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.     

Asymptotic behavior of the solutions of the p-Laplacian equation

The asymptotic behavior of the solutions for p-Laplacian equations as p→∞ is studied.     

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 Moments of the Modulus of Continuity of Itô Processes

The modulus of continuity of a stochastic process is a random element for any fixed mesh size. We provide upper bounds for the moments of the modulus of continuity of Itô processes with possibly unbounded coefficients, starting from the special case of Brownian motion. References to known results for the case of Brownian motion and Itô processes with uniformly bounded coefficients are included. As an application, we obtain the rate of strong convergence of Euler–Maruyama schemes for the approximation of stochastic delay differential equations satisfying a Lipschitz condition in supremum norm.     

Computation of the number representations of the elements of the ringℤ/dℤ as a sum of squares

The number of representations of the elements of the ringℤ/dℤ as a sum of invertible squares is computed, provided that each square occurs in the sum no more tha a fized number of times. For prime d an exhaustive answer is given in term of the class number and the fundamental unit of the real quadratic field . Biblography: 5 titles. Translted fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 5–8.     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

On the Complexity of the Julia Sets of Rational Functions

ＯｎｔｈｅＣｏｍｐｌｅｘｉｔｙｏｆｔｈｅＪｕｌｉａＳｅｔｓｏｆＲａｔｉｏｎａｌＦｕｎｃｔｉｏｎｓＱｉａｏＪｉａｎｙｏｎｇ（乔建永）（ＩｎｓｔｉｔｕｔｅｏｆＭａｔｈｅｍａｔｉｃｓ，ＡｃａｄｅｍｉａＳｉｎｉｃａ，Ｂｅｉｊｉｎｇ，１０００８０）Ｃｏｍｍｕｎ...     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

On the Gracefulness of Two Family of the Hexagonal Graphs

Thegracefulgraphswaspresentedinl966.Sincethattime,manyresultshavebeenobtainedongacefulgraphsandit'smanyapplicationsappear(see[2j,L4j).Thegraceful-nessofP.XP.andC.XP,areprovedbythepaper[1],[3j,[5j,[6]-Followingthis,inthispaper,itisobtainedthatthegracefu1nessofthefamiliesofthehexagonalgraphsG,(n)andG2(n).Fornotationsandteminologiesonthegracelulgraphnotdefinedhere,wefollowthereference[3j-NowletG(V,E)beagraph,6bethelabelingoftheverticesofG,andforanyedgeuv,define6(uv)=lo(u)-o(v)Iitistheedge's…     

On the Nontriviality of the Polynomial Invariant of Two-Component Link

ＯｎｔｈｅＮｏｎｔｒｉｖｉａｌｉｔｙｏｆｔｈｅＰｏｌｙｎｏｍｉａｌＩｎｖａｒｉａｎｔｏｆＴｗｏ－ＣｏｍｐｏｎｅｎｔＬｉｎｋ￥ＷａｎｇＸｉｎｇｙｕ；ＺｈａｏＧｕａｎｇｆｅｎｇ（ＨｅｎａｎＵｎｉｖｅｒｓｉｔｙ，Ｋａｉｆｅｎｇ，Ｈｅｎａｎ，４７５００１）（...     

On the points of order 3 of the period parallelogram of the Weierstrass elliptic function

We examine the significance of the values of the Weierstrass ℘ function at the points of order three in the period parallelogram. Based upon the well-known duplication formulae and the differential equation of the Weierstrass ℘ function, we derive a set of identities involving the values of ℘ at these points.      

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      

New Estimates for the Rate of Convergence of the Method of Subspace Corrections

We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections. We provide upper bounds and in a special case, a lower bound for preconditioners defined via the method of successive subspace corrections.     

On the topology of the set of singularities of a solution to the Hamilton–Jacobi equation

We address the topology of the set of singularities of a solution to a Hamilton–Jacobi equation. For this, we will apply the idea of the first two authors (Cannarsa and Cheng, Generalized characteristics and Lax–Oleinik operators: global result, preprint, arXiv:1605.07581, 2016) to use the positive Lax–Oleinik semi-group to propagate singularities.     

Arithmetics of the numbers of orbits of the Fermat–Euler dynamical systems

Given positive integers a and n with (a,n)=1, we consider the Fermat–Euler dynamical system defined by the multiplication by a acting on the set of residues modulo n relatively prime to n. Given an integer M>1, the integers n for which the number of orbits of this dynamical system is a multiple of M form an ideal in the multiplicative semigroup of odd integers. We provide new results on the arithmetical properties of these ideals by using the topological properties of some directed graphs (the monads).      

Approximation of the soluticon of the stochastic navier-stokes equation ∗

We consider a stochastic equation of the Navier-Stokes type containing a noise part given by a stochastic Ito integral with respect to a Wiener process. The purpose of this paper is to approximate the solution of this nonlinear equation by the solutions of a sequence of linear equations with additive noise. We prove the convergence in mean squares     

Estimate for the amplitude of the limit cycle of the Liénard equation

We consider the nonlinear Liénard equation \(\ddot x\left( t \right) + f\left( x \right)\dot x\left( t \right) + g\left( x \right) = 0\). Liénard obtained sufficient conditions on the functions f(x) and g(x) under which this equation has a unique stable limit cycle. Under additional conditions, we prove a theorem that permits one to estimate the amplitude (the maximum value of x) of this limit cycle from above. The theorem is used to estimate the amplitude of the limit cycle of the van der Pol equation \(\ddot x\left( t \right) + \mu \left[ {{x^2}\left( t \right) - 1} \right]\dot x\left( t \right) + x\left( t \right) = 0\).