arXiv subjects
Publications and source records attributed to Bipan Hazarika.
The objective of this paper is to construct separable Banach spaces $S{D^p}[\mathbb{R}^\infty]$ for $1\leq p \leq \infty$, each of which contains the $L^p[\mathbb{R}^\infty] $ spaces, as well as finitely additive measures, as compact dense embedding. Also these spaces contains Henstock-Kurzweil integrable functions.
The motivation of the article is to introduce Henstock-Orlicz space with non-absolute integrable functions. We prove $ C_{0}^{\infty} $ is dense in the Henstock-Orlicz space, which is not dense in the classical Orlicz space.
We introduce Kuelbs-Steadman-type spaces for real-valued functions, with respect to countably additive measures, taking values in Banach spaces. We investigate their main properties and embeddings in $L^p$-type spaces, considering both the norm associated to norm convergence of the involved integrals and that related to weak convergence of the integrals.
Given a real Banach space $\mathcal{X}$ and probability space $(Ω, Σ, μ)$ we characterize the countable additivity of Henstock-Dunford integral for Henstock integrable function taking values in $X$ as those weakly measurable function $ g: Ω\to \mathcal{X} $ for which $\{y^*g~: y^* \in B_\mathcal{X}^* \} $ is relatively weakly compact in some separable Orlicz space $ \mathcal{L}^{\overlineϕ}(μ) .$ We find relatively weakly compact in some Orlicz space with Henstock-Gel'fand integral.
Recently Ruckle \cite{RuckleArithmeticalSummability} introduced the theory of arithmetical summability suggested by the sum $ \sum_{k|m}f(k) $ as $ k $ ranges over the divisors of $m$ including $ 1 $ and $ m .$ Following Ruckle \cite{RuckleArithmeticalSummability} we construct the sequence space $ AS(G) $ and $ AC(G) $ of arithmetic summable and arithmetic convergent sequences in the sense of geometric calculus and derive interesting results in the geometric field.
The aim of the paper is to introduce the spaces $\ell_{\infty}^λ(\widehat{F})$ and $\ell_{p}^λ(\widehat{F})$ derived by the composition of the two infinite matrices $Λ=(λ_{nk})$ and $\widehat{F}=\left( f_{nk} \right),$ which are the $BK$-spaces of non-absolute type and also derive some inclusion relations. Further, we determine the $α$-, $β$-, $γ$-duals of those spaces and also construct the basis for $\ell_{p}^λ(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^λ(\widehat{F})$ and $\ell_{p}^λ(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$ Here we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators, where $X\in\{\ell_{\infty}^λ(\widehat{F}),\ell_{p}^λ(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\}$ by applying the Hausdorff measure of non-compactness, and $1\leq p<\infty.$
In this paper we introduce the notion of interval valued hesitant fuzzy soft topological space. Also the concepts of interval valued hesitant fuzzy soft closure, interior and neighbourhood are introduced here and established some important results.
A lacunary sequence is an increasing integer sequence $θ=(k_r)$ such that $k_r-k_{r-1}\rightarrow \infty$ as $r\rightarrow \infty.$ In this article we introduce arithmetic statistically convergent sequence space $ASC$ and lacunary arithmetic statistically convergent sequence space $ASC_θ$ and study some inclusion properties between the two spaces. Finally we introduce lacunary arithmetic statistical continuity and establish some interesting results.
In this article we introduce and study the lacunary arithmetic convergent sequence space $AC_θ$. Using the idea of strong Cesàro summable sequence and arithmetic convergence we define $AC_{σ_1}$ and study the relations between $AC_θ$ and $AC_{σ_1}$. Finally using modulus function we define $AC_θ(f)$ and study some interesting results.
Based on M. Grossman in \cite{Grossman83} and Grossman an Katz \cite{GrossmanKatz}, in this paper we discuss about the applications of bigeometric calculus in different branches of mathematics and economics.
Objective of this paper is to introduce a new type of calculus which will be called G-Calculus based on non-Newtonian calculus introduced by Grossman and Katz \cite{GrossmanKatz}. The basic difference between geometric calculus defined by Grossman and Katz and the present G-calculus is that Grossman took the values of the argument as $x, x+ h, x+2h,...$ but here in G-calculus we take the values as $x, x\oplus h, x\oplus e^2\odot h, x\oplus e^3\odot h....$ This calculus will have great deal with numerical analysis which are discussed in the last section of this paper.
The main purpose of this paper is to introduce the geometric difference sequence space $l_\infty^{G} (Δ_G)$ and prove that $l_\infty^{G} (Δ_{G})$ is a Banach space with respect to the norm $\left\|.\right\|^G_{Δ_G}.$ Also we compute the $α$-dual, $β$-dual and $γ$-dual spaces. Finally we obtain the Geometric Newton-Gregory interpolation formulae.
Objective of this paper is to introduce the generalized geometric difference sequence spaces $l_\infty^{G}(Δ^m_G), c^G(Δ^m_G), c_0^{G}(Δ^m_G)$ and to prove that these are Banach spaces. Then we prove some inclusion properties. Also we compute their dual spaces.
In this paper, we study composite mapping on hesitant fuzzy soft classes. We also establish some interesting properties of this notion and support them with examples. Also definitions of fuzzy soft image and fuzzy soft inverse image put forward by Kharal and Ahmad \cite{kharal} have been reintroduced in hesitant fuzzy soft classes.
The aim of the paper is to introduced the spaces $c_{0}^λ(\hat{F})$ and $c^λ(\hat{F})$ which are the BK-spaces of non-absolute type and also derive some inclusion relations. Further, we determine the $α-,β-,γ-$duals of those spaces and also construct their bases. We also characterize some matrix classes on the spaces $c_{0}^λ(\hat{F})$ and $c^λ(\hat{F}).$ Here we characterize the subclasses $\mathcal{K}(X,Y)$ of compact operators where $X$ is $c_{0}^λ(\hat{F})$ or $c^λ(\hat{F})$ and $Y$ is one of the spaces $c_{0},c, l_{\infty}, l_{1}, bv$ by applying Hausdorff measure of noncompactness.
The main objective of this paper is to introduced a new sequence space $l_{p}(\hat{F}(r,s)),$ $ 1\leq p \leq \infty$ by using the band matrix $\hat{F}(r,s).$ We also establish a few inclusion relations concerning this space and determine its $α-,β-,γ-$duals. We also characterize some matrix classes on the space $l_{p}(\hat{F}(r,s))$ and examine some geometric properties of this space.
The main aim of this paper is to introduced the operations "Union" and "Intersection," and four operators $O_1, O_2, O_3, O_4 $ on interval-valued hesitant fuzzy soft sets and discuss some of their properties.
An ideal $I$ is a family of subsets of positive integers $\mathbb{N}$ which is closed under taking finite unions and subsets of its elements. A sequence $(x_k)$ of real numbers is said to be lacunary $I$-convergent to a real number $\ell$, if for each $ \varepsilon> 0$ the set $$\left\{r\in \mathbb{N}:\frac{1}{h_r}\sum_{k\in J_r} |x_{k}-\ell|\geq \varepsilon\right\}$$ belongs to $I.$ The aim of this paper is to study the notion of lacunary $I$-convergence in probabilistic normed spaces as a variant of the notion of ideal convergence. Also lacunary $I$-limit points and lacunary $I$-cluster points have been defined and the relation between them has been established. Furthermore, lacunary-Cauchy and lacunary $I$-Cauchy sequences are introduced and studied. Finally, we provided example which shows that our method of convergence in probabilistic normed spaces is more general.