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Han Yu

Publications and source records attributed to Han Yu.

At least 271 records · Page 15Linked to original sources
arXiv2018

The Assouad spectrum and the quasi-Assouad dimension: a tale of two spectra

We consider the Assouad spectrum, introduced by Fraser and Yu, along with a natural variant that we call the `upper Assouad spectrum'. These spectra are designed to interpolate between the upper box-counting and Assouad dimensions. It is known that the Assouad spectrum approaches the upper box-counting dimension at the left hand side of its domain, but does not necessarily approach the Assouad dimension on the right. Here we show that it necessarily approaches the \emph{quasi-Assouad dimension} at the right hand side of its domain. We further show that the upper Assouad spectrum can be expressed in terms of the Assouad spectrum, thus motivating the definition used by Fraser-Yu. We also provide a large family of examples demonstrating new phenomena relating to the form of the Assouad spectrum. For example, we prove that it can be strictly concave, exhibit phase transitions of any order, and need not be piecewise differentiable.

math.CA↗
arXiv2018

Assouad dimension of random processes

In this paper we study the Assouad dimension of graphs of certain Lévy processes and functions defined by stochastic integrals. We do this by introducing a convenient condition which guarantees a graph to have full Assouad dimension and then show that graphs of our studied processes satisfy this condition.

math.PR↗
arXiv2018

Cubes, side lengths and centres

It is known that in $\mathbb{R}^n,n\geq 2$, a compact set which contains $n-1$ spheres with all radii in $[1/2,1]$ or with all possible centres in $[0,1]^n$ has full Hausdorff dimension. In fact the later set has positive Lebesgue measure. In this paper we consider a similar problem with sphere replacing by fractal cubes. The radii set and the centre set are also considered to be fractal sets. In addition we discuss the exceptional set in the setting of general largeness. In the end, an Furstenberg type example is discussed which can be somehow considered as the Furstenberg $\times 2$, $\times 3$ set conjecture (now theorem) in the setting of cubes/circles sets considered here.

math.CA↗
arXiv2018

Cube packings in Euclidean spaces

In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower box dimension at least $n-0.5$ and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper.

math.CA↗
arXiv2017

Exponential Structure of Income Inequality: Evidence from 67 Countries

Economic competition between humans leads to income inequality, but, so far, there has been little understanding of underlying quantitative mechanisms governing such a collective behavior. We analyze datasets of household income from 67 countries, ranging from Europe to Latin America, North America and Asia. For all of the countries, we find a surprisingly uniform rule: Income distribution for the great majority of populations (low and middle income classes) follows an exponential law. To explain this empirical observation, we propose a theoretical model within the standard framework of modern economics and show that free competition and Rawls' fairness are the underlying mechanisms producing the exponential pattern. The free parameters of the exponential distribution in our model have an explicit economic interpretation and direct relevance to policy measures intended to alleviate income inequality.

econ.GN↗
arXiv2017

Arithmetic patches, weak tangents, and dimension

We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including: the presence (or lack of) arithmetic progressions (or patches in dimensions $\geq 2$); the structure of tangent sets; and the Assouad dimension. We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to `asymptotically' containing arbitrarily large arithmetic patches. We present some applications of our results concerning sets of integers, which include a weak solution to the Erdös-Turán conjecture on arithmetic progressions.

math.CA↗
arXiv2017

On generalized trigonometric functions and series of rational functions

Here we introduce a way to construct generalized trigonometric functions associated with any complex polynomials, and the well known trigonometric functions can be seen to associate with polynomial $x^2-1$. We will show that those generalized trigonometric functions have algebraic identities which generalizes the well known $\sin^2(x)+\cos^2(x)=1$. One application of the generalized trigonometric functions is evaluating infinite series of rational functions.

math.CA↗
arXiv2017

Dimensions of sets which uniformly avoid arithmetic progressions

We provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions. More precisely, we say $F$ uniformly avoids arithmetic progressions of length $k \geq 3$ if there is an $ε>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $Δ>0$ inside the $εΔ$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $ε$. In the other direction, we provide examples of sets which uniformly avoid arithmetic progressions of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence we obtain a discretised version of a `reverse Kakeya problem': we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates arithmetic progressions in every direction.

math.CA↗
arXiv2017

Kakeya books and projections of Kakeya sets

Here we show some results related with Kakeya conjecture which says that for any integer $n\geq 2$, a set containing line segments in every dimension in $\mathbb{R}^n$ has full Hausdorff dimension as well as box dimension. We proved here that the Kakeya books, which are Kakeya sets with some restrictions on positions of line segments have full box dimension. We also prove here a relation between the projection property of Kakeya sets and the Kakeya conjecture. If for any Kakeya set $K\subset\mathbb{R}^n$, the Hausdorff dimension of orthogonal projections on $k\leq n$ subspaces is independent of directions then the Kakeya conjecture is true. Moreover, the converse is also true.

math.CA↗
arXiv2017

Root sets of polynomials and power series with finite choices of coefficients

Given $H\subseteq \mathbb{C}$ two natural objects to study are the set of zeros of polynomials with coefficients in $H$, $$\{z\in \mathbb{C}: \exists k>0,\, \exists (a_n)\in H^{k+1}, \sum_{n=0}^{k}a_{n}z^n=0\},$$ and the set of zeros of power series with coefficients in $H$, $$\{z\in\mathbb{C}: \exists (a_n)\in H^{\mathbb{N}}, \sum_{n=0}^{\infty} a_nz^n=0\}.$$ In this paper we consider the case where each element of $H$ has modulus $1$. The main result of this paper states that for any $r\in(1/2,1),$ if $H$ is $2\cos^{-1}(\frac{5-4|r|^2}{4})$-dense in $S^1,$ then the set of zeros of polynomials with coefficients in $H$ is dense in $\{z\in \mathbb{C}: |z|\in [r,r^{-1}]\},$ and the set of zeros of power series with coefficients in $H$ contains the annulus $\{z\in \mathbb{C}: |z|\in[r,1)\}$. These two statements demonstrate quantitatively how the set of polynomial zeros/power series zeros fill out the natural annulus containing them as $H$ becomes progessively more dense.

math.NT↗
arXiv2016

Surveillance Video Parsing with Single Frame Supervision

Surveillance video parsing, which segments the video frames into several labels, e.g., face, pants, left-leg, has wide applications. However,pixel-wisely annotating all frames is tedious and inefficient. In this paper, we develop a Single frame Video Parsing (SVP) method which requires only one labeled frame per video in training stage. To parse one particular frame, the video segment preceding the frame is jointly considered. SVP (1) roughly parses the frames within the video segment, (2) estimates the optical flow between frames and (3) fuses the rough parsing results warped by optical flow to produce the refined parsing result. The three components of SVP, namely frame parsing, optical flow estimation and temporal fusion are integrated in an end-to-end manner. Experimental results on two surveillance video datasets show the superiority of SVP over state-of-the-arts.

cs.CV↗
arXiv2016

Assouad type spectra for some fractal families

In a previous paper we introduced a new `dimension spectrum', motivated by the Assouad dimension, designed to give precise information about the scaling structure and homogeneity of a metric space. In this paper we compute the spectrum explicitly for a range of well-studied fractal sets, including: the self-affine carpets of Bedford and McMullen, self-similar and self-conformal sets with overlaps, Mandelbrot percolation, and Moran constructions. We find that the spectrum behaves differently for each of these models and can take on a rich variety of forms. We also consider some applications, including the provision of new bi-Lipschitz invariants and bounds on a family of `tail densities' defined for subsets of the integers.

math.CA↗
arXiv2016

Experimental validation of volume-based comparison for double-McCormick relaxations

Volume is a natural geometric measure for comparing polyhedral relaxations of non-convex sets. Speakman and Lee gave volume formulae for comparing relaxations of trilinear monomials, quantifying the strength of various natural relaxations. Their work was motivated by the spatial branch-and-bound algorithm for factorable mathematical-programming formulations. They mathematically analyzed an important choice that needs to be made whenever three or more terms are multiplied in a formulation. We experimentally substantiate the relevance of their main results to the practice of global optimization, by applying it to difficult box cubic problems (boxcup). In doing so, we find that, using their volume formulae, we can accurately predict the quality of a relaxation for boxcups based on the (box) parameters defining the feasible region.

math.OC↗
arXiv2016

A Survey on Artificial Intelligence and Data Mining for MOOCs

Massive Open Online Courses (MOOCs) have gained tremendous popularity in the last few years. Thanks to MOOCs, millions of learners from all over the world have taken thousands of high-quality courses for free. Putting together an excellent MOOC ecosystem is a multidisciplinary endeavour that requires contributions from many different fields. Artificial intelligence (AI) and data mining (DM) are two such fields that have played a significant role in making MOOCs what they are today. By exploiting the vast amount of data generated by learners engaging in MOOCs, DM improves our understanding of the MOOC ecosystem and enables MOOC practitioners to deliver better courses. Similarly, AI, supported by DM, can greatly improve student experience and learning outcomes. In this survey paper, we first review the state-of-the-art artificial intelligence and data mining research applied to MOOCs, emphasising the use of AI and DM tools and techniques to improve student engagement, learning outcomes, and our understanding of the MOOC ecosystem. We then offer an overview of key trends and important research to carry out in the fields of AI and DM so that MOOCs can reach their full potential.

cs.AI↗
arXiv2016

Building Robust Crowdsourcing Systems with Reputation-aware Decision Support Techniques

Crowdsourcing refers to the arrangement in which contributions are solicited from a large group of unrelated people. Due to this nature, crowdsourcers (or task requesters) often face uncertainty about the workers' capabilities which, in turn, affects the quality and timeliness of the results obtained. Trust is a mechanism used by people to facilitate interactions in human societies where risk and uncertain are common. The crucial challenge to building a robust crowdsourcing system is how to make trust-aware task delegation decisions to efficiently utilize the capacities of workers (or trustee agents) to achieve high social welfare? This book presents the research addressing this challenge. It goes beyond the existing trust management research framework by removing a widespread assumption implicitly adopted by existing research: that a trustee agent can process an unlimited number of interaction requests per discrete time unit without compromising its performance as perceived by the task requesters (or truster agents). Decision support in crowdsourcing is re-formalized as a multi-agent trust game based on the principles of the Congestion Game, which is solved by two trust-aware interaction decision-making approaches: 1) the Social Welfare Optimizing approach for Reputation-aware Decision-making (SWORD) approach, and 2) the Distributed Request Acceptance approach for Fair utilization of Trustee agents (DRAFT). SWORD is designed for centralized systems, while DRAFT is designed for fully distributed systems. Theoretical analyses have shown that the social welfare produced by these two approaches can be made closer to optimal by adjusting only one key parameter. With these two approaches, the framework of research for crowdsourcing systems can be enriched to handle more realistic scenarios where workers have varied and limited capabilities.

cs.MA↗
arXiv2014

Designing Socially Intelligent Virtual Companions

Virtual companions that interact with users in a socially complex environment require a wide range of social skills. Displaying curiosity is simultaneously a factor to improve a companion's believability and to unobtrusively affect the user's activities over time. Curiosity represents a drive to know new things. It is a major driving force for engaging learners in active learning. Existing research work pays little attention in curiosity. In this paper, we enrich the social skills of a virtual companion by infusing curiosity into its mental model. A curious companion residing in a Virtual Learning Environment (VLE) to stimulate user's curiosity is proposed. The curious companion model is developed based on multidisciplinary considerations. The effectiveness of the curious companion is demonstrated by a preliminary field study.

cs.HC↗
arXiv2014

Identifying Talented Software Engineering Students through Data-driven Skill Assessment

For software development companies, one of the most important objectives is to identify and acquire talented software engineers in order to maintain a skilled team that can produce competitive products. Traditional approaches for finding talented young software engineers are mainly through programming contests of various forms which mostly test participants' programming skills. However, successful software engineering in practice requires a wider range of skills from team members including analysis, design, programming, testing, communication, collaboration, and self-management, etc. In this paper, we explore potential ways to identify talented software engineering students in a data-driven manner through an Agile Project Management (APM) platform. Through our proposed HASE online APM tool, we conducted a study involving 21 Scrum teams consisting of over 100 undergraduate software engineering students in multi-week coursework projects in 2014. During this study, students performed over 10,000 ASD activities logged by HASE. We demonstrate the possibility and potentials of this new research direction, and discuss its implications for software engineering education and industry recruitment.

cs.SE↗