003-LeCroy unique function seminar_WaveScan

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参考点相依双曲绝对效用函数下语言型群决策方法

语言型决策方法的优点
能够充分反映决策者的主观意愿和偏好，具有较好的灵活性和适应性，可以针对不同问题进行定制和优化。
语言型决策方法的基本模型
基于参考点相依双曲绝对效用函数的语言型决策方法是一种常见的语言型决策模型，通过将决策问题转化为效用函数的形式进行求解。
语言型群决策方法基本流程
问题定义与问题建模
在许多实际决策问题中，决策者往往需要根据语言信息进行决策，例如在风险投资、医疗诊断、金融风险管理等领域。
针对这一问题，本研究旨在探讨参考点相依双曲绝对效用函数下语言型群决策方法，以更好地理解决策者在语言信息处理和决策制定过程中的心理行为特征。
研究内容与方法
研究内容
本研究将围绕参考点相依双曲绝对效用函数展开，研究内容包括：参考点相依双曲绝对效用函数的定义、性质和计算方法；语言型信息处理方法；基于参考点相依双曲绝对效用函数的群决策模型构建；实验设计和实证分析。
参考点相依双曲绝对效用函数下语言型群决策方法
汇报人： 2023-12-12
目录
• 引言 • 参考点相依双曲绝对效用函数 • 语言型群决策方法 • 基于参考点相依双曲绝对效用
函数的群决策方法 • 方法比较与展望
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引言
研究背景与意义
随着社会和经济的发展，人们越来越重视决策的合理性和科学性。
目前，对于语言型决策的研究主要集中在语言信息处理和决策模型构建上，而较少关注决策者心理因素对决策的影响。
风险分析
该函数可以用于评估风险偏好和风险厌恶程度，为决策者提供更加准确的决策支持。
行为经济学
参考点相依双曲绝对效用函数可以解释一些经济学中的行为现象，如参照依赖、损失厌恶等。
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FL-2000系列产品说明书

La Serie FL-2000 ofrece una amplia variedad de caudalímetros para utilizar en aplicaciones médicas, industriales, químicas y de laboratorio a un precio conveniente. Las unidades se venden con o sin válvula.CAUDALÍMETROS DE ACRÍLICOB-15aESpEciFicacionES precisión:Modelos FL-2001–FL-2025: ±5%de escala completa Modelos FL-2031–FL-2069: ±3% de escala completaModelos FL-2071–FL-2128: ±2% de escala completaFlotador: Acero inoxidable vidrio negro cuerpo: Acrílico transparente Sellos: Juntas tóricas de caucho sintético con adaptadores de latón o PVC, juntas tóricas de FKM con adaptadores de acero inoxidablepresión: 100 psig máx. a 21 °C (70 °F)Temperatura:65 °C (150 °F) máx. a 0 psigaccesorios: Latón estándar, acero inoxidable opcional salvo para el FL-2071 a FL-2128, que tienen accesorios de PVC de 1 NPT únicamenteVálvulas: Modelos FL-2001 a FL-2069: Latón estándar; tipo decartucho de acero inoxidable (opcional) FL-2071 a FL-2128: Puerta en línea de plástico opcionalU E scala métrica e inglesa de fácil lectura U R angos de agua desde 4 ccM a 20 GpM,rangos de aire de 40 cc a 4.000 LpM U i nserción de latónroscada para una rápida instalación U F ácil montajey desmontaje para su mantenimiento U c onstrucción de acrílico transparente en una pieza duradera U F lotador estable de fácil lectura U c alidad superiorapLicacionES U E quipo de muestreo de aireU a cuiculturaU Equipo de desalinización U a nalizador de gasU S istemas médicos U E quipo de procesamientofotográficoU S istemas de tratamiento y distribución de aguaEl modelo FL-2013 para aire se muestraen un tamaño inferior al real.El modelo FL-2066-nV para agua se muestra en un tamaño inferior al real.B(13,5)B-15bDimensiones de FL-2091 hasta FL-2128Para solicitar el producto con válvula de compuerta de plástico, añada el sufijo “-V” al número de modelo para ver el coste adicional para las Series FL-2090 y FL-2120.Para un certificado NIST de 10 puntos opcional, añada el sufijo “-NIST” al número de modelo, con coste adicional y dos semanas más al plazo de entrega estándar .Ejemplo de pedido: FL-2095, rotámetro, 100 a 1.400 LPM de aire FL-2127-V, rotámetro, 4 a 36 LPM de agua, con válvulas.El modelo FL-2097 se muestra en un tamaño inferior al real.Se trata de unidades estándar sin válvulas.Para solicitar el producto con válvula de compuerta de plástico, añada el sufijo “-V” al número de modelo para ver el coste adicional.Para un certificado NIST de 10 puntos opcional, añada el sufijo “-NIST” al número de modelo, con coste adicional y dos semanas más al plazo de entrega estándar.Ejemplos de pedidos: FL-2075, válvula de rotámetro, 100 a 1.400 LPM de aire.FL-2080, válvula de rotámetro, 2 a 19 LPM de agua.El modelo FL-2041-nV se muestra en un tamaño inferioral real.El modelo FL-2053 para agua se muestra en un tamaño inferioral real.El modelo FL-2066-nV se muestraen un tamaño inferior al real.Para solicitar el producto con válvula de acero inoxidable, añada el sufijo “-SS” al número de modelo para ver el coste adicional.Para solicitar el producto sin válvula, añada el sufijo “-NV” al número de modelo y descuéntelo del coste.Para un certificado NIST de 10 puntos opcional, añada el sufijo “-NIST ” al número de modelo, con coste adicional y dos semanas más al plazo de entrega estándar.Ejemplos de pedidos: FL-2036, rotámetro económico, con válvula de latón, 14 a 150 SCFH de aire.FL-2036-NV , rotámetro económico, sin válvula de latón, 14 a 150 SCFH de aire.BEl modelo FL-2060 para airese muestra en un tamaño inferior al real.Escalas dobles de modo estándar: SCFM/SCFH, GPM/GPH y LPM/LPH Para solicitar el producto con válvula de acero inoxidable, añada el sufijo “-SS” al número de modelo para ver el coste adicional.Para solicitar el producto sin válvula, añada el sufijo “-NV” al número de modelo y descuéntelo del coste.Para un certificado NIST de 10 puntos opcional, añada el sufijo “-NIST” alnúmero de modelo, con coste adicional y dos semanas más al plazo de entrega estándar.Ejemplos de pedidos: FL-2060, rotámetro con válvula de latón, 0,5 a 5 scfm. FL-2069-NV, rotámetro sin válvula, 2 a 20 LPM.El modelo FL-2091 paraaire se muestra en un tamaño inferior al real.B-15cEl modelo FL-2021-nV para agua se muestra en un tamaño superior al real.Para solicitar el producto con válvulas de acero inoxidable, añada el sufijo “-SS”al número de modelo para ver el coste adicional.Para solicitar el producto sin válvula, añada el sufijo “-NV” al número de modelo y descuéntelo del coste.Para un certificado NIST de 10 puntos opcional, añada el sufijo “-NIST” al número de modelo, con coste adicional y dos semanas más al plazo de entrega estándar.Ejemplos de pedidos: FL-2005, rotámetro económico con válvula de latón, 2 a 20 SCFH de aire.FL-2005-NV, rotámetro económico sin válvula, 2 a 20 SCFH de aire.B-15d。

VBOX The Racelogic IMU 03

The Racelogic IMU 03The IMU 03 provides highly accurate measurements ofvelocity, pitch, roll, and yaw, using three yaw rate sensorsand three accelerometers. It is a CAN based unit and istemperature compensated with improved calibration andstability.惯性测量装置通过使用三轴向偏航速率传感器和三轴向加速计，可以提供速度、倾斜、转动和偏航的高度精确值。

这是一个基本的CAN单元，它可以改进温度补偿的校准和稳定。

The IMU 03 is designed for use either as a stand-alonesensor with simple connection and configuration via theCAN bus interface, or for use with VBOX GPS data-loggers.惯性测量装置03通过简单的连接和CAN总线接口配置来作为独立操作的传感器使用，或者用于VBOXGPS的数据记录。

When used in conjunction with VBOX 3i, data from the IMU can be seamlessly integrated with GPS to produce pitch and roll values as well as smoother, more reliable position data. This ensures premium quality GPS data even when satellite reception is interrupted.连同VBOX 3i一起使用时，从惯性测量装置中获得的数据与GPS结合一起，来产生倾斜和转动值以及平滑值，更可靠的位置数据。

Skytech Sky200 Energy Monitoring System说明书

BOUTON MARCHE / ARRÊT :-Mise en marche : branchez le cordon d’alimentation sur une prise secteur.-Arrêt : débranchez le cordon d’alimentation de la prise secteur et appuyez sur le bouton Marche / Arrêt pendant plus de 2 s. REMARQUE : L ’appareil ne peut pas être arrêté lorsqu’il est branché sur le secteur ou qu’un enregistrement est en cours.Entrées de tensionEmplacement des pions de couleur Cordon secteur Logement Carte SD ConnexionUSB Ethernet RJ 45Entrées de courantPEL 103(PEL 103 uniquement) :Permet de parcourir et de sélectionner l’affichage des données.BOUTON SÉLECTION :Met en marche ou arrête la session d’enregistrement et active ou désactive la liaison Bluetooth.La fonction s’obtient par un appui de 2 s sur le bouton SÉLECTION , ce qui allume successivement le voyant REC pendant 3 s, puis le voyant Bluetooth.VOYANT REC (DÉCLENCHEMENT / ARRÊT)-Le relâchement du bouton lorsque le voyant est allumé lance l’enregistrement (s’il était arrêté) -Le relâchement du bouton lorsque le voyant est éteint arrête l’enregistrement (s’il était en cours)VOYANT BLUETOOTH (MARCHE / ARRÊT)-Le relâchement du bouton lorsque le voyant est allumé active Bluetooth (s’il était désactivé)-Le relâchement du bouton lorsque le voyant est éteint désactive Bluetooth (s’il était activé)754321698Installation de DataView ®NE CONNECTEZ PAS L’INSTRUMENT AU PC AVANT D’AVOIR INSTALLÉ LES LOGICIELS ET LES PILOTES.1.Branchez le CD dans son lecteur.Si l’exécution automatique est activée, le programme démarre automatiquement dans votre navigateur. Si l’exé-cution automatique n’est pas activée, sélectionnez Start.html dans D:\SETUP (si votre lecteur de CD-ROM est le lecteur D ; sinon, remplacez D par la lettre de lecteur appropriée).2. Sélectionnez votre langue et cliquez sur ENTRÉE . Autorisez votre navigateur à ouvrir le fichier.3. Sélectionnez la colonne Logiciel.4.Sélectionnez DataView ou PEL Transfer si vous ne souhaitez installer que PEL Transfer.5. Téléchargez le fichier et décompressez-le.6. Sélectionnez Setup.exe et suivez les instructions.REMARQUE : Pour des instructions d’installation complètes, reportez-vous au manuel fourni sur le CD-ROM.Carte SDIntroduisez la carte SD fournie dans le PEL.Le PEL prend en charge les cartes SD (jusqu’à 2 Go) et SDHC (entre 4 et 32 Go).• Une carte de 2 Go peut contenir 4 semaines d’enregistrements si vous n’enregistrez pas les harmoniques.• Lorsque la carte SD est dans l’appareil, il est possible de la formater, dans certaines conditions, lorsquevous êtes connecté à Dataview ®.• Un formatage est possible sans restriction si la carte est inséréedans un PC en utilisant le lecteur de carte fourni.• L’extraction à chaud est possible si aucun enregistrementn’est en cours.Ouverture de PEL Transfer• Branchez le cordon d’alimentation sur une prise secteur.L’appareil se met en marche.• Raccordez le PEL au PC avec le câble USB fourni. Attendez queles pilotes aient fini de s’installer avant de continuer.• Double-cliquez sur l’icône du PEL sur le bureau pour ouvrirPEL Transfer.• Sélectionnez l’icône Ajouter un appareil dans la barred’outil ou dans le menu principal Appareil .• Suivez les instructions de l’assistant Ajouter un appareil . SiPEL Transfer ne détecte pas l’appareil dans la liste déroulante, cliquez sur le bouton Actualiser ou débranchez, puis rebranchez le câble USB.• Lorsque la connexion avec l’appareil est établie, le nom de cedernier doit s’afficher sur le côté gauche de la fenêtre dans la branche Réseau PEL de l’arborescence.Configuration de l’appareilPour configurer votre PEL, sélectionnez l’appareil dans le répertoire Réseau PEL.Ouvrez la boîte de dialogue Configurer l’appareil en cliquant sur l’icône Configurer dans la barre d’outils,dans le menuAppareil ou dans la zone État .Cette boîte de dialogue comporte cinq onglets :• Général : Comporte des champs permettant d’attribuer des libellés à l’appareil, des options de commanded’arrêt automatique, de commande de l’afficheur LCD, de boutons de fonctionnement, de réglage de l’horloge et de formatage de la carte SD.• Communication : Options relatives à la liaison Bluetooth et au réseau LAN.• Mesure : Sélection du système de distribution, rapport des tensions, sélection de la fréquence et réglagedes capteurs de courant.• Enregistrement :Options de sélection des paramètres d’enregistrement.• Compteurs : Réinitialisation des compteurs et options de réinitialisation des compteurs d’énergie partielle.Cliquez sur le boutonpour transférer sur l’appareil la nouvelle configuration.Exemple de répertoire Réseau PELInstallation des sondes et des capteurs de courantDouze jeux de bagues et de pions de couleurs sont fournis avec l’appareil. Utilisez-les pour identifier les sondes et les bornes d’entrée.• Détachez les pions et placez-les dans les trous sous les bornes (les grands pour les bornes de courant, les petits pour les bornes de tension).• Clipsez une bague de la même couleur à l’extrémité de la sonde qui sera branchée sur la borne.• Mesure du courant : connecteurs 4 points I1, I2, I3• Mesure de la tension : bornes V1, V2, V3 et NLes sondes de mesure doivent être connectées au circuit à surveiller selon le schéma de branchement sélectionné. N’oubliez pas de définir le rapport de transformation lorsque nécessaire.MontageLe PEL comporte des aimants incorporés qui permettent de le fixer sur une surface magnétique.Lancement d’un enregistrement (Marche / Arrêt)Pour lancer un enregistrement, procédez de l’une des manières suivantes :• Dans PEL Transfer : Sélectionnez l’option appropriée dans l’onglet Enregistrement de la boîte de dialogue Configuration. L’appareil peut être configuré pour déclencher un enregistrement soit à une date et une heure future, soit immédiatement après écriture de la configuration sur l’appareil.• Sur l’appareil : Appuyez sur le bouton SÉLECTION et maintenez-le appuyé jusqu’à ce que le voyant vert s’allume, puis relâchez-le. L’appareil démarre l’enregistrement selon les réglages de configuration précédents. Pour arrêter un enregistrement, procédez de l’une des manières suivantes :• Dans PEL Transfer : Dans le menu, sélectionnez Appareil > Arrêter l’enregistrement .• Sur l’appareil : Appuyez sur le bouton SÉLECTION et maintenez-le enfoncé jusqu’à ce que le voyant vert s’allume, puis relâchez-le.Affichage de l’enregistrementLes données enregistrées peuvent être transférées de deux manières sur le PC pour y être affichées et pour générer des rapports :• La carte SD peut être retirée de l’appareil et branchée sur le PC via le lecteur de carte SD fourni. Lancez PEL Transfer, sélectionnez la commande Ouvrir dans le menu Fichier, pointez le fichier ICP portant le numéro de session souhaité sur la carte SD et sélectionnez Ouvrir.• Connexion directe entre le PC et le PEL (USB, réseau ou Bluetooth). Lancez PEL Transfer, ouvrez une connexion vers l’appareil, sélectionnez-le (veillez à ce qu’il soit connecté) dans l’arborescence, puis sélectionnez Sessions enregistrées. Double-cliquez sur la session d’enregistrement souhaitée.Le téléchargement terminé, sélectionnez le test téléchargé et cliquez sur le bouton Ouvrir dans la boîte de dialogue Téléchargement. Dans les deux cas, la session s’ajoute à Mes sessions ouvertes dans l’arborescence. Les données peuvent alors être affichées.09 - 2012Code 693779A01 - Ed. 1。

A Blow-Up Criterion for the Nonhomogeneous Incompressible Navier--Stokes Equations

SIAM J.M ATH.A NAL.c 2006Society for Industrial and Applied Mathematics Vol.37,No.5,pp.1417–1434A BLOW-UP CRITERION FOR THE NONHOMOGENEOUSINCOMPRESSIBLE NA VIER–STOKES EQUATIONS∗HYUNSEOK KIM†Abstract.Let(ρ,u)be a strong or smooth solution of the nonhomogeneous incompressible Navier–Stokes equations in(0,T∗)×Ω,where T∗is aﬁnite positive time andΩis a bounded domain in R3with smooth boundary or the whole space R3.We show that if(ρ,u)blows up at T∗,thenT∗0|u(t)|sL r w(Ω)dt=∞for any(r,s)with2s+3r=1and3<r≤∞.As immediate applications,we obtain a regularity theorem and a global existence theorem for strong solutions.Key words.blow-up criterion,nonhomogeneous incompressible Navier–Stokes equations AMS subject classiﬁcations.35Q30,76D05DOI.10.1137/S00361410044421971.Introduction.The motion of a nonhomogeneous incompressible viscousﬂuid in a domainΩof R3is governed by the Navier–Stokes equations(ρu)t+div(ρu⊗u)−Δu+∇p=ρf,(1.1)ρt+div(ρu)=0in(0,∞)×Ω,(1.2)div u=0(1.3)and the initial and boundary conditions(ρ,ρu)|t=0=(ρ0,ρ0u0)inΩ,u=0on(0,T)×∂Ω,(1.4)ρ(t,x)→0,u(t,x)→0as|x|→∞,(t,x)∈(0,T)×Ω.Here we denote byρ,u,and p the unknown density,velocity,and pressureﬁelds of theﬂuid,respectively.f is a given external force driving the motion.Ωis either a bounded domain in R3with smooth boundary or the whole space R3.Throughout this paper,we adopt the following simpliﬁed notation for standard homogeneous and inhomogeneous Sobolev spaces:L r=L r(Ω),D k,r={v∈L1loc(Ω):|v|D k,r<∞};H k,r=L r∩D k,r,D k=D k,2,H k=H k,2;D10={v∈L6:|v|D1<∞and v=0on∂Ω};H10=L2∩D10,,D10,σ={v∈D10:div v=0inΩ};|v|D k,r=|∇k v|L r and|v|D10=|v|D10,σ=|∇v|L2.Note that the space D10is the completion of C∞c(Ω)in D1,and thus there holds the following Sobolev inequality:|v|L6≤2√3|v|D1for all v∈D10.(1.5)∗Received by the editors March17,2004;accepted for publication(in revised form)May9,2005; published electronically January10,2006.This work was supported by the Japan Society for the Promotion of Science under the JSPS Postdoctoral Fellowship for Foreign Researchers./journals/sima/37-5/44219.html†School of Mathematics,Korea Institute for Advanced Study207-43Cheongnyangni2-dong, Dongdaemun-gu,Seoul,130-722,Korea(khs319@postech.ac.kr,khs319@kias.re.kr).14171418HYUNSEOK KIMFor a proof of (1.5),see sections II.5and II.6in the book by Galdi [11].The global existence of weak solutions has been established by Antontsev and Kazhikov [1],Fernandez-Cara and Guillen [10],Kazhikov [13],Lions [21],and Simon [26,27].From these results (see [21]in particular),it follows that for any data (ρ0,u 0,f )with the regularity0≤ρ0∈L 32∩L ∞,u 0∈L 6,and f ∈L 2(0,∞;L 2),there exists at least one weak solution (ρ,u )to the initial boundary value problem (1.1)–(1.4)satisfying the regularity ρ∈L ∞(0,∞;L 32∩L ∞),√ρu ∈L ∞(0,∞;L 2),and u ∈L 2(0,∞;D 10,σ)(1.6)as well as the natural energy inequality.Then an associated pressure p is determinedas a distribution in (0,∞)×Ω.But the global existence of strong or smooth solutions is still an open problem and only local existence results have been obtained for suﬃciently regular data satisfying some compatibility conditions.For details,we refer to the papers by Choe and the author [6],Kim [15],Ladyzhenskaya and Solonnikov [19],Okamoto [22],Padula [23],and Salvi [24].In particular,it is shown in [6](see also [7])that if the data ρ0,u 0,and f satisfy 0≤ρ0∈L 32∩H 2,u 0∈D 10,σ∩D 2,−Δu 0+∇p 0=ρ120g,(1.7)f ∈L 2(0,∞;H 1),and f t ∈L 2(0,∞;L 2)for some (p 0,g )∈D 1×L 2,then there exist a positive time T and a unique strongsolution (ρ,u )to the problem (1.1)–(1.4)such that ρ∈C ([0,T ];L 32∩H 2),u ∈C ([0,T ];D 10,σ∩D 2)∩L 2(0,T ;D 3),(1.8)u t ∈L 2(0,T ;D 10,σ),and √ρu t ∈L ∞(0,T ;L 2).Moreover,the existence of a pressure p in C ([0,T ];D 1)∩L 2(0,T ;D 2)can be deducedfrom (1.1)–(1.3).See [5]for a detailed argument.Let (ρ,u )be a global weak solution to the problem (1.1)–(1.4)with the data (ρ0,u 0,f )satisfying condition (1.7).Then from the above local existence result and weak-strong uniqueness results in [6]and [21],we conclude that the solution (ρ,u )satisﬁes the regularity (1.8)for some positive time T .One fundamental problem in mathematical ﬂuid mechanics is to determine whether or not (ρ,u )satisﬁes (1.8)for all time T .As an equivalent formulation,we may ask the following.Fundamental question 1.1.Does the solution (ρ,u )blow up at some ﬁnite time T ∗?Such a time T ∗,if it exists,is called the ﬁnite blow-up time of the solution (ρ,u )in the class H 2.In spite of great eﬀorts since the pioneering works by Leray [20]in 1930s,there have been no deﬁnite answers to the fundamental question even for the case of the homogenous Navier–Stokes equations with only some blow-up criteria available.The ﬁrst criterion is due to Leray [20]who proved,among other things,that if T ∗is the ﬁnite blow-up time of a strong solution u to the Cauchy problem for the homogeneous Navier–Stokes equations,then for each r with 3<r ≤∞,there exists a constant C =C (r )>0such that |u (t )|L r ≥C (T ∗−t )−12(1−3r )for all near t <T ∗.(1.9)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1419This estimate near the blow-up time was extended by Giga [12]to the case of bounded domains.An immediate consequence of (1.9)is the following well-known blow-up criterion in terms of the so-called Serrin class (see [9,25,29]):T ∗0|u (t )|s L r dt =∞for any (r,s )with 2s +3r =1,3<r ≤∞.(1.10)By virtue of Sobolev inequality (1.5),we deduce from (1.10)thatT ∗|∇u (t )|4L 2dt =∞.(1.11)Further generalizations of (1.10)and (1.11)have been obtained by Beirao da Veiga [2],Berselli [3],Chae and Choe [4],and Kozono and Taniuchi [17].The major purpose of this paper is to prove the blow-up criterion (1.10)for strong solutions of the nonhomogeneous Navier–Stokes equations (1.1)–(1.3).In fact,we establish a more general result.To state our main result precisely,we ﬁrst introduce the notion of the blow-up time of solutions in the class H 2m with m ≥1.Let (ρ,u )be a strong solution to the initial boundary value problem (1.1)–(1.4)with the regularityρ∈C ([0,T ];L 32∩H 2m ),u ∈C ([0,T ];D 10,σ∩D2m)∩L 2(0,T ;D 2m +1),∂j t u ∈C ([0,T ];D 10,σ∩D2m −2j)∩L 2(0,T ;D 2m −2j +1)for 1≤j <m,(1.12)∂m t u ∈L 2(0,T ;D 10,σ),and √ρ∂m t u ∈L ∞(0,T ;L 2)for any T <T ∗,where T ∗is a ﬁnite positive time.Then we can deﬁneΦm (T )=1+sup 0≤t ≤T|∇ρ(t )|H 2m −1+|u (t )|D 10∩D 2m + T|u (t )|2D 2m +1dt+sup1≤j<msup 0≤t ≤T|∂jt u (t )|D 10∩D 2m −2j +T|∂jt u (t )|2D 2m −2j +1dt(1.13)+ess sup 0≤t ≤T|√ρ∂mtu (t )|L 2+T|∂mt u (t )|2D 10dtfor any T <T ∗.Hereafter we use the obvious notation|·|X ∩Y =|·|X +|·|Yfor (semi-)normed spaces X,Y.Definition 1.2.A ﬁnite positive number T ∗is called the ﬁnite blow-up time ofthe solution (ρ,u )in the class H 2m ,provided thatΦm (T )<∞for0<T <T ∗andlim T →T ∗Φm (T )=∞.We are now ready to state the main result of this paper.Theorem 1.3.For a given integer m ≥1,we assume that∂mtf ∈L 2(0,∞;L 2)and∂jt f ∈L 2(0,∞;H 2m −2j −1)for 0≤j <m.Let (ρ,u )be a strong solution of the nonhomogeneous Navier–Stokes equations (1.1)–(1.3)satisfying the regularity (1.12)for any T <T ∗.If T ∗is the ﬁnite blow-up time of (ρ,u )in the class H 2m ,then we haveT ∗|u (t )|s L r wdt =∞for any (r,s )with 2s +3r =1,3<r ≤∞.(1.14)1420HYUNSEOK KIMHere L r w denotes the weak L r-space,that is,the space consisting of all vector ﬁelds v ∈L 1loc (Ω)such that |v |L r w=sup α>0α|{x ∈Ω:|v (x )|>α}|1r <∞for 3<r <∞and |v |L ∞w=|v |L ∞<∞.In the case when 3<r <∞,L ris a proper subspace of L r w (|x |−3/r ∈L r w (R 3)for instance)and so Theorem 1.3is in fact a generalizationof the blow-up criterion (1.10)due to Leray and Giga even for the homogeneous Navier–Stokes equations.Theorem 1.3is proved in the next two sections.In section 2,we provide a proof of the theorem for the very special case m =1.Our method of the proof is quite well known in the case of the homogeneous Navier–Stokes equations and was also applied in an earlier paper [6]by Choe and the author to the nonhomogeneous case:combining classical regularity results on the Stokes equations with H¨o lder and Sobolev inequal-ities,we show that Φ1(T )is bounded in a double exponential way by T0|u (t )|s L r wdt for any T less than the blow-up time T ∗.But the use of weak Lebesgue spaces in space variables makes it more diﬃcult to estimate the nonlinear convection term.To overcome this technical diﬃculty,we utilize some basic theory of the Lorenz spaces developed in [18]and [30].See the derivations of (2.6)and (2.7)for details.Concern-ing the proof for the general case m ≥2,the basic idea is also to show that Φm (T )isbounded in some speciﬁc way by T0|u (t )|s L r wdt for any T <T ∗.Such an approach is more or less standard in the case of the homogeneous Navier–Stokes equations,but its extension to the nonhomogeneous case is not straightforward and indeed much complicated due to the evolution of the density.A detailed argument is provided in section 3.Some corollaries of Theorem 1.3can be deduced from a local existence result on strong solutions in the class H 2m .For instance,as an immediate consequence of Theorem 1.3,the local existence result in the class H 2,and the weak-strong uniqueness result,we obtain the following regularity result whose obvious proof may be omitted.Corollary 1.4.Let (ρ,u )be a global weak solution to the initial boundary value problem (1.1)–(1.4)with the data satisfying condition (1.7).If there exists a ﬁnite positive time T ∗such thatu ∈L s (0,T ∗;L r w )for some (r,s )with2s +3r=1,3<r ≤∞,(1.15)then the solution (ρ,u )satisﬁes regularity (1.8)for some T >T ∗.A similar result was obtained by Choe and the author [6]assuming,however,astronger condition on u ,that is,u ∈L 4(0,T ∗;D 10).By virtue of Corollary 1.4,we may conclude that the class (which we call a weak Serrin class )in (1.15)is a regularity class for weak solutions of the nonhomogeneous Navier–Stokes equations,which was already proved by Sohr [28]for the homogeneous case.See also a local version of Sohr’s result in [14].Moreover,thanks to a recent result by Dubois [8],the weak Serrin class is a uniqueness class for the homogeneous Navier–Stokes equations.It is also noticed that the same results can be easily derived from regularity and uniqueness results due to Kozono by adapting the arguments in the remarks of Theorem 3in [16].Theorem 1.3and its proof can be used to obtain a global existence result on solutions in the class H 2under some smallness condition on u 0and f (but not on ρ0).Theorem 1.5.For each K >1,there exists a small constant ε>0,depending only on K and the domain Ω,with the following property:if the data ρ0,u 0,and f satisfy|ρ0|L 32∩L∞≤K,|u 0|D 10≤ε,and ∞|f (t )|2L 2dt ≤ε2(1.16)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1421in addition to condition (1.7),then there exists a unique global strong solution (ρ,u )to problem (1.1)–(1.4)satisfying regularity (1.8)for any T <∞.A rather simple proof of Theorem 1.5is provided in section 4.Finally,we recall that in the case when ρ0is bounded away from zero and Ωis a bounded domain in R 3with smooth boundary,Salvi [24]proved the local existence of strong solutions in the class H 2m for every m ≥1.Hence adapting the proofs of Corollary 1.4and Theorem 1.5,we can also prove analogous regularity and global existence results on strong solutions in every class H 2m provided that Ω⊂⊂R 3and ρ0>0on Ω.2.Proof of Theorem 1.3with m =1.In this section,we prove Theorem 1.3for the special case m =1.Let t 0be a ﬁxed time with 0<t 0<T ∗and let us denoteΦ0(T )= T|u (t )|s L r wdt for t 0≤T <T ∗,where (r,s )is any pair satisfying condition (1.14).To prove Theorem 1.3,we haveonly to show that Φ1(T )≤C exp (C exp (C Φ0(T )))for t 0≤T <T ∗.(2.1)Throughout this paper,we denote by C a generic positive constant depending onlyon r ,m ,Φm (t 0),T ∗,Ω,|ρ(0)|L 32∩L ∞,|u (0)|L 6,and the norm of f ,but independent of T in particular.To begin with,we recall from (1.6)that sup0≤t ≤T|ρ(t )|L 32∩L ∞+|√ρu (t )|L 2 +T|u (t )|2D 10dt ≤C(2.2)for t 0≤T <T ∗.2.1.Estimates for T 0|√ρu t (t )|2L 2dt and sup 0≤t ≤T |u (t )|D 10.Next,we willshow thatT 0|√ρu t (t )|2L 2+|u (t )|2D 10∩D 2 dt +sup 0≤t ≤T|u (t )|2D 10≤C exp (C Φ0(T ))(2.3)for t 0≤T <T ∗.To show this,we multiply the momentum equation (1.1)by u t andintegrate over Ω.Then using (1.2)and (1.3),we easily deriveρ|u t |2dx +12d dt |∇u |2dx = ρ(f −u ·∇u )·u t dx andρ|u t |2dx +ddt|∇u |2dx ≤2ρ|f |2dx +2ρ|u ·∇u |2dx.(2.4)On the other hand,since (u,p )is a solution of the stationary Stokes equations−Δu +∇p =Fanddiv u =0inΩ,where F =ρ(f −u ·∇u −u t ),it follows from the classical regularity theory that |∇u |H 1≤C (|F |L 2+|∇u |L 2)(2.5)≤C (|f |L 2+|√ρu t |L 2+|u ·∇u |L 2+|∇u |L 2).1422HYUNSEOK KIMTo estimate the right-hand sides of(2.4)and(2.5),weﬁrst observe that|u·∇u|L2=|u·∇u|L2,2≤C|u|L rw |∇u|L2rr−2,2,(2.6)which follows from H¨o lder inequality in the Lorenz spaces.See Proposition2.1in[18]. Next,we will show that|∇u|L2rr−2,2≤C|∇u|1−3rL2|∇u|3rH1.(2.7)If r=∞,then(2.7)is obvious.Assuming that3<r<∞,we choose r1and r2suchthat3<r1<r<r2<∞and2r =1r1+1r2.Then in view of H¨o lder and Sobolevinequalities,we have|∇u|L2r ir i−2≤|∇u|1−3r iL2|∇u|3r iL6≤|∇u|1−3r iL2(C|∇u|H1)3r ifor each i=1,2.Since L2r r−2,2is a real interpolation space of L2r2r2−2and L2r1r1−2,moreprecisely,L2r r−2,2=(L2r2r2−2,L2r1r1−2)12,2,it thus follows that|∇u|L2rr−2,2≤C|∇u|12L2r2r2−2|∇u|12L2r1r1−2≤C|∇u|1−3r2L2(C|∇u|H1)3r212|∇u|1−3r1L2(C|∇u|H1)3r112,which proves(2.7).For some facts on the real interpolation theory and Lorenz spaces used above,we refer to sections1.3.3and1.18.6in Triebel’s book[30].The estimates(2.6)and(2.7)yield|u·∇u|L2≤C|u|L rw |∇u|2sL2|∇u|3rH1≤η−3s2r C|u|s2L rw|∇u|L2+η|∇u|H1for any small numberη∈(0,1).Substituting this into(2.5),we obtain|∇u|H1≤C|f|L2+|√ρu t|L2+|u|s2L rw|∇u|L2+|∇u|L2,(2.8)and thus|u·∇u|L2≤η−3s2r C|u|s2L rw+1|∇u|L2+C|f|L2+η|√ρu t|L2.Therefore,substituting this estimate into(2.4)and choosing a suﬃciently smallη>0, we conclude that1 2|√ρu t(t)|2L2+ddt|∇u(t)|2L2≤C|f(t)|2L2+C|u(t)|s L rw+1|∇u(t)|2L2(2.9)for t0≤t<T∗.In view of Gronwall’s inequality,we haveT0|√ρu t(t)|2L2dt+sup0≤t≤T|∇u(t)|2L2≤C exp(CΦ0(T))for any T with t0≤T<T∗.Combining this and(2.8),we obtain the desired estimate (2.3).NONHOMOGENEOUS NAVIER–STOKES EQUATIONS14232.2.Estimates for ess sup 0≤t ≤T |√ρu t (t )|2L 2and T 0|u t (t )|2D 10dt .To de-rive these estimates,we diﬀerentiate the momentum equation (1.1)with respect to time t and obtainρu tt +ρu ·∇u t −Δu t +∇p t =ρt (f −u t −u ·∇u )+ρ(f t −u t ·∇u ).Then multiplying this by u t ,integrating over Ω,and using (1.2)and (1.3),we have12d dtρ|u t |2dx + |∇u t |2dx(2.10)=(ρt (f −u t −u ·∇u )+ρ(f t −u t ·∇u ))·u t dx.Note that since ρ∈C ([0,T ];L 32∩L ∞),ρt ∈C ([0,T ];L 32),and u t ∈L 2(0,T ;D 10)forany T <T ∗,the right-hand side of (2.10)is well deﬁned for almost all t ∈(0,T ∗).Hence using ﬁnite diﬀerences in time,we can easily show that the identity (2.10)holds for almost all t ∈(0,T ∗).In view of the continuity equation (1.2)again,we deduce from (2.10)that12ddtρ|u t |2dx + |∇u t |2dx≤2ρ|u ||u t ||∇u t |+ρ|u ||u t ||∇u |2+ρ|u |2|u t ||∇2u |(2.11)+ρ|u |2|∇u ||∇u t |+ρ|u t |2|∇u |+ρ|u ||u t ||∇f |+ρ|u ||f ||∇u t |+ρ|f t ||u t |dx ≡8 j =1I j .Following the arguments in [6],we can estimate each term I j :I 1,I 5≤C |ρ|12L ∞|∇u |L 2|√ρu t |L 3|∇u t |L 2≤C |ρ|34L ∞|∇u |L 2|√ρu t |12L 2|∇u t |32L 2≤C |∇u |4L 2|√ρu t |2L 2+116|∇u t |2L 2,I 2,I 3,I 4≤C |ρ|L ∞|∇u |2L 2|∇u t |L 2|∇u |H 1≤C |∇u |4L 2|∇u |2H 1+116|∇u t |2L 2,I 6,I 7≤C |ρ|L 6|∇u |L 2|f |H 1|∇u t |L 2≤C |∇u |2L 2|f |2H 1+116|∇u t |2L 2,and ﬁnallyI 8≤C |ρ|L 3|f t |L 2|∇u t |L 2≤C |f t |2L 2+116|∇u t |2L 2.Substitution of these estimates into (2.11)yieldsd dt |√ρu t |2L 2+|∇u t |2L 2≤C |∇u |4L 2 |√ρu t |2L 2+|∇u |2H 1+|f |2H 1 +C |f |2H 1+|f t |2L 2 .1424HYUNSEOK KIMTherefore,by virtue of estimate (2.3),we conclude that ess sup 0≤t ≤T|√ρu t (t )|2L 2+T|∇u t (t )|2L 2dt ≤C exp (C Φ0(T ))(2.12)for t 0≤T <T ∗.On the other hand,using the regularity theory of the Stokesequations again,we have|∇u |H 1≤C (|f |L 2+|√ρu t |L 2+|u ·∇u |L 2+|∇u |L 2)≤C |f |L 2+|√ρu t |L 2+|∇u |32L 2|∇u |12H 1+|∇u |L 2and|∇u |H 1,6≤C (|u t |L 6+|u ·∇u |L 6+|f |L 6+|∇u |L 6)≤C|∇u t |L 2+|∇u |2H1+|f |H 1+|∇u |H 1 .Hence it follows immediately from (2.3)and (2.12)that sup0≤t ≤T|u (t )|2D 10∩D 2+T|∇u (t )|2H 1,6dt ≤C exp (C Φ0(T ))(2.13)for t 0≤T <T ∗.2.3.Estimates for sup 0≤t ≤T |∇ρ(t )|H 1and T0|u (t )|2D 3dt .To derive these,we ﬁrst observe that each ρx j (j =1,2,3)satisﬁesρx jt +u ·∇ρx j =−u x j ·∇ρ.Then multiplying this by ρx j ,integrating over Ω,and summing up,we obtaind dt|∇ρ|2dx ≤C|∇u ||∇ρ|2dx ≤C |∇u |L ∞|∇ρ|2L 2.A similar argument shows thatddt|∇2ρ|2dx ≤C|∇u ||∇2ρ|2+|∇2u ||∇ρ||∇2ρ| dx ≤C |∇u |L ∞|∇2ρ|2L 2+|∇2u |L 6|∇ρ|L 3|∇2ρ|L 2.Hence using Sobolev embedding results and then Gronwall’s inequality,we derive thewell-known estimate|∇ρ(t )|H 1≤C exp C t 0|∇u (τ)|H 1,6dτ≤C exp tC |∇u (τ)|2H 1,6+1 dτ .Therefore by virtue of (2.13),we conclude that sup 0≤t ≤T|∇ρ(t )|H 1≤C exp (C exp (C Φ0(T )))(2.14)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1425for t 0≤T <T ∗.Finally,observing from the regularity theory on the Stokes equations that|u |D 3≤C (|ρu t |H 1+|ρu ·∇u |H 1+|ρf |H 1)≤C (|∇ρ|L 3+1) |∇u t |L 2+|∇u |2H 1+|f |H 1 ,we easily deduce from (2.3),(2.12),(2.13),and (2.14)thatT|u (t )|2D 3dt ≤C exp (C exp (C Φ0(T )))(2.15)for t 0≤T <T ∗.This completes the proof of (2.1)and thus the proof of Theorem 1.3with m =1.3.Proof of Theorem 1.3with m ≥ 2.Assume that m ≥ 2.Then to prove Theorem 1.3,it suﬃces to show that the following estimate holds for each k ,0≤k <m :Φk +1(T )≤C exp C expC Φk (T )10m for t 0≤T <T ∗.(3.1)The case k =0was already proved in section 2and so it remains to prove (3.1)forthe case 1≤k <m .Let k be a ﬁxed integer with 1≤k <m .From (1.13),we recall thatΦk (T )=1+sup0≤j<ksup0≤t ≤T|∂jt u (t )|D 10∩D 2k −2j +T|∂jt u (t )|2D 2k −2j +1dt(3.2)+sup 0≤t ≤T|∇ρ(t )|H 2k −1+ess sup 0≤t ≤T|√ρ∂kt u (t )|L 2+T|∂kt u (t )|2D 10dtfor any T <T ∗.3.1.Estimates for ∂j t(u ·∇u ),∂j +1t ρ,and ∂jt (ρu )with 0≤j ≤k .To estimate nonlinear terms,we will make repeated use of the following simple lemma whose proof is omitted.Lemma 3.1.If g ∈D 10∩D j ,h ∈H i ,0≤i ≤j ,and j ≥2,thengh ∈H iand|gh |H i ≤C |g |D 10∩D j |h |H i for some constant C >0depending only on j and Ω.Using this lemma together with the fact that∂j t (u·∇u )=j i =0j !i !(j −i )!∂itu ·∇∂j −i t u,we can estimate ∂jt(u ·∇u )as follows:for 0≤j <k ,|∂jt(u ·∇u )|H 2k −2j −1≤Cj i =0|∂it u ·∇∂j −i t u |H 2k −2j −1≤C j i =0|∂it u |D 10∩D 2k −2j |∇∂j −i tu |H 2k −2j −1≤Cj i =0|∂it u |D 10∩D 2k −2i |∂j −i tu |D 10∩D 2k −2(j −i )1426HYUNSEOK KIM and|∂k t(u·∇u)|L2≤C k−1i=0|∂i t u·∇∂k−itu|L2+|∂k t u·∇u|L2≤C k−1i=0|∂i t u|D1∩D2|∇∂k−itu|L2+|∂k t u|D1|∇u|H1≤C k−1i=0|∂i t u|D1∩D2k−2i|∂k−itu|D1+|∂k t u|D1|u|D1∩D2.Hence it follows from(3.2)thatsup 0≤j<ksup0≤t≤T|∂j t(u·∇u)(t)|H2k−2j−1+T|∂k t(u·∇u)(t)|2L2dt≤CΦk(T)4(3.3)for t0≤T<T∗.Applying Lemma3.1to the continuity equationρt=−div(ρu)=−u·∇ρ,(3.4)we also deduce thatsup 0≤t≤T |ρt(t)|H2k−1≤CΦk(T)2for t0≤T<T∗.(3.5)Using(3.4)and(3.5),we can show thatsup 1≤j<ksup0≤t≤T|∂j+1tρ(t)|H2k−2j+T|∂k+1tρ(t)|2L2dt≤CΦk(T)2k+4(3.6)for t0≤T<T∗.A simple inductive proof of(3.6)may be based on the observation that for1≤j<k,|∂j+1tρ|H2k−2j=|−∂j t(u·∇ρ)|H2k−2j≤Cji=0|∂j−itu·∇∂i tρ|H2k−2j≤Cji=0|∂j−itu|D1∩D2k−2j|∇∂itρ|H2k−2j≤Cji=0|∂j−itu|D1∩D2k−2(j−i)|∂itρ|H2k−2(i−1)and|∂k+1t ρ|L2≤Cki=0|∂k−itu·∇∂i tρ|L2≤Cki=0|∂k−itu|D1|∂i tρ|H2.Moreover,it follows easily from(3.6)thatsup 0≤j<ksup0≤t≤T|∂j t(ρu)(t)|H2k−2j+T|∂k t(ρu)(t)|2H1dt≤CΦk(T)4k+10(3.7)NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1427for t 0≤T <T ∗.Finally,recalling that∂k +1t f ∈L 2(0,∞;L 2)and ∂jt f ∈L 2(0,∞;H 2k −2j +1)for 0≤j ≤k,we deduce from standard embedding results that∂jt f ∈C ([0,∞);H 2k −2j )for0≤j ≤k.3.2.Estimates for T 0|√ρ∂k +1t u (t )|2L 2dt and sup 0≤t ≤T |∂k t u (t )|D 10.Fromthe momentum equation (1.1),we deriveρ ∂k t u t −Δ∂k t u +∇∂k t p =∂k t (ρf −ρu ·∇u )+ ρ∂k t u t −∂k t (ρu t ) .Hence multiplying this by ∂k +1t u and integrating over Ω,we haveρ|∂k +1t u |2dx +12d dt|∇∂k t u |2dx=∂k t (ρf −ρu ·∇u )+ ρ∂k t u t −∂kt (ρu t ) ·∂k +1tu dx (3.8)=I 0,1+k j =1k !j !(k −j )!(I j,1+I j,2),whereI j,1=∂j t ρ∂k −j t (f −u ·∇u )·∂k +1t u dx,I j,2=−∂j t ρ∂k −j t u t ·∂k +1t u dx.We easily estimate I 0,1as follows:I 0,1≤|ρ|12L ∞ |∂k t f |L 2+|∂k t (u ·∇u )|L 2 |√ρ∂k +1t u |L 2≤C |∂k t f |2L 2+|∂kt (u ·∇u )|2L2 +12|√ρ∂k +1t u |2L 2.To estimate I j,1for 1≤j ≤k ,we rewrite it asI j,1=d dt∂j t ρ∂k −j t (f −u ·∇u )·∂k t u dx − ∂j +1t ρ∂k −j t (f −u ·∇u )·∂kt u dx −∂j t ρ∂k −j +1t (f −u ·∇u )·∂kt u dx and observe that −∂j +1tρ∂k −j t (f −u ·∇u )·∂kt u dx ≤C |∂k −j t f |2H 1+|∂k −j t (u ·∇u )|2H 1 |∂j +1t ρ|2L 2+|∇∂k t u |2L 2and−∂j tρ∂k −j +1t (f −u ·∇u )·∂k t u dx ≤C |∂j t ρ|2H 1 |∂k −j +1t f |2L 2+|∂k −j +1t (u ·∇u )|2L 2 +|∇∂k t u |2L 2.1428HYUNSEOK KIMUsing the continuity equation (1.2),we can also estimate I j,2as follows:I 1,2=− ρt 12|∂k t u |2 t dx =−d dt ρt 12|∂k t u |2dx + ∂2t ρ12|∂k t u |2dx =−d dt ρu ·∇ 12|∂k t u |2 dx + ∂t (ρu )·∇ 12|∂k tu |2dx ≤−d dt (ρu ·∇∂k t u )·∂k t u dx +C |∂t (ρu )|H 1|∇∂k t u |2L 2and similarlyI j,2=−d dt ∂j t ρ∂k −j t u t ·∂k t u dx + ∂j +1t ρ∂k −j t u t +∂j t ρ∂k −j +1t u t ·∂kt u dx ≤−d dt ∂j −1t (ρu )·∇ ∂k −j +1t u ·∂kt u dx +C |∂j t (ρu )|H 1|∇∂k −j +1t u |L 2+|∂j −1t (ρu )|H 1|∇∂k −j +2t u |L 2 |∇∂kt u |L 2for 2≤j ≤k .Substituting all the estimates into (3.8),we have12 ρ|∂k +1t u |2dx +12d dt|∇∂k t u |2dx ≤d dt ⎛⎝k j =1k !j !(k −j )!∂j t ρ∂k −j t (f −u ·∇u )·∂k t u −(ρu ·∇∂k t u )·∂k t u ⎞⎠dx −d dt kj =2k !j !(k −j )!∂j −1t(ρu )·∇ ∂k −j +1t u ·∂kt u dx+Ck −1 j =1|∂jt (ρu )|2H 1+|∂j +1t ρ|2H 1|∂k −j t f |2H 1+|∂k −jt (u·∇u )|2H 1+|∂k −j +1t u |2D 1+C |∂k +1t ρ|2L 2|f |2H 1+|u |2D 10∩D 2+C 1+|∂t ρ|2H 1 |∂k t f |2L 2+|∂k t (u ·∇u )|2L2 +C |∂k t (ρu )|2H 1+C 1+|∂t (ρu )|2H 1+|∇∂t u |2L 2 |∇∂k t u |2L 2.Hence,integrating this in time over (t 0,T )and using (3.3),(3.5),(3.6),and (3.7)together with the estimates|∂j t ρ||∂k −j t (f −u ·∇u )||∂k t u |dx≤η−1|∂j t ρ|2H 1|∂k −j t (f −u ·∇u )|2L 2+η|∇∂k t u |L 2,ρ|u ||∇∂k t u ||∂k t u |dx ≤η−3C |ρ|3L ∞|∇u |4L 2|√ρ∂k t u |2L 2+η|∇∂k t u |2L 2and|∂j −1t (ρu )||∇∂k −j +1t u ||∂kt u |+|∂k −j +1t u ||∇∂kt u |dx≤η−1C |∂j −1t (ρu )|2H 1|∇∂k −j +1tu |2L 2+η|∇∂kt u |L 2,NONHOMOGENEOUS NAVIER–STOKES EQUATIONS1429 whereηis any small positive number,we deduce thatTt0|√ρ∂k+1tu(t)|2L2dt+|∇∂k t u(T)|2L2≤CΦk(T)20m+CTt01+|∂t(ρu)(t)|2H1+|u t(t)|2D1|∇∂k t u(t)|2L2dtfor t0≤T<T∗.Note thatT t01+|∂t(ρu)(t)|2H1+|u t(t)|2D1dt≤CΦk(T)10m.Therefore,in view of Gronwall’s inequality,we conclude that T0|√ρ∂k+1tu(t)|2L2dt+sup0≤t≤T|∂k t u(t)|2D1≤C expCΦk(T)10m(3.9)for any T with t0≤T<T∗.3.3.Estimates for ess sup0≤t≤T|√ρ∂k+1tu(t)|L2andT|∂k+1tu(t)|2D1dt.From the momentum equation(1.1),it follows thatρ∂k+1tut+ρu·∇∂k+1tu−Δ∂k+1tu+∇∂k+1tp=∂k+1t(ρf)+ρ∂k+1tu t−∂k+1t(ρu t)+ρu·∇∂k+1tu−∂k+1t(ρu·∇u).Multiplying this by∂k+1tu and integrating overΩ,we have1 2ddtρ|∂k+1tu|2dx+|∇∂k+1tu|2dx=∂k+1t(ρf)·∂k+1tu dx+ρ∂k+1tu t−∂k+1t(ρu t)·∂k+1tu dx(3.10)+ρu·∇∂k+1tu−∂k+1t(ρu·∇u)·∂k+1tu dx.This identity can be derived rigorously by using a standardﬁnite diﬀerence method because if0<T<T∗,then∂m tρ∈L2(0,T;L32∩L2)and∂j tρ∈C([0,T];L32∩L∞) for0≤j<m.Theﬁrst term of the right-hand side in(3.10)is bounded byCkj=0|∂j tρ||∂k−j+1tf||∂k+1tu|dx+|∂k+1tρ||f||∂k+1tu|dx≤Ckj=0|∂j tρ|2H1|∂k−j+1tf|2L2+C|∂k+1tρ|2L2|f|2H1+16|∇∂k+1tu|2L2.In view of the continuity equation(1.2),we can rewrite the second term as−k+1j=1(k+1)!j!(k−j+1)!∂j tρ∂k−j+1tu t·∂k+1tu dx=−k+1j=1(k+1)!j!(k−j+1)!∂j−1t(ρu)·∇∂k−j+2tu·∂k+1tudx,1430HYUNSEOK KIM which is bounded byC|ρ|L∞|u|2D10∩D2|√ρ∂k+1tu|2L2+Ckj=1|∂j t(ρu)|2H1|∂k−j+1tu|2D1+16|∇∂k+1tu|2L2.Finally,the last term is bounded byCkj=1|∂j t(ρu)||∇∂k−j+1tu||∂k+1tu|dx+|∂k+1t(ρu)||∇u||∂k+1tu|dx≤Ckj=1|∂j t(ρu)|2H1+|∂j tρ|2H1|u|2D1∩D2|∂k−j+1tu|2D1+C|∂k+1tρ|2L2|u|4D1∩D2+C|ρ|L∞|u|2D1∩D2|√ρ∂k+1tu|2L2+1|∇∂k+1tu|2L2.Hence substituting these estimates into(3.10),we haved dtρ|∂k+1tu|2dx+|∇∂k+1tu|2dx≤C1+|u|2D1∩D2|√ρ∂k+1tu|2L2+Ckj=0|∂j tρ|2H1|∂k−j+1tf|2L2+C|∂k+1tρ|2L2|f|2H1 +Ckj=1|∂j t(ρu)|2H1+|∂j tρ|2H1|u|2D1∩D2|∂k−j+1tu|2D1+C|∂k+1tρ|2L2|u|4D1∩D2.Therefore,by virtue of(3.5),(3.6),(3.7),and(3.9),we conclude thatess sup0≤t≤T |√ρ∂k+1tu(t)|L2+T|∂k+1tu(t)|2D1dt≤C expCΦk(T)10m(3.11)for any T with t0≤T<T∗.3.4.Estimates for sup0≤t≤T|∂j t u(t)|D10∩D2k−2j+2with0≤j≤k.Toderive these estimates,we observe that∂j t u∈C([0,T∗);D10,σ)and−Δ∂j t u+∇∂j t p=∂j t(ρf−ρu·∇u−ρu t) (3.12)for each j≤k.From(3.5),(3.6),(3.9),and(3.11),it follows easily thatess sup0≤t≤T|∂k t(ρu·∇u)(t)|L2+|∂k t(ρu t)(t)|L2≤C expCΦk(T)10mfor t0≤T<T∗.Hence applying the regularity theory of the Stokes equations to (3.12)with j=k,we obtainsup 0≤t≤T |∂k t u(t)|D1∩D2≤C expCΦk(T)10mfor t0≤T<T∗.It also follows from the Stokes regularity theory that for0≤j<k,|∂j t u|D10∩D2k−2j+2≤C|∂jt(ρf−ρu·∇u−ρu t)|H2k−2j+C|∂j t u|D1.(3.13)。

产业集群理论综述

产业集群理论综述当今世界经济中，产业集群的发展已成为一种最引人注目的现象，它既是产业组织方式的大提升，也是有效的经济发展战略。

众多的产业集群构成了色彩斑斓，块状明显的“经济马赛克”，世界财富的绝大部分都通过这些块状区域内制造出来的。

20世纪90年代中期，美国380个产业集群生产了全美接近60%的产值。

在意大利，形成了199个产业集群，每年200多亿美元的出口额主要由其中的66个产业集群生产。

印度在2000年就有350多个产业集群，产量占印度国内产量的75%－80%，创造了制造业出口额的60%。

法、英、德以及拉丁美洲等国都将产业集群作为一种行之有效的产业组织方式，列为主要的经济发展战略。

2001年，美国商务部发表了一篇《基于产业集群经济发展的州长指南》，分析了产业集群对地方经济发展的重要性，提出要利用政策有力地推动产业集群的发展。

在我国东南沿海（如浙江、广东及江苏省苏南等）不少发达地区，产业集群也取得了显著的经济绩效。

1产业集群的概念1.1.产业集群的概念1、波特给出的定义现在理论界对产业集群的公认的概念是由迈克尔.波特教授提出的。

波特首先于1990年的《国家竞争优势》一书中提出了“集群”（Clusters）的概念，某一特定领域内相互联系、在地理位置上集中的公司和机构的集合[1]。

在1998年发表的《集群于心经济竞争学》一文中，波特指出：产业集群包括一批对竞争起重要作用的、相互联系的实体和其他组织；产业集群还经常向下延伸到销售渠道和客户，并向侧面拓展至辅助性制造商，以及与技能技术或投入相关的产业公司；许多产业集群还包括提供专业化培训、教育、信息研究和技术支持的政府和其他结构[2]。

在其后的著作《竞争论》（2003）中波特将产业集群（Industry Cluster）的定义加以扩展，波特认为：产业集群是以某一个或几个相关产业为核心，以价值链为基础的地方生产系统，大量产业联系密切的企业（包含最终产品或服务厂商，专业元件、零部件、机器设备以及服务供应商、金融机构、及其相关产业的厂商）及相关支撑机构在空间上集聚，并形成强劲、持续竞争优势的现象 [3]。

T.W. ANDERSON (1971). The Statistical Analysis of Time Series. Series in Probability and Ma

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The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time seri

904N.E.Huang and others10.Discussion98711.Conclusions991References993 A new method for analysing has been devel-oped.The key part of the methodany complicated data set can be decomposed intoof‘intrinsic mode functions’Hilbert trans-This decomposition method is adaptive,and,highly eﬃcient.Sinceapplicable to nonlinear and non-stationary processes.With the Hilbert transform,Examplesthe classical nonlinear equation systems and dataare given to demonstrate the power new method.data are especially interesting,for serve to illustrate the roles thenonlinear and non-stationary eﬀects in the energy–frequency–time distribution.Keywords:non-stationary time series;nonlinear diﬀerential equations;frequency–time spectrum;Hilbert spectral analysis;intrinsic time scale;empirical mode decomposition1.Introductionsensed by us;data analysis serves two purposes:determine the parameters needed to construct the necessary model,and to conﬁrm the model we constructed to represent the phe-nomenon.Unfortunately,the data,whether from physical measurements or numerical modelling,most likely will have one or more of the following problems:(a)the total data span is too short;(b)the data are non-stationary;and(c)the data represent nonlinear processes.Although each of the above problems can be real by itself,the ﬁrst two are related,for a data section shorter than the longest time scale of a sta-tionary process can appear to be non-stationary.Facing such data,we have limited options to use in the analysis.Historically,Fourier spectral analysis has provided a general method for examin-the data analysis has been applied to all kinds of data.Although the Fourier transform is valid under extremely general conditions(see,for example,Titchmarsh1948),there are some crucial restrictions of Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis905the Fourier spectral analysis:the system must be linear;and the data must be strict-ly periodic or stationary;otherwise,the resulting spectrum will make little physicalsense.to the Fourier spectral analysis methods.Therefore,behoves us review the deﬁnitions of stationarity here.According to the traditional deﬁnition,a time series,X (t ),is stationary in the wide sense,if,for all t ,E (|X (t )2|)<∞,E (X (t))=m,C (X (t 1),X (t 2))=C (X (t 1+τ),X (t 2+τ))=C (t 1−t 2),(1.1)in whichE (·)is the expected value deﬁned as the ensemble average of the quantity,and C (·)is the covariance function.Stationarity in the wide sense is also known as weak stationarity,covariance stationarity or second-order stationarity (see,forexample,Brockwell &Davis 1991).A time series,X (t ),is strictly stationary,if the joint distribution of [X (t 1),X (t 2),...,X (t n )]and [X (t 1+τ),X (t 2+τ),...,X (t n +τ)](1.2)are the same for all t i and τ.Thus,a strictly stationaryprocess with ﬁnite second moments is alsoweakly stationary,but the inverse is not true.Both deﬁnitions arerigorous but idealized.Other less rigorous deﬁnitions have also beenused;for example,that is stationary within a limited timespan,asymptotically stationary is for any random variableis stationary when τin equations (1.1)or (1.2)approaches inﬁnity.In practice,we can only have data for ﬁnite time spans;these deﬁni-tions,we haveto makeapproximations.Few of the data sets,from either natural phenomena or artiﬁcial sources,can satisfy these deﬁnitions.It may be argued thatthe diﬃculty of invoking stationarity as well as ergodicity is not on principlebut on practicality:we just cannot have enough data to cover all possible points in thephase plane;therefore,most of the cases facing us are transient in nature.This is the reality;we are forced to face it.Fourier spectral analysis also requires linearity.can be approximated by linear systems,the tendency tobe nonlinear whenever their variations become ﬁnite Compounding these complications is the imperfection of or numerical schemes;theinteractionsof the imperfect probes even with a perfect linear systemcan make the ﬁnal data nonlinear.For the above the available data are ally of ﬁnite duration,non-stationary and from systems that are frequently nonlinear,either intrinsicallyor through interactions with the imperfect probes or numerical schemes.Under these conditions,Fourier spectral analysis is of limited use.For lack of alternatives,however,Fourier spectral analysis is still used to process such data.The uncritical use of Fourier spectral analysis the insouciant adoption of the stationary and linear assumptions may give cy range.a delta function will giveProc.R.Soc.Lond.A (1998)906N.E.Huang and othersa phase-locked wide white Fourier spectrum.Here,added to the data in the time domain,Constrained bythese spurious harmonics the wide frequency spectrum cannot faithfully represent the true energy density in the frequency space.More seri-ously,the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give theﬁnal delta function. Thus,the Fourier components might make mathematical sense,but do not really make physical sense at all.Although no physical process can be represented exactly by a delta function,some data such as the near-ﬁeld strong earthquake records areFourier spectra.Second,tions;wave-proﬁles.Such deformations,later,are the direct consequence of nonlinear eﬀects.Whenever the form of the data deviates from a pure sine or cosine function,the Fourier spectrum will contain harmonics.As explained above, both non-stationarity and nonlinearity can induce spurious harmonic components that cause energy spreading.The consequence is the misleading energy–frequency distribution forIn this paper,modemode functions The decomposition is based on the direct extraction of theevent on the time the frequency The decomposition be viewed as an expansion of the data in terms of the IMFs.Then,based on and derived from the data,can serve as the basis of that expansion linear or nonlinear as dictated by the data,Most important of all,it is adaptive.As will locality and adaptivity are the necessary conditions for the basis for expanding nonlinear and non-stationary time orthogonality is not a necessary criterionselection for a nonlinearon the physical time scaleslocal energy and the instantaneous frequencyHilbert transform can give us a full energy–frequency–time distribution of the data. Such a representation is designated as the Hilbert spectrum;it would be ideal for nonlinear and non-stationary data analysis.We have obtained good results and new insights by applying the combination of the EMD and Hilbert spectral analysis methods to various data:from the numerical results of the classical nonlinear equation systems to data representing natural phe-nomena.The classical nonlinear systems serve to illustrate the roles played by the nonlinear eﬀects in the energy–frequency–time distribution.With the low degrees of freedom,they can train our eyes for more complicated cases.Some limitations of this method will also be discussed and the conclusions presented.Before introducing the new method,we willﬁrst review the present available data analysis methods for non-stationary processes.Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis9072.Review of non-stationary data processing methodsWe willﬁrstgivea brief survey of themethodsstationary data.are limited to linear systems any method is almost strictly determined according to the special ﬁeld in which the application is made.The available methods are reviewed as follows.(a )The spectrogramnothing but a limited time window-width Fourier spectral analysis.the a distribution.Since it relies on the tradition-al Fourier spectral analysis,one has to assume the data to be piecewise stationary.This assumption is not always justiﬁed in non-stationary data.Even if the data are piecewise stationary how can we guarantee that the window size adopted always coincides with the stationary time scales?What can we learn about the variations longer than the local stationary time scale?Will the collection of the locally station-ary pieces constitute some longer period phenomena?Furthermore,there are also practical diﬃculties in applying the method:in order to localize an event in time,the window width must be narrow,but,on the other hand,the frequency resolu-tion requires longer time series.These conﬂicting requirements render this method of limited usage.It is,however,extremely easy to implement with the fast Fourier transform;thus,ithas attracted a wide following.Most applications of this methodare for qualitative display of speech pattern analysis (see,for example,Oppenheim &Schafer 1989).(b )The wavelet analysisThe wavelet approach is essentially an adjustable window Fourier spectral analysiswith the following general deﬁnition:W (a,b ;X,ψ)=|a |−1/2∞−∞X (t )ψ∗ t −b ad t,(2.1)in whichψ∗(·)is the basic wavelet function that satisﬁes certain very general condi-tions,a is the dilation factor and b is the translationof theorigin.Although time andfrequency do not appear explicitly in the transformed result,the variable 1/a givesthe frequency scale and b ,the temporal location of an event.An intuitive physical explanation of equation (2.1)is very simple:W (a,b ;X,ψ)is the ‘energy’of X ofscale a at t =b .Because of this basic form of at +b involvedin thetransformation,it is also knownas aﬃnewavelet analysis.For speciﬁc applications,the basic wavelet function,ψ∗(·),can be modiﬁed according to special needs,but the form has to be given before the analysis.In most common applications,however,the Morlet wavelet is deﬁned as Gaussian enveloped sine and cosine wave groups with 5.5waves (see,for example,Chan 1995).Generally,ψ∗(·)is not orthogonalfordiﬀerent a for continuous wavelets.Although one can make the wavelet orthogonal by selecting a discrete set of a ,thisdiscrete wavelet analysis will miss physical signals having scale diﬀerent from theselected discrete set of a .Continuous or discrete,the wavelet analysis is basically a linear analysis.A very appealing feature of the wavelet analysis is that it provides aProc.R.Soc.Lond.A (1998)908N.E.Huang and othersuniform resolution for all the scales.Limited by the size of thebasic wavelet function,the downside of the uniform resolution is uniformly poor resolution.Although wavelet analysis has been available only in the last ten years or so,it hasbecome extremelypopular.Indeed,it is very useful in analysing data with gradualfrequency changes.Since it has an analytic form for the result,it has attracted extensive attention of the applied mathematicians.Most of its applications have been in edge detection and image compression.Limited applications have also been made to the time–frequency distribution in time series (see,for example,Farge 1992;Long et al .1993)andtwo-dimensionalimages (Spedding et al .1993).Versatile as the wavelet analysis is,the problem with the most commonly usedMorlet wavelet is its leakage generated by the limited length of the basic wavelet function,whichmakesthe quantitativedeﬁnitionof the energy–frequency–time dis-tribution diﬃcult.Sometimes,the interpretation of the wavelet can also be counter-intuitive.For example,to deﬁne a change occurring locally,one must look for theresult in the high-frequencyrange,for the higher the frequency the more localized thebasic wavelet will be.If a local event occurs only in the low-frequency range,one willstill be forced to look for its eﬀects inthe high-frequencyrange.Such interpretationwill be diﬃcultif it is possible at all (see,for example,Huang et al .1996).Another diﬃculty of the wavelet analysis is its non-adaptive nature.Once the basic waveletis selected,one will have to use it to analyse all the data.Since the most commonlyused Morlet wavelet is Fourier based,it also suﬀers the many shortcomings of Fouri-er spectral analysis:it can only give a physically meaningful interpretation to linear phenomena;it can resolve the interwave frequency modulation provided the frequen-cy variationis gradual,but it cannot resolve the intrawave frequency modulation because the basic wavelet has a length of 5.5waves.Inspite of all these problems,wavelet analysisisstillthe bestavailable non-stationary data analysis method so far;therefore,we will use it in this paper as a reference to establish the validity and thecalibration of the Hilbert spectrum.(c )The Wigner–Ville distributionThe Wigner–Ville distribution is sometimes alsoreferred toas the Heisenberg wavelet.By deﬁnition,it is the Fourier transform of the central covariance function.For any time series,X (t ),we can deﬁne the central variance as C c (τ,t )=X (t −12τ)X ∗(t +12τ).(2.2)Then the Wigner–Ville distribution is V (ω,t )=∞−∞C c (τ,t )e −i ωτd τ.(2.3)This transform has been treated extensively by Claasen &Mecklenbr¨a uker (1980a ,b,c )and by Cohen (1995).It has been extremely popular with the electrical engi-neering community.The diﬃculty with this method is the severe cross terms as indicated by the exis-tence of negativepowerfor some frequency ranges.Although this shortcoming canbe eliminated by using the Kernel method (see,for example,Cohen 1995),the resultis,then,basically that of a windowed Fourier analysis;therefore,itsuﬀers all thelim-itations of the Fourier analysis.An extension of this method has been made by Yen(1994),who used the Wigner–Ville distribution to deﬁne wave packets that reduce Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis909 a complicated data set to aﬁnite number of simple components.This extension is very powerful and can be applied to a variety of problems.The applications to complicated data,however,require a great amount of judgement.(d)Evolutionary spectrumThe evolutionary spectrum wasﬁrst proposed by Priestley(1965).The basic idea is to extend the classic Fourier spectral analysis to a more generalized basis:from sine or cosine to a family of orthogonal functions{φ(ω,t)}indexed by time,t,and deﬁned for all realω,the frequency.Then,any real random variable,X(t),can beexpressed asX(t)= ∞−∞φ(ω,t)d A(ω,t),(2.4)in which d A(ω,t),the Stieltjes function for the amplitude,is related to the spectrum asE(|d A(ω,t)|2)=dµ(ω,t)=S(ω,t)dω,(2.5) whereµ(ω,t)is the spectrum,and S(ω,t)is the spectral density at a speciﬁc time t,also designated as the evolutionary spectrum.If for eachﬁxedω,φ(ω,t)has a Fourier transformφ(ω,t)=a(ω,t)e iΩ(ω)t,(2.6) then the function of a(ω,t)is the envelope ofφ(ω,t),andΩ(ω)is the frequency.If, further,we can treatΩ(ω)as a single valued function ofω,thenφ(ω,t)=α(ω,t)e iωt.(2.7) Thus,the original data can be expanded in a family of amplitude modulated trigono-metric functions.The evolutionary spectral analysis is very popular in the earthquake communi-ty(see,for example,Liu1970,1971,1973;Lin&Cai1995).The diﬃculty of its application is toﬁnd a method to deﬁne the basis,{φ(ω,t)}.In principle,for this method to work,the basis has to be deﬁned a posteriori.So far,no systematic way has been oﬀered;therefore,constructing an evolutionary spectrum from the given data is impossible.As a result,in the earthquake community,the applications of this method have changed the problem from data analysis to data simulation:an evo-lutionary spectrum will be assumed,then the signal will be reconstituted based on the assumed spectrum.Although there is some general resemblance to the simulated earthquake signal with the real data,it is not the data that generated the spectrum. Consequently,evolutionary spectrum analysis has never been very useful.As will be shown,the EMD can replace the evolutionary spectrum with a truly adaptive representation for the non-stationary processes.(e)The empirical orthogonal function expansion(EOF)The empirical orthogonal function expansion(EOF)is also known as the principal component analysis,or singular value decomposition method.The essence of EOF is brieﬂy summarized as follows:for any real z(x,t),the EOF will reduce it toz(x,t)=n1a k(t)f k(x),(2.8)Proc.R.Soc.Lond.A(1998)910N.E.Huang and othersin whichf j·f k=δjk.(2.9)The orthonormal basis,{f k},is the collection of the empirical eigenfunctions deﬁned byC·f k=λk f k,(2.10)where C is the sum of the inner products of the variable.EOF represents a radical departure from all the above methods,for the expansion basis is derived from the data;therefore,it is a posteriori,and highly eﬃcient.The criticalﬂaw of EOF is that it only gives a distribution of the variance in the modes deﬁned by{f k},but this distribution by itself does not suggest scales or frequency content of the signal.Although it is tempting to interpret each mode as indepen-dent variations,this interpretation should be viewed with great care,for the EOF decomposition is not unique.A single component out of a non-unique decomposition, even if the basis is orthogonal,does not usually contain physical meaning.Recently, Vautard&Ghil(1989)proposed the singular spectral analysis method,which is the Fourier transform of the EOF.Here again,we have to be sure that each EOF com-ponent is stationary,otherwise the Fourier spectral analysis will make little sense on the EOF components.Unfortunately,there is no guarantee that EOF compo-nents from a nonlinear and non-stationary data set will all be linear and stationary. Consequently,singular spectral analysis is not a real improvement.Because of its adaptive nature,however,the EOF method has been very popular,especially in the oceanography and meteorology communities(see,for example,Simpson1991).(f)Other miscellaneous methodsOther than the above methods,there are also some miscellaneous methods such as least square estimation of the trend,smoothing by moving averaging,and diﬀerencing to generate stationary data.Methods like these,though useful,are too specialized to be of general use.They will not be discussed any further here.Additional details can be found in many standard data processing books(see,for example,Brockwell &Davis1991).All the above methods are designed to modify the global representation of the Fourier analysis,but they all failed in one way or the other.Having reviewed the methods,we can summarize the necessary conditions for the basis to represent a nonlinear and non-stationary time series:(a)complete;(b)orthogonal;(c)local;and (d)adaptive.Theﬁrst condition guarantees the degree of precision of the expansion;the second condition guarantees positivity of energy and avoids leakage.They are the standard requirements for all the linear expansion methods.For nonlinear expansions,the orthogonality condition needs to be modiﬁed.The details will be discussed later.But even these basic conditions are not satisﬁed by some of the above mentioned meth-ods.The additional conditions are particular to the nonlinear and non-stationary data.The requirement for locality is the most crucial for non-stationarity,for in such data there is no time scale;therefore,all events have to be identiﬁed by the time of their occurences.Consequently,we require both the amplitude(or energy) and the frequency to be functions of time.The requirement for adaptivity is also crucial for both nonlinear and non-stationary data,for only by adapting to the local variations of the data can the decomposition fully account for the underlying physics Proc.R.Soc.Lond.A(1998)Nonlinear and non-stationary time series analysis911of the processes and not just to fulﬁl the mathematical requirements for ﬁtting the data.This is especially important for the nonlinear phenomena,for a manifestation of nonlinearity is the ‘harmonic distortion’in the Fourier analysis.The degree of distortion depends on the severity of nonlinearity;therefore,one cannot expect a predetermined basis to ﬁt all the phenomena.An easy way to generate the necessary adaptive basis is to derive the basis from the data.In this paper,we will introduce a general method which requires two steps in analysing the data as follows.The ﬁrst step is to preprocess the data by the empirical mode decomposition method,with which the data are decomposed into a number of intrinsic mode function components.Thus,we will expand the data in a basis derived from the data.The second step is to apply the Hilbert transform to the decomposed IMFs and construct the energy–frequency–time distribution,designated as the Hilbert spectrum,from which the time localities of events will be preserved.In other words,weneed the instantaneous frequency and energy rather than the global frequency and energy deﬁned by the Fourier spectral analysis.Therefore,before goingany further,we have to clarify the deﬁnition of the instantaneous frequency.3.Instantaneous frequencyis to accepting it only for special ‘monocomponent’signals 1992;Cohen 1995).Thereare two basicdiﬃculties with accepting the idea of an instantaneous fre-quency as follows.The ﬁrst one arises from the inﬂuence of theFourier spectral analysis.In the traditional Fourier analysis,the frequency is deﬁned for thesineor cosine function spanning the whole data length with constant ampli-tude.As an extension of this deﬁnition,the instantaneous frequencies also have torelate to either a sine or a cosine function.Thus,we need at least one full oscillationof a sineor a cosine wave to deﬁne the local frequency value.According to this logic,nothing full wave will do.Such a deﬁnition would not make sense forThe secondarises from the non-unique way in deﬁning the instantaneousfrequency.Nevertheless,this diﬃculty is no longer serious since the introduction ofthe meanstomakethedata analyticalthrough the Hilbert transform.Diﬃculties,however,still exist as ‘paradoxes’discussed by Cohen (1995).For an arbitrary timeseries,X (t ),we can always have its Hilbert Transform,Y (t ),as Y (t )=1πP∞−∞X (t )t −t d t,(3.1)where P indicates the Cauchy principal value.This transformexists forallfunctionsof class L p(see,for example,Titchmarsh 1948).With this deﬁnition,X (t )and Y (t )form the complex conjugate pair,so we can have an analytic signal,Z (t ),as Z (t )=X (t )+i Y (t )=a (t )e i θ(t ),(3.2)in which a (t )=[X 2(t )+Y 2(t )]1/2,θ(t )=arctanY (t )X (t ).(3.3)Proc.R.Soc.Lond.A (1998)912N.E.Huang andothers Theoretically,there are inﬁnitely many ways of deﬁning the imaginary part,but the Hilbert transform provides a unique way of deﬁning the imaginary part so that the result is ananalyticfunction.A brief tutorial on the Hilbert transform with theemphasis on its physical interpretation can be found in Bendat &Piersol is the bestlocal ﬁtan amplitude and phase varying trigonometric function to X (t ).Even with the Hilbert transform,there is still controversy in deﬁning the instantaneous frequency as ω=d θ(t )d t .(3.4)This leads Cohen (1995)to introduce the term,‘monocomponent function’.In prin-ciple,some limitations on the data are necessary,forthe instantaneous frequencygiven in equation (3.4)is a single value function of time.At any given time,thereis only one frequency value;therefore,it can only represent one component,hence ‘monocomponent’.Unfortunately,no cleardeﬁnition of the ‘monocomponent’signalwas given to judge whether a function is or is not ‘monocomponent’.For lack ofa precise deﬁnition,‘narrow band’was adopted a on the data for the instantaneous frequency to make sense (Schwartz et al .1966).There are two deﬁnitions for bandwidth.The ﬁrst one is used in the study of the probability properties of the signalsand waves,wherethe processes are assumed tobe stationary and Gaussian.Then,the bandwidth can be deﬁned in spectral moments The expected number of zero crossings per unit time is given byN 0=1π m 2m 0 1/2,(3.5)while the expected number of extrema per unit time is given byN 1=1π m 4m 2 1/2,(3.6)in which m i is the i th moment of the spectrum.Therefore,the parameter,ν,deﬁnedas N 21−N 20=1π2m 4m 0−m 22m 2m 0=1π2ν2,(3.7)oﬀers a standard bandwidth measure (see,for example,Rice 1944a,b ,1945a,b ;Longuet-Higgins 1957).For a narrow band signal ν=0,the expected numbers extrema and zero crossings have to equal.the spectrum,but in a diﬀerent way.coordinates as z (t )=a (t )e i θ(t ),(3.8)with both a (t )and θ(t )being functions of time.If this function has a spectrum,S (ω),then the mean frequency is given byω = ω|S (ω)|2d ω,(3.9)Proc.R.Soc.Lond.A (1998)Nonlinear and non-stationary time series analysis913which can be expressed in another way asω =z ∗(t )1i dd tz (t )d t=˙θ(t )−i ˙a (t )a (t )a 2(t )d t =˙θ(t )a 2(t )d t.(3.10)Based on this expression,Cohen (1995)suggested that ˙θbe treated as the instanta-neous frequency.With these notations,the bandwidth can be deﬁned asν2=(ω− ω )2 ω 2=1 ω 2(ω− ω )2|S (ω)|2d ω=1 ω 2z ∗(t ) 1i d d t− ω 2z (t )d t =1 ω 2 ˙a 2(t )d t +(˙θ(t )− ω )2a 2(t )d t .(3.11)For a narrow band signal,this value has to be small,then both a and θhave to begradually varying functions.Unfortunately,both equations (3.7)and (3.11)deﬁned the bandwidth in the global sense;they are both overly restrictive and lack preci-sion at the same time.Consequently,the bandwidth limitation on the Hilbert trans-form to give a meaningful instantaneous frequency has never been ﬁrmly established.For example,Melville (1983)had faithfully ﬁltered the data within the bandwidth requirement,but he still obtained many non-physical negative frequency values.It should be mentioned here that using ﬁltering to obtain a narrow band signal is unsat-isfactory for another reason:the ﬁltered data have already been contaminated by the spurious harmonics caused by the nonlinearity and non-stationarity as discussed in the introduction.In order to obtain meaningful instantaneous frequency,restrictive conditions have to be imposed on the data as discussed by Gabor (1946),Bedrosian (1963)and,more recently,Boashash (1992):for any function to have a meaningful instantaneous frequency,the real part of its Fourier transform has to have only positive frequency.This restriction can be proven mathematically as shown in Titchmarsh (1948)but it is still global.For data analysis,we have to translate this requirement into physically implementable steps to develop a simple method for applications.For this purpose,we have to modify the restriction condition from a global one to a local one,and the basis has to satisfy the necessary conditions listed in the last section.Let us consider some simple examples to illustrate these restrictions physically,by examining the function,x (t )=sin t.(3.12)Its Hilbert transform is simply cos t .The phase plot of x –y is a simple circle of unit radius as in ﬁgure 1a .The phase function is a straight line as shown in ﬁgure 1b and the instantaneous frequency,shown in ﬁgure 1c ,is a constant as expected.If we move the mean oﬀby an amount α,say,then,x (t )=α+sin t.(3.13)Proc.R.Soc.Lond.A (1998)。

法语综合教程1答案第14课

Le? on 14 Compréhension du texte zII ．Choisissez la r ponseé selon le texte :Charles (va / prend) d jeuneréau restaurant universitaire. Il y va ( / en) autobus.Ilà(aime / regarde) le menu. Il est alors (content / en col re) parce qu ’èmangeon( / veut) toujours du poisson et que le resto U (exag ère / est en gr ève). Mais il (attend / reste) son tour, pour (prendre / h siter)é de la glace au dessert.E xercices de grammairepl é tez avec les articles partitifs：1. de 6. de la, des, des, de l, du ’2. Du7. du3. des8. du4. du9. des, des, du5. de la10.dupl é tez les phrases avec ?de la ?, ? du ? , ? des ? , ? de l ’? ou ? de ? ：1.d’ 6.du, de la, del ’,des2.du7.de3.de, d’8.de4.de la9.de5.du, du10. du, de la, deIII. Complétez avec les articles qui conviennent :1.de la, de la , une2.des7.les, les3.le, une8.Le4.les, des9.une5.Le, une10. de, leE xercices de grammaireI. Compl étez avec les pronoms convenables :1.nous y 6.les y2.les lui3.nous en8.t ’ en4.t ’y9.me la5.nous le10. les yII. R é pondez d’ apr ès l’ exemple：1.Oui, je te les montre.2.Oui, elle nous le donne.3.Oui, je vous les passe.4.Oui, elle me le lit.5.Oui, je la lui dis.6.Oui, je la lui vends.7.Oui, il m y’attend.8.Oui, on les y voit.9.Oui, il nous en donne.10. Oui, il nous l’ explique bien.III.Mettez les doubles pronoms l à’ imp é ratif :1.Montre-les-lui. / Ne les lui montre pas.2.Donne-leur-en. / Ne leur en donne pas.3.Dites-le-lui. / Ne le lui dites pas.4.Lis-le-nous. / Ne nous le lis pas.5.Posons-la-lui. / Ne la lui posons pas.6.Passe-le-moi. / Ne me le passe pas.issez-la-nous. / Ne nous la laissez pas.8.Sers-m’ en/. Ne m’ ensers pas.9.Montre-la-leur. / Ne la leur montre pas.10.Donnez-lui-en un. / Ne lui en donnez pas.E xercices de vocabulaire qI. Mettez les verbes au présent de l’ indicatif:A. manger1. manger 3. mangeons 5. mange7. manger2. mangez 4. manger 6. mange8. mangentB. exag reré1. exagères 3. exagérez 5. exagère7. exagère2. exagérer 4. exagèrent 6. exagère8. exagèreII. Choisissez la bonne r ponseé :1. S’il (a faim / a une faim), il faut lui donner mangerà.2.Ils (ont une faim de loup / ont faim de loup), ils vont d?ner.3.Il (vient de / va) pleuvoir. La terre est mouill e（湿的）é.4.Isabelle n ’ aime(mp a ngers / prendre) le petit d jeuneré.5.(Venez / Venez de)m’ aider.6. Il (ne travaille plus / ne travaille pas), il est la retraiteà.7.Veux-tu (une tranche de pain / un pain) ?8.Le temps (change /passe) beaucoup en cette saison.III.Remplacez les points par les mots ou expressions du dialogue ou du texte :1. fini 5. plusieurs2. apporte 6. tour3. cuisine7. patiemment4. ne, plus8. hésitonsIV. Remplacez les points par une pr positioné :1. sur2. selon3.à7. de, aux4. dans8. à5.à+ le Au→9. en6.à10.avecE xercices de structureI. Posez des questions selon la partie soulign e : ément est la cuisine chinoise ?2.Que boivent-ils tousàles repas ?3.Avec quoi Luc prend-il souvent du riz ?4.Pourquoi te faut-il du courage ?5.Qu’ est-ce qui est app tissanté ?6. Le soir, quand d?nent les Fran?ais ? /àquelle heurenntd?les Fran?ais le soir ?7.Oùd éjeunes-tu ? / O prendsù-tu un sandwich ?8.Qu’ est-ce que Monsieur Dupont vient de faire ?II.R pondezé aux questions suivantes avec des pronoms convenables :1.- Oui, il faut aller en acheter.2.- Oui, vous devez la vendre.3.- Non, elles ne vont pas y aller.4.- Non, ils n ’ aiment pas en prendre.5.- Oui, je vais penser toi. à6. - Oui, je veux la d crire. é/nous voulons la d crire.é7.- Oui, elle commence la comprendreà.8.- Non, il ne va pas en tre contentê.III.Faites des phrases d’ apr ès les modè: lesModè le A :1. C’ estàvous que je parle.2. C’ est Juliette que Roméo aime.3.C’ est avec ses copains que Charles va allerjardinu du Luxembourg.4.C’ estmardi que tu vas venir.5. C’ est pour vivre qu’ on mange.6. C'est elle que Marc rencontre dans la rue.7. C’ est pour toi qu’ elle chante.8. C’ estàcause de sa maladie que mon grand-rep ne vaèpas voyager en Allemagne.Modè le B :1. Philippe vient d’ avoir une idée.2.Je viens de lui parler.3.Nous venons de rentrer la maisonà.4.Les enfants viennent de se lever.5.On vient de se promener sur le campus.6.Vous venez de visiter le mus e du Louvreé.7.Elle vient d’é crire une lettre.8.Les feuilles viennent de tomber.IV. Remettez les phrases suivantes dans un ordre logique :(1) ..........f..............(2) ............b...........(3) ..........d..............(4) ..........e..............(5) ..........g..............(6) ...........a............(7) ..........c..............(8) ..........h..............E xercices audio-orauxI. Ecoutez et remplissez les blancs :J’ aime fairela cuisine. Aujourd’ hui, je vais inviter mescopains d?nerà chez moi. Mais il me faut d’ abord faire des courses. Je dois acheterducaf . éEt je dois aussi acheter d’ autresboissons : du vin,du jus de pamplemousse(柚子)et une bouteille d’eau minérale(含矿物质的). Puis des produits laitiers : de la glace, du fromage, du lait et du beurre. Ensuite, des fruits et desl égumes : des pommes, 300 grammes de tomates, des choux et des carottes.Et oui, j’ ai envie(克 )de manger du poisson ce soir. Mais demain je voudrais manger du porc. Je vais aussi achetertrois tranches de jambon(火腿) et des desserts pour ce week-end.II. Choisissez la bonne r ponseé :1. b2. b3. a4. a5. bE xercices de traductionI. Traduisez les mots et locutions :非常乐意感到饿在校园里高中同学avec plaisir avoir faim sur le campus camarade de lyc e é愤怒绝食等待轮到他大学生食堂être en col re èfaire la gr ve deè la faim attendre son tour le resto UII. Rendez les phrases en fran?ais :1.- G néralement, que manges-tu comme petit d jeuner ?é/ D ’ habitude, qu -ce’queest tu prends au petit d jeuner?é- Mon petit d jeuneré est tr s simpleè : un verre de lait et deux petits pains cuits la vapeur.à2.Tous les jours, Camille va prendre le d jeuner laàcantineéavec ses camarades de classe. Elle trouve la cuisine de son universit bonne. / Elleé trouve que la cuisine y est pas mal.3.Elle a 45 ans. Elle n’ est plus. jeune4.Jacques va te tl éphoner,é soit avant les vacances, soit aprs les vacancesè.5.Je viens te chercher pour te dire au revoir. / C’ estpour te dire au revoir que je viens te chercher. /Je viens juste pour te dire au revoir.。

Lorex N842安装指南说明书

N842_QSG_FR_R1Série N8421 / 2Copyright © 2020 Lorex CorporationNos produits étant sans cesse améliorés, Lorex se réserve le droit de modifier la conception du produit, ses spécificationset son prix sans préavis et sans aucune obligation. E&OE. Tous droits réservés.Reportez-vous aux étapes ci-dessous (instructions détaillées à droite) pour terminer la configuration initiale de l’enregistreur :Lorex Fusion supporte la connectivité avec certains appareils sans fil au l’enregistreur. Pour en savoir plus sur la connectivité de fusion et les appareils Lorex compatibles avec cette fonctionnalité, visitez *Non inclus/vendus séparément.Pour en savoir plus sur la compatibilité des caméras, consultez le site /compatibility .Vue d’ensemble des ports supplémentaires :REMARQUE : Pour obtenir des instructions complètes sur l’utilisation des ports supplémentaires, veuillez consulter le manuel d’instructions de votre enregistreur de sécurité sur .Configuration de votre enregistreurLes panneaux arrière ci-dessous sont uniquement présentés à des fins d’illustration. Le panneau arrière de votre enregistreur peut paraître légèrement différent, avec tous lesmêmes ports à des endroits différents.ÉTAP 6 :Lorex Setup WizardNotez le mot de passe ci-dessous et gardez-le dans un endroit sûr :N842_QSG_FR_R1Ajouter des caméras à partir du LANSuivez les instructions ci-dessous pour ajouter des caméras qui ne sont pas directement connectées aux ports PoE à l’arrière de l’enregistreur.REMARQUE : Veuillez visiter Pour ajouter des caméras à partir du LAN :1. Connectez la caméra à un routeur ou à un commutateur branché sur le même réseau que l’enregistreur.2. Cliquez sur le bouton droit de la souris et sélectionnez l’aide du nom d’utilisateur du système (par défaut :3. Configurez les éléments suivants :a. Cliquez sur Camera Registration b. Cochez la/les caméra(s) à ajouter.c. Cliquez sur Add . L devient vert si la caméra est bien connectée.d. Les périphériques ajoutés apparaîtrontdans la liste Added Device le bouton droit de la souris pour quitter le Rechercher et lire des enregistrements vidéo depuis le disque dur.Pour rechercher et lire des enregistrements :Depuis le visionnement en direct, cliquez sur le bouton droit, puis sur Playback (lecture). Si vous y êtesinvité, connectez-vous à l’aide du nom d’utilisateur du système (par défaut : admin ) et votre nouveau Sauvegarder des enregistrements sur une clé USB (non fournie).Pour sauvegarder des enregistrements :Insérez une clé USB (non fournie) dans un port USB libre de l’enregistreur.Depuis le mode de visionnement en direct, cliquez avec le bouton droit de la souris, puis cliquez sur Main. Si vous y êtes invité, connectez-vous à l’aide du nom d’utilisateur du système (par défaut : admin ) et votre nouveau mot de passe sécurisé.Sélectionnezle canal d’unecaméra connectée avec détection de personnes et de Enable sous et/ou Vehicle . c. Cliquez sur Set à côté de Area pour définir des zones actives pour la détection despersonnes et/ou des véhicules. Consultez la Figure 1 ci-dessous pour plus de détails.d. Cliquez sur Set à côté de Schedule pour définir un calendrier hebdomadaire pour ladétection des personnes et/ou des véhicules. Consultez la Figure 2 ci-dessous pour plus de détails.e. Réglez les préférences pour la lumière d’avertissement et la sirène.f. Cliquez sur Apply .Pour déclencher les lumières d’avertissement et les sirènes de toutes les caméras de dissuasion connectées, appuyez sur le bouton du panneau avant et maintenez-le enfoncé pendant 3 secondes.Figure 2: CalendrierFigure 1: Zone de détection• Cliquez sur Add pour définir une zone de détection depersonnes ou de véhicules sur le canal sélectionné. Cliquez et faites glisser les coins pour redimensionner la zone.• Pour des résultats plus précis, définissez une zone où les objets d’intérêt se déplaceront à l’intérieur de la zone de délimitation ainsi qu’à l’entrée et à la sortie.• Cochez la Light à côté d’une règle pour faire clignoter la lumière d’avertissement de l’appareil lorsqu’un objet est détecté.• Consultez la documentation de votre caméra pour unpositionnement optimal de la caméra pour la détection des personnes et des véhicules.Option 1 : Caméras de détection avancée du mouvementOption 2 : Caméras de dissuasion active• L ’horaire par défaut, illustré à la Figure 2, est actif pendant la nuit, entre 17 h et 7 h. • Cliquez sur Set pour modifier l’horaire du jour de la semaine correspondant.• Cliquez sur OK lorsque vous avez terminé.Sélectionnez le canal d’une camérade dissuasion connectée.Enable .Set à côté de Area pourdéfinir des zones actives pour la détection des personnes et/ou des véhicules. Consultez la Figure 3 ci-dessous pour plus de détails.d. Cliquez sur Set à côté de Schedule pour définir un calendrierhebdomadaire pour la détection des personnes et/ou desvéhicules. Consultez la Figure 2 ci-dessous pour plus de détails.e. Réglez les préférences pour la lumière d’avertissement et la sirène.f.Réglez les niveaux de Sensitivity et de Threshold selon vos préférences.g. Cliquez sur Apply .• L ’image de la caméra apparaît avec une grillesuperposée. La zone verte est la zone active pour la dissuasion.• Cliquez ou cliquez et faites glisser pour ajouter/supprimer la zone de la grille rouge.• Dans la Figure 3, seul le mouvement autour de la porte déclenchera un voyant d’avertissement.• Cliquez à droite lorsque vous avez terminé.Figure 3: Zone de dissuasionModification de la résolution de sortie de l’enregistreurPour garantir la meilleure qualité d’image possible, réglez la résolution de sortie de l’enreg-istreur à la résolution la plus élevée prise en charge par votre moniteur.moniteur. Par exemple, sélectionnez Pour modifier la résolution de sortie de l’enregistreur :IMPORTANT : Si vous devez changer de moniteur, assurez-vous de régler l’enregistreur sur une résolution de sortie prise en charge par le nouveau moniteur avant de commuter.Pendant le visionnement en direct, passez le curseur de la souris au-dessus de l’écran pour ouvrir la barre de navigation. Déplacez le curseur de la souris en l’éloignant du dessus de l’écran pour fermer la barre de navigation.Lors du visionnement en direct :afin de faire un zoom avant et arrière.Utilisation du menu rapideCliquez avec le bouton droit n’importe où sur l’écran de visionnement en direct pour ouvrir le menu rapide.Ouvrir le menu principal.Rechercher et lire des enregistrements.Contrôle des caméras PTZ (nonabcabca b c defa c eb d fc d eb a gfab。

Logitech M330 Silent 无声鼠标及MK270 无线键盘说明书

Tracking Method: Optical Sensor Connection 2.4 GHz Radio Frequency Wireless Technology Working Range: 30 ft (10m)Adjustable DPI Resolutions: 3 levels (800, 1200, 1600 dpi)Navigation Hotkeys: Backward, Forward Hand Orientation: Right Hand Battery: (2) AAA Dimensions: 5.0 x 3.4 x 1.7" (127 x 85 x 45mm)Weight: 3.3 oz (94g)Key Layout 104-key US Layout Key Type Membrane Connection 2.4 GHz Radio Frequency Wireless Technology Working Range: 30 ft (10m)Battery: (2) AAAIndicators: Num Lock, Caps Lock & Low Battery Windows Hotkeys: 10 (Home, Back, Forward, Search, Email, Sleep, Wake up, Power, Media Player, Calculator Multimedia Hotkeys: 7 (Play/Pause, Stop, Previous Track, Next Track, Volume Up, Volume Down & Mute)Dimensions: 19.8 x 8.3 x 1" (503 x 212 x 25 mm)Weight: 1.87 lbs. (845 g)Requirements:Keyboard Specifications:Mouse Specifications:Operating SystemWindows® 10/ 8 / 7Connectivity InterfaceUSB Port for Wireless ReceiverIncludes:Wireless Ergo Keyboard Wireless Ergo MouseNano USB Wireless Receiver (4) AAA Batteries Quickstart GuideShipping Information:WKB-1600CB78375000869310Item UPC Code Box Dimensions Box Wt. Qty/Ctn TM1600Low-profile Chiclet KeycapsThe low-profile Chiclet Keys Provide a Quiet, Comfortable Typing Experience with its Sleek and Sturdy Design and a 5 million keystroke life.Split Ergonomic DesignThe design of this keyboard with splitted key zones and gently sloped shape encourages natural position of hands, wrists, and forearm in maximum comfort for long use.2.4 GHz RF WirelessTechnology22.8 x 13 x 2.2”3.2 lbsDPI Switch & Navigation HotkeysEnhanced Optical SensorQuickly and easily adjust mouse DPI resolution (800/ 1200/ 1600) for a faster response with the convenient DPI Switch located at the top of mouse. This mouse also includes two convenient side internet forward & back buttons for quick Internet navigationInternet & Multimedia HotkeysGet instant access to your commonly used Internet and multimedia tools with the 17 built-in hotkeys this keyboard has to offerOptical sensor technology improves precision and even lets you work on most surfaces like marble, wood and leather surfaces without the need of a mouse pad.Navigation Hotkeys 2.4 GHz RF Wireless Technology with On/Off SwitchThis keyboard and mouse work at any angle with a range of up to 30 ft. with no wires to get tangled. TheOn/Off switch helps conserve battery power on bothdevices, and the low battery indicator helps you know when it is time to change batteries.Integrated Palm RestSplit/Ergonomic Design Built-in Scroll Wheel Optical EngineIntegrated Palm RestDesigned to keep your hands in the natural resting position, this not only relieves joint pain and improves posture, but has also been shown to relieve back pain and spinal discomfort.The Built-in Scroll Wheel makes scrolling within documents a breeze. Users canachieve higher work efficiencies by elimina -ting the need to switch between their keyboard and mouse. Built-in Scroll Wheel。

Nonlinear dynamics, delay times, and embedding windows

49
time τd for attractor reconstruction using the method of delays has not been fully developed, many researchers use the autocorrelation function, which is computationally convenient and does not require large data sets. However, it has been pointed out [1] that the autocorrelation function is not appropriate for nonlinear systems, and, instead, τd should be chosen as the ﬁrst local minimum of the mutual information. Unfortunately, this approach is cumbersome computationally and requires large data sets [2]. According to Packard et al. [3] and Takens [4], the method of delays can be used to embed a scalar time series {xi }, i = 1, 2, . . . , into an m-dimensional space as follows: x i = (xi , xi +1 , . . . , xi +(m−1)t ), x i ∈ Rm , (1)
Keywords: Delay time; Correlation integral; Embedding; Time series PACS: 05.45. + b; 47.52. + j

three factors Linguistic Inquiry, Volume 36, Number 1, Winter 2005

Three Factors in LanguageDesignNoam ChomskyThe biolinguistic perspective regards the language faculty as an ‘‘organof the body,’’along with other cognitive systems.Adopting it,weexpect to find three factors that interact to determine (I-)languagesattained:genetic endowment (the topic of Universal Grammar),experi-ence,and principles that are language-or even organism-independent.Research has naturally focused on I-languages and UG,the problemsof descriptive and explanatory adequacy.The Principles-and-Param-eters approach opened the possibility for serious investigation of thethird factor,and the attempt to account for properties of language interms of general considerations of computational efficiency,eliminat-ing some of the technology postulated as specific to language andproviding more principled explanation of linguistic phenomena.Keywords:minimalism,principled explanation,Extended StandardTheory,Principles-and-Parameters,internal/external Merge,single-cycle derivation,phaseThirty years ago,in 1974,an international meeting took place at MIT,in cooperation with the Royaumont Institute in Paris,on the topic of ‘‘biolinguistics,’’a term suggested by the organizer,Massimo Piattelli-Palmarini,and the title of a recent book surveying the field and proposing new directions by Lyle Jenkins (2002).1This was only one of many such interactions in those years,including interdisciplinary seminars and international conferences.The biolinguistic perspective began to take shape over 20years before in discussions among a few graduate students who were much influenced by developments in biology and mathematics in the early postwar years,including work in ethology that was just coming to be known in the United States.One of them was Eric Lenneberg,whose seminal 1967study Biological Foundations of Language remains a basic document of the field.Many of the leading questions discussed at the 1974conference,and in the years leading up to it,remain very much alive today.One of these questions,repeatedly brought up in the conference as ‘‘one of the basic questions to be asked from the biological point of view,’’is the extent to which apparent principles of language,including some that had only recently come to light,are unique to this cognitive system or whether similar ‘‘formal arrangements’’are found in other cognitive domains in humans or This article is expanded from a talk presented at the annual meeting of the Linguistic Society of America,9January 2004.Thanks to Cedric Boeckx,Samuel David Epstein,Robert Freidin,Lyle Jenkins,Howard Lasnik,and Luigi Rizzi,among others,for comments on an earlier draft.1The conference,titled ‘‘A Debate on Bio-Linguistics,’’was held at Endicott House,Dedham,Massachusetts,20–21May 1974,and organized by the Centre Royaumont pour une science de l’homme,Paris.1Linguistic Inquiry,Volume 36,Number 1,Winter 20051–22᭧2005by the Massachusetts Institute of Technology2N O A M C H O M S K Yother organisms.An even more basic question from the biological point of view is how much of language can be given a principled explanation,whether or not homologous elements can be found in other domains or organisms.The effort to sharpen these questions and to investigate them for language has come to be called the‘‘Minimalist Program’’in recent years,but the questions arise for any biological system and are independent of theoretical persuasion,in linguis-tics and elsewhere.Answers to these questions are fundamental not only to understanding the nature and functioning of organisms and their subsystems,but also to investigating their growth and evolution.For any biological system,language included,the only general question that arises about the program is whether it can be productively pursued or is premature.In these remarks,I will try to identify what seem to me some of the significant themes in the past half-century of inquiry into problems of biolinguistics and to consider their current status. Several preliminary qualifications should be obvious.One is that the picture is personal;others would no doubt make different choices.A second is that things often seem clearer in retrospect than at the time,so there is some anachronism in this account,but not I think too much.A third is that I cannot even begin to mention the contributions of a great many people to the collective enterprise,particularly as the related fields have expanded enormously in the years since the1974 conference.The biolinguistic perspective views a person’s language as a state of some component of the mind,understanding‘‘mind’’in the sense of eighteenth-century scientists who recognized that after Newton’s demolition of the only coherent concept of body,we can only regard aspects of the world‘‘termed mental’’as the result of‘‘such an organical structure as that of the brain’’(Joseph Priestley).Among the vast array of phenomena that one might loosely consider language-related,the biolinguistic approach focuses attention on a component of human biology that enters into the use and acquisition of language,however one interprets the term‘‘language.’’Call it the ‘‘faculty of language,’’adapting a traditional term to a new usage.This component is more or less on a par with the systems of mammalian vision,insect navigation,and others.In many of these cases,the best available explanatory theories attribute to the organism computational systems and what is called‘‘rule-following’’in informal usage—for example,when a recent text on vision presents the so-called rigidity principle as it was formulated50years ago:‘‘if possible,and other rules permit,interpret image motions as projections of rigid motions in three dimensions’’(Hoffman1998:169).In this case,later work provided substantial insight into the mental computa-tions that seem to be involved when the visual system follows these rules,but even for very simple organisms,that is typically no slight task,and relating mental computations to analysis at the cellular level is commonly a distant goal.Adopting this conception,a language is a state of the faculty of language,an I-language,in technical usage.The decision to study language as part of the world in this sense was regarded as highly controversial at the time,and still is.A more careful look will show,I think,that the arguments advanced against the legitimacy of the approach have little force(a weak thesis)and that its basic assumptions are tacitly adopted even by those who strenuously reject them,and indeed must be, even for coherence(a much stronger thesis).I will not enter into this interesting chapter ofT H R E E F A C T O R S I N L A N G U A G E D E S I G N3 contemporary intellectual history here,but will simply assume that crucial aspects of language can be studied as part of the natural world,adopting the biolinguistic approach that took shape half a century ago and that has been intensively pursued since,along different paths.The language faculty is one component of what the cofounder of modern evolutionary theory, Alfred Russel Wallace,called‘‘man’s intellectual and moral nature’’:the human capacities for creative imagination,language and symbolism generally,mathematics,interpretation and record-ing of natural phenomena,intricate social practices,and the like,a complex of capacities that seem to have crystallized fairly recently,perhaps a little over50,000years ago,among a small breeding group of which we are all descendants—a complex that sets humans apart rather sharply from other animals,including other hominids,judging by traces they have left in the archaeological record.The nature of the‘‘human capacity,’’as some researchers now call it,remains a considera-ble mystery.It was one element of a famous disagreement between the two founders of the theory of evolution,with Wallace holding,contrary to Darwin,that evolution of these faculties cannot be accounted for in terms of variation and natural selection alone,but requires‘‘some other influence,law,or agency,’’some principle of nature alongside gravitation,cohesion,and other forces without which the material universe could not exist.Although the issues are framed differ-ently today within the core biological sciences,they have not disappeared(see Wallace1889: chap.15,Marshack1985).It is commonly assumed that whatever the human intellectual capacity is,the faculty of language is essential to it.Many scientists agree with paleoanthropologist Ian Tattersall,who writes that he is‘‘almost sure that it was the invention of language’’that was the‘‘sudden and emergent’’event that was the‘‘releasing stimulus’’for the appearance of the human capacity in the evolutionary record—the‘‘great leap forward’’as Jared Diamond called it,the result of some genetic event that rewired the brain,allowing for the origin of modern language with the rich syntax that provides a multitude of modes of expression of thought,a prerequisite for social development and the sharp changes of behavior that are revealed in the archaeological record, also generally assumed to be the trigger for the rapid trek from Africa,where otherwise modern humans had apparently been present for hundreds of thousands of years(Tattersall1998:24–25; see also Wells2002).Tattersall takes language to be‘‘virtually synonymous with symbolic thought.’’Elaborating,one of the initiators of the Royaumont-MIT symposia,Franc¸ois Jacob, observed that‘‘the role of language as a communication system between individuals would have come about only secondarily,as many linguists believe’’(1982:59),perhaps referring to discus-sions at the symposia,where the issue repeatedly arose,among biologists as well.In the1974 conference,his fellow Nobel laureate Salvador Luria was the most forceful advocate of the view that communicative needs would not have provided‘‘any great selective pressure to produce a system such as language,’’with its crucial relation to‘‘development of abstract or productive thinking’’(Luria1974:195).‘‘The quality of language that makes it unique does not seem to be so much its role in communicating directives for action’’or other common features of animal communication,Jacob continued,but rather‘‘its role in symbolizing,in evoking cognitive im-ages,’’in‘‘molding’’our notion of reality and yielding our capacity for thought and planning, through its unique property of allowing‘‘infinite combinations of symbols’’and therefore‘‘mental4N O A M C H O M S K Ycreation of possible worlds,’’ideas that trace back to the seventeenth-century cognitive revolution (1982:59).Jacob also stressed the common understanding that answers to questions about evolu-tion‘‘in most instances...can hardly be more than more or less reasonable guesses’’(1982: 31).We can add another insight of seventeenth-and eighteenth-century philosophy:that even the most elementary concepts of human language do not relate to mind-independent objects by means of some reference-like relation between symbols and identifiable physical features of the external world,as seems to be universal in animal communication systems.Rather,they are creations of the‘‘cognoscitive powers’’that provide us with rich means to refer to the outside world from certain perspectives,but are individuated by mental operations that cannot be reduced to a‘‘peculiar nature belonging’’to the thing we are talking about,as Hume summarized a century of inquiry.Those are critical observations about the elementary semantics of natural language, suggesting that its most primitive elements are related to the mind-independent world much as the internal elements of phonology are,not by a reference-like relation but as part of a considerably more intricate species of conception and action.It is for reasons such as these,though not clearly grasped at the time,that the early work in the1950s adopted a kind of‘‘use theory of meaning,’’pretty much in the sense of John Austin and the later Wittgenstein:language was conceived as an instrument put to use for various human purposes,generating expressions including arrange-ments of the fundamental elements of the language,with no grammatical-ungrammatical divide, each basically a complex of instructions for use(see Chomsky1955,hereafter LSLT).2 If this much is generally on the right track,then at least two basic problems arise when we consider the origins of the faculty of language and its role in the sudden emergence of the human intellectual capacity:first,the core semantics of minimal meaning-bearing elements,including the simplest of them;and second,the principles that allow infinite combinations of symbols, hierarchically organized,which provide the means for use of language in its many aspects.Accord-ingly,the core theory of language—Universal Grammar(UG)—must provide,first,a structured inventory of possible lexical items that are related to or perhaps identical with the concepts that are the elements of the‘‘cognoscitive powers,’’sometimes now regarded as a‘‘language of thought’’along lines developed by Jerry Fodor(1975);and second,means to construct from these lexical items the infinite variety of internal structures that enter into thought,interpretation, planning,and other human mental acts,and that are sometimes put to use in action,including the externalization that is a secondary process if the speculations just reviewed turn out to be correct.On the first problem,the apparently human-specific conceptual-lexical apparatus,there is important work on relational notions linked to syntactic structures and on the partially mind-internal objects that appear to play a critical role(events,propositions,etc.).3But there is little beyond descriptive remarks on the core referential apparatus that is used to talk about the world. The second problem has been central to linguistic research for half a century,with a long history before in different terms.2For later discussion,see among others Chomsky1966,2001b,McGilvray1999,Antony and Hornstein2003.3For insightful review and original analysis,see Borer2004a,b.T H R E E F A C T O R S I N L A N G U A G E D E S I G N5 The biolinguistic approach adopted from the outset the point of view that C.R.Gallistel (1997)calls‘‘the norm these days in neuroscience’’(p.86),the‘‘modular view of learning’’: the conclusion that in all animals,learning is based on specialized mechanisms,‘‘instincts to learn’’(p.82)in specific ways.We can think of these mechanisms as‘‘organs within the brain’’(p.86),achieving states in which they perform specific kinds of computation.Apart from‘‘ex-tremely hostile environments’’(p.88),they change states under the triggering and shaping effect of external factors,more or less reflexively,and in accordance with internal design.That is the ‘‘process of learning’’(Gallistel1997,1999),though‘‘growth’’might be a more appropriate term,avoiding misleading connotations of the term‘‘learning.’’The modular view of learning of course does not entail that the component elements of the module are unique to it:at some level,everyone assumes that they are not—the cellular level,for example—and the question of the level of organization at which unique properties emerge remains a basic one from a biological point of view,as it was at the1974conference.Gallistel’s observations recall the concept of‘‘canalization’’introduced into evolutionary and developmental biology by C.H.Waddington over60years ago,referring to processes‘‘adjusted so as to bring about one definite end result regardless of minor variations in conditions during the course of the reaction,’’thus ensuring‘‘the production of the normal,that is optimal type in the face of the unavoidable hazards of existence’’(Waddington1942).That seems to be a fair description of the growth of language in the individual.A core problem of the study of the faculty of language is to discover the mechanisms that limit outcomes to‘‘optimal types.’’It has been recognized since the origins of modern biology that such constraints enter not only into the growth of organisms but also into their evolution,with roots in the earlier tradition that Stuart Kauffman calls‘‘rational morphology’’(1993:3–5).4In a classic contemporary paper, John Maynard Smith and associates trace the post-Darwinian reformulation back to Thomas Huxley,who was struck by the fact that there appear to be‘‘predetermined lines of modification’’that lead natural selection to‘‘produce varieties of a limited number and kind’’for every species (Maynard Smith et al.1985:266).5They review a variety of such constraints in the organic world and describe how‘‘limitations on phenotypic variability’’are‘‘caused by the structure,character, composition,or dynamics of the developmental system,’’pointing out also that such‘‘develop-mental constraints...undoubtedly play a significant role in evolution’’though there is yet‘‘little agreement on their importance as compared with selection,drift,and other such factors in shaping evolutionary history’’(p.265).At about the same time,Jacob wrote that‘‘the rules controlling embryonic development,’’almost entirely unknown,interact with other constraints imposed by general body plan,mechanical properties of building materials,and other factors in‘‘restricting possible changes of structures and functions’’in evolutionary development(1982:21),providing ‘‘architectural constraints’’that‘‘limit adaptive scope and channel evolutionary patterns’’(Erwin 2003:1683).The best-known of the figures who devoted much of their work to these topics are 4For comment in a linguistic context,see Boeckx and Hornstein2003.For more general discussion,see Jenkins 2000.5For review of some of these topics,see Stewart1998.6N O A M C H O M S K YD’Arcy Thompson and Alan Turing,who took a very strong view on the central role of such factors in biology.In recent years,such considerations have been adduced for a wide range of problems of development and evolution,from cell division in bacteria to optimization of structure and function of cortical networks,even to proposals that organisms have‘‘the best of all possible brains,’’as argued by computational neuroscientist Christopher Cherniak(1995:522).6The prob-lems are at the border of inquiry,but their significance is not controversial.Assuming that the faculty of language has the general properties of other biological systems, we should,therefore,be seeking three factors that enter into the growth of language in the indi-vidual:1.Genetic endowment,apparently nearly uniform for the species,which interprets part ofthe environment as linguistic experience,a nontrivial task that the infant carries out reflexively,and which determines the general course of the development of the language faculty.Among the genetic elements,some may impose computational limitations that disappear in a regular way through genetically timed maturation.Kenneth Wexler and his associates have provided compelling evidence of their existence in the growth of language,thus providing empirical evidence for what Wexler(to appear)calls‘‘Lenne-berg’s dream.’’2.Experience,which leads to variation,within a fairly narrow range,as in the case of othersubsystems of the human capacity and the organism generally.3.Principles not specific to the faculty of language.The third factor falls into several subtypes:(a)principles of data analysis that might be used in language acquisition and other domains;(b)principles of structural architecture and developmental constraints that enter into canalization,organic form,and action over a wide range,including principles of efficient computation,which would be expected to be of particular significance for computational systems such as language.It is the second of these subcategories that should be of particular significance in determining the nature of attainable languages.Those exploring these questions50years ago assumed that the primitive step of analysis of linguistic experience would be feature-based phonetic analysis,along lines described by Roman Jakobson and his associates(see Jakobson,Fant,and Halle1953).We also tried to show that basic prosodic properties reflect syntactic structure that is determined by other principles,including crucially a principle of cyclic computation that was extended much more generally in later years (see Chomsky,Halle,and Lukoff1956).The primitive principles must also provide what George Miller called‘‘chunking,’’identification of phonological words in the string of phonetic units.In LSLT(p.165),I adopted Zellig Harris’s(1955)proposal,in a different framework,for identifying morphemes in terms of transitional probabilities,though morphemes do not have the required beads-on-a-string property.The basic problem,as noted in LSLT,is to show that such statistical 6See also Laughlin and Sejnowski2003,Cherniak et al.2004,and Physics News Update2001reporting Howard, Rutenberg,and de Vet2001.T H R E E F A C T O R S I N L A N G U A G E D E S I G N7 methods of chunking can work with a realistic corpus.That hope turns out to be illusory,as has recently been shown by Thomas Gambell and Charles Yang(2003),who go on to point out that the methods do,however,give reasonable results if applied to material that is preanalyzed in terms of the apparently language-specific principle that each word has a single primary stress.If so,then the early steps of compiling linguistic experience might be accounted for in terms of general principles of data analysis applied to representations preanalyzed in terms of principles specific to the language faculty,the kind of interaction one should expect among the three factors.In LSLT,it was assumed that the next step would be assignment of chunked items to syntactic categories,again by general principles of data analysis.A proposal with an information-theoretic flavor was tried by hand calculations in that precomputer age,with suggestive results,but the matter has never been pursued,to my knowledge.Surely what are called‘‘semantic properties’’are also involved,but these involve nontrivial problems at the most elementary level,as mentioned earlier.The assumption of LSLT was that higher levels of linguistic description,including mor-phemes,are determined by a general format for rule systems provided by UG,with selection among them in terms of a computational procedure that seeks the optimal instantiation,a notion defined in terms of UG principles of significant generalization.Specific proposals were made then and in the years that followed.In principle,they provided a possible answer to what came to be called the‘‘logical problem of language acquisition,’’but they involved astronomical calcula-tion and therefore did not seriously address the issues.The main concerns in those years were quite different,as they still are.It may be hard to believe today,but it was commonly assumed50years ago that the basic technology of linguistic description was available and that language variation was so free that nothing of much generality was likely to be discovered.As soon as efforts were made to provide fairly explicit accounts of the properties of languages,however,it became obvious how little was known,in any domain. Every specific proposal yielded a treasure trove of counterevidence,requiring complex and varied rule-systems even to achieve a very limited approximation to descriptive adequacy.That was highly stimulating for inquiry into language,but it also left a serious quandary,since the most elementary considerations led to the conclusion that UG must impose narrow constraints on possible outcomes—sometimes called‘‘poverty of stimulus’’problems in the study of language, though the term is misleading because this is just a special case of basic issues that arise universally for organic growth.A number of paths were pursued to try to resolve the tension.The most successful turned out to be efforts to formulate general principles,attributed to UG—that is,the genetic endow-ment—leaving a somewhat reduced residue of phenomena that would result,somehow,from experience.Early proposals were the A-over-A Principle,conditions on wh-extraction from wh-phrases(relatives and interrogatives),simplification of T-markers to base recursion(following observations by Charles Fillmore)and cyclicity(an intricate matter,as shown in an important paper of Robert Freidin’s(1978)and insightfully reviewed in a current paper of Howard Lasnik’s (to appear)which shows that many central questions remain unanswered),later John Robert Ross’s(1967)classic study of taxonomy of islands that still remains a rich store of ideas and observations to explore,then attempts to reduce islands to such properties as locality and structure8N O A M C H O M S K Ypreservation,and so on.These approaches had some success,but the basic tensions remained unresolved at the time of the1974conference.Within a few years,the landscape had changed considerably.In part this was because of great progress in areas that had hitherto been explored only in limited ways,including truth-and model-theoretic semantics and prosodic structures.In part it was the result of a vast array of new materials from studies of much greater depth than previously undertaken,and into a much wider variety of languages,much of it traceable to Richard Kayne’s work and his lectures in Europe, which inspired far-reaching inquiry into Romance and Germanic languages,later other languages, also leading to many fruitful ideas about the principles of UG.About25years ago,much of this work crystallized in a radically different approach to UG,the Principles-and-Parameters(P&P) framework,which for the first time offered the hope of overcoming the tension between descriptive and explanatory adequacy.This approach sought to eliminate the format framework entirely,and with it,the traditional conception of rules and constructions that had been pretty much taken over into generative grammar.That much is familiar,as is the fact that the new P&P framework led to an explosion of inquiry into languages of the most varied typology,yielding new problems previously not envisioned,sometimes answers,and the reinvigoration of neighboring disciplines concerned with acquisition and processing,their guiding questions reframed in terms of parameter setting within a fixed system of principles of UG with at least visible contours.Alternative paths, variously interrelated,were leading in much the same direction,including Michael Brody’s highly illuminating work(1995,2003).No one familiar with the field has any illusion today that the horizons of inquiry are even visible,let alone at hand,in any domain.Abandonment of the format framework also had a significant impact on the biolinguistic program.If,as had been assumed,acquisition is a matter of selection among options made available by the format provided by UG,then the format must be rich and highly articulated,allowing relatively few options;otherwise,explanatory adequacy is out of reach.The best theory of lan-guage must be a very unsatisfactory one from other points of view,with a complex array of conditions specific to human language,restricting possible instantiations.The only plausible theories had to impose intricate constraints on the permissible relations between sound and mean-ing,all apparently specific to the faculty of language.The fundamental biological issue of princi-pled explanation could barely be contemplated,and correspondingly,the prospects for serious inquiry into evolution of language were dim;evidently,the more varied and intricate the conditions specific to language,the less hope there is for a reasonable account of the evolutionary origins of UG.These are among the questions that were raised at the1974symposium and others of the period,but they were left as apparently irresoluble problems.The P&P framework offered prospects for resolution of these tensions as well.Insofar as this framework proves valid,acquisition is a matter of parameter setting and is therefore divorced entirely from the remaining format for grammar:the principles of UG.There is no longer a conceptual barrier to the hope that the UG might be reduced to a much simpler form,and that the basic properties of the computational systems of language might have a principled explanation instead of being stipulated in terms of a highly restrictive language-specific format for grammars. Within a P&P framework,what had previously been the worst theory—anything goes—might。

ctex数学学术报告的模板 -回复

ctex数学学术报告的模板-回复问题：如何使用ctex 模板写一份数学学术报告？在数学学术研究中，学术报告是一种重要的交流与展示方式。

使用ctex 模板可以在LaTeX 环境下撰写中文数学学术报告。

本文将一步一步地介绍如何使用ctex 模板来写一篇1500-2000字的数学学术报告。

第一步：安装LaTeX 发行版和编辑器首先，需要根据自己的操作系统选择一个合适的LaTeX 发行版，如TeX Live 或MiKTeX，并安装在计算机上。

其次，选择一个适合自己的LaTeX 编辑器，如TeXstudio、TeXworks 或Overleaf。

第二步：导入ctex 模板ctex 宏包是一个用于处理中文文档的LaTeX 宏包。

在LaTeX 文档的导言区（preamble）中导入ctex 宏包，可以使用以下代码：\documentclass{article}\usepackage{ctex}在导入ctex 宏包之后，可以使用ctexart 或ctexrep 文档类来设置文章的格式。

第三步：设置文章标题和作者在导入ctex 宏包之后，可以使用以下代码设置文章的标题和作者：\title{文章标题}\author{作者姓名}\date{}在花括号内分别填写文章的标题和作者的姓名。

使用date 命令可以设置日期，也可以将花括号内留空以使用当前日期。

第四步：撰写正文在设置完文章的标题和作者之后，就可以开始撰写文章的正文了。

在LaTeX 中，可以使用各种数学公式环境和命令来插入数学内容，如align、equation、theorem 等。

需要注意的是，中文文档中的数学公式可能需要设置字体、字号和样式。

可以使用各种命令来进行设置，如\mathbf、\textbf、\Large 等。

同时，不同的编辑器也有不同的快捷键和插入公式的方式，可以根据自己的需求进行调整。

第五步：添加参考文献数学学术报告中通常需要引用已有的文献，在LaTeX 中可以使用BibTeX 或者BibLaTeX 来管理和引用参考文献。

matlab中preview函数 -回复

matlab中preview函数-回复Matlab 中的preview 函数是一个非常有用的工具，用于预览图形和图像。

它允许用户在编辑和处理图像之前，对其进行快速检查和调整。

preview 函数提供了许多功能和选项，使它成为Matlab 中一个重要的图形处理工具。

本文将逐步回答关于preview 函数的一些常见问题，并介绍如何使用它来实现图像预览和调整的技术。

首先，让我们来了解preview 函数的基本概念。

preview 函数是Matlab 的图像处理工具箱中的一个函数，它允许用户在采集图像之前，预览来自图像设备（如相机）的图像。

这样，用户可以在选择图像捕捉参数和进行后续处理之前，对图像进行实时预览和调整。

使用preview 函数之前，我们首先需要连接到图像设备。

这可以通过使用Matlab 的Image Acquisition Toolbox 中的一些函数来实现，如imaqhwinfo、imaqmex、imaqvideo 等等。

这些函数允许我们查找和选择可用的图像设备，并为我们的程序提供与设备之间的连接。

完成设备连接后，我们可以使用preview 函数显示图像设备的预览窗口。

preview 函数的基本语法如下：preview(obj);在这里，obj 是一个图像设备对象，它代表已连接的图像设备。

preview 函数将自动打开一个新窗口，并在其中显示来自设备的实时图像。

这允许我们实时监视和调整图像捕捉参数以获得最佳结果。

在预览窗口中，我们还可以进行一些其他的调整。

例如，我们可以选择自动曝光功能，这将根据设备的当前环境自动调整曝光参数。

我们还可以通过设置窗口的大小和位置来调整预览窗口。

所有这些调整都可以通过preview 函数的参数进行实现。

另一个有用的功能是获取预览窗口中显示的图像数据。

为此，我们可以使用getsnapshot 函数，该函数使我们能够以图像格式获取当前在窗口中显示的图像。

此后，我们可以对获取的图像数据进行进一步的处理和分析。

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WaveScan是令人兴奋的功能，我很喜欢它！它是我的宝贝！
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Question & Answer

Apply a search/scan condition

>= <= In Range Out of Range

Table listing of “found” events Zoom to view an event Analyze further (WaveRunner Xi)
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WaveScanTM Locates Problems Faster
Locate unusual events in a single capture Scan for an event in many acquisitions More than 20 search/scan modes
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WaveScan – Finding the Shape of Rare Events.
In this example we see a zoom of a single rare event found by WaveScan. The user asked the scope to stop when it found a rare measurement of a risetime that was greater than normal
It’s impossible to capture it using hardware trigger
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WaveScan 在USB2.0 debug中的应用
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WaveScan 在USB2.0 debug中的应用
力科创新火炬传递之第3棒关键词：WaveScan
Advanced Search and Analysis Toolu need ! All just one touch away !
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WaveScan – Finding Problems

ScanHisto(gram) ScanOverlay
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WaveScanTM Locates Problems Faster
WaveScan makes it easy to find rare anomalies in circuit behavior �� Will isolate rare events, measure them and provide a view �� Can act as a “finder” for rare abnormalities in a long waveform that has already been acquired or as a “soft trigger” to find/view/measure abnormalities in incoming signals in real time mode
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WaveScan – Advanced Scan and Search
设置非常简单,使用cursor测量波形边沿的宽度 More than 23 parameters 像素或屏幕分辨率(量化误差)会降低精度 For choice in WaveRunner 只有单个波形,引入了触发抖动
Automatic using 10、90％ For runt gate
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WaveScan – Finding Intermittents that you cannot trigger on
This example finds rare edges that are nonmonotonic. The result is isolation of signal faults which could not be found using a hardware trigger unless you already knew the nature of the failure.
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More action for choice after acquisition
Search/Scan Condition
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WaveScan – Capture Runt Signal
Display runt signal’s amplitude
WaveScanTM Advance scan, search and analysis function. WaveScanTM A more powerful capability than hardware trigger. WaveScanTM Integrate advanced arithmetic to fast locate problems.
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WaveScan testing edge rates on a USB packet
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WaveScan testing edge rates on a USB packet
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WaveScan testing edge rates on a USB packet
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WaveScan testing edge rates on a USB packet
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来自工程师的声音