Non-integer Dimension Research Articles

APPROXIMATION OF THE SAME BOX DIMENSION IN CONTINUOUS FUNCTIONS SPACE

In this paper, we make research on the approximation of functions with fractal dimension in continuous functions space. We first investigate fractal dimension of the linear combination of continuous functions with different fractal dimensions. Then, fractal winding of continuous functions has been given. Furthermore, based on Weierstrass theorem and Weierstrass function, we establish a theorem that continuous functions with the non-integer fractal dimension can be approximated by the linear combination of trigonometric polynomials with the same fractal dimension. Condition about continuous functions with fractal dimension one and two has also been discussed elementary.

Two-body continuum states in noninteger geometry

Wave functions, phase shifts and corresponding elastic cross sections are investigated for two short-range interacting particles in a deformed external oscillator field. For this we use the equivalent $d$-method employing a non-integer dimension $d$. Using a square-well potential, we derive analytic expressions for scattering lengths and phase shifts. In particular, we consider the dimension, $d_E$, for infinite scattering length, where the Efimov effect occurs by addition of a third particle. We give explicitly the equivalent continuum wave functions in $d$ and ordinary three dimensional (3D) space, and show that the phase shifts are the same in both methods. Consequently the $d$-method can be used to obtain low-energy two-body elastic cross sections in an external field.

Study of scattering from a PEC strip placed at the planar interface of NID dielectric and dielectric-magnetic medium using the KP method

Mathematical expressions for electromagnetic scattering from a Perfect electric conductor strip positioned at the planar non-integer dimension (NID) dielectric- magnetic interface are determined by using the Kobayashi potential (KP) method. The Lorentz-Drude model is used to include dispersion and losses in both permeability and permittivity of the dielectric- magnetic medium. Based on the KP method, the unknown scattered fields are presumed in terms of the spectrum with an undetermined weighting function. By imposing the appropriate boundary conditions, the dual integral equations for undetermined weighting functions obtained, the integrals are further solved by utilizing the property of Weber-Schafheitilin integrals. Then, using the orthogonal properties of Jacobi polynomials, the set of matrix equations satisfying the WSIs along with edge boundary conditions are obtained. This set of equations is solved numerically to find the undetermined expansion coefficients. Far-field expressions are studied to note the impact of the NID parameter on the scattered field. Based on the operating frequency, dielectric-magnetic medium behaves as double positive/mu negative/double negative/epsilon negative. It is studied that how the type of medium for lower half-space along with the presence of NID medium in upper half-space affects scattering characteristics of the strip.

Comparative analysis of geometric forms and numerical calculation of fractals in Khayyam’s Mausoleum based on the Islamic architecture’s definitions and box-counting method

The interaction and balance between fractal components, which maintain to function in nature in perfect harmony with the highest efficiency, has been the source of inspiration for architects in different eras. Today, contemporary Iranian architecture needs a comprehensive study and familiarity with world issues, hence the Khayyam’s mausoleum was chosen. the main purpose of this article is to analyze the geometry and numerical calculation of the fractal size of the building and its semantic knowledge in the contemporary building, which leads to finding the architect’s goals of how to design these forms and common concepts with Islamic architecture. After achieving how to use Islamic elements in a contemporary monument, we come to the interrelationship between fractal geometry and architectural design.With the box-counting technique, the fractal dimension was calculated by checking the two-dimensional plates in 4 different scales from the largest to the smallest scale to prove its decimal dimension. Then, information was collected by descriptive-analytic method and after determining the common features between fractal forms with analytical and comparative process, the obtained quantitative data resulted in extracting similar cases from the building. Finally, the relationship between the characteristics of fractal geometry and the architecture of the tomb was proved and the obtained patterns were matched with the concepts of self-similarity, iteration (proportion and symmetry), micro scale and non-integer dimension. The elements used such as tiled ornaments, the Islamic star and Shamsa, domed-shaped roof, regular numerical pattern of geometric shapes with symbolism, unity in diversity, infinity and dynamism are reminiscent of Islamic architectural style. Furthermore, fractal has been used in three methods: conceptual, mathematical and intuitive, and has been manifested in both structural and aesthetic aspects.

Duals of Feynman integrals. Part I. Differential equations

We elucidate the vector space (twisted relative cohomology) that is Poincaré dual to the vector space of Feynman integrals (twisted cohomology) in general spacetime dimension. The pairing between these spaces — an algebraic invariant called the intersection number — extracts integral coefficients for a minimal basis, bypassing the generation of integration-by-parts identities. Dual forms turn out to be much simpler than their Feynman counterparts: they are supported on maximal cuts of various sub-topologies (boundaries). Thus, they provide a systematic approach to generalized unitarity, the reconstruction of amplitudes from on-shell data. In this paper, we introduce the idea of dual forms and study their mathematical structures. As an application, we derive compact differential equations satisfied by arbitrary one-loop integrals in non-integer spacetime dimension. A second paper of this series will detail intersection pairings and their use to extract integral coefficients.

Features of Spatial-Temporal Hierarchical Structures Formation

The degree of ordering of the structure of technologically important materials formed as a result of the evolution of complex physicochemical systems determines their physical properties, in particular optical. In this regard, the primary task for the theoretical study of methods for obtaining materials with predetermined physical properties is to develop approaches to describe the evolution of fractal (scale-invariant) objects in the formation of self-similar structures in systems exhibiting chaotic behavior. The paper forms an idea of the processes of evolution in materials formed as a result of stochastic processes. It is established that the conduct of ultrametrics in time space allows to characterize the time of the evolutionary process of fractal dimension, which is calculated either theoretically or model. The description of evolutionary processes in a condensed medium, accompanied by topological transformations, is significantly supplemented by the method of describing the stages of evolution of structures, which makes it possible to analyze a wide range of materials and can control their properties, primarily optical. It is shown that the most large-scale invariant structures, due to the investigated properties, can be used as information carriers. It is demonstrated that the presence in physical systems of fractal temporal dimension and generates a self-similar (consisting of parts in a sense similar to the whole object) evolutionary tree, which, in turn, generates spatial objects of non-integer dimension, observed in real situations. On the other hand, temporal fractality provides analysis of systems with dynamic chaos, leading to universal relaxation functions. In particular, in systems with a large-scale invariant distribution of relaxation characteristics, an algebraic law of relaxation is manifested, which leads to rheological models and equations of states, which are characterized by fractional derivatives. It is argued that the fractal dimension of time hierarchies stores information that determines the process of self-organization. Developed in the paper ideas about the processes of building the structure of materials, which lead to the fractal geometry of objects, can be used to predict their properties, in particular, optical.

Relativistic Fractional-Dimension Gravity

This paper presents a relativistic version of Newtonian Fractional-Dimension Gravity (NFDG), an alternative gravitational model recently introduced and based on the theory of fractional-dimension spaces. This extended version—Relativistic Fractional-Dimension Gravity (RFDG)—is based on other existing theories in the literature and might be useful for astrophysical and cosmological applications. In particular, in this work, we review the mathematical theory for spaces with non-integer dimensions and its connections with the non-relativistic NFDG. The Euler–Lagrange equations for scalar fields can also be extended to spaces with fractional dimensions, by adding an appropriate weight factor, and then can be used to generalize the Laplacian operator for rectangular, spherical, and cylindrical coordinates. In addition, the same weight factor can be added to the standard Hilbert action in order to obtain the field equations, following methods used for scalar-tensor models of gravity, multi-scale spacetimes, and fractional gravity theories. We then apply the field equations to standard cosmology and to the Friedmann-Lemaître-Robertson-Walker metric. Using a suitable weight vtt, depending on the synchronous time t and on a single time-dimension parameter αt, we extend the Friedmann equations to the RFDG case. This allows for the computation of the scale factor at for different values of the fractional time-dimension αt and the comparison with standard cosmology results. Future additional work on the subject, including studies of the cosmological late-time acceleration, type Ia supernovae data, and related dark energy theory will be needed to establish this model as a relativistic alternative theory of gravity.

Fractal analysis of the martian landscape: A study of kilometre-scale topographic roughness

Natural and engineered surfaces can be modelled as fractal objects described by a non-integer dimension (i.e., their fractal dimension). This approach has been used extensively to characterize surface complexity in terms of rugosity or roughness. In this paper, the local fractal dimension is proposed as a means of mapping the spatial variability of the surface roughness of the Martian landscape. The local fractal dimension complements measures of topographic roughness proposed by other authors and provides new images of the Martian surface texture at the kilometric scale that can be used to interpret geological events, geological processes and geological formations. It can also assist in geological mapping and in the detection and prioritisation of sites that merit further exploration. Mars Orbiter Laser Altimeter data were used to characterize the kilometre-scale terrain roughness of Mars. The procedure is general and can also be applied to digital elevation models with higher resolution (for example, digital elevation models generated by stereo-pair HiRISE images) to assess roughness at the metric scale. Sixteen test sites were selected and used to illustrate the patterns of the spatial variability of landscape roughness (local fractal dimension maps) at the kilometric scale. The results show that the mean of the local fractal dimension ranges from 2.036 in Argyre Planitia to 2.403 at the North Pole. The results also show that the mean roughness values of the Martian landscapes are less than those of the Earth's continents. Basically, the roughness of the Martian terrain at the kilometric scale depends, inter alia, on the number of kilometric-scale landforms present on the terrain. It is thus logical that the local fractal dimension is higher on Earth, a geologically active planet, than on Mars. Only the North Pole test site (an area larger than the North polar ice cap) shows a roughness histogram similar to those of terrestrial sites. The abundance of information embedded in the spatial variability patterns in the roughness maps provides a significant resource for planetary geology research.

Acoustics of Fractal Porous Material and Fractional Calculus

In this paper, we present a fractal (self-similar) model of acoustic propagation in a porous material with a rigid structure. The fractal medium is modeled as a continuous medium of non-integer spatial dimension. The basic equations of acoustics in a fractal porous material are written. In this model, the fluid space is considered as fractal while the solid matrix is non-fractal. The fluid–structure interactions are described by fractional operators in the time domain. The resulting propagation equation contains fractional derivative terms and space-dependent coefficients. The fractional wave equation is solved analytically in the time domain, and the reflection and transmission operators are calculated for a slab of fractal porous material. Expressions for the responses of the fractal porous medium (reflection and transmission) to an acoustic excitation show that it is possible to deduce these responses from those obtained for a non-fractal porous medium, only by replacing the thickness of the non-fractal material by an effective thickness depending on the fractal dimension of the material. This result shows us that, thanks to the fractal dimension, we can increase (sometimes by a ratio of 50) and decrease the equivalent thickness of the fractal material. The wavefront speed of the fractal porous material depends on the fractal dimension and admits several supersonic values. These results open a scientific challenge for the creation of new acoustic fractal materials, such as metamaterials with very specific acoustic properties.

Measure of unevenness in human genomes, described as a self-affine phase transition in a 'spin-chain’ model

Background: Non-Gaussian distribution of polymorphic positions across a genome can substantially influence the results of any approach to molecular evolution based on a 'classical' probability model. The infinite dispersion of non-Gaussian perturbations is a challenge in an attempt to accept it in a probability-based model of evolution. Methods: Here a model is proposed where non-Gaussian distribution is introduced to an exact solution of the 'Ising model'; it describes a behavior of one-dimensional chain of spins in an approaching to a phase transition. The distribution of fragments which are identical between two genomes is similar to distribution of islands of spins with the same orientation, in the model where non-integer dimension is introduced. Results: Application of this model allows us to compare the relative contributions of non-Gaussian perturbations for pairs of human genomes from different ethnic groups. An evolution of the three human races in a most compact presentation is considered, rates of development on the separated stages of the evolution are assumed to be proportional to a value of relative unevenness between the appropriate groups of genomes. In the resolved model, the meaningful details of the separation between Asian and European races are clarified, in a period around ten thousand years ago; a particular viewpoint to the separation of the African race is also presented. Conclusion: The proposed approximation of non-Gaussian perturbations in human genomes allows to support the statements which are otherwise missed in the scientific investigations of the early history of modern humans.

Newtonian fractional-dimension gravity and rotationally supported galaxies

ABSTRACT We continue our analysis of Newtonian fractional-dimension gravity, an extension of the standard laws of Newtonian gravity to lower dimensional spaces, including those with fractional (i.e. non-integer) dimension. We apply our model to three rotationally supported galaxies: NGC 7814 (bulge-dominated spiral), NGC 6503 (disc-dominated spiral), and NGC 3741 (gas-dominated dwarf). As was done in the general cases of spherically symmetric and axially symmetric structures, which were studied in previous work on the subject, we examine a possible connection between our model and modified Newtonian dynamics, a leading alternative gravity model that explains the observed properties of these galaxies without requiring the dark matter hypothesis. In our model, the modified Newtonian dynamics acceleration constant $a_{0} \simeq 1.2 \times 10^{ -10}\,\mbox{m}\, \mbox{s}^{ -2}$ can be related to a natural scale length l0, namely $a_{0} \approx GM/l_{0}^{2}$ for a galaxy of mass M. Also, the empirical radial acceleration relation, connecting the observed radial acceleration gobs with the baryonic one gbar, can be explained in terms of a variable local dimension D. As an example of this methodology, we provide detailed rotation curve fits for the three galaxies mentioned above.

Comb-like geometric constraints leading to emergence of the time-fractional Schrödinger equation

This paper presents an overview over several examples, where the comb-like geometric constraints lead to emergence of the time-fractional Schrödinger equation. Motion of a quantum object on a comb structure is modeled by a suitable modification of the kinetic energy operator, obtained by insertion of the Dirac delta function in the Laplacian. First, we consider motion of a free particle on two- and three-dimensional comb structures, and then we extend the study to the interacting cases. A general form of a nonlocal term, which describes the interactions of the particle with the medium, is included in the Hamiltonian, and later on, the cases of constant and Dirac delta potentials are analyzed. At the end, we discuss the case of non-integer dimensions, considering separately the case of fractal dimension between one and two, and the case of fractal dimension between two and three. All these examples show that even though we are starting with the standard time-dependent Schrödinger equation on a comb, the time-fractional equation for the Green’s functions appears, due to these specific geometric constraints.

Visual and Structural Analysis of Fractal Geometry in the Sheikh Lotfollah Mosque Ornaments (Isfahan-Iran)

The Islamic era is the period of new s‌tyles and beyond its own time. Shamsa, Muqarnas, S‌tar, geometric motifs, and decorative elements have fractal nature, which was used as a tool to express the architect’s idea by repeating the same components in a spatial dimension. We will reach the effect of Islamic impacts on fractal architecture and semantic cognition, and by finding the roots of theories about fractal with an analytical-descriptive approach, we will unders‌tand the relationship between fractals and how they are used by architects. Our focus is on comprehensive s‌tudy and calculation, not only in ornaments but also in s‌tructure, to find patterns of fractal form, to represent coordination between components. Firs‌t, we proved and extracted the fractal properties and patterns between the cons‌tituent elements, including self-similarity, repeatability, small-scale, and non-integer dimension, using visual analysis, and then we set up the Box-counting analysis technique with two purposes of calculating fractal dimension and finding their relationship. Aware of the mathematical proportions and relationships between the components of nature, the architect of the Sheikh Lotfollah Mosque has displayed the geometric sequence limit with an ascending equation in the interior ornaments of the dome. His goal is both to create the world of spirituality on a micro-scale and make it unders‌tandable for an observer. Thus, in this way, he has sometimes expressed his concept through mathematics, proportions, and sometimes by showing beliefs in concepts such as unity in diversity, all of which in their essence have concepts consis‌tent with fractal forms.

The fractal geometry of fitness landscapes at the local optima level

A local optima network (LON) encodes local optima connectivity in the fitness landscape of a combinatorial optimisation problem. Recently, LONs have been studied for their fractal dimension. Fractal dimension is a complexity index where a non-integer dimension can be assigned to a pattern. This paper investigates the fractal nature of LONs and how that nature relates to metaheuristic performance on the underlying problem. We use visual analysis, correlation analysis, and machine learning techniques to demonstrate that relationships exist and that fractal features of LONs can contribute to explaining and predicting algorithm performance. The results show that the extent of multifractality and high fractal dimensions in the LON can contribute in this way when placed in regression models with other predictors. Features are also individually correlated with search performance, and visual analysis of LONs shows insight into this relationship.

Information theory and dimensionality of space

We present an information-theoretic approach to the optimal representation of the intrinsic dimensionality of data and show it is a noninteger. Since optimality is accepted as a physical principle, this provides a theoretical explanation for why noninteger dimensions are useful in many branches of physics, where they have been introduced based on experimental considerations. Noninteger dimensions correlate with lesser density as in the Hausdorff dimension and this can have measurable effects. We use the lower density of noninteger dimension to resolve the problem of two different values of the Hubble constant obtained using different methods.

Newtonian Fractional-Dimension Gravity and MOND

This paper introduces a possible alternative model of gravity based on the theory of fractional-dimension spaces and its applications to Newtonian gravity. In particular, Gauss's law for gravity as well as other fundamental classical laws are extended to a $D$-dimensional metric space, where $D$ can be a non-integer dimension. We show a possible connection between this Newtonian Fractional-Dimension Gravity (NFDG) and Modified Newtonian Dynamics (MOND), a leading alternative gravity model which accounts for the observed properties of galaxies and other astrophysical structures without requiring the dark matter hypothesis. The MOND acceleration constant $a_{0} \simeq 1.2 \times 10^{ -10}\mbox{m}\thinspace \mbox{s}^{ -2}$ can be related to a natural scale length $l_{0}$ in NFDG, i.e., $a_{0} \approx GM/l_{0}^{2}$, for astrophysical structures of mass $M$, and the deep-MOND regime is present in regions of space where the dimension is reduced to $D \approx 2$. For several fundamental spherically-symmetric structures, we compare MOND results, such as the empirical Radial Acceleration Relation (RAR), circular speed plots, and logarithmic plots of the observed radial acceleration $g_{obs}$ vs. the baryonic radial acceleration $g_{bar}$, with NFDG results. We show that our model is capable of reproducing these results using a variable local dimension $D\left (w\right )$, where $w =r/l_{0}$ is a dimensionless radial coordinate. At the moment, we are unable to derive explicitly this dimension function $D\left (w\right )$ from first principles, but it can be obtained empirically in each case from the general RAR. Additional work on the subject, including studies of axially-symmetric structures, detailed galactic rotation curves fitting, and a possible relativistic extension, will be needed to establish NFDG as a viable alternative model of gravity.

Efimov effect in non-integer dimensions induced by an external field

The Efimov effect can be induced by means of an external deformed one-body field that effectively reduces the allowed spatial dimensions to less than three. To understand this new mechanism, conceptually and practically, we employ a formulation using non-integer dimension, which is equivalent to the strength of an external oscillator field. The effect most clearly appears when the crucial two-body systems are unbound in three, but bound in two, dimensions. We discuss energy variation, conditions for occurrence, and number of Efimov states, as functions of the dimension. We use practical examples from cold atom physics of 133Cs-133Cs-133Cs, 87Rb-87Rb-87Rb, 133Cs-133Cs-6Li, and 87Rb-87Rb-39K. Laboratory tests of the effect can be performed with two independent parameters, i.e. the external one-body field and the Feshbach two-body tuning. The scaling and (dis)appearance of these Efimov states occur precisely as already found in three dimensions.

The time-dependent Schrödinger equation in non-integer dimensions for constrained quantum motion

We propose a theoretical model, based on a generalized Schrödinger equation, to study the behavior of a constrained quantum system in non-integer, lower than two-dimensional space. The non-integer dimensional space is formed as a product space X×Y, comprising x-coordinate with a Hausdorff measure of dimension α1=D−1 (1<D<2) and y-coordinate with the Lebesgue measure of dimension of length (α2=1). Geometric constraints are set at y=0. Two different approaches to find the Green's function are employed, both giving the same form in terms of the Fox H-function. For D=2, the solution for two-dimensional quantum motion on a comb is recovered.

Three identical bosons: Properties in noninteger dimensions and in external fields

Three-body systems that are continuously squeezed from a three-dimensional (3D) space into a two-dimensional (2D) space are investigated. Such a squeezing can be obtained by means of an external confining potential acting along a single axis. However, this procedure can be numerically demanding, or even undoable, especially for large squeezed scenarios. An alternative is provided by use of the dimension $d$ as a parameter that changes continuously within the range $2\leq d \leq 3$. The simplicity of the $d$-calculations is exploited to investigate the evolution of three-body states after progressive confinement. The case of three identical spinless bosons with relative $s$-waves in 3D, and a harmonic oscillator squeezing potential is considered. We compare results from the two methods and provide a translation between them, relating dimension, squeezing length, and wave functions from both methods. All calculations are then possible entirely within the simpler $d$-method, but simultaneously providing the equivalent geometry with the external potential.

Conformal bootstrap signatures of the tricritical Ising universality class

We study the tricritical Ising universality class using conformal bootstrap techniques. By studying bootstrap constraints originating from multiple correlators on the CFT data of multiple OPEs, we are able to determine the scaling dimension of the spin field $\Delta_\sigma$ in various non-integer dimensions $2 \le d \le 3$. $\Delta_{\sigma}$ is connected to the critical exponent $\eta$ that governs the (tri-)critical behaviour of the two point function via the relation, $\eta = 2 - d + 2 \Delta_{\sigma}$. Our results for $\Delta_\sigma$ match with the exactly known values in two and three dimensions and are a conjecture for non-integer dimensions. We also compare our CFT results for $\Delta_\sigma$ with $\epsilon$-expansion results, available up to $\epsilon^3$ order. Our techniques can be naturally extended to study higher-order multi-critical points.