Self-similar Properties Research Articles

Exploring Fractal Quantum Field Theory in Higher-Order Dimensions Using the McGinty Equation

This hypothesis investigates the application of the McGinty Equation to quantum fields in higher-order dimensions, proposing that these fields exhibit self-similar fractal properties. The primary objective is to understand the implications of fractal geometry on quantum field interactions, coupling constants, and particle scattering processes, offering new insights into the fundamental forces and the structure of spacetime.

Self-similarity of quantum transport in graphene using electrostatic gate and substrate

Abstract A particular design for multibarrier structure in graphene, yielding a self-similar transport response, is proposed. The potential profile is based on rectangular wells and barriers, generated according independent n-th order scaling laws for their heights and widths. The barriers are constructed by means of two distinct approaches (electrostatic or substrate). Dirac equation and transfer matrix approach are used to calculate transmission properties which, in turn, allow to evaluate the conductance via Landauer-Büttiker formalism. It is found that self-similarity with determined scaling rules between n-th and (n + 1)-th generations of transport properties appears when the order of generating laws is equal or greater than n = 7. Our proposal would be the first in which the self-similarity property is transferred from geometry to the spectrum, and consequently, to the transport properties of a quantum heterostructure. Possible ways of practical realization for the proposed structures are commented.

Singular value decomposition: A useful technique for image denoising

A key function of image processing is picture denoising, which improves the quality of images by eliminating extraneous noise while keeping crucial information in tact. Singular Value Decomposition (SVD) is a linear algebraic technique that reduces the original datas complexity and scale by breaking down the matrices and extracting the important information. With the power of decomposition which utilizes the non-local self-similarity property of an image to achieve satisfactory denoising performance, SVD denoising has become a potent tool in image processing. In this paper, SVD is outlined and its working, applications, and challenges as a denoising technique in image denoising are discussed. The author discovered that Singular Value Decomposition can be a significant factor in image denoising by applying it to the image. As a result, Singular Value Decomposition could be thought as a helpful image denoising approach in the image processing sequence that will raise the images Peak Signal-to-Noise Ration (PSNR) and improve the quality of the image.

Local 2-connected bow-tie structure of the Web and of social networks

The explosive growth of the Web and of social networks motivates the need for analyzing the macroscopic structure of their underlying graphs. Although the characterization of the structure of a graph with respect to its pairwise connectivity has been known for over 15 years, just one subsequent study analyzed the world inside the giant strongly connected component, where it has been shown that the largest strongly connected component has its own microscopic bow-tie structure defined with respect to pairwise 2-connectivity among its vertices. In this paper, we introduce the local microscopic bow-tie structure of the largest strongly connected component, demonstrating its self-similarity property. Our experiments, conducted on the several Web graphs and social networks demonstrate clear structural differences between considered Web and social networks.

Torricelli’s Law in Fractal Space–Time Continuum

A new formulation of Torricelli’s law in a fractal space–time continuum is developed to compute the water discharge in fractal reservoirs. Fractal Torricelli’s law is obtained by applying fractal continuum calculus concepts using local fractional differential operators. The model obtained can be used to describe the behavior of real flows, considering the losses in non-conventional reservoirs, taking into account two additional fractal parameters α and β in the spatial and temporal fractal continuum derivatives, respectively. This model is applied to the flows in reservoirs with structures of three-dimensional deterministic fractals, such as inverse Menger sponge, Sierpinski cube, and Cantor dust. The results of the level water discharge H(t) are presented as a curve series, showing the impact and influence of fluid flow in naturally fractured reservoirs that posses self-similar properties.

A Minkowski-like fractal composite metamaterial for S-C band radar and infrared compatible stealth

Due to the opposite stealth mechanisms of radar and infrared waves, and the lack of narrow-band absorption at a single frequency point in metamaterials, it is difficult for low-frequency broadband radar and infrared waves to be compatibly stealthy at thin thickness. In order to solve the problem, Minkowski-like fractal metamaterial with frequency-selective surface of ''passing-low-frequency and blocking-high-frequency'' property is designed. The multiscale property of Minkowski-like fractal constructs multiple absorption frequency points and its suitable iteration factor achieves broadband reconfiguration of the absorption frequency points, which enables broadband absorption in S and C bands. Besides, the magneto-electric dual loss mechanism brought by the fractal and the intrinsic loss of the medium are the most fundamental reasons for the stealth being caused. In addition, the selection of iterative cell with infrared emissivity below 0.3 and the self-similarity property of Minkowski-like fractal make the metamaterial embody excellent infrared stealth performance. Ultimately, the 2.24 mm thick fractal metamaterial achieves cross-band radar wave absorption in the range of 3.41–5.82 GHz while being compatible with infrared wave stealth. The layered fractal model provides a new design idea for broadband stealth under multi-band compatible stealth.

Three-dimensional phononic crystals with self-similar structures

Acoustic metamaterials have the advantages of designability, strong pertinency, small size and good effect, and have good application value in solving the problem of sound insulation and noise reduction. Phononic crystals with wide bandgap and multi-bandgap can inhibit elastic wave propagation to some extent. In this study, a three-dimensional phononic crystal model with self-similar properties is designed by using fractal method. First, an initial unit is constructed, then the arm of the initial unit is replaced with the structure itself to form a self-similar structure. The self-similar model can block sound waves in the wide band and multi-band range. By changing the structure shape and size of phononic crystal, the sound wave blocking in different frequency range is also studied. At the same time of continuous optimization of the structure, the variation rules of the model band structure under different parameters are summarized. To find the good parameters of broadband and multi-band sound wave blocking, so as to achieve the effect of vibration isolation and noise reduction. The finite element method is used to simulate the vibration of the model to verify the existence of elastic wave bandgap. Phononic crystals have a good prospect in the field of sound insulation and noise reduction.

Discharge of rice-shaped particles from a monolayer flat-bottom silo.

In this work, we performed experiments regarding the outflow of spheres and two different types of rice-shaped particles in a quasi-two-dimensional monolayer silo with a flat bottom. We investigate the velocity and solid fraction profiles at the orifice and test whether the profiles for nonspherical particles have similar self-similar properties as in the spherical case. We find that the magnitude and shape of the velocity profiles for all three particle types are in a similar range. In contrast, the solid fraction at the orifice has a dome-shaped profile for both rice particles, whereas the profile for spherical particles is rather flat. The discharge rate determined from the velocity and solid fraction profiles describes the independently measured experimental discharge rate very well for all three investigated particle types.

Fundamental Dynamics of Popularity-Similarity Trajectories in Real Networks.

Real networks are complex dynamical systems, evolving over time with the addition and deletion of nodes and links. Currently, there exists no principled mathematical theory for their dynamics-a grand-challenge open problem. Here, we show that the popularity and similarity trajectories of nodes in hyperbolic embeddings of different real networks manifest universal self-similar properties with typical Hurst exponents H≪0.5. This means that the trajectories are predictable, displaying antipersistent or "mean-reverting" behavior, and they can be adequately captured by a fractional Brownian motion process. The observed behavior can be qualitatively reproduced in synthetic networks that possess a latent geometric space, but not in networks that lack such space, suggesting that the observed subdiffusive dynamics are inherently linked to the hidden geometry of real networks. These results set the foundations for rigorous mathematical machinery for describing and predicting real network dynamics.

Algebraic and toroidal representation of the genetic code

The genetic code is a set of regulatory principles that control the translation of information encoded in messenger RNA (mRNA) into a sequence of amino acids. This study proposes a model starting from the relationships between its physicochemical properties of nucleobase in the codons. We employed a binary metric and the Kronecker product to represent the physicochemical properties of nucleobases in a vector space. This state space has a hierarchical order, with self-similarity and symmetry properties in the hydrogen bonds of triplets. The state space can be mapped under linear transformations to a toroidal geometry with allometric properties. Furthermore, this geometric representation highlights a charge symmetry that exists in amino acids and in the distribution of essential and non-essential amino acids. These results describe a state space between codon-anticodon interactions that can be interpreted as Bell states. This suggests that there is a quantum phenomenon involved in the mechanisms of information storage in DNA. In addition, toroidal geometry can be used to represent the sequences of codons of the mRNA that encode the sequence of amino acids of the proteins to find similarities or homologies between evolutionary related species.

Similarity solutions in elastohydrodynamic bouncing

We investigate theoretically and numerically the impact of an elastic sphere on a rigid wall in a viscous fluid. Our focus is on the dynamics of the contact, employing the soft lubrication model in which the sphere is separated from the wall by a thin liquid film. In the limit of large sphere inertia, the sphere bounces and the dynamics is close to the Hertz theory. Remarkably, the film thickness separating the sphere from the wall exhibits non-trivial self-similar properties that vary during the spreading and retraction phases. Leveraging these self-similar properties, we establish the energy budget and predict the coefficient of restitution for the sphere. The general framework derived here opens many perspectives to study the lubrication film in impact problems.

Multiple Complementary Priors for Multispectral Image Compressive Sensing Reconstruction.

Compressive sensing (CS) techniques using a few compressed measurements have drawn considerable interest in reconstructing multispectral imagery (MSI). Nonlocal-based tensor methods have been widely used for MSI-CS reconstruction, which employ the nonlocal self-similarity (NSS) property of MSI to obtain satisfactory results. However, such methods only consider the internal priors of MSI while ignoring important external image information, for example deep-driven priors learned from a corpus of natural image datasets. Meanwhile, they usually suffer from annoying ringing artifacts due to the aggregation of overlapping patches. In this article, we propose a novel approach for highly effective MSI-CS reconstruction using multiple complementary priors (MCPs). The proposed MCP jointly exploits nonlocal low-rank and deep image priors under a hybrid plug-and-play framework, which contains multiple pairs of complementary priors, namely, internal and external, shallow and deep, and NSS and local spatial priors. To make the optimization tractable, a well-known alternating direction method of multiplier (ADMM) algorithm based on the alternating minimization framework is developed to solve the proposed MCP-based MSI-CS reconstruction problem. Extensive experimental results demonstrate that the proposed MCP algorithm outperforms many state-of-the-art CS techniques in MSI reconstruction. The source code of the proposed MCP-based MSI-CS reconstruction algorithm is available at: https://github.com/zhazhiyuan/MCP_MSI_CS_Demo.git.

Structured residual sparsity for video compressive sensing reconstruction

Recent advancements in the residual sparsity strategy have garnered widespread attention in video compressive sensing (CS) reconstruction. However, most of the existing residual sparsity-based video CS reconstruction methods usually suffer from some limitations that lead to undesired visual artifacts. Firstly, these methods only rely on a patch sparsity scheme that is limited by their focus on the local structures of each video frame, neglecting the nonlocal self-similarity (NSS) property inherent to each video frame. Secondly, these methods concentrate on utilizing the NSS property of external reference frames for multi-hypothesis (MH) prediction while disregarding the internal NSS property of the current frame. In this paper, we propose a new structured residual sparsity (SRS) approach for video CS reconstruction, which jointly exploits the NSS properties of the current frame and its reference frames. Specifically, due to the unavailability of the original video frames, we first devise an effective intraframe CS (EICS) reconstruction method that leverages the internal NSS property of each frame. This approach enables us to obtain initial recovery frames, which then facilitate the execution of MH prediction. Following this, we generate a residual frame for the current frame by employing the MH prediction. Then, we propose a novel SRS model jointly using the NSS properties of the current frame and its reference frames to explore both the correlations of intraframe and interframe for reconstructing the current frame. Furthermore, for the sake of optimization feasibility, we develop an effective alternating direction method of multipliers (ADMM) algorithm to address the objective. Our experimental findings reveal that the proposed SRS not only yields superior quantitative results, but also uncovers finer details and causes fewer visual artifacts compared to many popular or state-of-the-art video CS reconstruction approaches.

A Study on the Validity and Scope of Self-Similarity Property in Super-Resolution of Medical Images

A Study on the Validity and Scope of Self-Similarity Property in Super-Resolution of Medical Images

Distinguishing between fractional Brownian motion with random and constant Hurst exponent using sample autocovariance-based statistics.

Fractional Brownian motion (FBM) is a canonical model for describing dynamics in various complex systems. It is characterized by the Hurst exponent, which is responsible for the correlation between FBM increments, its self-similarity property, and anomalous diffusion behavior. However, recent research indicates that the classical model may be insufficient in describing experimental observations when the anomalous diffusion exponent varies from trajectory to trajectory. As a result, modifications of the classical FBM have been considered in the literature, with a natural extension being the FBM with a random Hurst exponent. In this paper, we discuss the problem of distinguishing between two models: (i) FBM with the constant Hurst exponent and (ii) FBM with random Hurst exponent, by analyzing the probabilistic properties of statistics represented by the quadratic forms. These statistics have recently found application in Gaussian processes and have proven to serve as efficient tools for hypothesis testing. Here, we examine two statistics-the sample autocovariance function and the empirical anomaly measure-utilizing the correlation properties of the considered models. Based on these statistics, we introduce a testing procedure to differentiate between the two models. We present analytical and simulation results considering the two-point and beta distributions as exemplary distributions of the random Hurst exponent. Finally, to demonstrate the utility of the presented methodology, we analyze real-world datasets from the financial market and single particle tracking experiment in biological gels.

MULTIFRACTAL CHARACTERISTICS OF THE MITOCHONDRIAL BK CHANNEL ACTIVITY IN THE PRESENCE OF THE CHANNEL-ACTIVATING FLAVONOIDS

Over the years, the (multi)fractal nature of signals describing the activity of ion channels has been observed both at the level of single-channel currents and the temporal scaling of the corresponding sojourns in conducting and non-conducting states. The recognized nonlinear and self-similar properties were interpreted regarding the underlying channel gating machinery. Nevertheless, the literature lacks in (multi)fractal description of the biochemical stimulation of ion channels. Therefore, in this work, we provide a multifractal description of the effects exerted by two different flavonoids — naringenin (Nar) and quercetin (Que) — being the regulators of the large-conductance voltage- and Ca[Formula: see text]-activated K[Formula: see text] channels from the inner mitochondrial membrane (mitoBK). Toward this aim, the focus-based multifractal detrended fluctuation analysis (MFDFA) was applied to the patch-clamp data obtained in the presence of Nar and Que in micromolar concentrations and the corresponding controls. Our results show that mitoBK channel activity has well-pronounced multifractal characteristics and long-range memory (measured by the generalized Hurst exponent) under biochemical stimulation by Nar and Que and in the absence of these substances. The spectral properties significantly change after the randomization of the analyzed series. Therefore, multifractality is an inherent feature of the mitoBK channel currents, which stems from their statistical distribution and temporal organization. Both flavonoids have similar structures and exert channel-activating effects, but they differently affect the parameters of the multifractal spectra of the corresponding patch-clamp signals. Coordination of Que leads to more prominent changes in signal complexity than Nar at membrane hyper- and depolarization. It suggests that this flavonoid has a stronger impact on the conformational dynamics of the mitoBK channels in comparison with naringenin.

Fractal representations in image processing of remote sensing of the earth

This article explores the application of fractal representations in the image processing of remote sensing data for Earth observation. Fractals, with their self-similar properties and complex patterns, offer a powerful mathematical framework for analyzing the intricate structures found in natural landscapes. The study highlights the advantages of using fractal-based methods over traditional image processing techniques, particularly in capturing the multifaceted textures and irregularities of Earth’s surface features. By leveraging fractal geometry, enhanced accuracy in the classification and interpretation of remote sensing images is achieved. This approach facilitates better monitoring and understanding of environmental changes, land use patterns, and natural disasters. The findings underscore the potential of fractal representations to significantly improve the quality and efficacy of remote sensing image analysis, providing a robust tool for Earth science research and practical applications in environmental management.

Latent Diffusion Enhanced Rectangle Transformer for Hyperspectral Image Restoration.

The restoration of hyperspectral image (HSI) plays a pivotal role in subsequent hyperspectral image applications. Despite the remarkable capabilities of deep learning, current HSI restoration methods face challenges in effectively exploring the spatial non-local self-similarity and spectral low-rank property inherently embedded with HSIs. This paper addresses these challenges by introducing a latent diffusion enhanced rectangle Transformer for HSI restoration, tackling the non-local spatial similarity and HSI-specific latent diffusion low-rank property. In order to effectively capture non-local spatial similarity, we propose the multi-shape spatial rectangle self-attention module in both horizontal and vertical directions, enabling the model to utilize informative spatial regions for HSI restoration. Meanwhile, we propose a spectral latent diffusion enhancement module that generates the image-specific latent dictionary based on the content of HSI for low-rank vector extraction and representation. This module utilizes a diffusion model to generatively obtain representations of global low-rank vectors, thereby aligning more closely with the desired HSI. A series of comprehensive experiments were carried out on four common hyperspectral image restoration tasks, including HSI denoising, HSI super-resolution, HSI reconstruction, and HSI inpainting. The results of these experiments highlight the effectiveness of our proposed method, as demonstrated by improvements in both objective metrics and subjective visual quality.

The Grounding Problem of Equal Respect

In this paper, I explore three theories of value to illuminate how nontheistic and theistic accounts may differ in grounding human dignity: neo- Aristotelian ethical naturalism, Kantian constructivism, and a theistic account of good simpliciter. The theistic account of good simpliciter that I offer adapts Robert Adams’s notion of the transcendent Good as the Excellent. In this account, I explain how Adams’ thesis that goodness is a property consisting in a sort of resemblance to God may be understood in a new way, using ideas drawn from contemporary mathematics and quantum mechanics. On my account, we must value human beings neither because such valuing would be beneficial or necessary for human flourishing nor because it is a logical outcome of anything we care to value. Rather it is because we recognize the property of self-similarity in all of us, which may be understood as a resemblance to God as good simpliciter.

УМОВИ ВИКОРИСТАННЯ КОНФОРМНИХ ВІДОБРАЖЕНЬ ДЛЯ ВИВЧЕННЯ ФРАКТАЛЬНОГО СТИСНЕННЯ ІНФОРМАЦІЇ

To describe the properties of self-similarity and invariance observed in various physical situations, the theory of fractals and multifractals is actively used in modern science. We consider a conformal mapping of the first kind, which is given by the entire linear function w = az + b, where w, z are complex variables, a, b are complex constants, a ≠ 0 . With its help, affine transformations are performed to construct the Sierpinski triangle and the Koch curve by specifying the functions w3(z) and finding the coefficients Re аі, Im аі, Re bi, Im bi, і = 1,2,3 or і = 1,2,3,4. These coefficients encode the image of the object and it can be unambiguously reconstructed by them. It is determined that the method in which self-similar regions in an object are detected and conformal mapping coefficients are found for them works under the condition that each such mapping is compressive. Only then will Banach’s fixed point theorem ensure image collection during decompression. This method of compressing graphic information is called fractal.