The Singularity Spectrum f(α) of Some Moran Fractals

We show that the multifractal decomposition behaves as expected for a family of sets E known as homogeneous Moran fractals associated with the Fibonacci sequence , using probability measures () associated with the Fibonacci sequence . For each value of a parameter (_min, _max), we define multifractal components E of E, and show that they are fractals in the sense of Taylor. We give the explicit formula for the dimension of E. Also our method can be used for the Moran fractals associated with some more general sequences.     

The recursive dimension formula for graded lie algebras and its applications

In this paper, we obtain a recursive dimension formula for all γ- graded Lie algebras 𝔏 = +_α∈γ𝔩_α with finite dimensional homogeneous sub-spaces, where T is a countable abelian semigroup satisfying a certain finiteness condition. We apply this dimension formula to several Lie algebras to obtain explicit and simple dimension formulas.     

Fractal systems of central places based on intermittency of space-filling

The central place models are fundamentally important in theoretical geography and city planning theory. The texture and structure of central place networks have been demonstrated to be self-similar in both theoretical and empirical studies. However, the underlying rationale of central place fractals in the real world has not yet been revealed so far. This paper is devoted to illustrating the mechanisms by which the fractal patterns can be generated from central place systems. The structural dimension of the traditional central place models is d = 2 indicating no intermittency in the spatial distribution of human settlements. This dimension value is inconsistent with empirical observations. Substituting the complete space filling with the incomplete space filling, we can obtain central place models with fractional dimension D < d = 2 indicative of spatial intermittency. Thus the conventional central place models are converted into fractal central place models. If we further integrate the chance factors into the improved central place fractals, the theory will be able to explain the real patterns of urban places very well. As empirical analyses, the US cities and towns are employed to verify the fractal-based models of central places.     

The quantization dimension of the self-affine measures on general Sierpi��ski carpets

Let??? be a self-affine measure on a general Sierpi??ski carpet E. We give a characterization for the upper and lower quantization dimension of??? in terms of revised cylinder sets. Using this characterization, we prove that the quantization dimension D _r (??) of??? exists for all r > 0 under an additional condition. We establish an explicit formula for D _r (??) and show that it increases to the box-counting dimension ${dim_B^* \mu}$ of??? as r tends to infinity. For a class of Sierpi??ski carpets E and the uniform measures??? on E, we show that the quantization dimension of??? coincides with its box-counting dimension and that the D _r (??)-dimensional upper and lower quantization coefficient of??? are both positive and finite.     

An improved estimation to make Markowitz’s portfolio optimization theory users friendly and estimation accurate with application on the US stock market investment

Using the Markowitz mean–variance portfolio optimization theory, researchers have shown that the traditional estimated return greatly overestimates the theoretical optimal return, especially when the dimension to sample size ratio p/n is large. Bai et al. (2009) propose a bootstrap-corrected estimator to correct the overestimation, but there is no closed form for their estimator. To circumvent this limitation, this paper derives explicit formulas for the estimator of the optimal portfolio return. We also prove that our proposed closed-form return estimator is consistent when n → ∞ and p/n → y ∈ (0, 1). Our simulation results show that our proposed estimators dramatically outperform traditional estimators for both the optimal return and its corresponding allocation under different values of p/n ratios and different inter-asset correlations ρ, especially when p/n is close to 1. We also find that our proposed estimators perform better than the bootstrap-corrected estimators for both the optimal return and its corresponding allocation. Another advantage of our improved estimation of returns is that we can also obtain an explicit formula for the standard deviation of the improved return estimate and it is smaller than that of the traditional estimate, especially when p/n is large. In addition, we illustrate the applicability of our proposed estimate on the US stock market investment.     

Peakon, loop and solitary travelling wave solutions for the general modified CH-DP equation

Travelling wave solutions for the general modified CH-DP equation u_t − u_xxt + αu²u_x − βu_xu_xx = uu_xxx are developed. By using the dynamical system method, a peakon and a dark soliton are found to coexist for the same wave speed. Exact explicit blow-up solutions are given. By using numerical simulation, a loop solution for a special case is discussed.     

The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations

Let ∥ · ∥ be the Frobenius norm of matrices. We consider (I) the set S_E of symmetric and generalized centro-symmetric real n × n matrices R_n with some given eigenpairs (λ_j, q_j) (j = 1, 2, … , m) and (II) the element in S_E which minimizes for a given real matrix R^∗. Necessary and sufficient conditions for S_E to be nonempty are presented. A general form of elements in S_E is given and an explicit expression of the minimizer is derived. Finally, a numerical example is reported.     

Weighted composition operators between weighted spaces of vector-valued holomorphic functions on Banach spaces

Let B(E, F) be the Banach Space of all continuous linear operators from a Banach Space E into a Banach space F. Let U_X and U_Y be balanced open subsets of Banach spaces X and Y, respectively. Let V and W be two Nachbin families of continuous weights on U_X and U_Y, respectively. Let HV(U_X, E) (or HV₀(U_X, E)) and HW(U_Y, F) (or HW₀(U_Y, F)) be the weighted spaces of vector-valued holomorphic functions. In this paper, we investigate the holomorphic mappings ? : U_Y → U_X and ψ : U_Y → B(E, F) which generate weighted composition operators between these weighted spaces.     

On the rates of convergence of BBH-Kantorovich operators and their Bézier variant

In the present paper we consider the Bézier variant of BBH-Kantorovich operators J_n,αf for functions f measurable and locally bounded on the interval [0, ∞) with α ? 1. By using the Chanturiya modulus of variation we estimate the rate of pointwise convergence of J_n,αf(x) at those x > 0 at which the one-sided limits f(x+), f(x−) exist. The very recent result of Chen and Zeng (2009) [L. Chen, X.M. Zeng, Rate of convergence of a new type Kantorovich variant of Bleimann-Butzer-Hahn Operators, J. Inequal. Appl. 2009 (2009) 10. Article ID 852897] is extended to more general classes of functions.     

齐次集及其拟Lipschitz等价

本文定义并研究一类齐次分形,该类分形包含所有的(拟)Ahlfors-David正则集和许多非正则的Moran集,这里如果一个分形的Hausdorff维数与packing维数不相等,则称它是非正则的.对于这类齐次分形,本文得出它们的分形维数,并且给出在适当分离条件下两个齐次分形拟Lipschitz等价的充要条件.随后,本文将这些结果应用到非正则的Moran集上.     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

The multifractal spectrum of some moran measures

The multifractal formalism is shown to hold for a class of Moran measures supported on the Moran fractals associated with the sequences of which the frequency of the letter exists.     

An approximate analytical solution of time-fractional telegraph equation

In this article, the powerful, easy-to-use and effective approximate analytical mathematical tool like homotopy analysis method (HAM) is used to solve the telegraph equation with fractional time derivative α (1 < α ? 2). By using initial values, the explicit solutions of telegraph equation for different particular cases have been derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical model.     

Multifractality in relativistic charged particles produced at SPS energies

The multifractal analysis of relativistic shower particles produced in ³²S–emulsion interactions at 200 AGeV has been investigated using the method of modified multifractal moments, G_q, in pseudo-rapidity space. The anomalous fractal dimension, d_q, and generalized fractal dimensions, D_q, are determined for the present data for different order of moment. The experimental data reflects multifractal geometry in a multipion production process. The downward concave shape of the multifractal spectral function, f(α_q), gives an evidence for self-similar cascade mechanism. The multifractal specific heat has also been evaluated for the present data using the generalized fractal dimensions, D_q. We compared our experimental results with those obtained from simulated events of the Lund Monte Carlo Code FRITIOF and uncorrelated Monte Carlo events, (MC-RAND) generated randomly in pseudorapidity space based on the assumption of independent emission of particles. The experimental data on multifractality has been found to exhibit a remarkable proximity to the analogous data obtained from the FRITIOF code and the uncorrelated Monte Carlo events.     

Eigenvalues and the One-Dimensional p-Laplacian

We consider the boundary value problem (?_p(u′))′ + λF(t, u) = 0, with p > 1, t ∈ (0, 1), u(0) = u(1) = 0, and with λ > 0. The value of λ is chosen so that the boundary value problem has a positive solution. In addition, we derive an explicit interval for λ such that, for any λ in this interval, the existence of a positive solution to the boundary value problem is guaranteed. In addition, the existence of two positive solutions for λ in an appropriate interval is also discussed.     

A note on the partition dimension of Cartesian product graphs

Let G = (V, E) be a connected graph. The distance between two vertices u, v ∈ V, denoted by d(u, v), is the length of a shortest u − v path in G. The distance between a vertex v ∈ V and a subset P ⊂ V is defined as , and it is denoted by d(v, P). An ordered partition {P₁, P₂, … , P_t} of vertices of a graph G, is a resolving partition of G, if all the distance vectors (d(v, P₁), d(v, P₂), … , d(v, P_t)) are different. The partition dimension of G, denoted by pd(G), is the minimum number of sets in any resolving partition of G. In this article we study the partition dimension of Cartesian product graphs. More precisely, we show that for all pairs of connected graphs G, H, pd(G × H) ? pd(G) + pd(H) and pd(G × H) ? pd(G) + dim(H), where dim(H) denotes the metric dimension of H. Consequently, we show that pd(G × H) ? dim(G) + dim(H) + 1.     

Structural stability of (C, A)-marked subspaces

Given an observable pair of matrices (C, A) we consider the manifold of (C, A)-invariant subspaces having a fixed Brunovsky-Kronecker structure. Using Arnold techniques we obtain the explicit form of a miniversal deformation of a marked (C, A)-invariant subspace with respect to the usual equivalence relation. As an application, we obtain the dimension of the orbit and we characterize the structurally stable subspaces (those with open orbit).     

Super connectivity of k-regular interconnection networks

Super connectivity is an important issue in interconnection networks. It has been shown that if a network possesses the super connectivity property, it has a high reliability and a small vertex failure rate. Many interconnection networks, like the hypercubes, twisted-cubes, crossed-cubes, möbius cubes, split-stars, and recursive circulant graphs, are proven to be super connected; and the augmented cubes are maximum connected. However, each network vertex has a higher degree as long as the number of vertices increases exponentially. For example, each vertex of the hypercube Q_n has a degree of n, and each vertex of the augmented cube AQ_n has a degree of 2n − 1. In this paper, we not only show that the augmented cube AQ_n is super connected for n = 1, 2 and n ? 4, but also propose a variation of AQ_n, denoted by AQ_n,i, such that V(AQ_n,i) = V(AQ_n), E(AQ_n,i) ⊆ E(AQ_n), and AQ_n,i is i-regular with n ? 3 and 3 ? i ? 2n − 1, in which AQ_n,i is also super connected. In addition, we state the diameter of AQ_n,i.     

Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations

In this paper, existence and attractiveness of solutions for quadratic Urysohn fractional integral equations on an unbounded interval are obtained by virtue of Tichonov fixed point theorem and suitable conjunction of the well known measure ω₀(X) and the spaces C(R⁺). Further, three certain solutions sets X_L,γ, X_1,α and X_{1,(1−(α+v))}, which tending to zero at an appropriate rate t^−ν (ν > 0), ν = γ (or α or 1 − (α + v)) as t → ∞, are introduced and stability of solutions for quadratic Urysohn fractional integral equations are obtained based on these solutions sets respectively by applying Schauder fixed point theorem via some easy checked conditions. An example is given to illustrate the results.