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1.
We considerC 2-solutionsf=u+iv+jw of the system
calledH-solutions introduced by H. Leutwiler. Iff is anH-solution in ω, thenf | Ω∩ℂ is holomorphic. SinceH-solutions are real analytic, a non-zeroH-solution cannot vanish in an open subdomain of ℝ3. Our object is, by the way of examples, to show that there are many kinds of null-sets ofH-solutions in ℝ3. This is in sharp contrast to a holomorphic functionf in ℂ, where the setf −1 ({0}) consists of discrete points only unlessf≡0. This research is supported by the Academy of Finland  相似文献   

2.
The main result of this work is the following theorem: LetP,QɛC[x, y] satisfy the Jacobian identity
  相似文献   

3.
  相似文献   

4.
We present a method for the solution of the Cauchy problem for three broad classes of nonlinear parabolic equations
and
with the infinite-dimensional Laplacian ΔL.  相似文献   

5.
The paper deals with localization properties of solutions to the Cauchy problem with the initial data u0(x) ∈ L2(ℝn) for a wide class of equations in the divergence form. This class contains, e.g., the following equation:
, Restrictions are obtained, sharp in a sense, on the behavior of the function ensuring the instantaneous compactification of the support of an arbitrary energy solution to the problem as well as the compactification of the support after a finite waiting-time. Translated from Trudy Seminara imeni l. G. Petrovskogo, No. 20, pp. 121–154, 1997.  相似文献   

6.
ＦＩＮＩＴＥＴＲＡＶＥＬＩＮＧＷＡＶＥＳＦＯＲＡＲＥＡＣＴＩＯＮ┐ＤＩＦＦＵＳＩＯＮＳＹＳＴＥＭＷＩＴＨＮ（３）ＣＯＭＰＯＮＥＮＴＳＷＡＮＧＳＨＵ,ＷＥＮＳＨＩＨＬＩＡＮＧＡＮＤＹＥＱＩＸＩＡＯＡｂｓｔｒａｃｔ．Ｉｎｔｈｉｓｐａｐｅｒ,ｔｈｅｅｘ...  相似文献   

7.
Let τ(n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
We also show that
where ω(n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9]. Received: 23 November 2006  相似文献   

8.
Let q ⩾ 3 be an integer, let χ denote a Dirichlet character modulo q. For any real number a ⩾ 0 we define the generalized Dirichlet L-functions
$ L(s,\chi ,a) = \sum\limits_{n = 1}^\infty {\frac{{\chi (n)}} {{(n + a)^s }},} $ L(s,\chi ,a) = \sum\limits_{n = 1}^\infty {\frac{{\chi (n)}} {{(n + a)^s }},}   相似文献   

9.
For γ≥1 we consider the solution u=u（x） of the Dirichlet boundary value problem Δu ＋ u^-γ=0 in Ω, u=0 on δΩ. For γ= 1 we find the estimate u（x）=p（δ（x））[1＋A（x）（log 1/δ（x）^-6], where p（r） ≈ r r√2 log（1/r） near r = 0,δ（x） denotes the distance from x to δΩ, 0 〈ε 〈 1/2, and A（x） is a bounded function. For 1 〈 γ 〈 3 we find u（x）=（γ＋1/√2（γ-1）δ（x））^2/γ＋[1＋A（x）（δ（x））2γ-1/γ＋1] For γ3= we prove that u（x）=（2δ（x））^1/2[1＋A（x）δ（x）log 1/δ（x）]  相似文献   

10.
The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special class of functionals:
with respect to the convergence un → u in measure, vn ⇀ v in Lp(Ω;ℝd) in W−1,p(Ω), and χn ⇀ χ in Lp(Ω), where χn ∈ Z:= {χ ∈ L∞(Ω): 0 ≤ χ(x) ≤ 1 for a.e. x}. Here is a constant-rank partial differential operator. The main result is that if the characteristic cone of has the full dimension, then the l.s.c. is equivalent to the fact that the F± are both -quasiconvex and
for a.e. x ∈ Ω and for all u ∈ ℝd. As a corollary, we obtain several results for the functional
with respect to the same convergence. We show that this functional is l.s.c. iff
Bibliography: 14 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 100–119.  相似文献   

11.
We study some arithmetic properties of the Ramanujan function τ(n), such as the largest prime divisorP (τ(n)) and the number of distinct prime divisors ω (τ (n)) of τ(n) for various sequences ofn. In particular, we show thatP(τ(n)) ≥ (logn)33/31+o(1) for infinitely many n, and
for every primep with τ(ρ) ≠ 0. Dedicated to T N Shorey on his sixtieth birthday  相似文献   

12.
In this paper the invariance criterion is applied for the nonlinear equation
(0.1)
where g(u) is a smooth function on u. Some particular set of Lie generators are given. In the case of inviscid Burger’s equation [1]
(0.2)
the Lie projectable symmetry algebra is determined, and the inviscid Burger’s equation will be connected to some order differential equations. The obtained differential equations are solved and some exact solutions of (2) are found. E.H. El Kinani, Junior Associate at The Abdus Salam ICTP, Trieste, Italy.  相似文献   

13.
§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＣｏｎｓｉｄｅｒｔｈｅｎｏｎａｕｔｏｎｏｍｏｕｓｄｅｌａｙｌｏｇｉｓｔｉｃｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎΔｙｎ =ｐｎｙｎ( 1 - ｙτ(ｎ) )　 ,ｎ =0 ,1 ,2 ,...,( 1 1 )ｗｈｅｒｅｐｎ ∞ｎ =0 ｉｓａｓｅｑｕｅｎｃｅｏｆｐｏｓｉｔｉｖｅｒｅａｌｎｕｍｂｅｒｓ ,τ(ｎ) ∞ｎ =0 ｉｓａｎｏｎｄｅｃｒｅａｓｉｎｇｓｅｑｕｅｎｃｅｏｆｉｎｔｅｇｅｒｓ,τ(ｎ) <ｎａｎｄｌｉｍｎ→∞τ(ｎ) =∞ ,Δｙｎ=ｙｎ +1- ｙｎ.Ｍｏｔｉｖａｔｅｄｂｙｐｌａｕｓｉｂｌｅａｐｐｌｉｃａｔｉｏｎｓ…  相似文献   

14.
具强迫力的奇数阶中立型微分方程的振动性   总被引：3，自引：0，他引：3  
Abstract. In this paper, the forced odd order neutral differential equations of the form are con-sidered  相似文献   

15.
Under certain conditions on continuous functions μ, λ, a, and f, we prove the inequality
and describe its application to the investigation of the problem of finding conditions under which Laplace integrals belong to a class of convergence. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1271–1275, September, 2006.  相似文献   

16.
A trace formula expressing the mean values of the form (k=2,3,...)
via certain arithmetic means on the group Г0(N1) is proved. Here the sum is taken over a normalized orthogonal basis in the space of holomorphic cusp forms of weight 2k with respect to Г0(N1). By H f (x) (s) we denote the Hecke series of the form f, twisted with the primitive character χ (mod N2), and λf(d), (d, N1N2)=1, are the eigenvalues of the Hecke operators
. The trace formula is used for obtaining the estimate
for the newform f for all ε>0, l=0,1,2,.... This improves the known result (Duke-Friedlander-Iwaniec, 1993) with upper bound (1+|t|)2N 2 1/2−1/22+ε on the right-hand side. As a corollary, we obtain the estimate
for the Fourier coefficients of holomorphic cusp forms of weight k+1/2, which improves Iwaniec' result (1987) with exponent 1/4–1/28+ε. Bibliography: 25 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 14–36.  相似文献   

17.
We study large time asymptotics of solutions to the Korteweg-de Vries-Burgers equation ut＋uux-uxx＋uxxx=0,x∈R,t〉0. We are interested in the large time asymptotics for the case when the initial data have an arbitrary size. We prove that if the initial data u0 ∈H^s （R）∩L^1 （R）, where s 〉 -1/2, then there exists a unique solution u （t, x） ∈C^∞ （（0,∞）;H^∞ （R）） to the Cauchy problem for the Korteweg-de Vries-Burgers equation, which has asymptotics u（t）=t^-1/2fM（（·）t^-1/2）＋0（t^-1/2） as t →∞, where fM is the self-similar solution for the Burgers equation. Moreover if xu0 （x） ∈ L^1 （R）, then the asymptotics are true u（t）=t^-1/2fM（（·）t^-1/2）＋O（t^-1/2-γ） where γ ∈ （0, 1/2）.  相似文献   

18.
Consider the two natural representations of the symmetric groupS n on the group algebra ℂ[S n ]: the regular representation and the conjugacy representation (acting on the basis by conjugation). Letm(λ) be the multiplicity of the irreducible representationS λ in the conjugacy representation and letf λ be the multiplicity ofS λ in the regular representation. By the character estimates of [R1] and [Wa] we prove
(1)  For any 1>ε>0 there exist 0<δ(ε) andN(ε) such that, for any partitionλ ofn>N(ε) with max ,
whereλ 1 is the size of the largest part inλ andλ′1 is the number of parts inλ.
(2)  For any fixed 1>r>0 and ε>0 there existκ=κ(ε, r) andN(ε, r) such that, for any partitionλ ofn>N(ε, r) with max ,
whereA is a constant which depends only on the fractions
This strengthens Adin-Frumkin’s result [AF] and answers a question of Stanley [St].
Partially sponsored by a Wolfson fellowship and the Hebrew University of Jerusalem.  相似文献   

19.
Let M be either the space of 2π-periodic functions Lp, where 1 ≤ p < ∞, or C; let ωr(f, h) be the continuity modulus of order r of the function f, and let
, where
, be the generalized Jackson-Vallée-Poussin integral. Denote
. The paper studies the quantity Km(f − Dn,r,l(f)). The general results obtained are applicable to other approximation methods. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 52–69.  相似文献   

20.
The paper considers the function defined by the Dirichlet series
, where τ(ν) is the νth Fourier coefficient of the cubic Kubota-Patterson theta function. The zeros of the function L(τ ·) in the domain |Im s| < 444 are computed. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 173–186.  相似文献   

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