Modules with Demazure Flags and Character Formulae

In this paper we study a family of finite-dimensional graded representations of the current algebra of $\mathfrak{sl}_2$ which are indexed by partitions. We show that these representations admit a flag where the successive quotients are Demazure modules which occur in a level $\ell$-integrable module for $A_1^1$ as long as $\ell$ is large. We associate to each partition and to each $\ell$ an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level $\ell$-Demazure module in the filtration. In the special case of the partition $1^s$ and $\ell=2$, we give a closed formula for the graded multiplicity of level two Demazure modules in a level one Demazure module. As an application, we use our result along with the results of Naoi and Lenart et al., to give the character of a $\mathfrak{g}$-stable level one Demazure module associated to $B_n^1$ as an explicit combination of suitably specialized Macdonald polynomials. In the case of $\mathfrak{sl}_2$, we also study the filtration of the level two Demazure module by level three Demazure modules and compute the numerical filtration multiplicities and show that the graded multiplicites are related to (variants of) partial theta series

A FOUNDATION FOR OPEN INFORMATION ENVIRONMENTS

Traditionally, information systems were developed within organizations for use by known audiences for known purposes. Advances in information technology have changed this landscape dramatically. The reach of information systems frequntly extends beyond organizational boundaries for use by unknown audiences and for purposes not originally anticipated. Individuals and informal communities can generate and use information in ways previously restricted to formal organizations. We term applications with these characteristics open information environments (OIEs). OIEs are marked by diversity of information available, flexibility in accommodating new sources, users and uses, and information management with minimal controls on structure, content, and access. This creates opportunities to generate new information and use it in unexpected ways. However, OIEs also come with challenges in managing the semantic diversity, flexibility of use, and information quality issus arising from the range of users and lack of controls. In this paper, we propose a set of principles for managing OIEs effectively. We outline a research program to examine the potential of OIEs, the challenges they present, and how to design OIEs to realize the benefits while mitigating the challenges. We highlight our ongoing research in this area, and conclude with a call for more research on this important phenomenon

Bringing a CURE into a Discrete Mathematics Course and Beyond

Course-based Undergraduate Research Experiences (CUREs) have been well developed in the hard sciences, but math CUREs are all but absent from the literature. Like biology and chemistry, math programs suffer from a lack of research experiences and many students are not able to participate in programs like REUs (Research Experiences for Undergraduates). CUREs are a great alternative, but the current definition of CURE (see [1]) has potential barriers when applied to mathematics (e.g. time, novelty of project). Our solution to these barriers was to develop a math CURE pathway in which students complete Math CUREs in targeted courses. After finishing the pathway (or part of the pathway), students complete a research project in at least one of the following areas: Lie theory, representation theory, or combinatorics. The focus of this paper is the math CURE implemented in a discrete mathematics course for math and computer science majors. We share our experiences with the development and implementation of this CURE over several iterations as well as the impact of the CURE on students experiences through participant survey data obtained from this CURE

Querying Instances – A Protocol Analysis Study

The instance-based paradigm – introduced as an alternative to traditional class-based database management methods – does not require imposing a well-defined schema over data, nor does it entail central control and planning. As a consequence, it supports information requirements agility, enables collection of higher quality data, and reduces the schema and database operation problems associated with traditional methods. This study investigates the ability of content-consumers to use instance-based representations effectively for information retrieval purposes. A visual representation of the instance-based data was created and empirically evaluated with 12 subjects using protocol analysis. Results show that instance-based users were able to retrieve the required information more accurately compared to users of the traditional representation. From a cognitive point of view, instance-based users were more efficient than class-based users – they experienced fewer breakdowns in their problem solving process and, when breakdowns occurred, were more successful in recovering from them

Sequential Testing in Reliability and Validity Studies With Repeated Measurements per Subject

In medical, health, and sports sciences, researchers desire a device with high reliability and validity. This article focuses on reliability and validity studies with n subjects and m ≥ 2 repeated measurements per subject. High statistical power can be achieved by increasing n or m, and increasing m is often easier than increasing n in practice unless m is too high to result in systematic bias. The sequential probability ratio test (SPRT) is a useful statistical method which can conclude a null hypothesis H0 or an alternative hypothesis H1 with 50% of the required sample size of a non-sequential test on average. The traditional SPRT requires the likelihood function for each observed random variable, and it can be a practical burden for evaluating the likelihood ratio after each observation of a subject. Instead, m observed random variables per subject can be transformed into a test statistic which has a known sampling distribution under H0 and under H1. This allows us to formulate a SPRT based on a sequence of test statistics. In this article, three types of study are considered: reliability of a device, reliability of a device relative to a criterion device, and validity of a device relative to a criterion device. Using SPRT for testing the reliability of a device, for small m, results in an average sample size of about 50% of the fixed sample size for a non-sequential test. For comparing a device to criterion, the average sample size approaches to 60% approximately as m increases. The SPRT tolerates violation of normality assumption for validity study, but it does not for reliability study

Macdonald Polynomials and Level Two Demazure Modules for Affine sln+1

We define a family of symmetric polynomials Gν,λ(z1, ...,zn+1, q) indexed by a pair of dominant integral weights for a root system of type An. The polynomial Gν,0(z, q) is the specialized Macdonald polynomial Pν(z, q, 0) and is known to be the graded character of a level one Demazure module associated to the affine Lie algebra sln+1. We prove that G0,λ(z, q) is the graded character of a level two Demazure module for sln+1. Under suitable conditions on (ν, λ) (which apply to the pairs (ν, 0) and (0, λ)) we prove that Gν,λ(z, q) is Schur positive, i.e., it can be written as a linear combination of Schur polynomials with coefficients in Z+[q]. We further prove that Pν(z, q, 0) is a linear combination of elements G0,λ(z, q) with the coefficients being essentially products of q-binomials. Together with a result of K. Naoi, a consequence of our result is an explicit formula for the specialized Macdonald polynomial associated to a non-simply laced Lie algebra as a linear combination of the level one Demazure characters of the non-simply laced algebra

A Tutorial of Bland Altman Analysis in A Bayesian Framework

There are two schools of thought in statistical analysis, frequentist, and Bayesian. Though the two approaches produce similar estimations and predictions in large-sample studies, their interpretations are different. Bland Altman analysis is a statistical method that is widely used for comparing two methods of measurement. It was originally proposed under a frequentist framework, and it has not been used under a Bayesian framework despite the growing popularity of Bayesian analysis. It seems that the mathematical and computational complexity narrows access to Bayesian Bland Altman analysis. In this article, we provide a tutorial of Bayesian Bland Altman analysis. One approach we suggest is to address the objective of Bland Altman analysis via the posterior predictive distribution. We can estimate the probability of an acceptable degree of disagreement (fixed a priori) for the difference between two future measurements. To ease mathematical and computational complexity, an interface applet is provided with a guideline

Frequency-tunable Pre-stabilized Lasers for LISA via Sideband-locking

Laser frequency noise mitigation is one of the most challenging aspects of the LISA interferometric measurement system. The unstabilized frequency fluctuations must be suppressed by roughly twelve orders of magnitude in order to achieve stability sufficient for gravitational wave detection. This enormous suppression will be achieved through a combination of stabilization and common-mode rejection. The stabilization component will itself be achieved in two stages: pre-stabilization to a local optical cavity followed by arm-locking to some combination of the inter-spacecraft distances. In order for these two stabilization stages to work simultaneously, the lock-point of the pre-stabilization loop must be frequency tunable. The current baseline stabilization technique, locking to an optical cavity, does not provide tunability between cavity resonances, which are typically spaced by 100s of MHz. Here we present a modification to the traditional Pound-Drever-Hall cavity locking technique that allows the laser to be locked to a cavity resonance with an adjustable frequency offset. This technique requires no modifications to the optical cavity itself, thus preserving the stability of the frequency reference. We present measurements of the system performance and demonstrate that we can meet implement the first two stages of stabilization

Male breast cancer in BRCA1 and BRCA2 mutation carriers : pathology data from the Consortium of Investigators of Modifiers of BRCA1/2

Background: BRCA1 and, more commonly, BRCA2 mutations are associated with increased risk of male breast cancer (MBC). However, only a paucity of data exists on the pathology of breast cancers (BCs) in men with BRCA1/2 mutations. Using the largest available dataset, we determined whether MBCs arising in BRCA1/2 mutation carriers display specific pathologic features and whether these features differ from those of BRCA1/2 female BCs (FBCs). Methods: We characterised the pathologic features of 419 BRCA1/2 MBCs and, using logistic regression analysis, contrasted those with data from 9675 BRCA1/2 FBCs and with population-based data from 6351 MBCs in the Surveillance, Epidemiology, and End Results (SEER) database. Results: Among BRCA2 MBCs, grade significantly decreased with increasing age at diagnosis (P = 0.005). Compared with BRCA2 FBCs, BRCA2 MBCs were of significantly higher stage (P for trend = 2 x 10(-5)) and higher grade (P for trend = 0.005) and were more likely to be oestrogen receptor-positive [odds ratio (OR) 10.59; 95 % confidence interval (CI) 5.15-21.80] and progesterone receptor-positive (OR 5.04; 95 % CI 3.17-8.04). With the exception of grade, similar patterns of associations emerged when we compared BRCA1 MBCs and FBCs. BRCA2 MBCs also presented with higher grade than MBCs from the SEER database (P for trend = 4 x 10(-12)). Conclusions: On the basis of the largest series analysed to date, our results show that BRCA1/2 MBCs display distinct pathologic characteristics compared with BRCA1/2 FBCs, and we identified a specific BRCA2-associated MBC phenotype characterised by a variable suggesting greater biological aggressiveness (i.e., high histologic grade). These findings could lead to the development of gender-specific risk prediction models and guide clinical strategies appropriate for MBC management.Peer reviewe