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.     

The umbral calculus and orthogonal polynomials

We determine all orthogonal polynomials having Boas-Buck generating functions g(t)(xf(t)), where% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHOo% qwcaGGOaGaamiDaiaacMcacqGH9aqpruqqYLwySbacfaGaa8hiamaa% BeaaleaacaaIWaaabeaakiaadAeacaqGGaWaaSbaaSqaaiaabgdaae% qaaOGaaeikaiaadggacaGGSaGaa8hiaiaadshacaqGPaGaaeilaiaa% bccacaqGGaGaaeiiaiaadggacqGHGjsUcaaIWaGaaiilaiaa-bcacq% GHsislcaaIXaGaaiilaiaa-bcacqGHsislcaaIYaGaaiilaiablAci% ljaacUdaaeaacqqHOoqwcaGGOaGaamiDaiaacMcacqGH9aqpcaWFGa% WaaSraaSqaaiaaicdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOm% aaqabaGccaGGOaWaaSqaaSqaaiaaigdaaeaacaaIZaaaaOGaaiilai% aa-bcadaWcbaWcbaGaaGOmaaqaaiaaiodaaaGccaGGSaGaa8hiaiaa% dshacaGGPaGaa8hiamaaBeaaleaacaaIWaaabeaakiaadAeacaqGGa% WaaSbaaSqaaiaabkdaaeqaaOGaaeikamaaleaaleaacaaIYaaabaGa% aG4maaaakiaacYcacaWFGaWaaSqaaSqaaiaaisdaaeaacaaIZaaaaO% Gaaiilaiaa-bcacaWG0bGaaiykaiaacYcacaWFGaWaaSraaSqaaiaa% icdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOmaaqabaGccaGGOa% WaaSqaaSqaaiaaisdaaeaacaaIZaaaaOGaaiilaiaa-bcadaWcbaWc% baGaaGynaaqaaiaaiodaaaGccaGGSaGaa8hiaiaadshacaGGPaGaai% 4oaaqaaiabfI6azjaacIcacaWG0bGaaiykaiabg2da9iaa-bcadaWg% baWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaacaqGZaaabe% aakiaacIcadaWcbaWcbaGaaGymaaqaaiaaisdaaaGccaGGSaGaa8hi% amaaleaaleaacaaIYaaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaS% qaaiaaiodaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGaaiykaiaa% -bcadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaaca% qGZaaabeaakiaabIcadaWcbaWcbaGaaGOmaaqaaiaaisdaaaGccaGG% SaGaa8hiamaaleaaleaacaaIZaaabaGaaGinaaaakiaacYcacaWFGa% WaaSqaaSqaaiaaiwdaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGa% aiykaiaacYcaaeaadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiam% aaBaaaleaacaqGZaaabeaakiaacIcadaWcbaWcbaGaaG4maaqaaiaa% isdaaaGccaGGSaGaa8hiamaaleaaleaacaaI1aaabaGaaGinaaaaki% aacYcacaWFGaWaaSqaaSqaaiaaiAdaaeaacaaI0aaaaOGaaiilaiaa% -bcacaWG0bGaaiykaiaacYcacaGGUaGaa8hiamaaBeaaleaacaaIWa% aabeaakiaadAeacaqGGaWaaSbaaSqaaiaabodaaeqaaOGaaeikamaa% leaaleaacaaI1aaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaSqaai% aaiAdaaeaacaaI0aaaaOGaaiilaiaa-bcadaWcbaWcbaGaaG4naaqa% aiaaisdaaaGccaGGSaGaa8hiaiaadshacaGGPaGaaiOlaaaaaa!C1F3!\[\begin{gathered}\Psi (t) = {}_0F{\text{ }}_{\text{1}} {\text{(}}a, t{\text{), }}a \ne 0, - 1, - 2, \ldots ; \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{2}} (\tfrac{1}{3}, \tfrac{2}{3}, t) {}_0F{\text{ }}_{\text{2}} {\text{(}}\tfrac{2}{3}, \tfrac{4}{3}, t), {}_0F{\text{ }}_{\text{2}} (\tfrac{4}{3}, \tfrac{5}{3}, t); \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{3}} (\tfrac{1}{4}, \tfrac{2}{4}, \tfrac{3}{4}, t) {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{2}{4}, \tfrac{3}{4}, \tfrac{5}{4}, t), \hfill \\{}_0F{\text{ }}_{\text{3}} (\tfrac{3}{4}, \tfrac{5}{4}, \tfrac{6}{4}, t),. {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{5}{4}, \tfrac{6}{4}, \tfrac{7}{4}, t). \hfill \\\end{gathered}\]We also determine all Sheffer polynomials which are orthogonal on the unit circle. The formula for the product of polynomials of the Boas-Buck type is obtained.     

Variants of the Maurey–Rosenthal Theorem for Quasi Köthe Function Spaces

The Maurey–Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some L_p()(0) which satisfies a vector-valued norm inequality

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!\[P_1 \left\{ \begin{gathered} \frac{{\partial \theta }}{{\partial t}} - \Delta \theta + V \cdot \nabla \theta - \frac{1}{h}\upsilon _3 = 0 \hfill \\ S\frac{{\partial V}}{{\partial t}} + V + \nabla \pi + Ra^* k\theta = 0 \hfill \\ {\text{div }}V = 0 \hfill \\ V \cdot n_{|\Gamma } = 0 \hfill \\ \theta (x,y,0) = \theta (x,y,h) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial x}}(0,y,z) = \frac{{\partial \theta }}{{\partial x}}(L,y,z) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial y}}(x,0,z) = \frac{{\partial \theta }}{{\partial y}}(x,l,z) = 0 \hfill \\ \end{gathered} \right.\]

---

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!\[P_1 \left\{ \begin{gathered} \frac{{\partial \theta }}{{\partial t}} - \Delta \theta + V \cdot \nabla \theta - \frac{1}{h}\upsilon _3 = 0 \hfill \\ S\frac{{\partial V}}{{\partial t}} + V + \nabla \pi + Ra^* k\theta = 0 \hfill \\ {\text{div }}V = 0 \hfill \\ V \cdot n_{|\Gamma } = 0 \hfill \\ \theta (x,y,0) = \theta (x,y,h) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial x}}(0,y,z) = \frac{{\partial \theta }}{{\partial x}}(L,y,z) = 0 \hfill \\ \frac{{\partial \theta }}{{\partial y}}(x,0,z) = \frac{{\partial \theta }}{{\partial y}}(x,l,z) = 0 \hfill \\ \end{gathered} \right.\]

     

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.     

Estimates for Multiplicative Functions on the Set of Shifted Primes

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

Problems on Extremal Decomposition of the Riemann Sphere

We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let be a system of distinct points on and let be the family of all systems of nonoverlapping simply connected domains on such that . Let

Generalization of some classical inequalities in the theory of orthogonal series

Let {X_i} _– be a sequence of random variables, E(X_i) 0. If 1, estimates for the -th moments can be derived from known estimates of the -th moment. Here we generalized the Men'shov-Rademacher inequality for =2 for orthonormal X_i, to the case 1 and dependent random variables. The Men'shov-Payley inequality >2 for orthonormal X_i) is generalized for >2 to general random variables. A theorem is also proved that contains both the Erdös -Stechkin theorem and Serfling's theorem withv > 2 for dependent random variables.Translated from Matematicheskie Zametki, Vol. 17, No. 2, pp. 219–230, February, 1975.This article was written while the author was working in the V. A. Steklov Mathematics Institute, Academy of Sciences of the USSR.     

Counting the centralizers of some finite groups

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is called n-centralizer if #Cent(G) =n, and primitiven-centralizer if # Cent(G)\text = # Cent\text(\fracGZ(G))\text = n\# Cent(G){\text{ = \# }}Cent{\text{(}}\frac{G}{{Z(G)}}){\text{ = }}n. In this paper we compute the number of distinct centralizers of some finite groups and investigate the structure of finite groups with exactly six distinct centralizers. We prove that ifG is a 6-centralizer group then % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% WGhbaabaGaamOwaiaacIcacaWGhbGaaiykaaaacaqGGaGaeyyrIaKa% aeiiaGqaciaa-readaWgaaWcbaGaa8hoaaqabaGccaGGSaGaaeiiai% aa-feadaWgaaWcbaGaa8hnaaqabaGccaGGSaGaaeiiaiaabQfadaWg% aaWcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaa% WcbaGaaeOmaaqabaGccaqGGaGaey41aqRaaeiiaiaabQfadaWgaaWc% baGaaeOmaaqabaGccaqGGaGaae4BaiaabkhacaqGGaGaaeOwamaaBa% aaleaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaa% leaacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaale% aacaqGYaaabeaakiaabccacqGHxdaTcaqGGaGaaeOwamaaBaaaleaa% caqGYaaabeaaaaa!62C4!\[\frac{G}{{Z(G)}}{\text{ }} \cong {\text{ }}D_8 ,{\text{ }}A_4 ,{\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ or Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} \] .     

Nonlinear optimization: Characterization of structural stability

We study global stability properties for differentiable optimization problems of the type: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbGaai% ikaGqaaiaa-jzacaGGSaGaamisaiaacYcacaqGGaGaam4raiaacMca% caGG6aGaaeiiaiaab2eacaqGPbGaaeOBaiaabccacaWFsgGaaeikai% aadIhacaqGPaGaaeiiaiaab+gacaqGUbGaaeiiaiaad2eacaGGBbGa% amisaiaacYcacaWGhbGaaiyxaiabg2da9iaacUhacaWG4bGaeyicI4% CeeuuDJXwAKbsr4rNCHbacfaGae4xhHe6aaWbaaSqabeaacaWGUbaa% aOGaaiiFaiaabccacaWGibGaaiikaiaadIhacaGGPaGaeyypa0JaaG% imaiaacYcacaqGGaGaam4raiaacIcacaWG4bGaaiykamaamaaabaGa% eyyzImlaaiaaicdacaGG9bGaaiOlaaaa!6B2E!\[P(f,H,{\text{ }}G):{\text{ Min }}f{\text{(}}x{\text{) on }}M[H,G] = \{ x \in \mathbb{R}^n |{\text{ }}H(x) = 0,{\text{ }}G(x)\underline \geqslant 0\} .\] Two problems are called equivalent if each lower level set of one problem is mapped homeomorphically onto a corresponding lower level set of the other one. In case that P(% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaaceWFsg% GbaGaacaWFSaGaa8hiaiqadIeagaacaiaacYcacaWFGaGabm4rayaa% iaaaaa!3EBF!\[\tilde f, \tilde H, \tilde G\]) is equivalent with P(f, H, GG) for all (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaaieaaceWFsg% GbaGaacaWFSaGaa8hiaiqadIeagaacaiaacYcacaWFGaGabm4rayaa% iaaaaa!3EBF!\[\tilde f, \tilde H, \tilde G\]) in some neighbourhood of (f, H, G) we call P(f, H, G) structurally stable; the topology used takes derivatives up to order two into account. Under the assumption that M[H, G] is compact we prove that structural stability of P(f, H, GG) is equivalent with the validity of the following three conditions:

1. The Mangasarian-Fromovitz constraint qualification is satisfied at every point of M[H, G].
2. Every Kuhn-Tucker point of P(f, H, GG) is strongly stable in the sense of Kojima.
3. Different Kuhn-Tucker points have different (f-)values.

     

Random graph approach to Gibbs processes with pair interaction

This study was motivated by the observation that, in a broad class of cases, the distribution of classical Gibbs point processes in R ^d governed by pair potential, can be obtained as the equilibrium distribution of a Markov chain of point processes in R ^d. Our analysis of this Markov chain is based on its imbedding in an infinite random graph. A condition of ergodicity of the chain is given in terms of the absence of percolation in the graph, and this can be checked in simpler cases. The embedding also suggests a stochastic construction for the equilibrium distribution in question.These constructions (which can also be of independent interest) are related to Gibbs processes by means of the results obtained in a recent paper of R. V. Ambartzumian and H. S. Sukiasian [1] where the existence of a new class of stationary point processes in R ^d was established which have density (correlation) functions of the form % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqdbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfiGae8NKbmOae8hkaGIae8hEaG3aaSba% a4qaaiaabgdaaeqaa0Gaaiilaiaac6cacaGGUaGaaiOlaiaacYcaca% WG4bWaaSbaa4qaaiaad6gaaeqaa0GaaiykaiaabccacqGH9aqpcaqG% GaGaamOyamaaCaaaoeqabaGaamOBaaaanmaarababaGaamiAaiaacI% cacaWG4bWaaSbaa4qaaiaadMgaaeqaaaqaaiaad6gaaeqaniabg+Gi% vdGaeyOeI0IaamiEamaaBaaaoeaacaWGQbaabeaaniaacMcaaaa!56B6!\[f(x_{\text{1}} ,...,x_n ){\text{ }} = {\text{ }}b^n \prod\nolimits_n {h(x_i } - x_j )\] (here and below, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaqdbaWaaebaaeaada% WgaaGdbaGaamOBaaqabaaabeqab0Gaey4dIunaaaa!38CA!\[\prod {_n } \] denotes a product taken over all two-subsets {i, j} {1,..., n}).     

Path integrals on finite sets

We construct an analogue of the Feynman path integral for the case of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% aaaeaacaaIXaaabaGaamyAaaaadaWcaaqaaiabgkGi2cqaaiabgkGi% 2kaadshaaaqeduuDJXwAKbYu51MyVXgaiuaacqWFvpGAcaWG0bGaey% ypa0JaamisamaaBaaaleaacaGGOaaabeaakmaaBaaaleaacaGGPaaa% beaakiab-v9aQjaadshaaaa!4A8D!\[ - \frac{1}{i}\frac{\partial }{{\partial t}}\varphi t = H_( _) \varphi t\] in which H ₍₎ is a self-adjoint operator in the space L ²(M)= % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSOaHmkaaa!3744!\[\mathbb{C}\], where _M is a finite set, the paths being functions of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa!375D!\[\mathbb{R}\] with values in M. The path integral is a family of measures F _t,t with values in the operators on L ²(M), or equivalently, a family of complex measures corresponding to matrix coefficients.It is shown that these measures on path space are in some sense dominated by the measure of a Markov process. This implies that F _t,t is concentrated on the set of step functions S[t,t].This allows one to make sense of, and prove, the analogue of Feynman's formula for the propagator of the Hamiltonian H=H ₀+V, where V is a potential, namely the formula: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzamaaCa% aaleqabaGaeyOeI0IaamyAaiaacIcacaWG0bGaai4jaiabgkHiTiaa% dshacaGGPaGaamisaaaakiabg2da9maapebabaGaaeyzamaaCaaale% qabaGaeyOeI0IaamyAamaapedabaGaamOvaiaacIcatCvAUfKttLea% ryqr1ngBPrgaiuGacqWF4baEcaGGOaGaam4CaiaacMcacaGGPaGaae% izaiaabohaaWqaaiaadshaaeaacaWG0bGaai4jaaGdcqGHRiI8aaaa% kiaadAeadaWgaaWcbaGaamiDaiaacEcacaGGSaGaamiDaaqabaGcca% GGOaGaaeizaiab-Hha4jaacMcaaSqaaiaadofacaGGBbGaamiDaiaa% cYcacaWG0bGaai4jaiaac2faaeqaniabgUIiYdaaaa!6410!\[{\text{e}}^{ - i(t' - t)H} = \int_{S[t,t']} {{\text{e}}^{ - i\int_t^{t'} {V(x(s)){\text{ds}}} } F_{t',t} ({\text{d}}x)} \]and the corresponding formulas for the matrix coefficients, in which the integral extends over the paths beginning and ending in the appropriate points. We show that the measures F _t,t are completely determined by these equations and by a certain multiplicative property.The path integral corresponding to a two-particle system without interaction is the direct product of the corresponding path integrals. The propagator for a two-particle system with interaction can be obtained by repeated integration.Finally, we show that the above integral formula can be generalized to the case where the potential is time dependent.     

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.     

Symmetric orbits of orthogonal Plücker actions and triviality of their second order envelopes

The orbits of Lie groups acting in Euclidean spaces by isometries are extrinsically symmetric iff they are parallel, i.e. satisfy % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0. Submanifolds characterized by the integrability condition \-R · h = 0 of this system % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]h = 0 are called semi-parallel; they are the second order envelopes of the symmetric orbits. Let the orbit set of an action of SO(n, R) in E ^{% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca% aIXaaabaGaaGOmaaaaaaa!3773!\[\frac{1}{2}\]n(n–1)} contain a Plücker submanifold. It is proved that 1) the only symmetric orbits are Plücker orbits and for n = 2 > 4 the unitary orbits, 2) each of their second order envelopes is trivial, i.e. is a single orbit or its open part.Partially supported by ESF Grant 139/305     

Geometric constant term functor(s)

We study the Eisenstein series and constant term functors in the framework of geometric theory of automorphic functions. Our main result says that for a parabolic \(P\subset G\) with Levi quotient M, the !-constant term functor

$$\begin{aligned}{\text {CT}}_!:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M)\end{aligned}$$

is canonically isomorphic to the *-constant term functor

$$\begin{aligned} {\text {CT}}^-_*:{\text {D-mod}}({\text {Bun}}_G)\rightarrow {\text {D-mod}}({\text {Bun}}_M), \end{aligned}$$

taken with respect to the opposite parabolic \(P^-\).

     

Estimation of a multivariate Box-Cox transformation to elliptical symmetry via the empirical characteristic function

Let X=(X ₁, X ₂,..., X _d ) ^t be a random vector of positive entries, such that for some =(₁,₂,..., _d ) ^t , the vector X ⁽⁾ defined by % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiwamaaDaaaleaamiaadMgaaSqaaWGaaiikaiabeU7aSnaaBaaa% baGaamyAaiaacMcaaeqaaaaakiabg2da9iaacIcadaWcgaqaaiaadI% fadaqhaaWcbaadcaWGPbaaleaamiabeU7aSnaaBaaabaGaamyAaaqa% baaaaOGaeyOeI0IaaGymaiaacMcaaeaacqaH7oaBdaWgaaWcbaadca% WGPbGaaiilaaWcbeaakiaadMgacqGH9aqpcaaIXaGaeSOjGSKaaiil% aiaadsgaaaaaaa!53BB!\[X_i^{(\lambda _{i)} } = ({{X_i^{\lambda _i } - 1)} \mathord{\left/ {\vphantom {{X_i^{\lambda _i } - 1)} {\lambda _{i,} i = 1 \ldots ,d}}} \right. \kern-\nulldelimiterspace} {\lambda _{i,} i = 1 \ldots ,d}}\]is elliptically symmetric. We describe a procedure based on the multivariate empirical characteristic function for estimating the _i's. Asymptotic results regarding consistency of the estimators are given and we evaluate their performance in simulated data. In a one-dimensional setting, comparisons are made with other available transformations to symmetry.Adolfo Quiroz and Miguel Nakamura's research was partially supported by CONACYT (Mexico) grants numbers 1858E9219 and 4224E9405, while Dr. Quiroz was visiting Centro de Investigación en Matemáticas at Guanajuato, Mexico.     

On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

On the Lipschitz Constant for a Nonisometric Bi-Lipschitz Transformation of a Cantor Set

A bi-Lipschitz continuous mapping of a space X is a bijection such that , where . We write if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by the set of all bi-Lipschitz mappings of X that are not isometry. Thus, if and blip . For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows: where Bibliography: 2 titles.     

K-Theory of Semi-local Rings with Finite Coefficients and étale Cohomology

Let A be a commutative semi-local ring containing 1/2. We construct natural isomor-phisms

The limiting case of the Marcinkiewicz integral: growth for convex sets

The Marcinkiewicz integral

plays a well-known and prominent role in harmonic analysis. In this paper, we estimate the growth of it in the limiting case . Throughout, we assume that is convex; it is interesting that this condition cannot be dropped.