Metrics on diagram groups and uniform embeddings in a Hilbert space

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Variational inequalities in Hilbert spaces with measures and optimal stopping problems

We study the existence theory for parabolic variational inequalities in weighted L2 spaces with respect to excessive measures associated with a transition semigroup. We characterize the value function of optimal stopping problems for finite and infinite dimensional diffusions as a generalized solution of such a variational inequality. The weighted L2 setting allows us to cover some singular cases, such as optimal stopping for stochastic equations with degenerate diffusion coeficient. As an application of the theory, we consider the pricing of American-style contingent claims. Among others, we treat the cases of assets with stochastic volatility and with path-dependent payoffs.

Explicit methods for Hilbert modular forms

We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.

On separation of a degenerate differential operator in Hilbert space

A coercive estimate for a solution of a degenerate second order di fferential equation is installed, and its applications to spectral problems for the corresponding dif ferential operator is demonstrated. The suffi cient conditions for existence of the solutions of one class of the nonlinear second order diff erential equations on the real axis are obtained.

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densitiesby generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight 2. Moreover, we determine the arithmetic selfintersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors.

Generalized Schmidt decomposition and classification of three-quantum-bit states

We prove for any pure three-quantum-bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the three-quantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties.

Mecánica clásica en el espacio de Hilbert

We continue our study of classical mechanics using the methods of quantum mechanics. A Hilbert space is introduced, new conservation laws deduced, and the possibility of representing by new methods the many body classical problem discussed.

Retribution in a cheap-talk experiment

We use a two-person 3-stage game to investigate whether people choose to punish or reward another player by sacrificing money to increase or decrease the other person’s payoff. One player sends a message indicating an intended play, which is either favorable or unfavorable to the other player in the game. After the message, the sender and the receiver play a simultaneous 2x2 game. A deceptive message may be made, in an effort to induce the receiver to make a play favorable to the sender. Our focus is on whether receivers’ rates of monetary sacrifice depend on the process and the perceived sender’s intention, as is suggested by the literature on deception and procedural satisfaction. Models such as Rabin (1993), Sen (1997), and Charness and Rabin (1999) also permit rates of sacrifice to be sensitive to the sender’s perceived intention, while outcome-based models such as Fehr and Schmidt (1999) and Bolton and Ockenfels (1997) predict otherwise. We find that deception substantially increases the punishment rate as a response to an action that is unfavorable to the receiver. We also find that a small but significant percentage of subjects choose to reward a favorable action choice made by the sender.

Asymptotics of orthogonal polynomials via the Koosis theorem

The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomi- als for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a SzegÄo measure on the unit circumference is perturbed by an arbitrary measure inside the unit disk and an arbitrary Blaschke sequence of point masses outside the unit disk.

Inequity version and team incentives

We study optimal contracts in a simple model where employees are averse to inequity as modelled by Fehr and Schmidt (1999). A "selfish" employer can profitably exploit such preferences among its employees by offering contracts which create inequity off-equilibrium and thus, they would leave employees feeling envy or guilt when they do not meet the employer's demands. Such contracts resemble team and relative performance contracts, and thus we derive conditions under which it may be beneficial to form work teams of employees with distributional concerns who were previously working individually. Similar results are obtained for status-seeking and efficiency concerns preferences.

Pure motives, mixed motives and extensions of motives associated to singular surfaces

We first recall the construction of the Chow motive modelling intersection cohomology of a proper surface X and study its fundamental properties. Using Voevodsky's category of effective geometrical motives, we then study the motive of the exceptional divisor D in a non-singular blow-up of X. If all geometric irreducible components of D are of genus zero, then Voevodsky's formalism allows us to construct certain one-extensions of Chow motives, as canonical subquotients of the motive with compact support of the smooth part of X. Specializing to Hilbert-Blumenthal surfaces, we recover a motivic interpretation of a recent construction of A. Caspar.

Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

We give sufficient conditions for existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Lévy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.