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Amit Priyadarshi

Publications and source records attributed to Amit Priyadarshi.

10 recordsLinked to original sources
arXiv2026

Dimension Spectrum of Continued fraction Expansions with Coefficients restricted to the Fibonacci Sequence

In this paper, we analyze the structure of the dimension spectrum of continued fraction expansions with coefficients restricted to the generalized Fibonacci sequence. Let $F_{(a_1,a_2)}$ denote the generalized Fibonacci sequence starting with the positive integers $a_1<a_2$. We prove that the continued fractions whose digits lie in $F_{(a_1,a_2)}$ have full dimension spectrum for every $(a_1,a_2)$ such that $a_1 \geq2$, or $a_1=1$ and $a_2\geq3$. On the other hand, using the numerical tools developed by Falk and Nussbaum, we show that the dimension spectrum has a gap for continued fractions with digits restricted to each of the sets $F_{(1,2)}$ and $F_{(2,1)}$, where $F_{(2,1)}$ denotes the set of Lucas numbers. Moreover, for $F_{(1,2)}$ and $F_{(2,1)}$, we prove that the dimension spectrum always contains a non-trivial interval.

math.DS↗
arXiv2025

Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems

Let $ν$ be a Borel probability measure on a $d$-dimensional Euclidean space $\mathbb{R}^d$, $d\geq 1$, with a compact support, and let $(p_0, p_1, p_2, \ldots, p_N)$ be a probability vector with $p_j>0$ for $0\leq j\leq N$. Let $\{S_j: 1\leq j\leq N\}$ be a set of contractive mappings on $\mathbb{R}^d$. Then, a Borel probability measure $μ$ on $\mathbb R^d$ such that $μ=\sum_{j=1}^N p_jμ\circ S_j^{-1}+p_0ν$ is called an inhomogeneous measure, also known as a condensation measure on $\mathbb{R}^d$. For a given $r\in (0, +\infty)$, the quantization dimension of order $r$, if it exists, denoted by $D_r(μ)$, of a Borel probability measure $μ$ on $\mathbb{R}^d$ represents the speed at which the $n$th quantization error of order $r$ approaches to zero as the number of elements $n$ in an optimal set of $n$-means for $μ$ tends to infinity. In this paper, we investigate the quantization dimension for such a condensation measure.

math.PR↗
arXiv2022

Fractal dimensions of fractal transformations and Quantization dimensions for bi-Lipschitz mappings

In this paper, we study the fractal dimension of the graph of a fractal transformation and also determine the quantization dimension of a probability measure supported on the graph of the fractal transformation. Moreover, we estimate the quantization dimension of the invariant measures corresponding to a weighted iterated function system consisting of bi-Lipschitz mappings under the strong open set condition.

math.DS↗
arXiv2022

Vector-valued fractal functions: Fractal dimension and Fractional calculus

There are many research available on the study of real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for vector-valued fractal interpolation function and its Riemann-Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of associated invariant measure supported on the graph of vector-valued fractal interpolation function. Next, We discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann-Liouville fractional integral of a vector-valued fractal interpolation function.

math.DS↗
arXiv2022

Fractal dimension for a class of complex-valued fractal interpolation functions

There are many research papers dealing with fractal dimension of real-valued fractal functions in the recent literature. The main focus of the present paper is to study fractal dimension of complex-valued functions. This paper also highlights the difference between dimensional results of the complex-valued and real-valued fractal functions. In this paper, we study the fractal dimension of the graph of complex-valued function $g(x)+i h(x)$, compare its fractal dimension with the graphs of functions $g(x)+h(x)$ and $(g(x),h(x))$ and also obtain some bounds. Moreover, we study the fractal dimension of the graph of complex-valued fractal interpolation function associated with a germ function $f$, base function $b$ and scaling functions $α_k$.

math.DS↗
arXiv2022

Graphs of continuous functions and fractal dimension

In this paper, we show that, for any $β\in [1,2]$, a given strictly positive real-valued continuous function on $[0,1]$ whose graph has upper box-counting dimension less than or equal to $β$ can be decomposed as a product of two real-valued continuous functions on $[0,1]$ whose graphs have upper box-counting dimension equal to $β$. We also obtain a formula for the upper box-counting dimension of every element of a ring of polynomials in finite number of continuous functions on $[0,1]$ over the field $\mathbb{R}.$

math.FA↗
arXiv2018

On the box dimension of graph of harmonic functions on the Sierpiński gasket

In this paper, we have obtained bounds for the box dimension of graph of harmonic function on the Sierpiński gasket. Also we get upper and lower bounds for the box dimension of graph of functions that belongs to $\text{dom}(\mathcal{E}),$ that is, all finite energy functionals on the Sierpiński gasket. Further, we show the existence of fractal functions in the function space $\text{dom}(\mathcal{E})$ with the help of fractal interpolation functions. Moreover, we provide bounds for the box dimension of some functions that belong to the family of continuous functions and arise as fractal interpolation functions.

math.MG↗
arXiv2018

A System of p-Laplacian Equations on the Sierpinski Gasket

In this paper we study a system of boundary value problems involving weak p-Laplacian on the Sierpiński gasket in $\mathbb{R}^2$. Parameters $λ, γ, α, β$ are real and $1 1$ we show the existence of at least two nontrivial weak solutions to the system of equations for some $(λ,γ) \in \mathbb{R}^2.$

math.AP↗
arXiv2018

Existence of multiple solutions of a p-Laplacian equation on the Sierpinski Gasket

In this paper we study the following boundary value problem involving the weak p-Laplacian. \begin{equation*} \quad -M(\|u\|_{\mathcal{E}_p}^p)Δ_p u = h(x,u) \; \text{in}\; \mathcal{S}\setminus\mathcal{S}_0; \quad u = 0 \; \mbox{on}\; \mathcal{S}_0, \end{equation*} where $\mathcal{S}$ is the Sierpiński gasket in $\mathbb{R}^2$, $\mathcal{S}_0$ is its boundary. $M : \mathbb{R} \to \mathbb{R}$ defined by $M(t) = at^k +b$ and $a,b,k >0$ and $h : \mathcal{S} \times \mathbb{R} \to \mathbb{R}.$ We will show the existence of two nontrivial weak solutions to the above problem.

math.AP↗