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The julia package integrates the Julia programming language into Stata. Users can transfer data between Stata and Julia, issue Julia commands to analyze and plot, and pass results back to Stata. Julia's econometric ecosystem is not as mature as Stata's or R's or Python's. But Julia is an excellent environment for developing high-performance numerical applications, which can then be called from many platforms. For example, the boottest program for wild bootstrap-based inference (Roodman et al. 2019) and fwildclusterboot for R (Fischer and Roodman 2021) can use the same Julia back end. And the program reghdfejl mimics reghdfe (Correia 2016) in fitting linear models with high-dimensional fixed effects while calling a Julia package for tenfold acceleration on hard problems. reghdfejl also supports nonlinear fixed-effect models that cannot otherwise be fit in Stata--though preliminarily, as the Julia package for that purpose is immature.
The study of dynamical systems involves analyzing how functions behave under iteration in different mathematical spaces. In the context of complex dynamics, tools such as Julia sets and filled Julia sets are used to understand the long-term behavior of functions in the complex Euclidean field. In this paper, we will present a review of Julia sets and filled Julia sets, provide an overview of the mathematical formulation of the alternated Julia sets introduced in the work of Danca-Romera-Pastor, extend it to the $p$-adic setting, and propose a tool that can potentially be used to study the arithmetic dynamics of various types of functions. Additionally, we will summarize key results on connectivity properties and visualization techniques as discussed in the work of Danca-Bourke-Romera and provide a visualization algorithm and pseudocode that enable the visualization of alternated Julia sets with various connectivity properties.
By a symmetry of the Julia set of a polynomial, also referred as polynomial Julia set, we mean an Euclidean isometry preserving the Julia set. Each such symmetry is in fact a rotation about the centroid of the polynomial. In this article, a survey of the symmetries of polynomial Julia sets is made. Then the Euclidean isometries preserving the Julia set of rational maps are considered. A rotation preserving the Julia set of a rational map is called a rotational symmetry of its Julia set. A sufficient condition is provided for a rational map to have rotational symmetries whenever the rational map has an exceptional point. Two classes of rational maps are provided whose Julia sets have rotational symmetries of finite orders. Using this, it is proved that $ z\mapsto μz$ where $μ^{m+n}=1$ is a rotational symmetry of the McMullen map $ z^m+\fracλ{z^n}$ for all $m,n$ with $m\geq 2$ and $λ\in \mathbb{C}\setminus \{0\}$. Assuming that a normalized polynomial has a simple root at the origin, it is shown that the groups of the rotational symmetries of the polynmial coincide with that of its Newton's method and Chebyshev's method.
It has been shown that Cantor bubble Julia sets can appear in the dynamics of polynomials and their singular perturbations. In this paper, we present a criterion that guarantees the existence of Cantor bubble Julia sets for certain rational maps with attracting or parabolic fixed points. Moreover, we construct other Cantor bubble Julia sets, including those with high-periodic attracting cycles and those with Hausdorff dimension two. Finally, we give a sufficient condition for Cantor bubble Julia sets to be quasisymmetrically equivalent to Cantor round bubbles.
We provide a complete quasisymmetric classification of the Julia sets of postcritically finite McMullen maps $f_λ(z)=z^n+λ/z^n$ with $λ\in\mathbb{C}^*$ and $n\geq 2$, and prove that the quasisymmetry group of each such Julia set is exactly the finite dihedral group generated by the natural symmetries of the map. These results establish quasisymmetric rigidity for all topological classes in this family, including Sierpiński-like carpets, necklaces, and clusters, and provide the first known examples of rigid Julia sets in each of the three classes.
This paper proposes integrating Aspect-oriented Programming (AOP) into Julia, a language widely used in scientific and High-Performance Computing (HPC). AOP enhances software modularity by encapsulating cross-cutting concerns, such as logging, caching, and parallelizing, into separate, reusable aspects. Leveraging Julia's powerful metaprogramming and abstract syntax tree (AST) manipulation capabilities, we introduce AspectJulia, an AOP framework designed to operate within Julia's runtime environment as a package. AspectJulia enables developers to define and apply aspects seamlessly, leading to more modular, maintainable, and adaptable code. We detail the implementation of AspectJulia and present diverse use cases, ranging from HPC and scientific computing to business applications, demonstrating its effectiveness in managing cross-cutting concerns. This integration simplifies application development and improves the adaptability of existing Julia modules and packages, paving the way for more efficient and maintainable software systems.
Julia is a mature general-purpose programming language, with a large ecosystem of libraries and more than 12000 third-party packages, which specifically targets scientific computing. As a language, Julia is as dynamic, interactive, and accessible as Python with NumPy, but achieves run-time performance on par with C/C++. In this paper, we describe the state of adoption of Julia in HEP, where momentum has been gathering over a number of years. HEP-oriented Julia packages can already, via UnROOT.jl, read HEP's major file formats, including TTree and RNTuple. Interfaces to some of HEP's major software packages, such as through Geant4.jl, are available too. Jet reconstruction algorithms in Julia show excellent performance. A number of full HEP analyses have been performed in Julia. We show how, as the support for HEP has matured, developments have benefited from Julia's core design choices, which makes reuse from and integration with other packages easy. In particular, libraries developed outside HEP for plotting, statistics, fitting, and scientific machine learning are extremely useful. We believe that the powerful combination of flexibility and speed, the wide selection of scientific
Like many groups considering the new programming language Julia, we faced the challenge of accessing the algorithms that we develop in Julia from R. Therefore, we developed the R package JuliaConnectoR, available from the CRAN repository and GitHub (https://github.com/stefan-m-lenz/JuliaConnectoR), in particular for making advanced deep learning tools available. For maintainability and stability, we decided to base communication between R and Julia on TCP, using an optimized binary format for exchanging data. Our package also specifically contains features that allow for a convenient interactive use in R. This makes it easy to develop R extensions with Julia or to simply call functionality from Julia packages in R. Interacting with Julia objects and calling Julia functions becomes user-friendly, as Julia functions and variables are made directly available as objects in the R workspace. We illustrate the further features of our package with code examples, and also discuss advantages over the two alternative packages JuliaCall and XRJulia. Finally, we demonstrate the usage of the package with a more extensive example for employing neural ordinary differential equations, a recent deep
We prove that every wandering exposed Julia component of a rational map is to a singleton, provided that each wandering Julia component containing critical points is non-recurrent. Moreover, we show that the Julia set contains only finitely many periodic complex-type components if each wandering Julia component containing critical values is non-recurrent.
The object of the paper is to characterize gasket Julia sets of rational maps that can be uniformized by round gaskets. We restrict to rational maps without critical points on the Julia set. Under these conditions, we prove that a Julia set can be quasiconformally uniformized by a round gasket if and only if it is a fat gasket, i.e., boundaries of Fatou components intersect tangentially. We also prove that a Julia set can be uniformized by a round gasket with a David homeomorphism if and only if every Fatou component is a quasidisk; equivalently, there are no parabolic cycles of multiplicity 2. Our theorem applies to show that gasket Julia sets and limit sets of Kleinian groups can be locally quasiconformally homeomorphic, although globally this is conjectured to be false.
The Julia programming language has gained acceptance within the High-Performance Computing (HPC) community due to its ability to tackle two-language problem: Julia code feels as high-level as Python but allows developers to tune it to C-level performance. But to squeeze every drop of performance, Julia needs to integrate with advanced performance analysis tools, also known as profilers. In this work, we present Extrae.jl, a Julia package to interface with the Extrae profiler.
The Julia programming language was designed to fill the needs of scientific computing by combining the benefits of productivity and performance languages. Julia allows users to write untyped scripts easily without needing to worry about many implementation details, as do other productivity languages. If one just wants to get the work done-regardless of how efficient or general the program might be, such a paradigm is ideal. Simultaneously, Julia also allows library developers to write efficient generic code that can run as fast as implementations in performance languages such as C or Fortran. This combination of user-facing ease and library developer-facing performance has proven quite attractive, and the language has increasing adoption. With adoption comes combinatorial challenges to correctness. Multiple dispatch -- Julia's key mechanism for abstraction -- allows many libraries to compose "out of the box." However, it creates bugs where one library's requirements do not match what another provides. Typing could address this at the cost of Julia's flexibility for scripting. I developed a "best of both worlds" solution: gradual typing for Julia. My system forms the core of a gradu
It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least 5. This paper introduce a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover this family contains some rational maps of degree 3 with explicit formula that answers a question McMullen raised.
The growing proliferation of FPGAs and High-level Synthesis (HLS) tools has led to a large interest in designing hardware accelerators for complex operations and algorithms. However, existing HLS toolflows typically require a significant amount of user knowledge or training to be effective in both industrial and research applications. In this paper, we propose using the Julia language as the basis for an HLS tool. The Julia HLS tool aims to decrease the barrier to entry for hardware acceleration by taking advantage of the readability of the Julia language and by allowing the use of the existing large library of standard mathematical functions written in Julia. We present a prototype Julia HLS tool, written in Julia, that transforms Julia code to VHDL. We highlight how features of Julia and its compiler simplified the creation of this tool, and we discuss potential directions for future work.
The primary aim of this paper is to give topological obstructions to Cantor sets in $\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our main tool is the genus of a Cantor set. We give a new construction of a genus $g$ Cantor set, the first for which the local genus is $g$ at every point, and then show that this Cantor set can be realized as the Julia set of a uniformly quasiregular mapping. These are the first such Cantor Julia sets constructed for $g\geq 3$. We then turn to our dynamical applications and show that every Cantor Julia set of a hyperbolic uniformly quasiregular map has a finite genus $g$; that a given local genus in a Cantor Julia set must occur on a dense subset of the Julia set; and that there do exist Cantor Julia sets where the local genus is non-constant.
The airplane, the basilica and the Douady rabbit (and, more generally, rabbits with more than two ears) are well-known Julia sets of complex quadratic polynomials. In this paper we study the groups of all homeomorphisms of such fractals and of all automorphisms of their laminations. In particular, we identify them with some kaleidoscopic group or universal groups and thus realize them as Polish permutation groups. From these identifications, we deduce algebraic, topological and geometric properties of these groups.
Fractals offer the ability to generate fascinating geometric shapes with all sorts of unique characteristics (for instance, fractal geometry provides a basis for modelling infinite detail found in nature). While fractals are non-euclidean mathematical objects which possess an assortment of properties (e.g., attractivity and symmetry), they are also able to be scaled down, rotated, skewed and replicated in embedded contexts. Hence, many different types of fractals have come into limelight since their origin discovery. One particularly popular method for generating fractal geometry is using Julia sets. Julia sets provide a straightforward and innovative method for generating fractal geometry using an iterative computational modelling algorithm. In this paper, we present a method that combines Julia sets with dual-quaternion algebra. Dual-quaternions are an alluring principal with a whole range interesting mathematical possibilities. Extending fractal Julia sets to encompass dual-quaternions algebra provides us with a novel visualize solution. We explain the method of fractals using the dual-quaternions in combination with Julia sets. Our prototype implementation demonstrate an effici
We develop a theory of quasisymmetries for finitely ramified fractals, with applications to finitely ramified Julia sets. We prove that certain finitely ramified fractals admit a naturally defined class of "undistorted metrics" that are all quasi-equivalent. As a result, piecewise-defined homeomorphisms of such a fractal that locally preserve the cell structure are quasisymmetries. This immediately gives a solution to the quasisymmetric uniformization problem for topologically rigid fractals such as the Sierpiński triangle. We show that our theory applies to many finitely ramified Julia sets, and we prove that any connected Julia set for a hyperbolic unicritical polynomial has infinitely many quasisymmetries, generalizing a result of Lyubich and Merenkov. We also prove that the quasisymmetry group of the Julia set for the rational function $1-z^{-2}$ is infinite, and we show that the quasisymmetry groups for the Julia sets of a broad class of polynomials contain Thompson's group $F$.
We continue the study of constructing invariant Laplacians on Julia sets, and studying properties of their spectra. In this paper we focus on two types of examples: 1) Julia sets of cubic polynomials $z^3 + c$ with a single critical point; 2) formal matings of quadratic Julia sets. The general scheme introduced in earlier papers in this series involves realizing the Julia set as a circle with identifications, and attempting to obtain the Laplacian as a renormalized limit of graph Laplacians on graphs derived form the circle with identifications model. In the case of cubic Julia sets the details follows the pattern established for quadratic Julia sets, but for matings the details are quite challenging, and we have only been completely successful for one example. Once we have constructed the Laplacian, we are able to use numerical methods to approximate the eigenvalues and eigenfunctions. One striking observation from the data is that for the cubic Julia sets the multiplicities of all eigenspaces (except for the trivial eigenspace of constants) are even numbers. Nothing like this is valid for the quadratic julia sets studied earlier. We are able to explain this, based on the fact tha
We prove that the Assouad dimension of a parabolic Julia set is $\max\{1,h\}$ where $h$ is the Hausdorff dimension of the Julia set. Since $h$ may be strictly less than 1, this provides examples where the Assouad and Hausdorff dimensions are distinct. The box and packing dimensions of the Julia set are also known to coincide with $h$ and, moreover, $h$ can be characterised by a topological pressure function. The distinctive behaviour of the Assouad dimension invites further analysis of the `Assouad type dimensions', including the lower dimension and the Assouad and lower spectra. We derive formulae for all of the Assouad type dimensions for parabolic Julia sets and the associated $h$-conformal measure. Further, we show that if a Julia set has a Cremer point, then the Assouad dimension is 2.