Uniqueness for weak solutions of parabolic equations with a fractional time derivative

We prove uniqueness for weak solutions to abstract parabolic equations with the fractional Marchaud or Caputo time derivative. We consider weak solutions in time for divergence form equations when the fractional derivative is transferred to the test function.Comment: Accepted version to appear in Contemporary Mathematics. Corrected inaccuracies regarding historical results in the introduction. Also changed the initial condition for the equation and fixed typo

Separation of a Lower Dimensional Free Boundary in a Two Phase Problem

This paper studies local properties of a two phase free boundary problem for the fractional Laplacian. The main result states that the two free boundaries of the positive and negativity sets cannot touch

Writers' Bloc: reading into late Soviet experience through Latvian artists' books.

Previously in the University eprints HAIRST pilot service at http://eprints.st-andrews.ac.uk/archive/00000364/Article 1 of 6 in an issue devoted to Scandinavian and Baltic visual cultureThis article focuses on book works by Latvian artists during the late-Soviet period, and also offers an initial discussion of the peculiarities of the Soviet publishing environment, as it existed shortly before the USSR’s annexation of Latvia at the end of World War II, and the roughly concurrent publication experiences of progressive artists in inter-bellum Latvia, the so-called First Republic. During its heyday in the 1960s and 70s the artist’s book was hailed by many practitioners in the West as the superlative democratic art form, due to the hypothetical possibility of the widespread ownership of the art object. An examination of how artist-authored books developed amid Latvian society's repeated, abrupt transitions between democracy and totalitarianism during the past century may further illuminate this concept of a democratic art medium.Postprin

Quantum Superpositions Cannot be Epistemic

Quantum superposition states are behind many of the curious phenomena exhibited by quantum systems, including Bell non-locality, quantum interference, quantum computational speed-up, and the measurement problem. At the same time, many qualitative properties of quantum superpositions can also be observed in classical probability distributions leading to a suspicion that superpositions may be explicable as probability distributions over less problematic states, that is, a suspicion that superpositions are \emph{epistemic}. Here, it is proved that, for any quantum system of dimension $d>3$, this cannot be the case for almost all superpositions. Equivalently, any underlying ontology must contain ontic superposition states. A related question concerns the more general possibility that some pairs of non-orthogonal quantum states $|\psi\rangle,|\phi\rangle$ could be ontologically indistinct (there are ontological states which fail to distinguish between these quantum states). A similar method proves that if $|\langle\phi|\psi\rangle|^{2}\in(0,\frac{1}{4})$ then $|\psi\rangle,|\phi\rangle$ must approach ontological distinctness as $d\rightarrow\infty$. The robustness of these results to small experimental error is also discussed.Comment: Updated to published version with slgihtly extended discussion and corrected mistakes. 6 + 7 pages, Quantum Studies: Mathematics and Foundations. Online First. (2015

Treating Time Travel Quantum Mechanically

The fact that closed timelike curves (CTCs) are permitted by general relativity raises the question as to how quantum systems behave when time travel to the past occurs. Research into answering this question by utilising the quantum circuit formalism has given rise to two theories: Deutschian-CTCs (D-CTCs) and "postselected" CTCs (P-CTCs). In this paper the quantum circuit approach is thoroughly reviewed, and the strengths and shortcomings of D-CTCs and P-CTCs are presented in view of their non-linearity and time travel paradoxes. In particular, the "equivalent circuit model"---which aims to make equivalent predictions to D-CTCs, while avoiding some of the difficulties of the original theory---is shown to contain errors. The discussion of D-CTCs and P-CTCs is used to motivate an analysis of the features one might require of a theory of quantum time travel, following which two overlapping classes of new theories are identified. One such theory, the theory of "transition probability" CTCs (T-CTCs), is fully developed. The theory of T-CTCs is shown not to have certain undesirable features---such as time travel paradoxes, the ability to distinguish non-orthogonal states with certainty, and the ability to clone or delete arbitrary pure states---that are present with D-CTCs and P-CTCs. The problems with non-linear extensions to quantum mechanics are discussed in relation to the interpretation of these theories, and the physical motivations of all three theories are discussed and compared.Comment: 20 pages, 4 figures. Edited in response to peer revie

Preprojective representations of valued quivers and reduced words in the Weyl group of a Kac-Moody algebra

This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words. The latter strengthens known results of Howlett, Fomin-Zelevinsky, and the authors