Some new formulas for π\pi

We show how to find series expansions for $\pi$ of the form $\pi=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$, $a=(-4)^k$, for any $k$, and give explicit examples for such expansions for small values of $m$, $p$ and $a$.Comment: 28 pages, LaTeX; some important references adde

Raw or Cooked? Object Detection on RAW Images

Images fed to a deep neural network have in general undergone several handcrafted image signal processing (ISP) operations, all of which have been optimized to produce visually pleasing images. In this work, we investigate the hypothesis that the intermediate representation of visually pleasing images is sub-optimal for downstream computer vision tasks compared to the RAW image representation. We suggest that the operations of the ISP instead should be optimized towards the end task, by learning the parameters of the operations jointly during training. We extend previous works on this topic and propose a new learnable operation that enables an object detector to achieve superior performance when compared to both previous works and traditional RGB images. In experiments on the open PASCALRAW dataset, we empirically confirm our hypothesis

Learning Future Object Prediction with a Spatiotemporal Detection Transformer

We explore future object prediction -- a challenging problem where all objects visible in a future video frame are to be predicted. We propose to tackle this problem end-to-end by training a detection transformer to directly output future objects. In order to make accurate predictions about the future, it is necessary to capture the dynamics in the scene, both of other objects and of the ego-camera. We extend existing detection transformers in two ways to capture the scene dynamics. First, we experiment with three different mechanisms that enable the model to spatiotemporally process multiple frames. Second, we feed ego-motion information to the model via cross-attention. We show that both of these cues substantially improve future object prediction performance. Our final approach learns to capture the dynamics and make predictions on par with an oracle for 100 ms prediction horizons, and outperform baselines for longer prediction horizons.Comment: 15 pages, 6 figure

You can have your ensemble and run it too -- Deep Ensembles Spread Over Time

Ensembles of independently trained deep neural networks yield uncertainty estimates that rival Bayesian networks in performance. They also offer sizable improvements in terms of predictive performance over single models. However, deep ensembles are not commonly used in environments with limited computational budget -- such as autonomous driving -- since the complexity grows linearly with the number of ensemble members. An important observation that can be made for robotics applications, such as autonomous driving, is that data is typically sequential. For instance, when an object is to be recognized, an autonomous vehicle typically observes a sequence of images, rather than a single image. This raises the question, could the deep ensemble be spread over time? In this work, we propose and analyze Deep Ensembles Spread Over Time (DESOT). The idea is to apply only a single ensemble member to each data point in the sequence, and fuse the predictions over a sequence of data points. We implement and experiment with DESOT for traffic sign classification, where sequences of tracked image patches are to be classified. We find that DESOT obtains the benefits of deep ensembles, in terms of predictive and uncertainty estimation performance, while avoiding the added computational cost. Moreover, DESOT is simple to implement and does not require sequences during training. Finally, we find that DESOT, like deep ensembles, outperform single models for out-of-distribution detection

Towards trustworthy multi-modal motion prediction: Holistic evaluation and interpretability of outputs

Predicting the motion of other road agents enables autonomous vehicles to perform safe and efficient path planning. This task is very complex, as the behaviour of road agents depends on many factors and the number of possible future trajectories can be considerable (multi-modal). Most prior approaches proposed to address multi-modal motion prediction are based on complex machine learning systems that have limited interpretability. Moreover, the metrics used in current benchmarks do not evaluate all aspects of the problem, such as the diversity and admissibility of the output. In this work, we aim to advance towards the design of trustworthy motion prediction systems, based on some of the requirements for the design of Trustworthy Artificial Intelligence. We focus on evaluation criteria, robustness, and interpretability of outputs. First, we comprehensively analyse the evaluation metrics, identify the main gaps of current benchmarks, and propose a new holistic evaluation framework. We then introduce a method for the assessment of spatial and temporal robustness by simulating noise in the perception system. To enhance the interpretability of the outputs and generate more balanced results in the proposed evaluation framework, we propose an intent prediction layer that can be attached to multi-modal motion prediction models. The effectiveness of this approach is assessed through a survey that explores different elements in the visualization of the multi-modal trajectories and intentions. The proposed approach and findings make a significant contribution to the development of trustworthy motion prediction systems for autonomous vehicles, advancing the field towards greater safety and reliability.Comment: 16 pages, 7 figures, 6 table

Repeated and Systematic Intimate Partner Violence in Rural Areas in Sweden

Violence against women lacks geographical boundaries, although research demonstrates higher rates of such violence in rural areas compared to urban areas. The repeated and systematic intimate partner violence (IPV) is especially problematic in isolated areas. This study aims to investigate how repeated and systematic IPV, was handled by the criminal justice system in rural areas in Sweden and how risk and victim vulnerability factors were related to recidivism in this longitudinal prospective study. The sample consisted of alleged perpetrators of repeated and systematic IPV who had been either reported, charged, or convicted of repeated and systematic IPV defined according to the Swedish Law Gross Violation of a Woman’s Integrity targeting such violence, in two rural Swedish police districts during 2011–2014 (N = 258). Results demonstrated that 30% of IPV perpetrators were charged with the Gross violation offense and 5% were charged for other IPV-related offenses. The conviction for the Gross violation offense was 11% and 24% for other IPV-related offenses. 56% were not charged or convicted of any IPV-related offenses. Perpetrators convicted of the Gross Violation offense were more likely to receive longer prison sentences than perpetrators convicted of other IPV-related offenses. Victim cooperation in the police investigation increased the likelihood for prosecution with 7.3 times and for a conviction with 6.1 times. In terms of recidivism 24% engaged in IPV towards the same victim and another 27% recidivated into general criminality. Recidivists had higher summary risk ratings and more individual risk factors than non-recidivists, such as general criminality, employment problems and mental health problems, and victim vulnerability factors including personal problems. To reduce re-victimization, risk and vulnerability factors and supporting victims to cooperate in the police investigation should be considered when forming risk management strategies to protect victims of repeated and systematic IPV in such rural areas

Qualitative Aspects of Nonlinear Parabolic Partial Differential Equations and Systems

This thesis contains four papers about some aspects of nonlinear parabolic equations and systems. Paper 1. A note on quenching for parabolic equations with dynamic boundary conditions We present a quenching result for semilinear parabolic equations with dynamic boundary conditions in bounded domains with a smooth boundary. Paper 2. On global existence for semilinear parabolic systems We present some results on global existence of classical solutions of certain semilinear parabolic systems with homogeneous Dirichlet boundary conditions in bounded domains with a smooth boundary, relaxing the usual monotonicity assumptions on the nonlinearities. Paper 3. Best constants for Gagliardo-Nirenberg inequalities on the real line We use a variational approach to find the best constants for certain Gagliardo-Nirenberg inequalities on the real line. To show the existence of a minimizer, we use the method of concentration-compactness. Paper 4. On the time evolution of extrema of solutions to nonlinear parabolic equations in unbounded domains We obtain a result about the time evolution of extrema that can be applied to the study of classical solutions to parabolic equations in unbounded domains with a smooth boundary

Identifying risk for recidivism among partner violent men reported to the Swedish police

Intimate partner violence (IPV) against women is a global public health issue, where every third woman has experienced such violence. Moreover, IPV recidivism rates are generally high. These figures indicate that the police need a better understanding of the risk factors related to those perpetrators who pose the highest risk of recidivating in IPV. To this end, research has found that IPV perpetrators who are violent towards their partner as well as others (referred to as the antisocial subtype) display more risk factors for IPV than those perpetrators who are violent only against their partner (referred to as the family-only subtype). However, there are still uncertainties whether these two subtypes differ in terms of characteristics related to recidivism (i.e., risk profile) and actual recidivism. Thus, this thesis aimed to examine differences in risk profiles and recidivism rates between the antisocial perpetrators and the family-only perpetrators. This thesis was based on a systematic literature review and three empirical studies. The empirical studies were based on data collected from the Swedish police and consisted of IPV risk assessments. The risk assessments were performed by the police using the Brief Spousal Assault Form for the Evaluation of Risk (B-SAFER). These empirical studies relied on a sample of 657 male perpetrators who had been reported to the police and subjected to a violence risk assessment for allegedly perpetrating IPV against a female partner. The results demonstrated that categorizing partner violent men as either antisocial or family-only can help identify the perpetrators most at risk to recidivate in IPV. As such, the antisocial perpetrators displayed a greater diversity as well as degree of risk factors for IPV, and were more likely to recidivate in IPV, despite legal interventions from the police. In contrast, the family-only perpetrators presented with fewer risk factors, were characterized as socially well-adjusted outside of the relationship, and less likely to recidivate in IPV. However, several family-only perpetrators recidivated in IPV, meaning that such perpetrators should not routinely be dismissed as low-risk perpetrators. In conclusion, the results of this thesis can be used to improve the ability of those assigned to assess risk for future IPV to identify those perpetrators most at risk to recidivate. In turn, this could enable a more informed and adequate response aiming to prevent, or at best reduce, this risk. Mäns partnervåld mot kvinnor är ett globalt folkhälsoproblem där uppskattningsvis var tredje kvinna någon gång utsatts för sådant våld. Dessutom är återfall i partnervåld vanligt förekommande. Detta indikerar på att polisen behöver mer kunskap om vilka förövare som utgör den högsta risken för att återfalla i nytt partnervåld. Tidigare forskning visar på att de partnervåldsförövare som är generellt våldsamma (d.v.s. våldsamma både mot sin partner och mot andra) uppvisar fler riskfaktorer för återfall i partnervåld än de partnervåldsförövare som är våldsamma enbart mot sin partner. Dock finns det otillräcklig kunskap huruvida dessa förövare uppvisar olika riskprofiler (d.v.s. riskfaktorer och karaktäristika relaterade till återfall i partnervåld) och återfallsbenägenhet. Syftet med föreliggande avhandling var således att undersöka skillnader i riskprofiler och återfall i partnervåld mellan generellt våldsamma och icke generellt våldsamma partnervåldsförövare. Föreliggande avhandling utgjordes dels av en systematisk litteraturöversikt, dels av tre empiriska vetenskapliga studier. De empiriska studierna baserades på riskbedömningar utförda av polisen på misstänkta partnervåldsförövare. Dessa riskbedömningar utfördes med hjälp av riskbedömningsinstrumentet Brief Spousal Assault Form for the Evaluation of Risk (B-SAFER). Urvalet för dessa empiriska studier bestod av 657 män som anmälts till polisen för partnervåld mot en kvinna. Resultaten påvisade att generellt våldsamma och icke generellt våldsamma partnervåldsförövare uppvisade olika riskprofiler och återfallsbenägenhet. Generellt våldsamma partnervåldsförövare uppvisade fler riskfaktorer för återfall i partnervåld och var mer benägna att återfalla i sådant våld. Icke generellt våldsamma partnervåldsförövare uppvisade färre riskfaktorer och återföll i mindre utsträckning. Dessa förövare var även mer socialt välanpassade utanför relationen. Dock fanns det flera icke generellt våldsamma partnervåldsförövare som återföll i partnervåld, vilket innebär att dessa förövare inte rutinmässigt bör betraktas som en låg-risk-grupp för återfall. Sammanfattningsvis kan möjligheten att identifiera de förövare som löper högst risk att återfalla i nytt partnervåld förbättras om hänsyn tas till huruvida dessa förövare är generellt eller icke generellt våldsamma. Detta kan i slutändan bidra till ökade möjligheter för polisen och övriga samhället att reducera de höga återfallssiffrorna och därmed förhindra partnervåld mot kvinnor.

On global existence for semilinear parabolic systems

We present some results on global existence of classical solutions of certain semilinear parabolic systems with homogeneous Dirichlet boundary conditions in bounded domains with a smooth boundary, relaxing the usual monotonicity assumptions on the nonlinearities