Positional Games

Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201

Can Positional Concerns Enhance the Private provision of Public Goods?

The social welfare effect of positional concerns over public goods is composed of two parts, a positional outcome and an outcome in terms of public goods provision. When agents have homogenous positional preferences over the public good, they overinvest in the positional public good, resulting in a zero-sum positional race with a higher provision of the public good. When agents differ in their positional preferences, the overall impact on social welfare is positive when endowments are homogenous and uncertain when endowments are heterogeneous. Given that the social loss from position-seeking is lower than the social gain from rank seeking, there is an increase of social welfare. If agents have different initial endowments, positional preferences might still be welfare enhancing as long as the positional loss does not exceed the gain in terms of public good provision.

Positional information, positional error, and read-out precision in morphogenesis: a mathematical framework

The concept of positional information is central to our understanding of how cells in a multicellular structure determine their developmental fates. Nevertheless, positional information has neither been defined mathematically nor quantified in a principled way. Here we provide an information-theoretic definition in the context of developmental gene expression patterns and examine which features of expression patterns increase or decrease positional information. We connect positional information with the concept of positional error and develop tools to directly measure information and error from experimental data. We illustrate our framework for the case of gap gene expression patterns in the early Drosophila embryo and show how information that is distributed among only four genes is sufficient to determine developmental fates with single cell resolution. Our approach can be generalized to a variety of different model systems; procedures and examples are discussed in detail

On the lexicographic representation of numbers

It is proven that, contrarily to the common belief, the notion of zero is not necessary for having positional representations of numbers. Namely, for any positive integer $k$, a positional representation with the symbols for $1, 2, \ldots, k$ is given that retains all the essential properties of the usual positional representation of base $k$ (over symbols for $0, 1, 2 \ldots, k-1$). Moreover, in this zero-free representation, a sequence of symbols identifies the number that corresponds to the order number that the sequence has in the ordering where shorter sequences precede the longer ones, and among sequences of the same length the usual lexicographic ordering of dictionaries is considered. The main properties of this lexicographic representation are proven and conversion algorithms between lexicographic and classical positional representations are given. Zero-free positional representations are relevantt in the perspective of the history of mathematics, as well as, in the perspective of emergent computation models, and of unconventional representations of genomes.Comment: 15 page

Positional Order and Diffusion Processes in Particle Systems

Nonequilibrium behaviors of positional order are discussed based on diffusion processes in particle systems. With the cumulant expansion method up to the second order, we obtain a relation between the positional order parameter $\Psi$ and the mean square displacement $M$ to be $\Psi \sim \exp(- {\bf K}^2 M /2d)$ with a reciprocal vector ${\bf K}$ and the dimension of the system $d$. On the basis of the relation, the behavior of positional order is predicted to be $\Psi \sim \exp(-{\bf K}^2Dt)$ when the system involves normal diffusion with a diffusion constant $D$. We also find that a diffusion process with swapping positions of particles contributes to higher orders of the cumulants. The swapping diffusion allows particle to diffuse without destroying the positional order while the normal diffusion destroys it.Comment: 4 pages, 4 figures. Submitted to Phys. Rev.

A Deep Architecture for Semantic Matching with Multiple Positional Sentence Representations

Matching natural language sentences is central for many applications such as information retrieval and question answering. Existing deep models rely on a single sentence representation or multiple granularity representations for matching. However, such methods cannot well capture the contextualized local information in the matching process. To tackle this problem, we present a new deep architecture to match two sentences with multiple positional sentence representations. Specifically, each positional sentence representation is a sentence representation at this position, generated by a bidirectional long short term memory (Bi-LSTM). The matching score is finally produced by aggregating interactions between these different positional sentence representations, through $k$-Max pooling and a multi-layer perceptron. Our model has several advantages: (1) By using Bi-LSTM, rich context of the whole sentence is leveraged to capture the contextualized local information in each positional sentence representation; (2) By matching with multiple positional sentence representations, it is flexible to aggregate different important contextualized local information in a sentence to support the matching; (3) Experiments on different tasks such as question answering and sentence completion demonstrate the superiority of our model.Comment: Accepted by AAAI-201

Language Modeling with Deep Transformers

We explore deep autoregressive Transformer models in language modeling for speech recognition. We focus on two aspects. First, we revisit Transformer model configurations specifically for language modeling. We show that well configured Transformer models outperform our baseline models based on the shallow stack of LSTM recurrent neural network layers. We carry out experiments on the open-source LibriSpeech 960hr task, for both 200K vocabulary word-level and 10K byte-pair encoding subword-level language modeling. We apply our word-level models to conventional hybrid speech recognition by lattice rescoring, and the subword-level models to attention based encoder-decoder models by shallow fusion. Second, we show that deep Transformer language models do not require positional encoding. The positional encoding is an essential augmentation for the self-attention mechanism which is invariant to sequence ordering. However, in autoregressive setup, as is the case for language modeling, the amount of information increases along the position dimension, which is a positional signal by its own. The analysis of attention weights shows that deep autoregressive self-attention models can automatically make use of such positional information. We find that removing the positional encoding even slightly improves the performance of these models.Comment: To appear in the proceedings of INTERSPEECH 201

Assessing the cervical range of motion in infants with positional plagiocephaly

Purpose: To determine if infants with positional plagiocephaly have limitations of active and passive cervical range of motion measured with simple and reliable methods. Methods: The examiners assessed bilateral active and passive cervical rotations and passive cervical lateral flexion. Cervical assessment was performed twice by 2 different physicians to assess intertester reliability. To assess intratester reliability the first investigator performed a second examination 48 hours after the first one. Results: One-hundred nine subjects were analyzed; 70.7% of the sample had head positional preference on the right, while 29.3% had head positional preference on the left (x2 35.52, P <0.001). Cervical rotations and lateral flexion showed reliable levels of agreement for intra and intertester reliability. Conclusions: The most limited range of motion in infants with positional plagiocephaly was cervical active rotation which affected more than 90% of patients. Passive cervical rotations and lateral flexion were limited in more than 60% of patient

Direct reconstruction of the two-dimensional pair distribution function in systems with angular correlations

An x-ray scattering approach to determine the two-dimensional (2D) pair distribution function (PDF) in partially ordered 2D systems is proposed. We derive relations between the structure factor and PDF that enable quantitative studies of positional and bond-orientational (BO) order in real space. We apply this approach in the x-ray study of a liquid crystal (LC) film undergoing the smectic-hexatic phase transition, to analyze the interplay between the positional and BO order during the temperature evolution of the LC film. We analyze the positional correlation length in different directions in real space.Comment: 23 pages, 8 figure

Positional Advantage within Small Farms: Evidence from Illinois

As the economic viability of small farms continues to be an issue facing policy makers and economists alike, a market orientation may be a valuable resource producers can develop as they compete in a marketplace dominated by larger firms. Marketing and strategy scholars have long established the importance of a market orientation in determining firm performance. More recently, scholars have studied the effect of these concepts in agriculture. Extending the literature of market orientation in agriculture, this study examines the concept of a positional advantage and its effect on performance using a sample of small farms in Illinois. Using a sample of 347 Illinois beef producers, we empirically measure and test the construct of positional advantage and test the relationship between positional advantage and subjective performance. Our results indicate that market orientation, entrepreneurship, innovation and learning are first-order indicators of positional advantage and that the positional advantage of a firm is positively related to firm performance.Agriculture, innovation, market orientation, positional advantage, Farm Management, Production Economics, L11, L25, L26,