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.     

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.     

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 threshold for the size of random caps to cover a sphere

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

Cross-monotone subsequences

Two finite real sequences (a ₁,...,a _k ) and (b ₁,...,b _k ) are cross-monotone if each is nondecreasing anda _i+1–a _i b _i+1–b _i for alli. A sequence (₁,..., _n ) of nondecreasing reals is in class CM(k) if it has disjointk-term subsequences that are cross-monotone. The paper shows thatf(k), the smallestn such that every nondecreasing (₁,..., _n ) is in CM(k), is bounded between aboutk ²/4 andk ₂/2. It also shows thatg(k), the smallestn for which all (₁,..., _n ) are in CM(k)and eithera _k b ₁ orb _k a ₁, equalsk(k–1)+2, and thath(k), the smallestn for which all (₁,..., _n ) are in CM(_k)and eithera ₁b ₁...a _k b _k orb ₁a ₁...b _k a _k , equals 2(k–1)²+2.The results forf andg rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangulark ²×k ² (0, 1)-matrix has eitherk 1's in consecutive columns, each below its predecessor, ork 0's in consecutive rows, each to the right of its predecessor, and the same conclusion is false whenk ² is replaced byk ²–1.     

Highly linked graphs

A graph with at least 2k vertices is said to bek-linked if, for any choices ₁,...,s _k ,t ₁,...,t _k of 2k distinct vertices there are vertex disjoint pathsP ₁,...,P _k withP _i joinings _i tot _i , 1ik. Recently Robertson and Seymour [16] showed that a graphG isk-linked provided its vertex connectivityk(G) exceeds . We show here thatk(G)22k will do.     

Solution of an extremal problem for sets using resultants of polynomials

A new, short proof is given of the following theorem of Bollobás: LetA ₁,..., A_h andB ₁,..., B_h be collections of sets with _i ¦A _i¦=r,¦B_i¦=s and ¦A _iB_j¦=Ø if and only ifi=j, thenh( _s ^r+s ). The proof immediately extends to the generalizations of this theorem obtained by Frankl, Alon and others.     

On a Turán type problem of Erdös

Let L ^k be the graph formed by the lowest three levels of the Boolean lattice B _k , i.e.,V(L ^k )={0, 1,...,k, 12, 13,..., (k–1)k} and 0is connected toi for all 1ik, andij is connected toi andj (1i<jk).It is proved that if a graph G overn vertices has at leastk ^3/2 n ^3/2 edges, then it contains a copy of L ^k .Research supported in part by the Hungarian National Science Foundation under Grant No. 1812     

Subelliptic operators on Lie groups: Variable coefficients

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

On the structure of singular sets of convex functions

Whenf is a convex function ofR ^h, andk is an integer with 0<k, then the set ^k (f)=x:dim(f(x)k may be covered by countably many manifolds of dimensionh–k and classC ² except an _h–k negligible subset.The author is supported by INdAM     

Sharp Bounds For Some Multicolor Ramsey Numbers

Let H ₁,H ₂, . . .,H _k+1 be a sequence of k+1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H ₁,H ₂,...,H _k+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of H _i in color i for some 1ik+1. We describe a general technique that supplies tight lower bounds for several numbers r(H ₁,H ₂,...,H _k+1) when k2, and the last graph H _k+1 is the complete graph K _m on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K ₃,K ₃,K _m ) = (m ³ poly logm), thus solving (in a strong form) a conjecture of Erdos and Sós raised in 1979. Another special case of our result implies that r(C ₄,C ₄,K _m ) = (m ² poly logm) and that r(C ₄,C ₄,C ₄,K _m ) = (m ²/log² m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.* Research supported in part by a State of New Jersey grant, by a USA Israeli BSF grant and by a grant from the Israel Science Foundation. Research supported by NSF grant DMS 9704114.     

On the product of sign vectors and unit vectors

For a fixed unit vectora=(a ₁,...,a _n )S ^n-1, consider the 2 ⁿ sign vectors=(₁,..., _n ){±1{ ⁿ and the corresponding scalar products^·a = _n ⁱ⁼¹ = _i a _i . The question that we address is: for how many of the sign vectors must^.a lie between–1 and 1. Besides the straightforward interpretation in terms of the sums ±a ₂ , this question has appealing reformulations using the language of probability theory or of geometry.The natural conjectures are that at least 1/2 the sign vectors yield |.a|1 and at least 3/8 of the sign vectors yield |.a|<1 (the latter excluding the case when |a _i |=1 for somei). These conjectured lower bounds are easily seen to be the best possible. Here we prove a lower bound of 3/8 for both versions of the problem, thus completely solving the version with strict inequality. The main part of the proof is cast in a more general probabilistic framework: it establishes a sharp lower bound of 3/8 for the probability that |X+Y|<1, whereX andY are independent random variables, each having a symmetric distribution with variance 1/2.We also consider an asymptotic version of the question, wheren along a sequence of instances of the problem satisfying ||a||0. Our result, best expressed in probabilistic terms, is that the distribution of .a converges to the standard normal distribution, and in particular the fraction of sign vectors yielding .a between –1 and 1 tends to 68%.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Inclusion-exclusion inequalities

Ifμ is a positive measure, andA ₂, ...,A _n are measurable sets, the sequencesS ₀, ...,S _n andP _[0], ...,P _[n] are related by the inclusion-exclusion equalities. Inequalities among theS _i are based on the obviousP _[k]≧0. Letting =the average average measure of the intersection ofk of the setsA _i , it is shown that (−1) ^k Δ ^k M _i ≧0 fori+k≦n. The casek=1 yields Fréchet’s inequalities, andk=2 yields Gumbel’s and K. L. Chung’s inequalities. Generalizations are given involvingk-th order divided differences. Using convexity arguments, it is shown that forS ₀=1, whenS ₁≧N−1, and for 1≦k<N≦n andv=0, 1, .... Asymptotic results asn → ∞ are obtained. In particular it is shown that for fixedN, for all sequencesM ₀, ...,M _n of sufficiently large length if and only if for 0<t<1.     

Estimating common parameters of growth curve models

Suppose that we have two independent random matrices X ₁ and X ₂ having multivariate normal distributions with common unknown matrix of parameters (q×m) and different unknown covariance matrices ₁ and ₂, given by N _{p1, N1} (B ₁ A ₁; ₁, I) and N _{p2, N2} (B ₂ A ₂; ₂, I) respectively. Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] (% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\]) be the MLE of based on X ₁ (X ₂) only. When q=1, necessary and sufficient conditions that a combined estimator of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] has uniformly smaller covariance matrix than those of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaigdaaaa!391C!\[\hat \xi 1\] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dbrpepeea0-rrpec8Ei0dbbf9q8WrFbJ8FeK8qq% -hc9Gqpee9FiuP0-is0dXdbba9pee9xq-Jbba9suk9fr-xfr-xfrpe% WZqaceaabiGaciaacaqabeaadaqaaqGaaOqaaiqbe67a4zaajaqcKf% aOaiaaikdaaaa!391D!\[\hat \xi 2\] are given. The k-sample problem as well as one-sample problem is also discussed. These results are extensions of those of Graybill and Deal (1959, Biometrics, 15, 543–550), Bhattacharya (1980, Ann. Statist., 8, 205–211; 1984, Ann. Inst. Statist. Math., 36, 129–134) to multivariate case.Dedicated to Professor Yukihiro Kodama on his 60th birthday.Bowling Green State UniversityVisiting Professor on leave from the University of Tsukuba, Japan. Now at Department of Mathematics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan.This research was partially supported by University of Tsukuba Project Research 1986.     

Recursive constructions for skew resolutions in affine geometries

A resolutionR inAG(n, q) is defined to be a partition of the lines into classesR ₁,R ₂, ...,R _t (t=(q ⁿ –1)/(q–1)) such that each point of the geometry is incident with precisely one line of each classR _l , 1it. Of course, the equivalence relation of parallelism defines a resolution in any affine geometry. A resolutionR is said to be a skew resolution provided noR _i , 1it, contains two parallel lines. Skew resolutions are useful for producing packings of lines in projective spaces and doubly resolvable block designs. Skew resolutions are known to exist inAG(n, q),n=2^t–1,i2,q a prime power. The entire spectrum is unknown. In this paper, we give two recursive constructions for skew resolutions. These constructions produce skew resolutions inAG(n, q) for infinietly many new values ofn.     

A Sharp Bound for the Number of Sets that Pairwise Intersect at <Emphasis Type="Italic">k</Emphasis> Positive Values

In this paper we prove that if is a set of k positive integers and {A ₁, ..., A _m } is a family of subsets of an n-element set satisfying , for all 1 i < j m, then . The case k = 1 was proven 50 years ago by Majumdar.     

Inclusion-exclusion: Exact and approximate

It is often required to find the probability of the union of givenn eventsA ₁ ,...,A _n . The answer is provided, of course, by the inclusion-exclusion formula: Pr(A _i )= _i – _i<j Pr(A _i A _j )±.... Unfortunately, this formula has exponentially many terms, and only rarely does one manage to carry out the exact calculation. From a computational point of view, finding the probability of the union is an intractable, #P-hard problem, even in very restricted cases. This state of affairs makes it reasonable to seek approximate solutions that are computationally feasible. Attempts to find such approximate solutions have a long history starting already with Boole [1]. A recent step in this direction was taken by Linial and Nisan [4] who sought approximations to the probability of the union, given the probabilities of allj-wise intersections of the events forj=1,...k. The developed a method to approximate Pr(A _i ), from the above data with an additive error of exp . In the present article we develop an expression that can be computed in polynomial time, that, given the sums _|S|=j Pr( _iS A _i ) forj=1,...k, approximates Pr(A _i ) with an additive error of exp . This error is optimal, up to the logarithmic factor implicit in the notation.The problem of enumerating satisfying assignments of a boolean formula in DNF formF=v _l ^m C _i is an instance of the general problem that had been extensively studied [7]. HereA _i is the set of assignments that satisfyC _i , and Pr( _iS A _i )=a _S /2ⁿ where _iS C _i is satisfied bya _S assignments. Judging from the general results, it is hard to expect a decent approximation ofF's number of satisfying assignments, without knowledge of the numbersa _S for, say, all cardinalities . Quite surprisingly, already the numbersa _S over |S|log(n+1)uniquely determine the number of satisfying assignments for F.We point out a connection between our work and the edge-reconstruction conjecture. Finally we discuss other special instances of the problem, e.g., computing permanents of 0,1 matrices, evaluating chromatic polynomials of graphs and for families of events whose VC dimension is bounded.Work supported in part by a grant of the Binational Israel-US Science Foundation.Work supported in part by a grant of the Binational Israel-US Science Foundation and by the Israel Science Foundation.     

Limit distributions of conditionalU-statistics

Let{(X_n, Y_n)}_n1 be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions ofU _n ^h (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h (t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k ) ^k whereE |h|<. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1ik, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.Research supported by National Board for Higher Mathematics, Bombay, India.     

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