for semilinear problems,Ph.D.thesis,Univer-sity of Bergen,2004.10.S.Nørsett,An A-stable modification of the Adams-Bashforth methods,Lect.NotesMath.109(1969),214–219.11.K.Strehmel and R.Weiner,B-convergence results for linearly implicit one stepmethods,BIT27(1987),264–281.。

3B SCIENTIFIC ® PHYSICS1Motore Stirling G 1002594Istruzioni per l'uso10/23 ALF/UD1 Volano con marcaturaper determinare il numero di giri2 Unità motore-generatorecon puleggia a due stadi 3 Interruttore 4 Lampadina5 Jack di sicurezza da 4mm6 Bruciatore ad alcol7 Prese di misura dellatemperatura 18 Pistone di compressione 9 Attacco del tubo contappo per la misurazione della pressione10 Prese di misura dellatemperatura 2 11 Pistone di lavoro 12 Asta filettata M3(collegata al pistone di lavoro)1. Norme di sicurezzaRiempire con attenzione il bruciatore ad alcol con alcol da ardere, in modo tale che non fuoriesca.Non riempire il bruciatore ad alcol se lostoppino sta ancora bruciando o se nelle vicinanze è presente un’altra fiamma aperta. Dopo l’uso chiudere immediatamente ilflacone di alcol.Non avvicinare le mani alla fiamma aperta. Attenzione! Spegnere la fiamma solo con ilcoperchio fissato.Il motore Stirling si riscalda durante il funzionamento con fiamma aperta.Non toccare il cilindro di compressionedurante e al termine del funzionamento del motore Stirling.Lasciare raffreddare il motore prima dirimuoverlo.2. DescrizioneIl motore Stirling permette l’analisi qualitativa e quantitativa del ciclo di Stirling. Può essere utilizzato in tre modalità diverse: motore termico, pompa di calore o macchina frigorifera.Il cilindro e il pistone di compressione sono realizzati in vetro resistente alle alte temperature; il cilindro di lavoro, il volano e le protezioni del cambio sono invece in vetro acrilico. In questo modo è possibile osservare molto bene i singoli movimenti in qualsiasi momento. Gli alberi a gomiti hanno cuscinetti a sfera e sono realizzati in acciaio temprato. Le bielle sono di plasticaresistente all’usura.L’unità motore-generatore incorporata, dotata di puleggia a due stadi consente di trasformare l’energia meccanica generata in energia elettrica. Con possibilità di commutazione per l’azionamento di una lampada incorporata o di carichi esterni, oppure per alimentare energia elettrica per il funzionamento in qualità di pompa di calore o macchina frigorifera.Fissando il filo in dotazione all'asta filettata del pistone di lavoro, è possibile misurarne la corsa.23. Dati tecniciUnità motore-generatore: max. 12 V CC Puleggia a due stadi: 30 mm Ø, 19 mm Ø Pistone di lavoro: 25 mm Ø Corsa pistone di lavoro: 24 mm Variazione del volume:24 mm 325mm 12cm 2Volume minimo: 32 cm³ Volume massimo: 44 cm³Potenza del motore Stirling: ca. 1 WDimensioni: ca. 300x220x160 mm³ Peso: ca. 1,65 kg4. Schema di funzionamentoIl ciclo Stirling ideale avviene in 4 fasi:1a fase: fase di espansione: cambiamento distato isotermico, l’aria si espande a temperatura costante2a fase: cambiamento di stato isocorico, l’ariasi raf fredda a volume costante nel rigeneratore3a fase: fase di compressione: cambiamentodi stato isotermico, l’aria viene compressa in modo isotermico4a fase: cambiamento di stato isocorico, l’ariaviene di nuovo riscaldata alla temperatura origina ria nel rigeneratoreNel motore Stirling, questo processo ideale è realizzato solo in maniera approssimativa, perché i quattro tempi si sovrappongono. Il gas passa da caldo a freddo già durante la fase di espansione e durante la fase di compressione l’aria non si trova ancora tutta nella parte fredda del motore.Fig. 1 Schema di funzionamento(A: Pistone di compressione, B: Pistone di lavoro)5. Utilizzo5.1 Il motore Stirling come motore termico Riempire il bruciatore ad alcool, inserirlonell’incavo della piastra di base, estrarre svitando lo stoppino di circa 1-2 mm e accenderlo.Portare i pistoni di compressione nellaposizioni più arretrata e dopo un breve periodo di riscaldamento (da 1 a 2 minuti circa) mettere in movimento il volano con una leggera pressione in senso orario (sguardo rivolto verso l'unità motore-generatore) (vedere fig. 2).Se necessario, impostare la tensione dellacinghia di trasmissione spostando l'unità motore-generatore. Accendere la lampadina spostandol'interruttore nella posizione “sopra”.In alternativa collegare il carico esternotramite le prese da 4 mm e mettere in funzione con l’interruttore in posizione "sotto".Numero di giri senza carico: ca. 1000 giri/min Numero di giri congeneratore come carico: ca. 650 giri/min Tensione generatore: ca. 6 V CC Scarto di pressione: +250 hPa/-150 hPa5.2 Il motore Stirling come pompa di calore omacchina frigoriferaDotazione supplementare necessaria:1 Alimentatore CC, 0 - 20 V, 0 - 5 A @ 230 V1003312oppure1 Alimentatore CC, 0 - 20 V, 0 - 5 A @ 115 V1003311Termometro digitale 1002794 Inserire i sensori di temperatura nelle presedi misura della temperatura e collegarli ad un misuratore (vedere fig. 3).Collegare la sorgente di corrente continuatramite le prese da 4 mm.Impostare al massimo 12 V e azionare ilmotore di Stirling con l'interruttore in posizione "sotto".Osservare l’aumento o la diminuzione ditemperatura.In modalità di funzionamento macchina frigorifera il volano ruota in senso orario (sguardo rivolto verso l'unità motore-generatore), in modalità pompa di calore in senso antiorario.Per cambiare la modalità difunzionamentoinvertire la polarità dei cavi di collegamento.3Scarto di pressione: +250 hPa/-150 hPa Tensione motore: 9 V Numero di giri: 600 giri/minDifferenza di temperatura (riferita a 21° C): Macchina frigorifera: -4 K (serbatoio: +6 K) Pompa di calore: +13 K (serbatoio: -1 K)5.3 Registrazione del diagramma pressione-volume del motore Stirling come pompa di caloreDotazione supplementare necessaria:1 Alimentatore CC, 0 - 20 V, 0 - 5 A @ 230 V1003312oppure1 Alimentatore CC, 0 - 20 V, 0 - 5 A @ 115 V10033111 Portasensori per motore Stirling G 1008500 1 Sensore di pressione relativa ±1000 hPa 1021533 1 Rilevatore di corsa FW 10215342 Cavos del sensore 1021514 1 Data logger 1 SoftwareUlteriori informazioni sulla misurazione digitale sono disponibili sul sito web del prodotto, nel webshop 3B.Fissare il portasensori alla piastra di base delmotore Stirling.Nel portasensori, montare il sensore dipressione relativa in basso e il rilevatore di corsa in alto in modo tale che il lato stampato sia rivolto verso l’alto.Collegare l’attacco del tubo “+” del sensore di pressione relativa e l’attacco tubo sul cilindro di lavoro del motore Stirling per mezzo del tubo flessibile fornito in dotazione con il portasensori (1008500).Avvitare il dado cieco fissato al filo (compreso nella dotazione del portasensori) sull’asta filettata del pistone di lavoro, passare il filo intorno alla puleggia del rilevatore di corsa e appendere la molla ad elica all’asta filettata. (Per una descrizione dettagliata relativa al montaggio dei sensori sul portasensori vedere le istruzioni per l’uso del portasensori 1008500.)Collegare il sensore di pressione relativa e il rilevatore di corsa al data logger.Collegare la sorgente di corrente continua tramite le prese da 4 mm.Impostare al massimo 12 V e azionare il motore di Stirling con l'interruttore in posizione "sotto".Avviare il software e registrare il diagramma pressione-volume.Fig.2Il motore Stirling come motore termico3B Scientific GmbH Ludwig-Erhard-Straße 20 20459 Amburgo • Germania • Con riserva di modifiche tecniche © Copyright 2023 3B Scientific GmbHFig. 3 Il motore Stirling come pompa di calore o macchina frigoriferaFig. 4 Registrazione del diagramma pressione-volume。

专利名称：PREPARATION OF MEDICAL HARD CAPSULE 发明人：ARAUME KIYOSHI,MUTOYASUAKI,NISHIYAMA YUICHI,CHIBA TORU申请号：JP14562689申请日：19890608公开号：JPH039755A公开日：19910117专利内容由知识产权出版社提供摘要：PURPOSE:To efficiently prepare a medical hard capsule whose wall part has a uniform thickness by coating a pin with an aqueous solution of hydroxypropyl methyl cellulose having structural viscosity in its flow characteristics and drying the coated pin to mold the prototype of the capsule and releasing said product to cut the same into a predetermined length. CONSTITUTION:The concn. of an aqueous solution of hydroxypropyl methyl cellulose is 15-25wt.% and the viscosity thereof at 40 deg.C is3,000-15,000cps and the ratio of the viscosity measured after standing for 30min at 40 deg.C and the viscosity measured at a shear rate of 1.5sec<-1> immediately after sufficient stirring is 130% or more. A pin having the shape corresponding to the respective body and cap of a medical hard capsule is immersed in the aqueous solution of hydroxypropyl methyl cellulose to be drawn up and the aqueous solution is applied to the pin and dried to mold the prototype of the capsule. The prototype is released from the pin to be cut into a predetermined length and the obtained body and cap are engaged with each other to make it possible to obtain a medical hard capsule whose wall part has a uniform thickness.申请人：SHIN ETSU CHEM CO LTD更多信息请下载全文后查看。

专利名称：Engagement device in automatictransmission发明人：Seitoku Kubo,Koujiro Kuramochi,Tatsuo Kyushima申请号：US06/190034申请日：19800923公开号：US04380179A公开日：19830419专利内容由知识产权出版社提供摘要：In an automatic transmission including sun gears, ring gears and a carrier for rotatably supporting a pinion on a shaft, the pinion being adapted to mesh with the sun gear and the ring gear; the improvements comprising a hub extending radially outside of the ring gear and having a friction surface in part of the outer peripheral wall thereof and external spline teeth provided in the other part of the outer peripheral wall, the hub being attached to the carrier; a friction element adapted to engage a cylindrical outer race fitted on internal spline-teeth of a transmission housing and the friction face of the hub; a plurality of first friction plates axially, slidably fitted on the external spline teeth of the hub; and a plurality of second friction plates axially, slidably fitted on the internal spline-teeth of the transmission housing, the first and second friction plates disposed one after another.申请人：TOYOTA JIDOSHA KOGYO KABUSHIKI KAISHA代理机构：Finnegan, Henderson, Farabow, Garrett & Dunner更多信息请下载全文后查看。