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By
Gordina, Maria; Laetsch, Thomas
8 Citations
We consider different subLaplacians on a subRiemannian manifold M. Namely, we compare different natural choices for such operators, and give conditions under which they coincide. One of these operators is a subLaplacian we constructed previously in Gordina and Laetsch (Trans. Amer. Math. Soc., 2015). This operator is canonical with respect to the horizontal Brownian motion; we are able to define this subLaplacian without some a priori choice of measure. The other operator is
$\operatorname {div}^{\omega } \operatorname {grad}_{\mathcal {H}}$
for some volume form ω on M. We illustrate our results by examples of three Lie groups equipped with a subRiemannian structure: SU(2), the Heisenberg group and the affine group.
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By
Chen, X. B.; Kostreva, M. M.
9 Citations
This paper introduces two new algorithms for finding initial feasible points from initial infeasible points for the recently developed normrelaxed method of feasible directions (MFD). Their global convergence is analyzed. The theoretical results show that both methods are globally convergent; one of them guarantees finding a feasible point in a finite number of steps. These two methods are very convenient to implement in the normrelaxed MFD. Numerical experiments are carried out to demonstrate their performance on some classical test problems and to compare them with the traditional method of phase I problems. The numerical results show that the methods proposed in this paper are more effective than the method of phase I problems in the normrelaxed MFD. Hence, they can be used for finding initial feasible points for other MFD algorithms and other nonlinear programming methods.
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By
Nehring, Thomas
1 Citations
In this article a priori estimates at the boundary for the second fundamental form of ndimensional convex hypersurfaces M with prescribed curvature quotient Sn (κM)/Sl (κM) in Riemannian manifolds are derived. A consequence of these estimates and other known results is an existence theorem for such hypersurfaces, which is a generalization of a recent result of Ivochkina and Tomi to the Riemannian case.
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By
Schlitt, H.
Zusammenfassung
Die statische Kennlinie des einfachen Schalters nach Bild XIV.1 hat die Form
$$
f(x) = h \cdot \operatorname{sgn} x
$$
, und für eine GAusssche Amplitudenverteilung
$$
p(x) = \frac{1}{{\sqrt {2\pi } \cdot {\sigma _x}}} \cdot {e^{  \frac{{{x^2}}}{{2{\sigma _{{x^2}}}}}}}
$$
des Eingangssignals erhält man mit
$$
K({\sigma _x}) = \frac{1}{{{\sigma _{{x^2}}}}} \cdot \int\limits_{  \infty }^{ + \infty } {x \cdot f(x)} \cdot p(x)dx
$$
den Ausdruck
$$
{\sigma _{{x^2}}} \cdot K({\sigma _x}) =  \frac{h}{{\sqrt {2\pi } \cdot {\sigma _x}}} \cdot \int\limits_{  \infty }^0 {x \cdot {e^{  \frac{{{x^2}}}{{2{\sigma _{{x^2}}}}}}}} dx + \frac{h}{{\sqrt {2\pi } \cdot {\sigma _x}}} \cdot \int\limits_0^\infty {x \cdot {e^{  \frac{{{x^2}}}{{2{\sigma _{{x^2}}}}}}}} dx = \frac{{2h}}{{\sqrt {2\pi } \cdot {\sigma _x}}} \cdot {\int\limits_0^\infty {x \cdot e} ^{  {x^2}/2{\sigma _{{x^2}}}}}dx
$$
.
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By
Novikov, A. A.; Kordzakhia, N. E.
7 Citations
In the context of dealing with financial risk management problems, it is desirable to have accurate bounds for option prices in situations when pricing formulae do not exist in the closed form. A unified approach for obtaining upper and lower bounds for Asiantype options is proposed in this paper. The bounds obtained are applicable to the continuous and discretetime frameworks for the case of timedependent interest rates. Numerical examples are provided to illustrate the accuracy of the bounds.
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By
Ernst, Oliver G.
29 Citations
Summary.
We introduce an algorithm for the efficient numerical solution of exterior boundary value problems for the Helmholtz equation. The problem is reformulated as an equivalent one on a bounded domain using an exact nonlocal boundary condition on a circular artificial boundary. An FFTbased fast Helmholtz solver is then derived for a finiteelement discretization on an annular domain. The exterior problem for domains of general shape are treated using an imbedding or capacitance matrix method. The imbedding is achieved in such a way that the resulting capacitance matrix has a favorable spectral distribution leading to mesh independent convergence rates when Krylov subspace methods are used to solve the capacitance matrix equation.
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By
Deroin, Bertrand; Kleptsyn, Victor; Navas, Andrés
3 Citations
We consider finitely generated groups of realanalytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due to É. Ghys, G. Hector and D. Sullivan.
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By
Petrykiewicz, Izabela
1 Citations
Let
$$k\in \mathbb {N}^*$$
be even. We consider two trigonometric series
$$ F_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k1}(n)}{n^{k+1}} \sin (2\pi n x)$$
and
$$G_k(x)= \sum _{n=1}^\infty \frac{\sigma _{k1}(n)}{n^{k+1}} \cos (2\pi n x),$$
where
$$\sigma _{k1}$$
is the divisor function. They converge on
$$\mathbb {R}$$
to continuous functions. In this paper, we examine the differentiability of
$$F_k$$
and
$$G_k$$
. These functions are related to Eisenstein series, and their (quasi)modular properties allow us to apply the method proposed by Itatsu in 1981 in the study of the Riemann series. We focus on the case
$$k=2$$
and we show that the sine series exhibits a different behaviour with respect to differentiability than the cosine series. We prove that the differentiability of
$$F_2$$
at an irrational x is related to the continued fraction expansion of x. We estimate the modulus of continuity of
$$F_2$$
. We formulate a conjecture concerning differentiability of
$$F_k$$
and
$$G_k$$
for any k even.
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By
Fournais, Søren; Helffer, Bernard
Using the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field
$$H_{C_3}$$
, describing the onset of superconductivity in type II superconductors. Furthermore, we prove that the definitions of this field corresponding to local minimizers and global minimizers coincide.
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