From Formulas to Functions through Geometry: A Path to Understanding Algebraic Computations

The teaching of algebra at the secondary school level has faced a great revolution during the last 50 years. While previously, it was focused on technicisms and pure syntactic rules, the most modern trends recommend using a functional approach to algebra and giving more prominence to conversions among different representation registers than treatments as simplifications and expansions. Nowadays, the daily practice in teaching algebra is still influenced by the traditional approach, and there is a need to offer teachers examples of activities that can give meaning to algebraic computations. This study proposes a set of interactive activities for eighth grade students, with a functional approach to formulas in a geometric context. The goal of the study is to investigate how similar activities can help students to develop multiple approaches to problems, understand algebraic formulas, and discern which main problems they face. The activities were tested with about 300 students, and qualitative and quantitative data were analyzed to answer the research questions.

Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation

This paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples.

Learning Multimodal Affinities for Textual Editing in Images

Nowadays, as cameras are rapidly adopted in our daily routine, images of documents are becoming both abundant and prevalent. Unlike natural images that capture physical objects, document-images contain a significant amount of text with critical semantics and complicated layouts. In this work, we devise a generic unsupervised technique to learn multimodal affinities between textual entities in a document-image, considering their visual style, the content of their underlying text, and their geometric context within the image. We then use these learned affinities to automatically cluster the textual entities in the image into different semantic groups. The core of our approach is a deep optimization scheme dedicated for an image provided by the user that detects and leverages reliable pairwise connections in the multimodal representation of the textual elements to properly learn the affinities. We show that our technique can operate on highly varying images spanning a wide range of documents and demonstrate its applicability for various editing operations manipulating the content, appearance, and geometry of the image.

Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations

<p style='text-indent:20px;'>The so-called method of phase synchronization has been advocated in a number of papers as a way of decoupling a system of linear second-order differential equations by a linear transformation of coordinates and velocities. This is a rather unusual approach because velocity-dependent transformations in general do not preserve the second-order character of differential equations. Moreover, at least in the case of linear transformations, such a velocity-dependent one defines by itself a second-order system, which need not have anything to do, in principle, with the given system or its reformulation. This aspect, and the related questions of compatibility it raises, seem to have been overlooked in the existing literature. The purpose of this paper is to clarify this issue and to suggest topics for further research in conjunction with the general theory of decoupling in a differential geometric context.</p>

Quantifying Intertidal Habitat Relative Coverage in a Florida Estuary Using UAS Imagery and GEOBIA

Intertidal habitats like oyster reefs and salt marshes provide vital ecosystem services including shoreline erosion control, habitat provision, and water filtration. However, these systems face significant global change as a result of a combination of anthropogenic stressors like coastal development and environmental stressors such as sea-level rise and disease. Traditional intertidal habitat monitoring techniques are cost and time-intensive, thus limiting how frequently resources are mapped in a way that is often insufficient to make informed management decisions. Unoccupied aircraft systems (UASs) have demonstrated the potential to mitigate these costs as they provide a platform to rapidly, safely, and inexpensively collect data in coastal areas. In this study, a UAS was used to survey intertidal habitats along the Gulf of Mexico coastline in Florida, USA. The structure from motion photogrammetry techniques were used to generate an orthomosaic and a digital surface model from the UAS imagery. These products were used in a geographic object-based image analysis (GEOBIA) workflow to classify mudflat, salt marsh, and oyster reef habitats. GEOBIA allows for a more informed classification than traditional techniques by providing textural and geometric context to habitat covers. We developed a ruleset to allow for a repeatable workflow, further decreasing the temporal cost of monitoring. The classification produced an overall accuracy of 79% in classifying habitats in a coastal environment with little spectral and textural separability, indicating that GEOBIA can differentiate intertidal habitats. This method allows for effective monitoring that can inform management and restoration efforts.

Spontaneous object-location memory based on environmental geometry is impaired by both hippocampal and dorsolateral striatal lesions

We examined the role of the hippocampus and the dorsolateral striatum in the representation of environmental geometry using a spontaneous object recognition procedure. Rats were placed in a kite-shaped arena and allowed to explore two distinctive objects in each of the right-angled corners. In a different room, rats were then placed into a rectangular arena with two identical copies of one of the two objects from the exploration phase, one in each of the two adjacent right-angled corners that were separated by a long wall. Time spent exploring these two objects was recorded as a measure of recognition memory. Since both objects were in different locations with respect to the room (different between exploration and test phases) and the global geometry (also different between exploration and test phases), differential exploration of the objects must be a result of initial habituation to the object relative to its local geometric context. The results indicated an impairment in processing the local geometric features of the environment for both hippocampus and dorsolateral striatum lesioned rats compared with sham-operated controls, though a control experiment showed these rats were unimpaired in a standard object recognition task. The dorsolateral striatum has previously been implicated in egocentric route-learning, but the results indicate an unexpected role for the dorsolateral striatum in processing the spatial layout of the environment. The results provide the first evidence that lesions to the hippocampus and dorsolateral striatum impair spontaneous encoding of local environmental geometric features.

An Original and Additional Mathematical Model Characterizing a Bayesian Approach to Decision Theory

We propose an original mathematical model according to a Bayesian approach explaining uncertainty from a point of view connected with vector spaces. A parameter space can be represented by means of random quantities by accepting the principles of the theory of concordance into the domain of subjective probability. We observe that metric properties of the notion of $\alpha$-product mathematically fulfill the ones of a coherent prevision of a bivariate random quantity. We introduce fundamental metric expressions connected with transformed random quantities representing changes of origin. We obtain a posterior probability law by applying the Bayes&#39; theorem into a geometric context connected with a two-dimensional parameter space.