2,748 research outputs found
Positive solutions of nonlinear problems involving the square root of the Laplacian
We consider nonlinear elliptic problems involving a nonlocal operator: the
square root of the Laplacian in a bounded domain with zero Dirichlet boundary
conditions. For positive solutions to problems with power nonlinearities, we
establish existence and regularity results, as well as a priori estimates of
Gidas-Spruck type. In addition, among other results, we prove a symmetry
theorem of Gidas-Ni-Nirenberg type
Numerical approximation of BSDEs using local polynomial drivers and branching processes
We propose a new numerical scheme for Backward Stochastic Differential
Equations based on branching processes. We approximate an arbitrary (Lipschitz)
driver by local polynomials and then use a Picard iteration scheme. Each step
of the Picard iteration can be solved by using a representation in terms of
branching diffusion systems, thus avoiding the need for a fine time
discretization. In contrast to the previous literature on the numerical
resolution of BSDEs based on branching processes, we prove the convergence of
our numerical scheme without limitation on the time horizon. Numerical
simulations are provided to illustrate the performance of the algorithm.Comment: 28 page
Branching diffusion representation of semilinear PDEs and Monte Carlo approximation
We provide a representation result of parabolic semi-linear PD-Es, with
polynomial nonlinearity, by branching diffusion processes. We extend the
classical representation for KPP equations, introduced by Skorokhod (1964),
Watanabe (1965) and McKean (1975), by allowing for polynomial nonlinearity in
the pair , where is the solution of the PDE with space gradient
. Similar to the previous literature, our result requires a non-explosion
condition which restrict to "small maturity" or "small nonlinearity" of the
PDE. Our main ingredient is the automatic differentiation technique as in Henry
Labordere, Tan and Touzi (2015), based on the Malliavin integration by parts,
which allows to account for the nonlinearities in the gradient. As a
consequence, the particles of our branching diffusion are marked by the nature
of the nonlinearity. This new representation has very important numerical
implications as it is suitable for Monte Carlo simulation. Indeed, this
provides the first numerical method for high dimensional nonlinear PDEs with
error estimate induced by the dimension-free Central limit theorem. The
complexity is also easily seen to be of the order of the squared dimension. The
final section of this paper illustrates the efficiency of the algorithm by some
high dimensional numerical experiments
THE QUADRATIC DYNATOMIC CURVES ARE SMOOTH AND IRREDUCIBLE
We reprove here the smoothness and the irreducibility of the quadratic dynatomic curves (c; z) 2 C2 j z is n-periodic for z2 + c . The smoothness is due to Douady-Hubbard. Our proof here is based on elementary calculations on the pushforwards of speci c quadratic di erentials, following Thurston and Epstein. This approach is a computational illustration of the power of the far more general transversality theory of Epstein. The irreducibility is due to Bousch, Morton and Lau-Schleicher with di erent ap- proaches. Our proof is inspired by the proof of Lau-Schleicher. We use elementary combinatorial properties of the kneading sequences instead of internal addresses
THE QUADRATIC DYNATOMIC CURVES ARE SMOOTH AND IRREDUCIBLE
We reprove here the smoothness and the irreducibility of the quadratic dynatomic curves (c; z) 2 C2 j z is n-periodic for z2 + c . The smoothness is due to Douady-Hubbard. Our proof here is based on elementary calculations on the pushforwards of speci c quadratic di erentials, following Thurston and Epstein. This approach is a computational illustration of the power of the far more general transversality theory of Epstein. The irreducibility is due to Bousch, Morton and Lau-Schleicher with di erent ap- proaches. Our proof is inspired by the proof of Lau-Schleicher. We use elementary combinatorial properties of the kneading sequences instead of internal addresses
Teichmüller spaces and holomorphic dynamics
One fundamental theorem in the theory of holomorphic dynamics is Thurston's topological characterization of postcritically finite rational maps. Its proof is a beautiful application of Teichmüller theory. In this chapter we provide a self-contained proof of a slightly generalized version of Thurston's theorem (the marked Thurston's theorem). We also mention some applications and related results, as well as the notion of deformation spaces of rational maps introduced by A. Epstein
Effects of isotonic drinks on academic performance for university students in Singapore
There are several factors that impact the academic performance of university students. The consumption of isotonic drinks is a factor that has not been explored enough to form a conclusive statement regarding its effects on academic performance. We will conduct an experiment in Singaporean universities to analyze the mean change in the scores of aptitude tests using a Paired Sample t-Test. We will also conduct a survey to compare how age, gender, study hours, sleep hours, smoking status, and exercise frequency correlate with measures of academic performance. Using regression analysis, we will determine which variables are associated with better levels of academic performance. Depending on the results, policies can be put in place to regulate isotonic drink consumption to ensure the health of the public is not compromised
Positive solutions of nonlinear problems involving the square root of the Laplacian
We consider nonlinear elliptic problems involving a nonlocal operator: the square root of the Laplacian in a bounded domain with zero Dirichlet boundary conditions. For positive solutions to problems with power nonlinearities, we establish existence and regularity results, as well as a priori estimates of Gidas-Spruck type. In addition, among other results, we prove a symmetry theorem of Gidas-Ni-Nirenberg type
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