Extended inverse theorems for hh-fold sumsets in integers

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for $hk-h+1<\left|hA\right| \leq hk+3h-4$. Furthermore, we give a range of $|hA|$ for which inverse problems are not possible.Comment: To be appear in contrib. discrete math., 17 pages, correct some typographical errors, Statement and proof of some results change

Generalized H-fold sumset and Subsequence sum

Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by $H^{(r)}A$, is the union of the sumsets $h^{(r)}A$ for $h\in H$ where, the sumset $h^{(r)}A$ is the set of all integers that can be represented as a sum of $h$ elements from $A$ with no summand in the representation appearing more than $r$ times. In this paper, we find the optimal lower bound for the cardinality of $H^{(r)}A$, i.e., for $|H^{(r)}A|$ and the structure of the underlying sets $A$ and $H$ when $|H^{(r)}A|$ is equal to the optimal lower bound in the cases $A$ contains only positive integers and $A$ contains only nonnegative integers. This generalizes recent results of Bhanja. Furthermore, with a particular set $H$, since $H^{(r)}A$ generalizes subsequence sum and hence subset sum, we get several results of subsequence sums and subset sums as special cases.Comment: To be appear in C. R. Math. Acad. Sci. Paris, 25 page

Extended inverse theorems for hh-fold sumsets in integers

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for hk-h+1 < \left|hA\right| \leq hk+2h-3. In this paper, we extend the work of Tang and Xing and study all possible inverse theorems for hk-h+1<\left|hA\right| \leq hk+2h +1. Furthermore, we give a range of $|hA|$ for which inverse problems are not possible

Electric service reliability cost/worth assessment in a developing country

Considerable work has been done in developed countries to optimize the reliability of electric power systems on the basis of reliability cost versus reliability worth. This has yet to be considered in most developing countries, where development plans are still based on traditional deterministic measures. The difficulty with these criteria is that they cannot be used to evaluate the economic impacts of changing reliability levels on the utility and the customers, and therefore cannot lead to an optimum expansion plan for the system. The critical issue today faced by most developing countries is that the demand for electric power is high and growth in supply is constrained by technical, environmental, and most importantly by financial impediments. Many power projects are being canceled or postponed due to a lack of resources. The investment burden associated with the electric power sector has already led some developing countries into serious debt problems. This thesis focuses on power sector issues facing by developing countries and illustrates how a basic reliability cost/worth approach can be used in a developing country to determine appropriate planning criteria and justify future power projects by application to the Nepal Integrated Electric Power System (NPS). A reliability cost/worth based system evaluation framework is proposed in this thesis. Customer surveys conducted throughout Nepal using in-person interviews with approximately 2000 sample customers are presented. The survey results indicate that the interruption cost is dependent on both customer and interruption characteristics, and it varies from one location or region to another. Assessments at both the generation and composite system levels have been performed using the customer cost data and the developed NPS reliability database. The results clearly indicate the implications of service reliability to the electricity consumers of Nepal, and show that the reliability cost/worth evaluation is both possible and practical in a developing country. The average customer interruption costs of Rs 35/kWh at Hierarchical Level I and Rs 26/kWh at Hierarchical Level II evaluated in this research work led to an optimum reserve margin of 7.5%, which is considerably lower than the traditional reserve margin of 15% used in the NPS. A similar conclusion may result in other developing countries facing difficulties in power system expansion planning using the traditional approach. A new framework for system planning is therefore recommended for developing countries which would permit an objective review of the traditional system planning approach, and the evaluation of future power projects using a new approach based on fundamental principles of power system reliability and economics

GRB000301C with peculiar afterglow emission

The CCD magnitudes in Johnson V and Cousins R and I photometric passbands are determined for GRB 000301C afterglow starting ~ 1.5 day after the gamma-ray burst. In fact we provide the earliest optical observations for this burst. Light curves of the afterglow emissions in U, B, V, R, I, J and K' passbands are obtained by combining the present measurements with the published data. Flux decay shows a very uncommon variation relative to other well observed GRBs. Overall, there is a steepening of the optical and near-infrared flux decay caused by a geometric and sideways expanding jet. This is superimposed by a short term variability especially during early time (Delta t < 8 days). The cause of variability is not well understood, though it has occurred simultaneously with similar amplitude in all the filters. We derive the early and late time flux decay constants using jet model. The late time flux decay is the steepest amongst the GRB OTs observed so far with alpha ~ 3. Steepening in the flux decay seems to have started simultaneously around Delta t ~ 7.6 day in all passbands. The value of spectral index in the optical-near IR region is ~ -1.0. Redshift determination with z=2.0335 indicates cosmological origin of the GRB having a luminosity distance of 16.6 Gpc. Thus it becomes the second farthest amongst the GRBs with known distances. An indirect estimate of the fluence > 20 keV indicates, if isotropic,> =10^53 ergs of release of energy. The enormous amount of released energy will be reduced, if the radiation is beamed which is the case for this event. Using a jet break time of 7.6 days, we infer a jet opening angle of ~ 0.15 radian. This means the energy released is reduced by a factor of ~ 90 relative to the isotropic value.Comment: LaTeX file, 11 pages including 4 figures, uses psfig.sty, Bull. Astron. Society of India(accepted, Sept, 2000 issue