Infinite Versions Research Articles

Circle packings from tilings of the plane

We introduce a new class of fractal circle packings in the plane, generalizing the polyhedral packings defined by Kontorovich and Nakamura. The existence and uniqueness of these packings are guaranteed by infinite versions of the Koebe–Andreev–Thurston theorem. We prove structure theorems giving a complete description of the symmetry groups for these packings. And we give several examples to illustrate their number-theoretic and group-theoretic significance.

A Shuffle Theorem for Paths Under Any Line

AbstractWe generalize the shuffle theorem and its$(km,kn)$version, as conjectured by Haglund et al. and Bergeron et al. and proven by Carlsson and Mellit, and Mellit, respectively. In our version the$(km,kn)$Dyck paths on the combinatorial side are replaced by lattice paths lying under a line segment whosexandyintercepts need not be integers, and the algebraic side is given either by a Schiffmann algebra operator formula or an equivalent explicit raising operator formula. We derive our combinatorial identity as the polynomial truncation of an identity of infinite series of$\operatorname {\mathrm {GL}}_{l}$characters, expressed in terms of infinite series versions of LLT polynomials. The series identity in question follows from a Cauchy identity for nonsymmetric Hall–Littlewood polynomials.

Reverse mathematics and Weihrauch analysis motivated by finite complexity theory

We extend a study by Lempp and Hirst of infinite versions of some problems from finite complexity theory, using an intuitionistic version of reverse mathematics and techniques of Weihrauch analysis.

Local Moderate and Precise Large Deviations via Cluster Expansions

We consider a system of classical particles confined in a box \(\varLambda \subset {\mathbb {R}}^d\) with zero boundary conditions interacting via a stable and regular pair potential. Based on the validity of the cluster expansion for the canonical partition function in the high temperature–low density regime we prove moderate and precise large deviations from the mean value of the number of particles with respect to the grand-canonical Gibbs measure. In this way we have a direct method of computing both the exponential rate as well as the pre-factor and obtain explicit error terms. Estimates comparing with the infinite volume versions of the above are also provided.

On infinite variants of De Morgan law in locale theory

A locale, being a complete Heyting algebra, satisfies De Morgan law (a∨b)⁎=a⁎∧b⁎ for pseudocomplements. The dual De Morgan law (a∧b)⁎=a⁎∨b⁎ (here referred to as the second De Morgan law) is equivalent to, among other conditions, (a∨b)⁎⁎=a⁎⁎∨b⁎⁎, and characterizes the class of extremally disconnected locales. This paper presents a study of the subclasses of extremally disconnected locales determined by the infinite versions of the second De Morgan law and its equivalents.

Convergence to the mean field game limit: A case study

We study the convergence of Nash equilibria in a game of optimal stopping. If the associated mean field game has a unique equilibrium, any sequence of $n$-player equilibria converges to it as $n\to\infty$. However, both the finite and infinite player versions of the game often admit multiple equilibria. We show that mean field equilibria satisfying a transversality condition are limit points of $n$-player equilibria, but we also exhibit a remarkable class of mean field equilibria that are not limits, thus questioning their interpretation as "large $n$" equilibria.

Positive speed for high-degree automaton groups

Mother groups are the basic building blocks for polynomial automaton groups. We show that, in contrast with mother groups of degree 0 or 1, any bounded, symmetric, generating random walk on the mother groups of degree at least 3 has positive speed. The proof is based on an analysis of resistance in fractal mother graphs. We give upper bounds on resistances in these graphs, and show that infinite versions are transient.

On the existence and distribution quality of hyperplane sequences

It is well known that digital (t,m,s)-nets and (T,s)-sequences over a finite field have excellent properties when they are used as underlying nodes in quasi-Monte Carlo integration rules. One very general sub-class of digital nets are hyperplane nets which can be viewed as a generalization of cyclic nets and of polynomial lattice point sets. In this paper we introduce infinite versions of hyperplane nets and call these sequences hyperplane sequences. Our construction is based on the recent duality theory for digital sequences according to Dick and Niederreiter. We then analyze the equidistribution properties of hyperplane sequences in terms of the quality function T and the star discrepancy.

Two infinite versions of the nonlinear Dvoretzky theorem

Two infinite versions of the nonlinear Dvoretzky theorem

Strategic best-response learning in multiagent systems

We present a novel and uniform formulation of the problem of reinforcement learning against bounded memory adaptive adversaries in repeated games, and the methodologies to accomplish learning in this novel framework. First we delineate a novel strategic definition of best response that optimises rewards over multiple steps, as opposed to the notion of tactical best response in game theory. We show that the problem of learning a strategic best response reduces to that of learning an optimal policy in a Markov Decision Process (MDP). We deal with both finite and infinite horizon versions of this problem. We adapt an existing Monte Carlo based algorithm for learning optimal policies in such MDPs over finite horizon, in polynomial time. We show that this new efficient algorithm can obtain higher average rewards than a previously known efficient algorithm against some opponents in the contract game. Though this improvement comes at the cost of increased domain knowledge, simple experiments in the Prisoner's Dilemma, and coordination games show that even when no extra domain knowledge (besides that an upper bound on the opponent's memory size is known) is assumed, the error can still be small. We also experiment with a general infinite-horizon learner (using function-approximation to tackle the complexity of history space) against a greedy bounded memory opponent and show that while it can create and exploit opportunities of mutual cooperation in the Prisoner's Dilemma game, it is cautious enough to ensure minimax payoffs in the Rock–Scissors–Paper game.

New structural properties of [formula omitted] policies for inventory models with lost sales

We revisit the classical inventory model for which ( s , S ) policies are optimal. We consider the finite and infinite horizon versions of the lost sales problem. New structural properties are developed for the optimal policy and cost function. These properties are then used to develop computational schemes for the lost sales problem with Erlang demands.

Partitions and indivisibility properties of countable dimensional vector spaces

We investigate infinite versions of vector and affine space partition results, and thus obtain examples and a counterexample for a partition problem for relational structures. In particular we provide two (related) examples of an age indivisible relational structure which is not weakly indivisible.

A nearfield-free definition of regular Hughes planes and some embeddings of PG(3,q) in Hughes planes of order q 4

We give a nearfield-free definition of some finite and infinite incidence systems by means of half-points and half-lines and show that they are projective planes. We determine a planar ternary ring for these planes and use it to determine the full collineation group and to demonstrate some embeddings of these planes among themselves. We show that these planes include all finite regular Hughes planes and many infinite ones. We also show that PG(3, q) embeds in Hu(q 4) (and show infinite versions of this embedding).

A statistical mechanics view on Kitaev's proposal for quantum memories

We compute rigorously the ground and equilibrium states for Kitaev's model in 2D, both the finite and infinite versions, using an analogy with the 1D Ising ferromagnet. Next, we investigate the structure of the reduced dynamics in the presence of thermal baths in the Markovian regime. Special attention is paid to the dynamics of the topological freedoms which have been proposed for storing quantum information.

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

Abstract: We study the stationary distribution of the standard Abelian sandpile model in the box $Lam_n = [-n, n]^d cap d$ for $d ge 2$. We show that as $n o infty$, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in $d$. This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on $d$. In the case $d > 4$, we also make use of Wilson's method.

Dynamic price competition

We consider the model of price competition for a single buyer among many sellers in a dynamic environment. The surplus from each trade is allowed to depend on the path of previous purchases, and as a result, the model captures phenomena such as learning by doing and habit formation in consumption. We characterize Markovian equilibria for finite and infinite horizon versions of the model and show that the stationary infinite horizon version of the model possesses an efficient equilibrium where all the sellers receive an equilibrium payoff equal to their marginal contribution to the social welfare.

The Cycle Space of an Infinite Graph

Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new ‘singular’ approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of $S^1$ in the space formed by the graph together with its ends.Our approach permits the extension to infinite graphs of standard results about finite graph homology – such as cycle–cocycle duality and Whitney's theorem, Tutte's generating theorem, MacLane's planarity criterion, the Tutte/Nash-Williams tree packing theorem – whose infinite versions would otherwise fail. A notion of end degrees motivated by these results opens up new possibilities for an ‘extremal’ branch of infinite graph theory.Numerous open problems are suggested.

Linear and DF Joint Detectors for DS-CDMA Communications Using Periodic Long Codes

Detection structures for a direct sequence (DS) code division multiple access (CDMA) multiuser uplink using codes with a period of several symbols, called long codes, are investigated. In a first part, we propose a multirate description of the uplink. Based on this formalism, we revisit the linear and decision feedback minimum mean square error (MMSE) detectors. In order to identify the best attainable performance, we focus on the infinite size versions. In a second part, reduced complexity solutions are put forward and discussed. Finally, the performance of the proposed detectors is investigated and compared with that of the classical rake receiver and the single-bit horizon detectors. The relevance of the different solutions is discussed in the light of the channel length.

Infinite Volume Limit for the Stationary Distribution of Abelian Sandpile Models

We study the stationary distribution of the standard Abelian sandpile model in the box Λn = [-n, n] d ∩ ℤ d for d≥ 2. We show that as n→ ∞, the finite volume stationary distributions weakly converge to a translation invariant measure on allowed sandpile configurations in ℤ d . This allows us to define infinite volume versions of the avalanche-size distribution and related quantities. The proof is based on a mapping of the sandpile model to the uniform spanning tree due to Majumdar and Dhar, and the existence of the wired uniform spanning forest measure on ℤ d . In the case d > 4, we also make use of Wilson’s method.

Memoryless determinacy of parity and mean payoff games: a simple proof

We give a simple, direct, and constructive proof of memoryless determinacy for parity and mean payoff games. First, we prove by induction that the finite duration versions of these games, played until some vertex is repeated, are determined and both players have memoryless winning strategies. In contrast to the proof of Ehrenfeucht and Mycielski, Internat. J. Game Theory, 8 (1979) 109–113, our proof does not refer to the infinite-duration versions. Second, we show that memoryless determinacy straightforwardly generalizes to infinite duration versions of parity and mean payoff games.