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1.
Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers
2 ≤ s < t let f
s,t
(n) = min{max{|S|: S ⊆ V (H) and H[S] contains no K
s
}}, where the minimum is taken over all K
t
-free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds
is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n
1/2+o(1)) ≤ f
s,s+1(n) ≤ O(n
1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f
s,s+1(n) ≤ O(n
2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ k ≪ s, Ω(n
1/2−ɛ
) ≤ f
s,s+k
(n) ≤ O(n
1/2+ɛ
. In addition, we also discuss some connections between the function f
s,t
and vertex Folkman numbers. 相似文献
2.
N. V. Lazakovich S. P. Stashulenok O. L. Yablonskii 《Lithuanian Mathematical Journal》1999,39(2):196-202
In this paper, we consider problems of approximation of stochastic θ-integrals (θ)∫
0
t
f(B(s))dB(s) with respect to a Brownian motion by sums of the form ∑
k=1
p
fn(B
n
θ
(tk-1))[B
n
θ
(tk)-B
n
θ
(tk-1], where the sequences {fn,n∈∕#x007D; and {[B
n
θ
,n∈∕} are convolution-type approximations of the functionf and Brownian motionB.
Belorussian State University, F. Skoryna ave. 4, 220050 Minsk, Belorus. Translated from Lietuvos Matematikos Rinkinys, Vol.
39, No. 2, pp. 248–256, April–June, 1999.
Translated by V. Mackevičius 相似文献
3.
We study the injectivity properties of the spherical mean value operators associated to the Gelfand pairs (H
n,K), whereK is a compact subgroup ofU(n). We show that these spherical mean value operators are injective onL
p Hn) for 1≤p<∞. Forp=∞, these operators are not injective. Nevertheless, if the spherical meansf*μ
i
overK-orbits of sufficiently many points (z
i,t
i) ∈H
n vanish, we identify a necessary and sufficient condition on the points (z
i,t
i) which guaranteesf=0. ForK=U(n), this is equivalent to the condition for the two-radius theorem.
Research supported by N.B.H.M. Research Grant, Govt. of India. 相似文献
4.
Der-Shin Chang Yuang-Chin Chiang 《Annals of the Institute of Statistical Mathematics》1980,32(1):275-281
Consider a realization of the process
on the intervalT=[0,1] for functionsf
1(t),f
2(t),...,f
n
(t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f
k
(t)}
k=1
n
to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD
s-optimal design problems are discussed. 相似文献
5.
V. A. Ustimenko 《Journal of Mathematical Sciences》2007,140(3):461-471
The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K,
i.e., a family F of nonlinear polynomial maps f
α : K
n
→ K
n
depending on “time” α ∈ {K − 0} such that f
α
−1 = f
−αM, the relation f
α1 (x) = f
α2 (x) for some x ∈ Kn implies α1 = α2, and each map f
α has no invariant points. The neighborhood {f
α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kn. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α1, υ, α2), s ≤ d, where αi + αi+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ
a = f
α1 × ⋯ × f
αs
(υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems Ln(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system Ln(q). If K = Fq, the graphs L(n, Fq) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative
ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops
to 4. We introduce some other families of graphs of large girth related to the dynamical systems Ln(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric
cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs
without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234. 相似文献
6.
Camino Balbuena Martín Cera Pedro García-Vázquez Juan Carlos Valenzuela 《数学学报(英文版)》2011,27(11):2085-2100
For a bipartite graph G on m and n vertices, respectively, in its vertices classes, and for integers s and t such that 2 ≤ s ≤ t, 0 ≤ m − s ≤ n − t, and m + n ≤ 2s + t − 1, we prove that if G has at least mn − (2(m − s) + n − t) edges then it contains a subdivision of the complete bipartite K
(s,t) with s vertices in the m-class and t vertices in the n-class. Furthermore, we characterize the corresponding extremal bipartite graphs with mn − (2(m − s) + n − t + 1) edges for this topological Turan type problem. 相似文献
7.
Letf(s, t; k) be the largest value ofm such that it is possible tok-color the edges of the complete graphK
m
so that everyK
s
⊆K
m
has exactlyt colors occuring on its edges. The main object of this paper is to describe the behavior of the functionf(s,t;k), usually thinking ofs andt fixed, and lettingk become large.
Dedicated to Paul Erdős on his seventieth birthday 相似文献
8.
Evangelos A. Latos Dimitrios E. Tzanetis 《NoDEA : Nonlinear Differential Equations and Applications》2010,17(2):137-151
We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ${u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}We investigate the behaviour of solution u = u(x, t; λ) at λ = λ* for the non-local porous medium equation ut = (un)xx + lf(u)/(ò-11 f(u)dx)2{u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2} with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s
n-1
f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution.
For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* = u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all x ? (-1,1){x\in(-1,1)}. 相似文献
9.
An Application of a Mountain Pass Theorem 总被引:3,自引:0,他引:3
We are concerned with the following Dirichlet problem:
−Δu(x) = f(x, u), x∈Ω, u∈H
1
0(Ω), (P)
where f(x, t) ∈C (×ℝ), f(x, t)/t is nondecreasing in t∈ℝ and tends to an L
∞-function q(x) uniformly in x∈Ω as t→ + ∞ (i.e., f(x, t) is asymptotically linear in t at infinity). In this case, an Ambrosetti-Rabinowitz-type condition, that is, for some θ > 2, M > 0,
0 > θF(x, s) ≤f(x, s)s, for all |s|≥M and x∈Ω, (AR)
is no longer true, where F(x, s) = ∫
s
0
f(x, t)dt. As is well known, (AR) is an important technical condition in applying Mountain Pass Theorem. In this paper, without assuming
(AR) we prove, by using a variant version of Mountain Pass Theorem, that problem (P) has a positive solution under suitable
conditions on f(x, t) and q(x). Our methods also work for the case where f(x, t) is superlinear in t at infinity, i.e., q(x) ≡ +∞.
Received June 24, 1998, Accepted January 14, 2000. 相似文献
10.
Z. Ditzian 《Israel Journal of Mathematics》1974,17(3):315-324
Saturation classes for the sequenceK
n
(f, x) = ∫f(x −t)dμ
n
(t) of linear operators whereK
n(f, x) is of the limited oscillation type, that is,μ
n
(t) is monotonic fort ≠ [−Aσ
n
,Aσ
n
],σ
n
=o(1),n → ∞ and ∫t
2m
dμ
n
(t), are obtained. Examples of applications to some sequences of non-positive operators are given. 相似文献
11.
12.
Constance van Eeden 《Annals of the Institute of Statistical Mathematics》1985,37(1):461-472
Summary Asymptotic properties of the mean integrated squared error (MISE) of kernel estimators of a density function, based on a sampleX
1, …,X
n, were obtained by Rosenblatt [4] and Epanechnikov [1] for the case when the densityf and its derivativef′ are continuous. They found, under certain additional regularity conditions, that the optimal choiceh
n0 for the scale factorh
n=Kn−α is given byh
n0=K0n−1/5 withK
0 depending onf and the kernel; they also showed that MISE(h
n0)=O(n−4/5) and Epanechnikov [1] found the optimal kernel.
In this paper we investigate the robustness of these results to departures from the assumptions concerning the smoothness
of the density function. In particular it is shown, under certain regularity conditions, that whenf is continuous but its derivativef′ is not, the optimal value of α in the scale factor becomes 1/4 and MISE(h
n0)=O(n−3/4); for the case whenf is not continuous the optimal value of α becomes 1/2 and MISE(h
n0)=O(n−1/2). For this last case the optimal kernel is shown to be the double exponential density.
Supported by the Natural Sciences and Engineering Research Council of Canada under Grant Nr. A 3114 and by the Gouvernement
du Québec, Programme de formation de chercheurs et d'action concertée. 相似文献
13.
A. A. Dovgoshei 《Mathematical Notes》1995,58(3):921-927
For an arbitrary compact setK⊂ℂ, we relate the order and the type of an entire functionf to the sequenceE
n
(f,K) of best polynomial approximations to this function onK.
Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 355–364, September, 1995. 相似文献
14.
Given two graphs G and H, let f(G,H) denote the minimum integer n such that in every coloring of the edges of Kn, there is either a copy of G with all edges having the same color or a copy of H with all edges having different colors. We show that f(G,H) is finite iff G is a star or H is acyclic. If S and T are trees with s and t edges, respectively, we show that 1+s(t?2)/2≤f(S,T)≤(s?1)(t2+3t). Using constructions from design theory, we establish the exact values, lying near (s?1)(t?1), for f(S,T) when S and T are certain paths or star‐like trees. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 1–16, 2003 相似文献
15.
Letf
t
be aC
2 Axiom A dynamical system on a compact manifold satisfying the transversality condition. We prove that ifB
x
(ε,t)=[y: dist (f
s
x,f
s
y)≤ε for all 0≤s≤t], then volB
x
(ε,t) has the order exp(∫
0
t
φ (f
s
x)ds) in the continuous time case and exp (Σ
s
t−1
φ (f
s
x)) in the discrete time case, whereφ is a Holder continuous extension from basic hyperbolic sets of the negative of the differential expansion coefficient in
the unstable direction. An application to the theory of large deviations is given.
Partially supported by US-Israel BSF.
Partially supported by a Darpa grant. 相似文献
16.
W. Luh T. Meyrath M. Niess 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2008,43(6):365-371
The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ n } ? ?, there exists a function ?, meromorphic on ?, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n k } of ? such that $ \left\{ {\varphi \left( {z + \lambda _{n_k } } \right)} \right\} The paper proves the following result on universal meromorphic approximation: Given any unbounded sequence {λ
n
} ⊂ ℂ, there exists a function ϕ, meromorphic on ℂ, with the following property. For every compact set K of rational approximation (i.e. Vitushkin set), and every function f, continuous on K and holomorphic in the interior of K, there exists a subsequence {n
k
} of ℕ such that
converges to f(z) uniformly on K.
A similar result is obtained for arbitrary domains G ≠ ℂ. Moreover, in case {λ
n
}={n} the function ϕ is frequently universal in terms of Bayart/Grivaux [3].
Original Russian Text ? W.Luh, T.Meyrath, M.Niess, 2008, published in Izvestiya NAN Armenii. Matematika, 2008, No. 6, pp.
66–75. 相似文献
17.
Transition waves and interactions between two kinds of instability—vortex shedding and transition wave in the near wake of
a circular cylinder in the Reynolds number range 3 000–10 000 are studied by a domain decomposition hybrid numerical method.
Based on high resolution power spectral analyses for velocity new results on the Reynolds-number dependence of the transition
wave frequency, i.e.f
t
/fs∼Re0.87 are obtained. The new predictions are in good agreement with the experimental results of Wei and Smith but different from
Braza s prediction and some early experimental resultsf
t/fs∼Re
0.5 given by Blooret al. The multi-interactions between two kinds of vortex are clearly visualized numerically. The strong nonlinear interactions
between the two independent frequencies (f
t,f
s
) leading to spectra broadening to form the couplingmf
s
+nf
t
are predicted and analyzed numerically, and the characteristics of the transition are described. Longitudinal variations
of the transition wave and its coupling are reported. Detailed mechanism of the flow transition in the near wake before occurrence
of the thedimensional evolution is provided.
Project supported by the National Natural Science Foundation of China, the LNM of Institute of Mechanics, and partially by
the National Basic Research Project. 相似文献
18.
For everyt>1 and positiven we construct explicit examples of graphsG with |V (G)|=n, |E(G)|c
t
·n
2–1/t
which do not contain a complete bipartite graghK
t,t
!+1 This establishes the exact order of magnitude of the Turán numbers ex (n, K
t,s
) for any fixedt and allst!+1, improving over the previous probabilistic lower bounds for such pairs (t, s). The construction relies on elementary facts from commutative algebra.Research supported in part by NSF Grants DMS-8707320 and DMS-9102866.Research supported in part by Hungarian National Foundation for Scientific Research Grant 相似文献
19.
We give a solution of the problem about trigonometric polynomials with a given leading harmonic and least deviating from zero
in measure; more precisely, with respect to the functional μ(f
n
) = mes {t ∈ [0, 2π]: |f
n
(t)| ≥ 1}. We give a solution of a related problem about the minimal value over compact sets (from the real line) of a given
measure of least uniform deviation from zero on a compact set for trigonometric polynomials with a fixed leading harmonic.
Published in Russian in Doklady Akademii Nauk, 2009, Vol. 425, No. 6, pp. 733–736.
Presented by Academician A.M. Il’in November 13, 2008
The article was translated by the authors. 相似文献
20.
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