Humans have difficulties grasping the notion of exponential growth and often underestimate the accumulated final value, a phenomenon called exponential growth bias (EGB). During the COVID-19 pandemic, this tendency led to the inaccurate judgment of the virus spread, ultimately making safety measures seem less important. In prospect theory, loss aversion refers to the tendency of perceiving loss as more severe than a gain of the same magnitude is perceived as good. The question addressed was whether loss aversion through the valence of framing influences the judgments of exponential changes within the context of COVID-19. Furthermore, the association between EGB and the individual characteristic Need for Cognition (NFC) was investigated. Participants (n=129) were randomized into one of the two framing conditions (Recovery or Infection) and were presented with six EGB problems with different change rates and the six-item version of the Need for Cognition Scale. The results confirmed the existence of EGB at all growth rates (+5%, +15%, +25%), while the effect of EGB was mixed for exponential decline. The framing did not show a considerable effect on the accuracy of judgments. Simple linear regression analyses indicated that NFC moderates the effect of EGB at higher growth rates (i.e., at +15% and +25%). Overall, the results were more consistent and clearer for exponential growth than for exponential decline. The underestimation of exponential growth in the context of COVID-19 is quite alarming as it entails the risk of insufficient behavioral changes, which can lead to serious consequences for both the individual and society.

The ping-pong lemma was introduced by Klein in the late 1800s to show that certain subgroups of isometries of hyperbolic 3-space are free and remains one of very few tools that certify when a pair of group elements generate a free subgroup or semigroup. Quantitatively applying the ping-pong lemma to more general group actions on metric spaces requires a blend of understanding the large-scale global geometry of the underlying space with local combinatorial and dynamical behavior of the action. In the 1980s, Gromov publish a sequence of seminal works introducing several metric notions of non-positive curvature in group theory where he asked which finitely generated groups have uniform exponential growth. We give an overview of various developments of non-positive curvature in group theory and past results related to building free semigroups in the setting of non-positive curvature. We highlight joint work with Radhika Gupta and Kasia Jankiewicz and with Carolyn Abbott and Davide Spriano that extends these tools and techniques to show several groups with that act on cube complexes and many hierarchically hyperbolic groups have uniform exponential growth. / Mathematics

We study stochastic heat equations of the forms $[\partial_t u-\sL u]\d t\d x=\lambda\int_\R\sigma(u,h)\tilde{N}(\d t,\d x,\d h),$ and $[\partial_t u-\sL u]\d t\d x=\lambda\int_{\R^d}\sigma(u,h)N(\d t,\d x,\d h)$. Here, $u(0,x)=u_0(x)$ is a non-random initial function, $N$ a Poisson random measure with its intensity $\d t\d x\nu(\d h)$ and $\nu(\d h)$ a L\'vy measure; $\tilde$ is the compensated Poisson random measure and $\sL$ a generator of a L\'{e}vy process. The function $\sigma:\R\rightarrow\R$ is Lipschitz continuous and $\lambda>0$ the noise level. The above discontinuous noise driven equations are not always easy to handle. They are discontinuous analogues of the equation introduced in \cite{Foondun} and also more general than those considered in \cite{Saint}. We do not only compare the growth moments of the two equations with each other but also compare them with growth moments of the class of equations studied in \cite{Foondun}. Some of our results are significant generalisations of those given in \cite{Saint} while the rest are completely new. Second and first growth moments properties and estimates were obtained under some linear growth conditions on $\sigma$. We also consider $\sL:=-(-\Delta)^{\alpha/2}$, the generator of $\alpha$-stable processes and use some explicit bounds on its corresponding fractional heat kernel to obtain more precise results. We also show that when the solutions satisfy some non-linear growth conditions on $\sigma$, the solutions cease to exist for both compensated and non-compensated noise terms for different conditions on the initial function $u_0(x)$. We consider also fractional heat equations of the form $ \partial_t u(t,x)=-(-\Delta)^{\alpha/2}u(t,x)+\lambda\sigma(u(t,x)\dot{F}(t,x),\,\, \text{for}\,\, x\in\R^d,\,t>0,\,\alpha\in(1,2),$ where $\dot{F}$ denotes the Gaussian coloured noise. Under suitable assumptions, we show that the second moment $\E|u(t,x)|^2$ of the solution grows exponentially with time. In particular we give an affirmative answer to the open problem posed in \cite{Conus3}: given $u_0$ a positive function on a set of positive measure, does $\sup_{x\in\R^d}\E|u(t,x)|^2$ grow exponentially with time? Consequently we give the precise growth rate with respect to the parameter $\lambda$.

Neste trabalho estudamos existência, multiplicidade e não existência de soluções não triviais para o seguinte problema elíptico: { - \'DELTA\' = f(x, u), em \'OMEGA\' u = 0, sobre \'\\PARTIAL\' \'OMEGA\', onde \'OMEGA\' é um conjunto limitado de \'R POT. 2\' com fronteira suave e a função f possui crescimento exponencial. Para a existência de soluções são aplicados métodos variacionais combinados com as desigualdades de Trudinger-Moser. O resultado de não-existência é restrito ao caso de soluções radiais positivas e \'OMEGA\' = \'B IND.1\'(0). A prova usa técnicas de equações diferenciais ordinárias / In this work we study the existence, multiplicity and non-existence of non-trivial solutions to the following elliptic problem: { - \'DELTA\' u = f(x; u); in \'OMEGA\', ; u = 0; on \'\\PARTIAL\' \'OMEGA\' where \"OMEGA\' is a bounded and smooth domain in \'R POT. 2\' and f possesses exponential growth. The existence results are proved by using variational methods and the Trudinger- Moser inequalities. The non-existence result is restricted to the case of positive radial solutions and \'OMEGA\' = \'B IND. 1\'(0). The proof uses techniques of the theory of ordinary differential equations.

Neste trabalho estudamos existência, multiplicidade e não existência de soluções não triviais para o seguinte problema elíptico: { - \'DELTA\' = f(x, u), em \'OMEGA\' u = 0, sobre \'\\PARTIAL\' \'OMEGA\', onde \'OMEGA\' é um conjunto limitado de \'R POT. 2\' com fronteira suave e a função f possui crescimento exponencial. Para a existência de soluções são aplicados métodos variacionais combinados com as desigualdades de Trudinger-Moser. O resultado de não-existência é restrito ao caso de soluções radiais positivas e \'OMEGA\' = \'B IND.1\'(0). A prova usa técnicas de equações diferenciais ordinárias / In this work we study the existence, multiplicity and non-existence of non-trivial solutions to the following elliptic problem: { - \'DELTA\' u = f(x; u); in \'OMEGA\', ; u = 0; on \'\\PARTIAL\' \'OMEGA\' where \"OMEGA\' is a bounded and smooth domain in \'R POT. 2\' and f possesses exponential growth. The existence results are proved by using variational methods and the Trudinger- Moser inequalities. The non-existence result is restricted to the case of positive radial solutions and \'OMEGA\' = \'B IND. 1\'(0). The proof uses techniques of the theory of ordinary differential equations.

Background: As more individuals continuously become more conscious of the external influences of their decisions, integrating social and ethical criteria and perceived non-monetary value in their investment decisions, the interest in socially responsible investments (SRI) has escalated in the past decade. Reflecting this shift, the Swedish Pension Agency continuously increases the requirements and sustainability demands for the funds available in the premium pension selection. To investigate the underlying variables affecting the decision to invest socially responsibly, the authors of this thesis studied Swedish pension beneficiaries’ demographics, attitudes and beliefs.  Purpose: The purpose of this thesis is to examine the socio-demographic and psychological determinants of pension beneficiaries' and the influence of these variables on the willingness to pay for socially responsible investments. The study will additionally explore the tendency to underestimate exponential growth in one’s pension savings.  Method: The study is conducted by collecting primary data in the form of quantitative research through an online questionnaire. Based on previous research, six hypotheses are developed. This in order to investigate the relationship between willingness to pay for socially responsible investments and several socio-demographic and psychographic variables. Additionally, to examine Swedish pension beneficiaries’ tendency to underestimate exponential growth. The data collected is analysed through a multiple linear regression model and other descriptive statistics to examine if the hypotheses are rejected or not.  Conclusion: The majority of the subjects in the study are willing to pay for SRI. Gender significantly impacts the willingness to pay for SRI, as men demonstrate a lower willingness to pay than women. Furthermore, altruistic values, concern for one’s pension savings, concern for ESG-related issues (environmental, social and governance) and perceived consumer effectiveness proves to have a significant impact on the willingness to pay for SRI. Further, the sample demonstrated a definite tendency to underestimate exponential growth.

Der verhaltensökonomischen Literatur entsprechend behandeln die drei Kapitel dieser Dissertation unterschiedliche Aspekte des menschlichen Verhaltens, welches als "nicht-rational" zu bezeichnen ist. Jedes dieser Kapitel leistet einen Beitrag zum aktuellen Stand der Forschung auf dem Gebiet der Verhaltensökonomik mit Hilfe von entweder experimentellen, empirischen oder methodischen Ansätzen. Das erste Kapitel schlägt ein einfaches verhaltensökonomisches Modell vor und unterzieht dieses einer Reihe von experimentellen Tests. Das Modell erweitert die Literatur zur Fehlwahrnehmung von multiplikativen Wachstumsprozessen und hilft somit typische Fehlinvestitionen in der langen Frist zu erklären. Im Rahmen des zweiten Kapitels werden Daten einer Online-Kreditbörse genutzt, um empirisch zu untersuchen, ob sich private Investoren entsprechend den Vorhersagen der standardmäßigen ökonomischen Fachliteratur verhalten und einzig die erwartete Rendite berücksichtigen oder ob sie von anderen nicht-finanztechnischen Attributen eines Schuldners beeinflusst werden. Der Schwerpunkt der Analyse liegt dabei auf Geschlechterdiskriminierung im Rahmen dessen unterschiedliche Diskriminierungskonzepte getestet werden. Das dritte Kapitel wählt einen methodischen Ansatz und schlägt ein innovatives Experiment-Design vor, welches den empirisch gut dokumentierten Schwierigkeiten bzgl. der Angabe von subjektiven Wahrscheinlichkeiten von Teilnehmern an Umfragen und Laborexperimenten Rechnung trägt. Ein Binary-Choice-Ansatz eingebettet in ein adaptives Experiment-Design minimiert den Aufwand für die Befragten und ermöglich somit eine praktikable und effiziente Elizitierung der subjektiven Meinungen. / In the line with the literature on behavioral economics, the three chapters of this dissertation shed light on different aspects of human behavior that are at odds with rationality. Each chapter contributes to the existing behavioral economic research using either experimental, empirical, or methodological tools. First, by proposing and experimentally testing a simple behavioral model that extends the literature on the misperception of multiplicative growth processes, Chapter 1 aims to explain common money mistakes that people often make with long-term investments such as retirement savings plans. Second, in Chapter 2, real-life investment data of an online-lending platform are used to empirically investigate if private investors behave as the standard economic literature would predict and solely consider an investment’s expected return or if they also care about other non-financial attributes of a debtor. The focus of the analysis is on gender discrimination, thereby defining and econometrically testing different concepts of how investors discriminate between male and female borrowers. Third, Chapter 3 takes a methodological path and proposes a novel experimental design that accounts for the empirically well-documented difficulties that survey respondents typically have when asked to state subjective probabilities. A binary choice approach embedded in an adaptive experimental design helps to minimize effort of the respondents, thus allowing for a more practical belief elicitation in both the lab and the field.

In this work we study the existence of nontrivial weak solutions for some Hamiltonian elliptic systems in dimension two, involving a potential function and nonlinearities which possess maximal growth with respect to a critical curve (hyperbola). We consider four different cases. First, we study Hamiltonian systems in bounded domains with potential function identically zero. The second case deals with systems of equations on the whole space, the potential function is bounded from below for some positive constant and satisfies some integrability conditions, while the nonlinearities involve weight functions containing a singulatity at the origin. In the third case, we consider systems with coercivity potential functions and nonlinearities with weight functions which may have singularity at the origin or decay at infinity. In the last case, we study Hamiltonian systems, where the potential can be unbounded or can vanish at infinity. To establish the existence of solutions, we use variational methods combined with Trudinger-Moser type inequalities for Lorentz-Sobolev spaces and a finite-dimensional approximation. / Neste trabalho estudamos a existência de soluções fracas não triviais para sistemas hamiltonianos do tipo elíptico, em dimensão dois, envolvendo uma função potencial e não linearidades tendo crescimento exponencial máximo com respeito a uma curva (hipérbole) crítica. Consideramos quatro casos diferentes. Primeiramente estudamos sistemas de equações em domínios limitados com potencial nulo. No segundo caso, consideramos sistemas de equações em domínio ilimitado, sendo a função potencial limitada inferiormente por alguma constante positiva e satisfazendo algumas de integrabilidade, enquanto as não linearidades contêm funções-peso tendo uma singularidade na origem. A classe seguinte envolve potenciais coercivos e não linearidades com funções peso que podem ter singularidade na origem ou decaimento no infinito. O quarto caso é dedicado ao estudo de sistemas em que o potencial pode ser ilimitado ou decair a zero no infinito. Para estabelecer a existência de soluções, utilizamos métodos variacionais combinados com desigualdades do tipo Trudinger-Moser em espaços de Lorentz-Sobolev e a técnica de aproximação em dimensão finita.

Disturbances introduced in wall-bounded flows can grow and lead to transition from laminar to turbulent flow. In order to reduce losses or enhance mixing in energy systems, a fundamental understanding of the flow stability and transition mechanism is important. In the present thesis, the stability, transition mechanism and early turbulent evolution of wall-bounded flows are studied. The stability is investigated by means of linear stability equations and the transition mechanism and turbulence are studied using direct numerical simulations. Three base flows are considered, the Falkner-Skan boundary layer, boundary layers subjected to wall suction and the Blasius wall jet. The stability with respect to the exponential growth of waves and the algebraic growth of optimal streaks is studied for the Falkner-Skan boundary layer. For the algebraic growth, the optimal initial location, where the optimal disturbance is introduced in the boundary layer, is found to move downstream with decreased pressure gradient. A unified transition prediction method incorporating the influences of pressure gradient and free-stream turbulence is suggested. The algebraic growth of streaks in boundary layers subjected to wall suction is calculated. It is found that the spatial analysis gives larger optimal growth than temporal theory. Furthermore, it is found that the optimal growth is larger if the suction begins a distance downstream of the leading edge. Thresholds for transition of periodic and localized disturbances as well as the spreading of turbulent spots in the asymptotic suction boundary layer are investigated for Reynolds number Re=500, 800 and 1200 based on the displacement thickness and the free-stream velocity. It is found that the threshold amplitude scales like Re^-1.05 for transition initiated by streamwise vortices and random noise, like Re^-1.3 for oblique transition and like Re^-1.5 for the localized disturbance. The turbulent spot is found to take a bullet-shaped form that becomes more distinct and increases its spreading rate for higher Reynolds number. The Blasius wall jet is matched to the measured flow in an experimental wall-jet facility. Both the linear and nonlinear regime of introduced waves and streaks are investigated and compared to measurements. It is demonstrated that the streaks play an important role in the breakdown process where they suppress pairing and enhance breakdown to turbulence. Furthermore, statistics from the early turbulent regime are analyzed and reveal a reasonable self-similar behavior, which is most pronounced with inner scaling in the near-wall region. / QC 20101025

<p>Disturbances introduced in wall-bounded flows can grow and lead to transition from laminar to turbulent flow. In order to reduce losses or enhance mixing in energy systems, a fundamental understanding of the flow stability and transition mechanism is important. In the present thesis, the stability, transition mechanism and early turbulent evolution of wall-bounded flows are studied. The stability is investigated by means of linear stability equations and the transition mechanism and turbulence are studied using direct numerical simulations. Three base flows are considered, the Falkner-Skan boundary layer, boundary layers subjected to wall suction and the Blasius wall jet. The stability with respect to the exponential growth of waves and the algebraic growth of optimal streaks is studied for the Falkner-Skan boundary layer. For the algebraic growth, the optimal initial location, where the optimal disturbance is introduced in the boundary layer, is found to move downstream with decreased pressure gradient. A unified transition prediction method incorporating the influences of pressure gradient and free-stream turbulence is suggested. The algebraic growth of streaks in boundary layers subjected to wall suction is calculated. It is found that the spatial analysis gives larger optimal growth than temporal theory. Furthermore, it is found that the optimal growth is larger if the suction begins a distance downstream of the leading edge. Thresholds for transition of periodic and localized disturbances as well as the spreading of turbulent spots in the asymptotic suction boundary layer are investigated for Reynolds number Re=500, 800 and 1200 based on the displacement thickness and the free-stream velocity. It is found that the threshold amplitude scales like Re^-1.05 for transition initiated by streamwise vortices and random noise, like Re^-1.3 for oblique transition and like Re^-1.5 for the localized disturbance. The turbulent spot is found to take a bullet-shaped form that becomes more distinct and increases its spreading rate for higher Reynolds number. The Blasius wall jet is matched to the measured flow in an experimental wall-jet facility. Both the linear and nonlinear regime of introduced waves and streaks are investigated and compared to measurements. It is demonstrated that the streaks play an important role in the breakdown process where they suppress pairing and enhance breakdown to turbulence. Furthermore, statistics from the early turbulent regime are analyzed and reveal a reasonable self-similar behavior, which is most pronounced with inner scaling in the near-wall region.</p>