Finding the coefficients in the new representation of the solution of the Riemann–Hilbert problem using the Lauricella function

The solution of the Riemann–Hilbert problem for an analytic function in a canonical domain for the case in which the data of the problem is piecewise constant can be expressed as a Christoffel–Schwartz integral. In this paper, we present an explicit expression for the parameters of this integral obtained by using a Jacobi-type formula for the Lauricella generalized hypergeometric function F _D ^(N). The results can be applied to a number of problems, including those in plasma physics and the mechanics of deformed solids.     

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

Note on the Norm Involved in the Definition of the ε-spectrum

There exist many definitions of the ε-spectrum. Unlike the spectrum, the ε-spectrum definitions depend on the choice of the norm. We propose to study the conditions on the norm that make all these definitions equivalent. The spectral radius of a related matrix is a useful tool in the eigenvalues perturbation analysis. The study of its connection with the ε-spectrum provides a new definition of the ε-spectrum. In most articles dealing with this subject, the chosen norm is the spectral one. In this paper, we intend to justify this choice.     

Why do the microstructures of the main pulse and the interpulse of the pulsar in the Crab nebula differ so dramatically?

Theoretical and Mathematical Physics - Pulsars are magnetized neutron stars. They are not resolved by modern radio telescopes and are studied only by radiation coming from the regions of the...     

On the sphericity of the subgroup PSL2(ℝ) in the group of diffeomorphisms of the circle

It is shown that the group PSL₂(?) is a spherical subgroup in the group of C³-diffeomorphisms of the circle.     

On the non-round points of the boundary of the numerical range ∗

Let A be a bounded linear operator acting on a Hilbert space. It is well known (Donoghue, 1957) that comer points of the numerical range W(A) are eigenvalues of A. Recently (1995), this result was generalized by Hiibner who showed that points of infinite curvature on the boundary of W(A) lie in the spectrum of A. Hübner also conjectured that all such points are either corner points or lie in the essential spectrum of A. In this paper, we give a short proof of this conjecture.     

On the Uniqueness of the ADS Spacetime

The uniqueness of the ADS spacetime among all static vacuum spacetimes with the same conformal infinity is proved for dimension n≤7. For dimension n 〉 7, the same result is established under the spin assumption.     

On the order of the operators in the Douglas–Rachford algorithm

The Douglas–Rachford algorithm is a popular method for finding zeros of sums of monotone operators. By its definition, the Douglas–Rachford operator is not symmetric with respect to the order of the two operators. In this paper we provide a systematic study of the two possible Douglas–Rachford operators. We show that the reflectors of the underlying operators act as bijections between the fixed points sets of the two Douglas–Rachford operators. Some elegant formulae arise under additional assumptions. Various examples illustrate our results.     

On the behavior of the solution of the dissipative Camassa–Holm equation with the arbitrary dispersion coefficient

In this paper, we consider the dissipative Camassa–Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated.     

Well-posedness for the system of the Hamilton–Jacobi and the continuity equations

Let H (t, x, p) be a Hamiltonian function that is convex in p. Let the associated Lagrangian satisfy the nonstandard minorization condition where m > 0, ω > 0, and C ≥ 0 are constants. Under some additional conditions, we prove that the associated value function is the unique viscosity solution of S _t + H(t, x, ∇S) = 0 in , without any conditions at infinity on the solution. Here ωT < π/2. To the Hamilton–Jacobi equation corresponding to the classical action integrand in mechanics, we adjoin the continuity equation and establish the existence and uniqueness of a viscosity–measure solution (S, ρ) of

On the solution of the Gellerstedt problem for the Lavrent’ev-Bitsadze equation

We study the solvability of the Gellerstedt problem for the Lavrent??ev-Bitsadze equation under an inhomogeneous boundary condition on the half-circle of the ellipticity domain of the equation, homogeneous boundary conditions on external, internal, and parallel side characteristics of the hyperbolicity domain of the equation, and the transmission conditions on the type change line of the equation.     

Refinements of the β-core and the strong equilibrium and the Aumann proposition

The Aumann proposition establishes that an outcome in the -core of the one-shot gameG is the same as the strong equilibrium utility allocations of the associated supergame. We refine the -core by requiring that the threat strategies used to deter deviations be credible and examine the consequences for the Aumann proposition in the case of the refinement. We show that a refinement of the strong-equilibrium notion is always in the refined -core. The converse is not necessarily true unless the utility allocations are linear.I am grateful to T. Ichiishi for several discussions that helped in the development of this paper. This is part of the author's Ph.D. dissertation submitted at the university of Iowa in December 1985.     

Solving the Ku–Wales conjecture on the eigenvalues of the derangement graph

We give a new recurrence formula for the eigenvalues of the derangement graph. Consequently, we provide a simpler proof of the Alternating Sign Property of the derangement graph. Moreover, we prove that the absolute value of the eigenvalue decreases whenever the corresponding partition with a fixed first part decreases in the dominance order. In particular, this settles affirmatively a conjecture of Ku and Wales [C.Y. Ku, D.B. Wales, Eigenvalues of the derangement graph, J. Combin. Theory Ser. A 117 (2010) 289–312] regarding the lower and upper bounds for the absolute values of these eigenvalues.     

On the Lévy Measure of the Lognormal and the LogCauchy Distributions

The densities of the Lévy measure and the Thorin measure of the standard lognormal distribution are approximated and presented in graphs. Moreover, the behavior of these densities at 0 and is studied. Laplace and saddlepoint approximations are used. The infinite divisibility of the standard logCauchy distribution is given some attention in passing.     

On the L∞ Norm of the First Eigenfunction of the Dirichlet Laplacian

We obtain the asymptotic behaviour for the L norm of the first eigenfunction of the Dirichlet Laplace operator on a conic sector over a geodesic disc in as . We are led to conjecture that for an open, bounded and convex set D with inradius and diameter d, where and      

On the regularity of the solutions to the 3D Navier–Stokes equations: a remark on the role of the helicity

We show that if velocity and vorticity are orthogonal at each point (and they become orthogonal fast enough) then solutions of the 3D Navier–Stokes equations are smooth. This condition implies that the helicity is identically zero and, in a certain sense, the flow resembles the 2D geometric situation. To cite this article: L.C. Berselli, D. Córdoba, C. R. Acad. Sci. Paris, Ser. I 347 (2009).