| A NOTE ON PO-EQUIVALENT TOPOLOGIES | 95--99 |
Abstract
Two topologies on a set $X$ are called PO-equivalent if their families of preopen sets concide. Let $P({\cal T})$ stand for the class of all topologies on X which are PO-equivalent to ${\cal T}$ and denote by ${\cal T}_M$ the topology on $X$ having for a base ${\cal T}_{\alpha}\cup \{\{x\}\mid \{x\}$ is closed-and-open in ${\cal T}_{\gamma}\}$. It was proved in [Andrijević, M. Ganster, \emph{On $PO$-equivalent topologies}, Suppl. Rend. Circ. Mat. Palermo, {\bf 24} (1990), 251--256] that the class $P({\cal T})$ does not have the largest member in general. Precisely, if $P({\cal T})$ has the largest member, say ${\cal U}$, then ${\cal U}={\cal T}_M$. On the other hand, it was shown that ${\cal T}_M$ does not necessarily belong to $P({\cal T})$. In this paper we are going to show that the topology ${\cal T}_M$ is actually the least upper bound of the class~$P({\cal T})$.
Keywords: Preopen set; topology ${\cal T}_{\alpha}$; topology ${\cal T}_{\gamma}$; topology ${\cal T}_M$.
MSC: 54A10
| ITERATIVE COMBINATIONS OF GENERALISED OPERATORS | 100--109 |
Abstract
In the present paper, We consider the generalised form of iterative combinations of positive linear operators with well-known Bernstein and Baskakov operators as its particular case. We have estimated the iterative operator's $r^{th} $ moment and found a recurrence relation between the central moments and their derivatives. We deduce the Voronovskaya type asymptotic formula and the relation between the error of a continuous function and its norm with restrictions on its higher derivatives.
Keywords: Bernstein operators; iterative operators; Voronovskaya type asymptotic formula; modulus of continuity.
MSC: 41A25, 41A36
| MULTIPLICATIVE ORDER CONTINUOUS OPERATORS ON RIESZ ALGEBRAS | 110--119 |
Abstract
In this paper, we investigate operators on Riesz algebras, which are continuous with respect to multiplicative modifications of order convergence and relatively uniform convergence. We also introduce and study $\mathbb{mo}$-Lebesgue, $\mathbb{mo}$-$KB$, and $\mathbb{mo}$-Levi operators.
Keywords: Riesz space; relatively uniform convergence; multiplicative order convergence; $\mathbb{mo}$-continuous operator; $\mathbb{omo}$-continuous operator; Riesz algebra; $d$-algebra; $\mathbb{mo}$-bounded operator; $\mathbb{mo}$-$KB$ operator.
MSC: 46A40, 46B42, 46J40, 47B65
| ON WARPED-TWISTED PRODUCT MANIFOLDS WITH GRADIENT SOLITONS | 120--130 |
Abstract
In this paper, we give new characterizations for warped-twisted product manifolds. We study gradient Riemann, gradient Ricci, gradient Yamabe solitons and quasi-Einstein case on warped-twisted product manifolds and we investigate the effect of a gradient soliton and quasi-Einstein case on such manifolds to their factor manifolds. We also get some results when the factor manifolds of the warped-twisted product manifolds are compact, the twisting and the warping functions are harmonic.
Keywords: Warped product; twisted product; Riemann soliton; Ricci soliton; Yamabe soliton; quasi-Einstein manifold.
MSC: 53C25, 53C20
| ON ERDŐS-LAX AND BERNSTEIN INEQUALITIES FOR GENERATING OPERATORS | 131--140 |
Abstract
In this paper, some generalizations and improved versions of classical Erdős-Lax inequality and Bernstein inequality are stated and proved.
Keywords: Polynomials; inequalities in complex plane; Rouche's theorem; generalized polar derivative; zeros.
MSC: 30A10, 30C10, 30C15
| AUTOMATIC CONTINUITY OF ALMOST DERIVATIONS ON LMC $\boldsymbol{Q}$-ALGEBRAS | 141--146 |
Abstract
In this article, almost derivation on LMC algebras is introduced. Also, it is proved that every almost derivation(or, surjective almost derivation) $T$ on semisimple LMC $Q$-algebras $\Gamma$ with an additional condition on $\Gamma$ has a closed graph. Moreover, it is derived that every almost derivation(or, surjective almost derivation) $T$ on semisimple commutative(or, non commutative) Fréchet $Q$-algebra $\Gamma$ with an additional condition on $\Gamma$ is continuous. To further illustrate our primary results, an example is provided.
Keywords: LMC-algebra; almost derivation; automatic continuity.
MSC: 46H40, 46H05
| GREEN FUNCTIONS FOR VARIOUS BLACK HOLE METRICS | 147--158 |
Abstract
A few models in general relativity concerning to black holes are considered. We studied the Green function for locally close points in the Schwarzschild (symmetric non-rotating and uncharged black hole), Reissner-Nordström (charged non-rotating black hole), Schwarzschild-de Sitter (black hole with a positive cosmological constant), Reissner-Nordström-de Sitter (when a constant electric charge is added to the cosmological term), Hayward (spherically symmetric non-rotating uncharged black hole having no singularity), Bardeen (for spherically symmetric black hole being a source of electric field which does not have a singularity but have the event horizon) metrics. The consideration is based on the Hadamard-WKB method. The Padé approximation is used for the Green function construction.
Keywords: Green's function; black hole; Hadamard-WKB method.
MSC: 83C57, 35J08
| ON SOME POLYNOMIAL OVERRINGS OF INTEGRAL DOMAINS | 159--172 |
Abstract
Let $D$ be an integral domain with quotient field $K$ and $X$ an indeterminate over $K$. A \emph{polynomial overring of} $D$ is a subring of $K[X]$ containing $D[X]$. The aim of this paper is to study some properties of the polynomial overrings of $D$, such as (faithful) flatness, locally freeness and Krull dimension.
Keywords: Integer-valued polynomials; ($w$-faithfully) flat module; Krull dimension.
MSC: 13F05, 13F20, 13B30, 16D40
| TOPOLOGICAL STUDY OF $g$-CONVERGENCE IN GENERALIZED 2-NORMED SPACES | 173--188 |
Abstract
Some topological properties of generalized $2$-normed (G2N) spaces have been studied in this article. The notion of $g$-convergence for sequences is introduced in general, and it is compared with the usual notion of convergence. It is shown that $g$-convergence is a more general idea, and under certain conditions $g$-convergence and convergence actually coincide. Using these concepts, a few fixed point theorems are developed.
Keywords: Generalized $2$-normed spaces; $g-3ps$ spaces; $G$-metric space; $S$-metric spaces; $g$-convergence.
MSC: 46B20, 47A30, 46S99, 55M20, 54H25, 47H10