Nice Curves Research Articles

Science, Seduction, and the Lure of Reality in Third Republic France

On June 8, 1905, a group of researchers representing the elite of the French scientific community gathered in the offices of the Institut general psychologique (IGP) on the rue Conde in Paris with the celebrated Italian spiritualist medium, Eusapia Palladino, and the goal of ascertaining the reality of mediumistic phenomena. In the course of forty-three seances stretched over three years, the members of the group witnessed Eusapia producing a variety of unexplainable phenomena. With her hands and feet carefully controlled by the attending scientists, Eusapia shattered drinking glasses without touching them, caused curtains and strips of fabric to billow and bulge, produced luminescent wisps of light, materialized “spirit” hands and arms, and caused tables to lift from the floor and fly about the room. The observers reported being tapped on the shoulder, poked in the back, and pinched by an unseen agent. They felt their hair being pulled and chairs suddenly jerked from underneath them and overturned by an invisible force. The movement of a small side table floating over his head caused Pierre Curie to exclaim: “What is astonishing is the precision with which the gueridon moved without touching anyone. It made a nice curve and did not touch me at all.” The physicist Jacques-Arsene d’Arsonval, seizing another table as it was floating from the floor, cried, “It is absolutely the resistance of the magnetic field!” In spite of these experiences, three years, and thousands of francs spent, the research of the IGP culminated in a disappointing lack of consensus among the attending scientists about the meaning of the phenomena. While d’Arsonval admitted in the discussions following the presentation of the written account that he and his fellow scientists had “observed phenomena” and “believed that they were protected against fraud,” in the final analysis he maintained that they could not be “scientifically certain.” The obstacle in the end proved to be the one element that the scrupulous measurements of the seances could not fathom. Like the purloined letter hidden in plain sight, subjectivity—both that of the medium and of the observers—remained

Singular integral operators in weighted spaces of continuous functions with non-equilibrated continuity modulus

We show that singular integral operators with piecewise continuous coefficients may gain massive spectra when considered in weighted spaces of continuous functions with a prescribed continuity modulus (generalized Hölder spaces Hω (Γ, ρ )), a fact known for example for Lebesgue spaces Lp (Γ, ρ ) in the case of general Muckenhoupt weights ρ or bad-behaved curves Γ. In the case under consideration the appearance of “lunes” generating massivity of the spectra is due to the presence of a general (non-equilibrated) continuity modulus ω . These lunes arise when the Boyd-type indices of the function ω (h ) do not coincide. Thus, the massive spectra may appear in the non-weighted case and on nice curves, a situation similar to Orlicz spaces. The main problems arising in the investigation are the nature of non-equilibrated continuity moduli ω and the failure of the density of “nice” functions in Hölder-type spaces. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Fredholmness of Singular Integral Operators with Piecewise Continuous Coefficients on Weighted Banach Function Spaces

We prove necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces $L^{p(\cdot)}_w$ (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever $w$ is a Khvedelidze weight, and the variable exponent $p(t)$ satisfies the estimate $|p(\tau)-p(t)|\le A/(-\log|\tau-t|)$.