The categorical theory of relations and quantizations

In this paper we develope a categorical theory of relations and use this formulation to define the notion of quantization for relations. Categories of relations are defined in the context of symmetric monoidal categories. They are shown to be symmetric monoidal categories in their own right and are found to be isomorphic to certain categories of $A-A$ bicomodules. Properties of relations are defined in terms of the symmetric monoidal structure. Equivalence relations are shown to be commutative monoids in the category of relations. Quantization in our view is a property of functors between monoidal categories. This notion of quantization induce a deformation of all algebraic structures in the category, in particular the ones defining properties of relations like transitivity and symmetry.Comment: corrected typo

Theory of linear G-difference equations

We introduce the notion of difference equation defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations,symmetries of an equation are module endomorphisms and conserved structures are invariants in the tensor algebra of the given equation. We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underluing set. We relate our notion of difference equations and solutions to systems of classical difference equations and their solutions and show that our notions include these as a special case.Comment: 34 page

How to Bootstrap Anonymous Communication

We ask whether it is possible to anonymously communicate a large amount of data using only public (non-anonymous) communication together with a small anonymous channel. We think this is a central question in the theory of anonymous communication and to the best of our knowledge this is the first formal study in this direction. To solve this problem, we introduce the concept of anonymous steganography: think of a leaker Lea who wants to leak a large document to Joe the journalist. Using anonymous steganography Lea can embed this document in innocent looking communication on some popular website (such as cat videos on YouTube or funny memes on 9GAG). Then Lea provides Joe with a short key $k$ which, when applied to the entire website, recovers the document while hiding the identity of Lea among the large number of users of the website. Our contributions include: - Introducing and formally defining anonymous steganography, - A construction showing that anonymous steganography is possible (which uses recent results in circuits obfuscation), - A lower bound on the number of bits which are needed to bootstrap anonymous communication.Comment: 15 page

Multimodality and literacy practices in English: Exploring the role of multimodal texts in English language teaching in Norway

This article-based doctoral dissertation investigates multimodality in English as a school subject in Norway, more specifically in relation to literacy. Calls for more research on multimodality in English language teaching in Norway motivated this study. Furthermore, the recent curriculum inclusion of multimodal texts in the English subject increases its relevance for both practice and research. The overarching research question addressed is as follows: What role does multimodality play in the literacy practices of the English subject in Norway? Taking its theoretical starting point in a social semiotic multimodal perspective on learning, this dissertation includes three qualitative case studies. The data material consists mainly of texts and observations of literacy events, collected in three different contexts. The main contribution of this dissertation is increased knowledge about multimodality as an inherent, but little researched phenomenon within the field of English subject didactics and English teaching in Norway. Findings indicate that the English subject has a multimodal literacy practice on the input end of teaching. This practice is, however, largely silent and bound to traditions of scaffolding and notions of motivation. The dissertation implies that the production and recognition of multimodal texts as output as well as input is a natural and necessary step forward for English language teaching, especially considering the recent curriculum.Denne artikkelbaserte doktoravhandlinga undersøker multimodalitet i engelsk som skolefag i Noreg, med særleg fokus på literacy/tekstkompetanse. Meir forsking på multimodal tekstkompetanse i engelskundervisninga i Noreg har vore etterlyst, og motiverte denne studien. I tillegg har den nye læreplanen, som inkluderer multimodale tekster i engelskfaget, aktualisert denne studiens relevans for både praksisfeltet og forsking. Det overordna forskingsspørsmålet avhandlinga stiller er som følger: Kva rolle speler multimodalitet i engelskfagets literacy-praksisar i Noreg? Med teoretisk utgangspunkt i eit sosialsemiotisk multimodalt perspektiv på læring, inneheld denne avhandlinga tre kvalitative case-studiar. Datamaterialet består hovudsakeleg av tekstar og observasjonar av teksthendingar, samla inn frå tre ulike kontekstar. Hovudbidraget i denne doktorgradsavhandlinga er auka kunnskap om multimodalitet som eit ibuande, men lite kjent fenomen innanfor fagdidaktikk og engelskundervisning i Noreg. Funna tyder på at engelskfaget har ein multimodal literacy-praksis på input-sida av undervisninga. Denne praksisen er likevel stort sett taus og bunden til tradisjonar for støttande stillas og motivasjon som drivkraft. Avhandlinga antydar at produksjon og anerkjenning av multimodale tekstar, som output så vel som input, er eit naturleg og nødvendig skritt framover for engelskundervisninga, spesielt med tanke på den nye læreplanen

Continuous dependence estimates for nonlinear fractional convection-diffusion equations

We develop a general framework for finding error estimates for convection-diffusion equations with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators that are generators of pure jump Levy processes (e.g. the fractional Laplacian). As an application, we derive continuous dependence estimates on the nonlinearities and on the Levy measure of the diffusion term. Estimates of the rates of convergence for general nonlinear nonlocal vanishing viscosity approximations of scalar conservation laws then follow as a corollary. Our results both cover, and extend to new equations, a large part of the known error estimates in the literature.Comment: In this version we have corrected Example 3.4 explaining the link with the results in [51,59

The Dynamics of Plant-Mediated Sediment Oxygenation in Spartina anglica Rhizospheres—a Planar Optode Study

Belowground sediment oxygenation in rhizospheres of wetland plants promotes nutrient uptake, serve as protection against toxic reduced compounds and play an important role in wetland nutrient cycling. The presence of ~1.5-mm-wide oxic zones around roots of the intertidal marsh grass Spartina anglica was demonstrated below the sediment surface using planar optode technology recording 2D images of the sediment oxygen distribution. Oxic root zones were restricted to the root tips stretching up to 16 mm along the roots with an oxygen concentration up to 85 μmol L−1 detected at the root surface. Radial oxygen loss across the root surface ranged from 250 to 300 nmol m−2 s−1, which is comparable to other wetland plants. During air exposure of the aboveground biomass, atmospheric oxygen was the primary source for belowground oxygen transport, and light availability only had a minor effect on the belowground sediment oxygenation. During inundations completely submerging the aboveground biomass cutting off access to atmospheric oxygen, oxic root zones diminished significantly in the light and were completely eliminated in darkness. Within the time frame of a normal tidal inundation (~1.5 h), photosynthetic oxygen production maintained the presence of oxic root zones in light, whereas oxic root zones were eliminated within 1 h in darkness. The results show that the sediment oxygenation in Spartina anglica rhizospheres is temporally dynamic as well as spatially variable along the roots

Covariant q-differential operators and unitary highest weight representations for U_q su(n,n)

We investigate a one-parameter family of quantum Harish-Chandra modules of U_q sl(2n). This family is an analog of the holomorphic discrete series of representations of the group SU(n,n) for the quantum group U_q su(n, n). We introduce a q-analog of "the wave" operator (a determinant-type differential operator) and prove certain covariance property of its powers. This result is applied to the study of some quotients of the above-mentioned quantum Harish-Chandra modules. We also prove an analog of a known result by J.Faraut and A.Koranyi on the expansion of reproducing kernels which determines the analytic continuation of the holomorphic discrete series.Comment: 26 page