The full-text may be used and/or reproduced, and given to third parties in any format or medium, ... more The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.
We study the e¤ects of corruption on equilibrium competition and social welfare in a public procu... more We study the e¤ects of corruption on equilibrium competition and social welfare in a public procurement auction. In our model, …rms are invited to the auction at positive costs, and a bureaucrat who runs the auction on behalf of a government may request a bribe from the winning …rm. We …rst present the over-invitation results in the absence of corruption, in which more than a socially optimal number of …rms will be invited. Second, we show that the e¤ects of corruption on equilibrium outcomes vary across di¤erent forms of bribery. For a …xed bribe, corruption has no e¤ect on equilibrium competition, although it does induce social welfare loss. For a proportional bribe, a corrupt bureaucrat may invite fewer or more …rms to the auction depending on how much he weights his personal interest relative to the government payo¤. Thus, corruption may result in either Pareto-improving or deteriorating allocations. Finally, we show that information disclosure may consistently induce more …rms to be invited, regardless of whether there is corruption.
We study the e¤ects of corruption on equilibrium competition and social welfare in a …rst-price p... more We study the e¤ects of corruption on equilibrium competition and social welfare in a …rst-price public procurement auction. In our model, …rms are invited into the auction at positive costs, and a bureaucrat runs the auction on behalf of a government, who may request a bribe from the winning …rm afterward. We …rst show that, in the absence of corruption, the bureaucrat will invite more than socially optimal number of …rms into the auction. Secondly, the e¤ects of corruption on equilibrium competition and social welfare vary across di¤erent forms of bribery. In the case of …xed bribe, corruption has no e¤ect on equilibrium competition, yet does induce social welfare loss due to the distortion cost of increased public spending. In the case of proportional bribe, the bureaucrat may invite fewer or more …rms into the auction, depending on how much the bureaucrat weights on his personal interest. Thirdly, we show that information disclosure may result in more dispersed distribution of …rms'cost estimates, and induce more …rms to be invited, no matter there is corruption or not. Finally, we also provide some discussions on the policy implications of our model.
Let ( N , v ) be a cooperative game with transferable utility and $$F\subseteq 2^N$$ F ⊆ 2 N an ... more Let ( N , v ) be a cooperative game with transferable utility and $$F\subseteq 2^N$$ F ⊆ 2 N an arbitrary set system, where F represents the set of feasible coalitions S whose worths v ( S ) are known. We introduce a game $$(N,v_F)$$ ( N , v F ) as follows. If $$S\in F$$ S ∈ F , then $$v_F(S)=v(S)$$ v F ( S ) = v ( S ) and otherwise $$v_F(S)$$ v F ( S ) is defined such that S has zero Harsanyi dividend. By taking different F , this model produces some well-known games directly or indirectly, such as hypergraph games. We characterize the Shapley value of $$(N,v_F)$$ ( N , v F ) on different domains similarly to that for the Myerson value.
Let (N, v, L) be the Myerson value for graph games (N, v, L). We call a link ij of a graph L safe... more Let (N, v, L) be the Myerson value for graph games (N, v, L). We call a link ij of a graph L safe if k (N, v, L) ≥ k (N, v, L ⧵ {ij}) for any k ∈ N , which means that none of players benefits from breaking the link ij. A link ij ∈ L is called a bridge if N splits into more components after ij is deleted. We show that if (N, v) is convex, then any bridge is safe. Furthermore, if (N, v) is strictly convex, then a link is safe if and only if it is a bridge.
Abstract In cooperative game theory the Shapley value is different from the egalitarian value, th... more Abstract In cooperative game theory the Shapley value is different from the egalitarian value, the latter of which allocates payoffs equally. The null player property and the nullifying player property assign zero payoff to each null player and each nullifying player, respectively. It is known that if the null player property for characterizing the Shapley value is replaced by the nullifying player property, then the egalitarian value is determined uniquely. We propose several properties to replace the nullifying player property to characterize the egalitarian value. Roughly speaking, the results in this note hint that equal division for players of certain types may lead to the egalitarian allocation.
We propose an efficient extension of the Myerson value for games with communication graph structu... more We propose an efficient extension of the Myerson value for games with communication graph structure. Define a quotient game on set of the components of the graph, in which each component acts as a component-player. Then, each player in a component receives his payoff according to the Myerson value and an equal share of the surplus of the Shapley value obtained by the component in the quotient game. We show that this efficient extension of the Myerson value can be characterized by quotient component efficiency, fair distribution of surplus within component and coherence with the Myerson value for connected graphs.
A directed graph game consists of a cooperative game with transferable utility and a digraph whic... more A directed graph game consists of a cooperative game with transferable utility and a digraph which describes limited cooperation and the dominance relation among the players. Under the assumption that only coalitions of strongly connected players are able to fully cooperate, we introduce the digraphrestricted game in which a non-strongly connected coalition can only realize the sum of the worths of its strong components. The Myerson value for directed graph games is defined as the Shapley value of the digraph-restricted game. We establish axiomatic characterizations of the Myerson value for directed graph games by strong component efficiency and either fairness or bi-fairness.
A cooperative game (N , v) is said to be monotone if v(S) ≥ v(T) for all T ⊆ S ⊆ N , and k-monoto... more A cooperative game (N , v) is said to be monotone if v(S) ≥ v(T) for all T ⊆ S ⊆ N , and k-monotone for k ≥ 2 if v(∪ k i=1 S i) ≥ I : ∅ =I ⊆{1,...,k} (−1) |I |−1 v(∩ i∈I S i) for all k subsets S 1 ,. .. , S k of N. Call a set function v totally monotone if it is monotone and k-monotone for all k ≥ 2. To generalize both of marginal contribution and Harsanyi dividend, we define derivatives of v as v (0) = v and for pairwise dis
We study the effects of corruption on equilibrium competition and social welfare in a public proc... more We study the effects of corruption on equilibrium competition and social welfare in a public procurement auction. A government pays costs to invite firms to the auction, and a bureaucrat who runs the auction may request a bribe from the winning firm. We first show that, with no corruption, the bureaucrat will invite more than the socially optimal number of firms to the auction. Second, the effects of corruption on equilibrium outcomes vary across different forms of bribery. For a fixed bribe, corruption does not affect equilibrium competition, yet it does induce social welfare loss. For a proportional bribe, the bureaucrat may invite either fewer or more firms, depending on how much he weights his private interest relative to the government payoff. Finally, we show that information disclosure may consistently induce more firms to be invited, regardless of whether there is corruption.
Journal of the Operations Research Society of China, 2019
Background: Aeration and water management increasing rhizosphere oxygen amount significantly prom... more Background: Aeration and water management increasing rhizosphere oxygen amount significantly promote rice (Oryza sativa) growth and yield, but the effect of root aeration on cadmium (Cd) toxicity and accumulation in rice seedlings under hydroponic culture remains unclear. Results: Results showed that aeration promoted rice seedling growth and alleviated Cd toxicity. Transverse section discovered that Cd accelerated root mature and senescence while aeration delayed the mature and senescence of roots. Non-invasive Micro-test Technology (NMT) showed that aeration increased net O 2 and Cd 2+ influxes on the surface of roots while decreased net Cd 2+ influx in xylem. Perls blue staining showed that aeration and Cd treatments increased iron plaque formation on the surface of roots. Results of metal concentration analysis showed that besides increasing Cd retention in iron plaque, aeration also increasing Cd retention in the cell wall of rice roots. Cell wall component analysis showed that aeration not only increased pectin content but also decreased pectin methylesterification degree (PMD) by increasing pectin methylesterase (PME) activity. Conclusions: All of these results indicate that aeration not only delays root mature and senescence but also increases Cd retention in roots by enhancing iron plaque formation and regulating pectin synthesis in the roots of rice seedlings.
The full-text may be used and/or reproduced, and given to third parties in any format or medium, ... more The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.
The position value is an allocation rule based on the Shapley value of the link game from the ori... more The position value is an allocation rule based on the Shapley value of the link game from the original communication situation, in which cooperation is restricted by a graph. In the link games, feasible coalitions are connected but their structures are ignored. We introduce structure functions to describe the structures of connected sets, and generalize the link game and the position value to the setting with local structures. We modify an axiomatic characterization for the position value by Slikker to the generalized position value by component efficiency and Balanced link contributions on local structures.
This paper explores the possibilities for sellers to usefully transmit product information to buy... more This paper explores the possibilities for sellers to usefully transmit product information to buyers by cheap talk public advertising. We explore two polar cases, contrasting vertically di¤erentiated products (à la Milgrom Weber's (1982) general symmetric model) with horizontally di¤erentiated products (à la Hotelling's (1929) line). We consider both the message only case and where reserve price-message pairs can be chosen by the seller. For horizontally di¤erentiated products partitional messageonly informative equilibria are shown to exist providing the number of bidders is su¢ ciently large. The equilibrium is characterized by more precise information provided for less popular product attributes. The seller optimal disclosure policy displays a complementarity relationship between the number of bidders and the amount of product information disclosed. In contrast, for the vertically di¤erentiated products benchmark, message-only informative equilibria do not exist. With reserve prices, informative equilibria exist in both cases. For the vertical case these equilibria yield lower seller revenue than uninformative equilibria. In the horizontal case with su¢ciently large number of bidders higher revenue is possible and full disclosure becomes feasible and seller optimal in the limit.
Let N = {1,. .. , n} be a set of customers who want to buy a single homogenous goods in market. L... more Let N = {1,. .. , n} be a set of customers who want to buy a single homogenous goods in market. Let q i > 0 be the quantity that i ∈ N demands, q = (q 1 ,. .. , q n) and q S = i∈S q i for S ⊆ N. If f (s) is a (increasing and concave) cost function, then it yields a cooperative game (N , f, q) by defining characteristic function v(S) = f (q S) for S ⊆ N. We now consider the way of taking packages of goods by customers and define a communication graph L on N , in which i and j are linked if they can take packages for each other. So if i and j are connected, then a package can be delivered from i to j by some intermediators. We thus admit any connected subset as a feasible coalition, and obtain a game (N , f, q, L) by defining characteristic function v L (S) = R∈S/L f (q R) for S ⊆ N , where S/L is the family of induced components (maximal connected subset) in S. It is shown that there is an allocation (cost shares) x = (x 1 ,. .. , x n) from the core for the game (x S ≤ v L (S) for any S ⊆ N) such that x satisfies Component Efficiency and Ranking for Unit Prices. If f (s) and q satisfy some further condition, then there is an allocation x from the core such that x satisfies Component Efficiency, and x i ≤ x j and x i q i ≥ x j q j if q i ≤ q j for i and j in the same component of N .
On the one hand, we generalize some results known for composition operators on Hardy spaces to th... more On the one hand, we generalize some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces H Ψ : construction of a "slow" Blaschke product giving a non-compact composition operator on H Ψ and yet "nowhere differentiable"; construction of a surjective symbol whose associated composition operator is compact on H Ψ and is, moreover, in all Schatten classes Sp(H 2), p > 0. On the other hand, we revisit the classical case of composition operators on H 2 , giving first a new, and simpler, characterization of composition operators with closed range, and then showing directly the equivalence of the two characterizations of membership in Schatten classes of Luecking, and Luecking-Zhu.
We produce several situations where some natural subspaces of classical Banach spaces of function... more We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c 0 .
Proceedings of the American Mathematical Society, 2008
We show how different random thin sets of integers may have different behaviour. First, using a r... more We show how different random thin sets of integers may have different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d'Analyse Mathématique 86 (2002), 105-138, namely that there exist 4 3-Rider sets which are sets of uniform convergence and Λ(q)-sets for all q < ∞ but which are not Rosenthal sets. In a second part, we show, using an older result of Kashin and Tzafriri, that, for p > 4 3 , the p-Rider sets which we had constructed in that paper are almost surely not of uniform convergence.
We show that the approximation numbers of a compact composition operator on the weighted Bergman ... more We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces B α of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example.
The full-text may be used and/or reproduced, and given to third parties in any format or medium, ... more The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.
We study the e¤ects of corruption on equilibrium competition and social welfare in a public procu... more We study the e¤ects of corruption on equilibrium competition and social welfare in a public procurement auction. In our model, …rms are invited to the auction at positive costs, and a bureaucrat who runs the auction on behalf of a government may request a bribe from the winning …rm. We …rst present the over-invitation results in the absence of corruption, in which more than a socially optimal number of …rms will be invited. Second, we show that the e¤ects of corruption on equilibrium outcomes vary across di¤erent forms of bribery. For a …xed bribe, corruption has no e¤ect on equilibrium competition, although it does induce social welfare loss. For a proportional bribe, a corrupt bureaucrat may invite fewer or more …rms to the auction depending on how much he weights his personal interest relative to the government payo¤. Thus, corruption may result in either Pareto-improving or deteriorating allocations. Finally, we show that information disclosure may consistently induce more …rms to be invited, regardless of whether there is corruption.
We study the e¤ects of corruption on equilibrium competition and social welfare in a …rst-price p... more We study the e¤ects of corruption on equilibrium competition and social welfare in a …rst-price public procurement auction. In our model, …rms are invited into the auction at positive costs, and a bureaucrat runs the auction on behalf of a government, who may request a bribe from the winning …rm afterward. We …rst show that, in the absence of corruption, the bureaucrat will invite more than socially optimal number of …rms into the auction. Secondly, the e¤ects of corruption on equilibrium competition and social welfare vary across di¤erent forms of bribery. In the case of …xed bribe, corruption has no e¤ect on equilibrium competition, yet does induce social welfare loss due to the distortion cost of increased public spending. In the case of proportional bribe, the bureaucrat may invite fewer or more …rms into the auction, depending on how much the bureaucrat weights on his personal interest. Thirdly, we show that information disclosure may result in more dispersed distribution of …rms'cost estimates, and induce more …rms to be invited, no matter there is corruption or not. Finally, we also provide some discussions on the policy implications of our model.
Let ( N , v ) be a cooperative game with transferable utility and $$F\subseteq 2^N$$ F ⊆ 2 N an ... more Let ( N , v ) be a cooperative game with transferable utility and $$F\subseteq 2^N$$ F ⊆ 2 N an arbitrary set system, where F represents the set of feasible coalitions S whose worths v ( S ) are known. We introduce a game $$(N,v_F)$$ ( N , v F ) as follows. If $$S\in F$$ S ∈ F , then $$v_F(S)=v(S)$$ v F ( S ) = v ( S ) and otherwise $$v_F(S)$$ v F ( S ) is defined such that S has zero Harsanyi dividend. By taking different F , this model produces some well-known games directly or indirectly, such as hypergraph games. We characterize the Shapley value of $$(N,v_F)$$ ( N , v F ) on different domains similarly to that for the Myerson value.
Let (N, v, L) be the Myerson value for graph games (N, v, L). We call a link ij of a graph L safe... more Let (N, v, L) be the Myerson value for graph games (N, v, L). We call a link ij of a graph L safe if k (N, v, L) ≥ k (N, v, L ⧵ {ij}) for any k ∈ N , which means that none of players benefits from breaking the link ij. A link ij ∈ L is called a bridge if N splits into more components after ij is deleted. We show that if (N, v) is convex, then any bridge is safe. Furthermore, if (N, v) is strictly convex, then a link is safe if and only if it is a bridge.
Abstract In cooperative game theory the Shapley value is different from the egalitarian value, th... more Abstract In cooperative game theory the Shapley value is different from the egalitarian value, the latter of which allocates payoffs equally. The null player property and the nullifying player property assign zero payoff to each null player and each nullifying player, respectively. It is known that if the null player property for characterizing the Shapley value is replaced by the nullifying player property, then the egalitarian value is determined uniquely. We propose several properties to replace the nullifying player property to characterize the egalitarian value. Roughly speaking, the results in this note hint that equal division for players of certain types may lead to the egalitarian allocation.
We propose an efficient extension of the Myerson value for games with communication graph structu... more We propose an efficient extension of the Myerson value for games with communication graph structure. Define a quotient game on set of the components of the graph, in which each component acts as a component-player. Then, each player in a component receives his payoff according to the Myerson value and an equal share of the surplus of the Shapley value obtained by the component in the quotient game. We show that this efficient extension of the Myerson value can be characterized by quotient component efficiency, fair distribution of surplus within component and coherence with the Myerson value for connected graphs.
A directed graph game consists of a cooperative game with transferable utility and a digraph whic... more A directed graph game consists of a cooperative game with transferable utility and a digraph which describes limited cooperation and the dominance relation among the players. Under the assumption that only coalitions of strongly connected players are able to fully cooperate, we introduce the digraphrestricted game in which a non-strongly connected coalition can only realize the sum of the worths of its strong components. The Myerson value for directed graph games is defined as the Shapley value of the digraph-restricted game. We establish axiomatic characterizations of the Myerson value for directed graph games by strong component efficiency and either fairness or bi-fairness.
A cooperative game (N , v) is said to be monotone if v(S) ≥ v(T) for all T ⊆ S ⊆ N , and k-monoto... more A cooperative game (N , v) is said to be monotone if v(S) ≥ v(T) for all T ⊆ S ⊆ N , and k-monotone for k ≥ 2 if v(∪ k i=1 S i) ≥ I : ∅ =I ⊆{1,...,k} (−1) |I |−1 v(∩ i∈I S i) for all k subsets S 1 ,. .. , S k of N. Call a set function v totally monotone if it is monotone and k-monotone for all k ≥ 2. To generalize both of marginal contribution and Harsanyi dividend, we define derivatives of v as v (0) = v and for pairwise dis
We study the effects of corruption on equilibrium competition and social welfare in a public proc... more We study the effects of corruption on equilibrium competition and social welfare in a public procurement auction. A government pays costs to invite firms to the auction, and a bureaucrat who runs the auction may request a bribe from the winning firm. We first show that, with no corruption, the bureaucrat will invite more than the socially optimal number of firms to the auction. Second, the effects of corruption on equilibrium outcomes vary across different forms of bribery. For a fixed bribe, corruption does not affect equilibrium competition, yet it does induce social welfare loss. For a proportional bribe, the bureaucrat may invite either fewer or more firms, depending on how much he weights his private interest relative to the government payoff. Finally, we show that information disclosure may consistently induce more firms to be invited, regardless of whether there is corruption.
Journal of the Operations Research Society of China, 2019
Background: Aeration and water management increasing rhizosphere oxygen amount significantly prom... more Background: Aeration and water management increasing rhizosphere oxygen amount significantly promote rice (Oryza sativa) growth and yield, but the effect of root aeration on cadmium (Cd) toxicity and accumulation in rice seedlings under hydroponic culture remains unclear. Results: Results showed that aeration promoted rice seedling growth and alleviated Cd toxicity. Transverse section discovered that Cd accelerated root mature and senescence while aeration delayed the mature and senescence of roots. Non-invasive Micro-test Technology (NMT) showed that aeration increased net O 2 and Cd 2+ influxes on the surface of roots while decreased net Cd 2+ influx in xylem. Perls blue staining showed that aeration and Cd treatments increased iron plaque formation on the surface of roots. Results of metal concentration analysis showed that besides increasing Cd retention in iron plaque, aeration also increasing Cd retention in the cell wall of rice roots. Cell wall component analysis showed that aeration not only increased pectin content but also decreased pectin methylesterification degree (PMD) by increasing pectin methylesterase (PME) activity. Conclusions: All of these results indicate that aeration not only delays root mature and senescence but also increases Cd retention in roots by enhancing iron plaque formation and regulating pectin synthesis in the roots of rice seedlings.
The full-text may be used and/or reproduced, and given to third parties in any format or medium, ... more The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.
The position value is an allocation rule based on the Shapley value of the link game from the ori... more The position value is an allocation rule based on the Shapley value of the link game from the original communication situation, in which cooperation is restricted by a graph. In the link games, feasible coalitions are connected but their structures are ignored. We introduce structure functions to describe the structures of connected sets, and generalize the link game and the position value to the setting with local structures. We modify an axiomatic characterization for the position value by Slikker to the generalized position value by component efficiency and Balanced link contributions on local structures.
This paper explores the possibilities for sellers to usefully transmit product information to buy... more This paper explores the possibilities for sellers to usefully transmit product information to buyers by cheap talk public advertising. We explore two polar cases, contrasting vertically di¤erentiated products (à la Milgrom Weber's (1982) general symmetric model) with horizontally di¤erentiated products (à la Hotelling's (1929) line). We consider both the message only case and where reserve price-message pairs can be chosen by the seller. For horizontally di¤erentiated products partitional messageonly informative equilibria are shown to exist providing the number of bidders is su¢ ciently large. The equilibrium is characterized by more precise information provided for less popular product attributes. The seller optimal disclosure policy displays a complementarity relationship between the number of bidders and the amount of product information disclosed. In contrast, for the vertically di¤erentiated products benchmark, message-only informative equilibria do not exist. With reserve prices, informative equilibria exist in both cases. For the vertical case these equilibria yield lower seller revenue than uninformative equilibria. In the horizontal case with su¢ciently large number of bidders higher revenue is possible and full disclosure becomes feasible and seller optimal in the limit.
Let N = {1,. .. , n} be a set of customers who want to buy a single homogenous goods in market. L... more Let N = {1,. .. , n} be a set of customers who want to buy a single homogenous goods in market. Let q i > 0 be the quantity that i ∈ N demands, q = (q 1 ,. .. , q n) and q S = i∈S q i for S ⊆ N. If f (s) is a (increasing and concave) cost function, then it yields a cooperative game (N , f, q) by defining characteristic function v(S) = f (q S) for S ⊆ N. We now consider the way of taking packages of goods by customers and define a communication graph L on N , in which i and j are linked if they can take packages for each other. So if i and j are connected, then a package can be delivered from i to j by some intermediators. We thus admit any connected subset as a feasible coalition, and obtain a game (N , f, q, L) by defining characteristic function v L (S) = R∈S/L f (q R) for S ⊆ N , where S/L is the family of induced components (maximal connected subset) in S. It is shown that there is an allocation (cost shares) x = (x 1 ,. .. , x n) from the core for the game (x S ≤ v L (S) for any S ⊆ N) such that x satisfies Component Efficiency and Ranking for Unit Prices. If f (s) and q satisfy some further condition, then there is an allocation x from the core such that x satisfies Component Efficiency, and x i ≤ x j and x i q i ≥ x j q j if q i ≤ q j for i and j in the same component of N .
On the one hand, we generalize some results known for composition operators on Hardy spaces to th... more On the one hand, we generalize some results known for composition operators on Hardy spaces to the case of Hardy-Orlicz spaces H Ψ : construction of a "slow" Blaschke product giving a non-compact composition operator on H Ψ and yet "nowhere differentiable"; construction of a surjective symbol whose associated composition operator is compact on H Ψ and is, moreover, in all Schatten classes Sp(H 2), p > 0. On the other hand, we revisit the classical case of composition operators on H 2 , giving first a new, and simpler, characterization of composition operators with closed range, and then showing directly the equivalence of the two characterizations of membership in Schatten classes of Luecking, and Luecking-Zhu.
We produce several situations where some natural subspaces of classical Banach spaces of function... more We produce several situations where some natural subspaces of classical Banach spaces of functions over a compact abelian group contain the space c 0 .
Proceedings of the American Mathematical Society, 2008
We show how different random thin sets of integers may have different behaviour. First, using a r... more We show how different random thin sets of integers may have different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d'Analyse Mathématique 86 (2002), 105-138, namely that there exist 4 3-Rider sets which are sets of uniform convergence and Λ(q)-sets for all q < ∞ but which are not Rosenthal sets. In a second part, we show, using an older result of Kashin and Tzafriri, that, for p > 4 3 , the p-Rider sets which we had constructed in that paper are almost surely not of uniform convergence.
We show that the approximation numbers of a compact composition operator on the weighted Bergman ... more We show that the approximation numbers of a compact composition operator on the weighted Bergman spaces B α of the unit disk can tend to 0 arbitrarily slowly, but that they never tend quickly to 0: they grow at least exponentially, and this speed of convergence is only obtained for symbols which do not approach the unit circle. We also give an upper bounds and explicit an example.
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