Near-optimal small-depth lower bounds for small distance connectivity

We show that any depth-$d$ circuit for determining whether an $n$-node graph has an $s$-to-$t$ path of length at most $k$ must have size $n^{\Omega(k^{1/d}/d)}$. The previous best circuit size lower bounds for this problem were $n^{k^{\exp(-O(d))}}$ (due to Beame, Impagliazzo, and Pitassi [BIP98]) and $n^{\Omega((\log k)/d)}$ (following from a recent formula size lower bound of Rossman [Ros14]). Our lower bound is quite close to optimal, since a simple construction gives depth-$d$ circuits of size $n^{O(k^{2/d})}$ for this problem (and strengthening our bound even to $n^{k^{\Omega(1/d)}}$ would require proving that undirected connectivity is not in $\mathsf{NC^1}.$) Our proof is by reduction to a new lower bound on the size of small-depth circuits computing a skewed variant of the "Sipser functions" that have played an important role in classical circuit lower bounds [Sip83, Yao85, H{\aa}s86]. A key ingredient in our proof of the required lower bound for these Sipser-like functions is the use of \emph{random projections}, an extension of random restrictions which were recently employed in [RST15]. Random projections allow us to obtain sharper quantitative bounds while employing simpler arguments, both conceptually and technically, than in the previous works [Ajt89, BPU92, BIP98, Ros14]

Comments: Eminent Domain: Private Corporations and the Public Use Limitation

Though no logical limits are evident in the cases, it nonetheless remains to be seen how far courts will be willing to go in allowing local development authorities to condemn property for commercial purposes

Linear Toric Fibrations

These notes are based on three lectures given at the 2013 CIME/CIRM summer school. The purpose of this series of lectures is to introduce the notion of a toric fibration and to give its geometrical and combinatorial characterizations. Polarized toric varieties which are birationally equivalent to projective toric bundles are associated to a class of polytopes called Cayley polytopes. Their geometry and combinatorics have a fruitful interplay leading to fundamental insight in both directions. These notes will illustrate geometrical phenomena, in algebraic geometry and neighboring fields, which are characterized by a Cayley structure. Examples are projective duality of toric varieties and polyhedral adjunction theory

Helicobacter pylori infection and gastric dysbiosis: Can probiotics administration be useful to treat this condition?

Helicobacter pylori (Hp) is responsible for one of the most common infections in the world. 'e prevalence exceeds 50% of the population in developing countries, and approximately one-third of the adults are colonized in North Europe and North America. It is considered a major pathogenic agent of chronic gastritis, peptic ulcer, atrophic gastritis, gastric cancer, and mucosa-associated lymphoid tissue lymphoma (MALT). Hp colonization modifies the composition of gastric microbiota that could drive the development of gastric disorders. Currently, an emerging problem in Hp treatment is represented by the increasing rate of antimicrobial therapy resistance. In this context, the search for adjuvant agents can be very useful to overcome this issue and probiotics administration can represent a valid option. The aim of this review is to describe the gastric microbiota changes during Hp colonization, the mechanisms of action, and a possible role of probiotics in the treatment of this infection

A data summary file structure and analysis tools for neutrino oscillation analysis at the NOνA experiment

The NuMI Off-axis Neutrino Experiment (NOvA) is designed to study neutrino oscillations in the NuMI beam at Fermilab. Neutrinos at the Main Injector (NuMI) is currently being upgraded to provide 700 kW for NOvA. A 14 kt Far Detector in Ash River, MN and a functionally identical 0.3 kt Near Detector at Fermilab are positioned 810 km apart in the NuMI beam line. The fine granularity of the NOvA detectors provides a detailed representation of particle trajectories. The data volume associated with such granularity, however, poses problems for analyzing data with ease and speed. NOvA has developed a data summary file structure which discards the full event record in favor of higher-level reconstructed information. A general- purpose framework for neutrino oscillation measurements has been developed for analysis of these data summary files. We present the design methodology for this new file format as well as the analysis framework and the role it plays in producing NOvA physics results

Inclusive electron-nucleus cross section within the Self Consistent Green's Function approach

We compute inclusive electron-nucleus cross sections using ab initio spectral functions of $^4$He and $^{16}$O obtained within the Self Consistent Green's Function approach. The formalism adopted is based on the factorization of the spectral function and the nuclear transition matrix elements. This allows to provide an accurate description of nuclear dynamics and to account for relativistic effects in the interaction vertex. Our calculations use a saturating chiral Hamiltonian in order reproduce the correct nuclear sizes. When final state interactions for the struck particle are accounted for, we find nice agreement between the data and the theory for the inclusive electron-$^{16}$O cross section. The results lay the foundations for future applications of the Self Consistent Green's Function method, in both closed and open shell nuclei, to neutrino data analysis. This work also presents results for the point-proton, charge and single-nucleon momentum distribution of the same two nuclei. The center of mass can affect these quantities for light nuclei and cannot be separated cleanly in most ab initio post-Hartree-Fock methods. In order to address this, we developed a Metropolis Monte Carlo calculation in which the center of mass coordinate can be subtracted exactly from the trial wave function and the expectation values. We gauged this effect for $^4$He by removing the center of mass effect from the Optimal Reference State wave function that is generated during the Self Consistent Green's Function calculations. Our findings clearly indicate that the residual center of mass contribution strongly modifies calculated matter distributions with respect to those obtained in the intrinsic frame. Hence, its subtraction is crucial for a correct description of light nuclei.Comment: 12 pages, 10 figure

Learning circuits with few negations

Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A. A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions)

Ab initio calculation of the electromagnetic and neutral-weak response functions of 4He and 12C

Precise measurement of neutrino oscillations, and hence the determination of their masses demands a quantitative understanding of neutrino-nucleus interactions. To this aim, two-body meson-exchange currents have to be accounted for along within realistic models of nuclear dynamics. We summarize our progresses towards the construction of a consistent framework, based on quantum Monte Carlo methods and on the spectral function approach, that can be exploited to accurately describe neutrino interactions with atomic nuclei over the broad kinematical region covered by neutrino experiments.Comment: 8 pages, 4 figure, Proceedings of the 21st International Conference on Few-Body Problems in Physics, Chicago, Illinois, US