On answering accusations in controversies

Accusations are a very frequent type of speech act both in everyday life and in formal controversies, and answering accusations is a sophisticated type of linguistic practice well worth analysing from a pragmatic point of view. In my paper I shall first describe some basic properties of accusations and characteristic types of reactions to accusations, i. e. denying the alleged fact, making excuses, and giving justifications. I then go on to describe some fundamental functions of accusations in controversies. Using the basic patterns of accusations and reactions to accusations as an object of comparison, I then analyse some relevant exchanges from historical controversies (l6th to 18th century), among them famous polemical interactions like the Hobbes-Bramhall controversy, but also less well-known debates from the fields of medicine and theology. The present paper is both a contribution to the theory of controversy and to the pragmatic history of controversies. Keywords: historical pragmatics, theory of controversy, ad hominem moves, dynamics of controvers

The sectorial projection defined from logarithms

For a classical elliptic pseudodifferential operator P of order m>0 on a closed manifold X, such that the eigenvalues of the principal symbol p_m(x,\xi) have arguments in \,]\theta,\phi [\, and \,]\phi, \theta +2\pi [\, (\theta <\phi <\theta +2\pi), the sectorial projection \Pi_{\theta, \phi}(P) is defined essentially as the integral of the resolvent along {e^{i\phi}R_+}\cup {e^{i\theta}R_+}. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that \P_{\theta, \phi}(P) is a \psi do of order 0; namely that p_m(x,\xi) cannot in general be modified to allow integration of (p_m(x,\xi)-\lambda)^{-1} along {e^{i\phi}R_+}\cup {e^{i\theta}R_+} simultaneously for all \xi . We show that the structure of \Pi_{\theta, \phi}(P) as a \psi do of order 0 can be deduced from the formula \Pi_{\theta, \phi}(P)= (i/(2\pi))(\log_\theta (P) - \log_\phi (P)) proved in an earlier work (coauthored with Gaarde). In the analysis of \log_\theta (P) one need only modify p_m(x,\xi) in a neighborhood of e^{i\theta}R_+; this is known to be possible from Seeley's 1967 work on complex powers.Comment: Quotations elaborated, 6 pages, to appear in Mathematica Scandinavic

A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras

Quantum theory's Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The reconstruction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.Comment: 19 page

Remarks on nonlocal trace expansion coefficients

In a recent work, Paycha and Scott establish formulas for all the Laurent coefficients of Tr(AP^{-s}) at the possible poles. In particular, they show a formula for the zero'th coefficient at s=0, in terms of two functions generalizing, respectively, the Kontsevich-Vishik canonical trace density, and the Wodzicki-Guillemin noncommutative residue density of an associated operator. The purpose of this note is to provide a proof of that formula relying entirely on resolvent techniques (for the sake of possible generalizations to situations where powers are not an easy tool). - We also give some corrections to transition formulas used in our earlier works.Comment: Minor corrections. To appear in a proceedings volume in honor of K. Wojciechowski, "Analysis and Geometry of Boundary Value Problems", World Scientific, 19 page

The Inner Life of the Kondo Ground State: An Answer to Kenneth Wilson's Question

The Kondo ground state has been investigated by numerical and exact methods, but the physics behind these results remains veiled. Nobel prize winner Wilson, who engineered the break through in his numerical renormalization group theory, commented in his review article "the author has no simple explanation ...for the crossover from weak to strong coupling". In this article a graphical interpretation is given for the extraordinary properties of the Kondo ground state. At the crossover all electron states in the low energy range of k_{B}T_{K} are synchronized. An internal orthogonality catastrophe is averted.Comment: 4 figure

Integration by parts and Pohozaev identities for space-dependent fractional-order operators

Consider a classical elliptic pseudodifferential operator $P$ on ${\Bbb R}^n$ of order $2a$ ($0<a<1)$ with even symbol. For example, $P=A(x,D)^a$ where $A(x,D)$ is a second-order strongly elliptic differential operator; the fractional Laplacian $(-\Delta )^a$ is a particular case. For solutions $u$ of the Dirichlet problem on a bounded smooth subset $\Omega \subset{\Bbb R}^n$, we show an integration-by-parts formula with a boundary integral involving $(d^{-a}u)|_{\partial\Omega }$, where $d(x)=\operatorname{dist}(x,\partial\Omega )$. This extends recent results of Ros-Oton, Serra and Valdinoci, to operators that are $x$-dependent, nonsymmetric, and have lower-order parts. We also generalize their formula of Pohozaev-type, that can be used to prove unique continuation properties, and nonexistence of nontrivial solutions of semilinear problems. An illustration is given with $P=(-\Delta +m^2)^a$. The basic step in our analysis is a factorization of $P$, $P\sim P^-P^+$, where we set up a calculus for the generalized pseudodifferential operators $P^\pm$ that come out of the construction.Comment: Final version to appear in J. Differential Equations, 42 pages. References adde