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167
The combinatorics of frieze patterns and Markoff numbers
, 2007
"... ... model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky’s cluster algebras of type A. One can derive from the matchings model an enumerativ ..."
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Cited by 35 (1 self)
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... model based on perfect matchings that explains the symmetries of the numerical arrays that Conway and Coxeter dubbed frieze patterns. This matchings model is a combinatorial interpretation of Fomin and Zelevinsky’s cluster algebras of type A. One can derive from the matchings model an enumerative meaning for the Markoff numbers, and prove that the associated Laurent polynomials have positive coefficients as was conjectured (much more generally) by Fomin and Zelevinsky. Most of this research was conducted under the auspices of REACH (Research Experiences in Algebraic Combinatorics at Harvard).
Distributing coalitional value calculations among cooperating agents
 In Proceedings of The Twentieth National Conference on Artificial Intelligence (AAAI05
, 2005
"... The process of forming coalitions of software agents generally requires calculating a value for every possible coalition which indicates how beneficial that coalition would be if it was formed. Now, instead of having a single agent calculate all these values (as is typically the case), it is more ef ..."
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Cited by 31 (20 self)
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The process of forming coalitions of software agents generally requires calculating a value for every possible coalition which indicates how beneficial that coalition would be if it was formed. Now, instead of having a single agent calculate all these values (as is typically the case), it is more efficient to distribute this calculation among the agents, thus using all the computational resources available to the system and avoiding the existence of a single point of failure. Given this, we present a novel algorithm for distributing this calculation among agents in cooperative environments. Specifically, by using our algorithm, each agent is assigned some part of the calculation such that the agents ’ shares are exhaustive and disjoint. Moreover, the algorithm is decentralized, requires no communication between the agents, has minimal memory requirements, and can reflect variations in the computational speeds of the agents. To evaluate the effectiveness of our algorithm, we compare it with the only other algorithm available in the literature for distributing the coalitional value calculations (due to Shehory and Kraus). This shows that for the case of 25 agents, the distribution process of our algorithm took less than 0.02 % of the time, the values were calculated using 0.000006 % of the memory, the calculation redundancy was reduced from 383229848 to 0, and the total number of bytes sent between the agents dropped from 1146989648 to 0 (note that for larger numbers of agents, these improvements become exponentially better). 1
The many faces of alternatingsign matrices
, 2008
"... I give a survey of different combinatorial forms of alternatingsign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as cornersum matrices, heightfunction matrices, threecolorings, monotone triangles, tetrahedral order ideals, square ice, gasketandbasket ..."
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Cited by 30 (0 self)
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I give a survey of different combinatorial forms of alternatingsign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as cornersum matrices, heightfunction matrices, threecolorings, monotone triangles, tetrahedral order ideals, square ice, gasketandbasket tilings and full packings of loops. (This article has been published in a conference edition of the journal Discrete Mathematics and Theoretical
A new applied approach for executing computations with infinite and infinitesimal quantities
 Informatica
, 2008
"... A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole ’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all ..."
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Cited by 19 (7 self)
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A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole ’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The new methodology has allowed us to introduce the Infinity Computer working with such numbers (its simulator has already been realized). Examples dealing with divergent series, infinite sets, and limits are given.
Selection Theorem for Systems With Inheritance
"... Abstract. The problem of finitedimensional asymptotics of infinitedimensional dynamic systems is studied. A nonlinear kinetic system with conservation of supports for distributions has generically finitedimensional asymptotics. Such systems are apparent in many areas of biology, physics (the the ..."
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Abstract. The problem of finitedimensional asymptotics of infinitedimensional dynamic systems is studied. A nonlinear kinetic system with conservation of supports for distributions has generically finitedimensional asymptotics. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. This conservation of support has a biological interpretation: inheritance. The finitedimensional asymptotics demonstrates effects of “natural ” selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t → ∞. The drift equations for peak motion are obtained. Various types of distribution stability are studied: internal stability (stability with respect to perturbations that do not extend the support), external stability or uninvadability (stability with respect to strongly small perturbations that extend the support), and stable realizability (stability with respect to small shifts and extensions of the density peaks). Models of selfsynchronization of cell division are studied, as an example of selection in systems with additional symmetry. Appropriate construction of the notion of typicalness in infinitedimensional space is discussed, and the notion of “completely thin” sets is introduced.
Numerical computations and mathematical modelling with infinite and infinitesimal numbers
 Journal of Applied Mathematics and Computing
"... Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the nonstandard analysis) is used to work with fi ..."
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Cited by 14 (6 self)
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Traditional computers work with finite numbers. Situations where the usage of infinite or infinitesimal quantities is required are studied mainly theoretically. In this paper, a recently introduced computational methodology (that is not related to the nonstandard analysis) is used to work with finite, infinite, and infinitesimal numbers numerically. This can be done on a new kind of a computer – the Infinity Computer – able to work with all these types of numbers. The new computational tools both give possibilities to execute computations of a new type and open new horizons for creating new mathematical models where a computational usage of infinite and/or infinitesimal numbers can be useful. A number of numerical examples showing the potential of the new approach and dealing with divergent series, limits, probability theory, linear algebra, and calculation of volumes of objects consisting of parts of different dimensions are given.
A new operation on sequences: the boustrophedon transform
 J. Combin. Th. Ser. A
, 1996
"... A generalization of the SeidelEntringerArnold method for calculating the alternating permutation numbers (or secanttangent numbers) leads to a new operation on sequences, the boustrophedon transform. This paper was published (in a somewhat different form) in J. Combinatorial Theory, Series A, 76 ..."
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A generalization of the SeidelEntringerArnold method for calculating the alternating permutation numbers (or secanttangent numbers) leads to a new operation on sequences, the boustrophedon transform. This paper was published (in a somewhat different form) in J. Combinatorial Theory, Series A, 76 (1996), pp. 44–54.
Free Deconvolution: from Theory to Practice
, 2010
"... In this paper, we provide an algorithmic method to compute the singular values of sum of rectangular matrices based on the free cumulants approach and illustrate its application to wireless communications. We first recall the algorithms working for sum/products of square random matrices, which hav ..."
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Cited by 13 (3 self)
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In this paper, we provide an algorithmic method to compute the singular values of sum of rectangular matrices based on the free cumulants approach and illustrate its application to wireless communications. We first recall the algorithms working for sum/products of square random matrices, which have already been presented in some previous papers and we then introduce the main contribution of this paper which provides a general method working for rectangular random matrices, based on the recent theoretical work of BenaychGeorges. In its full generality, the computation of the eigenvalues requires some sophisticated tools related to free probability and the explicit spectrum (eigenvalue distribution) of the matrices can hardly be obtained (except for some trivial cases). From an implementation perspective, this has led the community to the misconception that free probability has no practical application. This contribution takes the opposite view and shows how the free cumulants approach in free probability provides the right shift from theory to practice.