A point-line incidence identity in finite fields, and applications

Let $E \subseteq \mathbb{F}_q^2$ be a set in the 2-dimensional vector space over a finite field with $q$ elements. We prove an identity for the second moment of its incidence function and deduce a variety of existing results from the literature, not all naturally associated with lines in $\mathbb{F}_q^2$, in a unified and elementary way.Comment: 30 page

Products of Differences over Arbitrary Finite Fields

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then $$|(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| > \frac{q}{2}.$$ Any $\delta < 1/13,542$ suffices for sufficiently large $q$. This improves the condition $|A| > q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions. Our proof is based on a qualitatively optimal characterisation of sets $A,X \subseteq \mathbb{F}_q$ for which the number of solutions to the equation $$(a_1-a_2) = x (a_3-a_4) \, , \; a_1,a_2, a_3, a_4 \in A, x \in X$$ is nearly maximum. A key ingredient is determining exact algebraic structure of sets $A, X$ for which $|A + XA|$ is nearly minimum, which refines a result of Bourgain and Glibichuk using work of Gill, Helfgott, and Tao. We also prove a stronger statement for $$(A-B)(C-D) = \{ (a -b) (c-d) : a \in A, b \in B, c \in C, d \in D\}$$ when $A,B,C,D$ are sets in a prime field, generalising a result of Roche-Newton, Rudnev, Shkredov, and the authors.Comment: 42 page

Frand v. Compulsory Licensing: The Lesser of the Two Evils

This paper focuses on two types of licenses that can best be described as outliers—FRAND and compulsory licenses. Overall, these two specific forms of licenses share the objective of producing a fair and reasonable license of a technology protected by intellectual property. The comparable objective notwithstanding, each type of license achieves this end using different mechanisms. The FRAND license emphasizes providing the licensee with reasonable terms, e.g., by preventing a standard patent holder from extracting unreasonably high royalty rates. By contrast, compulsory licenses emphasize the public benefit that flows from enabling access to an otherwise inaccessible invention. Ultimately, both forms of license attempt to create a value for the licensed product that can be remarkably different from the product’s true market value. Nevertheless, both forms ultimately benefit the end-consumer who pays less to access a product subject to either of these forms of license. In comparing these two forms of licenses, the paper hopes to determine whether one form is better than the other, and if so, from whose perspective—the consumer, the licensor or the licensee. In doing so, this paper compares the different prevailing efforts to embrace such licenses as well as the impact of such licenses on the industry

A robust approach to model-based classification based on trimming and constraints

In a standard classification framework a set of trustworthy learning data are employed to build a decision rule, with the final aim of classifying unlabelled units belonging to the test set. Therefore, unreliable labelled observations, namely outliers and data with incorrect labels, can strongly undermine the classifier performance, especially if the training size is small. The present work introduces a robust modification to the Model-Based Classification framework, employing impartial trimming and constraints on the ratio between the maximum and the minimum eigenvalue of the group scatter matrices. The proposed method effectively handles noise presence in both response and exploratory variables, providing reliable classification even when dealing with contaminated datasets. A robust information criterion is proposed for model selection. Experiments on real and simulated data, artificially adulterated, are provided to underline the benefits of the proposed method

Variations on the sum-product problem II

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set $A \subset \mathbb R$, there exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i \in A \}| \gg \frac{|A|^2}{\log |A|}$. The latter result is optimal up to the logarithmic factor.Comment: This paper supersedes arXiv:1603.0682

Variable selection and updating in model-based discriminant analysis for high dimensional data with food authenticity applications

Food authenticity studies are concerned with determining if food samples have been correctly labelled or not. Discriminant analysis methods are an integral part of the methodology for food authentication. Motivated by food authenticity applications, a model-based discriminant analysis method that includes variable selection is presented. The discriminant analysis model is fitted in a semi-supervised manner using both labeled and unlabeled data. The method is shown to give excellent classification performance on several high-dimensional multiclass food authenticity datasets with more variables than observations. The variables selected by the proposed method provide information about which variables are meaningful for classification purposes. A headlong search strategy for variable selection is shown to be efficient in terms of computation and achieves excellent classification performance. In applications to several food authenticity datasets, our proposed method outperformed default implementations of Random Forests, AdaBoost, transductive SVMs and Bayesian Multinomial Regression by substantial margins

Canadian Import Demand, And The Effects Of Tariffs On The Prices Of Domestic Factors

Canada\u27s demand for imports is examined using an econometric model derived from production theory. The production theory approach provides a useful framework for examining two important issues: the effects of Canada\u27s tariff policies on wage and rental rates, and the effects of changes in the price of imports on the demand for imports.;Three important problems that arise in moving from the theoretical model of the production sector to the econometric model are examined in the empirical work: choosing a method of analysing the technology of the production sector; specifying a functional form for the technology; and specifying the speed with which the production sector adjusts following any change in exogenous variables.;The empirical results provide strong evidence that Canada\u27s tariff policies redistribute income from owners of labour to owners of capital. In fact, higher import prices--caused by (increases in) tariffs, for example--lead to an increase in the rental rate and a decrease in the wage rate. The empirical results also indicate that Canadian import demand is quite responsive ( elastic ) to changes in the price of imports.;The gradual adjustment of the production sector in response to an increase in the price of imports is analysed in the theoretical work. The analysis focuses on how domestic factor prices change during the various stages of the adjustment. One interesting result is that the short-run changes in the prices of some or all of the domestic factors could be reversed in the long run. Thus, there could be a conflict between the short-run and long-run interests of the owners of domestic factors