Open Learning Experience Bingo

Definitions of openness and open education abound, but with so many, how can we use them effectively to explore the openness of assignments, activities, classes, or programs? Open Learning Experience Bingo is a game that a group of collaborators have created to give people a way to surface and discuss the many different ways that educational experiences can “open” beyond traditional practices. Each bingo card includes boxes containing possible “ingredients” in a learning experience, and radiating from the center of each box, “dimensions” of openness along which an ingredient might be opened. You “play” bingo by reading or hearing about a learning experience and marking areas on the bingo card that you think the experience opens. The game incorporates broad concepts of openness and seeks not to measure the openness of learning experiences, but to identify and spark discussion about areas in which experiences are opening — or might be opened further. As artifacts, completed bingo cards display a sort of “heat map” of openness that can be used to compare and contrast bingo evaluations of various learning experiences.

Dimensions of Fractional Brownian Images

This paper concerns the intermediate dimensions, a spectrum of dimensions that interpolate between the Hausdorff and box dimensions. Potential-theoretic methods are used to produce dimension bounds for images of sets under Hölder maps and certain stochastic processes. We apply this to compute the almost-sure value of the dimension of Borel sets under index-$$\alpha $$ α fractional Brownian motion in terms of dimension profiles defined using capacities. As a corollary, this establishes continuity of the profiles for Borel sets and allows us to obtain an explicit condition showing how the Hausdorff dimension of a set may influence the typical box dimension of Hölder images such as projections. The methods used propose a general strategy for related problems; dimensional information about a set may be learned from analysing particular fractional Brownian images of that set. To conclude, we obtain bounds on the Hausdorff dimension of exceptional sets, with respect to intermediate dimensions, in the setting of projections.

INTERMEDIATE ASSOUAD-LIKE DIMENSIONS FOR MEASURES

The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the ‘thickest’ and ‘thinnest’ parts of the set. Less extreme versions of these dimensions for sets have been introduced, including the upper and lower quasi-Assouad dimensions, [Formula: see text]-Assouad spectrum, and [Formula: see text]-dimensions. In this paper, we study the analogue of the upper and lower [Formula: see text]-dimensions for measures. We give general properties of such dimensions, as well as more specific results for self-similar measures satisfying various separation properties and discrete measures.

Assouad Dimensions and Lower Dimensions of Some Moran Sets

We prove that the low dimensions of a class of Moran sets coincide with their Hausdorff dimensions and obtain a formula for the lower dimensions. Subsequently, we consider some homogeneous Cantor sets which belong to Moran sets and give the counterexamples in which their Assouad dimension is not equal to their upper box dimensions and packing dimensions under the case of not satisfying the condition of the smallest compression ratio c ∗ > 0 .

BOX DIMENSIONS OF THE RIEMANN–LIOUVILLE FRACTIONAL INTEGRAL OF HÖLDER CONTINUOUS MULTIVARIATE FUNCTIONS

If a continuous multivariate function satisfies a Lipschitz condition on its domain, Box dimension of its graph equals to the number of its arguments. Furthermore, Box dimension of the graph of its Riemann–Liouville fractional integral also equals to the number of its arguments.