MathSciDoc: An Archive for Mathematician ∫

We show that if E is an s-regular set in R^2 for which the triple integral \int_E \int_E \int_E c(x, y, z)^{2s} dH^sx dH^sy dH^sz of the Menger curvature c is finite and if 0 < s ≤ 1/2, then H^s almost all of E can be covered with countably many C^1 curves. We give an example to show that this is false for 1/2 <s< 1.

The problem is concerned about how large (e.g. the Hausdorff dimension) is Whitney's critical set contained in a given fractal. For this, we prove that the Moran arc, an arc containing a Moran set, is a Whitney's critical set. The excellent open set condition is defined, when the condition holds, the associated self-similar set contains a Whitney's critical subset of full dimension. As its application, the Sierpinski gasket and Koch curve have Whitney's critical subset of full dimension. Finally, we provide a self-similar tree which never contains any Whitney's critical set.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

A self-similar measure on R^n is defined to be a probability measure satisfying \mu = \sum_{j=1}^N p_j\mu\circ S_j^{-1} + \sum_{j=1}^M q_j(\mu*\mu)\circ T_j^{-1}, where S_j x = \rho_j R_j x + b_j, T_j x = \eta_j Q_j x + c_j are contractive similarities, 0 < \rho_j < 1, 0 < \eta_j < 1/2, 0 < p_j < 1, 0 < q_j < 1; \sum_{j=1}^N p_j + \sum_{j=1}^M q_j = 1, R_j, Q_j are orthogonal matrix and \mu*\mu is the convolution of two measures. When M = 0, \mu is a linear self-similar measure, we establish the asymptotic behavior of averages of the derivative of the Fourier transform of \mu, such as \int_{|x|\le R} |(\frac{\partial}{\partial x})^\alpha \hat\mu(x)|^2 dx = O(R^{n-\beta}) for any order derivation of \hat\mu(x) as R\to\infty under certain additional hypotheses. When M > 0, \mu is a nonlinear self-similar measure, we get some results of L^p boundedness for maximal operators of l, from the pointwise asymptotic estimate of the Fourier transform of \mu made by Strichartz.

This paper proves the local Lipschitz property for harmonic (or positive subharmonic) functions on graphs. An example is also obtained to show that the global Lipschitz property of harmonic function on graphs does not hold.

We modify the definition of Ricci curvature of Ollivier of Markov chains on graphs to study the properties of the Ricci curvature of general graphs, Cartesian product of graphs, random graphs, and some special class of graphs.

We explore the tunneling behavior of a quantum particle on a finite graph in the presence of an asymptotically large potential with two or three potential wells. The behavior of the particle is primarily governed by the local spectral symmetry of the graph around the wells. In the case of two wells the behavior is stable in the sense that it can be predicted from a sufficiently large neighborhood of the wells. However in the case of three wells we are able to exhibit examples where the tunneling behavior can be changed significantly by perturbing the graph arbitrarily far from the wells.

Many studies have shown that additional information can be gained on time series by investigating their associated complex networks. In this work, we investigate the multifractal property and Laplace spectrum of the horizontal visibility graphs (HVGs) constructed from fractional Brownian motions. We aim to identify via simulation and curve fitting the form of these properties in terms of the Hurst index H. First, we use the sandbox algorithm to study the multifractality of these HVGs. It is found that multifractality exists in these HVGs. We find that the average fractal dimension $\langle D(0)\rangle $ of HVGs approximately satisfies the prominent linear formula $\langle D(0)\rangle =2-H$ ; while the average information dimension $\langle D(1)\rangle $ and average correlation dimension $\langle D(2)\rangle $ are all approximately bi-linear functions of H when $H\geqslant 0.15$ . Then, we calculate the spectrum and energy for the general Laplacian operator and normalized Laplacian operator of these HVGs. We find that, for the general Laplacian operator, the average logarithm of second-smallest eigenvalue $\langle \ln \left({{u}_{2}}\right)\rangle $ , the average logarithm of third-smallest eigenvalue $\langle \ln \left({{u}_{3}}\right)\rangle $ , and the average logarithm of maximum eigenvalue $\langle \ln \left({{u}_{n}}\right)\rangle $ of these HVGs are approximately linear functions of H; while the average Laplacian energy $\langle {{E}_{\text{nL}}}\rangle $ is approximately a quadratic polynomial function of H. For the normalized Laplacian operator, $\langle \ln \left({{u}_{2}}\right)\rangle $ and $\langle \ln \left({{u}_{3}}\right)\rangle $ of these HVGs approximately satisfy linear functions of H; while $\langle \ln \left({{u}_{n}}\right)\rangle $ and $\langle {{E}_{\text{nL}}}\rangle $ are approximately a 4th and cubic polynomial function of H respectively.

Continuing our previous work (arXiv:1509.07981v1), we derive another global gradient estimate for positive functions, particularly for positive solutions to the heat equation on finite or locally finite graphs. In general, the gradient estimate in the present paper is independent of our previous one. As applications, it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs. These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al. [J. Differential Geom., 99, 359–409 (2015)]. In many topics, such as eigenvalue estimate and heat kernel estimate (not including the Liouville type theorems), replacing the Li–Yau inequality by the global gradient estimate, we can get similar results.

Let G=(V,E) be a connected finite graph and Δ be the usual graph Laplacian. Using the calculus of variations and a method of upper and lower solutions, we give various conditions such that the Kazdan–Warner equation Δu=c−he^u has a solution on V, where c is a constant, and h:V→R is a function. We also consider similar equations involving higher order derivatives on graph. Our results can be compared with the original manifold case of Kazdan and Warner (Ann. Math. 99(1):14–47, 1974).