1,548 research outputs found
PaL Diagrams: A Linear Diagram-Based Visual Language
No full textLinear diagrams have recently been shown to be
more effective than Euler diagrams when used
for set-based reasoning. However, unlike the
growing corpus of knowledge about formal aspects
of Euler and Venn diagrams, there has been no
formalisation of linear diagrams. To fill this
knowledge gap, we present and formalise Point
and Line (PaL) diagrams, an extension of simple
linear diagrams containing points, thus providing
a formal foundation for an effective visual
language.We prove that PaL diagrams are exactly
as expressive as monadic first-order logic with
equality, gaining, as a corollary, an equivalence
with the Euler diagram extension called spider
diagrams. The method of proof provides translations
between PaL diagrams and sentences of monadic
first-order logic
On Smarandache's form of the individual Fermat-Euler theorem
No full textIn the paper it is shown how a form of the classical FERMAT-EULER Theorem discovered by F. SMARANDACHE fits into the generalizations found by S.SCHWARZ, M.LASSAK and the author. Then we show how SMARANDACHE'S
algorithm can be used to effective computations of the so called group membership
Visualizing Sets: An Empirical Comparison of Diagram Types
No full textThere are a range of diagram types that can be used to visualize sets. However, there is a significant lack of insight into which is the most effective visualization. To address this knowledge gap, this paper empirically evaluates four diagram types: Venn diagrams, Euler diagrams with shading, Euler diagrams without shading, and the less well-known linear diagrams. By collecting performance data (time to complete tasks and error rate), through crowdsourcing, we establish that linear diagrams outperform the other three diagram types in terms of both task completion time and number of errors. Venn diagrams perform worst from both perspectives. Thus, we provide evidence that linear diagrams are the most effective of these four diagram types for representing sets
Almost sure exponential stability of the Euler–Maruyama approximations for stochastic functional differential equations
No full textBy the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately
Visualizing Sets with Linear Diagrams
No full textThis paper presents the first design principles that optimize the visualization of sets using linear diagrams. These principles are justified through empirical studies that evaluate the impact of graphical features on task performance. Linear diagrams represent sets using straight line segments, with line overlaps corresponding to set intersections. This work builds on recent empirical research which establishes that linear diagrams can be superior to prominent set visualization techniques, namely Euler and Venn diagrams. We address the problem of how to best visualize overlapping sets using linear diagrams. To solve the problem, we investigate which graphical features of linear diagrams significantly impact user task performance. To this end, we conducted seven crowd-sourced empirical studies involving a total of 1760 participants. These studies allowed us to identify the following design principles, which significantly aid task performance: use a minimal number of line segments, use guide-lines where overlaps start and end, and draw lines that are thin as opposed to thick bars. We also evaluated the following graphical properties which did not significantl impact task performance: colour, orientation, and set-order. The results are brought to life through a freely available software implementation that automatically draws linear diagrams with user-controlled graphical choices. An important consequence of our research is that users are now able to create effective visualizations of sets automatically, thus improving human-computer interaction
Assessing late-time singular behaviour in symmetry-plane models of 3D Euler flow
No full textMotivated by work on stagnation-point type exact solutions of the 3D Euler fluid equations by Gibbon [Gibbon et. al. Phys. D, 132, 497, (1999)] and the subsequent demonstration of finite-time blowup by Constantin [Constantin, Math. Res. Notices, 9, 455, (2000)] we introduce a one-parameter family of models of the 3D Euler equations on a 2D symmetry plane. These models provide a collection of blow-up scenarios which admit analytical solutions and are computationally inexpensive in comparison to the full 3D Euler equations. We take advantage of these features to examine the efficacy of novel methods which aid the assessment of finite-time blow-up in numerical simulations. The principal of these is the mapping to regular systems [Bustamante, Phys. D, 240, 1092, (2011)]; a bijective nonlinear mapping of time and the prognostic variables based on a Beale-Kato-Majda (BKM) type supremum norm regularity condition [Beale et. al. Commun. Math. Phys. 94, 61, (1984)]. We show a 3 order of magnitude increase of accuracy of the singularity time when employing the mapping with negligible additional computational expense. An investigation of the spectra of the primary field (vortex stretching rate) allows us to confirm a power law decrement of the analyticity-strip width with time in agreement with rigorous bounds bridging between the global spatial behaviour and BKM theorems [Bustamante & Brachet, Phys. Rev. E. 86, (2012)]
Artificial-Delay Adaptive Control for Under-actuated Euler-Lagrange Robotics
No full textArtificial-delay control is a method in which state and input measurements collected at an immediate past time instant (i.e. artificially delayed) are used to compensate the uncertain dynamics affecting the system at the current time. This work formulates an artificial-delay control method with adaptive gains in the presence of nonlinear (Euler-Lagrange) under-actuation. The appeal of studying Euler-Lagrange dynamics is to capture many robotics applications of practical interest, as demonstrated via stability and robustness analysis and via robotic ship and robotic aerial vehicle test cases.Accepted Author ManuscriptTeam Bart De Schutte
On the structure of minimizers of causal variational principles in the non-compact and equivariant settings
No full textWe derive the Euler-Lagrange equations for minimizers of causal variational
principles in the non-compact setting with constraints, possibly prescribing
symmetries. Considering first variations, we show that the minimizing measure is
supported on the intersection of a hyperplane with a level set of a function which
is homogeneous of degree two. Moreover, we perform second variations to obtain
that the compact operator representing the quadratic part of the action is positive
semi-definite. The key ingredient for the proof is a subtle adaptation of the Lagrange
multiplier method to variational principles on convex sets
Euler and Gravity
No full textIn this book chapter, the author explains Euler\u27s mathematical contributions to the theory of gravity
Adaptive single-stage control for uncertain nonholonomic Euler-Lagrange systems
No full textThis work introduces a new single-stage adaptive controller for Euler-Lagrange systems with nonholonomic constraints. The proposed mechanism provides a simpler design philosophy compared to double-stage mechanisms (that address kinematics and dynamics in two steps), while achieving analogous stability properties, i.e. stability of both original and internal states. Meanwhile, we do not require direct access to the internal states as required in state-of-the-art single-stage mechanisms. The proposed approach is studied via Lyapunov analysis, validated numerically on wheeled mobile robot dynamics and compared to a standard double-stage approach.Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Team Bart De Schutte
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