共查询到20条相似文献,搜索用时 953 毫秒
1.
Zachary Maddock 《Monatshefte für Mathematik》2010,160(2):167-186
The Takagi function τ(x) is a continuous non-differentiable function on the unit interval defined by Takagi in 1903. This paper studies level sets L(y) = {x : τ(x) = y} of the Takagi function and bounds their Minkowski dimensions and Hausdorff dimensions above by 0.668. There exist level sets with Minkowski dimension 1/2. The method of proof involves a multiscale analysis that relies upon the self-similarity of τ(x) up to affine shifts. 相似文献
2.
E. Preissmann 《Aequationes Mathematicae》1987,32(1):195-212
We solve independently the equations 1/θ(x)θ(y)=ψ(x)−ψ(y)+φ(x−y)/θ(x−y) and 1/θ(x)θ(y)=σ(x)−σ(y)/θ(x−y)+τ(x)τ(y), τ(0)=0. In both cases we find θ2=aθ4+bθ2+c. We deduce estimates for the spectral radius of a matrix of type(1/θ(x
r
−x
s
)) (the accent meaning that the coefficients of the main diagonal are zero) and we study the case where thex
r
are equidistant.
Dédié to à Monsieur le Professeur Otto Haupt à l'occasion de son cententiare avec les meilleurs voeux 相似文献
3.
The Takagi function ??: [0,1] ?? [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y)?=?{x : ??(x)?=?y} of the Takagi function ??(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a ??generic?? full Lebesgue measure set of ordinates y, the level sets are finite sets. In contrast, here it is shown for a ??generic?? full Lebesgue measure set of abscissas x, the level set L(??(x)) is uncountable. An interesting singular monotone function is constructed associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly ${\frac{3}{2}}$ . 相似文献
4.
M. Thamban Nair 《Advances in Computational Mathematics》2012,36(2):315-329
Problem of solving integral equations of the first kind, òab k(s, t) x(t) dt = y(s), s ? [a, b]\int_a^b k(s, t) x(t)\, dt = y(s),\, s\in [a, b] arises in many of the inverse problems that appears in applications. The above problem is a prototype of an ill-posed problem.
Thus, for obtaining stable approximate solutions based on noisy data, one has to rely on regularization methods. In practice,
the noisy data may be based on a finite number of observations of y, say y(τ
1), y(τ
2), ..., y(τ
n
) for some τ
1, ..., τ
n
in [a, b]. In this paper, we consider approximations based on a collocation method when the nodes τ
1, ..., τ
n
are associated with a convergent quadrature rule. We shall also consider further regularization of the procedure and show
that the derived error estimates are of the same order as in the case of Tikhonov regularization when there is no approximation
of the integral operator is involved. 相似文献
5.
Yu. K. Sabitova 《Russian Mathematics (Iz VUZ)》2009,53(12):41-49
We consider the equation y
m
u
xx
− u
yy
− b
2
y
m
u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u
y
(x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u
x
(0, y) = 0 or u
x
(0, y) = u
x
(1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems 相似文献
6.
7.
Zhiting Xu 《Monatshefte für Mathematik》2007,57(5):157-171
Some oscillation criteria are established by the averaging technique for the second order neutral delay differential equation
of Emden-Fowler type
(a(t)x¢(t))¢+q1(t)| y(t-s1)|a sgn y(t-s1) +q2(t)| y(t-s2)|b sgn y(t-s2)=0, t 3 t0,(a(t)x'(t))'+q_1(t)| y(t-\sigma_1)|^{\alpha}\,{\rm sgn}\,y(t-\sigma_1) +q_2(t)| y(t-\sigma_2)|^{\beta}\,{\rm sgn}\,y(t-\sigma_2)=0,\quad t \ge t_0,
where x(t) = y(t) + p(t)y(t − τ), τ, σ1 and σ2 are nonnegative constants, α > 0, β > 0, and a, p, q
1,
q2 ? C([t0, ¥), \Bbb R)q_2\in C([t_0, \infty), {\Bbb R})
. The results of this paper extend and improve some known results. In particular, two interesting examples that point out
the importance of our theorems are also included. 相似文献
8.
Strongly Closed Subgraphs in a Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 总被引:1,自引:1,他引:0
Akira Hiraki 《Graphs and Combinatorics》2008,24(6):537-550
Let Γ be a distance-regular graph of diameter d ≥ 3 with c
2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
Suppose that the condition (SC)
m
holds. Then it has been known that the condition (BB)
i
holds for all i with 1 ≤ i ≤ m. Similarly we can show that the condition (CA)
i
holds for all i with 1 ≤ i ≤ m. In this paper we prove that if the conditions (BB)
i
and (CA)
i
hold for all i with 1 ≤ i ≤ m, then the condition (SC)
m
holds. Applying this result we give a sufficient condition for the existence of a dual polar graph as a strongly closed subgraph
in Γ. 相似文献
(SC) m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. | |
(BB) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 1 and ∂ Γ (x, z) = ∂ Γ (y, z) = m. Then B(x, z) = B(y, z). | |
(CA) m : Let (x, y, z) be a triple of vertices with ∂ Γ (x, y) = 2, ∂ Γ (x, z) = ∂ Γ (y, z) = m and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z). |
9.
10.
We consider the delay differential equation [(x)\dot](t) = - mx(t) + f(x(t - t))\dot x(t) = - \mu x(t) + f(x(t - \tau )), where μ, τ are positive parameters and f is a strictly monotone, nonlinear C
1-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the
values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function
of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu
[14] about uniqueness and existence of periodic solutions of a system of delay differential equations. 相似文献
11.
P. K. SAHO 《数学学报(英文版)》2005,21(5):1159-1166
In this paper, we determine the general solution of the functional equation f1 (2x + y) + f2(2x - y) = f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4, f5 : R→R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) = g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszfi. The solution of this functional equation can also be determined in certain type of groups using two important results due to Szekelyhidi. 相似文献
12.
In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If
φ(x) is a polynomial, then φ(x) is called a polynomial solution. 相似文献
13.
For a positive integer n we let τ(n) denote the number of its positive divisors. In this paper, we obtain lower and upper bounds for the average value of the
ratio τ(n + 1)/τ(n) as n ranges through positive integers in the interval [1,x]. We also study the cardinality of the sets {τ(p − 1) : p ≤ x prime} and {τ(2n − 1) : n ≤ x}.
Authors’ addresses: Florian Luca, Instituto de Matemáticas, Universidad Nacional Autónoma'de'México, C.P. 58089, Morelia,
Michoacán, México; Igor E. Shparlinski, Department of Computing, Macquarie University, Sydney, NSW 2109, Australia 相似文献
14.
Akira Hiraki 《Graphs and Combinatorics》2009,25(1):65-79
Let Γ be a distance-regular graph of diameter d ≥ 3 with c
2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
In [12] we have shown that the condition (SC)
m
holds if and only if both of the conditions (BB)
i
and (CA)
i
hold for i = 1,...,m. In this paper we show that if a
1 = 0 < a
2 and the condition (BB)
i
holds for i = 1,...,m, then the condition (CA)
i
holds for i = 1,...,m. In particular, the condition (SC)
m
holds. Applying this result we prove that a distance-regular graph with classical parameters (d, b, α, β) such that c
2 > 1 and a
1 = 0 < a
2 satisfies the condition (SC)
i
for i = 1,...,d − 1. In particular, either (b, α, β) = (− 2, −3, −1 − (−2)
d
) or holds. 相似文献
(SC) m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them. | |
(BB) m : Let (x, y, z) be a triple of vertices with ∂Γ(x, y) = 1 and ∂Γ(x, z) = ∂Γ(y, z) = m. Then B(x, z) = B(y, z). | |
(CA) m : Let (x, y, z) be a triple of vertices with and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z). |
15.
ZHENGZUKANG 《高校应用数学学报(英文版)》1997,12(1):33-38
Let X,i.i.d. and Y1i. i.d. be two sequences of random variables with unknown distribution functions F(x) and G(y) respectively. X, are censored by Y1. In this paper we study the uniform consistency of the Kaplan-Meier estimator under the case ey=sup(t:F(t)<1)>to=sup(t2G(t)<1) The sufficient condition is discussed. 相似文献
16.
A class function φ on a finite group G is said to be an order separator if, for every x and y in G \ {1}, φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G\ {1}, φ(x) = φ(y) is equivalent to |C
G
(x)| = |C
G
(y)|. In this paper, finite groups whose nonlinear irreducible complex characters are all order separators (respectively, class-size
separators) are classified. In fact, a more general setting is studied, from which these classifications follow. This analysis
has some connections with the study of finite groups such that every two elements lying in distinct conjugacy classes have
distinct orders, or, respectively, in which disctinct conjugacy classes have distinct sizes.
Received: 10 April 2007 相似文献
17.
In this paper we study asymptotic properties of the third order trinomial delay differential equation (*) y‴(t) − p(t)y′(t) + g(t)y(τ(t)) = 0 by transforming this equation to the binomial canonical equation. The results obtained essentially improve known results
in the literature. On the other hand, the set of comparison principles obtained permits to extend immediately asymptotic criteria
from ordinary to delay equations.
Research was supported by S.G.A. No.1/003/09. 相似文献
18.
Dmitry Dordovskyi Oleksiy Dovgoshey Eugeniy Petrov 《P-Adic Numbers, Ultrametric Analysis, and Applications》2011,3(4):253-262
Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function τ: F(X) → ℝ there is an ultrametric on X such that τ(A) = diamA for every A ∈ F(X). For finite nondegenerate ultrametric spaces (X, d) it is shown that X together with the subset of diametrical pairs of points of X forms a complete k-partite graph, k ⩾ 2, and, conversely, every finite complete k-partite graph with k ⩾ 2 can be obtained by this way. We use this result to characterize the finite ultrametric spaces (X, d) having the minimal card{(x, y): d(x, y) = diamX, x, y ∈ X} for given card X. 相似文献
19.
We use the barrier strip method to prove sufficient conditions for the global solvability of the initial value problem f(t, x, x′) = 0, x(0) = A, including the case in which the function (t, x, y) → f(t, x, y) has a singularity at x = A. 相似文献
20.
Ján Ohriska 《Central European Journal of Mathematics》2008,6(3):439-452
The aim of this paper is to derive sufficient conditions for the linear delay differential equation (r(t)y′(t))′ + p(t)y(τ(t)) = 0 to be oscillatory by using a generalization of the Lagrange mean-value theorem, the Riccati differential inequality
and the Sturm comparison theorem.
相似文献