Inder Kumar Rana

• Indian Institute of Technology Bombay

In this paper, we extend the Hake's theorem over metric measure spaces. We provide its measure theoretic versions in terms of the Hen-stock variational measure VF.

We analyze the relationship between the absolute continuity of charges and the Henstock-Kurzweil integral on metric measure spaces. We also discuss a measure theoretic characterization of the Henstock-Kurzweil integral in terms of the Henstock variational measure, on such spaces.

We give a characterization of the Henstock-Kurzweil integral on \(\R^m\) in terms of variational measures. As an application of this we prove a generalization of the Hake’s theorem to \(\R^m\).

The aim of this paper is to give some equivalent ways of defining McShane integral for vector valued functions.

In this paper, we show that if an R-integrable (or BV integrable function) defined over a compact interval $[a,b]\times [c,d]\subset \mathbb R^2$ satisfies certain conditions, then over any subinterval of $[a,b]\times [c,d],$ the iterated integral exists and equals the double integral. We also present some examples relevant to our theory.

The aim of this note is to show that for a given continuous function $F$ on a set $E \subset \R,$ the associated interval function need not be continuous. We also give an example to show that the associated interval function can be continuous even if $F$ is not continuous.

Let G be a locally compact Hausdorif abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f: G → X, with the topology of uniform convergence on compact sets. Let X′ denote the dual of X with the weak* topology. Let Mc(G, X′) denote the space of all X′-valued compactly supported regular measures of...

Let $G$ be a topological group of second category, ${\cal B}_G$ be its Borel $\sigma$-algebra and ${\cal B}$ a $\sigma$-algebra of subsets of $G$ such that $(G,{\cal B})$ is a measurable group. For a probability measure $P$ on $(G,{\cal B}),$ let $P_g(E):=P(gE)$ for $g\in G, E\in{\cal B}.$ The aim of this note is to show that there exists an inner-...

The smoothing \(T_af\), for \(a \gt 0\), of a locally-integrable function \(f : \mathbb{R} \rightarrow \mathbb{R}\) is defined by \[(T_af)(x) := \frac{1}{2a} \int^{+a}_{-a} f(x+y) dy, \quad x \in \mathbb{R}.\] For a given \(g : \mathbb{R} \rightarrow \mathbb{R}\), any solution \(f\) of the equation \(T_af = g\) is called an unsmoothing of \(g\). In...

Let G be a locally compact abelian group with a Haar measure lambda(G). A function f on G is said to be mean-periodic if there exists a nonzero finite regular measure mu of compact support on G such that f*mu = 0. It is known that there exist no nontrivial integrable mean periodic functions on R(n) . We show that there exist nontrivial integrable m...

Let G be a locally compact abelian group with a Haar measure λG. A function f on G is said to be mean-periodic if there exists a nonzero finite regular measure μ of compact support on G such that f * μ = 0. It is known that there exist no nontrivial integrable mean periodic functions on Rn. We show that there exist nontrivial integrable mean period...

Let $B(a) = \left\{ {x \in IR^3 /\left| x \right| \leqslant a} \right\},a > 0.$B(a) = \left\{ {x \in IR^3 /\left| x \right| \leqslant a} \right\},a > 0. The main aim of the paper is to solve the integral equation: g(x) = òB(a) f(x + y)dy, x Î IR3 ,g(x) = \int_{B(a)} {f(x + y)dy, x \in IR^3 ,} for a given functiong. Following the ideas of F. Jo...

The smoothing Taf of a locally-integrable function f: ℝ→ℝ is defined by(Taf)(x)=12a∫−aaf(x+y)dy,where a>0. We show that Ta maps the space of locally-integrable functions on ℝ, Lloc(ℝ), onto the space ACloc(ℝ) of locally-absolutely continuous functions on ℝ. We construct a map Ra:ACloc(ℝ)→Lloc(ℝ) such that TaoRa = ld and such that Ra preserves the d...

Let $G$ be a topological group of second category and having cardinality at most that of the continuum. Let $\mathbf{B}$ be some $\sigma$-algebra of subsets of $G$ such that $(G, \mathbf{B})$ is a measurable group. For a probability measure $P$ on $(G, \mathbf{B})$, write $P_g$ for the measure defined by $P_g(E) = P(gE), E \in \mathbf{B}$. The aim...

Let G be a topological group of second category and having cardinality at most that of the continuum. Let B be some a-algebra of subsets of G such that (G, B) is a measurable group. For a probability measure P on (G, B), write P for the measure defined by Pg(E) = P(gE), E E B. The aim of this paper is to prove the following: if on (G, B) there exis...

Weanling rats were fed diets containing starch, sucrose, maltose, fructose or glucose for 50 days. When compared with the effects of the high starch diet, diets containing the sugars promoted a substantial and similar rise in plasma lipids. Hepatic lipogenesis was increased by all the sugars, with sucrose and maltose having an equal effect. Lipogen...

Diets rich in starch, sucrose or fat were fed to rats for various periods in order to investigate the relationship between plasma lipids and tissue lipogenesis. The high-fat diets supressed lipogenesis in the liver and adipose tissue, and reduced the plasma triglycerides (relative to values obtained in the starch-fed group). The high-sucrose diet g...

Education system in India is "10 + 2 + 3" model, and it is very examination-oriented. Education is a state matter and there are in all 28 states and 7 union territories. There is little scope for innovation and use of technology in regular teaching. In the second part of the paper, I will discuss the role of technology in teaching and a methodology...