Michal Johanis

Smooth analysis in Banach spaces

This book is about the subject of higher smoothness in separable real Banach spaces. It brings together several angles of view on polynomials, both in finite and infinite setting. Also a rather thorough and systematic view of the more recent results, and the authors work is given. The book revolves around two main broad questions: What is the best smoothness of a given Banach space, and its structural consequences? How large is a supply of smooth functions in the sense of approximating continuous functions in the uniform topology, i.e. how does the Stone-Weierstrass theorem generalize into infinite dimension where measure and compactness are not available? The subject of infinite dimensional real higher smoothness is treated here for the first time in full detail, therefore this book may also serve as a reference book.

Characterization of reflexivity by equivalent renorming

A new rotundity property of Days norm on c0(Γ) is introduced. This property provides in particular a renorming characterization of the class of all reflexive Banach spaces.

The effect of fight cost structure on fighting behaviour

A common feature of animal populations is the stealing by animals of resources such as food from other animals. This has previously been the subject of a range of modelling approaches, one of which is the so called “producer-scrounger” model. In this model a producer finds a resource that takes some time to be consumed, and some time later a (generally) conspecific scrounger discovers the producer with its resource and potentially attempts to steal it. In this paper we consider a variant of this scenario where each individual can choose to invest an amount of energy into this contest, and the level of investment of each individual determines the probability of it winning the contest, but also the additional cost it has to bear. We analyse the model for a specific set of cost functions and maximum investment levels and show how the evolutionarily stable behaviour depends upon them. In particular we see that for high levels of maximum investment, the producer keeps the resource without a fight for concave cost functions, but for convex functions the scrounger obtains the resource (albeit at some cost).

On uniformly Gâteaux smooth norms and normal structure

It is shown that every separable Banach space admits an equivalent norm that is uniformly Gâteaux smooth and yet lacks asymptotic normal structure.

Smooth approximations without critical points

In any separable Banach space containing c0 which admits a Ck-smooth bump, every continuous function can be approximated by a Ck-smooth function whose range of derivative is of the first category. Moreover, the approximation can be constructed in such a way that its derivative avoids a prescribed countable set (in particular the approximation can have no critical points). On the other hand, in a Banach space with the RNP, the range of the derivative of every smooth bounded bump contains a set residual in some neighbourhood of zero.

An Enhanced Model of a Two Player Singled Out Game

A singled out game for two players A and B is analysed. In this game, both players are trying to correctly guess answers to yes/no questions; the first player to answer n questions correctly wins the game. Player B has an advantage of knowing A’s answer before having to announce his own guess. We find the best strategy for both players and give formulas for winning probabilities. We also provide estimates of these probabilities for large n.

ISOMORPHIC EMBEDDINGS AND HARMONIC BEHAVIOUR OF SMOOTH OPERATORS

LetY be a Banach space, 1<p<∞. We give a simple criterion for embedding ℓp ⊂Y, namely it suffices that the positive cone ℓp+ ⊂Y. This result is applied to the study of highly smooth operators from ℓp intoY (p is not an even integer). The main result is that every such operator has a harmonic behaviour unless ℓp/K ⊂Y for someK ∈ ℕ.

The effect of fight cost structure on fighting behaviour involving simultaneous decisions and variable investment levels

In the “producer–scrounger” model, a producer discovers a resource and is in turn discovered by a second individual, the scrounger, who attempts to steal it. This resource can be food or a territory, and in some situations, potentially divisible. In a previous paper we considered a producer and scrounger competing for an indivisible resource, where each individual could choose the level of energy that they would invest in the contest. The higher the investment, the higher the probability of success, but also the higher the costs incurred in the contest. In that paper decisions were sequential with the scrounger choosing their strategy before the producer. In this paper we consider a version of the game where decisions are made simultaneously. For the same cost functions as before, we analyse this case in detail, and then make comparisons between the two cases. Finally we discuss some real examples with potentially variable and asymmetric energetic investments, including intraspecific contests amongst spiders and amongst parasitoid wasps. In the case of the spiders, detailed estimates of energetic expenditure are available which demonstrate the asymmetric values assumed in our models. For the wasps the value of the resource can affect the probabilities of success of the defender and attacker, and differential energetic investment can be inferred. In general for real populations energy usage varies markedly depending upon crucial parameters extrinsic to the individual such as resource value and intrinsic ones such as age, and is thus an important factor to consider when modelling.

A THREE PLAYER SINGLED OUT GAME

AbstractThe three-player singled out game is played in a series of rounds as follows: In each round, a coin is flipped and the result is kept secret. The players A, B, and C, in this order and one at a time, announce their guesses so that the others can hear them. After all players have guessed, the coin is revealed and all players who guessed correctly earn one point; incorrect guesses earn no points. The game continues until a player reaches n points and is declared the winner. For n = 5 and real questions instead of flipped coins, this game was first aired on an MTV show in 1995. We model the decision-making choices for players B and C. We determine Nash equilibria of the game and show that players B and C can always select the equilibrium with maximal payoffs for both of them. We also study the long-term behavior of the game and show that for large n, both B and C will win with probability almost