The limiting angle of certain riemannian brownian motions

Let M be a Cartan-Hadamard manifold of dimension d ≧ 3, let p ? M and x = exp {r(x)θ(x)} be geodesic polar coordinates with pole p and let X be the Brownian motion on M. Let Sect_M(x) denote the sectional curvature of any plane section in M_x. We prove that for each c > 2, there is a 0 < β < 1 such that if - L²r(x)^2β ≦ Sect_M(x) ≦ -cr(x)^?2 for all x in the complement of a compact set, then lim_{t → ∞} θ(X_t) exists a.s. and defines a nontrivial invariant random variable. The Dirichlet problem at infinity and a conjecture of Greene and Wu are also discussed.     

On perfect and multiply perfect numbers

Summary Denote by P(x) the number of integers n≤x satisfying σ(n)≡0 (mod n), and by P ₂ (x) the number of integers n ≤ x satisfying σ(n)=2n. The author proves that P(x)<x ^3/4+ɛ and P ₂ (x)<x ^(1−c)/2 for a certain c>0.     

Center conditions II: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). We introduce a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1). We show that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

On a method for interpolating functions on chaotic nets

Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x| ^m form odd,K(x)=|x| ^m In |x| form even (x∈ℝ ⁿ ),P is the set of real polynomials inn variables of total degree ≤m/2, andx ₁,...,x _N ∈ℝ ⁿ . We construct a function of the form

On levi–typ conditions for hypoellipticity of certain diffrential operators

The hypoelliticity is discussed for operators of the form P=D² _x+a(x)D² _y+b(x)D_ywhere a (x) and b (x) are real–valued C^∞ functions satisfying a(0)=0 and a(x) >0 for x≠0.We seek the conditions for P to be hypoelliptic, especially in the case where both a (x) and b(x) vanish to infinite order on x=0.     

THE EIGENVALUE DISTRIBUTION OF A HUBERT-SCHMIDT-TYPE OPERATOR

Abstract

We prove the following theorem in answer to a question raised by P Nowosad and R Tovar in [3]. If K is a kernel operator on L²(x,u) with kernel K(x, y) if P(x): = U_X |K(x, y)|² d μ(y))½ and Q(x): = (U_X |K (y, x)|² d μ(y))½ and if _x PQdμ < ∞, then σ|λ_i|² < ∫_X PQd μ where (λ_i) is the se = zuence of eigenvalues of K.     

Local solvability of partial differential operators on the Heisenberg group

In this article the following class of partial differential operators is examined for local solvability: Let P(X, Y) be a homogeneous polynomial of degree n ≥ 2 in the non-commuting variables X and Y. Suppose that the complex polynomial P(iz, 1) has distinct roots and that P(z, 0) = zⁿ. The operators which we investigate are of the form P(X, Y) where X = δ_x and Y = δ_y + xδ_w for variables (x, y, w) ∈ ?³. We find that the operators P (X, Y) are locally solvable if and only if the kernels of the ordinary differential operators P(iδ_x, ± x)* contain no Schwartz-class functions other than the zero function. The proof of this theorem involves the construction of a parametrix along with invariance properties of Heisenberg group operators and the application of Sobolev-space inequalities by Hörmander as necessary conditions for local solvability.     

On solvability of boundary value problems for systems of differential equations

We consider systems of differential equations (1)x=g(t, x) together with boundary conditions of the form (2)x(0)=x ₀,x(T)=x ₁; (3)x(0)=Qx(T), x(0)=Qx(T); and (4)B ₁ x(0)–B ₂ x(0)=0=C ₁ x(T)+C ₂ x(T). Herex=(x ₁,...,x _n )^T andg=g(t, x) are realn-vectors andQ, B _i ,C _i ,i=1, 2, denoten×n matrices withQ nonsingular andB ₂,C ₂ positive definite. We examine the existence of solutions of (1) satisfying (3) or (4) and which also stay in a certain regionin(t, x) space. Conditions in terms of the Jacobian matrixG (t, x)=g _x (t, x) and an auxiliary positive definite symmetric matrixP=P(t) C ² [0,T] are given which yield the existence of the desired solution of (1), (3) or (1), (4).Dedicated to Professor Hans W. Knobloch on the occassion of his sixtieth birthdayResearch supported by NSERC Canada Grant A7673 and NSF Grant DMS-8501311.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      10. Center conditions III: Parametric and model center problems      M. Briskin  J. -P. Francoise  Y. Yomdin 《Israel Journal of Mathematics》2000,118(1):83-108 We consider an Abel equation (*)y’=p(x)y 2 +q(x)y 3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty 0=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y 2 +εq(x)y 3 p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm k (1), wherem k (x)=∫ 0 x pk (t)q(t)(dt),P(x)=∫ 0 x p(t)dt. We investigate the structure of zeroes ofm k (x) and generalize a “canonical representation” ofm k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.      11. Einige unstetige stochastische Prozesse      Andreas Dalcher 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》1956,7(4):273-304 Summary Stochastic processes of the following type are considered. At random time points, the variablex(t) jumps fromy tox, say. The heightsx–y of the jumps have a given distributionG *(x–y) that may depend ony ort. Between the jumps,x(t) is a solution to a given differential equationdx/dt=x(x, t). We look for the distributionF(x, t) ofx at timet>0,F(x, 0) being given. In the stationary case, stable distributions are investigated.If there is a lower boundaryx 0 and ifF(x 0)>0, the problem is similar to the queueing problem. We solve it in the stationary case with integral equations of the Volterra type. Other problems can be transformed to differential equations for the moment generating functions. These equations are partial in the non stationary and ordinary in the stationary case.      12. Eine explizite Abschätzung für die Gitter-Diskrepanz von Rotationsellipsoiden      Ekkehard Krätzel  Werner Georg Nowak 《Monatshefte für Mathematik》2007,152(1):45-61 An explicit estimate for the lattice point discrepancy of ellipsoids of rotation. For the lattice point discrepancy (i.e., the number of integer points minus the volume) of the ellipsoid (u 1 2 + u 2 2)/a + a 2 u 3 2 ≤ x (a, x > 0), this paper provides an estimate of the form terms of smaller order in x. Die Autoren danken dem ?sterreichischen Fonds zur F?rderung der wissenschaftlichen Forschung (FWF) für finanzielle Unterstützung unter der Projekt-Nr. P18079-N12.      13. The greatest prime factor of the integers in a short interval (IV)      Jia Chaohua 《数学学报(英文版)》1996,12(4):433-445 LetP(x) denote the greatest prime factor of n. In this paper, we shall prove thatP(x)>x 0.728 holds true for sufficiently largex.Project supported by the Tian Yuan Item in the National Natural Science Foundation of China.      14. Removable singularities of weak solutions to the navier-stokes equations      Hideo Kozono 《偏微分方程通讯》2013,38(5-6):949-966 Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R3. We show that there is an absolute constant ε0 such that ever, y weak solution u with the property that Suptε(a,b)|u(t)|L(D)≤ε0 is necessarily of class C∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (xo, to) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|-1) as x→x 0 uniforlnly in t in some neighbourliood of to, then (xo,to) is actually a removable singularity of u.      15. Superstability of the generalized orthogonality equation on restricted domains      Soon-Mo Jung  Prasanna K. Sahoo 《Proceedings Mathematical Sciences》2004,114(3):253-267 Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD n be a suitable subset of ℝn. If a function f:D n → ℝn satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D n.      16. Purely imaginary eigenvalues of operator semigroups      W. M. Ruess 《Semigroup Forum》1995,51(1):335-341 For aC 0-contraction semigroup (S(t)) t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e iλt x, t≥0, for some λ∈ℝ, provided lim t→∞ |<S(t)x,x * >|=|<x,x * >| for allx *∈X*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.      17. Extended iterative methods for the solution of operator equations      M. A. Wolfe 《Numerische Mathematik》1978,31(2):153-174 Summary Given an iterative methodM 0, characterized byx (k+1=G 0(x( k )) (k0) (x(0) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM p characterized byx (k+1=G p (x( k )) (k0) (x(0) prescribed) with order of convergence higher than that ofM o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM 0 is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM p, which are referred to as extensions ofM 0. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.      18. On exponential sums over primes in short intervals      Guangshi Lü  Huixue Lao 《Monatshefte für Mathematik》2007,49(5):153-164 Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals S2(x,y;a)=?x < n ￡ x+yL(n)e(n2 a)S_2(x,y;{\alpha})=\sum_{x < n \le x+y}\Lambda(n)e(n^2 {\alpha}) for all α ∈ [0,1] whenever x\frac23+e ￡ y ￡ xx^{\frac{2}{3}+{\varepsilon}}\le y \le x . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.      19. On analytic microlocal hypoellipticity of linear partial differential operators of principal type      Alexandros T Himonas 《偏微分方程通讯》2013,38(14):1539-1574 It is shown that if P is a linear partial differential uperator with analytic coefficients defined near a point xo in Rn and if P in Rn - 0 is such that: the principal symbol pm,(x, ξ) vanishes at (x0. ξ0). the differential of pm, with respect to ξ is different from zero at (x0, ξ0). the Poisson bracket {Pm, Pm} is zero at (x0. ξ0) and the Poisson bracket {pm, {pm.pm }} is different from zero at (x0, ξ0), then P is analytic hypoelliptic at (x0, ξ0). It is also proved that P is analytic hypoelliptic under the assumption that the first non-vanishing repeated Poisson bracket of pm, and pm, is of odd length and under some additional hypothesis on the commutators of the Hamilton fields of Re pm, and Im pm,      20. On Sums of Two Coprime k-th Powers(II)      Wenguang Zhai  Xiaodong Cao 《Monatshefte für Mathematik》2006,2(4):153-172 For fixed k ≥ 3, let Ek(x) denote the error term of the sum ?n ￡ xrk(n)\sum_{n\le x}\rho_k(n) , where rk(n) = ?n=|m|k+|l|k, g.c.d.(m,l)=1\rho_k(n) = \sum_{n=|m|^k+|l|^k, g.c.d.(m,l)=1} 1. It is proved that if the Riemann hypothesis is true, then E3(x) << x331/1254+eE_3(x)\ll x^{331/1254+\varepsilon} , E4(x) << x37/184+eE_4(x)\ll x^{37/184+\varepsilon} . A short interval result is also obtained.

Center conditions III: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and generalize a “canonical representation” ofm _k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

Einige unstetige stochastische Prozesse

Summary Stochastic processes of the following type are considered. At random time points, the variablex(t) jumps fromy tox, say. The heightsx–y of the jumps have a given distributionG ^*(x–y) that may depend ony ort. Between the jumps,x(t) is a solution to a given differential equationdx/dt=x(x, t). We look for the distributionF(x, t) ofx at timet>0,F(x, 0) being given. In the stationary case, stable distributions are investigated.If there is a lower boundaryx ₀ and ifF(x ₀)>0, the problem is similar to the queueing problem. We solve it in the stationary case with integral equations of the Volterra type. Other problems can be transformed to differential equations for the moment generating functions. These equations are partial in the non stationary and ordinary in the stationary case.     

Eine explizite Abschätzung für die Gitter-Diskrepanz von Rotationsellipsoiden

An explicit estimate for the lattice point discrepancy of ellipsoids of rotation. For the lattice point discrepancy (i.e., the number of integer points minus the volume) of the ellipsoid (u ₁ ² + u ₂ ²)/a + a ² u ₃ ² ≤ x (a, x > 0), this paper provides an estimate of the form terms of smaller order in x. Die Autoren danken dem ?sterreichischen Fonds zur F?rderung der wissenschaftlichen Forschung (FWF) für finanzielle Unterstützung unter der Projekt-Nr. P18079-N12.     

The greatest prime factor of the integers in a short interval (IV)

LetP(x) denote the greatest prime factor of n. In this paper, we shall prove thatP(x)>x ^0.728 holds true for sufficiently largex.Project supported by the Tian Yuan Item in the National Natural Science Foundation of China.     

Removable singularities of weak solutions to the navier-stokes equations

Consider the Navier-Stokes equations in Ω×(0,T), where Ω is a domain in R³. We show that there is an absolute constant ε₀ such that ever, y weak solution u with the property that Sup_tε(a,b)|u(t)|_L(D)≤ε0 is necessarily of class C^∞ in the space-time variables on any compact suhset of D × (a,b) , where D?? and 0 a<b<T. As an application. we prove that if the weak solution u behaves around (x_o, t_o) εΩ×(o,T) 1ike u(x, t) = o(|x - xo|^-1) as x→x ₀ uniforlnly in t in some neighbourliood of t_o, then (x_o,t_o) is actually a removable singularity of u.     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

Extended iterative methods for the solution of operator equations

Summary Given an iterative methodM ₀, characterized byx ^(k+1=G ₀(x( ^k )) (k0) (x(⁰) prescribed) for the solution of the operator equationF(x)=0, whereF:XX is a given operator andX is a Banach space, it is shown how to obtain a family of methodsM _p characterized byx ^(k+1=G _p (x( ^k )) (k0) (x(⁰) prescribed) with order of convergence higher than that ofM _o. The infinite dimensional multipoint methods of Bosarge and Falb [2] are a special case, in whichM ₀ is Newton's method.Analogues of Theorems 2.3 and 2.36 of [2] are proved for the methodsM _p, which are referred to as extensions ofM ₀. A number of methods with order of convergence greater than two are discussed and existence-convergence theorems for some of them are proved.Finally some computational results are presented which illustrate the behaviour of the methods and their extensions when used to solve systems of nonlinear algebraic equations, and some applications currently being investigated are mentioned.     

On exponential sums over primes in short intervals

Let Λ(n) be the von Mangoldt function, x real and y small compared with x. This paper gives a non-trivial estimate on the exponential sum over primes in short intervals S₂(x,y;a)=?_{x < n ￡ x+y}L(n)e(n² a)S_2(x,y;{\alpha})=\sum_{x < n \le x+y}\Lambda(n)e(n^2 {\alpha}) for all α ∈ [0,1] whenever x^\frac23+e ￡ y ￡ xx^{\frac{2}{3}+{\varepsilon}}\le y \le x . This result is as good as what was previously derived from the Generalized Riemann Hypothesis.     

On analytic microlocal hypoellipticity of linear partial differential operators of principal type

It is shown that if P is a linear partial differential uperator with analytic coefficients defined near a point xo in Rⁿ and if P in Rⁿ - 0 is such that: the principal symbol p_m,(x, ξ) vanishes at (x₀. ξ⁰). the differential of p_m, with respect to ξ is different from zero at (x₀, ξ⁰). the Poisson bracket {P_m, P_m} is zero at (x₀. ξ⁰) and the Poisson bracket {p_m, {p_m.p_m }} is different from zero at (x₀, ξ⁰), then P is analytic hypoelliptic at (x₀, ξ⁰). It is also proved that P is analytic hypoelliptic under the assumption that the first non-vanishing repeated Poisson bracket of p_m, and p_m, is of odd length and under some additional hypothesis on the commutators of the Hamilton fields of Re p_m, and Im p_m,     

On Sums of Two Coprime k-th Powers(II)

For fixed k ≥ 3, let E_k(x) denote the error term of the sum ?_{n ￡ x}r_k(n)\sum_{n\le x}\rho_k(n) , where r_k(n) = ?_{n=|m|^k+|l|^k, g.c.d.(m,l)=1}\rho_k(n) = \sum_{n=|m|^k+|l|^k, g.c.d.(m,l)=1} 1. It is proved that if the Riemann hypothesis is true, then E₃(x) << x^331/1254+eE_3(x)\ll x^{331/1254+\varepsilon} , E₄(x) << x^37/184+eE_4(x)\ll x^{37/184+\varepsilon} . A short interval result is also obtained.