Quasirandom permutations are characterized by 4-point densities

For permutations π and τ of lengths |π|≤|τ| , let t(π,τ) be the probability that the restriction of τ to a random |π| -point set is (order) isomorphic to π . We show that every sequence {τj} of permutations such that |τj|→∞ and t(π,τj)→1/4! for every 4-point permutation π is quasirandom (that is, t(π,τj)→1/|π|! for every π ). This answers a question posed by Graham

Hfq binding changes the structure of Escherichia coli small noncoding RNAs OxyS and RprA, which are involved in the riboregulation of rpoS

OxyS and RprA are two small noncoding RNAs (sRNAs) that modulate the expression of rpoS, encoding an alternative sigma factor that activates transcription of multiple Escherichia coli stress-response genes. While RprA activates rpoS for translation, OxyS down-regulates the transcript. Crucially, the RNA binding protein Hfq is required for both sRNAs to function, although the specific role played by Hfq remains unclear. We have investigated RprA and OxyS interactions with Hfq using biochemical and biophysical approaches. In particular, we have obtained the molecular envelopes of the Hfq–sRNA complexes using small-angle scattering methods, which reveal key molecular details. These data indicate that Hfq does not substantially change shape upon complex formation, whereas the sRNAs do. We link the impact of Hfq binding, and the sRNA structural changes induced, to transcript stability with respect to RNase E degradation. In light of these findings, we discuss the role of Hfq in the opposing regulatory functions played by RprA and OxyS in rpoS regulation

Linear forms and quadratic uniformity for functions on ZN\mathbb{Z}_N

A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In $\mathbb{Z}_N$, substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm $\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].Comment: 66 page

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

SPICE compact modeling of bipolar/unipolar memristor switching governed by electrical thresholds

In this work we propose a physical memristor/resistive switching device SPICE compact model, that is able to accurately fit both unipolar/bipolar devices settling to its current-voltage relationship. The proposed model is capable of reproducing essential device characteristics such as multilevel storage, temperature dependence, cycle/event handling and even the evolution of variability/parameter degradation with time.The developed compact model has been validated against two physical devices, fitting unipolar and bipolar switching. With no requirement of Verilog-A code, LTSpice and Spectre simulations reproduce distinctive phenomena such as the preforming state, voltage/cycle dependent<br/