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  • Author or Editor: László Péter x
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Abstract  

We investigate the repeated and sequential portfolio St. Petersburg games. For the repeated St. Petersburg game, we show an upper bound on the tail distribution, which implies a strong law for a truncation. Moreover, we consider the problem of limit distribution. For the sequential portfolio St. Petersburg game, we obtain tight asymptotic results for the growth rate of the game.

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Abstract  

The aim of this paper is to continue our investigations started in [15], where we studied the summability of weighted Lagrange interpolation on the roots of orthogonal polynomials with respect to a weight function w. Starting from the Lagrange interpolation polynomials we constructed a wide class of discrete processes which are uniformly convergent in a suitable Banach space (C ρ, ‖‖ρ) of continuous functions (ρ denotes (another) weight). In [15] we formulated several conditions with respect to w, ρ, (C ρ, ‖‖ρ) and to summation methods for which the uniform convergence holds. The goal of this part is to study the special case when w and ρ are Freud-type weights. We shall show that the conditions of results of [15] hold in this case. The order of convergence will also be considered.

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Abstract  

Starting from the Lagrange interpolation on the roots of Jacobi polynomials, a wide class of discrete linear processes is constructed using summations. Some special cases are also considered, such as the Fejr, de la Valle Poussin, Cesro, Riesz and Rogosinski summations. The aim of this note is to show that the sequences of this type of polynomials are uniformly convergent on the whole interval [-1,1] in suitable weighted spaces of continuous functions. Order of convergence will also be investigated. Some statements of this paper can be obtained as corollaries of our general results proved in [15].

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The attempts to construct a counterexample to the Strong Perfect Graph Conjecture yielded the notion of partitionable graphs as minimal imperfect graphs; then near-factorizations of finite groups gained some interest since from any near-factorization some partitionable graphs can be constructed in a natural way. Recently, the proof of SPGC was declared by Chudnovsky, Robertson, Seymour and Thomas [3], but near-factorizations remain interesting on their own rights as (i) rare objects being “close” to factorizations of groups; and (ii) they yield graphs with surprising properties.

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Abstract  

Sealed bid auctions are a popular means of high-stakes bidding, as they eliminate the temporal element from the auction process, allowing participants to take less emotional, more thoughtful decisions. In this paper, we propose a digital communication protocol for conducting sealed bid auctions with high stakes, where the anonymity of bids as well as other aspects of fairness must be protected. The Dining Cryptographers’ Protocol (denoted by DC) was presented by David Chaum in 1988. The protocol allows the participants to broadcast a message anonymously. In a recent paper (Another Twist in the Dining Cryptographers’ Protocol, submitted to the Journal of Cryptology) the authors propose a variant of the original DC eliminating its main disadvantages. In this paper we present a cryptographic protocol realizing anonymous sealed bid auctions, such as first price or Vickrey auction, based on this variant. The proposed scheme allows to identify at least one dishonest participant violating the protocol without using of Trusted Third Parties. Additionally, we require that bids are binding. It is achieved by enabling all participants acting in concert (the so-called “angry mob”) to find out the identity of the winner, in case the winner fails to make the purchase.

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