Harmonic Shears and Numerical Conformal Mappings

In this article we introduce a numerical algorithm for finding harmonic mappings by using the shear construction introduced by Clunie and Sheil-Small in 1984. The MATLAB implementation of the algorithm is based on the numerical conformal mapping package, the Schwarz-Christoffel toolbox, by T. Driscoll. Several numerical examples are given. In addition, we discuss briefly the minimal surfaces associated with harmonic mappings and give a numerical example of minimal surfaces.Comment: 15 pages, 6 figure

Numeeriset konformikuvaukset ja kapasiteettilaskenta

Tämän diplomityön tarkoituksena on perehdyttää numeerisiin konformikuvauksiin ja nelikulmion konformisen modulin laskentaan. Konformikuvauksia tutkitaan sekä niiden kytköksistä fysikaalisiin sovellutuksiin, että matemaattisesta tärkeydestä. Laskennallista etua saavutetaan kuvaamalla konformisesti yhdesti yhtenäinen alue joko yksikkökiekolle, ylemmälle puolitasolle tai suorakaiteelle. Toisaalta kondensaattorin kapasitanssi voidaan karakterisoida nelikulmion konformisen modulin avulla. Tässä diplomityössä annamme kaksi tapaa modulin laskemiseen:. 1. Schwarz-Christoffelin kuvaukset. 2. Elementtimenetelmät. Saatuja tuloksia verrataan sekä keskenään, että tunnettuihin kirjallisuuden analyyttisiin ja referenssiarvoihin.The purpose of this thesis is to give an introduction to numerical conformal mappings and the computation of a conformal modulus of a quadrilateral. The theory of conformal mapping is studied because of its connections to physical applications and for its significance in mathematics. The computational advantage can be gain by conformally mapping a simply connected domain onto the unit disk, the upper half plane, or a rectangle. On the other hand the capacitance of a condenser can be characterized by the conformal modulus of a quadrilateral. In this thesis, we give two ways to compute the modulus, namely:. o Schwarz-Christoffel mappings. o finite element methods. The obtained results are compared to each other and to known analytic and reference values from literatures

Identifying shifts in multi-modal travel patterns during special events using mobile data: Celebrating Vappu in Helsinki

Large urban special events significantly contribute to a city's vibrancy and economic growth but concurrently impose challenges on transportation systems due to alterations in mobility patterns. This study aims to shed light on mobility patterns by utilizing a unique, comprehensive dataset collected from the Helsinki public transport mobile application and Bluetooth beacons. Earlier methods, relying on mobile phone records or focusing on single traffic modes, do not fully grasp the intricacies of travel behavior during such events. We focus on the Vappu festivities (May 1st) in the Helsinki Metropolitan Area, a national holiday characterized by mass gatherings and outdoor activities. We examine and compare multi-modal mobility patterns during the event with those during typical non-working days in May 2022. Through this case study, we find that people tend to favor public transport over private cars and are prepared to walk longer distances to participate in the event. The study underscores the value of using comprehensive multi-modal data to better understand and manage transportation during large-scale events.Comment: 6 pages, 12 figures, Submitted to ITSC202

Harmonic Shears of Slit and Polygonal Mappings

In this paper, we study harmonic mappings by using the shear construction, introduced by Clunie and Sheil-Small in 1984. We consider two classes of conformal mappings, each of which maps the unit disk D univalently onto a domain which is convex in the horizontal direction, and shear these mappings with suitable dilatations \omega. Mappings of the first class map the unit disk D onto four-slit domains and mappings of the second class take D onto regular n-gons. In addition, we discuss the minimal surfaces associated with such harmonic mappings. Furthermore, illustrations of mappings and associated minimal surfaces are given by using Mathematica.Comment: 17 pages, 6 figures, 2 table

Conjugate Function Method for Numerical Conformal Mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.Comment: 23 pages, 15 figures, 5 table

Graph of words for document classification

This FYP project is about the implementations and experimental studies of a novel framework for large data classifications of textual documents. Under this new framework, documents are first transferred from sentences into graph-of-words, so the original classification problem is then considered as graph classification and advanced representation learning (RL) model subgraph2vec can be applied. However, as shared by many other RL based methods, poor efficiency problem is serious because in general NLP dataset has a huge vocabulary. Thus, this project proposes hash embeddings version of subgraph2vec to significantly reduce required memory for training phase, make system become efficient without harming the quality of resultant representations. The approach is evaluated in terms of time, required memory, accuracy and f1 score with benchmark datasets on 3 domains (the first 2 are graph classification task and the last task is document classification). Through experiments, proposed approach outperforms other RL based methods and achieves comparable results with state-of-the-art method. Finally, the FYP project introduces semi supervised version of the method and observes the significant increases in sentimental analysis task.Bachelor of Engineerin

Histone Deacetylase Inhibitors: Novel advances in Cancer Therapy

Nowadays, it became evident that the onset of cancer and its progression may not occur only due to genetic mutations but also because of changes in the patterns of epigenetic modifications. In contrast to genetic mutations, which are almost impossible to reverse, epigenetic changes are potentially reversible. This implies that they are susceptible to pharmacological interventions. Therefore, a lot of work in recent years has focussed on the development of small molecule enzyme inhibitors like DNA-methyltransferase inhibitors or inhibitors of histone-modifying enzymes. Today, several epigenetic drugs are already approved by the FDA for cancer treatment and around ten histone deacetylase (HDAC) inhibitors are in clinical development. The purposes of this thesis are • Understand about mechanisms of action of HDAC inhibitors • Drugs are already approved by the FDA for cancer treatment and histone deacetylase (HDAC) inhibitors are in clinical developmentegységes, osztatlanáltalános orvosango