Discrete optimization problems with random cost elements

In a general class of discrete optimization problems, some of the elements mayhave random costs associated with them. In such a situation, the notion of optimalityneeds to be suitably modified. In this work we define an optimal solutionto be a feasible solution with the minimum risk. We focus on the minsumobjective function, for which we prove that knowledge of the mean values ofthese random costs is enough to reduce the problem into one with fixed costs.We discuss the implications of using sample means when the true means ofthe costs of the random elements are not known, and explore the relation betweenour results and those from post-optimality analysis. We also show thatdiscrete optimization problems with min-max objective functions depend moreintricately on the distributions of the random costs.

On solving discrete optimization problems with one random element under general regret functions

In this paper we consider the class of stochastic discrete optimization problems in which the feasibility of a solution does not depend on the particular values the random elements in the problem take. Given a regret function, we introduce the concept of the risk associated with a solution, and define an optimal solution as one having the least possible risk. We show that for discrete optimization problems with one random element and with min-sum objective functions a least risk solution for the stochastic problem can be obtained by solving a non-stochastic counterpart where the latter is constructed by replacing the random element of the former with a suitable parameter. We show that the above surrogate is the mean if the stochastic problem has only one symmetrically distributed random element. We obtain bounds for this parameter for certain classes of asymmetric distributions and study the limiting behavior of this parameter in details under two asymptotic frameworks. \u

On solving discrete optimization problems with multiple random elements under general regret functions

In this paper we attempt to find least risk solutions for stochastic discrete optimization problems (SDOP) with multiple random elements, where the feasibility of a solution does not depend on the particular values the random elements in the problem take. While the optimal solution, for a linear regret function, can be obtained by solving an auxiliary (non-stochastic) discrete optimization problem (DOP), the situation is complex under general regret. We characterize a finite number of solutions which will include the optimal solution. We establish through various examples that the results from Ghosh, Mandal and Das (2005) can be extended only partially for SDOPs with additional characteristics. We present a result where in selected cases, a complex SDOP may be decomposed into simpler ones facilitating the job of finding an optimal solution to the complex problem. We also propose numerical local search algorithms for obtaining an optimal solution. \u

Discrete optimization problems with random cost elements

In a general class of discrete optimization problems, some of the elements mayhave random costs associated with them. In such a situation, the notion of optimalityneeds to be suitably modified. In this work we define an optimal solutionto be a feasible solution with the minimum risk. We focus on the minsumobjective function, for which we prove that knowledge of the mean values ofthese random costs is enough to reduce the problem into one with fixed costs.We discuss the implications of using sample means when the true means ofthe costs of the random elements are not known, and explore the relation betweenour results and those from post-optimality analysis. We also show thatdiscrete optimization problems with min-max objective functions depend moreintricately on the distributions of the random costs

Anisotropic surface transport in topological insulators in proximity to a helical spin density wave

We study the effects of spatially localized breakdown of time reversal symmetry on the surface of a topological insulator (TI) due to proximity to a helical spin density wave (HSDW). The HSDW acts like an externally applied one-dimensional periodic(magnetic) potential for the spins on the surface of the TI, rendering the Dirac cone on the TI surface highly anisotropic. The decrease of group velocity along the direction $\hat{x}$ of the applied spin potential is twice as much as that perpendicular to $\hat{x}$. At the Brillouin zone boundaries (BZB) it also gives rise to new semi-Dirac points which have linear dispersion along $\hat{x}$ but quadratic dispersion perpendicular to $\hat{x}$. The group velocity of electrons at these new semi-Dirac points is also shown to be highly anisotropic. Experiments using TI systems on multiferroic substrates should realize our predictions. We further discuss the effects of other forms of spin density wave on the surface transport property of topological insulator.Comment: 8 pages, 8 figure

Radio and infrared study of the star forming region IRAS 20286+4105

A multi-wavelength investigation of the star forming complex IRAS 20286+4105, located in the Cygnus-X region, is presented here. Near-infrared K-band data is used to revisit the cluster / stellar group identified in previous studies. The radio continuum observations, at 610 and 1280 MHz show the presence of a HII region possibly powered by a star of spectral type B0 - B0.5. The cometary morphology of the ionized region is explained by invoking the bow-shock model where the likely association with a nearby supernova remnant is also explored. A compact radio knot with non-thermal spectral index is detected towards the centre of the cloud. Mid-infrared data from the Spitzer Legacy Survey of the Cygnus-X region show the presence of six Class I YSOs inside the cloud. Thermal dust emission in this complex is modelled using Herschel far-infrared data to generate dust temperature and column density maps. Herschel images also show the presence of two clumps in this region, the masses of which are estimated to be {\sim} 175 M{\sun} and 30 M{\sun}. The mass-radius relation and the surface density of the clumps do not qualify them as massive star forming sites. An overall picture of a runaway star ionizing the cloud and a triggered population of intermediate-mass, Class I sources located toward the cloud centre emerges from this multiwavelength study. Variation in the dust emissivity spectral index is shown to exist in this region and is seen to have an inverse relation with the dust temperature.Comment: 20 pages, 16 figures, accepted for publication in MNRA

Influence of Austempering Temperature on Microstructure and Mechanical Properties of Cast Fe-Si-Mn-VSteel

The present investigation was carried out to examine the influence of austempering temperature on the micro-structure and mechanical properties of a low carbon, high silicon (C-0.13%, Si-1.2%, Mn-1%, V-0.08%) cast steels. The Induction melted casting block was homogenized at 1000°C for 6 hrs. The samples for microstructure and the tens-ile specimens were prepared from the cast steel block according to ASTM standards and were imparted three diff-erent austempering heat treatments to produce different microstructure. The samples are austenitisation at 900°C for 30 minute and then rapidly quenched to a salt bath maintained at temperature 350/400/450°C for 10 minute and then finally air cooled. The microstructure were observed under optical and TEM microscopy. show that UTS decreases but % El increases with increasing apstempering tempe-rature. The 400°C austempering temperature exhibited the best combination of UTS and %EI at room temperature (UTS-663 MPa, EL-26%) with revealing microstructure of bainite and retained austenite