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Manuj Verma

Publications and source records attributed to Manuj Verma.

8 recordsLinked to original sources
arXiv2025

Hausdorff dimension of the Wedding cake type surfaces

In this paper, we study the Hausdorff dimension of fractal interpolation surfaces (FISs) over a triangular domain. These FISs are known as `wedding cake surfaces'. These surfaces are the attractor of some deterministic self-affine iterated function systems (IFS) on $\mathbb{R}^3$ generated by a fractal interpolation algorithm. Due to the recent seminal result of Rapaport (Adv. Math. 449 (2024) 109734), the dimension theory of self-affine IFS on $\mathbb{R}^3$ is known whenever the IFS is strongly irreducible and proximal. However, the self-affine IFSs associated with FIS are not strongly irreducible. We prove that the Hausdorff dimension of the self-affine set (or FIS) is the same as the affinity dimension outside a set of scaling parameters with zero Lebesgue measure. Lastly, by computing the overlapping number for the associated Furstenberg IFS, we determine the Hausdorff dimension for every type of scaling parameter in a certain range of parameters.

math.DS↗
arXiv2025

Hausdorff dimension of self-similar measures and sets with common fixed point structure

In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of some maps that share the same fixed point. In particular, we will show that out of a Hausdorff co-dimension one exceptional set of natural parameters, such systems satisfy a weak exponential separation. This significantly strengthens the previous result of the first author and Szv\'ak. As an application, we give the Hausdorff dimension of self-affine measures supported on the generalised 4-corner set.

math.DS↗
arXiv2023

Quantization dimensions for inhomogeneous bi-Lipschitz Iterated Function Systems

Let $\nu$ 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 $\mu$ on $\mathbb R^d$ such that $\mu=\sum_{j=1}^N p_j\mu\circ S_j^{-1}+p_0\nu$ 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 $D_r(\mu)$, of a Borel probability measure $\mu$ 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 $\mu$ 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 $\alpha_k$.

math.DS↗
arXiv2022

Graphs of continuous functions and fractal dimension

In this paper, we show that, for any $\beta \in [1,2]$, a given strictly positive real-valued continuous function on $[0,1]$ whose graph has upper box-counting dimension less than or equal to $\beta $ can be decomposed as a product of two real-valued continuous functions on $[0,1]$ whose graphs have upper box-counting dimension equal to $\beta$. 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↗