Cluster algebras of type A2(1)A_2^{(1)}

In this paper we study cluster algebras \myAA of type $A_2^{(1)}$. We solve the recurrence relations among the cluster variables (which form a T--system of type $A_2^{(1)}$). We solve the recurrence relations among the coefficients of \myAA (which form a Y--system of type $A_2^{(1)}$). In \myAA there is a natural notion of positivity. We find linear bases \BB of \myAA such that positive linear combinations of elements of \BB coincide with the cone of positive elements. We call these bases \emph{atomic bases} of \myAA. These are the analogue of the "canonical bases" found by Sherman and Zelevinsky in type $A_{1}^{(1)}$. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of \BB are parameterized by \ZZ^3 via their $\mathbf{g}$--vectors in every cluster. We prove that the denominator vector map in every acyclic seed of \myAA restricts to a bijection between \BB and \ZZ^3. In particular this gives an explicit algorithm to determine the "virtual" canonical decomposition of every element of the root lattice of type $A_2^{(1)}$. We find explicit recurrence relations to express every element of \myAA as linear combinations of elements of \BB.Comment: Latex, 40 pages; Published online in Algebras and Representation Theory, springer, 201

Quantum finite automata and linear context-free languages: a decidable problem

We consider the so-called measure once finite quantum automata model introduced by Moore and Crutchfield in 2000. We show that given a language recognized by such a device and a linear context-free language, it is recursively decidable whether or not they have a nonempty intersection. This extends a result of Blondel et al. which can be interpreted as solving the problem with the free monoid in place of the family of linear context-free languages. © 2013 Springer-Verlag

Minimal generating sets of non-modular invariant rings of finite groups

It is a classical problem to compute a minimal set of invariant polynomial generating the invariant ring of a finite group as an algebra. We present here an algorithm for the computation of minimal generating sets in the non-modular case. Apart from very few explicit computations of Groebner bases, the algorithm only involves very basic operations, and is thus rather fast. As a test bed for comparative benchmarks, we use transitive permutation groups on 7 and 8 variables. In most examples, our algorithm implemented in Singular works much faster than the one used in Magma, namely by factors between 50 and 1000. We also compute some further examples on more than 8 variables, including a minimal generating set for the natural action of the cyclic group of order 11 in characteristic 0 and of order 15 in characteristic 2. We also apply our algorithm to the computation of irreducible secondary invariants.Comment: 14 pages v3: Timings updated. One example adde

Multiscale simulations of sliding droplets

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Effective Invariant Theory of Permutation Groups using Representation Theory

Using the theory of representations of the symmetric group, we propose an algorithm to compute the invariant ring of a permutation group. Our approach have the goal to reduce the amount of linear algebra computations and exploit a thinner combinatorial description of the invariant ring.Comment: Draft version, the corrected full version is available at http://www.springer.com

Lunar Crater Identification in Digital Images

It is often necessary to identify a pattern of observed craters in a single image of the lunar surface and without any prior knowledge of the camera's location. This so-called "lost-in-space" crater identification problem is common in both crater-based terrain relative navigation (TRN) and in automatic registration of scientific imagery. Past work on crater identification has largely been based on heuristic schemes, with poor performance outside of a narrowly defined operating regime (e.g., nadir pointing images, small search areas). This work provides the first mathematically rigorous treatment of the general crater identification problem. It is shown when it is (and when it is not) possible to recognize a pattern of elliptical crater rims in an image formed by perspective projection. For the cases when it is possible to recognize a pattern, descriptors are developed using invariant theory that provably capture all of the viewpoint invariant information. These descriptors may be pre-computed for known crater patterns and placed in a searchable index for fast recognition. New techniques are also developed for computing pose from crater rim observations and for evaluating crater rim correspondences. These techniques are demonstrated on both synthetic and real images

Semi-invariants of symmetric quivers of finite type

Let $(Q,\sigma)$ be a symmetric quiver, where $Q=(Q_0,Q_1)$ is a finite quiver without oriented cycles and $\sigma$ is a contravariant involution on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form on a representation $V$ of $Q$. We shall call the representation orthogonal if is symmetric and symplectic if $$ is skew-symmetric. Moreover we can define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. For symmetric quivers of finite type, we prove that the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type $c^V$ and, in the case when matrix defining $c^V$ is skew-symmetric, by the Pfaffians $pf^V$

Patients' Post-/Long-COVID Symptoms, Vaccination and Functional Status-Findings from a State-Wide Online Screening Study.

(1) Background: Better understanding of post-/long-COVID and limitations in daily life due to the symptoms as well as the preventive potential of vaccinations is required. It is unclear whether the number of doses and timepoint interrelate with the trajectory of post-/long-COVID. Accordingly, we examined how many patients positively screened with post-/long-COVID were vaccinated and whether the vaccination status and the timepoint of vaccination in relation to the acute infection were related to post-/long-COVID symptom severity and patients' functional status (i.e., perceived symptom severity, social participation, workability, and life satisfaction) over time. (2) Methods: 235 patients suffering from post-/long-COVID were recruited into an online survey in Bavaria, Germany, and assessed at baseline (T1), after approximately three weeks (T2), and approximately four weeks (T3). (3) Results: 3.5% were not vaccinated, 2.3% were vaccinated once, 20% twice, and 53.3% three times. Overall, 20.9% did not indicate their vaccination status. The timepoint of vaccination was related to symptom severity at T1, and symptoms decreased significantly over time. Being vaccinated more often was associated with lower life satisfaction and workability at T2. (4) Conclusions: This study provides evidence to get vaccinated against SARS-CoV-2, as it has shown that symptom severity was lower in those patients who were vaccinated prior to the infection compared to those getting infected prior to or at the same time of the vaccination. However, the finding that being vaccinated against SARS-CoV-2 more often correlated with lower life satisfaction and workability requires more attention. There is still an urgent necessity for appropriate treatment for overcoming long-/post-COVID symptoms efficiently. Vaccination can be part of prevention measures, and there is still a need for a communication strategy providing objective information about the usefulness and risks of vaccinations

Spatial and temporal variability of soil fertility in relation to crop yield zones on hummocky terrain

Non-Peer ReviewedA field study was conducted on hummocky terrain at the Manitoba Zero Tillage Association Research Farm to determine the variability of crop yield as related to landscape position, soil properties, weed populations and plant disease. This information was also used to evaluate technology required for delineation of management units related to precision farming. Variable rate fertiliser management systems can improve efficiency of fertilizer use and environmental sustainability. Adoption of this technology has been hampered due to the difficulty of classifying fields into management units, the high cost of sampling soils on a grid basis, and the variability of soil and plant properties in the landscape. Technology for variable rate fertilizer systems is available, but there is little information available related to yield response in clay soils on hummocky terrain, and the relationship of plant tissue test levels in relation to soil fertility as measured by soil test nitrate nitrogen. Current soil test recommendations for nitrogen are based on soil test nitrate nitrogen from samples bulked from samples in several locations in the field preferably grouped according to topography. Yield data for 1997-2001 were classified into groups with the fuzzy k means, normal mixtures and self-organizing map variants of cluster analysis. Although fuzzy k means commonly used for classification of crop yield and soil properties, a method based on self-organizing maps provided consistent classes when compared across years. Soil nitrate nitrogen varied considerably across the landscape at the site, but was not significantly different (P<0.05) between classes based on crop yield. Yield data can be used to delineate zones for variable management, although fertilizer inputs may be a function of spring soil moisture, runoff and growing season precipitation as they affect seeding, crop emergence and establishment

The ‘Greenlandization’ of care: disability in postcolonial Greenland, 1950s-1980s

Political Culture and National Identit