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1.
Natalia Baldisserri 《Rendiconti del Circolo Matematico di Palermo》1999,48(2):299-308
We study the groupG
m of primitive solution of the diophantine equationx
2+my2=z2 (m>1, squarefree). Form∈3 this group is torsion free, form=3 it has a torsion element of order 3; moreover for a finite number of values ofm we prove thatG
m is a direct sum of infinite cyclic groups and we give the generators ofG
m in terms of the primes represented by the quadratic forms of discriminant Δ=−4m.
相似文献
2.
Quasilinear elliptic equations with boundary blow-up 总被引:2,自引:0,他引:2
Jerk Matero 《Journal d'Analyse Mathématique》1996,69(1):229-247
Assume that Ω is a bounded domain in ℝ
N
withN ≥2, which has aC
2-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δp
u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass
of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2]. 相似文献
3.
R. F. Millar 《Applicable analysis》2013,92(2):101-121
Two methods are described for the a priori location of singularities of solutions to exterior boundary value problems. One uses an expansion for the solution in a circle centered on a regular exterior point P. A singularity lies on the circle of convergence. The envelope of these circles, generated as P makes a circuit about the closed boundary, circumscribes the singularities. The radius of convergence depends on singularities of the solution u(s) and its normal derivative v(s) on the boundary. The second method employs complex characteristics to relate singularities of the boundary data to real singularities of the solution. Integral equations connecting (y), v(s) and the analytic boundary condition are used to continue the data into the complex s-plane and to locate their singularities. Explicit solution of the integral equations is unnecessary; some nonlinear boundary conditions can be handled. 相似文献
4.
We consider the differential operators Ψ
k
, defined by Ψ1(y) =y and Ψ
k+1(y)=yΨ
k
y+d/dz(Ψ
k
(y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ
k
F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z
2+β
z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ
k
(F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ
k
(f
′/f) =f
(k)/f, we deduce in particular that iff andf
(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f
′/f :f ∈F} is normal.
The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999,
and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank
Günter Frank for helpful discussions. 相似文献
5.
Konrad Neumann 《Israel Journal of Mathematics》1999,113(1):1-13
Proving primeness of an idealI=〈f
1, …,f
m〉 in a polynomial ringR=K[X
1, …,X
n]ofn indeterminates over an algebraically closed fieldK is a difficult task in general. Although there are straightforward algorithms that decide whetherI is prime or not, they are prohibitively lengthy if the number of indeterminates or the degrees of thef
iare large. In this paper we will give an easy criterion for the primeness ofI if thef
iare polynomials with separated variables, i.e. no mixed monomials occur in thef
i.
The work on this paper was done while the author was a MINERVA fellow at Tel Aviv University. 相似文献
6.
A classic theorem of Pólya shows that 2
z
is, in a strong sense, the “smallest” transcendental entire function that is integer valued on ℕ. An analogous result of
Gel’fond concerns entire functions that are integer valued on the setX
a={a
n:n ∈ ℕ}, wherea ∈ ℕ,|a|≥ 2. LetX=ℕ orX=X
a andκ ∈ ℕ orκ=∞. This paper pursues analogous results for entire functionsf having the following property: on any finite subsetD ofX with#D≤κ+1, the valuesf(z),z ∈D admit interpolation by an element of ℤ[z]. The results obtained assert that if the growth off is suitably restricted then the restriction off toX must be a polynomial. WhenX=X
a andκ<∞ a “smallest” transcendental entire function having the requisite property is constructed. 相似文献
7.
Bernard Pinchuk 《Israel Journal of Mathematics》1971,10(1):6-16
Classes of functionsU
k, which generalize starlike functions in the same manner that the classV
k of functions with boundary rotation bounded bykπ generalizes convex functions, are defined. The radius of univalence and starlikeness is determined. The behavior off
α(z) = ∫
0
z
[f'(t)]α
dt is determined for various classes of functions. It is shown that the image of |z|<1 underV
kfunctions contains the disc of radius 1/k centered at the origin, andV
k functions are continuous in |z|≦1 with the exception of at most [k/2+1] points on |z|=1. 相似文献
8.
Charles Horowitz 《Israel Journal of Mathematics》1978,30(3):285-291
LesB denote the class of functions analytic in the unit disc ofC which satisfy 0<|f(z)|<1. It is proved that there exists a numberc<1 such that iff∈B and iff(z)=Σ
n=0
∞
a
n
z
n
, then |a
n
|<c forn>=1. 相似文献
9.
E. G. Kir’yatskii 《Russian Mathematics (Iz VUZ)》2010,54(2):84-86
Let a set B have the following properties: if z ∈ B, then z ± 2π ∈ B and the intersection of B with the vertical strip 0 ≤ Re x ≤ π is a closed and bounded set. In this paper we study the approximation of a continuous on B and 2π-periodic function f(z) by trigonometric polynomials T
n
(z). We establish the necessary and sufficient conditions for the function f(z) to be entire and specify a formula for calculating its order. In addition, we describe some metric properties of periodic
sets in a plane. 相似文献
10.
Arthur G. Werschulz 《Journal of Complexity》1996,12(4):440-473
We study the complexity of 2mth order definite elliptic problemsLu=f(with homogeneous Dirichlet boundary conditions) over ad-dimensional domain Ω, error being measured in theHm(Ω)-norm. The problem elementsfbelong to the unit ball ofWr,p(Ω), wherep∈ [2, ∞] andr>d/p. Information consists of (possibly adaptive) noisy evaluations offor the coefficients ofL. The absolute error in each noisy evaluation is at most δ. We find that thenth minimal radius for this problem is proportional ton−r/d+ δ, and that a noisy finite element method with quadrature (FEMQ), which uses only function values, and not derivatives, is a minimal error algorithm. This noisy FEMQ can be efficiently implemented using multigrid techniques. Using these results, we find tight bounds on the ?-complexity (minimal cost of calculating an ?-approximation) for this problem, said bounds depending on the costc(δ) of calculating a δ-noisy information value. As an example, if the cost of a δ-noisy evaluation isc(δ) = δ−s(fors> 0), then the complexity is proportional to (1/?)d/r+s. 相似文献
11.
Fang Liping 《数学学报(英文版)》1998,14(1):139-144
Letf be a holomorphic self-map of the punctured plane ℂ*=ℂ\{0} with essentially singular points 0 and ∞. In this note, we discuss the setsI
0(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞} andI
∞(f)={z ∈ ℂ*:f
n
(z) → 0,n → ∞}. We try to find the relation betweenI
0(f),I
∞(t) andJ(f). It is proved that both the boundary ofI
0(f) and the boundary ofI
∞)f) equal toJ(f),I
0(f) ∩J(f) ≠ θ andI
∞(f) ∩J(f) ≠ θ. As a consequence of these results, we find bothI
0(f) andI
∞(f) are not doubly-bounded.
Supported by the National Natural Science Foundation of China 相似文献
12.
Assume thatf is an integer transcendental solution of the differential equationP
n
(z, f, f′)=P
n−1(z, f, f′, ... f
(p)), whereP
n
andP
n−1 are polynomials in all variables, the degree ofP
n
with respect tof andf′ is equal ton, and the degree ofP
n−1 with respect tof, f′, ... f
(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E
*, whereE
* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z
1/2 (β+o(1)), β∈C, holds forz=re
iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re
iν)‖=o(r
1/2),r→+∞,r>0,
, where Δ is a certain set on the semiaxisr>0 with mes Δ<∞.
“L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77,
January, 1999. 相似文献
13.
Miran Černe 《Arkiv f?r Matematik》2002,40(1):27-45
LetX(-ϱB
m
×C
n
be a compact set over the unit sphere ϱB
m
such that for eachz∈ϱB
m
the fiberX
z
={ω∈C
n
;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inC
n
. The polynomial hull
ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z
0,w
0)∈Int
there exists a smooth up to the boundary analytic discF:Δ→B
m
×C
n
with the boundary inX such thatF(0)=(z
0,w
0).
This work was supported in part by a grant from the Ministry of Science of the Republic of Slovenia. 相似文献
14.
A sequence (z
0,z
1,z
2,, ...,z
n, z
n+1) of points fromp=z
0 toq=z
n+1 in a metric spaceX is said to besequentially equidistant ifd(z
i−1,z
i)=d(z
i,z
i+1) for 1≦i≦n. If there is path inX fromp toq (or if a certain weaker condition holds), then such a sequence exists, with all points distinct, for every choice ofn, while ifX is compact and connected, then such a sequence exists at least forn=2. An example is given of a dense connected subspaceS ofR
m
,m≧2, and an uncountable dense subsetE disjoint fromS for which there is no sequentially equidistant sequence of distinct points (n ≧ 2) inS ∪E between any two points ofE. Techniques of dimension theory are utilized in the construction of these examples, as well as in the proofs of some of the
positive results.
Supported in part by NSF Grant DMS-8701666. 相似文献
15.
Jan Wiegerinck 《Arkiv f?r Matematik》2000,38(1):201-208
In this paper we show that the pluripolar hull ofE={(z, ω)∈C2:ω=e
−1/z,z≠0} is equal toE. This implies thatE is plurithin at 0, which answers a question of Sadullaev. The result remains valid ife
−1/z is replaced by certain other holomorphic functions with an essential singularity at 0. 相似文献
16.
In this paper the closed convex hulls of the compact familiesC
β(p), of multivalently close to convex functions of order β andV
0
k
(p), of multivalent functions of bounded boundary rotation, have been determined, respectively for β≥1 andk≥2(p+1)/p. Extreme points of these convex hulls are partially characterised. For a fixed pointz
0∈D={z:|z|<1}, a new familyC
β(p, z0) is defined through Montel normalisation and its closed convex hull is also foud. Sharp coefficient estimates for functions
subordinate to or majorised by some function inC
β(p) orC'
β(p) are discussed for β>0. It is shown that iff is subordinate to some function inC
β(p) then each Taylor coefficient off is dominated by the corresponding coefficient of the function
. 相似文献
17.
Dan Segal 《Israel Journal of Mathematics》1996,94(1):7-19
A groupG hasweak polynomial subgroup growth (wPSG) of degree ≤α if each finite quotient Ḡ ofG contains at most │Ḡ│
a
subgroups. The main result is that wPSG of degree α implies polynomial subgroup growth (PSG) of degree at mostf(α). It follows that wPSG is equivalent to PSG. A corollary is that if, in a profinite groupG, thek-generator subgroups have positive “density” δ, thenG is finitely generated (the number of generators being bounded by a function ofk and δ). 相似文献
18.
Willem Veys 《manuscripta mathematica》1995,87(1):435-448
Tof ∈ℂ[x
1…,x
n
] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f
−1{0} inf
−1{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local
zeta function. A major problem is the study of its poles.
In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x
1,x
2]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs
rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution.
The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research
N.F.W.O. 相似文献
19.
V. A. Solonnikov 《Journal of Mathematical Sciences》1998,92(6):4364-4385
The main result is an asymptotic formula for a solution to the conjugation problem for the Navier-Stokes equations describing
the slow motion of two immiscible liquids such that one of them occupies a bounded domain Ω1 ⊂ ℝ3, whereas the other occupies the exterior domain Ω2=ℝ4∖Ω. Such a formula was obtained for a solution to the exterior problem with sticking conditions on the boundary in the works
of Fischer, Hsiao, and Wendland. The result obtained is applied to the proof of the solvability of a free-boundary problem
describing a uniform drop in an infinite liquid. Bibliography: 10 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 208–238. 相似文献
20.
Abraham Zaks 《Israel Journal of Mathematics》1980,37(4):281-302
LetR be a commutative domain with 1. We termR an HFD (Half-Factorial-Domain) provided the equality Π
i=1
n
χi=Π{f=1/m}y
f impliesm=n, whenever thex’s and they’s are non-zero, non-unit and irreducible elements ofR. The purpose of this note is to study HFD’s, in particular, Krull domains that are HFD’s, and to provide examples of HFD’s,
that contradict a conjecture of Narkiewicz. 相似文献