The generalized faro shuffle

The standard faro shuffle, an idealized riffle shuffle, divides the deck into two equal portions, and perfectly interlaces them. The simple cut takes one card from the top to the bottom of the deck. It is known that for decks of even size, the faro shuffle and simple cut generate all possible permutations, while if the deck is of odu size, only a small fraction are available. This paper considers a generalized faro shuffle wherein the deck is divided into n rather than 2, portions and these portions are “interlaced” together. It is shown that the generalized faro shuffle and the simple cut generate either the symmetric group of the deck, the alternating group of the deck, or in one special case, only a small fraction of the possible permutations. Whether the symmetric group or alternating group is generated depends on the parity of the simple cut and the generalized faro shuffle as group operations.     

Single-suit two-person card play

We consider a two-person constant sum perfect information game, which we call theEnd Play Game, which arises from an abstraction of simple end play positions in card games of the whist family, including bridge. This game was described in 1929 by Emanuel Lasker, the mathematician and world chess champion, who called itwhistette. The game uses a deck of cards that consists of a single totally ordered suit of 2n cards. To begin play the deck is divided into two handsA andB ofn cards each, held by players Left and Right, and one player is designated as having thelead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a “trick” and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and thevalue of the game to each player is the number of tricks he takes. Despite its simple appearance, this game is quite complicated, and finding an optimal strategy seems difficult. This paper derives basic properties of the game, gives some criteria under which one hand is guaranteed to be better than another, and determines the optimal strategies and value functions for the game in several special cases.     

Reconstructing trees from two cards

Let ?? be the class of unlabeled trees. An unlabeled vertex‐deleted subgraph of a tree T is called a card. A collection of cards is called a deck. We say that the tree T has a deck D if each card in D can be obtained by deleting distinct vertices of T. If T is the only unlabeled tree that has the deck D, we say that T is ??‐reconstructible from D. We want to know how large of a deck D is necessary for T to be ??‐reconstructible. We define ??rn(T) as the minimum number of cards in a deck D such that T is ??‐reconstructible from D. It is known that ??rn(T)≤3, but it is conjectured that ??rn(T)≤2 for all trees T. We prove that the conjecture holds for all homeomorphically irreducible trees. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 243–257, 2010     

Staircase rook polynomials and Cayley’s game of Mousetrap

First introduced by Arthur Cayley in the 1850’s, the game of Mousetrap involves removing cards from a deck according to a certain rule. In this paper we find the rook polynomial for the number of Mousetrap decks in which at least two specified cards are removed. We also find a new expression for the rook polynomial for the number of decks in which exactly one specified card is removed and give expressions for counts of two kinds of Mousetrap decks in terms of other known combinatorial numbers.     

The mathematician who drove Whist forward: William Henry Whitfeld (1856–1915)

As a card game in which different partnerships play against each other, or each player plays with different partners, Whist is a popular social activity. The organization of whist drives where people move from table to table is a practical problem, and one that offers a mathematical challenge. How should a drive be organized? In solving such problems W H Whitfeld contributed to the development of the modern combinatorial theory of tournaments.     

Probabilistic and combinatorial aspects of the card‐cyclic to random insertion shuffle

Consider a permutation σ ∈ S_n as a deck of cards numbered from 1 to n and laid out in a row, where σ_j denotes the number of the card that is in the j‐th position from the left. We study some probabilistic and combinatorial aspects of the shuffle on S_n defined by removing and then randomly reinserting each of the n cards once, with the removal and reinsertion being performed according to the original left to right order of the cards. The novelty here in this nonstandard shuffle is that every card is removed and reinserted exactly once. The bias that remains turns out to be quite strong and possesses some surprising features. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 362–390, 2015     

Communication-Computation Trade-off in Executing ECDSA in a Contactless Smartcard

Emerging standards specify a communication rate between a contactless smartcard and a terminal that is of the same order of magnitude as the internal clock rate in the card. This gives a natural ground for the known card-terminal communication-computation trade-off, where non-secure operations should rather be performed by the terminal and not in the card. In this paper we treat an implementation of Elliptic Curve Digital Signature Algorithm (ECDSA), the most cost effective digital signature algorithm, which has a potential of being executed under the heavy constraints imposed by a contactless smartcard environment. This algorithm heavily relies on numerous calculations of modular multiplicative inverses. It is shown in this paper that, based on communicating with the terminal, each modular inverse operation needed to be executed in the card during ECDSA signature generation requires only two modular multiplications in the card. Each modular inverse operation performed during signature verification requires only one modular multiplication in the card. A complete ECDSA implementation over integers or over GF(2ⁿ) is then treated in detail AMS Classification: 94A62     

Strong authenticated key exchange with auxiliary inputs

In the Russian cards problem, Alice, Bob and Cath draw a, b and c cards, respectively, from a publicly known deck. Alice and Bob must then communicate their cards to each other without Cath learning who holds a single card. Solutions in the literature provide weak security, where Alice and Bob’s exchanges do not allow Cath to know with certainty who holds each card that is not hers, or perfect security, where Cath learns no probabilistic information about who holds any given card. We propose an intermediate notion, which we call \(\varepsilon \)-strong security, where the probabilities perceived by Cath may only change by a factor of \(\varepsilon \). We then show that strategies based on affine or projective geometries yield \(\varepsilon \)-strong safety for arbitrarily small \(\varepsilon \) and appropriately chosen values of a, b, c.     

An orthotropic plate model for decks of suspension bridges

The main purpose of the present paper is to compare two different kinds of approaches in modeling the deck of a suspension bridge: in the first approach we look at the deck as a rectangular plate and in the second one we look at the deck as a beam for vertical deflections and as a rod for torsional deformations. Throughout this paper we will refer to the model corresponding to the second approach as the beam-rod model. In our discussion, we observe that the beam-rod model contains a larger number of elastic parameters if compared with the isotropic plate model. For this reason the beam-rod model is supposed to be more appropriate to describe the behavior of the deck of a real suspension bridge. A possible strategy to make the plate model more efficient could be to relax the isotropy condition with a more general condition of orthotropy, which is expected to increase the number of elastic parameters. In this new setting, a comparison between the two approaches becomes now possible.Basic results are proved for the suggested problem, from existence and uniqueness of solutions to spectral properties. We suggest realistic values for the elastic parameters thus obtaining with both approaches similar responses in the static and dynamic behavior of the deck. This can be considered as a preliminary article since many work has still to be done with the perspective of formulating models for a complete suspension bridge which take into account not only the deck but also the action on it of cables and hangers. With this perspective, a section is devoted to possible future developments.     

The mathematics of perfect shuffles

There are two ways to perfectly shuffle a deck of 2n cards. Both methods cut the deck in half and interlace perfectly. The out shuffle O leaves the original top card on top. The in shuffle I leaves the original top card second from the top. Applications to the design of computer networks and card tricks are reviewed. The main result is the determination of the group 〈 I, O 〉 generated by the two shuffles, for all n. If 2n is not a power of 2, and if 2n ≠ 12,24, then 〈 I, O 〉 has index 1, 2, or 4 in the Weyl group B_n (the group of all 2ⁿn! signed n × n permutation matrices). If 2n = 2^k, then 〈 I, O 〉 is isomorphic to a semi-direct product of Z₂^k and Z_k. When 2n = 24, 〈 I, O 〉 is isomorphic to a semi-direct product of Z₂¹¹ and M₁₂, the Mathieu group of degree 12. When 2n = 12, 〈 I, O 〉 is isomorphic to a semi-direct product of Z₂⁶ and the group PGL(2,5) of all linear fractional transformations over GF(5).     

Some exploration of the game SET®

Since the game SET® was first introduced to the public in 1993, it has stimulated some interesting studies. While the game itself is rather straightforward, a plethora of decent mathematical questions lies beneath the surface. It is perhaps because the game ties in so closely with such an underlying mathematical term that its implications can be seen in many major fields of mathematics. This note introduces a generalized version of the game SET® in which there are n ^m cards, m???4,?n???3,?m???n. Each card has m features and each feature has n options. A precise formula is presented for calculating the total number of SETs in a general deck.     

Table algebras with central fusions

The character theory of table algebras is not as good as the character theory of finite groups. We introduce the notion of a table algebra with a central-fusion, in which the character theory has better properties. We study conditions under which a table algebra (A,B) has a central-fusion, and its central-fusion is exactly isomorphic to the wreath product of the central-fusion of a quotient table algebra of (A,B) and another table algebra. As a consequence, we obtain a complete characterization of table algebras with exactly one irreducible character whose degree and multiplicity are not equal. Applications to association schemes are also discussed.     

Wave propagation in a mixed convection boundary‐layer over a horizontal surface

The propagation of disturbances in a mixed convection boundary‐layer flow over a horizontal plate is described by a triple deck problem in the case of the buoyancy parameter being small. The pressure correction in the lower deck consists of two parts: One due to the buoyancy effects in the main deck and one due to the displacement of the outer flow field. The response of the boundary layer flow to an oscillator of frequency ω will be computed and upstream travelling waves will be identified.     

Analysis of the effectiveness of manufacturer-sponsored retailer gift cards in supply chains

Some manufacturers sponsor “free” retailer gift cards to be given to consumers who purchase their products. These gift cards are paid for by the manufacturer and are redeemable on all products at the retailer. We develop a model of such a supply chain. We analyze cases in which the gift cards’ redemption rate is constant or increasing in gift card value. The results indicate that in addition to the redemption rate and consumers’ valuation for gift card dollars, the profitability of manufacturer-sponsored gift cards depends on the average gross margin of the retailer and the type of purchases consumers make with gift cards. Furthermore, we show that under certain conditions, free gift cards will increase the expected profits of the retailer and manufacturer as well as decrease the retail price of the product. These conditions include a retailer with large average gross margin and consumers using gift cards to purchase products they would not buy with cash otherwise. Furthermore, all consumers, including those who do not redeem the gift card, are more likely to benefit from a reduced retail price when their probability of redeeming the gift card after purchase is equal to their estimated redemption probability at purchase time. We show the conditions under which gift cards are more profitable than cash mail-in rebates. We develop an incentive scheme to improve the performance of supply chains with gift cards.     

Single-suit two-person card play II. Dominance

This paper studies a two-person constant sum perfect information game, the End Play Game, arising from an abstraction of end play in bridge. This game was described by Emanuel Lasker who called it whistette. The game uses a deck of cards consisting of a single totally ordered suit of 2n cards. The deck is divided into two hands A and B of n cards each, held by players Left and Right, and one player is designated as having the lead. The player on lead chooses one of his cards, and the other player after seeing this card selects one of his own to play. The player with the higher card wins a trick and obtains the lead. The cards in the trick are removed from each hand, and play then continues until all cards are exhausted. Each player strives to maximize his trick total, and the value of the game to each player is the number of tricks he takes. The strategy of this game seems to be quite complicated, despite its simple appearance. This paper studies partial orderings on hands. One partial order recognizes regularities in the value function that persist when extra cards are added to hands. A pair of hands (A ^* , B ^* ) dominates a pair of hands (A, B) for Left, if for any set of extra cards (C ₁, C ₂) added to the deck such that A B (which equals A ^* B ^* ) is a block of consecutive cards in the expanded deck A B {C ₁ , C ₂} the value of (A C ₁, B C ₂) to Left always is at least as much as the value to Left of (A ^* C ₁, B ^* C ₂) both when Left has the lead in both games and when Right has the lead in both games. The main result is that ({4, 1}, {3, 2}) dominates ({3, 2}, {4, 1}). Note that with just four cards the hands {4, 1} and {3, 2} are of identical value — they both take one trick independent of the lead or how the hands are played. The dominance result shows that {4, 1} is preferable to {3, 2} when other cards are present. We show that the dominance relation gives a partial order that is not a total order on hands of 3 or more cards. We also study the total point count ordering, which gives a rough estimate for the value of a hand. We derive upper and lower bounds for the value of a hand with given point count.     

Aplicación de métodos computacionales en la evaluación de la respuesta aeroelástica de puentes soportados por cables

The possibilities of computational methods for assessing the response of cable supported bridges under wind action are considered in this work. The main objective is to study the possibilities of substituting wind tunnel campaigns by computer based analyses, particularly at the early design stage. The preliminary proposed design for a continuous cable-stayed bridge with two main spans of 650 m and a single box girder deck has been considered as a case study. The force coefficients of the deck cross-section have been computed and the unsteady response associated to vortex-shedding has been simulated using CFD commercial software. Furthermore, an in-house piece of software has been employed to obtain the response for flutter and buffeting phenomena adopting the hybrid approach; with that purpose the experimental flutter functions of a similar box girder deck were adopted. The computational results have been validated by comparison with similar experimental results published by other researchers. It has been verified that the set of adopted methods offers reliable results with moderate costs; therefore, the proposed approach is very suitable at the early design stage of long span bridges or at conceptual design works.     

The Fundamental Theorem of Card Counting with applications to Trente-et-Quarante and Baccarat

The Fundamental Theorem of Card Counting is a unifying principle for the analysis of card games of chance which are characterized by sampling without replacement. The Theorem says (roughly) the “spread” in distribution of player expectations for partially depleted card packs increases with depletion of the card pack. Furthermore, average player expectation is non-decreasing (increasing under suitable hypotheses) with increasing depletion. The Theorem is used to prove that significant favorable strategies based on card counting do not exist for Trente-et-Quarante or for “tie” bets in Nevada Baccarat. This is in sharp contrast with previous results for Blackjack and for Nevada Baccarat side bets on natural eight and natural nine.     

Weakness and improvement on Wang–Li–Tie’s user-friendly remote authentication scheme

In an open network environment, the remote authentication scheme using smart cards is a very practical solution to validate the legitimacy of a remote user. In 2003, Wu and Chieu presented a user-friendly remote authentication scheme using smart cards. Recently, Wang, Li, and Tie found that Wu–Chieu’s scheme is vulnerable to the forged login attack, and then presented an improvement to eliminate this vulnerability. In our opinion, the smart card plays an important role in those schemes. Therefore, we demonstrate that Wang–Li–Tie’s scheme is not secure under the smart card loss assumption. If an adversary obtains a legal user’s smart card even without the user’s corresponding password, he can easily use it to impersonate the user to pass the server’s authentication. We further propose an improved scheme to overcome this abuse of the smart card.     

Shock boundary layer interactions in dense gases

The concept of triple deck theory is applied to study laminar interacting boundary layers of dense gases in external purely supersonic flow. An impinging shock is generated at distances which are large compared to the upper deck thickness. As predicted by weakly nonlinear theory such a discontinuity may disintegrate into a sonic shock and an associated wave fan depending on its amplitude and the magnitude of the so called fundamental derivative. Incoming and outgoing waves are computed analytically by means of the method of multiple scales taking into account that mutual interaction effects between them are restricted to the linear upper deck region. The lower deck problem is solved numerically. The results show that it is possible to reduce the size of the separation bubble or even to avoid the occurrence of flow separation by choosing an optimal thermodynamic state. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)