Plane algebraic curves with many cusps,with an appendix by Eugenii Shustin

The maximum number \(k(d)\) of cusps on a plane algebraic curve of degree \(d\) is an open classical problem that dates back to the nineteenth century. A related open problem is the asymptotic value ( \(a.v.\) ) of the number of cusps on plane curves, that is \(a.v.=\limsup _{d\rightarrow \infty } k(d)/{d^2}\) . In this paper, we improve the best known lower bound for the asymptotic value by constructing curves with the largest known number of cusps for infinitely many degrees. Some particular curves of relatively low degree with many cusps are constructed too. The “Appendix” to this paper is devoted to the case of degree 11 and it is due to E. Shustin.     

A New Light on Nets of C*-Algebras and Their Representations

The present paper deals with the question of representability of nets of C*-algebras whose underlying poset, indexing the net, is not upward directed. A particular class of nets, called C*-net bundles, is classified in terms of C*-dynamical systems having as group the fundamental group of the poset. Any net of C*-algebras has a canonical morphism into a C*-net bundle, the enveloping net bundle, which generalizes the notion of universal C*-algebra given by Fredenhagen to nonsimply connected posets. This allows a classification of nets; in particular, we call injective those nets such that the canonical morphism is faithful. Injectivity turns out to be equivalent to the existence of faithful representations. We further relate injectivity to a generalized Čech cocycle of the net, and this allows us to give examples of nets exhausting the above classification.     

Representations of Nets of C*-Algebras over S 1

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C^*-algebras over S ¹ admits faithful representations, and when the net is covariant under Diff(S ¹), it admits representations covariant under any amenable subgroup of Diff(S ¹).     

Pion Compton scattering in perturbative QCD

Pion Campton scattering is studied in perturbative QCD for real and space-like initial photons. Different methods for the convolution of the hard amplitude with the pion wave-functions, which have in the past led to conflicting results, are compared.     

The universal C*-algebra of the electromagnetic field II. Topological charges and spacelike linear fields

Conditions for the appearance of topological charges are studied in the framework of the universal C*-algebra of the electromagnetic field, which is represented in any theory describing electromagnetism. It is shown that non-trivial topological charges, described by pairs of fields localised in certain topologically non-trivial spacelike separated regions, can appear in regular representations of the algebra only if the fields depend non-linearly on the mollifying test functions. On the other hand, examples of regular vacuum representations with non-trivial topological charges are constructed, where the underlying field still satisfies a weakened form of “spacelike linearity”. Such representations also appear in the presence of electric currents. The status of topological charges in theories with several types of electromagnetic fields, which appear in the short distance (scaling) limit of asymptotically free non-abelian gauge theories, is also briefly discussed.     

Dirichlet problem for a divergence form elliptic equation with unbounded coefficients in an unbounded domain

We prove existence and uniqueness of the solution of the Dirichlet problem for a class of elliptic equations in divergence form with discontinuous and unbounded coefficients in unbounded domains. Entrata in Redazione il 22 aprile 1999.     

Sigmoidal transformations and the Euler-Maclaurin expansion for evaluating certain Hadamard finite-part integrals

Summary. Starting with some results of Lyness concerning the Euler-Maclaurin expansion of Cauchy principal value integrals over it is shown how, by the use of sigmoidal transformations, good approximations can be found for the Hadamard finite-part integral where The analysis is illustrated by some numerical examples. Received March 13, 1996     

Quantitative studies on free radical reactions with the scavenger 1,1,3,3-tetramethylisoindolinyl-2-oxy

Previously unobserved reaction pathways for the attack of $\text{tert}$-butoxy radicals on vinyl monomers have been found by a technique which employs the title nitroxide as radical scavenging agent.     

A model to predict the steady-state concentration of hydroxyl radicals in the surface layer of natural waters

A model was developed to predict the steady-state [*OH] in the surface layer of natural waters as a function of nitrate, inorganic carbon (IC) and dissolved organic matter (DOM). The parameter values were studied in the range detected in shallow high-mountain lakes, to which the model results are most relevant. Calculations indicate that [*OH] increases with increasing nitrate and decreasing IC, and conditions are also identified where [*OH] is directly proportional, inversely proportional or independent of DOM. Based on the model results it is possible to predict the half-life time, due to reaction with *OH, of given dissolved compounds, including organic pollutants, from the water composition data.