Induced restricted Ramsey theorems for spaces

AbstractThe induced restricted versions of the vector space Ramsey theorem and of the Graham-Rothschild parameter set theorem are proved

Fluorescent non-toxic bait as a new method for black rat (Rattus rattus) monitoring

The detection of synathropic rodents may be difficult since they are animals with nocturnal activity. Methods of their detection and monitoring rely mostly on indirect signs of their activity such as the presence of faeces, urine, consumed foods and damaged materials. Our experimental hypothesis was that the production of fluorescent faeces - following consumption of fluorescent bait - may be used for rodent monitoring. For this purpose we studied the production of fluorescent faeces, temporal dynamics and detectability in wild black rat (Rattus rattus). Wild black rats were individually housed in experimental cages with the wire-mesh grid floor and faeces were collected in short-time intervals. The peak of fluorescent activity in faeces was detected 10-20 hours after bait ingestion. We found that there is only relatively short delay between bait consumption and defecation and fluorescent faeces are easily detectable at distance using an ultraviolet hand lamp. Thus, this method can contribute to effective monitoring of rodent pests.Keywords: Rattus rattus, Fluorescent bait, Monitoring, Rodent contro

Testing Linear-Invariant Non-Linear Properties

We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for Reed-Muller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for "triangle freeness": a function f:\cube^{n}\to\cube satisfies this property if $f(x),f(y),f(x+y)$ do not all equal 1, for any pair x,y\in\cube^{n}. Here we extend this test to a more systematic study of testing for linear-invariant non-linear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by $k$ points v_{1},...,v_{k}\in\cube^{k} and f:\cube^{n}\to\cube satisfies the property that if for all linear maps L:\cube^{k}\to\cube^{n} it is the case that $f(L(v_{1})),...,f(L(v_{k}))$ do not all equal 1. We show that this property is testable if the underlying matroid specified by $v_{1},...,v_{k}$ is a graphic matroid. This extends Green's result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of "1-complexity linear systems" of Green and Tao, and graphic matroids, to derive the results.Comment: This is the full version; conference version appeared in the proceedings of STACS 200

How Reasoning Aims at Truth

Many hold that theoretical reasoning aims at truth. In this paper, I ask what it is for reasoning to be thus aim-directed. Standard answers to this question explain reasoning’s aim-directedness in terms of intentions, dispositions, or rule-following. I argue that, while these views contain important insights, they are not satisfactory. As an alternative, I introduce and defend a novel account: reasoning aims at truth in virtue of being the exercise of a distinctive kind of cognitive power, one that, unlike ordinary dispositions, is capable of fully explaining its own exercises. I argue that this account is able to avoid the difficulties plaguing standard accounts of the relevant sort of mental teleology

Efficient O(N2)\mathcal{O}(N^2) approach to solve the Bethe-Salpeter equation for excitonic bound states

Excitonic effects in optical spectra and electron-hole pair excitations are described by solutions of the Bethe-Salpeter equation (BSE) that accounts for the Coulomb interaction of excited electron-hole pairs. Although for the computation of excitonic optical spectra in an extended frequency range efficient methods are available, the determination and analysis of individual exciton states still requires the diagonalization of the electron-hole Hamiltonian $\hat{H}$. We present a numerically efficient approach for the calculation of exciton states with quadratically scaling complexity, which significantly diminishes the computational costs compared to the commonly used cubically scaling direct-diagonalization schemes. The accuracy and performance of this approach is demonstrated by solving the BSE numerically for the Wannier-Mott two-band model in {\bf k} space and the semiconductors MgO and InN. For the convergence with respect to the \vk-point sampling a general trend is identified, which can be used to extrapolate converged results for the binding energies of the lowest bound states.Comment: 13 pages, 12 figures, 1 table, submitted to PR

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page