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Weighted Banzhaf power and interaction indexes through weighted approximations of games Marichal, Jean-Luc ; Mathonet, Pierre in European Journal of Operational Research (2011), 211(2), 352-358 The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside ... [more ▼] The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes. [less ▲] Detailed reference viewed: 119 (4 UL)Weighted Banzhaf power and interaction indexes through weighted approximations of games Marichal, Jean-Luc ; Mathonet, Pierre in Dubois, Didier; Grabisch, Michel; Mesiar, Radko (Eds.) et al 32nd Linz Seminar on Fuzzy Set Theory (LINZ 2011) - Decision Theory: Qualitative and Quantitative Approaches (2011) In cooperative game theory, various kinds of power indexes are used to measure the influence that a given player has on the outcome of the game or to define a way of sharing the benefits of the game among ... [more ▼] In cooperative game theory, various kinds of power indexes are used to measure the influence that a given player has on the outcome of the game or to define a way of sharing the benefits of the game among the players. The best known power indexes are due to Shapley [15,16] and Banzhaf [1,5] and there are many other examples of such indexes in the literature. When one is concerned by the analysis of the behavior of players in a game, the information provided by power indexes might be far insufficient, for instance due to the lack of information on how the players interact within the game. The notion of interaction index was then introduced to measure an interaction degree among players in coalitions; see [13,12,7,8,14,10,6] for the definitions and axiomatic characterizations of the Shapley and Banzhaf interaction indexes as well as many others. In addition to the axiomatic characterizations the Shapley power index and the Banzhaf power and interaction indexes were shown to be solutions of simple least squares approximation problems (see [2] for the Shapley index, [11] for the Banzhaf power index and [9] for the Banzhaf interaction index). We generalize the non-weighted approach of [11,9] by adding a weighted, probabilistic viewpoint: A weight w(S) is assigned to every coalition S of players that represents the probability that coalition S forms. The solution of the weighted least squares problem associated with the probability distribution w was given in [3,4] in the special case when the players behave independently of each other to form coalitions. In this particular setting we introduce a weighted Banzhaf interaction index associated with w by considering, as in [11,9], the leading coefficients of the approximations of the game by polynomials of specified degrees.We then study the most important properties of these weighted indexes and their relations with the classical Banzhaf and Shapley indexes. [less ▲] Detailed reference viewed: 135 (1 UL)Weighted Fair Multicast Multigroup Beamforming under Per-antenna Power Constraints Christopoulos, Dimitrios ; Chatzinotas, Symeon ; Ottersten, Björn in IEEE Transactions on Signal Processing (2014), 62(19), 5132-5142 A multi-antenna transmitter that conveys independent sets of common data to distinct groups of users is considered. This model is known as physical layer multicasting to multiple co-channel groups. In ... [more ▼] A multi-antenna transmitter that conveys independent sets of common data to distinct groups of users is considered. This model is known as physical layer multicasting to multiple co-channel groups. In this context, the practical constraint of a maximum permitted power level radiated by each antenna is addressed. The per-antenna power constrained system is optimized in a maximum fairness sense with respect to predetermined quality of service weights. In other words, the worst scaled user is boosted by maximizing its weighted signal-to-interference plus noise ratio. A detailed solution to tackle the weighted max-min fair multigroup multicast problem under per-antenna power constraints is therefore derived. The implications of the novel constraints are investigated via prominent applications and paradigms. What is more, robust per-antenna constrained multigroup multicast beamforming solutions are proposed. Finally, an extensive performance evaluation quantifies the gains of the proposed algorithm over existing solutions and exhibits its accuracy over per-antenna power constrained systems. [less ▲] Detailed reference viewed: 350 (40 UL)Weighted Fair Multicast Multigroup Beamforming Under Per-antenna Power Constraints Christopoulos, Dimitrios ; Chatzinotas, Symeon ; Ottersten, Björn in Multicast multigroup beamforming under per-antenna power constraints (2014, June) Linear precoding exploits the spatial degrees of freedom offered by multi-antenna transmitters to serve multiple users over the same frequency resources. The present work focuses on simultaneously serving ... [more ▼] Linear precoding exploits the spatial degrees of freedom offered by multi-antenna transmitters to serve multiple users over the same frequency resources. The present work focuses on simultaneously serving multiple groups of users, each with its own channel, by transmitting a stream of common symbols to each group. This scenario is known as physical layer multicasting to multiple co-channel groups. Extending the current state of the art in multigroup multicasting, the practical constraint of a maximum permitted power level radiated by each antenna is tackled herein. The considered per antenna power constrained system is optimized in a maximum fairness sense. In other words, the optimization aims at favoring the worst user by maximizing the minimum rate. This Max-Min Fair criterion is imperative in multicast systems, where the performance of all the receivers listening to the same multicast is dictated by the worst rate in the group. An analytic framework to tackle the Max-Min Fair multigroup multicasting scenario under per antenna power constraints is therefore derived. Numerical results display the accuracy of the proposed solution and provide insights to the performance of a per antenna power constrained system. [less ▲] Detailed reference viewed: 220 (7 UL)Weighted Group Algebras on Groups of Polynomial Growth ; ; et al in Mathematische Zeitschrift (2003), 245(4), 791-821 Detailed reference viewed: 103 (1 UL)Weighted lattice polynomials Marichal, Jean-Luc in Discrete Mathematics (2009), 309(4), 814-820 We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an ... [more ▼] We define the concept of weighted lattice polynomial functions as lattice polynomial functions constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a median based decomposition formula. [less ▲] Detailed reference viewed: 127 (7 UL)Weighted lattice polynomials Marichal, Jean-Luc in Proc. 11th Int. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU 2006), Paris, France, July 2-7, 2006 (2006, July) We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded ... [more ▼] We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula. [less ▲] Detailed reference viewed: 113 (4 UL)The weighted lattice polynomials as aggregation functions Marichal, Jean-Luc in Proc. 63rd Meeting of the Eur. Working Group "Multiple Criteria Decision Aiding" (MCDA 63) (2006, March) We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded ... [more ▼] We define the concept of weighted lattice polynomials as lattice polynomials constructed from both variables and parameters. We provide equivalent forms of these functions in an arbitrary bounded distributive lattice. We also show that these functions include the class of discrete Sugeno integrals and that they are characterized by a remarkable median based decomposition formula. [less ▲] Detailed reference viewed: 44 (1 UL)Weighted lattice polynomials of independent random variables Marichal, Jean-Luc in Discrete Applied Mathematics (2008), 156(5), 685-694 We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice ... [more ▼] We give the cumulative distribution functions, the expected values, and the moments of weighted lattice polynomials when regarded as real functions of independent random variables. Since weighted lattice polynomial functions include ordinary lattice polynomial functions and, particularly, order statistics, our results encompass the corresponding formulas for these particular functions. We also provide an application to the reliability analysis of coherent systems. [less ▲] Detailed reference viewed: 93 (4 UL)Weighted numbers: Commentary on “The Number Sense Represents (Rational) Numbers” by Sam Clarke and Jacob Beck Marinova, Mila ; ; in Behavioral and Brain Sciences (2021) Detailed reference viewed: 75 (4 UL)Weighted power variations of iterated Brownian motion Nourdin, Ivan ; Peccati, Giovanni in Electronic Journal of Probability (2008), 13 Detailed reference viewed: 164 (1 UL)Weighted Subspace Fitting for General Array Error Models ; ; Ottersten, Björn in IEEE Transactions on Signal Processing (1998), 46(9), 24842498 Detailed reference viewed: 47 (0 UL)Weighted sum rate maximization for MIMO broadcast channels using dirty paper coding and zero-forcing methods ; ; et al in IEEE Transactions on Communications (2013), 6(61), 2362-2373 We consider precoder design for maximizing the weighted sum rate (WSR) of successive zero-forcing dirty paper coding (SZF-DPC). For this problem, the existing precoder designs often assume a sum power ... [more ▼] We consider precoder design for maximizing the weighted sum rate (WSR) of successive zero-forcing dirty paper coding (SZF-DPC). For this problem, the existing precoder designs often assume a sum power constraint (SPC) and rely on the singular value decomposition (SVD). The SVD-based designs are known to be optimal but require high complexity. We first propose a low-complexity optimal precoder design for SZF-DPC under SPC, using the QR decomposition. Then, we propose an efficient numerical algorithm to find the optimal precoders subject to per-antenna power constraints (PAPCs). To this end, the precoder design for PAPCs is formulated as an optimization problem with a rank constraint on the covariance matrices. A well-known approach to solve this problem is to relax the rank constraints and solve the relaxed problem. Interestingly, for SZF-DPC, we are able to prove that the rank relaxation is tight. Consequently, the optimal precoder design for PAPCs is computed by solving the relaxed problem, for which we propose a customized interior-point method that exhibits a superlinear convergence rate. Two suboptimal precoder designs are also presented and compared to the optimal ones. We also show that the proposed numerical method is applicable for finding the optimal precoders for block diagonalization scheme. [less ▲] Detailed reference viewed: 210 (2 UL)Weighted Sum-SINR and Fairness Optimization for SWIPT-Multigroup Multicasting Systems with Heterogeneous Users Gautam, Sumit ; Lagunas, Eva ; Sharma, Shree Krishna et al in IEEE Open Journal of the Communications Society (2020) The development of next generation wireless communication systems focuses on the expansion of existing technologies, while ensuring an accord between various devices within a system. In this paper, we ... [more ▼] The development of next generation wireless communication systems focuses on the expansion of existing technologies, while ensuring an accord between various devices within a system. In this paper, we target the aspect of precoder design for simultaneous wireless information and power transmission (SWIPT) in a multi-group (MG) multicasting (MC) framework capable of handling heterogeneous types of users, viz., information decoding (ID) specific, energy harvesting (EH) explicit, and/or both ID and EH operations concurrently. Precoding is a technique well-known for handling the inter-user interference in multi-user systems, however, the joint design with SWIPT is not yet fully exploited. Herein, we investigate the potential benefits of having a dedicated precoder for the set of users with EH demands, in addition to the MC precoding. We study the system performance of the aforementioned system from the perspectives of weighted sum of signal-to-interference-plus-noise-ratio (SINR) and fairness. In this regard, we formulate the precoder design problems for (i) maximizing the weighted sum of SINRs at the intended users and (ii) maximizing the minimum of SINRs at the intended users; both subject to the constraints on minimum (non-linear) harvested energy, an upper limit on the total transmit power and a minimum SINR required to close the link. We solve the above-mentioned problems using distinct iterative algorithms with the help of semi-definite relaxation (SDR) and slack-variable replacement (SVR) techniques, following suitable transformations pertaining the problem convexification. The main novelty of the proposed approach lies in the ability to jointly design the MC and EH precoders for serving the heterogeneously classified ID and EH users present in distinct groups, respectively. We illustrate the comparison between the proposed weighted sum-SINR and fairness models via simulation results, carried out under various parameter values and operating conditions. [less ▲] Detailed reference viewed: 121 (18 UL)Weightwise almost perfectly balanced functions: secondary constructions for all n and better weightwise nonlinearities Gini, Agnese ; Meaux, Pierrick E-print/Working paper (2022) Detailed reference viewed: 30 (1 UL)Weightwise perfectly balanced functions and nonlinearity Gini, Agnese ; Meaux, Pierrick in Codes, Cryptology and Information Security (2022) In this article we realize a general study on the nonlinearity of weightwise perfectly balanced (WPB) <br />functions. First, we derive upper and lower bounds on the nonlinearity from this class of ... [more ▼] In this article we realize a general study on the nonlinearity of weightwise perfectly balanced (WPB) <br />functions. First, we derive upper and lower bounds on the nonlinearity from this class of functions for all n. Then, <br />we give a general construction that allows us to provably provide WPB functions with nonlinearity as low as <br />2 <br />n/2−1 <br />and WPB functions with high nonlinearity, at least 2 <br />n−1 − 2 <br />n/2 <br />. We provide concrete examples in 8 and <br />16 variables with high nonlinearity given by this construction. In 8 variables we experimentally obtain functions <br />reaching a nonlinearity of 116 which corresponds to the upper bound of Dobbertin’s conjecture, and it improves <br />upon the maximal nonlinearity of WPB functions recently obtained with genetic algorithms. Finally, we study the <br />distribution of nonlinearity over the set of WPB functions. We examine the exact distribution for n = 4 and provide <br />an algorithm to estimate the distributions for n = 8 and 16, together with the results of our experimental studies for <br />n = 8 and 16. [less ▲] Detailed reference viewed: 66 (8 UL)"Weil die meisten Comics einfach zu schwer sind!" - Zur Kategorie der Einfachheit beim Einsatz von Bandes dessinées im Fremdsprachenunterricht Französisch Morys, Nancy Scientific Conference (2014, September 26) Detailed reference viewed: 127 (0 UL)"... weil die meisten Comics einfach zu schwer sind!" - Zur Kategorie der Einfachheit beim Einsatz von Bandes dessinées im Fremdsprachenunterricht Französisch Morys, Nancy in Burwitz-Melzer, Eva; O´Sullivan, Emer (Eds.) Einfachheit in der Kinder- und Jugendliteratur - Ein Gewinn für den Fremdsprachenunterricht (2016, December 18) Detailed reference viewed: 163 (2 UL)The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds ; Schlenker, Jean-Marc in Handbook of Teichmüller theory. Volume III (2012) Detailed reference viewed: 144 (1 UL)Wéineg Diversitéit an de Schoulbicher ; Kerger, Sylvie ; Pianaro, Enrica et al Speeches/Talks (2023) Detailed reference viewed: 18 (0 UL) |
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