Vector-valued optimal Lipschitz extensions

Consider a bounded open set $U$ in $R^n$ and a Lipschitz function g from the boundary of $U$ to $R^m$. Does this function always have a canonical optimal Lipschitz extension to all of $U$? We propose a notion of optimal Lipschitz extension and address existence and uniqueness in some special cases. In the case $n=m=2$, we show that smooth solutions have two phases: in one they are conformal and in the other they are variants of infinity harmonic functions called infinity harmonic fans. We also prove existence and uniqueness for the extension problem on finite graphs.Comment: 24 pages, 10 figure

Discontinuous gradient constraints and the infinity Laplacian

Motivated by tug-of-war games and asymptotic analysis of certain variational problems, we consider a gradient constraint problem involving the infinity Laplace operator. We prove that this problem always has a solution that is unique if a certain regularity condition on the constraint is satisfied. If this regularity condition fails, then solutions obtained from game theory and $L^p$-approximation need not coincide

Metsien monet hyödyt ja taloudellinen arvottaminen

Tieteen tori: Tavoitteena metsien monet hyödyt201

Equivalence of Viscosity and Weak Solutions for the p(x)p(x)-Laplacian

We consider different notions of solutions to the $p(x)$-Laplace equation -\div(\abs{Du(x)}^{p(x)-2}Du(x))=0 with $1<p(x)<\infty$. We show by proving a comparison principle that viscosity supersolutions and $p(x)$-superharmonic functions of nonlinear potential theory coincide. This implies that weak and viscosity solutions are the same class of functions, and that viscosity solutions to Dirichlet problems are unique. As an application, we prove a Rad\'o type removability theorem

Russian loanwords in Skolt Saami

The distribution of the Russian loan vocabulary within the Saami languages centers on Skolt, Akkala, Kildin, and Ter Saami. In Skolt Saami, this loanword stratum forms the largest loanword stratum and contains more than 750 lexemes. Despite the significance of the loanword stratum, there has hardly been any actual analysis of the Russian loanwords in the Saami languages. This paper aims to fill this gap by presenting an overview of Russian vocabulary in Skolt Saami from a phonological, morphological, and semantic point of view. Besides analyzing the loanwords, approximately 150 new loan etymologies are discussed and some thirty new comparisons with Russian loanwords proposed in other Saami languages. It turns out that the Russian loan lexicon is relatively recent, and most if not all the words were borrowed from the Northwestern dialects of Russian between the beginning of the 17th century and 1920. Semantically the vocabulary is heterogenous. The most important semantic categories include religion, clothing, buildings and houses, diet, as well as administration and society

Sähköinen tilintarkastus

Opinnäytetyön tavoitteena on selvittää, miten taloushallinnon sähköistyminen on vai-kuttanut tilintarkastajien työhön, mitkä ovat sähköistyneen taloushallinnon edut ja haasteet tilintarkastustyön kannalta ja miten sähköistä taloushallintoa hyödyntävä tarkastuskohde eroaa perinteistä taloushallintoa käyttävästä. Teoreettinen viitekehys kuvaa tilintarkastuksen prosessin suunnittelusta toteutukseen ja päättämiseen sekä tilintarkastuksen sääntelyn. Lisäksi avataan sähköinen talous-hallinto käsitteenä. Tutkimus toteutettiin haastattelemalla tilintarkastajana työskente-leviä henkilöitä sähköisen Webrobol-kyselylomakkeen avulla. Täysin digitaalinen taloushallinto ei ole vielä yrityksissä käytössä, mutta suurin osa yrityksistä hyödyntää jo laajasti sähköistä taloushallintoa. Hyvin organisoitu ja toteu-tettu sähköinen taloushallinto nopeuttaa ja helpottaa tilintarkastusta. Sähköinen ym-päristö tuo omat haasteensa tilintarkastuksen suorittamiseen ja edellyttää tilintarkas-tajilta laajempaa tietoteknistä koulutusta. Koulutusta tulee uudistaa muokkaamalla tutkintojärjestelmää vastaamaan paremmin nykypäivän vaatimuksia.The objective of this thesis is to find out what the effects of digital financial man-agement to auditors job are, what the benefits and challenges of digital financial management are and how an enterprise that uses digital financial management differs from an enterprise which uses traditional financial management. The theoretical framework of this thesis describes auditing process from planning to execution and closing and law regulations concerning auditing. The term of digital financial management is also discussed. The research was executed by interviewing people who work as auditors. Interviews were made by using electronic Webropol-survey software. The results of this research indicate that enterprises do not use complete digital fi-nancial management systems yet. However, most of them are already utilizing widely digital financial management. Well organized and executed digital financial man-agement makes auditing faster and easier. Digital environment brings about its own challenges to auditing and requires that auditors must have wider knowledge on communications technology. Education and training must be enhanced by modifying the system of degrees to respond better the present day demands

THE TIME-VARYING SRI AND LUXURY GOODS

This thesis investigates the performance of Socially Responsible Investing (SRI). The thesis attempts to reveal whether it is still possible to reach abnormal returns by employing an SRI strategy based on com-panies’ nonfinancial ESG aspects. Previous academic literature displays mixed and varies evidence but also the connection between high ESG scores and positive abnormal returns have been announced by various studies. The thesis is inspired by the recent working paper of Bansal, Wu, and Yaron (2018) in which authors reveal results supporting the possibility to achieve abnormal returns. Authors find that SRI yields excess returns in wealth dependent manner leading to a conclusion that high responsible stocks behave in a fashion akin to luxury goods. Accordingly, the thesis targets to clarify if the performance of SRI is time-varying in a way suggested by Bansal et al. (2018). Utilizing ESG data from Thomson Reuters ASSET4 database, return data of S&P 500 firms from the Datastream database and factor data from Kenneth French data library this thesis evaluates the returns of SRI by applying the Carhart’s (1997) four-factor model. ASSET4 offers five annual ratings for all companies considering the Environmental, Social, Governance, Economical and Equally Weighted di-mensions. Within each category, the top, bottom and long-short portfolio referred to as the top-minus-bottom portfolio is formed based on annual ratings of the companies. The portfolios are rebalanced at the beginning of each year based on the ratings from the previous year. Portfolios are constructed based on the 10 % cut-off rate. The monthly performance of the portfolios is evaluated over the sample period of 16 years from January 2003 to December 2018. First, this thesis concludes that considering the whole sample period in question SRI generates significant-ly negative abnormal returns. The Equally Weighted top-minus-bottom portfolio yields a statistically significant alpha of -0.55 % per month. Secondly, to investigate the time variability of the returns the thesis employs 36-month rolling regression. Additionally, the sample period is divided into three subperi-ods including the times before, during and after the recession documented by the National Bureau of Economic Research. The thesis concludes that the performance of SRI is high time-varying and finds that the negative returns of the whole sample are driven by the significant underperformance during the post-recession era. Similar wealth dependent preference shifts in line with the findings of Bansal et al (2018) are not detected. In contrast, the evidence suggests that SRI significantly underperforms during the economically good post-recession era. Finally, the thesis examines the relationship between the per-formance of SRI and the consumption of luxury goods. The luxury goods consumption is measured by the U.S. personal consumption expenditures on jewelry and watches. The thesis concludes that a robust positive correlation between the growth of PCE on jewelry and watches and the 36-month rolling alpha of the Socially Responsible Investing is not discovered, instead the evidence rather suggests a negative correlation

Olosuhdeselvitys

Sosiaalilautakunnan selvitys eli olosuhdeselvitys on lakiin perustuva prosessi, jonka avulla on tarkoitus saada tietoa lapsen ja vanhempien olosuhteista. Selvityksen tarkoituksena on antaa tuomioistuimelle sellaista tietoa, mikä voidaan saada selville vain pyytämällä selvitys. Selvityspyynnön voi esittää ainoastaan tuomioistuin silloin, kun katsoo sen olevan tarpeen, jotta tuomioistuimella on käytössään kaikki tarvittava materiaali asian ratkaisemiseksi. Tämän opinnäytetyön tarkoituksena oli selvittää lakien ja kirjallisuuden avulla sosiaalilautakunnan laatimaa olosuhdeselvitystä. Olosuhdeselvitys on osa oikeudenkäyntimateriaalia, kun ratkaistaan lasta koskevaa huoltoriitaa. Työssä on tutkittu lapsen etua ja lapsen oikeuksia. Lapsen etu on ensisijainen päätöksentekoon vaikuttava seikka, joka tulee huomioida, kun tehdään lasta koskevia päätöksiä. Lapsen edun selvittämistä varten olosuhdeselvitys on usein tarpeen. Kirjallisuuden avulla on selvitetty, miten lapsen mielipiteet ja toiveet saadaan selville. Lasta ei kuulla tuomioistuimessa henkilökohtaisesti, mutta laki antaa kuulemiseen mahdollisuuden. Lapsen kuuleminen tapahtuu pääsääntöisesti olosuhdeselvitystyön yhteydessä. Opinnäytetyö toteutettiin tutkimalla oikeuskirjallisuutta. Tutkimusmenetelmänä työssä oli oikeusdogmatiikka eli lainoppi. Oikeusdogmatiikka selvittää miten lain mukaan toimitaan, joten tutkin lakeja ja asetuksia sekä lain esitöitä. Tutkin myös satunnaisotannalla valikoituja olosuhdeselvityksiä, niiden rakennetta ja sisältöä. Haastattelin Vantaan kaupungin sosiaalityöntekijöitä sekä Vantaan käräjäoikeuden tuomareita, jotta sain empiiristä näkökulmaa tutkimukseeni.A Statement by the Social Board is a process which is based on the law and with its help information about the conditions of a child and parents can be obtained. The purpose of a report is to give a court the information which is not found available without such a report. Only the court can make a request for such a report and have need of it. When considering the court will use all the necessary material to solve the matter. The purpose of this thesis was to clarify (with the help of laws and literature) the conditional report drawn up by the Social Board Statement. The statement will be a part of the trial material when a maintenance dispute concerning the child is solved. In this work the interests of the child and the child's rights have been studied. The advantage to the child is a primary point which affects decision-making and to which attention has to be paid when decisions concerning the child are made. For the clarifying of the interest of the child the conditional report is often necessary. With the help of the literature it has been clarified how the child's opinions and wishes are discovered. The child is not personally heard in the court but the law gives the opportunity for the hearing. The hearing of the child takes place in connection with the report work of the social welfare board as a rule. I interviewed the social workers of the town of Vantaa and the judges of the district court of Vantaa so that I got an empirical point of view to my study