Solutions fortes et comportement asymptotique pour un modèle de convection naturelle en milieu poreux

Résumé Nous étudions ici un système d'équations aux dérivées partielles qui gouverne la convection naturelle dans un milieu poreux soumis à un gradient de température T. Sous leur forme la plus générale, ces équations s'écrivent:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacqGHiiIZdaWcaaqaaiab% gkGi2kabeg8aYbqaaiabgkGi2kaadshaaaGaey4kaSIaaeizaiaabM% gacaqG2bGaaeiiaiabeg8aYjaadAfacqGH9aqpcaaIWaaabaWaaSaa% aeaacqaHbpGCaeaacqGHiiIZaaWaaSaaaeaacqGHciITaeaacqGHci% ITcaWG0baaaiabgUcaRiaabEgacaqGYbGaaeyyaiaabsgacaqGGaGa% amiCaiabgkHiTiabeg8aYjaadEgacqGHRaWkcqaH8oqBcaWGlbWaaW% baaSqabeaacqGHsislcaaIXaaaaOGaaeiiaiaadAfacqGH9aqpcaaI% WaaabaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaaiaacIcacq% aHbpGCcaWGJbGaaiykamaaCaaaleqabaGaaiOkaaaakiaadsfacqGH% sislcaqGGaGaaeizaiaabMgacaqG2bGaaeiiaiaabU5adaahaaWcbe% qaaiaabQcaaaGccaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfa% cqGHRaWkcaGGOaGaeqyWdiNaam4yaiaacMcadaWgaaWcbaGaamOzaa% qabaGccaWGwbGaaeiiaiabgwSixlabgEGirlaadsfacqGH9aqpcaaI% WaaabaGaeqyWdiNaeyypa0JaeqyWdi3aaSbaaSqaaiaadkhaaeqaaO% GaaiikaiaaigdacqGHsislcqaHXoqycaGGOaGaamivaiabgkHiTiaa% dsfadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiykaaaacaGL7baaaa% a!9527!\[P_1 \left\{ \begin{gathered} \in \frac{{\partial \rho }}{{\partial t}} + {\text{div }}\rho V = 0 \hfill \\ \frac{\rho }{ \in }\frac{\partial }{{\partial t}} + {\text{grad }}p - \rho g + \mu K^{ - 1} {\text{ }}V = 0 \hfill \\ \frac{\partial }{{\partial t}}(\rho c)^* T - {\text{ div \Lambda }}^{\text{*}} {\text{grad }}T + (\rho c)_f V{\text{ }} \cdot \nabla T = 0 \hfill \\ \rho = \rho _r (1 - \alpha (T - T_r )) \hfill \\ \end{gathered} \right.\] désigne la porosité, la masse volumique du fluide, V la vitesse, p la pression, T la température du fluide, la viscosité, K et % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4MdmaaCa% aaleqabaGaaeOkaaaaaaa!37E8!\[{\text{\Lambda }}^{\text{*}} \] sont les tenseurs respectifs de perméabilité et de conductivité thermique. La chaleur volumique du fluide est notée (c) _f , celle du solide (c) _s , et on définit alors la chaleur volumique équivalente par la relation: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiabeg% 8aYjaadogacaqGPaWaaWbaaSqabeaacaqGQaaaaOGaeyypa0Jaeyic% I4Saaiikaiabeg8aYjaadogacaGGPaWaaSbaaSqaaiaadAgaaeqaaO% Gaey4kaSIaaiikaiaaigdacaqGGaGaeyOeI0IaeyicI4Saaiykaiaa% cIcacqaHbpGCcaWGJbGaaiykaaaa!4C87!\[{\text{(}}\rho c{\text{)}}^{\text{*}} = \in (\rho c)_f + (1{\text{ }} - \in )(\rho c)\].De façon très classique, dans les problèmes de convection, on simplifie ce modèle en faisant l'approximation de Boussinesq qui consiste à négliger les variations de la masse volumique sauf dans le terme g, voir par exemple [6]. Ce modèle connu depuis longemps a été très étudié par de nombreux physiciens et numériciens depuis une dizaine d'années (voir par exemple [3–5, 7, 8, 18, 24]) mais à notre connaissance accune étude théorique n'a été entreprise jusqu'à aujourd'hui.On se limitera ici au cas d'un milieu homogène isotrope remplissant une cavité parallélépipédique dont l'un des axes a même direction que l'accélération de la pesanteur g. Sous forme adimensionnelle le système P ₂ s'écrit:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacqGHiiIZdaWcaaqaaiab% gkGi2kabeg8aYbqaaiabgkGi2kaadshaaaGaey4kaSIaaeizaiaabM% gacaqG2bGaaeiiaiabeg8aYjaadAfacqGH9aqpcaaIWaaabaWaaSaa% aeaacqaHbpGCaeaacqGHiiIZaaWaaSaaaeaacqGHciITaeaacqGHci% ITcaWG0baaaiabgUcaRiaabEgacaqGYbGaaeyyaiaabsgacaqGGaGa% amiCaiabgkHiTiabeg8aYjaadEgacqGHRaWkcqaH8oqBcaWGlbWaaW% baaSqabeaacqGHsislcaaIXaaaaOGaaeiiaiaadAfacqGH9aqpcaaI% WaaabaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaaiaacIcacq% aHbpGCcaWGJbGaaiykamaaCaaaleqabaGaaiOkaaaakiaadsfacqGH% sislcaqGGaGaaeizaiaabMgacaqG2bGaaeiiaiaabU5adaahaaWcbe% qaaiaabQcaaaGccaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfa% cqGHRaWkcaGGOaGaeqyWdiNaam4yaiaacMcadaWgaaWcbaGaamOzaa% qabaGccaWGwbGaaeiiaiabgwSixlabgEGirlaadsfacqGH9aqpcaaI% WaaabaGaeqyWdiNaeyypa0JaeqyWdi3aaSbaaSqaaiaadkhaaeqaaO% GaaiikaiaaigdacqGHsislcqaHXoqycaGGOaGaamivaiabgkHiTiaa% dsfadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiykaaaacaGL7baaaa% a!9527!\[P_1 \left\{ \begin{gathered} \in \frac{{\partial \rho }}{{\partial t}} + {\text{div }}\rho V = 0 \hfill \\ \frac{\rho }{ \in }\frac{\partial }{{\partial t}} + {\text{grad }}p - \rho g + \mu K^{ - 1} {\text{ }}V = 0 \hfill \\ \frac{\partial }{{\partial t}}(\rho c)^* T - {\text{ div \Lambda }}^{\text{*}} {\text{grad }}T + (\rho c)_f V{\text{ }} \cdot \nabla T = 0 \hfill \\ \rho = \rho _r (1 - \alpha (T - T_r )) \hfill \\ \end{gathered} \right.\]Dans % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCLaey% ypa0Jaai4EaiaacIcacaWG4bGaaiilaiaabccacaWG5bGaaeilaiaa% bccacaWG6bGaaiykaiabgIGiolaac2facaaIWaGaaiilaiaabccaca% WGmbGaai4waerbbjxAHXgaiuaacaWFfrGaaiyxaiaaicdacaGGSaGa% aeiiaiaadYgacaGGBbGaa8xreiaac2facaaIWaGaaiilaiaabccaca% WGObGaai4waiaac2haaaa!54B3!\[\Omega = \{ (x,{\text{ }}y{\text{, }}z) \in ]0,{\text{ }}L[]0,{\text{ }}l[]0,{\text{ }}h[\} \]: de frontière les conditions aux limites sont:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGub% GaaiikaiaadIhacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiaiaabcda% caGGPaGaeyypa0JaaGymaiaacYcacaqGGaGaaeiiaiaabccacaqGGa% GaaeiiaiaadsfacaGGOaGaamiEaiaacYcacaqGGaGaamyEaiaacYca% caqGGaGaamiAaiaacMcacqGH9aqpcaaIWaGaaiilaaqaamacmc4caa% qaiWiGcWaJaAOaIyRaiWiGdsfaaeacmcOamWiGgkGi2kacmc4G4baa% aiaacIcacaaIWaGaaiilaiaabccacaWG5bGaaiilaiaabccacaWG6b% Gaaiykaiabg2da9maalaaabaGaeyOaIyRaamivaaqaaiabgkGi2kaa% dIhaaaGaaiikaiaadYeacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcaWGubaabaGaeyOa% IyRaamiEaaaacaGGOaGaamiEaiaacYcacaqGGaGaamiBaiaacYcaca% qGGaGaamOEaiaacMcacqGH9aqpcaaIWaGaaiilaaqaaiaadAfacqGH% flY1caqGGaGaamOBamaaBaaaleaaruqqYLwySbacfaGaa8hFaiabgk% Gi2kabfM6axbqabaGccqGH9aqpcaaIWaaaaaa!8886!\[\begin{gathered} T(x,{\text{ }}y,{\text{ 0}}) = 1,{\text{ }}T(x,{\text{ }}y,{\text{ }}h) = 0, \hfill \\ \frac{{\partial T}}{{\partial x}}(0,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(L,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(x,{\text{ }}l,{\text{ }}z) = 0, \hfill \\ V \cdot {\text{ }}n_{|\partial \Omega } = 0 \hfill \\ \end{gathered} \], où n est la normale unitaire sortante à .Le vecteur adimensionnel k est pris égal à-e _z, Ra ^* est un paramètre proportionnel à la contrainte exercée au milieu et S un paramètre très petit [Smin(10^-6, 10^-6 Ra ^*)] que l'on fera tendre par la suite vers zéro.Dans [10, 11] nous avons étudié le problème bidimensionnel aussi bien d'évolution que stationnaire et nous avons montré, outre un théorème d'existence, d'unicité et de régularité, la présence de plusieurs solutions stationnaires. Le phénomène nous a incité à étudier le comportement asymptotique des solutions du problème d'évolution. Afin de rendre cette étude plus complète nous avons décidé de travailler en dimension 3 d'espace.Ce papier donne les résultats préliminaires à une étude un peu fine du comportement asymptotique. Nous allons en particulier établir un théorème de régularité et donner une majoration uniforme des dérivées secondes en espace des solutions dans le cas où S=0. Ces propriétés sont similaires à celles connues pour les équations de Navier-Stokes dans le cas bidimensionnel [13, 26] et généralisent à la dimension trois ceux que nous avons obtenus dans [10].La clef de le preuve du théorème d'existence et d'unicité est une estimation L en temps et en espace de la température T obtenue en découplant l'équation de l'énergie (0.3) et l'équation de Darcy (0.2). Ensuite on applique une méthode de point fixe. La régularité en espace est liée à la structure particulière de l'ouvert ainsi qu'à la nature des conditions limites. Cela étant acquis, les majorations uniformes en temps sont obtenues de façon assez classique. Nous étendons enfin à notre système les résultats obtenus par Foias et Temam [15] pour les équations de Navier-Stokes en dimension deux d'espace. Rappelons qu'il s'agit alors de montrer que la solution est parfaitement déterminée par ses valeurs prises sur un ensemble fini de points.Avant d'aller plus avant dans ce travail, signalons que l'on se ramène à des conditions aux limites homogènes en posant % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2% da9iabeI7aXjabgUcaRiaaigdacqGHsislcaGGOaGaamOEaiaac+ca% caWGObGaaiykaaaa!4004!\[T = \theta + 1 - (z/h)\]. Le système devient:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaadaWcaaqaaiabgkGi2kab% eI7aXbqaaiabgkGi2kaadshaaaGaeyOeI0IaeyiLdqKaeqiUdeNaey% 4kaSIaamOvaiabgwSixlabgEGirlabeI7aXjabgkHiTmaalaaabaGa% aGymaaqaaiaadIgaaaGaeqyXdu3aaSbaaSqaaiaaiodaaeqaaOGaey% ypa0JaaGimaaqaaiaadofadaWcaaqaaiabgkGi2kaadAfaaeaacqGH% ciITcaWG0baaaiabgUcaRiaadAfacqGHRaWkcqGHhis0cqaHapaCcq% GHRaWkcaWGsbGaamyyamaaCaaaleqabaGaaiOkaaaakiaadUgacqaH% 4oqCcqGH9aqpcaaIWaaabaGaaeizaiaabMgacaqG2bGaaeiiaiaadA% facqGH9aqpcaaIWaaabaGaamOvaiabgwSixlaad6gadaWgaaWcbaqe% feKCPfgBaGqbaiaa-XhacqqHtoWraeqaaOGaeyypa0JaaGimaaqaai% abeI7aXjaacIcacaWG4bGaaiilaiaadMhacaGGSaGaaGimaiaacMca% cqGH9aqpcqaH4oqCcaGGOaGaamiEaiaacYcacaWG5bGaaiilaiaadI% gacaGGPaGaeyypa0JaaGimaaqaamaalaaabaGaeyOaIyRaeqiUdeha% baGaeyOaIyRaamiEaaaacaGGOaGaaGimaiaacYcacaWG5bGaaiilai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcqaH4oqCaeaacqGH% ciITcaWG4baaaiaacIcacaWGmbGaaiilaiaadMhacaGGSaGaamOEai% aacMcacqGH9aqpcaaIWaaabaWaaSaaaeaacqGHciITcqaH4oqCaeaa% cqGHciITcaWG5baaaiaacIcacaWG4bGaaiilaiaaicdacaGGSaGaam% OEaiaacMcacqGH9aqpdaWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi% 2kaadMhaaaGaaiikaiaadIhacaGGSaGaamiBaiaacYcacaWG6bGaai% ykaiabg2da9iaaicdaaaGaay5Eaaaaaa!B7C4!\[P_1 \left\{ \begin{gathered} \frac{{\partial \theta }}{{\partial t}} - \Delta \theta + V \cdot \nabla \theta - \frac{1}{h}\upsilon _3 = 0 \hfill \\ S\frac{{\partial V}}{{\partial t}} + V + \nabla \pi + Ra^* k\theta = 0 \hfill \\ {\text{div }}V = 0 \hfill \\ V \cdot n_{|\Gamma } = 0 \hfill \\ \theta (x,y,0) = \theta (x,y,h) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial x}}(0,y,z) = \frac{{\partial \theta }}{{\partial x}}(L,y,z) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial y}}(x,0,z) = \frac{{\partial \theta }}{{\partial y}}(x,l,z) = 0 \hfill \\ \end{gathered} \right.\]

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Strong solutions and asymptotic behaviour for a natural convection problem in porous media

We discuss a system of partial differential equations which describes natural convection in a porous medium under a temperature gradient T. In their most general form these equations can be written% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacqGHiiIZdaWcaaqaaiab% gkGi2kabeg8aYbqaaiabgkGi2kaadshaaaGaey4kaSIaaeizaiaabM% gacaqG2bGaaeiiaiabeg8aYjaadAfacqGH9aqpcaaIWaaabaWaaSaa% aeaacqaHbpGCaeaacqGHiiIZaaWaaSaaaeaacqGHciITaeaacqGHci% ITcaWG0baaaiabgUcaRiaabEgacaqGYbGaaeyyaiaabsgacaqGGaGa% amiCaiabgkHiTiabeg8aYjaadEgacqGHRaWkcqaH8oqBcaWGlbWaaW% baaSqabeaacqGHsislcaaIXaaaaOGaaeiiaiaadAfacqGH9aqpcaaI% WaaabaWaaSaaaeaacqGHciITaeaacqGHciITcaWG0baaaiaacIcacq% aHbpGCcaWGJbGaaiykamaaCaaaleqabaGaaiOkaaaakiaadsfacqGH% sislcaqGGaGaaeizaiaabMgacaqG2bGaaeiiaiaabU5adaahaaWcbe% qaaiaabQcaaaGccaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfa% cqGHRaWkcaGGOaGaeqyWdiNaam4yaiaacMcadaWgaaWcbaGaamOzaa% qabaGccaWGwbGaaeiiaiabgwSixlabgEGirlaadsfacqGH9aqpcaaI% WaaabaGaeqyWdiNaeyypa0JaeqyWdi3aaSbaaSqaaiaadkhaaeqaaO% GaaiikaiaaigdacqGHsislcqaHXoqycaGGOaGaamivaiabgkHiTiaa% dsfadaWgaaWcbaGaamOCaaqabaGccaGGPaGaaiykaaaacaGL7baaaa% a!9527!\[P_1 \left\{ \begin{gathered} \in \frac{{\partial \rho }}{{\partial t}} + {\text{div }}\rho V = 0 \hfill \\ \frac{\rho }{ \in }\frac{\partial }{{\partial t}} + {\text{grad }}p - \rho g + \mu K^{ - 1} {\text{ }}V = 0 \hfill \\ \frac{\partial }{{\partial t}}(\rho c)^* T - {\text{ div \Lambda }}^{\text{*}} {\text{grad }}T + (\rho c)_f V{\text{ }} \cdot \nabla T = 0 \hfill \\ \rho = \rho _r (1 - \alpha (T - T_r )) \hfill \\ \end{gathered} \right.\]where represents the porosity, is the fluid density, T is the temperature, is the dynamic viscosity, K and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaae4MdmaaCa% aaleqabaGaaeOkaaaaaaa!37E8!\[{\text{\Lambda }}^{\text{*}} \] are, respectively, the tensor of permeability and of thermal conductivity. The heat capacity of fluid (resp., solid) is denoted by (c) _f (resp., (c) _s ). Thus, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeikaiabeg% 8aYjaadogacaqGPaWaaWbaaSqabeaacaqGQaaaaOGaeyypa0Jaeyic% I4Saaiikaiabeg8aYjaadogacaGGPaWaaSbaaSqaaiaadAgaaeqaaO% Gaey4kaSIaaiikaiaaigdacaqGGaGaeyOeI0IaeyicI4Saaiykaiaa% cIcacqaHbpGCcaWGJbGaaiykaaaa!4C87!\[{\text{(}}\rho c{\text{)}}^{\text{*}} = \in (\rho c)_f + (1{\text{ }} - \in )(\rho c)\] represents the equivalent heat capacity.As is usual in convection problems, we simplify the model by adopting the Boussinesq approximation which consists of neglecting the density variations except in the g term, (cf., for instance, [6]). This well-known model has often been studied by physicists and numerical analysts, but ([3–5, 7, 8, 18, 24]), as far as we know, it seems that a theoretical approach has not yet been developed. We shall restrict our study to the case of a homogeneous isotropic medium filling a parallelepipedic cavity, one of the axis of which is colinear to the gravitational acceleration g. In dimensionless form, the system P ₁ can be written% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaacaqGWaGaaeOlaiaabgda% caqGGaGaaeiiaiaabccacaqGKbGaaeyAaiaabAhacaqGGaGaamOvai% abg2da9iaaicdaaeaacaaIWaGaaiOlaiaaikdacaqGGaGaaeiiaiaa% bccacaWGtbWaaSaaaeaacqGHciITcaWGwbaabaGaeyOaIyRaamiDaa% aacqGHRaWkcaWGwbGaey4kaSIaae4zaiaabkhacaqGHbGaaeizaiaa% bccacaWGWbGaey4kaSIaamOuaiaadggadaahaaWcbeqaaiaacQcaaa% GccaWGRbGaamivaiabg2da9iaaicdaaeaacaaIWaGaaiOlaiaaioda% caqGGaGaaeiiaiaabccadaWcaaqaaiabgkGi2kaadsfaaeaacqGHci% ITcaWG0baaaiabgkHiTiabgs5aejaadsfacqGHRaWkcaqGGaGaamOv% aiaabccacaqGNbGaaeOCaiaabggacaqGKbGaaeiiaiaadsfacqGH9a% qpcaaIWaGaaiOlaaaacaGL7baaaaa!71EF!\[P_1 \left\{ \begin{gathered} {\text{0}}{\text{.1 div }}V = 0 \hfill \\ 0.2{\text{ }}S\frac{{\partial V}}{{\partial t}} + V + {\text{grad }}p + Ra^* kT = 0 \hfill \\ 0.3{\text{ }}\frac{{\partial T}}{{\partial t}} - \Delta T + {\text{ }}V{\text{ grad }}T = 0. \hfill \\ \end{gathered} \right.\]With boundary conditions in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdCLaey% ypa0Jaai4EaiaacIcacaWG4bGaaiilaiaabccacaWG5bGaaeilaiaa% bccacaWG6bGaaiykaiabgIGiolaac2facaaIWaGaaiilaiaabccaca% WGmbGaai4waerbbjxAHXgaiuaacaWFfrGaaiyxaiaaicdacaGGSaGa% aeiiaiaadYgacaGGBbGaa8xreiaac2facaaIWaGaaiilaiaabccaca% WGObGaai4waiaac2haaaa!54B3!\[\Omega = \{ (x,{\text{ }}y{\text{, }}z) \in ]0,{\text{ }}L[]0,{\text{ }}l[]0,{\text{ }}h[\} \]:% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGub% GaaiikaiaadIhacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiaiaabcda% caGGPaGaeyypa0JaaGymaiaacYcacaqGGaGaaeiiaiaabccacaqGGa% GaaeiiaiaadsfacaGGOaGaamiEaiaacYcacaqGGaGaamyEaiaacYca% caqGGaGaamiAaiaacMcacqGH9aqpcaaIWaGaaiilaaqaamacmc4caa% qaiWiGcWaJaAOaIyRaiWiGdsfaaeacmcOamWiGgkGi2kacmc4G4baa% aiaacIcacaaIWaGaaiilaiaabccacaWG5bGaaiilaiaabccacaWG6b% Gaaiykaiabg2da9maalaaabaGaeyOaIyRaamivaaqaaiabgkGi2kaa% dIhaaaGaaiikaiaadYeacaGGSaGaaeiiaiaadMhacaGGSaGaaeiiai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcaWGubaabaGaeyOa% IyRaamiEaaaacaGGOaGaamiEaiaacYcacaqGGaGaamiBaiaacYcaca% qGGaGaamOEaiaacMcacqGH9aqpcaaIWaGaaiilaaqaaiaadAfacqGH% flY1caqGGaGaamOBamaaBaaaleaaruqqYLwySbacfaGaa8hFaiabgk% Gi2kabfM6axbqabaGccqGH9aqpcaaIWaaaaaa!8886!\[\begin{gathered} T(x,{\text{ }}y,{\text{ 0}}) = 1,{\text{ }}T(x,{\text{ }}y,{\text{ }}h) = 0, \hfill \\ \frac{{\partial T}}{{\partial x}}(0,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(L,{\text{ }}y,{\text{ }}z) = \frac{{\partial T}}{{\partial x}}(x,{\text{ }}l,{\text{ }}z) = 0, \hfill \\ V \cdot {\text{ }}n_{|\partial \Omega } = 0 \hfill \\ \end{gathered} \], where n is the outward normal unit sector to .The dimensionless vector k stands for the unit gravitational acceleration vector and Ra ^* is a parameter which is proportional to the constraint acting on the medium. S is a small parameter (Smin{(10⁶, 10^-6 Ra ^*)}) which will eventually vanish to zero.In an earlier work [10, 11], we studied the two-dimensional case for both the evolution and stationary problem and showed the existence uniqueness and regularity of the evolution problem. However, we did show that several stationary solutions exist.We were then led to study the asymptotic behaviour of the solution of the evolution problem. To make this study more general we decided to work in three-dimensional space.This article contains the preliminary results to a somewhat fine study to an asymptotic behaviour. More precisely, we establish a regularity theorem and give a uniform estimation in time of second-order space derivatives of the solutions in the case S=0. These properties are similar to those found in two-dimensional Navier-Stokes equations and extend the solutions obtained in [10] to three dimensions.The key to the proof of the existence and uniqueness theorem is an L estimation in space and time of temperature T obtained by rendering the energy equation (0.3) and the Darcy equation (0.2) independent. Then a fixed point method is applied. Space regularity is related to a particular structure of the domain and also to the type of boundary conditions. Uniform time estimates can thus be obtained by a fairly classical method.In the spirit of the Foias and Temam paper [15], we extend some of their results to our system and show that the solution is completely determined by its nodal values on a finite set.Before proceding further, it should be pointed out that the change of the unknown % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamivaiabg2% da9iabeI7aXjabgUcaRiaaigdacqGHsislcaGGOaGaamOEaiaac+ca% caWGObGaaiykaaaa!4004!\[T = \theta + 1 - (z/h)\] leads to homogeneous boundary conditions. The system can then be written% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa% aaleaacaaIXaaabeaakmaaceaaeaqabeaadaWcaaqaaiabgkGi2kab% eI7aXbqaaiabgkGi2kaadshaaaGaeyOeI0IaeyiLdqKaeqiUdeNaey% 4kaSIaamOvaiabgwSixlabgEGirlabeI7aXjabgkHiTmaalaaabaGa% aGymaaqaaiaadIgaaaGaeqyXdu3aaSbaaSqaaiaaiodaaeqaaOGaey% ypa0JaaGimaaqaaiaadofadaWcaaqaaiabgkGi2kaadAfaaeaacqGH% ciITcaWG0baaaiabgUcaRiaadAfacqGHRaWkcqGHhis0cqaHapaCcq% GHRaWkcaWGsbGaamyyamaaCaaaleqabaGaaiOkaaaakiaadUgacqaH% 4oqCcqGH9aqpcaaIWaaabaGaaeizaiaabMgacaqG2bGaaeiiaiaadA% facqGH9aqpcaaIWaaabaGaamOvaiabgwSixlaad6gadaWgaaWcbaqe% feKCPfgBaGqbaiaa-XhacqqHtoWraeqaaOGaeyypa0JaaGimaaqaai% abeI7aXjaacIcacaWG4bGaaiilaiaadMhacaGGSaGaaGimaiaacMca% cqGH9aqpcqaH4oqCcaGGOaGaamiEaiaacYcacaWG5bGaaiilaiaadI% gacaGGPaGaeyypa0JaaGimaaqaamaalaaabaGaeyOaIyRaeqiUdeha% baGaeyOaIyRaamiEaaaacaGGOaGaaGimaiaacYcacaWG5bGaaiilai% aadQhacaGGPaGaeyypa0ZaaSaaaeaacqGHciITcqaH4oqCaeaacqGH% ciITcaWG4baaaiaacIcacaWGmbGaaiilaiaadMhacaGGSaGaamOEai% aacMcacqGH9aqpcaaIWaaabaWaaSaaaeaacqGHciITcqaH4oqCaeaa% cqGHciITcaWG5baaaiaacIcacaWG4bGaaiilaiaaicdacaGGSaGaam% OEaiaacMcacqGH9aqpdaWcaaqaaiabgkGi2kabeI7aXbqaaiabgkGi% 2kaadMhaaaGaaiikaiaadIhacaGGSaGaamiBaiaacYcacaWG6bGaai% ykaiabg2da9iaaicdaaaGaay5Eaaaaaa!B7C4!\[P_1 \left\{ \begin{gathered} \frac{{\partial \theta }}{{\partial t}} - \Delta \theta + V \cdot \nabla \theta - \frac{1}{h}\upsilon _3 = 0 \hfill \\ S\frac{{\partial V}}{{\partial t}} + V + \nabla \pi + Ra^* k\theta = 0 \hfill \\ {\text{div }}V = 0 \hfill \\ V \cdot n_{|\Gamma } = 0 \hfill \\ \theta (x,y,0) = \theta (x,y,h) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial x}}(0,y,z) = \frac{{\partial \theta }}{{\partial x}}(L,y,z) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial y}}(x,0,z) = \frac{{\partial \theta }}{{\partial y}}(x,l,z) = 0 \hfill \\ \end{gathered} \right.\]

     

Characteristic expansions and multiplicative addition theorems

The notions of -polynomial expansion % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqefeKCPfgBaG% qbciaa-DgacaGGOaWexLMBb50ujbqeguuDJXwAKbacgiGae4hEaGNa% ey4kaSsefCuzVj3zPfgaiCGacaqF5bGaaiykaiabg2da9iaa-Dgaca% GGOaGae4hEaGNaaiykaiabgUcaRiaa-DgacaGGOaGaa0xEaiaacMca% cqGHRaWkdaaeqbqaaiaadchadaWgaaWcbaGaamOBaaqabaGccaGGOa% Gaeqy1dOMaaiikaiab+Hha4jaacMcacaGGPaWaaSaaaeaacaqF5bWa% aWbaaSqabeaacaqFUbaaaaGcbaGaamOBaiaacgcaaaGaey4kaSIaam% OCaiaacIcacqGF4baEcqGFSaalcaqF5bGaaiykaiaacYcaaSqaaiaa% d6gatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaGacciab8v% MifkaaigdaaeqaniabggHiLdaaaa!7116!\[g(x + y) = g(x) + g(y) + \sum\limits_{n \geqslant 1} {p_n (\varphi (x))\frac{{y^n }}{{n!}} + r(x,y),} \] and multiplicative addition theorems % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% aaaeaacaaIXaaabaGaamyAaaaadaWcaaqaaiabgkGi2cqaaiabgkGi% 2kaadshaaaqeduuDJXwAKbYu51MyVXgaiuaacqWFvpGAcaWG0bGaey% ypa0JaamisamaaBaaaleaacaGGOaaabeaakmaaBaaaleaacaGGPaaa% beaakiab-v9aQjaadshaaaa!4A8D!\[ - \frac{1}{i}\frac{\partial }{{\partial t}}\varphi t = H_( _) \varphi t\] are introduced and characterization of some -polynomial expansions and multiplicative addition theorems are obtained.Sponsored by the International Science Foundation (Soros) Grant M3Z00 and by Russian Foundation of Fundamental Research 94-01-0144.     

Characterizations based on conditional expectations of the doubled truncated distribution

The expression of the continuous distribution function F(x) is obtained whenever % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaerbhv2BYDwAHbacfiGaa8xBaiaabIcacaWG4bGaaiilaiaadMha% caqGPaGaa8hiaiaab2dacaWFGaGaa8xraiaa-HcacaWFybGaa8hiai% aa-XhacaWFGaGaa8hEaiaa-bcacqGHKjYOcaWFGaGaa8hwaiaa-bca% cqGHKjYOcaWFGaGaa8xEaiaa-Lcaaaa!53EE!\[m{\rm{(}}x,y{\rm{)}} {\rm{ = }} E(X | x \le X \le y)\]is known. Moreover, we obtain the necessary and sufficient conditions so that any function m: ² is the conditional expectation % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiaadweacaGGOaGaamiwaerbhv2BYDwAHbacfiGaa8hiaiaacYha% caWFGaGaa8hEaiaa-bcacqGHKjYOcaWFGaGaa8hwaiaa-bcacqGHKj% YOcaWFGaGaa8xEaiaacMcaaaa!4D0D!\[E(X | x \le X \le y)\]of a random variable X with continuous distribution function. Furthermore, we relate m(x,y) to order statistics.     

Estimates for Multiplicative Functions on the Set of Shifted Primes

Let fi, i = 1, ... k, be complex-valued multiplicative functions satisfying the conditions

A global optimization approach for solving the convex multiplicative programming problem

We consider a convex multiplicative programming problem of the form% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG7bGaam% OzamaaBaaaleaacaaIXaaabeaakiaacIcacaWG4bGaaiykaiabgwSi% xlaadAgadaWgaaWcbaGaaGOmaaqabaGccaGGOaGaamiEaiaacMcaca% GG6aGaamiEaiabgIGiolaadIfacaGG9baaaa!4A08!\[\{ f_1 (x) \cdot f_2 (x):x \in X\} \]where X is a compact convex set of ⁿ and f ₁, f ₂ are convex functions which have nonnegative values over X.Using two additional variables we transform this problem into a problem with a special structure in which the objective function depends only on two of the (n+2) variables. Following a decomposition concept in global optimization we then reduce this problem to a master problem of minimizing a quasi-concave function over a convex set in ² ₂. This master problem can be solved by an outer approximation method which requires performing a sequence of simplex tableau pivoting operations. The proposed algorithm is finite when the functions f _i, (i=1, 2) are affine-linear and X is a polytope and it is convergent for the general convex case.Partly supported by the Deutsche Forschungsgemeinschaft Project CONMIN.     

Elliptic operators on Lie groups

We review the theory of strongly elliptic operators on Lie groups and describe some new simplifications. Let U be a continuous representation of a Lie group G on a Banach space and a ₁,...,a _d a basis of the Lie algebra g of G. Let A _i=dU(a _i) denote the infinitesimal generator of the continuous one-parameter group t U(exp(-ta _i)) and set % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaCaaale% qajeaObaGaeyySdegaaOGaeyypa0JaamyqamaaBaaajeaWbaGaaeyA% aaWcbeaajaaOdaWgaaqcbaAaamaaBaaajiaObaGaaiiBaaqabaaaje% aObeaakiaacElacaGG3cGaai4TaiaadgeadaWgaaqcbaCaaiaabMga% aSqabaGcdaWgaaWcbaWaaSbaaKGaahaacaGGUbaameqaaaWcbeaaaa% a!4897!\[A^\alpha = A_{\rm{i}} _{_l } \cdot\cdot\cdotA_{\rm{i}} _{_n } \], where =(i ₁,...,i _n) with _j and set ||=n. We analyze properties of mth order differential operators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2da9i% aabccadaaeqaqaaiaadogadaWgaaqcbaCaaiabgg7aHbWcbeaaaKqa% GgaacqGHXoqycaqG7aGaaeiiaiaabYhacqGHXoqycaqG8bGaeyizIm% QaaeyBaaWcbeqdcqGHris5aOGaamyqamaaCaaaleqajeaObaGaeyyS% degaaaaa!4A6C!\[H = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} \le {\rm{m}}} {c_\alpha } A^\alpha \] with coefficients c . If H is strongly elliptic, i.e., % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqFj0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciOuaiaacwgacq% GH9aqpcaqGGaWaaabeaeaacaGGOaaajeaObaGaeyySdeMaae4oaiaa% bccacaqG8bGaeyySdeMaaeiFaiabg2da9iaab2gaaSqab0GaeyyeIu% oakiaabMgacqaH+oaEcaGGPaWaaWbaaSqabKqaGgaacqGHXoqyaaGc% cqGH+aGpcaaIWaaaaa!4C40!\[{\mathop{\rm Re}\nolimits} = {\rm{ }}\sum\nolimits_{\alpha {\rm{; |}}\alpha {\rm{|}} = {\rm{m}}} ( {\rm{i}}\xi )^\alpha > 0\] for all % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqFfpeea0df9GqVa0-% aq0dXdarVe0-yr0RYxir-dbba9q8aq0-qq-He9q8qqQ8fq0-vr0-vr% Y-bdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOVdGNaeyicI4% SaeSyhHe6aaWbaaSqabeaacaWGKbaaaOGaaiixaiaacUhacaaIWaGa% aiyFaaaa!3EAA!\[\xi \in ^d \backslash \{ 0\} \], then we give a simple proof of the theorem that the closure of H generates a continuous (and holomorphic) semigroup on and the action of the semigroup is determined by a smooth, representation independent, kernel which, together with all its derivatives, satisfies mth order Gaussian bounds.     

A minimum average risk approach to shrinkage estimators of the normal mean

For the problem of estimating the normal mean based on a random sample X ₁,...,X _n when a prior value ₀ is available, a class of shrinkage estimators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqdbaGafqiVd0MbaKaaaaGccaqGGaGaaiikaiaa% dUgacaGGPaGaeyypa0Jaam4AaiaacIcadaqfqaqabKqaahaacaqGUb% aaleqaneaacaqGubaaaOGaaiykaiaabccadaqfqaqabKqaahaacaWG% UbaaleqaneaaceqGybGbaebaaaGccaqGGaGaey4kaSIaaeiiaiaacI% cacaaIXaGaaeiiaiabgkHiTiaabccacaWGRbGaaiikamaavababeqc% baCaaiaab6gaaSqab0qaaiaabsfaaaGccaGGPaGaaiykamaavababe% qcbaCaaiaad6gaaSqab0qaaiabeY7aTbaaaaa!5615!\[\mathop {\hat \mu }\nolimits_n {\rm{ }}(k) = k(\mathop {\rm{T}}\nolimits_{\rm{n}} ){\rm{ }}\mathop {{\rm{\bar X}}}\nolimits_n {\rm{ }} + {\rm{ }}(1{\rm{ }} - {\rm{ }}k(\mathop {\rm{T}}\nolimits_{\rm{n}} ))\mathop \mu \nolimits_n \] is considered, where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqcdawaaiaadsfaaaGccaqGGaGaaeypaiaabcca% caWGUbWaaWbaaSqabeaacaaIXaGaai4laiaaikdaaaGccaGGOaWaa0% aaaeaacaWGybaaamaaBaaajeaWbaGaamOBaaWcbeaakiaabccacqGH% sislcaqGGaWaaubeaeqajeaWbaGaaGimaaWcbeqdbaGaaeiVdaaaki% aacMcacaqGGaGaae4laiabeccaGiabeo8aZbaa!4C33!\[\mathop T\nolimits_n {\rm{ = }}n^{1/2} (\overline X _n {\rm{ }} - {\rm{ }}\mathop {\rm{\mu }}\nolimits_0 ){\rm{ /}} \sigma \] and k is a weight function. For certain choices of k, % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqdbaGafqiVd0MbaKaaaaGccaqGGaGaaiikaiaa% dUgacaGGPaaaaa!3CEE!\[\mathop {\hat \mu }\nolimits_n {\rm{ }}(k)\] coincides with previously studied preliminary test and shrinkage estimators. We consider choosing k from a natural non-parametric family of weight functions so as to minimize average risk relative to a specified prior p. We study how, by varying p, the MSE efficiency (relative to \-X) properties of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaubeaeqaje% aWbaGaamOBaaWcbeqdbaGafqiVd0MbaKaaaaGccaqGGaGaaiikaiaa% dUgacaGGPaaaaa!3CEE!\[\mathop {\hat \mu }\nolimits_n {\rm{ }}(k)\] can be controlled. In the process, a certain robustness property of the usual family of posterior mean estimators, corresponding to the conjugate normal priors, is observed.     

Subelliptic operators on Lie groups: Variable coefficients

Let G be a Lie group with Lie algebra g and a _i,...,a _d and algebraic basic of g. Futher, if A _i=dL(a_i) are the corresponding generators of left translations by G on one of the usual function spaces over G, let% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbciab-Heaijaab2dadaaeqbqa% aiaadogadaWgaaWcbaqedmvETj2BSbacgmGae4xSdegabeaakiaadg% eadaahaaWcbeqaaiab+f7aHbaaaeaacqGFXoqycaGG6aGaaiiFaiab% +f7aHjaacYhatuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0HgiuD3BaG% Wbbiab9rMiekaaikdaaeqaniabggHiLdaaaa!5EC1!\[H{\rm{ = }}\sum\limits_{\alpha :|\alpha | \le 2} {c_\alpha A^\alpha } \] be a second-order differential operator with real bounded coefficients c . The operator is defined to be subelliptic if% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiGacMgacaGGUbGaaiOzamXvP5wqonvsaeHbfv3ySLgzaGqbaKaz% aasacqWF7bWEcqWFTaqlkmaaqafabaGaam4yamaaBaaaleaarmWu51% MyVXgaiyWacqGFXoqyaeqaaaqaaiab+f7aHjaacQdacaGG8bGae4xS% deMaaiiFaiabg2da9iaaikdaaeqaniabggHiLdGccqWFOaakiuGacq% qFNbWzcqWFPaqkcqaH+oaEdaahaaWcbeqaamaaBaaameaacqGFXoqy% aeqaaaaakiaacUdacqqFNbWzcqGHiiIZcqqFhbWrcqqFSaalcqqFGa% aicqaH+oaEcqGHiiIZrqqtubsr4rNCHbachaGaeWxhHe6aaWbaaSqa% beaacqqFKbazcqqFNaWjcqaFaC-jaaGccaGGSaGaaiiFaiabe67a4j% aacYhacqGH9aqpjqgaGeGae8xFa0NccqGH+aGpcaaIWaGaaiOlaaaa% !7884!\[\inf \{ - \sum\limits_{\alpha :|\alpha | = 2} {c_\alpha } (g)\xi ^{_\alpha } ;g \in G, \xi \in ^{d'} ,|\xi | = \} > 0.\]We prove that if the principal coefficients {c ; ||=2} of the subelliptic operator are once left differentiable in the directions a ₁,...,a _d with bounded derivatives, then the operator has a family of semigroup generator extensions on the L _p-spaces with respect to left Haar measure dg, or right Haar measure d, and the corresponding semigroups S are given by a positive integral kernel,% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-HcaOGqbciab+nfatnaa% BaaaleaacaWG0baabeaaruqqYLwySbacgiGccaqFgpGae8xkaKIae8% hkaGIae43zaCMae8xkaKIae8xpa0Zaa8qeaeaacaqGKbaaleaacqGF% hbWraeqaniabgUIiYdGcceWGObGbaKaacaWGlbWaaSbaaSqaaiaads% haaeqaaOGae8hkaGIae43zaCMae43oaSJae4hAaGMae8xkaKIaa0NX% diab-HcaOiab+HgaOjab-LcaPiab-5caUaaa!5DFA!\[(S_t \phi )(g) = \int_G {\rm{d}} \hat hK_t (g;h)\phi (h).\]The semigroups are holomorphic and the kernel satisfies Gaussian upper bounds. If in addition the coefficients with ||=2 are three times differentiable and those with ||=1 are once differentiable, then the kernel also satisfies Gaussian lower bounds.Some original features of this article are the use of the following: a priori inequalities on L in Section 3, fractional operator expansions for resolvent estimates in Section 4, a parametrix method based on reduction to constant coefficient operators on the Lie group rather than the usual Euclidean space in Section 5, approximation theory of semigroups in Section 11 and time dependent perturbation theory to treat the lower order terms of H in Sections 11 and 12.     

Empirical Bayes with rates and best rates of convergence in u(x)C(θ) exp(-x/θ)-family: Estimation case

Let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiaacI% cacaWGybWaaSbaaSqaaiaadMgaaeqaaOGaaiilaiabeI7aXnaaBaaa% leaacaWGPbaabeaakiaacMcacaGG9baaaa!3ED1!\[\{ (X_i ,\theta _i )\} \] be a sequence of independent random vectors where X _i , conditional on % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS% baaSqaaiaadMgaaeqaaaaa!38BD!\[\theta _i \], has the probability density of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOzaiaacI% cacaWG4bGaaiiFaiabeI7aXnaaBaaaleaacaWGPbaabeaakiaacMca% cqGH9aqpcaWG1bGaaiikaiaadIhacaGGPaGaam4qaiaacIcacqaH4o% qCdaWgaaWcbaGaamyAaaqabaGccaGGPaGaaeyzaiaabIhacaqGWbGa% aiikaiabgkHiTiaadIhacaGGVaGaeqiUde3aaSbaaSqaaiaadMgaae% qaaOGaaiykaaaa!4FFF!\[f(x|\theta _i ) = u(x)C(\theta _i ){\text{exp}}( - x/\theta _i )\] and the unobservable % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS% baaSqaaiaadMgaaeqaaaaa!38BD!\[\theta _i \] are i.i.d. according to an unknown G in some class G of prior distributions on , a subset of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4EaiabeI% 7aXjabg6da+iaaicdacaGG8bGaam4qaiaacIcacqaH4oqCcaGGPaGa% eyypa0JaaiikaiaadAgacaWG1bGaaiikaiaadIhacaGGPaGaaeyzai% aabIhacaqGWbGaaeikaiabgkHiTiaadIhacaGGVaGaeqiUdeNaaiyk% aiaadsgacaWG4bGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaaaaki% abg6da+iaaicdacaGG9baaaa!54DE!\[\{ \theta > 0|C(\theta ) = (fu(x){\text{exp(}} - x/\theta )dx)^{ - 1} > 0\} \]. For a S(X ₁ , ..., X_n, X_n+1)-measurable function % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaOGaaiilaaaa!397F!\[\phi _n ,\] let % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa% aaleaacaWGUbaabeaakiabg2da9iaadweacaGGOaGaeqOXdy2aaSba% aSqaaiaad6gaaeqaaOGaeyOeI0IaeqiUde3aaSbaaSqaaiaad6gacq% GHRaWkcaaIXaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa!444A!\[R_n = E(\phi _n - \theta _{n + 1} )^2 \] denote the Bayes risk of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaaaa!38C5!\[\phi _n \] and let R(G) denote the infimum Bayes risk with respect to G. For each integer s>1 we exhibit a class of S(X ₁ , ..., X_n, X_n+1)-measurable functions % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaaaa!38C5!\[\phi _n \] such that for in [s ^–1, 1], % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4yamaaBa% aaleaacaaIWaaabeaakiaad6gadaahaaWcbeqaaiabgkHiTiaaikda% caWGZbGaai4laiaacIcacaaIXaGaey4kaSIaaGOmaiaadohacaGGPa% aaaOGaeyizImQaamOuamaaBaaaleaacaWGUbaabeaakiaacIcacqaH% gpGzdaWgaaWcbaGaamOBaaqabaGccaGGSaGaam4raiaacMcacqGHsi% slcaWGsbGaaiikaiaadEeacaGGPaGaeyizImQaam4yamaaBaaaleaa% caaIXaaabeaakiaad6gadaahaaWcbeqaaiabgkHiTiaaikdacaGGOa% Gaam4Caiabes7aKjabgkHiTiaaigdacaGGPaGaai4laiaacIcacaaI% XaGaey4kaSIaaGOmaiaadohacaGGPaaaaaaa!5F94!\[c_0 n^{ - 2s/(1 + 2s)} \leqslant R_n (\phi _n ,G) - R(G) \leqslant c_1 n^{ - 2(s\delta - 1)/(1 + 2s)} \] under certain conditions on u and G. No assumptions on the form or smoothness of u is made, however. Examples of functions u, including one with infinitely many discontinuities, are given for which our conditions reduce to some moment conditions on G. When is bounded, for each integer s>1 S(X ₁ , ..., X_n, X_n+1)-measurable functions % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdy2aaS% baaSqaaiaad6gaaeqaaaaa!38C5!\[\phi _n \] are exhibited such that for in % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4waiaaik% dacaGGVaGaam4CaiaacYcacaaIXaGaaiyxaiaadogadaqhaaWcbaGa% aGimaaqaaiaacEcaaaGccaWGUbWaaWbaaSqabeaacqGHsislcaaIYa% Gaam4Caiaac+cacaGGOaGaaGymaiabgUcaRiaaikdacaWGZbGaaiyk% aaaakiabgsMiJkaadkfadaWgaaWcbaGaamOBaaqabaGccaGGOaGaeq% OXdy2aaSbaaSqaaiaad6gaaeqaaOGaaiilaiaadEeacaGGPaGaeyOe% I0IaamOuaiaacIcacaWGhbGaaiykaiabgsMiJkaadogadaqhaaWcba% GaaGymaaqaaiaacEcaaaGccaWGUbWaaWbaaSqabeaacqGHsislcaaI% YaGaam4Caiabes7aKjaac+cacaGGOaGaaGymaiabgUcaRiaaikdaca% WGZbGaaiykaaaaaaa!637D!\[[2/s,1]c_0^' n^{ - 2s/(1 + 2s)} \leqslant R_n (\phi _n ,G) - R(G) \leqslant c_1^' n^{ - 2s\delta /(1 + 2s)} \]. Examples of functions u and class g are given where the above lower and upper bounds are achieved.Part of the research was carried out during R. S. Singh's visit to the University of Science and Technology of China.Research supported in part by a Natural Sciences and Engineering Research Council of Canada Grant No. #A4631.     

Higher order asymptotics in estimation for two-sided Weibull type distributions

We consider the estimation problem of a location parameter on a sample of size n from a two-sided Weibull type density % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOzaiaacIcacaWG4bGaeyOeI0IaeqiUdeNaaiykaiabg2da9iaa% doeacaGGOaGaeqySdeMaaiykaiGacwgacaGG4bGaaiiCaiaacIcacq% GHsislcaGG8bGaamiEaiabgkHiTiabeI7aXjaacYhadaahaaWcbeqa% aiabeg7aHbaakiaacMcaaaa!52AD!\[f(x - \theta ) = C(\alpha )\exp ( - |x - \theta |^\alpha )\] for –<x<, –<< and 1<a<3/2, where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Gaam4qaiaacIcacqaHXoqycaGGPaGaeyypa0JaeqySdeMaai4laiaa% cUhacaaIYaGaeu4KdCKaaiikaiaaigdacaGGVaGaeqySdeMaaiykai% aac2haaaa!4B0E!\[C(\alpha ) = \alpha /\{ 2\Gamma (1/\alpha )\} \]. Then the bound for the distribution of asymptotically median unbiased estimators is obtained up to the 2a-th order, i.e., the order % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamOBamaaCaaaleqabaGaeyOeI0IaaiikaiaaikdacqaHXoqycqGH% sislcaaIXaGaaiykaiaac+cacaaIYaaaaaaa!4444!\[n^{ - (2\alpha - 1)/2} \]. The asymptotic distribution of a maximum likelihood estimator (MLE) is also calculated up to the 2a-th order. It is shown that the MLE is not 2a-th order asymptotically efficient. The amount of the loss of asymptotic information of the MLE is given.     

Limit theorems for the maximum likelihood estimate under general multiply Type II censoring

Assume n items are put on a life-time test, however for various reasons we have only observed the r ₁-th,..., r _k-th failure times % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiEamaaBaaaleaamiaadkhadaWgaaqaaSGaaGymaiaacYcaaWqa% baGaamOBaiaacYcacaGGUaGaaiOlaiaac6caaSqabaGccaGGSaGaam% iEamaaBaaaleaamiaadkhadaWgaaqaaSGaam4AaiaacYcaaWqabaGa% amOBaaWcbeaaaaa!48BB!\[x_{r_{1,} n,...} ,x_{r_{k,} n} \]with % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaGimaiabgsMiJkaadIhadaWgaaWcbaadcaWGYbWaaSbaaeaaliaa% igdacaGGSaaameqaaiaad6gaaSqabaGccqGHKjYOcqWIVlctcqGHKj% YOcaWG4bWaaSbaaSqaaWGaamOCamaaBaaabaWccaWGRbGaaiilaaad% beaacaWGUbaaleqaaeXatLxBI9gBaGqbaOGae8hpaWJaeyOhIukaaa!521B!\[0 \le x_{r_{1,} n} \le \cdots \le x_{r_{k,} n} > \infty \]. This is a multiply Type II censored sample. A special case where each x _ri ,n goes to a particular percentile of the population has been studied by various authors. But for the general situation where the number of gaps as well as the number of unobserved values in some gaps goes to , the asymptotic properties of MLE are still not clear. In this paper, we derive the conditions under which the maximum likelihood estimate of is consistent, asymptotically normal and efficient. As examples, we show that Weibull distribution, Gamma and Logistic distributions all satisfy these conditions.This research was supported in part by the Designated Research Initiative Fund, University of Maryland Baltimore County.     

Consistency of a rank test against general alternatives of change points (surfaces) and continuous trend

Considered are modifications of a rank test of randomness for the one- and multi-dimensional regular design cases as well as for the one- and multi-dimensional random design cases. The null hypothesis is that all observations are independent and identically distributed. The main result is the proof of consistency of the test in each of the above cases against two general alternatives.Alternative 1: there exists a pairwise disjoint partion U _i =1 ^m D _i =D, where D d1, is a bounded domain inside which one makes observations, such that (1) if an observation point falls insideD _i , then the corresponding observed value is the realization of a random variable _i i = l,...,m; (2) there exists an ordering % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamXvP5wqonvsaeHbfv3ySLgzaGqbaiab-Tha7jabe67a4Hqbdiab% +LgaPnaaBaaaleaacaWGRbaabeaakiab-1ha9naaDaaaleaacaWGRb% Gaeyypa0JaaGymaaqaaiaad2gaaaaaaa!4C2D!\[\{ \xi i_k \} _{k = 1}^m \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaaaaa!454D!\[\xi i_k \] is stochastically smaller than % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabe67a4nXvP5wqonvsaeHbfv3ySLgzaGqbdiab-LgaPnaaBaaa% leaacaWGRbaabeaakmaaBaaaleaacqGHRaWkcaaIXaaabeaakiaacY% cacaWGRbGaeyypa0JaaGymaiaacYcacaGGUaGaaiOlaiaac6cacaGG% SaGaamyBaiabgkHiTiaaigdaaaa!509B!\[\xi i_k _{ + 1} ,k = 1,...,m - 1\], (3) the partition is independent of the number of observation points. Note thatm, this ordering, and the sets D _i are not known a priori: one tests only for the existence of such a partition. Note also that in the one-dimensional case the initial sequence need not be stochastically monotone under the alternative.Alternative 2: there exists an arbitrary asymptotically continuous trend in location. Asymptotically continuous means that the trend converges to some continuous, not identically constant function as the number of data points goes to infinity. This function need not be monotone.A numerical example illustrating the use of the obtained results for image analysis (edge detection) is presented.     

Contiguous alternatives which preserve Cramér-type large deviations for a general class of statistics

Let P _N and Q _N , N1, be two possible probability distributions of a random vector X _N =(X_N1,...,X_NN), whose components are independent. Suppose P _N and Q _N have respective densities % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiCamaaBaaaleaacaWGobaabeaakiabg2da9maaxadabaGaeuiO% dafaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaaaGccaWGMbGaai% ikaiaadIhadaWgaaWcbaGaamOtaiaadMgaaeqaaOGaeyOeI0YaaCbi% aeaacqaH4oqCaSqabeaacaGGFbaaaOWaaSbaaSqaaiaad6eaaeqaaO% Gaaiykaaaa!4DEC!\[p_N = \mathop \Pi \limits_{i = 1}^N f(x_{Ni} - \mathop \theta \limits^\_ _N )\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamyCamaaBaaaleaacaWGobaabeaakiabg2da9maaxadabaGaeuiO% dafaleaacaWGPbGaeyypa0JaaGymaaqaaiaad6eaaaGccaWGMbGaai% ikaiaadIhadaWgaaWcbaGaamOtaiaadMgaaeqaaOGaeyOeI0IaeqiU% de3aaSbaaSqaaiaad6eacaWGPbaabeaakiaacMcaaaa!4DA5!\[q_N = \mathop \Pi \limits_{i = 1}^N f(x_{Ni} - \theta _{Ni} )\], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbiaeaacqaH4oqCaSqabeaacaGGFbaaaOWaaSbaaSqaaiaad6ea% aeqaaOGaeyypa0JaamOtamaaCaaaleqabaGaeyOeI0IaaGymaaaakm% aaqahabaGaeqiUde3aaSbaaSqaaiaad6eacaWGPbaabeaaaeaacaWG% PbGaeyypa0JaaGymaaqaaiaad6eaa0GaeyyeIuoaaaa!4C75!\[\mathop \theta \limits^\_ _N = N^{ - 1} \sum\limits_{i = 1}^N {\theta _{Ni} } \], such that % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbeaeaacaqGTbGaaeyyaiaabIhaaSqaaiaaigdacqGHKjYOcaWG% PbGaeyizImQaamOtaaqabaGccaGG8bGaeqiUde3aaSbaaSqaaiaad6% eacaWGPbaabeaakiabgkHiTmaaxacabaGaeqiUdehaleqabaGaai4x% aaaakmaaBaaaleaacaWGobaabeaakiaacYhacqGH9aqpcaWGpbGaai% ikaiaad6eadaahaaWcbeqaaiabgkHiTiaaigdacaGGVaGaaGOmaaaa% kiaacMcaaaa!5647!\[\mathop {{\rm{max}}}\limits_{1 \le i \le N} |\theta _{Ni} - \mathop \theta \limits^\_ _N | = O(N^{ - 1/2} )\], f(x)>0 for almost every real x, f is absolutely continuous, and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaCbeaeaaciGGZbGaaiyDaiaacchaaSqaaiabeI7aXjaad+gacqGH% KjYOcqaH4oqCcqGHKjYOcqaH4oqCcaWGVbaabeaakmaapedabaGaai% 4waiaadAgaaSqaaiabg6HiLcqaaiabg6HiLcqdcqGHRiI8aOGaai4j% aiaacIcacaWG4bGaeyOeI0IaeqiUdeNaaiykamaaCaaaleqabaGaaG% Omaaaakiaac+cacaWGMbGaaiikaiaadIhacaGGPaGaamizaiaadIha% cqGH8aapcqGHEisPaaa!5ECE!\[\mathop {\sup }\limits_{\theta o \le \theta \le \theta o} \int_\infty ^\infty {[f} '(x - \theta )^2 /f(x)dx < \infty \] for some % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaeqiUde3aaSbaaSqaaiaaicdaaeqaaOGaeyOpa4JaaGimaaaa!3FD4!\[\theta _0 > 0\]. The contiguity of {q _N } to {p _N } is well known. In this paper it is proven that under these conditions {Q _N } preserves C.-T.L.D. (Cramér-type large deviation) from {P _N } for a general class of statistics which includes R-, U- and L-statistics as members. That means, for any {S _N =S_N(X_N)} from , a C.-T.L.D. theorem with range Cxo(N) (any C0), 0<4^-1, holds for {S _N } under {P _N }, implying that the same theorem holds for {S _N } under {Q _N }. It also provides a quick and simple way to establish C.-T.L.D. results for statistics under {Q _N }.Research supported in part by grant VE87080 from the National Science Council, Republic of China.Part of the research was done while the author was visiting the Institute of Statistical Science, Academia Sinica, Taipei, Taiwan.     

Intersections of random walks and Wiener sausages in four dimensions

Let {S ₁ (n)} _n0and {S ₂ (n)} _n0be independent simple random walks in Z ⁴ starting at the origin, and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaqGPbaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiaadggacaGGSaGaamOyaiab-LcaPiabg2da9i% ab-Tha7Hqbciab+Hha4jabgIGiolab+PfaAnaaCaaaleqabaGaaGin% aaaakiaacQdaieGacaqFtbWaaSbaaSqaaiaabMgaaeqaaOGae8hkaG% Iaa0xBaiab-LcaPiabg2da9iab+Hha4baa!5761!\[\Pi _{\rm{i}} (a,b) = \{ x \in Z^4 :S_{\rm{i}} (m) = x\]for the some % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaGqaciaa-1gacqGHiiIZtCvAUfKttLearyqr1ngBPrgaiuaacqGF% OaakcaWGHbGaaiilaiaadkgacqGFPaqkcqGF9bqFaaa!4936!\[m \in (a,b)\} \]. Let two integervalued sequences {a _n}_n0and {b _n}_n0be given, such that the limit lim_n a _nexists and lim _n b _n=+. In this paper, it is shown that the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaiaacYcaaeqaaOGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqGH% GjsUieaacaGFydaaaa!5904!\[\Pi _1 (0,\infty ) \cap \Pi _2 (a_{n,} a_n + b_n ) \ne \O \] is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaalaaabaGaaGymaaqaaiaaikdaaaGaciiBaiaac+gacaGGNbWe% xLMBb50ujbqeguuDJXwAKbacfaGae8hkaGIae8xmaeJae83kaSIaam% OyamaaBaaaleaacaWGUbaabeaakiaac+cacaWGHbWaaSbaaSqaaiaa% d6gaaeqaaOGae8xkaKIae83la8IaciiBaiaac+gacaGGNbGaamOyam% aaBaaaleaacaWGUbaabeaaaaa!5364!\[\frac{1}{2}\log (1 + b_n /a_n )/\log b_n \] if it tends to zero as n, and the probability of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaaiabfc6aqnaaBaaaleaacaaIXaaabeaatCvAUfKttLearyqr1ngB% Prgaiuaakiab-HcaOiab-bdaWiab-XcaSiabg6HiLkab-LcaPiabgM% Iihlabfc6aqnaaBaaaleaacaaIYaaabeaakiab-HcaOiaadggadaWg% aaWcbaGaamOBaaqabaGccaGGSaGaamyyamaaBaaaleaacaWGUbaabe% aakiabgUcaRiaadkgadaWgaaWcbaGaamOBaaqabaGccqWFPaqkcqWF% 9aqpieaacaGFydaaaa!583C!\[\Pi _1 (0,\infty ) \cap \Pi _2 (a_n ,a_n + b_n ) = \O \]is asymptotic to % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% abaeqabaaabaGaam4yaiaacUfaciGGSbGaai4BaiaacEgatCvAUfKt% tLearyqr1ngBPrgaiuaacqWFOaakcaWGHbWaaSbaaSqaaiaad6gaae% qaaOGaey4kaSIaamOyamaaBaaaleaacaWGUbaabeaakiab-LcaPiab% -9caViab-XgaSjab-9gaVjab-DgaNjab-HcaOiaadggadaWgaaWcba% GaamOBaaqabaGccqGHRaWkcaaIYaGae8xkaKIae8xxa01aaWbaaSqa% beaacqWFTaqlcqWFXaqmcqWFVaWlcqWFYaGmaaaaaaa!5BAC!\[\begin{array}{l} \Pi _1 (0,\infty ) \cap \Pi _2 (a_n ,a_n + b_n ) = \O \\ c[\log (a_n + b_n )/log(a_n + 2)]^{ - 1/2} \\ \end{array}\], for some constant c, if it tends to a finite constant (1) as n. These results extend some results obtained by G. F. Lawler about the intersection properties of simple random walks in Z ⁴. By using similar arguments, we also get corresponding results for the intersections of Wiener sausages in four dimensions. In particular, a conjecture suggested by M. Aizenman, which describes nonintersection of independent Wiener sausages in R ⁴, is proven.Partly supported by AvH Foundation.     

A threshold for the size of random caps to cover a sphere

Consider a unit sphere on which are placed N random spherical caps of area 4p(N). We prove that if % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% qGSbGaaeyAaiaab2gaaaWaaeWaaeaacaWGWbWaaeWaaeaacaWGobaa% caGLOaGaayzkaaGaai4Taiaad6eacaGGVaGaaeiBaiaab+gacaqGNb% Gaaeiiaiaad6eaaiaawIcacaGLPaaacqGH8aapcaaIXaaaaa!454E!\[\overline {{\rm{lim}}} \left( {p\left( N \right)\cdotN/{\rm{log }}N} \right) < 1\], then the probability that the sphere is completely covered by N caps tends to 0 as N , and if % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaWaaaeaaca% qGSbGaaeyAaiaab2gaaaWaaeWaaeaacaWGWbWaaeWaaeaacaWGobaa% caGLOaGaayzkaaGaai4Taiaad6eacaGGVaGaaeiBaiaab+gacaqGNb% Gaaeiiaiaad6eaaiaawIcacaGLPaaacqGH+aGpcaaIXaaaaa!4551!\[\underline {{\rm{lim}}} \left( {p\left( N \right)\cdotN/{\rm{log }}N} \right) > 1\], then for any integer n>0 the probability that each point of the sphere is covered more than n times tends to 1 as N .     

Forbidden paths and cycles in ordered graphs and matrices

Assume % MathType!End!2!1! and let Ω⊂R ^N(N≥4) be a smooth bounded domain, 0∈Ω. We study the semilinear elliptic problem: % MathType!End!2!1!. By investigating the effect of the coefficientQ, we establish the existence of nontrivial solutions for any λ>0 and multiple positive solutions with λ,μ>0 small.     

A class of consistent tests for exponentiality based on the empirical Laplace transform

The Laplace transform (t=E[exp(–tX)]) of a random variable with exponential density exp(–x), x0, satisfies the differential equation (+t)(t)+(t=0, t0). We study the behaviour of a class of consistent (omnibus) tests for exponentiality based on a suitably weighted integral of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGGBbGaaiikai% qbeU7aSzaajaWaaSbaaSqaaGqaciaa-5gaaeqaaOGaey4kaSIaamiD% aiaacMcacqaHipqEcaWFNaWaaSbaaSqaaiaad6gaaeqaaOGaaiikai% aadshacaGGPaGaey4kaSIaeqiYdK3aaSbaaSqaaiaad6gaaeqaaOGa% aiikaiaadshacaGGPaGaaiyxamaaCaaaleqabaGaaGOmaaaaaaa!4C69!\[[(\hat \lambda _n + t)\psi '_n (t) + \psi _n (t)]^2 \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacuaH7oaBgaqcam% aaBaaaleaaieGacaWFUbaabeaaaaa!3A66!\[\hat \lambda _n \] is the maximum-likelihood-estimate of and _n is the empirical Laplace transform, each based on an i.i.d. sample X ₁,...,X _n .     

A remark on the representation of vector lattices as spaces of continuous real-valued functions

The well-known Ogasawara-Maeda-Vulikh representation theorem asserts that for each Archimedean vector lattice L there exists an extremally disconnected compact Hausdorff space , unique up to a homeomorphism, such that L can be represented isomorphically as an order dense vector sublattice of the universally complete vector lattice C () of all extended-real-valued continuous functions f on for which % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiGc9yrFr0xXdbba91rFfpec8Eeeu0x% Xdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs% 0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaqaaqGaaO% qaamaacmqabaGaeqyYdCNaeyicI4SaeyyQdCLaeyOoaOJaaiiFaiab% gkzaMkabgIcaOiabgM8a3jabgMcaPiaacYhacqGH9aqpcqGHEisPai% aawUhacaGL9baaaaa!4E05!\[\left\{ {\omega \in \Omega :|f(\omega )| = \infty } \right\}\] is nowhere dense. Since the early days of using this representation it has been important to find conditions on L such that consists of bounded functions only.The aim of this short article is to present a simple complete characterization of such vector lattices.     

Building of mathematical models of continuum media on the basis of the invariance principle

The system of differential equations which describes the motion of continuum media of gas, liquid, Reiner-Rievling-type liquid, etc., is considered.% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHbp% GCdaWgaaWcbaGaamiDaaqabaGccqGHRaWkcaqGKbGaaeyAaiaabAha% caqGOaGaeqyWdiNaaeyDaiaabMcacaqG9aGaaeimaiaabUdaaeaacq% aHbpGCcaGGBbGaaeyDamaaBaaaleaacaqG0baabeaakiabgUcaRiaa% cIcacaqG1bGaeyyXICTaey4bIeTaaiykaiaabwhacaGGDbGaeyOeI0% IaamizaiaadMgacaWG2bGaey4dIuTaaiikaiabgEGirlaabwhacaGG% PaGaey4kaSIaey4bIeTaamiCaiaacUdaaeaacaWGWbWaaSbaaSqaai% aadshaaeqaaOGaey4kaSIaaeyDaiabgwSixlabgEGirlaadchacqGH% RaWkcaWGhbGaaeizaiaabMgacaqG2bGaaeiiaiaabwhacqGHRaWkca% WGibGaeqOXdyMaeyypa0JaaGimaiaac6caaaaa!7268!\[\begin{gathered} \rho _t + {\text{div(}}\rho {\text{u) = 0;}} \hfill \\ \rho [{\text{u}}_{\text{t}} + ({\text{u}} \cdot \nabla ){\text{u}}] - div\prod (\nabla {\text{u}}) + \nabla p; \hfill \\ p_t + {\text{u}} \cdot \nabla p + G{\text{div u}} + H\phi = 0. \hfill \\ \end{gathered} \]Solving the problem of its group classification, we obtained all the state equations which lead to the expansion of the main group assumed by the initial equations under the arbitrary elements , G, H.