scholarly journals Limiting probability measures

2020 ◽  
Vol 12 ◽  
Author(s):  
Irfan Alam
Keyword(s):  
Asymptotic Behavior ◽  
Classical Result ◽  
Nonstandard Analysis ◽  
High Dimensional ◽  
Probability Measures ◽  
Dimensional Sphere ◽  
Spherical Means ◽  
Fixed Direction ◽  
Set Up ◽  
Gaussian Moment

The coordinates along any fixed direction(s), of points on the sphere $S^{n-1}(\sqrt{n})$, roughly follow a standard Gaussian distribution as $n$ approaches infinity. We revisit this classical result from a nonstandard analysis perspective, providing a new proof by working with hyperfinite dimensional spheres. We also set up a nonstandard theory for the asymptotic behavior of integrals over varying domains in general. We obtain a new proof of the Riemann--Lebesgue lemma as a by-product of this theory. We finally show that for any function $f \co \mathbb{R}^k \to \mathbb{R}$ with finite Gaussian moment of an order larger than one, its expectation is given by a Loeb integral integral over a hyperfinite dimensional sphere. Some useful inequalities between high-dimensional spherical means of $f$ and its Gaussian mean are obtained in order to complete the above proof.

scholarly journals ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH SPATIALLY DEPENDENT RANDOM JUMPS

2011 ◽  
Vol 13 (06) ◽  
pp. 1077-1093
Author(s):  
NITAY ARCUSIN ◽  
ROSS G. PINSKY
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Large Time ◽  
Dirichlet Boundary ◽  
Principal Eigenvalue ◽  
Probability Measures ◽  
Rate Of Decay ◽  
Non Local ◽  
Random Jumps ◽  
Explicit Constant

Let D ⊂ Rd be a bounded domain and let [Formula: see text] denote the space of probability measures on D. Consider a Brownian motion in D which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity γV to a new point, according to a distribution [Formula: see text]. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator -Lγ,μ, defined by [Formula: see text] with the Dirichlet boundary condition, where Cμ is the "μ-centering" operator defined by [Formula: see text] The principal eigenvalue, λ0(γ, μ), of Lγ, μ governs the exponential rate of decay of the probability of not exiting D for large time. We study the asymptotic behavior of λ0(γ, μ) as γ → ∞. In particular, if μ possesses a density in a neighborhood of the boundary, which we call μ, then [Formula: see text] If μ and all its derivatives up to order k - 1 vanish on the boundary, but the kth derivative does not vanish identically on the boundary, then λ0(γ, μ) behaves asymptotically like [Formula: see text], for an explicit constant ck.

Response of a Rotating Beam Subject to an Accelerating Force

1991 ◽  
Author(s):  
A. Argento ◽  
R. A. Scott
Keyword(s):  
Constant Velocity ◽  
Integral Transforms ◽  
Beam Model ◽  
Rotating Beam ◽  
Velocity Function ◽  
Distributed Load ◽  
Fixed Direction ◽  
Convolution Integrals ◽  
Rotating Timoshenko Beam ◽  
Set Up

Abstract A method is given by which the response of a rotating Timoshenko beam subjected to an accelerating fixed direction force can be determined. The beam model includes the gyroscopically induced displacement transverse to the direction of the load. The solution for pinned supports is set up in general form using multi-integral transforms and the inversion is expressed in terms of convolution integrals. These are numerically integrated for a uniformly distributed load having an exponentially varying velocity function. Results are presented for the displacement under the load’s center as a function of position. Comparisons are made between the responses to a constant velocity load and a load which accelerates up to the same velocity.

scholarly journals Generalization of the Winfree model to the high-dimensional sphere and its emergent dynamics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Hansol Park
Keyword(s):  
Influence Function ◽  
Equilibrium Solution ◽  
Phase Locking ◽  
Small Diameter ◽  
High Dimensional ◽  
Dimensional Sphere ◽  
Sphere Model ◽  
Dirac Delta ◽  
Delta Distribution ◽  
Winfree Model

<p style='text-indent:20px;'>We present a high-dimensional Winfree model in this paper. The Winfree model is a mathematical model for synchronization on the unit circle. We generalize this model compare to the high-dimensional sphere and we call it the Winfree sphere model. We restricted the support of the influence function in the neighborhood of the attraction point to a small diameter to mimic the influence function as the Dirac delta distribution. We can obtain several new conditions of the complete phase-locking states for the identical Winfree sphere model from restricting the support of the influence function. We also prove the complete oscillator death(COD) state from the exponential <inline-formula><tex-math id="M1">\begin{document}$ \ell^1 $\end{document}</tex-math></inline-formula>-stability and the existence of the equilibrium solution.</p>

scholarly journals Three simple scenarios for high-dimensional sphere packings

2021 ◽  
Vol 104 (6) ◽  
Author(s):  
Patrick Charbonneau ◽  
Peter K. Morse ◽  
Will Perkins ◽  
Francesco Zamponi
Keyword(s):  
High Dimensional ◽  
Dimensional Sphere ◽  
Sphere Packings

scholarly journals Asymptotic Theory Of Stochastic Choice Functionals For Prospects With Embedded Comotonic Probability Measures

2021 ◽  
Author(s):  
Godfrey Cadogan
Keyword(s):  
Probability Measures ◽  
Value Functions ◽  
Stochastic Choice ◽  
Class Theory ◽  
Binary Operations ◽  
Choice Sets ◽  
Empirical Probability ◽  
Designed Experiment ◽  
Set Up ◽  
Gain Loss

We introduce a monotone class theory of Prospect Theory's value functions, which shows that they can be replaced almost surely by a topological lifting comprised of a class of compact isomorphic maps that embed weakly co-monotonic probability measures, attached to state space, in outcome space. Thus, agents solve a signal extraction problem to obtain estimates of empirical probability weights for prospects under risk and uncertainty. By virtue of the topological lifting, we prove an almost sure isomorphism theorem between compact stochastic choice operators, and well defined outcomes which, under Brouwer-Schauder theory, guarantees fixed point convergence in convex choice sets. Along the way we introduce a risk operator in the Hoffman-Jorgensen class of lifting operators, and value function [averaging] operators with respect to Radon measure. In that set up, suitable binary operations on gain-loss space show that our risk operator is isometric for gains and skewed for losses. The point spectrum from this operator constitutes the range of admissible observations for loss aversion index in a well designed experiment.

scholarly journals Asymptotic Theory Of Stochastic Choice Functionals For Prospects With Embedded Comotonic Probability Measures

2021 ◽  
Author(s):  
Godfrey Cadogan
Keyword(s):  
Probability Measures ◽  
Value Functions ◽  
Stochastic Choice ◽  
Class Theory ◽  
Binary Operations ◽  
Choice Sets ◽  
Empirical Probability ◽  
Designed Experiment ◽  
Set Up ◽  
Gain Loss

We introduce a monotone class theory of Prospect Theory's value functions, which shows that they can be replaced almost surely by a topological lifting comprised of a class of compact isomorphic maps that embed weakly co-monotonic probability measures, attached to state space, in outcome space. Thus, agents solve a signal extraction problem to obtain estimates of empirical probability weights for prospects under risk and uncertainty. By virtue of the topological lifting, we prove an almost sure isomorphism theorem between compact stochastic choice operators, and well defined outcomes which, under Brouwer-Schauder theory, guarantees fixed point convergence in convex choice sets. Along the way we introduce a risk operator in the Hoffman-Jorgensen class of lifting operators, and value function [averaging] operators with respect to Radon measure. In that set up, suitable binary operations on gain-loss space show that our risk operator is isometric for gains and skewed for losses. The point spectrum from this operator constitutes the range of admissible observations for loss aversion index in a well designed experiment.

Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications

2019 ◽  
Vol 69 (3) ◽  
pp. 707-720
Author(s):  
Haroon M. Barakat ◽  
M. A. Abd Elgawad
Keyword(s):  
Asymptotic Behavior ◽  
Weak Convergence ◽  
Sufficient Conditions ◽  
Record Values ◽  
Distribution Functions ◽  
Gaussian Sequence ◽  
Stationary Gaussian Sequence ◽  
Lower Record ◽  
Lower Record Values ◽  
Set Up

Abstract In this paper, we study the limit distributions of upper and lower record values of a stationary Gaussian sequence under an equi-correlated set up. Moreover, the class of limit distribution functions (df’s) of the joint upper (and the lower) record values of a stationary Gaussian sequence is fully characterized. As an application of this result, the sufficient conditions for the weak convergence of the record quasi-range, record quasi-mid-range, record extremal quasi-quotient and record extremal quasi-product are obtained. Moreover, the classes of the non-degenerate limit df’s of these statistics are derived.

scholarly journals Massively Parallel Tree Search for High-Dimensional Sphere Decoders

2019 ◽  
Vol 30 (10) ◽  
pp. 2309-2325 ◽  
Author(s):  
Konstantinos Nikitopoulos ◽  
Georgios Georgis ◽  
Chathura Jayawardena ◽  
Daniil Chatzipanagiotis ◽  
Rahim Tafazolli
Keyword(s):  
Massively Parallel ◽  
High Dimensional ◽  
Dimensional Sphere ◽  
Tree Search ◽  
Parallel Tree Search

scholarly journals A NOVEL FAST FRACTAL IMAGE COMPRESSION METHOD BASED ON DISTANCE CLUSTERING IN HIGH DIMENSIONAL SPHERE SURFACE

Fractals ◽  
2017 ◽  
Vol 25 (04) ◽  
pp. 1740004 ◽  
Author(s):  
SHUAI LIU ◽  
ZHENG PAN ◽  
XIAOCHUN CHENG
Keyword(s):  
Image Compression ◽  
High Dimensional ◽  
Dimensional Sphere ◽  
Compression Method ◽  
Sphere Surface ◽  
Fractal Image Compression ◽  
Fractal Compression ◽  
Fractal Image ◽  
Speed Up ◽  
Encoding Method

Fractal encoding method becomes an effective image compression method because of its high compression ratio and short decompressing time. But one problem of known fractal compression method is its high computational complexity and consequent long compressing time. To address this issue, in this paper, distance clustering in high dimensional sphere surface is applied to speed up the fractal compression method. Firstly, as a preprocessing strategy, an image is divided into blocks, which are mapped on high dimensional sphere surface. Secondly, a novel image matching method is presented based on distance clustering on high dimensional sphere surface. Then, the correctness and effectiveness properties of the mentioned method are analyzed. Finally, experimental results validate the positive performance gain of the method.

scholarly journals New Characterizations of the Clifford Torus and the Great Sphere

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1076 ◽  
Author(s):  
Sun Mi Jung ◽  
Young Ho Kim ◽  
Jinhua Qian
Keyword(s):  
Euclidean Space ◽  
Finite Type ◽  
Three Dimensional ◽  
Gauss Map ◽  
Ruled Surfaces ◽  
Dimensional Sphere ◽  
Round Sphere ◽  
Clifford Torus ◽  
Great Sphere ◽  
Set Up

In studying spherical submanifolds as submanifolds of a round sphere, it is more relevant to consider the spherical Gauss map rather than the Gauss map of those defined by the oriented Grassmannian manifold induced from their ambient Euclidean space. In that sense, we study ruled surfaces in a three-dimensional sphere with finite-type and pointwise 1-type spherical Gauss map. Concerning integrability and geometry, we set up new characterizations of the Clifford torus and the great sphere of 3-sphere and construct new examples of spherical ruled surfaces in a three-dimensional sphere.

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