Matrix approach to the Shapley value and dual similar associated consistency

Replacing associated consistency in Hamiache's axiom system by dual similar associated consistency, we axiomatize the Shapley value as the unique value verifying the inessential game property, continuity and dual similar associated consistency. Continuing the matrix analysis for Hamiache's axiomatization of the Shapley value, we construct the dual similar associated game and introduce the dual similar associated transformation matrix $M_\lambda^{DSh}$ as well. In the game theoretic framework we show that the dual game of the dual similar associated game is Hamiache's associated game of the dual game. For the purpose of matrix analysis, we derive the similarity relationship $M_\lambda^{DSh}=QM_\lambda Q^{-1}$ between the dual similar associated transformation matrix $M_\lambda^{DSh}$ and associated transformation matrix $M_\lambda$ for Hamiache's associated game, where the transformation matrix $Q$ represents the duality operator on games. This similarity of matrices transfers associated consistency into dual similar associated consistency, and also implies the inessential property for the limit game of the convergent sequence of repeated dual similar associated games. We conclude this paper with three tables summarizing all matrix results

Matrix analysis for associated consistency in cooperative game theory

Hamiache's recent axiomatization of the well-known Shapley value for TU games states that the Shapley value is the unique solution verifying the following three axioms: the inessential game property, continuity and associated consistency. Driessen extended Hamiache's axiomatization to the enlarged class of efficient, symmetric, and linear values, of which the Shapley value is the most important representative. In this paper, we introduce the notion of row (resp. column)-coalitional matrix in the framework of cooperative game theory. Particularly, both the Shapley value and the associated game are represented algebraically by their coalitional matrices called the Shapley standard matrix $M^{Sh}$ and the associated transformation matrix $M_\lambda,$ respectively. We develop a matrix approach for Hamiache's axiomatization of the Shapley value. The associated consistency for the Shapley value is formulated as the matrix equality $M^{Sh}=M^{Sh}·M_\lambda.$ The diagonalization procedure of $M_\lambda$ and the inessential property for coalitional matrices are fundamental tools to prove the convergence of the sequence of repeated associated games as well as its limit game to be inessential. In addition, a similar matrix approach is applicable to study Driessen's axiomatization of a certain class of linear values. Matrix analysis is adopted throughout both the mathematical developments and the proofs. In summary, it is illustrated that matrix analysis is a new and powerful technique for research in the field of cooperative game theory

Matrix approach to consistency of the additive efficient normalization of semivalues

In fact the Shapley value is the unique efficient semivalue. This motivated Ruiz et al. to do additive efficient normalization for semivalues. In this paper, by matrix approach we derive the relationship between the additive efficient normalization of semivalues and the Shapley value. Based on the relationship, we axiomatize the additive efficient normalization of semivalues as the unique solution verifying covariance, symmetry, and reduced game property with respect to the $p-$reduced game

Metal mask free dry-etching process for integrated optical devices applying highly photostabilized resist.

Photostabilization is a widely used post lithographic resist treatment process, which allows to harden the resist profile in order to maintain critical dimensions and to increase selectivity in subsequent process steps such as reactive ion etching. In this paper we present the optimization of deep UV-curing of 0,3-3.3 μm thick positive resist profiles followed by heat treatment up to 280 °C. The effectiveness of this resist treatment allows for metal mask free reactive ion etching with selectivity up to 6 for silicon structures, thermal silicon oxide and silicon oxynitride. This procedure is demonstrated by the results obtained in etching of various integrated optical structures

A solution defined by fine vectors

Bumb and Hoede have shown that a cooperative game can be split into two games, the reward game and the fine game, by considering the sign of quantities $c_v^S$ in the c-diagram of the game. One can then define a solution $x$ for the original game as $x = x_r - x_f$ , where $x_r$ is a solution for the reward game and $x_f$ is a solution for the fine game. Due to the distinction of cooperation rewards and fines, for allocating the fines one may use another solution concept than for the rewards. In this paper, a fine vector is introduced and a solution is defined by fine vectors. The structure and properties of this solution are studied. And the solution is characterized as the unique solution having efficiency and f-potential property (resp. f-balanced contributions property)

Deposition and characterization of PECVD phosphorus doped silicon oxynitride layers for integrated optics applications

Phosphorus-doped silicon oxynitride layers have been deposited by a Plasma Enhanced Chemical Vapor Deposition process from $N_20$, 2% $SiH_4/N_2$ and 5% $PH_3/Ar$ gaseous mixtures. The $PH_3/Ar$ flow rate was varied to investigate the effect of the dopant to the layer properties. As deposited and annealed (600, 800, 900 and 1000 °C) layers were characterized by Fourier transform infrared spectroscopy, Rutherford backscattering spectroscopy and spectroscopic ellipsometry. In this way the refractive index could be determined as well as the amount of hydrogen that is responsible for enhanced absorption in the 3rd telecommunication window around 1550 nm. The N-H bonds concentration was found to decrease with the phosphorus concentration. Furthermore the bonded hydrogen in the entire P-doped layers have been eliminated after annealing at 1000 °C, while undoped SiON layers require annealing at 1150 °C

Reduction of hydrogen-induced optical losses of plasma-enhanced chemical vapor deposition silicon oxynitride by phosphorus doping and heat treatment

Plasma enhanced chemical vapor deposition phosphoros-doped silicon oxynitride (SiON) layers with a refractive index of 1.505 were deposited from $N_{2}O$, 2% $SiH_{4}/N_{2}$, and 5% $PH_{3}/Ar$ gaseous mixtures. The $PH_{3}/Ar$ flow rate was varied to investigate the effect of the dopant to the layer properties. We studied the compositions and the chemical environment of phosphorus, silicon, oxygen, nitrogen and hydrogen in these layers by using x-ray photoelectron spectroscopy and Fourier transform infrared spectroscopy. The number of N-H and O-H bonds, which are responsible for optical losses around 1.55 and 1.3 μm, decreases in the as-deposited layers with increasing phosphorus concentration. Furthermore, the bonded hyrogen in all P-doped layers has been eliminated after annealing at a temperature significantly lower than required for undoped silicon oxynitride layers, that is so to say 1000°C instead of 1150°C. The resulting optical loss in the entire third telecommunication window was well below 0.2dB/cm, making P-doped SiON an attractive material for demanding integrated optics applications

Influence of phosphorus doping on hydrogen content and optical losses in PECVD silicon oxynitride

PECVD Phosphorus-doped silicon oxynitride layers (n=1.5) were deposited from N2O, 2%SiH4/N2, NH3 and 5%PH3/Ar gaseous mixtures. Chemical bonds were determined by Fourier transform infrared spectroscopy. N–H bond concentration of the layers decreased from 3.29×10-21 to 0.45×10-21 cm−3, as the 5%PH3/Ar flow rate increased from 0 to 60 sccm. A simultaneous decrease of O–H related bonds was also observed within the same phosphine flow range. The optical loss of slab-type waveguides at λ=1505 nm was found to decrease from 14.1 to 6.2 dB/cm as the 5%PH3/Ar flow rate increased from 0 to 30 sccm, respectively. Moreover, the optical loss values around λ=1400 and 1550 nm were found to decrease from 4.7 to below 0.2 dB/cm and from 1.8 to 1.0 dB/cm respectively. These preliminary results are very promising for applications in low-loss integrated optical devices

Fabrication of Polymeric Multimode Waveguides for Application in the Local Area Network and Optical Interconnects.

We report the fabrication of multimode polymeric waveguides using spin coating, photolithography, and reactive ion etching. Different layer structures have been used, e.g., a UV curable resin is used as a core layer and PMMA as a lower and upper cladding. The layer systems can be spun with good uniformity with thicknesses matching multimode fiber dimensions, and smooth waveguide side-walls have been obtained after etching. The measured performance of these waveguides demonstrates that low-loss waveguides can be fabricated cheaply. The waveguides are used in the realization of novel compact multimode power splitters and star couplers

B/P Doping in application of silicon oxynitride based integrated optics

In this paper, gaseous precursors containing boron or phosphorous were intentionally introduced in the deposition of SiON layers and upper SiO2 claddings. The measurements show that the as-deposited B/P-doped SiON layers contain less hydrogen than undoped layers. Furthermore, the necessary annealing temperature for elimination of hydrogen related absorption (propagation loss) is greatly reduced in B/P-doped layers