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1.
Let K be a field of characteristic 0 and let p, q, G
0
, G
1
, P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G
n
(x))
n=0
∞ be defined by the second order linear recurring sequence
In this paper we give conditions under which the diophantine equation G
n
(x) = G
m
(P(x)) has at most exp(1018) many solutions (n, m) ε ℤ2, n, m ⩾ 0. The proof uses a very recent result on S-unit equations over fields of characteristic 0 due to Evertse, Schlickewei and Schmidt [14]. Under the same conditions we
present also bounds for the cardinality of the set
In the last part we specialize our results to certain families of orthogonal polynomials.
This work was supported by the Austrian Science Foundation FWF, grant S8307-MAT.
The second author was supported by the Hungarian National Foundation for Scientific Research Grants No 16741 and 38225.
Received June 5, 2001; in revised form February 26, 2002
RID="a"
ID="a" Dedicated to Edmund Hlawka on the occasion of his 85th birthday 相似文献
2.
We consider solutions of functional-differential equations
in both real and complex variables. We characterize entire solutions g when f is a meromorphic function in the complex plane and a ≠ 0, b, c are constants or polynomials. We also examine questions of existence and uniqueness of the solutions in the real variable
for initial value problems and provide theorems that are valid “in the large”. 相似文献
3.
The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL
m
⊗GL
n
is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A
2 ≤ B ≤ A and nA,A
2 ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N
G
(E(m,n,R, A,B,C)). Bibliography: 46 titles. 相似文献
4.
For fixed generalized reflection matrix P, i.e. P
T
= P, P
2 = I, then matrix X is said to be generalized bisymmetric, if X = X
T
= PXP. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A
1
X
1
B
1 + A
2
X
2
B
2 + ⋯ + A
l
X
l
B
l
= C where [X
1,X
2, ⋯ ,X
l
] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When
the matrix equation is consistent, for any initial generalized bisymmetric matrix group , a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors,
and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized
bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized
bisymmetric matrix group in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation
, where . Given numerical examples show that the algorithm is efficient.
Research supported by: (1) the National Natural Science Foundation of China (10571047) and (10771058), (2) Natural Science
Foundation of Hunan Province (06JJ2053), (3) Scientific Research Fund of Hunan Provincial Education Department(06A017). 相似文献
5.
Donald I. Cartwright 《Monatshefte für Mathematik》2001,247(1):93-109
To any locally finite thick building of type there is naturally associated a commutative algebra of operators. When is constructed from a local field F with local ring , and , then is isomorphic to the convolution algebra of compactly supported bi-K-invariant functions on PGL(n+1,F). We give a proof, valid for any , that the multiplicative functionals on may all be expressed in terms of Hall–Littlewood polynomials. Regarding as a subalgebra of the C *-algebra of bounded operators on the space of square summable functions on the vertex set of , we find the spectrum of the C *-algebra , the closure of . This generalizes results obtained in [3] when n = 1 and in [5] when n = 2. 相似文献
6.
In this paper the invariance criterion is applied for the nonlinear equation
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]
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
(0.1) |
(0.2) |
7.
8.
Let k be a positive integer and α be a real number, and for if the fractional part of is <1/2 and e n =−1 if it is ≥1/2. The pseudorandom properties of the sequence are studied. As measures of pseudorandomness, the regularity of the distribution relative to arithmetic progressions and the correlation are used. Here the special cases k=1 and k=2 are studied (while the case k>2 will be studied in the sequel). 相似文献
9.
10.
The research of the Diophantine equation X~4-Dy~2=1 (1) was started by Ljunggren in 1942, where D>0 and is not a square integer. Since then thanks to the work of Ljunggren, Cohn, Bumby, and Ko Chao(柯召). Sun Chi(孙琦), many advances of the research have been made. One can refer to [1] and its references. 相似文献
11.
For the singular Cauchy problem, the authors find some sufficient conditions for the existence of continuously differentiable
solutions x: (0, ρ] → ℝ (ρ > 0 is sufficiently small) of the form
where m ≥ 2 and c
1,…, c
m
are definite constants.
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal
Conference–2006, Part 3, 2008. 相似文献
12.
13.
OnthePrimeSolutionsoftheDiophantineEquation(X~m-1)/(X-1)=Y~nLeMaohua(乐茂华)(Dept.ofMath.,ZhanjiangTeachersCollege,Zhanjian,Guangd... 相似文献
14.
Samuel A. Ilori 《K-Theory》1989,2(5):623-624
We use the work of Karoubi and Mudrinski on the real Grothendieck's groups of certain complex projective bundles to show that the torsion of the KO
–i
groups of G
3(
n
), n odd, are related to the known torsion of the KO
–i
groups of G2(
n
). 相似文献
15.
Sharp estimates are obtained for the convergence rate of double Fourier series in terms of general orthogonal polynomials
in some classes of functions and for the Kolmogorov N-widths of these classes. These results find applications in numerical analysis.
Original Russian Text ? V.A. Abilov, M.K. Kerimov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi
Fiziki, 2009, Vol. 49, No. 8, pp. 1364–1368. 相似文献
16.
Andreas Arvanitoyeorgos Ioannis Chrysikos Yusuke Sakane 《Annals of Global Analysis and Geometry》2010,38(4):413-438
We find the precise number of non-K?hler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U(p)×U(n−p) with n ≥ 4 and 2 ≤ p ≤ n−2. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems.
We also examine the isometric problem for these Einstein metrics. 相似文献
17.
David Soudry 《Israel Journal of Mathematics》2000,120(1):511-561
In this paper we prove the full multiplicativity (in both variables) of gamma factors for generic representations of SO2ℓ+1 × GL
n
. These gamma factors are initially defined as proportionality factors of local functional equations, derived from a corresponding
global theory of certain Rankin-Selberg integrals which interpolate standardL-functions for SO2ℓ+1 × GL
n
. 相似文献
18.
We prove that the quasi-Banach spaces 1 (p) and p (1), 0 < p < 1 have a unique unconditional basis up to permutation 相似文献
19.
We consider the projective line over the finite quotient ring R⋄ ≡ GF(2)[x]/〈x
3 − x〉. The line is endowed with 18 points, spanning the neighborhoods of three pairwise distant points. Because R⋄ is not a local ring, the neighbor (or parallel) relation is not an equivalence relation, and the sets of neighbors for two
distant points hence overlap. There are nine neighbors of any point on the line, forming three disjoint families under the
reduction modulo either of the two maximal ideals of the ring. Two of the families contain four points each, and they swap
their roles when switching from one ideal to the other, the points in one family merging with (the image of) the point in
question and the points in the other family passing in pairs into the remaining two points of the associated ordinary projective
line of order two. The single point in the remaining family passes to the reference point under both maps, and its existence
stems from a nontrivial character of the Jacobson radical
of the ring. The quotient ring
is isomorphic to GF(2) ⊗ GF(2). The projective line over
features nine points, each of them surrounded by four neighbors and four distant points, and any two distant points share
two neighbors. We surmise that these remarkable ring geometries are relevant for modeling entangled qubit states, which we
will discuss in detail in Part II of this paper.
__________
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 44–53, April, 2007. 相似文献