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.Comment: 7 page

On GILP's group theoretic approach to Falconer's distance problem

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf-Iosevich-Liu-Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP's theorem in [GILP15]. This allows us to extend the Wolff-Erdogan dimension bound for distance sets to finite points configurations with $k$ points for $k\in\{2,\dots,n+1\}.$ At the end of this paper, we extend this group-theoretic method and illustrate a `Fourier free' approach to Falconer's distance set problem for the Lebesgue measure. We explain how to use tubular incidence estimates in distance set problems. Curiously, tubular incidence estimates are also related to the Kakeya problem

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.Comment: 14 page

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.Comment: 9 pages. arXiv admin note: text overlap with arXiv:1711.0653

Dimensions of triangle sets

In this paper, we discuss some dimension results for triangle sets of compact sets in $\mathbb{R}^2$. In particular, we prove that for any compact set $F$ in $\mathbb{R}^2$, the triangle set $\Delta(F)$ satisfies $$\dim_{\mathrm{A}} \Delta(F)\geq \frac{3}{2}\dim_{\mathrm{A}} F.$$ If $\dim_{\mathrm{A}} F>1$ then we have $$\dim_{\mathrm{A}} \Delta(F)\geq 1+\dim_{\mathrm{A}} F.$$ If $\dim_{\mathrm{A}} F>4/3$ then we have the following better bound, $$\dim_{\mathrm{A}} \Delta(F)\geq \min\left\{\frac{5}{2}\dim_{\mathrm{A}} F-1,3\right\}.$$ Moreover, if $F$ satisfies a mild separation condition then the above result holds also for the box dimensions, namely, \[ \underline{\dim_{\mathrm{B}}} F\geq \frac{3}{2}\underline{\dim_{\mathrm{B}}} \Delta(F) \text{ and }\overline{\dim_{\mathrm{B}}} F\geq \frac{3}{2}\overline{\dim_{\mathrm{B}}} \Delta(F). \

Erd\H{o}s Semi-groups, arithmetic progressions and Szemer\'edi's theorem

In this paper we introduce and study a certain type of sub semi-group of $\mathbb{R}/\mathbb{Z}$ which turns out to be closely related to \sz's theorem on arithmetic progressions.Comment: 12 page

An Evolutionary Approach for Optimizing Hierarchical Multi-Agent System Organization

It has been widely recognized that the performance of a multi-agent system is highly affected by its organization. A large scale system may have billions of possible ways of organization, which makes it impractical to find an optimal choice of organization using exhaustive search methods. In this paper, we propose a genetic algorithm aided optimization scheme for designing hierarchical structures of multi-agent systems. We introduce a novel algorithm, called the hierarchical genetic algorithm, in which hierarchical crossover with a repair strategy and mutation of small perturbation are used. The phenotypic hierarchical structure space is translated to the genome-like array representation space, which makes the algorithm genetic-operator-literate. A case study with 10 scenarios of a hierarchical information retrieval model is provided. Our experiments have shown that competitive baseline structures which lead to the optimal organization in terms of utility can be found by the proposed algorithm during the evolutionary search. Compared with the traditional genetic operators, the newly introduced operators produced better organizations of higher utility more consistently in a variety of test cases. The proposed algorithm extends of the search processes of the state-of-the-art multi-agent organization design methodologies, and is more computationally efficient in a large search space

Assouad dimension of random processes

In this paper we study the Assouad dimension of graphs of certain L\'evy 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.Comment: 8 pages, 2 figures, to appear in Proceedings of the Edinburgh Mathematical Societ

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.Comment: Working Pape

Anomaly free cosmological perturbations with generalised holonomy correction in loop quantum cosmology

In the spatially flat case of loop quantum cosmology, the connection $\bar{k}$ is usually replaced by the holonomy $\frac{\sin(\bar{\mu}k)}{\bar{\mu}}$ in the effective theory. In this paper, instead of the $\bar{\mu}$ scheme, we use a generalised, undertermined function $g(\bar{k},\bar{p})$ to represent the holonomy and by using the approach of anomaly free constraint algebra we fix all the counter terms in the constraints and find the restriction on the form of $g(\bar{k},\bar{p})$, then we derive the gauge invariant equations of motion of the scalar, tensor and vector perturbations and study the inflationary power spectra with generalised holonomy correction.Comment: 26 pages,presentation improved and references adde