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.     

Optimum bandwidths and kernels for estimating certain discontinuous densities

Rosenblatt and Parzen proposed a well-known estimator f _n for an unknown density function f, and later Schuster suggested a modification % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybyaeqale% qabaGaaiOxaaqdbaGaamOzaaaaaaa!3851!\[\mathop f\limits^\^ \] _n to rectify certain drawbacks of f _n . This paper gives the asymptotically optimum bandwidth and kernel for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybyaeqale% qabaGaaiOxaaqdbaGaamOzaaaaaaa!3851!\[\mathop f\limits^\^ \] _n under the standard measure of IMSE when f is discontinuous at one or both endpoints of its support. We also consider an alternative definition of the IMSE under which the optimum bandwidths and kernels for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybyaeqale% qabaGaaiOxaaqdbaGaamOzaaaaaaa!3851!\[\mathop f\limits^\^ \] _n and f _n are derived. The latter supplement van Eeden's results.     

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     

The basic component of the mean curvature of Riemannian foliations

For a Riemannian foliation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\] on a compact manifold M with a bundle-like metric, the de Rham complex of M is C-splitted as the direct sum of the basic complex and its orthogonal complement. Then the basic component % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacciGae8NUdS% 2aaSbaaSqaaiaadkgaaeqaaaaa!38B9!\[\kappa _b \] of the mean curvature form of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\] is closed and defines a class (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\]) in the basic cohomology that is invariant under any change of the bundle-like metric. Moreover, any element in (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\]) can be realized as the basic component of the mean curvature of some bundle-like metric.It is also proved that (% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\]) vanishes iff there exists some bundle-like metric on M for which the leaves are minimal submanifolds. As a consequence, this tautness property is verified in any of the following cases: (a) when the Ricci curvature of the transverse Riemannian structure is positive, or (b) when % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFXeIraaa!4094!\[\mathcal{F}\] is of codimension one. In particular, a compact manifold with a Riemannian foliation of codimension one has infinite fundamental group.     

Riemannian submersions and the regular interval theorem of Morse theory

A generalized version of the regular interval theorem of Morse theory is proven using techniques from the theory of Riemannian submersions and conformal deformations. This approach provides an interesting link between Riemannian submersions (for real valued functions) and Morse theory.Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]: (M,) R be a smooth real valued function on a non-compact complete connected Riemannian manifold (M,g) such that df is bounded in norm away from zero. By pointwise conformally deforming g to pg, p = d% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]², we show that (M,pg) is a complete Riemannian manifold, and that % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]: (M,pg) R is a surjective Riemannian submersion and a globally trivial fiber bundle over R. In particular, all of the level hypersurfaces of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] are diffeomorphic, and M is globally diffeomorphic to the product bundle R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] ^–1(0) by a diffeomorphism F ⁰: R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0) M that straightens out the level hypersurfaces of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\].Moreover, we show that (F ⁰)^*(pg) is a parameterized Riemannian product manifold on R×% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), i.e., a product manifold with a metric that varies on the fibers {t} × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0). Also, F ⁰: (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0),(F ⁰)^*(pg)) (M,g) is a conformal diffeomorphism between the Reimannian manifolds (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), (F ⁰)^*(pg)) and (M,g),so that (M,g) is conformally equivalent to a parameterized Riemannian product manifold. The conformal diffeomorphism F ⁰ is an isometry between the Riemannian product manifold (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), 1 + g ₀) (where g ₀) is the metric induced by g on % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0) and (M,g) if and only if d% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] = 1 and Hess % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] = 0.     

A smoothing spline based test of model adequacy in polynomial regression

For the regression model % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa% aajeaWbaGaamyAaaWcbeaakiabeccaGiabg2da9iabeccaGiaabAga% caGGOaGaamiDamaaBaaajeaWbaGaamyAaaWcbeaakiaacMcacaqGGa% Gaey4kaSIaaeiiamaavababeqcbaCaaiaadMgaaSqab0qaaiabew7a% LbaakiaabccacaGGOaGaeqyTduMaai4jaiaadohacaqGGaGaamyAai% aadMgacaWGKbGaaeiiaiaad6eacaGGOaGaam4taiaacYcacaqGGaGa% eq4Wdm3aaWbaaSqabKqaGgaacaaIYaaaaOGaaiykaiaacMcaaaa!57B9!\[y_i = {\rm{f}}(t_i ){\rm{ }} + {\rm{ }}\mathop \varepsilon \nolimits_i {\rm{ }}(\varepsilon 's{\rm{ }}iid{\rm{ }}N(O,{\rm{ }}\sigma ^2 ))\], it is proposed to test the null hypothesis that f is a polynomial of degree less than some given value m. The alternative is that f is such a polynomial plus a scale factor b ^1/2 times an (m–1)-fold integrated Wiener process. For this problem, it is shown that no uniformly (in b) most powerful test exists, but a locally (at b=0) most powerful test does exist. Derivation and calculation of the test statistic is based on smoothing spline theory. Some approximations of the null distribution of the test statistic for the locally most powerful test are described. An example using real data is shown along with a computing algorithm.This author's research was supported by the National Science Foundation under grants numbered DMS-8202560 and DMS-8603083.     

Submanifolds of constant sectional curvature in Pseudo-Riemannian manifolds

The generalized equation and the intrinsic generalized equation are considered. The solutions of the first one are shown to correspond to Riemannian submanifolds M ⁿ (K) of constant sectional curvature of psedo-Riemannian manifolds % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaamaaDaaaleaaieGacaWFZbaabaGaaGOmaiaad6gacqGHsisl% caaIXaaaaOGaaiikamaanaaabaGaam4saaaacaGGPaaaaa!3D97!\[\overline M _s^{2n - 1} (\overline K )\] of index s, with % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgc% Mi5oaanaaabaGaam4saaaaaaa!3965!\[K \ne \overline K \], flat normal bundle and such that the normal principal curvatures are different from % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiabgk% HiTmaanaaabaGaam4saaaaaaa!388B!\[K - \overline K \]. The solutions of the intrinsic generalized equation correspond to Riemannian metrics defined on open subsets of R ⁿ which have constant sectional curvature. The relation between solutions of those equations is given. Moreover, it is proven that the submanifolds M under consideration are determined, up to a rigid motion of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \], by their first fundamental forms, as solutions of the intrinsic generalized equation. The geometric properties of the submanifolds M associated to the solutions of the intrinsic generalized equation, which are invariant under an (n – 1)-dimensional group of translations, are given. Among other results, it is shown that such submanifolds are foliated by (n – 1)-dimensional flat submanifolds which have constant mean curvature in M. Moreover, each leaf of the foliation is itself foliated by curves of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa0aaaeaaca% WGnbaaaaaa!36D0!\[\overline M \] which have constant curvatures.     

Designs as maximum codes in polynomial metric spaces

Finite and infinite metric spaces % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] that are polynomial with respect to a monotone substitution of variable t(d) are considered. A finite subset (code) W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is characterized by the minimal distance d(W) between its distinct elements, by the number l(W) of distances between its distinct elements and by the maximal strength (W) of the design generated by the code W. A code W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] is called a maximum one if it has the greatest cardinality among subsets of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with minimal distance at least d(W), and diametrical if the diameter of W is equal to the diameter of the whole space % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. Delsarte codes are codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with (W)2l(W)–1 or (W)=2l(W)–2>0 and W is a diametrical code. It is shown that all parameters of Delsarte codes W) % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are uniquely determined by their cardinality |W| or minimal distance d(W) and that the minimal polynomials of the Delsarte codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] are expansible with positive coefficients in an orthogonal system of polynomials {Q _i(t)} corresponding to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\]. The main results of the present paper consist in a proof of maximality of all Delsarte codes provided that the system {Q _i)} satisfies some condition and of a new proof confirming in this case the validity of all the results on the upper bounds for the maximum cardinality of codes W % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOHI0maaa!36D8!\[ \subseteq \]% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgzgj% xyRrxDYbqeguuDJXwAKbIrYf2A0vNCaGqbaiab-Xa8nbaa!427C!\[\mathfrak{M}\] with a given minimal distance, announced by the author in 1978. Moreover, it appeared that this condition is satisfied for all infinite polynomial metric spaces as well as for distance-regular graphs, decomposable in a sense defined below. It is also proved that with one exception all classical distance-regular graphs are decomposable. In addition for decomposable distance-regular graphs an improvement of the absolute Delsarte bound for diametrical codes is obtained. For the Hamming and Johnson spaces, Euclidean sphere, real and complex projective spaces, tables containing parameters of known Delsarte codes are presented. Moreover, for each of the above-mentioned infinite spaces infinite sequences (of maximum) Delsarte codes not belonging to tight designs are indicated.     

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.     

The thermodynamic limit of polygonal models

We discuss the basic models of polygonal Markov fields with a two-dimensional continuous parameter % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgI% GioNWaajaadsfaoiabgkOimlabl2riHoaaCaaabeqaaiaaikdaaaaa% aa!3DEB!\[x \in T \subset \mathbb{R}^2 \], which were introduced by Arak and studied later by Arak and surgailis. There are two types of polygonal models, either with a given initial distribution of the lines or of the points (vertices) of a random polygonal graph. The main result of this paper is proof of the existence of a thermodynamic limit (as % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbiqaaW2ajmaqca% WGubacciqceaQae8xKH0QdcqWIDesOdaahaaqabeaacaaIYaaaaaaa% !3C7D!\[T \uparrow \mathbb{R}^2 \]) for a class of polygonal models with a small contour length'. It is based on the study of Kirkwood-Salzburg-type equations for the correlation functions. We also discuss some examples of consistent polygonal models for which the existence of the thermodynamic limit is trivial.     

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

     

Statistical analysis of dyadic stationary processes

In this paper we consider a multiple dyadic stationary process with the Walsh spectral density matrix f(), where is an unknown parameter vector. We define a quasi-maximum likelihood estimator % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiUdyaaja% aaaa!377D!\[{\rm{\hat \theta }}\] of , and give the asymptotic distribution of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiUdyaaja% aaaa!377D!\[{\rm{\hat \theta }}\] under appropriate conditions. Then we propose an information criterion which determines the order of the model, and show that this criterion gives a consistent order estimate. As for a finite order dyadic autoregressive model, we propose a simpler order determination criterion, and discuss its asymptotic properties in detail. This criterion gives a strong consistent order estimate. In Section 5 we discuss testing whether an unknown parameter satisfies a linear restriction. Then we give the asymptotic distribution of the likelihood ratio criterion under the null hypothesis.This work is supported by Contract N00014-85-K-0292 of the Office of Naval Research and Contract F49620-85-C-0008 of the Air Force Office of Scientific Research. The United States Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notations hereon.The work of this author was done at the Center for Multivariate Analysis. His permanent address is Department of Mathematics, Hiroshima University, Hiroshima 730, Japan.     

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}).     

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.

     

Nonparametric confidence intervals for functions of several distributions

Let F=(F₁...F_k) denote k unknown distribution functions and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja% Gaeyypa0ZaaeWaaeaaceWGgbGbaKaadaWgaaWcbaGaaGymaaqabaGc% caGGUaGaaiOlaiaac6caceWGgbGbaKaadaWgaaWcbaGaam4Aaaqaba% aakiaawIcacaGLPaaaaaa!3E24!\[\hat F = \left( {\hat F_1 ...\hat F_k } \right)\] their sample (empirical) functions based on random samples from them of sizes n ₁, ..., n _k. Let T(F) be a real functional of F. The cumulants of T(% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja% aaaa!35B2!\[\hat F\]) are expanded in powers of the inverse of n, the minimum sample size. The Edgeworth and Cornish-Fisher expansions for both the standardized and Studentized forms of T(% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOrayaaja% aaaa!35B2!\[\hat F\]) are then given together with confidence intervals for T(F) of level 1–+O(n^-j/2) for any given in (0, 1) and any given j. In particular, confidence intervals are given for linear combinations and ratios of the means and variances of different populations without assuming any parametric form for their distributions.     

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.     

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!\[\frac{G}{{Z(G)}}{\text{ }} \cong {\text{ }}D_8 ,{\text{ }}A_4 ,{\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ or Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} {\text{ }} \times {\text{ Z}}_{\text{2}} \] .     

Finite-dimensional time-invariant linear dynamical systems: Algebraic theory

Willems' approach to dynamical systems without a priori distinguishing between inputs and outputs is accepted, and with this as a starting point, new linear dynamical systems are introduced and studied. It is proved in particular that (in the complex case) the set of isomorphism classes of completely observable (or completely reachable) linear systems with given input and output numbers and McMillan degree, has a natural structure of a compact algebraic variety. This variety is closely connected to the one constructed by Hazewinkel using the Rosenbrock linear systems % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]=Ax+Bu, v=Cx+D(·)u, where D is a polynomial matrix, and may be regarded as the most natural compactification of it. (The connection is very similar to that of Grass_m,mx+p() and Mat_m.p(). Input/output linear systems, i.e. linear systems equipped with an extra structure which distinguishes input and output signals, are also considered. It is shown that each of them may be represented by the equations K% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeiEayaaca% aaaa!35DB!\[{\rm{\dot x}}\]+L% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabeyDayaaca% aaaa!35D8!\[{\rm{\dot u}}\]=Fx+Gu, v=Hx+Ju (det(K–sF)0). Such systems clearly contain the so-called generalized linear systems. They also contain the Rosenbrock linear systems mentioned above and essentially coincide with them.     

A procedure for assessing vector correlations

Three known measures of multivariate relationship are presented. Under the null hypothesis of lack of multivariate relationship between K random vectors, the asymptotic joint distributions of the % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa% qabeGabaaabaGaam4saaqaaiaaikdaaaaacaGLOaGaayzkaaaaaa!390F!\[\left( {\begin{array}{*{20}c}K \\2 \\\end{array} } \right)\]values taken by these measures for all possible pairs (X ⁽ⁱ⁾, X^(j)), 1i<jK, is used to construct tests of the null hypothesis based on the maximum and more generally, on the greatest values of the measures. The asymptotic power of the tests is also obtained under a sequence of alternatives.The author was partially supported by a grant of the Natural Sciences and Engineering Research Council of Canada.     

Combinatorial Characterizations of Authentication Codes II

For any authentication code for k source states and v messages having minimum possible deception probabilities (namely, % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS% baaSqaaiaadsgaaeqaaOWaaSbaaSqaaiaaicdaaeqaaOGaeyypa0Za% aSGbaeaacaWGRbaabaGaamODaaaaaaa!3F28!\[P_d _0 = {k \mathord{\left/ {\vphantom {k v}} \right. \kern-\nulldelimiterspace} v}\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9qq-f0-yqaqVeLsFr0-vr% 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGqbWaaS% baaSqaaiaadsgaaeqaaOWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0Ze% xLMBb50ujbqeguuDJXwAKbacfaGae8hkaGIaam4AaiabgkHiTiaaig% dacqWFPaqkcqWFVaWlcqWFOaakcqWF2bGDcqGHsislcaaIXaGae8xk% aKIae8xkaKcaaa!4CD1!\[P_d _1 = (k - 1)/(v - 1))\], we show that there must be at least v encoding rules. (This can be thought of as an authentication-code analogue of Fisher's Inequality.) We derive several properties that an extremal code must satisfy, and we characterize the extremal codes for equiprobable source states as arising from symmetric balanced incomplete block designs. We also present an infinite class of extremal codes, in which the source states are not equiprobable, derived from affine planes.Dedicated to Gus Simmons