Pattern Avoidance Over a Hypergraph

A classic result of Marcus and Tardos (previously known as the Stanley-Wilf conjecture) bounds from above the number of $n$-permutations ($\sigma \in S_n$) that do not contain a specific sub-permutation. In particular, it states that for any fixed permutation $\pi$, the number of $n$-permutations that avoid $\pi$ is at most exponential in $n$. In this paper, we generalize this result. We bound the number of avoidant $n$-permutations even if they only have to avoid $\pi$ at specific indices. We consider a $k$-uniform hypergraph $\Lambda$ on $n$ vertices and count the $n$-permutations that avoid $\pi$ at the indices corresponding to the edges of $\Lambda$. We analyze both the random and deterministic hypergraph cases. This problem was originally proposed by Asaf Ferber. When $\Lambda$ is a random hypergraph with edge density $\alpha$, we show that the expected number of $\Lambda$-avoiding $n$-permutations is bounded (both upper and lower) as $\exp(O(n))\alpha^{-\frac{n}{k-1}}$, using a supersaturation version of F\"{u}redi-Hajnal. In the deterministic case we show that, for $\Lambda$ containing many size $L$ cliques, the number of $\Lambda$-avoiding $n$-permutations is $O\left(\frac{n\log^{2+\epsilon}n}{L}\right)^n$, giving a nontrivial bound with $L$ polynomial in $n$. Our main tool in the analysis of this deterministic case is the new and revolutionary hypergraph containers method, developed in papers of Balogh-Morris-Samotij and Saxton-Thomason.

A New Algorithm for Embedding Plane Graphs at Fixed Vertex Locations

We show that a plane graph can be embedded with its vertices at arbitrarily assigned locations in the plane and at most $6n-1$ bends per edge. This improves and simplifies a classic result by Pach and Wenger. The proof extends to orthogonal drawings.

A Bargaining Game with Proposers in the Hot Seat

This note reconsiders the Rubinstein bargaining game under the assumption that a rejected offer is only costly to the proposer who made the rejected offer. It is shown that then, the classic result of Shaked that, in the multilateral version of this game, every division of the good can be sustained in SPE no longer holds. Specifically, there are many SPE, but players’ (expected) payoffs in SPE are unique. The assumption further leads to a responder advantage.

Marginality and convexity in partition function form games

In this paper an order on the set of embedded coalitions is studied in detail. This allows us to define new notions of superaddivity and convexity of games in partition function form which are compared to other proposals in the literature. The main results are two characterizations of convexity. The first one uses non-decreasing contributions to coalitions of increasing size and can thus be considered parallel to the classic result for cooperative games without externalities. The second one is based on the standard convexity of associated games without externalities that we define using a partition of the player set. Using the later result, we can conclude that some of the generalizations of the Shapley value to games in partition function form lie within the cores of specific classic games when the original game is convex.

Distributions in CFT. Part II. Minkowski space

CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios ρ, $$ \overline{\rho} $$ ρ ¯ . We prove a key fact that |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ < 1 inside the forward tube, and set bounds on how fast |ρ|, $$ \left|\overline{\rho}\right| $$ ρ ¯ may tend to 1 when approaching the Minkowski space.We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).

Strategyproof Randomized Social Choice for Restricted Sets of Utility Functions

When aggregating preferences of multiple agents, strategyproofness is a fundamental requirement. For randomized voting rules, so-called social decision schemes (SDSs), strategyproofness is usually formalized with the help of utility functions. A classic result shown by Gibbard in 1977 characterizes the set of SDSs that are strategyproof with respect to all utility functions and shows that these SDSs are either indecisive or unfair. For finding more insights into the trade-off between strategyproofness and decisiveness, we propose the notion of U-strategyproofness which requires that only voters with a utility function in the set U cannot manipulate. In particular, we show that if the utility functions in U value the best alternative much more than other alternatives, there are U-strategyproof SDSs that choose an alternative with probability 1 whenever all but k voters rank it first. We also prove for rank-based SDSs that this large gap in the utilities is required to be strategyproof and that the gap must increase in k. On the negative side, we show that U-strategyproofness is incompatible with Condorcet-consistency if U satisfies minimal symmetry conditions and there are at least four alternatives. For three alternatives, the Condorcet rule can be characterized based on U-strategyproofness for the set U containing all equi-distant utility functions.

The Normalized Matching Property in Random and Pseudorandom Bipartite Graphs

A bipartite graph $G(X,Y,E)$ with vertex partition $(X,Y)$ is said to have the Normalized Matching Property (NMP) if for any subset $S\subseteq X$ we have $\frac{|N(S)|}{|Y|}\geq\frac{|S|}{|X|}$. In this paper, we prove the following results about the Normalized Matching Property. The random bipartite graph $\mathbb{G}(k,n,p)$ with $|X|=k,|Y|=n$, and $k\leq n<\exp(k)$, and each pair $(x,y)\in X\times Y$ being an edge in $\mathbb{G}$ independently with probability $p$ has $p=\frac{\log n}{k}$ as the threshold for NMP. This generalizes a classic result of Erdős-Rényi on the $\frac{\log n}{n}$ threshold for the existence of a perfect matching in $\mathbb{G}(n,n,p)$. A bipartite graph $G(X,Y)$, with $k=|X|\le |Y|=n$, is said to be Thomason pseudorandom (following A. Thomason (Discrete Math., 1989)) with parameters $(p,\varepsilon)$ if every $x\in X$ has degree at least $pn$ and every pair of distinct $x, x'\in X$ have at most $(1+\varepsilon)p^2n$ common neighbours. We show that Thomason pseudorandom graphs have the following property: Given $\varepsilon>0$ and $n\geq k\gg 0$, there exist functions $f,g$ with $f(x), g(x)\to 0$ as $x\to 0$, and sets $\mathrm{Del}_X\subset X, \ \mathrm{Del}_Y\subset Y$ with $|\mathrm{Del}_X|\leq f(\varepsilon)k,\ |\mathrm{Del}_Y|\leq g(\varepsilon)n$ such that $G(X\setminus \mathrm{Del}_X,Y\setminus \mathrm{Del}_Y)$ has NMP. Enroute, we prove an 'almost' vertex decomposition theorem: Every Thomason pseudorandom bipartite graph $G(X,Y)$ admits - except for a negligible portion of its vertex set - a partition of its vertex set into graphs that are spanned by trees that have NMP, and which arise organically through the Euclidean GCD algorithm.

Thick-Wall Elastic Collapse for Casing Design

Summary Elastic collapse is an important piece of the tubular collapse formulation in API TR 5C3 (2008) and ISO/TR 10400 (2007). Elastic collapse is significant because it is independent of the strength of the tubing, for example, K-55 and Q-125 have the same resistance to elastic collapse. Advanced collapse models, such as Klever and Tamano (2006), require a thick-wall collapse result as part of their formulation. What would the effect of a thick wall have on elastic collapse? There really is no way to tell from the classic formulation. The primary issue is whether the elastic collapse formula overpredicts or underpredicts collapse pressure. The developers of the API collapse equation thought the thin-wall equation overpredicted collapse pressure and put in terms to reduce the predictions. Other studies suggested the opposite effect. What is needed is a formulation that is based on an elastic solution for a thick-wall cylinder, but that can derive the classic solution for a thin wall. The elastic equations for a thick-walled cylinder exist, known as the Kirsch equations (Kirsch 1898). A new set of physically reasonable boundary conditions are proposed for the Kirsch equation, which was then used to determine the collapse resistance for a thick-wall pipe. This result also yielded the classic result in the limit because t/D is small. The thick-wall elastic collapse formula is then applied to the standard API TR 5C3 (2008) collapse formulation and to the Klever-Tamano formulation (Klever and Tamano 2006).

Design of a Computer-Aided Location Expert System Based on a Mathematical Approach

This article discusses how to calculate the location of a point on a surface using a mathematical approach on two levels. The first level uses the traditional calculation procedure via Cooper’s iterative method through a spreadsheet editor and a classic result display map. The second level uses the author-created computer-aided location expert system on the principle of calculation using Cooper’s iterative method with the direct graphical display of results. The problem is related to designing a practical computer location expert system, which is based on a new idea of using the resolution of a computer map as an image to calculate location. The calculated results are validated by comparing them with each other, and the defined accuracy for a particular example was achieved at the 32nd iteration with the position optima DC[x(32);y(32)] = [288.8;82.7], with identical results. The location solution in the case study to the defined accuracy was achieved at the 6th iteration with the position optima DC[x(6);y(6)] = [274;220]. The calculations show that the expert system created achieves the required parameters and is a handy tool for determining the location of a point on a surface.

Bounded Reasoning and Higher-Order Uncertainty

A standard assumption in game theory is that players have an infinite depth of reasoning: they think about what others think and about what others think that othersthink, and so on, ad infinitum. However, in practice, players may have a finite depth of reasoning. For example, a player may reason about what other players think, but not about what others think he thinks. This paper proposes a class of type spaces that generalizes the type space formalism due to Harsanyi (1967) so that it can model players with an arbitrary depth of reasoning. I show that the type space formalism does not impose any restrictions on the belief hierarchies that can be modeled, thus generalizing the classic result of Mertens and Zamir (1985). However, there is no universal type space that contains all type spaces.