Minimal winning coalitions and orders of criticality

In this paper, we analyze the order of criticality in simple games, under the light of minimal winning coalitions. The order of criticality of a player in a simple game is based on the minimal number of other players that have to leave so that the player in question becomes pivotal. We show that this definition can be formulated referring to the cardinality of the minimal blocking coalitions or minimal hitting sets for the family of minimal winning coalitions; moreover, the blocking coalitions are related to the winning coalitions of the dual game. Finally, we propose to rank all the players lexicographically accounting the number of coalitions for which they are critical of each order, and we characterize this ranking using four independent axioms.

Implementation of Adaptive Algorithm to Optimize the Placement of Wavelength Converters

Wavelength Converters in WDM Optical Network, is one the essential component in today’s advanced research, but at the same time they are very expensive and it should be used in limited number to make effective use of Wavelength Converters. The two main criteria to be achieved is to reduce the overall blocking probability at an affordable cost and the other performance measures are within reasonable limits. To equip the node with the Wavelength converter for any network is crucial. There should be a minimal blocking probability at those particular nodes chosen to place the Wavelength Converter. A large margin decrease in the blocking probability will be easy for the placement of Wavelength Converter. Compared with the previous method of allocation, the results of the proposed adaptive algorithm can significantly reduce the blocking probability at higher range. The implementation of Adaptive algorithm shows that it is more cost effective to optimize the blocking performance. The blocking performance improvement hence led to the comparison of performance and efficiency other conventional methods with the proposed Adaptive algorithm is done.

Optical Performance Modeling and Analysis of a Tensile Ganged Heliostat Concept

Designs of conventional heliostats have been varied to reduce cost, improve optical performance or both. In one case, reflective mirror area on heliostats has been increased with the goal of reducing the number of pedestals and drives and consequently reducing the cost on those components. The larger reflective areas, however, increase torques due to larger mirror weights and wind loads. Higher cost heavy-duty motors and drives must be used, which negatively impact any economic gains. To improve on optical performance, the opposite may be true where the mirror reflective areas are reduced for better control of the heliostat pointing and tracking. For smaller heliostats, gravity and wind loads are reduced, but many more heliostats must be added to provide sufficient solar flux to the receiver. For conventional heliostats, there seems to be no clear cost advantage of one heliostat design over other designs. The advantage of ganged heliostats is the pedestal and tracking motors are shared between multiple heliostats, thus can significantly reduce the cost on those components. In this paper, a new concept of cable-suspended tensile ganged heliostats is introduced, preliminary analysis is performed for optical performance and incorporated into a 10 MW conceptual power tower plant where it was compared to the performance of a baseline plant with a conventional radially staggered heliostat field. The baseline plant uses conventional heliostats and the layout optimized in System Advisor Model (SAM) tool. The ganged heliostats are suspended on two guide cables. The cables are attached to rotations arms which are anchored to end posts. The layout was optimized offline and then transferred to SAM for performance evaluation. In the initial modeling of the tensile ganged heliostats for a 10 MW power tower plant, equal heliostat spacing along the guide cables was assumed, which as suspected leads to high shading and blocking losses. The goal was then to optimize the heliostat spacing such that annual shading and blocking losses are minimized. After adjusting the spacing on tensile ganged heliostats for minimal blocking losses, the annual block/shading efficiency was greater than 90% and annual optical efficiency of the field became comparable to the conventional field at slightly above 60%.

Minimal Multiple Blocking Sets

We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $\frac{1}{2}(\sqrt{4tn - (3t + 1)(t - 1)} + t - 1) + 1$ points. We use this to show that for $n$ a prime power, equality can occur in our bound in exactly one of the following three cases: (a) $t = 1$, $n$ is a square and $S$ is a unital; (b) $t = n - \sqrt{n}$, $n$ is a square and $S$ is the complement of a Baer subplane; (c) $t = n$ and $S$ is equal to the set of all points except one. For a square prime power $q$ and $t \leq \sqrt{q} + 1$, we give a construction of a minimal $t$-fold blocking set $S$ in $\mathrm{PG}(2,q)$ with $|S| = q\sqrt{q} + 1 + (t - 1)(q - \sqrt{q} + 1)$. Furthermore, we obtain an upper bound on the size of minimal blocking sets in symmetric $2$-designs and use it to give new proofs of other known results regarding tangency sets in higher dimensional finite projective spaces. We also discuss further generalizations of our bound. In our proofs we use an incidence bound on combinatorial designs which follows from applying the expander mixing lemma to the incidence graph of these designs.

Blocking and Double Blocking Sets in Finite Planes

In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary, we obtain the existence of a minimal blocking set of a non-Desarguesian affine plane of order $q^2$ of size at most $4q^2/3+5q/3$, which is considerably smaller than $2q^2-1$, the Jamison bound for the size of a minimal blocking set in an affine Desarguesian plane of order $q^2$.We also consider particular André planes of order $q$, where $q$ is a power of the prime $p$, and give a construction of a small minimal blocking set which admits a secant line not meeting the blocking set in $1$ mod $p$ points. Furthermore, we elaborate on the connection of this problem with the study of value sets of certain polynomials and with the construction of small double blocking sets in Desarguesian projective planes; in both topics we provide some new results.