Narrow Results

We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

To determine the attractor dimension of chaotic dynamics, the box-counting method has the difficulty in getting accurate estimates because the boxes are not weighted by their relative probabilities. We present a new method to minimize this difficulty. The local box-counting method can be quite effective in determining the attractor dimension of high-order chaos as well as low-order chaos.

We consider a matrix space based on the spin degree of freedom, describing both a Hilbert state space and its corresponding symmetry operators. Under the requirement that the Lorentz symmetry be kept, at given dimension, scalar symmetries and their representations are determined. Symmetries are flavor or gauge-like, with fixed chirality. After spin 0, 1/2 and 1 fields are obtained in this space, we construct associated interactive gauge-invariant renormalizable terms, showing their equivalence to a Lagrangian formulation, using as example the previously studied (5+1)-dimensional case, with many standard-model connections. At 7+1 dimensions, a pair of Higgs-like scalar Lagrangian is obtained naturally producing mass hierarchy within a fermion flavor doublet.

Projective synchronization between two nonlinear systems with different dimension was investigated. The controllers were designed when the dimension of drive system greater than the one of response system. The opposite situation also was discussed. In addition, we found an approach to control the chaotic (hyperchaotic) system to exhibit the behaviors of hyperchaotic (chaotic) system. The numerical simulations were implemented on different chaotic (hyperchaotic) systems, and the results indicate that our methods are effective.

In this paper, we show that parasitic elements have a significant effect on the dynamics of memristor circuits. We first show that certain $2$-terminal elements such as memristors, memcapacitors, and meminductors can be used as nonvolatile memories, if the principle of conservation of state variables hold by open-circuiting, or short-circuiting, their terminals. We also show that a passive memristor with a strictly-increasing constitutive relation will eventually lose its stored flux when we switch off the power if there is a parasitic capacitance across the memristor. Similarly, a memcapacitor (resp., meminductor) with a positive memcapacitance (resp., meminductance) will eventually lose their stored physical states when we switch off the power, if it is connected to a parasitic resistance. We then show that the discontinuous jump that circuit engineers assumed to occur at impasse points of memristor circuits contradicts the principles of conservation of charge and flux at the time of the discontinuous jump. A parasitic element can be used to break an impasse point, resulting in the emergence of a continuous oscillation in the circuit. We also define a distance, a diameter, and a dimension, for each circuit element in order to measure the complexity order of the parasitic elements. They can be used to find higher-order parasitic elements which can break impasse points. Furthermore, we derived a memristor-based Chua’s circuit from a three-element circuit containing a memristor by connecting two parasitic memcapacitances to break the impasse points. We finally show that a higher-order parasitic element can be used for breaking the impasse points on two-dimensional and three-dimensional constrained spaces.

Definitions of Hausdorff–Lebesgue measure and dimension are introduced. Combination of Hausdorff and Lebesgue ideas are used. Methods for upper and lower estimations of attractor dimensions are developed.

This paper investigates the Weierstrass type function in local fields whose graph is a chaotic repelling set of a discrete dynamical system, and proves that their exists a linear connection between the orders of its p-adic calculus and the dimensions of the corresponding graphs.

The cello suites of Johann Sebastian Bach exhibit several types of power-law scaling, the best examples of which can be considered fractal in nature. This article examines scaling with respect to the characteristics of melodic interval and its derivative, melodic moment. A new and effective method for pitch-related analysis is described and then applied to a selection of the 36 pieces that comprise the six cello suites.

Fractal geometry can adequately represent many complex and irregular objects in nature. The fractal dimension is typically computed by the box-counting procedure. Here I compute the box-counting and the Kaplan-Yorke dimensions of the 14-dimensional models of the Drosophila circadian clock. Clockwork Orange (CWO) is transcriptional repressor of direct target genes that appears to play a key role in controlling the dynamics of the clock. The findings identify these models as strange attractors and highlight the complexity of the time-keeping actions of CWO in light-day cycles. These fractals are high-dimensional counterexamples of the Kaplan-Yorke conjecture that uses the spectrum of the Lyapunov exponents.

We consider some properties of the intersection of deleted digits Cantor sets with their translates. We investigate conditions on the set of digits such that, for any t between zero and the dimension of the deleted digits Cantor set itself, the set of translations such that the intersection has that Hausdorff dimension equal to t is dense in the set F of translations such that the intersection is non-empty. We make some simple observations regarding properties of the set F, in particular, we characterize when F is an interval, in terms of conditions on the digit set.

For a family of networks ${\{G_n\}}_{n\ge 1}$, we define the Hausdorff dimension of ${\{G_n\}}_{n\ge 1}$ inspired by the Frostman’s characteristics of potential for Hausdorff dimension of fractals on Euclidean spaces. We prove that our Hausdorff dimension of the touching networks is $logm/logN.$ Our definition is quite different from the fractal dimension defined for real-world networks.

In 2023, Xi et al. introduced the Hausdorff dimension of a family of networks which inspired by the potential theoretic methods in fractal geometry. In this paper, we will construct a class of colored substitution networks and obtain its Hausdorff dimension using the self-similarity.

The most common ways used to generate indistinguishability operators, namely as transitive closure of reflexive and symmetric fuzzy relation, via the Representation Theorem and as decomposable relations, are related for archimedean t-norms introducing the notion of length of indistinguishability operators.

Materialized view selection (MVS) improves the query processing efficiency and performance for making decisions effectively in a data warehouse. This problem is NP-hard and constrained optimization problem which involves space and cost constraint. Various optimization algorithms have been proposed in literature for optimal selection of materialized views. Few works exist for handling the constraints in MVS. In this study, authors have proposed the Cuckoo Search Algorithm (CSA) for optimization and Stochastic Ranking (SR) for handling the constraints in solving the MVS problem. The motivation behind integrating CS with SR is that fewer parameters have to be fine tuned in CS algorithm than in genetic and Particle Swarm Optimization (PSO) algorithm and the ranking method of SR handles the constraints effectively. For proving the efficiency and performance of our proposed algorithm Stochastic Ranking based Cuckoo Search Algorithm for Materialized View Selection (SRCSAMVS), it has been compared with PSO, genetic algorithm and the constrained evolutionary optimization algorithm proposed by Yu et al. SRCSAMVS outperforms in terms of query processing cost and scalability of the problem.

A novel approach called the YUV dimension method is proposed in this study to indirectly characterize fractured rock surface topography. This model is based on YUV color model theory in iconography and graphics. The process is described in detail as follows. A true-color photo with m pixels is selected. Y is denoted by gray scale, U by hue, and V by the saturation components of the pixel color. These components are applied to create the z, x and y coordinates of the point in the coordinate system that originates from the pixel. A similar method is applied in each pixel. M points are created in the coordinate space. The number of created points is equivalent to one of the pixels. The m points are then connected and a rough YUV surface is established. Otherwise, the calculation method for the self-affine dimension of a n - 1 ~ n-dimensional fractal body is presented by fractal Brownian motion theory and then degenerated to one between the 2D and 3D case. The approach is applied to evaluate the dimension of the YUV surface, i.e. the YUV dimension. To validate the feasibility of YUV dimension theory, numerical studies on the YUV dimension are conducted through a laser profilometer scanning experiment and scanning electron microscopy with the same specimens. The surface characteristics of similar samples are analyzed by probing into the YUV, general and grayscale dimensions of the specimens. The comparison shows that the YUV and general dimensions of similar specimens are fundamentally identical, and the complete trends of the YUV and gray dimensions remain consistent with changing specimens. The result indicates that YUV dimension theory is reasonable and feasible. In short, the YUV dimension is a new method that exhibits more advantages than the general and grayscale dimensions. This method characterizes surface configuration indirectly.

Studying topological characteristics of digital images is a fundamental issue in image analysis and understanding. In the present paper we first propose a linear time constant-working space algorithm for determining the genus of a connected digital image. The computation is based on a combinatorial relation for digital images that may be of independent interest as well. We also propose definitions of dimension for planar digital images. These definitions serve as an alternative to the one proposed by Mylopoulos and Pavlidis¹, and make up some of its shortcomings. We study various properties of the so-defined image dimension, in particular, characterization of dimension in terms of Euler characteristic. We also show that image dimension can be found within linear time and constant memory.

We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to products of sets in finite groups, in particular to word maps, and a generalization of Tao's Algebraic Regularity Lemma is noted.

We investigate geometrical properties and inequalities satisfied by the complex difference body, in the sense of studying which of the classical ones for the difference body have an analog in the complex framework. Among others we give an equivalent expression for the support function of the complex difference body and prove that, unlike the classical case, the dimension of the complex difference body depends on the position of the body with respect to the complex structure of the vector space. We use spherical harmonics to characterize the bodies for which the complex difference body is a ball, we prove that it is a polytope if and only if the two bodies involved in the construction are polytopes and provide several inequalities for classical magnitudes of the complex difference body, as volume, quermassintegrals and diameter, in terms of the corresponding ones for the involved bodies.

The tensor network variety is a variety of tensors associated to a graph and a set of positive integer weights on its edges, called bond dimensions. We determine an upper bound on the dimension of the tensor network variety. A refined upper bound is given in cases relevant for applications such as varieties of matrix product states and projected entangled pairs states. We provide a range (the “supercritical range”) of the parameters where the upper bound is sharp.

For an equilibrium measure of a Hölder potential, we prove an analogue of the Central Limit Theorem for the fluctuations of the logarithm of the measure of balls as the radius goes to zero.

A noticeable consequence is that when this measure is not absolutely continuous, the probability that a ball of radius ε chosen at random have a measure smaller (or larger) than ε^δ is asymptotically equal to 1/2, where δ is the Hausdorff dimension of the measure.

Our method applies to a class of non-conformal expanding maps on the d-dimensional torus. It also applies to conformal repellers and Axiom A surface diffeomorphisms and possibly to a class of one-dimensional non-uniformly expanding maps. These generalizations are presented at the end of the paper.