On the logarithm component in trace defect formulas

In asymptotic expansions of resolvent traces \Tr(A(P-\lambda)^{-1}) for classical pseudodifferential operators on closed manifolds, the coefficient $C_0(A,P)$ of $(-\lambda)^{-1}$ is of special interest, since it is the first coefficient containing nonlocal elements from $A$; on the other hand if $A=I$ and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of complex powers $P^s$ (or similar holomorphic families of order $s$), Okikiolu, Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace defects in terms of residues of operators defined from $A$, $A'$, $\log P$ and $\log P'$. The present paper has two purposes: One is to show how the trace defect formulas can be obtained from the resolvents in a simple way without use of the complex powers of $P$ as in the original proofs. We here also give a simple direct proof of a recent residue formula of Scott for $C_0(I,P)$. The other purpose is to establish trace defect residue formulas for operators on manifolds with boundary, where complex powers are not easily accessible; we do this using only resolvents. We also generalize Scott's formula to boundary problems.Comment: 41 page

Copyright

The purpose of copyright laws is discussed. Copyright is essentially about protecting the autonomy of authors

A Fair Use to Remember: Restoring Application of the Fair Use Doctrine to Strengthen Copyright Law and Disarm Abusive Copyright Litigation

The primary goal of copyright law is to benefit the public. By rewarding authors with exclusive rights, such as the power to enforce copyright infringement, copyright protection is the means through which copyright law accomplishes this goal. Another way that copyright law pursues its goal is through the fair use doctrine—an invaluable utilitarian limit on copyright protection. However, fair use is, among other things, vague. The current application of fair use as an affirmative defense magnifies the doctrine’s problems and makes copyright law hospitable to abusive copyright litigation. Current proposals in this area of reform target either fair use or abusive copyright litigation. This Note targets both problems with a single solution: applying fair use as a right. Applying fair use as a right alleviates some of the doctrine’s inherent problems and is the best long-term solution for eliminating abusive litigation from copyright law. As a right, fair use protects copyright’s core values and goals, alleviates the burden on courts, and cultivates creation. A review of the motivation behind fair use reveals that as a right, fair use is best able to serve the purpose for which it was designed

Copyright and truth

Copyright @ 2011 Berkeley Electronic PressThis Article calls into question the primary meaning of copyright law. It argues that copyright is not primarily a legal instrument, but rather a fundamental mode of human existence. The starting point of the analysis is Kant’s definition of a book as a “public address” and of author’s rights as ultimately being grounded in the furtherance and maintenance of truth. Building on Kant’s argument, the Article defines the copyright primary subject matter as the act of speaking publicly in one’s own name, and the copyright sphere as the author-public coalescence that such act of speaking generates. This enables reaching a proper understanding of the scope of copyright and to characterizing its specificity as compared to its “fellow rights,” patents and trademarks