Acute triangles in the n-ball

1982 ◽  
Vol 19 (3) ◽  
pp. 712-715 ◽  
Author(s):  
Glen Richard Hall
Keyword(s):  
Differential Equation ◽  
Lebesgue Measure ◽  
Integral Formula ◽  
Acute Triangle

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝn, with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

Acute triangles in the n-ball

1982 ◽  
Vol 19 (03) ◽  
pp. 712-715 ◽  
Author(s):  
Glen Richard Hall
Keyword(s):  
Differential Equation ◽  
Lebesgue Measure ◽  
Integral Formula ◽  
Acute Triangle

Using Baddeley's [1] extension of Crofton's differential equation we derive an elementary integral formula for the probability that three randomly chosen points in the unit n-ball in ℝ n , with respect to Lebesgue measure, form an acute triangle. When the dimension is 2 this probability is 4/π2 − 1/8, while when the dimension is 3 it is 33/70.

scholarly journals Joint distribution of a Lévy process and its running supremum

2018 ◽  
Vol 55 (2) ◽  
pp. 488-512 ◽  
Author(s):  
Laure Coutin ◽  
Monique Pontier ◽  
Waly Ngom
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Diffusion Process ◽  
Lebesgue Measure ◽  
Joint Distribution ◽  
Random Variable ◽  
Jump Diffusion ◽  
Marginal Density ◽  
The Law

Abstract Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

NON-LIPSCHITZ STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY MULTI-PARAMETER BROWNIAN MOTIONS

2006 ◽  
Vol 06 (03) ◽  
pp. 329-340 ◽  
Author(s):  
XICHENG ZHANG ◽  
JINGYANG ZHU
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lebesgue Measure ◽  
Successive Approximation ◽  
Existence And Uniqueness ◽  
Uniqueness Of Solution ◽  
Gronwall’S Inequality

By proving an extension of nonlinear Bihari's inequality (including Gronwall's inequality) to multi-parameter and non-Lebesgue measure, in this paper we first prove by successive approximation the existence and uniqueness of solution of stochastic differential equation with non-Lipschitz coefficients and driven by multi-parameter Brownian motion. Then we study two discretizing schemes for this type of equation, and obtain their L2-convergence speeds.

scholarly journals An integral formulation for the global error of Lie Trotter splitting scheme

2019 ◽  
Vol 9 (1) ◽  
pp. 36-40
Author(s):  
Muaz Seydaoğlu
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Numerical Method ◽  
Integral Formula ◽  
Numerical Error ◽  
Integral Formulation ◽  
Global Error ◽  
Splitting Scheme ◽  
Time Stepping ◽  
Error Expansion

An ordinary differential equation (ODE) can be split into simpler sub equations and each  of the  sub equations is  solved subsequently by a numerical method. Such a procedure  involves splitting error and numerical error caused by the time stepping methods applied to sub equations.  The aim of the paper is to present  an integral formula for the global error expansion of a splitting  procedure combined with any  numerical  ODE solver.

Weak star limits of probability measures of special type

1999 ◽  
Vol 127 (1) ◽  
pp. 173-191
Author(s):  
IOANNIS PAPADOPERAKIS
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lebesgue Measure ◽  
Probability Measures ◽  
Compact Set ◽  
Winding Number ◽  
Compact Sets ◽  
Partial Differential ◽  
Accumulation Points ◽  
Weak Star

We study the weak star accumulation points of sequences of probability measures of the form [XKn(x) ρn(x)dσ(x)]/ [∫Kn ρn(t)dσ(t)], where ρ(x)>0 is continuous on Rκ, σ denotes Lebesgue measure in Rκ and the sequence of compact sets Kn⊂Rκ converges in the sense of Hausdorff towards a compact set K. The motivation of our study was given by a result relating interval averages with the winding number. Similar probability measures are considered in partial differential equation problems and we extend our study to this case.

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