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1.
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
2.
In this paper, we study of Pólya urn model containing balls of (m+1) different labels under a general replacement scheme, which is characterized by an (m+1) × (m+1) addition matrix of integers without constraints on the values of these (m+1)2 integers other than non-negativity. LetX
1,X
2,...,X
n
be trials obtained by the Pólya urn scheme (with possible outcomes: “O”, “1”,...“m”). We consider the multivariate distributions of the numbers of occurrences of runs of different types arising from the various
enumeration schemes and give a recursive formula of the probability generating function. Some closed form expressions are
derived as special cases, which have potential applications to various areas. Our methods for the derivation of the multivariate
run-related distribution are very simple and suitable for numerical and symbolic calculations by means of computer algebra
systems. The results presented here develop a general workable framework for the study of Pólya urn models. Our attempts are
very useful for understanding non-classic urn models. Finally, numerical examples are also given in order to illustrate the
feasibility of our results.
This research was partially supported by the ISM Cooperative Research Program (2003-ISM·CRP-2007). 相似文献
3.
We prove that the sequence [ξ(5/4)n], n=1,2, . . . , where ξ is an arbitrary positive number, contains infinitely many composite numbers. A corresponding result for the sequences [(3/2)n] and [(4/3)n],n=1,2, . . . , was obtained by Forman and Shapiro in 1967. Furthermore, it is shown that there are infinitely many positive
integers n such that ([ξ(5/4)n],6006)>1, where 6006=2·3·7·11·13. Similar results are obtained for shifted powers of some other rational numbers. In particular,
the same is proved for the sets of integers nearest to ξ(5/3)n and to ξ(7/5)n, n∈ℕ. The corresponding sets of possible divisors are also described. 相似文献
4.
Daniel A. Nicks 《Journal d'Analyse Mathématique》2012,117(1):87-118
A real entire function belongs to the Laguerre-Pólya class LP if it is the limit of a sequence of real polynomials with real zeroes. By building upon results that resolved a long-standing
conjecture of Wiman, a number of conditions are established under which a real entire function f must belong to the class LP, or to one of the related classes U
2p
*. These conditions typically involve the non-real zeroes of f and its derivatives, or those of the differential polynomial f f″−a(f′)2. 相似文献
5.
We consider a class of fourth-order nonlinear difference equations of the form {fx006-01} where α, β are ratios of odd positive integers and {p
n}, {q
n} are positive real sequences defined for all n ∈ ℕ(n
0). We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic
behavior under suitable combinations of convergence or divergence conditions for the sums {fx006-02}.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 8–27, January, 2008. 相似文献
6.
This paper is concerned with the Shohat-Favard, Chebyshev and Modified Chebyshev methods for d-orthogonal polynomial sequences d∈ℕ. Shohat-Favard’s method is presented from the concept of dual sequence of a sequence of polynomials. We deduce the recurrence
relations for the Chebyshev and the Modified Chebyshev methods for every d∈ℕ. The three methods are implemented in the Mathematica programming language. We show several formal and numerical tests
realized with the software developed.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
7.
A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain
norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another
example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s
proof of Szemerédi’s Theorem) defined by the authors. For each integer k ≥ 1, we define seminorms on ℓ∞(ℤ) analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences,
which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences).
Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a
nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence.
As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic
theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem. 相似文献
8.
The well-known Taikov’s refined versions of the Hardy – Littlewood – Pólya inequality for the L
2 -norms of intermediate derivatives of a function defined on the real axis are generalized to the case of powers of self-adjoint
operators in a Hilbert space. 相似文献
9.
Tomasz Schoen 《Acta Mathematica Hungarica》2012,135(3):229-235
We prove two results concerning solvability of a linear equation in sets of integers. In particular, it is shown that for
every k∈ℕ, there is a noninvariant linear equation in k variables such that if A⫅{1,…,N} has no solution to the equation then
|A|\leqq 2-ck/(logk)2N|A|\leqq 2^{-ck/{(\log k)}^{2}}N, for some absolute constant c>0, provided that N is large enough. 相似文献
10.
We study equidistribution properties of nil-orbits (b
n
x)
n∈ℕ when the parameter n is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if X = G/Γ is a nilmanifold, b ∈ G is an ergodic nilrotation, and c ∈ ℝ \ ℤ is positive, then the sequence $
(b^{[n^c ]} x)_{n \in \mathbb{N}}
$
(b^{[n^c ]} x)_{n \in \mathbb{N}}
is equidistributed in X for every x ∈ X. This is also the case when n
c
is replaced with a(n), where a(t) is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials,
and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established
by Boshernitzan when X is the circle. 相似文献
11.
We define Wieferich numbers to be those odd integers w≥3 that satisfy the congruence 2
φ(w)≡1 (mod w
2). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and
we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numbers;
analogous results for the set of Wieferich primes remain out of reach. Finally, we consider several modifications of the above
definition and demonstrate that our methods apply to such sets of integers as well.
During the preparation of this paper, W.B. was supported in part by NSF grant DMS-0070628, F.L. was supported in part by grants
SEP-CONACYT 37259-E and 37260-E, and I.S. was supported in part by ARC grant DP0211459. 相似文献
12.
Julius Borcea 《Arkiv f?r Matematik》2006,44(2):211-240
The spectral order on R
n
induces a natural partial ordering on the manifold of monic hyperbolic polynomials of degree n. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony
property for infinite families of deformations of these operators parametrized by the space ℓ∞ of real bounded sequences. As a consequence, we deduce that the monoid of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya
type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global
minimum of its -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order. 相似文献
13.
JianYaLIU TaoZHAN 《数学学报(英文版)》2005,21(2):335-350
It is proved that with at most O(N^11/12 c) exceptions, all positive integers n ≤ N satisfying some necessary congruence conditions are the sum of three squares of primes. This improves substantially the previous results in this direction. 相似文献
14.
Jean-Philippe Furter 《manuscripta mathematica》1999,98(2):183-193
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 相似文献
15.
Some results about the continuity of special linear maps between F-spaces recently obtained by Drewnowski have motivated us to revise a closed graph theorem for quasi-Suslin spaces due to
Valdivia. We extend Valdivia’s theorem by showing that a linear map with closed graph from a Baire tvs into a tvs admitting
a relatively countably compact resolution is continuous. This also applies to extend a result of De Wilde and Sunyach. A topological
space X is said to have a (relatively countably) compact resolution if X admits a covering {A
α
:α ∈ ℕℕ} consisting of (relatively countably) compact sets such that A
α
⊆ A
β
for α ⩽ β. Some applications and two open questions are provided. 相似文献
16.
In this paper we give necessary and sufficient conditions for the block sequence of the set X = {x
1 < x
2 < … < x
n
< …} ⊂ ℕ to have an asymptotic distribution function in the form x
λ. 相似文献
17.
A stable set in a graph G is a set of pairwise nonadjacent vertices. The problem of finding a maximum weight stable set is one of the most basic ℕℙ-hard
problems. An important approach to this problem is to formulate it as the problem of optimizing a linear function over the
convex hull STAB(G) of incidence vectors of stable sets. Since it is impossible (unless ℕℙ=coℕℙ) to obtain a “concise” characterization of STAB(G) as the solution set of a system of linear inequalities, it is a more realistic goal to find large classes of valid inequalities
with the property that the corresponding separation problem (given a point x
*, find, if possible, an inequality in the class that x
* violates) is efficiently solvable.?Some known large classes of separable inequalities are the trivial, edge, cycle and wheel
inequalities. In this paper, we give a polynomial time separation algorithm for the (t)-antiweb inequalities of Trotter. We then introduce an even larger class (in fact, a sequence of classes) of valid inequalities,
called (t)-antiweb-s-wheel inequalities. This class is a common generalization of the (t)-antiweb inequalities and the wheel inequalities. We also give efficient separation algorithms for them.
Received: June 2000 / Accepted: August 2001?Published online February 14, 2002 相似文献
18.
For a set A, let P(A) be the set of all finite subset sums of A. We prove that if a sequence B={b
1<b
2<⋯} of integers satisfies b
1≧11 and b
n+1≧3b
n
+5 (n=1,2,…), then there exists a sequence of positive integers A={a
1<a
2<⋯} for which P(A)=ℕ∖B. On the other hand, if a sequence B={b
1<b
2<⋯} of positive integers satisfies either b
1=10 or b
2=3b
1+4, then there is no sequence A of positive integers for which P(A)=ℕ∖B. 相似文献
19.
Multiplicatively large sets are defined in (ℕ, ·) by an analogy to sets of positive upper density in (ℕ, +). By utilizing
various ergodic multiple recurrence theorems, we show that multiplicatively large sets have a rich combinatorial structure.
In particular, it is proved that for any multiplicatively large setE ⊂ ℕ and anyk ∈ ℕ, there existsa,b,c,d,e,q ∈ ℕ such that {fx23-1}
Dedicated to MORI VAHAVERI Hillel Furstenberg on the occasion of his retirement
The author acknowledges support received from the National Science Foundation (USA) via grants DMS-0070566 and DMS-0345350. 相似文献
20.
M. A. Nalbandyan 《Russian Mathematics (Iz VUZ)》2009,53(10):45-56
For any sequence {ω(n)}
n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n
2 + o(ω(n))}
n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $
\mathop \sum \limits_{k \in \Lambda }
$
\mathop \sum \limits_{k \in \Lambda }
α
k
ω
k
(x) that converges a. e. to this function, where {w
k
(x)}
k∈ℕ is the Walsh system. We find representation spectra in the form {n
l
+ o(n
l
)}
n∈ℕ, where l ∈ {2
k
}
k∈ℕ. 相似文献