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This paper investigates discrete-time Markov chains on non-negative integers whose transition probabilities are derived from the recurrence relation of ultraspherical polynomials. Generalizing prior single-time convergence results, our first main result is to establish the full weak convergence of the diffusively rescaled chains to a Bessel process of dimension d = 2α+ 2. The proof shows the tightness of the process sequence and the convergence of its finite-dimensional distributions. As a key application of this process-level convergence, our second main result shows that the rescaled first exit time of the chain from an interval converges in law to the corresponding exit time of the limiting Bessel process.

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In this paper, we have introduced and studied two new subdivision rules of triangles using incenter, namely, inward and outward binary incentric subdivisions, alongside the existing senary and ternary incentric subdivisions. To this end, we have used iterated function systems to establish approximation, convergence and denseness theorems in metric spaces as the main tools to study how diverse the shapes of the descendant triangles are upon repeated subdivisions of triangles using these old and new rules.

In the inward binary case, we have proved that the descendant triangles are dense in the space of triangles, similar to the senary case. We have also proved the distinctive behavior of the inward binary descendants that, in contrast to the senary case, they are spread over the space of triangle more and more evenly as subdivision keeps going, which is characterized by the weak convergence of their empirical measures to the uniform probability measure on the closure of the space of triangles.

In the outward binary case, we have proved that the descendant triangle branches all converge to a rather limited choice of triangles, similar to the ternary case. We have also followed up on connections between the ternary case and the Sierpiński triangle through formulating an old observation in precise terms and proved it.

We have also generalized both inward and outward binary incentric subdivisions to quasi-incentric sub-divisions, and discovered that they preserve well the behaviors of the descendant triangles in their incentric counterparts.

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The No-Three-in-Line problem asks how many points can be placed on an n × n integer grid so that no three are collinear. In this paper, we first generalise the problem to an m × n rectangular grid, establishing a construction method which we prove places the largest possible number of points on the grid for certain constraints on m and n. Then, using probabilistic heuristics, we estimate the maximum number of points that can be placed on the grid. Finally, we extend our analysis to the 3D No-Three-in-Line problem, deriving an asymptotic expression for the total number of collinear triples in an n × n × n grid as n → ∞. We demonstrate why the heuristic approach cannot be extended directly to 3D.

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Building on ‘What Can You Draw (2023)’ by Florian Frick and Fei Peng, this paper generalises ‘drawability’, a topological framework combining unions and exclusions of regions, to arbitrary brush shapes and infinite drawing sequences, enabling the study of more complex figures. ‘n-layer drawability’ is introduced as a new metric to quantify the complexity of undrawable sets. Our central theoretical result establishes that a figure is undrawable if and only if its boundary contains ‘containment loops’ (∈-loops). To analyse these loops, we developed the layer drawability graph and prove its chromatic number provides a lower bound on n-layer drawability. Finally, we prove this complexity metric is unbounded. Case studies of practical drawing are also discussed.

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As a complementary development of the nature of frieze structures, we partially solve the Fontaine-Plamondon conjecture on the enumeration of multiplicative E_n friezes through an elementary combinatorial approach. Incorporating with the concept of quiddity sequences, we explicitly show the number of distinct E₆ friezes are 70, with substantial evidence to support the number of E₇ and E₈ friezes to be 440 and 1694, respectively.

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While significant progress has been made on the question “How many shuffles are required to sufficiently randomize a deck of cards?”, the most influential work on the subject remains the 1992 paper by Bayer and Diaconis, which provided an answer for a standard 52-card deck, proving that seven shuffles suffice. However, many variations of this problem have yet to be extensively explored.

In particular, multiple decks of cards are commonly used in various card games, such as the popular Chinese game GuanDan (2 decks), making the randomness of a deck after shuffling highly relevant in both recreational and professional settings. In this paper, we provide a mathematical analysis on the randomness of riffle shuffling multiple decks of cards, extending the result of Bayer and Diaconis to the multi-deck setting. A key contribution of ours is the investigation of the randomness of a player’s hand after riffle shuffling in a deck, a question that has not been investigated at all previously, from which we are able to give explicit formulae for the separation distance for a player’s hand after riffle shuffling for both the single deck and double deck setting.

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Fractal interpolation functions were first introduced by Barnsley. In this paper, we investigate the fractal interpolation functions joining the points (0, 0),(x₁, y₁),(1, 1) for rational x₁, y₁ in the closed unit interval, and show that, when x₁ = 1/2, the function will always map rationals to rationals, but will not map any rational to 1/2 when y₁ has odd denominator. Moreover, we derive by similar considerations a result about a dynamical system on the closed unit interval.

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This paper investigates how varying the angle of contact α between a chalk and blackboard affects how annoying the sound produced is after drawing some distance d. Annoyance is quantified through Zwicker’s psychoacoustic annoyance model. Chalk is modeled as a cylinder and its crumbling is modeled by a downward translation “into” a chalkboard rotated at an angle α. This model is used to calculate the chalk’s area of contact with the board at any d and α. Next, the relationship between contact area and psychoacoustic annoyance is determined experimentally. The set of angles that minimize and maximize annoyance are then found, measured using two annoyance metrics: instantaneous and accumulated annoyance.

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