Indian Journal of Mathematics
Vol. 67, No.2 (2025)
Observability Study in Weighted and Directed Signed Multi-Agent Networks of Single- and Double-Integrator Dynamical Agents
Author: Athira V. S., Vijayakumar S. Muni, Muhammed Rafeek Kallu Vetty, Aadhithyan B., Gudala Janardhana Reddy
Pages: 173–205
Abstract:
In this study we examine observability in signed multi-agent networks with a directed, weighted, and time-invariant communication topology. The network follows linear heterogeneous dynamics, incorporating both first- and second-order integrator agents in continuous time. Agents are classified into sensors (whose state information is fully known beforehand) and non-sensors (whose state information is completely unknown). The objective is to determine whether it is possible to reconstruct the states of non-sensors using the knowledge of the sensors, a property referred to as the network's sensor--non-sensor observability. We formulate the necessary and sufficient mathematical conditions for analyzing this property by utilizing the rank and spectral characteristics of the matrices governing the dynamics. These conditions are applied to various networks to assess their observability, demonstrating the effectiveness of the results presented. Our findings indicate that the sensor--non-sensor observability property in signed, directed, and weighted networks generally differs from that of their unsigned, undirected, and unweighted counterparts. Read more
$s$-elementary super wavelets and super frame wavelets
Author: G. C. S. Yadav, Amita Dwivedi
Pages: 207–220
Abstract:
This paper presents a characterization of s-elementary super wavelets and super frame wavelets in $\bigoplus_{n} L^{2}(\mathbb{R})$, also some examples are constructed. Furthermore, it provides a characterization of s-elementary super wavelets and super frame wavelets for an n x n expansive matrix A in $\bigoplus _{n} L^{2}(\mathbb{R}^{n})$. Read more
Maximal Subgroups of the Multiplicative Semigroup of $2\times 2$ Supertropical Matrices
Author: Meenakshi , Pooja Devi
Pages: 221–252
Abstract:
We give a complete characterisation of Green’s relations $\mathscr{L},\mathscr{R},\mathscr{H}$, $\mathscr{D}$ and $\mathscr{J}$ in the multiplicative semigroup of $2\times 2$ supertropical matrices. We determine the maximal subgroups of the semigroup $M_2(\mathbb{T})$ by finding the $\mathscr{R}- classes$ containing idempotents, followed by the identification of $\mathscr{H}- classes$. Read more
A Comparative Study of Different Methods for Fractal Image Compression
Author: Subhash Chandra Shrivastava, Ritu Shrivastava
Pages: 253–266
Abstract:
Fractal image compression can be done by partitioning of an image into different domains, and for each domain there is a transformation into range element. The normal algorithm produces for this purpose is known as Recurrent Iterated Function System. The domain-range equivalence also applied to the general inverse problem of RIFS. The method of fractal image compression through RIFS is better than general fractal image compression technique. RIFS are improvements of IFS using elements of the theory of Marcovian stochastic processes which can produce more natural looking images. New RIFs consists of vertical contraction factor function and nonlinear transformations. A very fast fractal-based image compression encoding technique is the theory of IFS with probabilities. In this approach a Markov operator associated with the probability operator. One more extension of IFS theory for image compression is partitioned or local iterative function system (PIFS) for coding the gray level images. The difference between PIFS and IFS technique for image compression is in the application domain and computational cost. Read more
On a Class of Difference Sequences of Multiset Real Number Related to $\ell_p$-Space
Author: Amaresh Debnath, Runu Dhar, B.C. Tripathy
Pages: 267–279
Abstract:
In this article we introduce $m(\phi, n, \Delta)$, the class of differences sequences of multiset real numbers relating to $\ell_p$-space ($p \geq 1$). We have studied its different properties like linearity, completeness, solidity, symmetry, convexity, convergence free, sequence algebra etc. We have investigated its different properties. Also we have obtained some inclusion results involving this sequence space. Read more
On Almost Abelian Finite Type Transcendental Semigroups
Author: Dal Bahadur Saud, BIishnu Hari Subedi, Ajaya Singh
Pages: 281–305
Abstract:
Let $ S $ be a finite type semigroup generated by transcendental entire functions. Define escaping set of semigroup $ S $ by $ I(S)= \{ z \in \mathbb{C}:S \text{ is iteratively divergent at } z \} $. In this paper, we investigate the following results: Suppose $ I(S) = I(h) $ for generator $ h \in S $ such that there exists a conformal map $ \phi(z) = -z $ satisfying $ h = h \circ \phi $ and $ \phi \circ h = g $ with $ g \in S $ or there exists a conformal map $ \phi (z)$= $az+b $ for $a,b\in \mathbb{C}$ with $ a\neq 0 $ satisfying $ h \circ g = \phi \circ g \circ h $ for all generators $ h, g \in S $. Then the semigroup $ S=\langle h,g\rangle $ is almost abelian. Also under the similar condition involving finite generators of $ S $ and a conformal map, the subsemigroup generated by some generators among the finite generators is almost abelian. Moreover for every arbitrarily finitely generated semigroup $ S = \langle h_1, h_2, \ldots, h_n \rangle $ of finite type, neither the semigroup $S$ itself nor any of its subsemigroup(s) is necessarily an almost abelian. However, for every arbitrarily infinitely generated semigroup $ S = \langle h_1, h_2, h_3, \ldots \rangle $ or finitely generated semigroup $ S = \langle h_1, h_2, \ldots, h_n \rangle $ of finite type, if $ I(S) = I(h_i) $ for $ h_i \in S $ satisfying $ h_i = h_i \circ \phi $ and $ \phi \circ h_i = h_j \in S $ under the conformal map $ \phi(z) = -z $ or satisfying $ h_i \circ h_j = \phi \circ h_j \circ h_i $ for $ h_i, h_j \in S $, where $ h_i \neq h_j $ (for $ i, j\in\mathbb{N} : $ $ i \neq j $) under conformal map $ \phi (z)$= $az+b$ for $a,b\in \mathbb{C}$ with $a\neq 0 $, the full semigroup or its subsemigroup(s) or both become necessarily almost abelian. Read more