Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem

We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function $q$ from the class $C^2([0,1])$ and study the minimal number n(\e) of function evaluations or queries that are necessary to compute an \e-approximation of the smallest eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic) worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix $\exp(\tfrac12 {\rm i}M)$, where $M$ is an $n\times n$ matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in $n$ is an open issue.Comment: 33 page

An intractability result for multiple integration

Our aim is to show that in the worst case setting the integration problem is intractable. The implications of the intractability result for lattice methods are considered briefly in Section 3

Tractability of Integration in Non-periodic and Periodic Weighted Tensor Product Hilbert Spaces

AbstractWe study strong tractability and tractability of multivariate integration in the worst case setting. This problem is considered in weighted tensor product reproducing kernel Hilbert spaces. We analyze three variants of the classical Sobolev space of non-periodic and periodic functions whose first mixed derivatives are square integrable. We obtain necessary and sufficient conditions on strong tractability and tractability in terms of the weights of the spaces. For the three Sobolev spaces periodicity has no significant effect on strong tractability and tractability. In contrast, for general reproducing kernel Hilbert spaces anything can happen: we may have strong tractability or tractability for the non-periodic case and intractability for the periodic one, or vice versa

In search of the authentic nation: landscape and national identity in Canada and Switzerland

While the study of nationalism and national identity has flourished in the last decade, little attention has been devoted to the conditions under which natural environments acquire significance in definitions of nationhood. This article examines the identity-forming role of landscape depictions in two polyethnic nation-states: Canada and Switzerland. Two types of geographical national identity are identified. The first – what we call the ‘nationalisation of nature’– portrays zarticular landscapes as expressions of national authenticity. The second pattern – what we refer to as the ‘naturalisation of the nation’– rests upon a notion of geographical determinism that depicts specific landscapes as forces capable of determining national identity. The authors offer two reasons why the second pattern came to prevail in the cases under consideration: (1) the affinity between wild landscape and the Romantic ideal of pure, rugged nature, and (2) a divergence between the nationalist ideal of ethnic homogeneity and the polyethnic composition of the two societies under consideration

Herpesvirus Telomerase RNA (vTR) with a Mutated Template Sequence Abrogates Herpesvirus-Induced Lymphomagenesis

Telomerase reverse transcriptase (TERT) and telomerase RNA (TR) represent the enzymatically active components of telomerase. In the complex, TR provides the template for the addition of telomeric repeats to telomeres, a protective structure at the end of linear chromosomes. Human TR with a mutation in the template region has been previously shown to inhibit proliferation of cancer cells in vitro. In this report, we examined the effects of a mutation in the template of a virus encoded TR (vTR) on herpesvirus-induced tumorigenesis in vivo. For this purpose, we used the oncogenic avian herpesvirus Marek's disease virus (MDV) as a natural virus-host model for lymphomagenesis. We generated recombinant MDV in which the vTR template sequence was mutated from AATCCCAATC to ATATATATAT (vAU5) by two-step Red-mediated mutagenesis. Recombinant viruses harboring the template mutation replicated with kinetics comparable to parental and revertant viruses in vitro. However, mutation of the vTR template sequence completely abrogated virus-induced tumor formation in vivo, although the virus was able to undergo low-level lytic replication. To confirm that the absence of tumors was dependent on the presence of mutant vTR in the telomerase complex, a second mutation was introduced in vAU5 that targeted the P6.1 stem loop, a conserved region essential for vTR-TERT interaction. Absence of vTR-AU5 from the telomerase complex restored virus-induced lymphoma formation. To test if the attenuated vAU5 could be used as an effective vaccine against MDV, we performed vaccination-challenge studies and determined that vaccination with vAU5 completely protected chickens from lethal challenge with highly virulent MDV. Taken together, our results demonstrate 1) that mutation of the vTR template sequence can completely abrogate virus-induced tumorigenesis, likely by the inhibition of cancer cell proliferation, and 2) that this strategy could be used to generate novel vaccine candidates against virus-induced lymphoma