Higgledy-piggledy sets in projective spaces of small dimension

This work focuses on higgledy-piggledy sets of $k$-subspaces in $\text{PG}(N,q)$, i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these $k$-subspaces with any $(N-k)$-subspace $\kappa$ of $\text{PG}(N,q)$ spans $\kappa$ itself. We highlight three methods to construct small higgledy-piggledy sets of $k$-subspaces and discuss, for $k\in\{1,N-2\}$, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main result is the existence of six lines of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in $\text{PG}(5,q)$ consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding- and graph-theoretical context.Comment: [v1] 21 pages, 1 figure [v2] 21 pages, 1 figure: corrected minor details, updated bibliograph

Constructing saturating sets in projective spaces using subgeometries

A $\varrho$-saturating set of $\text{PG}(N,q)$ is a point set $\mathcal{S}$ such that any point of $\text{PG}(N,q)$ lies in a subspace of dimension at most $\varrho$ spanned by points of $\mathcal{S}$. It is generally known that a $\varrho$-saturating set of $\text{PG}(N,q)$ has size at least $c\cdot\varrho\,q^\frac{N-\varrho}{\varrho+1}$, with $c>\frac{1}{3}$ a constant. Our main result is the discovery of a $\varrho$-saturating set of size roughly $\frac{(\varrho+1)(\varrho+2)}{2}q^\frac{N-\varrho}{\varrho+1}$ if $q=(q')^{\varrho+1}$, with $q'$ an arbitrary prime power. The existence of such a set improves most known upper bounds on the smallest possible size of $\varrho$-saturating sets if $\varrho<\frac{2N-1}{3}$. As saturating sets have a one-to-one correspondence to linear covering codes, this result improves existing upper bounds on the length and covering density of such codes. To prove that this construction is a $\varrho$-saturating set, we observe that the affine parts of $q'$-subgeometries of $\text{PG}(N,q)$ having a hyperplane in common, behave as certain lines of $\text{AG}\big(\varrho+1,(q')^N\big)$. More precisely, these affine lines are the lines of the linear representation of a $q'$-subgeometry $\text{PG}(\varrho,q')$ embedded in $\text{PG}\big(\varrho+1,(q')^N\big)$.Comment: [v1] 25 pages, 1 figure [v2] 30 pages, 1 figure: added translation of the main results to the coding theoretical context and made a more thorough comparison with the existing literature [v3] 30 pages, 1 figure: fixed some details and minor grammar and spelling mistake

How does the Exchange Rate Movement Affect Macroeconomic Performance? A VAR Analysis with Sign Restriction Approach– Evidence from Turkey

In this paper, we assess the effect of exchange rate movement on macroeconomic performance by differentiating the source of exchange rate movement as either an expansionary monetary policy or a portfolio preference shock using quarterly data from Turkish economy for the period 1987:Q1 to 2008:Q3. Empirical evidence suggest that if the depreciation of the exchange rate stems from an expansionary monetary policy shock, then the effect of currency depreciation on the economy is expansionary. On the other hand, if currency depreciation comes from a portfolio choice allocation, then the effect of exchange rate deprecation on the economy is contractionary.Exchange Rates, Monetary Policy, Vector Autoregression and Sign Restrictions.

Small weight codewords of projective geometric codes II

The $p$-ary linear code $\mathcal C_{k}(n,q)$ is defined as the row space of the incidence matrix $A$ of $k$-spaces and points of $\text{PG}(n,q)$. It is known that if $q$ is square, a codeword of weight $q^k\sqrt{q}+\mathcal O \left( q^{k-1} \right)$ exists that cannot be written as a linear combination of at most $\sqrt{q}$ rows of $A$. Over the past few decades, researchers have put a lot of effort towards proving that any codeword of smaller weight does meet this property. We show that if $q \geqslant 32$ is a composite prime power, every codeword of $\mathcal C_k(n,q)$ up to weight $\mathcal O \left( {q^k\sqrt{q}} \right)$ is a linear combination of at most $\sqrt{q}$ rows of $A$. We also generalise this result to the codes $\mathcal C_{j,k}(n,q)$, which are defined as the $p$-ary row span of the incidence matrix of $k$-spaces and $j$-spaces, $j < k$.Comment: 22 page

Turkish Monetary Policy and Components of Aggregate Demand: A VAR Analysis with Sign Restrictions Model

Cataloged from PDF version of article.This article estimates the effects of monetary policy on components of aggregate demand using quarterly data on Turkish economy from 1987-2008 by means of structural Vector Autoregression (VAR) methodology. This study adopts Uhlig's (2005) sign restrictions on the impulse responses of main macroeconomic variables to identify monetary shock. This study finds that expansionary monetary policy stimulates output through consumption and investment in the short-run. However, expansionary monetary policy is ineffective in the long-run

Blocking subspaces with points and hyperplanes

In this paper, we characterise the smallest sets $B$ consisting of points and hyperplanes in $\text{PG}(n,q)$, such that each $k$-space is incident with at least one element of $B$. If $k > \frac {n-1} 2$, then the smallest construction consists only of points. Dually, if $k < \frac{n-1}2$, the smallest example consists only of hyperplanes. However, if $k = \frac{n-1}2$, then there exist sets containing both points and hyperplanes, which are smaller than any blocking set containing only points or only hyperplanes.Comment: 7 pages. UPDATE: After publication of this paper, we found out that in case $k = \frac{n-1}2$, the correct lower bound and a classification of the smallest examples was already established by Blokhuis, Brouwer, and Sz\H{o}nyi [A. Blokhuis, A. E. Brouwer, T. Sz\H{o}nyi. On the chromatic number of $q$-Kneser graphs. Des. Codes Crytpogr. 65:187-197, 2012

Assessing the Lexico-Semantic Relational Knowledge Captured by Word and Concept Embeddings

Deep learning currently dominates the benchmarks for various NLP tasks and, at the basis of such systems, words are frequently represented as embeddings --vectors in a low dimensional space-- learned from large text corpora and various algorithms have been proposed to learn both word and concept embeddings. One of the claimed benefits of such embeddings is that they capture knowledge about semantic relations. Such embeddings are most often evaluated through tasks such as predicting human-rated similarity and analogy which only test a few, often ill-defined, relations. In this paper, we propose a method for (i) reliably generating word and concept pair datasets for a wide number of relations by using a knowledge graph and (ii) evaluating to what extent pre-trained embeddings capture those relations. We evaluate the approach against a proprietary and a public knowledge graph and analyze the results, showing which lexico-semantic relational knowledge is captured by current embedding learning approaches.Comment: Accepted at the 10th International Conference on Knowledge Capture (K-CAP 2019