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Purbita Jana

Publications and source records attributed to Purbita Jana.

6 recordsLinked to original sources
arXiv2025

Continuous Algebra: Algebraic Semantics for Continuous Propositional Logic

We present algebraic semantics for Continuous Propositional Logic, CPL, introduced by Itai Ben Yaacov, viewed as {\L}ukasiewicz propositional logic with a reversed truth-falsity orientation and enriched by a unary halving connective. We introduce continuous algebras as MV-algebras together with an unary operator $\kappa$ analogous to the halving operator introduced in CPL and analyze their core structural properties, including ideals, quotient constructions, and subdirect representations. We further establish a correspondence between continuous algebras and the class of 2-divisible $\ell u$-groups, extending Mundici's representation theory to the continuous setting. This correspondence leads to a purely algebraic proof of the weak completeness theorem for CPL.

math.LO↗
arXiv2024

Boundary fractional Hardy's inequality in dimension one: The critical case

We prove fractional boundary Hardy's inequality in dimension one for the critical case $sp =1$. Optimality of the inequality is obtained for any $p$. The extra logarithmic correction term appears in usual fashion. We also provide a concrete (workable) example of a sequence of smooth functions that converges to constant function in $W^{s,p}((0,1))$ for $sp=1$ and $p=2$.

math.AP↗
arXiv2018

Fuzzy $α$-cut and related structures

This paper deals with a new notion called fuzzy $α$-cut and its properties. A notion called localic frame is also introduced. Algebraic structures arising out of the family of fuzzy $α$-cuts have been investigated. It will be seen that this family forms a localic frame. Some significance and usefulness of fuzzy $α$-cuts are discussed.

math.GM↗
arXiv2017

Categorical Accommodation of Graded Fuzzy Topological System, Graded Frame and Fuzzy Topological Space with Graded inclusion

A detailed study of graded frame, graded fuzzy topological system and fuzzy topological space with graded inclusion is already done in our earlier paper. The notions of graded fuzzy topological system and fuzzy topological space with graded inclusion were obtained via fuzzy geometric logic with graded con- sequence. As an off shoot the notion of graded frame has been developed. This paper deals with a detailed categorical study of graded frame, graded fuzzy topological system and fuzzy topological space with graded inclusion and their interrelation.

math.GM↗
arXiv2016

A study of the interrelation between fuzzy topological systems and logic

The major part of this thesis deals with fuzzy geometric logic and fuzzy geometric logic with graded consequence. The first chapter mainly contains the concept of topological system introduced by S. Vickers in 1989. In Chapter 2 the notion of fuzzy topological system is introduced and categorical relationship with fuzzy topology and frame is discussed in detail. Also this chapter contains some methodology to make new fuzzy topological systems from the old one.Chapter 3 provides a generalization of fuzzy topological system which shall be called L topological system and categorical relationships with appropriate topological space and frame. Furthermore, two ways of constructing subspaces and subsystems of an L topological space and an L topological system are respectively provided.Chapter 4 deals with the concept of variable basis fuzzy topological space on fuzzy sets and contains a new notion of variable basis fuzzy topological systems whose underlying sets are fuzzy sets. In this chapter categorical relationship between space and system is established. Chapter 5 contains a different proof of one kind of generalized stone duality, which was done directly by Maruyama, introducing a notion of n fuzzy Boolean system. The last two chapters, Chapter 6 and Chapter 7 deal the ultimate objective. Chapter 6 deals with the question- From which logic fuzzy topology can be studied?. To answer this, the notion of fuzzy geometric logic is invented.

math.GM↗