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2,178
result(s) for
"Parabola"
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Perception of the GeoGebra Software in learning the parabola on the cartesian plane for school students in Antofagasta, Chile, during 2024
This paper presents the main components of a qualitative research study, which includes three questions from semi-structured interviews and the conclusions drawn from the responses of five high school students from a school in the city of Antofagasta during the year 2024. Additionally, it presents a motivating example using ChatGPT for learning about the parabola. This research, in itself, promises to be a significant step forward in generating new studies on the use of GeoGebra and artificial intelligence.
Journal Article
Application of the Parabola Method in Nonconvex Optimization
We consider the Golden Section and Parabola Methods for solving univariate optimization problems. For multivariate problems, we use these methods as line search procedures in combination with well-known zero-order methods such as the coordinate descent method, the Hooke and Jeeves method, and the Rosenbrock method. A comprehensive numerical comparison of the obtained versions of zero-order methods is given in the present work. The set of test problems includes nonconvex functions with a large number of local and global optimum points. Zero-order methods combined with the Parabola method demonstrate high performance and quite frequently find the global optimum even for large problems (up to 100 variables).
Journal Article
Exponential integrators
In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in which exponential integrators are used.
Journal Article
Blood flow restriction exercise during microgravity exposure in parabolic flight
This case report evaluates whether it is possible to perform blood flow restriction (BFR) exercise during exposure to microgravity. The objectives were three‐fold: (1) to determine if a personalised tourniquet system (PTS) hardware technology performs nominally and enables BFR exercise in microgravity; (2) to determine if BFR augments the exercise stimulus in microgravity in a similar manner to its application on Earth; and (3) to evaluate tolerance and acceptability of performing BFR exercise and operating the PTS hardware in microgravity. Two participants performed resistance squat and deadlift exercises on a flywheel device (inertia of 0.01 kg m 2 ) with and without BFR during microgravity in parabolic flight onboard the National Research Council's Falcon 20 aircraft. Heart rate, perceived exertion and discomfort, and the participants’ tolerance and acceptability of performing BFR exercise in microgravity compared to exercise without BFR were measured. Performance of the PTS hardware technology was also evaluated. This case report demonstrates, for the first time, that it is possible to perform BFR exercise in microgravity in a manner that may augment the physiological stress of exercise in an acceptable and tolerable fashion. Importantly, the BFR hardware technology required to perform BFR exercise in an accurate, safe and effective manner performs nominally in microgravity. Future research should aim to conduct investigations during longer exposures to microgravity (i.e. during 3‐ to 6‐month missions on the International Space Station), providing a more comprehensive evaluation of the physiological stimulus provided and the tolerance and acceptability when performing BFR exercise in the space environment. What is the main observation in this case? It is possible to perform blood flow restriction (BFR) exercise in microgravity in a manner that may augment the physiological stress of exercise in an acceptable and tolerable fashion. The BFR hardware technology required to perform BFR exercise in an accurate, safe and effective manner performs nominally in microgravity. What insights does it reveal? This will support subsequent research exploring the use of BFR to optimise the current exercise countermeasure programme, and to provide a new modality of exercise for exploration‐class missions.
Journal Article
Solving high-dimensional partial differential equations using deep learning
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the “curse of dimensionality.” This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black–Scholes equation, the Hamilton–Jacobi–Bellman equation, and the Allen–Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their interrelationships.
Journal Article
Sharp asymptotic of solutions to some nonlocal parabolic equations
We show that if \\(u\\) solves the fractional parabolic equation \\((\\partial_t - \\Delta )^s u = Vu\\) in \\(B_5 \\times (-25, 0]\\) (\\(0
Paper
Brownian Gibbs property for Airy line ensembles
Consider a collection of
N
Brownian bridges
,
B
i
(−
N
)=
B
i
(
N
)=0, 1≤
i
≤
N
, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak limit as
N
→∞ of the collection of curves scaled so that the point (0,2
1/2
N
) is fixed and space is squeezed, horizontally by a factor of
N
2/3
and vertically by
N
1/3
. If a parabola is added to each of the curves of this scaling limit, an
x
-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and non-intersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers” and “outliers”. We formulate our results to treat these relatives as well.
Note that the law of the finite collection of Brownian bridges above has the property—called the Brownian Gibbs property—of being invariant under the following action. Select an index 1≤
k
≤
N
and erase
B
k
on a fixed time interval (
a
,
b
)⊆(−
N
,
N
); then replace this erased curve with a new curve on (
a
,
b
) according to the law of a Brownian bridge between the two existing endpoints (
a
,
B
k
(
a
)) and (
b
,
B
k
(
b
)), conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edge-scaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property.
An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the long-standing conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi-line Airy process, and readily yields several other interesting properties of this process.
Journal Article
Strong Characterization for the Airy Line Ensemble
In this paper we show that a Brownian Gibbsian line ensemble whose top curve approximates a parabola must be given by the parabolic Airy line ensemble. More specifically, we prove that if \\(\\boldsymbol{\\mathcal{L}} = (\\mathcal{L}_1, \\mathcal{L}_2, \\ldots )\\) is a line ensemble satisfying the Brownian Gibbs property, such that for any \\(\\varepsilon > 0\\) there exists a constant \\(\\mathfrak{K} (\\varepsilon) > 0\\) with $$\\mathbb{P} \\Big[ \\big| \\mathcal{L}_1 (t) + 2^{-1/2} t^2 \\big| \\le \\varepsilon t^2 + \\mathfrak{K} (\\varepsilon) \\Big] \\ge 1 - \\varepsilon, \\qquad \\text{for all \\(t \\in \\mathbb{R}\\)},$$ then \\(\\boldsymbol{\\mathcal{L}}\\) is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when \\(\\boldsymbol{\\mathcal{L}} (t) + 2^{-1/2} t^2\\) is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin-Hammond from 2014.
Paper
Bifurcation sets of families of reflections on surfaces in â,,3
We introduce a new affinely invariant structure on smooth surfaces in â,,3 by defining a family of reflections in all points of the surface. We show that the bifurcation set of this family has a special structure at ' points', which are not detected by the flat geometry of the surface. These points (without an associated structure on the surface) have also arisen in the study of the centre symmetry set; using our technique we are able to explain how the points are created and annihilated in a generic family of surfaces. We also present the bifurcation set in a global setting.
Journal Article