Motivated by an old paper of Wells [J. London Math. Soc. 2 (1970), 549-556] we define the space X... more Motivated by an old paper of Wells [J. London Math. Soc. 2 (1970), 549-556] we define the space X ⊗Y , where X and Y are "homogeneous" Banach spaces of analytic functions on the unit disk D, by the requirement that f can be represented as f = ∞ j=0 gn * hn, with gn ∈ X, hn ∈ Y and ∞ n=1 gn X hn Y < ∞. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula ((X ⊗ Y ), Z) = (X, (Y, Z)), where (U, V ) denotes the space of multipliers from U to V , and as a special case (X ⊗ Y ) * = (X, Y * ), where U * = (U, H ∞ ). We determine H 1 ⊗ X for a class of spaces that contains H p and p (1 ≤ p ≤ 2), and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.
We studied the effects of root zone temperature (RZT) and nutrient availability on free sterols a... more We studied the effects of root zone temperature (RZT) and nutrient availability on free sterols and phospholipids in the plasma membrane (PM) and on PM-ATPase activity in roots of 1-year-old Scots pine (Pinus sylvestris L.) seedlings during growth initiation in the spring. Seedlings were grown for 6 weeks in hydroponic cultures with low (0.5 mM N; LN) or high (3 mM N; HN) nutrient availability. The root zone was subjected to slow warming (SW) and fast warming (FW) treatments while maintaining similar air temperatures in both treatments. Decreases in the amount of phospholipids and in the phospholipid/free sterol ratio, an increase in the degree of saturation of phospholipid fatty acids and changes in free sterol composition were observed during root growth initiation. Changes in lipid composition of the PM associated with the cold deacclimation process were detected at RZTs above 9°C. Nutrient availability affected the lipid composition of the PM only when RZT was increased slowly. When RZT increased from 4 to 6°C in the SW treatment, the degree of saturation of phospholipid fatty acids decreased, especially in HN seedlings. The sitosterol/stigmasterol ratio remained higher in HN seedlings than in LN seedlings. After an RZT of 9°C had been reached in the SW treatment, HN caused increases in the saturation of phospholipid fatty acids and root PM-ATPase activity, and a decrease in the phospholipid/free sterol ratio. Possible effects of changes in PM lipid composition on root growth and PM-ATPase activity are discussed.
Let X be a metric space and define the category B(X) of manifolds bounded over X in the usual way... more Let X be a metric space and define the category B(X) of manifolds bounded over X in the usual way [FW]. The objects of B(X) are manifolds M equipped with proper 2 maps p : M → X. These maps p need not be continuous. A morphism (M 1 , p 1 ) → (M 2 , p 2 ) between objects of B(X) is a continuous map f : M 1 → M 2 which is bounded over X in the sense that there exists k > 0 for which d(p 1 (m), p 2 (f (m))) < k for all m ∈ M 1 . We can additionally form the bounded homotopy category over X by insisting that all relevant maps and homotopies be bounded over the metric space X. Of course, the bounded category is interesting only when X is a space of infinite diameter.
The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS... more The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS Univ. Lec. Ser. 26), the Specht module theory, which was developed by Dipper, James and Murphy in this case, and results from the theory of finite dimensional algebras.
Annales De L Institut Henri Poincare-analyse Non Lineaire, 2010
We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with b... more We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C2C2. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C2C2-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RNRN of some substance whose density is initially a characteristic function of the complement of a domain with bounded C2C2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SNSN and the hyperbolic space HNHN.Nous considérons la diffusion non linéaire d'une substance dans un récipient (pas nécessairement borné) avec frontière bornée de classe C2C2. Supposons qu'initialement, le récipient soit vide et, à sa frontière, la densité de la substance soit gardée à tout moment égale à 1. Nous montrons que, si le récipient contient un sous-domaine C2C2 propre à la frontière duquel la substance est gardée à tout moment à densité constante, alors la frontière du récipient doit être une sphère. Nous considérons aussi la diffusion non linéaire dans tout RNRN d'une substance dont la densité est initialement une fonction caractéristique du complémentaire d'un domaine ayant la frontière bornée et C2C2, et nous obtenons des résultats semblables. Ces résultats sont aussi généralisés au cas du flux de chaleur dans la sphère SNSN et l'espace hyperbolique HNHN.
Motivated by an old paper of Wells [J. London Math. Soc. 2 (1970), 549-556] we define the space X... more Motivated by an old paper of Wells [J. London Math. Soc. 2 (1970), 549-556] we define the space X ⊗Y , where X and Y are "homogeneous" Banach spaces of analytic functions on the unit disk D, by the requirement that f can be represented as f = ∞ j=0 gn * hn, with gn ∈ X, hn ∈ Y and ∞ n=1 gn X hn Y < ∞. We show that this construction is closely related to coefficient multipliers. For example, we prove the formula ((X ⊗ Y ), Z) = (X, (Y, Z)), where (U, V ) denotes the space of multipliers from U to V , and as a special case (X ⊗ Y ) * = (X, Y * ), where U * = (U, H ∞ ). We determine H 1 ⊗ X for a class of spaces that contains H p and p (1 ≤ p ≤ 2), and use this together with the above formulas to give quick proofs of some important results on multipliers due to Hardy and Littlewood, Zygmund and Stein, and others.
We studied the effects of root zone temperature (RZT) and nutrient availability on free sterols a... more We studied the effects of root zone temperature (RZT) and nutrient availability on free sterols and phospholipids in the plasma membrane (PM) and on PM-ATPase activity in roots of 1-year-old Scots pine (Pinus sylvestris L.) seedlings during growth initiation in the spring. Seedlings were grown for 6 weeks in hydroponic cultures with low (0.5 mM N; LN) or high (3 mM N; HN) nutrient availability. The root zone was subjected to slow warming (SW) and fast warming (FW) treatments while maintaining similar air temperatures in both treatments. Decreases in the amount of phospholipids and in the phospholipid/free sterol ratio, an increase in the degree of saturation of phospholipid fatty acids and changes in free sterol composition were observed during root growth initiation. Changes in lipid composition of the PM associated with the cold deacclimation process were detected at RZTs above 9°C. Nutrient availability affected the lipid composition of the PM only when RZT was increased slowly. When RZT increased from 4 to 6°C in the SW treatment, the degree of saturation of phospholipid fatty acids decreased, especially in HN seedlings. The sitosterol/stigmasterol ratio remained higher in HN seedlings than in LN seedlings. After an RZT of 9°C had been reached in the SW treatment, HN caused increases in the saturation of phospholipid fatty acids and root PM-ATPase activity, and a decrease in the phospholipid/free sterol ratio. Possible effects of changes in PM lipid composition on root growth and PM-ATPase activity are discussed.
Let X be a metric space and define the category B(X) of manifolds bounded over X in the usual way... more Let X be a metric space and define the category B(X) of manifolds bounded over X in the usual way [FW]. The objects of B(X) are manifolds M equipped with proper 2 maps p : M → X. These maps p need not be continuous. A morphism (M 1 , p 1 ) → (M 2 , p 2 ) between objects of B(X) is a continuous map f : M 1 → M 2 which is bounded over X in the sense that there exists k > 0 for which d(p 1 (m), p 2 (f (m))) < k for all m ∈ M 1 . We can additionally form the bounded homotopy category over X by insisting that all relevant maps and homotopies be bounded over the metric space X. Of course, the bounded category is interesting only when X is a space of infinite diameter.
The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS... more The proof uses the Fock space theory, which was developed for proving the LLT conjecture (see AMS Univ. Lec. Ser. 26), the Specht module theory, which was developed by Dipper, James and Murphy in this case, and results from the theory of finite dimensional algebras.
Annales De L Institut Henri Poincare-analyse Non Lineaire, 2010
We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with b... more We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C2C2. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C2C2-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RNRN of some substance whose density is initially a characteristic function of the complement of a domain with bounded C2C2 boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SNSN and the hyperbolic space HNHN.Nous considérons la diffusion non linéaire d'une substance dans un récipient (pas nécessairement borné) avec frontière bornée de classe C2C2. Supposons qu'initialement, le récipient soit vide et, à sa frontière, la densité de la substance soit gardée à tout moment égale à 1. Nous montrons que, si le récipient contient un sous-domaine C2C2 propre à la frontière duquel la substance est gardée à tout moment à densité constante, alors la frontière du récipient doit être une sphère. Nous considérons aussi la diffusion non linéaire dans tout RNRN d'une substance dont la densité est initialement une fonction caractéristique du complémentaire d'un domaine ayant la frontière bornée et C2C2, et nous obtenons des résultats semblables. Ces résultats sont aussi généralisés au cas du flux de chaleur dans la sphère SNSN et l'espace hyperbolique HNHN.
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